International audienceNonlinear elastic waveform inversion has advanced to the point where it is now possible to invert real multiple‐shot seismic data. The iterative gradient algorithm that we employ can readily accommodate robust minimization criteria which tend to handle many types of seismic noise (noise bursts, missing traces, etc.) better than the commonly used least‐squares minimization criteria. Although there are many robust criteria from which to choose, we have tested only a few. In particular, the Cauchy criterion and the hyperbolic secant criterion perform very well in both noise‐free and noise‐added inversions of numerical data. Although the real data set, which we invert using the sech criterion, is marine (pressure sources and receivers) and is very much dominated by unconverted P waves, we can, for the most part, resolve the short wavelengths of both P impedance and S impedance. The long wavelengths of velocity (the background) are assumed known. Because we are deriving nearly all impedance information from unconverted P waves in this inversion, data acquisition geometry must have sufficient multiplicity in subsurface coverage and a sufficient range of offsets, just as in amplitude‐versus‐offset (AVO) inversion. However, AVO analysis is implicitly contained in elastic waveform inversion algorithms as part of the elastic wave equation upon which the algorithms are based. Because the real‐data inversion is so large—over 230,000 unknowns (340,000 when density is included) and over 600,000 data values—most statistical analyses of parameter resolution are not feasible. We qualitatively verify the resolution of our results by inverting a numerical data set which has the same acquisition geometry and corresponding long wavelengths of velocity as the real data, but has semirandom perturbations in the short wavelengths of P and S impedance

International audienceThe aim of inverting seismic waveforms is to obtain the “best” earth model. The best model is defined as the one producing seismograms that best match (usually under a least‐squares criterion) those recorded. Our approach is nonlinear in the sense that we synthesize seismograms without using any linearization of the elastic wave equation. Since we use rather complete data sets without any spatial aliasing, we do not have the problem of secondary minima (Tarantola, 1986). Nevertheless, our gradient methods fail to converge if the starting earth model is far from the true earth (Mora, 1987; Kolb et al., 1986; Pica et al., 1989)

Interpretation of seismic waveforms can be expressed as an optimization problem based on a non‐linear least‐squares criterion to find the model which best explains the data. An initial model is corrected iteratively using a gradient method (conjugate gradient). At each iteration, computation of the direction of the model perturbation requires the forward propagation of the actual sources and the reverse‐time propagation of the residuals (misfit between the data and the synthetics); the two wave fields thus obtained are then correlated. An extra forward propagation is required to compute the amplitude of the perturbation along the conjugate‐gradient direction. The number of propagations to be simulated numerically in each iteration equals three times the number of shots. Since a 2-D finite‐difference code is employed to solve forward‐ and backward‐propagation problems, the method is general and can handle arbitrary 2-D source‐receiver configurations and lateral heterogeneities. Using conventional velocity analysis to derive an initial velocity model, the method is implemented on a real marine data set. The data set which has been selected corresponds approximately to a horizontally stratified medium. Consequently, a single‐shot gather has been used for inversion. In spite of some simplifying assumptions used for wave‐field propagation (acoustic approximation, point source), and using synthetics generated by a nearby sonic log to calibrate amplitudes, our final synthetics match the input data very well and the inversion result has clear similarities to the log.

Recorded seismic reflection waveforms contain information as to the small-scale variations of impedance and the large-scale variations of velocity. This information can be retrieved by minimizing the misfit between the recorded waveforms and synthetic seismograms as a function of the model parameters. Because of the different physical characters of the velocity and the impedance, we update these parameters in an alternating fashion, which amounts to a relaxation approach to the minimization of the waveform misfit. As far as the impedance is concerned, this minimization can be performed efficiently using gradient algorithms. For the inversion for seismic velocities, gradient methods do not work nearly as well; therefore, we use different minimization methods for INTRODUCTION