Patent Publication Number: US-10310112-B2

Title: Processing geophysical data using 3D norm-zero optimization for smoothing geophysical inversion data

Description:
BACKGROUND 
     In a geophysics inversion analysis, geophysical data is collected and analyzed to estimate the subsurface structure of the earth. Examples of geophysical data include impedance data, porosity data, and velocity data. In some cases, the geophysical data can be used to build a geographic model that assists in making drilling decisions. 
     SUMMARY 
     The present disclosure describes methods and systems, including computer-implemented methods, computer program products, and computer systems for processing geophysical data. One computer-implemented method for processing geophysical data includes obtaining a set of raw geophysical data, wherein the raw geophysical data include 3-Dimensional (3D) coordinates; grouping, by a data processing apparatus, the set of the raw geophysical data into a plurality of subsets; and processing, by the data processing apparatus, each subset of the raw geophysical data using a 3D norm zero objective energy function to generate a subset of smoothed geophysical data, wherein the smoothed geophysical data is used to build a subsurface model. 
     Other implementations of this aspect include corresponding computer systems, apparatuses, and computer programs recorded on one or more computer storage devices, each configured to perform the actions of the methods. A system of one or more computers can be configured to perform particular operations or actions by virtue of having software, firmware, hardware, or a combination of software, firmware, or hardware installed on the system that, in operation, causes the system to perform the actions. One or more computer programs can be configured to perform particular operations or actions by virtue of including instructions that, when executed by data processing apparatus, cause the apparatus to perform the actions. 
     The foregoing and other implementations can each optionally include one or more of the following features, alone or in combination: 
     A first aspect, combinable with the general implementation, wherein the raw geophysical data is at least one of impedance data, velocity data, or porosity data. 
     A second aspect, combinable with any of the previous aspects, wherein grouping the set of the raw geophysical data into a plurality of subsets includes determining geometry information for the set of the raw geophysical data; determining a memory size for processing a subset of the raw geophysical data by the data processing apparatus; and grouping the set of the geophysical data based on the geometry information and the memory size. 
     A third aspect, combinable with any of the previous aspects, wherein the geometry information includes a cross line direction of the raw geophysical data. 
     A fourth aspect, combinable with any of the previous aspects, wherein the 3D norm zero objective energy function is 
               obj   =       ∑     x   =   1     k     ⁢     (         (       d   ⁡     (   x   )       -     m   ⁡     (   x   )         )     2     +     λ   ·       ∑     i   =   1     3     ⁢     C   ⁡     (         ∂     m   ⁡     (   x   )           ∂     x   i         ≠   0     )             )         ,         
where d(x) represents the subset of the raw geophysical data, k represents the number of the raw geophysical data in the subset, i represents the dimension of the raw geophysical data, m(x) represents the subset of the smoothed geophysical data, C( . . . ) represents a counting function, and λ represents a constant weight.
 
     A fifth aspect, combinable with any of the previous aspects, wherein processing each subset of the raw geophysical data includes iteratively processing until an iterative condition is met, the iterative processing including computing an auxiliary value by solving a first sub energy function, wherein the first sub energy function is derived from the 3D norm zero objective energy function with a fixed value of an input data; computing a next input data by solving a second sub energy function, wherein the second sub energy function is derived from the 3D norm zero objective energy function with a fixed auxiliary value; and increasing a beta value that is included in the first and the second sub energy functions. 
     A sixth aspect, combinable with any of the previous aspects, wherein the iteration condition includes at least one of reaching a predetermined number of iterations or a beta value being over a predetermined threshold. 
     The subject matter described in this specification can be implemented in particular implementations so as to realize one or more of the following advantages. Using a 3D norm zero optimization process to smooth the raw data can enhance a seismic inversion image by removing noise and preserving the edges. This approach can provide a smooth model to better image complex geological structures. The resulted data model can help geological interpreters to build a more coherent reservoir model, improve drilling location accuracy, and save operational cost. The 3D norm zero optimization process described is also computation efficient because the solution uses a low term of multiplication and summation. Furthermore, the 3D norm zero function may be used as a global operator that may not be affected by local abnormality. In addition, the 3D norm zero function may not be limited by window sizes that may be required in algorithms using median filters. 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 below. 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  illustrates an example 3D norm zero optimization method according to an implementation. 
         FIG. 2  is an example Society of Exploration Geophysicists-Y (SEGY) file  200  storing the raw geophysical data according to an implementation. 
         FIG. 3  is a block diagram of an example grouping for a set of raw geophysical data according to an implementation. 
         FIG. 4  illustrates an example method of processing a block of geophysical data using 3D norm zero function according to an implementation. 
         FIG. 5  is a high-level architecture block diagram of a geophysical data processing system according to an implementation. 
         FIGS. 6A &amp; 6B  are example screenshots which illustrate velocity data according to an implementation. 
         FIGS. 7A &amp; 7B  are example screenshots which illustrate an inline section view of a 3D seismic survey according to an implementation. 
         FIGS. 8A &amp; 8B  are example screenshots which illustrate a 3D volume view of an acoustic impedance model according to an implementation. 
         FIGS. 9A &amp; 9B  are example screenshots which illustrate a 2D section view of an acoustic impedance model according to an implementation. 
         FIGS. 10A &amp; 10B  are example screenshots which illustrate a time slice view of an acoustic impedance model according to an implementation. 
         FIGS. 11A &amp; 11B  are example screenshots which illustrate a porosity model according to an implementation. 
     
    
    
     Like reference numbers and designations in the various drawings indicate like elements. 
     DETAILED DESCRIPTION 
     The following description is presented to enable any person skilled in the art to make and use the disclosed subject matter, and is provided 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 general principles defined herein may be applied to other implementations and applications without departing from scope of the disclosure. Thus, the present disclosure is not intended to be limited to the described and/or illustrated implementations, but is to be accorded the widest scope consistent with the principles and features disclosed herein. 
     This disclosure generally describes methods and systems, including computer-implemented methods, computer program products, and computer systems, for processing geophysical data. In seismic processing, geophysical data can be used to build geographic models of subsurface structures. In some cases, the geophysical data have the features of sparseness. These geophysical data can also be blocky and have natural edges. Examples of these geophysical data can include impedance data, porosity data, and velocity data. In these or other cases, the raw geophysical data can be processed by a computer system to optimize the smoothness of the data and, therefore, can provide a more accurate image for the development of geographic models. For example, velocity data can be used to determine seismic reflection through depth/time migration. The accuracy of the migration can depend on the accuracy of the velocity model, especially in a complex subsalt dome area. A smoothed velocity model can improve the migration image, and therefore can be used to build a more accurate reservoir model. In some cases, median filter or structure-oriented filter can be used to filter the raw data. However, these filters may not work well on geophysical data that represent blocky layered structure images. 
     A norm is a total size or length of all vectors in a vector space or matrices. A n th  norm, denoted as l n -norm, is defined as the n th -root of a summation of all elements to the n th  power. In some cases, l 1 -norm is also referred to as Euclidean distance, and l 2 -norm is also referred to as the Mean-squared Error. The following equation represents the definition of the l n -norm for a group of vectors x i : 
     
       
         
           
             
               
                 
                   
                     
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                           ∑ 
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                              
                             
                               x 
                               i 
                             
                              
                           
                           n 
                         
                       
                       n 
                     
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     Where 
                     ⁢ 
                     
                         
                     
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                     n 
                     ⁢ 
                     
                         
                     
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                     ϵ 
                     ⁢ 
                     
                         
                     
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                       ⁡ 
                       
                         ( 
                         
                           Real 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           number 
                         
                         ) 
                       
                     
                   
                 
               
               
                 
                   ( 
                   1 
                   ) 
                 
               
             
           
         
       
     
     The following represents a definition of norm-two, denoted as l 2 , for a set of numbers that are denoted as x(t):
 
 l   2 =Σ( x ( t )) 2   (2)
 
     The following represents a definition of norm-one, denoted as l 1  for x(t):
 
 l   1   =Σ∥x ( t )∥  (3)
 
     In some cases, a norm zero optimization can be used to find the sparsest solution of a linear system. The norm zero optimization can also be referred to as the l 0 -norm optimization or the l 0 -optimization. A norm zero optimization can be applied to geophysical data, e.g., impedance, porosity, or velocity, or any data that presents layer or blocky structure image. Applying a norm zero optimization to geophysical data with sparseness feature can smooth the data and, therefore, produces better geophysical models. In a norm zero optimization, the l 0 -norm of a vector can be minimized according to some constraints. The following represents an expression of the norm zero optimization problem: min∥x∥ 0  Subject to Ax=b. 
     In some cases, norm zero optimization can be used to determine the number of interfaces between lithological units based on a group of measured points. Initially, while the number of data points is much larger than the number of lithological units, every measured point could potentially be a major interface. The optimization procedure reduces the number of pseudo lithological interfaces from a large number to a smaller based on a least square error principle. A thicker lithological layer could contain hundreds of measured data points. Therefore, norm zero optimization can be used to smooth the data points and group them into segments to match the statistical characters of the layered earth model. 
     While l 2 -norm is commonly used in the field of engineering and science, l 0 -norm is much less applicable. Due to its complexity, l 0 -norm is often difficult to solve practically. The following equation represents the definition of the l 0 -norm for a group of vectors x i : 
     
       
         
           
             
               
                 
                   
                     
                        
                       x 
                        
                     
                     0 
                   
                   = 
                   
                     
                       
                         ∑ 
                         i 
                       
                       ⁢ 
                       
                         x 
                         i 
                         0 
                       
                     
                     0 
                   
                 
               
               
                 
                   ( 
                   4 
                   ) 
                 
               
             
           
         
       
     
     To avoid the presence of zero th -power and zero th -root in the above definition, in some cases, an alternative definition of l 0 -norm can be used. In the alternative definition, l 0 -norm is defined to represent a total number of non-zero elements in a vector. The following equation represents the alternative definition of the l 0 -norm for a vector x:
 
∥ x∥   0 =≠( i|x   i ≠0)  (5)
 
     The following represents a definition of norm zero, denoted as l 0 :
 
 l   0   =ΣC ( x ( t ))  (6)
 
     Where the counting function C is represented as the following: 
     
       
         
           
             
               
                 
                   
                     C 
                     ⁡ 
                     
                       ( 
                       x 
                       ) 
                     
                   
                   = 
                   
                     { 
                     
                       
                         
                           1 
                         
                         
                           
                             x 
                             ≠ 
                             0 
                           
                         
                       
                       
                         
                           0 
                         
                         
                           
                             x 
                             = 
                             0 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   7 
                   ) 
                 
               
             
           
         
       
     
     Unlike norm two, norm zero is not analytic. Therefore, its derivative cannot be obtained numerically. In some cases, norm zero optimization can be used to process 2-dimensional (2D) images. However, for geophysical data with 3-dimensional (3D) coordinates, a norm zero optimization for 2D data may not provide consistent images on cross line directions. 
     In some cases, a norm zero optimization process for 3D data can be used to smooth the raw geophysical data. The process can be applied on blocky structured geophysical data as an edge-preserving smoothing filter to build a better 3D model for geological interpretation. In some cases, as discussed in more detail in  FIGS. 1-11  and associated descriptions, the geophysical data are obtained, segmented, and processed based on a 3D norm zero objective energy function. In some cases, the 3D norm zero objective energy function is solved by using a two-step iterative process. The two-step iterative process includes fixing an input data to solve for an auxiliary value, and fixing the auxiliary value to solve for the next input data. The optimization process enhances image resolution by removing low amplitude components while preserving sharp edges, thus providing better geological interpretation models. The optimization process is computation efficient on improving data quality. 
       FIG. 1  illustrates an example 3D norm zero optimization method  100  according to an implementation. For clarity of presentation, the description that follows generally describes method  100  in the context of  FIGS. 2-11 . However, it will be understood that method  100  may be performed, for example, by any other suitable system, environment, software, and hardware, or a combination of systems, environments, software, and hardware as appropriate. In some implementations, various steps of method  100  can be run in parallel, in combination, in loops, and/or in any order. 
     At  102 , a set of raw geophysical data is obtained. In some cases, the raw geophysical data are 3D data that include 3D coordinates corresponding to a (x, y, z) coordinate system. In some cases, the data are stored in a Society of Exploration Geophysicists (SEG) format, e.g., the SEG-Y (SEGY) format. 
       FIG. 2  is an example SEGY file  200  storing the raw geophysical data according to an implementation. The file  200  includes a file header  210  and one or more trace files  220   a - b . The file header  210  can include an optional SEGY tape label, a textual file header, a binary file header, and one or more optional extended textual file headers. In some cases, the raw geophysical data are stored trace by trace. Each trace file, e.g., trace files  220   a - b , can include a trace header  222  and a trace data  224 . In some cases, the trace header  222  has 240 bytes. The trace header  222  can include the geometry information for the raw geophysical data in the trace. Examples of the geometry information include the inline number, the cross line number, sample rate, depth, and recorded time for the raw geophysical data. The trace data  224  can include the raw geophysical data in the trace. 
     Referring to  FIG. 1 , from  102 , method  100  proceeds to  104 , where the geometry information of the raw data are determined. The geometry information can include the inline number, the cross line number, sample rate, depth, or recorded time for the raw data. In some cases, the geometry information for the raw data in a trace can be determined based on a trace header of the trace in the SEGY file. From  104 , method  100  proceeds to  106 . 
     At  106 , a memory size for performing the optimization process is determined. In some cases, the memory size includes the size of memory that the computer system can use to store a block of data, perform the computation procedures described in  FIGS. 1-11  for the block of data, and output the data to a file. From  106 , method  100  proceeds to  108 . 
     At  108 , the set of raw data is grouped into one or more blocks. In some cases, the set of raw data can include a large number of data. In these or other cases, the set of raw data can be grouped into a plurality of blocks. Each block of raw data can be processed together. This approach enables the optimization process to be performed by a computer system with limited memory and processing power. In some cases, the grouping can be determined based on the geometry information of the raw data determined at  104 , the memory size for performing the optimization process determined at  106 , or a combination of both. For example, as illustrated in  FIG. 3 , the grouping can be determined based on the cross line directions of the data and the memory size. 
       FIG. 3  is a block diagram  300  of an example grouping for a set of raw geophysical data according to an implementation. As illustrated, the data are organized based on their inline and cross line numbers in a rectangle box. The width of the length represents the inline directions of the data. The length of the box represents the cross line directions of the data. The data are then grouped into blocks along the cross line directions. The size of the cross line direction for each block is determined based on the memory size for processing the block. In some cases, the raw data does not fit in a regular rectangle box, e.g., some data are missing in one or more blocks. In these or other cases, the raw data are regularized. For example, additional data with the value 0 can be added for each block to make the data in each block form a rectangle box. 
     In some cases, edge artifacts may be introduced by the segmentation of data into blocks and the processing of individual blocks. To smooth out these edge artifacts, the output data for each block is generated by processing a buffer block of data that includes data from adjacent blocks. In these or other cases, the size of each block is determined based on the memory size needed to process the data in a buffer block. For example, as shown in  FIG. 3 , the set of raw data is grouped into 4 blocks, block- 1   302 , block- 2   304 , block- 3   306 , and block- 4   308  along the cross line direction. When block- 1   302  is processed, the data in buffer block  1  input  312  is used as input. The buffer block  1  input  312  includes data in the block- 1   302  and data in the upper portion of the adjacent block- 1   304 . The optimization process can generate output data in block  1  output  322 , which is the same size as the block- 1   302 . The data in buffer block- 2  input  314  is used as input to produce output data in block- 2  output  324 . The buffer block- 2  input  314  includes data in the block- 2   304 , data in the lower portion of the adjacent block- 1   302 , and data in the upper portion of the adjacent block- 3   306 . Similarly, the data in buffer block- 3  input  316  and buffer block- 4  input  318  are used as inputs to produce output data in block- 3  output  326  and block- 4  output  328 , respectively. In some cases, the size of the last block in the set can be smaller than other blocks. 
     Referring to  FIG. 1 , from  108 , method  100  proceeds to  110 , where each block, or a buffer block as discussed above, of raw data is processed based on a 3D norm zero optimization process.  FIG. 4  and associated description provides additional details of these implementations. From  110 , method  100  proceeds to  112 . 
     At  112 , the smoothed data for each block is outputted into an output file. In some cases, the output data can be stored in a SEGY file. In some cases, the output data is de-regularized. In a de-regularization process, the size of output data for each block is restored to the original size of the input data. For example, if a number of additional data is added to make the data fit into a regular rectangle box in a block, the same number of output data is removed. The optimization step at  110  and the output step at  112  are repeated until the last block of data in the set is processed. 
       FIG. 4  illustrates an example method  400  of processing a block of geophysical data using 3D norm zero function according to an implementation. For clarity of presentation, the description that follows generally describes method  400  in the context of  FIGS. 1-3 and 5-11 . However, it will be understood that method  400  may be performed, for example, by any other suitable system, environment, software, and hardware, or a combination of systems, environments, software, and hardware as appropriate. In some implementations, various steps of method  400  can be run in parallel, in combination, in loops, and/or in any order. 
     In some cases, the 3D norm zero function can be solved in a two-step iterative process. The following represents the 3D norm zero objective energy function to be solved: 
     
       
         
           
             
               
                 
                   obj 
                   = 
                   
                     
                       ∑ 
                       
                         x 
                         = 
                         1 
                       
                       k 
                     
                     ⁢ 
                     
                       ( 
                       
                         
                           
                             ( 
                             
                               
                                 d 
                                 ⁡ 
                                 
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                               - 
                               
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                                 ⁡ 
                                 
                                   ( 
                                   x 
                                   ) 
                                 
                               
                             
                             ) 
                           
                           2 
                         
                         + 
                         
                           λ 
                           · 
                           
                             
                               ∑ 
                               
                                 i 
                                 = 
                                 1 
                               
                               3 
                             
                             ⁢ 
                             
                               C 
                               ⁡ 
                               
                                 ( 
                                 
                                   
                                     
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                                         m 
                                         ⁡ 
                                         
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                                           ) 
                                         
                                       
                                     
                                     
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                   8 
                   ) 
                 
               
             
           
         
       
     
     where d(x) enumerates k samples of 3D raw data, where x∈R 3 ; i enumerates the dimension of the raw data; m(x) represents the smooth version of the data; C( ) represents the l 0  function, which is a counting function that returns 0 if the input of the function is zero and returns 1 if the input of the function is not zero; λ represents a constant weight; and x enumerates all the pixels in the image. 
     The above function is neither smooth nor continuous and, therefore, not directly solvable if by close form or gradient based methods. To approximate the solution, an auxiliary variables {h i } is introduced. The following represents a definition of the auxiliary variables {h i }: 
     
       
         
           
             
               
                 
                   
                     h 
                     i 
                   
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                       ∂ 
                       m 
                     
                     
                       ∂ 
                       
                         x 
                         i 
                       
                     
                   
                 
               
               
                 
                   ( 
                   9 
                   ) 
                 
               
             
           
         
       
     
     Therefore, the objective energy function can be rewritten in the following: 
             obj   =       ∑     x   =   1     k     ⁢     (         (       d   ⁡     (   x   )       -     m   ⁡     (   x   )         )     2     +     β   ·       ∑     i   =   1     3     ⁢       (         h   i     ⁡     (   x   )       -       ∂     m   ⁡     (   x   )           ∂     x   i           )     2         +     λ   ·       ∑     i   =   1     3     ⁢     C   ⁡     (         h   i     ⁡     (   x   )       ≠   0     )             )             
(10)
 
     When the beta value {β} is large enough, the auxiliary variables {h i } approach the derivative of m(x). Thus, the model in the equation (10) approaches equation (8). 
     It may be difficult to analytically solve equation (10) because the two terms m(x) and h(x) model respectively the pixel-wise difference and global discontinuity statistically and, thus, are not suitable for traditional gradient descent or other discrete optimization methods. In some cases, an alternating optimization solution with half-quadratic splitting can be used. The solution can be performed in a loop of two alternating steps using an iteration approach: First, fix m and solve for h; and second, fix h and solve for m. Then, the value of β is increased before entering the next iteration. The iteration loop continues until an iterative condition is met. The following descriptions provide additional details of these implementations. 
     Referring to  FIG. 4 , at  402 , the parameters for the optimization process are initialized. The parameters can include an initial beta value β. For example, β can be initialized to 0.5. The parameters can also include a cut off ratio P. In some cases, a cut off ratio of 0.5 can create a minor smoothing effect, while a cut off ratio of 0.95 can generate heavily smoothed and blocky data. The parameters can also include the iterative condition. For example, the iterative condition can include a maximum number of iterations, a maximum number of β, a smoothing quality of the output data, or a combination of both. In some cases, some or all of the parameters can be initialized one time for raw data in different blocks or different traces; therefore, the same initiation values can be used. Alternatively or additionally, the parameters can be initialized each time the raw data in different blocks or traces are processed; therefore, different initiation values can be used. In some cases, the initial value of the output data m(x) can be set to d(x). In these or other cases, the derivative operator 
               ∂     m   ⁡     (   x   )           ∂     x   i             
is also computed. From  402 , method  400  proceeds to  404 .
 
     At  404  auxiliary variables {h i } is computed. In this step, m is fixed, and h can be solved using a sub energy optimization function described as the following: 
     
       
         
           
             
               
                 
                   
                     
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     The following represents the solution h: 
     
       
         
           
             
               
                 
                   { 
                   
                     
                       
                         
                           
                             
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                             ⁡ 
                             
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                           = 
                           
                             
                               
                                 
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                               x 
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                             otherwise 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   12 
                   ) 
                 
               
             
           
         
       
     
     This step removes small gradients and keeps the larger ones for future iterations. As illustrated in the equation, the power threshold λ/β can control the sparseness of the rock interfaces. A larger value λ/β can produce a blockier result. From  406 , method  400  proceeds to  408 . 
     In some cases, the constant weight λ can be computed by multiplying the cutoff ratio P and the power of 
                 ∂     m   ⁡     (   x   )           ∂     x   i         .         
The power of
 
               ∂     m   ⁡     (   x   )           ∂     x     i   ⁢                       
can be computed as following:
 
     power of 
                 ∂     m   ⁡     (   x   )           ∂     x   i         =           ∂     m   ⁡     (   x   )           ∂     x   1         *       ∂     m   ⁡     (   x   )           ∂     x   1           +         ∂     m   ⁡     (   x   )           ∂     x   2         *       ∂     m   ⁡     (   x   )           ∂     x   2           +         ∂     m   ⁡     (   x   )           ∂     x   3         *       ∂     m   ⁡     (   x   )           ∂     x   3                           where   ⁢           ⁢   x   ⁢           ⁢   1     ,     x   ⁢           ⁢   2     ,     x   ⁢           ⁢   3           
represents 3D space coordinate directions. From  404 , method  400  proceeds to  406 .
 
     At  406 , 3D discrete Fourier transform of the input data m(x) and the auxiliary variables {h i } are computed. As discussed in more detail below, the 3D discrete Fourier transform of the input data and the directive operators can be used to solve the frequency domain version of the function. In some cases, the 3D discrete Fourier transform can be performed using FFT. From  404 , method  400  proceeds to  406 . 
     At  408 , the next m is computed. In this step, h is fixed; therefore, m can be solved using another sub energy optimization function described as the following: 
     
       
         
           
             
               
                 
                   
                     arg 
                     ⁢ 
                     
                       { 
                       
                         min 
                         m 
                       
                       } 
                     
                   
                   = 
                   
                     
                       ∑ 
                       
                         x 
                         = 
                         1 
                       
                       k 
                     
                     ⁢ 
                     
                       ( 
                       
                         
                           
                             ( 
                             
                               
                                 d 
                                 ⁡ 
                                 
                                   ( 
                                   x 
                                   ) 
                                 
                               
                               - 
                               
                                 m 
                                 ⁡ 
                                 
                                   ( 
                                   x 
                                   ) 
                                 
                               
                             
                             ) 
                           
                           2 
                         
                         + 
                         
                           β 
                           · 
                           
                             
                               ∑ 
                               
                                 i 
                                 = 
                                 1 
                               
                               3 
                             
                             ⁢ 
                             
                               
                                 ( 
                                 
                                   
                                     
                                       h 
                                       i 
                                     
                                     ⁡ 
                                     
                                       ( 
                                       x 
                                       ) 
                                     
                                   
                                   - 
                                   
                                     
                                       ∂ 
                                       
                                         m 
                                         ⁡ 
                                         
                                           ( 
                                           x 
                                           ) 
                                         
                                       
                                     
                                     
                                       ∂ 
                                       
                                         x 
                                         i 
                                       
                                     
                                   
                                 
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                               2 
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   13 
                   ) 
                 
               
             
           
         
       
     
     According to Parseval&#39;s theorem, the energy in the time domain is equal to the energy in the frequency domain. Therefore, a Fourier transform can be applied to equation 10 to yield an object function in the frequency domain as described below:
 
arg{min M }=Σ x=1   k (( D−M ) 2 +β·Σ i=1   3 ( H   i   −W   i   T   ·M ) 2 )  (14)
 
     where D, M, {H i } represents the n-dim discrete Fourier transform of input data d, smoothed data m and auxiliary variables {h i }, the input W i  represents the 3D discrete Fourier transform of partial derivative operation along i-th dimension, W i   T  is the conjugation of W i . 
     This problem has a close form solution described below: 
     
       
         
           
             
               
                 
                   M 
                   = 
                   
                     
                       D 
                       + 
                       
                         β 
                         · 
                         
                           
                             ∑ 
                             
                               i 
                               = 
                               1 
                             
                             3 
                           
                           ⁢ 
                           
                             ( 
                             
                               
                                 W 
                                 i 
                                 T 
                               
                               · 
                               
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                             ) 
                           
                         
                       
                     
                     
                       1 
                       + 
                       
                         β 
                         · 
                         
                           
                             ∑ 
                             
                               i 
                               = 
                               1 
                             
                             3 
                           
                           ⁢ 
                           
                             ( 
                             
                               
                                 W 
                                 i 
                                 T 
                               
                               · 
                               
                                 W 
                                 i 
                               
                             
                             ) 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   15 
                   ) 
                 
               
             
           
         
       
     
     From  408 , method  400  proceeds to  410 , where an iterative condition is checked to determine whether the iterative condition has been met. As discussed above, the iterative condition can include a maximum number of iterations, a maximum number of β, a smoothing quality of the output data, or a combination of both. For example, the maximum number of iterations can be set to 5, or 8-10; the iteration can be determined if the number of iterations is equal to the maximum number of iterations. In some cases, the number of iterations affects the flatness between rock interfaces. For example, more iterations can result more flatness between rock interfaces. In some cases, the smoothing quality can be determined by empirical trial and testing. In some cases, the objective function may converge when β becomes large. In some cases, the smoothed output of each iteration may be examined to determine whether the results converges. In one example, the smoothed output does not change much after 10 iterations, and therefore 10 iterations may correspond to a predetermined smoothing quality. In some cases, the number of iterations may increase if the smoothing quality of the output is not satisfactory. 
     If the iterative condition is not met, method  400  proceeds from  410  to  412 , where the beta value is increased. In some cases, the beta value can double in each iteration. From  412 , method  400  proceeds to  404 , where the next h is computed. If the iterative condition is met, method  400  proceeds from  410  to  420 . 
     At  420 , the final output data m(x) is calculated by the inverse discrete Fourier transform of M. The inverse discrete Fourier transform can be performed using inverse FFT (IFFT). 
       FIG. 5  is a high level architecture block diagram of a geophysical data processing system  500  according to an implementation. At a high level, the illustrated system  500  includes a geological interpreter  570  that is communicably coupled with a geophysical data processing computer  502  through a network  530 . The described illustration is only one possible implementation of the described subject matter and is not intended to limit the disclosure to the single described implementation. Those of ordinary skill in the art will appreciate the fact that the described components can be connected, combined, and/or used in alternative ways consistent with this disclosure. 
     The geological interpreter  570  represents a person, an application, set of applications, software, software modules, hardware, or combination thereof that can be used to analyze the smoothed geophysical data and build reservoir model based on the smoothed data. In some cases, the geological interpreter  570  can also set the initialization parameters of the 3D norm zero optimization process discussed above. 
     The network  530  facilitates communications between the components of the system  500  (e.g., between the geological interpreter  570  and the geophysical data processing computer  502 ). In some cases, the geological interpreter  570  can access the geophysical data processing computer  502  from a remote network. In these or other cases, the network  530  can be a wireless or a wireline network. In some cases, the geological interpreter  570  can access the geophysical data processing computer  502  locally. In these or other cases, the network  530  can also be a memory pipe, a hardware connection, or any internal or external communication paths between the components. 
     The geophysical data processing computer  502  includes a computing system configured to process the geophysical data using the 3D norm zero optimization process. In some cases, the algorithm of the 3D norm zero optimization process can be implemented in an executable computing code, e.g., C/C++ executable codes. In some cases, the geophysical data processing computer  502  can include a standalone Linux system that runs batch applications. In some cases, the geophysical data processing computer  502  can include mobile or personal computers that have sufficient memory size to process each block of the geophysical data. 
     The computer  502  may comprise a computer that includes an input device, such as a keypad, keyboard, touch screen, microphone, speech recognition device, other device that can accept user information, and/or an output device that conveys information associated with the operation of the computer  502 , including digital data, visual and/or audio information, or a GUI. 
     The computer  502  can serve as a client, network component, a server, a database or other persistency, and/or any other component of the system  500 . In some implementations, one or more components of the computer  502  may be configured to operate within a cloud-computing-based environment. 
     At a high level, the computer  502  is an electronic computing device operable to receive, transmit, process, store, or manage data and information associated with the system  500 . According to some implementations, the computer  502  may also include or be communicably coupled with an application server, e-mail server, web server, caching server, streaming data server, business intelligence (BI) server, and/or other server. 
     The computer  502  can receive requests over network  530  from a client application (e.g., executing on another computer  502 ) and respond to the received requests by processing the said requests in an appropriate software application. In addition, requests may also be sent to the computer  502  from internal users (e.g., from a command console or by another 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  502  can communicate using a system bus  503 . In some implementations, any and/or all the components of the computer  502 , both hardware and/or software, may interface with each other and/or the interface  504  over the system bus  503  using an application programming interface (API)  512  and/or a service layer  513 . The API  512  may include specifications for routines, data structures, and object classes. The API  512  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  513  provides software services to the computer  502  and/or the system  500 . The functionality of the computer  502  may be accessible for all service consumers using this service layer. Software services, such as those provided by the service layer  513 , provide reusable, defined business 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  502 , alternative implementations may illustrate the API  512  and/or the service layer  513  as stand-alone components in relation to other components of the computer  502  and/or system  500 . Moreover, any or all parts of the API  512  and/or the service layer  513  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  502  includes an interface  504 . Although illustrated as a single interface  504  in  FIG. 5 , two or more interfaces  504  may be used according to particular needs, desires, or particular implementations of the computer  502  and/or system  500 . The interface  504  is used by the computer  502  for communicating with other systems in a distributed environment—including within the system  500 —connected to the network  530  (whether illustrated or not). Generally, the interface  504  comprises logic encoded in software and/or hardware in a suitable combination and operable to communicate with the network  530 . More specifically, the interface  504  may comprise software supporting one or more communication protocols associated with communications such that the network  530  or interface&#39;s hardware is operable to communicate physical signals within and outside of the illustrated system  500 . 
     The computer  502  includes a processor  505 . Although illustrated as a single processor  505  in  FIG. 5 , two or more processors may be used according to particular needs, desires, or particular implementations of the computer  502  and/or the system  500 . Generally, the processor  505  executes instructions and manipulates data to perform the operations of the computer  502 . Specifically, the processor  505  executes the functionality required for processing geophysical data. 
     The computer  502  also includes a memory  506  that holds data for the computer  502  and/or other components of the system  500 . Although illustrated as a single memory  506  in  FIG. 5 , two or more memories may be used according to particular needs, desires, or particular implementations of the computer  502  and/or the system  500 . While memory  506  is illustrated as an integral component of the computer  502 , in alternative implementations, memory  506  can be external to the computer  502  and/or the system  500 . 
     The application  507  is an algorithmic software engine providing functionality according to particular needs, desires, or particular implementations of the computer  502  and/or the system  500 , particularly with respect to functionality required for processing geophysical data. For example, application  507  can serve as one or more components/applications described in  FIGS. 1-4 and 6-11 . Further, although illustrated as a single application  507 , the application  507  may be implemented as multiple applications  507  on the computer  502 . In addition, although illustrated as integral to the computer  502 , in alternative implementations, the application  507  can be external to the computer  502  and/or the system  500 . 
     There may be any number of computers  502  associated with, or external to, the system  500  and communicating over network  530 . Further, the terms “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  502 , or that one user may use multiple computers  502 . 
       FIGS. 6A &amp; 6B  are example screenshots  600   a  &amp;  600   b , respectively, which illustrate velocity data according to an implementation. Screenshot  600   a  (in  FIG. 6A ) includes raw velocity data  610  and screenshot  600   b  (in  FIG. 6B ) includes smoothed velocity data  620 . The velocity data  610  and  620  show a complex velocity model with changing contrast and boundary. As illustrated, the smoothed velocity data  620  in  FIG. 6B  that is produced by the norm zero smoothing process removes small anomalies, keeps major boundary (edge preserve) while enhancing the boundary variations (gradual change of boundary). Although screenshots  600   a  and  600   b  of  FIGS. 6A and 6B , respectively, are illustrated in grayscale, in other implementations, the screenshots  600   a  and  600   b  can display the appropriate velocity data  610  and  620  in various layers of color, such as red, orange, yellow, green, blue, indigo, violet, and the like, and the like to help clearly distinguish different data layers. 
       FIGS. 7A &amp; 7B  are example screenshots  700   a  &amp;  700   b , respectively, which illustrate an inline section view of a 3D seismic survey according to an implementation. Screenshot  700   a  includes a raw data view  710  and Screenshot  700   b  includes a smoothed data view  720 . The views  710  and  720  represent recorded reflection seismic curves. As illustrated in  FIG. 7B , after norm zero optimization, the major reflection layers are kept but the small events between layers have been smoothed out. Therefore, in some cases, the norm-zero optimization process may improve sparse and layered data. However, the process may not be suitable to preserve small variations such as inter-bed reflections or thin layers. 
       FIGS. 8A &amp; 8B  are example screenshots  800   a  &amp;  800   b , respectively, which illustrate a 3D volume view of an acoustic impedance model according to an implementation. Screenshot  800   a  includes an impedance model  810  obtained based on raw data and screenshot  800   b  includes an impedance model  820  obtained based on data smoothed by 3D norm zero optimization. A zone  812   a  in  FIG. 8A  displays rough layers and defined edges. The optimization parameters include a cut off ratio of 80%, 5 iterations, and a beta value of 0.5. As illustrated in  FIG. 8B , after 3D norm zero smoothing, the layers of zone  812   b  are smooth and changes between layers are gradual with better continuation and clarity. Therefore, the 3D norm zero smoothing improves the event continuities while preserving edges and suppressing the undesired non-major events. In some implementations, the impedance model  810  and  820  can be displayed in various colors, such as red, orange, yellow, green, blue, indigo, violet and the like, based on impedance data, to help clearly distinguish different data layers. For example, impedance data around a value of 10000 can be displayed in indigo/violet while impedance data of around a value of 18000 can be in red. The zone  812   a  in  FIG. 8A  can be displayed in yellow to indicate impedance data of values between about 14000 and 16000. 
       FIGS. 9A &amp; 9B  are example screenshots  900   a  &amp;  900   b , respectively, which illustrate a 2D section view of an acoustic impedance model according to an implementation. The screenshot  900   a  includes a vertical directional view of the impedance model  910  obtained based on raw data and screenshot  900   b  includes a vertical directional view of the impedance model  920  obtained based on data smoothed by 3D norm zero optimization. Both views show the inline direction at one of the sublines. As illustrated, the 3D norm zero smoothing suppresses low amplitude images and provides better lateral variation layers. In some implementations, the impedance model  910  and  920  can be displayed in various colors, such as red, orange, yellow, green, blue, indigo, violet, and the like, based on impedance data, to help clearly distinguish different data layers. For example, impedance data of around a value of 10000 can be displayed in indigo/violet while impedance data of around a value of 18000 can be in red. 
       FIGS. 10A &amp; 10B  are example screenshots  1000   a  &amp;  1000   b , respectively, which illustrate a time slice view of an acoustic impedance model according to an implementation. The screenshot  1000   a  includes a time slice view of the impedance model  1010  obtained based on data smoothed by JASON software and screenshot  1000   b  includes a time slice view of the impedance model  1020  obtained based on data smoothed by 3D norm zero optimization. Both time slice views represent  444  MS. As illustrated, the 3D norm zero smoothing provides more natural varied changes that describe the subsurface geological information. In some implementations, the impedance model  1010  and  1020  can be displayed in various colors, such as red, orange, yellow, green, blue, indigo, violet, and the like, based on impedance data, to help clearly distinguish different data layers. For example, impedance data of around a value of 10000 can be displayed in indigo/violet while impedance data of around a value of 18000 can be in red. 
       FIGS. 11A &amp; 11B  are example screenshots  1100   a  &amp;  1100   b , respectively, which illustrate a porosity model according to an implementation. In some cases, geological interpretation can concentrate on a range of horizons where major geological layers (events) are present. In these or other cases, porosity can be computed from geophysical data of the interest zone to analyze the hydrocarbon mobility. The illustrated porosity model  1110  in  FIG. 11A  represents a zone of subline (110˜696), cross line (302˜1193), and time (0.5˜1.2 second) of a porosity model that is obtained by using support vector machine inversion out of a suit of seismic attributes in Saudi Arabia; and the porosity model  1120  in  FIG. 11B  is obtained based on the same inversion techniques using smoothed data. As illustrated, the porosity model  1120  is more consistent, and both the vertical and horizon gradient change becomes small and continuous compared to the model  1110 . In some implementations, the screenshot  1100   a  and  1100   b  can display the porosity model  1110  and  1120  in various colors, such as red, orange, yellow, green, blue, indigo, violet, and the like to help clearly distinguish different data layers. 
     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, i.e., one or more modules of computer program instructions encoded on a tangible, non-transitory computer-storage medium for execution by, or to control the operation of, data processing apparatus. Alternatively or in addition, the program instructions can be encoded on an artificially generated propagated signal, e.g., 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 one or more of them. 
     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, e.g., 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 and/or special purpose logic circuitry may be hardware-based and/or software-based. The apparatus can optionally include code that creates an execution environment for computer programs, e.g., code that constitutes processor firmware, a protocol stack, a database management system, an operating system, or a combination of one or more of them. 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, e.g., 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, e.g., 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. 
     The processes and 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 processes and logic flows can also be performed by, and apparatus can also be implemented as, special purpose logic circuitry, e.g., 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, e.g., magnetic, magneto-optical disks, or optical disks. However, a computer need not have such devices. Moreover, a computer can be embedded in another device, e.g., 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, e.g., 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 include all forms of non-volatile memory, media and memory devices, including by way of example semiconductor memory devices, e.g., erasable programmable read-only memory (EPROM), electrically erasable programmable read-only memory (EEPROM), and flash memory devices; magnetic disks, e.g., 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 business and/or 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, e.g., 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, e.g., 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, e.g., 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 operable by the business suite user. 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, e.g., as a data server, or that includes a middleware component, e.g., an application server, or that includes a front-end component, e.g., 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 and/or wireless digital data communication, e.g., 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 and/or 802.20, all or a portion of the Internet, and/or any other communication system or systems at one or more locations. The network may communicate with, for example, Internet Protocol (IP) packets, Frame Relay frames, Asynchronous Transfer Mode (ATM) cells, voice, video, data, and/or other suitable information 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. 
     In some implementations, any or all of the components of the computing system, both hardware and/or software, may interface with each other and/or the interface using an application programming interface (API) and/or a service layer. The API may include specifications for routines, data structures, and object classes. The API 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 provides software services to the computing system. The functionality of the various components of the computing system may be accessible for all service consumers via this service layer. Software services provide reusable, defined business 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. The API and/or service layer may be an integral and/or a stand-alone component in relation to other components of the computing system. Moreover, any or all parts of the service layer 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. 
     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 features may be described above 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 and parallel processing may be advantageous. 
     Moreover, the separation and/or integration of various system modules and components in the implementations described above should not be understood as requiring such separation and/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 above description of example implementations does not define or constrain this disclosure. Other changes, substitutions, and alterations are also possible without departing from the spirit and scope of this disclosure.