Patent Publication Number: US-8537638-B2

Title: Methods for subsurface parameter estimation in full wavefield inversion and reverse-time migration

Description:
CROSS-REFERENCE TO RELATED APPLICATION 
     This application claims the benefit of U.S. Provisional Application No. 61/303,148 filed Feb. 10, 2010, entitled Methods for Subsurface Parameter Estimation in Full Wavefield Inversion and Reverse-Time Migration, which is incorporated by reference, in its entirety, for all purposes. 
    
    
     FIELD OF THE INVENTION 
     This invention relates generally to the field of geophysical prospecting and, more particularly, to seismic data processing. Specifically, the invention is a method for subsurface parameter estimation in full wave field inversion and reverse-time migration. 
     BACKGROUND OF THE INVENTION 
     Full wavefield inversion (FWI) in exploration seismic processing relies on the calculation of the gradient of an objective function with respect to the subsurface model parameters [12]. An objective function E is usually given as an L 2  norm as 
                     E   =       1   2     ⁢     ∫     ∫     ∫                p   ⁡     (       r   g     ,       r   s     ;   t       )       -       p   b     ⁡     (       r   g     ,       r   s     ;   t       )              2     ⁢     ⅆ   t     ⁢     ⅆ     S   g       ⁢     ⅆ     S   s                   ,           (   1   )               
where p and p b  are the measured pressure, i.e. seismic amplitude, and the modeled pressure in the background subsurface model at the receiver location r g  for a shot located at r s . In iterative inversion processes, the background medium is typically the medium resulting from the previous inversion cycle. In non-iterative inversion processes or migrations, the background medium is typically derived using conventional seismic processing techniques such as migration velocity analysis. The objective function is integrated over all time t, and the surfaces S g  and S s  that are defined by the spread of the receivers and the shots. We define K d (r)=K(r)−K b (r) and ρ d (r)=ρ(r)−ρ b (r), where K(r) and ρ(r) are the true bulk modulus and density, and K b (r) and ρ b (r) are the bulk modulus and the density of the background model at the subsurface location r. We also define the difference between the measured and the modeled pressure to be p d (r g ,r s ;t)=p (r g ,r s ;t)−p b (r g ,r s ;t).
 
     The measured pressure p, satisfies the wave equation 
                           ⁢           ρ   ⁢           ⁢     ∇     ·     (       1   ρ     ⁢     ∇   p       )           -       ρ   K     ⁢     p   ¨         =       -     q   ⁡     (   t   )         ⁢     δ   ⁡     (     r   -     r   s       )           ,     
     ⁢           ⁢   or             (   2   )                   (       ρ   b     +     ρ   d       )     ∇     ⁣           ·     (       1       ρ   b     +     ρ   d         ⁢     ∇     (       ρ   b     +     ρ   d       )         )       -           ρ   b     +     ρ   d           K   b     +     K   d         ⁢     (         ρ   ¨     b     +       ρ   ¨     d       )         =       -     q   ⁡     (   t   )         ⁢     δ   ⁡     (     r   -     r   s       )           ,             (   3   )               
where q(t) is the source signature. By expanding the perturbation terms and keeping only the 1st order Born approximation terms, one can derive the Born scattering equation for the pressure p d ,
 
                           ρ   b     ⁢     ∇     ·     (       1     ρ   b       ⁢     ∇     p   d         )           -         ρ   b       K   b       ⁢       p   ¨     d         =     -     [             ρ   b     ⁢     K   d         K   b   2       ⁢       p   ¨     b       -       ρ   b     ⁢     ∇     ·     (         ρ   d       ρ   b   2       ⁢     ∇     p   b         )             ]         ,           (   4   )               
and so p d  satisfies
 
                         p   d     ⁡     (         r   g     ⁢     r   s       ;   t     )       =     ∫       [           ρ   b     ⁡     (     r   ′     )       ⁢         K   d     ⁡     (     r   ′     )           K   b   2     ⁡     (     r   ′     )         ⁢         p   ¨     b     ⁡     (       r   ′     ,       r   s     ;   t       )         -         ρ   b     ⁡     (     r   ′     )       ⁢     ∇     ·     (           ρ   d     ⁡     (     r   ′     )           ρ   b   2     ⁡     (     r   ′     )         ⁢     ∇       p   b     ⁡     (       r   ′     ,       r   s     ;   t       )           )             ]     *       g   b     ⁡     (       r   g     ,       r   ′     ;   t       )       ⁢     ⅆ     V   ′             ,           (   5   )               
where V′ is the volume spanned by r′, and g b  is the Green&#39;s function in the background medium.
 
     One can derive the equations for the gradients of p b  using Eq. (5) and by considering the fractional change δp b  due to fractional change δK b  and δρ b  over an infinitesimal volume dV, 
                         ∂     p   b         ∂       K   b     ⁡     (   r   )           =             ρ   b     ⁡     (   r   )       ⁢   dV         K   b   2     ⁡     (   r   )         ⁢       g   b     ⁡     (       r   g     ,     r   ;   t       )       *         p   ¨     b     ⁡     (     r   ,       r   s     ;   t       )           ,     
     ⁢   and           (   6   )                     ∂     p   b         ∂       ρ   b     ⁡     (   r   )           =       F     -   1       ⁢     {       dV       ρ   b     ⁡     (   r   )         ⁢       ∇       G   b     ⁡     (       r   g     ,     r   ;   f       )         ·     ∇       P   b     ⁡     (     r   ,       r   s     ;   f       )             }         ,           (   7   )               
where P b =F{x}, P d =F{p d }, G b =F{g b }, and F and F −1  are the Fourier transform and the inverse Fourier transform operators.
 
     By using Eqs. 6 and 7, and using the reciprocity relationship ρ b (r)G b (r g ,r)=ρ b (r g )G b (r,r g ), 
                               ∂   E       ∂       K   b     ⁡     (   r   )           =       ⁢     -     ∫     ∫     ∫       p   d     ⁢       ∂     p   b         ∂       K   b     ⁡     (   r   )           ⁢     ⅆ   t     ⁢     ⅆ     S   g       ⁢     ⅆ     S   s                             =       ⁢       -           ρ   b     ⁡     (   r   )       ⁢   dV         K   b   2     ⁡     (   r   )           ⁢     ∫     ∫     ∫         (     i   ⁢           ⁢   2   ⁢           ⁢   π   ⁢           ⁢   f     )     2     ⁢       P   b     ⁡     (     r   ,       r   s     ;   f       )                                 ⁢         G   b     ⁡     (       r   g     ,     r   ;   f       )       ⁢       P   d   *     ⁡     (       r   g     ,       r   s     ;   f       )       ⁢     ⅆ   f     ⁢     ⅆ     S   g       ⁢     ⅆ     S   s                     =       ⁢             ρ   b     ⁡     (   r   )       ⁢   dV         K   b   2     ⁡     (   r   )         ⁢     ∫     ∫           p   .     b     ⁡     (     r   ,       r   s     ;   t       )       ⁢     ∫           ρ   b     ⁡     (     r   g     )           ρ   b     ⁡     (   r   )         ⁢       g   b     ⁡     (     r   ,       r   g     ;     -   t         )       *                               ⁢             p   .     d     ⁡     (       r   g     ,       r   s     ;   t       )       ⁢     ⅆ     S   g       ⁢     ⅆ   t     ⁢     ⅆ     S   s         ,             ⁢     
     ⁢           ⁢   and           (   8   )                         ∂   E       ∂       ρ   b     ⁡     (   r   )           =       ⁢     ∫     ∫     ∫       p   d     ⁢       ∂     p   b         ∂       ρ   b     ⁡     (   r   )           ⁢     ⅆ   t     ⁢     ⅆ     S   g       ⁢     ⅆ     S   s                           =       ⁢       -     dV       ρ   b     ⁡     (   r   )           ⁢     ∫     ∫     ∫         P   d   *     ⁡     (       r   g     ,       r   s     ;   f       )       ⁢       ∇       P   b     ⁡     (     r   ,       r   s     ;   f       )         ·                               ⁢       ∇       G   b     ⁡     (         r   g     ⁢   r     ;   f     )         ⁢     ⅆ   f     ⁢     ⅆ     S   g       ⁢     ⅆ     S   s                     =       ⁢       -     dV       ρ   b     ⁡     (   r   )           ⁢     ∫     ∫       ∇       p   b     ⁡     (     r   ,       r   s     ;   t       )         ·                           ⁢       [     ∫       ∇     (         g   b     ⁡     (     r   ,       r   g     ;     -   t         )       *         ρ   b     ⁡     (     r   g     )           ρ   b     ⁡     (   r   )         ⁢       p   d     ⁡     (       r   g     ,       r   s     ;   t       )         )       ⁢     ⅆ     S   g           ]     ⁢     ⅆ   t     ⁢       ⅆ     S   s       .                     (   9   )               
One can then use Eqs. 8 and 9 to perform full wavefield inversion in an iterative manner.
 
     Reverse-time migration (RTM) is based on techniques similar to gradient computation in FWI, where the forward propagated field is cross-correlated with the time-reversed received field. By doing so, RTM overcomes limitations of ray-based migration techniques such as Kirchhoff migration. In RTM, the migrated image field M at subsurface location r is given as
 
 M ( r )=∫∫ p   b ( r,r   s   ;t )∫ g   b ( r,r   g   ;−t )* p ( r   g   ,r   s   ;t ) dS   g   dtdS   s ,  (10)
 
which is very similar to the gradient equation 8 of FWI.
 
     While Eqs. 8 and 9 provide the framework for inverting data into subsurface models, the convergence of the inversion process often is very slow. Also, RTM using Eq. 10 suffers weak amplitude in the deep section due to spreading of the wavefield. Many attempts have been made to improve the convergence of FWI or improve the amplitude of the reverse-time migration by using the Hessian of the objective function [9], i.e., a second derivative of the objective function. Computation of the Hessian, however, is not only prohibitively expensive in computational resources, but it requires prohibitively large storage space for a realistic 3-D inversion problem. Furthermore, FWI using the full Hessian matrix may result in suboptimal inversion [2]. 
     One may be able to perform more stable inversion by lumping non-diagonal terms of the Hessian into the diagonal terms [2]. These, however, still require computation of the full Hessian matrix or at least a few off-diagonal terms of the Hessian matrix, which can be costly computations. While one may choose to use the diagonal of the Hessian only [11], this is valid only in the high frequency asymptotic regime with infinite aperture [1, 7]. 
     Plessix and Mulder tried to overcome these difficulties by first computing an approximate diagonal Hessian, then by scaling these by z α ν p   β , where z is the depth and ν p  is the compressional wave velocity [7]. From numerical experiment, they have determined that the best scaling parameter is z 0.5 ν p   0.5 . This approach, however, does not provide quantitative inversion of the subsurface medium parameters with correct units, since only approximate scaling has been applied. Furthermore, this approach was applied to RTM where only variations in compressional wave velocity is considered, and so may not be applicable to FWI where other elastic parameters such as density and shear wave velocity vary in space. 
     SUMMARY OF THE INVENTION 
     In one embodiment, the invention is a method for determining a model of a physical property in a subsurface region from inversion of seismic data, acquired from a seismic survey of the subsurface region, or from reverse time migration of seismic images from the seismic data, said method comprising determining a seismic resolution volume for the physical property and using it as a multiplicative scale factor in computations performed on a computer to either 
     (a) convert a gradient of data misfit in an inversion, or 
     (b) compensate reverse-time migrated seismic images, 
     to obtain the model of the physical property or an update to an assumed model. 
     In some embodiments of the inventive method, the gradient of data misfit or the reverse time migrated seismic images are multiplied by additional scale factors besides seismic resolution volume, wherein the additional scale factors include a source illumination factor, a receiver illumination factor, and a background medium properties factor. This results in a model of the physical property or an update to an assumed model having correct units. 
     It will be obvious to those who work in the technical field that in any practical application of the invention, inversion or migration of seismic data must be performed on a computer specifically programmed to carry out that operation. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
       The present invention and its advantages will be better understood by referring to the following detailed description and the attached drawings in which: 
         FIG. 1  is a flowchart showing basic steps in one embodiment of the present inventive method; 
         FIGS. 2-5  pertain to a first example application of the present invention, where  FIG. 2  shows the gradient of the objective function with respect to the bulk modulus in Pa m 4  s, computed using Eq. 8; 
         FIG. 3  shows the bulk modulus update  K d  (r)  in Pa computed using Eq. 18 and the gradient in  FIG. 2 ; 
         FIG. 4  shows the bulk modulus update  K d (r)  in Pa computed using Eq. 24 and the gradient in  FIG. 2 ; 
         FIG. 5  shows the gradient of the objective function with respect to the density in Pa 2  m 7  s/kg, computed using Eq. 9; 
         FIGS. 6 and 7  pertain to a second example application of the present invention, where  FIG. 6  shows the density update  ρ d (r)  in kg/m 3  computed using Eq. 28 and the gradient in  FIG. 5 ; and 
         FIG. 7  shows the density update  ρ d (r)  in kg/m 3  computed using Eq. 34 and the gradient in  FIG. 5 . 
     
    
    
     The invention will be described in connection with example embodiments. However, to the extent that the following detailed description is specific to a particular embodiment or a particular use of the invention, this is intended to be illustrative only, and is not to be construed as limiting the scope of the invention. On the contrary, it is intended to cover all alternatives, modifications and equivalents that may be included within the scope of the invention, as defined by the appended claims. 
     DETAILED DESCRIPTION OF EXAMPLE EMBODIMENTS 
     We derive inversion equations for K d  and ρ d  in the present invention using Eqs. 8 and 9. This is done by first taking advantage of the fact that p d  in Eqs. 8 and 9 can also be expanded using the Born approximation in Eq. 5. Neglecting the crosstalk components between K d  and ρ d , Eqs. 8 and 9 can then be approximated as 
                       ∂   E       ∂       K   b     ⁡     (   r   )           ≈       -           ρ   b     ⁡     (   r   )       ⁢   dV         K   b   2     ⁡     (   r   )           ⁢     ∫     ∫     ∫     ∫               ρ   b     ⁡     (     r   ′     )       ⁢       K   d     ⁡     (     r   ′     )             K   b   2     ⁡     (     r   ′     )         ⁡     [         g   b     ⁡     (       r   g     ,       r   ′     ;   t       )       *         p   ¨     b     ⁡     (       r   ′     ,       r   s     ;   t       )         ]       ×               [         g   b     ⁡     (       r   g     ,     r   ;   t       )       *         p   ¨     b     ⁡     (     r   ,       r   s     ;   t       )         ]     ⁢     ⅆ   t     ⁢     ⅆ     S   g       ⁢     ⅆ     S   s       ⁢     ⅆ     V   ′         ,     
     ⁢           ⁢   and                           (   11   )                   ∂   E       ∂       ρ   b     ⁡     (   r   )           ≈       dV       ρ   b     ⁡     (   r   )         ⁢     ∫     ∫     ∫     ∫             ρ   d     ⁡     (     r   ′     )           ρ   b     ⁡     (     r   ′     )         ⁡     [         ∇   ′     ⁢       P   b   *     ⁡     (       r   ′     ,       r   s     ;   f       )         ·       ∇   ′     ⁢       G   b   *     ⁡     (       r   g     ,       r   ′     ;   f       )           ]       ×             [       ∇       P   b     ⁡     (     r   ,       r   s     ;   f       )         ·     ∇       G   b     ⁡     (       r   g     ,     r   ;   f       )           ]     ⁢     ⅆ   t     ⁢     ⅆ     S   g       ⁢     ⅆ     S   s       ⁢       ⅆ     V   ′       .                             (   12   )               
By changing the orders of the integral, Eq. 11 can be rewritten in the frequency domain as
 
                       ∂   E       ∂       K   b     ⁡     (   r   )           ≈             ρ   b     ⁡     (   r   )       ⁢   dV         K   b   2     ⁡     (   r   )         ⁢     ∫     ∫         (     2   ⁢           ⁢   π   ⁢           ⁢   f     )     4     ⁢           ρ   b     ⁡     (     r   ′     )       ⁢   dV   ⁢           ⁢       K   d     ⁡     (     r   ′     )             K   b   2     ⁡     (     r   ′     )         ×             {     ∫           ρ   b     ⁡     (     r   g     )           ρ   b     ⁡     (   r   )         ⁢       G   b     ⁡     (     r   ,       r   g     ;   f       )       ⁢       G   b   *     ⁡     (       r   g     ,       r   ′     ;   f       )       ⁢     ⅆ     S   g           }     ⁢     {     ∫         P   b     ⁡     (     r   ,       r   s     ;   f       )       ⁢       P   b   *     ⁡     (       r   ′     ,       r   s     ;   f       )       ⁢     ⅆ     S   s           }     ⁢     ⅆ     V   ′       ⁢       ⅆ   f     .                         (   13   )               
The first integral term
 
                   ∫           ρ   b     ⁡     (     r   g     )           ρ   b     ⁡     (   r   )         ⁢       G   b     ⁡     (     r   ,       r   g     ;   f       )       ⁢       G   b   *     ⁡     (       r   g     ,       r   ′     ;   f       )       ⁢     ⅆ     S   g                 (   14   )               
in Eq. 13 is the approximation to the time reversal backpropagation for a field generated by an impulse source at r′, measured over the surface S g , and then backpropagated to r (See, e.g., Refs. [8, 3]). The wavefield due to this term propagates back towards the impulse source location at r′, and behaves similar to the spatial delta function δ(r−r′) when t=0, if the integral surface S g  embraces the point r′. This wavefield is correlated with the wavefield due to the second term, ∫P b (r,r s ; f)P b *(r′,r s ; f)dS s , to form the gradient near r=r′. The correlation of the first and the second term then decays rapidly near r=r′. The present invention recognizes that the zone in which the amplitude of the correlation term is not negligible is determined by the seismic resolution of the survey. In the present invention, we make an approximation that
 
                       ∫         (     2   ⁢   π   ⁢           ⁢   f     )     4     ⁢     {     ∫           ρ   b     ⁡     (     r   g     )           ρ   b     ⁡     (   r   )         ⁢       G   b     ⁡     (     r   ,       r   g     ;   f       )       ⁢       G   b   *     ⁡     (       r   g     ,       r   ′     ;   f       )       ⁢     ⅆ     S   g           }     ⁢     {     ∫         P   b     ⁡     (     r   ,       r   s     ;   f       )       ⁢       P   b   *     ⁡     (       r   ′     ,       r   s     ;   f       )       ⁢     ⅆ     S   s           }     ⁢     ⅆ   f         ≈         I   K     ⁡     (   r   )       ⁢       V   K     ⁡     (   r   )       ⁢     δ   ⁡     (     r   -     r   ′       )           ,           (   15   )               
where
 
                         I   K     ⁡     (   r   )       =     ∫         (     2   ⁢   π   ⁢           ⁢   f     )     4     ⁢     {     ∫           ρ   b     ⁡     (     r   g     )           ρ   b     ⁡     (   r   )         ⁢       G   b     ⁡     (     r   ,       r   g     ;   f       )       ⁢       G   b   *     ⁡     (       r   g     ,     r   ;   f       )       ⁢     ⅆ     S   g           }     ×     {     ∫         P   b     ⁡     (     r   ,       r   s     ;   f       )       ⁢       P   b   *     ⁡     (     r   ,       r   s     ;   f       )       ⁢     ⅆ     S   s           }     ⁢     ⅆ   f           ,           (   16   )               
and V K  (r) is the seismic resolution at the subsurface location r. Equation 15 is equivalent to the mass lumping of the Gauss-Newton Hessian matrix by assuming that the non-diagonal components are equal to the diagonal components when the non-diagonal components are within the seismic resolution volume of the diagonal component, and those outside the resolution volume zero. In other words, Eq. 15 is equivalent to implicitly counting the number of non-diagonal components N i  of the Gauss-Newton Hessian matrix in each i-th row that are significant in amplitude by using the seismic resolution volume of the survey, then multiplying the diagonal component of the i-th row by N i .
 
     Seismic resolution volume may be thought of as the minimal volume at r that a seismic imaging system can resolve under given seismic data acquisition parameters. Two small targets that are within one seismic resolution volume of each other are usually not resolved and appear as one target in the seismic imaging system. The resolution volumes for different medium parameters are different due to the difference in the radiation pattern. For example, the targets due to a bulk modulus perturbation yield a monopole radiation pattern, while those due to a density perturbation yield a dipole radiation pattern. Seismic resolution volume V K (r) for bulk modulus can be computed, for example, using a relatively inexpensive ray approximation [6, 4]. Persons who work in the technical field may know other ways to estimate the resolution volume. For example, one may be able to empirically estimate the resolution volume by distributing point targets in the background medium, and by investigating spread of the targets in the seismic image. If the background medium contains discontinuity in velocity due to iterative nature of the inversion, the background medium may need to be smoothed for ray tracing. One may also make a simplifying assumption that the wavenumber coverage is uniform. The seismic resolution volume in this case is a sphere with radius σ≈(5/18π) 0.5 ν p (r)/f p , where f p  is the peak frequency [6]. One may also employ an approximation, σ≈ν p (r)T/4=ν p (r)/4B, where T and B are the effective time duration and the effective bandwidth of the source waveform, following the radar resolution equation [5]. 
     Equation 11 can then be simplified using Eq. 15 as 
                         ∂   E       ∂       K   b     ⁡     (   r   )           ≈       -           ρ   b   2     ⁡     (   r   )       ⁢   V         K   b   4     ⁡     (   r   )           ⁢     〈       K   d     ⁡     (   r   )       〉     ⁢       I   K     ⁡     (   r   )       ⁢       V   K     ⁡     (   r   )           ,           (   17   )               and   ⁢           ⁢   so                               〈       K   d     ⁡     (   r   )       〉     ≈       -         K   b   4     ⁡     (   r   )             ρ   b   2     ⁡     (   r   )       ⁢   dV         ⁢     1         I   K     ⁡     (   r   )       ⁢       V   K     ⁡     (   r   )           ⁢       ∂   E       ∂       K   b     ⁡     (   r   )               ,           (   18   )               
where  K d  (r)  is the spatial average of K d  over the seismic resolution at spatial location r.
 
     Equation 16 can be further simplified if we use free-space Green&#39;s function 
                       G   ⁡     (       r   g     ,     r   ;   f       )       =       1     4   ⁢   π   ⁢          r   -     r   g                ⁢     e     ⅈ   ⁢           ⁢   k   ⁢          r   -     r   g                    ,           (   19   )               
and assume that S g  subtends over half the solid angle. Equation 16 then simplifies to
 
                         I   K     ⁡     (   r   )       ≈         I     K   ,   s       ⁡     (   r   )       ⁢       I     K   ,   g       ⁡     (   r   )           ,     
     ⁢   where           (   20   )                     I     K   ,   s       ⁡     (   r   )       =     ∫                  p   ¨     b     ⁡     (     r   ,       r   s     ;   t       )            2     ⁢     ⅆ   t           ,     
     ⁢   and           (   21   )                   I     K   ,   g       ⁡     (   r   )       =       ∫           ρ   b     ⁡     (     r   g     )           ρ   b     ⁡     (   r   )         ⁢       G   b     ⁡     (     r   ,       r   g     ;   f       )       ⁢       G   b   *     ⁡     (       r   g     ,     r   ;   f       )       ⁢     ⅆ     S   g           ≈       1     8   ⁢   π       ⁢           ρ   b     ⁡     (     r   g     )           ρ   b     ⁡     (   r   )         .                 (   22   )               
The term I K,s (r) may be recognized as the source illumination in the background model, and I K,g (r) can be understood to be the receiver illumination. One may also be able to vary the solid angle of integral at each subsurface location r following the survey geometry. Equation 11 then becomes
 
     
       
         
           
             
               
                 
                   
                     
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                               ) 
                             
                           
                         
                       
                       . 
                     
                   
                 
               
               
                 
                   ( 
                   24 
                   ) 
                 
               
             
           
         
       
     
     Equations 18 and 24 show that one can convert the gradient ∂E/∂K b (r) into a medium parameter  K d (r)  by scaling the gradient by the source and receiver illumination, resolution volume, and the background medium properties. If the inversion process is not iterative, one should be able to use Eq. 24 for parameter inversion. If the inversion process is iterative, one can use  K d (r)  in Eq. 24 as a preconditioned gradient for optimization techniques such as steepest descent, conjugate gradient (CG), or Newton CG method. It is important to note that Eqs. (18) and (24) yield bulk modulus with the correct units, i.e. are dimensionally correct, because all terms have been taken into account, and none have been neglected to simplify the computation as is the case with some published approaches. The published approaches that neglect one or more of the terms source illumination, receiver illumination, background medium properties and seismic resolution volume, will not result in the correct units, and therefore will need some sort of ad hoc fix up before they can be used for iterative or non-iterative inversion. 
     For density gradient, we make an assumption similar to that in Eq. 15,
 
∫∫[∇′ P   b *( r′,r   s   ;f )·∇′ G   b *( r   g   ,r′;f )]×[∇ P   b ( r,r   s   ;f )·∇ G   b ( r   g   ,r;f )] dS   g   dS   s   df≈I   ρ ( r ) V   ρ ( r )δ( r−r ′),  (25)
 
where
 
 I   ρ ( r )=∫∫∫| ∇P   b ( r,r   s   ;f )· ∇G   b ( r   g   ,r;f )| 2   dS   g   dS   s   df,   (26)
 
and V ρ (r) is the seismic resolution for density ρ at subsurface location r. The resolution volume V ρ (r) differs from V K (r), since the wavenumbers are missing when the incident and the scattered field are nearly orthogonal to each other. This is due to the dipole radiation pattern of the density perturbation, as discussed previously. As was done for V K (r), one can employ ray tracing to compute the resolution volume V ρ (r) while accounting for these missing near-orthogonal wavenumbers. Alternatively, one may be able assume V K (r)≈V ρ (r) by neglecting the difference in the wavenumber coverage.
 
     The gradient equation 12 can then be rewritten as 
                         ∂   E       ∂       ρ   b     ⁡     (   r   )           ≈       -     dV       ρ   b   2     ⁡     (   r   )           ⁢     〈       ρ   d     ⁡     (   r   )       〉     ⁢       I   ρ     ⁡     (   r   )       ⁢       V   ρ     ⁡     (   r   )           ,           (   27   )               and   ⁢           ⁢   so                               〈       ρ   d     ⁡     (   r   )       〉     ≈       -         ρ   b   2     ⁡     (   r   )       dV       ⁢     1         I   ρ     ⁡     (   r   )       ⁢       V   ρ     ⁡     (   r   )           ⁢       ∂   E       ∂       ρ   b     ⁡     (   r   )               ,           (   28   )               
where  ρ d (r)  is the spatial average of ρ d (r) over the seismic resolution volume V ρ (r).
 
     We can further simplify Eq. 25 by using the vector identity (a·b)(c·d)=(a·d)(b·c)+(a×c)·(b×d) to obtain
 
[ ∇′P   b *( r′,r   s   ;f )· ∇′G   b *( r   g   ,r′;f )][ ∇P   b ( r,r   s   ;f )· ∇G   b ( r   g   ,r;f )]=[ ∇′P   b *( r′,r   s   ;f )· ∇P   b ( r,r   s   ;f )][ ∇′G   b *( r   g   ,r′;f )· ∇G   b ( r   g   ,r;f )]+[ ∇′P   b *( r′,r   s   ;f )× ∇G   b ( r   g   ,r;f )]·[ ∇′G   b *( r   g   ,r′;f )× ∇P   b ( r,r   s   ;f )].  (29)
 
The second term in the right-hand side of Eq. 29 is the correction term for the dipole radiation pattern of the scattered field, and so it reaches a maximum when ∇P b  and ∇G b  are orthogonal to each other. Neglecting this correction term,
 
[∇′ P   b *( r′,r   s   ;f )·∇′ G   b *( r   g   ,r′;f )][∇ P   b ( r,r   s   ;f )·∇ G   b ( r   g   ,r;f )]≈[∇′ P   b *( r′,r   s   ;f )·∇ P   b ( r,r   s   ;f )][∇′G b *( r   g   ,r′;f )·∇ G   b ( r   g   ,r;f )].  (30)
 
Then Eq. 12 can be approximated as
 
                       ∂   E       ∂       ρ   b     ⁡     (   r   )           ≈       -     dV       ρ   b     ⁡     (   r   )           ⁢     ∫     ∫           ρ   b     ⁡     (     r   ′     )           ρ   b     ⁡     (     r   ′     )         ⁢     {     ∫       [         ∇   ′     ⁢       G   b   *     ⁡     (       r   g     ,       r   ′     ;   f       )         ·     ∇       G   b     ⁡     (       r   g     ,     r   ;   f       )           ]     ⁢     ⅆ     S   g           }     ×     {     ∫       [         ∇   ′     ⁢       P   b   *     ⁡     (       r   ′     ,       r   s     ;   f       )         ·     ∇       P   b     ⁡     (     r   ,       r   s     ;   f       )           ]     ⁢     ⅆ     S   s           }     ⁢     ⅆ   f     ⁢     ⅆ     V   ′                       (   31   )               
We can approximate the integral over S g  in Eq. 26 by using free-space Green&#39;s function,
 
                       ∫         ∇       G   b   *     ⁡     (       r   g     ,     r   ;   f       )         ·     ∇       G   b     ⁡     (       r   g     ,     r   ;   f       )           ⁢     ⅆ     S   g           ≈           ρ   b     ⁡     (     r   g     )           ρ   b     ⁡     (   r   )         ⁢     ∫         ∇       G   b   *     ⁡     (       r   g     ,     r   ;   f       )         ·     ∇       G   b     ⁡     (     r   ,       r   g     ;   f       )           ⁢     ⅆ     S   g             ≈           ρ   b     ⁡     (     r   g     )           ρ   b     ⁡     (   r   )         ⁢         (     2   ⁢   π   ⁢           ⁢   f     )     2       8   ⁢   π   ⁢           ⁢       v   p   2     ⁡     (   r   )               ,           (   32   )               
The integral was performed over half the solid angle under the assumption that ∇(ρ b (r g )/ρ b (r))≈0, and ρ b  (r g ) is constant along S g .
 
     The gradient equation above can then be rewritten as 
                       ∂   E       ∂       ρ   b     ⁡     (   r   )           ≈       -     dV       ρ   b   2     ⁡     (   r   )           ⁢     〈       ρ   d     ⁡     (   r   )       〉     ⁢       I     ρ   ,   s       ⁡     (   r   )       ⁢       I     ρ   ,   g       ⁡     (   r   )       ⁢       V   ρ     ⁡     (   r   )                 (   33   )               and   ⁢           ⁢   so                               〈       ρ   d     ⁡     (   r   )       〉     ≈       -         ρ   b   2     ⁡     (   r   )       dV       ⁢     1         I     ρ   ,   s       ⁡     (   r   )       ⁢       I     ρ   ,   g       ⁡     (   r   )       ⁢       V   ρ     ⁡     (   r   )           ⁢       ∂   E       ∂       ρ   b     ⁡     (   r   )               ,           (   34   )             where                               I     ρ   ,   s       ⁡     (   r   )       =     ∫              ∇       p   .     ⁡     (     r   ,       r   s     ;   t       )              2     ⁢     ⅆ   t     ⁢     ⅆ     S   s             ,           (   35   )             and                             I     ρ   ,   s       ⁡     (   r   )       =           ρ   b     ⁡     (     r   g     )           ρ   b     ⁡     (   r   )         ⁢       1     8   ⁢   π   ⁢           ⁢       v   p   2     ⁡     (   r   )           .               (   36   )               
As was the case with  K d (r) , Eqs. 28 or 34 can be used as an inversion formula for non-iterative inversion, or as a preconditioned gradient equation for iterative inversion. It is important to note that these equations yield density with the correct units, i.e. are dimensionally correct, because all terms have been taken into account, and none have been neglected to simplify the computation as is the case with some published approaches. The same is true for Eqs. 18 and 24 for bulk modulus. The published approaches that neglect one or more of the terms source illumination, receiver illumination, background medium properties and seismic resolution volume, will not result in the correct units, and therefore will need some sort of ad hoc fix up before they can be used for iterative or non-iterative inversion.
 
     Since the first iteration of FWI is similar to RTM, the method provided here can be applied to analyze the amplitude term in RTM with little modification. Seismic migration including RTM is typically used to image the structure of the subsurface, and so amplitude information in the migrated image is often discarded. We show that the RTM amplitude, when properly scaled using the method provided here, represents the difference between the true compressional wave velocity of the subsurface and the velocity of the background model. 
     We note that the RTM equation 10 is missing the double derivative of the incident field in Eq. 8. This double derivative represents that high frequency components scatter more efficiently than the low frequency components in the classical Rayleigh scattering regime (Refs. [10, 13]). Therefore, we can consider Eq. 10 as the gradient computation operation in Eq. 8, partially neglecting the frequency dependence of the scattered field, 
                         ∂   E       ∂       K   b     ⁡     (   r   )           ≈       -         ρ   b     ⁢       dV   ⁡     (     ⅈ   ⁢           ⁢   2   ⁢   π   ⁢           ⁢     f   c       )       2           K   b   2     ⁡     (   r   )           ⁢     ∫     ∫     ∫         P   b     ⁡     (     r   ,       r   s     ;   f       )       ⁢       G   b     ⁡     (     r   ,       r   g     ;   f       )       ⁢       P   *     ⁡     (       r   g     ,       r   s     ;   f       )       ⁢     ⅆ   f     ⁢     ⅆ     S   g       ⁢     ⅆ     S   s                 ≈       -         ρ   b     ⁢       dV   ⁡     (     ⅈ   ⁢           ⁢   2   ⁢   π   ⁢           ⁢     f   c       )       2           K   b   2     ⁡     (   r   )           ⁢     M   ⁡     (   r   )           ,           (   37   )               
where f c  is the center frequency of the source waveform. Frequency dependence is partially neglected because, while the frequency dependence in the forward field p b  has been neglected, the frequency dependence implicit in the received field p s  cannot be neglected. Spatial variation of density is usually not considered in RTM, and so ρ b  is assumed to be constant in Eq. 37.
 
     One can now apply the same approximation used to derive Eqs. 17 and 23 to Eq. 10, 
                         ∂   E       ∂       K   b     ⁡     (   r   )           ≈       -         ρ   b   2     ⁢       I     K   ,   g       ⁡     (   r   )       ⁢       V   K     ⁡     (   r   )       ⁢   dV   ⁢     〈       K   d     ⁡     (   r   )       〉     ⁢       (     ⅈ2π   ⁢           ⁢     f   c       )     2           K   b   4     ⁡     (   r   )           ⁢     ∫     ∫         (     ⅈ   ⁢           ⁢   2   ⁢   π   ⁢           ⁢   f     )     2     ⁢       P   b     ⁡     (     r   ,       r   s     ;   f       )       ⁢       P   b   *     ⁡     (     r   ,       r   s     ;   f       )       ⁢     ⅆ   f     ⁢     ⅆ     S   s                 ,           (   38   )               
which, together with Eq. 37, yields
 
                     〈       K   d     ⁡     (   r   )       〉     ≈           K   b   4     ⁡     (   r   )           ρ   b     ⁢       I     K   ,   g       ⁡     (   r   )       ⁢       V   K     ⁡     (   r   )           ⁢         M   ⁡     (   r   )         ∫     ∫                  p   .     b     ⁡     (     r   ,       r   s     ;   t       )            2     ⁢     ⅆ   t     ⁢     ⅆ     S   s               .               (   39   )               
Equation 39 enables quantitative analysis of the amplitude in the reverse-time migrated image. More specifically, it enables inversion of the amplitude into the difference bulk modulus of the subsurface.
 
       FIG. 1  is a flowchart showing basic steps in one embodiment of the present inventive method. In step  103 , the gradient of an objective function is computed using an input seismic record ( 101 ) and information about the background subsurface medium ( 102 ). At step  104 , the source and receiver illumination in the background model is computed. At step  105 , the seismic resolution volume is computed using the velocities of the background model. At step  106 , the gradient from step  103  is converted into the difference subsurface model parameters using the source and receiver illuminations from step  104 , seismic resolution volume from step  105 , and the background subsurface model ( 102 ). If an iterative inversion process is to be performed, at step  107  the difference subsurface model parameters from step  106  are used as preconditioned gradients for the iterative inversion process. 
     EXAMPLES 
     We consider the case of a 30 m×30 m×30 m “perfect” Born scatterer in a homogeneous medium with K b =9 MPa and ρ b =1000 kg/m 3 . The target is centered at (x,y,z)=(0,0,250 m), where x and y are the two horizontal coordinates and z is the depth. The target may be seen in  FIGS. 2-7  as a 3×3 array of small squares located in the center of each diagram. We assume co-located sources and receivers in the −500 m≦x≦500 m and −500 m≦y≦500 m interval with the 10-m spacing in both x and y directions. We assume that the source wavelet has a uniform amplitude of 1 Pa/Hz at 1 m in the 1 to 51 Hz frequency band. 
     In the first example, we assume that the target has the bulk modulus perturbation given as K d =900 kPa.  FIG. 2  shows the gradient ∂E/∂K b (r) along the y=0 plane using Eq. 8. The scattered field p d  in Eq. 8 has been computed using Eq. 5. The gradient in  FIG. 2  has units of Pa m 4  s, and so cannot be directly related to K d . As described in the “Background” section above, this is a difficulty encountered in some of the published attempts to compute a model update from the gradient of the objective function. 
       FIG. 3  shows  K d (r)  using Eq. 18 of the present invention. It has been assumed that the seismic resolution volume V K (r) is a sphere with radius σ=ν p (r)/4B=15 m. One can see that  FIG. 3  is the smeared image of the target, since  K d (r)  is the averaged property over the seismic resolution volume.  FIG. 4  is  K d  (r)  when the less rigorous Eq. 24 is employed. One can see that  K d (r)  in  FIGS. 3 and 4  are in good agreement with each other. The value of  K d (r)  at the center of the target in  FIG. 3  is 752 kPa, and that in  FIG. 4  is 735 kPa, both of which are within 20% of the true value of 900 kPa. 
     The second example is the case where the target has a density perturbation of ρ d =100 kg/m 3 .  FIG. 5  shows the gradient ∂E/∂ρ b (r) along the y=0 plane using Eq. 9. The scattered field in Eq. 9 has been computed using Eq. 5. The gradient in  FIG. 5  has units of Pa 2  m 7  s/kg. As with the first example, the different units prevent the gradient from being directly related to a density update, again illustrating the problem encountered in published methods. 
       FIG. 6  shows  ρ d (r)  using Eq. 28. The seismic resolution volume V ρ (r) has been assumed to be identical to V K (r).  FIG. 7  is  ρ d (r)  when Eq. 34 is employed. Estimation of  ρ d (r)  using Eq. 34 results in less accurate inversion than that using Eq. 28 due to the neglected dipole illumination term in Eq. 31. 
     The foregoing patent application is directed to particular embodiments of the present invention for the purpose of illustrating it. It will be apparent, however, to one skilled in the art, that many modifications and variations to the embodiments described herein are possible. All such modifications and variations are intended to be within the scope of the present invention, as defined in the appended claims. Persons skilled in the art will readily recognize that in practical applications of the invention, at least some of the steps in the present inventive method are performed on or with the aid of a computer, i.e. the invention is computer implemented. 
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