We present a spectral–finite element approach to the forward problem of 3-Dglobal-scale electromagnetic induction in a heterogeneous conducting sphere excited by an external source current. It ...represents an alternative to a variety of numerical methods for 3-D global-scale electromagnetic induction modelling developed recently (the perturbation expansion approach and the finite element and finite difference schemes). Two possible formulations of electromagnetic induction boundary-value problem are introduced. The boundary data used in the Dirichlet boundary-value problem consist of the horizontal components of the total magnetic induction measured on the Earth's surface, whereas the mixed boundary-value problem makes use of the scalar spherical harmonic expansion coefficients of the normal component of total magnetic induction in a near-space atmosphere. The latter problem is then reformulated in a weak sense and parametrized by vector spherical harmonics in the angular direction, whereas piecewise linear finite elements span the radial direction. The solution is searched for using the Galerkin method, which leads to solving a system of linear algebraic equations. We employ the biconjugate gradient method with preconditioning to solve the Galerkin system numerically. Particular care is devoted to the construction of a preconditioner that stabilizes the solution and speeds up the convergence of iterations. The spectral–finite element method and associated numerical code have been tested for 2-D (azimuthally symmetric) and 3-D (off-axis) eccentrically nested spheres models, and good agreement has been obtained.
Wiener optimal filtering of GRACE data Sasgen, I; Martinec, Z; Fleming, K
Studia geophysica et geodaetica,
10/2006, Letnik:
50, Številka:
4
Journal Article
Recenzirano
We present a spatial averaging method for Gravity Recovery and Climate Experiment (GRACE) gravity-field solutions based on the Wiener optimal filtering.The optimal filter is designed from the ...least-square minimization of the difference between the desired and filtered signals. It requires information about the power spectra of the desired gravitational signal and the contaminating noise, which is inferred from the average GRACE degree-power spectrum. We show that the signal decreases with increasing spherical harmonic degree j with approximately j−b, where b = 1.5 for GRACE data investigations. This is termed the Second Kaula rule of thumb for temporal variations of the Earth’s gravity field. The degree power of the noise increases, in the logarithmic scale, linearly with increasing j.The Wiener optimal filter obtained for the signal model with b = 1.5 closely corresponds to a Gaussian filter with a spatial half width of 4° (∼440 km). We find that the filtered GRACE gravity signal is relatively insensitive to the exponent b of the signal model, which indicates the robustness of Wiener optimal filtering. This is demonstrated using the GFZ-GRACE gravity-field solution for April 2004.
This study is concerned with the influence of the glacial-isostatic adjustment caused by the last Pleistocene deglaciation on the present-day sea level. The viscoelastic deformation caused by the ...time-variable ice and ocean loads is simulated by computing the resulting perturbations for a spherical, self-gravitating, incompressible, Maxwell-viscoelastic earth model. The associated variation of the earth rotation is described in terms of the Liouville equation, which is solved by means of the MacCullagh formulae. This allows the determination of the vertical displacement and geoid height and, thus, the solution of the sea-level equation. We test several viscosity and ice models and evaluate them by comparison of the computed response with the Holocene relative sea-level record. Using the optimum combination of viscosity and ice models, we then estimate the influence of the last Pleistocene deglaciation on the tide-gauge measurements. A comparison between the observational and residual linear trends for the tide-gauge measurements shows a significant reduction of the variance and geographical variability for the latter, in particular for the formerly ice-covered regions of North America and Scandinavia. The favoured value determined for the global mean sea-level rise is (1.46±0.2) mm a^sup -1^.PUBLICATION ABSTRACT
Predictions are presented of secular changes in the geoid arising from glacial‐isostatic adjustment (GIA) following the Last Glacial Maximum and from present‐day mass changes in the Greenland Ice ...Sheet (GIS). Geoid displacement from ongoing GIA is dominated by ice‐load changes outside of Greenland at lower spherical‐harmonic degrees (<30), and modified at higher degrees by the recent (last few thousand years) GIS history. Ice‐margin mass changes dominate the present‐day GIS geoid response, although comparable signals are obtained when considering the uncertainty range in the higher‐elevation changes (>2000 m). Spatial variability is noted when the present‐day GIS response is expanded to degree and order 32. This is detectable by GRACE when assuming an optimistic accuracy, but is too small by a factor of ca. 3 for an alternate accuracy estimate. Present‐day GIS geoid displacement rates are generally less than the equivalent response from ice‐mass changes in Antarctica, Patagonia and Alaska.
Several alternative gravity forward modelling methodologies and associated numerical codes with their own advantages and limitations are available for the solid Earth community. With upcoming ...state-of-the-art lithosphere density models and accurate global gravity field data sets, it is vital to understand the opportunities and limitations of the various approaches. In this paper, we discuss the four widely used techniques: global spherical harmonics (GSH), tesseroid integration (TESS), triangle integration (TRI), and hexahedral integration (HEX). A constant density shell benchmark shows that all four codes can produce similar precise gravitational potential fields. Two additional shell tests were conducted with more complicated density structures: laterally varying density structures and a crust-mantle interface density. The differences between the four codes were all below 1.5 % of the modelled gravity signal suitable for reproducing satellite-acquired gravity data. TESS and GSH produced the most similar potential fields (<0.3 %).
We present a spectral-finite-element approach to the 2-D forward problem for electromagnetic induction in a spherical earth. It represents an alternative to a variety of numerical methods for 2-D ...global electromagnetic modelling introduced recently (e.g. the perturbation expansion approach, the finite difference scheme). It may be used to estimate the effect of a possible axisymmetric structure of electrical conductivity of the mantle on surface observations, or it may serve as a tool for testing methods and codes for 3-D global electromagnetic modelling. The ultimate goal of these electromagnetic studies is to learn about the Earth's 3-D electrical structure. Since the spectral-finite-element approach comes from the variational formulation, we formulate the 2-D electromagnetic induction problem in a variational sense. The boundary data used in this formulation consist of the horizontal components of the total magnetic intensity measured on the Earth's surface. In this the variational approach differs from other methods, which usually use spherical harmonic coefficients of external magnetic sources as input data. We verify the assumptions of the Lax-Milgram theorem and show that the variational solution exists and is unique. The spectral-finite-element approach then means that the problem is parametrized by spherical harmonics in the angular direction, whereas finite elements span the radial direction. The solution is searched for by the Galerkin method, which leads to the solving of a system of linear algebraic equations. The method and code have been tested for Everett & Schultz's (1995) model of two eccentrically nested spheres, and good agreement has been obtained.
We present a semi-analytic solution to the 3-D forward modelling of electromagnetic induction in a heterogeneous model consisting of multiple eccentrically nested spheres. A number of numerical ...methods for 2-D and 3-D global electromagnetic modelling have been applied recently, including thin-sheet, perturbation-expansion, finite-element and spectral–finite-element schemes. The present semi-analytical approach may be used as an aid for testing more general algorithms of electromagnetic induction modelling. The multiple eccentrically nested spheres solution has been tested by comparing against the analytical solution for electromagnetic induction in a uniform sphere and in two eccentrically nested spheres with an azimuthal structure of electrical conductivity, and good agreements have been obtained. We further solve the electromagnetic induction problem in three and four eccentrically nested spheres configurations and compute the global response function and the spherical components of magnetic intensity within the model.
Magnetometer data from recent low‐orbit satellite missions Ørsted and CHAMP offer an opportunity to improve our knowledge of electrical conductivity of the Earth's mantle. The forward modelling of ...electromagnetic induction (EM) is conventionally carried out in the frequency domain. We use a time‐domain technique to study the electromagnetic response of a realistic 3‐D electrical conductivity model based on seismic tomography and laboratory measurements. The model is excited by a Dst signal realistic in time evolution but simplified as spatial dipolar shape. The results show measurable response of simulated satellite time series to mid‐mantle conductivity inhomogeneities. The interpretation of satellite magnetic data for the determination of 3‐D mantle conductivity inhomogeneities will, however, require a more precise model of external ionospheric and magnetospheric currents. This can be obtained by simultaneous measurements by several satellites.
We present a spectral–finite element approach to the forward modelling of the visco-elastic response of a spherical earth with a 3-D viscosity structure to a surface mass load. It represents an ...alternative to a variety of numerical methods for 2-D and 3-D postglacial rebound modelling used recently (the finite element method, the perturbation method, the semi-analytical approach and the spectral–finite difference method). For a fixed time, the problem is reformulated in a weak sense and parametrized by tensor surface spherical harmonics in the angular direction, whereas piecewise linear finite elements span the radial direction. The solution is obtained with the Galerkin method, which leads to solving a system of linear algebraic equations. The time dependence of the problem is treated directly in the time domain (not in the Laplace domain) as a time evolution problem. The time derivative in the constitutive equation for a Maxwell viscoelastic body is approximated by the explicit Euler time-differencing scheme, which leads to time splitting of the stress tensor. The spectral–finite element method and the associated numerical code have been tested for 2-D (azimuthally symmetric) eccentrically nested spheres models, and good agreement has been obtained.
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