Clonal selection based intelligent parameter inversion algorithm for prestack seismic data

Information Sciences 517 (2020) 86–99

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Information Sciences journal homepage: www.elsevier.com/locate/ins

Clonal selection based intelligent parameter inversion algorithm for prestack seismic dataR Xuesong Yan a,∗, Pengpeng Li a, Ke Tang b, Liang Gao c, Ling Wang d a

School of Computer Science, China University of Geosciences, Wuhan, 430074 Hubei, PR China Shenzhen Key Lab of Computational Intelligence, Department of Computer Science and Engineering, Southern University of Science and Technology, Shenzhen, 518055 Guangdong, PR China c State Key Lab of Digital Manufacturing Equipment and Technology, Huazhong University of Science and Technology, Wuhan, 430074 Hubei, PR China d Department of Automation, Tsinghua University, Beijing 100084, PR China b

a r t i c l e

i n f o

Article history: Received 4 September 2019 Revised 23 December 2019 Accepted 30 December 2019 Available online 2 January 2020 MSC: 00-01 99-00 Keywords: prestack seismic data elastic parameter inversion clonal selection algorithm intelligent algorithms correlation coefficient

a b s t r a c t Amplitude variation with offset (AVO) elastic parameter inversion is an approach of oil exploration that employs seismic information, and it is a problem of non-linear optimization. When using a quasi-linear or linear approach to solve the problem, the inversion result is unreliable or inaccurate. Metaheuristic search methods, e.g., bio-inspired optimization algorithms such as genetic algorithms, are capable of handling highly non-linear optimization problems and thus provide a promising approach for oil and gas exploration. As one of the metaheuristic search approaches, the immune clone selection algorithm exhibits the property of fast convergence and strong global search capability. In this paper, the immune clone selection algorithm is used to address the problem of AVO elastic parameter inversion. This algorithm employs the specific initialization strategy of Aki as well as the approximation equation of Rechard, which is utilized in the elastic parameter inversion process to smooth the initialization parameter curve. Additionally, the genetic operation in the algorithm is improved in accordance. The results of multiple experiments demonstrate that the approach could significantly improve the inversion accuracy, and the correlation coefficient of the elastic parameters acquired via inversion is specifically high. © 2020 Elsevier Inc. All rights reserved.

1. Introduction Seismic exploration is an oil exploration approach that employs seismic information. As the seismic information can indicate the variation trend of reservoir parameters, the approach can be utilized for forecasting the reservoir parameters. Prestack seismic data includes a considerable amount of fluid information, and the prestack inversion approach exhibits significant benefits of strong controllability, high resolution and provision of stable results. Therefore, inversion on the basis of prestack seismic data has gradually become a research area of interest in the domain of seismic exploration. As a part of the seismic exploration, the elastic wave theory-based AVO technology is applied to study the variations of the seismic reflection amplitude with the distance between the receiver and gun point (or incident angle). Subsequently, the change in R ∗

Fully documented templates are available in the elsarticle package on http://www.ctan.org/tex-archive/macros/latex/contrib/elsarticle CTAN. Corresponding author. E-mail address: [email protected] (X. Yan).

https://doi.org/10.1016/j.ins.2019.12.083 0020-0255/© 2020 Elsevier Inc. All rights reserved.

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the reflection coefficient with the migration (or incident angle) is investigated, and the physical parameters as well as the lithological characteristics of underlying and overlying interfaces in the reflection interface are determined. The combination of this approach with logging, geological and seismic information has been successful for realizing petroleum-gas predictions. Based on the elastic wave theory, the AVO technology uses the common depth point (CDP) of the gathered prestack seismic data to analyse and research the change in the seismic reflection amplitude with the offset and acquires the correlation between the incident angle and reflection coefficient to analyse the physical parameters and lithological characteristics of the lower and upper reflection interfaces. Finally, the lithology and fluid properties of the oil and gas reservoirs are predicted and determined [1]. Prestack seismic data contain abundant helpful information that could be utilized to forecast the conditions of underground gas and oil [2]. The key parameters are the three elastic parameters of the density ρ , P-wave velocity Vp as well as S-wave velocity Vs . These elastic parameters can indicate the lateral saturation of the underground gas and oil. The inversion of the prestack AVO elastic parameters requires the formulation of a suitable objective function and subsequent optimization of the target, which is generally a non-linear function. When quasi-linear or linear approaches are utilized to address the problem, several limitations are encountered, such as a stable dependence on the original model. If the original model is incorrectly selected, the accuracy of the seismic inversion is reduced, and the inversion result is unreliable. In particular, these linear inversion approaches encounter bottlenecks when resolving non-linear inversions that have multi-extremum and multi-parameter characteristics. Therefore, the inversion of the AVO elastic parameters is a problem of nonlinear optimization. Accordingly, when utilizing a non-linear inversion approach, the state and nature of the solution space is superior to those of the linear inversion approach. However, the global optimal intelligent optimization algorithm has a strong global and local optimization ability, high computational efficiency as well as excellent convergence, and thus it is suitable for application in multi-extremum geophysical inversion, and non-linear as well as multi-parameter problems. Since the mid-1980s, non-linear global intelligent optimization inversion technology has attracted the attention of scholars and experts in the geophysics region. Several novel approaches and ideas pertaining to different fields are constantly introduced into the geophysics region. The non-linear global intelligent optimization inversion technology has been diffusely applied to a variety of inversion problems and several important research results have been obtained. Several scholars used the genetic algorithms for multi-parameter inversion [3–6]. Some other scholars used the particle swarm optimization (PSO) algorithms to solve the problem of elastic parameter inversion [7,8]. Other scholars using the differential evolution (DE) algorithms to perform seismic dislocation inversion in geophysical inversion [9–12]. Wu and Yan designed different intelligent optimization algorithms for solving prestack seismic AVO inversion problem [13–18]. Intelligent optimization algorithms comprise an approach for resolving sophisticated optimization problems on the basis of the computational intelligence mechanism. Intelligent optimization exposes the design principles of optimization algorithms through the understanding of the related rules, experiences, functions, behaviours as well as mechanisms of action in the systems or art, society, chemistry, physics, biology and a number of other fields. In contrast from the mathematical optimization algorithm, intelligent optimization algorithms do not rely on mathematical models of the problem, which include the decomposability, conductivity, and continuity of the objective function. Intelligent optimization algorithms have been extensively studied and diffusely utilized in multiple fields including architecture, artificial intelligence, power electronics, transportation, logistics dispatch, semiconductor manufacturing, steel production, chemical production, and machinery manufacturing [19–38]. Although intelligent optimization algorithms have become widely utilized for geophysical inversion and can obtain satisfactory results, they still involve certain limitations in solving the non-linear inversion of geophysics, such as premature convergence and lower convergence in the later period. Premature convergence is the most significant drawback of the genetic algorithm. Because of its limited ability to search for new spaces, the algorithm can easily converge to the local optimal solution. PSO also tends to prematurely converge, particularly when handling sophisticated multi-peak search problems with poor local optimization ability. In the solution process, the DE algorithm results in a smaller swarm diversity with an increase in the evolution algebra, thereby leading to premature convergence to the local optimal solution. These defects of intelligent optimization algorithms restrict the progress of the inversion problem. Therefore, the algorithm is required to be improved to ensure that an efficient solution can be proposed for the specific problem. De Castro proposed a clonal selection algorithm (CSA) based on the principle of clonal selection. The algorithm simulates the response of the human immune system to antigens and exhibits the properties of fast convergence as well as strong global optimization ability. Therefore, this algorithm can be diffusely applied to optimization as well as pattern recognition problems [39]. Maoguo Gong et al. proposed the Baldwinian clonal selection algorithm (BSCA) based on the Baldwin effect. The BCSA evolves and improves the antibody population using four operators: clonal proliferation, Baldwinian learning, hypermutation, and clonal selection [40]. Jie Feng et al. solved the problem of synthetic aperture radar (SAR) image classification by incorporating the clone selection algorithm into the bag-of-visual words (BOV) algorithm to optimize the prediction error [41]. Youssef Bouazza Elbenani and Bouchra Karoum proposed an improved clonal selection algorithm with a local search mechanism for the cell formation problem [42]. K. Vaisakh and B. Srinivasa Rao introduced a multi-objective adaptive clonal selection algorithm (MOACSA) with load uncertainty for solving the optimal power flow (OPF) problem [43]. R.K. Swain et al. proposed a CSA-based method with an efficient optimization procedure to solve the problem of hydrothermal scheduling [44]. Hamed Chitsaz et al. proposed an improved clonal selection algorithm to train a wavelet neural network model for wind power forecasting [45]. LS Sindhuja and G. Padmavathi using the clonal selection algorithm to detect clones

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by selecting the appropriate witness nodes; subsequently, this method was used to enhance the single hop detection (SHD) method in wireless sensor network replica nodes [46]. The remaining paper is organized as follows: Section 2 describes the problem of elastic parameter inversion along with its model; Section 3 introduces the improved clonal selection algorithm as well as its improvements; Section 4 describes the experimental results and presents the performance comparison of the algorithms; Section 5 provides the conclusions of this work. 2. AVO elastic parameter inversion problem The essential procedures to resolve the problems of AVO elastic parameter inversion are as follows. First, a battery of operations is performed to acquire the three parameters. Next, these parameters are substituted into an approximate equation to acquire a reflection coefficient. Subsequently, seismic data are acquired through the convolution of the reflection coefficient and seismic wavelet. Finally, the seismic data are compared with the actual seismic data. Once the data are similar, and the parameters fit the actual three parameters well, the inversion accuracy can be considered to be high. 2.1. Inversion steps One of the major processes in the inversion of the AVO elastic parameters is establishing the inverse convolution model. The establishment of the convolution model includes the following steps. First, the reflection coefficient Rpp is calculated. In this paper, the approximation equation of Aki and Rechard [2] is applied to calculate the reflection coefficient Rpp , as displayed in Eq. (1).

R pp (θ ) =

 Vp  2 2  Vs 1   ρ 1 2 1 + tan2 θ − 4γ sin θ + 1 − 4γ 2 sin θ 2 2 ρ Vp Vs

(1)

where Vp ,Vs , and ρ respectively denote the difference between Vp , Vs and ρ of the lower and upper layers, respectively. Vp , Vs and ρ denote the average value of Vp , Vs and ρ of the lower and upper layers, respectively. θ is the angle. γ = VVps can be calculated from the actual data. Rpp can be acquired as a constituent part of the seismic record convolution operation according to the equation. Second, the seismic wavelet is acquired, which is the other constituent part of the seismic record convolution model. The seismic record data are acquired through the convolution of the reflection coefficient and wavelet. These data are appropriate for establishing the forward model and fabricating the synthetic seismic records. In this work, the Ricker wavelet is utilized, which is a zero-phase seismic wavelet. The expression of this wavelet is displayed in Eq. (2).





f (t ) = 1 − 2π 2Vm t 2 e−π

2

Vm t 2

(2)

where Vm indicates the basic frequency, and t indicates the time, which can be set manually. Third, the reflection coefficient is convolved with the Ricker wavelet by utilizing Eq. (3).

s(θ ) = R pp (θ ) ∗ f (t ) + n(t )

(3)

where Rpp (θ ) indicates the reflection coefficient function, f(t) indicates the seismic wavelet and n(t) indicates the noise. In this study, the noise factors were not included. The calculated s(θ ) is utilized to establish the objective function. 2.2. Evaluation criteria In the paper, the problem of AVO elastic parameter inversion is converted to the problem of optimization, and the optimization algorithm is utilized in solving this problem. From an optimization point of view, the elastic parameter is deemed to be appropriate when the difference between the real seismic record data and inverted seismic data derived from the optimized elastic parameter is less than a certain threshold or zero. Since the optimization algorithm accesses an individual on the basis of the fitness function of the target function, the weaknesses and strengths of the target function of the inversion problem are the major elements that influence the inversion effect of the prestack AVO elastic parameters. Assuming that the number of sampling points is n, every sample point requires m divergent angles, and n × m seismic record data can be calculated. Finally, the m seismic records for each sampling point obtained through optimization as well as the real seismic record data are differentiated, squared, summed and divided by m. Subsequently, the data obtained from the n sampling points are added, and divided by n. After the extraction of the root, the desired result is obtained. On the basis of this equation, the inversion objective function can be established, as given in Eq. (4).



f (x ) =

n i=1

m j=1

(s(θi, j ) − s (θi, j ))2 n∗m

(4)

where s(θ i,j ) represents the forward seismic record, and s (θ i, j ) indicates the inversion seismic record. Because we calculated the reflection coefficient Rpp by considering the elastic parameters Vp , Vs and ρ , the ultimately calculated same set of inversion seismic data is the consequence of the combination of infinite Vp , Vs and ρ . Therefore,

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there exists a case in which the inversion seismic data calculated by utilizing these three parameters with error is the same as that calculated using one parameter with error and two parameters without error. To evaluate the influence of the optimization algorithm on the AVO elastic parameter inversion results, the Pearson product-moment correlation coefficient (PCC) is used to determine the correlation among three real and three inverted parameters. In this paper, the correlation coefficient function is obtained using Eq. (5).

r=

n

(Xi − X )(Yi − Y )  n 2 2 i=1 (Xi − X ) ∗ i=1 (Yi − Y )

n

i=1

(5)

where Xi is the standard value of the three parameters’, Yi is the corresponding inversion value, X and Y are the corresponding averages of a set of values. Because the process of resolving the seismic data is super intricate, a smaller value of the objective function does not necessarily mean that the three parameters have higher correlation coefficient. Moreover, a higher correlation coefficient of the three parameters does not necessarily correspond to a smaller value of the objective function. Nevertheless, when the objective function attains the theoretical optimal value of 0, the correlation coefficients of Vp , Vs and ρ can attain the theoretical optimal values of 1. Accordingly, this paper combines the value of the correlation coefficient and objective function to evaluate the advantages and disadvantages of the inversion results. The final objective is to reduce the value of the objective function acquired through the inversion and increase the correlation coefficient of the three parameters’. In the entire process of the algorithm, the population is generated according to the threshold values of Vp , Vs and ρ , and each individual is a group of values of Vp , Vs , ρ . When the fitness value is calculated, the value of the reflection coefficient Rpp is calculated according to Eq. (1), and then the seismic wavelet is obtained according to Eq. (2). The reflection coefficient and seismic wavelet are deconvoluted according to Eq. (3) to obtain the seismic record of our performance. According to the above calculated seismic data and actual data, the value obtained by Eq. (4) is the fitness value. Eventually, the optimal solution is selected according to the fitness value, and the correlation coefficient is evaluated according to Eq. (5). 3. Improved clonal selection algorithm for inversion problem The essential process of resolving the problem of AVO elastic parameter inversion, involving the intelligent optimization algorithm is as follows. First, the value range of the three elastic parameters is determined on the basis of the current empirical information and initialized separately. Subsequently, the reflection coefficients are calculated based on the approximation equation of Aki and Rechard. After performing the convolution with the wavelet, the difference between the seismic record data and result is calculated, according to which individual screening is performed. The fitness value is refreshed through a battery of optimization operations, and the best result is exported while the termination condition of the algorithm is implemented. 3.1. Improved initialization strategy In the proposed algorithm, the elastic inversion parameters compose the individual. In terms of the real logging curve model, every sampling point is a layer, which is a dimension. The individuals length is three times that of the sampling point because of the three parameters inversion investigated in this work. If n sampling points are contained along with 3∗ n solution model parameters,the relevant single coding mode is:



Gi = Vp1 , Vs1 , ρ1 , · · · , Vp j , Vs j , ρ j , · · · , Vpn , Vsn , ρn



In the population space considered in this work, the population individual (chromosomes) is designed by utilizing traditional real number coding. The population individuals are initialized via random initialization in a certain range. Every chromosome is made up of a set of real numbers. It is considered that the N, where Vpj , Vsj , ρ j represent the values of the three parameters corresponding to jth sampling point of individual Gi , and the variation range is set according to the real recorded data. The amount of the population individuals is N. Each individual is represented in a one-dimensional array, and thus the array length can be expressed as 3∗ n. The individuals original values are chosen within a limited range of experience (bound function constraints). Next, each parameter is optimized using the genetic algorithm. Finally, the optimal individual is output as the optimal set of elastic parameter solutions. The bound range constraint formula is shown in Eq. (6).

0.8 ∗ Vpwell ≤ Vp ≤ 1.2 ∗ Vpwell 0.8 ∗ Vswell ≤ Vs ≤ 1.2 ∗ Vswell 0.9 ∗ ρwell ≤ ρ ≤ 1.1 ∗ ρwell

(6)

In the analysis, we noted that in the process of calculating Rpp by using the approximate equations of Aki and Rechard, every Rpp was acquired on the basis of the elastic parameters of the lower and upper layers. Nevertheless, the approximate equations of Aki and Rechard can be represented in terms of the difference  among the lower and upper elastic parameters and the upper elastic parameters after the mathematical transformation. The conversion process is indicated in Eq. (7).

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R pp (θ ) = =

V 1 V 1 ρ (1 + tan2 θ ) p − (4γ 2 sin2 θ ) s + (1 − 4γ 2 sin2 θ ) 2 2 ρ Vp Vs Vp 1 Vs 2 (1 + tan2 θ ) − (4γ 2 sin θ ) 2 (2 ∗ Vp1 + Vp )/2 (2 ∗ Vs1 + Vs )/2 +

1 ρ (1 − 4γ 2 sin2 θ ) 2 ( 2 ∗ ρ1 + ρ ) / 2

(7)

Suppose that Vp1 and Vp are nearly equivalent to the true value. Consequently, Vp2 is also similar to the true value. According to the equivalence of reasoning, Vp3 , Vp4 , . . . , Vpn are similar to the true value, and this aspect is also true for Vs and ρ . The reflection coefficients Rpp calculated in this way are also considerably accurate. To achieve such an effect, the first three elastic parameters must be close to the true value. Therefore, the following strategy is employed for the initialization. The bound constraints of the first three parameters are shown in Eq. (8).

0.9 ∗ Vp1well ≤ Vp1 ≤ 1.1 ∗ Vp1well 0.9 ∗ Vs1well ≤ Vs1 ≤ 1.1 ∗ Vs1well 0.95 ∗ ρ1well ≤ ρ1 ≤ 1.05 ∗ ρ1well

(8)

The bound constraints of the three parameters from the second to n groups are shown in Eq. (9).

⎧ V = V + V ⎪ ⎪Vpi+1 = V pi+ V pi ⎪ ⎪ si +1 si si ⎨ ρi+1 = ρi + ρi i = 1, 2, . . . , n − 1 0.8 ∗ (Vpi+1well − Vpiwell ) ≤ Vpi ≤ 1.2 ∗ (Vpi+1well − Vpiwell ) ⎪ ⎪ ⎪ ⎪ ⎩0.8 ∗ (Vsi+1well − Vsiwell ) ≤ Vsi ≤ 1.2 ∗ (Vsi+1well − Vsiwell ) 0.9 ∗ (ρi+1well − ρiwell ) ≤ ρi ≤ 1.1 ∗ (ρi+1well − ρiwell )

(9)

3.2. Clonal selection strategy An antigen usually corresponds to a problem and its constraints existing in an artificial immune system. Specifically, an antigen is the objective function of the problem to be solved. Antibodies often refer to the problems candidate solutions, which correspond to the individual genes in an evolutionary algorithm. A collection of antibodies is known as the antibody group. The antibody-antigen fitness reflects the degree of matching between the antibodies and antigens, which corresponds to the value of the objective function in an evolutionary algorithm. Burnet et al. first introduced the cloning selection theory in 1959 [47]. The core idea is that an antibody exists in the form of receptors on the cell surface, and the antigen can selectively react with it. The mutual stimulation between the antigen and the corresponding antibody receptor can lead to the clonal proliferation of the cells. Throughout the process, the human immune system activates, differentiates and proliferates the immune cells by cloning to increase the number of antibodies, thereby eliminating the antigens in vivo. According to Burnet’s theory of antibody clonal selection, De Castro simulated the mechanism of clonal selection of the biological antibodies described above and proposed the clonal selection algorithm, which is applied in the cloning and proliferation of the dominant population. After the population proliferation, genetic manipulation is performed to generate a new population, and the population after the genetic manipulation is subject to the selection operation to finally achieve the goal of optimization. The values of the objective function differ for different antibodies (i.e., the fitness of the antibody antigens differ). To realize a larger clone size for superior antibodies, the population is ranked prior to cloning with the strategy proposed by De Castro, as given in Eq. (10).

Nc =

n  i=1



round

β ·N

i

(10)

where Nc represents the total size of the clonal population, round(·) is the rounding function, N represents total number of antibodies, also, β indicates the cloning coefficient used to control the size of the clones. As seen from the above equation, the ith antibody will clone round ( β i·N ) antibodies of the same type. In other words, a better antibody fitness results in a larger clone size. Thus, the excellent genes in a fit individual can be better preserved and developed. In this manner, the purpose of optimization is achieved. The cloned population is subject to genetic manipulation. Suppose that the original population size is n, the population size after the cloning operation is m, and the size of the clonal population is still m after the genetic manipulation of the clonal population. The original population of size n is combined with the genetically manipulated clonal population of size m to select the next generation of population size n. The final population size remains n. 3.3. Improved genetic manipulation Van Wahlen, an evolutionary biologist, introduced the Red Queen hypothesis in 1973 [48] and suggested that species maintain a dynamic balance among them. This hypothesis assumes that the individuals in a population are divided into

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dominant individuals and inferior individuals according to the Darwinian approach of the survival of the fittest. Since inferior individuals also have the desire to survive and multiply, they must continue to evolve to compete with dominant individuals for survival. Therefore, with the constant evolution, the overall fitness of the population is continuously increased; however, the differences in the fitness of dominant individuals do not expand further and remain relatively constant. According to the Red Queen hypothesis, an inferior population is also involved in the evolutionary process. Therefore, after the cloning operation, the average of individual fitness values in the clonal population is calculated, the individuals whose fitness values are higher than the mean value (that is, inferior individuals) are marked as 1 and the others are marked as 0. The cloned individuals are combined with the individuals previously marked as 1 to generate a temporary population for the next crossover operation. Assuming that the size of the cloned population is m, and the number of inferior individuals is set as k, the population size generated after the merging crossover operations is still m, and the operation is completed according to Eq. (11).

Fig. 1. Original logging curves for the theoretical model.

Fig. 2. Seismic records for theory model.

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n ave f it = f lagi =

i=1



1 0

f itnessi n f itnessi > ave f it f itnessi <= ave f it

(11)

In the immunology theory, mutation plays a dominant role. However, in this study, based on the elastic parametric inversion problem, the genetic manipulation includes both crossover and mutation. Traditional crossover methods, such as single-point crossover, have weaker search capabilities. Therefore, targeting this problem, this work adopts a crossover operator which is more appropriate for the real coding, and an arithmetic crossover strategy is adopted. This strategy produces two fresh individuals by utilizing a linear combination of 2 individuals. Using λ as random distribution number in the interval [0,1], the following expressions are obtained:

Fig. 3. Logging curves of the inversion generated using the CSA.

Fig. 4. Seismic records of the inversion generated using the CSA.

X. Yan, P. Li and K. Tang et al. / Information Sciences 517 (2020) 86–99

child1 = child2 =

λ × parent2 + (1 − λ ) × parent1 λ × parent1 + (1 − λ ) × parent2

93

(12)

Furthermore, a non-uniform and adaptive mutation operation is adopted in this study. When performing the mutation operation, if the current evolution number is pi , the maximum evolution algebra of the population is p, the gene whose genetic algorithm is selected for the mutation is v, the range of v is [bound1, bound2], and the range is given before the algorithm starts. Assume v1 and v2 as:

v1 = v − bound1 v2 = bound2 − v

(13)

The mutation begins by first generating a random number λ. When λ > 0.5, the following operation is performed on the mutated gene:

Fig. 5. Logging curves of the inversion generated using the GA.

Fig. 6. Seismic records of the inversion generated using the GA.

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v2 ∗ (1 − λ(1−pi /p) ) = v + v

v =

vnew

2

(14)

When λ < 0.5, the following operation is performed for the mutated gene:

 2 v1 ∗ 1 − λ(1−pi /p) = v − v

v =

vnew

(15)

The particular steps of the cloning selection algorithm are as follows: 1. The parameter initialization initializes the antibody group Pa. 2. The antibody-antigen fitness of antibody group Pa is calculated. 3. It is determined whether the maximum number of iterations is achieved; if so, the iteration is ended. Otherwise, the next step is performed until the iteration ends.

Fig. 7. Logging curves of the inversion generated using the PSO.

Fig. 8. Seismic records of the inversion generated using the PSO.

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Table 1 Related parameters of the immune clonal selection algorithm. N

β

Cross_ probability

Mutation_ probability

Max_iteration

40

0.9

0.7

0.05

5000

Table 2 Experimental environment parameters. Simulation environment

Parameter description

Java version Compiling environment Processor Memory(RAM) Operating system

1.8.0_111-b14 eclipse − jee − luna − SR1a − win32 − x86_64 Int el (R )C ore(T M )i5 − [email protected] 8.00GB 64 − bit operating system

4. Performing sorting according to the antibody-antigen fitness. A higher antibody-antigen fitness results in a higher antibody ranking 5. The sorted antibody group is cloned according to formula (10) to generate the antibody group Pa’. 6. The antibody group Pa’ is subject to cross mutation and the genetic operation to generate the antibody group Pa”. 7. The antibody group Pa” is merged with antibody group Pa and their antibody-antigen fitness is calculated. 8. The tournament selection is performed on the merged population, and the next generation antibody group Pa is selected; return to the third step. The clonal selection algorithm clones all the antibodies and determines the number of clones of the antibody based on the location of the different antibodies in the antibody group. The dimension of the clonal population Nc is also defined, given the size of the antibody group N and the clonal coefficient β . For the inversion problem, the value of the cloning coefficient β in this experiment is 0.9, and the value of the population size N is 40. 4. Experimental simulation and analysis 4.1. Algorithm parameter setting To validate the effectiveness of the proposed algorithm, that is the improved clonal selection algorithm (CSA), we validated the algorithm using the logging data and compared the inversion result with that obtained using the particle swarm optimization (or PSO for short), differential evolution algorithm (or DE for short), and genetic algorithm (or GA for short). The parameters of the clonal selection algorithm employed in this experiment are listed in Table 1. The experimental environment parameters are listed in Table 2:

Fig. 9. Logging curves of the inversion generated using the DE.

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X. Yan, P. Li and K. Tang et al. / Information Sciences 517 (2020) 86–99 Table 3 Mean correlation coefficient of the three parameters.

Vp Vs

ρ

GA

PSO

DE

CSA

0.556364 0.650805 0.462346

0.765481 0.832094 0.649089

0.701792 0.787196 0.606702

0.951875 0.925657 0.948136

Fig. 10. Seismic records of the inversion generated using the DE.

Fig. 11. Logging curves of the inversion generated using the Bayesian.

4.2. Experimental results and analysis The log curve data in the data set is derived from two hundred and forty-one sample points, which include the Swave velocity Vp , P-wave velocity Vs and the density ρ . Every sampling point corresponds to eight divergent angles: [0◦ , 6◦ , 11◦ , 17◦ , 23◦ , 29◦ , 34◦ , and 40◦ ]. Every data set utilizes the above mentioned 8 angles. The formula of Rechard and Aki is applied in calculating the theoretical log model, which is used to calculate the reflection coefficient as well as the

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Fig. 12. Seismic records of the inversion generated using the Bayesian.

convolution reflection coefficient with the wavelet. The generation of the seismic records requires the relationship among groups of up and down sampling points, thus, the seismic data include 240∗ 8 data points. The initial log data are displayed in Figs. 1 and 2. The experimental results obtained using different intelligent optimization algorithms to solve this problem are shown in the Fig. 3 to Fig. 10. We also compared our proposed algorithm with classical Bayesian linearized inversion method [49], the experimental result shown in Fig. 11 and Fig. 12. In Table 3 the mean correlation coefficients of the three parameters Vp , Vs and ρ acquired from the four algorithms are compared. The comparison data presented in Figs. 4–11 and Table 3 indicates that the CSA is evidently better than GA, PSO and DE. After being combined with the improved initialization strategy, the effect is considerably better than that of the original algorithm, and the correlation coefficient is greatly enhanced. The comparison given in Table 3 indicates that, the CSA algorithm can obtain better correlation coefficients of the elastic parameters, allowing the inverted seismic data to be consistent with the real seismic data. Fig. 13 shows the values of the objective function of the inversion for the different algorithms. The DE algorithm exhibits the worst effect. The GA has an effect superior to that of the differential evolution algorithm, and the value of target function

Fig. 13. Convergence curve of algorithm.

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reaches approximately 0.0035. The value of the objective function value of the PSO remains approximately 0.0018, which is better than that for the genetic algorithm. The value of the objective function of the CSA is 0.0 0 085, which is increased by one magnitude compared with that of other algorithms. The CSA can significantly reduce the objective function of the inversion. The comparison shown in Fig. 13 indicates that, the CSA exhibits faster convergence speed and better optimization efficiency than those of the GA, PSO and DE. 5. Conclusion In recent decades, inversion on the basis of prestack seismic data has gradually become a popular topic in the domain of seismic exploration. The prestack seismic data contain a large amount of information, which can reflect the characteristics of the substructure. By utilizing the AVO information to solve the approximation formula of the Zoeppritz equation, the inversion of the prestack can directly acquire the elastic parameters (including the density, S-wave velocity and P-wave velocity) that reflect the underground rock characteristics. Therefore, the inversion can be utilized to forecast the conditions of underground gas and oil. The AVO elastic parameter inversion is a problem of nonlinear optimization. Accordingly, if the nonlinear inversion approach is employed, the state and property of the solution space are better than those of the linear inversion approach. Furthermore, the global optimal intelligent optimization algorithm has a strong global as well as local optimization ability, which is appropriate to solve the considered problem. In this thesis, the clonal selection optimization algorithm, which is more appropriate to solve problem of AVO elastic parameter inversion is introduced. The inversion is optimized by cloning the antibody group and improving the initialization strategy to improve the inversion accuracy. The results of several perform experiments indicate that the clone selection algorithm can improve the inversion accuracy and make the inverted seismic data more consistent with the actual seismic data. Declaration of Competing Interest The authors declared that they have no conflicts of interest to this work. We declare that we do not have any commercial or associative interest that represents a conflict of interest in connection with the work submitted Acknowledgments This paper is supported by Natural Science Foundation of China. 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