Klaus Spitzer

Bio: Klaus Spitzer is an academic researcher from Freiberg University of Mining and Technology. The author has contributed to research in topics: Finite element method & Adaptive mesh refinement. The author has an hindex of 16, co-authored 53 publications receiving 1399 citations. Previous affiliations of Klaus Spitzer include École Polytechnique.

Three‐dimensional modelling and inversion of dc resistivity data incorporating topography – II. Inversion

Abstract: SUMMARY We present a novel technique for the determination of resistivity structures associated with arbitrary surface topography. The approach represents a triple-grid inversion technique that is based on unstructured tetrahedral meshes and finite-element forward calculation. The three grids are characterized as follows: A relatively coarse parameter grid defines the elements whose resistivities are to be determined. On the secondary field grid the forward calculations in each inversion step are carried out using a secondary potential (SP) approach. The primary fields are provided by a one-time simulation on the highly refined primary field grid at the beginning of the inversion process. We use a Gauss‐Newton method with inexact line search to fit the data within error bounds. A global regularization scheme using special smoothness constraints is applied. The regularization parameter compromising data misfit and model roughness is determined by an L-curve method and finally evaluated by the discrepancy principle. To solve the inverse subproblem efficiently, a least-squares solver is presented. We apply our technique to synthetic data from a burial mound to demonstrate its effectiveness. A resolution-dependent parametrization helps to keep the inverse problem small to cope with memory limitations of today’s standard PCs. Furthermore, the SP calculation reduces the computation time significantly. This is a crucial issue since the forward calculation is generally very time consuming. Thus, the approach can be applied to large-scale 3-D problems as encountered in practice, which is finally proved on field data. As a by-product of the primary potential calculation we obtain a quantification of the topography effect and the corresponding geometric factors. The latter are used for calculation of apparent resistivities to prevent the reconstruction process from topography induced artefacts.

Three-dimensional adaptive higher order finite element simulation for geo-electromagnetics—a marine CSEM example

Abstract: SUMMARY We present a new 3-D vector finite element code and demonstrate its strength by modelling a realistic marine CSEM scenario. Unstructured tetrahedral meshes easily allow for the inclusion of arbitrary seafloor bathymetry so that natural environments are mapped into the model in a close-to-reality way. A primary/secondary field approach, an adaptive mesh refinement strategy as well as a higher order polynomial finite element approximation improve the solution accuracy. A convergence study strongly indicates that the use of higher order finite elements is beneficial even if the solution is not globally smooth. The marine CSEM scenario also shows that seafloor topography gives an important response which needs to be reproduced by numerical modelling to avoid the misinterpretation of measurements.

The three‐dimensional DC sensitivity for surface and subsurface sources

Abstract: SUMMARY Four diierent methods of determining the DC sensitivity in three dimensions are presented: three numerical approaches for arbitrary conductivity structures and an analytical one for a homogeneous case using the sensitivity theorem. Since the sensitivity is a very important and indicative property in any interpretation process, its spatial distribution is shown as an overview for commonly used pole^pole, pole^dipole, and dipole^dipole arrangements at the surface and subsurface. Distinct regions of negative sensitivities appear for any con¢guration. For horizontal subsurface pole^pole con¢gurations, they assume tube-like, cylindrical shapes stretching from the electrode locations towards the surface and yielding a circular sign-reversal pattern at the surface. These shapes and sign reversals occur as long as the electrodes are located at a ¢nite depth. Similar forms occur for subsurface pole^dipole and dipole^dipole arrangements. A series of model studies are carried out to examine the validity of the homogeneous responses for more realistic inhomogeneous media. Generally, the spatial sensitivity patterns for homogeneous environments are good approximations for moderate conductivity contrasts not exceeding 1:10 if the source is located within conductive material. If the source is buried within a resistor, conductive structures perturb the homogeneous pattern more signi¢cantly. Finally, a crosshole model study reveals signi¢cant diierences between 2-D and 3-D conductive bodies, suggesting a need to examine the target very carefully before approximating some structures in two dimensions.

Fast 3-D simulation of transient electromagnetic fields by model reduction in the frequency domain using Krylov subspace projection

Abstract: SUMMARY We present an efficient numerical method for the simulation of transient electromagnetic fields resulting from magnetic and electric dipole sources in three dimensions The method we propose is based on the Fourier synthesis of frequency domain solutions at a sufficient number of discrete frequencies obtained using a finite element (FE) approximation of the damped vector wave equation obtained after Fourier transforming Maxwell's equations in time We assume the solution to be required only at a few points in the computational domain, whose number is small relative to the number of FE degrees of freedom The mapping which assigns to each frequency the FE approximation at these points of interest is a vector-valued rational function known as the transfer function Its evaluation is approximated using Krylov subspace projection, a standard model reduction technique Computationally, this requires the FE discretization at a small number of reference frequencies and the generation of a sufficiently large Krylov subspace associated with each reference frequency Once a basis of this subspace is available, a sufficiently accurate rational approximation of the transfer function can be evaluated at the remaining frequencies at negligible cost These partial frequency domain solutions are then synthesized to the time evolution at the points of interest using a fast Hankel transform To test the algorithm, responses obtained by2-D and 3-D FE formulations have been calculated for a layered half-space and compared with results obtained analytically, for which we observed a maximum deviation of less than 2 per cent in the case of transient EM modelling We complete our model studies with a number of comparisons with established numerical approaches A first implementation of our new numerical algorithm already gives very good results using much less computational time compared with time stepping methods and comparable times and accuracy compared with the Spectral Lanczos Decomposition Method (SLDM)

Three‐dimensional DC resistivity forward modelling using finite elements in comparison with finite‐difference solutions

Abstract: SUMMARY A 3-D finite-element scheme for direct current resistivity modelling is presented. The singularity is removed by formulating the problem in terms of the secondary potential, which improves the accuracy considerably. The resulting system of linear equations is solved using the conjugate gradient method. The incomplete Cholesky preconditioner with a scaled matrix has been proved to be faster than the symmetric successive overrelaxation preconditioner. A compact storage scheme fully utilizes the sparsity and symmetry of the system matrix. The finite-element (FE) and a previously developed finite-difference (FD) scheme are compared in detail. Generally, both schemes show good agreement, the relative error in apparent resistivity for a vertical dike model presented in this paper is less than 0.5 per cent overall. The FD scheme produces larger errors near the conductivity contrast, whereas the FE scheme requires approximately 3.4 times as much storage as the FD scheme and is less robust with respect to coarse grids. As an improvement to the forward modelling scheme, a modified singularity removal technique is presented. A horizontally layered earth or a vertical contact is regarded as the normal structure, the solution of which is the primary potential. The effect of this technique is demonstrated by two examples: a cube in two-layered earth and a cube near a vertical contact.

DC Resistivity and Induced Polarization Methods

Abstract: Direct current (DC) resistivity (here referred to as resistivity) and induced polarization (IP) methods allow, respectively, the determination of the spatial distribution of the low-frequency resistive and capacitive characteristics of soil. Since both properties are affected by lithology, pore fluid chemistry, and water content (see Chapter 4 of this volume), these methods have significant potential for hydrogeophysical applications. The methods can be applied at a wide range of laboratory and field scales, and surveys may be made in arbitrary geometrical configurations (e.g., on the soil surface and down boreholes). In fact, resistivity methods are one of the most widely used sets of geophysical techniques in hydrogeophysics. These surveys are relatively easy to carry out, instrumentation is inexpensive, data processing tools are widely available, and the relationships between resistivity and hydrological properties, such as porosity and moisture content, are reasonably well established. In contrast, applications of induced polarization methods in hydrogeophysics have been limited. As noted by Slater and Lesmes (2002), this is partly because of the more complex procedure for data acquisition, but also because the physicochemical interpretation of induced polarization parameters is not fully understood.

Three‐dimensional modelling and inversion of dc resistivity data incorporating topography – II. Inversion

Abstract: SUMMARY We present a novel technique for the determination of resistivity structures associated with arbitrary surface topography. The approach represents a triple-grid inversion technique that is based on unstructured tetrahedral meshes and finite-element forward calculation. The three grids are characterized as follows: A relatively coarse parameter grid defines the elements whose resistivities are to be determined. On the secondary field grid the forward calculations in each inversion step are carried out using a secondary potential (SP) approach. The primary fields are provided by a one-time simulation on the highly refined primary field grid at the beginning of the inversion process. We use a Gauss‐Newton method with inexact line search to fit the data within error bounds. A global regularization scheme using special smoothness constraints is applied. The regularization parameter compromising data misfit and model roughness is determined by an L-curve method and finally evaluated by the discrepancy principle. To solve the inverse subproblem efficiently, a least-squares solver is presented. We apply our technique to synthetic data from a burial mound to demonstrate its effectiveness. A resolution-dependent parametrization helps to keep the inverse problem small to cope with memory limitations of today’s standard PCs. Furthermore, the SP calculation reduces the computation time significantly. This is a crucial issue since the forward calculation is generally very time consuming. Thus, the approach can be applied to large-scale 3-D problems as encountered in practice, which is finally proved on field data. As a by-product of the primary potential calculation we obtain a quantification of the topography effect and the corresponding geometric factors. The latter are used for calculation of apparent resistivities to prevent the reconstruction process from topography induced artefacts.

Computational recipes for electromagnetic inverse problems

Abstract: SUMMARY The Jacobian of the non-linear mapping from model parameters to observations is a key component in all gradient-based inversion methods, including variants on Gauss–Newton and non-linear conjugate gradients. Here, we develop a general mathematical framework for Jacobian computations arising in electromagnetic (EM) geophysical inverse problems. Our analysis, which is based on the discrete formulation of the forward problem, divides computations into components (data functionals, forward and adjoint solvers, model parameter mappings), and clarifies dependencies among these elements within realistic numerical inversion codes. To be concrete, we focus much of the specific discussion on 2-D and 3-D magnetotelluric (MT) inverse problems, but our analysis is applicable to a wide range of active and passive source EM methods. The general theory developed here provides the basis for development of a modular system of computer codes for inversion of EM geophysical data, which we summarize at the end of the paper.