Using an Artificial Neural Network to Assess Several Rainfall Estimation Algorithms Based on X-Band Polarimetric Variables in West Africa

Abstract

Quantitative precipitation estimation using polarimetric radar in attenuation-prone frequency (X-band) in tropical regions characterized by convective rain systems with high intensities is a major challenge due to strong attenuations that can lead to total signal extinction over short distances. However, some authors have addressed this issue in Benin since 2006 in the framework of the African Monsoon Multidisciplinary Analysis program. Thus, with an experimental setup consisting of an X-band polarimetric weather radar (Xport) and a network of rain gauges, investigations have started on the subject with the aim of improving rainfall estimates. Based on simulated polarimetric variables and using a Multilayer Perceptron artificial neural network, several bi-variable and tri-variable algorithms were assessed in this study. The data used in this study are of two categories: (i) simulated polarimetric variables (Rayleigh reflectivity $Z$, horizontal attenuation $A_h$, horizontal reflectivity $Z_h$, differential reflectivity $Z_{dr}$, and specific differential phase $K_{dp}$) and rainfall intensity (R) obtained from Rain Drop Size Distribution (DSD) measurements used for algorithm evaluation (training and testing); (ii) polarimetric variables measured by the Xport radar and rainfall intensity measured by rain gauges used for algorithm validation. The simulations are performed using the T-matrix code, which leverages the scattering properties of spheroidal particles. The DSD measurements taken in northwest Benin were used as input for this code. For each spectrum, the T-matrix code simulates multiple variables. The simulated data (first category) were divided into two parts: one for training and one for testing. Subsequently, the best algorithms were validated with the second category of data. The performance of the algorithms during training, testing, and validation was evaluated using metrics. The best selected algorithms are $A1:R(Z,K_{dp})$ and $A12:R(Z_{dr},K_{dp})$ (among the bi-variable); $B2:R(Z_h,Z_{dr},K_{dp})$ and $B3:R(A_h,Z_{dr},K_{dp})$ (among the tri-variable). Tri-variable algorithms outperform bi-variable algorithms. Validation with observation data (Xport measurements and rain gauge network) showed that the algorithm $B3:R(A_h,Z_{dr},K_{dp})$ performs better than $B2:R(Z_h,Z_{dr},K_{dp})$.

1. Introduction

The emissions from the production systems, such as agricultural, livestock, and industrial sectors, are the main reason for extreme weather events such as floods and drought [1,2,3,4]. In particular, West Africa faces the challenges of climate change, manifested by the recurrence of floods and droughts [5,6,7]. Furthermore, the low density of rain gauge networks and the absence or insufficient maintenance make it difficult to access qualitative (or high-resolution) rainfall information, which is a critical variable in understanding and mitigating floods. Despite its acquisition and maintenance costs, meteorological radar remains an important tool for fine-scale spatiotemporal rainfall estimation. The use of X-band polarimetric radar in tropical regions presents significant challenges due to severe signal attenuation by heavy precipitation, reducing effective range and complicating the detection of intense rainfall [8]. Frequent lightning strikes can interfere with radar signals, requiring additional filtering techniques [9]. However, utilizing radar measurements for hydrometeorological purposes requires two essential upstream steps: correcting raw data and designing effective algorithms to convert corrected measurements into rainfall quantities. This work addresses the issue of designing algorithms for quantitative rainfall estimation based on meteorological radar measurements.

Several studies have shown that polarimetric-based techniques are particularly effective for rainfall estimation using X-band radar, exploiting observables such as differential phase [10,11,12], differential reflectivity [13], or various combinations of reflectivity and other polarimetric variables. Several authors [8,9,14] have recommended the assessment of rainfall measurement by X-band polarimetric radar in a tropical environment. Thus, during the intensive measurement phase (from 2005 to 2007) of the African Monsoon Multidisciplinary Analysis (AMMA) program, an experimental rainfall measurement device was deployed for the first time in West Africa, in northern Benin, at the Upper Oueme Hydro-meteorological observatory [15]. This device consisted of an X-band polarimetric radar named Xport, three optical spectropluviometers, and a network of fifty-four (54) rain gauges.

Studies have demonstrated that most rainfall in this region is generated by mesoscale convective systems [16], frequently organized into squall lines with a leading convective cells front followed by a stratiform trailing region [17]. In this context, quantitative precipitation estimation based on attenuated radar frequency, as is the case for X-band, can be challenging. However, in the study region, Zahiri et al. [18] conducted tests on several X-band polarimetric radar rainfall estimation algorithms. These algorithms were designed using the linear regression method based on squall line simulations using a high-resolution atmospheric model (Meso-NH). Subsequently, Gosset et al. [19] simulated X-band polarimetric variables from measured Rain Drop Size Distribution (DSD) in the region using the T-matrix code [20]. The simulated variables were used to design algorithms, either by simple or multiple linear regressions, for quantitative rainfall estimation. This simulation approach allowed the assessment of the intrinsic performances of each algorithm used. After correcting Xport radar measurement attenuation, Koffi et al. [21] tested the algorithms designed by Gosset et al. [19] by comparing the rainfall intensities retrieval by the Xport radar with the rainfall intensities measured by rain gauges in the radar-covered area. They found that these algorithms provided good estimates of rainfall intensity but with a precision that could be improved. Due to measurement problems affecting radar observables, especially in light rain, they concluded that the single-parameter algorithm $R\left(K_{dp}\right)$ made better estimates than the multi-parameter $R(Z_h, K_{dp}, Z_{dr})$, whereas intrinsically, one would expect opposite results since multi-parameters would benefit from information redundancy ($R$ is the rainfall intensity estimated from the polarimetric variables $Z_h$, $K_{dp}$, and $Z_{dr}$). Such a conclusion illustrates the difficulty encountered in applying algorithms, sometimes designed from simulated data to real radar data.

Additionally, empirical relationships (or algorithms) obtained from parametric methods are sensitive to DSD variability, and determining their coefficients assumes a linear correlation between variables, which is not always established depending on the type of rainfall considered. Thus, some authors [22,23] have proposed non-parametric methods, such as ANNs, to design rainfall retrieval algorithms from radar measurements. Other authors, such as Brandes et al. [24], Gorgucci et al. [25], and Zhang et al. [26], have shown that non-parametric methods produce more efficient algorithms than parametric methods.

Following the work of Gosset et al. [19] and Koffi et al. [21], this present study aims to assess rainfall retrieval algorithms from measurements of an X-band polarimetric radar using an artificial neural network (ANN). Specifically, (i) based on simulated X-band polarimetric variables, this study will identify the best combinations (bi and tri-variables) to retrieve rainfall intensity using the ANN method; (ii) examine the accuracy of the best algorithms designed with ANN with their empirical counterparts designed with multiple linear regression; (iii) and validate the best algorithms identified from Xport radar measurements.

2. Data Sets

2.1. Measurement Setup

The study relies on three types of measurements: (i) Measurements of DSD obtained using optical disdrometers [27,28,29]; (ii) measurements from the Xport radar [21]; (iii) measurements from the rain gauge network. These measurements were carried out from 2006 to 2007 during the intensive phase of the AMMA campaigns. The measurement setup is illustrated in Figure 1, indicating the positions of the Xport radar in Djougou, the disdrometers in Copargo, Djougou, and Nangatchiori, and the rain gauges in the Donga supersite. The supersite, located in the Donga basin, covers an area of $600 {\mathrm{k}\mathrm{m}}^2$ and is situated at the heart of the Upper Ouémé Valley Hydrometeorological Observatory (OHHVO is its abbreviation in French), which spans $\mathrm{14,600} {\mathrm{k}\mathrm{m}}^2$ and is centered at $2° \mathrm{E}$ and $9.5° \mathrm{N}$.

2.2. Simulated Polarimetric Variables

The various steps and modules used to produce the simulated polarimetric variables are thoroughly described in the work of Zahiri et al. [18]. The simulation of polarimetric variables is typically conducted using the numerical code known as the T-matrix [20]. This code is based on a precise method for determining light scattering by nonspherical particles. It has been used by several authors in various regions of the planet [30,31,32,33,34,35], including our study area [18,19,21]. The input parameters for this code include the following:

• Polarimetric parameters (here, those of the X-band);
• Radar elevation angle (fixed at 2° in this study);
• Temperature of raindrops (ranging from 0 °C to 35 °C in 5 °C increments);
• Raindrop shape model (those used in this study are presented at the end of this section);
• Analytical model of the DSD or DSD measurements.

Zahiri [15] demonstrated, using synthetic radar data (utilizing the T-matrix code), the variability of relationships between different radar variables and their relationship to rainfall intensity. Among the sources of influence, temperature and models of raindrop flattening can be noted. Since these variables are used in combination to estimate rainfall, it is essential to examine the behavior of these different algorithms in the face of these influences. The temperature ranges and raindrop flattening models used are those studied by Zahiri [15].

In this study, the measured DSDs described by Moumouni et al. [29] are used for simulation. This dataset consists of 12,342 DSD spectra (at one-minute time intervals) obtained from 93 rainfall events sampled between 2005 and 2007 using optical spectropluviometer-type disdrometers. For each DSD spectrum, rainfall intensity (R [$\mathrm{m}\mathrm{m}·{\mathrm{h}}^{-1}$]) is calculated, and the simulated polarimetric variables are as follows:

• Rayleigh reflectivity (Z [${\mathrm{m}\mathrm{m}}^6·{\mathrm{m}\mathrm{m}}^{-3}$]);
• Horizontal reflectivity (Z_ℎ) [${\mathrm{m}\mathrm{m}}^6·{\mathrm{m}\mathrm{m}}^{-3}$];
• Vertical reflectivity (Z_v) [${\mathrm{m}\mathrm{m}}^6·{\mathrm{m}\mathrm{m}}^{-3}$];
• Differential reflectivity (Z_dr) [dimensionless];
• Horizontal specific attenuation (A_ℎ [$\mathrm{d}\mathrm{B}·{\mathrm{k}\mathrm{m}}^{-1}$]);
• Vertical specific attenuation (A_v) [$\mathrm{d}\mathrm{B}·{\mathrm{k}\mathrm{m}}^{-1}$];
• Specific differential phase shift (K_dp [$\mathrm{d}\mathrm{e}\mathrm{g}·{\mathrm{k}\mathrm{m}}^{-1}$]).

The raindrop shape models used are as follows:

• S1: Andsager et al. [36]: $a/b=1.012-0.144 D-1.03{ D}^2$, $D$ in $\mathrm{c}\mathrm{m}$;
• S2: Pruppacher et Beard [37]: $a/b=1.030-0.62 D$, $D$ in $\mathrm{c}\mathrm{m}$;
• S3: Goddard et al. [38]: $a/b=1.075-0.065 D-0.0036 D^2+0.0004 D^3$, $D$ in $\mathrm{m}\mathrm{m}.$

$a/b$ represents the axial ratio of drop shapes and $D$ represents the diameter of drop.

These drop flattening models are used to account for a wide range of realistic drop shapes during the simulation by modeling the drops as oblate spheroids.

2.3. Xport Radar Data and Rain Gauge

The data used for algorithm validation consists of 5-min time step measurements conducted by the Xport radar and the rain gauge network during the AMMA campaign in the years 2006 and 2007. These data have been pre-processed by Koffi et al. [21]. Specifically, horizontal and differential reflectivities have been corrected for attenuation using the method proposed by Carey et al. [39], which relies on a linear relationship between total attenuation along a given radial and the total differential phase shift (${\Phi }_{dp}$), the integral of specific differential phase shift ($K_{dp}$). The correction coefficients used are 0.285 dB/deg and 0.051 dB/deg for horizontal and differential reflectivities, respectively.

To detect and correct the radar calibration issue, we exploit the consistency between radar variables as proposed by various authors [40,41]. From numerical simulations based on actual (or realistic) DSDs, we can establish intrinsic and expected relationships between pairs of variables of interest [21,42], standing for reference. The consistency analysis involves comparing the observed radar variables with each other and checking conformity with the reference theoretical curve. Since $K_{dp}$ is calibration-independent, the radar variable pairs ($K_{dp}-Z_h$) and ($Z_h-Z_{dr}$) are used. To carry out this analysis, the radar variables $K_{dp}$ and $Z_{dr}$ are plotted against $Z_h$ using a 2D histogram. A comparison of the $Z_h-K_{dp}$ and $Z_h-Z_{dr}$ radar scatterplots with theoretical curves obtained from simulation can provide information on the validity of taking a calibration offset into account.

3. Methods

3.1. Assessed Algorithms

The choice of the ANN architecture is based on the problem to be solved. In our case study, we use two or three polarimetric variables to estimate the rainfall intensity. Therefore, we can identify the polarimetric variables as the inputs (two or three inputs) and the rainfall intensity as the output. We decided to use the multilayer perceptron (MLP) architecture, inspired by the work of Orlandini and Morlini [22], who used this architecture to estimate rainfall intensity from multiple inputs. Regarding the number of layers and neurons, there is no fixed rule. It all depends on the complexity of the problem to be solved. Thus, we conducted some preliminary tests by varying the number of layers and neurons and then evaluated the learning of all models using several metrics. After this preliminary work, we fixed the number of layers and neurons for the evaluation of the algorithms.

The assessed algorithms include both bi-variable and tri-variable algorithms, as presented in

Table 1. List of assessed algorithms with ANN.

Bi-variable Algorithms	$A1:R(Z,K_{dp})$	$A2:R(Z,Z_h)$	$A3:R(Z,A_h)$
	$A4:R(Z_h,A_h)$	$A5:R(Z_h,Z_{dr})$	$A6:R(Z_h,K_{dp})$
	$A7:R(Z_v,A_v)$	$A8:R(Z_v,Z_{dr})$	$A9:R(Z_v,K_{dp})$
	$A10:R(A_h,Z_{dr})$	$A11:R(A_v,K_{dp})$	$A12:R(Z_{dr},K_{dp})$
Tri-variable Algorithms	$B1:R(Z,Z_{dr},K_{dp})$	$B2:R(Z_h,Z_{dr},K_{dp})$	$B3:R(A_h,Z_{dr},K_{dp})$
	$B4:R(Z_h,A_v,K_{dp})$	$B5:R(Z_v,A_h,K_{dp})$

. Since multivariable algorithms are inherently more effective than single-variable algorithms, this study focused on assessing these combined estimators. Using simulated polarimetric variables and rainfall intensity measured by the disdrometers, these algorithms were developed from an ANN and assessed through metrics described in the following sections. Some combined estimators, such as $R(Z,Z_v)$, $R(Z_h,Z_v)$, or $R(Z,Z_v,A_h)$, did not converge during ANN training and were thus excluded from the list of assessed algorithms. Indeed, for these combinations, the training process of the ANN did not reach a state where it could make accurate estimations. The errors between estimated and actual values during training remained high. This could be explained by an ANN architecture or hyperparameters that are not suitable for these variable combinations. It is also possible that these variable combinations are not adequate for accurate rainfall estimation.

Following the selection of the best algorithms by the ANN, the parametric forms (multivariable regressions) of these algorithms were established using the same data. Finally, the best algorithms (both non-parametric and parametric forms) were validated using data measured by the Xport radar and the rain gauges. The polarimetric variables serve as input variables, also known as explanatory or predictor variables, while rainfall intensity R serves as the output variable, also known as the explained or predicted variable.

3.2. Artificial Neural Network Method

Formal neural networks possess the remarkable property of being universal and parsimonious approximators [43]. ANNs are widely used for prediction in hydrometeorology [44,45,46,47,48], particularly for rainfall estimation using meteorological radar measurements [22,23,49,50,51].

ANN codes execute in three main steps: learning, testing, and prediction. During the learning phase, input variables (in this study, a pair or triplet of polarimetric variables) and output variables (here, rainfall rate R) are provided to the ANN. In supervised learning, the ANN attempts to obtain a better approximation of the output variables by minimizing a cost function to determine the best weighting coefficients for the input variables. These coefficients are saved for the testing and prediction phases. During testing and prediction steps, only the input variables are provided to the ANN, which estimates the output variables. During testing, the estimated values by the ANN are compared to the expected values. Once testing is successful, the algorithm is used for prediction. In our case, we have 12,342 DSD spectra. Each spectrum was used to simulate polarimetric variables and calculate rainfall rates; 75% of these were used for learning, and the remainder for assessment of the method. For prediction, the polarimetric variables measured by the Xport radar and the rainfall rate measured by the rain gauges are used. In practice, the input variables for the ANN are normalized according to the formula:

$$x_r={\displaystyle \frac{x-\overline{x}}{{\sigma }_x}} ,$$

(1)

$x_r$ represents the normalized input variable, $x$ is the input variable, $\overline{x}$ denotes the mean, and ${\sigma }_x$ stands for the standard deviation of $x$.

The issue of estimating rainfall from polarimetric radar variables is an approximation problem (estimation of one variable from others). There are several architectures of artificial neural networks suitable for this problem: a deep learning approach using ANN [49], recurrent neural networks [52], or MLP neural networks [23]. The architecture used in this study is the MLP [53], encoded in the Tensorflow and Keras libraries under the Python language in the Jupyter environment. The MLP is organized into several layers: an input layer, one or more hidden layers, and an output layer. The presence of the hidden layer is crucial as it facilitates the modeling of the nonlinear relationships between the inputs and the output.

The MLP has several types of activation functions, both in the output layer and in the hidden layers. The linear activation function (also called the identity function) was used in the output layer. It does not alter the output variables, which is practical for the approximation problem in this study. Regarding the hidden layers, the ReLU (Rectified Linear Unit) activation function was used to filter out negative values before the output layer. Krizhevsky et al. [54] demonstrated that neural networks with large capacities in terms of neurons and hidden layers yield better performance for complex tasks. This study chose one (01) output layer, six (06) hidden layers, each containing one thousand five hundred (1500) neuron units, and one (01) input layer. For training, the Adam optimizer (Adaptive Moment Estimation) was selected due to its ability to dynamically adapt to various data characteristics and maintain an adaptive learning rate [55].

3.3. Regression Metrics

The various metrics used to assess the performance of the algorithms aim to compare the estimated rainfall intensity values by the algorithms with the original rainfall intensity values (measured by the disdrometers or pluviographs). With a large number of available metrics, it is not always straightforward to choose the one that best suits the problem at hand. The final analysis can be significantly different depending on the metric used to assess the models. Five of the most commonly used metrics have been chosen to assess the algorithms. Let $R_{orig}$ be the original rainfall rate and $R_{est}$ be the rainfall rate estimated by the algorithms; $E\left[R_{orig}\right]$ and $E\left[R_{est}\right]$ denote their respective means; ${\sigma }_{orig}$ and ${\sigma }_{est}$ denote their respective standard deviations. The different metrics used are:

• The coefficient of linear correlation of Pearson measures the strength and direction of the linear relationship between the estimated rainfall intensity values by the algorithms and the original rainfall intensity values. Its expression is:

$$\rho ={\displaystyle \frac{E\left[\right(R_{orig}-E\left[R_{orig}\right]\left)\right(R_{est}-E\left[R_{est}\right]\left)\right]}{{\sigma }_{orig}{\sigma }_{est}}}.$$

(2)

A coefficient close to 1 or −1 indicates a strong positive or negative linear relationship, respectively, and thus precise estimation. A coefficient close to 0 indicates a weak linear relationship and, therefore, poor estimation performance.

• The Nash coefficient [56] measures the accuracy of predictions relative to the mean of the observations and is defined by:

$$Nash=1-{\displaystyle \frac{E\left[{\left(R_{est}-R_{orig}\right)}^2\right]}{{E\left[\right(R_{orig}-E\left[R_{orig}\right])}^2]}}.$$

(3)

A Nash value close to 1 indicates a perfect match between observed and estimated values, while a value close to 0 indicates that the estimation is no better than the mean of the observations.

• The efficiency coefficient KGE [57] evaluates the overall performance of the model. It is defined by:

$$KGE=1-\sqrt{{\left(\rho -1\right)}^2+{\left({\displaystyle \frac{{\sigma }_{est} }{{\sigma }_{orig}}}-1\right)}^2+{\left({\displaystyle \frac{E\left[R_{est}\right]}{E\left[R_{orig}\right]}}-1\right)}^2}.$$

(4)

A KGE value close to 1 indicates perfect performance and precise estimation, while a lower value indicates poorer performance.

• The mean relative error measures the average of the relative errors between the estimated and real values. Its expression is:

$$MRE= E\left[{\displaystyle \frac{(R_{est}-R_{orig})}{R_{orig}}}\right].$$

(5)

A lower MRE indicates better accuracy of the estimations compared to the real values, signifying more precise estimation.

• Standard deviation of fractional error measures the dispersion of errors around the mean error. Its expression is:

$$SDFE= \sqrt{E\left[{\left({\displaystyle \frac{R_{est}-R_{orig})}{R_{orig}}}\right)}^2\right]}.$$

(6)

A lower SDFE indicates less dispersion of errors, leading to better consistency and precision in the estimations.

3.4. Multivariable Regression Method

Multivariable regression is a mathematical method used to explain the various values taken by a variable based on other explanatory variables. Mathematically, for a quantitative variable $Y$ to be explained by $p$ explanatory variables $X^1\cdots X^p$, a linear model is defined as follows:

$$Y_i={\beta }_0+{\beta }_1{X}_i^1+{\beta }_2{X}_i^2+\cdots +{\beta }_p{X}_i^p+{\epsilon }_i , i=1, 2, \dots , n.$$

(7)

With ${\epsilon }_i$ being the independent and identically distributed error terms; $n$ is the number of quantitative variables; and the unknown parameters ${\beta }_0, \dots ,{\beta }_p$ assumed to be constants to be determined. However, empirical relationships between rainfall intensity and polarimetric variables are power functions [19,21,58] in the form:

$$Y={\alpha }_0{\left(X^1\right)}^{{\alpha }_1}\cdots {\left(X^p\right)}^{{\alpha }_p},$$

(8)

with ${\alpha }_0, \dots ,{\alpha }_p$ assumed to be constants to be determined. By applying the logarithm function, we obtain a linear model:

$${\log }_{10}\left(Y\right)={\log }_{10}\left({\alpha }_0\right)+{\alpha }_1{\log }_{10}\left(X^1\right)+\cdots +{\alpha }_p{\log }_{10}\left(X^p\right).$$

(9)

In this study, multiple linear regression using the least squares method was employed.

4. Results

4.1. Assessment of the Algorithms Using ANN

The assessed algorithms are those presented in Table 1. They fall into two categories: bi-variable and tri-variable algorithms. The data used consist of simulated polarimetric variables and rainfall intensities calculated from 12,342 DSD spectra. The method employed is MLP. Approximately 75% (9262) of the data are used for ANN training, with the remaining 25% (3080) used for testing. The presented results are from the testing phase.

4.1.1. Preselection of Algorithms

In the study region, daily minimum temperatures close to the ground are rarely below 20 °C. For simulation purposes, the temperature is fixed at 25 °C, and the drop flattening model used is S1 [36], as performed by Zahiri et al. [18], Gosset et al. [19], and Koffi et al. [21]. Figure 2 and Figure 3 present the test results: Scatterplot of estimated versus original rainfall rate. The analysis of the scatterplot concerning the first bisector and the metric values led to the preselection of algorithms from

Table 2. List of preselected algorithms with ANN.

Bi-variable Algorithms	$A1:R(Z,K_{dp})$	$A9:R(Z_v,K_{dp})$	$A11:R(A_v,K_{dp})$
	$A12:R(Z_{dr},K_{dp})$
Tri-variable Algorithms	$B1:R(Z,Z_{dr},K_{dp})$	$B2:R(Z_h,Z_{dr},K_{dp})$	$B3:R(A_h,Z_{dr},K_{dp})$
	$B4:R(Z_h,A_v,K_{dp})$	$B5:R(Z_v,A_h,K_{dp})$

. The same analysis conducted at 25 °C, using drop flattening models S2 [37] and S3 [38], resulted in the preselection of the same algorithms (algorithm scores with all drop flattening models at 25 °C are presented in

Table A1. S1 [36], S2 [37], and S3 [38] at a temperature of 25 °C.

S1 Bi-Variables
Algorithms	Ro	Nash	KGE	MRE	SDFE
A1	0.9985	0.9969	0.9891	0.1129	0.3436
A2	0.9909	0.9817	0.9739	0.4329	1.0515
A3	0.9834	0.9667	0.9621	0.5013	1.1623
A4	0.9871	0.9742	0.9712	0.268	0.6325
A5	0.9822	0.958	0.9207	0.1378	0.5432
A6	0.9981	0.9961	0.9903	0.1672	0.4663
A7	0.9705	0.9404	0.9527	0.2834	0.778
A8	0.9813	0.9562	0.9218	0.1186	0.4953
A9	0.9974	0.9949	0.9913	0.1756	0.4893
A10	0.9926	0.9847	0.9789	0.0221	0.1923
A11	0.9978	0.9955	0.9911	0.1419	0.4131
A12	0.9978	0.9956	0.997	0.0865	0.4065
S1 Tri-Variables
Algorithms	Ro	Nash	KGE	MRE	SDFE
B1	0.9988	0.9975	0.99	0.0875	0.3352
B2	0.9984	0.9968	0.9877	0.1097	0.4702
B3	0.9978	0.9956	0.9959	0.0214	0.1948
B4	0.9985	0.9969	0.994	0.1333	0.3978
B5	0.9979	0.9957	0.9841	0.1528	0.4574
S2 Bi-Variables
Algorithms	Ro	Nash	KGE	MRE	SDFE
A1	0.996	0.9921	0.9926	0.3875	1.0921
A2	0.9926	0.9852	0.9874	0.3729	1.0469
A3	0.984	0.9673	0.9428	0.1188	0.4217
A4	0.9874	0.9746	0.965	0.1759	0.5454
A5	0.9858	0.9711	0.9562	0.076	0.3475
A6	0.9956	0.9912	0.9919	0.4126	1.1575
A7	0.9757	0.95	0.9176	0.1463	0.6359
A8	0.9847	0.9693	0.9597	0.083	0.3524
A9	0.9951	0.9902	0.9896	0.471	1.3139
A10	0.9918	0.9837	0.9806	0.0372	0.2515
A11	0.997	0.994	0.9905	0.2544	0.7564
A12	0.9972	0.9944	0.9907	0.1085	0.5866
S2 Tri-Variables
Algorithms	Ro	Nash	KGE	MRE	SDFE
B1	0.9977	0.9955	0.9944	0.0161	0.3801
B2	0.9981	0.9962	0.9906	0.0503	0.3306
B3	0.9975	0.9949	0.9929	0.0353	0.1861
B4	0.9973	0.9946	0.9916	0.2971	0.8769
B5	0.9958	0.9917	0.9949	0.0937	0.3723
S3 Bi-Variables
Algorithms	Ro	Nash	KGE	MRE	SDFE
A1	0.999	0.9979	0.9941	0.0725	0.3267
A2	0.9906	0.9813	0.9871	0.3102	0.9033
A3	0.9845	0.969	0.9764	0.0896	0.3609
A4	0.9885	0.9771	0.9816	0.2047	0.5404
A5	0.9778	0.9549	0.973	0.0493	0.5802
A6	0.9988	0.9976	0.9951	0.0791	0.3489
A7	0.9807	0.9618	0.9745	0.256	0.7869
A8	0.978	0.9557	0.9753	0.0544	0.416
A9	0.9988	0.9976	0.9956	0.0795	0.3475
A10	0.9914	0.9829	0.9896	0.0263	0.1713
A11	0.9988	0.9976	0.9964	0.038	0.2459
A12	0.9988	0.9976	0.9958	0.0244	0.2026
S3 Tri-Variables
Algorithms	Ro	Nash	KGE	MRE	SDFE
B1	0.9992	0.9984	0.9933	0.0185	0.2152
B2	0.999	0.998	0.9968	0.036	0.297
B3	0.9989	0.9978	0.9967	0.0238	0.2211
B4	0.999	0.9979	0.9944	0.0342	0.2479
B5	0.9987	0.9974	0.9952	0.0557	0.3245

in the Appendix A).

4.1.2. Selection of the Best Algorithms

With the preselected algorithms, the previous analysis is repeated by varying the temperature from 0 °C to 35 °C in steps of 5 °C. Figure 4 (for bi-variable algorithms) and Figure 5 (for tri-variable algorithms) show the variations of the metrics as a function of droplet temperature using the drop shape model S1 [36]. The selection of the best algorithms is based on three principles. The first principle is that the best algorithms should be less sensitive to temperature compared to others (comparison of bi-variable algorithms among themselves and tri-variable algorithms among themselves). The second principle focuses on the best correlations found for the Ro, Nash, and KGE metrics between the actual rainfall intensity and that estimated by the algorithms, taking into account the different temperatures. The last principle, in agreement with the second, identifies the algorithms that present fewer errors in rainfall estimation at different temperatures, considering the MRE and SDFE criteria. The best bi-variable algorithms are $A1:R(Z,K_{dp})$ and $A12:R(Z_{dr},K_{dp})$. The best tri-variable algorithms are $B2:R(Z_h,Z_{dr},K_{dp})$ and $B3:R(A_h,Z_{dr},K_{dp})$. The presence of these variables in the best algorithms is confirmed by the information they provide about rainfall: Rayleigh reflectivity ($Z$), horizontal attenuation ($A_h$), horizontal reflectivity ($Z_h$), differential reflectivity ($Z_{dr}$), and specific differential phase ($K_{dp}$). Raleigh reflectivity ($Z$), which is proportional to the sixth power of the diameter of raindrops, is crucial for estimating precipitation rates but may be insufficient on its own. Horizontal attenuation ($A_h$) measures the weakening of the radar signal due to rain, thereby correcting other radar measurements. Horizontal reflectivity ($Z_h$), the most common radar measurement, is sensitive to the size and concentration of raindrops. Differential reflectivity ($Z_{dr}$) compares horizontal and vertical reflectivities, providing information on the shape of the raindrops. Specific differential phase ($K_{dp}$) measures the phase difference between polarizations through a precipitation layer and is particularly useful for heavy rainfall. By using these variables, algorithms can provide more accurate and reliable estimates of precipitation, taking into account temperature variations and raindrop characteristics. Similar results are obtained when simulating with drop shape models S2 [37] and S3 [38].

4.1.3. Comparison Between the Best Bi-Variable and Tri-Variable Algorithms

With the drop shape model S1 [36], the best algorithms (bi-variable and tri-variable) are compared against each other. The variations of metrics as a function of temperature are plotted in the same figure (Figure 6). The bi-variable algorithms demonstrating good performance are A1 and A12, with metrics around 0.99 for Ro and Nash, and 0.97 for KGE. The MRE and SDFE values are 0.05 and 0.14 for A1, and 0.04 and 0.26 for A12, respectively. The algorithms with poorer performance are A7 and A2. Their Ro, Nash, and KGE are approximately 0.98, 0.94, and 0.80, respectively. The MRE and SDFE values are 0.38 and 1.13 for A2, and 0.23 and 0.77 for A7. Regarding the tri-variable algorithms, their performances are generally close to each other. The best ones are B2 and B3, with metrics around 0.99 for Ro, Nash, and KGE, approximately 0.14 for MRE, and around 0.5 for SDFE.

Algorithms $A12:R(Z_{dr},K_{dp})$ and $B2:R(Z_h,Z_{dr},K_{dp})$ appear to have similar performance. This is likely due to the correlation between $Z_{dr}$ and $Z_h$. Ultimately, algorithm $B3:R(A_h,Z_{dr},K_{dp})$ outperforms algorithms $A12:R(Z_{dr},K_{dp})$ and $B2:R(Z_h,Z_{dr},K_{dp})$, which in turn outperform algorithm $A1:R(Z,K_{dp})$. Tri-variable algorithms outperform bi-variable algorithms because they provide more information to the ANN, which is essential for effective learning. By increasing the number of radar observables, we can take advantage of the redundancy of information provided by each radar variable in the characterization of rainfall. This additional information allows for better adjustment of the coefficients during training, captures additional aspects of the rainfall phenomenon that may not be observable with two variables, and better captures variability. Furthermore, it helps correct potential errors in the bi-variable algorithms, thus contributing to the optimization of the performance of these algorithms.

4.2. The Empirical Relationships of the Best Algorithms Selected by the ANN

The best algorithms selected in the previous section are $B3:R(A_h,Z_{dr},K_{dp})$, $B2:R(Z_h,Z_{dr},K_{dp})$, $A12:R(Z_{dr},K_{dp})$, and $A1:R(Z,K_{dp})$. This section aims to establish the empirical relationships of these algorithms and perform inter-comparison. The simulated polarimetric variables with the 12,342 measured DSD spectra are used to establish the empirical relationships. Simulations are performed with the drop flattening model S1 [36] at a temperature of 25 °C. The different coefficients of the empirical relationships of the best algorithms are determined by multiple linear regressions using the method of least squares described in Section 3.4. The 25% of the simulated data used for the testing phase of the ANN algorithms are also used to test the empirical algorithms. Specifically, the polarimetric variables present in each empirical algorithm were utilized to calculate the rainfall rate (R) using various empirical relations. The calculated rainfall rate was then compared to the expected intensity through scatter plot representation and by calculating the values of different metrics. The comparison between the ANN algorithms and the empirical algorithms is performed pairwise, observing how the scatter plots are distributed around the first bisector and identifying the algorithms with the best metric values. The results are presented in

Table 3. Expressions of empirical relationships.

ANN-Derived Algorithms	Empirical Relationships
$A1:R(Z,K_{dp})$	$R=2033.20 Z^{-0.541}{K}_{dp}^{1.421}$
$A12:R(Z_{dr},K_{dp})$	$R=17.03 Z_{dr}^{-0.892}{K}_{dp}^{0.867}$
$B3:R(A_h,Z_{dr},K_{dp})$	$R=118.60 A_h^{0.896}{Z}_{dr}^{-2.147}{K}_{dp}^{0.049}$
$B2:R(Z_h,Z_{dr},K_{dp})$	$R=0.06 Z_h^{0.676}{Z}_{dr}^{-3.069}{K}_{dp}^{0.208}$

. The obtained relationships were tested with the same portion of the data used to test the ANN algorithms (i.e., 3080 spectra). Figure 7 (bi-variables) and Figure 8 (tri-variables) present the test results: scatter plot of the estimated rainfall intensity versus the original rainfall intensity. It can be observed that each algorithm obtained with the ANN outperforms its counterpart obtained by the empirical method.

4.3. Validation of the Best Algorithms Derived from MLP with Measurements from Xport Radar

This section aims to validate the algorithms designed with the ANN using measurements from the Xport radar from 2006 to 2007. Since the Xport radar does not measure Rayleigh reflectivity (Z), the validated algorithms are $B3:R(A_h,Z_{dr},K_{dp})$, $B2:R\left(Z_h,Z_{dr},K_{dp}\right),$ and $A12:R(Z_{dr},K_{dp})$.

At this stage, validation involves using real measurements from the Xport radar and rain gauges to assess the performance of the selected best algorithms. Our database consists of Xport sampling points located above active rain gauges. In total, 54 active rain gauges are distributed across the $\mathrm{14,600} {\mathrm{k}\mathrm{m}}^2$ of the OHHVO basin. As all sensors are synchronous, we have a set of variables ($R, A_h, A_v, {Z_h, Z_v, Z}_{dr}, K_{dp},$ etc.) at each sampling interval of 5 min, referred to as a dataset. There are 13,291 datasets in total (7632 for 2006 and 5659 for 2007). In this study, only datasets where the rainfall intensity measured by the rain gauge is greater than or equal to $0.1 \mathrm{m}\mathrm{m}·{\mathrm{h}}^{-1}$ are considered. After this thresholding, we have 9575 datasets remaining (5077 for 2006 and 4498 for 2007). Since the data used to select the algorithms do not have the same configuration as the data used for validation, the ANN required retraining (the validation data are sampled every 5 min, in contrast to the simulated data from DSD measurements, which are sampled every minute). For this purpose, 7127 datasets (from 2006 and part of 2007) are used for the new training, and the remaining 2448 are used for testing. First, the database was divided into two parts. The ANN training was carried out on about 75% (7127) of the data, using the same parameters used to select the best performing algorithms: polarimetric variables as input and rainfall intensity measured by the rain gauge network as output. After successful training, evaluated with the different metrics mentioned in the methodology section, we saved the model for each algorithm. From the remaining 25% (2448) of the validation data, we extracted the polarimetric variables measured by the Xport radar and applied them as inputs to the ANN according to each algorithm, then retrieved the output representing the rainfall intensity estimated by the ANN. This estimated rainfall intensity, denoted as $R_{est}$, is compared with the rainfall intensity measured by the corresponding rain gauge, denoted as $R_{orig}$.

Figure 9 depicts the test results for each algorithm: a scatter plot of the estimated rainfall intensity against the original rainfall intensity. From the analysis of this figure, it is observed that the tri-variable algorithms outperform the bi-variable algorithm, and that algorithm $B3:R(A_h,Z_{dr},K_{dp})$ is the best among the three. The performance of the algorithms was evaluated by assessing their ability to adequately reproduce the dynamics of a rainfall event. For this purpose, two rainfall events (on 4 July and 22 July 2007, at the Copargo site) from the test data were selected. The observed hyetograph and those estimated by the algorithms are represented in Figure 10. The metrics are presented in

Table 4. Metrics from the comparison between the observed hyetograph and those estimated by the algorithms.

Metrics		Ro	Nash	KGE	MRE	SDFE
Event of 4 July 2007	$A12:R(Z_{dr},K_{dp})$	0.841	0.6731	0.6175	0.796	3.0649
	$B3:R(A_h,Z_{dr},K_{dp})$	0.9909	0.9818	0.9787	0.2349	2.8802
	$B2:R(Z_h,Z_{dr},K_{dp})$	0.9548	0.9089	0.8939	0.3719	2.6975
Event of 22 July 2007	$A12:R(Z_{dr},K_{dp})$	0.5772	0.1292	0.1666	−0.0636	0.7486
	$B3:R(A_h,Z_{dr},K_{dp})$	0.9643	0.9285	0.9299	0.0754	0.5158
	$B2:R(Z_h,Z_{dr},K_{dp})$	0.8644	0.6394	0.7612	−0.0481	0.3946

. It is noted that the tri-variable algorithms reproduce the dynamics of the events well, and that algorithm $B3:R(A_h,Z_{dr},K_{dp})$ remains the best.

4.4. Dynamics of Some Rainfall Events

This section presents the hyetograph observed by the rain gauge and those estimated by the best-selected algorithms A12, B2, and B3 from two case events extracted from the test validation data (Figure 10). We note that the bi-variable algorithm A12 does not accurately estimate the rainfall peaks in either event, unlike the two tri-variable algorithms B2 and B3. Additionally, the tri-variable algorithms provide a better fit for the rainfall than the A12 algorithm. Between the two tri-variable algorithms, B3 provides the best estimation for both rainfall events.

5. Discussion

The main limitations of this study may be related to the spatial variability between rain gauge measurements, which are point measurements, and Xport radar measurements, which cover a wide area. This difference in resolution can lead to discrepancies. Additionally, missing or erroneous data were removed from the database used, which may contain information that affects the learning and predictions of the algorithms. Generally, ANN models often make simplifying assumptions about the input data, which may not fully account for the real complexity of meteorological phenomena. These ANN trained on this specific dataset may also not generalize well to other climatic or geographical conditions. The variability of the rain drop size distribution, the attenuation and calibration effects of X-band radars, and the influence of environmental factors can all affect the performance of the algorithms. Despite the various limitations and potential sources of error, the results of this study remain significant.

This study has shown that rainfall rate retrieval algorithms, based on measurements from an X-band polarimetric radar and designed using an ANN, outperform those designed by empirical methods (multivariable linear regression), confirming the results of Xiao et Chandrasekar [59], Zhang et al. [26], Yo et al. [49], Zhang et al. [50], and Cheng et al. [51]. The proposed best algorithms consist of specific differential phase shift ($K_{dp}$), differential reflectivity ($Z_{dr}$), Rayleigh reflectivity ($Z$), horizontal reflectivity ($Z_h$), and horizontal specific attenuation ($A_h$). This agrees with the results of Gosset et al. [19], who suggested these variables or their combinations for accurate rainfall intensity retrieval using X-band polarimetric radar measurements in the study area.

The study also demonstrated that the best tri-variable algorithms outperform the best bi-variable algorithms since using three variables reduces both the uncertainty associated with the DSD and the variability in droplet shape [33,34,60]. The two best tri-variable algorithms proposed contain the pair $\left(Z_{dr},K_{dp}\right)$. Several authors [19,21,33,60] have shown that algorithms containing this pair of variables perform better in rainfall intensity retrieval. One advantage of using $Z_{dr}$ in algorithms is its independence from the particle concentration parameter in DSD models [15]. Additionally, Seliga et Bringi [61] demonstrated that $Z_{dr}$ can be related to the median volumetric diameter of target particles. Gosset et al. [19] also explained that raindrops tend to flatten as they grow, leading to an increase in $Z_{dr}$ with rainfall rate. Regarding $K_{dp}$, some advantages of its use in algorithms are listed by Zahiri [15]. These include (i) its independence from attenuation effects of radar waves by the rainy medium; (ii) its independence from radar calibration issues since it is not derived from power measurements; and (iii) its low sensitivity to the presence of hail in rainfall.

The selected best tri-variable algorithms are $B2:R\left(Z_h,Z_{dr},K_{dp}\right)$ and $B3:R\left(A_h,Z_{dr},K_{dp}\right)$. One associates horizontal reflectivity $Z_h$ with the pair $\left(Z_{dr},K_{dp}\right)$, and the other associates horizontal specific attenuation $A_h$ with the pair $\left(Z_{dr},K_{dp}\right)$. The measurement of $Z_h$ is affected by attenuation $A_h$, which is a function of rainfall intensity [15]. Attenuation is also dependent on the radar wavelength, especially short-wavelength radars, such as X-band radars, which are heavily attenuated. Although attenuation $A_h$ is detrimental to $Z_h$ measurement, its quasi-linear relationship with rainfall intensity [15] makes it a good estimator of rainfall intensity. This explains why the algorithm $B3:R\left(A_h,Z_{dr},K_{dp}\right)$ outperforms algorithm $B2:R\left(Z_h,Z_{dr},K_{dp}\right)$.

In the study area, using the same data, Koffi et al. [21] established algorithms using the empirical method, using simulations or a portion of Xport and pluviograph measurements. They noted that the algorithm established with the triplet $\left(Z_h,Z_{dr},K_{dp}\right)$ from measurements is more efficient than other algorithms. This algorithm estimated instantaneous accumulations (5-min time step) from 2007 measurements, with $Ro=0.80$ and $KGE=0.54$. In this study, the algorithm $B2:R\left(Z_h,Z_{dr},K_{dp}\right)$ designed with ANN estimated instantaneous intensities (5-min time step) from a portion of the 2007 measurements, had $Ro=0.92$ and $KGE=0.81$. Hence, algorithm $B2:R\left(Z_h,Z_{dr},K_{dp}\right)$ is more efficient than the best one proposed by Koffi et al. [21]. Furthermore, the algorithm $B3:R\left(A_h,Z_{dr},K_{dp}\right)$ was designed with ANN estimated instantaneous intensities (5-min time step) from a portion of the 2007 measurements, with $Ro=0.98$ and $KGE=0.94$. Koffi et al. [21] fitted the $\left(Z_h,Z_{dr},K_{dp}\right)$ algorithm to six rainfall events in 2006 with a $Ro=0.79$. Algorithm $B3:R\left(A_h,Z_{dr},K_{dp}\right)$ assessed on the same database gives $Ro=0.97$.

Findings from the research highlighted the use of an ANN and optimal combinations of polarimetric variables for a more accurate estimation of rainfall intensity. Once we have an operational X-band radar, applying these results will provide precise rainfall information, which is essential for improving precipitation forecasts, mitigating the adverse effects of flash floods, and enhancing hydrological modeling. This information can be integrated into hydrological models to better understand and manage water resources, predict floods, and optimize watershed management systems. Deploying our algorithms in real-time weather radar systems presents several challenges.

The computational power required for real-time processing can be an issue, especially with limited hardware capabilities. The quality of training data is crucial to avoid inaccurate predictions, including the effective management of missing and noisy data. Additionally, the models must be robust to meteorological variations and radar data anomalies. Finally, regular maintenance and updates of the models are essential to improve performance and incorporate new data.

To apply our study to other regions with different climatic conditions, it is necessary to use a dataset specific to that region, revisit the selection of algorithms while optimizing the various ANN parameters, and refine the training process to achieve accurate rainfall intensity estimates. Additionally, it is crucial to validate the selected algorithms to ensure their relevance.

6. Conclusions

Following these studies, the present work focused on (i) identifying the best combinations of simulated polarimetric variables that can produce a better estimation of rainfall intensity R using an ANN (non-parametric method), (ii) examining the accuracy of the best algorithms designed with ANN with their empirical counterparts designed with multiple linear regression and (iii) validating the algorithms using measurements from the Xport radar and the rain gauge network.

The polarimetric variables studied include Rayleigh reflectivity ($Z$), horizontal reflectivity ($Z_h$), vertical reflectivity ($Z_v$), differential reflectivity ($Z_{dr}$), horizontal specific attenuation ($A_h$), vertical-specific attenuation ($A_v$), and specific differential phase shift ($K_{dp}$). The ANN used is the MLP. All possible combinations of bi-variable and tri-variable algorithms were experimented with. Some did not converge during the ANN training. Thus, twelve bi-variable algorithms and five tri-variable algorithms were assessed. Following the assessment, two bi-variable algorithms ($1:R(Z,K_{dp})$ and $A12:R(Z_{dr},K_{dp})$) and two tri-variable algorithms ($B2:R(Z_h,Z_{dr},K_{dp})$ and $B3:R(A_h,Z_{dr},K_{dp})$) were selected. The empirical relationship of these algorithms was established and compared to the ANN method. It was found that algorithms derived from ANN are more effective than those obtained using the empirical method. The comparison between the best algorithms derived from ANN showed that tri-variable algorithms outperform bi-variable algorithms. Using the empirical method, Koffi et al. [21] showed that the triplet $(Z_h,Z_{dr},K_{dp})$ was the best estimator of rainfall quantity. This study demonstrated that this algorithm is less effective than the $B2:R(Z_h,Z_{dr},K_{dp})$ algorithm derived from ANN. Finally, validation with measurements (Xport and rain gauge network) showed that the $B3:R(A_h,Z_{dr},K_{dp})$ algorithm is more effective than $B2:R(Z_h,Z_{dr},K_{dp})$.

In the future, it would be interesting to study the sensitivity of different algorithms to the variability of DSD and their parameters. It would also be beneficial to conduct this study using other ANN architectures or other non-parametric methods to identify the most optimal models for these algorithms. The problem of estimating the parameters of DSD from polarimetric variables could also be addressed using ANN.