Nonlinearity And Uncertainty Analysis Based On Bayeisan MT Inversion

The work presented in this thesis partly supported by Chinese Scholarship Council[2007U18089]. In order to make some effort toward higher-dimensional Bayesian inversion, some primary researches on improving the modeling efficiency have been carried out. We applied Fourier analysis (including local Fourier analysis and general Fourier analysis) to multi-grid method for discretized Helmholtz equations with complex entries, aroused in the magnetotelluric problem, with a purpose to predict and analyze its convergence. The Fourier spectra are usually in complex domain for the complex-valued matrix, however, visualizing the convergence behavior, it is necessary to project the complex spectra to real domain. The eigenvector spectra in real domain can explain the slow convergence observed in two-grid method with Gauss-Seidel solver on the coarsest grid. Since without considering boundary conditions and variant coefficients, local Fourier analysis (e.g., one-grid Fourier analysis and two-grid Fourier analysis) is hardly to give a reasonable asymptotic convergence estimate of two-grid method with a direct solver on the coarsest grid, whereas two-grid general Fourier analysis can. Fourier analysis for five-grid method using a direct solver on the coarsest grid, shows as the general higher-grid Fourier analysis is applied, the general Fourier analysis gets close to the numerical asymptotic convergence. Based on a set of general different-grid Fourier analysis results, we derived the empirical formula to approximate the asymptotic convergence for the five-grid method. Thus, we just need to carry out the general low-grid Fourier analysis(two-grid and three-grid) on coarsest grids and approximate the asymptotic convergence with low cost.Then boundary truncation was applied for homogeneous half-space model and three layer model based on the multigrid method. With a comparison to traditional coarse grid approximations, an average weighted method was used to generate the coarse grid approximation and general Fourier analysis was carried out for convergence analysis. Utilizing cycles or component of multi-grid method, the modeling area was shrunk to improve modeling efficiency. Numerical results show that the weighted average method has a better convergence behavior than Galerkin method and slightly poor convergence over geometric method. General Fourier analysis is not sensitive to conductivity discontinuities and can not demonstrate the convergence differences caused by the discontinuities. The numerical results for boundary truncation were compared with numerical results by direct solver. As the truncation increases, deviation from the theoretical results becomes more obvious. When the boundary is truncated to around twice the skin depth, the results are well approximated with theoretical results. It is interesting to note, in the log space (or just log frequency), when the truncated boundary value cannot well represent the real value, the calculated absolute electrical field (or phase) tend to move up and down at the same level with respect to frequency.Then an efficient algorithm is presented in this paper to improve the perturbation efficiency of the adaptive downhill simplex simulated annealing (ASSA) method for magnetotelluric inversion with oblique correlated misfit valleys. The correlated model space is rotated to a less dependent space which is defined by the eigenvectors of the parameter covariance updated by the inversion itself progressively. The downhill simplex step and rotation work together to decorrelate the correlated model space. The application to two synthetic cases and real data indicates that for trongly correlated model space, ASSA in the rotated space generally has a better convergence and efficient behaviour. In the rotated space, the high rejection rates, happened in the unrotated space, are avoided. At low temperature, the estimated covariance can be used to approximate the global covariance. For all the cases, ASSA in the rotated space gives better inversion results (in terms of statistic behavior).Finally, this paper employs Bayesian inference theory to study model parameterisation, parameter uncertainties, and nonlinearity for the one-dimensional magnetotelluric (MT) inverse problem. In the Bayesian formulation, the data and the model parameters are considered random variables and the multi-dimensional posterior probability density(PPD), combining data and prior information, is interpreted in terms of parameter estimates, uncertainties, and inter-relationships, which requires optimizing and integrating the PPD. In the nonlinear formulation developed here, optimization is carried out using an adaptive-hybrid algorithm that combines very-fast simulated annealing and the downhill simplex method. Integration applies Metropolis-Hastings sampling (considering nonunity sampling temperatures to ensure sample completeness), rotated to a principal-component parameter space for efficient sampling of correlated parameters. Since appropriate model parameterisations are generally not known a priori, both under- and over-parameterized approaches are considered. For under-parameterisation, the maximum a posteriori (MAP) solution is determined for a sequence of inversions with an increasing number of layers, and the appropriate parameterisation is chosen using the Bayesian information criterion for model selection. For over-parameterisation, prior information is included which favours simple structure in a manner similar to regularized (Occam's) inversion. The data error variance and the tradeoff parameter regulating data and prior information are included as nuisance parameters in the PPD sampling. Nonlinear and (approximate) linearised inversion results are compared for synthetic test cases and for the COPROD1 MT data set by considering one- and two-dimensional marginal probability distributions and marginal probability profiles. In some cases, important differences are indicated between the nonlinear and linearised approaches, including multi-modal PPDs which cannot be addressed within a linearised approximation.