About the stability of the inverse problem in 1-D wave equations—application to the interpretation of seismic profiles

Abstract

This paper is devoted to the study of the following inverse problem: Given the 1-D wave equation:

$$\begin{gathered} \rho (z)\frac{{\partial ^2 y}}{{\partial t^2 }} - \frac{\partial }{{\partial z}}\left( {\mu (z)\frac{{\partial y}}{{\partial z}}} \right) = 0 z > 0,t > 0 \hfill \\ + boundary excitation at z = 0 + zero initial conditons \hfill \\ \end{gathered} $$

((1))

how to determine the parameter functions (ρ(z),μ(z)) from the only boundary measurementY(t) ofy(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 constructequivalence classes σ(x) of couples (ρ(z),ρ(z)) that areundistinguishable from the givenboundary measurements Y(t).

Then we give in §2 existence theorems of the solutiony of the state equations (1), and study the mappingσ→Y: We define on the set of equivalence classes Σ={σ(x)|σ _min ⩽σ(x) ⩽ σ _max for a.e.x} (σ _min andσ _max a priori given) a distanced which is weak enough to make Σ compact, but strong enough to ensure the (lipschitz) continuity of the mappingσ→Y. This ensures the existence of a solution to the inverse problem set as an optimization problem on Σ. The fact that this distanced is much weaker than the usualL ² 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 forσ) are given for the interpretation of seismic profiles with the above method.