# Layer-cake model

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Series Geophysical References Series Digital Imaging and Deconvolution: The ABCs of Seismic Exploration and Processing Enders A. Robinson and Sven Treitel 8 http://dx.doi.org/10.1190/1.9781560801610 9781560801481 SEG Online Store

What is a reflectivity function? A reflectivity function is a time series intended to represent reflecting interfaces by their reflection coefficients, which usually are defined for the normal-incidence case. A reflectivity section is a display of the reflectivity functions versus their locations on a seismic line.

A multiple reflection represents seismic energy that has been reflected more than once. A primary reflection represents energy that has been reflected only once and hence is not a multiple. All recorded seismic energy involves multiples.

An important distinction is between long-path and short-path multiples. A long-path multiple arrives as a distinct event, whereas a short-path multiple arrives so soon after the primary that it merely adds a tail to the primary. Short-path multiples can obscure stratigraphic detail even when structural aspects are not affected significantly. The moveout behavior of long-path multiples might not be representative of the portion of the section associated with their arrival times. Usually, long-path multiples have traveled more in the slower (shallower) part of the section than have the primaries with the same normal-incidence arrival times, so long-path multiples generally show greater normal moveout and thus can be attenuated by common-midpoint (CMP) stacking.

The main purpose of a seismic signal model is to explain the seismic-wave propagation phenomenon. The most valuable models are three dimensional. Such models tend to be numerical because the mathematics of a theoretical 3D model is much too involved to produce closed-form solutions except in the simplest cases. The most pronounced variations in the earth layering are usually along the vertical direction, so a 1D vertical model often is adequate. The foremost 1D model — the so-called stratified, or layered-earth, or layer-cake model — is mathematically identical to the lattice model for electric transmission lines. The model is also mathematically identical both to the acoustic tube model used in speech processing and to the thin-film model used in optics. In such a 1D model, the earth is sliced mathematically into many thin horizontal layers that are normal to the vertical z direction. Such a theoretical division of the earth into thin layers produces a stratified medium characterized by the interfaces between the layers. Thus, a vector of the discrete amplitudes of the signal thus can represent a digital signal. One vector results if the amplitude of the signal is measured by a geophone in terms of particle velocity. Another vector results if the amplitude of the signal is measured by a hydrophone in terms of pressure. The inner (or dot) product of these two vectors gives the energy of the signal. In the basic model treated here, there is no allowance for dissipation of kinetic energy into heat. Thus, all the source energy imparted into the body can be accounted for, over time, in terms of the resulting elastic wave motion.

A synthetic seismogram is an artificial seismic-reflection record made by assuming that a wavelet travels through an assumed earth model (Anstey, 1960[1]; Kelly et al., 1976[2]; Shtivelman and Loewenthal, 1989[3]). A 1D synthetic seismogram (without multiple reflections) can be obtained by convolving a wavelet with a reflectivity function. A reflectivity function consists of a series of spikes that indicates the sign and magnitude of the reflection coefficient of each interface. The reflectivity usually is calculated for normal-incidence wave motion on the basis of changes in velocity and in density. Often, only velocity changes are considered because density information frequently is unavailable. Alternatively, some empirical relationship between density and velocity can be assumed.

The interfaces usually are identified by their normal-incidence two-way traveltime from source to receiver and back to the source. The wavelet sometimes is an assumed waveform such as a Ricker wavelet and sometimes is a waveform resulting from analysis of actual seismic data. The most common type of synthetic seismogram involves primary reflections only. Other synthetics include the contribution of short-path multiples in addition to the primaries. Even more sophisticated synthetic seismograms include only certain selected multiples or all possible multiples. Often, a synthetic seismogram includes various earth-filtering effects (such as attenuation resulting from geometric divergence and attenuation resulting from absorption) as well as distortions from instrumental filtering effects.

What are the limitations of a 1D synthetic seismogram? A 1D synthetic seismogram is the result of a single-channel convolution of a wavelet with a computed impulse response of a layer-cake medium. This impulse response can include either primary reflections only or multiples as well. Only vertical travel paths are used in this model. However, the layer-cake model can be varied in a lateral direction, and successive 1D synthetic traces can be displayed side by side to simulate a seismic section. Such a synthetic seismic section can be compared with an actual seismic section to help identify events and to see how variations in the model might appear on the synthetic section. Usually, the models assume that the source and receiver are coincident, and sometimes offset-dependent effects can be included. Some models include head waves, surface waves, and other wave modes. Calculation of a synthetic seismogram is a direct or forward problem, as opposed to the inverse problem, which produces estimates of the seismic wavelet along with the reflection coefficients characterizing layer interfaces and their depths in terms of their two-way traveltimes.

Why should one study the layer-cake model at all? Certainly, stratigraphic traps can occur in flat-lying strata, which fit the layer-cake model. In addition, many great oil fields are found in huge, gently sloping anticlines, where the layer-cake approximation serves quite well. However, exploration cannot be restricted to the use of such a simple model. It is essential that geophysicists have the tools to model regions of highly complex geologic structure. Reflections come from interfaces, and the observed reflections are used to reveal structure.

The layer-cake model is the simplest expression of a sequence of interfaces. The corresponding 1D synthetic seismogram without multiples (equation 3), based on (1) a fixed source wavelet and (2) only primary reflections is the simplest expression of the linear time-invariant convolutional model. Such a relationship between geologic model and synthetic seismogram is so basic that it cannot be disregarded. We cannot stop at this point, however. We also must understand the corresponding 1D synthetic seismogram with multiples (equation 2), which is based on (1) a fixed-source wavelet, (2) primary reflections, and (3) multiple reflections.

The synthetic seismogram can be represented more exactly by the dynamic convolutional model (equation 1). Again, such a relationship between geologic model and synthetic seismogram is so basic that it cannot be disregarded. In this respect, we owe a debt of gratitude to the layer-cake model for being amenable to such an elegant mathematical solution. From a learning point of view, the layer-cake model serves as an indispensable pathway to the discovery of the intricacies of multiple reflections by mathematics and not just by computations.

As soon as we start to ease some of the restrictions present in the layer-cake case, the simplicity rapidly dissipates. A 2D synthetic seismogram can simulate such effects as reflections from dipping reflectors and diffractions from sharp discontinuities. However, a 2D model has a shortcoming in that it cannot adequately handle energy coming from outside the plane of the model. A 3D model is required to take into account all such effects. Attempts to find closed mathematical expressions that model the wave motion in 3D complex geologic structures soon encounter insurmountable difficulties, so numerical methods on computers must be used instead. The ingenuity displayed in geophysics in the construction of computer-based models is indeed impressive (Hilterman et al., 1998[4]).

Geophysics has become even more exciting as it has evolved. By the 1970s, geophysicists confirmed that seismic data could reveal information not only about geologic structure but also about lithology. By using such things as wave patterns, frequency content, and strengths as well as reflection continuity and terminations, geophysicists could find important clues about rock types and depositional environments. Thus, the disciplines known as seismic stratigraphy (Vail, 1977[5]) and sequence stratigraphy (Van Wagoner et al., 1988[6]) were born. Geology and geophysics had found common ground on which exploration could and did achieve new heights (Sangree and Widmier, 1979[7]; Mallick, 2007[8]).

## References

1. Anstey, N. A., 1960, Attacking the problems of the synthetic seismogram: Geophysical Prospecting, 8, 242–259.
2. Kelly, K., R. Ward, S. Treitel, and R. Alford, 1976, Synthetic seismograms, a finite difference approach: Geophysics, 41, 2–27.
3. Shtivelman, V., and D. Loewenthal, 1989, Construction of the generalized one-dimensional synthetic seismograms by a three-step extrapolation procedure: Geophysics, 54, no. 8, 1050–1053.
4. Hilterman, F., J. W. C. Sherwood, R. Schellhorn, B. Bankhead, and B. DeVault, 1998, Identification of lithology in the Gulf of Mexico: The Leading Edge, 17, no. 2, 215–222.
5. Vail, P. R., R. M. Mitchum, and S. Thompson, 1977, Seismic stratigraphy and global changes of sea level, part 3: Relative changes of sea level from coastal onlap, in C. E. Payton, ed., Seismic stratigraphy — Applications to hydrocarbon exploration: AAPG Memoir 26, 63–81.
6. Van Wagoner, J. C., H. W. Posamentier, R. M. Mitchum, P. R. Vail, J. F. Sarg, T. S. Loutit, and J. Hardenbol, 1988, An overview of the fundamentals of sequence stratigraphy and key definitions, in C. K. Wilgus, C. A. Ross, and H. Posamentier, eds., Sea-level changes: An integrated approach: SEPM Special Publications 42, 39–46.
7. Sangree, J. B., and J. M. Widmier, 1979, Interpretation of depositional facies from seismic data: Geophysics, 44, no. 2, 131–160.
8. Mallick, S., 2007, Amplitude-variation-with-offset, elastic-impedance, and wave-equation synthetics — A modeling study: Geophysics, 72, no. 1, C1–C7.