Resolution in seismic trace inversion by parameter estimation

An alternative to the conventional time series approach to single-trace modeling and inversion by convolution and inverse filtering is a parametric approach. To obtain insight into the potential of the parametric approach, the solution of the single-trace forward problem is formulated in matrix terms. For the nonlinear reflector lag time parameters this is achieved by linearization, which is shown to be a valid approximation over a sufficiently large region.The matrix forward operators are analyzed by means of the singular value decomposition (SVD). The SVD can be considered a generalization of the Fourier transform of convolution operators. On the basis of the SVD analysis, inverse operators are designed which combine stability with high resolving power. A method to determine the resolving power of the parametric inverse operators is presented. Several examples show how wavelet bandwidth, data noise level, and model complexity influence the resolving power of the data for the reflection coefficient and the lag-time parameters.The most important result is that the resolution obtained in parametric inversion is, in most cases, superior to and, at worst, equal to the resolution obtained with wavelet inverse filtering. The explanation is that in parametric inversion a different representation of the reflectivity function is used which, in practical situations, involves fewer unknowns. In wavelet inverse filtering the reflectivity function is represented as a regularly sampled function where every sample point represents an unknown. In practical applications of parametric inversion the reflectivity function is represented as a model with a limited number of reflectors as unknowns. To formulate parametric models, a priori information is required. The effort of collecting sufficient a priori information is the cost of increasing resolution beyond the resolution offered by wavelet inverse filtering.