# Duration of Rainfall Fades in GeoSurf Satellite Constellations

^{*}

## Abstract

**:**

## 1. GeoSurf Satellite Constellations at Millimeter Wavelengths

## 2. Fade Duration Processes A and B

## 3. Worldwide Sites Investigated with the Synthetic Storm Technique

## 4. Fade Duration Experimental Results

- a.
- As threshold increases, fade duration largely decreases, as physically expected.
- b.
- A fade duration $D$ is exceeded with very different probability at the different sites; or, at the same probability, $D$ is diverse.
- c.
- The probability distributions tend to overlap at the largest thresholds. In other words, the same fade duration is also substantially found at higher thresholds, as shown in Figure 1.

- (a)
- As threshold increases, the occurrences largely decrease for any $D,$ as physically expected.
- (b)
- As fade duration $D$ increases, sharp peaks are clearly evident in many sites.
- (c)
- Peaks tend to occur at lower thresholds.
- (d)
- As fade duration $D$ increases, curves tend to collapse, in agreement with the probability distributions ${P}_{S,B}(D)$ shown in Figure 4a,b. For example, the curves regarding $D=50~60$ min tend to coincide.

## 5. Processes A and B Interdependence

## 6. Uniformity Index

- (a)
- $U(S)>~0.4$ for all sites.
- (b)
- For $S=0$ dB, $U(0)$ ranges from $~0.42$ to $~0.58$. Since this value refers to the single rain events (i.e., the time series $R(t)$ of each rain event), then the duration of rain events does change in a large range. In fact, if all rain events were of equal duration $D$, they would give ${P}_{S,B}(D)={P}_{S,A}(D)=\delta (1)$, $U(0)=1$. This case would be represented by the point (1,1) in Figure 9, with probability $\delta (1)$, a Dirac impulse of unit area.
- (c)
- Some sites show marked dips at low thresholds. In other words, at these thresholds the two processes are the furthest away from the uniformity model (Spino d’Adda, Gera Lario, Madrid, Vancouver).
- (d)
- As the threshold increases, $U(S)$ increases; therefore, the two processes tend to be closer to the uniformity model at larger rain attenuation. This is a sound physical result because fades tend to be more similar in duration at large rain attenuation (see, for example, Figure 1). $U(S)$ approaches 1 at very large thresholds (Madrid, Vancouver).
- (e)
- The red line, Equation (3), gives $U=0.2605/0.5=0.521$ (a line $y=0.521$ in Figure 10) practically the mid–range value at $S=0$ dB: $(0.42+0.58)/2=0.5$.

## 7. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Data Availability Statement

## Conflicts of Interest

## Appendix A

**Figure A1.**Gera Lario. (

**a**) Probability distribution ${P}_{S,A}(D)$ of exceeding the value indicated in abscissa at the following thresholds: 0 dB, blue; 3 dB, green; 6 dB, cyan; 10 dB, magenta; 20 dB, red; 29$~40$ dB, cyan and black; (

**b**) ${P}_{S,B}(D)$, same legend.

**Figure A2.**Fucino. (

**a**) Probability distribution ${P}_{S,A}(D)$ of exceeding the value indicated in abscissa at the following thresholds: 0 dB, blue; 3 dB, green; 6 dB, cyan; 10 dB, magenta; 20 dB, red; 29$~40$ dB, cyan and black; (

**b**) ${P}_{S,B}(D)$, same legend.

**Figure A3.**Madrid. (

**a**) Probability distribution ${P}_{S,A}(D)$ of exceeding the value indicated in abscissa at the following thresholds: 0 dB, blue; 3 dB, green; 6 dB, cyan; 10 dB, magenta; 20 dB, red; 29$~40$ dB, cyan and black; (

**b**) ${P}_{S,B}(D)$, same legend.

**Figure A4.**Prague. (

**a**) Probability distribution ${P}_{S,A}(D)$ of exceeding the value indicated in abscissa at the following thresholds: 0 dB, blue; 3 dB, green; 6 dB, cyan; 10 dB, magenta; 20 dB, red; 29$~40$ dB, cyan and black; (

**b**) ${P}_{S,B}(D)$, same legend.

**Figure A5.**Norman. (

**a**) Probability distribution ${P}_{S,A}(D)$ of exceeding the value indicated in abscissa at the following thresholds: 0 dB, blue; 3 dB, green; 6 dB, cyan; 10 dB, magenta; 20 dB, red; 29$~40$ dB, cyan and black; (

**b**) ${P}_{S,B}(D)$, same legend.

**Figure A6.**Vancouver. (

**a**) Probability distribution ${P}_{S,A}(D)$ of exceeding the value indicated in abscissa at the following thresholds: 0 dB, blue; 3 dB, green; 6 dB, cyan; 10 dB, magenta; 20 dB, red; 29$~40$ dB, cyan and black; (

**b**) ${P}_{S,B}(D)$, same legend.

**Figure A7.**White Sands. (

**a**) Probability distribution ${P}_{S,A}(D)$ of exceeding the value indicated in abscissa at the following thresholds: 0 dB, blue; 3 dB, green; 6 dB, cyan; 10 dB, magenta; 20 dB, red; 29$~40$ dB, cyan and black; (

**b**) ${P}_{S,B}(D)$, same legend.

## Appendix B

**Figure A8.**Annual cumulative number of outage occurrences at the threshold indicated in abscissa, 80 GHz, circular polarization, zenith paths: (

**a**) Gera Lario; (

**b**) Fucino. The lines refer to constant fade duration, in 5 min steps, from $\tau =5$ min to $\tau =60$. The $y-$axis has different range.

**Figure A9.**Annual cumulative number of outage occurrences at the threshold indicated in abscissa, 80 GHz, circular polarization, zenith paths: (

**a**) Madrid; (

**b**) Prague. The lines refer to constant fade duration, in 5 min steps, from $\tau =5$ min to $\tau =60$. The $y-$axis has different range.

**Figure A10.**Annual cumulative number of outage occurrences at the threshold indicated in abscissa, 80 GHz, circular polarization, zenith paths: (

**a**) Tampa; (

**b**) Norman. The lines refer to constant fade duration, in 5 min steps, from $\tau =5$ min to $\tau =60$. The $y-$axis has different range.

**Figure A11.**Annual cumulative number of outage occurrences at the threshold indicated in abscissa, 80 GHz, circular polarization, zenith paths: (

**a**) White Sands; (

**b**) Vancouver. The lines refer to constant fade duration, in 5 min steps, from $\tau =5$ min to $\tau =60$. The $y-$axis has different range.

## Appendix C

**Figure A12.**${P}_{S,B}(D)$ ($y-$axis) versus ${P}_{S,A}(D)$ $x-$axis, for thresholds from 0 to 40 dB, 80 GHz, circular polarization, zenith paths: (

**a**) Gera Lario; (

**b**) Fucino. The red line is drawn from Equation (3); the cyan line is drawn from Equation (5).

**Figure A13.**${P}_{S,B}(D)$ ($y-$axis) versus ${P}_{S,A}(D)$ $x-$axis, for thresholds from 0 to 40 dB, 80 GHz, circular polarization, zenith paths: (

**a**) Madrid; (

**b**) Prague. The red line is drawn from Equation (3); the cyan line is drawn from Equation (5).

**Figure A14.**${P}_{S,B}(D)$ ($y-$axis) versus ${P}_{S,A}(D)$ $x-$axis, for thresholds from 0 to 40 dB, 80 GHz, circular polarization, zenith paths: (

**a**) Tampa; (

**b**) Norman. The red line is drawn from Equation (3); the cyan line is drawn from Equation (5).

**Figure A15.**${P}_{S,B}(D)$ ($y-$axis) versus ${P}_{S,A}(D)$ $x-$axis, 80 GHz, circular polarization, zenith paths: (

**a**) White Sands, for thresholds from 0 to 40 dB; (

**b**) Vancouver, for thresholds from 0 to 20 dB. The red line is drawn from Equation (3); the cyan line is drawn from Equation (5).

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**Figure 1.**Rain attenuation time series (blue line), 80 GHz circular polarization, zenith path, at Spino d’Adda, 6 May 2000, rain event starts at 17:30 local time. The red lines are drawn at thresholds 3, 10, 20 and 30 dB.

**Figure 2.**Annual probability distribution (%) $P\left(R\right)$ of exceeding the value indicated in abscissa at the indicated sites. Spino d’Adda: continuous blue line; Gera Lario: continuous black line; Fucino: continuous red line; Madrid: continuous green line; Prague: continuous magenta line; Tampa: dashed red line; Norman: dashed magenta line; White Sands: dashed green line; Vancouver: dashed blue line.

**Figure 3.**Annual probability distribution (%) $P\left(A\right)$ of exceeding the value indicated in abscissa–80 GHz, circular polarization, zenith paths–at the indicated sites. Spino d’Adda: continuous blue line; Gera Lario: continuous black line; Fucino: continuous red line; Madrid: continuous green line; Prague: continuous magenta line; Tampa: dashed red line; Norman: dashed magenta line; White Sands: dashed green line; Vancouver: dashed blue line.

**Figure 4.**Spino d’Adda. (

**a**) Probability distribution ${P}_{S,A}(D)$ of exceeding the value indicated in abscissa at the following thresholds: 0 dB, blue; 3 dB, green; 6 dB, cyan; 10 dB, magenta; 20 dB, red; 29$~40$ dB, cyan and black; (

**b**) ${P}_{S,B}(D)$, same legend.

**Figure 5.**Tampa. (

**a**) Probability distribution ${P}_{S,A}(D)$ of exceeding the value indicated in abscissa at the following thresholds: 0 dB, blue; 3 dB, green; 6 dB, cyan; 10 dB, magenta; 20 dB, red; 29$~40$ dB, cyan and black; (

**b**) ${P}_{S,B}(D)$, same legend.

**Figure 6.**Median fade duration in process B ($y-$axis) versus the median fade duration in process A ($x-$axis) at the indicated sites. Spino d’Adda: continuous blue line; Gera Lario: continuous black line; Fucino: continuous red line; Madrid: continuous green line; Prague: continuous magenta line; Tampa: dashed red line; Norman: dashed magenta line; White Sands: dashed green line; Vancouver: dashed blue line.

**Figure 7.**Average annual number of outage occurrences at the threshold indicated in abscissa—80 GHz, circular polarization, zenith paths–at the indicated sites. Spino d’Adda: continuous blue line; Gera Lario: continuous black line; Fucino: continuous red line; Madrid: continuous green line; Prague: continuous magenta line; Tampa: dashed red line; Norman: dashed magenta line; White Sands: dashed green line; Vancouver: dashed blue line.

**Figure 8.**Annual cumulative number of outage occurrences at the threshold indicated in abscissa, 80 GHz, circular polarization, zenith paths at: (

**a**) Spino d’Adda; (

**b**) Tampa. The continuous lines are drawn at equal fade duration, in 5 min steps from $\tau =5$ min to $\tau =60$. The y–axis has different range.

**Figure 9.**${P}_{S,B}(D)$($y-$axis) versus ${P}_{S,A}(D)$ $x-$axis, for thresholds from 0 to 40 dB, 80 GHz. The continuous red line is drawn from Equation (3); the continuous cyan line is drawn from Equation (5). (

**a**) Spino d’Adda; (

**b**) All sites.

**Figure 10.**Uniformity index versus the threshold indicated in abscissa at the indicated sites. Spino d’Adda: continuous blue line; Gera Lario: continuous black line; Fucino: continuous red line; Madrid: continuous green line; Prague: continuous magenta line; Tampa: dashed red line; Norman: dashed magenta line; White Sands: dashed green line; Vancouver: dashed blue line.

**Table 1.**Geographical coordinates, altitude (km), number of years of continuous rain rate time series measurements at the indicated sites.

Site | Latitude N (°) | Longitude E (°) | $\mathbf{Altitude}{\mathit{H}}_{\mathit{S}}$ (m) | Rain Rate Observation Time (Years) |
---|---|---|---|---|

Spino d’Adda (Italy) | 45.4 | 9.5 | 84 | 8 |

Gera Lario (Italy) | 46.2 | 9.4 | 210 | 5 |

Fucino (Italy) | 42.0 | 13.6 | 680 | 5 |

Madrid (Spain) | 40.4 | 356.3 | 630 | 8 |

Prague (Czech Republic) | 50.0 | 14.5 | 250 | 5 |

Tampa (Florida) | 28.1 | 277.6 | 50 | 4 |

Norman (Oklahoma) | 35.2 | 262.6 | 420 | 4 |

White Sands (New Mexico) | 32.5 | 253.4 | 1463 | 5 |

Vancouver (British Columbia) | 49.2 | 236.8 | 80 | 3 |

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**MDPI and ACS Style**

Matricciani, E.; Riva, C.
Duration of Rainfall Fades in GeoSurf Satellite Constellations. *Appl. Sci.* **2024**, *14*, 1865.
https://doi.org/10.3390/app14051865

**AMA Style**

Matricciani E, Riva C.
Duration of Rainfall Fades in GeoSurf Satellite Constellations. *Applied Sciences*. 2024; 14(5):1865.
https://doi.org/10.3390/app14051865

**Chicago/Turabian Style**

Matricciani, Emilio, and Carlo Riva.
2024. "Duration of Rainfall Fades in GeoSurf Satellite Constellations" *Applied Sciences* 14, no. 5: 1865.
https://doi.org/10.3390/app14051865