## 3 Methodology
### LEO/GEO-Integrated AOD Fusion Process
#### 3.2.3 Spatiotemporal Covariance Modeling  
For any variable \(Z(p)\) in the space/time random field (S/TRE), \(p\) = (\(s\), \(t\)) denotes a spatiotemporal point (\(s\) is the geographic location, and \(t\) is time). It can be expressed using the following equation:  
\[Z(p)=X(p)+m(p) \tag{12}\]  
where the mean trend model \(X(p)\) is a function of parameter \(\beta\). The zero-mean autocorrelated Gaussian process \(m(p)\) has covariance with parameter \(\theta\) that varies at spatial lag \(\gamma\) = \(|s-s^{\prime}|\) and temporal lag \(\tau\) = \(|t-t^{\prime}|\) (i.e., space/time distance between points \(p\) = (\(s\), \(t\)) and \(p^{\prime}\)= (\(s^{\prime}\), \(t^{\prime}\))). The fixed parameters \(\beta\) and \(\theta\) can assist in describing the statistical distribution of variable \(Z\)[69]. In the BME method, all variables must be spatially homogenous and temporally stationary with constant mean and variance [70]. In this study, the spatiotemporal mean trend component is estimated and separated from the preprocessed satellite-based AOD datasets using a moving filter window of size 99 (pixels along-track) \(\times\) 99 (pixels cross-track) \(\times\) 3 (10 min). The remaining isotropic components are then employed to describe the spatiotemporal autocorrelation by modeling the covariance function. Figure 4a,b and Table 2 present the covariance modeling results for three theoretical fitting functions (i.e., exponential, spherical, and Gaussian models). It is demonstrated that the exponential fitting curve coincides the most with the experimental covariance, with the square of the Pearson correlation coefficient (R\({}^{2}\)) of 0.94 and 0.90, the relative mean bias (RMB) of 1.110 and 1.112, and the root-mean-square error (RMSE) of 0.0015 and 0.0016 under the condition of spatial lag and temporal lag, respectively. To sufficiently describe the sophisticated spatiotemporal variation process, we ultimately adopt the nested spatiotemporal covariance model with the superposition of two exponential functions [71; 72]. Every model characterizes spatiotemporal dependency and is parameterized by partial sill variance (\(c1\) or \(c2\)), spatial range (\(\alpha_{\gamma 1}\) or \(\alpha_{\gamma 2}\)), and temporal range (\(\alpha_{\tau 1}\) or \(\alpha_{\tau 2}\)):  
\[C(\gamma\,,\tau)=c1\text{exp}\bigg{(}-\frac{3\gamma}{a_{\gamma 1}}\bigg{)} \text{exp}\bigg{(}-\frac{3\tau}{a_{\tau 1}}\bigg{)}+c2\text{exp}\bigg{(}-\frac{3 \gamma}{a_{\gamma 2}}\bigg{)}\text{exp}\bigg{(}-\frac{3\tau}{a_{\tau 2}}\bigg{)} \tag{13}\] \begin{table}
\begin{tabular}{l c c c c c} \hline \hline \multirow{2}{*}{**Type/Parameters**} & \multicolumn{3}{c}{**Covariance vs. Spatial Lag**} & \multicolumn{3}{c}{**Covariance vs. Temporal Lag**} \\ \cline{2-7}  & **R\({}^{2}\)** & **RMB** & **RMSE** & **R\({}^{2}\)** & **SSE** & **RMSE** \\ \hline Exponential Model & 0.94 & 1.110 & 0.0015 & 0.90 & 1.112 & 0.0016 \\ Spherical Model & 0.91 & 1.117 & 0.0017 & 0.86 & 1.116 & 0.0016 \\ Gaussian Model & 0.92 & 1.116 & 0.0016 & 0.82 & 1.114 & 0.0017 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Fitting results of the experimental covariance for the spatial and temporal lags based on three classical models (exponential, spherical, and Gaussian models). Figure 3: Gaussian probability density functions of the random discrepancy between (**a**) Himawari-8 AHI AODs, (**b**) MODIS AODs, and (**c**) CALIOP and AERONET AODs. In this study, the interior point penalty function method is introduced to obtain the optimal parameter estimation of the above nested exponential models by constructing an unconstrained objective function as per our previous research [73]. Moreover, the nested spatiotemporal covariances are fitted for each month in this study due to the relatively large amount of 10-min AOD datasets to ensure the stability of the spatiotemporal covariance model and improve the efficiency of model fitting. Figure 4c shows a case of modeling the spatiotemporal covariance function over the study area.