# Seismic Inversion Problem Using a Multioperator Whale Optimization Algorithm

Authors: Rui Ni; Xiaodan Liang
Journal: Discrete Dynamics in Nature and Society
 (2022)

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## Abstract

The whale optimization algorithm (WOA) is a metaheuristic algorithm based on swarm intelligence and it mimics the hunting behavior of whales. It has the imperfection of premature convergence into local optima. In order to overcome this disadvantage, a multioperator WOA (MOWOA) is proposed. Four main strategies are introduced to the MOWOA to heighten the search capacity of WOA. The strategies include nonlinear adaptive parameter design, an exploration mechanism of honey badger, Cauchy factor strategy, and greedy strategy. This paper tests the versatility of MOWOA with three different types of benchmark functions, and a kind of seismic inversion problem are trialed run. From the experimental results, the performance of MOWOA outperforms the compared algorithms in global optimization.


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## Body

## 1. Introduction

Metaheuristic algorithms are becoming increasingly popular, in which swarm intelligence (SI), a kind of metaheuristic, has also received attention. SI is a random optimization algorithm created by people based on the behavior of animals or plants. Some popular SI algorithms include artificial bee colony algorithm (ABC) [1], brain storm optimization (BSO) [2], particle swarm optimization (PSO) [3], chicken swarm optimization (CSO) [4], bacteria foraging optimization (BFO) [5], ant colony optimization (ACO) [6], and whale optimization (WOA) [7].WOA was first proposed in 2016, based on the hunting and social behavior of humpback whales. The characteristics of WOA include simple principles and short time consumption, and many studies have been presented. They are generally divided into two diffident kinds of questions. One is to introduce some strategies to improve the algorithm. The other one is to solve some practical problems by using WOA. An improved WOA called opposition-based WOA (OWOA) was proposed by Alamri et al. in 2018 [8]. The OWOA introduced the opposition-based method to enhance the performance, which searches the solution in the direction opposite to the initial population. An improved WOA consisting of three methods is proposed to enhance the explorative ability [9]. The WOA based on the Lévy flight trajectory was proposed to enhance the search performance [10]. Mostafa Bozorgi and Yazdani proposed IWOA based on differential evolution [11]. Based on this, a variant (IWOA+) was proposed to enhance IWOA which introduced reinitialization and adaptive parameters. In [12], chaotic-based WOA (CWOA) was presented. The chaotic theory was used to tune the main parameter of WOA and control the exploration and exploitation phase by a chaotic map. A hybrid WOA (GWOA-TEO) was proposed in 2021 [13]. This algorithm introduced a genetic and thermal exchange optimization-based to enhance search optimum. Wu et al. used a nonlinear arcsine function (NCS-Arcsine) to improve WOA [14]. This method can balance the parameter a during the iteration and enhance the capability of WOA. A named WOAmM was proposed in 2020 [15]. A strategy of modified mutualism phase was used to balance search space and avoid premature convergence. For solving global optimization problems, a hybrid WOA (ESSAWOA) was proposed [16]. Zhang and Liu used Lamarckian learning to enhance WOA (WOALam) [17] and got better adaptability by this learning mechanism on the high-dimensional optimization problems. In [18], a modified WOA (MWOA) was proposed to solve large-scale global optimization problems. By comparison, the results show MWOA was superior to the other variant. The MMWOA introduced the niching and the Gaussian sampling technique into WOA for application in the multimodal optimization [19], which enhances the multimodal search ability and improves the local search capability. An improved WOA named PDWOA was proposed in 2018 [20]. It is worth mentioning that this algorithm used the equal pitch Archimedes spiral curve and the perceptual perturbation mechanism further enhances WOA. The WOA based on quadratic interpolation (QIWOA) was proposed [21]. From the results, the QIWOA’s performance was better than several other algorithms. In [22], an adaptive localized decision variable analysis approach was proposed and its effectiveness and efficiency are verified on the objective function.Furthermore, WOA can also solve some engineering problems. Kaveh et al. utilized the hybrid WOA-CBO to solve the problem of layout planning, colliding bodies optimization are mixed into WOA, and the result shows that it had better advantages compared with other algorithms [23]. Selim et al. proposed a hybrid WOA with sine cosine (SC) [24]. The results testified the effectiveness and the proposed algorithm optimized the voltage profile in the system. Chinamalli and Sasikaa introduced the cross assist into WOA (CWOA) for controlling permanent-magnet synchronous generators in wind power generation [25]. Compared with other intelligent algorithms, the performance of CWOA had better effectiveness. Khurshaid et al. mixed WOA and simulated annealing (SA) to propose a hybrid algorithm (HWOA) that enhances the global search in the exploitation phase for solving the problem of the directional overcurrent relays [26]. Abd Elaziz and Oliva introduced the OBL strategy into WOA (OBWOA) to optimize the problem of the solar cell diode model [27]. For solving the problem of industrial copper burdening optimization, an adaptive reference vector reinforcement learning approach was proposed in [28], and its effectiveness was verified.As the difficulty of exploration and development continues to increase, the development of prestack seismic inversion technology has an important significance for predicting the distribution of reservoir lithology and oil and gas exploration. Prestack seismic inversion provides abundant physical properties of underground features, but this technology is not mature and the process is relatively cumbersome, and limited by various factors such as the high resolution, low speed, and poor stability to affect the accuracy. Currently, the technology needs to solve some inevitable technical problems, but it is undeniable that the prestack inversion technology is still a very challenging and attractive research subject.Many research teams have done a lot of research on solving the prestack inversion problem with optimization algorithms. Xiao et al. proposed a nonlinear method based on lithology constraints and introduced a Gaussian distribution parameter into the simulated annealing algorithm (SA) [29]. The results show that this proposed method has the better resolution and stability performance. Peng-Fei et al. utilized a modified SA algorithm to solve the problems of Poisson ratio, P-wave velocity, and density in the prestack inversion [30]. The results demonstrated the effectiveness. Gao et al. introduced a new multimutation strategy into the differential evolution (DE) algorithm [31], which generates better mutant operation for the parameter’s iteration of the prestack inversion problem. From experiment results, this algorithm is faster convergence than the present algorithm for the target problem. In [32], an improved differential evolution algorithm was utilized to calculate the oil-gas exploration, and the result indicated the effectiveness. An adaptive genetic algorithm (GA) was proposed to deal with the tradition algorithm’s weakness in the seismic inversion problem [33]. In [34], a novel algorithm was proposed to design a ZVD shaper and solved the real problems.Although WOA is popular due to its good convergence rates, it still has many shortcomings such as the multimodal problems with local optimal solutions which do not solve global optimum well. In this paper, the proposed MOWOA including four strategies is proposed to enhance the search capacity and solve a kind of seismic inverse problem. Nonlinear adaptive parameter design is utilized to balance search proportion. An exploration mechanism of honey badger is introduced to speed up the rate of convergence. The Cauchy factor strategy is utilized to develop the capacity of jumping out of the local optimum. A greedy strategy is utilized to compare the original solution with the solution of added Cauchy operator and choose the solution with the best fitness. To validate the proposed algorithm’s capacity to find an optimal solution, we conduct a series of experiments, and MOWOA is applied to find the solution of well-known nineteen benchmark functions including three types and solve a kind of seismic inverse problem, respectively. Finally, the proposed MOWOA is demonstrated to be superior to WOA and its three variants.The main contributions of this paper are as follows: (1) nonlinear adaptive parameter design and honey badger search mechanism are introduced into WOA, which improves its search performance. (2) The Cauchy factor and greedy strategy are introduced to improve the optimization capacity. (3) A kind of seismic inverse problem is solved with the proposed MOWOA. The rest of this paper is organized as follows. Section2 introduces the WOA and illustrates the MOWOA and its strategies. Section 3 evaluates MOWOA with three different types of benchmark functions, presents and analyzes the experiment results with other algorithms, and then tries to solve a kind of seismic inverse problem with MOWOA. Finally, Section 4 concludes the study.


## 2. The Proposed Algorithm

### 2.1. Whale Optimization Algorithm (WOA)

The whale optimization algorithm is a metaheuristic algorithm based on swarm intelligence and it mimics the hunting behavior of humpback species of whales in the ocean [7]. These patterns are divided into the exploitation phase which encircles prey and bubble-net attack and the exploration phase which searches for prey.The mathematical expression of the exploitation phase is shown as follows:(1)A⟶=2a⟶⋅r⟶−a⟶,C⟶=2⋅r⟶,D⟶=C⟶⋅X∗⟶−X⟶t,D′⟶=X∗⟶t−X⟶t,(2)X⟶t+1=X∗⟶t−A⟶⋅D⟶,D′⟶⋅ebl⋅cos2πl+X∗t.The mathematical expression of the exploration phase is shown as follows:(3)D⟶=C⟶⋅X⟶randt−X⟶t,X⟶t+1=X⟶randt−A⟶⋅D⟶.In the above equation,A⟶ and C⟶ are the coefficient vectors where a⟶∈0,2 andr⟶∈0,1. X⟶, X∗⟶, and X⟶rand are the current solution, the best solution, and the random solution, respectively, and t denotes the number of iterations. D⟶ and D′⟶ represent the distance vector for the used method. b and l are a constant and a random number, respectively, where l∈−1,1.In WOA, the values ofA⟶ and p are set to balance the proportion of two search phases, where A⟶∈−1,1 and p∈0,1.


### 2.2. Multioperator Whale Optimization Algorithm (MOWOA)

In this section, nonlinear adaptive parameter design, honey badger mechanism, Cauchy factor, and greedy strategy will be introduced into WOA. Nonlinear adaptive parameter design is used to balance the proportion of exploitation and exploration phase. An exploration mechanism of honey badger is introduced to speed up the rate of convergence. The Cauchy factor strategy is utilized to develop the capacity of jumping out of the local optimum. Finally, greedy strategy is utilized to compare the original solution with the solution of added Cauchy operator and choose the solution with the best fitness. The pseudoalgorithm of the MOWOA is given in Algorithm 1.Algorithm 1: MOWOA.
Initialization phase:(1)
Initialize the populationXi of the MOWOA(2)
Calculate the fitness values(3)
FindXbest∗ according to the fitness values(4)
whilet ≤ Max_iter do(5)
Update the nonlinear parameter a(6)
fori = 1, 2, …, Ndo(7)
Update the parameters (i.e.,A, C, l, p, p∗, F, and m⟶n⟶)(8)
forj = 1, 2, …, Mdo(9)
ifp < p∗then(10)
if |A| < 1 then(11)
D=C⋅X∗j−Xj(12)
X∗j+1=X∗j−AD(13)
V∗j+1=V∗j−AD−AD⋅m⟶n⟶(14)
elseif |A| ≥ 1 then(15)
Select a random solutionXrand(16)
D=C⋅Xrandj−Xj(17)
Xj+1=Xrandj+F⋅r⋅a⋅D(18)
Vj+1=Xrandj+F⋅r⋅a⋅D⋅m⟶n⟶(19)
end if(20)
else ifp ≥ p∗then(21)
ifp<0.5then(22)
D=C⋅X∗j−Xj(23)
X∗j+1=X∗j−AD(24)
V∗j+1=V∗j−AD−AD⋅m⟶n⟶(25)
else then(26)
D′=X∗j−Xj(27)
Xj+1=D′⋅ebl⋅cos2πl+X∗j(28)
Vj+1=m⟶n⟶⋅D′⋅ebl⋅cos2πl+X∗j(29)
end if(30)
end if(31)
end for(32)
end for(33)
Calculate the fitness values(34)
if fit (V(t)) < fit (X(t)) then(35)
Select the fitness fit (V(t))(36)
else then(37)
Select the fitness fit (X(t))(38)
end if(39)
end whileTo recap the pseudo-algorithm 1, the fitness of each generation solution and theXbest∗ will give (see lines 2–3). Nonlinear parameter a and other parameters (i.e., A, C, l, p, p∗, F and m⟶n⟶) are updated (see lines 5 and 7), introduce the nonlinear adaptive parameter design to replace the value of a in equation (1) and reinstall a new parameter p∗. The exploration mechanism of the honey badger is introduced into the proposed algorithm to replace equation (3) (see line 17). The Cauchy factor is introduced (see lines 13, 18, and 24 lines) to calculate the fitness together (see lines 33). The greedy strategy is used to compare the fitness of V(t) and X(t), and the smallest is selected as the fitness of the iteration (see lines 34–39). The original three WOA operators are used in lines 11–12, 15–16, and 26–27, respectively.


### 2.3. Nonlinear Adaptive Parameter Design

A cosine functiona⟶ is utilized to improve WOA. a⟶ is an important vector that determines the change of the coefficient vector A⟶ that regulates the proportion of exploitation and exploration phase. The mathematical expression of a⟶ is given as(4)a=2⋅cosπ2⋅tMax_iter,where t and Max_iter are the number of current and the maximum iterations, respectively.In the early iteration, the large value ofa⟶ can fully explore the overall situation and decreases slowly in the exploitation phase. In the later iteration, the value of a⟶ decreases rapidly to perform the local search.In order to synchronize two methods that mimic the whale attacking, an adaptive threshold of the logarithmic formp∗ is utilized to balance the global and local development. The adaptive parameter design p∗ is designed as follows:(5)p∗=1−log21+tMax_iter.Whenp < p∗, the algorithm will perform the encircled search. When p ≥ p∗ and p>0.5, the algorithm will perform the spiral search. This strategy is given as (6)X⟶t+1=X∗⟶t−A⟶⋅D⟶ifp<p∗,D′⟶⋅ebl⋅cos2πl+X∗⟶tifp≥p∗andp≥0.5.


### 2.4. An Exploration Mechanism of Honey Badger

Due to the slow convergence rate of WOA in the exploration phase, a search mechanism—honey badger [35]—is introduced into the improved WOA, and this search mechanism is given as(7)X⟶t+1=X⟶randt+F⋅r⟶⋅a⟶⋅D⟶,where r⟶ is a random vector between 0 and 1 and the vector of D⟶ refers to equation (3). F is the flag that changes search direction and is described as follows:(8)F=1,ifr≤0.5,−1,else,where r is a random number between 0 and 1.


### 2.5. Cauchy Factor Strategy

The Cauchy factor is introduced into the improved WOA. Because of the crest of the standard Cauchy distribution at the zero point is low and the downward trend on both sides of the zero point is slower, the Cauchy mutation has a stronger perturbation ability, which can make WOA obtain a better capacity to jump out of the local optimum. This Cauchy factor strategy is shown as follows:(9)V⟶t+1=X⟶randt+F⋅r⟶⋅a⟶⋅D⟶⋅m⟶n⟶,ifp<p∗andA≥1,X∗⟶t−A⟶⋅D⟶−A⟶⋅D⟶⋅m⟶n⟶,ifp<p∗A<1,D′⟶⋅ebl⋅cos2πl⋅m⟶n⟶+X∗j,ifp≥p∗andp≥0.5.The vectorm⟶ is a Cauchy and exponent mixing factor and has better perturbation ability, and n⟶ is a flag vector. They are calculated by equations as follows:(10)m⟶=Cauchy0,1⋅exp−10tMax_iter,(11)n⟶=2∗r1−12∗12−0.25tMaxiter,ifr1>r2,2∗r1−12∗2∗12−0.25tMaxiter,ifr1≤r2,where r1 and r2 are random numbers between 0 and 1.


### 2.6. Greedy Strategy

A greedy strategy is introduced into the improved WOA, which is a select operation. This selection operation is to compare the mutated new individual with the original individual to determine whether its fitness value is better and keep only the individual with the better fitness. The greedy strategy is mathematically described as follows:(12)X⟶newt+1=V⟶t+1,if fitV⟶t+1<fitX⟶t+1,X⟶t+1,if fitV⟶t+1≥fitX⟶t+1.


## 2.1. Whale Optimization Algorithm (WOA)

The whale optimization algorithm is a metaheuristic algorithm based on swarm intelligence and it mimics the hunting behavior of humpback species of whales in the ocean [7]. These patterns are divided into the exploitation phase which encircles prey and bubble-net attack and the exploration phase which searches for prey.The mathematical expression of the exploitation phase is shown as follows:(1)A⟶=2a⟶⋅r⟶−a⟶,C⟶=2⋅r⟶,D⟶=C⟶⋅X∗⟶−X⟶t,D′⟶=X∗⟶t−X⟶t,(2)X⟶t+1=X∗⟶t−A⟶⋅D⟶,D′⟶⋅ebl⋅cos2πl+X∗t.The mathematical expression of the exploration phase is shown as follows:(3)D⟶=C⟶⋅X⟶randt−X⟶t,X⟶t+1=X⟶randt−A⟶⋅D⟶.In the above equation,A⟶ and C⟶ are the coefficient vectors where a⟶∈0,2 andr⟶∈0,1. X⟶, X∗⟶, and X⟶rand are the current solution, the best solution, and the random solution, respectively, and t denotes the number of iterations. D⟶ and D′⟶ represent the distance vector for the used method. b and l are a constant and a random number, respectively, where l∈−1,1.In WOA, the values ofA⟶ and p are set to balance the proportion of two search phases, where A⟶∈−1,1 and p∈0,1.


## 2.2. Multioperator Whale Optimization Algorithm (MOWOA)

In this section, nonlinear adaptive parameter design, honey badger mechanism, Cauchy factor, and greedy strategy will be introduced into WOA. Nonlinear adaptive parameter design is used to balance the proportion of exploitation and exploration phase. An exploration mechanism of honey badger is introduced to speed up the rate of convergence. The Cauchy factor strategy is utilized to develop the capacity of jumping out of the local optimum. Finally, greedy strategy is utilized to compare the original solution with the solution of added Cauchy operator and choose the solution with the best fitness. The pseudoalgorithm of the MOWOA is given in Algorithm 1.Algorithm 1: MOWOA.
Initialization phase:(1)
Initialize the populationXi of the MOWOA(2)
Calculate the fitness values(3)
FindXbest∗ according to the fitness values(4)
whilet ≤ Max_iter do(5)
Update the nonlinear parameter a(6)
fori = 1, 2, …, Ndo(7)
Update the parameters (i.e.,A, C, l, p, p∗, F, and m⟶n⟶)(8)
forj = 1, 2, …, Mdo(9)
ifp < p∗then(10)
if |A| < 1 then(11)
D=C⋅X∗j−Xj(12)
X∗j+1=X∗j−AD(13)
V∗j+1=V∗j−AD−AD⋅m⟶n⟶(14)
elseif |A| ≥ 1 then(15)
Select a random solutionXrand(16)
D=C⋅Xrandj−Xj(17)
Xj+1=Xrandj+F⋅r⋅a⋅D(18)
Vj+1=Xrandj+F⋅r⋅a⋅D⋅m⟶n⟶(19)
end if(20)
else ifp ≥ p∗then(21)
ifp<0.5then(22)
D=C⋅X∗j−Xj(23)
X∗j+1=X∗j−AD(24)
V∗j+1=V∗j−AD−AD⋅m⟶n⟶(25)
else then(26)
D′=X∗j−Xj(27)
Xj+1=D′⋅ebl⋅cos2πl+X∗j(28)
Vj+1=m⟶n⟶⋅D′⋅ebl⋅cos2πl+X∗j(29)
end if(30)
end if(31)
end for(32)
end for(33)
Calculate the fitness values(34)
if fit (V(t)) < fit (X(t)) then(35)
Select the fitness fit (V(t))(36)
else then(37)
Select the fitness fit (X(t))(38)
end if(39)
end whileTo recap the pseudo-algorithm 1, the fitness of each generation solution and theXbest∗ will give (see lines 2–3). Nonlinear parameter a and other parameters (i.e., A, C, l, p, p∗, F and m⟶n⟶) are updated (see lines 5 and 7), introduce the nonlinear adaptive parameter design to replace the value of a in equation (1) and reinstall a new parameter p∗. The exploration mechanism of the honey badger is introduced into the proposed algorithm to replace equation (3) (see line 17). The Cauchy factor is introduced (see lines 13, 18, and 24 lines) to calculate the fitness together (see lines 33). The greedy strategy is used to compare the fitness of V(t) and X(t), and the smallest is selected as the fitness of the iteration (see lines 34–39). The original three WOA operators are used in lines 11–12, 15–16, and 26–27, respectively.


## 2.3. Nonlinear Adaptive Parameter Design

A cosine functiona⟶ is utilized to improve WOA. a⟶ is an important vector that determines the change of the coefficient vector A⟶ that regulates the proportion of exploitation and exploration phase. The mathematical expression of a⟶ is given as(4)a=2⋅cosπ2⋅tMax_iter,where t and Max_iter are the number of current and the maximum iterations, respectively.In the early iteration, the large value ofa⟶ can fully explore the overall situation and decreases slowly in the exploitation phase. In the later iteration, the value of a⟶ decreases rapidly to perform the local search.In order to synchronize two methods that mimic the whale attacking, an adaptive threshold of the logarithmic formp∗ is utilized to balance the global and local development. The adaptive parameter design p∗ is designed as follows:(5)p∗=1−log21+tMax_iter.Whenp < p∗, the algorithm will perform the encircled search. When p ≥ p∗ and p>0.5, the algorithm will perform the spiral search. This strategy is given as (6)X⟶t+1=X∗⟶t−A⟶⋅D⟶ifp<p∗,D′⟶⋅ebl⋅cos2πl+X∗⟶tifp≥p∗andp≥0.5.


## 2.4. An Exploration Mechanism of Honey Badger

Due to the slow convergence rate of WOA in the exploration phase, a search mechanism—honey badger [35]—is introduced into the improved WOA, and this search mechanism is given as(7)X⟶t+1=X⟶randt+F⋅r⟶⋅a⟶⋅D⟶,where r⟶ is a random vector between 0 and 1 and the vector of D⟶ refers to equation (3). F is the flag that changes search direction and is described as follows:(8)F=1,ifr≤0.5,−1,else,where r is a random number between 0 and 1.


## 2.5. Cauchy Factor Strategy

The Cauchy factor is introduced into the improved WOA. Because of the crest of the standard Cauchy distribution at the zero point is low and the downward trend on both sides of the zero point is slower, the Cauchy mutation has a stronger perturbation ability, which can make WOA obtain a better capacity to jump out of the local optimum. This Cauchy factor strategy is shown as follows:(9)V⟶t+1=X⟶randt+F⋅r⟶⋅a⟶⋅D⟶⋅m⟶n⟶,ifp<p∗andA≥1,X∗⟶t−A⟶⋅D⟶−A⟶⋅D⟶⋅m⟶n⟶,ifp<p∗A<1,D′⟶⋅ebl⋅cos2πl⋅m⟶n⟶+X∗j,ifp≥p∗andp≥0.5.The vectorm⟶ is a Cauchy and exponent mixing factor and has better perturbation ability, and n⟶ is a flag vector. They are calculated by equations as follows:(10)m⟶=Cauchy0,1⋅exp−10tMax_iter,(11)n⟶=2∗r1−12∗12−0.25tMaxiter,ifr1>r2,2∗r1−12∗2∗12−0.25tMaxiter,ifr1≤r2,where r1 and r2 are random numbers between 0 and 1.


## 2.6. Greedy Strategy

A greedy strategy is introduced into the improved WOA, which is a select operation. This selection operation is to compare the mutated new individual with the original individual to determine whether its fitness value is better and keep only the individual with the better fitness. The greedy strategy is mathematically described as follows:(12)X⟶newt+1=V⟶t+1,if fitV⟶t+1<fitX⟶t+1,X⟶t+1,if fitV⟶t+1≥fitX⟶t+1.


## 3. Experiment Results

### 3.1. Parameter Setting and Benchmark Functions

In this part, the proposed MOWOA is appraised by nineteen benchmark functions F1– F19. F1– F7 are the unimodal functions that are given in Table1 to test the MOWOA’s exploitation ability in the case of only one global optimum. F8–F12 are the multimodal functions that are given in Table 2 to examine the MOWOA’s exploration tendency in the case of many local optima. These two sets of functions are employed in 30 dimensions. F13–F19 are the fixed-dimension multimodal functions that are given in Table 3 to evaluate the MOWOA’s convergence in the problem of the low dimensions.Table 1
Expression of F1–F7 benchmark functions.
FunctionsDimRangefminF1=∑i=1nxi230[−100, 100]0F2=∑i=1nxi+∏i=1nxi30[−10, 10]0F3=∑i=1n∑j=1ixj230[−100, 100]0F4=maxixi,1≤i≤n30[−100, 100]0F5=∑i=1n−1100xi+1−xi22+xi−1230[−30, 30]0F6=∑i=1nxi+0.5230[−100, 100]0F7=∑i=1nixi4+random0,130[−1.28, 1.28]0Table 2
Expression of F8–F12 benchmark functions.
FunctionsDimRangefminF8=∑i=1nxi2−10cos2πxi+1030[−5. 12. 5. 12]0F9=−20exp−0.21/n∑i=1nxi2−exp1/n∑i=1ncos2πxi+20+e30[−32, 32]0F10=1/4000∑i=1nxi2+∏i=1ncosxi/i+130[−600, 600]0F11=π/n10sinπy1+∑i=1n−1yi−121+10sin2πyi+1+yn−12+∑i=1nuxi,10,100,4yi=1+xi+1/4uxi,a,k,m=kxi−ama<xi0−a<xi<ak−xi−amxi<−a30[−50, 50]0F12=0.1sin23πxi+∑i=1n−1xi−121+sin23πxi+1+xn−12×1+sin22πxn+∑i=1nuxi,5,100,430[−50, 50]0Table 3
Expression of F13–F19 benchmark functions.
FunctionDimRangefminF13=1/500+∑j=1251/j+∑i=12xi−aij6−12[−65, 65]0F14=∑i=111ai+x1bi2+bix2/bi2+bix3+x424[−5, 5]0.0003F15=4x12−2.1x14+1/3x16+x1x2−4x22+4x242[−5, 5]−1.0316F16=x2−5.1/4π2x12+5/πx1−62+101−1/8πcosx1+102[−5, 5]0.398F17=1+x1+x2+1219−14x1+3x12−14x2+6x1x2+3x22×30+2x1−3x22×18−32x1+12x12+48x2−36x1x2+27x222[−2, 2]3F18=−∑i=14ciexp−∑j=13aijxj−pij23[1, 3]−3.86F19=−∑i=14ciexp−∑j=16aijxj−pij26[0, 3]−3.12where Dim, Range, andfmin are the dimension, the lower and upper bound, and the minimum value of the function, respectively.The proposed MOWOA will be compared with WOA [7] and its three variants OWOA [8], OEWOA [9], and DEWOA [11]. For the algorithms tested above, the number of population sizes and maximum iterations has been tuned as 30 and 500, respectively, while all tested algorithms are run independently 30 times with benchmark functions in Tables 1–3.


### 3.2. Result and Discussion

Table4 depicts the experimental results of WOA, OWOA, OEWOA, DEWOA, and MOWOA obtained among the nineteen functions, where Std is the standard deviation.Table 4
Experimental results of WOA, OWOA, OEWOA, DEWOA, and MOWOA on benchmark function.
FunctionAlgorithmBestMeanStdF1WOA2.83E − 844.28E − 742.29E − 73OWOA4.59E − 867.95E − 752.66E − 74OEWOA8.76E − 1035.13E − 881.59E − 87DEWOA6.77E − 311.94E − 284.52E − 28MOWOA4.98E − 992.15E − 838.78E − 83F2WOA8.71E − 572.85E − 519.40E − 51OWOA1.53E − 571.89E − 524.72E − 52OEWOA2.93E − 667.49E − 611.83E − 60DEWOA4.31E − 201.19E − 181.72E − 18MOWOA1.07E − 766.88E − 663.59E − 65F3WOA11389.118450275.467116562.2253OWOA26046.721747505.576413886.3143OEWOA12028.025939227.165917413.6989DEWOA0.002001407610.648203636.5865551MOWOA347.5717955457.181884555.64762F4WOA3.523351.610127.7077OWOA5.272845.476129.8808OEWOA4.549859.49726.6399DEWOA0.000304510.004807350.00478835MOWOA0.00711592.321512.72084F5WOA27.571928.14580.400962OWOA27.686128.23820.372579OEWOA27.752328.45840.308603DEWOA24.947228.816312.0021MOWOA25.643627.20241.04652F6WOA0.519651.16060.34138OWOA0.13640.596480.27012OEWOA0.157760.854470.26204DEWOA0.245730.821210.41509MOWOA0.000153930.337030.23659F7WOA3.35E − 050.00553250.0081194OWOA3.15E − 050.00166680.00317OEWOA4.25E − 050.00197360.0054556DEWOA0.00131550.00688320.0039475MOWOA1.85E − 050.00367180.0035788F8WOA03.78E − 152.07E − 14OWOA01.89E − 151.03E − 14OEWOA00.5282632.46821DEWOA5.6843E − 1434.412229.1644MOWOA04.9545317.8488F9WOA8.88E − 164.79E − 152.52E − 15OWOA8.88E − 164.79E − 153.14E − 15OEWOA8.88E − 164.79E − 152.52E − 15DEWOA5.77E − 142.26E − 132.53E − 13MOWOA8.88E − 164.55E − 152.37E − 15F10WOA00.00431590.023639OWOA000OEWOA00.0134090.073444DEWOA00.00748470.013641MOWOA00.000867610.003396F11WOA0.0116720.0643820.11014OWOA0.00778610.0328890.03719OEWOA0.0204970.0549750.047825DEWOA0.0187520.0764520.10195MOWOA0.00184310.03020.025889F12WOA0.473431.01960.34191OWOA0.232390.62260.32139OEWOA0.206960.785040.3463DEWOA0.36841.04570.2804MOWOA0.123390.598670.29991F13WOA0.9980038383.09043.4404OWOA0.9980038382.14961.9772OEWOA0.9980038382.34811.9499DEWOA0.9980038386.14014.9939MOWOA0.9980038382.05111.8885F14WOA0.000311470.000661580.00047928OWOA0.000315620.000699090.00048553OEWOA0.000307660.000774450.00056134DEWOA0.000307540.0101880.015558MOWOA0.000307530.000584420.00023566F15WOA−1.031628453−1.0316284534.06E − 09OWOA−1.031628453−1.0316284531.78E − 08OEWOA−1.031628453−1.0316284536.47E − 09DEWOA−1.031628453−1.0316284534.12E − 10MOWOA−1.031628453−1.0316284535.16E − 16F16WOA0.3978873580.39792442705.23E − 05OWOA0.3978873580.39790299102.65E − 05OEWOA0.3978873580.39797670400.00022822DEWOA0.3978873580.39788759834.54E − 07MOWOA0.3978873580.39788735770F17WOA33.00111920030.00520741OWOA33.00017804370.00025126OEWOA33.00018061540.00022324DEWOA36.600000309915.4264MOWOA33.00000014321.99E − 07F18WOA−3.862776732−3.8469485730.024831771OWOA−3.862782093−3.8518077120.025891037OEWOA−3.862781853−3.8378838760.054873368DEWOA−3.862782128−3.8614826920.002955750MOWOA−3.862782147−3.8619881950.002403123F19WOA−3.320772006−3.2376834240.14504OWOA−3.321595755−3.2425737150.091649OEWOA−3.319657692−3.2092441750.09022DEWOA−3.321993785−3.2582888270.087085MOWOA−3.321995171−3.2494305500.080041From experiment results, the performance of MOWOA is absolutely best for F2, F6, F9, and F11–F18. For F1, F3, F4, and F10, MOWOA obtains the second rank. F1 and F2 are simple functions without any local solution, MOWOA is only worse than OEWOA for F1, but better than other algorithms for F2. F3, F4, and F5 are the nonseparable unimodal functions, MOWOA is only worse than DEWOA for F3 and F4, and the results obtained by four group algorithms are comparable for F5. For F7 and F19, the result of MOWOA is approximately equal to OWOA and OEWOA, respectively.For OWOA and OEWOA, the results of F8 and F1 are better than MOWOA, respectively. Because both OWOA and OEWOA introduce into the opposition-based learning method, which expends the search space of the initial population, but as the complexity increases, the stability of OWOA and OEWOA decreases. For DEWOA, the results of F3 and F4 are better than MOWOA. However, the results of other functions are also not ideal. Because of DE operator has better self-adaptive optimization ability, its stability is affected for other functions with few local solutions. In MOWOA, an exploration mechanism of the honey badger and greedy strategy enhances the convergence speed and selection and is introduced into the nonlinear adaptive parameter and Cauchy factor to improve the case of falling in the local solution while making up for weakness.The Wilcoxon test results of MOWOA are given in Table5, where symbols “+,” “=,” and “−” represent that the proposed MOWOA is significantly better, similar, or worse than WOA, OWOA, OEWOA, and DEWOA, respectively. The significance level is set to 0.05. According to Table 5 data, the proposed MOWOA is better or statistically similar to WOA or its variants for most benchmark functions.Table 5
The Wilcoxon test results of MOWOA and WOA, OWOA, OEWOA, and DEWOA.
AlgorithmMOWOA-WOAMOWOA-OWOAMOWOA-OEWOAMOWOA-DEWOASymbol+/=/−+/=/−+/=/−+/=/−Result15/4/013/5/112/6/114/3/2To more intuitively validate and represent the performance of MOWOA, the convergence curves and boxplots are given in Figures1 and 2, respectively.Figure 1
Convergence graphs of WOA, OWOA, OEWOA, DEWOA, and MOWOA.
(a)(b)Figure 2
Boxplot of WOA, OWOA, OEWOA, DEWOA, and MOWOA.
(a)(b)In Figure1, the convergence curves of MOWOA are generally good for most benchmark functions. For F3 and F4, the convergence curves of MOWOA are gentler than that of DEWOA, but the rates are relatively slower. In Figure 2, MOWOA is generally stable for most benchmark functions. Only for F3 and F4, the upper and lower limits and median of MOWOA are worse than those of DEWOA. The above simulation results can demonstrate that MOWOA has the versatility and stability. The Y-axis labels 1–5 are WOA, OWOA, OEWOA, DEWOA, and MOWOA, respectively, in Figure 2.


### 3.3. Application of Seismic Inversion Problem

The prestack seismic inversion technology has an important significance for predicting the distribution of reservoir lithology and oil and gas exploration. This technique has some advantages that the prestack data can be made relatively undistorted and intact and contain richer information, the obtained accuracy is higher, and the P wave and S wave can be fully utilized. In recent years, with the development of swarm intelligence optimization algorithms, it has been more and more widely used in prestack seismic inversion.


#### 3.3.1. Seismic Inversion Problem

This section describes an 11-layer theoretical geological model, where there are 3 types of parameters in each layer, namely, velocity of longitudinal wave (VP), velocity of shear wave (VS), and density (ρ), respectively, and their values are given in Table 6.Table 6
Values of the velocity of the longitudinal and shear wave and density in each layer.
Layer numberVP (m·s−1)VS (m·s−1)ρ (g·cm−3)1198085122218516052.23198385624220016222.25243715102.36266313802.47198385128243715012.391980856210265613802.41119838512The zoeppritz equation of reflection coefficients is shown as follows:(13)RPRSTPTS=−sinθ1−cosφ1cosθ1−sinφ1sinθ2cosφ2cosθ2−sinφ2sin2θ1VP1VS1cos2φ1−cos2φ1VS1VP1sin2φ1ρ2VS22VP1ρ1VS12VP2cos2φ1ρ2VS2VP1ρ1VS12cos2φ2ρ2VP2ρ1VP1cos2φ2−ρ2VS2ρ1VP1sin2φ2−1sinθ1cosθ1sin2θ1cos2φ1,where VP1, VS1, and ρ1 are the up-layer parameters. VP2, VS1 and ρ2 are the low-layer parameters. θ1 and θ2, φ1 and φ2 represent the incident and refraction angle of the longitudinal wave and shear ware, respectively.φ1, θ2, and φ2 are as follows:(14)φ1=arcsinθ1×VS1VP1,(15)θ2=arcsinθ1×VS2VP1,(16)φ2=arcsinθ1×VP2VP1.The range of incident angleθ1 is between 1° and 50°. The mathematical expression of 50 seismic records calculated by convolution is given as (17)S=R∗Rw,where S, R, and Rw are the seismic record, the reflection coefficient, and the 35 HZ Ricker wavelet, respectively.We will divide the seismic records into three classes. Class-No. 1, Class-No. 2, and Class-No. 3 are used to record small incident angle from 1° to 15°, medium incident angle from 16° to 32°, and large incident angle from 33° to 50°, respectively. The three classes of information are synthesized to represent the superimposed seismic records byS1, S2, and S3, respectively. The fitness of this problem is mathematically presented as follows:(18)FX=13S1X−S12+S2X−S22+S3X−S32,where X is the solution of the superimposed seismic records Si(X) and the models are the same.


#### 3.3.2. Experiment Results

To validate the effectiveness of MOWOA on the seismic inverse problem, five algorithms are run independently 20 times. The experiment results are given as in Table7, and the convergence graph and boxplots are given as in Figure 3.Table 7
Experiment results of WOA, OWOA, OEWOA, DEWOA, and MOWOA on seismic inversion problem.
ProblemAlgorithmBestMeanStdSeismic inversionWOA0.0062781100.0147276220.004562819OWOA0.0035360200.0126910740.005694382OEWOA0.0052792860.0121866220.003115750DEWOA0.0062553600.0130829800.004423114MOWOA0.0017389710.0045028750.0018956875Figure 3
Results of WOA, OWOA, OEWOA, DEWOA, and MOWOA. (a) Convergence graph. (b) Boxplots.
(a)(b)The data values of MOWOA are all obviously smaller than other algorithms in Table7. In Figure 3(a), the convergence rate and curve of MOWOA are faster and more stable than other algorithms. And in Figure 3(b), MOWOA is more stable than other algorithms in terms of upper and lower limits and median of results. It can be demonstrated the usability and effectiveness of the MOWOA for the seismic inverse problem.


## 3.1. Parameter Setting and Benchmark Functions

In this part, the proposed MOWOA is appraised by nineteen benchmark functions F1– F19. F1– F7 are the unimodal functions that are given in Table1 to test the MOWOA’s exploitation ability in the case of only one global optimum. F8–F12 are the multimodal functions that are given in Table 2 to examine the MOWOA’s exploration tendency in the case of many local optima. These two sets of functions are employed in 30 dimensions. F13–F19 are the fixed-dimension multimodal functions that are given in Table 3 to evaluate the MOWOA’s convergence in the problem of the low dimensions.Table 1
Expression of F1–F7 benchmark functions.
FunctionsDimRangefminF1=∑i=1nxi230[−100, 100]0F2=∑i=1nxi+∏i=1nxi30[−10, 10]0F3=∑i=1n∑j=1ixj230[−100, 100]0F4=maxixi,1≤i≤n30[−100, 100]0F5=∑i=1n−1100xi+1−xi22+xi−1230[−30, 30]0F6=∑i=1nxi+0.5230[−100, 100]0F7=∑i=1nixi4+random0,130[−1.28, 1.28]0Table 2
Expression of F8–F12 benchmark functions.
FunctionsDimRangefminF8=∑i=1nxi2−10cos2πxi+1030[−5. 12. 5. 12]0F9=−20exp−0.21/n∑i=1nxi2−exp1/n∑i=1ncos2πxi+20+e30[−32, 32]0F10=1/4000∑i=1nxi2+∏i=1ncosxi/i+130[−600, 600]0F11=π/n10sinπy1+∑i=1n−1yi−121+10sin2πyi+1+yn−12+∑i=1nuxi,10,100,4yi=1+xi+1/4uxi,a,k,m=kxi−ama<xi0−a<xi<ak−xi−amxi<−a30[−50, 50]0F12=0.1sin23πxi+∑i=1n−1xi−121+sin23πxi+1+xn−12×1+sin22πxn+∑i=1nuxi,5,100,430[−50, 50]0Table 3
Expression of F13–F19 benchmark functions.
FunctionDimRangefminF13=1/500+∑j=1251/j+∑i=12xi−aij6−12[−65, 65]0F14=∑i=111ai+x1bi2+bix2/bi2+bix3+x424[−5, 5]0.0003F15=4x12−2.1x14+1/3x16+x1x2−4x22+4x242[−5, 5]−1.0316F16=x2−5.1/4π2x12+5/πx1−62+101−1/8πcosx1+102[−5, 5]0.398F17=1+x1+x2+1219−14x1+3x12−14x2+6x1x2+3x22×30+2x1−3x22×18−32x1+12x12+48x2−36x1x2+27x222[−2, 2]3F18=−∑i=14ciexp−∑j=13aijxj−pij23[1, 3]−3.86F19=−∑i=14ciexp−∑j=16aijxj−pij26[0, 3]−3.12where Dim, Range, andfmin are the dimension, the lower and upper bound, and the minimum value of the function, respectively.The proposed MOWOA will be compared with WOA [7] and its three variants OWOA [8], OEWOA [9], and DEWOA [11]. For the algorithms tested above, the number of population sizes and maximum iterations has been tuned as 30 and 500, respectively, while all tested algorithms are run independently 30 times with benchmark functions in Tables 1–3.


## 3.2. Result and Discussion

Table4 depicts the experimental results of WOA, OWOA, OEWOA, DEWOA, and MOWOA obtained among the nineteen functions, where Std is the standard deviation.Table 4
Experimental results of WOA, OWOA, OEWOA, DEWOA, and MOWOA on benchmark function.
FunctionAlgorithmBestMeanStdF1WOA2.83E − 844.28E − 742.29E − 73OWOA4.59E − 867.95E − 752.66E − 74OEWOA8.76E − 1035.13E − 881.59E − 87DEWOA6.77E − 311.94E − 284.52E − 28MOWOA4.98E − 992.15E − 838.78E − 83F2WOA8.71E − 572.85E − 519.40E − 51OWOA1.53E − 571.89E − 524.72E − 52OEWOA2.93E − 667.49E − 611.83E − 60DEWOA4.31E − 201.19E − 181.72E − 18MOWOA1.07E − 766.88E − 663.59E − 65F3WOA11389.118450275.467116562.2253OWOA26046.721747505.576413886.3143OEWOA12028.025939227.165917413.6989DEWOA0.002001407610.648203636.5865551MOWOA347.5717955457.181884555.64762F4WOA3.523351.610127.7077OWOA5.272845.476129.8808OEWOA4.549859.49726.6399DEWOA0.000304510.004807350.00478835MOWOA0.00711592.321512.72084F5WOA27.571928.14580.400962OWOA27.686128.23820.372579OEWOA27.752328.45840.308603DEWOA24.947228.816312.0021MOWOA25.643627.20241.04652F6WOA0.519651.16060.34138OWOA0.13640.596480.27012OEWOA0.157760.854470.26204DEWOA0.245730.821210.41509MOWOA0.000153930.337030.23659F7WOA3.35E − 050.00553250.0081194OWOA3.15E − 050.00166680.00317OEWOA4.25E − 050.00197360.0054556DEWOA0.00131550.00688320.0039475MOWOA1.85E − 050.00367180.0035788F8WOA03.78E − 152.07E − 14OWOA01.89E − 151.03E − 14OEWOA00.5282632.46821DEWOA5.6843E − 1434.412229.1644MOWOA04.9545317.8488F9WOA8.88E − 164.79E − 152.52E − 15OWOA8.88E − 164.79E − 153.14E − 15OEWOA8.88E − 164.79E − 152.52E − 15DEWOA5.77E − 142.26E − 132.53E − 13MOWOA8.88E − 164.55E − 152.37E − 15F10WOA00.00431590.023639OWOA000OEWOA00.0134090.073444DEWOA00.00748470.013641MOWOA00.000867610.003396F11WOA0.0116720.0643820.11014OWOA0.00778610.0328890.03719OEWOA0.0204970.0549750.047825DEWOA0.0187520.0764520.10195MOWOA0.00184310.03020.025889F12WOA0.473431.01960.34191OWOA0.232390.62260.32139OEWOA0.206960.785040.3463DEWOA0.36841.04570.2804MOWOA0.123390.598670.29991F13WOA0.9980038383.09043.4404OWOA0.9980038382.14961.9772OEWOA0.9980038382.34811.9499DEWOA0.9980038386.14014.9939MOWOA0.9980038382.05111.8885F14WOA0.000311470.000661580.00047928OWOA0.000315620.000699090.00048553OEWOA0.000307660.000774450.00056134DEWOA0.000307540.0101880.015558MOWOA0.000307530.000584420.00023566F15WOA−1.031628453−1.0316284534.06E − 09OWOA−1.031628453−1.0316284531.78E − 08OEWOA−1.031628453−1.0316284536.47E − 09DEWOA−1.031628453−1.0316284534.12E − 10MOWOA−1.031628453−1.0316284535.16E − 16F16WOA0.3978873580.39792442705.23E − 05OWOA0.3978873580.39790299102.65E − 05OEWOA0.3978873580.39797670400.00022822DEWOA0.3978873580.39788759834.54E − 07MOWOA0.3978873580.39788735770F17WOA33.00111920030.00520741OWOA33.00017804370.00025126OEWOA33.00018061540.00022324DEWOA36.600000309915.4264MOWOA33.00000014321.99E − 07F18WOA−3.862776732−3.8469485730.024831771OWOA−3.862782093−3.8518077120.025891037OEWOA−3.862781853−3.8378838760.054873368DEWOA−3.862782128−3.8614826920.002955750MOWOA−3.862782147−3.8619881950.002403123F19WOA−3.320772006−3.2376834240.14504OWOA−3.321595755−3.2425737150.091649OEWOA−3.319657692−3.2092441750.09022DEWOA−3.321993785−3.2582888270.087085MOWOA−3.321995171−3.2494305500.080041From experiment results, the performance of MOWOA is absolutely best for F2, F6, F9, and F11–F18. For F1, F3, F4, and F10, MOWOA obtains the second rank. F1 and F2 are simple functions without any local solution, MOWOA is only worse than OEWOA for F1, but better than other algorithms for F2. F3, F4, and F5 are the nonseparable unimodal functions, MOWOA is only worse than DEWOA for F3 and F4, and the results obtained by four group algorithms are comparable for F5. For F7 and F19, the result of MOWOA is approximately equal to OWOA and OEWOA, respectively.For OWOA and OEWOA, the results of F8 and F1 are better than MOWOA, respectively. Because both OWOA and OEWOA introduce into the opposition-based learning method, which expends the search space of the initial population, but as the complexity increases, the stability of OWOA and OEWOA decreases. For DEWOA, the results of F3 and F4 are better than MOWOA. However, the results of other functions are also not ideal. Because of DE operator has better self-adaptive optimization ability, its stability is affected for other functions with few local solutions. In MOWOA, an exploration mechanism of the honey badger and greedy strategy enhances the convergence speed and selection and is introduced into the nonlinear adaptive parameter and Cauchy factor to improve the case of falling in the local solution while making up for weakness.The Wilcoxon test results of MOWOA are given in Table5, where symbols “+,” “=,” and “−” represent that the proposed MOWOA is significantly better, similar, or worse than WOA, OWOA, OEWOA, and DEWOA, respectively. The significance level is set to 0.05. According to Table 5 data, the proposed MOWOA is better or statistically similar to WOA or its variants for most benchmark functions.Table 5
The Wilcoxon test results of MOWOA and WOA, OWOA, OEWOA, and DEWOA.
AlgorithmMOWOA-WOAMOWOA-OWOAMOWOA-OEWOAMOWOA-DEWOASymbol+/=/−+/=/−+/=/−+/=/−Result15/4/013/5/112/6/114/3/2To more intuitively validate and represent the performance of MOWOA, the convergence curves and boxplots are given in Figures1 and 2, respectively.Figure 1
Convergence graphs of WOA, OWOA, OEWOA, DEWOA, and MOWOA.
(a)(b)Figure 2
Boxplot of WOA, OWOA, OEWOA, DEWOA, and MOWOA.
(a)(b)In Figure1, the convergence curves of MOWOA are generally good for most benchmark functions. For F3 and F4, the convergence curves of MOWOA are gentler than that of DEWOA, but the rates are relatively slower. In Figure 2, MOWOA is generally stable for most benchmark functions. Only for F3 and F4, the upper and lower limits and median of MOWOA are worse than those of DEWOA. The above simulation results can demonstrate that MOWOA has the versatility and stability. The Y-axis labels 1–5 are WOA, OWOA, OEWOA, DEWOA, and MOWOA, respectively, in Figure 2.


## 3.3. Application of Seismic Inversion Problem

The prestack seismic inversion technology has an important significance for predicting the distribution of reservoir lithology and oil and gas exploration. This technique has some advantages that the prestack data can be made relatively undistorted and intact and contain richer information, the obtained accuracy is higher, and the P wave and S wave can be fully utilized. In recent years, with the development of swarm intelligence optimization algorithms, it has been more and more widely used in prestack seismic inversion.


### 3.3.1. Seismic Inversion Problem

This section describes an 11-layer theoretical geological model, where there are 3 types of parameters in each layer, namely, velocity of longitudinal wave (VP), velocity of shear wave (VS), and density (ρ), respectively, and their values are given in Table 6.Table 6
Values of the velocity of the longitudinal and shear wave and density in each layer.
Layer numberVP (m·s−1)VS (m·s−1)ρ (g·cm−3)1198085122218516052.23198385624220016222.25243715102.36266313802.47198385128243715012.391980856210265613802.41119838512The zoeppritz equation of reflection coefficients is shown as follows:(13)RPRSTPTS=−sinθ1−cosφ1cosθ1−sinφ1sinθ2cosφ2cosθ2−sinφ2sin2θ1VP1VS1cos2φ1−cos2φ1VS1VP1sin2φ1ρ2VS22VP1ρ1VS12VP2cos2φ1ρ2VS2VP1ρ1VS12cos2φ2ρ2VP2ρ1VP1cos2φ2−ρ2VS2ρ1VP1sin2φ2−1sinθ1cosθ1sin2θ1cos2φ1,where VP1, VS1, and ρ1 are the up-layer parameters. VP2, VS1 and ρ2 are the low-layer parameters. θ1 and θ2, φ1 and φ2 represent the incident and refraction angle of the longitudinal wave and shear ware, respectively.φ1, θ2, and φ2 are as follows:(14)φ1=arcsinθ1×VS1VP1,(15)θ2=arcsinθ1×VS2VP1,(16)φ2=arcsinθ1×VP2VP1.The range of incident angleθ1 is between 1° and 50°. The mathematical expression of 50 seismic records calculated by convolution is given as (17)S=R∗Rw,where S, R, and Rw are the seismic record, the reflection coefficient, and the 35 HZ Ricker wavelet, respectively.We will divide the seismic records into three classes. Class-No. 1, Class-No. 2, and Class-No. 3 are used to record small incident angle from 1° to 15°, medium incident angle from 16° to 32°, and large incident angle from 33° to 50°, respectively. The three classes of information are synthesized to represent the superimposed seismic records byS1, S2, and S3, respectively. The fitness of this problem is mathematically presented as follows:(18)FX=13S1X−S12+S2X−S22+S3X−S32,where X is the solution of the superimposed seismic records Si(X) and the models are the same.


### 3.3.2. Experiment Results

To validate the effectiveness of MOWOA on the seismic inverse problem, five algorithms are run independently 20 times. The experiment results are given as in Table7, and the convergence graph and boxplots are given as in Figure 3.Table 7
Experiment results of WOA, OWOA, OEWOA, DEWOA, and MOWOA on seismic inversion problem.
ProblemAlgorithmBestMeanStdSeismic inversionWOA0.0062781100.0147276220.004562819OWOA0.0035360200.0126910740.005694382OEWOA0.0052792860.0121866220.003115750DEWOA0.0062553600.0130829800.004423114MOWOA0.0017389710.0045028750.0018956875Figure 3
Results of WOA, OWOA, OEWOA, DEWOA, and MOWOA. (a) Convergence graph. (b) Boxplots.
(a)(b)The data values of MOWOA are all obviously smaller than other algorithms in Table7. In Figure 3(a), the convergence rate and curve of MOWOA are faster and more stable than other algorithms. And in Figure 3(b), MOWOA is more stable than other algorithms in terms of upper and lower limits and median of results. It can be demonstrated the usability and effectiveness of the MOWOA for the seismic inverse problem.


## 3.3.1. Seismic Inversion Problem

This section describes an 11-layer theoretical geological model, where there are 3 types of parameters in each layer, namely, velocity of longitudinal wave (VP), velocity of shear wave (VS), and density (ρ), respectively, and their values are given in Table 6.Table 6
Values of the velocity of the longitudinal and shear wave and density in each layer.
Layer numberVP (m·s−1)VS (m·s−1)ρ (g·cm−3)1198085122218516052.23198385624220016222.25243715102.36266313802.47198385128243715012.391980856210265613802.41119838512The zoeppritz equation of reflection coefficients is shown as follows:(13)RPRSTPTS=−sinθ1−cosφ1cosθ1−sinφ1sinθ2cosφ2cosθ2−sinφ2sin2θ1VP1VS1cos2φ1−cos2φ1VS1VP1sin2φ1ρ2VS22VP1ρ1VS12VP2cos2φ1ρ2VS2VP1ρ1VS12cos2φ2ρ2VP2ρ1VP1cos2φ2−ρ2VS2ρ1VP1sin2φ2−1sinθ1cosθ1sin2θ1cos2φ1,where VP1, VS1, and ρ1 are the up-layer parameters. VP2, VS1 and ρ2 are the low-layer parameters. θ1 and θ2, φ1 and φ2 represent the incident and refraction angle of the longitudinal wave and shear ware, respectively.φ1, θ2, and φ2 are as follows:(14)φ1=arcsinθ1×VS1VP1,(15)θ2=arcsinθ1×VS2VP1,(16)φ2=arcsinθ1×VP2VP1.The range of incident angleθ1 is between 1° and 50°. The mathematical expression of 50 seismic records calculated by convolution is given as (17)S=R∗Rw,where S, R, and Rw are the seismic record, the reflection coefficient, and the 35 HZ Ricker wavelet, respectively.We will divide the seismic records into three classes. Class-No. 1, Class-No. 2, and Class-No. 3 are used to record small incident angle from 1° to 15°, medium incident angle from 16° to 32°, and large incident angle from 33° to 50°, respectively. The three classes of information are synthesized to represent the superimposed seismic records byS1, S2, and S3, respectively. The fitness of this problem is mathematically presented as follows:(18)FX=13S1X−S12+S2X−S22+S3X−S32,where X is the solution of the superimposed seismic records Si(X) and the models are the same.


## 3.3.2. Experiment Results

To validate the effectiveness of MOWOA on the seismic inverse problem, five algorithms are run independently 20 times. The experiment results are given as in Table7, and the convergence graph and boxplots are given as in Figure 3.Table 7
Experiment results of WOA, OWOA, OEWOA, DEWOA, and MOWOA on seismic inversion problem.
ProblemAlgorithmBestMeanStdSeismic inversionWOA0.0062781100.0147276220.004562819OWOA0.0035360200.0126910740.005694382OEWOA0.0052792860.0121866220.003115750DEWOA0.0062553600.0130829800.004423114MOWOA0.0017389710.0045028750.0018956875Figure 3
Results of WOA, OWOA, OEWOA, DEWOA, and MOWOA. (a) Convergence graph. (b) Boxplots.
(a)(b)The data values of MOWOA are all obviously smaller than other algorithms in Table7. In Figure 3(a), the convergence rate and curve of MOWOA are faster and more stable than other algorithms. And in Figure 3(b), MOWOA is more stable than other algorithms in terms of upper and lower limits and median of results. It can be demonstrated the usability and effectiveness of the MOWOA for the seismic inverse problem.


## 4. Conclusions

In this paper, a multioperator WOA (MOWOA) has been proposed. This algorithm introduces four strategies into WOA to enhance performance, balance the proportion of search phase with nonlinear adaptive parameter design, improve convergence speed with honey badger, and jump out of the local optimum with Cauchy factor and greedy strategy. To validate the proposed MOWOA’s capacity to find optimal solutions, simulation experiments are carried out with algorithms by using three different types of benchmark functions. Through data analysis and comparison, the performance of MOWOA is better or approximately equal to WOA, OWOA, OEWOA, and DEWOA, which demonstrate its versatility and stability. Moreover, for the seismic inversion problem, the experimental result of MOWOA is superior to WOA, OWOA, OEWOA, and DEWOA, and it demonstrates MOWOA’s usability and effectiveness in the seismic inversion problem. Our future direction will be to study more operators or strategies to investigate the effect of MOWOA and further enhance the performance of our proposed algorithm.



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*Source: 1966054-2022-07-15.xml*