Patent Publication Number: US-2013245954-A1

Title: Seismic imaging system using cosine transform in logarithmic axis

Description:
CROSS-REFERENCE TO RELATED APPLICATION 
     This application claims the benefit under 35 U.S.C. §119(a) of a U.S. Provisional Patent Application No. 61/610,074, No. 61/610,078 and No. 61/610,082, all filed on Mar. 13, 2012, the entire disclosures of which are incorporated herein by reference for all purposes. 
    
    
     BACKGROUND 
     1. Field 
     The following description relates to seismic imaging technology. 
     2. Description of the Related Art 
     Waveform inversion and reverse time migration are well known in the seismic imaging field. These algorithms are defined based on time or frequency domain. Frequency-domain based algorithms are based on Fourier-transformed measured data and a Fourier-transformed waveform equation. Another waveform inversion and migration algorithms based on Laplace-transformed measured data and a Laplace-transformed waveform equation have been proposed by inventors of the present invention, which enable the estimation of deeper parts of the subsurface with a smaller number of streamers. However, reflected waves still cause artifacts in the imaging output and make the algorithms less efficient. 
     SUMMARY 
     The following description relates to a technique for improving the resolution of reverse-time migration through source estimation. 
     According to an aspect of embodiment, there is provided a seismic imaging system including: a measured data processing unit converting measured data from receiver to cosine transformed data in logarithmic scale axes; a subsurface structure estimating unit calculating subsurface modeling parameters with measured data transformed in the measured data processing unit based on a cosine transformed acoustic waveform equation defined along logarithmic axes; and an image output unit converting modeling parameters calculated in the subsurface structure estimating unit and outputting the converted image data. 
     The subsurface structure estimating unit may include a waveform inversion unit updating initial modeling parameters of the cosine transformed acoustic waveform equation defined along logarithmic axes so as to minimize an objective function related to a residual between the measured data calculated in the measured data processing unit and modeling data calculated from a waveform equation with previous modeling parameters. 
     The waveform inversion unit may include a modeling data calculating unit calculating modeling data to be detected in each of receivers when a waveform propagates through subsurface structure defined by modeling parameters of cosine transformed waveform equation defined along logarithmic scale axes, an objective function calculating unit calculating an objective function related to a residual between measured data processed in the measured data processing unit and modeling data processed in the modeling data computation unit, comparing gradient of the objective function with a predetermined value and outputting the gradient if the gradient is larger than a predefined value and outputting the modeling parameters then if the gradient is smaller than the predefined value, and a modeling parameter updating unit updating modeling parameters to a direction to which the objective function decreases and outputting the result to the modeling data calculating unit. 
     The subsurface structure estimating unit may include a back-propagation unit configured to back-propagate the measured data based on a modeling parameter of a cosine transformed acoustic waveform equation defined along logarithmic-scaled axes, a virtual source estimator configured to estimate virtual sources, and a convolution unit configured to convolve the back-propagated measured data with the virtual source and to output the results of the convolution. 
     The seismic imaging system may further include a back-propagation unit configured to back-propagate the measured data based on a modeling parameter of a cosine transformed acoustic waveform equation defined along logarithmic-scaled axes; a virtual source estimator configured to estimate virtual sources; and a convolution unit configured to convolve the back-propagated measured data with the virtual source and to output the results of the convolution. 
     Other features and aspects will be apparent from the following detailed description, the drawings, and the claims. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         FIG. 1  is a block diagram illustrating an example of a seismic imaging system according to an exemplary embodiment. 
         FIG. 2  is a block diagram illustrating an example of a seismic imaging system using waveform inversion according to an exemplary embodiment. 
         FIG. 3  is a block diagram illustrating an example of a seismic imaging system using reverse-time migration according to an exemplary embodiment. 
     
    
    
     Throughout the drawings and the detailed description, unless otherwise described, the same drawing reference numerals will be understood to refer to the same elements, features, and structures. The relative size and depiction of these elements may be exaggerated for clarity, illustration, and convenience. 
     DETAILED DESCRIPTION 
     The following description is provided to assist the reader in gaining a comprehensive understanding of the methods, apparatuses, and/or systems described herein. Accordingly, various changes, modifications, and equivalents of the methods, apparatuses, and/or systems described herein will be suggested to those of ordinary skill in the art. Also, descriptions of well-known functions and constructions may be omitted for increased clarity and conciseness. 
       FIG. 1  is a block diagram illustrating an example of a seismic imaging system according to an exemplary embodiment. As illustrated in  FIG. 1 , the seismic imaging system includes a measured data processing unit  110  to convert measured data from a receiver into cosine transformed data in logarithmic scale axes and a subsurface structure estimating unit  300  to calculate subsurface modeling parameters with measured data transformed in the measured data processing unit, based on a cosine transformed acoustic waveform equation defined along logarithmic axes. 
     In one example, cosine transform is used as a modeling operator. An x-axis as a propagation direction and a z-axis as a depth direction in a modeling space are transformed into new logarithmic scale X-axis and Z-axis, respectively. 
     In the embodiment described herein, the modeling domain and operator is different from the original one. Cosine transform is used, instead of Fourier transform. Also, linear scale axis x and z are transformed into new logarithmic scale axis X and Z. 
     The acoustic wave equation in the time domain can be written as follows: 
     
       
         
           
             
               
                 
                   
                     
                       1 
                       
                         c 
                         2 
                       
                     
                      
                     
                       
                         
                           ∂ 
                           2 
                         
                          
                         
                           u 
                            
                           
                             ( 
                             
                               x 
                               , 
                               z 
                               , 
                               t 
                             
                             ) 
                           
                         
                       
                       
                         ∂ 
                         
                           t 
                           2 
                         
                       
                     
                   
                   = 
                   
                     
                       
                         
                           ∂ 
                           2 
                         
                          
                         
                           u 
                            
                           
                             ( 
                             
                               x 
                               , 
                               z 
                               , 
                               t 
                             
                             ) 
                           
                         
                       
                       
                         ∂ 
                         
                           x 
                           2 
                         
                       
                     
                     + 
                     
                       
                         
                           ∂ 
                           2 
                         
                          
                         
                           u 
                            
                           
                             ( 
                             
                               x 
                               , 
                               z 
                               , 
                               t 
                             
                             ) 
                           
                         
                       
                       
                         ∂ 
                         
                           z 
                           2 
                         
                       
                     
                     + 
                     
                       f 
                        
                       
                         ( 
                         
                           x 
                           , 
                           z 
                           , 
                           t 
                         
                         ) 
                       
                     
                   
                 
               
               
                 
                   ( 
                   1 
                   ) 
                 
               
             
           
         
       
     
     where c is the velocity, u is the wavefield in the time domain and f is the source wavelet. 
     A frequency-domain wave equation may be obtained by applying Cosine transform to the equation (1) as follows: 
     
       
         
           
             
               
                 
                   
                     
                       
                         
                           - 
                           
                             
                               w 
                               2 
                             
                             
                               c 
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                          
                         
                           
                             u 
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                            
                           
                             ( 
                             
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                             ) 
                           
                         
                       
                       = 
                       
                         
                           
                             
                               ∂ 
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                              
                             
                               
                                 u 
                                 ~ 
                               
                                
                               
                                 ( 
                                 
                                   x 
                                   , 
                                   z 
                                   , 
                                   t 
                                 
                                 ) 
                               
                             
                           
                           
                             ∂ 
                             
                               x 
                               2 
                             
                           
                         
                         + 
                         
                           
                             
                               ∂ 
                               2 
                             
                              
                             
                               
                                 u 
                                 ~ 
                               
                                
                               
                                 ( 
                                 
                                   x 
                                   , 
                                   z 
                                   , 
                                   t 
                                 
                                 ) 
                               
                             
                           
                           
                             ∂ 
                             
                               z 
                               2 
                             
                           
                         
                         + 
                         
                           
                             f 
                             ~ 
                           
                            
                           
                             ( 
                             
                               x 
                               , 
                               z 
                               , 
                               t 
                             
                             ) 
                           
                         
                       
                     
                      
                     
                       
 
                     
                      
                     where 
                      
                     
                       
 
                     
                      
                     
                       
                         u 
                         ~ 
                       
                        
                       
                         ( 
                         
                           x 
                           , 
                           z 
                           , 
                           t 
                         
                         ) 
                       
                     
                     = 
                     
                       
                         ∫ 
                         0 
                         ∞ 
                       
                        
                       
                         
                           u 
                            
                           
                             ( 
                             
                               x 
                               , 
                               z 
                               , 
                               t 
                             
                             ) 
                           
                         
                          
                         
                           cos 
                            
                           
                             ( 
                             
                               w 
                                
                               
                                   
                               
                                
                               t 
                             
                             ) 
                           
                         
                          
                         
                             
                         
                          
                         
                            
                           t 
                         
                       
                     
                   
                    
                   
                     
 
                   
                    
                   
                     
                       
                         f 
                         ~ 
                       
                        
                       
                         ( 
                         
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                           , 
                           z 
                           , 
                           t 
                         
                         ) 
                       
                     
                     = 
                     
                       
                         ∫ 
                         0 
                         ∞ 
                       
                        
                       
                         
                           f 
                            
                           
                             ( 
                             
                               x 
                               , 
                               z 
                               , 
                               t 
                             
                             ) 
                           
                         
                          
                         
                           cos 
                            
                           
                             ( 
                             
                               w 
                                
                               
                                   
                               
                                
                               t 
                             
                             ) 
                           
                         
                          
                         
                             
                         
                          
                         
                            
                           t 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   2 
                   ) 
                 
               
             
           
         
       
     
     Also, the x-axis and z-axis are transformed into new axis X and Z. Assuming that the wave propagates along the new axes X (X=g(x)) and Z (Z=h(z)), equation (5) may be defined as follows: 
     
       
         
           
             
               
                 
                   
                     
                       - 
                       
                         
                           w 
                           2 
                         
                         
                           c 
                           2 
                         
                       
                     
                      
                     
                       
                         u 
                         ~ 
                       
                        
                       
                         ( 
                         
                           X 
                           , 
                           Z 
                           , 
                           t 
                         
                         ) 
                       
                     
                   
                   = 
                   
                     
                       
                         
                           ∂ 
                           2 
                         
                          
                         
                           
                             u 
                             ~ 
                           
                            
                           
                             ( 
                             
                               X 
                               , 
                               Z 
                               , 
                               t 
                             
                             ) 
                           
                         
                       
                       
                         ∂ 
                         
                           x 
                           2 
                         
                       
                     
                     + 
                     
                       
                         
                           ∂ 
                           2 
                         
                          
                         
                           
                             u 
                             ~ 
                           
                            
                           
                             ( 
                             
                               X 
                               , 
                               Z 
                               , 
                               t 
                             
                             ) 
                           
                         
                       
                       
                         ∂ 
                         
                           z 
                           2 
                         
                       
                     
                     + 
                     
                       
                         f 
                         ~ 
                       
                        
                       
                         ( 
                         
                           X 
                           , 
                           Z 
                           , 
                           t 
                         
                         ) 
                       
                     
                   
                 
               
               
                 
                   ( 
                   3 
                   ) 
                 
               
             
           
         
       
     
     In the equation above, the partial derivative with respect to x and z should be replaced as follows: 
     
       
         
           
             
               
                 
                   
                     
                       
                         
                           ∂ 
                           2 
                         
                          
                         
                           
                             u 
                             ~ 
                           
                            
                           
                             ( 
                             
                               X 
                               , 
                               Z 
                               , 
                               t 
                             
                             ) 
                           
                         
                       
                       
                         ∂ 
                         
                           x 
                           2 
                         
                       
                     
                     = 
                     
                       
                         
                           
                             
                               ∂ 
                               2 
                             
                              
                             
                               u 
                                
                               
                                 ( 
                                 
                                   X 
                                   , 
                                   Z 
                                   , 
                                   t 
                                 
                                 ) 
                               
                             
                           
                           
                             ∂ 
                             
                               X 
                               2 
                             
                           
                         
                          
                         
                           
                             ( 
                             
                               
                                 ∂ 
                                 X 
                               
                               
                                 ∂ 
                                 x 
                               
                             
                             ) 
                           
                           2 
                         
                       
                       + 
                       
                         
                           
                             
                               ∂ 
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                              
                             
                               u 
                                
                               
                                 ( 
                                 
                                   X 
                                   , 
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                                 ) 
                               
                             
                           
                           
                             ∂ 
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                          
                         
                           
                             
                               ∂ 
                               2 
                             
                              
                             X 
                           
                           
                             ∂ 
                             
                               x 
                               2 
                             
                           
                         
                       
                     
                   
                    
                   
                     
 
                   
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                           ∂ 
                           2 
                         
                          
                         
                           
                             u 
                             ~ 
                           
                            
                           
                             ( 
                             
                               X 
                               , 
                               Z 
                               , 
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                             ) 
                           
                         
                       
                       
                         ∂ 
                         
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                           2 
                         
                       
                     
                     = 
                     
                       
                         
                           
                             
                               ∂ 
                               2 
                             
                              
                             
                               u 
                                
                               
                                 ( 
                                 
                                   X 
                                   , 
                                   Z 
                                   , 
                                   t 
                                 
                                 ) 
                               
                             
                           
                           
                             ∂ 
                             
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                               2 
                             
                           
                         
                          
                         
                           
                             ( 
                             
                               
                                 ∂ 
                                 Z 
                               
                               
                                 ∂ 
                                 z 
                               
                             
                             ) 
                           
                           2 
                         
                       
                       + 
                       
                         
                           
                             ∂ 
                             
                               u 
                                
                               
                                 ( 
                                 
                                   X 
                                   , 
                                   Z 
                                   , 
                                   t 
                                 
                                 ) 
                               
                             
                           
                           
                             ∂ 
                             Z 
                           
                         
                          
                         
                           
                             
                               ∂ 
                               2 
                             
                              
                             Z 
                           
                           
                             ∂ 
                             
                               z 
                               2 
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   4 
                   ) 
                 
               
             
           
         
       
     
     If the logarithmic function is used as a new axis, the wave propagates along the logarithmic axis. Then a large modeling domain may be constructed without computational overburden. Exemplary new axes for 2-D may be defined as follow: 
     
       
         
           
             
               
                 
                   X 
                   = 
                   
                     { 
                     
                       
                         
                           
                             
                               
                                 log 
                                  
                                 
                                   ( 
                                   
                                     x 
                                     + 
                                     1 
                                   
                                   ) 
                                 
                               
                             
                             
                               
                                 x 
                                 ≥ 
                                 0 
                               
                             
                           
                           
                             
                               
                                 - 
                                 
                                   log 
                                    
                                   
                                     ( 
                                     
                                       
                                         - 
                                         x 
                                       
                                       + 
                                       1 
                                     
                                     ) 
                                   
                                 
                               
                             
                             
                               
                                 x 
                                 ≤ 
                                 0 
                               
                             
                           
                         
                          
                         
                           
 
                         
                          
                         Z 
                       
                       = 
                       
                         
                           
                             log 
                              
                             
                               ( 
                               
                                 z 
                                 + 
                                 1 
                               
                               ) 
                             
                           
                            
                           
                               
                           
                            
                           z 
                         
                         ≥ 
                         0 
                       
                     
                   
                 
               
               
                 
                   ( 
                   5 
                   ) 
                 
               
             
           
         
       
     
     Then, a new 2-D waveform equation which provides modeling of wavelet propagating along logarithmic axes in a cosine transformation domain can be expressed as: 
     
       
         
           
             
               
                 
                   
                     
                       - 
                       
                         
                           w 
                           2 
                         
                         
                           c 
                           2 
                         
                       
                     
                      
                     
                       
                         u 
                         ~ 
                       
                        
                       
                         ( 
                         
                           X 
                           , 
                           Z 
                           , 
                           w 
                         
                         ) 
                       
                     
                   
                   = 
                   
                     
                       
                         
                           
                             
                               ∂ 
                               2 
                             
                              
                             
                               
                                 u 
                                 ~ 
                               
                                
                               
                                 ( 
                                 
                                   X 
                                   , 
                                   Z 
                                   , 
                                   w 
                                 
                                 ) 
                               
                             
                           
                           
                             ∂ 
                             
                               X 
                               2 
                             
                           
                         
                          
                         
                           1 
                           
                              
                             
                               2 
                                
                               X 
                             
                           
                         
                       
                       - 
                       
                         
                           
                             ∂ 
                             
                               u 
                                
                               
                                 ( 
                                 
                                   X 
                                   , 
                                   Z 
                                   , 
                                   w 
                                 
                                 ) 
                               
                             
                           
                           
                             ∂ 
                             X 
                           
                         
                          
                         
                           1 
                           
                              
                             
                               2 
                                
                               X 
                             
                           
                         
                       
                       + 
                       
                         
                           
                             
                               ∂ 
                               2 
                             
                              
                             
                               
                                 u 
                                 ~ 
                               
                                
                               
                                 ( 
                                 
                                   X 
                                   , 
                                   Z 
                                   , 
                                   w 
                                 
                                 ) 
                               
                             
                           
                           
                             ∂ 
                             
                               Z 
                               2 
                             
                           
                         
                          
                         
                           1 
                           
                              
                             
                               2 
                                
                               Z 
                             
                           
                         
                       
                       + 
                       
                         
                           
                             f 
                             ~ 
                           
                            
                           
                             ( 
                             
                               X 
                               , 
                               Z 
                               , 
                               t 
                             
                             ) 
                           
                         
                          
                         
                             
                         
                          
                         
                           ( 
                           
                             x 
                             ≥ 
                             0 
                           
                           ) 
                         
                       
                        
                       
                         
 
                       
                       - 
                       
                         
                           
                             w 
                             2 
                           
                           
                             c 
                             2 
                           
                         
                          
                         
                           
                             u 
                             ~ 
                           
                            
                           
                             ( 
                             
                               X 
                               , 
                               Z 
                               , 
                               w 
                             
                             ) 
                           
                         
                       
                     
                     = 
                     
                       
                         
                           
                             
                               ∂ 
                               2 
                             
                              
                             
                               
                                 u 
                                 ~ 
                               
                                
                               
                                 ( 
                                 
                                   X 
                                   , 
                                   Z 
                                   , 
                                   w 
                                 
                                 ) 
                               
                             
                           
                           
                             ∂ 
                             
                               X 
                               2 
                             
                           
                         
                          
                         
                           1 
                           
                              
                             
                               
                                 - 
                                 2 
                               
                                
                               X 
                             
                           
                         
                       
                       + 
                       
                         
                           
                             ∂ 
                             
                               u 
                                
                               
                                 ( 
                                 
                                   X 
                                   , 
                                   Z 
                                   , 
                                   w 
                                 
                                 ) 
                               
                             
                           
                           
                             ∂ 
                             X 
                           
                         
                          
                         
                           1 
                           
                              
                             
                               
                                 - 
                                 2 
                               
                                
                               X 
                             
                           
                         
                       
                       + 
                       
                         
                           
                             
                               ∂ 
                               2 
                             
                              
                             
                               
                                 u 
                                 ~ 
                               
                                
                               
                                 ( 
                                 
                                   X 
                                   , 
                                   Z 
                                   , 
                                   w 
                                 
                                 ) 
                               
                             
                           
                           
                             ∂ 
                             
                               Z 
                               2 
                             
                           
                         
                          
                         
                           
                               
                           
                           ε 
                         
                       
                       + 
                       
                         
                           
                             f 
                             ~ 
                           
                            
                           
                             ( 
                             
                               X 
                               , 
                               Z 
                               , 
                               t 
                             
                             ) 
                           
                         
                          
                         
                             
                         
                          
                         
                           ( 
                           
                             x 
                             ≤ 
                             0 
                           
                           ) 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   6 
                   ) 
                 
               
             
           
         
       
     
     In one example, the seismic imaging system may further include an image output unit  500  to convert modeling parameters calculated in the subsurface structure estimating unit and outputting the converted image data. The image output unit  500  may convert modeling parameters, for example, velocity distribution or mass distribution, into an image file for the visible display of the parameters. 
       FIG. 2  is a block diagram illustrating an example of a seismic imaging system using waveform inversion according to an exemplary embodiment. Similar to the seismic imaging system illustrated in  FIG. 1 , the seismic imaging system of  FIG. 2  includes a measured data processing unit  110  and a subsurface structure estimating unit  300 . In one example, the subsurface structure estimating unit  300  may include a waveform inversion unit  310  to update initial modeling parameters of the cosine transformed acoustic waveform equation defined along logarithmic axes so as to minimize an objective function related to a residual between the measured data calculated in the measured data processing unit and modeling data calculated from a waveform equation with previous modeling parameters. 
     The waveform inversion unit may include a modeling data calculating unit  313  to calculate modeling data to be detected in each of receivers when a waveform propagates through subsurface structure defined by modeling parameters of a cosine transformed waveform equation defined along logarithmic scale axes, an objective function calculating unit  315  calculating an objective function related to a residual between measured data processed in the measured data processing unit  110  and modeling data processed in the modeling data computation unit  313 , compare gradient of the objective function with a predetermined value and output the gradient if the gradient is larger than a predefined value and output the modeling parameters then if the gradient is smaller than the predefined value, and a modeling parameter updating unit  311  to update modeling parameters to a direction to which the objective function decreases and output the result to the modeling data calculating unit  313 . 
     The seismic imaging system illustrated in  FIG. 2  conforms to the general structure and operation of the conventional waveform inversion, except for the cosine transformation and logarithmic scale axes. Source estimation, gradient calculation, regularization and velocity update are similar to the conventional full-waveform inversion. 
       FIG. 3  is a block diagram illustrating an example of a seismic imaging system using reverse-time migration according to an exemplary embodiment. Referring to  FIG. 3 , a subsurface structure estimating unit  300  of the seismic imaging system includes a back-propagation unit  331  configured to back-propagate measured data converted by the measured data processing unit  110 , based on a modeling parameter of a cosine transformed acoustic waveform equation defined along logarithmic-scaled axes, a virtual source estimator  333  configured to estimate virtual sources, and a convolution unit  335  configured to convolve the back-propagated measured data with the virtual source and to output the results of the convolution. 
     A reverse-time migration (RTM) using cosine transform and the axis transformation technique are suggested. The frequency domain RTM image can be written by using a Fourier transformation as follows: 
     
       
         
           
             
               
                 
                   
                     
                       Φ 
                       k 
                     
                     = 
                     
                       
                         ∑ 
                         
                           s 
                           = 
                           1 
                         
                         
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                             max 
                           
                         
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                                         u 
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                                        
                                       
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                                         ) 
                                       
                                     
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                                     * 
                                   
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                                     ( 
                                     
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                         k 
                       
                       = 
                       
                         
                           
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                               k 
                             
                           
                         
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                                             ∂ 
                                             
                                               
                                                 u 
                                                 s 
                                               
                                                
                                               
                                                 ( 
                                                 ω 
                                                 ) 
                                               
                                             
                                           
                                           
                                             ∂ 
                                             
                                               p 
                                               k 
                                             
                                           
                                         
                                         ] 
                                       
                                       T 
                                     
                                      
                                     
                                       
                                         d 
                                         s 
                                         * 
                                       
                                        
                                       
                                         ( 
                                         
                                           r 
                                           , 
                                           ω 
                                         
                                         ) 
                                       
                                     
                                   
                                   ) 
                                 
                               
                                
                               
                                   
                               
                                
                               
                                  
                                 ω 
                               
                             
                           
                         
                       
                     
                     , 
                   
                 
               
               
                 
                   1 
                   ) 
                 
               
             
           
         
       
     
     where ω is the angular frequency, u s  and d s  are the modeled and observed data in the frequency domain, * indicates the complex-conjugate, and T indicates the matrix transposition. In one example, the above equation may be resolved by back-propagation algorithm. Such method is similar to the conventional reverse-time migration. 
     Generally, reverse-time migration is applied to waveform inversion data so as to correct distortion, and therefore more accurate result can be obtained. Hence, the embodiment illustrated in  FIG. 3  may be applied to the system illustrated in  FIG. 2  in a recursive manner. 
     A number of examples have been described above. Nevertheless, it will be understood that various modifications may be made. For example, suitable results may be achieved if the described techniques are performed in a different order and/or if components in a described system, architecture, device, or circuit are combined in a different manner and/or replaced or supplemented by other components or their equivalents. Accordingly, other implementations are within the scope of the following claims.