Robert Sun

Prestack scalar reverse-time depth migration of 3D elastic seismic data

Using two independent, 3D scalar reverse-time depth migrations, we migrate the reflected P- and S-waves in a prestack 3D, three-component (3-C), elastic seismic data volume generated with a P-wave source in a 3D model and recorded at the top of the model. Reflected P- and S-waves are extracted by divergence (a scalar) and curl (a 3-C vector) calculations, respectively, during shallow downward extrapolation of the elastic seismic data. The imaging time for the migrations of both the reflected P- and P-S converted waves at each point is the one-way P-wave traveltime from the source to that point.The divergence (the extracted P-waves) is reverse-time extrapolated using a finite-difference solution of the 3D scalar wave equation in a 3D P-velocity modeland is imaged to obtain the migrated P-image. The curl (the extracted S-waves) is first converted into a scalar S-wavefield by taking the curl’s absolute value as the absolute value of the scalar S-wavefield and assigning a positive sign if the curl is counterc...

Separating P- and S-waves in prestack 3D elastic seismograms using divergence and curl

The reflected P‐ and S‐waves in a prestack 3D, three‐component elastic seismic section can be separated by taking the divergence and curl during finite‐difference extrapolation. The elastic seismic data are downward extrapolated from the receiver locations into a homogeneous elastic computational model using the 3D elastic wave equation. During downward extrapolation, divergence (a scalar) and curl (a three‐component vector) of the wavefield are computed and recorded independently, at a fixed depth, as a one‐component seismogram and a three‐component seismogram, respectively. The P‐ and S‐velocities in the elastic computational model are then split into two independent models. The divergence seismogram (containing P‐waves only) is then upward extrapolated (using the scalar wave equation) through the P‐velocity model to the original receiver locations at the surface to obtain the separated P‐waves. The x‐component, y‐component, and z‐component seismograms of the curl (containing S‐waves only) are upward ex...

Amplitude balancing in separating P- and S-waves in 2D and 3D elastic seismic data

The reflected P- and S-waves in elastic displacement component data recorded at the earth’s surface are separated by reverse-time (downward) extrapolation of the data in an elastic computational model, followed by calculations to give divergence (dilatation) and curl (rotation) at a selected reference depth. The surface data are then reconstructed by separate forward-time (upward) scalar extrapolations, from the reference depth, of the magnitude of the divergence and curl wavefields, and extraction of the separated P- and S-waves, respectively, at the top of the models. A P-wave amplitude will change by a factor that is inversely proportional to the P-velocity when it is transformed from displacement to divergence, and an S-wave amplitude will change by a factor that is inversely proportional to the S-velocity when it is transformed from displacement to curl. Consequently, the ratio of the P- to the S-wave amplitude (the P-S amplitude ratio) in the form of divergence and curl (postseparation) is different...

3-D prestack full-wavefield inversion

A full-wavefield inversion algorithm for direct imaging of a 3-D compressional wave velocity distribution is based on the full 3-D scalar wave equation and operates on common-source data recorded by areal arrays. For each source, the method involves reverse-time extrapolation of the residual wavefield. Application of the image condition by crosscorrelation with the source wavefield at each time step produces a 3-D image whose amplitude at each point is proportional to the required velocity update at that point. Convergence to local minima is mitigated against by gradually increasing the wavenumber bandwidth in the estimated 3-D velocity distribution as iterations proceed, starting from the smallest wavenumber. The algorithm is illustrated by successful application to synthetic data for a multilayered monocline, and for a multilayered structure with the geometry of the standard French model. The latter demonstrates good performance with noisy, unequally spaced data with significant elevation statics.

Finite-difference modeling of borehole resonances

Abstract Most attempts at numerical synthesis of vertical seismic profiles ( VSPs) do not explicitly include the interaction of seismic waves with the borehole, the casing, or instruments located within the hole. Thus, borehole resonances, a type of tube wave that is prominent in many real VSPs, are typically missing in synthetic VSPs. Characteristics of generation and propagation of borehole resonances can be studied by including details of the borehole environment in acoustic and elastic finite-difference simulations. Specific models presented incorporate a variety of hole diameters, cased and uncased holes, the presence of geophones within a hole, a variety of source positions, the effects of local changes in hole diameter and lithology, and the mutual interaction of the response of two neighboring wells. The main conclusion is that the hole and the medium should be considered as a single coupled system as there is a continuous mutual transfer of energy between them.

Amplitude effect of the free surface in elastic reverse-time extrapolation

We evaluate the physical validity of surface boundary conditions of the computational model in reverse-time extrapolation of 3D, three-component (3-C) elastic seismic data acquired at the earth’s free surface by using mathematical derivations and numerical simulations. Reverse-time extrapolation of elastic data assumes that only the incident P- or S-waves are reconstructed during extrapolation into the computational grid. However, superposition of the (upgoing) incident waves and the (downgoing) reflected and converted waves generated at the free surface also is recorded in data acquisition and is input into reverse-time extrapolation. In elastic reverse-time extrapolation, the computational model needs to have an absorbing top boundary. When the 3D, 3-C elastic data are inserted into the computational model during reverse-time extrapolation, the originally incident P- or S-wave is reconstructed. In addition, the free-surface P-to-P reflected and P-to-S converted waves recombine to reconstruct a second in...

Nonlinear Reverse-time Inversion of Elastic Offset VSP Data

An iterative nonlinear inversion algorithm for two-dimensional elastic media gives estimates of P-velocity and S-velocity distributions from synthetic offset vertical seismic profiles. The algorithm is a hybrid of inversion, augmented with principles borrowed from reverse-time migration. Gradients of the misfit function are dynamically determined by cross-correlations of the incident wavefields with the scattered compressional and shear wavefields. Model perturbations are defined in the steepest descent direction. In order to optimize the sensitivity of the inversion to both compressional and shear velocity distributions, two data collection experiments are required, one with a compressional wave source and the other with a shear wave source.

Line sources for seismic modeling by finite differences in inhomogeneous media

Abstract Initial (source) conditions for second-order finite-difference seismic-wave equation computations are usually implemented by inserting an analytic solution of the wave equation into the finite difference grid at two successive time steps. This approach is difficult to use in an arbitrarily inhomogeneous model and when free-surface boundary conditions are a required component due to proximity of the source to that surface. These problems can be overcome by noting that a small aperture behaves as a point source. In this algorithm, an aperture placed at the free-surface admits energy to the inhomogeneous structure from a small portion (a ‘line’) of a wavefront propagating in an auxiliary homogeneous medium defined above the free-surface. After the source pulse has passed through the aperture, the aperture is closed and is replaced by the free-surface boundary conditions. Examples include both acoustic and elastic sources initiated by both spherical and plane waves in both homogeneous and heterogeneous media.