The identification of the source signature is a major calibration problem in reflection seismology. Conventional approaches to this problem involve statistical methods, direct measurement of the source, or the location of a clean reflection in the seismic section. As an alternative, we investigate the possibility of recovering the source and velocity from the data by means of nonlinear least squares.
We model the earth as a linearly acoustic layered fluid. The problem of recovering the source and velocity then becomes that of identifying a boundary term and a velocity coefficient in an initial-boundary-value problem for the one-dimensional wave operator.
We examine two formulations of the source-velocity identification problem in terms of nonlinear least squares. The first of these is output least squares, which in recent years has received attention as a method for velocity inversion. One question which heretofore has gone unanswered is that of the differentiability of the impulse response forward map, which relates the sound velocity and the seismogram in the case that the source is an impulse $\delta(t)$. The resolution of this issue is our first major result. We prove that the forward map will be differentiable at a velocity $c$ provided $c$ has three derivatives in L\sp2. Moreover, this result is optimal.
Using this regularity theorem and a result due to Bube, Lailly, Sacks, Santosa, and Symes, we then prove a local uniqueness result for the output least squares source-velocity identification problem. We assume that the source is quasi-impulsive; i.e., the source $f$ has the form $f = \delta + h$, where h \in L\sp2. We show that locally the source and velocity are uniquely determined by boundary measurements.
Recent research has shown that for a number of reasons, the output least squares approach is unsuitable for numerical computation. Moreover, realistic source models and seismograms are band-limited, so that results concerning quasi-impulsive sources have no immediate practical consequences. Thus, we will also examine the source-velocity problem in the setting of coherency optimization, an alternative to output least squares proposed by Symes. We will present a result which indicates that we should be able to stably recover band-limited components of the source and velocity