Patent Publication Number: US-10775524-B2

Title: Simultaneous wavefield reconstruction and receiver deghosting of seismic streamer data using an L1 inversion

Description:
CROSS REFERENCE TO RELATED APPLICATION(S) 
     This application claims the benefit of priority to U.S. Provisional Application Ser. No. 62/312,934, filed on Mar. 24, 2016, the contents of which are hereby incorporated by reference. 
    
    
     BACKGROUND 
     Receiver deghosting for seismic streamer data is a data processing challenge in geophysical prospecting. As the air-water interface is a strong reflector for pressure waves, upgoing seismic waves are reflected downward at the air-water interface and then further propagate back into the water where they interfere with an existing seismic wavefield at the seismic streamer locations. This effect is referred to as the “Receiver Ghost” effect in marine streamer data acquisition, causes distortions of both phase and amplitude, and normally a representative notch can be observed in the frequency spectrum of the recorded signals. 
     SUMMARY 
     The present disclosure describes simultaneous wavefield reconstruction and receiver deghosting of three-dimensional (3D) seismic streamer data using an L 1  inversion. 
     In an implementation, raw 3D seismic streamer wavefield data is received as a receiver-ghosted shot gather. The received receiver-ghosted shot gather shot gather is processed into a normalized form as normalized data. The normalized data is partitioned into a plurality of user-defined sub-gathers and processed to generate a complete receiver-deghosted shot gather. Output of the complete receiver-deghosted shot gather is initiated. 
     The previously described implementation is implementable using a computer-implemented method; a non-transitory, computer-readable medium storing computer-readable instructions to perform the computer-implemented method; and a computer-implemented system comprising a computer memory interoperably coupled with a hardware processor configured to perform the computer-implemented method/the instructions stored on the non-transitory, computer-readable medium. 
     The subject matter described in this specification can be implemented in particular implementations so as to realize one or more of the following advantages. First, the described receiver deghosting method has no limitations on seismic streamer data acquisition schemes (for example, horizontal, slanted or curved), and can be applied on any type of measurements (for example, pressure or components of particle velocity). Second, a new redundant, hybrid, apex-shifted Radon dictionary (RHARD) is used as a basis dictionary for reconstructing a dense up-going wavefield at the surface of the water. The RHARD makes it possible to ideally build a complex wavefield (that is, with a mix of linear and curved events) with a few basic functions; making the use of sparsity inversion a suitable scheme in the described receiver deghosting method. Third, the use of the RHARD permits intelligent selection of equation terms to tradeoff between deghosting functionality and consumption of computing resources. Fourth, the described receiver deghosting method provides robust to realistic sparse data acquisition, and final output is a completely reconstructed and deghosted wavefield which can be located at arbitrary positions. Other advantages will be apparent to those of ordinary skill in the art. 
     The details of one or more implementations of the subject matter of this specification are set forth in the accompanying drawings and the description. Other features, aspects, and advantages of the subject matter will become apparent from the description, the drawings, and the claims. 
    
    
     
       DESCRIPTION OF DRAWINGS 
         FIG. 1  is a diagram illustrating seismic streamer data acquisition, according to an implementation. 
         FIG. 2  is a diagram illustrating a proposed solution to the reflection illustrated in  FIG. 1 , according to an implementation. 
         FIG. 3  is an illustration of a tailored three-dimensional (3D) EAGE/SEG overthrust model, according to an implementation. 
         FIG. 4A  is a plot illustrating a horizontal streamer configuration, according to an implementation. 
         FIG. 4B  is a plot illustrating a denser grid which the deghosted wavefield is output onto in the horizontal streamer configuration of  FIG. 4A , according to an implementation. 
         FIG. 5A  is a plot illustrating a quasi-slanted streamer configuration, according to an implementation. 
         FIG. 5B  is a plot illustrating a deghosted wavefield output to a denser grid for the quasi-slanted streamer configuration of  FIG. 5A , according to an implementation. 
         FIGS. 6A / 6 B and  10 A/ 10 B are plots illustrating different crossline sections before and after deghosting and reconstruction manipulation in the horizontal streamer situation and in the quasi-slanted streamer situation, respectively, according to an implementation. 
         FIGS. 7A / 7 B,  8 A/ 8 B,  11 A/ 11 B, and  12 A/ 12 B are plots illustrating different inline sections before and after receiver deghosting in both acquisition scenarios, according to an implementation. 
         FIGS. 9A-9C and 13A-13C  are plots illustrating trace comparisons, both before and after execution, of the described receiver deghosting method, according to an implementation. 
         FIG. 14  illustrates a flowchart of an example method for simultaneous wavefield reconstruction and receiver deghosting of seismic streamer data using an L 1  inversion, according to an implementation. 
         FIG. 15  is a block diagram of an example computer system  1500  used to provide computational functionalities associated with described algorithms, methods, functions, processes, flows, and procedures, as described in the instant disclosure, according to an implementation. 
     
    
    
     Like reference numbers and designations in the various drawings indicate like elements. 
     DETAILED DESCRIPTION 
     The following detailed description describes simultaneous wavefield reconstruction and receiver deghosting of seismic streamer data using an L 1  inversion and is presented to enable any person skilled in the art to make and use the disclosed subject matter in the context of one or more particular implementations. Various modifications to the disclosed implementations will be readily apparent to those skilled in the art, and the defined general principles may be applied to other implementations and applications without departing from the scope of the disclosure. The present disclosure is not intended to be limited to the described or illustrated implementations, but to be accorded the widest scope consistent with the described principles and features. 
     Receiver deghosting for seismic streamer data is a data processing challenge in geophysical prospecting. As the air-water interface is a strong reflector for pressure waves, upgoing seismic waves are reflected downward at the air-water interface and then further propagate back into the water where they interfere with an existing seismic wavefield at the seismic streamer locations. This effect is referred to as the “Receiver Ghost” effect in marine streamer data acquisition, causes distortions of both phase and amplitude, and normally a representative notch can be observed in the frequency spectrum of the recorded signals. Receiver deghosting aims at removal of the receiver ghost reflection from the marine seismic streamer data. With the advancement of marine broadband data acquisition and processing, it is becoming more crucial to apply proper three-dimensional (3D) receiver deghosting technologies to better preserve both bandwidth and resolution in the final receiver ghost-free signals. 
     To date, various receiver deghosting methods have been proposed, including or variants of: 1) use of Green&#39;s theorem as a general theoretical framework in deghosting; 2) use of a filter in the F-K domain; 3) use of a twin seismic streamer configuration; 4) use of both a hydrophone and geophone as a robust deghosting tool for ocean bottom cable (OBC) data (generally referred to as PZ-summation, which exploits the unique physical characteristics of the air-water interface reflected pressure and particle velocity signals); 5) consideration of multi-component seismic streamer data as an input for deghosting; 6) τ−p inversion; 7) use of an approximated pressure gradient to remove ghost signals; 8) application of a deconvolution method to suppress ghost signals; 9) deghosting based on building mirror data first using ray tracing; 10) consideration of similarity between deghosting and de-blending to handle deghosting as de-blending; 11) deghosting with over/under source acquisition; 12) developing an optimal stacking procedure that can help minimize ghost effects in a final image; 13) use of joint interpolation-deghosting to achieve genuine 3D deghosting; and 14) a method to exploit the causality property of the ghost effect by honoring, as much as possible, wave propagation. 
     Regardless of the listed and other receiver ghosting methods, a robust and physical 3D receiver deghosting utility is still lacking. Almost all of the proposed methods require dense wavefield sampling, but in real data acquisition a crossline interval is normally much larger than an inline interval (for example, typically, the ratio of the inline interval to the crossline interval varies between 1:4 and 1:8), which violates this fundamental assumption. As a trade-off solution, some methods are configured to work only on densely sampled two-dimensional (2D) inline data or even to make a bold one-dimensional (1D) propagation assumption. Other proposed methods involve interpolation explicitly as an independent operation. However, for real data situations, interpolation by itself is not a trivial problem, making optimal application of both steps difficult without propagation of errors. As a result, for 3D real data, the receiver deghosting challenge includes: 1) receiver ghost signal removal and 2) data sparsity. 
     Described is a receiver deghosting method capable of simultaneously deghosting receiver ghost signals and reconstructing a deghosted wavefield for seismic streamer data. The described method rigorously honors wave propagation phenomena during the receiver ghost process in a real 3D sense, and a new redundant, hybrid, apex-shifted Radon dictionary (RHARD) is used as a basis dictionary for reconstructing a dense up-going wavefield at the surface of water. The described receiver deghosting method frames the receiver deghosting problem as a Lasso problem, with a goal to minimize a mismatch between actual measurements and a simulated wavefield with an L 1  constraint in the extended Radon space exploited in order to handle an under-determination challenge in an inversion. The receiver deghosting algorithm can be demonstrated, for example, on a realistic tailored 3D EAGE/SEG Overthrust model. 
     Theory and Method 
     In marine data acquisition, watercraft towed seismic streamers, which can be in any form (for example, horizontal, slanted, or curved), are normally situated from a few meters to several tens of meters below the surface of the water. As a result, an incident wavefield will pass the seismic streamers twice—once upward and once downward after being reflected by the surface of the water. Certain wavefield frequency components are suppressed or even cancelled due to the interference of the upgoing and downgoing seismic wavefields. 
       FIG. 1  is a diagram  100  illustrating seismic streamer data acquisition, according to an implementation. As illustrated seismic streamer  102  is situated below the surface of the water  104 . Seismic streamer  104  can be in any form (for example, curved, horizontal, or slanted). Upgoing incident wavefield  106  passes seismic streamer  102  and reflects of the surface of the water  104  to generate downgoing incident wavefield  108 . 
       FIG. 2  is a diagram  200  illustrating a proposed solution to the reflection illustrated in  FIG. 1 , according to an implementation. In the described implementation, the desired upgoing incident wavefield  106  is parameterized at the surface of the water  104  using dense sampling (dense wavefield)  202  and both forward propagated with water surface reflection considered ( 108 ) and backwardly propagated incident wavefield  204  to form measured data. Performing incident wavefield parameterization at the surface of the water  104  better meets the plane integral surface requirement in a Rayleigh integral, and realistic seismic streamer movements due to water currents in field data acquisition become a reduced challenge. 
     Data sparsity must also be considered in seismic streamer data processing. Normally, the inline direction of the seismic streamer is a dense data acquisition direction, while the crossline direction has a much larger interval (for example, the ratio of inline interval to crossline interval is typically between 1:4 and 1:8—a ratio of 1:4 is used in  FIGS. 4A and 5A ). As a result, a final acquired seismic wavefield data is usually aliased in the crossline direction, which complicates receiver deghosting. 
     In order to optimally handle both the receiver deghosting problem and data sparsity, both challenges should be simultaneously addressed while rigorously honoring actual physics principles. Therefore, the aim is finding a densely sampled upgoing wavefield that, after including the correct physics of a free water surface, matches measured data at receiver locations. The previously described receiver ghosting process happens in water, which here is treated as an isotropic and homogeneous medium. Therefore, acoustic wave propagation theory is already sufficient, and the Rayleigh integral is chosen to physically describe the receiver ghosting effect. 
     In order for the Rayleigh integral to function correctly, two conditions must be fulfilled: 1) the integral surface must be a plane and 2) wavefield sampling on the plane must be dense. The dense wavefield sampling requirement is automatically met in the described method, as a dense wavefield being targeted as the final solution, but the integral plane condition must be handled with care in order for the method to be versatile and robust for any possible streamer shapes, even including feathering effects in real data acquisition scenarios. 
     The surface of the water  104 , which, in reality, is roughly planar, is used as an integral plane, and the dense upgoing incident wavefield  106  at the surface of the water is treated as a final solution target. The receiver ghost process can be described as a summation of the backwardly propagated incident wavefield  204  (from the water surface to the receiver locations) and the water surface reflected and forwardly propagated (downgoing) incident wavefield  108  (also from the surface of the water  104  to seismic receiver locations—not illustrated). 
     Mathematically, the receiver ghosting process can be described as:
 
 b=SP   −   y+SP   +   Ry=S ( P   −   +P   +   R ) y,   (1)
 
where:
 
     b is a vector containing measured data, y is a dense wavefield vector at the surface of the water, S is a sub-sampling matrix corresponding to a real, sparse data acquisition scheme in both the inline and the crossline directions, P −  and P +  are one-way wavefield propagation matrices (backward and forward, respectively) from the surface of the water to pre-defined dense target locations, and R is a water surface reflectivity matrix. 
     The dense wavefield  202  at the surface of the water  104  can be considered a 3D seismic data cube ƒ(g x , g y , t), where g x  and g y  are x position and y position of the receiver at the surface, and t is time. This dense wavefield  202  is packed into the vector y of Equation (1). 
     With S (indicating that the measured data is sparse (that is, dim(b)&lt;dim(y)) in Equation (1)), S(P − +P + R) is generally not mathematically invertible. In order to overcome this mathematical problem, the receiver deghosting problem is framed as: 
     
       
         
           
             
               
                 
                   
                     
                       
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     Equation (2) is the Lasso problem in mathematics. In Equations (2) and (3), x is a vector containing an encoded dense wavefield at the surface of the water and D is a corresponding transformation matrix, which ensures that y=Dx. The key to the success of Equation (2) is to find a suitable dictionary so that the encoded wavefield x in this dictionary is indeed a sparse representation of the dense wavefield y at the surface of the water. 
     In the described method, this dense wavefield is encoded per crossline slice, and every crossline slice has its own particular representation. Together with the transformation matrix D, this encoding step can be expressed mathematically as: 
                     y   =       [           y     slice   ⁢           ⁢   1                 y     slice   ⁢           ⁢   2               ⋮             y       slice   ⁢           ⁢   n     -   1                 y     slice   ⁢           ⁢   n             ]     =     Dx   =       [           D     2   ⁢           ⁢   d                       0       …       0                   0           0                     D     2   ⁢           ⁢   d                       0                   0                       ⋮                   ⋱                   ⋮                       0                   0       …         D     2   ⁢           ⁢   d                       0           0                   0                   0                     D     2   ⁢           ⁢   d             ]     ⁡     [           x     slice   ⁢           ⁢   1                 x     slice   ⁢           ⁢   2               ⋮             x       slice   ⁢           ⁢   n     -   1                 x     slice   ⁢           ⁢   n             ]             ,           (   4   )               
where:
 
     x slice i  is a vector containing wavefield representations of the crossline slice i of the dense wavefield and D 2d  is the corresponding matrix that transforms the encoded wavefield x slice i  to wavefield y slice i . As previously discussed, in order for x slice i  to be sparse, a redundant dictionary should be used. An explanation of building an appropriate redundant dictionary follows. 
     The method starts with the known Taylor Series for function ƒ of variable ξ: 
                       f   ⁡     (   ξ   )       =       ∑     n   =   0     ∞     ⁢           ⁢           f     (   n   )       ⁡     (     ξ   0     )         n   !       ⁢       (     ξ   -     ξ   0       )     n           ,           (   5   )               
where:
 
     ƒ (n)  is the n th  order derivative function and ξ 0  is a constant apex value. Equation (5) indicates that any smooth 2D curve can be described by representations 
             (           f     (   0   )       ⁡     (     ξ   0     )         0   !       ,         f     (   1   )       ⁡     (     ξ   0     )         1   !       ,         f     (   2   )       ⁡     (     ξ   0     )         2   !       ,   …   ⁢           ,         f     (   n   )       ⁡     (     ξ   0     )         n   !       ,   …     ⁢           )         
with the corresponding dictionary ((ξ−ξ 0 ) 0 , (ξ−ξ 0 ) 1 , (ξ−ξ 0 ) 2 , . . . , ξ−ξ 0 ) n , . . . ) used. However, if Equation (5) is scrutinized, the dictionary is actually sparse—for every term containing ξ n , there is only one building block available, (ξ−ξ 0 ) n , meaning that for any curve with the apex far away from ξ 0 , many terms will be required for an accurate approximation of ƒ(ξ).
 
     Therefore, this kind of sparse dictionary cannot meet the fundamental requirement for sparse representation, so modifications to Equation (5) are necessary. As a result, manual introduction of more apex shifts are made into Equation (5). The modified equation becomes:
 
ƒ(ξ)=τ+ pξ+Σ   n=2   ∞ Σ i     n     =A     n     B     n     q   i     n   (ξ−ξ i     n   ) n ,  (6)
 
where:
 
     the 0 th  order and 1 st  order term are untouched compared to Equation (5), but for the n th  order term (n&gt;1), there are multiple building blocks available, (ξ−ξ i     n   ) n , where ξ i     n    is a manually introduced apex and q i     n    is the corresponding representation. This extra redundancy is controllable in Equation (6) using control parameters A n  and B n . Equation (6) reflects thinking regarding redundancy building, implemented in the Radon domain as a dictionary and, as previously described, is referred to as a Redundant, Hybrid, Apex-Shifted Radon Dictionary (RHARD). 
     Using the RHARD, D 2d  in Equation (4) can be expressed as: 
                         D     2   ⁢           ⁢   d       =       F     -   1       ·     O     -   1       ·     [       L   1     ⁢     L     2   ,   1       ⁢     L     2   ,   2       ⁢           ⁢   …   ⁢           ⁢     L     2   ,     n   2         ⁢     L     3   ,   1       ⁢           ⁢   …   ⁢           ⁢     L     3   ,     n   3         ⁢           ⁢   …   ⁢           ⁢     L     p   ,     n   p           ]     ·   O   ·   F       ,     
     ⁢     where   ⁢     :         ⁢                   (   7   )                   L   1     =     [           L     1   ,     ω   1                         0       …                   0                   0                       0                     L     1   ,     ω   2                                     0                   0                                   ⋮                   ⋱                               ⋮                                   0                   0       …         L     1   ,     ω       N   max     -   1                                                   0           0                   0                   0                                             L     1   ,     ω     N   max                 ]       ,     
     ⁢     with   ⁢     :               (   8   )                   L     1   ,     ω   i         =     [           e       -   i     ⁢           ⁢     ω   i     ⁢     x   1     ⁢     p   1                           e       -   i     ⁢           ⁢     ω   i     ⁢     x   1     ⁢     p   2             …         e       -   i     ⁢           ⁢     ω   i     ⁢     x   1     ⁢     p       M   1     -   1                             e       -   i     ⁢           ⁢     ω   i     ⁢     x   1     ⁢     p     M   1                     e       -   i     ⁢           ⁢     ω   i     ⁢     x   2     ⁢     p   1                           e       -   i     ⁢           ⁢     ω   i     ⁢     x   2     ⁢     p   2                           e       -   i     ⁢           ⁢     ω   i     ⁢     x   2     ⁢     p       M   1     -   1                             e       -   i     ⁢           ⁢     ω   i     ⁢     x   2     ⁢     p     M   1                               ⋮                   ⋱                   ⋮                         e       -   i     ⁢           ⁢     ω   i     ⁢     x     N   -   1       ⁢     p   1                           e       -   i     ⁢           ⁢     ω   i     ⁢     x     N   -   1       ⁢     p   2             …         e       -   i     ⁢           ⁢     ω   i     ⁢     x     N   -   1       ⁢     p       M   1     -   1                             e       -   i     ⁢           ⁢     ω   i     ⁢     x     N   -   1       ⁢     p     M   1                     e       -   i     ⁢           ⁢     ω   i     ⁢     x   N     ⁢     p   1                           e       -   i     ⁢           ⁢     ω   i     ⁢     x   N     ⁢     p   2                           e       -   i     ⁢           ⁢     ω   i     ⁢     x   N     ⁢     p       M   1     -   1                             e       -   i     ⁢           ⁢     ω   i     ⁢     x   N     ⁢     p     M   1                 ]       ,     1   ≤   i   ≤     N   max       ,     
     ⁢     and   ⁢           ⁢   where   ⁢     :               (   9   )                   L     s   ,     n   s         =     [           L     s   ,     n   s     ,     ω   1                         0       …                   0                   0                       0                     L     s   ,     n   s     ,     ω   2                                     0                   0                                   ⋮                   ⋱                               ⋮                                   0                   0       …         L     s   ,     n   s     ,     ω       N   max     -   1                                                   0           0                   0                   0                                             L     s   ,     n   s     ,     ω     N   max                 ]       ,     2   ≤   s   ≤     n   p       ,     
     ⁢     with   ⁢     :               (   10   )                   L     s   ,     n   s     ,     ω   i         =     [           e       -   i     ⁢           ⁢         ω   i     ⁡     (       x   1     -     a     n   s         )         s     q   1                               e       -   i     ⁢           ⁢         ω   i     ⁡     (       x   1     -     a     n   s         )         s     q   2                 …         e       -   i     ⁢           ⁢         ω   i     ⁡     (       x   1     -     a     n   s         )         s     q       M   s     -   1                                 e       -   i     ⁢           ⁢         ω   i     ⁡     (       x   1     -     a     n   s         )         s     q     M   s                         e       -   i     ⁢           ⁢         ω   i     ⁡     (       x   2     -     a     n   s         )         s     q   1                               e       -   i     ⁢           ⁢         ω   i     ⁡     (       x   2     -     a     n   s         )         s     q   2                               e       -   i     ⁢           ⁢         ω   i     ⁡     (       x   2     -     a     n   s         )         s     q       M   s     -   1                                 e       -   i     ⁢           ⁢         ω   i     ⁡     (       x   2     -     a     n   s         )         s     q     M   s                                   ⋮                   ⋱                   ⋮                         e       -   i     ⁢           ⁢         ω   i     ⁡     (       x     N   -   1       -     a     n   s         )         s     q   1                               e       -   i     ⁢           ⁢         ω   i     ⁡     (       x     N   -   1       -     a     n   s         )         s     q   2                 …         e       -   i     ⁢           ⁢         ω   i     ⁡     (       x     N   -   1       -     a     n   s         )         s     q       M   s     -   1                                 e       -   i     ⁢           ⁢         ω   i     ⁡     (       x     N   -   1       -     a     n   s         )         s     q     M   s                         e       -   i     ⁢           ⁢         ω   i     ⁡     (       x   N     -     a     n   s         )         s     q   1                               e       -   i     ⁢           ⁢         ω   i     ⁡     (       x   N     -     a     n   s         )         s     q   2                               e       -   i     ⁢           ⁢         ω   i     ⁡     (       x   N     -     a     n   s         )         s     q       M   s     -   1                                 e       -   i     ⁢           ⁢         ω   i     ⁡     (       x   N     -     a     n   s         )         s     q     M   s                     ]       ,     1   ≤   i   ≤     N   max       ,           (   11   )               
where:
 
     ω i  is the angular frequency. 
     In Equation (7), F is a Fourier transformation matrix; F −1  is an inverse Fourier transformation matrix; [L 1  L 2,1  L 2,2  . . . L 2,n     2    L 3,1  . . . L 3,n     3    . . . L p,n     p   ] is a RHARD matrix which reflects the idea of Equation (6), where L 1  is the linear Radon transformation matrix as detailed in Equations (8) and (9), and L p,n     p    is the p th  order Radon transformation matrix with the curve apex at a n     p   , as detailed in Equations (10) and (11); O is a matrix that reorders the vector so that the RHARD matrix can be used; O −1  is an inverse manipulation corresponding to O which reorders the vector back to the original order so that the F −1  matrix can be used. 
     Equations (2)-(11) form the backbone of the described deghosting method, and once x is solved, then either in-situ deghosted wavefield, SP − Dx, or deghosted and reconstructed wavefield, P − Dx, can be calculated in a straightforward manner. Further, since the upgoing dense wavefield at the surface of the water, y=Dx, is available, the wavefield at any position in the water can be calculated provided that a corresponding propagation matrix is available. 
     Equation (2) is the Lasso problem in Mathematics, for which various solvers can be used. In one implementation, a spectral projected gradient (SPGL 1 ) solver can be is used. In other implementations, other solvers (for example, public, proprietary, or commercial) can be used. 
     In order to achieve satisfactory results, it is crucial to build P −  and P +  in Equation (3) suitably. Theoretically speaking, in the Rayleigh theory, the integral aperture should be infinite, but in reality this requirement is not practical. As a result, P −  and P 3+  must take this limited aperture effect into consideration, and different choices with respect to these two matrices bring slightly different results. Experimental evidence suggests optimally to use smooth, weighted, least-squares optimized, one-way wavefield propagation operators to build P −  and P 3+ . An advantage of using limited aperture is that the wavefield calculation can be decoupled: for a wavefield within a certain area, only limited integral area in the integral plane needs to be considered. Therefore, in the described calculations, the complete solution area is divided into overlapping smaller spatial-time windows which are separately processed. All the results of the separate processing are averaged and spliced together to form a final result. Two benefits can be obtained from this approach: 1) although the complete solution area may be large, a sub-area can be configured to be of a size to be handled with limited computational resources and 2) due to edge effects of the integral aperture, computational artifacts exist but can be largely suppressed using averaging since the sub-areas overlap. 
     R in Equation (3) should also be handled with care. Although in most reported methods R is always assumed to be −I, in reality its behavior is much more complex since the surface of the water cannot be perfectly planar. Fluctuations of the surface of the water mainly influence reflection behavior of high-frequency components. Tuning the value of R can provide better results in realistic field data situations, especially for high frequency signals. Note that R is set to −I in the following example. 
       FIG. 3  is an illustration of a tailored 3D EAGE/SEG overthrust model  300 , according to an implementation. An example using a tailored 3D European Association of Geoscientists (EAGE)/Society of exploration Geophysicists (SEG) overthrust model is now described. To generate the tailored overthrust model, a 500 m thick water layer  302  is placed on top of a base EAGE/SEG overthrust model. A subset of this modified base model (crossline direction  304  (X direction) between 10 km and 11.5 km, inline direction  306  (Y direction) between 7.5 km and 14.5 km, and depth  308  (Z direction) between 0 and 5 km) is selected as the tailored SEG overthrust model  300 . The velocity (m/s)  310  indicates the medium velocity of this tailored model. In this implementation, a 3D acoustic finite-difference time-domain (FDTD) simulation package is used as a forward modelling engine. 
     A corresponding density model is also built based upon this tailored velocity model, and density values are connected with the velocity values using the Gardner Equation relating P-wave velocity to bulk density (except for the water layer, where the density value is 1000 kg/m 3 ):
 
ρ=α V   P   β ,  (12)
 
where:
 
     ρ is bulk density given in kg/m 3 , V p  is P-wave velocity given in m/s, and α and β are empirically derived constants that depend on geology. 
     Regarding the data acquisition schemes, two situations are considered: 1) a horizontal streamer situation and 2) a slanted streamer situation (as illustrated in  FIG. 2 ). In both situations, the source wavelet is a Ricker wavelet with the dominant frequency set at 20 Hz, and the source position  402  is located at (crossline (X) position (m)  404  at 750 m, inline (Y) position (m)  406  at 750 m, and depth (m)  408  at 5 m). The maximum frequency for the inversion is set at 75 Hz, so the inversion is band limited. The streamer data acquisition covers an area of 500 m (X direction, between 500 m and 1000 m) by 6 km (Y direction, between 500 m and 6500 m). The inline (Y direction) interval is set at 12.5 m, and the crossline (X direction) interval is set at 50 m. 
       FIG. 4A  is a plot  400   a  illustrating a horizontal streamer configuration, according to an implementation. The red star designates the source position, (crossline (X) position (m)  402  at 750 m, inline (Y) position (m)  404  at 750 m, and depth (m)  406  at 5 m). Inline (Y direction) interval is 12.5 m, and crossline (X direction) interval is 50 m. Graphed points represent receiver locations and illustrate where a wavefield was sampled. For the horizontal streamer situation, the streamer depth is set at 30 m. 
       FIG. 4B  is a plot  400   b  illustrating a denser grid which the deghosted wavefield is output onto in the horizontal streamer configuration of  FIG. 4A , according to an implementation. After reconstruction of the deghosted wavefield, both inline interval and crossline interval are 12.5 m, and the receiver depth remains at 30 m. Graphed points represent receiver locations and illustrate where a wavefield was sampled. 
       FIG. 5A  is a plot  500   a  illustrating a quasi-slanted streamer configuration, according to an implementation. Graphed points represent receiver locations and illustrate where a wavefield was sampled. Inline (Y direction) interval is 12.5 m, and crossline (X direction) interval is 50 m. For the slanted streamer situation, due to the fact that FDTD is the simulation method, continuously varying depth is not possible, so as a trade-off solution a quasi-slanted data-acquisition scheme is used as shown in  FIG. 5A . In this quasi-slanted data-acquisition scheme, the crossline direction receivers are all at the same depth, while in the inline direction the receiver depth within the initial 500 m is set at 25 m, and after every 1 km the receiver depth is further increased by 2.5 m, until the receiver depth reaches 40 m at the boundary of the data acquisition area. After reconstruction of the deghosted wavefield, both inline interval and crossline interval are 12.5 m, and the receiver depth geometry remains the same as before the reconstruction as shown in  FIG. 5B . Before and after reconstruction, the receiver depth geometry follows the same design rules. 
     In order to suppress computation artifacts, in one particular implementation, thirty-four overlapped spatial windows are used to carry out the described deghosting procedure on data: in the time direction the whole window being taken; xϵ[500 m, 1000 m], yϵ[500+200*im, 500+200*i+500 m], 0≤i≤27, and yϵ[500*im, 500*(i+1)m], i=1, 3, 5, 7, 9, 11. The final deghosted result is the arithmetic average of all the spatial windows. A square Rayleigh integral aperture with side length of 250 m is used. 
     In Equation (7), the linear Radon term plus three parabolic Radon terms are used, and, in an implementation, apex value selection can be as follows: for each spatial window, a sparse central crossline image is selected; the parabolic Radon contribution is calculated with the apex scanned through the whole crossline range, xϵ[500 m, 1000 m], and an apex value is selected that leads to the smallest residual on this crossline image. After the apex value is singled out, the residual image is used as a new crossline image and the process repeated until all needed apex values have been selected. The apex value selection process reflects the known concept of Matching Pursuits. 
       FIG. 5B  is a plot  500   b  illustrating a denser grid which the deghosted wavefield is output onto in the quasi-slanted streamer configuration of  FIG. 5A , according to an implementation. Graphed points represent receiver locations and illustrate where a wavefield was sampled. 
       FIGS. 6A / 6 B and  10 A/ 10 B are plots ( 600   a / 600   b  and  1000   a / 1000   b , respectively) illustrating different crossline sections before and after deghosting and reconstruction manipulation in the horizontal streamer situation and in the quasi-slanted streamer situation, respectively, according to an implementation. 
     As illustrated,  FIG. 6A  illustrates 500 m crossline sections  602   a - 602   f  (at 1 km inline increments and 4 s time duration) of a ghosted wavefield in the horizontal streamer situation with the previously described, tailored 3D EAGE/SEG overthrust model.  FIG. 6B  illustrates corresponding crossline sections of  FIG. 6A  of the deghosted and reconstructed wavefield in the horizontal streamer situation using the described receiver deghosting method. 
     Similar to  FIG. 6A ,  FIG. 10A  illustrates 500 m crossline sections  1002   a - 1002   f  (at 1 km inline increments and 4 s time duration) of a ghosted wavefield in the quasi-slanted streamer situation with the previously described, tailored 3D EAGE/SEG overthrust model. Likewise,  FIG. 6B ,  FIG. 10B  illustrates corresponding crossline sections of  FIG. 10A  of the deghosted and reconstructed wavefield in the quasi-slanted streamer situation using the described receiver deghosting method. Wavefield reconstruction effects can be observed in both  FIG. 6B  and  FIG. 10B  as events are continuous in the X direction. 
       FIGS. 7A / 7 B,  8 A/ 8 B,  11 A/ 11 B, and  12 A/ 12 B are plots ( 700   a / 700   b ,  800   a / 800   b ,  1100   a / 1100   b , and  1200   a / 1200   b , respectively) illustrating different inline sections before and after deghosting in both acquisition scenarios, according to an implementation.  FIG. 7A  illustrates a single inline section  702   a  (x=750m and of 4s time duration) of the ghosted wavefield in the horizontal streamer situation with the previously described, tailored 3D EAGE/SEG overthrust model.  FIG. 7B  illustrates a corresponding deghosted inline section  702   b  using the described receiver deghosting method, and it can be observed at  704   b  that receiver ghosting has been removed when compared to the corresponding portion of  FIG. 7A . 
       FIG. 8A  illustrates a single inline section  802   a  (x=650m and of 4s time duration) of the ghosted wavefield in the horizontal streamer situation with the previously described, tailored 3D EAGE/SEG overthrust model.  FIG. 8B  illustrates a corresponding deghosted inline section  802   b  using the described receiver deghosting method, and it can be observed at  804   b  that receiver ghosting has been removed when compared to the corresponding portion of  FIG. 8A . 
       FIG. 11A  illustrates a single inline section  1102   a  (x=750 m and of 4 s time duration) of the ghosted wavefield in the quasi-slanted streamer situation with the previously described, tailored 3D EAGE/SEG overthrust model.  FIG. 11B  illustrates a corresponding deghosted inline section  1102   b  using the described receiver deghosting method, and it can be observed at  1104   b  that receiver ghosting has been removed when compared to the corresponding portion of  FIG. 11A . 
       FIG. 12B  illustrates a graph  1200   b  of a corresponding in-situ deghosted inline section, according to an implementation.  FIG. 12A  illustrates a single inline section  1202   a  (x=650m and of 4 s time duration) of the ghosted wavefield in the quasi-slanted streamer situation with the previously described, tailored 3D EAGE/SEG overthrust model.  FIG. 12B  illustrates a corresponding deghosted inline section  1202   b  using the described receiver deghosting method, and it can be observed at  1204   b  that receiver ghosting has been removed when compared to the corresponding portion of  FIG. 12A . 
       FIGS. 9A-9C and 13A-13C  are plots ( 900   a - 900   c  and  1300   a - 1300   c , respectively) illustrating comparison of seismic traces (amplitude vs. time (s)), both before and after execution, of the described receiver deghosting method, according to an implementation.  FIGS. 9A-9C  represent the described horizontal streamer case.  FIGS. 13A-13C  represent the described quasi-slanted streamer case. Amplitude is in Arbitrary Unit (AU) as the data has been normalized. The illustrated results demonstrate the effectiveness of the described receiver deghosting method. While crossline wavefield reconstruction capability is demonstrated in  FIGS. 6A-6B and 10A-10B ,  FIGS. 9A-9C and 13A-13C  demonstrate the deghosting effect (indicated by arrows) and that better event recognition is possible (indicated by circles). 
       FIG. 9A  illustrates comparison of seismic traces  902   a  (ghosted) and  904   a  (deghosted) in the horizontal streamer situation with the previously described, tailored 3D EAGE/SEG overthrust model and the trace located at x=1050 m and y=750 m. It can be observed that receiver ghosting has been removed in corresponding portions of the deghosted wavefield  904   a  when compared to the ghosted wavefield  902   a.    
       FIG. 9B  illustrates comparison of seismic traces  902   b  (ghosted) and  904   b  (deghosted) in the horizontal streamer situation with the previously described, tailored 3D EAGE/SEG overthrust model and the trace located at x=3000 m and y=650 m. It can be observed that receiver ghosting has been removed in corresponding portions of the deghosted wavefield  904   b  when compared to the ghosted wavefield  902   b.    
       FIG. 9C  illustrates comparison of seismic traces  902   c  (ghosted) and  904   c  (deghosted) in the horizontal streamer situation with the previously described, tailored 3D EAGE/SEG overthrust model and the trace located at x=5000 m and y=950 m. It can be observed that receiver ghosting has been removed in corresponding portions of the deghosted wavefield  904   c  when compared to the ghosted wavefield  902   c . Also, it can be observed that events  908   c  and  912   c  are better defined in the deghosted wavefield  904   c  when compared to the corresponding events  906   c  and  910   c , respectively, of the ghosted wavefield  902   c.    
       FIG. 13A  illustrates comparison of seismic traces  1302   a  (ghosted) and  1304   a  (deghosted) in the quasi-slanted streamer situation with the previously described, tailored 3D EAGE/SEG overthrust model and the trace located at x=1050 m and y=750 m. It can be observed that receiver ghosting has been removed in corresponding portions of the deghosted wavefield  1304   a  when compared to the ghosted wavefield  1302   a.    
       FIG. 13B  illustrates comparison of seismic traces  1302   b  (ghosted) and  1304   b  (deghosted) in the quasi-slanted streamer situation with the previously described, tailored 3D EAGE/SEG overthrust model and the trace located at x=3000 m and y=650 m. It can be observed that receiver ghosting has been removed in corresponding portions of the deghosted wavefield  1304   b  when compared to the ghosted wavefield  1302   b.    
       FIG. 13C  illustrates comparison of seismic traces  1302   c  (ghosted) and  1304   c  (deghosted) in the quasi-slanted streamer situation with the previously described, tailored 3D EAGE/SEG overthrust model and the trace located at x=5000 m and y=950 m. It can be observed that receiver ghosting has been removed in corresponding portions of the deghosted wavefield  1304   c  when compared to the ghosted wavefield  1302   c . Also, it can be observed that events  1308   c  and  1312   c  are better defined in the deghosted wavefield  1304   c  when compared to the corresponding events  1306   c  and  1310   c , respectively, of the ghosted wavefield  1302   c.    
       FIG. 14  is a flowchart of an example method  1400  for simultaneous wavefield reconstruction and receiver deghosting of seismic streamer data using an L 1  inversion, according to an implementation. For clarity of presentation, the description that follows generally describes methods  1400  in the context of the other FIGS. in this description. However, it will be understood that method  1400  may be performed, for example, by any suitable system, environment, software, and hardware, or a combination of systems, environments, software, and hardware as appropriate. In some implementations, various steps of method  1400  can be run in parallel, in combination, in loops, or in any order. 
     At  1402 , 3D ghosted raw seismic streamer wavefield data is received (for example, using push, pull, or both push and pull operations). In typical implementations, the raw seismic streamer wavefield data is a receiver-ghosted shot gather. From  1402 , method  1400  proceeds to  1404 . 
     At  1404 , the received 3D receiver-ghosted shot gather is normalized and regularized. For example, if the raw data is structured in an irregular grid, each trace can be binned to a user-defined dense regular grid for further processing. Normalization operations can be based on any criteria consistent with this disclosure (for example, amplitude normalization by the total energy of the shot gather). From  1404 , method  1400  proceeds to  1406 . 
     At  1406 , the regularized data is partitioned for processing. For example, the normalized data can be partitioned into a plurality of user-defined, overlapping sub-gathers based on time, geographic location, or other criteria. The partition method can be influenced by signal-to-noise ratio in the regularized data, but typically, the complete crossline range as a whole is used in the user-defined partition scheme. From  1406 , method  1400  proceeds to  1408 . 
     At  1408 , a determination is made whether there are remaining partitions to process. If there are no remaining partitions to process, method  1400  proceeds back to  1416 . If there are remaining partitions to process, method  1400  proceeds to process a remaining partition using the described simultaneous wavefield reconstruction and receiver deghosting method. From  1408 , method  1400  proceeds to  1410 . 
     At  1410 , apex positions are determined for the current partition to decide where the apexes, ξ i     n   , of different Radon curves should be using Equation (6). From  1410 , method  1400  proceeds to  1412 . 
     At  1412 , a sparseness inversion is performed on the current partition using Equation (2). From  1412 , method  1400  proceeds to  1414 . 
     At  1414 , the processed partition is stored for output. For example, the processed partition can be stored in RAM memory or a database. From  1414 , method  1400  proceeds back to  1408  to determine whether there are remaining partitions to process. 
     At  1416 , the processed partition data is recovered from storage and combined to form a receiver-deghosted shot gather. For example, in an implementation, the processed partition data can be mathematically averaged with a user-defined weight function (for example, a trapezoidal weight function) and combined to form the complete receiver-deghosted shot gather. From  1416 , method  1400  proceeds to  1418 . 
     At  1418 , output of the receiver-deghosted shot gather is initiated. For example, the receiver-deghosted shot gather can be output as a data set for storage in a computer data store or displayed on a graphical user interface. After  1418 , method  1400  stops. 
       FIG. 15  is a block diagram of an example computer system  1500  used to provide computational functionalities associated with described algorithms, methods, functions, processes, flows, and procedures, as described in the instant disclosure, according to an implementation. The illustrated computer  1502  is intended to encompass any computing device such as a server, desktop computer, laptop/notebook computer, wireless data port, smart phone, personal data assistant (PDA), tablet computing device, one or more processors within these devices, or any other suitable processing device, including physical or virtual instances (or both) of the computing device. Additionally, the computer  1502  may comprise a computer that includes an input device, such as a keypad, keyboard, touch screen, or other device that can accept user information, and an output device that conveys information associated with the operation of the computer  1502 , including digital data, visual, or audio information (or a combination of information), or a graphical user interface (GUI). 
     The computer  1502  can serve in a role as a client, network component, a server, a database or other persistency, or any other component (or a combination of roles) of a computer system for performing the subject matter described in the instant disclosure. The illustrated computer  1502  is communicably coupled with a network  1530 . In some implementations, one or more components of the computer  1502  may be configured to operate within environments, including cloud-computing-based, local, global, or other environment (or a combination of environments). 
     At a high level, the computer  1502  is an electronic computing device operable to receive, transmit, process, store, or manage data and information associated with the described subject matter. According to some implementations, the computer  1502  may also include or be communicably coupled with an application server, e-mail server, web server, caching server, streaming data server, or other server (or a combination of servers). 
     The computer  1502  can receive requests over network  1530  from a client application (for example, executing on another computer  1502 ) and respond to the received requests by processing the received requests using an appropriate software application(s). In addition, requests may also be sent to the computer  1502  from internal users (for example, from a command console or by other appropriate access method), external or third-parties, other automated applications, as well as any other appropriate entities, individuals, systems, or computers. 
     Each of the components of the computer  1502  can communicate using a system bus  1503 . In some implementations, any or all of the components of the computer  1502 , hardware or software (or a combination of both hardware and software), may interface with each other or the interface  1504  (or a combination of both), over the system bus  1503  using an application programming interface (API)  1512  or a service layer  1513  (or a combination of the API  1512  and service layer  1513 ). The API  1512  may include specifications for routines, data structures, and object classes. The API  1512  may be either computer-language independent or dependent and refer to a complete interface, a single function, or even a set of APIs. The service layer  1513  provides software services to the computer  1502  or other components (whether or not illustrated) that are communicably coupled to the computer  1502 . The functionality of the computer  1502  may be accessible for all service consumers using this service layer. Software services, such as those provided by the service layer  1513 , provide reusable, defined functionalities through a defined interface. For example, the interface may be software written in JAVA, C++, or other suitable language providing data in extensible markup language (XML) format or other suitable format. While illustrated as an integrated component of the computer  1502 , alternative implementations may illustrate the API  1512  or the service layer  1513  as stand-alone components in relation to other components of the computer  1502  or other components (whether or not illustrated) that are communicably coupled to the computer  1502 . Moreover, any or all parts of the API  1512  or the service layer  1513  may be implemented as child or sub-modules of another software module, enterprise application, or hardware module without departing from the scope of this disclosure. 
     The computer  1502  includes an interface  1504 . Although illustrated as a single interface  1504  in  FIG. 15 , two or more interfaces  1504  may be used according to particular needs, desires, or particular implementations of the computer  1502 . The interface  1504  is used by the computer  1502  for communicating with other systems that are connected to the network  1530  (whether illustrated or not) in a distributed environment. Generally, the interface  1504  comprises logic encoded in software or hardware (or a combination of software and hardware) and is operable to communicate with the network  1530 . More specifically, the interface  1504  may comprise software supporting one or more communication protocols associated with communications such that the network  1530  or interface&#39;s hardware is operable to communicate physical signals within and outside of the illustrated computer  1502 . 
     The computer  1502  includes a processor  1505 . Although illustrated as a single processor  1505  in  FIG. 15 , two or more processors may be used according to particular needs, desires, or particular implementations of the computer  1502 . Generally, the processor  1505  executes instructions and manipulates data to perform the operations of the computer  1502  and any algorithms, methods, functions, processes, flows, and procedures as described in the instant disclosure. 
     The computer  1502  also includes a database  1506  that can hold data for the computer  1502  or other components (or a combination of both) that can be connected to the network  1530  (whether illustrated or not). For example, database  1506  can be an in-memory, conventional, or other type of database storing data consistent with this disclosure. In some implementations, database  1506  can be a combination of two or more different database types (for example, a hybrid in-memory and conventional database) according to particular needs, desires, or particular implementations of the computer  1502  and the described functionality. Although illustrated as a single database  1506  in  FIG. 15 , two or more databases (of the same or combination of types) can be used according to particular needs, desires, or particular implementations of the computer  1502  and the described functionality. While database  1506  is illustrated as an integral component of the computer  1502 , in alternative implementations, database  1506  can be external to the computer  1502 . As illustrated, the database  1506  holds the previously described received receiver-ghosted shot gather  1516 , partition data  1518 , and receiver-deghosted shot gather  1520 . 
     The computer  1502  also includes a memory  1507  that can hold data for the computer  1502  or other components (or a combination of both) that can be connected to the network  1530  (whether illustrated or not). For example, memory  1507  can be random access memory (RAM), read-only memory (ROM), optical, magnetic, and the like, storing data consistent with this disclosure. In some implementations, memory  1507  can be a combination of two or more different types of memory (for example, a combination of RAM and magnetic storage) according to particular needs, desires, or particular implementations of the computer  1502  and the described functionality. Although illustrated as a single memory  1507  in  FIG. 15 , two or more memories  1507  (of the same or combination of types) can be used according to particular needs, desires, or particular implementations of the computer  1502  and the described functionality. While memory  1507  is illustrated as an integral component of the computer  1502 , in alternative implementations, memory  1507  can be external to the computer  1502 . 
     The application  1508  is an algorithmic software engine providing functionality according to particular needs, desires, or particular implementations of the computer  1502 , particularly with respect to functionality described in this disclosure. For example, application  1508  can serve as one or more components, modules, or applications. Further, although illustrated as a single application  1508 , the application  1508  may be implemented as multiple applications  1508  on the computer  1502 . In addition, although illustrated as integral to the computer  1502 , in alternative implementations, the application  1508  can be external to the computer  1502 . 
     The computer  1502  can also include a power supply  1514 . The power supply  1514  can include a rechargeable or non-rechargeable battery that can be configured to be either user- or non-user-replaceable. In some implementations, the power supply  1514  can include power-conversion or management circuits (including recharging, standby, or other power management functionality). In some implementations, the power-supply  1514  can include a power plug to allow the computer  1502  to be plugged into a wall socket or other power source to, for example, power the computer  1502  or recharge a rechargeable battery. 
     There may be any number of computers  1502  associated with, or external to, a computer system containing computer  1502 , each computer  1502  communicating over network  1530 . Further, the term “client,” “user,” and other appropriate terminology may be used interchangeably, as appropriate, without departing from the scope of this disclosure. Moreover, this disclosure contemplates that many users may use one computer  1502 , or that one user may use multiple computers  1502 . 
     Described implementations of the subject matter can include one or more features, alone or in combination. 
     For example, in a first implementation, a computer-implemented method, comprising: receiving raw 3D seismic streamer wavefield data as a receiver-ghosted shot gather; processing the received receiver-ghosted shot gather into a normalized form as normalized data; partitioning the normalized data into a plurality of user-defined sub-gathers; processing, by a computer, each of the plurality of user-defined sub-gathers to generate a complete receiver-deghosted shot gather; and initiating output of the complete receiver-deghosted shot gather. 
     The foregoing and other described implementations can each, optionally, include one or more of the following features: 
     A first feature, combinable with any of the following features, further comprising binning the normalized data into a user-defined, dense, regular grid if the receiver-ghosted shot gather is in an irregular grid. 
     A second feature, combinable with any of the previous or following features, further comprising determining apex positions of Radon curves associated with a particular user-defined sub-gather. 
     A third feature, combinable with any of the previous or following features, wherein the determination of the apex positions uses a Redundant, Hybrid, Apex-Shifted Radon Dictionary such that a function of variable ξ is expressed as:
 
ƒ(ξ)=τ+ pξ+Σ   n=2   ∞ Σ i     n     =A     n     B     n     q   i     n   (ξ−ξ i     n   ) n .
 
     A fourth feature, combinable with any of the previous or following features, wherein ξ i     n    is manually introduced as an additional apex shift, q i     n    is a corresponding representation, and A n  and B n  are control parameters. 
     A fifth feature, combinable with any of the previous or following features, further comprising performing a sparseness inversion using the determined apex positions and solving for: 
     
       
         
           
             
               
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     A sixth feature, combinable with any of the previous or following features, further comprising combining the processed partition data corresponding to each of the plurality of user-defined, sub-gathers to generate the receiver-deghosted shot gather. 
     In a second implementation, a non-transitory, computer-readable medium storing one or more instructions executable by a computer system to perform operations comprising: receiving raw 3D seismic streamer wavefield data as a receiver-ghosted shot gather; processing the received receiver-ghosted shot gather into a normalized form as normalized data; partitioning the normalized data into a plurality of user-defined sub-gathers; processing each of the plurality of user-defined sub-gathers to generate a complete receiver-deghosted shot gather; and initiating output of the complete receiver-deghosted shot gather. 
     The foregoing and other described implementations can each, optionally, include one or more of the following features: 
     A first feature, combinable with any of the following features, further comprising binning the normalized data into a user-defined, dense, regular grid if the receiver-ghosted shot gather is in an irregular grid. 
     A second feature, combinable with any of the previous or following features, further comprising one or more instructions to determine apex positions of Radon curves associated with a particular user-defined, sub-gather. 
     A third feature, combinable with any of the previous or following features, wherein the determination of the apex positions uses a Redundant, Hybrid, Apex-Shifted Radon Dictionary such that a function of variable ξ is expressed as:
 
ƒ(ξ)=τ+ pξ+Σ   n=2   ∞ Σ i     n     =A     n     B     n     q   i     n   (ξ−ξ i     n   ) n .
 
     A fourth feature, combinable with any of the previous or following features, wherein ξ i     n    is manually introduced as an additional apex shift, q i     n    is a corresponding representation, and A n  and B n  are control parameters. 
     A fifth feature, combinable with any of the previous or following features, further comprising one or more instructions to perform a sparseness inversion using the determined apex positions and solving for: 
     
       
         
           
             
               
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     A sixth feature, combinable with any of the previous or following features, further comprising one or more instructions to combine the processed partition data corresponding to each of the plurality of user-defined sub-gathers to generate the receiver-deghosted shot gather. 
     In a third implementation, a computer system, comprising: a computer memory; and a hardware processor interoperably coupled with the computer memory and configured to perform operations comprising: receiving raw 3D seismic streamer wavefield data as a receiver-ghosted shot gather; processing the received receiver-ghosted shot gather into a normalized form as normalized data; partitioning the normalized data into a plurality of user-defined sub-gathers; processing each of the plurality of user-defined sub-gathers to generate a complete receiver-deghosted shot gather; and initiating output of the complete receiver-deghosted shot gather. 
     The foregoing and other described implementations can each, optionally, include one or more of the following features: 
     A first feature, combinable with any of the following features, further comprising binning the normalized data into a user-defined, dense, regular grid if the receiver-ghosted shot gather is in an irregular grid. 
     A second feature, combinable with any of the previous or following features, further comprising one or more instructions to determine apex positions of Radon curves associated with a particular user-defined sub-gather. 
     A third feature, combinable with any of the previous or following features, wherein the determination of the apex positions uses a Redundant, Hybrid, Apex-Shifted Radon Dictionary such that a function of variable ξ is expressed as:
 
ƒ(ξ)=τ+ pξ+Σ   n=2   ∞ Σ i     n     =A     n     B     n     q   i     n   (ξ−ξ i     n   ) n .
 
     A fourth feature, combinable with any of the previous or following features, wherein ξ i     n    is manually introduced as an additional apex shift, q i     n    is a corresponding representation, and A n  and B n  are control parameters. 
     A fifth feature, combinable with any of the previous or following features, further comprising one or more instructions to perform a sparseness inversion using the determined apex positions and solving for: 
     
       
         
           
             
               
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     A sixth feature, combinable with any of the previous or following features, further configured to combine the processed partition data corresponding to each of the plurality of user-defined sub-gathers to generate the receiver-deghosted shot gather. 
     Implementations of the subject matter and the functional operations described in this specification can be implemented in digital electronic circuitry, in tangibly embodied computer software or firmware, in computer hardware, including the structures disclosed in this specification and their structural equivalents, or in combinations of one or more of them. Implementations of the subject matter described in this specification can be implemented as one or more computer programs, that is, one or more modules of computer program instructions encoded on a tangible, non-transitory, computer-readable computer-storage medium for execution by, or to control the operation of, data processing apparatus. Alternatively, or additionally, the program instructions can be encoded in/on an artificially generated propagated signal, for example, a machine-generated electrical, optical, or electromagnetic signal that is generated to encode information for transmission to suitable receiver apparatus for execution by a data processing apparatus. The computer-storage medium can be a machine-readable storage device, a machine-readable storage substrate, a random or serial access memory device, or a combination of computer-storage mediums. 
     The term “real-time,” “real time,” “realtime,” “real (fast) time (RFT),” “near(ly) real-time (NRT),” “quasi real-time,” or similar terms (as understood by one of ordinary skill in the art), means that an action and a response are temporally proximate such that an individual perceives the action and the response occurring substantially simultaneously. For example, the time difference for a response to display (or for an initiation of a display) of data following the individual&#39;s action to access the data may be less than 1 ms, less than 1 sec., or less than 5 secs. While the requested data need not be displayed (or initiated for display) instantaneously, it is displayed (or initiated for display) without any intentional delay, taking into account processing limitations of a described computing system and time required to, for example, gather, accurately measure, analyze, process, store, or transmit the data. 
     The terms “data processing apparatus,” “computer,” or “electronic computer device” (or equivalent as understood by one of ordinary skill in the art) refer to data processing hardware and encompass all kinds of apparatus, devices, and machines for processing data, including by way of example, a programmable processor, a computer, or multiple processors or computers. The apparatus can also be or further include special purpose logic circuitry, for example, a central processing unit (CPU), an FPGA (field programmable gate array), or an ASIC (application-specific integrated circuit). In some implementations, the data processing apparatus or special purpose logic circuitry (or a combination of the data processing apparatus or special purpose logic circuitry) may be hardware- or software-based (or a combination of both hardware- and software-based). The apparatus can optionally include code that creates an execution environment for computer programs, for example, code that constitutes processor firmware, a protocol stack, a database management system, an operating system, or a combination of execution environments. The present disclosure contemplates the use of data processing apparatuses with or without conventional operating systems, for example LINUX, UNIX, WINDOWS, MAC OS, ANDROID, IOS, or any other suitable conventional operating system. 
     A computer program, which may also be referred to or described as a program, software, a software application, a module, a software module, a script, or code can be written in any form of programming language, including compiled or interpreted languages, or declarative or procedural languages, and it can be deployed in any form, including as a stand-alone program or as a module, component, subroutine, or other unit suitable for use in a computing environment. A computer program may, but need not, correspond to a file in a file system. A program can be stored in a portion of a file that holds other programs or data, for example, one or more scripts stored in a markup language document, in a single file dedicated to the program in question, or in multiple coordinated files, for example, files that store one or more modules, sub-programs, or portions of code. A computer program can be deployed to be executed on one computer or on multiple computers that are located at one site or distributed across multiple sites and interconnected by a communication network. While portions of the programs illustrated in the various figures are shown as individual modules that implement the various features and functionality through various objects, methods, or other processes, the programs may instead include a number of sub-modules, third-party services, components, libraries, and such, as appropriate. Conversely, the features and functionality of various components can be combined into single components, as appropriate. Thresholds used to make computational determinations can be statically, dynamically, or both statically and dynamically determined. 
     The methods, processes, or logic flows described in this specification can be performed by one or more programmable computers executing one or more computer programs to perform functions by operating on input data and generating output. The methods, processes, or logic flows can also be performed by, and apparatus can also be implemented as, special purpose logic circuitry, for example, a CPU, an FPGA, or an ASIC. 
     Computers suitable for the execution of a computer program can be based on general or special purpose microprocessors, both, or any other kind of CPU. Generally, a CPU will receive instructions and data from a read-only memory (ROM) or a random access memory (RAM), or both. The essential elements of a computer are a CPU, for performing or executing instructions, and one or more memory devices for storing instructions and data. Generally, a computer will also include, or be operatively coupled to, receive data from or transfer data to, or both, one or more mass storage devices for storing data, for example, magnetic, magneto-optical disks, or optical disks. However, a computer need not have such devices. Moreover, a computer can be embedded in another device, for example, a mobile telephone, a personal digital assistant (PDA), a mobile audio or video player, a game console, a global positioning system (GPS) receiver, or a portable storage device, for example, a universal serial bus (USB) flash drive, to name just a few. 
     Computer-readable media (transitory or non-transitory, as appropriate) suitable for storing computer program instructions and data includes all forms of non-volatile memory, media and memory devices, including by way of example semiconductor memory devices, for example, erasable programmable read-only memory (EPROM), electrically erasable programmable read-only memory (EEPROM), and flash memory devices; magnetic disks, for example, internal hard disks or removable disks; magneto-optical disks; and CD-ROM, DVD+/−R, DVD-RAM, and DVD-ROM disks. The memory may store various objects or data, including caches, classes, frameworks, applications, backup data, jobs, web pages, web page templates, database tables, repositories storing dynamic information, and any other appropriate information including any parameters, variables, algorithms, instructions, rules, constraints, or references thereto. Additionally, the memory may include any other appropriate data, such as logs, policies, security or access data, reporting files, as well as others. The processor and the memory can be supplemented by, or incorporated in, special purpose logic circuitry. 
     To provide for interaction with a user, implementations of the subject matter described in this specification can be implemented on a computer having a display device, for example, a CRT (cathode ray tube), LCD (liquid crystal display), LED (Light Emitting Diode), or plasma monitor, for displaying information to the user and a keyboard and a pointing device, for example, a mouse, trackball, or trackpad by which the user can provide input to the computer. Input may also be provided to the computer using a touchscreen, such as a tablet computer surface with pressure sensitivity, a multi-touch screen using capacitive or electric sensing, or other type of touchscreen. Other kinds of devices can be used to provide for interaction with a user as well; for example, feedback provided to the user can be any form of sensory feedback, for example, visual feedback, auditory feedback, or tactile feedback; and input from the user can be received in any form, including acoustic, speech, or tactile input. In addition, a computer can interact with a user by sending documents to and receiving documents from a device that is used by the user; for example, by sending web pages to a web browser on a user&#39;s client device in response to requests received from the web browser. 
     The term “graphical user interface,” or “GUI,” may be used in the singular or the plural to describe one or more graphical user interfaces and each of the displays of a particular graphical user interface. Therefore, a GUI may represent any graphical user interface, including but not limited to, a web browser, a touch screen, or a command line interface (CLI) that processes information and efficiently presents the information results to the user. In general, a GUI may include a plurality of user interface (UI) elements, some or all associated with a web browser, such as interactive fields, pull-down lists, and buttons. These and other UI elements may be related to or represent the functions of the web browser. 
     Implementations of the subject matter described in this specification can be implemented in a computing system that includes a back-end component, for example, as a data server, or that includes a middleware component, for example, an application server, or that includes a front-end component, for example, a client computer having a graphical user interface or a Web browser through which a user can interact with an implementation of the subject matter described in this specification, or any combination of one or more such back-end, middleware, or front-end components. The components of the system can be interconnected by any form or medium of wireline or wireless digital data communication (or a combination of data communication), for example, a communication network. Examples of communication networks include a local area network (LAN), a radio access network (RAN), a metropolitan area network (MAN), a wide area network (WAN), Worldwide Interoperability for Microwave Access (WIMAX), a wireless local area network (WLAN) using, for example, 802.11 a/b/g/n or 802.20 (or a combination of 802.11x and 802.20 or other protocols consistent with this disclosure), all or a portion of the Internet, or any other communication system or systems at one or more locations (or a combination of communication networks). The network may communicate with, for example, Internet Protocol (IP) packets, Frame Relay frames, Asynchronous Transfer Mode (ATM) cells, voice, video, data, or other suitable information (or a combination of communication types) between network addresses. 
     The computing system can include clients and servers. A client and server are generally remote from each other and typically interact through a communication network. The relationship of client and server arises by virtue of computer programs running on the respective computers and having a client-server relationship to each other. 
     While this specification contains many specific implementation details, these should not be construed as limitations on the scope of any invention or on the scope of what may be claimed, but rather as descriptions of features that may be specific to particular implementations of particular inventions. Certain features that are described in this specification in the context of separate implementations can also be implemented, in combination, in a single implementation. Conversely, various features that are described in the context of a single implementation can also be implemented in multiple implementations, separately, or in any suitable sub-combination. Moreover, although previously described features may be described as acting in certain combinations and even initially claimed as such, one or more features from a claimed combination can, in some cases, be excised from the combination, and the claimed combination may be directed to a sub-combination or variation of a sub-combination. 
     Particular implementations of the subject matter have been described. Other implementations, alterations, and permutations of the described implementations are within the scope of the following claims as will be apparent to those skilled in the art. While operations are depicted in the drawings or claims in a particular order, this should not be understood as requiring that such operations be performed in the particular order shown or in sequential order, or that all illustrated operations be performed (some operations may be considered optional), to achieve desirable results. In certain circumstances, multitasking or parallel processing (or a combination of multitasking and parallel processing) may be advantageous and performed as deemed appropriate. 
     Moreover, the separation or integration of various system modules and components in the previously described implementations should not be understood as requiring such separation or integration in all implementations, and it should be understood that the described program components and systems can generally be integrated together in a single software product or packaged into multiple software products. 
     Accordingly, the previously described example implementations do not define or constrain this disclosure. Other changes, substitutions, and alterations are also possible without departing from the spirit and scope of this disclosure. 
     Furthermore, any claimed implementation is considered to be applicable to at least a computer-implemented method; a non-transitory, computer-readable medium storing computer-readable instructions to perform the computer-implemented method; and a computer system comprising a computer memory interoperably coupled with a hardware processor configured to perform the computer-implemented method or the instructions stored on the non-transitory, computer-readable medium.