Joint inversion for magnetotelluric data based on cuckoo search algorithm and least squares method

Joint inversion for magnetotelluric data based on cuckoo search algorithm and least squares method | Research Square window.SnipcartSettings = { analytics: { enabled: false } }; (function() { var accessVector = localStorage.getItem('access_vector') || ''; window.dataLayer = window.dataLayer || []; if (accessVector) { window.dataLayer.push({ user: { profile: { profileInfo: { snid: accessVector } } } }); } })(); (function(w,d,s,l,i){w[l]=w[l]||[];w[l].push({'gtm.start':new Date().getTime(),event:'gtm.js'});var f=d.getElementsByTagName(s)[0],j=d.createElement(s),dl=l!='dataLayer'?'&l='+l:'';j.async=true;j.src='https://www.googletagmanager.com/gtm.js?id='+i+dl;f.parentNode.insertBefore(j,f);})(window,document,'script','dataLayer','GTM-K279D39R'); Browse Preprints In Review Journals COVID-19 Preprints AJE Video Bytes Research Tools Research Promotion AJE Professional Editing AJE Rubriq About Preprint Platform In Review Editorial Policies Our Team Advisory Board Help Center Sign In Submit a Preprint Cite Share Download PDF Research Article Joint inversion for magnetotelluric data based on cuckoo search algorithm and least squares method Ruiyou Li, Yong Zhang, Xiaohui Ding, Min Li, Long Zhang This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-4023456/v1 This work is licensed under a CC BY 4.0 License Status: Posted Version 1 posted You are reading this latest preprint version Abstract The main drawback of using traditional linear or quasi-linear methods for magnetotelluric (MT) data inversion is its strong dependence on the initial model, which leads to its tendency to fall into a local optimum. However, the lack of sufficient a priori information makes it difficult to obtain suitable initial models in most geophysical inversions. To solve this problem, a joint approach (CS-MLS) based on the cuckoo search (CS) algorithm and the moving least squares (MLS) method is proposed to perform MT data inversion. The CS algorithm has a good global search performance without being sensitive to the initial model, which is used for the initial search of the stratum model parameter space. The search results are then used as the initial model to explore the quasi-linear inversion based on MLS. The synthetic and field data experiments show that the reliable initial model provided by the CS algorithm has greatly refined the results of the MLS inversion. The joint CS-MLS algorithm has great inversion performance, with significant improvement in inversion efficiency and accuracy, and is especially suitable for MT data inversion without initial information. Magnetotelluric cuckoo search algorithm least squares method inversion Figures Figure 1 Figure 2 Figure 3 Figure 4 Figure 5 Figure 6 Figure 7 Figure 8 Figure 9 1 Introduction The magnetotelluric (MT) method is widely used in geothermal resource exploration [ 1 ], oil and gas exploration [ 2 ], mineral exploration [ 3 ], and engineering geological survey [ 4 ], which is one of the most important techniques in deep earth studies [ 5 ]. Inversion algorithms have been developed to perform MT data preprocessing [ 6 – 8 ], which is a key aspect for MT exploration data interpretation. Currently, most MT inversion methods are based on linear or nonlinear iterative inversion, and the least squares method (LSM) inversion is one of the most widely used traditional algorithms with fast speed in the late stage [ 9 – 11 ], but the LSM inversion is heavily dependent on the initial model [ 12 ]. An efficient initial model gives high quality results; conversely, an irrational initial model can cause the inversion iterations to fall into the local optimum. Therefore, the initial model quality affects the LSM inversion results. Cuckoo search (CS) algorithm is a novel intelligent optimization algorithm proposed by Yang [ 13 ], which is inspired by the parasitic breeding habit of cuckoos, and it adopts the Lévy flight mode to update the individuals, which can effectively jump out of the local optimum. With the advantages of few parameters, simple operation, fast convergence and strong global optimization ability, the CS algorithm has been successfully applied to a variety of fields, such as multi-objective optimization [ 14 , 15 ], image processing [ 16 , 17 ], resource allocation [ 18 , 19 ], control problems [ 20 , 21 ], and computer vision [ 22 ]. Recently, the CS algorithm has been successfully applied in the geophysical inversion [ 23 – 24 ], which does not depend on either the initial model or the gradient information. Satisfactory inversion results can be obtained by the CS algorithm with a stochastic initial model, but at the cost of problems such as cumbersome computation and slow convergence. Therefore, the CS method is not suitable for practical applications involving a large number of model parameters. In order to obtain satisfactory inversion results in a shorter time, a joint inversion method combining the CS algorithm and LSM inversion has been proposed, in which the stability and non-uniqueness of the inversion can be improved by the a priori constraints provided by the CS algorithm to reduce the dependence of the MT data inversion on the initial model. The traditional LSM is a global approximation approach, which is not applicable to problems with large data volume and irregular or scattered distribution. Therefore, a moving least squares (MLS) method with local approximation is studied for MT data inversion, which has a fast convergence rate in the later stage. In our paper, a joint CS-MLS inversion method is constructed, which consists of two main steps: first, the CS algorithm is utilized to search the model parameter within a limited iterations; second, the CS search results are used as the initial model to perform MLS inversion to achieve better MT data inversion effects than a single MLS algorithm [ 25 ]. As a global nonlinear optimization algorithm, the CS does not need to give the initial value manually and has strong optimization adaptability, but its fitting efficiency decreases in the late stage. The MLS method is more stable and faster in the late stage, but it needs to give the initial value manually. To fully combine the advantages of the two algorithms, a joint CS-MLS inversion approach is proposed to improve the inversion efficiency. Based on the joint and single algorithms, synthetic MT data inversion is performed to verify the inversion accuracy and anti-noise performance of the CS-MLS method, and then field data inversion interpretation is performed via the joint algorithm to prove its practicality and effectiveness. 2. Methodology 2.1 MT forward modeling theory MT sounding is an electromagnetic exploration method that uses natural electromagnetic fields as field source signals, which is also required to satisfy the a priori assumptions of Carnia's classical theory [36]. Assuming that the earth structure is composed of horizontally layered media and the layer model profile is uniformly distributed, a 1-D model with N -layered media is shown in Fig. 1 , from which it can be seen that the 1-D media model can be represented by 2 N -1 dimensional model vectors: $${\mathbf{\vec {m}}}={({\rho _1},{\rho _2}, \cdots ,{\rho _N},{h_1},{h_2}, \cdots ,{h_{N - 1}})^T}$$ 1 where \({\rho _i}\) and \({h_i}\) are resistivity and thickness, respectively. For the layered media model, the apparent resistivity and impedance phase at the surface observation points with the following equations: $${\rho _a}(\omega )=\frac{{{{\left| {Z(\omega )} \right|}^2}}}{{\omega \mu }},{\text{ }}{\phi _a}=\arctan \frac{{\operatorname{Im} (Z)}}{{\operatorname{Re} (Z)}}$$ 2 where the angular frequency is denoted as \(\omega =2\pi T\) , the magnetic permeability is taken as \(\mu\) , and the surface wave impedance represents \(Z(\omega )\) , which is calculated by the following recursive formula: $${Z_i}={Z_{0i}}\frac{{{Z_{0i}}(1 - {e^{ - 2{k_i}{h_i}}})+{Z_{i+1}}(1+{e^{ - 2{k_i}{h_i}}})}}{{{Z_{0i}}(1+{e^{ - 2{k_i}{h_i}}})+{Z_{i+1}}(1 - {e^{ - 2{k_i}{h_i}}})}},{\text{ }}{{\text{Z}}_N}=\frac{{\omega \mu }}{{{k_N}}}={Z_{0N}}$$ 3 where \({k_i}=\sqrt {\frac{{i\omega \mu }}{{{\rho _i}}}}\) is the i -th layer complex wave number, \({Z_{0i}}\) is the i -th layer characteristic impedance, and \({Z_i}\) is the i -th layer wave impedance of the top surface. For the N -layer media model, the observed data as \({d^{obs}}\) , which is a 2 K dimensional vector as \({({\rho _1},{\rho _2}, \cdots ,{\rho _K},{\phi _1},{\phi _2}, \cdots ,{\phi _K})^T}\) ( K represents the number of frequency points observed on the surface), where \({\rho _i}\) and \({\phi _i}\) are the apparent resistivity and impedance phase from the i -th frequency point, respectively. To summarize, the forward modeling described by Eqs. ( 2 ) and ( 3 ) can be expressed as the following operator: $${\mathbf{\vec {d}}}=F({\mathbf{\vec {m}}})$$ 4 where \({\mathbf{\vec {m}}}\) denotes a 2 N -1 dimensional model parameters, \({\mathbf{\vec {d}}}\) represents a 2 K dimensional observation, and is a mapping operator from model space to data space. In addition, based on the given model parameters given as \({\mathbf{\vec {m}}}\) , the observed data \({\mathbf{\vec {d}}}\) can be calculated according to the recursive formulas of Eqs. ( 2 ) and ( 3 ). 2.2 CS-MLS inversion algorithm (1) The implementation of the newly proposed CS-MLS inversion algorithm is described in detail in this section. The CS algorithm is a swarm intelligence nonlinear global optimization technique [ 13 ], in which the positions of bird nests are initialized (the position represents a possible optimal solution), and then all the nests tend to move to the optimal position based on certain rules. For the MT inversion problem, the distance between the bird nest position and the optimal solution is taken as the objective function of the inversion [ 26 ]. First, the CS algorithm randomly initializes n nests in the search space as initial values, and then CS updates the nest positions in real time as follows: $${\mathbf{x}}_{i}^{{(t+1)}}={\text{ }}{\mathbf{x}}_{i}^{{(t)}}+\alpha {\mathbf{L}}\left( \beta \right)$$ 5 Then, Eq. ( 5 ) is the basic formula for the Lévy flight mode, where the number of nests is \(i=1,2, \cdots ,n\) , the number of iterations is \(t=1,2, \cdots ,ite{r_{max}}\) ( \(ite{r_{max}}\) denotes the maximum iterations), \({\mathbf{x}}_{i}^{{(t)}}\) and \({\mathbf{x}}_{i}^{{(t+1)}}\) present the nest positions in the t and t + 1 generations, respectively; \(\alpha\) denotes the step scaling factor, with a default value of 0.01; \(\beta\) represents the randomized step, which is usually assumed to be 1.5; and \({\mathbf{L}}\left( \beta \right)\) is the Lévy stochastic path, which is as follows: $${\mathbf{L}}\left( \beta \right)=\frac{{\vartheta {\mathbf{u}}}}{{{{\left| {\mathbf{v}} \right|}^{{1 \mathord{\left/ {\vphantom {1 \beta }} \right. \kern-0pt} \beta }}}}}$$ 7 where u and \(\nu\) are a random variable obeying a standard normal distribution and \(\vartheta\) is defined as follows. $$\vartheta ={\left[ {\frac{{\Gamma (1+\beta )\sin \frac{{\pi \beta }}{2}}}{{{2^{\frac{{\beta - 1}}{2}}}\Gamma (\frac{{1+\beta }}{2})\beta }}} \right]^{{1 \mathord{\left/ {\vphantom {1 \beta }} \right. \kern-0pt} \beta }}}$$ 8 where \(\Gamma (\cdot )\) is the gamma function. There are two ways for the CS algorithm to update the position of the bird's nest: The first is the Lévy flight model, which is a random walk that simulates the foraging process of flying animals. During the flight process, a large number of short-range searches alternate with occasional long-range jumps, in which the short-range searches are conducive to improving the local search ability, while the occasional long-range jumps make the CS algorithm less likely to fall into the local optimum [ 27 ]. The other is random search, which is updated by generating a random value between [0, 1] as \(\varepsilon\) for each nest, comparing it with the discovery probability P , and then determining whether to generate a new nest, as shown below: $${\mathbf{x}}_{i}^{{(t+1)}}={\text{ }}{\mathbf{x}}_{i}^{{(t)}}+\gamma H(P - \varepsilon )[{\mathbf{x}}_{j}^{{(t)}} - {\mathbf{x}}_{k}^{{(t)}}]$$ 9 where \(\gamma\) is a random number between [0, 1], \({\mathbf{x}}_{i}^{{(t)}}\) , \({\mathbf{x}}_{j}^{{(t)}}\) , \({\mathbf{x}}_{k}^{{(t)}}\) are the positions of the three random bird's nests in t generations. \(H(\cdot )\) is the Heaviside function, which has a value of 0 at \(P - \varepsilon >0\) , that \({\mathbf{x}}_{i}^{{(t+1)}}\) remains unchanged; the function value of 1 at \(P - \varepsilon <0\) , that \({\mathbf{x}}_{i}^{{(t+1)}}\) is randomly changed; the function value of 0.5 at \(P - \varepsilon =0\) , that \({\mathbf{x}}_{i}^{{(t+1)}}\) is slightly randomly changed. The CS algorithm is derived from biological group intelligence and its mathematical expression is simple and straightforward [ 28 ]. It does not need to manually specify the initial model in the optimization process, but it has poor convergence performance in the later stage. (2) The MLS algorithm was improved by Lancaster and Salkauskas [ 29 ], which is a locally approximated moving least squares method. The main difference between the traditional LSM and MLS methods are the coefficients, which vary with the variable parameters in MLS and are constant in LSM, making the MLS method particularly suitable for MT data fitting applications. In this study, an MLS inversion method optimized by the CS algorithm is proposed. The MLS method is computationally fast but requires suitable initial values; the CS algorithm does not require a given initial value but converges slowly in the later stages of iteration.The joint CS-MLS approach takes full advantage of both algorithms and improves the optimization efficiency. It is worth noting that the traditional CS requires a large number of iterations (possibly hundreds), while the joint inversion requires only a few iterations to ensure that the fitting error can be quickly reduced, further improving the inversion efficiency. The flowchart of CS-MLS is shown in Fig. 2 below. The inversion results via the CS algorithm are used as the initial model of MLS, then the fine inversion is performed, and finally the inversion is stopped by reaching a given fitting error ( E ) or maximum iterations ( T ). 3. Inversion of synthetic data 3.1 Objective function construction The MT inversion is an optimization problem whose objective is to predict a parametric model close to the real geoelectric structure based on the observed data. In other words, the parametric model \({{\mathbf{\vec {m}}}^{est}}\) is continuously estimated, so that the fitting error between the theoretical data \({{\mathbf{\vec {d}}}^{obs}}\) that is calculated by MT forward modeling and the actual observations \({{\mathbf{\vec {d}}}^{obs}}\) is minimized. Therefore, the objective function is constructed as follows: $$f({{\mathbf{\vec {m}}}^{est}})={\left\| {{{{\mathbf{\vec {d}}}}^{obs}} - F({{{\mathbf{\vec {m}}}}^{est}})} \right\|^2} \to \hbox{min}$$ 10 Since the objective function is highly nonlinear and multipolar, the CS algorithm with great global search capability is used to solve the MT inversion problem. The fitness in the CS optimization refers to the objective function in MT inversion, and usually the L 2 paradigm is adopted to define the misfit between the observed data and the predicted data. The fitness can be expressed as: $$fit={c_{rho}}fi{t_{rho}}+{c_{phi}}fi{t_{phi}}={c_{rho}}{\left\| {1 - {{{\rho _{pred}}} \mathord{\left/ {\vphantom {{{\rho _{pred}}} {{\rho _{obs}}}}} \right. \kern-0pt} {{\rho _{obs}}}}} \right\|^2}+{c_{phi}}{\left\| {1 - {{{\varphi _{pred}}} \mathord{\left/ {\vphantom {{{\varphi _{pred}}} {{\varphi _{obs}}}}} \right. \kern-0pt} {{\varphi _{obs}}}}} \right\|^2}$$ 11 where the total fitness consists of the resistivity fitness and the phase fitness ( \({c_{rho}}\) and \({c_{phi}}\) represent the weighting coefficients), which greatly reduces the non-uniqueness and improves the inversion accuracy and reliability. \({\rho _{pred}}\) and \({\rho _{obs}}\) are the predicted apparent resistivity and observed apparent resistivity, respectively; \({\varphi _{pred}}\) and \({\varphi _{obs}}\) are the predicted phase and observed phase, respectively. In addition, the observed and predicted data are normalized. It is worth noting that MLS inversion is a quasi linear iterative method whose objective function is also expressed as Eq. ( 11 ). 3.2 1-D MT model inversion To verify the superiority of the joint CS-MLS inversion method, typical three-layer (H-type) and five-layer (HKH-type) geoelectric models were developed for noise-free and noise-contaminated synthetic data inversion using the joint algorithm. CS inversion parameters: the number of nests as 40, the discovery probability as P = 0.25, and the maximum iterations as 30; MLS inversion parameters: the fitting error as E = 1.0e-5, and the maximum iterations as T = 30. All algorithms are simulated in Matlab R2022a, and the PCs are powered by an Intel(R) Core(TM) i7-12700F processor with speeds of 2.10 GHz and 16.0 G of RAM. (1)Noise-free model test To validate the effectiveness of the joint CS-MLS inversion method, noise-free experiments were conducted for two typical geoelectric models. Model 1 is a three-layer (H-type) model, which is shown as \({\rho _1}=100\) \(\Omega \cdot {\text{m}}\) , \({\rho _2}=20\) \(\Omega \cdot {\text{m}}\) , \({\rho _3}=100\) \(\Omega \cdot {\text{m}}\) ; \({h_1}=100\) m, \({h_2}=200\) m; Model 2 is a five-layer (HKH-type) model, which is shown as \({\rho _1}=100\) \(\Omega \cdot {\text{m}}\) , \({\rho _2}=20\) \(\Omega \cdot {\text{m}}\) , \({\rho _3}=200\) \(\Omega \cdot {\text{m}}\) , \({\rho _4}=50\) \(\Omega \cdot {\text{m}}\) , \({\rho _5}=100\) \(\Omega \cdot {\text{m}}\) ; \({h_1}=1000\) m, \({h_2}=500\) m, \({h_3}=1000\) m, \({h_4}=2000\) m. First, the joint algorithm inversion is performed on the noise-free synthetic data. The inversion results of the CS algorithm and random values within a certain range are used as initial models, respectively, and then the MLS inversion is performed until a certain fitting error or maximum iterations are reached. It should be noted that the MLS method is a local approximation algorithm, and an inappropriate initial model will lead to inversion distortion phenomena. In the single MLS inversion, the inversion is successful every time for the three-layer model; the inversion is successful 3 out of 10 times for the five-layer model. However, in the joint CS-MLS algorithm, the inversion is successful every time for both the three-layer and five-layer models. In the joint inversion algorithm, the CS algorithm inversion is performed first to obtain a good initial model for the subsequent MLS inversion, and the inversion results of the three-layer model and the five-layer model are shown in Figs. 3 and 4 . As can be seen from Figs. 3 (a) and 4(a), the results of the CS algorithm (blue dashed line) can provide an initial model that is roughly similar to the real model. Due to the reliable initial model, the CS-MLS inversion exhibits an excellent result (red dashed line with dots) that is quite close to the real model. Moreover, accurate inversion results are obtained for the low resistance layer of the three-layer model ( \({\rho _2}\) ) and the low-height-low resistance layer of the five-layer model ( \({\rho _2}\) , \({\rho _3}\) , \({\rho _4}\) ). The single MLS inversion results (green dashed line) show a large deviation from the true model, especially for the middle resistance layer. As shown in Figs. 3 (b)(c) and 4(b)(c), the synthetic data corresponding to the real model (apparent resistance and phase) and those calculated by the joint inversion model are in good agreement. However, the synthetic data calculated by the single CS and MLS inversion models deviate more slightly from the real model. Compared with the single CS and MLS algorithms, the joint CS-MLS inversion approach achieves a significant advantage, due to the fact that the CS algorithm does not require an initial model and has a strong global search ability, but has a lower fitting efficiency in the late stage, while the MLS algorithm is a local approximation method with fast convergence in the late stage, but requires an initial model manually. The joint inversion method fully utilizes the advantages of the two algorithms, which can improve the inversion efficiency. 2. Noise-contaminated model test To simulate the real field data containing noise, Gaussian noise of 5%, 10% and 20% was added to the synthetic MT data. The inversion error results by the CS-MLS method for noise-contaminated data are saved in Table 1 , where ε is the random noise added to the synthetic data d . As shown in Table 1 , the estimation error between the synthetic data and the predicted data \(F({\mathbf{m}})\) increases with the percentage of noise data, but the estimation error always remains in a small range. Therefore, the presence of noise data does not significantly affect the inversion performance of the joint algorithm. Table 1 Inversion error results for noise-contaminated data Model H-type HKH-type Noise \({{\left\| {F({\mathbf{m}}) - {\mathbf{d}}} \right\|} \mathord{\left/ {\vphantom {{\left\| {F({\mathbf{m}}) - {\mathbf{d}}} \right\|} {\left\| {\mathbf{d}} \right\|}}} \right. \kern-0pt} {\left\| {\mathbf{d}} \right\|}}\) \({{\left\| {\mathbf{\varepsilon }} \right\|} \mathord{\left/ {\vphantom {{\left\| {\mathbf{\varepsilon }} \right\|} {\left\| {\mathbf{d}} \right\|}}} \right. \kern-0pt} {\left\| {\mathbf{d}} \right\|}}\) \({{\left\| {F({\mathbf{m}}) - {\mathbf{d}}} \right\|} \mathord{\left/ {\vphantom {{\left\| {F({\mathbf{m}}) - {\mathbf{d}}} \right\|} {\left\| {\mathbf{d}} \right\|}}} \right. \kern-0pt} {\left\| {\mathbf{d}} \right\|}}\) \({{\left\| {\mathbf{\varepsilon }} \right\|} \mathord{\left/ {\vphantom {{\left\| {\mathbf{\varepsilon }} \right\|} {\left\| {\mathbf{d}} \right\|}}} \right. \kern-0pt} {\left\| {\mathbf{d}} \right\|}}\) 5% 0.0153 0.0664 0.0097 0.0713 10% 0.0398 0.1784 0.0770 0.1357 20% 0.0882 0.3241 0.1685 0.2725 The CS-MLS inversion results of the synthetic data with 5%, 10% and 20% noise are shown in Fig. 5 . As seen in Fig. 5 (a), the estimated model with different noise levels agrees well with the theoretical model for the three-layer geoelectric model; as seen in Fig. 5 (b), with the increase of noise data, the deviation of the estimated model from the theoretical model gradually increases for the five-layer geoelectric model, especially the high resistance layer, but it can basically reflect the real geoelectric model structure. The inversion results illustrate that the joint inversion method achieves a good suppression effect on the noise data and can basically find the global optimal solution. In addition, as the complexity of the geoelectric model increases, the degree of model mismatch increases, but it does not affect the global optimization performance of the CS-MLS inversion method. 3.3 Pseudo 2-D model inversion To further validate the feasibility of the CS-MLS algorithm for MT inversion, two typical pseudo 2-D geoelectric models are designed for testing: Model 1 is a horizontal double anomalous body model, and Model 2 is a vertical double anomalous body model. To simulate real field data, 5% random noise was added to the observed MT data obtained at each measurement point. ① Horizontal double anomalous body model The theoretical model and the location of the survey points are shown in Fig. 6 (a), with the horizontal direction as the survey line and the vertical direction as the survey depth. There are 10 survey points with a line length of 900 m and an interval of 100 m between the survey points. Both anomalous bodies have a shape of 100 m×100 m, the low-resistivity body as 20 Ω·m, and the high-resistivity body as 200 Ω·m. They are buried at a depth of 100 m from the top and both are located in a homogeneous half-space with 100 Ω·m. The inversion is performed using the proposed CS-MLS algorithm, and the resistivity profile of the obtained pseudo 2-D model is shown in Fig. 7(a). As can be seen, the anomalous bodies by the joint algorithm are basically consistent with the theoretical model, which has an accurate resistivity and position, an uniform resistivity distribution, and a significant boundary effect. Therefore, the inversion results show that the joint CS-MLS approach has great efficiency in the horizontal double anomalous body model. ② Vertical double anomalous body model The theoretical model and the location of the survey points are shown in Fig. 6 (b), with the horizontal direction as the survey line and the vertical direction as the survey depth. There are 22 survey points with a line length of 2100 m and an interval of 100 m between the survey points. Both anomalous bodies have a shape of 200 m×200 m, the high-resistivity body as 200 Ω·m buried at a depth of 200 m from the top; the low-resistivity body as 20 Ω·m buried at a depth of 700 m from the top, and they are located in a homogeneous half-space of 100 Ω·m. The inversion is performed using the proposed CS-MLS algorithm, and the resistivity profile of the obtained pseudo 2-D model is shown in Fig. 7(b). As can be seen, the anomalous bodies by the joint algorithm are roughly close to the theoretical model, the high-resistivity model has an uniform resistivity distribution, with accurate locations and clearer boundaries; the low-resistivity model has accurate locations and clearer boundaries, while the resistivity distributions are less uniform. Therefore, the inversion results show that the joint CS-MLS approach has great efficiency in the vertical double anomalous body model, which can be suitable for practical MT inversion application. The feasibility of the CS-MLS method in MT inversion is further verified by the theoretical layer model and the pseudo 2-D model example. The proposed approach achieves a high inversion efficiency, mainly due to the use of the global CS algorithm to obtain the initial value of MLS, which can effectively avoid it from falling into the local optimum, and has a strong global search capability and robustness. In addition, the joint inversion of CS and MLS methods will fully combine the advantages of the two algorithms, which can improve the inversion accuracy as well as the computational efficiency, and will show great potential in the MT inversion. 4. Field data study The COPROD2 dataset is a public dataset for testing the effectiveness of MT inversion, which contains field observation data [ 30 ]. However, the subsurface structures have not been accurately confirmed. The inversion effect can be evaluated by the difference in fitting between the predicted and measured data, and the prediction effect of the joint CS-MLS algorithm is compared with that of the single MLS algorithm. The COPROD2 data contains about 30 observations, four of which were randomly selected as test data. According to the polarization mode, the observations can be classified into XY mode and YX mode, and the prediction results for the two modes are shown in Fig. 8 and Fig. 9 . The prediction results of the joint CS-MLS method are significantly better than those of the single MLS algorithm in the four observations, especially the apparent resistivity inversion effects. For the fluctuating flat observation data, the prediction data by the joint method fit well with the measured data, but the prediction results by the single MLS algorithm have a large deviation and are only roughly consistent with the measured data in terms of tendency, as shown in Fig. 8 (a)(d)(e)(h). For the more fluctuating observation data, the prediction results of the two methods differ significantly from the measured data, in which the single MLS algorithm suffers from distortion, as shown in Fig. 8 (b)(c)(f)(g). Compared with the XY polarization model, the inversion of the YX polarization model is better, especially the apparent resistivity results, as shown in Fig. 9 . Considering the volume effect of the electromagnetic wave and the static effect near the surface, the response curve of the 1-D geoelectric model is difficult to fit perfectly with the measured data curve. In addition, the large fluctuation of field data is also affected by anthropogenic noise, magnetic storms and substation interference, which non-random noise increases the difficulty of inversion. Overall, the effectiveness of the joint CS-MLS inversion method in interpreting real MT data has been verified by the COPROD2 field dataset. 5 Conclusion The joint inversion method developed from the CS and MLS algorithms is suitable for the MT data inversion, which successfully utilizes the advantages of the two algorithms and solves the problems of the CS algorithm's poor convergence speed in the late stage and the MLS algorithm's heavy dependence on the initial model. In the joint inversion, the CS algorithm provides a reliable initial model, and then MLS is applied to the MT data inversion to obtain the final result. Thus, the accurate inversion result is obtained by the proposed approach even if an arbitrary initial model is provided initially. The inversion experiments show that the joint algorithm for MT data inversion is more efficient than the single algorithm, which exhibits excellent results in inversion accuracy and convergence speed. Additionally, the synthetic data and field data inversion confirm the high robustness and feasibility. Therefore, the joint CS-MLS approach is of great importance for actual data inversion without proper prior knowledge. Considering that 2-D or 3-D layer models are more consistent with the actual geology, the joint algorithm can be extended to high-dimensional inversion in the future. Declarations Author Contributions Ruiyou Li: Writing - original draft, Methodology and Data curation; Yong Zhang: Supervision and Validation; Xiaohui Ding: Methodology and Investigation; Min Li: Writing - original draft and Software; Long Zhang: Validation, Format modification, Manuscript layout and correction. Conflicts of Interests The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Data Availability Statement The data used to support the findings of this study are included within the article. Acknowledgments This work was supported by the National Natural Science Foundation of China (No.61762043), the China Postdoctoral Science Foundation (No.2023M741480), and the Scientific Research Foundation of Jiangxi Provincial Education Department, China (No. GJJ2200528). References Ardid A, Dempsey D, Bertrand E, et al. Bayesian magnetotelluric inversion using methylene blue structural priors for imaging shallow conductors in geothermal fields[J]. Geophysics, 2021, 86(3): E171-E183. Miri H, Dehkordi B H, Payrovian G. Oil field imaging on the Sarab Anticline, southwest of Iran, using magnetotelluric data[J]. Journal of Petroleum Science and Engineering, 2021, 202: 108497. 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Zhang L, Yu Y, Luo Y, et al. Improved cuckoo search algorithm and its application to permutation flow shop scheduling problem[J]. Journal of Algorithms & Computational Technology, 2020, 14: 1748302620962403. Nguyen T T, Pham L H, Mohammadi F, et al. Optimal scheduling of large-scale wind-hydro-thermal systems with fixed-head short-term model[J]. Applied Sciences, 2020, 10(8): 2964. Fan J, Xu W, Huang Y, et al. Application of chaos cuckoo search algorithm in computer vision technology[J]. Soft Computing, 2021, 25(18): 12373-12387. Poormirzaee R, Fister J I. Model-based inversion of Rayleigh wave dispersion curves via linear and nonlinear methods[J]. Pure and Applied Geophysics, 2021, 178: 341-358. Liu S, Duan Z, Zhou F, et al. Simultaneous inversion of petrophysical parameters of reservoir based on cuckoo search algorithm[J]. Oil Geophysical Prospecting, 2022, 57(3): 638-646. Tang B Q, Han J. Evaluation model of project investment risk based on particle swarm optimization improved least squares support vector machine[J]. Journal of Civil Engineering and Management, 2019 36 (2), 98-103. Marini F, Walczak B. Particle swarm optimization (PSO): A tutorial[J]. Chemometrics and Intelligent Laboratory Systems, 2015, 149: 153-165. Qi X, Yuan Z, Song Y. An integrated cuckoo search optimizer for single and multi-objective optimization problems[J]. PeerJ Computer Science, 2021, 7: 370. Santilano A, Godio A, Manzella A. Particle swarm optimization for simultaneous analysis of Magnetotelluric and time domain EM data[J]. Geophysics, 2018, 83(3):1-48. Salkauskas P L. Surfaces generated by moving least squares methods[J]. Math Compt, 1981, 37(155):141-158. Beusekom A, Parker R L, Bank R E, et al. The 2-D magnetotelluric inverse problem solved with optimization[J]. Geophysical Journal International, 2011, 184(2), 639-650. Additional Declarations No competing interests reported. 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Also discoverable on Platform About Our Team In Review Editorial Policies Advisory Board Help Center Resources Author Services Accessibility API Access RSS feed Manage Cookie Preferences © Research Square 2026 | ISSN 2693-5015 (online) Privacy Policy Terms of Service Do Not Sell My Personal Information {"props":{"pageProps":{"initialData":{"identity":"rs-4023456","acceptedTermsAndConditions":true,"allowDirectSubmit":true,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":277388374,"identity":"62c8bf62-91bc-4faa-9b2d-1b3498afd8a9","order_by":0,"name":"Ruiyou Li","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAA40lEQVRIie3RsQrCMBCA4SuFumi7nhT1FRRBHPowKUJnJ3EQLAhdXetbFArOFwpxCc51dHEXFwcRE8G1ZhTMDylJuY8UCmCz/WC+qx5MrQAIYKlf0RfifUg3VaPSiHw2Qz1pRlqdC56zKC65nCAtoe/XzLnNGz+sNcE4S+I9iQRJwrhbMzfMG4nnKVIpUonwmkFc1Mxz2yakTHmG/Alrc1JAJZCnwIYGxJ2yYzLOScymJHC0k+dN2ESCQDin+yLqbXM5qmkVDfzDjN+ayDtH/x0kvUV9TL8B1UPfZzJos9lsf9kLYhBL7rkR/lMAAAAASUVORK5CYII=","orcid":"","institution":"Jiangxi University of Finance and Economics","correspondingAuthor":true,"prefix":"","firstName":"Ruiyou","middleName":"","lastName":"Li","suffix":""},{"id":277388375,"identity":"c76dad83-c32a-488c-ae97-a0b29b1951a8","order_by":1,"name":"Yong Zhang","email":"","orcid":"","institution":"Jiangxi University of Finance and Economics","correspondingAuthor":false,"prefix":"","firstName":"Yong","middleName":"","lastName":"Zhang","suffix":""},{"id":277388376,"identity":"de5d0db7-fc37-495a-a925-d43d988ac960","order_by":2,"name":"Xiaohui Ding","email":"","orcid":"","institution":"Jiangxi University of Finance and Economics","correspondingAuthor":false,"prefix":"","firstName":"Xiaohui","middleName":"","lastName":"Ding","suffix":""},{"id":277388377,"identity":"edb3546d-e1b7-479e-a155-a7ec69705857","order_by":3,"name":"Min Li","email":"","orcid":"","institution":"Jiangxi University of Finance and Economics","correspondingAuthor":false,"prefix":"","firstName":"Min","middleName":"","lastName":"Li","suffix":""},{"id":277388378,"identity":"eb9567d6-aa31-4d1d-bb10-5826782346d2","order_by":4,"name":"Long Zhang","email":"","orcid":"","institution":"Jiangxi University of Finance and Economics","correspondingAuthor":false,"prefix":"","firstName":"Long","middleName":"","lastName":"Zhang","suffix":""}],"badges":[],"createdAt":"2024-03-07 08:00:34","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-4023456/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-4023456/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":52602553,"identity":"42915beb-0da5-40ec-9443-5f185e0e821d","added_by":"auto","created_at":"2024-03-13 13:08:45","extension":"png","order_by":1,"title":"Figure 1","display":"","copyAsset":false,"role":"figure","size":8935,"visible":true,"origin":"","legend":"\u003cp\u003e1-D layered media model\u003c/p\u003e","description":"","filename":"1.png","url":"https://assets-eu.researchsquare.com/files/rs-4023456/v1/3783e1cc01881fb1cadb2418.png"},{"id":52604608,"identity":"b7a1d282-db0f-4693-8f94-c03aed6d71f9","added_by":"auto","created_at":"2024-03-13 13:24:45","extension":"png","order_by":2,"title":"Figure 2","display":"","copyAsset":false,"role":"figure","size":672577,"visible":true,"origin":"","legend":"\u003cp\u003eFlowchart of the CS-MLS algorithm\u003c/p\u003e","description":"","filename":"2.png","url":"https://assets-eu.researchsquare.com/files/rs-4023456/v1/ff93b5e3613ee36dfb91f41b.png"},{"id":52602555,"identity":"e861e6ef-9745-4b25-a25a-7ba898c0b222","added_by":"auto","created_at":"2024-03-13 13:08:45","extension":"png","order_by":3,"title":"Figure 3","display":"","copyAsset":false,"role":"figure","size":94788,"visible":true,"origin":"","legend":"\u003cp\u003eInversion results calculated by the CS, MLS and CS-MLS. (a) Inversion structure of geoelectric model for three-layer (H-type) . (b) Apparent resistivity responses. (c) Phase responses.\u003c/p\u003e","description":"","filename":"3.png","url":"https://assets-eu.researchsquare.com/files/rs-4023456/v1/227ffc5d8ccb207521226efe.png"},{"id":52604609,"identity":"7711115d-8385-47af-9792-fd6422154e2b","added_by":"auto","created_at":"2024-03-13 13:24:45","extension":"png","order_by":4,"title":"Figure 4","display":"","copyAsset":false,"role":"figure","size":104791,"visible":true,"origin":"","legend":"\u003cp\u003eInversion results calculated by the CS, MLS and CS-MLS. (a) Inversion structure of geoelectric model for five-layer (HKH-type) . (b) Apparent resistivity responses. (c) Phase responses.\u003c/p\u003e","description":"","filename":"4.png","url":"https://assets-eu.researchsquare.com/files/rs-4023456/v1/a58690ca8bc9dc139d9ff9b2.png"},{"id":52603820,"identity":"1f090b2e-a7e2-494a-b84b-e23ee657a044","added_by":"auto","created_at":"2024-03-13 13:16:45","extension":"png","order_by":5,"title":"Figure 5","display":"","copyAsset":false,"role":"figure","size":73777,"visible":true,"origin":"","legend":"\u003cp\u003eCS-MLS inversion results with different noise levels. (a) Three-layer (H-type) geoelectric model; (b) Five-layer (HKH-type) geoelectric model\u003c/p\u003e","description":"","filename":"5.png","url":"https://assets-eu.researchsquare.com/files/rs-4023456/v1/ba951af1de757cbd4f82c9df.png"},{"id":52603822,"identity":"cf5a6e4f-f89d-48b4-acf4-64f4929bee56","added_by":"auto","created_at":"2024-03-13 13:16:45","extension":"png","order_by":6,"title":"Figure 6","display":"","copyAsset":false,"role":"figure","size":85400,"visible":true,"origin":"","legend":"\u003cp\u003eSchematic diagrams of the quasi 2-D model and measurement positions\u003c/p\u003e","description":"","filename":"6.png","url":"https://assets-eu.researchsquare.com/files/rs-4023456/v1/50010d8c44ad9e3ab64a6ad6.png"},{"id":52602554,"identity":"2411397d-2707-42db-8191-053ca3320875","added_by":"auto","created_at":"2024-03-13 13:08:45","extension":"png","order_by":7,"title":"Figure 7","display":"","copyAsset":false,"role":"figure","size":107203,"visible":true,"origin":"","legend":"\u003cp\u003eComparison of inversion results by the joint CS-MLS algorithm\u003c/p\u003e","description":"","filename":"7.png","url":"https://assets-eu.researchsquare.com/files/rs-4023456/v1/c2126e246f7b709280eb4118.png"},{"id":52602561,"identity":"eb43face-be44-4bc8-bc3e-679238dec6f3","added_by":"auto","created_at":"2024-03-13 13:08:45","extension":"png","order_by":8,"title":"Figure 8","display":"","copyAsset":false,"role":"figure","size":270569,"visible":true,"origin":"","legend":"\u003cp\u003eComparison of predicted and field data in XY mode. (a-d) Apparent resistivity data curves; (e-h) Phase data curves; (a, e) Data curves for the 1st observation; (b, f) Data curves for the 2nd observation; (c, g) Data curves for the 3rd observation; (d, h) Data curves for the 4th observation.\u003c/p\u003e","description":"","filename":"8.png","url":"https://assets-eu.researchsquare.com/files/rs-4023456/v1/3aa0d71a636194fd84bc4ed7.png"},{"id":52603825,"identity":"6cf48c6a-ad94-4859-9412-19e0a808bd40","added_by":"auto","created_at":"2024-03-13 13:16:45","extension":"png","order_by":9,"title":"Figure 9","display":"","copyAsset":false,"role":"figure","size":289900,"visible":true,"origin":"","legend":"\u003cp\u003eComparison of predicted and field data in YX mode. (a-d) Apparent resistivity data curves; (e-h) Phase data curves; (a, e) Data curves for the 1st observation; (b, f) Data curves for the 2nd observation; (c, g) Data curves for the 3rd observation; (d, h) Data curves for the 4th observation.\u003c/p\u003e","description":"","filename":"9.png","url":"https://assets-eu.researchsquare.com/files/rs-4023456/v1/b8bc2e3b168a532693325f09.png"},{"id":64246287,"identity":"b0b187c7-730f-4ffa-a78b-d7fb44902a92","added_by":"auto","created_at":"2024-09-10 19:53:52","extension":"pdf","order_by":0,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":2057628,"visible":true,"origin":"","legend":"","description":"","filename":"manuscript.pdf","url":"https://assets-eu.researchsquare.com/files/rs-4023456/v1/887aa4fc-a4d3-4bb8-9b95-927e9091f789.pdf"}],"financialInterests":"No competing interests reported.","formattedTitle":"Joint inversion for magnetotelluric data based on cuckoo search algorithm and least squares method","fulltext":[{"header":"1 Introduction","content":"\u003cp\u003eThe magnetotelluric (MT) method is widely used in geothermal resource exploration [\u003cspan citationid=\"CR1\" class=\"CitationRef\"\u003e1\u003c/span\u003e], oil and gas exploration [\u003cspan citationid=\"CR2\" class=\"CitationRef\"\u003e2\u003c/span\u003e], mineral exploration [\u003cspan citationid=\"CR3\" class=\"CitationRef\"\u003e3\u003c/span\u003e], and engineering geological survey [\u003cspan citationid=\"CR4\" class=\"CitationRef\"\u003e4\u003c/span\u003e], which is one of the most important techniques in deep earth studies [\u003cspan citationid=\"CR5\" class=\"CitationRef\"\u003e5\u003c/span\u003e]. Inversion algorithms have been developed to perform MT data preprocessing [\u003cspan additionalcitationids=\"CR7\" citationid=\"CR6\" class=\"CitationRef\"\u003e6\u003c/span\u003e\u0026ndash;\u003cspan citationid=\"CR8\" class=\"CitationRef\"\u003e8\u003c/span\u003e], which is a key aspect for MT exploration data interpretation. Currently, most MT inversion methods are based on linear or nonlinear iterative inversion, and the least squares method (LSM) inversion is one of the most widely used traditional algorithms with fast speed in the late stage [\u003cspan additionalcitationids=\"CR10\" citationid=\"CR9\" class=\"CitationRef\"\u003e9\u003c/span\u003e\u0026ndash;\u003cspan citationid=\"CR11\" class=\"CitationRef\"\u003e11\u003c/span\u003e], but the LSM inversion is heavily dependent on the initial model [\u003cspan citationid=\"CR12\" class=\"CitationRef\"\u003e12\u003c/span\u003e]. An efficient initial model gives high quality results; conversely, an irrational initial model can cause the inversion iterations to fall into the local optimum. Therefore, the initial model quality affects the LSM inversion results.\u003c/p\u003e \u003cp\u003eCuckoo search (CS) algorithm is a novel intelligent optimization algorithm proposed by Yang [\u003cspan citationid=\"CR13\" class=\"CitationRef\"\u003e13\u003c/span\u003e], which is inspired by the parasitic breeding habit of cuckoos, and it adopts the L\u0026eacute;vy flight mode to update the individuals, which can effectively jump out of the local optimum. With the advantages of few parameters, simple operation, fast convergence and strong global optimization ability, the CS algorithm has been successfully applied to a variety of fields, such as multi-objective optimization [\u003cspan citationid=\"CR14\" class=\"CitationRef\"\u003e14\u003c/span\u003e, \u003cspan citationid=\"CR15\" class=\"CitationRef\"\u003e15\u003c/span\u003e], image processing [\u003cspan citationid=\"CR16\" class=\"CitationRef\"\u003e16\u003c/span\u003e, \u003cspan citationid=\"CR17\" class=\"CitationRef\"\u003e17\u003c/span\u003e], resource allocation [\u003cspan citationid=\"CR18\" class=\"CitationRef\"\u003e18\u003c/span\u003e, \u003cspan citationid=\"CR19\" class=\"CitationRef\"\u003e19\u003c/span\u003e], control problems [\u003cspan citationid=\"CR20\" class=\"CitationRef\"\u003e20\u003c/span\u003e, \u003cspan citationid=\"CR21\" class=\"CitationRef\"\u003e21\u003c/span\u003e], and computer vision [\u003cspan citationid=\"CR22\" class=\"CitationRef\"\u003e22\u003c/span\u003e].\u003c/p\u003e \u003cp\u003eRecently, the CS algorithm has been successfully applied in the geophysical inversion [\u003cspan citationid=\"CR23\" class=\"CitationRef\"\u003e23\u003c/span\u003e\u0026ndash;\u003cspan citationid=\"CR24\" class=\"CitationRef\"\u003e24\u003c/span\u003e], which does not depend on either the initial model or the gradient information. Satisfactory inversion results can be obtained by the CS algorithm with a stochastic initial model, but at the cost of problems such as cumbersome computation and slow convergence. Therefore, the CS method is not suitable for practical applications involving a large number of model parameters. In order to obtain satisfactory inversion results in a shorter time, a joint inversion method combining the CS algorithm and LSM inversion has been proposed, in which the stability and non-uniqueness of the inversion can be improved by the a priori constraints provided by the CS algorithm to reduce the dependence of the MT data inversion on the initial model.\u003c/p\u003e \u003cp\u003eThe traditional LSM is a global approximation approach, which is not applicable to problems with large data volume and irregular or scattered distribution. Therefore, a moving least squares (MLS) method with local approximation is studied for MT data inversion, which has a fast convergence rate in the later stage. In our paper, a joint CS-MLS inversion method is constructed, which consists of two main steps: first, the CS algorithm is utilized to search the model parameter within a limited iterations; second, the CS search results are used as the initial model to perform MLS inversion to achieve better MT data inversion effects than a single MLS algorithm [\u003cspan citationid=\"CR25\" class=\"CitationRef\"\u003e25\u003c/span\u003e].\u003c/p\u003e \u003cp\u003eAs a global nonlinear optimization algorithm, the CS does not need to give the initial value manually and has strong optimization adaptability, but its fitting efficiency decreases in the late stage. The MLS method is more stable and faster in the late stage, but it needs to give the initial value manually. To fully combine the advantages of the two algorithms, a joint CS-MLS inversion approach is proposed to improve the inversion efficiency. Based on the joint and single algorithms, synthetic MT data inversion is performed to verify the inversion accuracy and anti-noise performance of the CS-MLS method, and then field data inversion interpretation is performed via the joint algorithm to prove its practicality and effectiveness.\u003c/p\u003e"},{"header":"2. Methodology","content":"\u003cdiv id=\"Sec3\" class=\"Section2\"\u003e \u003ch2\u003e2.1 MT forward modeling theory\u003c/h2\u003e \u003cp\u003eMT sounding is an electromagnetic exploration method that uses natural electromagnetic fields as field source signals, which is also required to satisfy the a priori assumptions of Carnia's classical theory [36].\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003eAssuming that the earth structure is composed of horizontally layered media and the layer model profile is uniformly distributed, a 1-D model with \u003cem\u003eN\u003c/em\u003e-layered media is shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig1\" class=\"InternalRef\"\u003e1\u003c/span\u003e, from which it can be seen that the 1-D media model can be represented by 2\u003cem\u003eN\u003c/em\u003e-1 dimensional model vectors:\u003cdiv id=\"Equ1\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ1\" name=\"EquationSource\"\u003e\n$${\\mathbf{\\vec {m}}}={({\\rho _1},{\\rho _2}, \\cdots ,{\\rho _N},{h_1},{h_2}, \\cdots ,{h_{N - 1}})^T}$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e1\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003ewhere \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({\\rho _i}\\)\u003c/span\u003e\u003c/span\u003e and \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({h_i}\\)\u003c/span\u003e\u003c/span\u003e are resistivity and thickness, respectively.\u003c/p\u003e \u003cp\u003eFor the layered media model, the apparent resistivity and impedance phase at the surface observation points with the following equations:\u003cdiv id=\"Equ2\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ2\" name=\"EquationSource\"\u003e\n$${\\rho _a}(\\omega )=\\frac{{{{\\left| {Z(\\omega )} \\right|}^2}}}{{\\omega \\mu }},{\\text{ }}{\\phi _a}=\\arctan \\frac{{\\operatorname{Im} (Z)}}{{\\operatorname{Re} (Z)}}$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e2\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003ewhere the angular frequency is denoted as \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\omega =2\\pi T\\)\u003c/span\u003e\u003c/span\u003e, the magnetic permeability is taken as \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\mu\\)\u003c/span\u003e\u003c/span\u003e, and the surface wave impedance represents \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(Z(\\omega )\\)\u003c/span\u003e\u003c/span\u003e, which is calculated by the following recursive formula:\u003cdiv id=\"Equ3\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ3\" name=\"EquationSource\"\u003e\n$${Z_i}={Z_{0i}}\\frac{{{Z_{0i}}(1 - {e^{ - 2{k_i}{h_i}}})+{Z_{i+1}}(1+{e^{ - 2{k_i}{h_i}}})}}{{{Z_{0i}}(1+{e^{ - 2{k_i}{h_i}}})+{Z_{i+1}}(1 - {e^{ - 2{k_i}{h_i}}})}},{\\text{ }}{{\\text{Z}}_N}=\\frac{{\\omega \\mu }}{{{k_N}}}={Z_{0N}}$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e3\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003ewhere \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({k_i}=\\sqrt {\\frac{{i\\omega \\mu }}{{{\\rho _i}}}}\\)\u003c/span\u003e\u003c/span\u003e is the \u003cem\u003ei\u003c/em\u003e-th layer complex wave number, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({Z_{0i}}\\)\u003c/span\u003e\u003c/span\u003e is the \u003cem\u003ei\u003c/em\u003e-th layer characteristic impedance, and \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({Z_i}\\)\u003c/span\u003e\u003c/span\u003e is the \u003cem\u003ei\u003c/em\u003e-th layer wave impedance of the top surface.\u003c/p\u003e \u003cp\u003eFor the \u003cem\u003eN\u003c/em\u003e-layer media model, the observed data as \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({d^{obs}}\\)\u003c/span\u003e\u003c/span\u003e, which is a 2\u003cem\u003eK\u003c/em\u003e dimensional vector as \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({({\\rho _1},{\\rho _2}, \\cdots ,{\\rho _K},{\\phi _1},{\\phi _2}, \\cdots ,{\\phi _K})^T}\\)\u003c/span\u003e\u003c/span\u003e (\u003cem\u003eK\u003c/em\u003e represents the number of frequency points observed on the surface), where \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({\\rho _i}\\)\u003c/span\u003e\u003c/span\u003e and \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({\\phi _i}\\)\u003c/span\u003e\u003c/span\u003e are the apparent resistivity and impedance phase from the \u003cem\u003ei\u003c/em\u003e-th frequency point, respectively.\u003c/p\u003e \u003cp\u003eTo summarize, the forward modeling described by Eqs.\u0026nbsp;(\u003cspan refid=\"Equ2\" class=\"InternalRef\"\u003e2\u003c/span\u003e) and (\u003cspan refid=\"Equ3\" class=\"InternalRef\"\u003e3\u003c/span\u003e) can be expressed as the following operator:\u003cdiv id=\"Equ4\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ4\" name=\"EquationSource\"\u003e\n$${\\mathbf{\\vec {d}}}=F({\\mathbf{\\vec {m}}})$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e4\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003ewhere \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({\\mathbf{\\vec {m}}}\\)\u003c/span\u003e\u003c/span\u003e denotes a 2\u003cem\u003eN\u003c/em\u003e-1 dimensional model parameters, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({\\mathbf{\\vec {d}}}\\)\u003c/span\u003e\u003c/span\u003e represents a 2\u003cem\u003eK\u003c/em\u003e dimensional observation, and \u003cspan class=\"InlineEquation\"\u003e\u003c/span\u003e is a mapping operator from model space to data space. In addition, based on the given model parameters given as \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({\\mathbf{\\vec {m}}}\\)\u003c/span\u003e\u003c/span\u003e, the observed data \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({\\mathbf{\\vec {d}}}\\)\u003c/span\u003e\u003c/span\u003e can be calculated according to the recursive formulas of Eqs.\u0026nbsp;(\u003cspan refid=\"Equ2\" class=\"InternalRef\"\u003e2\u003c/span\u003e) and (\u003cspan refid=\"Equ3\" class=\"InternalRef\"\u003e3\u003c/span\u003e).\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec4\" class=\"Section2\"\u003e \u003ch2\u003e2.2 CS-MLS inversion algorithm\u003c/h2\u003e \u003cp\u003e(1) The implementation of the newly proposed CS-MLS inversion algorithm is described in detail in this section. The CS algorithm is a swarm intelligence nonlinear global optimization technique [\u003cspan citationid=\"CR13\" class=\"CitationRef\"\u003e13\u003c/span\u003e], in which the positions of bird nests are initialized (the position represents a possible optimal solution), and then all the nests tend to move to the optimal position based on certain rules. For the MT inversion problem, the distance between the bird nest position and the optimal solution is taken as the objective function of the inversion [\u003cspan citationid=\"CR26\" class=\"CitationRef\"\u003e26\u003c/span\u003e].\u003c/p\u003e \u003cp\u003eFirst, the CS algorithm randomly initializes \u003cem\u003en\u003c/em\u003e nests in the search space as initial values, and then CS updates the nest positions in real time as follows:\u003cdiv id=\"Equ5\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ5\" name=\"EquationSource\"\u003e\n$${\\mathbf{x}}_{i}^{{(t+1)}}={\\text{ }}{\\mathbf{x}}_{i}^{{(t)}}+\\alpha {\\mathbf{L}}\\left( \\beta \\right)$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e5\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003eThen, Eq.\u0026nbsp;(\u003cspan refid=\"Equ5\" class=\"InternalRef\"\u003e5\u003c/span\u003e) is the basic formula for the L\u0026eacute;vy flight mode, where the number of nests is \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(i=1,2, \\cdots ,n\\)\u003c/span\u003e\u003c/span\u003e, the number of iterations is \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(t=1,2, \\cdots ,ite{r_{max}}\\)\u003c/span\u003e\u003c/span\u003e(\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(ite{r_{max}}\\)\u003c/span\u003e\u003c/span\u003e denotes the maximum iterations), \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({\\mathbf{x}}_{i}^{{(t)}}\\)\u003c/span\u003e\u003c/span\u003e and \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({\\mathbf{x}}_{i}^{{(t+1)}}\\)\u003c/span\u003e\u003c/span\u003e present the nest positions in the \u003cem\u003et\u003c/em\u003e and \u003cem\u003et\u0026thinsp;+\u003c/em\u003e\u0026thinsp;1 generations, respectively; \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\alpha\\)\u003c/span\u003e\u003c/span\u003e denotes the step scaling factor, with a default value of 0.01; \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\beta\\)\u003c/span\u003e\u003c/span\u003e represents the randomized step, which is usually assumed to be 1.5; and \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({\\mathbf{L}}\\left( \\beta \\right)\\)\u003c/span\u003e\u003c/span\u003e is the L\u0026eacute;vy stochastic path, which is as follows:\u003cdiv id=\"Equ6\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ6\" name=\"EquationSource\"\u003e\n$${\\mathbf{L}}\\left( \\beta \\right)=\\frac{{\\vartheta {\\mathbf{u}}}}{{{{\\left| {\\mathbf{v}} \\right|}^{{1 \\mathord{\\left/ {\\vphantom {1 \\beta }} \\right. \\kern-0pt} \\beta }}}}}$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e7\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003ewhere \u003cb\u003eu\u003c/b\u003e and \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\nu\\)\u003c/span\u003e\u003c/span\u003e are a random variable obeying a standard normal distribution and \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\vartheta\\)\u003c/span\u003e\u003c/span\u003e is defined as follows.\u003cdiv id=\"Equ7\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ7\" name=\"EquationSource\"\u003e\n$$\\vartheta ={\\left[ {\\frac{{\\Gamma (1+\\beta )\\sin \\frac{{\\pi \\beta }}{2}}}{{{2^{\\frac{{\\beta - 1}}{2}}}\\Gamma (\\frac{{1+\\beta }}{2})\\beta }}} \\right]^{{1 \\mathord{\\left/ {\\vphantom {1 \\beta }} \\right. \\kern-0pt} \\beta }}}$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e8\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003ewhere \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\Gamma (\\cdot )\\)\u003c/span\u003e\u003c/span\u003e is the gamma function.\u003c/p\u003e \u003cp\u003eThere are two ways for the CS algorithm to update the position of the bird's nest: The first is the L\u0026eacute;vy flight model, which is a random walk that simulates the foraging process of flying animals. During the flight process, a large number of short-range searches alternate with occasional long-range jumps, in which the short-range searches are conducive to improving the local search ability, while the occasional long-range jumps make the CS algorithm less likely to fall into the local optimum [\u003cspan citationid=\"CR27\" class=\"CitationRef\"\u003e27\u003c/span\u003e]. The other is random search, which is updated by generating a random value between [0, 1] as \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\varepsilon\\)\u003c/span\u003e\u003c/span\u003e for each nest, comparing it with the discovery probability \u003cem\u003eP\u003c/em\u003e, and then determining whether to generate a new nest, as shown below:\u003cdiv id=\"Equ8\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ8\" name=\"EquationSource\"\u003e\n$${\\mathbf{x}}_{i}^{{(t+1)}}={\\text{ }}{\\mathbf{x}}_{i}^{{(t)}}+\\gamma H(P - \\varepsilon )[{\\mathbf{x}}_{j}^{{(t)}} - {\\mathbf{x}}_{k}^{{(t)}}]$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e9\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003ewhere \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\gamma\\)\u003c/span\u003e\u003c/span\u003e is a random number between [0, 1], \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({\\mathbf{x}}_{i}^{{(t)}}\\)\u003c/span\u003e\u003c/span\u003e, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({\\mathbf{x}}_{j}^{{(t)}}\\)\u003c/span\u003e\u003c/span\u003e, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({\\mathbf{x}}_{k}^{{(t)}}\\)\u003c/span\u003e\u003c/span\u003e are the positions of the three random bird's nests in \u003cem\u003et\u003c/em\u003e generations. \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(H(\\cdot )\\)\u003c/span\u003e\u003c/span\u003e is the Heaviside function, which has a value of 0 at \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(P - \\varepsilon \u0026gt;0\\)\u003c/span\u003e\u003c/span\u003e, that \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({\\mathbf{x}}_{i}^{{(t+1)}}\\)\u003c/span\u003e\u003c/span\u003e remains unchanged; the function value of 1 at \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(P - \\varepsilon \u0026lt;0\\)\u003c/span\u003e\u003c/span\u003e, that \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({\\mathbf{x}}_{i}^{{(t+1)}}\\)\u003c/span\u003e\u003c/span\u003e is randomly changed; the function value of 0.5 at \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(P - \\varepsilon =0\\)\u003c/span\u003e\u003c/span\u003e, that \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({\\mathbf{x}}_{i}^{{(t+1)}}\\)\u003c/span\u003e\u003c/span\u003e is slightly randomly changed.\u003c/p\u003e \u003cp\u003eThe CS algorithm is derived from biological group intelligence and its mathematical expression is simple and straightforward [\u003cspan citationid=\"CR28\" class=\"CitationRef\"\u003e28\u003c/span\u003e]. It does not need to manually specify the initial model in the optimization process, but it has poor convergence performance in the later stage.\u003c/p\u003e\u003cp\u003e(2) The MLS algorithm was improved by Lancaster and Salkauskas [\u003cspan class=\"CitationRef\"\u003e29\u003c/span\u003e], which is a locally approximated moving least squares method. The main difference between the traditional LSM and MLS methods are the coefficients, which vary with the variable parameters in MLS and are constant in LSM, making the MLS method particularly suitable for MT data fitting applications.\u003c/p\u003e\n\u003cp\u003eIn this study, an MLS inversion method optimized by the CS algorithm is proposed. The MLS method is computationally fast but requires suitable initial values; the CS algorithm does not require a given initial value but converges slowly in the later stages of iteration.The joint CS-MLS approach takes full advantage of both algorithms and improves the optimization efficiency. It is worth noting that the traditional CS requires a large number of iterations (possibly hundreds), while the joint inversion requires only a few iterations to ensure that the fitting error can be quickly reduced, further improving the inversion efficiency. The flowchart of CS-MLS is shown in Fig. \u003cspan class=\"InternalRef\"\u003e2\u003c/span\u003e below. The inversion results via the CS algorithm are used as the initial model of MLS, then the fine inversion is performed, and finally the inversion is stopped by reaching a given fitting error (\u003cem\u003eE\u003c/em\u003e) or maximum iterations (\u003cem\u003eT\u003c/em\u003e).\u003c/p\u003e"},{"header":"3. Inversion of synthetic data","content":"\u003cdiv id=\"Sec6\" class=\"Section2\"\u003e \u003ch2\u003e3.1 Objective function construction\u003c/h2\u003e \u003cp\u003eThe MT inversion is an optimization problem whose objective is to predict a parametric model close to the real geoelectric structure based on the observed data. In other words, the parametric model \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({{\\mathbf{\\vec {m}}}^{est}}\\)\u003c/span\u003e\u003c/span\u003e is continuously estimated, so that the fitting error between the theoretical data \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({{\\mathbf{\\vec {d}}}^{obs}}\\)\u003c/span\u003e\u003c/span\u003e that is calculated by MT forward modeling and the actual observations \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({{\\mathbf{\\vec {d}}}^{obs}}\\)\u003c/span\u003e\u003c/span\u003e is minimized. Therefore, the objective function is constructed as follows:\u003cdiv id=\"Equ9\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ9\" name=\"EquationSource\"\u003e\n$$f({{\\mathbf{\\vec {m}}}^{est}})={\\left\\| {{{{\\mathbf{\\vec {d}}}}^{obs}} - F({{{\\mathbf{\\vec {m}}}}^{est}})} \\right\\|^2} \\to \\hbox{min}$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e10\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003eSince the objective function is highly nonlinear and multipolar, the CS algorithm with great global search capability is used to solve the MT inversion problem. The fitness in the CS optimization refers to the objective function in MT inversion, and usually the \u003cem\u003eL\u003c/em\u003e\u003csub\u003e2\u003c/sub\u003e paradigm is adopted to define the misfit between the observed data and the predicted data. The fitness can be expressed as:\u003cdiv id=\"Equ10\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ10\" name=\"EquationSource\"\u003e\n$$fit={c_{rho}}fi{t_{rho}}+{c_{phi}}fi{t_{phi}}={c_{rho}}{\\left\\| {1 - {{{\\rho _{pred}}} \\mathord{\\left/ {\\vphantom {{{\\rho _{pred}}} {{\\rho _{obs}}}}} \\right. \\kern-0pt} {{\\rho _{obs}}}}} \\right\\|^2}+{c_{phi}}{\\left\\| {1 - {{{\\varphi _{pred}}} \\mathord{\\left/ {\\vphantom {{{\\varphi _{pred}}} {{\\varphi _{obs}}}}} \\right. \\kern-0pt} {{\\varphi _{obs}}}}} \\right\\|^2}$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e11\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003ewhere the total fitness consists of the resistivity fitness and the phase fitness (\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({c_{rho}}\\)\u003c/span\u003e\u003c/span\u003e and \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({c_{phi}}\\)\u003c/span\u003e\u003c/span\u003e represent the weighting coefficients), which greatly reduces the non-uniqueness and improves the inversion accuracy and reliability. \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({\\rho _{pred}}\\)\u003c/span\u003e\u003c/span\u003e and \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({\\rho _{obs}}\\)\u003c/span\u003e\u003c/span\u003e are the predicted apparent resistivity and observed apparent resistivity, respectively; \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({\\varphi _{pred}}\\)\u003c/span\u003e\u003c/span\u003e and \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({\\varphi _{obs}}\\)\u003c/span\u003e\u003c/span\u003e are the predicted phase and observed phase, respectively. In addition, the observed and predicted data are normalized. It is worth noting that MLS inversion is a quasi linear iterative method whose objective function is also expressed as Eq.\u0026nbsp;(\u003cspan refid=\"Equ10\" class=\"InternalRef\"\u003e11\u003c/span\u003e).\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec7\" class=\"Section2\"\u003e \u003ch2\u003e3.2 1-D MT model inversion\u003c/h2\u003e \u003cp\u003eTo verify the superiority of the joint CS-MLS inversion method, typical three-layer (H-type) and five-layer (HKH-type) geoelectric models were developed for noise-free and noise-contaminated synthetic data inversion using the joint algorithm. CS inversion parameters: the number of nests as 40, the discovery probability as \u003cem\u003eP\u003c/em\u003e\u0026thinsp;=\u0026thinsp;0.25, and the maximum iterations as 30; MLS inversion parameters: the fitting error as \u003cem\u003eE\u003c/em\u003e\u0026thinsp;=\u0026thinsp;1.0e-5, and the maximum iterations as \u003cem\u003eT\u003c/em\u003e\u0026thinsp;=\u0026thinsp;30. All algorithms are simulated in Matlab R2022a, and the PCs are powered by an Intel(R) Core(TM) i7-12700F processor with speeds of 2.10 GHz and 16.0 G of RAM.\u003c/p\u003e \u003cp\u003e(1)Noise-free model test\u003c/p\u003e \u003cp\u003eTo validate the effectiveness of the joint CS-MLS inversion method, noise-free experiments were conducted for two typical geoelectric models. Model 1 is a three-layer (H-type) model, which is shown as \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({\\rho _1}=100\\)\u003c/span\u003e\u003c/span\u003e \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\Omega \\cdot {\\text{m}}\\)\u003c/span\u003e\u003c/span\u003e, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({\\rho _2}=20\\)\u003c/span\u003e\u003c/span\u003e \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\Omega \\cdot {\\text{m}}\\)\u003c/span\u003e\u003c/span\u003e, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({\\rho _3}=100\\)\u003c/span\u003e\u003c/span\u003e \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\Omega \\cdot {\\text{m}}\\)\u003c/span\u003e\u003c/span\u003e; \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({h_1}=100\\)\u003c/span\u003e\u003c/span\u003e m, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({h_2}=200\\)\u003c/span\u003e\u003c/span\u003e m; Model 2 is a five-layer (HKH-type) model, which is shown as \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({\\rho _1}=100\\)\u003c/span\u003e\u003c/span\u003e \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\Omega \\cdot {\\text{m}}\\)\u003c/span\u003e\u003c/span\u003e, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({\\rho _2}=20\\)\u003c/span\u003e\u003c/span\u003e \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\Omega \\cdot {\\text{m}}\\)\u003c/span\u003e\u003c/span\u003e, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({\\rho _3}=200\\)\u003c/span\u003e\u003c/span\u003e \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\Omega \\cdot {\\text{m}}\\)\u003c/span\u003e\u003c/span\u003e, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({\\rho _4}=50\\)\u003c/span\u003e\u003c/span\u003e \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\Omega \\cdot {\\text{m}}\\)\u003c/span\u003e\u003c/span\u003e, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({\\rho _5}=100\\)\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\Omega \\cdot {\\text{m}}\\)\u003c/span\u003e\u003c/span\u003e; \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({h_1}=1000\\)\u003c/span\u003e\u003c/span\u003e m, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({h_2}=500\\)\u003c/span\u003e\u003c/span\u003e m, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({h_3}=1000\\)\u003c/span\u003e\u003c/span\u003e m, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({h_4}=2000\\)\u003c/span\u003e\u003c/span\u003e m.\u003c/p\u003e \u003cp\u003eFirst, the joint algorithm inversion is performed on the noise-free synthetic data. The inversion results of the CS algorithm and random values within a certain range are used as initial models, respectively, and then the MLS inversion is performed until a certain fitting error or maximum iterations are reached. It should be noted that the MLS method is a local approximation algorithm, and an inappropriate initial model will lead to inversion distortion phenomena. In the single MLS inversion, the inversion is successful every time for the three-layer model; the inversion is successful 3 out of 10 times for the five-layer model. However, in the joint CS-MLS algorithm, the inversion is successful every time for both the three-layer and five-layer models.\u003c/p\u003e \u003cp\u003eIn the joint inversion algorithm, the CS algorithm inversion is performed first to obtain a good initial model for the subsequent MLS inversion, and the inversion results of the three-layer model and the five-layer model are shown in Figs.\u0026nbsp;\u003cspan refid=\"Fig3\" class=\"InternalRef\"\u003e3\u003c/span\u003e and \u003cspan refid=\"Fig4\" class=\"InternalRef\"\u003e4\u003c/span\u003e. As can be seen from Figs.\u0026nbsp;\u003cspan refid=\"Fig3\" class=\"InternalRef\"\u003e3\u003c/span\u003e(a) and 4(a), the results of the CS algorithm (blue dashed line) can provide an initial model that is roughly similar to the real model. Due to the reliable initial model, the CS-MLS inversion exhibits an excellent result (red dashed line with dots) that is quite close to the real model. Moreover, accurate inversion results are obtained for the low resistance layer of the three-layer model (\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({\\rho _2}\\)\u003c/span\u003e\u003c/span\u003e) and the low-height-low resistance layer of the five-layer model (\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({\\rho _2}\\)\u003c/span\u003e\u003c/span\u003e,\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({\\rho _3}\\)\u003c/span\u003e\u003c/span\u003e,\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({\\rho _4}\\)\u003c/span\u003e\u003c/span\u003e). The single MLS inversion results (green dashed line) show a large deviation from the true model, especially for the middle resistance layer. As shown in Figs.\u0026nbsp;\u003cspan refid=\"Fig3\" class=\"InternalRef\"\u003e3\u003c/span\u003e(b)(c) and 4(b)(c), the synthetic data corresponding to the real model (apparent resistance and phase) and those calculated by the joint inversion model are in good agreement. However, the synthetic data calculated by the single CS and MLS inversion models deviate more slightly from the real model.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003eCompared with the single CS and MLS algorithms, the joint CS-MLS inversion approach achieves a significant advantage, due to the fact that the CS algorithm does not require an initial model and has a strong global search ability, but has a lower fitting efficiency in the late stage, while the MLS algorithm is a local approximation method with fast convergence in the late stage, but requires an initial model manually. The joint inversion method fully utilizes the advantages of the two algorithms, which can improve the inversion efficiency.\u003c/p\u003e\u003cp\u003e2. Noise-contaminated model test\u003c/p\u003e\n\u003cp\u003eTo simulate the real field data containing noise, Gaussian noise of 5%, 10% and 20% was added to the synthetic MT data. The inversion error results by the CS-MLS method for noise-contaminated data are saved in Table \u003cspan class=\"InternalRef\"\u003e1\u003c/span\u003e, where \u003cstrong\u003e\u0026epsilon;\u003c/strong\u003e is the random noise added to the synthetic data \u003cstrong\u003ed\u003c/strong\u003e. As shown in Table \u003cspan class=\"InternalRef\"\u003e1\u003c/span\u003e, the estimation error between the synthetic data and the predicted data \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(F({\\mathbf{m}})\\)\u003c/span\u003e\u003c/span\u003e increases with the percentage of noise data, but the estimation error always remains in a small range. Therefore, the presence of noise data does not significantly affect the inversion performance of the joint algorithm.\u003c/p\u003e\u0026nbsp;\u003ctable id=\"Tab1\" border=\"1\"\u003e\n \u003ccaption language=\"En\"\u003e\n \u003cdiv class=\"CaptionNumber\"\u003eTable 1\u003c/div\u003e\n \u003cdiv class=\"CaptionContent\"\u003e\n \u003cp\u003eInversion error results for noise-contaminated data\u003c/p\u003e\n \u003c/div\u003e\n \u003c/caption\u003e\n \u003cthead\u003e\n \u003ctr\u003e\n \u003cth align=\"left\"\u003eModel\u003cbr\u003e\u003c/th\u003e\n \u003cth align=\"left\" colspan=\"3\"\u003eH-type\u003cbr\u003e\u003c/th\u003e\n \u003cth align=\"left\" colspan=\"3\"\u003eHKH-type\u003cbr\u003e\u003c/th\u003e\n \u003c/tr\u003e\n \u003c/thead\u003e\n \u003ctbody\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003eNoise\u003cbr\u003e\u003c/td\u003e\n \u003ctd align=\"left\"\u003e\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({{\\left\\| {F({\\mathbf{m}}) - {\\mathbf{d}}} \\right\\|} \\mathord{\\left/ {\\vphantom {{\\left\\| {F({\\mathbf{m}}) - {\\mathbf{d}}} \\right\\|} {\\left\\| {\\mathbf{d}} \\right\\|}}} \\right. \\kern-0pt} {\\left\\| {\\mathbf{d}} \\right\\|}}\\)\u003c/span\u003e\u003c/span\u003e\u003cbr\u003e\u003c/td\u003e\n \u003ctd align=\"left\"\u003e\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({{\\left\\| {\\mathbf{\\varepsilon }} \\right\\|} \\mathord{\\left/ {\\vphantom {{\\left\\| {\\mathbf{\\varepsilon }} \\right\\|} {\\left\\| {\\mathbf{d}} \\right\\|}}} \\right. \\kern-0pt} {\\left\\| {\\mathbf{d}} \\right\\|}}\\)\u003c/span\u003e\u003c/span\u003e\u003cbr\u003e\u003c/td\u003e\n \u003ctd align=\"left\"\u003e\u0026nbsp;\u003cbr\u003e\u003c/td\u003e\n \u003ctd align=\"left\"\u003e\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({{\\left\\| {F({\\mathbf{m}}) - {\\mathbf{d}}} \\right\\|} \\mathord{\\left/ {\\vphantom {{\\left\\| {F({\\mathbf{m}}) - {\\mathbf{d}}} \\right\\|} {\\left\\| {\\mathbf{d}} \\right\\|}}} \\right. \\kern-0pt} {\\left\\| {\\mathbf{d}} \\right\\|}}\\)\u003c/span\u003e\u003c/span\u003e\u003cbr\u003e\u003c/td\u003e\n \u003ctd align=\"left\"\u003e\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({{\\left\\| {\\mathbf{\\varepsilon }} \\right\\|} \\mathord{\\left/ {\\vphantom {{\\left\\| {\\mathbf{\\varepsilon }} \\right\\|} {\\left\\| {\\mathbf{d}} \\right\\|}}} \\right. \\kern-0pt} {\\left\\| {\\mathbf{d}} \\right\\|}}\\)\u003c/span\u003e\u003c/span\u003e\u003cbr\u003e\u003c/td\u003e\n \u003ctd align=\"left\"\u003e\u0026nbsp;\u003cbr\u003e\u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e5%\u003cbr\u003e\u003c/td\u003e\n \u003ctd align=\"left\"\u003e0.0153\u003cbr\u003e\u003c/td\u003e\n \u003ctd align=\"left\" colspan=\"2\"\u003e0.0664\u003cbr\u003e\u003c/td\u003e\n \u003ctd align=\"left\"\u003e0.0097\u003cbr\u003e\u003c/td\u003e\n \u003ctd align=\"left\" colspan=\"2\"\u003e0.0713\u003cbr\u003e\u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e10%\u003cbr\u003e\u003c/td\u003e\n \u003ctd align=\"left\"\u003e0.0398\u003cbr\u003e\u003c/td\u003e\n \u003ctd align=\"left\" colspan=\"2\"\u003e0.1784\u003cbr\u003e\u003c/td\u003e\n \u003ctd align=\"left\"\u003e0.0770\u003cbr\u003e\u003c/td\u003e\n \u003ctd align=\"left\" colspan=\"2\"\u003e0.1357\u003cbr\u003e\u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e20%\u003cbr\u003e\u003c/td\u003e\n \u003ctd align=\"left\"\u003e0.0882\u003cbr\u003e\u003c/td\u003e\n \u003ctd align=\"left\" colspan=\"2\"\u003e0.3241\u003cbr\u003e\u003c/td\u003e\n \u003ctd align=\"left\"\u003e0.1685\u003cbr\u003e\u003c/td\u003e\n \u003ctd align=\"left\" colspan=\"2\"\u003e0.2725\u003cbr\u003e\u003c/td\u003e\n \u003c/tr\u003e\n \u003c/tbody\u003e\n\u003c/table\u003e\n\u003cp\u003e\u003c/p\u003e\n\u003cp\u003eThe CS-MLS inversion results of the synthetic data with 5%, 10% and 20% noise are shown in Fig. \u003cspan class=\"InternalRef\"\u003e5\u003c/span\u003e. As seen in Fig. \u003cspan class=\"InternalRef\"\u003e5\u003c/span\u003e(a), the estimated model with different noise levels agrees well with the theoretical model for the three-layer geoelectric model; as seen in Fig. \u003cspan class=\"InternalRef\"\u003e5\u003c/span\u003e(b), with the increase of noise data, the deviation of the estimated model from the theoretical model gradually increases for the five-layer geoelectric model, especially the high resistance layer, but it can basically reflect the real geoelectric model structure. The inversion results illustrate that the joint inversion method achieves a good suppression effect on the noise data and can basically find the global optimal solution. In addition, as the complexity of the geoelectric model increases, the degree of model mismatch increases, but it does not affect the global optimization performance of the CS-MLS inversion method.\u003c/p\u003e\n\u003cdiv id=\"Sec8\" class=\"Section2\"\u003e\n \u003ch2\u003e3.3 Pseudo 2-D model inversion\u003c/h2\u003e\n \u003cp\u003eTo further validate the feasibility of the CS-MLS algorithm for MT inversion, two typical pseudo 2-D geoelectric models are designed for testing: Model 1 is a horizontal double anomalous body model, and Model 2 is a vertical double anomalous body model. To simulate real field data, 5% random noise was added to the observed MT data obtained at each measurement point.\u003c/p\u003e① Horizontal double anomalous body model\u003cp\u003eThe theoretical model and the location of the survey points are shown in Fig. \u003cspan class=\"InternalRef\"\u003e6\u003c/span\u003e(a), with the horizontal direction as the survey line and the vertical direction as the survey depth. There are 10 survey points with a line length of 900 m and an interval of 100 m between the survey points. Both anomalous bodies have a shape of 100 m\u0026times;100 m, the low-resistivity body as 20 Ω\u0026middot;m, and the high-resistivity body as 200 Ω\u0026middot;m. They are buried at a depth of 100 m from the top and both are located in a homogeneous half-space with 100 Ω\u0026middot;m. The inversion is performed using the proposed CS-MLS algorithm, and the resistivity profile of the obtained pseudo 2-D model is shown in Fig.\u0026nbsp;7(a). As can be seen, the anomalous bodies by the joint algorithm are basically consistent with the theoretical model, which has an accurate resistivity and position, an uniform resistivity distribution, and a significant boundary effect. Therefore, the inversion results show that the joint CS-MLS approach has great efficiency in the horizontal double anomalous body model.\u003c/p\u003e\n \u003cp\u003e② Vertical double anomalous body model\u003c/p\u003e\n \u003cp\u003eThe theoretical model and the location of the survey points are shown in Fig. \u003cspan class=\"InternalRef\"\u003e6\u003c/span\u003e(b), with the horizontal direction as the survey line and the vertical direction as the survey depth. There are 22 survey points with a line length of 2100 m and an interval of 100 m between the survey points. Both anomalous bodies have a shape of 200 m\u0026times;200 m, the high-resistivity body as 200 Ω\u0026middot;m buried at a depth of 200 m from the top; the low-resistivity body as 20 Ω\u0026middot;m buried at a depth of 700 m from the top, and they are located in a homogeneous half-space of 100 Ω\u0026middot;m. The inversion is performed using the proposed CS-MLS algorithm, and the resistivity profile of the obtained pseudo 2-D model is shown in Fig. 7(b). As can be seen, the anomalous bodies by the joint algorithm are roughly close to the theoretical model, the high-resistivity model has an uniform resistivity distribution, with accurate locations and clearer boundaries; the low-resistivity model has accurate locations and clearer boundaries, while the resistivity distributions are less uniform. Therefore, the inversion results show that the joint CS-MLS approach has great efficiency in the vertical double anomalous body model, which can be suitable for practical MT inversion application.\u003c/p\u003e\n \u003cp\u003eThe feasibility of the CS-MLS method in MT inversion is further verified by the theoretical layer model and the pseudo 2-D model example. The proposed approach achieves a high inversion efficiency, mainly due to the use of the global CS algorithm to obtain the initial value of MLS, which can effectively avoid it from falling into the local optimum, and has a strong global search capability and robustness. In addition, the joint inversion of CS and MLS methods will fully combine the advantages of the two algorithms, which can improve the inversion accuracy as well as the computational efficiency, and will show great potential in the MT inversion.\u003c/p\u003e\n\u003c/div\u003e"},{"header":"4. Field data study","content":"\u003cp\u003eThe COPROD2 dataset is a public dataset for testing the effectiveness of MT inversion, which contains field observation data [\u003cspan citationid=\"CR30\" class=\"CitationRef\"\u003e30\u003c/span\u003e]. However, the subsurface structures have not been accurately confirmed. The inversion effect can be evaluated by the difference in fitting between the predicted and measured data, and the prediction effect of the joint CS-MLS algorithm is compared with that of the single MLS algorithm. The COPROD2 data contains about 30 observations, four of which were randomly selected as test data. According to the polarization mode, the observations can be classified into XY mode and YX mode, and the prediction results for the two modes are shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig7\" class=\"InternalRef\"\u003e8\u003c/span\u003e and Fig.\u0026nbsp;\u003cspan refid=\"Fig8\" class=\"InternalRef\"\u003e9\u003c/span\u003e.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003eThe prediction results of the joint CS-MLS method are significantly better than those of the single MLS algorithm in the four observations, especially the apparent resistivity inversion effects. For the fluctuating flat observation data, the prediction data by the joint method fit well with the measured data, but the prediction results by the single MLS algorithm have a large deviation and are only roughly consistent with the measured data in terms of tendency, as shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig7\" class=\"InternalRef\"\u003e8\u003c/span\u003e(a)(d)(e)(h). For the more fluctuating observation data, the prediction results of the two methods differ significantly from the measured data, in which the single MLS algorithm suffers from distortion, as shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig7\" class=\"InternalRef\"\u003e8\u003c/span\u003e(b)(c)(f)(g). Compared with the XY polarization model, the inversion of the YX polarization model is better, especially the apparent resistivity results, as shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig8\" class=\"InternalRef\"\u003e9\u003c/span\u003e. Considering the volume effect of the electromagnetic wave and the static effect near the surface, the response curve of the 1-D geoelectric model is difficult to fit perfectly with the measured data curve. In addition, the large fluctuation of field data is also affected by anthropogenic noise, magnetic storms and substation interference, which non-random noise increases the difficulty of inversion. Overall, the effectiveness of the joint CS-MLS inversion method in interpreting real MT data has been verified by the COPROD2 field dataset.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e"},{"header":"5 Conclusion","content":"\u003cp\u003eThe joint inversion method developed from the CS and MLS algorithms is suitable for the MT data inversion, which successfully utilizes the advantages of the two algorithms and solves the problems of the CS algorithm's poor convergence speed in the late stage and the MLS algorithm's heavy dependence on the initial model. In the joint inversion, the CS algorithm provides a reliable initial model, and then MLS is applied to the MT data inversion to obtain the final result. Thus, the accurate inversion result is obtained by the proposed approach even if an arbitrary initial model is provided initially. The inversion experiments show that the joint algorithm for MT data inversion is more efficient than the single algorithm, which exhibits excellent results in inversion accuracy and convergence speed. Additionally, the synthetic data and field data inversion confirm the high robustness and feasibility. Therefore, the joint CS-MLS approach is of great importance for actual data inversion without proper prior knowledge.\u003c/p\u003e \u003cp\u003eConsidering that 2-D or 3-D layer models are more consistent with the actual geology, the joint algorithm can be extended to high-dimensional inversion in the future.\u003c/p\u003e"},{"header":"Declarations","content":"\u003cp\u003e\u003cstrong\u003eAuthor Contributions\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eRuiyou Li: Writing - original draft, Methodology and Data curation; Yong Zhang: Supervision and Validation; Xiaohui Ding: Methodology and Investigation; Min Li: Writing - original draft and Software; Long Zhang: Validation, Format modification, Manuscript layout and correction.\u003c/p\u003e\n\u003cp\u003e\u0026nbsp;\u003cstrong\u003eConflicts of Interests\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eThe authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.\u003c/p\u003e\n\u003cp\u003e\u0026nbsp;\u003cstrong\u003eData Availability Statement\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eThe data used to support the findings of this study are included within the article.\u003c/p\u003e\n\u003cp\u003e\u0026nbsp;\u003cstrong\u003eAcknowledgments\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eThis work was supported by the National Natural Science Foundation of China (No.61762043), the China Postdoctoral Science Foundation (No.2023M741480), and the Scientific Research Foundation of Jiangxi Provincial Education Department, China (No. GJJ2200528).\u003c/p\u003e"},{"header":"References","content":"\u003col\u003e\n\u003cli\u003eArdid A, Dempsey D, Bertrand E, et al. Bayesian magnetotelluric inversion using methylene blue structural priors for imaging shallow conductors in geothermal fields[J]. Geophysics, 2021, 86(3): E171-E183.\u003c/li\u003e\n\u003cli\u003eMiri H, Dehkordi B H, Payrovian G. Oil field imaging on the Sarab Anticline, southwest of Iran, using magnetotelluric data[J]. Journal of Petroleum Science and Engineering, 2021, 202: 108497.\u003c/li\u003e\n\u003cli\u003eJiang W, Brodie R C, Duan J, et al. Probabilistic inversion of audio-frequency magnetotelluric data and application to cover thickness estimation for mineral exploration in Australia[J]. Journal of Applied Geophysics, 2023, 208: 104869.\u003c/li\u003e\n\u003cli\u003eLiu W, L\u0026uuml; Q, Yang L, et al. Application of sample-compressed neural network and adaptive-clustering algorithm for magnetotelluric inverse modeling[J]. IEEE Geoscience and Remote Sensing Letters, 2021, 18(9): 1540-1544.\u003c/li\u003e\n\u003cli\u003eChao G, Osella A. 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PeerJ Computer Science, 2021, 7: 370.\u003c/li\u003e\n\u003cli\u003eSantilano A, Godio A, Manzella A. Particle swarm optimization for simultaneous analysis of Magnetotelluric and time domain EM data[J]. Geophysics, 2018, 83(3):1-48.\u003c/li\u003e\n\u003cli\u003eSalkauskas P L. Surfaces generated by moving least squares methods[J]. Math Compt, 1981, 37(155):141-158.\u003c/li\u003e\n\u003cli\u003eBeusekom A, Parker R L, Bank R E, et al. The 2-D magnetotelluric inverse problem solved with optimization[J]. Geophysical Journal International, 2011, 184(2), 639-650.\u003c/li\u003e\n\u003c/ol\u003e"}],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":true,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":false,"hideJournal":true,"highlight":"","institution":"","isAcceptedByJournal":false,"isAuthorSuppliedPdf":false,"isDeskRejected":"","isHiddenFromSearch":false,"isInQc":false,"isInWorkflow":false,"isPdf":false,"isPdfUpToDate":true,"isWithdrawnOrRetracted":false,"journal":{"display":true,"email":"[email protected]","identity":"researchsquare","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":true,"externalIdentity":"","sideBox":"","snPcode":"","submissionUrl":"/submission","title":"Research Square","twitterHandle":"researchsquare","acdcEnabled":true,"dfaEnabled":false,"editorialSystem":"","reportingPortfolio":"","inReviewEnabled":false,"inReviewRevisionsEnabled":true},"keywords":"Magnetotelluric, cuckoo search algorithm, least squares method, inversion","lastPublishedDoi":"10.21203/rs.3.rs-4023456/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-4023456/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eThe main drawback of using traditional linear or quasi-linear methods for magnetotelluric (MT) data inversion is its strong dependence on the initial model, which leads to its tendency to fall into a local optimum. However, the lack of sufficient a priori information makes it difficult to obtain suitable initial models in most geophysical inversions. To solve this problem, a joint approach (CS-MLS) based on the cuckoo search (CS) algorithm and the moving least squares (MLS) method is proposed to perform MT data inversion. The CS algorithm has a good global search performance without being sensitive to the initial model, which is used for the initial search of the stratum model parameter space. The search results are then used as the initial model to explore the quasi-linear inversion based on MLS. The synthetic and field data experiments show that the reliable initial model provided by the CS algorithm has greatly refined the results of the MLS inversion. The joint CS-MLS algorithm has great inversion performance, with significant improvement in inversion efficiency and accuracy, and is especially suitable for MT data inversion without initial information.\u003c/p\u003e","manuscriptTitle":"Joint inversion for magnetotelluric data based on cuckoo search algorithm and least squares method","msid":"","msnumber":"","nonDraftVersions":[{"code":1,"date":"2024-03-13 13:08:40","doi":"10.21203/rs.3.rs-4023456/v1","editorialEvents":[{"type":"communityComments","content":0}],"status":"published","journal":{"display":true,"email":"[email protected]","identity":"researchsquare","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":true,"externalIdentity":"","sideBox":"","snPcode":"","submissionUrl":"/submission","title":"Research Square","twitterHandle":"researchsquare","acdcEnabled":true,"dfaEnabled":false,"editorialSystem":"","reportingPortfolio":"","inReviewEnabled":false,"inReviewRevisionsEnabled":true}}],"origin":"","ownerIdentity":"d5325b27-f85e-4d97-9baa-2597d16fd97d","owner":[],"postedDate":"March 13th, 2024","published":true,"recentEditorialEvents":[],"rejectedJournal":[],"revision":"","amendment":"","status":"posted","subjectAreas":[],"tags":[],"updatedAt":"2024-09-10T19:45:44+00:00","versionOfRecord":[],"versionCreatedAt":"2024-03-13 13:08:40","video":"","vorDoi":"","vorDoiUrl":"","workflowStages":[]},"version":"v1","identity":"rs-4023456","journalConfig":"researchsquare"},"__N_SSP":true},"page":"/article/[identity]/[[...version]]","query":{"redirect":"/article/rs-4023456","identity":"rs-4023456","version":["v1"]},"buildId":"_2-kVJe1T_tPrBINL-cwx","isFallback":false,"isExperimentalCompile":false,"dynamicIds":[84888],"gssp":true,"scriptLoader":[]}