1. Introduction

2. Materials and Methods

2.3. Dew Evolution

2.3.1. Energy Model

In order to compute the dewfall potential, a physical energy balance model (or “physical model” developed by [35] is used. The model needs only a few classical meteorological data collected at a given time step ∆t (here every hour): T_a (°C), RH (%) or equivalently dew point temperature T_d (°C), cloud cover (N, oktas), wind speed at 10 m elevation (V₁₀, m.s^-1). Dew yields in mm per unit of ∆t, noted $h_{\mathsf{\Delta }t}$ (mm. ∆t^-1), are calculated on the time step ∆t (expressed in h.) from the following formulation:

$$h_{\mathsf{\Delta }t}=\left(\frac{\mathsf{\Delta }t}{12}\right)\times \left(HL+RE\right)$$

(1)

The numerical factor ∆t/12 = 1/12 corresponds here to the data time step ∆t = 1h. Events with $h_{\mathsf{\Delta }t}$ > 0 correspond to condensation and $h_{\mathsf{\Delta }t}$ < 0 to evaporation. The latter are rejected. The quantity HL represents the convective heat losses between air and condenser, with a cut-off for wind speed V > V₀ = 4.4 m.s^-1 where condensation vanishes: HL = 0 if V > V₀. When V ≤ V₀, HL is expressed as:

$$HL=0.06\left(T_d-T_a\right)$$

(2)

The quantity RE in Eq. 1 is the available cooling energy by radiative deficit. Depending on the water content of air (measured by T_d, in °C), the site elevation H (in km) and the cloud cover N (in oktas), RE is evaluated by the following expression:

$$RE=0.37\times \left(1+0.204323H-0.0238893H^2-\left(18.0132-1.04963H+0.2189H²\right)\times {10}^3\times T_d\right)\left({\left(T_d+273.15\right)/285)}^4\times (1-N/8\right)$$

(3)

Daily time series corresponding to $h_{\mathsf{\Delta }t}$ > 0 are built after removing all data where rain events are present. The calculated cumulative yields can then be obtained by summing the data e.g on daily, monthly or yearly bases.

2.3.2. Perceptron Analysis for Extrapolation

Concerning the extrapolation of data one considers an approach by Multi-Layer Perceptron Artificial Neural Networks (MLP-ANN). It is a type of artificial neural network inspired by the functioning of the human brain [44,45]. Other methods are available such as Long Short-Term Memory (LSTM) Networks [46,47] or Spiking Neural Networks with Spike-Timing-Dependent Long Short-Term Memory (SNN-STLSTM) [48,49].

Each methods exhibit strengths and weaknesses. The MLP-ANN method is simple to implement and understand, can approximate any continuous function given enough neurons and data and is effective for a large variety of tasks such as classification and regression. However, they are not well-suited for tasks involving temporal sequences or data with time dependencies since they do not have a memory mechanism. They can also easily overfit to the training data if not properly regularized. The LSTM method is capable of learning long-term dependencies, making them worthy for time-series prediction, speech recognition, and natural language processing; it also mitigates the vanishing gradient problem, allowing for better learning over longer sequences. However, it exhibits more complex and computationally intensive work compared to simpler models like MLPs. Training times are longer due to their complexity and the need for sequence processing. The SNN-STLSTM approach can well mimic biological neural processing, which can be more efficient for certain tasks. It can also precisely capture timing information, which is critical for tasks requiring high temporal resolution. It shows a better energy efficiency, especially when implemented on neuromorphic hardware. It is, however, more challenging to implement and train compared to traditional neural network. Training SNNs often requires specialized algorithms and can be less straightforward. In addition, it exhibits some hardware dependency concerning the benefits in energy efficiency.

One here uses the simplest MLP-ANN approach by sake of simplicity and also because such an approach is currently used to predict meteorological variables such as solar radiation prediction [50], rainfall / evapotranspiration [51], air quality monitoring [52] or temperature [53]. More important, the method was already used to specifically estimate dew [54]. The multi-layer perceptron is thus a set of interconnected neurons [55,56,57,58]. Information flows from input to output without backtracking [58]. It is composed of three distinct layers (Figure 2). The first layer or the input layer is formed by the input data on a hourly basis: T_a (°C), T_d (°C), RH (%), V (m.s^-1) and N (oktas). These data, known to correspond to the parameters that determine the dew amount [4,35,54], are introduced in the MLP-ANN on a monthly basis. The use of T_a, RH and T_d is somewhat redundant as they are related by analytical equations but it increases the accuracy. The second layer is the hidden layer to prepare the data using activation functions in their neuron to present it in the last layer, the output layer, which represents the dew yield h (mm.mth^-1) output.

For the MLP-ANN network, a back-propagation algorithm is used. The back-propagation algorithm consists in forward-flowing the input data until a network-calculated input is obtained, and then comparing the calculated output to the known actual output. The weights are modified such that at the next iteration the error made between the calculated output is minimized. Taking into consideration the presence of the hidden layers, the error is back propagated backwards to the layer input while changing the weights. The process is repeated on all the data until the output error can be considered as negligible [59]. The package R interface for 'H2O' [60] has been used in this study, with the hyperbolic tangent function as an activation function. This function is indeed monotonic and having an identity for 0. It also allows normalization to be applied to the input values. This improves the conditioning of the optimization problem. If some inputs are systematically too large compared to others, they will have a disproportionate contribution to the error gradient, which will prevent the network from using the other variables. In addition, these variables will tend to saturate the hidden units at the start of training, which slows it down. This activation function also allows rapid learning to be performed because it initializes the weights randomly, which makes the model more efficient.

Figure 2. MLP-ANN configurations for monthly dew yields prediction. When using a MLP-ANN, it is customary to add a value “1” called ‘neuron bias’ to the input of a neuron. The bias is a kind of local weight that is used in several activation functions [61].

Figure 2. MLP-ANN configurations for monthly dew yields prediction. When using a MLP-ANN, it is customary to add a value “1” called ‘neuron bias’ to the input of a neuron. The bias is a kind of local weight that is used in several activation functions [61].

The multi-layer perceptron can give a projection from 2023 to 2033. The corresponding approach is summarized in Figure 3.

To predict the future value of the rain, we used the package ‘nnfor’ in R software (latest version 0.9.9, published on 2023-11-15). The package is designed for time series and univariate data like rain [62,63]. It is an automatic time series modeling, that is it automatically adds an activation function and other neurons to the input layer. With such a package, time series forecasting can be performed with multilayer perceptrons and extreme learning machines. Different model architectures were tried, from no hidden layers to 10 hidden layers. The minimum number of hidden layers to fit data was found to be one.

3. Results

3.3. Rain Evolution

3.3.1. Years 1/1991-7/2023

The rainfall evolution in the period 1/1991-7/2023 is shown in Figure 10 for Toliara and Ifaty (same data, see section 2.2) and Andremba. Without surprises, the rain amplitudes and evolutions are similar in all sites. One notes obvious differences between the dry and rainy seasons (analyzed below in Figure 11) and the presence of large peaks related to cyclone events. Between 1991 and 2000 is observed an important rise from ~ 25 to ~ 60 mm.mth^-1, then a long decrease until 2018-2020 with a rise until 2023. As noted in section 3.2.1 (Figure 5), dew and rain are seen to follow similar evolutions. Rain (together with nearby sea evaporation) indeed provides the atmosphere with large relative humidity needed to condense water vapor [66].

Figure 10. Evolution of the monthly rainfalls. (a,b): Training, validation and extrapolation of rain data. The black bold curves are data smoothening and the black lines are linear fits to all data in the period 1/1991-8/2033 (x = date-1/1/1904 in s). (a) Ifaty and Toliara, (b) Andremba. Grey lines: Actual rain data 01/1991 – 07/2023. Blue lines: ANN training period 01/1991 – 12/2016; red lines: ANN validation period 01/2017 – 07/2023; green lines: ANN extrapolation period 08/2023 – 07/2033. (a’, b’): Difference between ANN and actual values data values for (a’) Ifaty and Toliara, (b’) Andremba.

Figure 10. Evolution of the monthly rainfalls. (a,b): Training, validation and extrapolation of rain data. The black bold curves are data smoothening and the black lines are linear fits to all data in the period 1/1991-8/2033 (x = date-1/1/1904 in s). (a) Ifaty and Toliara, (b) Andremba. Grey lines: Actual rain data 01/1991 – 07/2023. Blue lines: ANN training period 01/1991 – 12/2016; red lines: ANN validation period 01/2017 – 07/2023; green lines: ANN extrapolation period 08/2023 – 07/2033. (a’, b’): Difference between ANN and actual values data values for (a’) Ifaty and Toliara, (b’) Andremba.

Table 6. Statistical analysis of the rain data according to the Mann-Kendall (MK) and the Sen's slope tests.

Table 6. Statistical analysis of the rain data according to the Mann-Kendall (MK) and the Sen's slope tests.

Rain	Data	mths.	Min (mm.mth-1)	Max (mm.mth-1)	Mean (mm.mth-1)	SD (mm.mth-1)	p-value*	MK mea- ningful trend$	Sen’s slope (×10-5 mm.mth-2)	Sen’s con- stant
Ifaty Toliara	Meas.	391	0	455.6	42.343	73.426	0.244	No	-11.9	15.791
	Extrap.	120	0	307.0	53.265	71.611	0.738	No	0	19.793
	All	511	0	455.6	44.907	73.081	0.496	No	5.1	10.790
Andr-emba	Meas.	391	0	435.5	48.998	74.108	0.377	No	-11.8	19.225
	Extrap.	120	0	227.0	56.489	61.977	0.819	No	45.8	4.643
	All	511	0	435.5	50.757	71.457	0.102	No	27.0	6.937

^* Fraction of tied observations. ^$ p < 0.05.

Figure 11 presents the rainfall yields in the dry season (Apr.-Oct.) and rainy season (Nov.-Mar.). There is obviously a large difference in the volumes of the precipitations, with a ratio ~8 , the mean precipitation rate being in the dry season 61 mm.season^-1 and 485 mm.season^-1 for the rainy season. The minimum rainfall is 18 mm in 2021 (dry season, all sites) and the maximum is 850 mm in 1998 (Andremba) and 2022 (Ifaty-Toliara).

As for dew (see section 3.2.1), the frequency of consecutive rainy days without rain can be well represented by an exponential (Eq. 4; Figure 12). For all sites, the amplitude $f_0$is much larger in the rainy seasons (~ 300-350) than in the dry season (~ 45-100). Unsurprisingly, the average number of days without rain N_c0 is also much greater in the dry season (6-10 days) than in the rainy seasons (~ 2.45 days).

The evolution of N_c0, is reported in Figure 13 for the dry season (Apr. - Oct.) and the rainy season (Nov. - March). One obviously observes a much smaller value (about a factor of 1:3.3) in the rainy season (~ 4 days in Ifaty-Toliara, ~ 3 days in Andremba, see Table 7) than in the dry season (~ 14 d in Ifaty-Toliara, ~ 9.5 d in Andremba, see Table 7). The statistical tests (Table 7) do not allow trends to be detected in the observation period 1/1991-7/2023.

3.3.2. Extrapolation 8/2023-7/2033

The monthly data of the period 1/1991 - 7/2023 are extrapolated to the period 8/2023 -7/2033 according to the procedure using MLP-ANN, as described in section 2.3.2 and more precisely the package ‘nnfor’ in R software. This package was used because it is specially designed for time series and univariate data like rain.

As for dew, the procedure follows a period of training with 77% of the data (1/1991 – 12/2016) and a period of validation corresponding to 23% of the data (1/2017 – 7/2023). In Figure 10aa’bb’ the training and validation data at Ifaty-Toliara and Andremba compare well with the measured rain data.

The monthly dew yields are reported in Figure 10, with smoothening curves for visual aid. In order to determine a trend over the observed period (1/1991-7/2023), the extrapolated period (8/2023-7/2033) and the whole period (1/1991-7/2033), the MK and Sen’s slope statistical methods are applied in Table 7. Trends are found to be not statistically valid for any periods at all sites. Linear fits of the data (Figure 9d) on the whole period 1991 – 2033 are in agreement with this analysis, giving the slopes with a large SD (2.3 ± 8.5)×10^-9 mm.mth^-1.s^-1 (Ifaty-Toliara ), and (1 ± 8)×10^-9 mm.mth^-1.s^-1 (Andremba). The standard deviations of the values are quite large, which cast some doubts about the actuality of the slopes. As a matter of fact, the statistical quality of this trend is not assessed by the MK and Sen’s slopes methods (Table 6) in any sites. While the observed positive and negative evolutions during the period are clearly observed (see Figure 10), they cancel each other when looking to a mean trend.

4. General Discussion

5. Conclusions

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