Zugl.: Stuttgart, Univ., Diss., 2000.

A three-dimensional inversion was implemented for magnetotellurics, which is a passive electromagnetic method in geophysics. It exploits natural electromagnetic fields of the Earth, which function as sources. Their interaction with the conductive parts of the subsurface are registered when components of the electric and the magnetic field are measured and evaluated. The all-at-once approach is an inversion scheme that is relatively new to geophysics. In this approach, the objective function – the basis of each inversion – is called the Lagrangian. It consists of three parts: (i) the data residual norm, (ii) the regularisation part, and (iii) the forward problem. The latter is the significant difference to conventional inversion approaches that are built up of a forward calculation part and an inversion part. In the case of all-at-once, the forward problem is incorporated in the objective function and is therefore already taken into account in each inversion iteration. Thus, an explicit forward calculation is obsolete. As an objective function, the Lagrangian shall reach a minimum and therefore its first and second derivatives are evaluated. Hence, the gradient of the Lagrangian and its Hessian are constituent parts of the KKT system – the Newton-type system that is set up in the all-at-once inversion. Conventional inversion approaches avoid the Hessian because it is a large, dense, not positive definite matrix that is challenging to handle. However, it provides additional information to the inversion, which raises hope for a high quality inversion result. As a first step, the inversion was programmed for the more straightforward one-dimensional magnetotelluric case. This was particularly suitable to become familiar with sQMR – a Krylov subspace method which is essential for the three-dimensional case to be able to work with the Hessian and the resulting KKT system. After the implementation and validation of the one-dimensional forward operator, the Lagrangian and its derivatives were set up to complete the inversion, which successfully solved the KKT system. Accordingly, the three-dimensional forward operator also needed to be implemented and validated, which was done using published data from the 3D-2 COMMEMI model. To realise the inversion, the Lagrangian was assembled and its first and second derivatives were validated with a test that exploits the Taylor expansion. Then, the inversion was initially programmed for the Gauss-Newton approximation where second order information is neglected. Since the system matrix of the Gauss-Newton approximation is positive definite, the solution of this system of equations could be carried out by the conventional solver pcg. Based on that, the complete KKT system (Newton\\\'s method) was set up and preconditioned sQMR solved this system of equations.

A three-dimensional inversion was implemented for magnetotellurics, which is a passive electromagnetic method in geophysics. It exploits natural electromagnetic fields of the Earth, which function as sources. Their interaction with the conductive parts of the subsurface are registered when components of the electric and the magnetic field are measured and evaluated. The all-at-once approach is an inversion scheme that is relatively new to geophysics. In this approach, the objective function – the basis of each inversion – is called the Lagrangian. It consists of three parts: (i) the data residual norm, (ii) the regularisation part, and (iii) the forward problem. The latter is the significant difference to conventional inversion approaches that are built up of a forward calculation part and an inversion part. In the case of all-at-once, the forward problem is incorporated in the objective function and is therefore already taken into account in each inversion iteration. Thus, an explicit forward calculation is obsolete. As an objective function, the Lagrangian shall reach a minimum and therefore its first and second derivatives are evaluated. Hence, the gradient of the Lagrangian and its Hessian are constituent parts of the KKT system – the Newton-type system that is set up in the all-at-once inversion. Conventional inversion approaches avoid the Hessian because it is a large, dense, not positive definite matrix that is challenging to handle. However, it provides additional information to the inversion, which raises hope for a high quality inversion result. As a first step, the inversion was programmed for the more straightforward one-dimensional magnetotelluric case. This was particularly suitable to become familiar with sQMR – a Krylov subspace method which is essential for the three-dimensional case to be able to work with the Hessian and the resulting KKT system. After the implementation and validation of the one-dimensional forward operator, the Lagrangian and its derivatives were set up to complete the inversion, which successfully solved the KKT system. Accordingly, the three-dimensional forward operator also needed to be implemented and validated, which was done using published data from the 3D-2 COMMEMI model. To realise the inversion, the Lagrangian was assembled and its first and second derivatives were validated with a test that exploits the Taylor expansion. Then, the inversion was initially programmed for the Gauss-Newton approximation where second order information is neglected. Since the system matrix of the Gauss-Newton approximation is positive definite, the solution of this system of equations could be carried out by the conventional solver pcg. Based on that, the complete KKT system (Newton\\\'s method) was set up and preconditioned sQMR solved this system of equations.

This thesis provides a feasibility study on the determination of formation anisotropy parameters from logging-while-drilling (LWD) borehole acoustic measurements. For this reason, the wave propagation in fluid-filled boreholes surrounded by transverse isotropic (TI) formations is investigated in great detail using the finite-difference method. While the focus is put on quadrupole waves, the sensitivities of monopole and flexural waves are evaluated as well. All three wave types are considered with/without the presence of an LWD tool. Moreover, anisotropy-induced mode contaminants are discussed for various TI configurations. In addition, the well-known plane wave Alford rotation has been generalized to cylindrical borehole waves of any order, except for the monopole. This formulation has been extended to allow for non-orthogonal multipole firings, and associated inversion methods have been developed to compute formation shear principal velocities and accompanying polarization directions, utilizing various LWD (cross-) quadrupole measurements.:1 Introduction 1.1 Borehole acoustic configurations 1.2 Wave propagation in a fluid-filled borehole in the absence of a logging tool 1.3 Wave propagation in a fluid-filled borehole in the presence of a logging tool 1.4 Anisotropy 2 Theory 2.1 Stiffness and compliance tensor 2.1.1 Triclinic symmetry 2.1.2 Monoclinic symmetry 2.1.3 Orthotropic symmetry 2.1.4 Transverse isotropic (TI) symmetry 2.1.5 Isotropy 2.2 Reference frames 2.3 Seismic wave equations for a linear elastic, anisotropic medium 2.3.1 Basic equations 2.3.2 Integral transforms 2.3.3 Christoffel equation 2.3.4 Phase slowness surfaces 2.3.5 Group velocity 2.4 Solution in cylindrical coordinates for the borehole geometry 2.4.1 Special case: vertical transverse isotropy (VTI) 2.4.2 General case: triclinic symmetry 3 Finite-difference modeling of wave propagation in anisotropic media 3.1 Finite-difference method 3.2 Spatial finite-difference grids 3.2.1 Standard staggered grid 3.2.2 Lebedev grid 3.3 Heterogeneous media 3.4 Finite-difference properties and grid dispersion 3.5 Initial conditions 3.6 Boundary conditions 3.7 Parallelization 3.8 Finite-difference parameters 4 Wave propagation in fluid-filled boreholes surrounded by TI media 4.1 Vertical transverse isotropy (VTI) 4.1.1 Monopole excitation 4.1.2 Dipole excitation 4.1.3 Quadrupole excitation 4.1.4 Summary 4.2 Horizontal transverse isotropy (HTI) 4.2.1 Monopole excitation 4.2.2 Theory of cross-multipole shear wave splitting 4.2.3 Dipole excitation 4.2.4 Quadrupole excitation 4.2.5 Hexapole waves 4.2.6 Summary 4.3 Tilted transverse isotropy (TTI) 4.3.1 Monopole excitation 4.3.2 Dipole excitation 4.3.3 Quadrupole excitation 4.3.4 Summary 4.4 Anisotropy-induced mode contaminants 4.4.1 Vertical transverse isotropy (VTI) 4.4.2 Horizontal transverse isotropy (HTI) 4.4.3 Tilted transverse isotropy (TTI) 4.4.4 Summary 5 Inversion methods 5.1 Vertical transverse isotropy (VTI) 5.2 Horizontal transverse isotropy (HTI) 5.2.1 Inverse generalized Alford rotation 5.2.2 Inversion method based on dipole excitations 5.2.3 Inversion method based on quadrupole excitations 5.3 Tilted transverse isotropy (TTI) 5.4 Challenges in real measurements 5.4.1 Signal-to-noise ratio (SNR) 5.4.2 Tool eccentricity 6 Conclusions References List of Abbreviations and Symbols List of Figures List of Tables A Integral transforms A.1 Laplace transform A.2 Spatial Fourier transform A.3 Azimuthal Fourier transform A.4 Meijer transform B Stiffness and compliance tensor B.1 Rotation between reference frames B.2 Cylindrical coordinates C Christoffel equation C.1 Cartesian coordinates C.2 Cylindrical coordinates D Processing of borehole acoustic waveform array data D.1 Time-domain methods D.2 Frequency-domain methods D.2.1 Weighted spectral semblance method D.2.2 Modified matrix pencil method