Linearized inversion of 3-D multi-offset data: background reconstruction and AVO inversion

Abstract

SUMMARY We present a linearized inversion algorithm for 3-D multi-offset seismic reflection data. The fundamental assumptions are that the background medium is only depth-dependent and that the 3-D multi-offset seismic data are regularly sampled as a series of 2-D multi-offset profiles. Our algorithm can be performed in the following three steps. (1) The pressure field P(x, y, h, t) is 4-D Fourier transformed to P(kx, ky, kh, ω) with respect to the midpoint coordinates (x, y), the half-offset coordinates (h, 0), and the time t. (2) A 2-D inversion operator is applied to P(kx, ky, kh, ω) for each given midpoint wavenumber ky. This computation can be carried out in parallel for all the ky wavenumbers so that this step becomes numerically equivalent to a 2-D multi-offset inversion in the Fourier domain. (3) A deconvolution by the source and for spatial effects is applied to correct for geometrical spreading and to separate the elastic parameters. About 98 per cent of the computation time of this algorithm takes place in the second step. Therefore it is worthwhile to note that this step is suitable for a data-parallel computing model. Most of the central memory and/or storage device required in this algorithm is used to store the 4-D FFT of the pressure field. By taking advantage of the limited frequency content of the seismic data, we can reduce its storage to about 4 Gbyte for a section of 256 lines, each containing up to 256 CMP. The input for this algorithm is a smooth, depth-dependent, background velocity model (the low spatial frequencies), and the results are the contrasts of P-wave impedance and acoustic velocity (the high spatial frequencies). The inversion for high spatial frequencies of P-wave impedance is essentially based on small-offset data, while that of acoustic velocity is essentially based on large-offset data. By using the equivalence between acoustic scattering and P-P scattering, the algorithm can also output the classical AVO parameters: P-wave intercept and gradient coefficients. This algorithm can alternatively be used to estimate the background velocity as follows. First, the pressure field is 4-D Fourier transformed, and inversions with various background velocity models are performed. For isotropic media, the good models for the background are then extracted by analysing outputs of P-wave impedance or acoustic velocity, as both yield the same background velocity. The anisotropic behaviour of the background model is depicted by differences between the background model obtained from the analysis of P-wave impedance outputs and that obtained from the analysis of acoustic velocity outputs. The computation time, in this process, can be substantially reduced either by wavenumber filtering because we are seeking a coarse grid model or by using analytic background velocity models to simplify the inversion solution further.