Geometric spreading correction
A field record represents a wavefield that is generated by a single shot. Conceptually, a single shot is thought of as a point source that generates a spherical wavefield. The earth has two effects on a propagating wavefield:
- In a homogeneous medium, energy density decays proportionately to 1/r2, where r is the radius of the wavefront. Wave amplitude is proportional to the square root of energy density; it decays as 1/r. In practice, velocity usually increases with depth, which causes further divergence of the wavefront and a more rapid decay in amplitudes with distance.
- The frequency content of the initial source signal changes in a time-variant manner as it propagates. In particular, high frequencies are absorbed more rapidly than low frequencies. This is because of the intrinsic attenuation in rocks.
Attenuation mechanisms still are the subject of extensive research. However, one plausible mechanism for attenuation is related to pore fluids. As the wavefront passes through rocks, the fluids that are present in the pores are disturbed. This disturbance is greater in partially saturated rocks than fully saturated rocks. Pore fluids consume part of the energy of the propagating wavefield, which causes a frequency-dependent decay.
From Figure 1.4-1, note the wavefront divergence and frequency absorption on the field record. The first panel represents field data without any gain recovery function applied. Note the gradual decay in amplitude at later times. This record was filtered with a series of 10-Hz-wide band-pass filters. The signal in the 10-to-20-Hz panel exists down to about 6 s. On the 20-to-30-Hz panel, however, signal is visible only down to about 4 s. Moving to the higher frequency panels, note that the signal level is confined mainly to increasingly shallower times. Now apply the geometric spreading correction to the original field record in the far left panel of Figure 1.4-1. The result is shown in the far left panel of Figure 1.4-2. The amplitude level has been restored at late traveltimes. Filter panels of this record also are shown in Figure 1.4-2. When the filter panels in Figures 1.4-1 and 1.4-2 are compared with the same pass band, we see that the geometric spreading correction brought up some of the signal level at late times. However, note that the geometric spreading correction did not restore the amplitudes of the high frequencies as much as it restored the low frequencies, since the high frequencies were subject to stronger attenuation.
Figure 1.4-3 shows a portion of a CMP-stacked section and its filtered versions using narrow band-pass filters. Note that signal bandwidth only up to 20 Hz is observed down to 5 s, and frequencies up to 36 Hz are observed down to 3.5 s. The erosional unconformity just below 3.5 s constitutes a boundary with large frequency absorption. This results in attenuation of much of the high-frequency signal below this boundary. Frequencies above 36 Hz are confined to the shallow portion of the stacked section down to 2.5 s.
The effect of attenuation must be removed by modifying the amplitude spectrum of the signal, thereby making it broader. Deconvolution is one process that is used to achieve this goal. Alternative methods to compensate for frequency attenuation are time-variant spectral whitening and inverse-Q filtering. All three processes are described in deconvolution.
The factor 1/r that describes the decay of wave amplitudes as a function of the radius of the spherical wavefront is valid for a homogeneous medium without attenuation. For a layered earth, amplitude decay can be described approximately by 1/[ν2 (t) t] . Here, t is the two-way traveltime and ν(t) is the root-mean-squared (rms) velocity (normal moveout) of the primary reflections (those reflected only once) averaged over a survey area. Therefore, the gain function for geometric spreading compensation is defined by
where ν0 is the reference velocity at specified time t0. A more rigorous offset-dependent and time-dependent description of the geometric spreading correction function also can be used.
Signal-level decay is evident in the field records in Figure 1.4-4. Note the weak appearance of reflections, particularly below 1 s. This does not mean that there are no strong reflections below this time. Because of the amplitude decay resulting from wavefront divergence, no signal is seen at late times. As stated previously, this earth effect must be removed to bring up any signal that may be present in the deep portion of the record.
The same shot records after geometric spreading correction are shown in Figure 1.4-5. While reflections have been brought up in strength, noise components in the data also have been boosted. This is one undesirable aspect of any type of gain application.
Besides ambient noise, coherent noise in the data may be boosted as shown in Figure 1.4-6. By using the primary velocity function in correcting for geometric spreading, the amplitudes of the dispersive coherent noise and multiples have been overcorrected. Another example of overcorrected multiples is shown in Figure 1.4-2. (Compare the far left panel with its equivalent in Figure 1.4-1.)
To prevent overcorrection of amplitudes of multiple reflections, a velocity-independent scaling function, such as
where α usually is set to 2, can be used for geometric spreading correction . Figure 1.4-7 shows a marine record after muting the guided waves and applying t-squared scaling. Note from the amplitude spectrum averaged over the shot record that geometric spreading correction does not restore frequency components of the wavefield which are subject to absorption effects in rocks. Nevertheless, by correcting for amplitude decay caused by wavefront divergence, the autocorrelogram better describes the reverberations across all offsets. Actually, t-squared scaling now is a commonly used scaling function for geometric spreading correction.
Figure 1.4-6 (a) A raw field record from a marine survey. Before correcting for geometric spreading, refraction and guided wave energy dominate the record. (b) After the geometric spreading correction, while reflection amplitudes have been restored, multiples and coherent noise also have been boosted.
- Newman, 1973, Newman, P., 1973, Divergence effects in a layered earth: Geophysics, 38, 481–488.
- Claerbout, 1985, Claerbout, J. F., 1985, Imaging the earth’s interior: Blackwell Scientific Publications.