Scaling geology and smooth solutions to inverse problems

Abstract

Geophysical inverse problems are often ill-posed and require regularization or smoothing to obtain a physically realistic solution. Methods which determine the amount of smoothing to incorporate into the inversion are generally subjective, either through the choice of mathematical technique or the interpreter’s prior beliefs. (1985) for reflection sequences. Todoeschuck and Jensen (1988,1989) have presented wavelet estimation methods taking this into account which can improve deconvolution of seismograms. Pilkmgton and Todoeschuck (19%) examined a variety of geophysical parameters from well logs, concluded they showed scaling behaviour and discussed some implications for inversion. Here we focus on the question of smoothness. Using stochastic inversion in its complete form, that is, incorporating both a priori parameter and data covariances, answers the question of the degree and the character of smoothing to be used. Geophysical welllogs suggest a scaling noise model where power is proportional to some power of frequency. Knowing the power spectrum allows us to specify a priori parameter covariances that provide the appropriate amount of structure in the solution to a given inverse problem, Imposing smoothness constraints