Model-driven deep learning-based seismic super-resolution inversion method

Abstract

A model-driven deep learning-based seismic super-resolution inversion method includes the following steps: 1) mapping each iteration of a model-driven alternating direction method of multipliers (ADMM) into each layer of a deep network, and learning proximal operators by using a data-driven method to complete the construction of a deep network ADMM-SRINet; 2) obtaining label data used to train the deep network ADMM-SRINet; 3) training the deep network ADMM-SRINet by using the obtained label data; and 4) inverting test data by using the deep network ADMM-SRINet trained at step 3). The method combines the advantages of a model-driven optimization method and a data-driven deep learning method, and therefore the network has the interpretability; and meanwhile, due to the addition of physical knowledge, the iterative deep learning method lowers requirements for a training set, and therefore an inversion result is more reliable.

Claims

1. A model-driven deep learning-based seismic super-resolution inversion (SRI) method, comprising the following steps: 1) obtaining seismic data from an oil field and model data and using the seismic data and the model data to produce label data; 2) mapping each iteration of a model-driven alternating direction method of multipliers (ADMM) to each layer of a deep network, and learning proximal operators by using a data-driven method to complete a construction of a deep network ADMM-SRI; 3) training the deep network ADMM-SRI by using the label data; 4) inverting test data by using the deep network ADMM-SRI trained at step 3); 5) inverting the seismic data from the oil field using the deep network ADMM-SRI; and 6) selecting an oil or gas reservoir using the inverted seismic data from the oil field and performing exploration and/or development of the selected oil or gas reservoir in the oil field.

2. The model-driven deep learning-based seismic super-resolution inversion (SRI) method according to claim 1, wherein at step 2), a proximal form of the model-driven ADMM is shown as formula (1): $\begin{array}{cc}\{\begin{array}{c}{r}^k={\mathrm{prox}}_{\rho f}\left(x^{k-1}-{\beta }^{k-1}\right),\\ x^k={\mathrm{prox}}_{\rho g}\left(r^k+{\beta }^{k-1}\right),\\ {\beta }^k={\beta }^{k-1}+{\eta }^k\left(r^k-x^k\right),\end{array}& \left(1\right)\end{array}$ wherein two modules r.sup.k and x.sup.k in formula (1) are respectively learned by using a residual convolutional network block, and for a third module β.sup.k in formula (1), only a parameter η needs to be determined in each iteration; and each iteration process in formula (1) is unrolled to construct the deep network, wherein prox.sub.ρf( ) and prox.sub.μg( ) are proximal operators.

3. The model-driven deep learning-based seismic super-resolution inversion (SRI) method according to claim 2, wherein at step 2), the deep network ADMM-SRI comprises three stages, wherein in a k-th stage, the deep network is composed of three modules comprising the module r.sup.k, the module x.sup.k, and the module β.sup.k, wherein the module r.sup.k and the module x.sup.k are configured to calculate values of r.sup.k and x.sup.k by using learning operators; the module β.sup.k is configured to calculate a value of β.sup.k, and nodes of the three modules are connected via straight lines with a directionality.

4. The model-driven deep learning-based seismic super-resolution inversion (SRI) method according to claim 2, wherein a first learning module in formula (1) is as follow:
r.sup.k=Γ.sub.Θ.sub.k.sub.f(r.sup.k-1,x.sup.k-1,β.sup.k-1,W.sub.TWr.sup.k-1,y),  (2) wherein Γ.sub.Θ.sub.k.sub.f represents a learning operator in a k-th iteration, and wherein a parameter set in the learning operator is represented by Θ.sub.k.sup.f; r.sup.k-1, x.sup.k-1, β.sup.k-1, W.sup.TWr.sup.k-1, and y are inputs of the learning operator Γ.sub.Θ.sub.k.sub.f, the learning operator is learned by the residual convolutional network block composed of two “Conv+ReLu” layers and one “Conv” layer, wherein “Conv” represents a convolution operator, and “ReLu” represents a linear rectification activation function; in a residual convolutional network of the residual convolutional network block, each layer comprises 32 convolution kernels with a size of 3×1, and a last layer comprises only one convolution kernel with a size of 3×1×32; an input of the residual convolutional network comprises physical knowledge in a forward model and gradient information of a loss function in the model-driven deep learning-based seismic super-resolution inversion method.

5. The model-driven deep learning-based seismic super-resolution inversion (SRI) method according to claim 2, wherein a second learning module in formula (1) is configured to learn a mapping relationship γ.sub.Θ.sub.k.sub.g between x.sup.k and p.sup.k as follows:
x.sup.k=γ.sub.Θ.sub.k.sub.g(p.sup.k),  (3) wherein Θ.sub.k.sup.g represents a network parameter in a k-th iteration of the deep network, p.sup.k=r.sup.k+β.sup.k-1, the second learning module is learned by using three residual convolutional network blocks, wherein a structure of each of the three residual convolutional network blocks is equal to a structure of a residual convolutional network configured to learn a first learning operator.

6. The model-driven deep learning-based seismic super-resolution inversion (SRI) method according to claim 2, wherein for the third module in formula (1), only the parameter η needs to be determined in each iteration, η is considered as a weight in the deep network, and η is learned together with other parameters in the deep network from training data.

7. The model-driven deep learning-based seismic super-resolution inversion (SRI) method according to claim 1, wherein at step 2), the label data comprises the model data and field data, wherein: for the model data, a known velocity model is configured to perform a forward modeling to obtain seismic super-resolution data used as the label data; and for the field data, an acquisition procedure is as follows: the seismic data is first subjected to a denoising preprocessing and then subjected to a non-stationary correction to obtain stationary seismic data; a reflection coefficient profile is obtained by using an alternating iterative inversion method, and the reflection coefficient profile is filtered through a wide-band Gaussian or a Yu wavelet to obtain band-limited super-resolution data used as the label data.

8. The model-driven deep learning-based seismic super-resolution inversion (SRI) method according to claim 7, wherein step 3) of training the deep network ADMM-SRI by using the label data specifically comprises: preliminarily training the deep network by using the model data to enable parameters in the deep network to have an initial value; specifically obtaining the label data from the field data in different target areas, and performing a fine adjustment on the deep network by a transfer learning strategy, and configuring the deep network to invert the field data.

9. The model-driven deep learning-based seismic super-resolution inversion (SRI) method according to claim 8, wherein during the training of the deep network ADMM-SRI, a loss function is as follow:
E(Θ)=1/NΣ.sub.i=1.sup.N∥{circumflex over (r)}.sub.i−r.sub.i.sup.gt∥.sub.2.sup.2,  (4) wherein a network parameter is Θ={Θ.sub.k.sup.g,Θ.sub.k.sup.f,η.sup.k}.sub.k=1.sup.N.sup.t, N represents a number of training data pairs, Θ.sub.k.sup.g, Θ.sub.k.sup.f are network parameters in a k-th iteration, {circumflex over (r)}.sub.i is an output of the deep network corresponding to an i-th training data pair, and r.sub.i.sup.gt is a ground truth solution corresponding to {circumflex over (r)}.sub.i.

10. The model-driven deep learning-based seismic super-resolution inversion (SRI) method according to claim 9, wherein in formula (4), an optimization is performed by a batch stochastic gradient descent (SGD) method, and a fixed number of iterations is set as a stop condition.

Description

BRIEF DESCRIPTION OF THE DRAWINGS

(1) FIG. 1 shows a residual convolutional network structure used to learn an operator Γ.sub.Θ.sub.k.sub.f;

(2) FIG. 2 shows a residual convolutional network structure used to learn an operator γ.sub.Θ.sub.k.sub.g;

(3) FIG. 3 shows a complete deep network ADMM-SRINet structure;

(4) FIG. 4A to FIG. 4B show a wavelet and its corresponding amplitude spectrum thereof in a model example; wherein FIG. 4A shows a Ricker wavelet and a Yu wavelet in a time domain, and FIG. 4B shows amplitude spectra of the Ricker wavelet and the Yu wavelet;

(5) FIG. 5 shows band-limited super-resolution data, wherein data in a dotted block is used to generate a training data set, and other data is used to generate a test data set;

(6) FIG. 6 shows a wavelet library used to generate a training data set, wherein wavelets in the wavelet library have different peak frequencies and rotation phases;

(7) FIG. 7A to FIG. 7C show an inversion result display 1 of a test data set, data at a dotted line position being assumed to be well-logging data; wherein FIG. 7A shows observed data, FIG. 7B shows a genuine solution of a super-resolution inversion, and FIG. 7C shows an inversion result of the present disclosure;

(8) FIG. 8A to FIG. 8C show an inversion result display 2 of a test data set, data at a dotted line position being assumed to be well-logging data; wherein FIG. 8A shows observed data, FIG. 8B shows a genuine solution of a super-resolution inversion, and FIG. 8C shows an inversion result of the present disclosure;

(9) FIG. 9A to FIG. 9E show an inversion result display of model single trace data; wherein FIG. 9A to FIG. 9E are respectively correspond to the data at the dotted line position in FIG. 7A to FIG. 7C and FIG. 8A to FIG. 8C, a gray solid line represents observed data, a black dotted line represents a genuine solution of an inversion, and a gray dotted line represents an inversion result;

(10) FIG. 10 shows network training and an inversion process of field data;

(11) FIG. 11A to FIG. 11B show a wide-band Gaussian wavelet used to invert field data; wherein FIG. 11A shows the wide-band Gaussian wavelet, and FIG. 11B shows an amplitude spectrum of the wide-band Gaussian wavelet;

(12) FIG. 12A to FIG. 12C show an inversion result display of field data; wherein FIG. 12A shows field observed data, FIG. 12B shows an inversion result obtained by a model-trained network, and FIG. 12C shows an inversion result obtain by a finely adjusted network; and

(13) FIG. 13A to FIG. 13C show an amplitude spectrum display of an inversion result of field data; wherein FIG. 13A shows an amplitude spectrum of the data in FIG. 12A, FIG. 13B shows an amplitude spectrum of the data in FIG. 12B, and FIG. 13C shows an amplitude spectrum of the data in FIG. 12C.

DETAILED DESCRIPTION OF THE EMBODIMENTS

(14) In order to enable those skilled in the art to better understand solutions of the present disclosure, the technical solutions in embodiments of the present disclosure will be clearly and completely described below in conjunction with the drawings in the embodiments of the present disclosure. Apparently, the described embodiments are only some embodiments but not all embodiments of the present disclosure. All other embodiments obtained by those of ordinary skill in the art based on the embodiments of the present disclosure without creative work shall fall within the scope of protection of the present disclosure.

(15) It should be noted that the terms “first”, “second”, and the like in the description, claims and drawings of the present disclosure are used to distinguish similar objects, and are not necessarily used to describe a specific sequence or order. It should be understood that data used in this way may be interchanged under appropriate conditions so that the embodiments of the present disclosure described here can be implemented in a sequence other than those illustrated or described here. In addition, the terms “include” and “have” and any variation thereof are intended to cover non-exclusive inclusions. For example, a process, method, system, product, or device that includes a series of steps or units is not necessarily limited to those steps and units clearly listed, but may include other steps or units that are not clearly listed or are inherent to these processes, methods, products, or devices.

(16) The present disclosure will be further described below in detail with reference to the drawings.

(17) The present disclosure proposes a model-driven deep learning method to implement a seismic super-resolution inversion, which is called ADMM-SRINet. The method combines a model-driven alternating direction method of multipliers (ADMM) with a data-driven deep learning method to construct a deep network structure. Specifically, according to the method, each iteration of ADMM is mapped to each layer of a network, and proximal operators are learned by using the data-driven method. All parameters such as a regularization parameter and a transformation matrix in the network may be implicitly learned from a training data set, and are not limited to a form of regularization term. In addition, for complex field data, the present disclosure designs a set of processes for obtaining label data and a novel solution for network training, so ADMM-SRINet may be used to better invert the field data. Finally, the network is configured to invert synthetic and field data, which verifies the effectiveness of the present disclosure.

(18) 1. ADMM-Based Seismic Super-Resolution Inversion

(19) Based on a conventional convolution model, seismic records may be modeled by using the following mathematical framework:
y=Wr,  (1)

(20) y∈R.sup.n represents observed data, W∈R.sup.n×m is a convolution matrix composed of seismic wavelets, and r∈R.sup.m is a super-resolution result to be solved. A main objective of the seismic super-resolution inversion is to optimize an objective function in the following formula:

(21) $\begin{array}{cc}\hat{r}=\underset{r}{\mathrm{argmin}}\frac{1}{2}{.Math.y-\mathrm{Wr}.Math.}_2^2+\lambda {.Math.\mathrm{Dr}.Math.}_p,& \left(2\right)\end{array}$

(22) D represents a sparse transformation matrix, λ is a regularization parameter, and ∥•∥.sub.P represents a norm of l.sub.p(0≤p≤1). In order to solve formula (2), various iterative optimization algorithms such as an alternating direction method of multipliers (ADMM) and an iterative shrinkage threshold algorithm (ISTA) have been proposed. In the present disclosure, ADMM is adopted.

(23) ADMM is also known as a Douglas-Rachford splitting algorithm that may be used to split an objective function into multiple sub-problems and then perform alternating solution. ADMM may be interpreted by using a thought of augmented Lagrangian. First an auxiliary variable x is introduced, and formula (2) may be written as the following form:

(24) $\begin{array}{cc}\begin{array}{cc}\underset{r|}{\min }\frac{1}{2}{.Math.y-\mathrm{Wr}.Math.}_2^2+\lambda {.Math.\mathrm{Dx}.Math.}_p& s.t.x=r,\end{array}& \left(3\right)\end{array}$

(25) An augmented Lagrangian form of formula (3) is as follow:

(26) $\begin{array}{cc}{L}_{\rho }\left(r,x,\alpha \right)=\frac{1}{2}{.Math.y-\mathrm{Wr}.Math.}_2^2+\lambda {.Math.\mathrm{Dx}.Math.}_p+{\alpha }^T\left(r-x\right)+\frac{\rho }{2}{.Math.r-x.Math.}_2^2,& \left(4\right)\end{array}$

(27) where ρ is a penalty parameter, and α is a dual variable. Based on three variables in formula (4), formula (4) is split into the following three sub-problems:

(28) $\begin{array}{cc}\{\begin{array}{c}\underset{r}{\mathrm{argmin}}\frac{1}{2}{.Math.y-\mathrm{Wr}.Math.}_2^2+\frac{\rho }{2}{.Math.r+\beta -x.Math.}_2^2,\\ \underset{x}{\mathrm{argmin}}\frac{\rho }{2}{.Math.r+\beta -x.Math.}_2^2+\lambda {.Math.\mathrm{Dx}.Math.}_p,\\ \beta \leftarrow \beta +\eta \left(r-x\right),\end{array}& \left(5\right)\end{array}$

(29) where

(30) $\beta =\frac{\alpha }{\rho },$
and η is an updated parameter. A first formula and a third formula in formula (5) are easy to solve, but a solution of a second formula is challenging, specially under a condition that the regularization term is nonconvex, it is difficult for researchers who are not engaged in algorithm optimization to solve the non-convex problem. Generally, when ∥Dx∥.sub.p=∥Dx∥.sub.1 and D is an orthogonal matrix, a solution of each sub-problem in formula (5) is as follow:

(31) $\begin{array}{cc}\{\begin{array}{c}{r}^k={\left(W^{T}W+\rho I\right)}^{-1}\left(W^{T}y+\rho \left(x^{k-1}-{\beta }^{k-1}\right)\right),\\ x^k=D^T{S}_{\lambda /\rho }\left\{D\left(r^k+{\beta }^{k-1}\right)\right\},\\ {\beta }^k={\beta }^{k-1}+\eta \left(r^k-x^k\right),\end{array}& \left(6\right)\end{array}$

(32) where S.sub.λ/ρ is a soft threshold function with a threshold of Alp. Other threshold functions may be selected to replace the soft threshold function.

(33) ADMM described above is a model-driven method, where the regularization term λ, the parameter ρ, the sparse matrix D, and some other hyper-parameters in ADMM need to be determined in advance. In addition, ADMM needs to be subjected to multiple iterations to achieve a satisfied result, which brings a big challenge to an inversion of high-dimensional data. Moreover, the non-orthogonal matrix D and 0≤p<1 make formula (5) difficult to be solved.

(34) 2. Model-Driven ADMM-SRINet

(35) In order to solve the limitation of ADMM, a model-driven deep network is designed to implement a seismic super-resolution inversion, which is called ADMM-SRINet. In order to introduce the proposed network structure, formula (5) is written as the following proximal form:

(36) $\begin{array}{cc}\{\begin{array}{c}{r}^k=pro{x}_{\rho f}\left(x^{k-1}-{\beta }^{k-1}\right),\\ x^k=pro{x}_{\rho g}\left(r^k+{\beta }^{k-1}\right),\\ {\beta }^k={\beta }^{k-1}+{\eta }^k\left(r^k-x^k\right),\end{array}& \left(7\right)\end{array}$

(37) where η is a parameter varying with the number of iterations, and prox.sub.ρf(•) and prox.sub.ρg (•) are proximal operators and are defined as follow:

(38) 0$\begin{array}{cc}pro{x}_{\rho f}\left(z\right)=\underset{\hat{z}}{\mathrm{argmin}}f\left(\hat{z}\right)+\frac{\rho }{2}{.Math.\hat{z}-z.Math.}_2^2,& \left(8\right)\end{array}$

(39) and

(40) $\begin{array}{cc}pro{x}_{\rho g}\left(z\right)=\underset{\hat{z}}{\mathrm{argmin}}g\left(\hat{z}\right)+\frac{\rho }{2}{.Math.\hat{z}-z.Math.}_2^2,& \left(9\right)\end{array}$

(41) For an input variable {circumflex over (z)},

(42) $f\left(\hat{z}\right)=\frac{1}{2}{.Math.y-W\hat{z}.Math.}_2^2$
and g({circumflex over (z)})=λ∥D{circumflex over (z)}∥.sub.p. By observing formula (7), it can be seen that the proximal operators prox.sub.ρf (•) and prox.sub.ρg(•) are the keys to solve the inverse problems and may be replaced with other operators such as a denoising operator. In the present disclosure, based on the inspiration of the strong learning capacity of the deep learning, a learning operator is used to replace the proximal operator according to Adler's work in 2018, wherein parameters in the learning operator may be obtained through training. Therefore, a relatively optimal solution may be obtained for the proximal problem under a relatively small number of iterations, which avoids the determination of some parameters. Although there is no explicit expression for learning the proximal operator, a universal approximation property of a neural network ensures that these operators may be arbitrarily approximated. The following is a detailed description of ADMM-SRINet.

(43) A. First Learning Operator

(44) Although there is an analytical solution for the first proximal problem in formula (7), the calculation of the matrix increases the calculation cost, and the selection of the parameters increases the difficulty of solution. Therefore, different form Yang's work in 2018, one residual convolutional network block is used to replace the first proximal operator prox.sub.ρf(•) so as to learn a mapping relationship between r.sup.k and (x.sup.k-1−β.sup.k-1), which is expressed by the following formula:
r.sub.k=Γ.sub.Θ.sub.k.sub.f(r.sup.k-1,x.sup.k-1,β.sup.k-1,W.sup.TWr.sup.k-1,y),  (10)

(45) where Γ.sub.Θ.sub.k.sub.f represents a learning operator in a kth iteration, and a parameter in the learning operator is represented by Θ.sub.k.sub.f. r.sup.k-1, x.sup.k-1, β.sup.k-1, W.sup.TWr.sup.k-1, and y are input of the learning operator Γ.sub.Θ.sub.k.sub.f. FIG. 1 shows a residual convolutional block used to learn the operator Γ.sub.Θ.sub.k.sub.f, the network block is composed of two “Conv+ReLu” layers and one “Conv” layer, wherein “Conv” represents a convolution operator, and “ReLu” represents a linear rectification activation function. In the residual convolutional block, there are 32 convolution kernels with a size of 3×1 on each layer, and there is only one convolution kernel with a size of 3×1×32 on the last layer. A dark gray arrow represents input and output of the network, and a light gray arrow represents a processing relationship between different network layers. It should be noted that the input of the network includes physical knowledge in the forward model and gradient information of the loss function in the model-driven method.

(46) B. Second Learning Operator

(47) The second proximal problem in formula (7) is usually non-convex, and it is difficult to select an appropriate normal form to obtain an optimal result, and therefore it is a novel way to solve the limitation of the conventional method by using the currently rapidly developed deep learning method. In order to design a network structure to learn the proximal operator prox.sub.ρg(•), the second formula in formula (5) is rewritten as the following form:

(48) $\begin{array}{cc}{x}^k=\underset{x}{\mathrm{argmin}}\frac{\rho }{2}{.Math.p^k-x.Math.}_2^2+\lambda {.Math.F\left(x\right).Math.}_p,& \left(11\right)\end{array}$

(49) where p.sup.k=r.sup.k+β.sup.k-1, and F represents a nonlinear sparse function. Based on theorem 1 in the work of Zhang, et al. in 2018, the following approximation expression may be obtained:
∥F(x)−F(p.sup.k)∥.sub.2.sup.2≈υ∥x−p.sup.k∥.sub.2.sup.2,  (12)

(50) where p.sup.k and F(p.sup.k) are assumed to be mean values of x and F(x). Therefore, formula (11) may be rewritten as follow:

(51) $\begin{array}{cc}{x}^k=\underset{x}{\mathrm{argmin}}\frac{\rho }{2}{.Math.{F\left(p\right)}^k-F\left(x\right).Math.}_2^2+ϛ{.Math.F\left(x\right).Math.}_p=\mathrm{pro}{x}_{{\mathrm{ϛg}}^{\prime }}\left(F\left(p^k\right)\right),& \left(13\right)\end{array}$

(52) where g′(•)=ç∥F(•)∥.sub.p, and ç=υλ. By using the residual convolutional network block, a solution of formula (13) may be obtained through one learning operator, which is expressed as follow:
F(x.sup.k)=Λ.sub.Θ.sub.k.sub.g′(F(p.sup.k)),  (14)

(53) where ΛΘ.sub.k.sub.g′ is a learning operator used to replace the proximal operator. Formula (14) is an inverse problem solution in a sparse domain. Assuming that the transformation function F has an inverse transformation function, a solution of formula (13) is as follow:
x.sup.k=F.sup.H.sup.kΛ.sub.Θ.sub.k.sub.g′(F.sup.k(p.sup.k)),  (15)

(54) where similarly the learning operator is used to transform function F and its inverse function, and F.sup.k and F.sup.H.sup.k are used to represent learning operators in the kth iteration. If an enhanced version is considered, x.sup.k may be updated through the following formula:
x.sup.k=p.sup.k+Q.sup.k(F.sup.H.sup.kΛ.sub.Θ.sub.k.sub.g′(F.sup.k(J.sup.k(p.sup.k)))),  (16)

(55) where a learning operator R.sup.k=Q.sup.koJ.sup.k is used to extract a lost high frequency component from x.sup.k, that is, w.sup.k=R.sup.k(x.sup.k), so under a noiseless condition, x.sup.k=p.sup.k+w.sup.k.

(56) By observing formula (16), it can be seen that if operators Q.sup.kF.sup.H.sup.k and F.sup.kJ.sup.k are learned by using the same network structure, the second term in the formula has the symmetry. Therefore, the operators Q.sup.kF.sup.H.sup.k, F.sup.kJ.sup.k, and Λ.sub.Θ.sub.k.sub.g′ in formula (16) may be learned by using three residual convolutional blocks, wherein each residual convolutional block has the same structure as the network in FIG. 1, and there is a large skip connection between input and output. In addition, a first residual convolutional block shares parameters with a third residual convolutional block, which not only reduces the number of parameters in the network, but also reflects the symmetry of the network. It is worthwhile to note that the form of formula (16) only provides one thought for constructing the network, but each part in the network not strictly corresponds to each operator in formula (16). Similarly, other more effective network structures may be designed to implement formula (16).

(57) Based on the above derivation, FIG. 2 shows three designed residual convolutional blocks. A main objective of the network is to learn a mapping relationship γ.sub.Θ.sub.k.sub.g between x.sup.k and p.sup.k, that is:
x.sub.k=γ.sub.Θ.sub.k.sub.g(p.sup.k),  (17)

(58) where Θ.sub.k.sup.g represents a network parameter in the kth iteration of the network.

(59) C. Network Structure of ADMM-SIRNet

(60) Based on the above descriptions of A and B, it can be seen that the proximal operator in formula (7) may be replaced with a residual convolutional network. For the third part in formula (7), only one parameter η needs to determine in each iteration. η may be considered as a weight in the network, which is learned together with other parameters in the network from training data. Finally, a complete network structure of ADMM-SRINet is shown in FIG. 3. In the network, an iteration parameter is set as 3, and in order to distinguish it from the number of iterations used in a minimizing loss function during the training, each iteration of the network structure is named as a stage, that is, a structure in each stage approximately corresponds to each iteration process in the iterative optimization algorithm ADMM. In a kth stage, the network is composed of three modules, i.e. a module r.sup.k, a module x.sup.k, and a module β.sup.k. The modules r.sup.k and x.sup.k are configured to calculate values of r.sup.k and x.sup.k by using a learning operator. The module β.sup.k is configured to calculate a value of β.sup.k. Nodes of the three modules are connected via straight lines with directionality, which illustrate a relationship between the nodes. Finally, observed data, model information, and some initial values are given, and then the network ADMM-SRINet may be used to implement a seismic super-resolution inversion.

(61) D. Network Optimization of ADMM-SRINet

(62) In order to obtain all parameters in ADMM-SRINet, the following function is minimized:

(63) $\begin{array}{cc}E\left(\Theta \right)=\frac{1}{N}\underset{i=1}{\overset{N}{.Math.}}{.Math.{\hat{r}}_i-r_i^{gt}.Math.}_2^2,& \left(18\right)\end{array}$

(64) where a network parameter is Θ={Θ.sub.k.sup.g,Θ.sub.k.sup.f,η.sup.k}.sub.k=1.sup.N.sup.t, N represents the number of training data pairs, {circumflex over (r)}.sub.i is output of the network corresponding to a ith training data pair, and r.sub.i.sup.gt is a genuine solution corresponding to {circumflex over (r)}.sub.i. Formula (18) can be optimized by a small batch stochastic gradient descent (SGD) method. For ease of the optimization, a fixed number of iterations is set as a stop condition for the iterations. In addition, an initial value of a convolution kernel in the network is set as a random number, the number of each batch is 10, r.sup.0 is set as a least squares solution, and x.sup.0 and β.sup.0 are set as zero. Because TensorFlow can automatically calculate a gradient and optimize a parameter, all source codes are implemented through TensorFlow.

(65) A material basis of the present disclosure is a seismic data volume, and a trace-by-trace processing method is used. Specific steps may refer to a process framework in FIG. 10, and includes the following steps:

(66) step 1: each iteration of a model-driven alternating direction method of multipliers (ADMM) is mapped to each layer of a deep network, and proximal operators are learned by using a data-driven method to complete construction of a deep network ADMM-SRINet (see FIG. 3);

(67) step 2: label data used to train the deep network ADMM-SRINet is obtained;

(68) step 3: the deep network ADMM-SRINet is trained by using the obtained label data; and

(69) step 4: test data is inverted by using the deep network ADMM-SRINet trained at step 3).

(70) Effectiveness Analysis:

(71) 1. Synthetic Data Example

(72) Due to a relatively low signal to noise ratio of field seismic data, it is usually difficult to recover an underground full-band reflection coefficient. In contrast, an inversion result of a band-limited reflection coefficient is more reliable. Therefore, in the present disclosure, the super-resolution inversion method is mainly used to invert a band-limited reflection coefficient from seismic data. First, in order to train the proposed network ADMM-SRINet, a Marmousi II velocity model was used to generate a reflection coefficient profile, and then the reflection coefficient module was convoluted with a Yu wavelet (a dotted line in FIG. 4A) to obtain required band-limited super-resolution data. FIG. 5 shows the band-limited super-resolution data, wherein data in a dotted block is used to generate a training data set, and other data is used to generate a test data set. In addition, a Ricker wavelet library with different peak frequencies and phases was designed for generating training data, as shown in FIG. 6. The design of the wavelet library may be understood as one data augmentation manner, which may not only increase the number of the training data sets, but also enhance the generalization of the network. It should be noted that in each training of the network, input of the network was generated by convoluting any wavelet in the wavelet library with any super-resolution data, that is, the training data was generated. Finally, the network was trained by using input data and label data.

(73) In order to verify the effectiveness of ADMM-SRINet, the network was used to invert a model test data set. The test set was generated by convoluting the data outside the dotted block in FIG. 5 with a test wavelet (a solid line in FIG. 4A). Inversion results of the test data set are shown in FIGS. 7A-7C and FIGS. 8A-8C, wherein FIG. 7A and FIG. 8A show observed data, FIG. 7B and FIG. 8B show a genuine solution of the inversion, and FIG. 7C and FIG. 8C show an inversion result. Through visual observation, it can be seen that the inversion result is very approximate to the genuine solution, and a resolution is higher than that of the original observed data. In order to clearly observe the inversion result, FIGS. 9A-9E show data at a dotted line position in FIGS. 7A-7C and FIGS. 8A-8C, wherein a solid line represents the observed data, a black dotted line represents the genuine solution of the inversion, and a gray dotted line represents the inversion result. It can be observed that the gray dotted line is almost the same as the black dotted line, which indicates the accuracy of the inversion result. In order to quantificationally analyze the inversion result, correlation coefficients between the observed data and the genuine solution of the inversion were first calculated, which were respectively 0.1712, 0.1907, 0.2239, 0.1973, and 0.1764, and then correlation coefficients between the inversion result and the genuine solution were calculated, which were respectively 0.9967, 0.9969, 0.9952, 0.9936, and 0.9960. Through comparative analysis, it can be seen that there were higher correlation coefficients between the inversion result and the genuine solution, which indicates high inversion accuracy of the method of the present disclosure. In addition, the inversion result was better matched with the genuine solution (may be understood as well data), which indicates that the resolution of the inversion result is improved.

(74) 2. Field Seismic Data Example

(75) One section of a post-stack three dimensional data volume of a certain oil field was selected to test the method of the present disclosure. Due to the complex field seismic data, in order to obtain a relatively good inversion result, an inversion process was designed, as shown in FIG. 10. For acquisition of label data, relatively good label data may also be obtained by using a non-learning method processing process besides a model data. Specially for field data, if only a model data-trained network is used to invert the field data, details of an inversion result will be missing or some pseudomorphism will be generated, a main reason is that the model data cannot simulate the same background environment, noise environment, etc. as those of the field data, and a reflection coefficient distribution of the model is not necessarily the same as a reflection coefficient distribution in the filed data, which may affect the inversion result of the field data. Therefore, in order to improve the inversion result of the filed data, a network training strategy was adjusted: the parameters of the model-trained network were used as an initial value, and then the network was finely adjusted by using the label data obtained by the non-learning method, which not only reduces the training time, but also avoids an error brought by the model training.

(76) In order to enhance the reliability of the label data, super-resolution data was obtained by using an alternating iterative reflection coefficient inversion method proposed by Wang in 2016, so not only relatively reliable label data may be obtained, but also wavelet information may be obtained. For an inversion of the field data, similarly a band-limited reflection coefficient was inverted. In order to enable the obtained inversion result to have a higher resolution, a wide-band Gaussian wavelet shown in FIG. 11A and FIG. 11B was designed, and different from a Yu wavelet, the wavelet may not only keep low frequency information, but also recover high frequency information to a certain extent. The reflection coefficient obtained by the alternating iterative inversion method was filtered by using the wide-band Gaussian wavelet to obtained band-limited super-resolution data. Because the network had a relatively good initial value, in the present disclosure, 100 traces of super-resolution data were selected as label data, and then a relatively good inversion result may be obtained. FIGS. 12A-12C show an inversion result of the field data; wherein FIG. 12A shows field observed data, there were total 1001 traces, and data of each trace included 551 time sampling points, FIG. 12B shows an inversion result obtained by the model-trained network, and FIG. 12C shows an inversion result obtained by the network finely adjusted by using the field data. Through visual observation, it can be found that compared to the observed data, resolutions of the inversion results in FIG. 12B and FIG. 12C are both improved, and a description of a geological structure is clearer. But details of the inversion result in FIG. 12B are missing, and a main reason is that the sparseness of the model super-resolution data is different from that of the field data, which causes the inversion result obtained by the model-trained network is sparser. The inversion result in FIG. 12C is more detailed due to a fact that the label data incorporates information of the field data. In order to further compare the inversion results, FIGS. 13A-13C show amplitude spectra of the three inversion results in FIGS. 12A-12C, and it can be found that the inversion results may all expand a frequency band range of the seismic data, and the frequency band range of FIG. 13B is very close to that of FIG. 13C, and close to that of the amplitude spectrum of the Gaussian wavelet in FIG. 11A and FIG. 11B, which indicates that the inversion result of the network has a close relationship with the design of label data. Finally, in order to verify the reliability of the inversion result, an impedance well-logging curve was inserted to the three inversion results in FIGS. 12A-12C, and the well-logging curve was filtered by using a low-pass filter with a cut-off frequency of 250 Hz. Through observation, it can be found that two inversion results of near-well seismic data both match the well-logging curve closely, and the result of FIG. 12C matches the well more closely, which indicates that the reliability of the inversion of the method of the present disclosure is relatively high.

(77) In conclusion, according to the present disclosure, each iteration of ADMM is mapped into each layer of a network, and the proximal operators are learned by using a data-driven method. The method fully combines advantages of a model-driven alternating direction method of multipliers (ADMM) and a data-driven deep learning method, avoids the design of regularization terms and then implements a fast calculation of high-dimensional data. In addition, the present disclosure designs a set of process for obtaining label data and a novel solution for network training, and mainly uses model label data and field label data to train the deep network through a transfer learning strategy, and thus ADMM-SRINet can be used to better invert field data. Finally, the network is used to invert synthetic and filed data, which verifies the effectiveness of the present disclosure.

(78) The above content is only to illustrate the technical ideas of the present disclosure, and cannot be used to limit the scope of protection of the present disclosure. Any changes made on the basis of the technical solutions based on the technical ideas proposed by the present disclosure shall fall within the scope of protection of the claims of the present disclosure.