CN104730572A - Diffracted wave imaging method and device based on L0 semi-norm - Google Patents

Abstract

The invention discloses a diffracted wave imaging method based on the L0 semi-norm. The method comprises the steps that seismic data with no reflected waves are obtained to be used as input data; discretization is carried out on underground imaging space, any imaging point is selected as the current diffracted imaging point, and a green function of any diffracted imaging point is calculated according to a given speed model and the relation between short points and wave detection points in the input seismic data; other imaging points which are not selected are executed in a loop mode, and other corresponding green functions are calculated until the green functions of all the imaging points in the underground imaging space are obtained; a diffracted wave imaging model based on the L0 semi-norm is built; the homotopy analysis iterative algorithm is used for solving the model, and a diffracted wave imaging result is obtained. The invention further discloses a diffracted wave imaging device based on the L0 semi-norm. By means of the diffracted wave imaging method and device based on the L0 semi-norm, the seismic data imaging resolution ratio can be improved, the diffracted wave imaging signal to noise ratio is increased, and therefore a small-scale geological unit related to reservoir space connectivity can be recognized more easily.

Description

A kind of diffracted wave formation method based on L0 semi-norm and device

Technical field

The invention belongs to seismic exploration technical field, relate to a kind of diffracted wave formation method based on L0 semi-norm, the invention still further relates to a kind of diffracted wave imaging device based on L0 semi-norm.

Background technology

In Oil And Gas Exploration And Development process, effectively identify structure and lithologic anomalous body, as solution cavity, crack, stratigraphic pitch-out, weathering crust etc. are most important to reservoir understanding.Above-mentioned small scale geologic body is usually less than or much smaller than seismic wavelet wavelength, therefore exists with the form of diffracted wave in the seismic data on spatial.In the wild in operation process, the seismic signal that wave detector receives comprises reflection wave and diffracted wave, and reflection wave imaging condition supposes infinitely great smooth interface, therefore generally can only reflect large scale geologic background.Comparatively speaking, diffracted wave is then the reflection of geology details, is the important information carrier improving seismic resolution.But the diffraction energy that geologic body produces is about reflected energy 0.1-0.01 doubly, for weak signal, often be submerged in reflection wave imaging result, therefore need to remove reflection wave, the Harlan that has that conventional method comprises signal decomposition converts (Harlan et at., 1984), Radon conversion (Zhang, 2005), plane wave destroy filtering (Taner et al., 2006; Fomel et al., 2006,2007) etc.

Through patent retrieval and domestic and foreign literature investigation, find the diffracted wave formation method carried out at present, focus majority is placed on reflection wave and removes, and object is with outstanding diffracted wave by depressor reflex ripple.As offset-inclining angle gathers reflection wave removal method (Klokov and Fomel, 2012), focus on-excision-anti-focusing (Khaidukov et al., 2004), CRS stack (Dell and Gajewski, 2011; Asgedomet al., 2011), multi-focus (Berkovitch et al., 2009) etc.

Above-mentioned diffracted wave formation method does not launch research for diffracted wave imaging operator, and in fact, diffraction information has sparse uncontinuity in space distribution, and mathematical L0 semi-norm can be utilized completely to describe.

Therefore, the present invention constructs the diffracted wave imaging model based on L0 semi-norm according to Green function, and this model is openness by constraint diffraction information, can reach high precision, high s/n ratio diffracted wave imaging results.Compared with general diffracted wave formation method, based on the diffracted wave formation method of L0 semi-norm, while raising seismic signal resolution, the signal to noise ratio (S/N ratio) of imaging data can be ensured further, and reduces multi-solution.

Summary of the invention

The object of this invention is to provide a kind of diffracted wave formation method based on L0 semi-norm, solve in existing diffracted wave imaging technique, cannot take into account the problem of seismic data signal to noise ratio (S/N ratio) while improving imaging resolution, this technology can realize the small scale geological anomalous body meticulous depictions such as minor fault, crack, solution cavity.

Another object of the present invention is to provide a kind of diffracted wave imaging device based on L0 semi-norm.

The technical solution adopted in the present invention is, a kind of diffracted wave formation method based on L0 semi-norm, comprises the following steps:

Step 101: obtain the geological data that is removed reflection wave, as input data;

Step 102: discretize is carried out to underground imaging space, choose any one imaging point as current Diffraction Imaging point, and according to the shot point in input geological data and geophone station relation and given rate pattern, calculate the Green function of above-mentioned any Diffraction Imaging point;

Step 103: circulation performs other imaging point do not chosen in subsurface imaging space in above-mentioned steps 102, and calculates its corresponding Green function, until draw all imaging point Green functions in subsurface imaging space;

Step 104: the Green function of all imaging points of imaging space and initial imaging model under base area, build the diffracted wave imaging model based on L0 semi-norm;

Step 105: solve this model by homotopy analysis iterative algorithm, draw diffracted wave imaging results.

Another technical scheme of the present invention is, a kind of diffracted wave imaging device based on L0 semi-norm, comprises the seismic data acquisition module connected successively, for inputting the geological data that is removed reflection wave; Diffraction Imaging clicks delivery block, calculates position for being selected to picture point from the subsurface imaging space of discretize as Green function; Green function computing module, for calculating by shot point through Diffraction Imaging point to during the walking of seismic detection point and amplitude compensation item; L0 semi-norm model construction module, builds the solving model based on L0 semi-norm according to all imaging point Green functions of the underground space; Model solution device, utilizes homotopy analytical algorithm to solve L0 model by Green function and Diffraction Point model construction, draws diffracted wave imaging results.

The invention has the beneficial effects as follows, to the geological data removing reflection wave, carry out Green function calculating, thus build L0 semi-norm inverse model.Under normal circumstances, it is not exclusive that inverse problem solves, unless added that qualifications is to reduce hunting zone, consider diffraction information space uncontinuity, in an embodiment, define model openness, make the more realistic geological condition of solving model, and noise can be suppressed to a certain extent by iterative approach, improve diffracted wave imaging data signal to noise ratio (S/N ratio).Improve imaging of seismic data resolution, strengthen diffracted wave imaging signal to noise ratio (S/N ratio), thus more easily identify relevant small scale geologic unit connective with reservoir space.

Accompanying drawing explanation

Fig. 1 is the process flow diagram of the inventive method.

Fig. 2 is the structured flowchart of apparatus of the present invention.

Wherein, 201. seismic data acquisition module, 202. Diffraction Imagings click delivery block, 203. Green function computing modules, 204.L0 semi-norm model construction module, 205. model solution devices.

Embodiment

Below in conjunction with the drawings and specific embodiments, the present invention is described in detail.

As shown in Figure 1, a kind of diffracted wave formation method based on L0 semi-norm that the present invention proposes, comprises the following steps:

Step 101: obtain the geological data that is removed reflection wave, as input data; Corresponding geological data, can be prestack big gun collection geological data or post-stack seismic data, on reflection wave minimizing technology, adopts plane wave to destroy filtering method;

Step 102: discretize is carried out to underground imaging space, and choose any one imaging point as current Diffraction Imaging point, and according to the shot point in input geological data and geophone station relation and given rate pattern, calculate the Green function of above-mentioned any Diffraction Imaging point;

Step 103: circulation performs other imaging point do not chosen in subsurface imaging space in above-mentioned steps 102, and calculates its corresponding Green function, until draw all imaging point Green functions in subsurface imaging space;

Step 104: the Green function of all imaging points of imaging space and initial imaging model under base area, build the diffracted wave imaging model based on L0 semi-norm;

Step 105: solve this model by homotopy analysis iterative algorithm, draw diffracted wave imaging results.

Green function computing method in above-mentioned steps 102,103, calculate by shot point when imaging point is walked to geophone station according to ray tracing, and store amplitude weight item.

In above-mentioned steps 104, set up the diffracted wave imaging model based on L0 semi-norm, design as follows:

$\min J_{\mathrm{\α}}\left(m\right):=\frac{1}{2}{\left|\right|\mathrm{Gm}-d\left|\right|}_{l_2}^2+\mathrm{\α}{\left|\right|m\left|\right|}_{l_0}$

Wherein, min represents and minimizes, J _αm () is objective function, m is the diffraction model solved, mathematic sign :=expression is defined as, and G is Green function, and d is the geological data removing reflection wave, and α is just

Then change the factor, represent l ₂norm, for l ₀semi-norm.

By Proximal point method, above formula can be written as:

$\min H_{\mathrm{\β},\mathrm{\α}}(m_0,m):=(G^T{\mathrm{Gm}}_0-G^{T}d+\mathrm{\α}\frac{d}{\mathrm{dm}}{\left|\right|m\left|\right|}_{l_0,m=m_0},m-m_0)+\frac{\mathrm{\β}}{2}{\left|\right|m-m_0\left|\right|}_{l_2}^2+\mathrm{\α}{\left|\right|m\left|\right|}_{l_0}$

Wherein, H _{β, α}(m ₀, m) be J _αm () is approximant, m ₀for initial model, β is adjustable regular parameter, and () represents inner product.

Above formula minimum problems, tries to achieve by hard-threshold operator, as follows:

$V_{\mathrm{\β}}\left(m_0\right)=\underset{m}{\arg \min }{H}_{\mathrm{\β},\mathrm{\α}}(m_0,m)$

Wherein, $s_{\mathrm{\β}}\left(m\right)=m-\frac{1}{2\mathrm{\β}}\▿{\left|\right|\mathrm{Gm}-d\left|\right|}_{l_2}^2,$ [] _irepresent vector the _iindividual element.

In above-mentioned steps 105, to L0 semi-norm model construction module, take model solution device to realize, its iterative process is completed by homotopy analytical algorithm, and detailed process is as follows:

Step 1: input Lipschitz parameter beta ₀, regularization parameter ₀, and requirement

β ₀∈ [β _min, β _max], β _min, β _maxbe respectively Pu Xici parameter beta ₀bound, initiation parameter, k=0, ρ ∈ (0,1), initialization model m ⁰;

Step 2: setting, i=0, m ^{k, 0}=m ^k, β _{k, 0}=β _k;

Step 3: $m^{k,i+1}=V_{{\mathrm{\β}}_k,i}\left(m^{k,i+1}\right);$

When $J_{{\mathrm{\&alpha;}}_k}\left(m^{k,i}\right)-J_{{\mathrm{\&alpha;}}_k}\left(m^{k,i+1}\right)<\frac{\mathrm{\&eta;}}{2}\left|\right|m^{k,i}-m^{k,i+1}\left|\right|,$

Perform β _k,i=min{ γ β _k,i, β _max,

β _k,i+1＝β _k,i,i＝i+1；

Return step 3, until, || m ^k,i-m ^{k, i+1}|| _∞≤ ε ₀

Wherein, terminal parameter ε ₀be generally ε ₀=10 ^-1;

Step 4:m ^k+1=m ^k,i, β _k+1=β _k,i, α _k+1=ρ α _k, k=k+1;

Return step 2, until, || m ^k,i-m ^{k, i+1}|| _∞≤ ε;

Wherein, ε is generally ε=10 ^-5;

Step 5: export final iteration result, m ^*=m ^k.

Based on same inventive concept, present invention also offers a kind of diffracted wave imaging device based on L0 semi-norm.Because a kind of principle of dealing with problems based on the diffracted wave imaging device of L0 semi-norm is similar to a kind of diffracted wave formation method based on L0 semi-norm, therefore the enforcement based on the diffracted wave imaging device of L0 semi-norm see an a kind of enforcement of the diffracted wave formation method based on L0 semi-norm, can repeat part and repeats no more.Following used, term " unit " or " module " can realize the software of predetermined function and/or the combination of hardware.Although the device described by following examples preferably realizes with software, hardware, or the realization of the combination of software and hardware also may and conceived.

Fig. 2 is a kind of structured flowchart of a kind of diffracted wave imaging device based on L0 semi-norm of the present invention, comprise that the seismic data acquisition module 201, the Diffraction Imaging that connect successively click delivery block 202, Green function computing module 203, L0 Norm Model build module 204 and model solution device 205, below this structure is described.

Seismic data acquisition module 201, for inputting the geological data that is removed reflection wave;

Diffraction Imaging clicks delivery block 202, calculates position for being selected to picture point from the subsurface imaging space of discretize as Green function;

Green function computing module 203, for calculating by shot point through Diffraction Imaging point to during the walking of seismic detection point and amplitude compensation item;

L0 semi-norm model construction module 204, builds the solving model based on L0 semi-norm according to all imaging point Green functions of the underground space;

Model solution device 205, utilizes homotopy analytical algorithm to solve L0 model by Green function and Diffraction Point model construction, draws diffracted wave imaging results.

Wherein, Green function computing module 203, calculates by shot point when imaging point is walked to geophone station according to ray tracing, and stores amplitude weight item.

L0 semi-norm model construction module 204, designs as follows:

$\min J_{\mathrm{\&alpha;}}\left(m\right):=\frac{1}{2}{\left|\right|\mathrm{Gm}-d\left|\right|}_{l_2}^2+\mathrm{\&alpha;}{\left|\right|m\left|\right|}_{l_0}$

Wherein, min represents and minimizes, J _αm () is objective function, m is the diffraction model solved, mathematic sign :=expression is defined as, and G is Green function, and d is the geological data removing reflection wave, and α is regularization factors, represent l ₂norm, for l ₀semi-norm.

By Proximal point method, above formula can be written as:

$\min H_{\mathrm{\&beta;},\mathrm{\&alpha;}}(m_0,m):=(G^T{\mathrm{Gm}}_0-G^{T}d+\mathrm{\&alpha;}\frac{d}{\mathrm{dm}}{\left|\right|m\left|\right|}_{l_0,m=m_0},m-m_0)+\frac{\mathrm{\&beta;}}{2}{\left|\right|m-m_0\left|\right|}_{l_2}^2+\mathrm{\&alpha;}{\left|\right|m\left|\right|}_{l_0}$

Wherein, H _{β, α}(m ₀, m) be J _αm () is approximant, m ₀for initial model, β is adjustable regular parameter, and () represents inner product.

Above formula minimum problems, tries to achieve by hard-threshold operator, as follows:

$V_{\mathrm{\&beta;}}\left(m_0\right)=\underset{m}{\arg \min }{H}_{\mathrm{\&beta;},\mathrm{\&alpha;}}(m_0,m)$

Wherein, $s_{\mathrm{\&beta;}}\left(m\right)=m-\frac{1}{2\mathrm{\&beta;}}\&dtri;{\left|\right|\mathrm{Gm}-d\left|\right|}_{l_2}^2,$ [] _irepresent vector i-th element.

Optionally, to L0 semi-norm model construction module, model solution device is taked to realize, its iteration

Process is completed by homotopy analytical algorithm, and process is as follows:

Step 1: input Lipschitz parameter beta ₀, regularization parameter ₀, and requirement

β ₀∈ [β _min, β _max], β _min, β _maxbe respectively Pu Xici parameter beta ₀bound, initiation parameter, k=0, ρ ∈ (0,1), initialization model m ⁰;

Step 2: setting, i=0, m ^{k, 0}=m ^k, β _{k, 0}=β _k;

Step 3: $m^{k,i+1}=V_{{\mathrm{\&beta;}}_k,i}\left(m^{k,i+1}\right);$

When $J_{{\mathrm{\&alpha;}}_k}\left(m^{k,i}\right)-J_{{\mathrm{\&alpha;}}_k}\left(m^{k,i+1}\right)<\frac{\mathrm{\&eta;}}{2}\left|\right|m^{k,i}-m^{k,i+1}\left|\right|,$

Perform β _k,i=min{ γ β _k,i, β _max,

β _k,i+1＝β _k,i,i＝i+1；

Return step 3, until, || m ^k,i-m ^{k, i+1}|| _∞≤ ε ₀

Wherein, terminal parameter ε ₀general setting ε ₀=10 ^-1;

Step 4:m ^k+1=m ^k,i, β _k+1=β _k,i, α _k+1=ρ α _k, k=k+1;

Return step 2, until, || m ^k,i-m ^{k, i+1}|| _∞≤ ε;

Wherein, ε is generally ε=10 ^-5;

Step 5: export final iteration result, m ^*=m ^k.

In another embodiment, additionally provide a kind of software, this software is for performing the technical scheme described in above-described embodiment and preferred implementation.

In another embodiment, additionally provide a kind of storage medium, store above-mentioned software in this storage medium, this storage medium includes but not limited to: CD, floppy disk, hard disk, scratch pad memory etc.From above description, can find out, the embodiment of the present invention achieves following technique effect: a kind of diffracted wave formation method based on L0 semi-norm and device, openness by limiting model, make the more realistic geological condition of solving model, and noise can be suppressed to a certain extent by iterative approach, improve diffracted wave imaging data signal to noise ratio (S/N ratio), this technology can realize the small scale geological anomalous body meticulous depictions such as minor fault, crack, solution cavity, in oil-gas exploration reservoir study, have important using value.

Finally it should be noted that: above-mentioned only in order to the present invention is described and and unrestricted technical scheme described in the invention; Although this instructions is to present invention has been detailed description, but, those skilled in the art still can modify to the present invention or equivalent replacement, and all do not depart from technical scheme and the improvement thereof of the spirit and scope of the present invention, and it all should be encompassed in right of the present invention.

Claims (9)

1., based on a diffracted wave formation method for L0 semi-norm, it is characterized in that, comprise the following steps:

Step 101: obtain the geological data that is removed reflection wave, as input data;

Step 102: discretize is carried out to underground imaging space, and choose any one imaging point as current Diffraction Imaging point, and according to the shot point in input geological data and geophone station relation and given rate pattern, calculate the Green function of above-mentioned any Diffraction Imaging point;

Step 103: circulation performs other imaging point do not chosen in subsurface imaging space in above-mentioned steps 102, and calculates its corresponding Green function, until draw all imaging point Green functions in subsurface imaging space;

Step 104: the Green function of all imaging points of imaging space and initial imaging model under base area, build the diffracted wave imaging model based on L0 semi-norm;

Step 105: solve this model by homotopy analysis iterative algorithm, draw diffracted wave imaging results.

2. a kind of diffracted wave formation method based on L0 semi-norm according to claim 1, is characterized in that, the geological data in described step 101, is prestack big gun collection geological data or post-stack seismic data; The minimizing technology of reflection wave, adopts plane wave to destroy filtering method.

3. a kind of diffracted wave formation method based on L0 semi-norm according to claim 1, it is characterized in that, the computing method of the Green function in described step 102,103, calculate by shot point when imaging point is walked to geophone station according to ray tracing, and store amplitude weight item.

4. a kind of diffracted wave formation method based on L0 semi-norm according to claim 1, is characterized in that, in described step 104, builds the diffracted wave imaging model based on L0 semi-norm, designs as follows:

$\min J_{\mathrm{\&alpha;}}\left(m\right):=\frac{1}{2}{\left|\right|\mathrm{Gm}-d\left|\right|}_{l_2}^2+\mathrm{\&alpha;}{\left|\right|m\left|\right|}_{l_0}$

Wherein, min represents and minimizes, J _αm () is objective function, m is the diffraction model solved, mathematic sign :=expression is defined as, and G is Green function, and d is the geological data removing reflection wave, and α is regularization factors, represent l ₂norm, for l ₀semi-norm.

By Proximal point method, above formula is written as:

$\min H_{\mathrm{\&beta;},\mathrm{\&alpha;}}(m_0,m):=(G^{T}G{m}_0-G^{T}d+\mathrm{\&alpha;}\frac{d}{\mathrm{dm}}{\left|m\right||}_{l_0,m=m_0},m-m_0)+\frac{\mathrm{\&beta;}}{2}{\left|\right|m-m_0\left|\right|}_{l_2}^2+\mathrm{\&alpha;}{\left|\right|m\left|\right|}_{l_0}$

Wherein, H _{β, α}(m ₀, m) be J _αm () is approximant, m ₀for initial model, β is adjustable regular parameter, and () represents inner product,

Minimum problems, try to achieve by hard-threshold operator:

$V_{\mathrm{\&beta;}}\left(m_0\right)=\underset{m}{\arg \min }{H}_{\mathrm{\&beta;},\mathrm{\&alpha;}}(m_0,m)$

Wherein, $s_{\mathrm{\&beta;}}\left(m\right)=m-\frac{1}{2\mathrm{\&beta;}}\&dtri;{\left|\right|\mathrm{Gm}-d\left|\right|}_{l_2}^2,$ [] _irepresent vector the _iindividual element.

5. a kind of diffracted wave formation method based on L0 semi-norm according to claim 1, is characterized in that, in described step 105, homotopy analysis iterative algorithm detailed process is as follows:

Step 1: input Lipschitz parameter beta ₀, regularization parameter ₀, and require β ₀∈ [β _min, β _max], β _min, β _maxbe respectively Pu Xici parameter beta ₀bound, initiation parameter, k=0, ρ ∈ (0,1), initialization model m ⁰;

Step 2: setting, i=0, m ^{k, 0}=m ^k, β _{k, 0}=β _k;

Step 3:m ^{k, i+1}=V _{β k, i}(m ^{k, i+1});

When $J_{{\mathrm{\&alpha;}}_k}\left(m^{k,i}\right)-J_{{\mathrm{\&alpha;}}_k}\left(m^{k,o=i+1}\right)<\frac{\mathrm{\&eta;}}{2}\left|\right|m^{k,i}-m^{k,i+1}\left|\right|,$

Perform β _k,i=min{ γ β _k,i, β _max, m ^{k, i+1}=T _{β k, i}(m ^{k, i+1});

β _k,i+1＝β _k,i,i＝i+1；

Return step 3 to start, until, || m ^k,i-m ^{k, i+1}|| _∞≤ ε ₀;

Wherein, terminal parameter ε ₀for ε ₀=10 ^-1;

Step 4:m ^k+1=m ^k,i, β _k+1=β _k,i, α _k+1=ρ α _k, k=k+1;

Return step 2, until, || m ^k,i-m ^{k, i+1}|| _∞≤ ε;

Wherein, ε is ε=10 ^-5;

Step 5: export final iteration result, m ^*=m ^k.

6. based on a diffracted wave imaging device for L0 semi-norm, it is characterized in that, comprise the seismic data acquisition module (201) connected successively, for inputting the geological data that is removed reflection wave;

Diffraction Imaging clicks delivery block (202), calculates position for being selected to picture point from the subsurface imaging space of discretize as Green function;

Green function computing module (203), for calculating by shot point through Diffraction Imaging point to during the walking of seismic detection point and amplitude compensation item;

L0 semi-norm model construction module (204), builds the solving model based on L0 semi-norm according to all imaging point Green functions of the underground space;

Model solution device (205), utilizes homotopy analytical algorithm to solve L0 model by Green function and Diffraction Point model construction, draws diffracted wave imaging results.

7. a kind of diffracted wave imaging device based on L0 semi-norm according to claim 6, it is characterized in that, described Green function computing module (203), calculates by shot point when imaging point is walked to geophone station according to ray tracing, and stores amplitude weight item.

8. a kind of diffracted wave imaging device based on L0 semi-norm according to claim 6, is characterized in that, described L0 semi-norm model construction module (204) is designed as follows:

$\min J_{\mathrm{\&alpha;}}\left(m\right):=\frac{1}{2}{\left|\right|\mathrm{Gm}-d\left|\right|}_{l_2}^2+\mathrm{\&alpha;}{\left|\right|m\left|\right|}_{l_0}$

Wherein, min represents and minimizes, J _αm () is objective function, m is the diffraction model solved, mathematic sign :=expression is defined as, and G is Green function, and d is the geological data removing reflection wave, and α is regularization factors, represent l ₂norm, for l ₀semi-norm;

By Proximal point method, above formula is written as:

$\min H_{\mathrm{\&beta;},\mathrm{\&alpha;}}(m_0,m):=(G^{T}G{m}_0-G^{T}d+\mathrm{\&alpha;}\frac{d}{\mathrm{dm}}{\left|m\right||}_{l_0,m=m_0},m-m_0)+\frac{\mathrm{\&beta;}}{2}{\left|\right|m-m_0\left|\right|}_{l_2}^2+\mathrm{\&alpha;}{\left|\right|m\left|\right|}_{l_0}$

Wherein, H _{β, α}(m ₀, m) be J _αm () is approximant, m ₀for initial model, β is adjustable regular parameter, and () represents inner product,

Minimum problems, tried to achieve by hard-threshold operator:

$V_{\mathrm{\&beta;}}\left(m_0\right)=\underset{m}{\arg \min }{H}_{\mathrm{\&beta;},\mathrm{\&alpha;}}(m_0,m)$

Wherein, $s_{\mathrm{\&beta;}}\left(m\right)=m-\frac{1}{2\mathrm{\&beta;}}\&dtri;{\left|\right|\mathrm{Gm}-d\left|\right|}_{l_2}^2,$ [] _irepresent vector i-th element.

9. a kind of diffracted wave imaging device based on L0 semi-norm according to claim 6, is characterized in that, described model solution device (205) iterative process is completed by homotopy analytical algorithm, and process is as follows:

Step 1: input Lipschitz parameter beta ₀, regularization parameter ₀, and require β ₀∈ [β _min, β _max], β _min, β _maxbe respectively Pu Xici parameter beta ₀bound, initiation parameter, k=0, ρ ∈ (0,1), initialization model m ⁰;

Step 2: setting, i=0, m ^{k, 0}=m ^k, β _{k, 0}=β _k;

Step 3:m ^{k, i+1}=V _{β k, i}(m ^{k, i+1});

When $J_{{\mathrm{\&alpha;}}_k}\left(m^{k,i}\right)-J_{{\mathrm{\&alpha;}}_k}\left(m^{k,i+1}\right)<\frac{\mathrm{\&eta;}}{2}\left|\right|m^{k,i}-m^{k,i+1}\left|\right|,$

Perform β _k,i=min{ γ β _k,i, β _max, m ^{k, i+1}=T _{β k, i}(m ^{k, i+1});

β _k,i+1＝β _k,i,i＝i+1；

Return step 3, until, || m ^k,i-m ^{k, i+1}|| _∞≤ ε ₀

Wherein, terminal parameter ε ₀setting ε ₀=10 ^-1;

Step 4:m ^k+1=m ^k,i, β _k+1=β _k,i, α _k+1=ρ α _k, k=k+1;

Return step 2, until, || m ^k,i-m ^{k, i+1}|| _∞≤ ε;

Wherein, ε is ε=10 ^-5;

Step 5: export final iteration result, m ^*=m ^k.