anisotropy in the flexural response of the indian shield
TRANSCRIPT
Tectonophysics 532–535 (2012) 193–204
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Anisotropy in the flexural response of the Indian Shield
Rajesh R. Nair a,⁎, Yudhvir Singh a, Deshraj Trivedi b, Suresh Ch. Kandpal b
a Department of Ocean Engineering, IIT Madras, Indiab Department of Geology and Geophysics, IIT Kharagpur, India
⁎ Corresponding author.E-mail address: [email protected] (R.R. N
0040-1951/$ – see front matter © 2012 Elsevier B.V. Alldoi:10.1016/j.tecto.2012.02.006
a b s t r a c t
a r t i c l e i n f oArticle history:Received 19 May 2011Received in revised form 2 January 2012Accepted 5 February 2012Available online 20 February 2012
Keywords:Elastic thicknessMechanical strengthMechanical anisotropyIndian ShieldHermite tapers
We seek to determine the strain field which has accumulated in the Indian Shield due to the continental driftof Gondwanaland. We have used a method which involves the calculation of the 2D isostatic coherence re-sponse function between Bouguer gravity and topography as a function of azimuth by way of multispectro-gram analysis. The average coherence is maximum consistently in the Indian Shield in a direction, which is atan angle of 45° to the major trend of suture zones within the shield, a result which is in good agreement withthe strain inferred from absolute plate motion (APM) in a hot spot reference frame. This directionality of me-chanical plate weakness suggests that all paleostress fields were erased due to the movement of the Indianplate during the Himalayan orogeny.
© 2012 Elsevier B.V. All rights reserved.
1. Introduction
Mechanical anisotropy can be detected in the response of the lith-osphere to long-term load emplaced as topography. The integratedmechanical strength of the plate is expressed as an effective elasticthickness (Te) (Audet and Mareschal, 2004; Braitenberg et al., 2003;Burov and Diament, 1995; Forsyth, 1985; Pérez-Gussinyé et al.,2004; Simons and van der Hilst, 2003; Watts, 2001). The continentallithosphere is old in contrast to oceanic lithosphere. It has multilayerrheology and its effective elastic thickness does not depend on a sin-gle controlling mechanism nor its thermal age, but is an integrated ef-fect of a number of other factors that include the geotherm, crustalthickness, composition, and its variability with azimuthal direction,or look-angle, which depends on its stress/strain distribution.(McNutt et al., 1988; Watts, 1992; Watts and Burov, 2003).
Previous coherence estimates for Te in cratonic regions that allowfor both surface and subsurface loads are 65–120 km,which are compa-rable to those obtained from forward models of rift and foreland basins(Djomani et al., 1995; Ebinger et al., 1989; Pérez-Gussinyé et al., 2009).In the context of the Indian peninsular shield, the earlier estimatesbased on different methods show variations ranging from 10 to110 km (Jordan and Watts, 2005; Karner and Watts, 1983; Lyon-Caenand Molnar, 1985; McKenzie and Fairhead, 1997; Rajesh and Mishra,2004; Tiwari and Mishra, 1999). Lyon-Caen and Molnar (1983) andKarner and Watts (1983) used forward modeling techniques andshowed that Bouguer gravity anomaly over the Ganges basin could be
air).
rights reserved.
explained by the flexure of the Indian continental lithosphere. The esti-mated Te value of the continental lithosphere was in the range of80–110 km. McKenzie and Fairhead (1997) questioned the validity ofcontinental Te values >25 km, especially those based on the Bouguercoherence spectral technique (Bechtel et al., 1990; Ussami et al., 1993;Zuber et al., 1989) and estimated Te value of 24 km (for the IndianPeninsula) and 42 km (for profiles across the Himalayan foredeep) re-spectively, using spectral estimates of the free-air admittance and anon-spectral free-air gravity anomaly profile shape fitting technique.Tiwari and Mishra (1999) estimated Te to be 10±2 km under the Dec-can Volcanic Province based on admittance and coherence between thegravity field and topography and concluded that the value estimatedfrom Bouguer coherence and free-air admittance gave almost same Tefor the region but a considerable change in the compensation depthwas noticed. Rajesh and Mishra (2004) considered the possibility ofspatial variations in rigidity and found Te values ranging between 11and 16 km for the Indian cratonic region by a robust coherence methodbased on multitaper spectral analysis on overlapping windows of equalsize. Further, in this context of the India–Eurasia collisional system,Jordan andWatts (2005) estimated a high Te value of 70 km in the cen-tral region and a range of 30–50 km in the east and west. Their studyemployed forward modeling and results were verified with a 2-Dnon-spectral modeling technique using Bouguer gravity anomaly andtopography data. According to Forsyth (1985), a flexural isostaticmodel must include both the surface as well as subsurface loads inorder to accurately estimate Te. The coherence method based onBouguer gravity yields an estimate of Te that is less biased by the pres-ence of subsurface load than the free-air admittance method. In otherwords, it enables us to incorporate the effects of subsurface loading,though it is well known that coherence is not overly sensitive to the
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precise ratio or relative importance of those loads at either interface(Forsyth, 1985). Some of the previous works which investigated aniso-tropic mechanical response of lithosphere using spectral methods in-clude Simons et al. (2003), Audet and Mareschal (2004), Stephen et al.(2003) and Rajesh et al. (2003). Audet and Mareschal (2004) estimatedthe anisotropy of flexural response of the lithosphere in the CanadianShield using themultitapermethod and obtained results that show aver-age coherence, a proxy for plate weakness, increasing in the directionperpendicular to the main tectonic boundaries. Their inference, basedon correlation with seismic fast axes, was that anisotropy reflects thesame stress field possibly due to the last tectonic event. Stephen et al.(2003) estimated coherence anisotropy in the South Indian Shieldusing only the Slepian multitaper method and could capture only littlemechanical anisotropy. Using large window sizes based on Slepian func-tions, Rajesh et al. (2003) obtained anisotropy in Eastern Himalayasaligned with maximum compression direction. Simons et al. (2003)computed themechanical anisotropy of the Australian shield by calculat-ing the coherence function between topography and Bouguer gravityanomalies. This procedure ensures spectral leakage control and spatialselectivity on the coherence estimation was provided using Hermitemultitapers. Nair et al. (2010) estimated the mechanical strength of theIndonesian active continental margin by applying azimuthally averagedcoherence measurements between Bouguer gravity and topographyusing a multitaper method with Hermite tapers. Simons et al. (2003)compared the merits of Slepian wavelets and Hermite windows bytheir average Wigner-Ville transformation and concluded that the sym-metry and smoothness of the Hermite function kernels make them at-tractive for use in coherence analysis. In the context of the abovestudies, coherence estimation between Bouguer gravity anomaly and to-pography using the orthonormalized multispectogram method withHermite circular function remains our method of choice. The geologyand some major tectonic features are shown in Fig. 1. Topography (m)
Fig. 1. Map of Indian Shield showing different geological formations with some major tectoFront(HF), Godavri Rift(G), Narmada–Son Rift(NS), Mahanadi Rift(M) and Satpura Range(S
and Bouguer gravity (mGal) map of the Indian Shield, used in thisstudy, is shown in Figs. 2(a) and (b) respectively.
Seismic anisotropy ismainly produceddue to the lattice preferred ori-entation (LPO) of olivinemineral as a result of plate deformation aided byincreasedmelt productionwithin the uppermantle; at the edges of fault-bounded rift valleys. Seismic anisotropy can be due to the alignment ofinclusions such as crack-like melt inclusions (Kaminski, 2006, Kendallet al., 2005, 2006). The asthenospheric flow in the direction of absoluteplate motion (APM) and the flow in oceanic ridges can be responsiblefor induced lattice preferred orientation (Kaminski and Ribe, 2001;Kendall et al., 2005; Ribe, 1989). For isotropic media, seismic SKS wavesshould exhibit linear particle motion. However, in case of anisotropy,this phase splits into a fast and slow shear-wave and produces ellipticalparticle motion. The hypothesis of upwelling mantle material beneathhotspots predicts that the ascending hot material is deflected as well assheared by the lithosphere. This phenomenon is apparent in shear-wave splitting studies (Walker et al. (2005). Surface wave studies indi-cate that the lithospheric thickness of the Indian subcontinent varies be-tween 160 and 280 km (Pasyanos, 2008). The analysis of SKS splittingcan be used for the study of upper mantle anisotropy (Savage, 1999;Silver, 1996). The reviewof the relationship between elastic thickness es-timates and their relationship to that of the seismogenic layer is not con-sistent. Therefore, much controversy still exists on whether elasticthickness represents only the crustal strength. Kendall et al. (2005,2006) presented the evidence of anisotropy in the upper mantle of theNorthern Ethiopian Rift using shear-wave splitting in the teleseismicphases, SKS, SKKS and PKS. Their results indicate that the magnitude ofSKS splitting correlates with the degree of magmatism along the rift val-ley, and that, for the uppermost 75 km, it is primarily due to melt align-ment. Although away from the rift valley, the anisotropy is due to thepre-existing lithospheric fabric. However Burov and Watts (2006) useda numerical thermo mechanical model and argued that irrespective of
nic features, viz., Aravalli Range(A), Dharwar Craton(DC), Deccan Trap(DT), Himalayan). The geology of the Indian subcontinent is from Dasgupta et al. (2000).
70 75 80 85 90 955
10
15
20
25
30
-4500 -3000 -1500 0 1500 3000 4500
Topography (m)
B1
B2
B3
B4
70 75 80 85 90 955
10
15
20
25
30
-300 -200 -100 0 100 200 300(mGal)
Bouguer anomaly
B1
B2
B3
B4
a
b
Fig. 2. Topography (m) and Bouguer gravity (mGal) map of the Indian Shield indicatingthe different window locations used in the study. Black squares indicate the centre ofthe windows of size 550×550 km2. B1, B2, B3 and B4 indicate different window sizes:750×750 km2, 850×850 km2, 650×650 km2 and 550×550 km2 respectively. The legend(square boxes) at the right hand bottom corner of the figure indicates the window sizesmentioned above and the color of legend is same for respective window color.
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the actual crustal strength, the “jelly sandwich model” is found to be ap-plicable in various tectonic settings, rather than a “crème brulée model”.All of these current controversies and ambiguities concerning thestrength of continental lithosphere are a matter of debate, and our resultprovides additional evidence to nuance it. In the case of coherent or
isotropic deformation, the fast polarization direction of SKS splitting,i.e., the seismic fast axis is either parallel or perpendicular to the trendsof the structural features (Silver, 1996). The seismic fast axis is perpendic-ular to the direction of maximum deformation, i.e., the mechanically“weak” direction. Isostatic compensation is another tool in the determi-nation of the nature of deformation. Anisotropic isostatic compensationshows that the complete lithosphere has not undergone deformation ina coherent fashion as there is a varying degree of correlation between to-pography and Bouguer gravity depending on direction. This is alsoreflected in the structural trends (strikes of faults, stress directions,etc.). In case of isotropic-isostatic compensation the topography andBou-guer gravity are uniformly correlated. In this study, we examinewhetheror not the fast seismic axes and the mechanically “weak” direction areperpendicular, parallel, or at any other angle to each other. It is significantto note that recrystallization might alter the relationships mentionedabove (Kaminski and Ribe, 2001). Seismic anisotropy can be character-ized by either “frozen” anisotropy or “present day” anisotropy or a com-bination of both with a transitional boundary between the two. It hasbeen a matter of debate to actually find this boundary. For example, inthe Canadian Shield, Audet and Mareschal (2004) showed that, alongthe seismic fast axis, the average coherence value increased dramatically,which suggested that the entire lithosphere was affected by that samestress field. The variations of the directions of the coherence obtainedfor each wave vector are averaged with an azimuthal wedge to obtainthe average coherence. Simons and van der Hilst (2003) suggested thatin theAustralian Shield, the seismic fast axis is at large angles to the direc-tion of principal shortening and that the seismic anisotropy obtained inthat case could be considered as “frozen”. However at depth, the fastaxes are more commonly reflecting present deformation in alignmentwith absolute platemotion. Our present studieswillfill the gap on the es-timation of coherence anisotropy with seismic anisotropy in the IndianShield because our method has the ability to map the spatial variationsof the coherence function in great detail.
The Indian Shield is an amalgamation of Archean cratons andProterozoic mobile belts. The rift zones in East Antarctica exhibitclose similarity to the rift zones in the Indian Shield, namely Godavari,Narmada and Mahanadi (Subrahmanyam and Chand, 2006). A geo-logical map of Indian subcontinent is depicted in Fig. 1, with majortectonic features (Dasgupta et al., 2000). The map depicts the broadstratigraphic classification of lithostratigraphic units of India. ThePre-Cambrian rocks occur predominantly in the Dharwar Craton inthe south India, in small patches in north-east and north-westAravalli Ranges. Deccan trap, a conspicuous geological feature, is amassive Cretaceous flood basalt region in the western part of theIndian Shield. A huge Himalayan Front, narrow belt in north-east andEastern Ghats are regions of sedimentation in recent and Pleistoceneage. Tertiary occurs in some part of north-east while the Godwanaand Vindhvan is distributed in central India as some patches. Fig. 1shows the distribution of Granites and Cuddapah. Aravali range,Dharwar Caton, Godavri Rift, Narmada son Rift, Mahanadi Rift andSatpura Ranges are some of the major tectonic features in Indian sub-continent. Several observations have been made to reconstruct theIndian Shield drifting with respect to the Gondwanaland. Studingerand Miller (1999) obtained the value of effective elastic thicknessTe=35±5 km from coherence spectra and a crustal thickness of27 km for the Ronne shelf, Antarctica and inferred its thermal age tobe between 165 and 230 Ma, which supposedly marks the last signifi-cant tectonic event and is coeveal with the breakup of Gondwanaland.The mobility of the Indian region, with velocities ranging up to 20 cm/yr after it separated from the mainland around 130 Ma ago, makes itquite unique in nature with respect to other parts of Gondwanalandand in comparison to the other shield regions. The drift of the IndianShield over the Reunion, Krozet Kerguelen and Marion hotspots culmi-nated in enormous magmatic episodes like the Deccan and Rajmahaltraps. The Indian Shield is still considered to be quite a fast movingplate despite the fact that it slowed down because of its collision with
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Asia. The surface geology of the Indian Shield is an amalgamation of Ar-chean cratons that got accreted andwelded bymobile belts. In this con-text, it is interesting to investigate the distinct anisotropic character ofthe geologic provinces, whether or not the fast moving plate causes de-formation in the platemotion direction, and if remnants of fossilized an-isotropy exist.
Audet and Mareschal (2004) compared the seismic structure,electrical conductivity, and flexural response of the lithosphere inthe Canadian Shield and obtained a strong correlation with the tec-tonic features throughout the shield. Our present estimates basedon gravity–topography coherence and seismic anisotropy provideunique constraints on the rheology and deformation of the IndianShield. The present study is based on independent observations ofmechanical and seismic anisotropy, and we believe that our approachwill fill a lacuna by providing answers to several questions about rhe-ology that might not have been interpreted fully due to lack ofcomplete data. Simons and van der Hilst (2003) compared the azi-muthally anisotropic shear-wave speed of the Australian upper man-tle with mechanical anisotropy as a proxy for the fossil strain field.They further jointly interpreted alignment of the fast axis and tecton-ic data (the absolute plate motion direction) in a framework consis-tent with the deformation of a dry olivine mantle.
We will analyze the mechanical anisotropy of the Indian Shield bycalculating coherence functions between topography and Bouguergravity anomalies using the circular Hermite functions. The mechanicalproperties of the Indian lithosphere derived from our findings will beverified with the fast directions obtained from high-resolution seismicsurface wave tomography with azimuthal anisotropy (Kumar andSingh, 2008). Following this, we will identify directions that have accu-mulated more than the isotropic average of gravitational anomalies fora given amount of topography and test the hypothesis as to whether ornot seismic anisotropy is reflected in the anisotropy of fossilized strain.
2. Mechanical anisotropy analysis
The present study uses the multitaper method (Percival andWalden, 1993; Thomson, 1982) with Hermite functions applied totwo-dimensional fields (Simons et al., 2003).
2.1. Coherence
The assumptions on the estimation of plate flexure and loading ratiovia parameterized fitting of the coherence imply that the surface and in-ternal loads should not be statistically related to each other. The funda-mental sensitivity in the estimation of the rigidity of the plate lies in thetransition wavelength from low to high coherence where in the case oflonger wavelengths the Bouguer anomaly is coherent with the topogra-phy (as surface loads are fully compensated). On the other hand, forshorter wavelengths, Bouguer anomaly and topography lack coherence(as the loads are supported by the strength of the lithosphere) (Audetand Mareschal, 2004; Bechtel et al., 1987; Forsyth, 1985). For two non-stationary random processes {X} (gravity) and {Y} (topography), de-fined on r in the spatial domain and on k in the Fourier domain, thecoherence-square function relating both fields, γ2
XY, is defined as theratio of their cross-spectral density, SXY, normalized by the individualpower spectral densities, SXX and SYY (Bendat and Piersol, 2000)
γ2XY r; kð Þ ¼ SXY r; kð Þj j2
SXX r; kð ÞSYY r; kð Þ
¼bE ~X r; kð Þ~Y � r; kð Þn o
>��� ���2
b E ~X r; kð Þ~X � r; kð Þn o
> b E ~Y r; kð Þ~Y � r; kð Þn o
>
ð1Þ
Here, E denotes an expectation operator, tildes refer to theFourier-transformed signal, and the asterisk refers to the complex
conjugate. The periodogram ~X ~X � is a direct spectral estimator of X, al-though not a particularly accurate one. Windowing is a way of modi-fying the periodogram by multiplying the spatial data by a suitablychosen window function. The ideal data window has a narrow spec-tral response and low sidelobes, and it is able to isolate geographicalfeatures spatially with little sensitivity to the regions outside of thedomain of interest. Slepian functions are one way to accomplish thisgoal (Simons et al., 2000). In this study we use Hermite functions,which share many of the desirable properties, but which are also iso-tropically sensitive in the wave number domain (Simons et al., 2003).In the most recent literature (Simons and Wang, 2011), Slepian func-tions have been designed to maximize the spatial concentration toarbitrally irregular data field, while maintaining isotropic spectral fea-ture extraction. Each of the windowing functions discussed aboveprovides control on the spectral bias and leakage. Reduction of theestimation variance is achieved by averaging the results of many indi-vidually windowed modified periodogram estimates. If those win-dows, as is our case, are orthonormal, the resulting estimates areapproximately statistically uncorrelated, and the variance reductionwith the inclusion of each additional window is swift and scales in-versely with the number of data windows considered in the average.The unavoidable trade-off lies in that spectral bias degrades with in-creasing bandwidth of windows, while variance improves (Percivaland Walden, 1993).
The coherence value (0bγ2b1) is a measure of how well the twofields namely Bouguer gravity and topography are correlated with eachother. Fields having γ2=0 suggest that they are not “well-correlated”and are simply randomly distributed. On the contrary, high values ofγ2 are yielded by “well-correlated” fields as their cross spectra resultsin constructive interference. Values of γ2 are obtained by averagingover all azimuths leading to loss in directional information. ObservedCoherence (γ2|k|) is, shown in Fig. 3, as a function of wave number (k)and wavelength (λ) for windows B1, B2, B3 and B4 for varying valuesof R, viz., R=2, 3 and 4. In the multitaper method (MTM), we dealwith data consisting of spectra with multiple orthogonal 2D windows,as described above. Thus, the MTMmethod can be used to calculate co-herence without loss of directional information. As with admittance,the information on the flexural rigidity lies in the relationship betweentopography and gravity. For example, in the case of Bouguer coherencefor a surface load,Phb, is given by :
Phb ¼ HνtWt ð2Þ
Ht is frequency domain topography from surface loading; Wt isthe flexural response as Wt ¼ Htρcφ k
� �= ρm−ρcð Þ where φ(k)=
[Dk4(ρm−ρc)g+1]−1,D is flexural rigidity related to Te as: D=ETe3/12(1−σ2)where E and σ are Young's modulus and Poisson'sratio respectively. Further k is wave number g is acceleration due togravity, ρc and ρm are the crustal and mantle densities respectively.The Te value is obtained when the root mean square error goes to athreshold value (approximately zero) (Fig. 4e). Flexural response isone manifestation of rigidity of plate in response to surface loadssuch as ice, volcanoes, sediments and subsurface loads such as thoseassociated with tectonic emplacement and magmatic intrusions, andtectonic boundary loads . It is possible to determine the flexural rigid-ity of the lithosphere and its spatial and temporal variation by com-paring the observed flexure of the lithosphere and predicted flexureof simple plate models (Watts and Burov, 2003). These long-termflexural and mechanical responses are reflected in the coherence(Forsyth, 1985) and admittance (McKenzie and Bowin, 1976) be-tween gravity anomalies and topography. In the case of low signal-to-noise ratio, the coherence method using multitaper spectral esti-mation yields significant underestimates of Te value. The solution isto use data windows several times larger than the true flexural wave-length (Swain and Kirby, 2003). In the present study, signal-to-noise
Fig. 3. Observed coherence (γ2|k|) shown as a function of wave number (k) and wavelength (λ) for windows B1, B2, B3 and B4 for varying values of R, viz., R=2, 3 and 4.
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ratio is sufficiently good, and we also used data windows having sizesseveral times larger than the true flexural wavelength.
2.1.1. Spatiospectral localization properties
2.1.1.1. Hermite tapers. The multitaper method (MTM) involves thecalculation of multiple orthogonal window spectra which are usedas data tapers, and forming the spectra, followed by their averagingover other independent subsets of the data (Percival and Walden,1993). The spectral resolution of Hermite functions is identical in allpossible azimuths without the introduction of spurious anisotropy(Simons et al., 2003), but in the case of alternative tonsorial combina-tion of 1D Slepian functions, this is not achieved due to rectangulartiling (Hanssen, 1997; Parks and Shenoy, 1990; Simons and Wang,2011). We have thus used orthonormal Hermite circular functionsas data tapers. In space wave number domain applying circular sym-metry the resulting spherical concentration domains allow the esti-mation of spatially localized directional spectral properties withoutanisotropic bias. In circular symmetry, these are the eigenfunctionsof the operators concentrating in a disk: t2+(2πf)2≤R2
, where t, fand R stand for time, frequency, and radius of concentration regionrespectively. Their analytic form is given by (Olhede and Walden,2002)
hj tð Þ ¼ Hj tð Þe−t2=2
π1=4ffiffiffiffiffiffiffi2jj!
q ; ð3Þ
i.e., they are Hermite polynomials, Hj, modulated by a Gaussianfunction.
The present method optimizes the retrieval of coherence in thespectral domain and its localization in the spatial domain. This method-ology retrieves the effective elastic thickness assuming isotropic re-sponse of spectral estimation in all azimuths in order to get anunbiased estimation of directional dependence. See Simons et al.(2003) for a description of the MTM applied to flexural studies. Fig. 3shows our tests with various windows B1, B2, B3 and B4 retrieving ob-served coherence with respect to varying R values from 2 to 4. Fig. 4d
shows calculated coherence value for R=3 and predicted coherencefor a set of Te values. The number of tapers used for each value of R isequal to R2 in each dimension. Simons et al. (2003) demonstratedwith synthetic data that R=3 gives the best results. Increasing thevalue of R decreases the resolution confirming choice of R=3 as provid-ing the best trade-off between resolution and variance in our presentanalysis. We notice that that the transitional coherence wavelength cal-culated for R=3 lies in between R=2 and R=4. This analysismakes usconsider R=3 and use 9 tapers in each dimension (total 81 differentHermite tapers) for our coherence study. The width of the central lobeof the periodogramof the Hermitewindowing functions provides an ef-fective measure of the resolution with which any spectral informationcan be attributed to a particular wave number. It is important to notethat this uncertainty can be large. We also have demonstrated thepower spectral estimator in 2-D space using discrete prolate spheroidalSlepian sequences with a fixed time–bandwidth product (NW=3) andShannon number equal to 5. Fig. 5 shows the comparison of azimuthalcoherences and averages of different window sizes. It also shows theobserved and predicted isotropic coherence curves and RMS error var-iation as the Te value changes. Here different window sizes taken areas: 550×550 km2, 650×650 km2, 750×750 km2 and 850×850 km2.
3. Coherence anisotropy measurements in the Indian Shield
In our previous work (Rajesh andMishra, 2004), we determined thevariations in the isotropic elastic thickness Te in the Indian Shield usinga Slepianwindowmultitaper method. In this study, we calculate the 2Dcoherence anisotropy with a Hermite window multitaper methodorthonormalized averaged MTM coherence to estimate Te, using theBouguer gravity and topography data from published gravity maps(NGRI, 1975). The gravity data is further merged with published dataof different tectonic provinces (Mishra et al., 1987, 2000; Rajesh,2004; Singh et al., 2003; Subrahmanyam and Verma, 1982; Verma andBanerjee, 1992) and the cumulative error in the Bouguer gravity field,including errors in elevation, is estimated to be 1–2 mGal (Rajesh andMishra, 2004). Oceanic bathymetry [GEBCO] was added to provide con-tinuity to the continental topography, and Andersen data (Andersenand Knudsen, 1998; Andersen et al., 2008) is used in marine regions.
Fig. 4. A few illustrations of the determination of the flexural anisotropy and estimation of Te from isotropic coherence using Hermite multitaper. (a) Compares the azimuthal co-herences and averages of varying window sizes (550×550 km2, 650×650 km2, 750×750 km2 and 850×850 km2). (b) Shows the power spectra of gravity and topography.(c) Shows the variations with direction of coherences averaged within an azimuthal wedge. (d) Observed (black circles) and predicted (colored lines) isotropic coherence curvesfor different elastic thicknesses. (e) RMS Errors used to estimate Te.
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Weused theDNSC07model (Andersen et al., 2008)whose overall accu-racy of the model is 2.78 mGal. Geoidal corrections have been incorpo-rated to account for the central Indian Ocean geoidal low (Savage,1999), which if neglected, will cause a very large negative bias in thepeninsular Indian gravity field (Rajesh and Mishra, 2004). Geoidal un-dulations over the study region vary from −105 m to−10 m presum-ably due to a deep mantle source. Since oceanic bathymetry is rarelyindependently observed from the gravity signal, we restricted our anal-ysis to data windows well within the continents. Moreover, in regions
with an oceanic component, the free-air gravity fields determinedfrom satellite altimetry (Andersen and Knudsen, 1998) are convertedinto Bouguer gravity anomaly ΔGb using the slab formula:
ΔGb ¼ ΔGf þ 2πΔρGH ð4Þ
where Δρ (ρc-ρw) is the difference between mean crustal density ( ρc)and water density (ρw), H is the bathymetry (in meters) and G is thegravitational constant. This Bouguer data is then merged with the
Fig. 5. Results of tests with a different spectral estimator with power spectrum estimated in 2-D space using discrete prolate spheroidal Slepian sequences with a fixed time-bandwidth product (NW=3) and Shannon number equal to 5. (a) Compares the azimuthal coherences and averages of varying window sizes (550×550 km2, 650×650 km2,750×750 km2 and 850×850 km2). (b) Observed (circles) and predicted (colored line) isotropic coherence curves for different elastic thicknesses shown in the figure. (c) RMSErrors used to estimate Te.
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continental Bouguer data. The equivalent topography is the height ordepth the crust will assume in the absence of ice or water present andunder isostatic conditions this is (Daly et al., 2004) given by
h xð Þ ¼ ρc−ρw
ρc
� �� d ð5Þ
where h(x) is the equivalent topography, d is the bathymetry (in m), ρcand ρw are the mean crustal density (2800 kg m-3 ) and the waterdensity( 1030 kg m−3), respectively (Curray et al., 1982; Grevemeyeret al., 2001).
We extracted subgrids of size 550×550 km2 overlapped in thecomplete Indian Shield and also extracted varying size windows(650×650 km2, 750×750 km2, 850×850 km2) to confirm that thecoherence anisotropy measured is not a function of the windowsizes chosen. For each window, we estimated the coherence and di-rection of highest coherence (the weak axis). For estimating Te, weassumed Poissons ratio ν=0.25 and Young's modulus E=1011 Pa(Curray et al., 1982; Grevemeyer et al., 2001). The elastic model pa-rameter, i.e., crustal thickness used for the present analysis isobtained from several sources. In the South Indian Shield, the con-straint comes from Kumar et al. (2001) and Rai et al. (2003) and inthe North Indian Shield it comes from Rai et al. (1999), Jagadeesh
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and Rai (2008), Kaila et al. (1987) and Kaila et al. (1990). We invertedthe predicted coherence for each iteration of Te value incrementallyuntil we attained the convergence of a solution with an optimizationbased on least square minimization. The Te value is presumablyobtained from a best fit between observed and predicted coherencevalue. The general form of coherence-square function, γ2 (Eq. (1)),for gravity and topography for a model of single density contrast(Δρ) can be rewrite as a function of subsurface to surface loadingratio ‘f’ as well as function of other elastic parameters (Eq. (6)).
γ2 ¼ξΔρ2 þ f 2ρ2ϕ
� �2
ξΔρ2 þ f 2ρ2�
Δρ2 þ f 2ρ2ϕ2� where
ξ ¼ 1þ Dk4
Δρgandϕ ¼ 1þ Dk4
ρg
ð6Þ
So the Te value (i.e. D=ETe3/12(1−σ2)) depends on number ofelastic parameters as well as on subsurface to surface loading ratio.Further the loading ratio (and hence Te) will be a function ofwavelength, i.e. f= f(k) and is influenced by the assumed depth tothe loading interface. The loading ratio increases with the wavenumber and that is inversely correlate with the elastic thickness(Forsyth, 1985). The insignificance and dominance of subsurfaceloads resembles in the much lower and higher value of f. Fig. 6shows the spatial variation of the loading ratio in the Indiansubcontinent. From Figs. 6 and 7 one can see that there are patchesof high f(k) where Te value is very low, except some exceptionswhich impose restrictions on the interpretation of the spatialvariation of f(k). Tassara et al. (2007) did detailed analysis forflexural loading ratio (which is average of f(k) for three wavenumbers near the corresponding flexural wavelength) andobtained similar correlation with the Te value in South America.
We perform additional test with a different spectral estimator es-timating the 2-D coherence function γ2(k) calculated 2D MTM
Fig. 6. Shows the spatial variation of s
Slepian sequences. The 2-D coherence anisotropy plot (Fig. 5a) forB1, B2, B3, B4 shows the weakness direction of lithosphere. These co-herence anisotropy directions are in agreement with the resultsyielded by Hermite multitaper method. In addition the lower Tevalue (Fig. 5b) obtained is in agreement with similar results yieldfrom Bouguer coherence based Hermite multitaper method. TheRMS error plotted in Fig. 5c suggests that the Te results are obtainedwhen the error goes to minimum.
The obtained Te value using Hermite multitaper method ismapped in Fig. 7, where the direction of the weak axis is superim-posed on the map of Te, obtained from orthonormalized Hermite ta-pers. The direction of SKS splitting (φ) for each window is obtainedby averaging the values falling within that particular window, thosewhich are given by Kumar and Singh (2008). The estimated Te andφ values, with average error, for each window are tabulated inTable 1. As an additional verification to the direction of anisotropyobtained, several random tests are undertaken by applying rotationsto the data sets from 5° to 90°. These tests confirmed that the ob-served anisotropy direction rotates with it. Debayle et al. (2005)have published worldwide maps of anisotropy from surface wavestudies, quite different from SKS studies. Their map may not havesame resolution but they give a North direction for the weak axis.These results also results in agreement with our conclusion that flex-ural anisotropy recorded in the Indian plate is influenced by APMstresses rather than past tectonic history.
We did not try to calculate physical parameters from the anisotropic2D coherence, but only determined the direction of maximum coher-ence sincewe are only interested in the directionality ofmechanical an-isotropy, rather than the values of the physical parameters. However,we have seen that the Te estimates are obtained only when the solutionconverges with a minimum error value, mostly global minimum, andthe anisotropy directions are retrieved at such cases within its uncer-tainty. The procedure is illustrated in (Fig. 4a–e), which shows thetwo-dimensional coherences (Fig. 4a), the power spectra of gravity
ubsurface to surface loading ratio.
Fig. 7.Map of elastic thickness of the Indian Shield showing variations in isotropic Te and the direction of maximum coherence in different wavelength regimes. The major tectonicfeatures are also included, viz., Aravalli Range(A), Dharwar Craton(DC), Deccan Trap(DT), Himalayan Front(HF) and Satpura Range(S). Black squares indicate the centre of the re-gions where Te is estimated. Black bars along with red circles indicate the fast seismic axis (Kumar and Singh, 2008). Orange arrows indicate the APM direction (DeMets et al.,1994).
201R.R. Nair et al. / Tectonophysics 532–535 (2012) 193–204
and topography (Fig. 4b), and the variations with direction of coher-ences averaged within an azimuthally wedge (Fig. 4c). We interpretthe increased coherence due to the lithosphere as being the weakestin that direction. Thus, the weak axis in Fig. 4a is the direction ofmaximum coherence. In Fig. 2, white triangles indicate the centre ofoverlapping windows with size 850×850 km2. Brown, red and blacktriangles represent centres of varying window sizes: 750×750 km2,650×650 km2, 550×550 km2 respectively. Although we computedthe Te values for the entire Indian Shield by merging with the land–ocean scheme, here we chose to show only those window centreswhose corresponding windows lie completely within the land. Hermitetapers are preferred for coherence anisotropy measurements instead oftensorial combinations of 1D Slepian tapers for the reasons mentionedin the earlier sections.
The obtained values for effective elastic thickness (Te) in IndianShield are indeed low, when compared to the values obtained from
Table 1Listing of the coherent anisotropy direction, Elastic thickness (Te) and error, and ratioof (Max−Min)/(Max+Min) Coherence Anisotropy for each window. The SKS splittingdirection is taken from Kumar and Singh (2008) and averaged for all stations falling ina particular window.
Window no. Coherenceanisotropydirection
Te(km) (Max−Min)/(Max+Min)(coherence anisotropy)
SKS splittingdirection (φ°)
1 45° 3±1 0.96 25.3±14.92 45° 6±2 0.875 33.1±13.63 45° 11±2 0.91 38.1±13.54 45° 12±2 0.96 34.1±13.25 45° 10±1 0.875 30.0±13.46 45° 8±1 0.91 28.8±13.37 45° 5±2 0.84 26.8±13.28 45° 5±1 0.96 28.1±14.29 45° 7±2 0.91 24.5±11.810 45° 3±1 0.96 24.6±12.011 45° 5±1 0.91 29.8±13.312 45° 10±1 0.875 43.1±14.113 90° 12±2 0.96 46.4±14.014 90° 19±3 0.96 46.6±14.3
other Shield regions. Tiwari and Mishra (1999) used free-air admit-tance and computed similar low elastic thickness under the Deccanvolcanic province. Their estimated admittance function shows agood fit with the regional compensation model with and effectiveelastic thickness of 10±2 km. In this study we chose the Bouguer co-herence method of Forsyth (1985) with the coherence estimated viaHermite windowmultitapers method of Simons et al. (2003). The lat-ter provides low variance estimates of the coherence between gravityand topography with a fairly uniform average over a selection of wavenumber ranges: the former attempts to find Te by fitting a parameter-ized curve through it. This is especially possible due to the loaddeconvolution that is implicitly in the Forsyth (1985) method as weuse it, the deliberate spectral bias, introduced by the multitaper anal-ysis of data thus influencing the predicted coherence leading to rea-sonably faithful recovery of elastic thickness in synthetic/real data.In addition McKenzie (2010) suggested that the elastic thickness ofcraton is indeed comparable to the elastic thickness of the ocean(also normally low Te values). In marine realms since the free-air co-herence is high the estimated Te from free-air admittance andBouguer coherence will be nearly similar. However cratonic regions(as in Indian Shield) the coherence is reduced by subsurface loadwithout topographic expression (McKenzie, 2003, McKenzie, 2010;McKenzie and Fairhead, 1997). This could be the possible reason forthe low coherence and low Te value. We do caution that the relativeestimation of elastic thickness is more reliable than the absolutevalues whichmay contain additional bias. Either admittance or coher-ence method uses only information about the Earth's gravity and to-pography and both form different ratio of their spectral energydensity. Stephen et al. (2003) used multitaper method by way of dis-crete prolate spheroidal Slepian sequences in the south Indian Shieldand obtain low Te values of 11–16 km and shows how the result cor-relates with max. horizontal stress orientation and heat flow mea-surements. Thus as a priori there is no reason to doubt that valuesobtained from one method with their correct uncertainties shouldbe compatible with the value obtained using the other methods.There are uncertainty studies that argue the difference in absolutevalues of the Te values between competing analysis methods. This
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comparison is significant with proper accounting of the statistics ofthe estimates. We remain confident that the results obtained via Her-mite window multitaper method leads to results which are quantita-tively interpretable to a level that is attempted in this paper.
Stephen et al. (2003) in south Indian Shield used window sizes of750×750 km2, 620×620 km2 and 450×450 km2 to confer a test onthe ability of these window sizes in capturing the elastic behavior ofLithosphere. Our further tests with window sizes of 650×650 km2,750×750 km2, 850×850 km2 in the present study demonstrate thatthese window sizes are capable of retrieving the relative estimatesof effective elastic thickness. Results of tests with a different spectralestimator with power spectrum estimated in 2-D space using discreteprolate spheroidal Slepian sequences with a fixed time-bandwidthproduct (NW=3) and Shannon number equal to 5 are shown inFig. 5 in comparison to Fig. 4 where the B1, B2, B3 and B4 correlateswell with the measured anisotropy. In Fig. 5 the minima for RMSerror is more clear and the corresponding Te values matches withthe values obtained due to Hermite multitaper method (Fig. 4) inwhich we have taken minimum threshold RMS error.
Mechanical anisotropy was found consistently aligned at N 45° Ethroughout the Indian Shield with an exception of N–S orientationin Himalayan Front (HF) and in a part of the Narmada-Son lineament(NS). Fig. 7 shows that there are distinct variations in isotropic elasticthickness with values as high as on the order of 20–25 km in the Indo-Gangetic alluvial planes (Himalayan Front (HF) in Fig. 1) and 0–7 kmin the South Indian Shield (the area includes the Deccan Trap (DT)and the Dharwar Craton (DC) shown in Fig. 1) and the North-East re-gion (Fig. 1), and elastic thickness of 10–13 km on the eastern bound-ary of Deccan traps. Fig. 4c compares the azimuthally coherences andaverages of varying window sizes as mentioned above. On the 2Dspectrum, the anisotropy appears to be better defined in all the win-dows and the azimuthal averages show a clear maximum consistentlyfor all regions (Table 1).
4. Discussion
We have used flexural anisotropy estimated from topography andgravity anomalies to understand the continental deformation. Thewavelength-dependent coherence function captures the magnitudeof flexure experienced by density interfaces in the lithosphere dueto loading by topography. The elastic strength or weakness of theplate is retrieved from the 2-D coherence. The lithosphere is theweakest in the direction of the maximum coherence. We comparethe seismic fast axis with respect to the flexural anisotropy andAPM. The seismic fast axis is dependent on the preferred orientationof anisotropic minerals such as olivine, developed in response to de-formation (Park and Levin, 2002). Various factors such as strainrate, stress, temperature and water content determine the orientationof the seismic fast axes relative to the strain directions (Karato andWu, 1993). Although the presence of water can alter the relationshipssignificantly (Jung and Karato, 2001; Kaminski, 2002), it is observedthat the fast polarization azimuth of seismic waves may align eitherwith the direction of maximum extension (perpendicular to the di-rection of principal horizontal shortening) (Ribe, 1992; Wenk et al.,1991), or with the shear direction (the direction of flow) (Zhangand Karato, 1995). Kumar and Singh (2008) obtained fast axis azi-muths from 182 measurements of azimuthal anisotropy by analyzingthe SKS and SKKS waveforms from 85 earthquakes recorded at 35broadband stations located at various geological units of the IndianShield. They do not reveal coherent patterns, and the northern partof the lithosphere is thick showing a strong correlation betweenplate motion and azimuths of fast polarization, interpreted as beingdue to the effect of basal cratonic keel ploughing through the sub-lithospheric mantle. In this particular tectonic setting, we calculatedabsolute Te values of up to a maximum of 25 km, which is withinthe limit of seismogenic thickness of crust for Indo-Gangetic planes.
This implies that elastic strength resides in the crustal layer and the“crème brulée” model is applicable. Similarly, the values that weobtained for isostatic coherence anisotropy measurements vary be-tween 45° and 90°. We interpreted the increased coherence as dueto the lithosphere being the weakest in that direction. Deformationis assumed to take place in a coherent fashion. If we consider thatall the SKS splitting is from the mantle, then during a coherent defor-mation, the splitting direction should be either parallel or perpendic-ular to the tectonic trends (Silver, 1996). If we apply the same logic tothe isostatic coherence, then the deformation in the mantle shouldhave an effect on the crustal scale. We found that the coherence an-isotropy is aligned in the direction of APM (DeMets et al., 1994),thus providing a strong evidence for late plate motion related straindue to the last major tectonic event. We interpret the exceptional90° anisotropy obtained in Indo-Gangetic plains as being due to theedge flow associated with a transition from a thick (Himalayas) to arelatively thin lithosphere (Gangetic plains), as well as a transitionfrom high mechanical strength to low mechanical strength (Fig. 7).It is well understood that the low heat flow in Deccan province sug-gest that lithosphere was not heated and magma came to the surfacethrough a few narrow conduits without causing a major thermal per-turbation. Our flexure anisotropy measurements suggest that even insuch regions the average anisotropy is positively correlated with theabsolute plate motion of plate. This suggested that although such sig-natures are present they are not predominant in the context of flexur-al anisotropy measurements. The physical mechanism underlying theanisotropy in elastic thickness and in seismic velocities is not likely tobe same in different geological regions as they involve different depthranges. As for example in the Canadian Shield the anisotropy in elasticthickness positively related to the dominant direction of faults ratherthan the physical properties in the mantle (Audet and Mareschal,2004). The effective elastic thikness has been studied in Shield re-gions with different methods that yield different values owing tothe methodologies used. Thus focus is rather on relative variation inelastic thickness than absolute values. We caution that the valuesobtained for one method should be compared with the same methodin different geological regimes. Some high values of heat flow havebeen observed in Shield regions (Gawler craton in Australia). Thehigh heat flow values in the Indian Shield and elsewhere are certainlycorrelated with high surface heat production and not with mantleheat flow(Roy and Mareschal, 2011). Zoback et al. (1989) showsthat the tectonic stress orientations and rheology is positevly corre-lated in the Indian plate. We infer that the continuing plate motioncan result in weaking of the plate to a large extent with perdominentintra-plate seismicity. These are some indications that is consistentwith low effective thickness observations in the Indian Shield in ourpresent study. Perez-Gussinye and Watts (2005) demonstrated theretrieval of spatial strength variations in Europe and suggested thatthese variations were due to changes in lithospheric plate structure.Stephenson and Beaumont (1980) hypothesized that flexural anisot-ropy can be related to mantle convection. Our study, which is the firstof its kind in the Indian Shield, supports the mantle flow hypothesis.In the Canadian Shield, mantle flow occurs beneath the cratonicroot which is much thicker in than Te (Polet and Anderson, 1995;van der Lee and Nolet, 1997). Our observations reveal that the direc-tion of anisotropy remains consistently the same and therefore mightfit with the mantle flow lines. Throughout the Indian Shield, the flex-ural anisotropy is found to have a weak correlation with tectonic fea-tures, fair correlation with seismic anisotropy, and a good correlationwith APM (Fig. 7). In the western, southern, and northern parts of theIndian Shield, the seismic fast axis, APM, and coherence weak axis areall nearly parallel to each other. In the Indo-Gangetic alluvial plains,however, the seismic fast axis and APM are nearly parallel to eachother, though the coherence weak axis is oriented nearly vertically(in the N-S direction). A good correlation of the coherence weakaxis with APM suggests that the crust and mantle were affected by
203R.R. Nair et al. / Tectonophysics 532–535 (2012) 193–204
the same stress field during the last major tectonic episode (Rondenayet al., 2000). We now hypothesize that any fossil anisotropy beforethe Himalayas formed was erased by the movement of the Indianplate over various hot spots.
Comparing the coherence anisotropy directions and values of theelastic thickness in the Indian Shield with reference to Figs. 1, 4aand Table 1, we observe that in the southern part of the Indian Shield,in Dharwar Craton (window 10), the coherence anisotropy directionis 45°, and the Te value is 3 km. Coming to the western part ofIndia, in the Deccan Traps (window 8), the coherence anisotropy di-rection is 45° and the Te value is 5 km. In the central part of the shield,that is, in the Narmada–Son lineament (window 5 and 13) which in-cludes a part of the Satpura, the coherence anisotropy directions are45° and 90° respectively, and the Te values are 10 km and 12 km re-spectively. Moving on to the eastern part of the shield, in the Himala-yan Front (window 14), the coherence anisotropy direction is 90° andthe Te value is 19 km. Finally, in the in the Aravalli range (window 2)in the northern part of the shield, the coherence anisotropy directionis 45° and the Te value is 6 km. Thus, it can be observed that mechan-ical anisotropy is found throughout the Indian Shield as nearly all thecoherence anisotropy directions are at an angle of 45° with the excep-tion of Himalayan Front (window 14) and a part of the Narmada–SonLineament (window 13) for which the coherence anisotropy direc-tions are at an angle of 90°.
Some of the shear-wave splitting observations (Kumar and Singh,2008) as shown in Fig. 7 indicate that in Dharwar Craton, most of thestations reveal anisotropic characteristics with delay time ~1 s. Thenorth-east orientations of fast polarization directions at two stationsin Dharwar Craton show N 44°E and N 30°E which is nearly close tothe APM directions (N 43°E and N 44°E). There are some exceptions ina few other stations in Dharwar Craton showing N–S directions. Seismicstations close to the Narmada–Son lineament (NS) show consistent fastpolarization directions N 37°E, N 41°E, and N 47°E. A few other seismicstations to the north-west of Narmada–Son lineament (NS) characterizeanisotropy of N 62°E. The Indo-Gangetic plains and Himalayan foothillsshow polarization directions ranging fromN 36°E to N 46°E. In total, thefast azimuths are aligned in either the N–E or N–S directions.
5. Conclusion
Coherence anisotropy pertaining to the mechanically weak direc-tions is mostly oriented along N 45° E as described above. The averageabsolute plate motion (APM) of the Indian plate in a no-net rotationframe is consistently in the N–E direction (DeMets et al., 1994). Itcan thus be seen that the coherence anisotropy direction is in verygood agreement with the absolute plate motion (APM), which sug-gests that the strain fields accumulating in the Indian Shield overthe years must have been due to the last major tectonic episode (Hi-malayan orogeny). Our study confirms that the lithosphere in the In-dian Shield is mechanically anisotropic. The mechanical properties ofthe shallow lithosphere recorded in the flexural anisotropy arealigned consistently with the APM direction and satisfactorily withseismic anisotropy, suggesting that they all recorded the same stressfield during the last major tectonic episode.
Acknowledgements
We sincerely thank DST (SR/FTP/ES-35/2007) for necessary fund-ing. The paper benefited immensely from the constructive commentsof the anonymous reviewers of the journal. We would also like tothank Tanmay K. Maji, R.K. Reddy and Sharath Shekhar for their tech-nical help and useful discussions. We sincerely thank an anonymousreviewer, who provided several suggestions that improved the man-uscript in its totality.
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