## 2 Methods
### Problem Setting  
The problem is formulated as a typical supervised machine learning estimation, where inputs \(X\) are the variables in \(col\_sat\_pos\) and \(col\_det\_pos\) and the outputs \(Y\) are the variables in \(col\_range\). The model, denoted as \(F\), is trained to provide an estimation such that \(Y\approx F(X)\). In this multi-output regression, the model \(F\) approximates the background photon count rates based on the variables of the satellite and its detectors. The model employed is a feed forward neural network with 3 hidden dense layers and 3 dropouts, 1 per each hidden layer; see Figure 1.  
During the pre-processing phase, the input training dataset is standardized, and data collected while Fermi passes through the high radiation environment of the South Atlantic Anomaly (SAA) are removed. The SAA, characterized by reduced geomagnetic intensity, facilitates the closer penetration of charged particles from space to the Earth's surface. These particles interact with the detectors, increasing the background count rate and posing challenges to accurate estimation. Consequently, data collected within the SAA are excluded to ensure the reliability of estimation. \begin{table}
\begin{tabular}{c l} \hline
**Feature Label** & **Description** \\ \hline \(pos\_x\), \(pos\_y\), \(pos\_z\) & position of Fermi in Earth inertial coordinates \\ \(a\), \(b\), \(c\), \(d\) & Fermi attitude quaternions \\ \(lat\) & Fermi geographical latitude \\ \(lon\) & Fermi geographical longitude \\ \(alt\) & Fermi orbital altitude \\ \(vx\), \(vy\), \(vz\) & velocity of Fermi in Earth inertial coordinates \\ \(w1\), \(w2\), \(w3\) & Fermi angular velocity \\ \(sun\_vis\) & Sun’s visibility boolean flag \\ \(sun\_ra\) & Sun’s right ascension \\ \(sun\_dec\) & Sun’s declination \\ \(earth\_r\) & Earth’s apparent radius \\ \(earth\_ra\) & Earth center right ascension \\ \(earth\_dec\) & Earth center declination \\ \(sa\) & SAA transit boolean flag \\ \(l\) & approximate McIlwain L value \\ \hline \end{tabular}
\end{table} 
Table 2: A table containing 24 orbital features, derived from the POSHIST files and processed using the Fermi GBM Data Tools library, serves as the input for the neural network (NN). These features pertain to satellite position, velocity, Sun and Earth visibility, passage through the South Atlantic Anomaly (SSA) and the local McIlwain L value (\(l\)), which is determined based on Fermi orbital position and corresponds to a geomagnetism parameter that describes the magnetic field intensity along a specific magnetic field line in the Earth’s magnetosphere. Reproduced with permission from ([PERSON] et al., 2013) ©[PERSON] et al. (2023), CC BY 4.0. \begin{table}
\begin{tabular}{c l} \hline
**Feature Label** & **Description** \\ \hline ni\_ra & \(i\)-labeled detector pointing right ascension \\ ni\_dec & \(i\)-labeled detector pointing declination \\ ni\_vis & \(i\)-labeled detector Earth occultation \\ boolean flag \\ \hline \end{tabular}
\end{table} 
Table 3: A table comprising 36 detector features is utilized to construct the neural network input table, with the detector label denoted as \(i\in\{0,1,2,3,4,5,6,7,8,9,a,b\}\). These features are extracted from the POSHIST files and processed using the Fermi GBM Data Tools library. The detector features encompass pointing coordinates in equatorial coordinates and indicate whether a detector Field of View (FOV) is occulted by the Earth. Reproduced with permission from ([PERSON] et al., 2013) ©[PERSON] et al. (2023), CC BY 4.0. The splitting procedure divides the dataset into train and test sets; part of the training set is further kept as the validation set. The loss function is the Mean Absolute Error  
\[\text{MAE}(y,\hat{y})=\frac{1}{n}\sum_{i=1}^{n}\lvert y_{i}-\hat{y}_{i}\rvert \tag{1}\]  
where \(n\) is the total number of samples in the training set, \(i\) refers to the specific sample, \(y_{i}\) the target value (the observed count rate) and \(\hat{y}_{i}\) is the estimated value (the estimated count rate).