- Open Access
Crustal thickness recovery using an isostatic model and GOCE data
© The Society of Geomagnetism and Earth, Planetary and Space Sciences (SGEPSS); The Seismological Society of Japan; The Volcanological Society of Japan; The Geodetic Society of Japan; The Japanese Society for Planetary Sciences; TERRAPUB. 2012
- Received: 12 September 2011
- Accepted: 20 April 2012
- Published: 26 November 2012
One of the GOCE satellite mission goals is to study the Earth’s interior structure including its crustal thickness. A gravimetric-isostatic Moho model, based on the Vening Meinesz-Moritz (VMM) theory and GOCE gradiomet ric data, is determined beneath Iran’s continental shelf and surrounding seas. The terrestrial gravimetric data of Iran are also used in a nonlinear inversion for a recovering-Moho model applying the VMM model. The newly-computed Moho models are compared with the Moho data taken from CRUST2.0. The root-mean-square (RMS) of differences between the CRUST2.0 Moho model and the recovered model from GOCE and that from the terrestrial gravimetric data are 3.8 km and 4.6 km, respectively.
- Tikhonov regularization
- nonlinear ill-posed problem
The Mohorovičić discontinuity usually called the Moho, is the boundary between the Earth’s crust and mantle. This boundary can be determined by isostatic/gravimetric and seismic methods. Several isostatic hypotheses and seismic models exist for estimating the crustal thickness/Moho. The isostatic models are well-known from the literature (see e.g. Heiskanen and Moritz, 1967, p. 133; Moritz, 1990, chapter 8; Sjöberg, 2009; Bagherbandi, 2011). A comparison between different classical Moho models, and a Moho model determined from seismic data, was presented in Bagherbandi (2011). The advantage of using an isostatic/gravimetric model to determine the crustal thickness is the uniform coverage and relatively-detailed resolution of the currently-available global geopotential models and satellite data, especially over large areas of the world where seismic data are not available or their spatial coverage is not sufficient.
According to our knowledge, there exist few studies, based on satellite data, to determine the crustal thickness. In Shin et al. (2007), a recovery of the Moho depth using a geopotential model obtained from a Gravity Recovery and Climate Experiments (GRACE) satellite mission (Tapley et al., 2005) was presented to determine the Moho depth beneath Tibet. Sampietro (2009) considered the local inversion of Satellite Gravity Gradiometry (SGG) data by simulating a Moho surface and generating the SGG data based on that. Bagherbandi (2011) studied a Moho model obtained from the Vening Meinesz-Moritz (VMM) model (Sjöberg, 2009) and simulated SGG data by EGM08 (Pavlis et al., 2008) in the presence of a white noise of 10 mE (1 mE = 0.0001 mGal/km).
The Gravity field and steady-state Ocean Circulation Explorer (GOCE) (see ESA, 1999) could deliver Earth’s gravity fields to degree and order 250 in its spherical harmonic expression. Here, our purpose is to use real GOCE data directly for determining a regional Moho model, and not their products as the geopotential models. The downward continuation of the GOCE gradiometric data and the regional recovery of the Moho depth is performed simultaneously using the VMM model (Sjöberg, 2009) through a nonlinear integral inversion procedure.
As observed, the unknown parameter s is inside the kernel function of the integral equations. Therefore, Eq. (3a) is categorized in the nonlinear integral equations requiring the approximate values of s. In fact, the Moho undulations will be added to these approximate values by iterating the inversion. This is the reason for the appearance of in the right-hand side of Eq. (3a), if the integral equation was linear, this term would not appear.
Here, we select Iran, restricted between latitudes 19° and 46°N and longitudes 19° and 46°E as our study area. Values of 650 and 430 kg m−3 were taken for the crust-mantle density contrast in land and sea areas, according to Sjöberg and Bagherbandi (2011). Mean Moho depths of 37 km and 23 km were obtained from the spatial averaging of the 2° × 2° CRUST2.0 (Bassin et al., 2000) Moho depths for the corresponding areas, respectively (see, for example, the discussion concerning the seismic method and its accuracy in Nakamura and Umedu, 2009). Our goal is to determine 2° × 2° Moho models with the same resolution as that of CRUST2.0, from 1° × 1° GOCE and terrestrial gravimetric data. Two gravimetric models for Moho beneath this area are computed. One based on Sjöberg’s (2009, equation 53b) direct integral approach using terrestrial data, and the other one based on solving the nonlinear inversion method presented in the previous section. Finally, both models are compared to each other and to that with CRUST2.0 data. Here, a larger area by 5° is considered for the recovery which is required to reduce the effect of the truncation error of the integral formula in the inversion method (Eq. (4)) but the results of the central part are selected. Due to the nonlinear nature of the problem the approximate value outside the central area are not updated for the reduction of this truncation error. According to Bagherbandi (2011), three iterations are required to reach the acceptable convergence level, i.e. 50 m, and removing the bias of regularization, Eq. (5b), has an essential role for solving the problem.
Statistics of Moho depths and their differences, Sjöberg’s direct solution ( ), the Moho recovered from GOCE data (TGOCE) and the seismic Moho (TCRUST2.0). Unit: 1 km.
TGOCE − TCRUST2.0
We have used the Vening Meinesz-Moritz theory and GOCE data, to compute and analyze the crust thickness beneath Iran’s continental shelf and surrounding areas. Here, the Vening Meinesz-Moritz theory has been further developed so that the satellite gravity gradiometry (SGG) data can be used for recovering the Moho depth through a nonlinear integral inversion procedure. The kernels of its forward and inverse problems showed that the inversion should be performed in an area larger by 5° than the desired one to reduce the effect of the spatial truncation error of the integral formula. The results were compared with the CRUST2.0 data. Our numerical study showed that the effect of the truncation error on the recovered Moho depths can attain 6 km in Iran, and this is very significant. The iterative Tikhonov regularization in combination with either the generalized cross-validation, or quasi-optimal, criterion of estimating the regularization parameter seems to be suitable and the solution is semi-convergent up to the third iteration. The Moho depth recovered from GOCE data was the same as that obtained from the terrestrial data with a root-mean-square error of 4.56 km. The results revealed the significant correlation of the Moho geometry with the seismic model, CRUST2.0. The root-mean-square error of the recovered Moho from GOCE with CRUST2.0 is 3.78 km.
The authors would like to thank Professor Lars E. Sjöberg for his guidance and help. The unknown reviewers are cordially thanked for their constructive comments on the manuscript. Mohammad Bagherbandi and Mehdi Eshagh were supported by Projects no. 76/10:1 and 98/09:1 of the Swedish National Space Board (SNSB), respectively. Mr. Mohsen Romeshkani and Mr. Makan Abdolahzadeh are thanked for preparing the SGG data and terrestrial gravity data for this work.
- Abdollahzadeh, M. and M. Najafi Almdari, Detection of Gross-Errors into Gravity Data of IRAN by Kriging Approach, SGM 2009 Abstract Vol Book V, presented in 6th Swiss Geoscience Meeting (2008), Lugano, Switzerland, pp. 297–298, 2008.Google Scholar
- Bagherbandi, M., An isostatic Earth crustal model and its application, Ph.D. Dissertation, Royal Institute of Technology (KTH), Stockholm, Sweden, 207 pp., 2011.Google Scholar
- Bassin, C., G. Laske, and T. G. Mastersm, The current limits of resolution for surface wave tomography in North America, EOS Trans. AGU, 81, F897, 2000.Google Scholar
- Bouman, J., Quality of regularization methods, DECS Report no. 98.2, Delft University Press, 1998.Google Scholar
- ESA, Gravity field and steady-state ocean circulation mission, ESA SP-1233(1), Report for Mission Selection of the Four Candidate Earth Explorer Missions, pp. 217, ESA Publications Division, July 1999, 1999.Google Scholar
- ESA, GOCE Level 2 Product Data Handbook, Doc. No.: GO-MA-HPF-GS-0110. Issue: 4 Date: 09/06/2008, prepared by: The European GOCE Gravity Consortium EGG-C, 2008.Google Scholar
- Eshagh, M., On satellite gravity gradiometry, Doctoral dissertation in Geodesy, Royal Institute of Technology (KTH), Stockholm, Sweden, 2009.Google Scholar
- Eshagh, M., The effect of spatial truncation error on integral inversion of satellite gravity gradiometry data, Adv. Space Res., 47, 1238–1247, 2011.View ArticleGoogle Scholar
- Eshagh, M. and L. E. Sjöberg, Impact of topographic and atmospheric masses over Iran on validation and inversion of GOCE gradiometric data, Earth Space Phys., 34, 15–30, 2008.Google Scholar
- Hansen, P. C., Rank-deficient and discrete ill-posed problems: numerical aspects of linear inversion, SIAM, Philadelphia, 1998.Google Scholar
- Hansen, P. C., Regularization Tools version 4.0 for Matlab 7.3, Numer. Algorithms, 46, 189–194, 2008.View ArticleGoogle Scholar
- Heiskanen, W. and H. Moritz, Physical Geodesy, W.H. Freeman and company, San Francisco and London, 1967.Google Scholar
- Martinec, Z., The minimum depth of compensation of topographic masses, Geophys. J. Int., 117, 545–554, 1994.View ArticleGoogle Scholar
- Moritz, H., The Figure of the Earth, 277 pp., H Wichmann, Karlsruhe, 1990.Google Scholar
- Nakamura, M. and N. Umedu, Crustal thickness beneath the Ryukyu arc from travel-time inversion, Earth Planets Space, 61, 1191–1195, 2009.View ArticleGoogle Scholar
- Pavlis, N. K., J. K. Factor, and S. A. Holmes, R. Forsberg, Proceedings of the 1st International Symposium of the International Gravity Field Service (IGFS), Harita Dergisi, Special Issue No. 18, General Command of Mapping, Ankara, Turkey, 2007Google Scholar
- Pavlis, N., S. A. Holmes, S. C. Kenyon, and J. K. Factor, An Earth Gravitational model to degree 2160: EGM08, presented at the 2008 General Assembly of the European Geosciences Union, (Vienna, Austria), April 13–18, 2008.Google Scholar
- Sampietro, D., An inverse gravimetric problem with GOCE data, Ph.D. thesis, Politecnico di Milano Polo regionale di Como. 2009.Google Scholar
- Shin, Y. H., H. Xu, C. Braitenberg, J. Fang, and Y. Wang, Moho undulations beneath Tibet from GRACE-integrated gravity data, Geophys. J. Int., 170, 971–985, 2007.View ArticleGoogle Scholar
- Sjöberg, L. E., Solving Vening Meinesz-Moritz inverse problem in isostasy, Geophys. J. Int., 179 (3), 1527–1536, 2009.View ArticleGoogle Scholar
- Sjöberg, L. E. and M. Bagherbandi, A method of estimating the Moho density contrast with a tentative application by EGM08 and CRUST2.0, Acta Geophys., 59 (3), 502–525, 2011.View ArticleGoogle Scholar
- Tapley, B., J. Ries, S. Bettadpur, D. Chambers, M. Cheng, F. Condi, B. Gunter, Z. Kang, P. Nagel, R. Pastor, T. Pekker, S. Poole, and F. Wang, GGM02-An improved Earth gravity field model from GRACE, J Geod., 79, 467–478, 2005.View ArticleGoogle Scholar
- Tikhonov, A. N., Solution of incorrectly formulated problems and regularization method, Soviet Math. Dokl., 4, 1035–1038, English translation of Dokl. Akad. Nauk. SSSR, 151, 501–504, 1963.Google Scholar
- ai]Wild, F. and B. Heck, A comparison of different isostatic models applied to satellite gravity gradiometry, Gravity, Geoid and Space Missions GGSM 2004 IAG International Symposium Porto, Portugal August 30–September 3, 2004a.Google Scholar
- Wild, F. and B. Heck, Effects of topographic and isostatic masses in satellite gravity gradiometry, in Proc. Second International GOCE User Workshop GOCE. The Geoid and Oceanography, ESA-ESRIN, Frascati/Italy, March 8–10, 2004 (ESA SP–569, June 2004), CD-ROM, 2004b.Google Scholar
- Xu, P., Truncated SVD methods for discrete linear ill-posed problems, Geophys. J. Int., 135, 505–514, 1998.View ArticleGoogle Scholar
- Yoshida, M., Influence of two major phase transitions on mantle convection with moving and subducting plates, Earth Planets Space, 56, 1019–1033, 2004.View ArticleGoogle Scholar