K.1 Zero-offset traveltime modeling
Seismic modeling essentially is a simulation of a recorded seismic wavefield, seismic amplitudes, or seismic traveltimes. The input to seismic modeling is a representation of the earth’s reflectivity and a velocity-depth model. Seismic migration is a process of estimating earth’s reflectivity from a recorded seismic wavefield using a velocity-depth model. Therefore, seismic wavefield modeling may be viewed as the reverse process of seismic migration. As such, both seismic migration and seismic wavefield modeling algorithms are based on the wave equation.
Given a seismic wavefield P(x, z = 0, t) recorded over time t, at the surface z = 0, and along the spatial axis x, seismic migration yields the earth’s reflectivity P(x, z, t = 0) based on a process of wavefield extrapolation in depth z and collecting the image at time t = 0 (migration principles). Conversely, the fundamental ingredient of the modeling process is wavefield extrapolation in time t and collecting the result at depth z = 0. Both processes use wave equation as the basis for wave extrapolation.
Seismic modeling is different from data modeling (Appendix J). The latter involves, given an observed data set d, an estimation of a set of parameters p that are used to construct a model d′ of the observed data set d, such that the difference between the observed data set d and the modeled data set d′ is minimum based on a specific mathematical norm.
Throughout this book, we have seen numerous examples of seismic modeling:
- to explain a process such as deconvolution (Figure 2.1-3) or migration (Figure 4.0-8),
- to test an algorithm such as predictive deconvolution (Figure 2.4-13) or implicit frequency-space 3-D poststack time migration (Figure 8.4-2), or
- to understand a structural or stratigraphic phenomenon that may be of interest in exploration.
Just as there are several approaches to solving the wave equation for migration, there also are several types of modeling techniques. There are modeling techniques based on the Kirchhoff integral , finite-difference , and f − k domain  solutions to the wave equation. The algorithms based on the scalar (acoustic) wave equation (Section D.1), which describes P-wave propagation, are suitable for structural modeling in which amplitudes are not as important as traveltimes. The algorithms based on the elastic wave equation (Section L.2), which describes both P- and S-wave propagation, are suitable for detailed stratigraphic modeling in which amplitudes are as important as traveltimes. Modeling based on one-way wave equations does not include multiples, while modeling based on two-way wave equations includes multiples in the simulated wavefields. In this appendix, we shall not discuss the details of specific algorithms for seismic modeling which span a broad range of applications including 2-D and 3-D, zero-offset and nonzero-offset, acoustic and elastic simulation. Instead, we shall provide examples of the most common seismic modeling strategies — zero-offset traveltime modeling, zero-offset and nonzero-offset acoustic wavefield modeling, and elastic modeling.
Shown in Figure K-1a is a velocity-depth model for a salt diapir with an overhang structure. The zero-offset traveltime response shown in Figure K-1b is created by normal-incidence ray tracing. Note that the top-salt boundary has given rise to a complex and multivalued traveltime trajectory. Note also that the base-salt boundary is flat and continuous in the velocity-depth model (Figure K-1a), whereas the reflection traveltime follows a discontinuous and multivalued trajectory (Figure K-1b).
Zero-offset traveltime modeling using normal-incidence rays is a very useful and trivially simple tool for understanding the complexity of a reflection traveltime in field data. The disruptive behavior in traveltime trajectory associated with a layer boundary below a complex overburden as in Figure K-1b is observed also in real data (Figure 10.1-1).
Figure 2.1-3 A wavelet (a) traveling in the earth repeats itself when it encounters a reflector along its path (b, c, d, e, f). The left column represents the reflection coefficients, while the right column represents the response to the wavelet. Amplitudes of the response are scaled by the reflection coefficient. The resulting seismogram (bottom right) represents the composite response of the earth’s reflectivity (bottom left) to the wavelet (top right).
Figure 4.0-8 Exploding-reflector modeling of zero-offset traveltimes associated with (a) a water bottom, (b) a flat, and (c) a dipping reflector. (d) The superposition of the normal-incidence traveltime responses in (a), (b), and (c). Shown on the velocity-depth models in the left-hand column are the normal-incidence rays used to compute the traveltime trajectories. The time sections shown on the right-hand column are equivalent to a zero-offset traveltime section with the vertical axis in two-way time.
K.2 Zero-offset wavefield modeling
Recall from introduction to migration that a stacked section often is assumed to be a close representation of a zero-offset wavefield. A modeled zero-offset wavefield therefore can be used to test poststack migration algorithms. Zero-offset wavefields can be simulated very efficiently using the exploding reflectors, also discussed in introduction to migration.
Wave-equation datuming , which was described in layer replacement, can be used to perform the simulation based on exploding reflectors. In particular, the datuming approach can propagate a wavefield from one irregular interface to another. Consider zero-offset modeling using the datuming technique of the velocity-depth model shown in Figure K-2a. Horizons 2 and 3 are the top and base of a salt dome. Start with the receivers situated along horizon 3. The corresponding zero-offset section (Figure K-2b) contains the reflection from the bottom of the velocity-depth model at z = 4000 m (not shown in Figure K-2a). Take this wavefield and extrapolate it to a new datum, horizon 2, using the salt velocity (5000 m/s). The resulting zero-offset section (Figure K-2c) contains the reflection (the deeper one) from the bottom of the velocity-depth model (z = 4000 m) and the reflection (the shallow one) from the base of the salt (horizon 3). Finally, extrapolate this wavefield (Figure K-2c) from horizon 2 to the surface (horizon 1 at z = 0) using the overburden velocity (3000 m/s) to get the 2-D zero-offset section in Figure K-2d. This section contains reflections from both the top and base of the salt. (The reflection from the bottom of the model arrives after the latest time shown on this section.) Note the velocity pull-up along the reflection from the base of the salt dome. Proper imaging of the top of the salt dome can be achieved by time migration (introduction to migration), while proper imaging of the base of the salt requires depth migration (introduction to earth imaging in depth).
K.3 Nonzero-offset wavefield modeling
Understanding complexities of recorded wavefields clearly requires nonzero-offset wavefield modeling. A finite-difference technique for modeling acoustic and elastic wavefields is described by Kelly . Figure K-3 shows an example of acoustic modeling of a complex structure associated with overthrust tectonics. A seismic line is simulated over a 2-D complex structure (Figure K-3a). Selected common-shot (Figure K-3b) and CMP gathers (Figure K-3c) from this simulation show the many complexities in the arrivals. Since this is a two-way acoustic simulation, the modeled gathers contain not only primaries but also multiples. The zero-offset and stacked sections associated with this nonzero-offset data are shown in Figure K-4. Note the broad traveltime trajectories associated with the tight imbricate structures in the velocity-depth model (Figure K-3a).
An example of a nonzero-offset modeling application of wave-equation datuming is provided in Figure K-5. The shot gathers in Figure K-3b are computed to a flat datum level z = 0. Better simulation of the actual field conditions requires that the gathers be computed using an irregular topography. To do this, we can upward continue the shots and receivers to the new irregular datum represented by the topography shown in Figure K-5a, then compute the shot gathers in Figure K-5b and sort them to the CMP gathers shown in Figure K-5c. Compare Figures K-3b and K-3c with Figures K-5b and K-5c, and note the traveltime distortions.
Figure K-6 shows a velocity-depth model associated with a salt sill structure caused by salt tectonics in the Gulf of Mexico. Note that velocity variations in some parts of the sedimentary section are structure independent and represent overpressured zones. Selected common-shot gathers shown in Figure K-7 have been created by two-way acoustic wavefield modeling ; therefore, they contain both primaries and multiples. Each shot gather represents a modeled wavefield. Note the complex events in the gathers above and in the vicinity of the salt sill shown in Figure K-6.
Shown in Figure K-8 are selected CMP gathers sorted from the modeled shot gathers as in Figure K-7. Observe the events with complex moveout in the gathers above and in the vicinity of the salt sill. The zero-offset section obtained by collecting the zero-offset traces from the modeled shot gathers is shown in Figure K-9a, and the stacked section obtained from the CMP gathers as in Figure K-8 is shown in Figure K-9b. A zero-offset wavefield simulated by exploding reflectors does not include multiples, because the exploding reflectors are associated with simulation based on one-way wave equation . When the simulation is based on the two-way acoustic wave-equation as in Figure K-9, the zero-offset and stacked sections both include primary and multiple reflections. Since it is wavefield modeling, not just traveltime modeling, the simulated shot gathers (Figure K-7), the associated CMP gathers (Figure K-8), and sections (Figure K-9) all contain the diffractions caused by the reflector discontinuities in the velocity-depth model (Figure K-6).
Figure K-5 Upward continuation using wave-equation datuming of shots and receivers from a flat datum at z = 0 to an irregular topography shown above the velocity-depth model in (a). Selected input shot gathers are shown in Figure K-3b. (b) Shot gathers and (c) CMP gathers along the irregular datum. (Data courtesy Amoco Production Company.)
Figure K-6 A velocity-depth model of a salt sill structure commonly encountered in the Gulf of Mexico .
Figure K-7 Selected modeled common-shot gathers using the velocity-depth model shown in Figure K-6 .
Figure K-8 Part 1: Selected CMP gathers associated with the stacked section in Figure K-9b, the modeled shot records as in Figure K-7, and the velocity-depth model shown in Figure K-6 .
Figure K-8 Part 2: Selected CMP gathers associated with the stacked section in Figure K-9b, the modeled shot records as in Figure K-7, and the velocity-depth model shown in Figure K-6 .
Figure K-9 (a) Modeled zero-offset wavefield , and (b) stacked section derived from CMP gathers as in Figure K-8 which are associated with the modeled shot records as in Figure K-7 and the velocity-depth model shown in Figure K-6.
Figure K-10 Elastic modeling of a water layer on top of an earth model represented by a vertically varying velocity. The water depths are (from left to right) 5, 10, 15, 20, and 50 m. Identify multiples (both reflected and refracted) and guided waves. The linear features below 3 s are artifacts of the modeling program.
K.4 Elastic wavefield modeling
Elastic wavefield modeling primarily is used to understand the effect of lithology and pore fluids on seismic amplitudes (seismic resolution and analysis of amplitude variation with offset). Sherwood  developed an f − k method for nonzero-offset modeling of elastic waves in a 2-D horizontally layered medium. Figure K-10 shows five shot gathers derived from an earth model represented by a vertically varying velocity function that includes a water layer with five different thicknesses. The water depths are 5, 10, 15, 20, and 50 m. Note the guided wave energy, which is especially prominent in gathers corresponding to water depths of 5, 10, and 15 m. These gathers contain all primaries, both P-waves and S-waves, as well as all possible multiples and converted modes. By examining such modeled data, we can better understand the nature of coherent noise (guided waves and multiples) in both land and marine environments.
A more interpretive application of elastic modeling is shown in Figure K-11 . The synthetic shot gather on the left in Figure K-11a is from a clastic section with a shallow layer of low P-wave velocity. This layer has been replaced with a fast-velocity limestone for the record on the right. Note the invasion of the large-offset primary reflection data with coherent noise that is associated with this limestone layer. In the field, limestone on the surface often generates a large amount of coherent noise.
Figures K-11b and K-11c show two synthetic shot gathers for which the upper part of the depth model consists of a 50-ft water layer underlain by a 1020-ft shale section with a P-wave velocity of 5500 ft/s. The deeper portion of the model for Figure K-11b is an all-shale section with P-wave velocity increasing from 5600 ft/s on the top to 7700 ft/s on the bottom of the layer. The primary reflections PP between 380 and 850 ms are associated with this all-shale sequence. Figure K-11c shows the effects of including a 30 percent sand layer between 660 and 850 ms. The P-wave reflections PP in Figure K-11c show stronger amplitudes at large offsets. An analysis of amplitudes as a function of offset can provide hints for determining the sand-shale ratio, as well as fluid content, in some cases. Because the effects are complex, this type of modeling can be helpful in analyzing amplitude variations with offset. Also note the relatively strong converted PS and SP waves on the record from the sand-shale model. Modeling of this type also is useful when analyzing converted waves in multicomponent reflection data (4-C seismic method).
- Hilterman, 1970, Hilterman, F. J., 1970, Three-dimensional seismic modeling: Geophysics, 35, 1020–1037.
- Kelly et al., 1976, Kelly, K. R., Ward, R. W., Treitel, S. and Alford, R. M., 1976, Synthetic seismograms: A finite-difference approach: Geophysics, 41, 2–27.
- Sherwood et al., 1983, Sherwood, J. W. C., Hilterman, F. J., Neale, R. N. and Chen, K. C., 1983, Synthetic seismograms with offset for a layered elastic medium; Offshore Technology Conference, Paper 4508.
- Berryhill, 1979, Berryhill, J., 1979, Wave-equation datuming: Geophysics, 44, 1329–1344.
- O’Brien and Gray, 1996, O’Brien, M. and Gray, S., 1996, Can we image beneath salt?: The Leading Edge, 17–22.
- Claerbout, 1985, Claerbout, J. F., 1985, Imaging the earth’s interior: Blackwell Scientific Publications.
- Introduction to structural inversion
- Subsalt imaging in the North Sea
- Subsalt imaging in the Gulf of Mexico
- Imaging beneath irregular water bottom in the Northwest Shelf of Australia
- Imaging beneath volcanics in the West of the Shetlands of the Atlantic Margin
- Imaging beneath shallow gas anomalies in the Gulf of Thailand
- 3-D structural inversion applied to seismic data from the Southern North Sea
- 3-D structural inversion applied to seismic data from the Central North Sea
- 3-D structural inversion applied to seismic data from offshore Indonesia
- 3-D structural inversion applied to seismic data from the Northeast China