Prestack seismic migration

A method of prestack time migration based on the principles of prestack Kirchhoff time migration that can be applied to both 2-D and 3-D data. Common scatter point (CSP) gathers are created as an intermediate step for each output migrated trace. Normal moveout (NMO) and stacking of CSP gathers is performed to complete the prestack migration process. The method of the present invention allows the CSP gathers to be formed at any arbitrary location for velocity analysis, or to prestack migrate a 2-D line from a 3-D volume. The CSP gather is similar to a CMP gather as both contain offset traces, and both represent a vertical array of scatter points or reflectors. The CSP gather is formed from all the input traces within the migration aperture. Samples in the input traces are assigned an equivalent offset for each CSP location, then copied into the appropriate offset bin of the CSP gather in an efficient manner. The input time samples remains at the same time when copied to the CSP gather. Data in the CSP gathers may be scaled, filtered, or have noise attenuation processes applied. The equivalent offset may be derived in a simplified form as a combination NMO and post-stack migration or in a more complex form based on the double square root (DSR) equation of prestack migration. The prestack migration form of the CSP gather enables it to correctly image scattered and dipping events prior to NMO. The prestack nature, high fold, and large offsets aid in providing a better focus of the semblance plot for an improved velocity analysis.

FIELD OF THE INVENTION 
This invention relates to the field of obtaining scatter point gather 
information for accurately determining properties of an interior portion 
of a body. More particularly, this invention relates to prestack seismic 
migration by equivalent offset and common scatter point gather. 
BACKGROUND OF THE INVENTION 
Migration is a process that attempts to reconstruct an image of an original 
reflecting structure from energy recorded an the surface in seismic 
traces. 
Prior art migration processing techniques expend a great deal of effort to 
produce a snacked section from common mid-point (CMP) gathers, followed by 
a post-stack migration based on the stacking velocities. Stacking 
velocities however, require higher velocities for dipping events than 
those required for post stack migration, and some form of migration 
velocity estimation is required. Further advances in the prior art 
recognized that smearing from dip compensated velocities could be 
corrected by the inclusion of dip moveout (DMO) and prestack migration. 
The use of these prestack processes in velocity analysis loops enabled a 
more accurate estimate of the subsurface velocities and improved 
subsurface images. 
DMO and post stack migration is currently more economical than conventional 
prestack migration, consequently he DMO method tends to be the current 
processing standard. However, the use of DMO is generally restricted to 
areas with smoothly varying velocities. In areas where the smooth velocity 
criteria fails, prestack migration is currently the preferred processing 
method. Typical prestack migration methods include migration of source (or 
shot) records (shown in Schultz, P. and Sherwood, 1980, Depth migration 
before sack, Geophysics, Vol. 45, pp. 376-393); migration of constant (or 
limited) offset sections (shown in Sattlegger, J. W., et al., 1980, Common 
offset plane migration (COPMIG), Geophys. Prosp., Vol. 28, pp. 859-871, 
Deregowski, S. M., 1990, Common-offset migrations and velocity analysis, 
First Break, Vol. 8, pp. 225-234); and migration by alternating downward 
continuation between shot and geophone (S-G) gathers (shown in Diet, J. P. 
et al., 1993, Velocity analysis with prestack time migration using the S-G 
method: A unified approach, Technical Program, Soc. Expl. Geophys. 63rd 
Ann. Int. Mtg., Washington, D.C., pp. 957-960). 
Kirchhoff's method is a common prestack migration procedure that can be 
used in many of the above recited migration algorithms. Full prestack 
Kirchhoff migration is a stand-alone process described by summation of 
input samples directly to the output migrated sample (described in Yilmaz, 
O. 1987, Seismic data processing, SEG, pp. 331-334). The common use of 
full prestack migrations is limited by computer hardware long on run 
times. 
SUMMARY OF THE INVENTION 
An object of the present invention is to provide a time domain method of 
prestack migration that is faster, simpler, and more flexible than 
conventional prior art methods. 
In accordance with one aspect of the present invention there is provided a 
method of obtaining a common scatter point gather for determing properties 
of an interior portion of a body having a plurality of scatter points, 
comprising the steps of: (a) retrieving a plurality of input traces where 
each input trace is a sequential record of energy originating from a 
source and reflecting from the scatter points to a receiver, each one of 
said input traces having a plurality of data samples representing energy 
amplitudes; (b) defining a common scatter point location at a point within 
an area of interest, said common scatter point location represents a 
subset of the scatter points proximate to the common scatter point 
location; (c) defining a common scatter point gather for said common 
scatter point location based on geometry of the source and the receiver 
for the input traces relative to the scatter point location; (d) 
calculating a plurality of equivalent offset locations for said common 
scatter point gather based on: (i) geometry of the source and the receiver 
for each one of the input traces relative to said common scatter point 
location, (ii) position of the data samples in each input trace, and (iii) 
phsyical characteristics of the body, each one of said equivalent offset 
locations defining a distance between said common scatter point location 
and a location o a generally collocated model source and model receiver 
where travel time from said source to a selected one of said scatter 
points to said receiver is approximately equal to the travel Time from The 
model source to said selected scatter point to the model receiver; and (e) 
mapping each one of The data samples of the input traces to the equivalent 
offset locations in said common scatter point gather for said common 
scatter point location, wherein said common scatter point gather defines 
properities of the interior of the body.

DETAILED DESCRIPTION OF THE EMBODIMENTS OF THE INVENTION 
The Kirchhoff approach to prestack migration is based on a model of scatter 
points that will scatter (or reelect) energy from any source to any 
receiver. A reflecting event is defined at an organized arrangement of 
scatter points that produce a diffuse reflection. The surface position of 
a vertical array (or trace) of scatter points is referred to as a common 
scatter point (CSP) location. The objective of prestack migration is to 
gather all the scattered energy and relocate it at the position of the 
scatter points. The Lime of scattered energy in each input trace (relative 
to a scatter point) is identified by the travel time along the raypath 
between the source, scatter point, and receiver. 
FIG. 1 shows a simplified schematic of prestack Kirchhoff time migration 
geometry. The travel time computation is simplified to assume linear ray 
paths from a source 10 to a scatter point 12, and from the scatter point 2 
to a receiver 14. Travel times are computed using the geometry and a 
velocity that is defined for each scatter point. The migration velocity is 
generally equated to a root-mean-square (RMS) velocity. Results of the 
calculation are referred to as a time migration. After time migration the 
samples in a trace follow the time and path of an image ray. Image rays 
enable a conversion of the time migration to an estimate of the depth 
migration. Consequently, time migrations can be used to aid in defining 
the structure and velocity fields used in depth migration. 
The total travel time T is estimated from a source to scatter point time 
T.sub.s, and the scatter point to a receiver time T.sub.r by: 
EQU T=T.sub.s +T.sub.r (1) 
T.sub.0 is defined as the two-way zero offset time, to represent the zero 
offset travel time from a position immediately above the scatter point 12. 
However, for prestack computations, it is preferable to use one half the 
value of T.sub.0 (1/2T.sub.0) to simplify prestack equations. 
Source and receiver distances h.sub.s and h.sub.r, respectively, are 
measured from a common scatter point (CSP) location 16, and the migration 
velocity V.sub.rms (T.sub.0) is defined at 1/2T.sub.0 and is independent 
of h.sub.s and h.sub.r. Equation (1) then becomes a double square root 
(DSR) equation: 
##EQU1## 
Kirchhoff's migration algorithm generally processes one scatter point at a 
time, and searches all the input traces for energy that has been 
scattered. The travel time T is computed for each input trace within a 
migration aperture. The energy at those times is filtered, scaled, and 
then summed into a migrated sample. The energy gathered into the migrated 
sample will constructively (or destructively) sum o recreate the structure 
of the subsurface. 
The filtering of the input traces is usually necessary to prevent aliasing 
noise in the reconstructed image. Antialias filtering will vary for the 
same sample of a given input trace depending on the relative location of 
the scatter point. The amplitude weighting of the input samples also 
varies with each input trace depending on the relative location of the 
scatter point. 
FIG. 2 illustrates a perspective view of a Cheop's pyramid (an impulse 
response of a point scatterer in a prestack volume) in which the two-way 
travel time T is computed from one scatter point to a continuum of 2-D 
source and receiver locations. The pyramid is displayed as a surface 30 in 
a three dimensional volume 32 defined by a common mid-point (CMP) axis 34, 
a source/receiver half offset (h) axis 36, and a time (t) axis 38. The 
Cheop's pyramid has two hyperbolic planes, one at zero offset location 
(i.e. in the CMP-t plane), and the other at the scatter point location 
with variable offset. The hot plane on the side 32 of FIG. 2 illustrates a 
CMP gather with energy coming from a scatter point out of the plane of the 
CMP gather. This energy is distributed with non-hyperbolic moveout 30 and 
cannot be accurately focused by NMO and stacking. 
For real data, examples provided hereinbelow, the location of energy on the 
pyramid surface 30 will be determined from the reflection strength of the 
neighboring scatter points. Horizontal linear reflections will end to 
concentrate energy at the top portion of the pyramid, while dipping 
reflectors will tend to have more energy at the corresponding seismic dip. 
Energy from edges, such as faults, will be dispersed more evenly over he 
entire surface. The velocity for data on the pyramid is defined at the 
apex, or scatter point location, even though it may extend into areas with 
complex velocity structures. The pyramid 30 shown in FIG. 2 was defined 
for 2-D data where the top surface of the volume is equivalent to a 
stacking chart. Energy from a scatter point in 3-D data may also be mapped 
into a Cheop's pyramid by substituting other parameters known in the art 
for the CMP location and half source/receiver offset. 
The method of the present invention collapses the energy of the Cheop's 
pyramid surface 30 to a scatter point 12 to emulate full prestack 
Kirchhoff time migration (discussed above) where time T is computed from 
the DSR equation (2). 
The method of the present invention generates prestack migration gathers, 
termed common scatter point (CSP) gathers, that are based on the scatter 
point principles shown in FIG. 1. The formation of CSP gathers can be 
created at any arbitrary location within a 3-D volume for velocity 
analysis, or may be used to quickly extract an arbitrarily located 
prestack migrated 2-D line from the 3-D volume. 
CSP gathers are formed when samples in the input traces are assigned an 
offset that is based on the distance between the source/receiver positions 
relative to each CSP location. This offset is referred to as the 
equivalent offset. Once this offset is assigned, the input trace is then 
summed directly into the offset bins of the CSP gather. No time shifting 
of the input data is required, however, time shifting can still be 
accommodated if required. 
EQUIVALENT OFFSET DEFINITIONS 
In conventional normal moveout (NMO) and post-stack migration, energy from 
each input trace is spread to all output traces similar to prestack 
migration. Energy is first moved from the original two way time position 
T, by NMO, to the zero offset two way time T.sub.0-nm0 : 
##EQU2## 
, where h is half the source/receiver offset. 
Energy may then be moved from the zero offset at two-way time T.sub.0-nm0 
to the migrated two-way time T.sub.0-mig by the kinematic Kirchhoff 
migration equation: 
##EQU3## 
where X.sub.off is the offset between the CMP and migrated position. 
These two equations (3 and 4) can be combined to move energy directly from 
the input trace to the migrated position by substituting (4) into (3) to 
give: 
##EQU4## 
assigning an initial estimate of the equivalent offset term (h.sub.e) as: 
EQU h.sub.e =h.sup.2 +x.sub.off.sup.2 (6) 
FIG. 3 schematically illustrates a process that combines normal moveout 
(NMO) and post-stack migration into one NMO step with a new offset defined 
in equation (6). This process is essentially prestack migration utilizing 
Kirchhoff migration principles, requiring the use of scaling, antialiasing 
filters, and phase compensation filters. 
A prestack volume 50 is defined by a half source/receiver offset axis 52, a 
common mid-point (CMP) trace position axis 54, and a time t axis 56. 
Although FIG. 3 relates to 2-D data with one axis defining the CMP 
location, similar concepts apply for 3-D data. A shaded plain region 58 
represents a CMP gather and includes one example of an input trace 60 at a 
CMP location with offset h. Hyperbolic curves 62 on the CMP gather 
represent the time shifting paths taken by data during NMO and stacking. 
The front surface of the volume 50 represents a stacked or zero offset 
section. The zero offset section contains an arbitrarily located migrated 
trace 64 with a plurality of hyperbolas 66, and also shows the time 
shifting paths for migration. 
The equivalent offset h.sub.e allows all the input traces to be gathered 
prior to NMO for a given migrated position as shown in FIG. 4. FIG. 4 
shows the same input trace 60 copied into a bin of CSP gather 80 (shaded 
region) at the offset defined by h.sub.e. NMO curves 82 required for 
prestack migration to the output location (i.e. the CSP location) are also 
shown in the shaded migration gather 80. 
Intersection points 84 of a grid 86 on the top surface of volume 50 are 
used to identify the position of other input traces positioned by CMP and 
offset h. All these traces may be assigned an equivalent offset and copied 
onto the prestack migration gather. These equivalent offsets may be much 
larger than the maximum source receiver offset that is illustrated by 
extending the shaded area 80 of the gather beyond the maximum source 
receiver offset. When the migration position is moved, the input traces 
will be assigned different equivalent offsets, 
The definition of equivalent offset provided above may be further improved 
by incorporating the DSR equation (2). This is accomplished by defining a 
new source and receiver that are collocated at an equivalent offset 
position. 
FIG. 5 illustrates prestack migration containing an equivalent offset 
position at E for a collocated source and receiver. The equivalent offset 
is chosen to maintain the same total travel times as the original paths 
100 and 102 (dotted lines) by a single two way path 104 (solid line) 
defined as 2T.sub.e. In forcing the source and receiver to be collocated, 
and positioned at the equivalent offset E, he reflection time is now 
located on an hyperbola (the zero offset NMO curve) centered at the 
scatter point as shown in FIG. 5. 
The NMO and stacking of the energy on the hyperbola complete the prestack 
migration. Equation (1) is modified to include the equivalent offset 
one-way time T.sub.e to become: 
EQU 2T.sub.e =+T.sub.s T.sub.r =T (7) 
Substituting the raypath parameters and using one-way times gives: 
##EQU5## 
The RMS velocities are all defined at the scatter point 106. Solving for 
h.sub.e we obtain: 
EQU h.sub.s.sup.2 =0.25[(.sub.1/2 T.sub.0.sup.2 V.sub.rms.sup.2 +h.sub.2.sup.2) 
.sup.1/2 +(.sub.1/2 T.sub.0.sup.2 V.sub.rms.sup.2 +h.sub.r.sup.2) .sup.1/2 
].sup.2 -.sub.1/2 T.sub.0.sup.2 V.sub.rms.sup.2 (9) 
When the source and receiver are both on the same side of the CSP location 
(2-D data) as in FIG. 1, then the half source/receiver offset h and the 
CMP to CSP distance x.sub.off, are defined by: 
##EQU6## 
Input 2-D traces in which the source and receiver straddle the CSP, and all 
input 3-D traces, may be converted into a geometry similar to FIG. 1 by 
rotating the raypaths to be in a common vertical plane on the same side of 
the CSP location. Equations (10) and (11) are used as a general definition 
of h and are used instead of x.sub.off, and the actual 3-D surface 
geometry. The relative position of the source and receiver may alter the 
signs in equations (11) and (13), but will have no effect on the following 
computation for the equivalent offset. Azimuthal information of the source 
and receiver rays should still be preserved and used for azimuthal 
stacking. 
Substituting equations (12) and (13) into equation (8) yields a more useful 
full definition of the equivalent offset: 
##EQU7## 
The equivalent offset h.sub.e defined in equation (14) is time or depth 
varying, and also a function of velocity. As a consequence, an input trace 
may have its samples spread over a number of offset bins. An example of 
the time varying equivalent offset is shown in FIG. 6A, where the source s 
and receiver r positions are located relative to CSP location 130. The 
first useful energy in the input traces comes at a time t.sub..alpha. 
defined as the reflection from a scatter point at the surface of the CSP 
location 130 and is given by: 
##EQU8## 
with an offset of this first point h.sub.e.alpha. is defined by: 
EQU h.sub.e.alpha. =x.sub.off (16) 
Energy at this point will migrate to the surface of the CSP migrated output 
trace with a dip of 90 degrees. The energy contribution to the CSP remains 
below T.sub..alpha., where the offset increases slightly and tends to an 
offset asymptote h.sub.e.omega. given by: 
EQU h.sub.e.omega..sup.2 =x.sub.off.sup.2 +h.sup.2 (17) 
which is also the same offset given by equation (6), derived from NMO and 
post-stack migration. 
Although it appears from equation (14) that the equivalent offset h.sub.e 
needs to be computed for each input sample, this is not the case since it 
is only actually necessary when the input samples start in a new bin. The 
first useful sample on an input trace will come from a reelection off the 
scatter point at the surface. The travel time to this scatter point is 
T.sub..alpha., and these times lie at a 45 degree angle on the CSP gather 
as illustrated in FIGS. 4 and 6. 
An initial equivalent offset h.sub.e.alpha. may be computed from this time 
and assigned to an appropriate offset bin with central offset h.sub.e (n) 
and incremental offset .delta.h.sub.e. The following samples are added to 
this bin until the equivalent offset increases to h.sub.e 
(n)+1/2.delta.h.sub.e, at which point the samples are then added to the 
next offset bin h.sub.e (n+1). The time at which this occurs is T(m+1) and 
may be found by rearranging equation (14) to give: 
##EQU9## 
In a similar manner, the transition times for all the offset bins may be 
computed, to allow efficient copying of the input trace samples into the 
respective bins. The above assumption of using T.sub..alpha. as the 
starting time for an input trace assumes that the energy will propagate to 
time zero and correspond to a 90 degree migration. Time migrations to this 
steep angle may be impractical and include a great deal of unwanted noise. 
Limiting the extent of the migration angle is a popular feature of 
Kirchhoff migrations and it may also be applied to the formation of the 
CSP gathers of the present invention. 
The pre-migration dip angles on the CSP gather may also be limited to 
reduce the noise, with an additional benefit of reducing the number of 
offset bins used by an input trace. In many cases, all samples in an input 
trace will map into one offset bin at offset h.sub.e.delta.. This benefit 
also occurs when half offset h is small relative to x.sub.off, or when the 
first useful scatter points are below the surface as in marine data. 
FIG. 4 also illustrates the migration aperture or the amount of useful 
information contributed by each input trace relative to its equivalent 
offset. The NMO curves 82 on he shaded area 80 may be used to project the 
input data to the final position at zero offset. For this CSP location, 
only a small portion to the bottom of the input trace will contribute 
energy to the top half of the migrated trace. The 45 degree diagonal line 
shows the time T.sub..alpha. of the first useful input energy of the 
input traces. This line can be observed going to the maximum time and thus 
define the maximum offset for the migration aperture. Traces with offset 
beyond this range should not be considered for the respective CSP gather. 
In some applications, the maximum useful offset of the migration aperture 
may be further reduced when the offset apply to data below the zone of 
interest. 
The illustration in FIG. 4 uses a source/receiver offset that is relatively 
large when compared to the trace length, and may imply that the equivalent 
offset only contributes a few additional offsets with limited energy 
contribution. For real data, useful equivalent offsets are often three 
times the maximum source/receiver half offset. 
Scaling and filtering operations can be applied efficiently before NMO and 
summation with little extra overhead. Application of a 3-D differential 
filter (root differential filter for 2-D) before NMO and summation aids in 
focusing the velocity energy on semblance plots to improve the accuracy of 
velocity analysis. In addition, other parameters such as dip range may 
also be applied to the CSP gathers and tested to optimize their effect on 
the final migration. The size of the bin spacing .delta.h in a CSP gather 
may be determined from the maximum allowable time shift which results from 
applying NMO to neighboring bins. This value will depend on the frequency 
content of the data and the acquisition geometry. The effect of finite bin 
size with the high fold, and an assumed linear distribution, is to box car 
filter the binned data with period equal to the time shift. This filter 
has the same parameters as one required for antialiasing and in effect 
provides a natural filter for the data in a CSP gather. 
In practice, the bin spacing in the CSP. gathers is less than that of the 
3-D subsurface grid (or CMP spacing for 2-D) that require a more severe 
antialiasing filter criteria. The bin size has little effect on the time 
required for acquiring the CSP gathers as the same number of input samples 
are summed. Smaller bin sizes require more offset bins, so the memory 
requirements to save the offset will however increase. The offset bin size 
should not confused with the offset bin sizes in constant offset migration 
where the number of offset bins tend to be small to expedite the 
algorithm. 
After the gathering, scaling, and filtering of the CSP gathers, the 
remaining processing steps to complete the prestack migration are NMO and 
stacking. Regular processing software may be used, along with conventional 
velocity analysis techniques. The only new software required is a routine 
to define the equivalent offset, and a routine to add the input energy to 
the appropriate offset bin. 
A comparison between the fold of the CMP gather and the prestack migration 
gather may be visualized from FIG. 4. A CMP gather contains 9 traces from 
the grid, while the CSP gather could contain 225 traces. In real 2-D data, 
the CMP gather may contain only 15 live traces for a maximum possibility 
of 60 bins, while the corresponding CSP gather may contain tens of 
thousands of input traces spread over a few hundred bins. A CSP gather 
from 3-D data may contain hundreds of thousands of input traces. This 
extremely high fold and large offsets enable accurate velocity analysis at 
each migrated position. In contrast several CMP gathers may be required to 
obtain a single fold coverage for conventional velocity analysis. 
An important property of the CSP gather is that many traces are summed 
using only the asymptotic equivalent offset h.sub.e.delta.. This data is 
independent of both time and velocity, providing stability to the CSP 
gather when the velocities are unknown. The CSP gathers may be formed with 
an arbitrary velocity, and the gather used o define a more accurate 
velocity. The iterations of this process converge very rapidly, and 
usually only one or two iterations are required. Velocities derived from 
the CSP gather are RMS type velocities, and can be converted to velocities 
and depths that will tie reasonably with sonic logs. 
The increased offset range of the CSP gather, and its high fold improve the 
resolution of velocity analysis. Consequently, the velocities on semblance 
plots focus to smaller points than on conventional semblance plots formed 
from a CMP gather. The better focusing of the data on semblance plots 
illustrates the improved resolving power of prestack migration, and 
illustrates that velocities must be quite accurate to enhance the signal 
and reduce the energy of multiples. This feature of the CSP gather may 
have substantial implications for acquisition designs that are based on 
CMP gathers. In addition to the benefits of prestack migration, the 
standard algorithms for noise or multiple removal, designed for use with 
CMP gathers, may also be applied directly to CSP gathers. 
The above discussion of the present invention has used prestack migration 
to define CSP gathers based on equivalent offsets. However, other criteria 
for forming the gathers are available. For example, azimuth restrictions 
may be applied to 2-D or 3-D data. Additional applications allow the 
inclusion of converted wave velocity analysis for prestack migration, 
crooked line processing.sub.-- and prestack migration in areas of rugged 
topography. 
Equation (2) gives the kinematic or timing solution for Kirchhoff prestack 
time migration. In conventional prestack time migrations this equation is 
evaluated many times since each input sample contributes energy to many 
migrated samples. The number of times the DSR equation must be evaluated 
depends on the number of traces in the x direction n.sub.x, the number of 
traces in the y direction n.sub.y, the average fold n.sub.f, and the 
number of samples in a trace n.sub.s. The total number of input samples is 
approximately n.sub.x .times.n.sub.y .times.n.sub.f .times.n.sub.s, and 
many of these samples must be copied to every migrated trace n.sub.x 
.times.n.sub.y. Input traces that are more distant from a CSP location 
contribute less energy to form a cone shape of input data. This cone shape 
reduces the number of DSR's by two thirds. For a small 3-D, the total 
number of DSR computations N.sub.DSR may be approximated from: 
EQU N.sub.DSR .apprxeq.n.sub.x.sup.2 .times.n.sub.y.sup.2 .times.n.sub.f 
.times.n.sub.s .div.3 (19) 
For a small 3-D project with traces n.sub.x =115 and n.sub.y =120, an 
average fold n.sub.f =20, and n.sub.s =1000 samples, the total number of 
DSR computations is approximately 1.27.times.10.sup.12. This number must 
also be increased to get the actual number of floating point operations. 
These computation are only for the kinematic or time considerations. 
Consideration for scaling and filtering of each input sample relative to 
each output scatter point to prevent aliasing can also be included 
according to principles known in the art. Even with megaflop processors, 
these numbers become quite formidable. For large 3-D's, the number of 
input traces contributing energy to a CSP location becomes limited by the 
migration aperture and significantly reduce the size of N.sub.DSR that is 
estimated from equation (3). 
An earlier section specified equation (3) for estimating the number of DSR 
computations for full prestack Kirchhoff time migration in a small 3-D. A 
numerical comparison with the equivalent offset and CSP gather may now be 
made. Defining the number of offset bins for each CSP gather to be 
n.sub.bin, the number of NMO computations N.sub.nm0 may be estimated from: 
EQU N.sub.nm0 .apprxeq.n.sub.x .times.n.sub.y .times.n.sub.bin .times.n.sub.s 
.div.2 (20) 
,where the division by two compensates for the cone input data projected 
onto a 2-D plane. 
The speed of evaluating the DSR and NMO equations may be significantly 
increased by using look up tables, and it is difficult to estimate the 
actual number of floating point computations. As a result a comparison 
will only be made between the number of DSR computations N.sub.dsr with 
the number of NMO computations N.sub.nm0, and is found by dividing 
equation (20) into equation (3): 
##EQU10## 
Using the values defined earlier for a small 3-D project, and using a 
number of offset bins n.sub.b =150, the above ratio of equation (21) 
becomes approximately 1227. The resulting computer time for the NMO 
computation may be less than the time required for creating the CSP 
gathers, and becomes a small part of the overall processing time. 
Consequently changes in the number of CSP offset bins has very little 
effect on the overall computation time for prestack migration. This 
overall improvement in speed ie in the order of 100's for a 3-D survey 
relative to full prestack Kirchhoff time migration. 
Static analysis is an area in which the CSP gather may also be of 
significant benefit. When conventional 2-D lines are recorded with source 
points at four-station intervals, four independent (de coupled) surface 
consistent solutions are obtained. Each receiver only contributes to every 
fourth CMP, requiring filtering techniques to combine he solutions. In 
addition, the static solutions are obtained on NMO'ed data by correlating 
each input trace with a model trace that is typically from a smoothed 
brute section. 
The CSP gather, in contrast to a CMP gather, contains many contributions 
from all sources and all receivers within the prestack migration aperture. 
This greatly increases the number of correlations and ensures the coupling 
of all sources and receivers with all CSP's. The high fold of the CSP 
gather may enable it to serve as a model for the input traces before NMO, 
to give statics that are independent of the stretching due to NMO. The 
success may possibly depend on removing coherent noise to create a 
suitable CSP gather model for correlating input traces. 
Many traces in a CSP gather are positioned with offsets close to the 
asymptote h.sub.e.delta. and are therefore independent of time and 
velocity. When CSP gathers are produced independent of time and velocity, 
the potential applications may exceed those of prestack time migration. 
Time migrations with more complex moveout equations are possible and may 
be a necessity for taking advantage of the long offsets. In addition the 
accurate velocities derived from the CSP gathers may allow an accurate 
estimation of average velocities to produce an approximate depth 
migration. 
CSP gathers may be formed to bias the azimuth of a ray path from either the 
source to scatter point, or the scatter point to receiver. Ray paths with 
a desired azimuth of one ray path leg may be collected with the azimuth of 
the other leg left to vary randomly. Thus, a number of CSP gathers may be 
formed at the same CSP location, each with a different azimuth. Comparison 
between azimuthal gathers may allow a better estimate of anisotropic 
velocities, and produce a better prestack migration. The processing time 
of azimuthal gathers will only double the stack time required to create 
the azimuthal CSP gathers at a given CSP location because each input trace 
will only sum into two azimuthal gathers, one for each leg of the total 
ray path. 
The step of computing the equivalent offset for rugged topography wall be 
discussed in conjunction with FIG. 6B. 
FIG. 6B illustrates an example of rugged topography geometry. A datum 
correction for a source 150 is t.sub.s. A datum correction for a receiver 
152 is t.sub.r. Accurate prestack migrations can be achieved by migrating 
from a surface 154, i.e. from the elevation of the source 150 and the 
elevation of the receiver 152. The velocity V.sub.srs of the source 150 is 
defined from the elevation of the source 150, with the velocity at a 
scatter point 156 defined by V.sub.srs (T.sub.0 +t.sub.s) and a similar 
receiver velocity V.sub.rec (T.sub.0 +t.sub.r). The velocity at the 
scatter point 156 from a datum 158 is given by V(T.sub.0). The replacement 
velocity is given by V.sub.rep. The total travel time T from the source 
150 to the scatter point 156 T.sub.s and from the scatter point 156 to he 
receiver 152 T.sub.r is: 
EQU T=T.sub.s +T.sub.r (22) 
The travel times Ts and Tr are defined by the double square root equation 
(DSR) modified for rugged topography: 
##EQU11## 
These times equate the travel time of the collocated source and receiver 
located on the datum 158 for the common scatter point gather: 
##EQU12## 
giving the rugged topography equivalent offset: 
##EQU13## 
The velocities V.sub.srs for a source ray path 160 and V.sub.rec for a 
receiver raypath 162 are based on the velocity at the scatter point 156 V, 
and modified using the Dix equation to account the different elevation: 
##EQU14## 
The process of moving input data to the common scatter point gather begins 
with T.sub.0 equal to zero. The following three steps are performed in a 
loop with incremental increases in T.sub.0 storing the results for T(m) 
and h.sub.e (m) in arrays, until T(m) exceeds the maximum time on the 
input trace: 
1. calculate V.sub.srs and V.sub.rec at T.sub.0 using equations (26); 
2. calculate T using equation (24); and 
3. calculate the equivalent offset h.sub.e3 from equation (25). 
The data from the input trace is copied to the offset bins of he CSP gather 
using the arrays T(m) and h.sub.e (m). 
Converted wave processing assumes the downward propagating ray path is a P 
wave, and the reflection converts some P wave energy into shear wave 
energy that propagates to the surface. Recording of this shear wave energy 
with 3 component receivers provides additional information about the 
reflecting or converting surface, and allows the estimation of properties 
such as Poisson's ratio. The principle of equivalent offsets may be 
applied to the processing of converted wave data. CSP gather processing 
provides better velocity analysis, simplifies he process, reduces the 
computation time, and gives improved results. The processing method starts 
with the DSR equation using the appropriate P and S velocities for each 
leg of the ray path going to and from the equivalent offset. The 
subscripts P and S are used with the velocity V to indicate the use of the 
appropriate velocity, giving: 
##EQU15## 
The equation may be solved to find the equivalent offset h.sub.e in terms 
of the pseudo depth Z.sub.0, and the Vp/Vs ratio .gamma. 
The solution: 
##EQU16## 
shows that the equivalent offset will enable the formation of a common 
conversion scatter point (CCSP) gather. Use of the equivalent offset, 
transforms the input converted wave data to be hyperbolic in the CCSP 
gathers and therefore allows conventional velocity analysis and 
processing. Specifically, the process of moving samples from input traces 
to the offset bins in the CSP gather is similar to the process described 
above in conjunction with FIG. 6B. More generally, the rugged topography 
considerations can be included in the converted wave process discussed 
above. 
Other applications of CSP gathers according to the present invention 
include the formation of an unmigrated image in which data in the CSP 
gathers contain zero offset information required to migrate the energy to 
the trace at the CSP gather. With the appropriate azimuth sorting, energy 
in these gathers may be rotated back to the zero offset plane to form an 
unmigrated zero offset stacked section. In addition, in the field of 
surface consistent statics conventional processing usually forms a brute 
stack to serve as a model for the evaluation of surface consistent 
statics. However, use of the CSP gather can form the basis of a model that 
will allow statics to be evaluated prior to NMO. 
EXAMPLES 
Synthetic Model 
FIGS. 7a-f illustrate various synthetic models illustrating a hockey stick 
model (a), the source record (b), the CSP gather (c), the CSP gather with 
NMO (d), the equivalent offset migration (e), and the close up CSP gather 
(f). 
The model was created to evaluate the performance of the migration. The 2-D 
model consists of one scatter point and two linear reflectors. One linear 
reflector is short and horizontal, while the other dips steeply with one 
end meeting the horizontal reflector. The dipping reflector has a gap 
close to its middle. Simple source records were created from the model by 
estimating the travel times from each reflector and placing a wavelet 
centered at the travel times. No attempt was made to model amplitude 
variations. No diffractions were included in the model except for the 
impulse response from the one scatter point. 
The model was used to create 101 source records which were collected into 
CSP gathers spanning the reflectors. One of these CSP gathers is shown in 
FIG. 7c at a location directly above the scatter point. The result of 
normal moveout applied to this CSP gather is shown in FIG. 7d. Note the 
horizontal alignment of energy for the scatter point and for the left end 
of the horizontal reflector. Stacking the NMO'ed CSP completes the 
prestack migration process, and this trace may be seen as the stacked 
trace is the central trace in FIG. 7e. The result of this equivalent 
offset migration compares identically with a full Kirchhoff prestack 
migration. 
FIG. 7f shows a part of a CSP gather created from one source record located 
directly above the scatter point. The location of the CSP gather is also 
above the scatter point. A small offset interval helps to illustrate,the 
time-varying equivalent offset variations that may occur for input traces. 
The variations are most apparent at shallow times on the traces on the 
right side of the figure. 
EXAMPLES 
Land Environment 
A land example is illustrated in FIGS. 8a-d, showing an example of (a) CMP; 
(b) CSP gathers and semblance; (c) DMO and post-stack migration; and (d) 
CSP migration. 
The real data example was acquired in the Foothills west of Calgary, 
Canada. This line was conventionally processed using DMO and post stack 
migration to obtain the best section. A CMP super gather and semblance 
plot are shown in FIG. 8a. A CSP gather and the corresponding semblance 
plot are shown in FIG. 8b. The final DMO and migrated section is shown in 
FIG. 8c, and the corresponding equivalent offset migration in FIG. 8d. 
Note the improvement in the semblance of the CSP gather over the CMP super 
gather. The peaks of the velocities are more accurately defined. This 
enabled rapid picking of the velocities which significantly reduced the 
processing time. The statics solution derived from conventional processing 
of CMP gathers was used to create the CSP gathers. 
EXAMPLES 
Marine Environment 
An example of a super CMP gather and a CSP gather from a marine line is 
shown in FIGS. 9a-d. The super CMP spanned eight CMP's to achieve full 
trace coverage in the offset bins. The CSP gather is displayed as a two 
sided plot and has an average fold of fifteen. The CSP gather was formed 
using h.sub.e.delta. at a constant equivalent offset for each input 
trace. Consequently, the CSP gather is independent of time and velocity, 
and thus not limited by the constraints of prestack time migration. 
The CSP gather shows many coherent reflections that extend to far offsets, 
along with other reflections hat become visible at offsets beyond the 
range of the CMP gather. High order NMO equations may be required to 
obtain optimum migration of this data. 
In summary, the prestack migration method of the present invention is 
simpler and faster than conventional methods. The method of the present 
invention modifies the conventional Kirchhoff prestack time migration 
process by gathering, performing NMO, and stacking. The present method 
correctly maps energy from prestack traces to equivalent offsets in common 
scatter point (CSP) gathers. Conventional velocity analysis tools may be 
used on the CSP gathers to accurately determine RMS velocities.