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.
The computation of synthetic seismograms can be linearized with respect to a reference medium that issno close to the actual medium. Using a least‐squares formulation, the inverse problem can then be set up as a problem of quadratic optimization. The inverse problem is greatly simplified if the reference medium is symmetric. For a homogeneous reference medium, a rigorous and economic solution can be obtained by Fourier transforming all spatial variables. In particular, the solution can be obtained through an explicit formula that does not require the resolution of any linear system (as is the case when not working in the Fourier domain). However, the assumption of a homogeneous reference medium is generally not realistic. In some situations, the reference medium can be depth‐dependent. It can then be shown that by Fourier transforming time and all spatial variables except depth, the inverse problem also has an elegant and economic solution. If [Formula: see text] is the (unknown) difference between the reference medium and the true (2-D) medium, the Fourier‐transformed solution [Formula: see text] can be obtained by solving, for each value of the horizontal wavenumber [Formula: see text] a linear system whose dimension equals the number of depth samples.
In the acoustic approximation, the Earth is described using only density and bulk modulus. Assuming smooth density variations, reflections can be described using a single function—the velocity of compressional waves. If a reference model which is close enough to the actual Earth is known, the problem of estimating the medium velocity from the observed data can be linearized. Using a least‐squares formulation and working in the ω-k domain, the linearized inverse problem for a homogeneous reference medium can be solved by a noniterative algorithm which is economically competitive with prestack migration. Numerical tests with synthetic and real data demonstrate the feasibility and the numerical stability of the method. The numerical results compare well with those obtained by migration of unstacked data, although superior results will only be obtained when the physics of the problem (including elastic versus acoustic effects, three‐dimensional propagation, and accurate source estimation) will realistically be taken into account.
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