(College of Marine Earth Sciences, Qingdao Ocean University, Qingdao 266003)

He Qiao Deng

(Department of Geophysics, Changchun Institute of Geology, Changchun 130026)

Parameter inversion of anisotropic media is usually a nonlinear optimization problem. Nonlinear inversion methods can be divided into two categories: random search methods, such as Monte Carlo method, simulated annealing and genetic algorithm, and nonlinear analytical inversion methods based on nonlinear least square theory. Genetic algorithm can find the global optimal solution, but it is a time-consuming method. The nonlinear analytical inversion method can give the local optimal solution related to the initial model. However, this method converges quickly. The inversion method combining genetic algorithm with nonlinear analytical inversion method makes use of the advantages of these two inversion methods and overcomes their disadvantages. Therefore, the combined inversion method can not only converge quickly, but also find the global optimal solution. How to combine genetic algorithm with nonlinear analytical inversion method reasonably is very important. In this paper, a combination scheme is proposed, that is, nonlinear analytical inversion is carried out after multiple iterations of genetic algorithm. Theoretical examples show that the combined inversion method has the above characteristics.

Genetic algorithm; Nonlinear analytical inversion; Nonlinear combination inversion of anisotropic media

1 Introduction

Genetic algorithm is a random search method based on probability theory, which is used to solve multi-extreme complex optimization problems.

Gradient is an element in the dual space of model space. The model space and its dual space are related to the covariance operator Cm=Diag(Cp, Cc) of model parameters by formula (4d). The expressions of and will be given later.

Equation (4) is the basic formula of gradient inversion method. When each quantity in the formula is known, it can be iterated. Among these variables, the gradient vector is the most critical.

2.2 objective function

In the least squares theory, the weight function is the core of covariance operator inverse. Assuming that the error in the data set is irrelevant, it only depends on the time or the position of the source and receiver, then there is, satisfying the end time condition.

3 Combined inversion method

3. 1 Advantages and disadvantages of genetic algorithm and nonlinear analytical inversion method

Genetic algorithm (GA) is a random search method which uses probability theory to solve multi-extreme complex optimization problems. It starts with a group of randomly selected models and samples the optimal part of the model space widely and effectively without more prior information. Although genetic algorithm is based on natural selection and genetic law, searching the optimal part of model space to get the optimal solution, it is a method with a large amount of calculation. Because of the large space of seismic model, it is very time-consuming to estimate the parameters of anisotropic media by global optimization method.

The gradient information of objective function is the basis of modifying model parameters by nonlinear analytical inversion method, which can give a local optimal solution close to the initial model. If the initial model is selected properly, that is, when the initial model is at the low point of the objective function where the global optimal solution is located, the nonlinear analytical inversion method can converge to the global optimal solution. However, the probability of only giving an initial model close to the global optimal solution is very small, especially without any prior information of model parameters. However, it should be emphasized that the nonlinear analytical inversion method has a fast convergence speed.

It is of great significance to give full play to the advantages of fast convergence of nonlinear analytical inversion method and global optimal solution of genetic algorithm, and overcome the shortcomings of the former that only local optimal solution can be found and the latter that requires a large amount of calculation. The combination of nonlinear analytical inversion method and genetic algorithm can achieve the above objectives. In the combined inversion method, the function of genetic algorithm is to provide a model close to the global optimal solution, and the function of nonlinear analytical inversion is to find the global optimal solution as soon as possible. Therefore, the combined inversion method has the ability to search the global optimal solution and the characteristics of faster convergence than genetic algorithm.

3.2 Combination scheme

Genetic algorithm constantly searches the whole model space in the optimization process. At the end of each iteration, the optimal model of this generation is obtained. According to the mathematical principle of genetic algorithm [3], the number of optimal models is increased in the next generation, and new models are generated through crossover and mutation. In the next generation population, the optimal model may be the same as or worse than the previous generation. All these optimal models may be at the same low point of the objective function or at other low points. Genetic algorithm needs many iterations to find the optimal model to determine the extreme value. The randomness of genetic algorithm makes genetic algorithm a time-consuming method. However, it is the randomness of genetic algorithm that ensures that it can search for the global optimal solution.

If the optimal solution of each iteration of genetic algorithm is taken as the initial model of nonlinear analytical inversion, the local optimal solution adjacent to the initial model can be found through nonlinear analytical inversion. Because nonlinear analytical inversion is a deterministic method, it modifies the model according to the gradient direction of the objective function, so the nonlinear analytical inversion method can converge only in a few iterations. Whether the solution obtained by nonlinear analytical inversion is the global optimal solution cannot be guaranteed by the nonlinear analytical inversion method itself. Only when the genetic algorithm provides an initial model close to the global optimal solution can the nonlinear analytical inversion converge to the global optimal solution.

It is very important to combine the matching method of genetic algorithm with the nonlinear analytical inversion method in the inversion method. After the nonlinear analytical inversion method obtains the local optimal solution close to the initial model provided by genetic algorithm, due to the randomness of genetic algorithm, its optimal solution is the same as the local optimal solution of future generations. If nonlinear analytical inversion is carried out after each iteration of genetic algorithm, the results of several generations of combined inversion are the same. Obviously, some nonlinear analytical inversion is unnecessary. Therefore, the combination method should be nonlinear analytical inversion after multiple iterations of genetic algorithm, and then the results of nonlinear analytical inversion should be used as the female parent model in the next generation population. Figure 1 is a block diagram of combinatorial inversion.

Figure 1 combined inversion block diagram

Four examples

In order to verify the superiority of the joint inversion method, an example of one-dimensional multi-layer transversely isotropic medium parameter inversion theory is analyzed.

Fig. 2 is a graph showing the relationship between the objective function value and the number of iterations. In this example, a nonlinear analytical inversion iteration is performed after each iteration of the genetic algorithm. The error of joint inversion decreases rapidly in the previous iterations, especially in the first three iterations. Combined with the small error achieved by the inversion method in the 10 iteration, the genetic algorithm only achieved it in the 42nd iteration. The error of combined inversion is greater than that of genetic algorithm. This is because the model obtained by nonlinear analytical inversion participates in reproduction as a parent in genetic algorithm. Because of the randomness of genetic algorithm, this model is often replaced by new models. These two models may be located in two different valleys of the objective function, so the results of nonlinear analytical inversion are different.

Although the objective function combined with inversion oscillates a little, there is also a phenomenon that the objective function is almost unchanged after several iterations. This means that the optimal models of these iterations are very close. In this case, nonlinear analytical inversion can not provide great improvement. Therefore, there is no need for nonlinear analytical inversion at this time, otherwise it will only increase the amount of calculation.

Fig. 2 shows the relationship between the errors of inversion (solid line) and genetic algorithm (dotted line) and the number of iterations.

In the inversion, nonlinear analytical inversion iteration is performed after each iteration of genetic algorithm.

Fig. 3 is another example. In this combined inversion example, a nonlinear analytical inversion is performed every five iterations of the genetic algorithm. Genetic algorithm is dominant here. At this time, the error function of combined inversion is obviously smaller than that of genetic algorithm. Combining the sudden decrease of inversion error at the end of the fifth iteration and the small error at the 10 iteration, the genetic algorithm was only achieved in the 42nd generation. Genetic algorithm has never reached the minimum error of combinatorial inversion. The error of combined inversion decreases steadily in the later iteration, which is the reason why genetic algorithm is dominant.

It can be seen from this example that if the genetic algorithm is reasonably combined with the nonlinear analytical inversion method, the convergence speed of the combined inversion method is much faster than that of the genetic algorithm.

5 conclusion

The nonlinear combination inversion method improves the advantages of genetic algorithm and nonlinear analytical inversion method, and suppresses its disadvantages. This is a global inversion method with fast convergence speed.

The combination of genetic algorithm and nonlinear analytical inversion method is an important content in combinatorial inversion. It can be seen from the example that the effects of nonlinear analytical inversion and combined inversion after five iterations of genetic algorithm are obviously better than those after each iteration of genetic algorithm. However, the number of continuous iterations of genetic algorithm in combinatorial inversion and the variability of continuous iterations in the whole iteration process need to be further studied.

Fig. 3 shows the relationship between the errors of inversion (solid line) and genetic algorithm (dotted line) and the number of iterations.

Combined with inversion, nonlinear analytical inversion iteration is performed after every five iterations of genetic algorithm.

The function of genetic algorithm in combined inversion is to provide an initial model close to the global optimal solution. The operation speed of joint inversion mainly depends on the operation speed of genetic algorithm. Uniform design theory can be applied to genetic algorithm to speed up random search.

Like genetic algorithm, other random search methods can also be used to form an inversion method combined with nonlinear analytical inversion method.

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