Geoid modelling based on EGM08 and recent Earth gravity models of GOCE

In this paper an estimator for geoid is presented and applied for geoid computation which considers the topographic and atmospheric effects on the geoid. The total atmospheric effect is mathematically developed in terms of spherical harmonics to degree and order 2,160 based on a recent static atmospheric density model. Also the contribution of its higher degrees is formulated. Another idea of this paper is to combine one of the recent Earth gravity models (EGMs) of the Gravity field and steady-state Ocean Circulation Explorer (GOCE) mission with EGM08 and the terrestrial gravimetric data of Fennoscandia in an optimum way. To do so, the GOCE EGMs are compared with the Global Positioning System (GPS)/levelling data over the area for finding the most suited one. This comparison is done in two different ways: with and without considering the errors of the EGMs. Comparison of the computed geoids with the GPS/levelling data shows that a) considering the total atmospheric effect will improve the geoid by about 5 mm, b) GOCO03S is the most suited GOCE EGM for Fennoscandia, c) the errors of some of the GOCE EGMs are optimistic and far from reality. Combination of GOCO03S from degree 120 to 210 and EGM08 for the rest of degrees shows its good quality in these frequencies.     

Alternative expressions for gravity gradients in local north-oriented frame and tensor spherical harmonics

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 north-oriented 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.     

Local Recovery of Sub-Crustal Stress Due to Mantle Convection from Satellite-to-Satellite Tracking Data

Two integral transformations between the stress function, differentiation of which gives the meridian and prime vertical components of the sub-crustal stress due to mantle convection, and the satellite-to-satellite tracking (SST) data are presented in this article. In the first one, the SST data are the disturbing potential differences between twin-satellites and in the second one the line-of-sight (LOS) gravity disturbances. It is shown that the corresponding integral kernels are well-behaving and therefore suitable for inversion and recovery of the stress function from the SST data. Recovery of the stress function and the stress components is also tested in numerical experiments using simulated SST data. Numerical studies over the Himalayas show that inverting the disturbing potential differences leads to a smoother stress function than from inverting LOS gravity disturbances. Application of the presented integral formulae allows for recovery of the stress from the satellite mission GRACE and its planned successor.     

From Satellite Gradiometry Data to Subcrustal Stress Due to Mantle Convection

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

Topographic and atmospheric effects on goce gradiometric data in a local north-oriented frame: A case study in Fennoscandia and Iran

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.     

The Modified Best Quadratic Unbiased Non-Negative Estimator (MBQUNE) of variance components

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.     

Comparison of Different Methods for a Moho Modeling Under Oceans and Marginal Seas: A Case Study for the Indian Ocean

Surveys in Geophysics - Since marine seismic studies are relatively sparse and unevenly distributed, detailed tomographic images of the Moho geometry under large parts of the world’s oceans...     

EGMlab, a scientific software for determining the gravity and gradient components from global geopotential models

Nowadays, Global Geopotential Models (GGMs) are used as a routine stage in the procedures to compute a gravimetric geoid. The GGMs based geoidal height also can be used for GPS/levelling and navigation purposes in developing countries which do not have accurate gravimetric geoid models. Also, the GGM based gravity anomaly including the digital elevation model can be used in evaluation and outlier detections schemes of the ground gravity anomaly data. Further, the deflection of vertical and gravity gradients components from the GGMs can be used for different geodetic and geophysical interpretation purposes. However, still a complete and user-friendly software package is not available for universities and geosciences communities. In this article, first we review the procedure for determination of the basic gravity field and gradient components from the GGMs, then general MATLAB based software is presented and applied as a sample case study for determination of gravity components based on the most recent EIGEN-GL04C GRACE model in Sweden. Electronic supplementary material  The online version of this article (doi:) contains supplementary material, which is available to authorized users.     

Comparison of two approaches for considering laterally varying density in topographic effect on satellite gravity gradiometric data

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 Sjöberg as well as Novák and Grafarend to consider the laterally varying density for topographic masses in formulation of topographic potential in spherical harmonics.