## 2 Materials and Methods
### Downscaling Model  
We propose an Artificial Neural Network Sequential Downscaling Method (ASDM) with Transfer Learning Enhancement (ASDMTE) to generate fine-scale (FS) data from coarse-scale (CS) data. The method can be formulated as follows:  
Let \(y_{i,j,t}\) denote the FS AOD referenced at \(i,j,t\), where \(i\in\{1,2,\cdots,h\},j\in\{1,2,\cdots,w\}\), \(t\in\{1,2,\ldots,d\};h\) and \(w\) index latitude and longitude over the study domain and \(d\) is the time index. Similarly, we define the CS AOD referenced at \(x_{i^{\prime},j^{\prime},t^{\prime}}\), where \(i^{\prime}\in\{1,2,\cdots,h^{\prime}\}\), \(j^{\prime}\in\{1,2,\cdots,w^{\prime}\}\), \(t^{\prime}\in\{1,2,\cdots,d^{\prime}\};h^{\prime}\), \(w^{\prime}\) and \(d^{\prime}\) are latitude, longitude and time indices, respectively. Although the CS data have a longer overall period of temporal coverage, the FS and CS data have the same time step (day). The estimated downscaling model \(\hat{f}\) can then be denoted as:  
\[y_{i,j,t} =\hat{f}(\mathbf{y}_{(i,j,t-1),\mathbf{n}},x_{i^{\prime},i^{\prime},j^{ \prime},t},Ele_{i,j},Lat_{i},\mathit{Lon}_{j},Day_{t}) \tag{1}\] \[\mathbf{y}_{(i,j,t-1),\mathbf{n}} =y_{i,j,t-1},\cdots,y_{i,j,t-n},\]  
where \(Ele_{i,j},Lat_{i},\mathit{Lon}_{j},Day_{t}\) are elevation, latitude, longitude and day of the year at \(i,j,t\), respectively; \(x_{i^{\prime},i^{\prime},j^{\prime},t}\) represents CS AOD that spatially covers \(y_{i,j,t}\) (at the same time t); \(\mathbf{y}_{(i,j,t-1),\mathbf{n}}\) is a list of \(n\) temporal lagging variables at location \(i,j\). Through \(\hat{f}\), we not only learned empirical associations between the CS and FS variables, \(x_{i^{\prime},i^{\prime},t}\) and \(y_{i,j,t}\), but also short-term temporal associations within the FS data by including \(n=25\) time lags of the fine-scale variables, \(\mathbf{y}_{(i,j,t-1),\mathbf{n}}\). In the model we also adjusted for location (latitude, \(Lat_{i}\) and longitude, \(\mathit{Lon}_{j}\)), long-term time (day of the year, \(Day_{t}\)), and geographic information (elevation, \(Ele_{i,j}\)) making \(\hat{f}\) a function of space and time. This also enabled the use of data at different locations and times to train our model, which provided more information for training and partially alleviated the issue of having limited overlapping (in time) data. The larger the spatial area and temporal range, the more data we had for training; however, at the same time, the model \(\hat{f}\) became more complex. This increasing complexity in the target model is equivalent to adding difficulty in the learning process, thus we made the decision to trade off between data availability and model complexity. To enhance the performance of \(\hat{f}\), we incorporated transfer learning [41] into ASDM. Machine learning methods traditionally solve isolated tasks from scratch, which make them data hungry. Transfer learning attempts to solve this problem by developing methods to transfer knowledge learned in other sources and use it to improve the learning performance in a related target task [42]. The formal definition of transfer learning can be expressed as [41]:  
**Definition 1** (Transfer Learning). : _Given a source domain \(\mathcal{D}_{S}\) and learning task \(\mathcal{T}_{S}\), a target domain \(\mathcal{D}_{T}\) and learning task \(\mathcal{T}_{T}\), transfer learning aims to help improve the learning of the target predictive function \(h_{T}(\cdot)\) in \(\mathcal  
For model fitting, longitude, latitude, day of the year and elevation were normalized to a range \([0,1]\). The CS and FS AOD variables, \(X,Y\), have natural range \([0,6]\) which is approximately the same scale as \([0,1]\) and thus they were kept on their original scale. 'Input I' used all available features except lagging variables, \(X_{i^{\prime},j^{\prime},t}\), \(Ele_{i,j}\), \(Lat_{i}\), \(Lon_{j}\), \(Day_{i}\), and was processed by 'Process Block I'. 'Input II' was composed of the 25 FS lags \(\mathbf{y}_{(i,j,t-1),25}\) and went through 'Temporal Block I' and 'Temporal Block II' in ASDM.