CA2568103A1 - Removal of noise from seismic data using improved radon transformations - Google Patents

Abstract

Methods of processing seismic data to remove unwanted noise from meaningful reflection signals are provided for. The seismic data is transformed from the offset-time domain to the time-slowness domain using a Radon transformation.
Preferably, the Radon transformation is applied within defined slowness limits p min and p max, where p min is a predetermined minimum slowness and p max is a predetermined maximum slowness, and an offset weighting factor x" is applied to the amplitude data, wherein 0 < n < 1. A , high resolution Radon transformation preferably is used, where the transformation is performed according to an index j of the slowness set and a sampling variable .DELTA.p ; wherein .DELTA.p is from about 0.5 to about 4.0 µsec/m, p max is a predetermined maximum slowness, and p min is a predetermined minimum slowness.
A high-low, preferably a time variant, high low corrective filter is then applied to enhance the primary reflection signal content of the data and to eliminate unwanted noise events. The tau-P filter is defined by reference to the velocity function of the primary reflection signals, and preferably is expressed as:
where r1 and r2 are percentages expressed as decimals. The slowness limits typically are set within ~ 15% of the velocity function, and r1 and r2 may be set accordingly.
After filtering, the enhanced signal content is inverse transformed from the time-slowness domain back to the offset-time domain using an inverse Radon transformation.

Description

REMOVAL OF NOISE FROM SEISMIC DATA
z USING IMPROVED RADON TRANSFORMATIONS
BACKGROUND OF THE INVENTION
a The present invention relates to processing of seismic data representative of s subsurface features in the earth and, more particularly, to improved methods of and 6 apparatus for processing seismic data using improved Radon transformations to ~ remove unwanted noise from meaningful reflection signals.
a Seismic surveys are one of the most important techniques for discovering the 9 presence of oil and gas deposits. If the data is properly processed and interpreted, a ~o seismic survey can give geologists a picture of subsurface geological features, so that > > they may better identify those features capable of holding oil and gas.
Drilling is ~z extremely expensive, and ever more so as easily tapped reservoirs are exhausted and 3 new reservoirs are harder to reach. Having an accurate picture of an area's subsurface ~a features can increase the odds of hitting an economically recoverable reserve and is decrease the odds of wasting money and effort on a nonproductive well.
The principle behind seismology is deceptively simple. As seismic waves travel through the earth, portions of that energy are reflected back to the surface as the ~s energy waves traverse different geological layers. Those seismic echoes or reflections 9 give valuable information about the depth and arrangement of the formations, some of zo which hopefully contain oil or gas deposits.
z~ A seismic survey is conducted by deploying an array of energy sources and an zz array of sensors or receivers in an area of interest. Typically, dynamite charges are z3 used as sources for land surveys, and air guns are used for marine surveys.
The za sources are discharged in a predetermined sequence, sending seismic energy waves zs into the earth. The reflections from those energy waves or "signals" then are detected ~ by the array of sensors. Each sensor records the amplitude of incoming signals over z time at that particular location. Since the physical location of the sources and 3 receivers is known, the time it takes for a reflection signal to travel from a source to a a sensor is directly related to the depth of the formation that caused the reflection.
s Thus, the amplitude data from the array of sensors can be analyzed to determine the 6 size and location of potential deposits.
This analysis typically starts by organizing the data from the array of sensors s into common geometry gathers. That is, data from a number of sensors that share a 9 common geometry are analyzed together. A gather will provide information about a ~o particular spot or profile in the area being surveyed. Ultimately, the data will be i i organized into many different gathers and processed before the analysis is completed ~z and the entire survey area mapped.
The types of gathers typically used include: common midpoint, where the 14 sensors and their respective sources share a common midpoint; common source, i s where all the sensors share a common source; common offset, where all the sensors 6 and their respective sources have the same separation or "offset"; and common m receiver, where a number of sources share a common receiver. Common midpoint i 8 gathers are the most common gather today because they allow the measurement of a 19 single point on a reflective subsurface feature from multiple source-receiver pairs, zo thus increasing the accuracy of the depth calculated for that feature.
zi The data in a gather is typically recorded or first assembled in the offset-time zz domain. That is, the amplitude data recorded at each of the receivers in the gather is 23 assembled or displayed together as a function of offset, i.e., the distance of the za receiver from a reference point, and as a function of tithe. The time required for a zs given signal to reach and be detected by successive receivers is a function of its z6 velocity and the distance traveled. Those functions are referred to as kinematic travel z~ time trajectories. Thus, at least in theory, when the gathered data is displayed in the zs offset-time domain, or "X-T" domain, the amplitude peaks corresponding to reflection z9 signals detected at the gathered sensors should align into patterns that mirror the 3o kinematic travel time trajectories. It is from those trajectories that one ultimately may ;~ determine an estimate of the depths at which formations exist.

t A number of factors, however, make the practice of seismology and, z especially, the interpretation of seismic data much more complicated than its basic s principles. First, the reflected signals that indicate the presence of geological strata a typically are mixed with a variety of noise.
s The most meaningful signals are the so-called primary reflection signals, those 6 signals that travel down to the reflective surface and then back up to a receiver. When ~ a source is discharged, however, a portion of the signal travels directly to receivers a without reflecting off of any subsurface features. In addition, a signal may bounce off 9 of a subsurface feature, bounce off the surface, and then bounce off the same or to another subsurface feature, one or more times, creating so-called multiple reflection t t signals. Other portions of the signal turn into noise as part of ground roll, refractions, tz and unresolvable scattered events. Some noise, both random and coherent, is r3 generated by natural and mananade events outside the control of the survey.
to All of this noise is occurring simultaneously with the reflection signals that is indicate subsurface features. Thus, the noise and reflection signals tend to overlap t6 when the survey data is displayed in X-T space. The overlap can mask primary m reflection signals and make it difficult or impossible to identify patterns in the display is upon which inferences about subsurface geology may be drawn. Accordingly, various i9 mathematical methods have been developed to process seismic data in such a way that zo noise is separated from primary reflection signals.
21 Many such methods seek to achieve a separation of signal and noise by zz transforming the data from the X-T domain to other domains. In other domains, such z3 as the frequency-wavenumber (F-K) domain or the time-slowness (tau-P), there is less za overlap between the signal and noise data. Once the data is transformed, various zs mathematical filters are applied to the transformed data to eliminate as much of the zs noise as possible and, thereby, to enhance the primary reflection signals.
The data z~ then, is inverse transformed back into the offset-time domain for interpretation or za further processing.
z9 For example, so-called Radon filters are commonly used to attenuate or so remove multiple reflection signals. Such methods rely on Radon transformation 3t equations to transform data from the offset-time (X-T) domain to the time-slowness sz (tau-P) where it can be filtered. More specifically, the X-T data is transformed along ~ kinematic travel time trajectories having constant velocities and sIownesses, where z slowness p is defined as reciprocal velocity (or p =1 l v ).
Such prior art Radon methods, however, typically first process the data to a compensate for the increase in travel time as sensors are further removed from the s source. This step is referred to as normal moveout or "NMO" correction. It is s designed to eliminate the differences in time that exist between the primary reflection ~ signals recorded at close-in receivers, i.e." at near offsets, and those recorded at s remote receivers, i.e., at far offsets. Primary signals, after NMO
correction, generally s will transform into the tau-P domain at or near zero slowness. Thus, a mute filter may ~o be defined and applied in the tau-P domain. The filter mutes, i.e., excludes all data, > > including the transformed primary signals, below a defined slowness value p"",re.
12 The data that remains after applying the mute filter contains a substantial 3 portion of the signals corresponding to multiple reflections. That unmuted data is then ~a transformed hack into offset-time space and is subtracted from the original data in the ~s gather. The subtraction process removes the multiple reflection signals from the data 6 gather, leaving the primary reflection signals more readily apparent and easier to interpret.
~a It will be appreciated, however, that in such prior art Radon filters, noise and 9 multiple reflection signals recorded by receivers close to the gather reference point zo ("near trace multiples") are not as effectively separated from the primary reflections.
z~ The lack of separation in large part is an artifact of the NMO correction performed zz prior to transformation. Because NMO correction tends to eliminate offset time z3 differences in primary signals, primary signals and near trace multiple signals both za transform at or near zero slowness in the tau-P domain. When the mute filter is zs applied, therefore, the near trace multiples.are muted along with the primary signal.
z6 Since they are muted, near trace multiples are not subtracted from and remain in the z~ original data gather as noise that can mask primary reflection signals.
zs Radon filters have been developed for use in connection with common source, z9 common receiver, common offset, and common midpoint gathers. They include those 3o based on linear slant-stack, parabolic, and hyperbolic kinematic travel time ~ trajectories. The general case forward transformation equation used in Radon z filtration processes, R(p,z)[d(x,t)], is set forth below:
zc(p, z) = f dx f dtd (x, t)~[ f (t, x, z, p)] (forward transformation) a where s zc(p,z) = transform coefficient at slowness p and zero-offset time z d(x, t) = measured seismogram at offset x and two-way time t p = slowness s t = two-way travel time z= two-way intercept time at p = 0 ~o x = offset > > 8 = Dirac delta function iz f(t,x,zp) = forward transform function ~s The forward transform function for linear slant stack kinematic travel time ~a trajectories is as follows:
~s f~t,x,z,P~=t-z-Px 6 where ~z S[f(t,x,z,P)] _ ~(t-z-Px) = l, when t = z + px , and 19 = 0, elsewhere.
zo Thus, the forward linear slant stack Radon transformation equation becomes z~ u(p,z) = Jdxd(x,z+ px) zz The forward transform function for parabolic kinematic trajectories is as z3 follows:
za f~t,x,z,P~=t-z-px~
zs where z6 S[f (t, x, z, P)] = S(t - z - Px2 ) = l, when t = z + pxz , and z8 = 0, elsewhere.

~ Thus; the forward parabolic Radon transformation equation becomes z u(p,z) = f dxd(x,z+ px') The forward transform function for hyperbolic kinematic travel time a trajectories is as follov,~s:
f.(t~x~z~p)-t - zz +pZx?
6 where z S[.f (t~ x~z~ P)] = d(t -1 zz + Pzx2 ) s ~ = l, when t = z2 +p~x2 , and = 0, elsewhere.
~o Thus, the forward hyperbolic Radon transformation equation becomes ~ i u(p, z) _ ~dxd (x, zz + p2x2 ) iz A general case inverse Radon transformation equation is set forth below:
i3 . d(x,t~= jdp Jdzp(z)*u(p,z)8[g(t,x,z,p)] (inversetransformation) 4 where ~s g(t,x,zp) = inverse transform function, and p(z) * = convolution of rho filter.
m The inverse transform function for linear slant stack kinematic trajectories is ~ s as follows:
g(t,x,z,p)=z-t+px zo Thus, when z = t - px , the inverse linear slant stack Radon transformation equation z~ becomes zz d(x~t)= ~dPP(z)'~(P~t-Px) z3 The inverse transform function for parabolic trajectories is as follows:
za g(t,x,z,p)=z-t+px2 ~ Thus, when r = t - px' , the inverse parabolic stack Radon transformation eduation z becomes ~(x~t)= ~dPP(z)*u(P~t-Px-) a The inverse transform function for hyperbolic trajectories is as follows:
s grt ~ r pl - ~ - t~ - p2x~
6 Thus, when z = t' - p'x' , the inverse hyperbolic Radon transformation equation ~ becomes s d(x~t)= f dPP(z)'~(P~ t2 -Pzx~) 9 The choice of which form of Radon transformation preferably is guided by the ~o travel time trajectory at which signals of interest in the data are recorded. Common > > midpoint gathers, because they offer greater accuracy by measuring a single point ~z from multiple source-receiver pairs, are preferred over other types of gathers. Primary ~3 reflection signals in a common midpoint gather generally will have hyperbolic ~a kinematic trajectories. Thus, it would be preferable to use hyperbolic Radon ~s transforms.
To date, however, Radon transformations based on linear slant stack and parabolic trajectories have been more commonly used. The transform function for ~ s hyperbolic trajectories contains a square root whereas those for linear slant stack and ~9 parabolic transform functions do not. Thus, the computational intensity of hyperbolic zo Radon transformations has in large part discouraged their use in seismic processing.
z~ It has not been practical to accommodate the added complexity of hyperbolic zz Radon transformations because the computational requirements of conventional z3 processes are already substantial. NMO correction involves a least-mean-squares za ("LMS") energy minimization calculation. Forward and inverse Radon transforms zs also are not exact inverses of each other. Accordingly, an additional LMS
calculation, z6 conjugate gradient and/or sparsity constraints are often used in the transformation.
z~ Those calculations in general and, especially LMS analyses, are complex and require zs substantial computing time, even on advanced computer systems.

Moreover, a typical seismic survey may involve hundreds of sources and z receivers, thus generating tremendous quantities of raw data. The data may be 3 subjected to thousands of different data gathers. Each gather is subjected not only to a filtering processes as described above, but in all likelihood to many other s enhancement processes as well. For example, data is typically processed to ~ compensate for the diminution in amplitude as a signal travels through the earth z ("amplitude balancing"). Then, once the individual gathers have been filtered, they s must be "stacked", or compiled with other gathers, and subjected to further processing 9 in order to make inferences about subsurface formations. Seismic processing by prior ~o art Radon methods, therefore, requires substantial and costly computational resources, ~ i and there is a continuing need for more efficient and economical processing methods.
~z It is understood, of course, that the transformation, NMO correction, and 13 various other mathematical processes used in seismic processing do not necessarily ~a operate on all possible data points in the gather. Instead, and more typically, those ~s processes may simply sample the data. For example, the transform functions in 6 Radon transformations are based on variables t, x, z, and p. The transformations, however, will not necessarily be performed at all possible values for each variable.
~ a Instead, the data will be sampled a specified number of times, where the number of 9 samples for a particular variable may be referred to as the index for the variable set.
zo For example, k, l, m, and j may be defined as the number of samples in, or the index z~ of, respectively, the time (t), offset (x), tau (z), and slowness (p) sets.
The samples zz will be taken at specified intervals, ~t , ~x , D z , and 0p . Those intervals are z3 referred to as sampling variables.
za The magnitude of the sampling variables, the variable domain and all other zs factors being equal, will determine the number of operations or samples that will be z6 performed in the transformation and the accuracy of the transformation. As the z~ sampling variables become finer, i.e., smaller, the number of samples increases and so za does the accuracy. By making the sampling variables larger, or coarser, the number of z9 samples is reduced, but the accuracy of the transformation is reduced as well.
3o In large part because of the computational intensity of those processes, 3~ however, the sampling variables used in prior art Radon processes can be relatively sz coarse. In NMO correction, for example, industry typically samples at Ot values ~ several time greater than that at which the field data was recorded, i.e., in the range of z 20-40 milliseconds, and samples at ,w values of from about 1~ to about 120 m/sec.
To the extent that the sampling variables are greater than the data acquisition sample a rates, data is effectively discarded and the accuracy of the process is diminished. In s the Radon transformation itself, prior art methods typically set dt, di, and dx at the ~ corresponding sample rates at which the data was recorded, i.e., in the range of 2 to 4 ~ milliseconds and 25 to 400 meters or more, respectively. 0p is relatively coarse, s however, typically being set at values in the range of 4-24 ,usec/m. Also, the index j ~ of the slowness (p) set usually is equal to the fold of the offset data, i.e., the number of ~o source-receiver pairs in the survey, which Typically ranges from about 20 to about 120.
~ i Accordingly, prior art methods may rely on additional computations in an ~2 attempt to compensate for the loss of accuracy and resolution inherent in coarse ~3 sampling variables. For example, the LMS analysis applied in NMO correction and ~a prior art Radon transformations are believed to obviate the need for finer sampling ~ s variables. Certain prior art processes utilize trace interpolation, thereby nominally i6 reducing ~.a: values, but the additional "data" is an approximation of what would be a recorded at a particular offsets between the offsets where receivers were actually ~s located. Relative inaccuracy in processing individual data gathers also is believed to ~9 be acceptable because the data from a large number of gathers ultimately are zo combined for analysis, or "stacked". Nevertheless, there also is a continuing need for zi more accurate processing of seismic data and processes having finer resolutions.
zz The velociTy at which primary reflection signals are traveling, what is referred z3 to as the stacking velocity, is used in various analytical methods that are applied to za seismic data. For example, it is used in determining the depth and lithology or zs sediment type of geological .formations in the survey area. It also is used various zb seismic attribute analyses. A stacking velociTy function may be determined by what is z~ referred to as a semblance analysis. Which such methods have provided reasonable zs estimations of the stacking velociTy function for a data set, given the many z9 applications based thereon, it is desirable to define the stacking velocity function as 3o precisely as possible.

An object of this invention, therefore, is to provide a method for processing z seismic data that more effectively removes unwanted noise from meaningful reflection 3 signals.
a It also is an object to provide such methods based on hyperbolic Radon s transformations.
6 Another object of this invention is to provide methods for removal of noise ~ from seismic data that are comparatively simple and require relatively less computing a time and resources.
Yet another object is to provide uethods for more accurately defining the ~o stacking velocity fimction for a data set.
> > It is a further object of this invention to provide such methods wherein all of ~z the above-mentioned advantages are realized.
Those and other objects and advantages of the invention will be apparent to 4 those skilled in the art upon reading the . following detailed description and upon ~ s reference to the drawings.
SUMMARY OF THE INVENTION
m The subject invention provides for methods of processing seismic data to ~ s remove unwanted noise from meaningful reflection signals indicative of subsurface 19 formations. The methods comprise the steps of obtaining field records of seismic data zo detected at a number of seismic receivers in an area of interest. The seismic data zi comprises signal amplitude data recorded over time by each of the receivers and zz contains both primary reflection signals and unwanted noise events. The seismic data z3 is assembled into common geometry gathers in an offset-time domain.
za The amplitude data is then transformed from the offset-time domain to the zs time-slowness domain using a Radon transformation. Preferably, the Radon z6 transformation is applied within defined slowness limits p",;" and p",ax, where p",;" is a z~ predetermined minimum slowness and p",p, is a predetermined maximum slowness, z8 and an offset weighting factor x" is applied to the amplitude data, wherein 0 < n < 1.
z9 Such Radon transformations include the following continuous transform 3o equation, and discrete versions thereof that approximate the continuous transform 3~ equation:

u(p,z) = f dx Jdtd(x,t)x"S[f (t, x, z, p)]
v, -z Radon transformations based on linear slant stack, parabolic, and other non-3 hyperbolic kinematic travel time trajectories may be used, but those based on n hyperbolic Radon kinematic trajectories are preferred. Suitable hyperbolic Radon s transformations include the following continuous transform equation, and discrete 6 versions thereof that approximate the continuous transform equation:
u(p,z) = jdax"d(x, zz + pZx' ) s A high resolution Radon transformation preferably is used, where the 9 transformation is performed according to an index j of the slowness set and a ~o sampling variable 0p ; wherein Pmax - Pmin + l,u sec/ m j - DP
~z Op is from about 0.5 to about 4.0 psec/m, p",~ is a predetermined maximum t3 slowness, and p",;" is a predetermined minimum slowness.
A corrective filter is then applied to the transformed data in order to enhance ~ s the primary reflection signal content of the data and to eliminate unwanted noise 6 events. Preferably, the corrective filter is a high-low filter, and most preferably a time variant, high-low ftlter, and the filter is determined by reference to the velocity to function derived from performing a semblance analysis or a pre-stack time migration i9 analysis on the amplitude data.
zo After filtering, the enhanced signal content is inverse transformed from the z~ time-slowness domain back to the offset-time domain using an inverse Radon zz transformation, and, if necessary, an inverse of the offset weighting factor pn is z3 applied to the inverse transformed data, wherein 0 < n < l . Such Radon z4 transformations include the following continuous inverse transform equation, and zs discrete versions thereof that approximate the continuous inverse transform equation;
z6 d(x~ t) _ [dp ~dzP~tP(z)'ru(P~ z)S~g(t~ x~ z~ P)j _J~

Suitable hyperbolic inverse Radon transformations include the following a continuous inverse transform equation, and discrete versions thereof that approximate 3 the continuous inverse transform equation:
a d(x,t) = f dpp"p(T) *u(p, t'' - p'x' ) s The signal amplitude data for the enhanced signal content is thereby restored.
6 The processed and filtered data may then be subject to further processing by which ~ inferences about the subsurface geology of the survey area may be made.
a It will be appreciated that primarily because they utilize an offset weighting 9 factor and its inverse, because they limit the Radon transformation, and because they ~o avoid more complex mathematical operations required by prior art processes, the > > novel methods require less total computation time and resources.
i2 It will be appreciated that by limiting the transformation within defined ~3 slowness limits p",;" and p",px, the novel processes provide increased efficiency by i4 reducing the amount of data transformed and increase efficacy by decimating the noise ~ s signals that fall outside those limits. In general, p",px is greater than the slowness of 6 reflection signals from the shallowest reflective surface of interest. The lower i~ transformation limit, p",in, generally is less than the slowness of reflection signals from ~ s the deepest reflective surface of interest. Typically, the upper and lower i9 transformation limits pm;" and pM~ will be set within 20% below (for lower limits) or zo above (for upper limits) of the slownesses of the pertinent reflection signals, or more z~ preferably, within 10% or 5% thereof.
zz At the same time, because overall they are computationally less intensive, the z3 sampling variables used in the transformations and semblance analyses may be set z4 much finer than are typically used in prior art processes. That is, the Radon zs transformations utilized in the subject invention offer higher resolution as compared z6 to prior art Radon filters, thereby enhancing. the accuracy of the process.
~p typically z~ is from about 0.5 to about 4.0 Psec/m. A 0p of about 1.0 ,usec/m is most preferably zs used. Slowness limit p",Q~, which is used to determine the index j of the slowness set, z9 generally is greater than the slowness of reflection signals from the shallowest so reflective surface of interest. The minimum slowness limit, p",;", generally is less than 3i the slowness of reflection signals from the deepest reflective surface of interest.

r Typically, the minimum and maximum limits p",;" and p",~. will be sat within 20%
z below (for minimum limits) or above (for maximum limits) of the slownesses of the 3 pertinent reflection signals, or more preferably, within 10% or 5% thereof.
The a specific values for the slowness limits p",;" and p",4X will depend on the data recorded s in the survey, but typically, therefore, j preferably is from about 125 to about 1000, s most preferably from about 250 to about 1000.
It will be appreciated that the improved tau-P filters utilized in the subject a invention more effectively enhance primary reflection signals and attenuate noise as 9 compared to prior art Radon filters. The high-low filter is defined by reference to the io velocity function for primary reflection signals, as determined, for example, through a 11 semblance analysis. The velocity function and pass region will transform to a lz slowness function and pass region. Preferably, the slowness pass region is defined as:

Va,~l+YZ~ C p C vr~l Yl la where n, and Y~ are percentages expressed as decimals. The slowness limits typically 1 s are set within ~ 15%, preferably within ~ 10%, and most preferably within t5% of the 16 velocity function, and ri and r, may be set accordingly.
1~ Since the velocity function for the primary reflection signals is time variant, 1 a the filter limits preferably are time variant as well. In that manner, the filter may more 19 closely fit the reflection's velocity function and, therefore, more effectively isolate zo primary reflection signals and water bottom reflection signals and remove multiple zl reflection signals and other noise. Moreover, since the strongest noise signals zz typically occur at velocities slower than primary reflection signals, it is preferable that z3 v",;" lower limit more closely approximate the velocity function of the primary as reflection signals. A
zs Thus, even though hyperbolic Radon transformations heretofore have z6 generally been avoided because of their greater complexity relative to linear slant z~ stack and parabolic Radon transformations, the novel processes are able to more zs effectively utilize hyperbolic Radon transformations and take advantage of the greater z9 accuracy they can provide, especially when applied to common midpoint geometry 3o gathers.

The methods of the subject invention preferably are implemented by computers and other conventional data processing equipment. Accordingly, the subject invention also provides for methods of processing seismic data in a computer system and for generating an output signal to provide a display of the processed data and apparatus for processing seismic data. Because the methods of the subject invention remove noise that otherwise might interfere with accurate interpretation of seismic data, the subject invention also provides for methods for selecting a drilling site which comprises processing and analyzing the seismic data to search for subsurface formations of interest and drilling in the location of interest.
The subject invention also provides for methods for determining a stacking velocity function. The methods comprise the steps of processing and filtering the data as described above. A semblance analysis is performed on the restored data to generate a refined semblance plot, and a velocity analysis is performed on the refined semblance plot to define a refined stacking velocity function. Preferably, an offset weighting factor x" is first applied to the restored amplitude data, wherein 0<n<1.
It will be appreciated that, because the semblance and velocity analysis is performed on data that has been more accurately processed and filtered, that the resulting stacking velocity function is more accurate. This enhances the accuracy of the many different seismic processing methods that utilize stacking velocity functions.
Ultimately, the increased accuracy and efficiency of the novel processes enhances the accuracy of surveying underground geological features and, therefore, the likelihood of accurately locating the presence of oil and gas deposits.
According to one aspect of the present invention, there is provided a method of processing seismic data to remove unwanted noise from meaningful reflection signals indicative of subsurface formations, wherein said seismic data comprise amplitude data recorded over time at a number of seismic receivers in an area of interest, said amplitude data are assembled into common geometry gathers in an offset-time domain, and said amplitude data comprise primary reflection signals and coherent noise that overlap in an offset-time domain; said method comprising the step of enhancing the separation between primary reflection signals and coherent noise by transforming said data from the offset-time domain to the time-slowness domain, wherein:
(a) an offset weighting factor x" is applied to said amplitude data, wherein 0<n<1; and (b) said offset weighted amplitude data are transformed from the offset-time domain to the time-slowness domain using a Radon transformation;
(c) whereby said primary reflection signals and said coherent noise transform into different regions of the time-slowness domain to enhance the separation therebetween.
According to another aspect of the present invention, there is provided an apparatus for processing seismic data to remove unwanted noise from meaningful reflection signals indicative of subsurface formations; said apparatus comprising a storage device and a processor connected to the storage device, the storage device storing a program for controlling the processor, and the processor operative with the program to: (a) receive computer seismic data detected at a number of seismic receivers in an area of interest, said seismic data comprising signal amplitude data 14a recorded over time and containing both primary reflection signals and unwanted noise events; (b) assemble said seismic data into common geometry gathers in an offset-time domain;
(c) apply an offset weighting factor x° to said amplitude data, wherein 0 < n < 1 ; (d) transform said offset weighted amplitude data from the offset-time domain to the time-slowness domain using a Radon transformation; (e) apply a corrective filter to enhance the primary reflection signal content of the data and to eliminate unwanted noise events;
(f) inverse transform said enhanced signal content from the time-slowness domain back to the offset-time domain using an inverse Radon transformation; and (g) apply an inverse of the offset weighting factor p" to said inverse transformed data, wherein 0<n<1, thereby restoring the signal amplitude data for said enhanced signal content.
According to still another aspect of the present invention, there is provided a method of selecting a drilling site to access a subsurface formation, the method comprising the steps of: (a) obtaining field records of seismic data detected at a number of seismic receivers in an area of interest, said seismic data comprising signal amplitude data recorded over time and containing both primary reflection signals and unwanted noise events; (b) processing the seismic data to search for the presence of a subsurface formation of interest, said processing removing unwanted noise from meaningful reflection signals indicative of subsurface formations and comprising the steps of: i) assembling said seismic data into common geometry gathers in an offset-time domain; ii) applying an offset weighting factor x" to said amplitude data, wherein 0 < n < 1 ; iii) transforming said offset weighted amplitude data from the offset-time domain to the time-slowness domain using a Radon transformation; iv) applying a corrective filter to enhance 14b the primary reflection signal content of the data and to eliminate unwanted noise events;-v) inverse transforming said enhanced signal content from the time-slowness domain back to the offset-time domain using an inverse Radon transformation; and vi) applying an inverse of the offset weighting factor p" to said inverse transformed data, wherein 0<n<l, thereby restoring the signal amplitude data for said enhanced signal content; and (c) drilling at a location likely to access the subsurface formations indicated by said processing steps.
BRIEF DESCRIPTION OF THE DRAWINGS
FIG. 1 is a schematic diagram of a preferred embodiment of the methods of the subject invention showing a sequence of steps for enhancing the primary reflection signal content of seismic data and for attenuating unwanted noise events, thereby rendering it more indicative of subsurface formations.
FIG. 2 is a schematic diagram of a two-dimensional seismic survey in which field records of seismic data are obtained at a number of seismic receivers along a profile of interest.
FIG. 3 is a schematic diagram of a seismic survey depicting a common midpoint geometry gather.
14c FIG. 4 is a plot or display of model seismic data in the offset-time domain of a z common midpoint gather containing overlapping primary reflections, a water-bottom 3 reflection, and undesired multiple reflections.
n FIG. 5 is a plot of the model seismic data of FIG. 4 after transformation to the s time-slowness domain using a novel high resolution hyperbolic Radon transform in 6 accordance with the subject invention that shows the separation of primary and water-~ bottom reflections from multiple reflections.
s FIG. 6 is a plot or display in the offset-time domain of the model seismic data 9 of FIG. 4 after applying a time variant, high-low filter in the time-slowness domain ~o and inverse transforming the filtered data that shows the retention of primary and ~ r water-bottom reflections and the elimination of multiple reflections.
~z FIG. 7 is a plot in the offset-time domain of field records of seismic data ~3 recorded along a two-dimensional profile line in the West Texas region of the United ~a States and assembled in a common midpoint geometry gather.
is FIG. 8 is a plot in the offset-time domain of the seismic data of FIG. 7 after i6 having been processed and filtered with a high resolution Radon transformation and a m time variant, high-low filter in accordance with the subject invention.
~ s FIG. 9 is a plot in the offset-time domain of data processed and filtered with a i9 high resolution Radon transformation and a time variant, high-low filter according to 2o the subject invention, including the data of FIG. 8, that have been stacked in z~ accordance with industry practices.
zz FIG. 10 (prior art) is in contrast to FIG. 9 and shows a plot in the offset-time z3 domain of unfiltered, conventionally processed data, including the unprocessed data of 2a FIG. 7, that have been stacked in accordance with industry practices.
zs FIG. 11 is a plot in the offset-time domain of field records of seismic data z6 recorded along a two-dimensional profile line in a West African production area and z~ assembled in a common midpoint geometry gather.
zs FIG. 12 is a semblance plot of the seismic data of FIG. 11.
z9 FIG. 13 is a plot in the offset-time domain of the seismic data of FIG. 11 after 3o having been processed and filtered with a limited, high resolution Radon 3~ transformation and a time variant, high-low filter in accordance with the subject 3z invention.

FIG. 14 is a semblance plot of the processed seismic data of FIG. 13.
z FIG. 15 is a plot in the offset-time domain of data processed and filtered with 3 a limited, high resolution Radon transformation and a time variant, high-low filter a according to the subject invention, including the data of FIG. 13, that have been s stacked in accordance with industry practices using a stacking velocity function 6 determined from the semblance plot of FIG. 14.
FIG. 16 (prior art) is in contrast to FIG. 15 and shows a plot in the offset-8 time domain of unfiltered, conventionally processed data, including the unprocessed 9 data of FIG. 11, that have been stacked in accordance with industry practices.
~o FIG. 17 is a plot in the offset-time domain of field records of seismic data > > recorded along a two-dimensional profile line in the Viosca Knoll region of the United z States and assembled in a common midpoint geometry gather.
FIG. 18 is a semblance plot of the seismic data of FIG. 17.
~a FIG. 19 is a plot in the offset-time domain of the seismic data of FIG. 17 after is having been processed and filtered with a limited, high resolution Radon 6 transformation and a time variant, high-low filter in accordance with the subject invention.
~ s FIG. 20 is a semblance plot of the processed seismic data of FIG. 19.
19 FIG. 21 is a plot in the offset-time domain of data processed and filtered with zo a limited, high resolution Radon transformation and a time variant, high-low filter zi according to the subject invention, including the data of FIG. 19, that has been zz stacked in accordance with industry practices using a stacking velocity function z3 determined from the semblance plot of FIG. 20.
za FIG. 21A is an enlargement of the area shown in box 21A of FIG. 21.
is FIG. 22 (prior art) is in contrast to FIG. 21 and shows a plot in the offset-z6 time domain of data, including the data of FIG. 17, that was processed and filtered in z~ accordance with conventional parabolic Radon methods and then stacked in zs accordance with industry practices.
z9 FIG. 22A (prior art) is an enlargement of the area shown in box 22A of FIG.
30 22.

DESCRIPTION OF ILLUSTRATIVE EMBODIMENTS
2 The subject invention is directed to improved methods for processing seismic 3 data to remove unwanted noise from meaningful reflection signals and for more a accurately determining the stacking velocity function for a set of seismic data.
s Obtaining and Preparing Seismic Data for Processing s More particularly, the novel methods comprise the step of obtaining field ~ records of seismic data detected at a number of seismic receivers in an area of interest.
s The seismic data comprises signal amplitude data recorded over time and contains 9 both primary reflection signals and unwanted noise events.
~o By way of example, a preferred embodiment of the methods of the subject > > invention is shown in the flow chart of FIGURE 1. As shown therein in step 1, ~z seismic amplitude data is recorded in the offset-time domain. For example, such data 3 may be generated by a seismic survey shown schematically in FIG. 2.
~a The seismic survey shown in FIG. 2 is a two-dimensional survey in offset-is time (X-T) space along a seismic line of profile L. A number of seismic sources S"
~s and receivers R~ are laid out over a land surface G along profile L. The seismic m sources S" are detonated in a predetermined sequence. As they are discharged, ~ s seismic energy is released as energy waves. The seismic energy waves or "signals"
9 travel downward through the earth where they encounter subsurface geological zo formations, such as formations F~ and Fz shown schematically in FIG. 2. As they do, z1 a portion of the signal is reflected back upwardly to the receivers R~. The paths of zz such primary reflection signals from S1 to the receivers R" are shown in FIG. 2.
z3 The receivers R" sense the amplitude of incoming signals and digitally record za the amplitude data over time for subsequent processing. Those amplitude data zs recordations are referred to as traces. It will be appreciated that the traces recorded by zb receivers R" include both primary reflection signals of interest, such as those shown in z~ FIG. 2, and unwanted noise events.
za It also should be understood that the seismic survey depicted schematically in z9 FIG. 2 is a simplified one presented for illustrative purposes. Actual surveys typically so include a considerably larger number of sources and receivers. Further, the survey 3~ may taken on land or over a body of water. The seismic sources usually are dynamite 3z charges if the survey is being done on land, and geophones are used. Air guns are ~ typically used for marine surveys along with hydrophones. The survey may also be a z three-dimensional survey over a surface area of interest rather than a two-dimensional s survey along a profile as shown.
a In accordance with the subject invention, the seismic data is assembled into s common geometry gathers in an offset-time domain. For example, in step 2 of the 6 preferred method of FIG. 1, the seismic amplitude data is assembled in the offset-time ~ domain as a common midpoint gather. That is, as shown schematically in FIG.
3, s midpoint CMP is located halfway between source sl and receiver rl. Source s2 and 9 receiver rZ share the same midpoint CMP, as do all pairs of sources s" and receivers ~o r" in the survey. Thus, it will be appreciated that all source s~ and receiver r~ pairs are > > measuring single points on subsurface formations F". The traces from each receiver 12 r" in those pairs are then assembled or "gathered" for processing.
It will be appreciated, however, that other types of gathers are known by 4 workers in the art and may be used in the subject invention. For example, the seismic ~ s data may be assembled into common source, common receiver, and common offset 6 gathers and subsequently processed to enhance signal content and attenuate noise.
It will be appreciated that the field data may be processed by other methods for ~ s other purposes before being processed in accordance with the subject invention as 9 shown in step 3 of FIG. 1. The appropriateness of first subjecting the data to zo amplitude balancing or other conventional pre-processing, such as spherical zi divergence correction and absorption compensation, will depend on various geologic, zz geophysical, and other conditions well known to workers in the art. The methods of zs the subject invention may be applied to raw, unprocessed data or to data preprocessed 24 by any number of well-known methods.
zs As will become apparent from the discussion that follows, however, it is 26 important that the data not be NMO corrected, or if preprocessed by a method that z~ relies on NMO correction, that the NMO correction be reversed, prior 'to zs transformation of the data. Otherwise, it is not practical to design and apply the high-z9 low filters of the subject invention. Importantly, however, because the methods of the so subject invention do not contemplate the use of NMO correction, the overall ~ efficiency of the novel processes is improved. NMO correction requires an LMS
sz analysis, and typically is followed by another LMS analysis in the Radon ~ transformation, both of which require a large number of computations. It also is z possible to avoid the loss of resolution caused by the use of coarse sampling variables 3 in NMO correcting.
a Transformation of Data Once the data is assembled, the methods of the subject invention further 6 comprise the step of transforming the offset weighted amplitude data from the offset-~ time domain to the time-slowness domain using a Radon transformation.
Preferably, a an offset weighting factor x" is applied to the amplitude data to equalize amplitudes 9 in the seismic data across offset values and to emphasize normal amplitude moveout ~o differences between desired reflection signals and unwanted noise, wherein 0 < n < 1 .
> > It also is preferred that the Radon transformation be applied within defined iz slowness limits p",;" arid p",ax, where p",;" is a predetermined minimum slowness and 3 p",pa is a predetermined maximum slowness. By limiting the transformation in that ~a manner, it is more efficient and effective. The slowness limits p",;" and p",~ preferably ~s are determined by reference to a velocity function derived by performing a semblance 6 analysis or a pre-stack time migration analysis on the amplitude data.
a The transformation also is preferably performed with a high resolution Radon ~ a transformation having an index j of the slowness set and a sampling variable ~p , i9 wherein zo __ pm~ - pma + l~ sec/ m J
zWp is from about 0.5 to about 4.0 ~sec/m, p",px is a predetermined maximum zz slowness, and p",;" is a predetermined minimum slowness. Op preferably is from z3 about 0.5 to about 3.0 ,usec/m, and more preferably, is from about 0.5 to about 2.0 za ,usec/m. Op of about 1.0 ,usec/m is most preferably used. This provides a finer zs resolution transformation and, therefore, better resolution in the filtered data.
z6 For example, in step 4 of the exemplified method of FIG. 1, an offset z~ weighting factor is applied to the data that was assembled in step 2. A
semblance za analysis is performed on the offset weighted amplitude data to generate a semblance z9 plot, as shown in step 6. Then, in step 17 the semblance plot is used to define the 3o slowness limits p",;" and p",px that will be applied to the transformation.
The offset r weighted data is then transformed with a high resolution hyperbolic Radon z transformation according to the transform limits in step 5. It will be appreciated, 3 however, that the Radon transformation of the offset weighted data may be may be a based on linear slant stack, parabolic, or other non-hyperbolic kinematic travel time s trajectories.
s The slowness transformation limits p",;" and p",Q, are defined by reference to ~ the velocity function for primary reflection signals as determined, for example, a through the semblance analysis described above. High and low transformation limits, ~ i.e., a maximum velocity (v",~~) and minimum velocity (v",;"), are defined on either side ~o of the velocity function. The velocity limits then are converted into slowness limits > > p",;" and p",pa which will limit the slowness domain for the transformation of the data iz from the offset-time domain to the tau-P domain, where p",;" = 1/v",ax and p",Gx = 1/v",;".
13 When the semblance analysis is performed on amplitude data that has been is offset weighted, as described above, appropriate slowness limits p""" and P",~ may be is more accurately determined and, therefore, a more efficient and effective 6 transformation may be defined. It will be appreciated, however, that the semblance m analysis may be performed on data that has not been offset weighted.
Significant to improvements in computational efficiency still will be achieved by the subject 9 invention.
zo In general, p",a~ is greater than the slowness of reflection signals from the zt shallowest reflective surface of interest. In marine surveys, however, it typically is z2 desirable to record the water bottom depth. Thus, p",~ may be somewhat greater, i.e., z3 greater than the slowness of reflective signals through water in the area of interest.
za The lower transformation limit, p",;", generally is less than the slowness of reflection zs signals from the deepest reflective surface of interest. Typically, the upper and lower 26 transformation limits p",;" and p",~ will be set within 20% below (for lower limits) or z~ above (for upper limits) of the slownesses of the pertinent reflection signals, or more zs preferably, within l0% or 5% thereof. While the specific values will depend on the z9 data recorded in the survey, therefore, p",;" generally will be less than about 165 30 ,usec/m and, even more commonly, less than about 185 ,usec/m. Similarly, p",ax 3t generally will be greater than about 690 ,usec/m, and even more commonly, greater 3z than about 655 ,usec/m for marine surveys. For land surveys, p",~,.
generally will be ~ greater than about 3,125 psec/m, and even more commonly, greater than about z ~csec/m .
3 It will be appreciated that by limiting the transformation, the novel processes a provide increased efficiency by reducing the amount of data transformed and increase s efficacy by decimating the noise signals that fall outside those limits.
Prior art Radon 6 methods do not incorporate any effective limits to their transformations. An upper ~ limit is necessarily set, but it is typically well beyond the limits of the data and, a a fortioni, even further beyond the slowness of the shallowest reflective surface of 9 interest. Since they rely on mute filters in the tau-P domain, such prior art methods ~o prefer to transform any and all signals that transform into higher slowness regions.
> > Those higher slowness signals correspond in large part to multiple reflection signals ~z and other noise. That noise will be preserved outside the mute filter and removed ~3 from the original data set in the subtraction process. It is important, therefore, that ra there be no effective upper slowness limits on the transform since that will ensure the is removal of the greatest amount of noise in the subtraction process.
~s Moreover, prior art radon transformations do not apply a lower slowness limit to the transformation. Those processes design their mute filter in the tau-P
domain by ~a reference to the transformed primary reflection signals and the high amplitude noise 9 transforming in the vicinity thereof. Thus, prior art methods prefer to transform as Zo much data as possible so that they may more accurately define the mute filter.
When the data is not subject to NMO correction, as contemplated by the zz subject invention, signals for primary reflection events do not transform at or near z3 zero slowness, i.e., they transform above a definable p",;". It is possible, therefore, to za apply a lower limit to the transformation that will preserve primary reflections signals.
zs Moreover, the strongest noise signals typically transform at slowness greater than z6 those of primary reflection signals, and corrective filters that are applied to z~ transformed data in the tau-P domain may not completely attenuate high amplitude zs noise. In the practice of the subject invention, however, all noise above the defined zy upper transformation limit p",~ or below the defined lower transformation limit p""" is 3o decimated regardless of its amplitude.

A general mathematical formulation utilizing the offset weighting factor and z encompassing the linear, parabolic, and hyperbolic forward Radon transforms is as 3 follows:
a (1) u(p,z) = jdx ~dtd(x,t)x"d[f (t,x,z, p)] (forward transformation) -d, s where .
b u(p,i) = transform coefficient at slowness p and zero-offset time z d(x, t) = measured seismogram at offset x and two-way time t s x" = offset weighting factor ( 0 < n < 1 ) 8 = Dirac delta function ~o f(t,x,z p) = forward transform function > > The forward transform function for hyperbolic trajectories is as follows:
~z f~t,x,z,p~=t- zz+pzxz 3 Thus, when t = zz +pzxz , the forward hyperbolic Radon transformation equation ~a becomes ~s u(p,z) = Jdxx"d(x, zz + pzxz) The forward transform function for linear slant stack kinematic trajectories is as follows:
Is f~t,x,z,p)=t-z-px i9 Thus, when t = z + px , the forward linear slant stack Radon transformation equation zo becomes 2 i u( p, z) = f dxx"d (x, z + px) zz The forward transform function for parabolic trajectories is as follows:
zs f(t,x,z,p)=t-z-pxz za Thus, when t = z+ pxz, the forward parabolic Radon transformation equation zs becomes m(p,r) = f dxx"d(x,r+ px-) z As with conventional Radon transformations, the function _f (t, x, z, p~
allows 3 kinematic travel time trajectories to include anisotropy, P-S converted waves, wave-a field separations, and other applications of current industry that are used to refine the s analysis consistent with conditions present in the survey area. Although hyperbolic 6 travel time trajectories represent more accurately reflection events for common ~ midpoint gathers in many formations, hyperbolic Radon transformations to date have s not been widely used. Together with other calculations necessary to practice prior art 9 processes, the computational intensity of hyperbolic Radon transforms tended to make io such processing more expensive and less accurate. Hyperbolic Radon ~ i transformations, however, are preferred in the context of the subject invention because ~z the computational efficiency of the novel processes allows them to take advantage of 3 the higher degree of accuracy provided by hyperbolic travel time trajectories.
~a It will be noted, however, that the novel Radon transformations set forth above is in Equation 1 differ from conventional Radon transformations in the application of 6 an offset weighting factor x", where x is offset. The offset weighting factor m emphasizes amplitude differences that exist at increasing offset, i.e., normal amplitude ~s moveout differences between desired primary reflections and multiples, linear, and 9 other noise whose time trajectories do not fit a defined kinematic function.
Since the zo offset is weighted by a factor n that is positive, larger offsets receive preferentially z~ larger weights. The power n is greater than zero, but less than 1.
Preferably, n is zz approximately 0.5 since amplitudes seem to be preserved better at that value. While z3 the power n appears to be robust and insensitive, it probably is data dependent to some za degree.
zs While the application of an offset weighting factor as described above .is z6 preferred, it will be appreciated that other methods of amplitude balancing may be z~ used in the methods of the subject invention. For example, automatic gain control za (AGC) operators may be applied. A gain control operator tnay be applied where the z9 gain control operator is the inverse of the trace envelope for each trace as disclosed in ~ U.S. Patent No. 5, I 89,644 to Lawrence C. Wood and entitled "Removal of Amplitude z Abasing Effect From Seismic Data".
3 As will be appreciated by workers in the art, execution of the transformation a equations discussed above involve numerous calculations and, as a practical matter, s must be executed by computers if they are to be applied to the processing of data as 6 extensive as that contained in a typical seismic survey. Accordingly, the ~ transformation equations, which are expressed above in a continuous form, preferably s are translated into discrete transformation equations which approximate the solutions 9 provided by continuous transformation equations and can be encoded into and ~o executed by computers.
> > For example, assume a seismogram d(x,t) contains 2L+I traces each having N
~z time samples, i.e., 13 l = 0,~1,...,~L and 14 k = I,..., N
~ s and that ~ 6 x_L < x_L+~ < ... < xL-, < xL
a A discrete general transform equation approximating the continuous general transform ~s Equation 1 set forth above, therefore, may be derived as set forth below:
L N
~ 9 (2) u(p, z) _ ~ ~ d (xn tk )x~"sL.f (tk ~ xn z~ P)~~r~tk I=-L A=1 zo where zW;r = x~+' 2 x~-' for l = 0,~1,...,~(L-I) zz dxL = xL - xL_, 23 ~_L - x_L+I ~-L
za 4tk = tk+' 2 tk-' for k = 2,..., N -1 25 Ol~ = h -l~
26 ~IN = tN - tN-~
z~ By substituting the hyperbolic forward transform function set forth above, the za discrete general forward transformation Equation 2 above, when t = .~TZ +
p2x' , ~ may be reduced to the discrete transformation based on hyperbolic kinematic travel z time trajectories that is set forth below:
u(p,z) _ ~xl~~d(xl, z'' + p'xl'-)~.r r- L
a Similarly, when t = z+ px , the discrete linear slant stack forward s transformation derived from general Equation 2 is as follows:
L
u(p,z) - ~xr"d(xr,z+ pxr)y I=-L
When t = z + px' , the discrete parabolic forward transformation is as follows:
L
s ar(p,z) _ ~x,'~d(xl,z+ pxr )Oxr r=_L
Those skilled in the art will appreciate that the foregoing discrete ~o transformation Equation 2 sufficiently approximates continuous Equation 1, but still ~ i may be executed by computers in a relatively efficient manner. For that reason, the ~z foregoing equation and the specific cases derived therefrom are preferred, but it is ~3 believed that other discrete transformation equations based on Radon transformations ~a are known, and still others may be devised by workers of ordinary skill for use in the ~s subject invention.
ib It will be appreciated, however, that the subject invention provides for the o transformation of data with a high resolution Radon filter. That is, preferably the ~s transformation is performed according to an index j of the slowness set and a ~9 sampling variable 0p , wherein Pm~ - p~n~n + l,u sec/ m zo j = Qp , z~ p",p, is a predetermined maximum slowness, and p",;" is a predetermined minimum zz slowness. Thus, the novel methods provide finer resolution transformations and, zs therefore, better resolution in the filtered data.
za More specifically, Op typically is from about 0.5 to about 4.0 ,usec/m. 4p zs preferably is from about 0.5 to about 3.0 ,usec/m, and more preferably, is from about z6 0.5 to about 2.0 ,usec/m. A 4p of about 1.0 ,usec/m is most preferably used.
z~ Slowness limits p",;" and p",~" which are used to determine the index j of the slowness zs set, are generally set in~ the same manner as the slowness limits that preferably are ~ applied to limit the transformation as described above. That is, p",Q~.
generally is z greater than the slowness of reflection signals from the shallowest reflective surface of 3 interest, although in marine surveys it may be somewhat greater, i.e., greater than the a slowness of reflective signals through water in the area of interest, so as to record the s water bottom depth. The minimum slowness limit, p",;", generally i s less than the 6 slowness of reflection signals from the deepest reflective surface of interest.
~ Typically, the minimum and maximum limits p",;" and p",~ will be set within 20%
s below (for minimum limits) or above (for maximum limits) of the slownesses of the 9 pertinent reflection signals, or more preferably, within 10% or 5% thereof.
The ~o specific values for the slowness limits p",;" and p",px will depend on the data recorded a in the survey, but typically, therefore, j preferably is from about 125 to about 1000, ~z most preferably from about 250 to about 1000.
~s Such values are significantly finer than prior art Radon processes, where 0p is ~a typically set at values in the range of 4-24 ,usec/m. Also, the index j of the slowness ~ s set usually is equal to the fold of the offset data, which typically ranges from about 20 ~6 to about 120. The novel processes, therefore, provide a much higher resolution transformation and, to that extent, increased accuracy in the filtering of the data.
a s It also will be appreciated that this increased resolution and accuracy may be i9 achieved without reliance on computationally intensive processing steps, such as trace zo interpolation, LMS analysis in the transformation, or NMO correction of the data prior z~ to transformation, which also requires a LMS analysis. Moreover, it is possible to zz avoid the loss of resolution caused by the use of coarse sampling variables in NMO
zs correcting, i.e., 0t values in the range of 20-40 milliseconds and 0v values of from za about 15 to about 120 m/sec.

Filterinb of Data z The methods of the subject invention further comprise the step of applying a 3 corrective filter to enhance the primary reflection signal content of the data and to a eliminate unwanted noise events. Preferably, the corrective filter is a high-low filter, s most preferably, a time variant, high-low filter. The corrective filter preferably is 6 determined by performing a semblance analysis on the amplitude data to generate a ~ semblance plot and performing a velocity analysis on the semblance plot to define a s stacking velocity function and the corrective flter. The corrective filter also may be 9 determined by performing a pre-stack time migration analysis on the amplitude data to ~o define a velocity function and the corrective filter.
> > For example, in step 6 of the preferred method of FIG. l, a semblance analysis iz is performed on the offset weighted amplitude data to generate a semblance plot.
:3 Then, in step 7 a velocity analysis is performed on the semblance plot to define a ~a stacking velocity function and a time variant, high-low corrective filter.
The ~s corrective filter then is applied to the transformed data as shown in step 8.
i6 When the semblance analysis is performed on amplitude data that has been offset weighted, as described above, the resulting velocity analysis will be more is accurate and, therefore, a more effective filter may be defined. It will be appreciated, i9 however, that the semblance analysis may be performed on data that has not been zo offset weighted. Significant improvements in accuracy and computational efficiency ~ still will be achieved by the subject invention.
zz The high-low filter is defined by reference to the velocity function for primary zs reflection signals, as determined, for example, through the semblance analysis z4 described above. The stacking velocity function, [t~,vs(t~)], describes the velocity of zs primary reflections signals as a function of to. High and low filter limits, i.e., a z6 maximum velocity, v",aa.(to), and a minimum velocity, v"""(to), are defined on either z~ side of the velocity function, for example, within given percentages of the velocity zs function. In such cases, the velocity pass region at a selected time to corresponds to z9 v,,. (1- ~; ) <- v <_ v,~. (1 + rZ ) , where r, and r~ are percentages expressed as decimals. The 3o velocity function will transform to a slowness function, [z, p(T)] , in the tau-P domain.
~ Similarly, the velocity pass region will map into a slowness pass region, namely:

1 1~
r v.,.(1+r;) ~ p ~ v~.(1-r) z for application to the transformed data in the tau-P domain. The slowness limits s typically are set within ~ 1 S%, preferably within ~ 10%, and most preferably within a ~5% of the velocity function, and r, and rz may be set accordingly.
s Since the velocity function for the primary reflection signals is time variant, 6 the filter limits preferably are time variant as well. In that manner, the filter may more ~ closely fit the reflection's velocity function and, therefore, more effectively isolate a primary reflection signals and water bottom reflection signals and remove multiple 9 reflection signals and other noise. Moreover, since the strongest noise signals ro typically occur at velocities slower than primary reflection signals, it is preferable that r r v""" Lower limit more closely approximate the velocity function of the primary is reflection signals.
r3 It will be appreciated, however, that one or both of the upper or lower filter ra limits may, if desired, be made time invariant. Although such filters in general will r s not enhance the primary signal content or attenuate noise as effectively as a suitably r6 designed filter where both the upper and lower limits are time variant, they may be r~ adapted for use in the subject invention.
r s In contrast to prior art Radon methods that rely on mute filters and subtraction r9 processes, the methods and filters of the subject invention are more effective. Mute zo filters are applied to all data below a defined, time invariant slowness value p",ute zr greater than the slownesses of the primary reflections. The slowness of the primary zz reflections is essentially zero when data is NMO corrected prior to transformation, so zs p"""~ typically is set relatively close to zero. Nevertheless, while most of the multiple za reflections signals will have slowness greater than p"",,e and, therefore, ultimately will zs be subtracted froth the data, the near trace reflection signals that are muted and remain zb in the data after subtraction can have greater amplitude than the primary signals. They z~ also represent a disproportionately greater share of the total amplitude of the reflection za signals. To that extent, then, mute filters are ineffective in removing multiple z9 reflection signals.
3o Moreover, since such prior art methods typically subject the data to NMO
3r correction prior to transforming the data, it is not possible to apply a high-low mute ~ filter. Primary reflection signals transform at zero slowness when the data has been NMO corrected. Thus, the application of any lower mute limit, other than zero, would 3 not mute the primary signals. They would be preserved in the tau-P domain along a with multiples and, therefore, subtracted along with the multiples from the original s data set in the time-space domain. In essence, the primary signals would be f ltered 6 out by any lower mute limit.
The methods of the subject invention, however, are more effective at filtering s . all multiples, including those recorded at near offsets. When the data is not subject to 9 NMO correction, signals for multiple reflection events, including those recorded at ~o near traces, will transform into regions of greater slownesses than those for primary o signals, i.e., above a definable p",pa.. They then will be filtered out, along with other ~z noise at greater slownesses, by the upper limit p",U~. of the novel filters. Moreover, 3 because primary reflection signals do not transform at or near zero slowness, i.e., they 14 transform above a defmablep",;", the filters of the subject invention are able to remove ~s noise transforming below p""".
Though they usually will not be as effective as when the filter limits are time o variable, however, high-low filters may be used where one or both of the filter limits ~s are time invariant. Mute filters based on a defined p,nu~e also may be used, and, if the i9 data has been NMO corrected prior to transformation, generally will be required.
zo Other tau-P filters are known by workers in the art and may be used.
z~ Finally, it will be appreciated that while the subject invention contemplates the zz application of a filter to the transformed data in the tau-P domain, the filter may be z3 applied before transforming the data. Additional filters may be applied to the data in za the time-space domain before or after application of the Radon transformations, or in zs other domains by other filtration methods, if desired or appropriate.
z6 The design of a corrective filter is guided by the following principles.
Seismic z~ waves can be analyzed in terms of plane wave components whose arrival times in X-T
za space are linear. Seismic reflection times are approximately hyperbolic as a function z9 of group offset X in the following relations:
x_ 3o t~ = t~ +
~_ ., where x is the group offset, z t,t. is the reflection time at offset x;
t" is the reflection time at zero offset; and a y,. is the stacking velocity.
s The apparent surface velocity of a reflection at any given offset relates to the 6 slope of the hyperbola at that offset. Those slopes, and thus, the apparent velocities of ~ reflections are bounded between infinite velocity at zero offset and the stacking s velocity at infinite offset. Noise components, however, range from infinite apparent ~ velocity to about one-half the speed of sound in air.
~o Impulse responses for each Radon transform can be readily calculated by > > setting g(t, x, T, , p, ) = 0 or f (t, , x, , T, p) = 0 in the respective domains, for example, ~2 by substituting a single point sample such as (x"t,) or (T,, pl) in each domain.
3 Substituting d(x,,t,)=A8(x-x,)8(t-t~) Is into the linear slant stack equation yields ~6 u(p,T) = A8(t, -z- px,) .
i ~ Thus it shows that a point of amplitude A at (x" tl ) maps into the straight line ~ s r + px, = t, .
On the other hand, substituting (p"T,) of amplitude B yields 2o u(p,T) = B~(TI -t + p,x) 2~ showing that a point (p~,T,) in tau-P space maps into the line zz T, =t-p,x.
2s In a similar manner it can be shown that a hyperbola x~
2a t2 = t~ + 2 v zs maps into the point 26 (T, p) _ (t~,l ~ l') , z~ and vice-versa.
zs In the practice of the subject invention a point (x,, t, ) maps into the ellipse 1 - z, + p_xi_ l~' t~
z (X-T domain impulse response) under the hyperbolic Radon transform. Thus, 3 hyperbolas representing signals map into points.
4 In addition straight lines s t=mx ~ representing organized source-generated noise also map into points (z~ p) _ (o~ ~) s in the tau-P domain. These properties form the basis for designing tau-P
filters that 9 pass signal and reject noise in all common geometry domains.
io In summary then, in a common midpoint gather prior to NMO correction, the > > present invention preserves the points in tau-P space corresponding to the hyperbolas ~z described by the stacking velocity function (t,,v), where v is the dip corrected ~s stacking velocity. A suitable error of tolerance such as 5-20% is allowed for the ~a stacking velocity function. Similar procedures exist for each of the other gather is geometries.
The signal's characterization is defined by the Canonical equations that are a used to design two and three dimensional filters in each common geometry gather.
~ s Filters designed to pass signal in the tau-P domain for the common geometry gathers 9 for both two-and-three dimensional seismic data are derived from the following basic zo considerations. In the X-T domain the kinematic travel time for a primary reflection z~ event's signal in a common midpoint ("CMP") gather is given by the previously zz described hyperbolic relationship z x tXz = t~Z + z za Under the hyperbolic Radon transformation, this equation becomes z z z z zs t, =z +p x .
z6 The source coordinate for two-dimensional seismic data may be denoted by I
z~ (Initiation) and the receiver coordinate by R (receiver) where z = t~, and p =1 /v,~ .
zs The signal slowness in the tau-P domain for common source data is given by ar._ ar,.ac alai - acaR + axaR
aR I=I
where the CMP coordinate (C~ and offset (x) are as follows:
C=I+R

a x=R-I
s Defining common source slowness ( p,l ), common offset slowness ( pc ), and ~ common midpoint slowness ( pa. ) as follows:
as x Prr =
aR I
ar x Pc =
ac x at 9 P - x x ax C
to we obtain a general (canonical) relationship for signal slowness in the tau-P domain:
i I Pn = z Pc + Px Iz Common offset slowness pc is given by acx at~, acx P~ = araac - aco p°
la where p~ is the slowness or "dip" on the CMP section as given by ac~
IS p° = ac is Since z z JC' 17 la. = t~~ -I-l~.s I a we obtain atx h xza at" fx txVs ~ where acceleration (a) a = ~~° .
ar~
3 Thus, we obtain that common offset slowness ( p~ ) t~, ax z 4 pc = pa 1 _ tav.s3 s Under a hyperbolic Radon transformation where 6 Z=l~~
7 tx = tz + p'xz , a we determine that the common midpoint slowness ( pT ), common source slowness 9 ( p~z ), common receiver slowness ( p, ), and common offset slowness ( pc ) may be to defined as follows:
x ~ i pX =
~~,z zz + pzxz ., ~z pe = z pc +pX
13 pi = z pc - Px T axz 14 pC - po Tz + pzx2, 1 -to Vs is In common source, receiver and offset gathers, the pass regions or filters in 6 tau-P space are defined as those slowness (p's) not exceeding specified percentages of ~ slownesses p,z , p, , and p~ , respectively:
~ s p <_ 1 pa (common source) 1+r3 p <_ p, (common receiver) 1+~4 zo p <_ 1 pc (common offset) 1+r5 zi where r3, r4 , and rs are percentages expressed as decimals.
zz Slownesses pR, p,, and pc are expressed, respectively, as follows:

p,z = ~ p~ + p~. (common source) z p, _ '-, p~ - p.~. ~ (common receiver) s p~. = t-°° p~. 1- ax~3 (common offset) tz. / 1~
o .s a where x pa = z tA. V.S.
a = al~' (velocity acceleration) at"
p" _ ~~ (dip on CMP stack) x s ta, _ , to +
v., 9 z = l ~o The value for offset x in these equations is the worse case maximum offset determined i i by the mute function.
iz Inverse Transforming the Data ~3 After the corrective filter is applied, the methods of the subject invention 4 further comprise the steps of inverse transforming the enhanced signal content from ~s the time-slowness domain back to the offset-time domain using an inverse Radon ~s transformation. If, as is preferable, an offset weighting factor x" was applied to the transformed data, an inverse of the offset weighting factor p" is applied to the inverse s transformed data. Similarly, if other amplitude balancing operators, such as an AGC
i9 operator or an operator based on trace envelopes, were applied, an inverse of the zo amplitude balancing operator is applied. The signal amplitude data is thereby restored z~ for the enhanced signal content.
zz For example, in step 9 of the method of FIG. 1, an inverse hyperbolic Radon z3 transformation is used to inverse transform the enhanced signal from the time-za slowness domain back to the offset-time domain. An inverse of the offset weighting ~ factor p" then is applied to the inverse transformed data, as shown in step 10. The z signal amplitude data for the enhanced signal content is thereby restored.
3 A general mathematical formulation utilizing the inverse offset weighting 4 factor and encompassing the linear, parabolic, and hyperbolic inverse Radon s transforms is as follows:
6 (3) d(x,t)= Jdp jdzp"p(z)*u(p,z)8[g(t,x,z,p)] (inversetransformation) ~ where s u(p,z) = transform coefficient at slowness p and zero-offset time z 9 d(x, t) = measured seismogram at offset x and two-way time t ~o p" = inverse offset weighting factor ( 0 c n c 1 ) > > p(z) * = convolution of rho filter z ~ = Dirac delta function i3 g(t,x,zp) = inverse transform function ~a The inverse transform function for hyperbolic trajectories is as follows:
~s g(t~x~z~P~=z- tz-Pzxz 6 Thus, when z = tz - pzxz , the inverse hyperbolic Radon transformation equation becomes is d(x~t)= f dPPnP(z)*zz(p, tz-pzxz) The inverse transform function for linear slant stack trajectories is as follows:
zo g(t,x,z,p)=z-t+px z~ Thus, when z = t-pa: , the inverse linear slant stack Radon transformation equation zz becomes z3 d(x,t) = f dPP"p(z)'~(P~t-Px) za The inverse transform function for parabolic trajectories is as follows:
zs g~t,x,z, p)= z-t+ pxz 3~

~ Thus, when z = t - px'' , the inverse parabolic Radon transformation equation z becomes v 3 d (.~, t) _ ~dpp"p(z) 'ku(P, t - Px' ) a As with the forward transform function f ~t,x, z, p~ in conventional Radon s transformations, the inverse travel-time function g(t,x,z,p) allows kinematic travel 6 time trajectories to include anisotropy, P-S converted waves, wave-field separations, ~ and other applications of current industry that are used to refine the analysis consistent a with conditions present in the survey area. ~ It will be noted, however, that the novel 9 inverse Radon transformations set forth above in Equation 3 differ from conventional ~o inverse Radon transformations in the application of an inverse offset weighting factor > > p" , where p is slowness.
lz The inverse offset weighting factor restores the original amplitude data which ~s now contains enhanced primary reflection signals and attenuated noise signals.
14 Accordingly, smaller offsets receive preferentially larger weights since they received is preferentially less weighting prior to filtering. The power n is greater than zero, but 16 less than 1. Preferably, n is approximately 0.5 because amplitudes seem to be n preserved better at that value. While the power n appears to be robust and insensitive, ~a it probably is data dependent to some degree.
As discussed above relative to the forward transformations, the continuous zo inverse transformations set forth above preferably are translated into discrete z~ transformation equations which approximate the solutions provided by continuous zz transformation equations and can be encoded into and executed by computers.
z3 For example, assume a transform coefficient u(p,z) contains 2,T+1 discrete za values of the parameter p and M discrete values of the parameter i, i. e., zs j = 0,~1,...,~J and z6 m =1,..., M
z~ and that za P_,, < P_.~+~ < ... < p~_~ < P.i z9 A discrete general transform equation approximating the continuous general transform 3o Equation 3 set forth above, therefore, may be derived as set forth below:

d(x~t)= ~ ~u(P;~z,rr)Pj"PST)*d(g(t~x~zn~~Pj)j~P;~z,rr J---J n i=1 z where ~p~ - Pj+I 2 pr-I for j = 0,~1,...,~(J-1) 4 DPJ = PJ - PJ-I
~P-J - P-,I i-I P-J
Zm = Zm+I 2 T nr-1 for na = 2, ..., M -1 7 Otl = TZ -TI
8 O TM = TM - ZM-I
9 By substituting the hyperbolic inverse transform function set forth above, the ~o discrete general inverse transformation Equation 4 above, when z = t'' -p'x'' , may ~ i be reduced to the discrete inverse transformation based on hyperbolic kinematic travel ~z time trajectories that is set forth below:
:3 d(x~t)= ~PjnP(z)*u(P~~ tZ-PjZxz)~Pi j J
~a Similarly, when z=t-px, the discrete linear slant stack inverse ~s transformation derived from the general Equation 4 is as follows:
J
~s d(x,t)= ~pjnp(z)*u(p~,t-pjx)Opj j- J
n When z = t - px' , the discrete parabolic inverse transformation is as follows:
d(x~t)= ~ pJnP(Z)*u(pJ~t- pJxz)Opi .1 J
Those skilled in the art will appreciate that the foregoing inverse zo transformation Equations 3 and 4 are not exact inverses of the forward transformation zt Equations 1 and 2. They are sufficiently accurate, however, and as compared to more zz exact inverse equations, they are less complicated and involve fewer mathematical z3 computations. For example, more precise inverse transformations could be za formulated with Fourier transform equations, but such equations require an extremely as large number of computations to execute. For that reason, the foregoing equations are i preferred, but it is believed that other inverse transformation equations are known, and z still others may be devised by workers of ordinary skill for use in the subject 3 invention.
a The transformations and semblance analyses described above, as will be s appreciated by those skilled in the art, are performed by sampling the data according 6 to sampling variables ~t , da , 0z , and t1p . Because the novel methods do not ~ require NMO correction prior to transformation or a least mean square analysis, s sampling variables may be much finer. Specifically, 4t and 0z may be set as low as 9 the sampling rate at which the data was collected, which is typically 1 to 4 ~o milliseconds, or even lower if lower sampling rates are used. 0p values as small as > > about 0.5 ,usec/m may be used. Preferably, 4p is from about 0.5 to 4.0 ~csec/m, more ~z preferably from about 0.5 to about 3.0 ,usec/m, and even more preferably from about 0 0.5 to about 2.0 ,usec/m. Most preferably Op is set at 1.0 ,usec/m. Since the sampling ~a variables are finer, the calculations are more precise. It is preferred, therefore, that the is sampling variables 0t, dx, and Oz be set at the corresponding sampling rates for the ~s seismic data field records and that ~p be set at 1 ~sec/m. In that manner, all of the ~~ data recorded by the receivers is processed. It also is not necessary to interpolate ~ a between measured offsets.
i9 Moreover, the increase in accuracy achieved by using finer sampling variables zo in the novel processes is possible without a significant increase in computation time or zi resources. Although the novel transformation equations and offset weighting factors zz may be operating on a larger number of data points, those operations require fewer zs computations that prior art process. On the other hand, coarser sampling variables za more typical of prior art processes may be used in the novel process, and they will zs yield accuracy comparable to that of the prior art, but with significantly less z6 computational time and resources.
zz For example, industry commonly utilizes a process known as constant velocity zs stack (CVS) panels that in some respects is similar to processes of the subject z9 invention. CVS panels and the subject invention both are implemented on common 3o mid-point gathers prior to common mid-point stacking and before NMO
correction.
31 Thus, the starting point of both processes is the same.

i In the processes of the subject invention, the discrete hyperbolic radon z transform u~p,z~ is calculated by summing seismic trace amplitudes d(x,t) along a 3 hyperbolic trajectory 4 t = z2 ~ pzx~
s in offset-time (x,~~ space for a particular tau-p (z,p~ parameter selection.
6 Amplitudes d(x,i) are weighted according to a power n of the offset x, i.e., by the ~ offset weighting factor x" . The summation and weighting process is described by the s discrete forward transform Equation 2. In~comparison, a CVS panel does not apply 9 any offset dependent weighting.
~o In the processes of the subject invention, the calculation consists of first > > scanning over all desired zero-offset times i for a given specification of p, and then ~z changing the value of p and repeating the x-t summation for all desired taus (i's).
3 Thus, all desired values of z and p are accounted for in the summations described by ~a Equation 2 above.
is A given choice of p describes a family of hyperbolas over the common 6 midpoint gather's x-t space. A fixed value of p is equivalent to fixing the stacking o velocity V~ since p is defined as its reciprocal, viz., p = I lVS . By varying the zero-~ s offset time i over the desired i-scan range for a fixed value of p =1 /
Vy. , seismic 29 amplitudes are summed in x-t space over a family of constant velocity hyperbolic zo trajectories. Thus, the novel Radon transforms u(p, z) can be thought of as z~ successively summing amplitudes over constant velocity hyperbolic time trajectories zz that have not been NMO corrected.
z3 In constructing a CVS panel the starting point is a common midpoint gather.
za As in the case of the novel processes, successive scans over stacking velocity V~ and zs zero-offset time T~, are implemented for hyperbolic trajectories defined by the z6 equation z x z~ t = T" z +
v.,. -zs The equivalence of T~ = z and p =1/Vf in defining the same families of z9 hyperbolas is obvious. The difference between the novel processes and CVS
is in the ~ implementation of amplitude summations. In both cases, a I y. =1/p is first held z constant, and the T~ = z is varied. A new V,. = I l p is then held constant, and the 3 T~, = z scan repeated. In this manner, the same families of hyperbolas are defined.
a However, an NMO correction is applied to all the hyperbolas defined by the V, =1 / p > choice prior to summing amplitudes. A CVS trace is calculated for each constant 6 velocity iteration by first NMO correcting the traces in the common midpoint gather ~ and then stacking (i.e., summing) the NMO corrected traces. Thus, one CVS
trace in s a CVS panel is almost identical to a u(p=constant,i) trace in the novel processes.
~ The difference lies in whether amplitudes are summed along hyperbolic trajectories ~o without an NMO correction as in the novel process or summed (stacked) after NMO
1 ~ correction.
tz The novel processes and CVS differ in the interpolations required to estimate t3 seismic amplitudes lying between sample values. In the novel processes amplitude to interpolations occur along hyperbolic trajectories that have not been NMO
corrected.
is In the CVS procedure amplitude interpolation is an inherent part of the NMO
t6 correction step. The novel processes and CVS both sum over offset x and time t after interpolation to produce a time trace where p =1 / VS is held constant over time t s T~ = z . Assuming interpolation procedures are equivalent or very close, then a single 9 CVS time trace computed from a particular common midpoint gather with a constant zo V, =1 / p is equivalent in the novel processes to a time trace u(p, z) where p = I lVS
zt is held constant. Thus, in actual practice the velocity ensemble of CVS
traces for a zz single common midpoint gather corresponds to a coarsely sampled transformation of z3 the same common midpoint gather according to the novel processes. As noted, za however, the novel processes are capable of using finer sampling variables and zs thereby of providing greater accuracy, z6 Refining of the Stacking I~elocity Function z~ The subject invention also encompasses improved methods for determining a za stacking velocity function which may be used in processing seismic data.
Such z9 improved methods comprise the steps of performing a semblance analysis on the 3o restored data to generate a second semblance plot. Though not essential, preferably an ~ offset weighting factor x" , where 0 < N < 1, is f rst applied to the restored data. A
z velocity analysis is then performed on the second semblance plot to define a second stacking velocity function. It will be appreciated that this second stacking velocity, a because it has been determined based on data processed in accordance with the subject s invention, more accurately reflects the true~stacking velocity function and, therefore, 6 that inferred depths and compositions of geological formations and any other ~ subsequent processing of the data that utilities a stacking velocity function are more s accurately determined.
9 For example, as shown in step I1 of FIG. 1, an offset weighting factor x" is io applied to the data that was restored in step 10. A semblance plot is then generated i ~ from the offset weighted data as shown in step 12. The semblance plot is used to ~z determine a stacking velocity function in step 13 which then can be used in further is processing as in step 14. Though not shown on FIG. 1, it will be appreciated that the ~a semblance analysis can be performed on the inverse transformed data from step 9, is thereby effectively applying an offset weighting factor since the effect of offset i6 weighting the data prior to transformation has not been inversed.
Display and Further Processing of Data ~ s After the data is restored, it may be displayed for analysis, for example, as 9 shown in step 15 of FIG. 1. The restored data, as discussed above, may be used to zo more accurately define a stacking velocity function. It also may subject to further z1 processing before analysis as shown in step 16. Such processes may include pre-stack zz time or depth migration, frequency-wave number filtering, other amplitude balancing zs methods, removal of abasing effects, seismic attribute analysis, spiking z4 deconvolution, data stack processing, and other methods known to workers in the art.
zs The appropriateness of such further processing will depend on various geologic, z6 geophysical, and other conditions well known to workers in the art.
z~ Invariably, however, the data in the gather, after it has been processed and za filtered in accordance with the subject invention, will be stacked together with other z9 data gathers in the survey that have been similarly processed and filtered.
The stacked 3o data is then used to make inference about the subsurface geology in the survey area.

The methods of the subject invention preferably are implemented by z computers and other conventional data processing equipment. Such data processing equipment, as appreciated by workers in the art, will typically comprise a storage a device and a processor connected to the storage device, wherein the storage device s stores a software program for controlling the processor to execute the novel methods.
6 An output signal for displaying the processed data will be provided to a printer or ~ other display device. Suitable software for doing so may be written in accordance s with the disclosure herein. Such software also may be designed to process the data by 9 additional methods outside the scope of, but complimentary to the novel methods.
io Accordingly, it will be appreciated that suitable software will include a multitude of > > discrete commands and operations that may combine or overlap with the steps as iz described herein. Thus, the precise structure or logic of the software may be varied 3 considerably while still executing the novel processes.
Examples is The invention and its advantages may be further understood by reference to the 6 following examples. It will be appreciated, however, that the invention is not limited thereto.
i s Example 1- Model Data Seismic data was modeled in accordance with conventional considerations and zo plotted in the offset-time domain as shown in FIG. 4. Such data displays a common z~ midpoint gather which includes desired primary reflections Re" and undesired zz multiples reflections M~ beginning at 1.0, 1.5, 2.0, 2.5, 3.0 and 3.5 seconds. It also z3 includes a desired water-bottom reflection WR at 0.5 seconds. It will be noted in za particular that the signals for the desired reflections Re~ and WR overlap with those zs of the undesired multiple reflections M".
z6 The modeled seismic data was then transformed into the time-slowness z~ domain utilizing a high resolution, hyperbolic Radon transformation of the subject za invention and an offset weighting factor x° 5. As shown in FIG. 5, the reflections Re", z9 WR, and M" all map to concentrated point-like clusters in time-slowness space.
3o Those points, however, map to different areas of time-slowness space, and thus, the s~ points corresponding to undesired multiple reflections M" can be eliminated by 3z appropriate time-slowness filters, such as the time variant, high-low slowness filter ~ represented graphically by lines 20. Thus, when the model data is inverse transformed z back to the offset-time domain by a novel hyperbolic Radon inverse transformation, as shown in FIG. 6, only the desired primary reflections Re" and the water-bottom a reflection WR are retained.
s Example 2 - West Texas Production Area 6 A seismic survey was done in the West Texas area of the United States. FIG.
~ 7 shows a portion of the data collected along a two-dimensional survey line.
It is a displayed in the offset-time domain and has been gathered into a common midpoint ~ geometry. It will be noted that the data includes a significant amount of unwanted io noise, including muted refractions appearing generally in an area 30, aliased ground > > rol l occurring in area 31, and apparent air wave effects at areas 32.
a In accordance with the subject invention, a conventional semblance analysis 3 was performed on the data. The semblance plot was then used to identify a stacking 4 velocity function and, based thereon, to define a filter that would preserve primary ~s reflection events and eliminate noise.
The data of FIG. 7 was also transformed into the time-slowness domain with a high resolution, hyperbolic Radon transformation of the subject invention using an ~ a offset weighting factor x°'S. A time variant, high-low filter defined through the i9 semblance analysis was applied to the transformed data, and the data was transformed Zo back into the offset-time domain using a hyperbolic Radon inverse transform and an z~ inverse of the offset weighting factor p°-5. The data then was displayed, which display zz is shown in FIG. 8.
z3 In FIG. 8, seismic reflection signals appear as hyperbolas, such as those za indicated at 33 and 34 (troughs). It will be noted that the unwanted noise that was ?s present in the unprocessed data of FIG. 7 as muted refractions 30, abased ground roll z~ 31, and air wave effects 32 is no longer present in the processed display of FIG. 8.
The data of FIG. 8 was then stacked along with other processed and filtered z8 CMP gathers from the survey in accordance with industry practice. A plot of the z9 stacked data is shown in FIG. 9. Seismic reflection signals appear as horizontal lines, 3o e.g., those lines appearing at approximately 0.45 sec and 1.12 sec, from which the 31 depth of geological formations may be inferred.

By way of comparison, the data of FIG. 7, which had not been subject to z filtering by the subject invention, and from other CMP gathers from the survey were 3 conventionally processed and stacked in accordance with industry practice.
The a unfiltered, stacked data is shown in FIG. 10. It will be appreciated therefore that the s stacked filter data of FIG. 9 show reflection events with much greater clarity due to ~ the attenuation of noise by the novel methods. For example, the reflection signals ~ appearing at 0.45 and 1.12 sec in FIG. 9 are not clearly present in FIG. 10.
x Example 3 - West African Production Area A marine seismic survey was done in a West African offshore production area.
~o FIG. 11 shows a portion of the data collected along a two-dimensional survey line. It > > is displayed in the offset-time domain and has been gathered into a common midpoint ~z geometry. It will be noted that the data includes a significant amount of unwanted m noise.
~a In accordance with the subject invention, a conventional semblance analysis Is was performed on the data. A plot of the semblance analysis is shown in FIG. 12.
6 This semblance plot was then used to identify a stacking velocity function and, based thereon, to define a filter that would preserve primary reflection events and eliminate ~x noise, in particular a strong water bottom multiple present at approximately 0.44 sec 9 in FIG. 12. The semblance analysis was also used to define lower and upper slowness zo limits p",;" and p",ax for transformation of the data.
zi The data of FIG. 11 was also transformed, in accordance with the slowness zz limits p",;" and p",az, into the time-slowness domain with a high resolution, hyperbolic zs Radon transformation of the subject invention using an offset weighting factor x°'S. A
za time variant, high-low filter defined through the semblance analysis was applied to the zs transformed data, and the data was transformed back into the offset-time domain using z6 a hyperbolic Radon inverse transform and an inverse of the offset weighting factor z~ p°'S. The data then was displayed, which display is shown in FIG.
13.
zx In FIG. 13, seismic reflection signals appear as hyperbolas, such as those z9 starting from the near offset (right side) at approximately 1.7 sec and 2.9 sec. It will 3o be noted that the unwanted noise that was present in the unprocessed data of FIG. 11 was severely masking the reflection hyperbolas and is no longer present in the 3z processed display of FIG. 13.

The processed data of FIG. 13 then was subject to a conventional semblance z analysis. A plot of the semblance analysis is shown in FIG. 14. This semblance plot 3 was then used to identify a stacking velocity function. By comparison to FIG. 12, it a will be noted that the semblance plot of FIG. 14 shows the removal of various noise s events, including the strong water bottom multiple present at approximately 0.44 sec ~ in FIG. 12. Thus, it will be appreciated that the novel processes allow for a more precise definition of the stacking velocity function which then can be used in s subsequent processing of the data, for example, as described below.
The data of FIG. 13, along with processed and filtered data from other CMP
~o gathers in the survey, were then stacked in accordance with industry practice except > > that the stacking velocity function determined through the semblance plot of FIG. 14 ~z was used. A plot of the stacked data is shown in FIG. 15. Seismic reflection signals 3 appear as generally horizontal lines, from which the depth of geological formations 4 may be inferred. For example, the reflective signals starting on the right side of the is plot at approximately 0.41, 0.46, 1.29, and 1.34 sec. It will be noted, however, that i6 the reflection signals in FIG. 15 slope and have slight upward curves from le .ft to m right. This indicates that the reflective formations are somewhat inclined, or what it ~ s referred to in the industry as "dip".
is By way of comparison, the unprocessed data of FIG. 11, along with data from zo other CMP gathers, were conventionally processed and then stacked in accordance z~ with industry practice. The unfiltered, stacked data is shown in FIG. 16.
It will be zz appreciated therefore that the stacked filter data of FIG. 15 show reflection events z3 with much greater clarity due to the attenuation of noise by the novel methods. In za particular, it will be noted that the reflection signals at 0.41 and 0.46 sec in FIG. 15 zs are not discernable in FIG. 16, and that the reflection signals at 1.29 and 1.34 sec are z~ severely masked.
z~ Example 4 - Viosca Knoll Production Area zs A marine seismic survey was done in the Viosca Knoll offshore region of the z9 United States. FIG. 17 shows a portion of the data collected along a two-dimensional 3o survey line. It is displayed in the offset-time domain and has been gathered into a ~~ common midpoint geometry. It will be noted that the data includes a significant ~ amount of unwanted noise, including severe multiples created by short cable z acquisition.
3 In accordance with the subject invention, a conventional semblance analysis a was applied to the data. A plot of the semblance analysis is shown in FIG.
18. This s semblance plot was then used to identify a stacking velocity function and, based s thereon, to define a filter that would preserve primary reflection events and eliminate ~ noise, in particular the multiples caused by short cable acquisition that are present a generally from approximately 2.5 to 7.0 sec. The stacking velocity function is 9 represented graphically in FIG. 18 as line 40. The semblance analysis was also used ~o to define lower and upper slowness limits p~,;" and p",p, for transformation of the data.
> > The data of FIG. 17 was also transformed, in accordance with the slowness tz limits p",;" and p",ox, into the time-slowness domain with a high resolution, hyperbolic 13 Radon transformation of the subject invention using an offset weighting factor x°'S. A
~a time variant, high-low filter defined through the semblance analysis was applied to the ~s transformed data, and the data was transformed back into the offset-time domain using is a hyperbolic Radon inverse transform and an inverse of the offset weighting factor p -5. The data then was displayed, which display is shown in FIG. 19.
is In FIG. 19, seismic reflection signals appear as hyperbolas, such as those 9 beginning at the near offset (right side) below approximately 0.6 sec. It will be noted zo that the unwanted noise that was present in the unprocessed data of FIG. 17 below 0.6 z~ sec and severely masking the reflection hyperbolas is no longer present in the zz processed display of FIG. 19. In particular, it will be noted that the reflection signals z3 beginning at the near offset at approximately 2.0 and 2.4 sec in FIG. 19 are not za discernable in the unfiltered data of FIG. 17.
zs The processed data of FIG. 19 then was subject to a conventional semblance zs analysis. A plot of the semblance analysis is shown in FIG. 20. By comparison to z~ FIG. 18, it will be noted that the semblance plot of FIG. 20 shows the removal of zs various noise events, including the strong multiples located generally from 2.5 to 7.0 z9 sec in FIG. 18 that were created by short cable acquisition. The semblance plot was 3o then used to identify a stacking velociy function, which is represented graphically in s~ FIG. 20 as line 41. Thus, it will be appreciated that the novel processes allow for a more precise definition of the stacking velocity function which then can be used in z subsequent processing of the data, for example, as described below.
The data of FIG. 19, along with the data from other CMP gathers that had a been processed and filtered with the novel methods, were then stacked in accordance s with industry practice except that the stacking velocity function determined through 6 the semblance plot of FIG. 20 was used. A plot of the stacked data is shown in FIG.
~ 21. Seismic reflection signals appear as horizontal lines, from which the depth of s geological formations may be inferred. In particular, the presence of various traps o potentially holding reserves of oil or gas appear generally in areas 42, 43, 44, 45, and ~0 46, as seen in FIG. 21A, which is an enlargement of area 21A of FIG. 21.
i ~ By way of comparison, the data of FIG. 17 and data from other CMP gathers 12 In the survey was processed in accordance with conventional methods based on 13 parabolic Radon transformations. In contrast to the subject invention, the data were is first NMO corrected. The NMO corrected data then were transformed, without any Is transformation limits, to the tau-P domain using a low resolution parabolic Radon 6 transformation that did not apply an offset weighting factor x". A
conventional mute filter then was applied, and the preserved data were transformed back into the offset-~ a time domain using a conventional parabolic Radon inverse transform that did not i9 apply an inverse offset weighting factor p". The NMO correction was reversed to zo restore the preserved data and it was subtracted from the original data in the respective z~ gathers.
zz The data that had been processed and filtered with the parabolic Radon z3 forward and inverse transforms was then stacked in accordance with industry practice.
za A plot of the stacked data is shown in FIG. 22. Seismic reflection signals appear as zs horizontal lines, from which the depth of geological formations may be inferred. It z6 will be noted, however, that as compared to the stacked plot of FIG. 21, reflection z~ events are shown with much less clarity. As may best be seen in FIG. 22A, which is zs an enlargement of area 22A in FIG. 22, and which corresponds to the same portion of z9 the survey profile as shown in FIG. 21A, traps 42, 43, 44, 45, and 46 are not revealed 3o as clearly by the conventional process.

The foregoing examples demonstrate the improved attenuation of noise by the novel methods and thus, that the novel methods ultimately allow for more accurate inferences about the depth and composition of geological formations.
a While this invention has been disclosed and discussed primarily in terms of s specific embodiments thereof, it is not intended to be limited thereto.
Other 6 modifications and embodiments will be apparent to the worker in the art.

Claims (35)

1. A method of processing seismic data to remove unwanted noise from meaningful reflection signals indicative of subsurface formations, wherein said seismic data comprise amplitude data recorded over time at a number of seismic receivers in an area of interest, said amplitude data are assembled into common geometry gathers in an offset-time domain, and said amplitude data comprise primary reflection signals and coherent noise that overlap in an offset-time domain; said method comprising the step of enhancing the separation between primary reflection signals and coherent noise by transforming said data from the offset-time domain to the time-slowness domain, wherein:
(a) an offset weighting factor x n is applied to said amplitude data, wherein 0 < n < 1; and (b) said offset weighted amplitude data are transformed from the offset-time domain to the time-slowness domain using a Radon transformation;
(c) whereby said primary reflection signals and said coherent noise transform into different regions of the time-slowness domain to enhance the separation therebetween.

2. The method of claim 1, wherein said method comprises the steps of:
(a) applying a corrective filter to the offset-weighted, transformed data to enhance the primary reflection signal content of said data and to eliminate unwanted noise events;
(b) inverse transforming said enhanced signal content from the time-slowness domain back to the offset-time domain using an inverse Radon transformation;
and (c) applying an inverse of the offset weighting factor p n to said inverse transformed data, wherein 0 < n < 1, thereby restoring the signal amplitude data for said enhanced signal content.

3. The method of claim 1 or 2, wherein said method comprises the steps of:
(a) inputting said seismic data into a computer system; and (b) operating said computer system in accordance with a computer program to perform said method steps and to generate an output signal for providing a display of said processed data.

4. An apparatus for processing seismic data to remove unwanted noise from meaningful reflection signals indicative of subsurface formations; said apparatus comprising a storage device and a processor connected to the storage device, the storage device storing a program for controlling the processor, and the processor operative with the program to:
(a) receive computer seismic data detected at a number of seismic receivers in an area of interest, said seismic data comprising signal amplitude data recorded over time and containing both primary reflection signals and unwanted noise events;
(b) assemble said seismic data into common geometry gathers in an offset-time domain;
(c) apply an offset weighting factor x n to said amplitude data, wherein 0 < n < 1;
(d) transform said offset weighted amplitude data from the offset-time domain to the time-slowness domain using a Radon transformation;
(e) apply a corrective filter to enhance the primary reflection signal content of the data and to eliminate unwanted noise events;
(f) inverse transform said enhanced signal content from the time-slowness domain back to the offset-time domain using an inverse Radon transformation; and (g) apply an inverse of the offset weighting factor p n to said inverse transformed data, wherein 0 < n < 1, thereby restoring the signal amplitude data for said enhanced signal content.

5. A method of selecting a drilling site to access a subsurface formation, the method comprising the steps of:
(a) obtaining field records of seismic data detected at a number of seismic receivers in an area of interest, said seismic data comprising signal amplitude data recorded over time and containing both primary reflection signals and unwanted noise events;
(b) processing the seismic data to search for the presence of a subsurface formation of interest, said processing removing unwanted noise from meaningful reflection signals indicative of subsurface formations and comprising the steps of:

i) assembling said seismic data into common geometry gathers in an offset-time domain;
ii) applying an offset weighting factor x n to said amplitude data, wherein 0 < n < 1;
iii) transforming said offset weighted amplitude data from the offset-time domain to the time-slowness domain using a Radon transformation;
iv) applying a corrective filter to enhance the primary reflection signal content of the data and to eliminate unwanted noise events;
v) inverse transforming said enhanced signal content from the time-slowness domain back to the offset-time domain using an inverse Radon transformation; and vi) applying an inverse of the offset weighting factor p n to said inverse transformed data, wherein 0 < n < 1, thereby restoring the signal amplitude data for said enhanced signal content; and (c) drilling at a location likely to access the subsurface formations indicated by said processing steps.

6. The method or apparatus of any one of claims 2 to 5, wherein n in the offset weighting factor x n and the inverse of the offset weighting factor p n is approximately 0.5.

7. The method or apparatus of any one of claims 2 to 6, wherein the forward and inverse transform functions in said Radon transformations are selected from the transform functions for linear slant stack, parabolic, and hyperbolic kinematic travel time trajectories.

8. The method or apparatus of any one of claims 2 to 7, wherein said offset weighting factor x" is applied and said amplitude data is transformed with a continuous Radon transformation defined as follows:
or a discrete Radon transformation approximating said continuous Radon transformation, and said enhanced signal content is inversed transformed and said inverse offset weighting factor p" is applied with a continuous inverse Radon transformation defined as follows:

or a discrete Radon transformation approximating said continuous inverse Radon transformation, where u(p,~) =transform coefficient at slowness p and zero-offset time ~
d(x,t) = measured seismogram at offset x and two-way time t x n = offset weighting factor p(~)* = inverse offset weighting factor p(~) * = convolution of rho filter .delta. = Dirac delta function f(t,x,~,p) = forward transform function g(t,x,~,p) = inverse transform function

9. The method or apparatus of claim 8, wherein where said forward transform function, f(t,x,~,p), and said inverse transform function, g(t,x,~,p), are selected from the transform functions for linear slant stack, parabolic, and hyperbolic kinematic travel time trajectories, which functions are defined as follows:

(a) transform functions for linear slant stack:

f(t,x,~,p)=t-~-px g(t, x, z, p) = z - t + px (b) transform functions for parabolic:

f(t,x,~,p)=~-t- px2 g(t,x,~,p)=~-t+ px2 (c) transform functions for hyperbolic:

10. The method or apparatus of any one of claims 2 to 7, wherein said offset weighting factor x" is applied and said amplitude data is transformed with a continuous hyperbolic Radon transformation defined as follows:

or a discrete hyperbolic Radon transformation approximating said continuous hyperbolic Radon transformation, and said enhanced signal content is inversed transformed and said inverse offset weighting factor p n applied with a continuous inverse hyperbolic Radon transformation defined as follows:
or a discrete inverse hyperbolic Radon transformation approximating said continuous inverse hyperbolic Radon transformation, where u(p,.tau.) = transform coefficient at slowness p and zero-offset time .tau.
d(x,t) = measured seismogram at offset x and two-way time t x n = offset weighting factor p n = inverse offset weighting factor p(.tau.)* = convolution of rho filter .delta. = Dirac delta function

11. The method or apparatus of any one of claims 2 to 7, wherein said offset weighting factor x n is applied and said amplitude data is transformed with a discrete Radon transformation defined as follows:
where u(p,.tau.) = transform coefficient at slowness p and zero-offset time .tau.
d(x l,t k) = measured seismogram at offset x l and two-way time t k x l n = offset weighting factor at offset x l .delta. = Dirac delta function .function.(t k,x 1,.tau.,p) = forward transform function at t k and x l and said enhanced signal content is inversed transformed and said inverse offset weighting factor p n is applied with a discrete inverse Radon transformation defined as follows:
where u(p j,.tau.m) = transform coefficient at slowness p j and zero-offset time .tau.m d(x,t) = measured seismogram at offset x and two-way time t p j n = inverse offset weighting factor at slowness p j p(.tau.)* = convolution of rho filter .delta. = Dirac delta function g(t,x,.tau.m,p j) = inverse transform function at .tau.m and p j

12. The method or apparatus of claim 11, wherein where said forward transform function, .function.(t k,x 1,.tau.,p), . and said inverse transform function, g(t,x,.tau.m,p j), are selected from the transform functions for linear slant stack, parabolic, and hyperbolic kinematic travel time trajectories, which functions are defined as follows:
(a) transform functions for linear slant stack:
.function.(t k,x l,.tau.,p,) = t k - .tau. - px l g(t,x,.tau.m,p j) = .tau.m - t + p j x (b) transform functions for parabolic:
.function.(t k,x,.tau.,p) = t k - .tau. - px l2 g(t,x,.tau.m,p j) = .tau.m - t + p j x2 (c) transform functions for hyperbolic:

13. The method or apparatus of any one of claims 2 to 7, wherein said offset weighting factor x n is applied and said amplitude data is transformed with a discrete hyperbolic Radon transformation defined as follows:
where u(p,.tau.) = transform coefficient at slowness p and zero-offset time .tau.
x l n = offset weighting factor at offset x l and said enhanced signal content is inversed transformed and said inverse offset weighting factor p n applied with a discrete inverse hyperbolic Radon transformation defined as follows:
where d(x,t) = measured seismogram at offset x and two-way time t p j n = inverse offset weighting factor at slowness p j p(.tau.)* = convolution of rho filter

14. The method or apparatus of any one of claims 1 to 5, wherein n in said offset weighting factor x n is approximately 0.5.

15. The method or apparatus of any one of claims 1 to 5 and 14, wherein said assembled amplitude data are transformed using a hyperbolic Radon transformation.

16. The method or apparatus of any one of claims 1 to 5 and 14 to 15, wherein said offset weighting factor x n is applied and said amplitude data are transformed with a continuous Radon transformation defined as follows:
or a discrete Radon transformation approximating said continuous Radon transformation, where u(p,.tau.) = transform coefficient at slowness p and zero-offset time .tau.
d(x,t) = measured seismogram at offset x and two-way time t x n = offset weighting factor .delta. = Dirac delta function .function.(t,x,.tau.,p) = forward transform function

17. The method or apparatus of claim 16, wherein said forward transform function, .function.(t,x,.tau.,p), is selected from the transform functions for linear slant stack, parabolic, and hyperbolic kinematic travel time trajectories, which functions are defined as follows:
(a) transform function for linear slant stack:
.function.(t,x,.tau.,p) = t - .tau. - px (b) transform function for parabolic:
.function.(t,x,.tau.,p) = t - .tau. - px2 (c) transform function for hyperbolic:

18. The method or apparatus of any one of claims 1 to 5 and 14 to 15, wherein said offset weighting factor x n is applied and said amplitude data are transformed with a continuous hyperbolic Radon transformation defined as follows:
or a discrete hyperbolic Radon transformation approximating said continuous hyperbolic Radon transformation, where u(p,.tau.) = transform coefficient at slowness p and zero-offset time .tau.
x n = offset weighting factor

19. The method or apparatus of any one of claims 1 to 5 and 14 to 15, wherein said offset weighting factor x n is applied and said amplitude data are transformed with a discrete Radon transformation defined as follows:
where u(p,.tau.) = transform coefficient at slowness p and zero-offset time .tau.
d(x l,t k) = measured seismogram at offset x l and two-way time t k x l n = offset weighting factor at offset x l .delta. = Dirac delta function .function.(t k,x l,.tau.,p) = forward transform function at t k and x l .DELTA.t1 = t2 - t1 .DELTA.t N = t N - t N-1

20. The method or apparatus of claim 19, wherein said forward transform function, .function.(t k,x l,.tau.,p), is selected from the transform functions for linear slant stack, parabolic, and hyperbolic kinematic travel time trajectories, which functions are defined as follows:
(a) transform function for linear slant stack:
.function.(t k,x l,.tau.,p) = t k - .tau. - px l (b) transform function for parabolic:
.function.(t k,x l,.tau.,p) = t k - .tau. - px l2 (c) transform function for hyperbolic:

21. The method or apparatus of any one of claims 1 to 5 and 14 to 15, wherein said offset weighting factor x n is applied and said amplitude data are transformed with a discrete hyperbolic Radon transformation defined as follows:
where u(p,.tau.) = transform coefficient at slowness p and zero-offset time .tau.
x l n = offset weighting factor at offset x l

22. The method or apparatus of any of claims 2 to 13, wherein said corrective filter is a high-low corrective filter.

23. The method or apparatus of claim 22, wherein said high-low corrective filter is determined by:

(a) performing a semblance analysis on said amplitude data to generate a semblance plot; and (b) performing a velocity analysis on said semblance plot to define a stacking velocity function and said high-low corrective filter.

24. The method or apparatus of claim 22, wherein said high-low corrective filter is determined by performing a pre-stack time migration analysis on said amplitude data to define a velocity function and said high-low corrective filter.

25. The method or apparatus of any one of claims 22 to 24, wherein said high-low corrective filter is time variant.

26. The method or apparatus of any one of claims 22 to 25, wherein said corrective filter is defined as follows:
where r1 and r2 are percentages expressed as decimals.

27. The method or apparatus of any one of claims 1 to 26, wherein said Radon transformation is applied within defined slowness limits p min and p max, where p min is a predetermined minimum slowness and p max is a predetermined maximum slowness.

28. The method or apparatus of claim 27, wherein p min is less than the slowness of reflection signals from the deepest reflective surface of interest.

29. The method or apparatus of claim 27, wherein p min is up to 20% less than the slowness of reflection signals from the deepest reflective surface of interest.

30. The method or apparatus of any one of claims 27 to 29, wherein said field records are obtained from a marine survey and p max is greater than the slowness of reflective signals through water in the area of interest.

31. The method or apparatus of any one of claims 27 to 29, wherein said field records are obtained from a marine survey and p max is up to 20% greater than the slowness of reflective signals through water in the area of interest.

32. The method or apparatus of any one of claims 27 to 29, wherein p max is greater than the slowness of reflection signals from the shallowest reflective surface of interest.

33. The method or apparatus of any one of claims 27 to 29, wherein p max is up to 20% greater than the slowness of reflection signals from the shallowest reflective surface of interest.

34. The method or apparatus of any one of claims 1 to 33, wherein said assembled amplitude data are uncorrected for normal moveout.

35. The method or apparatus of any one of claims 1 to 34, further comprising the step of forming a display of said processed data.