Joint Inversion Method of Gravity and Magnetic Data with Adaptive Zoning Using Gramian in Both Petrophysical and Structural Domains

Abstract

Different observation data are utilized to obtain a unified geophysical model based on the correlations of underground geological bodies in joint inversions. By specifying a type of Gramian constraints, Gramian as a coupling term can link geophysical models through relationships of physical properties or structural similarities. Considering the complex relationships of physical properties of underground geological bodies, we proposed an adaptive zoning method to automatically divide the whole inversion area into subregions with different relationships of physical properties and to determine the number and range of subregions that utilized correlation between geophysical data before joint inversions. On this basis, we considered the use of a combination of Gramian coupling terms rather than one term to link petrophysical and structural domains during joint inversions. Synthetic tests showed that the algorithm is capable of having a robust estimate of the spatial distribution and relationships between density and magnetization intensity of geological bodies. The idea was also applied to the ore concentration area in the middle and lower reaches of the Yangtze River to obtain the three-dimensional (3-D) distribution model of magnetite-bearing rocks within 5 km underground, which corresponds well with the existing shallow ore sites and demonstrates the existence of available deep resources in the study area.

Appendices

Appendices

Appendix A: Pseudo-Code for Adaptive Zoning Method

The pseudo-code of the proposed adaptive zoning method is as follows:

Appendix B: Conjugate Gradient Algorithm

The objective functions can be minimized by an iterative algorithm. We used a subscript on variable to represent the value of that variable at iteration k, with k increasing from 0. The kth iterative conjugate gradient algorithm is as follows.

Step 1. Calculate the initial residual error

$${\text{r}}_{k} = {\mathbf{A}}^{{\text{T}}} ({\mathbf{Am}}_{k} - {\mathbf{d}}_{{{\text{obs}}}} )$$

(B1)

Step 2. Calculate the search direction of the Gramian constraints

$$\begin{aligned} (\ell_{G}^{(1)} )_{k}^{i} = & (\tau_{{{\text{A}}_{{1}} }} )_{k}^{i} \left( {\left\| {{\mathbf{k}}_{\sigma }^{i} {\mathbf{M}}_{k} } \right\|_{2}^{2} {\mathbf{k}}_{\sigma }^{i} ({\mathbf{m}}_{(1)} )_{k} - ({\mathbf{k}}_{\sigma }^{i} {\mathbf{M}}_{k} ,{\mathbf{k}}_{\sigma }^{i} ({\mathbf{m}}_{(1)} )_{k} ){\mathbf{k}}_{\sigma }^{i} {\mathbf{M}}_{k} } \right) \\ & \quad + \, (\tau_{{\text{B}_{1} }} )_{k}^{i} (\left\| {{\mathbf{k}}_{\sigma }^{i} \nabla {\mathbf{M}}_{k} } \right\|_{2}^{2} {\mathbf{k}}_{\sigma }^{i} \nabla^{2} ({\mathbf{m}}_{(1)} )_{k} - ({\mathbf{k}}_{\sigma }^{i} \nabla {\mathbf{M}}_{k} ,{\mathbf{k}}_{\sigma }^{i} \nabla ({\mathbf{m}}_{(2)} )_{k} ){\mathbf{k}}_{\sigma }^{i} \nabla^{2} {\mathbf{M}}_{k} ) \\ \end{aligned}$$

(B2)

$$\begin{aligned} (\ell_{G}^{(2)} )_{k}^{i} = & (\tau_{{\text{A}_{2} }} )_{k}^{i} \left( {\left\| {{\mathbf{k}}_{{\text{M}}}^{i} ({\mathbf{m}}_{\sigma } )_{k} } \right\|_{2}^{2} {\mathbf{k}}_{{\text{M}}}^{i} ({\mathbf{m}}_{(2)} )_{k} - ({\mathbf{k}}_{{\text{M}}}^{i} ({\mathbf{m}}_{(2)} )_{k} ,{\mathbf{k}}_{{\text{M}}}^{i} ({\mathbf{m}}_{\sigma } )_{k} ){\mathbf{k}}_{{\text{M}}}^{i} ({\mathbf{m}}_{\sigma } )_{k} } \right) \\ & \quad + (\tau_{{\text{B}_{2} }} )_{k}^{i} \left( {\left\| {{\mathbf{k}}_{{\text{M}}}^{i} \nabla ({\mathbf{m}}_{\sigma } )_{k} } \right\|_{2}^{2} {\mathbf{k}}_{{\text{M}}}^{i} \nabla^{2} ({\mathbf{m}}_{(2)} )_{k} - ({\mathbf{k}}_{{\text{M}}}^{i} \nabla ({\mathbf{m}}_{(2)} )_{k} ,{\mathbf{k}}_{{\text{M}}}^{i} \nabla ({\mathbf{m}}_{\sigma } )_{k} ){\mathbf{k}}_{{\text{M}}}^{i} \nabla^{2} ({\mathbf{m}}_{\sigma } )_{k} } \right) \\ \end{aligned}$$

(B3)

Step 3. Calculate the gradient of the objective function

$${\mathbf{\ell }}_{k} = {\mathbf{r}}_{k} + \alpha_{k} {\mathbf{m}}_{k} + \sum\limits_{i = 1}^{{\text{N}}} {({\mathbf{\ell }}_{G} )_{k}^{i} }$$

(B4)

Step 4. Calculate the conjugate gradient direction

$${\mathbf{p}}_{k} = {\mathbf{\ell }}_{k} + \frac{{\left\| {{\mathbf{\ell }}_{k} } \right\|^{2} }}{{\left\| {{\mathbf{\ell }}_{k - 1} } \right\|^{2} }}{\mathbf{p}}_{k - 1} , \, {\mathbf{p}}_{0} = {\mathbf{\ell }}_{0}$$

(B5)

Step 5. Calculate the model iteration step

$${\varvec{s}}_{k} = {{\left( {{\mathbf{p}}_{k} ,{\mathbf{\ell }}_{k} } \right)} \mathord{\left/ {\vphantom {{\left( {{\mathbf{p}}_{k} ,{\mathbf{\ell }}_{k} } \right)} (}} \right. \kern-0pt} (}\left\| {{\mathbf{A}}_{k} {\mathbf{p}}_{k} } \right\|^{2} + \alpha_{k} \left\| {{\mathbf{p}}_{k} } \right\|^{2} + \tau_{k} {\text{S}}_{k} )$$

(B6)

Step 6. Update model

$${\mathbf{m}}_{k + 1} = {\mathbf{m}}_{k} - {\varvec{s}}_{k} {\mathbf{p}}_{k}$$

(B7)

where $({\mathbf{m}}_{(1)} )_{k} = \left[ {({\mathbf{M}}_\text{x} )_{k} ,({\mathbf{M}}_\text{y} )_{k} ,({\mathbf{M}}_\text{z} )_{k} } \right]$, ${(}{\mathbf{M}})_{k} = \sqrt {({\mathbf{M}}_\text{x} )_{k}^{2} + ({\mathbf{M}}_\text{y} )_{k}^{2} + ({\mathbf{M}}_\text{z} )_{k}^{2} }$, which represents the value of magnitude of magnetization vector at iteration k, ${\mathbf{k}}_{{\text{M}}}^{i} { = }\left[ {\begin{array}{*{20}c} {{\mathbf{k}}_{\sigma }^{i} } & {{\mathbf{k}}_{\sigma }^{i} } & {{\mathbf{k}}_{\sigma }^{i} } \\ \end{array} } \right]^{{\text{T}}}$, and ${(}{\mathbf{m}}_{\sigma } )_{k} { = }\left[ {\begin{array}{*{20}c} {{(}{\mathbf{m}}_{(1)} )_{k} } & {{(}{\mathbf{m}}_{(1)} )_{k} } & {{(}{\mathbf{m}}_{(1)} )_{k} } \\ \end{array} } \right]^{{\text{T}}}$.When we find the minimum of the objective function of Method A, ${(}\tau_{{{\text{A}}_{1} }} )_{k}^{i} = 0$.Similarly, ${(}\tau_{{{\text{B}}_{1} }} )_{k}^{i} = 0$ for Method B.

Appendix C: L ₁ Norms

We have adopted L₂ norms for regularization terms in objective functions. Inversions with L₂ norms usually produce a smooth image of the subsurface structure, while inversions with L₁ norms provide sparse reconstructed solutions with sharp boundaries (Ghalehnoee and Ansari 2022). We have demonstrated through the preceding synthetic tests that algorithm of joint inversion method with adaptive zoning using Gramian in petrophysical and structural domains (named as Method C in Fig. 2) can enhance unknown property relationships and spatial characteristic by using Gramian to link petrophysical and structural domains after adaptive zoning. We now investigate how the algorithm would behave on model recovery through the use of L₁ norms. The objective functions of Method C with L₁ norms can be written as:

$$\left\{ {\begin{array}{*{20}c} {\varphi_{{{\text{C}}_{1} }} = \left\| {{\mathbf{W}}_{{{\text{d}}_{{1}} }} ({\mathbf{A}}_{{\text{g}}} {\mathbf{W}}_{{\text{G}}} {\mathbf{W}}_{{\text{G}}}^{ - 1} \rho - {\mathbf{d}}^{{\text{g}}} )} \right\|_{2}^{2} + \alpha_{1} \left\| {{\mathbf{W}}_{{\text{G}}}^{ - 1} {\mathbf{W}}_{{{\text{L}}_{{1}} }}^{{\text{G}}} \rho } \right\|_{1} + S_{{{\text{C}}_{1} }} } \\ {\varphi_{{{\text{C}}_{2} }} = \left\| {{\mathbf{W}}_{{{\text{d}}_{2} }} \left( {\left[ {{\mathbf{A}}_{{{\text{M}}_\text{x} }} {\mathbf{W}}_{{\text{M}}} {, }{\mathbf{A}}_{{{\text{M}}_\text{y} }} {\mathbf{W}}_{{\text{M}}} {, }{\mathbf{A}}_{{{\text{M}}_\text{z} }} {\mathbf{W}}_{{\text{M}}} } \right]\left[ {\begin{array}{*{20}l} {{\mathbf{W}}_{{\text{M}}}^{ - 1} {\mathbf{M}}_\text{x} \, } \\ {{\mathbf{W}}_{{\text{M}}}^{ - 1} {\mathbf{M}}_\text{y} \, } \\ {{\mathbf{W}}_{{\text{M}}}^{ - 1} {\mathbf{M}}_\text{z} } \\ \end{array} } \right] - {\mathbf{d}}^{{\text{m}}} } \right)} \right\|_{2}^{2} + \alpha_{2} \left\| {\left[ {\begin{array}{*{20}l} {{\mathbf{W}}_{{\text{M}}}^{ - 1} {\mathbf{W}}_{{L_{1} }}^{{\text{M}}} {\mathbf{M}}_\text{x} \, } \\ {{\mathbf{W}}_{{\text{M}}}^{ - 1} {\mathbf{W}}_{{L_{1} }}^{{\text{M}}} {\mathbf{M}}_\text{y} \, } \\ {{\mathbf{W}}_{{\text{M}}}^{ - 1} {\mathbf{W}}_{{L_{1} }}^{{\text{M}}} {\mathbf{M}}_\text{z} \, } \\ \end{array} } \right]} \right\|_{1} + {\text{S}}_{{{\text{C}}_{{2}} }} } \\ \end{array} } \right.$$

(C1)

where ${\mathbf{W}}_{{L_{1} }}^{j} = {\text{diag}}\left( {\frac{1}{{(({\mathbf{m}}_{(j)} )^{2} + \varepsilon^{2} )^{1/4} }}} \right),j = 1,2$, and we use $\varepsilon^{2} = 1e^{ - 10}$.

We applied Method C with L₁ norms to invert gravity and magnetic anomalies produced by Combined Model, and the cross-sections of inversion results are shown in Fig. 21. The RMS data misfits  are also calculated to monitor inversion, and it can be found that the error no longer decreases, but the iteration termination condition is not reached after about 50 iterations (Fig. 21c and d). Though the recovered density value is closer to the true value in the cross-section of the density (Fig. 21a) compared with that obtained by Method C with L₂ norms, but the performance of recovering deep geological body is not good. Inversion methods with L₁ norms can recover physical properties which are about the same as the true values and obtain inverted models with sharp boundaries. However, the choose of upper and lower bounds faces challenge to handle the absence of prior information.

Fig. 21

Cross-sections of the inversion results of a density ρ and b magnetization intensity M obtained by method C with L₁ norms; curves of RMS data misfit of inversions of gravity data c and of magnetic data d by method C with L₁ norms

Imprecise boundary information is unavoidable because the actual measured petrophysical properties from the rock samples are not fixed values but vary within intervals, which poses a challenge in setting appropriate upper and lower bounds in the inversion stage. We believe Method C with L₂ norms is more appropriate to handle unknown condition in areas covered by sediments where the priori information is difficult to obtain and not often sufficient.