Stacking wavefields

The time domain stacking shown in equation 5 that is implied by correlating field data with equation 3 can be explored by considering two transmission wavefields, $a({\bf x}_r,t)$ and $b({\bf x}_r,t)$, from individual sources. When placed randomly on the field record with wait-times $\tau_a$ and $\tau_b$

$$T_f(\tau)=a*\delta(t-\tau_a)+b*\delta(t-\tau_b).$$ (6)

Correlation in the Fourier domain by equation 3 yields  

$$T_fT_f^*= AA^* + BB^* +AB^*\mbox{e}^{-i\varpi(\tau_a-\tau_b)} +BA^*\mbox{e}^{-i\varpi(\tau_b-\tau_a)} \;.$$ (7)

The sum of the first two terms is the result dictated by equation 1. The second two are extra. If $\vert\tau_b-\tau_a\vert\gt\max(t)$, one term will be purely acausal, and the other at very late lags that can be windowed away in the time domain. However, if the operation is performed in the Fourier domain, circular correlation is actually implemented and energy from the cross-terms may not so easily be avoided. If $\vert\tau_b-\tau_a\vert<\max(t)$, the cross-terms are included in the correlated gathers.

Redefine A and B as the impulse response of the earth, I_e, convolved with source functions, F which now contain their phase delays. As such, the cross-terms of equation 7 are  

$$AB^*=(F_aI_e)(F_bI_e)^*= F_aF_b^* \; I_e^2 = F_c \; I_e^2 \;.$$ (8)

Like the first two terms in equation 7, the cross-terms have the desired information about the earth. However, the source function f_c included is not zero phase. These terms are the other-terms or virtual multiples in Schuster et al. (2004). If the source functions are random series, terms with residual phase (such as F_cI_e² above) within the gathers will decorrelate and diminish in strength as the length of f and the number of cross-terms increases. While we may hope to collect a large number of sources, it is probably unreasonable to expect many of them to be random series of great length.

The inclusion of these cross-terms in the correlation output produces a data volume

$$\tilde{R}({\bf x}_r, {\bf x}_s,\varpi) \not= R({\bf x}_r, {\bf x}_s,\omega)\;.$$

The ratio of desirable zero-phase terms to cross-terms containing residual phase decreases as 1/(n_s-1). $\tilde{R}$ is not especially useful however. The inclusion of the cross-terms between the experiments returns a data volume that may not be more interpretable than the raw data. This will be the case if the individual source functions are correlable or not conveniently located along the $\tau$-axis such that all of their correlated phase terms, F_c in equation 8, are zero. These virtual multiple events will likely be more problematic than conventional multiples as every reflector can be repeated n_s!/(n_s-2)! times.

Figure  shows the effect of the cross terms expanded in equation 8. The figure is directly analogous to Figure , though with two important differences. First, there are overlapping source function-reflection pairs. Second, the direct arrivals are spaced randomly along the time axis rather than engineering them to lie at the first sample of one of the short subsections. The second source arrives at the receiver before the reflection from the first source. The third source is randomly placed at the far end of the trace. The traces on the right are zoomed versions of their counterparts to the left. The result desired by a passive seismologist trying to produce a zero-offset trace from R, bottom trace Figure , can not be produced. The middle trace was correlated in the Fourier domain and transformed back to time. The bottom trace was computed by stacking eight windows from the input before correlation. The difference between the two output traces is not from reordering the summation for the Fourier transform in equation 5. Actually, this is the aliasing of the autocorrelation result itself. Both methods produce the wrong result at almost all times. They are however correct and identical at one location: zero-lag.

 

freq2 Figure 2 Right panel is 32x zoom of left. (top) Idealized signal of three identical subsurface sources. First two direct-reflection pairs overlap. (middle) Autocorrelation. (bottom) Autocorrelation performed after stacking 8 constituent windows. Zero values are padded on the bottom trace to facilitate plotting.

Figure shows a more complicated example of the effective summation of source terms during the Fourier transform. The first two panels are individual transmission wavefields from sources at the left side of the model and just to the right of the center of the model. Notice that the wavefields have been carefully windowed to assure that the minimum traveltime for the two wavefields is the same. The model is a constant velocity medium with two diffractors in the center. The right panel is the sum of 225 similar sources covering the bottom of the velocity model. Summing the many shots has created a zero-offset data volume that could be migrated with a planewave migration algorithm. Cross-correlating this data to produce shot-gathers produces several dozen flat plane-waves and only the faintest hint of a ringing train of diffraction hyperbolas. $\tilde{R} \not=R$ due to processing T_f rather than the individual $T(\xi)$ records.

 

diff.noshift
Figure 3 (a) Transmission wavefield from a subsurface source below x=1200m in a model containing two diffractors. (b) Transmission wavefield from source below x=5000m. (c) Sum of 225 modeled wavefields.

Figure mimics Figure directly without having taken care to align the direct arrivals to the same time sample. The summation of all 225 wavefields gives the result on the right. Cross-correlating this data to produce shot-gathers makes a very messy plot.

 

diff.shift
Figure 4 (a) Transmission wavefield from a source below x=1200m in a model containing two diffractors. (b) Transmission wavefield from source below x=5000m. (c) Sum of all wavefields.