This project deals with the inversion of reflection seismic data, in other words attempting to find the subsurface geological parameters that best explain the surface recorded seismic traces.

The subsurface is approximated as an acoustic medium where wave propagation is described by only two parameters: velocity of propagation and acoustic impedance (for 1D-models only the latter is involved).

These two parameters are the unknowns for the inverse problem.

The first objective of this project is to study the feasibility of several inversion type problems (depending on the geometrical hypotheses : 1D, 2D horizontally stratified, heterogeneous 2D).

The second goal is to write software for processing large field data volume.

Part A of the project, involving 1D inversion, has been developed. Numerous theoretical as well as pratical results can be found in the literature, describing the possibilities and the difficulties of this problem.

From these results, after the construction of a solid modelling and gradient calculation software package we were able to quickly set a path towards its practical application.

A software package was created for stratigraphic extrapolation of well data using 1D inversion. A seismic profile of zero offset traces is constructed, after which each trace is inverted by using the low frequency impedance trend of the previous trace. The first inversed trace uses the trend from the well.

This tool appears to be a very useful on for nearly horizontal stratigraphies. Nevertheless, field acquisition problems have been highlighted. These problems have to be solved before the tool can acquire a certain credibility.

Parts B and C of this project related to 2D inversion have needed theinvention of new techniques to solve the problems associated with finding a velocity field. These techniques (use of travel time instead of depth, progressive downward continuation in time, progressive increase in frequencies, adapted unknowns) were successfully implemented in the problem B. This has led to a satisfying solution for synthetic data of large dimensions.

For heterogenous 2D media, these new techniques are much too heavy to be implemented at the moment. There is however, one aspect that has been solved in a very successful fashion : that is finding the impedance reflectors when the velocity field is approximately known. The results consist in a refinement of a migration before stack using the wave equation.

The reflection seismic method can be briefly described by a source near the surface which produces a propagation wave in the substratum. The resulting perturbation which is reflected, refracted or diffracted is measured at the surface by hydrophones (Gi) as pressure (or displacement) Pobs (Gi,t) as a function of time.

The forward problem involves the computation of synthetic seismograms Pm (Gi,t) that correspond to a given subsurface model m. Acoustic wave propagation is assumed.

The inverse problem, briefly explained, will involve the search of an earth model m that produces seismograms Pm (Gi,t) as close as possible to the observed seismograms Pobs (Gi,t).

The first step is then to construct a tool that efficiently resolves the forward problem, inversion requiring the computation of J(m) for a large number of models m.

The next step will require the construction of a second basic tool which can calculate the gradient of J with respect to the model m. An optimization strategy must then be defined, entirely automatic or at the contrary interactive, conveniently integrating a priori information of the model.

A sensitivity study has had to be done, with two objectives: one was to determine the space of admissible models in which the search of the solution was stable; the second objective was to find the best optimization variables along with a correct preconditioning of the problem to increase the stability and the convergence rate.

Many software tests were required, first with noisy synthetic data, and later with field data.

Three inversion problems have been studied:

A 1D inversion : plane waves modelling in an horizontally stratified media. Acoustic impedance as a function of travel-time is sought. B 2D inversion of horizontally stratified media: waves propagation is 2D, the model to be found is 1D. The information available for different offsets allows the search for both velocity and acoustic impedance.

C Heterogeneous 2D inversion : the geometry of theentire problem is 2D, which means that there will be a considerable number of unknowns, and that a large number of seismic records is required to supply enough information.

This problem is therefore very complex as well as very costly to study.