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Epilogue

Four-dimensional reservoir monitoring gives us an important new capability for understanding the earth and the seismic tools we use to image it and even for understanding ourselves, the practitioners. Below I list some lessons we can take from this brief review of 4D surveys. Monitoring in 4D is not a new branch of technology with laws and theorems. It does not even have special software. It is a collection of methods with a definite goal. A remarkable and interesting feature of 4D monitoring is that the quality of results usually can be recognized instantly and objectively. Data differences should be concentrated where production activity is occurring. We easily can recognize poor results that indicate changes everywhere. In our business, objective measures of quality are rare and valuable. In school, we learned that the scientific method requires us to use carefully controlled repeat measurements to understand things. We should admit that until the advent of 4D monitoring, exploration geophysicists were not very scientific. Geophysicists always tried to make new results distinctively different from earlier measurements, and they hoped that such results would be improved by changing nearly everything. In exploration that uses 3D surveys, the result we judge as best is often the result closest to our expectation or to what we wish to see. This is a special case of a more general law — we favor the models and explanations that best support our intuition. We usually change our intuition only when we are confronted by new measurements that unambiguously contradict our assumptions, and even then we change it reluctantly. Four-dimensional monitoring has been very good at helping us to reevaluate our assumptions. Reservoir models with connectivity that we have defined by intuition nearly always are shown to be wrong. People are beginning to believe and understand this, which is a significant advance in itself. Until recently, uncertainty was perceived as weakness. Now uncertainty is considered an important quantity to manage. The magnitude of our seismic nonrepeatability is shocking to the seismologist. If we compare two optimized, independently and expertly processed 3D surveys, their differences will be enormous. We are far from being able to measure meaningful, quantitatively absolute amplitudes from our data. Currently, we can measure only locally relative amplitudes from conformable reflectors. We should rethink our notions of noise and how to combat it. We can use data we have considered to be noisy to closely monitor changes in reservoirs that we cannot even see on 3D images. The power of the scientific principle of repeat measurements has many more lessons to teach. We will learn what responses constitute the real results of pressure changes. We will learn the geomechanical effects of small changes. We will learn how absorption really alters with fluid changes and saturation changes. We will learn what parts of reservoirs produce and what the relations are between recovery factors and reservoir property distributions. We will develop an ability to test exploration fluid-prediction techniques. A working model is a good thing, but we must be willing to change it as soon as it ceases to work.

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Sensors and data handling recently have been improved greatly. Now we should consider measuring some of the things we cleverly derive and model from data. The measurements might be better, and they certainly will speed the processing. • We should measure reservoir changes rather than merely modeling them. • We should measure static differences in water rather than trying to estimate them. • We should measure 3C receiver orientations rather than estimating them. • We should consider measuring Green’s functions for imaging rather than trying to estimate them from models. If we are going to adapt and prosper in any situation, we must measure or observe, update our model if necessary, make decisions on the basis of our observations, and then act and measure again. That is the secret of life. It is the way any viable organism must evolve, whether it is a bacterium, a single cell, an animal, an organization, or even an oilfield asset-management team.

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Appendix A

ACQUISITION/PROCESSING Coordinated by Ali Tura

Seismic repeatability, normalized rms, and predictability

ED KRAGH and PHIL CHRISTIE, Schlumberger Cambridge Research, Cambridge, England, U.K.

Time-lapse data are increasingly used to study production- induced changes in the seismic response of a reservoir as part of a reservoir management program. However, residual differences in the repeated time-lapse data that are independent of changes in the subsurface geology impact the effectiveness of the method. These differences depend on many factors such as signature control, streamer positioning, and recording fidelity differences between the two surveys. Such factors may be regarded as contributing to the time-lapse noise and any effort designed to improve the time-lapse signal-to-noise ratio must address the quantifi- able repeatability of the seismic survey. Although there are counter-examples (for example, Johnston et al., 2000), minimization of the acquisition foot- print and repeatability of the geometry to equalize residual footprints in both surveys are considered important. This has been a key objective in the development of point receiver acquisition systems. In this study, which develops the analysis from Kragh and Christie (2001), we examine the use of two repeatability metrics in assessing the similarity of two sets of repeat 2D lines acquired with a marine point receiver system. In one repeat set, no streamer positioning control was in use; in the other repeat set, positioning differences were mini- Figure 1. Gulf of Mexico stacks. First pass (upper panel) and second mized using the streamer positioning control. pass (lower panel). summed product of the autocorrelations, expressed as a Repeatability metrics. There does not appear to be a stan- percentage: dard measure of repeatability, defined as a metric, to quantify the likeness of two traces. One commonly used metric is the normalized rms difference of the two traces, at and bt within a given window t1-t2: the rms of the difference divided by the average rms of the inputs, and expressed as a per- φ centage: where ab denotes the crosscorrelation between traces at and bt computed within the time window t1-t2. When expressed as a percentage, predictability values lie in the range 0-100%. Predictability is sensitive to the length of the where the rms operator is defined as: correlation time window and to the number of lags in the correlations, so absolute numbers are not meaningful. In our examples, the summation intervals for the dot products are fixed at ±16 samples from zero lag. Predictability is not sensitive to overall static, phase, or amplitude differences (unlike and N is the number of samples in the interval t1-t2. nrms). It is sensitive to noise and to changes in the earth The values of nrms are not intuitive and are not limited reflectivity. If both traces are uncorrelated, predictability is to the range 0-100%. For example, if both traces contain ran- zero. If both traces anticorrelate, predictability is 100%. If dom noise, the nrms value is 141% (2). If both traces anti- one trace is half the amplitude of the other, the predictabil- correlate (i.e., 180° out of phase, or if one trace contains only ity is 100%. Conversely, nrms is extremely sensitive to the zeros) the nrms error is 200%, the theoretical maximum. If smallest of changes in the data. For example, a 10° phase one trace is half the amplitude of the other, the nrms error shift, which is equivalent to a 0.55 ms shift at 50 Hz (or 1.1 is 66.7%. ms at 25 Hz), gives rise to a 17.4% nrms residual. Predictability (Kristiansen et al., 2000) is another mea- Several authors quote values of nrms from time-lapse sure of repeatability and is equivalent to the coherence of case studies. Koster et al. (2000) reported a typical nrms of White (1980), who used it to quantify the spectral match 35% in Draugen Field where they successfully observed between synthetic seismograms and seismic traces as the water replacing oil. With careful matching, Eiken et al. (1999) proportion of power on the seismic trace that can be pre- improved from 15% to 6% the nrms between two densely dicted by linear filtering of the synthetic trace. Here, it is sampled surveys acquired a few days apart. Kristiansen et defined in terms of correlations. It is the summed squared al. (2000) applied both metrics to the towed and seabed 4D crosscorrelation within a time window divided by the data in Foinaven and observed that, while the seabed data

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in both passes. Data were acquired from point receivers and processed using source signature deconvolution and digital group forming (Jenkerson et al., 2000; Özbek, 2000). The output group interval was 12.5 m. Both lines were processed identically. No active streamer positioning control was used when acquiring these data. Figure 1 shows the first pass (upper panel) and second pass (lower panel). No migration has been performed. Figure 2 shows the difference image after subtracting the two processed stacked lines. There are clear features across much of the difference image that relate to subsurface structure. Between CMPs 1400 and 1700 the difference is much qui- eter; the seismic is, visually, highly repeatable. Below the seismic difference image, a number of attributes are plotted. The second panel down shows the nrms and predictability values computed for each CMP over the time window 2-4 s. There is a clear negative correlation between these two metrics. Between CMPs 1400 and 1700 (where the data appear highly repeatable), nrms values are typically 18-30% with predictability values of 99-93%. Where the data are most predictable (just over 99%), the corresponding nrms residual is just over 20%. The third panel down in Figure 2 shows CMP tracks for every 30th shot position along the line. Note that the vertical axis (cross-line offset) is highly exaggerated. The black lines show the first pass line 1, and the red lines show the second pass line 2. The plots essen- tially represent the streamer positions for these shots (all Figure 2. Gulf of Mexico data. Top down: Difference image, nrms and source locations were within 10 m, cross-line). predictability, CMP tracks for every 30th shot position along the line The data are most repeatable where the streamer posi- (black = first pass and red = second pass), nrms and mean midpoint error. tions most closely coincide, suggesting that the chief cause of the residual error is nonrepeatability of the streamer positioning. We note that, for this test, streamer positioning control (Bittleston et al., 2000) was not used and strong currents made it difficult to replicate the streamer position on the second pass. The lowest panel in Figure 2 plots the nrms values (as in the second panel down) with rms error of the midpoint locations (in meters) in red. A roughly linear correlation is evident, with the most repeatable data corresponding to rms errors in midpoint location of 15-20 m, which is mostly a cross-line error. The Gulf of Mexico data suggest a strong correlation of nrms with the positioning error. Figure 3 shows a plot of the nrms values against the midpoint error (i.e, a plot of data Figure 3. Gulf of Mexico data; nrms values versus rms midpoint error. Modeled data (Morice et al., 2000), are plotted as magenta stars. in the lowest panel in Figure 2). An approximately linear correlation is evident (the magenta colored stars are modeled data explained in the following paragraph). The red line is placed fiducially to indicate this correlation. It does not represent any kind of best fit, or expectation. We would, in fact, not expect this correlation to pass through the origin, as even with zero positioning error, we would still expect some residual differences due to other sources of noise. It is possible that we can see this tail-off toward finite nrms at zero positioning error on the real data values. However, the approximate one-to-one correlation of nrms against midpoint error is clear, out to about 70 m. It is interesting that Morice et al. (2000) also found a lin- Figure 4. Gulf of Mexico data; nrms values versus rms midpoint error, as shown in Figure 3, but plotted on a log-log axis. Modeled data ear trend in the nrms errors associated with positioning errors (Morice et al., 2000), are plotted as magenta stars. using DMO illumination. They used a modeling approach, with typical 3D geometries, and the positioning errors they sets were visually more repeatable than the towed data sets, considered were much smaller than those observed in the Gulf the metrics showed a more marginal difference in pre- of Mexico data (they considered errors of the size expected dictability (93% for the towed data and 96% for the seabed when streamer positioning control is in use). The results from data). Morice et al. (2000) are plotted on Figure 3 as the magenta- colored stars. Note that the 40-Hz modeled values are plot- Data example 1, Gulf of Mexico. A repeat 2D seismic line ted, which matches the real data central frequency. The was acquired in the Gulf of Mexico. The time interval modeled data lie almost exactly on the fiducial line. Figure 4 between acquisitions was two days, and sea state was calm shows exactly the same data but plotted on a log-log scale.

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Appendix A

Figure 7. North Sea data. First pass (upper panel) with signal and noise estimates in dB (lower panel). second pass (line 065, lower panel). No migration has been performed; the stacks are log-stretch DMO stacks. Figure 6 shows the difference image (upper panel) over the window 2.5-3.5 s obtained after subtracting the two processed stacked lines. The lower panel shows the nrms and predictability values (one for every CMP) computed from 2.5-3.5 s. The difference is, visually, highly repeatable over the whole line, Figure 5. North Sea stacks. First pass (upper panel) and second pass though the nrms values are large, ranging from 20% to over (lower panel). 60% in places, with corresponding predictability values of over 99% to around 85%. Where the data are most predictable (over 99%), the corresponding nrms residual is just over 20% (exactly as in the Gulf of Mexico example). The negative correlation between the metrics is, again, also clear. The high values of nrms are surprising, given that the data are visually very repeatable. The large range in the repeatability metrics is also interesting. We note that this does not correlate with the acquisition geometry in any way. Visually, from Figure 6, the fluctuations in nrms and predictability appear to inversely correlate with the seismic repeatability; where there is greater residual signal (as marked by the red arrows), the nrms values are smallest (largest predictability). Conversely, the largest nrms values (smallest predictability) appear to correlate with areas of less residual signal. This is not what we expect from our repeatability metrics, as we would like these to be a measure of the seismic repeatability. Signal and noise estimates were computed for both lines from 2.5 s to 3.5 s. The signal estimate was computed from the zero lag of the cross-correlation between neighboring Figure 6. North Sea data. Difference image (upper panel), nrms and CMPs, and the noise estimate was computed from the zero predictability values (lower panel). The red arrows indicate where lag of the average auto-correlation of the two CMPs, minus visually less repeatable seismic correlates with lower values of nrms the signal estimate. This method estimates the uncorrelated and higher values of predictability. noise and assumes that the signal does not change across Data example 2, North Sea. The second data example is a neighboring CMPs. This is a reasonable assumption given single 2D line taken from the central streamer of a repeat the small CMP interval of 3.125 m. We note that because the swath of six streamers acquired in the North Sea during 2001. data are unmigrated stacks, diffraction energy will partly Data were acquired from point receivers and processed contribute to this noise estimate. using digital group forming. The output group interval was Figure 7 shows the signal and noise estimates below the 6.25 m. Both lines were processed identically. Active streamer first pass line 034. The values are plotted in dB with respect positioning control was used when acquiring these data, and to the peak signal level. The black line is the signal estimate positioning repeatability was extremely high. In-line posi- for line 034 and the green line is for the second pass, line tioning errors for source and receivers are within 3 m and 065. They are very similar. The red line is the noise estimate 5 m, respectively. Cross-line positioning errors are within 5 for 034 and the blue line is for 065. They are, again, very m and 30 m (source and receivers), and are generally less similar. Although there is some mild correlation between the than this with an rms midpoint error of around 5 m. signal and noise estimates, the signal is generally fluc- Figure 5 shows the first pass (line 034, upper panel) and tuating more with the noise at a more constant level along

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the line. This implies the noise is not purely due to perturbations because the noise would then be at a fixed level below the signal. Figure 7 shows that there are a number of places where the noise level does change. To the left of the 4000-m mark, the noise level drops slightly on both passes. We have no explanation for this. However, the significant noise reduction to the left of 2000 m (and which is observable in the seismic image) correlates with the position of the first shot. To the left of this, increasingly fewer near offsets contribute to the stack. At the very left edge, the noise increases again where the fold of stack progressively decreases. The opposite effect is observed on the right of Figure 7. The noise rapidly increases after the point where we lose the far offsets from the stack (to the right of the 10 000-m mark). These observations imply that the noise is present on the near offsets to a greater degree, which might be expected. However, it is believed caused by the DMO operator—the noise-atten- uating effect of the DMO operator is a function of offset Figure 8. North Sea data. Difference image (upper panel), nrms and because the operator aperture increases with offset. This is predictability values (lower panel), as shown in Figure 6, with the a subject for further investigation. average noise-to-signal ratio plotted in red. From the signal and noise estimates, we compute the noise-to-signal ratio, and we average the values from both lines as they are so similar. Figure 8 shows the same difference image and repeatability metrics as Figure 6, but with the noise-to-signal ratio also shown (red curve). The correlation with the repeatability metrics is startling. Clearly, fluctuations observed in nrms (or predictability) have little to do with signal “repeatability” but directly relate to the noise-to-signal ratio in the data, before any differencing. Because the noise level is relatively constant compared to the signal level (Figure 7), the repeatability metrics are simply showing the seismic signal strength. The correlation of nrms with the noise-to-signal ratio (N2S) is explainable. Nrms equals the rms of the difference between the two repeat lines (the time-lapse noise) divided by the average rms of the signal and noise in the two lines. For small noise values, nrms is approximately the rms of the time-lapse noise divided by the rms of the signal. The noise-to-signal ratio is the mean square of the noise divided by the mean square of the signal. If the noise is purely random, then the time-lapse noise rms amplitude equals 2 Figure 9. Synthetic data, noise free. Baseline (top left), repeat (bottom times the noise in either input trace. We then end up with left), difference (top right), repeatability metrics nrms and predictabil- the relationship: ity (bottom right).

for random noise and noise << signal.

This will not be an exact relationship with real data as there is residual signal and the noise is not random or, necessarily, small. However, the observed correlation between nrms and N2S tells us that noise-to-signal ratio is driving the repeatability metrics.

Synthetic example. Figure 9 shows a simple synthetic example. The data are composed of a single trace replicated a number of times with a fixed time window scaling down the amplitudes by 0.6 and to zero over a portion of the trace. The few traces to the right are all zero valued. Data in the top left panel represent the baseline survey, and data in the bottom left panel represent the repeat survey. All we have done is scale the amplitudes by 0.9 for the repeat survey (for example, a simple approximation to a source strength difference). The difference data are shown in the panel to the top right. The panel to the bottom right shows the repeatability metrics nrms and predictability. Figure 10. Synthetic data with added white noise. Baseline (top left), Predictability is 100% for the whole line and nrms is con- repeat (bottom left), difference (top right), repeatability metrics nrms stant (about 10.5%) for the whole line. Data are noise free. and predictability (bottom right).

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Repeatability or predictability? From the Gulf of Mexico data example, it appears that the streamer position repeatability governs data repeatability; this appears to be the overriding factor. From the North Sea data example, it appears that signal-to-noise governs data repeatability, the data acquisition geometry being highly repeatable. In both cases, there seems to be a relationship between the repeatability metrics, nrms and predictability. As we have already noted, nrms is very sensitive to any change in the data or noise, whereas predictability is sensitive to reflectivity changes and to noise. Either metric used alone can be misleading. In this section, we examine plots of the predictability and nrms values to gain an insight into their relationship. Figure 12 shows plots of predictability and nrms for the real data examples, Gulf of Mexico (left) and North Sea (right). The negative correlation between the metrics is clear and, in both cases, there is a sharp bounding edge to the lower left of the cross-plot values. Data points in red are for the central full-fold traces, which tend to lie toward the top Figure 11. Synthetic data with constant signal and increasing noise left of the plot (more repeatable data). levels. Baseline (top left), repeat (bottom left), difference (top right), The sharp bounding edge is intriguing and some sim- repeatability metrics nrms and predictability (bottom right). ple modeling suggests an explanation, the results of which are shown in Figure 13 (left panel). Two initially equal traces were perturbed and the repeatability metrics were calculated for the resulting difference trace. Traces were perturbed in amplitude, static, and phase, and by adding increasing amounts of random noise to the traces. Amplitude, static, and phase perturbations result in the expected straight line parallel to the abscissa; predictability is not sensitive to these changes. Random noise perturbations result in a bell curve similar to the bounding edge of the real data plots (Figure 12). At the peak of the bell, a large change in nrms corresponds to a small change in predictability, supporting the assertion that nrms is sensitive to small perturbations at high Figure 12. Nrms predictability crossplots. Gulf of Mexico data (left), repeatabilities (also observed in our modeling examples). North Sea data (right). As we move down this curve, data are less correlated; at 141% (2) nrms and 0% predictability, the data are com- Figure 10 shows exactly the same synthetic example but pletely uncorrelated. Where there is a combination of these with white noise added to both the baseline and repeat sur- perturbations (as expected on real data) we would expect vey. The standard deviation of the white noise is 5% of the many points to lie to the right of this bounding line, as in peak signal amplitude. The effect on the repeatability met- the real data. rics (bottom right) clearly demonstrates what is happening The relationship between nrms and predictability can be in the real data; the repeatability metrics are directly related worked out theoretically for the random noise case. If the to the signal-to-noise level, and because the noise level is ratio of the white noise amplitude to initial trace amplitude constant, they relate directly to signal strength. For the zero is λ, then we obtain the following relationships, expressed signal strength traces on the far right, nrms values are around as percentages: 141% and predictability values are near zero (theoretically zero for truly random noise). and The synthetic results suggest that nrms is more sensitive to changes in the signal-to-noise level than predictability (at these high values of predictability). This is what we To work out this relationship for predictability, we have observed on the real North Sea data example (Figures 6 and considered only the zero lags of the correlations, whereas 8). However, because the noise level is constant in this syn- values of predictability presented in this report use 16 pos- thetic example, the varying signal level may be biasing this itive and 16 negative lags (as defined by the WesternGeco observation. The following final synthetic example demon- data processing software). The inclusion of more lags essen- strates the effect of varying noise levels. In Figure 11, the tially dampens, or smoothes, the predictability computation signal is constant for all traces. 5% white noise has been (if all lags were considered predictability would always be added to the central traces, and 10% white noise has been 100%), and we can simulate this by adding a damping term, added to traces to the right (the standard deviation of the d, to the equation: white noise is related to the peak signal amplitude). The repeat data set is simply the same with different seeds for the noise generation. The metrics (bottom right) clearly show that nrms is more sensitive to changes in the noise These theoretical curves are in the right panel of Figure level than predictability is, for these high levels of pre- 13 as plots of nrms and predictability. The blue curve is for dictability; we note that, for typical repeat seismic data, pre- the undamped case, and the red curve is for the damped dictability values will generally be quite high. case. The modeled points for the random noise perturbations, as shown in the left panel of Figure 13, are also plot-

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15, the blue curve is for broadband noise, and the green curve (left, GoM only) is for 5-70 Hz. The modeled curves show an excellent fit to the real data in both cases; the hard bounding edge appears to define the residual noise field, which will vary with the bandwidth and with the form of the noise.

A matching methodology. We make the assertion that the hard leading edge observed in the real data defines the random time-lapse noise field. Nrms is sensitive to amplitude, Figure 13. Nrms predictability crossplots. Synthetic modeled data phase, and static change, whereas predictability is not; it (left), theoretical curves for random noise (right). reflects changes in the reflectivity and in the noise. Matching two data sets with linear filters should move points on the plot to the left, toward the leading edge, predictability is unaffected. We suggest a matching methodology of (1) plotting predictability versus nrms, (2) identifying data that trail away Figure 14. Nrms predictability from the leading edge of the plot, (3) using the sensitivity crossplots. Synthetic modeled of nrms to minimize the perturbations in static, amplitude, random noise data. Broadband and phase to bring the data back to the leading edge of the (black) as shown in Figure 13. plot (the leading edge can be seen as representing the match- 10-125 Hz (red), 20-125 Hz ing goal for nrms for a given noise level), (4) correlating nrms (green), 0-80 Hz (blue), and 0-40 Hz (magenta). and noise-to-signal ratio; a positive correlation implies that noise is the dominating factor driving the metrics, and (5) application of noise reduction methods then move the data up this leading edge.

Discussion and conclusions. We have analyzed two different repeat seismic lines in terms of repeatability metrics, nrms, and predictability. The results are very different. In one case, residual differences correlate with streamer position repeatability. Where the streamer positions closely coincide, data are highly repeatable, demonstrating that repeatability errors in the source position and source signature, after shot-by-shot signature deconvolution, are much less than those due to streamer position nonrepeatability. The nrms error correlates with CMP position error and is consistent with the results of Morice et al. (2000). No active streamer Figure 15. Nrms predictability crossplots, as shown in Figure 12. Gulf of Mexico data (left), North Sea data (right). The modeled curve for positioning was used during acquisition of these data. broadband random noise is blue. On the Gulf of Mexico data, the mod- In the second case, active streamer positioning was used eled curve for 5-70 Hz is green. during the acquisition and the positioning repeatability was very good. Data are highly repeatable along the whole line, ted. The theoretical curves exhibit the bell-like shape seen though trace residuals are large (larger than in the first case), in the real data and modeled data, and a value of d=0.75 and the residual differences anticorrelate with the signal- seems to accurately represent the modeled points, i.e. d=0.75 to-noise ratio in the data. The background random noise level compensates for the 16-lag smoothing of the predictability appears to be high (and fairly constant) in these data, and computation for the (broadband) random noise case. leads to the nrms error correlating with the signal strength. In this case (where noise is dominant), variations in nrms The effect of bandwidth. The synthetic modeling was tell us nothing about the signal repeatability. This correla- repeated using random noise, but after applying varying tion of nrms with signal-to-noise ratio tells us that noise to- band-limiting filters. Figure 14 shows the resulting plots. The signal ratio is driving the repeatability metrics, not the signal broadband results of Figure 13 are black, along with 10-125 repeatability. Hz (red), 20-125 Hz (green), 0-80 Hz (blue), and 0-40 Hz The metrics of nrms and predictability alone do not lead (magenta). As the low end is cut, there is a slight increase to an intuitive understanding of the data repeatability, and in predictability with little change in nrms; hence, the curve they require interpretation of their values. We observe pre- shifts slightly up. As the high end is cut (which we might dictability values of 90-(almost) 100%, which suggest high often expect on real data), predictability still slightly repeatability, but corresponding nrms residuals of 60-18% increases (it will always increase as the bandwidth is lim- suggest data that are not so repeatable. We also observe high ited), but the overriding effect is a reduction in nrms; hence nrms values when the data are, visually, highly repeatable. the curves shift to the left. These observations may help to explain why Koster et al. Figure 15 shows the real data plots (as in Figure 12) but (2000) distinguish water replacing oil in Draugen (an imped- with the modeled curves for random noise (which agree with ance change of 10%, but an unstated reflectivity change) with theoretical curves for the broadband case) also plotted. nrms of 35%. It may also explain why Kristiansen et al. Average noise spectra were computed for the two real data (2000) observe a visually better repeatability on the Foinaven sets where there was least residual signal. The Gulf of Mexico seabed data (predictability, 96%) than the towed data (pre- data show the noise falling away after 70 Hz. The North Sea dictability, 93%). We may also infer in the Foinaven case that data show more broadband noise up to 125 Hz. In Figure the unpredictable power to total power ratios are 4% and

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Appendix A

7%, respectively. Therefore, the seabed data have just over cessing, are needed to assess overall uncertainty and repeata- half the time-lapse noise power of the towed data. However, bility of the two surveys and their relation to the time-lapse relying solely on predictability can be misleading, as it is signal and to the signal-to-noise levels. insensitive to amplitude, static, and phase changes. Plots of nrms and predictability values show a sharp Suggested reading. “Marine seismic cable steering and control” bounding edge, which is interpreted as a reflection on the resid- by Bittleson et al. (EAGE 2000 Expanded Abstracts). “Aproven con- ual time-lapse noise field, which is bandwidth dependent. We cept for acquiring highly repeatable towed streamer seismic data” note also that nrms is more sensitive to changes in random by Eiken et al. (EAGE 1999 Expanded Abstracts). “Signal preserv- background noise when predictability values are high. These ing swell noise attenuation using point receiver seismic data” by observations and sensitivity differences suggest a matching Jenkerson et al. (SEG 2000 Expanded Abstracts). “Using legacy methodology of (1) plotting predictability and nrms; (2) iden- seismic data in an integrated time-lapse study: Lena Field, Gulf tifying the data which trail away from the leading edge of the of Mexico” Johnson et al. (TLE, 2000). “Time-lapse seismic sur- crossplot; (3) using the sensitivity of nrms to minimize the per- veys in the North Sea and their business impact” by Koster et al. turbations in static, amplitude, and phase to bring the data (TLE, 2000). “Seismic repeatability, normalized rms and pre- back to the leading edge of the crossplot (the leading edge can dictability” by Kragh and Christie (SEG 2001 Expanded Abstracts). be seen as representing the matching goal for nrms for a given “Foinaven 4D: processing and analysis of two designer 4Ds” by noise level); (4) correlating nrms and noise-to-signal ratio; a Kristiansen et al. (SEG 2000 Expanded Abstracts). “The impact of positive correlation implies that noise is the dominating fac- positioning differences on 4D repeatability” by Morice et al. (SEG tor driving the metrics, and (5) application of noise reduction 2000 Expanded Abstracts). “Adaptive beamforming with general- methods then move the data up this leading edge. ized linear constraints” by Özbek (SEG 2000 Expanded Abstracts). Typically, noise reduction will occur prior to final signal “Partial coherence matching of synthetic seismograms with seis- matching as part of processing in order to optimize the sig- mic traces” by White (Geophysical Prospecting, 1980). TLE nal-to-noise ratio in the final images. Because the data presented here are unmigrated stacks, the crossplot metrics reside Acknowledgments: We thank the many colleagues at WesternGeco who were lower down the leading edge than could be ultimately involved with this work. However, the responsibility for errors remains with achieved. us. The combination of repeatability metrics should assist in matching time-lapse data sets, but metrics of the prior uncer- Corresponding author: [email protected]; [email protected] tainty in the data, together with careful analysis during pro-

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Appendix B

GEOPHYSICS, VOL. 69, NO. 6 (NOVEMBER-DECEMBER 2004); P. 1425–1442, 19 FIGS. 10.1190/1.1836817

Seismic imaging of reservoir ﬂow properties: Time-lapse amplitude changes

Don W. Vasco1, Akhil Datta-Gupta2, Ron Behrens3, Pat Condon3, and James Rickett3

1998; Smith et al., 2001), including stress changes (Watts et al., ABSTRACT 1996; Guilbot and Smith, 2002). Time-lapse monitoring is ma- turing as a technology and guidelines for its application are Asymptotic methods provide an efficient means by available (Lumley et al., 1997). Thus, time-lapse monitoring is which to infer reservoir flow properties, such as perme- becoming a tool in reservoir management (Fanchi 2001). Still, ability, from time-lapse seismic data. A trajectory-based the methodology is advancing, and new techniques are under methodology, similar to ray-based methods for medical development which allow for discrimination between pressure and seismic imaging, is the basis for an iterative inver- and fluid saturation changes over time (Brevik, 1999; Tura and sion of time-lapse amplitude changes. In this approach, a Lumley, 1999; Landro, 2001). single reservoir simulation is required for each iteration To date, time-lapse seismic has chiefly served as a monitoring of the algorithm. A comparison between purely numer- tool. That is, time-lapse seismic observations are used to map ical and the trajectory-based sensitivities demonstrates changes in reservoir saturation and pressure. The next logical their accuracy. Analysis of a set of synthetic amplitude step is to use time-lapse seismic data to characterize the reser- changes indicates that we are able to recover large-scale voir, to infer reservoir permeability and porosity heterogeneity. reservoir permeability variations from time-lapse ampli- Such time-lapse–generated models of reservoir permeability tude data. In an application to actual time-lapse ampli- aid in optimizing secondary recovery of bypassed oil and gas. tude changes from the Bay Marchand field in the Gulf The additional information provided by the time-lapse data of Mexico, we are able to reduce the misfit by 81% in will result in more accurate predictions of the performance of 12 iterations. The time-lapse observations indicate lower future production efforts. Currently, there have been very few permeabilities are required in the central portion of the attempts at formal reservoir characterization using 3D time- reservoir. lapse observations (He et al., 1998; Huang et al., 1998). By formal reservoir characterization, we mean some manner of inversion of the time-lapse field data for reservoir-flow properties. INTRODUCTION Time-lapse reservoir characterization is hampered by computational difficulties. Typically, finding a reservoir model which It has been appreciated for some time that saturation and is compatible with a set of saturation and pressure changes, as fluid pressure changes in a reservoir can lead to detectable would be derived from the time-lapse data, requires a signifi- changes in seismic attributes (Domenico, 1974; Nur, 1989). Cor- cant number of reservoir simulations. Because each reservoir respondingly, there has been a gradual advancement in the use simulation may take hours, if not days, of CPU time on a work- of time-lapse seismic observations to monitor reservoir pro- station, formal inversion can be prohibitively expensive. For cesses. Early studies, such as the work of Greaves and Fulp example, stochastic methods, such as the simulated annealing (1987) on the monitoring of a fireflood, focused on detection approach used by Huang et al., (1998), require hundreds, if not and the estimation of basic features such as the propagation thousands, of reservoir simulations. direction of the fireflood. Time-lapse seismic is now a useful In this paper we introduce a new trajectory-based approach tool for monitoring thermal processes (Eastwood et al., 1994; for time-lapse reservoir characterization. The methodology, Lee et al., 1995; Mathisen et al., 1995), CO2 flooding (Lazaratos which is akin to ray methods used in medical and seismic to- and Marion, 1997; Benson and Davis, 2000), gas-driven produc- mographic imaging, is extremely efficient. In fact, just a sin- tion (Burkhart et al., 2000), water-driven production (Behrens gle reservoir simulation is required to compute the trajecto- et al., 2002), and more complex combinations (Johnston et al., ries necessary to take a step in the inversion algorithm. The

Manuscript received by the Editor April 23, 2003; revised manuscript received June 17, 2004. 1Lawrence Berkeley Laboratory, Earth Sciences Division, Berkeley, California 94720. E-mail: [email protected]. 2Texas A&M University, Department of Petroleum Engineering, College Station, Texas 77843-3116. E-mail: [email protected]. 3Chevron Texaco Petroleum Technology Company, San Ramon, California 94503. E-mail: [email protected]; Pat.Condon@ chevrontexaco.com; [email protected]. c 2004 Society of Exploration Geophysicists. All rights reserved.

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1426 Vasco et al.

trajectories, which are closely related to streamlines, may be The composite fluid bulk modulus is computed from the indi- computed directly from the output of a numerical reservoir vidual components using the simple mixing rule simulator. The trajectory-based approach for inverting time- 1 S 1 − S lapse observations is an extension of techniques developed for = + , (2) the inversion of dynamic reservoir engineering data such as K fluid Kw Ko tracer data (Vasco and Datta-Gupta, 1999), two-phase flow data (Vasco et al., 1999), and transient pressure data (Vasco where Kw and Ko are the bulk moduli of the water and oil, et al., 2000). Here, we implement the methodology for time- respectively. The shear velocity is relatively unaffected by lapse seismic data and illustrate the approach by an application changes in the composition of the pore fluid according to to synthetic time-lapse amplitude changes. Finally, we apply Gassmann’s derivation. However, we should note that the dry the technique to a set of time-lapse amplitude changes from frame moduli of both the shear velocity (Gdry) and the bulk the Bay Marchand field in the Gulf of Mexico. sound speed (Kdry) are functions of the porosity and the differential (lithologic-pore) pressure (Pdif f ). METHODOLOGY We assume that, as is the case at Bay Marchand (Behrens et al., 2002), the dependence of the dry moduli on porosity In this section, we outline our approach for imaging reservoir and effective pressure are determined from laboratory obser- flow properties using time-lapse seismic observations. First, vations made on cores. The linearized functional forms used to we describe the method used to compute changes in elastic fit the observations are properties for a poroelastic medium induced by changes in fluid saturations and pressure. For this, we adopt the com- = + · φ, monly used Gassmann’s model (Gassmann, 1951), simple mix- Kdry a b (3)

ing laws, and laboratory-derived relationships. The method- Gdry = c + d · φ, (4) ology for computing the seismic response of the reservoir interval and surrounding layers involves an approximation to the response of stratification, as given by Kennett (1983). We where compute the seismic response of the reservoir in a piecewise fashion. That is, we compute a 1D response for each column a = a0 + a1 · Pdif f , of cells in the reservoir model. The novel aspect of our ap- b = b + b · P , proach concerns the relationship between reservoir saturation 0 1 dif f and pressure changes and reservoir flow properties. We use a c = c0 + c1 · Pdif f , trajectory-based technique, which is somewhat akin to seismic and electromagnetic ray methods, to relate reservoir porosi- d = d0 + d1 · Pdif f , ties and permeabilities to saturation and pressure variations. In many cases, the trajectories may be identified with streamlines, and the numerical values of a0, a1, b0, b1, c0, c1, d0, and d1 (e.g., flow lines through the reservoir) (King and Datta-Gupta, are the result of a nonlinear regression based upon the lab- 1998). oratory measurements. We should note that the relationship between measured laboratory compressional velocity changes and actual in-situ changes is imperfect. Due to changes as- Petrophysical model and seismic amplitude calculations sociated with coring and pressure release upon extraction, The key reservoir properties influencing its elastic response the cores will not necessarily produce accurate pressure re- are the saturations of the various fluids in the reservoir, the sponses. In practice, the compressional velocity is often found pressure in the reservoir, and the porosity within the reservoir. to be less sensitive to differential pressure variations than is We only consider the variations in water and oil saturation, indicated by laboratory measurements. Thus, there is consid- denoted by Sw and So, respectively. This simplifies the presen- erable uncertainty associated with the relationship between tation and, in our application to Bay Marchand, the influence reservoir stress changes and velocity and impedance changes. of gas in the reservoir is small and may be neglected. Because There also is the confounding effect of reservoir subsidence the saturations sum to unity in the pore spaces, we just con- which often accompanies stress induced changes in elastic mod- sider the water saturation, which is denoted in what follows uli. Recently, some of these issues were examined using cou- by S. The oil saturation is then given by 1 − S. We denote the pled fluid flow and geomechanical modeling (Minkoff et al., reservoir pore pressure by P(x, t) and the porosity by φ(x). 2004). Such an approach complements existing laboratory de- For the frequency range and the reservoir conditions of in- rived relationships between deformation and elastic moduli terest Gassmann’s equation (Gassmann, 1951) is adequate. changes. Gassmann’s equation relates the bulk modulus of the fluid sat- The density of the saturated rock is given by the weighted urated rock (Ksat) to the moduli of the fluid mixture (K fluid), average of the densities of the components: the moduli of the dry rock (Kdry), the moduli of the rock grains (Kgrains), and the porosity ρ = φSρw + φ(1 − S)ρo + (1 − φ)ρgrains, (5) 2 (1 − Kdry/Kgrains) Ksat = Kdry + . ρ ρ ρ φ/ + − φ / − / 2 where w, o, and grains are the respective densities for wa- K fluid (1 ) Kgrains Kdry Kgrains ter, oil, and the component grains. The compressional (Vp) (1) and shear velocities (Vs ) are then computed for an isotropic,

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Appendix B

Seismic Imaging of Flow Properties 1427

layered, elastic medium (Kennett 1983): method is quite efficient and allows us to compute the seismic response for each column of the reservoir model in a fraction of a CPU second. + 4 Ksat Gdry V = 3 , p ρ (6) Reservoir saturation and pressure variations and reservoir flow properties Gdry The relationships given above enable us to relate reservoir Vs = . (7) ρ saturation and pressure changes to seismic-amplitude changes. These relationships form the basis for most studies using time- Figure 1 displays the variation in Vp as a function of satura- lapse seismic observations to monitor reservoir production. If, tion and differential pressure for conditions similar to those at however, we wish to infer reservoir flow properties themselves, the Bay Marchand (Behrens et al., 2002). In this figure, we see such as porosity and permeability, from time-lapse seismic ob- that the compressional velocity is relatively insensitive to dif- servations, we must relate saturation and pressure changes to ferential pressure. The compressional velocity is more strongly flow properties. In general, the relationship between reservoir influenced by variations in water saturation. This is particularly flow properties and saturation changes is nonlinear and in- important at Bay Marchand, where there is a strong water drive volves the use of a complex numerical flow simulator. To date, and the pressure does not vary by more than 100–200 psi. The this has limited the direct use of time-lapse seismic observa- compressional impedance, which is just the product of the com- tions in reservoir characterization (He et al., 1998; Huang et al., pressional velocity and the density, (Vp · ρ), displays a similar 1998). In this subsection we use a trajectory-based technique behavior as the differential pressure and water saturation are which provides an analytic relationship between reservoir flow varied (Figure 1). properties and saturation changes. The approach is the basis of In order to compute the seismic response of the reservoir an efficient formalism for the inversion of reservoir production and variations in seismic amplitudes induced by production data such as water-cut observations (Vasco et al., 1999; Vasco related processes, we adopt the approach of Kennett (1983). and Datta-Gupta, 2001a), as well as tracer (Vasco and Datta- In general, the layers in our reservoir model can be quite thin Gupta, 1999) and transient pressure (Vasco et al., 2000; Vasco (a fraction of a meter) relative to the dominant seismic wave- and Datta-Gupta, 2001b) data. lengths (tens of meters). In addition, the lateral dimension of The essential ideas underlying this approach are similar to the cells in our reservoir model are of the order of tens to a those used for high-frequency approximations in seismic wave hundred meters. We compute the seismic amplitudes for each propagation. There are a number of ways to motivate this tech- vertical column of the reservoir model, approximating the re- nique, but the main idea is the same (Anile et al., 1993). That sponse for a stack of horizontal layers. The approach involves is, the temporal and spatial variations associated with a propa- a partial expansion of reverberation operators (Kennett, 1983, gating front, in this case a two-phase fluid front, are much more p. 217) and allows for reflections at nonzero offsets, internal rapid than the variations in the background saturation distri- multiples, and tuning effects within the reservoir interval. The bution (S0). Stated another way, the jump in saturation across

Figure 1. (Left) Variationin compressional wave velocity as a function of water saturation and differential pressure. (Right) Variation in compressional wave impedance as a function of water saturation and differential pressure.

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the two-phase front is “fast” with respect to the changes in the quantity. A perturbation in σ is related to a perturbation in background saturation. Indeed, the advancing water can form saturation δS by a self-sharpening front in which the water saturation jumps, σ 1 σ almost discontinuously, from the background value to a large δS = S δσ, (10) value. In the Appendix, the idea of “rapid” and “slow” scales t t t is expressed in the form of an asymptotic expansion in a scale where the prime denotes differentiation with respect to the ε parameter . As in elastic and electromagnetic wave propa- argument σ/t. The quantity δσ follows from a perturbation of gation (Kline and Kay, 1965), this approach leads to solutions the integral 8: which are defined along trajectories or rays (Chapman et al., 1999). As indicated by equation A-7 in the Appendix, the tra- δσ = δp(r)dr, (11) jectories depend upon the flow velocity field U, which in turn is a function of the pressure distribution in the reservoir. Thus, the trajectories are similar to streamlines which are increas- where p(r) is the integrand in the integral 8: ingly used to model reservoir flow (King and Datta-Gupta, φ(r) p(r, t) = . (12) 1998). κ K (r)|∇ P(r, t)| There are several advantages associated with our trajectory- based approach. One advantage is that we obtain an analytic For a total mobility (κ) which does not vary significantly, the expression for the traveltime (σ) of the two-phase front along perturbation of p(r, t) is of the form a trajectory , ∂p ∂p ∂p δp r, t = δφ r + δK r + δ|∇ P r, t |. ( ) ∂φ ( ) ∂ ( ) ∂|∇ | ( ) φ(r) K P σ = dr, (8) (13) κ K (r)|∇ P| The partial derivatives may be calculated directly from the an- where φ(r) is the porosity, K (r) is the absolute permeability, P alytic form for p(r, t) given above (see equations A-19 in the is the pore pressure, and r is the distance along the trajectory Appendix). It is clear from equation 12 that the porosity, φ(r), (see equation A-11 in the Appendix). The variable κ is the and permeability, K (r), can trade-off. That is, we can only re- total mobility, a function of the relative permeabilities kro, krw solve their ratio unambiguously. In order to isolate a single and the viscosities µo, µw: property, such as permeability, we must make additional assumptions. κ = kro + krw . µ µ o w Sensitivity of time-lapse amplitude changes In equation 8, we have an expression for the traveltime along to reservoir flow properties a trajectory in terms of fluid properties, reservoir flow prop- Here, we combine the results of the previous two subsec- erties, and the pressure distribution. This expression can be tions and obtain a linear expression relating perturbations in interpreted in terms of the physical processes at play in multi- reservoir properties to perturbations in time-lapse seismic- phase flow. For example, increasing porosity results in a larger amplitude changes. This linear relationship forms the basis for volume which the fluid must fill as it propagates through the an iterative inversion. According to our petrophysical model, rock. Thus, it will take longer for the fluid to travel from an in- the seismic amplitude response of a column of cells in our reser- jection well to a producing well. Increasing permeability results σ voir model is a function of the saturation, pressure, and porosity in a greater flow velocity, reducing the travel time . Similarly, in each cell of the column. For the ijth column of the reservoir a larger pressure gradient produces more rapid flow, causing model (where i is the index for the east-west direction and j the fluid front to arrive sooner. The traveltime also depends is the index for the north-south direction) let us write the am- on the fluids present ahead of the two-phase front, specifically plitude response as A(Sij, Pij, Φij), where the vectors Sij, Pij, their viscosities and saturations. and Φij denote vectors containing the saturations, pressures, By transforming the two-phase flow problem into charac- and porosities for all k cells in the ijth column, respectively. teristic coordinates (coordinates oriented with respect to the For a given location (ij value), a perturbation in the amplitude trajectories) we obtain a semi-analytic expression for the sat- response is the sum over the perturbations in each of the k cells uration history at a point on the trajectory (equation A-15 in in the column: the Appendix): ∂ A ∂ A σ δ A(Sij, Pij, Φij) = δSijk + δPijk. ∂ S ∂ P S(t,σ) = S , (9) k ijk k ijk t ∂ + A δ . σ ijk (14) where is given by equation 8. The sensitivities required to ∂ ijk solve the inverse problem follow from the form of the solu- k tion (equation 9), defined along the trajectories. Specifically, in For the following derivation, we neglect the pressure terms order to better fit the observations, we must relate perturba- and only concentrate on saturation and porosity changes. This tions in the model parameters (the reservoir flow properties) simplifys the calculations and, as indicated above, is an appro- to perturbations in the observations (the saturations within priate approximation for our Bay Marchand field application. the reservoir). Because the flow properties enter equation 9 Note that this is not always the case; in many situations we through the variable σ, we can consider a perturbation in this must take pressure into consideration and work with the full

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Appendix B

Seismic Imaging of Flow Properties 1429

expression 14. Neglecting the effect of pressure perturbations, (e.g., more producers or injectors are added), during the time the expression for the amplitude perturbation reduces to interval we will have to recompute the pressure field when ap- 0 ∂ ∂ propriate. The paths thus computed define the trajectories ijk A A 1 δ A , Φ = δS + δΦ . and ijk implicitly contained in equation 16. We use the sat- (Sij ij) ∂ ijk ∂ ijk (15) k Sijk k ijk uration history to compute the time derivative in equation 17 and the background flow properties, pressures, and saturations For time-lapse amplitude data, we need to consider the state to calculate the sensitivities in equation 13 (see equations A-19 of the reservoir at two distinct times T0 and T1. We denote quan- in the Appendix). Given the saturation distribution in the col- tities associated with each time by superscripts. We assume that umn of cells within the reservoir model at times T 0 and T 1,we the surveys have been cross-equalized such that the amplitudes use numerical differencing to compute the seismic amplitude and frequency content of the seismic traces are roughly equiv- partial derivatives in equation 16. Thus, in a single reservoir δ 1 Φ − δ 0 alent. Then, in forming the difference A (Sij, ij) A (Sij, simulation, we have all the quantities necessary for one step of Φ ij), the porosity term will cancel for reservoirs which have not an iterative linearized inversion algorithm. undergone compaction. Thus, the time-lapse difference takes the form of a single summation over the perturbations in satu- NUMERICAL CALCULATIONS ration within each of the k cells of the ijth column: In this section, we demonstrate the correctness and util- 1 0 ity of our trajectory-based approach. First, we compare the δ A (Sij, Φij) − δ A (Sij, Φij) trajectory-based sensitivity estimates with purely numerical ∂ A1 ∂ A0 = δ 1 − δ 0 . results. Sensitivities, relating perturbations in model param- S ijk S ijk (16) eters to perturbations in observations, are the basis for our ∂ Sijk ∂ Sijk k iterative inversion algorithm. Therefore, it is important to ver- The partial derivatives in equation 16 are computed by numer- ify their correctness. Next, we illustrate the usefulness of the ical differencing, that is, by computing the amplitude using the methodology by inverting a set of synthetic time-lapse am- method of Kennett (1983) at a given value of saturation and plitudes for reservoir permeability variations. The iterative then perturbing the saturation and recomputing the amplitude. inversion enables us to image large-scale permeability varia- Differencing these amplitudes and dividing by the saturation tions in the reservoir, based upon time-lapse seismic amplitude perturbation produces a numerical estimate of the derivative. changes. 1 0 The quantities δS ijk and δS ijk are given by combining equations 10–13. For example, δS1 is given by ijk Sensitivity computations 1 1 1 Amplitude sensitivity to saturation changes. — We first ex- δS ijk = S (T ) δp(x)dr, (17) 1 1 T ijk amine the sensitivity of time-lapse seismic-amplitude changes to perturbations in water saturation within the reservoir. As 1 where ijk denotes the trajectory from the ijkth cell to a point stated above, we compute sensitivities on a column-by-column on the initial position of the water front. Equations 16 and 17, basis, treating each column as a stratified medium and com- when combined with equations 10–13, relate perturbations in puting the approximate seismic response using a partial expan- time-lapse amplitudes to perturbations in reservoir flow prop- sion of reverberation operators (Kennett, 1983). Our reservoir erties. The sensitivities are trajectory-based, computed as line model is based upon a geostatistical realization intended to 1 0 integrals over the paths ijk and ijk in equation 17. Note mimic the Bay Marchand reservoir (Behrens et al., 2002). In that when there is no significant variation in reservoir pres- particular, we modeled over 14 000 days of production using a sure during the time interval between the seismic surveys, the reservoir simulator. The saturation distribution within an arbi- trajectories will be virtually identical. trary column of cells in the reservoir model is shown in Figure 2 Let us describe how we evaluate the expressions 16 and 17 for three time periods (2000, 10 000, and 14 100 days). The cor- in practice. We begin with a reservoir model, consisting of responding compressional velocity is also presented in Figure 2. nx × n y × nz grid blocks. For example, we might have an ini- The reservoir model consists of 23 layers approximately 30 m tial distribution of porosity and permeability determined from in thickness. For this particular column of cells, the reservoir well logs, cores, and seismic attributes. Given the initial reser- spans a depth range of 2089 to 2115 m. The thickness of the voir model and well schedules, we conduct a reservoir simula- reservoir should be compared with a quarter wavelength of the tion and save the saturation and pressure histories for each grid seismic data, about 17 m. Thus, a seismic reflection off the top block of the model. The nature of the reservoir simulator is not of the reservoir will be influenced nonuniformly by saturation important, we simply postprocess the saturation and pressure changes within the reservoir interval itself. histories to define the trajectories. Consider a point at the top We examined the sensitivity associated with the reflection of the reservoir where we observe changes in the amplitude off the top of the reservoir as this formed our basic data set of a reflection over some time interval T 1 − T 0. The point will for the Bay Marchand field case, presented below. Other at- be located at the top of some column of cells in the reservoir tributes may be used to constrain saturation variations over model, say the ijth column, containing k cells in depth. For time. For example, it is possible to use the entire waveform each cell, we define the trajectories by moving up the pressure to determine saturation changes. This may be preferable for gradient from the cell center. We will end up at the edge of the reservoirs of intermediate thickness. Alternatively, a seismic reservoir model, at a water injector, or at the initial position impedance-based approach may have some advantages over of an aquifer. If there are changes in the pressure conditions layer reflectivity.

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As mentioned above, we rely on a straight-forward numeri- are accurate and are reasonable approximations to the ex- cal difference to compute amplitude sensitivities to saturation act sensitivities. Here, we compare our trajectory-based esti- variations within the column of reservoir grid blocks. Figure 3 mates to sensitivities computed using a numerical differencing displays the sensitivities corresponding to the saturation technique. variations in Figure 2. Note how the sensitivity of time-lapse The numerical results are obtained by, first, perturbing the amplitude variations to changes in water saturation decreases reservoir porosities and permeabilities. Next, we run a reser- dramatically at a depth of around 2104 m, almost 15 m into the voir simulation to compute saturation and pressure, changes. reservoir. This is approximately one-quarter of a wavelength Based upon the saturations and pressures we compute the elas- from the top of the reservoir. The limited sensitivity with depth tic moduli at two times of interest. Then, we calculate the into the reservoir agrees with the findings of Behrens et al. (2002). They found that the best correlation coefficient was obtained when properties were averaged over the top 17 m of the reservoir. Note also that the sensitivities depend on the water saturation within the reservoir. In particular, the sensitivity of time-lapse amplitude changes to water saturation grows with increasing water content. In an effort to examine the numerical stability of our finite- difference approximations, we varied the magnitude of the perturbations used to compute the derivatives. In Figure 4, three sets of estimates are shown, corresponding to the saturation distribution at 14 100 days. It is clear that for saturation perturbations from 2% to 10%, the overall behavior is the same. For example, all the curves have the rapid decay at about 2104 m in depth. Though the magnitude of the perturbation changes by five times, the estimates are all within approximately 5% of each other. Thus, the difference estimates appear to be relatively stable.

Amplitude sensitivity to porosity and permeability changes. — In several respects, the trajectory-based technique is quite different from other methods for computing sensitivities. First, we invoke a “high frequency” approximation in Figure 3. Sensitivity of time-lapse amplitude differences to sat- the derivation. Second, we neglect the contribution of pres- uration changes in a column of the reservoir model. The verti- sure variations to the amplitude changes. Finally, we ignore cal scale is unitless, the ratio of the fractional change in amplitude to the fractional change in saturation. The sensitivities are any shifting of the trajectories induced by perturbations in computed with respect to three background saturation distri- porosity and permeability. It is important to verify that the butions. The column and times correspond to those in Figure 2. sensitivities computed using our trajectory-based approach The horizontal line indicates an amplitude sensitivity of zero.

Figure 2. (Left) Saturation distribution in a column of our reservoir model for three different times: 2000, 10 000, and 14 100 days. (Right) Compressional wave (P-wave) velocity for the corresponding column of reservoir model and three time intervals.

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Appendix B

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amplitudes of a reflection at these two particular times, sim- Sensitivities were computed for reflections from three locations ulating a time-lapse experiment. By differencing the results off the top of the reservoir. of the two computations, we compute the amplitude changes The left side of Figure 5 displays the results of the numeric over the time interval. The ratio of the change in the differential perturbation sensitivity estimates; on the right the trajectory- amplitude to the change in porosity or permeability provides based, or semi-analytic, estimates are shown. The numeric a numerical estimate of the sensitivity. Note, this sensitivity is perturbation estimates required 962 reservoir simulations, different from a purely static sensitivity of the seismic ampli- whereas the analytic estimates result from a single simulation, tude to permeability, which is typically quite small. Rather, this almost three orders of magnitude less computation. The close is a time-lapse sensitivity, which depends on porosity and per- agreement between the sensitivity estimates that result from meability via the saturation and pressure changes induced dur- these very different approaches is clear from Figure 5. The ing the flow simulations. Thus, the porosity and permeability sensitivity is largest along a narrow zone extending from the sensitivities are primarily controlled by their influence on the central water injector to the point of reflection, which is de- multiphase flow, as expressed in equations 10–13. By the chain noted by a star. The nature of the sensitivities makes physical rule, the total sensitivity will be the product of the sensitiv- sense; we would expect the porosities along the flow path from ity of the seismic amplitude change to saturation and pressure the water injector to the reflection point to have the greatest changes, multiplied by the sensitivity of saturation and pres- influence on the evolution of water saturation, and hence on sure changes to flow properties, as expressed in equation 16. the amplitude changes. The differencing methodology is simple to implement and ac- The sensitivities of time-lapse amplitude changes to reser- curate but computationally intensive, requiring N + 1 reservoir permeabilities are shown in Figure 6. The pattern of sen- voir simulations, where N is the number of grid blocks in the sitivities in Figure 6 is similar to the porosity sensitivities, an reservoir model. However, the results of the numerical dif- elongated region extending from the reflection point to the wa- ferencing provide an important check on the trajectory-based ter injector. However, the sign of the permeability sensitivities results. is opposite to that of the porosity sensitivities. This sign change We computed sensitivities with respect to reservoir porosi- agrees with the analytic expressions given by equations A-19 ties and permeabilities (Figures 5 and 6). The well configuration in the Appendix. The permeability sensitivities are about three was that of a five-spot with a central water injection well and orders of magnitude smaller than the porosity sensitivities. This four producing wells at the corners. The central water injector is also in accordance with equations A-19 and the values of is indicated by the open circles in Figures 5 and 6. The reservoir porosity (0.1) and permeability (150 md) in the layer. Specif- model is a homogeneous layer of 31 × 31 grid blocks. The spa- ically, from equations 12 and 13, note that the partial deriva- tial dimensions of the layer are 1 km × 1 km in area and 30-m tive with respect to porosity is of order 1, whereas the partial thick. The permeability of the layer is 150 md, and the poros- derivative with respect to permeability is of order 1/K 2. Thus, ity is 10%. The incoming wave has a dominant frequency of because we express K in millidarcys for a background perme- 100 Hz, and we treat the reflection of the compressional wave. ability of 150 mD, the sensitivity to K can be many orders of We consider amplitude changes for seismic reflection surveys magnitude smaller than the sensitivity to φ. However, this ap- occurring 180 and 270 days after the initiation of production. parent difference in sensitivity magnitude is mostly due to the chosen units and the representation of the unknown parameters. If we work in terms of 1/K rather than K , the squared term in the denominator vanishes. Furthermore, if the permeabilities are measured in darcys rather than millidarcys, the sensitivities are closer in magnitude to the porosity sensitivities. Such issues become important when we solve the inverse problem, that is, when we actually try and match the time-lapse observations. There are some small discrepancies between the numerical and analytic sensitivities. These are thought to be due to slight precision problems in computing the numeric perturbation estimates. However, they may be due to the influence of pressure variations and the shifting of the trajectories when the permeabilities are perturbed. Regardless, the differences are second-order effects, and there is excellent overall agreement between the two estimates.

Inversion of synthetic time-lapse amplitude changes We now consider a numerical illustration of the inversion of time-lapse amplitude changes. We wish to image the spatial dis- Figure 4. Amplitude sensitivities for saturations in a column of tribution of permeability within the reservoir based upon the the reservoir model. The vertical scale is unitless, the ratio of time-lapse data. The reference permeability for the synthetic the fractional change in amplitude to the fractional change in saturation. The background saturation distribution is that of test is shown in Figure 7, along with the location of ﬁve pro- 14 100 days in Figure 2. The sensitivities are computed using ducing wells (circles) and a single water injector (star). Using a three different saturation perturbations: 0.02, 0.05, and 0.10. reservoir simulator, we modeled 1000 days of production from

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the layer. The initial oil saturation was 0.91%, and the reser- tude variations (for example, permeability can vary by many voir porosity was fixed at 10%. The distribution of water within orders of magnitude and can control flow within a reservoir). the layer is shown after 180 and 670 days of oil production The latter approach is adopted for all that follows. That is, in in Figure 8. The water advances along the high-permeability this numerical example and the application below, we only con- east-central zone, extending to the southeast. The traveltime sider variations in the inverse of permeability, K −1. As stated of the advancing water front is shown in Figure 9. While water above, using K −1 normalizes the sensitivities, eliminating the reaches some wells after less than 200 days, it takes more than K −2 which appears when the partial derivative of p(r, t) with 700 days to reach other producers. respect to K is calculated (see equation 12). This is similar to Using the method of Kennett (1983), we computed ampli- the use of “slowness” rather than velocity in seismic traveltime tudes for reflections from the top of the reservoir for two times: tomography. Given a collection of amplitude changes (denoted 180 days and 670 days. The amplitude changes during this time by the vector δA1−0) by combining equations 12, 13, 16, and 17, interval are shown in Figure 10. The largest amplitude changes we arrive at a system of linear equations relating perturbations occur between the location of the front at 180 days and at 670 days (Figure 8). We only consider the largest fractional amplitude changes in our inversion, changes greater than 0.015. Such changes have a larger signal-to-noise ratio and are thus more reliable. Implementing a cutoff is similar to what is done in practice. In Fig- ure 10, we also indicate the trajectories associated with the 105 largest amplitude changes. The trajectories extend from the reflection points to the water injector, trav- eling up the pressure gradient, which is also shown in the figure. The trajectories, which are computed after a single reservoir simulation, are the critical elements needed to take a step in the inversion algorithm. In particular, the trajectories define 0 1 the path of integration ijk and ijk in equations 16 and 17. The 105 amplitude changes form our basic set of observations which we use to infer reservoir permeabilities. We adopt an iterative linearized inverse method to match the observed time-lapse amplitude changes. That is, we start with an initial reservoir model and iteratively update the permeabilities in order to better fit the data. At each step, we solve a penalized least squares problem for the updates to the permeability model (Parker, 1994). The sensitivities are crucial in this iterative algorithm, for they indicate the manner in which we should modify the permeabilities in order to reduce the misfit. The sensitivities are obtained by combining equations 12, 13, 16, and 17. The result is a linearized expression relating perturbations in time-lapse amplitude changes to perturbations in reservoir flow properties. Because of the tradeoff between porosity and permeability, we cannot resolve both parameters unambiguously. Thus, we must either express one parameter in terms of another (e.g. permeability as a function of porosity), or we assume that the variation Figure 5. Numeric (left) and trajectory-based semi-analytic (right) amplitude sensitiv- of one parameter is dominating the flow ity to changes in reservoir porosity. The scales are unitless, the ratio of the fractional and hence the saturation changes which change in amplitude to the fractional change in saturation. The amplitudes correspond are responsible for the time-lapse ampli- to reflections from three different points, denoted by the stars.

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Appendix B

Seismic Imaging of Flow Properties 1433

in time-lapse amplitude changes to perturbations in inverse to be underdetermined. Typically, there are many more un- permeability: known reservoir parameters than there are datapoints. This is particularly true when the full 3D inverse problem is con- δA1−0 = Mδk−1, (18) sidered. The trajectory-based sensitivities indicate that we can adjust the permeability anywhere along the trajectory in order where δk−1 denotes a vector containing inverse permeabilities to fit the observations. The regularization is designed to bias as elements, and M is a matrix of sensitivity coefficients. We the updates towards smoothly varying permeability variations. solve equation 18 using a least-squares algorithm that is ap- That is, because we cannot resolve small-scale heterogeneity, propriate for sparse matrices (Paige and Saunders, 1982). we chose to distribute the permeability updates smoothly over The inverse problem is regularized through the addition of the entire trajectory path if possible. The norm penalty term bi- roughness and model norm penalty terms. Such regularization ases the result in the direction of a prior model. The prior model is important because, in most cases, the inverse problem is likely should be based upon all available geologic and geophysical information. The penalty terms are in the form of quadratic forms, defined over the model space (Parker, 1994). The exact penalized misfit function is of the form −1 1−0 −1 −1 P(δk ) =δA − Mδk + Wnδk −1 −1 − δk 0+Wr ∇δk , (19)

where ·signifies the L2 vector norm, and Wn and Wr are the norm and roughness penalty weights, respectively (Parker 1994). The weights determine the importance of satisfying the regularization relative to fitting the observations. We chose the regularization weights Wn and Wr by trial and error. That is, we conducted a number of inversions with differing values of Wn and Wr . Other approaches, such as constructing a tradeoff curve are possible (Parker, 1994). For each iteration of the algorithm, we conduct a reservoir simulation to recompute the trajectories and redefine the pressure and saturation histories in each grid block. Our starting model is a homogeneous layer with an initial permeability of 100 md. The misfit reduction as a function of the number of iterations is shown in Fig- ure 11. The misfit is reduced by about two orders of magnitude after 30 iterations. However, because the problem is nonlinear, the reduction is not monotonic, and some steps result in a larger misfit. That is, the model updates reduce the misfit for the linearized problem but, when the model is updated and the reservoir simulator rerun, the misfit actually increases. Overall, the misfit is substantially reduced after 15 iterations, equivalent to 15 reservoir simulations. The initial and final fits to the amplitude changes are shown in Figure 12. For the most part, the initial predicted amplitude changes are much lower than the actual amplitude changes. This is most likely due to the fact that the two-phase front has either arrived too quickly or too slowly at Figure 6. Numeric (left) and trajectory-based analytic (right) amplitude sensitivity to changes in reservoir permeability. The scales have units of per millidarcy (md−1), the the observation points. In either case, the ratio of the fractional amplitude change to the change in permeability. The amplitudes saturation change (and the corresponding correspond to reflections from three different points, denoted by the stars. amplitude change) can be lower because

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the slope approaches zero ahead and behind the jump in sat- require additional time-lapse surveys. One may think of each uration. After the inversion, we match all but one amplitude time-lapse survey as a snapshot of a migrating wavefront. By ac- change. cumulating information on the position of the wavefront over The final reservoir permeability model is shown in Figure 13. time, we can image the velocity in the region over which the The model contains the large-scale features of our reference wavefront propagates. Similarly, we can iteratively construct model (Figure 7). In particular, there is a high-permeability a permeability model by imaging the movement of saturation anomaly running to the southeast and lower permeability to changes over time. Finally, note that we can use the trajecto- the west of the injection well. In general, the resolution of het- ries to efficiently compute the resolution and uncertainty as- erogeneity is limited along the trajectories (that is, we cannot sociated with our estimates (Parker, 1994; Datta-Gupta et al., localize features on the trajectories). Rather, the anomalies are 2002). smeared out along the flow paths. In addition, we cannot resolve flow properties beyond the edge of the final location of the water front. In order to resolve more detailed features, we

Figure 9. The traveltime of the water phase as it ﬂows from the injection well (star) to the producing wells (circles).

Figure 7. Reference permeability ﬁeld used to compute synthetic time-lapse amplitude changes.

Figure 10. (a) Pressure in the reservoir after 180 days of production. (b) Amplitude changes between 180 and 670 days of production. The curves in this ﬁgure are the trajectories used for the inversion of the amplitude values. They travel up the Figure 8. Water saturation distributions after (a) 180 and pressure gradient from the reﬂection points to the injection (b) 670 days of oil production from the reservoir. well.

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Appendix B

Seismic Imaging of Flow Properties 1435 APPLICATION: BAY MARCHAND Bay Marchand field and time-lapse surveys Bay Marchand field is a mature field characterized by complex geologic structure. Production began in 1949 and has con- tinued up to the present. We shall concentrate on the shallowest producing reservoir, contained within a single fault block at the northern flank of the field. The reservoir is a regressive marine sequence of sands and shales which is bounded to the south by a major counter-regional growth fault. The particular interval we shall work with is the 7100-ft sand, which is roughly 20–30 m thick. The mean porosity of the sand is 30% and the concen- tration of channel sands diminishes to the west. Strong water drive from a downdip aquifer to the north and east maintains the pressure in the reservoir and assists in the production. The 7100-ft sand is underlain by several productive reservoirs. Nu- merous wells penetrate the 7100-ft sand on their way to these deeper sands, providing valuable log data for reservoir characterization. In all, seven wells were active at various times in the 7100-ft sand from 1949 until 2000 (Figure 14a). The well Figure 11. Squared misfit as a function of the number of itera- configuration changes over the life of the field as indicated by tions of the inversion algorithm. the water-cut history. An initial 3D survey was conducted in 1987 as part of a program to increase production and improve the stratigraphic model of the reservoir. Subsequently, a regional 3D seismic survey extending across Bay Marchand field was conducted. The geometries of the two surveys are quite different, complicating the interpretation of the time-lapse changes. The first (1987) survey used oriented lines (046◦) of seabed hydrophones, with a bin spacing of 17 × 17 m. The 1987 data were only available as poststack legacy data and could not be reprocessed to enhance the time-lapse signal (Behrens et al., 2002). The subsequent 1998 Geco survey was part of a larger nonexclusive survey and was not designed for the purposes of time-lapse monitoring. The survey lines were oriented at 030◦ with a bin spacing of 34 × 34 m. The two surveys were reinterpolated onto a common geometry and crossequalized in order to enhance the time-lapse effects (Behrens et al., 2002). Crossequaliza- tion is an important procedure in the analysis of time-lapse seismic observations (Rickett and Lumley, 2001). The cross- Figure 12. Observed time-lapse amplitude changes plotted equalization was applied to the stacked data and primarily in- against the calculated amplitude changes. The initial fits (open volved static and phase shift corrections and trace equalization circles) are based on the initial homogeneous reservoir model. in a 400–1200 ms time window. The final fits (filled squares) are computed using the final iteration of the inversion algorithm. Some idea of the repeatability or similarity of the surveys is apparent in Figure 14b, where we superimpose 14 traces from the two surveys. These traces are remarkable similar, suggesting that the repeatability is quite good overall. A more quantitative comparison was undertaken by Behrens et al. (2002), who examined the signal-to-noise ratio, the root-mean-square differences, and the correlation between the surveys. They found that, prior to cross-equalization, the signal-to-noise- ratio was about 1.2. Following cross-equalization, the signal-to- noise-ratio increased to 1.50, 8.24, and 2.44 for three different techniques. Amplitudes were extracted for a reflection from the top of the 7100-ft sand. The horizon is denoted by the stars in Figure 14b. The amplitude change is defined as the amplitude of the 1998 survey subtracted from the amplitude of the 1987 survey. The peak amplitude change in a 20-ms window Figure 13. Final permeability model which results from an in- was taken as the amplitude difference. We averaged the am- version of the synthetic time-lapse amplitude changes. plitude changes within the lateral boundaries of our reservoir

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1436 Vasco et al. model to further enhance the signal. The results are shown in The reservoir model at Bay Marchand field Figure 15. In Figure 15 we only show nonzero values for those cells which do not lie within the aquifer. The aquifer is defined As discussed by Behrens et al. (2002), a reservoir model was by a depth of 2144 m, and corresponds to the zone of higher carefully constructed for the 7100-ft sand at Bay Marchand water saturation in Figure 16b. We observe significant time- field. Because of the deeper producing sands, there were some lapse amplitude changes near the near the southern edge of the 50 logged wells penetrating the 7100-ft reservoir yet not pro- model where production is taking place. Fractional amplitude ducing from it. Detailed facies logs and maps were used to sim- changes of over 5% are observed over the reservoir proper. The ulate petrophysical facies and generate geostatistical reservoir amplitude changes are mostly positive because this is a system models. The reservoir model consisted of 23 layers, and predic- where the impedance stiffens with water encroachment and the tions based upon it generally match the cumulative production initial reservoir impedance is greater than that of the overbur- data. den. Furthermore, because the strong aquifer drive keeps the For this preliminary study, we only consider a single-layer reservoir pressure relatively constant, gas plays a very minor reservoir model. Our focus here is on the correctness of the role at Bay Marchand field. algorithm and our ability to match actual field observations. This is not intended to be an in-depth case study of the Bay Marchand reservoir. In a future publication, we shall consider a multilayer reservoir model and a coupled inversion of time- lapse amplitude and water cut. We averaged the reservoir properties of the 23 layers to produce an initial one-layer reservoir model. The porosity and saturations in each layer were linearly averaged together while the logarithm of the permeabilities were averaged. The resulting permeabilities and initial water saturations are shown in Figure 16. The initial water saturations were produced by a reservoir simulation of field production using the initial reservoir model. As noted above, the quality of the reservoir diminishes to the west, this is notable in Figure 16a as a sharp decrease in permeability. In addition, the southern edge of the model is truncated by a fault, which is represented by a sharp decrease in permeability. The initial location of the aquifer is indicated by the high water saturation in Figure 16b.

Inversion of the time-lapse data Using the methodology outlined above, we inverted the amplitude changes between the two seismic surveys. Due to the nature of the surveys (the initial survey consists of legacy data gathered after the onset of production), the amplitude changes

Figure 14. (a) Water cut from the seven wells producing from the 7100-ft sand at Bay Marchand field. (b) Selected traces from a base survey (solid) shot in 1987 at Bay Marchand field. Corresponding traces from a regional survey (dashed) shot in Figure 15. Peak amplitude changes between 1987 and 1998 for 1998 for a region encompassing Bay Marchand field. the 7100-ft sand at Bay Marchand field.

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Appendix B

Seismic Imaging of Flow Properties 1437

are rather noisy. Therefore, we used the largest amplitude in general. Another reason that we could not reduce the misfit changes in the inversion: values larger than 0.05. In particu- further may be due to simplifying assumptions in our modeling, lar, 21 of the largest amplitude changes comprised our basic in particular, the fact that we are using a single layer to model data set. Starting from the initial permeability model shown in the reservoir permeability variations. Reservoir modeling of Figure 16a, we iteratively updated the permeabilities in order the production at Bay Marchand field and a time-lapse pulsed to better fit the amplitude changes. The misfit reduction as a neutron log indicate that the water saturation in not uniform function of the number of iterations is shown in Figure 17. The with depth (Behrens et al., 2002). Rather, there is a progressive squared misfit is reduced by 81% in 12 iterations. The final fit filling of the lower stratigraphic section over time. Thus, we ex- to the amplitude changes is indicated in Figure 18. Because we pect the fit to improve when additional layers are added to the did not consider amplitudes smaller than 0.05 in order to in- reservoir model. Our final model (shown in Figure 19a) con- crease the signal-to-noise ratio, all the points lie to the right of tains generally lower permeabilities in the central region of the the cutoff. Initially, the amplitude changes are systematically reservoir. The lower permeabilities are required to slow down underpredicted. The final fit still contains considerable scat- the arrival of the water in order to produce the largest changes ter. One reason for this is the nature of the time-lapse data within the time interval between the two seismic surveys. from Bay Marchand field. Because the two surveys were not The predicted amplitude changes, based upon the perme- designed for time-lapse monitoring, the data are rather noisy ability model in Figure 19a are show in Figure 19b. Many

Figure 17. Squared misﬁt as a function of the number of iterations of the inversion algorithm.

Figure 16. Initial permeability model for the 7100-ft sand at Bay Marchand field. Initial water saturation prior to the start Figure 18. Initial (open circles) and final (solid squares) fits to of production at the field. the Bay Marchand time-lapse amplitude changes.

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of the general features of the observed time-lapse amplitude ments. Another disadvantage of these data is the lack of a pre- changes (Figure 15) are reproduced by the updated model of production survey. Thus, we could not infer the initial reservoir flow properties. In particular, the location of the largest am- conditions, and we had to work with saturation changes rather plitude changes coincides in both models. Furthermore, the than the saturations themselves. Despite these limitations, the largest amplitude changes form an arcuate pattern extending time-lapse amplitude data indicate that permeabilities in the to the westernmost producing wells. The detailed amplitude central portion of the reservoir are too large to match the ob- variation is not reproduced by the updated reservoir model. servations. This finding was consistent with an initial inversion For example, the amplitude changes to the north of the arcu- of the water-cut observations. ate amplitude changes are not found in the predicted amplitude changes. But, as noted above, because the legacy data could CONCLUSIONS not be reprocessed and the second survey was not designed for time-lapse monitoring, the observed amplitude variations An asymptotic approach to the modeling of two-phase flow are rather noisy and, thus, detailed interpretation may not be provides an efficient formalism with which to invert time-lapse warranted. Also, we have adopted a simple one-layer reservoir seismic-amplitude changes. Specifically,the formulation results model in this initial attempt to match the time-lapse measure- in a trajectory-based algorithm for using time-lapse observations to update reservoir flow properties, such as permeability. Synthetic testing and comparisons with numerical computations indicate that the inversion scheme is both efficient and accurate. We find that time-lapse seismic-amplitude changes may be used to infer the large-scale permeability variations in a reservoir. Our application to actual time-lapse data from Bay Marchand field indicates that the method is robust in the presence of noise. As is true for the vast majority of inverse problems, there are issues of uniqueness and uncertainty associated with permeability estimates based upon time-lapse observations. For example, depending on the type of observation, there may be trade-offs in depth associated with the use of seismic reflection data. Many of these issues may be addressed using available tools, such as the computation of resolution and covari- ance matrices (Parker, 1994). The efficiency of our approach and the explicit expressions for model parameter sensitivities should help in this regard. Furthermore, we can take advantage of sparse-matrix methods for approximating resolution and model parameter variance (Vasco et al., 2003). There is also the issue of the dependence of the solution on the starting model caused by the nonlinearity of the inverse problem. This question is difficult to address is a satisfactory manner and un- derscores the fact that we should begin with the best possible model derived from 3D seismic and well log data. However, we can also again take advantage of the efficiency of the inversion algorithm to explore the range of possible solutions. Specifically, we can conduct a number of inversions, starting from various plausible initial models. By examining the common elements of the solutions, we gain an understanding of those features which are robust. In favorable situations, such as when preproduction survey data are available, it might be possible to infer the movement of a two-phase front. Estimates of front-arrival times, rather than saturation-amplitude changes, are quasi-linearly related to flow properties (Vasco and Datta- Gupta, 2001a). Thus, an inversion based upon arrival times is much less sensitive to the initial reservoir model. This work can be extended in several respects. In this paper, we investigated the inversion of time-lapse amplitude data for reservoir permeabilities. In order to keep the pre- sentation focused on the correctness and efficiency of our approach, we did not include other types of data. In the future, we would like to integrate both reservoir production observa- Figure 19. Final permeability model resulting from an inver- tions (such as water-cut and well pressure data) with seismic sion of the time-lapse amplitude changes. Fractional amplitude time-lapse measurements, as advocated by Landa and Horne changes predicted by the final inversion results. (1997) and Gosselin et al. (2001). It would also be interesting to

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Appendix B

Seismic Imaging of Flow Properties 1439 incorporate pressure estimates derived from time-lapse seismic Governing equations for two-phase flow amplitude variation, with offset (Landro, 2001). While the sensitivity of the compressional wave velocity to pressure changes We begin with the pair of partial differential equations de- is weak relative to saturation changes at the Bay Marchand scribing the flow of a wetting (water) phase and a nonwetting field, this is not always the case. Furthermore, the relative sen- (oil) phase (Peaceman, 1977): sitivity of the shear velocity to pressure changes is more sig- ρw K (x)krw ∂(ρw Sw) ∇· ∇(Pw(x, t) − ρw gz) = φ(x) , nificant. Hence, using multiple offset data, one can estimate µw ∂t pressure changes in a reservoir (Landro, 2001). In our sensitivity computations, synthetic illustration, and field application, ρo K (x)kro ∂(ρo So) ∇· ∇(Po(x, t) − ρogz) = φ(x) , we used amplitude changes associated with reflections off the µo ∂t top of the reservoir. Our approach is not limited to amplitudes from specific horizons. Because we compute the seismic sensi- where the variables Sw and So denote the saturations of the tivities numerically, we can use the entire waveform associated water and oil phases, respectively. The relative permeabilities with the reservoir interval or even surrounding reflectors. We of the water and oil phases are represented by krw and kro, can also work in the frequency domain, using particular fre- whereas the absolute permeability is given by K (x). The den- ρ ρ quency components of the traces. sities of the water and oil are w and o, respectively, g denotes We would also like to extend the trajectory-based method- the gravitational constant, and the spatially varying porosity is φ , ology to more general settings, for example, when capillary ef- (x). The pressure associated with the wetting phase is Pw(x t), , fects are present. We are beginning to generalize the trajectory- and Po(x t) is the pressure associated with the oil phase, The µ µ based approach to three-phase flow, by including gas. This is viscosities for the water and oil phases are w and o. The two an important extension of the methodology, particularly with equations are coupled because the saturations are constrained respect to time-lapse monitoring where the effects of gas can to sum to unity: be significant. Although the trajectory-based methodology is Sw + S = 1. still valid in the presence of gas, new phases introduce corre- o sponding nonuniqueness in relating amplitude changes to satu- If capillary forces are small relative to other factors, we may ration and pressure variations. Thus, more comprehensive data derive a single equation describing the evolution of the satu- sets (perhaps including electromagnetic observations) may be ration of the wetting phase, which shall be denoted by S(x, t) required (Hoversten et al., 2003) The dramatic impact of gas (Peaceman, 1977; Bedrikovetsky, 1993): may provide a sensitive indicator of reservoir pressure changes ∂ S which can be used to image flow properties directly, based upon φ(x) + U ·∇S + C(S) = 0, (A-1) a linear inversion of the estimated pressure changes (Vasco, ∂t 2004). Finally, we want to apply the methodology to additional where U is the flow velocity and C(S) is a term related to data sets, including data from surveys designed specifically for gravity: time-lapse monitoring. (ρw − ρo) kro(S) ∂ K (x) C S = gλw . ( ) λ + λ µ ∂ ACKNOWLEDGMENTS o w o z The spatial variations in the relative permeability parameters This work was supported by the Assistant Secretary for Fossil k and k w are assumed to occur due to variations in saturation. Energy, Office of Oil, Gas and Shale Technologies, of the U.S. ro r Department of Energy under contract DE-AC03-76SF00098, John Rogers, Program Manager. Support was also provided Asymptotic solutions for two-phase flow by the Assistant Secretary, Office of Basic Energy Sciences of the U.S. Department of Energy. We thank Bill Haworth, Mark The motivation for our asymptotic solution is based upon a Bergeron, and ChevronTexaco North America Upstream Gulf variation in scale (Whitham, 1974; Jeffrey, 1976; Anile et al., of Mexico Business Unit for providing the data and information 1993). That is, we assume that the initial saturation distribu- concerning Bay Marchand field. We also thank WesternGeco tion is a relatively slowly varying function of space and time for providing 3D seismic data for use in this study. All com- when compared to the jump in saturation across the two-phase putations were carried out at the Center for Computational front. In effect, there is a scale L describing the variation in

Seismology. background saturation in time and space and a scale describing the spatial and temporal variation in saturation across APPENDIX A the propagating two-phase front. We are assuming that L holds in the domain of interest. If we denote /L by , the ASYMPTOTIC SOLUTION condition is 0 < 1. FOR TWO-PHASE FLOW An asymptotic expansion is the representation of the solution as a formal series in powers of the parameter (Anile This Appendix outlines the derivation of an asymptotic so- et al., 1993): lution for the propagation of a two-phase fluid front. In partic- ∞ ular, we derive an analytic relationship between reservoir flow S(x, t) = S (x, t) + n S (x, t,ω), (A-2) properties and the arrival time and amplitude variation of a 0 n n=1 propagating two-phase fluid front. This asymptotic solution is the basis for an semi-analytic relationship between time-lapse where S0(x, t) represents the background variation in satura- seismic amplitude changes and reservoir flow properties. tion, and ω is the frequency of the wave. The frequency ω is

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assumed to have the form where is the trajectory from an initial point on the water σ , front to the final point. Note that this expression for phase is ω = (x t) (A-3) similar to that in Vasco and Datta-Gupta (2001a) if we incorporate generalized Darcy’s law U = κ K ∇ P, where κ is the total (Anile et al., 1993; Prasad, 2001), where σ(x, t) is the phase mobility: variation of the wave, a function describing the geometric con- k k w figuration of the propagating multiphase front in space and κ = ro + r . µ µ time. Note that the methodology is similar to asymptotic meth- ro rw ods in electromagnetic (Kline and Kay, 1965) and seismic (Aki Equation A-10 then becomes and Richards, 1980) wave propagation. φ(r) In constructing our asymptotic representation, we substitute σ = dr (A-11) the expansion A-2 of S(x, t) into the various terms of equation κ K |∇ P| A-1. For example, consider the components of the vector U(S) or which may be represented as a power series in S. The expansion is given by σ = p(r, t)dr (A-12) ∂ U 2 U(S, x, t) = U(S0, x, t) + S1 + O( ), (A-4) if we write the integral in terms of ∂ S φ(r) where O( 2) denotes terms of order 2 and higher. Substituting p(r, t) = , (A-13) κ K (r)|∇ P(r, t)| the expansions into equation A-1 produces an equation containing an infinite sequence of terms. Each term in the sequence which we call the front slowness. Note that the front slowness contains to some power as a factor. depends on time through the time dependence of the pressure gradient. Arrival time of the two-phase front. — As shown in Vasco and Datta-Gupta (2001a), neglecting terms containing of or- Amplitude of the two-phase front. — We may write the equa- der one or greater produces the equation tion governing two-phase flow (equation A-1) in terms of the ∂σ characteristic coordinates (e.g. a coordinate system defined by φ(x) + U(S , x, t) ·∇σ = 0. (A-5) ∂t 0 the characteristic curves associated with equations A-7 and A-8 (Vasco et al., 2001a). The result is a a first-order, quasi- The quantity σ(x, t) governs the propagation or kinematics linear, hyperbolic equation for S(t,σ) of the form (King and of the multiphase front. Note that the velocity vector U only Datta-Gupta 1998) depends on the background saturation. ∂ ∂ F(S) + S = , Using the implicit function theorem, we may write equa- ∂σ ∂ 0 (A-14) tion A-5 in the form ϕ(X, T ) = T − σ(X), where σ only depends t on position. Then, equation A-5 reduces to where F(S) is the fractional flow function mw ∇σ · U(S0, x, t) = φ(x), (A-6) F(S) = mo + mw

a ﬁrst-order linear partial differential equation, governing the for mw = k w/µ w and m = k /µ . Equation A-14 is invariant σ r r o ro ro distribution of . We may solve this equation directly, using with respect to coordinate scalings of the type the method of characteristics (Courant and Hilbert, 1962, p. 70). In the method of characteristics, solutions are developed t = εt,σ = εσ, ε > 0, along particular trajectories, the characteristic curves, which are denote by X(), where is a parameter signifying position which requires the solution to take the general form (set ε = / along the curve. The equations for the characteristic curves are 1 t) a set of four ordinary differential equations: σ S(t,σ) = S (A-15) dX t = S , , t , U( 0 x ) (A-7) d if it is to be unique (Chorin and Marsden, 1990; Bedrikovetsky, dσ = φ(x) (A-8) 1993). A speciﬁc form for S(t,σ) was derived by Buckley and d Leverett (1942). (Courant and Hilbert, 1962, p. 70). For a coordinate system with one axis oriented along U, we can write equation A-7 as Amplitude sensitivities: Analytic expressions. — From the dr general form of the solution for saturation (equation A-15), = U, (A-9) d we may compute analytic expressions for saturation amplitude sensitivities to changes in reservoir properties (Vasco et al., =| | where U U , and r denotes the distance along the axis 1999). That is, we wish to determine how perturbations in reser- aligned with U. Combining equations A-8 and A-9, we may voir properties (such as porosity and permeability) map into σ express as the integral perturbations in the saturation at a point in the reservoir. The φ σ = (r) , important point to note is that the reservoir properties enter dr (A-10) σ U expression A-15 through the variable . As noted in Vasco

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Seismic Imaging of Flow Properties 1441

and Datta-Gupta (2001a), a perturbation in σ is related to a Domenico, S. N., 1974, Effect of water saturation on seismic reflectivity δ of sand reservoirs encased in shale: Geophysics, 39, 759–769. perturbation in saturation S by Eastwood, J., J. P. Lebel, A. Dilay, and S. Blakeslee, 1994, Seismic σ 1 σ monitoring of steam-based recovery of bitumen: The Leading Edge, δS = S δσ, (A-16) 4, 242–251. t t t Fanchi, J. R., 2001, Time-lapse seismic monitoring in reservoir management: The Leading Edge, 20, 1140–1147. where σ is given by integral A-11. The quantity δσ follows from Gassmann, F., 1951, Elastic waves through a packing of spheres: Geo- a perturbation of integral A-11: physics, 16, 673–685. Gosselin, O., S. van den Berg, and A. Cominelli, 2001, Integrated δσ = δ , history-matching of production and 4D seismic data: paper SPE p(r)dr (A-17) 71599 presented at the 2001 Annual Technical Conference, Society of Petroleum Engineers. where Greaves, R. J., and T. Fulp, 1987, Three dimensional seismic monitoring of an enhanced oil recovery process: Geophysics, 52, 1175–1187. ∂p ∂p ∂p δp r = δφ r + δK r + δ|∇ P r |. Guilbot, J., and B. Smith, 2002, 4-D constrained depth conversion for ( ) ∂φ ( ) ∂ ( ) ∂|∇ | ( ) reservoir compaction estimation: Application to Ekofisk field: The K P Leading Edge, 21, 302–308. (A-18) He, W., G. Guerin, R. N. Anderson, and U. T. Mello, 1998, Time- The partial derivatives may be calculated directly from equa- dependent reservoir characterization of the LF sand in the South tion A-13: Eugene Island 330 field, Gulf of Mexico: The Leading Edge, 17, ∂ 1434–1438. p = p(r), Hoversten, G. M., R. Gritto, J. Washbourne, and T. Daley, 2003, Pres- ∂φ φ sure and fluid saturation prediction in a multicomponent reservoir (r) using combined seismic and electromagnetic imaging: Geophysics, ∂p p r 68, 1580–1591. =− ( ) , Huang, X., L. Meister, and R. Workman, 1998, Improving production ∂ K K (r) history matching using time-lapse seismic data: The Leading Edge, 17, 1430–1433. ∂p p(r) Jeffrey, A., 1976, Quasilinear hyperbolic systems and waves: Pitman =− . (A-19) Publishing. ∂|∇ P| |∇ P(r)| Johnston, D. H., R. S. McKenny, J. Verbeek, and J. Almond, 1998, Time-lapse seismic analysis of Fulmar field: The Leading Edge, 17, If the trajectories are not significantly perturbed by the passage 1420–1428. of the saturation front, then the perturbed trajectory may be Kennett, B. L. N., 1983, Seismic wave propagation in stratified media: Cambridge University Press. replaced by the unperturbed trajectory 0. This approximation King, M. J., and A. Datta-Gupta, 1998, Streamline simulation: A cur- is thought to be a good one for waterflood fronts (Datta-Gupta rent perspective: In Situ, 22, 91–140. Kline, M., and I. W.Kay, 1965, Electromagnetic theory and geometrical and King, 1995) and appropriate for the oil field application optics: John Wiley and Sons. considered. Landa, J. L., and R. N. Horne, 1997, A procedure to integrate well test data, reservoir performance history, and 4-D seismic information into a reservoir description: paper SPE 38653 presented at the 1997 REFERENCES Annual Technical Conference, Society of Petroleum Engineers. Landro, M., 2001, Discrimination between pressure and fluid satura- Aki, K., and P.G. Richards, 1980, Quantitative Seismology: W.H. Free- tion changes from time-lapse seismic data: Geophysics, 66, 836–844. man and Co. Lazaratos, S. K., and B. P. Marion, 1997, Crosswell seismic imaging of Anile, A. M., J.K. Hunter, P.Pantano, and G. Russo, 1993, Ray methods reservoir changes caused by CO2 injection: The Leading Edge, 16, for nonlinear waves in fluids and plasmas: Longman Scientific and 1300–1307. Technical Publishing. Lee, D. S., V. M. Stevenson, P. F. 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Guinzy, 1995, Time-lapse crosswell seismic tomogram inter- and dynamic reservoir characterization, Central Vacuum unit, Lea pretation: Implications for heavy oil reservoir characterization, ther- County, New Mexico: SPE Reservoir Evaluation and Engineering, mal recovery process monitoring, and tomographic imaging technol- 3, 88–97. ogy: Geophysics, 60, 631–650. Brevik, I., 1999, Rock model based inversion of saturation and pressure Minkoff, S. E., C. M. Stone, S. Bryant, and P.Peszynska, 2004, Coupled changes from time lapse seismic data: 69th Annual International geomechanics and flow simulation for time-lapse seismic modeling: Meeting SEG, Expanded Abstracts, 1044–1047. Geophysics, 69, 200–211. Burkhart, T., A. R. Hoover, P. B. Flemings, 2000, Time-lapse (4D) Nur, A., 1989, Four-dimensional seismology and (true) direct detec- seismic monitoring of primary production of turbidite reservoirs tion of hydrocarbon: The petrophysical basis: The Leading Edge, 8, as South Timbalier Block 295, offshore Louisiana, Gulf of Mexico: 30–36. Geophysics, 65, 351–367. Paige, C. C., and M. A. Saunders, 1982, LSQR: An algorithm for sparse Buckley, S. E., and M. C. Leverett, 1942, Mechanism of fluid displace- linear equations and sparse linear systems: ACM Transactions on ment in sands: Transactions of the American Institute of Mining, Mathematical Software, 8, 195–209. Metallurgical, and Petroleum Engineering, 146, 107–116. Parker, R. L., 1994, Geophysical inverse theory: Princeton University Chapman, S. J., J. M. H. Lawry, and J. R. Ockendon, 1999, Ray the- Press. ory for high-peclet-number convection-diffusion: SIAM Journal of Peaceman, D. W., 1977, Fundamentals of numerical reservoir simula- Applied Mathematics, 60, 121–135. tion: Elsevier Scientific Publishing Co. Chorin, A. J., and J. E. Marsden, 1990, A mathematical introduction Prasad, P., 2001, Nonlinear hyperbolic waves in multi-dimensions: to fluid mechanics: Springer-Verlag. Chapman and Hall. Courant, R., and D. Hilbert, 1962, Methods of mathematical physics: Rickett, J. E., and D. E. Lumley, 2001, Cross-equalization data process- Interscience. ing for time-lapse seismic reservoir monitoring: A case study from Datta-Gupta, A., and M. J. King, 1995, A semianalytic approach to the Gulf of Mexico: Geophysics, 66, 1015–1025. tracer flow modeling in heterogeneous permeable media: Advances Smith, P., J. I. Berg, S. Eidsvig, I. Magnus, F. Verhelst, and J. Helgesen, in Water Resources, 18, 9–24. 2001, 4-D seismic in a complex fluvial reservoir: The Snorre feasibil- Datta-Gupta, A., S. Yoon, D. W. Vasco, and G. A. Pope, 2002. Inverse ity study: The Leading Edge, 20, 270–276. modeling of partitioning tracer tests: A streamline approach: Water Tura, A., and D. E. Lumley, 1999, Estimating pressure and satura- Resources Research, 38, No. 6, 10.1029. tion changes from time-lapse AVO data: 61st Annual Conference,

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European Association of Geoscientists and Engineers, Extended certainty, and whole Earth tomography: Journal of Geophysical Re- Abstracts, 1-38. search, 108, 9-1/9-26. Vasco, D. W., 2004, Seismic imaging of reservoir flow properties: Time- Vasco, D. W., K. Karasaki, and H. Keers, 2000. Estimation of reservoir lapse pressure changes: Geophysics, 69, 511–521. properties using transient pressure data: An asymptotic approach: Vasco, D. W., and A. Datta-Gupta, 1999, Asymptotic solutions for so- Water Resources Research, 36, 3447–3465. lute transport: A formalism for tracer tomography: Water Resources Vasco, D. W., S. Yoon, and A. Datta-Gupta, 1999. Integrating dy- Research, 35, 1–16. namic data into high-resolution reservoir models using streamline- ——— 2001a, Asymptotics, saturation fronts, and high resolu- based analytic sensitivity coefficients: Society of Petroleum Engi- tion reservoir characterization: Transport in Porous Media, 42, neers Journal, 4, 389–399. 315–350. Watts, G. F. T., D. Jizba, D. E. Gawith, and P. Gutteridge, 1996, Reser- ——— 2001b, Asymptotics, streamlines, and reservoir modeling: voir monitoring of the Magnus field through 4D time-lapse seismic A pathway to production tomography: The Leading Edge, 20, analysis: Petroleum Geoscience, 2, 361–372. 1164–1171 Whitham, G. B., 1974, Linear and nonlinear waves: John Wiley and Vasco, D. W., L. R. Johnson, and Q. Marques, 2003, Resolution, un- Sons.

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Appendix C

Geophysical Prospecting, 2002, 50, 195±208

Rough seas and time-lapse seismic Robert Laws and Ed Kragh Schlumberger Cambridge Research Ltd, High Cross, Madingley Road, Cambridge CB3 0EL, UK

Received December 2000, revision accepted July 2001

ABSTRACT Time-lapse seismic surveying has become an accepted tool for reservoir monitoring applications, thus placing a high premium on data repeatability. One factor affecting data repeatability is the influence of the rough sea-surface on the ghost reflection and the resulting seismic wavelets of the sources and receivers. During data analysis, the sea-surface is normally assumed to be stationary and, indeed, to be flat. The non- flatness of the sea-surface introduces amplitude and phase perturbations to the source and receiver responses and these can affect the time-lapse image. We simulated the influence of rough sea-surfaces on seismic data acquisition. For a typical seismic line with a 48-fold stack, a 2-m significant-wave-height sea introduces RMS errors of about 5±10% into the stacked data. This level of error is probably not important for structural imaging but could be significant for time-lapse surveying when the expected difference anomaly is small. The errors are distributed differently for sources and receivers because of the different ways they are towed. Furthermore, the source wavelet is determined by the sea shape at the moment the shot is fired, whereas the receiver wavelet is time-varying because the sea moves significantly during the seismic record.

image. Successful time-lapse studies have been reported INTRODUCTION where the RMS difference, away from the position of the The problem of a seismic source in a rough sea was studied anomaly, is as high as 35% (Koster et al. 2000). by Jovanovich, Sumner and Easterlin (1983), who discussed Let us consider the relative importance of phase and ampli- the superiority of deterministic de-ghosting relative to statis- tude perturbations, and how errors expressed as a percentage tical deconvolution. Later, however, Dragoset, Hargreaves relate to errors expressed as decibels. If we take two identical and Larner (1987) concluded that the rough-sea error was sine waves, perturb one in amplitude or phase and subtract unimportant over the typical seismic band (5±80 Hz, wave- them, sample by sample, we generate a difference signal. We length 20±300 m) for structural imaging. More recently, can express the level of this difference signal, which is also a Eiken et al. (1999) studied a repeated seismic line where a sine wave, as a proportion of the unperturbed signal level or, heavy emphasis was placed on reproducing the same shot equivalently, in dB relative to the unperturbed signal. For positions and interpolating (between streamers) to the same example, if the two signals are at amplitude 1 (unperturbed) receiver positions. The test was performed in a relatively and 1.06 (perturbed) then the amplitude perturbation is 0.5 dB

calm sea with a significant wave height (SWH) of ( 20 log10(1.06)). The resulting difference signal has ampli-

0.5±1.2 m. SWH is a subjective peak-to-trough wave height, tude 0.06 (6%) and therefore a level of À24.4 dB ( 20 log10 and is typically defined to be four times the RMS wave (0.06)). A pure phase perturbation of 3.48 also leads to a 0.06 amplitude (Carter et al. 1986). Eiken et al. (1999) found (À24.4 dB) level of difference signal (sin3.4 0.06). that the RMS amplitude of the difference image was in the The statistics of the sea-surface shape have been discussed region of 5±10% of the RMS amplitude of the complete by many authors; Kinsman (1983) and Carter et al. (1986)

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are good general references. Pierson and Moskowitz (1964) Kirchhoff approximation by comparing it with a specialized gave expressions for the average spectrum of the sea-surface finite-difference code. These two methods differ in their fun- as a function of weather conditions. Their work described an damental approximations but yielded similar response wave- isotropic spectrum and used experimental measurements of forms. the sea height as a function of time at one point; no direc- We assessed the size of the rough-sea errors by Monte- tional information was included. Later studies by Hassel- Carlo methods; we made many realizations of the rough mann, Dunckel and Ewing (1980) and Mitsuyasu et al. sea-surface and analysed the spectra statistically. To do this (1975) extended this to include directional effects through we needed to develop simple models for the suspension the use of a buoy sensitive to tilt and angular acceleration as methods and shapes of the seismic source and streamer. We well as height. The average surface spectrum depends, inter assessed the importance of the rough-sea error to time-lapse alia, on the wind speed, the wind direction, the fetch (the surveying by simulating a seismic line shot twice. The seismic distance over which the wind has blown), the duration for propagation was modelled with a 2D finite-difference code which the wind has been blowing, and the water depth. A using realistic acquisition geometry and realistic data technique for generating 1D sea-surface realizations, by volumes. The rough-sea errors were introduced to both the multiplying the average spectrum by Gaussian complex source and receiver wavelets taking into account the proper random numbers, was described by Thorsos (1987). spatial coherence between the source and the receivers. A The reflection response of a curved surface can be com- different sea-surface realization was used for each shot. puted by various methods. Jovanovich et al. (1983) used a We found that the errors caused by the rough seas are Kirchhoff approach, whereas Dragoset et al. (1987) used the typically 1±3 dB in amplitude for 2±4-m SWH seas, and small wave-height approximation of Labianca and Harper 10±208 in phase at 50 Hz. Simulations suggest that the re- (1977). Kirchhoff methods can be used, provided that mul- sidual error (RMS amplitude of difference/RMS amplitude of tiple scattering and shadowing are not significant and that image) of a repeated line is in the region of 5±10% when the the surface radius of curvature is large compared with the first survey is made in a flat sea and the second in a 2-m sound wavelength. These conditions are typically met for SWH sea. near-normal-incidence scattering from sea-surface waves We first describe how we simulate ensembles of typical (see below). Care must be taken to include the Kirchhoff sea-surface shapes. We show how the reflection responses near-field term because the towed marine source or receiver of these surfaces are computed by Kirchhoff methods. After is typically only a few metres below the surface. A finite- this we show, using Monte-Carlo methods, the 95% confi- difference method can be used, provided it uses a grid refine- dence limits imposed on the source and receiver wavelets by ment method to cope with the relatively small but important the rough sea-surface. Next we use simulated seismic data to height variations present in the sea-surface (Robertsson, assess the importance of the rough-sea error. We `shot' the Blanch and Symes 1994; Robertsson 1996). The finite- same line twice, first with a flat sea-surface and then with difference approach has the advantage of avoiding the vari- moderately rough sea. Finally, we show that, in this example, ous approximations in the Kirchhoff and small-height a 2-m SWH sea leads to an RMS amplitude error in the methods, but it suffers from being practical only for 2D seismic difference image of about 5±10%. problems (with 1D sea-surfaces). There is a large literature on the scattering of sound by rough sea-surfaces but most of GENERATING THE AVERAGE SEA-SURFACE it concentrates on near-grazing incidence where the difficul- SPECTRUM ties are greater (see, e.g., Thorsos 1990). We have studied these rough-sea errors by computer simu- To study rough-sea effects it is necessary to be able to simulation. We simulated realistic, moving sea-surfaces using late realistic, moving sea-surfaces and to quantify their influ- straightforward extensions of established techniques. We ence on seismic data. We have adopted the following placed synthetic sources and receivers under the simulated approach. The average spectrum of the sea-surface is based sea-surfaces and we computed the reflection responses for on the work of Pierson and Moskowitz (1964) which has each gun and individual receiver using the Kirchhoff method also been well described by Kinsman (1983) and Carter et al. (including both near- and far-field terms). We incorporated (1986). We add the directivity of the sea-surface waves by the array responses by summation of the single gun and using the results of Mitsuyasu et al. (1975) and Hasselmann single sensor responses. We checked the accuracy of the et al. (1980).

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The Pierson±Moskowitz (1964) power spectrum of the experiment recorded data using a buoy that was sensitive not height at one position in a fully developed sea has the form, only to height but also to tilt and angular acceleration. Has- selmann et al. (1980) developed directivity corrections to the ag2 S f df exp Àbg4 2pf À4UÀ4df ; 1 isotropic Pierson±Moskowitz spectrum. They multiplied the 2p4f 5 isotropic spectrum by a directional term of the form, where f denotes the frequency, a and b are constants, g is the y D yN cos2s ; 4 acceleration due to gravity and U is the wind speed measured 2 at a height of 19.5 m. The frequency at which (1) has a where y denotes the azimuthal angle relative to the wind maximum value is called the modal frequency, fm. A `fully developed' sea means that the wind has been blowing for a direction, N is a normalization factor such that N D(y) long time duration over a large distance (the fetch), and the dy 1, and s is an empirical spreading function derived sea has built up to a steady state. Hasselmann et al. (1973) from a match to the experimental data (see below). discussed the growth and decay of the swell; it would be Hasselmann et al. (1980) expressed s as a function of f/fm: possible to include the effects of limited duration and fetch f m in the model if this were of particular interest. S Sm ; 5 fm This spectrum was based on elevation measurements made

as a function of time at a single position on the ocean surface; where fm is the modal frequency; sm 9.77 for f >1.05fm and

they included no directional information. It is conventional 6.97 otherwise; m À2.33 for f <1.05fm and 4.06 other- to express isotropic surface spectra in terms of frequency wise. The function s peaks near the modal frequency. That is, rather than wavenumber because the expression is derived, waves with frequencies near the modal frequency are very in part, from measurements of elevation made over time at a directional but those with other frequencies are less direc- single point. tional. The constants in (5) come from the best fit by Has- In deep water, the wavenumber and frequency are related selmann et al. (1980) to the experimental data. by the dispersion relationship (see, e.g., Carter et al. 1986), The full expression for the directional wavenumber spectrum is obtained by multiplying (3) and (4) together: 2pf 2 gk; 2 a y S k; ydkdy exp Àb2g2kÀ2UÀ4N cos2s dkdy: 6 where g is the acceleration due to gravity and k is the wave- 2k3 2 number in radians per metre. The wave-speed (f/k)isin- versely proportional to the frequency; long-period waves With a change of variable from polar to rectangular co- travel faster than short-period waves. ordinates, (6) can be written in a form that is more suitable

The modal frequency, fm (peak of the spectrum), occurs for for use on a rectangular grid: a wave whose speed is slightly less than the wind speed. That a S k ; k dk dk exp Àb2g2kÀ2UÀ4 is to say, for a given wind speed, the waves that are most xy x y x y 3 2k strongly excited are those which travel at a speed slightly less y 1 N cos2s dk dk : 7 than the wind speed. As the wind speed is increased, not only 2 k x y does the wave power spectrum increase in overall level, but Figures 1±3 show the appearance of the average directional the peak moves to longer wavelengths. This selection of a wavenumber spectrum for SWHs of 2, 4 and 8 m. All three dominant wavelength occurs because the waves are disper- graphs are plotted on the same scale, normalized so that the sive. It gives the sea its characteristic appearance with fast, peak of the 8-m SWH spectrum is 0 dB. The wind is blowing long-wavelength waves moving under a `skin' of slower- along the Y-axis. It can be seen at once that no waves travel moving, short-wavelength waves. Using (2), we can rewrite directly into the wind but that there is a general background (1) in terms of the absolute wavenumber, of high wavenumber waves travelling in a wide range of a S kdk exp Àb2g2kÀ2UÀ4dk: 3 directions. In addition to the general background there is an 2k3 elevated narrow arc in the spectrum centred on the origin. A major step forward in ocean-wave measurement was This corresponds to the main wind-driven waves and achieved with the JONSWAP experiment (Joint North Sea is predominantly in the wind direction. As the SWH is Wave Project), described by Hasselmann et al. (1973). This increased, this arc becomes higher, corresponding to

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Figure 1 Average sea-surface wavenumber spectrum for a 2-m SWH sea. The left side of the figure corresponds to waves travelling away from the wind.

Figure 2 Average sea-surface wavenumber spectrum for a 4-m SWH sea. The left side of the figure corresponds to waves travelling away from the wind.

higher-amplitude waves, and it also moves closer to the spectrum (e.g. Fig. 1) in the wavenumber domain by a con- origin, corresponding to longer wavelengths; as the sea gets jugate symmetric array of complex random numbers drawn rougher, the wave height and wavelength of the dominant from a Gaussian distribution. The resulting spectrum is in- waves both increase. verse Fourier transformed to the spatial domain, producing a realization of the sea-surface. The method has been used in the 1D case by, for example, Thorsos (1987) and Hastings, GENERATING INDIVIDUAL SEA-SURFACE Schneider and Broschat (1995). The generalization to 2D is REALIZATIONS straightforward, with the conjugate symmetry of the array The generation of realizations of a sea-surface with a defined required so that the resulting surface is real. Each realization average spectrum is achieved by multiplying the average of the surface is obtained by multiplying the spectrum by a

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Rough seas and time-lapse seismic 199

Figure 3 Average sea-surface wavenumber spectrum for an 8-m SWH sea. The left side of the figure corresponds to waves travelling away from the wind.

Figure 4 Snapshot from one realization of a simulated sea-surface. The SWH is 2 m and the wind is blowing along the Y-axis from Y.

different array of random numbers. Figures 4±6 show typical spectrum. Each component of the wavenumber spectrum is realizations of the sea-surface corresponding to the spectra phase delayed by an angle equal to the desired time shift given in Figs 1±3. Only a small patch of the simulated sea- divided by the angular frequency corresponding to the wave- surface is shown. The patch is taken from the centre of a number through the dispersion relationship. In our simula- 4-km square area. tions we use a different realization for each shot, but when The short-term time development of any realization sur- we study the temporal coherence of the wavelet over short face is generated by applying a phase-roll to the wavenumber time intervals, we use the phase-rolling.

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Figure 5 Snapshot from one realization of a simulated sea-surface. The SWH is 4 m and the wind is blowing along the Y-axis from Y.

Figure 6 Snapshot from one realization of a simulated sea-surface. The SWH is 8 m and the wind is blowing along the Y-axis from Y.

where t t À (R R )/c, t is the time, c is the speed of THE ROUGH-SEA GHOST REFLECTION 1 2 sound, r is the density, R and R are the distances to source RESPONSE 1 2 and receiver from the elementary area dA, and y1 and y2 are A standard Kirchhoff method is used to compute the re- the incidence and scattering angles to the surface at dA. S(t) sponse of each simulated rough sea-surface. is the monopole source function of the source. Clay and Medwin (1977) is a good reference on this topic. The near- field term in the Kirchhoff formulation (which is often omit- Z dS t cos y1cos y2 S t cos y1 cos y2 r dt c R1 R2 ted in seismic applications) is included. p r; t dA; 4p R1R2 The Kirchhoff method includes three approximations. First, 8 the plane-wave, plane-reflector reflection coefficient is used to

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Rough seas and time-lapse seismic 201

obtain an approximation to the reflected wave even though 3 Hz, so we can calculate the reflection response using a the surface is curved. Second, the reflection coefficient is as- `patch' of sea that is approximately 100 m in radius. Of sumed to be constant with respect both to position and to course, in order to obtain the correct spatial coherence be- angle of incidence. Third, multiple reflections are ignored. tween the source and all the receivers, a single, large sea- The first approximation is typically considered to be satis- surface must be used for each shot; the response of each factory when the Kirchhoff parameter, 2kRsin3f, is much receiver or source, however, can be computed from that larger than unity. R is the radius corresponding to the RMS patch of sea nearby. curvature of the surface, k is the wavenumber (radians/metre) Figure 8 shows typical rough-sea impulse responses com- of the wavefield and f is the angle to the surface of the puted at a point located at 6 m depth below the mean sea- specularly reflected ray. For a detailed discussion of this, see level, representing a single receiver or point source. The Thorsos (1987). For our simulated sea-surfaces, R is about direct (upward travelling) wave is unperturbed, but the 80 m. We are considering the near-vertical response, and the ghost reflections are perturbed in both amplitude and arrival surfaces have an RMS slope of about 48,sof is about 868. time. The data have been filtered to 200 Hz. The relatively This suggests that the first approximation will be satisfactory high cut-off was chosen so that the ghost notch is clearly for frequencies that are large compared with approximately visible. The corresponding amplitude spectra are shown on 1.5 Hz. The second approximation is exact for an air/water the right. In addition to the flat-sea response, two rough-sea interface because the reflection coefficient does not depend examples are shown; in one there is an increase in both the on position or on angle of incidence for such an interface. ghost arrival time and amplitude; in the other there is a The third approximation, omission of the multiple scattering, decrease. Other sea-surface shapes above the receiver (or is known to be a problem at specular reflection near-grazing source) would give rise to different perturbations (the signs incidence (Thorsos 1987, 1990). However, at the near- of the arrival time and amplitude perturbations are not ne- normal-incidence and small-slope conditions we are dealing cessarily coupled). The pulse shape is also perturbed, al- with, this is not expected to be a problem. though this is not easy to see in the figure. Note that there Figure 7 shows the radius of the quarter-wave Fresnel zone is a trailing coda at later times resulting from scattered for point sources or receivers placed 6 m and 12 m below the energy from increasingly distant parts of the surface. This surface. The curves indicate that the zones are quite small, gives rise to the `ripples' on the amplitude spectra. The ghost being less than 100 m for frequencies above approximately notches are perturbed in frequency and no longer go exactly

Figure 7 Radius of the quarter-wave Fresnel 200 zone as a function of frequency for two different receiver depths (12 and 6 m). At 180 3 Hz the Fresnel zone has a radius of approximately 100 m. 160

140

120

100

Radius of Fresnel zone [m] 60

20 12 metres deep

0 6 metres deep 0 20 40 60 80 100 120 140 160 180 200 Frequency of the sound wave in water [Hz]

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Figure 8 Typical rough-sea impulse re- 1 0 sponses (left) and the corresponding power 0.5 spectra (right) for a deployment fixed at 6 m –10 below mean sea-level. The ghost reflection is 0 perturbed in both amplitude and arrival Flat-sea ghost [dB] –20 –0.5 time, and there is a trailing coda of scattered

Amplitude Rough-sea ghosts energy from increasingly distant parts of the –30 –1 surface.

–1.5 –40 020406080 0 50 100 150 Time [ms] Frequency [Hz]

Figure 9 Typical rough-sea impulse re- 1 0 sponses (left) and the corresponding power 0.5 spectra (right) for a buoyed suspension at –10 6 m depth below the surface immediately 0 above (typical for a source). The ghost re- Flat-sea ghost [dB] –20 –0.5 flection is perturbed in both amplitude and

Amplitude Rough-sea ghosts arrival time, the direct wave has a static time –1 –30 shift, and there is a trailing coda of scattered energy from increasingly distant parts of the –1.5 –40 020406080 0 50 100 150 surface. The sea-surface realizations used to compute these two impulse responses are the Time [ms] Frequency [Hz] same as those in Fig. 8.

to zero amplitude. Notice the large effect of the rough sea on response of each. We estimate the 95% confidence limits of the upper part of the spectrum approaching 100 Hz. the amplitude and phase by statistical analysis of this ensem- The impulse responses shown in Fig. 8 are computed for a ble, each frequency being analysed separately. Log-frequency single point at a nominal depth of 6 m. The point is located at and phase are used for the analysis. This is done because a fixed position relative to the mean sea-level. This `fixed' errors caused by convolution with a perturbed wavelet are deployment is typical for a receiver or VSP rig source. How- additive when expressed in terms of log-frequency and phase. ever, a typical towed marine source has a `buoyed' deploy- Figure 10 shows these errors for 2-m, 4-m and 8-m SWH ment. This is represented by suspending each gun at its seas, for single receivers/sources with two different suspen- nominal depth measured relative to the sea-surface immedi- sion methods: fixed position relative to the mean sea-level ately above the gun. Both these two deployment models, (typical for a towed streamer receiver) and buoyed (typical `fixed' and `buoyed', are used in this study. for a towed source). The nominal depth is 6 m. Typically Figure 9 shows the resulting impulses under the same two seismic data acquisition would not currently take place in a sea-surface realizations used in Fig. 8, but using a buoyed sea with an SWH greater than approximately 4 m. suspension model; again the nominal depth is 6 m. For The errors rise with increasing wave height, though more this suspension model, the ghost notch is more stable and so for the fixed position geometry. Even for relatively calm time shifts are present on the direct wavelet. The amplitudes seas of 2 m, the amplitude error is around 1 dB (12% RMS) of the ghost notches are similar, and the frequency shift of the and the phase error is around 108 at 50 Hz, for both types of ghost notch is smaller. suspension. Comparing the two suspension methods under the same sea conditions, we find that the amplitude errors for buoyed suspensions are generally smaller, but the phase QUANTIFICATION OF THE ROUGH-SEA errors are larger. For example, in a 4-m SWH sea, the total ERROR BY MONTE-CARLO METHODS RMS error of an image with bandwidth 3±70 Hz is 17% for We quantify the size of the rough-sea error statistically using the fixed position deployment and 15% for the buoyed de- a Monte-Carlo method. Using this approach, we simulate ployment. Table 1 gives the RMS errors for 2-m, 4-m and an ensemble of random sea-surfaces and we compute the 8-m seas, for both suspension mechanisms.

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Figure 10 Rough-sea error quantification 95% Amplitude Error [dB] 95% Phase Error [degrees] for 2-m, 4-m and 8-m SWH seas. Top: for Fixed depth suspension Fixed depth suspension a deployment fixed at 6 m below mean sea- 4 60 level (typical for a streamer). Bottom: for a 8m [dB] suspension 6 m below the surface immedi- 8m 4m 40 ately above (typical for a source). The curves 2 4m 2m

show the 95% amplitude (left) and 95% degrees 2m 20 phase (right) spectral errors. 0 0 050100 050100 Buoyed suspension Buoyed suspension 4 60 [dB] 8m 2 40 8m 4m

degrees 20 4m 2m 2m 0 0 050100 050100 Frequency [Hz] Frequency [Hz]

Table 1 RMS error averaged over 3±70 Hz bandwidth for 2-m, 4-m and 8-m SWH rough seas for both fixed position relative to mean sea-level (typical for a streamer) and buoyed suspension (typical for a towed source). The nominal depth is 6 m

RMS error (%): Deployment: fixed position RMS error (%): Significant wave height (SWH) relative to mean sea-level Deployment: buoyed suspension

2m 9 9 4 m 17 15 8 m 29 26

We have assumed that for the streamer, the receiver pos- above the streamer (2D slice from a 3D surface), and the ition is fixed with respect to mean sea-level. In other words, rough-sea impulses. The wind is blowing directly against the the streamer shape is approximated as a horizontal line. sail direction, and the impulses are computed every 12 m. Kragh and Combee (2000) used reflection data from the The perturbations on the ghost reflection vary smoothly salinity interface in the Orca Basin in the NW Gulf of along the streamer because the dominant wavelengths in the Mexico to show that the streamer's position does indeed sea-surface are long (100±200 m for 2±4 m SWH). The ar- remain substantially unaffected by the sea-surface waves rival times of the ghost (top panel) generally mirror the sea- that move above it. surface heights (third panel from top); the ghost arrival time is directly related to the height of the water column above the receiver (or source). Of course, if the sea were flat, the ghost SPATIAL CORRELATION, ARRAY SIZE AND arrival time would correspond exactly to the height of the WIND DIRECTION water column above the receiver (or source). The amplitude In addition to the suspension mechanism considered above, it perturbations vary more rapidly along the streamer than do is necessary to consider the geometry of the receiver (or the ghost arrival-times; they are related more to the curvature source) array and its orientation with respect to the wind of the sea-surface, which has a shorter coherence length than direction. These factors will involve some averaging of the does the height. The scattering from more distant parts of the rough-sea error. First, we show an example of the spatial surface appears as a quasi-random background noise. correlations involved by computing the rough-sea impulses From Fig. 11, we can infer that group length, which is along 400 m of streamer deployed at 6 m below mean sea- short compared with sea-surface wavelengths, will not have a level. Figure 11 shows, from the top downwards: the ghost major influence on the rough-sea impulse. We quantify this arrival time, the ghost amplitude, the sea-surface height error by comparing rough-sea impulses computed at a single

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Figure 11 Rough-sea impulses along 400 m Ghost arrival time [ms] of a streamer deployed at a depth of 6 m 14 below mean sea-level. From the top down- 16 wards: the ghost arrival time, the ghost 18 Ghost amplitude amplitude, the sea-surface height above the –0.8 streamer (2D slice from a 3D surface), and –1 the rough-sea impulses. The wind is blowing –1.2 Sea-surface slice, SWH = 2m along the streamer. 2 0 –2 Rough-sea impulses, d = 6m 0 Direct arrival 10 Perturbed ghost arrival 20

30 Scattering noise 40

Time [ms] 50 60 70 0 100 200 300 400 Distance [m]

point with rough-sea impulses averaged over a typical 12-m because typical record lengths are of several seconds, during group-length, using the same Monte-Carlo approach. For which the sea-surface moves substantially. It is not necessary typical seismic frequencies (say up to 70 Hz), the error is to model the time variance in order to quantify the magni- small ± less than 0.5 dB in amplitude and less than 28 in tude and character of the rough-sea impulse, but time vari- phase. This averaging by the group gives only a small reduc- ance is important when considering windowed processing tion in error when using arrays as opposed to single elements. such as deconvolution, or any other process that attempts We also studied the averaging action of a typical marine to correct or match the seismic wavelet. It is also important source array. For source arrays, it is necessary to apply when considering multiple reflections; they bounce from a weights to represent the different gun volumes. We studied different sea-surface on each multiple leg, accruing further different suspension mechanisms for the airguns and found perturbations on each bounce. that the rough-sea error depends little on the details of the For time lags of, say, 10 s, the sea-surface shows little correl- buoying mechanism or on the wind direction. The smallness ation. So for modelling sources, we can use sea-surfaces that of the influence of the wind direction is interesting; the sea is are uncorrelated from one shot to the next. Over shorter time directive and the source is not small compared with all the spans, a smoothly varying change is observable in the rough- sea-surface wavelengths so we might expect to see a direc- sea impulse. Figure 12 shows an example of time variance tional effect. However, the sea is primarily directive at long under a 4-m SWH sea. The impulses are computed every wavelengths but the source is relatively small in comparison 100 ms. The smooth variation with time is apparent. Typically, with them, so at short sea wavelengths, where the source this variation is about 0.5 dB in amplitude and 58 in phase at dimension is significant, the sea is non-directive. The overall 50 Hz, for every 200 ms of elapsed time in a 4-m SWH sea. The averaging effect of the source array is similar to that dis- short coherence time of the rough-sea wavelet means that it cussed above for receiver arrays of comparable spatial extent. would not be easy to estimate the wavelet from the data itself.

TIME VARIANCE SYNTHETIC ROUGH-SEA TIME-LAPSE SEISMIC The sea-surface is not stationary, so its reflection response is time-variant. For the source impulse, the sea-surface is con- In order to assess the size and manifestation of the rough-sea sidered frozen, but for the receiver, this is not the case error on a typical seismic image with typical seismic

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Rough seas and time-lapse seismic 205

Figure 12 Example of time variance of the −1 rough-sea impulse under a 4-m SWH sea. Seven seconds of impulses are shown 0 computed at 100-ms intervals for a fixed position 6 m below mean sea-level. 1

5 Impulse onset time (s) 6

8 0 1020304050607080 Impulse duration time (ms)

acquisition geometry, we carried out the following model In a second modelling study, a time-lapse change was intro- study. We modelled with 2D finite-difference methods a syn- duced into the subsurface model and the finite-difference data thetic towed-streamer data set with an offset range of remodelled. The time-lapse change was achieved by shifting 100±3000 m and 160 shots. Subsurface parameters were the oil/water contact upward by 6 m and adding a 6-m thick, taken from a North Sea log, and simple structures represent- partially depleted zone above this (the oil/water contact is at ing a fault, a channel and a pinch-out were added. Rough-sea about 2000 m depth to the right of the fault and at about impulses for 2-m SWH seas were included in the synthetic 2150 m depth to the left of the fault). This change results in a seismic data at each single receiver (at 1.5-m spacing). Time small acoustic impedance change and a time shift in the pos- variance of the rough-sea impulse was not accounted for in ition of various reflectors. The 3-m grid size used for this this exercise, and no data-adaptive processing was used. The finite-difference modelling precluded the inclusion of more individual receiver responses were simulated and the data realistic production scenarios. The data were again processed were then grouped to form typical 12-m length groups, both with and without inclusion of the rough-sea impulses. processed through to stack followed by post-stack time- The data were also reprocessed at half-fold (24). migration. The data were also processed without the rough- The results are shown in Fig. 15. The inclusion of the rough- sea impulses, i.e. using a flat sea-surface to supply the ghost sea impulses confuses interpretation of the time-lapse image. reflection. The nominal stack fold is 60, but within the mute With rough seas included on both the baseline and time-lapse zone around 2-s two-way time, the actual stack fold is surveys, and the data processed at half-fold, the spurious around 48. No multiples were included in the modelling. events caused by the level of rough sea are about the same The 2D migrated results are shown in Fig. 13. The figure size as the time-lapse anomaly in this particular example. depicts a time-lapse scenario of a baseline survey acquired in Clearly, the extent to which the time-lapse anomaly is masked a flat sea, with a repeat survey acquired immediately after- depends upon the magnitude of the anomaly itself, which will wards, but in a 2-m sea (there is no time-lapse change in the be different in each situation. The approach adopted here subsurface in this case). The rough sea introduces an error allows the rough-sea error to be included in a survey design into the seismic signal, giving rise to spurious events in the in order to assess whether or not it will be problematic. difference image that `mimic' the real structure in the section. It should be noted that the modelling exercise carried out The RMS amplitude of this difference with respect to the flat- here does not include free-surface multiples, because we used a sea baseline survey is shown in Fig. 14. It is between 5 and convolution approach to add the rough sea-surface impulse. 10% (À20 to À26 dB below the signal). Water-layer multiples are often high-amplitude events, and

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Figure 13 Synthetic stacked and time- Flat sea 2m SWH rough sea 1.4s migrated data. Top left: the unperturbed flat-sea result. Top right: the perturbed 2-m 1.6s SWH rough-sea result. Bottom left: the subsurface model. Bottom right: the difference 1.8s result scaled by a factor of 2. There is no time-lapse anomaly in this example. The 2.0s difference image is due solely to the rough sea.

2.2s

2.4s

Model Difference x 2 1200m

1400m

1600m

1800m

2000m

2200m

2400m 2500m 3500m 4500m 5500m

Figure 14 RMS amplitude trace residuals RMS trace differences, 2m SWH % dB after the inclusion of 2-m SWH rough-sea 15 16.5 perturbations. The 2-m SWH sea intro- 10 20 duces errors of about 5±10% RMS. 5 26 0 2500 3000 3500 4000 4500 5000 5500 m

perturbation noise caused by the rough sea will be high when processed data depends upon stack fold and migration aper- compared with the (normally) weaker underlying primary ture as well as sea-state, but even for relatively calm seas with signal; the rough-sea effect on field data is likely to be larger 2-m SWH and stacking folds of around 48, the error is than predicted by this primary-only modelling exercise. between 5 and 10% RMS. Although many successful time-lapse studies have been reported, the rough-sea effect could be important for studies CONCLUSIONS where the time-lapse anomalies being imaged are of a similar We have introduced a method for modelling the moving size to the rough-sea error. The techniques described here can rough sea and computing its time-varying reflection response. be used as input to survey evaluation and design procedures We have shown that the rough sea imparts amplitude and for predicting minimum observable time-lapse anomalies phase errors to the reflected ghost, including an extended under different weather conditions. coda caused by surface scattering. For each seismic receiver Time-lapse data are often processed using matching, or the error is time-variant because of movement of the sea- cross-equalization, techniques, to reduce systematic differ- surface waves during the seismic record. On raw data, the ences between the data vintages (e.g. Koster et al. 2000). It spectral errors caused by seas with SWH of 2±4 m are typic- may be possible to reduce the rough-sea errors using these ally 1±3 dB in amplitude, and 10±208 in phase at 50 Hz. techniques. However, the time-variant nature of the rough- In time-lapse difference data, the rough-sea response mani- sea receiver ghosts is problematic and reduces their effective- fests itself as spurious structure. The final error level in the ness (Kragh and Laws 2001).

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Rough seas and time-lapse seismic 207

Figure 15 Time-lapse modelling. Top: base- Baseline, Flat sea Time-lapse, Flat sea line flat sea (as Fig. 14) and time-lapse flat sea. Middle: difference scaled by a factor of 2. Bottom: differences scaled by a factor of 6m movement of 2; left: with a 2-m SWH rough sea on base- oil-water contact line; right: with a 2-m SWH rough sea on plus 6m partial both baseline and time-lapse and processed depletion zone at half-fold (24).

Difference x 2

2m SWH rough sea baseline only 2m SWH rough sea on both, half fold 1.4s

1.6s

1.8s

2.0s

2.2s

2.4s 2500m 3500m 4500m 5500m

Hasselmann D.E., Dunckel M. and Ewing J.A. 1980. Directional ACKNOWLEDGEMENT wave spectra observed during JONSWAP 1973. Journal of Phys- ical Oceanography 10, 1264±1280. We thank Johan Robertsson for his help with the finite- Hasselmann K., Barnett T.P., Bouws E., Carlson H., Cartwright D.E., difference calculations. Enke K. et al. 1973. Measurements of wind-wave growth and swell decay during the joint North Sea wave project (JONSWAP). Dt. Hydrogr. Z., Reihe A (8), 12, 95. REFERENCES Hastings F.D., Schneider J.B. and Broschat S.L. 1995. A Monte Carlo FDTD technique for rough surface scattering. IEEE Transactions Carter D.J.T., Challenor P.G., Ewing J.A., Pitt E.G., Srokosz M.A. on Antennae and Propagation 43, 1183±1191. and Tucker M.J. 1986. Estimating Wave Climate Parameters for Jovanovich D.B., Sumner R.D. and Akins-Easterlin S.L. 1983. Engineering Applications. HMSO OTH 86228. Ghosting and marine signature deconvolution: a prerequisite for Clay S.C. and Medwin H. 1977. Acoustical Oceanography, p. 544. detailed seismic interpretation. Geophysics 48, 1468±1485. Wiley Interscience. Kinsman B. 1983. Wind Waves: Their Generation and Propagation Dragoset B., Hargreaves N. and Larner K. 1987. Air-gun source on the Ocean Surface, 2nd edn. Dover Publications, Inc. instabilities. Geophysics 52, 1229±1251. Koster K., Gabriels P., Hartung M., Verbeek J., Deinum G. and Eiken O., Waldemar P., Chonewille M., Ultveit G. and Duijndam A. Staples R. 2000. Time-lapse seismic surveys in the North Sea 1999. A proven concept for acquiring highly repeatable towed and their business impact. The Leading Edge 19, 286±293. streamer data. 61st EAGE conference, Helsinki, Finland, Extended Kragh E. and Combee L. 2000. Using a seismic reflector for resolving Abstracts, 1-40. streamer depth and sea-surface profiles. First Break 18, 463±467.

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Kragh E. and Laws R.M. 2001. Rough seas and statistical deconvo- S.A. Kitaigorodskii. Journal of Geophysical Research 69, lution. 63rd EAGE conference, Amsterdam, The Netherlands, 5181±5190. Extended Abstracts, A022. Robertsson J.O.A. 1996. A numerical free-surface condition for Labianca F.M. and Harper E.Y. 1977. Connection between various elastic/viscoelastic finite-difference modeling in the presence of small-waveheight solutions to the problem of scattering from the topography. Geophysics 61, 1921±1934. ocean surface. Journal of the Acoustical Society of America 62, Robertsson J.O.A., Blanch J.O. and Symes W.W. 1994. Viscoelastic 1144±1157. finite-difference modelling. Geophysics 59, 1444±1456. Mitsuyasu H.M., Tasai F., Suhara T., Mizuno S., Ohkusu M., Honda Thorsos E.I. 1987. The validity of the Kirchhoff approximation for T. and Rikiishi K. 1975. Observations of the directional spectrum rough surface scattering using a Gaussian roughness spectrum. of ocean waves using a clover-leaf buoy. Journal of Physical Journal of the Acoustical Society of America 83, 78±92. Oceanography 5, 750±761. Thorsos E.I. 1990. Acoustic scattering from a `Pierson±Moskowitz' Pierson W.J. and Moskowitz L. 1964. A proposed spectral form sea-surface. Journal of the Acoustical Society of America 88, for fully developed wind seas based on the similarity theory of 335±349.

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Appendix D

GEOPHYSICS, VOL. 68, NO. 4 (JULY-AUGUST 2003); P. 1303–1309, 17 FIGS. 10.1190/1.1598123

A proven method for acquiring highly repeatable towed streamer seismic data

Ola Eiken∗, Geir Ultveit Haugen‡, Michel Schonewille∗∗, and Adri Duijndam§

and processing. Lack of repeatability either limits the applica- ABSTRACT bility of the monitoring or how frequently we can monitor a change. Seismic reservoir monitoring has become an impor- Seismic reservoir monitoring is still immature, and the in- tant tool in the management of many fields. Monitoring dustry’s experience on repeatability is sparse. The qualitative subtle changes in the seismic properties of a reservoir results are mixed, ranging from carefully designed experiments caused by production places strong demands on seis- (Lumley, 1995; Ross and Altan, 1997) to the use of standard mic repeatability. A lack of repeatability limits how fre- imaging methods with larger discrepancies (Johnston et al., quently reservoir changes can be monitored or the ap- 1998; Landrø et al., 1999). The factors contributing to nonre- plicability of seismic monitoring at all. In this paper we peatable noise seem to be poorly understood, and little has show that towing many streamers with narrow separa- been published on the subject. Variations in recording position, combined with cross-line interpolation of data onto tions (streamer feathering, shot positions, and acquisition di- predefined sail lines, can give highly repeatable marine rection) from one pass to another limit the repeatability of seismic data. recent multistreamer technology. Permanent sea-floor sensors Results from two controlled zero time lag monitoring may eliminate these positional variations on the receiver side, experiments in the North Sea demonstrate high sensi- though there is a high initial cost. Horizontal steering of stream- tivity to changing water level and variations in lateral ers (Bittleston et al., 2000) may improve future monitoring positions. After corrections by deterministic tidal time efforts. shifts and spatial interpolation of the irregularly sampled This paper presents an alternative way to improve the re- streamer data, relative rms difference amplitude levels peatability in towed streamer data. Improved repeatability can are as low as 12% for a deep, structurally complex field be obtained by towing streamers with narrower than normal and as low as 6% for a shallow, structurally simple field. cross-line separation (50 m or less), followed by spatial inter- Reducing the degree of nonrepeatability to as low polation in processing. The method works well in two North as 6% to 12% allows monitoring of smaller reflectiv- Sea examples. ity changes. In terms of reservoir management this has three important benefits: (1) reservoirs with small seismic changes resulting from production can be moni- SENSITIVITY TO TIMING AND POSITION VARIATIONS tored, (2) reservoirs with large seismic changes can be monitored more frequently, and (3) monitoring data can Small shifts in time or lateral position are often of minor im- be used more quantitatively. portance for seismic imaging applied to exploration or reservoir characterization but can produce strong artifacts in time- lapse seismic imaging, e.g., in difference sections. The effect is INTRODUCTION more pronounced for higher frequencies and/or steeper dips. This can be quantified by shifting a seismic section u(t, x) When looking for subtle production-related changes in the to u(t + t, x + x) and calculating the difference. In the seismic properties of a reservoir, our analysis becomes highly frequency–wavenumber domain the difference section is trans- sensitive to changes in the acquisition system, environment, formed to U( f, k)(1 − e−i2π( f t−kx)) with an absolute value

Manuscript received by the Editor July 24, 2000; revised manuscript received March 13, 2002. ∗Statoil Research Centre, Rotvoll, N-7005 Trondheim, Norway. E-mail: [email protected]. ‡Formerly Statoil Research Center, Trondheim, Norway; presently GE Vingmed Ultrasound AS, Gaustadallen 21, N-0349 Oslo, Norway. E-mail: [email protected]. ∗∗Formerly Delft University, The Netherlands; presently PGS Research, PGS Court, Halfway Green, Walton-on-Thames, Surrey KT12 1RS, United Kingdom. E-mail: [email protected]. §Formerly Delft University, Netherlands; presently Philips Medical Systems, PMG Magnetic Resonance, P.O. Box 10000, 5680 DA Best, The Netherlands. c 2003 Society of Exploration Geophysicists. All rights reserved.

1303

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relative to the first section U( f, k)of fast Fourier transform (Duijndam et al., 1999). It is important to remember that regularization/interpolation techniques do − −i2π( f t−kx) = π| − | . 1 e 2 sin( f t k x ) (1) not replace proper sampling. In many cases adequate sampling requires spacing as low as 10–15 m. These dimensions are com- This shows how the difference section scales with f and k for monly acquired along the streamers but not in the cross-line a given time and spatial shift, t and x. Figure 1 shows this direction. Interpolation of coarser sampled data leads to sig- relationship in a separable manner. Figure 1a shows how re- nificant errors. peatability worsens with increasing frequency for a given time shift (assuming no spatial error). Figure 1b shows how a lat- THE ACQUISITION CONCEPT eral shift decreases repeatability for increased dips expressed in wavenumber (assuming no additional time shifts). Recent improvements in streamer towing technology allow A more direct way of illustrating sensitivity to changes in several contractors to offer nominal (front-end) streamer sep- the source and/or receiver positions is to apply a lateral shift to arations of 40 m or less for full streamer lengths (2000–4000 m) a seismic section and calculate the difference. Figure 2 shows with eight streamers or more. Crosscurrents inevitably cause how a 25-m lateral shift causes significant difference ampli- feathering and varying streamer positions behind the vessel. tudes predominantly related to dipping events and diffraction For positions in between this irregular receiver coverage, we tails. A change in position of a few meters creates significant can reconstruct the wavefield by interpolation. If that can be difference amplitudes, as shown in Figure 3. Note that it is done accurately, we will be independent of feathering and ef- not straightforward to assess and quantify repeatability. Dif- fectively have a receiver array covering the whole area between ference rms amplitudes relative to input data rms level in a the outer streamers. To obtain repeatable coverage, it would time window comprising the target is a robust measure, which is used throughout this paper. Proper spatial interpolation may correct mispositioning if the data are sufficiently densely sampled with adequate navigation accuracy. However, interpolation is not straightforward since the receiver spacing is often irregular—especially in the cross-line direction. Most common interpolation techniques are based on a regular grid. This leads to simple interpolation schemes such as sinc interpolation in the time–space domain. When the sampling is irregular, the interpolation can be formu- lated as an inverse problem: Using a least-squares approach, the signal is estimated in the Fourier domain from the observed data at the irregular positions. The data are subsequently transformed to a regular grid in the spatial domain using an inverse

FIG. 2. The sensitivity of a difference to a change in position can be illustrated by taking a seismic section (a), shifting it lat- erally 25 m, and subtracting it from itself (b). The difference amplitude level averages about 40% of the input amplitudes, and dipping events (such as diffraction tails) dominate the resulting difference section.

FIG. 1. Nonrepeatability expressed as the difference amplitude relative to original amplitude (percentage) plotted as a function of frequency and wavenumber. (a) A time shift leads to a diminishing repeatability for increasing frequency (assuming horizontal events, i.e., k = 0). (b) A spatial shift leads to diminishing repeatability for increasing wavenumber (assuming no FIG. 3. Difference (rms) amplitudes as a function of lateral shift time shift). (procedure described in Figure 2).

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Appendix D

Highly Repeatable Streamer Seismic Data 1305

then be sufficient to cover the same line or gate within the outer The vessel Geco Beta acquired the data. For the shallow field, streamers on subsequent repeats and repeat the shotpoints (the the crew skillfully operated eight 1800-m-long cables with a sparse spatial shot sampling hinders cross-line interpolation in front-end separation of 25 m—probably the narrowest sepa- the common receiver domain). The concept is illustrated in ration ever towed for such cable lengths. For the deeper field, Figure 4 and can be applied in a 2D sense with one pass for eight 3600-m-long cables were towed at 50 m front-end separa- each survey or in a 3D sense by shooting dense parallel lines tion. In the in-line direction the data were well sampled, with for each survey. 12.5/18.75 m shotpoint interval and 12.5 m receiver interval. Repeated shot positions from survey to survey are obtained The weather was good throughout the acquisition period, with by active steering of the seismic boat, and possibly the source only 0.5–1.2 m significant wave heights. The acquisition sys- subarrays themselves, along predefined lines. Although shot tem was well controlled with near-field hydrophones for each position variations may be compensated for on the receiver side air gun/cluster. The estimated source signature [common mid- to yield repeated midpoints, this will cause azimuth variations point (CMP) average] varied within a few percent in rms am- which have an unknown effect on repeatability. plitudes, as shown in Figure 5 for the repeated line from the This method differs from the common 3D approach of steer- shallow field. ing for coverage and binning that requires repeated shot posi- Navigation accuracy was estimated to be better than 5 m tions and fully repeated spatial receiver sampling after inter- for all elements. The shot cross-line positions deviated only a polation. A main challenge is the interpolation step because few meters from the pre-plotted line for the shallow field. De- seismic data are commonly coarse and irregularly sampled in viations were up to 10–15 m for the deeper field (Figure 6). the cross-line direction. As expected, the receiver coverage varied considerably. When the feather angles were more than about 3◦, coverage of the FIELD TESTS

To test this concept and acquire high-quality 2D baseline data for later monitoring, data were acquired in 1997 over two producing North Sea fields. For both surveys, one of the lines was shot twice a few days apart in an almost zero time lag repeatability test. The two reservoirs differ in that one has a simple structure at a shallow depth with good data quality, while the other is more complex and deeper with moderate data quality. These two experiments are henceforth referred to as the shallow field and the deeper field.

FIG. 5. Amplitude variation (percentage rms) of estimated source output (from near-ﬁeld hydrophones) for the two runs. Each value is an average of 100 shots, approximately as many traces as are contributing to each CMP in the stack.

FIG. 4. Illustration of the acquisition concept. Good receiver FIG. 6. Shotpoint position in cross-line distances off the pre- coverage is obtained between the outer streamers (and at a predefined line for the test runs on the (top) shallow field and defined line) by towing many streamers with dense separation. (bottom) deeper field.

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preplotted line was lost at the far offsets and acquisition was predefined line (for each shot and offset). As shown in the up- halted. Several such situations occurred during the 400-km- per part of Figure 9, this sequence gives a good image with high long acquisition route. In some situations with weak crosscur- s/n ratio. Further comparisons were mainly made on difference rents, the propeller-induced turbulence (prop wash) caused a sections, since visual comparison of the individual images does gap between the midstreamers, as shown in Figure 7. The dis- not reveal significant variations in repeatability. tance from the predefined line to the nearest streamer, averaged over all offsets within each CMP, is shown in Figure 8 for the two passes. Tidal correction Varying sea level clearly influences differences. A portion of DATA PROCESSING the line was shot at different periods in the ocean tidal cycle, We will concentrate on the results from the shallow field, resulting in height differences of 0.4 to 0.6 m (corresponding where data are best sampled and best illustrate some promi- to time shifts of about 0.6 to 0.8 ms). Sea level was measured nent nonrepeatable factors. Data from the deeper field are used by sea-floor pressure gauges and by downward-focused wave to show the robustness of the concept. The simple processing radar from a nearby production platform resting on the sea sequence used comprises a fixed gain (spherical divergence cor- floor. In areas with well repeated streamer positions, the dif- rection +5 dB/s), NMO correction, and an inner-trace mute ference rms amplitudes at target level reduced from about 15% excluding offsets less than 500 m at target depth (to attenu- to about 6% by tidal correction. This is shown in Figure 10 for ate water layer multiples) followed by CMP stack. The data a portion of the stack section (left), difference section without initially were stacked after picking the traces closest to the tidal correction (middle), and difference with tidal correction (right). This is in accordance with the theory described above where time shifts of 0.6 to 0.8 ms lead to nonrepeatability of about 15% at 35 to 40 Hz (Figure 1a). In regions with poorly repeated streamer positions, the difference level remains at about 20%.

Receiver positioning/binning/interpolation Selecting recorded traces closest to the predeﬁned line is equivalent to ﬂexi-binning with fully uniform offset coverage. The resulting difference stack between the two passes is shown in Figure 9b. Notice how the vertical bands of higher amplitudes in this section correlate well with variations in the receiver position, as shown in Figure 11. In regions where the cross-line positions were close for the two runs, the rms difference amplitudes are about 6%, whereas amplitudes increase to above 20% in regions with larger variations in the coverage

FIG. 7. Example of streamer positions for the same shotpoint, two different runs.

FIG. 8. Cross-line distance from the predefined line to the near- FIG. 9. Stack section after picking the nearest traces est streamer, averaged over all traces within each CMP. Vari- (flexi-binning) (top), with resulting difference section between ations for the two runs along the repeated line are shown for the two runs (bottom). The difference section has been gained the shallow field. to the same level as the input stack in the plot.

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Appendix D

Highly Repeatable Streamer Seismic Data 1307

(more than 20 m for some receivers). These figures are in ac- the regions where the receiver positions more or less coincide cordance with the theory described above. Most of the energy (Figure 12). A more advanced method of interpolating irreg- is at wavenumbers less than 4 cycles/km in the stack (midpoint ularly sampled data has been published by Duijndam et al. domain estimated in the in-line direction, but assuming similar (1999). In this method the frequency and wavenumber compo- geology a similar value can be expected in the cross-line direc- nents are estimated with a least-squares procedure for a certain tion.) Figure 1b indicates a maximum of 25% nonrepeatability bandwidth. Data at any desired spatial locations are then re- for cross-line midpoint offsets of 8 to 10 m. This demonstrates constructed by an inverse Fourier transform (Figure 13). When the large sensitivity of the seismic method to positional vari- this method is applied to this data set, the best result is ob- ations. The low noise level in the well repeated regions also tained when a quite restricted cross-line spatial bandwidth is demonstrates that modern marine navigation is accurate to estimated in the reconstruction. As shown in Figure 12, the within a few meters. The remaining 6% level of nonrepeatabil- resulting difference section is greatly improved, and the non- ity may be the result of limits in this accuracy, as well as source repeatability is down to about 6% for the regions with more stability, ocean waves, background noise, and other causes. irregular coverage. Rather than picking the nearest traces, interpolation in the For the deeper field, tidal variations are only 10–20 cm and cross-line direction is preferred. Simple linear interpolation thus are barely observed in the data. Short period multiples reduces the nonrepeatability in areas of positional variations are more pronounced in this case. Predictive deconvolution significantly, but not to the low level of about 6% found in helps to improve the image, but at the cost of lower repeatability. Once again we keep the processing sequence simple. In this case, with longer cables and larger cable separation, the

FIG. 12. Difference amplitudes (rms) along the test line for different binning/interpolation algorithms: picking nearest trace FIG. 10. (a) Portion of the stacked section from the shallow (blue), using linear interpolation (green), using Fourier recon- ﬁeld. (b) Difference between the two runs without tidal cor- struction (red). Note how Fourier reconstruction reduces the rection. (c) Difference after tidal correction. nonrepeatability in the regions with large cross-line position errors.

FIG. 11. Comparison of position error (red line) with repeatability error (black line) for the two runs (shallow field). Position error refers to cross-line error difference (CMP average) for the two runs. Repeatability error refers to rms amplitude of FIG. 13. Example of a cross-line gather (one shot and offset) the difference section derived using the trace nearest to the prior to and after Fourier reconstruction. Data have here been predefined line (flexi-binning) in the processing. regularized with a much denser sampling interval.

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distribution of receivers is statistically more uniform. A corre- experiment, these differences could easily be misinterpreted lation between position variations and repeatability is not as as true reservoir changes. clear as the shallow field example. Linear interpolation gave In addition to the original data, a decimated subset with a significant improvement compared to selecting the nearest every other cable omitted was processed, giving a front-end traces (flexi-binning), reducing difference amplitudes by 1/3— separation of 50 m instead of 25 m. This increased the difference down to about 12%. The more advanced method of interpo- amplitudes by a factor of 3/2 for all processing sequences— lating irregularly sampled data was not tested on this data set. even for the one applying the proper regularization scheme (Figure 14, white bars). This illustrates the importance of being CONTRACTOR COMPARISON well sampled in the cross-line direction. Interpolation helps but does not replace proper spatial sampling. To quantify the repeatability processing skills of some seismic contractors, we gave the shallow and less complex data set to four different contractors. They were asked to make images from the two passes of the line as repeatable as possible without applying data-dependent local matching in the reservoir zone. This can obviously reduce the differences toward zero for such a zero time lag experiment where no significant reservoir changes can be expected to take place. There were surprisingly large variations between the results, ranging from 6% to 16% (Figure 14, gray bars), indicating that the industry has not converged to a common satisfactory approach. We prescribe these differences to the processing sequences. The best result benefited from a proper irregular interpolation, while interpolation assuming regular sampling, as well as data- dependent trim statics, seemed to compromise the repeatability. No one trusted the interpolation capabilities of their dip moveout algorithms. Quality control procedures varied considerably, and the contractors with the best results quantified FIG. 15. (a) Stack and difference sections obtained by contrac- repeatability throughout all processing steps. tors A (b) and D (c). Note how the best difference section The significance of having good repeatability is illustrated (b) shows the noise from a cargo vessel passing during acqui- in Figures 15 and 16. During one of the passes along the line, sition of one of the repeated lines. a cargo vessel entered the area, causing interference noise of about 10 µbar, well below what can be seen in Figure 15a). These signals are only observed on the difference section from the best contractor that used proper regularization. (Since the noise from the cargo vessel arrived horizontally from the side, it corresponded to large wavenumbers in the cross-line direction. A high quality regularization scheme could also attenuate that energy.) As shown in Figure 16, the sections with poor repeatability display significant energy in the differences at both the top reservoir and the flat-spot levels. In a real time-lapse

FIG. 14. Contractor comparison. Four contractors were given FIG. 16. Targetzone comparison of the difference sections from data from the shallow field with 25 m nominal streamer sepa- contractors A (b) and D (c). Note how the 16% nonrepeata- ration, and their average repeatabilities expressed as rms dif- bility result from contractor D display significant energy in the ference amplitudes are shown as gray bars. Dropping each sec- differences at both top reservoir and flat-spot level. In a real ond streamer, the test was also performed with 50 m nominal time-lapse experiment, these differences could easily be mis- streamer separation (white bars). interpreted as true reservoir changes.

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Appendix D

Highly Repeatable Streamer Seismic Data 1309

CONCLUSIONS

This paper shows some prominent factors affecting repeatability. It clearly demonstrates the importance of properly hon- oring the lateral irregular positions and correct timing. The new method offers a high-fidelity alternative to conventional streamer or sea-floor receiver surveys for time-lapse work. Two controlled time-lapse tests show the ability to reach repeatability error levels as low as 6%. This is a much lower level than reported from other published marine streamer experiments. The main key to obtaining such a high repeatability is to tow the cables with sufficient narrow cross-line separation to be able to benefit from proper interpolation/regularization. The high level of repeatability also implies that other factors of nonrepeatability—variations in source output, sea waves (at such low sea states), and ambient noise—were less important (below ∼6% level for these tests). We expect that use of this method will increase the fidelity in seismic monitoring projects. Used in a 2D sense (one sail FIG. 17. Repeatability error from the best difference section line with many streamers), the method may be well suited for obtained through Fourier reconstruction, shown as a function frequent seismic monitoring of well defined targets. Used in a of frequency and wavenumber. A data window from 1.4 to 2.0 s 3D sense (parallel sail lines with overlap), the method can be and containing CMPs 2400–3600 was used. The color code is expected to give high-quality time-lapse data with increased labelled with percentage difference amplitudes. Note the reduced repeatability at nonflat events. cost of acquisition time compared to standard 3D surveys because of narrower sail-line separation. In terms of reservoir management, reducing the degree of nonrepeatability to such FREQUENCY AND WAVENUMBER DEPENDENCE low levels opens the possibilities of (1) monitoring reservoirs with small seismic changes attributable to production, such as Repeatability is strongly frequency and wavenumber de- low-porosity zones, deep reservoirs, and reservoirs containing pendent. Figure 17 illustrates how repeatability varies with heavy oil, (2) monitoring more frequently in calendar time, and frequency and wavenumber for the best difference section (3) obtaining a more quantitative use of time-lapse data. obtained with Fourier reconstruction. Note how 12% nonre- The method would benefit from further developments in peatability (plotted in yellow) is reached at 60 Hz for k = 0 streamer towing capabilities, increasing number of streamers (flat events), while it is reached at 35 Hz for a wavenumber of at decreasing cross-line separations, and possibly horizontal −8 cycles/km. The strong event in the lower right-hand corner steering of the streamers (Bittleston et al., 2000). of the figure corresponds to a region with very little energy in both the original and the difference section; thus, energy like ACKNOWLEDGMENTS additive noise may give a high value of the nonrepeatability measure. We thank Statoil and partners Shell, Exxon, Norsk Hydro, Time shifts are likely to lead to such a frequency depen- Saga, Elf, Conoco, and Total for permission to publish the dence (see Figure 1a) and can be caused by changing water work; the crew onboard Geco Beta for good work at sea; velocities, incorrect tidal corrections, ocean waves, or varying Ensign Geophysical for good trial processing work; and our tow depths. Wavenumber dependency can be expected for po- Statoil colleagues Patrick Waldemar, Terje Tollefsen, Hans sitional variations (see Figure 1b). Spatial averaging improves Aronsen, Paul Meldahl, and Martin Landrø for help and repeatability significantly, both in the in-line and cross-line di- inspiring discussions. rections. In this study we used a standard 25-m in-line averaging; longer smoothing operators may be fully adequate for REFERENCES monitoring areally extensive changes. Bittleston, S., Canter, P., Hillesund, Ø., and Welker, K., 2000, Marine Low-pass filtering improves the repeatability at the cost of seismic cable steering and control: 62nd Conf., Eur. Assn. Geosci. Engin., Extended Abstracts, 1–16. resolution. For example, the usable frequency and wavenum- Duijndam, A. J. W., Schonewille, M. A., and Hindriks, C. O. H., ber bandwidth determine the resolving limit of small 1999, Reconstruction of band-limited signals: Geophysics 64, 524– 535. saturation-front movements. For a repeatability behavior as Johnston, D. H., Mckenny, R. S., and Burkhart, T. D., 1998, Time-lapse shown in Figure 17, frequencies and wavenumbers well below seismic analysis of the North Sea Fulmar field: Offshore Tech. Conf., the maximum are likely to give the best time-lapse resolution Soc. Petr. Eng., Proceedings, 93–96. Landrø, M., Solheim, O. A., Hilde, E., Ekren, B. O., and Strønen, L. K., because of their superior repeatability. The time-lapse reso- 1999, The Gullfaks 4D seismic study: Petr. Geosci., 5, 213–226. lution may be significantly better than the λ/4 usually consid- Lumley, D. E., 1995, Seismic time-lapse monitoring of subsurface fluid ered in imaging because well constrained monitoring removes flow: Ph.D. thesis, Stanford Univ. Ross, C. P., and Altan, M. S., 1997, Time-lapse seismic monitoring: uncertainties in the geology. A detectable difference may be Some shortcomings in nonuniform processing: The Leading Edge, readily interpreted in terms of reservoir changes. 16, 931–937.

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