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  • 1.
    Alizadeh Khameneh, Mohammad Amin
    et al.
    KTH.
    Eshagh, Mehdi
    University West, Department of Engineering Science, Division of Computer, Electrical and Surveying Engineering.
    Sjöberg, Lars E.
    KTH.
    Optimisation of Lilla Edet Landslide GPS Monitoring Network2015In: Journal of Geodetic Science, ISSN 2081-9919, E-ISSN 2081-9943, Vol. 5, no 1, p. 57-66Article in journal (Refereed)
    Abstract [en]

    Since the year 2000, some periodic investigations have been performed in the Lilla Edet region to monitor and possibly determine the landslide of the area with GPS measurements. The responsible consultant has conducted this project by setting up some stable stations for GPS receivers in the risky areas of Lilla Edet and measured the independent baselines amongst the stations according to their observation plan. Here, we optimise the existing surveying network and determine the optimal configuration of the observation plan based on different criteria.We aim to optimise the current network to become sensitive to detect 5 mm possible displacements in each net point. The network quality criteria of precision, reliability and cost are used as object functions to perform single-, bi- and multi-objective optimisation models. It has been shown in the results that the single-objective model of reliability, which is constrained to the precision, provides much higher precision than the defined criterion by preserving almost all of the observations. However, in this study, the multi-objective model can fulfil all the mentioned quality criteria of the network by 17% less measurements than the original observation plan, meaning 17%of saving time, cost and effort in the project.

  • 2.
    Alizadeh-Khameneh, Mohammad Amin
    et al.
    KTH Royal Institute of Technology, Division of Geodesy and Satellite Positioning, Stockholm, Sweden.
    Eshagh, Mehdi
    University West, Department of Engineering Science, Division of Mathematics, Computer and Surveying Engineering.
    Jensen, Anna O.
    KTH Royal Institute of Technology, Division of Geodesy and Satellite Positioning, Stockholm, Sweden.
    Optimization of deformation monitoring networks using finite element strain analysis2018In: Journal of Applied Geodesy, ISSN 1862-9016, E-ISSN 1862-9024, Vol. 2, no 2, p. 187-197Article in journal (Refereed)
    Abstract [en]

    An optimal design of a geodetic network can fulfill the requested precision and reliability of the network, and decrease the expenses of its execution by removing unnecessary observations. The role of an optimal design is highlighted in deformation monitoring network due to the repeatability of these networks. The core design problem is how to define precision and reliability criteria. This paper proposes a solution, where the precision criterion is defined based on the precision of deformation parameters, i. e. precision of strain and differential rotations. A strain analysis can be performed to obtain some information about the possible deformation of a deformable object. In this study, we split an area into a number of three-dimensional finite elements with the help of the Delaunay triangulation and performed the strain analysis on each element. According to the obtained precision of deformation parameters in each element, the precision criterion of displacement detection at each network point is then determined. The developed criterion is implemented to optimize the observations from the Global Positioning System (GPS) in Skåne monitoring network in Sweden. The network was established in 1989 and straddled the Tornquist zone, which is one of the most active faults in southern Sweden. The numerical results show that 17 out of all 21 possible GPS baseline observations are sufficient to detect minimum 3 mm displacement at each network point. © 2018 Walter de Gruyter GmbH, Berlin/Boston.

  • 3.
    Alizadeh-Khameneh, Mohammad Amin
    et al.
    Division of Geodesy and Satellite Positioning, Royal Institute of Technology (KTH), Stockholm, Sweden.
    Eshagh, Mehdi
    University West, Department of Engineering Science, Division of Computer, Electrical and Surveying Engineering.
    Sjöberg, Lars
    Division of Geodesy and Satellite Positioning, Royal Institute of Technology (KTH), Stockholm, Sweden.
    The Effect of Instrumental Precision on Optimisation of Displacement Monitoring Networks2016In: Acta Geodaetica et Geophysica Hungarica, ISSN 1217-8977, E-ISSN 1587-1037, Vol. 51, no 4, p. 761-772Article in journal (Refereed)
    Abstract [en]

    In order to detect the geo-hazards, different deformation monitoring networks are usually established. It is of importance to design an optimal monitoring network to fulfil the requested precision and reliability of the network. Generally, the same observation plan is considered during different time intervals (epochs of observation). Here, we investigate the case that instrumental improvements in sense of precision are used in two successive epochs. As a case study, we perform the optimisation procedure on a GPS monitoring network around the Lilla Edet village in the southwest of Sweden. The network was designed for studying possible displacements caused by landslides. The numerical results show that the optimisation procedure yields an observation plan with significantly fewer baselines in the latter epoch, which leads to saving time and cost in the project. The precision improvement in the second epoch is tested in several steps for the Lilla Edet network. For instance, assuming two times better observation precision in the second epoch decreases the number of baselines from 215 in the first epoch to 143 in the second one.

  • 4.
    Bagherbandi, Mohammad
    et al.
    Royal Institute of Technology (KTH),Division of Geodesy and Geoinformatics .
    Eshagh, Mehdi
    Royal Institute of Technology (KTH),Division of Geodesy and Geoinformatics, .
    Crustal thickness recovery using an isostatic model and GOCE data2012In: Earth Planets and Space, ISSN 1343-8832, E-ISSN 1880-5981, Vol. 64, no 11, p. 1053-1057Article in journal (Refereed)
    Abstract [en]

    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 gradiometric 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.

  • 5.
    Bagherbandi, Mohammad
    et al.
    Royal Institute of Technology (KTH), Division of Geodesy and Geoinformatics.
    Eshagh, Mehdi
    Department of Surveying, Islamic Azad University, Shahr-e-Rey Branch, Tehran, Iran.
    Recovery of Moho's undulations based on the Vening Meinesz-Moritz theory from satellite gravity gradiometry data: A simulation study2012In: Advances in Space Research, ISSN 0273-1177, E-ISSN 1879-1948, Vol. 49, no 6, p. 1097-1111Article in journal (Refereed)
    Abstract [en]

    In the gravimetric approach to determine the Moho depth an isostatic hypothesis can be used. The Vening Meinesz–Moritz isostatic hypothesis is the recent theory for such a purpose. Here, this theory is further developed so that the satellite gravity gradiometry (SGG) data are used for recovering the Moho depth through a nonlinear integral inversion procedure. The kernels of its forward and inverse problems show that the inversion should be done in a larger area by 5° than the desired one to reduce the effect of the spatial truncation error of the integral formula. Our numerical study shows that the effect of this error on the recovered Moho depths can reach 6 km in Persia and it is very significant. The iterative Tikhonov regularization in a combination with either 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. Also the Moho depth recovered from the simulated SGG data will be more or less the same as that obtained from the terrestrial gravimetric data with a root mean square error of 2 km and they are statistically consistent.

  • 6.
    Eshagh, Mehdi
    Royal Institute of Technology (KTH),Division of Geodesy and Geoinformatics, .
    A strategy towards an EGM08-based Fennoscandian geoid model2012In: Journal of Applied Geophysics, ISSN 0926-9851, E-ISSN 1879-1859, Vol. 87, p. 53-59Article in journal (Refereed)
    Abstract [en]

    Today, the recent global Earth's gravity model, EGM08, is successfully utilised for different purposes in geosciences. Here, EGM08 is used to compute a geoid model for Fennoscandia and since it is restricted to degree and order 2160, the higher frequencies of the geoid, or the truncation bias, is recovered directly from terrestrial gravity anomalies using a simple formula. The total topographic and atmospheric effects are computed and added to the derived geoid as well. A very simple EGM08-based non-integral geoid estimator is developed and applied for computing the geoid of Fennoscandia. The outcome of the estimator is compared with the Global Positioning System (GPS)/levelling data of Sweden, Denmark, Finland and Norway. Numerical results show the successful performance of the presented estimator as the geoid become closer to GPS/levelling data than the one computed solely with EGM08. This study will show that considering the truncation bias of EGM08 will reduce the root mean square error (RMSE) of the differences between the geoid and GPS/levelling data by about 1.3 cm and the additive topographic and atmospheric corrections by 1 cm further. It is shown that the correlations among the data have no significant influence on the estimated geoid.

  • 7.
    Eshagh, Mehdi
    University West, Department of Engineering Science, Division of Computer, Electrical and Surveying Engineering.
    A theoretical discussion on Vening Meinesz-Moritz inverse problem of isostasy2016In: Geophysical Journal International, ISSN 0956-540X, E-ISSN 1365-246X, Vol. 207, no 3, p. 1420-1431Article in journal (Refereed)
    Abstract [en]

    The Moho surface can be determined according to isostatic theories and the recent Vening Meinesz-Moritz (VMM) theory of isostasy has been successful for this purpose. In this paper, we will study this method from a theoretical prospective and try to find its connection to the Airy-Heiskanen (AH) and Vening Meinesz original theories. We develop Jeffrey’s inverse solution to isostasy according to the recent developments of the VMM method and compare both methods in similar situations. We will show that they are generalisations of the AH model in a global and continuous domain. In the VMM spherical harmonic solution for Moho depth, the mean Moho depth contributes only to the zero-degree term of the series, whilst in Jeffrey’s solution it contributes to all frequencies. We improve the VMM spherical harmonic series further so that the mean Moho can contribute to all frequencies of the solution. This modification makes the VMM global solution superior to the Jeffrey one, but in a global scale, the difference between both solutions is less than 3 km. Both solutions are asymptotically-convergent and we present two methods to obtain smooth solutions for Moho from them. 

  • 8.
    Eshagh, Mehdi
    University West, Department of Engineering Science.
    A theoretical study on terrestrial gravimetric data refinement by earth gravity models2014In: Geophysical Prospecting, ISSN 0016-8025, E-ISSN 1365-2478, Vol. 62, no 1, p. 158-171Article in journal (Refereed)
    Abstract [en]

    The idea of this paper is to present estimators for combining terrestrial gravity data with Earth gravity models (EGMs) and produce a high-quality source of the Earth's gravity field data through all wavelengths. To do so, integral and point-wise estimators are mathematically developed, based on the spectral combination theory, in such a way that they combine terrestrial data with one and/or two Earth gravity models. The integral estimators are developed so that they become biased or unbiased to a priori information. For testing the quality of the estimators, their global mean square errors (MSEs) are generated using an Earth gravity model08 model and one of the recent products of the gravity field and steady-state ocean circulation explorer (GOCE) mission. Numerical results show that the integral estimators have smaller global root mean square errors (RMSEs) than the point-wise ones but they are not efficient practically. The integral estimator of the biased type is the most suited due to its smallest global root mean square error comparing to the rest of the estimators. Due largely to the omission errors of Earth gravity models the point-wise estimators are not sensitive to the Earth gravity model commission error; therefore, the use of high-degree Earth gravity models is very influential for reduction of their root mean square errors. Also it is shown that the use of the ocean circulation explorer Earth gravity model does not significantly reduce the root mean square errors of the presented estimators in the presence of Earth gravity model08. All estimators are applied in the region of Fennoscandia and a cap size of 2° for numerical integration and a maximum degree of 2500 for generation of band-limited kernels are found suitable for the integral estimators.

  • 9.
    Eshagh, Mehdi
    University West, Department of Engineering Science, Division of Natural Sciences and Electrical and Surveying Engineering.
    An integral approach to regional gravity field refinement using Earth gravity models2013In: Journal of Geodynamics, ISSN 0264-3707, E-ISSN 1879-1670, Vol. 68, p. 18-28Article in journal (Refereed)
    Abstract [en]

    The idea of this paper is to refine the terrestrial gravimetric data with the Earth's gravity models (EGMs) and produce a high quality source of gravity data. For this purpose, biased and unbiased integral estimators are presented. These estimators are used to refine gravimetric data over Fennoscandia with the ITG-GRACE2010s and GO_CONS_GCF_2_DIR_R2 EGMs, which are the recent products of the gravity field and climate experiment (GRACE) and the gravity field and steady-state ocean circulation explorer (GOCE) satellite missions. Numerical results show that the biased integral estimator has smaller global root mean square error (RMSE) than the unbiased one. Also a simple strategy is presented to down-weight the low-frequencies the terrestrial data in spectral combination. The gravity anomalies, computed by EGM08, are compared to the refined anomalies for evaluation purpose. In the case of using a cap size of 1° for integration the EGM08 gravity anomalies are more correlated with the refined ones. Also the band-limited kernels can simply be generated to maximum degree of the used EGMs for both estimators. Comparisons of the combined anomalies and those of EGM08 show insignificant differences between the biased and unbiased estimators in practice. However, the biased estimator seems to be proper one for gravity data refinement due to its smaller global RMSE.

  • 10.
    Eshagh, Mehdi
    University West, Department of Engineering Science, Division of Natural Sciences and Electrical and Surveying Engineering.
    Determination of Moho Discontinuity from Satellite Gradiometry Data: Linear Approach2014In: Geodynamics Reseach International Bulletin, ISSN 2345-4997, Vol. 1, no 2, p. 1-13Article in journal (Other academic)
    Abstract [en]

    The satellite gradiometry data (SGD) can be used for studying the crustal structure in addition to the Earth’s gravity field. This paper will show how this type of data is related to the Moho discontinuity or the boundary between the Earth’s crust and mantle. Here, the Vening Meinesz-Moritz (VMM) theory of isostasy is used and its mathematical formulae are modified to use the SGD instead of the Earth gravity models. A linear integral equation with a well-behaving kernel is presented by approximating the Moho depth formula derived based on the VMM theory. The error of this approximation is less than 300 m in Iran as the study area. Furthermore, this paper shows that the contribution of the higher degree harmonics than 215 is less than 1% with respect to the total signal of Moho undulations. This means that the use of SGD is meaningful as they sense the harmonics of the Earth’s gravity field to this degree. Two methods of one-step and two-step are proposed for Moho determination and applied in Iran.  It is shown that to reduce the effect of spatial truncation error of the integral formulae of both methods the central area should be smaller by 6 than the inversion area. Numerical studies show that the two-step approach is superior to the other one and the root mean squared error of differences between the Moho model recovered by an Earth gravity model and SGD is about 1.5 km in Iran.

  • 11.
    Eshagh, Mehdi
    University West, Department of Engineering Science, Division of Mathematics, Computer and Surveying Engineering.
    Elastic thickness determination based on Vening Meinesz-Moritz and flexural theories of isostasy2018In: Geophysical Journal International, ISSN 0956-540X, E-ISSN 1365-246X, Vol. 213, no 3, p. 1682-1692Article in journal (Refereed)
    Abstract [en]

    Elastic thickness (Te) is one of mechanical properties of the Earth's lithosphere. The lithosphere is assumed to be a thin elastic shell, which is bended under the topographic, bathymetric and sediment loads on. The flexure of this elastic shell depends on its thickness or Te. Those shells having larger Te flex less. In this paper, a forward computational method is presented based on the Vening Meinesz–Moritz (VMM) and flexural theories of isostasy. Two Moho flexure models are determined using these theories, considering effects of surface and subsurface loads. Different values are selected for Te in the flexural method to see by which one, the closest Moho flexure to that of the VMM is achieved. The effects of topographic/bathymetric, sediments and crustal crystalline masses, and laterally variable upper mantle density, Young's modulus and Poisson's ratio are considered in whole computational process. Our mathematical derivations are based on spherical harmonics, which can be used to estimate Te at any single point, meaning that there is no edge effect in the method. However, the Te map needs to be filtered to remove noise at some points. A median filter with a window size of 5° × 5° and overlap of 4° works well for this purpose. The method is applied to estimate Te over South America using the data of CRUST1.0 and a global gravity model.

  • 12.
    Eshagh, Mehdi
    University West, Department of Engineering Science, Division of Natural Sciences and Electrical and Surveying Engineering.
    From satellite gradiometry data to the sub-crustal stress due to the mantle convection2014In: Pure and Applied Geophysics, ISSN 0033-4553, E-ISSN 1420-9136, Vol. 171, no 9, p. 2391-2406Article in journal (Refereed)
    Abstract [en]

    Subcrustal stress induced by mantle convection can be determined by the Earth's gravitational potential. In this study, the spherical harmonic expansion of the simplified Navier–Stokes equation is developed further so satellite gradiometry data (SGD) can be used to determine the subcrustal stress. To do so, we present two methods for producing the stress components or an equivalent function thereof, the so-called S function, from which the stress components can be computed numerically. First, some integral estimators are presented to integrate the SGD and deliver the stress components and/or the S function. Second, integral equations are constructed for inversion of the SGD to the aforementioned quantities. The kernel functions of the integrals of both approaches are plotted and interpreted. The behaviour of the integral kernels is dependent on the signal and noise spectra in the first approach whilst it does not depend on extra information in the second method. It is shown that recovering the stress from the vertical–vertical gradients, using the integral estimators presented, is suitable, but when using the integral equations the vertical–vertical gradients are recommended for recovering the S function and the vertical–horizontal gradients for the stress components. This study is theoretical and numerical results using synthetic or real data are not given.

  • 13.
    Eshagh, Mehdi
    University West, Department of Engineering Science, Division of Natural Sciences and Electrical and Surveying Engineering.
    From tensor to vector of gravitation2014In: Artificial Satellites, ISSN 2083-6104, Vol. 49, no 2, p. 63-80Article in journal (Refereed)
    Abstract [en]

    Different gravitational force models are used for determining the satellites’ orbits. The satellite gravity gradiometry (SGG) data contain this gravitational information and the satellite accelerations can be determined from them. In this study, we present that amongst the elements of the gravitational tensor in the local north-oriented frame, all of the elements are suitable for this purpose except Txy. Three integral formulae with the same kernel function are presented for recovering the accelerations from the SGG data. The kernel of these integrals is well-behaving which means that the contribution of the far-zone data is not very significant to their integration results; but this contribution is also dependent on the type of the data being integrated. Our numerical studies show that the standard deviations of the differences between the accelerations recovered from Tzz, Txz and Tyz  and those computed by an existing Earth´s gravity model reduce by increasing the cap size of integration. However, their root mean squared errors increase for recovering Tyfrom Tyz.  Larger cap sizes than 5   is recommended for recovering Tx and Tz  but smaller ones for Ty.

  • 14.
    Eshagh, Mehdi
    University West, Department of Engineering Science, Division of Natural Sciences and Electrical and Surveying Engineering.
    Integral Approaches to Determine Sub-Crustal Stress from Terrestrial Gravimetric Data2016In: Pure and Applied Geophysics, ISSN 0033-4553, E-ISSN 1420-9136, Vol. 173, no 3, p. 805-825Article in journal (Refereed)
    Abstract [en]

    The spherical harmonic expressions of the horizontal sub-crustal stress components induced by the mantle convection are convergent only to low degrees. In this paper, we use the method of stress (S) function with numerical differentiation and present a formula for determining the degree of convergence from the mean Moho depth. We found that for the global mean Moho depth, 23 km, this convergence degree is 622 and for Iran, 35 km, it is 372. Also, three methods are developed and applied for computing the sub-crustal stress, (1) direct integration with a spectral kernel limited up to the degree of convergence, (2) integral inversion with a kernel having closed-form formula without any frequency limit, and (3) solving an integral equation with limited spectral kernel to the convergence degree. The second method has no divergence problem and its kernel function is well behaving so that the system of equations from which the S function is determined is stable, and no regularisation is needed to solve it. It should be noted that for using this method the resolution of the recovery should be higher than 0.5° × 0.5°, otherwise the recovered S function and correspondingly the stress components will have smaller magnitude than those derived from the other two methods. Our numerical studies for stress recovery in Iran and its surrounding areas show that the methods, which use the limited spectral kernels to the convergence degree, deliver consistent results to that of the spherical harmonic expansion.

  • 15.
    Eshagh, Mehdi
    University West, Department of Engineering Science, Division of Natural Sciences and Electrical and Surveying Engineering.
    Integral developments of Vening Meinesz-Moritz formula for local determination of Moho discontinuity and their applications in Iran2014In: Geodynamics Research International Bulletin, ISSN 2345-4997, Vol. 2, no 3, p. I-IXArticle in journal (Refereed)
    Abstract [en]

    Global models of the Earth gravity field and topographic/bathymetric data can be used for the gravimetric determination of the Moho discontinuity based on the Vening Meinesz-Moritz theory. In this paper, we mathematically develop this method in such a way that the local data can be used for Moho modelling. Two integral formulae are presented, one for integrating the data and one for their inversion. The kernels of both integrals are well-behaving meaning that the contribution of far-zone quantities being integrated are not very significant in the results. Both of these methods are applied for computing the Moho model of Iran and their results are compared to the Moho model determined based on the global models. Consistency of the computed Moho models from the simulated data and the global models verifies the correctness of both approaches. The presented methods are consistent even for the case of using real data. Numerical results show that the minimum value of the Moho models derived by the simulated data and global models are about 31 km, whilst those derived from the real data are about 3 km smaller. Similarly, the mean value of Moho depths derived from real data is about 1 km smaller than that from the global models.

  • 16.
    Eshagh, Mehdi
    University West, Department of Engineering Science, Division of Computer, Electrical and Surveying Engineering.
    Local recovery of lithospheric stress tensor from GOCE gravitational tensor2017In: Geophysical Journal International, ISSN 0956-540X, E-ISSN 1365-246X, Vol. 209, no 1, p. 317-333Article in journal (Refereed)
    Abstract [en]

    The sub-lithospheric stress due to mantle convection can be computed from gravity data and propagated through the lithosphere by solving the boundary-value problem of elasticity for the Earth's lithosphere. In this case, a full tensor of stress can be computed at any point inside this elastic layer. Here, we present mathematical foundations for recovering such a tensor from gravitational tensor measured at satellite altitudes. The mathematical relations will be much simpler in this way than the case of using gravity data as no derivative of spherical harmonics or Legendre polynomials is involved in the expressions. Here, new relations between the spherical harmonic coefficients of the stress and gravitational tensor elements are presented. Thereafter integral equations are established from them to recover the elements of stress tensor from those of the gravitational tensor. The integrals have no closed-form kernels, but they are easy to invert and their spatial truncation errors are reducible. The integral equations are used to invert the real data of the gravity field and steady-state ocean circulation explorer (GOCE) mission, in November 2009, over the South American plate and its surroundings to recover the stress tensor at a depth of 35 km. The recovered stress fields are in good agreement with the tectonic and geological features of the area.

  • 17.
    Eshagh, Mehdi
    University West, Department of Engineering Science, Division of Computer, Electrical and Surveying Engineering.
    Non-singular expressions for vector and gradient tensor of gravitation in a geocentric spherical frame2008In: Computers & Geosciences, ISSN 0098-3004, E-ISSN 1873-7803, Vol. 34, no 12, p. 1762-1768Article in journal (Refereed)
    Abstract [en]

    The traditional expressions of the gravitational vector (GV) and the gravitational gradient tensor (GGT) have complicated forms depending on the first- and the second-order derivatives of associated Legendre functions (ALF), and also singular terms when approaching the poles. This article presents alternative expressions for the GV and GGT, which are independent of the derivatives, and are also non-singular. By using such expressions, it suffices to compute the ALF to two additional degrees and orders, instead of computing the first and the second derivatives of all the ALF. Therefore, the formulas are suitable for computer programming. Matlab software as well as an output of a numerical computation around the North Pole is also presented based on the derived formulas.

  • 18.
    Eshagh, Mehdi
    University West, Department of Engineering Science, Division of Mathematics, Computer and Surveying Engineering.
    On the approximations in formulation of the Vening Meinesz-Moritz theory of isostasy2017In: Geophysical Journal International, ISSN 0956-540X, E-ISSN 1365-246X, Vol. 210, no 1, p. 500-508Article in journal (Refereed)
    Abstract [en]

    Different approximations are used in Moho modelling based on isostatic theories. The well-known approximation is considering a plate shell model for isostatic equilibrium, which is an oversimplified assumption for the Earth’s crust. Considering a spherical shellmodel, as used in the Vening Meinesz-Moritz (VMM) theory, is a more realistic assumption, but it suffers from different types of mathematical approximations. In this paper, the idea is to investigate such approximations and present their magnitudes and locations all over the globe. Furthermore, we show that the mathematical model of Moho depth according to the VMM principle can be simplified to that of the plate shell model after four approximations. Linearisation of the binomial term involving the topographic/bathymetric heights is sufficient as long as their spherical harmonic expansion is limited to degree and order 180. The impact of the higher order terms is less than 2 km. The Taylor expansion of the binomial term involving the Moho depth (T) up to second order with the assumption of T-2 = TT0, T-0 is the mean compensation depth, improves this approximation further by up to 4 km over continents. This approximation has a significant role in Moho modelling over continents; otherwise, loss of frequency occurs in the Moho solution. On the other hand, the linear approximation performs better over oceans and considering higher order terms creates unrealistic frequencies reaching to a magnitude of 5 km in the Moho solution. Involving gravity data according to the VMM principle influences the Moho depth significantly up to 15 km in some areas.

  • 19.
    Eshagh, Mehdi
    University West, Department of Engineering Science, Division of Natural Sciences and Electrical and Surveying Engineering.
    On the relation between Moho and sub-crustal stress induced by mantle convection2015In: Journal of Geophysics and Engineering, ISSN 1742-2132, E-ISSN 1742-2140, Vol. 12, no 1, p. 1-11Article in journal (Refereed)
    Abstract [en]

    The sub-crustal stress components due to mantle convection have a direct relation with the spherical harmonic coefficients of the Earth's disturbing potential like those of the Moho model, developed by the Vening–Meinesz–Moritz theory. In this paper, the relation between the stress components and the global and local models of Moho is mathematically developed in three different ways. Here, we present the S function (S) with a numerical differentiation approach to generate the stress components and we show that its spherical harmonic series is convergent to a degree of about 600 based on a mean global Moho depth of 23 km. An integral approach is developed for integration of a local Moho model for the stress recovery, but the kernels of this integral are not likely to be convergent and should be generated by their spectral forms to a limited degree. Another method is developed based on integral inversion, which is free of any mathematical problem and suitable for recovering S from an existing model of Moho. Our numerical presentation shows that the stress has a good agreement with the tectonic boundaries and the places at which the curvature of the Moho surface changes.

  • 20.
    Eshagh, Mehdi
    University West, Department of Engineering Science, Division of Natural Sciences and Electrical and Surveying Engineering.
    On the reliability and error calibration of some recent Earth's gravity models of GOCE with respect to EGM082013In: Acta Geodaetica et Geophysica Hungarica, ISSN 1217-8977, E-ISSN 1587-1037, Vol. 48, no 2, p. 199-208Article in journal (Refereed)
    Abstract [en]

    The Gravity field and steady-state Ocean Circulation Explorer (GOCE) mission is dedicated to recover spherical harmonic coefficients of the Earth's gravity field to degree and order of about 250 using its satellite gradiometric data. Since these data are contaminated with coloured noise, therefore, their inversion will not be straightforward. Unsuccessful modelling of this noise will lead to biases in the harmonic coefficients presented in the Earth's gravity models (EGMs). In this study, five of the recent EGMs of GOCE such as two direct, two time-wise and one space-wise solution are used to degree and order 240 and their reliability is investigated with respect to EGM08 which is assumed as a reliable EGM. The detected unreliable coefficients and their errors are replaced by the corresponding ones from EGM08 as a combination strategy. A condition adjustment model is organised for each two corresponding coefficients of GOCE EGMs and EGM08; and errors of the GOCE EGMs are calibrated based on a scaling factor, obtained from a posteriori variance factor. When the factor is less than 2.5 it will be multiplied to the error otherwise the error of EGM08 coefficient will be considered as the calibrated one. At the end, a simple geoid estimator is presented which considers the EGMs and their errors and its outcomes are compared with the corresponding geoid heights derived from the Global Positioning System (GPS) and the levelling data (GPS/levelling data), over Fennoscandia. This comparison shows that some of the combined-calibrated GOCE EGMs are closer to the GPS/levelling data than the original ones.

  • 21.
    Eshagh, Mehdi
    University West, Department of Engineering Science, Division of Computer, Electrical and Surveying Engineering.
    On Vening Meinesz-Moritz and flexural theories of isostasy and their comparison over Tibet Plateau2016In: Journal of Geodetic Science, ISSN 2081-9919, E-ISSN 2081-9943, Vol. 6, p. 139-151Article in journal (Refereed)
    Abstract [en]

    Gravity and topographic/bathymetric data are used for gravimetric modelling of Moho discontinuity by hydrostatic or flexural theories of the isostasy. Here, two hydrostatic models, based on the Vening Meinesz-Moritz (VMM) principle, and two based on the loading theories and flexural isostasy are compared over Tibet Plateau. It is shown that the Moho models generated based on the VMM theory and flexural isostasy have very good agreements if the mean compensation depth and the mean elastic thickness are selected properly. However, the model computed based on the flexural isostasy is smoother. A more rigorous flexural model, which considers the membrane stress and curvature of the lithosphere, is used to model the Moho surface over the study area. It is shown that the difference between the Moho models, derived by considering and ignoring these parameters, is not significant. By combination of the flexural and VMM hydrostatic models new mathematical formulae for crustal gravity anomalies are provided and it is shown that the crustal gravity anomalies produced by them are also equivalent.

  • 22.
    Eshagh, Mehdi
    University West, Department of Engineering Science, Division of Natural Sciences and Electrical and Surveying Engineering. Department of Geodesy, K.N.Toosi University of Technology, Tehran.
    Spectral combination of spherical gradiometric boundary-value problems: a theoretical study2012In: Pure and Applied Geophysics, ISSN 0033-4553, E-ISSN 1420-9136, Vol. 169, p. 2201-2215Article in journal (Refereed)
    Abstract [en]

    The Earth’s gravity potential can be determined from its second-order partial derivatives using the spherical gradiometric boundary-value problems which have three integral solutions. The problem of merging these solutions by spectral combination is the main subject of this paper. Integral estimators of biased- and unbiased-types are presented for recovering the disturbing gravity potential from gravity gradients. It is shown that only kernels of the biased-type integral estimators are suitable for simultaneous downward continuation and combination of gravity gradients. Numerical results show insignificant practical difference between the biased and unbiased estimators at sea level and the contribution of far-zone gravity gradients remains significant for integration. These contributions depend on the noise level of the gravity gradients at higher levels than sea. In the cases of combining the gravity gradients, contaminated with Gaussian noise, at sea and 250 km levels the errors of the estimated geoid heights are about 10 and 3 times smaller than those obtained by each integral

  • 23.
    Eshagh, Mehdi
    University West, Department of Engineering Science, Division of Land Surveying and Mathematics.
    The effect of constraints on bi-objective optimization of geodetic networks2014In: Acta Geodaetica et Geophysica Hungarica, ISSN 1217-8977, E-ISSN 1587-1037Article in journal (Refereed)
    Abstract [en]

    One of the problems in the single-objective optimisation models (SOOMs) for optimising geodetic networks is the contradiction of the controlling constraints, which may lead to their violation or infeasibility in the optimisation process. One way to solve this problem is to use a bi-objective optimisation model (BOOM) instead of SOOMs. In this paper, we will use the BOOM of precision and reliability and investigate the influence of the controlling constraints in a two-dimensional simulated network. Our studies show that the unconstrained BOOM is a good model, which almost fulfils our precision and reliability demands of the network. This model is also economical as more observables are removed from the plan whilst adding the controlling constraints leads to including more observables, which have no significant role.

  • 24.
    Eshagh, Mehdi
    et al.
    University West, Department of Engineering Science, Division of Computer, Electrical and Surveying Engineering.
    Alizadeh Khameneh, Mohammad AMin
    KTH.
    Two-epoch doptimal design of displacement monitoring networks2015In: Boletim de Ciências Geodésicas, ISSN 1413-4853, E-ISSN 1982-2170, Vol. 21, no 3, p. 484-497Article in journal (Refereed)
    Abstract [en]

    In the traditional method of optimal design of displacement monitoring networks a higher precision, times better than the desired accuracy of displacements, is considered for the net points in such a way that the accuracy of the detected displacements meets the desired one. However, in this paper, we develop an alternative method by considering the total number of observations in two epochs without such a simple assumption and we call it two-epoch optimisation. This method is developed based on the Gauss-Helmert adjustment model and the variances of the observations are estimated instead of the weights to optimise the observation plan. This method can deliver the same results as the traditional one, but with less required observations in each epoch

  • 25.
    Eshagh, Mehdi
    et al.
    University West, Department of Engineering Science, Division of Natural Sciences and Electrical and Surveying Engineering.
    Alizadeh-Khameneh, Mohammad. Amin
    Royal Institute of Technology (KTH), Division of Geodesy and Geoinformatics.
    The effect of constraints on bi-objective optimisation of geodetic networks2015In: Acta Geodaetica et Geophysica Hungarica, ISSN 1217-8977, E-ISSN 1587-1037, Vol. 50, no 4, p. 449-459Article in journal (Refereed)
    Abstract [en]

    One of the problems in the single-objective optimisation models (SOOMs) for optimising geodetic networks is the contradiction of the controlling constraints, which may lead to their violation or infeasibility in the optimisation process. One way to solve this problem is to use a bi-objective optimisation model (BOOM) instead of SOOMs. In this paper, we will use the BOOM of precision and reliability and investigate the influence of the controlling constraints in a two-dimensional simulated network. Our studies show that the unconstrained BOOM is a good model, which almost fulfils our precision and reliability demands of the network. This model is also economical as more observables are removed from the plan whilst adding the controlling constraints leads to including more observables, which have no significant role

  • 26.
    Eshagh, Mehdi
    et al.
    University West, Department of Engineering Science, Divison of Natural Sciences, Surveying and Mechanical Engineering.
    Bagherbandi, Mohammad
    University of Gävle, Department of Industrial Development.
    Combined Moho Estimators2014In: Geodynamics Research International Bulletin, ISSN 2345-4997, Vol. 1, no 3, p. 1-11Article in journal (Other academic)
    Abstract [en]

    In this study, we develop three estimators to optimally combine seismic and gravimetric models of Moho surface. The first estimator combines them by their special harmonic coefficients; the second one uses the spherical harmonic coefficients of the seismic model and use integral formula for the gravimetric one. The kernel of the integral terms of this estimator shows that a cap size of 20 is required for the integration, but since this integral is presented to combine the low frequencies of the gravimetric model, a low resolution model is enough for the integration. The third estimator uses the gravity anomaly and converts its low frequencies to those of the gravimetric Moho model, meanwhile combining them with those of seismic one. This integral requires an integration domain of 30 for the gravity anomalies but since the maximum degree of this kernel is limited to a specific degree, the use of its spectral form is recommended. The kernel of the integral involving the gravity anomalies, developed for recovering high frequencies of Moho, is written in a closed-from formula and its singularity is investigated. This kernel is well-behaving and decreases fast, meaning that it is suitable for recovering the high frequencies of Moho surface.

  • 27.
    Eshagh, Mehdi
    et al.
    University West, Department of Engineering Science, Division of Natural Sciences and Electrical and Surveying Engineering. K N Toosi University of Technology, Department of Geodesy, Tehran, Iran.
    Bagherbandi, Mohammad
    University of Gävle, Department of Industrial Development, IT and Land Management .
    Quality description for gravimetric and seismic Moho models of Fennoscandia through a combined adjustment2012In: Acta Geodaetica et Geophysica Hungarica, ISSN 1217-8977, E-ISSN 1587-1037, Vol. 47, no 4, p. 388-401Article in journal (Refereed)
    Abstract [en]

    The gravimetric model of the Moho discontinuity is usually derived based on isostatic adjustment theories considering floating crust on the viscous mantle. In computation of such a model some a priori information about the density contrast between the crust and mantle and the mean Moho depth are required. Due to our poor knowledge about them they are assumed unrealistically constant. In this paper, our idea is to improve a computed gravimetric Moho model, by the Vening Meinesz-Moritz theory, using the seismic model in Fennoscandia and estimate the error of each model through a combined adjustment with variance component estimation process. Corrective surfaces of bi-linear, bi-quadratic, bi-cubic and multi-quadric radial based function are used to model the discrepancies between the models and estimating the errors of the models. Numerical studies show that in the case of using the bi-linear surface negative variance components were come out, the bi-quadratic can model the difference better and delivers errors of 2.7 km and 1.5 km for the gravimetric and seismic models, respectively. These errors are 2.1 km and 1.6 km in the case of using the bi-cubic surface and 1 km and 1.5 km when the multi-quadric radial base function is used. The combined gravimetric models will be computed based on the estimated errors and each corrective surface.

  • 28.
    Eshagh, Mehdi
    et al.
    University West, Department of Engineering Science, Division of Natural Sciences and Electrical and Surveying Engineering.
    Ebadi, Sahar
    Department of Geodesy, K.N.Toosi University of Technology.
    Geoid modelling based on EGM08 and the recent Earth gravity models of GOCE2013In: Earth Science Informatics, ISSN 1865-0481, Vol. 6, no 3, p. 113-125Article in journal (Refereed)
  • 29.
    Eshagh, Mehdi
    et al.
    University West, Department of Engineering Science, Division of Computer, Electrical and Surveying Engineering.
    Ebadi, Sahar
    University of Tehran, Department of Surveying Engineering, Iran.
    Tenzer, Robert
    University of West Bohemia, New Technologies for the Information Society (NTIS), Czech Republic.
    Isostatic GOCE Moho model for Iran2017In: Journal of Asian Earth Sciences, ISSN 1367-9120, E-ISSN 1878-5786, Vol. 138, p. 12-24Article in journal (Refereed)
    Abstract [en]

    One of the major issues associated with a regional Moho recovery from the gravity or gravity-gradient data is the optimal choice of the mean compensation depth (i.e., the mean Moho depth) for a certain area of study, typically for orogens characterised by large Moho depth variations. In case of selecting a small value of the mean compensation depth, the pattern of deep Moho structure might not be reproduced realistically. Moreover, the definition of the mean compensation depth in existing isostatic models affects only low-degrees of the Moho spectrum. To overcome this problem, in this study we reformulate the Sjöberg and Jeffrey’s methods of solving the Vening-Meinesz isostatic problem so that the mean compensation depth contributes to the whole Moho spectrum. Both solutions are then defined for the vertical gravity gradient, allowing estimating the Moho depth from the GOCE satellite gravity-gradiometry data. Moreover, gravimetric solutions provide realistic results only when a priori information on the crust and upper mantle structure is known (usually from seismic surveys) with a relatively good accuracy. To investigate this aspect, we formulate our gravimetric solutions for a variable Moho density contrast to account for variable density of the uppermost mantle below the Moho interface, while taking into consideration also density variations within the sediments and consolidated crust down to the Moho interface. The developed theoretical models are applied to estimate the Moho depth from GOCE data at the regional study area of the Iranian tectonic block, including also parts of surrounding tectonic features. Our results indicate that the regional Moho depth differences between Sjöberg and Jeffrey’s solutions, reaching up to about 3 km, are caused by a smoothing effect of Sjöberg’s method. The validation of our results further shows a relatively good agreement with regional seismic studies over most of the continental crust, but large discrepancies are detected under the Oman Sea and the Makran subduction zone. We explain these discrepancies by a low quality of seismic data offshore.

  • 30.
    Eshagh, Mehdi
    et al.
    University West, Department of Engineering Science, Division of Natural Sciences and Electrical and Surveying Engineering.
    Ghorbannia, Morteza
    K.N. Toosi University of Technology, Department of Geodesy, Tehran.
    The effect of spatial truncation error on the variance of gravity anomalies derived from inversion of satellite orbital and gradiometric data2014In: Advances in Space Research, ISSN 0273-1177, E-ISSN 1879-1948, Vol. 54, no 2, p. 261-271Article in journal (Refereed)
    Abstract [en]

    The spatial truncation error (STE) is a significant systematic error in the integral inversion of satellite gradiometric and orbital data to gravity anomalies at sea level. In order to reduce the effect of STE, a larger area than the desired one is considered in the inversion process, but the anomalies located in its central part are selected as the final results. The STE influences the variance of the results as well because the residual vector, which is contaminated with STE, is used for its estimation. The situation is even more complicated in variance component estimation because of its iterative nature. In this paper, we present a strategy to reduce the effect of STE on the a posteriori variance factor and the variance components for inversion of satellite orbital and gradiometric data to gravity anomalies at sea level. The idea is to define two windowing matrices for reducing this error from the estimated residuals and anomalies. Our simulation studies over Fennoscandia show that the differences between the 0.5°×0.5°0.5°×0.5° gravity anomalies obtained from orbital data and an existing gravity model have standard deviation (STD) and root mean squared error (RMSE) of 10.9 and 12.1 mGal, respectively, and those obtained from gradiometric data have 7.9 and 10.1 in the same units. In the case that they are combined using windowed variance components the STD and RMSE become 6.1 and 8.4 mGal. Also, the mean value of the estimated RMSE after using the windowed variances is in agreement with the RMSE of the differences between the estimated anomalies and those obtained from the gravity model.

  • 31.
    Eshagh, Mehdi
    et al.
    University West, Department of Engineering Science, Division of Natural Sciences and Electrical and Surveying Engineering.
    Ghorbannia, Morteza
    Department of Geodesy, K.N.Toosi University of Technology.
    The use of Gaussian equations of motions of a satellite for local gravity anomaly recovery2013In: Advances in Space Research, ISSN 0273-1177, E-ISSN 1879-1948, Vol. 52, no 1, p. 30-38Article in journal (Refereed)
    Abstract [en]

    The orbital elements of a low Earth orbiting satellite and their velocities can be used for local determination of gravity anomaly. The important issue is to find direct relations among the anomalies and these parameters. Here, a primary theoretical study is presented for this purpose. The Gaussian equations of motion of a satellite are used to develop integral formulas for recovering the gravity anomalies. The behaviour of kernels of the integrals are investigated for a two-month simulated orbit similar to that of the Gravity field and steady-state ocean circulation explorer (GOCE) mission over Fennoscandia. Numerical investigations show that the integral formulas have neither isotropic nor well-behaved kernels. In such a case, gravity anomaly recovery is not successful due to large spatial truncation error of the integral formulas. Reformulation of the problem by combining the orbital elements and their velocities leads to an integral with a well-behaved kernel which is suitable for our purpose. Also based on these combinations some general relations among the orbital elements and their velocities are obtained which can be used for validation of orbital parameters and their velocities

  • 32.
    Eshagh, Mehdi
    et al.
    University West, Department of Engineering Science, Division of Computer, Electrical and Surveying Engineering.
    Hussain, Matloob
    University West, Department of Engineering Science. Earth Sciences department, Quaid-i-Azam University, Islamabad, Pakistan.
    Relationship amongst gravity gradients, deflection of vertical, Moho deflection and the stresses derived by mantle convections: a case study over Indo-Pak and surroundings2015In: Geodynamics research international bulletin, ISSN 2345-4997, Vol. 3, no 4, p. I-XIII, article id 12Article in journal (Refereed)
    Abstract [en]

    All  quantities,  which are  measured  in  the  gravity  field  of  the  Earth are affected by  the  field,  therefore,  there should  be correlations  amongst  them.  Here,  we  focus  on  some  gravimetrically-determined  quantities  like  deflections  of  vertical, deflections of Moho, vertical-horizontal gravity gradients and the shear sub-lithospheric stress components due to mantle convection. We show that how these quantities are related to each other mathematically so that one of them can be written in term of another. This somehow proves the presence of the mentioned correlations theoretically. Also, we generate the maps of these quantities over the Indo-Pak and surrounding areas and show how similar they are. Thereafter, they are explained and  interpreted  geologically.  Our  investigations  show  that  the  maps  of  these  quantities  are  in  good  agreements  with topographic and geological features. The map of the vertical-horizontal gravity gradients shows more detailed information of the  gravity field due to  signal amplification at high  degrees,  that of Moho deflection shows sub-surface features due to reduction of the effect of topographic masses. The map of the shear sub-lithospheric stress components is much smoother than the gradients, as expected, and has good agreement with the collisional and subduction zones as well. 

    Relationship amongst gravity gradients, deflection of vertical, Moho deflection and the stresses derived by mantle convections-a case study over Indo-Pak and surroundings. Available from: https://www.researchgate.net/publication/292538682_Relationship_amongst_gravity_gradients_deflection_of_vertical_Moho_deflection_and_the_stresses_derived_by_mantle_convections-a_case_study_over_Indo-Pak_and_surroundings [accessed Feb 1, 2016].

  • 33.
    Eshagh, Mehdi
    et al.
    University West, Department of Engineering Science, Division of Computer, Electrical and Surveying Engineering.
    Hussain, Matloob
    University West, Department of Engineering Science, Division of Computer, Electrical and Surveying Engineering.
    Tenzer, Robert
    Wuhan University, The Key Laboratory of Geospace Environment and Geodesy, Wuhan 430079, China.
    Romeshkani, Mohsen
    University of Tehran, School of Surveying and Geospatial Engineering, College of Engineering, Tehran 14395-515, Iran.
    Moho density contrast in central Eurasia from GOCE gravity gradients2016In: Remote Sensing, ISSN 2072-4292, E-ISSN 2072-4292, Vol. 8, no 5, p. 1-18, article id 418Article in journal (Refereed)
    Abstract [en]

    Seismic data are primarily used in studies of the Earth's inner structure. Since large partsof the world are not yet sufficiently covered by seismic surveys, products from the Earth's satellite observation systems have more often been used for this purpose in recent years. In this study we use the gravity-gradient data derived from the Gravity field and steady-state Ocean Circulation Explorer (GOCE), the elevation data from the Shuttle Radar Topography Mission (SRTM) and other global datasets to determine the Moho density contrast at the study area which comprises most of the Eurasian plate (including parts of surrounding continental and oceanic tectonic plates). A regional Moho recovery is realized by solving the Vening Meinesz-Moritz's (VMM) inverse problem of isostasy and a seismic crustal model is applied to constrain the gravimetric solution. Our results reveal that the Moho density contrast reaches minima along the mid-oceanic rift zones and maxima under the continental crust. This spatial pattern closely agrees with that seen in the CRUST1.0 seismic crustal model as well as in the KTH1.0 gravimetric-seismic Moho model. However, these results differ considerably from some previously published gravimetric studies. In particular, we demonstrate thatt here is no significant spatial correlation between the Moho density contrast and Moho deepening under major orogens of Himalaya and Tibet. In fact, the Moho density contrast under most of the continental crustal structure is typically much more uniform.

  • 34.
    Eshagh, Mehdi
    et al.
    University West, Department of Engineering Science, Division of Computer, Electrical and Surveying Engineering.
    Hussain, Mutloob
    Department of Earth Sciences, Quad-i-Azam University, Islamabad 45320, Pakistan.
    An approach to Moho discontinuity recovery from on-orbit GOCE data with application over Indo-Pak region2016In: Tectonophysics, ISSN 0040-1951, E-ISSN 1879-3266, Vol. Part B, p. 253-262Article in journal (Refereed)
    Abstract [en]

    In this research, a modified form of Vening Meinesz-Moritz (VMM) theory of isostasy for the second-order radial derivative of gravitational potential, measured from the Gravity field and steady-state Ocean Circulation Explorer (GOCE), is developed for local Moho depth recovery. An integral equation is organised for inverting the GOCE data to compute a Moho model in combination with topographic/bathymetric heights of SRTM30, sediment and consolidated crystalline basement and the laterally-varying density contrast model of CRUST1.0. A Moho model from EGM2008 to degree and order 180 is also computed based on the same principle for the purpose of comparison. In addition, we compare both of them with the 3 available seismic Moho models; two global and one regional over the Indo-Pak region. Numerical results show that our GOCE-based Moho model is closer to the all seismic models than that of EGM2008. The model is closest to the regional one with a standard deviation of 5.5 km and a root mean squares error of 7.8 km, which is 2.3 km smaller than the corresponding one based on EGM2008.

  • 35.
    Eshagh, Mehdi
    et al.
    University West, Department of Engineering Science, Division of Computer, Electrical and Surveying Engineering.
    Hussain, Mutloob
    Quaid-i-Azam University, Department of Earth Sciences, 45320 Islamabad, Pakistan.
    Tiampo, Kristy F.
    University of Colorado at Boulder, Department of Geological Sciences and CIRES, USA.
    Towards sub-lithospheric stress determination from seismic Moho, topographic heights and GOCE data2016In: Journal of Asian Earth Sciences, ISSN 1367-9120, E-ISSN 1878-5786, Vol. 129, p. 1-12Article in journal (Refereed)
    Abstract [en]

    Sub-lithospheric stresses can be estimated by analysis of gravity field measurements. Depending on the measured gravimetric quantity, different methods can be employed to estimate those sub-lithospheric stresses. Here, we further develop the Runcorn's theory for estimation of mantle stresses (1967) such that a Moho model and full topographic information are used to recover the function from which the stress can be computed by taking derivatives northwards and eastwards. We develop new integral equations for such a purpose and recover this function by solving those integral equations locally over the Indo-Pak (India-Pakistan) region from (1) a gravimetric Moho model computed from the SRTM (Shuttle Radar Topography Mission) and the Earth gravity model EGM2008, (2) SRTM and the seismic Moho model of CRUST1.0 and (3) data and measurements of the GOCE (Gravity field and steady-state Ocean Circulation Explorer) mission. Finally, we perform a joint inversion of seismic and GOCE data for the same purpose. The numerical results show that the use of a seismic Moho model recovers information about the stress field which is not seen in the results derived from a gravimetric Moho model. A combination of the seismic Moho model, SRTM and GOCE yields a better stress field than that of either the seismic and/or gravimetric data alone. The magnitudes of the sub-lithospheric stress are computed from the shear stress components over the area and good agreement is seen between the recovered combined stress field, the regional tectonic boundaries and the seismicity of the World Stress Map 2008 database.

  • 36.
    Eshagh, Mehdi
    et al.
    Royal Institute of Technology, Division of Geodesy, Stockholm, Sweden .
    Lars E., Sjöberg
    Royal Institute of Technology, Division of Geodesy, Stockholm, Sweden .
    Impact of topography and atmosphere over Iran on validation and inversion of GOCE gradiometric data2008In: Journal of the Earth and Space Physics, ISSN 1025-8647, Vol. 34, no 3, p. 15-30Article in journal (Refereed)
    Abstract [en]

    The dedicated satellite mission GOCE will sense various small mass variations along its path around the Earth. Here we study the effect of the Earth's topography and atmosphere on GOCE data. The effects depend on the magnitude of topographic height, and they will therefore vary by region. As the effect of the atmosphere and topography must be removed from the total gravity anomaly prior to geoid determinations, these effects should also be removed to simplify the downward continuation of the GOCE data to the sea level. The main goal of this article is to investigate the direct topographic and atmospheric effects in a rough region like Iran. Maps of the direct effects and their statistics are presented and discussed. Numerical results show maximum direct topographic and atmospheric effects on the GOCE data can reach 2.64 E and 5.53 mE, respectively, when the satellite flies over Iran. The indirect effect of the atmospheric and topographic masses are also formulated and presented.

  • 37.
    Eshagh, Mehdi
    et al.
    University West, Department of Engineering Science, Division of Computer, Electrical and Surveying Engineering. Royal Institute of Technology, Division of Geodesy, Stockholm, Sweden .
    Lars E:, Sjöberg
    Royal Institute of Technology, Division of Geodesy, Stockholm, Sweden .
    The modified best quadratic unbiased non-negative estimator (MBQUNE) of variance components2008In: Studia Geophysica et Geodaetica, ISSN 0039-3169, E-ISSN 1573-1626, Vol. 52, no 3, p. 305-320Article in journal (Refereed)
    Abstract [en]

    Estimated variance components may come out as negative numbers without physical meaning. One way out of this problem is to use non-negative methods. Different approaches have been presented for the solution. Sjöberg presented a method of Best Quadratic Unbiased Non-Negative Estimator (BQUNE) in the Gauss-Helmert model. This estimator does not exist in the general case. Here we present the Modified BQUNE (MBQUNE) obtained by a simple transformation from the misclosures used in the BQUE to residuals. In the Gauss-Markov adjustment model the BQUNE and MBQUNE are identical, and they differ in condition and Gauss-Helmert models only by a simple transformation. If the observations are composed of independent/disjunctive groups the MBQUNE exists in any adjustment model and it carries all the properties of the BQUNE (when it exists). The presented variance component models are tested numerically in some simple examples. It is shown that the MBQUNE works well for disjunctive groups of observations.

  • 38.
    Eshagh, Mehdi
    et al.
    University West, Department of Engineering Science, Division of Computer, Electrical and Surveying Engineering. Division of Geodesy, Royal Institute of Technology, Stockholm, Sweden.
    Lars E., Sjöberg
    Division of Geodesy, Royal Institute of Technology, Stockholm, Sweden.
    Ramin, Kiamehr
    Department of Geodesy and Geomatics, Zanjan University, Zanjan, Iran.
    Evaluation of robust techniques in suppressing the impact of outliers in a deformation monitoring network – A case study on the Tehran Milad tower network2007In: Acta Geodaetica et Geophysica Hungarica, ISSN 1217-8977, E-ISSN 1587-1037, Vol. 42, no 4, p. 449-463Article in journal (Refereed)
    Abstract [en]

    The problem of handling outliers in a deformation monitoring network is of special importance, because the existence of outliers may lead to false deformation parameters. One of the approaches to detect the outliers is to use robust estimators. In this case the network points are computed by such a robust method, implying that the adjustment result is resisting systematic observation errors, and, in particular, it is insensitive to gross errors and even blunders. Since there are different approaches to robust estimation, the resulting estimated networks may differ. In this article, different robust estimation methods, such as the M-estimation of Huber, the “Danish”, and the L 1-norm estimation methods, are reviewed and compared with the standard least squares method to view their potentials to detect outliers in the Tehran Milad tower deformation network. The numerical studies show that the L 1-norm is able to detect and down-weight the outliers best, so it is selected as the favourable approach, but there is a lack of uniqueness. For comparison, Baarda’s method “data snooping” can achieve similar results when the outlier magnitude of an outlier is large enough to be detected; but robust methods are faster than the sequential data snooping process.

  • 39.
    Eshagh, Mehdi
    et al.
    Royal Institute of Technology (KTH),Division of Geodesy and Geoinformatics.
    Lemoine, Jean-Michel
    Department of Space Geodesy (GRGS), French Space Agency (CNES), Toulouse, France.
    Gegout, Pascal
    Department of Space Geodesy (GRGS), French Space Agency (CNES), Toulouse, France.
    Biancale, Richard
    Department of Space Geodesy (GRGS), French Space Agency (CNES), Toulouse, France.
    On regularized time varying gravity field models based on GRACE data and their comparisons with hydrological models2013In: Acta Geophysica, ISSN 1895-6572, Vol. 61, no 1, p. 1-17Article in journal (Refereed)
    Abstract [en]

    Determination of spherical harmonic coefficients of the Earth's gravity field is often an ill-posed problem and leads to solving an ill-conditioned system of equations. Inversion of such a system is critical, as small errors of data will yield large variations in the result. Regularization is a method to solve such an unstable system of equations. In this study, direct methods of Tikhonov, truncated and damped singular value decomposition and iterative methods of ν, algebraic reconstruction technique, range restricted generalized minimum residual and conjugate gradient are used to solve the normal equations constructed based on range rate data of the gravity field and climate experiment (GRACE) for specific periods. Numerical studies show that the Tikhonov regularization and damped singular value decomposition methods for which the regularization parameter is estimated using quasioptimal criterion deliver the smoothest solutions. Each regularized solution is compared to the global land data assimilation system (GLDAS) hydrological model. The Tikhonov regularization with L-curve delivers a solution with high correlation with this model and a relatively small standard deviation over oceans. Among iterative methods, conjugate gradient is the most suited one for the same reasons and it has the shortest computation time

  • 40.
    Eshagh, Mehdi
    et al.
    University West, Department of Engineering Science, Division of Computer, Electrical and Surveying Engineering. Royal Institute of Technology, Stockholm, Sweden.
    Mehdi, Najafi-Alamdari
    KN Toosi University of Technology,Tehran, Iran.
    Perturbations in orbital elements of a low earth orbiting satellite2007In: Journal of the Earth and Space Physics, Vol. 33, no 1, p. 1-12Article in journal (Refereed)
    Abstract [en]

    The main point of this paper is to evaluate the perturbations in orbital elements of a low Earth orbiting satellite. The outcome of a numerical orbit integration process is the position and velocity vectors of satellite in an inertial coordinate system. The velocity and position vectors are converted into the corresponding orbital elements. Perturbations in a satellite motion affect the orbital elements in the sense of Keplerian motion. In this paper after introducing the perturbing forces acting on a satellite, the method of converting the position and velocity into the orbital elements is presented, and finally the perturbations in orbital elements of the low Earth orbiting satellite of CHAMP are evaluated. The numerical results show that, disregarding the geopotential perturbing forces, the air drag is the most predominant among other perturbing forces: rotational deformation, solar radiation, third body effect, solid Earth tide, ocean tide, and general relativity arranged by their magnitude respectively.

  • 41.
    Eshagh, Mehdi
    et al.
    University West, Department of Engineering Science, Division of Computer, Electrical and Surveying Engineering.
    Mohsen, Romeshkani
    Islamic Azad University, Qazvin Branch Iran.
    Determination of sub-lithospheric stress due to mantle convection using GOCE gradiometric data over Iran2015In: Journal of Applied Geophysics, ISSN 0926-9851, E-ISSN 1879-1859, Vol. 122, p. 11-17, article id 2807Article in journal (Refereed)
    Abstract [en]

    Sub-lithospheric stress due to mantle convection can be determined from gravimetric data based on Runcorn’s theory. In this paper, the satellite gradiometric data of the recent European satellite mission, the Gravity field and steady-state Ocean Circulation Explorer (GOCE) is used to determine the sub- lithospheric stress locally in Iran. The method of S function (SF) with numerical differentiation is developed further and an integral equation connecting satellite gradiometric data to SF is presented. The integral equation will be used to invert the real gradiometric data of GOCE to recover the SF. Later on, the sub-lithospheric shear stresses, which are the northwards and eastwards derivatives of the SF, are computed numerically. Our numerical results show that the mean square error of the recovered SF is smaller than the values of the SF meaning that the recovery process is successful. Also, the recovered stress has a good agreement with the tectonic boundaries and active seismic points of the world stress map (WSM) database. This stress reaches amplitude of 100 MPa in the territory.

  • 42.
    Eshagh, Mehdi
    et al.
    Division of Geodesy, Royal Institute of Technology, Stockholm.
    Ramin, Kiamehr
    Division of Geodesy, Royal Institute of Technology, Stockholm.
    A strategy for optimum designing of the geodetic networks from the cost, reliability and precision views2007In: Acta Geodaetica et Geophysica Hungarica, ISSN 1217-8977, E-ISSN 1587-1037, Vol. 42, no 3, p. 297-308Article in journal (Refereed)
    Abstract [en]

    There are different criteria for designing a geodetic network in an optimal way.An optimum network can be regarded as a network having high precision, reliabilityand low cost. Accordingly, corresponding to these criteria different single-objectivemodels can be defined. Each one can be subjected to two other criteria as constraints.Sometimes the constraints can be contradictory so that some of the constraints areviolated. In this contribution, these models are mathematically reviewed. It is numericallyshown how to prepare these mathematical models for optimization processthrough a simulated network. We found that the reliability model yields small positionchanges between those obtained using precision respectively. Elimination ofsome observations may happen using precision and cost model while the reliabilitymodel tries to save number of observations. In our numerical studies, no contradictionscan be seen in reliability model and this model

  • 43.
    Eshagh, Mehdi
    et al.
    Department of Geodesy, K.N.Toosi University of Technology, Tehran, Iran.
    Romeshkani, Mohsen
    Department of Geodesy, K.N.Toosi University of Technology, Tehran, Iran.
    Quality assessment of terrestrial gravity anomalies from GOCE gradiometric data and Earth's gravity models using variance component estimation2013In: Studia Geophysica et Geodaetica, ISSN 0039-3169, E-ISSN 1573-1626, Vol. 57, no 1, p. 67-83Article in journal (Refereed)
    Abstract [en]

     The satellite gravity gradiometry (SGG) data of the recent European satellite mission, the Gravity field and steady-state Ocean Circulation Explorer (GOCE), can be used as an external source for quality description of terrestrial gravity anomalies and the Earth's gravity models (EGMs). In this study integral estimators are provided and modified in a least-squares sense to regenerate the SGG data of GOCE from terrestrial gravity anomalies and an existing EGM. Based on the differences between the generated and real GOCE SGG data, condition adjustment models are constructed and variance component estimation (VCE) is applied for balancing the a priori errors of data with these differences. Here, a 1-month orbit of GOCE is considered over Iran and the condition adjustment models and VCE process are used to calibrate the errors of the GOCE data, terrestrial gravity anomalies of the area and the EGM. Numerical studies over Iran show that the a priori errors of the GOCE data and the EGM were properly presented. Also the average error of the terrestrial gravity anomalies, with a resolution of 0.5° × 0.5°, after condition adjustment and VCE process using Tzz, Tx, Tyz and −Txx −Tyy is about 30 mGal.

  • 44.
    Eshagh, Mehdi
    et al.
    University West, Department of Engineering Science, Division of Natural Sciences and Electrical and Surveying Engineering.
    Sprlak, Michal
    University of West Bohemia, NTIS - New Technologies for the Information Society, Faculty of Applied Sciences, Plzeň, Czech Republic.
    On the integral inversion of satellite-to-satellite velocity differences for local gravity field recovery: A theoretical study2016In: Celestial mechanics & dynamical astronomy, ISSN 0923-2958, E-ISSN 1572-9478, Vol. 124, no 2, p. 124-144Article in journal (Refereed)
    Abstract [en]

    The gravity field can be recovered locally from the satellite-to-satellite velocity differences (VDs) between twin-satellites moving in the same orbit. To do so, three different integral formulae are derived in this paper to recover geoid height, radial component of gravity anomaly and gravity disturbance at sea level. Their kernel functions contain the product of two Legendre polynomials with different arguments. Such kernels are relatively complicated and it may be impossible to find their closed-forms. However, we could find the one related to recovering the geoid height from the VD data. The use of spectral forms of the kernels is possible and one does not have to generate them to very high degrees. The kernel functions are well-behaving meaning that they reduce the contribution of far-zone data and for example a cap margin of 7∘ is enough for recovering gravity anomalies. This means that the inversion area should be larger by 7∘ from all directions than the desired area to reduce the effect of spatial truncation error of the integral formula. Numerical studies using simulated data over Fennoscandia showed that when the distance between the twin-satellites is small, higher frequencies of the anomalies can be recovered from the VD data. In the ideal case of having short distance between the satellites flying at 250 km level, recovering radial component of gravity anomaly with an accuracy of 7 mGal is possible over Fennoscandia, if the VD data is contaminated only with the spatial truncation error, which is an ideal assumption. However, the problem is that the power of VD signal is very low when the satellites are close and it is very difficult to recognise the signal amongst the noise of the VD data. We also show that for a successful determination of gravity anomalies at sea level from an altitude of 250 km mean VDs with better accuracy than 0.01 mm/s are required. When coloured noise at this level is used for the VDs at 250 km with separation of 300 km, the accuracy of recovery will be about 11 mGal over Fennoscandia. In the case of using the real velocities of the satellites, the main problems are downward/upward continuation of the VDs on the mean orbital sphere and taking the azimuthal integration of them.

  • 45.
    Eshagh, Mehdi
    et al.
    University West, Department of Engineering Science, Division of Mathematics, Computer and Surveying Engineering.
    Steinberger, Bernhard
    GFZ, Germany.
    Tassara, Anders
    Comparison of gravimetricand mantle flow solutions for lithospheric stress modelling and theircombinationIn: Geophysical Journal International, ISSN 0956-540X, E-ISSN 1365-246XArticle in journal (Refereed)
  • 46.
    Eshagh, Mehdi
    et al.
    University West, Department of Engineering Science, Division of Mathematics, Computer and Surveying Engineering.
    Steinberger, Bernhard
    Helmholtz Centre Potsdam, GFZ German Research Centre for Geosciences, Telegrafenberg, Potsdam, Germany & Centre for Earth Evolution and Dynamics (CEED), University of Oslo, Postboks 1028 Blindern, Oslo, Norway.
    Tenzer, Robert
    Hong Kong Polytechnic University, Department of Land Surveying and Geo-Informatics, 11 Yuk Chai Rd, Hung Hom, Hong Kong.
    Tassara, Andrés
    Universidad de Concepción, Departamento de Ciencias de la Tierra, Facultad de Ciencias Químicas, Victor Lamas 1290, Concepción, Chile.
    Comparison of gravimetric and mantle flow solutions for sub-lithopsheric stress modeling and their combination2018In: Geophysical Journal International, ISSN 0956-540X, E-ISSN 1365-246X, Vol. 213, no 2, p. 1013-1028Article in journal (Refereed)
    Abstract [en]

    Based on Hager and O’Connell’s solution to mantle flow equations, the stresses induced by mantle convection are determined using the density and viscosity structure in addition to topographic data and a plate velocity model. The solution to mantle flow equations requires the knowledge of mantle properties that are typically retrieved from seismic information. Large parts of the world are, however, not yet covered sufficiently by seismic surveys. An alternative method of modeling the stress field was introduced by Runcorn. He formulated a direct relation between the stress field and gravity data, while adopting several assumptions, particularly disregarding the toroidal mantle flow component and mantle viscosity variations. A possible way to overcome theoretical deficiencies of Runcorn’s theory as well as some practical limitations of applying Hager and O’Connell’s theory (in the absence of seismic data) is to combine these two methods. In this study, we apply a least-squares analysis to combine these two methods based on the gravity data inversion constraint on mantle flow equations. In particular, we use vertical gravity gradients from the Gravity field and steady state Ocean Circulation Explorer that are corrected for the gravitational contribution of crustal density heterogeneities prior to applying a localized gravity-gradient inversion. This gravitational contribution is estimated based on combining the Vening Meinesz-Moritz and flexural isostatic theories. Moreover, we treat the non-isostatic effect implicitly by applying a band-limited kernel of the integral equation during the inversion. In numerical studies of modeling, the stress field within the South American continental lithosphere we compare the results obtained after applying Runcorn and Hager and O’Connell’s methods as well as their combination. The results show that, according to Hager and O’Connell’s (mantle flow) solution, the maximum stress intensity is inferred under the northern Andes. Additional large stress anomalies are detected along the central and southern Andes, while stresses under most of old, stable cratonic formations aremuch less pronounced or absent. A prevailing stress-vector orientation realistically resembles a convergent mantle flow and downward currents under continental basins that separate Andean Orogeny from the Amazonian Shield and adjacent cratons. Runcorn’s (gravimetric) solution, on the other hand, reflects a tectonic response of the lithosphere to mantle flow, with the maximum stress intensity detected along the subduction zone between the Nazca and Altiplano plates and along the convergent tectonic margin between the Altiplano and South American plates. The results also reveal a very close agreement between the results obtained from the combined and Hager and O’Connell’s solutions. © The Author(s) 2018. Published by Oxford University Press on behalf of The Royal Astronomical Society.

  • 47.
    Eshagh, Mehdi
    et al.
    University West, Department of Engineering Science, Division of Computer, Electrical and Surveying Engineering.
    Tenzer, Robert
    University of West Bohemia Plzen,New Technologies for the Information Society (NTIS), Czech Republic.
    Lithospheric stress tensor from gravity and lithospheric structure models2017In: Pure and Applied Geophysics, ISSN 0033-4553, E-ISSN 1420-9136, Vol. 174, no 7, p. 2677-2688Article in journal (Refereed)
    Abstract [en]

    n this study we investigate the lithospheric stresses computed from the gravity and lithospheric structure models. The functional relation between the lithospheric stress tensor and the gravity field parameters is formulated based on solving the boundary-value problem of elasticity in order to determine the propagation of stresses inside the lithosphere, while assuming the horizontal shear stress components (computed at the base of the lithosphere) as lower boundary values for solving this problem. We further suppress the signature of global mantle flow in the stress spectrum by subtracting the long-wavelength harmonics (below the degree of 13). This numerical scheme is applied to compute the normal and shear stress tensor components globally at the Moho interface. The results reveal that most of the lithospheric stresses are accumulated along active convergent tectonic margins of oceanic subductions and along continent-to-continent tectonic plate collisions. These results indicate that, aside from a frictional drag caused by mantle convection, the largest stresses within the lithosphere are induced by subduction slab pull forces on the side of subducted lithosphere, which are coupled by slightly less pronounced stresses (on the side of overriding lithospheric plate) possibly attributed to trench suction. Our results also show the presence of (intra-plate) lithospheric loading stresses along Hawaii islands. The signature of ridge push (along divergent tectonic margins) and basal shear traction resistive forces is not clearly manifested at the investigated stress spectrum (between the degrees from 13 to 180).

  • 48.
    Eshagh, Mehdi
    et al.
    University West, Department of Engineering Science, Divison of Natural Sciences, Surveying and Mechanical Engineering.
    Tenzer, Robert
    Wuhan University, China.
    Sub-crustal stress determined using gravity and crust structure models2015In: Computational Geosciences, ISSN 1420-0597, E-ISSN 1573-1499, p. 115-125Article in journal (Refereed)
    Abstract [en]

    The sub-crustal stress induced by mantle convection has been traditionally computed using the Runcorn formulae of solving the Navier-Stokes problem. The main disadvantage of this method is a limited spectral resolution (up to degree 25 of spherical harmonics) due to a divergence of the spherical harmonic expression. To improve the spectral resolution, we propose a new method of computing the horizontal components of the sub-crustal stress based on utilising the stress function with a numerical differentiation. According to the proposed method, the stress function is functionally related to the gravity and crust structure models expressed in terms of spherical harmonics, instead of directly relating the stress components with partial derivatives of these spherical harmonics. The stress components are then computed from the stress function by applying a numerical differentiation. This modification increases the degree-dependent convergence domain of the asymptotically convergent series and consequently allows computing the stress components to a higher spectral resolution, which is compatible with currently available global crustal models. We further utilise the solution to the Vening Meinesz-Moritz inverse problem of isostasy in definition of the stress function. This definition facilitates a variable crustal thickness instead of assuming only a constant value adopted in the Runcorn formulae. The crustal thickness and sub-crustal stress are then determined directly from gravity data and a crustal structure model. We apply this numerical approach to compute the sub-crustal stress globally. Regional results are also presented and discussed over study areas of oceanic subduction zones, convergent continent-to-continent collision zones and hotspots. We demonstrate that the largest (in magnitude) sub-crustal stress occurs mainly along seismically active convergent tectonic plate boundaries.

  • 49.
    Eshagh, Mehdi
    et al.
    University West, Department of Engineering Science, Division of Computer, Electrical and Surveying Engineering.
    Zoghi, Sedigheh
    Division of Geodesy and Satellite Positioning, Royal Institute of Technology (KTH), Stockholm, Sweden.
    Local error calibration of EGM08 geoid using GNSS/levelling data2016In: Journal of Applied Geophysics, ISSN 0926-9851, E-ISSN 1879-1859, Vol. 130, no July, p. 209-217Article in journal (Refereed)
    Abstract [en]

    The geoid error, computed from EGM08, is unrealistically large due to the continuation of the spherical harmonic coefficient errors down to the surface of the reference ellipsoid. In this study, we try to calibrate such an error by the differences between the EGM08 and GNSS/levelling geoids over Fennoscandia. We use the variance component estimation procedure through combined adjustments of the geoid and GNSS/levelling heights using corrector surfaces of 4-, 5- and 7-parameter. We also develop a simple iterative method to calibrate the geoid error from the a posteriori variance factor and the errors of GNSS/levelling geoid. Our numerical investigations show that performing the separate adjustment and variance component estimation for each country with a two-component stochastic model is more successful than performing it in the whole area with a five-component model. The number of GNSS/levelling data over Sweden and Norway are much larger than those in Denmark and Finland. This causes that the corrector surfaces are fitted better in these countries and consequently the estimated errors for the geoid become larger than what they should be in the others. Based on a 7-parameter corrector surface model, the average error of the EGM08 geoid becomes 12, 17, 51 and 34 mm, in Sweden, Denmark, Norway and Finland, respectively. If the two-component stochastic model is used in a combined adjustment over Fennoscandia this average error will be 48 mm.

  • 50.
    Hussain, Matloob
    et al.
    Earth Sciences department, Quaid-i-Azam University, Islamabad, Pakistan.
    Eshagh, Mehdi
    University West, Department of Engineering Science, Division of Computer, Electrical and Surveying Engineering.
    Zulfiqar, Ahmed
    Quaid-i-Azam University, Department of Earth Sciences, Islamabad, Pakistan.
    Sadiq, Mohammed
    Quaid-i-Azam University, Department of Earth Sciences, Islamabad, Pakistan.
    Fatolazadeh, Farzam
    K. N. Toosi University of Technology, Department of Geodesy, Iran.
    Changes in gravitational parameters inferred from time-variable GRACE data- A case study for October 2005 Kashmir Earthquake2016In: Journal of Applied Geophysics, ISSN 0926-9851, E-ISSN 1879-1859, Vol. 132, p. 174-183Article in journal (Refereed)
    Abstract [en]

    The earth's gravity changes are attributed to the redistribution of masses within and/or on the surface of the earth, which are due to the frictional sliding, tensile cracking and/or cataclastic flow of rocks along the faults and detectable by earthquake events. Inversely, the gravity changes are useful to describe the earthquake seismicity over the active orogenic belts. The time variable gravimetric data are hardly available to the public domain. However, Gravity Recovery and Climatic Experiment (GRACE) is the only satellite mission dedicated to model the variation of the gravity field and an available source to the science community. Here, we have tried to envisage gravity changes in terms of gravity anomaly (Δg), geoid (N) and the gravity gradients over the Indo-Pak plate with emphasis upon Kashmir earthquake of October 2005. For this purpose, we engaged the spherical harmonic coefficients of monthly gravity solutions from the GRACE satellite mission, which have good coverage over the entire globe with unprecedented accuracy. We have analysed numerically the solutions after removing the hydrological signals, during August to November 2005, in terms of corresponding monthly differentials of gravity anomaly, geoid and the gradients. The regional structures like Main Mantle Thrust (MMT), Main Karakoram Thrust (MKT), Herat and Chaman faults are in closed association with topography and with gravity parameters from the GRACE gravimetry and EGM2008 model. The monthly differentials of these quantities indicate the stress accumulation in the northeast direction in the study area. Our numerical results show that the horizontal gravity gradients seem to be in good agreement with tectonic boundaries and differentials of the gravitational elements are subtle to the redistribution of rock masses and topography caused by 2005 Kashmir earthquake. Moreover, the gradients are rather more helpful for extracting the coseismic gravity signatures caused by seismicity over the area. Higher positive values of gravity components having higher terrain elevations are more vulnerable to the seismicity and lower risk of diastrophism otherwise.

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