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  • 1.
    Abdollahzadeh, Makan
    et al.
    K.N.Toosi University of Technology, Faculty of Geodesy and Geomatic Engineering, Tehran, Iran.
    Eshagh, Mehdi
    KTH, School of Architecture and the Built Environment (ABE), Transport and Economics (closed 20110301), Geodesy (closed 20110301).
    Najafi, Mehdi
    K.N.Toosi University of Technology, Faculty of Geodesy and Geomatic Engineering, Tehran, Iran.
    A semi-vectorization algorithm to synthesis of gravitational anomaly quantities on the Earth's surface2009Conference paper (Refereed)
    Abstract [en]

    The Earth’s gravitational potential can be expressed by the well-known spherical harmonic expansion. The computationaltime of summing up this expansion is an important practical issue which can be reduced by an efficientnumerical algorithm. This paper proposes such a method for block-wise synthesizing the anomaly quantities onthe Earth surface using vectorization.Fully-vectorization means transformation of the summations to the simple matrix and vector products. It is not apractical for the matrices with large dimensions. Here a semi-vectorization algorithm is proposed to avoid workingwith large vectors and matrices. It speeds up the computations by using one loop for the summation either ondegrees or on orders. The former is a good option to synthesize the anomaly quantities on the Earth surfaceconsidering a digital elevation model (DEM). This approach is more efficient than the two-step method whichcomputes the quantities on the reference ellipsoid and continues them upward to the Earth surface. The algorithmhas been coded in MATLAB which synthesizes a global grid of 50 x 50 (corresponding 9 million points) of gravityanomaly or geoid height using a geopotential model to degree 360 in 10000 seconds by an ordinary computer with2G RAM.

  • 2.
    Alizadeh Khameneh, Mohammad Amin
    et al.
    KTH, School of Architecture and the Built Environment (ABE), Urban Planning and Environment, Geodesy and Satellite Positioning.
    Eshagh, Mehdi
    KTH, School of Architecture and the Built Environment (ABE), Urban Planning and Environment, Geodesy and Satellite Positioning. University West, Division of Surveying Engineering.
    Sjöberg, Lars E.
    KTH, School of Architecture and the Built Environment (ABE), Urban Planning and Environment, Geodesy and Satellite Positioning.
    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 the 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.

  • 3.
    Alizadeh Khameneh, Mohammad Amin
    et al.
    KTH, School of Architecture and the Built Environment (ABE), Urban Planning and Environment, Geodesy and Satellite Positioning.
    Eshagh, Mehdi
    KTH, School of Architecture and the Built Environment (ABE), Urban Planning and Environment, Geodesy and Satellite Positioning. Department of Engineering Science, University West, Trollhättan, Sweden.
    Sjöberg, Lars E.
    KTH, School of Architecture and the Built Environment (ABE), Urban Planning and Environment, Geodesy and Satellite Positioning.
    The Effect of Instrumental Precision on Optimisation of Displacement Monitoring Networks2016In: Acta Geodaetica et Geophysica, ISSN 2213-5820, 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, M.
    et al.
    KTH, School of Architecture and the Built Environment (ABE), Urban Planning and Environment, Geodesy and Geoinformatics.
    Eshagh, Mehdi
    KTH, School of Architecture and the Built Environment (ABE), Urban Planning and Environment, 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 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.

  • 5.
    Bagherbandi, Mohammad
    et al.
    KTH, School of Architecture and the Built Environment (ABE), Urban Planning and Environment, Geodesy and Geoinformatics.
    Eshagh, Mehdi
    KTH, School of Architecture and the Built Environment (ABE), Urban Planning and Environment, Geodesy and Geoinformatics.
    Sjöberg, Lars E.
    KTH, School of Architecture and the Built Environment (ABE), Urban Planning and Environment, Geodesy and Geoinformatics.
    Multi-objective versus single-objective models in geodetic network optimization2009In: Nordic Journal of Surveying and Real Estate Research, ISSN 1459-5877, E-ISSN 2341-6599, Vol. 6, no 1, p. 7-20Article in journal (Refereed)
  • 6.
    Eshagh, Medhi
    KTH, School of Architecture and the Built Environment (ABE), Transport and Economics, Geodesy.
    INVERSION OF GRAVITY GRADIENTS FOR DETERMINATION OF GRAVITY ANOMALY IN THE POLAR GAPS2010In: ACTA GEOD GEOPHYS HUNG, ISSN 1217-8977, Vol. 45, no 4, p. 440-451Article in journal (Refereed)
    Abstract [en]

    The Polar Regions are not covered by satellite gravity gradiometry data if the orbital inclination of the satellite is not equal to 90 degrees. This paper investigates the feasibility of determining gravity anomaly (at sea level) by inversion of satellite gravity gradiometry data in these regions. Inversion of each element of tensor of gravitation as well as their joint inversion are investigated. Numerical studies show that gravity anomaly can be recovered with an error of 3 mGal in the north polar gap and 5 mGal in south polar gaps in the presence of 1 mE white noise in the satellite data. These errors can be reduced to 1 mGal and 3 mGal, respectively, by removing the regularization bias from the recovered gravity anomalies.

  • 7.
    Eshagh, Mehdi
    KTH, School of Architecture and the Built Environment (ABE), Urban Planning and Environment, 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.

  • 8.
    Eshagh, Mehdi
    KTH, School of Architecture and the Built Environment (ABE), Transport and Economics, Geodesy.
    Alternative Expressions for Gravity Gradients in Local North-Oriented Frame and Tensor Spherical Harmonics2010In: ACTA GEOPHYS, ISSN 1895-6572, Vol. 58, no 2, p. 215-243Article in journal (Refereed)
    Abstract [en]

    The traditional expressions for gravity gradients in local north-oriented frame and tensor spherical harmonics have complicated forms involved with first- and second-order derivatives of spherical harmonics and also singular terms. In this paper we present alternative expressions for these quantities, which are simpler and contain no singular terms. The presented formulas are useful for those disciplines of geosciences which are involved with potential theory, tensor spherical harmonics and second-order derivatives of spherical harmonic series in the local northoriented frame. A simple numerical test on the solution of the gradiometric boundary value problems presents the correctness of these new expressions and ability of the solutions to continue the gravity gradients from satellite level down to sea level using spherical harmonics.

  • 9.
    Eshagh, Mehdi
    KTH, School of Architecture and the Built Environment (ABE), Urban Planning and Environment, Geodesy and Geoinformatics. Department of Engineering Science, University West, Trollhättan, Sweden .
    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 degrees 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
    KTH, School of Architecture and the Built Environment (ABE), Urban Planning and Environment, Geoinformatik och Geodesi.
    Bi-Objective models of Geodetic Network Optimization2006Conference paper (Refereed)
  • 11.
    Eshagh, Mehdi
    KTH, School of Architecture and the Built Environment (ABE), Transport and Economics, Geodesy.
    Comparison of two approaches for considering laterally varying density in topographic effect on satellite gravity gradiometric data2010In: ACTA GEOPHYS, ISSN 1895-6572, Vol. 58, no 4, p. 661-686Article in journal (Refereed)
    Abstract [en]

    The satellite gravity gradiometric data are influenced by laterally varying density in topographic masses, while in most of studies a constant density for the masses was considered. This assumption causes an error in estimating the topographic effect. This paper theoretically and numerically investigates the methods of Sjoberg as well as Novak and Grafarend to consider the laterally varying density for topographic masses in formulation of topographic potential in spherical harmonics.

  • 12.
    Eshagh, Mehdi
    KTH, School of Architecture and the Built Environment (ABE), Urban Planning and Environment, Geoinformatics.
    Complementary studies in Satellite Gravity Gradiometry2009Report (Other (popular science, discussion, etc.))
    Abstract [en]

    The gravity field and steady-state ocean circulation explorer (GOCE) (ESA 1999,Albertella et al. 2002, Balmino et al. 1998 and 2001) was finally launched on 17thMarch in 2009. In this satellite mission the second-order derivatives of the Earth’sgravitational potential are measured based on differential accelerometer at a satellitebornegradiometer. It is expected to recover the geopotential coefficients to higherdegrees and orders than those were obtained from the former satellite missions; sayup to degree and order 300. Such an Earth’s gravity model will have an accuracy of 1cm in global geoid height and of 1 mGal for the gravity anomalies, which areextremely good accuracies of the long-wavelength structure of the gravity field.

    The present report is a summary of the studies of Mehdi Eshagh when a postdoct/research associate position in division of Geodesy was available for him. Theresearch work consists of the studies continuing his thesis work and completingproject no. 63/07:1, funded by the Swedish National Space Board (SNSB). Thereport is of collective papers type with a series of Papers A-Q. In the following wepresent summaries of his complementary studies. We just present the background ofeach study and the author’s contributions in comparing his research to others’ works.The details about the methodology, theory, numerical investigations and conclusionsare given in the corresponding papers of each subject

  • 13.
    Eshagh, Mehdi
    KTH, School of Architecture and the Built Environment (ABE), Transport and Economics (closed 20110301), Geodesy (closed 20110301).
    Contribution of 1st-3rd order terms of a binomial expansion of topographic heights in topographic and atmospheric effects on satellite gravity gradiometric data2009In: Artificial Satellites: journal of planetary geodesy, ISSN 0208-841X, Vol. 44, no 1, p. 21-31Article in journal (Refereed)
    Abstract [en]

    In mathematical modeling of the topographic and atmospheric potentials in spherical harmonics, the topographic heights can binomially be expanded a certain order, usually to the third order. Some studies have been done on the effect of each order on geoid and gravity anomaly. However similar study on the satellite gravity gradiometric data is missed yet. This paper will investigate this matter globally. It presents that the contribution of the second- and third-order topographic terms is within 0.08 E and 2 mE, respectively on satellite gravity gradiometric data at 250 km level. Also the contribution of these terms is within 0.5 mE and 0.08 mE for the atmospheric effect.

  • 14.
    Eshagh, Mehdi
    KTH, School of Architecture and the Built Environment (ABE), Transport and Economics (closed 20110301), Geodesy (closed 20110301).
    Error calibration of quasi-geoidal, normal and ellipsoidal heights of Sweden using variance component estimation2010In: Contributions to Geophysics and Geodesy, ISSN 1335-2806, Vol. 40, no 1, p. 1-30Article in journal (Refereed)
    Abstract [en]

    Errors of estimated parameters in an adjustment process should be scaled according to the size of the estimated residuals or misclosures. After computing a quasigeoid (geoid), its biases and tilts, due to existence of systematic errors in the terrestrial data, are removed by fitting a corrective surface to the misclosures of the differences between the GNSS/levelling data and the quasi-geoid (geoid). Variance component estimation can be used to re-scale or calibrate the error of the GNSS/levelling data and the quasi-geoid (geoid) model. This paper uses this method to calibrate the errors of the recent quasi-geoid model, the GNSS and the normal heights of Sweden. Different stochastic models are investigated in this study and based on a 7-parameter corrective surface model and a three-variance component stochastic model, the calibrated error of the quasi-geoid and the normal heights are 6 mm and 5 mm, respectively and the re-scaled error of the GNSS heights is 18 mm.

  • 15.
    Eshagh, Mehdi
    KTH, School of Architecture and the Built Environment (ABE), Urban Planning and Environment, Geoinformatik och Geodesi.
    Generalized Least Squares Adjustment of Gauss-Helmert Model2005Conference paper (Refereed)
  • 16.
    Eshagh, Mehdi
    KTH, School of Architecture and the Built Environment (ABE), Transport and Economics (closed 20110301), Geodesy (closed 20110301).
    Impact of vectorization on global synthesis and analysis in gradiometry2009In: Acta Geodaetica et Geophysica Hungarica, ISSN 1217-8977, Vol. 44, no 3, p. 323-342Article in journal (Refereed)
    Abstract [en]

    Computational time is an important matter in numerical aspects and it depends on the algorithm and computer that is used. An inappropriate algorithm can increase computation time and cost. The main goal of this paper is to present a vectorization algorithm to speed up the global gradiometric synthesis and analysis. The paper discusses details of this technique and its very high capabilities. Numerical computations show that the global gradiometric synthesis with 0.5 degrees x 0.5 degrees resolution can be done in a few minutes (6 minutes) by vectorization, which is considerable less compared to several hours (9 hours) by an inappropriate algorithm. The global gradiometric analysis of representation by spherical harmonics up to degree and order of 360, can be performed within one hour using vectorization, but if an inconvenient algorithm is used it can be delayed more than 1 day. Here we present the vectorization technique to gradiometric synthesis and analysis, but it can also be used in many other computational aspects and disciplines.

  • 17.
    Eshagh, Mehdi
    KTH, School of Architecture and the Built Environment (ABE), Urban Planning and Environment, Geoinformatik och Geodesi.
    Inversion of satellite gradiometry data using statistically modified integral formulas for local gravity field recovery2011In: Advances in Space Research, ISSN 0273-1177, E-ISSN 1879-1948, Vol. 47, no 1, p. 74-85Article in journal (Refereed)
    Abstract [en]

    The satellite gravity gradiometric data can be used directly to recover the gravity anomaly at sea level using inversion of integral formulas. This approach suffers by the spatial truncation errors of the integrals, but these errors can be reduced by modifying the formulas. It allows us to consider smaller coverage of the satellite data over the region of recovery. In this study, we consider the second-order radial derivative (SORD) of disturbing potential (T-rr) and determine the gravity anomaly with a resolution of 1 degrees x 1 degrees at sea level by inverting the statistically modified version of SORD of extended Stokes' formula. Also we investigate the effect of the spatial truncation error on the quality of inversion considering noise of T-rr. The numerical investigations show satisfactory results when the area of T-rr coverage is the same with that of the gravity anomaly and the integral formula is modified by the biased least-squares modification. The error of recovery will be about 6 mGal after removing the regularization bias in the presence of 1 mE noise in T-rr measured on the orbit. (c) 2010 COSPAR. Published by Elsevier Ltd. All rights reserved.

  • 18.
    Eshagh, Mehdi
    KTH, School of Architecture and the Built Environment (ABE), Transport and Economics (closed 20110301), Geodesy (closed 20110301).
    Least-squares modification and satellite gravity gradiometry2009Conference paper (Refereed)
    Abstract [en]

    Satellite gravity gradiometry is a technique to determine a precise high-resolution geopotential model based on spatial-differential accelerometry. The satellite gradiometric data plays an important role in this respect and they must be validated before doing any computation. One way of validating such a data is to use the second-order partial derivatives of the extended Stokes formula to generate the gradients at satellite level, from terrestrial gravimetric data. A global coverage of the terrestrial data is required to perform the integration, but having such coverage is neither practical nor reasonable, and the integrals should be modified. The integrals’ kernel is not isotropic (except for second-order radial derivative) and modification of such integrals will not be easy task. Here, general integral estimators for vertical-vertical, vertical-horizontal and horizontal-horizontal gradients are presented, based on combination of the gradients, so that that the estimators become modifiable. Least-squares modification minimizes not only the truncation error of the integral, but the errors of global gravitational model and the terrestrial data. Elements of the system of equations, from which the modification parameters based on biased, unbiased and optimum least-squares modification is derived, are mathematically formulated.

  • 19.
    Eshagh, Mehdi
    KTH, School of Architecture and the Built Environment (ABE), Transport and Economics, Geodesy.
    Least-squares modification of extended Stokes' formula and its second-order radial derivative for validation of satellite gravity gradiometry data2010In: Journal of Geodynamics, ISSN 0264-3707, E-ISSN 1879-1670, Vol. 49, no 2, p. 92-104Article in journal (Refereed)
    Abstract [en]

    The gravity anomalies at sea level can be used to validate the satellite gravity gradiometry data. Validation of such a data is important prior to downward continuation because of amplification of the data errors through this process. In this paper the second-order radial derivative of the extended Stokes' formula is employed and the emphasis is on least-squares modification of this formula to generate the second-order radial gradient at satellite level. Two methods in this respect are proposed: (a) modifying the second-order radial derivative of extended Stokes' formula directly, and (b) modifying extended Stokes' formula prior to taking the second-order radial derivative. Numerical studies show that the former method works well but the latter is very sensitive to the proper choice of the cap size of integration and degree of modification.

  • 20.
    Eshagh, Mehdi
    KTH, School of Architecture and the Built Environment (ABE), Urban Planning and Environment, Geoinformatik och Geodesi.
    Least-Squares Modification of Stokes' Formula with EGM082009In: Geodesy and Cartography, ISSN 1648-3502, Vol. 35, no 4, p. 111-117Article in journal (Refereed)
    Abstract [en]

    Least-squares modification is an optimal method of modifying Stokes' formula. This method can be categorized as a generalization of the spectral combination methods as it considers the truncation error of the integral formulas in its combination process. In short, this method involves the modification parameters based on minimizing the error of terrestrial gravimetric data, satellite data and the truncation error of the integral. In this respect, the choice of the geopotential model definitely plays an important role. This paper uses the recent combined geopotential model EGM08 for generating the spectra of gravity anomaly and its error. Numerical results show that EGM08 improves least-squares modification by about 10 cm comparing to the traditional way.

  • 21.
    Eshagh, Mehdi
    KTH, School of Architecture and the Built Environment (ABE), Urban Planning and Environment, Geoinformatik och Geodesi.
    Long Wavelength Geoid Determination of Iran2006Conference paper (Refereed)
  • 22.
    Eshagh, Mehdi
    KTH, School of Architecture and the Built Environment (ABE), Urban Planning and Environment, Geoinformatik och Geodesi.
    Multi-Objective Models of Geodetic Network Optimization2006Conference paper (Refereed)
  • 23.
    Eshagh, Mehdi
    KTH, School of Architecture and the Built Environment (ABE), Urban Planning and Environment, Geoinformatics (closed 20110301).
    Non-singular expressions for the 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 gravitational gradient tensor (GGT) have complicated forms depending on the first and second order derivatives of associated Legendre functions (ALF), and also singular terms when approaching the poles. The 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 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.

  • 24.
    Eshagh, Mehdi
    KTH, School of Architecture and the Built Environment (ABE), Urban Planning and Environment, Geodesy and Geoinformatics.
    Numerical aspects of EGM08-based geoid computations in Fennoscandia regarding the applied reference surface and error propagation2013In: Journal of Applied Geophysics, ISSN 0926-9851, E-ISSN 1879-1859, Vol. 96, p. 28-32Article in journal (Refereed)
    Abstract [en]

    So far the recent Earth's gravity model, EGM08, has been successfully applied for different geophysical and geodetic purposes. In this paper, we show that the computation of geoid and gravity anomaly on the reference ellipsoid is of essential importance but error propagation of EGM08 on this surface is not successful due to downward continuation of the errors. Also we illustrate that some artefacts appear in the computed geoid and gravity anomaly to lower degree and order than 2190. This means that the role of higher degree harmonics than 2160 is to remove these artefacts from the results. Consequently, EGM08 must be always used to degree and order 2190 to avoid the numerical problems.

  • 25.
    Eshagh, Mehdi
    KTH, School of Architecture and the Built Environment (ABE), Urban Planning and Environment, Geoinformatik och Geodesi.
    On BQUE and BQUNE of variance-covariance components2005In: Proceedings of the Geomatics 84 Conference, National Cartographic Center, Tehran, Iran, Iranian National Cartographic Center, 2005Conference paper (Refereed)
  • 26.
    Eshagh, Mehdi
    KTH, School of Architecture and the Built Environment (ABE), Transport and Economics, Geodesy.
    On integral approach to regional gravity field modelling from satellite gradiometric data2011In: ACTA GEOPHYS, ISSN 1895-6572, Vol. 59, no 1, p. 29-54Article in journal (Refereed)
    Abstract [en]

    Solution of the gradiometric boundary value problems leads to three integral formulas. If we are satisfied with obtaining a smooth solution for the Earth's gravity field, we can use the formulas in regional gravity field modelling. In such a case, satellite gradiometric data are integrated on a sphere at satellite level and continued downward to the disturbing potential (geoid) at sea level simultaneously. This paper investigates the gravity field modelling from a full tensor of gravity at satellite level. It studies the truncation bias of the integrals as well as the filtering of noise of data. Numerical studies show that by integrating T (zz) with 1 mE noise and in a cap size of 7A degrees, the geoid can be recovered with an error of 12 cm after the filtering process. Similarly, the errors of the recovered geoids from T (xz,yz) and T (xx-yy, 2xy) are 13 and 21 cm, respectively.

  • 27.
    Eshagh, Mehdi
    KTH, School of Architecture and the Built Environment (ABE), Transport and Economics, Geodesy.
    On Satellite Gravity Gradiometry2009Doctoral thesis, monograph (Other academic)
  • 28.
    Eshagh, Mehdi
    KTH, School of Architecture and the Built Environment (ABE), Transport and Economics (closed 20110301), Geodesy (closed 20110301).
    On the convergence of spherical harmonic expansion of topographic and atmospheric biases in gradiometry2009In: Slovak Academy of Sciences. Geophysical Institute. Contributions to Geophysics and Geodesy, ISSN 1335-2806, Vol. 39, no 4, p. 273-299Article in journal (Refereed)
    Abstract [en]

    The gravity gradiometric data are affected by the topographic and atmospheric masses. In order to fulfill Laplace-Poisson’s equation and to simplify the downward continuation process, these effects should be removed from the data. However, if the analytical downward continuation is considered, the gravity gradients can be continued downward disregarding such effects but the result will be biased. The topographic and atmospheric biases can be expressed in terms of spherical harmonics and studying these biases gives some ideas about analytical downward continuation of these quantities to sea level. In formulation of harmonic coefficients of the topographic and atmospheric biases, a truncated binomial expansion of topographic height is used. In this paper, we show that the harmonics are convergent to the third term of this binomial expansion. The harmonics of the biases on Vzz are convergent to the first term and they are convergent in Vxy for all the terms. The harmonics of the other components of the gravity gradient tensor are convergent to the second terms, while the third terms are only symptotically convergent. This means that in terrestrial and airborne gradiometry the biases should be computed just to the second order term, while in satellite gravity gradiometry, e.g. GOCE, the third term can also be considered.

  • 29.
    Eshagh, Mehdi
    KTH, School of Architecture and the Built Environment (ABE), Urban Planning and Environment, Geoinformatik och Geodesi.
    ON THE ESTIMATION OF VARIANCE IN UNSTABLE CONDITION ADJUSTMENT MODELS2011In: ACTA GEOD GEOPHYS HUNG, ISSN 1217-8977, Vol. 46, no 1, p. 71-83Article in journal (Refereed)
    Abstract [en]

    Estimation of variance in an ordinary adjustment model is straightforward, but if the model becomes unstable or ill-conditioned its solution and the variance of the solution will be very sensitive to the errors of observations. This sensitivity can be controlled by stabilizing methods but the results will be distorted due to stabilization. In this paper, stabilizing an unstable condition model using Tikhonov regularization, the estimations of variance of unit weight and variance components are investigated. It will be theoretically proved that the estimator of variance or variance components has not the minimum variance property when the model is stabilized, but unbiased estimation of variance is possible. A simple numerical example is provided to show the performance of the theory.

  • 30.
    Eshagh, Mehdi
    KTH, School of Architecture and the Built Environment (ABE), Urban Planning and Environment, Geodesy and Geoinformatics.
    On the reliability and error calibration of some recent Earth's gravity models of GOCE with respect to EGM082013In: ACTA GEOD GEOPHYS, ISSN 2213-5812, 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.

  • 31.
    Eshagh, Mehdi
    KTH, School of Architecture and the Built Environment (ABE), Transport and Economics (closed 20110301), Geodesy (closed 20110301).
    Optimal combination of integral solutions of gradiometric boundary value problem using variance component estimation in Earth gravitational modelling2010In: Earth Planets and Space, ISSN 1343-8832, E-ISSN 1880-5981, Vol. 62, no 5, p. 437-448Article in journal (Refereed)
    Abstract [en]

    Optimal estimation of geopotential coefficients is an important aspect of gravitational field recovery using satellite gravity gradiometry. The combination of gradiometric data and the use of tensor spherical harmonics is useful in this field. Here, we present a new strategy for combining different spectral solutions of the gradiometric boundary value problem by defining and formulating degree-order variance components and using the condition adjustment model. Numerical results show that the spectral combination of considering one degree-order variance component for each type of observation yields better results than the case where one degree-order variance component is estimated for each integral solution of the gradiometric boundary value problem. In this study, the estimates of the variance components are not considered in the standard way; rather, these components are mainly used to absorb the discretization error of the integral solutions. This method is capable of combining integrals in geosciences disciplines.

  • 32.
    Eshagh, Mehdi
    KTH, School of Architecture and the Built Environment (ABE), Urban Planning and Environment, Geoinformatik och Geodesi.
    Orbit integration in non-inertial frame2009In: Journal of the Earth and Space Physics, ISSN 0378-1046, Vol. 35, no 1, p. 1-8Article in journal (Refereed)
  • 33.
    Eshagh, Mehdi
    KTH, School of Architecture and the Built Environment (ABE), Urban Planning and Environment, Geoinformatik och Geodesi.
    Robust maximum Likelihood Estimation of Huber in geodetic Networks2006Conference paper (Refereed)
  • 34.
    Eshagh, Mehdi
    KTH, School of Architecture and the Built Environment (ABE), Urban Planning and Environment, Geoinformatik och Geodesi.
    Scalar Risk functions as Criteria for datum Definition of Geodetic Networks2006Conference paper (Refereed)
  • 35.
    Eshagh, Mehdi
    KTH, School of Architecture and the Built Environment (ABE), Urban Planning and Environment, Geoinformatik och Geodesi.
    Semi-stochastic modification of second-order radial derivative of Abel-Poisson's formula for validating satellite gravity gradiometry data2011In: Advances in Space Research, ISSN 0273-1177, E-ISSN 1879-1948, Vol. 47, no 4, p. 757-767Article in journal (Refereed)
    Abstract [en]

    The geoid can be used to validate the satellite gravity gradiometry data. Validation of such data is important prior to their downward continuation because of amplification of the data errors through this process. In this paper, the second-order radial derivative of Abel-Poisson's formula is modified stochastically to reduce the effect of the far-zone geoid and generate the second-order radial derivative of geopotential at 250 km level. The numerical studies over Fennoscandia show that this method yields the gradients with an error of 10 mE and when the long wavelength of geoid is removed from the estimator and restored after the computations (remove compute restore) the error will be in 1 mE level. We name this method semi-stochastic modification. The best case scenario is found when the degree of modification of the integral formula is 200 and the long wavelength geoid to degree 100 is removed and restored. In this case the geoid should have a resolution of 15' x 15' and the integration should be performed over a cap size of 3 degrees.

  • 36.
    Eshagh, Mehdi
    KTH, School of Architecture and the Built Environment (ABE), Urban Planning and Environment, Geodesy and Geoinformatics.
    Sequential Tikhonov Regularization: an alternative way for inverting satellite gradiometric data2011In: Zfv, ISSN 1618-8950, Vol. 136, no 2, p. 113-121Article in journal (Refereed)
    Abstract [en]

    Numerous regularization methods exist for solving the ill-posed problem of downward continuation of satellite gravity gradiometry (SGG) data to gravity anomaly at sea level. Generally, the use of a dense set of data is recommended in the downward continuation. However, when such dense data are used some of the regularization methods are not efficient and applicable. In this paper, a sequential way of using the Tikhonov regularization is developed for solving large systems and compared to methods of direct truncated singular value decomposition and iterative methods of range restricted minimum residual, algebraic reconstruction technique,  and conjugate gradient for recovering gravity anomaly at sea level from the SGG data. Numerical studies show that the sequential Tikhonov regularization is comparable to the conjugate gradient and yields similar result.

     

  • 37.
    Eshagh, Mehdi
    KTH, School of Architecture and the Built Environment (ABE), Urban Planning and Environment, Geoinformatik och Geodesi.
    Single-Objective Models of Geodetic Network Optimization2006Conference paper (Refereed)
  • 38.
    Eshagh, Mehdi
    KTH, School of Architecture and the Built Environment (ABE), Urban Planning and Environment, Geoinformatik och Geodesi.
    Spatially Restricted Integrals in Gradiometric Boundary Value Problems2010In: Artificial Satellites, ISSN 2083-6104, Vol. 44, no 4, p. 131-148Article in journal (Refereed)
    Abstract [en]

    The spherical Slepian functions can be used to localize the solutions of the gradiometric boundary value problems on a sphere. These functions involve spatially restricted integral products of scalar, vector and tensor spherical harmonics. This paper formulates these integrals in terms of combinations of the Gaunt coefficients and integrals of associated Legendre functions. The presented formulas for these integrals are useful in recovering the Earth's gravity field locally from the satellite gravity gradiometry data.

  • 39.
    Eshagh, Mehdi
    KTH, School of Architecture and the Built Environment (ABE), Urban Planning and Environment, Geodesy and Geoinformatics.
    Spectral Combination of Spherical Gradiometric Boundary-Value Problems: A Theoretical Study2012In: Pure and Applied Geophysics, ISSN 0033-4553, E-ISSN 1420-9136, Vol. 169, no 12, 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.

  • 40.
    Eshagh, Mehdi
    KTH, School of Architecture and the Built Environment (ABE), Urban Planning and Environment, Geoinformatik och Geodesi.
    Spectral combination of vector gravimetric boundary value problems2011In: Engineering Journal of Geospatial Information system, Vol. 1, no 3, p. 23-50Article in journal (Refereed)
    Abstract [en]

    f there are more than a unique type of boundary value problem, so there may not be just one solution for problem. The vector gravimetric boundary value problem is one of the types of such problems which include two integral solutions. In this paper, this problem is solved in spectral domain, and then the solutions will be converted to integrals in spatial domain. The kernels of these integrals are divergent but by using spectral combination they become convergent and even they will have the downward continuation property. To do so, different stochastic estimators for recovering the disturbing potential at sea level are presented, and for each one of them the spectral coefficients are derived. Numerical computations show that the convergent kernels have the property of modifying the integral formulas in addition to the downward continuation and Wiener filtering, so that the kernels are well-behaved and reduce the contributions of far-zone data easily. The method presented in this paper can be applied for combination of satellite or air-borne vector gravimetric data.

  • 41.
    Eshagh, Mehdi
    KTH, School of Architecture and the Built Environment (ABE), Urban Planning and Environment, Geoinformatik och Geodesi.
    Spherical harmonics expansion of the atmospheric gravitational potential based on exponential and power models of atmosphere2008In: Artificial Satellites, ISSN 0208-841X, Vol. 43, no 1, p. 26-43Article in journal (Refereed)
  • 42.
    Eshagh, Mehdi
    KTH, School of Architecture and the Built Environment (ABE), Transport and Economics (closed 20110301), Geodesy (closed 20110301).
    Step Variable Numerical Orbit Determination of a Low Earth Orbit Satellite2005In: Journal of Earth and Space Physics, ISSN 1025-8647, Vol. 31, no 1, p. 1-12Article in journal (Refereed)
  • 43.
    Eshagh, Mehdi
    KTH, School of Architecture and the Built Environment (ABE), Transport and Economics (closed 20110301), Geodesy (closed 20110301).
    The Effect Of Lateral Density Variations Of Crustal And Topographic Masses On Goce Gradiometric Data - A Study In Iran And Fennoscandia2009In: ACTA GEODAETICA ET GEOPHYSICA HUNGARICA, ISSN 1217-8977, Vol. 44, no 4, p. 399-418Article in journal (Refereed)
    Abstract [en]

    The topographic and atmospheric masses influence the satellite gravity gradiometry data, and it is necessary to remove these effects as precise as possible to make the computational space harmonic and simplify the downward continuation of such data. The topographic effects have been formulated based on constant density assumption for the topographic masses. However in this paper we formulate and study the effect of lateral density variation of crustal and topographic masses on the satellite gravity gradiometry data. Numerical studies over Fennoscandia and Iran show that the lateral density variation effect of the crust on GOCE data can reach to 1.5 E in Fennoscandia and 1 E in Iran. The maximum effect of lateral density variation of topography is 0.1 E and 0.05 E in Iran and Fennoscandia, respectively.

  • 44.
    Eshagh, Mehdi
    KTH, School of Architecture and the Built Environment (ABE), Transport and Economics (closed 20110301), Geodesy (closed 20110301).
    The Effect of Polar Gaps on the Solutions of Gradiometric Boundary Value Problems2008In: Artificial Satellites, ISSN 0208-841X, Vol. 43, no 3, p. 97-108Article in journal (Refereed)
    Abstract [en]

    The lack of satellite gravity gradiometric data, due to inclined orbit, in the Polar Regions influences the geopotential coefficients obtained from the solutions of gradiometric boundary value problems. This paper investigates the polar gaps effect on these solutions and it presents that the near zero-, first- and second-order geopotential coefficients are weakly determined by the vertical-vertical, vertical-horizontal and horizontal solutions, respectively. Also it shows that the vertical-horizontal solution is more sensitive to the lack of data than the other solutions.

  • 45.
    Eshagh, Mehdi
    KTH, School of Architecture and the Built Environment (ABE), Urban Planning and Environment, Geoinformatik och Geodesi.
    The effect of spatial truncation error on integral inversion of satellite gravity gradiometry data2011In: Advances in Space Research, ISSN 0273-1177, E-ISSN 1879-1948, Vol. 47, no 7, p. 1238-1247Article in journal (Refereed)
    Abstract [en]

    The satellite gravity gradiometry (SGG) data can be used for local modelling of the Earth's gravity field. In this study, the SGG data in the local north-oriented and orbital frames are inverted to the gravity anomaly at sea level using the second-order partial derivatives of the extended Stokes formula. The emphasis is on the spatial truncation error and the kernel behaviour of the integral formulas in the aforementioned frames. The paper will show that only the diagonal elements of gravitational tensor at satellite level are suitable for recovering the gravity anomaly at sea level. Numerical studies show that the gravity anomaly can be recovered in Fennoscandia with an accuracy of about 6 mGal directly from on-orbit SGG data.

  • 46.
    Eshagh, Mehdi
    KTH, School of Architecture and the Built Environment (ABE), Urban Planning and Environment, Geoinformatik och Geodesi.
    Towards Validation of Satellite Gradiometric Data Using Modified Version of 2nd Order Partial Derivatives of Extended Stokes' Formula2010In: Artificial Satellites, ISSN 2083-6104, Vol. 44, no 4, p. 103-129Article in journal (Refereed)
    Abstract [en]

    The satellite gradiometric data should be validated prior to being used. One way of such a validation process is to use some integral estimators which are the second-order partial derivatives of the extended Stokes formula to regenerate the data from the gravity anomaly at the topographic surface. In this paper, we present how least-squares modification methods are used to modify such integral estimators. Our concentration will be on validation of the vertical-horizontal and horizontal-horizontal elements of the gravitational tensor at satellite level. The paper will formulate the elements of the system of equations from which the modification parameters are derived based on all types of least-squares modification. The truncation and Paul's coefficients will also be modelled.

  • 47.
    Eshagh, Mehdi
    KTH, School of Architecture and the Built Environment (ABE), Transport and Economics, Geodesy.
    Variance component estimation in linear ill-posed problems: TSVD issue2010In: Acta Geodaetica et Geophysica Hungarica, ISSN 1217-8977, Vol. 45, no 2, p. 184-194Article in journal (Refereed)
    Abstract [en]

    An ill-posed problem which involves heterogonous data can yield good results if the weight of observations is properly introduced into the adjustment model. Variance component estimation can be used in this respect to update and improve the weights based on the results of the adjustment. The variance component estimation will not be as simple as that is in an ordinary adjustment problem, because the result of the solution of an ill-posed problem contains a bias due to stabilizing the adjustment model. This paper investigates the variance component estimation in those ill-posed problems solved by the truncation singular value decomposition. The biases of the variance components are analyzed and the biased-corrected and the biased-corrected non-negative estimators of the variance components are developed. The derivations show that in order to estimate unbiased variance components, it suffices to estimate and remove the bias from the estimated residuals.

  • 48.
    Eshagh, Mehdi
    et al.
    KTH, School of Architecture and the Built Environment (ABE), Transport and Economics (closed 20110301), Geodesy (closed 20110301).
    Abdollahzadeh, M.
    THE EFFECT OF GEOPOTENTIAL PERTURBATIONS OF GOCE ON ITS OBSERVATIONS - A NUMERICAL STUDY2009In: ACTA GEODAETICA ET GEOPHYSICA HUNGARICA, ISSN 1217-8977, Vol. 44, no 4, p. 385-398Article in journal (Refereed)
    Abstract [en]

    In solution of gradiometric boundary value problem in space a regular grid of satellite gravity gradiometry data is required. This grid is considered on a sphere with radius of the mean Earth sphere and altitude of satellite. However, the gravitational gradients are measured by a gradiometer mounted on GOCE satellite and orbital perturbations of the satellite influence GOCE observations as well. In this study we present that these effects are about 2 E on GOCE data. Also numerical studies on the gravitational gradients in orbital frame show that the perturbations of co-latitude are more significant than that of inclination. The effect of perturbed inclination is less than -9 mE while the effect of perturbed co-latitude is within -173 mE in one day revolution of GOCE.

  • 49.
    Eshagh, Mehdi
    et al.
    KTH, School of Architecture and the Built Environment (ABE), Transport and Economics, Geodesy.
    Abdollahzadeh, Makan
    KTH, School of Architecture and the Built Environment (ABE), Transport and Economics, Geodesy.
    Semi-vectorization: an efficient technique for synthesis and analysis of gravity gradiometry data2010In: EARTH SCI INFORM, ISSN 1865-0473, Vol. 3, no 3, p. 149-158Article in journal (Refereed)
    Abstract [en]

    The harmonic synthesis and analysis of the elements of gravitational tensor can be done in few minutes if a suitable programming algorithm is used. Vectorization is an efficient technique for such processes, but the size of matrices will increase when the resolution of synthesis or analysis is high; say higher than 0.5 degrees x 0.5 degrees. Here, we present a technique to manage the computer memory and computational time by excluding one computational loop from the matrix products and we call this method semi-vectorization. Based on this technique, we synthesize the gravitational tensor using the EGM96 geopotential model and after that we analyze the tensor for recovering the geopotential coefficients. MATLAB codes are provided which are able to analyze 224 millions gradiometric data, corresponding to a global grid of 2.5' x 2.5' on a sphere in 1,093 s by a personal computer with 2 Gb RAM.

  • 50.
    Eshagh, Mehdi
    et al.
    KTH, School of Architecture and the Built Environment (ABE), Urban Planning and Environment, Geoinformatik och Geodesi.
    Abdollahzadeh, Makan
    KTH, School of Architecture and the Built Environment (ABE), Urban Planning and Environment, Geoinformatik och Geodesi.
    Software for generating gravity gradients using a geopotential model based on an irregular semivectorization algorithm2012In: Computers & Geosciences, ISSN 0098-3004, E-ISSN 1873-7803, Vol. 39, p. 152-160Article in journal (Refereed)
    Abstract [en]

    The spherical harmonic synthesis of second-order derivatives of geopotential is a task of major concern when the spatial resolution of synthesis is high and/or a high-resolution Earth's gravity model is used. Here, a computational technique is presented for such a process. The irregular semivectorization is introduced as a vectorization technique in which one loop is excluded from matrix-vector products of mathematical models in order to speed up the computation and manage the computer memory. The proposed technique has the capability of considering heights of computation points on a regular grid. MATLAB-based software is developed, which can be used for generating gravity gradients on an ordinary personal computer. The numerical results show that irregular semivectorization significantly reduces the computation time to 1 h for synthesis of these data with global coverage and resolution of 5′×5′ on an elevation model. In addition, a numerical example is presented for testing satellite gravity gradiometry data of the recent European Space Agency satellite mission, the gravity field and steady-state ocean circulation explorer (GOCE), using an Earth's gravity model.

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