should generate a better signal-to-noise ratio pilot trace than the individual traces. The time .... function. ε is the damping factor which stabilizes the solution.
Robust estimation of large surface-consistent residual statics Side Jin and Shuki Ronen Veritas DGC, 10300 Town Park Drive, Houston, TX 77072, USA
Abstract Surface-consistent residual static correction is applied to land data on a routine basis. However, if the statics are large, conventional methods often break down due to, among other reasons, the difficulty to construct a good pilot trace. We developed a large static correction method that constructs the pilot trace by a local robust L1-norm inversion at each CDP location. The local inversion calculates the relative time shifts of the traces within a CDP gather by minimizing the trace-to-trace time difference. A pilot trace is obtained by stacking the CDP gather after aligning the reflectors by applying the time shifts to the traces. Finally, a global L1-norm inversion resolves the source and receiver statics from the time shifts of traces of all CDP gathers relative to their pilot traces. The method is demonstrated with synthetic and real data examples.
Introduction The effect of a highly variable near-surface weathering zone can be approximated by uniform time shifts that are called statics. Failure to correct the statics leads to poor quality of seismic images. Surface-consistent static correction is therefore routinely applied to land data. The source and receiver statics can be estimated from the time shifts relative to some reference represented by pilot traces. In practice, a pilot trace is often constructed by stacking CDP gathers. The common tool for obtaining the time shift is to cross-correlate the individual traces against the pilot trace. Seismic exploration has been moving into more difficult areas with complex terrain and tough geological conditions, which gives rise to more serious statics problems such as large magnitude of the static values. Shear velocity variation in the weathering zone causes large statics, and static correction of converted PS waves is more challenging and requires more robust methods than before. Most conventional static correction methods fail to solve large statics problems. One of the reasons for the failure is the difficulty to obtain a good pilot trace. In order to avoid the pilot trace problem, Kirchheimer (1986) proposed a scheme that does not use pilot traces at all. His method resolves the statics directly from the trace-to-trace time differences obtained by cross-correlating the individual traces against each other. However, this method often results in poor cross-correlations because both traces are noisy, and thus leads to many wrong time picks. In this paper, we propose a robust technique to tackle the large residual statics problem. The technique constructs the pilot trace by a local robust L1-norm inversion at each CDP location. The inversion estimates the trim statics to align the reflections on the CDP gather. The pilot trace is then obtained by a weighted stack of the gather. The local inversion minimized the trace-to-trace time differences, so the stack should generate a better signal-to-noise ratio pilot trace than the individual traces. The time shift of a trace relative to the pilot trace can now be calculated by cross-correlating the trace to the pilot trace. Finally a global L1 inversion resolves the source and receiver statics from the time shifts of all CDP gathers. The method is tested on synthetic and field data.
Theory and method Surface-consistent residual statics Under the assumption that the residual statics are surface-consistent and subsurface consistent, the total time shift or trim static Tijk for a given reflection event on a particular trace can be written as the sum of four terms (Wiggins el al, 1976): (1) T ijk = S i + R j + G k + Q k X ij2 , where Si and Rj are the statics at the ith source position and the jth receiver position, respectively. Gk is the structure term at the kth CDP position. Qk is the residual NMO component at the kth CDP. Xij is the source-to-receiver distance normalized by the maximum source-to-receiver distance of the survey. The four terms Si, Rj, Gk and Qk constitute the unknowns of the statics inversion problem. They are traditionally solved by least-squares methods (Taner et al., 1974). To set up the equation system, the key step is to compute the total time shifts Tijk with respect to some datum plane represented by pilot traces. The time shifts are usually picked from the cross-correlation between the trace ij and a pilot trace. An initial pilot trace is often constructed by stacking the CDP gather. However, in the presence of large statics in the input data, such initial pilot trace may be too poor to provide any useful cross-correlations with individual traces. One natural way to improve the pilot trace quality is to adjust the individual traces so that a constructive stack can be obtained. The adjustment is often referred to as trim static correction. Adjusting the individual traces needs a provisional pilot trace. The selection of the provisional pilot trace itself is a question when the statics are large. To overcome this, we propose a trim statics based on a local inversion at each CDP location without using a provisional pilot trace.
Local inversion for trim statics In order to construct a reliable pilot trace, we perform a trim static correction on each CDP gather by an inversion process. This inversion is algorithmically similar to the inversion developed by Jin et al., (2004) for large converted-wave receiver statics, with the difference that now we apply it to prestack data and not to common receiver stacks. The new method is better than conventional methods for large statics and good signal/noise ratio. To shun a poor provisional pilot trace, the inversion minimizes the trace-to-trace time differences to find the trim statics. We index the traces with numbers from 1 to N, where N is the total number of traces of the CDP gather, and denote Tm as the trim static of the mth trace of the CDP gather. Tm and Tijk in equation (1) are the same except the way of indexing: if i and j are, respectively, the indices of the source and receiver corresponding to the mth trace of the kth CDP gather, then Tijk = Tm. The reason we introduce Tm is only for mathematical formulation simplicity of the local inversion. The time difference between the traces q and r, 1≤q≤N, 1≤r≤N, of the CDP gather can be expressed as, (2) D l = D l (q, r ) = Tq − Tr , where l is the index for the trace pair (q, r). Dl can be obtained by picking the peak of the cross-correlation between the traces q and r. Inverse problems are usually solved by the least-squares or L2 method. However, the leastsquares solution is sensitive to large input error, which is always there for large statics problem. Instead, we adopt an L1-norm minimization to reduce the influence related to the use of incorrect peaks. The solution of the local inverse problem is obtained by minimizing the L1-norm cost function, S =
∑
w l | D l (q, r ) − Tq + Tr | +
l
N
∑
m =1
ε |Tm |,
(3)
where wl is the cross-correlation coefficient of the peak where the time difference Dl is picked. The system (2) is under-constrained, so a damping term is added into the cost function. ε is the damping factor which stabilizes the solution. The use of wl as a weight reduces the influence of very noisy traces which generally have low cross-correlation coefficients. The L1-norm solution can be obtained by the IRLS (Iteratively Reweighted Least Squares) algorithm. The reader is referred to Jin et al., (2004) and references therein for the detail about IRLS algorithm. Pilot trace construction The local inversion has already found the time shifts Tijk for the resolution of the linear system (1) for source and receiver statics. The construction of pilot trace seems unnecessary. Indeed, if the signal quality of the data is reasonably good, the pilot trace is not needed and the system (1) can be readily solved. However, it is desirable to use a pilot trace in the cases with strong noise. The time shifts from the local inversion are obtained from the cross-correlations of individual traces and inevitably contain picking errors. When a good quality pilot trace can be constructed, the individual traces can be cross-correlated with the pilot trace. This should yield more reliable time shifts, because the cross-correlations are improved in quality. The cross-correlation coefficients can be used again as a measure of the reliability of the picks of time shifts. They can be used to weight the resolution of the system (1) to ensure that good cross-correlations contribute more to the final solutions. The local inversion generates time shifts Tm to align the traces in the CDP gather. An initial pilot trace p0(t) now can be constructed by stacking the time shifted seismic trace dm(t): p0(t) =Σ dm(t + Tm ) where t is the travel time. We then update the initial pilot to build the final pilot trace. Cross-correlating the individual traces with the initial pilot trace, we obtain new time shifts Tm and associated cross-correlation coefficients wm. The weighted stack yields the final pilot trace p(t): p (t ) =
N
∑
m =1
w m d m (t + T m )
N
∑
m =1
wm
(4)
Global inversion for source and receiver statics As the last step, a global inversion is applied to resolve the source and receiver statics using the equation system (1). The time shifts Tijk in equation (1) can be obtained from the crosscorrelation between the individual traces and the pilot traces. The normal procedure for solving a set of simultaneous equations (1) is to find the best least-squares, or L2-norm, solution (Taner et al., 1974, and Wiggins et al., 1976). Like the local inversion, we prefer L1norm solution for its insensibility against noise. The source and receiver statics are obtained by minimizing the L1-norm cost function, (5) S = ∑ w ijk T ijk − S i − R j − G k − Q k X ij + ∑ ε s S i + ∑ ε r R j + ∑ ε G G k + ∑ ε Q Q k i , j ,k
i
j
k
k
where wijk is the cross-correlation coefficient at which the time shift is picked. εs, εr, εG and εQ are damping factors. The L1 solution can also be obtained by the IRLS algorithm as used by the local inversion.
Examples We first test the method on synthetic data that contain noise and large statics. The model is composed of three reflectors. Figure 1a shows one of NMO-corrected CDP gathers contaminated by statics, random noise and ground roll. The statics are so large that a stack of the gather will not produce a good pilot trace. The random noise and the ground roll generate multiple peaks in the trace-to-trace cross-correlations so the erroneous picks of time shifts are inevitable. Figure 1b shows the CDP gather after the trim static corrections using time shifts from the local inversion. Although the local inversion does not perfectly align the reflectors due to the very poor signal-to-noise ratio of the data, it is good enough to generate a pilot trace by stacking the gather. The stack also attenuates the ground roll and random noise, so there are fewer peaks in the trace-to-pilot cross-correlations than in trace-to-trace crosscorrelations. This greatly reduces the errors related to the wrong picks. Figure 1c shows the final surface-consistent static correction by the global inversion using the time shifts obtained from trace-to-pilot cross-correlations. It now perfectly aligns the three reflectors. 0.0
0.5
1.0
(a)
(b)
(c)
Figure 1: Synthetic CDP gathers: (a) raw gather; (b): gather obtained by shifting traces of gather (a) with the time shifts computed by the local inversion; (c): final surface-consistent static correction using pilot trace constructed by stacking gather (b). We now test the method on high-resolution land data acquired for heavy oil reservoir characterization. Figure 2 shows the raw stack without residual static correction. The presence of statics makes the amplitude weak in the middle and produces anomalies which could be mistaken for gas or steam clouds. Figure 3 illustrates the stack after the surface consistent residual static correction. The technique presented in this paper significantly improved the continuity of the reflections. After the removal of the effect of the statics, the amplitude
information of the reflections reveals more reliably the changes of rock properties that are vital for reservoir characterization.
Figure 2: CDP stacked section before surface-consistent residual static correction. The energy of the reflections is weak in the central part of the section due to the presence of statics.
Figure 3: CDP stacked section after surface-consistent residual static correction. The continuity of the reflectors is significantly improved.
Conclusions We have developed an inversion method for large surface-consistent residual static correction. It automatically constructs a pilot trace by using a robust L1 norm inversion at each CDP gather. The quality of the pilot trace is enhanced by weighted stack of trim-staticcorrected gather. Cross-correlating individual traces against the pilot trace yields more reliable time shifts because the cross-correlations are improved in quality. Finally, a global L1 inversion resolves the source and receiver statics from the time shifts of traces of all CDP gathers relative to their pilot traces. The technique is tested on synthetic and field data and produced satisfactory results.
References Jin, S., Li, J., and Ronen, S., 2004, Robust inversion for converted wave receiver statics: 64th Ann. Internat. Mtg., Soc. Expl. Geophys., Expanded Abstracts, MC 2.8. Kirchheimer, F., 1986, Robust residual statics by means of intertrace lag estimates: 46th Ann. Internat. Mtg., Soc. Expl. Geophys., Expanded Abstracts, S 15.2. Taner, M.T., Koehler, F., and Alhilali, A., 1974, Estimation and correction of near-surface time anomalies: Geophysics, 39, 441-463. Wiggins, R. A., Larner, K. L., and Wisecup, R. D., 1976, Residual statics analysis as a general inverse problem: Geophysics, 41, 922-938.
