## Case Study: Data Analysis and Results
### Bayesian Statistics and Hierarchical Model Fitting
#### 4.3.2 Hierarchical Bayesian Model  
The Bayesian model requires the definition of priors, _i.e._, probability distributions for all model parameters. These shall be defined in the following. Figure 9: Estimated transponder drift and temperature stability (main influence for gain drift) for the overpass on 21 April 2013. The transponder operates with its nominal loop gain if a relative drift of 0 dB is detected. The loop gain was adjusted at 16:30 (red dotted line), after which the drift was monitored (blue circles) until the overpass (dashed green line). After drift correction, the drift is estimated to be with high probability within the range \([-0.02,0.02]\) dB at 17:03 UTC. For the eight overpass days \(d\), eight different daily RADARSAT-2 drifts \(r_{d}\) need to be determined. It is estimated from Figure 8 that the daily drift, expressed as a scaling factor, will certainly be in the range 0.4 to 1.6 (_i.e._, \(-4\) dB to 2 dB). The prior can thus be written as \(r_{d}\sim\mathrm{U}(0.4,1.6)\), where \(\mathrm{U}(a,b)\) describes a uniform distribution with lower and upper bounds \(a\) and \(b\), respectively. Besides the RADARSAT-2 drift, the transponder drift needs to be modeled. The values in column three of Table 4 define maximum error bounds, _i.e._, a lower bound \(a\) and an upper bound \(b\). According to [7], this information can be converted to a standard uncertainty and therefore a normal distribution, where the standard deviation \(\sigma\) is given as  
\[\sigma=\frac{1}{\sqrt{12}}(b-a) \tag{5}\]  
Hence, the daily transponder drift \(s_{d}\) is modeled as a normal distribution \(\mathrm{N}(\mu,\sigma)\), where \(\mu\) describes its mean and \(\sigma\) its standard deviation: \(s_{d}\sim\mathrm{N}(\mu_{s,d},\sigma_{s,d})\). The best estimate \(\mu_{s,d}\) is taken from Table 4, where also the \(\sigma_{s,d}\) (resulting from Equation (5)) are listed. It is assumed that all measured point target powers of all targets within one target group \(g\) (transponder \(t\), 3.0 \"30\", and 1.5 corners \"15\") belong to the same population, which can be described by a normal distribution \(\mathrm{N}(\mu_{g},\sigma_{g})\), having the group location (mean) \(\mu_{g}\) and group scale (standard deviation) \(\sigma_{g}>0\). A normal distribution was chosen because it is symmetric (no plausible reason can be found for an asymmetric distribution), and because the distribution of the measured values results from several physical effects (satellite thermal drift, satellite and target alignment errors, target stability, clutter, _etc._) so that the central limit theorem suggests a normal distribution as well. Each \(\mu_{g}\) and \(\sigma_{g}\) are modeled to originate from uniform distributions: \(\mu_{g}\sim\mathrm{U}(10^{1.5},10^{7})\) (_i.e._, \(\mathrm{U}(15\,\mathrm{dB},70\,\mathrm{dB})\)) and \(\sigma_{g}\sim\mathrm{U}(0,10^{6})\). In a way, the \(\sigma_{g}\) are nuisance parameters which are not required to derive the transponder ERCS, but they are nevertheless necessary in order to describe the normal distributions \(\mathrm{N}_{g}\). Now, for every overpass \(d\), the data \(y_{d,g}\) is fitted depending on its group \(g\in[t,15,30]\):  
\[y_{d,30} \sim\mathrm{N}(r_{d}\mu_{30},\sigma_{30})\] \[y_{d,15} \sim\mathrm{N}(r_{d}\mu_{15},\sigma_{15})\] \[y_{d,t} \sim\mathrm{N}(r_{d}s_{d}\mu_{t},\sigma_{t}) \tag{6}\]  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Overpass Date** & **Estimated Drift \(\mu_{s}\) (dB)** & **Maximal Error (dB)** & **Resulting \(\sigma_{s}\) (dB)** \\ \hline
2013-04-07 & 0.00 & 0.05 & 0.03 \\
2013-04-08 & 0.00 & 0.02 & 0.01 \\
2013-04-14 & 0.02 & 0.03 & 0.02 \\
2013-04-15 & \(-0.01\) & 0.03 & 0.02 \\
2013-04-18 & 0.00 & 0.07 & 0.04 \\
2013-04-21 & 0.00 & 0.02 & 0.01 This way, the daily drift parameter \(r_{d}\) is estimated from all available data, exploiting the fact that the drift should be equal for data across all three groups.