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Lars E Sjoberg - One of the best experts on this subject based on the ideXlab platform.
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the topographic bias in Stokes Formula vs the error of analytical continuation by an earth gravitational model are they the same
Journal of Geodetic Science, 2015Co-Authors: Lars E SjobergAbstract:Geoid determination below the topographic surface in continental areas using analytical continuation of gravity anomaly and/or an external type of solid spherical harmonics determined by an Earth G ...
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Computation of the Gravimetric Quasigeoid Model over Uganda Using the KTH Method
2015Co-Authors: Ronald Ssengendo, Lars E Sjoberg, Anthony GiduduAbstract:The gravimetric quasigeoid can be determined either directly by Stokes Formula or indirectly by computing the geoid first and then determining the quasigeoid-to-geoid separation which is then used ...
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on the topographic effects by Stokes Formula
Journal of Geodetic Science, 2014Co-Authors: Lars E SjobergAbstract:Traditional gravimetric geoid determination re- lies on Stokes' Formula with removal and restoration of the topographic effects. It is shown that this solution is in error of the order of the quasigeoid-to-geoid difference, which is mainly due to incomplete downward continuation (dwc) of gravity from the Earth's surface to the geoid. A slightly improved estimator, based on the surface Bouguer gravity anomaly, is also biased due to the imperfect harmonic dwc the Bouguer anomaly. Only the third estimator, which uses the (harmonic) surface no-topography gravity anomaly, is consistent with the boundary condition and Stokes' for- mula, providing a theoretically correct geoid height. The difference between the Bouguer and no-topography gravity anomalies (on the geoid or in space) is the "sec- ondary indirect topographic effect", which is a necessary correction in removing all topographic signals.
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The new gravimetric quasigeoid model KTH08 over Sweden
Journal of Applied Geodesy, 2012Co-Authors: Jonas Agren, Lars E Sjoberg, Ramin KiamehrAbstract:The least squares modification of Stokes Formula has been developed in a series of papers published in Journal of Geodesy between 1984 and 2008. It consists of a least squares (stochastic) Stokes k ...
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comparison of remove compute restore and least squares modification of Stokes Formula techniques to quasi geoid determination over the auvergne test area
Journal of Geodetic Science, 2012Co-Authors: Hasan Yildiz, Jonas Agren, Rene Forsberg, C C Tscherning, Lars E SjobergAbstract:The remove-compute-restore (RCR) technique for regional geoid determination implies that both topography and low-degree global geopotential model signals are removed before computation and restored after Stokes' integration or Least Squares Collocation (LSC) solution. The Least Squares Modification of Stokes' Formula (LSMS) technique not requiring gravity reductions is implemented here with a Residual Terrain Modelling based interpolation of gravity data. The 2-D Spherical Fast Fourier Transform (FFT) and the LSC methods applying the RCR technique and the LSMS method are tested over the Auvergne test area. All methods showed a reasonable agreement with GPS-levelling data, in the order of a 3-3.5 cm in the central region having relatively smooth topography, which is consistent with the accuracies of GPS and levelling. When a 1-parameter fit is used, the FFT method using kernel modification performs best with 3.0 cm r.m.s difference with GPS-levelling while the LSMS method gives the best agreement with GPS-levelling with 2.4 cm r.m.s after a 4-parameter fit is used. However, the quasi-geoid models derived using two techniques differed from each other up to 33 cm in the high mountains near the Alps. Comparison of quasi-geoid models with EGM2008 showed that the LSMS method agreed best in term of r.m.s.
Artu Ellmann - One of the best experts on this subject based on the ideXlab platform.
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two deterministic and three stochastic modifications of Stokes s Formula a case study for the baltic countries
Journal of Geodesy, 2005Co-Authors: Artu EllmannAbstract:In regional gravimetric geoid determination, it is customary to use the modified Stokes Formula that combines local terrestrial data with a global geopotential model. This study compares two deterministic and three stochastic modification methods for computing a regional geoid over the Baltic countries. The final selection of the best modification method is made by means of two accuracy estimates: the expected global mean square error of the geoid estimator, and the statistics of the post-fit residuals between the computed geoid models and precise GPS-levelling data. Numerical results show that the modification methods tested do not provide substantially different results, although the stochastic approaches appear formally better in the selected study area. The 2.8-5.3 cm (RMS) post-fit residuals to the GPS-levelling points indicate the suitability of the new geoid model for many practical applications. Moreover, the numerical comparisons reveal a one-dimensional offset between the regional vertical datum and the geoid models based upon the new GRACE-only geopotential model GGM01s. This gives an impression of a greater reliability of the new model compared to the earlier, EGM96-based and somewhat tilted regional geoid models for the same study area.
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a numerical comparison of different ellipsoidal corrections to Stokes Formula
2005Co-Authors: Artu EllmannAbstract:This paper reviews the new method (recently derived by L.E. Sjoberg) of the ellipsoidal correction for Stokes’ Formula. Importantly, the correction can also be expressed in a series of spherical harmonics of the disturbing potential, which is a considerable computational advantage. In order to assess the applicability of this approach, it is numerically compared with two other methods. The results reveal that the new method practically coincides with one of the earlier methods, although their derivational approaches and resulting expressions are considerably different. The new method is adapted for the modified Stokes Formula, which combines regional terrestrial gravity data with a global geopotential model. The magnitude of the ellipsoidal correction in the modified Stokes Formula does not exceed cm level, globally.
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on the numerical solution of parameters of the least squares modification of Stokes Formula
General Assembly of the International-Association-of-Geodesy Location: Sapporo JAPAN Date: JUN 30-JUL 11 2003, 2005Co-Authors: Artu EllmannAbstract:In regional gravimetric geoid determination, it has become customary to utilize the modified Stokes Formula, which combines local terrestrial data with a global geopotential model (GGM). A modification method, proposed by L.E. Sjoberg in 1984 (with later developments), allows least squares minimization of the influence of any error source in geoid modelling. In this approach, depending on the local gravity data quality, the chosen radius of integration, and the characteristics of the used GGM, the modification parameters s n , vary. New satellite gravity missions are expected to improve significantly the accuracy of geopotential models. Of particular interest of this study is to evaluate the impact of future (i.e. post-GOCE) geopotential coefficients. A set of least squares modification parameters is determined from the system of linear equations, aiming at minimizing the global mean square error of geoid estimator. Some difficulties may be encountered when practically computing the modification parameters. In particular, for certain parameters the design matrix suffers from numerical ill-conditioning. Importantly, Tikhonov regularization is satisfactory in providing a solution for the modification parameters. Numerical results are presented to illustrate the applicability of the obtained parameters in geoid modelling by comparing with GPS-levelling data.
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ellipsoidal correction for the modified Stokes Formula
Bollettino di geodesia e scienze affini, 2004Co-Authors: Artu Ellmann, Lars E SjobergAbstract:In regional gravimetric geoid determination it has become customary to utilize the modified Stokes Formula, which combines local terrestrial data with a global geopotential model. A correction to the geoidal height for the Earth's ellipticity is often assumed to be negligible in practical computations. In this paper the ellipsoidal correction for the modified Stokes Formula is reviewed. In particular, the approaches utilizing the modified Stokes function are considered. Approximate Formulas for the ellipsoidal correction are developed and numerically verified over an area, where the largest correction values occur. The ellipsoidal correction is depending on the variations in modification parameters and the degree of modification, although the maximum range of correction remains within a few centimetres. We also attempt to estimate the omission error committed by the truncation of the harmonic series at some limits of contemporary geopotential models.
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the geoid for the baltic countries determined by the least squares modification of Stokes Formula
2004Co-Authors: Artu EllmannAbstract:Precise knowledge of the geoid contributes to the studies ofthe Earths interior, the long-term geophysical processesand to oceanography. An accurate regional geoid model, inparticular, enables the ...
L.e. Sjöberg - One of the best experts on this subject based on the ideXlab platform.
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Considering data gaps in geoid modelling by modifying Stokes’ Formula
Acta Geodaetica et Geophysica Hungarica, 2010Co-Authors: L.e. Sjöberg, M. EshaghAbstract:There are numerous methods to modify Stokes’ Formula with the usually common feature of reducing the truncation error committed by the lack of gravity data in the far-zone, resulting in an integral Formula over the near-zone combined with an Earth Gravity Model that mainly contributes with the long-wavelength information. Here we study the reverse problem, namely to estimate the geoid height with data missing in a cap around the computation point but available in the far-zone outside the cap. Secondly, we study also the problem with gravity data available only in a spherical ring around the computation point. In both cases the modified Stokes Formulas are derived using Molodensky and least squares types of solutions. The numerical studies show that the Molodensky type of modification is useless, while the latter method efficiently depresses the various errors contributing to the geoid error. The least squares methods can be used for estimating geoid heights in regions with gravity data gaps, such as in Polar Regions, over great lakes and in some developing countries with lacking gravity data.
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An optimum way to determine a precise gravimetric geoid model based on the least-squares modification of Stokes’ Formula — A case study of Sweden
Acta Geodaetica et Geophysica Hungarica, 2010Co-Authors: R. Kiamehr, L.e. SjöbergAbstract:The modification of Stokes’ Formula allows the user to compensate the lack of a global coverage of gravity data by a combination of terrestrial gravity and a global geopotential model. The minimization of the errors of truncation gravity data and potential coefficients could be treated in a least-squares sense as is the basic ingredient in the Royal Institute of Technology (KTH) approach as proposed by Sjöberg in 1984. This article presents the results from a joint project between KTH and the National Land Survey of Sweden, whose main purpose is to evaluate the KTH approach numerically and to compute a gravimetric geoid model for Sweden. The new geoid model (KTH06) was computed based on the least-squares modification of Stokes’ Formula, the GRACE global geopotential model, a high-resolution digital terrain model and the NKG gravity anomaly database. The KTH06 was fitted to 1162 GPS/levelling points by a 7-parameter transformation, yielding an all-over fit of 19 mm and 0.17 ppm. The fit is even smaller than the estimated internal accuracy for the geoid model (28 mm). If we assume that the accuracy of the GPS and levelling heights are 10 mm and 5 mm, respectively, it follows that the accuracy of the expected gravimetric geoid heights are of the order of 11 mm. Also, we found a significant expected difference between the KTH06 and NKG2004 models in rough topographic areas (up to 36 cm). As the major ground data and global geopotential model were almost same in the two models, we believe that there are different reasons that come into play for interpreting the discrepancies between them, as the method for eliminating outliers from the gravity database, the interpolated denser gravity observations using the high-resolution digital elevation model before Stokes’ integration, the potential of the LSM kernel, which matches the errors of the terrestrial gravity data, GGM and the truncation error in an optimum way, and the effect of applying more precise correction terms in the KTH approach compared to the remove-compute-restore method. It is concluded that the least-squares modification method with additive corrections is a very promising alternative for geoid computation.
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A Local Least-Squares Modification of Stokes’ Formula
Studia Geophysica et Geodaetica, 2005Co-Authors: L.e. SjöbergAbstract:The combination of Stokes’ Formula and an Earth Gravity Model (EGM) for geoid determination has become a standard procedure. However, the way of modifying Stokes’ Formula vary from author to author, and numerous methods of modification exist. Most methods are deterministic, with the primary goal of reducing the truncation bias committed by limiting the area of Stokes’ integration around the computation point, but there are also some stochastic methods with the explicit goal to reduce the global mean square error of the geoid height estimator stemming from the truncation bias as well as the random errors of the EGM and the gravity data. The latter estimators are thus, at least from a theoretical point of view, optimal in a global mean sense, but in a local sense they may be far from optimality. Here we take advantage of the error variance-covariance matrices of the EGM and the terrestrial gravity data to derive the modification parameters of Stokes’ kernel in a local least-squares sense. The solution is given for the unbiased type of modification of Stokes’ Formula of Sjöberg (1991).
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A spherical harmonic representation of the ellipsoidal correction to the modified Stokes Formula
Journal of Geodesy, 2004Co-Authors: L.e. SjöbergAbstract:Ellipsoidal corrections to order e ^2 of the original and the modified Stokes Formulas are presented in series of spherical harmonics. Numerical tests show that the ellipsoidal effect can reach several decimetres for an integration cap size of 10^∘ in Stokes’ Formula.
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Ellipsoidal corrections to order e ^2 of geopotential coefficients and Stokes’ Formula
Journal of Geodesy, 2003Co-Authors: L.e. SjöbergAbstract:Assuming that the gravity anomaly and disturbing potential are given on a reference ellipsoid, the result of Sjöberg (1988, Bull Geod 62:93–101) is applied to derive the potential coefficients on the bounding sphere of the ellipsoid to order e ^2 (i.e. the square of the eccentricity of the ellipsoid). By adding the potential coefficients and continuing the potential downward to the reference ellipsoid, the spherical Stokes Formula and its ellipsoidal correction are obtained. The correction is presented in terms of an integral over the unit sphere with the spherical approximation of geoidal height as the argument and only three well-known kernel functions, namely those of Stokes, Vening-Meinesz and the inverse Stokes, lending the correction to practical computations. Finally, the ellipsoidal correction is presented also in terms of spherical harmonic functions. The frequently applied and sometimes questioned approximation of the constant m , a convenient abbreviation in normal gravity field representations, by e ^2/2, as introduced by Moritz, is also discussed. It is concluded that this approximation does not significantly affect the ellipsoidal corrections to potential coefficients and Stokes’ Formula. However, whether this standard approach to correct the gravity anomaly agrees with the pure ellipsoidal solution to Stokes’ Formula is still an open question.
Mehdi Eshagh - One of the best experts on this subject based on the ideXlab platform.
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On quality of NKG2015 geoid model over the Nordic countries
Journal of Geodetic Science, 2020Co-Authors: Mehdi Eshagh, Jenny BerntssonAbstract:The NKG2015 geoid model covers the Nordic and Baltic countries and has been computed based on the least-squares modification of Stokes’ Formula with additive corrections method. New and precise ter ...
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least squares modification of extended Stokes Formula and its second order radial derivative for validation of satellite gravity gradiometry data
Journal of Geodynamics, 2010Co-Authors: Mehdi EshaghAbstract:Abstract 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.
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least squares modification of Stokes Formula with egm08
Geodesy and Cartography, 2009Co-Authors: Mehdi EshaghAbstract:Abstract Least‐squares modifcation is an optimal method of modifying Stokes’ Formula. Tis 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 modifcation 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 defnitely plays an important role. Tis 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 modifcation by about 10 cm comparing to the traditional way.
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towards validation of satellite gradiometric data using modified version of 2nd order partial derivatives of extended Stokes Formula
Artificial Satellites, 2009Co-Authors: Mehdi EshaghAbstract: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.
Ramin Kiamehr - One of the best experts on this subject based on the ideXlab platform.
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The new gravimetric quasigeoid model KTH08 over Sweden
Journal of Applied Geodesy, 2012Co-Authors: Jonas Agren, Lars E Sjoberg, Ramin KiamehrAbstract:The least squares modification of Stokes Formula has been developed in a series of papers published in Journal of Geodesy between 1984 and 2008. It consists of a least squares (stochastic) Stokes k ...
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an optimum way to determine a precise gravimetric geoid model based on the least squares modification of Stokes Formula a case study of sweden
Acta Geodaetica Et Geophysica Hungarica, 2010Co-Authors: Ramin Kiamehr, Lars E SjobergAbstract:The modification of Stokes’ Formula allows the user to compensate the lack of a global coverage of gravity data by a combination of terrestrial gravity and a global geopotential model. The minimization of the errors of truncation gravity data and potential coefficients could be treated in a least-squares sense as is the basic ingredient in the Royal Institute of Technology (KTH) approach as proposed by Sjoberg in 1984. This article presents the results from a joint project between KTH and the National Land Survey of Sweden, whose main purpose is to evaluate the KTH approach numerically and to compute a gravimetric geoid model for Sweden. The new geoid model (KTH06) was computed based on the least-squares modification of Stokes’ Formula, the GRACE global geopotential model, a high-resolution digital terrain model and the NKG gravity anomaly database. The KTH06 was fitted to 1162 GPS/levelling points by a 7-parameter transformation, yielding an all-over fit of 19 mm and 0.17 ppm. The fit is even smaller than the estimated internal accuracy for the geoid model (28 mm). If we assume that the accuracy of the GPS and levelling heights are 10 mm and 5 mm, respectively, it follows that the accuracy of the expected gravimetric geoid heights are of the order of 11 mm. Also, we found a significant expected difference between the KTH06 and NKG2004 models in rough topographic areas (up to 36 cm). As the major ground data and global geopotential model were almost same in the two models, we believe that there are different reasons that come into play for interpreting the discrepancies between them, as the method for eliminating outliers from the gravity database, the interpolated denser gravity observations using the high-resolution digital elevation model before Stokes’ integration, the potential of the LSM kernel, which matches the errors of the terrestrial gravity data, GGM and the truncation error in an optimum way, and the effect of applying more precise correction terms in the KTH approach compared to the remove-compute-restore method. It is concluded that the least-squares modification method with additive corrections is a very promising alternative for geoid computation.
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least squares modification of Stokes Formula vs remove compute restore technique
The EGU General Assembly, 2009Co-Authors: Lars E Sjoberg, Ramin Kiamehr, Jonas AgrenAbstract:Today's applications of Stokes' Formula combine the classical Formula with an Earth Gravity model (EGM). In the remove-compute-restore technique this is performed by removing the EGM from the gravi ...
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The impact of lateral density variation model in the determination of precise gravimetric geoid in mountainous areas: a case study of Iran
Geophysical Journal International, 2006Co-Authors: Ramin KiamehrAbstract:The existence of topography above the geoid violates the basic assumption of Stokes' Formula for the determination of the geoid. Usually a constant density of 2.67 g cm(-3) is used in the determi ...
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hybrid precise gravimetric geoid model for iran based on recent grace and srtm data and the least squares modification of Stokes Formula
J. Physics of Earth and Space, 2006Co-Authors: Ramin KiamehrAbstract:Since 1986, several gravimetric geoid models have been published in the Iran region. It was found thatthe standard deviation of fitting between these models versus GPS/levelling data in most cases ...