Patent Publication Number: US-9423518-B2

Title: Method for processing dual-sensor streamer data with anti-alias protection

Description:
BACKGROUND 
     In the oil and gas industry, geophysical prospecting is commonly used to aid in the search for and evaluation of subsurface earth formations. Geophysical prospecting techniques yield knowledge of the subsurface structure of the earth, which is useful for finding and extracting valuable mineral resources, particularly hydrocarbon deposits such as oil and natural gas. A well-known technique of geophysical prospecting is a seismic survey. In a land-based seismic survey, a seismic signal is generated on or near the earth&#39;s surface and then travels downward into the subsurface of the earth. In a marine seismic survey, the seismic signal may also travel downward through a body of water overlying the subsurface of the earth. Seismic energy sources are used to generate the seismic signal which, after propagating into the earth, is at least partially reflected by subsurface seismic reflectors. Such seismic reflectors typically are interfaces between subterranean formations having different elastic properties, specifically sound wave velocity and rock density, which lead to differences in acoustic impedance at the interfaces. The reflected seismic energy is detected by seismic sensors (also called seismic receivers) at or near the surface of the earth, in an overlying body of water, or at known depths in boreholes. The seismic sensors generate signals, typically electrical or optical, from the detected seismic energy, which are recorded for further processing. 
     The appropriate seismic sources for generating the seismic signal in land seismic surveys may include explosives or vibrators. Marine seismic surveys typically employ a submerged seismic source towed by a ship and periodically activated to generate an acoustic wavefield. The seismic source generating the wavefield may be of several types, including a small explosive charge, an electric spark or arc, a marine vibrator, and, typically, a gun. The seismic source gun may be a water gun, a vapor gun, and, most typically, an air gun. Typically, a marine seismic source consists not of a single source element, but of a spatially-distributed array of source elements. This arrangement is particularly true for air guns, currently the most common form of marine seismic source. 
     The appropriate types of seismic sensors typically include particle velocity sensors, particularly in land surveys, and water pressure sensors, particularly in marine surveys. Sometimes particle displacement sensors, particle acceleration sensors, or pressure gradient sensors are used in place of or in addition to particle velocity sensors. Particle velocity sensors and water pressure sensors are commonly known in the art as geophones and hydrophones, respectively. Seismic sensors may be deployed by themselves, but are more commonly deployed in sensor arrays. Additionally, pressure sensors and particle motion sensors may be deployed together in a marine survey, collocated in pairs or pairs of arrays. 
     In a typical marine seismic survey, a seismic survey vessel travels on the water surface, typically at about 5 knots, and contains seismic acquisition equipment, such as navigation control, seismic source control, seismic sensor control, and recording equipment. The seismic source control equipment causes a seismic source towed in the body of water by the seismic vessel to actuate at selected times. Seismic streamers, also called seismic cables, are elongate cable-like structures towed in the body of water by the seismic survey vessel that tows the seismic source or by another seismic survey ship. Typically, a plurality of seismic streamers are towed behind a seismic vessel. The seismic streamers contain sensors to detect the reflected wavefields initiated by the seismic source and returning from reflective interfaces. 
     It is well known in the art that pressure and particle motion signals can be combined to derive both the up-going and the down-going wavefield. For sea floor recordings, the up-going and down-going wavefields may subsequently be combined to remove the effect of the surface reflection and to attenuate water borne multiples in the seismic signal. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
       The invention and its advantages may be more easily understood by reference to the following detailed description and the attached drawings, in which: 
         FIG. 1  is a diagram of an exemplary system for acquiring seismic data that can be used with seismic data processing methods according to the invention; 
         FIG. 2  is a diagram illustrating, by way of example, one of many different types of computer systems that can be used with seismic data processing methods according to the invention; 
         FIG. 3  is a plot of two plane wave events in a f-k x  domain; 
         FIG. 4  is a plot of two plane wave events in a f-k x  domain with no backscattered energy; 
         FIG. 5  is a flowchart illustrating an embodiment of the invention for geophysical prospecting; 
         FIG. 6  is a seismic section of the modeled pressure data. 
         FIG. 7  is a seismic section of the modeled vertical particle velocity data; 
         FIG. 8  is a seismic section of the modeled up-going pressure field that serves as the reference; 
         FIG. 9  is a seismic section of the up-going pressure field obtained from the input data in  FIGS. 6 and 7  using standard wavefield separation; 
         FIG. 10  is the difference between the up-going pressure wavefield for the standard wavefield separation in  FIG. 9  and for the reference in  FIG. 8 ; 
         FIG. 11  shows the up-going pressure field obtained from the input data in  FIGS. 6 and 7  using wavefield separation applying the method of the invention; 
         FIG. 12  is the difference between the up-going pressure wavefield for the wavefield separation applying the method of the invention in  FIG. 11  and for the reference in  FIG. 8 ; 
         FIG. 13  is the modeled pressure data in a f-k x  domain, which corresponds to the seismic section shown in  FIG. 6 ; 
         FIG. 14  is the modeled vertical particle velocity data in a f-k x  domain, which corresponds to the seismic section shown in  FIG. 7 ; 
         FIG. 15  is the modeled up-going pressure field reference in a f-k x  domain, which corresponds to the seismic section in  FIG. 8 ; 
         FIG. 16  shows the up-going pressure wavefield in a f-k x  domain obtained from standard wavefield separation, corresponding to the seismic section in  FIG. 9 ; and 
         FIG. 17  is the up-going pressure wavefield in the f-k x  domain obtained from wavefield separation applying the method of the invention, corresponding to the seismic section in  FIG. 11 . 
     
    
    
     While the invention will be described in connection with its preferred embodiments, it will be understood that the invention is not limited to these. On the contrary, the invention is intended to cover all alternatives, modifications, and equivalents that may be included within the scope of the invention, as defined by the appended claims. 
     DETAILED DESCRIPTION 
       FIG. 1  is a diagram of an exemplary system for acquiring seismic data that can be used with seismic data processing methods according to the invention. In various embodiments, a single seismic sensor cable (also called a seismic streamer) or a single ocean bottom cable are shown for simplicity of illustration. This illustration of one cable is only meant to more clearly demonstrate principles of the invention and is not intended as a limitation of the invention. 
     In  FIG. 1 , the seismic acquisition system is designated generally as  100 . A seismic vessel  101  is disposed in a body of water  102  and carries equipment  103  for navigation, seismic source control, and seismic sensor recording. The seismic vessel  101  or another service vessel (not shown) tows seismic sources  104  through the body of water  102  below the surface  105  of the water. Two seismic sources  104  at different depths are illustrated here in  FIG. 4 . The seismic sources  104  comprise any appropriate type of source, typically in arrays. The configuration of seismic sources  104  illustrated is not intended to be a limitation of the invention. 
     In one embodiment, the seismic vessel  101  or another service vessel (not shown) tows a seismic streamer  106  through the body of water  102 . The seismic streamer  106  comprises seismic sensors  107  at spaced apart positions along the seismic streamer  106 , so that the seismic streamer  106  containing the seismic sensors  107  is disposed in the body of water  102 . The seismic sensors  107  are typically pressure sensors, such as hydrophones. In another embodiment, the seismic streamer  106  comprises a dual-sensor streamer, in which the seismic sensors  107  comprise pairs of collocated pressure and particle motion sensors. The particle motion sensors are typically particle velocity sensors, such as geophones, or accelerometers. The seismic sensors  107  typically comprise arrays of sensors at each spaced apart position. An alternative to having the pressure and particle motion sensors co-located is to have sufficient spatial density of sensors so that the respective wavefields recorded by the pressure and particle motion sensors can be interpolated or extrapolated to produce the two wavefield signals at the same location. 
     In another embodiment, the seismic vessel  101  or another service vessel (not shown) disposes an ocean bottom cable  108  on the water bottom  109 . The ocean bottom cable  108  also comprises seismic sensors  107  at spaced apart positions along the cable, also typically in arrays of sensors at each spaced apart position. The seismic sensors  107  in the ocean bottom sensor  108  can also comprise pairs of pressure and particle motion sensors. In yet another embodiment, both seismic streamers  106  and ocean bottom cable  108  are employed. The type of sensors illustrated in the seismic acquisition system  100  is not intended to be a limitation of the invention. For example, in other embodiments, discrete seismic sensors  107  located at ocean bottom nodes (not shown) could be included in the seismic acquisition system  100 . 
     When the seismic sources  104  are activated, acoustic energy travels downwardly, at  110 , through the body of water  102  and the water bottom  109  to layer boundaries, such as  111  and  112 , surrounding a subterranean formation layer, such as  113 . A portion of the acoustic energy is reflected from the layer boundary at  111  and travels upwardly, at  114 . The upwardly traveling acoustic energy  114  is detected at seismic sensors  107  on the ocean bottom cable  108  or the seismic streamer  106 . The upwardly traveling acoustic energy continues upward, at  115 , until reflecting off the water surface  105  and then travels downwardly again, at  116 . The downwardly traveling acoustic energy  116  may be detected again by seismic sensors  107  on the seismic streamer  106  or the ocean bottom cable  108 , resulting in a ghost signal. The acoustic energy detected at the seismic sensors  107  may be recorded onto any type of appropriate storage media at any location, such as, but not restricted to, at the seismic streamer  106  or the ocean bottom cable  108 , on the seismic vessel  101  or another service vessel, or onshore. 
       FIG. 2  is a diagram illustrating, by way of example, one of many different types of computer systems that can be used with seismic data processing methods according to the invention. A central processor  20  is coupled to user input devices, such as a keyboard  21  (wired or wireless) and a mouse  22  (wired or wireless). The processor  20  is further coupled to a display, such as a monitor  23 . A computer program according to the invention may reside on any of a number of computer readable media, such as a disk  24  insertable into a disk drive  25  or on an internal or external hard drive (not shown). 
     A recorded seismic wavefield is typically sampled at discrete locations which results in spatial aliasing. Energy at spatial wavenumbers higher than the Nyquist wavenumber, which is determined by the sensor separation in each direction independently, will be mapped to an incorrect wavenumber when multi-channel processing methods are applied. Consequently spatially aliased energy will not generally be handled correctly by such processing methods due to this ambiguity in the spatial wavenumber. Common methods for dealing with this problem include assuming that there is no aliased energy, or identifying and removing the aliased energy before applying multi-channel processing. Both of these approaches degrade the accuracy of the output. Thus, a need exists for a method for handling spatial aliasing in seismic data processing; especially a method that can take advantage of dual-sensor data. 
     Processing of seismic data from dual-sensor streamers, which includes low-frequency reconstruction of geophone signals (LFC), wavefield separation, and redatuming, is typically performed in the frequency-wavenumber (f-k) domain in order to deal with seismic events with any emergence angle, and in particular, high emergence angles. Each point in f-k space corresponds to a particular emergence angle. However, for aliased energy this emergence angle is in error. The problem is illustrated in  FIG. 3  for a 2-D acquisition geometry, which shows a plot of two plane wave events in a f-k x  domain. 
     Solid black lines  30  are events with velocity 1500 m/s, the nominal speed of sound in water. In practice, a user-specified water velocity based on the best available measurement would be used to identify the region where aliasing occurs rather than the nominal 1500 m/s used here in  FIG. 3  for merely illustrative purposes. Numbers in boxes  31  are the maximum order of aliasing in each region. A first event  32  and a second event  33  cross at a point  34  where the maximum aliasing order is 1. The first event  32  is not aliased, and so will be processed correctly. However, the second event  33  will only be processed correctly up to the temporal frequency that corresponds to the Nyquist wavenumber for an event at that particular emergence angle. In processing data from dual-sensor streamers, the obliquity term is typically parameterized as a k z  term, where k z  is the vertical component of the angular wavenumber vector. 
     When performing wavefield separation or other seismic processing, different values of k z  should be used for the two events  32 ,  33  for correct scaling at the cross-over point  34 . For example, the obliquity factor for a constant velocity water column can be defined in the frequency-wavenumber domain as 
                 ρ   ⁢           ⁢   ω       k   z       ,         
where ρ is water density, ω is circular frequency, and k z  is the vertical component of the circular wavenumber vector.
 
     The vertical wavenumber k z  is calculated at each point in f-k space. However, if spatially aliased energy is present at a particular f-k location, the value of k z  that is used is systematically too large for the aliased part. This leads to incorrect treatment of the aliased energy in dual-sensor data processing. 
     For the 2-D example illustrated in  FIG. 3 , at the point  34  where the first event  32  and the second event  33  cross, the obliquity term that will be used in conventional dual-sensor streamer would use the conventional vertical wavenumber, designated here as k z0 : 
                       k     z   ⁢           ⁢   0       =           (     ω   c     )     2     -     k   x   2           ,           (     1   ⁢   a     )               
where c is the speed of sound in water and k x  is a horizontal wavenumber. Conventionally, k x  is in the inline direction, that is, parallel to the towed streamers. In the 3-D case, the expression corresponding to Equation (1a) is given by:
 
                       k     z   ⁢           ⁢   0       =           (     ω   c     )     2     -     k   x   2     -     k   y   2           ,           (     1   ⁢   b     )               
where k y  is a horizontal wavenumber orthogonal to k x . Thus, conventionally, k y  is in the crossline direction, that is, perpendicular to the towed streamers.
 
     Equations (1a) and (1b) are correct for the first event  32 , but the obliquity factor that should be used for the second event  33  should use the following alternative vertical wavenumber of the invention, designated here as k z1 : 
                       k     z   ⁢           ⁢   1       =           (     ω   c     )     2     -       (       2   ⁢           ⁢     k   Ny       -          k   x            )     2           ,           (     2   ⁢   a     )               
where k Ny  is the Nyquist wavenumber for the recorded seismic data. In the 3-D case, the expression corresponding to Equation (2a) is given by:
 
                       k     z   ⁢           ⁢   1       =           (     ω   c     )     2     -       (       2   ⁢           ⁢     k     Ny   -   x         -          k   x            )     2     -       (       2   ⁢           ⁢     k     Ny   -   y         -          k   y            )     2           ,           (     2   ⁢   b     )               
where k Ny-x  and k Ny-y  are Nyquist wavenumbers in the inline and crossline directions, respectively. These Nyquist wavenumbers may take different values if the inline and crossline sampling intervals are different, as they typically are.
 
     The invention is a method for producing correct emergence angles for seismic processing in the presence of aliased energy. In the method of this invention, recorded seismic energy in the pressure and particle velocity records that maps to a particular point in f-k space is separated into aliased and unaliased parts. In particular, recorded seismic energy in the regions labeled 1 is separated into an unaliased part (designated by subscript “0”) and a first-order aliased part (designated by subscript “1”). This separation is discussed below. 
     The recorded pressure data P can be expressed as a sum of an unaliased part P 0  and an aliased part P 1  by:
 
 P=P   0   +P   1   (3)
 
and the vertical particle velocity data V can be expressed as a sum of an unaliased part V 0  and an aliased part V 1  by:
 
 V=V   0   +V   1 .  (4)
 
     The free surface condition is applied to the unaliased and aliased parts in Equations (3) and (4) independently, using the obliquity factors given by Equations (1a) or (1b), and (2a) or (2b), respectively. The free surface condition assumes that the pressure at the sea surface is zero. For illustrative purposes, the free surface condition will be applied to the case in which both the sea surface and the recording surface (cable containing the receivers) are flat and a known distance apart. By making this simple assumption, it is possible to illustrate an embodiment of the invention using simple algebraic expressions in the f-k domain. However, the invention does not require these assumptions about the sea surface and recording surface shape, as long as their relative positions are known. Applying the free surface condition yields: 
                           k     z   ⁢           ⁢   0         ω   ⁢           ⁢   ρ       ⁢     (     1   +     exp   ⁡     [       -   2     ⁢           ⁢   ⅈ   ⁢           ⁢     k     z   ⁢           ⁢   0       ⁢     z   R       ]         )     ⁢     P   0       =       -     (     1   -     exp   ⁡     [       -   2     ⁢           ⁢   ⅈ   ⁢           ⁢     k     z   ⁢           ⁢   0       ⁢     z   R       ]         )       ⁢     V   0         ⁢     
     ⁢   and           (   5   )                       k     z   ⁢           ⁢   1         ω   ⁢           ⁢   ρ       ⁢     (     1   +     exp   ⁡     [       -   2     ⁢           ⁢   ⅈ   ⁢           ⁢     k     z   ⁢           ⁢   1       ⁢     z   R       ]         )     ⁢     P   1       =       -     (     1   -     exp   ⁡     [       -   2     ⁢           ⁢   ⅈ   ⁢           ⁢     k     z   ⁢           ⁢   1       ⁢     z   R       ]         )       ⁢     V   1         ,           (   6   )               
respectively, where z R  is receiver depth.
 
     Equations (3) to (6) yield a fully determined system of equations that can be solved to split the pressure and particle velocity records into aliased and unaliased parts. In other words, Equations (3) to (6) yield four linearly independent equations in the four unknowns P 0 , P 1 , V 0 , and V 1  and thus can be solved uniquely. 
     The solution to the system of Equations (3) to (6) can be simplified by using the variable substitution:
 
 {tilde over (V)}={tilde over (V)}   0   +{tilde over (V)}   1   =−iωρV.   (7)
 
The substitution in Equation (7) allows one to rewrite the free surface condition in Equations (5) and (6) as:
 
 k   z0  cos( k   z0   z   R ) P   0 =sin( k   z0   z   R ) {tilde over (V)}   0   (8)
 
and
 
 k   z1  cos( k   z1   z   R ) P   1 =sin( k   z1   z   R ) {tilde over (V)}   1 .  (9)
 
     The solution to the four unknowns P 0 , P 1 , {tilde over (V)} 0 , and {tilde over (V)} 1  in the four Equations (3), (7), (8), and (9) can then be expressed concisely in matrix notation as: 
                       (           P   0               P   1                 V   ~     0                 V   ~     1           )     =       1   D     ⁢     (             k     z   ⁢           ⁢   1       ⁢     sin   ⁡     (       k     z   ⁢           ⁢   0       ⁢     z   R       )       ⁢     cos   ⁡     (       k     z   ⁢           ⁢   1       ⁢     z   R       )                 -     sin   ⁡     (       k     z   ⁢           ⁢   0       ⁢     z   R       )         ⁢     sin   ⁡     (       k     z   ⁢           ⁢   1       ⁢     z   R       )                     -     k     z   ⁢           ⁢   0         ⁢     cos   ⁡     (       k     z   ⁢           ⁢   0       ⁢     z   R       )       ⁢     sin   ⁡     (       k     z   ⁢           ⁢   1       ⁢     z   R       )                 sin   ⁡     (       k     z   ⁢           ⁢   0       ⁢     z   R       )       ⁢     sin   ⁡     (       k     z   ⁢           ⁢   1       ⁢     z   R       )                     k     z   ⁢           ⁢   0       ⁢     k     z   ⁢           ⁢   1       ⁢     cos   ⁡     (       k     z   ⁢           ⁢   0       ⁢     z   R       )       ⁢     cos   ⁡     (       k     z   ⁢           ⁢   1       ⁢     z   R       )                 -     k     z   ⁢           ⁢   0         ⁢     cos   ⁡     (       k     z   ⁢           ⁢   0       ⁢     z   R       )       ⁢     sin   ⁡     (       k     z   ⁢           ⁢   1       ⁢     z   R       )                     -     k     z   ⁢           ⁢   0         ⁢     k     z   ⁢           ⁢   1       ⁢     cos   ⁡     (       k     z   ⁢           ⁢   0       ⁢     k   R       )       ⁢     cos   ⁡     (       k     z   ⁢           ⁢   1       ⁢     z   R       )                 k     z   ⁢           ⁢   1       ⁢     sin   ⁡     (       k     z   ⁢           ⁢   0       ⁢     z   R       )       ⁢     cos   ⁡     (       k     z   ⁢           ⁢   1       ⁢     z   R       )               )     ⁢     (         P             V   ~           )         ,           (   10   )               
where the denominator factor D in Equation (10) is given by:
 
 D=k   z1  sin( k   z0   z   R )cos( k   z1   z   R )− k   z0  cos( k   z0   z   R )sin( k   z1   z   R ).  (11)
 
     From Equation (11), when the denominator factor D=0, then:
 
 k   z1  sin( k   z0   z   R )cos( k   z1   z   R )= k   z0  cos( k   z0   z   R )sin( k   z1   z   R ).  (12)
 
It follows from Equation (12) that:
 
 k   z1  cot( k   z1   z   R )= k   z0  cot( k   z0   z   R ).  (13)
 
     The representation of the five surface condition given by Equations (8) and (9) can then be rewritten as:
 
 k   z0  cot( k   z0   z   R ) P   0   ={tilde over (V)}   0   (14)
 
and
 
 k   z1  cot( k   z1   z   R ) P   1   ={tilde over (V)}   1 .  (15)
 
By Equation (13), the cotangent term in either of Equations (14) or (15) can then be substituted for the cotangent term in the other equation.
 
     Thus, when D=0, the relationship between pressure and particle velocity is the same for both the unaliased and aliased parts of the seismic energy. Hence it is not possible to use the free surface condition to separate them. This means that at these locations in f-k space, the system of equations being solved is underdetermined. 
     For values of the denominator factor D in Equation (11) close to zero, the solution of Equation (10) becomes unstable. In one embodiment, an attempt to prevent division by D≈0 in Equation (10) is given by:
 
for | D |&lt;ε, all energy is treated as unaliased.  (16)
 
     In an alternative embodiment, the following is used:
 
for | D |&lt;ε, all energy is treated as aliased,  (17)
 
where ε is a small number chosen to provide a stable separation of aliased and unaliased energy. Exemplary appropriate values for r are 0.1 or 0.01.
 
     A further effect that should be taken into account is the effect of receiver array responses, which are different for hydrophone and geophone arrays, since, for example, it is not possible to locate a hydrophone and a geophone at the same physical location in the cable. In one embodiment, a spatial filter is applied to the geophone data to match it to the hydrophone data. In an alternative embodiment, a spatial filter is applied to the hydrophone data to match it to the geophone data. In yet another embodiment, spatial filters are applied to both the geophone and hydrophone data to match them to each other. Equations (5) and (6), from which all subsequent relationships are derived, assume that suitable spatial filters have been applied to match the responses of the two sensors. However, this filter is correct for unaliased energy only. For 12.5 m trace spacing, aliasing begins at 0.04/m. Hence a correction factor must be built in to correct the array response for aliased energy. 
     For illustrative purposes, the method of accounting for differences in receiver array responses will be applied in the case in which both hydrophone and geophone receiver arrays comprise regularly spaced arrays of sensors. By making this simplification, it is possible to illustrate the invention using simple algebraic expressions. However, the invention does not require these assumptions about the array geometries, provided that the locations of the sensors making up each array are known, so that their transfer functions, and hence their responses, can be determined. Furthermore, for illustrative purposes, it is assumed that suitable spatial filters have been applied to one, the other, or both of the geophone and hydrophone data, to match the responses of the two sensor arrays. 
     The response A of an array of N receivers with regular receiver separation Δx can be expressed as a function of horizontal wavenumber k x  as follows: 
     
       
         
           
             
               
                 
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     When the pressure sensor (P) and velocity sensor (V) arrays comprise different numbers of sensors N p  and N v , respectively, with different receiver spacing Δx p  and Δx v , respectively, then the filter F that is applied to match the response of the velocity sensor array A v  to the response of the pressure sensor array A p  for unaliased energy is given by: 
                   F   =         A   p       A   v       =           N   v       N   p       ⁡     [         sin   ⁡     (         N   p     ⁢           ⁢   Δ   ⁢           ⁢     x   p     ⁢           ⁢     k   x       2     )       ⁢     sin   ⁡     (       Δ   ⁢           ⁢     x   v     ⁢     k   x       2     )             sin   ⁡     (         N   v     ⁢   Δ   ⁢           ⁢     x   v     ⁢           ⁢     k   x       2     )       ⁢     sin   ⁡     (       Δ   ⁢           ⁢     x   p     ⁢     k   x       2     )           ]       .               (   19   )               
In the embodiment illustrated in Equation (19), it is assumed that the matching spatial filter has been applied to the geophone data.
 
     For aliased energy, the filter F′ that should have been applied is: 
     
       
         
           
             
               
                 
                   
                     
                       
                         
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                                       ] 
                                     
                                   
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                                     sin 
                                     ⁡ 
                                     
                                       [ 
                                       
                                         
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                                           ⁢ 
                                           
                                               
                                           
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                             . 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   20 
                   ) 
                 
               
             
           
         
       
     
     Hence the correction factor F c  that should be applied to the aliased part of the particle velocity data to account for the incorrect treatment of aliased energy by the array response matching step is the ratio given by: 
     
       
         
           
             
               
                 
                   
                     
                       
                         
                           F 
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                         = 
                           
                         ⁢ 
                         
                           
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                                       sin 
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                           . 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   21 
                   ) 
                 
               
             
           
         
       
     
     To account for array response matching, the set of equations, analogous to Equations (3) and (7) to (9) above, that must be solved is revised as follows: 
     
       
         
           
             
               
                 
                   P 
                   = 
                   
                     
                       P 
                       0 
                     
                     + 
                     
                       P 
                       1 
                     
                   
                 
               
               
                 
                   ( 
                   22 
                   ) 
                 
               
             
             
               
                 
                   
                     V 
                     ~ 
                   
                   = 
                   
                     
                       
                         V 
                         ~ 
                       
                       0 
                     
                     + 
                     
                       
                         
                           V 
                           ~ 
                         
                         1 
                       
                       
                         F 
                         c 
                       
                     
                   
                 
               
               
                 
                   ( 
                   23 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       k 
                       
                         z 
                         ⁢ 
                         
                             
                         
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                     ⁢ 
                     
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                       ⁡ 
                       
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                         ) 
                       
                     
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                       ⁡ 
                       
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                   = 
                   
                     
                       sin 
                       ⁡ 
                       
                         ( 
                         
                           
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                           V 
                           ~ 
                         
                         1 
                       
                       . 
                     
                   
                 
               
               
                 
                   ( 
                   25 
                   ) 
                 
               
             
           
         
       
     
     The solution, analogous to Equation (10) above, to the four unknowns P 0 , P 1 , {tilde over (V)} 0 , and {tilde over (V)} 1  in Equations (22) to (25) is: 
                       (           P   0               P   1                 V   ~     0                 V   ~     1           )     =       1     D   ′       ⁢     (             k     z   ⁢           ⁢   1       ⁢     sin   ⁡     (       k     z   ⁢           ⁢   0       ⁢     z   R       )       ⁢     cos   ⁡     (       k     z   ⁢           ⁢   1       ⁢     z   R       )                 -     F   c       ⁢     sin   ⁡     (       k     z   ⁢           ⁢   0       ⁢     z   R       )       ⁢     sin   ⁡     (       k     z   ⁢           ⁢   1       ⁢     z   R       )                     -     F   c       ⁢     k     z   ⁢           ⁢   0       ⁢     cos   ⁡     (       k     z   ⁢           ⁢   0       ⁢     z   R       )       ⁢     sin   ⁡     (       k     z   ⁢           ⁢   1       ⁢     z   R       )                 F   c     ⁢     sin   ⁡     (       k     z   ⁢           ⁢   0       ⁢     z   R       )       ⁢     sin   ⁡     (       k     z   ⁢           ⁢   1       ⁢     z   R       )                     k     z   ⁢           ⁢   0       ⁢     k     z   ⁢           ⁢   1       ⁢     cos   ⁡     (       k     z   ⁢           ⁢   0       ⁢     z   R       )       ⁢     cos   ⁡     (       k     z   ⁢           ⁢   1       ⁢     z   R       )                 -     F   c       ⁢     k     z   ⁢           ⁢   0       ⁢     cos   ⁡     (       k     z   ⁢           ⁢   0       ⁢     z   R       )       ⁢     sin   ⁡     (       k     z   ⁢           ⁢   1       ⁢     z   R       )                     -     F   c       ⁢     k     z   ⁢           ⁢   0       ⁢     k     z   ⁢           ⁢   1       ⁢     cos   ⁡     (       k     z   ⁢           ⁢   0       ⁢     z   R       )       ⁢     cos   ⁡     (       k     z   ⁢           ⁢   1       ⁢     z   R       )                 F   c     ⁢     k     z   ⁢           ⁢   1       ⁢     sin   ⁡     (       k     z   ⁢           ⁢   0       ⁢     z   R       )       ⁢     cos   ⁡     (       k     z   ⁢           ⁢   1       ⁢     z   R       )               )     ⁢     (         P             V   ~           )         ,           (   26   )               
where the denominator factor D in Equation (11) is replaced by the analogous denominator factor D′, given by:
 
 D′=k   z1  sin( k   z0   z   R )cos( k   z1   z   R )− F   c   k   z0  cos( k   z0   z   R )sin( k   z1   z   R ).  (27)
 
     For values of the denominator factor D′ in Equation (27) close to zero, the solution of Equation (26) becomes unstable. An approach for stabilizing the result can be adopted that is similar to that described above for Equation (10). This approach is discussed above with respect to Equations (16) and (17). 
     For 12.5 m channel separation and 1500 m/s sound propagation velocity, spatial aliasing starts at 60 Hz. In practice, this means that low-frequency geophone signal reconstruction is mostly immune to aliasing effects, but wavefield separation and redatuming are affected. 
     The invention allows first-order aliased energy to be treated correctly. Higher aliasing orders, denoted by numbers greater than 1 in  FIG. 3  will not be treated correctly. However, a further simplifying assumption that can be made is to assume all the signal energy is scattered forward. This assumption means that there will not be aliasing orders higher than 1 below about 180 Hz. This is shown in  FIG. 4 , which is a plot of two plane wave events  32 ,  33  in a f-k x  domain with no backscattered energy ( FIG. 4  uses the same reference numerals as  FIG. 3 ). Although any aliased backscattered energy will not be treated correctly, for the case of towed marine streamer acquisition, there is likely to be very little of this energy in practice. Using the convention that forward scattered energy maps to positive wavenumbers, the expression for k z1  is changed from that in Equation (2) to the following: 
                     k     z   ⁢           ⁢   1       =           (     ω   c     )     2     -         (       2   ⁢     k   Ny       +     k   x       )     2     .                 (     28   ⁢   a     )               
If there is assumed to be no aliased back scattered energy, then Equation (28) is correct in the regions with aliasing orders up to 1, numbered 0 and 1 (designated by numbered boxes  31 ) in  FIG. 4 . Note that these areas in  FIG. 4  represent a greater portion of the data than the corresponding areas in  FIG. 3 . In the 3-D case, the expression corresponding to Equation (28) is given by:
 
     
       
         
           
             
               
                 
                   
                     k 
                     
                       z 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       1 
                     
                   
                   = 
                   
                     
                       
                         
                           
                             ( 
                             
                               ω 
                               c 
                             
                             ) 
                           
                           2 
                         
                         - 
                         
                           
                             ( 
                             
                               
                                 2 
                                 ⁢ 
                                 
                                   k 
                                   
                                     Ny 
                                     - 
                                     x 
                                   
                                 
                               
                               + 
                               
                                 k 
                                 x 
                               
                             
                             ) 
                           
                           2 
                         
                         - 
                         
                           
                             ( 
                             
                               
                                 2 
                                 ⁢ 
                                 
                                   k 
                                   
                                     Ny 
                                     - 
                                     y 
                                   
                                 
                               
                               + 
                               
                                 k 
                                 y 
                               
                             
                             ) 
                           
                           2 
                         
                       
                     
                     . 
                   
                 
               
               
                 
                   ( 
                   
                     28 
                     ⁢ 
                     b 
                   
                   ) 
                 
               
             
           
         
       
     
       FIG. 5  is a flowchart illustrating an embodiment of the invention for geophysical prospecting. 
     At block  50 , dual-sensor seismic data are separated into aliased and unaliased parts in a frequency-wavenumber domain. In one embodiment, the dual-sensor seismic data comprises recorded pressure and particle velocity data. In a first embodiment, the separation is given by Equations (10) and (11). In a second embodiment, the separation is given by Equations (21), (26), and (27). 
     At block  51 , the seismic data in the unaliased part from block  50  are processed using a conventional vertical wavenumber. In one embodiment, the conventional vertical wavenumber is given by Equation (1). 
     At block  52 , the seismic data in the aliased part from block  50  are processed using an alternative vertical wavenumber additionally based on a Nyquist wavenumber for the seismic data. In one embodiment, the alternative vertical wavenumber is given by Equation (2). In an alternative embodiment, where it is assumed that all signal energy is scattered forward, the alternative vertical wavenumber is given by Equation (28). In various embodiments, the processing includes, but is not limited to, wavefield separation, redatuming, and applying a dip filter. 
     At block  53 , the processed unaliased seismic data from block  51  and the processed aliased seismic data from block  52  are combined. The combined processed data may be transformed back into the space-time (x-t) domain for viewing and further processing. 
       FIGS. 6 to 17  illustrate the application of the method of the invention to wavefield separation for a synthetic dataset.  FIGS. 6 to 12  show the results as seismic sections, while  FIGS. 13 to 17  show the same results in the frequency-wavenumber domain. 
     The synthetic dataset generated comprised three planar events with emergence angles of 0°, 30°, and 60° simulated. Data were simulated for 12.5 m receiver separation and a 120 Hz low-pass filter, applied to ensure that there is no more than one order of aliasing (see  FIG. 3 ), so that the assumptions of the method are satisfied. 
     Three datasets were created. First, a reference dataset with no ghost was created. Second, a hydrophone dataset was created by convolution of the reference dataset with a ghost function with a reflection coefficient of −1. Third, a geophone dataset was similarly created by convolution of the reference dataset with a ghost function with a reflection coefficient of +1 and further application of a scalar multiplier of cos(θ), for emergence angle θ, to simulate the vertical particle velocity component. The ghost period was calculated based on a receiver depth of 15 m for each emergence angle θ. 
     The hydrophone and geophone data were processed using both standard wavefield separation and wavefield separation according to the method of the invention, which handles aliasing. The up-going pressure field was produced by each method and then compared to a modeled reference for the up-going pressure field. 
       FIGS. 6 and 7  show seismic sections of the modeled pressure and vertical particle velocity data, respectively.  FIG. 8  shows a seismic section of the modeled up-going pressure field that serves as the reference. The three planar events with emergence angles of 0°, 30°, and 60° are designated by reference numerals  60 ,  61 , and  62 , respectively, in  FIGS. 6 to 17 . 
       FIG. 9  shows a seismic section of the up-going pressure field obtained from the input data in  FIGS. 6 and 7  using standard wavefield separation techniques, i.e. no aliasing protection.  FIG. 10  shows the residual, the difference between the up-going pressure wavefield for the standard wavefield separation and for the reference ( FIG. 9  minus  FIG. 8 ). Apart from f-k edge effects, the dominant feature is residual energy associated with the steepest event at 60°. 
       FIG. 11  shows the up-going pressure field obtained from the input data in  FIGS. 6 and 7  using wavefield separation applying the method of the invention.  FIG. 12  shows the residual when the method of the invention is used ( FIG. 11  minus  FIG. 8 ), which demonstrates that the residual is significantly suppressed. 
     The reason for the residual in  FIG. 10  is more apparent in the f-k domain, in the results as shown in  FIGS. 13 to 17 .  FIGS. 13 and 14  show the modeled pressure and vertical particle velocity data, respectively, in the f-k domain.  FIGS. 13 and 14  correspond to the seismic sections shown in  FIGS. 6 and 7 , respectively.  FIG. 15  shows the modeled up-going pressure field reference in the f-k domain, which corresponds to the seismic section in  FIG. 8 . 
       FIG. 16  shows the up-going pressure wavefield in the f-k domain obtained from standard wavefield separation, corresponding to the seismic section in  FIG. 9 . The up-going pressure field is derived by combining the hydrophone data (unsealed) with geophone data that has been scaled by an obliquity factor. For spatially aliased energy, this obliquity factor is incorrect. 
       FIG. 16  illustrates that the events  60 ,  61  with 0° and 30° emergence angle do not suffer from spatial aliasing below 120 Hz, hence the standard wavefield separation gives a near perfect result for these events. For the 60° event  62 , the particle velocity data is scaled wrongly for the aliased part  133  when using the standard wavefield separation. 
     The result is errors in the up-going pressure field which are at a maximum at the hydrophone notch  134  in the aliased part  133 , where the energy comes entirely from the geophone. At geophone notches ( 145  in  FIG. 14 ), the energy is derived entirely from the hydrophone, so there is no error at these locations. At the other hydrophone notches  135  in the unaliased parts, there is also no error. 
       FIG. 17  shows the up-going pressure wavefield in the f-k domain obtained from wavefield separation applying the method of the invention, corresponding to the seismic section in  FIG. 11 . When the method of the invention is used, the error in the hydrophone notch  134  in the aliased part  133  is greatly reduced compared to the standard wavefield separation shown in  FIG. 16 . These synthetic results indicate that the method of the invention handles first-order aliased energy better than standard processing. 
     For up to two orders of aliasing, the set of equations, analogous to Equations (3) and (7) to (9) above, that would be solved is revised as follows:
 
 P=P   0   +P   1   +P   2   (29)
 
 {tilde over (V)}={tilde over (V)}   0   +{tilde over (V)}   1   +{tilde over (V)}   2 ,  (30)
 
 k   z0  cos( k   z0   z   R ) P   0 =sin( k   z0   z   R ) {tilde over (V)}   0 ,  (31)
 
 k   z1  cos( k   z1   z   R ) P   1 =sin( k   z1   z   R ) {tilde over (V)}   1 ,  (32)
 
 k   z2  cos( k   z2   z   R ) P   2 =sin( k   z2   z   R ) {tilde over (V)}   2 .  (33)
 
In Equation (33), k z2  is an obliquity factor that is correct for second order aliased energy, analogous to those expressions for k z1  in Equations (2a) or (2b) and (28a) or (28b). Similar obliquity factors can be determined for higher aliasing orders.
 
     Equations (29) to (33) now yield a set of five equations in the six unknowns P 0 , P 1 , P 2 , {tilde over (V)} 0 , {tilde over (V)} 1 , and {tilde over (V)} 2 , and thus is an underdetermined system of equations. Thus, unfortunately, for each extra order of aliasing, only one extra equation and two extra unknowns are added, so that the problem becomes progressively more underdetermined. Thus, the method of the invention is a treatment of first order spatially aliased energy unless additional constraints (such as, for example, a minimum energy solution) can be applied to obtain a physically plausible separation of energy between different aliasing orders. 
     The seismic data obtained in performing a seismic survey, representative of earth&#39;s subsurface, are processed to yield information relating to the geologic structure and properties of the subsurface earth formations in the area being surveyed. The processed seismic data are processed for display and analysis of potential hydrocarbon content of these subterranean formations. The goal of seismic data processing is to extract from the seismic data as much information as possible regarding the subterranean formations in order to adequately image the geologic subsurface. In order to identify locations in the earth&#39;s subsurface where there is a probability for finding petroleum accumulations, large sums of money are expended in gathering, processing, and interpreting seismic data. The process of constructing the reflector surfaces defining the subterranean earth layers of interest from the recorded seismic data provides an image of the earth in depth or time. 
     The image of the structure of the earth&#39;s subsurface is produced in order to enable an interpreter to select locations with the greatest probability of having petroleum accumulations. To verify the presence of petroleum, a well must be drilled. Drilling wells to determine whether petroleum deposits are present or not, is an extremely expensive and time-consuming undertaking. For that reason, there is a continuing need to improve the processing and display of the seismic data, so as to produce an image of the structure of the earth&#39;s subsurface that will improve the ability of an interpreter, whether the interpretation is made by a computer or a human, to assess the probability that an accumulation of petroleum exists at a particular location in the earth&#39;s subsurface. The processing and display of acquired seismic data facilitates more accurate decisions on whether and where to drill, and thereby reduces the risk of drilling dry holes. 
     The invention has been discussed above as a method, for illustrative purposes only, but can also be implemented as a system. The system of the invention is preferably implemented by means of computers, in particular digital computers, along with other conventional data processing equipment, as illustrated in  FIG. 2 , above. Such data processing equipment, well known in the art, will comprise any appropriate combination or network of computer processing equipment, including, but not limited to, hardware (processors, temporary and permanent storage devices, and any other appropriate computer processing equipment), software (operating systems, application programs, mathematics program libraries, and any other appropriate software), connections (electrical, optical, wireless, or otherwise), and peripherals (input and output devices such as keyboards, pointing devices, and scanners; display devices such as monitors and printers; computer readable storage media such as tapes, disks, and hard drives, and any other appropriate equipment). 
     In another embodiment, the invention could be implemented as the method described above, specifically carried out using a programmable computer to perform the method. In another embodiment, the invention could be implemented as a computer program stored in a computer readable medium, with the program having logic operable to cause a programmable computer to perform the method described above. In another embodiment, the invention could be implemented as a computer readable medium with a computer program stored on the medium, such that the program has logic operable to cause a programmable computer to perform the method described above. 
     It should be understood that the preceding is merely a detailed description of specific embodiments of this invention and that numerous changes, modifications, and alternatives to the disclosed embodiments can be made in accordance with the disclosure here without departing from the scope of the invention. The preceding description, therefore, is not meant to limit the scope of the invention. Rather, the scope of the invention is to be determined only by the appended claims and their equivalents.