The dynamics of statics

Transcription

1 ACQUISITION/PROCESSING Coordinated by Jeff Deere The dynamics of statics M. TURHAN TANER, Rock Solid Images, Houston, USA A. J. BERKHOUT, Technical University of Delft, The Netherlands SVEN TREITEL, TriDekon, Tulsa, Oklahoma, USA PANOS G. KELAMIS, Saudi Aramco, Dhahran, Saudi Arabia The statics problem, whether short wavelength, long wavelength, residual, or trim, has always been one of the more time-consuming and problematic steps in seismic data processing. We routinely struggle with issues such as poor signal-to-noise (S/N) ratio, cycle skipping, truncated refractors, wavelets with ambiguous first arrival times, etc. Elevation variations create their own problems and impact the choice of datum floating, phantom or recourse to a zero-velocity layer. Even if we can overcome some of these problems, we still have a catch 22 situation in which accurate velocity estimation requires good statics, while good statics estimation requires accurate velocities. To characterize these ambiguities, we have come up the oxymoron time-varying statics. Here we construct a more effective model of the statics phenomenon by introducing the concept of a near-surface propagation operator. We argue that statics, as a physical phenomenon, does not really exist; the problem lies in our inability to define the velocity field with the necessary precision. The downgoing source wavefront illuminates the subsurface, but suffers little distortion from the presence of near-surface heterogeneities (the far-field effect). On the other hand, the reflected upward traveling wavefield is subject to distortion by these near-surface heterogeneities, and therefore the reflected wavefield reaches the geophone arrays before it has had a chance to heal (the near-field effect). We show that shot and receiver statics can be determined directly from arrival time differences, without velocity information. Shot and receiver statics share many similar characteristics but also differ in some of their far-field as well as near-field effects. We show that receiver statics represent a major portion of the short wavelength time perturbation in the recorded wavefield. The near surface is not simulated with a velocity model but rather with a distribution of Green s functions that so often better represent the observed traveltimes. Such a change in perspective implies the need for a different class of time corrections, namely those resulting from recourse to Berkhout s double focusing method (Berkhout, 1995). Double focusing technology formulates the migration process as a cascade of two focusing steps: focusing in emission and focusing in detection. It requires no explicit knowledge of the velocity field. Instead, wavefield operators, directly estimated from the seismic data, are used to perform the focusing. We remark that images can be generated from the surface data without need to apply either long-wavelength or elevation statics corrections. We provide a new data processing sequence that produces more accurate and higher-resolution seismic images. Our proposed technique is equally applicable to 2D, crooked line, or 3D recording geometries. There is a vast literature on the subject of statics. A few of the significant contributions are listed in suggested reading at the end of this article. Cox s excellent monograph, Static Corrections for Seismic Reflection Surveys, does a fine job Figure 1. Source wavefront healing after propagation through the weathered layer. of summarizing our present knowledge and experience. It sometimes seems as if all there is to know about statics has already been published. However, we feel that this means that the end of this one particular road has now been reached, and that the time is right to look at this problem from a fresh vantage point. Near-surface problem. Our underlying model is based on the detail-hiding matrix operator notation of Berkhout (1982). One can express the influence of a complex near surface on the seismic reflection data in the form of vectors and matrices where the vector P i is the i th shot record containing the reflections from depth levels below the near surface (z z 1 ), while the vector δp i is the i th shot record containing the diffractions generated by upward-traveling reflections at the irregular boundary of the near surface, respectively. In more detail, (2) Here the vector P i- (z 1, z 0 ) represents reflections traveling upward, being incident to the irregular lower boundary of the near surface, described by the diagonal matrix δ(z 1, z 1 ). Further, the matrices W(z 0, z 1 ) and W(z 1, z 0 ) are the wavefield operators respectively carrying out up- and downgoing propagation in the complex near surface. According to the theory of wave propagation, we have (1) 396 THE LEADING EDGE APRIL 2007

2 W(z 0, z 1 ) = W T (z 1, z 0 ) where the superscript T denotes the matrix transpose. This implies that traveltimes for up- and downgoing waves must be surface-consistent for any propagation angle. Finally, the matrix X(z 1, z 1 ) represents the multichannel spatial impulse response from below the near surface (z z 1 ), the vector S i (z 0 ) is the ith source (array) located on the acquisition surface z 0, and the matrix D i (z 0, z 0 ) represents the distribution of detectors at depth z 0 for the ith source. The spatial sampling of W(z 0, z 1 ) is determined from the known detector geometry in the field, namely D i (z 0, z 0 ). Note that we generally have the situation z=z(x,y). Our concept and solution. We do not estimate the detailed near-surface velocity model, rather we estimate the near-surface propagation operator, W(z 0, z 1 ). This implies that the complex near-surface behavior can be described in terms of the properties of this operator, which is determined by the nearsurface velocity distribution as well as by the shape of the near-surface layering. In other words, W(z 0, z 1 ) is determined from the near-surface velocity distribution and from the shape of the near-surface layers. Further, δ i (z 1 ) represents the irregular base of the near-surface layer z 1. It results in diffractions due to the short-wavelength variations. We estimate W(z 0, z 1 ) from the field data, followed by removing the effects of W and W T from the data and replacing them by homogeneous propagators. This operator-driven process will be called near-surface inversion. At the present time we identify all effects acting in a timeconstant manner as statics. We divide such statics into two fuzzy classes, namely short- and long-wavelength statics. We adopt such a fuzzy classification scheme because we do not have an absolute metric to differentiate between the two. We also use this term only as it refers to time delays. It is well known that the near surface is the most weathered and heterogeneous part of the subsurface and that, therefore, waves propagating through it will be affected by time delays as well as amplitude and phase distortions. For the sake of simplicity, this article will restrict itself to time delay aspects only. A more general formulation can be developed with the WRW matrix notation introduced by Berkhout (1995). In conventional processing we compute long-wavelength statics from the reflection geometry, which follows from the first-arrival times of refracted events. These static corrections serve to replace up to several hundred meters of lowervelocity, heterogeneous layering by a set of higher-velocity, homogeneous layers. Unfortunately, these arrival times are not sufficient to determine overburden velocities directly. In most instances, we have to guess at these velocities. The static correction values and the long-wavelength refractor geometries we obtain in this way are the result of this guesswork. It is often possible to compute the refractor depth more accurately from reflected data. In most cases the near-offset traces that could contain reflected arrivals from near-surface refractors are not recorded. Applied static corrections are computed from differences between vertical traveltimes through the layers as obtained from estimated velocities of consolidated layers below the refractors, rather than from the actual travel path differences. This naturally creates traveltime errors. If we include large elevation variations, the situation becomes even worse. A number of approximations are available to handle the problem. We can use a floating datum and correct the data so that each trace of a CDP gather is reduced to the same elevation. In this way elevations change from one CDP gather to the next. If the elevation change is severe, the corrections within one CDP gather become very large and vary significantly with time. Truncated refractors also create problems. For example, refraction statics will have to be computed with different velocities on either side of a truncated refractor. Problems arise when the first refractor is shallow, while the second is deep, so that it may not even be recorded. In other instances there could be many thin highand low-velocity layers, which can give rise to a series of refracted arrivals and make it hard to construct a simple nearsurface model. Often refractor signatures are weak or buried in noise. The determination of actual traveltimes, which are required for the refraction computations, may sometimes be a serious challenge. In conventional processing we need to determine both short- and long-wavelength statics. In practice, shorterwavelength statics are computed from differences in NMOcorrected reflection arrival times. Here we face cycle-skip problems due to uncertainties in trace-to-trace traveltime differences, inaccurate NMO velocity estimates, and poor S/N ratios. Some of these inconsistencies are mitigated by the computation of surface-consistent statics. The resulting surface-consistency equations yield solutions valid up to a cable length. Because computed statics are influenced by inaccuracies of the applied NMO corrections, a new set of NMO velocities must be determined after the application of a statics correction step. The procedure can be repeated until an acceptable solution has been achieved. Thus far, we have reviewed problems encountered while trying to find and correct for time anomalies generated by the near surface. Now we take yet another look at the problem, and sketch some simpler, more general solutions. The rest of this article consists of two parts. In the first we review near-surface effects and derive some general conclusions. The second part then allows us to describe processing sequences based either on more conventional approaches or on the more recent double focusing methods. A simple surface statics model. It is generally accepted that four data samples per cycle are needed to define the amplitude and the phase of a wavefield frequency component. This holds for both temporal and spatial sampling. The limitations inherent in temporal sampling are better understood than those in spatial sampling. We can determine wave numbers properly for wavelengths greater than four times the receiver interval. All wavelengths shorter than this will be recorded, but not properly recognized. This also means that we need at least four recorded samples for each Fresnel zone. Diffracted waves arriving from shallower interfaces will be sharper and have shorter Fresnel zones. Thus they will tend to be undersampled. On the other hand, events arriving from deeper reflectors are associated with an expanding wavefront along with wider Fresnel zones. Thus, except for near-surface effects, the reflected wavefield will be adequately sampled. To account for all traveltimes in the recorded wavefield, we must properly sample the velocity/depth field. This sampling, of course, depends on how we sample the velocity table. Typical sampling intervals are of the order of several hundred meters, which allows us to reconstruct just the longer-wavelength portion of the recorded wavefield. The shorter-wavelength portions of the recorded wavefield cannot be so determined, and this is a real problem. Finally, upward-traveling reflected and refracted waves will also exhibit short wavelength, near-surface-induced undulations. The nature of the statics corrections can be discussed by pointing out details about the generation and the arrival of a seismic wavefront. Figure 1 shows the initiation of a seis- APRIL 2007 THE LEADING EDGE 397

3 Figure 2. A smoothed reflected wavefront impinging from below, scattered by the weathered layer. Figure 3. The near-surface model. Vertical and horizontal distances are in meters. mic wavefront. We assume that the initial wavefront geometry is hemispherical as it originates in a near-surface layer. It is then perturbed by the near-surface heterogeneities, and will break down into many, smaller radius diffraction patterns containing short-wavelength delay patterns. However, as propagation occurs over longer distances, the radii of the diffraction patterns increase and the perturbation wavelengths become longer (with an increase of the size of the Fresnel zone). This is the well-known wavefront healing process, one which cannot be modeled by any ray-tracing method. Thus only reflections arriving from shallow depths will experience the influence of near-surface effects. The reflections from deeper layers will not display this effect, so that we see only an average delay for the arriving wavefront. Ironically, the so-called shot statics and their effective wavelengths are both time and space varying. This variation is most often observed in the shallow parts of the seismic data, and manifests itself as receiver-dependent delays, because each receiver records arrivals from a different propagation direction. Because we are generally imaging deeper targets, we need consider only the average time delays resulting from the combined effect of all near-source surface inhomogeneities. These are the source statics. Figure 2 sketches the near-surface effects on upwardtraveling waves before they arrive at the receivers. Here again a number of diffraction patterns are generated by inhomogeneities in the near surface. These patterns arrive at the receivers with various wavelengths, which tend to be proportional to the distance between a subsurface inhomogeneity and a receiver. Thus, most short-wavelength time disturbances are generated by the near surface around and below the receiver positions. Because all upward-traveling wavefronts must pass through the near surface, their effect will be nearly constant with time, so that we can also treat this effect as static. Figure 2 shows that the upward-traveling short-wavelength time disturbances existing on common source records result mostly from the influence of near-surface inhomogeneities on the upward-traveling wavefront. This effect is most pronounced immediately before it is recorded by the receivers. These phenomena, then, give rise to receiver statics. Figure 4. Surface recording of near-surface location 2500 from a shot at the bottom of the model. Horizontal coordinate is distance (m) and vertical coordinate is time (s). Figure 5. Bottom recording from a shot near surface location Horizontal coordinate is distance (m), and vertical coordinate is time (s). A synthetic data example. For a more realistic example, we constructed the simple near-surface model shown in Figure 3. We generated synthetic seismograms for two different recording conditions. The surface of the model is horizontal, as are the deeper reflectors. Only the base of the near-surface, low-velocity layer is corrugated. In the first experiment we placed receivers at the surface and a source on the deep reflector. Figure 4 shows the wavefield recorded by the receivers at the surface. The data contain a large number of diffraction patterns with small Fresnel zones, as has already been remarked. The entire wavefield can be viewed as the envelope of all these small diffraction patterns. Nevertheless, its general shape indicates that it is an arrival from a deeper source. The statics problem is confined to the short wavelengths. 398 THE LEADING EDGE APRIL 2007

4 Figure 6. Overlay of two reciprocal traces recorded by exchange of their source and receiver positions (one trace has been shifted by one time sample for better visibility). Horizontal coordinate is time (s). Figure 8. Synthetic data recorded at the surface from a shot near surface location Figure 7. Synthetic data recorded at the surface from a shot near surface location In Figures 7 10, vertical coordinate is time (s), and horizontal coordinate is distance (m). In the next test, we exchanged the positions of shot and receiver. The source was placed at the surface, and the waves were recorded at a deeper reflector level. This resulted in the seismogram shown in Figure 5. Again we see a number of diffraction patterns, but the Fresnel zones are now much wider than the ones of Figure 4. The first arrivals are much smoother and do not contain the short wavelength disturbances arising from near-surface effects. While the shapes of the upward- and downward-traveling waves are different, the reciprocity principle is not violated. Figure 6 shows an overlay of two reciprocal traces, recorded by exchanging their source and receiver positions. To appreciate the similarities, we have shifted one trace by one pixel. It is clear from Figure 6 that both traces are identical. Next, we generated several records with sources at various positions along the deep reflector, while the receivers remained at their same surface positions. Figures 7 10, synthetic data recorded at the surface, clearly show the surface consistency of the short-wavelength perturbations. By surface consistency in this context we mean that the salient features of the delay patterns are similar, although there are small differences caused by the different travel paths at or near the surface. To render these similarities more visible, Figure 9. Synthetic data recorded at the surface from a shot near surface location we have applied approximate NMO to the first arrival times, and display them with their receiver positions in vertical alignment. We now observe that the similarities between the short wavelength delays are much more apparent in the corrected NMO display (Figure 11). It is interesting that, even though the amplitudes are now lower, the refracted events also exhibit similar short-wavelength perturbations. To summarize, we account for the long-wavelength portion of the traveltime-distance relationships by parameterizing in terms of average or rms velocity. This implies that for CDP processing, statics must be removed up to the wavelength equivalent of the maximum offset (i.e., one cable length). Therefore, the boundary between short- and longwavelength statics is one cable length. This is not the case for the double focusing method. In this approach each path between an image position and the surface must be determined and used separately, thus associating all delay effects APRIL 2007 THE LEADING EDGE 399

5 Figure 10. Synthetic data recorded at the surface from a shot near surface location Figure 11. A surface-consistent composite display of linear moveoutcorrected data shown in Figures with the near surface and with the deeper layers. We therefore will not have to contend with two separate statics problems (Kelamis et al., 2002). Short-wavelength statics can be computed from differential arrival times. Because the long-wavelength portion of the traveltimes can be handled with the familiar time-distance approximations, the statics problem is not confined to the short-wavelength portion of the spectrum. A short wavelength implies that we can utilize differential arrival times rather than actual arrival times. This overcomes many problems associated with refraction statics computations. Picking differential delay times, especially when considering surface consistency, is much simpler than first break picking. Figure 11 shows the degree of similarity of the delay patterns between several common source records. The picked differential times are integrated to obtain the actual statics correction times. Due to picking errors and inaccuracies, integration may produce long-wavelength undulations. These can be removed by appropriate long-wavelength cut filtering. For CDP processing, such filtering must remove all undulations up to and including one cable length. In the double focusing method, these results are used as initial approximations of the required focusing operators. Such initial estimates are then improved by an iterative process described by Berkhout (1997) and by Kelamis et al. (2002). The longer wavelength traveltime versus distance relations are parameterized in terms of conventionally obtained stacking, or migration velocities. Most procedures used in conventional processing, like CDP stacking and time or depth migration, are based on far-field imaging principles and on parameterization of traveltime/depth relations. In practice, each portion of the wavefront propagates with a different speed. This speed depends on the dip and azimuth of a local portion of the wavefront, as well as on local geologic conditions. We obtain theoretical traveltimes by basing them on assumed simple Earth models, parameterized in the form of velocities. Such models, no matter how complicated they may be (homogenous, isotropic, anisotropic) are much simplified by our choice of the velocity fields. Therefore, structural configurations and individual bedding geometries are generally modeled as longer wavelength approximations of the actual formations. The short-wavelength time differences are the ones most damaging to our image construction procedure. We must therefore determine these differences, and adjust the appropriate arrival times. If the near-surface variations are handled adequately, we can then obtain reasonably good images of the deeper structures. These shorter-wavelength portions thus become the basis for our statics correction method. Large surface elevation differences can cause additional problems. Another issue concerns how to handle surface topography. A number of methods are known, but most tend to be oversimplified. Seismic data are usually generated and recorded at the surface. In order to visualize a section that approximately resembles an equivalent section recorded at given subsurface positions, we use an imaginary datum plane and adjust traveltimes from the surface to the datum. This datum could be a planar surface or it could be a smoothly varying surface that follows the surface topography. In any case, appropriate corrections can be determined for vertical traveltimes in a homogeneous medium. Conventional methods introduce errors which become more serious with increasing elevation variations. It is a fact that the thing we know best about the subsurface is the surface. Thus, after the proper handling of short-wavelength near-surface effects, imaging can then begin at the surface. For the double focusing method, such short-wavelength statics corrections are incorporated into the source or receiver focusing operators. As a result, images can be formed directly from the surface. Following time migration, adjustments to a datum plane can be made in the form of simple time shifts. This is both simpler and more accurate. Examples. We computed short wavelength statics from the picks obtained with a commercially available first-break picking program. We produced statics with two maximum wavelengths, namely 750 and 1500 m. The center panel in Figure 12 shows the original common source gathers; the left and right panels show statics-corrected data for the and 750-m wavelength cases, respectively. Longerwavelength statics give smoother arrival times and are more suitable for the more conventional hyperbolic velocity scans as well as for NMO correction. Because in this case the actual cable length was 3000 m, a statics correction solution designed for a 3000-m wavelength would have been more appropriate. Figures 13 and 14 show one of the benefits of stacking from the surface. Figure 13 is a conventional stack section generated with a smooth floating datum, as indicated on the section. Figure 14 shows the stack section generated with reference to the actual surface topography. Better over- 400 THE LEADING EDGE APRIL 2007

6 Figure 12. A real shot record (center), statics-corrected up to wavelength = 750 m (right), and up to wavelength 1500 m (left). Figure 13. Conventional stack generated with a smooth floating datum. Figure 14. Stack generated with reference to actual surface topography. all reflector resolution is now apparent. Because the surface is at zero time, the section reflects delays due to surface topography. This section can now be used directly as input for a given migration procedure. To summarize: Traveltime perturbations on common source records are usually identifiable with short-wavelength near-surface (receiver) statics. These statics are nearly time invariant and should account for most observed shortwavelength time delays. The remaining delays are due to geometrical inaccuracies in the CDP gathers. These should be taken into consideration if the goal is improved stacking resolution. Shot statics can be viewed as averaged delays introduced by the near surface. These may be determined from traveltime differences obtained from common receiver trace gathers. Conversely, common source records provide data for the measurement of differential receiver statics. In conventional processing for statics corrections, the input is in the form of picked differential arrivals of reflected data. For noisy data, such differential arrivals can be checked for surface consistency, picking accuracy, and bad pick detection. Differential arrival times can be integrated to determine a first statics estimate. This estimate will contain some slowly varying trends due to picking inaccuracies or some form of bias. Such trends can be removed with a long-wavelength suppressing filter. Its length must be selected to correspond with the spatial resolution capability of the recording geometry. Such a process is similar for both source and receiver statics. The method we have outlined here completely decomposes shot and receiver statics without the need to determine velocities. The resulting statics-corrected data will exhibit improved spatial resolution and provides an adequate input for both NMO velocity determination or for prestack migration. Long-wavelength trend removal filtering produces zero mean statics over the design wavelength. By this term we imply that some statics are positive (advances) and some are negative (delays). Because actual statics are time delays (they cannot be time advances), we will have to add a bias so that most of the computed statics become time delays. Such a step will remove most of the delay bias, leaving some residual time shifts on the final section. We compute all velocities by taking the surface topography into account explicitly. This will eliminate the need for elevation corrections. Double focusing operators will contain all near-surface effects as a natural part of the traveltimes between the focal point and the surface. In the double focusing method, all traveltimes are defined between the surface and the subsurface focusing points. These traveltimes incorporate both the short- and APRIL 2007 THE LEADING EDGE 401

7 long-wavelength portions of the time-distance relations, as well as elevation effects. The images are directly formed from the surface. However, short wavelengths will still be a problem if not handled properly. Here, again, the problem can be solved by picking the differential arrival times in a surface-consistent manner. Integration and suppression of long wavelengths of the integrated results provides first-order estimates of short-wavelength perturbations. These can in turn be incorporated into an initial estimate of the traveltimes between the surface and the focal points. The traveltimes are optimized with use of focusing analysis. Once the short-wavelength delay patterns to a first major boundary have been established, we will be free of near-surface disturbances. From then on we can compute partial focusing operators to the first major boundary, which can be specifically chosen to have an uncomplicated geometry. Conclusions. We have presented a simple model of the near surface. Based on this model we have shown that shortwavelength undulations in the statics correction curves are due to near-surface effects on the upward-traveling reflected waves. These short-wavelength undulations cannot be explained by a computed velocity field, nor by the coarseness of the receiver spacing. We have shown that in conventional processing all statics up to the cable length must be eliminated. Based on these observations, we have presented a statics correction method compatible with the production of sections from surface topography. This procedure eliminates the need to pick actual first breaks, the uncertainty associated with precursors of vibratory signals. It also handles difficulties arising from echeloning and from truncated refractor problems, from large elevation corrections, and from cycle skips of the residual static corrections. It also mitigates the effects of uncertainties in the near-surface velocities and in the corresponding long-wavelength statics. In the case of other (longer wavelength) arrival time versus distance relationships, these too are specifiable in terms of computed (parametric) velocities. Stacked and migrated sections are easily and more accurately generated by starting from the surface topography downward. Because statics are computed independently of velocities, both can be determined separately, thus eliminating the need for classic iterative velocity/statics computations. This approach is more economical and at the same time more accurate. We have shown that the sampling interval in the time and space direction gives rise to the wavelength limitation for proper measurement and reconstruction of the reflected wavefield. On the other hand, short-wavelength disturbances are generated in the near-surface layers. Thus, they are nearly time invariant and can be computed from differential arrival times with the desired wavelength. We have shown that the dividing line between statics and dynamics is controlled by the parameterization of the velocity/traveltime relationship. Conventional velocity analysis requires this separation to occur around a cable length. The double focusing method reduces this separation down to a geophone group interval. Therefore, wavelengths for the required corrections can be established deterministically and can be based on an imaging method. This can be done even before the statics corrections are determined. In summary: When the subsurface is illuminated by a wavefront, it is healed from near surface irregularities by an expanding Fresnel zone. The reflected wavefield is recorded in the presence of near-surface irregularities before any appreciable expansion of the Fresnel zone occurs. Events reflected from shallow zones exhibit shorter Fresnel zones, hence require shorter receiver intervals for proper sampling and imaging. Deeper reflectors will be recorded with wider Fresnel zones, which are more than adequately sampled. Our recording intervals and our inability to construct more detailed velocity fields for CDP type methods create the need for additional time correction terms under the general rubric of statics. Our difficulties stem from both wide shot and wide receiver distances. Thus, we cannot adequately represent the short-wavelength portion of a propagating wavefield. Because we are dealing with a short-wavelength problem, it is sufficient to observe and pick differential arrival times to determine the necessary corrections. Statics correction wavelengths will depend on the type of imaging process used. Conventional hyperbolic scans for velocity analyses require statics corrections up to a cable length. The double focusing method does not need a separate static correction, because each focusing operator includes statics shifts as segments of the actual traveltimes. Static corrections should be applied only as time delays. We propose to form images starting from the surface, where we know elevations from actual measurement. Apparently, what we know most about the subsurface is the surface. The remaining imaging steps can be implemented with respect to the surface elevations. Depth migration will require no further modification. Time migration will require time correction with respect to a well established datum plane. This correction is applied after migration. Imaging from the surface eliminates the elevation correction problem. Our method aims to minimize the confusion introduced by various statics solutions namely, short and long wavelength, reflection or residual, elevation or datum correction, etc. It also eliminates the need for first-break picking and the need for statics/velocity iteration. There are no statics problems associated with the double focusing method. A given traveltime represents the traveltime between a surface point and a subsurface focus point, without parametric approximation. The results are similar to those achieved with time migration, but with better resolution. Depth images can also be obtained via tomographic inversion (Zhu et al., 1992). Suggested reading. Seismic Migration by Berkhout (Handbook of Geophysical Exploration, Volume 14a, Elsevier, 1982). Prestack migration in terms of double focusing by Berkhout (Journal of Seismic Exploration, 1995). Imaging and characterization with CFP technology, an overview by Berkhout (SEG 1997 Expanded Abstracts). Static Corrections for Seismic Reflection Surveys by Cox (SEG, 1999). Velocity-independent redatuming: A new approach to the near-surface problem in land seismic data processing by Kelamis et al. (TLE, 2002). Tomostatics: Turning-ray tomography + static corrections by Zhu et al. (SEG 1992 Expanded Abstracts). TLE Acknowledgments: We thank Gareth Taylor (in person), and Graham Jago (in spirit) for their valuable comments and editing contributions. We also are grateful to Eric Verschuur for providing the synthetic data examples. Corresponding author: Sven Treitel, streitel@tridekon.net 402 THE LEADING EDGE APRIL 2007