A non-linear regularized constrained impedance inversion

Inversion for seismic impedance is an inherently complicated problem. It is ill-posed and band-limited. Thus the inversion results are non-unique and the process is unstable. Combining regularization with constraints using sonic and density log data can help to reduce these problems. To achieve this, we developed an inversion method by constructing a new objective function, including edge-preserving regularization and a soft constraint based on a Markov random field. The method includes the selection of proper initial values of the regularization parameters by a statistical method, and it adaptively adjusts the regularization parameters by the maximum likelihood method in a fast simulated-annealing procedure to improve the inversion result and the convergence speed. Moreover, the method uses two kinds of regularization parameter: a 'weighting factor' lambda and a 'scaling parameter' delta. We tested the method on both synthetic and field data examples. Tests on 2D synthetic data indicate that the inversion results, especially the aspects of the discontinuity, are significantly different for different regularization functions. The initial values of the regularization parameters are either too large or too small to avoid either an unstable or an over-smoothed result, and they affect the convergence speed. When selecting the initial values of lambda, the type of the regularization function should be considered. The results obtained by constant regularization parameters are smoother than those obtained by adaptively adjusting the regularization parameters. The inversion results of the field data provide more detailed information about the layers, and they match the impedance curves calculated from the well logs at the three wells, over most portions of the curves.