## Case Study: Data Analysis and Results
### Bayesian Statistics and Hierarchical Model Fitting
#### 4.3.2 Hierarchical Bayesian Model
Estimating parameters with the model Equation (6) results in estimates for the (relative) point target powers for each group (\(\mu_{t}\), \(\mu_{30}\), and \(\mu_{15}\)) after drift compensation. The next step is to relate the point target powers to ERCS. For this, a reference ERCS is necessary. It was chosen to be the group ERCS of the 1.5 corners. (The not more than 7 years old 1.5 corner reflectors were, mechanically speaking, in a better shape than the more than 20 years old 3.0 corners, which show apparent deformations due to damages and their continuous exposition on a field. Mechanical deformations result in a reduction of a corners monostatic ERCS because some of the incident power is reflected away from the incident beam direction. The visual observation was confirmed by looking at the ERCS dispersion within each group: \(\sigma_{15}=0.15\,\mathrm{dBm^{2}}\) and \(\sigma_{30}=0.41\,\mathrm{dBm^{2}}\). These observations lead to the conclusion that the ERCS of the utilized 3.0 corners should not be used as an absolute reference, and that the 1.5 corners provide a better link to an absolute ERCS. Nevertheless, the 3.0 corners and their large ERCS helped in determining more accurately the daily RADARSAT-2 drift.) The value of the reference ERCS is modeled as \(\varsigma_{15}\sim\mathrm{N}(10^{3.838}\,\mathrm{m^{2}},10^{0.02}\,\mathrm{m^ {2}})\) (_i.e._, \(\mathrm{N}(38.38\,\mathrm{dBm^{2}},0.2\,\mathrm{dBm^{2}})\)). Its location is defined by Equation (3). The standard deviation, or standard uncertainty according to [7], characterizes the state of knowledge about the reference ERCS. The statement that the ERCS of 1.5 corners can be determined with Equation (3) up to a standard uncertainty of \(0.2\,\mathrm{dBm^{2}}\) is certainly the weakest point in the argument. It is based on previous experience gained from numerical field simulations on corners of the same size at X-band, and on plausibility. (A standard uncertainty of \(0.2\,\mathrm{dBm^{2}}\) is plausible because with it the theoretical RCS difference of 3.0 and 1.5 corners can be (well) explained. The theoretical difference, according to Table 2, is \(12.05\,\mathrm{dBm^{2}}\). The difference between the estimated mean target powers (in MCMC parameters: \(\mu_{30}-\mu_{15}\)) is \(11.92\,\mathrm{dB}\), _i.e._, \(0.13\,\mathrm{dB}\) away from the predicted value despite the already discussed deformation of the 3.0 corners. It is still possible though that the RCS of all corners is, due to deformation and the approximate nature of Equation (3), lower than assumed. Nevertheless, Equation (3) seems to characterize the absolute RCS of trihedral corners despite some mechanical deformations with a standard uncertainty of not more than \(0.2\,\mathrm{dBm^{2}}\).) Nevertheless, it cannot be proofed and further work should be conducted in determining the absolute knowledge of a trihedral corner reflector's ERCS. Now, the final transponder ERCS is deterministically related to the already derived model parameters through  
\[\varsigma_{t}=\frac{\mu_{t}}{\mu_{15}}\varsigma_{15} \tag{7}\]  
according to the measurement model Equation (4). The complete Bayesian model described above is visualized in Figure 10.