Synthetic Wavefields Research Articles

Inversion of Seismic Wavefield Data for P- and S-wave Velocity

Described and illustrated are seismic inversion formulas derived by Tarantola (1987) and Mora (1987d). The formulas are based on nonlinear least squares iterations to find the elastic properties of the Earth corresponding to the synthetic wavefield that best matches the seismic observations. Primary P- and S-wave reflections, mode converted waves and Rayleigh waves are all theoretically useful in the inversions. Beginning with a starting guess, the Earth properties are iteratively updated using a preconditioned conjugate?gradient algorithm. The gradient direction is cast in terms of two wave propagations which can be easily implemented on fast fine grain parallel computers such as the 'Connection Machine' (see Hillis (1986)). The method is tested by inverting a shot profile. The wave propagations are done/using elastic finite differences to allow for Earth models of realistic complexity. The results verify the theoretical predictions of Mora (1987c) that the iterative elastic inversion formulas are capable of obtaining all wavenumbers of the compressional and shear wavespeeds that are resolvable separately by prestack elastic migration and elastic reflection tomography.

A comparison of common-midpoint, single-shot, and plane-wave depth migration

A comparison of common-midpoint (CMP), single-shot, and plane-wave migration was made for simple two-dimensional structures such as a syncline and a horizontal reflector with a laterally variable reflection coefficient by using synthetic seismograms. The seismograms were calculated employing the finite-difference technique. CMP sections were simulated by 18-fold stacking and plane-wave sections by slant stacking. By applying a finite-difference scheme, the synthetic wave field was continued downward. The usual imaging condition of CMP migration was extended in order to carry out migration of single-shot and plane-wave sections. The reflection coefficient was reconstructed by comparing the migrated wave field with the incident wave field at the reflector. The results are: (1) all three migration techniques succeeded in reconstructing the reflector position; (2) as a consequence of the finite aperture of the geophone spread, only segments of the reflector could be reconstructed by single-shot and planewave migration; (3) for single-shot and plane-wave migration the reflection coefficient could be obtained; and (4) CMP migration may lead to incorrect conclusions regarding the reflection coefficient.

A comparison of common‐midpoint, single‐shot, and plane‐wave depth migration

A comparison of common‐midpoint (CMP), single‐shot, and plane‐wave migration was made for simple two‐dimensional structures such as a syncline and a horizontal reflector with a laterally variable reflection coefficient by using synthetic seismograms. The seismograms were calculated employing the finite‐difference technique. CMP sections were simulated by 18-fold stacking and plane‐wave sections by slant stacking. By applying a finite‐difference scheme, the synthetic wave field was continued downward. The usual imaging condition of CMP migration was extended in order to carry out migration of single‐shot and plane‐wave sections. The reflection coefficient was reconstructed by comparing the migrated wave field with the incident wave field at the reflector. The results are: (1) all three migration techniques succeeded in reconstructing the reflector position; (2) as a consequence of the finite aperture of the geophone spread, only segments of the reflector could be reconstructed by single‐shot and plane‐wave migration; (3) for single‐shot and plane‐wave migration the reflection coefficient could be obtained; and (4) CMP migration may lead to incorrect conclusions regarding the reflection coefficient.

A New Method for Measuring the Directional Wave Spectrum. Part I. Description

A new method for measuring the directional wave spectrum is introduced, and the usefulness of the method is examined through numerical analysis of synthetic wave fields. The principle of the method consists of the direct application of the fast Fourier transform technique to the two-dimensional spectrum estimation. Unlike other known methods, the procedure is not based explicitly on the evaluation of the cross-covariance function or the cross spectrum. Two examples of wave gage arrangement for the method have been presented. The first army (Array A) consists of a set of twelve wave detectors equally spaced around a circle, and the second (Army B) consists of eight detectors on an outer circle and four detectors on an inner circle with half the radius of the outer. Denoting the nondimensional scale of the wave detector array by = 2r/L [where r is the radius of the (outer) wave detector circle and L the wavelength], the ranges in which the method is useful turn out to be 1.0 ≤ ≤ 1.18 and 1.25 ≤ ≤ 1.60 for Array A and 0.6 ≤ ≤ 1.7 for Array B. In this method, the use of an improper dispersion relation for actual wind waves is a possible factor in methodological error. Many of the spurious estimates obtained can be ascribed to the use of improper dispersion relations. Using this fact, it is possible to determine a reasonable dispersion relation for actual wind waves.