Prediction of Daily River Discharge to Estuaries Based on Meteorological Data

Abstract

A methodology is proposed to predict the daily river discharge (RD) to estuaries from rivers draining in similar temperate zones. Multiple regression models are proposed to estimate RD using only available meteorological data. The models are based on monthly air temperature (T) and recent (P_R) and non-recent (P_NR) atmospheric precipitation (rainfall). They consist of the linear and nonlinear terms of T, P_R, and P_NR, without interaction terms between them. Four rivers located in the north and centre of Portugal (flowing to the Atlantic Ocean) are used in this study—Vouga, Antuã, Neiva, and Mondego. The optimal period used to compute the recent precipitation history is between 4 and 7 days for Vouga, Antuã, and Mondego and is 11 days for Neiva. The recommended lag to compute the non-recent precipitation history is between 50 and 90 days. The optimisation of the lengths of recent and non-recent periods improved the model performance, compared with previously proposed models with interaction terms between the meteorological variables. The obtained models provide a clear interpretation of the impact that meteorology has on RD. All rivers showed similar responses, but the flows of bigger rivers (Vouga, Mondego) were more significantly affected by precipitation and temperature. The proposed models are useful for analysing biogeochemical processes in rivers and estuaries, as well as for assessing flood and drought risks in sensitive areas.

1. Introduction

River fluxes to estuaries are responsible for the transport of particulate matter as well as the riverine export of contaminants to the coastal ocean [1]. Daily river discharge (RD) recordings may be interrupted, and consequently, recent data on RD are often not available, posing great challenges in hydrological studies. The accurate simulation of RD demands a thorough characterisation of the most significant hydrological processes acting in catchment areas and therefore requires a substantial amount of data on topography, geology, hydropedology, land cover, and climate [2,3,4]. Also, a solid conceptual model is necessary to obtain realistic results. However, detailed information regarding the catchment features may not be available. For example, the application of fully physical-based models may require extensive field surveys to quantify soil, vegetation, and geological features. Moreover, such surveys are frequently only feasible in small and extensively instrumented experimental catchments [5].

The above arguments have motivated the development of baseline models, which are less input-demanding but are still capable of providing satisfactory results [6,7,8]. In these models, RD is estimated from historical data of inputs (precipitation, air temperature) and past values of the output (RD) if they are available [9].

However, on a daily time scale, RD may have large variations, and its estimation could be a challenging task. This aspect may be critical if the discharge measurements are made once a day [10]. To overcome this difficulty, one may use larger periods, such as monthly RD. For instance, monthly RD was correlated with monthly precipitation in the Yangtze river [11]. Similarly, monthly flow was related to temperature and precipitation through linear, quadratic, and interaction terms [12].

The previous works that address this topic may assume that historical RD data are continuously available (through measurements). Regression models counting on historical RD data as explanatory variables are more accurate; however, the installation and maintenance of costly complex sensor systems are required.

In order to overcome these limitations, in our previous works [1,8], we proposed a methodology (multiple regression models) to estimate highly variable daily downstream flows from the Vouga and Antuã rivers based on daily precipitation and average monthly temperatures (T) and omitted the use of historic RD as an explanatory variable. We proposed a separation of the precipitation history into recent and non-recent events prior to the day of RD estimation, which can account for the high and low values of RD. The proposed model is simple and robust; however, the interaction terms between the input variables limit its physical interpretation.

To make the model more transparent and explainable to its final users, in the present study, we propose a modified model for daily RD forecasting based on meteorological data only and without interaction terms between T and recent (P_R) and non-recent (P_NR) precipitation events. The hypothesis is that by omitting interaction terms, the model forecasting may deteriorate; however, it will be compensated by optimising the length of recent (R) and non-recent (NR) precipitation periods. The RD prediction models were tuned with data from four rivers in the north and centre of Portugal, selected as case studies. The proposed models can be useful in biogeochemical studies of coastal contamination and the export of pollutants from rivers to the ocean (e.g., for the Vouga, Antuã, and Mondego rivers) or in ecological studies of pristine systems (e.g., for the Neiva river).

2. Methods

2.1. Study Regions

All rivers have their estuaries on the Atlantic coast of Portugal. Two relatively large rivers (Vouga, Mondego) and two small rivers (Antuã, Neiva) were selected as case studies. According to Agencia Estatal de Meteorologia (Spain) and Instituto de Meteorologia (Portugal) [13], the four studied rivers correspond to Csb catchment Köppen–Geiger climate classification, which corresponds to a temperate dry warm summer climate.

The Vouga river (Centre of Portugal) is the main freshwater source of a biologically productive lagoon (Aveiro Lagoon). The average daily RD is 50 m³ s⁻¹, which represents 2/3 of the freshwater entering the lagoon [14]. The Vouga river catchment area is around 3362 km². The Vouga river source is located at 930 m a.s.l. in the Lapa Mountain and has a total length of around 141 km.

The Antuã river (Centre of Portugal) is the second largest source of the Aveiro Lagoon, with an average flow of only 4–5 m³ s⁻¹. The drainage area is only 149 km², the river length is 38 km and its source in Romariz is at 400 m a.s.l. [8,15]. Although the Antuã river is a less significant freshwater source of the lagoon, it is more important than the Vouga river for the export of contaminants from Laranjo Bay [1]. Laranjo Bay is part of the Aveiro Lagoon, which is industrially contaminated, namely with mercury/methylmercury [16,17] and other heavy metals [18]. The Antuã river is also a significant contributor of nutrients [15] and sewage [19] to the Aveiro Lagoon.

The Neiva river (North of Portugal) is a small river with a length of 46 km, having a 240 km² drainage area and an average RD of only 3.3 m³ s⁻¹ [20,21]. Its source (Serra de Oural) is at 720 m a.s.l. The estuary is situated in Castelo de Neiva, between Esposende and Viana do Castelo. There are only agriculture and forest areas and minimal industry at its margins [22]. The river is clean and hosts different populations of species previously considered extinct in Portugal [22,23].

The Mondego river (Centre of Portugal) is the longest (227 km) river situated entirely in Portuguese territory. Its source is in Serra da Estrela at 1547 m a.s.l. and its estuary is situated in Figueira da Foz, with an average discharge of 80 m³ s⁻¹ [24]. The Mondego river has a large drainage area (6670 km²). The Mondego estuary is impacted by agriculture runoffs from upstream farmlands that could lead to eutrophication [19,24].

2.2. Data Collection

All historical data were taken from the Sistema Nacional de Informação de Recursos Hídricos, Portugal (http://snirh.pt/, accessed on 30 November 2013) and summarised in

Table 1. Hydrological (for RD) and meteorological stations (for P and T) used for the Vouga, Antuã, Neiva, and Mondego rivers (data collection).

River	Stations for RD	Stations for P	Stations for T
Vouga	09H/01H; 10F/02H	08G/01C	08G/01C
Antuã	09F/01H	08G/01C	08G/01C
Neiva	04E/06H	04E/03UG	04G/06C; 04F/01C
Mondego	12E/01H; 13E/04H; 13F/02H; 13F/04H; 13F/01H; 12G/04H	11J/02C; 12F/02C; 13H/06C; 09K/01G; 10M/01G; 11I/01G; 13F/01G; 13I/01G; 09L/01UG; 10I/01UG; 10K/01UG; 11H/01UG; 11J/01UG; 11K/01UG; 11L/01UG; 12F/01UG; 13G/01UG; 13J/01UG; 14F/01UG; 11L/07CG	12F/02C; 13H/04C; 13H/06C

Note: Data from the Sistema Nacional de Informação de Recursos Hídricos (http://snirh.pt/, accessed on 30 November 2013).

. The downstream RD of the Vouga river was determined as the sum of the discharge information from Pedre Ribeiradio (09H/01H station located on the Vouga river) and Ponte Requeixo (10F/02H station located on the Agueda river) for the period of 19 February 1978–30 September 1980 (see Figure 1). The RD of the Antuã river was extracted from 09F/01H station for the period of 1 January 1985–30 April 1989. Meteorological data for P (mm) and T (°C) were taken from Barragem de Castelo Burgães (08G/01C station) for the same periods. The RD for the Neiva river was extracted from 04E/06H station for the period of 1 October 1984–30 September 1989. Data from the meteorological station 04E/03UG (for P) and stations 04G/06C or 04F/01C (for T) was used in model fitting. Having a large catchment area, the RD for the Mondego river was estimated for the period of 10 November 1972–31 July 1976 as the sum of RD data from the Mondego river and some of its tributaries (stations 12E/01H, 13E/04H, 13F/02H, 13F/04H, 13F/01H, 12G/04H).

Precipitation data (P) was obtained as an average from the stations 11J/02C, 12F/02C, 13H/06C, 09K/01G, 10M/01G, 11I/01G, 13F/01G, 13I/01G, 09L/01UG, 10I/01UG, 10K/01UG, 11H/01UG, 11J/01UG, 11K/01UG, 11L/01UG, 12F/01UG, 13G/01UG, 13J/01UG, 14F/01UG, and 11L/07CG. Temperature data (T) was obtained as an average from the following stations: 12F/02C, 13H/04C, and 13H/06C. In the case of missing values (for example, 3 missing values for station 11L/07CG), these were estimated using data from other stations based on the highest correlation. All data for RD, T and P for the four studied rivers are stored in Supplementary Materials (Excel files) and publicly available.

2.3. Statistical Methods

Statistical data treatment was carried out using R software 3.4.2 [25]. In the regression models, the precipitation history is separated into recent and not recent events. The recent precipitation events (P_R) represent daily precipitations P, averaged over R days before the day for which the RD is predicted. The not-recent precipitation events (P_NR) are daily precipitations P, averaged over NR days prior to the P_R period.

The dependent variable (RD) is represented as a function of the independent (explanatory) variables P_R, P_NR, and T. Preliminary data analysis and the graphical visualisation of density functions show that T has a normal distribution; however, P_R, P_NR, and RD are left-skewed to small values. Since linear regression models are sensitive to not-normal data distribution [26,27], we applied natural logarithm (for RD) and square root transformations (for P_R, P_NR).

$$\ln \mathrm{R}\mathrm{D}=a_0+a_1{\mathrm{P}}_{\mathrm{R}}^{1/2}+a_2{\mathrm{P}}_{\mathrm{N}\mathrm{R}}^{1/2}+a_3\mathrm{T}+a_{\mathrm{1,1}}{\left({\mathrm{P}}_{\mathrm{R}}^{1/2}\right)}^2+a_{\mathrm{2,2}}{\left({\mathrm{P}}_{\mathrm{N}\mathrm{R}}^{1/2}\right)}^2+a_{\mathrm{1,1},1}{\left({\mathrm{P}}_{\mathrm{R}}^{1/2}\right)}^3+a_{\mathrm{2,2},2}{\left({\mathrm{P}}_{\mathrm{N}\mathrm{R}}^{1/2}\right)}^3$$

(1)

In Equation (1), $a_0$ is the intercept coefficient, and $a_1$, $a_2$, and $a_3$ are the coefficients of the simple (linear) effects of the transformed precipitation variables (${\mathrm{P}}_{\mathrm{N}\mathrm{R}}^{1/2}$, ${\mathrm{P}}_{\mathrm{R}}^{1/2}$) and the temperature T. There are also quadratic (with coefficients $a_{\mathrm{1,1}}$ and $a_{\mathrm{2,2}}$) and cubic (with coefficients $a_{\mathrm{1,1},1}$ and $a_{\mathrm{2,2},2}$) terms with respect to the transformed precipitation variables (only in case of significant simple effects). The models were fitted using all available data or only a training subset consisting of the first half of the studied temporal period, while the second half was left for model validation. The variance inflation factor (vif) for T, ${\mathrm{P}}_{\mathrm{N}\mathrm{R}}^{1/2}$, ${\mathrm{P}}_{\mathrm{R}}^{1/2}$ in the fitted models is less than 3, which means the transformed explanatory variables are slightly to moderately correlated.

The precipitation terms (P_R and P_NR) were calculated for different lengths of recent (R, 1–20 days) and non-recent (NR, 10–90 days) periods, preceding the day for which RD is to be estimated. The optimal lengths of R and NR were determined by comparison of the model predictions (RD_MODEL) and the real measurements (RD) for the complete time series and for the training subset. The evaluation metrics were the maximum of adjR² and the minimum of root mean square deviation (RMSD) of the log-transformed RD. Furthermore, the linear correlation between the model and the real measurements of RD was assessed through the slope c₁ (which has to be approximately 1) and the intercept c₀ (which has to be close to 0). After the optimal lengths of R and NR were determined, the models were fitted and simplified by leaving only the terms with coefficients significantly different from 0 (p < 0.05) by the gradual removal of the non-significant terms and starting with the higher order (quadratic and cubic) effects [26]. The stability of the coefficients of the minimally adequate models obtained was studied using the bootstrap method with row resampling. If the coefficient does not change the sign between P_2.5 and P_97.5 percentiles, it means that the effect of the associated explanatory variable on RD is stable [27]. As an example of the computation method, a txt file with R code (the Mondego river) is provided in the Supplementary Materials.

2.4. Validation of the Results

The best models were compared in terms of their success and failure characteristics. The criteria for successes (S) and fails (F) of the model were determined from the ratio between the maximum and minimum value of RD in the dataset and were different for large RD variation (the Vouga and Mondego rivers) and small RD variation (the Antuã and Neiva rivers) scenarios. For large RD variation, the model is successful if the predicted RD values are not more than two times higher or smaller than the real values. The model fails if the predicted RD values are more than four times higher or smaller than the real values. For small RD variations, the model is successful if the model RD values are not more than 1.5 times higher or smaller than the real values. The model fails if the model RD values are more than three times higher or smaller than the real values.

The root mean square deviation (RMSD) values for the log-transformed RD and the original RD were calculated as follows (n is the number of examples, i.e., the number of data rows):

$${RMSD}_T=\sqrt{\sum {\left(\ln {\mathrm{R}\mathrm{D}}_{\mathrm{M}\mathrm{O}\mathrm{D}\mathrm{E}\mathrm{L}}-\ln \mathrm{R}\mathrm{D}\right)}^2/n}$$

(2)

$$RMSD=\sqrt{\sum {\left({\mathrm{R}\mathrm{D}}_{\mathrm{M}\mathrm{O}\mathrm{D}\mathrm{E}\mathrm{L}}-\mathrm{R}\mathrm{D}\right)}^2/n}$$

(3)

The methodology is summarised in Algorithm 1:

Algorithm 1: Methodology Flow Chart

1. Collect data (RD, T, P) 2. Compute recent and non-recent variables (PR, PNR) 3. Normalise PR, PNR, and RD 4. Divide data (train, validation) 5. Model training (choose the model hyperparameters—recent and non-recent periods) 6. Removal of nonsignificant terms 7. Model Validation

3. Results and Discussion

Average values, median, quartiles (q₁, q₃) and range for T, P_R, P_NR and RD are given in

Table 2. Average values (bold), median (m), quartiles (q₁, q₃), and range (in italic) for monthly air temperature (T), recent (P_R) and non-recent (P_NR) precipitation, river discharge (RD), and increase (or decrease) in RD on a given day compared to previous day, either in absolute terms |ΔRD| or relative terms |ΔRD/RD|. Statistics regarding training and validation set for the Vouga, Antuã, Neiva and Mondego rivers.

	T (°C)	PR (mm Day−1)	PNR (mm Day−1)	RD (m3 s−1)	|ΔRD| (m3 s−1)	|ΔRD/RD|
Vouga, Training a 27 March 1978–21 April 1979	14.2; m: 12.8 (9.1–22.3) q1: 10.6, q3: 17.0	6.91; m: 1.72 (0–48.32) q1: 0, q3: 9.40	6.97; m: 5.68 (0–19.42) q1: 1.77, q3: 13.06	99.8; m: 34.0 (0.71–1491.3) q1: 3.24, q3: 91.9	24.9; m: 1.25 (0–735.9) q1: 0.18, q3: 10.2	0.123; m: 0.072 (0–1.39) q1: 0.041, q3: 0.137
Vouga, Validation a 12 June 1979–30 September 1980	16.0; m: 14.7 (9.7–22.3) q1: 11.1, q3: 19.7	3.64; m: 0.76 (0–27.22) q1: 0, q3: 4.66	3.80; m: 3.88 (0.10–10.61) q1: 1.20, q3: 5.75	32.8; m: 17.8 (0.44–267.4) q1: 1.96, q3: 47.6	4.40; m: 0.77 (0–119.4) q1: 0.10, q3: 3.53	0.090; m: 0.058 (0–0.61) q1: 0.034, q3: 0.113
Antuã, Training b 16 March 1985–12 January 1987	15.0; m:15.4 (8.3–22.4) q1: 10.1, q3: 18.5	3.98; m: 0.40 (0–42.25) q1: 0, q3: 4.53	4.15; m: 3.26 (0.28–13.00) q1: 1.53, q3: 7.06	2.84; m: 1.79 (0.17–29.7) q1: 0.76, q3: 3.76	0.349; m: 0.070 (0–16.2) q1: 0, q3: 0.200	0.088; m: 0.043 (0–8.07) q1: 0, q3: 0.099
Antuã, Validation b 28 March 1987–30 April 1989	15.2; m: 14.0 (9.5–22.4) q1: 12.0, q3: 18.5	4.63; m: 0.90 (0–40.33) q1: 0, q3: 6.15	4.59; m:4.57 (0.34–12.46) q1: 1.86, q3: 6.58	3.86; m: 2.88 (0.17–45.1) q1: 1.35, q3: 4.82	0.383; m: 0.100 (0–25.5) q1: 0.030, q3: 0.240	0.069; m: 0.039 (0–1.53) q1: 0.014, q3: 0.077
Neiva, Training c 24 October 1984–14 January 1987	14.2; m: 13.6 (8.6–20.5) q1: 10.4, q3: 19.3	4.97; m: 2.73 (0–27.00) q1: 0.27, q3: 7.14	4.99; m: 4.67 (0.36–11.84) q1: 2.36, q3: 7.92	3.18; m: 2.92 (0.08–14.5) q1: 1.68, q3: 4.20	0.330; m: 0.130 (0–8.07) q1: 0.070, q3: 0.280	0.117; m: 0.055 (0–5.20) q1: 0.027, q3: 0.130
Neiva, Validation c 9 April 1987–30 September 1989	15.9; m: 15.5 (8.7–22.7) q1: 12.2, q3: 19.8	3.10; m: 1.32 (0–21.09) q1: 0, q3: 4.80	3.32; m: 3.10 (0.11–8.57) q1: 1.43, q3: 4.62	2.74; m: 2.30 (0.06–20.9) q1: 1.26, q3: 3.46	0.267; m: 0.090 (0–16.9) q1: 0.040, q3: 0.210	0.098; m: 0.047 (0–3.50) q1: 0.021, q3: 0.111
Mondego, Training d 10 November 1972–11 July 1974	12.8; m: 11.9 (6.9–21.4) q1: 9.3, q3: 17.4	3.34; m: 1.26 (0–24.28) q1: 0.07, q3: 4.67	3.35; m: 2.96 (0.16–8.81) q1: 1.77, q3: 4.74	78.7; m: 36.4 (0.26–1160.7) q1: 11.6, q3: 91.0	18.3; m: 2.13 (0–918.2) q1: 0.464, q3: 10.4	0.153; m: 0.086 (0–2.93) q1: 0.040, q3: 0.184
Mondego, Validation d 06 September 1974–31 July 1976	13.5; m: 12.2 (7.4–20.9) q1: 9.9, q3: 17.2	1.93; m: 0.57 (0–17.82) q1: 0.02, q3: 2.56	1.91; m: 1.76 (0.04–6.77) q1: 0.67, q3: 2.63	34.2; m: 17.0 (0.41–445.7) q1: 4.82, q3: 29.2	5.51; m: 0.674 (0–308.3) q1: 0.250, q3: 2.13	0.096; m: 0.062 (0–0.963) q1: 0.029, q3: 0.123

Note: ^a R = 5 days; NR = 47 days; ^b R = 4 days; NR = 71 days; ^c R = 11 days; NR = 74 days; ^d R = 6 days; NR = 51 days.

for the training set and the validation set. The change in RD on a given day compared to the previous day is presented either in absolute RD difference |ΔRD| or as a relative deviation |ΔRD/RD|. Despite being in the same climatic zone, slight differences between the studied rivers are observed. The average temperature values are between 14.2 and 16.0 °C for the Vouga, Antuã and Neiva rivers, but lower T values are measured (12.8–13.5 °C) near the Mondego river. Similarly, higher average daily precipitations were found in the drainage basin of the Vouga, Antuã and Neiva rivers (3.1–7.0 mm) than near the Mondego river (1.9–3.4 mm). A possible reason for such a difference is that the data for Mondego belongs to periods considered wet in Europe (e.g., as in the Rhine basin) and the periods studied for the Antuã, Vouga and Neiva rivers are considered dry in Europe based on a global climate study [28]. The weather during the training set was wetter for the Neiva, Mondego and, in particular, Vouga rivers than for the validation set.

Average daily RD (m³ s⁻¹) for the Vouga (32.8–99.8), Antuã (2.84–3.86), Neiva (2.74–3.18) and Mondego (34.2–78.7) rivers are similar to average RD reported previously for these rivers [14,15,21,24], The values mainly depended on the drainage basin size [29]. The ratio between RD during flood events (RD_MAX) and medium RD (Table 2) is higher for the larger rivers Vouga (15.0–43.9) and Mondego (26.2–31.9) than for the smaller Antuã (15.7–16.6) and, in particular, Neiva rivers (5.0–9.1). Generally, that ratio is between 5 and 30, as expected for rivers in a temperate environment [29].

Huge variation in RD is observed even on a day-to-day basis, as marked by the large change in RDs on a given day compared to the previous day (ΔRD). On average, ΔRD is between 9.0% and 15.3% from RD for the Vouga, Neiva, and Mondego rivers (Table 2) and slightly lower for the Antuã river (6.9–8.8%). High values of |ΔRD/RD| occur at high RD values for the Vouga during winter (|ΔRD| is up to 1.4 times larger than RD) and at low RD values during summer for the Neiva (|ΔRD| is up to 5.2 times larger than RD), although the dependence for the Antuã and Mondego rivers is not clear (Figure 2).

3.1. Multiple Regression Model

Figure 3 shows the models vs. the actual RD measurements (validation set) for the four rivers, demonstrating that the predictions differ on average between 60% and 72% from the real values. The model accounts for the huge variation (more than two or even three orders of magnitude) in the daily RD for the validation set well, but is less reliable for explaining low RD, especially for the Neiva and Mondego rivers.

The coefficient estimates with bootstrap ranges, RMSD and RMSD_T values are presented in support materials provided online (ESM_Equations.doc file). The obtained models, fitted with either all data or the training subset only, are summarised in

Table 3. Multiple regression models (Equation (1), fitted with all data or only with the training set) for RD of the Vouga, Antuã, Neiva and Mondego rivers. adjR², the slope (c₁) and the intercept (c₀) for the dependence between model values and experimental RD values.

	Final Fitted Models	adjR2	c1	c0
Vouga All data	$\ln \mathrm{R}\mathrm{D}=a_0+a_1{\mathrm{P}}_{\mathrm{R}}^{1/2}+a_2{\mathrm{P}}_{\mathrm{N}\mathrm{R}}^{1/2}-\left|a_3\right|\mathrm{T}+a_{\mathrm{1,1}}{\mathrm{P}}_{\mathrm{R}}-\left|a_{\mathrm{2,2},2}\right|{\mathrm{P}}_{\mathrm{N}\mathrm{R}}^{3/2}$	0.919	1.179 ± 0.019	−7.31 ± 2.98
Vouga Training	$\ln \mathrm{R}\mathrm{D}=a_0+a_1{\mathrm{P}}_{\mathrm{R}}^{1/2}+a_2{\mathrm{P}}_{\mathrm{N}\mathrm{R}}^{1/2}+a_{\mathrm{1,1}}{\mathrm{P}}_{\mathrm{R}}+a_{\mathrm{2,2}}{\mathrm{P}}_{\mathrm{N}\mathrm{R}}-\left|a_{\mathrm{2,2},2}\right|{\mathrm{P}}_{\mathrm{N}\mathrm{R}}^{3/2}$	0.949	0.987 ± 0.021	2.39 ± 4.75
Antuã All data	$\ln \mathrm{R}\mathrm{D}=a_0+a_1{\mathrm{P}}_{\mathrm{R}}^{1/2}+a_2{\mathrm{P}}_{\mathrm{N}\mathrm{R}}^{1/2}-\left|a_3\right|\mathrm{T}+a_{\mathrm{1,1},1}{\mathrm{P}}_{\mathrm{R}}^{3/2}-\left|a_{\mathrm{2,2},2}\right|{\mathrm{P}}_{\mathrm{N}\mathrm{R}}^{3/2}$	0.810	0.675 ± 0.013	0.887 ± 0.065
Antuã Training	$\ln \mathrm{R}\mathrm{D}=a_0+a_1{\mathrm{P}}_{\mathrm{R}}^{1/2}+a_2{\mathrm{P}}_{\mathrm{N}\mathrm{R}}^{1/2}-\left|a_3\right|\mathrm{T}+a_{\mathrm{1,1},1}{\mathrm{P}}_{\mathrm{R}}^{3/2}-\left|a_{\mathrm{2,2}}\right|{\mathrm{P}}_{\mathrm{N}\mathrm{R}}$	0.864	0.776 ± 0.017	0.495 ± 0.076
Neiva All data	$\ln \mathrm{R}\mathrm{D}=a_0+a_1{\mathrm{P}}_{\mathrm{R}}^{1/2}+a_2{\mathrm{P}}_{\mathrm{N}\mathrm{R}}^{1/2}-\left|a_3\right|\mathrm{T}+a_{\mathrm{1,1}}{\mathrm{P}}_{\mathrm{R}}-\left|a_{\mathrm{2,2},2}\right|{\mathrm{P}}_{\mathrm{N}\mathrm{R}}^{3/2}$	0.794	0.778 ± 0.012	0.578 ± 0.043
Neiva Training	$\ln \mathrm{R}\mathrm{D}=a_0+a_1{\mathrm{P}}_{\mathrm{R}}^{1/2}-\left|a_2\right|{\mathrm{P}}_{\mathrm{N}\mathrm{R}}^{1/2}-\left|a_3\right|\mathrm{T}-\left|a_{\mathrm{1,1}}\right|{\mathrm{P}}_{\mathrm{R}}+a_{\mathrm{1,1},1}{\mathrm{P}}_{\mathrm{R}}^{3/2}+a_{\mathrm{2,2}}{\mathrm{P}}_{\mathrm{N}\mathrm{R}}-\left|a_{\mathrm{2,2},2}\right|{\mathrm{P}}_{\mathrm{N}\mathrm{R}}^{3/2}$	0.854	0.917 ± 0.015	0.229 ± 0.056
Mondego All data	$\ln \mathrm{R}\mathrm{D}=a_0+a_1{\mathrm{P}}_{\mathrm{R}}^{1/2}+a_2{\mathrm{P}}_{\mathrm{N}\mathrm{R}}^{1/2}-\left|a_3\right|\mathrm{T}+a_{\mathrm{1,1},1}{\mathrm{P}}_{\mathrm{R}}^{3/2}$	0.802	0.950 ± 0.014	0.086 ± 1.560
Mondego Training	$\ln \mathrm{R}\mathrm{D}=a_0+a_1{\mathrm{P}}_{\mathrm{R}}^{1/2}+a_2{\mathrm{P}}_{\mathrm{N}\mathrm{R}}^{1/2}-\left|a_3\right|\mathrm{T}+a_{\mathrm{1,1},1}{\mathrm{P}}_{\mathrm{R}}^{3/2}-\left|a_{\mathrm{2,2}}\right|{\mathrm{P}}_{\mathrm{N}\mathrm{R}}+a_{\mathrm{2,2},2}{\mathrm{P}}_{\mathrm{N}\mathrm{R}}^{3/2}$	0.788	0.962 ± 0.020	0.695 ± 2.87
Vouga, All data R = 5 days; NR = 80 days; S = 84.9%, F = 1.24%; Training R = 5 days; NR = 47 days; S = 89.8%, F = 0.51%; Validation: S = 70.0%, F = 2.94% Antuã, All data R = 6 days; NR = 90 days; S = 66.2%, F = 1.14%; Training R = 4 days; NR = 71 days; S = 73.7%, F = 1.95%; Validation: S = 54.4%, F = 4.18% Neiva, All data R = 11 days; NR = 80 days; S = 79.0%, F = 1.95%; Training R = 11 days; NR = 74 days; S = 85.0%, F = 1.35%; Validation: S = 58.5%, F = 2.43% Mondego, All data R = 7 days; NR = 50 days; S = 72.5%, F = 5.37%; Training R = 6 days; NR = 51 days; S = 63.7%, F = 9.0%; Validation: S = 55.0%, F = 17.55% The sign in front of the absolute values of the coefficients shows the direction of the effect of respective explanatory variables on the dependent variable (RD).

. The length R for the recent precipitation history was between 4 and 7 days for the Vouga, Antuã and Mondego rivers and was 11 days for the Neiva river. The lengths NR of non-recent precipitation are in an order of 50 to 90 days. The success for the models is between 60% and 70% (for the Antuã and Mondego rivers) and between 80% and 90% (for the Vouga and Neiva rivers). Failure rates are approximately 1–2% (for the Vouga, Antuã, Neiva rivers) and 5–9% for the Mondego. A decrease in the model’s performance was observed for the validation set, with success rates of about 55% (for the Antuã, Neiva, and Mondego rivers) and 70% (for Vouga). The fail rates for the validation sets are approximately 2.4–4.2% (for the Vouga, Antuã, Neiva rivers) and 17.6% for the Mondego. The high fail rate of the Mondego is assumed to be due to variables not accounted for in the model [12]. Thus, the Mondego flow may be influenced by some other processes in its large drainage area. Such processes could be snowmelt in Serra da Estrela, affecting upper Mondego/left-side tributaries (e.g., the Alva river) [30,31], or anthropogenic influence from the construction of dams. For example, monthly discharge could be influenced by such human activities, decreasing the correlation with monthly precipitation [11,32]. The time series for actual RD and RD_MODEL (training and validation sets) for the four rivers is presented in Figure 4.

3.2. Model Limitations

Despite showing results with physical meaning, the multiple regression model presented here has important limitations, and analysis of fail situations for different rivers could shed light on the conditions under which these failures occur. For the validation sets, the more problematic failure situations for the Vouga and Antuã rivers occur when model values are lower than actual RDs (Figure 5). This situation occurs for high RD > 63.2 m³ s⁻¹ (Vouga) and medium to high RD > 1.69 m³ s⁻¹ (Antuã). In such conditions, fail rates increase to 15.9% (Vouga) and 8.2% (Antuã). On rare occasions, fails for both Vouga and Antuã occur for model values higher than actual RD, occurring at a low change in RD on a given day compared to the previous day (Figure 5).

For the Neiva, the model fails at low RD < 0.31 m³ s⁻¹ (resulting in higher model predictions than actual RD measurements), leading to a fail rate under these conditions of 56.7%. The same (Neiva) model fails at a high RD > 3.25 m³ s⁻¹, resulting in lower model predictions than an actual RD and leading to a fail rate of 19.2% under these conditions.

The Mondego only presented problems for lower model predictions than actual RD measurements. They occurred at low values of P_NR < 1.17 mm day⁻¹ leading to three times higher fail rates in these conditions (52.1%). Nevertheless, success and fail rates for all data of the Vouga river were better (adjR² = 0.919, S = 85%; F = 1.24%), compared to values of previous models with interaction terms [8] using the same data (adjR² = 0.888, S = 79%; F = 2.4% for R = 5 days and NR = 30 days). Not accounting for interaction terms between the explanatory variables improved the physical interpretation of the current model. Additionally, the optimisation of R and NR length periods enhanced the prediction capacity of the model.

Overall, the exclusive focus on meteorological factors may hinder precision in extreme conditions in the context of climate change, which we recognise as a limitation of the proposed model.

3.3. Physical Interpretation

For physical interpretation, the multiple regression models with predictions from all data (Table 3) are considered. The contour plot of the RD prediction presented as a non-decreasing function of P_R and P_NR for low- and high-temperature situations is shown in Figure 6. For the Blue River (Oklahoma), at higher soil dryness and water evaporation in the hot season, a significant reduction in monthly RD for the same monthly precipitation was identified [33]. For the selected Portuguese rivers, a higher variation in RD (higher possible effects of T) is observed for larger rivers. Thus, for the Vouga and Mondego rivers, a decrease in T by one degree would lead to an increase in RD by 23.5% and 18.1%, respectively. For much smaller rivers, such as the Antuã and Neiva rivers, each degree decrease in T would result in an increase in RD by 6.1% and 4.8%, respectively.

The effect of precipitation terms P_R and P_NR on the RD model predictions is shown in Figure 7. Here, larger rivers, such as the Vouga and Mondego, are also more influenced by both P_R and P_NR. For all rivers, an increase in P_R (precipitation history for brief recent periods before the day for which RD is to be estimated, 5–11 days for different rivers) leads to an exponential increase in RD, potentially responsible for catastrophic flood events. This is especially evident in the Mondego which, having the largest drainage area, is the most strongly influenced by recent precipitation events. Thus, the area of the Mondego river is one of the most affected by flood regions in Portuguese history. For more distant time periods (50–90 days depending on the river), the average precipitation P_NR only has an effect on low values of RD during the hot months (determining possible drought conditions) and tend to saturate at P_NR values above 7–10 mm day⁻¹ for all the studied rivers. Therefore, a large quantity of precipitation impacts RD (specifically determined for each river based on its drainage area size) if it has occurred recently but, for all the studied rivers, it does not have a large influence on RD if it occurs in non-recent periods. Note that, additional data (not readily available) regarding infiltration, groundwater recharge, or delays in catchment response may shed further light on the observed effects.

3.4. Disscusion of the Results

As observed (Figure 6 and Figure 7), larger rivers are very strongly influenced by meteorological conditions such as precipitation and air temperature. Therefore, although the four studied systems are situated in the same temperate microclimate zone and show similar shape responses, important differences in effect magnitude still exist between them. Success and fail analysis show that the models for the Neiva and Mondego rivers suffer from poor fits during draughts. Additionally, the higher fail rate for the Mondego might show the influence of snowmelt and dam construction on model equations for RD variation. The developed method is important for studying biogeochemical processes and transport of contaminants in rivers or estuaries (e.g., the Vouga, Antuã and Mondego rivers) and for evaluating flood and drought risks in ecologically sensitive areas (e.g., the Neiva river). It can be extended to other rivers, which are similarly influenced by seasonal changes in precipitation and temperature.

4. Conclusions

Regression analysis was applied in order to study the capacity of the meteorological data (monthly air temperature (T) and atmospheric precipitation history) to predict the hydrological variable with large variation, namely the daily river discharge (RD). We propose a multiple regression model where recent (R), non-recent (NR) precipitation variables, and temperature terms are combined but restricted so that there is no interaction between them. This model was different from previous models proposed by the same authors where no restrictions were posed with respect to the interaction between meteorological variables.

The added value of the proposed approach is to allow for a better explanation of the relationships, providing clear and cost-effective physical interpretation and evaluation of the effects that the meteorological variables have on RD. Furthermore, the optimisation of R and NR precipitation period lengths greatly improves the model performance compared to the previous models with interaction terms.

The focus solely on meteorological factors makes the models understandable and useful even in areas with limited data. This strategy increases the practicality of the models across various global situations. Incorporating delay periods for both recent and non-recent rainfall (4–7 days and 50–90 days, respectively) allows the models to effectively account for the timing of the effect of rainfall on streamflow. Employing various rivers as examples, particularly with diverse catchment sizes, adds richness to the results and promotes their applicability to other contexts. Emphasising that larger rivers are more reactive to meteorological changes offers important insights into the hydrology of such systems.