The convolutional model in the time domain
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A convolutional model for the recorded seismogram now can be proposed. Suppose a vertically propagating downgoing plane wave with source signature (Figure 2.1-3a) travels in depth and encounters a layer boundary at 0.2-s two-way time. The reflection coefficient associated with the boundary is represented by the spike in Figure 2.1-3b. As a result of reflection, the source wavelet replicates itself such that it is scaled by the reflection coefficient. If we have a number of layer boundaries represented by the individual spikes in Figures 2.1-3b through 2.1-3f, then the wavelet replicates itself at those boundaries in the same manner. If the reflection coefficient is negative, then the wavelet replicates itself with its polarity reversed, as in Figure 2.1-3c.
Now consider the ensemble of the reflection coefficients in Figure 2.1-3g. The response of this sparse spike series to the basic wavelet is a superposition of the individual impulse responses. This linear process is called the principle of superposition. It is achieved computationally by convolving the basic wavelet with the reflectivity series (Figure 2.1-3g). The convolutional process already was demonstrated by the numerical example in the 1-D Fourier transform.
The response of the sparse spike series to the basic wavelet in Figure 2.1-3g has some important characteristics. Note that for events at 0.2 and 0.35 s, we identify two layer boundaries. However, to identify the three closely spaced reflecting boundaries from the composite response (at around 0.6 s), the source waveform must be removed to obtain the sparse spike series. This removal process is just the opposite of the convolutional process used to obtain the response of the reflectivity series to the basic wavelet. The reverse process appropriately is called deconvolution.
Figure 2.1-1 (a) A segment of a measured sonic log, (b) the reflection coefficient series derived from (a), (c) the series in (b) after converting the depth axis to two-way time axis, (d) the impulse response that includes the primaries (c) and multiples, (e) the synthetic seismogram derived from (d) convolved with the source wavelet in Figure 2.1-4. One-dimensional seismic modeling means getting (e) from (a). Deconvolution yields (d) from (e), while 1-D inversion means getting (a) from (d). Identify the event on (a) and (b) that corresponds to the big spike at 0.5 s in (c). Impulse response (d) is a composite of the primaries (c) and all types of multiples.
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 2.1-4 The top frame is the same as in Figure 2.1-1d. The asterisk denotes convolution. The recorded seismogram (bottom frame) is the sum of the noise-free seismogram and the noise trace. This figure is equivalent to equation (2a).
Figure 2.1-5 A random signal with infinite length has a flat amplitude spectrum and an autocorrelogram that is zero at all lags except the zero lag. The discrete random series with finite length shown here seems to satisfy these requirements. What distinguishes a random signal from a spike (1, 0, 0, …)?
The principle of superposition now is applied to the impulse response derived from the sonic log in Figure 2.1-1d. Convolution of a source signature with the impulse response yields the synthetic seismogram shown in Figure 2.1-4. The synthetic seismogram also is shown in Figure 2.1-1e. This 1-D zero-offset seismogram is free of random ambient noise. For a more realistic representation of a recorded seismogram, noise is added (Figure 2.1-4).
The convolutional model of the recorded seismogram now is complete. Mathematically, the convolutional model illustrated in Figure 2.1-4 is given by
where x(t) is the recorded seismogram, w(t) is the basic seismic wavelet, e(t) is the earth’s impulse response, n(t) is the random ambient noise, and * denotes convolution. Deconvolution tries to recover the reflectivity series (strictly speaking, the impulse response) from the recorded seismogram.
An alternative to the convolutional model given by equation (2a) is based on a surface-consistent spectral decomposition . In such a formulation, the seismic trace is decomposed into the convolutional effects of source, receiver, offset, and the earth’s impulse response, thus explicitly accounting for variations in wavelet shape caused by near-source and near-receiver conditions and source-receiver separation. The following equation describes the surface-consistent convolutional model:
where is a model of the recorded seismogram, sj(t) is the waveform component associated with source location j, gi(t) is the component associated with receiver location i, and hl(t) is the component associated with offset dependency of the waveform defined for each offset index l = |i − j|. As in equation (2a), ek(t) represents the earth’s impulse response at the source-receiver midpoint location, k = (i + j)/2. By comparing equations (2a) and (2b), we infer that w(t) represents the combined effects of s(t), h(t), and g(t).
The assumption of surface-consistency implies that the basic wavelet shape depends only on the source and receiver locations, not on the details of the raypath from source to reflector to receiver. In a transition zone, surface conditions at the source and receiver locations may vary significantly from dry to wet surface conditions. Hence, the most likely situation where the surface-consistent convolutional model may be applicable is with transition-zone data. Nevertheless, the formulation described in this section is the most accepted model for the 1-D seismogram.
The random noise present in the recorded seismogram has several sources. External sources are wind motion, environmental noise, or a geophone loosely coupled to the ground. Internal noise can arise from the recording instruments. A pure-noise seismogram and its characteristics are shown in Figure 2.1-5. A pure random-noise series has a white spectrum — it contains all the frequencies. This means that the autocorrelation function is a spike at zero lag and zero at all other lags. From Figure 2.1-5, note that these characteristic requirements are reasonably satisfied.
Now examine the equation for the convolutional model. All that normally is known in equation (2a) is x(t) — the recorded seismogram. The earth’s impulse response e(t) must be estimated everywhere except at the location of wells with good sonic logs. Also, the source waveform w(t) normally is unknown. In certain cases, however, the source waveform is partly known; for example, the signature of an air-gun array can be measured. However, what is measured is only the waveform at the very onset of excitation of the source array, and not the wavelet that is recorded at the receiver. Finally, there is no a priori knowledge of the ambient noise n(t).
We now have three unknowns — w(t), e(t), and n(t), one known — x(t), and one single equation (2a). Can this problem be solved? Pessimists would say no. However, in practice, deconvolution is applied to seismic data as an integral part of conventional processing and is an effective method to increase temporal resolution.
To solve for the unknown e(t) in equation (2a), further assumptions must be made.
- Assumption 4. The noise component n(t) is zero.
- Assumption 5. The source waveform is known.
Under these assumptions, we have one equation,
and one unknown, the reflectivity series e(t). In reality, however, neither of the above two assumptions normally is valid. Therefore, the convolutional model is examined further in the next section, this time in the frequency domain, to relax assumption 5.
If the source waveform were known (such as the recorded source signature), then the solution to the deconvolution problem is deterministic. In inverse filtering, one such method of solving for e(t) is considered. If the source waveform were unknown (the usual case), then the solution to the deconvolution problem is statistical. The Wiener prediction theory (optimum wiener filters) provides one method of statistical deconvolution.
- ↑ Taner and Coburn, 1981, Taner, M. T. and Coburn, K., 1981, Surface-consistent deconvolution: Presented at the 51st Ann. Internat. Mtg., Soc. Expl. Geophys.