Abstract:
A method for moveout-correcting CDP gathers of seismic data that takes nonhyperbolic moveout at large offsets into account. The entire range of offsets is divided into smaller ranges. Within each smaller range, a curve is fitted to the reflection time vs. offset data, and these separate curves are made to be continuous at range boundaries. The curves may be the hyperbolic or quartic options available in commercial seismic data processing packages. Additional computation time required due to lack of a single velocity function can be compensated for by interpolation techniques. The method has application to, inter alia, AVO analysis.

Description:
FIELD OF THE INVENTION 
     This invention relates to the field of seismic data processing and in particular to a method for moveout-correcting seismic data having non-hyperbolic moveout at longer offsets. Uses of the invention include, without limitation, pre-stack data analysis for AVO applications. 
     BACKGROUND OF THE INVENTION 
     AVO is an abbreviation for Amplitude versus Offset, or, as it is sometimes written, amplitude variation with offset. AVO is a seismic data analysis method based on studying the variation in the amplitude of reflected waves with changes in the distance (offset) between the seismic source and receiver. The AVO response of the reflection events associated with the boundaries between the reservoir rock and the surrounding sealing materials often depends on the properties of the fluid stored in the reservoir pore space. Because of this property, AVO analysis is often used as a tool for reservoir fluid prediction. 
     AVO analysis is performed on Common-Depth-Point (CDP) gathers. Seismic events observed in CDP gathers exhibit a curvilinear shape (moveout). In order for the AVO method to be practically applicable to large data volumes, the gather has to be transformed (moved-out), so that the reflection events become horizontal (flat). In standard practice, this is performed using the Normal Moveout (NMO) method, which assumes that the original moveout of the reflection events is hyperbolic. This assumption works well for a small range of recorded offsets (usually up to about 3km). But for longer-offset gathers that are now commercially feasible to acquire and necessary for effective AVO analysis and reservoir properties prediction, the assumption is not adequate; events exhibit non-hyperbolic moveout over the longer offset ranges. An example is provided in FIGS. 1A and 1B. The picked reflection event  10  in FIG. 1A is flattened only over a limited range of offsets after NMO in FIG.  1 B. In analyzing the AVO behavior of such data, the valuable far offsets have to be discarded; if they are included without proper flattening, there is a risk of wrong fluid type prediction. There is, therefore, a need for an efficient method that takes into account the non-hyperbolic moveout of the reflection events and flattens them over the complete range of offsets. 
     Non-hyperbolic reflection moveout has been recognized as a significant problem for a long time, and a large body of geophysical literature has been devoted to it. Current solutions fall into the following categories. 
     Different Approximations Used for the Moveout of Reflection Events 
     Normal-moveout is based on the hyperbolic approximation          t   2     =       t   0   2     +       (     X   V     )     2                              
     where t is the reflection time at an offset X, t 0  the zero-offset time for the same reflection and V the Normal-Moveout (NMO) velocity. (Notwithstanding the preceding common terminology, it should be understood that V and t 0  are not necessarily physically meaningful, but rather merely parameters that characterize the observations.) This approach is implemented in standard velocity-analysis packages where measures of reflection signal coherency (semblance) are calculated along hyperbolic trajectories through the seismic data. Trajectories corresponding to maximum coherency (semblance peaks) correspond to seismic events. Identifying the semblance peaks on 2-dimensional velocity analysis displays allows determination of the Normal-Moveout velocity V. If the parameter t 0  is changed, then a different value of V will correspond to the semblance peaks. Thus V can be considered to be a function of t 0 , or V(t 0 ). 
     A commonly adopted method for extending the validity of this equation to longer offsets is the use of the truncated 4-th order Taylor series expansion. (See, for example, Gidlow, P. M., and Fatti, J. L., 1990, “Preserving far offset seismic data using non-hyperbolic moveout correction”, Expanded Abstracts of 60th Ann. Int. SEG Mtg., 1726-1729.) This approach is commercially available in the data processing systems of most seismic contractors. The equation used is:          t   2     =       t   0   2     +       (     X   V     )     2     -       (     X   W     )     4                              
     For each zero-offset time to, two parameters, V and W, need to be defined. Extending the standard semblance-based velocity analysis method to this case would imply picking peaks in 3-dimensional semblance panels (one for each CDP location, the three axes being time, V and W). This would be cumbersome and require special graphics. For this reason the most common approach is to first get an estimate of V, using the near offset data, then fix V and estimate W and then repeat the process until satisfactory alignment of the events has been achieved. The disadvantage of such an approach is that its application is time-consuming, because several iteration steps may be required for converging to a good set of coefficients for 2-nd and 4-th order terms. 
     Tsvankin and Thomsen derived an expression which they claimed to be an improvement upon the Taylor expansion formula in the presence of velocity anisotropy (wave velocity different for different propagation directions). (Tsvankin, I., and Thomsen, L., 1994, “Nonhyperbolic reflection moveout in anisotropic media”,  Geophysics , 59, 1290-1304.) This expression was re-written by Alkhalifah. (Alkhalifah, T., 1997, “Velocity analysis using nonhyperbolic moveout in transversely isotropic media”,  Geophysics , 62, 1839-1854.) Their expression is:          t   2     =       t   0   2     +       X   2       V   2       -       2      η                   X   4           V   2          [         t   0   2          V   2       +       (     1   +     2      η       )          X   2         ]                                  
     where η is an effective anisotropy parameter and V is the NMO velocity. 
     In addition to the above formulas, other types of equations have been proposed. Byun et al. introduced a “skewed” hyperbolic moveout formula. (Byun, B. S., Corrigan, D., and Gaiser, J. E., 1989, “Anisotropic velocity analysis for lithology discrimination”,  Geophysics , 54, 1564-1574.) Castle and de Bazelaire proposed the use of a shifted-hyperbola equation. (Castle, R. J., 1994, “A theory of normal moveout”,  Geophysics , 59, 983-999; de Bazelaire, E., 1988, “Normal moveout revisited: Inhomogeneous media and curved interfaces”,  Geophysics , 53, 143-157.) Yilmaz proposes the use of parabolas to describe the residual moveout of reflections after NMO has been applied. (Yilmaz, O., 1989, “Velocity-stack processing”,  Geophysical Prospecting , 37, 351-382.) 
     All of the above methods suffer from the following shortcomings: 
     Non-hyperbolic moveout can be caused by vertical velocity heterogeneity, curvature of the reflecting horizon, lateral velocity variation and velocity anisotropy. Given this number of factors and the geologic complexity of the subsurface, it is unlikely that a single mathematical expression can provide a universal description applicable to all situations. Locations where reflection events exhibit anomalous moveout are often the places where hydrocarbon traps exist (vicinity of faults, salt domes). Failure of moveout algorithms to work adequately under such situations is a significant defect. 
     Even in cases when such algorithms can work, their application can be cumbersome and time-consuming, because several iteration steps may be required for determining the parameters included in the expressions (see the comments on the truncated Taylor series approach above). 
     Depth Migration with Detailed Subsurface Velocity Models 
     Since non-hyperbolic moveout is caused by geologic complexity and heterogeneity, an imaging method that can handle those complications accurately should be able to produce CDP gathers with flattened reflection events. Depth migration is the most accurate imaging method available today, and algorithms are available that can deal with velocity anisotropy and heterogeneity. Yet, depth migration is extremely sensitive to the accuracy of the velocity model used for imaging and very expensive to do in a manner that preserves AVO information. Numerous publications exist on methods for determining velocity models for depth migration. The basic scheme in such methods is to depth migrate with an initial velocity model, inspect the results, modify the velocity model and migrate again, performing a number of iterations until reflection events in CDP gathers are flat. This process is extremely time consuming and usually requires geologic input for constructing adequate velocity models. 
     Near-surface Correction Models 
     In situations where non-hyperbolic moveout is caused by topographic relief or near-surface heterogeneity (for example presence of shallow gas, permafrost), methods are available to address the problem. Of course, such methods will be effective only if near-surface heterogeneity is the only cause of non-hyperbolic moveout and if the near-surface properties are known with a high degree of accuracy. 
     Automatic Event Detection Methods 
     Such methods do not assume any model for the moveout of reflection events. Rather, they use pattern recognition techniques to track the events in the data. Once the events are tracked, they can be flattened using time shifting. An example of such a method is the one proposed by Boyd et al. (See U.S. Pat. No. 5,128,899.) The problem with such methods is that they fail to work adequately under the presence of noise, since noise can easily cause erroneous event tracking. 
     In view of the preceding description of current technology, a need exists for a method that: 
     Is generally applicable for addressing non-hyperbolic moveout problems, independent of the geologic factor causing non-hyperbolic moveout. The method should work adequately even for areas of large geologic complexity, large source-to-receiver offsets and large amounts of velocity anisotropy. 
     Is efficient to apply, because it is not based on an iterative procedure. 
     Is based on familiar velocity analysis tools and displays, readily available in most commercial processing packages. 
     The present invention satisfies these needs. 
     SUMMARY OF THE INVENTION 
     In one embodiment, the invention is a method for moveout-correcting a common-depth-point (CDP) gather of seismic data where the range of offsets is large enough that the moveout is nonhyperbolic for the longer offsets, which method comprises the steps of (a) dividing the offsets into discrete ranges, (b) fitting an analytical function for reflection time as a function of offset within each offset range, (c) combining these functions, enforcing continuity at range boundaries, into a single, piecewise-continuous function, and (d) moveout-correcting the seismic data using the piecewise-continuous function. 
     Hyperbolic functions are used in some embodiments of the invention; others use quartic polynomials or a combination of hyperbolic and quartic. Higher-degree polynomials or other functions may be used. Preferred embodiments of the invention use hyperbolic or quartic velocity analysis in forms currently available in standard seismic processing packages. The invention includes methods for flattening the reflection events over all offsets with a minimum number of offset ranges, thereby speeding up the process further. 
     The invention also includes methods for efficiently moving out the seismic data to offset the increase in computation time due to using more than one velocity function for moveout. Curves called pseudo-horizons are generated from the piecewise-continuous velocity function, and moveout is accomplished in some embodiments by interpolating between the pseudo-horizons. 
     In another embodiment, before horizon-based moveout, the input trace is interpolated to a finer sampling interval using high precision interpolation such as frequency-domain, sinc-function interpolation. 
     Potential uses of the present invention include AVO (Amplitude vs. Offset) applications. 
    
    
     BRIEF DESCRIPTION OF THE DRAWINGS 
     The present invention and its advantages will be better understood by referring to the following detailed description and the attached drawings in which: 
     FIGS. 1A and 1B illustrate the problem of nonhyperbolic moveout; 
     FIGS. 2A,  2 B, and  2 C illustrate the improvement obtained by piecewise hyperbolic fit (FIG. 2B) and piecewise quartic fit (FIG. 2C) compared to NMO (FIG.  2 A); 
     FIG. 3 is a flowchart showing the broad features of some embodiments of the present invention; 
     FIG. 4 is a flow chart that expands upon the velocity analysis part (step  31  of FIG. 3) of some embodiments of the invention; 
     FIGS. 5A and 5B show, respectively, actual semblance plots, and velocity functions picked from them, for three offset ranges, with the flattening of velocity events by such velocity functions also shown for each offset range; 
     FIG. 6 is a flow chart showing the standard quartic velocity analysis process; 
     FIG. 7 is a flow chart showing pseudo-horizon moveout techniques and data interpolation techniques used in some embodiments of the present invention; 
     FIGS. 8A and 8B illustrate horizon-based moveout on an actual dataset; 
     FIGS. 9A,  9 B, and  9 C compare a 2-pass hyperbolic application (FIG. 9B) to a 2-pass quartic application (FIG. 9C) for one embodiment for the present invention, with NMO also shown (FIG.  9 A); 
     FIG. 10B illustrates a 2-pass quartic application of one embodiment of the present invention, compared to NMO as shown in FIG. 10A; and 
     FIG. 11B illustrates a 3-pass quartic application of one embodiment of the present invention, compared to NMO as shown in FIG.  11 A. 
    
    
     The invention will be described in connection with its preferred embodiments. However, to the extent that the following detailed description is specific to a particular embodiment or a particular use of the invention, this is intended to be illustrative only, and is not to be construed as limiting the scope of the invention. On the contrary, it is intended to cover all alternatives, modifications, and equivalents which may be included within the spirit and scope of the invention, as defined by the appended claims. 
     DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS 
     The main concept underlying the present invention is described in FIGS. 2A-2C, which show reflection time graphs in a time-squared vs. offset-squared representation. The vertical axis represents the value of the function−(t 2 −X 2 /1.75 2 ), where t is the reflection time in seconds and X the offset in kilometers. The horizontal axis represents the value of X 2 . This is a rotated version of a standard t 2  vs. X 2  plot. On such a display hyperbolic events plot as straight lines. Curve  20  is the same on all three graphs and represents the times for the picked reflection event  10  shown in FIGS. 1A and 1B. This curve is not a straight line because the event moveout is not hyperbolic. NMO attempts to approximate the reflection times with a straight line  21 , as shown in FIG.  2 A. Clearly, in this case, this is not a good approximation; the shape of the real reflection event cannot be described with a hyperbolic equation. 
     A fundamental concept of the present invention is the following: instead of attempting to use a single functional form (such as hyperbolic) to describe the reflection times over the entire offset range, the range is split into sub-ranges. A separate function such as a hyperbola or a higher-order polynomial is used within each of the sub-ranges to approximate the reflection times. Thus, the present invention uses a piecewise curvilinear approximation of the reflection times to get a better fit than would be possible using a single function over the entire offset range. This requires a mechanism to: 
     Fit curves to reflection times. For the hyperbolic case, this can be done with standard velocity analysis. Velocity analysis based on fourth-order polynomial functions (rather than hyperbolas) is also available in several current processing systems. Velocity analysis based on higher-order polynomial curves or different types of functions can be implemented in a way similar to the hyperbolic or fourth-order cases described herein. 
     Combine the curvilinear pieces defined for the different offset sub-ranges, enforcing continuity of the composite event at the boundaries between sub-ranges. A procedure for achieving this will be described later. Two implementations of the present invention, hyperbolic and quartic, are described below. 
     i. Piecewise hyperbolic fit, or piecewise linear in the t 2  vs. X 2  domain. This approximation is shown in FIG.  2 B. Reflection times are approximated by three different hyperbolas, each providing a close approximation over a different offset range. The hyperbolas graph as straight lines ( 22 ,  23 , and  24 ) in this rotated t2 vs. X 2  plot. Lines  23  and  24  are denoted by their dotted-line extensions. 
     The offset ranges used for this piecewise hyperbolic representation may differ somewhat for different areas. The following values are reasonable defaults: 2000-2500 meters for the near-offset range upper boundary; approximately 5000 meters for the next boundary; and approximately 6000 meters for a third boundary. On the scale used for the horizontal axis of FIG. 2B, these boundaries located at 4-6.25; are  16 , and  36 , respectively. In the particular instance of FIG. 2B, better choices would be  11 ,  18 , and  25 . 
     ii. Piecewise quartic fit. The hyperbolic approximation of FIG. 2B is used for the near offsets, because moveout for small offsets will always be closely approximated by a hyperbola. Other offsets ranges are approximated by 4-th order polynomials. For the example shown in FIG. 2C, two offset ranges provide a close fit. The near offsets are approximated by hyperbola  22  (straight line in the t 2  vs. X 2  domain); the far offsets are approximated by the curve  25  for which the dotted-line extension is shown. This curve is a 4-th order (quartic) polynomial. 
     Typical offset ranges for the quartic approximation where only two offset ranges are used are 2000-2500 meters for the near-offset range upper boundary and approximately 5000 meters for the far-offset range upper boundary. In FIG. 2C, these boundaries correspond to 4-6.25 and 25, respectively. In this particular instance,  11  is a better choice for the first boundary. 
       
     A broad overview of the present inventive method is provided in FIG.  3 . 
     At step  31 , velocity analysis is performed to determine the boundaries of the offset ranges and the velocity-time function for each offset range. The remaining steps of FIG. 3 will be discussed after the detailed discussion of step  31  which follows. 
     A flowchart outlining the velocity analysis process for some embodiments of the present invention is shown in FIG.  4 . After tentative range offset boundaries are chosen, hyperbolic velocity analysis is performed for the near-offset range at step  41 . At step  42 , the entire gather is moveout-corrected using the velocity function determined from the near offset range, and at step  42   a  the result is examined. If the reflection events are flat for all offsets, the present invention is not needed, and the process is stopped. If they depart from flatness at longer offsets, however, a hyperbolic velocity function will not be adequate for flattening them over the entire offset range. This determination is part of step  42   a . At step  43 , the first offset boundary is reset to that offset past which the events tend to deviate from flatness. At step  44 , either a second hyperbolic or a quartic velocity analysis is performed for the next offset range. At step  45 , events for all offsets larger than the previous offset boundary are moveout-corrected using this second hyperbolic or quartic velocity , and at step  45   a  it is determined whether this second hyperbolic or quartic velocity is adequate for flattening these events. If so, step  31  (FIG. 3) is finished. Otherwise, a new offset boundary is set and the process is repeated until the reflection events are flat over the entire range of offsets. 
     Although FIG. 4 illustrates the case where a combination of hyperbolic and quartic curves are used to describe the reflection event moveout, the present invention is not limited to these types of curves. Other functional forms can also be used. 
     When hyperbolic velocity analysis is used in the present invention, it may be performed by any of the standard methods known to those trained in the art. FIG. 5 shows how this is done in one embodiment of the present invention where hyperbolic analysis is performed over each of three offset ranges using a measure of coherence called semblance. Values of semblance are calculated for each offset range using the equation          t   2     =       t   0   2     +       (     X   V     )     2                              
     to determine V(t 0 ) that maximizes the semblance. In this equation, t represents reflection time, t 0  the reflection time at zero offset, X the offset, and V=V(t 0 ) is the velocity function for the offset range being considered. The left third of FIG. 5A shows the semblance panel  51  for the offset range 0-2000 m, and next to it ( 52 ) the corresponding position of the gather after hyperbolic moveout with the velocity function determined from semblance panel  51 . The middle third of FIG. 5A shows the semblance panel  51   a  for the offset range 2000-4000 m, and at  56  the gather is shown over the same offset range after hyperbolic moveout correction using the velocity function determined from semblance panel  51   a . Similarly, the semblance panel for the offset range 4000-5100 m is at 51 b , and at  57  the gather is shown over the same offset range after hyperbolic moveout correction using the velocity function determined from semblance panel  51   b . It can be seen that the combination of three ranges and hyperbolic velocity analysis is sufficient to flatten the reflection events over the entire offset range. FIG. 5B shows the three velocity functions determined from the semblance panels plotted together for comparison. The velocity function for the offset range 0-2000 m is shown at  55 , for 2000-4000 m at  54 , and for 4000-5000 m at  53 . 
     FIG. 6 is a flow chart of the quartic velocity analysis process for some embodiments of the present invention. As with hyperbolic velocity analysis, persons trained in the art will know variations of the method shown for quartic analysis, and all such variations are embraced within the present invention. FIG. 6 shows how quartic velocity analysis is applied to the second offset range after the first offset range, i.e., the range for which moveout is hyperbolic, has been determined. Quartic analysis for offsets beyond the second offset range, as necessary, is performed in the same way. 
     Looking at FIG. 6 in more detail, at step  61 , standard hyperbolic velocity analysis is performed on the near-offset range, and the near-offset velocity function V 1  is determined. Since hyperbolic velocity analysis will be the first step in the velocity analysis process in most embodiments of the present invention (see FIG.  4 ), this step will have already been completed. 
     At step  62 , the full gather (the whole range of offsets) is moveout corrected using the velocity fiction V 1  picked in step  61 . The effect of this step is that an event at time t 2  moves to time t 1 , where          t   2   2     =       t   1   2     +         (     X       V   1          (     t   1     )         )     2     .                              
     At step  63 , inverse quartic moveout is applied to the full gather using a constant pseudo-velocity V c . An event, a time t 1  moves to time t in accordance with the equation          t   1     =     t   -         (     X     2      Vc       )     4     .                              
     The factor V c  has units of (distance)/(time) ¼ . Hence, it is called a pseudo-velocity since it does not have units of velocity. The exact value of the constant V c  is not important. A typical choice is the average velocity for the region. 
     At step  64 , quartic velocity analysis is performed to determine the necessary velocities flattening the portions of the reflection events for the given offset range. This can be done by calculating semblances based on the equation:        t   =       t   0     +       (     X     2        V   2         )     4                              
     and determining the pseudo-velocity function V 2 (t 0 ) that maximizes the semblances. As with hyperbolic analysis, if the quartic fit is not satisfactory for the entire remaining range of offset values, a boundary for the second offset range is set based on where the fit becomes unsatisfactory, and the process is repeated on a third offset range, and so on until a piecewise fit has been achieved. 
     Having performed the hyperbolic or quartic velocity analyses to complete step  31  of FIG. 3, the events for the different offset ranges can be flattened, as is shown, for example, in FIG.  5 A. Yet, events will usually line up at different zero-offset times (read off the vertical axis of FIG. 5A) for the different offset ranges. This can be seen in the example of FIG.  5 A: if the three pieces of the gather  52 ,  56  and  57  corresponding to the different offset ranges are put together, the reflection events (broad, dark, horizontal lines), have discontinuities at the boundaries between the different ranges. This same effect is also shown in FIGS. 2B and 2C, by the dotted lines  23  and  24  extending the hyperbolic pieces or the dotted curve  25  extending the quartic piece to zero offset. The times at which these lines intersect the X=0 axis are the zero-offset times at which the different pieces of the event will be lined up after application of hyperbolic (NMO) or quartic moveout correction. 
     To solve this problem, the present invention imposes the condition that the reflection times at offset boundaries must be continuous. This uniquely determines all parameters in the reflection time as a function of offset over the full offset range, whether the function is hyperbolic, quartic, or some other. This is part of step  32  of FIG. 3, where the velocity functions from step  31  are used to calculate a continuous function approximating reflection time as a function of offset. Then at step  33 , this continuous function is used to moveout−correct the seismic data. 
     The necessary formulas for the calculation of the reflection times in one preferred embodiment of the present invention are given below for the hyperbolic and quartic cases. Other mathematically equivalent, or approximately equivalent, expressions may be used in other embodiments.                  t   i   2          (   X   )       =       t     0   ,   i     2     +       [     X       V   i          (     t     0   ,   i       )         ]     2               Hyperbolic                           t   i   2          (   X   )       =         t     nmo   ,   i     2          (   X   )       +       [     X       V   i          (     t     nmo   ,   i       )         ]     2                       t     nmo   ,   i            (   X   )       =       t     0   ,   i       +       [     X     2          V   i          (     t     0   ,   i       )           ]     4     -       (     X     2        V   c         )     4               }           Quartic                                
     where: 
     t i =reflection time for the approximating curve used for the i-th offset range, i=1 denoting the near-offset range, etc.; 
     X=offset within i-th offset range; 
     t 0,i =zero-offset time for the i-th approximating curve; 
     V i =velocity (or pseudo-velocity) function for the i-th offset range (V 1  is the hyperbolic velocity function for the near-offset range); and 
     V c =a constant pseudo-velocity, chosen as described above. 
     The parameter t 0,1  will still be determined, to use FIG. 5A as an example, from where the reflection event lines  52  intersect the time axis. The other necessary parameters t 0,i , i=2, 3, . . . , are determined by enforcing the continuity conditions for the events at the boundary offsets, i.e.: 
     Continuity at X 1 : t 1 (X 1 )=t 2 (X 1 )→find t 0,2    
     Continuity at X 2 : t 2 (X 2 ) =t 3 (X 2 )→find t 0,3  etc. 
     With the parameters t 0,i  thus determined, the preceding equations are combined to yield the approximate reflection time as a continuous function of offset throughout all the offset ranges. 
     With the reflection times thus defined for all offsets, applying moveout correction to the seismic data (step  33  of FIG. 3) is a straightforward matter: an event at a time t i (X) is moved to the corresponding zero-offset time t 0,i . This can be achieved in the present invention in a variety of ways. A preferred embodiment is shown in the flow chart of FIG.  7 . 
     At step  71 , the reflection times are calculated for a set of fictitious reflectors called pseudo-horizons, located at pre-determined zero-offset time increments. The pseudo-horizons are generated by using the piecewise continuous velocity function determined as outlined immediately above. FIG. 8A shows a set of such pseudo-horizons located at 200 ms increments beginning at t 0 =2000 ms and superimposed on the CDP gather for which they were determined. This is the same gather shown for the example of FIGS. 5A and 5B, and the pseudo-horizons were determined from the three hyperbolic velocity functions shown there. Therefore, the pseudo-horizons are piecewise hyperbolic. These horizons do not necessarily coincide with real reflection events, since they are constructed at regular zero-offset time increments (200 milliseconds in this display); they are hence called pseudo-horizons. Being constructed from the velocity functions that were fit to the data, they naturally tend to parallel nearby reflection events. For an actual application of the present invention, the pseudo-horizons would normally not be plotted or otherwise shown in an output display. They would be calculated only internally within the computer program automation of the present invention. 
     Once pseudo-horizons are defined, flattening the reflection events in the gather is simply achieved through a generalization of the normal-moveout approach, which can be called horizon-based moveout. Horizon-based moveout amounts to transforming the gather so that data samples falling on the horizons are shifted up and lined up at the corresponding zero-offset times. Data samples between horizons are shifted up by intermediate amounts. FIG. 8B illustrates the result of applying horizon-based moveout to the gather depicted in FIG.  8 A. 
     The horizon-based moveout process (step  73  in FIG. 7) is performed with the following steps: 
     For a given output zero-offset time, the corresponding input data time T is found. This is performed by linear interpolation, based on the input horizon times. 
     The output data value (amplitude) is calculated from the input data samples neighboring time T, using interpolation. For example, a four-point polynomial interpolation may be used here. 
     In another embodiment of the present invention, prior to the application of horizon-based moveout at step  73 , the input trace is interpolated at step  72  to a finer sampling interval using high-precision interpolation, such as frequency-domain sinc-function interpolation. This is done to improve the performance of the interpolation between data samples in the second sub-step immediately above. This is particularly important for the interpolation of high frequencies in 4-ms data. After application of horizon-based moveout, the trace may be desampled at step  74  to its original sampling rate. 
     Besides the use of pseudo-horizons and various interpolation methods known to persons skilled in the art, the present invention includes embodiments such as those in which the moveout correction is calculated directly from the piecewise-continuous velocity function that the present invention generates; however, the use of such approximating techniques as pseudo-horizons and interpolation are usually preferable in non-hyperbolic situations because of the need for great savings in computing time as contrasted with NMO where direct calculations using a single hyperbola equation can be made quickly. Persons trained in the art will also recognize that the present inventive method can be readily automated in computer programs. 
     In general, the quartic option may allow the flattening of the events over a larger range of offsets for the same number of velocity analysis passes. This is illustrated in FIGS. 9A-9C. FIG. 9A shows certain data after normal moveout correction. FIGS. 9B and 9C show the same data after moveout correction by the present invention. In FIG. 9B, two offset ranges are used, with hyperbolic velocity analysis used in each (“2-pass hyperbolic”). In FIG. 9C, 2-pass quartic correction is used. It is significant that the three events  91 ,  92 , and  93  are better flattened using the quartic option. Use of the quartic option may often lead to considerable time savings by reducing the number of velocity analysis passes (i.e., the number of offset ranges) that will need to be performed. In many instances, the 2-pass quartic option may be almost equivalent to the 3-pass hyperbolic option. 
     On the other hand, since the physical meaning of hyperbolic moveout velocities is better understood, hyperbolic velocity analysis provides some feedback to the processor about the correctness of the velocity picks. For example, FIG. 5B shows how the shapes of the three velocity functions are similar, yet they become progressively faster for longer offsets. If the functions appear to cross or are very dissimilar, the processor should be alerted to re-examine the velocity picks. 
     FIGS. 5A and 8B illustrate examples that demonstrate the effectiveness of the present invention using the hyperbolic option. FIG. 9B enables the 2-pass hyperbolic option to be compared to the 2-pass quartic option in FIG.  9 C. FIGS. 10A and 10B and FIGS. 11A and 11B demonstrate the application of 2-pass and 3-pass quartic velocity analysis, respectively, to the same dataset. FIGS. 10A and 11 A are identical and show the data after NMO. FIG. 10B shows the improved flatness achieved by use of the 2-pass quartic option of the present invention, and FIG. 11B shows the result of using the 3-pass quartic option. It can be seen that 2-pass quartic is very nearly as good as 3-pass quartic in this example. 
     The foregoing description is directed to particular embodiments of the present invention for purposes of illustrating it. It will be apparent, however, to one skilled in the art that many modifications and variations to the embodiments described herein are possible. For example, although use of the invention may be primarily for AVO applications, these are not the only potential uses. Others include (1) residual post-migration event alignment to achieve better stacks; (2) converted-wave processing; and (3) alignment of noise before filtering. All such modifications and variations are intended to be within the scope of the present invention as defined in the appended claims.