Satellite Gradiometry Data Research Articles

A Recursive Probabilistic Inversion of Satellite Gradiometry Data for the Upper Mantle Density Variation

A Recursive Probabilistic Inversion of Satellite Gradiometry Data for the Upper Mantle Density Variation

Applying an Expansion of the Second Derivatives of the Earth’s Gravitational Potential in Modified Spherical Harmonics for Constructing and Estimating Its Models

Second-order derivatives of the Earth’s potential in a local north-oriented coordinate system are expanded in series of modified spherical harmonics. Linear relations are derived between the spectral coefficients of these series and the spectrum of the geopotential. Based on these relations, recurrent procedures are developed for estimating the geopotential coefficients from the spectrum of each derivative and, conversely, for simulating the spectrum from a known geopotential model. The very simple structure of the expressions for the derivatives is convenient for estimating the coefficients of the geopotential by the least squares method at a certain step of processing satellite gradiometry data. Since the new series are orthogonal, the method with a quadrature formula can be applied, which helps avoid aliasing errors caused by the truncation of the series. The spectral coefficients of the derivatives are estimated using the derived relations for different models on an average orbital sphere of the GOCE satellite and at other altitudes above the Earth’s surface.

Letter from the editor

Letter from the editor

Rosborough approach for the determination of regional time variability of the gravity field from satellite gradiometry data

It is well-known that Rosborough approach, based on spherical harmonics, can be efficiently applied to determine the unknown global gravitational field parameters in the framework of the so-called space-wise approach. Usually, for regional gravity field modeling radial basis functions instead of spherical harmonics are used as basis functions. Using the strong relationship between radial basis functions and spherical harmonics, the Rosborough counterparts of radial basis functions are developed. Based on these regionally supported Rosborough basis functions, all components of gravity-gradient tensor are expressed as series in regionally supported Rosborough basis functions. In a simulation study the feasibility of regional static and time-variable gravity field recovery from satellite gradiometry data using regionally supported Rosborough functions is tested.

Технология определения параметров гравитационного поля Земли по градиентометрическим измерениям. 2. Системы координат и времени в спутниковой градиентометрии

Abstract: This paper is the second of paper series devoted to the technology description of the gravity field determination using satellite gradiometry data. The objective of this paper is the description of reference frames and time systems used for mathematical processing gradiometry data of mission GOCE.

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.

Using modern geopotential models in studying vertical deflections in the Arctic

The paper continues analyzing the results of the mathematical modeling of the Earth’s anomalous gravity field in high-latitude regions which are of interest from the viewpoint of Arctic exploration. The structure and the accuracy characteristics of vertical deflections in the Arctic have been studied. The study was conducted with the use of 16 modern Russian and foreign models of spherical harmonics of the Earth’s gravity potential, including the ultrahigh degree model EGM-2008, the first models based on satellite gradiometry data, and the latest Russian model GAO-2012.

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.

Development of the second-order derivatives of the Earth’s potential in the local north-oriented reference frame in orthogonal series of modified spherical harmonics

The second-order derivatives of the Earth’s potential in the local north-oriented reference frame are expanded in series of modified spherical harmonics. Linear relations are derived between the spectral coefficients of these series and the spectrum of the geopotential. On the basis of these relations, recurrence procedures are developed for evaluating the geopotential coefficients from the spectrum of each derivative and, inversely, for simulating the latter from a known geopotential model. Very simple structure of the derived expressions for the derivatives is convenient for estimating the geopotential coefficients by the least-squares procedure, at a certain step of processing satellite gradiometry data. Due to the orthogonality of the new series, the quadrature formula approach can be also applied, which allows avoidance of aliasing errors caused by the series truncation. The spectral coefficients of the derivatives are evaluated on the basis of the derived relations from the geopotential models EGM96 and EIGEN-CG01C at a mean orbital sphere of the GOCE satellite. Various characteristics of the spectra are studied corresponding to the EGM96 model.

New relations among associated Legendre functions and spherical harmonics

Several new relations among associated Legendre functions (ALFs) are derived, most of which relate a product of an ALF with trigonometric functions to a weighted summation over ALFs, where the weights only depend on the degree and order of the ALF. These relations are, for example, useful in applications such as the computation of geopotential coefficients and computation of ellipsoidal corrections in geoid modelling. The main relations are presented in both their unnormalised and fully normalised (4π-normalised) form. Several approaches to compute the weights involved are discussed, and it is shown that the relations can also be applied in the case of first- and second-order derivatives of ALFs, which may be of use in analysis of satellite gradiometry data. Finally, the derived relations are combined to provide new identities among ALFs, which contain no dependency on the colatitudinal coordinate other than that in the ALFs themselves.

The effect of topography on the determination of the geoid using analytical downward continuation

The method of analytical downward continuation has been used for solving Molodensky’s problem. This method can also be used to reduce the surface free air anomaly to the ellipsoid for the determination of the coefficients of the spherical harmonic expansion of the geopotential. In the reduction of airborne or satellite gradiometry data, if the sea level is chosen as reference surface, we will encounter the problem of the analytical downward continuation of the disturbing potential into the earth, too. The goal of this paper is to find out the topographic effect of solving Stoke’sboundary value problem (determination of the geoid) by using the method of analytical downward continuation.