Three-Dimensional Joint Inversion of the Resistivity Method and Time-Domain-Induced Polarization Based on the Cross-Gradient Constraints

Abstract

The resistivity method and time-domain-induced polarization (TDIP) are two branches of electric exploration that are used to solve problems in mineral exploration, hydrogeology and engineering geology. In recent years, integrating different physical parameters for joint inversion to improve the accuracy of inversion results has been extensively examined; however, three-dimensional joint inversion of the two methods above has not been realized. To further address this issue, in this research, we used the limited-memory BFGS (L-BFGS) method to develop a three-dimensional joint inversion algorithm of the resistivity method and TDIP based on the cross-gradient constraints. In the new algorithm, the resistivity method and TDIP inversion were iteratively updated alternately to ensure that the inversion results can simultaneously meet the two conditions of obtaining minimum data misfits and finding structural similarity. The three-dimensional synthetic dataset inversion results showed that the models obtained by joint inversion are more accurate in the recovery of both the boundaries and the values of the anomalies. Especially in the background of high noise, joint inversion has higher resolution for the target body. The joint inversion algorithm was also successfully applied to a groundwater detection practice in Beijing, China, in which the practicability of the algorithm was confirmed.

1. Introduction

Electrical exploration is one of the main methods of geophysical exploration. The resistivity method and TDIP are two branch methods of electrical exploration. The resistivity method studies the differences in electrical conductivity (expressed as resistivity) between different underground rocks, while TDIP studies the difference in the IP effect (expressed as polarizability) of electric field changes in a certain time span under the excitation of the stable current of different rocks and ore, and the polarizability of the medium is expressed through the ratio between the potential measured just after canceling the external field (secondary potential) and the potential measured just before canceling the applied field (primary potential). With the innovation of methods and technologies, the resistivity method and TDIP have been widely used in geological structure exploration, environmental pollution investigation, geological disaster warning and other fields [1,2,3].

With the continuous development of separate method inversion algorithms and the advancement of computing science, the accuracy of geophysical inversion has been increasingly scrutinized strictly. Vozoff and Jupp proposed the integration of various geophysical data for joint inversion [4]. After more than 40 years of development, joint inversion has been shown to play a vital role in suppressing the ambiguity of geophysical inversion and improving the accuracy of inversion. Joint inversion has become an important quantitative interpretation method in comprehensive geophysical research, and has promising developmental and application prospects [5].

Joint inversion refers to the simultaneous inversion of a variety of geophysical data, using different physical parameters to comprehensively interpret the same target and finally obtaining an inverted model that represents the actual geological situation more closely. Joint inversion is mainly divided into two categories: physical coupling and structural coupling. Joint inversion based on physical property coupling takes advantage of the connections between different known geophysical parameters (velocity, density, resistivity) or deduced parameters (saturation, permeability, reservoir porosity) and unknown geophysical parameters, and strives to strengthen the linear or nonlinear correlations of the physical properties to be inverted by incorporating all kinds of known physical parameters into an objective function. Joint inversion based on the coupling of physical properties has been applied in practice [6,7]. However, it is a complicated and time-consuming process to obtain prior information on rock physical properties, and the multimodal nature of rock physical data makes it almost impossible to accurately determine the spatial applicability of these rock physical relationships in complicated underground geophysical cases [8]. Although physical coupling applies stronger constraints than structural coupling, the inversion results will be undermined if the physical constraint conditions of rock and the true geological conditions of the study area are not well matched [9].

Structural coupling joint inversion is based on the premise that the underground rock mass structures and boundaries of different physical properties are identical or partially identical. The coupling is derived from the spatial structure similarity of different physical properties; therefore, the prior information of certain known physical properties is no longer mandatory, which makes it more applicable in practice. Gallardo and Meju proposed a scheme of joint inversion with the introduction of the cross-gradient constraint and carried out a study on the two-dimensional cross-gradient joint inversion of seismic travel time data and the resistivity data [10,11]. The core concept is to minimize the data misfit and the cross gradient of the models of different physical properties at the same time. When the cross gradient of the models is close to zero, the two models are structurally similar. Since then, many other joint inversion studies based on the cross-gradient constraint theory have been carried out. Joint inversion algorithms of more than two physical properties have been realized and widely used. Fregoso and Gallardo extended cross-gradient joint inversion to three-dimensional gravity and magnetic data [12]. Moorkamp et al. proposed a joint inversion framework containing multiple datasets of three-dimensional magnetotelluric, gravity and seismic refraction data, and summarized the advantages and disadvantages of both cross-gradient structural coupling and physical coupling [13]. Peng et al. developed a joint inversion algorithm of receiver functions and magnetotelluric data to determine the crustal and mantle structure beneath central Namche Barwa in the eastern Himalayan syntaxis [14]. Scholl et al. applied the cross-gradient constraint to the joint inversion of three-dimensional airborne electromagnetic data and gravity data [15]. Peng et al. combined magnetotelluric and wide-angle seismic reflection/refraction inversion to detect crustal structures and Moho discontinuities [16].

However, the three-dimensional joint inversion of the resistivity method and TDIP has not been realized. In this research, a three-dimensional joint inversion algorithm of the resistivity method and TDIP based on the cross-gradient constraint was developed by utilizing L-BFGS as the inversion tool. In the joint inversion process, the resistivity and polarizability models were iteratively updated alternately, and the cross gradient was used to constrain the inversion, thus improving the reliability and consistency of the inversion results. The stability and reliability of the algorithm were tested in three synthetic inversion cases. Then, the algorithm was applied to groundwater detection work, and geologically sound results were obtained, which verifies the practicability of the algorithm.

2. Three-Dimensional Forward Modeling Algorithm

2.1. Three-Dimensional Forward Modeling of the Resistivity Method

We used the finite difference method to realize the three-dimensional forward simulation of the resistivity method. According to forward modeling of the resistivity method derived by Ma et al. [17], the partial differential equation satisfied by the point-source resistivity method is as follows:

$$\nabla ·\left[\sigma \left(x,y,z\right)\nabla \phi \left(x,y,z\right)\right]=-j\delta (x-x_0)\delta (y-y_0)\delta (z-z_0),$$

(1)

where $\sigma$ is the electrical conductivity of rock ore (S·m⁻¹), $\phi$ is the potential array (V), $j$ is the electric current density (A·m⁻²) and $\delta$ is a Dirac function, which specifies the position of the supply point.

The finite difference method is used to discretize (1), forming large sparse linear equations [18], whose matrix form can be expressed as follows:

$$A\phi =b,$$

(2)

where $A$ is the coefficient matrix and $b$ is a vector that contains information with respect to the field source.

In this study, the biconjugate gradient stabilized method (BICGSTAB) based on incomplete LU decomposition was introduced to solve the equation to obtain the potential [19]. After the potential was obtained, the apparent resistivity was calculated according to (3):

$$\{\begin{array}{l}U=\alpha \phi \\ {\rho }_a=K\frac{\Delta U}{I}\end{array},$$

(3)

where $\alpha$ is the vector that is designed to select the potential at the receiver among all grid node potentials; $\Delta U$ is the potential difference between the receiving electrodes (V); $I$ is the electric current (A); and $K$ is the geometric factor.

2.2. Three-Dimensional Forward Modeling of TDIP

The difference between the forward modeling of TDIP and that of the resistivity method lies in whether the IP effect is taken into consideration. Therefore, the total field of TDIP is a combination of the fields, with and without the IP effect. The calculation of the total field was carried out in two steps: first, the potential or potential difference without the polarization effect is obtained by forward modeling; second, the conductivity $\sigma$ in (1) is replaced by the conductivity ${\sigma }_{\eta }$ with the IP effect in (4), and then the potential or potential difference $u_{\eta }$ in which the IP effect is considered is derived forward, and the apparent polarizability ${\eta }_a$ is calculated with (5) [20]:

$${\sigma }_{\eta }=\sigma (1-\eta ),$$

(4)

where $\eta$ is the polarizability of the anomaly, which is nondimensional.

$${\eta }_a=\frac{u_{\eta }-u}{u_{\eta }},$$

(5)

where $u$ is the potential or potential difference in which the IP effect is not considered.

3. Three-Dimensional Joint Inversion Algorithm

3.1. Cross-Gradient Function

The cross-gradient function defined by Gallardo and Meju [10] is as follows:

$$t(x,y,z)=\nabla m_a(x,y,z)\times \nabla m_b(x,y,z)=(t_x,t_y,t_z),$$

(6)

where $m_a$ and $m_b$ are models of different physical parameters. The specific forms of $t_x$, $t_y$ and $t_z$ are given as follows:

$$\{\begin{array}{l}{t}_x(m_a,m_b)=\frac{𝜕m_a(y,z)}{𝜕y}\frac{𝜕m_b(y,z)}{𝜕z}-\frac{𝜕m_b(y,z)}{𝜕y}\frac{𝜕m_a(y,z)}{𝜕z}\\ t_y(m_a,m_b)=\frac{𝜕m_a(x,z)}{𝜕x}\frac{𝜕m_b(x,z)}{𝜕z}-\frac{𝜕m_b(x,z)}{𝜕x}\frac{𝜕m_a(x,z)}{𝜕z}\\ t_z(m_a,m_b)=\frac{𝜕m_a(x,y)}{𝜕x}\frac{𝜕m_b(x,y)}{𝜕y}-\frac{𝜕m_b(x,y)}{𝜕x}\frac{𝜕m_a(x,y)}{𝜕y}\end{array},$$

(7)

The expression of the cross-gradient term is the following:

$${\mathrm{\Phi }}_{cg}=t^{T}t={\left[\begin{array}{l}{t}_x\\ t_y\\ t_z\end{array}\right]}^T\left[\begin{array}{l}{t}_x\\ t_y\\ t_z\end{array}\right]=t_x^T{t}_x+t_y^T{t}_y+t_z^T{t}_z,$$

(8)

By performing a Taylor expansion of $t$ against model $m$ ($m$ represents $m_a$ or $m_b$) at $m_0$, and ignoring partial derivatives of more than one order, we can obtain the following:

$$t=t_0+\frac{𝜕t}{𝜕m}(m-m_0),$$

(9)

Under the condition of homogeneous half space, $t_0=0$. Yan et al. used the finite difference method to discretize grids at different positions [21]. According to the formula he derived, $\raisebox{1ex}{$𝜕t$}\!\left/ \!\raisebox{-1ex}{$𝜕m$}\right.$ was discretized, making the discretization matrix $B_x=\raisebox{1ex}{$𝜕t_x$}\!\left/ \!\raisebox{-1ex}{$𝜕m$}\right.$, $B_y=\raisebox{1ex}{$𝜕t_y$}\!\left/ \!\raisebox{-1ex}{$𝜕m$}\right.$, $B_z=\raisebox{1ex}{$𝜕t_z$}\!\left/ \!\raisebox{-1ex}{$𝜕m$}\right.$ and $B={\left(B_x B_y B_z \right)}^T$, and we can obtain (10):

$$\frac{𝜕{\mathrm{\Phi }}_{cg}}{𝜕m}=2\left[{\left(B_x\right)}^T{B}_x+{\left(B_y\right)}^T{B}_y+{\left(B_z\right)}^T{B}_z\right]\left(m-m_0\right)=2B^{T}B\left(m-m_0\right)$$

(10)

In the joint inversion, we strengthened the structural similarity between different physical models by minimizing the cross-gradient term. The joint inversion based on the cross-gradient theory has another advantage: when there is no correlation between $m_a$ and $m_b$, the cross-gradient term equals zero, which can avoid artifacts that may arise during the joint inversion.

3.2. Three-Dimensional Joint Inversion Objective Function

The strategy of constructing two joint inversion objective functions was adopted [22], in which the conductivity and polarizability models are updated alternately. The weighted cross-gradient constraint term was added into both objective functions. The objective function of the joint inversion is composed of the data misfit term, the model smoothing constraint and the cross-gradient constraint. The objective functions of the resistivity method and TDIP inversion are defined as follows:

$$\{\begin{array}{l}\mathrm{\Phi }(m_{\sigma })={\mathrm{\Phi }}_d(m_{\sigma })+{\lambda }_{\sigma }{\mathrm{\Phi }}_m(m_{\sigma })+{\beta }_{\sigma }{\mathrm{\Phi }}_{cg}(m_{\sigma },m_{\eta })\\ \mathrm{\Phi }(m_{\eta })={\mathrm{\Phi }}_d(m_{\eta })+{\lambda }_{\eta }{\mathrm{\Phi }}_m(m_{\eta })+{\beta }_{\eta }{\mathrm{\Phi }}_{cg}(m_{\eta },m_{\sigma })\end{array}$$

(11)

where $m_{\sigma }$ and $m_{\eta }$ are the natural logarithm of the conductivity model and polarizability model, respectively, which can avoids negative conductivity or polarizability during inversion; ${\mathsf{\Phi }}_d$ is the data misfit term, ${\mathsf{\Phi }}_m$ is the model smoothness constraint, ${\mathsf{\Phi }}_{cg}$ is the cross-gradient constraint; ${\lambda }_{\sigma }$ and ${\lambda }_{\eta }$ are the regularization weighting factors of the model smoothness constraints, and ${\beta }_{\sigma }$ and ${\beta }_{\eta }$ are the regularization weighting factors of the cross gradients.

${\mathsf{\Phi }}_d$, ${\mathsf{\Phi }}_m$ and ${\mathsf{\Phi }}_{cg}$ were expanded to obtain the specific form of the objective function:

$$\{\begin{array}{l}\mathrm{\Phi }(m_{\sigma })={\left[d_{\rho ,obs}-F(e^{m_{\sigma }})\right]}^T{V}_{\rho }^{-1}\left[d_{\rho ,obs}-F(e^{m_{\sigma }})\right]+{\lambda }_{\sigma }{(m_{\sigma }-m_{\sigma ,0})}^T{C}_{m,\sigma }{}^{-1}(m_{\sigma }-m_{\sigma ,0})+{\beta }_{\sigma }{t}^T(m_{\sigma },m_{\eta })t(m_{\sigma },m_{\eta })\\ \mathrm{\Phi }(m_{\eta })={\left[d_{\eta ,obs}-F(e^{m_{\eta }})\right]}^T{V}_{\eta }^{-1}\left[d_{\eta ,obs}-F(e^{m_{\eta }})\right]+{\lambda }_{\eta }{(m_{\eta }-m_{\eta ,0})}^T{C}_{m,\eta }{}^{-1}(m_{\eta }-m_{\eta ,0})+{\beta }_{\eta }{t}^T(m_{\eta },m_{\sigma })t(m_{\eta },m_{\sigma })\end{array},$$

(12)

where $d_{\rho ,obs}$ and $d_{\eta ,obs}$ are the apparent resistivity and apparent polarizability data, respectively; $F\left(m\right)$ is the forward response; $V_{\sigma }$ and $V_{\eta }$ are the data covariance matrices, mainly to avoid overfitting the observed data with poor quality in the inversion process; $C_{m,\sigma }{}^{-1}$ and $C_{m,\eta }{}^{-1}$ are the model covariance matrices of conductivity and polarizability, respectively.

The partial derivative of the joint objective function with respect to model $m$ is as follows:

$$\{\begin{array}{l}\frac{𝜕\mathrm{\Phi }(m_{\sigma })}{𝜕m_{\sigma }}=-2J_{\sigma }^T{V}_{\sigma }^{-1}\left[d_{\rho ,obs}-F(e^{m_{\sigma }})\right]+2{\lambda }_{\sigma }{C}_{m,\sigma }{}^{-1}(m_{\sigma }-m_{\sigma ,0})+2{\beta }_{\sigma }{B}_{\sigma }^T{B}_{\sigma }(m_{\sigma }-m_{\sigma ,0})\\ \frac{𝜕\mathrm{\Phi }(m_{\eta })}{𝜕m_{\eta }}=-2J_{\eta }^T{V}_{\eta }^{-1}\left[d_{\eta ,obs}-F(e^{m_{\eta }})\right]+2{\lambda }_{\eta }{C}_{m,\eta }{}^{-1}(m_{\eta }-m_{\eta ,0})+2{\beta }_{\eta }{B}_{\eta }^T{B}_{\eta }(m_{\eta }-m_{\eta ,0})\end{array},$$

(13)

where $J_{\sigma }$ and $J_{\eta }$ are the sensitivity matrices of conductivity inversion and polarizability inversion, respectively, Ma derived the calculation of the sensitivity matrices of the resistivity method and TDIP [18].

Egbert and Kelbert proposed a simple and efficient method to smooth the gradient of model parameters [23]. First, an affine linear transformation of model parameters is defined:

$$\tilde{m}=C_m{}^{-1/2}\left(m-m_0\right),$$

(14)

Equation (13) can be rewritten as follows:

$$\{\begin{array}{l}\frac{𝜕\mathrm{\Phi }(m_{\rho })}{𝜕{\tilde{m}}_{\sigma }}=-2C_{m,\sigma }{}^{1/2}{J}_{\sigma }^T{V}_{\sigma }^{-1}\left[d_{\rho ,obs}-F(e^{m_{\sigma }})\right]+2{\lambda }_{\sigma }{\tilde{m}}_{\sigma }+2{\beta }_{\sigma }{C}_{m,\sigma }{}^{1/2}{B}_{\sigma }^T{B}_{\sigma }{C}_{m,\sigma }{}^{1/2}{\tilde{m}}_{\sigma }\\ \frac{𝜕\mathrm{\Phi }(m_{\eta })}{𝜕{\tilde{m}}_{\eta }}=-2C_{m,\eta }{}^{1/2}{J}_{\eta }^T{V}_{\eta }^{-1}\left[d_{\eta ,obs}-F(e^{m_{\eta }})\right]+2{\lambda }_{\eta }{\tilde{m}}_{\eta }+2{\beta }_{\eta }{C}_{m,\eta }{}^{1/2}{B}_{\eta }^T{B}_{\eta }{C}_{m,\eta }{}^{1/2}{\tilde{m}}_{\eta }\end{array},$$

(15)

In traditional methods, the calculation of the Jacobi matrix and Hessian matrix requires a large amount of storage space. L-BFGS is used to solve the objective function and its gradient, and double cyclic recursive implicit is used to avoid the storage of the Hessian matrix [24]. After obtaining $\tilde{m}$, the following inverse transformation equation is used to solve m:

$$m=C_m{}^{1/2}\tilde{m}+m_0$$

(16)

3.3. Process of Three-Dimensional Joint Inversion

The typical routine of the conventional TDIP inversion is to first invert the conductivity, and then the inversion of polarizability is fulfilled based on the conductivity model, which usually leads to the excessive dependence of the polarizability result on the conductivity result. In this study, the alternating iteration method to invert the conductivity and polarizability models was applied in the algorithm, and the inversion results of the two parameters were mutually constrained by means of the cross gradient. Thus, on the basis of minimizing the data misfit of separate inversions, a model in which the conductivity and polarizability share a structurally similar distribution can be found. The process of joint inversion is shown in Figure 1, which can be briefly summarized in the following steps:

(1)

Input the initial conductivity and polarizability model.

(2)

Run the separate conductivity inversion. After several iterations, the conductivity model with a certain abnormal form is obtained, and then separate TDIP inversion is performed based on the conductivity model to update the polarizability model. The cross-gradient term calculated by the models derived from a separate inversion, which is substituted into (11).

(3)

Start joint inversion, and the conductivity and polarizability models with cross-gradient constraints are calculated using several iterations through joint inversion.

(4)

Repeat step (3) several times and update the conductivity and polarizability models iteratively, in which the cross-gradient term is calculated by the latest models that derived from joint inversion. Check whether the iteration termination criteria are met. If so, the iteration is terminated; if not, the alternating iteration continues.

(5)

The termination criteria of inversion are that the data misfit is less than a predefined threshold or the number of iterations reaches a set value. The formula for calculating the misfit (root mean square, RMS) is the following:

$$RMS=\sqrt{\frac{{\left[d_{obs}-F(m)\right]}^T{V}^{-1}\left[d_{obs}-F(m)\right]}{N}},$$

(17)

where $N$ is the number of data points.

(6)

If the convergence criterion established for one of the properties is reached, then the renewal of this property is stopped, while the other property continues to be updated until both properties meet the convergence condition.

4. Synthetic Examples Tests

The three-dimensional joint inversion algorithm of the resistivity method and TDIP was implemented based on the theory above, and the theoretical model synthesis data were used for inversion trial calculation to verify the correctness of the joint inversion algorithm.

4.1. Example I: A Single Prism Model

The true model in example I is a prismatic body with a length, width and height of 10 m, 10 m and 7 m, respectively, and the buried depth of the top surface is 5 m. The resistivity of the abnormal body is ρ1 = 10 Ω·m, and the polarizability is η1 = 0.2. The abnormal body is buried in the uniform half space with resistivity ρ2 = 100 Ω·m and polarizability η2 = 0.01. Figure 2a,b show slices of the model at X = 0 m and Z = 8 m. The observation system is shown in Figure 2c: the electrodes are arranged in a rectangular square matrix. When each transmitting electrode transmits, all adjacent receiving electrodes in the X direction receive at the same time to ensure that enough information can be obtained in three-dimensional space. The model response was obtained through forward calculation, and a total of 22,819 data points were chosen as the observed data.

A Gaussian random error of 5% was added to the observed data, which was used to test the algorithms of separate inversion and joint inversion. A uniform half-space with a resistivity of 100 Ω·m and a polarizability of 0.01 was designed as the initial model for inversion. In the inversion, the regularization factor was set to a constant. To obtain a better result, we used multiple sets of different values of ${\lambda }_{\sigma }$, ${\lambda }_{\eta }$, ${\beta }_{\sigma }$ and ${\beta }_{\eta }$ for testing. Ultimately, it was found that the inversion results are best when the values of ${\mathsf{\Phi }}_m$ and ${\mathsf{\Phi }}_{cg}$ are about one-tenth of ${\mathsf{\Phi }}_d$. In this case, the value range of $\lambda$ is 0.1~10, and the value range of $\beta$ is 10~1000. Larger $\lambda$ and $\beta$ will result in the RMS not converging, and smaller $\lambda$ and $\beta$ will result in ${\mathsf{\Phi }}_m$ and ${\mathsf{\Phi }}_{cg}$ being unable to play the role of constrained inversion. The final inversion results are shown in Figure 3.

The inversion results were analyzed with horizontal and vertical slicing diagrams of the resistivity and polarizability models. As shown in Figure 3a,c, the separate inversion results differed from the true model in terms of convergence degree and boundary recovery. Compared with the single inversion, the results of joint inversion are closer to the true model and converge better at the boundary (Figure 3b,d). Joint inversion overcomes some shortcomings of separate inversion, and improves the accuracy of inversion to a certain extent.

The RMS values of the resistivity method and TDIP obtained in separate inversion were 0.724 and 0.849, respectively, and that in joint inversion were 0.722 and 0.802, respectively. The variation in the RMS value of the resistivity method is not obvious, while that of TDIP had a small decrease, which indicates that the TDIP results of joint inversion are better than separate inversion.

Since there was little difference in RMS values of the resistivity method, to quantitatively evaluate the inversion results, we evaluated the model misfit between the true model and the joint inversion model. The calculation of model perturbation RMS refers to the method proposed by Moorkamp [13], which is defined as follows:

$$\zeta =\frac{1}{M_p}\sqrt{{{\displaystyle \sum _{i=1}^{M_p}\left(\frac{m_i^{inv}-m_i^{true}}{m_i^{true}}\right)}}^2},$$

(18)

where $m_i^{true}$ is the true model, $m_i^{inv}$ is the result of the separate method or joint inversion, and $M_p$ is the total number of grid cells where the prism is located. In the calculation, only the grid cell where the prism is located was considered, rather than all of the grid cells. In this way, the recovery degree of the prism can be reflected more intuitively.

The values for separate inversion were ${\zeta }_{\rho }$ = 7.59 and ${\zeta }_{\eta }$ = 0.64, while those for joint inversion were ${\zeta }_{\rho }$ = 4.02 and ${\zeta }_{\eta }$ = 0.47, which indicates that the results of the joint inversion are more similar to the true model.

As shown in Figure 3a,c, there are structural differences between the resistivity and polarizability models of separate inversion, which are reduced in the joint inversion (Figure 3b,d). To verify the degree of structural similarity between the models with different physical properties, we also compared the cross-gradient values of the separate inversion model and the joint inversion model (Column 3 in Figure 3). The cross-gradient values obtained by single inversion were higher, while the values were significantly decreased by joint inversion. This observation was also confirmed by a comparison of the average cross-gradient values of the final model:

$$\overline{t}=\frac{1}{M}\sqrt{{\displaystyle \sum _{i=1}^M\left[{(t_{x,i})}^2+{(t_{y,i})}^2+{(t_{z,i})}^2\right]}},$$

(19)

where M represents the number of cells in the grid.

The value of $\overline{t}$ obtained in separate inversion was 5.74 × 10⁻⁴, while it was 2.34 × 10⁻⁴ in joint inversion, indicating that the resistivity and polarizability models obtained by the joint inversion are more similar in structure.

4.2. Example II: A Single Prism Model in a High Noise Background

The true model in example II is the same as the model in example I. A 20% Gaussian random error was added to the observed data to form the synthesized data under the high noise background. The initial conditions of inversion were the same as those of example I, and the inversion results are shown in Figure 4.

As shown in Figure 4a,c, under the background of high noise, the model of separate inversion is distorted on the boundary. Compared with separate inversion, joint inversion can suppress the effect of high noise and recover the shape and value of abnormal bodies better, especially near the boundary of anomalies (Figure 4b,d). The cross-gradient value of the joint inversion is significantly reduced compared with the separate inversion (column 3 in Figure 4).

The RMS values of the resistivity method and TDIP obtained in separate inversion were 1.762 and 1.821, respectively, and that in joint inversion were 1.764 and 1.801, respectively. The variation in RMS values of the resistivity method and TDIP is not obvious, because RMS represents the data misfit, but the response data cannot reflect the real information of the underground model in the high noise background. Therefore, $\zeta$ is more suitable for evaluating inversion results than the RMS.

The values obtained for separate inversion were ${\zeta }_{\rho }$ = 17.89 and ${\zeta }_{\eta }$ = 0.65, while those for joint inversion were ${\zeta }_{\rho }$ = 3.59 and ${\zeta }_{\eta }$ = 0.44. The $\overline{t}$ in separate inversion was 4.67 × 10⁻⁴, while it was 3.73 × 10⁻⁴ in joint inversion. The values of $\zeta$ and $\overline{t}$ were smaller in the joint inversion than in the separate inversion, especially the value of ${\zeta }_{\rho }$, which decreased significantly.

4.3. Example III: Double Prisms Model in High Noise Background

The true model in synthetic example III is composed of two prism bodies with a length, width and height of 6 m, 5 m and 5 m, respectively. The burial depth of the top surface of the two abnormal bodies is 5 m, and the interval is 6 m. The resistivity of the abnormal body is ρ1 = 10 Ω·m, and the polarizability is η1 = 0.2. The abnormal body is buried in the uniform half space with resistivity ρ2 = 100 Ω·m and polarizability η2 = 0.01. Figure 5a,b show slices of the model at X = 0 m and Z = 8 m, respectively. The observation system used was the same as in example I (Figure 5c).

A 20% Gaussian random error was added to the observed data derived from the model forward. A uniform half-space with a resistivity of 100 Ω·m and a polarizability of 0.01 was designed as the initial model for inversion, and the inversion results are shown in Figure 6.

As shown in Figure 6a,c, the separate inversion results of resistivity are distorted in the background of high noise, and a large number of false anomalies appear around the abnormal bodies, making it difficult to recover the abnormal body to the true model, and difficult to distinguish the two adjacent abnormal bodies. The separate inversion results of polarizability have a small deviation to the left and right. Compared with separate inversion, joint inversion can suppress the influence of high noise, recover the shape and value of the anomalies better, identify the location of the anomalies effectively, and accurately distinguish the two adjacent anomalous bodies (Figure 6a,b).

The RMS values of the resistivity method and TDIP obtained in separate inversion were 1.765 and 1.807, respectively, and that in joint inversion were 1.764 and 1.790, respectively. The variation in RMS values of the resistivity method and TDIP is not obvious, which is the same as example II.

The values obtained for separate inversion were ${\zeta }_{\rho }$ = 15.03 and ${\zeta }_{\eta }$ = 0.45, while those for joint inversion were ${\zeta }_{\rho }$ = 12.3 and ${\zeta }_{\eta }$ = 0.4. The value of $\overline{t}$ in separate inversion was 1.09 × 10⁻⁴, while it was 0.28 × 10⁻⁴ in joint inversion. The values of $\zeta$ and $\overline{t}$ decreased to different degrees in the joint inversion.

The statistical results of $\zeta$ and $\overline{t}$ of separate inversion and joint inversion for examples I, II and III are shown in

Table 1. $\zeta$ and $\overline{t}$ values of separate and joint inversion.

Parameter	Example	Separate Inv		Joint Inv
		ρ	η	ρ	η
$\zeta$	I	7.59	0.64	4.02	0.47
	II	17.89	0.65	3.59	0.44
	III	15.03	0.45	12.3	0.4
$\overline{t}$	I	5.74 × 10−4		2.34 × 10−4
	II	4.67 × 10−4		3.73 × 10−4
	III	1.09 × 10−4		0.28 × 10−4

. As seen in the table, the joint inversion is superior to the separate inversion in the degree of model misfit, and the average cross-gradient value of the joint inversion results decreases. Thus, the effectiveness and superiority of joint inversion is verified.

5. Application

The three-dimensional joint inversion algorithm was applied to groundwater detection. The Miyun District of Beijing is the area with the most abundant groundwater resources, which plays an important role in the urban water supply system of Beijing. The Quaternary pore water in the region mainly exists of sand and gravel formed by the impact of the Chaobai River, which is a single aquifer. There are many monitoring wells for regularly monitoring groundwater levels and electrical conductivity (EC).

The area between monitoring well MY2 (40°21′51″, 116°48′9″) and the river was chosen as the study area.

Table 2. Groundwater level and EC values of borehole MY2.

Data	October 2021	November 2021	December 2021	January 2022	April 2022	June 2022	August 2022
water level (m)	17.68	16.7	19.37	16.54	16.88	17.35	17.4
EC (μs/cm)	866	842	832	657	827	868	765

shows the statistical table of the groundwater level and EC value of MY2 in the past year. As shown in Table 2, the variation in the groundwater table is not regular, and the EC value is nonlinearly correlated with the groundwater table value. The project has two purposes: (1) to determine the groundwater level between the MY2 monitoring well and Chaobai River; (2) to determine whether there are abnormal bodies that cause changes in the EC value of groundwater.

To achieve the purposes above, five measuring lines (L1~L5) were arranged in the research area. To display the research results more intuitively, the projection of the midpoint of the measuring line at the riverbank was taken as the origin O. The direction of the river is positive on the X-axis, and the direction on the right bank of the river is positive on the Y-axis. A rectangular coordinate system was established to obtain the distribution of the measuring lines (Figure 7). L1 to L5 correspond to Y = 30 m, 70 m, 100 m, 130 m and 170 m, respectively. The receiving electrodes were spaced at 3 m equal intervals along the black measuring line.

The GDD full waveform high power IP equipment was used for data acquisition, which can measure the apparent resistivity and apparent polarizability. The technical specifications of the measuring equipment are as follows: steady current accuracy 0.001; 10 A transmission current; 150–2400 V voltage; resolution 1 μV; accuracy 0.8%; operating temperature range: −45 °C to ~60 °C. In the process of data acquisition, we used a three-pole device for measurement. When each transmitting electrode transmits, all receiving electrodes located on the same line receive simultaneously; the power supply cycle is 8 s; each data point is superimposed at least 10 times; 20 windows are used in decay curve reconstruction.

After preprocessing the collected data, a total of 24,540 data points were chosen as the measured data, and then separate inversion and joint inversion were carried out. The inversion results of separate inversion and joint inversion are shown in Figure 8. The first and second columns of Figure 8 show the resistivity results of separate and joint inversion, respectively, and the third and fourth columns show the polarizability results of separate and joint inversion, respectively.

The separate inversion results of resistivity had good stratification, and showed that the farther away from the river, the thicker the high-resistivity layer and the larger the resistivity value. The separate inversion achieved good results, while there was little difference between separate inversion and joint inversion. The separate inversion results of polarizability show a low layer on the surface, which increases with increasing depth, but it does not present a layered distribution with depth. In particular, as the groundwater level becomes increasingly deeper in the area far from the river, the IP effect becomes weaker, the value of polarizability obtained by inversion is too small, the polarizability model appears distorted, and it is difficult to reflect the information of the groundwater level. Compared with the separate inversion results of the polarizability, the polarizability model obtained by the joint inversion results has clearer morphology, more obvious interface, and better stratification, and can more directly reflect the layered distribution of groundwater.

The RMS values of the resistivity method and TDIP obtained in separate inversion were 2.351 and 2.72, respectively, and that in joint inversion were 2.278 and 2.68, respectively. The variation in RMS values of the resistivity method and TDIP is not obvious, and the $\zeta$ cannot be calculated since the true model is unknown. Therefore, we compared the separate inversion and the joint inversion through the value of $\overline{t}$ and the groundwater level of the borehole.

The value of $\overline{t}$ obtained in separate inversion was 6.14 × 10⁻⁶, while it was 5.77 × 10⁻⁶ in joint inversion, which indicates that the resistivity and polarizability models obtained by the joint inversion are more similar in structure. After joint inversion, the average cross-gradient value decreased, and the cross-gradient values of the five sections in Figure 8 were calculated, as shown in Figure 9. As shown in Figure 9, the cross-gradient value is larger at the surface and does not decrease in joint inversion, which is caused by the influence of various factors, such as silty clay, corrosive material, pebbles and water stagnation in the upper layer. However, below 10 m underground, the cross-gradient values decreased significantly, and the farther from the river, the more obvious the cross-gradient value decreased. The polarizability results of joint inversion have fewer false anomalies, and can better reflect the groundwater level.

Combined with local geological data, geological interpretation of the inversion results was made (Figure 10). As shown in Figure 10, the surface layer of 0~3 m is mainly a silty clay layer. Irrigation and vegetation cover on the surface resulted in good water retention and resistivity of approximately 100 Ω·m. The strata below 3 m are all gravel beds, but due to the influence of groundwater, there is a great difference in resistivity. The resistivity of the non-water-containing gravel layer was up to 1000 Ω·m, while that of the water-containing gravel layer was down to 10~100 Ω·m.

Finally, we obtained the groundwater level in the study area, and the following conclusions can be drawn: the stratification of the strata in the study area is good, no obvious abnormal body was found, and the change in EC in the monitoring well is mainly affected by the recharge water of the river.

6. Conclusions

In this research, the three-dimensional joint inversion of the resistivity method and TDIP is realized based on a cross-gradient coupling constraint.

Separate inversion and joint inversion trial calculations were carried out on three examples, the results showed that the resistivity and polarizability models obtained by joint inversion can improve the recovery of model values and boundaries. Especially in high background noise, joint inversion can suppress the influence of background noise and identify the location of abnormal bodies more accurately with higher resolution. Through the analysis of three synthetic examples, the average cross-gradient value of joint inversion is smaller, which indicates that the resistivity and polarizability models are more similar in structure, and the resistivity method and TDIP can achieve the effect of mutual constraint and complementary advantages.

Joint inversion was applied to groundwater detection, and information on the groundwater level in the study area was finally determined. We also confirmed that the soluble ions in the monitoring well mainly come from river water. Compared with separate inversion, joint inversion can obtain more accurate information underground, which provides a new concept for solving practical problems in geophysical exploration and geologic mapping.