Under the condition of ignoring the electromagnetic coupling effect, the complex resistivity forward calculation is similar to the DC method. The Cole-Cole model is introduced to characterize the induced polarization effect. After Fourier transform, 2D complex potential boundary value problem in wavenumber domain is [12]:

{ ∇ ⋅ ( σ ∇ U ) − k 2 σ U = − I δ ( A ) ∈ Ω ∂ U ∂ n = 0 ∈ Γ s ∂ U ∂ n + k K 1 ( k r ) K 0 ( k r ) c o s ( r , n ) U = 0 ∈ Γ ∞ (1)

In Equation (1), the area Ω is a three-dimensional study area, Boundary Γ s , Γ ∞ is the boundary of a two-dimensional region. Among them, Γ s is the surface boundary of area Ω, and Γ ∞ is the underground boundary of area Ω, As shown in Figure 1, σ 2 is the complex conductivity of the polarized medium, r is the distance from the power source to the boundary, n is the outer normal direction of the infinity boundary, k is the wavenumber, K 0 is the zero-order Bessel function, and K 1 is the second kind of first-order Bessel function. The functional of Equation (1) is:

F ( U ) = ∫ Ω [ σ 2 ( ∇ U ) 2 + 1 2 k 2 σ U 2 − I δ ( A ) U ] d Ω       + 1 2 ∫ Γ ∞ σ k K 1 ( k r ) K 0 ( k r ) cos ( r , n ) U 2 d Γ (2)

Discrete the calculation area and construct an interpolation function. The discrete form of Equation (2) is:

F ( U ) = 1 2 U Τ K U − U Τ P (3)

where P is the column vector related to the excitation point power. Solving the variation of the Equation (3) and making it equal to zero, then we can get the system of linear equations:

K U = P (4)

The wavenumber domain potential U of each point is obtained by solving the linear Equation (4), and the potential is obtained by inverse Fourier transform [13].

u ( r ) = ∑ j = 1 n U ( r , k j ) g j (5)

where r is the position of the measuring point, k j is the wavenumber and g j is the weighting coefficient, the values of both are shown in Table 1 [14].

The complex resistivity produced by the induced polarization effect of rock and ore is expressed by Cole-Cole model [15]:

ρ ( i ω ) = ρ 0 { 1 − m [ 1 − 1 1 + ( i ω τ ) c ] } (6)

where, ρ 0 is resistivity (0 Hz); m is charge ability; c is relaxation-constant; τ is a time constant. The complex resistivity forward calculation can be carried out by replacing the conductivity in Equation (1) and Equation (2) with the complex conductivity.

In order to verify the accuracy of the CR forward algorithm, the two-layer horizontal complex resistivity model is used to verify the calculation accuracy.

For the two-layer horizontal formation, by solving the simplified Laplace equation of the cylindrical coordinate system and power series expansion, the calculation expression of the complex potential U ˜ at each point on the surface can be obtained as:

U ˜ = I ρ 1 ( i ω ) 2 π [ 1 r + ∑ n = 1 ∞ K 12 n r 2 + ( 2 n h 1 ) 2 ] (7)

In Equation (7), K 12 = [ ρ 2 ( i ω ) − ρ 1 ( i ω ) ] / [ ρ 2 ( i ω ) + ρ 1 ( i ω ) ] , ρ 1 ( i ω ) , ρ 2 ( i ω ) is the complex resistivity calculated by Equation (6) for the first and second layers, r is the distance between the measuring point and the origin, h 1 is the thickness of the first stratum. The calculation accuracy of Equation (7) depends on the size of the number of terms n. The larger the n is, the higher accuracy of the calculation is. This calculation takes n = 200. The parameters of the two-layer polarization model are shown in Table 2.

The numerical solution and analytical solution of 2.5D complex potential of the two-layer horizontal layer model are shown in Figure 2(a) and Figure 2(b). Both the amplitude and phase curves of complex potential are consistent with the analytical solution, which shows the accuracy of the finite element algorithm. The finite element algorithm provides a reliable basis for constructing BP neural network training database.

Quantum particle swarm optimization (QPSO) absorbs the characteristics and advantages of PSO algorithm and quantum computing, combines the two methods, and presents the solution of the problem to be optimized in the form of qubit probability amplitude [16].

Initialization: The qubit phase θ in the range of [ 0 , 2 π ] is randomly generated by using a random number function, and the qubit can be obtained, which can be expressed as a probability amplitude [ sin θ , cos θ ] Τ . After that, the solution space is transformed by combining the upper and lower limits of each variable set by the algorithm, and the qubit is transformed into the solution space to calculate the appropriate value.

Update: In the QPSO algorithm, with the help of quantum rotation gates operation, the individual and group optimal positions (set as cosine positions) found by the particle p i in the current solution space can be expressed as:

p i l = [ cos ( θ i l 1 ) , cos ( θ i l 2 ) , ⋯ , cos ( θ i l D ) ] p g = [ cos ( θ g 1 ) , cos ( θ g 2 ) , ⋯ , cos ( θ g D ) ] (8)

The updated rules of particle state are as follows:

The incremental update of qubit angle on particles is:

Δ θ i j ( t + 1 ) = ω Δ θ i j ( t ) + c 1 r 1 ( Δ θ l ) + c 2 r 2 ( Δ θ g ) (9)

where

Δ θ l = { 2 π + θ i l j − θ i j ( θ i l j − θ i j < − π ) θ i l j − θ i j ( π ≤ θ i l j − θ i j ≤ π ) θ i l j − θ i j − 2 π ( θ i l j − θ i j > π ) , Δ θ g = { 2 π + θ g j − θ i j ( θ g j − θ i j < − π ) θ g j − θ i j ( π ≤ θ g j − θ i j ≤ π ) θ g j − θ i j − 2 π ( θ g j − θ i j > π )

And ω is a random number with inertia weight; c 1 , c 2 is a self-factor and a global factor respectively; r 1 , r 2 is a random number of (0, 1).

The probability amplitude of qubit on particles is updated as:

[ cos ( θ i j ( t + 1 ) ) sin ( θ i j ( t + 1 ) ) ] = [ cos ( Δ θ i j ( t + 1 ) ) − sin ( Δ θ i j ( t + 1 ) ) sin ( Δ θ i j ( t + 1 ) ) cos ( Δ θ i j ( t + 1 ) ) ] [ cos ( θ i j ( t ) ) sin ( θ i j ( t ) ) ] = [ cos ( θ i j ( t ) + Δ θ i j ( t + 1 ) ) sin ( θ i j ( t ) + Δ θ i j ( t + 1 ) ) ] (10)

Among them i = 1 , 2 , ⋯ , n ; j = 1 , 2 , ⋯ , d , n and d are the number of population and the dimension of unknown variables, respectively.

Mutation: In the QPSO algorithm, quantum non-gate is used to mutate particles. First, the mutation probability K m is set, and each particle is given a random number r a n d i between (0, 1). If r a n d i < K m , then [n/2] qubits are randomly selected, and the mutation operation is carried out by using Equation (11), and the memorized optimal position and rotation angle vector remains unchanged.

[ 0 1 1 0 ] [ cos ( θ i j ) sin ( θ i j ) ] = [ sin ( θ i j ) cos ( θ i j ) ] = [ cos ( θ i j + π 2 ) sin ( θ i j + π 2 ) ] (11)

where i = 1 , 2 , ⋯ , n ; j = 1 , 2 , ⋯ , d .

Weight adjustment: In particle swarm optimization, inertia weight ω is an important parameter. If the value of ω is increased, the global search ability will be enhanced. If the value of ω is decreased, the local search ability will be enhanced. Equation (12) is used to realize the nonlinear dynamic adjustment of inertia weight:

ϖ = { ϖ min − ( ϖ max − ϖ min ) ∗ ( f − f min ) f a v g − f min , f ≤ f avg ϖ max , f > f a v g (12)

where f represents the real-time objective function value of particles; f min and f avg are the minimum and average moderate values of all particles.

BP neural network modeling method: take the horizontal position, vertical position, amplitude and phase information of measuring apparent complex resistivity as the input information of neural network, and take the parameter information of model as the output information of the neural network.

In order to adapt to the multi-parameter characteristics of complex resistivity inversion, the BP neural network model obtained through numerical test is shown in Figure 3. The overall neural network has 12 input nodes in the input layer, 56 hidden neurons in two hidden layers (the first hidden layer has 46 neurons and the second hidden layer has 10 neurons), 4 output nodes in the output layer, and the overall topology is 12-46-10-4.

QPSO-BP algorithm mainly uses the global optimization ability of QPSO to optimize the initial weight and threshold of BP neural network. In order to obtain the best overall effect, BP neural network uses the optimized initial weight and threshold to continue training.

The inversion algorithm flow is as follows:

Step 1: Determine the topology of BP neural network and the number of parameters to be optimized, and set the particle population dimension. Each individual qubit phase θ is a D-dimensional parameter vector corresponding to the weights and thresholds of BP neural network. Initialize the particle population and set the basic parameters of the algorithm, such as ϖ adjustment range [ ω min , ω max ] , self-factor c 1 , global factor c 2 , mutation probability K m .

Step 2: According to the upper and lower limits of each particle, the solution space is transformed. Then, according to Equation (13), calculate the particle fitness value, select and record the individual optimal position and group optimal position found in the solution space.

F k = 1 n ∑ j = 1 l ( e k j − y k j ) 2 f i = 1 c n ∑ k = 1 c n ( F k ) 2 (13)

where, cn is the total number of inversion parameters, k is the number of inversion parameters, k = 1 , 2 , ⋯ , c n , n is the total number of training samples, j is the number of training samples, j = 1 , 2 , ⋯ , n . e k j and y k j are the expected output and actual neural network output of the j-th training sample of the k-th parameter, respectively. F k is the mean square error of the k-th parameter. f i is the fitness value of the i-th particle.

Step 3: particle state update, nonlinear dynamic adjustment of inertia weight, and qubit mutation calculation operations.

Step 4: Compare the appropriate value of each particle in order. Then the optimal particle position is recorded and updated as an individual optimal position. At the same time, the optimal position of each generation is sorted forward in the current evolution generation, and the optimal position of the individual is recorded and updated as the global optimal position.

Step 5: Identify whether the iteration meets the end condition, if it is satisfied, exit the algorithm and output the result, otherwise return to step 2 to continue the iterative calculation.

Step 6: The global optimal solution is used as the initial weight and threshold of BP neural network. Then we continue to train the neural network by using the BP elastic algorithm (RPROP). When the number of algorithm iterations reaches the maximum or the output error satisfies the requirements, output and save the network. At the last, we input the data volume which needs to be inverted into the network to obtain the QPSO-BP algorithm inversion result.

In this paper, we construct an apparent complex resistivity database with 48 single-polarization anomaly models. The forward simulation of the database model was calculated out by the finite element method described in Section 2. All the data are used for QPSO-BP neural network training. There are two sizes of abnormal models in the database; each of them has six different spatial locations. We also set a variety of contrast characteristics for these abnormal bodies, such as low resistance with high polarization, high resistance with low polarization. Five independent models are used to test the generalization ability of the algorithm. The test data is not involved in neural network training. Part of the sample model used for training is shown in Figure 4.

At the same time, QPSO algorithm and basic PSO algorithm are used to optimize the initial value and threshold of BP neural network. Set the particle population size of QPSO algorithm to 80, set the evolution generation to 100, ω max = 0.6 , ω min = 0.4 , c 1 = c 2 = 1.2 , K m = 0.35 . Set the particle population

size of basic PSO algorithm to 80, set the evolution generation to 100, ω = 0.5 , c 1 = 1.5 , c 2 = 2.5 . The fitness decline curves of the two algorithms are shown in Figure 5(a).

QPSO algorithm is easier to jump out of the local extreme value in the early calculation and converge to the global extreme value in the later calculation than the basic PSO algorithm with fixed weight because it has more particle number (sine solution and cosine solution) and nonlinear dynamic adjustment weight.

Using the global optimal solution as the initial weight and threshold of the BP neural network, and using the BP elastic algorithm (RPROP) to continue training the neural network, finally, the convergence curve of algorithm training error is shown in Figure 5(b). The optimized BP neural network can converge to the global optimal value more quickly, and its final training MSE is lower than that of the non-optimized network. At the same time, Figure 6 shows the difference of MSE distribution histogram between training and test samples under the two algorithms.

The BP neural network optimized by QPSO algorithm performs better than the BP neural network without initial optimization on the inversion training and test models, and the mean square error distribution is better. This shows that the BP neural network optimized by QPSO has a better training effect.