## 2 Methodology
### Theoretical Principle of OBD  
As illustrated in Figure 2, the fundamental distinction between pixel-based (PBA) and object-based (OBD) downscaling approaches lies in their thermal property treatment during resolution enhancement. The upper section of Figure 2 demonstrates the conventional PBA methodology, which processes thermal data at the individual pixel level by establishing statistical correlations between coarse-resolution land surface temperatures and spectral indices from auxiliary datasets. In contrast, the lower schematic presents our proposed OBD framework, which implements object-oriented thermal conservation  
Figure 1: The location of the data used in the approach: (**a**) the MODIS image (RGB: 321) and its geographical location in the eastern part of the USA; (**b**) the ETM+ (RGB: 321) and ASTER (RGB: 3N21) image; (**c**) the ETM+ subset covering the Lancaster and York region; and (**d**) the ASTER subset covering the same region.  
principles. Here, multiscale segmentation generates homogeneous units (labeled \"LST objects\" in Figure 2) that serve as fundamental thermal redistribution units. This approach maintains the physical consistency of landcover thermal characteristics through object-level radiation constraints while effectively eliminating the pixel-scale discontinuities that become apparent when comparing PBA and OBD results.  
The LST retrieved from the MODIS thermal infrared band had a relatively high temporal resolution and a low spatial resolution. Many PBA downscaling approaches could be expressed as Equation (1), using the LST and RSI relationships (\(f(RSI_{low}-LST_{low})\)) to downscale the MODIS LST with high-resolution VNIR data (eg: Landsat or ASTER) as assistance. LST and RSI relationships may be linear, nonlinear, or non-uniformly generated from different window sizes or whole images. Figure 2 illustrates the main process of the OBD method. Some constraints, like maintaining the TR unchanged after downscaling, as shown in Equation (2), were used.  
\[LST_{high}=f(RSI_{low}-LST_{low})f(RSI_{high}) \tag{1}\]  
Subject to  
\[TR_{low\_pixel}=\sum_{k=1}^{n}TR_{high} \tag{2}\]  
The OBD approach:  
\[TR_{low\_obj}=\sum_{k=1}^{m}TR_{high} \tag{3}\]  
Enhancing the spatial resolution of MODIS land surface temperature (LST) data necessitates the incorporation of auxiliary datasets. The methodology involved comprehensive pre-processing procedures: dataset format conversion, geometric correction, the calculation of effective thermal radiance, and the derivation of essential spectral indices (including the NDVI, NDBI, and MNDWI). Subsequently, a multi-sensor fusion technique was implemented to achieve the improved LST resolution. Figure 3 illustrates the main process of the approach for LST decomposition. Mainly, two parts are involved in the process: the thermal radiance (TR) estimation of objects from MODIS LST products and the weight calculation of sub-objects or subpixels. The TR estimations of objects are determined by the region covered by the object of the MODIS TR. The objects were generated by OBIA  
Figure 2: The process of the OBD method.  
methods with high-resolution multispectral data, and some remote sensing indices (RSIs) had fine relationships with the LST (e.g., the NDVI, the NDBI, and fractions of different endmembers). The MODIS TRs of the pixels were computed by their LST and broadband emissivity (BBE).  
Finally, the TR of the objects was downscaled into a fine spatial resolution with the weights of subpixels in each object and maintain the thermal radiance of the object unchanged after it is disaggregated into the subpixels in the resulted image. The weights of the subpixels in each object were determined through the initial downscaled LSTs that were estimated by the RSI auxiliary data and the LST-RSI relationships.