## 2 RFM Optimization  
### Method  
The image orientation model, which relates the image and ground coordinates, can be based on the rigorous Sensor Model (RSM) of the imaging system or can be approximated by a generic sensor model known as a rational functional model (RFM). The flowchart in Figure 1 shows schematically how these models might be determined and optimized. Figure 1: Workflow for image geometry refinement including physical model (Rigorous Sensor Model) and rational function model (RFM) approaches. The blue color pathway shows the processing path described in this study and is added as an additional module in the COSI-Corr package. The rigorous sensor model (RSM) accounts for the interior orientation (defined by the focal length, principal point location, pixel size, and the lens distortion) and the external orientation (defined by the position and attitude of the satellite during the image acquisition) [8]. This formalism is based on the extended collinearity equations for the external orientation model,  
\[\left[ \begin{array}{c}X\\ Y\\ Z\end{array} \right]=P(t)+\lambda.\,\mathbf{R}(t).\,\mathbf{\mathrm{K}}\!\left[ \begin{array} []{c}x\\ y\end{array} \right], \tag{1}\]  
where \([X,Y,Z]\) is the ground coordinate of the pixel \([x,y]\) in the image geometry after correction for the distortions represented by the internal orientation model. \(\lambda\) and \(K\) denote the unknown scaling factor and the calibration matrix, respectively. \(P(t)\) indicates the satellite position, and \(R(t)\) is the satellite attitude rotation matrix. These parameters represent the exterior orientation and they are functions of time, \(t\). The internal and external orientation parameters can be determined from a field calibration or from field-independent block adjustment. In the first case, the parameters are solved by using ground control points (GCPs) measured in the field or extracted from existing maps [14; 15; 16]. In the second case, often called the \"auto-calibration\" method, the parameters are solved from a block adjustment using a large set of overlapping images [17; 18; 19]. The RSM is, for example, implemented in the Coregistration of Optical Sensed Images and Correlation software (COSI-Corr) [14; 15]. The main difficulty with the RSM is that the model cannot be standardized as it must be tailored to the technical characteristics of each imaging system. In addition, the information needed for its implementation is not always provided by the operators [8]. This issue has become more acute with the increasing number of Earth Observation satellites. By contrast, the rational function model (RFM) is sensor agnostic and allows for a standardization of the metadata. The RFM relates image pixel coordinates to object coordinates in the form of rational functions expressed as the ratios of polynomials, usually of the third order [7; 20; 21],  
\[ \begin{array}{c}c_{n}=\frac{p_{1}(lat_{n},lon_{n},alt_{n})}{p_{2}(lat_{n},lon _{n},alt_{n})},\\ r_{n}=\frac{p_{2}(lat_{n},lon_{n},alt_{n})}{p_{4}(lat_{n},lon_{n},alt_{n})}, \end{array}  \tag{2}\]  
\[ \begin{array}{c}p_{l}(lat_{n},lon_{n},alt_{n})=\sum\limits_{i=0}^{3}\sum \limits_{j=0}^{i}\sum\limits_{k=0}^{j}c_{m}^{i}lat_{n}^{i-j}lon_{n}^{j-k}alt_{ n}^{k},\\ m=\frac{i(i+1)(i+2)}{6}+\frac{j(j+1)}{2}+k,\end{array}  \tag{3}\]  
where \((c_{n},r_{n})\) is the normalized values of the image point coordinates \((c,r)\), and \((lat_{n},lon_{n},alt_{n})\) is the normalized latitude, longitude and altitude of the corresponding ground point, respectively. The normalization of the coordinates is performed according to the process described in [7]. \(c_{m}^{l}\)\((l=1,2,3)\) refers to the coefficients of the polynomials \(p_{l}\). There are two ways to determine the RFM (Figure 1). In the terrain-independent approach, the physical model is used to generate virtual GCPs, which are then used to compute the parameters of the RFM [7]. In the terrain-dependent approach, the RFM is based on GCPs measured in the field or extracted from an existing reference map [7; 20].