CN112698389B - Inversion imaging method and device for seismic data - Google Patents
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Abstract
The invention relates to a seismic data inversion imaging method and device, comprising the following steps: acquiring seismic data, wherein the seismic data comprises offset imaging results, offset velocity fields, source wavelets and observation data; calculating anti-migration simulation data according to the migration imaging result, the migration velocity field, the source wavelet and the selected coding function; constructing functional residual errors in logarithmic form according to the anti-offset simulation data and super gun data combined by the observation data; and deriving the functional residual error to obtain a gradient, and continuously and iteratively updating the offset imaging result according to the gradient until the residual error is smaller than a set low value to obtain a final imaging result. The invention greatly improves the tolerance and adaptability of the inversion imaging algorithm to the actual seismic data noise and improves the accuracy of the inversion imaging result.
Description
Technical Field
The invention relates to a seismic data inversion imaging method and device, and belongs to the technical field of oil-gas geophysical prospecting engineering.
Background
With the deep development of oil and gas exploration, the requirements on the imaging precision of seismic data are higher and higher, and the conventional offset imaging method cannot meet the requirements of exploration and development. The inversion imaging method of the seismic data is continuously approximate to the inverse of the Hessian matrix in an iterative mode, the adverse effects of an observation system, a wavelet frequency band, a complex structure and the like are gradually eliminated from an imaging result, and compared with a traditional migration imaging algorithm, the inversion imaging method of the seismic data has higher resolution, amplitude balance and fidelity, is more and more focused in the industry, and is the next development direction of seismic imaging.
The seismic data inversion imaging method has many advantages, but the excessive calculated amount is an unavoidable defect. In order to improve the calculation efficiency of the seismic data inversion imaging algorithm, a multi-source strategy is often adopted to combine individual cannons into a super cannon, and the calculation time is greatly saved due to the reduction of the calculation cannon number. However, the multi-source strategy introduces more crosstalk noise, and the conventional pressing strategy (such as dynamic coding) theory is too simple and has no obvious effect, and needs to be further improved. In addition, the suitability of the inversion imaging method for actual data and the tolerance of the inversion imaging method for seismic noise are directly determined by the functional residual error of the seismic data inversion imaging, the current target functional is based on the L2 mode expansion of the residual error, the theoretical derivation is established under a Bayesian framework, the noise is required to be accordant with Gaussian distribution, namely, only random noise can be processed, so that the tolerance of the seismic noise is low, the inversion imaging is inaccurate, the complex situation of an actual site cannot be adapted, and the popularization and the application of the inversion imaging method are greatly limited.
Disclosure of Invention
The invention aims to provide a seismic data inversion imaging method and device, which are used for solving the problem that the inversion imaging result is inaccurate due to low tolerance of functional residual errors adopted by the existing inversion imaging method to seismic noise.
In order to solve the technical problems, the invention provides a seismic data inversion imaging method, which comprises the following steps:
acquiring seismic data, wherein the seismic data comprises an offset imaging result, an offset velocity field, a seismic source wavelet and observation data;
calculating anti-migration simulation data according to the migration imaging result, the migration velocity field, the source wavelet and the selected coding function;
constructing functional residual errors according to the anti-offset simulation data and super gun data combined by the observation data, wherein the functional residual errors are in a logarithmic form;
and deriving the functional residual error to obtain a gradient, and continuously and iteratively updating the offset imaging result according to the gradient until the residual error is smaller than a set low value to obtain a final imaging result.
In order to solve the technical problem, the invention also provides a seismic data inversion imaging device, which comprises a processor and a memory, wherein the processor is used for processing instructions stored in the memory to realize the following method:
acquiring seismic data, wherein the seismic data comprises an offset imaging result, an offset velocity field, a seismic source wavelet and observation data;
calculating anti-migration simulation data according to the migration imaging result, the migration velocity field, the source wavelet and the selected coding function;
constructing functional residual errors according to the anti-offset simulation data and super gun data combined by the observation data, wherein the functional residual errors are in a logarithmic form;
and deriving the functional residual error to obtain a gradient, and continuously and iteratively updating the offset imaging result according to the gradient until the residual error is smaller than a set low value to obtain a final imaging result.
The beneficial effects of the invention are as follows: by constructing the functional residual in logarithmic form, the residual term appears in both the numerator and denominator in the process of acquiring the gradient after derivation, so that even if the data contains larger noise, the gradient can still be unaffected, and the larger the noise is, the more obvious the suppression is. Therefore, the invention greatly improves the tolerance and adaptability of the inversion imaging algorithm to the actual seismic data noise and improves the accuracy of the inversion imaging result.
As a further improvement of the method and the device, in order to construct a proper functional residual to reduce the influence of noise on the imaging result, the functional residual is calculated by the following formula:
where J (m) is a functional residual, m is an imaging result, s is a data weighting parameter, σ is a weight adjustment factor, L (m, ES) =ed cal For the reverse offset simulation data, L is a forward operator simulation operator, lambda is a regularization parameter, ed obs For super gun data combined by observation data, C is a sparse transformation operator, ES is a super gun source function, and Cm is 1 L as a sparse coefficient 1 Norm constraint term.
As a further improvement of the method and apparatus, in order to obtain an accurate imaging result, an update formula corresponding to the offset imaging result is:
wherein C is -1 Is sparse and inverseThe operator, m k For the imaging result of the kth iteration, m k+1 Alpha is the imaging result of the (k+1) th iteration k Update step length for the kth iteration, g is gradient, L T For the transposition of the forward modeling operator, abs () is a function taking absolute value.
As a further improvement of the method and the device, in order to obtain an optimal coding function, the selected method of the selected coding function is: and respectively encoding the observation data with the set proportion by adopting N coding functions to obtain super gun data, and performing imaging test on the super gun data to obtain corresponding imaging signal to noise ratio, wherein the coding function corresponding to the maximum imaging signal to noise ratio is the selected coding function.
As a further improvement of the method and apparatus, the sparse transform operator C is a dictionary learning basis function in order to adequately suppress crosstalk noise introduced by the multi-source strategy.
Drawings
FIG. 1 is a flow chart of a seismic data inversion imaging method of the present invention;
FIG. 2 is a schematic representation of a depression model velocity field in accordance with the present invention;
FIG. 3 is a schematic diagram of gun record data obtained by simulating a field blasting forward calculation in the invention;
FIG. 4 is a schematic diagram of the final determined optimal coding function of the present invention;
FIG. 5 is a schematic diagram of the inversion imaging result in the present invention;
FIG. 6 is a schematic diagram of the L2 mode inversion imaging results of the prior art;
fig. 7 is a schematic diagram of a convergence curve of the functional residual of the present invention.
Detailed Description
In order to make the objects, technical solutions and advantages of the present invention more apparent, the present invention will be further described in detail with reference to the accompanying drawings and specific embodiments.
Example 1:
the embodiment provides a seismic data inversion imaging method, which is mainly designed to construct a functional residual in a logarithmic form, so that in the process of deriving the functional residual to obtain a gradient, residual items can be simultaneously appeared in a numerator and a denominator, further offset noise introduced by abnormal noise in seismic data can be suppressed, and the tolerance and adaptability of an inversion imaging algorithm to actual seismic data noise are improved. In addition, in the iterative updating imaging result process, dictionary learning is utilized to sparsely express gradients, crosstalk noise introduced by multiple seismic sources is further suppressed, and therefore a high-quality gradient updating direction is obtained, and the purpose of steady, rapid and high-quality imaging for actual materials is achieved.
Specifically, a flow chart corresponding to the seismic data inversion imaging method is shown in fig. 1, and specifically includes the following contents:
1) And obtaining conventional migration imaging results, migration velocity fields, seismic source wavelets, field acquired observation data and other seismic data, and presetting a control parameter-residual threshold (setting a low value).
Wherein conventional offset imaging results may be obtained by a single pass wave offset or reverse time offset imaging process. In practice, if the conventional offset imaging process is not performed, the conventional offset imaging result may be made 0. In this embodiment, the shot record data obtained by simulating the field blasting forward modeling calculation is shown in fig. 3, the bar chart on the right side of the graph shows the amplitude of the seismic data, the set low value is set to be 1%, and the conventional offset imaging result is 0.
2) Selecting data with a set proportion (for example, about 5%) from the observed data in the step 1), and combining the data into super gun data by using five different coding functions. The five different coding functions are respectively polarity coding, frequency division coding, plane wave coding, amplitude coding and random time shift coding. And then, adopting a reverse time migration imaging test to obtain 5 migration sections, respectively calculating imaging signal to noise ratios of the 5 migration sections, and determining a coding function corresponding to the maximum signal to noise ratio as an optimal coding function E under the data.
In this embodiment, the speed field of the depression model selected for the imaging test is shown in fig. 2, the size of the model mesh is 260 x 188, the pitch of the longitudinal and transverse meshes is 8m, the bar graph on the right side in fig. 2 shows the speed value, the unit m/s, and fig. 4 shows the optimal coding function determined in this embodiment, that is, the polarity coding function. The calculation formula of the imaging signal-to-noise ratio adopted in the embodiment is as follows:
wherein SNR is imaging signal-to-noise ratio, m ref M is the reference model mig And (5) imaging results of the super gun data.
When the optimal encoding function E is obtained, the type and data of the encoding function may be selected autonomously according to the need, and the method is not limited to the above-listed five encoding functions.
3) All observed data d are obtained by using the optimal coding function E obs Combined into super gun Ed obs Then obtaining anti-migration simulation data Ed according to the conventional migration imaging result, migration velocity field, seismic source wavelet and optimal coding function cal . Due to the composition of super cannon Ed obs Obtaining anti-offset analog data Ed cal The specific process of (2) belongs to the prior art and is not described in detail herein.
4) According to the anti-offset simulation data Ed cal And all observed data d obs Combined super gun Ed obs Constructing a functional residual (namely a target functional), wherein the corresponding calculation formula is as follows:
where J (m) is a functional residual, m is an imaging result, s is a data weighting parameter, σ is a weight adjustment factor, L (m, ES) =ed cal For the reverse offset simulation data, L is a forward operator simulation operator, lambda is a regularization parameter, ed obs For super gun data combined by observation data, C is a sparse transformation operator, ES is a super gun source function, and Cm is 1 L as a sparse coefficient 1 Norm constraint term.
Unlike the target functional of the L2 mode used in conventional seismic inversion imaging, the data of the target functional in equation (2)The fitting term (i.e. the first term at the right end of equation 2) is in weighted logarithmic form, and the derivative calculated data residual term (i.e. L (m, ES) -Ed obs ) The method can be simultaneously present in a numerator and a denominator, so that even if the data contains larger noise, the gradient can still be unaffected, and the larger the noise is, the more obvious the suppression is, so that the method greatly improves the adaptability of the algorithm to the noise of the actual data.
The present embodiment uses a dictionary learning basis function based on K-SVD as the sparse transform operator C, which has better sparsity and is more adaptive to each imaging profile than a fixed basis function (e.g., DCT, curvelet, shearlet) or the like. The imaging result is transformed into a sparse domain through a sparse transformation operator C, and the sparsity is continuously constrained in the inversion iteration process, so that crosstalk noise which cannot be well represented by the sparse transformation operator C is suppressed, and a high-quality clean imaging section is obtained.
In this embodiment, the dictionary learning adopts the K-SVD method, but is not limited to the K-SVD method, and as other embodiments, the dictionary learning may also adopt a low-rank graph holding method, a matching pursuit method, a distributed dictionary learning method, or the like.
5) And deriving the functional residual error to obtain a gradient, and then continuously and iteratively updating the offset imaging result based on the gradient and the sparsity thereof until the residual value is smaller than a set low value to obtain a final imaging result.
The calculation process corresponding to the gradient g is obtained according to the functional residual error:
the better gradient profile can be obtained through the formula (3), the influence of noise in the seismic data can be effectively removed, and crosstalk noise which is introduced by multiple seismic sources still exists. In order to improve the calculation efficiency, a second item of processing, namely a sparse constraint item of a gradient, is required to be carried out, the display solution of the item is complex, and the item is blended into the updating process of a conjugate gradient method of an imaging section to improve the algorithm simplicity, wherein the specific process is as follows:
existing general iterative threshold contraction algorithms are used to solve the following formula:
J(x)＝||Lx-y||+λ||x|| 1
the iterative threshold shrink solution is:
wherein x is a model variable, y is a data variable, L is a forward modeling operator, lambda is a regularization parameter, and g is a gradient.
Let x=cm, and substituting the above formula (4), there is:
Cm k+1 ＝T h (C(m k -α k g),λ)
and C is moved to the right end of the formula, so that a final imaging result updating formula is obtained as follows:
wherein C is -1 For sparse inverse transform operator, m k For the imaging result of the kth iteration, m k+1 Alpha is the imaging result of the (k+1) th iteration k The update step length for the kth iteration can be obtained by a parabolic fitting or linear searching method, g is gradient, L T For the transposition of the forward modeling operator, abs () is a function taking absolute value.
The application process of the gradient second term is to extract a basis function from an imaging section by using K-SVD dictionary learning, and continuously restrict the sparsity of the imaging section by using an iterative shrinkage threshold value method to realize suppression of crosstalk noise. And finally, iteratively updating the imaging result by a conjugate gradient method until the residual threshold set in the step 1) is met, namely, the residual threshold is reduced to below 1%, and ending updating the imaging result. In this embodiment, the final inversion imaging result is shown in fig. 5.
Fig. 6 shows the inversion imaging result of the L2 mode (i.e. conventional least squares shift) commonly used in the prior art, and it can be seen by comparing fig. 5 and fig. 6 that the final imaging result (fig. 5) of the present invention is greatly improved compared with the inversion imaging result (fig. 6) in the prior art, and the specific improvement is as follows:
(1) The low-frequency noise is well suppressed, and the low-frequency noise does not contain shallow high-frequency noise introduced by filtering in conventional offset;
(2) The longitudinal resolution is greatly improved, especially in the vicinity of complex reverse-masked faults;
(3) The amplitude is more balanced, and illumination compensation is better at two sides of the section.
FIG. 7 is a functional residual convergence curve, wherein the dashed line represents the functional residual convergence curve obtained by using the L2 mode commonly used in the prior art, and is denoted by old, and the solid line represents the functional residual convergence curve obtained by using the inversion imaging method of the present invention, and is denoted by new. As can be seen from fig. 7, the functional residual of the present invention can converge to a smaller value, indicating that the data fit is better and less affected by seismic noise.
Example 2:
the embodiment provides a seismic data inversion imaging method, which specifically comprises the following steps:
(1) And obtaining conventional migration imaging results, migration velocity fields, seismic source wavelets, field acquired observation data and other seismic data, and presetting a control parameter-residual threshold (setting a low value).
The seismic data acquired in step (1) are the same as the seismic data acquired in step 1) of embodiment 1, and will not be described here.
(2) Selecting data with a set proportion (for example, about 5%) from the observed data in the step (1), and combining the data into super gun data by using four different coding functions. The four different coding functions are polarity coding, frequency division coding, plane wave coding and amplitude coding respectively. And then, adopting a reverse time migration imaging test to obtain 4 migration sections, respectively calculating imaging signal to noise ratios of the 4 migration sections, and determining a coding function corresponding to the maximum signal to noise ratio as an optimal coding function E under the data.
The process of obtaining the optimal encoding function E in the step (2) is the same as the process of obtaining the optimal encoding function E in the step 2) of embodiment 1, and the difference is only that the number of the employed encoding functions is different, which is not described here again.
(3) All observed data d are obtained by using the optimal coding function E obs Combined into super gun Ed obs Then obtaining anti-migration simulation data Ed according to the conventional migration imaging result, migration velocity field, seismic source wavelet and optimal coding function cal . Due to the composition of super cannon Ed obs Obtaining anti-offset analog data Ed cal The specific process of (2) belongs to the prior art and is not described in detail herein.
(4) According to the anti-offset simulation data Ed cal And all observed data d obs Combined super gun Ed obs Constructing a functional residual (namely a target functional), wherein the corresponding calculation formula is as follows:
where J (m) is a functional residual, m is an imaging result, s is a data weighting parameter, σ is a weight adjustment factor, L (m, ES) =ed cal For the reverse offset simulation data, L is a forward operator simulation operator, lambda is a regularization parameter, ed obs For super gun data combined by observation data, C is a sparse transformation operator, ES is a super gun source function, and Cm is 1 L as a sparse coefficient 1 Norm constraint term.
In step (4), the analog data Ed is simulated according to the anti-offset cal And all observed data d obs Combined super gun Ed obs For a specific process of constructing the functional residual, reference may be made to step 4) in embodiment 1, and details are not repeated here.
(5) And deriving the functional residual error to obtain a gradient, and then continuously and iteratively updating the offset imaging result based on the gradient and the sparsity thereof until the residual value is smaller than a set low value to obtain a final imaging result.
In step (5), the process of obtaining the gradient and the process of obtaining the final imaging result according to the gradient and the sparsity thereof can refer to step 5) in embodiment 1, namely, the calculation formula of the gradient g is as follows:
the imaging result updating formula is as follows:
example 3:
the embodiment provides a seismic data inversion imaging method, which specifically comprises the following steps:
(A) And obtaining conventional migration imaging results, migration velocity fields, seismic source wavelets, field acquired observation data and other seismic data, and presetting a control parameter-residual threshold (setting a low value).
The seismic data acquired in step (a) are the same as the seismic data acquired in step 1) of embodiment 1, and will not be described here.
(B) And (3) selecting data with a set proportion (for example, about 5%) from the observed data in the step (A), and combining the data into super gun data by using five different coding functions. The five different coding functions are respectively polarity coding, frequency division coding, plane wave coding, amplitude coding and random time shift coding. And then, adopting a reverse time migration imaging test to obtain 5 migration sections, respectively calculating imaging signal to noise ratios of the 5 migration sections, and determining a coding function corresponding to the maximum signal to noise ratio as an optimal coding function E under the data.
The process of obtaining the optimal encoding function E in the step (B) is the same as the process of obtaining the optimal encoding function E in the step 2) of embodiment 1, and will not be repeated here.
(C) All observed data d are obtained by using the optimal coding function E obs Combined into super gun Ed obs Then obtaining anti-migration simulation data Ed according to the conventional migration imaging result, migration velocity field, seismic source wavelet and optimal coding function cal . Due to the composition of super cannon Ed obs Obtaining anti-offset analog data Ed cal The specific process of (2) belongs to the prior art and is not described in detail herein.
(D) According to the anti-offset simulation data Ed cal And all observed data d obs Combined super gun Ed obs Constructing a functional residual (namely a target functional), wherein the corresponding calculation formula is as follows:
where J (m) is a functional residual, m is an imaging result, s is a data weighting parameter, σ is a weight adjustment factor, L (m, ES) =ed cal For the reverse offset simulation data, L is a forward operator simulation operator, lambda is a regularization parameter, ed obs For super gun data combined by observation data, C is a sparse transformation operator, ES is a super gun source function, and Cm is 1 L as a sparse coefficient 1 Norm constraint term.
In this embodiment, a fixed basis function (such as DCT, curvelet, shearlet) is used as a sparse transform operator C, an imaging result is transformed into a sparse domain through the sparse transform operator C, and sparsity is continuously constrained in an inversion iteration process, so that crosstalk noise which cannot be well represented by the sparse transform operator C is suppressed, and a high-quality clean imaging section is obtained.
(E) And deriving the functional residual error to obtain a gradient, and then continuously and iteratively updating the offset imaging result based on the gradient and the sparsity thereof until the residual value is smaller than a set low value to obtain a final imaging result.
In step (E), the process of obtaining the gradient and the process of obtaining the final imaging result according to the gradient and the sparsity thereof can refer to step 5) in embodiment 1, namely, the calculation formula of the gradient g is as follows:
the imaging result updating formula is as follows:
example 4:
the present embodiment provides a seismic data inversion imaging apparatus including a processor and a memory, the processor being configured to process instructions stored in the memory to implement the seismic data inversion imaging method of embodiment 1 described above. For those skilled in the art, according to the seismic data inversion imaging method in embodiment 1, a corresponding computer instruction may be generated to obtain the seismic data inversion imaging apparatus in this embodiment, which will not be described herein.
Example 5:
the present embodiment provides a seismic data inversion imaging apparatus, including a processor and a memory, the processor being configured to process instructions stored in the memory to implement the seismic data inversion imaging method in embodiment 2 described above. For those skilled in the art, according to the seismic data inversion imaging method in embodiment 2, a corresponding computer instruction may be generated to obtain the seismic data inversion imaging apparatus in this embodiment, which will not be described herein.
Example 6:
the present embodiment provides a seismic data inversion imaging apparatus, including a processor and a memory, the processor being configured to process instructions stored in the memory to implement the seismic data inversion imaging method in embodiment 3 above. For those skilled in the art, according to the seismic data inversion imaging method in embodiment 3, a corresponding computer instruction may be generated to obtain the seismic data inversion imaging apparatus in this embodiment, which will not be described herein.
Finally, it should be noted that the foregoing embodiments are merely for illustrating the technical solution of the present invention and not for limiting the scope of protection thereof, and although the present application has been described in detail with reference to the foregoing embodiments, it should be understood by those skilled in the art that various changes, modifications or equivalents may be made to the specific embodiments of the application after reading the present application, and these changes, modifications or equivalents are within the scope of protection of the claims of the present invention.
Claims (8)
1. The seismic data inversion imaging method is characterized by comprising the following steps of:
acquiring seismic data, wherein the seismic data comprises an offset imaging result, an offset velocity field, a seismic source wavelet and observation data;
calculating anti-migration simulation data according to the migration imaging result, the migration velocity field, the source wavelet and the selected coding function;
constructing a functional residual according to the anti-offset simulation data and super gun data combined by the observation data, wherein the calculation formula of the functional residual is as follows:
where J (m) is a functional residual, m is an imaging result, s is a data weighting parameter, σ is a weight adjustment factor, L (m, ES) =ed cal For the reverse offset simulation data, L is a forward operator simulation operator, lambda is a regularization parameter, ed obs For super gun data combined by observation data, C is a sparse transformation operator, ES is a super gun source function, and Cm is 1 L as a sparse coefficient 1 A norm constraint term;
and deriving the functional residual error to obtain a gradient, and continuously and iteratively updating the offset imaging result according to the gradient until the residual error is smaller than a set low value to obtain a final imaging result.
2. The seismic data inversion imaging method of claim 1, wherein the updated formula corresponding to the migration imaging result is:
wherein C is -1 For sparse inverse transform operator, m k Is the kthImaging results of multiple iterations, m k+1 Alpha is the imaging result of the (k+1) th iteration k Update step length for the kth iteration, g is gradient, L T For the transposition of the forward modeling operator, abs () is a function taking absolute value.
3. The seismic data inversion imaging method of any of claims 1-2, wherein the selected method of selecting the selected encoding function is: and respectively encoding the observation data with the set proportion by adopting N coding functions to obtain super gun data, and performing imaging test on the super gun data to obtain corresponding imaging signal to noise ratio, wherein the coding function corresponding to the maximum imaging signal to noise ratio is the selected coding function.
4. The seismic data inversion imaging method of claim 1 or 2, wherein the sparse transform operator C is a dictionary learning basis function.
5. A seismic data inversion imaging apparatus comprising a processor and a memory, the processor configured to process instructions stored in the memory to implement the method of:
acquiring seismic data, wherein the seismic data comprises an offset imaging result, an offset velocity field, a seismic source wavelet and observation data;
calculating anti-migration simulation data according to the migration imaging result, the migration velocity field, the source wavelet and the selected coding function;
constructing a functional residual according to the anti-offset simulation data and super gun data combined by the observation data, wherein the calculation formula of the functional residual is as follows:
where J (m) is a functional residual, m is an imaging result, s is a data weighting parameter, σ is a weight adjustment factor, L (m, ES) =ed cal For the reverse offset simulation data, L is the forward modeling operator, and lambda isRegularization parameter, ed obs For super gun data combined by observation data, C is a sparse transformation operator, ES is a super gun source function, and Cm is 1 L as a sparse coefficient 1 A norm constraint term;
and deriving the functional residual error to obtain a gradient, and continuously and iteratively updating the offset imaging result according to the gradient until the residual error is smaller than a set low value to obtain a final imaging result.
6. The seismic data inversion imaging apparatus of claim 5, wherein the updated formula for the offset imaging result is:
wherein C is -1 For sparse inverse transform operator, m k For the imaging result of the kth iteration, m k+1 Alpha is the imaging result of the (k+1) th iteration k Update step length for the kth iteration, g is gradient, L T For the transposition of the forward modeling operator, abs () is a function taking absolute value.
7. The seismic data inversion imaging apparatus of any of claims 5-6, wherein the selected encoding function is selected by the method of: and respectively encoding the observation data with the set proportion by adopting N coding functions to obtain super gun data, and performing imaging test on the super gun data to obtain corresponding imaging signal to noise ratio, wherein the coding function corresponding to the maximum imaging signal to noise ratio is the selected coding function.
8. The seismic data inversion imaging apparatus of claim 5 or 6, wherein the sparse transform operator C is a dictionary learning basis function.
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