Correlation processing experiments

INTRODUCTION

Cross-correlation of traces from transmission wavefields, followed by summing the individual results, yields a data volume with the same kinematics as active seismic experiments if sufficient coverage of subsurface sources was captured in the transmission wavefields. The success of the effort to image the subsurface with truly passive data is predicated on the strength, number and distribution of the sources captured during the recording campaign. Therefore, the most important investigations into how well such an experiment can work in practice deal with the distribution and character of the subsurface sources within the data.

To analyze the efficacy of correlation processing under various source configurations, a suite of 2D synthetic transmission wavefields were generated. The data were generated with a finite-difference two-way operator. All surface-related and internal multiples are included, and the transmission wavefields have been flux-normalized (?). The model contains two synclines in an otherwise flat three layer velocity model. Subsurface impulsive sources were modeled every 25 m across the bottom of the model at the same depth level. The data contain 280 receivers at the surface, spaced 20 m apart. For comparison, Figure 1 shows two modeled surface seismic shot gathers at locations in the model from which the forthcoming synthesized gathers are extracted.

Figure 1: Two active reflection seismic shot gathers, corresponding to the locations of the synthesized gathers in the next figures. actshots [CR]

Figure 2 shows two representative transmission wavefields and the velocity model with which they were generated. The left panel shows data from a source at the left edge of the model domain, while the center panel is a transmission wavefield due to a source directly beneath the center of the model domain. The first hyperbolas are direct arrivals and the rest are due to the structure within the velocity model.

Figure 2: Representative transmission wavefields from impulsive subsurface sources through a three layer velocity model with two synclines (right). All orders of surface-related and internal multiples are included. data [ER]

Figure 3 shows two slices through a 3D volume of synthesized gathers produced by correlating all traces using a single transmission wavefield (left panel of Figure 2). Thus, this volume represents a partial processing of the source summation required to synthesize active data from transmission data. The front face of the left cube shows the first trace in the transmission data correlated with all other traces. The correlation does not dramatically change the kinematics of the events seen in Figure 2. The first-order effect is a simple subtraction of the time to the direct arrival in the source trace from each subsequent trace. The right face of the panels have axes time and source position. Roughly hyperbolic acausal movement of the reflectors is clear. As traces further to the right in the transmission wavefield are used as source functions for correlation, the hyperbolic increase in the delay time of the direct arrival causes most of the energy in the gathers to be at negative time after source position x D 4000 m. The front panel of the right cube used the trace from the transmission wavefeild located under the black vertical line on the section. The section contains only the late-time multiples from the two layer model. In real data, it is likely that these high-order multiples would be much weaker than in this model or completely absent. Clearly, this partial result does not accurately synthesize active shot gathers.

Figure 4 shows synthesized gathers at locations corresponding to Figure 3 produced by summing the correlations from subsurface sources across the entire bottom of the model. The left front face is not substantially different from the correlations of a single transmission wavefield above. There is some increased noise at early time. Despite using transmission wavefields Figure 3: Two correlated shot gathers from the volume generated with a single source below the left edge of the model. shot1200 [CR]

from 225 sources across the bottom of the model, the summation has not perfectly cancelled this noise through destructive interference. Modeling transmission wavefields only from the bottom of the model ignores the energy contributions from locations at the sides of the model needed for a perfect result. However, the receiver and shot axes are much more symmetric than previously, which is appropriate for this model with synclines in otherwise flat layers.

Comparing Figure 1 with Figure 4 highlights several key differences between surface siesmic data and the synthesized gathers. Obvious amplitude differences should probably be ignored, as this work is focused on recovering the kinematics of the active experiment. The direct arrival in the surface seismic gathers interfers with events after about 2000 m offset, and is largely absent in the synthesized gathers. Because the transmission wavefields do not contain much energy with high ray parameters, the direct arrivals and refractions are not well represented in the synthesized gathers. The correlations above also have events that would be acausal relative to the direct arrival of the surface seismic. This energy can easily be removed by muting before further processing. The main events in Figures 4 and 1 look very similar. However, the synthesized gathers suffer from substantial coherent noise due to not using subsurface sources completely surrounding the domain of interest. In the Chapter ??, these data volumes will are migrated to explore these effects in the image space.

MULTIPLE SOURCES

If multiple sources are captured in a passive recording, processing them together violates the rigorous theory of interferometry, and the resulting volume of correlations can be more complicated than the desired shot gathers. It may be impossible to separate passive data into transmis-Figure 4: Two correlated shot gathers from the volume generated by summing correlations from 225 sources across the bottom of the model. allshots [CR]

sion wavefields due to only a single subsurface source if sources are weak and unidentifiable, fire simultaneously, or when attempting to process long passive records as a single wavefield.

Consider field data, T_{f} , composed of transmission wavefields a(x,t) and b(x,t) from individual sources. These individual transmission wavefields are placed on the field record at unknown times τ_{a} and τ_{b} such that

T_{f} (t) D a ∗ δ(t − τ_{a}) C b ∗ δ(t − τ_{b}).  (1)

Correlation in the Fourier domain yields

T_{f} T_{f}^{∗} D AA^{∗}C BB^{∗}C AB^{∗}e^{−iω(τ_{a}−τ_{b})} C BA^{∗}e^{−iω(τ_{b}−τ_{a})} .  (2)

The sum of the first two terms is the desired result. The last two terms are cross-talk. Though passive recordings are likely very long, the two-way traveltime to the deepest reflector of interest, max(t), is the latest lag that need be maintained after correlation. If |τ_{b}−τ_{a}| > max(t), one cross-talk term is acausal, and the other is at late lags that can be windowed away in the time domain. If |τ_{b}− τ_{a}| < 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 2 are

AB^{∗} D (F_{a}I_{e})(F_{b}I_{e})^{∗} D F_{a}F_{b}^{∗} I_{e}^{2} D F_{c} I_{e}^{2} .  (3)

Like the first two terms in equation 2, the cross terms contain the desired information about the Earth. However, the source function f_{c} included is not zero phase. If the source functions are random series, terms with residual phase (F_{c}I_{e}^{2}) within the gathers will decorrelate and diminish in strength as the length of f and the number of cross terms increases. Designing uncorrelable functions F_{i} is the chief concern to effectively process sums of phase-encoded gathers. For passive recordings however, the summation of transmission wavefields can be a natural phenomenon. While we may hope to collect a large number of sources, it is probably unreasonable to expect many of them to be perfectly random series of great length. This is not probable if the noise sources have similar source mechanisms.

Figure 5 represents the case where the energy from two subsurface sources sources (x D 1200,3500 m) is processed together. In contrast, Figure 6 shows the sum of correlations of the two transmission wavefields. Thus, Figure 6 is a partial result of the summation of correlations presented in Figure 4.

Figure 5: Synthesized shot gathers produced by correlating the sum of transmission wavefields due to sources beneath the left edge and middle of the model. shot1200.3500.shift [CR]

Both the time sections and time slices in Figures 5 and 6 show the differences associated with the improper processing of correlating the sum of two transmission wavefields. There are ringing concave-up half hyperbolas in the top-left quadrant of the time-slices in Figure 5 that are not present in Figure 6, as well as a strong concave-right hyperbola at shot position x D 3000 m. There are also strong half hyperbolas on the left edge of the front face of the right panel in Figure 5 that are not present in Figure 6. Correlating the sum of the transmission wavefields produces a result that contains all the events of both transmission wavefields spread throughout the volume. Both shot posistions in Figure 5 look very similar, and the noise is symmetric across the zero-offset diagonal in the time-slices.

As more sources are processed together, such as a storm of small subsurface sources during natural or induced fracturing, this problem could overwhelm the correlations. Figure 7 shows the result of simultaneously processing 25 sources spaced every 125 m across the bottom of the model. Although the entire model is illuminated by this distribution, correlating the sum of transmission wavefields leads to so much cross-talk noise that the gathers are uninterpretable.

Figure 6: Synthesized shot gathers produced by summing the correlations from sources beneath the left edge and middle of the model. shot1200csum3500 [CR]

Figure 8 was produced with 30 shots every 25 m clustered around the center of the model. There is little to no energy in the first gather. The gather in the midst of the swarm has events, but the produce a ringing time section that looks nothing like the desired result.

The total correlation volume produced by a single source, Figure 3, could be confusing as it represents an incomplete data set. However, the synthesized gather immediately above the subsurface source is fairly representative of the complete summation at the same location. Processing summed transmission wavefields copies each shot panel throughout the volume. This fact can be observed by noting the similarities in Figures 3 and 5. When more than two sources are captured in the passive data the resulting correlations, Figures 7 and 8, become completely uninterpretable.

The cross-talk introduced by processing the reverbrations from multiple sources simultaneously takes the form of the convolution of the Earth’s impulse response (all reflections and multiples) with the correlation function of the two sources, F_{c}I_{e}. The cross-talk events due to n_{s} sources will likely be more problematic than conventional multiples as every multiple and reflector can be repeated n_{s}!/(n_{s} − 2)! times. The ratio of desirable zero phase terms, I_{e}, to cross terms, F_{j} I_{e}^{2}, decreases as 1/(n_{s} − 1) if the source terms are maximally correlable.

INTRODUCING RANDOMNESS

In the previous examples concerning processing transmission wavefields containing multiple subsurface sources, the same wavelet was used in every case. Therefore, events in summed transmission wavefields are perfectly correlable. While local sources probably share similar wavelet characteristics, it is also informative to explore the effects of varying the source Figure 7: Synthesized shot gathers with 25 subsurface sources spaced evenly across the bottom of the model. shotsmall [CR]

Figure 8: Synthesized shot gathers with 30 subsurface sources clustered under the center of the model. shotsmallsmall [CR] wavelets used to model the transmission wavefields. For this exercise, I convolve the individual transmission wavefields with a random number sequence before correlation. Random source wavelets will act to decrease the correlation cross-talk when it is not possible or practical to perfectly separate the passive data into individual transmission wavefields.

Figures 9 - 12 recapitulate Figures 5 - 8 with the addition of convolving the modeled data with a random source wavelet before further processing. Previous authors have convolved impulsive transmission wavefields with long random time series to simulate passive data. Such modeling implies long, continuously ringing, source functions whose energy is distributed evenly over the time axis. I believe it is more appropriate to restrict the duration of the wavelet (no longer than 1 s), use random amplitude terms, and apply a bulk phase shift to the entire record. This modeling simulates finite duration, random phase source functions, that can happen at any moment within the entire recording interval.

Figure 9: Synthesized shot gathers produced by correlating the sum of transmission wavefields due to randomized sources beneath the left edge and middle of the model. shot1200.3500.rand [CR]

In Figure 9, introducing the randomized source function, many of the inappropriate features in the time slices have been eliminated in comparison to Figure 5. However, the randomized wavelet associated with the transmission wavefield for the source on the left side of the model also included a nearly 2 s time delay. Therefore, the cross-talk visible in the front face of the right panel is still present, though shifted to later time. Interpreting this gather without such apriori knowledge, one might assume that there is a diffractor (or a series of diffractors if not aware of the many orders of multiples included in the data) on the left edge of subsurface domain.

Figure 10 still contains all of the features of Figure 6, but now has an extended coda on the wavelet and some amplitude variation due to the modeling parameters of the individual sources. The longer wavelets may result in overall less energy within their correlation func-Figure 10: Synthesized shot gathers produced by summing the correlations from randomized sources beneath the left edge and middle of the model. shot1200csum3500rand [CR]

tions, but this energy is also distributed in time as diminishing correlation side-lobes. Therefore, though the randomized wavelets decreased the total power of the cross-talk, it has also created more events. If the wavelet codas are truly random and long, the ringing in this result would be attenuated by the square root of the length of the wavelet. Limiting the random wavelet to 1 s in duration does not allow for much cancelation of the correlation side-lobes.

Figures 11 and 12 add the random wavelet to the summed transmission wavefields before correlation. Comparing Figures 7 and 11 shows that the randomized wavelet has made the correlation result of the passive data (25 sources across the bottom of the model) less interpretable.

Figure 12 shows an contrary image. Whereas Figure 8 is full of ringing uninterpretable energy, Figure 12 has events that are better differentiated and seem more "seismic-like". Whether these more clear events are appropriately interpreted at subsurface information will be explored in the next chapter in which the correlation volumes above are migrated assuming that the synthesized shot gathers contain the kinematics of the active seismic reflection survey.

CONCLUSION

Synthetic transmission wavefields due to impulsive sources across the bottom of an arbitrary velocity model show that correlation of every trace with every other trace does synthesize a volume close to that produced by modeling the active surface seismic experiment. However, if only sources directly beneath the array are captured during the recording, some coherent nose events are included in the result due to incomplete destructive interference that would be enjoyed by a complete distribution of source locations surrounding the subsurface volume of Figure 11: Synthesized shot gathers with 25 randomized subsurface sources spaced evenly across the bottom of the model. shotsmallrand [CR]

Figure 12: Synthesized shot gathers with 30 randomized subsurface sources clustered under the center of the model. shotsmallsmallrand [CR] interest. Positively however, direct arrivals and most of the refracted energy is not synthesized, which is often problematic in the conventional processing strategies used for active seismic data.

If it is not possible to correlate individual transmission wavefields before summation, the resulting volume of synthesized shot gathers will contain coherent artifacts. These artifacts can not be distinguished as such and add coherent noise to the result that can quickly overwhelm the subsurface structure one is attempting to synthesize. If the source functions within the transmission wavefields are identical and impulsive, the resulting synthesized shot gathers are sometimes clear or ringing depending on the subsurface distribution of source locations. However believable the gathers are in either case, interpretation of these volumes as if they were exactly conventional shot gathers would likely lead to misinterpreting the subsurface.

Modeling the potential difference between local source functions as finite, random perturbations, does not seem to alleviate the problems identified above. Depending on the distribution of the subsurface sources, the correlation volume can seem more or less realistic. Despite the appearance of realism however, the inclusion of multiples sources with or without random wavelets does not visibly approach the kinematics of the desired active data.