## 3 Case Study
### Spatiotemporal Field Estimation  
For the spatiotemporal field estimation, we used the mean square error (MSE) as in Equation (17) to evaluate the proposed field estimation method.  
\[\text{MSE}=\frac{1}{MN}\sum\limits_{m=1}^{M}\sum\limits_{i=1}^{N}\Bigl{(}Y( \mathbf{s}_{i},t_{m})-\hat{Y}(\mathbf{s}_{i},t_{m})\Bigr{)}^{2}, \tag{17}\]  
where \(Y(\mathbf{s}_{i},t_{m})\) denotes a sensing observation in the test data set, and \(\hat{Y}(\mathbf{s}_{i},t_{m})\) denotes the estimated temperature value of \(Y(\mathbf{s}_{i},t_{m})\). We tested the proposed field estimation method by estimating the grain temperature at the sensor sites of test data set by Equations (15) and (16). The MSE is equal to 0.04, which indicates the effectiveness of the estimation results using the proposed field estimation method. We show the examples of temperature profiles by the proposed field estimation method in Figure 7. The blue-star dots represent the training data, the red-plus dots represent the test data, and the blue lines denotes the estimated temperature profiles using our proposed method. The results show both interpolation and prediction capability of the proposed model with acceptable confidence intervals.  
To show the superiority of our approach, which integrates the physics-specific model and the spatiotemporal Gaussian process model, we compared it with other existing methods. In Table 3, we compared the model performance by changing mean functions proposed in the relevant literature, that is, a constant and a sinusoidal model. The result shows the superiority of the thermodynamic  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline \(\mathbf{\sigma}^{2}\) & \(\mathbf{a}_{x}\) & \(\mathbf{a}_{y}\) & \(\mathbf{a}_{z}\) & \(\mathbf{\varphi}\) \\ \hline
0.1289 & 10.4956 & 10.1552 & 2.9974 & 0.0154 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Estimated parameter values of \(\mathbf{\theta}^{*}\).  
Figure 6: Selection of the range parameter.  
model as the mean function. In Table 4, our proposed method delivers a better result than the thermodynamic model and consumes less than 50% of the computational time than the conventional Gaussian process model.  
Figure 8 shows an example of temperature profiles at different locations estimated by the proposed field estimation method. The grain temperature field obtained by our model changes with seasonal effects. Specifically, grain temperature inside the granary varies more slowly over time and has a smaller fluctuating range than grain temperature at the outer locations. This finding is consistent with the grain temperature fields in most of the granaries in China. Moreover, it provides a new quantitative understanding of temperature field evaluation in granaries [34].  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Model** & **MSE** & **Computational Time/s** \\ \hline Thermodynamic model & 0.45 & - \\ Thermodynamic model + conventional Gaussian process & 0.04 & 129.23 \\ Thermodynamic model + Gaussian process with tapering covariance function & 0.04 & 63.81 \\ \hline \hline \end{tabular}
\end{table}
Table 4: Comparison of model performance using various Gaussian process models.  
Figure 7: Examples of estimated temperature profiles: **(a)** coordinate (10.5, 20.5, 3.9), **(b)** coordinate (15.5, 5.5, 3.9), **(c)** coordinate (15.5, 40.5, 2.1), **(d)** coordinate (20.5, 25.5, 2.1). (Note: the blue dashed area represents the 95% confidence interval).