## 2 Methodology: Computation of the Altimetry Performance
### Platform Error Sources in GNSS-R Altimetry
#### 2.4.2 Impact of ISS Shadowing and Multiple Scattering  
The ISS has a number of advantages for experimental payload installation, power availability etc., but it also has some drawbacks that have to be taken into account in a payload like this one. The potential shadowing and multiple scattering have to carefully analyzed because they can seriously distort the observations. In order to do that, a full model of the ISS has been used [16], which consists of 551,748 triangles. Since this was a high number for a detailed electromagnetic simulation, a simplified model eliminating hidden parts, with just 11,620 triangles, more amenable for the analysis was used instead. Large flat areas were grouped, and hidden parts removed, as shown in Figure 8. The software package used is newFASANT [17], a commercial one, with custom made tuning for improved computational speed in our particular scenario. Multiple reflections over flat and curved surfaces up to third order (two reflections) were found. Using the reciprocity theorem, calculations have assumed a transmitting isotropic antenna located in the position of the receiving antenna, and for each direction in the space (far-field), the different contributions (direct, the second order, and third order reflections) are computed. Simulation results are shown in Figures 9 and 10, where the notations of dot, circle o, and cross x indicate the directions of arrival of the direct signal, the second, and third order reflections, respectively (i.e., arrive to antenna after one, or two reflections). Empty areas correspond to shadowed directions blocked by the ISS structure. Note that two of the latest second order reflections occur at delays of  
Figure 7: Evolution of the ISS orbital height during one year: periodic impulses to increase ISS height, and gradual fall between them are clearly visible. Gradual fall is caused by the atmospheric drag. Non-constant descent rate is due to changes in the upper atmosphere density, mainly due to solar activity [[[http://www.heavens-above.com/IssHeight.aspx](http://www.heavens-above.com/IssHeight.aspx)]([http://www.heavens-above.com/IssHeight.aspx](http://www.heavens-above.com/IssHeight.aspx))]. \(\sim\)150 ns and (\(\theta\), \(phi\)) \(\approx\) (100\({}^{\circ}\), \(\pm\)90\({}^{\circ}\)), from the closest solar panels. The latest third order reflection occurs at a delay of \(\sim\)300 ns and (\(\theta\), \(\phi\)) \(\approx\) (100\({}^{\circ}\), 90\({}^{\circ}\)), again from the closest solar panel. The impact in the waveforms (or delay Doppler maps) is tricky to imagine intuitively, because of the different amplitude (in case of diffraction), and phases of each reflection, and because the scattered fields' polarization changes, so it is collected through the co- or cross-polar antenna patterns. Figure 8: Simplified ISS model and reference frame used. Figure 10: **(a)** Delays for the second order rays, and **(b)** for the third order rays. Direction of Arrival (\(\theta\), \(\phi\)) in the antenna far field. Co-polar antenna pattern used for the direct and third order rays, cross-polar pattern used for the second order rays. Top: first matrix and bottom: second matrix, refer to the fact that there are second order and third order rays that have the same far field direction, but enter through different directions in the antenna pattern reference frame. Color code corresponds to the delay in ns. Figure 9: Graphical representation of the directions of arrival of direct signal or first order rays (no reflections) (-), second order rays (one reflection) (**o**), and third order rays (two-reflections) (**x**). Empty areas correspond to shadowed directions blocked by the ISS structure. The total electric field at a given time instant is the sum of the first, second, and third order rays that reach the antenna with different time delays, and through different directions (and polarizations) of the antenna pattern.