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  • 1. Abbak, Ramazan A.
    et al.
    Sjöberg, Lars E.
    KTH, School of Architecture and the Built Environment (ABE), Urban Planning and Environment, Geodesy and Geoinformatics.
    Ellmann, Artu
    Ustun, Aydin
    A precise gravimetric geoid model in a mountainous area with scarce gravity data: a case study in central Turkey2012In: Studia Geophysica et Geodaetica, ISSN 0039-3169, E-ISSN 1573-1626, Vol. 56, no 4, p. 909-927Article in journal (Refereed)
    Abstract [en]

    In mountainous regions with scarce gravity data, gravimetric geoid determination is a difficult task that needs special attention to obtain reliable results satisfying the demands, e.g., of engineering applications. The present study investigates a procedure for combining a suitable global geopotential model and available terrestrial data in order to obtain a precise regional geoid model for Konya Closed Basin (KCB). The KCB is located in the central part of Turkey, where a very limited amount of terrestrial gravity data is available. Various data sources, such as the Turkish digital elevation model with 3 '' x 3 '' resolution, a recently published satellite-only global geopotential model from the Gravity Recovery and Climate Experiment satellite (GRACE) and the ground gravity observations, are combined in the least-squares sense by the modified Stokes' formula. The new gravimetric geoid model is compared with Global Positioning System (GPS)/levelling at the control points, resulting in the Root Mean Square Error (RMS) differences of +/- 6.4 cm and 1.7 ppm in the absolute and relative senses, respectively. This regional geoid model appears to he more accurate than the Earth Gravitational Model 2008, which is the best global model over the target area, with the RMS differences of +/- 8.6 cm and 1.8 ppm in the absolute and relative senses, respectively. These results show that the accuracy of a regional gravimetric model can be augmented by the combination of a global geopotential model and local terrestrial data in mountainous areas even though the quality and resolution of the primary terrestrial data are not satisfactory to the geoid modelling procedure.

  • 2.
    Bagherbandi, Mohammad
    et al.
    KTH, School of Architecture and the Built Environment (ABE), Urban Planning and Environment, Geoinformatik och Geodesi.
    Sjoberg, Lars E.
    KTH, School of Architecture and the Built Environment (ABE), Urban Planning and Environment, Geoinformatik och Geodesi.
    Comparison of crustal thickness from two gravimetric-isostatic models and CRUST2.02011In: Studia Geophysica et Geodaetica, ISSN 0039-3169, E-ISSN 1573-1626, Vol. 55, no 4, p. 641-666Article in journal (Refereed)
    Abstract [en]

    The MohoroviiA double dagger discontinuity is the boundary between the Earth's crust and mantle. Several isostatic hypotheses exist for estimating the crustal thickness and density variation of the Earth's crust from gravity anomalies. The goal of this article is to compare the Airy-Heiskanen and Vening Meinesz-Moritz (VMM) gravimetric models for determining Moho depth, with the seismic Moho (CRUST2.0 or SM) model. Numerical comparisons are performed globally as well as for some geophysically interesting areas, such as Fennoscandia, Persia, Tibet, Canada and Chile. These areas are most complicated areas in view of rough topography (Tibet, Persia and Peru and Chile), post-glacial rebound (Fennoscandia and Canada) and tectonic activities (Persia). The mean Moho depth provided by CRUST2.0 is 22.9 +/- 0.1 km. Using a constant Moho density contrast of 0.6 g/cm(3), the corresponding mean values for Airy-Heiskanen and VVM isostatic models become 25.0 +/- 0.04 km and 21.6 +/- 0.08 km, respectively. By assuming density contrasts of 0.5 g/cm(2) and 0.35 g/cm(3) for continental and oceanic regions, respectively, the VMM model yields the mean Moho depth 22.6 +/- 0.1 km. For this model the global rms difference to CRUST2.0 is 7.2 km, while the corresponding difference between Airy-Heiskanen model and CRUST2.0 is 11 km. Also for regional studies, Moho depths were estimated by selecting different density contrasts. Therefore, one conclusion from the study is that the global compensation by the VMM method significantly improves the agreement with the CRUST2.0 vs. the local compensation model of Airy-Heiskanen. Also, the last model cannot be correct in regions with ocean depth larger than 9 km (e.g., outside Chile), as it may yield negative Moho depths. This problem does not occur with the VMM model. A second conclusion is that a realistic variation of density contrast between continental and oceanic areas yields a better fit of the VMM model to CRUST2.0. The study suggests that the VMM model can primarily be used to densify the CRUST2.0 Moho model in many regions based on separate data by taking advantage of dense gravity data. Finally we have found also that the gravimetric terrain correction affects the determination of the Moho depth by less than 2 km in mean values for test regions, approximately. Hence, for most practical applications of the VMM model the simple Bouguer gravity anomaly is sufficient.

  • 3.
    Bagherbandi, Mohammad
    et al.
    KTH, School of Architecture and the Built Environment (ABE), Urban Planning and Environment, Geodesy and Geoinformatics.
    Sjöberg, Lars E.
    University of Gävle, Sweden.
    A synthetic Earth gravity model based on a topographic-isostatic model2012In: Studia Geophysica et Geodaetica, ISSN 0039-3169, E-ISSN 1573-1626, Vol. 56, no 4, p. 935-955Article in journal (Refereed)
    Abstract [en]

    The Earth's gravity field is related to the topographic potential in medium and higher degrees, which is isostatically compensated. Hence, the topographic-isostatic (TI) data are indispensable for extending an available Earth Gravitational Model (EGM) to higher degrees. Here we use TI harmonic coefficients to construct a Synthetic Earth Gravitational Model (SEGM) to extend the EGMs to higher degrees. To achieve a high-quality SEGM, a global geopotential model (EGM96) is used to describe the low degrees, whereas the medium and high degrees are obtained from the TI or topographic potential. This study differes from others in that it uses a new gravimetric-isostatic model for determining the TI potential. We test different alternatives based on TI or only topographic data to determine the SEGM. Although the topography is isostatically compensated only to about degree 40-60, our study shows that using a compensation model improves the SEGM in comparison with using only topographic data for higher degree harmonics. This is because the TI data better adjust the applied Butterworth filter, which bridges the known EGM and the new high-degree potential field than the topographic data alone.

  • 4.
    Bagherbandi, Mohammad
    et al.
    KTH, School of Architecture and the Built Environment (ABE), Urban Planning and Environment, Geodesy and Geoinformatics.
    Tenzer, Robert
    Sjöberg, Lars E.
    KTH, School of Architecture and the Built Environment (ABE), Urban Planning and Environment, Geodesy and Geoinformatics.
    Moho depth uncertainties in the Vening-Meinesz Moritz inverse problem of isostasy2014In: Studia Geophysica et Geodaetica, ISSN 0039-3169, E-ISSN 1573-1626, Vol. 58, no 2, p. 227-248Article in journal (Refereed)
    Abstract [en]

    We formulate an error propagation model based on solving the Vening Meinesz-Moritz (VMM) inverse problem of isostasy. The system of observation equations in the VMM model defines the relation between the isostatic gravity data and the Moho depth by means of a second-order Fredholm integral equation of the first kind. The corresponding error model (derived in a spectral domain) functionally relates the Moho depth errors with the commission errors of used gravity and topographic/bathymetric models. The error model also incorporates the non-isostatic bias which describes the disagreement, mainly of systematic nature, between the isostatic and seismic models. The error analysis is conducted at the study area of the Tibetan Plateau and Himalayas with the world largest crustal thickness. The Moho depth uncertainties due to errors of the currently available global gravity and topographic models are estimated to be typically up to 1-2 km, provided that the GOCE gravity gradient observables improved the medium-wavelength gravity spectra. The errors due to disregarding sedimentary basins can locally exceed similar to 2 km. The largest errors (which cause a systematic bias between isostatic and seismic models) are attributed to unmodeled mantle heterogeneities (including the core-mantle boundary) and other geophysical processes. These errors are mostly less than 2 km under significant orogens (Himalayas, Ural), but can reach up to similar to 10 km under the oceanic crust.

  • 5.
    Eshagh, Mehdi
    et al.
    KTH, School of Architecture and the Built Environment (ABE), Transport and Economics, Geodesy.
    Sjöberg, Lars E.
    KTH, School of Architecture and the Built Environment (ABE), Transport and Economics, Geodesy.
    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.

  • 6.
    Eshagh, Mehdi
    et al.
    KTH, School of Architecture and the Built Environment (ABE), Transport and Economics (closed 20110301), Geodesy (closed 20110301).
    Sjöberg, Lars E.
    KTH, School of Architecture and the Built Environment (ABE), Transport and Economics (closed 20110301), Geodesy (closed 20110301).
    Topographic and atmospheric effects on goce gradiometric data in a local north-oriented frame: A case study in Fennoscandia and Iran2009In: Studia Geophysica et Geodaetica, ISSN 0039-3169, E-ISSN 1573-1626, Vol. 53, no 1, p. 61-80Article in journal (Refereed)
    Abstract [en]

    Satellite gradiometry is an observation technique providing data that allow for evaluation of Stokes' (geopotential) coefficients. This technique is capable of determining higher degrees/orders of the geopotential coefficients than can be achieved by traditional dynamic satellite geodesy. The satellite gradiometry data include topographic and atmospheric effects. By removing those effects, the satellite data becomes smoother and harmonic outside sea level and therefore more suitable for downward continuation to the Earth's surface. For example, in this way one may determine a set of spherical harmonics of the gravity field that is harmonic in the exterior to sea level. This article deals with the above effects on the satellite gravity gradients in the local north-oriented frame. The conventional expressions of the gradients in this frame have a rather complicated form, depending on the first-and second-order derivatives of the associated Legendre functions, which contain singular factors when approaching the poles. On the contrary, we express the harmonic series of atmospheric and topographic effects as non-singular expressions. The theory is applied to the regions of Fennoscandia and Iran, where maps of such effects and their statistics are presented and discussed.

  • 7.
    Sjöberg, Lars E.
    KTH, School of Architecture and the Built Environment (ABE), Transport and Economics, Geodesy.
    A local least-squares modification of Stokes' formula2005In: Studia Geophysica et Geodaetica, ISSN 0039-3169, E-ISSN 1573-1626, Vol. 49, no 1, p. 23-30Article in journal (Refereed)
    Abstract [en]

    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.

  • 8.
    Sjöberg, Lars E.
    KTH, School of Architecture and the Built Environment (ABE), Transport and Economics, Geodesy.
    A refined conversion from normal height to orthometric height2006In: Studia Geophysica et Geodaetica, ISSN 0039-3169, E-ISSN 1573-1626, Vol. 50, no 4, p. 595-606Article in journal (Refereed)
    Abstract [en]

    The difference between orthometric and normal heights (or the height anomaly and the geoid height) is usually approximated by a term consisting of the Bouguer anomaly times elevation divided by normal gravity. We derive an improved formula, which includes a topographic roughness term (terrain correction) and a term due to the lateral variation of topographic density, for the practical application of this conversion. It is shown that for high mountainous areas with rough topography these two terms are of the same order as the Bouguer anomaly related term. Already for elevations of a few hundred metres they could reach the order of a centimetre. In addition, for the more precise computations in high mountainous areas, a term related with the downward continuation of topographic potential from the surface to sea level could be significant.

  • 9.
    Sjöberg, Lars E.
    KTH, School of Architecture and the Built Environment (ABE), Urban Planning and Environment, Geoinformatik och Geodesi.
    Geoid determination by spectral combination of an Earth gravitational model with airborne and terrestrial gravimetry - a theoretical study2011In: Studia Geophysica et Geodaetica, ISSN 0039-3169, E-ISSN 1573-1626, Vol. 55, no 4, p. 579-588Article in journal (Refereed)
    Abstract [en]

    Today air-gravimetry is a versatile technique to quickly collect gravity data over large regions, where terrestrial gravity data are sparse and/or of poor quality. The method requires the data to be downward continued to sea level for use in geoid determination, an inverse problem operation that calls for smoothing of the data and/or the kernel function involved (in either spectral or space domain). In this purely theoretical study we avoid this separate computational step by performing direct geoid estimation by so-called spectral combination/filtering of the data, which includes terrestrial gravimetry, airgravimetry, an Earth Gravitational Model (EGM) as well as their signal and error degree variances. Each derived geoid estimator is presented as the sum of one or two integral formulas and the harmonic series of the EGM together with the expected mean square error of the estimator. The article is limited to a theoretical study, leaving its practical tests for future investigation.

  • 10.
    Sjöberg, Lars E.
    KTH, School of Architecture and the Built Environment (ABE), Urban Planning and Environment, Geodesy and Geoinformatics.
    The geoid-to-quasigeoid difference using an arbitrary gravity reduction model2012In: Studia Geophysica et Geodaetica, ISSN 0039-3169, E-ISSN 1573-1626, Vol. 56, no 4, p. 929-933Article in journal (Refereed)
    Abstract [en]

    Recently it was proved that the classical formula for computing the geoid to quasigeoid separation (GQS) by the Bouguer gravity anomaly needs a topographic correction. Here we generalize the modelling of the GQS not only to Bouguer types of anomalies, hut also to arbitrary reductions of topographic gravity. Of particular interest for practical applications should be isostatic and Helmert types of reductions, which provide smaller and smoother components, more suitable for interpolation and calculation, than the Bouguer reduction.

  • 11.
    Sjöberg, Lars E.
    KTH, School of Architecture and the Built Environment (ABE), Urban Planning and Environment, Geodesy and Satellite Positioning.
    The secondary indirect topographic effect in physical geodesy2015In: Studia Geophysica et Geodaetica, ISSN 0039-3169, E-ISSN 1573-1626, Vol. 59, no 2, p. 173-187Article in journal (Refereed)
    Abstract [en]

    The use of Stokes’ integral by the remove-compute-restore technique is the most common way to determine the geoid today. The method includes direct, primary and secondary indirect topographic effects. This article is mainly devoted to the secondary indirect topographic effect (SITE), which reaches the extreme values of 265 mGal and -0.6 mGal for the uncompensated and Helmert condensation compensated gravity anomalies, respectively. The corresponding effects on the geoid height reach the magnitudes of 328 m and - 0.5 m, respectively. Here we emphasize that the SITE is a direct effect, needed in a rigorous gravity anomaly. For surface as well as for classical gravity anomalies, located at the geoid, the SITE can be interpreted as a shift in the normal gravity along the ellipsoidal normal to the point where the normal potential equals the topographically reduced geopotential at the computation point. We show that it may yield a bias of the order of - 0.9 m in the Himalayas if not properly considered in the surface anomaly. This bias does not change when using a topographic compensation model, e.g., by Helmert condensation of the topography. The problem is avoided when using the no-topography gravity anomaly with or without compensation.

  • 12.
    Sjöberg, Lars Erik
    et al.
    KTH, School of Architecture and the Built Environment (ABE), Transport and Economics, Geodesy.
    Eshagh, Mehdi
    KTH, School of Architecture and the Built Environment (ABE), Transport and Economics, Geodesy.
    A geoid solution for airborne gravity data2009In: Studia Geophysica et Geodaetica, ISSN 0039-3169, E-ISSN 1573-1626, Vol. 53, no 3, p. 359-374Article in journal (Refereed)
    Abstract [en]

    Airborne gravity data is usually attached with satellite positioning of data points, which allow for the direct determination of the gravity disturbance at flight level. Assuming a suitable gridding of such data, Hotine's modified integral formula can be combined with an Earth Gravity Model for the computation of the disturbing potential (T) at flight level. Based on T and the gravity disturbance data, we directly downward continue T to the geoid, and we present the final solution for the geoid height, including topographic corrections. It can be proved that the Taylor expansion of T converges if the flight level is at least twice the height of the topography, and the terrain potential will not contribute to the topographic correction. Hence, the simple topographic bias of the Bouguer shell yields the only topographic correction. Some numerical results demonstrate the technique used for downward continuation and topographic correction.

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