https://orcid.org/0000-0002-8277-2068
Abstract
A low complexity accurate model for precipitation estimation is crucial for monitoring several hydrological and water resource applications. Based on the R-k empirical power-law relation described by the P.838-3 ITU recommendation, rainfall rate can be predicted based on specific attenuation of microwave links. The accuracy of this method is impacted by several ambiguities and errors. In order to overcome these limitations, numerous highly complex pre-treatment and post-processing methods should be used. As an alternative method of low complexity, a supervised learning algorithm using a single-layer neural network (the perceptron) is suggested in this paper. Optimal weights parameters were obtained based on the minimization of the mean square error (MSE). A case study was carried out using 40 days of data gathered from two commercial microwave links (CMLs) and one rain gauge. Experimental results showed that this machine learning-supervised approach performed better than the R-k-based method. The mean square error of the path-averaged rainfall rate was reduced from 0.13 mm2 h-1 to 0.08 mm2 h-1 for training data, and from 0.2 mm2 h-1 to 0.1 mm2 h-1 for test data. This promising alternative method for rainfall estimation could considerably improve the efficiency of many applications, such as those developed for real-time urban flood risk management.
HIGHLIGHTS
Rainfall can be estimated based on commercial microwave links (CMLs) power attenuations.
The empirical R-k power relationship is tested and discussed.
A single-layer neural network (ANN)-based technique is developed.
INTRODUCTION
Real-time accurate rainfall estimation is critical for water management, weather forecasting, flood warnings, agriculture, as well as a range of other applications such as the prediction of river runoff (Tabari 2016). Precipitation is usually estimated directly using ground-based instruments, such as rain gauges and disdrometers, or indirectly using remote sensing techniques, such as satellite sensors and weather radars. These measurements come up against several limits, including poor temporal or spatial resolution of some of them, high cost or difficulty of implementation of others (Allerup & Madsen 1980; Seed et al. 1996; Habib et al. 2008; Hossain & Huffman 2008). To increase their accuracy and availability, data fusion techniques that combine measurements from many sources can be used.
In order to overcome some limitations of the methods cited above, an innovative opportunistic sensing (OS) approach for rainfall estimation has recently been developed by using received signal levels (RSLs) data of commercial microwave link (CML) networks (Messer 2018; Uijlenhoet et al. 2018; Chwala & Kunstmann 2019). This technique was widely investigated and tested in many countries such as the Netherlands (Leijnse et al. 2007; Overeem et al. 2011, 2016a), the Czech Republic (Fencl et al. 2017), Italy (Roversi et al. 2020), Germany (Chwala et al. 2012, 2016; Smiatek et al. 2017; Graf et al. 2020), Switzerland (Bianchi et al. 2013), and Lebanon (Daher & Al Sakka 2021). This approach does not require any additional installation since RSL data is usually recorded by mobile operators in order to monitor the quality and stability of their networks. Most operators log the minimum and maximum power data over a 15-minute sampling period, while a few others record 15-minute average power data or, in some cases, 15-minute instantaneous power data. Using a large network of CMLs with statistical spatial interpolation methods, high temporal and spatial resolution precipitation estimates can be provided.
Despite its benefits, the 
-based approach for rainfall estimation is affected by various uncertainties. Several experimental studies have been carried out to better understand the underlying uncertainty at various stages of the rainfall retrieval technique (Fenicia et al. 2012; Van Leth et al. 2018). Using the minimum and maximum RSL data with a constant weighted average approach yields a significant source of uncertainties since the distribution of precipitation or attenuation is rarely consistent. Another common source of inaccuracy is the bias due to the effect of wet antenna attenuation (WAA), fog, and dew formation. In Leijnse et al. (2008), a semi-empirical model of WAA, dependent on rainfall intensity, was proposed as a function of the thickness of the water film on the antenna surface. In Overeem et al. (2011), a constant WAA value during wet times is considered. In Schleiss et al. (2013), compensation of the maximum value of WAA by an exponential model is suggested. The efficiency of direct antenna shielding compared with post-processing approaches in minimizing WAA is examined by Fencl et al. (2014). Results showed that shielding did not outperform model-based corrections, as signals from shielded antennas were still attenuated. Rainy and wet weather may affect the scattering and reflection behavior of the surrounding buildings, such as nearby walls, roofs, or impervious surfaces. Indeed, more delicate processing and filtering are necessary to undergo various types of deformations and distortions. Recently, a power relationship between the WAA and rainfall intensity has been described by Valtr et al. (2019). Another challenge is the determination of the reference or baseline power, such as in the case of links with variable transmitted power. In Overeem et al. (2016a), a method based on the calculation of the median value of past measurements, classified as dry, in a window of 24 h, was used to determine the reference signal level. Reliable classification of wet and dry periods is needed to prevent rainfall overestimation (Overeem et al. 2016b). Wet–dry classifiers based on neural network approaches have been explored using communication satellite data by Mishra et al. (2018) and RSL CML data by Habi & Messer (2018) and Polz et al. (2020). In order to enhance precipitation retrievals based on RSL data, a recurrent neural network (RNN) has been recently developed using disdrometer reference data in Pudashine et al. (2020).
Although the RNN-based method and all the studies cited above can provide good accuracy based on the 
-based method, their computational costs are too high. The aim of this research is to develop a low-complexity highly accurate method for rainfall estimation based on RSL data. An optimal single-layer neural network (perceptron) is investigated. Appropriate error indices including correlation coefficient (R), Mean Square Error (MSE), and Mean Absolute Error (MAE) for training and test data were calculated, and the values of the parameters related to the model with the lowest value of the MSE were considered as the optimal model optimal.
Our paper is organized as follows. Section 2 discusses data collection and engineering. Rainfall estimation using the 
power relationship is described in Section 3. The single-layer neural network-based technique for rainfall estimation using CML RSL data is described in Section 4. Experimental results are presented in Section 5. Finally, the conclusion and perspective are drawn in Section 6.
DATA ENGINEERING
and 
(Figure 1). Hourly rainfall precipitation values denoted, by 
(in 
), were gathered by the Lebanese Agricultural Research Institute (LARI). LARI is a governmental organization under the Minister of Agriculture Supervision. It has a network of more than 60 automated agro-weather stations across Lebanon, giving weather forecasts, early alerts on major pests, and pest management advice. The institute also conducts applied and basic scientific research for the development and advancement of the agricultural sector in Lebanon. 
Rain gauge location.
On the other hand, RSL data was collected from two active CMLs, vertically polarized and located in Tyre. 15-minute mean, minimum, and maximum received powers were gathered for each link. These measurements were provided in collaboration with Touch, a mobile telecommunication operator in Lebanon. The first link, Link1, has a length of 3.09 km and operates at a frequency of 23.43 GHz while the second link, Link 2, of length 5.54 km operates at 19.15 GHz. Due to some regulations at the mobile operator Touch, we have been informed that these two CMLs are near the rain gauge but we have not been authorized to obtain information on their precise locations.
To evaluate the accuracy, CML-based rainfall estimates were compared with rain gauge measurements considered as reference values. Our optimal estimator is based on the minimization of the 
(Mean Square Error). Other metrics like 
(Mean Absolute Error), R (Correlation) and 
(R-squared) have been defined. The bias (accuracy) and variability (precision) of the estimators have been evaluated.
, 
, and 
to be produced, as described below:
, 
, and 
are the functions that return the mean, minimum, and maximum values, respectively. 
, 
, and 
, 
stand for the mean, minimum, and maximum received powers which correspond to the first, second, third, and fourth quarter of each hour.R-k POWER-LAW-BASED METHOD
relationship between specific attenuation k (
) and rainfall rate R (
) was described by the P.838-3 ITU recommendation (Itu 2005). It is given by:
(
) and 
(
) are the coefficient and exponent parameters, respectively. 
and 
(and thus a and 
) are influenced by the rain drop-size distribution, the microwave frequency and the polarization (horizontal, vertical) (Olsen et al. 1978; Leijnse et al. 2008). For frequencies between 15 and 40 GHz, the exponent parameter b is very close to unity.
can be estimated by:
(
) is a correction term that represents wet antenna attenuation and L is the path length. The average attenuation, 
, is obtained by:
is the reference (baseline) signal power.
can be obtained indirectly, as described in Overeem et al. (2016b). Indeed, maximum and minimum attenuations, 
and 
, are obtained by:
and 
, are then obtained by:
is obtained by a weighted average as follows:
is the weight relative to the maximum path-averaged rainfall intensity (
).Optimal values of 
and 
can be obtained by training the model and optimizing CML rainfall estimates compared with the rain gauge measurements.
With a network of CMLs whose positions are known, it is possible to apply statistical spatial interpolation methods in order to estimate the rainfall in each point of the 
space. Therefore, high temporal and spatial resolution precipitation estimates can be provided by this technique.
SINGLE-LAYER NEURAL NETWORK-BASED METHOD
Neural networks provide robust solutions to a wide range of problems in many disciplines, particularly areas involving classification, prediction, filtering, optimization, pattern recognition, and function approximation. An artificial neural network (ANN) is represented by a layered and interconnected group of nodes containing an input layer, one or more hidden layers, and an output layer. A neural network consisting of more than three layers, including the inputs and the output layers, is called a deep neural network. A simple or basic neural network is defined as a network with only two or three layers.
Perceptron model
In order to develop a low-complexity highly accurate method for rainfall estimation based on RSL data, a single-layer neural network-based model was investigated in this paper.
defined in Equation (1), a 1-input perceptron model was developed as illustrated in Figure 2. We have added the unit vector 
in order to be able to estimate the constant coefficient 
(bias term). 
1-input perceptron ANN architecture with ReLU activation function.
given by:
Optimization and performance
The evaluation and optimization of these models are performed using the adaptive moment (Adam) algorithm and the MSE as a cost (loss) function. Adam is an optimization solver for the neural network algorithm that is computationally efficient, requires little memory, and is well suited for problems that are large in terms of data or parameters. It is a combination of Root Mean Square Propagation (RMSprop) and Stochastic Gradient Descent with Momentum (SGDM) optimization algorithms.
In this study, data were available for only 40 days, which can be considered a small dataset. The two subsets used to train (fit the optimal parameters) and test the model (provide the model evaluation) can be considered as small. Consequently, a simple split cannot reveal the confidence level of the results. Since that, a basic approach called k-fold cross-validation has been used. Indeed, the initial set was split into k smaller sets called folds. The model was trained using k − 1 folds and tested on the remaining fold. This procedure was repeated such that different training and testing sets were obtained each time. Statistical performance metrics were collected for each repetition and then aggregated in order to provide an estimate of the variability of the model's statistical performance. In this study, k is assumed to be 
. Therefore, at each iteration, 
of the data were considered in the training subset, while the remaining 
belongs to the test data.
R-k parameters by the perceptron-based method
RESULTS
In our simulation, Python programming language (VanderPlas 2016) was used with several libraries, mainly TensorFlow (Deep Learning Library), pandas (Data Analysis), matplotlib (plotting), scikit-learn (machine learning), NumPy (numerical computing), and seaborn (data visualization). We started pre-processing RSL data by eliminating samples with incomplete data (samples with missing values due to some technical reasons) and samples with erroneous data (
, 
, or 
).
, 
, and 
collected from CMLs Link 1 (left side) and Link 2 (right side). The rain gauge data reveals a total precipitation of 
with the highest hourly rainfall of 
and the highest daily rainfall of 
. Correlation values between the rain gauge and CML RSL data are illustrated in Table 1. They reveal strong correlations, mainly between the rain gauge precipitation data and 
. Therefore, we can conclude that this CML RSLs data has the potential to estimate rainfall. In addition, correlation values between the RSL data of the two CMLs are shown. They admit strong relationships, mainly between the mean powers, 
, and between the minimum powers, 
. These correlation values allow us to conclude that these two links are located in the same region of the rain gauge under almost the same precipitation conditions. Correlations between 
and RSL data
 |  |  | ||||
|---|---|---|---|---|---|---|
| Link 1 | Link 2 | Link 1 | Link 2 | Link 1 | Link 2 | |
| Rain gauge – Link | −0.87 | −0.9 | −0.89 | −0.87 | −0.35 | −0.23 |
| Link 1 − Link 2 | + 0.91 | + 0.94 | + 0.62 | |||
| | | ||||
|---|---|---|---|---|---|---|
| Link 1 | Link 2 | Link 1 | Link 2 | Link 1 | Link 2 | |
| Rain gauge – Link | −0.87 | −0.9 | −0.89 | −0.87 | −0.35 | −0.23 |
| Link 1 − Link 2 | + 0.91 | + 0.94 | + 0.62 | |||
Temporal variation of 
, 
, 
, and 
for (a) Link 1 and (b) Link 2.
, as a function of hourly average powers, 
, is shown in Figure 4. We can notice that, for Link 1, an almost linear relationship exists for values of 
below 
, while precipitation is mostly zero for 
. Similar results are obtained for Link 2 with a threshold of 
. 
Precipitation 
in terms of mean received power 
for (a) Link 1 and (b) Link 2.
and 
while they are 
and 
for Link 2. Theoretical values of a and b coefficients have been determined as a function of the frequency and polarization only, while in reality, they also depend on the raindrop size distribution. Figure 5 illustrates the difference in precipitation estimates between the CML 
power-law-based method and the rain gauge measurements, considered as the reference measurement. The linear regression model will result in an optimal estimation of 
. However, negative outcomes may occasionally be produced as shown. 
CML linear regression: (a) Link 1 and (b) Link 2.
CML linear regression with ReLU: (a) Link 1 and (b) Link 2.
CML linear regression with ReLU, CML estimates versus rain gauge precipitations: (a) Link 1 and (b) Link 2.
CML linear regression with ReLU, CML estimates versus rain gauge precipitations: (a) Link 1 and (b) Link 2.
The different CML-based methods are evaluated based on the estimation of different parameters (MSE, MAE, R, and r2). Their corresponding mean and standard deviation are summarized in Table 2. In these calculations, we have considered that the two CMLs are equally weighted. These results showed that our perceptron-based methods with the ReLU activation function lead to better accuracy estimates.
Overall results of rainfall estimation Link 1 and Link 2
| MSE | MAE | R | r2 | |||||
|---|---|---|---|---|---|---|---|---|
| Dataset | Training | Test | Training | Test | Training | Test | Training | Test |
| R-k method |  |  |  |  |  |  |  |  |
| Linear regression |  |  |  |  |  |  |  |  |
| Perceptron with ReLU |  |  |  |  |  |  |  |  |
| MSE | MAE | R | r2 | |||||
|---|---|---|---|---|---|---|---|---|
| Dataset | Training | Test | Training | Test | Training | Test | Training | Test |
| R-k method | | | | | | | | |
| Linear regression | | | | | | | | |
| Perceptron with ReLU | | | | | | | | |
Temporal variation of precipitations obtained by rain gauge (Top), CML perceptron ReLU (Middle) and error (Bottom).
Temporal variation of precipitations obtained by rain gauge (Top), CML perceptron ReLU (Middle) and error (Bottom).
CONCLUSION AND PERSPECTIVES
In this study, we have tested and analyzed the simple use of the empirical 
power relationship that derives rainfall estimates based on CMLs power attenuations. In order to overcome its limitations and enhance its accuracy, we have proposed a single-layer neural network-based method. Experimental results have proven the high accuracy of this new technique. This may help to develop real-time applications for rainfall monitoring, forecasting, flood warnings, and many other hydrological domains. More precise and consistent estimates of precipitation can be reached by exploiting more evolved ANN architectures with deeper network topology and longer-term CML data, at the cost of higher complexity. This can be useful mainly for cases where the coefficient b of the 
relation is somewhat far from unity.
We hope that this work inspires the Lebanese Ministry of Telecoms, as well as Lebanon's mobile telecommunications and data operators, to continue and strengthen their grateful cooperation with us in the near future. By delivering data from all the microwave links covering Lebanon, with their precise geographical locations, we will be able to draw an accurate series of rainfall maps of the country.
ACKNOWLEDGEMENTS
The authors would like to thank the Deanship of Scientific Research at Umm Al-Qura University for supporting this work by Grant Code: (22UQU4361048DSR02). This work was also supported by the Lebanese University (project entitled ‘Artificial intelligence-based techniques for rainfall estimation using microwave links attenuation data from cellular communication networks: A case study in Lebanon’). We gratefully acknowledge Touch, the leading mobile telecommunications and data operator in Lebanon, and the Lebanese Agricultural Research Institute (LARI) for providing us with the requested data for our project. Finally, we would like to express our gratitude to Ir. Aart Overeem, a research scientist, R&D observations and data technology, at the Royal Netherlands Meteorological Institute (KNMI), for his intellectual and scientific assistance.
DATA AVAILABILITY STATEMENT
Data cannot be made publicly available; readers should contact the corresponding author for details.
CONFLICT OF INTEREST
The authors declare there is no conflict.
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