As one of the most active nonlinear inversion methods in transient electromagnetic (TEM) inversion, the back propagation (BP) neural network has high efficiency because the complicated forward model calculation is unnecessary in iteration. The global optimization ability of the particle swarm optimization (PSO) is adopted for amending the BP's sensitivity to its initial parameters, which avoids it falling into a local optimum. A chaotic-oscillation inertia weight PSO (COPSO) is proposed for accelerating convergence. The COPSO-BP algorithm performance is validated by two typical testing functions, two geoelectric models inversions and a field example. The results show that the COPSO-BP method is more accurate, stable and needs relatively less training time. The proposed algorithm has a higher fitting degree for the data inversion, and it is feasible to use it in geophysical inverse applications.

The transient electromagnetic (TEM) method applies the secondary receiving voltage induced by the rapid switching of pulse current, and it then deduces the geoelectrical parameters consisting of the resistivities and thicknesses of the layers. The later is a typical TEM inversion issue with nonlinear features. The linear inversion method was simple and widely used through linearization processes, yet it is extremely dependent on the selection of initial parameters and results in poor inversion accuracy. Hence, the nonlinear inversion methods have attracted more geophysicists' attention in recent years.

The artificial neural network (ANN) is one of the most active nonlinear inversion methods, and it has very high computation efficiency because the complicated forward model calculation is unnecessary in this iteration. All the geoelectrical parameters and the forward model relations are implied in the weight and threshold parameters of ANN. And it is different from the nonlinear Monte Carlo method with a global space search solution (He et al., 2018; Jha et al., 2008; Pekşen et al., 2014; Sharma, 2012; Tran and Hiltunen, 2012). Srinivas et al. (2012) compared the inversion performance of back propagation (BP), the radial basis function (RBF) and the generalized regression neural network (GRNN) in vertical electrical sounding data, then established a 1-D inversion model with BP and finally realized the parameter inversion. Maiti et al. (2012) proposed a Bayesian neural-network training method in 1-D electrical sounding. Jiang et al. (2018) improved the training method for the kernel principal-component wavelet neural network and achieved the resistivity imaging. Jiang et al. (2016a) produced a learning algorithm based on information criterion (IC) and particle swarm optimization for RBF network which improves the global search ability. Johnson and Aizebeokhai (2017) utilized neural-network method to invert multi-layer georesistivity sounding. Jiang et al. (2016b) presented a pruning Bayesian neural-network (PBNN) method for resistivity imaging and solved the instability and local minimization problems. Raj et al. (2014) solved nonlinear apparent resistivity inversion problems with ANN. The ANN has been widely applied in electric-prospecting data interpretation for its powerful fitting ability. However, the neural-network method is sensitive to its initial parameter settings and falls easily into a local minimum. Many improved methods were proposed for balancing the convergence rate and inversion quality. Zhang and Liu (2011) proposed ant colony optimization for ANN, and they applied high-density resistivity and acquired smaller inversion errors and higher determinant coefficients. Dai et al. (2014) suggested a differential evolution (DE) for BP which enhanced the global search ability. Rosas-Carbajal et al. (2014) introduced the genetic algorithm for ANN.

Inertial weight curves comparison.

The particle swarm optimization (PSO) has a simple structure, fast convergence rate, high accuracy and global optimization ability. Fernández et al. (2010) successfully introduced the PSO in a 1-D resistivity inversion. Godio and Santilano (2018) applied it in a geophysical inversion and deduced a depth resistivity earth model. Due to the PSO's global searching performance, the BP's initial weights and thresholds can be trained by the PSO, and the BP's global optimization ability can be improved. Compared to the standard PSO (SPSO), a chaotic-oscillation inertia weight PSO (COPSO) can accelerate the convergence rate in the early stage, and it was proposed naturally (Shi et al., 2009).

The paper structure is as follows: the principles of the PSO algorithm with different inertia weights schemes, the BP neural network and the proposed COPSO-BP algorithm are given in Sect. 2. Then, the COPSO-BP algorithm performance is validated by two typical testing functions in Sect. 3. And in a later section, inversion simulations of three-layer and five-layer geoelectric models are carried out; the hidden-layer neuron numbers determining method is put forward; and algorithm performance is compared.

For an

Three-layer BP neural-network structure.

Training error curves of the SPSO-BP and COPSO-BP algorithms.

The inertia weight parameter

The BP neural network has a multi-layer feed-forward structure, and a typical three-layer network is shown in Fig. 2 (Li et al., 2009).

For Fig. 2,

The initial parameters are chosen randomly, which affects the convergence rate, learning efficiency and perhaps falling into a local minimum. The chaotic-oscillation PSO (COPSO) has a much better global optimization capability; therefore, the COPSO algorithm is proposed to optimize the initial weight and threshold of the BP. The COPSO-BP pseudo-codes are briefly described in Algorithm 1.

The formula for calculating the

In order to investigate the COPSO-BP performance and reliability, Rosenbrock and Bohachevsky testing functions were adopted, which are typical non-convex functions and mainly evaluate the performance of unconstrained algorithms. However, due to the random nature of the function, it is not easy to solve and has a global minimum function value of zero.

Comparison of the SPSO-BP and COPSO-BP algorithms for testing functions.

Fitting curves of the COPSO-BP algorithm.

Influence of hidden-layer nodes on

Distribution of resistivity

It can be seen in Table 1 that although both the SPSO-BP and COPSO-BP algorithms can acquire relatively high accuracy for testing functions, the COPSO-BP algorithm is slightly better than the SPSO-BP algorithm. However, the COPSO-BP algorithm has a better convergence rate and optimization efficiency in the early stage in Fig. 3. Therefore, the SPSO-BP and COPSO-BP algorithms have a strong learning ability, good stability and generalization ability, which will be suitable for TEM inversion.

According to the derivation of Kaufman and Keller (1983), the frequency response of the central
loop source for the layered model takes the following Hankel transform:

The ramp excitation current of TEM is

However, the vertical magnetic field

For BP structure, the output nodes are determined by
the number of inversion geoelectrical parameters, the input
nodes are determined by the samples number of

It can be seen that the optimal neural-network
structures were 10-2-5 and 10-5-9 for three- and five-layer
models based on the maximum

Comparison of different inertia weights in PSO algorithms (

The simulation was implemented on a Core (TM) i5-7500 processor with 8 GB of memory. It is obviously found in Table 2 that the COPSO algorithm has a much faster convergence rate and a lower iteration number and is time consuming.

A three-layer and five-layer geoelectric models were investigated, and the PSO parameter values are the same as those of the “Algorithm testing” section in this paper. In order to simulate actual TEM applications, the ramp turn-off is taken into account. Considering the probability distribution characteristic of the above algorithms, the average of 20 simulation results was chosen. The BP, SPSO-BP and COPSO-BP algorithms and a nonlinear programming genetic algorithm (NPGA) (Li et al., 2017) were compared.

The central loop TEM parameters were set as follows: the transmitting coil
radius was set to

The BP training samples, which are a series of

The inversion results were shown in Table 3 and Figs. 7–8. The BP type algorithms were superior to the NPGA inversion in Table 3. Moreover, the inversion accuracy, convergence rate and optimization ability of the COPSO-BP algorithm were better than others.

Inversion comparison of three-layer H-type geoelectric models.

Fitness curves of SPSO-BP and COPSO-BP.

Comparison of mean square error curves.

Additional results showed that the solution range of

A five-layer KHK-type geoelectric model was adopted, and its resistivities were

The training samples with parameter ranges were

Inversion comparison of five-layer KHK-type geoelectric models.

Fitness curves of SPSO-BP and COPSO-BP.

Comparison of mean square error curves.

Three kinds of BP methods, the traditional BP, SPSO-BP and COPSO-BP algorithms, were compared in Table 5. Hence, the training times of COPSO-BP were obviously less than those of SPSO-BP and were almost equal to BP; it can obtain better precision especially for its global optimization performance.

Simulation comparison of different algorithms.

Inversion comparison of different geoelectric models.

Forward data of Hz and data with 5 % noise.

The inversion of COPSO-BP and NPGA were compared in Fig. 11. The fitting ability of COPSO-BP was much better than NPGA.

Comparison of inversion results of three-layer H-type (with noise) models.

1-D inversion of forward results for

Inversion results for BP

In order to verify the algorithm robustness, 5 % (26 dB) and 10 % (20 dB)
Gaussian random noise was added in TEM data for the three-layer geoelectric
model. Three kinds of inversions were implemented respectively. The results
and a comparison were shown in Table 6. The

As can be seen in Table 2, after applying 5 % and 10 % Gaussian noise, the COPSO-BP inversion has a higher robust ability. The accuracy was obviously improved based on the total relative-error data.

In order to test the effectiveness of the method, a transient
electromagnetic vertical magnetic field (Hz) with 10 measuring points at
380 to 1280 m of the no. 1 line from a mining area in Anhui Province were
selected. After the data processing, the inversion was performed using the
three-layer neural-network model in the previous section, and the results of the BP
and COPSO-BP inversion were compared. Figure 13 shows the comparison between
the surveyed data and the inversion data at 380 m of the no. 2 line in the
mining area. Figure 14 displays the pseudo-sections of the 10 sets of
inversion data combined with the geological data interpolation smoothing. It
can be seen from Fig. 14 that the first layer has low resistivity
(100–200

The inversion is performed for three-layer (H-type) and five-layer (KHK-type) geoelectric models in this paper. The results show that the BP neural network is better than the NPGA algorithm because the BP method does not need to use the forward algorithm repeatedly, and its calculation time is short, which is different from the nonlinear heuristic method based on a global space search solution.

The main advantage of the BP is that it can interpret the transient electromagnetic sounding results quickly after training the network. Furthermore, the BP algorithm could automatically obtain the “reasonable rules” between input and output data by learning, and it can adaptively store the learning content in the network weight, for which the BP neural network has a high self-learning and self-adaptation ability. In addition, the superior simulation results of the test function indicate that the BP algorithm can approximate any nonlinear continuous function with arbitrary precision, which means it has a strong nonlinear mapping ability; the inversion results of the layered geoelectric model with uncorrelated noise data prove that the BP algorithm has strong robustness, which means it has the ability to apply learning results to new knowledge. However, the BP neural-network weight is gradually adjusted by the direction of local improvement, which causes the algorithm to fall into a local extremum, and the weight converges to a local minimum that leads to the network training failure. Moreover, the BP is very sensitive to the initial network weight, and the initialization network with different weight values tends to converge at different local minimums, so it obtains different results each time. In addition, the BP algorithm is essentially a gradient descent method, which leads to a slow convergence rate.

From the results of the layered-model and parametric analysis part, it can be seen that the single BP algorithm has a higher error value than SPSO-BP because the BP method is sensitive to initial weight and easy to fall into local minimum values; thus a heuristic global search particle swarm optimization algorithm with a simple structure, rapid convergence and high precision is applied to optimize the weight and threshold of the BP neural network, which improves the global optimization performance of the algorithm. Furthermore, the PSO algorithm adjusts the inertia weight adaptively based on the chaotic-oscillation curve that is similar to the annealing process in the simulated annealing algorithm (SA), which jumps out the local extremum faster in the early stage and accelerates the convergence and reduces the training times. Therefore, compared with the SPSO-BP and BP algorithms, the inversion results of COPSO-BP are closer to the theoretical data with smaller error fluctuations, stronger anti-noise controls, a better generalization of performance and higher stability, which it is effective in solving geophysical inverse problems.

From the simulation experiment, it is not clear how the weight organization affects the BP neural-network weight learning process. It is necessary to conduct a more systematic study on this problem to improve our understanding of how the BP neural network handles training data.

The nonlinear COPSO-BP method was proposed for TEM inversion. The BP's initial weight and threshold parameters were trained by the COPSO algorithm, which makes it easy to not fall into a local optimum. The chaotic-oscillation inertia weight for PSO was proposed so as to improve the PSO's global optimization ability and fast convergence in the early stage. The layered geoelectric model inversion showed that the COPSO-BP method is more accurate and stable and requires relatively less training time.

The PSOBP code was developed by Huaiqing Zhang and Ruiyou Li of Chongqing University in China, who are able to be contacted at the telephone number 13752954568 or e-mail address zhanghuaiqing@cqu.edu.cn. A computer with MATLAB R2016a is required is to run this code. The programming language of this code is C

HZ conceived this paper. RL and HZ developed the main algorithmic idea and the mathematical part. RL and NY carried out the simulation and data analysis. QZ completed the writing and interpretation of this paper. All authors contributed to the writing of the paper and approved the final paper.

The authors declare that they have no conflict of interest.

This work was partly supported by the National Natural Science Foundation of China (grant no. 51377174).

This research has been supported by the National Natural Science Foundation of China (grant no. 51377174).

This paper was edited by Norbert Marwan and reviewed by two anonymous referees.