Synthetic seismogram without multiples
The layer-cake model gives us a means for discovering firsthand why deconvolution works. Let us introduce an analogy. A trace with primaries and all multiples corresponds to whole milk. A deconvolved trace, which we hope has primaries only, corresponds to skim milk. In other words, the multiples correspond to the cream, which is bad because it contains cholesterol. The purpose of deconvolution is to remove the cream from the whole milk or, in other words, to remove the multiples from the whole trace (i.e., the trace with primaries and multiples). The layer-cake model gives us an opportunity to show that the deconvolution operation indeed can be done.
The time unit on a seismic trace is the time spacing of the sampling — for example, one time unit could be equal to 4 ms. This time unit sets the resolution of the seismic model in terms of the layering. In other words, the thinnest layer that can be distinguished has a two-way traveltime of 4 ms. Thus, the thickness of each layer is chosen so that the two-way traveltime (downward traveltime plus upward traveltime) in each layer is equal to one time unit. Each interface separating two adjacent layers with different impedances has a nonzero reflection coefficient. Of course, several thin layers can be lumped together to form a thicker uniform layer by setting the intermediate reflection coefficients equal to zero. The greater the impedance contrast is between the two adjacent layers, the greater is the magnitude of the reflection coefficient. For computation, the wave motion is digitized so that a signal becomes a discrete sequence (that is, a time series with discrete values separated by the given unit time interval).
In the layered or stratified model, the boundary layers are the air (on the top) and the basement rock (on the bottom). Let N + 1 be the number of interfaces, with interface 0 the ground surface and interface N the deepest interface.
A plane wave is the simplest form of a propagating wave. We consider plane waves traveling normal to the interfaces — that is, waves traveling up and down in the vertical z direction. A pulse that is normally incident on interface i is split into a reflected pulse and a transmitted pulse. Energy is conserved. As a consequence, the magnitude of the reflection coefficient must be less than or equal to one.
The magnitude of the reflection coefficient does not depend on the direction in which the plane wave travels through the interface, but the sign of the reflection coefficient does. For a given interface, the reflection coefficient for an upgoing pulse is the negative of the reflection coefficient for a downgoing pulse. As was pointed out above, the sequence of the downgoing reflection coefficients is called the reflectivity function or simply the reflectivity. The reflectivity represents the internal structure of the earth and is an unknown quantity in the remote-detection problem faced in seismic prospecting.
We suppose that the subsurface consists of many ideal layers, each of which has a two-way traveltime . Let the top interface (the surface of the ground or water, as the case may be) be called interface 0, let the next interface down be interface 1, and so on, to the bottom interface N. Denote the reflection coefficient of interface n by . In practice, many of these interfaces will be nonexistent. Any nonexistent interface has the reflection coefficient 0. Normal incidence is assumed, so the source and receiver coincide. This source-receiver point is chosen as a point just below the surface, so it is just barely inside the top layer (Figure 5).
An impulsive source is activated that emits a one-way downgoing spike at time 0. The receiver is a one-way receiver; that is, the receiver picks up all of the upgoing waves and none of the downgoing waves. The approximate impulse response, consisting of primary reflections only (without transmission losses), is given by the sequence of reflection coefficients (i.e., the reflectivity function ). In Figure 5, for clarity, the raypaths have been drawn as slanting lines, although in our normal-incidence model they are perpendicular to the interfaces.
The synthetic seismic trace consists of the primary reflections from the underground interfaces. Because the source point is below the surface of the ground, the reflection coefficient of the surface does not generate a primary reflection. The approximate impulse response (i.e., the reflectivity without the ) has the Z-transform
This is the Z-transform of an Nth-order causal feedforward filter with the subsurface reflection coefficients on the feedforward loops. For example, the filter in the case of just three primary reflections is depicted by the pure feedforward block diagram shown in Figure 6.
Instead of a pure impulse, an arbitrary source wavelet or signature s can be used as the input wavelet to the synthetic seismogram generator, whose output is the synthetic trace x. The synthetic trace without multiples is the convolution of this signature with the reflectivity (without the term). The result is , which we recognize as the multiple-free convolutional model (equation 3) with which geophysicists are so familiar. From downhole surveys, we can obtain the (approximate) reflection coefficient series as a function of seismic traveltime. This signal represents the reflectivity .
In addition, an estimate of the seismic wavelet s must be found. An example of a reflectivity function with a seismic wavelet is shown in Figure 7, in which we also show the characteristic impedance from which the reflectivity function is computed. The characteristic impedance is defined as the product of layer velocity and layer density.
In summary, when we convolve a wavelet with a reflectivity function, we obtain the synthetic multiple-free trace shown in Figure 7. In other words, the reflectivity represents the primaries-only impulse response of the subsurface. The synthetic trace (without multiples) is the convolution of this impulse response with the seismic wavelet.
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Also in this chapter
- Reflection coefficients and transmission coefficients
- Ghost reflection
- Layer-cake model
- Water reverberations
- Synthetic seismogram with multiples
- Small and white reflection coefficients
- Appendix H: Exercises