# Patent

## Title
Moveout correction of seismic data

## Abstract

A method of processing seismic data including pressure data and pressure gradient data to obtain gradients of moveout corrected pressure data, comprising applying a move-out correction function to the pressure gradient data; and adding a correction term dependent on a moveout corrected time-derivative of the pressure data and a spatial derivative of the moveout correction function. The method may be used prior to any data processing algorithm which uses pressure gradient data in which moveout correction is applied prior to the algorithm, either because the algorithm makes a zero offset assumption or because it is beneficial for the algorithm to operate on moveout corrected data to reduce aliasing.

## Background
FIELD OF THE INVENTION
The present invention relates to the application of moveout correction to seismic data.
BACKGROUND TO THE INVENTION
Seismic data are typically gathered using an array of detectors. In the case of marine data, hydrophones measure pressure fluctuations in the water caused by incoming seismic waves. Geophones measure vector quantities such as displacement, velocity or acceleration. In the case of marine data, a plurality of cables or streamers, which are spaced apart typically by about 100 meters, are towed behind a boat. Each cable has detectors spaced along the cable at intervals. In the case of land data, a geophone array is laid out on the ground with the geophones in an approximate grid formation. The detector array detects seismic signals from reverberations of a signal from a seismic source, such as an airgun for marine data. In Ocean Bottom (OBC or OBS) acquisition, a detector array is fixed, on the sea bed. The source may be an airgun mounted on a boat.
The distance between a seismic source and a receiver is known as the offset. A flat reflector will generate an event of an approximately hyperbolic form as a function of offset. Moveout correction can be viewed as a process that compensates for offset by distorting the traces along the time axis in a manner that depends on a velocity function. The aim may be to simulate zero offset data and thus “flatten” the hyperbolic event, using a velocity function selected to match the velocity of the seismic energy through the water and the earth. This may be because the algorithm involves a zero offset assumption, in which case an accurate velocity function is desirable. One example of a data processing algorithm of this type is migration. However, very often moveout correction is applied preceding another data processing algorithm, typically to reduce aliasing effects. If the purpose is simply to reduce the slope of the hyperbolic events to avoid aliasing, an accurate velocity function is not required. In this case, the data are often purposely under corrected using a velocity function faster than one which would create zero offset data. After the data processing algorithm is applied, the moveout correction is reversed or “backed off”. One example of a type of data processing algorithm in which moveout correction is used in this way is interpolation.
Recent developments in marine acquisition record not only the pressure using hydrophones but also record particle velocity or acceleration in directions parallel to the surface. This allows the gathering of pressure gradient (or spatial derivative) data, in, addition to pressure data. New algorithms are being developed which utilise both pressure data and pressure gradient data. However, if moveout correction is required it is not correct simply to apply moveout correction to the gradient, because this does not yield the derivative of the moveout-corrected data. Moveout-correction and differentiation do not, in general, commute.
SUMMARY OF THE INVENTION
According to the present invention, a method of processing seismic data including pressure data and pressure gradient data to obtain gradients of moveout corrected pressure data, comprises:

    
    
        applying a moveout correction function to the pressure gradient data; and
        adding a correction term dependent on a moveout corrected time-derivative of the pressure data and a spatial derivative of the moveout correction function.
    
    


In some data processing algorithms, pressure data and pressure gradients are used. It is also desirable or necessary in many algorithms for moveout correction to be applied prior to the application of the algorithm. Moveout correction is when a velocity function is used to transform the seismic traces along a time axis, to compensate for an offset between source and receiver. Moveout correction is often applied before another processing algorithm, either because zero offset assumptions are made by that algorithm (for example, in migration) or to reduce aliasing. If the purpose is to reduce aliasing, it is not necessary that the moveout function completely compensates for the offset, just that the data are transformed sufficiently to reduce the aliasing effects. As the moveout is “backed off” after the processing algorithm, there is no need for the function to compensate accurately for offset. If both pressure data and pressure gradient data are required by the algorithm, then it is necessary to obtain both the moveout corrected pressure data and the gradients of the moveout corrected data. The present invention allows the calculation of the latter from the pressure data and the pressure gradient data.
Preferably, the method comprises the steps of:

    
    
        a) computing a time-derivative of the pressure data by differentiation of the pressure data;
        b) applying the moveout correction function to the time-derivative of the pressure data;
        c) computing a spatial derivative of the moveout correction function;
        d) multiplying the results of steps (b) and (c);
        e) applying the moveout correction function to the pressure gradient data; and
        f) adding the results of steps (d) and (e) to obtain the gradients of the moveout corrected pressure data.
    
    


In one embodiment, the moveout correction function is a hyperbolic function.
The present invention may be used in combination with a method further comprising spatially interpolating values of moveout corrected pressure data using the moveout corrected pressure data and the gradients of the moveout corrected pressure data. In the interpolation, by use of the gradients of the pressure rather than the pressure data alone, the aliasing problem can be greatly reduced. The pressure values may be derived from hydrophone readings (for marine data), and the spatial derivatives of pressure may be derived from measurements of particle velocity or acceleration in the direction of interpolation.
Preferably, the interpolation comprises applying an interpolation operator calculated by assuming that an interpolated pressure value comprises a linear combination of the known pressure values at locations xi and the spatial derivatives at locations xα, with operator coefficients wi and wα, respectively, and calculating the coefficients by minimising an error function. Preferably, minimising an error function comprises using a least squares method. However, for example, an L-1 method may be used instead of an L-2 (least squares) method.
The interpolation operator may include a factor relating to the noise level in the known values. In the final operator, the effect of this is that the higher the noise in the pressure data, the more weight is given to the gradient data in performing the interpolation and vice versa.
In one embodiment, the direction of interpolation is substantially perpendicular to a cable connecting a plurality of geophones/hydrophones. Thus, the method can be used to interpolate extra lines of data, thus simulating the use of more closely spaced cables. In the case of marine data acquired using a plurality of substantially parallel streamers having hydrophones spaced along each streamer, and the method is used to interpolate data points between the streamers, thus simulating the use of more closely spaced streamers in the acquisition.
Preferably, the seismic data are marine data acquired using a plurality of substantially parallel streamers having hydrophones spaced along each streamer, and the method is used to interpolate data points between the streamers.
Preferably, the pressure data is in the form of shot gathers and that all the data used for interpolation of any one point comes from the same shot. Preferably the pressure data are calculated from raw measurements of particle velocity or acceleration.
In one embodiment of the present invention, the processed seismic data may be used to determine physical properties of the earth's interior.



BRIEF DESCRIPTION OF THE DRAWINGS
Preferred embodiments of the present invention will now be described with reference to the accompanying drawings, in which:
FIG. 1 shows schematically the layout of a marine seismic data acquisition system;
FIG. 2 illustrates moveout of seismic data;
FIG. 3 is a flow diagram illustrating a method for obtaining gradients of moveout corrected data;
FIG. 4 illustrates a system for carrying out the method of the invention;
FIG. 5 shows an example of the process applied to a synthetic dataset;
FIG. 6 shows a comparison of interpolated functions using data only and using data and gradients;
FIG. 7 shows a comparison of gradients of interpolated functions using operators calculated using data only and data with gradients;
FIG. 8 shows a comparison of errors within an ensemble for data only and data with gradient operators;
FIG. 9 illustrates the dependence of ensemble averaged errors on the noise levels in the data when the gradients are noise free; and
FIG. 10 shows the dependence of ensemble averaged errors on the noise levels in the gradients when the data is noise free.



DETAILED DESCRIPTION
FIG. 1 illustrates a typical layout of an acquisition system for marine seismic data. A plurality of streamers, in this case five streamers A, B, C, D, E are towed behind a boat 1. The typical spacing S between the streamers A, B, C, D, E is 50 to 100 meters although the streamers A, B, C, D, E are only fixed at the end adjacent to the boat and therefore will drift so that spacing S becomes non-uniform, particularly towards the free ends of the streamers A, B, C, D, E. Each streamer A, B, C, D, E has a plurality of sensors 2 equally spaced along the length of the streamer. At each sensor position xn, a hydrophone records pressure changes which correspond to seismic signals from reverberations from a shot fired by a source. At each sensor position xn, at least one sensor records particle velocity or acceleration. There may be multiple velocities/acceleration sensors recording the velocity/acceleration in different directions. In this case, the value of interest is the velocity/acceleration reading in the x direction, i.e. substantially perpendicular to the streamer direction.
Velocity or acceleration readings can readily be converted to pressure gradient, in this case the pressure gradient of interest being ∂P/∂x. This can be calculated from acceleration data by the simple application of Newton's second law:

∂P/∂x=−a 

where  is the water density and a is the x-component of acceleration.

If the sensors measure the particle velocity these can readily be converted to acceleration values because the data is typically well sampled (for example every 2 ms) in the time direction and so the acceleration can readily be derived by differentiation.
The locations of the pressure readings and the pressure gradients may be the same. However, even if the sensors are located at the same location, in the case of instrument failures, pressure and/or gradient information may be missing at one or more locations and therefore the locations of the pressure values and the pressure gradient values are not necessarily the same. Furthermore, it may be desirable to place the velocity/acceleration sensors more closely than the pressure sensors to remove noise that would otherwise be aliased.
FIG. 2 illustrates the concept of moveout. A source S0 generates a seismic signal which is reflected from a reflector R and received at a plurality of receivers at offsets yn, the offset being the distance from source to receiver. Each receiver will detect the reflection at an arrival time which is dependent on the offset and the velocity at which the seismic energy travels through the medium (which will be partly water, partly the earth). The moveout is the difference in arrival time between the actual arrival time and a theoretical arrival time for a receiver at zero offset. For a theoretical flat reflector and a uniform velocity medium, the moveout will be a hyperbolic function of offset.
For some seismic data processing algorithms, such as some migration algorithms, it is necessary for the algorithm to operate on zero offset data, so before application of the algorithm the data are corrected using a moveout correction function based on the velocity of the medium, such that all of the data are transformed to appear as if they were recorded at zero offset. It is also common to apply moveout correction to seismic data before some data processing algorithms in order to reduce the slope of events in shot gathers to reduce aliasing, and then to remove this correction afterwards. In this case an accurate velocity function is not required. Indeed, the data is often purposely under corrected using a velocity function faster than one which would create zero offset data. In this case, the application of the moveout correction function reduces the slope of the hyperbolic events but does not completely flatten them. After the data processing algorithm is applied, the moveout correction is reversed or “backed off”. One example of a type of data processing algorithm in which moveout correction is used in this way is interpolation.
Whilst the application of moveout correction is straightforward for interpolation algorithms that involve only data, it is more complicated for algorithms that make use of spatial derivatives of those data. It is not correct simply to apply moveout correction to the derivatives, because this does not yield the derivative of the moveout-corrected data. Moveout-correction and differentiation do not, in general, commute.
Suppose we have recorded data p(x, y, t), where x and y are the components of the offset vector, and t is two-way travel time at that offset. Suppose also that we have the spatial derivative of those data, px(x, y, t). The moveout-corrected data, M(p)(x, y, t0) are given by

M(p)(x,y,t0)=p(x,y,tm(t0,x,y))

where t0 is the zero-offset time, and tm(x, y, t0) may be the moveout function, which gives the time at offset (x,y) corresponding to t0. However, the moveout function need not move the data completely to zero offset. As the purpose is simply to reduce the slope of the events to avoid aliasing, a very approximate moveout function can be used, or one which under-corrects the data (ie the velocity function is slightly faster than for zero offset correction. The above formula assumes that moveout correction does not include stretch compensation, which would bring in an extra factor of ∂tm/∂t0.

The spatial derivative of M(p)(x, y, t0) is given by




  
    
      
        
          
            
              M
              ⁡
              
                (
                p
                )
              
            
            x
          
          ⁢
          
            (
            
              x
              ,
              y
              ,
              
                t
                0
              
            
            )
          
        
        =
        
        ⁢
        
          
            
              p
              x
            
            ⁡
            
              (
              
                x
                ,
                y
                ,
                
                  
                    t
                    m
                  
                  ⁡
                  
                    (
                    
                      x
                      ,
                      y
                      ,
                      
                        t
                        0
                      
                    
                    )
                  
                
              
              )
            
          
          +
        
      
    
  
  
    
      
        
        ⁢
        
          
            
              p
              t
            
            ⁡
            
              (
              
                x
                ,
                y
                ,
                
                  
                    t
                    m
                  
                  ⁡
                  
                    (
                    
                      x
                      ,
                      y
                      ,
                      
                        t
                        0
                      
                    
                    )
                  
                
              
              )
            
          
          ⁢
          
            
              t
              
                m
                ,
                x
              
            
            ⁡
            
              (
              
                x
                ,
                y
                ,
                
                  t
                  0
                
              
              )
            
          
        
      
    
  
  
    
      
        =
        
        ⁢
        
          
            
              M
              ⁡
              
                (
                
                  p
                  x
                
                )
              
            
            ⁢
            
              (
              
                x
                ,
                y
                ,
                
                  t
                  0
                
              
              )
            
          
          +
          
            
              M
              ⁡
              
                (
                
                  p
                  t
                
                )
              
            
            ⁢
            
              (
              
                x
                ,
                y
                ,
                
                  t
                  0
                
              
              )
            
            ⁢
            
              
                t
                
                  m
                  ,
                  x
                
              
              ⁡
              
                (
                
                  x
                  ,
                  y
                  ,
                  
                    t
                    0
                  
                
                )
              
            
          
        
      
    
  




The first term on the right is simply the moveout-corrected derivative. The second term is a correction to this that depends on both the time-derivative of the recorded data and the spatial derivative of the moveout function. Fortunately, both of these are generally easy to compute because the data are well-sampled in time, and the moveout function is an analytical function of offset. If the moveout correction is perfect, in the sense that it renders the data independent of offset, then the second term cancels the first to yield a zero derivative. In general, however, there will be inaccuracies in the moveout function, and moveout correction will create offset-dependent wavelet stretch, such that the derivative will be non-zero.
Hyperbolic Moveout
For the specific case of hyperbolic moveout, we have

t2m(x,y,t0)=t20+(x2+y2)/v2(t0)

where v(t0) is the rms velocity. Differentiating this gives

tm,x=x/v2tm 

in which it should be noted that the numerator is the component of the offset in the direction in which the gradient is required.

Linear Moveout

The corresponding equations for linear moveout are

tm(x,y,t0)=t0+√(x2+y2)/v(t0)

and

tm,x=x/(v√(x2+y2))

FIG. 3 illustrates a flow diagram for the calculation of the gradient of the moveout corrected data.
In step S1, the pressure data p(x, y, t) is input and in step S2 the time derivative pt(x, y, t) is computed. In step S3, the moveout correction is applied to pt(x, y, t) to give M(pt)(x, y, t0).
In step S4, the spatial derivative tm,x(x, y, t0) of the moveout function v(t0) is calculated using the hyperbolic formula.
In step S5, the results of steps S3 and S4 are multiplied. In step S6, the pressure gradient data is input and the moveout correction is applied to give M(px)(x, y, t0). In step S7, the results of steps S5 and S6 are added to give the gradients of the moveout corrected pressure values.
FIG. 4 illustrates a typical system for processing seismic data. Acquired seismic data is stored in a storage means 13, which may be a hard drive, tape or disk or any other form of data storage. A user terminal 11 is used by an operator to input parameters and instructions to build a suitable processing flow for data, which is submitted to a processor 10, which loads the appropriate data from the storage means 13 and carries out the processing steps in accordance with the user's instruction's. The processor 10 then outputs processed data to the storage means 13 (or to a separate storage device). The processor may also send results to the user terminal 11, and/or a notification that the processing is complete. The processing means 10 may also send output data to a printer 12.
FIG. 5 shows an example of the process applied to a simple synthetic dataset. The first five gathers are the raw pressure gradients. The centre five are the gradients of NMO corrected pressure calculated in accordance with the invention, and the final five are pressure gradients derived by direct differentiation of pressure data.
Application to Spatial Interpolation
One example of an algorithm where the present invention is applicable is spatial interpolation using both pressure values and pressure gradient values. With reference to FIG. 1, it may be desirable to interpolate additional data values between the streamers eg interpolating between points x1 and x2. In this case, moveout correction is applied to the pressure data and the gradients of the moveout corrected pressures are obtained by the above method before interpolation in order to reduce aliasing, and then the moveout correction is reversed after interpolation.
In the discussion below, the pressure data and the pressure gradient data are the moveout corrected pressure data and the gradients of the moveout corrected pressures obtained by the above method.
The locations of the known pressure readings and the pressure gradients detected by the hydrophones and the velocity/acceleration sensors may be the same. However, as discussed above, the locations of the pressure values and the pressure gradient values are not necessarily the same. It may be desirable to place the velocity/acceleration sensors more closely than the pressure sensors to remove noise that would otherwise be aliased. In the processing, interpolation may use the finely spaced values, or the velocity/acceleration value may be interpolated to the same spacing as the pressure values, and then the pressure data may be interpolated using co-located measurements. The following treatment therefore does not assume the locations are the same.
Suppose (in 1D) that we have pressure data at locations {xp; i=1, . . . , nd} and gradients at locations {xα; α=1, . . . , ng}. Note the use of different subscripts for the locations of the data and gradients, respectively, so that the locations where the gradients are recorded need not be the same as those where the data are recorded.
Suppose we wish to interpolate the data to a location y. Assume that the interpolated datum at y is a linear combination of the irregular data and the gradients, with coefficients wi and wα respectively (the operator). Consider the interpolation of a function d(x) in the presence of noise. The operator is designed such that the error,




  
    ɛ
    n
  
  =
  
    
      
        ∑
        i
      
      ⁢
      
        
          w
          i
        
        ⁡
        
          [
          
            
              d
              ⁡
              
                (
                
                  x
                  i
                
                )
              
            
            +
            
              n
              i
            
          
          ]
        
      
    
    +
    
      
        ∑
        α
      
      ⁢
      
        
          w
          α
        
        ⁡
        
          [
          
            
              d
              ⁡
              
                (
                
                  x
                  α
                
                )
              
            
            +
            
              n
              α
            
          
          ]
        
      
    
    -
    
      d
      ⁡
      
        (
        y
        )
      
    
  




is small for all functions d(x) of interest. In the above, ni and nα are the noise in the recorded data and derivatives, respectively.

Consider the family of functions:

d(x)=√2 cos(kx+φ)

such that

εn=wi(Ci+ni)+wα(−kSα+nα)−C 

where Ci=√2 cos(k,xi+φ), Sα=√2 sin(kxα+φ), C=√2 cos(ky+φ) and summation is implied by repeated subscripts.

The expected value of the squared error is given by

ε2(k,φ)=E[εn2]=E[wiwi(Ci+ni)(Cj+nj)+wαwβ(−kSα+nα)(−kSβ+nβ+C2+2wiwα(Ci+ni)(−kSα+n−2wi(Ci+ni)C−2wα(−kSα+nα)C]

where E[ . . . ] denotes expected value. Assuming that the noise is uncorrelated at different locations, and that the noise in the data is uncorrelated with that in the derivative even at the same location, then

ε2(k,φ)=wiwj(CiCj+Nδαβ)+wαwβ(k2SαSβ+λNδαβ)+C2−2wiwαkCiSα−2wiCiC+2wαkSαC 

where N and λN are the noise variances in the data and derivatives, respectively.

Given the identities

∫ cos(a+φ)cos(b+φ)dφ=π cos(a−b)

∫ cos(a+φ)sin(b+φ)dφ=−π sin(a−b)

∫ sin(a+φ)sin(b+φ)dφ=π cos(a−b)

wherein the range of integration is 0≦φ≦2π, the phase-averaged error spectrum (defined as previously) is given by





  
    
      
        
          
            ɛ
            ⁡
            
              (
              k
              )
            
          
          2
        
        =
        
        ⁢
        
          
            
              (
              
                2
                ⁢
                π
              
              )
            
            
              -
              1
            
          
          ⁢
          
            ∫
            
              
                
                  ɛ
                  2
                
                ⁡
                
                  (
                  
                    k
                    ,
                    φ
                  
                  )
                
              
              ⁢
              
                d
                φ
              
            
          
        
      
    
  
  
    
      
        =
        
        ⁢
        
          
            
              w
              i
            
            ⁢
            
              w
              j
            
            ⁢
            
              {
              
                
                  cos
                  ⁡
                  
                    (
                    
                      k
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Δ
                        ij
                      
                    
                    )
                  
                
                +
                
                  N
                  ⁢
                  
                    
                  
                  ⁢
                  
                    δ
                    ij
                  
                
              
              }
            
          
          +
          
            
              w
              α
            
            ⁢
            
              w
              β
            
            ⁢
            
              {
              
                
                  
                    k
                    2
                  
                  ⁢
                  
                    cos
                    ⁡
                    
                      (
                      
                        k
                        ⁢
                        
                          
                        
                        ⁢
                        
                          Δ
                          αβ
                        
                      
                      )
                    
                  
                
                +
                
                  λ
                  ⁢
                  
                    
                  
                  ⁢
                  N
                  ⁢
                  
                    
                  
                  ⁢
                  
                    δ
                    αβ
                  
                
              
              }
            
          
          +
        
      
    
  
  
    
      
        
        ⁢
        
          1
          +
          
            2
            ⁢
            
              w
              i
            
            ⁢
            
              w
              α
            
            ⁢
            k
            ⁢
            
              
            
            ⁢
            
              sin
              ⁡
              
                (
                
                  k
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Δ
                    
                      i
                      ⁢
                      
                        
                      
                      ⁢
                      α
                    
                  
                
                )
              
            
          
          -
          
            2
            ⁢
            
              w
              i
            
            ⁢
            
              cos
              ⁡
              
                (
                
                  k
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Δ
                    i
                  
                
                )
              
            
          
          +
          
            2
            ⁢
            
              w
              α
            
            ⁢
            k
            ⁢
            
              
            
            ⁢
            
              sin
              ⁡
              
                (
                
                  k
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Δ
                    α
                  
                
                )
              
            
          
        
      
    
  




where Δij=xi-xj and Δi=xi-y, etc and the range of integration is 0≦φ≦π.

With wavenumber-averages also defined as previously, we have

km−1∫ cos(kx)dk=(kmx)−1[sin(kx)]02π=sin c(kmx)

km−1∫k sin(kx)dk=−∂x{km−1∫ cos(kx)dk}=−km sin c′(kmx)

km−1∫k2 cos(kx)dk=∂x{km−1∫k sin(kx)dk}=−km2 sin c′′(kmx)

where ∂x denotes differentiation with respect to x, and sin c′(x) and sin c′′(x) are the first and second derivatives of the sin c function respectively, both of which are non-singular at x=0.

The wavenumber-averaged error, is given by




  
    
      
        
          ɛ
          2
        
        =
        
        ⁢
        
          
            k
            m
            
              -
              1
            
          
          ⁢
          
            ∫
            
              
                
                  ɛ
                  ⁡
                  
                    (
                    k
                    )
                  
                
                2
              
              ⁢
              
                d
                k
              
            
          
        
      
    
  
  
    
      
        =
        
        ⁢
        
          
            
              w
              i
            
            ⁢
            
              w
              j
            
            ⁢
            
              {
              
                
                  
                    
                      
                        sin
                        ⁢
                        
                          
                        
                        ⁢
                        c
                        ⁢
                        
                          (
                          
                            
                              k
                              m
                            
                            ⁢
                            
                              Δ
                              ij
                            
                          
                          )
                        
                      
                      +
                    
                  
                
                
                  
                    
                      N
                      ⁢
                      
                        
                      
                      ⁢
                      
                        δ
                        ij
                      
                    
                  
                
              
              }
            
          
          +
          
            
              w
              α
            
            ⁢
            
              w
              β
            
            ⁢
            
              {
              
                
                  
                    
                      
                        
                          -
                          
                            k
                            m
                            2
                          
                        
                        ⁢
                        sin
                        ⁢
                        
                          
                        
                        ⁢
                        
                          
                            c
                            ′′
                          
                          ⁡
                          
                            (
                            
                              
                                k
                                m
                              
                              ⁢
                              
                                Δ
                                αβ
                              
                            
                            )
                          
                        
                      
                      +
                    
                  
                
                
                  
                    
                      λ
                      ⁢
                      
                        
                      
                      ⁢
                      N
                      ⁢
                      
                        
                      
                      ⁢
                      
                        δ
                        αβ
                      
                    
                  
                
              
              }
            
          
          +
          1
          -
        
      
    
  
  
    
      
        
        ⁢
        
          
            2
            ⁢
            
              w
              i
            
            ⁢
            
              w
              α
            
            ⁢
            
              k
              m
            
            ⁢
            sin
            ⁢
            
              
            
            ⁢
            
              
                c
                ′
              
              ⁡
              
                (
                
                  
                    k
                    m
                  
                  ⁢
                  
                    Δ
                    
                      i
                      ⁢
                      
                        
                      
                      ⁢
                      α
                    
                  
                
                )
              
            
          
          -
          
            2
            ⁢
            
              w
              i
            
            ⁢
            sin
            ⁢
            
              
            
            ⁢
            
              c
              ⁡
              
                (
                
                  
                    k
                    m
                  
                  ⁢
                  
                    Δ
                    i
                  
                
                )
              
            
          
          -
          
            2
            ⁢
            
              w
              α
            
            ⁢
            
              k
              m
            
            ⁢
            sin
            ⁢
            
              
            
            ⁢
            
              
                c
                ′
              
              ⁡
              
                (
                
                  
                    k
                    m
                  
                  ⁢
                  
                    Δ
                    α
                  
                
                )
              
            
          
        
      
    
  
  
    
      
        =
        
        ⁢
        
          
            
              w
              T
            
            ⁢
            Sw
          
          -
          
            2
            ⁢
            
              w
              T
            
            ⁢
            g
          
          +
          1
        
      
    
  




Integrating from 0≦k≦km 
Where:




  
    w
    =
    
      (
      
        
          
            
              w
              i
            
          
        
        
          
            
              w
              α
            
          
        
      
      )
    
  
  ,
  
    S
    =
    
      (
      
        
          
            
              s
              ij
            
          
          
            
              s
              
                i
                ⁢
                
                  
                
                ⁢
                β
              
            
          
        
        
          
            
              s
              
                α
                ⁢
                
                  
                
                ⁢
                j
              
            
          
          
            
              s
              αβ
            
          
        
      
      )
    
  
  ,
  
    g
    =
    
      (
      
        
          
            
              g
              i
            
          
        
        
          
            
              g
              α
            
          
        
      
      )
    
  




and

Sij=sin c(kmΔij)+NδijSia=−km sin c′(kmΔiα)

Sai=−km sin c′(kmΔiα)Sαβ=−km2 sin c′′(kmΔαβ)+λNδαβ

gi=sin c(kmΔi)

gα=km sin c′(kmΔα)

In accordance with previous notation, the argument k on ε is dropped to indicate the averaging over k. S is symmetrical and depends only on the maximum wavenumber km and the irregular locations.
Differentiating the wavenumber-averaged error with respect to w and setting the derivatives equal to zero gives a simple linear system for the unknown weights, namely:

Sw=g

Which can be solved simply by inverting S. When the operator w is designed such that it satisfies the equation Sw=g then the equation for the wavenumber averaged error reduces to:



  
    
      
        
          
            ɛ
            2
          
          =
          
          ⁢
          
            1
            -
            
              
                w
                T
              
              ⁢
              g
            
          
        
      
    
    
      
        
          =
          
          ⁢
          
            1
            -
            
              
                g
                T
              
              ⁢
              
                S
                
                  -
                  1
                
              
              ⁢
              g
            
          
        
      
    
  




  Because




  
    
      
        
          sin
          ⁢
          
            
          
          ⁢
          
            c
            ⁡
            
              (
              x
              )
            
          
        
      
      
        
          =
          
            
              sin
              ⁡
              
                (
                x
                )
              
            
            /
            x
          
        
      
      
        
          =
          
            1
            -
            
              
                x
                2
              
              /
              6
            
            +
            
              
                x
                4
              
              /
              120
            
            -
            
              
                x
                6
              
              /
              840
            
            +
            ...
          
        
      
    
    
      
        
          sin
          ⁢
          
            
          
          ⁢
          
            
              c
              ′
            
            ⁡
            
              (
              x
              )
            
          
        
      
      
        
          =
          
            
              {
              
                
                  cos
                  ⁡
                  
                    (
                    x
                    )
                  
                
                -
                
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    c
                    ⁡
                    
                      (
                      x
                      )
                    
                  
                
              
              }
            
            /
            x
          
        
      
      
        
          =
          
            
              
                -
                x
              
              /
              3
            
            +
            
              
                x
                3
              
              /
              30
            
            -
            
              
                x
                5
              
              /
              140
            
            +
            ...
          
        
      
    
    
      
        
          sin
          ⁢
          
            
          
          ⁢
          
            
              c
              ′′
            
            ⁡
            
              (
              x
              )
            
          
        
      
      
        
          =
          
            
              {
              
                
                  -
                  
                    sin
                    ⁡
                    
                      (
                      x
                      )
                    
                  
                
                -
                
                  2
                  ⁢
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    
                      c
                      ′
                    
                    ⁡
                    
                      (
                      x
                      )
                    
                  
                
              
              }
            
            /
            x
          
        
      
      
        
          =
          
            
              
                -
                1
              
              /
              3
            
            +
            
              
                x
                2
              
              /
              10
            
            -
            
              
                x
                4
              
              /
              28
            
            +
            ...
          
        
      
    
  



the diagonal elements of S are 1+N (first nd terms) and km2/3+λN (remaining ng terms), respectively. N is essentially equivalent to the usual white noise parameter, and λ defines the relative noise level in the derivatives compared to that in the data. For large λ, the interpolated data will tend to depend on the data, rather than the derivatives.

When km=0, we have Sij=1+Nδij, Sia=Sai=0, Sαβ=λNδαβ, gi=1 and gα=0, such that the solution is wi=1/(nd+N) and vα=0, as expected.
In general, in nD, d(x) is a product of cosine functions in each direction, and each derivative datum represents a derivative of this in a specific direction, which may or may not be aligned with a principle axis. The derivative of d(x) in this direction therefore consists of a sum of terms, each one being a product of (n−1) cosines and a single sine. It is possible to extend the above treatment to a case when both the inline and crossline derivatives are used, to take into account curving of the cable. The matrices above will contain terms each of which is a linear combination of products of sin c functions and their derivatives.
It is further possible to treat the pressure and its derivative as complex numbers, thus taking into account both phase and amplitude. In this case, we consider basis functions ei(kx+φ), allow the operators to be complex and use an asymmetric wavenumber range for the wavenumber averaging (i.e. averaging from k1 to k2 rather than 0 to km).
FIG. 6 compares interpolation results for data-only and with-gradient operators. The irregular locations were computed by adding random variations with a uniform distribution on the interval [−0.5,0.5] to regular locations, x=0, 1, . . . , 10. Noise-free data and gradient information were supplied at the same locations, indicated by the markers. The true curve shows the underlying function being interpolated, which is a sin c function with maximum wavenumber (kf) equal to 1.5 kNyq, where kNyq is the Nyquist corresponding to the average sample interval. The peak of the sin c function is located at a random location taken from a uniform distribution on the interval [4.5,5.5]. The reason for using a random location for this peak is to avoid biases in the ensemble averaging (described later) associated with a correspondence between the peak location and regular data locations.
Interpolation operators were designed using km=1.6 kNyq. The “data only” curve shows the interpolating function for the data-only operator. As expected, there are significant errors because the data are aliased. Note though that the interpolating function does interpolate the supplied data. The “with-gradient” curve is the corresponding function for the with-gradient operator, and it is near perfect.
FIG. 7 shows the gradients of the interpolated functions. The “true” curve is the true gradient and the markers show the supplied gradient information. The “data only” and “with-gradient” curves show the gradients of the interpolated functions for the data-only and with-gradient algorithms respectively. The “data only” curve shows significant error because it does not use the gradient information. The “with-gradient” curve does use the gradient information and properly interpolates the gradients.
In order to remove the impact of having a specific set of irregular locations on the results, ensemble averaging was performed. A specific output location, indicated by the vertical bars in FIGS. 6 and 7, was chosen. In order to avoid bias, the output location was chosen to be displaced from the peak of the sin c function by a random amount with uniform distribution on the interval [−0.5,0.5]. The errors at this location, indicated by the horizontal bars in FIG. 6, were computed for each of the 1000 instances in the ensemble and are plotted in FIG. 8 for the first 200 instances. The benefits of using the gradient information are clear.
It should be noted that the errors plotted in FIG. 8 (actual errors) are not the same as the error, ε, defined above. In particular, the actual errors depend on the function being interpolated, whereas ε depends only on the maximum wavenumber used for the operator design. The reason for using actual errors in this example is that, when the data or gradients are noisy, the actual errors give a true measure of the error in the interpolated function. In contrast, ε only knows about the noise levels in the data and gradients through the supplied values for the noise variances (N and λ), which may be in error.
The impact of noise on the accuracy of the interpolation is extremely important. FIG. 9 shows the dependence of the interpolation error on the standard deviation of the noise level in the data, σd, for kf=1.5 kNyq. The noise was random and Gaussian with zero mean. The gradients are noise-free. In all cases, the noise variance was assumed known for the operator design, such that the operator should be optimal in terms of ε. As expected, the interpolation error increases as the noise increases and the gradient information provide a significant reduction in the error.
FIG. 10 shows the corresponding graph when the errors are in the gradients, rather than the data. The error for the data-only operator is constant. This is to be expected because the gradients are not used. The error for the with-gradient operator grows with increasing noise level, but remains below that for the data-only operator. If the gradients are made very noisy, then the errors would become equal because the gradients would not be used in the interpolation. This last statement assumes that it is known that the gradients are very noisy, otherwise the with gradient operator may be worse than the data-only operator because the former operator would try too hard to honour the gradients.
The present invention has been described particularly with regard to spatial interpolation of marine seismic data. However, the present invention may be used with any data processing algorithm which uses pressure gradient data in which moveout correction is applied prior to the algorithm, either because the algorithm makes a zero offset assumption or because it is beneficial for the algorithm to operate on moveout corrected data to reduce aliasing. In the latter case, the moveout correction need not be full correction to zero offset.
In the foregoing description, for the purposes of illustration, various methods and/or procedures were described in a particular order. It should be appreciated that in alternate embodiments, the methods and/or procedures may be performed in an order different than that described. It should also be appreciated that the methods described above may be performed by hardware components and/or may be embodied in sequences of machine-executable instructions, which may be used to cause a machine, such as a general-purpose or special-purpose processor or logic circuits programmed with the instructions, to perform the methods. These machine-executable instructions may be stored on one or more machine readable media, such as CD-ROMs or other type of optical disks, floppy diskettes, ROMs, RAMs, EPROMs, EEPROMs, magnetic or optical cards, flash memory, or other types of machine-readable media suitable for storing electronic instructions. Merely by way of example, some embodiments of the invention provide software programs, which may be executed on one or more computers, for performing the methods and/or procedures described above. In particular embodiments, for example, there may be a plurality of software components configured to execute on various hardware devices. Alternatively, the methods may be performed by a combination of hardware and software.
Hence, while detailed descriptions of one or more embodiments of the invention have been given above, various alternatives, modifications, and equivalents will be apparent to those skilled in the art without varying from the scope of the invention. Moreover, except where clearly inappropriate or otherwise expressly noted, it should be assumed that the features, devices and/or components of different embodiments can be substituted and/or combined. Thus, the above description should not be taken as limiting the scope of the invention, which is defined by the appended claims.

## Claims

1. A method of processing seismic data, comprising:
receiving from a plurality of seismic receivers in a seismic survey measurements of pressure data and at least one of particle velocity data and particle acceleration data;
using a non-transitory computer readable medium containing computer instructions stored therein, for causing a computer processor to perform the following processes:
processing pressure gradient data using the measured pressure data and at least one of the measured particle velocity data and the measured particle acceleration data;
computing a time-derivative of the measured pressure data by differentiating the measured pressure data;
applying a moveout correction function to the time-derivative of the measured pressure data to obtain a corrected time-derivative result;
computing a spatial derivative of the moveout correction function;
multiplying the corrected time-derivative result and the spatial derivative to obtain a correction factor;
applying the moveout correction function to the processed pressure gradient data to obtain moveout pressure gradient data;
adding the correction factor and the moveout pressure gradient to obtain moveout corrected pressure gradient data, wherein the moveout corrected pressure gradient data comprises the processed pressure gradient data corrected for moveout; and
using the moveout corrected pressure gradient data to process physical properties of the earth's interior.

2. The method of claim 1, further comprising the processes of:
applying the moveout correction function to the measured pressure data to determine moveout corrected pressure data; and
spatially interpolating additional moveout corrected pressure data values using the moveout corrected pressure gradient data and the moveout corrected pressure data, wherein the additional moveout corrected pressure data comprises processed pressure values for locations where the pressure was not measured by the plurality of seismic receivers.

3. The method of claim 2, wherein the interpolation comprises applying an interpolation operator to the moveout corrected pressure data, wherein the interpolation operators is calculated by assuming that an interpolated pressure value comprises a linear combination of known pressure values at locations xi and spatial derivatives at locations xα, and operator coefficients wi and wα respectively, and wherein the coefficients wi and wα are calculated by minimizing an error function.

4. The method of claim 3, wherein minimizing an error function comprises using a least squares method.

5. The method of claim 3, wherein the operator includes a factor relating a noise level in the known pressure values and spatial derivatives.

6. The method of claim 3, wherein a direction of interpolation is substantially perpendicular to a line formed by two or more of the plurality of seismic receivers.

7. The method of claim 3, wherein the seismic data is marine data acquired using a plurality of substantially parallel streamers having hydrophones spaced along each streamer, and the method is used to interpolate data points between the streamers.

8. The method of claim 1, wherein the moveout correction function is a hyperbolic function.

9. The method of claim 1, wherein the process of using the moveout corrected pressure gradient data to process physical properties of the earth's interior comprises applying a data processing algorithm utilizing the moveout corrected pressure gradient-data and moveout corrected pressure data.

10. The method of claim 1, wherein the pressure data is derived from hydrophone readings, and the pressure gradient data derived from measurements of particle velocity or acceleration made by geophones.

11. The method claim 1, wherein the pressure data is produced from shot gathers.

12. The method of claim 1, wherein the measured pressure data is calculated from the particle velocity data or particle acceleration data.