Wave‐equation traveltime + waveform inversion

Abstract

A seismic inversion method is presented which minimizes the misfit functions for both travel times and wave forms using a least-squares criterion. This method, designated as WlW inversion, is a hybrid combination of wave equation travel time inversion (Luo and Schuster, 199Oa) and full wave inversion (Tarantola, 1987). The WTW method retains the advantages of both full wave inversion and traveltime inversion; i.e., it is characterized by rapid convergence that is somewhat independent of the initial model and it can resolve fine features of the velocity model. The computational cost of the WTW method is only 510% more than that for full wave inversion. INTRODUCTION There are three criteria which are sometimes used to evaluate the merits of an inversion method; rejection of data noise, degree of model resolution, and convergence rate with respect o different starting models. Among the various inversion methods there are two extremes, travel time inversion ((Dines and Lytle, 1979; Paulsson et al., 1985; Ivansson, 1987; Bishop et al., 1985; Lines, 1988; Justice et al., 1989 and many others) and full wave inversion (I’arantola, 1987; and others). Travel time inversion usually assumes a high frequency approximation of the data and can therefore fail when the earth’s velocity variations are nearly the same wavelength as the source wavelet. In addition, the model resolution of travel timeinversion is less than that of full wave inversion. On the other hand, Luo and Schuster (1990a) showed that the traveltime misfit function (normed squared error between observed and calculated travel times) can be shown to be pseudo-linear with respect o the normed difference between the starting and actual velocity model. Hence, successful inversion can be achieved even if the starting model is far from the actual model. Moreover, traveltime inversion is more stable than amplitude inversion for media with random impedance perturbations (which always exist in the real earth). The model resolution characteristics of full wave inversion are almost complementary to that of traveltime inversion. While very sensitive to the choice of starting model or noisy amplitudes, full wave inversion can sometimes achieve a very high resolution of the model. This is because there are no approximations to the data, and all seismic events are included in the minimization of the misfit function. The problem with full wave inversion, however, is that its misfit function (normed difference between observed and synthetic seismograms) is hrghly IW~-OWX~ with respect to the actual and assumed velocity models (Gauthier et al., 1986; Luo and Schuster, 199Oa). In this case, a gradient method will tend to get stuck in local minima if the starting model is moderately fat from the actual model. Both traveltime and full wave inversion methods have complementary strengths and weaknesses. TO exploit these strengths and eliminate the weaknesses, this paper presents a hybrid invasion method which minimizes a weighted combination of traveltime (Luo and Schuster, 199Ob) and seismogram (Tarantola, 1987) misfit functions. The main benefits are a convergence rate which is somewhat insensitive to the starting model, high model resolution, no approximations to the data and a robustness in the presence of data noise. Synthetic tests show this new method, designated wave equation traveltime and wave form inversion (WTW), is significantly superior to standard full wavefield inversion. THEORY The following analysis assumes that the propagation of seismic waves honors the 2-D acoustic wave equation. Let the pressure observed at the receiver location x, (t=1,2...Nr) due to a source at x, (s=1,2,...Ns) be denoted by p(x,,r; xs)obs. The source is always assumed to be initiated at time t=O. For a given model, p Cx, t ; x,)~, denotes the computed seismograms which honor the wave equation 1 J2P(X,,c x 1 K(x) cJr* v. [~VP(r,l; .I]