**1. Introduction**

The magnetotelluric (MT) technology is a geophysical electromagnetic detection method that uses electromagnetic induction signals to detect underground electrical structures [1,2]. The horizontal magnetic field is vertically incident into the Earth, which produces a time-harmonic changing induced electromagnetic field in the ground. When the excitation field source is constant, the electromagnetic field induced in the Earth is determined by the underground electrical structure and frequency [3]. Calculating the induced electromagnetic signal based on the electrical structure and frequency constitutes MT forward modeling, and this process satisfies the Maxwell equations. The process of calculating the geoelectric structure according to the induced electromagnetic signal and frequency is the MT inversion, which is implemented by the optimization method [4].

In the optimization process, the electrical structure is used as the optimization parameter to find the smallest objective function, and the difference between the predicted electromagnetic signal and the observed signal is evaluated by the objective function [5]. When only surface electromagnetic signals can be obtained, the inversion problem is severely underdetermined and has multiple solutions. Model roughness is commonly added as a Lagrangian penalty term to the objective function to address ambiguity [6,7].

**Citation:** Li, R.; Gao, L.; Yu, N.; Li, J.; Liu, Y.; Wang, E.; Feng, X. Memetic Strategy of Particle Swarm Optimization for One-Dimensional Magnetotelluric Inversions. *Mathematics* **2021**, *9*, 519. http://doi.org/10.3390/math9050519

Academic Editor: Amir H. Alavi

Received: 23 January 2021 Accepted: 24 February 2021 Published: 2 March 2021

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However, due to the serious nonlinearity of the MT inversion problem, the commonly used gradient optimization method is slow in the optimization process, and the optimal solution is not accurate. Nonlinear optimization methods based on intelligent algorithms often have better results in solving such nonlinear problems [8,9].

Heuristic algorithms are commonly used to solve such nonlinear problems [10,11]. Several common algorithms, including the simulated annealing method, the Bayesian inversion method and genetic algorithm, have been able to initially solve the MT inversion problem and determine the underground electrical structure through the electromagnetic response signal of the MT method [12,13]. Among these heuristic swarm intelligence algorithms, the particle swarm optimization (PSO) algorithm is widely used in the MT inversion due to its simple implementation and less adjustment parameters [14,15]. With the introduction of the inertia weight factor, the time-varying acceleration factor strategy and the strategy based on reproduction and subgroup hybridization, the shortcomings of PSO—that it easily falls into local extremes and has slow convergence in the later stages of evolution—are gradually improved [16–18]. However, these algorithms still have not overcome the shortcomings of the lack of population diversity and the uncoordination of individual cognition and social cognition capabilities.

With the development of memetic strategies, which take the process of memetic evolution as inspiration, using interactions between intelligent individuals to achieve population evolution and memetic evolution has become an important tool for enhancing population diversity and coordinating individual cognition and social cognition [19,20]. For the MT inversion problem, our strategy is to calculate the cognitive attraction coefficient through sine-cosine mapping to balance individual cognition and social cognition in the optimization process. Then, to further improve convergence in the optimization process and the ability to escape local extremes, we use dynamic inertia weights (DIWs) to integrate the previous experience of the population into the evolutionary process, and we use genetic mutations to enrich the diversity of the population.

Our contributions to the MT inversion with PSO optimization are as follows:


In the remainder of this paper, we first review the background of MT inversions based on PSO in Section 2. Then, we present the proposed memetic strategy in detail in Section 3. This section mainly focuses on the main framework of the memetic strategy, introduces population initialization, uses DIWs to integrate empirical cognition, and uses the cognitive attraction coefficient to accelerate population evolution and population mutation (PM). In Section 4, the inversion effects of the proposed memetic strategy on different geoelectric models are presented. Subsequently, in Section 5, we evaluate the stability of the memetic strategy using a noise immunity test and an actual data test. Finally, conclusions are drawn in Section 6.

#### **2. PSO for 1D MT Inversions**
