A multimetric evaluation method for comprehensively assessing the influence of the icosahedral diamond grid quality on SCNN performance

ABSTRACT

The increasing availability of global observational data has sparked a demand for deep learning algorithms on spherical grids to enable intelligent analysis at a global scale. However, a spherical surface cannot be subdivided into completely identical grid cells through recursive division, and its nonuniformity and irregular deformations lead to uncertainties in the spherical convolutional neural network (SCNN). This paper proposes a multimetric evaluation method to assess the impact of the icosahedral diamond grid quality on the performance of the SCNN by introducing the random forest algorithm to establish nonlinear relationships between multiple grid quality metrics and the SCNN performance and using feature importance analysis to assign impact weights to each grid quality metric considering the SCNN performance. The results show an R2 score of 0.80 for the evaluation method, with four indicators having different weights: cell wall midpoint ratio (0.47), distance between grid points and neighbouring points (0.29), zone standardized compactness (0.13), and angle between a grid point and its two neighbours (0.11). The cell wall midpoint ratio indicator has the most significant impact on the SCNN performance among all grid indicators.

KEYWORDS:

• Discrete global grid
• grid indicators
• icosahedral diamond grid
• weight calculation
• spherical deep learning

1. Introduction

With the rapid development of Earth observation technologies, the scale and diversity of data have experienced explosive growth. Earth observation data are gradually becoming an integral part of the broader domain of big data research and applications (Yao et al. 2019). The wealth of available Earth observation data has created valuable opportunities for applying deep learning algorithms, with convolutional neural networks (CNN) emerging as a prominent approach for extracting meaningful information from such datasets (Reichstein et al. 2019). However, traditional CNN frameworks are designed for Euclidean planes. Applying traditional CNN frameworks to global or large-scale geographic data by projecting it onto a plane can result in data fragmentation, geometric distortions, and topological inconsistencies, leading to decreased prediction accuracy in deep learning models (Rasp et al. 2020; da Silveira et al. 2022).

The discrete global grid system (DGGS) serves as a next-generation digital earth framework, providing a spherical (or ellipsoidal) grid that can be subdivided infinitely without altering its inherent shape (Gibb et al. 2022). DGGS exhibit distinct features in terms of their cell structure, geo-encoding, quantization strategy, and associated mathematical functions (Purss et al. 2016). The OGC defined DGGS as ‘a spatial reference system that uses a hierarchical tessellation of cells to partition and address the globe’(OGC 2017). Therefore, the hierarchical and globally continuous nature of DGGS addresses the limitations associated with planar projections (Hojati et al. 2022). At the same time, the global discrete grid provides a unified and continuous global analytical framework to address large regional or global issues, such as global environmental modelling and analysis (Robertson et al. 2020), spatial data management and computation (Zhou et al. 2020; Kim et al. 2021), terrain data analysis (Liao et al. 2022, 2023), open-source web service (Bowater and Stefanakis 2020), spatial analysis and prediction (Wang et al. 2023; Jendryke and McClure 2021) and earth system modelling (Engwirda and Liao 2021). In particular, DGGS, as a data analysis framework, has been used in various fields of spherical deep learning, such as global weather forecasting (Weyn et al. 2021; Kurth et al. 2023), cortical surface-based analysis (Zhao, Wu, and Li 2023), and omnidirectional vision (da Silveira et al. 2022).

Theoretically, a spherical surface cannot be subdivided into identical grid units through recursive division (Wang et al. 2021; Kimerling et al. 1999). The nonuniformity of spherical grid units and the irregular distribution of deformations significantly increase the uncertainty of the application results of DGGS in deep learning (Lee et al. 2020). Therefore, it is necessary to assess the impact of the discrete global grid quality on the SCNN performance, thereby enhancing the stability of the SCNN.

Current research endeavours are primarily focused on developing more efficient and powerful SCNN architectures. However, there is a lack of research on the impact of the grid quality on the SCNN performance. Jacquemont et al. (2019) presented indexed convolution and pooling for the hexagonal grids. Luo et al. (2019) proposed an SCNN based on the icosahedral hexagonal grid using D6 group convolution. (Jiang et al. 2019) proposed that using an icosahedral grid, which is the most uniform and precise regular polyhedron discretization of a sphere, can enhance the predictive performance of the SCNN. Therefore, the authors replaced the latitude and longitude grid with an icosahedral triangular grid to implement the SCNN, achieving improved predictive performance in the field of climate model segmentation. Subsequent research findings have also confirmed that the use of the spherical icosahedral grid effectively enhances the predictive performance of SCNN (Zhang et al. 2023; Yoon et al. 2022; Eder et al. 2020). These studies only noted that, through experimental validation, icosahedron with uniform grid quality can improve the prediction performance of the SCNN. However, they did not investigate how grid quality affects the predictive performance of SCNN. In addition, other scholars have studied the impact of the spherical grid quality on the performance of the SCNN. For example, (Lee et al. 2020) generated spherical MNIST datasets by projecting original MNIST images onto an icosahedral triangular grid. The authors observed that after training, SCNNs based on the icosahedral triangular grid exhibited varying classification accuracy on the spherical MNIST test set in response to changes in latitude and longitude positions. However, this article did not delve into the reasons for this phenomenon. (Cho, Jung, and Kwon 2022) evaluated the effect of the grid distance inhomogeneity on the spherical deep learning performance. The authors found that, in comparison to ERP and cube grids, icosahedral triangles exhibit the most uniform distance index, resulting in superior SCNN prediction performance. However, the authors exclusively considered distance metrics and did not account for the influence of other grid parameters on the SCNN performance.

Some researchers have proposed a series of evaluation metrics based on the ‘Goodchild Criterion’ (Goodchild 1994) for assessing the geometric attributes of grids. For example,(Heikes et al. 1995a, 1995b) proposed the ‘cell wall midpoint ratio’ index, specifically designed for atmospheric modelling applications. White et al. (1998) defined the ‘zone standardized compactness’ index as the ratio of the grid's perimeter to its area. Zhang et al. (2015) developed a fuzzy similarity index that utilizes the principle of similar triangles to assess the geometric shape and area deformations of the quaternary triangular mesh (QTM) grid. Ming et al. (2007) constructed two indices to measure the uniform distribution of grid points, that is, the great circle distance between a grid point and its neighbouring point and the angle between adjacent central points. Wang et al. (2021) proposed a comprehensive evaluation system for assessing the quality of discrete global grids based on the QTM grid. The authors considered the correlation among indices derived from the ‘Goodchild Criterion’ to improve the credibility of the quality assessment results by eliminating redundant indices. Luo et al. (2023) employed a method similar to Wang et al. (2021) to propose a comprehensive evaluation system based on icosahedral diamond grids.

In the literature, scholars only use a single metric to assess the impact of the DGGS quality on SCNN. Furthermore, since a sphere cannot be recursively subdivided into exactly the same grid units as the plane (Kimerling et al. 1999), some researchers have developed various metrics based on different attributes of the DGGS to describe its quality. Since different grid metrics represent different attributes, they have varying effects on the SCNN performance. Therefore, relying solely on a single grid metric does not adequately capture the influence of the grid quality on the SCNN performance. As a result, the current body of research lacks a more comprehensive method to evaluate the impact of multiple grid indicators on the SCNN performance, which hinders the application of the DGGS in the field of the SCNN. To address this gap, for the first time, in this study, a multimetric evaluation method was developed based on a random forest algorithm for evaluating the influence of the icosahedral diamond grid quality on the SCNN performance, using the SCNN constructed with an icosahedron diamond grid (SCNN-IDG) based on a Fuller projection (Gray 1994). The main contributions are as follows:

1. For the first time, this study employs a multimetric evaluation method to assess the impact of multiple grid quality metrics on the SCNN performance.

2. This article introduces the random forest algorithm to establish nonlinear relationships between multiple grid quality metrics and the SCNN-IDG performance.

3. Feature importance analysis was employed to attribute impact weights to individual grid quality metrics, taking into account the performance of SCNN-IDG. Among all grid indicators, the cell wall midpoint ratio indicator emerged as the most influential factor affecting SCNN performance, with a weight value of 0.47.

4. An analysis is performed to evaluate the relationship between the predictive performance and geometric quality of the SCNN-IDG model based on three different projections: Fuller, Snyder, and Gnomonic. It is observed that the SCNN-IDG model utilizing Gnomonic projection exhibits the most uniform cwm metric, thereby demonstrating the best predictive capability. This confirms the effectiveness of the comprehensive evaluation model and weighted grid quality assignment.

2. Method

To create a multimetric evaluation method for assessing the impact of the icosahedral diamond grid quality on the SCNN performance, we propose a multimetric evaluation method using a random forest algorithm to establish the nonlinear relationships between grid quality metrics and the SCNN performance, with an icosahedral diamond grid based on the Fuller projection as an example, and then assign impact weights to each grid quality metric considering the SCNN performance. The process is illustrated in Figure 1, and the key steps are as follows:

1. Train and test the SCNN-IDG. In this study, an icosahedral diamond grid was built using the Fuller projection with a 1-to-4 refinement approach. Diamond grids are similar to planar square grids, and many advanced planar CNN modules can be directly extended to icosahedral diamond grids, such as Swin-Transformer and ASPP modules. Then, spherical convolution and spherical pooling modules are designed based on the characteristics of the icosahedral diamond grid, and these modules are stacked to build the SCNN-IDG model following the structure of ResNet (He et al. 2016). Finally, the SCNN-IDG model's performance is assessed on the SPH-UC dataset. For detailed information, please refer to Section 2.1.

2. Create a sample set for the random forest evaluation method. The independent variables of the sample set consist of the mean and standard deviation of the grid quality metrics within regions with a field of view (FOV) of 100 degrees, as well as the mean and standard deviation of the ratio between the maximum and minimum values. In this study, four grid quality metrics are selected following the principle of describing point uniformity, totalling fourteen independent variables. Additionally, the test set results of SCNN-IDG on the SPH-UC dataset serve as the dependent variable. For specific details, please refer to Section 2.2.

3. Create a multimetric evaluation method to evaluate the impact of the grid quality indicators on the SCNN performance. Initially, we removed independent variables with a correlation value of less than 0.4. Subsequently, we introduced the random forest algorithm to establish nonlinear relationships between the independent variables and the dependent variable. We employed feature importance analysis to assign impact weights to the independent variables. For detailed information, please refer to Section 2.3.

Figure 1. The research methodology and steps.

2.1. Construction of the SCNN-IDG

The icosahedron based discrete global grid cells mainly include triangles, hexagons and diamonds. The shapes of triangular and hexagonal grids, are different from those of the square grids used by traditional CNNs. As a result, many complex CNN algorithms based on square grids, such as Swin-Transformer and ASPP modules, face challenges in regard to adaptation. This limitation hinders the applications of the SCNN based on the icosahedral grid of triangles or the hexagonal grid. On the other hand, diamond grids are similarity to planar square grids, and many advanced planar deep learning modules can be directly extended to icosahedron diamond grids. Therefore, in this study, a spherical convolutional neural network framework is built based on icosahedral diamond grid, which is also the first application of icosahedral diamond grids in spherical deep learning.

2.1.1. Initial icosahedron diamond subdivision and encoding

The icosahedral diamond grid used in this study adopts the Fuller projection and a 1-to-4 refinement approach (Amiri, Harrison, and Samavati 2016). In considering that most existing convolutional neural network architectures utilize$2\times 2$ upsampling and downsampling (Chen et al. 2021), the 1-to-4 refinement approach is selected in this paper.

Initial Subdivision: First, two vertices of the icosahedron are placed at the North and South poles, and the third vertex is located at (0°, 25.56505°N). The icosahedron is then unfolded, and each vertex is assigned a number following a specific rule. The rule is as follows: the South and North pole vertices are assigned numbers 0 and 11, respectively. Vertex 1 is positioned on the prime meridian, and the remainder of the vertices are numbered sequentially. Next, the triangles in the regular icosahedron that have vertices at the pole locations are merged with the adjacent triangles along their shared edges. This process results in the subdivision of the regular icosahedron into 10 initial diamonds grid units (as shown in Figure 2).

Figure 2. The positions of the twelve vertices of the icosahedron and the ten initial diamonds formed by two adjacent triangles.

Encoding: Z-order filling curve encoding is employed for indexing sample data (Li et al. 2022).

For the generation and encoding of the grid, the open-source code library DGGRID is utilized (Kmoch et al. 2022; Sahr 2019). An example of grid generation is illustrated in Figure 3.

Figure 3. Example of grid generation.

2.1.2. Construction of the SCNN-IDG module

Based on the residual structure (He et al. 2016) and an icosahedron diamond grid at the fifth level, the SCNN-IDG is constructed as described in Figure 4. In the network architecture, L_n represents the icosahedral diamond grid of the nth level. The SConv2D operation is implemented using Algorithm A.1 (Table A.1), and the SMaxPool2D operation is implemented using Algorithm A.2 (Table A.2). The global average pooling (GAP) layer and the multilayer perceptron (MLP) layer are used for output prediction. The ResBlock employs the BasicBlock structure, which consists of two SConv2D operations with a kernel equal to 9 and one SConv2D operation with a kernel equal to 1 (SConv1 × 1). Each ResBlock has an input channel size of a, an intermediate channel size of b, and an output channel size of c. The SCNN-IDG architecture combines and stacks these components to efficiently train and test the SCNN-IDG using open deep learning libraries.

Figure 4. SCNN-IDG network structure.

This study primarily focuses on the influence of the discrete global grid quality on the SCNN performance. Therefore, in addition to the basic residual structure, no other deep learning operations are extended to the SCNN-IDG.

2.1.3. Training and testing of the SCNN-IDG

Data Processing: In this study, we construct the Spherical UC (SPH-UC) dataset based on the UC Merced Land Use dataset (Yang and Newsam 2010). This dataset replaces the Spherical MNIST (SPH-MNIST) dataset used in previous literature for training and testing the SCNN (Lee et al. 2020). SPH-MNIST, a new dataset constructed by projecting MNIST onto the sphere for evaluating and comparing the SCNN performance, has been widely used in SCNN performance comparison and evaluation (Lee et al. 2020; Gerken et al. 2022; Zhang et al. 2023). Lee et al. (2020) generated an SPH-MNIST dataset by projecting original MNIST images onto an icosahedral triangular grid. They observed that following training, the SCNN based on the icosahedral triangular grid exhibited varying classification accuracy at different latitudinal and longitudinal positions within the spherical MNIST test set. However, the SPH-MNIST dataset only contains simple black and white pixel handwritten digits 0–9 and does not include geographical features. Additionally, its prediction difficulty is relatively low. The SPH-UC dataset considers geographical features more comprehensively, ensuring the reliability and applicability of this study. Furthermore, to enhance the data processing flexibility, we employ online data processing by embedding the steps for creating the SPH-UC dataset into the deep learning data preprocessing process using the torchvision transforms class in PyTorch. This enables end-to-end SCNN.

Training of the SCNN-IDG Model: The training set is randomly projected onto a spherical region with a field of view (FOV) of 100 degrees, following the approach described in Lee et al. (2020). Fivefold cross-validation is employed to train the SCNN-IDG model. Figure 5 provides an illustrative example of randomly projecting UC dataset samples onto different FOV regions of the globe.

Figure 5. An example of the SPH-UC dataset. The UC data are projected onto a spherical surface with an FOV of 100 degrees (nonblack regions in the image). The red colour represents the initial icosahedron, and the white numbers show the latitude and longitude of the centre of view.

Testing of the SCNN-IDG Model: All samples in the test set are projected to an area with an FOV of 100 degrees using the centre point of each cell of the icosahedral diamond grid as the viewpoint. This process generates$10\ast 4^l$ ($l$ is grid cell subdivision level) groups of the testing set located in different regions. The grid quality varies within each group's FOV region, and the results obtained from SCNN-IDG also differ. Therefore, the grid indicators within the FOV region of each group in the testing set are considered independent variables, while the SCNN-IDG predictions for different groups of the testing set are considered dependent variables. This allows for the construction of a dataset for evaluating the impact of grid quality on SCNN performance.

Result Validation of the SCNN-IDG: In this study, accuracy is adopted as the validation metric for the SCNN-IDG results. Accuracy is defined as follows: (1) $\mathrm{acc}(fD)={\displaystyle \frac{1}{m}\sum _{i=1}^m\mathbb{I}(f({\mathit{x}}_i)=y_i)}$(1)

2.2. Construction of the sample set

2.2.1. Construction of the independent variables

Following the principle of describing point uniformity, grid metric indicators are selected for the multimetric evaluation of the impact of the grid quality on the SCNN performance. The selected indicators include the distance between neighbouring central points (dis), the angle between adjacent centre points (angle)(Ming et al. 2007), the cell wall midpoint ratio (cwm)(Gregory et al. 2008), and the zone standardized compactness (zsc)(Kimerling et al. 1999). Based on these selected grid metric indicators, the construction of the sample set independent variables is performed in two steps:

1. Calculate the quality indicators for each grid cell.

In the first step, we calculate the zsc of each grid cell and the mean and the maximum minimum ratio (mmr) for the dis, angle, and cwm of each grid cell. The detailed descriptions of the indicators are as follows:

angle: angle represents the dihedral angle between a grid point and its two neighbours. As shown in Figure 6, $\alpha$ can be calculated using the sine and cosine formulas for spherical triangles (Wang et al. 2021). The computed metrics include the mean value of all angles corresponding to the grid point (angle_m) and the ratio of the maximum and minimum values of all angles corresponding to the grid point (angle_mmr).

Figure 6. Schematic diagram of angle and dis indicators. The dis indicators are shown as a dotted line between P₀ and the adjacent point P_i, and the angle indicators are shown as the angle α.

dis: dis represents the great circle distance between a grid point and one of its neighbouring points. As shown in Figure 6, the great circle distances can be calculated using the sine and cosine formulas for spherical triangles. The computed metrics include the mean value of all distances corresponding to the grid point (dis_m) and the ratio of the maximum and minimum values of all distances corresponding to the grid point (dis_mmr).

cwm: cwm represents the great circle distance between the midpoint of the line connecting the centre points of two adjacent grid cells and the midpoint of their shared edge, as shown in Figure 7. The calculated indicators include the mean value of all cwm values corresponding to the grid point (cwm_m) and the ratio of the maximum and minimum values of all cwm values corresponding to the grid point (cwm_mmr).

Figure 7. Cell wall midpoint ratio.

In particular, there are nine neighbours or seven neighbours in the cell at a special position of the icosahedral diamond grid, and these cells are distributed around the vertices of the initial subdivision of the icosahedron diamond grid, as shown in Figure 8. In this study, the corresponding nine neighbours and seven neighbours are used to calculate the angle, dis and cwm indicators.

Figure 8. There are nine adjacent cells and seven adjacent cells distributed around the vertices of the initial subdivision of the icosahedral diamond grid. The red line is the initial diamond, and the black line is level 3 the icosahedral diamond grid.

zsc: zsc describes the degree of shape regularity of spherical polygons. It is calculated as the ratio of the perimeter of a spherical polygon to the perimeter of a spherical circle with the same area. zsc is a comprehensive indicator that incorporates both the area and perimeter and represents the regularity of the ‘influence region’ of a pixel. The uniformity of the regularity also represents the uniformity of the point set. The calculation method is shown in Equation (2(2) $\mathrm{zsc}={\displaystyle \frac{\sqrt{4\mathrm{\pi A}-A^2/r^2}}{\mathrm{Per}}}$(2) ), where A represents the area of the spherical polygon, and per represents the perimeter of the spherical polygon (Wang et al. 2021). (2) $\mathrm{zsc}={\displaystyle \frac{\sqrt{4\mathrm{\pi A}-A^2/r^2}}{\mathrm{Per}}}$(2)

(2)	Calculate the mean and standard deviation for the grid indicators within the FOV region

For any given test set, to calculate the mean and standard deviation statistics of the indicators within the FOV region (as shown in Figure 5), follow these steps:

Step 1. Identify the grid cells within the FOV region.

Step 2. For each selected grid cell, collect the values of the desired indicators within that cell.

Step 3. Calculate the mean (average) and standard deviation of the indicator values within the FOV region.

Repeat steps 2 and 3 for all desired indicators within the FOV region. We can obtain a total of 14 independent variables, and the abbreviations and meanings for all the independent variable names can be found in Appendix C.

2.2.2. Construction of the dependent variables

As noted in Section2.1.3, the accuracy of the 10240 sets of test results generated by the SCNN-IDG is considered the dependent variable in this study's sample set. The dependent variable can also be understood as a comprehensive evaluation index, which represents the comprehensive influence of various grid indices on the prediction results of the spherical CNN.

2.3. Multimetric evaluation method for the impact of the grid quality on the SCNN performance

Correlation Filtering: By calculating the correlation between the independent variables and the dependent variable, we removed the independent variables with an absolute correlation value of less than 0.4; only independent variables with strong correlations were retained to enhance the accuracy of the multimetric evaluation method and improve the result interpretation accuracy.

Method Construction for Multimetric Evaluation: A random forest (RF) builds upon a bagging ensemble by introducing random attribute selection during the training process and has achieved strong performance in various tasks (Breiman 2001). In this study, multiple grid quality metrics are chosen as independent variables, and the testing accuracy of the SCNN-IDG in different regions is considered the dependent variable. By fitting a random forest model to capture the nonlinear relationships between these variables, the impact of grid metrics on the SCNN performance is quantified. Consequently, a multimetric evaluation method is constructed to assess the discrete global grid quality's influence on SCNN performance. Furthermore, feature importance evaluation (Ye et al. 2019) is utilized to assign weights considering the impact of SCNN performance.

Accuracy Verification of the Multimetric Evaluation Method: The coefficient of determination, R², is utilized as the accuracy verification metric for the multimetric evaluation method. The R² reflects the goodness of fit between the simulated values and the observed values, with a value closer to 1 indicating a higher level of model fitting accuracy. The calculation method is defined by Formula (3) as follows. (3) $R^2={\displaystyle \frac{{\sum }_{i=1}^n{(x_i-\overline{x})}^2\times {(y_i-\overline{y})}^{2{\sum }_{}^{}}}{{\sum }_{i=1}^n{(x_i-\overline{x})}^2\times {\sum }_{i=1}^n{(y_i-\overline{y})}^2}}$(3)

3. Results and analysis

3.1. SCNN-IDG result verification

The SCNN-IDG was trained using fivefold. For the parameter configuration of the training process, please refer to Appendix B.1. The validation set accuracy and loss curves for each of the five training iterations are illustrated in Figure 9. The accuracy of the verification set is shown on the left side of the figure, and the accuracy index gradually increased and tended to stabilize. The loss function result of the verification set is shown on the right side of the figure, and the loss function result gradually decreased and tended to stabilize. The validation set accuracies consistently remained at approximately 90% throughout the five iterations. Both the accuracy and loss on the validation set stabilized after approximately 150 epochs, triggering early stopping before 200 epochs. This indicates that the optimal performance is achieved in all five training instances of SCNN-IDG.

Figure 9. SCNN-IDG five-fold cross-validation results.

3.2. Results of constructing the sample set

3.2.1. Results of constructing the dependent variable

Based on the trained SCNN-IDG model, the accuracies of 10240 test set samples located in different FOV regions were calculated. The accuracy values were then visualized by assigning them as pixel values to the grid cells at the centre of each FOV region, as shown in Figure 10. From the figure, it can be observed that the maximum accuracy of the test set is 90.1%, while the minimum accuracy is 86.71%. The difference in accuracy between these values is 3.4%, indicating significant variations in accuracy across different regions of the global discrete grid. Furthermore, through observation, it was found that the accuracy of the symmetric position of the initial diamond face of the icosahedron is the same. However, the accuracy was lower around the vertices where five initial diamond faces intersected (corresponding to the North or South Pole), while the interior regions of the diamond faces exhibited higher accuracy values. The deformation ellipses of the Fuller projection indicate that grid quality varies more significantly closer to the vertices (Gregory et al. 2008). The North or South Pole, formed by the intersection of five initial diamond-shaped grids, exhibits relatively substantial changes in grid quality, resulting in lower predictive accuracy. In contrast, the grid quality variations in the interior regions are comparatively minor, leading to higher predictive accuracy.

Figure 10. The left side of the figure shows the histogram and descriptive statistics of the dependent variable, and the right side shows the spatial distribution of the dependent variable. The black line segment represents the initial diamonds, and the centre point of the image represents the north or south pole.

3.2.2. Results of constructing the independent variable

1. Results of the correlation analysis

The independent variables were analyzed in terms of their correlation with the dependent variable, and the results are presented in Table 1. Variables with a correlation coefficient of less than 0.4 were removed, resulting in 12 variables that showed a high correlation with the dependent variable. The scatter plot depicting these variables is shown in Figure 11. Interestingly, it can be observed that the standard deviation (std) of all grid indicators exhibits a negative correlation with the dependent variable. This indicates that as the grid indicator variation decreases and the grid points become more uniform, the predictive accuracy of the SCNN model increases.

(2)	Spatial distribution of the independent variables

Figure 11. Scatterplots of independent and dependent variables. The y-axis represents the accuracy of SCNN-IDG, while the x-axis represents the grid indicators of the FOV region.

Table 1. The correlation coefficient between the independent variable and dependent variable.

	angle	angle_mmr	dis	dis_mmr	cwm	cwm _mmr	zsc
mean	0.64**	0.69**	0.76**	0.69**	−0.80**	−0.77**	−0.70**
std	−0.08*	−0.17**	−0.78**	−0.44**	−0.76**	−0.75**	−0.54**

*p < 0.01,**p < 0.001

The histograms and spatial distribution maps of the independent variables are shown in Figure 12. It is important to note that these maps differ from those presented in the literature (Gregory et al. 2008; Kimerling et al. 1999). In the figure, the pixel value of each cell represents the mean or standard deviation of all cell quality indicators in the FOV of 100 degrees with the centre point of the cell as the centre of view (as shown in Figure 5).

Figure 12. Histogram and spatial distribution of each independent variable.

By conducting a comprehensive analysis of Figure 10 and Figure 12, we can conclude that the values of the independent and dependent variables are completely identical at the symmetric positions of the ten initial diamonds on the icosahedron (as shown in Figure 2). In other words, even if the FOV region is located differently when the grid metrics of the FOV region are the same, the corresponding performance of the SCNN is also the same. This indicates a strong correlation between the predictive performance of the SCNN and the grid metrics.

3.3. Multimetric evaluation method accuracy verification

For the parameter configuration of the random forest evaluation method, please refer to Appendix B.2. The values of the sample set are the same at the symmetrical positions of the ten initial diamonds. Therefore, to avoid repeated values, the independent variable and dependent variable in any initial diamond are selected to construct a multimetric evaluation method. The results are shown in Figure 13. On the multimetric training set, the method achieved an R² value of 0.82 and an RMSE of 0.26, and on the testing set, the method achieved an R² value of 0.80 and an RMSE of 0.30. The multimetric evaluation method effectively captures the nonlinear relationship between the discrete global grid quality and the accuracy of the SCNN.

Figure 13. Multimetric evaluation method accuracy verification.

In this study, weights are assigned to grid quality metrics using the feature importance of the random forest model. As shown in Figure 14, among the indicators, the cwm indicator has the highest impact on the performance of the SCNN-IDG, contributing 47% of the importance, with a weight of 0.47. The dis indicator contributes 29% of the importance, with a weight of 0.29. The zsc and angle indicators contribute 13% and 11% of the importance, respectively, with weights of 0.13 and 0.11.

Figure 14. Impact of the grid quality indicators weight on the SCNN-IDG learning performance.

In this paper, the residual structure of ResNet (He et al. 2016) is adopted to construct the SCNN. Some researchers have noted that this structure can be considered a variant of finite differencing. Stacking ResNet with these residual structures can be seen as a simplified version of global atmospheric models (Clare, Jamil, and Morcrette 2021). Additionally, the cwm metric evaluates the accuracy of finite differencing when using the DGGS within global atmospheric models (Heikes and Randall 1995a; 1995b). A smaller cwm value corresponds to a higher differencing accuracy, which is also associated with higher accuracy in ResNet. This explains why the cwm metric has the highest impact weight on the SCNN performance based on the icosahedral diamond grid proposed in Figure 14.

Partial dependency plots (PDPs) are graphical tools used to visualize the relationship between a predictor variable and the response variable in a machine learning method. These plots show the change in a response variable while controlling for other features when a single feature is varied (Agarwal and Das 2020). Figure 15 presents the PDP after constructing the multimetric evaluation method. The y-axis represents the accuracy of SCNN-IDG, while the x-axis represents the grid indicators of the FOV region. By observing the plot, the following results can be obtained:

1. The relationship between the SCNN-IDG results and discrete global grid quality metrics is nonlinear. Therefore, compared with the linear regression model, the random forest method can more accurately capture the complex correlation between the quality of the grid and the performance of SCNN-IDG.

2. For the standard deviation(std) statistics of all grid metrics (subplots a-e), there is a decreasing trend in the performance of SCNN-IDG as the standard deviation value increases. The std statistic represents the uniformity of grid indicators, and a higher value indicates poorer grid uniformity, leading to worse performance of the spherical CNN. For example, in subplot (a), as cwm_std increases, the prediction results of the SCNN rapidly decline, and this metric exhibits the largest decrease among all discrete global grid quality metrics, indicating that the uniformity of the cwm has the greatest impact on the spherical CNN. This is also supported by the highest weight assigned to the cwm_std indicator, with a weight value of 0.22, as shown in Figure 14.

3. Different trends are observed in terms of the impact of the mean statistic of each indicator on the SCNN-IDG results. For example, in subplot (f), as the angle_mean increases, the accuracy of SCNN-IDG shows a slow growth trend. angle_mean represents the mean value of all dihedral angles corresponding to each grid point within the FOV region. For the icosahedral diamond grid, when the FOV region does not include the initial diamond vertex (as shown in Figure 8), each grid element always has eight neighbours, resulting in a constant angle_mean value of $2\times \pi /8=0.7859$. Therefore, the angle_mean can only reflect the variation in the SCNN accuracy when the FOV region includes the initial diamond vertex. This explains why the angle_mean has the smallest weight of only 1.6% in Figure 14.

Figure 15. Partial dependency plots of the multimetric evaluation method.

3.4. The impact of icosahedron diamond grids with different projections on SCNN performance

To compare the impact of different projection methods on the performance of SCNN-IDG, we employed the same methodology to construct SCNN-IDG models using two other different projections: the Icosahedral Snyder Equal Area Projection (ISEA) and the Gnomonic Projection (Gnomonic). SCNN-IDG models were trained using a fivefold cross-validation approach, maintaining identical deep learning parameter settings across both projection methods. Figure 16 depicts the accuracy of the validation set for each of the five training iterations. Optimal performance was consistently achieved across all five training instances of SCNN-IDG models.

Figure 16. The results of five-fold cross-validation for SCNN-IDG based on different projections. The left side displays the validation set accuracy using the ISEA projection, while the right side presents the validation set accuracy using the Gnomonic projection.

Subsequently, the spatial distribution maps of the dependent variable were constructed using the same methodology, as shown in Figure 17, akin to the outcomes depicted in Figure 10 in the article, significant differences in accuracy were observed across different regions of the discrete global grid, with the accuracy remaining consistent for symmetric positions on the initial diamond face of the icosahedron. Notably, lower accuracy was observed around the vertices where five initial diamond faces intersected, corresponding to the North or South Pole. Conversely, higher accuracy values were evident in the interior regions of the diamond faces.

Figure 17. Statistical and spatial distribution of SCNN-IDG test set accuracy based on ISEA and Gnomonic projections.

Furthermore, we conducted an analysis of SCNN-IDGs with different projections after five-fold cross-validation training, evaluating their accuracy on the test set, as depicted in Table 2. Observing the mean accuracy of models on the test set, it's evident that the Gnomonic projection yields the highest accuracy, followed by the ISEA projection, while the Fuller projection exhibits the lowest accuracy. Based on these results, we compared the ratio of minimum value (RMV) of grid indicators for different projections of SCNN-IDGs, as shown in Figure 18. For a grid indicator m, the calculation method for RMV is as Equation (4(4) $\mathrm{RM}{V}_m=\{m/\min (m_i)|i=(\mathrm{ISEA},\mathrm{Fuller},\mathrm{Gnomonic})\}$(4) ). (4) $\mathrm{RM}{V}_m=\{m/\min (m_i)|i=(\mathrm{ISEA},\mathrm{Fuller},\mathrm{Gnomonic})\}$(4) From Figure 18, it can be observed that all indicators in the Gnomonic projection are relatively smaller, followed by ISEA. Fuller projection exhibits the poorest uniformity across indicators. Specifically, within the cwm_mmr_std sub-indicator, the RMV of Fuller projection is 14 times greater than that of Gnomonic projection, while the RMV of ISEA projection is 4 times greater than that of Gnomonic projection.

Figure 18. the RMV of grid indicators for different projections of SCNN-IDGs. It primarily highlights the two indicators, cwm and dis, whose combined weights constitute 73% of the total.

Table 2. Accuracy of SCNN-IDGs with different projections on the test set. As five-fold cross-validation was conducted, resulting in five sets of test accuracies, the average is taken as the final accuracy of the SCNN-IDGs models on the test set.

	Fold 1	Fold 2	Fold 3	Fold 4	Fold 5	Mean
Fuller	87.90	88.27	88.07	88.71	90.71	88.73
ISEA	87.81	88.67	89.62	88.47	90.58	89.03
Gnomonic	88.95	88.75	89.18	90.81	89.62	89.46

Finally, combined with Table 2 and Figure 18, the following conclusions can be drawn:

1. Although the Gnomonic projection results in deformation of angles and areas, its uniformity in the cwm index, particularly in the uniformity of the maximum-to-minimum value ratio, leads to the highest accuracy of SCNN-IDG based on Gnomonic projection.

2. The ISEA projection, while ensuring equal area, sacrifices the uniformity of the maximum-to-minimum ratio between adjacent grid cells, resulting in lower SCNN-IDG accuracy based on the ISEA projection compared to that based on the Gnomonic projection.

3. Although the Fuller projection performs the best in the cwm_mean and cwm_std sub-indices, this comes at the cost of sacrificing the uniformity of the maximum-to-minimum value ratio between adjacent grid cells. Therefore, the SCNN-IDG accuracy based on the Fuller projection is the lowest.

4. Analyzing SCNN-IDGs composed of different projections validated the correctness of the indicator weight allocation in this paper and also illustrated that a single indicator cannot entirely represent the impact of DGGS grid quality on SCNN performance. It is necessary to consider multiple indicators to comprehensively assess the influence of grid quality on SCNN performance.

3.5. Comparative analysis with existing achievements

1. Compared to the results obtained by Lee et al. (2020), which proposed using the mean of the area as an indicator to assess the impact of grid quality on SCNN performance, this study observed that the performance of SCNN is influenced by the quality of the grid within the specific geographical area where the data are situated. The study further analyzed the correlation between area metric and the performance of SCNN-IDGs with different projections, as depicted in Figure 19. It was observed that the SCNN models composed of three different projections all showed no correlation with area metric. Hence, area metric might not be crucial grid indicators affecting spherical deep learning performance. Moreover, this paper introduces the utilization of the random forest method to comprehensively assess the impact of multiple grid quality metrics on SCNN grid performance for the first time. Impact weights are assigned to each grid quality metric concerning the SCNN performance.

2. In comparison to the study by Cho, Jung, and Kwon (2022), which only relied on distance metrics to assess the influence of the grid quality on spherical deep learning performance, it is demonstrated in this paper that through the assigned weights (Figure 14), the cwm indicator has the highest impact on the performance of the spherical CNN, with a weight of 0.47. In contrast, the weight for the distance metric is 0.29. Therefore, it is evident that a single distance indicator cannot provide a comprehensive evaluation of the impact of the DGGS quality on the SCNN performance. The method proposed in this paper enables a more comprehensive assessment of the impact of the DGGS quality on the SCNN performance and can also serve as a constraint for the development of more reliable DGGS tailored for SCNN applications.

Figure 19. The correlation between the test set accuracy of SCNN-IDGs based on three different projections and their corresponding area metrics. From left to right, these are the Fuller projection, the ISEA projection (which is an equal-area projection, hence represented as a straight line parallel to the y-axis in the graph), and the Gnomonic projection.

4. Discussion

4.1. Application scenario

DGGS serve as a spatial partitioning method and can also serve as geographic information carriers for natural, social, and economic attributes within specific spatial ranges (Li, Shao, and Ding 2014). However, existing research lacks a multimetric evaluation of the impact of grid quality on the SCNN performance when grids are used as information carriers for artificial intelligence analysis. This study establishes a multimetric evaluation method for assessing the impact of grid quality metrics on the SCNN performance and provides weights for these grid quality metrics. Through the established evaluation method and the weight allocation system, reliable support can be provided for intelligent geographic feature recognition using discrete global grids in fields such as meteorological forecasting and disaster prediction. Furthermore, this method offers constraints for optimizing discrete global grids. For example, when establishing an intelligent disaster prediction model for large regions using deep learning based on discrete global grids, the multimetric evaluation model proposed in this paper can be used as a primary constraint. The method presented by (Zhou et al. 2020) can be employed to optimize the positions and orientations of the vertices of a regular polyhedron, ensuring that the disaster prediction region achieves the highest comprehensive indicator, which corresponds to the region with the best performance for spherical deep learning prediction. In addition, the prediction accuracy of spherical deep learning can be improved by optimizing discrete global grids using the influence weights of grid metrics; for example, when using deep learning for global weather forecasting (Bi et al. 2023; Kurth et al. 2023), the optimized discrete global grid can be used as a framework to build a global deep learning weather forecasting model. A more uniform discrete global grid can help improve the accuracy of weather forecasting. These will be the next steps of this research.

4.2. Residual structure

The SCNN-IDG framework constructed in this study is based on the residual structure of ResNet (Figure 20). The skip connection mechanism in ResNet avoids the vanishing gradient problem, making ResNet the first neural network capable of achieving ultradeep convolution and exhibiting higher prediction accuracy (He et al. 2016). Consequently, the residual structure has been widely applied as a foundational architecture in various deep learning algorithms, such as transformer and graph convolutional neural networks (Li et al. 2022; Dosovitskiy et al. 2021). Moreover, researchers have discovered that neural network modules composed of residual structures can be regarded as discrete forms of ordinary differential equations (Sander, Ablin, and Peyré 2022) or as a variant of the forward Euler finite difference, representing a highly simplified version of global atmospheric numerical models (Clare, Jamil, and Morcrette 2021). For atmospheric modelling, (Heikes and Randall 1995a; 1995b) proposed the cwm indicator and indicated that a smaller cwm value indicates a higher accuracy of finite difference schemes in global atmospheric numerical models. Therefore, the cwm indicator has a significant impact on the prediction performance of spherical deep learning networks composed of residual structures. This explains why the cwm indicator carries the highest weight in terms of the influence on the SCNN in Figure 14.

Figure 20. Residual structure.

4.3. Expanding the global discrete grid deep learning framework to other fields

Spherical deep learning frameworks based on global discrete grids can be extended to various fields beyond geospatial data. Similar spherical data structures can be found in domains such as autonomous driving, augmented reality (AR) / virtual reality (VR), robot navigation, and cortical surface parcellation for brain segmentation. These domains also face challenges related to uneven geometric properties that can hinder the performance of spherical deep learning (Zhao, Wu, and Li 2023; da Silveira et al. 2022). In the future, it would be beneficial to extend the development and optimization of uniform global discrete grids to these domains to enhance the reliability and accuracy of SCNN predictions, addressing the challenges posed by uneven geometric properties in these fields.

4.4. Multicollinearity of independent variables

We categorize the multicollinearity among independent variables into four main types:

1. Strong correlation among independent variables, but different weights and impact trends on the dependent variable. Figure 21 shows, from left to right, the correlation between the cwm_mean and cwm_std indicators, the trend of the cwm_mean indicator on the dependent variable (from Figure 15), and the trend of the cwm_std indicator on the dependent variable (from Figure 15). We can observe that although there is a very strong correlation between these two variables, the trend of their influence on the dependent variable and the weights of the dependent variables are not the same when considered together with the weighting information in Figure 14.

2. Strong correlation among independent variables, with approximately equal weights and similar impact trends on the dependent variable. When considering Figure 22 in conjunction with Figure 14, it becomes evident that the variables zsc_mean and dis_std exhibit a strong correlation. Moreover, they impact the dependent variable with approximately equal weights and similar impact trends.

3. Weak correlation among independent variables, but with similar trends in their impact on the dependent variable. When considering Figure 23 in conjunction with the relative magnitudes of weights in Figure 14, it can be observed that the two independent variables, dis_mmr_mean and dis_mmr_std, although having a weak correlation, lead to approximately equal weights and similar impact trends on the dependent variable.

4. Weak correlation among independent variables, with different trends in their impact on the dependent variable. When considering Figure 24 in conjunction with the magnitudes of weights in Figure 14, it becomes evident that the two independent variables, angle_mean and dis_mmr_std, have a weak correlation, and their weights and impact trends on the dependent variable are also dissimilar.

Figure 21. Correlation between the two independent variables, cwm_std and cwm_mean, and their effects on the dependent variable.

Figure 22. Correlation between two independent variables, zsc_mean and dis_std, and their effects on the dependent variable.

Figure 23. Correlation between two independent variables, dis_mmr_mean and dis_mmr_std, and their effects on the dependent variable.

Figure 24. Correlation between two independent variables, angle_mean and dis_mmr_std, and their effects on the dependent variable.

Therefore, due to the nonlinear relationship between independent variables and the dependent variable, independent variables with multicollinearity may exhibit different impact trends and weights on the dependent variable. However, the main purpose of this article is to establish a multimetric assessment method to evaluate the impact of icosahedral diamond grid quality on the performance of SCNN. By introducing the random forest method, we model the nonlinear relationships between grid quality indicators as independent variables and SCNN performance as the dependent variable. The random forest algorithm is an ensemble machine learning technique that combines multiple decision trees to make accurate predictions while mitigating overfitting and handling a wide range of data types and complexities. Random forests are generally robust to multicollinearity among independent variables, which is one of its advantages over traditional linear regression models (Fawagreh, Gaber, and Elyan 2014; Lindner, Puck, and Verbeke 2022). The main reasons for this robustness are summarized as follows:

1. Ensemble of Decision Trees: Random forests are an ensemble of decision trees, and each tree considers only a subset of features for splitting at each node. This means that even if two variables are highly correlated (multicollinear), different trees in the forest may choose different variables for splitting at different nodes, effectively reducing the impact of multicollinearity.

2. Voting and Averaging: In random forests, predictions are made by averaging or voting on the outputs of multiple decision trees. The ensemble nature of random forests helps to ‘smooth out’ the impact of multicollinearity because the predictions are based on a combination of different trees, each potentially handling multicollinear variables differently.

3. Bootstrap Aggregation: Random forests build multiple decision trees using bootstrapped samples of the data and then combine their predictions. This ensemble approach tends to reduce the overall sensitivity to noise and collinearity in the data.

Based on these reasons, this paper, following references (Wang et al. 2022; Li et al. 2020; Guo, Hu, and Zheng 2023), chooses not to consider multicollinearity among independent variables.

4.5. Future work

4.5.1. Performance comparison of the SCNN-IDG framework with other spherical deep learning frameworks.

In this study, the SCNN-IDG framework is built based on the icosahedral diamond grid and focuses on the multimetric evaluation method of the grid quality impact on the performance of the SCNN. However, the predictive performance of the SCNN-IDG has not been fully explored. Therefore, in future research, advanced planar deep learning modules, such as Swin Transformer and ASPP modules, will be introduced into the SCNN-IDG framework. Furthermore, a performance comparison will be conducted between the SCNN-IDG framework and other spherical deep learning methods to validate its superiority.

4.5.2. The impact of the grid quality on the SCNN performance for different types of DGGS

This study evaluates the impact of the grid quality on SCNN performance for the icosahedral discrete grid but does not consider the influence of different types of DGGS on the performance of the SCNN. Such as cubed-sphere grid (Weyn, Durran, and Caruana 2020), QTM grid (Wang et al. 2021), HEALPix grid (Defferrard et al. 2019), rhombic triacontahedron grid (Liang et al. 2022), and disdyakis triacontahedron grid (Hall et al. 2020). Therefore, in future work, we will try to use the evaluation method proposed in this article to explore the impact of different types of DGGS and different types of grid conversion on SCNN performance. This will help enable a comprehensive understanding of the suitability of various grid types in SCNN applications and provide guidance for selecting the optimal grid.

4.5.3. Evaluate the impact of icosahedron grids with different cell shapes on SCNN performance

In future work, we will evaluate the impact of the quality of different icosahedral grids with varying cell shapes on the performance of SCNN, and compare the similarities and differences in the impact of icosahedral grids with different properties on SCNN performance.

4.5.4. The impact of grid metrics on the SCNN performance at multiple resolutions

This study proposes a multimetric evaluation method that considers the impact of grid metrics at a single resolution on SCNN performance. However, SCNN is a multiresolution grid model, and it is necessary to consider the influence of grid metrics at multiple resolutions on SCNN performance. One of the reasons why the R² value in Figure 13 is only 0.8 and not higher is the lack of information on grid metrics at multiple resolutions. The next step should involve comprehensively considering the influence of grid metrics at multiple resolutions, providing a more comprehensive and accurate grid quality assessment method, and offering guidance for the design and optimization of spherical deep learning grids.

5. Conclusion

In this study, an SCNN framework is proposed based on an icosahedral diamond grid, and a sample set is constructed using the prediction results of the SCNN and grid quality indicators. A multimetric evaluation method is developed using the random forest method to assess the impact of the discrete global grid quality on the performance of SCNN and assign quantitative weights to each grid indicator. The main contributions are as follows:

1. This study presents the first SCNN framework based on an icosahedral diamond grid, providing a new architecture for spherical deep learning.

2. The random forest algorithm is applied to capture the complex nonlinear relationship between the grid quality and SCNN performance, and a multimetric evaluation method is constructed for assessing the impact of the global discrete grid quality on the SCNN performance.

3. Quantitative weights are assigned to each grid indicator based on the feature importance evaluation. Specifically, the weight for the cwm indicator is 0.47, the weight for the dis indicator is 0.29, the weight for the zsc indicator is 0.13, and the weight for the angle indicator is 0.11. Among all grid metrics, the cwm indicator has the greatest impact on the SCNN. These results provide an important weight reference for optimizing grid metrics to improve the prediction accuracy of the SCNN.

In conclusion, this study presents a multimetric evaluation method for assessing the impact of the grid quality on the SCNN performance. The weights proposed in this study can be applied in fields such as weather forecasting and disaster prediction to enhance the reliability and performance of intelligent geographic feature identification. Furthermore, the multi-metric evaluation method proposed in this paper can be used as a constraint or objective function. By iteratively optimizing the grid metrics of the icosahedral diamond grid through fine-tuning the grid positions, a more uniform icosahedral diamond grid can be obtained, which can enhance the performance of the SCNN. This is conducive to the application and promotion of DGGS in deep learning.

Acknowledgement

The authors would like to thank the editors and the anonymous reviewers for their constructive comments and suggestions, which greatly helped to improve the quality of the manuscript.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Data availability statement

All grids in this study are generated by the open-source DGGS implementation – DGGRID, which can be retrieved from https://www.discreteglobalgrids.org/software/. SCNN-IDG code and data can be obtained from https://github.com/Seraph0317/Data-Multi-Metric-Evaluation-Method-.git.

Additional information

Funding

This research was supported by the general program of National Natural Science Foundation of China [grant number: 42371412 and 42271435].

Appendix A

The SCNN-IDG module is designed to perform deep learning tasks on the icosahedral diamond grid. It consists of the following key components:

Spherical Convolutional Layer: For a 2D planar image I, if we perform convolution using a two-dimensional kernel K, the convolution operation can be represented by Equation (A-1), where i and j represent the range of the image, and m and n represent the range of the convolutional kernel. Typically, m and n are much smaller than i and j. (A-1) $S(i,j)=(K\ast I)(i,j)=\sum _m\sum _{n}I(m,n)K(i-m,j-n)$(A-1)

Another form of convolution used in deep learning is known as cross-correlation, which is represented by Equation (A-2). Cross-correlation is similar to convolution but does not involve flipping the kernel. It is considered more intuitive than true convolution operations (Goodfellow, Bengio, and Courville 2016).

(A-2) $S\left(i,j\right)=\left(I\ast K\right)\left(i,j\right)=\sum _m\sum _{n}I\left(i+m,j+n\right)K\left(m,n\right)$(A-2) According to the planar definition, the spherical convolution operations of the SCNN-IDG can be decomposed into two steps, i.e. neighbourhood sampling and weighted summation.

Neighbourhood Sampling: In deep learning, to make full use of computing resources such as GPUs, the input of two-dimensional convolution is a multidimensional matrix composed of multiple feature maps of the same size. Therefore, the icosahedral data need to be processed into matrix form of the same size. In this study, the spherical data are flattened into one-dimensional data in encoding order, and a neighbourhood index table (NI-table) is constructed for these data (Lin 2014; White 2000) Subsequently, the unfolded spherical data are processed into matrix form of the same size using the NI-table as an index and used as the input of the two-dimensional convolution (The line marked [2] in Algorithm A.1). Specifically, there will be seven neighbours and nine neighbours in the grid at the ten initial diamond special positions (as shown in Figure A.1). When a grid cell has seven neighbours, we pad the last element with zeros. When a grid element has nine neighbours, we average the corresponding data from the last two neighbours. The reasons for using this method to process irregular neighbourhoods of special positions are twofold. First, this approach ensures that the icosahedral data can be stacked into matrices of the same size. On the other hand, this method ensures that any position of the input in SCNN-IDG uses the same convolution kernel K for the convolution operation, that is, parameter sharing, which is an important idea that can help improve a deep learning system (Goodfellow, Bengio, and Courville 2016).

Figure A1. There are nine adjacent cells and seven adjacent cells distributed around the vertices of the initial subdivision of the icosahedral diamond grid. The red line is the initial diamond, and the black line is the level 3 icosahedral diamond grid.

Weighted Summation: A two-dimensional convolution operation is performed with a convolution kernel of size [$1$, $N$] (The line marked [3] in Algorithm A.1), where $N$ represents the convolution kernel size. He et al. (2019) demonstrated that employing a 7 × 7 convolutional kernel at the inception stage of ResNet leads to a 5.4-fold escalation in computational expenditure compared to a 3 × 3 convolutional kernel. This increase in computational cost is accompanied by a marginal 0.2% decrement in prediction accuracy on the ImageNet dataset. In essence, utilizing multiple stacked 3 × 3 convolutional layers prove more efficient than singularly employing larger convolutional kernels (e.g. 5 × 5 or 7 × 7). This approach not only reduces computational complexity and memory consumption but also facilitates the realization of deeper convolutional networks to achieve higher predictive performance. Therefore, following the construction principles of ResNet (Agrawal et al. 2023), we employed a 3 × 3 convolution kernel for convolutions, which means that N remains constant at 9 and is the first-order neighbour of the grid point. The pseudocode of the spherical convolution algorithm is shown in Table A.1.

Table A1. SCNN-IDG Convolution Operation.

Algorithm A.1 SCNN-IDG Convolution Operation

Input: One-dimensional spherical feature map unfolded by encoding $F_{\mathrm{in}}$ of the shape is [$B,C_{\mathrm{in}},10\times 4^l$], $\mathrm{NI}-\mathrm{table}$ $N{T}_{\mathrm{in}}$ of the shape [$10\times 4^l,N$] Output: One-dimensional spherical feature map$F_{\mathrm{out}}$ of the shape [$B,C_{\mathrm{out}},10\times 4^l$] Where: $B$ represents the batch;$C$ represents the current number of channels; $l$ represents the subdivision level of the diamond grid; $N$represents the convolution kernel size. The convolution kernel size of the ResNet architecture used in this article is $3\times 3$, so N is always equal to 9. [1]: begin SConv2D ($F_{\mathrm{in}},$ $N{T}_{\mathrm{in}},C_{\mathrm{out}}$): [2]: $F_{\mathrm{nt}}$ = Gather ($F_{\mathrm{in}}$, $N{T}_{\mathrm{in}}$,axis = 1) // $F_{\mathrm{nt}}\in R^{B\times c_{\mathrm{in}}\times (10\times 4^l)\times N}$. [3]: $F_{\mathrm{con}}$ = Conv2D ($F_{\mathrm{in}}$, kernel_size = [ $1$, $N$]) // $F_{\mathrm{con}}\in R^{B\times c_{\mathrm{out}}\times (10\times 4^l)\times 1}$ [4]: $F_{\mathrm{out}}$ = Squeeze ($F_{\mathrm{con}}$, axis = $-1$) // $F_{\mathrm{out}}\in R^{B\times c_{\mathrm{out}}\times (10\times 4^l)}$ [5]: end

Figure A2. Schematic diagram of 2 × 2 maximum pooling.

Spherical Pooling Layer: The pooling layer is responsible for downsampling the feature maps. Common pooling operations include max pooling and average pooling (Chen et al. 2021), as shown in Figure A.2. Based on the spatial aggregation property of the Z-order curve encoding, during the pooling process, we reshape the feature map matrix to [$C_{\mathrm{in}},10\times 4^{l-1},4$] and perform pooling along the last dimension. The max pooling operation selects the maximum value within each pooling region. The algorithmic details of the max pooling operation are described in Table A.2.

Table A2. SCNN-IDG MaxPool Operation

Algorithm A.2 SCNN-IDG MaxPool Operation

Input: One-dimensional spherical feature map unfolded by encoding $F_{\mathrm{in}}$ of the shape is [$B,C_{\mathrm{in}},10\times 4^l$] Output: One-dimensional spherical feature map$F_{\mathrm{out}}$ of shape [$B,C_{\mathrm{in}},10\times 4^{l-1}$] Where: $l$ represents the subdivision level of the diamond grid; $B$ represents the batch;$C$ represents the current number of channels; [1]: begin SMaxPool2D ($F_{\mathrm{in}}$): [2]: $F_{\mathrm{re}}$ = Reshape ($F_{\mathrm{in}}$,$\mathrm{shape}=[-1,4]$) // $F_{\mathrm{re}}\in R^{B\times c_{\mathrm{in}}\times (10\times 4^{l-1})\times 4}$ [3]: $F_{\mathrm{pool}}$ = MaxPool2d ($F_{\mathrm{in}}$, kernel_size = [1,4]) // $F_{\mathrm{pool}}\in R^{B\times c_{\mathrm{in}}\times (10\times 4^{l-1})\times 1}$ [4]: $F_{\mathrm{out}}$ = Squeeze ($F_{\mathrm{pool}}$, axis = $-1$) // $F_{\mathrm{out}}\in R^{B\times c_{\mathrm{in}}\times (10\times 4^{l-1})}$ [5]: end

Appendix B

For this study, the SCNN-IDG model was built and trained using the PyTorch 2.0 open-source library (Paszke et al. 2019). The programming languages employed were Python 3.10.6 and GCC 11.3.0. The software utilized was VSCode 1.77.3, and the operating system employed was Ubuntu 22.04.2 LTS. The hardware configuration included a 12th generation Intel® Core™ i7-12700F CPU, a GeForce RTX 3090 24G GPU, and 32 GB of RAM.

B.1 Parameter configuration for training and testing the SCNN-IDG

In this study, a fifth-level icosahedral diamond grid with a total of $10\times 4^5$=  10240 grid centre points were utilized. During the training phase of the SCNN-IDG, a fivefold cross-validation approach was employed. The initial learning rate was set to 0.001, and the CosineAnnealingLR learning rate update strategy was adopted. The value of T_max was set to 195, eta_min was set to 1e-05, and weight_decay was set to 0.0001. The batch was set to 200, and the Adam optimizer was used for backpropagation. The training was stopped if there was no improvement in performance on the validation set for 30 consecutive iterations. During the model testing phase, the test set was input into the five cross-validated models, and the accuracy of the test set was calculated for each model. The average accuracy across the five models was used as the final prediction accuracy for the test set.

B.2 Multimetric evaluation method setting

To establish a multimetric evaluation method for assessing the impact of global discrete grid quality on the SCNN performance, the random forest algorithm was employed. The parameter settings were determined based on the methodology suggested by (Rawson, Sabeur, and Brito 2021; Ye et al. 2019).

Appendix C

1. angle_mean: the mean of angle_m within the FOV region, where angle_m represents the mean value of all angles corresponding to each grid point.

2. angle_std: the standard deviation of angle_m within the FOV region.

3. dis_mean: the mean of dis_m within the FOV region, where dis_m represents the mean value of all distances corresponding to each grid point.

4. dis_std: the standard deviation of dis_m.

5. cwm_mean: the mean of cwm_m within the FOV region, where cwm_m represents the mean value of all cwm values corresponding to each grid point.

6. cwm_std: the standard deviation of cwm_m within the FOV region.

7. zsc_mean: the mean of the zone standardized compactness within the FOV region.

8. zsc_std: the standard deviation of the zone standardized compactness within the FOV region.

9. angle_mmr_mean: the mean of the ratio of the maximum and minimum values of all angles corresponding to the grid point within the FOV region.

10. angle_mmr_std: the standard deviation of the ratio of the maximum and minimum values of all angles corresponding to the grid point within the FOV region.

11. dis_mmr_mean: the mean of the ratio of the maximum and minimum values of all distances corresponding to the grid point within the FOV region.

12. dis_mmr_std: the standard deviation of the ratio of the maximum and minimum values of all angles corresponding to the grid point within the FOV region.

13. cwm_mmr_mean: the mean of the ratio of the maximum and minimum values of all cwm values corresponding to the grid point within the FOV region.

14. cwm_mmr_std: the standard deviation of the ratio of the maximum and minimum values of all cwm values corresponding to the grid point within the FOV region.