DETERMINATION OF EFFECTIVE ELASTIC THICKNESS OF THE ... · DETERMINATION OF EFFECTIVE ELASTIC THICKNESS OF THE COLOMBIAN ANDES USING SATELLITE-DERIVED GRAVITY DATA Remy A. Galán

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EARTH SCIENCES

RESEARCH JOURNAL

Earth Sci. Res. J. Vol. 14, No. 1 (June 2010): 7-16

DETERMINATION OF EFFECTIVE ELASTIC THICKNESS

OF THE COLOMBIAN ANDES USING SATELLITE-DERIVED

GRAVITY DATA

Remy A. Galán1, Iván F. Casallas1

1Universidad Distrital Francisco José de Caldas, Facultad de Ingeniería.

Kr 7 40-53. Bogotá. Colombia

[email protected]

ABSTRACT

Gravity anomaly values derived from Global Geopotential Models (calculated from the CHAMP and GRACE satellite mis-sions), are compared with free air ground gravity data to find the best representation of surface data. Using these values andtopographical heights extracted from digital topography models, we applied the isostatic response function (admittance) to acollection of profiles, to find an average of elastic thickness for the Colombian Andes.

Key words: isostasy, elastic thickness, satellite, admittance, flexure, gravity, topography.

RESUMEN

Se extraen valores de anomalía de gravedad de aire libre derivadas de Modelos Geopotenciales Globales, (calculados de lasmisiones satelitales CHAMP y GRACE) los cuales son comparados con datos de gravedad terrestre para encontrar entre estosmodelos la mejor representación de los datos de superficie. Usando estos valores de anomalía y valores de alturas topográficasextraídos de un modelo de topografía digital, se aplica la función de respuesta isostática (admitancia) a un conjunto de perfiles,para hallar un promedio del espesor elástico de los Andes colombianos.

Palabras clave: isostasia, espesor elástico, satélite, admitancia, flexura, gravedad, topografía.

Introduction

This paper essentially makes use of gravity anomalies andtopographical heights to obtain an average of the effectiveelastic thickness (Te) for the Colombian Andes. The gravityanomalies are derived from data obtained from satellite mis-sions, which are a recent technology that is revolutionizingthe world of the geosciences. To get this purpose, geophysi-

cal and geodetics concepts are used. In this way, obtainingthe value of elastic thickness becomes a technical task, underthe principle of a methodological application (Admittanceanalysis), but in the scientific world the debate about the se-lection of the method to find the Te even continuous open;under this context, two schools of thought can be find: Theschool that accepts Admittance analysis developed by

Manuscript received: 04/01/2010

Accepted for publication: 18/03/2010

AGOSTO 9-GEOCIENCIAS-VOL 14-1 2010.prnD:\GEOCIENCIAS-JUNIO 2010\GEOCIENCIAS-VOL 14-1 2010.vplunes, 09 de agosto de 2010 13:45:03

p p pComposite 133 lpi at 45 degrees

Dorman and Lewis (Dorman and Lewis, 1970) as the func-tion that shows the better fit between Topography and Grav-ity functions and argues that Coherence analysis functionoverestimates the values of Te, and in the same way, the Co-herence’s school (Forsyth, 1985) argues that Admittanceanalysis does not take into account of subsurface loads incomputes of Te, and for this reason this latest subestimatesTe value. The authors with this work do not pretend to takesides in this debate, because this work seeks to be a furthercontribution to the knowledge of the Colombian Andes, aswell as to contribute to the understanding of the new tech-nologies such as satellital geophysics.

Isostatic model

Isostatic models can be classified into two categories: Localand Regional models. In the isostatic local model, compen-sation occurs directly beneath the load, which is supportedby materials which have a behavior similar to liquids and donot have rigidity. In the isostatic compensation regionalmodel, the load is supported by a material that presents a cer-tain degree of rigidity and hence their behavior is similar toan elastic plate that bends to support the load. Among theisostatic local models can found the hypothesis proposed byAiry and Pratt, whereas the most representative isostatic re-gional model is proposed by Vening Meinesz.

The concept of lithospheric elastic behavior is devel-oped inside the context of Regional isostatic model, in thisway the present work takes the theoretical frame of VeningMeinesz model.

Gravity data

The analytical representation of the ground gravity field is oneof the main aims of Geodesy; this is work carried out throughanalysis of different measurements on the Earth’s surface (val-ues of gravity, topographical heights, etc.). This analysis leadsto the formulation of the equation DV =0 known also asBoundary value problem which is treated in the branch ofPhysical Geodesy under the topic of Potential Theory.

The representation of this phenomenon is more under-standable when a reference figure that represents the Earth istaken. Geodesy takes several reference surfaces of represen-tation of the Earth’s shape, a physical shape known as geoidand a geometrical-mathematical shape known as ellipsoid;the geoid is the equipotential surface most similar to the sealevel mean at rest and it is represented by W = W0. This po-tential W0 is called real gravity potential, and with the nor-

mal gravity potential U0 generated by ellipsoidal surface,make up the theoretical basis of mathematical BoundaryValue Problem.

The difference between these two potentials is known asperturbation potential or just as potential difference T:

T = W - U (1)

When atmospheric attraction disregards, T is harmonicoutside Earth and satisfies Laplace’s equation:

DT = 0 (2)

These parameters allow obtaining several functions ofthe gravity field (gravity anomalies, geoid undulations, ver-tical deflections, etc.) through its relationship with theanomalous potential T, whose series development is accord-ing to (Heiskanen and Moritz 1967, Torge 2001):

T rGM

r

a

nnm

n

n

m

n

( , ( )) (cos )q l q=æ

èçç

ö

ø÷÷

=

¥

=å å

2 0

(3)

Analysis of perturbations of artificial satellite orbits hasmade the largest contribution to global determination of thelong-wavelength components of the Earth’s gravitationalfield. The resulting global geopotential models (GGM), areusually provided as truncated series of spherical harmonics.Due to several limiting factors these satellite-only GGMs areof a limited spherical harmonics degree (typically 20-30),hence spatial resolution.(Featherstone W. E., 2003).

Among the largest list of available global geopotentialmodels (GGM), three models with several properties havebeen selected to achieve the purpose of obtaining a surfaceof gravity anomaly that represents the gravity function in ad-mittance analysis.

Obtaining gravity surface for Colombian Andes

The global geopotential models offer a uniform coverage ofthe study area, so it is possible to obtain the required term inthe gravity field for a particular study. Data from groundgravity do not have a uniform distribution, although the gen-eration of maps of gravity anomaly is possible thanks to theapplications of several interpolation methods. The selectionof the final model that best represents the field of groundgravity, takes place through the analysis of the correlationparameters between the map of ground gravity anomaly andeach of the different maps resulting from the models.

To achieve the purpose of obtaining a gravity field rep-resentation for Colombian Andes that best fits with groundgravity data, it is necessary to obtain data from sampling

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REMY A. GALÁN, IVÁN F. CASALLAS



grids of free air gravity anomalies from global geopotentialmodels (GGM), with a spacing resolution of approximately0.5 degrees or 30 arcmin, and subsequently to obtain theanomaly maps through interpolation method. The availabledata of ground gravity anomaly are the third order gravitynetwork of IGAC showed in the figure 1. The GGM derived

gravity anomalies can be computed from spherical harmon-ics coefficients to degree nmax using:

DgGM

Rn

a

RnmGGM

n

m

n

n

n

= -æ

èçç

ö

ø÷÷

==åå( ) (cos )

max

102

q (4)

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DETERMINATION OF EFFECTIVE ELASTIC THICKNESSOF THE COLOMBIAN ANDES USING SATELLITE-DERIVED GRAVITY DATA

Figure 1: Available free air ground gravity data, Source: División de Geodesia, Instituto Geográfico Agustíýn Codazzi (IGAC).



The selection of the final model that best representsthe Earth gravity field is done by analyzing the parametersof correlation between the ground gravity map and each ofthe maps resulting from different models. Table 1 showsthe obtained correlations. According to data in the Table1, there is a high level of correlation existing betweenTEG4 and GGM02 models, while the correlation betweenthese models and EIGEN-CG03C model is the lowest. Onthe other hand, the same table shows the correlation be-tween GGM models and ground gravity data maps andsuggests that the average of correlation which only is upto 55 percent. This is the result from low and non uniformground gravity data coverage over Colombia’s continen-tal territory (i.e. Colombian Amazonia). Finally, this sug-gests that the model that best represent the terrestrialgravity field in Colombian continental crust isEIGEN-CG03C (Figure 2) with a correlation coefficient of0.58921.

Effective elastic thickness determination

On large time scales the Earth’s lithosphere exhibit a re-gional behavior, thus it tends to experiment flexure, due tothe loads. It can be assumed that the lithosphere presents thebehavior of a filter which removes large amplitudes, i.e.,short wavelengths associated with local isostasy models al-low the pass of small amplitudes, or long wavelengths areassociated with flexural models (Watts, 2001). In this filter,load h1(x) produces a deflection y1(x) and load h2(x) pro-duces a deflection y2(x) then load h1(x) + h2(x) produce aflexure y1(x) + y2(x). The filters that have this kind of behav-ior are called Linear Space Invariant (LSI) and are character-ized because when they are subjected to periodic loads itsoutput is also periodic (Watts, 2001).

In the Earth’s internal structure the part which supportsthe deflection is called Effective elastic thickness (Te).“This is defined as the thickness of the crust that behaveselastically and that supports some or all topographical load”.(Burov and Diament, 1995). It lies on a fluid asthenosphere;therefore the largest value of elastic thickness increments thecapacity of the lithosphere to support topographical loadswithout having deflection. Airy’s model represents a specialcase in which the value of Te is null. To calculate the effec-tive elastic thickness, there are several methods that are mostbased on spectral and spatial relationships between topogra-phy and gravity, which are obtained through the use of mapsor profiles. The approach to be adopted in this work is to usea technique within the framework of spectral methods, usinginformation from profiles.

Admittance analysis

Admittance analysis or also known as isostatic responsefunction was developed by LeRoy M. Dorman and Brian T.R. Lewis in the year 1970. The Gravitational admittance “is

the wave number parameter that modifies the topography so

as to produce gravity anomaly”. (Watts, 2001). This allows“expresses the dependence of the gravity anomaly on topog-

raphy as a function of several physical parameters (plate

thickness, plate rigidity, density distribution) and wave-

length”. (Billen, 2001).

Admittance function Z(k) is defined as:

Z kG k

H k( )

( )

( )= (5)

Where k is wavenumber, G(k) and H(k) are Fouriertransforms of gravity and topography respectively. Crossspectrum C(k) is given by:

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Table 1: Correlation Matrix between GGM and Ground Gravity.

Correlation matrix

Ground GGM02C TEG4 EIGEN-CG03C

Ground 1.00000 0.57244 0.58255 0.58921

GGM02C 0.57244 1.00000 0.96028 0.93157

TEG4 0.58255 0.96028 1.00000 0.89357

EIGEN-CG03C 0.58921 0.93157 0.89357 1.0000



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Figure 2: EIGEN-CG03C free air gravity anomaly map. EIGEN Models (from 01 to 04) are derived from CHAMP and GRACE observations andpresent two versions, accompanied by a S only has the satellite component (n = 120) and the accompanying by a C has two components (Sat-ellite and Terrestrial), this last includes the same terrestrial data contained in the model EGM96 and presents until n = 360; these models aremaintained and updated by the GFZ (GeoForschungsZentrum).



C kN

G k H kr rr

N

( ) ( ) ( )= *=å1

1

(6)

Here * denotes conjugate complex and N is the totalnumber of profiles employed.

The power spectrum of topography E(k) is given as:

E kN

H k H kr rr

N

( ) ( ) ( )= *=å1

1

(7)

Finally, Observed Admittance average Zobs(k) is com-puted from:

Z kC k

E kobs ( )

( )

( )= (8)

The observed admittance curve is compared with a setof theoretical admittance curves for several values of effec-tive elastic thickness. The final value of Te is obtainedthrough the selection of the lowest mean square error be-tween the observed admittance curve and each one of thetheoretical curves.

Theoretical Admittance is defined as:

Z k GkT

Atheo c

c( )exp( )

=- -æ

èçç

ö

ø÷÷2

1p r (9)

With:

ADk

g m c

= +-

14

( )r r(10)

And D (flexural rigidity):

DETe=

-

3

212 1( )r(11)

Where:

K =2p

lWavenumber

G Nm

Kg= * -6 67 10 11

2

2. Universal gravity constant

E Pa= 1011 Young modulus

r = 0 25. Poisson rate

ggr

cm= 9 8

3. Gravity acceleration

rc

gr

cm=2 8

3. Average Crustal density

rm

gr

cm= 3 3

3. Average Mantle density

Tc = 33 km Contrast Density Layer Depth(average crustal thickness)

Te Effective elastic thickness

The mean square error is calculated using:

RMSpi ri

Ni

N

=-

=å ( )2

1

(12)

Here:

p Projected value.

r Observed value.

N Sample size

To obtain the observed admittance curve, data wereused from the free air gravity anomaly map produced fromthe model EIGEN-CG03C (Figure 2) and the digital terrainmodel constructed with data from SRTM (Shuttle Radar To-pography Mission) show in Figure 3.

A set of seven (7) profiles was drawn, both in the map offree air anomaly, as in the digital terrain model, crossing theAndes on a perpendicular path and covers it homoge-neously, Figure 3. Profiles were sampled each 1800 m tak-ing the ratio of DTM spatial resolution and free air gravityanomaly map resolution. To employ the Fourier analysis thisprofiles should complete a number of 2n data, where n = 8

because the total distance of the profiles have a range be-tween 400 and 700 km approximately. Theoretical admit-tance curves were computed for Te values of 10, 15, 20, 25and 30 km, in the same way observed admittance curve andits square medium errors were computed too. The resultsare showed in Figure 4 and Table 2.

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Table 2: Admittance Mean Square Errors.

MEAN SQUARE ERRORS

Te=10 km Te=15 km Te=20 km Te=25 km Te=30 km

Teobs 0.0002084 0.0000705 0.0000505 0.0001410 0.0002977



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Figure 3: DTM and Profiles used in Admittance Analysis. The profiles were used to extract both gravity and topographical data from maps.



In conclusion according to Table 2, the average of effec-tive elastic thickness for Colombian Andes is 20 km. Takinginto account this value, theoretical admittance curve and ob-served admittance values with its standard deviations arerepresented in Figure 5.

Discussion

Effective elastic thickness computes studies made in Andesmountain belt inside Colombia, have employed several tech-

niques and also have obtained several values. Some of thisworks were made for South American continental surfaceand some others specifically in Colombian Andes employ-ing different methods.

Kellogg et al. (1995) used flexural models 2D employ-ing a fractured plate, obtaining a value of effective elasticthickness between 20 and 25 km below volcanic arc and 60km for mountain belt placed at north of volcanic arc.Quintero (1998) used admittance method with profiles em-ploying Bouguer anomaly and topographical heights, in the

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Figure 4: Theoretical admittance curves for Te = 10, 15, 20, 25 and 30 km, and observed admittance curve. All the curves were calculatedfrom topographical heights and free air gravity anomalies.

Figure 5: Theoretical admittance curve (dashed curve) for Te = 20 km and observed admittance values (solid circle) with its respected stan-dard deviations (vertical bars).



same way used flexural modeling using a continuous plate;in West and Central mountain belt found an elastic thicknessvalue of 10 km, while Eastern mountain belt does not haveelastic support, this means that it has a null value of elasticthickness.

Londoño (2004) also employed flexural modeling withcontinuous plate finding for Putumayo Basin, an ElasticThickness value between 20 and 40 km. Recently Cerón et al

(2007) found for Plato Basin an elastic thickness value of 27km, this value was computed from Flexural Analysis withcontinuous plate. Finally, Stewart and Watts (1997) throughcontinuous plate modeling obtained a value of Te for Co-lombian Andes of 45 km with 40 km bias.

In South America have been conducted several studies,with the main aim of obtaining elastic thickness variationsmaps, although there are works that provide specific values.Ojeda (2000) made his research for Andes north region; thisregion includes Colombian East Cordillera and Sierra delMérida in Venezuela. To find the value of Elastic Thicknesswas used Coherence technique employing Hanning andMultitaper spectral estimators, obtaining an average value of30 km. Montavani et al (2001) get a spatial variations mapusing empirical correlations between tidal gravity anomaliesand elastic thickness values, finding for Colombian Andes avalue between 69 and 79 km. Into recent studies and thathave employed satellite-derived data too, there are twoworks Tassara et al (2007) and Perez-Gusinye et al (2007).Tassara et al (2007) has used topographical heights andEIGEN-CG03C model data to derive Bouguer anomalies,which were applied Coherence method using continuousWavelets transform’s Morlet family, getting an ElasticThickness structure map for all South America, in Colom-bian Andes found a 40 km value with 15 km bias.Perez-Gusinye et al (2007) has used topographical heightsand EIGEN-CG03C model data and also ground gravity data,which were applied Coherence technique employingmultitaper spectral estimators, obtaining the elastic thick-ness value between 25 and 30 km.

In the field of studies that employ spectral methods tofind elastic thickness value, all of these make use of Coher-ence technique with Bouguer anomalies and several esti-mators like Hanning, Multitaper o more recently likeWavelet Morlet family. Starting with this approach, theobtained value of 20 km, in comparison with Ojeda (2000)and Perez-Gusinye (2007) works, is low; although withTassara’s research, there is more agreement, because if thelowest limit value of this study is taken, it will be 25 km ofTe. Finally making a comparison with Montavani et al

(2001) study, it is noticeable that the value that it gets ismarkedly high, but is important to highlight thatPerez-Gusinye et al (2007) argue that this values cannot becompared with studies that make use of gravity anomaliesand topographical height, because while the tidal loadshave a short duration, the elastic thickness responses to adeformation process that only can be measured in geologi-cal time.

The average value of Elastic Thickness for the Colom-bian Andes obtained in this work, it means, 20 km compar-ing it with other studies, suggests that this one is similar tothe values obtained for the Putumayo Basin and the Volca-nic Arc. Now, comparing this work with Quintero’s (1998)study, a value of 20 km is equivalent to twice the obtainedvalue for this author, but, the 10 km value is the lowest in-side the totally of known studies. If the same confrontationis made with the obtained value from Plato Basin study, 20km value is low too, in the same way that occurs with Stew-ard and Watts (1997) in which this value is only a half.

Conclusions

The Te value obtained in this work is relatively low with themost studies, like it could be expected. The main cause is thekind of method that was used, because other techniques likeCoherence suggest that Admittance gets an underestimationor lower limit of the elastic thickness true value, argumentthat can be debated, because researchers that have employedAdmittance argue that processes as erosion and sedimenta-tion tends to thin the topographical surface, and in this waythe response of Crust is reduced in presence of subsurfaceloads.

Although 20 km value is low in relation to other works,it is totally congruent with the results that have to be ex-pected in Admittance studies, that make use of free airanomalies, because in studies like McKenzie and Fairhead(1997), it has been found that for continental lithosphere theelastic thickness value, with some exceptions, do not exceed25 km, since, it is usually close to the seismogenic thicknessvalue, this argument is still in discussion.

Acknowledgments

We thank Laura Sánchez (DGF - München), Miguel Ávila(Universidad Distrital) and very especially to AndrésTassara (Universidad de Concepción) for their suggestions,collaboration, and encouragement in the elaboration of thisproject.

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DETERMINATION OF EFFECTIVE ELASTIC THICKNESS OF THE ... · DETERMINATION OF EFFECTIVE ELASTIC THICKNESS OF THE COLOMBIAN ANDES USING SATELLITE-DERIVED GRAVITY DATA Remy A. Galán