## 1. Introduction

The observation, monitoring, and estimation of rainfall are matters of major concern for a number of reasons. For instance, public administrators aim at guaranteeing an adequate level of security for people living and operating on a given territory: for this purpose, rainfall forecasting is key for the assessment of hydrogeological risks due to landslides and floods. An accurate measurement of total rainfall amount is also critical in climatological studies, since it represents one of the main climate indices. Moreover, the knowledge of the real-time rain rate during precipitation events is useful in urban areas to estimate water runoff from the drainage system.

A very high spatial and temporal variability characterizes rainfall [

1,

2,

3]: it is then difficult to obtain accurate measurements with proper spatial density and temporal detail. Several instruments and techniques have been developed to gauge this quantity (e.g., the use of weather radars, rain gauges, infrared satellite imagery, microwave links, etc.). Each of them presents advantages and limitations [

4]: for example, remote sensing measurements aim at estimating the precipitation field over wide areas, but typically with limited spatial resolution and accuracy; on the contrary, point observations, e.g., collected by rain gauges and disdrometers, deliver precise local measurements, but they hardly provide large and dense coverage.

Recently, increasing attention has been paid to the exploitation of satellite downlink signals for the estimation of rainfall, as most of the commercial satellite communication systems currently operate in the Ku band (12–18 GHz) and the K band (18–26 GHz): in fact, in such a frequency range, rain significantly impairs the propagation of electromagnetic waves [

5,

6]. Some approaches have been developed for estimating the intensity of precipitation on a large scale from space-borne signals, with a spatiotemporal resolution between that of rain gauges and of weather radars. In fact, the opportunistic use of satellite downlink signals to estimate rainfall combines the benefit of covering areas where ground instrumentation or radar networks are not available, and the requirement of high temporal resolution measurements, which is in the order of one minute [

7]: this approach would allow providing two-dimensional maps of the rain intensity, even across continental regions (e.g., Europe) by using a sufficiently dense network of ground receivers.

The methodologies that are currently available to assess rainfall rate via satellite links are of limited accuracy and reliability and, in some cases, of complex implementation [

8,

9,

10,

11]: as a consequence, it becomes necessary to investigate alternative estimation methods, which only require few measurable physical parameters as input.

This contribution presents a novel estimation methodology that aims to estimate the rain rate by opportunistically exploiting local downlink satellite communication signals and by making use of local ancillary meteorological data, namely the hourly zero-degree isotherm height and the monthly convectivity index. The model also includes different expressions to take the different impact of stratiform and convective rain events on the link into due account. More specifically, the remainder of the paper is structured as follows.

Section 2 presents the Earth-space propagation experiment carried out at Politecnico di Milano, along with the experimental setup used in this work.

Section 3 describes the experimental datasets.

Section 4 describes the proposed prediction methodology in detail, while

Section 5 summarizes the full procedure for rain rate estimation. Lastly,

Section 6 discusses the performance of the proposed model (as well as of other methodologies that are available in the literature) and

Section 7 draws some conclusions.

## 6. Performance Evaluation

The calibration of the model’s parameters was carried out on a statistical basis, i.e., by maximizing the agreement between the measured and estimated $P\left(R\right)s.$ However, it is worth verifying the accuracy in estimating the rain rate applicability when the model is used on an instantaneous basis. To this aim, we have resorted to the following indicators of the rain event:

total rainfall accumulation, ${R}_{T}$ [mm];

maximum rain rate, ${R}_{M}$ [mm/h]; and,

average rain rate, $\overline{R}$ [mm/h].

For each indicator, we have defined the difference between the daily estimated (denoted with the subscript

$est$) and measured (denoted with the subscript

$meas$) quantity of interest, for each of the 70 considered rain events:

Table 5 lists the associated average (on all the events) mean and RMS values of the residuals.

The results obtained are satisfactory, as the indicators reveal that the proposed method allows (on average) for estimating the rainfall accumulation with an error in the order of few millimeters. Concerning the detection of peak rain rates, the resulting value of the RMS error is considered to be acceptable, given the difficulty in estimating extreme rain rate values, which might even range between 150 mm/h and 200 mm/h.

Besides the overall statistical indicators, it is worth further investigating the model accuracy by selecting some sample events.

Figure 11 depicts the trend of the rain rate during a typical Spring rain event, which is characterized by moderate rain rates and occasional more intense peaks. The equivalent rain height is calculated while using (10) since the event occurred in April (

$m=4)$: the variability of the precipitation is tracked with a significant degree of accuracy, leading to an estimate of the total rainfall accumulation that closely matches the measurements (see

Figure 12).

Figure 13 and

Figure 14 refer to a typical event in Autumn (November,

$m=11$): in this case, the equivalent rain height is calculated while using (8). The model accuracy is lower than in the previous event, but the results are still quite satisfactory.

The correct evaluation of the equivalent rain height plays a key role in the prediction of the rain rate, as also shown by the curves in

Figure 7.

Figure 15 depicts the trend of

${h}_{R}$ for the previously mentioned convective and stratiform events, as reported in

Figure 11 and

Figure 13, respectively, and points out the importance of using local forecast values of

${h}_{R}$ with high temporal resolution. The values in

Figure 15 indicate the opposite for the two events, although the rain height is generally higher for convective events than for stratiform ones; in this case, using monthly values of

${h}_{R}$ (or even worst, a single yearly value) would have led to a much lower prediction accuracy than that shown in

Figure 11 and

Figure 13.

## Author Contributions

Conceptualization, L.L. and C.G.R.; methodology, R.A.G., L.L. and C.G.R.; software, R.A.G., L.L. and C.G.R.; validation, R.A.G. and L.L.; formal analysis, R.A.G. and L.L.; investigation, L.L. and C.G.R.; writing—original draft preparation, R.A.G.; writing—review and editing, R.A.G., L.L. and C.G.R.; visualization, R.A.G.; supervision, L.L. and C.G.R.; project administration, L.L. and C.G.R. All authors have read and agreed to the published version of the manuscript.

## Funding

This work was partially supported by the Air Force Office of Scientific Research, Air Force Material Command, United States Air Force, European Office of Aerospace Research and Development under Grant FA9550-17-1-0335.

## Acknowledgments

The Authors want to thank NASA for making the propagation data available.

## Conflicts of Interest

The authors declare no conflict of interest. The funders had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript, or in the decision to publish the results.

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**Figure 1.**
Experimental setup on DEIB rooftop in Milan, Italy.

**Figure 2.**
Low-pass filtered signal (black line) of the Ka-band received signal (orange curve) on 7th September 2017.

**Figure 3.**
Detection of the rain event on 7th September 2017.

**Figure 4.**
Reference power level, obtained through linear interpolation during precipitation, for the rain event on 7th September 2017.

**Figure 5.**
Rain attenuation time series on 7th September 2017.

**Figure 6.**
Specific rain attenuation as a function of the rain rate in Milan (19.701 GHz).

**Figure 7.**
Comparison between the inverse functions

${g}^{-1}$ (dotted lines) and

$q$ (continuous lines), using the

$k$ and

$\alpha $ values reported in the third column of

Table 1.

**Figure 8.**
Trend of $\beta $ with the month $m$ for Milan.

**Figure 9.**
Comparison between annual rain rate Complementary Cumulative Distribution Functions (CCDFs).

**Figure 10.**
Trend of ${\widehat{\tau}}_{0}\left(m\right)$ and $\beta \left(m\right)$ as a function of the month.

**Figure 11.**
Rain rate time series for the event on 26th April 2017.

**Figure 12.**
Accumulated rainfall for the event on 26th April 2017.

**Figure 13.**
Rain rate time series for the event on 25th November 2017.

**Figure 14.**
Accumulated rainfall for the event on 25th November 2017.

**Figure 15.**
Trend of the equivalent rain height for the events represented in

Figure 11 and

Figure 13.

**Figure 16.**
Yearly P(R): comparison of different models.

**Table 1.**
$k$ and $\alpha $ fitting coefficients at 19.701 GHz.

Coefficient | Recommendation ITU-R P.838-3 | Local DSD Data |
---|

$k$ | 0.0924 | 0.0495 |

$\alpha $ | 0.9989 | 1.0870 |

**Table 2.**
Fitting accuracy for function $q\left({A}_{R},{h}_{R}\right),a\left({h}_{R}\right),b\left({h}_{R}\right)$ and $c\left({h}_{R}\right)$.

Function | ${\mathit{R}}^{2}$ | $\mathit{E}\left(\mathbf{mm}/\mathbf{h}\right)$ | $\mathit{R}\mathit{M}\mathit{S}\left(\mathbf{mm}/\mathbf{h}\right)$ |
---|

$q\left({A}_{R},{h}_{R}\right)$ | $0.99$ | $-0.03$ | $0.46$ |

$a\left({h}_{R}\right)$ | $0.99$ | $0$ | $0$ |

$b\left({h}_{R}\right)$ | $0.99$ | $0$ | $0$ |

$c\left({h}_{R}\right)$ | $1$ | $0$ | $0$ |

**Table 3.**
Comparison between different values of $\tau \left(m\right)$.

$\mathit{m}$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ |
---|

${\widehat{\tau}}_{0}\left(m\right)$ | $1.100$ | $1.200$ | $1.450$ | $1.500$ | $-$ | $1.000$ |

${\widehat{\tau}}_{p}\left(m\right)$ | $1.099$ | $1.202$ | $1.448$ | $1.501$ | $1.552$ | $1.000$ |

**Table 4.**
Mean and RMS values for the comparison between the curves reported in

Figure 9.

| $\mathit{E}\left\{{\mathit{\u03f5}}_{\mathit{r}}\left(\mathit{P}\right)\right\}$ | $\mathit{R}\mathit{M}\mathit{S}\left\{{\mathit{\u03f5}}_{\mathit{r}}\left(\mathit{P}\right)\right\}$ |
---|

${\widehat{\tau}}_{0}\left(m\right)$ | $0.97$ mm/h | $1.50$ mm/h |

${\widehat{\tau}}_{p}\left(m\right)$ | $0.62$ mm/h | $1.19$ mm/h |

**Table 5.**
Average (on all the events) of the mean and Root Mean Square (RMS) values of the residuals in (21)–(23).

Residual | Mean Value | $\mathbf{RMS}\text{}\mathbf{Value}$ |
---|

${\u03f5}_{r,T}$ | $1.39$ mm/h | $5.34$ mm/h |

${\u03f5}_{r,M}$ | $-2.15$ mm/h | $11.83$ mm/h |

${\overline{\u03f5}}_{r}$ | $0.04$ mm/h | $1.52$ mm/h |

**Table 6.**
Mean and RMS values corresponding to the comparison between the measurement and the estimated curves reported in

Figure 16.

Model | $\mathit{E}\left\{{\mathit{\u03f5}}_{\mathit{r}}\left(\mathit{P}\right)\right\}$ | $\mathit{R}\mathit{M}\mathit{S}\left\{{\mathit{\u03f5}}_{\mathit{r}}\left(\mathit{P}\right)\right\}$ |
---|

Proposed Model | $0.62$ mm/h | $1.19$ mm/h |

$NF$ | $3.82$ mm/h | $4.79$ mm/h |

$CM1$ | $0.14$ mm/h | $2.03$ mm/h |

$CM2$ | $-0.29$ mm/h | $4.24$ mm/h |

$BM$ | $0.70$ mm/h | $1.57$ mm/h |

**Table 7.**
Average (on all the events) of the mean and RMS values of ${\u03f5}_{r,T}$.

Model | $\mathit{E}\left\{{\mathit{\u03f5}}_{\mathit{r},\mathit{T}}\right\}$ | $\mathit{R}\mathit{M}\mathit{S}\left\{{\mathit{\u03f5}}_{\mathit{r},\mathit{T}}\right\}$ |
---|

Proposed model | $1.39$ mm | $5.34$ mm |

$NF$ | $10.70$ mm | $38.64$ mm |

$CM1$ | $1.12$ mm | $10.45$ mm |

$CM2$ | $0.94$ mm | $9.04$ mm |

$BM$ | $2.42$ mm | $12.75$ mm |

**Table 8.**
Average (on all the events) of the mean and RMS values of ${\u03f5}_{r,M}.$

Model | $\mathit{E}\left\{{\mathit{\u03f5}}_{\mathit{r},\mathit{M}}\right\}$ | $\mathit{R}\mathit{M}\mathit{S}\left\{{\mathit{\u03f5}}_{\mathit{r},\mathit{M}}\right\}$ |
---|

Proposed model | $-2.15$ mm/h | $11.83$ mm/h |

$NF$ | $-3.58$ mm/h | $28.22$ mm/h |

$CM1$ | $-8.45$ mm/h | $31.69$ mm/h |

$CM2$ | $-11.79$ mm/h | $35.88$ mm/h |

$BM$ | $-6.86$ mm/h | $30.69$ mm/h |

**Table 9.**
Average (on all the events) of the mean and RMS values of ${\overline{\u03f5}}_{r}$.

Model | $\mathit{E}\left\{{\overline{\mathit{\u03f5}}}_{\mathit{r}}\right\}$ | $\mathit{R}\mathit{M}\mathit{S}\left\{{\overline{\mathit{\u03f5}}}_{\mathit{r}}\right\}$ |
---|

Proposed model | $0.04$ mm/h | $1.52$ mm/h |

$NF$ | $0.95$ mm/h | $2.75$ mm/h |

$CM1$ | $-0.21$ mm/h | $2.02$ mm/h |

$CM2$ | $-0.40$ mm/h | $2.79$ mm/h |

$BM$ | $0.04$ mm/h | $1.92$ mm/h |

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