Joe Dellinger

Kirchhoff modeling, inversion for reflectivity, and subsurface illumination

Because of its computational efficiency, prestack Kirchhoff depth migration is currently one of the most popular algorithms used in 2-D and 3-D subsurface depth imaging. Nevertheless, Kirchhoff algorithms in their typical implementation produce less than ideal results in complex terranes where multipathing from the surface to a given image point may occur, and beneath fast carbonates, salt, or volcanics through which ray‐theoretical energy cannot penetrate to illuminate underlying slower‐velocity sediments. To evaluate the likely effectiveness of a proposed seismic‐acquisition program, we could perform a forward‐modeling study, but this can be expensive. We show how Kirchhoff modeling can be defined as the mathematical transpose of Kirchhoff migration. The resulting Kirchhoff modeling algorithm has the same low computational cost as Kirchhoff migration and, unlike expensive full acoustic or elastic wave‐equation methods, only models the events that Kirchhoff migration can image. Kirchhoff modeling is also...

Do traveltimes in pulse-transmission experiments yield anisotropic group or phase velocities?

The elastic properties of layered rocks are often measured using the pulse through‐transmission technique on sets of cylindrical cores cut at angles of 0, 90, and 45 degrees to the layering normal (e.g., Vernik and Nur, 1992; Lo et al., 1986; Jones and Wang, 1981). In this method transducers are attached to the flat ends of the three cores (see Figure 1), the first‐break traveltimes of P, SV, and SH‐waves down the axes are measured, and a set of transversely isotropic elastic constants are fit to the results. The usual assumption is that frequency dispersion, boundary reflections, and near‐field effects can all be safely ignored, and that the traveltimes measure either vertical anisotropic group velocity (if the transducers are very small compared to their separation) or phase velocity (if the transducers are relatively wide compared to their separation) (Auld, 1973).

Modeling elastic fields across irregular boundaries

Geologists often see the earth as homogeneous blocks separated by smoothly curving boundaries. In contrast, computer modeling algorithms based on finite‐difference schemes require elastic constants to be specified on the vertices of a regular rectangular grid. How can we convert a continuous geological model into a form suitable for a finite‐difference grid? One common way is to lay the finite‐difference grid down on the continuous geological model and use whatever elastic constants happen to lie beneath each of the grid points.

Imaging reflections in elliptically anisotropic media

In an isotropic medium, waves reflected from a mirror form a virtual image of their source. This property of planar reflectors is generally not true in the presence of anisotropy. In their short note, Blair and Korringa (1987) show that for the special case of SH waves from a point source in a transversely isotropic medium, an aberration‐free image is formed for any orientation of the mirror. While their proof is mathematical, we show the same result in an intuitive, pictorial fashion and in the process discover that although the image is indeed aberration free, it is still distorted.

Anisotropic finite‐difference traveltimes using a Hamilton‐Jacobi solver

Van Trier and Symes (1991) introduced a simple method for calculating first-arrival traveltimes based on recasting the eikonal equation in flux-conservative form. In that form, the eikonal equation can be efficiently solved using the first-order Engquist-Osher scheme (Engquist and Osher, 1980). Dellinger (1991b) later showed how to generalize their method to allow for transverse isotropy aligned with Cartesian coordinate axes. Unfortunately, the method does not easily generalize to allow general anisotropy, high-order numerical accuracy, or threedimensional propagation.

Computing the optimal transversely isotropic approximation of a general elastic tensor

Mathematically, 21 stiffnesses arranged in a 6 × 6 symmetric matrix completely describe the elastic properties of any homogeneous anisotropic medium, regardless of symmetry system and orientation. However, it can be difficult in practice to characterize an anisotropic mediums properties merely from casual inspection of its (often experimentally measured) stiffness matrix. For characterization purposes, it is better to decompose a measured stiffness matrix into a stiffness matrix for a canonically oriented transversely isotropic (TI) medium (whose properties can be readily understood) plus a generally anisotropic perturbation (representing the mediums deviation from perfect symmetry), followed by a rotation (giving the relationship between the mediums natural coordinate system and the measurement coordinate system). To accomplish this decomposition, we must find the rotated symmetric medium that best approximates a given stiffness matrix. An analytical formula exists for calculating the distance between...

Efficient Estimates of Subsurface Illumination for Kirchhoff Prestack Depth Migration

Summary Because of its computational efficiency, prestack Kirchhoff depth migration is currently the most popular algorithm used in 2-D and 3-D subsurface imaging. Nevertheless, Kirchhoff algorithms in their typical embodiment produce less than ideal results in complex terranes where we may encounter multipathing from the surface to a given image point, and beneath fast carbonates, salt or volcanics where ray theoretical energy cannot penetrate through to illuminate slower velocity sediments. When faced with particularly difficult to understand illumination problems, we might exploit full acoustic or elastic wave equation forward modeling to evaluate the effectiveness of a proposed seismic acquisition program. Unfortunately, seismic events that are predicted to reach the earth’s surface with sufficient amplitude may not be among those that can be imaged by our production Kirchhoff imaging scheme. Worse yet, full wave equation prestack modeling may well cost more than the field acquisition itself, and several orders of magnitude more than the Kirchhoff migration step that produces a useful image. We show here how Kirchhoff modeling, the mathematical adjoint of Kirchhoff migration, can be most useful in determining which components of signal and noise, including diffractions, can be imaged by Kirchhoff migration before acquisition begins. Kirchhoff modeling is a necessary element of least square Kirchhoff migration that produces an improved estimate of reflectivity that compensates for irregularities in surface sampling, including missing data, as well as for irregularities in ray coverage due to strong lateral variations in velocity. As a by-product we also obtain an image of subsurface illumination that is a measure of our confidence in our least square reflectivity estimate.

Eisner’s reciprocity paradox and its resolution

The principle of reciprocity says that when a vertical source (a vibrator) and a vertical receiver (a geophone) are interchanged, the same seismogram will be recorded in each case. In a field study conducted by D. Fenati and F. Rocca in 1984, it was found that the reciprocal principle also applies surprisingly well even when the required conditions are technically violated, such as when an isotropic source (dynamite) is used with a vertical receiver (again a geophone).

Low frequencies with a “dense” OBS array: The Atlantis‐Green Canyon earthquake data set

We extracted 2.5 hours of continuous data from the Atlantis OBSnode array around the time of the 10 February, 2006 magnitude-5.2 Green-Canyon earthquake, for the purpose of characterizing the location, mechanism, and significance of this unusual event. The array, consisting of about 500 active 4C nodes, was several tens of kilometers to the South of the earthquake. Typical earthquakes radiate the bulk of their energy at frequencies much lower than the 10Hz geophones used in the nodes were designed to record, so the dataset also served as a testbed for understanding how standard exploration-seismic geophones might be used at frequencies below 5Hz. At traditional frequencies of 10Hz and above the Atlantis airgun signal completely dominates the data. However, filtering away frequencies above 2Hz removes the Atlantis airgun signal and reveals a strong series of arrivals from the earthquake. At “earthquake” frequencies of 2Hz and below, the Atlantis array becomes densely sampled in space (i.e., with more than 2 receivers per wavelength), allowing it to be formed into a powerful directional antenna. Beam steering the array reveals many interesting signals in the data. Examining our own airguns, we find that ”inconsequential” differences in our airgun arrays had unexpected and significant effects. Beam steering resolves a complex sequence of distinct arrivals from the Green-Canyon earthquake spanning 8 minutes of time, and two other mysterious events subsequent to the Green-Canyon earthquake that appear to be temporally related to it, but cannot be traditional aftershocks because they arrive from different azimuths.

A crossed‐dipole reciprocity “paradox”

The principle of seismic reciprocity states that the roles of sources and receivers in seismic experiments can be interchanged, and the same signal as a function of time will be recorded in both cases. In a multicomponent zero‐offset shear‐wave experiment, the sources and receivers are at the same location, and the reciprocal experiment is the same as the original. Reciprocity thus predicts that the recorded XY (inline source to crossline receiver) and YX (crossline source to inline receiver) sections should be identical. This should be true regardless of anisotropy, attenuation, or heterogeneity in the Earth.