Crustal rheology and postseismic deformation: …Crustal rheology and postseismic deformation: modeling and application to the Apennines Proefschrift ter verkrijging van de graad van

Crustal rheology and postseismicdeformation: modeling and

application to the Apennines

Crustal rheology and postseismicdeformation: modeling and

application to the Apennines

Proefschrift

ter verkrijging van de graad van doctoraan 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 maandag 8 november 2004 om 10.30 uur

door

Riccardo Emilio Maria RIVA

dottore in Fisica, Universiteit van Milano, Italiegeboren te Milano, Italie

Dit proefschrift is goedgekeurd door de promotor:

Prof. ir. B.A.C. Ambrosius

Samenstelling promotiecommissie:

Rector Magnificus voorzitterProf. ir. B.A.C. Ambrosius, Technische Universiteit Delft, promotorProf. dr. R. Sabadini, Univesita degli Studi di Milano, ItalieProf. dr. W. Spakman, Universiteit UtrechtProf. dr. ir. C.P.A. Wapenaar, Technische Universiteit DelftProf. ir. K.F. Wakker, Technische Universiteit DelftDr. R.M.A. Govers, Universiteit UtrechtDr. L.L.A. Vermersen, Technische Universiteit Delft

Published and distributed by: DUP Science

DUP Science is an imprint ofDelft University PressP.O.Box 982600 MG DelftThe NetherlandsTelephone: +31 15 278 5678Telefax: +31 15 278 5706E-mail: [email protected]

ISBN 90-407-2547-0

Keywords: postseismic deformation, viscoelastic relaxation, normal modes

Copyright c© 2004 by R.E.M. Riva

All rights reserved. No part of the material protected by this copyright notice maybe reproduced or utilized in any form or by any means, electronic or mechanical,including photocopying, recording or by any information storage and retrievalsystem, without the prior permission of the author.

Printed in The Netherlands

To Rebecca

Contents

1 Introduction 11.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 The 1997 Umbria-Marche seismic sequence in the Central Apennines 21.3 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2 Normal modes modeling 52.1 Theory of relaxation . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.1.1 Fundamental equations . . . . . . . . . . . . . . . . . . . . . 52.1.2 Normal modes approach . . . . . . . . . . . . . . . . . . . . . 6

2.2 Approximation method for high-degree harmonics . . . . . . . . . . 112.2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.2.2 Mathematical structure of the solution . . . . . . . . . . . . . 122.2.3 Approximation of the solution . . . . . . . . . . . . . . . . . 132.2.4 Rescaling the solution . . . . . . . . . . . . . . . . . . . . . . 152.2.5 Accuracy of the approximation . . . . . . . . . . . . . . . . . 162.2.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.3 Compressible, self-gravitating relaxation . . . . . . . . . . . . . . . . 202.3.1 Spherical Bessel functions . . . . . . . . . . . . . . . . . . . . 202.3.2 Inverse matrix . . . . . . . . . . . . . . . . . . . . . . . . . . 212.3.3 Extra analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 222.3.4 Results: elastic response . . . . . . . . . . . . . . . . . . . . . 242.3.5 Results: stable relaxation . . . . . . . . . . . . . . . . . . . . 292.3.6 Incompressible vs compressible stable relaxation . . . . . . . 37

3 Finite Elements Method 413.1 Basic concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 413.2 Tecton, the Govers version . . . . . . . . . . . . . . . . . . . . . . . . 423.3 Validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

3.3.1 Internal convergence . . . . . . . . . . . . . . . . . . . . . . . 453.3.2 Comparison with analytical models . . . . . . . . . . . . . . . 48

3.4 2D vs. 3D representation . . . . . . . . . . . . . . . . . . . . . . . . 50

viii Contents

4 Fault depth, compressibility and styles of relaxation 554.1 Vertical relaxation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 564.2 Horizontal relaxation . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

4.2.1 Horizontal spheroidal component . . . . . . . . . . . . . . . . 584.2.2 Toroidal component . . . . . . . . . . . . . . . . . . . . . . . 60

4.3 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 614.4 Addenda . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

4.4.1 The double fault . . . . . . . . . . . . . . . . . . . . . . . . . 624.4.2 Semi-compressible model . . . . . . . . . . . . . . . . . . . . 63

5 Central Apennines: constraints on viscosity and fault geometry 655.1 Crustal vs. upper mantle relaxation . . . . . . . . . . . . . . . . . . 655.2 GPS campaign results . . . . . . . . . . . . . . . . . . . . . . . . . . 66

5.2.1 Yearly solutions . . . . . . . . . . . . . . . . . . . . . . . . . . 675.2.2 Rates of baseline change: network consistency . . . . . . . . . 70

5.3 Viscosity models for relaxation in the transition zone and lower crust 735.4 Channel relaxation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 765.5 Mechanical model vs. GPS results . . . . . . . . . . . . . . . . . . . 78

5.5.1 Viscosity models validation . . . . . . . . . . . . . . . . . . . 785.5.2 Model baseline variations . . . . . . . . . . . . . . . . . . . . 805.5.3 Constraints on viscosity values and fault models . . . . . . . 825.5.4 Deformation map views . . . . . . . . . . . . . . . . . . . . . 83

6 Central Apennines: effect of lateral heterogeneities 876.1 Non-linear rheologies: controls on viscosity . . . . . . . . . . . . . . 87

6.1.1 Temperature dependence . . . . . . . . . . . . . . . . . . . . 886.1.2 Effect of pre-stress . . . . . . . . . . . . . . . . . . . . . . . . 906.1.3 Conditions for a weak upper crust (or transition zone) . . . . 91

6.2 Laterally heterogeneous crustal structure . . . . . . . . . . . . . . . . 926.2.1 Realistic temperature profile . . . . . . . . . . . . . . . . . . 946.2.2 Realistic viscosity profile . . . . . . . . . . . . . . . . . . . . . 96

6.3 Model results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 976.3.1 Effect of different viscosity structures . . . . . . . . . . . . . 98

Afterword 103

Bibliography 105

Acknowledgments 111

Summary 113

Samenvatting 115

Curriculum Vitae 117

Chapter 1

Introduction

The description of the physics of the Earth relies on many different techniques,each one dealing with particular temporal and spatial scales and appropriate to thedescription of certain phenomena.

The study of postseismic deformation, i.e. the Earth’s response to the stressperturbation induced by an earthquake, focuses on very specific deformation signals:magnitude of a few millimeters per year, spatial extent of a few tens of kilometersand lasting for several years. Those characteristics are matched by recent majorimprovements in space-geodetic techniques, in particular the Global PositioningSystem (GPS), that allow the measurement of surface deformation signals at thekilometer scale with a precision of a few millimeters in the horizontal direction.

The availability of reliable measurements has opened the way to the observa-tional validation of long-existing mathematical models to describe the continuousdeformation of the Earth’s crust. The increased knowledge of the physical processes,in turn, leads to the creation of new families of numerical approaches.

1.1 Motivation

This study wants to address the compelling question about the actual possibilityof investigating the structure and rheology of the Earth’s crust by detection ofpostseismic deformation at the surface.

Rheology, which describes the relation between stress and strain, i.e. betweenforces and displacements, requires the observation of a dynamical process. It ispossible to perform laboratory experiments to simulate pressure and temperatureconditions of rocks at depth, but precious information is provided by the observationof natural phenomena.

Attention is here focused on the specific postseismic process of viscoelastic re-laxation, which is particularly indicated to investigate crustal and upper mantleregions, due to its characteristic temporal and spatial scales. In particular, crustal

2 Introduction

viscosity is still largely unknown, because only recently relaxation models and geode-tic techniques have reached the precision necessary to retrieve such information.

The motivation for this research, thus, lies in the need of bringing togetherthe newly available observational evidence and the physical interpretation, in thespecific case represented by a mathematical model. Particular attention is focusedon the determination of the crustal properties and structures that can effectivelybe recognized and on the respective resolution and accuracy.

1.2 The 1997 Umbria-Marche seismic sequence in

the Central Apennines

The 1997 Umbria-Marche sequence represents the strongest seismic activity in Italyin the last two decades and caused damage to a wide area in the Central Apennines.

The region is part of a post-collisional segment of the Mediterranean Africa-vergent mountain system: the Umbria-Marche belt formed during late Mioceneto middle Pliocene compression and has been subsequently affected by extensionaltectonics related to the opening of the Tyrrhenian Sea [Doglioni, 1991; Cinti etal., 1999]. In this high-topography zone of the central Apennines, late Quaternaryextension has been accommodated by NW-SE trending normal faults, which over-print and/or invert older structures [Cello et al., 1997]. The normal fault system,dipping to the SW, dominates the regional geomorphology and controlled the for-mation of intermontane extensional basins [D’Agostino et al., 2001; Galadini andGalli, 2000]. The area is characterized by a high level of background seismicity andby frequent sequences with moderate magnitude (Mw ≤ 5.0). Several historicalearthquakes occurred around the zone of the 1997 seismic sequence, with intensitiesranging from VIII to XI MCS [Boschi et al., 1997].

Two moderate size earthquakes struck the Colfiorito area on September 26th: alarge foreshock (Mw 5.7, 0:33 UTC) was followed by the main shock (Mw 6.0, 9:40UTC). A number of aftershocks continued into October and a third major eventtook place around Sellano on October 14th (Mw 5.6, at 15 km SE of Colfiorito).All the mentioned events, together with several aftershocks (including three withMw ≥ 5.0), had NW trending normal fault focal mechanisms and relatively shallowhypocentral depths (less than 7 km) [Amato et al., 1998]. We will focus ourattention on the main shock (M0 = 1.0− 1.2 · 1018 N m), which accounts for morethan 50% of the total moment release of the whole sequence [Ekstrom et al., 1998].

For the purpose of studying postseismic relaxation, which is the main interestof this thesis, the information about focal mechanism and energy release has to becompleted with the knowledge of fault geometry and slip distribution. There aretwo main family of datasets available to obtain a dislocation model: seismologi-cal data, consisting in records gathered at seismic stations located at any distancefrom the earthquake, or geodetic data, represented by measurements of coseismic

1.2 The 1997 Umbria-Marche seismic sequence in the Central Apennines 3

displacements obtained with space (GPS, InSAR...) or terrestrial (leveling) tech-niques.

The only seismological model available for the earthquake under study has beenobtained from the inversion of near source strong motion records and publishedby Zollo et al. [1999]. S-wave polarizations, apparent source time duration andwaveforms are used to constrain the location of the fracture origin point, the faultgeometry, the final slip distribution, size and mechanism of the event. The sourceparameters provided for the main shock describe a rupture plane that is 12 km long,dipping 38o and approximatively rupturing between 4 and 8 km depth. Maximumslip amounts to about 55 cm and is located in the southern part of the rupture.

The geodetic approach is based on the reproduction of the coseismic signal ob-served at the surface. Comparison between the geodetic measurements and a syn-thetic displacement field, commonly obtained with the analytical half-space modeldeveloped by Okada [1985], leads to the determination of the best dislocation modelparameters. The initial guess, supported by the moment tensor solution and by ge-ological information, is usually refined with the help of a minimization technique[Briole et al., 1986; Ihmle and Ruegg, 1997].

Between 1999 and 2003, a number of papers has been published with rupturemodels obtained from the inversion of geodetic data, represented by the motion ofa few GPS sites (two to six statistically significant, depending on the publication)and by three SAR interferometric images. Early solutions [Hunstad et al., 1999;Stramondo et al., 1999] where later refined in Salvi et al. [2000], publication thatwe have used as reference in the present study. The main rupture is 10 km long,dipping 45o and almost reaching the surface. Most of the slip concentrates between2.5 and 5.5 km depth, with a maximum value reaching 77 cm. A partially differentsolution was published by Lundgren and Stramondo [2002], but still confirmed thefundamentally shallow slip distribution already proposed by Salvi et al. [2000].

A last alternative approach was followed by Basili and Meghraoui [2001], whocorrected the model of Zollo et al. [1999] with data coming from leveling linesalong a deformed aqueduct located in the epicentral area. Their dislocation modelsuggests the presence of a high angle (80o) upper fault branch almost reachingthe surface and characterized by about 60% of the slip on the lower branch (thusaccommodating an extra 30% moment release).

Large differences are present between the two families of dislocation models,where the main discrepancy regards the depth at which most of the slip is concen-trated. In general, it is very difficult to provide a reliable estimate of the accuracy ofthe obtained solutions and only a few papers directly address the problem. Cattin etal. [1999] discussed the effect of surface layers on coseismic displacement and showedthat a contrast in Young’s modulus between a top layer and the half-space below iscapable of perturbing fault depth and slip up to 1 km and 25 per cent respectively.Belardinelli et al. [2003], performed a specific analysis for the Umbria-Marche caseand confirmed both the general features of the dislocation model proposed by Salviet al. [2000] and the conclusions of Cattin et al. [1999], which means that large

4 Introduction

sigmas are accompanying those results. Moreover, the use of a half-space modelneglects the effect of topography, which is another factor capable of major pertur-bations on the coseismic displacement signal as observed at the surface [Tinti andArmigliato, 2002].

As a general rule, we prefer to use the seismological dislocation model. Thischoice is mainly motivated by a wavelength factor: the seismic stations are locatedat distance from the fault that is comparable to the scale of postseismic deformation,whereas the coseismic signal reproduced by the geodetic models is only covering asmall area around the rupture. However, no definite choice can be made in advance:the impact of the two dislocation models on postseismic deformation will be objectof extended discussion in Chapter 4 and a comparison with GPS measurements willbe presented in Section 5.5.

1.3 Outline

This chapter concludes with an outline of the structure of this thesis.In the second chapter we introduce the basic formulation of the normal modes

approach for the determination of viscoelastic relaxation on a spherically layeredEarth, together with two major advances. The first is represented by an approxima-tion method for high-degree harmonics, published by Riva and Vermeersen [2002],which allows to apply the normal modes technique for a self-gravitating earth tothe study of shallow earthquakes. The second is a theoretical study on compressiblemodels, which consists in the numerical implementation and testing of the theorypublished by Vermeersen and Sabadini [1997].

In the third chapter, we present the validation of the three-dimensional FiniteElements Method (FEM) developed by Dr. Rob Govers at Utrecht University, withthe addition of a discussion about the difference between 2D and 3D models.

According to results already published by Riva et al. [2000], the quality of post-seismic deformation is largely controlled by the depth of the earthquake rupture. Inchapter four, we analyse in a systematic way how fault depth controls the deforma-tion pattern at the surface; the issue is discussed in combination with the effect ofcompressibility, because we have realized that the two problems are closely related.

The last two chapters are devoted to the study of postseismic deformation afterthe 1997 Umbria-Marche seismic sequence.

In chapter five we present and discuss the available GPS results, which are thenused to obtain information about the rupture mechanisms and the regional earthstructure, by means of comparison with predictions coming from the normal modesapproach.

In the last chapter, we use the Finite Elements Method to provide a thermo-mechanical justification of the preferred earth structure. After that, we discus theimpact on postseismic deformation of the lateral heterogeneities which are possiblypresent in the region under study.

Chapter 2

Normal modes modeling

2.1 Theory of relaxation

2.1.1 Fundamental equations

The equation of motion for a continuum in quasi-static approximation reads

∂σij∂xj

+ fi = 0 (2.1)

where σij is the stress tensor and fi the body force [Ranalli, 1995].The infinitesimal strain tensor is

εij =1

2

(∂ui∂xj

+∂uj∂xi

)(2.2)

where ui is the displacement vector.The constitutive equation for Newtonian viscosity for an isotropic and homoge-

neous body reads

σij = −p δij −2

3η θ δij + 2 η εij (2.3)

where p is the pressure, δij the Kronecker delta, η the viscosity, εij the strain rate

tensor ( ˙= ∂∂t ) and θ = εkk .

In case of power-law creep, the constitutive equation assumes the general form

εij = Aσ′(n−1)E σ′ij (2.4)

where σ′ij is the deviatoric stress, σ′E = ( 12σijσij)

1/2 is the effective stress and Ais a function of pressure, temperature and material parameters. We see that therelation between stress and strain rate is not linear (with the stress exponent n > 1)and depending on the state of stress.

6 Normal modes modeling

Further, we have the equation of continuity and the Poisson’s equation for thegravitational potential φ

∂ρ

∂t+∇ · (ρui) = 0 (2.5)

∇2φ = 4πGρ (2.6)

where ρ is the density and G is the universal gravitation constant.

2.1.2 Normal modes approach

We follow the presentation of normal mode theory as published by Sabadini et al.[1982], Piersanti et al. [1995], Vermeersen and Sabadini [1997] and Sabadini andVermeersen [1997] for a spherical, multi-layered, self-gravitating, viscoelastic earthwith linear (Maxwell) rheology. The constitutive equation is represented by (2.3).

Working in the Laplace transformed domain, the equation of motion is solvedtogether with the constitutive equation, the continuity equation, the Poisson’sequation for the gravitational potential and the strain-displacement relationship.

This leads to the solution of a set of linear differential equations which, for eachlayer i, can be written in the compact form

dyidr

= Ai yi (2.7)

where yi are the solution vectors containing deformation, stresses, potential andpotential gradient, and Ai a square matrix. The set (2.7) can be solved analyticallyfor both incompressible and compressible linear rheologies and expressed in theform

yi = Yj cj (2.8)

where Yj contains the fundamental solutions and where ci are vector constants. Thefundamental solutions are different for an incompressible or a compressible earthmodel and will be briefly discussed in the following sections.

The inverse relaxation times sj of the modes j follow as roots of the secularequation

Det[M(R1)BIc] = 0 (2.9)

in which M is Y without the rows corresponding to the deformations and thegravitational potential, Ic is a rectangular matrix containing the conditions at thecore-mantle boundary given by Eq. (63) in Sabadini et al. [1982] and B is given by

B = Y −11 (R2)

NL−1∏

i=2

Yi(Ri)Y−1i (Ri+1) (2.10)

where Ri is the external radius of the layer i and NL the total number of layers.

2.1 Theory of relaxation 7

The residues ~zj , i.e. the amplitudes of the modes j, for each of the roots sj areobtained by complex contour integration along the Bromwich path in the s-plane[Wu, 1978], which in case of surface loading result in

~zj =

N(R1)BIc · [M(R1)BIc]

†

ddsDet[M(R1)BIc]

s=sj

·~b (2.11)

where N is Y without the rows corresponding with the stresses and the potentialgradient), ~b is a vector representing the forcing term and the † symbol denotesthe (−1)i+j-transposed of the minor determinants of the matrix between squarebrackets, with i and j denoting the number of rows and columns. The elasticresponse is given as the limit for s going to infinity of the same relation (2.11),without the s−derivative.

In case of relaxation for an internal loading and extra term M(R1)Y −1(RS)has to be added at the numerator of (2.11), where RS locates the source.

Last we write the adimensionalised residues, the so-called Love numbers, in caseof surface loading

kj = −RGzj,3 ke = −1− R

Gze,3

hj = g0R

Gzj,1 he = g0

R

Gze,1

lj = g0R

Gzj,2 le = g0

R

Gze,2

(2.12)

where zj,i is the i-component of the residual vector of Eq. 2.11 and ze,i the elasticcontribution; further, k stands for the gravitational potential, h the radial displace-ment and l the horizontal one; the suffix e stands for the elastic response and j fora single relaxation mode; g0 is the adimensionalised gravity at the Earth’s surface.

Incompressible, self-gravitating

When the hypothesis of incompressibility is introduced, the continuity equationsimplifies to

∇ · u = 0 (2.13)

and the Poisson’s equation for the gravitational potential reduces to the Laplaceequation

∇2φ = 0 (2.14)

Thanks to (2.13) is it possible to decompose the displacement vector in aspheroidal and a toroidal component, which in the Laplace transformed domainread

u = uS + uT = ∇×∇×[S (r) er

]+∇×

[T (r) er

](2.15)

where S(r) and T (r) are scalar functions which can be expressed as summation ofspherical harmonics and er is the radial unit vector.


Using the zonal harmonic expansion in Legendre polynomials Pn it is possibleto write the displacement vector as

u =∞∑

n=0

[Un(r, s)Pn(cos θ)er + Vn(r, s)

∂

∂θPn(cos θ)eθ

](2.16)

The solution of equations (2.1)-(2.3) together with (2.13) and (2.14) leads tothe fundamental matrix Y , which is given in explicit form by Sabadini et al. [1982],and the inverse matrix Y −1, published by Spada et al. [1992] (print corrections inVermeersen et al. [1996a]).

The main characteristic of the fundamental solutions is that the dependence onthe radius at which solutions are evaluated (the Ri in 2.10) is in terms of powersof the radius itself. This fact is crucial for the practical evaluation of the analyticalsolution.

Compressible, self gravitating

For the general theory we refer to Vermeersen et al. [1996b]. Here we just writethe final equations implemented in the code and necessary to discuss the results,together with some corrections.

Using the zonal harmonic expansion in Legendre polynomials Pn, the divergenceof the displacement vector and the gravitational potential can be written as

∇ · u =

∞∑

n=0

χn(r, s)Pn(cos θ) (2.17)

φ =∞∑

n=0

Φn(r, s)Pn(cos θ) (2.18)

with

χn =dUndr

+2

rUn −

n(n+ 1)

rVn (2.19)

As a consequence, the Laplace transformed differential equations for conserva-tion of momentum and Poisson’s equation in the case of quasi-static deformationscan be rewritten in the Laplace domain, respectively as

[∇2r + k2(s)][∇2

r + q2(s)]χn = 0 (2.20)

∇2rΦn = −4πGρχn (2.21)

where k2(s) and q2(s) are two wave numbers.

The solutions of the fourth-order equation (2.20) are spherical Bessel func-tions of first order, regular and called jn, and of second order, irregular andcalled yn. Thus the fundamental solutions Y1,Y2,Y4 and Y5 result from χn =

2.1 Theory of relaxation 9

jn[k(s)r], jn[q(s)r], yn[k(s)r], yn[q(s)r] respectively, and their prototype is given by

− 1w2(s)rn(n+ 1)Cw(s)bn[w(s)r] + w(s)rb′n[w(s)r]

− 1w2(s)r[1 + Cw(s)]bn[w(s)r] + Cw(s)w(s)rb′n[w(s)r]

λ(s)bn[w(s)r] + 2µ(s)n(n+ 1)Cw(s)w(s)r

(1

w(s)r bn[w(s)r] − b′n[w(s)r])

+

−b′′n[w(s)r]

−µ(s)Cw(s)bn[w(s)r] + 2µ(s) 1+Cw(s)w(s)r

(1

w(s)r bn[w(s)r] − b′n[w(s)r])

+

−Cw(s)b′′n[w(s)r]

4πGρ 1w2(s)bn[w(s)r]

4πGρ[1− nCw(s)](1 + n) 1w2(s)bn[w(s)r]

(2.22)where

µ(s) =µs

s+ µ/ν, λ(s) =

λs+ κµ/ν

s+ µ/ν, b′n =

dbndz

, b′′n =d2bndz2

(2.23)

and

Cw(s) = − ρ

µ(s)· ξ

w2(s), ξ = g/r, β = λ+ 2µ, κ = λ+ 2µ/3 (2.24)

The appropriate functions for the wave number w(s) = k(s), q(s) and sphericalBessel function bn = jn, yn have to be chosen depending on the solution.

The determination of the eigenmodes involves the separation into five cases asthe arguments of the spherical Bessel functions are generally complex. Dependingon the case, namely every time that w2(s) < 0, w(s) has to be replaced by x(s) withx2(s) = −w2(s) and the spherical Bessel functions bn[w(s)r] have to be replacedby the spherical modified Bessel functions mn[x(s)r]. Further, the elements of thesame fundamental solutions which adopt x(s) are also subjected to sign change: the1st, 2nd, 5th and 6th elements change the global sign, whereas the 3rd and the 4thchange the sign in front of 2µ(s).

The five cases are as follows:

(1) − n(n+1)κµ/ν(4πGρ+ξ)2

4ξ2µ+n(n+1)β

< s < 0, k2(s) and q2(s) complex

(2) − κβµ/ν < s < − n(n+1)κµ/ν

(4πGρ+ξ)2

4ξ2µ+n(n+1)β

, k2(s) > 0 and q2(s) > 0

(3) − µ/ν < s < −κβµ/ν, k2(s) < 0 and q2(s) > 0

(4) s < −µ/ν, k2(s) > 0 and q2(s) < 0(5) s > 0, k2(s) > 0 and q2(s) < 0

(2.25)


and from these we see that Y1 and Y4 change in region 3, whereas Y2 and Y5 inregion 4 and 5. In region 1 the fundamental solutions Y1 and Y2 must be combinedto obtain the two new linearly independent solutions

Y′1 =1

2(Y1 + Y2) = Re(Y1) = Re(Y2)

Y′2 =i

2(Y2 − Y1) = Im(Y1) = −Im(Y2)

(2.26)

and the same applies to Y4 and Y5.

The solutions Y3 and Y6 result from solving χn = 0, which according to (2.17)amounts to incompressibility, leading to ∇2

rΦn = 0 which has the regular solutionrn and the irregular one 1/rn+1. Their explicit form is

Y3 =

nrn−1

rn−1

2µ(s)n(n− 1)rn−2

2µ(s)(n− 1)rn−2

−nξrn−n[(2n+ 1)ξ − 4πGρ]rn−1

and Y6 =

−n+ 1

rn+2

1

rn+2

2µ(s)(n+ 1)(n+ 2)

rn+3

−2µ(s)n+ 2

rn+3

ξn+ 1

rn+1

−4πGρn+ 1

rn+2

(2.27)

Compressible, approximated gravitation

An alternative solution to the one proposed by Piersanti et al. [1995] has been firstproposed by Pollitz [1992] and further refined for the elastic case in Pollitz [1996]and for relaxation in Pollitz [1997].

In Sabadini et al. [1982] the normal modes approach was applied to the study ofpolar wandering; later the method was used to model glacial isostatic adjustment,with a final re-formalization published by Vermeersen and Sabadini [1997].

In both cases, the normal modes technique was applied to the study of globalphenomena on long time scales (compared to the problem of postseismic relaxation),where the hypothesis of incompressibility is a second order approximation and self-gravitation an essential condition. Those fundamental characteristics were inheritedby the work of Piersanti et al. [1995] and Sabadini and Vermeersen [1997].

On the contrary, Pollitz directly focused on the problem of postseismic defor-mation, which is in general interested in the near field and short term response tothe loading. For this reason, it can be advisable to ”include compressibility, ne-glect perturbation terms proportional to gravitational constant G in the equationof quasi-static equilibrium, and assume that the product ρ0(r)g0(r) in a particular

2.2 Approximation method for high-degree harmonics 11

layer varies inversely with radius” [Pollitz, 1997]. In other words, Pollitz’ approachsacrifices self-gravitation to include compressibility. In this way, the fundamentalsolutions are still expressed in powers of the radius and allow a fast and accuratenumerical evaluation.

In addiction, Pollitz [1997] integrates (2.7) by means of the method of second-order minors [Takeuchi and Saito, 1972; Woodhouse, 1980], which allows a stablesolution up to the short wavelengths required by shallow seismic events.

2.2 Approximation method for high-degree

harmonics

This section has been published in Riva and Vermeersen [2002].

For some loading applications, the normal modes approach to the viscoelasticrelaxation of a spherical earth requires the use of spherical harmonics up to a highdegree. Examples include postseismic deformation (internal loading) and sea-levelvariations due to glacial isostatic adjustment (external loading). In the case ofpostseismic modeling, the convergence of the solution, given as a spherical har-monic expansion series, is directly dependent on loading depth and requires severalthousands of terms for shallow earthquake sources. The particular structure of theanalytical fundamental solutions used in normal mode techniques usually does notallow a straightforward calculation, since numerical problems can readily occur dueto the stiffness of the matrices used in the propagation routines. Here we show away of removing this stiffness problem by approximating the fundamental matrixsolutions, followed by a rescaling procedure, in this way we can virtually go up towhatever harmonic degree is required.

2.2.1 Introduction

When using spherical harmonics to describe earth surface deformations we have toface the problem of how many terms we should sum up in order to obtain conver-gence of the solution. Every harmonic represents a standing wave on the earth’s sur-face. Concerning pointlike seismic sources we find that this wavelength is uniquelyrelated to the source depth for the elastic response, and to the thickness of theelastic layer for (viscoelastic) relaxation. E.g., a shallow earthquake which can berepresented by sources located at one kilometer below the surface thus requires thesummation of more than 40,000 harmonics in order to get saturated convergence ofthe coseismic deformation [Pollitz, 1996].

The normal mode approach used to describe the viscoelastic relaxation of aspherical, multi-layered, self-gravitating earth with a Maxwell rheology makes useof six fundamental solutions for the spheroidal part, and two fundamental solutionsfor the toroidal part [Piersanti et al., 1995]. Due to the stiffness of the fundamental


matrices, numerical algorithms do not allow to reach degrees higher than a fewthousands in practice.

One way to cope with this problem is to use minors of the fundamental matrices[Woodhouse, 1980]. This technique is commonly used in seismic applications thatemploy numerical integration by means of propagator matrices. Here we present anew, alternative, approximation technique that can be used for analytical propaga-tor matrix models.

In this paper we will first demonstrate, by means of matrix algebra, that itis possible to obtain a first order approximation of the fundamental solutions byneglecting the irregular part of the fundamental matrix from a certain degree on-wards. We will then determine the accuracy of this approximation and describehow to apply a reliable rescaling procedure.

2.2.2 Mathematical structure of the solution

Let us now focus on the incompressible case for which the explicit form of the Ymatrix and its inverse are known.

The problem we have to deal with in calculating residuals given by (2.11), isthat the fundamental matrix Y contains three regular basis functions Rl and threeirregular R−l. Non-dimensionalised (from now on in lower case), all the radii aredivided by the average earth radius RT and are thus represented by a numberbetween zero and one, it is obvious that

rll→∞−→ 0 and r−l

l→∞−→ +∞ (2.28)

Especially for deep layers (r ' 1/2 at the CMB), due to the stiffness of thematrices, numerical evaluation of the fundamental solutions can represent a seriousproblem for high-degree harmonics: for practical purposes, then, we need to find away to rescale (2.9) and (2.11).

However, the propagation matrix technique, which is at the basis of the explicitform for B in (2.10), does not allow a simple rescaling due to the non-commutativityof the matrix algebra. The fundamental matrix is stiff and this stiffness, in turn,would require rescaling by means of a transformation like Y ′ = K Y , with K beinga 6×6 linear operator. But once we put all the Y ′ and (Y ′)−1 into (2.9) and (2.11)we have no way, at the end of the calculations, to find an inverse transformationwhich can produce the actual values for the relaxation times and their residuals.

The explicit analytical form of the layer propagator matrix Y Y −1 can be foundin Martinec and Wolf [1998]; note, however, that use of the explicit analytical formof the propagator matrix in the propagator matrix technique does not solve thestiffness problem.

However, there is an approximation method that does solve the stiffness problem,as discussed in the following section.


2.2.3 Approximation of the solution

The basic idea underlying the approximation procedure is to consider the 6 × 6matrix Y as formed by two 6× 3 matrices, one regular (YR) and one irregular (YI)

Y = [ YR YI ] (2.29)

This splitting can also be found in Sabadini et al. [1982]. Also the projectors Nand M of (2.11), evaluated at the surface, keep the same structure as Y (left partregular, right irregular).

Similarly, the inverse fundamental matrix Y −1 can be split into two parts, Y −1U

(up) and Y −1D (down), both 3× 6

Y −1 =

[Y −1U

Y −1D

](2.30)

and from the structure of the fundamental matrix and the fact that Y Y −1 = I , itis clear that Y −1

U is irregular and Y −1D regular. We finally split B into two parts

B =

[BUBD

](2.31)

and we can demonstrate that BU is irregular and BD regular, independently on thenumber of layers. It is possible to rewrite (2.10) by grouping the matrices evaluatedat the same radii and by isolating the one at the CMB:

B =

[NL−1∏

i=2

Y −1i−1(ri)Yi(ri)

]Y −1NL−1(rc) (2.32)

It is evident, then, that all couples of matrices evaluated at the same radiicompensate each rli term with an analogous r−li term: the only net dependenceon the radius is coming from the inverse fundamental matrix at the CMB. As aconsequence, the structure of B is the same as Y −1(rc), thus characterized by anirregular upper part and a regular lower part.

At this point we can perform the first product in (2.11), resulting in

NBIc = [NR NI ]

[BUBD

]Ic = NRBU Ic +NIBDIc (2.33)

In this way we have separated NBIc into the two terms NRBU Ic and NIBDIcwhich present a different behaviour, since

NRBU Icl→∞−→ +∞ ,whereas NIBDIc

l→∞−→ 0 (2.34)

The same holds for MBIc in (2.11) and we are thus allowed to write

NBIcl→∞−→ NRBUIc and MBIc

l→∞−→ MRBU Ic (2.35)


Consequentially, when we determine the products in (2.11) by taking into account(2.35), we obtain

~zjl→∞−→

NRBU Ic · (MRBU Ic)

†

ddsDet(MRBU Ic)

s=sj

·~b (2.36)

The relaxation times result from the approximated form of the secular equation(2.9):

Det(MRBUIc) = 0 (2.37)

with BU defined as:

B =

[NL−1∏

i=2

Y −1i−1,U (ri)Yi,R(ri)

]Y −1NL−1,U (rc) (2.38)

A further refinement of this last result is also possible: by splitting the matrixproduct of (2.38) into two parts, we are allowed to apply only a partial approxima-tion to (2.10), instead of the full one in (2.38):

B =

[NA−1∏

i=2

Y −1i−1(ri)Yi(ri)

NL−1∏

i=NA

Y −1i−1,U (ri)YiR(ri)

]Y −1NL−1,U (rc) (2.39)

where NA is the first boundary where approximated solutions are used. Inthis way a good compromise can be sought between approximation accuracy andnumerical stability. As a general rule, the resolution required by shallow seismicevents can be reached by approximating boundaries only within the lower mantle,typically represented by the CMB itself.

The possibility of a partial approximation becomes crucial to implement the caseof an internal mantle loading as a pointlike seismic source. A new term appears inequation (2.11) which becomes

zj =

NBIc · [MBIc]

†M(r1)Y −1(rs)ddsDet(MBIc)

s=sj

·~b (2.40)

where Y −1(rs) represents the inverse fundamental matrix evaluated at the source

depth and~b is now a 6-element vector. Another extra term appears when computingthe elastic response which requires the limit for s→ −∞ of

N(r1)Y −1(rs)−

NBIc · [MBIc]†M(r1)Y −1(rs)

Det(MBIc)

·~b (2.41)

All the arguments previously presented keep on holding and no further demonstra-tion is required when the approximation is applied below the source: the new termsonly consist of matrices evaluated at the surface and at the source depth and arethus not affected by the approximation.


On the contrary, if we force that procedure up to the surface then the twoterms in (2.41) exactly reach the same value and give no net contribution. We caneasily demonstrate this fact once we recall that keeping only the regular part of thesolution leads to deal with square matrices (the two projectors N and M and theproduct matrix BIc, all of them 3 × 3) and this allows some extra manipulations.Starting from the well known relation (2.42) (e.g., Lang [1971], chapter 32) we canin fact demonstrate that

(MBIc)†

Det(MBIc)= (MBIc)

−1 (2.42)

(MBIc)†

Det(MBIc)= (BIc)

−1M−1 (2.43)

BIc(MBIc)†M

Det(MBIc)= I (2.44)

and from this last relation we find that (2.41) becomes exactly zero. A null solutionis also obtained when the approximation is started from the surface in (2.40), butthe reason is in the way the s−derivative is discretized.

2.2.4 Rescaling the solution

We can further exploit the fact that, as we see from B in (2.32), the only netdependence of the amplitudes on the basis functions is coming from the inversefundamental matrix evaluated at the CMB.

Moreover, the two matrices NR(r1) and MR(r1) are evaluated at the surface,whose radius is normalised (r1 = 1), and we can thus avoid to take into accounttheir dependence on rl1. We will show later that, anyway, the presence of rl1 termsis not affecting the validity of our results.

For the evaluation of equations (2.36)-(2.38), we thus have to consider the prod-uct of two matrices, namely Y −1

N−1,U (rc) and Ic, whose dimensions are respectively

3× 6 and 6× 3. All the elements of Y −1N−1,U depend on r−lc , whereas only the first

column of Ic is depending on powers of the radius, due to the free-slip boundarycondition at the interface with the fluid core (see Eq. (63) in Sabadini et al. [1982]).

Performing the products we get

NRBU IcMRBU Ic

(:) Y −1

N−1,U (rc) Ic (:)

r−lc . . . r−lcr−lc . . . r−lcr−lc . . . r−lc

rlc 1 1...

......

rlc 1 1

(:)

1 r−lc r−lc1 r−lc r−lc1 r−lc r−lc

(2.45)

Something is changing when computing (MRBUIc)† since determinants enter

the definition of †; proceeding from (2.45) with the calculation we obtain

(MRBU Ic)† (:)

r−2lc r−2l

c r−2lc

r−lc r−lc r−lcr−lc r−lc r−lc

(2.46)


We are now able to compute the numerator in (2.36), which gives

NRBUIc · (MRBUIc)† (:)


r−2lc r−2l

c r−2lc

r−lc r−lc r−lcr−lc r−lc r−lc

(:)

r−2lc r−2l

c r−2lc

r−2lc r−2l

c r−2lc

r−2lc r−2l

c r−2lc

(2.47)and we see that (2.47) is homogeneously dependent on r−2l

c .On the other side, due to the properties of the determinant and the structure of

MRBUIc given in (2.45), also for the determinant (2.37) we obtain

Det(MRBU Ic) (:) Det


(:) r−2l

c + r−lcl→∞−→ r−2l

c (2.48)

and again this r−dependence is not affected by the supplementary s−derivativeentering the fluid residues formula.

It is so clear, once we use (2.47) and (2.48) to evaluate (2.36), that the depen-dence on r−2l

c compensates between the numerator and the denominator.The key point consists of the possibility to take the r±l dependence out of the

computation at the very beginning, still obtaining the same residues. This can bedone by normalizing YR by means of rl and Y −1

U by means of r−l. In this way thestiffness problem is avoided.

To be complete, if we want to take into account rl1 coming from NR and MR,we obtain a factor r3l

1 multiplying all the elements of (2.47), and this is in turncompensated by the same term r3l

1 coming out of the determinant (2.48).

2.2.5 Accuracy of the approximation

We will now study the accuracy of this approximation for the value of elastic andfluid Love numbers for an incompressible Earth in the case of surface loading. Allthe results are expressed as normalized residuals

knorm.res. =

∣∣∣∣k − kapp.

k

∣∣∣∣ (2.49)

with k the Love number without the approximation and kapp. the approximatedLove number. In the figures, a square represents the radial displacement number,a triangle the tangential displacement and a circle the gravitational ones.

The Earth model taken into consideration is characterized by five layers: anelastic lithosphere 120 km thick, a viscoelastic mantle with chemical discontinuitiesat 420 km and 660 km and, finally, an inviscid core. Values for rigidity and densityare PREM averaged [Dziewonski and Anderson, 1981] and viscosity in the mantleis fixed at 1021Pa · s.

In Figure 2.1 we show the elastic k normalized residues of the Earth, when theapproximation procedure is started at different depths. In panel (a) we take the


0.0

0.1

0.2

0.3

0.4

0.5re

scal

ed a

t CM

B

0 5 10 15

gravitational

vertical

horizontal

(a)

0.0

0.5

1.0

1.5

resc

aled

from

420

km

0 25 50 75 100

(b)

0.0

0.5

1.0

1.5

fully

res

cale

d

0 100 200 300 400Harmonic degree

(c)

Figure 2.1 Normalized residual elastic Love numbers


0.00

0.05

0.10

0.15

resc

aled

at C

MB

0 5 10 15

gravitational

vertical

horizontal

(a)

0.0

0.1

0.2

0.3

0.4

0.5

resc

aled

from

420

km

0 10 20 30 40 50Harmonic degree

(b)

Figure 2.2 Normalized residual fluid Love numbers

regular part of the solution only at the core-mantle boundary: differences betweenexact and approximated Love numbers are up to 40%, but affect only the first fewterms and are negligible for degree 10 and higher. In panel (b) we rescale all theboundaries from the 420 km discontinuity downwards (i.e. also at 660 km and theCMB): the vertical and gravitational responses are highly affected (more than 100%difference), but still converge quite rapidly to the exact solution (1% difference atdegree 50); the horizontal Love number is less affected, but convergence is slower(less than 1% at degree 70). Finally, in panel (c), full approximation is applied (B asin (2.38)): more terms are required before reaching convergence between the exactand the approximated solution (1% difference around degree 160 for the verticaland the gravitational numbers), in particular for the horizontal Love number whichremains significantly different for the first 200 terms and reaches the 1% level arounddegree 330.

In Figure 2.2 we show the effect of the approximation for the fluid response:results are not significantly different from the elastic case. When only the CMB isapproximated, as in panel (a), convergence is very fast and the maximum difference

2.3 Compressible, self-gravitating relaxation 19

is limited (only the gravitational number is above 10% for degree 2). When approx-imation is started at 420 km, as in panel (b), both vertical and gravitational Lovenumbers show a discrepancy from the exact solution smaller than 20% and again afast convergence (below 1% at degree 25), whereas a few more horizontal terms arehighly affected. This last feature, however, is due to the fact that the horizontalfluid number is changing sign around degree 10 and the normalization (2.49) is notappropriate.

The only limit we found is the impossibility to apply the approximation at aboundary with an elastic layer when the fluid response is evaluated: null residualsare obtained, in that case. This fact, however, doesn’t represent a real problem,since several thousands of terms are already reached by just rescaling the solutionat deeper boundaries.

Another important issue is the numerical computation of spherical harmonicsby use of the recursive relation for the Legendre polynomials: results up to degree60,000 have been compared after working with both double- and quadruple-precisionreals and the relation appears to be stable.

2.2.6 Conclusions

In this paper we have presented a method to remove the stiffness problem by apply-ing a successful approximation and rescaling procedure to the analytical propagatormatrix technique commonly employed in normal mode models.

It has been mathematically demonstrated that the irregular fundamental solu-tions in non-homogeneous earth models can be neglected, the weight getting smallerand smaller with increasing harmonic degree. Keeping only the regular part of thefundamental solution allows to by-pass the non-commutativity problem of the ma-trix algebra and to rescale the solution within the propagation procedure. Thus,the method presented here is an alternative to the minors-only method commonlyemployed in (seismic) numerical normal mode modeling and in Pollitz [1997].

The results we have shown give a quantitative estimate of the applicability ofthe approximation procedure. In practice, in most cases it is necessary to rescaleonly the CMB, which means that just the first few terms need the full solution inorder to obtain adequate results. However, we have also shown that even when thesolution is fully approximated, good results are obtained after a few hundreds ofterms.

This approach has been explicitly discussed for the case of an incompressible lin-ear rheology, due to the availability of a short explicit form for both the fundamentalmatrix and its inverse. This assumption is not necessary for the demonstration, asthe character of the fundamental solutions (regular/irregular) is the only informa-tion required.


2.3 Compressible, self-gravitating relaxation

The general theory of viscoelastic relaxation in a spherical and self-gravitating earthis widely discussed in Vermeersen et al. [1996b]. The essential equations and thecorrected fundamental solutions have been presented in section 2.1.2.

In this section we will discuss some issues related to the numerical realizationof the semi-analytical solution, together with a systematic analysis of results for anumber of earth models.

2.3.1 Spherical Bessel functions

The solution of the equation of motion (2.20) can be be expressed in terms ofspherical Bessel functions, which are a particular kind of Bessel function of fractionalorder which solve the differential equation

z2w′′ + 2zw′ + [z2 − n(n+ 1)]w = 0, with n = 0,±1,±2, . . . (2.50)

and can be of first kind

jn(z) =

√1

2

π

zJn+ 1

2(z) (2.51)

or of second kind

yn(z) =

√1

2

π

zYn+ 1

2(z) (2.52)

with Jν and Yν being ordinary Bessel functions [Abramowitz and Stegun, 1964]; athird kind also exists, but they are not related to our problem.

In principle, those functions and their derivatives can be easily evaluated byupward recurrence of the relations

fn+1 =2n+ 1

zfn − fn−1

d

dzfn = −n+ 1

zfn + fn−1

d2

dz2fn =

(n+ 1)(n+ 2)

z2fn −

2n+ 1

zfn−1 − fn−2

(2.53)

together with the initial terms

j0(z) =sin z

z, j1(z) =

sin z

z2− cos z

z

y0(z) = −j−1(z) = −cos z

z, y1(z) = j−2(z) = −cos z

z2− sin z

z

(2.54)

Now, when we want to use the first of (2.53) to obtain the value of (2.51)or (2.52), only for yn(z) it is numerically stable in the direction of increasing nfor z < n. This happens because a three-term linear recurrence relation has twolinearly independent solutions, but only one of them corresponds to the sequence


of functions we are trying to generate: if the other one is exponentially growing, asit happens for (2.51), then the recurrence relation is of no practical use. Luckilythe same relation turns out to be stable if used backwards, since the undesiredsolution will decrease very rapidly in that direction and this fact, together with anormalization procedure, allows to use Miller’s algorithm and get in a robust wayalso (2.51) (see Press et al. [1992], §5.4).

The method works as follows: the recurrence is started with any seed for thetwo consecutive terms fm+1 and fm, with m > n, and propagated downwards to f0

remembering to store the value of fn; since the initial guess is fm = kjm and therecurrence formula is linear, the only thing needed is a relation to obtain the valueof the constant k. This is given, in the particular case, by the explicit form for j0as in (2.54): it simply gives k = f0/j0, hence

jn =j0f0fn =

sin z

z

fnf0

(2.55)

and accuracy can be increased by choosing a larger value for m.All the same holds for the spherical modified Bessel functions, which solve

z2w′′ + 2zw′ − [z2 − n(n+ 1)]w = 0, with n = 0,±1,±2, . . . (2.56)

and are based on the hyperbolic functions sinh and cosh. Their recurrence relationis

fn+1 = −2n+ 1

zfn + fn−1 (2.57)

with the differentiation formulas in (2.53) still holding and initial terms

j(m)0 (z) =

sinh z

z, j

(m)1 (z) = − sinh z

z2+

cosh z

z

y(m)0 (z) =

cosh z

z, y

(m)1 (z) = −cosh z

z2+

sinh z

z

(2.58)

Last, in region 1 of (2.25) complex spherical Bessel functions are required, butthose are just obtained in the same way as (2.51) and (2.52), only taking care ofdefining all the variables as complex.

Here we have to point out that alternative and more elaborate ways of obtainingthe spherical Bessel functions are discussed in the literature (see for example Presset al. [1992], §6.7). Nonetheless, the proposed algorithm is already capable to reacha degree n higher than what is practically manageable in the following steps of thecomputation and it is thus fitting our necessities.

2.3.2 Inverse matrix

A crucial step in the analytical solution of this deformation model in representedby the propagator matrix technique which allows to manage a spherically symmetriclayering for the Earth and leads to the construction of matrix B according to (2.10).


In the incompressible case a compact explicit form for the inverse of the fun-damental matrix is available [Spada et al., 1992], whereas in case of compressiblemodels this has to be numerically evaluated, since the analytical inverse is onlyavailable in the form of formal terms.

Unluckily the fundamental matrix itself has the characteristic to be highly stiff,where stiffness comes from the fact that the first three fundamental solutions areregular and the other three irregular: this means that for increasing order n, fixingthe radius, the fundamental matrix gets more and more split into two parts, onegrowing and the other one going to zero. This, from a numerical point of view,leads to a difficult evaluation of the inverse which prevents from using very fastalgorithms.

Thus Gauss-Jordan elimination with full pivoting appears to be the best choice:quoting Press et al. [1992], (§2.1), ”it is straightforward, understandable, solidas a rock, and an exceptionally good psychological backup for those times thatsomething is going wrong and you think it might be your linear-equation solver”. Inany case, it proved to be faster and more accurate that a straightforward inversionby use of the Cramer’s method.

The use of pivoting, which is at the root of the high robustness of the proposedalgorithm, requires some care: in certain occasions, due to the above mentionednumerical instabilities, the fundamental matrix appears to be singular and thesearch for the pivoting element fails. In that case inversion is automaticallyskipped by the algorithm and it is then very important to also skip all the rest ofthe propagator matrix process (namely, the building of B, as in Eq. 2.10), thusavoiding to misuse previously computed values.

In spite of all those precautions, the computation of the inverse matrix is stillthe weakest point in the whole procedure: as it could be expected, problems ininversion arise earlier than overflow in the construction of the Bessel functions.

If compared to the incompressible case, where the elements of the fundamentalmatrix grow much less dramatically with the order and its inverse is analyticallybuilt, it is clear that the task of practically using this fully analytical approach tosolve the problem of compressible deformation turns out to be a challenge towardsextreme numerical computation.

2.3.3 Extra analysis

Some extra studies have been performed while debugging the program, since manyoptions were open regarding the final task of obtaining an efficient and stable code.

A first, practical, issue is the root-finding procedure.As already mentioned, the eigenmodes of the Earth are determined as roots of

the secular equation (2.9) and this is the only step in the whole process which isperformed numerically, apart from the inversion procedure. A standard bisectionalgorithm appears as the most robust approach due the sharp variations of the sec-


ular determinant, which looks very much as a step function with absolute valuessuddenly dropping by several orders of magnitude in proximity of a root. Thismethod has the limitation to detect false roots in presence of vertical asymptotes,which are difficult to eliminate, but which give very weak residuals and don’t rep-resent a real problem. Moreover, their number is of the same order as the numberof layers in the model, thus quite small.

Another disadvantage in using a bisection algorithm is its slowness, if comparedto other methods: in the incompressible case, the problem is more stable and, mostof all, the roots are in a finite number. Hence it is possible to use the propertyof the roots to vary slowly with the degree: for every new degree the roots areonly searched in a small interval around the ones just found and this ’trick’ highlyaccelerates the whole process, making the algorithm quite competitive, if not evenoptimal.

In the compressible case, in addiction to the above mentioned false roots, adenumerably infinite set of roots (the so called D-modes) appears in region 2 of(2.25), as discussed in Vermeersen et al. [1996b]. Since for different layers thisregion is located in different parts of the spectrum, the final result with a multi-layered model is that the D-modes eventually mix with the other principal modesand practically frustrate any attempt to keep on tracing the modes singularly. Inother words the root-finding procedure has to be performed anew for every degreewith a strong impact on the computation time.

Considering the issue of obtaining a fast, and thus practically useful, algorithmand the fact that the effective weight of the D-modes is so small to result negligible,we tried to eliminate them from the whole procedure.

A first simple trial has been the one in which the value of all the Bessel functionsin the D-region was put equal to zero, but it just returned wrong values for bothinverse relaxation times and Love numbers.

A second more reasonable approach has been the substitution of the funda-mental matrix Y with the identity matrix I , again in region 2: the idea was tolet the propagator matrix technique work without the disturbances coming fromthe strongly oscillating values of the spherical Bessel functions in the region of theD-modes.

Nonetheless the exact value for the Bessel functions turns out to be really nec-essary for root determination. First of all a huge number of false roots is detected:this is a problem which could be solved with some programming efforts, but whichwould probably foil the attempt to reduce the computation time.

Second and most relevant, some important roots are missing, where we call’important’ those which give strong Love numbers: as it could be expected, zerosin region 2 for certain layers can give small residuals by themselves, but are stillcapable to give life to roots that, thanks to the contribution of other layers, playan important role in the relaxation process.

In other words, the analysis of the secular determinant cannot be performed fora multi-layer mode in a way as easy as it is for a homogeneous Earth, and this fact


Model Layer Radius ρ µ λ η(km) (kgm−3) (GPa) (GPa) (Pa s)

I Mantle 6371-0 5488 145 361 1021

II Mantle 6371-3480 4428 172 247 1021

Core 3480-0 10932 - - -

III Lithosphere 6371-6251 3234 61 77 1050

Mantle 6251-3480 4511 180 259 1021

Core 3480-0 10932 - - -

IV Upper mantle 6371-5701 3548 92 120 1021

Lower mantle 5701-3480 4878 218 313 1021

Core 3480-0 10932 - - -

V Lithosphere 6371-6251 3234 61 77 1050

Upper mantle 6251-5701 3631 87 131 1021

Lower mantle 5701-3480 4878 218 313 1021

Core 3480-0 10932 - - -

Table 2.1 Earth models used in this chapter, parameters are PREM-averaged.

is reflected into the practical impossibility to perform many simplifications whentranslating the analytical formulation into a computer code.

2.3.4 Results: elastic response

In this section we compare the elastic response coming from our compressible modelwith the one obtained with the fully incompressible one: here we remind thatthe formalism of the two models is the same, but the incompressible fundamentalsolutions don’t contain any Bessel function and in that case also the analytical formof the inverse fundamental matrix is known.

We study the effects of compressibility over the three Love numbers (vertical,horizontal and gravitational) for different Earth models and depending on the an-gular order. In this way we try to obtain a general estimation of the impact ofcompressibility which can be used above the practical limitations of the model dis-cussed in this paper. Earth parameters, listed in Table 2.1, are PREM averagedvalues [Dziewonski and Anderson, 1981].

In figures from 2.3 to 2.5 the first three panels show Love numbers, with solidsymbols for the compressible model and open symbols for the incompressible one;in the bottom panel normalized residuals are shown and they are obtained by sub-tracting the two above results and then normalising with the compressible one (thusa unitary value means a difference of one hundred percent).

We begin with a homogeneous Earth, in the left column of Figure 2.3: in the


-3

-2

-1

0

1

2

3

h Lo

ve

-0.3

-0.2

-0.1

-0.0

0.1

0.2

0.3

l Lov

e

-0.3

-0.2

-0.1

-0.0

0.1

0.2

0.3

k Lo

ve

10-4

10-310-3

10-2

10-1

100

101

Nor

mal

ised

Res

idua

ls

100 101101 102 103

Degree

-3

-2

-1

0

1

2

3

h Lo

ve

-0.3

-0.2

-0.1

-0.0

0.1

0.2

0.3

l Lov

e

-0.3

-0.2

-0.1

-0.0

0.1

0.2

0.3k

Love

10-4

10-310-3

10-2

10-1

100

101

Nor

mal

ised

Res

idua

ls

100 101101 102 103

Degree

Figure 2.3 Elastic love numbers: homogeneous Earth (left), core-mantle model (right).

upper panel the vertical Love number h is shown and we see that its strengthslowly grows with the order for the first few terms and then stabilizes to an almostconstant value. Compressibility causes an off-set which starts from about 40 percentdifference for the first terms and leads to an asymptotic limit of 22 percent.

In the second panel the horizontal Love number l is plotted. This is the termwhich shows the biggest discrepancy between the two models: apart from the n = 2term, all the others have a different sign and a completely different magnitude.The residual between the two models amounts to almost one hundred percent sincethe compressible response is bigger by more than two orders of magnitude for highdegrees and is also used for the renormalisation, thus leading to a unitary value.

This completely different behaviour has been tested by means of a set of in-creasing values of the Lame parameter λ and appears to be an effective charac-teristic of compressibility: the larger the value of λ, i.e. the more incompressiblethe model, the higher the degree when the Love number changes sign. In other


words, increasing the angular degree requires a larger value of λ in order to reachthe incompressible limit.

In the third panel the gravitational Love number k is plotted and we see that, inthis case, a very small difference is present between the two models: it amounts toabout 10 percent for degree 2 and reduces linearly (in bi-logarithmic representation)with increasing order, going below the 0.1 percent level around degree 300. For allthose applications that require the summation of several angular terms, then, wecan say that compressibility doesn’t affect the gravitational component.

The second Earth model is a simple core-mantle one, shown in the right columnof Figure 2.3: physically it is quite different from the homogeneous one, since it alsotakes into account the effect of the irregular fundamental solutions.

Surprisingly the new results don’t show a great difference from the previousones: only the first few terms are actually affected by the change, but they stillremain within the same order of magnitude. In particular the vertical Lovenumber shows larger values for the first terms and a slightly smaller asymptoticlimit, with a difference between the two models of about 30 percent; whereasthe horizontal Love number shows an increase in magnitude for the compressibleresponse, again only for the first few terms, but an almost unchanged relativedifference between the two models. The gravitational Love number is slightlysmaller for the first one hundred terms, but again almost no effect of compressibility.

The third model, shown in the left column of Figure 2.4, includes the introduc-tion of a 120 km thick lithosphere and this fact leads to some major differences withrespect to the two previous cases. Here we need to mention the matter of modelresolution: in order to be able to see, on the surface of a sphere with a diameterof about 40.000 km, the effect of a discontinuity at a certain depth, it is neces-sary to reach a harmonical degree with a wavelength at least equal to that depth(wavelength = circumference/degree). In the particular case, a layer of 120 kmis expected to affect the first 300 harmonics (40.000/120' 330).

The term which shows the largest effect is clearly the vertical one: thoughthe starting values don’t change much, the asymptotic ones are almost threetimes larger and are only reached around degree 200. Nonetheless the impact ofcompressibility remains the same, with a relative difference between the two modelsof about 30 percent. The horizontal Love number shows a different behaviour forthe incompressible model, with a region of small variation centered around degree50, whereas the compressible model behaves as in the previous cases: thus, whenthe difference between the two models is computed, a relative maximum appearsin the plot; around degree 200 the situation is again the same as in the previousresults. The gravitational Love number begins as in the previous case, but then,up to degree 100, the difference between the compressible and the incompressiblemodel remains almost constant, as is shown in the bottom panel. Later, it reachesthe behaviour of the two-layer model around degree 300, accordingly to the


-3

-2

-1

0

1

2

3

h Lo

ve

-0.3

-0.2

-0.1

-0.0

0.1

0.2

0.3

l Lov

e

-0.3

-0.2

-0.1

-0.0

0.1

0.2

0.3

k Lo

ve

10-4

10-310-3

10-2

10-1

100

101

Nor

mal

ised

Res

idua

ls

100 101101 102 103

Degree

-3

-2

-1

0

1

2

3

h Lo

ve

-0.3

-0.2

-0.1

-0.0

0.1

0.2

0.3

l Lov

e

-0.3

-0.2

-0.1

-0.0

0.1

0.2

0.3k

Love

10-4

10-310-3

10-2

10-1

100

101

Nor

mal

ised

Res

idua

ls

100 101101 102 103

Degree

Figure 2.4 Elastic love numbers: core-mantle-lithosphere model (left), core-LM-UM model(right).

expected sensitivity of this model.

The fourth model, shown in the right column of Figure 2.4, is also composedby three layers, but this time upper and lower mantle are differentiated, with adiscontinuity at a depth of 670 km (which should affect the first 60 harmonics).

Once again the component which experiences the largest effect is the verticalone: the asymptotic value is almost double than the one of the core-mantle modeland it is reached around degree 40. The horizontal number shows a relativedifference between the incompressible and compressible model which is largerthan in the previous case, mainly for the first 30 terms. Also the vertical Lovenumber presents a region of slightly larger response (and bigger relative differ-ence) around degree 10, but then reaches again the values of the core-mantle model.


-3

-2

-1

0

1

2

3

h Lo

ve

-0.3

-0.2

-0.1

-0.0

0.1

0.2

0.3

l Lov

e

-0.3

-0.2

-0.1

-0.0

0.1

0.2

0.3

k Lo

ve

10-4

10-310-3

10-2

10-1

100

101

Nor

mal

ised

Res

idua

ls

100 101101 102 103

Degree

Figure 2.5 4 layers model, elastic Love numbers

Finally we study a four layers model, with the presence of both a lithosphereand an upper mantle; results are shown in Figure 2.5.

Due to the different spatial scales of the effects produced by the two dis-continuities at 120 km and 670 km depth, the behaviour of this last model isthe summation of the two previous ones, where those discontinuities have beenseparately analyzed. In the same way for all the Love numbers, though with moreevidence for the vertical one, the final asymptotic values are reached with thepassage through two different steps. Almost the same happens with the residualdifference between the compressible and the incompressible models, this time moreevident for the gravitational component.

We don’t analyse here Earth models with a higher number of layers, since themean features caused by the introduction of discontinuities at different depths al-ready appear in the simple cases proposed. Moreover, high numbers of layers are


mainly used to introduce variations in viscosity and are thus not strongly affectingthe elastic response.

2.3.5 Results: stable relaxation

In this section we are going to discuss viscous relaxation, focusing our attentionon the gravitational Love number kj defined in (2.12), separately for each eigenfre-quency sj .

With the term stable relaxation we refer to all those modes which are charac-terised by a negative relaxation time, i.e. in regions from 1 to 4 of (2.25), henceleading to a steady final state.

An exhaustive discussion about unstable modes (especially about the so-calledRayleigh-Taylor of ’RT’ modes) can be found in Vermeersen and Mitrovica [2000],where the role of unstable relaxation has been significantly reduced with respect toprevious studies.

Problems of numerical instability (Bessel functions for the homogeneous case,matrix inversion for the other ones) only allow to reach a maximum angular degreebetween 150 and 200, depending on the Earth model. Thus all the tests have beenperformed up to degree 100, stable enough to provide accurate results in most ofthe situations analysed.

The five different earth models are the same as for the study of the elasticresponse, with parameters listed in Table 2.1 on page 24.

In pictures from 2.6 to 2.11, eigenfrequencies are represented in inverse kyear andLove numbers are multiplied by harmonic degree and relaxation time (i.e. modalstrength is plotted).

Homogeneous Earth

We begin our discussion with the case of a homogeneous Earth. Eigenfrequenciesand modal strength for the gravitation Love number are shown in the left columnof Figure 2.6: in the incompressible case only a single mode would be present, M0,which is due to the presence of a free surface.

In the compressible case a denumerable set of roots, the so-called D-modes,appears in region 2 and 3. We name D0 the stronger of those modes, located inregion 3, D1 the mode at the boundary between region 2 and region 3 (carrying nostrength) and with growing numbers all the other modes present in region 2.

In the right panel of Figure 2.6, in order to allow a detailed analysis of degree2, we show a plot of the secular determinant: paying attention to the reversedfrequency scale (so that from left to right modes with long relaxation time appearfirst) we clearly see the mode M0, followed by the still quite isolated mode D1 andfinally the set of D-modes from D2 onwards. The secular determinant in region2 gets smaller and smaller while approaching the boundary with region 3. Thisgeneral behaviour is not changing for increasing degree which simply causes M0 tomove to the left (longer times) and D1 to the right (then shrinking region 2, sincethe boundary with region 3 is not degree dependent). Region 3 contains the single


10-8

10-710-7

10-6

10-5

10-4

10-3

10-2

10-1

100

101

102

103

k Lo

ve (

Am

pl.*

Deg

./Eig

enfr

eq.)

100 101101 102

Degree

M0

D0

D2

10-2

10-110-1

100

101

Eig

enfr

eque

ncy

(1/k

yr)

100 101101 102

M0

D1D2,...

D0 D0bound. reg.4

-20

-15

-10

-5

0

5

10

15

20

Det

R

-7-6-5-4-3-2-10s [1/kyr]

M0D1

D2 D0

Figure 2.6 Homogeneous Earth: viscous k-Love numbers (left), secular determinant fordegree 2 (right)

mode D0 and region 4 has no roots: the boundary between those two regions isa vertical asymptote only for degree 2 and 3, whereas it turns into an horizontalinflection point from degree 4 on (but the associated mode carries no strength).

If we go back to the left column of Figure 2.6 we can study the behaviour ofall those modes: M0 has a frequency which gets smaller and smaller for increasingorder (one order of magnitude less after 100 degrees) and a modal strength linearlygrowing (two orders of magnitude bigger after 100 degrees). All the D-modes havean approximately constant frequency, coming from the fixed boundaries delimitingregion 3, and a strength which is several orders of magnitude smaller than M0. Inparticular D0 is far the strongest mode and it is slowly reducing with the order,whereas all the others get small very quickly and are already difficult to detect afterdegree 20.


10-8

10-710-7

10-6

10-5

10-4

10-3

10-2

10-1

100

101

102

103

k Lo

ve (

Am

pl.*

Deg

./Eig

enfr

eq.)

100 101101 102

Degree

M0

C0

D1

D0

D2

C0

D0

M0

10-2

10-110-1

100

101

Eig

enfr

eque

ncy

(1/k

yr)

100 101101 102

M0

C0D2,...

D1D0

-50

-40

-30

-20

-10

0

10

20

30

40

50

Det

S

-7-6-5-4-3-2-10s [1/kyr]

M0 C0 D2 D1 D0

Figure 2.7 Core-mantle model: viscous k-Love numbers (left), secular determinant fordegree 2 (right)

Core-mantle model

The introduction of an inviscid core changes the physics of relaxation, since alsothe three irregular solutions are taken into account; nonetheless the model is stillcharacterized by the presence of a single viscous layer and is thus not dramaticallydifferent from the previous case.

When we look at the left column of Figure 2.7 we see the presence of a newmode, C0, which is due to the boundary between the core and the mantle and ispresent also in the incompressible case. Its relaxation times are much shorter thanthe ones of M0 for the first few degrees and then keep just above them. Howeverthe strength of this mode is comparable with the one of M0 just for the abovementioned first degrees and after that becomes extremely small in a few steps.

More differences in the D-modes appear when we look at the secular determi-nant, plotted as usual for degree 2, which we have to divide into three pictures due


-200000

-100000

0

100000

200000

Det

S

-3.33-3.32-3.31-3.30s [1/kyr]

D2 D3 D4

-20000

-10000

0

10000

20000

Det

S

-6-5-4-3s [1/kyr]

D2 D1 D0

Figure 2.8 Core-mantle model, secular determinant for degree 2: zoom on region 2 (left)and region 3 (right).

to the different magnitudes from region to region. In the right panel of Figure 2.7we see M0 and the new mode C0, while all the other modes are associated to largevariations of the secular determinant; we can in any case notice that there is nomode at the boundary between region 1 and region 2 (for degree 2 and with thoseEarth parameters it is s ' −2.9 kyr−1) and that all the D-modes of region 2 areconcentrated in a narrow band around s ' 3.3 kyr−1.

In the left panel of Figure 2.8 we show a zoom on the denumerable set ofD-modes(with vertical scale adapted to the region) and we see that, differently from thehomogeneous case, the secular determinant gets larger and larger when approachingthe right boundary of the region: this effect is due to the presence of the irregularfunctions, hence gradually vanishing for increasing order.

In the right panel of Figure 2.8 we show more in detail region 3 and we seethat it is delimited by two vertical asymptotes and contains two roots: the one onthe right is again D0 and we will call D1 the other one, since that mode is notpresent anymore between region 1 and 2. Looking back to Figure 2.7 we see thatthe strength of D0 is approximately the same as for a homogeneous Earth and theone of D1 keeps always below C0 (very fast decrease with the order).

No significant changes in this structure occur for increasing degree, apart froma general amplification of the secular determinant (i.e. the problem is more andmore numerically unstable).

Core-mantle-lithosphere model

The last Earth model with a single viscous layer is the one which also includes thepresence of an elastic lithosphere, 120 km thick, on top of the mantle.

In the left column of Figure 2.9 we show the modes of relaxation for this model:first of all we see the new mode L0, due to the boundary between the mantle andthe lithosphere, which is the one with longer times for the first few modes, but withtimes shorter than M0 and C0 from degree 10 onwards. Secondly, the lithosphereaffects also the M0 mode (as it could have been expected, since the mantle loosesthe free surface): for degree 26 the eigenfrequency of M0 is almost equal to the


10-8

10-710-7

10-6

10-5

10-4

10-3

10-2

10-1

100

101

102

103

k Lo

ve (

Am

pl.*

Deg

./Eig

enfr

eq.)

100 101101 102

Degree

M0C0

L0D1

D0

D2C0

D0

M0

L0

10-2

10-110-1

100

101

Eig

enfr

eque

ncy

(1/k

yr)

100 101101 102

L0

M0

C0

D2,...D1D0

L0

M0

C0

10-8

10-710-7

10-6

10-5

10-4

10-3

10-2

10-1

100

101

102

103k

Love

(A

mpl

.*D

eg./E

igen

freq

.)

100 101101 102

Degree

M0

M1

T1

T2

D1(U

M)

D0(U

M)D

0(LM)C0

10-2

10-110-1

100

101

Eig

enfr

eque

ncy

(1/k

yr)

100 101101 102

M1

M0

C0D2(UM)

T1D0(UM)

T2D2(LM)D1(LM)D0(LM)

C0

D1(UM)

Figure 2.9 Viscous k-Love numbers: core-mantle-lithosphere model (left), core-LM-UMmodel (right)

one of C0 (and the detection of both modes fails) and after that point we see aninversion in the trend of M0 which reaches smaller times for increasing order. Noimpact on C0.

If we now look at the bottom panel of the same figure, we see that the strength ofM0 gets smaller for high degrees (mainly due to the change in the eigenfrequencies),the strength of L0 is the second one, but remains several orders of magnitude smallerthan the one of M0 and, last, both L0 and D0 suffer from local fluctuations instrength (around degree 40 for L0 and around degree 10 and 50 for D0) and thisis due to the fact that the two residues change sign, but are plotted as absolutevalues.

The D-modes behave as in the previous model.


Core - lower mantle - upper mantle model

We now consider the presence of two viscous layers, introducing the discontinuityat 670 km depth: all the parameters are PREM averaged and thus the differencebetween the two mantle layers is in both the elastic properties (the two Lameconstants) and the density. Results are shown in the right column of Figure 2.9.

About the fundamental modes (present also in the incompressible case), thedensity discontinuity is at the origin of the M1 mode, characterised by long timesand a considerable strength, whereas the difference in the Maxwell times (τ = ν/µ)between the two layers causes the presence of the modes T1 and T2, with shortertimes and a smaller strength than the M -modes.

Apart from the still simple structure of the three modes with longer relaxationtimes (C0, M0 and M1), all the other ones become very complicated since thefeatures of the various regions of (2.25) occur at different times for the two layers.More in detail, all of the four regions for the upper mantle are in the first regionfor the lower mantle.

In order to understand the complicate structure of relaxation for short timeswhich we see in the upper-right panel of Figure 2.9, we have to give a look at a fullplot of the secular determinant.

In the left panel of Figure 2.10, the case for degree 2 is shown. From right toleft we find all the D-modes of the lower mantle: the two in region 3 (D0 and D1)and all of region 2, which is reduced to a very narrow band; all those modes arein region 4 of the upper mantle. The rest of the spectrum on the left is dividedinto three regions by the two asymptotes surrounding region 3 of the upper mantle(s2 ' −1.73 kyr−1 and s3 ' −2.90 kyr−1). Continuing from right to left we findthe following roots: T2 and T1, then a vertical asymptote at s = s3 (not plottedsince too close to the following root), D0 from the upper mantle, C0, the asymptoteat s = s2 (again not plotted), region 2 from the upper mantle and, finally, M0 andM1. The only root missing is D1 from the upper mantle, probably not detecteddue to the closeness to the asymptote at s = s2.

In the right panel of Figure 2.10 we show the same plot, this time for degree 3,

-2000

-1000

0

1000

2000

Det

S

-10-9-8-7-6-5-4-3-2-10s [1/kyr]

M1

M0

D2(

UM

) C0

D0(

UM

) T1

T2 D2(

LM

)

D1(LM) D0(LM)

-2000

-1000

0

1000

2000

Det

S

-10-9-8-7-6-5-4-3-2-10s [1/kyr]

M1

M0 C0

D2(

UM

)D

1(U

M)

T1

D0(

UM

)

T2 D2(

LM

)

D1(LM) D0(LM)

Figure 2.10 Core-LM-UM model: secular determinant for degree 2 (left) and degree 3(right).


and we see that the two asymptotes from the upper mantle are moved with respectto the roots: C0 is now on the left of s2, whereas s3 is between T1 and T2. Thesituation is not changing for increasing degree: only the two regions 2 become smalland are thus more and more difficult to detect, whereas mode D1 from the uppermantle moves away from the vertical asymptote and gets more evident.

We can now go back to the right column of Figure 2.9 and study the strengthof all the modes. The only ones that remain almost the same as in the core-mantlemodel are M0 and C0; M1 is the second stronger mode, up to degree 50, but afterthat it rapidly degrades, probably due to numerical problems; the second modedominating relaxation for high degrees is T1, which has a strength quite slowlydecreasing, but in any case several orders of magnitude smaller than M0; we thenfind two couples of modes with quite similar strength, namely T2 with D1 from theupper mantle and the two D0; all of them become rapidly very small after degree20.

Four layers model

In Figure 2.11 we show relaxation when both an elastic lithosphere and a secondmantle layer are present: the result is the combination of the effects of the twoprevious models, with the major change being the one on M0 and the fluctuatingstrength of some of the other modes (e.g. L0 and T1). As expected, no new featuresappear, since the elastic lithosphere and the viscous upper mantle are affectingcompletely different time-scales.

Conclusions

We have so far discussed only simple models, with no more than two viscous layers,but this is enough to draw some conclusions.

The homogeneous model, presents a M0 mode dominating relaxation with acharacteristic time above 1 kyear (10 kyears for degree 100) and a strength sev-eral orders of magnitude bigger than all the other modes. Short time relaxation(characteristic times of a few hundreds of year) is also affected by the compressibleD-modes, especially by D0 for high degrees.

The introduction of an inviscid core, which allows to take into account also thethree irregular fundamental solutions, has the effect of adding a new mode, C0,with a significant strength (about the same as M0, but on a time scale one ordersmaller) only for the first few modes; in this same region it also causes an increase inthe relaxation times of M0. The inviscid core is in any case essential to the model,in order to provide the necessary boundary conditions required by a multi-layeredEarth.

An elastic lithosphere on top of the mantle causes the presence of a new mode, L0(the second stronger one with characteristic times of about 1 kyear and a strengthjust above the one of D0) and a radical change in M0 for high degrees, with char-acteristic times now decreasing with the order.


10-8

10-710-7

10-6

10-5

10-4

10-3

10-2

10-1

100

101

102

103

k Lo

ve (

Am

pl.*

Deg

./Eig

enfr

eq.)

100 101101 102

Degree

M0

M1

L0

T1

T2

D1(U

M)

D0(U

M)D

0(LM)C0

10-2

10-110-1

100

101

Eig

enfr

eque

ncy

(1/k

yr)

100 101101 102

M1

L0

M0

C0D2(UM)

T1D0(UM)

T2D2(LM)D1(LM)D0(LM)

C0

D1(UM)

M0

L0

Figure 2.11 4 layers model, viscous k-Love numbers

The introduction of a discontinuity in the mantle adds three new modes (M1with characteristic times larger than 10 kyears, T1 and T2 with times of a fewhundreds of years) and the doubling of the D-modes, now repeated at differenttimes (but all of them between 200 and 500 years). M0 remains the same as before,M1 is the second stronger mode for the first few tens of degrees and T1 is the onlyother mode which maintains its strength for high degrees.

The features described above are then entering the relaxation process of morecomplex Earth models, as can be seen when four layers are taken into account. Forall tested viscosity profiles, C0, L0 and all the M -modes, i.e. the ones with longcharacteristic times, remain substantially isolated and thus easy to detect. On thecontrary, the modes with relaxation times smaller than 1 kyear (i.e. the T - and theD-modes) mix together an can be at the origin of serious numerical problems incase of Earth models with a large number of layers.

We conclude by stressing the fact that all those considerations about relativestrength of the various modes are primarily concerned with relaxation above degree10, where the different trends become more evident: for low order relaxation many


more modes share comparable strengths and in that case a degree by degree studyis required.

2.3.6 Incompressible vs compressible stable relaxation

In this section we will give a comparison between the compressible model and theincompressible one, this time for relaxation. All values are presented as percentageof difference with respect to the compressible model. A positive residue for theeigenfrequencies means that the compressible model is relaxing over larger timesand for the modal strength that it shows a larger relaxation.

In the left column of Figure 2.12 we show the results for the homogeneousEarth, when only mode M0 is present. Eigenfrequencies for the two models arevery similar, with a difference always below 1%, a maximum at degree 7 and

0

1

2

3

4

5

Love

str

engt

h (%

diff

.)

100 101101 102

Degree

h,M0

k,M0

l,M0

0.00

0.25

0.50

0.75

1.00

Eig

enfr

eque

ncy

(% d

iff.)

100 101101 102

M0

0

1

2

3

4

5

Love

str

engt

h (%

diff

.)

1010 20 50 100Degree

h,M0

k,M0

l,M0

0

10

20

30

40

50

Eig

enfr

eque

ncy

(% d

iff.)

1010 20 50 100

M0

C0

Figure 2.12 Compressible vs. incompressible viscous Love numbers: homogeneous Earth(left) and core-mantle model (right).


rapidly decreasing with the order (below 0.2% around degree 100). Also thedifference in modal strength is small, with the three components showing a similarbehaviour: the vertical Love number has the maximum discrepancy, starting with adifference of about 5% which is, then, rapidly decreasing (about 0.5% at degree 100).

When we pass on to the core-mantle model, shown in the right column of Figure2.12, we get a small problem in comparing low-order (less than 10) residues: whilethe compressible model, in fact, has smallest eigenfrequencies for mode M0 (seeFigure 2.7), the incompressible one presents the same for mode C0. This apparentlysurprising fact has, on the contrary, a simple explanation: C0 is originated by thedensity contrast at the CMB, whereas M0 by the one at the surface, and the largerthe contrast, the larger the eigenfrequency (faster relaxation). When one of thosecontrasts is much bigger than the other one, the associated mode shows a fasterrelaxation (easy to see with exaggerated density values). For standard (PREM)density values for core and mantle, we are close to the critical point when the twoprincipal modes switch their positions; it is noteworthy that this critical point isslightly different for the two models. The final result is that, for all the multi-layercases we analyse, huge differences are present in the first few terms (before degree10) between the compressible and incompressible model. A detailed analysis ofthose differences requires a term-to-term comparison, but that is above the level ofour general study.

Back to the core-mantle model, the presence of the C0 mode forces a changein the scale of the top picture, since the effect of compressibility amounts, for thismode, to a value between 20 and 25 %. Moreover, the impact on the strength ofthis mode is not plotted, since its absolute value is rapidly decreasing after degree10 and the datum would not be very significant. Almost unchanged, as it could beexpected, is the impact on mode M0.

In the left column of Figure 2.13 we show the effect of the presence of an elasticlithosphere on top of the mantle: from Figure 2.9 we remember that relaxation issignificantly changing with respect with the 2-layer Earth model.

Looking at the frequency plot, we see that the behaviour of the C0 mode isalmost unchanged. The L0 mode is characterised by an initial (degree 10) differenceof almost 40%, to end below the 30% level for high orders. Unluckily it’s notpossible to show the result of a direct comparison of the amplitudes for mode L0,since the compressible model presents a behaviour much more unstable than theincompressible one, with regions where the residues change sign (see comments toFigure 2.9).

The only mode for which a full study of the impact of compressibility ispossible, then, is again mode M0: in opposition to the previous cases, we see thatthe difference in the frequencies is gradually increasing with the order, reaching40% around degree 70 and 50% at degree 100. In particular, a compressible Earthis relaxing over longer times (smaller eigenfrequencies). Also the difference instrength of the Love numbers for the M0 mode is much larger than in the previous


0

10

20

30

40

50

Love

str

engt

h (%

diff

.)

1010 20 50 100Degree

l,M0

h,M0

k,M0

0

10

20

30

40

50

Eig

enfr

eque

ncy

(% d

iff.)

1010 20 50 100

L0

C0

M0

-100

-75

-50

-25

0

25

50

75

100Lo

ve s

tren

gth

(% d

iff.)

100 101101 102

Degree

k,M1

l,M1

h,M1

-100

-75

-50

-25

0

25

50

75

100

Eig

enfr

eque

ncy

(% d

iff.)

100 101101 102

M1

Figure 2.13 Compressible vs. incompressible viscous Love numbers: core-mantle-lithospheremodel (left) and core-LM-UM model (right).

cases, with the vertical and gravitational components behaving in a similar way:residues grow with the order, reach a maximum (almost 25%) around degree 80and than the tendency changes with gradual reduction in the difference betweenthe two models. The horizontal Love number shows a large difference for low orders(45% for degree 10), a minimum around degree 26 (not detected due to intersectionbetween relaxation times of M0 and C0) and a later convergence to the trend of theother two numbers. Due to computational limits (evident in some sparse data in thepicture), it’s not possible to reach the order required to see an asymptotic behaviour.

The last case under analysis is the one showing mode M1, originated by thedensity discontinuity at 670 km depth. Results are in the right column of Figure2.13. We remind that for this Earth model also two T -modes are excited (caused bythe discontinuity in rigidity), but their analysis is here skipped, due to the closenesswith some of the D-modes from the compressible Earth.


This is the only case which presents also negative differences, meaning thatthere is a region where the incompressible model shows larger times and/orstronger relaxation. Eigenfrequencies begin with a rather faster relaxation forthe compressible case (more than 50% difference for degree 2), followed by a fastconvergence between the two models (almost no difference at degree 11) and thenby a progressive slowness in compressible relaxation (more than 20% difference atdegree 100). On the side of the modal strength comparison is only possible up todegree 50, because numerical problems arise after that point (see Figure 2.9). Thecurves of the gravitational and the horizontal Love numbers follow the same trend,with the gravitational one shifted upwards: the maximum positive difference isthen shown by the gravitational Love number (about 70% for degree 2) and themaximum negative one by the horizontal component (again almost 70%, this timearound degree 50). The vertical Love number shows an intermediate behaviour,with small differences (below 10%) for the first 20 terms, then converging to thesame trend as the other two for higher orders.

In conclusion, compressibility is marginally effecting the M0 mode when nocrust is present, but it provides a longer and stronger relaxation when the free sur-face is represented by a purely elastic layer. Mode M1 is significantly different inboth eigenfrequencies and modal strengths, but no asymptotic behaviour can befound numerically. Mode C0 doesn’t fit into a general discussion, since it carries asignificant strength only for very low order degrees: however, large differences arisebetween the two models. Mode L0 shows a significant difference in the eigenfre-quencies, but again a direct comparison of modal amplitudes is prevented by thehighly varying behaviour shown by the compressible model.

In the end, we can say that significant differences are surely arising about re-laxation, especially when high order terms are playing a role, since the M0 mode isdominating the relaxation process. On the other hand, numerical limitations don’tallow to give a quantitative estimate about the impact of compressibility on shorttime scales, when the D- and T -modes play a relevant role.

Chapter 3

Finite Elements Method

The Finite Elements Method (FEM) represents an existing numerical technique tosolve the differential equations that govern postseismic relaxation, in particular theequation of motion (Eq. 2.1).

In this chapter we will provide a simple introduction to the principles of theFEM approach, followed by a validation study of the specific implementation usedin this thesis (TECTON, The Govers version, references in Section 3.2) and by adiscussion about the differences between two- and three-dimensional models.

3.1 Basic concepts

For a general and exhaustive discussion of the Finite Elements Method we refer toFelippa [2003] and to references therein: here we will just synthesize a few funda-mental aspects.

A mathematical model is often represented by a differential equation in spaceand time which represents a simplified version of the physical problem. In the caseof continuum mechanics, such a model has an infinite number of degrees of freedom:to make numerical simulations practical it is necessary to reduce them to a finitenumber in a process called ’discretization’. This process represents the concept ofmathematical FEM. A more rigorous definition says that the solution of a boundaryvalue problem posed over a domain Ω is obtained by the local approximation of thefunction over disjoint subdomains Ω(e) called finite elements.

Equivalently, the physical FEM consists in the subdivision of a complex sys-tem into disjoint (non-overlapping) components of simple geometry called finiteelements. The strategy adopted for such an approach is divide et impera (Latinfor divide and conquer), which consists in dividing the system into more manage-able subsystems: the total behaviour is that of the individual elements plus theirinteraction.

The practical advantage of such a numerical approach, with respect to theuse of an analytical solution, consists in the possibility of dealing with complex

42 Finite Elements Method

geometries and with different kinds of boundary conditions.

For the practical realization of a FEM model, the degrees of freedom are collectedin a column vector u, called state vector. In analytical mechanics, each degree offreedom has a corresponding dual term, which represents a generalized force; theseforces are collected in a column vector called f. The inner product fTu has themeaning of external energy or work.

In the simplest case, the relation between u and f is assumed to be linear andhomogeneous and it is expressed by the master stiffness equation:

Ku = f (3.1)

where K is called the stiffness matrix. The same formalism is adopted alsofor non-mechanical applications: in a thermal problem, for example, u representstemperature and f the heat flux.

The basic structure of Eq. 3.1 remains characteristic of more complex models,where the various features enter the definition of the stiffness matrix K (as thecase of a specific rheology) or of the forcing therm f (as for the introduction of afault).

Two aspects are of crucial importance in the practical realization of a FEMmodel: the mesh and the boundary conditions (BCs).

For the specific problem under study, i.e. the visco-elastic response of the earthafter an earthquake, the only necessary BCs are represented by displacement con-ditions at the extremes of the domain (to avoid displacement and rotation of thewhole model).

We didn’t introduce in the model any extra tectonic feature, like a constantvelocity at the boundaries, and we chose the model dimension to be larger than thearea affected by deformation: in this way, it has been possible to experimentallydemonstrate that results are independent from the specific configuration of thedisplacement BCs.

For this reason, in the validation section 3.3 we discuss only the problem ofmesh refinement and convergence of the solution, without any further reference tothe impact of specific BCs.

3.2 Tecton, the Govers version

The FEM package used for this study is a further development of the original codeby Melosh and Raefsky [1980, 1981] realized by Rob Govers at Utrecht University(Utrecht, The Netherlands).

Here is a list of fundamental publications where new features were first presented:i- extensive documentation: Govers [1993];ii- short documentation: Govers and Wortel [1995];iii- thermal code: Govers and Wortel [1993];

3.3 Validation 43

iv- linked nodes technique: Govers and Wortel [1999];v- 2,5D version (out-of-plane): Govers [1998];vi- spherical plane stress technique: Govers and Meijer [2001];vii- 3D version, based on quads: Govers et al. [2000];The 3D version based on tetrahedra is used for the first time in the present

study.

3.3 Validation

The validation of the 3D FEM code has been performed for elastic and viscoelasticrelaxation due to a normal fault earthquake. Internal convergence has been studiedand converged results have been compared with other existing deformation models.

The whole domain (Figure 3.1) is represented by a box with dimensions200x200x100 km, showing a portion of the Earth around the earthquake. Dimen-sions have been chosen to be large enough to make the deformation field not affectedby the specific choice of boundary conditions (namely, zero displacement at the cor-ners). The domain has been layered, with the introduction of an elastic upper crust(first 8 km), a crustal transition zone (12 km), a lower crust (15 km) and the restof the lithosphere.

Figure 3.1 Whole domain.


Figure 3.2 Finer box around fault, zoom.

Figure 3.3 Expanded fault plane. The fault is representer by the light central area.

The main attention in this section is on a small box around the fault (Figure3.2), where mesh density will be varied. Limitations to the size of this box (about30 km in the horizontal direction, embracing only the elastic crust) are coming fromthe necessity of working with a reasonable number of elements (already hundredsof thousands).

The fault is 10 km long, with the top at 1.5 km depth, dipping 45 degrees andreaching a depth of 5.5 km. The slip on the fault is homogeneous and amountsto 50 cm. In Figure 3.3, we show an example of how the surfaces are meshedwith triangles: the picture represents the fault plane and its prolongation, i.e. thediagonal plane of Figure 3.2.

The operation of meshing is performed with the publicly available softwareGmsh by C. Geuzaine and JF. Remacle (version 1.45.1, Copyright(c) 1997-2003).The software proceeds in a hierarchical way, by meshing first lines, than planes withtriangles and eventually volumes with tetrahedra. The key parameter controlling

3.3 Validation 45

mesh density is represented by a characteristic length, which is freely assigned tothe points defining the initial geometry: points constrain lines, bounding lines con-strain surfaces and bounding surfaces constrain volumes. The final mesh is thenbuilt in a way to guarantee a smooth change in elements size when passing betweenregions with different characteristic lengths.

3.3.1 Internal convergence

In this subsection, we discuss the effect of a change in mesh density on elastic andviscoelastic deformation. We will vary the characteristic length of the mesh at thesurface and at the bottom of the box around the fault and on the fault itself.

Test are performed on the box of Figure 3.2, which extends 10 km from thefault in each direction (so it measures 30 km along fault and 24 km across fault),the fault mesh has a characteristic length of 500 m and slip tapering is applied on aframe of 500 m width. By slip tapering we refer to the presence of a frame of fixedsize around the fault plane (visible in Figure 3.3) where slip is linearly reduced tozero, from the fault until the external boundary of the frame itself. In this way,the effective rupture surface (and the total applied forcing) is not affected by themesh density around the fault edges.

All pictures represent sections of surface deformation, taken along a line per-pendicular to the fault and crossing in its middle. Deformations are in millimetersand relaxation is intended as the deformation accumulated in the first year afterthe earthquake (thus subtracting the elastic signal, showed in a different panel).

The box meshes that are represented have a characteristic length varying be-tween 3000 m (dotted line) and 500 m (solid line). In Figure 3.4 we show verticaldisplacement. The elastic deformation (upper panel) can already be consideredconverged whit a mesh of 2000 m (dash-dotted line), the finer models representingonly a second-order correction. As far as relaxation is concerned (bottom panel), onthe contrary, there is very little improvement in convergence with respect to meshrefinement. The rougher model (3000 m at both surface and bottom) is alreadycapable of providing a first-order solution. Finer models only have the effect ofslightly modifying the deformation signal.

Similar considerations are valid for the case of horizontal displacements (Figure3.5), which is however showing a more clear behaviour toward convergence ofthe solution. Interesting is the fact that the elastic signal is characterized bysmaller wavelength features than in the vertical case and this requires a finermeshing. Notably, a definite increase in the sharpness of the deformation patternis visible up to the finest models: the small difference between the dashed andthe solid line, however, is enough to prove the first-order convergence (furthertested with even finer models, not shown here). In the case of relaxation, a slightlyincrease in the accuracy of the solution is reached with a meshing of at least 1000 m.


-300

-200

-100

0

100

elas

tic (

mm

)

-30 -20 -10 0 10 20 30

-100

-50

0

50

100

rela

xatio

n (m

m)

-30 -20 -10 0 10 20 30km

Figure 3.4 Internal convergence for vertical displacement. Box meshing: 3000 m (dotted),2000 m (dash-dotted), 1000 m (dashed), 500/1000 m (solid).

Another test which has been performed is the effect of mesh density on the faultplane.

For the finest box meshing (500 m at the surface and 1000 m at the bottom),the initial fault grid of 500 m has been refined to 200 m and 100 m, withoutany noticeable effect on both elastic and relaxation signals. For this reason,no pictures of the convergence test are presented here. It is worth noticingthat this comparison is made possible by the presence of tapering at the faultedges, which allows a constant forcing independently on the grid size. For anon-tapered fault, the position where the imposed slip reaches a null value isdependent on the size of the elements connected to the most external faulted nodes.

One important consideration has to be made, concerning fault refinement.The way the mesh is build, i.e. starting from the boundaries and then smoothingthe internal mesh, and the way we have performed our test, by simply refining adefinite volume around the fault, don’t allow a rigorous convergence test. Namely,

3.3 Validation 47

-100

-50

0

50

100

elas

tic (

mm

)

-30 -20 -10 0 10 20 30

-100

-50

0

50

100

rela

xatio

n (m

m)

-30 -20 -10 0 10 20 30km

Figure 3.5 Internal convergence for horizontal displacement. Box meshing: 3000 m(dotted), 2000 m (dash-dotted), 1000 m (dashed), 500/1000 m (solid).

the refined box has a limited size which is smaller than the volume affected bydeformation. This means that the peripheral areas will have a gradually increasingmesh size (in order to match the rough grid at the domain boundary), which isnot only related to the mesh size inside the box. On the other side, the size of therefined box is limited by necessity of keeping the total number of elements withina reasonable value (with respect to computer resources and computation time).To give a rough idea, the mesh with a characteristic length of 1000 m is composedby 110,000 elements and requires 76 MB of static memory, whereas a mesh with asize of 500 m needs 480,000 elements and 334 MB of memory. If the linear size ofthe fine box were increased by a factor two, that would approximately increase theelement number by a factor eight, making the problem only solvable with the useof a supercomputer.

Summarizing, the study of elastic deformation is constrained by the horizontalsignal, which requires at least a mesh with characteristic length of 500 m at the


surface, whereas relaxation already provides a first order approximation with themesh of 3000 m.

Those values are intended as a rule-of-thumb, the effect of mesh density beingcontrolled by a number of factors, first of all by the fault geometry and location.However, modelling co- and post-seismic deformation requires the definition of anumber of parameters, even in a first order approach. The convergence of the meshis an important (numerical) issue, but extending it to a second order precision isan operation that needs to be tailored to each specific case, if ever necessary.

3.3.2 Comparison with analytical models

In this subsection we want to compare the results of the finite elements modelagainst the two more widely used analytical models. The first of those model,published by Okada in 1985, is capable to compute the elastic deformation for afault embedded in a uniform half-space and it is widely used to compare the effect of

-300

-200

-100

0

100

elas

tic (

mm

)

-30 -20 -10 0 10 20 30

-100

-50

0

50

100

rela

xatio

n (m

m)

-30 -20 -10 0 10 20 30km

Figure 3.6 Vertical displacement. Comparison between Finite Elements (solid), analyticalhalf-space (Okada, dashed) and normal modes (Pollitz, dotted).

3.3 Validation 49

fault models against measured surface deformations. We use the implementation byFeigl and Dupre [1998]. Pollitz’ model, published in [Pollitz, 1996] (elastic solution)and [Pollitz, 1997] (viscoelastic relaxation), is based on a normal modes approach,solving the problem in a spherical radially stratified earth. This code has beenwidely used, mainly for the interpretation of measured postseismic deformation.

Fault model and rheology are the same for all three models, where for the half-space, the elastic parameters of the upper crust have been used (thus the other twomodels actually have a slightly different lower crustal rheology).

As we can see from the upper panel of Figure 3.6, the agreement between allmodels for vertical elastic deformation is reasonably good. However, the finite el-ements solution predicts about 10% more subsidence than the other two models:this fact is mainly, if not entirely, due to the wide frame adopted for fault patching.The position of the upper fault limit, in fact, is located at 1.5 km depth for theanalytical models, whereas tapering is applied up to a depth of 1 km for the numer-

-100

-50

0

50

100

elas

tic (

mm

)

-30 -20 -10 0 10 20 30

-100

-50

0

50

100

rela

xatio

n (m

m)

-30 -20 -10 0 10 20 30km

Figure 3.7 Horizontal displacement. Comparison between Finite Elements (solid),analytical half-space (Okada, dashed) and normal modes (Pollitz, dotted).


ical model. It doesn’t surprise that this difference in fault extension is introducingquite some effect on the final deformation pattern.

Tests with a smaller region of tapering have been performed and they indeedlead to a definite convergence of the numerical solution toward the analytical ones.However, for sake of consistency, we have chosen to display has a reference the sameresults that have been used for the convergence study of the previous subsection.

Similar considerations hold for horizontal elastic deformation (upper panel ofFigure 3.7), where the finite element solution predicts a somehow larger displace-ment, together with a less marked convergence toward the fault in the near-fieldregion on the hanging wall.

In the case of relaxation, not provided by Okada’s model, we see that Pollitz’solution is relaxing slightly faster (more deformation accumulated in the first year)in vertical direction (lower panel of Figure 3.6), whereas agreement is remarkablygood for the horizontal response (lower panel of Figure 3.7). The reason for thisdifference is not completely clear: the fact that the loading is not exactly the same,as demonstrated by the observed discrepancy in the elastic solution, is definitelyaccountable for a partial explanation. On the other side, a larger difference in thevertical signal suggests some depth dependent effect, probably related to meshingin the viscous layers.

For the purpose of this section, however, we can be satisfied with the fact thatthe numerical solution converges to the analytical result. Again, a second order fitcan only be attained by tailoring the meshing to the specific geometry, fact whichis of limited relevance for the purpose of validation of the finite elements code.

3.4 2D vs. 3D representation

In the last section of this chapter we want to address a problem which is directly con-nected with the validation of a numerical code, namely the use of a two-dimensional(2D) model.

Approaching a problem in a 2D approximation presents major numerical advan-tages:

- from the computational side, the number of equations to be solved is muchless, leading to a smaller solution matrix, which means a minor memory use and afaster inversion;

- from the technical side, it is much easier to mesh a plane with triangles thana volume with tetrahedra;

- last, but not least, a 2D finite elements code has a more simple structure anda consequently easier debugging.

In theory, a two-dimensional approximation means that the model is infinitelylong in the third direction: this is practically equivalent to say that the describedentity, in this case the fault, is much longer in the third direction than in the othertwo. Physically, the solution obtained is only valid in proximity of the forcing term(the fault).

3.4 2D vs. 3D representation 51

-150

-100

-50

0

50

vert

ical

ela

stic

(m

m)

-50 -25 0 25 50km

Figure 3.8 Vertical elastic deformation for the 3D (solid) and 2D (dashed) model.

In a practical situation, however, it is difficult to predict what is the actuallimitation of a 2D model, or to give a quantitative estimate of the error introduced.

The specific case of the main event of the 1997 Umbria-Marche earthquakesequence is a potential candidate to show the impact of a 2D approximation, witha proposed rupture which is 10-12 km long and 6-8 km wide [Zollo et al., 1999;Salvi et al., 2000; Basili and Meghraoui, 2001].

In Figure 3.8, we first compare a section of vertical elastic deformation for thetwo approaches. The fault model follows the double rupture solution proposed byBasili&Meghraoui (2001). The agreement between the models is rather good, withthe only difference that the 2D solution predicts a larger uplift in the peripheralareas. The off-set amounts to less than one centimeter, about 6% of the total peak-to-peak displacement of almost 16 cm, and it can easily appear like a small deviation.This might be true in proximity of the fault, but the difference becomes of majorimportance in the far field: the 3D model, in fact, reaches a null deformation withina range of about 60 km, whereas the same happens for the 2D model over almost800 km. The vertical signal is smaller than 2 mm at a distance of about 100 kmfrom the fault, so the deviation is quantitatively rather small, but it suggests thatsomething could be different in the physics of 2D deformation. From the modellingside, the fact of working in a 2D domain which is ’only’ 1000 km wide results inthe high sensitivity of the deformation pattern to the boundary conditions, whichshowed up dramatically in the validation phase (not discussed here because of minorinterest after availability of the 3D model).

The comparison of vertical relaxation, displayed in Figure 3.9, shows a differencebetween the two approaches beyond any expectation. For the three-dimensionalmodel (upper panel), deformation accumulated in the first five years after the earth-quake presents a fast central uplift (not showing significant variation after the firstyear) and a more gradual subsidence on the hanging wall (left of the fault). Maxi-mum peak-to-peak displacement amounts to about 45 mm.


-80

-40

0

40

80

rela

xatio

n (m

m)

- 3D

-150 -100 -50 0 50 100 150

-80

-40

0

40

80

rela

xatio

n (m

m)

- 2D

-150 -100 -50 0 50 100 150km

Figure 3.9 Vertical relaxation after 1 (solid), 2 (dashed) and 5 (dotted) years. Upperpanel: 3D model. Lower panel: 2D model.

A very different pattern characterizes the result from the two-dimensional model(lower panel of Figure 3.9), where vertical relaxation is dominated by a large centralsubsidence and peripheral uplifting bulges. However, a short wavelength signal isclearly present in the center of the domain, in form of a marked local peak resemblingthe 3D result.

In order to separate the two signals, characterized by different wavelengths,we need a band-pass filter: after a trial-and-error search, we found that the bestfilter is represented by a Fourier polynomial model up to degree 25 (the modeldomain extends from -250 to 250 km and it is scaled to [−π, π] for the Fourier basisfunctions). The filter, i.e. the subtracted signal, is shown in the bottom panel ofFigure 3.10.

The remarkable result is represented by the very good agreement betweenthe upper panel of Figure 3.10, showing the filtered 2D model, and the formerlydiscussed 3D relaxation pattern (upper panel of Figure 3.9). In particular, the tworesults match both the fast central uplift and the slower subsidence on the hanging

3.4 2D vs. 3D representation 53

-80

-40

0

40

80

rela

xatio

n (m

m)

- 2D

filte

red

-150 -100 -50 0 50 100 150

-80

-40

0

40

80

rela

xatio

n (m

m)

- 2D

filte

r

-150 -100 -50 0 50 100 150km

Figure 3.10 Vertical relaxation after 1 (solid), 2 (dashed) and 5 (dotted) years for the 2Dmodel. Upper panel: residual deformation after subtraction of the longwavelength signal. Lower panel: subtracted long wavelength signal.

wall. The moderate discrepancy still present is probably due to a partial couplingbetween the short and the long wavelength signals.

It is evident from those simple tests how the effect of a two-dimensional repre-sentation of a fault is not appropriate to model postseismic relaxation, at least forthe case of a moderate size rupture.

In particular, the assumption of an infinitely long fault implies the represen-tation of a very different physical entity, as it is qualitatively and quantitativelydemonstrated by the presence of a major long-wavelength signal in the relaxationpattern.

Moreover, the rather small discrepancy between the 2D and 3D elastic responsescan easily mislead the modeler in the validation phase: when numerical results arecompared with those from other (analytical) models, a separate benchmarking ofviscoelastic relaxation is a necessary step.

Chapter 4

Fault depth, compressibilityand styles of relaxation

In the introduction we have already discussed the fact that for the main shock of the1997 earthquake two main fault models have been presented: a family of geodeticmodels which lead to a rather shallow event (rupturing from about 5 km depth andalmost reaching the surface) and a family of seismological models which depict amuch deeper event (most of the slip is localized between a depth of 8 and 4 km).

If, as in Riva et al. [2000] and Aoudia et al. [2003], we assume that the thicknessof the elastic crust is about 8 km, then the two models locate most of the sliprespectively in the upper [Salvi et al., 2000] and lower [Zollo et al., 1999] part ofthe elastic layer. In those papers, it has already been shown that this difference inthe fault location has a large impact on the expected postseismic relaxation pattern.

In this chapter we want to address this issue in a more systematic way, bypresenting horizontal and vertical surface deformation patterns from numericalmodels for the two possible fault locations and discussing the spheroidal/toroidaldecomposition of the horizontal deformation.

Other crucial parameters controlling the relaxation process are represented bythe elastic properties of the earth model, in particular by the level of compressibility.As we have already discussed in Section 2.1, the choice between compressibility andincompressibility is usually made in the preliminary stage of the solution of themomentum equation. Further, in Section 2.3, we have shown how compressibilityis capable of largely affecting the magnitude and scale of relaxation.

We want to address this issue in the present chapter, because we have realizedthat the effect of compressibility is strongly dependent on the styles of relaxationas determined by the fault location.

In the next sections, we will make use of the non-gravitational version of thecode by F. Pollitz [Pollitz, 1997], which is capable of dealing with both compressibleand incompressible linear rheologies. In the cited paper, the author presents a

56 Fault depth, compressibility and styles of relaxation

Layer Depth (km) κ (GPa) µ (GPa) η (Pa s)

Upper crust 0-8 44.2 32.5 1030

Transition zone 8-20 56.2 33.7 1018

Lower crust 20-35 59.2 35.5 1018

Mantle 35-880 122.5 73.5 1021

Table 4.1 Compressible Earth model used in this chapter. The incompressible limit isobtained by multiplying the bulk modulus κ by a factor 100.

comprehensive discussion of the physics of relaxation for a compressible earth, buthe doesn’t address any of the issues we are about to discuss.

In order to show the extreme cases, two earth models have been used: in thefirst the earth is a Poissonian solid (i.e. the Poisson’s ratio equals 0.25), whereas inthe second the earth is assumed to be incompressible (Poisson’s ratio almost 0.5).Layering and elastic parameters are listed in Table 4.1.

In the coming sections, all the pictures are aerial views of the deformationpattern (spatial scale in km), where the fault is in the center, dipping to the left.

4.1 Vertical relaxation

We begin our analysis from the vertical relaxation, which in a spherical harmonicsrepresentation is defined by a projection of the spheroidal component. Results areplotted in Figure 4.1.

The shallow fault (Salvi FM, left panels) shows the strongest signal centered withrespect to the fault, with a dominant uplift on the foot-wall and a much smallerregion of subsidence on the hanging wall. The only difference between a Poissonian(upper-left panel) or an incompressible (lower-left panel) earth is that the latter isrelaxing faster, with a slightly more marked area of subsidence.

The deep fault (Zollo FM, right panels), on the contrary, shows only subsidence.In case of an incompressible earth (lower-right panel), the pattern somehow resem-bles the shallow source, with the presence of a relative minimum at each side ofthe fault, but the marked central maximum is disappeared and the whole area ischaracterized by a diffused subsidence. The maximum signal is on the hanging wall.

The deep fault in a Poissonian earth (upper-right panel) shows a completelydifferent relaxation style: not only the magnitude is much smaller than in theincompressible case, but also the deformation pattern is very different from all theprevious results, with most of the deformation located at the ends of the fault.

4.2 Horizontal relaxation 57

-20

-10

0

10

20

km

-30 -20 -10 0 10 20 30km

04

-20

-10

0

10

20

km

-4

04

8

-6 -4 -2 0 2 4 6 8 10 12mm

-30 -20 -10 0 10 20 30km

-4

-4

-4

0

-12

-8

-4

-4

0

-14 -12 -10 -8 -6 -4 -2 0 2mm

Figure 4.1 Vertical relaxation after 1 year. Left column: Salvi FM. Right column: ZolloFM. Upper line: Poisson’s earth. Lower line: incompressible earth.

4.2 Horizontal relaxation

In a similar way, also horizontal deformation is affected by differences in fault depthand earth rheology. Results are plotted in Figure 4.2.

The shallow fault in a Poissonian earth (upper-left panel) shows two distinctlobes, moving away from the fault and with slightly larger deformations on thehanging wall. Again, the only effect of incompressibility (lower-left panel) is toincrease the amount of deformation, maintaining the same surface pattern.

For the case of a deep fault (right panels), the pattern looses symmetry: onlythe foot-wall exhibits some relaxation in the near field, whereas a considerableamount of deformation is observed in the far field. In particular, the Poissonianearth (upper-right) has a distinct area of maximum deformation on the foot-wall,


-20

-10

0

10

20

km

-30 -20 -10 0 10 20 30km

4

4

-20

-10

0

10

20

km

4

4

0 2 4 6 8 10mm

-30 -20 -10 0 10 20 30km

4

4

0 2 4 6 8 10mm

Figure 4.2 Horizontal relaxation after 1 year. Left column: Salvi FM. Right column: ZolloFM. Upper line: Poisson’s earth. Lower line: incompressible earth.

whereas in the incompressible case (lower-right) that area exhibits the sameamount of relaxation as the peripheral lobes.

It is important to recall that horizontal deformation in a spherical harmonicrepresentation is constituted by the contribution of a spheroidal and a toroidalcomponent, which are responding in a different way to the issues under analysis.For this reason it is useful to study them separately.

4.2.1 Horizontal spheroidal component

The spheroidal contribution for the shallow fault (left column of Figure 4.3) presentsa regular pattern: it is characterized by a generalized displacement away from the

4.2 Horizontal relaxation 59

-20

-10

0

10

20

km

-30 -20 -10 0 10 20 30km

2

4

4

-20

-10

0

10

20

km

2

4

46

0 2 4 6 8 10mm

-30 -20 -10 0 10 20 30km

2

1

1

1

2

0 1 2 3 4 5mm

Figure 4.3 Spheroidal horizontal relaxation after 1 year. Left column: Salvi FM. Rightcolumn: Zollo FM. Upper line: Poisson’s earth. Lower line: incompressibleearth.

middle point of the fault, with two lobes where most of the deformation localizes.Again, the only impact of compressibility is in the magnitude of the displacement.

The behaviour of the deep fault is very different (right column of Figure 4.3).The moderate contribution of the spheroidal component shows a regular pattern fora Poissonian earth (upper-right panel), with most of the displacement in the nearfield on the foot-wall. For an incompressible earth (lower-right panel), only the farfield is dominated by a regular motion away from the fault, whereas a large areadirectly around the fault shows a few inversions in the direction of displacement,with the highest magnitude on the hanging wall.


4.2.2 Toroidal component

Differently from the previous case, the toroidal component is not affected by thedegree of compressibility. For this reason only one image per each fault model isshown in Figure 4.4.

For the case of a shallow fault (left panel), we observe the appearance of fourmain lobes: the two normal to the fault show a spreading movement, whereas thetwo along the fault converge toward the center. When this component is addedtogether with the spheroidal parts shown in the left column of Figure 4.3, the twocontributions work constructively across the fault and destructively along the fault:for this reason, the final deformation pattern shows such marked lobes as in the leftcolumn of Figures 4.2. Moreover, the spheroidal contribution is larger.

The toroidal deformation pattern is slightly different for the deep fault (rightpanel of Figure 4.4): four main lobes are still present, but the highest displacementis on the foot-wall, with an additional considerable amount of deformation along thefault, in the two converging lobes. With respect to the total horizontal displacement,the largest contribution is in this case given by the toroidal component, whichis in turn not affected by compressibility. For this reason, the largely differentdeformation patterns of the spheroidal components (right column of Figure 4.3) arenot directly evident in the final deformation patterns (right column of Figure 4.2).

-20

-10

0

10

20

km

-30 -20 -10 0 10 20 30km

-30 -20 -10 0 10 20 30km

2 3

4

0 1 2 3 4 5mm

Figure 4.4 Toroidal relaxation after 1 year. Left panel: Salvi FM. Right panel: Zollo FM.

4.4 Addenda 61

4.3 Conclusions

A shallow and a deep fault have different styles of relaxation, both in patternand in response to changes in the rheology of the earth model. In particular,incompressibility has no effect on the toroidal components and shows the largesteffect on the deep fault.

For the shallow source, postseismic vertical deformation is characterized by alarge uplift on the foot-wall and a modest subsidence on the hanging wall. In thehorizontal direction, relaxation is concentrated in two lobes spreading away fromthe center of the fault, along the dip direction. An incompressible earth simplyexhibits a faster deformation.

The deep source, on the other hand, is always subsiding, but in a more diffusedway (and with a larger magnitude) for an incompressible earth, and only in twolobes at the ends of the fault for a Poissonian earth. In the horizontal direction,the area of largest deformation is on the foot-wall, but a considerable signal is alsopresent in the far-field, both along- and across-fault.

From this analysis, it is clear how the thickness of what is modelled as elasticcrust is able to control the relaxation process. The upper crust is not only acting asa low bandpass filter, by imposing a minimum wavelength at the postseismic signal,but it’s also characterizing the pattern and direction of the displacement observedat the surface.

This fact is of particular importance when we consider that most geodeticallyderived rupture models are obtained from inversions of coseismic signals, which arenot sensitive to the depth of the viscous layers.

We are thus able to affirm that it is in principle possible to constrain the thick-ness of the upper crust by observing postseismic relaxation. Requirement for such astudy, however, is the availability of a good fault model and a large geodetic dataset.

This study has also indicated how the level of compressibility is not only affectingthe deformation pattern in a quantitative way, acting as some kind of dumpingdevice, but it is also changing the style of relaxation.

4.4 Addenda

Discussion so far has been focused on end-member situations: shallow vs. deepfault and compressible vs. incompressible earth model. In general, extreme casesare useful to show the basic features of a physical process, but they are often farfrom being representative of real phenomena.

In this section we want to describe two intermediate cases, both about faultlocation (deep fault with shallow extension) and rheology of the earth model (semi-compressible case).


4.4.1 The double fault

A situation which is often occurring in reality is that an earthquake is rupturingtwo distinct fault planes, as proposed by Pollitz and Burgmann [1998] for the 1989Loma Prieta earthquake where the faulting process is expected to have rupturedboth a low angle shallow fault and a high angle deep fault.

Similarly, about the 1997 Umbria-Marche earthquake, Basili and Meghraoui[2001] published a refined version of the fault model proposed by Zollo et al. [1999],based on leveling measurements effectuated along a transect across the fault. Theresult of their inversion of coseismic deformation suggests the presence of a highangle (80o) upper fault branch, accommodating about 60% of the slip of the lowerfault and almost reaching the surface.

-20

-10

0

10

20

km

-30 -20 -10 0 10 20 30km

0

0

-20

-10

0

10

20

km

-4

-8 -6 -4 -2 0 2 4mm

-30 -20 -10 0 10 20 30km

4

4

0 2 4 6 8 10mm

Figure 4.5 Relaxation after 1 year for a Poisson’s earth. Left column: vertical component.Right column: horizontal component. Lower line: Basili-Zollo FM. Upper line:contribution of the steep-shallow rupture added by Basili&Meghraoui.

4.4 Addenda 63

The Basili&Meghraoui fault model provides a good opportunity to study therelaxation process for a complex faulting structure, when we can observe the com-bination the two end members (shallow/deep source) presented so far. For brevitysake, in Figure 4.5 we only show results for a compressible earth.

The deformation pattern induced by the shallow fault is different from the pre-vious case, due to the high value of the deep angle. In particular, the verticaldeformation (upper-left panel) is more symmetric, whereas the horizontal signal(upper-right panel) is concentrated in a single small lobe moving westwards. Mag-nitudes are smaller due to the lower amount of slip.

When both deep and shallow contributions are added together, the final defor-mation shows a remarkably different pattern from the cases previously analyzed(upper-left panels of Figures 4.1 and 4.2). Vertical displacement (lower-left panel)is characterized by a large area of subsidence embracing the hanging wall and thesides of the fault. Horizontal deformation (lower-right panel) maintains a four-lobespattern, but displacement on the foot-wall is distributed over a wider area and asmall maximum appears in the far field, west of the fault.

4.4.2 Semi-compressible model

An important consideration about most deformation models, including the normalmodes approach by Pollitz which has been used in this chapter, regards an as-sumptions used to obtain the generalized constitutive equation (see Ranalli [1995],Chapter 4). Namely, flow is usually supposed to be incompressible, also in mostso-called compressible models. The compressible and self-gravitating technique dis-cussed in the Section 2.3 represents a rare exception, but application to earthquakesis not yet completely feasible.

A possible approximated approach to obtain compressible relaxation consists inlaying a compressible elastic upper crust on top of an incompressible visco-elastic

-20

-10

0

10

20

km

-30 -20 -10 0 10 20 30km

-4

-4

0

0

0

-6

-4

-2

0

2

mm

Figure 4.6 Zollo FM: vertical relaxation after 1 year, obtained with a semi-compressibleearth model (compressible elastic layer, incompressible visco-elastic layers).


earth. In Figure 4.6, we display the result of this hybrid model, for the case ofvertical deformation due to a deep source, where the largest difference betweencompressible and incompressible models was observed. Results have to be comparedwith the right column of Figure 4.1, on page 57.

The test clearly provides a positive result: the deformation pattern of the semi-compressible model is very close to the full compressible case, displayed in theupper-right panel of Figure 4.1, and completely different from the incompressiblemodel, shown in the lower-right panel of the same figure. The magnitude of subsi-dence is slightly underestimated, but the main features are present, including theconcentration of deformation at the sides of the fault and the presence of a moderaterelative maximum in the center.

This behaviour implies that compressibility plays a minor role for the viscoelasticlayers, fact which is not a surprise: viscous flow is anyway modelled as incompress-ible and the elastic response is only important on a short time scale (deformationis here computed at one year after the earthquake, comparable with the Maxwelltime τM ).

On the other side, when this test was first performed it was not clear what theimpact of a compressible elastic crust would be on postseismic deformation. Fromthis simple example, it is possible to state that the level of compressibility of theupper layer is enough to completely modify the deformation pattern as observedat the surface.

The last statement has two important consequences: from the physical point ofview, it increases the importance of the upper crust in characterising postseismicrelaxation; from the technical side, it allows an easier implementation of a (semi-)compressible model based on the self-gravitating approach developed from Sabadiniet al. [1982] until Vermeersen and Sabadini [1997].

Gravitational effects play a minor role for the time and scale of the Umbria-Marche earthquake, discussed as a study case in the present thesis, but they mightbecome relevant for larger seismic events. For this reason, a fully operational spher-ical, self-gravitating and semi-compressible viscoelastic model is presently underdevelopment.

Chapter 5

Central Apennines: constraintson viscosity and fault geometry

5.1 Crustal vs. upper mantle relaxation

Several studies of postseismic deformation have been dealing with relaxation in thelower crust and the mantle [Piersanti et al., 1997; Pollitz and Burgmann, 1998;Pollitz et al., 2000]. The interplay between crustal and mantle relaxation appearsto be very different, depending on the specific location, magnitude and kind ofearthquake.

Piersanti et al. [1997] studied postseismic deformation for the 1964 Alaska earth-quake and combination with VLBI data: due to the large size of this event, theymodelled a completely elastic crust (100 km thick) and studied the effect of man-tle relaxation. The observed deformation pattern can be reproduced by relaxationin a weak region in the upper mantle (between 100 and 300 km, viscosity around1019 Pa s).

Pollitz and Burgmann [1998] focused on the first five years after the 1989 LomaPrieta earthquake, covered by GPS data. They came to the conclusion, alreadyproposed by Savage et al. [1994] for the first three years after the event, that post-seismic deformation is mainly dominated by after-slip. The smaller contributionof viscoelastic relaxation doesn’t allow to distinguish between a pure lower crustalflow or a combined lower crustal and mantle flow (viscosities around 1019 Pa s).

Pollitz et al. [2000] studied postseismic deformation during the first threeyears after the 1992 Landers earthquake and compared their results with GPSand InSAR data. Viscoelastic relaxation is accounting for most of the observeddeformation and it’s mainly concentrated in the mantle (Moho at 30 km), withviscosity between 1018 and 1019 Pa s. Relaxation is also expected in the lower crust(between 14 and 30 km depth), but viscosity values are poorly constrained by thedata. A model with low viscosity in the lower crust (1018 Pa s) and a hard mantle,

66 Central Apennines: constraints on viscosity and fault geometry

as proposed by Deng et al. [1998], provides a much poorer fit to the data.

As we can already see from those studies, which don’t cover the whole literature,but are widely cited examples for postseismic deformation, crustal relaxation isoften considered either not relevant or a co-player with mantle relaxation. Thoseconclusions are supported by high quality data and hold for both dip-slip and strike-slip events.

Nonetheless, it has already been shown that for the 1997 Umbria-Marche earth-quake, the only condition to have some detectable postseismic deformation signalsis to have low viscosities (1017 − 1019 Pa s) in the crust [Riva et al., 2000]. Ver-tical deformation rates of the order of few millimeters per year on a spatial scaleof 50 km are expected for an earth model with a low viscosity zone decoupling theelastic upper crust and the lower crust. On the contrary, a model with a hardercrust (1021 Pa s) on top of a weaker mantle (1019 Pa s) provides extremely smalldeformation rates (of the order of one tenth of a millimeter per year) spread over amuch broader area.

At the time of publication, when GPS campaigns in the earthquake area hadjust started, crustal flow was proposed as a candidate for postseismic relaxation.Data collected in the following years supported this initial statement: the outcomeof the first two campaigns was published by Aoudia et al. [2003], whereas a completediscussion of the results follows in this chapter.

5.2 GPS campaign results

Four GPS campaigns, between October 1999 and May 2003, have been carried outalong a 30 km cross-section perpendicular to the principal fault, comprising sixobservation points. The cross-section goes through the area of maximum release ofthe 09:40 earthquake fault.

The network geometry is shown in Figure 5.1, where the surface projection ofthe fault trace is represented as a solid black line. Site locations were chosen inorder to be as much as possible aligned perpendicularly to the fault. However, thefinal choice has been strongly influenced by the rapid change in local topography,together with the need to place monuments on stable ground (where the bed-rockis exposed) and in open locations (where the view of the satellites constellation ismaximal).

Monumentation was carefully performed to ensure a sub-millimeter centering.At each site, a 25 cm long steel rod was fixed in solid rock and the antenna wasmounted using a forced stationing device. During all campaigns, data were collectedover a period of four consecutive days, with daily sessions of eight hours. Thesampling rate was fixed at 15 s and the cut-off angle at 15o. The measurementswere collected with Trimble 4000 and Trimble 4700 receivers and a combination ofTrimble Geodetic and Trimble Micro-Centered antennas.

5.2 GPS campaign results 67

12˚ 36' 12˚ 48' 13˚ 00'

43˚ 00'

43˚ 12'

0

250

500

750

1000

1250

1500

m

SEFR

MONT

COLL

CERE VALL

SPEL

Figure 5.1 Topography, fault location and GPS network configuration. The black solid linerepresents the surface projection of the fault trace.

Data analysis was performed by Dr. Alessandra Borghi at Politecnico di Mi-lano with the software package Bernese 4.2 and by Ir. Saskia Matheussen at DelftUniversity of Technology with the software package Gamit 10.07. Tropospheric pa-rameters were estimated on a two-hourly basis. Wet zenith delays were modeledas stochastic parameters and ambiguity fixing was applied. In both data analysisschemes, the observations were first analyzed on a daily basis, yielding full networksolutions. Subsequently, the daily network solutions were combined into multi-daysolutions for each year.

The unpublished results discussed in this chapter were made available underthe umbrella of a collaboration within different institutions (further including thedepartments of Earth Sciences of Milano University and of Trieste University).

5.2.1 Yearly solutions

In Figure 5.2, we show an example of time-series for two baselines, namely COLL-CERE (left) and CERE-VALL (right). From the top, panels show the three com-ponents (North, East and Up) and the change in baseline length, with respect tothe estimated trend. Values on top of each panel are referred to the interpolatedline and show velocities (in mm/yr) with formal sigmas and scattering (r.m.s.). Inthose time-series, it is important to stress that error bars represent formal sigmas,which are usually over optimistic. A more realistic estimate of the uncertainties iscoming from the weighted r.m.s. (w.r.m.s.) values.

For the baseline COLL-CERE (describing how CERE is moving w.r.t. COLL),we see that yearly solutions are evenly spread around the interpolated line, with all


Figure 5.2 Time-series for the baselines COLL-CERE (left) and CERE-VALL (right).

the significant horizontal motion in the North component, unluckily almost perpen-dicular to the baseline itself, and a rather large signal in the vertical direction. Thecombination of the three components leads to the baseline change, which doesn’tshow a significant variation through the years.

The baseline CERE-VALL is shown here as an example of a worse solutionthan the case above. With respect to the estimated trend, the deformation ratebetween the first two campaigns is different in both magnitude and direction: thisfact is the main reason for the large w.r.m.s. values. If more campaigns wereavailable, it might have been possible to prove that the second year is an out-lierand remove it from the dataset. Unluckily, also the remaining three campaigns arenot completely in agreement with each other: the most reasonable choice, in thiscase, is to keep all campaigns and take into account the fact that accuracy on thisbaseline is poor.

In Figure 5.3 we show the baseline changes for all campaigns, with respect tothe position of SPEL in 1999. Error bars represent one sigma. The solution hasbeen provided by Alessandra Borghi from Polimi. Differences with respect to theprevious two pictures are due to the fact that the former results have been obtainedby Saskia Matheussen at DEOS, by applying slightly different precessing strategies.


-15

-12

-9

-6

-3

0

3

6

9

12

15B

asel

ine

varia

tion

(mm

)

0 5 10 15 20 25Baseline length (km)

99-0099-0199-03

SPEL VALL CERE COLL MONT SEFR

Figure 5.3 GPS baselines changes 1999-2003 w.r.t. SPEL

However, both solutions agree within the uncertainty level. The surface extensionof the fault is located at about 18 km from SPEL.

GPS results show displacements in the order of a few millimeters per year:the baseline SPEL-VALL is the only one showing shortening, while all the otherpoints move away from SPEL. Results for the first two campaigns (1999-2000)are represented as diamonds connected by a solid line, for the first and the thirdcampaign (1999-2001) squares connected by a dashed line and for the largest time-span (1999-2003) stars connected by a dotted line. The use of joining the differentbaseline changes with lines is purely for clarity of representation: in reality, as it ispossible to see in Figure 5.1, the sites are not perfectly aligned and baseline changesdon’t allow the description of a continuous deformation pattern.

In general, GPS results show a progressive increase in displacement, consistentwith expectations for viscoelastic relaxation. We remind that the first campaigntook place two years after the earthquake, so that most transitory relaxation pro-cesses, like after-slip, are not expected to be relevant.

The behaviour of single sites is different, but the relatively large sigmas, in theorder of 2 mm, can account for most of the inconsistencies. In particular, VALLappears to move very little in the first two years, followed by a large displacementbetween the last two campaigns. CERE remains around the position assumed afterthe second campaign. COLL shows a progressive increase in deformation, reachinga final extension of 9 mm. MONT shows large displacements in the first two years,


0

100

200

300R

adio

Flu

x (1

0.7

cm)

2001.50 2001.75 2002.00year

Figure 5.4 Solar activity in the second half of 2001 (source: U.S. Dept. of Commerce,NOAA, Space Environment Center).

followed by a null motion between the last two campaigns. SEFR has an oscillatorybehaviour, with a final relative displacement of about 5 mm.

Overall, the displacement pattern of the second and the fourth campaign, withrespect to position in 1999, are in very good agreement with expectations, whereasresults for 2001 present some odd behaviours. However, during that campaignan intense solar activity has taken place (radio flux above 150, see Figure 5.4), afact which is further supported by the large impact that different corrections forionospheric delay have on the final results.

5.2.2 Rates of baseline change: network consistency

The baseline changes showed in Figure 5.3 present a general agreement with the ex-pected extensional regime due to postseismic relaxation, together with some incon-sistencies. Though within the uncertainty level, facts like the sudden acceleration ofVALL and COLL between the last two campaigns, contemporary to a null motionof MONT, suggest that year-to-year estimates of displacement are not completelyreliable.

A major difficulty in understanding if the relative motion of the GPS sites hasan internal consistency comes from the use of one of the six sites as reference point.This choice is forced by the small size of the network and the lack of an externalreliable reference station at a useful distance. In order to overcome this difficulty,it is possible to examine the relative motions of all fifteen possible combinationsbetween the six sites.


Baseline ∆l (mm) r.m.s. n.r.m.s

MONT-VALL* 11.0 1.4 0.5CERE-SPEL* 2.9 2.8 1.0COLL-SPEL* 8.6 3.0 1.1SEFR-COLL* -7.2 3.1 1.1VALL-SPEL* -8.8 3.5 1.4SEFR-CERE 1.8 4.4 1.6COLL-CERE 1.4 5.0 1.8SEFR-MONT -2.7 6.3 2.3SEFR-SPEL 3.7 6.7 2.4MONT-CERE 4.2 8.2 2.9MONT-SPEL 7.7 8.3 3.0

MONT-COLL -1.7 10.5 3.8CERE-VALL 6.0 13.7 4.9SEFR-VALL 9.9 14.1 5.0COLL-VALL 7.5 15.6 5.6

Table 5.1 Baselines variation and standard deviations between 1999 and 2003.

For the specific time-span under analysis, i.e. between 2 and 5.5 years afterthe earthquake, viscoelastic relaxation follows an almost linear behaviour. Thus,the best way to obtain some robust estimate of the measured deformations is toperform a linear regression through all the campaign years.

In Table 5.1 we list the resulting deformation accumulated during whole cam-paign period (Sept. 1999 - May 2003) for all fifteen combinations, together with thestandard deviation as it results from the interpolation (r.m.s) and in a renormalizedversion (n.r.m.s.) meant to give a less conservative measure of the error. Resultsare sorted according to the level of scattering (best on top).

From the analysis of Table 5.1, it is possible to make some considerations aboutthe reliability of the campaign results: first of all, it is evident how some baselinespresent an abnormal level of scattering, with respect to the rest of the network.A line has been drawn to cut off the four combinations with a standard deviationabove the centimeter level: it is useful to point out how VALL is present in threeof those baselines, a fact which hints that some site effect could be present. Thesecond ”worst performing” site is clearly MONT, with three r.m.s. values above8 mm: curiously, the combination MONT-VALL is providing the over-all best fit.There is no clear explanation for this fact, most likely due to some error cancellationeffect.

The other sites behave in a homogeneous way, with standard deviations between3 and 6 millimeters. Those error values seem acceptable for campaign results, butthey turn out to be very pessimistic when they are compared with the magnitudeof the measured deformations, also of the order of a few millimeters. If the scat-


-15

-12

-9

-6

-3

0

3

6

9

12

15B

asel

ine

varia

tion

(mm

)

0 5 10 15 20 25Baseline length(km)

SPEL VALL CERE COLL

FAU

LT

MONT SEFR

Figure 5.5 Interpolated GPS baselines changes 1999-2003 w.r.t. SPEL (black dots) andCERE (gray stars). Error bars represent the n.r.m.s. values listed in Table 5.1.

tering is considered as the real error, then only a few baselines are statisticallysignificant (they have been marked with an asterisk in Table 5.1), which wouldtechnically prevent any further use of the GPS measurements. For this reason, itseemed reasonable to use the r.m.s. as a scaling factor to make the formal sigmasmore realistic. The baseline CERE-SPEL, the second best one, has been chosen asreference and its r.m.s. has been renormalized to a unitary value; consequently, allother r.m.s. entries have been divided by a factor 2.8. In this way, baselines can becharacterized by an error which is consistent with the level of scattering, but whichis also small enough to allow some use of the measurements.

The discussion about the real uncertainty of GPS measurements, especially whentalking about campaign data, is far from a general agreement and beyond thepurpose of this study. The choice of rescaling the errors is a common practice andin the specific case is supported by the comparison between the GPS results justdiscussed and the model predictions which are presented in the following section.

Results for the whole campaign period, with respect to SPEL (black dots) andCERE (gray stars), are shown in Figure 5.5. Error bars represent the n.r.m.s. valueslisted in Table 5.1.

5.3 Viscosity models for relaxation in the transition zone and lower crust 73

5.3 Viscosity models for relaxation in thetransition zone and lower crust

In this section, relaxation for a number of different crustal viscosity models willbe analyzed. As in Chapter 4, the solution of the fundamental equation will makeuse of Pollitz’s non-gravitational version of the analytical approach discussed inthe second chapter, for a spherically layered, compressible, viscoelastic Earth withMaxwell rheology, Comparison with the GPS data will be the object of the lastsection of this chapter.

According to the results published by Chimera et al. [2003], discussed in moredetail in Section 6.2, the crust will be divided into three layers:

- an elastic upper crust (UC), 8 km thick;- a visco-elastic transition zone (TZ) between 8 and 20 km depth;- a visco-elastic lower crust (LC) between 20 km and 35 km depth.

Viscosity in both the TZ and the LC has been varied between 1017 Pa s and1019 Pa s, whereas the mantle has been represented as a single homogeneous layerwith viscosity 1021 Pa s.

Both the geodetic [Salvi et al., 2000] and the seismological [Zollo et al., 1999]fault model have been used, according to the evidence presented in the previouschapter about the impact that different fault locations have on the relaxationprocess.

For clarity of representation and discussion, results will be presented only forthe four most representative earth models, summarized in Table 5.2. The charac-teristic relaxation time (or Maxwell time, τM = η/µ) for the elastic parametersused in this study is about one year for a viscosity of 1018 Pa s: this means thatthere is no reason to extend the analysis beyond the limit of 1019 Pa s (τM ' 10a). As a lower limit, we chose a viscosity of 1017 Pa s (τM ' 0.1 a) because weexperimentally saw that the most relevant modes for the analysis of postseismicrelaxation (spatial scale extending to a few tens of kilometers, i.e. high degreeharmonics) have characteristic relaxation times close to the Maxwell time.

Sections of surface deformation perpendicular to the fault and accumulated be-tween 2 and 5.5 years after the earthquake (time-span of the GPS campaigns) are

Layer MOD1 (Pa s) MOD2 (Pa s) MOD3 (Pa s) MOD4 (Pa s)

TZ 1018 1018 1019 1017

LC 1018 1017 1017 1018

Table 5.2 Viscosity models.


-20

-10

0

10

20

vert

(m

m)

TZ18-LC18TZ18-LC17TZ19-LC17TZ17-LC18

-20

-10

0

10

20

horiz

(m

m)

-30 -20 -10 0 10 20km


-20

-10

0

10

20

vert

(m

m)


-20

-10

0

10

20

horiz

(m

m)

-30 -20 -10 0 10 20km


Figure 5.6 Sections of vertical (top panels) and horizontal (bottom panels) relaxationbetween 1999 and 2003. Two fault models (Salvi, left, and Zollo, right) andfour viscosity models (MOD1, solid; MOD2, dashed; MOD3, dotted; MOD4,dash-dotted).

shown in Figure 5.6, where the fault is dipping to the left and the zero positionmarks the top of the rupture.

In the upper-left panel, vertical deformation for the shallow fault model (Salvi,geodetic) is presented: MOD1 and MOD2 are both characterized by a predominantpositive central peak, bounded by an area of moderate subsidence. It is worthnoticing that the model with a viscosity of 1017 Pa s in the LC (MOD2) shows amore positive deformation: this is due to the fact that most of the relaxation insuch a low viscosity layer takes place in the first few months after the earthquake(τM ' 0.1 a), so that almost complete relaxation has already taken place before theGPS campaigns and the positive contribution of the LC is missing. If viscosity inthe TZ is set to a higher value (1019 Pa s, MOD3), the deformation pattern becomescompletely positive and the magnitude is reduced to about one third of the previouscase. The last earth model, MOD4, is characterized by a considerable signal witha much longer wavelength: in this case it is the TZ which has relaxed completely,so that deformation is taking place only in the LC.

The case of horizontal deformation for a shallow fault, presented in the lower-

5.4 Channel relaxation 75

left panel of Figure 5.6, leads to similar considerations: MOD1 and MOD2 remainthe cases with the largest deformation and MOD3 presents a similar pattern, but areduced magnitude. It is interesting to observe how almost no difference is presentbetween MOD1 and MOD2, which suggests that horizontal flow is less sensitive todeeper layers: this consideration is also supported by the result for MOD4, which isagain characterized by a longer wavelength, but with a relatively smaller magnitudethan for the vertical case.

The situation is very different when the fault is located in the lower half ofthe elastic upper crust (Zollo, seismological). As already presented in the previouschapter, vertical deformation (upper-right panel of Figure 5.6) becomes dominatedby subsidence, with different specific patterns depending on the viscosity structure.MOD1 and MOD2 present a marked area of subsidence bounded by peripheraluplifting bulges, with the addition of a small relative minimum in the center. MOD3is affected by a very small deformation and, noteworthy, no significant subsidence,whereas MOD4 shows a wide central subsidence and lateral uplift, but it’s missingthe small central peak which characterizes all the other models.

Last, horizontal deformation for a deep source (lower-bottom panel of Figure5.6) follows a double-peaked pattern for the first three models, whereas MOD4appears to behave in a very different way. In reality, this effect is again due tothe fact that a dominantly lower crustal relaxation is happening on a much longerwavelength, so that only the first two central peaks are visible in the picture.

From the four cases presented here, it is possible to draw a few remarks aboutthe style of relaxation for an earth model with very shallow low-viscosity zones.

In general, the shortest wavelength signals depend on the shallowest viscouslayer, which in principle allows to distinguish whether relaxation is taking place inthe TZ or in the LC.

Further, since the main purpose of this chapter is to use GPS measurementsto retrieve information about the viscosity structure in the Central Apennines, thetime span of the observations (2 to 5.5 years after the earthquake) puts a limitto the viscosity range which can be detected. In particular, it is possible to testonly the presence of layers with an average viscosity around the value of 1018 Pa s:lower viscosities lead to deformation concentrated in the first few months after theseismic sequence, whereas higher viscosities generate too little displacements to bedetectable over a time span of three and half years.

The last issue concerns how stress flows through two consecutive layers withdifferent viscosities: from the examples shown in Figure 5.6, it strikes how relaxationfor MOD1 is very different from the combined effect of MOD2 and MOD4. Thisphenomenon is consistent with the fact that different layers are coupled with eachother, as it was already known from the mathematical formulation of the normalmodes approach, in a way that relaxation is also depending on the viscosity contrastbetween two consecutive viscoelastic layers.


-20

-10

0

10

20

vert

(m

m)

8-18 km8-13 km8-11 km13-18 km

-20

-10

0

10

20

horiz

(m

m)

-30 -20 -10 0 10 20km

8-18 km8-13 km8-11 km13-18 km

-20

-10

0

10

20

vert

(m

m)

8-18 km8-13 km8-11 km13-18 km

-20

-10

0

10

20

horiz

(m

m)

-30 -20 -10 0 10 20km

8-18 km8-13 km8-11 km13-18 km

Figure 5.7 Sections of vertical (top panels) and horizontal (bottom panels) relaxationbetween 1999 and 2003 for the two fault models (Salvi, left, and Zollo, right).Results for different thicknesses and locations of the TZ (see legend in figure).

5.4 Channel relaxation

The discussion of the previous section has been focused on an earth model with twoconsecutive viscous layers, namely the crustal transition zone (TZ) and the lowercrust (LC). Considerations about composition and thermal structure for the crust,which will be the object of discussion in the next chapter, suggest the possibilityof a very different scenario, in which the lower crust is dominated by an elasticbehaviour.

The object of this section is to perform a simple sensitivity study about theeffect of changes in location and thickness of the TZ, when this is the only viscouscrustal layer. The earth model will consist of an elastic upper crust, a low viscositytransition zone (1018 Pa s, the reference value in the previous section) and an elasticlower crust. Viscosity in the mantle is fixed to 1021 Pa s, but the effect is irrelevant.

As in the previous section, both the shallow (Salvi, geodetic) and the deep (Zollo,seismological) fault model have been analyzed. Results are plotted in Figure 5.7,following the same graphical composition as in Figure 5.6.

In all panels, the first three models have an 8-km-thick elastic upper crust, while

5.4 Channel relaxation 77

the depth of the bottom of the TZ is respectively set at 18 km (10-km-thick TZ,solid line), at 13 km (5-km-thick TZ, dashed line) and at 11 km (3-km-thick TZ,dotted line). The fourth model has a 13-km-thick UC, followed by a 5-km-thick TZ(dash-dotted line).

As is visible in the top left panel, lowering the bottom of the TZ has a doubleeffect on the vertical relaxation signal for a shallow earthquake: the central upliftincreases substantially and the deformation area broadens. Noteworthy, the amountof subsidence remains almost unchanged. The last model, where the viscous channelhas a deeper location, shows a reduced amount of deformation, both for the centraluplift and the lateral subsidence, at a longer wavelength. A similar behaviour ispresent in the case of horizontal deformation, plotted in the left-bottom panel ofFigure 5.7.

The case of a fault closer to the bottom of the elastic upper crust is shown inthe right column of Figure 5.7. When the UC is only 8 km thick, the central areais dominated by a large subsidence and further characterized by some degree ofasymmetry between the two sides of the fault (vertical deformation, upper-rightpanel). The relative behaviour of the first three models is rather different fromthe case of a shallow fault: deformation on the hanging wall (left side) remainsalmost unchanged, whereas more subsidence appears on the foot-wall (right side)when the TZ thickens. The 3-km-thick TZ (dotted line) shows almost no trace ofa second maximum of deformation on the foot-wall, whereas the 10-km-thick TZ isclose to the case where also the LC is viscous, as in Figure 5.6. Interesting is thebehaviour of the fourth model (dash-dotted line), where deformation approachesthe case of a shallow source, with a dominant central uplift and minor lateralsubsidence. Again, a similar relative behaviour is present for the case of horizontaldeformation (bottom-right panel).

In general, it is possible to state that for the case of a shallow fault, or moreprecisely the case of a rupture which is relatively far from the bottom of the elasticupper crust, the pattern of relaxation is mainly controlled by the location of thetop of the viscous layer, whereas the magnitude of deformation is largely affectedalso by the thickness of the relaxing layer.

When the fault is rupturing the base of the elastic layer, on the contrary, thestress gradient in the first few kilometers of the viscous layer is very large: this factis responsible for both the specific deformation pattern and the reduced sensibilitywith respect to the thickness of the relaxing layer.

Those considerations are supported by an additional test which has been per-formed: relaxation due to the 10-km-thick TZ has been compared with the con-tributions of the other two 5-km-thick TZs. With respect to Figure 5.7, the dashand dash-dotted lines have been subtracted from the solid line and the absolutedifferences have been normalized with respect to the 13-18-km model. Residualsare plotted in Figure 5.8. Apart from the sharp peaks, due to the fact that the


0.0

0.5

1.0

1.5

2.0

Nor

mal

ized

res

idua

ls

-30 -20 -10 0 10 20km

0.0

0.5

1.0

1.5

2.0

Nor

mal

ized

res

idua

ls

-30 -20 -10 0 10 20km

Figure 5.8 Difference between relaxation due to a single 8-18 km channel and two distinctchannels (8-13 km and 13-18 km). Residuals for vertical (solid) and horizontal(dashed) deformation for the two fault models (Salvi, left, and Zollo, right).Absolute residuals are normalized with respect to the 13-18-km model.

reference curve used for normalization presents zeros, it is evident how the level ofcoupling between the two viscous layers is much lower for the shallow fault.

Again, this effect can be explained in terms of stress gradient, which is thefundamental reason for viscous flow to occur. For a shallow fault, the difference instress gradient between the layer starting at 8 km and the layer starting at 13 km isnot extreme: the rate of flow for the two layers, which are further characterized bythe same rheological properties, will be similar and for this reason their combinedbehaviour is close to a single 8-18 km layer. On the contrary, flow in the upperviscous layer will be much faster when the rupture is close, as in the case of a deepfault, in a way that increases the level of coupling between the two parts in whichthe 10-km-thick TZ has been divided.

5.5 Mechanical model vs. GPS results

In this section, surface horizontal deformation expected from a number of relaxationmodels will be tested against the GPS results presented in Table 5.1 on page 71and plotted in Figure 5.5 on page 72.

In the following sub-section, attention will be focused on the results alreadydiscussed earlier in the present chapter, whereas further refinements on the faultmodels will be discussed later.

5.5.1 Viscosity models validation

The first step to compare model deformations with GPS data is to put both of themin the same reference frame: for this reason, the sections represented in the bottom

5.5 Mechanical model vs. GPS results 79

-20

-10

0

10

20

horiz

(m

m)

-20 -10 0 10km


-20

-10

0

10

20

horiz

(m

m)

-20 -10 0 10km


Figure 5.9 Sections of horizontal deformation between 1999 and 2003 w.r.t. Spello. Dotsrepresent GPS baseline variations. Model results for two fault models (Salvi,left, and Zollo, right) and four viscosity models (see legend in figure).

panels of Figure 5.6 and 5.7 have here been referred to SPEL, the westernmost GPSsite located at about 19 km from the fault.

GPS measurement results have been superimposed to the pictures, with errorbars representing the renormalized standard deviation (n.r.m.s. of Table 5.1).

It is very important to stress once more that the GPS network is not perfectlyaligned across the fault, as is visible in Figure 5.1, and for this reason it is not com-pletely correct to compare measurements of baseline length variation with sectionsof model deformation. Nonetheless, the purpose of the current discussion is onlyto provide a first order comparison between model and data: a real baseline-to-baseline validation will follow in the next subsection, together with the discussionabout heterogeneous slip fault models.

In Figure 5.9, GPS measurements are compared with deformation resultingfrom relaxation of the four models with low viscosity in both TZ and LC (as inFigure 5.6). The Salvi fault model, in the left panel, shows a general agreementwith the data when viscosity in the TZ has a value of 1018 Pa s. More in detail,the model underestimates the shortening between SPEL and VALL and fails toreproduce the short wavelength behaviour between CERE and SEFR. As far asthe Zollo fault model is concerned, there is a complete disagreement with respectto the baseline SPEL-VALL, but a rather good fit for the rest of the network,especially when both TZ and LC have a viscosity of 1018 Pa s.

For the case of channel relaxation, showed in Figure 5.10 for the same modelsalready presented in Figure 5.7, no substantial differences arise. Results for theSalvi fault model tend to underestimate the motions, but are characterized by ashorter wavelength that better fits the general picture. Almost unchanged is theresult about the Zollo fault model, coherently with the consideration that most ofthe relaxation for this case is taking place in the shallow part of the TZ.

In general, all earth models with a TZ starting at a depth of 8 km and with


-20

-10

0

10

20

horiz

(m

m)

-20 -10 0 10km

8-18 km8-13 km8-11 km13-18 km

-20

-10

0

10

20

horiz

(m

m)

-20 -10 0 10km

8-18 km8-13 km8-11 km13-18 km

Figure 5.10 Sections of horizontal deformation between 1999 and 2003 w.r.t. Spello. Dotsrepresent GPS baseline variations. Model results for two fault models (Salvi,left, and Zollo, right) and four TZ channels (see legend in figure).

a viscosity of 1018 Pa s succeed in reproducing the general pattern of deformationrevealed by the GPS measurements. Pure lower crustal relaxation, as well as a deepTZ, completely fail in reproducing both the wavelength and the magnitude of themeasured signal.

As far as the fault model is concerned, Zollo provides a better fit to the foureasternmost sites, but predicts extension between SPEL and VALL. On the con-trary, Salvi produces a moderately worse fit to the data, but it also predicts someamount of shortening in the western part of the transect.

5.5.2 Model baseline variations

The final considerations of the previous sub-section are based on a comparisonbetween model sections and GPS measurements of baseline length variation, per-formed to provide a first order validation of model results. The main reason todiscuss model sections is a more clear discussion of the physics of relaxation, with-out the bias introduced by the projection of three-dimensional deformation patternsinto one-dimensional information (baselines).

The evaluation of baseline change from modelled deformation represents a fur-ther step in the search for the best fitting earth and fault models. In Figure 5.11,we present again a section of horizontal deformation for an earth model where bothTZ and LC have a viscosity of 1018 Pa s (solid lines of Figures 5.9 and 5.6) for bothfault models. Black dots with error bars represent the GPS measurements.

Open circles are the baseline length variations predicted by the same modelas for the solid line. For the case of the shallow fault (left panel), the projectionreduces the expected deformation: the effect amounts to about one millimeter forthe three western sites and becomes dramatic for the baseline SPEL-VALL, whichreduces from about 2.5 mm to less than 1 mm. The deep fault, on the contrary,


-15

-10

-5

0

5

10

15

Bas

elin

e va

riatio

n (m

m)

0 10 20 30Baseline length (km)

SPEL VALL CERE COLL

FAU

LT

MONT SEFR-15

-10

-5

0

5

10

15

Bas

elin

e va

riatio

n (m

m)


SPEL VALL CERE COLL

FAU

LT

MONT SEFR

Figure 5.11 Baseline length variations between 1999 and 2003, w.r.t. SPEL and for bothfault models (Salvi, left, and Zollo, right). Black dots represent GPSmeasurements, open circles are model predictions with uniform slip distributionon the fault and open diamonds model predictions with heterogeneous slipdistribution. The solid lines are re-proposed from Figure 5.9, as a reference.

is only slightly affected by the projection, apart from the baseline SPELL-COLLwhich reduces from 6 mm to about 4.5 mm.

The previous considerations about comparison with the GPS measurementsremain thus almost unchanged, with the only important exception of the baselineSPEL-VALL no longer extremely favouring the Salvi fault model.

All the results presented so far have been obtained by assuming a uniform slipdistribution over the whole rupture, because a symmetric displacement field is moreappropriate when comparing sections of deformation for different earth models.Nonetheless, both Salvi and Zollo present a rupture solution which concentratesmost of the slip on the southern end of the fault, where rupture initiated.

The impact of the asymmetric slip distribution on the baseline variations hasbeen tested for the same earth model as above, where both TZ and LC have aviscosity of 1018 Pa s. Results are presented in Figure 5.11 as open diamonds. Thealmost perfect agreement between circles and diamonds shows how the refinedrupture models do not influence the expected motion of the GPS sites: for allpractical purposes, thus, a homogeneous slip distribution can be considered to bea very good approximation.

Another point of attention regards the possibility that the choice of a differentreference point could influence the way model results are validated by GPS mea-surements. We have already discussed how the error distribution over the variousbaselines is not homogeneous: according to the results listed in Table 5.1, CEREqualifies as the second best site after SPEL (for the second largest number of base-lines with small errors). For this reason, in Figure 5.12 model predictions of baselinelength change with respect to CERE have been compared with GPS measurements:


-15

-10

-5

0

5

10

15

Bas

elin

e va

riatio

n (m

m)

-10 0 10Baseline length (km)

SPEL VALL CERE COLL

FAU

LT

MONT SEFR-15

-10

-5

0

5

10

15

Bas

elin

e va

riatio

n (m

m)


SPEL VALL CERE COLL

FAU

LT

MONT SEFR

Figure 5.12 Baseline length variations between 1999 and 2003, w.r.t. CERE and for bothfault models (Salvi, left, and Zollo, right). Black dots represent GPSmeasurements and open circles are model predictions with uniform slipdistribution on the fault.

the earth model is the same as for Figure 5.11 and the slip distribution on the faultsis homogeneous. The effect of setting CERE as reference site is to favour Zollo’sfault solution: with attention on the right panel of Figure 5.12, it is possible tosee how all sites show a very good agreement between measurements and modelpredictions, with the exception of the baseline CERE-VALL, which is anyway char-acterized by a very large degree of uncertainty. On the other hand, Salvi faultsolution (left panel) fails in reproducing any observed motion: it under-estimatesthe first two sites on the left and over-estimates the rest of the network.

5.5.3 Constraints on viscosity values and fault models

The most important result is found in the good overall agreement between mea-sured and modelled displacements, maintained over the whole campaign period:it supports the initial geophysical considerations which suggested the presence offairly low viscosity values in shallow crustal layers.

Though measured displacements are rather small (less than one centimeter overthe whole campaign period) and characterized by relatively large uncertainties, theinternal deformation revealed by the GPS network shows a signal consistent withpostseismic viscoelastic relaxation.

The pattern of deformation requires the presence of a layer with low viscosityat a depth of about 8 km and the magnitude of the signal constraints viscosityin the layer to a value around 1018 Pa s. Lower crustal relaxation is a possiblecoexistent process, but the resolution of the data is not high enough to put furtherconstraints on the actual viscosity values.

The second issue, namely the fault model problem, cannot be addressed withsuch a certainty.


-15

-10

-5

0

5

10

15

Bas

elin

e va

riatio

n (m

m)


SPEL VALL CERE COLL

FAU

LT

MONT SEFR-15

-10

-5

0

5

10

15

Bas

elin

e va

riatio

n (m

m)


SPEL VALL CERE COLL

FAU

LT

MONT SEFR

Figure 5.13 Baseline length variations between 1999 and 2003, w.r.t. SPEL (left) andCERE (right). The fault model follows the refinement of Zollo proposed byBasili&Meghraoui.

The seismological fault model proposed by Zollo appears to be favoured by thedata, especially when baseline length variations are referred to CERE. The levelof agreement between measurements and model predictions is remarkable whenviscosity is fixed to a value of 1018 Pa s between a depth of 8 km and the Moho at35 km (right panel of Figure 5.12).

Nonetheless, the geodetic fault model proposed by Salvi could provide anequally good fit in a possible alternate scenario where low viscosity is concentratedin a narrower channel, as visible in Figure 5.10.

The third fault model, proposed by Basili and Meghraoui [2001] as a geodeticrefinement of Zollo’s seismological model, is tested against GPS measurements inFigure 5.13. Results in the left panel, referred to SPEL, have to be compared withFigure 5.11: the level of fit is poorer than the other two fault models, especiallyfor the two sites COLL and MONT, the closest to the rupture. Deformation withrespect to CERE, in the right panel and to be compared to Figure 5.12, shows aworse fit than the original Zollo model due to the lack of a short wavelength signalbetween CERE and SEFR.

In conclusion, the shallow extension of the rupture proposed byBasili&Meghraoui to provide a better fit to coseismic deformation doesn’tseem to be supported by the observed postseismic relaxation signal. The levelof uncertainty in the data and in the earth structure, however, doesn’t allow todefinitely discard the model.

5.5.4 Deformation map views

We close this chapter with map views of horizontal deformation for the threecandidate fault models. Figures show relaxation accumulated during the wholeGPS campaign period for the preferred earth model (both TZ and LC with a


12˚ 36' 12˚ 48' 13˚ 00'

43˚ 00'

43˚ 12'

2

4

4

6

SEFR

MONT

COLL

CERE VALL

SPEL

0

2

4

6

8

mm

Figure 5.14 Salvi FM.

viscosity of 1018 Pa s) and with homogeneous slip on the fault.

Results are analogous to what discussed in Chapter 4, with the only differencethat the deformations represented there were accumulated in the first year afterthe earthquake, instead of during the time-span of the GPS campaigns, and thecoordinates were given in a local kilometric system.

In particular, Figures 5.14 and 5.15 repeat the features of the upper line ofFigure 4.2 (left and right panels respectively), whereas Figure 5.16 is analogous tothe bottom-right panel of Figure 4.5.

The qualitative features of those deformation patterns have already been dis-cussed: results are reproposed here to clarify the relation between the horizontaldeformation pattern, the perpendicular sections and the baseline variations dis-cussed all through the current chapter.


12˚ 36' 12˚ 48' 13˚ 00'

43˚ 00'

43˚ 12'

22

2

4

SEFR

MONT

COLL

CERE VALL

SPEL

0

2

4

6

8

mm

Figure 5.15 Zollo FM.

12˚ 36' 12˚ 48' 13˚ 00'

43˚ 00'

43˚ 12'

2

2

2

2

4

SEFR

MONT

COLL

CERE VALL

SPEL

0

2

4

6

8

mm

Figure 5.16 Basili&Meghraoui FM.

Chapter 6

Central Apennines: effect oflateral heterogeneities

In the last chapter, GPS measurements have been compared with model predictionsobtained by means of a vertically stratified earth model, where viscosity values wereintroduced as a priory data.

The purpose of this chapter is to go a step further into the realization of a morerealistic deformation model. This objective will be pursued in two steps:

- first, viscosity will result from non-linear flow laws, which will lead to a con-sistent relation between rheology, temperature profile and state of stress;

- second, lateral variations in crustal structure and isotherm will be introducedin the model, according to tomographic results for the area under study.

Surface deformation obtained with a realistic earth model will be then comparedto the previous results and conclusions will be drawn about the impact of lateralheterogeneities on postseismic relaxation processes in the Central Apennines.

6.1 Non-linear rheologies: controls on viscosity

The simplified version of the empirical flow law for rocks reads [Ranalli, 1995]

η = (2A)−1σ(1−n)E exp

(E

RT

)(6.1)

where the dependence of the viscosity η on the temperature T is a function of therock type through the pre-exponent A, the activation energy E and the stress powern. If the rheology is non-linear (i.e. n 6= 1), then viscosity is also dependent on theeffective stress (σE). R is the universal gas constant.

The final aim of getting a realistic viscosity profile, thus, is controlled by threeindependent factors: rheology, temperature and state of stress. The first step inthe procedure consisted in choosing a rheology: though aware that this choice

88 Central Apennines: effect of lateral heterogeneities

E n A Rock type ReferenceLayer (kJ mol−1) (Pa−n s−1)

UC 150.1 1.80 1.075e-13 wet qtz-rich Jaoul et al., 1984LC 219.0 2.40 5.000e-18 quartz diorite Kirby, 1983UM 510.0 3.00 7.000e-14 olivine Kirby, 1983

Table 6.1 Rheological parameters used in this study.

alone can strongly influence the final results, we decided to use a single set ofrheological parameters. This choice has been motivated by the necessity of limitingthe number of parameters and by the fact that there is little evidence about theactual composition of the crust in the region of interest. Specifically, we assumedquartzite-rich rocks in the upper crust (UC), quartz diorite in the lower crust (LC)and olivine in the upper mantle (UM). Parameters and references are listed in Table6.1.

6.1.1 Temperature dependence

The temperature profile, or geotherm, is the second ingredient necessary to deter-mine viscosity values according to Eq. (6.1).

The steady-state model proposed by Chapman [1986] describes temperaturevariation with depth, once some constraints on surface heat flux are given.

In the left panel of Figure 6.1, we show geotherms for an earth model with a 15-km-thick UC and the Moho at 30 km, when surface heat flux assumes values between50 mW m−2 and 90 mW m−2, with an exponential distribution of heat sources inthe upper crust. It is possible to observe how geotherms follow an approximativelylinear trend, with different gradients in the various layers.

In the right panel of Figure 6.1, it is possible to observe the viscosity profilesresulting from the same geotherms, according to Eq. 6.1, with the rheological pa-rameters of Table 6.1 and a uniform stress of 1 MPa in the whole domain. TheUC has a much softer rheology than the LC: the viscosity jump at the boundarybetween the two layers amounts to about six orders of magnitude, for the warmestgeotherm. The contrast at the Moho is slightly larger, about seven orders of magni-tude, but the rheology parameters in the UM refer to a rather dry olivine and haveto be considered an extreme situation. For the earthquake under study, as alreadydiscussed in Section 5.1, mantle relaxation is irrelevant.

Important is the relative difference between two consecutive profiles: a changein surface heat flux of 20 mW m−2 induces a temperature delta of about 110 oCat the bottom of the UC and a difference in viscosity of almost two orders ofmagnitude. Moreover, a different surface heat flux has also a qualitative effect: awarmer geotherm presents a steeper viscosity gradient with respect to depth. Inother words, the width of a low viscosity channel is larger for a colder geotherm,as it is observable in Figure 6.2, where viscosity curves have been normalized with

6.1 Non-linear rheologies: controls on viscosity 89

0

10

20

30

40

50

Dep

th (

km)

0 250 500 750 1000Temperature (C)

UC

LC

UM

15 18 21 24 27 30Viscosity (Log(Pa s))

UC

LC

UM

Figure 6.1 Left panel: steady state geotherm according to Chapman (1986) for threesurface heat flows (q0): 50 mW m−2 (solid line), 70 mW m−2 (dashed) and90 mW m−2 (dotted). Right panel: viscosity profiles for the three givengeotherms, with the rheological parameters listed in Table 6.1 andσE = 1 MPa.

UC

LC

0

10

20

30

Dep

th (

km)

1.0 1.5 2.0Relative viscosity

Figure 6.2 Relative logarithmic viscosity. Curves from the right panel of Figure 6.1 havebeen normalized with respect to the minimum value for each layer andgeotherm.


respect to the minimum value reached in each layer, independently for each line.A channel where viscosity in the UC increases by 20% reaches a depth of 7 km forq0 = 50 mW m−2 (solid line) and only 9 km when q0 = 90 mW m−2 (dotted line).

6.1.2 Effect of pre-stress

Given a non-linear rheology, the state of stress (σE) represents an important factorin controlling the actual viscosity values. Nonetheless, the real state of stress in thelithosphere is largely unknown, with estimates ranging a few orders of magnitude(typically between 100 kPa and 100 MPa), depending on the specific location andtectonic setting.

In Figure 6.3, we show the impact of a uniform change in pre-stress on theviscosity profile, as determined by Eq. 6.1, when the geotherm is controlled by asurface heat flux q0 of 70 mW m−2 and rheology is taken from Table 6.1.

At the bottom of the UC, increasing σE by one order of magnitude lowersviscosity by almost one order of magnitude: compared to the effect of a variationin surface heat flow, thus, the state of stress has a minor impact on the assessmentof the crustal viscosity structure.

Stress lateral variations, probably present in tectonically active regions as theCentral Apennines, could be the reason for a localization of deformation. However,available constraints on crustal structure are too loose to allow such a detailedgeodynamical description.

UC

LC

0

10

20

30

Dep

th (

km)


Figure 6.3 Viscosity profile for q0 = 70 mW m−2 and three different values of pre-stress(σE): 1 MPa (solid line), 10 MPa (dashed) and 100 MPa (dotted).

6.1 Non-linear rheologies: controls on viscosity 91

6.1.3 Conditions for a weak upper crust (or transition zone)

The viscosity profiles described so far have covered a range of ”reasonable” condi-tions as far as surface heat flux and pre-stress are concerned (q0 = 50−90 mW m−2

and σE = 1− 100 MPa).Results from the previous chapter suggest the presence of a region with very low

viscosity located at the bottom of the UC (the so-called transition zone, or TZ, inAoudia et al. [2003], with η ≤ 1019 Pa s from a depth of about 8-10 km). However,the lowest viscosities obtained so far at 10 km depth remain around the value of1021 Pa s, which corresponds to a characteristic Maxwell time (τM ) of about 1000years.

A viscosity profile with the presence of a TZ is plotted in the left panel ofFigure 6.4: it requires rather high values for both surface heat flux and pre-stress(q0 = 100 mW m−2 and σE = 100 MPa). The proposed conditions are unlikelyto be found around the area of the 1997 earthquake sequence: a surface heat fluxof 100 mW m−2 and higher is observed only in the western part of the CentralApennines and a stress value of 100 MPa can occur in specific locations, but it isunrealistic for the whole upper crust.

Another possible way of obtaining low viscosities in the UC, with reasonablevalues of q0 and σE , is to change some of the rheological parameters listed in Table6.1.

The choice of a different rock-type or reference doesn’t represent a useful op-tion: parameters determined by Jaoul [1984] for a wet quartzite-rich rock are veryclose to the results published by Kirbi [1983] for wet granite and by Hansen and

UC

LC

0

10

20

30

Dep

th (

km)

15 18 21 24 27 30Viscosity (Log (Pa s))

UC

LC

0

10

20

30

Dep

th (

km)


Figure 6.4 Left panel: viscosity profile for q0 = 100 mW m−2 and σE = 100 MPa. Rightpanel: viscosity profile for q0 = 70 mW m−2 and σE = 10 MPa, for threedifferent values of activation energy in the UC: 150 kJ mol−1 (solid line), 130kJ mol−1 (dashed) and 110 kJ mol−1 (dotted).


Carter [1982] for wet westerly granite. Moreover, the three mentioned rocks alreadyrepresent the softest possible documented rheology for upper crustal material.

However, laboratory derived parameters are only representative of a limitednumber of samples and are not expected to cover all possible conditions. A majorsource of uncertainty, in particular, is provided by the water content: the label”wet”, in fact, only means that samples have not been dried before experiments.It is therefore reasonable to affirm that rocks rich in quartz, such as quartzite andgranite, can have activation energies of the order of 100-150 kJ mol−1 (Ranalli[1995], 10.4).

In the right panel of Figure 6.4, we plot viscosity profiles for reasonable stressand heat flow conditions (σE = 10 MPa and q0 = 70 mW m−2) and for progres-sively lower values of activation energy in the UC. The rest of the parametersremain as indicated in Table 6.1. The dotted line, relative to E = 110 kJ mol−1,reaches a viscosity of 1019 Pa s at a depth of 9.4 km. With respect to the leftpanel, the viscosity variation with depth is slower, due to the fact that a smallerheat flows implies a milder thermal gradient: this is the reason for the presence ofa thicker low-viscosity channel.

In the previous chapter, the presence of a shallow low-viscosity crustal zonehad been inferred from the comparison between GPS measurements and modelpredictions. In the present section, it has been demonstrated how an earth modelwith the possibility of a significant crustal flow on the year time-scale is compatiblewith laboratory derived flow-laws.

Results will be applied to the definition of a realistic and consistent earth modelfor the Central Apennines, focused on the Umbria-Marche geological domain, the-ater of the 1997 seismic sequence.

6.2 Laterally heterogeneous crustal structure

The purpose of the present section is to discuss a temperature profile consistentwith lateral crustal variations possibly present in the Central Apennines and toobtain a realistic viscosity structure for non-linear rheologies.By means of a 3D finite elements approach, we model a volume extending for 100km at each side of the fault. Lateral variations are introduced along a SW-NEsection, represented as a solid line in Figure 6.5, and follow the results publishedby Chimera et al. [2003], here reproduced in Figure 6.6. No lateral variations arespecified in a direction perpendicular to the transect: the section of Figure 6.7 isthus representative of the whole domain. In particular, the bottom of the uppercrust (UC) is located at a depth of 13 km under the fault, at 18 km under theAdriatic Sea (on the right) and it reaches a minimum depth of 11.5 km at about 55km west of the fault.

A large level of heterogeneity characterizes also the topography of the Moho,which is located at a depth of 30 km in the center of the model and at 38 km under

6.2 Laterally heterogeneous crustal structure 93

10˚ 12˚ 14˚42˚

44˚

Figure 6.5 Central Apennines: location of the section reproduced in this chapter. A graystar indicates the fault location.

Figure 6.6 Figure 12 from Chimera et al (2003). This sketch displays the lithosphere-asthenosphere system from the Tyrrhenian to the Adriatic sea. The fault ishere located at about 175 km from the origin.


-100

-50

0

Dep

th (

km)

-100 -50 0 50 100km

UC

LC

UM

SW NE

Figure 6.7 Central Apennines: simplified upper crustal and Moho topography. The smallsegment in the center of the UC represents the fault.

the Adriatic Sea, whereas on the left side it presents a minimum of 28 km at about40 km from the fault and finally deepens to 33 km before the end of the domain.

6.2.1 Realistic temperature profile

A standard approach to define the geotherm consists in constraining the surfaceheat flux, as already discussed about the Chapman [1986] steady-state model.

A detailed map of surface heat flux for Italy has been recently published byDella Vedova et al. [2000]: unluckily, deep meteoric water infiltration affect a largearea of the Central Apennines and bias the measurements. Data in the region ofthe earthquake, in fact, show a very low heat flux (about 30 mW m−2) which isabsolutely not compatible with the presence of low-viscosity crustal zones.

We decided, consequently, to follow a different approach to constrain thegeotherm, assuming a simple profile characterized by a single discontinuity. Ac-cording to Chimera et al. [2003], a 20-km-thick layer of hot lithospheric material,denominated mantle wedge, underlies the Moho in most of the Central Apennines.This layer breaks before reaching the Adriatic coast, where a colder thermal profileis expected. For this reason, the model domain has been divided into two regions:the warm Apennines and the cold Adriatic, with a gradual transition in correspon-dence of the thickening of the crust, at the right side of the fault.

A plot of surface heat flow is displayed in the top panel of Figure 6.8, togetherwith measured data from Della Vedova et al. [2000]. The solid line shows model heatflux, with values of 80 mW m−2 over most of the Apennines and of 50 mW m−2 forthe Adriatic. Gray diamonds represent measured data and, as already anticipated,they are in clear disagreement with model results almost everywhere. Coherently

6.2 Laterally heterogeneous crustal structure 95

0

50

100

q0 (

mW

/m2)

-100

-50

0

Dep

th (

km)

-100 -50 0 50 100km

200400

600

800

1000

0

200

400

600

800

1000

1300

deg

Figure 6.8 Upper panel: model (solid line) and data (gray diamonds) surface heat flux.Lower panel: thermal profile and crustal topography.

with the map published by Della Vedova et al. [2000], modelled and measured heatflux come to an agreement at the extremes of the domain, in regions where waterinfiltration is expected to have a minor effect.

The rather arbitrary value of 80 mW m−2 for the Apennines has been chosento provide the warmest thermal structure in the crust, compatible with laboratoryderived flow laws: temperature at the Moho remains within about 700 oC, the limitof phase changes for crustal material.

The resulting thermal profile is shown in the bottom panel of Figure 6.8, togetherwith the crustal topography already presented. Temperatures in the upper crustare below 400 oC; moreover, a temperature of about 200 oC is observed aroundthe fault at a depth of 8 km, in correspondence with the region of aftershocks cut-off. Moho temperature is about 680 oC in the center of the domain and graduallyreduces to about 520 oC, moving eastwards from the fault.

A last comment concerns the fact that upper crustal topography has little impacton the thermal profile, only visible under the largest heterogeneity, at about 60 kmwest of the fault. The reason lies in the exponential distribution of heat sourcesproposed by Chapman [1986]: the decay depth has been fixed at 8 km, so that onlya small fraction of the heat sources are located at depth below 13 km, where UCtopography begins.


6.2.2 Realistic viscosity profile

Now that the thermal structure is set, it is possible to make a choice for a specificrheology, by keeping as a requirement the presence of a low-viscosity channel at thebottom of the UC.

Pre-stress is not capable of determining large variations in viscosity and for thisreason it has been set to a value of 10 MPa in all the crust.

The only way to obtain the required low viscosities in the UC, as already dis-cussed in Section 6.1.3, is to lower the activation energy. For upper crustal materi-als, values between 100 kJ mol−1 and 150 kJ mol−1 are considered acceptable: asa further constraint, thus, we have put the condition that viscosity at the boundarybetween UC and LC has to remain above a value of 1016 Pa s. The final activa-tion energy chosen for the wet quartzite-rich UC is E = 118 kJ mol−1, about 20%less than the value published by Jaoul [1984]. A plot of the final viscosity profileis presented in Figure 6.9. In this way, under the fault it is possible to obtainη = 1019 Pa s at 9.3 km depth and η = 1017 Pa s at the bottom of the UC, whileregions with η < 1017 Pa s are limited to a narrow band, where the UC is thicker.Under the Adriatic Sea, where temperatures are lower, viscosity remains above the1018 Pa s level.

Similarly, activation energy in the LC has been lowered by about 10%, so toremain within the acceptable value of 200 kJ mol−1: in this way, viscosity in thecentral and left parts of the domain reaches 1018 Pa s, even if only in the neighbor-hoods of the Moho. The choice of a weaker rheology, as the wet diorite publishedby Hansen and Carter [1982], would lower viscosities in the LC by about two or-ders of magnitude. However, regions where flow occurs on the year time-scale(η ≤ 1018 Pa s) would only appear in the lower half of the layer, where stress dif-ferences are small. The rheology of the lower crust, thus, has a small influence onsurface deformation.

-100

-50

0

Dep

th (

km)

-100 -50 0 50 100km

16

18

20

22

24

36

Pa s

Figure 6.9 Viscosity profile.

6.3 Model results 97

In the mantle, not important for the present study, pre-stress has been fixed to1 MPa and laboratory flow parameters for olivine have been maintained: in thisway it is possible to obtain a viscosity of about η = 1019 Pa s in the warmest partsof the domain, where maximum temperature is limited to 1200 oC.

6.3 Model results

In this section we discuss surface deformation resulting from the viscosity structureof Figure 6.9, when the fault model is taken from Zollo et al. [1999].

In Figure 6.10, we show a map view of horizontal relaxation accumulated duringthe whole time-span of the GPS campaigns. GPS sites are indicated with labelleddots and a solid line represents the surface projection of the fault top. Maximumdeformation reaches a value of 1.5 mm and it localizes in a few distinct regions. Inthe near field, displacement mainly occurs across fault, with a slightly dominantSW motion on the hanging wall. A considerable amount of deformation also takesplace in the far field, with two lateral lobes converging toward the hanging wall andsecond set of across-fault lobes (located out of the picture, but represented in theleft panel of Figure 6.14).

The relaxation pattern maintains the general features of the preferred mechan-ical model, displayed in Figure 5.15 on page 85, with the exception of the markedlobe on the hanging-wall, completely absent in the former results. The presenceof this lobe, characteristic of relaxation for a shallow fault, is probably due to thefact that the low-viscosity channel is located further away from the rupture. A sec-

12˚ 36' 12˚ 48' 13˚ 00'

43˚ 00'

43˚ 12'

1

11

1

SEFR

MONT

COLL

CERE VALL

SPEL

0.0

0.5

1.0

1.5

2.0

2.5

mm

Figure 6.10 Map view of horizontal deformation accumulated between the time-span of theGPS campaigns (from 2 to 5.5 years after the earthquake). GPS sites arelabelled and a black segment represents the surface projection of the rupturetop.


-15

-10

-5

0

5

10

15

Bas

elin

e va

riatio

n (m

m)


SPEL VALL CERE COLL

FAU

LT

MONT SEFR-15

-10

-5

0

5

10

15

Bas

elin

e va

riatio

n (m

m)


SPEL VALL CERE COLL

FAU

LT

MONT SEFR

Figure 6.11 Baseline length variations between 1999 and 2003, w.r.t. SPEL (left) andCERE (right). Full dots with error-bars represent GPS measurements, opencircles are model predictions.

ond important difference regards the magnitude of deformation, reduced by a factorthree with respect to the laterally homogeneous mechanical model of Section 5.5 (inparticular, Figure 5.15). The new viscosity profile, in fact, presents a much smallerlow-viscosity channel than the mechanical case: as it has already been observed inSection 5.4, this fact leads to a general decrease in the magnitude of deformation.

A comparison between GPS measurements and the new model results is pre-sented in Figure 6.11. The model highly underestimates all the baselines w.r.t.SPEL (left panel), with the exception of SPEL-SEFR, falling within the uncer-tainty limit. On the contrary, a good agreement is reached in the eastern part ofthe transect (sites from CERE to SEFR), when baseline variations are computedwith CERE as reference point. The level of agreement between model and mea-surements, thus, is close to the results for the preferred mechanical model (rightpanels of Figure 5.11 and 5.12, respectively on page 81 and 82), with the exceptionof the baselines SPEL-CERE and SPEL-MONT, where agreement reduces.

It strikes how two models with large differences in the surface deformation signal(both in pattern and magnitude) give rise to a relatively small difference in the de-tectable GPS motions. This fact is mainly due to the unfortunate non-alignment ofthe GPS sites and to the limited extension of the transect, covering only the centralpart of the deformation area. For this reason, in the next Section we will discusshow different viscosity structures affect the whole deformation pattern, withoutfurther reference to the GPS results.

6.3.1 Effect of different viscosity structures

In this Section, we will discuss the effect of different viscosity structures on viscoelas-tic relaxation. In all results, the geometries of the UC and the Moho topographyare as in Figure 6.7 and viscosity is determined by the thermal structure, accordingto Eq. 6.1.


-100

-50

0

Dep

th (

km)

-100 -50 0 50 100km

2

3

4

5

6

7

Log Pa

Figure 6.12 Section of elastic effective stress. Zollo FM.

The stress perturbation induced by the faulting process represents a fundamentalaspect to understand the physical link between viscosity structure and viscoelasticrelaxation. A section of the elastic effective stress is presented in Figure 6.12, wherepre-stress has been subtracted (it amounts to 10 MPa in all the crust and 1 MPain the mantle); the scale is logarithmic, a black segment represents the ruptureand layer boundaries are indicated with a white solid line. The contour line forσE = 10 kPa (Log σE = 4) is plotted as reference: it depicts the region where mostof the deformation takes place for the proposed viscosity structure.

A rather symmetric pattern radiates from the fault: stress above 1 MPa isreached only in a very narrow region around the rupture and it is barely visible inthe picture, whereas a larger band, about 10 km wide, is characterized by a stresslarger than 100 kPa.

Lateral heterogeneous viscosity

The laterally heterogeneous viscosity structure discussed in Section 6.2 results froma combination of two main factors: a heterogeneous geometry, taken from Chimeraet al. [2003], and a laterally heterogeneous thermal structure (Figure 6.8).

Results from Chimera et al. [2003] are obtained by a combination of differentdatasets and techniques (deep crustal reflection studies, s- and p-wave tomography),whereas the thermal structure has been deduced from rather weak constraints, asdiscussed in Section 6.2.1.

In this sub-section, we want to show the different styles of relaxation when thelaterally heterogeneous viscosity structure is purely due to the presence of a crustaltopography, or to the further addition of a heterogeneous thermal structure.

In Figure 6.13, we present sections of two different viscosity structures andrespective effective strain rates, computed for a time of two years after the seismicevent. In the upper-left panel, for the purpose of an easier comparison, we represent


-50

-25

0

1919

19 19

1919

15

17

19

21

23

35

Log Pa

-50

-25

0

-50 -25 0 25 50km

-22

-50 -25 0 25 50km

-22

-25

-24

-23

-22

-21

-20Log 1/s

Figure 6.13 Laterally heterogeneous (left column) vs. homogeneous (right column) thermalmodel. Viscosities (upper panels) and effective strain rates (lower panels) twoyears after the earthquake (Sept. 1999, start of GPS campaigns).

a zoom on the results of Figure 6.9 (surface heat flow is equal to 80 mW m−2 onthe left and to 50 mW m−2 on the right), whereas in the upper-right panel theviscosity structure results from a laterally homogeneous thermal structure (q0 =70 mW m−2, the highest value compatible with realistic viscosities in all the UC).The major difference between the two models consists in the location of the regionwith the lowest viscosity: the heterogeneous model presents it at the left side ofthe fault, where the UC is slightly thicker, whereas in the homogeneous model it issituated at the right side of the fault, where the UC reaches the maximum thickness.

As a result, relaxation localizes in different ways, as it is possible to see inthe bottom panels of Figure 6.13. The heterogeneous model shows the largeststrain rates, mainly due to the fact that viscosity is lower below the fault (whereq0 = 80 mW m−2). The most important aspect regards the spatial distribution ofthe region with large strain rates values (marked by the −22 s−1 contour line): theheterogeneous model shows a tendency of relaxation to be more concentrated onthe left side, with an opposite behaviour of the homogeneous model.

Even if differences in strain rates distribution seem moderate, the resulting defor-mation pattern at the surface is largely affected by the change in viscosity structure.In Figure 6.14, we show map views of modelled horizontal relaxation, accumulatedbetween 2 and 5.5 years after the earthquake. The left panel is a larger view ofthe deformation pattern already discussed in Figure 6.10, whereas the right panelregards the model with a laterally homogeneous thermal structure. The most evi-


12˚ 24' 12˚ 36' 12˚ 48' 13˚ 00' 13˚ 12'42˚ 48'

43˚ 00'

43˚ 12'

1

1

11

1

2

SEFR

MONT

COLL

CERE

VALL SPEL

0.0 0.5 1.0 1.5 2.0 2.5mm

12˚ 24' 12˚ 36' 12˚ 48' 13˚ 00' 13˚ 12'42˚ 48'

43˚ 00'

43˚ 12'

1

1

1

SEFR

MONT

COLL

CERE

VALL SPEL

0.0 0.5 1.0 1.5 2.0 2.5mm

Figure 6.14 Map views of horizontal deformation between 2 and 5.5 years after theearthquake. Laterally heterogeneous (left panel) vs. homogeneous (right panel)thermal structure.

dent qualitative difference between the two models is in the absence of a significantnear-field motion on the hanging wall for the homogeneous model, recalling theresults for the mechanical model of Figure 5.15 on page 85. Furthermore, displace-ment on the side-fault lobes always points to the region of maximum deformation,respectively SW and NE from the fault, for the heterogeneous and the homogeneousmodel. Also relaxation magnitudes are different, but this is mainly due to the factthat the heterogeneous model presents low viscosities at a shallower depth.

A laterally homogeneous thermal structure is not realistic for the area understudy, where a contrast is expected between the warmer Apenninic region and thecolder Adriatic, and it is discussed here as an end-member case. On the other side,the level of heterogeneity proposed represents a possible opposite extreme, but spaceis left for speculation on the actual quantitative thermal variation moving acrossthe Apennines.

GPS data in the area are not capable of distinguishing between the proposed sce-narios. The observed transect is short compared to the area where relaxation takesplace and deformation magnitudes are small, in a way that relative site motionsdue to the different models are almost null.

Upper crustal rheology

A further assumption discussed in Section 6.2.2 regards the activation energy chosenfor the crustal material. The value of 150 kJ mol−1, published by Jaoul [1984] for awet quartzite-rich UC, has voluntarily been reduced by 20% (E = 118 kJ mol−1),in order to obtain a low-viscosity channel at a shallow depth. In this subsection,we want to study the effect on surface deformation of a different activation energy


12˚ 24' 12˚ 36' 12˚ 48' 13˚ 00' 13˚ 12'42˚ 48'

43˚ 00'

43˚ 12'

1

1

1

1

SEFR

MONT

COLL

CERE

VALL SPEL

0.0 0.5 1.0 1.5 2.0 2.5mm

12˚ 24' 12˚ 36' 12˚ 48' 13˚ 00' 13˚ 12'42˚ 48'

43˚ 00'

43˚ 12'

1

SEFR

MONT

COLL

CERE

VALL SPEL

0.0 0.5 1.0 1.5 2.0 2.5mm

Figure 6.15 Map views of horizontal deformation between 2 and 5.5 years after theearthquake. Alternative UC rheologies: softer (E = 110 kJ mol−1, left panel)and harder (E = 127.5 kJ mol−1 right panel).

for the UC, where all the other parameters remain fixed and the thermal profile islaterally heterogeneous.

In the left panel of Figure 6.15, we present the case where E = 110 kJ mol−1

(about 25% less than the published value), which lowers viscosity by about oneorder of magnitude. Notably, the main effect is in a modification of the deformationpattern, in comparison with the reference model, already presented in the left panelof Figure 6.14. Overall magnitudes remain almost unchanged, whereas in a smallarea SW of the fault (i.e. on the hanging wall) the direction of displacement reverses,leading to a moderate motion toward the fault itself.

Differently, when activation energy is only lowered by 15% (E = 127.5 kJ mol−1,right panel of Figure 6.14), both pattern and magnitude of deformation changedrastically. In particular, relaxation reduces everywhere and almost disappears atNE of the fault, whereas the area of maximum deformation concentrates on thehanging wall.

From those examples it follows that the range of possible values for the acti-vation energy in the LC is rather limited, if we expect deformation rates in theorder of a millimeter per year. The cases discussed here already represent extremesituations: an even harder rheology would further slow down the process to an un-detectable level, whereas a softer rheology, with the given thermal profile, wouldlead to unrealistic low viscosities at the bottom of the UC (below 1016 Pa s).

Unluckily, the GPS data available are not able to put a further constraint onthe upper-crustal rheology.

Further, since a minor amount of flow occurs in the lower crust, a change in theLC activation energy only leads to a small variation in the magnitude of relaxation,with almost no effect on the global pattern.

Afterword

A peculiarity of geophysical research consists in the fact that, differently from manyother branches of science, it often misses both repeatability and falsifiability: theexperimental opportunity can be neither programmed nor repeated. In the specificcase, an earthquake is always a unique event, whose occurrence and location areout of control.

This practically means that data are collected according to criteria based onthe analysis of earlier events and on a priori predictions of the behaviour of thephysical system. Thanks to the information obtained from the comparison betweenthe new data and the prediction of the existing models, the knowledge about thephysics of Earth’s processes can be refined.

Once all the available information has been extracted from the ”old” models,the next step consists in refining the model itself, with the development of newfeatures whose importance has an experimental or theoretical foundation. In themost lucky case, the whole procedure of modeling and comparing with the datacan be iterated a few times with success. Rather often, however, the new modelrequires a different or larger dataset that cannot be collected for the same event,which has already taken place.

When this happens, a choice has to be made between moving to a differenttest case, or to go a step forward in synthetic experiments, with the intention ofclarifying the physics of the process, even without the availability of experimentaldata.

Data (with a certain availability and quality) and models (from formulation toimplementation) are two necessary ingredients for the development of science. Bothare subject to interpretation, which is a necessity and a right of the scientist. How-ever, the more theoretical person will trust the model (”the physics of the processis clear”), whereas the experimental person will trust the data (”measurements arereal”).

While doing the research described in this book, I have tried to keep a balancebetween a theoretical and an experimental approach: the modeling side representsmost of the work done during this five years period and it’s largely present in almostevery chapter. Synthetic experiments are given a considerable importance, because

104 Afterword

I’m convinced that they represent a useful way of understanding the behaviour ofany model.

At the same time, the interface with geodesy has revealed to me the majoreffort that is often necessary to obtain significant displacement measurements ofthe Earth’s surface. In the specific case, the GPS observation of a few pointson the site of a moderate size earthquake requires the collection of data throughmany years of campaigns, carried on with the hope that the actual motions willreach the detectability level. For this reason, sometimes it’s wise for the theoreticalgeophysicist to subdue the theory to the evidence, i.e. to leave the first word to theobservations and let the available geodetic data control the modeling strategy.

Bibliography

Abramowitz, M., and I.A. Stegun, 1964. Handbook of Mathematical Functions, Ap-plied Mathematic Series, Volume 55, Washington: National Bureau of Stan-dards, reprinted 1968 by Dover Publications, New York (USA).

Amato, A., R. Azzara, C. Chiarabba, G.B. Cimini, M. Cocco, M. Di Bona, L.Margheriti, S. Mazza, F. Mele, G. Selvaggi, A. Basili, E. Boschi, F. Courboulex,A. Deschamps, S. Gaffet, G. Bittarelli, L. Chiaraluce, D. Piccinini, and M.Ripepe, 1998. The 1997 Umbria-Marche, Italy, earthquake sequence: a first lookat the main shocks and aftershoks, Geophys. Res. Lett., 25 (15), 2861-2864.

Aoudia, A., A. Borghi, R. Riva, R. Barzaghi, B.A.C. Ambrosius, R. Sabadini, L.L.A.Vermeersen, and G.F. Panza, 2003. Postseismic deformation following the 1997Umbria-Marche (Italy) moderate normal faulting earthquakes, Geophys. Res.Lett., 30 (7), 1390.

Barba, S., and R. Basili, 2000. Analysis of seismological and geological observationsfor moderate-size earthquakes: the Colfiorito Fault System (Central Apennines,Italy), Geophys. J. Int., 141, 241-252.

Basili, R., and Meghraoui, M., 2001. Coseismic and postseismic displacements re-lated with the 1997 earthquake sequence in Umbria-Marche (central Italy), Geo-phys. Res. Lett., 28 (14), 2695-2698.

Belardinelli, M.E., L. Sandri, and P. Baldi, 2003. The major event of the 1997Umbria-Marche (Italy) sequence: what could we learn from DInSAR and GPSdata?, Geophys. J. Int., 153, 242-252.

Boschi, E., G. Ferrari, P. Gasperini, E. Guidoboni, and G. Valensise, 1997. Catalogodei forti terremoti in Italia dal 461 a.C. al 1990, volume 2, Istituto Nazionale diGeofisica, 643 pp.

Briole, P., G. De Natale, R. Gaulon, F. Pingue and R. Scarpa, 1986. Inversionof geodetic data and seismicity associated with the Friuly earthquake sequence(1976-1977), Annales Geophysicae, 4, 481-492.

Cattin, R., P. Briole, H. Lyon-Caen, P. Bernard and P. Pinettes, 1999. Effects ofsuperficial layers on coseismic displacements for a dip-slip fault and geophysicalimplications, Geophys. J. Int., 137, 149-158.

106 Bibliography

Cello, G., S. Mazzoli, E. Tondi, and E. Turco, 1997. Active tectonics in the centralApennines and possible implications for seismic hazard analysis in peninsularItaly, Tectonophysics, 272, 43-68.

Chapman, D.S., 1986. Geol. Soc. Lond. Spec. Pub., 24, 63.

Chimera, G., A. Aoudia, A. Sarao, and G.F. Panza, 2003. Active tectonics in CentralItaly: constraints from surface wave tomography and source moment tensorinversion, Phys. Earth Planet. Inter., 138, 241-262.

Cinti, F.R., L. Cucci, F. Marra, and P. Montone, 1999. The 1997 Umbria-Marche(Italy) earthquake sequence: relationship between ground deformation and seis-mogenic structure, Geophys. Res. Lett., 26 (7), 895-898.

D’Agostino, N., R. Giuliani, M. Mattone, and L. Bonci, 2001a. Active crustal ex-tension in the central Apennines (Italy) inferred from GPS measurements in theinterval 1994-1999, Geophys. Res. Lett., 28 (10), 2121-2124.

D’Agostino, N., J.A. Jackson, F. Dramis, and R. Funicello, 2001b. Interaction be-tween mantle upwelling, drainage evolution and active normal faulting: andexample from the central Apennines (Italy), Geophys. J. Int., 147, 475-497.

Della Vedova, B., Bellani S., Pellis G. and Squarci P., 2000. Deep Temperatures andSurface Heat-Flow Distribution. In: G.B. Vai & L.P. Martini (editors), Anatomyof an orogen: the Apennines and adjacent Mediterranean basins, Kluwer Aca-demic Publishers, Dordrecht, 2000.

Deng, J., M. Gurnis, H. Kanamori, and E. Hauksson, 1998. Viscoelastic flow inthe lower crust after the 1992 Landers, California, earthquake, Science, 282,1689-1692.

Doglioni, C., 1991. A proposal for the kinematic modelling of W-dipping subduc-tions - possible applications to the Tyrrhenian-Apennines system, Terra Nova,3, 423-434.

Dziewonski, A.M., and Anderson, D.L., 1981. Preliminary reference Earth model,Phys. Earth Planet. Inter., 25, 297-356.

Ekstrom, G., A, Morelli, E. Boschi, and A.M. Dziewonski, 1998. Moment tensoranalysis of the central Italy earthquake sequence of September-October 1997,1998. Geophys. Res. Lett., 25 (11), 1971-1974.

Finetti, I.R., M. Boccaletti, M. Bonini, A. Del Ben, R. Geletti, M. Pipan, andF. Sani, 2001. Crustal section based on CROP seismic data across the NorthTyrrhenian-Northern Apennines-Adriatic Sea, Tectonophysics, 343, 135-163.

Felippa, C.A., 2003. Introduction to Finite Element Methods, Department ofAerospace Engineering Sciences and Center for Areospace Structures, Universityof Colorado, Boulder, Colorado 80309-0429, USA.

Feigl, K. L. and E. Dupre, 1998. RNGCHN: a program to calculate displacementcomponents from dislocations in an elastic half-space with applications for mod-eling geodetic measurements of crustal deformation, revised for Computers andGeosciences.

Bibliography 107

Galadini, F., and P. Galli, 2000. Active Tectonics in the Central Apennines (Italy)- Input Data for Seismic Hazard Assessment, Natural Hazards, 225-270.

Govers, R.M.A., 1993. Dynamics of lithospheric extension: A modeling study, Ge-ologica Ultraiectina, 105, 240 pp.

Govers, R., Wortel, M.J.R., 1993. Initiation of asymmetric extension in continentallithosphere, Tectonophysics, 223, 75-96.

Govers, R., Wortel, M.J.R., 1995. Extension of stable continental lithosphere andthe initiation of lithosphere scale faults, Tectonics, 14(4), 1041-1055.

Govers, R., 1998. Lithosphere Scale Geodynamic Simulations of Transpression, EOSTrans. AGU, 103, F903, Fall Meeting Suppl., San Francisco (USA).

Govers, R., and M.J.R. Wortel, 1999. Some remarks on the relation between verticalmotions of the lithosphere during extension and the ”necking depth” parameterinferred from kinematic modeling studies, J. Geophys. Res., 104, 23,245-23,254.

Govers, R., Wortel, M.J.R., and Buiter, S.J.H., 2000. Surface expressions of slabbreak-off: results from 3D numerical models, Annales Geophysicae Supplement,EGS Meeting, Nice (France).

Govers, R., Meijer, P. Th., 2001. On the dynamics of the Juan de Fuca plate EarthPlanet. Sc. Lett., 189, 115-131.

Hansen, F.D., and N.L. Carter, 1982. EOS Trans. AGU, 63, 437.

Hunstad, I., M. Anzidei, M. Cocco, P. Baldi, A. Galvani and A. Pesci, 1999. Mod-elling coseismic displacements during the 1997 Umbria-Marche earthquake (cen-tral Italy), Geophys. J. Int., 139, 283-295.

Ihmle, P.F., and J.-C. Ruegg, 1997. Source tomography by simulated annealingusing broad-band surface waves and geodetic data: Application to the Mw = 8.1Chile 1995 event, Geophys. J. Int., 131, 146-158.

Jaoul, O., 1984. J. Geophys. Res., 89, 4298.

Kirby, S.H., 1983. Rev. Geophys., 21, 1458.

Lang, S., 1971. Linear algebra, Reading, MA, Addison-Wesley, 400 pp.

Lundgren, P., and S. Stramondo, 2002. Slip distribution of the 1997 Umbria-Marcheearthquake sequence: Joint inversion of GPS and synthetic aperture radar inter-ferometry data, J. Geophys. Res., 107 (B11), 2316, doi:10.1029/2000JB000103.

Martinec, Z., and Wolf, D., 1998. Explicit form of the propagator matrix for amulti-layered, incompressible viscoelastic sphere, Scientific Technical ReportSTR98/08, GeoForschungsZentrum Potsdam, Germany.

Melosh, H.J., and Raefsky, A., 1980. The dynamical origin of a subduction zonetopography, Geophys. J. R. astr. Soc., 60, 333-354.

Melosh, H.J., and Raefsky, A., 1981. A simple and efficient method for introducingfaults into finite element computations, Bull. Seism. Soc. Am., 71, 5, 1391-1400.

108 Bibliography

Okada, Y., 1985. Surface deformation due to shear and tensile faults in a half-space,Bull. Seism. Soc. Am., 75, 1135-1154.

Pialli, G., M. Barchi, and G. Minelli (Eds.), 1998. Results of the CROP-03 DeepSeismic Reflection Profile, Mem. Soc. Geol. Ital., LII, 657 pp.

Piersanti, A., Spada, G., Sabadini, R., and Bonafede, M., 1995. Global post-seismicdeformation, Geophys. J. Int., 120, 544-566.

Piersanti, A., G. Spada, and R. Sabadini, 1997. Global postseismic rebound of aviscoelastic Earth: Theory for finite faults and application to the 1964 Alaskaearthquake, J. Geophys. Res., 102 (B1), 477-492.

Pollitz, F.F., 1992. Postseismic relaxation theory on the spherical earth, Bull. Seism.Soc. Am., 82 (1), 422-453.

Pollitz, F.F., 1996. Coseismic deformation from earthquake faulting on a layeredspherical earth, Geophys. J. Int., 125, 1-14.

Pollitz, F.F., 1997. Gravitational viscoelastic postseismic relaxation on a layeredspherical Earth, J. Geophys. Res., 102 (B8), 17921-17941.

Pollitz, F.F., and R. Burgmann, 1998. Joint estimation of afterslip rate and post-seismic relaxation following the 1989 Loma Prieta earthquake, J. Geophys. Res.,103 (B11), 26975-26992.

Pollitz, F.F., G. Peltzer, and R. Burgmann, 2000. Mobility of continental man-tle: Evidence for postseismic geodetic observations following the 1992 Landersearthquake, J. Geophys. Res., 105 (B4), 8035-8054.

Press, W.H., S.A. Teukolsky, W.T. Vetterling, and B.P. Flannery, 1992. Numericalrecipies in Fortran 77: the art of scientific computing, Cambridge UniversityPress, Cambridge (UK).

Ranalli, G., 1995. Rheology of the earth - second edition, Kluwer Academic Pub-lishers, Dordrecht, 432pp.

Riva, R., A. Aoudia, L.L.A. Vermeersen, R. Sabadini and G.F. Panza, 2000. Crustalversus asthenospheric relaxation and post-seismic deformation for shallow nor-mal faulting earthquakes: the Umbria-Marche (central Italy) case, Geophys. J.Int., 141, F7-F11.

Riva, R.E.M., and L.L.A. Vermeersen, 2002. Approximation method for high-degreeharmonics in normal mode modelling, Geophys. J. Int., 151, 309-313.

Sabadini, R., Yuen, D.A., and Boschi, E., 1982. Polar wandering and the forcedresponses of a rotating, multilayered, viscoelastic planet, J. geophys. Res., 87,2885-2903.

Sabadini, R., and Vermeersen, L.L.A., 1997. Influence of lithospheric and mantlelayering on global post-seismic deformation, Geophys. Res. Lett., 24, 2075-2078.

Salvi, S., S. Stramondo, M. Cocco, M. Tesauro, I. Hunstad, M. Anzidei, P. Briole, P.Baldi, E. Sansosti, G. Fornaro, R. Lanari, F. Doumaz, A. Pesci, and A. Galvani,

Bibliography 109

2000. Modeling coseismic displacements resulting from SAR interferometry andGPS measurements during the 1997 Umbria-Marche seismic sequence, Journalof Seismology, 4, 479-499.

Savage, J.C., M. Lisowski, and J.L. Svarc, 1994. Postseismic deformation followingthe 1989 (M=7.1) Loma Prieta, California, earthquake, J. Geophys. Res., 99,13757-13765.

Spada, G., Sabadini, R., Yuen, D.A., and Ricard, Y., 1992. Effects on post-glacialrebound from the hard rheology in the transition zone, Geophys. J. Int., 109,683-700.

Stramondo, S., M. Tesauro, P. Briole, E. Sansosti, S. Salvi, R. Lanari, M. Anzidei,P. Baldi, G. Fornaro, A. Avallone, M.F. Buongiorno, G. Franceschetti, and E.Boschi, 1999. The September 26, 1997 Colfiorito, Italy, earthquakes: modeledcoseismic surface displacement from SAR interferometry and GPS, Geophys.Res. Lett., 26 (7), 883-886.

Takeuchi, H., and M. Saito, 1972. Seismic surface waves, In: it Methods in Compu-tational Physics: Seismology, edited by B. Bolt, S. Fernbach, and M. Rotenberg,pp. 217-294, Academic Press, San Diego, California.

Tinti, S., and A. Armigliato, 2002. A 2-D hybrid technique to model the effectof topography on coseismic displacements. Application to the Umbria-Marche(central Italy) 1997 earthquake sequence, Geophys. J. Int., 150, 542-557.

Vermeersen, L.L.A., Sabadini, R., and Spada, G., 1996a. Analytical visco-elasticrelaxation models, Geophys. Res. Lett., 23, 697-700.

Vermeersen, L.L.A., Sabadini, R., and Spada, G., 1996b. Compressible rotationaldeformation, Geophys. J. Int., 126, 735-761.

Vermeersen, L.L.A., and Sabadini, R., 1997. A new class of stratified viscoelasticmodels by analytical techniques, Geophys. J. Int., 129, 531-570.

Vermeersen, L. L. A., and Mitrovica, J. X., 2000. Gravitational stability of sphericalself-gravitating relaxation models, Geophys. J. Int., 142 (2), 351-360.

Wells, D.L., and K.J. Coppersmith, 1994. New empirical relationships among mag-nitude, rupture length, rupture width, rupture area , and surface displacement,Bull. Seism. Soc. Am., 84, 974-1002.

Westaway, R., 1992. Seismic moment summation for hystorical earthquakes in Italy:tectonic implications, J. Geophys. Res., 97, 15437-15464.

Woodhouse, J.H., 1980. The calculation of eigenfrequencies and eigenfunctions ofthe free oscillations of the Earth and the Sun, in: Seismological Algorithms, pp.321-370, London, ed. Doornbos D., Academic Press.

Wu, P., 1978. The response of a Maxwell Earth to applied surface mass loads:Glacial isostatic adjustment, M. Sc. thesis, University of Toronto, Canada.

Zollo, A., Marcucci, S., Milana, G., and Capuano, P., 1999. The 1997 Umbria-Marche (central Italy) earthquake sequence: Insights on the mainshock rupturesfrom near source strong motion records, Geophys. J. Int., 26 (20), 3165-3168.

Acknowledgments

Since the beginning of my Ph.D. research, in 1999, I have had contacts with manypeople, who have helped and supported me in different ways.

First of all I would like to thank Roberto Sabadini, for introducing me to theworld of Solid-Earth Geophysics during my undergraduate studies in Milano andfor all the scientific discussions which have followed since those days. A specialthanks also goes to my direct supervisors in the Netherlands: Bert Vermeersen, whogave me the opportunity to come to Delft and has supervised my whole research;Boudewijn Ambrosius, my dear promotor, a valuable supporter and friend; andRob Govers, who put a major amount of time into helping me during the last twoyears of my research. Further, I would like to thank two colleagues in Italy: KarimAoudia, for his enthusiasm in research and critical viewpoint on many issues, andAlessandra Borghi, for her help with the GPS measurements in the Apennines.

A dear word of thanks goes to my colleague Ph.D. students and best friendsSaskia Matheussen and Rui Fernandes, for their friendship, moral support, companyand scientific input from the very beginning of my staying in Delft and I’m sure formany years to come; and to Stefania, for the company and help while sharing theoffice at TU. My friends from IFD, Carmen, Raji, Yadira, Dedy and Firas, deserve aspecial word of gratitude for sharing their thoughts and opening their hearts aboutspirituality: our meetings represent one of my nicest experiences in Delft.

Many colleagues and friends met while in the Netherlands, part of them nowspread all over the world, largely contributed to the success of my Ph.D. Thank youMarco, Ingrid, Menno, Edith, Antonio, Heert, Nacho, Ron, Wim, Vero, Magda, Eve,Eefje, Margreet, Erik, Ben, Dani, Fabi, Luca, the Locos, and many many others!Further, I would like to thank my old friends in Italy and my family, for alwaysbeing a close presence and a precious support, even when geographically distant.

My last and dearest thanks go to Rebecca, my lovely wife, to whom I am proudto dedicate this book: since the day we met, she has always been at my side, takingcare of me, cheering my days and giving me the first reason to accomplish this work.

Summary

The study of postseismic deformation, i.e. the Earth’s response to the stress per-turbation induced by an earthquake, focuses on very specific deformation signals:magnitude of a few millimeters per year, spatial extent of a few tens of kilometersand lasting for several years. Those characteristics are matched by recent majorimprovements in space-geodetic techniques, in particular the Global PositioningSystem (GPS), that allow the measurement of surface deformation signals at thekilometer scale with a precision of a few millimeters in the horizontal direction.The availability of reliable measurements has opened the way to the observationalvalidation of long-existing mathematical models to describe the continuous defor-mation of the Earth’s crust. The increased knowledge of the physical processes, inturn, leads to the creation of new generations of numerical approaches.

This study wants to address the compelling question about the actual possibilityof investigating the structure and rheology of the Earth’s crust by detection ofpostseismic deformation at the surface. Attention is here focused on the specificpostseismic process of viscoelastic relaxation, which is particularly indicated toinvestigate crustal and upper mantle regions, due to its characteristic temporaland spatial scales. In particular, crustal viscosity is still largely unknown, becauseonly recently relaxation models and geodetic techniques have reached the precisionnecessary to retrieve such information.

Postseismic deformation is first analytically modeled by means of a normalmodes approach, solving the problem on a spherically layered and self-gravitatingearth [Sabadini and Vermeersen, 1997]. A major effort has been put into the re-finement of the existing model, by developing an approximation technique crucialto extend the results to shallow earthquakes and by removing the hypothesis ofincompressibility.

Application to the moderate-size normal faulting event of Umbria-Marche(26.9.1997, Central Italy), combined with the results of four GPS campaigns heldbetween 1999 and 2003, shows that observed postseismic displacements can beexplained by viscoelastic relaxation, provided that a low-viscosity layer underliesthe seismogenic zone (η ' 1018 Pa s from 8-10 km depth).

114 Summary

Later, we use a numerical (Finite Elements) method to provide a thermo-mechanical justification of the preferred earth structure. Similarly to earth materialproperties, mechanical properties in the model depend non-linearly on temperature,pressure and rock type. We use laboratory derived flow laws to infer the regionaldistribution of brittle and viscous properties and we show that the earth structurepreviously obtained for the Central Apennines is reasonable.

After that, we discuss the effect of lateral heterogeneities which are possiblypresent in the region under study. The lateral variations in viscosity are supportedby tomographic imaging of crustal thickness and to the rather sharp heat flowdifferences observed in the area. Our finite element model results show that lateralvariations in the mechanical properties of the crust have a significant effect onpostseismic signals recorded at the earth surface

Samenvatting

De studie naar postseismische deformatie, dat wil zeggen de reactie van de aarde opde spanningsstoring die door een aardbeving wordt veroorzaakt, concentreert zichop zeer specifieke deformatiesignalen: signalen van een paar millimeter per jaar overeen paar tientallen kilometers en een duur van een aantal jaar. Deze kenmerkenworden gaan gepaard met recente belangrijke verbeteringen van ruimte-geodetischetechnieken, in het bijzonder het Global Positioning System (GPS), die de meting vande signalen van de oppervlaktedeformatie op de kilometerschaal met een precisievan een paar millimeter in de horizontale richting toestaan. De beschikbaarheidvan betrouwbare metingen heeft de manier voor de valitatie van reeds bestaandewiskundige modellen geopend om de continue deformatie van de korst van de aardete beschrijven. De verhoogde kennis van de natuurkundige processen leidt tot deverwezenlijking van nieuwe generaties van numerieke benaderingen.

Deze studie gaat in op de dwingende vraag over de daadwerkelijke mogelijkheidom de structuur en de rheologie van de korst van de aarde door opsporing vanpostseismische deformatie aan de oppervlakte te onderzoeken. De aandacht wordthier gericht op het specifieke postseismische proces van viscoelastische relaxatie,dat in het bijzonder gebruikt wordt om de korst en hogere mantelgebieden teonderzoeken wegens zijn kenmerkende tijds- en ruimteschalen. In het bijzonderis viscositeit van de korst nog grotendeels onbekend, omdat slechts onlangs deontspanningsmodellen en geodetische technieken de precisie hebben bereikt dienoodzakelijk is om dergelijke informatie te verkrijgen.

Postseismische deformatie wordt eerst analytisch gemodelleerd door middel vaneen normaalmode benadering die het probleem oplost op een sferisch gelaagde enzelfgraviterende aarde [Sabadini and Vermeersen, 1997]. Met name is gewerkt aande verbetering van het bestaande model, door een benaderingstechniek te ontwikke-len die essentieel is om de resultaten uit te breiden tot ondiepe aardbevingen endoor de hypothese van incompressibiliteit te verwijderen.

De toepassing op de gematigde ”normal fault” aardbeving van Umbria-Marche(26.9.1997, Centraal Italie), gecombineerd met de resultaten van vier GPScampagnes tussen 1999 en 2003 gehouden, toont aan dat de waargenomen postseis-mische verplaatsingen door viscoelastische relaxatie kunnen worden verklaard, op

116 Samenvatting

voorwaarde dat een lage viscositeitslaag aan de seismogenisch streek ten grondslagligt (η ' 1018 Pa s van 8-10 km diepte).

Later gebruiken wij een numerieke (eindige elementen) methode om een thermo-mechanische rechtvaardiging van de aangewezen aardstructuur te geven. Net alsaardmaterialen, hangen de mechanische eigenschappen in het model niet-lineairvan temperatuur, druk en gesteentetype af. Wij gebruiken laboratorium afgeleidestromingswetten om de regionale distributie van brosse en kleverige eigenschappenaf te leiden en wij tonen aan dat de aardstructuur die eerder voor de CentraleApennijnen werd verkregen redelijk is.

Hierna bespreken wij het effect van laterale variaties die in het studiegebiedmisschien aanwezig is. De laterale variaties in viscositeit worden door tomografis-che weergave van de korstdikte en de tamelijk scherpe verschillen van de hitte-stroom die in het gebied worden waargenomen gesteund. Onze ”eindige elementen”-modelresultaten tonen aan dat de laterale variaties in de mechanische eigenschappenvan de korst een significant effect op postseismische signalen hebben die aan hetaardoppervlak worden geregistreerd.

Curriculum Vitae

Riccardo Riva was born in Milano, Italy, on the 21st of February 1975. From 1989he attended highschool (’Liceo Scientifico’) at the ’Istituto Leone XIII’ in Milano,where he obtained his ’Maturita Scientifica’ diploma in 1994.

Between 1994 and 1999 he was an undergraduate student at the University ofMilano, where he got the title of ’Dottore’ (M.Sc.) in Physics with a thesis onpostseismic deformation, under the supervision of prof. R. Sabadini (Dep. of EarthSciences, University of Milano).

In 1999 he started working on his Ph.D. research on geodetic measurements andmodelling of postseismic deformation within the group Astrodynamics & SatelliteSystems of the faculty of Aerospace Engineering, at Delft University of Technology.

Since May 2004 he is employed as a Research Fellow at the Department ofEarth Sciences of the University of Trieste, Italy, where he is involved in a Euro-pean project for the creation of a transnational GPS network to monitor crustaldeformation in the Alps (EU program Alpine Space, project Alps-GPSQuakeNet).

Documents

Crustal rheology and postseismic deformation: …Crustal rheology and postseismic deformation: modeling and application to the Apennines Proefschrift ter verkrijging van de graad van