THE AIM OF THIS PROJECT IS TO DEVELOP NON-LINEAR INVERSE METHODS FOR A QUANTITATIVE INTERPRETATION OF SEISMIC REFLECTION DATA TO BE IMPLEMENTED ON VECTOR SUPERCOMPUTERS AND RENDERED INDUSTRIALLY USABLE.

The aim of the project was to demonstrate the interpretation of seismic reflection data by modelling the propagation of elastic waves inside the Earth, and to identify the model which predicts the best synthetic seismograms. It is shown that it is possible to invert real data and to obtain images of the Earth's crust that are better than those obtained through conventional techniques.

Full nonlinear methods of inversion are used. The forward and inverse problems are solved without using the Born approximation, ie without neglecting multiply reflected energy or evanescent waves and without assuming good starting models. 2 very smooth images are retrieved at wavelengths greater than 300 m, corresponding to the seismic velocities, and 2 more rough images at wavelengths in the range 20 to 80 m, corresponding to the contrasts of seismic impedances.

The seismic inverse problem is formulated as an optimisation problem. This is elastic rather than acoustic, and may account for attenuation. Considerable theoretical work on the solution of highly nonlinear inverse problems using simulated annealing techniques is included.

The best way of interpreting seismic reflection data is by accurate modelling of the propagation of elastic waves inside the Earth, by obtaining, among all conceivable Earth models, the one predicting the best synthetic seismograms.

Using conventional seismic reflection data it was possible to retrieve 4 images of the underground, 2 very smooth images and 2 rough images. The 2 smooth images corresponded to the seismic velocities (P and S), while the rough images corresponded to the contrasts of seismic impedances (product of density by velocity). Outside these wavelength windows, no parameter can be resolved using seismic data alone.

The seismic inverse problem can be formulated as an optimization problem, namely the problem of obtaining the Earth model predicting the best synthetic seismograms. Using gradient methods, a very elegant formulation was found, where the updating of the current Earth model was performed via the correlation of 2 displacement fields: the field predicted using the actual source and the field obtained using the current data residuals as sources. This formulation is elastic (as opposed to acoustic) and may even account for attenuation.

Gradient methods of optimization work well for obtaining the image of impedance contrasts, but performed poorly for obtaining the smooth image of the background velocities. Ad hoc methods of optimization have been developed to solve that problem. They range from flexible polyhedron methods to simulated annealing methods.

The theory of solving highly nonlinear inverse problems using simulated annealing techniques has been developed. In particular, Monte Carlo simulations may provide, besides the optimum Earth model, a complete (nonlinear) estimation of uncertainties in the solution.

Linearized methods of inversion, based on Ray and Fourier domain techniques have been developed and compared with full nonlinear methods, based on a finite difference approximation to the elastic wave equation. While l inearized methods are directly comparable to migration methods, they do not give any clear computational advantage. The recent advent of massively parallel computers will still favour finite difference modelling.

A THREE YEAR RESEARCH PROGRAMME IS ENVISAGED TO ATTAIN THE FINAL GOAL.

FIRST, THE MOST MODERN RAY-TRAXING TECHNIQUES, INCLUDING THE PARABOLIC APPROXIMATION AND GAUSSIAN BEAMS, WILL BE APPLIED FOR SOLVING THE ELASTIC PROPAGATION EQUATION. COMPUTER PROGRAMMES USING THESE METHODS AND BASED ON BORN APPROXIMATION WILL BE DEVELOPED AND TESTED. A GENERALISATION TO THE ELASTODYNAMIC CASE WILL BE CONDUCTED ON ALGORITHMS ALREADY DEVELOPED IN THE ACOUSTIC APPROXIMATION, REQUIRING SMALL COMPUTING TIMES. THE RECENTLY DEVELOPED ALGORITHM FOR SOLVING THE ELASTODYNAMIC EQUATION BY THE IMPLICIT METHOD WILL BE ADAPTED TO THE TYPICAL GEOMETRIES OF SEISMIC REFLECTION DATA. THE FOLLOWING POINTS WILL BE EXAMINED FOR MODELLING PURPOSES: LOW VELOCITY ZONE, RADIATION DIAGRAMS OF REAL SOURCES, DIRECTIONAL DIAGRAMS OF RECEIVER ARRAYS, AQUATIC PROPAGATION.

NEXT, THE LINEARIZED INVERSE PROBLEM WILL BE STUDIED BY USING RAY-TRACING METHODS IN THE ELASTODYNAMIC CONTEXT. THE PROPER CHOICE OF THE PARAMETERS TO BE CONSIDERED WILL ALLOW DRAMATIC SAVINGS IN COMPUTING TIME. SEVERAL METHODS WILL BE TESTED: STEEPEST DESCENT, CONJUGATE GRADIENT, VARIABLE METRICS, ETC.

A STRATEGY WILL BE CHOSEN FOR THE SEQUENCE OF OPTIMIZATION OF THE PHYSICAL PARAMETERS. TECHNIQUES OF LINEARIZED INVERSION WILL THEN BE DEVELOPED BY USING RAY- AND K-W SPACE METHODS. THEN SEVERAL OPTIMIZATION TECHNIQUES FITTED TO THE NON-LINEAR PROBLEM WILL BE EXPERIENCED.

A PARTICULAR ATTENTION WILL BE PAID TO A NUMBER OF POINTS SUCH AS: AVOIDING CONSIDERING SECONDARY MINIMA BY USING PREVIOUS INFORMATION ON THE MEDIUM, AVOIDING DISTURBANCES FROM FREE SURFACE EFFECTS, REDUCING THE SHOT NUMBERS. ALGORITHMS USING THE ELASTIC APPROXIMATION WILL BE DEVELOPED. AS TRUE 3D MODELLING WOULD BE TOO EXPENSIVE ON THE PRESENT GENERATION COMPUTERS, EVALUATIONS WILL BE MADE OF THE BIASING- AND POSSIBLY OF ITS CORRECTION - ARISING FROM THE 2D MODELLING. THEORETICAL INVESTIGATIONS WILL BE MADE IN MULTIDIMENSIONAL INVERSION THEORY INCLUDING ATTENUATION. THE NON-LINEAR INVERSION OF REAL DATA WILL BE TESTED BY USING ANELASTIC MODELLING BY F.D.

THE LINEARIZED INVERSION PROGRAMMES WILL BE OPTIMIZED FOR IMPLEMENTATION ON VECTOR SUPERCOMPUTERS MULTIPROCESSORS (VSCMP) AND RENDERED INDUSTRIALLY USABLE.

3D WAVE PROPAGATIONS WILL BE TESTED ON SECOND GENERATION PARALLEL COMPUTERS (MIPAX, AMETEX, INTEL) WITH LIKELY INTERESTING PERFORMANCES. FINALLY THE NON-LINEAR INVERSION PROGRAMMES WILL BE OPTIMIZED FOR VSCMP AND RENDERED INDUSTRIALLY USABLE. EACH METHOD WILL BE TESTED ON THIRD GENERATION PARALLEL COMPUTERS AS SOON AS THEY WILL BECOME AVAILABLE.