Analysis of one-dimensional seismic waveform inversion by Regularized Global Approximation

Direct analysis of normal incidence seismogram inversion with respect to a velocity profile is now available due to application of a new global optimization algorithm. The latter is based on regularized global approximation of an objective function which is not supposed to re differentiable. The new technique allows one to see clearly a nonuniqueness of the inversion problem, no matter how high the quality of the input data may be. It is induced by a few factors: a source wavelet is a function of a finite frequency band, an effective wave length of the sounding signal is increasing jointly with the velocity, and the power of a media response is decreasing with respect to the depth. The nonuniqueness means that no inversion/processing is capable of solving the problem if it does not take into account a priori information about the recovered velocity profile. It is shown how an a priori assumption about a trend of the profile can essentially reduce the nonuniqueness of the problem. The corresponding regularization has the form of a soft constraint on the misfit function and leads to an unbiased estimation of the velocity profile when the latter is a monotonous function with respect to the depth. On the other hand, the regularization suggested allows of reconstructing nonmonotonous functions as well, which leads to a biased estimation of the velocity profile as any conventional regularized inversion also does. Examples of computer experiments are given that yield an opportunity of reconstructing the images of nonregularized and regularized objective functions as well as determining the accuracy of corresponding solutions of the inverse problem.