Shallow-Earth Rheology from Glacial Isostasy and Satellite ......We start with an introduction on shallow-earth rheology (Section 1.1), glacial isostasy (Section 1.2) and satellite

Shallow-Earth Rheologyfrom Glacial Isostasyand Satellite Gravity

a sensitivity analysis for GOCE

Proefschrift

ter verkrijging van de graad van doctor

aan de Technische Universiteit Delft,

op gezag van de Rector Magnificus prof. dr. ir. J.T. Fokkema,

voorzitter van het College voor Promoties,

in het openbaar te verdedigen op donderdag 26 juni 2008 om 12:30 uur

door Hugo Herman Anthony SCHOTMAN

natuurkundig & geodetisch ingenieur

geboren te Steenderen

Dit proefschrift is goedgekeurd door de promotoren:

Prof. ir. B.A.C. Ambrosius

Prof. dr. P. Wu

Toegevoegd promotor:

Dr. L.L.A. Vermeersen

Samenstelling promotiecommissie:

Rector Magnificus, voorzitter

Prof. ir. B.A.C. Ambrosius, Technische Universiteit Delft, promotor

Prof. dr. P. Wu, University of Calgary, promotor

Dr. L.L.A. Vermeersen, Technische Universiteit Delft, toegevoegd promotor

Prof. dr.-ing. R. Rummel, Technische Universität München

Prof. dr. S.B. Kroonenberg, Technische Universiteit Delft

Dr. R. Govers, Universiteit Utrecht

Dr. R. Koop, Grotius College te Delft

Prof. dr. ir. drs. H. Bijl, Technische Universiteit Delft, reservelid

Dit onderzoek is financieel ondersteund door SRON Netherlands Institute for Space

Research en het Nationale Programma Gebruikersondersteuning 1996-2005 (GO-

2, project nummer: EO-064) van de Nederlandse Organisatie voor Wetenschap-

pelijk Onderzoek (NWO).

Dit proefschrift is gedrukt door drukkerij Mostert & Van Onderen! te Leiden,

met een financiële bijdrage van de vakgroep Astrodynamica en Satellietsystemen,

faculteit Luchtvaart- en Ruimtevaarttechniek, Technische Universiteit Delft.

Foto omslag: ESA - AOES Medialab

ISBN 978-90-9023097-9

Acknowledgements

This thesis reflects the outcome of many years of research and I would like to

thank the following people for their support during this period:

• all at EOS, SRON, especially Avri Selig, Jennifer Serkei, Sietse Rispens, Sander

de Witte, Martijn Smit, Johannes Bouman, Femke Vossepoel, Annemarie Bos,

Gabriele Marquart and Daphne Stam;

• all at AS, DEOS, especially Boudewijn Ambrosius, Ellie Verbarendse, Pieter

Visser, Ejo Schrama, Ron Noomen, Marc Naeije, Ge van Geldrop, Eelco Doorn-

bos, Nacho Andrés, Sander Goossens, Riccardo Riva, Jef van Hove, Bert Wouters

and ’my’ MSc students, Maud van den Broek and Niek van Dael;

• at IVAU: Rob Govers, Martyn Drury, Hans de Bresser, Paul Meijer, Ildiko Csikos

and Thomas Geenen;

• at IMAU: Roderik van de Wal, Jojanneke van den Berg, Richard Bintanja and

Laura de Steur;

• the Geoqus working group, especially Oliver Heidbach, Holger Steffen, Kasper

Fischer, Wouter van der Zee, Andrea Hampel and Paola Ledermann;

• from the scientific community: Reiner Rummel, Salomon Kroonenberg, Hester

Bijl, Erik Ivins, Jerry Mitrovica, Mark Drinkwater, Scott King, Giorgio Spada,

Detlef Wolf, Kurt Lambeck, Hansheng Wang and Glenn Milne;

• everyone who helped me getting back on track after my illness, especially Gerard

Bunschoten and all at P&O, SRON.

I would also like to thank my family, Inez’ family and my friends, especially Fred,

Dirk, Erik-Jan, Menno and Robbert.

Special thanks to Radboud Koop, Bert Vermeersen, Patrick Wu, John van Wester-

laak, Wouter van der Wal, José van den IJssel, my mother, Inez and Igone, without

whom this thesis would not have been.

Hugo Schotman

Leiden, May 2008

to the memory of Arno

’In a world of steel-eyed death, and men

who are fighting to be warm.

"Come in," she said, "I’ll give you

shelter from the storm."’

from Shelter from the Storm by Bob Dylan (1975)

’Probably Scholars, I reckon. Only the Masters get coffins.

There’s probably been so many Scholars all down the centuries

that there wouldn’t be room to bury the whole of ’em,

so they just cut their heads off and keep them.

That’s the most important part of ’em anyway.’

Lyra in Northern Lights by Philip Pullman (1995)

Contents

Acknowledgements iii

1 Introduction 1

1.1 Shallow-Earth Rheology . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Glacial Isostasy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.3 Satellite Gravity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.4 Rationale and Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2 Isostatic Adjustment Theory 13

2.1 Governing Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.2 Spectral Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.3 Finite-Element Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

3 Mechanical Model and Satellite Gravity Data 27

3.1 Earth Stratification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

3.2 Ice-Load History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

3.3 Ocean-Load History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

3.4 Satellite Gravity Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

4 Gravity Field Perturbations due to Low-Viscosity Zones 35

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

4.2 Relaxation Spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

4.3 Geoid Heights and Gravity Anomalies . . . . . . . . . . . . . . . . . . . 40

4.4 Sensitivity to the Properties of LVZs . . . . . . . . . . . . . . . . . . . . 44

4.5 Role of the Background Model . . . . . . . . . . . . . . . . . . . . . . . . 47

4.6 Present-Day Sea-Level Change . . . . . . . . . . . . . . . . . . . . . . . 54

4.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

5 Sensitivity to the Load History 57

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

5.2 Geoid Heights and Perturbations . . . . . . . . . . . . . . . . . . . . . . 63

x Contents

5.3 Sensitivity to the Load History . . . . . . . . . . . . . . . . . . . . . . . 66

5.4 Spatial and Spectral Signatures . . . . . . . . . . . . . . . . . . . . . . . 70

5.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

6 Regional Perturbations in a Global Background Model 77

6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

6.2 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

6.3 Model Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

6.4 Test Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

6.5 Example for Northern Europe . . . . . . . . . . . . . . . . . . . . . . . . 94

6.6 Realistic Ocean-Load History . . . . . . . . . . . . . . . . . . . . . . . . 99

6.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

7 Thermomechanical Models of the Shallow Earth 103

7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

7.2 Thermomechanical Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

7.3 Composition and Creep Parameters . . . . . . . . . . . . . . . . . . . . . 114

7.4 Predictions for Northern Europe . . . . . . . . . . . . . . . . . . . . . . 122

7.5 Constraints from Future GOCE Data . . . . . . . . . . . . . . . . . . . 131

7.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136

8 Conclusions 141

A Crustal Low Viscosity and Lithospheric Strength 147

B Thermal Model 149

C Test Results from Thermomechanical Models 151

C.1 Shallow Upper Mantle Perturbations . . . . . . . . . . . . . . . . . . . . 152

C.2 Crustal Perturbations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154

D Surface Velocities from Thermomechanical Models 157

E Paper from "GOCE, the Geoid and Oceanography" 161

Bibliography 169

Summary 179

Samenvatting 183

List of Publications 187

Curriculum Vitae 189

Chapter 1

Introduction

In this thesis we compare gravity signatures of shallow low-viscosity layers in-

duced by the glacial isostatic adjustment (GIA) process with the expected perfor-

mance of the future ESA Gravity field and steady-state Ocean Circulation Ex-

plorer (GOCE) satellite mission. The rationale for this research and the outline of

this thesis are described in Section 1.4. We start with an introduction on shallow-

earth rheology (Section 1.1), glacial isostasy (Section 1.2) and satellite gravity

(Section 1.3).

1.1 Shallow-Earth Rheology

Since the general acceptance of the plate-tectonics hypothesis in the 1960s, the

solid earth is thought to consist of a strong outer shell divided in a number of

plates, on top of a weaker substratum. The plates are created from mantle mate-

rial at spreading ridges and disappear into the mantle at subduction zones. The

strong shell is called the lithosphere and consists of the crust and the lithospheric

part of the mantle. Below the lithosphere, the almost globally present astheno-

sphere is thought to be weaker and to be able to flow on relatively short timescales

(∼ 100−1000 yrs) due to its low viscosity. The average viscosity of the whole up-

per mantle below the lithosphere to a depth of 670 km is estimated to be about

1020−1021 Pas (e.g. Haskell, 1935; Cathles, 1975; Mitrovica, 1996; Lambeck et al.,

1998; Milne et al., 2001; Peltier, 2004), whereas the asthenosphere can have vis-

cosities that are more than an order of magnitude smaller. Indications for such an

asthenospheric low-viscosity zone (ALVZ) come from seismic data, which shows,

except for old and cold cratonic areas, a low-velocity zone (e.g. Stein & Wysession,

2003, p.170). This low-velocity zone is associated with ductile flow, as creep laws,

2 Chapter 1. Introduction

(a) cold continental (b) hot continental (c) oceanic

Figure 1.1: Yield-strength envelopes for cold continental lithosphere (a), hot continentallithosphere (b) and oceanic lithosphere (c). Plots are from Ranalli & Murphy (1987).

the convergence of the geotherm and the solidus, and the strong vertical advection

of deep heat from mantle convection predict low-viscosity material in this area (e.g.

Stein & Wysession, 2003, p.204). The viscosity of the asthenosphere is strongly de-

pendent on thermal regime, with no evidence for an ALVZ in old and cold cratonic

areas as the Baltic Shield (e.g. Steffen & Kaufmann, 2005), which has a surface

heatflow of 20−40 mW/m2. From postseismic deformation studies in the western

United States, which has a relatively high heatflow (60−90 mW/m2), viscosities in

the range of 1017 −1019 Pas are found (Pollitz, 2003; Dixon et al., 2004).

The crust is chemically different from the mantle and seismically separated by the

Mohorovicic discontinuity (Moho). The thickness of the crust in general increases

with the age of the crust, from zero at spreading ridges to more than 60 km in old

cratons. In areas of relatively high heatflow, the lower crust might be ductile and

have low-viscosity (< 1020 Pas) layers. Indications for such crustal low-viscosity

zones (CLVZs) come from intraplate earthquakes, which are mainly confined to

the upper crust (0−20 km, Watts & Burov, 2003; Ranalli & Murphy, 1987) and

the subcrustal mantle part of the lithosphere (Ranalli & Murphy, 1987), which

are thought to be brittle. The lack of seismicity in the lower crust is associated

with ductile flow, with a viscosity that depends on composition, fluid content and

geotherm (Watts & Burov, 2003; Ranalli & Murphy, 1987). This is supported by the

high seismic reflectivity of the lower crust in areas with relatively high heatflow

(> 70 mW/m2, Meissner & Kusznir, 1987), which indicates lamination supported

by ductile flow. From mining-induced (Klein et al., 1997) and postseismic (Hearn,

2003) deformation studies, viscosities as low as 1017 Pas are deduced. CLVZs can

be expected in continental regions with relatively high heatflow, which excludes

for example the Baltic Shield (e.g. Wu & Mazotti, 2007). In oceanic lithosphere,

earthquakes occur from a depth of 5 to 40 km (Watts & Burov, 2003; Wiens &

Stein, 1983), which indicates brittle behavior of both the crustal and mantle part

of the lithosphere and a lack of low-viscosity.

1.1. Shallow-Earth Rheology 3

Estimates of the viscosity in the shallow earth can be obtained from thermome-

chanical models, in which the mechanical behavior of the earth depends on a

laboratory-derived creep law for a certain (synthetic) earth material and on a

model of the change of temperature in the earth, the geotherm, which is con-

strained by certain thermal data such as the surface heatflow. Typical yield-

strength envelopes (YSEs), which result from this kind of modeling, are shown

in Figure 1.1, which is taken from Ranalli & Murphy (1987). A YSE indicates for

which stress level rocks will yield at a certain depth. The depth ranges that are

accompanied by a vertical bar will yield in the brittle regime, whereas in the other

parts the shallow earth will yield in the ductile regime, i.e. starts to flow. The

viscosity associated with this ductile behavior is related to the ratio of the stress

level and strain rate (10−14 s−1 for these figures) and is about 5 ·1019 Pas for 1

MPa. Both for cold (a) and hot (b) continental lithosphere, relatively weak, quartz-

rich rocks are used, whereas for oceanic lithosphere relatively strong mafic rock is

used. For this choice of parameters, even cold continental lithosphere shows lower

crustal flow, however, with a high viscosity (> 1021 Pas). For hot continental litho-

sphere such a high viscosity is predicted if stronger materials in the lower crust

are used (e.g. diabase, Ranalli & Murphy, 1987; Kaikkonen et al., 2000). Note that

for a thicker crust, the viscosity at the Moho will be lower. The thickness of the

lithosphere is larger than the depth to the bottom of the brittle part, which is at

80 km, and depends, among others, on the time scale of loading. Hot continental

lithosphere shows a CLVZ with a viscosity of about 1019 Pas or larger and an ALVZ

below a very thin lithosphere (∼ 40 km). An ALVZ can also be found below oceanic

(c) lithosphere, but no CLVZ is expected.

Regional studies in this thesis focus mostly on Northern Europe, which ranges

roughly from the British Isles in the southwest to Nova Zembla in the north-

east. This area is characterized by large differences in surface heatflow, with val-

ues smaller than 40 mW/m2 in an area in Finland and heatflows larger than 80

mW/m2 in the oceans and some continental areas, see Figure 1.2a. Crustal thick-

nesses range from about 50 km under the Baltic Shield to smaller than 10 km in

oceanic areas (Figure 1.2b). In the next section we will show that during the last

glacial cycle (120 kyrs BP to the present), the ice sheet over Scandinavia was more

or less centered in the area with the lowest heatflow and thickest crust. This is

not a coincidence, but related to the fact that, due to the lower density of the crust

compared to the mantle, a thicker than average crust is elevated (Airy-Heiskanen

model of isostasy, e.g. Watts, 2001, p. 20) and that ice sheets start to grow in high

places (e.g. Oerlemans, 2003). However, parts of the ice sheet were also situated

in hotter and thinner areas as e.g. the Barents Sea.


(a) surface heatflow

40 60 80surface heatflow [mW/m^2]

(b) crustal thickness

10 20 30 40 50crustal thickness [km]

Figure 1.2: Surface heatflow (a) and crustal thickness (b) in Northern Europe. In (a), surfaceheatflow values are from Pollack et al. (1993), with an update for continental Europe byArtemieva & Mooney (2001). In (b), crustal thickness values are from CRUST2.0 (Bassinet al., 2000). The datasets are discretized to five levels and interpolated to a 1×1 grid.

1.2 Glacial Isostasy

Glacial isostasy or glacial isostatic adjustment1 (GIA) refers to the geophysical

process in which the solid earth is deformed by changes in continental ice masses

and resulting changes in ocean load. Solid-earth deformation is mainly guided

by the thickness of the elastic lithosphere (∼ 100 km) and the viscosity of the

mantle (∼ 1021 Pas, Haskell, 1935; Cathles, 1975). The process is illustrated in

Figure 1.3a, which is originally by Daly (1934), but is here reproduced from Watts

(2001). Initially (’1.’, before loading), the earth is in isostatic equilibrium, which

means that there are no residual forces in the earth. Upon loading the crust (2.),

there is an instantaneous deformation of the elastic lithosphere (labeled ’Crust’

in the figure, but consisting actually of the crust and the lithospheric part of the

mantle) and the mantle (’Substratum’) followed by outward viscoelastic mantle

material flow which creates bulges just outside the loading areas. The instanta-

neous deformation of the mantle will only occur if it is viscoelastic, and not viscous

as suggested in the figure. Also on longer timescales viscoelastic behavior deviates

1we do not use the term "postglacial rebound" here, as it suggests only uplift of areas that were once

glaciated as e.g. Scandinavia, while some areas are actually subsiding due to glacial isostasy.

1.2. Glacial Isostasy 5

(a) GIA process (b) RSES ice-load distribution at LGM

1000 2000 3000 4000 5000ice height [m]

Figure 1.3: Illustration of the GIA process, originally from Daly (1934), this picture repro-duced from Watts (2001) (a) and the RSES ice-load distribution of Lambeck et al. (1998)at LGM (b).

from purely viscous behavior, because of the advection of pre-stress in a viscoelas-

tic material, which can be regarded as the restoring force of isostasy (Wu, 1992b,

2004). When the ice sheet melts (’3.’), the bulges collapse, both decreasing in height

and moving inwards.

Studies of GIA focus mainly on the last glacial cycle, which started about 120 kyrs

BP and showed a maximum continental ice mass at the last glacial maximum

(LGM, ∼ 21 kyrs BP). The largest ice masses could be found on the northern hemi-

sphere, especially centered over the Hudson Bay area in Canada (Laurentide ice

sheet) and the Gulf of Bothnia (Fennoscandian ice sheet, see Figure 1.3b). Studies

of GIA provide estimates of the viscosity of the earth’s mantle and of the thick-

ness of the lithosphere, and on especially the thickness of the different ice sheets

(van Bemmelen & Berlage, 1935; Haskell, 1935; Vening Meinesz, 1937; Critten-

den, 1963; McConnell, 1965; Cathles, 1971; Walcott, 1972; Parsons, 1972; Peltier,

1974, 1976; Clark et al., 1978; Wu & Peltier, 1982, 1983; Wolf, 1987). The esti-

mation process is in general an inversion of especially Holocene relative sea-level

(RSL) data (e.g. Cathles, 1975; Peltier, 1976; Wu & Peltier, 1982, 1983; Tushing-

ham & Peltier, 1991; Lambeck et al., 1998; Peltier, 2004), though also constraints

from earth rotation (e.g. O’Connell, 1971; Nakiboglu & Lambeck, 1980; Sabadini &

Peltier, 1981; Wu & Peltier, 1984; Vermeersen et al., 1997, 1998), VLBI-baselines

(e.g. Giunchi et al., 1997; Peltier, 2004), static gravity (e.g. O’Connell, 1971; Wu

& Peltier, 1983; Mitrovica & Peltier, 1989), gravity change (e.g. Peltier, 2004;


Tamisiea et al., 2007) and GPS (e.g. Milne et al., 2001) have been used. In most

studies, an existing ice-load history is used, though one should keep in mind that

these ice-load histories contain information on the viscosity structure of the earth,

as both the viscosity stratification and the ice-load history are usually estimated in

the same inversion process (e.g. Wu & Peltier, 1983; Tushingham & Peltier, 1991;

Lambeck et al., 1998; Peltier, 2004).

One of the earliest estimates of viscosity, based on Holocene RSL curves in Scan-

dinavia, is from Haskell (1935). He finds an average mantle viscosity of 1021 Pas

from the base of the lithosphere to a resolving depth of about 1400 km (Mitro-

vica, 1996). Another classical study by Cathles (1975) finds a similar viscosity

value for both the upper (from the bottom of the lithosphere to a depth of 670 km)

and lower mantle (from a depth of 670 km). The much-used ICE-3G deglaciation

history (Tushingham & Peltier, 1991), constrained by geomorphological data and

Holocene sea-level curves, was inferred using a slight deviation from the above-

mentioned univiscous model, by increasing the viscosity of the lower mantle (νLM)

to 2·1021 Pas, using a lithospheric thickness (LT) of 120 km (’VM1’, Tushingham &

Peltier, 1991). The ice-load history was further developed into ICE-4G, for which

a new, more detailed, viscosity stratification was derived with an average upper

mantle viscosity νUM of 4 ·1020 Pas and an average viscosity for the upper 500

km of the lower mantle of about 2 ·1021 Pas (’VM2’, Peltier, 1998). For ICE-5G

(Peltier, 2004), which also uses information from VLBI and changes in gravity, the

same viscosity stratification is used, though Peltier (2004) states that the litho-

spheric thickness of 120 km is excessive for the British Isles, and puts it closer to

90 km. Upon reconstructing a deglaciation history for Northern Europe (RSES,

Figure 1.3b), Lambeck et al. (1998) find lower LT values for the British Isles (65

km) and Fennoscandia (75 km). Though their estimate for Northern Europe of

νUM (3−4 ·1020 Pas) is close to the value of VM2, they find especially larger values

for νLM (4 ·1021−3 ·1022 Pas). They also state that if LT is fixed to relatively large

values, values for νUM are relatively high and for νLM are relatively low, which

could explain the differences with VM1 and VM2. The regional study of Milne et

al. (2004) for Northern Europe, using GPS data and a comparable loading history

as Lambeck et al. (1998), also stresses that a large LT (e.g. 120 km) with a high

νUM (e.g. 1 ·1021 Pas) gives a comparable quality of fit as a small LT (e.g. 80 km)

and a low νUM (e.g. 5 ·1020 Pas). Their results mainly confirm the results of Lam-

beck et al. (1998), with a similar νLM (5 ·1021−1 ·1022 Pas) and a somewhat larger

LT (> 90 km) and νUM (5 ·1020−1 ·1021 Pas).

The last decade has seen the development of 3D-stratified earth models, both for

regional studies (e.g. Kaufmann & Wu, 1998, 2002; Wu, 2005; Steffen et al., 2006,

2007) and recently also for global studies (e.g. Wu & van der Wal, 2003; Zhong

et al., 2003; Wu et al., 2005; Latychev et al., 2005; Wang & Wu, 2006), and more

realistic models of the ocean-load history. In the simplest case, the ocean load

1.3. Satellite Gravity 7

(a) time-dependent coastlines (b) meltwater influx

Figure 1.4: Effect of time-dependent coastlines (a) and meltwater influx (b), from Milne etal. (1999) and Mitrovica & Milne (2003).

is equal to the eustatic load, which is a geographically uniform change with a

mass equal to the change in ice mass. Most studies however consider the effect of

self-gravitation in the oceans (Farrell & Clark, 1976; Mitrovica & Peltier, 1991),

in which the change in ocean load is equal to the difference between the change

in geoid and in ocean bathymetry. In the last 15 years, some studies have also in-

cluded time-dependent coastlines and the influx of meltwater in formerly glaciated

areas (e.g. Johnston, 1993; Milne et al., 1999; Mitrovica & Milne, 2003; Lambeck

et al., 2003), as illustrated in Figure 1.4. In (a) we see how the coastline mi-

grates from time t j−1 to t j as sea level rises and falls. We see also that only due

to this effect and not considering solid-earth deformation, the paleotopography

changes. The paleotopography can be estimated from the present-day topogra-

phy and changes in self-gravitating ocean load during the last glacial cycle, and is

needed to model the influx of meltwater into areas of negative topography during

ice retreat (Figure 1.4b).

1.3 Satellite Gravity

The equipotential surfaces of the earth’s gravitational field have a dominantly

spherical shape. As a result, the gravitational accelerations, being the gradient

of the potential, are dominantly in the radial direction. The main deviation from

sphericity is due to the rotation of the earth, which leads to an elliptical shape of

the equipotential surfaces of the gravity field. The differences between the geoid

(equipotential surface of the true earth at mean sea level) and the equipotential

surface of a perfect ellipsoid with homogeneous mass distribution (normal earth)

are called geoid heights. These and gravity anomalies, which are differences with

gravity accelerations of a normal earth, provide information on mass anomalies in

and on the surface of the earth. The mass anomalies can be due to, for example,


(a) GGM02S

−60 −40 −20 0 20 40 60geoid height [m]

(b) gradiometer

Figure 1.5: Geoid heights as measured by the GRACE satellite mission (a) and the gra-diometer onboard the future GOCE mission (b, source: ESA - AOES Medialab).

deformation-induced density changes, chemical or thermal anomalies in the earth,

topography, ocean tides and temporal variations in continental water storage and

atmospheric pressure. An example of geoid heights as measured by the recent

Gravity Recovery and Climate Experiment (GRACE) satellite mission is given in

Figure 1.5a. A study of Tamisiea et al. (2007) uses the rate of change in grav-

ity over a 4-year period as measured by GRACE to estimate a viscosity profile for

Northern America. This must be considered a preliminary study, as the uncertain-

ties in the hydrological contribution to the gravity field are large (e.g. Rangelova

et al., 2007) and because Tamisiea et al. (2007) use an existing ice-load history

(ICE-5G) and a fixed value for the lithosphere. Using the estimated viscosity pro-

file they find that GIA can explain 25−45% of the static geoid low under Canada

(see Figure 1.5a), which is somewhat less than the estimate of Simons & Hager

(1997) of 50%. The remaining part of the low is mostly due to mantle convection

(e.g. Simons & Hager, 1997; Tamisiea et al., 2007). From GIA studies, a somewhat

smaller low is expected in Scandinavia, however, this low is not visible due to the

large positive effect of the Iceland plume, but can be retrieved by removing the

long-wavelengths (down to ∼ 4000 km, see e.g. Vermeersen & Schotman, 2008).

Until recently, the global gravity field was known with a very heterogeneous ac-

curacy. Terrestrial gravity measurements have a very high accuracy, but provide

only point measurements, and, due to the costs of campaigns in sometimes re-

mote areas, are limited to certain regions. Using shipborne and airborne gravity

measurements, larger areas can be covered, but maintaining a high accuracy on

a moving platform is difficult and, again, campaigns are expensive and thus lim-

ited. Using altimeter measurements from satellites, the geoid can be recovered

over the oceans, but with considerable lower accuracy compared to geoid informa-

1.3. Satellite Gravity 9

Table 1.1: Characteristics of the CHAMP, GRACE and GOCE satellite missions

Mission Accuracy Maximum Resolution Launch

degree l (half-wavelength)

CHAMP 5 cm 50 400 km 2000

GRACE 1−5 cm 80−100 250−200 km 2002

GOCE 1−2 cm 200 100 km 2008

tion from modern terrestrial measurements. The global gravity field can be re-

trieved by satellite orbit analysis using satellite laser ranging (SLR), though only

the long wavelengths (down to ∼ 2000 km). This is mainly due to the attenuation

of earth’s gravity field with height, which increases with decreasing wavelength of

this field2. This attenuation with height can be counteracted by using low-earth

orbiters (LEOs), for which the lowest altitude is, due to an increase in atmospheric

drag, limited to ∼ 200 km.

Using the Global Positioning System (GPS), the use of high-low satellite-to-satellite

tracking (SST) can increase the resolution (half-wavelength3) of the gravity field

to about 400 km (l ∼ 50) with cm-accuracy (Visser et al., 2002). This is proven

by the Challenging Minisatellite Payload (CHAMP) satellite mission of the Ger-

man agencies DLR and GFZ, launched in July 2000. It provides a gravity field

with an accuracy of 5 cm at 400 km resolution (EIGEN-CHAMP03S, Reigber et

al., 2004) by measuring directly the gravity acceleration, i.e. the first derivative

of the gravity potential. Using two LEOs in an identical orbit and separated by

only a few hundred kilometers, the concept of low-low SST can be implemented,

which provides finite differences in the first derivative of the gravity potential by

measuring acceleration differences over the intersatellite distance. This concept

was used for the joint NASA/DLR GRACE mission, which was launched in March

2002 and provides a static gravity field with an accuracy of 1−2 cm in geoid height

for a resolution of about 250 km (l ∼ 80, GGM02S, Tapley et al., 2005).

The future Gravity field and steady-state Ocean Circulation Explorer (GOCE)

satellite mission, planned for launch by ESA in the summer of 2008, is expected

to provide estimates of the static gravity field with an accuracy of 1−2 cm in geoid

height and a few mGal in gravity anomaly for a resolution smaller than 100 km

(l > 200, Visser et al., 2002). GOCE uses the technique of satellite gradiometry,

which provides direct measurements of gravity gradients, the second derivatives

of the gravity potential. Due to the sensitivity of gradient gradients to especially

the higher harmonics, this counteracts, together with the low (250−300 km) satel-

2as (R/r)l+1, with R the radius of the earth, r the distance from the center of the earth and l the

harmonic degree, see Eq. (2.23) in Chapter 2.3the resolution is often taken as half-wavelength, where one wavelength is roughly equal to the

circumference of the earth (∼ 40,000 km) divided by the spherical harmonic degree l.


lite orbit, the attenuation with height. The gradient tensor is measured by a gra-

diometer (Figure 1.5b), which consists of three orthogonal pairs of accelerometers.

Note that also CHAMP and GRACE have accelerometers onboard to measure non-

gravitational accelerations and that for both CHAMP and GOCE high-low SST is

used to provide estimates of the long-wavelength part of the gravity field. The

main characteristics of the CHAMP, GRACE and GOCE missions are summarized

in Table 1.1.

1.4 Rationale and Outline

One of the first GIA studies to use asthenospheric low-viscosity layers is by Cath-

les (1975). He finds an average viscosity of 4 ·1019 Pas and a thickness of 75 km,

but with large regional variations (Cathles, 1975, p. 270). Regionally, a study of

Kaufmann & Wu (1998) for the Barents Sea assumes a 110 km thick lithosphere

and a lateral variation in asthenospheric viscosity from 1018 to 1021 Pas. A study

for Northern Europe of Kaufmann & Wu (2002) assumes, based on seismological

evidence and estimates from seismic tomography, variations in lithospheric thick-

ness (from 90 km in oceanic areas to 170 km under cratons) and a low-viscosity

(1018 Pas) asthenosphere in the oceanic areas. They show that RSL data have

difficulties detecting these ALVZs. Steffen & Kaufmann (2005) find some evidence

for a low-viscosity region (1019-1020 Pas) in the Barents Sea area in a depth range

of 160 to 200 km. However, no such region is found in Scandinavia, and no clear

indication for such a region is found under the British Isles. Note that the regional

studies of Lambeck et al. (1998) and Milne et al. (2004) do not exclude the possibil-

ity of an ALVZ, but merely state that such a low-viscosity layer is not warranted

by the data.

The first studies on crustal low-viscosity layers in the GIA process are by Wu

(1997) on the effect of a CLVZ on GIA-induced seismicity and by Klemann & Wolf

(1999) on the effect on vertical displacements. After that, studies have shown the

effect of laterally homogeneous CLVZs on present-day (Di Donato et al., 2000a) and

late-Holocene (Kendall et al., 2003) sea-level change, and the global gravity field

as expected from GOCE (Vermeersen, 2003; van der Wal et al., 2004). The depth

of the CLVZ used in these studies varies between 20 and 35 km, the thickness

between 10 and 15 km and the viscosity between 1017 and 1019 Pas.

In this thesis we investigate the sensitivity of GOCE to gravity signatures of shal-

low low-viscosity layers induced by the GIA process. We focus on the crust, but

also consider the effect of asthenospheric low-viscosity. Ultimately, we try to an-

swer the following three questions:

1.4. Rationale and Outline 11

I. What are the amplitudes and distributions (spatial, spectral) of geoid height

perturbations due to low-viscosity layers and how are these compared to the

performance of GOCE?

II. Is GOCE sensitive to the properties of LVZs and the properties of the back-

ground model (earth stratification, ice-load history), and are there signa-

tures (spatial, spectral) that are robust to these properties?

III. Which unique information does GOCE provide on the rheology of the shallow

earth and which additional datasets can we use to constrain this rheology?

To answer these questions we first give a basic description of the theory (Chap-

ter 2), and models and data (Chapter 3) used in this thesis. The sensitivity analy-

sis for GOCE consists of four chapters:

1. In Chapter 4 we study the patterns generated by shallow LVZs in a radially

stratified earth and the sensitivity to the background earth stratification.

Here we also introduce the concept of spatial signatures;

2. In Chapter 5 we study the sensitivity of these patterns to the ice-load history

in a radially stratified earth. Here we also introduce the concept of spectral

signatures;

3. In Chapter 6 we study the effect of lateral heterogeneities on the patterns

and the incorporation of a regional model in a global background model.

Here we also investigate the use of surface velocities as an additional con-

straint on the shallow viscosity structure;

4. In Chapter 7 we study the effect of laboratory-derived creep laws on the

patterns and thus the use of heatflow data as an additional constraint on the

shallow low-viscosity structure. Here we also introduce a concept to invert

GOCE data for the properties of LVZs.

We answer the questions I, II and III in the Conclusions (Chapter 8), where we

also make recommendations for further research.

Chapter 2

Isostatic Adjustment Theory

Here we describe the theory needed to model the process of glacial-isostatic ad-

justment (GIA), which consists essentially of continuum mechanics to describe

the behavior of the solid earth and potential theory do describe the changes in

the gravity field. The governing equations are described in Section 2.1, while in

Sections 2.2 and 2.3 we describe how these are respectively implemented in the

spectral (SP) and finite-element (FE) method.

2.1 Governing Equations

2.1.1 Momentum and Laplace’s Equation

For problems in which inertia can be neglected and in which the body force is due

to gravity, conservation of linear momentum gives (e.g. Sabadini & Vermeersen,

2004, p. 4):

~∇·~~σ+ρ~g = 0 (2.1)

with ~~σ the stress tensor, related to the strain tensor by a constitutive equation

(Section 2.1.2), ρ the density and ~g the gravity acceleration.

Though we are mainly dealing with viscoelastic materials, we derive here theelastic

equation of momentum, the reason being that the two solution strategies we use

(normal modes, Section 2.2.3 and finite elements, Section 2.3) both start with this

equation, but use a different approach to deal with viscoelasticity. From Eq. (2.1)

we can derive, using linear perturbation theory and noticing that the initial hy-

drostatic pressure (pre-stress) will not change (Cathles, 1975, p. 13), the linearized,

14 Chapter 2. Isostatic Adjustment Theory

elastic equation of motion (Wu & Peltier, 1982; Wu, 2004; Sabadini & Vermeersen,

2004, p. 5):

~∇·~~σδ−~∇(~u ·ρ0 g0~er)−ρδg0~er +ρ0~gδ = 0 (2.2)

with ~u the displacement vector and where the subscripts (’0’, ’δ’) denote the ini-

tial and incremental state (Wolf, 1998), respectively. The second term represents

the advection of pre-stress and the third term internal buoyancy due to material

compressibility. For an incompressible material (Poisson’s ratio ν = 0.5, see Sec-

tion 2.1.2), ρδ = 0 and the third term vanishes. The fourth term describes the

effect of the incremental gravity field on the solid earth (self-graviation), where

the incremental gravity vector ~gδ is related to the incremental gravity potential φ

as:

~gδ =−~∇φ (2.3)

The incremental potential φ can be found from Poisson’s equation:

∇2φ= 4πGρδ (2.4)

which reduces for an incompressible earth model to Laplace’s equation:

∇2φ= 0 (2.5)

2.1.2 Constitutive Equations

Viscoelastic behavior can be simulated in a number of ways. The most popular is

Maxwell viscoelasticity, in which the total strain rate ǫi j is the sum of the rate of

change of the elastic strain ǫEi j

and creep strain ǫCi j

:

ǫi j = ǫEi j + ǫC

i j (2.6)

A linear relationship for ǫEi j

and the elements σi j of the stress tensor ~~σ is given by

Hooke’s Law, which is for an incompressible material equal to:

σi j =σ0δi j +2µǫEi j (2.7)

with σ0 the mean normal stress and µ the shear modulus or rigidity. For a linear

viscous material a similar relation holds:

σi j =σ0δi j +2ηǫCi j (2.8)

2.1. Governing Equations 15

(a) Maxwell model (b) Kelvin model

(c) Burgers model

Figure 2.1: Spring-dashpot analogues and relaxation curves for the Maxwell (a), Kelvin (b)and Burgers (c) model. The springs represent the elastic behavior (proportional to Young’smodulus E = 2µ+ν, with µ the rigidity and ν Poisson’s ratio) and dashpots the viscousbehavior (proportional to the viscosity η).


with η the linear or Newtonian viscosity. Using the definition of the deviatoric

stress (σ′i j=σi j −σ0

i j, with σ0

i j=σ0δi j) in Eq. (2.6) gives:

ǫi j =σ′

i j

2µ+

σ′i j

2η(2.9)

which is the constitutive equation for a Maxwell viscoelastic body. For a body that

is only linear in the elastic limit, i.e. the viscous part shows a non-linear relation

between stress and strain, a similar relation holds, with the linear viscosity η

replaced by the effective viscosity η∗. We refer to Chapter 7 (Section 7.2.2) for a

deriviation and an explicit formula for the effective viscosity.

A Maxwell body can be schematically represented by a combination of a spring

(elastic behavior) and dashpot (viscous behavior) in series (Figure 2.1.2a). Note

that the Maxwell model can both describe steady-state creep behavior on long

timescales and elastic behavior as found from seismic measurements for very short

timescales (though not the anelasticity associated with seismic attenuation, e.g.

Wu & Peltier, 1982). The characteristic timescale for stress relaxation is called

the Maxwell time τM , which is equal to the ratio of viscosity η to rigidity µ (e.g.

Ranalli, 1995, p. 222):

τM =η

µ(2.10)

On timescales shorter than the Maxwell time, deformation will be predominantly

elastic and for timescales larger than the Maxwell time predominantly viscous.

The Maxwell model may not always be suitable for processes on transient timescales

(Ivins & Sammis, 1996), as for example post-seismic deformation, which can be

described with the Burgers model (Maxwell and Kelvin model in parallel, Fig-

ure 2.1.2c). We thus assume by using a Maxwell rheology that the earth is de-

forming in a steady-state manner. The Kelvin model, which can be represented

as a spring and dashpot in parallel, and in which the elastic behavior delays the

viscous behavior (Figure 2.1.2b), cannot account for the elastic behavior of the

earth. Note that the E in the figure generally stands for Young’s modulus, which

is related to the rigidity µ by E = 2µ(1+ν), where ν is Poisson’s ratio.

In this thesis we will almost exclusively assume that the earth is incompressible,

in which case ν= 0.5 and E = 3µ, the only exception being Section 6.3 in Chapter 6.

The assumption of incompressibility is largely for convenience, as semi-analytical

normal-mode techniques (Section 2.2.3) require very high accuracy of the Bessel

functions arising for a compressible earth model (Wu & Peltier, 1982) and because

root-finding is complicated due to excitation of an infinite set of dilatation modes

(Vermeersen et al., 1995). However, though the effect of compressiblity on hor-

izontal and vertical velocities is large, the effect on geoid heights is small (see

Chapter 6).

2.2. Spectral Method 17

2.2 Spectral Method

In this thesis, the spectral (SP) method is used for computations on spherical, self-

gravitating earth models, which are radially stratified and have a linear rheology.

The basis for this method is the use of a spherical harmonic expansion of the gov-

erning equations, which follows naturally if Laplace’s equation (Eq. 2.5) is solved

by separation of variables in a spherical coordinate system (with latitude θ and

longitude λ). We use the following spherical harmonic expansion (Heiskanen &

Moritz, 1967, p. 20):

f (θ,λ) =∞∑

l=0

l∑

m=−l

FlmYlm(θ,λ) (2.11)

in which Flm are complex coefficients of degree l and order m and Ylm(θ,λ) are

surface spherical harmonics, fully normalized such that their power is unity over

the sphere (Mitrovica et al., 1994a; Heiskanen & Moritz, 1967, p. 31):

∫∫

Ω

Y ∗l′m′(θ,λ)Ylm(θ,λ)dΩ = 4πδll′δmm′ (2.12)

with dΩ = sinθdθdλ,∫∫

Ωdenoting integration over the entire solid angle, δi j

the Kronecker delta and where the asterisk denotes complex conjungation. The

summation in Eq. (2.11) is general cut-off at a maximum degree of expansion lmax

and written in short-hand as:

lmax∑

l=0

l∑

m=−l

→∑

l,m

(2.13)

The SP method is mainly employed in Chapters 4 and 5 and is fast, but is in

principle limited to radially stratified earth models with a linear rheology. This

is because upon the introduction of lateral variations in earth properties or non-

linearity in rheology, the principle of superposition no longer applies and the gov-

erning equations cannot longer be solved separately for each harmonic degree due

to mode coupling (Wu, 2002). We therefore use in Chapters 6 and 7, where we in-

troduce lateral heterogeneities and non-linear rheologies, the finite-element (FE)

method, see Section 2.3.

2.2.1 Love Numbers

On a radially stratified earth with mass M and radius R, the elastic response (in

θ,λ) at an angular distance ψ from a unit point load (in θ′,λ′), can be written


as a series of Legendre polynomials Pl(cosψ) as (Longman, 1963; Farrell, 1972;

Mitrovica et al., 1994a):

GU (ψ)= R ·1

M

∑

l

hl Pl(cosψ)

GV (ψ)= R ·1

M

∑

l

l l

∂Pl(cosψ)

∂ψ~eψ (2.14)

Gφ(ψ)=GM

R·

1

M

∑

l

(1+kl)Pl(cosψ)

where GU , GV and Gφ are the Green’s functions for respectively the radial dis-

placement U, the tangential displacement V and the incremental gravity poten-

tial φ. G is Newton’s gravitational constant and the dimensionless numbers hl ,

l l and kl are named load Love numbers (Longman, 1963; Farrell, 1972; Lambeck,

1988, p. 92) after A.E.H. Love, who introduced them in the beginning of the 20th

century.

To compute the response ℜ to a general surface load density ρLL(θ′,λ′) we convo-

lute the Green’s function with the load as (Mitrovica et al., 1994a; Lambeck, 1988,

p. 99):

ℜ(θ,λ) = R2

∫∫

Ω

Gℜ(ψ)ρLL(θ′,λ′)dΩ (2.15)

We now expand the load height L(θ′,λ′) in spherical harmonics as:

L(θ′,λ′)=∑

l,m

LlmYlm(θ′,λ′) (2.16)

and use the decomposition formula (Heiskanen & Moritz, 1967, p. 33) or addi-

tion theorem (e.g. Mitrovica et al., 1994a) to decompose the Legendre polynomials

Pl(cosψ) into spherical harmonics:

Pl(cosψ)=1

2l+1

l∑

m=−l

Y ∗lm(θ′,λ′)Ylm(θ,λ) (2.17)

Combining Eqs (2.15-2.17) and making use of Eq. (2.12), we can write for the total

response:

U(θ,λ) = R ·4πR2ρL

ME

∑

l,m

hl

2l+1LlmYlm(θ,λ)

~V (θ,λ) = R ·4πR2ρL

ME

∑

l,m

l l

2l+1Llm

~∇Ylm(θ,λ) (2.18)

φ(θ,λ) =GM

R·4πR2ρL

ME

∑

l,m

1+kl

2l+1LlmYlm(θ,λ)


with ~∇= (∂/∂θ)~eθ + (1/sinθ)(∂/∂λ)~eλ (Mitrovica et al., 1994a)

Eq. (2.18) shows that, if we know the surface load distribution and the properties

of the earth as represented by the Love numbers, we can make predictions of radial

and tangential displacements and the incremental gravity potential.

Satellite gravity missions as GRACE and GOCE deliver dimensionless (complex)

potential coefficients Clm defined as (Heiskanen & Moritz, 1967, p. 35):

φ=GM

R·∑

l,m

ClmYlm (2.19)

Comparing this with the expression for the incremental gravity potential of Eq. (2.18)

we see that:

Clm =4πR2ρL

ME

1+kl

2l+1Llm (2.20)

This means that from potential coefficients as measured by GRACE and GOCE

we can obtain information on the (radially stratified) solid earth, as represented

by kl , and the distribution of the load Llm. In the next section we show how we can

derive geoid heights N and gravity anomalies ∆g from these potential coefficients.

Note that here we have only considered an elastic earth and a static load. For

GIA, we have to consider the viscoelastic behavior of the earth, which is repre-

sented by a time-dependent Love number (kl (t) for the incremental gravity poten-

tial) and the history of the loading (Llm(t)). In Section 2.2.3 we show how these

time-dependent Love numbers can be computed using normal-mode techniques

and that the multiplication of the Love numbers with the surface load becomes a

convolution in time.

2.2.2 Geoid Heights and Gravity Anomalies

The geoid height N, the distance between the geoid and the undeformed surface,

can be computed using Brun’s formula (Heiskanen & Moritz, 1967, p. 85):

N =φ

γ0(2.21)

where γ0 = g0(R) = GM/R2 is the magnitude of the initial gravity at the unde-

formed surface. Using Eq. (2.19) we can write for the spherical harmonic compo-

nents Nlm of N (compare with Wahr et al., 1998, Equation 12):

Nlm = R · Clm = R ·4πR2ρL

ME

1+kl

2l+1Llm (2.22)


The gravity anomaly is defined as the difference between the incremental gravity

(Eq. 2.3) on the geoid and the initial gravity γ0 on the undeformed surface, and is

equal to (Heiskanen & Moritz, 1967, p. 85):

∆g =−∂φ

∂r+

1

γ0

∂g

∂rφ (2.23)

The term ∂g/∂r can be regarded as the free-air correction and is on the undeformed

surface equal to:

(

∂g

∂r

)

r=R

=(

∂(GM/r2)

∂r

)

r=R

=−2γ0

R(2.24)

This gives for Eq. (2.23) (Heiskanen & Moritz, 1967, p. 89):

∆g =−∂φ

∂r−

2

Rφ (2.25)

If there are no masses outside the geoid we can use the following expansion for φ

(Heiskanen & Moritz, 1967, p. 35):

φ=GM

R·∑

l,m

(

R

r

)l+1

ClmYlm (2.26)

which gives for the coefficients ∆glm of the gravity anomaly:

∆glm = γ0 · (l−1)Clm (2.27)

which can be used to compute the gravity anomaly from static satellite gravity

data, or, by replacing Clm with the rate-of-change ˙Clm of the dimensionless co-

efficients, the rate-of-change of the gravity anomaly from time series of satellite

gravity data.

To predict gravity anomalies from GIA models, however, we have to consider that

in Eq. (2.20) the direct attraction due to the surface mass load (the 1-term) is

initially outside the geoid. Therefore we have to use for this part of the incremental

potential φ the expansion (Heiskanen & Moritz, 1967, p. 34):

φ=GM

R·∑

l,m

( r

R

)l

ClmYlm (2.28)

We can now compute the gravity anomaly from the incremental potential φ (Eq. 2.18)

by using the appropriate expansions (Eq. (2.28) for the direct term and Eq. (2.26)

for the mass redistribution term proportional to kl), and their gradients in Eq. (2.25):

∆glm = γ0 ·4πR2ρL

M

−(l+2)+ (l−1)kl

2l+1Llm (2.29)


To show the difference with Eq. (2.27), we write, using Eq. (2.20), this expression

as a function of the potential coefficients :

∆glm = γ0 ·−(l+2)+ (l−1)kl

1+kl

Clm (2.30)

which shows that in general we cannot use Eq. (2.27) to directly convert potential

coefficients predicted from GIA models to gravity anomalies.

Eq. (2.29) is essentially the same as Equation 5 in Mitrovica & Peltier (1989),

however, they take a somewhat more complicated route to arrive at this equation.

They start with Longman (1963), who finds, using the appropriate expansions for

Eq. (2.20), that the incremental gravity at the undeformed surface is:

∆gUlm = γ0 ·

4πR2ρL

M

−(l+2)+ (l−1)kl

2l+1Llm (2.31)

Then Longman (1963) and Mitrovica & Peltier (1989) compute the incremental

gravity at the deformed surface, which consists of the change in acceleration from

moving through the incremental gravity field, proportional to the radial displace-

ment Love number hl , the direct attraction of the mass load and the effect of

mass redistribution, proportional to kl (Farrell, 1972). Using the free-air correc-

tion (Eq. 2.24) as an approximation for the gravity change in the incremental field,

and using the appropriate expansions, Longman (1963) andMitrovica & Peltier

(1989) find from Eq. (2.31):

∆gDlm = γ0 ·

4πR2ρL

M

−l+ (l+1)kl −2hl

2l+1Llm (2.32)

which is the incremental gravity at the deformed surface. To find the gravity

anomaly (at the geoid), Mitrovica & Peltier (1989) transform Eq. (2.32) to the

geoid, which gives a change in acceleration proportional to hl − (1+ kl ), to obtain

Eq. (2.29).

2.2.3 Normal-Mode Relaxation Theory

In the SP method, it is common to compute load-Love numbers in the Laplace-

transformed domain for an elastic earth. An inverse Laplace transformation then

yields viscoelastic Love numbers in the time domain, according to the correspon-

dence principle (see e.g. Peltier, 1974). To compute the Love numbers, we use a

semi-analytical normal-mode relaxation model (Peltier, 1974; Wu & Peltier, 1982;

Vermeersen & Sabadini, 1997). First, we transform the governing equations from

Sections 2.1.1 and 2.1.2 to the Laplace domain. Upon expansion of the variables in

Legendre polynomials (e.g. U =∑

l Ul Pl cosψ for the radial displacement), we can


write the governing equations as a system of ordinary differential equations (Wu

& Peltier, 1982; Sabadini & Vermeersen, 2004, p. 12):

d~y

dr= ~~A ·~y (2.33)

with the components of ~y equal to (Wu & Peltier, 1982; Sabadini & Vermeersen,

2004, p. 11):

y1 =Ul

y2 =Vl

y3 = Trl (2.34)

y4 = Tθl

y5 =−φl

y6 =−∂φl

∂r−

l+1

rφl +4πGρ0Ul

Note that the components are dependent on the harmonic degree l, the radial dis-

tance from the center of the earth r and the Laplace variable s, which has the

dimension of inverse time. Trl and Tθl are the components of the radial and tan-

gential stress, respectively, and y6 is chosen such that it zero at the free surface

of the earth and continuous at internal boundaries. This can be understood by

considering the boundary condition at the free surface (Wu, 2004; Sabadini & Ver-

meersen, 2004, p. 19):

∂φel

∂r−

∂φl

∂r=−4πGρ0Ul (2.35)

with φel, φl the incremental potential above, respectively below, the free surface

(compare with Eq. (6.7) in Chapter 6). Outside the free surface, the potential

obeys Eq. (2.26) and we find for the gravity gradient:

∂φel

∂r=−

l+1

rφe

l (2.36)

As φel=φl at the surface of the earth, we find that:

y6 =−∂φl

∂r−

l+1

rφl +4πGρ0Ul = 0 (2.37)

at the free surface.

For a multilayer model, the boundary conditions at the surface of the earth can

be coupled to the boundary conditions at the core-mantle boundary (CMB) using a


propagator matrix technique (Sabadini & Vermeersen, 2004, p. 19). For a free sur-

face, the radial and tangential stress vanish at this surface (y3 = y4 = 0), together

with y6. As the boundary conditions at the CMB can be written as linear combina-

tions of three constants (Wu & Peltier, 1982; Sabadini & Vermeersen, 2004, p. 21),

it can be shown that only for certain values of the Laplace variable s the homo-

geneous system of differential equations (Eq. 2.33) has non-zero solutions. These

values of s are the inverse relaxation times si of relaxation modes i. The number

of modes M depends on the number of layers, and their rheology and density. For

an example, see Figure 4.1a in Chapter 4, which is based on the 8-layer model of

Table 3.1 in Chapter 3.

A homogeneous viscoelastic earth generates one buoyancy mode (M0) and each

additional viscoelastic layer, with a different density and Maxwell time (Eq. 2.10),

generates one buoyancy mode (e.g. M1) and two viscoelastic modes (e.g. T1(2), Wu

& Ni, 1996). In normal-mode methods an elastic layer is simulated by not taking

into account the buoyancy mode and one of the two viscoelastic modes of a vis-

coelastic layer, for which the relaxation time increase with viscosity and for which

the relaxation time is larger than 1 Myrs for a viscosity of 1025 Pas. The remain-

ing viscoelastic mode is then due to the difference in (infinite) Maxwell time of the

upper and (finite) Maxwell time of the lower crust (Wu & Ni, 1996) and labeled

L0 for the lithosphere. Note that no buoyancy mode and no viscoelastic modes are

generated for two elastic layers on top of each other (i.e. crust on lithosphere in

Table 3.1). For an inviscid layer (e.g. the fluid outer core) only a buoyancy mode,

in the case of a density constrast, is generated (e.g. C0, Wu & Ni, 1996)

Upon loading the earth, an inhomogeneous system of differential equations has to

be solved for Ul , Vl and φl . The boundary conditions at the surface of the earth

are then (Wu & Peltier, 1982; Sabadini & Vermeersen, 2004, p. 31):

y3(R)=−g(R)(2l+1)

4πR2, y4(R)= 0, y6(R)=−

G(2l+1)

R2(2.38)

From solving the inhomogeneous system of differential equations the residuals r i

or strengths r i /si of each mode can be found (see Figure 4.1b in Chapter 4), from

which the viscoelastic Love number can be calculated.

For harmonic degree one, only two of the three boundary conditions (Eq. 2.38) are

needed and the consistency relation (Farrell, 1972; Greff-Lefftz & Legros, 1997):

y3(R)+2y4(R)−g(R)

4πGy6(R)= 0 (2.39)

ensures automatically that the third boundary condition is met. With the two

boundary conditions, the response to a surface load for degree one can be found,

except for a shift of the origin or center of mass. To conserve the center of mass, it


can be shown that the degree-one surface potential has to be zero (Greff-Lefftz &

Legros, 1997):

y5 = 0 (2.40)

Note that our implementation of degree one differs from the implementation of

Farrell (1972) and Mitrovica et al. (1994a), who use a center of earth rather than

a center of mass definition (Blewitt, G., 2003). In this thesis we in general do

not consider degree one deformation, except in Chapter 6, and we put the Love

numbers h1 and l1 to zero and k1 =−1, compare with Eq. (2.18).

The response in the time domain can now be described with viscoelastic load Love

numbers defined as (Peltier, 1974):

kl (t)= kEl δ(t)+

∑

i

rli exp(sl

i t) (2.41)

where kEl

is the elastic Love number of degree l for the incremental gravity po-

tential φ, the summation is over all modes i of the particular model, and rli

is the

residual for the inverse relaxation time sli

of the incremental gravity potential φ.

For a Heaviside kind of loading H(t− t0), with the t0 the time that the step load is

applied, the viscoelastic Love numbers become (Wu & Peltier, 1982):

kHl (t) = kE

l +∑

i

rli

sli

[

1−exp(sli t)

]

(2.42)

where the ratiorl

i

sli

is called the modal strength.

For the total viscoelastic response to a load that is stepped in time Llm(t)= Lnlm

H(t−tn), we can now write Eq. (2.18) as (Wu & Peltier, 1982; Mitrovica & Peltier, 1991;

Mitrovica et al., 1994a):

φ(θ,λ, t) =GM

R·4πR2ρL

ME

∑

l,m

(1+kEl

)Llm(t)+∑

nβ(l, tn, t)Lnlm

2l+1Ylm(θ,λ) (2.43)

with β(l, tn, t) =∑

i

rli

sli

[

1−exp(sli(t− tn)

]

. Similar expressions can be derived for the

radial and tangential displacement load Love numbers hl and l l .

2.3 Finite-Element Method

In this thesis, the finite-element (FE) method is used for computations on flat, non-

self-gravitating earth models, which can include lateral heterogeneities (Chap-

ter 6) and non-linear rheologies (Chapter 7). In Section 2.3.1 we give a short

2.3. Finite-Element Method 25

derivation of the way Eq. (2.1) in the absence of a gravity field is solved with the

FE method in one dimension. In Section 2.3.2 we then describe shortly how we

compute the gravity potential when using an FE model and in Section 2.3.3 how

viscoelasticity is treated in the FE method.

2.3.1 General Theory

In one dimension1 and in the absence of a gravity field, Eq. (2.1) becomes, using

the deviatoric part of Eq. (2.7) and ǫ11 =∇u, with ∇= d/dx and u the displacement

in the x-direction:

2µ∇·∇u= f (2.44)

If we now multiply both sides with an arbitrary trial function v and integrate over

the complete interval (taken for convenience to be [0, 1]) we get:

2µ

∫1

0(∇·∇u)vdx =

∫1

0f vdx (2.45)

Applying integration by parts to the left-hand side and assuming the vanishing of

the derivative of u at the boundaries we can write this as:

2µ

∫1

0∇u∇vdx =

∫1

0f vdx (2.46)

We approximate the unknown u by a linear combination u(x)=∑N

i=1biφi(x), where

the coefficients are given by bi = u(xi), and the basis functions by φi(xi) = 1 and

φi(x j) = 0. As the trial functions are arbitray, we can take them to be equal to the

basis functions, v = φk (Galerkin method, Zienkiewicz, 1977, p. 50) and replacing

u with u we get:

2µN∑

i=1

bi

∫1

0∇φi∇φkdx =

∫1

0f φkdx (2.47)

which can be written as the following matrix equation:

bi Aik = gk (2.48)

with Aik =∫1

0 ∇φi∇φkdx the elements of the stiffness matrix and in which the

source term gk = (1/2µ)∫1

0 f φkdx can be modified to include the boundary condi-

tions. The solution for the unknown bi = u(xi) is then:

bi = A−1ki gk (2.49)

1this derivation is from a course of Heiner Igel, Ludwig Maximilian University of Munich, for a

more general treatment we refer to Zienkiewicz (1977, Chapter 3).


2.3.2 Laplace’s Equation

Laplace’s equation (Eq. 2.5) can be solved in the same way as the momentum equa-

tion (Eq. 2.1), with the displacement u replaced by the incremental gravity poten-

tial φ, and gk determined by boundary conditions only. This would be advanta-

geous if we could solve Eqs (2.1) and (2.5) at the same time, which is, however, not

possible with the FE package we use (ABAQUS, see next chapter). A more practical

method is therefore to solve Laplace’s equation by separation of variables, which

leads to a 2D Fourier transform on the horizontal coordinates:

f =∞∑

kx=0

∞∑

ky=0

F(kx,ky)ei(kx x+ky y) (2.50)

with F(kx,ky) complex coefficients and kx and ky the wavenumbers in the x- and

y-direction, respectively. See Section 6.2 for a further derivation.

2.3.3 Viscoelasticity

For the viscoelastic problem, we can derive a time scheme by rearranging and

discretizing the constitutive equation for a Maxwell body (Eq. 2.9) as (Martinec,

2000):

σ′k+1−σ′

k = 2µ (ǫk+1−ǫk)− (∆t/τM )σ′m (2.51)

with ∆t = tk+1 − tk, τM = η/µ the Maxwell time and the choice of m determining

the time scheme. For m = k an explicit (or forward) Euler time scheme is defined

and for m = k+1 an implicit (or backward) Euler time scheme (Press et al., 1992,

p. 728). We find for the explicit scheme:

σ′k+1 = 2µǫk+1+

[

(1−∆t/τM )σ′k −2µǫk

]

(2.52)

and for the implicit scheme:

σ′k+1 =

1

1+∆t/τM

2µǫk+1+[

1

1+∆t/τM

(

σ′k −2µǫk

)

]

(2.53)

The terms between square brackets are the history terms. We see that for the

explicit scheme ∆t should be smaller than a certain critical value (say τM /2) for

the scheme to be stable. The implicit scheme is unconditionally stable, although

for large time steps iterations are required (Press et al., 1992, p. 730) to establish

force equilibrium.

Chapter 3

Mechanical Model and

Satellite Gravity Data

We will use two mechanical earth models, both Maxwell viscoelastic and incom-

pressible: A spherical, radially (1D) stratified, self-gravitating model based on the

spectral (SP) method, as described in the previous chapter (Section 2.2), and a

flat, 3D-stratified, non-self-gravitating model based on finite elements (FE, Sec-

tion 2.3). The SP method is implemented in a code developed by Sabadini et al.

(1982), Spada et al. (1992) and Vermeersen & Sabadini (1997), with refinements

by the master-students Mark-Willem Janssen and Wouter van der Wal of the Fac-

ulty of Aerospace Engineering, Delft University of Technology, and by the author

of this thesis. It is extensively benchmarked and used as a benchmark for other

studies (Wu et al., 2005; Wang et al., 2006; Spada & Boschi, 2006).

The FE implementation of the commercial package ABAQUS is used, which already

exists since the 1970s and which is constantly being developed and tested. The

accuracy of computations in ABAQUS is controlled by comparing the error in creep

strain increment, which is the change in creep strain rate over an interval times

the length of the interval, to a predefined tolerance (CETOL in ABAQUS). If the

accuracy is too low, the time step is decreased. ABAQUS generally starts with an

explicit scheme and switches to an implicit time scheme (Section 2.3.3) if the time

step is limited by accuracy only. For a further description of the FE model we

refer to Section 6.3, where we also discuss the use of Winkler foundations to model

advection of pre-stress.

Both models use a certain earth stratification (Section 3.1) and are forced by a cou-

pled ice- and ocean-load history (Sections 3.2 and 3.3). In Section 3.4 we describe

the satellite gravity data used in this thesis.

28 Chapter 3. Mechanical Model and Satellite Gravity Data

Table 3.1: Standard earth stratification

Layer Depth Densitya ρ Rigiditya µ Viscosity η

[km] [kg/m3] [GPa] [Pas]

STD UNI

crust 0 2700 27 elastic elastic

lithosphere 32b 3380 68 : :

asthenosphere 80c : : 5 ·1020 1 ·1021

low-velocity zone 115d : : : :

upper mantle 220 3480 77 : :

transition zone 400 3870 108 : :

lower mantle 670 4890 221 5 ·1021 1 ·1021

core 2891 10925 0 0 0

aDensity ρ and rigidity µ are volume-averaged from PREM (Dziewonski & Anderson, 1981)

b30 km in Chapters 6 and 7

c100 km in Chapter 7

d140 km in Chapter 6, 160 km in Chapter 7

3.1 Earth Stratification

In this study, we are mainly interested in perturbations due to low viscosity and

lateral heterogeneity in the shallow earth, down to about 200 km. Perturbations

are the differences between a earth stratification which includes low viscosity and

lateral heterogeneity and a background stratification. The background stratifica-

tion is only of importance insofar it influences the perturbations. Therefore we

will use relatively simple reference background models and test the sensitivity to

changes in the parameters. Our reference background stratifications have an or-

der of magnitude increase in viscosity across the 670 km boundary (ηUM = 5 ·1020,

ηLM = 5·1021 Pas, ’STD’), in line with Lambeck et al. (1998) and Milne et al. (2004),

or are univiscous (ηUM = ηLM = 1021 Pas, ’UNI’). Both have a lithospheric thickness

LT of 80 km, except in Chapter 7 where it is 100 km (Table 3.1). For consistency,

we will also use in Chapter 5 the ice-load histories with their preferred earth strat-

ification, see Section 3.2. Note that when using the flat FE model, the lower mantle

extends to a depth of 10,000 km and no inviscid core is assumed (see Section 6.3).

We choose a reference model for a crustal low-viscosity zone (CLVZ) with a lower

crust starting at a depth of 20 km, with a thickness of 12 km and a viscosity

of 1018 Pas (Chapters 4 and 5) or a thickness of 10 km and a viscosity of 1019

Pas (Chapters 6 and 7). We have modelled an asthenospheric low-viscosity zone

(ALVZ) below the lithosphere of 80 km, with a thickness of 35 km and a viscosity

of 1018 Pas (Chapters 4 and 5) or a thickness of 60 km and a viscosity of 1019 Pas

3.2. Ice-Load History 29

(a) ice-sheet profiles (b) deglaciation histories

051015202530

0

20

40

60

80

100

120

time [kyrs BP]

ice

equi

vale

nt s

ea le

vel [

m]

RSES, totalICE5G, totalRSES, Northern EuropeICE5G, Northern Europe

Figure 3.1: Theoretical ice-sheet profiles (a) and the deglaciation history for the RSES andICE-5G ice-load histories (b). In (b), ice equivalent sea level is equal to eustatic sea level asdefined in Section 3.3.

below an 80 km lithosphere (Chapter 6) or a 100 km lithosphere (Chapter 7).

We take the density ρ and rigidity µ from the Preliminary Reference Earth Model

(PREM, Dziewonski & Anderson, 1981), which is based on seismic measurements.

This model consists of 94 layers, which we volume-average to models of three lay-

ers (crust/lithosphere, mantle, core) to nine layers (upper crust, lower crust, litho-

sphere, asthenosphere, low-velocity zone, upper mantle, transition zone, lower

mantle, core).

3.2 Ice-Load History

To test the accuracy of the FE method (Section 6.4 in Chapter 6) and to test the

thermomechanical earth model of Chapter 7 (Appendix C), we use an elliptic ice-

load profile, defined as:

h(θ) = H

√

1− (θ/θM )2 (3.1)

with H the height at the center of the ice sheet and θM the extent of the ice sheet.

In this thesis we take H = 2500 m and θM = 8 (about 900 km), which gives an ice

sheet that is comparable to the former Fennoscandian ice sheet (e.g. Amelung &

Wolf, 1994; Wu, 1992a, 1995). We implement this profile as a Heaviside load, i.e.

we apply the load at t= t0, keep it constant and look at the predictions of our GIA

model after 10 kyrs of loading.

An elliptical profile is close to theoretical predictions of the equilibrium shape of

ice sheets based on the mechanism of internal deformation (Weertman profile) or

basal sliding (Vialov profile, van Veen, 1999, p. 156). Thermomechanical ice-sheet


(a) Modified ICE-3G

1000 2000 3000 4000 5000ice height [m]

(b) ICE-5G

1000 2000 3000 4000 5000ice height [m]

Figure 3.2: Modified ICE-3G (a) and ICE-5G (b) ice-load distribution at LGM.

models take mechanisms of internal deformation and basal sliding into account

and can simulate ice-load histories from paleo-temperatures for the atmosphere

and a global circulation model (e.g. Bintanja et al., 2002). These models mostly

use very simplified earth models (see Le Meur & Huybrechts, 1996, for a discus-

sion), but recently thermomechanical ice-load histories have been generated on

spherical, viscoelastic earth models (Tarasov & Peltier, 2002; van den Berg et al.,

2008). In Schotman & Vermeersen (2005) we have used an ice-load history from a

thermomechanical model, comparable to the one in Bintanja et al. (2002), to gen-

erate LVZ-induced perturbations, see also Appendix E. However, due to the lack

of geomorphological constraints in that model, we have not used it in this thesis.

Lambeck et al. (1998) use the Weertman profile to constrain an ice-load history

for Fennoscandia. We will use a global extension of this ice-load history as our

reference (denoted ’RSES’ in this thesis, see Figure 1.3b for the distribution at

LGM and Figure 3.1b for the deglaciation history). We take earth parameters

from our standard earth model (STD, LT = 80 km, ηUM = 5 ·1020 Pas and ηLM =5 ·1021), which are close to the values found by Lambeck et al. (1998) for Northern

Europe. The combination of ice and earth is referred to as ’RSES(STD)’ (’STD’ in

Chapter 7).

We will use the recent ICE-5G ice-load history (see Figures 3.1b and 3.2b, Peltier,

2004), which includes ice heights predicted from thermomechanical ice-sheet mod-

els in Greenland and Northern America, to test the sensitivity of our GIA model

to changes in the ice-load history. ICE-5G also includes information about the last

glaciation phase. However, because the details of the glaciation phase have in gen-

3.3. Ocean-Load History 31

eral not a large influence on predictions, and for ease of implementation, we will

use only the information from 30 kyrs BP to the present (Figure 3.1b) and use a

linear increase from zero ice height at 120 kyrs BP to the ice heights at 30 kyrs

BP. The preferred earth stratification of ICE-5G is VM2 (see Section 1.2), which

we have approximated to LT = 120 km, ηUM = 4 ·1020 Pas and ηLM = 2 ·1021 Pas

(’ICE-5G(VM2)’). To test as much as possible the sensitivity to the ice-load history

only, we also use ICE-5G with the STD model (’ICE-5G(STD)’, ’STDi’ in Chapter 7).

In Chapters 4 and 5 we use a modified version of the ICE-3G ice-load history (Fig-

ure 3.2a, Tushingham & Peltier, 1991). This ice-load history is not directly com-

parable with the above-mentioned histories as ICE-3G is given in uncalibrated

kyrs. ICE-3G has however has been until recently extensively used with a defini-

tion of ’kyr’ that agrees with ’calibrated’ and not ’uncalibrated’ (e.g. Tushingham &

Peltier, 1991; Mitrovica et al., 1994b; Vermeersen et al., 1998; Milne et al., 1999).

Following Milne et al. (2002) we have increased the volume of ICE-3G with a fac-

tor 1.2. Furthermore, we have smoothed the ICE-3G model to remove holes that

arise because of the finite-disc definition of this model, using a Gaussian filter

(e.g. Wahr et al., 1998) with a halfwidth of 200 km. To show the sensitivity to

the filtering we also use a halfwidth of 100 km. We use this ice model with its

preferred earth model VM1 (Section 1.2, ’ICE-3G(VM1)’) and the standard model

STD (’ICE-3G(STD)’).

3.3 Ocean-Load History

In the simplest case, the ocean-load history is obtained by distributing the change

in continental ice mass uniformly (eustatically, Mitrovica & Peltier, 1991) over the

ocean basins:

∆SE =−∆MI

AoρW

(3.2)

with ∆SE the eustatic change in ocean height, ∆MI the change in ice mass, Ao the

ocean area and ρW the density of sea water. However, due to self-gravitation (see

also the introduction to Chapter 6), the actual change in ocean load will deviate

from this eustatic load and will be equal, over the ocean basins, to the change in

sea level ∆SL, which is also defined over land:

∆SL =φ

γ0−U +CSL (3.3)

This sea level equation (Farrell & Clark, 1976) states that the incremental sea level

∆SL is equal to the geoid height (N =φ/γ0+CSL, with φ the incremental potential

and where CSL is independent of geographical location) minus the incremental


Table 3.2: Operator F j that describes the grounding and floating of ice

F j−1 = 0 F j−1 = 1

F j = 0 ice growth/melt grounding of ice

F j = 1 influx of water no ice growth/melt

radial displacement U. This equation is solved in the spectral domain, except for

the mapping of ∆SL on the ocean basins, which is performed in the spatial domain

(pseudo-spectral method, Mitrovica & Peltier, 1991). From this mapping CSL is

obtained, because the average of ∆SL over the ocean basins should be equal to

∆SE (⟨∆SL⟩o/Ao =∆SE , with ⟨⟩o denoting integration over the ocean area, Farrell

& Clark, 1976; Mitrovica & Peltier, 1991), which gives for CSL:

CSL =∆SE −1

Ao

⟨

φ

γ0−U

⟩

o

(3.4)

The sum of CSL and the incremental potential φ divided by γ0 gives then the geoid

height N, which is the equipotential surface at sea level.

To include time-dependent coastlines and meltwater influx into formerly glaciated

areas that are now below sea level (Section 1.2), we have implemented the theory

as described by Milne et al. (1999) and more recently by Mitrovica & Milne (2003).

From Mitrovica & Milne (2003) we use equation (39), which is in somewhat differ-

ent notation (and where the geographical coordinates are omitted):

∆S j =[

SL j −SL j−1

]

O jF j +T j−1

[

O j−1F j−1−O j F j

]

(3.5)

with ∆S j the incremental change in ocean height at timestep j, SL j the change

in global sea level from the onset of loading to time t j , T j the topography at time

t j , and where O j describes the time-dependent ocean function (which is equal

to one over the oceans and zero over land) and F j the grounding of ice (which

is equal to one if ice is grounded and zero if not). In Table 3.2 we have shown

possible combinations of the latter for subsequent times t j−1, t j . To compute the

time-dependent ocean function O j we need the paleotopography at time t j . This

is estimated from the present-day topography, as derived from the ETOPO5 data

set (NOAA, 1988), and the sea level change from that time to the present. As the

latter is not known, but is computed in the process, the computation is iterated

over a full glacial cycle. The operator F j is determined by comparing the mass per

unit area of the ocean at a certain grid point with that of the prescribed ice-load

(as taken from a certain ice-load history) at that point.

In this thesis we will in general assume a eustatic ocean load, because computa-

tion times are considerably smaller when neglecting the effect of self-gravitation,

changing coastlines and meltwater influx. We test this assumption in Chapters 5

and 6.

3.4. Satellite Gravity Data 33

(a) degree amplitudes

0 50 100 150 20010

−4

10−3

10−2

10−1

100

harmonic degree

geoi

d he

ight

deg

ree

ampl

itude

[m]

σ = 1.5 mEσ = 3.0 mEGOCEGGM02SRSES(STD)

(b) recovery errors for 1.5 mE

−10 −5 0 5 10geoid height perturbation [cm]

Figure 3.3: Geoid height degree amplitudes of the expected GOCE and realized GRACE(GGM02S) performance (a) and an example of recovery errors from GOCE in NorthernEurope (b).

3.4 Satellite Gravity Data

ESA’s upcoming GOCE satellite gravity mission, which is planned for launch in

the summer of 2008, is designed to map the static global gravity field with cen-

timeter accuracy in geoid height at a resolution of one hundred kilometer or better

(Visser et al., 2002). To counteract the effect of attenuation of especially the higher

harmonics of the gravity field with altitude (Section 1.3), GOCE uses a gradiome-

ter (Figure 1.5b), consisting of three orthogonal pairs of accelerometers. The gra-

diometer measures directly gravity gradients by differing the accelerations along

an axis (differential mode), except for a scale-factor due to an unknown instrument

gain and a possible misalignment of the accelerometers. To counteract the attenu-

ation, GOCE will moreover fly at a very low altitude of 250−300 km. A drag-free

control compensates for the relatively large atmospheric drag at this altitude, us-

ing non-gravitational accelerations as estimated from the common mode (sum of

accelerations along an axis) of the gradiometer.

To compare predictions from our GIA models with satellite gravity data as ex-

pected from GOCE, we will use degree amplitudes. Degree amplitudes are the

square-root of the degree variances of the dimensionless potential coefficients as

expected from GOCE and are for geoid height perturbations equal to (compare

with Eq. 2.22):

al = R ·

√

√

√

√

l∑

m=0

ClmC∗lm

(3.6)


and for gravity anomaly perturbations (compare with Eq. 2.27):

bl = γ0 · (l−1)

√

√

√

√

l∑

m=0

ClmC∗lm

(3.7)

where R is the radius of the earth, γ0 the gravity at the undeformed surface of the

earth, Clm a set of fully normalized, dimensionless potential coefficients of degree

l and order m, and where the asterisk denotes complex conjungation.

In Figure 3.3a we give the geoid height degree amplitudes of predictions from our

reference earth-ice model combination RSES(STD) and the estimated GOCE per-

formance based on formal errors (orbit accuracy, measurement noise and down-

ward continuation, Visser et al., 2002). Note that the harmonic degree l corre-

sponds, as a rule of thumb, with a resolution (half-wavelength) of 20,000/l km

(footnote 2 in Section 1.3) and that we are using only part of the information that

is available in our predictions, because for each degree we sum over all orders. This

simplifies comparison but neglects a large amount of longitudinal information.

In Chapter 7 (Section 7.5) we compare predictions with the expected performance

of GOCE in the spatial domain. To simulate the performance of GOCE, we add

white noise with a standard deviation of σ = 1.5 mE (1 Eötvös = 10−9 s−2) to one

month of gravity gradients computed along a circular orbit (no polar gap) at 250

km altitude. We recover the gravity field using an iterative block-diagonal solu-

tion method as described in Klees et al. (2000). We see from Figure 3.3a that the

spectrum of this simulated recovery error (’1.5 mE’) corresponds well with the es-

timated GOCE performance computed from the more extensive set-up of Visser et

al. (2002). In Figure 3.3b we have plotted an example of the spatial distribution of

the recovery error in Northern Europe. Note the increase with error for decreas-

ing latitude due to the increase of the distance between the groundtracks of the

satellite (lower data density). In Chapter 7 we will, as a worst-case scenario, also

use measurement noise with a standard deviation of σ= 3.0 mE, which leads to a

factor 2 increase in amplitudes.

We also use results of the GRACE satellite gravity mission (Tapley et al., 2004),

based on the performance over a 363-day period (model GGM02S, Tapley et al.,

2005). GGM02S is based on GRACE measurements only and not complemented

with other data, but the difference with a combined gravity model (GGM02C) is

only visible from about degree 80 (Tapley et al., 2005). From Figure 3.3a we see

that GRACE performs better than GOCE for the long wavelengths (below degree

∼ 20), but that the error increases more rapidly with increasing harmonic degree.

Chapter 4

Gravity Field Perturbations

due to Low-Viscosity Zones

In van der Wal et al. (2004) we have shown that in the glacial isostatic adjustment process,

a crustal low-viscosity zone (CLVZ) can introduce variations in geoid height up to several

decimeters with spatial scales down to hundred kilometers underneath and just outside for-

merly glaciated areas. Here we investigate the underlying physics using relaxation spectra,

and extent the work to asthenospheric low-viscosity zones (ALVZs). We show that, whereas

a CLVZ increases the relaxation times due to an extra buoyancy mode, an ALVZ in general

enhances material flow. We compare the predictions of LVZ-induced perturbations with the

expected performance of GOCE and the realized performance of GRACE (GGM02S). We show

that both GOCE and GRACE are sensitive to an ALVZ up to harmonic degree 60, but that

especially GOCE is predicted to provide information on CLVZs. GOCE is also sensitive to

properties of the LVZ, though this sensitivity is comparable to the sensitivity to the ice-load

history and earth stratification of the background model. We therefore focus on the detection

of LVZs and generate predictions of LVZ-induced perturbations that are insensitive to varia-

tions in the properties of the LVZ and in the upper mantle viscosity of the background model

(spatial signatures). The upper mantle viscosity seems, together with uncertainties in the ice-

load history, to be the largest background model error source. We estimate that uncertainties

in the earth stratification of the background model restrict the lower bound of recoverable

harmonic degrees to 30-50. The upper bound is estimated from the GOCE performance to be

around 120 for a CLVZ and about 65 for an ALVZ. Finally we show the effect of the inclusion

of an LVZ on estimates of present-day sea level change in western Europe.

This chapter and the next chapter are based on Schotman & Vermeersen (2005) and Schotman et al.

(2007), but are reworked for consistency and to include recent developments (GGM02S) and research

(role of the background earth stratification, spatial signatures).

36 Chapter 4. Low-Viscosity Zones

4.1 Introduction

As described in Section 3.4, the GOCE satellite gravity mission is designed to map

the static global gravity field with centimeter accuracy in geoid height and mgal

accuracy in gravity anomaly, at a resolution of one hundred kilometer or better.

Features in such a high-resolution gravity field can be associated with geophysical

processes that act on a regional scale. An example of such a process is the response

of shallow low-viscosity zones (LVZs) to loading and unloading of the crust, as for

example in glacial isostatic adjustment (GIA).

Some GIA studies, focussing on the vertical displacement (Klemann & Wolf, 1999),

present-day (Di Donato et al., 2000a) and late-Holocene (Kendall et al., 2003) sea-

level change, and the global gravity field as expected from GOCE (Vermeersen,

2003), have included a CLVZ at a depth of 20 to 35 km, with a thickness of 10

to 15 km and a viscosity of 1017 to 1019 Pas. We choose a model with a lower

crust starting at a depth of 20 km, with a thickness of 12 km and a viscosity of

1018 Pas, in a lithosphere of 80 km thickness (Table 4.1). This model seems to

be especially appropriate for the coastal shelf areas of western Europe for which

we show in Section 4.6 the influence of LVZs on estimates of present-day sea-level

change. These areas have a crustal thickness of 30−35 km (Mooney et al., 1998)

and a relatively high heat flow of 60−80 mW/m2 (Pollack et al., 1993; Pasquale et

al., 2001), which makes a CLVZ of 1018 Pas possible, see Section 1.1.

In van der Wal et al. (2004) we have shown that a CLVZ introduces variations in

geoid heights typically up to a few decimeters with spatial scales down to hundred

kilometers underneath and just outside formerly glaciated areas. Here we will

focus on the underlying physics of flow in low-viscosity channels using relaxation

spectra (Section 4.2) both for a CLVZ and an asthenospheric low-viscosity zone

(ALVZ). We have modeled an ALVZ below a fully elastic lithosphere (i.e. without a

CLVZ) of 80 km, with a thickness of 35 km and a viscosity of 1018 Pas, see Table 4.1.

We then show the effect of a CLVZ and ALVZ on geoid heights using perturbations,

which are the difference between predictions for a model with and without an LVZ

(Section 4.3.1). To compare our predictions with the performance of GOCE and

GRACE (GGM02S, see Section 3.4), we compute degree amplitudes and show that

for a CLVZ the perturbations are above the expected GOCE performance up to

spherical harmonic degree 120 and for an ALVZ up to degree 65.

In Section 4.4 we show the sensitivity to different properties of the LVZ, where the

sensitivity is defined as the difference between LVZ-induced perturbations from

the reference model and an LVZ model with different properties (as thickness or

viscosity). As the sensitivity is relatively small, we will focus on the detectability of

LVZs by computing perturbations that are robust to variations in the properties of

the LVZ. We then consider the role of the background model, especially the radial

4.2. Relaxation Spectra 37

Table 4.1: Viscosity stratification of the background (BG) model and the models with LVZs(crustal: CLVZ, asthenospheric: ALVZ)

Layer Depth [km] Viscosity η [Pas]

BG (STD) CLVZ ALVZ

upper crust 0 elastic elastic elastic

lower crust 20 : 1 ·1018 :

lithosphere 32 : elastic :

asthenosphere 80 5 ·1020 5 ·1020 1 ·1018

upper mantle 115 : : 5 ·1020

transition zone 400 : : :

lower mantle 670 5 ·1021 5 ·1021 5 ·1021

core 2891 0 0 0

earth stratification, on the predictions. We show that the predictions are sensitive

to the background earth stratification (Section 4.5.1) and that uncertainties in the

background earth stratification introduce errors (Section 4.5.2). This limits the

recoverable degrees to a lower bound of 30 (only uncertainties in the lower mantle

viscosity ηLM) to 60 (also uncertainties in the lithospheric thickness LT and upper

mantle viscosity1 ηUM).

As a reference background earth stratification, we take a model with LT = 80 km,

ηUM = 5 ·1020 Pas and ηLM = 5 ·1021 Pas (’STD’, see Section 3.1 and Table 4.1). We

use a modified version of the ICE-3G ice-load history as described in Section 3.2

and a eustatic ocean-load history (Section 3.3). Note that the earth model we use

in this chapter is laterally homogeneous, which is not very realistic, as it can for

example be expected that there are no CLVZs in old and cold areas as the Baltic

Shield (see e.g. Pasquale et al., 2001). Also, in the continental shelf area off the

coast of Norway earthquakes as deep as 25−30 km are found (Byrkjeland et al.,

2000). ALVZs can be expected more globally, especially below oceanic lithosphere,

though with variable thickness. In Chapters 6 and 7 we will show the effect of

lateral heterogeneities on our predictions.

4.2 Relaxation Spectra

The modal relaxation times and strengths (Section 2.2.3) for a model with a CLVZ

(dots •) and the background model (without a CLVZ, squares ) are plotted in Fig-

ure 4.1a. The introduction of a CLVZ induces 1 extra buoyancy mode and 3 extra

viscoelastic modes, as already demonstrated by Klemann & Wolf (1999) for their

1this includes the asthenosphere, the upper mantle and the transition zone as given in Table 4.1


(a) CLVZ, modal relaxation times

0 50 100 150 200 250

100

102

106

104

10−2

harmonic degree

rela

xatio

n tim

e [k

yr]

TC(2)

LC

MC

M3

C0

M2

M1

L0M0

T1(2),T2(2),T3(2)

squares : no CLVZdots : CLVZ

(b) ALVZ, modal relaxation times

0 50 100 150 200 250

10−2

100

102

104

106

harmonic degree

rela

xatio

n tim

e [ k

yr]

M2

M1

M3

C0

M0 (no ALVZ)L0 (no ALVZ)

T1(2),T2(2),T3(2)M0 (ALVZ)

squares : no ALVZdots : ALVZ

T4(2)

L0 (ALVZ)

(c) CLVZ, modal strengths

0 50 100 150 200 25010

−3

10−2

10−1

100

harmonic degree

mod

al s

tren

gth squares : no CLVZ

dots : CLVZline : difference intotal modal strengthbetween CLVZ andno CLVZM0

(no CLVZ)

M0(CLVZ)

MC

LC

L0

(d) ALVZ, modal strengths

0 50 100 150 200 25010

−3

10−2

10−1

100

harmonic degree

mod

al s

tren

gth squares : no ALVZ

dots : ALVZline : difference intotal modal strengthbetween ALVZ andno ALVZ

M0

M0(ALVZ)

M0(no ALVZ)

L0 (ALVZ)

L0(no ALVZ)

T4

(e) CLVZ, k-Love number

0 50 100 150 200 250

−1

−0.9

−0.8

−0.7

−0.6

−0.5

−0.4

−0.3

−0.2

−0.1

0

harmonic degree

k−Lo

ve n

umbe

r

10 kyrs (no CLVZ)10 kyrs (CLVZ)difference 10 kyrs100 kyrs (no CLVZ)100 kyrs (CLVZ)difference 100 kyrsfluid (no CLVZ)fluid (CLVZ)difference fluid

(f) ALVZ, k-Love number

0 50 100 150 200 250

−1

−0.9

−0.8

−0.7

−0.6

−0.5

−0.4

−0.3

−0.2

−0.1

0

harmonic degree

k−Lo

ve n

umbe

r

10 kyrs (no ALVZ)10 kyrs (ALVZ)difference 10 kyrs100 kyrs (no ALVZ)100 kyrs (ALVZ)difference 100 kyrsfluid limit

Figure 4.1: Modal relaxation times (a, b) and strengths (c, d), and k-Love numbers (e, f)for a model with a CLVZ (left) and an ALVZ (right).

4.2. Relaxation Spectra 39

halfspace model. The extra buoyancy mode, labelled MC, has significant strength

and is even stronger than M0 above degree 50 (Figure 4.1c), where the associated

relaxation time is tens of kyrs. The typical behavior of the MC mode, with large

relaxation times for small harmonic degree, is probably because adjustment for

long-wavelength loads to isostasy is slower for channel flow than for mantle flow

(see e.g. Cathles, 1975, p. 158). Note also the increase in the relaxation time of

the M0 mode. The viscoelastic LC mode, due to the difference in (infinite) Maxwell

time of the upper and (finite) Maxwell time of the lower crust (Wu & Ni, 1996),

and the two transient TC modes, due to the difference in Maxwell time of the

lower crust and the lithosphere (Wu & Ni, 1996), have strengths up to an order

of magnitude smaller than the dominant modes and might be interesting for fast,

high degree relaxation as in postseismic deformation.

In Figure 4.1c we have plotted the absolute value of the difference in total strength

of the k-Love number for a model with a CLVZ and the background model, which

shows that the difference in response is mainly confined to degree 50 to 150. It

also shows that our normal-mode method has very high accuracy, as the curve is

smooth, which indicates that probably all relevant modes have been found with

the correct strength. This is not trivial because for large relaxation times (or low

viscosities), the system of differential equations (Eq. 2.35 in Chapter 2) is unstable.

This is because in the fluid (or inviscid) limit, the viscoelastic shear modulus µ(s)2

(Wu & Peltier, 1982; Sabadini & Vermeersen, 2004, p. 17), goes to zero, and the

sixth-order system degenerates to a second order system (Wu & Peltier, 1982).

The (difference in) fluid limit (see Figure 4.1e) is equal to the plotted (difference

in) total strength (Figure 4.1c) plus the (difference in) elastic limit, see Eq. (2.42).

The modal relaxation times and strengths for a model with an ALVZ (dots •) and

without an ALVZ (squares ) are given in Figure 4.1b and d. In contrast to the

model with a CLVZ, the behavior seems mainly to be guided by changes in both

the relaxation time and strength of the M0 and L0 modes. As in the fluid limit

all viscoelastic material has relaxed, irrespective of the viscosity, the difference in

total strength between a model with an ALVZ and the background model is zero

in this limit and therefore not shown in Figure 4.1d.

To understand the different behavior of a CLVZ and an ALVZ, it is useful to look

at the (Heaviside) k-Love number (Eq. 2.42) at different time scales. For a CLVZ,

we see in Figure 4.1e that the difference between the models with and without a

CLVZ is confined to high (> 50) harmonic degrees, and that the difference increases

with increasing timescale. This behavior is due to the decoupling of the crust

and the lithosphere for long timescales, in which the behavior of a model with a

CLVZ is comparable to a model without a CLVZ, but with a lithosphere consisting

2The viscoelastic shear modulus µ(s) is equal to µs/(s+µ/η), with s the Laplace variable, µ the

(elastic) shear modulus (Eq. 2.7) and η the viscosity (Eq. 2.8), and is used to write the Laplace transform

of the constitutive equation for a Maxwell viscoelastic body (Eq. 2.9) in the Hookean form of Eq. (2.7).


(a) geoid heights

−16 −12 −8 −4 0 4geoid height [m]

(b) gravity anomalies

−12 −8 −4 0 4gravity anomaly [mgal]

Figure 4.2: Total GIA-induced geoid heights (a) and gravity anomalies (b) as predicted bythe background model ICE-3G(STD).

of only the upper crust. On short timescales, the CLVZ will behave effectively

elastic due to the long relaxation times of especially the low degree harmonics.

This means that the strength of the lithosphere on long timescales and in the case

of a ductile lower crust only resides in the upper crust. This in turn should lead

to very small estimates of the elastic thickness of the lithosphere, as found from

Bouguer coherence or free-air admittance studies (see e.g. Pérez-Gussinyé & Watts

(2005) and Figure 7.1a in Chapter 7), see Appendix A for a further discussion.

For an ALVZ (Figure 4.1f) we see that the difference in response between a model

with and without an AVLZ is only significant between degree 20 and 60, and for

time scales of 10 kyrs. For longer time scales the difference decreases towards the

zero difference in the fluid limit.

4.3 Geoid Heights and Gravity Anomalies

Geoid height predictions of our GIA model are dominated by lows at the centers of

rebound. We see from Figure 4.2a that the low due to the former Laurentide ice

sheet has a maximum of about −15 m in the Hudson Bay area and that the low

due to the former Fennoscandian ice sheet is about −6 m in the Gulf of Bothnia.

Mantle material has been pushed away from these areas towards the peripheral

bulges, which show a positive geoid height. This is even more clear from the free-

air gravity anomaly as plotted in Figure 4.2b, which shows a much more detailed

4.3. Geoid Heights and Gravity Anomalies 41

(a) CLVZ

−10 0 10 20 30 40 50geoid height perturbation [cm]

(b) ALVZ

−180 −120 −60 0 60 120 180geoid height perturbation [cm]

Figure 4.3: CLVZ- (a) and ALVZ- (b) induced geoid height perturbations.

pattern. This is due to the additional dependence on the harmonic degree l relat-

ing the gravity anomalies to the potential coefficients, which amplifies the high-

degree harmonics, see Eq. (2.30) in Section 2.2.2. Lows for the Laurentide and

Fennoscandian ice sheet are here −15 and −9 mgal, respectively. Note that these

values depend on the ice-load history and the earth stratification (LT, ηUM, ηLM).

As we are interested in perturbations due to an LVZ, we subtract from the pre-

dictions for a model with an LVZ the predictions of the background model. The

perturbations are then due to the difference in viscosity of the lower crust (1018

Pas versus effectively elastic) for a CLVZ, and the viscosity of the asthenosphere

(1018 Pas versus 5 ·1020 Pas) for an ALVZ.

4.3.1 Perturbations due to Low-Viscosity Zones

The geoid height perturbations induced by a CLVZ are shown in Figure 4.3a. The

induced signal shows amplitudes of tens of centimeters with spatial scales down

to one hundred kilometer. The pattern is mainly explained by the extra flow away

from the formerly glaciated areas through the low-viscosity channel during the

glaciation phase, leading to perturbative forebulges around the ice sheets. Due

to the relatively long relaxation time of the MC mode (Figure 4.1a) and the short

deglaciation period, these peripheral areas have only slightly adjusted to isostasy

and thus show still a positive perturbation. The position of especially the smaller

ice sheets (e.g. the British Isles and Iceland) can clearly be distinguished in these


(a) CLVZ

−2.4 −1.8 −1.2 −0.6 0.0 0.6 1.2gravity anomaly perturbation [mgal]

(b) ALVZ

−2 −1 0 1 2 3gravity anomaly perturbation [mgal]

Figure 4.4: CLVZ- (a) and ALVZ- (b) induced gravity anomaly perturbations.

figures, which already indicates the sensitivity to the ice-load history. An impor-

tant far-field phenomenon is the extra tilting of the coastlines due to enhanced

material flow forced by the ocean load (Di Donato et al., 2000a; Kendall et al.,

2003). This leads to a relative low on the land-side of the coastline and a relative

high on the sea-side (see e.g. the eastern part of the Iberian peninsula and the

Adriatic Sea), due to the negative sea-load in the glaciation period and the much

faster deglaciation.

The effect of an ALVZ is shown in Figure 4.3b. Due to the larger depth of the

LVZ, the pattern is much less dependent on the fine-scale structure of the ice load.

Moreover, instead of extra material flow away from the formerly glaciated areas,

the perturbed geoid signal is dominated by extra material flow towards the center

of rebound, thus accelerating the process of isostatic adjustment. This can be

explained by the shorter relaxation times of the M0 and L0 mode for the model

with an ALVZ compared to the background model, see Figure 4.1b.

Next we show gravity anomaly perturbations in Figure 4.4. Perturbation ampli-

tudes are up to a few mgal which is not small in comparison to the total signal

(Figure 4.2b), but seems to be small compared to the expected performance of

GOCE (1− 2 mgal for a resolution of 100 km, see Section 3.4). Note moreover

that the signal is very sensitive to the detailed ice-load history. The ALVZ-induced

gravity anomaly perturbations show comparable amplitudes as for a CLVZ, but

with much longer wavelengths.

4.3. Geoid Heights and Gravity Anomalies 43

(a) geoid height perturbations

0 25 50 75 100 125 150 175 20010

−4

10−3

10−2

10−1

100

harmonic degree

geoi

d he

ight

deg

ree

ampl

itude

[m]

GOCEGGM02SCLVZALVZ

(b) gravity anomaly perturbations

0 25 50 75 100 125 150 175 20010

−4

10−3

10−2

10−1

100

harmonic degree

grav

ity a

nom

aly

degr

ee a

mpl

itude

[mga

l]

GOCEGGM02SCLVZALVZ

Figure 4.5: Geoid height (a) and free-air gravity anomaly (b) perturbation degree amplitudescompared with an existing (GGM02S) and a future (GOCE) gravity model.

4.3.2 Comparison with the Expected GOCE Performance

To compare the computed geoid height and gravity anomaly perturbations with

GOCE and GRACE we use degree amplitudes, which are the square roots of the

spherical harmonic degree variances of the perturbed field (Section 3.4). In Figure

4.5a we have plotted the CLVZ (dots •) and ALVZ (pluses +) induced geoid height

perturbation degree amplitudes together with the error characteristics of the ex-

pected performance of GOCE based on formal errors (Section 3.4, solid –) and the

realize performance of GRACE (GGM02S, Section 3.4, dashed - - ).

The degree amplitudes of the CLVZ induced perturbations are above the perfor-

mance of GOCE up to degree 120 (above GGM02S up to degree 70) and of the

ALVZ induced perturbations up to degree 65 (GGM02S: 55). The same values hold

for the gravity anomaly perturbations, see Figure 4.5b. This means that a large

part of the information about LVZs is expected in the GOCE signal and, to a lesser

degree, in the GRACE signal. From the figures we see that GOCE, compared to

GRACE, is especially predicted to deliver additional information on CLVZs.

The error characteristics of satellite gravity data thus only provide an upper bound

to the recoverable harmonic degrees. In the next section we will show that a lower

bound is provided by uncertainties in the background earth stratification. We will

also show that the perturbations are not only sensitive to properties of the LVZ,

but also to the background earth stratification and the ice-load history.

As the predictions for geoid heights and gravity anomalies lead mainly to the same

conclusions, we focus on geoid heights only from here on.


(a) CLVZ

−12 −8 −4 0 4 8geoid height sensitivity [cm]

(b) ALVZ

−60 −40 −20 0 20 40 60geoid height sensitivity [cm]

Figure 4.6: Sensitivity of CLVZ- (a) and ALVZ- (b) induced geoid height perturbations tothe viscosity of the LVZ.

4.4 Sensitivity to the Properties of LVZs

Up to here we have only used the standard properties of the LVZs as given in Ta-

ble 4.1. If we want to be able to constrain the properties of LVZs from GOCE, it is

important to know how sensitive these predictions are to changes in these prop-

erties. We define the sensitivity as the difference in LVZ-induced perturbations

generated with different properties of the LVZ.

In Figure 4.6 we show the sensitivity to an order of magnitude increase in the

viscosity of the LVZ (from 1018 to 1019 Pas) for a CLVZ and an ALVZ. Note that the

sensitivity is relatively small compared to the perturbations (Figure 4.3), which

suggests that it will be difficult to differentiate between an order of magnitude

change in viscosity of the LVZ. This can be seen more clearly when we compare

the sensitivity to the LVZ-induced perturbations and the GOCE performance in

Figure 4.7. For a CLVZ, the sensitivity to a order of magnitude increase in viscosity

(’viscosity sens.’) is small, except for a peak around degree 75, where already for

a viscosity of 1019 Pas the perturbations due to a CLVZ are small (compare with

Figure 4.13b, from which we can also see that these perturbations cross the GOCE

performance at lower degree, around 90, than for the reference CLVZ of 1018 Pas).

The peak can be associated with the small patches of highs and lows in Figure 4.6a

as e.g. on Iceland. For a thicker CLVZ (20 km instead of 12 km, ’thickness sens.’),

the situation looks more promising, with a high peak around degree 60. For an

4.4. Sensitivity to the Properties of LVZs 45

(a) CLVZ

0 25 50 75 100 125 150 175 200

10−3

10−2

harmonic degree

geoi

d he

ight

deg

ree

ampl

itude

[m]

GOCECLVZ perturb.viscosity sens.thickness sens.

(b) ALVZ

0 25 50 75 100 125 150 175 200

10−3

10−2

10−1

harmonic degree

geoi

d he

ight

deg

ree

ampl

itude

[m]

GOCEALVZ perturb.viscosity sens.thickness sens.

Figure 4.7: Geoid height perturbation degree amplitudes for different properties of a CLVZ(a) and an ALVZ (b).

ALVZ, the sensitivity to the viscosity seems to be larger, but we will show further

on that the sensitivity to the upper mantle viscosity is even larger. For a thicker

ALVZ (60 km instead of 35 km), the sensitivity is also large, especially for low

degree harmonics (l ∼ 10−20).

If we shift our focus to the detection of LVZs, it is interesting to investigate if the

perturbative predictions are in certain areas insensitive or robust to variations in

properties of the LVZ. As we are interested in relatively high harmonic signals,

it is useful to remove the low degree harmonics (< 10−20) from the predictions,

and to focus on spatial patterns it is useful to remove an offset and scale factor

(using least-squares, see Section 7.4.1 in Chapter 7). If we then compute the mean

of predictions for different properties of the LVZ (higher viscosity, thicker chan-

nel, see Figure 4.8a for a CLVZ) and the standard deviation of these predictions

(Figure 4.8b) and we choose those areas with a relatively large mean (> 10 cm,

one-scale interval) and small standard deviation (< 5 cm, half-scale interval), we

find patches as shown in Figure 4.8c. These predictions are robust to (small) varia-

tions in the properties of the CLVZ (spatial signatures), however, not to variations

in the properties of the ice-load history. The low in eastern Siberia for example

(see Figure 4.3a) is due to an ice sheet in ICE-3G that is not present in more re-

cent loading histories as ICE-5G, see Figure 3.2 in Chapter 3. Moreover, our earth

stratification is laterally homogeneous, which means that at least part of the low

in Scandinavia is unlikely, due to the old and cold crust there (compare with Fig-

ure 1.2a), though we will show in Chapter 6 that the lows and highs along the

Norwegian coast are also present for a laterally heterogeneous model.

In Figure 4.8d we show robust ALVZ-induced perturbations, where we have used

means larger than 30 cm and standard deviations smaller than 15 cm. The quite

distinctive highs in Northern Europe (Greenland and Gulf of Bothnia) are also


(a) mean CLVZ

−30 −20 −10 0 10 20geoid height perturbation [cm]

(b) standard deviation CLVZ

0.0 2.5 5.0 7.5 10.0geoid height perturbation [cm]

(c) robust CLVZ


(d) robust ALVZ

−60 −30 0 30 60 90geoid height perturbation [cm]

Figure 4.8: Mean (a) and standard deviation (b) of CLVZ-induced geoid height perturbationsfor different properties (higher viscosity and thicker CLVZ) of the CLVZ, and CLVZ- (c)and ALVZ- (d) induced geoid height perturbations that are robust to variations in theproperties (viscosity, thickness) of the LVZ. The perturbations are filtered to remove lowdegree (< 20 for CLVZ, < 10 for ALVZ) harmonics and subsequently linearly fitted to thereference perturbations of Figure 4.3.

4.5. Role of the Background Model 47

(a) CLVZ

−40 −20 0 20 40geoid height sensitivity [cm]

(b) ALVZ


Figure 4.9: Sensitivity to the ice-load history for CLVZ- (a) and ALVZ- (b) induced geoidheight perturbations.

robust to changes in the ice-load history (see Figure 5.10b in Chapter 5 and to the

upper mantle viscosity of the background model, as show in Section 4.5.1.

4.5 Role of the Background Model

The role of the background model can be clarified by considering hypothetical

GOCE data, in which the effect of all mass anomalies are removed error-free, ex-

cept for GIA. Of course, this is in reality not possible for all other processes or

structures that induce gravity signals, see for a further discussion Section 7.1.

If we want to extract perturbations from this hypothetical data, we have to sub-

tract a background model, which consists of an earth stratification and ice-load

history. If we do not subtract the correct (i.e. real) background model, we induce

errors in the perturbations. To fit the extracted perturbations we need to generate

predictions of these perturbations, for which we again need a background model.

If we use the same background model to generate the predictions, we get an ad-

ditional error in the predicted perturbations, i.e. the perturbations are sensitive

to the background model. We will consider both effects here, i.e. the sensitivity of

our predictions to the background model (Section 4.5.1) and the error introduced

by uncertainties in the real background model (Section 4.5.2). For the former, to

highlight the differences between predictions based on different background mod-

els, we will again use sensitivities, i.e. the difference between LVZ-induced per-


turbations generated with different background earth stratifications or ice-load

histories. For the errors, we consider the difference between total predictions from

different background models.

In Chapter 5 we will look at the sensitivity to variations in the ice-load history,

here we will mainly consider the background earth stratification. Note that the

sensitivity to the ice-load history and earth stratification cannot be completely

decoupled, as the ice-load history is contaminated by its preferred earth stratifi-

cation (see Section 3.2). To give an idea of the sensitivity to the ice-load history,

we show here results for CLVZ- and ALVZ-induced geoid heights when we use a

shorter halfwidth of the Gaussian filter with which we smooth the ice-load history

(100 km instead of 200 km, see Section 3.2).

The results for CLVZ- and ALVZ-induced sensitivities in the geoid height are given

in Figure 4.9. For a CLVZ the sensitivity is large, with amplitudes as large as, or

even larger than, the CLVZ-induced perturbations (compare with Figure 4.3a).

The peaks near the edges of the former ice sheets are important, because they

show that the predictions are very sensitive to the extent of the ice sheets. The

extent is mostly derived from geomorphological data and quite well constrained

on land. However, in sea areas the extent is much less well known, compare e.g.

the discussion on the glaciation of the North Sea in Carr et al. (2005) and van den

Broek (2006). The peaks in North America are however largely related to holes in

the original ICE-3G due to its definition in finite discs, and can thus be regarded

as artefacts.

The sensitivity for an ALVZ to the filtering is much smaller, with amplitudes that

are only a fraction of the ALVZ-induced perturbations, compare with Figure 4.3b.

However, in realistic ice-load histories, there are also long- to medium-wavelength

differences and then the effect can be significant, see Chapter 5.

4.5.1 Sensitivity to the Background Earth Stratification

To test the sensitivity to the background earth stratification, we vary one at a

time the upper mantle and lower mantle viscosities ηUM and ηLM, and the thick-

ness of the lithosphere LT (only for a CLVZ, not for an ALVZ), to respectively

ηUM = 1 ·1021 Pas, ηLM = 1 ·1022 and LT = 120 km. Note that the viscosity values

are in line with the upper bounds found by Milne et al. (2004) and that the val-

ues of our reference model are in line with the lower bounds found by Milne et al.

(2004), see Section 1.2. In Figure 4.10 we show the sensitivity to ηUM of CLVZ-

and ALVZ-induced geoid height perturbations. The sensitivity to ηUM is especially

large for an AVLZ, with amplitudes larger than the perturbations itself, compare

with Figure 4.3b. Moreover, the wavelengths of the sensitivities seem to be compa-

rable to the wavelengths of the perturbations, which means that it will not be easy


(a) CLVZ


(b) ALVZ

−300 −200 −100 0 100geoid height sensitivity [cm]

Figure 4.10: Sensitivity of CLVZ- (a) and ALVZ- (b) induced geoid height perturbations tothe upper mantle viscosity ηUM.

to remove this sensitivity by filtering the perturbations. The sensitivity to ηLM is

also quite large (not plotted), however, now only the long-wavelength components

are affected, so the sensitivity to ηLM can largely be removed. We can show this by

looking at the degree amplitudes of the geoid height sensitivities.

The degree amplitudes for the sensitivity of CLVZ-induced geoid height perturba-

tions are plotted in Figure 4.11a, together with the CLVZ-induced perturbations

(dots •). The sensitivity to the ice-load history (’ice sens.’) is large up to degree 130,

due the smaller halfwidth of the filter. For realistic differences in ice-load histo-

ries the sensitivity will even be larger and broader, see Chapter 5. This also means

that the perturbations for a different ice-load history (e.g. ICE-5G) are above the

GOCE performance up to higher degree (∼ 130−140, see Figure 5.5 in Chapter 5)

than estimated in Section 4.3.2 (l = 120).

The sensitivity to variations in LT (’LT sens.’) and ηUM (’ηUM sens.’) is smaller,

though significant up to degree 75−90. The somewhat similar sensitivity for vari-

ations in LT and ηUM is due the dampening effect of both a larger LT (thicker

lithosphere) and smaller ηUM (weaker upper mantle, see Sections 1.2 and 7.1).

Together with the ice-load history, uncertainties in LT and ηUM seem to be the

major uncertainty in constraining properties of CLVZs, as GOCE is sensitive to

these uncertainties with significant strength up to high degree (Figure 4.11a). The

sensitivity to ηLM is, as expected, confined to the low degree harmonics, and will

not be of much importance, because, as we will show in Section 4.5.2, we have


(a) CLVZ

0 25 50 75 100 125 150 175 200

10−3

10−2

harmonic degree

geoi

d he

ight

deg

ree

ampl

itude

[m]

GOCECLVZ perturb.ice sens.LT sens.η

UM sens.

ηLM

sens.

(b) ALVZ

0 25 50 75 100 125 150 175 200

10−3

10−2

harmonic degree

geoi

d he

ight

deg

ree

ampl

itude

[m]

GOCEALVZ perturb.ice sens.η

UM sens.

ηLM

sens.

Figure 4.11: Degree amplitudes of the sensitivity for CVLZ- (a) and ALVZ (b) -inducedgeoid height perturbations.

to filter the perturbations at least up to degree 30 to remove uncertainties in the

background earth stratification.

We can filter the perturbations to remove a large part of the sensitivity, but as

the sensitivity to LT and ηUM is within the range of harmonic degrees where we

expect sensitivity to properties of the CLVZ (see Figure 4.7a), we have to take into

account the uncertainties in LT and ηUM. Note that especially the sensitivities to

LT and ηUM are worst cases, as it should be possible to know LT with a maximum

uncertainty of 20 km, and ηUM with a maximum uncertainty of 2−3 ·1020 Pas, see

Section 1.2.

From Figure 4.11b we see that for an ALVZ the sensitivity to ηUM (’ ηUM sens.’)

is larger than the ALVZ-induced perturbations (pluses +). This will make it very

difficult to constrain properties of the ALVZ only, i.e. we must at the same time

constrain ηUM. However, if there are patterns that are insensitive to variations in

ηUM, we can at least detect ALVZs. From Figure 4.12b we see that some pertur-

bations are robust to changes in ηUM (from 5 ·1020 to 1 ·1021 Pas), with a mean

larger than 30 cm and a standard deviation smaller than 15 cm. More impor-

tantly, some patterns are also robust to variations in the properties of the ALVZ,

e.g. the highs over Greenland and the Gulf of Bothnia (compare with Figure 4.8d).

This is also the case for a CLVZ, for example along the Norwegian coast, compare

Figures 4.12a and Figure 4.8c.

As GOCE is especially predicted to add information on CLVZs (see Figure 4.5a), we

will for the remainder of this chapter concentrate on perturbations due to CLVZs

only.


(a) CLVZ


(b) ALVZ


Figure 4.12: CLVZ- (a) and ALVZ- (b) induced geoid height perturbations, robust to vari-ations in ηUM.

4.5.2 Errors Introduced by the Background Earth Stratifi-

cation

In Section 4.3.2 we have shown that satellite gravity data, due to the decrease of

the performance with harmonic degree, only limits the recovery of the high har-

monic signal induced by CLVZs. In Section 4.5.1 we have shown that predictions of

CLVZ-induced perturbations are sensitive to the background earth stratification.

Here we will show that uncertainties in the background earth stratification limit

the recovery of the low degree harmonics. To obtain perturbations from gravity

data from which all non-GIA-related signals are removed (see beginning of Sec-

tion 4.5), we have to subtract a GIA background model, which in principle consists

of an earth stratification and an ice-load history. In the next chapter we will in-

vestigate in more detail the effect of uncertainties in the ice-load history, here we

will focus on the error introduced by uncertainties in the background earth strat-

ification. As in the previous section, we will consider a model with a larger LT

(120 vs. 80 km), a model with a larger ηUM (1 ·1021 vs. 5 ·1020 Pas), and a model

with a larger ηLM (1 ·1022 vs. 5 ·1021 Pas). Note that errors are generated by sub-

tracting total predictions from a certain background earth model from a reference

background earth model, which is here ICE-3G(STD).

In Figure 4.13a we have plotted total (i.e. not perturbative) geoid height degree

amplitudes for models with a CLVZ (with a viscosity of 1018 Pas and of 1019 Pas)

and background models with different values for LT, ηUM and ηLM. The effect of


(a) total geoid heights

0 25 50 75 100 125 150 175 200

10−3

10−2

10−1

100

harmonic degree

geoi

d he

ight

deg

ree

ampl

itude

[m]

GOCE

CLVZ (1018 Pas)

CLVZ (1019 Pas)BG, STDBG, LT=120BG, η

UM=1021 Pas

BG, ηLM

=1022 Pas

(b) perturbations and errors

0 25 50 75 100 125 150 175 200

10−3

10−2

10−1

100

harmonic degree

geoi

d he

ight

deg

ree

ampl

itude

[m]

GOCE

CLVZ (1018 Pas)

CLVZ (1019 Pas)BG err. LTBG err. η

UM

BG err. ηLM

Figure 4.13: Degree amplitudes of models with a CLVZ and background models (a) andof CLVZ-induced perturbations and errors introduced by uncertainties in the backgroundmodel (b).

the CLVZ is only visible from about degree 25, so up to this degree the behavior

is largely due to the background model. The difference between a CLVZ of 1018

Pas and 1019 Pas is significant from degree 50. Due to the quite large differences

between the background models, the different models can be separated quite well

(LT from degree 20, ηUM from degree 10), except for the model with a larger ηLM,

which only influences the low harmonics (degree < 10). This is even more clear if

we look at degree correlation coefficients, defined as (O’Connell, 1971; Mitrovica &

Peltier, 1989):

ρl =

l∑

m=0

AlmBlm

√

√

√

√

l∑

m=0

Alm A∗lm

l∑

m=0

BlmB∗lm

(4.1)

with Alm and Blm complex sets of spherical harmonic coefficients. If ρl is equal to

1 or −1, then the sets A and B are linearly dependent for a certain degree, and if

ρl = 0 then the two sets are uncorrelated.

For a CLVZ with a viscosity of 1018 Pas the degree correlation coefficients for a

model with a CLVZ and the background model (without a CLVZ) is plotted in Fig-

ure 4.14a and for a CLVZ with a viscosity of 1019 Pas in Figure 4.14b. We see

that for these variations in the background model, the correlation coefficients fa-

vor the reference background model, because the correlation is closer to one than

the other background models up to harmonic degree 40, where we expect the back-

ground model to have its largest strength (see Figure 4.13a). Note however that

for ηLM the differences are very small, and that for ηUM we have even for degree

60 a good correlation. This is even more true for the CLVZ with 1019 Pas (Fig-


(a) 1018 Pas

0 20 40 60 80

0.7

0.75

0.8

0.85

0.9

0.95

1

harmonic degree

degr

ee c

orre

latio

n co

effic

ient

STDLT = 120 kmη

UM = 1021 Pas

ηLM

= 1022 Pas

(b) 1019 Pas

0 20 40 60 80

0.7

0.75

0.8

0.85

0.9

0.95

1

harmonic degree

degr

ee c

orre

latio

n co

effic

ient

STDLT = 120 kmη

UM =1021 Pas

ηLM

= 1022 Pas

Figure 4.14: Degree correlation coefficients for different background models and a modelwith a CLVZ of 1018 Pas (a) and 1019 Pas (b).

ure 4.14b), and together with for example smaller differences in ηUM, it is possible

that, from degree correlations, not the reference background model is favored.

In Figure 4.13b we show which errors are introduced by not using the reference

background model. We do this by subtracting from predictions generated with

the reference background model a different background model, thus simulating

errors due to uncertainties in the background model. Comparing these errors with

CLVZ-induced perturbations, we see that for a part of the spectrum, the degree

amplitudes of the errors are larger than the perturbations. It thus seems that,

after subtracting a background model, we at least have to filter the data and the

predictions up to degree 30, where the errors due to uncertainties in ηLM cross

the GOCE performance. However, the errors due to uncertainties in ηUM suggest

that we might have to filter up to degree 50−70, in which case we do not have to

subtract a background model first, because from degree 50 the geoid heights are

almost completely determined by the CLVZ-induced perturbations, compare with

Figure 4.13a.

From Figure 4.13b we see that uncertainties in ηLM only affect the lower (< 30)

degree harmonics and from Figure 4.14a that uncertainties in LT and ηUM can

to a large extent be removed using degree correlations. However, for increasing

viscosity of the CLVZ, we see from Figure 4.14b that it will be more difficult to

reduce the uncertainty in LT and ηUM. We thus conclude that uncertainties in

the background earth stratification give a lower bound of degree 50, where the er-

rors due to uncertainties in ηUM cross the perturbations, on recoverable spherical

harmonic coefficients of CLVZ-induced perturbations. This means that we have

to filter the data and predictions up to degree 30−50, depending on how well the

correct background LT and ηUM can be estimated from degree correlations.


(a) no LVZ

−20˚

−10˚

0˚ 10˚ 20˚

30˚

40˚

40˚

45˚

45˚

50˚

50˚

55˚

55˚

60˚

60˚

65˚

Ne

Br

MaGe

Tr

Ca

−0.75 −0.50 −0.25 0.00 0.25 0.50 0.75rsl rate [mm/yr]

(b) CLVZ

−20˚

−10˚

0˚ 10˚ 20˚

30˚

40˚

40˚

45˚

45˚

50˚

50˚

55˚

55˚

60˚

60˚

65˚

Ne

Br

MaGe

Tr

Ca

−0.75 −0.50 −0.25 0.00 0.25 0.50 0.75rsl rate [mm/yr]

(c) ALVZ

−20˚

−10˚

0˚ 10˚ 20˚

30˚

40˚

40˚

45˚

45˚

50˚

50˚

55˚

55˚

60˚

60˚

65˚

Ne

Br

MaGe

Tr

Ca

−0.75 −0.50 −0.25 0.00 0.25 0.50 0.75rsl rate [mm/yr]

Figure 4.15: GIA-induced rate of RSL change for our background model (no LVZ, a), amodel with a CLVZ (b) and a model with an ALVZ (c). The white line indicates zero RSLrate of change.

4.6 Present-Day Sea-Level Change

In global warming studies, estimates of present-day relative sea level (RSL) change

from tide gauges are corrected for ongoing GIA, see e.g. Douglas (1991), Mitrovica

& Davis (1995), Douglas (2001) and Peltier (2001). If a certain GIA model de-

creases the standard deviation globally or in a certain area, then the corrected

estimate is regarded as better than the original. In this way, the estimates can

be improved without knowledge about other processes that influence estimates of

present-day RSL change as tectonics. The correction is very sensitive to the ice-

load history and the earth stratification. Here we will not argue for a specific earth

and ice model combination, but merely show the effect of an LVZ on the predicted

rate of present-day RSL change in western Europe (see Di Donato et al., 2000a, for

a similar analysis for the U.S. east coast) . We show that for this area the standard

deviation of the corrected tide gauge measurements can be smaller for a model

with an LVZ than for a model without an LVZ. As we are using a relatively small

set of tide gauge measurements, these results should be regarded as a possible

way to explain short-scale differences in tide gauge measurements, not as better

corrections than proposed by other authors as e.g. Peltier (2001).

In Figure 4.15a we show the predicted rate of RSL change due to GIA in western

Europe, computed using our background model. The main features are a large

positive rate of RSL change (> 0.75 mm/yr, white area in Northern Atlantic) due to

collapse of the forebulge, a large negative rate of RSL change (up to −10 mm/yr in

Scandinavia) due to rebound of the formerly glaciated areas, and a small positive

or negative rate near the coasts depending on the location of a site relative to the

long-wavelength trend of the coastline (see e.g. Kendall et al., 2003).

In Figure 4.15 we have also indicated the 6 tide gauge stations that we use (Ne:

4.6. Present-Day Sea-Level Change 55

Table 4.2: Rates of present-day RSL change in western Europe corrected for GIA (in mm/yr)

Trend Trend-GIA No LVZ CLVZ ALVZ

MEAN 1991 1.42 1.65 1.32 1.31 1.18

SIGMA 1991 0.22 0.19 0.21 0.19 0.24

MEAN 2001 1.35 1.24 1.26 1.21 1.13

SIGMA 2001 0.24 0.36 0.25 0.23 0.22

Newlyn, Br: Brest, Ma: Marseille, Ge: Genoa, Tr: Trieste and Ca: Cascais).

RSL rates estimated from these tide gauges are taken from two studies of Dou-

glas (’1991’: Douglas (1991), ’2001’: Douglas (2001)). As the standard deviation

is a measure for the appropriateness of the GIA correction, we work with the

mean (’MEAN’) of the 6 estimates and the standard deviation (’SIGMA’), which

are shown in the column ’Trend’ of Table 4.2. In the next column (’Trend-GIA’) we

show the estimates of RSL change after correction for ongoing GIA using a certain

earth and ice model (’1991’: ICE3G(VM1), Tushingham & Peltier (1991); Douglas

(1991), ’2001’: ICE4G(VM2), Peltier (2001)). Note that for ’2001’ the standard de-

viation is increased, meaning that the correction is not appropriate for this region.

In the third column (’No LVZ’) of Table 4.2 we use the values from Figure 4.15a

and see that our correction gives a smaller mean with a comparable (’1991’) or

smaller (’2001’) standard deviation. This indicates that for ’2001’ the STD model

(LT = 80 km, ηUM = 5 ·1020 Pas, ηUM = 5 ·1020 Pas) with the modified ICE-3G load

history (ICE-3G(STD), see Section 3.2), is better suited for correction of tide-gauge

data in western Europe than the VM2 model (LT = 120 km, ηUM = 4 ·1020 Pas and

ηLM = 2 ·1021 Pas) with the ICE-4G load history.

Figure 4.15b shows the GIA-induced RSL change if a CLVZ is incorporated in our

earth model, showing a pattern of relatively large gradients along coastlines due

to extra tilting (see e.g. Kendall et al., 2003, and Section 4.3.1). The effect on

the rate of RSL change in Europe is to lower both the mean and the standard

deviation (’CLVZ’, Table 4.2), the first with a maximum of 4% (from 1.26 to 1.21

mm/yr) and the latter with 10%. The effect of an ALVZ is smoother and tends

to increase the GIA-induced rate of RSL change, due to the faster collapse of the

bulge area (Figure 4.15c). The effect of an ALVZ is larger than that of a CLVZ,

with a reduction in the mean rate of about 10% (’ALVZ’, Table 4.2). For the tide

gauge estimates of ’2001’ this leads to a mean RSL rate of change corrected for GIA

of 1.13 mm/yr (1.26 mm/yr for the background model) and a standard deviation of

0.22 mm/yr (0.25 mm/yr for the background model), which is the best estimate for

’2001’ in terms of standard deviation.


4.7 Conclusions

We have shown that the response of a shallow low-viscosity zone (LVZ) to loading

and unloading during the last glacial cycle is a complex function of spherical har-

monic degree and time. The effect of a crustal LVZ (CLVZ) is to decouple the crust

and lithosphere, so that on large time scales (> 1 Myrs) all strength resides in the

upper crust. For a CLVZ, the relaxation time and strength are increased for all de-

grees due to the extra buoyancy mode MC. This leads to extra material flow away

from the (formerly) loaded areas, especially for degrees 50 to 150. Coastlines expe-

rience additional tilting, thus lowering estimates of present-day sea level rise. For

an asthenospheric LVZ (ALVZ) the relaxation time is decreased for all degrees,

leading to faster adjustment to isostasy and lower estimates of present-day sea

level rise. For both LVZs, the standard deviation of the estimates is decreased for

a region as western Europe.

We have compared our results with the expected performance of GOCE and the

realized performance of GRACE (GGM02S). Degree amplitudes for CLVZ-induced

perturbations are above the former up to degree 120 and for the latter up to degree

70. For an ALVZ this is up to degree 65 for GOCE and 55 for GGM02S. GOCE is

thus expected to be especially useful in detecting the presence of CLVZs. GOCE is

found to be also sensitive to changes in the properties of a CLVZ, which means that

in principle GOCE should be able to constrain the rheology. However, this sensi-

tivity is small and GOCE is also predicted to be sensitive to uncertainties in the

ice-load history and background earth stratification. We have generated plots of

CLVZ- and ALVZ-induced perturbations that are robust to variations in the prop-

erties of the LVZs (spatial signatures). Comparison with the robustness against

variations in the upper mantle viscosity of the background model shows that some

CLVZ- and ALVZ-induced perturbations are robust to both, which means that de-

tection of these features with e.g. GOCE is more likely.

The sensitivity to especially the ice-load history, the lithospheric thickness and the

upper mantle viscosity of the background model is large, which means in principle

that these have to be known with sufficient accuracy. Moreover, uncertainties in

the background model provide a lower bound on the range of recoverable degree

harmonics of CLVZ-induced perturbations, restricting it to degree 30 due to un-

certainties in the lower mantle viscosities up to degree 50 due to uncertainties in

the upper mantle viscosity. This means that we have to filter the data and pre-

dictions up to degree 30−50, depending on how well the background model can

be estimated from degree correlations. In the next chapter we will especially look

at the ice-load history and we estimate there that background model errors intro-

duced by uncertainties in the ice-load history are larger, limiting the lower bound

of recoverable spherical harmonic coefficients to degree 40−60.

Chapter 5

Sensitivity to the Ice- and

Ocean-Load History

In the previous chapter we have shown that the response to low-viscosity zones (LVZs) in the

shallow earth is sensitive to the properties of an LVZ and to the background earth stratifi-

cation. In this chapter we focus on Northern Europe and look more closely at the sensitivity

to the ice-load history by using three different ice-load histories: RSES, ICE-5G and the

modified ICE-3G history of the previous chapter. We test the sensitivity of LVZ-induced per-

turbations to the different ice-load histories, using a standard earth stratification to focus

as much as possible on the ice-load history only, and to more realistic ocean-load histories.

We show that GOCE is predicted to be especially sensitive to differences in the ice-load his-

tory, up to degree 140 for a CLVZ and degree 70 for an ALVZ. This means that GOCE could

provide information on the ice-load history in the presence of an LVZ, if our knowledge of

the earth model is sufficient. However, if we want to extract information about an LVZ from

degree amplitudes, we first have to remove part of the uncertainty in the ice-load history.

This is especially important because the uncertainties in the ice-load history give, compared

to uncertainties in the earth stratification, a larger lower bound of recoverable harmonic de-

grees of 40 to 60. We show in which regions the LVZ-induced perturbations are insensitive to

variations in the ice-load history, and that the ALVZ-induced perturbations are at the same

time also robust for variations in the properties of the LVZ. We show for a CLVZ that it is

possible to estimate a preferred ice-load history from degree correlation coefficients. We can

then normalize the geoid height perturbations with the ice heights at LGM, thus obtaining

spectral signatures for different properties of the CLVZ.

This chapter and the previous chapter are based on Schotman & Vermeersen (2005) and Schot-

man et al. (2007), but are reworked for consistency and to include recent developments (ICE-5G) and

research (comparison of total GIA predictions with GPS and gravity data, spatial signatures).

58 Chapter 5. Sensitivity to the Load History

5.1 Introduction

From this chapter on, we will concentrate on Northern Europe. The main reason

is that in this area low-viscosity zones (LVZs) are not unlikely in areas just in- and

outside the formerly glaciated areas, as the North Sea basin, the Barents Sea area

and western Norway (Pollack et al., 1993; Pasquale et al., 2001). Another reason is

that the background earth stratification is quite well constrained in this area, both

from Holocene sea-level curves (Lambeck et al., 1998) and GPS-measurements

(Milne et al., 2001). We will mainly focus on the sensitivity to the load history of

LVZ-induced perturbations in geoid height.

However, we start here with a small excursion and show three-dimensional rates

of deformation (3D velocities) for different ice-load histories and their laterally ho-

mogeneous earth stratifications in Figure 5.1. The ice-load histories are modified

ICE-3G with earth model VM1 (ICE-3G(VM1), Tushingham & Peltier, 1991, see

Section 3.2), ICE-5G(VM2) (Peltier, 2004) and RSES(STD) (Lambeck et al., 1998),

see Section 3.2. To highlight the effect of the former Fennoscandian ice sheet,

we remove the lower degree harmonics (< 9), which are mainly due to the former

Laurentide ice sheet. We then very effectively decouple North America and Eu-

rope, thus simulating the weak lithosphere along the Mid-Atlantic Ridge (see e.g.

Whitehouse et al. (2006), who use laterally heterogeneous or 3D-stratified earth

models).

The resulting predictions compare especially well in the southern part, where we

simulate the outward movement as measured in the BIFROST network (Milne et

al. (2001), Figure 5.1d). Reported accuracies for this network are 0.8−1.3 mm/yr

in the vertical direction and 0.2−0.4 mm/yr in the horizontal direction (Johansson

et al., 2002). Note that Milne et al. (2004), who derive a 1D-viscosity profile for

Northern Europe, have difficulties fitting the outward pattern in the southern part

of Scandinavia, because they (have to) include the effect of the Laurentide ice

sheet. As they use the same RSES ice-load history as Lambeck et al. (1998), who

derives a 1D-viscosity profile based on RSL-curves, they should in principle find

the same earth stratification as Lambeck et al. (1998). As explained in Section 7.1,

differences either reflect different a priori assumptions or a different sampling in

space or time. Steffen et al. (2007) for example cannot find a better fit to the

BIFROST data using a laterally heterogeneous (3D) earth model than Milne et

al. (2004), probably at least partially due to the use of the existing RSES ice-load

history that was constructed on a radially symmetric earth.

Note that the vertical rate of displacement is about the negative of the present-day

rate of change of relative sea-level, because the rate of geoid height change is in

this area very small compared to vertical displacement rate. For example, in the

center of rebound, the geoid rate is about 0.3 mm/yr, while in the peripheral areas

5.1. Introduction 59

(a) ICE-3G(VM1)

1 mm/yr

−4 −2 0 2 4 6 8uplift rate [mm/yr]

(b) ICE-5G(VM2)

1 mm/yr


(c) RSES(STD)

1 mm/yr


(d) BIFROST data

Figure 5.1: Filtered 3D-velocity predictions for modified ICE-3G(VM1) (a), ICE-5G(VM2)(b) and RSES(STD) (c), and 3D-velocites as measured with GPS in the BIFROST project(d, left: vertical velocities, right: horizontal velocities, plot reproduced from Milne et al.(2001)).


(a) CLVZ

1 mm/yr

−2 −1 0 1 2 3vertical deformation rate [mm/yr]

(b) ALVZ

1 mm/yr

−2.4 −1.6 −0.8 0.0 0.8 1.6vertical deformation rate [mm/yr]

Figure 5.2: Filtered CLVZ- (a) and ALVZ- (b) induced perturbations in 3D-velocity predic-tions using RSES(STD).

the rate is about −0.3 mm/yr. The GRACE solutions seem currently be able to

see such small effects (see Tamisiea et al., 2007, for Northern America), however,

interpretation of the data is complicated because other effects as e.g. hydrology

also contribute to the signal (see e.g. Rangelova et al., 2007, for Northern America).

LVZ-induced perturbations, which are the differences between predictions from a

model with and without an LVZ, in these rates of geoid height change are much

smaller, so we will not consider temporal gravity here.

LVZ-induced perturbations in 3D-velocities are however not small, as shown in

Figure 5.2, where we have subtracted from a model with a crustal low-viscosity

zone (CLVZ) or asthenospheric LVZ (ALVZ), a background earth stratification (STD)

without LVZs. All models in this chapter are laterally homogeneous (1D-stratified)

and given in Table 4.1. We have used the RSES ice-load history and we have

again removed all harmonics below degree 9 to highlight the effect of the former

Fennoscandian ice sheet. For vertical deformation rates, the effect of a CLVZ is

mainly to enhance the uplift rate below the former ice sheet (compare with Fig-

ure 5.1c) due to the longer relaxation times associated with channel flow. An ALVZ

reduces the uplift rate, because the faster adjustment to isostasy due to the LVZ

results in a smaller present-day depression. In this light it is interesting to see

that the pattern of horizontal deformation rate is very similar, and thus not obey-

ing for a CLVZ the general pattern of outward horizontal velocities in uplifting

5.2. Geoid Heights and Perturbations 61

areas, see e.g. the southern part of Scandinavia. From tests with different upper

and lower mantle viscosities, we have found that for relatively high values of the

upper mantle viscosity and relatively low values of the lower mantle viscosity (e.g.

univiscous mantle of 1021 Pas), the pattern is also partly inverted. We will return

to this in the next chapter. Finally, note that the effect of a CLVZ is as large as,

or larger than, the effect of an ALVZ for 3D-velocities, and that, in principle, the

perturbations have rates that are large enough to be detected by time series of

GPS, however, only in certain areas. Note that in these areas, that are in gen-

eral cratonic, we do not expect LVZs, but that we cannot regionally exclude them

because the earth model is laterally homogeneous. Moreover, GPS provides point

measurements and thus makes the recognition of spatial signatures, as defined in

Section 4.4, difficult.

We will therefore look at geoid heights and geoid height perturbations (Section 5.2)

and show the background model error introduced by uncertainties in the ice-load

history. We then focus on the sensitivity of LVZ-induced geoid height perturbations

to the load history. The sensitivity is defined as the difference between perturba-

tions generated with different load histories. We will consider both the sensitivity

to the ice-load history and the coupled ocean-load history (Section 5.3.1). For the

latter we solve the sea-level equation (Eq. 3.3) to include self-gravitation, and use

time-dependent coastlines and meltwater influx (Sections 1.2 and 3.3). In the pre-

vious chapter we have shown, using spherical harmonic degree amplitudes, that

for a CLVZ the perturbations are above the expected GOCE performance up to

spherical harmonic degree 120 and for an ALVZ up to degree 65. Here we show

(Section 5.3.2) that the sensitivity to different load-histories is above the GOCE

performance up to degree 140 for a CLVZ and degree 75 for an ALVZ. This means

that GOCE could provide information on the ice-load history in the presence of an

LVZ. If we are however interested in constraining properties of an LVZ, uncertain-

ties in the ice-load history are an error source. In Section 5.4.1 we show in which

areas LVZ-induced perturbations are robust to variations in the ice-load history

and to both the ice-load history and the properties of the LVZ. The latter can be

considered as spatial signatures for the detection of LVZs. To constrain proper-

ties of LVZs, we have to remove part of the uncertainty in the ice-load history. In

Section 5.4.2 we show that for a CLVZ this is largely possible using correlations

between ice heights and geoid heights in the spectral domain. By normalizing the

geoid height perturbations we can then compute spectral signatures for different

properties of the CLVZ.


(a) ICE-3G(VM1)

−3 −2 −1 0 1geoid height [m]

(b) ICE-5G(VM2)


(c) RSES(STD)


(d) GRACE (GGM02S) data

−18 −12 −6 0 6 12geoid height [m]

Figure 5.3: Filtered geoid height predictions for modified ICE-3G(VM1) (a), ICE-5G(VM2)(b) and RSES(STD) (c), and filtered geoid heights as measured with GRACE (GGM02S,d).


(a) CLVZ

−80 −60 −40 −20 0 20 40geoid height perturbation [cm]

(b) ALVZ

−120 −80 −40 0 40 80 120geoid height perturbation [cm]

Figure 5.4: CLVZ- (a) and ALVZ- (b) induced perturbations in geoid height usingRSES(STD).

5.2 Geoid Heights and Perturbations

We show geoid height predictions in Figure 5.3 for ice-load histories ICE-3G(VM1),

ICE-5G(VM2) and RSES(STD). To highlight the effect of the Fennoscandian ice

sheet, and to make comparisons with satellite gravity data possible, we have again

removed all harmonics lower than degree 9. Note that this is roughly in line with

the filtering by Mitrovica & Peltier (1989), who only retain the signal from de-

gree 10. The GGM02S solution from GRACE (Section 3.4 in Chapter 3) has to be

filtered to remove especially the high due to the Iceland plume, the result is plot-

ted in Figure 5.3d. Note that there is a broad similarity in pattern between the

predictions and the data, especially for RSES(STD) and ICE-5G(VM2). However,

note also that the amplitudes in the data are up to 10 times larger. To confirm

that the signal in the data is not (solely) due to GIA, we also used RSES with

an earth model with a very stiff lower mantle (∼ 1023 Pas, as e.g. derived in the

global inversion of Kaufmann & Lambeck (2002), see Section 1.2). Maximum uplift

rates are then 12 mm/yr, which is still acceptable when compared to the BIFROST

data (Figure 5.1d), but even then, the low in the Gulf of Bothnia is not larger than

about −7 m. As the pattern in the data, which is also visible in earlier gravity data

sets, also roughly corresponds with the thickness of the crust, compare with Fig-

ure 1.2b, Mitrovica & Peltier (1989) suggest, based on a study of Anderson (1984a),


that a large part of the anomaly is provided by the depression of the Moho. An-

derson (1984b) compute, based on the assumption of Pratt isostasy, a theoretical

geoid anomaly of −0.85 m per km depth of the Moho is derived (this value can

be computed from Equation 5-154 in Turcotte & Schubert, 2002, p. 221, with the

density of water replaced by the crustal density and assuming a depth of compen-

sation of 100 km). However, Anderson (1984b) does not provide any argumenta-

tion for using Pratt isostasy, which implies a compensation of the relatively small

crustal density with an anomalously high density of the shallow mantle, though it

is probably based on the lack of topography in large parts of eastern Scandinavia,

which seems to exclude Airy-type of compensation. From correlations between

geoid data and crustal thickness data, Anderson (1984b) finds a value of about −1

m/km. However, Sjöberg et al. (1994) estimate, after removing the harmonics be-

low degree 10 and thus using comparable geoid heights as in Figure 5.3d, a geoid

anomaly of −0.11 m/km. We thus conclude that the interpretation of the data is

not unambiguous and that at least part of the low is due to GIA (Vermeersen &

Schotman, 2008).

From here on we will use the RSES ice-load history as our reference, and use

the modified ICE-3G and the ICE-5G ice-load histories to test the sensitivity of

LVZ-induced perturbations in geoid height. We will use all histories with the STD

earth model (Table 4.1), to concentrate as much as possible on the effect of the

ice-load history only. Note that all three ice-load histories contain information on

the earth stratification, as the ice-load history is, through the use of for example

Holocene sea-level curves, dependent on the earth stratification (Section 3.2). The

use of an ice-load history with a different earth model makes the combination thus

inconsistent with e.g. sea-level curves. However, for our purpose, this is of minor

importance. We compute predictions of LVZ-induced geoid height perturbations

as plotted in Figure 5.4. The pattern is quite similar to the vertical deformation

rates, compare with Figure 5.2. However, because we have not filtered these geoid

height perturbations, the patterns for especially an ALVZ are much broader. Dif-

ferences with the LVZ-induced perturbations in Chapter 4 (Figure 4.3) are due

to differences in the ice-load history only, as we use the same background earth

stratification and properties of the LVZs.

Before we look at the sensitivity to the ice-load history, we repeat the exercise from

Section 4.5.2 where we showed background model errors introduced by uncertain-

ties in the earth stratification, for uncertainties in the ice-load history. In Fig-

ure 5.5a we show the degree amplitudes of two models with a CLVZ (the reference

CLVZ from Table 4.1, ’CLVZ (1018 Pas)’, an a CLVZ with a higher viscosity of 1019

Pas, ’CLVZ (1019 Pas)’), both using RSES(STD) as a background model, and differ-

ent background ice-load histories: RSES, which is the same as used in the models

with a CLVZ, ICE-3G and ICE-5G. We can show that the background ice-load his-

tories are sufficiently different to be able to separate them, either by correlating


(a) total geoid heights

0 25 50 75 100 125 150 175 200

10−3

10−2

10−1

100

harmonic degree

geoi

d he

ight

deg

ree

ampl

itude

[m]

GOCEGGM02SCLVZ (1018 Pas)CLVZ (1019 Pas)BG, RSESBG, ICE−3GBG, ICE−5G

(b) perturbations and errors

0 25 50 75 100 125 150 175 200

10−3

10−2

10−1

100

harmonic degree

geoi

d he

ight

deg

ree

ampl

itude

[m]

GOCEGGM02SCLVZ (1018 Pas)

CLVZ (1019 Pas)BG err. ICE−3GBG err. ICE−5G

Figure 5.5: Degree amplitudes of the performance of GOCE and GRACE (GGM02S), and ofpredictions of geoid heights for models with a CLVZ and background models (no LVZ, a) andCLVZ-induced geoid height perturbations and errors due to uncertainties in the backgroundmodel (b).

geoid height predictions of the different background models with the data (which

are here the models with a CLVZ, compare with Section 4.5.2) or by correlating

the ice-heights at LGM of the different background models with the data (see Sec-

tion 5.4.2). The errors introduced by using ICE-3G or ICE-5G as the background

ice-load history (Figure 5.5b) are thus an upper bound, which we estimate to be

at degree 40 (where the errors cross the perturbations) to 60 (where the errors

are smaller than 10% of the perturbations). When we also include uncertainties

in the earth stratification (i.e. when we use ICE-3G(VM1) and ICE-5G(VM2)) the

induced errors are about the same, so the errors are controlled by uncertainties in

the ice-load history.

If we take the harmonic degree where the errors cross the CLVZ-induced perturba-

tions (∼ 40 in Figure 5.5b) as the lower bound of possibly recoverable harmonic de-

grees and the harmonic degree where the GRACE performance (GGM02S) crosses

the perturbations (∼ 80) as the upper bound, we can compare predictions of CLVZ-

induced perturbations with GGM02S by filtering both so as only to retain the har-

monic degrees between 40 and 80. We have plotted the predictions and measure-

ments in Figure 5.6. It is clear from this simple exercise that it is not straightfor-

ward to retrieve information about CLVZs from static satellite gravity data only,

as the amplitudes in the filtered data are up to 10 times as large, which means

that any information on CLVZs that might be present in the data is well hidden

by signals due to other processes or structures. However, if it is possible to partly

remove these signals by using models for processes and structures in this area,

then the distinctive spatial and spectral patterns induced by CLVZs (Section 5.4)

might at least enable detection of CLVZs.


(a) predictions


(b) measurements

−450 −300 −150 0 150 300geoid height [cm]

Figure 5.6: Filtered (degrees < 40 and > 80) predictions of CLVZ-induced geoid heightperturbations (a) and measured GGM02S geoid heights (b).

5.3 Sensitivity to the Load History

5.3.1 Spatial Domain

We test the sensitivity of our GIA model with an LVZ to the ice-load history by

subtracting the geoid height perturbations predicted by RSES from the perturba-

tions predicted by ICE-5G. Results are then geoid height sensitivities as plotted

in Figure 5.7. For both CLVZ- and ALVZ-induced perturbations, the sensitivity

is large. As expected, the CLVZ-induced perturbations are especially sensitive to

the detailed geometry, with for example large differences between RSES and ICE-

5G just off the coast of Scotland and Norway (see Figure 7.7 in Chapter 7). For

ALVZ-induced perturbations, we see clear large-scale differences between RSES

and ICE-5G in the Barents Sea, where ICE-5G defines a large ice mass (posi-

tive sensitivity), and between RSES and ICE-5G around the Baltic states, where

ICE-5G defines substantially less ice mass (negative sensitivity, compare with Fig-

ure 7.7).

As shown in the previous chapter (Section 4.3.1), the effect of a CLVZ outside the

formerly glaciated areas is additional tilting of the coastlines, resulting, due to

the pertaining negative ocean load, in a relative low on the land-side of the coast-

line and a relative high on the sea-side. For this result we used an eustatic ocean

5.3. Sensitivity to the Load History 67

(a) CLVZ, ice-load sensitivity


(b) ALVZ, ice-load sensitivity

−40 −20 0 20 40 60 80geoid height sensitivity [cm]

(c) CLVZ, ocean-load sensitivity


(d) ALVZ, ocean-load sensitivity


Figure 5.7: Sensitivity to differences in the ICE-5G and RSES ice-load history (a, b) and todifferences in a realistic and a eustatic ocean-load history (c, d) of CLVZ- (left) and ALVZ-(right) induced geoid height perturbations.


load, and we did not consider the effect of self-gravitation in the oceans, time-

dependent coastlines and meltwater influx, together referred to as the effect of

realistic oceans (ROs, Section 3.3). The effect of ROs on CLVZ-induced perturba-

tions is to reduce the tilting near coastlines, i.e. the sensitivity shows a relative

high on the land-side and relative low on the sea-side, which can be seen in Fig-

ure 5.7c along the coasts of Ireland and The Netherlands. This can be understood

if we look at the effect of ROs on relative sea levels and paleotopography (Fig-

ure 5.8a for the RSES ice-load history), in which white areas are either above sea

level or ice-covered (within the thick black contours). We see that a large part

of the North Sea and the Celtic Sea (below Ireland) were above sea level at LGM

and thus experienced a smaller (negative) ocean load than eustatic, which is about

−140 m at LGM. The relatively large negative sensitivity in the Gulf of Bothnia in

Figure 5.7c can be explained by the influx of meltwater during deglaciation, which

provides a relatively large positive load after the ice has disappeared, which leads

to outward flow through the CLVZ in roughly the last 10 kyrs.

The negative sensitivities for an ALVZ over the northern Atlantic (Figure 5.7d)

are due to the effect of self-gravitation in the oceans. Due to self-graviation, the

(negative) ocean load at LGM is larger than the eustatic load in the northern At-

lantic (Figure 5.8a), while near the ice sheets, the relative sea level becomes less

negative. As a result, the present-day ocean load for the self-gravitating ocean

is larger relative to LGM than the eustatic load, resulting in additional material

flow from under the northern Atlantic, and thus in a relative low compared to the

eustatic case.

The effect of three glacial pre-cycles (PCs) is not shown, but is comparable to the

effect of ROs for a CLVZ (as in Figure 5.7c) and negligibly small for an ALVZ. This

can be explained by the large relaxation times induced by a CLVZ and the short

relaxation times for an ALVZ (Section 4.2), i.e. after each glacial cycle the effect of

an ALVZ has largely disappeared, while the effect of a CLVZ due to a glacial cycle,

especially the glaciation phase, persists well into the following glacial cycle. The

main effect of PC is to enhance the perturbations as predicted in Figure 5.4.

The reason we consider the sensitivity to the ocean-load history is that the use

of ROs and PCs significantly increases the computation time. The effect of self-

gravitating oceans in ROs requires 3 to 5 iterations for each time step and the

effect of time-dependent coastlines and meltwater influx requires 3 to 5 iterations

over a complete glacial cycle. The inclusion of three glacial pre-cycles increases,

for a reasonable timestep, the computation time with a factor two. For a later-

ally homogeneous earth, for which we can use the computational efficient spectral

method (Section 2.2), this is in most cases acceptable. However, if we use for ex-

ample the computational-intensive finite-element method (Section 2.3), this is not

feasible anymore, see Chapter 6.

5.3. Sensitivity to the Load History 69

(a) relative sea level at LGM

−225 −200 −175 −150 −125 −100 −75relative sea level [m]

(b) ice height degree amplitudes

0 25 50 75 100 125 150 175 200

101

102

harmonic degree

ice

heig

ht d

egre

e am

plitu

de [m

]

RSESICE−3GICE−5G

Figure 5.8: Relative sea level at LGM compared to present-day (a) and degree amplitudesof the RSES, ICE-3G and ICE-5G ice-load histories at LGM (b). In (a), all white areas areeither above sea-level at LGM (positive paleotopography) or, if they are within the thickblack contour, covered with ice. The eustatic sea level at LGM for the RSES ice-load historyis about −140 m.

5.3.2 Spectral Domain

In Section 5.2 and Chapter 4 (Figure 4.5) we have shown that LVZ-induced per-

turbations are to a certain degree above the performance of GOCE and GRACE.

In the previous chapter we have also indicated that GOCE is sensitive to the ice-

load history by using different filters for our modified ICE-3G ice-load history (see

Figure 4.11). Before we compare the sensitivity in the spectral domain, we first

have a look at the spectral distribution of the ice-load histories at LGM. From Fig-

ure 5.8b we see that the degree amplitudes of the (modified) ICE-3G ice heights

drop off faster with increasing harmonic degree than those of RSES, due to the

use of a Gaussian filter with 200 km halfwidth (Section 3.2). The degree ampli-

tudes of the ICE-5G ice heights are larger than those of RSES for high harmonic

degree. These differences between ICE-3G or ICE-5G with RSES can largely ex-

plain the high sensitivity to both the ICE-3G (’ICE-3G sens.’) and ICE-5G (’ICE-5G

sens.’) ice-load history in Figure 5.9a. We have also plotted the sensitivity to the

inclusion of RO- and PC-effects in the ocean-load history, and for comparison, the

CLVZ-induced perturbations (’CVLZ perturb.’). We see that the sensitivity induced

by the ice-load history is as large as, or larger than, the CLVZ-induced perturba-

tions. This means, on one hand, that GOCE should be able to distinguish between

different ice-load histories in the presence of a CLVZ. On the other hand, uncer-

tainties in the ice-load history will make it very hard to extract information about


(a) CLVZ

0 25 50 75 100 125 150 175 200

10−3

10−2

harmonic degree

geoi

d he

ight

deg

ree

ampl

itude

[m]

GOCECLVZ perturb.ICE−3G sens.ICE−5G sens.RO sens.PC sens.

(b) ALVZ

0 25 50 75 100 125 150 175 200

10−3

10−2

harmonic degree

geoi

d he

ight

deg

ree

ampl

itude

[m]

GOCEALVZ perturb.ICE−3G sens.ICE−5G sens.RO sens.PC sens.

Figure 5.9: Degree amplitudes of CLVZ- (a) and ALVZ- (b) induced sensitivities to theice-load history.

properties of a CLVZ from GOCE data only. We will show in Section 5.4.2 that it

is possible to remove part of the uncertainty in the ice-load history by computing


GOCE is sensitive to the detailed ocean load-history, as both the sensitivity to the

ROs and PCs are above the GOCE performance. Moreover, both sensitivities peak

broadly in the same domain where we expect the largest sensitivity to changes

in the properties of the CLVZ, i.e. between harmonic degrees 40 and 120. The

effect of PCs has a sharp peak around degree 50, which suggests that especially

amplitudes in this area are enhanced due to this effect. The sensitivity to ROs has

a much broader spectrum, with smaller amplitudes, and almost affects the whole

range of recoverable harmonic degrees. The main message here is that, based

on these simulations, both ROs and PCs have to be included in the simulations

to make full use of the accuracy of GOCE. In the presence of an ALVZ (Figure

5.9b) the sensitivity to the ocean-load history is confined to the lower degrees, and

does not seem to be important when compared to ALVZ-induced perturbations

(’ALVZ perturb.’). In the next chapter (Section 6.6) we show that ignoring ROs is

in general acceptable for ALVZ-induced perturbations, but not for CLVZ-induced

perturbations. The sensitivity to the ice-load history is for an ALVZ again large.

5.4 Spatial and Spectral Signatures

We have shown in the previous sections that LVZ-induced perturbations are es-

pecially sensitive to uncertainties in the ice-load history. This sensitivity makes

it difficult to constrain properties of LVZs in the presence of these uncertainties.

In the next section we show areas in which predictions are insensitive to the ice-

load history. These spatial signatures can then be used to detect LVZs. To constrain

5.4. Spatial and Spectral Signatures 71

(a) robust CLVZ


(b) robust ALVZ


Figure 5.10: Predictions of CLVZ- (a) and ALVZ- (b) induced perturbations that are robustto variations in the ice-load history.

properties of LVZs, we show in Section 5.4.2 that for a CLVZ it is possible to largely

remove uncertainties in the ice-load history in the spectral domain and to generate


5.4.1 Spatial Signatures

In Figure 5.10 we show predictions for CLVZ- and ALVZ-induced perturbations

that are robust to variations in the ice-load history. We do this by computing the

mean and the standard deviation of the predictions for different ice-load histories,

see Section 4.4 in Chapter 4. We consider only areas where the mean is larger

than one scale interval (8 cm for a CLVZ and 20 cm for an ALVZ) and the stan-

dard deviation is smaller than half a scale interval (4 cm for a CLVZ and 10 cm

for an ALVZ). From Figure 5.10a we see that we then get for a CLVZ only very

small patches just in- and outside the formerly glaciated areas. For an ALVZ (Fig-

ure 5.10b) we see a clear consistent high at the center of rebound. This means

that, independent of the ice-load history, the effect of an ALVZ on predictions in

the central region is large.

When we compare this with the features that are robust to changes in the prop-

erties of the LVZ and to changes in the upper mantle viscosity of the background


(a) perturbations

0 50 100 150 200

0

0.2

0.4

0.6

0.8

1

harmonic degree

degr

ee c

orre

latio

n co

effic

ient

v19(RSES)⊗ v18(RSES)v18(ICE−3G)⊗ v18(RSES)v18(ICE−5G)⊗ v18(RSES)v19(ICE−5G)⊗ v18(RSES)

(b) ice heights

0 50 100 150 200

−1

−0.8

−0.6

−0.4

−0.2

0

harmonic degree

degr

ee c

orre

latio

n co

effic

ient

ICE(RSES)⊗ GEC(RSES)ICE(ICE−3G)⊗ GEC(RSES)ICE(ICE−5G)⊗ GEC(RSES) ICE(RSES)⊗ GEE(RSES)

Figure 5.11: Degree correlation coefficients between CLVZ-induced geoid height perturba-tions and geoid height predictions of different test models (a), and ice heigts at LGM andgeoid height predictions (b).

model (Figures 4.8c-d and 4.12a-b in Chapter 4), we see that CLVZ-induced pertur-

bations are more robust to changes in the properties of the CLVZ than to changes

in the ice-load history and the upper mantle viscosity. For an ALVZ, we see that

most features (e.g. in northern Greenland and the Gulf of Bothnia) are robust

to both changes in the properties of the ALVZ and of the background model (ice-

load history, upper mantle viscosity), which makes these features more likely to

be detectable from data.

5.4.2 Spectral Signatures

We have shown in Figure 5.9a that if we know the properties of the CLVZ, we

should be able to discriminate between different ice-load histories using GOCE.

However, we only have limited information on properties of CLVZs (see Section 1.1)

and moreover, from a geophysical point of view, we are more interested in con-

straining the properties of the CLVZ using GOCE. The differences between ice-

load histories in CLVZ-induced geoid height perturbations can then be regarded

as an error source. We will show here that it is possible to extract information

from GOCE measurements about a possible CLVZ without exact information on

the ice-load history, under the assumption that the CLVZ is laterally homogenous.

This assumption seems to be quite restrictive, however, we will show in the next

chapter that the effect of realistic lateral heterogeneities is not always large.

First we will show, using degree correlation coefficients (Eq. (4.1) in Section 4.5.2),

that the uncertainty in the ice-load history makes it, for example, impossible to

constrain the viscosity of the CLVZ within an order of magnitude. In Figure 5.11a

we have plotted the degree correlation coefficients between CLVZ-induced geoid

height perturbations (properties for the CLVZ from Table 4.1, i.e. with a viscosity

5.4. Spatial and Spectral Signatures 73

of 1018 Pas, and ice-model RSES, ’v18(RSES)’) and different test model values. If

we use in our test model the ICE-3G or ICE-5G ice-load history, then the correla-

tion between the reference and test model is very poor (’v18(ICE-3G)⊗v18(RSES)’

and ’v18(ICE-5G)⊗v18(RSES)’). If we know the ice-load history, then we can dis-

tinguish between a CLVZ with a viscosity of 1018 Pas and with a viscosity of 1019

Pas (’v19(RSES)⊗v18(RSES)’), because for high harmonic degree the correlation

deviates significantly from one. If we do not know the ice-load history, and corre-

late for example predictions of our reference CLVZ (’v18(RSES)’) with a test model

with both a different viscosity of the CLVZ (e.g. 1019 Pas) and a different ice-load

history (e.g. ICE-5G), then the degree correlation will be poor due to the bad cor-

relation for different ice-load histories (’v19(ICE-5G)⊗v18(RSES)’). This indicates

that we cannot differentiate between an order of magnitude change in viscosity

of the CLVZ in the presence of uncertainties in the ice-load history, because the

correlation for a CLVZ with a viscosity of 1019 Pas (’v19(ICE-5G)⊗v18(RSES)’) is

equally poor as for a viscosity of 1018 Pas (’v18(ICE-5G)⊗v18(RSES)’). At the end

of this thesis (Section 7.5 in Chapter 7) we return to this issue when we try to

recover properties of a CLVZ from synthetic GOCE data.

The poor correlation between models with different ice-load histories suggests

that we might learn something about the ice-load history by correlating the geoid

height data (in this case predictions of geoid heights for a model which includes

a CLVZ) with a certain spectrum of the ice-load history. This is justified further

on by using analogies from signal processing theory, here we will just show that

if we use the spectrum of the ice heights at LGM as representative for the ice-

load history, we can clearly see (Figure 5.11b) that the actual ice-load history that

generated the predictions has a significant higher (negative) correlation with the

geoid heights (’ICE(RSES)⊗GEC(RSES)’, where ICE denotes ice heights at LGM

and GEC denotes geoid heights at present for a model with a CLVZ) than the other

ice-load histories (’ICE(ICE-3G)⊗GEC(RSES)’ and ’ICE(ICE-5G) ⊗GEC(RSES)’).

Note that the large differences are due to the presence of a CLVZ, as the corre-

lation is significantly smaller for geoid heights computed from a model without a

CLVZ (’ICE(RSES)⊗GEE(RSES)’, where GEE are geoid heights at present for a

model with a fully elastic crust). In reality, we do not know the actual ice-load

history, but we can state that the history which has the highest correlation with

the measured geoid heights is the most likely actual history.

If we normalize the geoid height perturbations with the degree amplitudes of the

most likely actual history the resulting curves are very close together (Figure

5.12a), i.e. we have largely removed the influence of (uncertainties in) the ice-load

history. Note that the result for ICE-3G is less convincing because of the sup-

presion of the high harmonics due to the long halfwidth (200 km) of the Gaussian

filter, see Figure 5.8b. If we use a shorter halfwidth (100 km), we see that again we

obtain the same profile for this CLVZ. Finally, we have plotted estimates of spectral


(a)

0 25 50 75 100 125 150 175 200

10−5

10−4

10−3

harmonic degree

norm

aliz

ed d

egre

e am

plitu

de

RSESICE−3GICE−5GICE−3G*

(b)

0 25 50 75 100 125 150 175 200

10−5

10−4

10−3

harmonic degree

norm

aliz

ed d

egre

e am

plitu

de

d20t12v18d20t12v19d20t20v18d32t08v18

ICE-3G*: ICE-3G filtered with halfwidth of 100 km instead of 200 km

Figure 5.12: Normalized geoid height degree amplitudes for CLVZ-induced perturbations(a) and spectral signatures for different properties of the CLVZ (b).

signatures for different properties of the CLVZ in Figure 5.12b, where ’d20t20v18’

has a thicker CLVZ (20 km), ’d20t12v19’ has a higher viscosity CLVZ (1019 Pas)

and ’d32t08v18’ both a deeper (at 32 km) and thinner (8 km) CLVZ. As the sig-

natures differ significantly, it should be possible to extract information about the

properties of a CLVZ from GOCE measurements by following the above-mentioned

procedure. That is, estimate a most likely actual ice-load history from degree cor-

relations with the measured geoid heights, use this information to largely remove

the influence of the load history to obtain a spectral signature, and compare this

measured signature with a set of signatures for different properties of the CLVZ.

Note that the spectral signature for our reference model (’d20t12v18’) is roughly

the same as the total difference in modal strength as discussed in Section 4.2 (solid

line in Figure 4.1c). Actually, if we put a load on the earth for an infinitely long

time, the spectral signature will be equal to the fluid Love number. Equally, for

transient time scales the spectral signature will be equal to the Love number on

transient time scales. This can be understood if we view our GIA model as a lin-

ear shift-invariant, discrete-space system (compare with Oppenheimer & Schafer,

1975, p. 11):

y(n) = h(n)∗ x(n) (5.1)

where ∗ denotes convolution of the unit-sample response h(n) (i.e. the Green’s

function, Eq. 2.15) with the input x(n) (i.e. the load history) to give the output y(n)

(e.g. geoid heights), with n a set of geographical coordinates. The spectral form of

this equation is (Oppenheimer & Schafer, 1975, p. 68):

Y (ν)= H(ν)X (ν) (5.2)

5.5. Conclusions 75

where ν is a frequency associated with a certain base function (in this case degree

and order of a spherical harmonic expansion) and H(ν) is called the system func-

tion. As our system is both a function of geographical coordinates and of time, we

can regard the Love number as the system function of our GIA model for a certain

time interval. For a load that is constant in time, the spectral signature is also

equal to the system function H(ν) for a certain time interval as it is the square-

root of the response power density φyy(ν) (i.e. degree variance of the geoid height

perturbations) divided by the forcing power density φxx(ν) (i.e. degree variance of

the load, compare with Oppenheimer & Schafer, 1975, p. 393):

|H(ν)|2 =φyy(ν)/φxx(ν) (5.3)

As the load that we have used in this study is phased in time, not constant, we

can regard the spectral signature as an approximation of the system function of

our GIA model for a certain time interval. In turn, the constant load is approxi-

mated by the ice load at LGM, which is justified by the constant extent of the ice

load in the glaciation phase in our implementation, and the much longer period of

glaciation compared to deglaciation.

5.5 Conclusions

In comparing predictions of 3D-rates of displacement and geoid heights for dif-

ferent ice-load histories in Northern Europe, we have shown the importance of

filtering the predictions before comparison with data. For 3D-displacement rates

this is useful because the effect of the former Laurentide ice sheet is largely re-

moved, simulating the weak lithosphere along the Mid-Atlantic Ridge. For geoid

heights this is necessary to remove especially the effect of the Iceland plume domi-

nating the GRACE data in Northern Europe. We have filtered the predictions and

data by removing all harmonics below degree 9, which seems to give optimum re-

sults. LVZ-induced perturbations in 3D velocities are larger than the accuracy of

the BIFROST network, but because GPS-measurements are point-measurements,

they are not well suited to recognize the spatial signatures of LVZs, which seems

to be especially important for CLVZs. As CLVZ-induced perturbations have scales

down to 100 km, the resolution of the GPS-measurements should be comparable

to recognize patterns.

We have tested the sensitivity to the load-history of LVZ-induced perturbations by

using three different ice-load histories, and the implementation of a realistic ocean

function and the addition of glacial pre-cycles. The sensitivity to a realistic ocean

function is for a CLVZ dominated by the effect of time-dependent coastlines and

meltwater influx, and is confined to shallow seas as the North Sea and sea areas


that were once glaciated, as the Gulf of Bothnia. For an ALVZ, especially self-

gravitation in the oceans has a large effect. The sensitivity to the ice-load history

is mainly due to ice sheet geometry and the effect for a CLVZ is confined to an area

of a few hundred kilometers around the formerly glaciated areas. There are only

a few patches in this area that are robust to changes in the ice-load history, which

means that detection of CLVZs will be difficult if the assumption on the uncertainty

in ice-load history is correct. The sensitivity in the presence of an ALVZ indicates

more fundamental differences in ice-load histories as e.g. the way the Barents Sea

was glaciated. However, the main low centered on the Gulf of Bothnia is robust to

changes in the ice-load history.

To constrain properties of LVZs, the sensitivity to the ice-load history should at

least be smaller than the sensitivity to properties of the LVZ, or be limited to a

different part of the harmonic spectrum, which is in general not the case. For a

CLVZ, GOCE will be sensitive to differences in the ice-load history up to degree 140

and for an ALVZ up to 70. This means that GOCE should be able to discriminate

between different ice-load histories if we know the properties of the LVZ. How-

ever, this also means that in the presence of uncertainties in the ice-load history,

we cannot recover the properties of the LVZ, which we have shown using degree

correlation coefficients. From the degree correlations between geoid heights gen-

erated with a certain ice-load history and ice heights at LGM we can determine

the most likely ice-load history. If we then normalize the degree amplitudes of

CLVZ-induced geoid height perturbations with the most likely ice-load history, we

can show that the results are almost independent of the ice-load history. We can

thus compute spectral signatures for different properties of the CLVZ.

In reality however, we first have to extract the signal due to an LVZ from the

measurements. We have already shown in Chapter 4 that uncertainties in the

background earth stratification limit the lower bound of recoverable harmonics to

degree 30−50. Here we have shown that the lower bound due to uncertainties in

the ice-load history is larger, around degree 40−60. Moreover, there are a number

of other processes that give rise to mass anomalies in the measured gravity field.

For example, the gravity field in Northern Europe shows a close resemblance to

geoid height predictions of GIA, however, with much larger amplitudes, which

suggests that the signal is at least partly due to e.g., variations in the depth of

the Moho. For CLVZ-induced perturbations we do no find any resemblance with a

filtered version of GGM02S (from degree 40 to 80), where amplitudes are up to 10

times as large. Note, moreover, that this study is valid for the limiting case of a

laterally homogeneous earth, which is clearly not realistic everywhere with regard

to the presence and properties of LVZs. In the following chapter (Chapter 6) we

will therefore introduce lateral heterogeneities in our earth models.

Chapter 6

Regional Perturbations in a

Global Background Model

In Chapters 4 and 5 the effect of crustal and asthenospheric low-viscosity zones (LVZs) on

geoid heights was shown, as predicted by models of glacial-isostatic adjustment. The govern-

ing equations were solved analytically in the spectral domain, which makes the method used

accurate and fast. However, it does not allow for (large) lateral variations in earth stratifica-

tion. As the properties of shallow LVZs can be expected to vary laterally, we have developed a

finite-element model based on ABAQUS. Global (spherical-3D) finite-element models are cur-

rently not capable of providing high-resolution predictions, which we expect due to the shal-

lowness of the LVZs. We therefore use a regional (flat-3D) model and compute geoid heights

from the predicted displacements at density boundaries by solving Laplace’s equation in the

Fourier-transformed domain. The finite-element model is not self-gravitating, but we com-

pare the results with a self-gravitating spectral model under the assumption that the lack

of self-gravitation is partly compensated by the lack of sphericity, and that long-wavelength

differences largely cancel out when using perturbations, which are the difference between a

model with and without an LVZ. We show that geoid height perturbations due to an LVZ

can be computed accurately, though the accuracy deteriorates somewhat with the depth of

the LVZ. Moreover, we show that horizontal rates of displacement, though not accurate for

total displacements, are accurate for perturbations in the near field. We show the effect of

lateral variations in the properties of the LVZ and in lithospheric thickness, and compute

geoid height perturbations for Northern Europe based on a simple laterally heterogeneous

model. The model is forced with a realistic ice-load history and a eustatic ocean-load history.

The errors introduced by using a eustatic instead of realistic ocean-load history are generally

smaller than 10%, but might be critical for perturbations due to crustal LVZs.

This chapter is published in Physics of the Earth and Planetary Interiors (Schotman et al., 2008a).

78 Chapter 6. Regional Perturbations

6.1 Introduction

The process of glacial-isostatic adjustment (GIA), in which the solid earth is de-

formed by load changes due to variations in the volume of continental ice sheets,

is commonly associated with long-wavelength phenomena (spherical harmonic de-

gree <20) as land-uplift in Scandinavia ("postglacial rebound", e.g. Milne et al.,

2001), part of the geoid low in Hudson Bay-area (e.g. Tamisiea et al., 2007), and

variations in the position and speed of the earth’s axis of rotation (polar wan-

der and changes in length-of-day, e.g. Mitrovica et al., 2005). These phenomena

are mainly influenced by the thickness of the lithosphere and the viscosity of the

mantle, and can to a large extent be explained using spherical, radially stratified

(laterally homogeneous or 1D) viscoelastic earth models, which often use a spher-

ical harmonic expansion to solve the governing equations in the spectral (SP) do-

main. In the case of lateral variations, the transformed differential equations in

SP methods can no longer be solved separately for each degree due to mode cou-

pling (Wu, 2002), which complicates the use of the SP method, especially for large

viscosity gradients. Recently, laterally heterogeneous (3D) spherical earth models

based on the finite-element (FE) method have been developed, showing the effect

of lateral variations in lithospheric thickness and mantle viscosity (Wu & van der

Wal, 2003; Zhong et al., 2003; Wu et al., 2005; Latychev et al., 2005).

The upcoming GOCE satellite gravity mission, planned for launch by ESA sum-

mer 2008, is predicted to measure the static gravity field with centimeter accuracy

at a resolution of 100 km or less (Visser et al., 2002, and Section 3.4). This makes

the detection of high-resolution signals as expected from shallow earth structures

possible. In GIA, high-harmonic signatures (spherical harmonic degree > 20) can

be induced by shallow layers (depth < 200 km) with low viscosity (∼ 1018 − 1020

Pas, see Chapter 4). Crustal low-viscosity zones (CLVZs) are layers in the lower

crust which have viscosities that are significantly smaller than the high-viscosity,

for GIA studies effectively elastic, upper crust and lithospheric mantle. As the

presence and properties of CLVZs depend strongly on thermal regime (Meissner

& Kusznir, 1987; Ranalli & Murphy, 1987; Watts & Burov, 2003), these can in

principle not be modeled using 1D-stratified earth models. In Northern Europe

for example, CLVZs are not expected in the cold and thick crust of the Baltic

Shield, but might be present in surrounding younger continental shelves (Sec-

tions 1.1 and 4.1). Asthenospheric LVZs (ALVZs) can probably be found below the

oceanic lithosphere and perhaps also below the continental shelves, but not below

the Baltic Shield (Steffen & Kaufmann, 2005, and Sections 1.1, 4.1).

As these shallow low-viscosity layers induce high-resolution signals, the use of a

3D-spherical FE model is considered to be not feasible yet. We therefore test if pre-

dictions from a 3D-flat FE model, based on the commercial package ABAQUS (Wu,

2004), are accurate enough for computing geoid height perturbations. Perturba-


(a) near-field

8 16 24 32 40

−100

−50

0

50

100

co−latitude [°]

disp

lace

men

t [m

]

self−gravitating RSLeustatic RSLradial displacement

(b) far-field

40 60 80 100 120 140 160 180

−10

−5

0

5

10

co−latitude [°]

disp

lace

men

t [m

]

self−gravitating RSLeustatic RSLradial displacement

Figure 6.1: Near- (a) and far- (b) field effect of an elliptical iceload (8 radius) on the NorthPole after 10 kyrs of loading.

tions are differences between predictions from a perturbed background model and

predictions from the background model itself. Here we assume that we know the

background model, consisting of a laterally homogeneous earth stratification and

ice-load history, from previous GIA studies. As we concentrate on Northern Eu-

rope, this background model will be mainly based on Fennoscandian studies (Lam-

beck et al., 1998; Milne et al., 2001). The presence and properties of LVZs are, via

thermal regime, closely related to the thickness of the lithosphere. We are there-

fore also interested in constraining both the properties of LVZs and the thickness

of the lithosphere simultaneously. As an additional constraint, apart from high-

resolution satellite-gravity data, we will consider the use of 3D-velocities as for

example measured by the BIFROST network (Milne et al., 2001), which has an

accuracy of 0.8− 1.3 mm/yr in the vertical direction and 0.2− 0.4 mm/yr in the

horizontal direction (Johansson et al., 2002).

A change in the volume of continental ice sheets leads to a global change in relative

sea level (RSL), which is defined as the difference between a change in the ocean

bottom topography and in the equipotential surface that coincides with mean sea

level, the geoid. This makes GIA essentially a global process. In this global pro-

cess, mass is conserved and redistributed gravitationally self-consistently, i.e. the

mass redistribution on the surface and in the interior of the earth affect the grav-

itational potential, which in turn affects the mass redistribution. This is called

self-gravitation and leads for example to a higher than eustatic (= uniformly dis-

tributed ice-mass equivalent) RSL near an ice sheet, because the mass of the ice

sheet attracts the nearby ocean. Another effect is the long-wavelength effect in the

solid earth: If we place an ice mass on the North Pole, not only the top of the earth

will depress, but also the opposite side. This can be understood by considering the

northward movement of the center of mass due to accumulation of mass at the

North Pole, the subsequent flattening of the North Pole due to outward movement


of the solid earth and the resulting change in the gravitational potential. Both

effects are illustrated in Figure 6.1 for an elliptical ice load, with a height at the

center of 2500 m and a radius of 8, which is on the North Pole for 10 kyrs.

In Section 6.6 we test if we can use a eustatic ocean-load history to force the

flat, non-self-gravitating FE model. The reason we do not want to include self-

gravitation in the flat model is that we then have to iterate several times over a

glacial cycle (see Section 6.2, and Wu (2004) for a spherical model), which is very

demanding in terms of CPU time. We assume that we can neglect self-gravitation

in the solid earth, because in a flat model the lack of self-gravitation is partly

compensated by the lack of sphericity (Amelung & Wolf, 1994) and because we

assume that some long-wavelength phenomena largely cancel out when using per-

turbations. Using an axisymmetric Heaviside load, we test in Section 6.4 the accu-

racy of predictions from the FE model by comparing differences with a spherical,

self-gravitating SP model with the expected accuracies of the GOCE mission and

the realized accuracy of the BIFROST network. Using the RSES ice-load history

(Lambeck et al., 1998, and Section 3.2) and a simple model of Northern Europe,

with CLVZs in relatively young continental crust and ALVZs below young con-

tinental and oceanic areas, we try to deduce in Section 6.5 if it is important to

consider lateral heterogeneities in LVZs. We start with a short description of the

theory, especially how we compute geoid heights from the output of the flat FE

model (Section 6.2), followed by a description of the FE and SP model used in this

study (Section 6.3).

6.2 Theory

The linearized, elastic equation of motion for geophysical problems in which inertia

can be neglected is (Eq. (2.2), Section 2.1.1):

~∇·~~σδ−~∇(~u ·ρ0 g0~er)−ρδg0~er −ρ0~gδ = 0 (6.1)

with ~~σ the stress tensor, ~u the displacement vector, ρ the density, ~g the gravity

acceleration and g its magnitude, and where the subscripts (’0’, ’δ’) denote the

initial and incremental state (Wolf, 1998), respectively. The incremental gravity

acceleration ~gδ is related to the incremental gravity potential φ as ~gδ =−~∇φ. For

an incompressible material (Poisson’s ratio ν = 0.5), ρδ = 0 and the third term

(internal buoyancy due to material compressibility) vanishes. In this case, the

incremental gravity potential φ can be found from Laplace’s equation (Eq. (2.5),

Section 2.1.1):

∇2φ= 0 (6.2)

6.2. Theory 81

The viscoelastic problem, mostly using Maxwell rheology, is generally solved us-

ing normal-mode techniques which involves a Laplace transformation of Eq. (6.1)

and expanding the system of equations (6.1) and (6.2) in spherical harmonics (Sec-

tion 2.2). We will call these therefore spectral (SP) methods. The system of equa-

tions is solved from the core-mantle boundary upwards using a propagator matrix

technique (Section 2.2.3). For a surface load, boundary conditions at the surface

of the earth are that the normal stress is equal to the load-induced pressure and

that the shear stress is zero (Section 2.2.3). The boundary conditions for the grav-

ity potential are treated further on in Eq. (6.7).

The finite-element (FE) method solves the stiffness equation, which is by the prin-

ciple of virtual work equivalent to the equation of motion (Wu, 2004):

~∇·~~σδ = 0 (6.3)

It thus neglects the advection of pre-stress (second term in Eq. 6.1), the effect of in-

ternal buoyancy (third term) and the effect of self-gravitation (fourth term). In Sec-

tion 6.3 we describe how we will handle these terms when using the FE method.

To compute geoid heights with the flat FE model, we solve Eq. (6.2) by separation

of variables. This leads for planar boundaries to a 2D-Fourier transform (Eq. 2.50)

of the (incremental) gravity potential φ on the horizontal (x− and y−) coordinates:

∂2

∂z2Φ(kx,ky, z)= (k2

x +k2y)Φ(kx,ky, z) (6.4)

with kx = 2π/λx, ky = 2π/λy, and λx, λy the wavelength in the x− and y−direction,

respectively. The general solution to this differential equation is

Φ(kx,ky, z)= A(kx,ky)ekz z +B(kx,ky)e−kz z (6.5)

with kz =√

k2x +k2

y.

Suppose we have a finite earth layer with the surface at z = 0 and the bottom

at z = z1, overlying an infinite substratum. In the substratum, B2(kx,ky) should

be zero to ensure that the potential is zero for z →−∞. At the boundary with the

finite layer (z = z1), we have the following boundary conditions (Wu, 2004; Cathles,

1975, p. 19):

Φ1(z1)=Φ2(z1), ∂zΦ1(z1)−∂zΦ2(z1)=−4πG(ρ2 −ρ1)U1 (6.6)

with Φ1, Φ2 the potential in the finite layer, respectively the substratum, ρ1, ρ2

the respective densities, and U1 the vertical displacement at the boundary. From

this we find that B1(kx,ky)= (2πG/kz)(ρ2 −ρ1)U1ekz z1 in the finite layer.


At the surface of the earth (z = 0) the transformed boundary conditions are (Wu,

2004; Cathles, 1975, p. 19):

Φ0(0)=Φ1(0), ∂zΦ0(0)−∂zΦ1(0)=−4πG(Γ+ρ1U0) (6.7)

with Φ0 the potential above the surface and Φ1 below the surface, and Γ the trans-

formed surface density layer (i.e. water or ice).

As the solution should be finite outside the earth (A0(kx,ky) = 0), we find from

Eq. (6.7) that A1(kx,ky) = (2πG/kz)(Γ+ρ1)U0 in the finite layer, so that the total

solution becomes:

Φ(kx,ky, z) =2πG

kz

(

(Γ+ρ1U0)ekz z + (ρ2 −ρ1)U1ekz(z1−z))

(6.8)

or, at the surface of the earth:

Φ(kx,ky,0) =2πG

kz

(

Γ+ρ1U0 + (ρ2 −ρ1)U1ekz z1

)

(6.9)

For a multilayer model with N layers, this can be generalized to:

Φ(kx,ky,0) =2πG

kz

(

Γ+N∑

n=0

(ρn+1−ρn)Unekz zn

)

(6.10)

We will refer to the individual parts in the summation as displacement potentials.

To make the FE earth model self-gravitating, the displacement potential has to

be computed at every density contrast and applied as an extra load on the corre-

sponding boundaries, after which the FE model has to be updated, and so on (see

Wu, 2004, for a spherical FE model). This is very time-consuming, and is shown

below not to be necessary for a flat model of Northern Europe, because of the rel-

atively small size of the former Fennoscandian ice sheet and the relatively small

density contrasts at shallow depths (see Table 6.1), and because the effect of self-

gravitation is partly compensated by the lack of sphericity in the flat FE model

(Amelung & Wolf, 1994). As the neglect of self-gravitation increases the rate of

displacement and the lack of sphericity seems to increase the displacement (see

Figure 7 of Amelung & Wolf, 1994), the two effects seem to partly compensate each

other under the load only upon unloading the earth (positive rate of displacement,

negative displacement) and enhance each other upon loading (negative displace-

ment rate and negative displacement).

6.3 Model Description

The 3D-flat FE model used in this study is based on the commercially available

package ABAQUS. The area of loading is 5,840x5,840 km with a resolution of 80

6.3. Model Description 83

km (73x73 elements), while the total surface area of the model is 10,000x10,000

km (97x97 elements) to minimize edge effects. The model consists of 14 finite ele-

ments in the vertical direction, of which 11 are used to model the crust and upper

mantle (to 670 km depth) and 3 to simulate the lower mantle (to 10,000 km depth).

This gives a total of 131,726 elements, all 8-node linear bricks. We fix the bottom

of the model and restrict the movement of the edges to the vertical direction. Com-

putations which include LVZs are very time-consuming: A model with an LVZ of

1019 Pas takes about two weeks of CPU-time on a single core 2.8 GHz processor,

which gives a computation time of more than four days on a parallel machine with

two dual core 2.8 GHz INTEL XEON processors. For comparison, the SP model we

use takes less than half an hour on a single core processor, independent of the vis-

cosity stratification. As we are mainly interested in perturbations due to shallow

LVZs, which might make the exact modeling of the lower mantle of minor impor-

tance, we test if we can approximate the lower mantle with infinite elements or

with dashpots. The latter model is based on ideas of Hetzel & Hampel (2005) and

Hampel & Hetzel (2006) for earthquake triggering by GIA.

To test the accuracy of the FE model, we compare predictions with results from

an analytical, spectral model, based on normal-mode analysis (Section 2.2). This

model provides predictions on an incompressible (ρ1 = 0 in Eq. 6.1), Maxwell vis-

coelastic, spherical, self-gravitating earth and is benchmarked with other GIA

models (see Spada & Boschi, 2006, for comparison with an independent method).

To compare the output of the SP model with the output of the flat FE model (Ivins

& James, 1999), we map the former to a Cartesian grid using the Lambert equal

area projection. We take the FE model also to be incompressible, and moreover, to

be not self-gravitating, as explained in Section 6.2. We simulate the advection of

pre-stress by attaching Winkler foundations to each density boundary (Ricchio &

Cozzarelli, 1980; Williams & Richardson, 1991; Wu, 2004). The Winkler founda-

tions acts like springs with a spring constant equal to the product of the gravity

acceleration at and the density contrast across the boundary. Upon loading, the

foundations act instantaneously, i.e. in the elastic limit of a viscoelastic material.

In a purely viscous or inviscid material, the foundations only work after a cer-

tain amount of time, acting as buoyancy forces, as there can be no instantaneous

displacement in a viscous material (see the footnote in Cathles, 1975, p. 14).

The radial earth stratification used is given in Table 6.1, where the background

model consists of a 30 km fully elastic crust, a lithospheric thickness LT = 80 km,

and an upper1 and lower mantle viscosity ηUM = ηLM = 1021 Pas (model UNI, Sec-

tion 3.1). The perturbed model can have a lower crust or asthenosphere with a low

viscosity of 1019 Pas (CLVZ respectively ALVZ), either laterally homogeneous or

1this includes the asthenosphere, the shallow upper mantle, the upper mantle and the transition

zone as defined in Table 6.1.


Table 6.1: Radial earth stratification in which the lower crust and the asthenosphere canhave an LVZ of 1019 Pas

Layer Depth Density ρ Rigidity ν Viscosity η

[km] [kg/m3] [GPa] [Pas]

upper crust 0 2700 27 elastic

lower crust 20 : : elastic or 1019

lithosphere 30 3380 68 elastic

asthenosphere 80 : : 1021 or 1019

shallow upper mantle 140 : : 1021

upper mantle 220 3480 77 :

transition zone 400 3870 108 :

lower mantle 670 4890 221 1021

core 2891 10925 0 0

heterogeneous. Moreover, the thickness of the lithosphere can increase to 140 km

for a cratonic area, and the crustal thickness can vary from 10 km in oceanic areas

to 50 km in cratonic areas. Values for the density ρ and rigidity µ are volume-

averaged from PREM (Dziewonski & Anderson, 1981).

6.4 Test Results

To test the validity of using a flat FE model for low-viscosity layers in GIA studies,

the results are compared with the output of a spherical SP model for a Heaviside

loading (Wolf, 1985; Ivins & James, 1999). The Heaviside load is put on the North

Pole at t = 0 s and stays there infinitely long. In most cases, we evaluate predic-

tions after 10 kyrs. The load consists of an elliptical ice load, with a maximum

height (at the center) of 2500 m, and a radius of 8 (∼ 890 km, comparable with

the dimensions of the former Fennoscandian ice sheet), complemented with a eu-

static ocean load of about −7.5 m. We will first consider total predictions from the

background model, after which we will concentrate on perturbations due to CLVZs

and ALVZs. We model a CLVZ from a depth of 20 to 30 km (10 km thickness) and a

viscosity of 1019 Pas. The perturbations are the difference between the predictions

of a model with a CLVZ and the background model, which has a fully elastic crust.

An ALVZ is modeled at the bottom of the lithosphere (80 km) to 140 km, with a

viscosity of 1019 Pas. Perturbations are now due to the difference in viscosity with

the background model, which has a viscosity of 1021 Pas in this depth range, see

Table 6.1.

In Figure 6.2a we show a comparison of the vertical displacement at t = 0 s as

6.4. Test Results 85

(a) E, FE vs. SP

0 500 ice 1500 2000 2500−45

−40

−35

−30

−25

−20

−15

−10

−5

0

vert

ical

dis

plac

emen

t [m

]

distance from load center [km]

SPFEdiff. (FE−SP) [dm]

(b) VE, FE vs. SP

0 500 ice 1500 2000 2500−500

−400

−300

−200

−100

0

vert

ical

dis

plac

emen

t [m

]



(c) E, deep vs. shallow

0 500 ice 1500 2000 2500

−40

−30

−20

−10

0

vert

ical

dis

plac

emen

t [m

]


crust (deep)crust (infinite)crust (dashpot)670 (deep)670 (infinite)670 (dashpot)

(d) VE, deep vs. shallow

0 500 ice 1500 2000 2500

−500

−400

−300

−200

−100

0

vert

ical

dis

plac

emen

t [m

]


crust (deep)crust (infinite)crust (dashpot)670 (deep)670 (infinite)670 (dashpot)

Figure 6.2: Elastic (t= 0 s, ’E ’, left) and viscoelastic (t> 0 s, total response minus responseat t = 0 s, ’VE ’, right) response for the SP and FE model (a, b) and a deep model (lowermantle to 10,000 km), and shallow models using infinite elements for the lower mantle, orusing dashpots for the lower mantle (c, d).


(a) geoid heights

0 500 ice 1500 2000 2500

−50

−40

−30

−20

−10

0

geoi

d he

ight

per

turb

atio

n [m

]



(b) uplift rates

0 500 ice 1500 2000 2500−12

−10

−8

−6

−4

−2

0

2

4

uplif

t rat

e [m

m/y

r]


SPFEdiff. (FE−SP)

Figure 6.3: Comparison of geoid heights (a) and uplift rates (b) as predicted by the SP andFE method.

predicted by the background model for the SP and FE method. Also the difference

between the two methods is plotted, in decimeters. The deformation is determined

completely by the density and rigidity of the earth model and shows a relatively

large difference (∼ 1 m) below the center of loading. This difference can probably be

attributed to the lack of self-gravitation in our implementation of the FE method,

which leads to a larger gravitational downpull (Wu & Ni, 1996). The effect of in-

compressibility (Poisson’s ratio ν= 0.5) is large in the elastic limit. If we take the

earth to be a Poisson solid (Haskell, 1953; Dahlen and Tromp, 1998, p. 4, 297),

which is an approximation often used to simplify seismological problems and in

which the Lamé parameter λ=µ with the result that ν= 0.25 (Stein & Wysession,

2003, p. 51), the vertical displacement will be about 50% larger. Note that we did

only test material compressibility and that we did not include internal buoyancy

(third term in Eq. (6.1), see for a short discussion Wu, 2004). The difference be-

tween the FE and SP model for the subsequent (t> 0 s) viscoelastic deformation is

smaller, see Figure 6.2b, where the total vertical displacement after 10 kyrs minus

the elastic displacement is plotted. Besides a long-wavelength difference, there are

some smaller differences in the steep part of the curve, which might be due to a

lack of bending in the FE model. This suggests the use of a higher resolution grid

or quadratic elements, but, because the induced errors are very small, only linear

elements are used in this study. The effect of material compressibility is small for

the viscoelastic deformation, decreasing the vertical displacement with only 5%

under the center of the load.

If now infinite elements are used in the lower mantle, instead of 3 layers of ele-

ments for the lower mantle as above (from 670 km to 10,000 km, deep model), we

see from Figure 6.2c that the response is reasonably good in the elastic limit, only

showing deviation in the center. The viscoelastic response however is not large

enough, see Figure 6.2d. In contrast, the model with dashpots for the lower mantle


(a) univiscous

0 500 ice 1500 2000 2500−2.5

−2

−1.5

−1

−0.5

0

0.5

1

1.5

horiz

onta

l dis

plac

emen

t rat

e [m

m/y

r]


SPFEdiff. (FE−SP)inward gradient uplift ratematerial compressible

(b) order of magnitude

0 500 ice 1500 2000 2500−3.5

−3

−2.5

−2

−1.5

−1

−0.5

0

0.5

horiz

onta

l dis

plac

emen

t rat

e [m

m/y

r]


SPFEdiff. (FE−SP)inward gradient uplift ratematerial compressible

Figure 6.4: Horizontal velocities (positive is outward motion) as predicted by the SP andFE method for a univiscous earth model (a) and an earth model with an order of magnitudeincrease in viscosity from the upper to lower mantle (b).

does not show enough response in the elastic limit. This is because there is no re-

sponse of the upper mantle-lower mantle boundary at 670 km, as a dashpot, which

represents a purely viscous material, only acts after a certain amount of time (see

the discussion about Winkler foundations in Section 6.3). The model shows how-

ever a reasonable viscoelastic response. For this work it is especially important

if perturbations generated with these models are accurate enough, which we will

consider further on.

Continuing with the deep model, we see from Figure 6.3a that total geoid heights

can be computed very accurately with the FE method and the use of displacement

potentials as described in Section 6.2. Note that the differences are again given

in decimeters and that the direct effect of the load on the geoid heights (the Γ-

term in Eq. 6.10) is not included. The comparison for vertical velocities (or uplift

rates, Figure 6.3b) is also excellent, and from tests (not shown) we have found

that for both geoid heights and uplift rates the effect of material compressibility is

negligibly small.

For horizontal velocities (Figure 6.4a), the quality of the FE predictions deterio-

rates. This is probably at least partly due to the horizontal boundary conditions

in the FE model and partly due to the lack of sphericity. However, the FE model

shows the behavior that is generally expected (Wu, 2005; Mitrovica et al., 1994b,

Figure 1) and found (Milne et al., 2001) for loading of the earth in GIA studies,

i.e. an inward velocity near the center of the load and an outward velocity just

outside the loading area. This corresponds with a pattern in which the peaks in

the horizontal velocities correspond to the steepest parts in the uplift rates and

small horizontal velocities correspond to the flat parts in the uplift rates, i.e. the

pattern of horizontal velocities can for a large extent be explained by the (inward)

gradient of the uplift rates, as also plotted Figure 6.4a. A strong change in verti-


cal motion thus induces a large horizontal velocity in the direction of the largest

stress, which is under the load and at the forebulge, due to buoyancy forces on

upper mantle material.

For a compressible material, the FE model is in much closer agreement with the

(incompressible) SP model, which seems to indicate that the lack of sphericity

is partly compensated by material compressibility. However, for an earth model

which has an order of magnitude increase in viscosity from the upper to the lower

mantle (ηUM = 0.5 ·1021 Pas, ηLM = 5.0 ·121 Pas, model STD in Section 3.1), the

effect of compressibility is smaller, see Figure 6.4b. Note that now also the SP

model predicts inward motion under the load, but that the differences with the

FE model are still large. We have tested if also part of the differences might be

due to the neglect of degree 1 in the SP model, which changes the reference frame

from the center of mass of the initial earth to the center of mass of the incre-

mental earth (Cathles, 1975, p. 101). However, including degree 1 (following the

implementation of Greff-Lefftz & Legros, 1997) does only significantly affect the

far field (> 2,500 km) predictions, preventing the outward motion to change to an

inward motion from about 7,500 to 20,000 km. We will show below that differ-

ences in horizontal velocity perturbations are much smaller, suggesting that it is

preferable to compute perturbations on a spherical background model. Finally, it

might be of interest to mention that in the elastic limit (as in Figure 6.2a), the

model predicts inward motion only for a compressible material (O’Keefe & Wu,

2002). For an incompressible material, only outward motion is predicted due to

the horizontal elongation of elements under the load.

Differences in vertical displacement perturbations due to a CLVZ do not show the

long-wavelength behavior, because it is largely differenced out, see Figure 6.5a.

The short-wavelength difference is still present, though the results of the SP

method look suspicious near the edge of the load, which might indicate an in-

accuracy due to Gibbs’ phenomenon. The shape of the displacement perturbations

can be attributed to channel flow, in which, opposite to deep flow where relax-

ation times are inversely proportional to the wavelength (Cathles, 1975, p. 43),

relaxation times are proportional to the square of the wavelength (Cathles, 1975,

p. 49), see the relaxation spectrum of the MC-mode in Section 4.2 and the results

in Klemann et al. (2007). This leads to a lack of outward material flow below the

center of the load and a relatively large deformation at the edge of the load (see

Figure IV-21 in Cathles, 1975, p. 158) and in some cases even to a tilting of the

upper crust under the center of the load (e.g. in the presence of a weaker upper

mantle, compare the geoid height perturbations in Figure 6.7a). The comparison

with Figure IV-21 in Cathles (1975) needs some comment, as there total flow of a

channel is shown, whereas we consider perturbations due to a channel in a back-

ground model that also consists of a lithosphere and mantle. We argue however

that the comparison is valid, as a large part of the response of the lithosphere and


(a) CLVZ, vertical displ. perturbations

0 500 ice 1500 2000 2500−10

−5

0

5

10

15

vert

ical

dis

plac

emen

t per

turb

atio

n [m

]


SPFEdiff. (FE−SP)

(b) ALVZ, vertical displ. perturbations

0 500 ice 1500 2000 2500−20

−15

−10

−5

0

5

10

15

20

radi

al d

ispl

acem

ent p

ertu

rbat

ion

[m]


SPFE (2 el. AS)diff. (FE − SP)FE (4 el. AS)diff. (FE − SP)

(c) CLVZ, geoid height perturbations

0 500 ice 1500 2000 2500−20

−15

−10

−5

0

5

10

15

20

geoi

d he

ight

per

turb

atio

n [c

m]


SPFEdiff. (FE − SP)

(d) ALVZ, geoid height perturbations

0 500 ice 1500 2000 2500−60

−40

−20

0

20

40

60

geoi

d he

ight

per

turb

atio

n [c

m]


SPFE (4 el. AS)diff. (FE−SP)

Figure 6.5: CLVZ- (left) and ALVZ- (right) induced vertical displacement perturbations (a,b) and geoid height perturbations (c, d) due to a laterally homogeneous LVZ.


(a) vertical displ. perturbations

0 500 ice 1500 2000 2500−10

−5

0

5

10ve

rtic

al d

ispl

acem

ent p

ertu

rbat

ion

[m]


deepinfinitedashpot

(b) geoid height perturbations

0 500 ice 1500 2000 2500−20

−15

−10

−5

0

5

10

15

20

geoi

d he

ight

per

turb

atio

n [c

m]


deepinfinitedashpot

Figure 6.6: CLVZ-induced vertical displacement (a) and geoid height (b) perturbations fora deep model (lower mantle to 10,000 km), a model using infinite elements for the lowermantle, and a model using dashpots for the lower mantle.

mantle is differenced out, and because the sensitivity to the background model is

not changing the pattern of the perturbations, see Figure 6.7a.

The computed geoid height perturbations agree very well, smoothing the short-

wavelength differences visible for the vertical displacement perturbations, see Fig-

ure 6.5c. Amplitudes are small, but significantly larger than the expected accuracy

of the GOCE mission, which is smaller than 1 cm for wavelengths longer than a

few 100 km. The maximum difference with the SP model is a few centimeters,

which is comparable to the accuracy of GOCE, especially if we take into account

the suspicious behavior of the SP method near the edge of the ice load (see Fig-

ure 6.5a). From Figure 6.5b we see that for an ALVZ, the quality of the predictions

of the FE model seems to deteriorate. We found that at least 4 element-layers

are needed to model the 60 km thick ALVZ and that the improvement using 6

layers is only marginal. The remaining discrepancy can probably be attributed to

the lack of self-gravitation in the flat model, as the difference for the 4 element-

layer shows a trend from negative under the load towards zero in the far field, as

was also found for the elastic limit of the total displacements (Figure 6.2a). The

short wavelength difference is probably due to the use of linear elements, as ex-

plained in discussing the differences between the results of the FE and SP method

in Figure 6.2b. The same conclusions seem to hold for ALVZ-induced geoid height

perturbations as shown in Figure 6.5d.

As discussed before, it would be useful if we could replace the layering of the lower

mantle with either infinite elements or dashpots in computing perturbations. In

Figure 6.6a we compare the perturbations in vertical displacement due to a CLVZ

computed with a deep model (lower mantle to 10,000 km), a model with infinite

elements for the lower mantle, and a model with dashpots for the lower man-

tle. Differences are small, though in general larger than the differences between


(a) CLVZ

0 500 ice 1500 2000 2500−30

−20

−10

0

10

20

30

geoi

d he

ight

per

turb

atio

n [c

m]


80/ 1.0/ 1.0 80/ 1.0/ 5.0 80/ 0.5/ 5.0100/ 0.5/ 5.0

(b) ALVZ

0 500 1000 1500 2000 2500−300

−250

−200

−150

−100

−50

0

50

100


geoi

d he

ight

per

turb

atio

n [c

m]

80/ 1.0/ 1.0 80/ 1.0/ 5.0 80/ 0.5/ 5.0100/ 0.5/ 5.0

Figure 6.7: Effect of the background earth model on CLVZ- (a) and ALVZ- (b) inducedperturbations (the legend gives: LT [km]/ ηUM [1021 Pas]/ ηLM [1021 Pas]).

the deep model and the SP model. However, for geoid height perturbations (Fig-

ure 6.6b), differences are larger than a few centimeters and it seems necessary for

our purposes to use the deep model. The differences can probably be explained by

the contribution of the displacement potential (see Eq. (6.10) and the discussion

there) of the upper mantle-lower mantle (670 km) boundary to the geoid height

perturbations, of which the movement is not modeled very accurately for espe-

cially the model with infinite elements (see Figure 6.2d). The contribution of this

boundary is small compared to the surface contribution, but not much smaller in

magnitude than the contribution of the much shallower (30 km) crust-lithosphere

boundary. This is due to the larger density contrast at 670 km (1020 vs. 680

kg/m3 at 30 km, see Table 6.1) and due to the longer wavelength induced at 670

km (about five times larger than at 30 km). For an ALVZ, the displacement po-

tential of the upper mantle-lower mantle boundary has, due to the large effect of

an ALVZ on the movement of this boundary (amplitude of ∼ 10 m to ∼ 2 m for

a CLVZ), a significant contribution to the geoid height perturbations, being only

smaller than the surface contribution. Predictions using a model with the lower

mantle replaced by dashpots or infinite elements are thus worse than for a CLVZ,

and are therefore not shown.

The perturbations are sensitive to the background model, see Figure 6.7a, where

we have plotted the CLVZ-induced perturbations for our reference background

model (LT = 80 km, ηUM = ηLM = 1.0 ·1021 Pas, ’80/ 1.0/ 1.0’), and models with

changes in ηLM, ηUM and LT. The effect of a weaker upper mantle (compare ’80/

0.5/ 5.0’ and ’80/ 1.0/ 5.0’) is largest, because of additional outward material flow

in the upper mantle and larger bulges just outside of the load, i.e. the depres-

sion under the load is deeper and steeper. This increases the stress in the CLVZ

just inside and outside the loading area (in the steep part of the depression and

the bulge), enhancing mass flow through the CLVZ from the edges outwards, and


(a) CLVZ

0 500 ice 1500 2000 2500−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

horiz

onta

l vel

ocity

per

turb

atio

n [m

m/y

r]


SPFEdiff. (FE−SP)

(b) ALVZ

0 500 ice 1500 2000 2500−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

horiz

onta

l vel

ocity

per

turb

atio

n [m

m/y

r]


SPFEdiff. (FE−SP)

Figure 6.8: CLVZ- (a) and ALVZ- (b) induced horizontal velocity perturbations (positivedirection is outward motion).

tilting of the upper crust in the center. The background model does not change

the pattern of channel flow, which is not the case for ALVZ-induced perturbations,

see Figure 6.7b. Where the perturbations for our reference background model are

determined by channel flow (small perturbations under the center of the load),

the perturbations for a weaker upper mantle (ηUM = 0.5 ·1021 Pas) show deep flow

behavior. This effect is, somewhat less pronounced, also present for only a stiffer

lower mantle (ηLM = 5.0 ·1021 Pas).

Perturbations in horizontal velocities show (see Figure 6.8), in contrast with total

horizontal velocities (Figure 6.4a), the same behavior for the FE and SP method.

Both for a CLVZ and an ALVZ, the quality deteriorates towards the far field, at

least partly due to the boundary conditions in the FE model, which are fixed in the

horizontal direction at 10,000 km. If we compare this with the SP model, in which,

due to symmetry considerations, the horizontal direction is fixed at the south pole

(∼ 20,000 km), we argue that part of the difference can be explained by the larger

horizontal elongation or stretching of the crust/lithosphere for a shorter distance to

a fixed point, i.e. for the FE model. However, due to the large differences between

the FE and SP model for total horizontal velocities (Figure 6.4a), other causes

such as the lack of sphericity cannot be ruled out. The patterns can again for

a large part be explained by the inward gradient of the uplift rates (not shown).

The noisy character of the perturbations, especially for a CLVZ, are due to the

double differencing and the relatively low precision (’single’, default) used for the

output of ABAQUS, which can be changed to ’double’ in the execution procedure.

Note that the differences with the SP model are small compared to the accuracy

of the BIFROST network (∼ 0.3 mm/yr), however, so are the perturbations. This

means that LVZs can probably not be detected by BIFROST, especially because

high-resolution spatial information is not provided.


(a) CLVZ

0 500 ice 1500B 2000 2500A−20

−15

−10

−5

0

5

10

15

geoi

d he

ight

per

turb

atio

n [c

m]


homogeneous CLVZCLVZ > A (670 km)CLVZ > B (1110 km)

(b) LT and CLVZ

0 500 ice BA 1500 2000 2500

−20

0

20

40

60

80

100

geoi

d he

ight

per

turb

atio

n [c

m]


craton < A (670 km)craton < B (1110 km)craton < A, CLVZ > Acraton < B, CLVZ > B

Figure 6.9: Geoid height perturbations due to a laterally heterogeneous CLVZ (a) and dueto a laterally heterogeneous LT and CLVZ (b).

In Figure 6.9a perturbations are shown due to a laterally heterogeneous CLVZ,

that extends from just under the load (’A’, 670 km) outward. This means that the

laterally heterogeneous model is equal to the background model from the center of

the load to ’A’, and equal to the model with a laterally homogeneous CLVZ from ’A’

outwards. Because under the load the induced stresses are largest, the response

is comparable to the homogeneous case from 670 km outwards. Inwards, there

seems to be some tilting of the crust relative to the displacement of the crust in

the background model, due to the indentation of the crust from ’A’ outwards. For a

CLVZ from 1110 km (’B’) outwards, the effect of the lateral heterogeneity is much

more pronounced, and leads to a significant relative tilting of the crust. For an

ALVZ the effect of lateral heterogeneities is larger, substantially weakening the

amplitudes of the homogeneous case, but mainly showing the same effect as for

a CLVZ and therefore not shown. As shallow viscosity and lithospheric thickness

LT are via thermal regime closely related, we show in Figure 6.9b the effect of

the inclusion of a larger LT (’craton’) under the load only, and the inclusion of

both a larger LT under the load and a laterally heterogeneous CLVZ. The effect

of a craton under the load is to strengthen the lithosphere, thus de-amplifying

the vertical displacement in the center, and thus inducing a positive geoid height

perturbation. If the craton extends to outside the load (’craton < B (1110 km)’),

there is also a negative perturbation outside the load, because, due to the larger

lithospheric strength, the bulge area is broader and smoother. If we now compare

the perturbations due to variations in LT only with perturbations due to both a

laterally heterogeneous LT and CLVZ, we see that only for the case that part of the

ice load is in an area of small LT and a CLVZ (’craton < A, CLVZ > A’), differences

are as large as 10 cm. Due to the different scales of deformation for variations in

lithospheric thickness (wavelengths mainly larger than 1000 km) and for a CLVZ

(mainly smaller than 1000 km), the two effects might be separated using medium-


to high-resolution gravity field information as provided by GOCE.

6.5 Example for Northern Europe

We will now investigate the effect of lateral heterogeneities in LVZs and LT using

a simple regional model for Northern Europe, in which we assume 3 different types

of areas, see also Figure 6.10a-c:

I. Continental areas with a 50 km thick elastic crust in a thick elastic litho-

sphere (LT = 140 km), on top of an upper mantle with a viscosity of 1021 Pas,

which represent old, cratonic areas that are mainly Archean and Early Pro-

terozoic (> 1.5 Gyrs, Pérez-Gussinyé & Watts, 2005). This type corresponds

well with estimates of thermal thickness larger than 140 km (Artemieva &

Mooney, 2001);

II. Continental areas with 20 km elastic upper crust and a 10 km lower crust

that is either elastic or has low viscosity (1019 Pas, CLVZ). The total litho-

spheric thickness is thinner now (LT = 80 km) and overlies a 60 km thick

asthenosphere that has either the viscosity of the upper mantle (1021 Pas)

or a low viscosity (1019 Pas, ALVZ). These areas are younger (< 1.5 Gyrs) and

have a thermal thickness smaller than 140 km;

III. Oceanic areas with a 10 km crust in a lithosphere of LT = 80 km, overlying

a 60 km thick asthenosphere with a viscosity of 1021 Pas or 1019 Pas (ALVZ).

The different areas are derived from CRUST2.0 (Bassin et al., 2000, and see Fig-

ure 1.2b), in which an area with a crustal thickness between 15 and 35 km is

considered as Type II. Areas with a crustal thickness thinner than 15 km are then

Type III and thicker than 35 km Type I. The different areas are indicated in Fig-

ures 6.10d. Note that this is a first approximation, which means that some areas

in Type II are oceanic and that some areas in Type I are not cratonic. We use

the ice-load history as derived by Lambeck (RSES, Lambeck et al., 1998), which

is given from 30 kyrs BP to present for Fennoscandia, the Barents Sea area and

the British Isles. We complement this with the eustatic ocean-load history of the

global RSES model, which is about −130 m from 30 kyrs BP to the last glacial

maximum (LGM, 21 kyrs BP), and include a linear glaciation phase of 90 kyrs

to have a glacial cycle of 120 kyrs. We have plotted the ice heights at LGM in

Figure 6.10e.

In Figure 6.11a we show total (i.e. background-induced) 3D-velocity predictions for

Northern Europe computed with the FE model. Contour lines show the difference

in uplift rates with the SP model, the black arrows are horizontal velocities com-

puted with the FE model and the white arrows are computed with the SP model.

6.5. Example for Northern Europe 95

(a) type I (b) type II (c) type III

(d) areas

−1500

−750

0

750

1500

[km

]

−1500 −750 0 750 1500[km]

I II IIItypes

(e) RSES ice heights

−1500

−750

0

750

1500

[km

]

−1500 −750 0 750 1500[km]

500 1000 1500 2000 2500 3000ice height [m]

Figure 6.10: Earth stratification for the 3 areas: type I (a), type II (b) and type III (c),corresponding geographical areas based on CRUST2.0 (d), and RSES ice heights at LGM(e).


(a) 3D-velocities

−1500

−750

0

750

1500

[km

]

−1500 −750 0 750 1500[km]

1 mm/yr

−0.2

−0.2

−0.2

−4 −2 0 2 4 6uplift rate [mm/yr]

(b) geoid heights

−1500

−750

0

750

1500

[km

]

−1500 −750 0 750 1500[km]

−0.

02

−0.02

0.02

0.02

0.02

0.02

−3.2 −2.4 −1.6 −0.8 0.0 0.8geoid height [m]

Figure 6.11: Total (no LVZ) 3D-velocities (a) and geoid heights (b) at present, computedwith FE (white arrows are computed with SP, contour lines are difference with SP).

As expected from Figure 6.3b, the uplift rates compare well, with only some small

areas where the difference is larger than 0.2 mm/yr (which is much smaller than

the accuracy of about 1 mm/yr as reported for the BIFROST network, Johansson et

al., 2002). Note that the predicted uplift rate is sensitive to the earth stratification,

which explains differences with e.g. Milne et al. (2001), who use the same ice-load

history. The horizontal rates compare less well, though not as bad as expected

from Figure 6.4a. In Figure 6.11b we show the total geoid height predictions. Con-

tour lines are again differences with the SP model, which are generally smaller

than 2 cm, which is comparable to the accuracy of the GOCE mission. Note that

we have subtracted the mean from both fields, as we have in our FE computations

no self-gravitating sea level that can serve as a reference for the geoid height.

Also for LVZ-induced perturbations in 3D-velocities we see some differences be-

tween FE and SP (Figure 6.12), though the general pattern agrees well. Note that

perturbations in horizontal velocities are up to 0.5 mm/yr, which is slightly larger

than the accuracy of the BIFROST network of about 0.3 mm/yr. However, due

to the point-like character of GPS measurements, which makes the extraction of

relatively high-resolution spatial signatures difficult, and the fact that part of the

perturbative signal is generated by assuming LVZs below Scandinavia, we expect

that BIFROST cannot add information on LVZs. This is confirmed by Milne et al.

(2001), who indicate that the radial resolving power of their inversion is ∼200 km

in the shallow earth. In Figure 6.13a we show geoid height perturbations due to

6.5. Example for Northern Europe 97

(a) CLVZ

−1500

−750

0

750

1500

[km

]

−1500 −750 0 750 1500[km]

1 mm/yr

−0.1

−0.1

−0.1

−0.1

0.1

0.1

0.1

0.1

0.1

0.1

0.1

−0.8 −0.4 0.0 0.4 0.8 1.2uplift rate [mm/yr]

(b) ALVZ

−1500

−750

0

750

1500

[km

]

−1500 −750 0 750 1500[km]

1 mm/yr

−0.1

0.1

0.1

0.1

−1.2 −0.6 0.0 0.6 1.2 1.8uplift rate [mm/yr]

Figure 6.12: CLVZ- (a) and ALVZ- (b) induced 3D-velocity perturbations (white arrows arecomputed with SP, contour lines are difference with SP).

a laterally homogeneous CLVZ, in which contour lines indicate again differences

with the SP model. The pattern can be explained by lower crustal material flow

through the low-viscosity channel from under the ice load outwards during glacia-

tion, and the long relaxation times of crustal channel flow (Section 4.2). Due to

the long glaciation period (∼ 100 kyrs), the earth is close to isostatic equilibrium

at LGM and during the much shorter deglaciation period (∼ 20 kyrs) only part of

the crustal material has returned from the perturbative bulges. For a homogenous

ALVZ the differences are also small, see Figure 6.13b. The pattern is broader, and

moreover of opposite sign. This is due to the fact that an ALVZ accelerates the

deformation process, as described in Section 4.2.

To highlight the effect of lateral heterogeneities in the LVZ, we show in Figure 6.13c

and 6.13d the difference between a model which includes a laterally heteroge-

neous LVZ and a model which includes a laterally homogeneous LVZ (as shown

in Figure 6.13a and 6.13b). To focus on the LVZ only, we have not yet included

variations in the lithospheric thickness. For a CLVZ the differences are relatively

small, though still with amplitudes up to 20 cm. Of course, the differences are

large in the Baltic Shield, but also in southern Scandinavia, at the edge of the

cratonic area, and off the Norwegian coast, where the geoid high predicted by the

model with a homogeneous CLVZ is for the model with a laterally heterogeneous

CLVZ significantly smaller. For an ALVZ, the effect of lateral heterogeneities is

somewhat larger, showing a clear difference under the main domes of the RSES


(a) CLVZ, laterally homogeneous

−1500

−750

0

750

1500

[km

]

−1500 −750 0 750 1500[km]

−2

−2

−2

−2

−2

−2

−2

2

2

2

2

2

2

2

2

2


(b) ALVZ, laterally homogeneous

−1500

−750

0

750

1500

[km

]

−1500 −750 0 750 1500[km]

−5

5

5

5

5

−90 −60 −30 0 30geoid height perturbation [cm]

(c) CLVZ, effect of lateralheterogeneities

−1500

−750

0

750

1500

[km

]

−1500 −750 0 750 1500[km]


(d) ALVZ, effect of lateralheterogeneities

−1500

−750

0

750

1500

[km

]

−1500 −750 0 750 1500[km]

−16 0 16 32 48geoid height perturbation [cm]

Figure 6.13: Geoid height perturbations for a laterally homogeneous CLVZ (a) and ALVZ(b), and differences between a laterally heterogeneous and homogeneous CLVZ (c) andALVZ (d).

6.6. Realistic Ocean-Load History 99

(a) laterally heterogeneousLT

−1500

−750

0

750

1500

[km

]

−1500 −750 0 750 1500[km]


(b) laterally heterogeneousLT and CLVZ

−1500

−750

0

750

1500

[km

]

−1500 −750 0 750 1500[km]


(c) laterally heterogeneousLT and ALVZ

−1500

−750

0

750

1500

[km

]

−1500 −750 0 750 1500[km]


Figure 6.14: Geoid height perturbations due to a laterally heterogeneous LT in the absenceof an LVZ (a), and combined effect of a laterally heterogeneous LT and CLVZ (b) or ALVZ(c).

ice-load history, but also further outwards, especially to the east, due to the larger

depth of the ALVZ compared to the CLVZ.

Before we look at the simultaneous effect of lateral variations in LT and LVZs, we

show the effect of variations in LT in the absence of an LVZ. The geoid height per-

turbations in Figure 6.14a are thus only due to the thicker lithosphere (LT = 140

km) of the Baltic Shield. The effect is largest over the Baltic Sea, as there the

ice load is largest, see Figure 6.10e. In the perturbations due to lateral hetero-

geneities in both LT and a CLVZ (Figure 6.14b), we can still see the effect of

the CLVZ around the formerly glaciated area (geoid highs of 20−30 cm, see Fig-

ure 6.13a) and the effect of the large LT below the largest ice load (geoid high of

50 cm, compare with Figure 6.14a). Due to the different spatial signatures of both

effects, it seems to be possible to separate the two and extract information on both

LT and crustal low-viscosity. For an ALVZ (Figure 6.14c) the separation of the

two effects seems to be difficult, as the effect of variations in LT (Figure 6.14a)

is very similar to ALVZ-induced perturbations (Figure 6.13b), especially along the

Norwegian coast and under the major ice domes.

6.6 Realistic Ocean-Load History

For loading the FE model for Northern Europe in Section 6.5, we used the eustatic

ocean-load output of the SP model. Here we show which error this introduces in

geoid height predictions compared with predictions of a self-gravitating model that

includes realistic oceans (ROs, which includes the effect of self-gravitation in the

oceans, but also the effect of time-dependent coastlines and meltwater influx, see


(a) background, error ingeoid heights

−1500

−750

0

750

1500

[km

]

−1500 −750 0 750 1500[km]

−20 −10 0 10 20geoid height [cm]

(b) CLVZ, error in geoidheight perturbations

−1500

−750

0

750

1500

[km

]

−1500 −750 0 750 1500[km]


(c) ALVZ, error in geoidheight perturbations

−1500

−750

0

750

1500

[km

]

−1500 −750 0 750 1500[km]


Figure 6.15: Errors introduced by using a eustatic ocean load in geoid heights for thebackground model (a), and in geoid height perturbations for a CLVZ (b) and ALVZ (c).

Sections 3.3 and 5.3.1). For total geoid heights predicted with the RSES ice-load

history, the errors introduced by using the eustatic load output of the SP model

is shown in Figure 6.15a. The maximum error is about 20 cm, which is small

compared to the total amplitudes of several meters (Figure 6.11b). Moreover, the

relatively large error over the Northern Atlantic (geoid height predictions in this

area are about zero) can be removed if we use the realistic ocean-load output of

the SP model. However, the errors in areas that were once glaciated and are

now ocean covered increase to 30 cm (Barents Sea, North Sea) and 50 cm (Gulf of

Bothnia). For CLVZ-induced perturbations, the error is generally smaller than 5

cm (Figure 6.15b), compared to CLVZ-induced geoid height perturbations of tens

of centimeters (Figure 6.13a). Note the clear effect of time-dependent coastlines

and meltwater influx in the the Gulf of Bothnia, which is also present if we use the

realistic ocean-load output of the SP model. Overall errors are somewhat smaller

then, but probably not negligibly small. For an ALVZ, the errors are up to 10 cm

(Figure 6.15c), but ALVZ-induced geoid height perturbations have amplitudes that

are about 5 times as large, compare Figure 6.13b, so these errors can probably be

neglected.

6.7 Conclusion

We have shown that a 3D-flat, non-self-gravitating finite-element (FE) model of

glacial-isostatic adjustment (GIA) can deliver accurate predictions of radial dis-

placements, uplift rates and geoid heights when compared to conventional spher-

ical, self-gravitating spectral (SP) models. Perturbations in these predictions due

to shallow low-viscosity zones (LVZs) can also be computed accurately, though the

6.7. Conclusion 101

Table 6.2: Error estimates for the use of a regional model in a global background model.GEO are geoid heights and VEL are horizontal velocities, L indicates that the error has along-wavelength (>∼ 1000 km) signature and S a short-wavelength (<∼ 1000 km).

Source Quantity total CLVZ pert. ALVZ pert.

FE vs. SP GEO 1% (L) 5% (Sa) 10% (Sa) + 10% (L)

VEL 50% (L) 5% (Lb) 25% (Lb)

EU vs. RO GEO 1% (L) 25% (Sc) 10% (L)

acan probably be removed using smaller gridsize or quadratic elements (Section 6.4)

bcan probably be reduced by increasing the horizontal extent of the FE model (Section 6.4)

ccan be somewhat reduced using the RO instead of the EU load output of the SP model (Section 6.6)

accuracy seems, due to the neglect of self-graviation, to decrease somewhat with

increasing depth, and moreover, the sensitivity to the background model increases

considerably. Predictions of horizontal velocities are less accurate, showing con-

siderable differences with the predictions of an SP model, probably due to the lack

of sphericity of the model. We have to keep in mind however that horizontal ve-

locities are very sensitive to model parameters in general. We have shown that

material compressibility can have a significant effect on horizontal velocities. The

effect on uplift rates and geoid heights is found to be negligibly small.

For a simple model of Northern Europe, horizontal and vertical velocity pertur-

bations due to shallow LVZs are only slightly above the current BIFROST per-

formance and due to the relatively low spatial resolution of BIFROST, we do not

expect BIFROST to add information on LVZs. We have found that lateral hetero-

geneities do not necessarily have a large influence on CLVZ-induced perturbations,

though amplitudes are still up to 20 cm in geoid height, which is an order of mag-

nitude larger than the expected GOCE performance. In reality, however, we also

have to consider unmodeled thermally- and chemically-induced shallow mass in-

homogeneities, that are expected to be effective in masking the gravity signal due

to CLVZs. We find, for example, from the GRACE satellite gravity mission (solu-

tion GGM02S, Tapley et al., 2005), only retaining the spherical harmonic coeffi-

cients from degree 40 to 80 (Section 5.2), where CLVZs have their largest signal,

geoid heights of a few meters. Effective spatio-spectral filtering tools have to be

developed to extract the GIA-related signal (see the introduction of Chapter 7 for

a discussion), in which we expect the specific signatures of CLVZs (Section 5.4) to

be useful. For example, because of the different signature of perturbations due to

a CLVZ and due to variations in lithospheric thickness, it seems possible to solve

for both lithospheric thickness and CLVZs at the same time.

Forcing the FE model with a eustatic ocean load gives in general acceptable re-

sults, which can be improved somewhat using the realistic load output of the self-


gravitating SP model. Errors in CLVZ-induced geoid height perturbations are

about 5 cm, which is small but not negligible compared to the amplitudes of the

perturbations (tens of centimers). For an ALVZ, the errors are somewhat larger,

but small compared to the perturbations. Based on the above we have given in

Table 6.7 overview of error estimates using the flat, not self-gravitating FE model

and the spherical, self-gravitating SP model (’FE vs. SP’), for geoid heights and

geoid height perturbations (’GEO’) and horizontal velocities and velocity pertur-

bations (’VEL’). We have also given the error estimates of using a eustatic instead

of realistic ocean load (’EU vs. RO’). If we accept errors equal to or smaller than

10%, we have to be critical when computing (perturbations in) horizontal velocities

and when using a eustatic ocean load for computing CLVZ-induced geoid height

perturbations. For this error bound, we cannot use infinite or dashpot elements

to replace the deep layering of the lower mantle (to 10,000 km), as the errors are

larger than 10% for LVZ-induced perturbations.

Acknowledgements

We thank Taras Gerya, Erik Ivins and an anonymous reviewer for their useful

comments, Kurt Lambeck and co-workers (RSES, Australian National Univer-

sity, Canberra) for providing the global RSES ice-load history, Marianne Greff-

Lefftz (IPGP, Paris) for benchmarking degree-1 deformation, Pieter Visser (DEOS,

Delft University of Technology, Delft) for providing estimates of the GOCE perfor-

mance, and Radboud Koop (NIVR, Delft, previously at SRON, Utrecht), Rob Gov-

ers (IVAU, Utrecht University, Utrecht) and the Geoscientific ABAQUS User Group

(Geoqus), especially Andrea Hampel (Institut für Geologie, Mineralogie und Geo-

physik, Ruhr-Universität Bochum), for discussions.

Chapter 7

Thermomechanical Models

of the Shallow Earth

Current constraints on the glacial-isostatic adjustment process in Northern Europe are main-

ly provided by relative sea level data and GPS measurements. Due to a lack of resolving

power in the shallow earth (down to ∼ 200 km), these datasets only provide weak con-

straints on the shallow viscosity structure and the thickness of the lithosphere. Future high-

resolution gravity data, as expected from ESA’s Gravity field and steady-state Ocean Circu-

lation Explorer (GOCE) to be launched summer 2008, are predicted to provide additional

information on the shallow earth, especially the viscosity structure. However, mass inhomo-

geneities due to chemical and thermal anomalies are expected to interfere with the gravity

signals induced by shallow low-viscosity structures. We test therefore if heatflow data and

laboratory-derived creep laws for the crust (plagioclase feldspars) and shallow upper man-

tle (olivine) can provide additional information on the shallow earth. We show estimates

of lithospheric thickness and viscosity that can be expected in the shallow earth. Using a

mechanical model based on the commercially available finite-element package ABAQUS and

representative creep laws, we generate predictions for Northern Europe, in which lateral het-

erogeneities are induced based on heatflow data. We show that perturbations, which are

differences with a background model, due to shallow low-viscosity structures are one to two

orders of magnitude larger than the predicted accuracy of GOCE. Moreover, some features in

geoid height perturbations are robust to changes in composition and creep regime, and have

therefore a spatial signature that is representative for low-viscosity structures. We argue that

these signatures are more likely to be detectable by GOCE. Finally, we show that GOCE is

predicted to be sensitive to the creep regime in the lower crust, but not to the composition, at

least not for the plagioclase feldspars used here.

This chapter is submitted to Geophysical Journal International (Schotman et al., 2008b).

104 Chapter 7. Thermomechanical Models

7.1 Introduction

During the last glacial maximum (LGM, ∼ 21 kyrs BP) large parts of Northern

Europe were glaciated. From field evidence and relative sea-level (RSL) curves,

ice-load histories have been constructed (e.g. Lambeck et al., 1998; Peltier, 2004)

that cover the British Isles, Scandinavia and the Barents Sea, and parts of con-

tinental Europe (e.g. Denmark, Poland) and the Kara Sea. This ice-load caused

a sea-level drop of about 15 m, out of a total of about 130 m, and a maximum

depression of the Earth’s surface of about 600 m in Northern Europe. Nowadays

almost the complete area is ice-free, but the land is still rebounding due to mantle

flow towards the formerly glaciated areas. Using models of this process of glacial

isostatic adjustment (GIA) and using RSL curves and geodetic data (especially

GPS), inferences about the thickness of the lithosphere and the viscosity of the

mantle have been derived for this area, either estimating the ice-load history in

the same inversion (e.g. Lambeck et al., 1998) or using an existing ice-load history

(e.g. Milne et al., 2004). For the latter it should be kept in mind that the ice-load

history is contaminated by information on the viscosity structure of the mantle,

and that, in principle, the recovered viscosity structure is contaminated by the

assumed ice-load history. Differences either reflect different a priori assumptions

(e.g. thickness of the lithosphere, number of mantle layers) or data that samples a

different part in space or time. The difference in upper mantle viscosity estimated

by Lambeck et al. (1998) (3−4 ·1020 Pas) and Milne et al. (2004) (5 ·1020−1 ·1021

Pas), using the same RSES ice-load history of Lambeck et al. (1998), might for

example be caused by non-linearity of the rheology, leading to a higher estimate of

viscosity for inversions from present-day GPS-data (Milne et al., 2004) than from

Holocene RSL-curves (Lambeck et al., 1998, Kurt Lambeck, personal communica-

tion, 2006). Results from high-pressure and -temperature laboratory experiments

show that for high stress levels (> 1−10 MPa) and/or large grainsizes (> 100−1000

µm) most earth materials show non-linear creep (e.g. Ranalli & Murphy, 1987;

Karato & Wu, 1993), in which strain rate and stress are related by a power-law

and the viscosity is stress-dependent. The reason most GIA studies still employ

a linear rheology is that it greatly simplifies the mathematical formulation of the

problem and that it is successful in explaining different observations simultane-

ously (Wu, 1992a).

Another reason for the discrepancies between the result of Lambeck et al. (1998)

and Milne et al. (2004) might be the different values they find for the lithopheric

thickness (65− 85 km, Lambeck et al. (1998) versus larger than 90 km, Milne

et al. (2004)), because a combined thicker lithosphere and larger upper mantle

viscosity give a comparable quality of fit as a thinner lithosphere and smaller vis-

cosity (Lambeck et al., 1998; Milne et al., 2004). This trade-off is partly due to the

dampening effect of both a thicker lithosphere and a weaker shallow upper mantle


and partly due to the relatively low spatial resolution of the RSL- and GPS-data,

which thus only probe the medium- to long-wavelength signature of the induced

field. This is confirmed by the low radial resolving power (∼ 200 km) in the shallow

upper mantle (Milne et al., 2004) and by a reply of Lambeck & Johnston (2000) to

a comment by Fjeldskaar (2000) on the lack of a low-viscosity asthenosphere in

the stratification of Lambeck et al. (1998). In this reply, Lambeck & Johnston

(2000) state that such a low-viscosity region did not significantly improve the fit

to the data in an earlier study for the British Isles, and that this, together with

the relatively large uncertainties in the ice-load history, does not warrant a finer

stratified earth model than the used three-layer model. The difficulty of constrain-

ing the viscosity of asthenosphere with RSL data in the presence of uncertainty

in the thickness of the lithosphere is also found in laterally heterogeneous studies

for Northern Europe (Kaufmann & Wu, 2002; Martinec & Wolf, 2005). The exis-

tence of asthenospheric low-viscosity zones (ALVZs) is predicted from seismology,

which show a low-velocity zone (see e.g. Stein & Wysession, 2003, p. 170) below the

lithosphere. This zone is associated with ductile flow, because deformation laws,

the convergence of the geotherm and the solidus, and the strong vertical advection

of deep heat from mantle convection predict low-viscosity material in this area

(e.g. Stein & Wysession, 2003, p. 204). From postseismic deformation studies in

the western US shallow upper mantle viscosities in the range of 1017 −1019 Pas

are found (Pollitz, 2003; Dixon et al., 2004). Steffen & Kaufmann (2005), using

a comparable loading history as Lambeck et al. (1998), find some evidence for a

low-viscosity region (1019 −1020 Pas) in the Barents Sea area in a depth range of

160−200 km. However, no such region is found in Scandinavia, and no clear indi-

cation for such a region is found under the British Isles. Values for the lithospheric

thickness range from 60−70 km beneath the Barents Sea and the British Isles to

larger than 120 km beneath Scandinavia.

The thickness of the lithosphere and the viscosity of the asthenosphere are closely

related to the thermal regime, which can for the lithospheric thickness be shown

by comparing estimates of the thermal thickness (Figure 7.1a, Artemieva, 2006)

and effective elastic thickness (EET, Figure 7.1b, Pérez-Gussinyé & Watts, 2005).

The former is determined from heatflow data, and defined as the depth to 1300C,

which is equal to the crossing of the conductive profile and the mantle adiabat

(Artemieva & Mooney, 2001, see also Section 7.2.1). The latter is determined from

Bouguer gravity anomalies and flexure models of the lithosphere (Pérez-Gussinyé

et al., 2004). From Figure 7.1 it is clear that both definitions show a large corre-

lation, especially in Baltica, though in some areas there are large discrepancies

(e.g. Avalonia, see also Pérez-Gussinyé & Watts, 2005). In areas of relative high

heatflow (or small thermal thicknesses) not only the asthenosphere can show low-

viscosity, also the crust. Indications for such crustal low-viscosity zones (CLVZs)

come from seismology, which shows lower crustal reflectivity in continental re-


(a) thermal thickness

Avalonia

Baltica

120 160 200 240thermal thickness [km]

(b) effective elastic thickness

Avalonia

Baltica

20 40 60elastic thickness [km]

Figure 7.1: Thermal thickness of the lithosphere (from Artemieva (2006), a) and effectiveelastic thickness (from Pérez-Gussinyé & Watts (2005), b).

gions with relatively high (> 70 mW/m2) heatflow (Meissner & Kusznir, 1987),

which indicates lamination supported by ductile flow. The viscosity depends on

composition, fluid content and geotherm (Watts & Burov, 2003; Ranalli & Mur-

phy, 1987) and can be as low as 1017 Pas, as derived from postseismic (e.g. Hearn,

2003) and mining-induced (e.g. Klein et al., 1997) deformation studies. Another

indication is the lack of seismicity in the lower crust, which means that intraplate

earthquakes are mainly confined to the upper crust (0−20 km, Watts & Burov,

2003; Ranalli & Murphy, 1987) and the subcrustal mantle part of the lithosphere

(Ranalli & Murphy, 1987), which are thought to be brittle. Recently, this ’jelly-

sandwich’ model has been questioned in favor of a model in which all strength

resides in the crust (see the discussion in De Meer et al., 2002; Jackson, 2002),

though this model seems only be possible for a strong (dry mafic) lower crust and

high surface heatflux (Afonso & Ranalli, 2004). Some GIA studies, focussing on

the vertical displacement (Klemann & Wolf, 1999), present-day (Di Donato et al.,

2000a) and late-Holocene (Kendall et al., 2003) sea-level change, and the global

gravity field as expected from GOCE (Vermeersen, 2003; van der Wal et al., 2004),

have included a CLVZ at a depth of 20−35 km, with a thickness of 10−15 km and

a viscosity of 1017 to 1019 Pas.

In recent GIA studies that include a laterally heterogeneous lithospheric thick-

ness, the thickness variations are either deduced from seismic tomography models


(e.g. Wu et al., 2005; Wang & Wu, 2006) or derived from EET-estimates by mul-

tiplying with a constant factor (Zhong et al., 2003) or adding a constant factor

(Whitehouse et al., 2006, Pippa Whitehouse 2007, personal communication). Lat-

eral variations in mantle viscosity are usually derived from seismic tomography

models and a certain scaling law (Latychev et al., 2005; Wu et al., 2005; Wang &

Wu, 2006; Whitehouse et al., 2006; Steffen et al., 2006). Here we derive the rheol-

ogy of the crust and shallow upper mantle, to a depth of 220 km, from laboratory-

derived creep laws for plagioclase feldspars (Rybacki & Dresen, 2004) and olivine

(Hirth & Kohlstedt, 2003), with either a linear or non-linear rheology. Using creep

laws the distinction between ’lithospheric thickness’ and ’asthenospheric viscosity’

is rather arbitrary and we base the lithospheric thickness on a comparison of the

timescale of the process and the Maxwell time of the earth material, which is de-

fined as the ratio of the viscosity to the rigidity (e.g. Ranalli, 1995, p. 222). For

timescales smaller than the Maxwell time, deformation is dominated by the elas-

tic response, for timescales larger than the Maxwell time by the viscous response.

If we assume timescales for GIA of 10− 100 kyrs, we find that, using a rigidity

of 68 GPa (Table 7.4), the shallow upper mantle (to a depth of ∼ 200 km) will be

effectively elastic for a viscosity larger than about 1023 Pas. This viscosity-based

lithospheric thickness estimate is closely related to thickness of the rheological

lithosphere (e.g. Kukkonen & Peltonen, 1999; Pasquale et al., 2001; Ranalli, 1995,

p. 231), which is defined as the depth to a certain strength (e.g. 1 or 10 MPa)

of the shallow upper mantle. Both definitions are not sensitive to the inclusion

of shallow LVZs, in contrast to the EET, which is an integrated measure of the

strength of the lithosphere. The EET can be estimated from GIA inversions (e.g.

Lambeck et al., 1998; Milne et al., 2004), in which case it defines the short-term

(10− 100 kyrs) elastic thickness (Martinec & Wolf, 2005), or from inversions of

gravity data (Bouguer coherence or free-air admittance studies, see Figures 7.1b

and Pérez-Gussinyé et al., 2004; Pérez-Gussinyé & Watts, 2005), in which case it

defines the long-term (> 1 Myrs) elastic thickness of the lithosphere. The short-

term elastic thickness is larger than the long-term elastic thickness, because on

the short timescales (10−100 kyrs) involved in GIA, a high-viscosity (e.g. > 1023

Pas) shallow mantle will not relax and is effectively elastic. In the absence of shal-

low LVZs the short-term elastic thickness should be comparable to the viscosity-

based lithospheric thickness, if the assumption of the threshold viscosity of 1023

Pas is correct. Note that the viscosity-based lithospheric thickness is smaller than

the thermal thickness, because a viscosity smaller than 1023 Pas can be reached

in the conductive area.

The upcoming Gravity field and steady-state Ocean Circulation Explorer (GOCE)

satellite mission, planned for launch by ESA in the summer of 2008, is predicted

to measure the static gravity field with centimeter accuracy at a resolution of 100

km or less (Visser et al., 2002). In Chapter 4 we have shown that geoid heights


due to a CLVZ or ALVZ are above the performance of GOCE, and that GOCE is es-

pecially sensitive to the properties of a CLVZ, which shows maximum amplitudes

from spherical harmonic degree 40 to 120 (∼ 500−150 km). The models we used

were for radially stratified (i.e. laterally homogeneous) earth models and linear

rheology, which could be solved using a semi-analytical normal-mode technique

(Section 2.2.3). Upon the introduction of lateral heterogeneities and non-linearity,

the governing equations can no longer be solved using normal-mode techniques

and therefore we employ a FE model, based on the commercial package ABAQUS

(e.g. Wu, 2004; Steffen et al., 2006, and Chapter 6). As we are interested in high-

resolution signals, the use of a global 3D FE model (e.g. Wu et al., 2005; Wang &

Wu, 2006) is considered to be not feasible yet. In Chapter 6 we have shown that

geoid height perturbations computed from a flat 3D FE model are very accurate.

Perturbations are differences between predictions from a perturbed background

model and predictions from the background model itself. The background model

consists of a certain earth stratification and loading history. Here we assume that

we know the laterally homogeneous background earth stratification from 220 km

downwards from other GIA studies (e.g. Lambeck et al., 1998; Milne et al., 2004)

and we test the sensitivity to the loading history by using two ice-load histories:

RSES of Lambeck et al. (1998) and ICE-5G of Peltier (2004). We then consider

perturbations due to creep laws for olivine in the shallow upper mantle and espe-

cially due to creep laws for plagioclase feldspars in the crust. For computation of

the latter, we will use a suitable creep law for olivine in the shallow upper mantle

of the background model.

To extract information on CLVZs from high-resolution static gravity information

as measured by GOCE, all other structures and processes that give rise to mass

anomalies have to be modelled and removed. This is however not feasible at the

moment, as most structures and processes are not sufficiently well known (see e.g.

Velicogna & Wahr, 2002). Even in the presence of unmodelled errors, it is possible

to extract information from the static field by some sort of spatio-spectral filtering

(e.g. Simons & Hager, 1997). Here we show an approach to detect CLVZs by con-

sidering features that are robust to variations in composition or flow regime of the

lower crust (spatial signatures). To estimate if GOCE can also constrain properties

(composition, creep regime, grainsize) of CLVZs, we compute prediction errors for

different test predictions and synthetic data. The synthetic data is generated by

considering recovery errors from GOCE only, due to measurement noise of the gra-

diometer onboard the satellite, and downward continuation. We do, for example,

not consider gravity field signals due to chemically- or thermally-induced crustal

mass inhomogeneities, which we expect to be effective in masking gravity field

variations due to CLVZs. We also do not consider uncertainties in the deep (> 200

km) background model of the earth, which in general, however, can be largely re-

moved by estimating the best GIA background model from RSL-curves and GPS

7.2. Thermomechanical Model 109

data, and filtering the residual low-degree harmonic signal (Chapter 4). Filter-

ing of the low-degree harmonics is also needed to remove, for example, the strong

gravity signal in Northern Europe due to the Iceland Plume and the North At-

lantic Ridge, as shown in Chapter 5 for data from GRACE satellite gravity mission.

Results from this mission seem promising for the long-wavelength signal of GIA

(Tamisiea et al., 2007), however, because we are interested in short-wavelength

phenomena, we will not consider GRACE in this study. Tests have shown that

errors in correcting for the amount of isostatic adjustment of topography, using

the Airy-Heiskanen and Vening-Meinesz models (Watts, 2001, p. 20, 65), only af-

fect the high spherical harmonic degrees (> 100 or spatial scales < 200 km, see

Appendix E). These high harmonics have to be removed anyway because of the

increase in recovery error with harmonic degree (see Section 7.5). Errors in mod-

elling time-variable processes (ocean tides, atmospheric and hydrological mass

variations) are in general smaller than the measurement noise (Han et al., 2006)

and can therefore be neglected.

In Section 7.2, we start with describing the thermal and mechanical model. In

Section 7.3 we discuss the laboratory-derived creep laws we use for the crust and

mantle and estimate effective viscosities and lithospheric thicknesses. In Sec-

tion 7.4 we apply the creep laws and predict geoid height perturbations for North-

ern Europe using the RSES ice-load history (Lambeck et al., 1998) and investigate

the sensitivity to the ice-load history by using the ICE-5G load history of Peltier

(2004). In Section 7.5 we then investigate to what extent GOCE is sensitive to

composition and creep regime of the lower crust in Northern Europe, using esti-

mated recovery errors for GOCE.

7.2 Thermomechanical Model

7.2.1 Thermal Model

We estimate the temperature T in the crust and lithosphere by solving the 1D

steady-state heat conduction equation (Eq. (B.1), Appendix B), for which the solu-

tion for a multilayer model is (Liu & Zoback, 1997):

Ti+1(z)= Ti(zi)+qi

ki+1

(z− zi)−Ai+1

2ki+1

(z− zi)2 (7.1)

with Ti, qi , zi the temperature in [K], heatflow in [W/m2] and depth (i.e. positive

in the downward direction) in [m] at the top of layer i+1, and ki+1, Ai+1 the ther-

mal conductivity in [Wm−1K−1] and heat production in [W/m3] in layer i+1. The

heatflow qi can be computed from from Fourier’s law (Turcotte & Schubert, 2002,


p. 132) and Eq. (7.1) as:

qi = ki

(

dTi

dz

)

z=zi

= qi−1 − Ai(zi − zi−1) (7.2)

We compute the temperature downwards from the surface heatflow q0 and surface

temperature T0 using Eqs (7.1) and (7.2). When the temperature in the mantle be-

comes close to the mantle solidus TM , heat transport is not controlled by conduc-

tion but by convection. For a reversible, adiabatic process, the entropy change is

zero and the relation between temperature T and pressure P is given by (Turcotte

& Schubert, 2002, p. 187):

dT

dp=

αvT

ρcP

(7.3)

where αv is the volumetric coefficient of thermal expansion (3·10−5 K−1 for olivine,

Inoue et al., 2004), ρ is the density and cP is the specific heat at constant pressure

(8 ·102 J kg−1K−1 for olivine, Osako et al., 2004). Using the lithostatic pressure

gradient dP/dy = ρg we find the following adiabatic temperature gradient:

dT

dy=

αv gT

cP

(7.4)

If we solve this equation for T = T1 at z = z1 we get:

T(z)= T1eαv g(z−z1)/cP (7.5)

We assume that heat transport is by conduction until the temperature is 0.85TM

(Pollack & Chapman, 1977) and that heat transport is by convection from T1 =1300C. In between, we assume that the temperature follows the 0.85TM contour,

with the solidus TM in [C] given by (Hirschmann, 2000, Table 2, Recommended

Fit):

TM(P)=−5.14P2 +132.9P +1121 (7.6)

with P the lithostatic pressure in [GPa].

This might seem a rather crude assumption, but it compares well with a bound-

ary layer between the purely conductive lithosphere and purely convecting as-

thenosphere as proposed by Jaupart & Mareschal (1999). Following Jaupart &

Mareschal (1999) and Artemieva & Mooney (2001) we can now define three litho-

spheric thicknesses (Figure 7.2a): The boundary of the conductive layer (z1, where

the conductive geotherm crosses 0.85 times the solidus), the depth where the

conductive profile crosses the adiabat at 1300C (z2, thermal thickness, see Fig-

ure 7.1a in Section 7.1), and the depth where the geotherm crosses the adiabat

(z3, seismic thickness), below which is a fully convective mantle.


(a) thermal definitions (b) geotherms

0 200 400 600 800 1000 1200 1400−200

−180

−160

−140

−120

−100

−80

−60

−40

−20

0

temperature [°C]

dept

h [k

m]

40 mW/m2

60 mW/m2

80 mW/m2

Figure 7.2: Thermal definitions of lithospheric thickness (a) and computed geotherms usedin this study (b). In (a), ’T_M’ is the solidus TM as given by Eq. (7.6).

Table 7.1: Parameters for the thermal model

layer depth k A

[km] [Wm−1K−1] [µW/m3]

upper crust 10−15 2.5 1

middle crust 20−30 2.2 0.3-0.6

lower crust 30−50 2.0 0.07-0.2

lithosphere 3.5 0.01

Due to the depth-dependence of heat generation and thermal conductivity, the

results are sensitive to the assumed thickness of the crustal layers. By comparing

heatflow measurements (Figure 1.2a in Chapter 1, based on the dataset of Pollack

et al. (1993) and for continental Europe on the extended set of Artemieva (2006))

and estimates of crustal thickness (Figure 1.2b, from CRUST2.0, Bassin et al.,

2000) we estimate a representative crustal thickness of 40−50 km for low heatflow

(40 mW/m2), 30−40 km for average heatflow (60 mW/m2) and 20−30 km for high

heatflow (80 mW/m2). In the computations of the geotherms we take the upper

values for each heatflow value, because the effect of the thickness within the limits

is small. The values we use for heat production A and thermal conductivity k

are given in Table 7.1. The values are based on a number of studies (Clauser &

Huenges, 1995; Jaupart & Mareschal, 1999; Kaikkonen et al., 2000; Pasquale et

al., 2001; Artemieva & Mooney, 2001) and for heat production A on an exponential

decay with depth in the crust (Turcotte & Schubert, 2002, p. 141), see for a further

discussion Appendix B.

The computed geotherms (Figure 7.2b) for low, average and high heatflow compare

well with Artemieva (2006). For example, our results show a thermal thickness

(i.e. depth of the conductive profile to 1300C) for low heatflow of about 200−220

km, which also nicely fits the xenolith data of Kukkonen & Peltonen (1999). We


expect the chosen heatflow values to be representative for our purposes, because

values lower than 40 mW/m2 are restricted to a small area in Finland (see Fig-

ure 1.2a), and values larger than 80 mW/m2 are restricted to oceanic areas, where

the effect of changes in rheology are much smaller than in and just outside the ice-

load areas. A notable exception is the Barents Sea, which is an area that has been

glaciated in the past, though with large differences in the amplitude of the load

between the RSES (Lambeck et al., 1998) and ICE-5G (Peltier, 2004) ice-load his-

tories (> 1000 m, see Figure 7.7b in Section 7.4), and where the value of 80 mW/m2

is on basis of the available heatflow data too low (Figure 1.2b). In oceanic areas,

which have a thin and mafic crust, we do not expect a ductile lower crust and as

the effect of higher heatflows is small on mantle temperatures (see Figure 7.2b),

we can also use there a heatflow of 80 mW/m2.

7.2.2 Mechanical Model

Experimentally-derived creep laws for deformation in uniaxial stress have the

form (Ranalli, 1995, p. 77):

ǫC1 = Aσn

1 (7.7)

where σ1 is the uniaxial stress and ǫC1

the corresponding creep strain rate. Note

that A is in general a function of pressure, temperature and material properties,

see Section 7.3. Using the definition of the effective deviatoric stress σ′E

and strain

rate ǫ′E

and their relation to the uniaxial stress σ1 and strain rate ǫ1 (Ranalli,

1995, p. 76):

σ′E =

√

1

2σ′

i jσ′

i j=

1p

3σ1, ǫ′E = ǫE =

√

1

2ǫi j ǫi j =

p3

2ǫ1 (7.8)

we can write Eq. (7.7) as (Ranalli, 1995, p. 76):

ǫCE =

3(n+1)/2

2Aσ′

En (7.9)

If we now assume a linear dependence of the components of strain rate and stress

deviator then we can rewrite this equation to the following tensor form (Ranalli,

1995, p. 77):

ǫCi j =

3(n+1)/2

2Aσ′

E(n−1)σ′

i j (7.10)

Using this relation in Eq. (2.6) the constitutive equation for a material that is only

linear in the elastic limit can now be written as:

ǫi j =σ′

i j

2µ+

σ′i j

2η∗(7.11)


with η∗ the effective viscosity defined as:

η∗ =1

3(n+1)/2Aσ′E

(n−1)=

1

3Aσ(n−1)1

(7.12)

Note that this combination of elastic and viscous behavior can in principle only de-

scribe a steady-state process (Section 2.1.2), but that in GIA also transient effects

might play a role (Ivins & Sammis, 1996). Note moreover that we only consider the

GIA-induced stress and not the ambient stress, which might for certain tectonic

stress levels have a significant influence on the viscosity (e.g. Wu, 1995, 2001).

For a triaxial deformation experiment we can use the same definition for the effec-

tive viscosity with the uniaxial stress σ1 replaced by the stress difference σ1 −σ3.

For a deformation experiment in simple shear with shear stress σs we get the

following relation for the effective viscosity:

η∗ =1

2Asσ′E

(n−1)=

1

2Asσ(n−1)s

(7.13)

so that from comparing Eq. (7.13) with Eq. (7.12) it follows that As = 3(n+1)/2

2A.

In this study we use a flat, 3D-stratified deformation model based on the com-

mercially available finite-element code ABAQUS. For implementation of creep law

parameters, ABAQUS uses Eq. (7.7) and the fact that the uniaxial stress is equal

to the von Mises stress σ =√

32σ′

i jσ′

i j. With σ1 replaced by σ, we can estimate

effective viscosities from Eq. (7.12) for applications that are not purely uniaxial

as GIA. Note that Wu (1992a) uses σ′E

instead of σ1 (which differ by a factorp

3,

see Eq. 7.8) in Eq. (7.12), but this equation is not used in any computations with

ABAQUS.

The model has an 80 km horizontal resolution in the central area of 2,920×2,920

km2, with a decreasing resolution outward to the edges at 10,000 km. In the ver-

tical, the first 220 km is finely stratified (14 element layers), below which are five

layers to model the deeper upper mantle and the lower mantle to a total depth

of 10,000 km. The bottom of the model is fixed in all directions and the sides of

the model are fixed in the horizontal direction. The advection of pre-stress, which

describes the restoring force of isostasy, is simulated using Winkler foundations

as in Wu (2004). We neglect the effect of self-graviatation, which is partly com-

pensated by the lack of sphericity (Amelung & Wolf, 1994) and which is shown to

have a negligible effect on the accuracy of predictions for Northern Europe (see

Chapter 6). We compute geoid heights by solving Laplace’s equation in the 2D-

Fourier transformed domain, which gives accurate results compared with analyti-

cal, normal-mode methods (Chapter 6). We refer to Chapter 6 for a validation and

more extensive description of the model.


7.3 Composition and Creep Parameters

From high-pressure and -temperature deformation experiments, rock materials

show mainly two regimes of creep: grainsize sensitive diffusion creep for low stress

levels (< 1−10 MPa) and/or small grainsizes (< 100−1000 µm), and grains size in-

sensitive dislocation creep for high stress levels and/or large grainsize (Ranalli &

Murphy, 1987; Karato & Wu, 1993). Diffusion (Nabarro-Herring or Coble) creep

shows a linear relation between stress and strain rate and a strong dependence on

grainsize. In practice, diffusion creep is accommodated by grain boundary sliding

which leads to a slight dependence on stress (with a stress exponent n= 1−2). Dis-

location creep shows a power-law relation (with the power n generally larger than

3, e.g. Evans & Kohlstedt, 1995; Kohlstedt et al., 1995) between stress and strain

rate, but is not dependent on grainsize. Grain boundary sliding accommodated by

dislocation creep can result in a power law relation with n= 1.5−2 (McDonnell et

al., 1999). Both diffusion creep and dislocation creep can operate at the same time,

in which the mechanism with the higher creep rate will be dominant (e.g. Karato

& Wu, 1993). Combined activity of both mechanisms may result in apparent n-

values measured in laboratory experiments in between 1 and 3. Both diffusion

and dislocation creep are very sensitive to water content, a wet material being

weaker than a dry material. Note that the stress levels induced by the Fennoscan-

dian ice sheet (a few MPa, with a maximum larger than 10 MPa, see Wu, 1992a,

1995) are in the regime where the transition from diffusion to dislocation creep

occurs. Here we either either use a linear (n = 1) or non-linear (n > 1) rheology

and not a composite rheology (e.g. Giunchi & Spada, 2000; Dal Forno et al., 2005).

In Sections 7.3.2 and 7.3.3 we show effective viscosities and yield strengths, where

the latter is in the brittle regime equal to (Sibson, 1974; Ranalli, 1995, p. 248):

σB1 =αP(1−λ) (7.14)

with P the lithostatic pressure at depth and λ the ratio of pore-fluid pressure

to lithostatic pressure, taken to be 0.35 (Kaikkonen et al., 2000; Pasquale et al.,

2001). This is based on the assumption that the pore-fluid pressure is equal to

the hydrostatic pressure, which results in λ = ρW /ρE (Ranalli, 1995, p. 248), with

ρW the density of water and ρE the density of overlying material, assumed to

be ρE = 2850 kg/m3. The factor α depends on the assumed kind of faulting and

following Pasquale et al. (2001), we assume that in the Baltic shield strike-slip

faults are the most dominant, in which case α= 1.2 (Ranalli, 1995, p. 248). This is

intermediate between normal (minimum strength, α= 0.75) and thrust (maximum

strength, α = 3) faulting (Ranalli, 1995, p. 248). In the ductile regime, the yield

strength σD1

is related to the (effective) viscosity η∗ by:

σD1 = 3ǫη∗ (7.15)

7.3. Composition and Creep Parameters 115

which follows from Eq. (7.7) and Eq. (7.12). In intraplate regions strain rates are

typically larger than 10−18 s−1 (∼ 10−10 yr−1, Wu & Mazotti, 2007; Pérez-Gussinyé

et al., 2004). Typical present-day rebound strain rates can be computed from mea-

sured horizontal and vertical velocities in Fennoscandia, which are in the order of

1 mm/yr and 1 cm/yr, respectively (Milne et al., 2001). Using typical horizontal

scales of 100 km and vertical scales of 1000 km, we arrive at maximum present-

day strain rates of 1 [mm/yr]/100 [km] = 1 [cm/yr]/1000 [km] = 10−17 s−1 (∼ 10−9

yr−1), which is in agreement with findings of Wu & Mazotti (2007). As strain

rates during glaciation and deglaciation might be higher, the value of 10−15 s−1

estimated by Karato (1998) can be considered as an upper bound for GIA. We take

therefore values of 3·10−15−3·10−21 s−1, with 3·10−18 s−1 as a typical value, where

the factor 3 (actually 3.33) is included to simplify the translation from viscosity to

strength with Eq. (7.15) in the figures in Sections 7.3.2 and 7.3.3.

We first discuss in Section 7.3.1 the definition of the viscosity-based lithospheric

thickness we use in this study. We then (Section 7.3.2) consider olivine creep laws

for the shallow upper mantle and show effective viscosities and lithospheric thick-

nesses. In Section 7.3.3 we concentrate on plagioclase feldspars for the crust and

show effective viscosities for different creep parameters and heatflow values.

7.3.1 Lithospheric Thickness Definitions

The viscoelastic strength of the lithosphere depends, amongst others, on the time-

scale of the loading. The long-term strength is associated with the effective elas-

tic thickness as estimated from Bouguer coherence or free-air admittance studies

(e.g. Pérez-Gussinyé et al., 2004; Pérez-Gussinyé & Watts, 2005). It gives a thick-

ness varying between a few km to thicknesses larger than 60 km, which is the

upper bound with which the long-term elastic thickness can be derived with con-

fidence (Pérez-Gussinyé & Watts, 2005). In Section 7.1, Figure 7.1b we showed

results obtained by Pérez-Gussinyé & Watts (2005) for Northern Europe using

Bouguer gravity anomalies. As already mentioned in Section 7.1, the (short-term)

elastic thickness in GIA is larger than the long-term elastic thickness as we con-

sider shorter timescales. For the timescales we are considering we derived in Sec-

tion 7.1, based on the Maxwell time, a viscosity at which the crust and mantle

can be considered elastic of 1023 Pas. For geological timescales (> 1 Myrs), this

viscosity is larger than 1025 Pas. The viscosity-based lithospheric thickness thus

defined is closely related to the rheological thickness, which is defined based on

certain yield stress (e.g. 1 or 10 MPa, Ranalli, 1995, p. 231) and a certain strain

rate. From Eq. (7.15) we see that for a yield strength of 1 MPa the same viscos-

ity (1023 Pas) is obtained using a strain rate of 3 ·10−18 s−1, which nicely fits the

assumption of this strain rate to be representative for GIA. Pasquale et al. (2001)

estimate a rheological thickness of 110−140 km for the Baltic Shield based on a


yield strength of 1 MPa and a strain rate of 10−15 s−1, which corresponds, using

Eq. (7.15), to a viscosity of about 5 ·1020 Pas. This seems to be a very low vis-

cosity for the lithosphere, except for processes with timescales smaller than a few

hundred years as e.g. (post)seismic deformation. This is confirmed by the much

smaller strain rates (< 10−17 s−1) assumed in shield areas by some other authors

(e.g. Pérez-Gussinyé et al., 2004; Wu & Mazotti, 2007), which corresponds to vis-

cosities larger than 5 ·1022 Pas and a difference in rheological thickness of 20−40

km, as estimated from the variation of strength with depth as in Section 7.3.2.

Kukkonen & Peltonen (1999) use the same strain rate as Pasquale et al. (2001),

and a yield strength of 1− 10 MPa to arrive at lithospheric thickness estimates

of 130− 185 km. They estimate a thickness variation of 15 km for one order of

magnitude change in strain rate. The uncertainty in the rheological thickness of

the lithosphere is thus dependent on the uncertainty in yield strength and strain

rate, where for estimates based on viscosity it is dependent on uncertainty in the

timescale of the process. Note that the Maxwell time is a characteristic timescale

of stress relaxation, that only provides a measure for which deformation mech-

anism (elastic, viscous) dominates for the assumption of a Maxwell viscoelastic

earth. This means that there could be, for example, also viscous deformation on

timescales shorter than the Maxwell time. Moreover, deviations from a Maxwell

viscoelastic earth will lead to deviations in the characteristic timescale of stress

relaxation.

7.3.2 Shallow Upper Mantle

For the upper mantle, the pre-exponential factor A in Eq. (7.7) generally depends

on temperature T and pressure P, assuming negligible melt fractions, as (Hirth &

Kohlstedt, 2003):

A = A∗d−mCrOH e(E∗+PV∗)/RT (7.16)

with E∗ and V∗ the activation energy in [kJ/mol] and volume in [10−6m3], re-

spectively, A∗ a pre-exponential factor, d the grainsize in [µm], COH the water

concentration as H/(106Si), and m and r sensitivity factors.

Material parameters can be found in literature (e.g Karato & Wu, 1993; Hirth &

Kohlstedt, 2003). We will use the recent compilation of Hirth & Kohlstedt (2003)

for dry (COH = 0) and wet (COH = 1000) olivine in both the diffusion and dislocation

creep regime (Table 7.2). The water content of COH = 1000 is taken from the

estimate of Hirth & Kohlstedt (1996) for the sub-lithospheric mantle. For diffusion

creep we take 8 mm as a typical grainsize in the upper mantle, and 1 mm as a

lower and 15 mm as an upper boundary (Ave Lallemant et al., 1980; Dijkstra et

al., 2002).


Table 7.2: Creep parameters for olivine (from Hirth & Kohlstedt, 2003)

mechanism n E∗ V∗ A∗ m COH r

[kJ/mol] [10−6m3/mol] [MPa−(n+r)µmm/s]

dry diffusion 1.0 375 6 1.5 ·109 3 0 -

wet diffusion 1.0 335 4 1.0 ·106 3 1000 1.0

dry dislocation 3.5 530 16 1.1 ·105 0 0 -

wet dislocation 3.5 480 11 9.0 ·101 0 1000 1.2

(a) diffusion

18 19 20 21 22 23 24 25 26 2717−200

−175

−150

−125

−100

−75

−50

viscosity [log(Pas)]

dept

h [k

m]

strength [log(MPa)] −3 −2 −1 0 1 2 3

brittledrywet

158

1581

1

(b) dislocation

18 19 20 21 22 23 24 25 26 2717−200

−175

−150

−125

−100

−75

−50


dept

h [k

m]

brittledrywet

2115 18

15 18 21

1815

Figure 7.3: Viscosity estimates for low heatflow (40 mW/m2) for olivine in the diffusion (a)and dislocation (b) creep regime. In (a), values in the figure are grainsize in [mm], strengthestimates are for ǫ= 3 ·10−18 s−1. In (b), values a in the figure are strain rates as 3 ·10−a

s−1. The brittle curves are computed from Eq. (7.14) and the temperature profile used isfrom Figure 7.2b (’40 mW/m2’).


For low heatflow, in which the geotherm reaches the adiabat just below a depth of

200 km (compare with Figure 7.2b), viscosities in the diffusion creep regime are

given in Figure 7.3a. These curves can be computed using Eq. (7.16) in Eq. (7.7)

and solving for the stress, and computing the viscosity from Eq. (7.15). If we com-

pare the viscosity curves of Figure 7.3a with results of GIA studies for Fennoscan-

dia, we see that a grainsize of 1 mm is not very likely in this area, as predicted

viscosities are one to more than two orders of magnitudes smaller than the gen-

erally expected value of about 5 ·1020 Pas for the upper mantle (Lambeck et al.,

1998; Milne et al., 2001) and a low-viscosity asthenosphere is not likely for cold

areas as e.g. the Baltic Shield (Steffen & Kaufmann, 2005). For larger grainsize,

a dry rheology seems to give too large viscosity values. However, if we define the

lithospheric thickness as the depth where the viscosity is 1023 Pas (see previous

section) we find a thickness for a dry rheology of 175−200 km, which is used for the

Baltic Shield in some GIA studies (e.g. Wang & Wu, 2006; Martinec & Wolf, 2005)

and which is close to the upper limit of 170 km found by Milne et al. (2001). For

wet rheology and a grainsize of 8 mm we seem to get both a reasonable lithospheric

thickness (140 km) and upper mantle viscositiy (3 ·1020 Pas).

For geological timescales (> 1 Myrs) the viscosity for which the mantle is effec-

tively elastic was estimated in Section 7.3.1 to be larger than 1025 Pas, which

gives Figure 7.3a thicknesses smaller than 110 km, which is not in contradiction

with the values found by Pérez-Gussinyé & Watts (2005) for the long-term elastic

thickness (Figure 7.1b). If we base the lithosperic thickness on the strength rather

than the viscosity, the thickness of the lithosphere becomes strain-rate dependent.

We then find from Figure 7.3a that for a yield strength of 1 MPa (e.g. Pasquale et

al., 2001) or 10 MPa (e.g. Kukkonen & Peltonen, 1999) the rheological lithosphere

has a thickness of 120−135 km. For a strain rate of 3 ·10−15 s−1 (not shown) this

increases to 185− 210 km and for a strain rate of 3 ·10−21 s−1 this decreases to

80− 90 km. We see from Figure 7.3a that the rheological lithosphere compares

well with the viscosity-based thickness, i.e. the depth at which the viscosity is

1023 Pas, for a strain rate of 3 ·10−18 s−1 and with the long-term elastic thickness

(> 1025 Pas) for 3 ·10−21 s−1.

For dislocation creep (Figure 7.3b) it is difficult to combine a reasonable upper

mantle viscosity (1020 −1021 Pas) with a reasonable lithospheric thickness (> 90

km). Reasonable upper mantle viscosities are only achieved for high strain rate (3·10−15 s−1, ’15’ in Figure 7.3b), but then the viscosity-based lithospheric thickness

(depth to 1023 Pas) is only realistic for a dry material (90 km, found as a lower

boundary by Milne et al., 2001). However, for a strain rate of 3 ·10−18 s−1, which

is more common for present-day GIA, upper mantle viscosities are very high (∼6 ·1021 Pas for wet olivine), though then the viscosity-based lithospheric thickness

of 125 km seems to be acceptable. The long-term thickness (depth to > 1025 Pas)

is smaller than 65 km, which is too small if we compare estimates of long-term


(a) diffusion creep

17 18 19 20 21 22 23 24 25 26 27−200

−175

−150

−125

−100

−75

−50


dept

h [k

m]

strength [log(MPa)] −3 −2 −1 0 1 2 3

brittledrywet

15

8 15

81

1

(b) dislocation creep

1817 19 20 21 22 23 24 25 26 27−200

−175

−150

−125

−100

−75

−50


dept

h [k

m]

brittledrywet

2115 18

15 18 21

1815

Figure 7.4: Viscosity estimates for olivine and average heatflow (60 mW/m2) for diffusion(a) and dislocation (b) creep. In (a), values in the figure are grainsize in [mm], strengthestimates are for ǫ= 3 ·10−18 s−1. In (b), values a in the figure are strain rates as 3 ·10−a

s−1. The brittle curves are computed from Eq. (7.14) and the temperature profile used isfrom Figure 7.2b (’60 mW/m2’).

elastic thickness (Figure 7.1b) in areas of low heatflow (Figure 1.2a).

For average heatflow (60 mW/m2, Figure 7.4), mantle viscosity shows a minimum

where the temperature profile reaches the mantle adiabat at 120 km (compare

with Figure 7.2b). For diffusion creep (Figure 7.4a) and small grainsize (d = 1

mm), viscosities in the upper mantle seem again to be too low, especially for a wet

rheology. Even if a viscosity of 5 ·1018 Pas, as for dry composition, is reasonable,

then the very thin lithospheric thickness (as determined by the depth to a viscosity

of 1023 Pas) of 40 km seems unrealistic. For larger grainsizes (d = 8−15 mm), a

dry rheology seems to predict too high values of viscosity. For a wet rheology and

average to large grainsize, the viscosity-based lithospheric thickness is about 60

km, which is not unrealistic for this heatflow when compared with the results

for the British Isles and the Barents Sea of Lambeck et al. (1998) and Steffen &

Kaufmann (2005). The long-term thickness is then smaller than 50 km (end of the

scale) and probably resides in the crust only. This would give, depending on the

composition of the lower crust, long-term elastic thickness estimates of 30−40 km,

which seems quite realistic (compare Figures 7.1b and 1.2b). Note that in this case

we do not have a ’jelly-sandwich’ because all long-term strength would reside in

the crust, which is in line with predictions from Afonso & Ranalli (2004) for high

heatflow. In Figure 7.4b we show the viscosity for dislocation creep. The shallow

mantle has viscosities as low as 1019 Pas, but only for high strain rate (3·10−15 s−1),

whereas for lower strain rates the viscosities seem to become too high. For high

heatflow (80 mW/m2) the profiles are comparable to those for average heatflow,

though with a somewhat lower viscosity to 75 km depth.

We conclude that the creep law for a wet rheology in the diffusion creep regime


Table 7.3: Creep parameters for plagioclase feldspars (from Rybacki & Dresen, 2004)

Name n Q∗ A∗ m

[kJ/mol] [MPa−nµmm /s]

An100 1 170 5.0·101 3

An60 1 153 1.3·101 3

Ab100 1 193 7.9·103 3

An100 3 356 4.0·102 0

An60 3 235 3.2·10−2 0

Ab100 3 332 2.5·103 0

with an average grainsize is the most representative for Northern Europe. We

will therefore use this creep law (model DIF, see Table 7.4) as a reference for re-

alistic predictions (Section 7.4) and use a wet rheology in the dislocation creep

regime (model DIS, Table 7.4) to test the sensitivity to the background model of

perturbations due to a ductile crustal layer.

7.3.3 Crust

For the crust, the pre-exponential factor A as defined in Section 7.2.2, Eq. (7.7),

generally depends on temperature T and grainsize d (in [µm]), as:

A = A∗d−m e−Q∗/RT (7.17)

with Q∗ the activation energy, A∗ a pre-exponential factor and m a sensitivity

factor.

Values for Q∗, A∗ and the stress exponent n in the dislocation creep regime (m = 0)

for different natural rocks can be found in literature (e.g. Ranalli & Murphy, 1987;

Carter & Tsenn, 1987; Wilks & Carter, 1990). These values should however be

treated with care, as the deforming mechanism is often not ductile, but (semi-

)brittle (Carter & Tsenn, 1987; Wilks & Carter, 1990). In that case, the values

cannot be extrapolated to conditions in the lower crust, as studies of natural lower

crust materials show that the lower crust most likely deforms in the ductile regime

(Carter & Tsenn, 1987; Rutter & Brodie, 1992; Kohlstedt et al., 1995). As the duc-

tile strength of deformed natural lower crustal rocks seems to be taken up by pla-

gioclase feldspars (Wilks & Carter, 1990) and because plagioclase feldspars are the

most abundant mineral in lower crustal rocks, with values of 30−60 vol.% found in

high-temperature mylonites (Rybacki & Dresen, 2004), we will use experimental

values of Rybacki & Dresen (2004) on synthetic plagioclase feldspars (anorthite

(An100), labradorite (An60) and albite (Ab100), both in the diffusion (n = 1, m > 0)

and dislocation (m = 0, n > 1) creep regime, see Table 7.3). The materials shown

are all wet in that they contain trace amounts of water larger than 0.07 wt.%


(a) effect of composition

16 17 18 19 20 21 22 23 24 25 26−50

−40

−30

−20

−10

0


dept

h [k

m]

strength [log(MPa)] −3 −2 −1 0 1 2 3

brittleAn

100

An60

Ab100

40 mW/m2

60 mW/m2

80 mW/m2

(b) effect of grainsize

16 17 18 19 20 21 22 23 24 25 26−50

−40

−30

−20

−10

0


dept

h [k

m]

strength [log(MPa)] −3 −2 −1 0 1 2 3

brittle1 mm0.1 mm0.01 mm

40 mW/m2

60 mW/m2

80 mW/m2

Figure 7.5: Viscosity estimates for diffusion creep for different composition (d = 100 µm, a)and grainsize (An60, b). On the upper axis we have also shown the strength for a strainrate of 3 ·10−18 s−1. The brittle curves are computed from Eq. (7.14) and the temperatureprofiles used are from Figure 7.2b.

H2O, whereas natural feldspars contain trace amounts ranging from 0.02 to 0.5

wt.% H2O (Rybacki & Dresen, 2004). As diffusion creep is grainsize dependent,

we chose two end-members of possible grainsize: 10 and 1000 µm. The lower

value can be found in shear zones, but is probably localized and not representa-

tive for bulk behavior (Steffen et al., 2001; Kenkmann & Dresen, 2002; Rosenberg

& Stünitz, 2003; Waters-Tormey & Tikoff, 2007). From 100 to 1000 µm a transi-

tion to dislocation creep is possible, depending on temperature (Rybacki & Dresen,

2004). We take 100 µm as a representative value for diffusion creep.

In Figure 7.5a we show the effect of composition in the diffusion creep regime

(d = 100 µm) for low (40 mW/m2), average (60 mW/m2) and high (80 mW/m2) heat-

flow. As with the computation of the geotherms (Section 7.2.1), we use assumptions

on the thickness of the crust for different heatflow values, i.e. 50 km for low, 40

km for average and 30 km for high heatflow. The effect of composition is not very

large, although albite (Ab100) is somewhat weaker than labradorite (An60) and

anorthite (An100). For low heatflow, the viscosity of the lower crust is larger than

about 3 ·1020 Pas and for a channel of 10 km thickness the viscosity is about 1022

Pas. This indicates that deformation on glacial timescales is small, as a viscosity

of 1022 Pas corresponds for the crust (µ= 27 GPa, Table 7.4) to a Maxwell time of

10 kyrs. However, if the assumption of lower crust consisting largely of plagioclase

feldspars is true for cratonic areas, it shows that ductile deformation is also possi-

ble for cold areas as the Baltic Shield. For high heatflow, the viscosity of the lower

crust decreases to 1017 −1018 Pas, although a reasonable channel is only formed

for viscosities higher than about 1019 Pas. For small grainsize (d = 10 µm for the

crust) viscosities become smaller than 1017 Pas (see Figure 7.5b, for labradorite)

and are probably only realistic for localized shear zones. For large grainsize (1000


(a) effect of composition

16 17 18 19 20 21 22 23 24 25 26−50

−40

−30

−20

−10

0


dept

h [k

m]

strength [log(MPa)] −3 −2 −1 0 1 2 3

brittleAn

100

An60

Ab100

40 mW/m2

60 mW/m2

80 mW/m2

(b) effect of strain rate

16 17 18 19 20 21 22 23 24 25 26−50

−40

−30

−20

−10

0


dept

h [k

m]

brittle3⋅10−21 s−1

3⋅10−18 s−1

3⋅10−15 s−1

40 mW/m2

60 mW/m2

80 mW/m2

3⋅10−18 s−13⋅10−15 s−1

Figure 7.6: Viscosity estimates for dislocation creep for different composition (ǫ= 3 ·10−18

s−1, a) and strain rates (An60, b). The brittle curves are computed from Eq. (7.14) andthe temperature profiles used are from Figure 7.2b.

µm) the lower crust becomes effectively elastic, but for this grainsize, dislocation

creep might be the dominant mechanism (e.g. Rybacki & Dresen, 2004).

In Figure 7.6a we show the effect of composition on the viscosity and strength in

the dislocation creep regime for a strain rate of 3 ·10−18 s−1. The effect of com-

position is somewhat larger than for diffusion creep, especially in the thickness

of the channels, though for this strain rate there is low viscosity. For high strain

rates (3 ·10−15 s−1), viscosities decrease drastically (Figure 7.6b, for labradorite),

although viscosities are still larger than for diffusion creep with average grainsize.

We conclude that albite is generally the weakest composition and anorthite the

strongest. Low-viscosity (∼ 1019 Pas) is only reached in the diffusion creep regime

for average grainsize. For realistic perturbations (Section 7.4) we use albite in the

diffusion creep regime with average grainsize as a reference and use anorthite to

test the sensitivity to composition. To test the sensitivity to grainsize, we use albite

with large grainsize, as viscosities for small grainsize seem to be not realistic. We

use albite in the dislocation creep regime to test the sensitivity to creep regime.

7.4 Predictions for Northern Europe

In this section we show the effect of the creep regime in the shallow upper mantle,

and composition and creep regime in the crust on predictions for Northern Europe.

As a loading history we use the RSES ice-load history (Lambeck et al., 1998) for

the British Isles, Scandinavia and the Barents Sea, which is given from 30 kyrs

BP to present and of which we have plotted the ice heights at the last glacial

maximum (LGM, 21 kyrs BP) in Figure 7.7a. We add a 90 kyrs linear glaciation

phase (from 120 to 30 kyrs BP) and we complement the ice-load history with a

7.4. Predictions for Northern Europe 123

(a) RSES

−1500

−750

0

750

1500

[km

]

−1500 −750 0 750 1500[km]

500 1000 1500 2000 2500 3000ice height [m]

(b) [ICE-5G]-[RSES]

−1500

−750

0

750

1500

[km

]

−1500 −750 0 750 1500[km]

−1000 −500 0 500 1000 1500ice height [m]

Figure 7.7: RSES ice heights (a) and differences between ICE-5G and RSES ice heights (b)at LGM.

eustatic ocean-load history, which has a maximum of about −140 m at LGM.

Total GIA-induced geoid heights at present, predicted from the laterally homoge-

neous model STD (RSES ice-load history, Figure 7.8a) show a deep low of −4 to

−5 m in the Gulf of Bothnia and bulge areas from 1000 km of the center of load-

ing outwards. Note that to compare this with global predictions of GIA, the low

harmonics (up to about degree 10) have to be filtered out to remove the effect of

the former Laurentide ice sheet. We show the sensitivity to the ice-load history

using the ICE-5G ice-load history (Peltier, 2004), for which we have plotted the

differences with the RSES ice heights at LGM in Figure 7.7b. The differences

are especially large off the coast of Scotland and Norway and in the Barents Sea,

where the ICE-5G model defines substantially larger ice masses. Geoid height

perturbations due to the ICE-5G ice-load history, i.e. differences between predic-

tions using ICE-5G and RSES and the same earth model (STDi-STD), show a deep

perturbative low over the Barents Sea of a few meter (Figure 7.8b). As the surface

heatflow in the Barents Sea is relatively high (Figure 1.2a), we expect crustal and

asthenospheric low-viscosity in this area. In the presence of a weak lower crust

or asthenosphere, we expect that the large differences in assumed ice mass will

then lead in the following sections to relatively large differences in perturbations

in this area.

In Sections 7.4.1 and 7.4.2 we use creep laws for the shallow upper mantle (mod-

els DIF and DIS from Table 7.4) and crust (models Ab100, Ab1100, Abn

100and An100,


(a) STD

−1500

−750

0

750

1500

[km

]

−1500 −750 0 750 1500[km]

−4.5 −3.0 −1.5 0.0 1.5geoid height [m]

(b) STDi-STD

−1500

−750

0

750

1500

[km

]

−1500 −750 0 750 1500[km]

−240 −160 −80 0 80geoid height perturbation [cm]

Figure 7.8: Total geoid heights as predicted by the laterally homogeneous earth model STD(RSES ice-load history, a) and geoid height height perturbations due to different ice-loadhistories (b). Perturbations are computed by subtracting predictions generated with RSESfrom predictions generated with ICE-5G (STDi-STD).

see footnote c in Table 7.4). As these creep laws are temperature-dependent (Sec-

tions 7.3.2 and 7.3.3), we use the temperature profiles from Figure 7.2b. Lateral

heterogeneities are introduced by the dependence of the temperature profile on the

surface heatflow and the crustal thickness (see Section 7.2.1). We use three areas

of surface heatflow (40, 60 and 80 mW/m2, Figure 1.2a) based on the database of

Pollack et al. (1993) and the extension of Artemieva & Mooney (2001) and five ar-

eas of crustal thickness (10, 20, 30, 40 or 50 km, Figure 1.2b) based on CRUST2.0

(Bassin et al., 2000).

7.4.1 Shallow Upper Mantle Perturbations

In this section we consider perturbations due to a certain viscosity stratification

(model ALVZ) or creep law in the shallow upper mantle (model DIF or DIS, Ta-

ble 7.4), in which the perturbations are computed by subtracting predictions of

background model STD (e.g. DIF-STD or DIS-STD). We also show perturbations

using the ICE-5G ice-load history, in which case we subtract from predictions gen-

erated with model DIF and ICE-5G (DIFi) predictions of background model STD

and ICE-5G (STDi).

Geoid height perturbations due a laterally homogeneous ALVZ (in which the as-


(a) ALVZ-STD

−1500

−750

0

750

1500

[km

]

−1500 −750 0 750 1500[km]

−40 0 40 80 120 160geoid height perturbation [cm]

(b) DIS-STD

−1500

−750

0

750

1500

[km

]

−1500 −750 0 750 1500[km]


Figure 7.9: Geoid height height perturbations for a laterally homogeneous ALVZ (a) andfor model DIS (wet olivine in the dislocation creep regime) (b). Both models are givenin Table 7.4) and perturbations are the difference with predictions from background modelSTD as shown in Figure 7.8a.

thenosphere, from 100 to 160 km depth, has a constant linear viscosity of 1019 Pas,

see Table 7.4) are shown in Figure 7.9a. Note that these predictions can also be

computed with analytical normal-mode codes as used in Chapters 4 and 5. The

ALVZ tends to decrease the total geoid height as predicted from STD (Figure 7.8a)

due to acceleration of the adjustment process. Because the pattern is very similar

to the total geoid heights, we do not expect that GOCE can add information on

the asthenosphere, because it will be difficult to separate the total geoid height

from the perturbations. The similarity in the spatial pattern indicates that the

ALVZ shows the same deep flow behavior (e.g. Cathles, 1975, p. 43 and Chap-

ter 6) as the deeper mantle. This is not generally true for an ALVZ, because for

a stiffer upper mantle or weaker lower mantle, the ALVZ can show channel flow

behavior (e.g. Cathles, 1975, p. 49 and Chapter 6). In the next section we discuss

the difference between the two when we compare the response of a CLVZ to an

ALVZ. Perturbations due to wet olivine in the dislocation creep regime (DIS-STD,

Figure 7.9b) show broad similarities with perturbations due to a constant linear

ALVZ. Amplitudes are comparable, but the center of the perturbative high has

shifted somewhat to the area of relatively high heatflow southwest of the Baltic

Shield (compare with Figure 1.2a).

In Figure 7.10a we show the effect of diffusion creep in the shallow mantle, using a

model with wet olivine and average grainsize (model DIF, Table 7.4) and subtract-


(a) DIF-STD

−1500

−750

0

750

1500

[km

]

−1500 −750 0 750 1500[km]


(b) DIFi-STDi

−1500

−750

0

750

1500

[km

]

−1500 −750 0 750 1500[km]

40

40

40


Figure 7.10: Geoid height perturbations due to diffusion creep in the shallow upper mantlefor the RSES ice-load history (DIF-STD, a) and the ICE-5G ice-load history (DIFi-STDi,b). In (b), the area within the contour is robust to changes in the ice-load history with astandard deviation (of predictions DIF-STD and DIFi-STDi) smaller than 20 cm and a meanlarger than 40 cm.

ing again model STD. Compared with model DIS (Figure 7.9b), amplitudes are

somewhat smaller, but the pattern is in general the same. This quite distinctive

pattern, in which the high shifts to the region of average heatflow, is very sensi-

tive to the shallow geotherm, so it seems necessary to use in the future tighter

constraints on the geotherms, for example by using shallow mantle temperatures

derived from seismic tomography models (e.g. Goes et al., 2000).

The positive perturbations under the load for wet olivine (Figures 7.9b and 7.10a)

indicate that either the effective viscosity in the shallow upper mantle is lower

than that of model STD (i.e. < 5 ·1020 Pas, compare with the predictions for a

constant linear ALVZ, Figure 7.9a) or that the lithospheric thickness is larger (i.e.

> 100 km, compare with Figure 6.9b in Chapter 6), or both. We cannot sepa-

rate between the two, because both induce perturbative highs with comparable

wavelengths, see Figure C.2 in Appendix C. This conclusion is strongly dependent

on the choice of composition (wet olivine), as from results for a dry rheology (not

shown) the strong increase in viscosity in the shallow upper mantle leads to a

much slower adjustment to isostasy and a perturbative low under the load.

The predictions are not only sensitive to lateral heterogeneities, but also to the

ice-load history. In Figure 7.10b we show again the effect of diffusion creep in


the shallow mantle, but now using ice-load history ICE-5G (DIFi-STDi). We see

some large differences, especially in the Barents Sea area, but also some features

that seem to be quite robust to changes in the ice-load history. These features are

useful to detect, rather than constrain, shallow low-viscosity regions, because these

features are not dependent on some uncertainties in the models (in this case the

ice-load history), which means that they can be expected with a larger certainty in

the measured gravity field.

These features we will call spatial signatures, which are thus geographical pat-

terns that are associated with shallow low-viscosity regions that are only loosely

constrained. To compare spatial patterns in two datasets, it is convenient to re-

move an off-set and scale factor. We estimate these from a least-squares fit of a

test set of predictions to a reference set of predictions, which gives mostly a negli-

gibly small offset and a scale-factor of 0.5−2. For the uncertainty in the ice-load

history, for example, the least-square fit of the test set DIFi-STDi to the reference

set DIF-STD gives a scale factor of 0.7. We then compute for each geographical

location the mean and the standard deviation of the scaled perturbations (DIFi-

STDi) and reference perturbations (DIF-STD). Those features that have a rela-

tively small standard deviation are then robust to changes in the ice-load history.

To only select those areas that have a distinct pattern, we choose a subset with a

relatively large mean, which works well in most cases. The threshold levels are

somewhat arbitrary and for consistency with some examples further on we take

half a scale interval (in this case 20 cm) for the standard deviation and one scale

interval (1/6 of the full scale, 40 cm) for the mean. In Figure 7.10b, the area within

the contour labeled ’40’ is therefore robust to changes in the ice-load history for

perturbations due to diffusion creep in the shallow upper mantle with a standard

deviation smaller than 20 cm and mean amplitudes larger than 40 cm.

7.4.2 Crustal Perturbations

In Figure 7.11a we show perturbations due to a laterally homogeneous CLVZ, in

which the lower crust from 20 to 30 km depth has a constant (i.e. temperature-

independent), linear (in rheology) viscosity of 1019 Pas and the background model

is STD (CLVZ-STD, see Table 7.4). Comparison with perturbations due to an ALVZ

(Figure 7.9a) shows mainly two things: Due to the larger depth of the ALVZ com-

pared to the CLVZ, the picture for an ALVZ is much smoother than for a CLVZ,

and an ALVZ accelerates the process of GIA (e.g. smaller smaller geoid low under

the load), whereas a CLVZ delays the process (larger geoid low). The second differ-

ence is mainly due to the difference between deep flow for the ALVZ and channel

flow for the CLVZ (Cathles, 1975, p. 219, and Section 4.2 in Chapter 4). If flow is

constrained to a channel, first the channel at the edge of the load will relax and,

due to very long relaxation of the low harmonics, the relaxation under the center of


(a) CLVZ-STD

−1500

−750

0

750

1500

[km

]

−1500 −750 0 750 1500[km]


(b) Ab100(DIS)-DIS

−1500

−750

0

750

1500

[km

]

−1500 −750 0 750 1500[km]


Figure 7.11: Geoid height height perturbations due to a laterally homogeneous CLVZ (Ta-ble 7.4, a) and for albite (Ab100) in the diffusion creep regime (d = 100 µm) with DIS(Table 7.4) as a background model (Ab100(DIS), b). Perturbations are the difference with(a) predictions from background model STD as shown in Figure 7.8a (CLVZ-STD) and (b)predictions from background model DIS as shown in Figure 7.9b (Ab100(DIS)-DIS).

the load will be delayed (Cathles, 1975, p. 158 and Section 4.2). For deep flow the

long wavelengths relax fastest, which results in a broad perturbation from under

the load outwards (Cathles, 1975, p. 219 and Section 4.2). Deep flow behavior can

be seen for the geoid height perturbation due to a constant linear ALVZ (model

ALVZ, Figure 7.9a), which shows a broad perturbative high with a maximum of

about 1.5 m under the load.

To have a consistent shallow earth model, in which viscosity is controlled by creep

laws and heatflow (and which is thus laterally heterogeneous), we will from here

on use either model DIF or DIS as our background model. First we show pertur-

bations for albite (Ab100) in the diffusion creep regime, with a grainsize of d = 100

µm and DIS (i.e. dislocation creep in the shallow upper mantle, Table 7.4) as the

background model, denoted as Ab100(DIS). Amplitudes of geoid height perturba-

tions due to Ab100(DIS) (Ab100(DIS)-DIS, Figure 7.11b) are comparable to those

due to a laterally homogeneous CLVZ, but the pattern of perturbations differs sig-

nificantly. The low under the largest dome (around the Gulf of Bothnia), due to

the long relaxation times induced by channel flow, is still visible, indicating that

lower crustal flow seems to be possible in the Baltic Shield based on the adapted

heatflow values and creep laws. The highs are now distributed around this low

and occur mainly in areas south and west of the Baltic Shield that experienced


(a) Ab100(DIF)-DIF

−1500

−750

0

750

1500

[km

]

−1500 −750 0 750 1500[km]


(a) Ab100(DIFi)-DIFi

−1500

−750

0

750

1500

[km

]

−1500 −750 0 750 1500[km]

8

8

8

8

8


Figure 7.12: Geoid height perturbations for Ab100 in the diffusion creep regime (d = 100

µm) with DIF (RSES ice-load history) as a background model (Ab100(DIF)-DIF, a) and DIFi(ICE-5G ice-load history) as a background model (Ab100(DIFi)-DIFi, b). In (b), the areawithin the contour is robust to changes in the ice-load history with a standard deviation (ofscaled test predictions Ab100(DIFi)-DIFi and reference predictions Ab100(DIF)-DIF) smallerthan 4 cm and a mean larger than 8 cm.

significant loading and have average heatflow.

To show the sensitivity to the background model, we keep the same composition,

creep regime and composition in the crust (Ab100), but change the background

model from dislocation creep in the shallow upper mantle (model DIS) to diffu-

sion creep in the shallow upper mantle (model DIF). If we compare Figure 7.12a

(Ab100(DIF)-DIF) with Figure 7.11b (Ab100(DIS)-DIS), we see that the perturba-

tions are very sensitive to changes in the background model as both figures show

a significantly different spatial pattern, although with similar amplitudes. The

perturbations are also sensitive to the ice-load history, which can be shown by com-

paring Figure 7.12a and 7.12b. The latter is generated with ICE-5G (Ab100(DIFi)-

DIFi) and shows a somewhat different response in e.g. the Barents Sea area. In

Figure 7.12b we have also given contours for which the predictions are robust to

changes in the ice-load history, with a threshold level for the standard deviation of

4 cm (half scale interval) and mean of 8 cm (one scale interval). As in Section 7.4.1,

the contours are computed by scaling the test set (Ab100(DIFi)-DIFi) to the refer-

ence set (Ab100(DIF)-DIF, scale facter: 0.6, see ’ICE’ in Table 7.6) and computing

for each geographical location the mean and standard deviation of the scaled test

set and the reference set. Robust features are most likely to be recoverable from a


(a) An100(DIF)-DIF

−1500

−750

0

750

1500

[km

]

−1500 −750 0 750 1500[km]

8

8

88

8

8


(b) Abn100

(DIF)-DIF

−1500

−750

0

750

1500

[km

]

−1500 −750 0 750 1500[km]

8

8

8

88


Figure 7.13: Geoid height perturbations for anorthite in the diffusion creep regime (d = 100

µm, An100(DIF-DIF), a) and for albite in the dislocation creep regime (Abn100

(DIF)-DIF,b). Areas within contour lines are robust for changes in the composition (a) or creep regime(b) with a standard deviation smaller than 4 cm and a mean larger than 8 cm.

high-resolution gravity field as, for example, expected from GOCE, as they do not

depend on certain assumptions, as in this case on the ice-load history.

In Figure 7.13a we show perturbations due to different composition of the lower

crust by using anorthite (An100) instead of albite (Ab100). For different composi-

tion the pattern is similar, although with smaller amplitudes, because An100 is

somewhat stronger than Ab100 (Section 7.3.3). We have again indicated which ar-

eas are robust to changes in the composition by linearly fitting An100(DIF)-DIF

to Ab100(DIF)-DIF (scale factor: 1.3, see ’COMposition’ in Table 7.6), which shows

that the main lows and highs in the perturbations due to diffusion creep in the

crust are robust to changes in composition. In Figure 7.13b we show perturba-

tions due to different creep regime in the lower crust by using dislocation creep

(Abn100

) instead of diffusion creep (Ab100). For changes in creep regime (scale fac-

tor: 1.9, see ’REGime’ in Table 7.6) only the highs seem to be robust (Figure 7.13b),

which implies that these highs are robust to both changes in composition and creep

regime. As mentioned before, these features are most likely to be recoverable with

GOCE, as these do not depend on composition nor creep regime in the crust.

In the next section we will test if GOCE can constrain the properties of the lower

crust, i.e. can discriminate between different compositions and creep regimes.

From Figure 7.13 we can already predict that it will be easier to discriminate

7.5. Constraints from Future GOCE Data 131

between creep regimes than compositions, because the latter is more robust to

changes in parameters. In the next section we will also estimate if the conclusions

depend on the background model (DIF versus DIS and DIFi).

7.5 Constraints from Future GOCE Data

We expect GOCE to especially add information on the viscosity of the crust, be-

cause of the high-resolution gravity field expected from GOCE and the distinct

spatial patterns generated by crustal low-viscosity layers, as shown in the previ-

ous section. The latter is important to distinguish the signal due to low-viscosity

layers from other signals due to, for example, thermally or chemically-induced

mass inhomogeneities in the crust.

In the previous section we have also shown which parts of the predictions are

robust to changes in composition and creep regime of the lower crust and there-

fore facilitate the detection of low-viscosity layers. Here we estimate the extent

to which high-resolution gravity field information, as expected from GOCE, can

constrain the composition and creep regime of the lower crust. This involves the

prediction of geoid height perturbations and testing which predictions have the

best fit to the (synthetic) satellite gravity data. The data is able to constrain prop-

erties of the lower crust if it is able to distinguish between different predictions,

i.e. if there are significant differences in prediction errors. Here we will use the

normalized prediction error (NPE, Hengl et al., 2004):

NPE =

√

√

√

√

∑

i(di − p(1)i

)2

∑

i d2i

·100% (7.18)

with p(1)i

a test prediction, di = p(0)i

+ e i synthetic data, with p(0)i

the reference

prediction that generates the data, and e i the errors. Both the test prediction p(1)i

and the prediction p(0)i

that generates the synthetic data are perturbations (i.e. the

background model is subtracted) due to a certain creep law in the crust. We will

for example use predictions from albite in the diffusion creep regime with average

grainsize (Ab100, footnote c in Table 7.4) in a background model with diffusion

creep in the shallow upper mantle (DIF, Table 7.4), subtract the predictions from

background model DIF and denote p(0)i

as Ab100(DIF). Hengl et al. (2004) take an

NPE smaller than 40%, in which (1−0.42) ·100%≈ 85% or more of the variance of

the data d is explained by a certain prediction p(1), as an acceptable prediction.

For us, an acceptable test prediction p(1) is not distinguishable from the synthetic

data d, i.e. from the reference prediction p(0) in the presence of errors e. We take

an NPE larger than 60%, in which case 65% or less of the variance is explained by

the data, for a prediction that GOCE can distinguish from the data.


(a) 2D PSD

0 0.005 0.01 0.0150

0.005

0.01

0.015

k y [rad

/km

]

kx [rad/km]

5

10

15

20

25

(b) 1D PSD

0.005 0.01 0.015 0.02 0.025 0.0310

0

101

102

pow

er (

arbi

trar

y un

its)

kz [rad/km]

recovery error

CLVZ (1019 Pas)Ab

100(DIF)

An100

(DIF)−Ab100

(DIF)

Ab100

n(DIF)−Ab100

(DIF)

←3000 km 100 km→

3.0 mE

1.5 mE

Figure 7.14: 2D power spectral density (PSD) of the perturbations due to a constant linearCLVZ as shown in Figure 7.11a (CLVZ-STD, a) and the radially averaged 1D PSD (b).

We simulate the error introduced by the noise behavior of the gradiometer onboard

GOCE and downward continuation (recovery error) by adding random noise with

a standard deviation of 1.5 mE (1 Eötvös = 10−9 s−2) to one month of gravity gra-

dients computed from a reference prediction p(0) along a circular orbit (no polar

gap) at 250 km altitude. For the reference prediction p(0) we take perturbations

due to a certain composition, creep regime and grain size of the lower crust (e.g.

An100(DIF) with the background model DIF subtracted, see Table 7.6). We recover

the gravity field (i.e. geoid heights) from the gravity gradients using an iterative

block-diagonal solution method, as described in Klees et al. (2000). The simulated

error corresponds well with the expected GOCE performance computed from a

more extensive simulation set-up (Visser et al., 2002), as shown in Vermeersen &

Schotman (2008) and Section 3.4. As a worst-case scenario, we also simulate the

recovery error for a noise level of 3.0 mE. As explained in Section 7.1 we do not

include any other error sources.

To obtain spectral information from our model, we use a 2D FFT, as also used

to compute geoid heights from the flat model (see Section 6.2). In Figure 7.14a

we have plotted the 2D spectrum for the CLVZ-induced geoid height perturba-

tions from Figure 7.11a. The wavenumbers kx, ky are defined as 2π/λ, with λ the

wavelength in [km] corresponding to spatial scales (λ/2) =π/k. The power spectral

density (PSD) peaks for a wavenumber-distance kz =√

k2x +k2

y from the center of

about 0.002 to 0.005 rad/km (spatial scales of ∼ 1500−600 km) and in a band be-

tween 0.01 and 0.015 rad/km (∼ 300−200 km). This can be seen more clearly from

a 1D spectrum, in which the 2D spectrum is radially averaged (i.e. we compute

the total power in spherical bands and divide by the area of the band) and plotted

as a function of wavenumber kz (Figure 7.14b). Next to the CLVZ-induced per-

turbations from the 2D spectrum, we have plotted the spectrum of the recovery


(a) effect of filtering on NPE (b) NPE for different test predicitions

Figure 7.15: Effect of filtering on the NPE for REF (p(1) = p(0) = Ab100(DIF), a) and theNPE for different test predictions p(1) with the data d = p(0) + e generated with p(0) =Ab100(DIF), b). For an explanation of ’REF’, ’COM’, ’REG’, ’GRA’, ’AST’ and ’ICE’, seeTable 7.6.

error for low (1.5 mE) and high (3.0 mE) noise level and the perturbations due to

albite Ab100 in the diffusion creep regime, with model DIF as a background model

(’Ab100(DIF)’, Figure 7.12a). We have also plotted in Figure 7.14b the sensitivity

to the compostion of the lower crust (’An100(DIF)-Ab100(DIF)’) and the sensitivity

to the creep regime (’Abn100

(DIF)-Ab100(DIF)’), where the superscript n indicates

that Ab100 is in the dislocation creep regime. We see that GOCE is sensitive to

both composition and regime, though the sensitivity to composition seems to be

low, which was already expected from Figure 7.13a, where we showed that the

perturbations are quite robust to changes in composition. This is confirmed below

when using the NPE to show the sensitivity of GOCE to composition and creep

regime of the lower crust. We have applied a Bartlett window (e.g. Press et al.,

1992, p. 547), which is a pyramid in 2D, to the data, which surpresses a large part

of the spectral leakage. This leakage is due to the inherent windowing of the data

with a box when computing the 2D Fourier-transform, which results in a convo-

lution of the spectrum with a sinc-function (e.g. Press et al., 1992, p. 546). Some

leakage is, however, still visible for low wavenumbers (< 0.005 rad/km) of the re-

covery error. Before we proceed, we first analyse the synthetic data. For this we

can use the NPE, because for test predictions that are the same as the reference

predictions that generated the synthetic data (p(1) = p(0)), the NPE gives (in %)

the square-root of the ratio of the power of the recovery error and the power of

the data, i.e. the square root of the noise-to-signal (N/S) ratio. For the data we

use perturbations from model Ab100(DIF), which has a crust consisting of Ab100 in

the diffusion creep regime with a grainsize of 100 µm, and DIF as a background

model. From Figure 7.15a we see that the NPE for unfiltered data and predictions

is quite high, which means that a considerable part of the variance of the data

is due to noise. If we start filtering the data by removing the high wavenumbers


(short wavelengths), the NPE drops to a minimum if we remove all wavenumbers

larger than 0.015 rad/km. For 1.5 mE measurement noise, the NPE is about 33%

(N/S ratio of 0.1, see ’REF’ in Table 7.6), which means that the recovery error only

contributes 10% to the variance of the data. For 3.0 mE however, the NPE is close

to 60% (N/S ratio of 0.3, see ’REF’ in Table 7.6) so that the error contributes al-

most 35% to the data. From here on we remove all wavenumbers larger than 0.015

rad/km.

We now investigate if the correct prediction (’REF’) has the smallest NPE for all

noise levels, and if other predictions can be distinguished from the data. To do

this, we compute NPEs for changes in the composition of the lower crust (to An100,

’COM’), in the creep regime (to Abn100

, ’REG’) and in the grainsize (to An1100

, grain-

size of 1000 µm, ’GRA’), see Table 7.6. We also change the background model,

either the creep regime of the asthenosphere (to DIS, ’AST’ in Table 7.6) or the ice-

load history (to DIFi, ’ICE’ in Table 7.6). From Figure 7.15b we see that the correct

predictions always give the smallest NPE, also for high measurement noise level

(3.0 mE). The NPEs for ’COM’ are, however, close to the NPEs for ’REF’, which in-

dicates that GOCE can probably not discriminate between the two compositions.

This was already expected from the 1D PSD, where the sensitivity to composition

was only slightly above the noise level, see Figure 7.14b. All other predictions have

NPEs larger than 60%, which means that GOCE can discriminate between creep

regime in the lower crust (’REG’) and grainsize (’GRA’).

For changes in the background model, dislocation creep in the shallow upper man-

tle (’AST’) or the ICE-5G load history (’ICE’), this merely shows that, in the pres-

ence of albite in the lower crust, GOCE is predicted to be sensitive to both the

creep regime in the shallow upper mantle and the ice-load history. This, however,

also shows that, if we want to constrain properties of the crust, a priori informa-

tion on the background model is needed, because we will not predict the correct

answer with a sufficiently small NPE, i.e. a NPE smaller than 60%. Because the

influence of the recovery error is small, we can conclude that the difference in test

and reference predictions is dominating, which is confirmed by the relatively low

N/S ratios for Ab100(DIF) (Table 7.6). The increase of the NPE for increasing noise

level moreover shows that there is a correlation between the predictions and the

data, and that the NPEs are not generated by chance.

It is of course important to know if the conclusions we have drawn from Fig-

ure 7.15b also hold for other synthetic data. Is GOCE, for example, also sensi-

tive to the background model for dislocation creep or anorthite in the crust? Yes,

also for dislocation creep regime or anorthite in the crust, GOCE is sensitive to

the background model. But is it, for example, also true that GOCE cannot dis-

criminate between albite and anorthite in the crust for other assumptions on the

crust (creep regime, grainsize) or background model (creep regime upper mantle,


(a) composition (b) creep regime

Figure 7.16: NPE for changes in composition (a) and in creep regime (b).

ice-load history)? We test this by computing NPEs for Ab100 and An100 in the crust

for different assumptions on creep regime and grainsize in the crust, and different

background models (Figure 7.16a). Note that we now change for each test both

the test prediction p(1) and the reference prediction p(0) that generates the syn-

thetic data, see ’COM(REG)’ to ’COM(ICE)’ in Table 7.6. In the presence of error

sources (’1.5 mE’, ’3.0 mE’), the conclusion that GOCE cannot discriminate be-

tween composition of the crust seems only to hold for changes in the background

models (to DIS, ’COM(AST)’ and DIFi, ’COM(ICE)’), because for changes in the

creep regime(’COM(REG)’) or grainsize (’COM(GRA)’) of the crust, the NPE in-

creases to about 80%. The strong sensitivity of the NPE to the measurement noise

indicates that the N/S ratio, as shown for ’REF’ in Figure 7.15a, is large for both

dislocation creep and large grainsize in the crust (see Table 7.6). This means that

the data for ’COM(REG)’ and ’COM(GRA)’ is largely controlled by the recovery

error and that the correct prediction is not acceptable (NPE larger than 60%).

To see if the conclusion that GOCE can discriminate between diffusion creep and

dislocation creep in the crust is robust, we change the composition (’REG(COM)’)

and grainsize (’REG(GRA)’) of the crust, and the background model (to DIS, ’REG

(AST)’ and DIFi, ’REG(ICE)’), see Figure 7.16b. For a different composition of the

crust (’REG(COM)’) GOCE can still discriminate between diffusion and disloca-

tion creep, but for different grainsize (’REG(GRA)’) the strong sensitivity of the

NPE to the measurement noise indicates that the data is controlled by the recov-

ery error (see Table 7.6), and that, if the crust shows diffusion creep with large

grainsize, GOCE measurements merely show noise. Note that the scale factor for

’REG(GRA)’ is substantially different from the other scale factors of this figure,

which is because we cannot change the grainsize of the prediction, as it is in the

dislocation creep regime. For ’REG(AST)’ and ’REG(ICE)’ the NPEs are smaller,

indicating that it will be more difficult to discriminate between creep regime in

the crust if the background model shows dislocation creep in the shallow upper


mantle (model DIS) or if the ice-load history is ICE-5G (model DIFi).

7.6 Conclusions

Using a thermomechanical model (Section 7.2), we have estimated effective vis-

cosities for plagioclase feldspars in the crust and olivine in the shallow (to a depth

of 220 km) upper mantle (Section 7.3). We have shown the effect of creep laws on

GIA-induced geoid height predictions in Northern Europe (Section 7.4) and have

tested the sensitivity of future GOCE data to creep laws in the crust (Section 7.5).

The main conclusions are:

• Based on comparison with GIA studies for Northern Europe and effective

elastic thickness studies, the creep law for wet olivine with average grain

size (8 mm) provides the most reasonable results. For diffusion creep, the

viscosity at 200 km depth is 3 ·1020 Pas for low heatflow (40 mW/m2) and

6 ·1019 Pas for average to high (60−80 mW/m2) heatflow;

• Based on the Maxwell time, we estimate a viscosity of 1023 Pas for which

the shallow upper mantle is effectively elastic in the process of GIA. For wet

olivine with average grainsize in the diffusion creep regime this corresponds

to a lithospheric thickness for the process of GIA of 140 km for low heatflow

and 60 km for average heatflow;

• For the long-term (> 1 Myrs) strength, we estimate a viscosity larger than

> 1025 Pas, which gives lithospheric thicknesses smaller than 110 km for

low heatflow and smaller than 30−40 km, depending on the strength of the

lower crust, for average heatflow;

• For the crust we use wet plagioclase feldspars, with albite in general the

weakest and anorthite the strongest. Viscosities for average grainsize (100

µm) and average to high heatflow can be as low as 1019 Pas, and even for low

heatflow the lower crust seems to show some ductile behavior;

• For a realistic load case, using the RSES ice-load history, amplitudes of per-

turbations due to diffusion or dislocation creep in the shallow upper man-

tle are comparable to a laterally homogeneous asthenospheric low-viscosity

zone at a depth of 100 km, with a thickness of 60 km and a viscosity of 1019

Pas. The spatial pattern is, however, more complicated, with the pertur-

bative high shifting towards an area of average heatflow southwest of the

center of loading (see Figure 7.9);

• Perturbations due to a weak lower crust are sensitive to the composition and

creep regime of the crust, and the background model (diffusion or dislocation

7.6. Conclusions 137

creep in the shallow upper mantle, ice-load history). The sensitivity to the

background model is larger than to composition, but smaller than to creep

regime;

• Some features of perturbations due to a weak lower crust are robust to

changes in certain parameters (composition, creep regime, ice-load history.

Robust features are features that differ less than 4 cm between a test and

a reference prediction, and that have a relatively large amplitude (> 8 cm).

These features are most likely to be detectable by GOCE. We find for exam-

ple that some perturbative lows and highs are robust to the composition of

the lower crust and some perturbative highs are robust to the creep regime;

• From normalized prediction errors (NPEs), we have found that the synthetic

data, consisting of a reference prediction and recovery errors as expected

from GOCE, needs to be filtered to minimize the noise-to-signal ratio. For

different test predictions (changing the compostion, grainsize, creep regime

or background model), the reference prediction always gives the lowest NPE,

independent of the assumed measurement noise level;

• However, for a change in composition, the NPE is also small, which indicates

that GOCE can probably not constrain the composition of the lower crust, in

contrast to creep regime and grainsize;

• GOCE is also predicted to be sensitive to the background model in the pres-

ence of a crust consisting of plagioclase feldspars, irrespective of the creep

regime. This also means if the wrong background model is assumed, we can

no longer predict the correct properties of the lower crust, because prediction

errors are larger than 60%.

We plan to improve the constraints on the temperature profile in the earth, us-

ing temperature at depth from seismic data (e.g. Goes et al., 2000), because the

geoid height predictions are quite sensitive to the thermally-induced lateral het-

erogeneities in our model. As the adapted ice-load history is contaminated by the

assumed earth stratification, we plan plan to couple our thermomechanical earth

model to a thermomechanical ice sheet model (e.g. Bintanja et al., 2002) and es-

timate the ice-load history and earth stratification at the same time. Finally, we

have assumed in this study that the only error source in the GOCE data is the

recovery error. However, especially unmodelled shallow mass inhomogeneities are

expected to mask gravity signals due to shallow low viscosity, and effective spatio-

spectral filtering tools have to be developed to deal with this.


Acknowledgements

We are grateful to Irina Artemieva (Geological Institute, University of Copen-

hagen, Copenhagen) for providing thermal datasets and heatflow data for conti-

nental Europe, Marta Pérez-Gussinyé (Institute of Earth Sciences ’Jaume Almera’,

CSIC, Barcelona) for providing datasets of effective elastic thickness estimates of

Northern Europe, Pieter Visser (DEOS, Delft University of Technology, Delft) for

providing the software to estimate predicted recovery errors of GOCE, Kurt Lam-

beck (RSES, Australian National University, Canberra) for providing the RSES

ice-load history, and Radboud Koop (NIVR, Delft, previously at SRON, Utrecht)

and Rob Govers (IVAU, Utrecht University, Utrecht) for discussions over the past

few years which have lead to this paper. The geographical plots in this paper have

been generated with GMT (Wessel & Smith, 1991).

7.6

.Conclu

sions

139

Table 7.4: Earth stratification of the background models

Layer Depth Densitya ρ Rigiditya ν STDe DIFe DIS

[km] [kg/m3] [GPa]

upper crust 0 2700 27 elastic elastic elasic

lower crust 20 : : : b : c : c

lithosphere 30 3380 68 elastic wet diffus.d wet disloc.d

asthenosphere 100 : : 5 ·1020 Pase : :

shallow upper mantle 160 : : : : :

upper mantle 220 3480 77 : 5 ·1020 Pas 5 ·1020 Pas

transition zone 400 3870 108 : : :

lower mantle 670 4890 221 5 ·1021 Pas 5 ·1021 Pas 5 ·1021 Pas

aDensity and rigidity are volume-averaged from PREM (Dziewonski & Anderson, 1981).

bModel CLVZ is equal to model STD, except for the lower crust which has a viscosity of 1019 Pas.

cIf we include creep laws in the crust, the crustal thickness is derived from CRUST2.0 and thus varies. We use the following creep

laws: Ab100 (albite in the diffusion creep regime with average (100 µm) grainsize), Ab1100

(idem, but with large (1000 µm) grainsize),

Abn100

(albite in the dislocation creep regime) and An100 (anorthite in the diffusion creep regime with average grainsize). These creep

laws are used in combination with either diffusion (model DIF) or dislocation (model DIS) creep in the shallow upper mantle, denoted

as e.g. Ab100(DIF) and Ab100(DIS), respectively.d

Grainsize for wet diffusion (model DIF) is d = 8 mm, the stress exponent for wet dislocation (model DIS) is n = 3.5, see also Table 7.2.e

Model ALVZ is equal to model STD, except for the asthenosphere (from 100 to 160 km) which has a viscosity of 1019 Pas.f

If we use in Section 7.4 ice-load history ICE-5G instead of RSES, we attach an ’i’ (e.g. STDi, DIFi).

140

Chapte

r7.T

her

mom

echanic

alM

odel

sTable 7.5: Predictions and Data

Name p(1) (test pred.a) scaleb p(0) (data pred.a) N/Sc (1.5 mE) N/Sc (3.0 mE)

REFerence Ab100(DIF) 1.0 Ab100(DIF) 0.1 0.3

COMposition An100(DIF) 1.3 Ab100(DIF) 0.1 0.3

COM(REG) Ann100

(DIF)d 1.3 Abn100

(DIF) 0.4 0.7

COM(GRA) An1100

(DIF)d 1.3 Ab1100

(DIF) 0.5 0.8

COM(AST) An100(DIS) 1.4 Ab100(DIS) 0.1 0.2

COM(ICE) An100(DIFi) 1.3 Ab100(DIFi) 0.1 0.2

REGime Abn100

(DIF) 1.9 Ab100(DIF) 0.1 0.3

REG(COM) Ann100

(DIF) 2.0 An100(DIF) 0.2 0.5

REG(GRA) Abn100

(DIF) 0.9e

Ab1100

(DIF) 0.5 0.8

REG(AST) Abn100

(DIS) 1.9 Ab100(DIS) 0.1 0.2

REG(ICE) Abn100

(DIFi) 2.2 Ab100(DIFi) 0.1 0.2

GRAinsize Ab1100

(DIF) 2.0 Ab100(DIF) 0.1 0.3

AST Ab100(DIS) 0.6 Ab100(DIF) 0.1 0.3

ICE Ab100(DIFi) 0.6 Ab100(DIF) 0.1 0.3

ap(1) are test predictions, whereas p(0) are predictions that generate the data. These are all perturbations, which means that the

background model, as given in brackets, is subtracted. This is especially important when the background model of the predictions

and data differ (see ’AST’ and ’ICE’), because otherwise the NPE is dominated by (long-wavelength) differences in the background

model. A description of the background models can be found in Table 7.4.b

Scale factor determined from a least-squares fit of predictions to data, which is applied to the predictions before computing the

NPE.c

Noise-to-signal ratio of the data for different noise levels, estimated as N/S=(NPE/100%)2 , see Figure 7.15a and discussion there.d

The superscripts n, 1 indicate that the crust is respectively in the dislocation creep regime or in the diffusion creep regime with

large grainsize.e

This odd scale factor is due to the fact that we cannot change the prediction here to investigate the sensitivity to grainsize, see also

Figure 7.16b and the discussion there.

Chapter 8

Conclusions

For a radially (1D-) stratified earth, we have shown in Chapter 4 the patterns

generated by shallow low-viscosity zones (LVZs), both crustal (CLVZs) and as-

thenospheric (ALVZs). In Chapter 4 we have also tested the sensitivity to the

background earth stratification and in Chapter 5 we have tested the sensitivity to

the load history. For laterally heterogeneous (3D-stratified) earth models, we have

shown in Chapter 6 the effect of lateral heterogeneities on LVZ-induced patterns

and in Chapter 7 the effect of laboratory-derived creep laws. In the latter chapter

we also introduced concepts to invert GOCE data for the properties of LVZs.

Below we answer the questions (I-III) we posed in Chapter 1 (Section 1.4).

I. What are the amplitudes and distributions (spatial, spectral) of geoid height

perturbations due to low-viscosity layers and how are these compared to the

performance of GOCE?

CLVZ-induced geoid height perturbations show distinctive spectral patterns,

which depend on the properties of the CLVZ (Section 5.4.2). Spectral ampli-

tudes peak at harmonic degree 40−80 (∼ 500−250 km, Sections 4.3.2 and 5.2),

but amplitudes are larger than the expected GOCE performance up to degree

120−140 (∼ 150 km). As the amplitudes are only larger than the realized

GRACE performance (GGM02S) up to degree 70−80 (∼ 500−250 km), GOCE

is predicted to provide unique information on especially short-scale features

(< 250 km).

Other conclusions are:

• The response of a shallow LVZ to loading and unloading during the last

glacial cycle is a complex function of spherical harmonic degree and

time (Section 4.2);

142 Chapter 8. Conclusions

• The effect of a CLVZ (1018−1019 Pas) is to decouple the crust and litho-

sphere, so that on large time scales (> 1 Myrs) all strength resides in

the upper crust (Section 4.2 and Appendix A);

• For such a CLVZ, the relaxation time and strength are increased for

all degrees due to the extra buoyancy mode MC, which has its largest

strength from degree 50 to 150 (channel flow, Section 4.2);

• For an ALVZ, the relaxation time is decreased for all degrees, leading

to faster adjustment to isostasy (deep flow, Section 4.2);

• Geoid height perturbations due to a CLVZ have amplitudes of 30 cm

(for 1019 Pas, e.g. Section 7.4.2) to 60 cm (for 1018 Pas, e.g. Section 5.2)

with scales down to hundred km, and due to an ALVZ have amplitudes

of more than 1 meter (e.g. Sections 5.2 and 7.4.1) with spatial scales

larger than 1000 km;

• Degree amplitudes for ALVZ-induced perturbations are above the ex-

pected GOCE perfomance to degree 65 and above GGM02S to degree

55 (Section 4.3.2).

II. Is GOCE sensitive to the properties of LVZs and the properties of the back-

ground model (earth stratification, ice-load history), and are there signa-

tures (spatial, spectral) that are robust to these properties?

The sensitivity of GOCE to changes in the properties of a CLVZ is small (Sec-

tion 4.4) and, moreover, GOCE is also predicted to be sensitive to uncertain-

ties in the ice-load history (Section 5.3.2) and background earth stratification

(Section 4.5.1). The small sensitivity to the properties indicates that some

spatial features of CLVZ-induced perturbations are robust to certain changes

in the properties of the CLVZ (spatial signatures) and are thus most likely

to be detectable by GOCE. We find some predictions, for example along the

Norwegian coast, that are robust to changes in the viscosity and thickness of

the CLVZ (Section 4.4) and to changes in the composition and creep regime

(Section 7.4.2). Some patterns are also robust to changes in the background

model, as upper mantle viscosity (Section 4.5.1) and ice-load history (Section

7.4.2).


• The sensitivity to the ice-load history is mainly due to ice sheet geom-

etry and is for a CLVZ mainly confined to an area of a few hundred

kilometers around the formerly glaciated areas. The sensitivity in the

presence of an ALVZ is larger, and indicates more fundamental differ-

ences in ice-load histories as e.g. the way the Barents Sea was glaciated

(Sections 5.3.1 and 7.4);

143

• For a CLVZ, GOCE will be sensitive to differences in the ice-load history

up to degree 140 and for an ALVZ up to 70, which means that GOCE

should be able to discriminate between different ice-load histories if we

know the properties of the LVZ. However, this also means that in the

presence of uncertainties in the ice-load history, we cannot recover the

properties of the LVZ, as shown using degree correlation coefficients

(Section 5.3.2);

• From degree correlations between geoid heights generated with a cer-

tain ice-load history and ice heights at LGM, we can determine a most

likely ice-load history and use this to normalize the degree amplitudes

of CLVZ-induced perturbations to obtain spectra for different proper-

ties of the CLVZ that are almost independent of the ice-load history

(spectral signatures, Section 5.4.2);

• The sensitivity to a realistic ocean function is for a CLVZ dominated

by the effect of time-dependent coastlines and meltwater influx, and is

confined to shallow seas as the North Sea and sea areas that were once

glaciated, as the Gulf of Bothnia (Sections 5.3.1 and 6.6).

• For an ALVZ, the effect of time-dependent coastlines and meltwater

influx seems to be small, however self-gravitation in the oceans has a

large effect (Sections 5.3.1 and 6.6).

• Errors introduced by using a eustatic ocean load can be larger than

10% for CLVZ-induced perturbations, which means that in some cases

a realistic ocean function has to be used, and are in general smaller

than 10% for ALVZ-induced perturbations (Section 6.6);

• CLVZ-induced perturbations have to be filtered up to degree 30 for un-

certainties in the viscosity of the lower mantle and to degree 50 for

uncertainties in the upper mantle viscosity, though the latter depends

on how well the background model can be estimated from degree corre-

lations (Section 4.5.2);

• The lower bound due to uncertainties in the ice-load history is larger,

around degree 40−60 (Section 5.2);

• 3D-flat, non-self-gravitating models can deliver accurate predictions of

radial displacements, uplift rates and geoid heights when compared to

conventional spherical, self-gravitating models (Section 6.4);

• LVZ-induced perturbations can also be computed accurately, though

the accuracy seems to decrease somewhat with increasing depth (Sec-

tion 6.4);

• Lateral heterogeneities do not necessarily have a large influence on

CLVZ-induced perturbations, though amplitudes are still up to 20 cm in


geoid height, which is an order of magnitude larger than the expected

GOCE performance (Section 6.5);

• Because of the different signature of perturbations due to a CLVZ and

due to variations in lithospheric thickness, it seems possible to solve for

both lithospheric thickness and CLVZs at the same time (Section 6.5);

• For synthetic GOCE data, the reference prediction (that generates the

synthetic data) always gives the lowest prediction error, independent of

the assumed measurement noise level (Section 7.5);

• However, prediction errors are comparably small for different composi-

tion, which indicates that GOCE can probably not constrain the com-

position of the crust, in contrast to creep regime and grainsize (Sec-

tion 7.5);

• GOCE is also predicted to be sensitive to the background model in the

presence of a crust consisting of plagioclase feldspars, irrespective of

the creep regime. This also means that if the wrong background model

is assumed, we can no longer predict the correct properties of the lower

crust, because prediction errors are larger than 60% (Section 7.5);

III. Which unique information does GOCE provide on the rheology of the shallow

earth and which additional datasets can we use to constrain this rheology?

In contrast to Holocene sea-level curves and GPS measurements, which have

in the past provided constraints on the rheology of especially the deeper upper

(> 200 km) and lower mantle in e.g. Northern Europe (Section 1.2), satellite

gravity data provides a spatial continuous field instead of a network of point

measurements. As GOCE provides a high-resolution gravity field, it can be

used to detect spatial signatures with scales down to 100 km, as predicted

for CLVZs. As an additional dataset we have considered, among others, the

combination of laboratory-derived creep laws and surface heatflow data, for

which however more geological field data on especially composition is needed

to constrain the rheology of the shallow earth.


• Tide gauges: Both for a CLVZ, due extra coastline tilting, and an ALVZ,

due to faster adjustment to isostasy, estimates of present-day sea level

rise (∼ 1.3 mm/yr) are lowered (with 0.01−0.14 mm/yr) while decreasing

the standard deviation (∼ 0.2 mm/yr) with about 10% for a region as

western Europe (Section 4.6).

• For individual tide gauges the effect on present-day sea level change

can be larger (∼ 1 mm/yr along coastlines), especially in formerly glacia-

ted areas (Section 5.1). However, it will be difficult to separate the

145

signal due to LVZs from the background GIA signal and other contribu-

tions to present-day sea level change as e.g., present-day ice melt and

local tectonics;

• GRACE data: For regional applications, we have shown that the large

positive effect of the Iceland Plume on geoid heights in Northern Eu-

rope, as seen in the GRACE data (GGM02S), can be removed by filter-

ing all harmonics below degree 9 (Section 5.2);

• For CLVZ-induced perturbations, we do no find any resemblance with

a filtered (Clm = 0 for l < 40 and l > 80) version of GGM02S, where

amplitudes are up to 10 times as large (Section 5.2);

• GPS: In comparing predictions of 3D-rates of horizontal displacement

in Northern Europe, we have shown the importance of filtering the pre-

dictions before comparison with data to remove the effect of the former

Laurentide ice sheet, simulating the weak lithosphere along the Mid-

Atlantic Ridge (Section 5.1).

• LVZ-induced perturbations in 3D velocities are slightly larger than the

accuracy of the BIFROST network, but because GPS-measurements

are point measurements, they are not well suited to recognize the spa-

tial signatures of LVZs (Sections 5.1 and 6.5);

• Moreover, predictions of horizontal velocities from a flat 3D-stratified

earth model, which can include lateral heterogeneities and non-linear

rheology, show considerable differences with the predictions of an SP

model (up to 50%, Section 6.4);

• However, predictions of horizontal velocity perturbations due to LVZs

are accurate (error < 10%) in the near field (Section 6.4);

• Laboratory-derived creep laws and heatflow data: Creep laws provide

only a weak constraint on the rheology of the shallow earth and we

need existing GIA studies and elastic thickness studies to estimate the

most reasonable creep law (Section 7.3);

• For this we estimate, based on the Maxwell time, a viscosity of 1023 Pas

for which the shallow upper mantle is effectively elastic in the process

of GIA and viscosities larger than > 1025 Pas for the long-term (> 1

Myrs) strength (Section 7.3.1);

• For the shallow upper mantle, the creep law for wet olivine in the dif-

fusion creep regime with average grain size (8 mm) gives a viscosity at

200 km depth of 3·1020 Pas for low heatflow (40 mW/m2) and 6·1019 Pas

for average to high (60−80 mW/m2) heatflow, an effective lithospheric

thickness of 140 km for low heatflow and 60 km for average heatflow,

and a long-term thickness smaller than 110 km for low heatflow and


smaller than 30−40 km, depending on the strength of the lower crust,

for average heatflow (Section 7.3.2);

• For the crust we use wet plagioclase feldspars, with albite in general

the weakest and anorthite the strongest. Viscosities for average grain-

size (100 µm) and average to high heatflow can be as low as 1019 Pas,

and even for low heatflow the lower crust seems to show some ductile

behavior (Section 7.3.3).

Summarizing, GOCE is predicted to add especially information on the rheology

of the crust, due to the specific high-harmonic spatial and spectral signature of

the induced gravity signal. The signatures are largely insensitive to uncertainties

in the background earth stratification, as these are confined to low-degree har-

monics, but not to uncertainties in the ice-load history. A low-viscosity astheno-

sphere induce gravity signals with comparable spatial and spectral signatures to

the lithosphere and the deeper mantle. Recovery is therefore strongly affected by

uncertainties in the background earth stratification and ice-load history.

In this thesis we have only considered errors induced by uncertainties in the back-

ground earth stratification (Section 4.5.2) and ice-load history (Section 5.2), and

recovery errors from GOCE (e.g. Sections 4.3.2 and 7.5). However, there are a

number of other processes and structures that give rise to mass anomalies to the

measured gravity field. For example, the gravity field in Northern Europe shows

a close resemblance to geoid height predictions of GIA, but with much larger am-

plitudes. This suggests that at least part of the signal is due to e.g., the anoma-

lously large depth of the Moho in Scandinavia (Section 5.2). We expect especially

unmodeled thermally- and chemically-induced shallow mass inhomogeneities to

be effective in masking the gravity signal due to CLVZs (Section 7.1). Effective

spatio-spectral filtering tools have to be developed to extract the GIA-related sig-

nal in which we expect the specific signatures of CLVZs (Section 5.4) to be useful.

As the adapted ice-load history is contaminated by the assumed earth stratifi-

cation, the thermomechanical earth model of Chapter 7 should be coupled to a

thermomechanical ice sheet model (Section 3.2 and e.g. van den Berg et al., 2008)

and estimate both the ice-load history and the shallow earth stratification. At

the same time, because geoid height predictions are sensitive to the thermally-

induced lateral heterogeneities in our model (Section 7.4.1), tighter constraints on

the temperature profile with depth are needed (from e.g. seismic data, Goes et al.,

2000). The GIA model for Northern Europe thus developed can then be used in

other studies that involve solid-earth deformation and sea-level change, as studies

of present-day sea level change (e.g. Douglas, 2001) and river-delta dynamics (e.g.

Whitehouse et al., 2007), or for other areas that, for example, experience present-

day ice mass changes as Antarctica (e.g. Kaufmann et al., 2005).

Appendix A

Crustal Low Viscosity and

Lithospheric Strength

From the k-load Love number (see Sections 2.2.1 and 4.2), used in the spectral

method (Section 2.2) to compute the incremental potential due to a surface load,

we get an interesting insight in the difference between a ’jelly-sandwich’ model (a

soft lower crust sandwiched between a strong upper crust and a strong lithospheric

mantle) and a model with a strong crust on top of a weak mantle, as discussed in

Section 7.1. The introduction of a crustal low-viscosity layer has an effect on the

strength of the lithosphere. The total strength can be defined as an integral over

the yield strength profile to a certain depth at which the yield strength becomes

lower than e.g. 1 MPa and where there are no more jumps in strength below

(Ranalli, 1995, p. 364). However, this definition does not take into account de-

coupling of the upper crust and the lithosphere in the presence of a ductile lower

crust.

This decoupling can be shown by looking at the fluid limit (i.e. for infinitely long

timescales) of the incremental potential (k-) Love number as a function of har-

monic degree (Figure A.1a). This fluid limit for a model with only an elastic upper

crust of 20 km thickness and no lithosphere (’+ 20 km’ in Figure A.1a) is equal to

that for a model with a 10 km thick crustal low-viscosity zone (CLVZ) between an

upper crust of 20 km thickness and an elastic lithosphere of 50 km thickness (’o,

20 + 50 km’) and substantially different from a model with a fully elastic crust (no

CLVZ) and a lithosphere with a total of thickness of 80 km thickness (’− 80 km’).

We have to keep in mind that in the fluid limit all viscoelastic material has relaxed,

independent of the viscosity. For shorter timescales, the decoupling depends on the

viscosity. For a timescale of 100 kyrs, we see in Figure A.1b that the response for

148 A. Crustal Low Viscosity and Lithospheric Strength

(a) fluid limit

0 50 100 150 200−1

−0.8

−0.6

−0.4

−0.2

0

harmonic degree

k−Lo

ve n

umbe

r

80 km20 + 50 km20 km

(b) 100 kyrs

0 50 100 150 200−1

−0.8

−0.6

−0.4

−0.2

0

harmonic degree

k−Lo

ve n

umbe

r

80 km20 + 50 km20 km

Figure A.1: Potential perturbation Love numbers in the fluid limit (a) and after 100 kyrs(b).

the model with a CLVZ (’o’) is for the long wavelengths comparable to the response

of a fully elastic crust and lithosphere (’−’). This can be explained by the long re-

laxation times of the low harmonics for flow confined to a channel (Sections 4.2

and 6.4). For the high harmonics we see that the pattern is closer to the model

without a lithosphere (’+’). We can also see that on this timescale there is differ-

ence between a ’jelly-sandwich’ model and a model with a strong crust and weak

shallow upper mantle, which cannot be seen in the fluid limit. The latter model

shows deep flow, in which the low harmonics relax fastest (Sections 4.2 and 6.4).

The viscosity for which decoupling is effective should be much smaller than the

viscosity for which the earth is effectively elastic (estimated to be 1023 Pas, see

Section 7.1). The viscosity of the CLVZ is 1019 Pas, i.e. four orders of magnitude

smaller than this viscosity and still the decoupling is weak. We therefore assume

that decoupling only occurs at geological timescales.

The decoupling of the crust and lithosphere in the presence of a CLVZ might ex-

plain the small elastic thicknesses found by Pérez-Gussinyé & Watts (2005) on e.g.

the southern tip of Scandinavia (Figure 7.1b in Chapter 7), which are equal to,

or smaller than, the crustal thicknesses in that area (Figure 1.2b in Chapter 1).

However, this feature can as well be explained by a strong crust on top of a weak

mantle, as on these timescales the difference in response between the two will be

very small.

Appendix B

Thermal Model

In Chapter 7 we use laboratory-derived creep laws for crustal and upper mantle

rocks, which are in general temperature-dependent. Here we describe the basic

equations to model heat conduction in the crust and the lithosphere and the choice

of parameters for heat conduction. The theory for modeling heat convection in the

asthenosphere is treated in Chapter 7 (Section 7.2.1).

We estimate the temperature in the crust and lithosphere by solving the 1D steady-

state heat conduction equation (e.g. Meissner (1986, p. 126), Turcotte & Schubert

(2002, p. 139)):

kd2T

dz2+ A = 0 (B.1)

with k the thermal conductivity and A the radioactive heat production. The sim-

plest solution to this problem is for a half-space with constant k and A (Meissner

(1986, p. 127), Turcotte & Schubert (2002, p. 139)):

T(z)= T0+q0

kz−

A

2kz2 (B.2)

where T0, q0 are the temperature and heatflow at the surface of the earth, respec-

tively. For a multilayer model see Eq. (7.1) in Section 7.2.1.

When the temperature in the mantle becomes close to the mantle solidus TM ,

heat transport is not controlled by conduction but by convection. In that case the

temperature follows an adiabatic temperature gradient, see Section 7.2.1.

To compute the temperature profile in the conductive part (Eq. B.2), we have to

make some assumptions on the variation of heat generation A and thermal con-

ductivity k with depth. We estimate heat generation A in the crust by assuming

150 B. Thermal Model

that the heat production decays exponentially with depth (Turcotte & Schubert,

2002, p. 141):

A = A0e−z/H (B.3)

where A0 is the heat production at the surface and H is a characteristic depth. We

take A0 = 2 ·10−6 W/m3 and H = 12 km, which gives values of A for the crustal

layers that agree well with values used in other studies (e.g. Kaikkonen et al.,

2000; Pasquale et al., 2001; Artemieva & Mooney, 2001). For heat production in

the mantle, which does not have a large influence on the computed geotherms,

we assume a value of 0.01 W/m3, which is also in line with the above-mentioned

studies. The values for A we use can be found in Chapter 7, Table 7.1.

There is no univocal relation for the temperature-dependence of the thermal con-

ductivity (see Clauser & Huenges, 1995), and Jaupart & Mareschal (1999) take the

thermal conductivity in the crust to be constant at k = 2.5 Wm−1K−1. Some other

studies use a temperature-dependent thermal conductivity for the crust (Kaikko-

nen et al., 2000; Pasquale et al., 2001; Artemieva & Mooney, 2001), which gives

mainly typical values for the upper crust (∼ 200C) of k ∼ 2.5 Wm−1K−1 and some-

what smaller values for the middle crust (∼ 300C, k ∼ 2.2 Wm−1K−1). Where

Pasquale et al. (2001) assume an increase of conductivity for the lower crust, we

follow Kaikkonen et al. (2000) and Artemieva & Mooney (2001), who use values

of k ∼ 2.0 Wm−1K−1. Note that the effect of realistic changes in this value have

a smaller effect on the geotherm than uncertainties in crustal heat generation.

For the upper mantle, the variation in thermal conductivity for different stud-

ies is large. Jaupart & Mareschal (1999), Kaikkonen et al. (2000) and Pasquale

et al. (2001) use temperature-dependent conductivities which for shallow mantle

temperatures (1000−1200C) are around 3.0−3.5 Wm−1K−1. Clauser & Huenges

(1995) finds a temperature-dependent conductivity with much higher values of

4−5 Wm−1K−1. Finally, Artemieva & Mooney (2001) use a constant value of 4.0

Wm−1K−1 and a value of 3.3 Wm−1K−1 for a sensitivity analysis. We assume a

constant value of k = 3.5 Wm−1K−1. The values for k we use can be found in

Table 7.1, Chapter 7.

Geotherms computed for this thesis can be found in Chapter 7, Figure 7.2b.

Appendix C

Test Results from


To better understand the effect of the different creep laws used in Chapter 7 on

GIA predictions we use an elliptical ice load with a radius of 8 (∼ 900 km) and

a height at the center of 2500 km, as described in Section 3.2 in Chapter 3. We

apply it as a Heaviside load at t = 0 kyrs and consider the results after 10 kyrs of

loading. For both the shallow upper mantle (Section C.1) and crust (Section C.2)

we will show first vertical displacement perturbations due to a certain creep law,

which are the difference with total predictions from background model STD. Model

STD has a fully elastic crust and lithosphere to 100 km depth and a mantle with

a linear rheology that is not explicitly dependent on temperature and pressure

(referred to as constant from here on, see Table 7.4).

Vertical displacements and velocities (Figure C.1a) show a deep low under the cen-

ter of the load and a small bulge just outside the ice sheet, which is representative

for the process of GIA. The predictions for vertical velocities here are the negative

of the predictions that would follow from predictions using an infinitely long glacia-

tion phase (i.e. isostatic equilibrium is reached at LGM) and removal of the total

load at LGM. Compared to realistic predictions for Northern Europe, in which the

glaciation phase is not infinitely long and the load is removed gradually, the am-

plitudes are about two times too large (compare with Figure D.1a in Appendix D).

Geoid heights consist of 2 parts: a direct part due to the (ice) mass and an indirect

part due to GIA-induced solid-earth deformation (see Eq. 2.18 in Section 2.2.1).

As a reference, we show in Figure C.1b only the indirect part, with a low of −45

m under the center of the load, because in realistic predictions (e.g. Section 7.4)

the direct effect is very small (the ice load has disappeared, so only the relatively

152 C. Test Results from Thermomechanical Models

(a) vertical displacements & velocities

−600

−500

−400

−300

−200

−100

0

100

vert

ical

dis

plac

emen

t [m

]

0 500 ice 1500 2000 2500−18

−15

−12

−9

−6

−3

0

3

vert

ical

vel

ocity

[mm

/yr]


velocity

displacement

(b) geoid heights

0 500 ice 1500 2000 2500−50

−40

−30

−20

−10

0

geoi

d he

ight

[m]


Figure C.1: Total vertical displacements and velocities (a) and geoid heights (b) computedfrom background model STD (Table 7.4), after 10 kyrs of loading.

small ocean-load contributes), and moreover, in computing perturbations the di-

rect effect of the load disappears. Note that in contrast to vertical displacements

and velocities, geoid heights are also sensitive to the movement of deeper layers,

through its dependence on the displacement of density boundaries, although the

contribution of the surface is largest by far, see Section 6.4 in Chapter 6.

C.1 Shallow Upper Mantle Perturbations

In this section we compare the effect of diffusion creep in shallow upper mantle

(below the crust to a depth of 220 km), with average grainsize (8 mm, model DIF),

and dislocation creep, with a stress exponent of3.5 (model DIS) with the effect of

a constant linear asthenospheric low-viscosity zone (ALVZ, see for all models Ta-

ble 7.4 in Chapter 7). From Figure C.2a we see that model ALVZ shows deep flow,

whereas for model DIF and average heatflow (60 mW/m2) we see a pattern that

is a combination of both channel and deep flow. The reason for this might be the

somewhat higher viscosity of the asthenosphere for model DIF (∼ 1020 Pas, Fig-

ure 7.4a, dashed curve labeled ’8’) than for model ALVZ (1019 Pas) and the smaller

thickness of the lithosphere (∼ 60 km, Figure 7.4a, dashed curve labeled ’8’) than

for an ALVZ (100 km), which amplifies the high harmonics. This combination of

both channel and deep flow is also clearly visible for model DIS, which shows how-

ever much larger amplitudes especially under the load than model DIF due to the

high strain rates (>∼ 10−15 s−1) induced upon loading, and the relatively short

time interval (10 kyrs) after which we consider the response. The patterns are

comparable to the results of e.g. Figure 13 in Wu (1992a), though with smaller

amplitudes as we consider much thinner channels. In the next section we will see

that crustal low viscosity layers show predominantly channel flow behavior.

C.1. Shallow Upper Mantle Perturbations 153

(a) vertical displacement perturbations

0 500 ice 1500 2000 2500−120

−100

−80

−60

−40

−20

0

20

40

60

vert

ical

dis

plac

emen

t per

turb

atio

n [m

]


ALVZ40 mW/m2

60 mW/m2DIF

DIS

DIF

DIS

(b) geoid height perturbations

0 500 ice 1500 2000 2500−750

−600

−450

−300

−150

0

150

300

geoi

d he

ight

per

turb

atio

n [c

m]


ALVZ40 mW/m2

60 mW/m2

DIF

DIS

DIF

DIS

Figure C.2: Vertical displacement (a) and geoid height (b) perturbations for wet olivine inthe diffusion creep regime (d = 8 mm, model DIF) and dislocation creep regime (model DIS).ALVZ denotes a model with a constant linear channel of 1019 Pas just below a lithosphere of100 km, and a thickness of 60 km. Perturbations are the difference with background modelSTD, generated with a Heaviside elliptical load with a radius of 890 km and the results areafter 10 kyrs of loading.

(a) vertical velocity perturbations

0 500 ice 1500 2000 2500−1.5

−1

−0.5

0

0.5

1

vert

ical

vel

ocity

per

turb

atio

n [m

m/y

r]


ALVZDIFDIS

(b) horizontal velocities

0 500 ice 1500 2000 2500−3.5

−3

−2.5

−2

−1.5

−1

−0.5

0

horiz

onta

l vel

ocity

[mm

/yr]


STDALVZDIFDIS

Figure C.3: Vertical velocity perturbations (a) and horizontal velocities (b) for low heatflow(40 mW/m2) and diffusion (DIF) or dislocation (DIS) creep in the shallow upper mantle.


The response for model DIF and low heatflow (40 mW/m2, Figure C.2a) is almost

opposite to the response for average heatflow, which is due to the large thickness

of the lithosphere (see Figure 7.3a and compare with Figure 6.9b) and the lack of a

low-viscosity asthenosphere. Note that this difference is specific to this load-case,

as for a realistic load-case, which involves loading and unloading, the perturba-

tions due to both a thicker lithosphere and a weaker shallow upper mantle are

positive at present (see Section 7.4.1). For the latter this is only true if deep flow is

dominating, because then the process of GIA is accelerated. For dislocation creep

in the shallow upper mantle (model DIS, Table 7.4) and low heatflow the perturba-

tions are almost zero, indicating that differences with the background model are

small at present.

From geoid height perturbations (Figure C.2b) we can clearly distinguish between,

for example, model DIF for low heatflow, model DIS and model ALVZ. This is how-

ever not the case for vertical velocity perturbations and horizontal velocities, as

shown in Figure C.3, which all largely show the same behavior. From the vertical

velocity perturbations (Figure C.3a) we see that all three models decrease the total

vertical velocity as plotted in Figure C.1a, which can for model ALVZ be explained

by the enhanced flow through the low-viscosity channel (i.e. model ALVZ is closer

to isostatic equilibrium than model STD) and for model DIF by the thicker litho-

sphere which dampens the high harmonics. The similarity in the patterns for the

ALVZ, DIF and DIS models is related to the non-uniquness of an inversion for

both lithospheric thickness and upper mantle viscosity as discussed in Section 7.1,

i.e. a weak shallow upper mantle gives comparable predictions to a thick litho-

sphere. Perturbations for model DIS can be explained by the stress-dependence

of the viscosity, which leads to a faster initial adjustment to isostasy and a slower

adjustment after 10 kyrs of loading (Wu, 1992a), thus decreasing the vertical ve-

locities. Horizontal velocities (Figure C.3b) show an inward pattern, due to the

strong negative total vertical velocity under the load, and peak where the gradi-

ent in vertical velocity is largest. Horizontal velocities due to model ALVZ, DIF

and DIS are smaller, either due to a lower viscosity in the shallow upper mantle

(ALVZ), a thicker lithosphere (DIF) or stress-dependence of the viscosity (DIS).

C.2 Crustal Perturbations

In this section we show the effect of diffusion and dislocation creep in the crust

for different heatflow values, using labradorite (An60, Table 7.3), which is stronger

than anorthite (An100) and somewhat weaker than albite (Ab100) (Section 7.3.3).

In Figure C.4a we show vertical displacement perturbations in the diffusion creep

regime (d = 100 µm) for different heatflow values. We have computed perturba-

tions using the appropriate crustal thickness for each heatflow, see Sections 7.2.1

C.2. Crustal Perturbations 155

(a) diffusion creep

0 500 ice 1500 2000 2500−20

−15

−10

−5

0

5

10

15

20

vert

ical

dis

plac

emen

t per

turb

atio

n [m

]


CLVZ40 mW/m2

60 mW/m2

80 mW/m2

(b) dislocation creep

0 500 ice 1500 2000 2500−20

−15

−10

−5

0

5

10

15

20

vert

ical

dis

plac

emen

t per

turb

atio

n [m

]


CLVZ40 mW/m2

60 mW/m2

80 mW/m2

Figure C.4: Vertical displacement perturbations for labradorite (An60) in the diffusion creepregime (d = 100 µm, a) and dislocation creep regime (b), for different heatflow values.CLVZ denotes a model with a constant linear channel of 1019 Pas at 20 km depth, and athickness of 10 km. Perturbations are the difference with background model STD.

and 7.3.3. As a reference, we have also included perturbations due to a constant

linear crustal low-viscosity zone (CLVZ) of 1019 Pas, at a depth of 20 km and with

a thickness of 10 km (Table 7.4). Perturbations are computed by subtracting back-

ground model STD. All perturbations show channel flow behavior, in which de-

formation occurs first at the edge of the load, creating a perturbative depression

and bulge there. The deformation under the center of the load is initially small,

because the channel looks infinitely long for short wavelengths, but then starts

to show tilting of the upper crust. The largest perturbations are for a heatflow of

60 mW/m2, which does not generate the lowest viscosities, but, due to the thicker

crust, has the thickest channel (compare with Figure 7.5a). The response for 80

mW/m2 is very similar to the response of the constant linear CLVZ, which was to be

expected, as a channel of 10 km thick at 1019 Pas is generated for this heatflow (see

’An60’ in Figure 7.5a). The response for 40 mW/m2 is small, though not zero, which

indicates that even shield areas can in principle show lower crustal flow when the

assumption on composition (plagioclase feldspars) is valid. For dislocation creep

(Figure C.4b), results are comparable to diffusion creep, which indicates that aver-

age strain rates are higher than 3·10−15 s−1 (compare with Figure 7.6b), though an

alternative explanation for the relatively large response is that for non-linear rhe-

ology, the channel flow behavior is emphasized by the stress-induced low viscosity

near the edge of the ice load. As for diffusion creep (Figure C.4a), the response is

comparable to the response of the linear CLVZ for 80 mW/m2 (Figure C.4b), which

shows that the effect of non-linearity is small for predictions after 10 kyrs of load-

ing, at least for this load-case. Note also that from the results for both diffusion

and dislocation creep we can conclude that viscosities lower than 1019 Pas in the

lower crust are not realistic in most areas, because heatflow values larger than 80


(a) geoid height perturbations

0 500 ice 1500 2000 2500−50

−40

−30

−20

−10

0

10

20

30ge

oid

heig

ht p

ertu

rbat

ion

[cm

]


CLVZdiffusiondislocation

(b) horizontal velocities

0 500 ice 1500 2000 2500−3.5

−3

−2.5

−2

−1.5

−1

−0.5

0

horiz

onta

l vel

ocity

[mm

/yr]


STDCLVZdiffusiondislocation

Figure C.5: Geoid height perturbations (a) and horizontal velocities (b) for An60 and averageheatflow (60 mW/m2). In (a), perturbations are the difference with background model STD.

mW/m2 would be required in continental areas, or crustal thicknesses larger than

40 km with average to high heatflow (compare with Figure 1.2in Chapter 1).

For average heatflow, geoid height perturbations for both diffusion and dislocation

creep show a comparable response as for model CLVZ, with amplitudes of 20 to

40 cm (Figure C.5b), which is small compared to the total geoid height (∼ 10 m,

Figure C.1b), but large compared to the expected performance of GOCE (∼ 1−2 cm

for wavelengths longer than ∼ 100−200 km). The pattern for a CLVZ and diffusion

creep is very similar, but for dislocation creep wavelengths are somewhat shorter,

probably due to the stress-dependency of the viscosity. The amplitudes confirm

that for 60 mW/m2, diffusion creep shows somewhat lower viscosities than 1019

Pas and dislocation creep shows somewhat higher viscosities. Horizontal veloci-

ties due to low viscosity in the crust (Figure C.5b) are significantly smaller than

for the background model, though differences between the models (CLVZ, diffusion

creep, dislocation creep) are small. Note that the pattern is largely the same as for

horizontal velocities for models ALVZ, DIF and DIS (Figure C.3b), which means

that horizontal velocities cannot discriminate between the two. For vertical veloc-

ities the opposite is true: Where shallow upper mantle-induced perturbations are

positive under the load and negative outside the load (Figure C.3a), the crustal-

induced perturbations show a similar pattern as the vertical displacement pertur-

bations of Figure C.4, i.e. a negative perturbation under the load and a positive

perturbation outside the load. Amplitudes of the perturbations are comparable to

perturbations in the shallow upper mantle (0.5−1.0 mm/yr).

Appendix D

Surface Velocities from


As an additional future constraint on the rheology of the shallow earth, apart from

high-resolution satellite-gravity data, we consider the use of surface velocities as

for example measured by the BIFROST network (Milne et al., 2001), which has

an accuracy of 0.8−1.3 mm/yr in the vertical direction and 0.2−0.4 mm/yr in the

horizontal direction (Johansson et al., 2002). We will see that perturbations due to

shallow low-viscosity structures as predicted from a thermomechanical model (as

in Chapter 7) are above this accuracy. The relatively low spatial resolution, how-

ever, does not allow for the use of spatial signatures as in Chapter 7, Section 7.4.

For horizontal velocities we do not compute perturbations, but plot velocities of the

background model and the perturbed model in the same figure, because we think

that this more clearly shows differences, especially for the vector predictions given

below.

Total present-day surface velocities predicted from model STD (RSES ice-load his-

tory, Figure D.1a) are comparable to the results of Steffen et al. (2007), who use

a similar earth stratification and load history. Differences with the predictions of

Milne et al. (2001) can be largely explained by the effect of the former Laurentide

ice sheet, which is not present in our predictions. This is the main reason that

our model predicts significant southward motion, which is also partly present in

the BIFROST data, probably because of a weak zone along the Atlantic Ridge, see

Section 5.1 in Chapter 5. Note that the predicted amplitudes are too large because,

as shown in Chapter 6, total horizontal velocities predicted from a flat model are

not very accurate and exaggerate the motion near the ice load. In Figure D.1b

we show vertical velocity perturbations (STDi-STD) and horizontal velocities due

158 D. Surface Velocities from Thermomechanical Models

(a) STD

−1500

−750

0

750

1500

[km

]

−1500 −750 0 750 1500[km]

1 mm/yr

−2.5 0.0 2.5 5.0 7.5 10.0vertical velocity [mm/yr]

(b) STDi-STD

−1500

−750

0

750

1500

[km

]

−1500 −750 0 750 1500[km]

1 mm/yr

−1.2 0.0 1.2 2.4 3.6vertical velocity perturbation [mm/yr]

Figure D.1: Total surface velocities as predicted by earth model STD (RSES ice-load history,a), and horizontal velocities and vertical velocity perturbations due to different ice-loadhistories (b). In (b), white arrows are predictions from ICE-5G (STDi) and black arrowsfrom RSES (STD, same as in a). Perturbations are computed by subtracting predictionsgenerated with RSES from predictions generated with ICE-5G (STDi-STD). These figuresaccompany the geoid height predictions of Figure 7.8 in Chapter 7.

159

to different ice-load histories, with white vectors for ICE-5G and black vectors for

RSES. The effect is largest in the Barents Sea area, where ICE-5G assumes a sub-

stantially larger ice mass than RSES (see Figure 7.7 in Chapter 7). As the vertical

velocity in the center is larger, horizontal velocities due to ICE-5G (white) show an

additional outward pattern, increasing the horizontal velocities as predicted from

RSES (black).

The ALVZ-induced horizontal velocities and vertical velocity perturbations (Fig-

ure D.2a) mainly show that the amplitudes of surface velocities, as predicted from

background model STD, are decreased, as already expected from Appendix C.1 for

a simple load-case. Note that the sign of the surface velocities is opposite as pre-

dicted there, because here we consider loading and subsequent unloading of the

earth. As for geoid height perturbations (Figure 7.9b in Chapter 7), there is for

model DIS a shift in vertical velocity perturbations to the southeast (Figure D.2b),

which results in changes in the northeast and southwest directions of the horizon-

tal velocities.

From Figure D.2c, in which we plot horizontal velocities and vertical velocity

perturbations for a laterally homogeneous CLVZ, we see that the effect on hor-

izontal velocities is similar for a CLVZ and an ALVZ, as already explained in

Appendix C.2. For albite in the lower crust and DIS as a background model

(Ab100(DIS)), both vertical velocity perturbations and differences in horizontal ve-

locities with the background model are small (Figure D.2d), which indicates that

the much larger effect of a laterally homogeneous CLVZ as shown in Figure D.2c

is not very realistic as it does not consider lateral heterogeneity.

160 D. Surface Velocities from Thermomechanical Models

(a) ALVZ-STD

−1500

−750

0

750

1500

[km

]

−1500 −750 0 750 1500[km]

1 mm/yr

−1.0 −0.5 0.0 0.5vertical velocity perturbation [mm/yr]

(b) DIS-STD

−1500

−750

0

750

1500

[km

]

−1500 −750 0 750 1500[km]

1 mm/yr

−2.4 −1.6 −0.8 0.0 0.8vertical velocity perturbation [mm/yr]

(c) CLVZ-STD

−1500

−750

0

750

1500

[km

]

−1500 −750 0 750 1500[km]

1 mm/yr

−0.5 0.0 0.5 1.0vertical velocity perturbation [mm/yr]

(d) Ab100(DIS)-DIS

−1500

−750

0

750

1500

[km

]

−1500 −750 0 750 1500[km]

1 mm/yr

−1.0 −0.5 0.0 0.5vertical velocity perturbation [mm/yr]

Figure D.2: Horizontal velocities and vertical velocity perturbations for a laterally homo-geneous ALVZ (a), for model DIS (wet olivine in the dislocation creep regime, Table 7.4,b), a laterally homogeneous CLVZ (c), and for albite (Ab100) in the diffusion creep regime(d = 100 µm) with model DIS as a background model (Ab100(DIS), d). White arrows arepredictions from a model with an ALVZ (a), from model DIS (b), a model with an CLVZ(c), or from Ab100(DIS) (d). Black arrows are prediction from from model STD (a-c, asin D.1a), or model DIS (d, as in b). Perturbations are the difference with predictions frommodel STD (a-c) or model DIS (d). Figures (a, b) accompany the geoid height predictionsof Figure 7.9 and figures (c, d) of Figure 7.11 in Chapter 7.

Appendix E

"Recovery of the gravity field

signal due to a low viscosity

crustal layer in GIA models from

simulated GOCE data"

In this appendix we include a paper that was presented at the Second Interna-

tional GOCE User Workshop, 8-10 March 2004, ESA-ESRIN, Frascati:

Schotman, H.H.A., Visser, P.N.A.M., Vermeersen, L.L.A. & Koop, R., 2004. Recov-

ery of the gravity field signal due to a low viscosity crustal layer in GIA models

from simulated GOCE data, in GOCE, the Geoid and Oceanography, Proceedings

of the Second International GOCE User Workshop, 8-10 March 2004, ESA-ESRIN,

Frascati (ESA SP-569), accessible at http://earth.esa.int/workshops/goce04/ partic-

ipants/31/paper_frascati_paper3.pdf

As this paper is not peer-reviewed, we have not included it in the Bibliography or

the List of Publications. However, we think it is of interest as a reference, because

here we investigate the effect of uncertainties in the correction for the amount of

isostatic compensation of topography (as mentioned in Section 7.1) and because we

show some geoid height predictions from thermomechanical ice sheet models (see

Section 3.2). We have not attempted to update this paper, which means that, for

example, the scales of the figures, which were originally in color, are not suitable

for grayscale and that the models and synthetic data used are not necessarily in

line with the description in Chapter 3.

162 E. Paper from "GOCE, the Geoid and Oceanography"

163


165


167

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Summary

Shallow-Earth Rheology from Glacial Isostasy and Satellite Gravity:

a sensitivity analysis for GOCE

In the past few years, the CHAMP and GRACE satellite missions have probed

the earth’s long- to medium-wavelength (> 500 km) gravity field. ESA’s upcom-

ing Gravity field and steady-state Ocean Circulation Explorer (GOCE) mission is

predicted to measure geoid heights with centimeter accuracy at spatial scales of

100 km. Such a high-resolution gravity field contains information on mass inhomo-

geneities in the shallow earth, which can be partly due to the creep of low-viscosity

material in response to the growth and decay of ice sheets during the last glacial

cycle. In this glacial-isostatic adjustment (GIA) process, crustal low-viscosity zones

(CLVZs) can for example introduce perturbations in the geoid height up to several

decimeters with spatial scales down to hundred kilometers. In this thesis we in-

vestigate the sensitivity of GOCE to these perturbations and to perturbations in-

troduced by asthenospheric low-viscosity zones (ALVZs). We test the sensitivity to

properties of the LVZ and of the background model, and try to answer the question

which unique information GOCE can provide on the rheology of the shallow earth.

Since the general acceptance of the plate-tectonics hypothesis in the 1960s, the

solid earth is thought to consist of a strong outer shell, the lithosphere, divided

in a number of plates, on top of a weaker substratum, the asthenosphere. The

lithosphere consists of the crust and the strong part of the mantle, which differ

chemically and are seismically separated by the Mohorovicic discontinuity (Moho).

The average viscosity of the whole upper mantle below the lithosphere to a depth

of 670 km is generally estimated to be about 1020−1021 Pas, whereas the astheno-

sphere can have viscosities that are more than an order of magnitude smaller.

The crust is mainly strong, but parts of the lower crust might be ductile and have

low-viscosity (< 1020 Pas). Both CLVZs and ALVZs can be expected in continental

regions with relatively high heatflow (> 60 mW/m2), whereas ALVZs can also be

expected below oceanic lithosphere.

Estimates of the viscosity of the mantle are mainly provided by studies of GIA,

180 Summary

the geophysical process of solid-earth deformation due to changes in continental

ice masses and resulting changes in ocean load. These studies focus on the last

glacial cycle, which started about 120 kyrs BP and showed a maximum continen-

tal ice mass at about 21 kyrs BP. The largest ice masses could be found on the

northern hemisphere, especially centered over the Hudson Bay area in Northern

America and the Gulf of Bothnia in Northern Europe. Information about the rhe-

ology of the earth and the volumes of former ice sheets found from these studies

are used, for example, to estimate the contribution of GIA to present-day sea-level

change. Current constraints on the GIA process are mainly provided by relative

sea-level data and, in Northern Europe, by GPS measurements, which have a lack

of resolving power in the shallow earth (down to ∼ 200 km).

In this thesis we show that the spectral amplitudes of geoid height perturbations

due to a CLVZ peak betwee harmonic degrees 40 and 80, which corresponds to a

spatial resolution between ∼ 500 and ∼ 250 km. The amplitudes remain larger

than the expected GOCE performance up to degree 120−140 (∼ 150 km). As the

amplitudes are only larger than the performance of the GRACE satellite grav-

ity mission up to degree 70− 80 (∼ 250 km), we predict that GOCE can provide

unique information on especially short-scale features (< 250 km). The sensitivity

of GOCE to an order of magnitude variation in the properties of the CLVZ is small,

which indicates that constraining the properties of CLVZs from GOCE is difficult.

It also indicates that we are likely to find geoid height patches that are robust to

variations in the properties of the CLVZ and that are thus more likely to be de-

tectable by GOCE. We find for example patches that are robust to variations in the

viscosity and thickness of the CLVZ along the Norwegian coast.

We predict that GOCE is sensitive to uncertainties in the earth stratification and

ice-load history of the background model, though also here we find patches, around

the formerly glaciated areas, that are robust to realistic variations in upper mantle

viscosity and the ice-load history. The sensitivity to uncertainties in the ice-load

history is above the predicted GOCE performance up to degree 140 in the presence

of a CLVZ, which means that GOCE in principle could provide information on the

ice-load history if our knowledge of the earth model is sufficient. To extract infor-

mation about a CLVZ, we have to largely remove the uncertainty in the ice-load

history, which is to some extent possible using degree correlations. Uncertainties

in the ice-load history prevent the recovery by GOCE of spherical harmonic coef-

ficients below degree 40−60. Perturbations due to an ALVZ are above the GOCE

and GRACE performance up to degree 60, but as an ALVZ induces gravity sig-

nals with signatures that are comparable to those induced by the deeper mantle,

recovery is strongly affected by uncertainties in the background model.

As the semi-analytical normal-mode model of the solid earth that is used in the

first part of our research cannot handle large lateral viscosity contrasts, we have

181

developed a finite-element model based on ABAQUS. The model has a flat geom-

etry and geoid heights are computed from the predicted displacements at density

boundaries by solving Laplace’s equation in the Fourier-transformed domain. We

show that geoid height and surface velocity perturbations due to an LVZ can be

computed accurately, at least in the near field. The model is forced with a realistic

ice-load history and a eustatic ocean-load history. The errors introduced by using

a eustatic instead of realistic ocean-load history are generally smaller than 10%,

but might be critical for perturbations due to CLVZs.

As an additional dataset to constrain the rheology of the shallow earth, we have

considered the combination of laboratory-derived creep laws and surface heatflow

data. Using plagioclase feldspars for the crust and olivine for the shallow upper

mantle, geoid height perturbations due to shallow low-viscosity structures are in-

duced that are one to two orders of magnitude larger than the predicted accuracy

of GOCE for scales of a few hundred kilometers. GOCE is predicted to be sensitive

to the creep regime in the lower crust, but not to the composition, at least not for

the plagioclase feldspars used here. To constrain the rheology of the shallow earth,

tighter constraints on the temperature profile with depth and more geological field

data on especially composition are needed.

We expect that GOCE can make an important contribution to the understanding

of the rheology of the shallow earth and the small-scale ice-load distribution dur-

ing the last glacial cycle. To achieve this, effective spatio-spectral filtering tools

have to be developed to separate the GIA-related signal from other shallow mass

inhomogeneities, in which we expect the specific signatures of CLVZs to be use-

ful. Furthermore, we recommend a coupling of the thermomechanical earth model

described in this thesis to a thermomechanical ice sheet model and the use of

geological and heatflow data. The information about the shallow earth and ice-

load history thus obtained can for example be used to improve estimates of the

contribution of GIA to present-day sea-level change. Finally, we expect that the

developed thermomechanical earth model can also be used for other phenomena

as relative sea-level change due to present-day ice mass changes.

Hugo Schotman

Samenvatting

Rheologie van de Ondiepe Aarde uit Postglaciale Opheffing en Satelliet-

Zwaartekrachtwaarnemingen: een gevoeligheidsanalyse voor GOCE

De afgelopen jaren hebben de satellietmissies CHAMP en GRACE de lange en

middellange ruimtelijke golflengtes van het aardse zwaartekrachtveld in kaart

gebracht. De toekomstige ESA Gravity field and steady-state Ocean Circulation

Explorer (GOCE) missie is ontworpen om centimeternauwkeurige geoidehoogten

te meten met een resolutie van 100 km. Deze metingen bevatten informatie over

massa-inhomogeniteiten in de ondiepe aarde, die gedeeltelijk veroorzaakt kun-

nen worden door de kruip van laag-visceus materiaal. In dit postglaciale op-

heffingsproces kunnen laag-visceuze zones (LVZs) in de korst bijvoorbeeld geoide-

hoogteverstoringen met amplitudes van enkele decimeters en met ruimtelijke scha-

len van 100 km introduceren. In dit proefschrift onderzoeken wij de gevoeligheid

van GOCE voor deze geoidehoogteverstoringen en voor verstoringen door LVZs in

de asthenosfeer. We testen de gevoeligheid voor variaties in de eigenschappen van

LVZs en van het achtergrondmodel, en we proberen een antwoord te vinden op de

vraag welke unieke informatie GOCE kan verschaffen over de ondiepe aarde.

Sinds de algemene acceptatie van de platentectoniek-hypothese in de jaren 60

wordt de ondiepe aarde gezien als een combinatie van een sterke buitenste laag,

de lithosfeer, verdeeld in een aantal tectonische platen, boven een zwakkere laag,

de asthenosfeer. De lithosfeer bestaat uit de korst en het sterke gedeelte van de

mantel, die chemisch verschillen en seismisch gescheiden zijn door de Mohorovicic-

discontinuïteit (Moho). De asthenosfeer kan viscositeiten hebben die meer dan

een orde kleiner zijn dan de gemiddelde viscositeit (1020−1021 Pas) van de gehele

bovenmantel. De korst is over het algemeen sterk, maar delen van de onderkorst

kunnen ductiel zijn en een lage viscositeit hebben (< 1020 Pas). LVZs zijn met een

zekere mate van waarschijnlijkheid aanwezig in continentale gebieden met een re-

latief grote aardwarmtestroom (> 60 mW/m2), terwijl onder oceanische lithosfeer

waarschijnlijk LVZs in de asthenosfeer aanwezig zijn.

This section contains a translation of the Summary in Dutch

184 Samenvatting

Schattingen van de mantelviscositeit komen met name uit studies naar postglacia-

le opheffing, het geofyische proces waarin de vaste aarde vervormt ten gevolge van

veranderingen in de continentale ijsbelasting en resulterende veranderingen in de

oceaanbelasting. Deze studies concentreren zich op de laatste glaciale cyclus, die

120.000 jaar geleden begon en 21.000 jaar geleden een maximale ijsbelasting had.

De grootste ijskappen waren gecentreerd op de Hudsonbaai in Canada en de Bot-

nische Golf in Noord-Europa. Informatie uit deze studies over de rheologie van de

aarde en de volumes van voormalige ijskappen kan, bijvoorbeeld, gebruikt worden

om het aandeel van postglaciale opheffing in hedendaagse zeespiegelverandering

te schatten. De belangrijkste gegevens voor het postglaciale opheffingsproces, re-

latieve zeespiegelveranderingen en, in Noord-Europa, GPS-metingen, hebben een

laag oplossend vermogen in de ondiepe aarde (tot een diepte van ∼ 200 km).

In dit proefschrift laten we zien dat de spectrale amplitudes van geoidehoogtever-

storingen ten gevolge van LVZs in de korst pieken tussen de harmonische graden

40 en 80, wat overeenkomt met ruimtelijke resoluties van ∼ 500 tot ∼ 250 km.

De amplitudes blijven groter dan de verwachte nauwkeurigheid van GOCE tot

graad 120− 140 (∼ 150 km). Omdat de amplitudes slechts groter zijn dan de

nauwkeurigheid van GRACE tot graad 70− 80 (∼ 250 km), verwachten wij dat

GOCE met name nieuwe informatie kan verschaffen over kleinschalige (< 250 km)

eigenschappen. De gevoeligheid van GOCE voor een ordegrootte variatie in de

eigenschappen van een LVZ in de korst is klein, wat betekent dat het moeilijk zal

zijn deze eigenschappen te bepalen. Het betekent ook dat er gebieden zijn waarin

de geoidehoogteverstoringen robuust zijn voor variaties, zoals voor de kust van

Noorwegen voor variaties in de viscositeit en dikte van de LVZ.

Wij voorspellen dat GOCE gevoelig is voor onzekerheden in de aardstratificatie en

ijsbelastingsgeschiedenis van het achtergrondmodel, hoewel we ook hier gebieden

vinden die robuust zijn voor realistische variaties in de ijsbelasting en viscositeit

van de bovenmantel. In het bijzijn van een LVZ in de korst is de gevoeligheid

voor onzekerheden in de ijsbelasting groter dan de nauwkeurigheid van GOCE tot

graad 140. Dit betekent dat GOCE in principe informatie kan verschaffen over de

voormalige ijsbelasting als onze kennis van de vaste aarde voldoende is. Om infor-

matie over een LVZ in de korst te kunnen krijgen moeten we een groot deel van de

onzekerheid in de ijsbelasting verwijderen, wat met behulp van graadcorrelaties

gedeeltelijk mogelijk is. De onzekerheid in de voormalige ijsbelasting zorgt ervoor

dat GOCE niet de sferische harmonische coëfficiënten onder graad 40− 60 kan

bepalen. Verstoringen ten gevolge van een LVZ in de asthenosfeer zijn groter dan

de nauwkeurigheid van GOCE en GRACE tot graad 60, maar omdat deze LVZ

zwaartekrachtsignalen introduceert met vergelijkbare signaturen als de diepere

mantel, is het moeilijk het signaal te scheiden van het signaal ten gevolge van

onzekerheden in het achtergrondmodel.

185

Het semi-analystische normal-mode model van de vaste aarde, dat gebruikt is in

het eerste deel van ons onderzoek, mag geen grote laterale viscositeitscontrasten

bevatten en daarom gebruiken we in het tweede deel een eindig-elementen model

gebaseerd op ABAQUS. Dit model heeft een platte geometrie en geoidehoogten

worden berekend uit de voorspelde verplaatsingen van dichtheidscontrasten door

het oplossen van Laplace’s vergelijking in het Fouriergetransformeerde domein.

We laten zien dat geoidehoogte- en korstsnelheidverstoringen ten gevolge van een

LVZ nauwkerig berekend kunnen worden, in ieder geval in het nabije veld. We

laten zien dat fouten door het gebruik van een geografisch uniforme oceaanbelas-

ting in plaats van een realistische oceaanbelasting in het algemeen kleiner zijn

dan 10%, maar kritisch kunnen zijn voor verstoringen ten gevolge van LVZs in de

korst.

Als aanvullende gegevens om de rheologie van de ondiepe aarde te bepalen ge-

bruiken we vloeiwetten en aardwarmtestroomgegevens. Als we veldspaten (pla-

gioklazen) voor de korst kiezen en olivijn voor de ondiepe bovenmantel, dan zijn

de verstoringen door deze structuur in de ondiepe aarde één tot twee ordes groter

dan de verwachte nauwkeurigheid van GOCE voor ruimtelijke schalen van enkele

honderden kilometers. Wij voorspellen dat, als de onderkorst bestaat uit plagio-

klazen, GOCE gevoelig is voor het kruipgedrag in de onderkorst, maar niet voor

de compositie. Meer geologische veldgegevens en betere schattingen van het tem-

peratuurprofiel in de aarde zijn nodig om de rheologie van de ondiepe aarde te

bepalen.

Wij verwachten dat GOCE een belangrijke bijdrage kan leveren aan onze ken-

nis van de rheologie van de ondiepe aarde en de korte ruimtelijke golflengten

van de voormalige ijsbelasting. Hiertoe dienen effectieve filtergereedschappen on-

twikkeld te worden om de zwaartekrachtverstoringen ten gevolge van het post-

glaciale opheffingsproces te kunnen onderscheiden van andere ondiepe inhomoge-

niteiten, waarbij wij verwachten dat de specifieke signaturen van LVZs een belang-

rijke bijdrage leveren. Daarnaast doen wij de aanbeveling het door ons ontwikkel-

de thermomechanisch aardmodel te koppelen aan een thermomechanisch ijsmodel

en naast GOCE metingen ook aardwarmtestroommetingen en geologische veldge-

gevens te gebruiken. De informatie over de ondiepe aarde en voormalige ijsbelas-

ting die zo verkregen wordt, kan dan bijvoorbeeld gebruikt worden om schattingen

van de bijdrage van postglaciale opheffing aan hedendaagse zeespiegelverander-

ing te verbeteren. Ten slotte verwachten wij dat het ontwikkelde model gebruikt

kan worden voor andere fenomenen als bijvoorbeeld relatieve zeespiegelverander-

ing ten gevolge van hedendaagse ijsafsmelting.

Hugo Schotman

List of Publications

Schotman, H.H.A., Vermeersen, L.L.A., Wu, P., Drury, M. & de Bresser, H., 2008

Constraints of Future GOCE Data on Thermomechanical Models of the Shallow

Earth: A Sensitivity Study for Northern Europe, Geophys. J. Int., submitted.

Schotman, H.H.A., Wu, P. & Vermeersen, L.L.A., 2008

Regional Perturbations in a Global Background Model of Glacial Isostasy, Phys.

Earth Planet. Inter., in press, 10.1016/j.pepi.2008.02.010.

Schotman, H.H.A., Vermeersen, L.L.A. & Visser, P.N.A.M., 2007

High-Harmonic Gravity Signatures Related to Postglacial Rebound, in Dynamic

Planet, International Association of Geodesy Symposia 130, pp. 103–111, eds P.

Tregoning and C. Rizos, Springer, Heidelberg.

Schotman, H.H.A. & Vermeersen, L.L.A., 2005

Sensitivity of glacial isostatic adjustment models with shallow low-viscosity earth

layers to the ice-load history in relation to the performance of GOCE and GRACE,

Earth Planet. Sci. Lett., 236, 10.1016/j.epsl.2005.04.008.

Vermeersen, L.L.A. & Schotman, H.H.A., 2008

Constraints on glacial isostatic adjustment from satellite gravity data, Special

Issue on "Geophysical Geodesy", eds E. Grafarend, Z. Martinec and D. Wolf, J.

Geodesy, submitted.


Shallow earth rheology from glacial isostatic adjustment constrained by GOCE,

Boll. Soc. Geol. It., 127, in press.


High-harmonic geoid signatures related to glacial isostatic adjustment and their

detectability by GOCE, J. Geodyn., in press.

Wang, H., Wu, P., Schotman, H. & Wang, Y., 2006

Validation of the coupled Laplace-finite-element method for a laterally heteroge-

neous spherical earth, Chinese Journal of Geophysics, 49, 1515–1523.

188 List of Publications

Wu, P., Wang, H. & Schotman, H., 2005

Postglacial induced surface motions, sea-levels and geoid rates on a spherical, self-

gravitating, laterally heterogeneous Earth, J. Geodyn., 39, 127–142.

van der Wal, W., Schotman, H.H.A. & Vermeersen, L.L.A., 2004

Geoid heights due to a crustal low viscosity zone in glacial isostatic adjustment

modeling; a sensitivity analysis for GOCE, Geophys. Res. Lett., 31, 10.1029/

2003GL019139.

Curriculum Vitae

Hugo Schotman was born in Steenderen on January 10, 1972. From 1984 to 1990

he attended secondary school in Zutphen (Baudartius College), after which he

moved to Enschede to study Applied Physics at the University of Twente. He

carried out his master’s research project on amorphous silicon in solar cells at

Utrecht University and graduated in 1996. In 1997 he joined Unisource, an inter-

national collaboration of KPN, and in 1998 he became a product manager at KPN

and started a master in Geodesy at Delft University of Technology. In 2000 he

moved to Dhaka, Bangladesh, where his girlfriend was posted at the Royal Nether-

lands Embassy, from where he finished his MSc thesis on satellite gravity in 2002.

He then returned to The Netherlands to start a PhD at SRON Netherlands Insti-

tute for Space Research in Utrecht, mainly doing his research at DEOS, Faculty

of Aerospace Engineering, Delft University of Technology. Since April 2008, Hugo

Schotman is employed as a senior member of staff at the Office of Energy Regula-

tion, Netherlands Competition Authority (NMa) in The Hague.

Documents

Shallow-Earth Rheology from Glacial Isostasy and Satellite ......We start with an introduction on shallow-earth rheology (Section 1.1), glacial isostasy (Section 1.2) and satellite