Are our parameter estimators biased? The significance of finite-difference regularization operators

The resolution of a particular solution to a given inverse problem may be quantified by resorting to a posteriori resolution indicators. In constrained least-squares inversions, for weakly nonlinear problems, the authors compute resolution kernels that relate the true model parameters to the estimated ones. These kernels reflect the significant effect on resolution that regularizing operators have. Since these operators are normally chosen to control the effect of noise, the kernels predict the degradation of the solutions in the presence of noise. For real problems with data errors, the knowledge of the resolving kernels enables us to quantify confidence in any parameter estimate, and gives a clear physical picture of the resolution; the resolution (indirectly) depends on the noise level in the data. The resolving kernels are an indication of the averaging of the true model parameters. The question arises as to whether these averages are biased or not. Popular methods, such as SVD truncation, or the use of a priori covariance operators, produce biased averages. In contrast, finite-difference regularizing operators systematically produce unbiased averages and prove to be a superior regularization technique in this respect. The authors tested the use of finite-difference regularizing operators using a simulation case study, in which they synthesized full waveforms for a simulated multi-offset 'VSP' survey, and inverted the picked traveltimes for 1D anisotropic parameters. The model resolution matrix for this case study was shown to be a reliable resolution indicator and it gave invaluable information on the essence of this inverse problem.