G. Ryzhikov et al., ANALYSIS OF ONE-DIMENSIONAL SEISMIC WAVE-FORM INVERSION BY REGULARIZED GLOBAL APPROXIMATION, Journal of seismic exploration, 5(4), 1996, pp. 349-362
Direct analysis of normal incidence seismogram inversion with respect
to a velocity profile is now available due to application of a new glo
bal optimization algorithm. The latter is based on regularized global
approximation of an objective function which is not supposed to re dif
ferentiable. The new technique allows one to see clearly a nonuniquene
ss of the inversion problem, no matter how high the quality of the inp
ut 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 s
ounding signal is increasing jointly with the velocity, and the power
of a media response is decreasing with respect to the depth. The nonun
iqueness means that no inversion/processing is capable of solving the
problem if it does not take into account a priori information about th
e recovered velocity profile. It is shown how an a priori assumption a
bout a trend of the profile can essentially reduce the nonuniqueness o
f 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 re
gularized inversion also does. Examples of computer experiments are gi
ven that yield an opportunity of reconstructing the images of nonregul
arized and regularized objective functions as well as determining the
accuracy of corresponding solutions of the inverse problem.