Space-associated domain adaptation for three-dimensional mineral prospectivity modeling

Figures & data

Figure 1. Pipeline and architecture of spatial-associated domain adaptation network for 3D MPM. Given the 3D geological model and a target voxel, we generate a multi-channel image for the voxel via descriptor extraction and projection and input it to the deep neural network. The backbone of the network is a convolution neural network consisting of five convolutional layers and three fully-connected layers. The mineralization-associated features are extracted from fully-connected layers for domain adaptation. To extract the transferable features across geospace, we achieve domain adaptation by using the SAMMD module, which measures the spatial-associated domain discrepancy between shallow and deep zones according to spatial distance and feature dissimilarity. The transferable features learned by minimizing the SAMMD are finally used to predict the ore-bearing probability for the target voxel.

Algorithm 1 Framework of domain alg:Framwork adaptation with SAMMD.

Input: The feature representations and space position for source and target domain observation data, $(X^s,Z^s)$ and $(X^t,Z^t)$; the label for source domain observation data, $Y^s$; learning rate η; penalty parameter λ;

Output: Network parameters Θ, multi-kernel coefficients β and matrix Σ.

1. Initialize ${\mathrm{\Theta }}^{\left(0\right)}$ randomly, ${\text{β}}^{\left(0\right)}=\frac{1}{U}{\mathbf{1}}^{\mathrm{T}}$ and ${\mathrm{\Sigma }}^{\left(0\right)}={\mathbf{I}}_{3\times 3}$, where $\mathbf{1}$ is an all-ones vector and ${\mathbf{I}}_{3\times 3}$ is an identity matrix;

2. for $epoch\tau =1\to E$ do

3.  Update Θ according to the gradient $\frac{\mathrm{\partial }\mathrm{L}\mathrm{o}\mathrm{s}\mathrm{s}(\mathrm{\Theta },\text{β},\mathrm{\Sigma })}{\mathrm{\partial }{\mathrm{\Theta }}^l}$ in Eq.(12(12) $\frac{\mathrm{\partial }\mathrm{L}\mathrm{o}\mathrm{s}\mathrm{s}\left(\mathrm{\Theta }\right)}{\mathrm{\partial }{\mathrm{\Theta }}^l}=-\frac{\mathrm{\partial }J\left(v_i\right)}{\mathrm{\partial }{\mathrm{\Theta }}^l}+\frac{2\lambda }{m}\sum _{i=1}^{m/2}\frac{\mathrm{\partial }{h}_{\mathrm{S}\mathrm{A}\mathrm{M}\mathrm{M}\mathrm{D}}\left(v_i\right)}{\mathrm{\partial }{\mathrm{\Theta }}^l}$(12) ): ${\mathrm{\Theta }}^{\left(\tau \right)}\leftarrow {\mathrm{\Theta }}^{(\tau -1)}-\eta \frac{\mathrm{\partial }\mathrm{L}\mathrm{o}\mathrm{s}\mathrm{s}({\mathrm{\Theta }}^{(\tau -1)},{\text{β}}^{(\tau -1)},{\mathrm{\Sigma }}^{(\tau -1)})}{\mathrm{\partial }\mathrm{\Theta }};$

4.  Optimize β by solving the QP problem in Equation (14(14) $\underset{{\text{β}}^{\mathrm{T}}\zeta =1,\text{β}\ge 0}{min}{\text{β}}^{\mathrm{T}}\mathrm{cov}\left(h_{\text{β}}\right)\text{β}.$(14) ): ${\text{β}}^{\left(\tau \right)}=\underset{{\text{β}}^{\mathrm{T}}\zeta =1,\text{β}\ge 0}{\arg min}{\text{β}}^{\mathrm{T}}\mathrm{cov}(h_{\text{β}}({\mathrm{\Theta }}^{\left(\tau \right)},{\mathrm{\Sigma }}^{(\tau -1)}))\text{β}$

5.  $t=0;$ ${\mathrm{\Sigma }}^{\left(t\right)}={\mathrm{\Sigma }}^{(\tau -1)}$;

6.  while not converged do

7.   Update Σ according to Equation (18(18) ${\mathrm{\Sigma }}^{\left(t\right)}={\left(\frac{1}{\alpha }\sum _{i\ne j}{\omega }_{ij}\right({\mathrm{\Sigma }}^{(t-1)}\left)\right(z_i-z_j\left)\right(z_i-z_j{)}^{\mathrm{T}})}^{\frac{1}{2}}$(18) ): ${\mathrm{\Sigma }}^{\left(t\right)}={\left(\frac{1}{\alpha }\sum _{i\ne j}{\omega }_{ij}\right({\mathrm{\Sigma }}^{(t-1)};{\mathrm{\Theta }}^{\left(\tau \right)},{\text{β}}^{\left(\tau \right)}\left)\right(z_i-z_j\left)\right(z_i-z_j{)}^{\mathrm{T}})}^{\frac{1}{2}}$

8.   t = t + 1;

9.  end while

10.  ${\mathrm{\Sigma }}^{\left(\tau \right)}={\mathrm{\Sigma }}^{\left(t\right)}$;

11.  end for

Figure 2. Geological map of the Sanshandao gold deposit (adapted from Song et al. 2021).

Figure 3. Mineralization-alteration zoning pattern (at section line # 96) of the Sanshandao gold deposit (adapted from Hu et al. 2013).

Figure 4. 3D models of Sanshandao faults (a), orebodies (Au grade ≥ 1g/t) (b), and drillholes (c). Left column: down-plunge views, right column: top views.

Table 1. Summary of data characteristics.

Data type	Resolution	Analytic/Construction method
Geological profile	∼100 m	Drill hole / CSAMT
Gold assay data	Sampling resolution ∼1 m at drill holes	Fire assay, followed by flame atomic absorption analysis
3D geologic model	Triangular mesh resolution ∼30 m	Implicit modeling
3D voxel model	$25\mathrm{m}\times 25\mathrm{m}\times 25\mathrm{m}$	Voxelization of 3D geologic model / Kriging interpolation

Figure 5. Shape descriptors of Laplace–Beltrami eigenfunctions (a) and surface normals (b) of the Sanshandao fault.

Figure 6. Projection results of the shape descriptors for target voxels: Laplace–Beltrami eigenfunctions (a) and surface normals (b).

Figure 7. ROC curves (a) and success-rate curves (b) for the five evaluated prospectivity models. The SAMMD-based model attains the best performance among the five tested methods.

Figure 8. Cross entropy and SAMMD loss resulting from CNN, MMD and SAMMD models.

Figure 9. Variograms corresponding to the SAMMDs for 3D MPM at Sanshandao.

Table 2. MYIs and the corresponding thresholds for the SAMMD and the MMD-based models.

Prospectivity model	MYI	Probability
$\mathrm{S}\mathrm{A}\mathrm{M}\mathrm{M}\mathrm{D}$	0.6251	0.4546
MMD	0.5197	0.4065

Figure 10. The high-potential areas resulting from the SAMMD (a) and the MMD (b) models.

Figure 11. Two mineral exploration targets identified by the SAMMD model.

Song, Ming-chun, Zheng-jiang Ding, Jun-jin Zhang, Ying-xin Song, Jun-wei Bo, Yu-qun Wang, and Hong-bo Liu. 2021. “Geology and Mineralization of the Sanshandao Supergiant Gold Deposit (1200 T) in the Jiaodong Peninsula, China: A Review.” China Geology 4:686–719. https://doi.org/10.31035/cg2021070.

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Hu, F., Hongrui Fan, X. Jiang, X. Li, K. Yang, and Terrence Mernagh. 2013. “Fluid Inclusions At Different Depths in the Sanshandao Gold Deposit, Jiaodong Peninsula, China.” Geofluids 13 (4): 528–541. https://doi.org/10.1111/gfl.12065.

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