A Multi-Convolutional Autoencoder Approach to Multivariate Geochemical Anomaly Recognition

Abstract

The spatial structural patterns of geochemical backgrounds are often ignored in geochemical anomaly recognition, leading to the ineffective recognition of valuable anomalies in geochemical prospecting. In this contribution, a multi-convolutional autoencoder (MCAE) approach is proposed to deal with this issue, which includes three unique steps: (1) a whitening process is used to minimize the correlations among geochemical elements, avoiding the diluting of effective background information embedded in redundant data; (2) the Global Moran’s I index is used to determine the recognition domain of the background spatial structure for each element, and then the domain is used for convolution window size setting in MCAE; and (3) a multi-convolutional autoencoder framework is designed to learn the spatial structural pattern and reconstruct the geochemical background of each element. Finally, the anomaly score at each sampling location is calculated as the difference between the whitened geochemical features and the reconstructed features. This method was applied to the southwestern Fujian Province metalorganic belt in China, using the concentrations of Cu, Mn, Pb, Zn, and Fe₂O₃ measured from stream sediment samples. The results showed that the recognition domain determination greatly improved the quality of anomaly recognition, and MCAE outperformed several existing methods in all aspects. In particular, the anomalies from MCAE were the most consistent with the known Fe deposits in the area, achieving an area under the curve (AUC) of 0.89 and a forecast area of 17%.

1. Introduction

Geochemical anomaly recognition is a key task in mineral exploration [1,2] for the alleviation of the current shortages of mineral resources [3]. A large number of geochemical anomaly recognition methods have been developed in the past few decades [4,5,6]. The recognition of geochemical anomalies is to discover the prospection information contained in geochemical data by identifying the anomalies deviating from the normal geochemical samples (i.e., geochemical background) [3]. Full exploitation of geochemical information can improve the performance of anomaly recognition. Regional geochemical samples demonstrate clear signs of spatial autocorrelation/dependence, meaning that they are not independent of each other [7,8]. The spatial structure of a geochemical element is influenced by the combination of a variety of geological processes, including diagenesis and mineralization [9,10]. Therefore, the analysis of spatial structure information derived from geochemical data can facilitate the study on regional metallogenic mechanism and the recognition of geochemical anomalies for ore prospection.

Geochemical anomaly recognition methods that consider the spatial distribution of geochemical samples include spatial analysis methods (e.g., moving average technique and spatial fractal analysis) [8,11,12,13] and geostatistical methods (e.g., geographically weighted principal components analysis, Kriging, multivariate statistics, and inverse distance-weighted regression) [14,15,16]. These methods facilitate the detection and understanding of geochemical spatial patterns and local spatial structures.

In 2006, the Deep Learning (DL) model proposed by Hinton and colleagues inspired a surge in artificial neural network (ANN) application research [17]. A variety of ANNs, such as deep belief network (DBN) and convolutional neural network (CNN) have been successfully applied in various fields, such as speech recognition [18], image recognition [19], and information retrieval [20]. In the field of geochemical exploration, DL models have also been applied for geochemical anomaly recognition [3,21,22]. For example, DBN has been used to extract the characteristics of the relationships between the elements related to the Fe ore, and to reconstruct the backgrounds to separate anomalies from the background [23]. Chen et al. used the continuous restricted Boltzmann machine (CRBM) model to learn the relationship between known deposits and an evidence map [24]. In short, applications of ANN/DL models in geochemical prospecting have only focused on the relationships among the geochemical elements and/or the correlations between known deposits and geochemical elements. The spatial distributions of geochemical elements have rarely been taken into account in the existing ANN/DL-based models.

CNN, a typical DL model, is capable of extracting the patterns of spatial arrangement of features [25,26,27]. Therefore, CNN has become a widely used technique for image pattern recognition such as handwriting and facial recognition [28,29]. A classical CNN combines multiple convolution layers and pooling layers to create a map between the input features and output targets [30]. A convolution layer extracts local spatial patterns, and a pooling layer reduces the amount of data while retaining useful information. CNN has been used for outlier detection of spatial data [31,32]. Convolutional autoencoder (CAE) is an autoencoder (AE) neural network that uses convolution layers and pooling layers to extract the hidden patterns of input features (i.e., encoding), and deconvolution layers and unpooling layers to reconstruct the features from the hidden patterns (i.e., decoding). By integrating convolutional and deconvolutional layers in an AE structure, CAE is capable of learning the spatial structure of input features, and reconstructing these features while taking into account their spatial structural patterns [33,34,35]. Chen et al. used a CAE model to achieve the unsupervised extraction of spatial structural patterns of color images [36]. Baccouche et al. achieved local two-dimensional (2D) feature acquisition using CAE for video sequence classification [37]. Chen et al. measured the similarity of images using CAE for medical image analysis [38]. Given that the geochemical background is represented by the majority of geochemical samples and the much fewer anomalies pose little impact on the general spatial pattern of geochemistry, CAE provides a promising approach to reconstruct the geochemical background by learning the spatial structural pattern of samples, and can thus identify those anomalous samples by their relatively larger differences from the reconstructed values.

In regional geochemical explorations, the concentrations of multiple elements are often measured at each sampling location. Multivariate analysis usually provides more robust and indicative results by identifying the combined anomalies of multiple elements, because the source or process that has generated the anomalies commonly has an association with a suite of elements (i.e., target and pathfinder elements) [39]. When applying CAE in the anomaly recognition of multiple geochemical elements in a region, two issues must be dealt with: (1) the geochemical backgrounds of different elements can be quite different [1,40]; therefore, it is necessary to reconstruct the backgrounds of elements separately in order to avoid mutual interference. Furthermore, multiple elements at a certain location are inevitably correlated [41], leading to redundant information contained in the raw multivariate geochemical data. Such redundancy must be minimized through a pre-processing procedure such that the background of each pre-processed element can be effectively learned and reconstructed. (2) For a geochemical element, the spatial autocorrelation exhibits various patterns within different distance ranges. Within a small range, the concentration values are often highly similar with each other, exhibiting a clustered spatial pattern. As the range increases, the degree of clustering decreases and the spatial distribution changes towards a random pattern. Therefore, the size of convolution window has a great impact on the effectiveness of spatial pattern learning [28], and thus on the performance of anomaly recognition. Currently, there is no unified method, and the window size is usually set based on experience [42]. A method is needed for determining the proper convolutional window size based on the quantitative analysis of spatial autocorrelation ranges of geochemical elements.

In this study, a multi-convolutional autoencoder (MCAE) approach to multivariate geochemical anomaly recognition is proposed. Specifically, a whitening process based on the zero-phase component analysis (ZCA) is used to minimize the correlations among geochemical elements. In a multi-CAE model, each CAE is trained using the spatial distribution map of a corresponding whitened element, and generates a reconstructed map that represent the geochemical background of such element. For the convolution window size, the Global Moran’s I, a measurement of spatial correlation that has been used to explore the spatial correlation range of the mineralization elements [43], is used to determine the size of recognition domain (i.e., convolution window) of each whitened element. Finally, the anomaly score at each sampling location is calculated as the difference (measured by Euclidean distance) between the original values of whitened elements and the reconstructed values, and an anomaly map of the region is produced.

The MCAE was applied to the stream sediment survey data of the southwestern Fujian province, which is one of the most important iron polymetallic metallogenic regions in China. The results showed that MCAE outperformed several existing methods in all aspects. Additionally, setting the convolution window size using Global Moran’s I effectively helped improve the anomaly recognition accuracy in the experiments.

2. Multi-Convolutional Autoencoder (MCAE) Approach

As shown in Figure 1, the proposed MCAE includes the following steps. Firstly, a whitening algorithm is used to reduce the correlations among multiple geochemical elements. Secondly, the Global Moran’s I values of each element with different spatial ranges (also termed bandwidths) are calculated, and the convolution window size for an element is determined based on the Moran’s I values. Thirdly, multiple CAEs are used to learn and reconstruct the backgrounds of multiple elements separately. Finally, the anomaly score at each sampling location is calculated as the multivariate difference between whitened values and reconstructed values, and a geochemical anomaly map is generated.

2.1. Decorrelation of Geochemical Elements

In the MCAE, the geochemical background of each element is learned and reconstructed by a CAE separately. The correlations among elements need to be minimized to avoid mutual interference. The decorrelation of multiple variables is also termed whitening, and the common whitening methods include principal component analysis (PCA) and zero-phase component analysis (ZCA) [44,45]. The basic principle of ZCA whitening is to rotate the results of PCA whitening so that the data dimension after whitening is closer to the original data. ZCA whitening is characterized by a whitened variable that remains highly correlated with the original variable, maintaining its original interpretation [46]. PCA whitening has better compression capability, and the whitening variable has no interpretable relation to the original data [47]. To avoid the loss of the original meaning of the variables, ZCA whitening is used in MCAE.

In ZCA, n samples with m variables are formed into an n × m matrix Ψ. Through the eigenvalue decomposition of the covariance matrix C = cov(Ψ), the eigenvalues and the eigenvectors are obtained, and then the input data is scaled with the eigenvalues factor:

$${\mathsf{\Psi }}_{PCAwhite}=\frac{\left(\mathsf{\Psi }-\mathrm{mean}\left(\mathsf{\Psi }\right)\right)·V}{\sqrt{diag\left(D\right)+\xi }}$$

(1)

where ξ is the whitening factor, mean(Ψ) is the matrix averaging function, D is a diagonal matrix containing the eigenvalues, and V is a matrix whose columns correspond to the appropriate eigenvectors. To avoid numerical instability or data overflow caused by the eigenvalue diag(D) approaching 0, ξ can be set as a small normal number. Based on this, Equation (2) is used to for further processing of PCA whitening results to obtain element values close to the original data and with low correlation:

$${\mathsf{\Psi }}_{ZCAwhite}={\mathsf{\Psi }}_{PCAwhite}·V^T$$

(2)

2.2. Determination of the Recognition Domain of the Background Spatial Structure

The concentrations of geochemical elements are the results of the superposition of multiple geological processes and often exhibit certain forms of spatial structures, mainly influenced by the spatial autocorrelations of elements [48,49]. Within a certain distance, the change of element concentration in space shows a degree of regularity, and the spatial structure can provide key information for the recognition of geochemical background and anomalies. Beyond this distance, randomness becomes dominant in the spatial distribution of element concentrations, where weak regularity or no regularity is found [50].

In geochemical data analysis, the method based on spatial autocorrelation theory, such as the Global Moran’s I, can reveal the spatial structure of regional geochemical variables [51,52,53]. Generally, the Global Moran’s I is between −1.0 and 1.0. A positive correlation among spatial samples is found when the Moran’s I is close to 1, while a negative correlation is found when I is close to −1. Both indicate regular spatial changes of observed values, that is, a spatial structure. A value of 0 or close to 0 means that the observed values are randomly distributed in space and no spatial structure is found [54,55].

Both the background spatial structure and anomaly spatial structure can be found for a certain element. They have different recognition domains (Figure 2). Generally, background recognition domains are larger than anomaly recognition domains. In the background recognition domains, the spatial structure is stable, meaning the structure does not change greatly with the increase of recognition domain. For example, as shown in Figure 2, as the domain increases from 45 km × 45 km to 50 km × 50 km, the spatial structure still shows a linear pattern. On the other hand, the anomaly shows the opposite behavior and the spatial structure changes greatly with the increase of the recognition domain.

In MCAE, the Global Moran’s I is used to measure the degrees of spatial autocorrelation among samples within different distance bands, and eventually to determine the recognition domain of a whitened element. If the distance between two samples is less than a distance band, these two samples are considered to be adjacent in the Global Moran’s I. The specific process is as follows: First, the Global Moran’s I values with different distance bands are calculated for a certain whitened element, and a relationship curve between the I values and the distance bands is generated. The I value usually decreases as the distance band increases, since concentrations of an element are often highly similar within a short distance and the similarity reduces as the distance is larger. After the dropping in the early stages, the relationship curve gradually becomes stable as the distance band increases. Second, the distance bands corresponding to the stable section of the curve with I values greater than 0 are selected. As mentioned before, a smaller absolute value of Moran’s I corresponds to stronger randomness. To capture the background spatial structure of the element and avoid the interference of randomness, the distance band of the maximum absolute value of Moran’s I is selected from prior selected distance bands as the background recognition domains.

The Moran’s I is defined as:

$$\mathrm{I}=\frac{n\times {\sum }_{i=1}^n{\sum }_{j=1}^n{w}_{ij}\left(x_i-x_m\right)\left(x_j-x_m\right)}{{\sum }_{i=1}^n{\sum }_{j=1}^n{w}_{ij}\times {\sum }_{i=1}^n{\left(x_i-x_m\right)}^2}$$

(3)

where $x_i,x_j$ are the whitened values at sampling location i and j; $x_m$ is the mean value of the whitened values at all locations; $w_{ij}$ are the weights representing the proximity relationships between location i and j, and is often a function of distance; and n is the total number of samples.

2.3. Multi-CAE

A convolutional autoencoder (CAE) integrates the merits of a convolutional neural network (CNN) and an autoencoder neural network (AE) [37,56]. The convolutional layer in CNN is used for sharing the local weights, and AE is employed for capturing the high probability information [56], i.e., the geochemical backgrounds. Based on CAE, the multi-convolutional autoencoder (MCAE) is the model for multiple geochemical elements. A CAE corresponding to an element extracts the background spatial structure features and reconstructs the background with these features (as shown in Figure 3). MCAE has the following characteristics: (1) the convolution windows (or filters and kernels) can be used to identify the structure features of the spatial distribution; (2) unsupervised training can avoid the omission of valid information caused by supervised learning; (3) CAEs are mutually independent, avoiding the mutual interference of the spatial characteristics of different elements; and (4) the method of parallel handling of multiple geochemical elements considerably accelerates the training time.

MCAE includes two stages: CAEs and anomaly score calculation. CAEs complete the extraction of the spatial structure features and background reconstruction, and the anomaly score calculation is used to calculate the difference between the input and reconstruction.

2.3.1. Convolutional Autoencoders (CAEs)

A CAE is composed of one or more Encoder–Decoder groups, as shown in Figure 4. The composition of multiple groups is similar to the embedded one. Thus, the extracted spatial structure is cut by the newly added encoder to meet certain application requirements. An encoder includes a convolutional layer and a pooling layer, and the corresponding decoder includes an unpooling layer and a deconvolution layer (as shown in Figure 5).

• Spatial information encoder part

The purpose of the encoder component is to capture the spatial structure features of geochemical backgrounds. The encoder involves the convolutional layer and the pooling layer.

In the convolutional layer, convolutional windows carrying the weight parameters slide in an element map from the left-upper corner to right-lower corner for detecting the local spatial structure. Then, the feature maps are produced by convolutional calculations, as shown in Figure 5.

In the pooling layer, the output of the convolutional layer is used as the input of pooling. The purpose of pooling is to abandon the local spatial details and preserve the most representative local spatial information. This can prevent the model from overfitting. Pooling is carried out using two methods, namely, mean-pooling and max-pooling. Max-pooling is currently the most widely used method. When dealing with regional geochemical data, pooling capturing the maximum help model avoids learning all of the neurons of the feature map, leading to the prevention of anomaly learning. Similar to the convolution layer, the max-pooling calculation is carried out using a sliding filter to operate the maximum. In this process, the model records the maximum value and the position information of the maximum value in the area covered by the filter. These data are used for the un-pooling of the decoder part.

• Spatial information decoder component

The decoder component involving the de-convolutional layer and the un-pooling layer is used for reconstructing the input with the features from the encoder (Figure 5). Generally, the max-pooling operation is non-invertible in CNN. However, we can obtain an approximate inverse by using the maximum value and its position recorded in the pooled process to obtain the inverse pooled operation. Thus, in the unpooling layer, we use the unpooling filter to decode the information from the pooling encoder. Specifically, in the unpooling process, according to the location information recorded during pooling, the maximum value is placed in the corresponding position, and the other positions are filled with 0 (Figure 5). The preservation of the structure of the stimulus is an advantage of this process [57].

In the de-convolutional layer, the transposed convolution is used to restore the data compressed by convolution (Figure 5). Previous research has proven that transposed convolution can generate the convolutional image with the original input resolution [57]. Here, similar to the image convolution, de-convolution also employs a sliding filter to complete the matrix operation. De-convolution uses the transposed convolution to restore the input before convolution, which is a decoding process [35].

• Objective function of CAE neural network

Each CAE is separately trained, and the CAEs do not interfere with each other. Here, we take the mean square error (MSE) as the cost function of CAE:

$$\mathrm{L}=\frac{1}{wh}{\displaystyle \sum }_{i=1}^w{\displaystyle \sum }_{j=1}^h|\left|D\left(i,j\right)-X\left(i,j\right)\right|{|}^2$$

(4)

where w and h are the width and height of each sample, respectively; D is the data passing through the model at position (i,j); and X is the data input to the model at position (i,j). We use the stochastic gradient descent (SGD) method to train the model and randomly initialize the weight parameters of each layer.

2.3.2. Anomaly Score Calculation and Anomalies Map Generation

The Euclidean distance is used to measure the difference between two multi-feature samples. A larger distance corresponds to a more anomalous sample. Euclidean distance can reflect the absolute difference of individual numerical characteristics, and therefore, it is commonly used for analyzing numerical difference in m dimensional space, for example, as a measure of anomaly scores; here, we number the elements from 1 to n. The outliers of each sample are calculated as follows:

$${\mathrm{d}}_i=\sqrt{{\displaystyle \sum }_{k=1}^n{\left(x_{ik}-x_{ik}^{\prime }\right)}^2}$$

(5)

Here, $x_{ik}$ denotes the k-th element attribute value of sample i and $x_{ik}^{\prime }$ represents the reconstitution of $x_{ik}$.

Finally, the anomaly scores ${\mathrm{d}}_i$ are projected to the corresponding geographical locations to generate an anomaly map.

3. Experiment and Evaluation

3.1. Study Area and Data

To assess its performance for geochemical anomaly recognition, the proposed MCAE approach was applied to the Fe metallogenic belt located in the southwest Fujian depression center. This region is characterized by large areas of Paleozoic stratigraphy and intermediate acidic magmatic rocks from the Yanshan period that provide a material base and metallogenic environment for the formation of iron, copper, and zinc polymetallic ore deposits [58,59]. Several skarn-type Fe deposits have been found in this area (Figure 6).

To examine the performance of MCAE, the stream sediment geochemical dataset at a scale of 1:200,000 collected at 2 km × 2 km grids in a regional geochemical survey as part of the Chinese National Geochemical Mapping (CNGM) project [60] was used in the experiment. The dataset includes 39 content values for each sample. Previous studies have indicated that Cu, Mn, Pb, Zn, and Fe₂O₃ are most strongly related to the Fe deposits in the region [61,62], and therefore, their concentration values were used for the anomaly recognition experiment. All of the concentration values were transformed into the range of [0, 1] using min–max normalization, as shown in Figure 7. Figure 7 shows that the spatial distribution of the five elements presents a certain regularity. The distribution of geochemical elements in hypergenic media is the result of the superposition of many complex geological processes [63]. The spatial regularity of the elements is due to the combination of the changes of the main geological action, while the secondary geological action is manifested by the randomness of the elements [64].

3.2. Specific Implementation Process

3.2.1. ZCA Whitening

Pretreatment whitening makes CAE learn the unique spatial structure information of an element, rather than the repeated structure between elements. We generated the correlation matrices of the samples before and after whitening using the Spearman method (see

Table 1. Correlation matrix of not-whitened elements.

Element	Cu	Mn	Pb	Zn	Fe2O3
Cu	1
Mn	−0.01 **	1
Pb	−0.65 **	0.01 **	1
Zn	−0.53 **	−0.54 **	−0.03 **	1
Fe2O3	−0.41 **	−0.21 **	−0.05 **	0.19 **	1

** Correlation is significant at the 0.01 level (2-tailed).

and

Table 2. Correlation matrix of whitened elements.

Element	Cu	Mn	Pb	Zn	Fe2O3
Cu	1
Mn	−0.21 **	1
Pb	−0.33 **	−0.28 **	1
Zn	−0.29 **	−0.26 **	−0.32 **	1
Fe2O3	−0.13 **	−0.12 **	−0.18 **	−0.16 **	1

** Correlation is significant at the 0.01 level (2-tailed).

below). In the entire study area, the correlations between Cu and Pb, Mn and Zn, Zn and Cu, and Fe₂O₃ and Cu were moderate (0.3 < |r| < 0.7), while other correlations were low. After whitening, the correlation coefficients of the elements were all below 0.35.

3.2.2. Global Moran’s I

Global Moran’s I values were calculated with Equation (3) using ESRI@ ArcGIS software. The function of distance was defined using inverse distance weighting (IDW) based on the Euclidian distance. Distance bands were used for distinguishing between “close” and “not so close” locations. The “not so close” elements were ignored when calculating the Moran’s I value. To determine the background recognition domains of Cu-Fe₂O₃-Mn-Pb-Zn in the study area, we calculated the global Moran’s I values in different distance bands.

As shown in Figure 8, the global Moran’s I values decreased when the distance band increased, and the five curves were similar. When the distance band was less than 48 km, the Moran’s I value was between 0.1 and 0.7, showing that the Moran’s I values differed significantly between different distance bands, indicating the unstable spatial structures in these recognition domains. When the distance band was between 48 and 64 km, the Moran’s I value decreased slowly with the increasing distance bands, with smaller differences at different distance bands. This demonstrates that the spatial structures in these recognition domains were relatively stable. When the distance band was greater than 64 km, the Moran’s I value started to approach 0, the spatial structure was unclear, and the regular patterns of the background spatial changes were difficult to find. Consequently, the background recognition domain of each element was set to 48–64 km, in which case CAE could easily extract the background spatial structure. Finally, those recognition domains were set to 50 km.

3.2.3. MCAE Structure and Training

The MCAE included five CAEs corresponding to five elements (Cu-Fe₂O₃-Mn-Pb-Zn). Each CAE was independently trained and included one Encoder–Decoder group. The convolution window, acting as the spatial structure feature extraction tool, was key for the entire modeling process. The hyper-parameter settings of the convolution window were highly important and could directly affect the accuracy of the background reconstruction. The hyper-parameter included mainly the size and number. The background recognition domain obtained through Global Moran’s I was used for setting the size. This number is generally set based on experience. A greater number of convolution windows will lead to more details in the reconstruction. This means that the reconstruction may include much invalid information that will interfere with anomaly recognition. If the number of the convolution windows is too small, the spatial structure feature extraction will be insufficient, also causing the background reconstruction to be inaccurate. To find a suitable value, we examined the variation in the reconstruction error by using a variety of CAE models with a different number of convolution windows (Figure 9).

Figure 9 shows the total errors as the sum of the five element reconstruction errors calculated when the reconstruction error curve was trained to be stable (Figure 9A). A stable reconstruction error meant that the model was considered "nearly well-trained". When CAE had 1–15 convolution windows, the total reconstruction error decreased faster with an increasing number of convolution windows. When the number was between 16 and 35, the error fluctuated smoothly. This shows that in the investigated area, when the number of convolution windows was greater than 15, the CAEs were not being underfit. To avoid the excessive convolution windows resulting in overfitting and increasing computational complexity, each CAE had 16 convolution windows.

The CAE network structure was based on Keras, which is a Python library for deep learning that can run on top of TensorFlow frameworks.

The CAE network parameters are as follows:

The encoder:

• Input: 78 × 88 geochemical element concentration map.
• C1: convolution filter was 25 × 25, the number of convolution kernels was 16, and the nonlinear function was Rectified Linear Unit (ReLU).
• P1: max pooling filter was 2 × 2.

The decoder:

• Un-P1: the size of un-pooling area was 2 × 2.
• De-C1: the de-convolution filter was 25 × 25, the number of convolution kernels was 16, and the nonlinear function was ReLU.
• Output: the geochemical background map had 78 × 88 grids of 2 km * 2 km resolution, and the nonlinear function was Sigmoid.

The background recognition domain of each element was 50 km × 50 km from the Moran value (as 3.2(2)) and the input was a 2 km × 2 km grid. Therefore, the convolution window size was set to 25 × 25. The five CAEs had the same network structure but different initial weights (w,b). The weights were randomly derived from a zero mean unit variance Gaussian N (0,1). After CAE training, reconstruction of the five elements was completed.

3.2.4. Anomalies Map Generation

Using Equation (5), we obtained the difference between the input and reconstructed data of each sample that was considered as the anomaly score of each sample. Then, these anomaly scores were projected to the corresponding geographical locations to generate an anomaly map (Figure 10).

Figure 10 shows that most of the known Fe deposits were in areas of high anomaly scores, and a few were close to the high anomaly area (marked as C in Figure 10). This suggests that these anomalies may be related to iron mineralization, and other areas with higher anomaly scores indicate the possibility of finding Fe deposits in the studied area.

3.3. Performance Evaluation

3.3.1. Receiver Operating Characteristic Curve

The Receiver Operating Characteristic (ROC) Curve is often used to measure the predictive ability of a model [65]. A ROC curve that is steep and toward the upper left corner of the plot indicates good performance of a model. Previous studies showed that an Area Under the Curve (AUC) greater than 0.5 and close to 1 often indicates that the prediction model of metalorganic prognosis achieved an acceptable accuracy [66]. In our experiment, ROC was used to measure the coincidence degree between the areas with high anomaly scores and the known Fe deposits.

The anomaly recognition results from MCAE, Mahalanobis Distance (MAHAL) [67], Spectrum–area model (SA) [68], and deep belief nets (DBN) [23] were compared. As shown in Figure 11, the AUC of MCAE was considerably higher than those of other methods, indicating MCAE outperformed other methods in geochemical anomaly recognition.

3.3.2. Weights-of-Evidence and Student’s t-Value

The weights-of-evidence (WofE) is a multivariable information synthesis method based on the Bayesian probability theory that has been used to investigate the significance of spatial correlation between multiclass patterns of a geological feature and a set of mineral deposits [69]. In WofE, each metallogenic variate served as a layer of evidence that had a positive weight value (W⁺) and a negative weight value (W⁻), representing the influences of the presence and absence of the metallographic evidence layer [70]. The difference between the positive and negative weights was defined as the contrast difference in WofE and was denoted as C. Student’s t-values were used to measure the significance of the contrast difference C, and was calculated as:

t = C/S(C)

(6)

where S(C) stands for the standard deviation of C.

A larger t value corresponded to a more significant C and stronger controlling effect of the ore controlling factors on the ore deposits. It was generally believed that there was significant spatial correlation for t > 1.96 and 95% confidence [71].

In

Table 3. Comparison of various models.

Model	t-Value	AUC	Area of Forecasting	Correct Ratio
MAHAL	3.044	0.800	0.117	0.632
SA	3.668	0.816	0.211	0.790
DBN	3.519	0.776	0.309	0.737
MCAE	4.949	0.894	0.174	0.895

, the first column (t-value) shows whether a result was spatially correlated with known Fe deposits. The second column (AUC) indicates the correct identification probability of the known Fe deposits. The third column shows the range of anomaly recognition, denoting the proportion of the identified anomaly grids [24]. The fourth column shows the ratio of the number of Fe deposits falling in the forecast area to the total number of known Fe deposits [24]. It is clear that MCAE outperformed the other methods in all aspects.

4. Discussion

The key innovation of MCAE is the use of spatial similarity theory to determine the recognition domains of spatial structures of geochemical backgrounds. If the domain is not clear, the size of the convolution window of MCAE will be set to an incorrect value, leading to inaccurate background reconstruction. To explore this problem in greater depth, we set up MCAEs with different convolution window sizes for observing their anomaly recognition results. According to the Global Moran’s I value curve (Figure 8), we divide the models into three groups: Group 1 (0–48 km), Group 2 (48–64 km), and Group 3 (64–108 km). The MCAE convolution window sizes were set to 5 × 5 grids (Group 1), 10 × 10 grids (Group 1), 30 × 30 grids (Group 2), 40 × 40 grids (Group 3), and 45 × 45 grids (Group 3), and the window sizes of all of the pooling layers were 2 × 2. The samples in the study area were collected in a 2 km × 2 km grid. Therefore, the corresponding domains of these convolution window sizes were 10 km × 10 km (Group 1), 20 km × 20 km (Group 1), 60 km × 60 km (Group 2), 80 km × 80 km (Group 3), and 90 km × 90 km (Group 3). After the three sets of models were trained, we evaluated their anomaly recognition results, as shown in Figure 12.

Figure 12 shows significant differences of the evaluation indicators. Group 2 was superior to Group 1 and Group 3 in all aspects. Figure 13 shows their anomaly recognition results. It was observed that the anomaly areas of the 10 km × 10 km, 20 km × 20 km, 80 km × 80 km, and 90 km × 90 km models were relatively large, and the recognized anomalies were less consistent with the known Fe deposits. MCAE with the convolution window size of 60 km × 60 km (Group 2) performed better; the recognized anomalies were more relevant to the known Fe deposits and the anomaly areas were smaller. This result indicated that the distance band defined by stable values above 0 along the Moran’s I curve can be used for the convolution size setting of MCAE.

It was necessary to explore whether multiple Encoder–Decoder groups could help improve the extraction ability of spatial structure features of MCAE. We analyzed the outputs of MCAEs composed of one, two, and three Encoder–Decoder groups. To eliminate the impact of the weight parameter differences, the modeling process of the three MCAEs was as follows: first an MCAE including an Encoder–Decoder group was trained and its anomaly result (R1) was obtained. Then, a new Encoder–Decoder group was added to the MCAE that was trained to obtain the new anomaly result (R2). In this process, MCAE retained the previously trained weights and only trained the weights of the newly added group. MCAE including three Encoder–Decoder groups generated the anomaly result (R3). Generally, multiple encoders enable more precise feature extraction, easily obtaining the reconstruction that is highly similar to the input. For example, in image processing, an increased number of encoders leads to an increased level of detail of the reconstructed image. This happens because the latter encoder breaks the spatial features extracted by the previous encoder, forming a smaller reconstruction unit. However, in geochemical background reconstruction, this would destroy the background spatial structure that had been extracted previously, hindering the accurate reconstruction of the background. In addition, for the window size setting of multiple convolution layers, we used the conventional approach, with each convolution layer half the size of the previous convolution layer, which was beneficial for reducing the computational complexity.

The network structure with one Encoder–Decoder group was set to 25-2-2-25, that with two Encoder–Decoder groups was set to 25-2-13-2-2-13-2-25, and that with three Encoder–Decoder groups was set to 25-2-13-2-7-2-2-7-2-13-2-25. The layer with a size of two was the pooling layer or unpooling layer. For more than three groups, the excessive number of layers led to over-fitting, thus, these are not discussed in the comparison with the experimental data.

Figure 14 shows that as the number of the Encoder–Decoder groups increased, the reconstruction error was gradually reduced, indicating that the reconstructed data became more similar to the original data with a greater number of Encoder–Decoder groups. However, the reduction of the reconstruction error did not represent the accuracy of background reconstruction. The smaller units reconstructed unnecessary details, resulting in the reduction of the accuracy of background reconstruction.

Table 4. Comparison of various structures.

Model	t-Value	AUC	Area of Forecasting	Correct Ratio
25-2-2-25	4.949	0.894	0.174	0.895
25-2-13-2-2-13-2-25	3.670	0.841	0.217	0.794
25-2-13-2-7-2-2-7-2-13-2-25	3.666	0.831	0.243	0.768

shows the evaluation of the results. It was observed that the correlation between the identified anomalies and the known Fe deposits decreased with increasing number of the Encoder–Decoder groups.

It was necessary to explore the influence of element decorrelation on MCAE. We mapped the multi-geochemical anomaly map with MCAE for the whitened elements and the not-whitened elements. To avoid the interference of different structures, the network structures of MCAE were randomly selected and were 10-2-5-2-2-5-2-10, 25-2-2-25, and 40-2-20-2-10-2-2-10-2-20-2-40. We evaluated the results of these structures with whitening and with not-whitened (Figure 15). From the AUC and T-value indicators, it is clear that whitening had a significant impact on the results, and the results more closely fit the known Fe deposits. This suggests that whitening reduced useful information resulting from redundancy.

5. Conclusions

The spatial structure of regional geochemical elements represents the synthesis of the regular changes of various main geological processes [72]. We used the spatial structure features as the reconstruction component to reconstruct the geochemical background accurately, providing a new concept and method for the identification of geochemical anomalies. The determination of the recognition domains of the background spatial structure of geochemical elements was the focus of this paper. Generally, the size of the convolution window is set to a default value that is usually set to 3 × 3, 5 × 5 [56,73]. For instance, in image processing, the convolution window with 3 × 3, 5 × 5 can extract the boundary of the graph well and achieve good image reconstruction [19,28]. However, in geochemical exploration, this approach led to unsatisfactory performance of CAE because the convolution window sizes were not set as the background domain recognition (see Section 4). In this work, the use of Moran’s I index to determine the recognition domains of background spatial structure helped MCAE to accurately extract the background spatial structure features and reconstruct the backgrounds. This method was applied to the analysis of the data from the southwestern Fujian province (China). The anomaly result exhibited a strong spatial correlation with the known Fe deposits.

The method used in this study still has some limitations as follows: (1) the method is limited to the application of regular grid sampling of geochemical samples; and (2) the lack of consideration of the spatial structure extraction of multi-source spatial data (such as remote sensing data and geophysical data). Therefore, future research will focus on improving the MCAE in the following respects: (1) the limitation of regular grid sampling for the input data must be eliminated, and the method of mapping multivariate anomalies for irregular sampled data should be developed; and (2) for multiple sources of spatial data, a suitable spatial structure extraction and fusion method should be developed.