Jacob T. Fokkema

Green's function representations for seismic interferometry

The term seismic interferometry refers to the principle of generating new seismic responses by crosscorrelating seismic observations at different receiver locations. The first version of this principle was derived by Claerbout (1968), who showed that the reflection response of a horizontally layered medium can be synthesized from the autocorrelation of its transmission response. For an arbitrary 3D inhomogeneous lossless medium it follows from Rayleighs reciprocity theorem and the principle of time-reversal invariance that the acoustic Greens function between any two points in the medium can be represented by an integral of crosscorrelations of wavefield observations at those two points. The integral is along sources on an arbitrarily shaped surface enclosing these points. No assumptions are made with respect to the diffusivity of the wavefield. The Rayleigh-Betti reciprocity theorem leads to a similar representation of the elastodynamic Greens function. When a part of the enclosing surface is the earths free surface, the integral needs only to be evaluated over the remaining part of the closed surface. In practice, not all sources are equally important: The main contributions to the reconstructed Greens function come from sources at stationary points. When the sources emit transient signals, a shaping filter can be applied to correct for the differences in source wavelets. When the sources are uncorrelated noise sources, the representation simplifies to a direct crosscorrelation of wavefield observations at two points, similar as in methods that retrieve Greens functions from diffuse wavefields in disordered media or in finite media with an irregular bounding surface.

Retrieving the Green’s function in an open system by cross correlation: A comparison of approaches (L)

We compare two approaches for deriving the fact that the Green’s function in an arbitrary inhomogeneous open system can be obtained by cross correlating recordings of the wave field at two positions. One approach is based on physical arguments, exploiting the principle of time-reversal invariance of the acoustic wave equation. The other approach is based on Rayleigh’s reciprocity theorem. Using a unified notation, we show that the result of the time-reversal approach can be obtained as an approximation of the result of the reciprocity approach.

Three-dimensional imaging of multicomponent ground-penetrating radar data

Scalar imaging algorithms originally developed for the processing of remote sensing measurements (e.g., the synthetic‐aperture radar method) or seismic reflection data (e.g., the Gazdag phase‐shift method) are commonly used for the processing of ground‐penetrating radar (GPR) data. Unfortunately, these algorithms do not account for the radiation characteristics of GPR source and receiver antennas or the vectorial nature of radar waves. We present a new multicomponent imaging algorithm designed specifically for vector electromagnetic‐wave propagation. It accounts for all propagation effects, including the vectorial characteristics of the source and receiver antennas and the polarization of the electromagnetic wavefield. A constant‐offset source‐receiver antenna pair is assumed to overlie a dielectric medium. To assess the performance of the scalar and multicomponent imaging algorithms, we compute their spatial resolution function, which is defined as the image of a point scatterer at a fixed depth using a ...

Rayleigh Hypothesis in the Theory of Reflection by a Grating

In this paper, the Rayleigh hypothesis in the theory of reflection by a grating is investigated analytically. Conditions are derived under which the Rayleigh hypothesis is rigorously valid. A procedure is presented that enables the validity of the Rayleigh hypothesis to be checked for a grating whose profile can be described by an analytic function. As examples, we consider some grating profiles described by a finite Fourier series. Numerical results are then presented.

The Rayleigh hypothesis in the theory of diffraction by a cylindrical obstacle

The Rayleigh hypothesis in the theory of scattering by a cylindrical obstacle of arbitrary cross section is investigated analytically. The hypothesis asserts that outside and on the obstacle the scattered field may be expanded in terms of outward-going wave functions of the circular cylinder. As such, it is analogous to the assumption made by Lord Rayleigh in his treatment of diffraction by a reflection grating. We show that the validity of the Rayleigh hypothesis is governed by the distribution of singularities in the analytic continuation of the exterior scattered field. Conditions are derived under which the Rayleigh hypothesis is rigorously valid. As examples, the elliptic cylinder and the perturbed circular cylinder are considered in detail.

Removal of surface-related wave phenomena—The marine case

Removal of the effects of the free surface from seismic reflection data is an essential preprocessing step before prestack migration. The problem can be formulated by means of Rayleigh’s reciprocity theorem which leads to an integral equation of the second kind for the desired pressure field that does not include these free‐surface effects. This integral equation can be solved numerically, both in the spatial domain and in the double Radon domain. Solving the integral equation in the double Radon domain has the advantage of reducing the computation time significantly since the kernel of the integral equation becomes dominant diagonally. Two methods are proposed to solve the integral equation: direct matrix inversion and a recursive subtraction of the free‐surface multiples using a Neumann series. Both methods have been developed and tested on a synthetic data set, which was computed with the help of an independent forward‐modeling scheme.

Plane-wave depth migration

We present fast and efficient plane-wave migration methods for densely sampled seismic data in both the source and receiver domains. The methods are based on slant stacking over both shot and receiver positions (or offsets) for all the recorded data. If the data-acquisition geometry permits, both inline and crossline source and receiver positions can be incorporated into a multidimensional phase-velocity space, which is regular even for randomly positioned input data. By noting the maximum time dips present in the shot and receiver gathers and constant-offset sections, the number of plane waves required can be estimated, and this generally results in a reduction of the data volume used for migration. The required traveltime computations for depth imaging are independent for each particular plane-wave component. It thus can be used for either the source or the receiver plane waves during extrapolation in phase space, reducing considerably the computational burden. Since only vertical delay times are required, many traveltime techniques can be employed, and the problems with multipathing and first arrivals are either reduced or eliminated. Further, the plane-wave integrals can be pruned to concentrate the image on selected targets. In this way, the computation time can be further reduced, and the technique lends itself naturally to a velocity-modeling scheme where, for example, horizontal and then steeply dipping events are gradually introduced into the velocity analysis. The migration method also lends itself to imaging in anisotropic media because phase space is the natural domain for such an analysis.

Elastodynamic Diffraction by a Periodic Rough Surface (Stress-Free Boundary)

The reflection of an elastic wave by a rough stress‐free surface with a periodic profile has been investigated rigorously. The problem is formulated in terms of an integral equation for the particle displacement at a single period of the boundary surface. Numerical results pertaining to the reflection of either an incident P wave or an incident SV wave for a sinusoidal profile are presented.

Reciprocity Theorems for Diffusion, Flow, and Waves

Diffusion, flow, and wave phenomena can each be captured by a unified differential equation in matrix-vector form. This equation forms the basis for the derivation of unified reciprocity theorems for diffusion, flow and wave phenomena.

Monitoring the width of hydraulic fractures with acoustic waves

During scaled hydraulic fracturing experiments in our laboratory, the fracture growth process is monitored in a time‐lapse experiment with ultrasonic waves. We observe dispersion of compressional waves that have propagated across the hydraulic fracture. This dispersion appears to be related to the width of the hydraulic fracture. This means that we can apply the dispersion measurements to monitor the width of the hydraulic fracture in an indirect manner. For a direct determination of the width, the resolution of the signal is required to distinguish the reflections that are related with two distinct fluid/solid interfaces delimiting the hydraulic fracture from its solid embedding. To make this distinction, the solid/fluid interfaces must be separated at least one eighth of a wavelength and represent sufficient impedance contrast. The applicability of the indirect dispersion measurement method however, extends to a fracture width that is in the order of 1% of the incident wavelength. The time‐lapse ultraso...