The convolutional model
A sonic log segment is shown in Figure 2.1-1a. The sonic log is a plot of interval velocity as a function of depth based on downhole measurement using logging tools. Here, velocities were measured between the 1000- to 5400-ft depth interval at 2-ft intervals. The velocity function was extrapolated to the surface by a linear ramp. The sonic log exhibits a strong low-frequency component with a distinct blocky character representing gross velocity variations. Actually, it is this low-frequency component that normally is estimated by velocity analysis of CMP gathers (velocity analysis).
In many sonic logs, the low-frequency component is an expression of the general increase of velocity with depth due to compaction. In some sonic logs, however, the low-frequency component exhibits a blocky character (Figure 2.1-1a), which is due to large-scale lithologic variations. Based on this blocky character, we may define layers of constant interval velocity (Table 2-1), each of which can be associated with a geologic formation (Table 2-2).
|Layer Number||Interval Velocity,* ft/s||Depth Range, ft|
|1||21 000||1000 – 2000|
|2||19 000||2000 – 2250|
|3||18 750||2250 – 2500|
|4||12 650||2500 – 3775|
|5||19 650||3775 – 5400|
|* The velocity in Layer 2 gradually decreases from the top of the layer to the bottom.|
|Layer Number||Lithologic Unit|
|2||Shaly limestone with gradual increase in shale content|
The sonic log also has a high-frequency component superimposed on the low-frequency component. These rapid fluctuations can be attributed to changes in rock properties that are local in nature. For example, the limestone layer can have interbeddings of shale and sand. Porosity changes also can affect interval velocities within a rock layer. Note that well-log measurements have a limited accuracy; therefore, some of the high-frequency variations, particularly those associated with a first arrival that is strong enough to trigger one receiver but not the other in the log tool (cycle skips), are not due to changes in lithology.
Well-log measurements of velocity and density provide a link between seismic data and the geology of the substrata. We now explain the link between log measurements and the recorded seismic trace provided by seismic impedance — the product of density and velocity. The first set of assumptions that is used to build the forward model for the seismic trace follows:
- Assumption 1. The earth is made up of horizontal layers of constant velocity.
- Assumption 2. The source generates a compressional plane wave that impinges on layer boundaries at normal incidence. Under such circumstances, no shear waves are generated.
Assumption 1 is violated in both structurally complex areas and in areas with gross lateral facies changes. Assumption 2 implies that our forward model for the seismic trace is based on zero-offset recording — an unrealizable experiment. Nevertheless, if the layer boundaries were deep in relation to cable length, we assume that the angle of incidence at a given boundary is small and ignore the angle dependence of reflection coefficients. Combination of the two assumptions thus imply a normal-incidence one-dimensional (1-D) seismogram.
Based on assumptions 1 and 2, the reflection coefficient c (for pressure or stress), which is associated with the boundary between, say, layers 1 and 2, is defined as
where I is the seismic impedance associated with each layer given by the product of density ρ and compressional velocity v.
From well-log measurements, we find that the vertical density gradient often is much smaller than the vertical velocity gradient. Therefore, we often assume that the impedance contrast between rock layers is essentially due to the velocity contrast, only. Equation (1a) then takes the form:
If v2 is greater than v1, the reflection coefficient would be positive. If v2 is less than v1, then the reflection coefficient would be negative.
The assumption that density is invariant with depth or that it does not vary as much as velocity is not always valid. The reason we can get away with it is that the density gradient usually has the same sign as the velocity gradient. Hence, the impedance function derived from the velocity function only should be correct within a scale factor.
For vertical incidence, the reflection coefficient is the ratio of the reflected wave amplitude to the incident wave amplitude. Moreover, from its definition (equation 1a), the reflection coefficient is seen as the ratio of the change in acoustic impedance to twice the average acoustic impedance. Therefore, seismic amplitudes associated with earth models with horizontal layers and vertical incidence (assumptions 1 and 2) are related to acoustic impedance variations.
The reflection coefficient series c(z), where z is the depth variable, is derived from sonic log v(z) and is shown in Figure 2.1-1b. We note the following:
- The position of each spike gives the depth of the layer boundary, and
- the magnitude of each spike corresponds to the fraction of a unit-amplitude downward-traveling incident plane wave that would be reflected from the layer boundary.
To convert the reflection coefficient series c(z) (Figure 2.1-1b) derived from the sonic log into a time series c(t), select a sampling interval, say 2 ms. Then use the velocity information in the log (Figure 2.1-1a) to convert the depth axis to a two-way vertical time axis. The result of this conversion is shown in Figure 2.1-1c, both as a conventional wiggle trace and as a variable area and wiggle trace (the same trace repeated six times to highlight strong reflections). The reflection coefficient series c(t) (Figure 2.1-1c) represents the reflectivity of a series of fictitious layer boundaries that are separated by an equal time interval — the sampling rate . The major events in this reflectivity series are from the boundary between layers 2 and 3 located at about 0.3 s, and the boundary between layers 4 and 5 located at about 0.5 s.
The reflection coefficient series (Figure 2.1-1c) that was constructed is composed only of primary reflections (energy that was reflected only once). To get a complete 1-D response of the horizontally-layered earth model (assumption 1), multiple reflections of all types (surface, intrabed and interbed multiples) must be included. If the source were unit-amplitude spike, then the recorded zero-offset seismogram would be the impulse response of the earth, which includes primary and multiple reflections. Here, the Kunetz method  is used to obtain such an impulse response. The impulse response derived from the reflection coefficient series in Figure 2.1-1c is shown in Figure 2.1-1d with the variable area and wiggle display.
The characteristic pressure wave created by an impulsive source, such as dynamite or air gun, is called the signature of the source. All signatures can be described as band-limited wavelets of finite duration — for example, the measured signature of an Aquapulse source in Figure 2.1-2. As this waveform travels into the earth, its overall amplitude decays because of wavefront divergence. Additionally, frequencies are attenuated because of the absorption effects of rocks (see gain applications). The progressive change of the source wavelet in time and depth also is shown in Figure 2.1-2. At any given time, the wavelet is not the same as it was at the onset of source excitation. This time-dependent change in waveform is called nonstationarity.
Wavefront divergence is removed by applying a spherical spreading function (the 2-D Fourier transform). Frequency attenuation is compensated for by the processing techniques discussed in the problem of nonstationarity. Nevertheless, the simple convolutional model discussed here does not incorporate nonstationarity. This leads to the following assumption:
- Assumption 3. The source waveform does not change as it travels in the subsurface — it is stationary.
- Goupillaud, 1961, Goupillaud, P., 1961, An approach to inverse filtering of near-surface layer effects from seismic records: Geophysics, 26, 754–760.
- Claerbout, 1976, Claerbout, J. F., 1976, Fundamentals of geophysical data processing: McGraw-Hill Book Co.
- The convolutional model in the time domain
- The convolutional model in the frequency domain
- Mathematical foundation of deconvolution