ABOUT THE STABILITY OF THE INVERSE PROBLEM IN 1-D WAVE-EQUATIONS - APPLICATION TO THE INTERPRETATION OF SEISMIC PROFILES

This paper is devoted to the study of the following inverse problem: Given the 1-D wave equation: {Mathematical expression} how to determine the parameter functions (ρ(z),μ(z)) from the only boundary measurement Y(t) of y(z, t)/z=0. This inverse problem is motivated by the reflection seismic exploration techniques, and is known to be very unstable. We first recall in §1 how to construct equivalence classes σ(x) of couples (ρ(z),ρ(z)) that are undistinguishable from the given boundary measurements Y(t). Then we give in §2 existence theorems of the solution y of the state equations (1), and study the mapping σ→Y: We define on the set of equivalence classes Σ={σ(x)|σmin ≤σ(x) ≤ σmax for a.e. x} (σmin and σmaxa priori given) a distance d which is weak enough to make Σ compact, but strong enough to ensure the (lipschitz) continuity of the mapping σ→Y. This ensures the existence of a solution to the inverse problem set as an optimization problem on Σ. The fact that this distance d is much weaker than the usual L2 norm explains the tendency to unstability reported by many authors. In §3, the case of piecewise constant parameter is carefully studied in view of the numerical applications, and a theorem of stability of the inverse problem is given. In §4, numerical results on simulated but realistic datas (300 unknown values for σ) are given for the interpretation of seismic profiles with the above method. © 1979 Springer-Verlag New York Inc.