# On the Relationship between Suspended Sediment Concentration, Rainfall Variability and Groundwater: An Empirical and Probabilistic Analysis for the Andean Beni River, Bolivia (2003–2016)

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## Abstract

**:**

^{2}= 0.73 (p < 0.05). With regard to water discharge components, a linear function relates the direct surface flow Q

_{s}–SSSC, and a hysteresis is observed in the relationship between the base flow Q

_{b}and SSSC. A higher base flow index (Q

_{b}/Q) is related to lower SSSC and vice versa. This article highlights the role of base flow on sediment dynamics and provides a method to analyze it through a seasonal empirical model combining the influence of both Q

_{b}and Q

_{s}, which could be employed in other watersheds. A probabilistic method to examine the SSSC relationship with R and Q is also proposed.

## 1. Introduction

^{−2}·year

^{−1}have been estimated [5,6,7,8]. In fact, many geomorphological, biochemical and ecological features of the Amazon plain are linked to the magnitude and variability of water and sediment supplied from the Andes [9].

## 2. Material and Methods

#### 2.1. Study Area

^{2}, which represents 25% of the Beni sub-basin and 6.3% of Bolivia’s territory (Figure 1a). McQuarrie [34] divided the sub-basin into four tectono-structural zones: (1) the Altiplano, a low-relief plateau in Quaternary rocks; (2) the Eastern Andean Cordillera, a thrust belt that deforms lower Paleozoic rocks; (3) the Inter-Andean zone, with a structure similar to that of the Eastern Cordillera, but with younger rocks and deformation at higher structural levels and (4) the Sub-Andes, consisting of faulted folds within Tertiary foreland sedimentary rocks.

^{−1}[35,36,37]. For the 2003–2016 period at RU, the mean annual rainfall was estimated at 1230 mm by the satellite-based precipitation product CHIRPS.v2 (Climate Hazard Group Infrared Precipitation with Stations in its second version) [38], with more than 50% (770 mm) of the annual rainfall occurring during the austral summer (December–March, DJFM; Figure 1b) [35].

^{−1}[40] (Figure 1a), inferring that groundwater supports a significant part of the total streamflow, as described by [15].

^{2}·year

^{−1}over two different periods, 1984–1989 and 2002–2011, respectively. Suspended sediment loads of nearly 212 and 192 Mt·year

^{−1}and mean annual water discharge (Q) of 1990 and 2014 m

^{3}·s

^{−1}were estimated for the same locations and periods. Erosion rates estimated by Guyot et al. [6] at Angosto del Bala and three upstream stations are presented in Figure 1a. The largest erosion rate was estimated at 18,250 t·km

^{2}·s

^{−1}for the La Paz River sub-basin at the Cajetillas station to the southwest of the Andean Beni River sub-basin. Suspended sediment load at RU represents 55% of the Beni River sediment load estimated at the Cachuela Esperanza outlet gauging station for the 2002–2011 period [3]. Regarding particle size distribution, suspended material seems to be dominated by fine sands and silts, with a clay fraction that exceeds 10% [39].

^{3}·s

^{−1}), SSSC decreases from a mean value of 3100 mg·L

^{−1}in February to 1600 mg·L

^{−1}in March (Figure 1c,d).

#### 2.2. Data and Methodology

_{t−1}) to January to August (year

_{t0}).

_{s}) and base flow (Q

_{b;}Table 1). The discharge components were separated at daily time step through the recursive digital filter, and then averaged to the monthly scale. The recursive filter uses a frequency analysis separating the Q

_{s}with the following equation [47]:

_{s,t}is the filtered surface discharge at the time step t; Q

_{s,t−1}is the filtered surface discharge at the t–1 time step; $\propto $ is the filter parameter and Q

_{t}is the total discharge at t and Q

_{t}

_{−1}is the total discharge at t–1. Hence the base flow is Q

_{b}= Q

_{t}− Q

_{s}and the base flow index (BFI) is the relationship between the base flow and the total discharge (Q

_{b}/Q). Q

_{s}and Q

_{b}separation was achieved using the free Web-based Hydrograph Analysis Tool (WHAT) from Purdue University [48]. The $\propto $ filter parameter ranges from 0.90 to 0.95. It was set to 0.93 in this study using the master recession curve (MRC) [47,49]. The MRC represents the summary of all the hydrograph’s recession fragments (i.e., period of no or reduced recharge but with groundwater drainage, in this case from May to August), and relates these fragments through concatenation so that a main curve can be established with $\mathrm{MRC}={\mathrm{Q}}_{\mathrm{o}}{\mathrm{e}}^{-\left(\frac{\mathrm{t}}{\mathrm{k}}\right)}$, $\propto ={\mathrm{e}}^{-\left(\frac{1}{\mathrm{k}}\right)}$ and with k as a recession constant.

_{1},u

_{2}), and bivariate distribution is provided by Sklar’s theorem, with the Equation (5) presented in Table 1 [50]. In this equation, F(x

_{1},x

_{2}) is the joint cumulative distribution function with the continuous marginal distribution functions (i.e., each univariate probability distribution) of the random variables: F

_{1}(x

_{1}) and F

_{2}(x

_{2}).

^{2}), Kendall’s τ and the maximum under- and over-estimation range (Table 2). PBIAS measures the tendency of simulated data to be larger or smaller than the observed counterpart. The NSE is a normalized statistic for determining the relative magnitude of the residual variance compared with the measured data variance [67], indicating the agreement between observed versus simulated data. The coefficient of determination (R

^{2}) describes the proportion of the variance in measured data explained by the model. It ranges from 0 to 1, with values closer to 1 indicating less error variance.

## 3. Results and Discussions

#### 3.1. Multivariate Time-Series

^{2}= 0.73, p < 0.05) over the annual scale with fitted parameters (a and b) that differ when evaluating the R–Q hysteresis period (Figure 2c). As observed by Andermann et al. [29] in the Himalayas, the fact that water discharge displays a clockwise hysteresis within the SSSC suggests that material supply varies between seasons. The authors also observed a counter-clockwise hysteresis between R–Q, which was related to the transient storage and release of water in the basin from the basement aquifer, which in turn could control the sediment concentration transported by the river because of the dilution differential.

#### 3.2. The Relationship of Water Discharge Components (Q_{s} and Q_{b}) with SSSC

_{s}) and base flow (Q

_{b}) from the total water discharge, a hysteresis pattern emerges from the relationship between temporal Q

_{b}–SSSC series, and a linear relation between Q

_{s}–SSSC (Figure 4a,b). The hysteresis could be distinguished in the following three periods: the first from the start of the rainfall in September until its peak in January (SONDJ) when concentration increased from the previous dry season (SSSC = 0.90Q

_{s}, Figure 4b); during the second period, R and Q

_{s}decreased, which generated the start of SSSC decline from February until April SSSC = 0.76 Q

_{s,}(Figure 4b); this reduction was further pronounced for the following months (May–August dry season), which was the third hysteresis period until the next rainy season. We observed this hysteresis throughout all the fourteen years analyzed, including the heavy rainfall event of the austral summer of 2013–2014 [70], with an extended influence in 2015 (Figure 4a). Regarding the BFI, the relationship with SSSC shows an exponential decay (R

^{2}= 0.76), with SSSC less than 1000 mg·L

^{−1}for a ratio larger than 0.60. A very dominant base flow (BFI > 0.90) was associated with a very low mean SSSC value of 140 mg·L

^{−1}and a coefficient of variability (CV) of 0.81 (Figure 4c). Conversely, the diminution of base flow was accompanied by a varied increase of sediment concentration. The SSSC shows a high variability (CV of 0.99) at the start of the rainy season, which decreased to 0.72 after its February peak. The Q

_{s}contribution thus presents a linear relationship for the annual cycle within the suspended sediments (R

^{2}= 0.66). However, if we consider the hysteresis periods and their relationships with the discharge components, we see the following fitted parameters:

_{s}and by the effect of groundwater Q

_{b}. Groundwater generated the dilution of sediment concentrations and increased the river transport capacity. This suggests that the suspended matter in this sub-basin was not dependent on the magnitude of water discharge but on the transport capacity that exceeds the material mobilized by the direct flow. As described by Santini et al. [71] on the Peruvian Andean Amazon Rivers (Ucayali and Marañon), when the fine suspended fraction is dominant, the wash load depends on different factors such as the matter availability, rainfall and sediment entrainment processes that occur on the hillslopes, which can act alone or in combination.

#### 3.3. Bivariate Copula Functions

_{re}). As a result, the Kendall’s correlation coefficients for the dependence between the variables were τ = 0.35 (p < 0.0001) for the R–SSSC relationship and τ = 0.34 (p < 0.0001) for Q

_{re}–SSSC.

_{re,}and Weibull 3 for the SSSC (NSE = 0.99; BIAS = 0.006). The joint functions were then determined, and in both cases a Gaussian Copula was selected after a minimum AIC value, –52.64 and –46.72, for R–SSSC and Q

_{re}–SSSC, respectively. For the R–SSSC function the Gaussian parameter = 0.54 was determined and for the Q

_{re}–SSSC, 0.52. With the defined Copula functions, we made the two simulations of 20,000 pseudo Copula anomalies. From each simulation, we estimated the mean value and transformed it to the standardized SSSC values, which were later converted to SSSC (estimated SSSC in Section 3.4).

#### 3.4. SSSC Estimations Assessments Performance

^{2}and NSE, varying the range of under- and over-estimation (Table 4). What is of interest is that even the Copula functions had very good performance ratings over the complete time-series estimation, but they underestimated the two highest SSSC months (i.e., January and February). The observed mean peak in 2005 and 2008, 4660 and 4350 mg·L

^{−1}, respectively were poorly estimated by the bivariate Copula functions, at nearly –36% and –31%, while the lowest observed mean peak in 2009 of 1780 mg·L

^{−1}was overestimated by more than +68%. Indeed, none of the functions were able to estimate the highest sediment peak during 2008. It is important to point out that the study period was too short, and that of the 176 months available of SSSC, only 10 months had concentrations larger than 4000 mg·L

^{−1}, which made it impossible for all the functions to represent these concentrations. Between January and February 2014 alone, instant measurements were larger than 5300 mg·L

^{−1}, while in February 2013 a 15,100 mg·L

^{−1}was reported. Though values larger than 4000 mg·L

^{−1}might be considered as outliers, they could represent flash floods that can occur over this piedmont station, affecting the daily time-series interpolations and monthly aggregation [3].

## 4. Conclusions and Perspectives

^{2}, NSE, Kendall’s τ and under/overestimation range). We also compared the results based on the model’s ability to achieve satisfactory annual behavior.

- Although the power rating curve presented a satisfactory R
^{2}= 0.73, it was insufficient to estimate SSSC, because it did not account for scatters along the regression curve, caused by events with high SSSC and the hysteresis between the variables (rainfall vs. water discharge and base flow vs. SSSC). - The evaluation of the SSSC differentiating the contribution of direct surface and base flows from total water discharge allowed us to see the role these components had in the sediment dynamics in this sub-basin. Thus SSSC was estimated through a sum of seasonal functions based on surface and base flow contributions. Considering other floodplain areas and aquifers, mainly at the sub-basin’s headwaters in the Amazon, future sediment dynamic research could take into account the potential role of base flow in the suspended sediment concentration.
- By considering the time-series’ marginal distributions in a bivariate Copula function, we reduced the PBIAS significantly to less than 6% and achieved a very good NSE of 0.83. Furthermore, the annual cycle could be reproduced satisfactorily. However, it is not only the stability of the time-series, which must be continuously evaluated to search for changes in the distributions’ parameters that can modify the Copula function; it is also necessary to consider that as a probabilistic technique it still fail to establish the physical understanding that relates the variables. Moreover, a further evaluation could consider using a multivariate Copula to estimate SSSC based on both (rainfall and discharge) in the same joint function.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**(

**a**) Andean Beni sub-basin at the Rurrenabaque outlet gauge station (RU). Annual cycle (2003–2016) for (

**b**) basin mean rainfall (R) at RU; (

**c**) water discharge (Q) and (

**d**) surface suspended sediment concentration (SSSC). Sources: for (

**a**) Elevation data from Shuttle Radar Topography Mission (STRM) [41], sediment yields from Guyot et al. [6], and groundwater recharge from Bundesanstalt für Geowissenschaften und Rohstoffe (BGR) [40], (

**b**) CHIRPS.v2 [38], (

**c**) and (

**d**) Hydrogéochimie du Bassin Amazonien (HYBAM), [42]. In (

**c**) the 12,230 m

^{3}·s

^{−1}measured during the extreme rainfall event in February 2014 is not plotted for visual purpose of the y-axis.

**Figure 2.**Monthly time-series relationships 2003–2016: (

**a**) mean Andean Beni sub-basin R in mm vs. Q at RU in m

^{3}·s

^{−1}; (

**b**) as (

**a**) between R–SSSC at RU in mg·L

^{−1}and (

**c**) as (

**a**) between Q–SSSC. First R

^{2}representing September–January and second R

^{2}for February–August periods.

**Figure 3.**CHIRPS’ rainfall spatial r-Pearson correlation (α = 0.05) with monthly SSSC measured at RU in the Andean Beni River from December to April.

**Figure 4.**(

**a**) Base flow (Q

_{b}) vs. SSSC at RU in the Andean Beni River (2003–2016) at monthly scale. (

**b**) Surface discharge (Q

_{s}) vs. SSSC; here black (red) line for the linear relationship and dashed black (red) lines for the 95% confidence interval for September to January—SONDJ (February to March, FMA). (

**c**) Base Flow Index (BFI) vs. SSSC.

**Figure 5.**Observed SSSC (red line) at RU from September 2003 to September 2016 with the four models: (

**a**) here Q^ stands for the power functions; (

**b**) Qs_Qb for the seasonal discharge components function and (

**c**) BICOP-R and BICOP-Q for the two bivariate Copula functions, while CDF stands for the cumulative distribution function.

**Table 1.**Functions and dependences evaluated. Here Q

_{s}stands for the surface component, Q

_{b}for the baseflow and ε for the error.

Dependence | Type of Function Analyzed | Equation |
---|---|---|

$\mathrm{SSSC}=\mathrm{f}\left(\mathrm{Q}\right)$ | $\mathrm{SSSC}={\mathrm{aQ}}^{\mathrm{b}}+\mathsf{\epsilon}$ $\mathrm{SSSC}={\mathrm{aQ}}_{\mathrm{s}}+{\mathrm{bQ}}_{\mathrm{b}}+\text{}\mathsf{\epsilon}$ | Equation (3) Equation (4) |

$\mathrm{SSSC}=\mathrm{f}\left(\mathrm{R}\right)$ $\mathrm{SSSC}=\mathrm{f}\left(\mathrm{Q}\right)$ Bivariate Copula | $\mathrm{F}\left({\mathrm{x}}_{1},{\mathrm{x}}_{2}\right)=\mathrm{C}\left[{\mathrm{F}}_{1}\left({\mathrm{x}}_{1}\right),{\mathrm{F}}_{2}\left({\mathrm{x}}_{2}\right)\right]$ | Equation (5) |

**Table 2.**Statistical criteria used for evaluation. Performance ratings based on Moriasi et al. [68]. Here $\mathrm{xobs}\left(\mathrm{i}\right)$ refers to observed data at month i, with $\overline{\mathrm{xobs}}$ as the mean and $\mathrm{ymod}\left(\mathrm{i}\right)$ for model estimation with $\overline{\mathrm{ymod}}$ as the mean. For Kendall’s Tau: ${\mathrm{P}}_{\mathrm{n}}$ number of concordant pairs, with $\mathrm{n}$ as the number of pairs; two pairs (X

_{i}, Y

_{i}),(X

_{j}, Y

_{j}) being concordant if (X

_{i}− X

_{j})·(Y

_{i}− Y

_{j}) > 0.

Percent Bias Equation (6) | $\mathrm{PBIAS}=\left[\raisebox{1ex}{${{\displaystyle \sum}}_{\mathrm{i}=1}^{\mathrm{n}}\left(\mathrm{xobs}\left(\mathrm{i}\right)-\mathrm{ymod}\left(\mathrm{i}\right)\right)$}\!\left/ \!\raisebox{-1ex}{${{\displaystyle \sum}}_{\mathrm{i}=1}^{\mathrm{n}}\left(\mathrm{xobs}\left(\mathrm{i}\right)\right)$}\right.\right]\times 100$ | ||

Nash-Sutcliffe Efficiency Equation (7) | $\mathrm{NSE}=1-\left[\raisebox{1ex}{${{\displaystyle \sum}}_{\mathrm{i}=1}^{\mathrm{n}}{\left(\mathrm{xobs}\left(\mathrm{i}\right)-\mathrm{ymod}\left(\mathrm{i}\right)\right)}^{2}$}\!\left/ \!\raisebox{-1ex}{${{\displaystyle \sum}}_{\mathrm{i}=1}^{\mathrm{n}}{\left(\mathrm{xobs}\left(\mathrm{i}\right)-\overline{\mathrm{xobs}}\right)}^{2}$}\right.\right]$ | ||

Linear Coefficient of determination Equation (8) | ${\mathrm{R}}^{2}=\raisebox{1ex}{${\left[{{\displaystyle \sum}}_{\mathrm{i}}^{\mathrm{n}}\left(\mathrm{xobs}\left(\mathrm{i}\right)-\overline{\mathrm{xobs}}\right)(\mathrm{ymod}\left(\mathrm{i}\right)-\overline{\mathrm{ymod})}\right]}^{2}$}\!\left/ \!\raisebox{-1ex}{${{\displaystyle \sum}}_{\mathrm{i}}^{\mathrm{n}}{\left(\mathrm{xobs}\left(\mathrm{i}\right)-\overline{\mathrm{xobs}}\right)}^{2}{{\displaystyle \sum}}_{\mathrm{i}}^{\mathrm{n}}{\left(\mathrm{ymod}\left(\mathrm{i}\right)-\overline{\mathrm{ymod}}\right)}^{2}$}\right.$ | ||

Kendall’s Tau Equation (9) | ${\mathsf{\tau}}_{\mathrm{n}}=\frac{4}{\mathrm{n}\left(\mathrm{n}-1\right)}{\mathrm{P}}_{\mathrm{n}}-1$ | ||

Performance Rating | PBIAS | NSE | R^{2} |

Very good | <±10 | 0.75 < NSE ≤ 1.00 | ≥0.60 |

Good | ±10 ≤ PBIAS < ±15 | 0.65 < NSE ≤ 0.75 | |

Satisfactory | ±15 ≤ PBIAS < ±25 | 0.50 < NSE ≤ 0.65 | |

Unsatisfactory | ≥±25 | ≤0.50 |

**Table 3.**Kendall’s τ for rainfall (R), discharge (Q), discharge after one-month lag (Q-Lag) and surface suspended sediment concentration (SSSC) at monthly scale (2003–2016).

Kendall’s τ | R | Q | Q-Lag | SSSC |
---|---|---|---|---|

R | - | 0.57 | 0.70 | 0.72 |

Q | 0.57 | - | 0.61 | 0.65 |

Q-Lag | 0.70 | 0.61 | - | 0.65 |

SSSC | 0.72 | 0.65 | 0.65 | - |

**Table 4.**Statistical criteria obtained for the different functions (2003–2016). In all cases Kendall’s τ was significant at the 99% level.

Function | Kendall’s τ (–) | Linear R^{2} (–) | Max Underestimation/Max Overestimation (mg·L^{−1}) | PBIAS (%) | NSE (–) |
---|---|---|---|---|---|

$\mathrm{SSSC}=0.003{\mathrm{Q}}^{1.60}+\mathsf{\epsilon}$ | |||||

0.66 | 0.65 | −2400/4500 | 17.7 | 0.58 | |

$\mathrm{SSSC}=\left(0.90{\mathrm{Q}}_{\mathrm{s}-\mathrm{SONDJ}}+160\right)+\left(0.75{\mathrm{Q}}_{\mathrm{s}-\mathrm{FMA}}+346\right)+\left(0.034{\mathrm{Q}}_{\mathrm{b}-\mathrm{MJJA}}^{1.20}\right)+\text{}\mathsf{\epsilon}$ | |||||

0.72 | 0.72 | −2130/4140 | 5.0 | 0.71 | |

$\mathrm{SSSC}=\mathrm{Gaussian}\text{}\mathrm{Bivariate}\text{}\mathrm{Copula}\text{}+\mathsf{\epsilon}$ | |||||

$\mathrm{SSSC}\_1=\left(\mathrm{R}-\mathrm{SSSC}\right)$ | 0.75 | 0.85 | −1600/2500 | 4.6 | 0.84 |

$\mathrm{SSSC}\_1=\left(\mathrm{Q}-\mathrm{SSSC}\right)$ | 0.75 | 0.83 | −1400/2600 | 5.2 | 0.83 |

**Table 5.**Cross-validation statistical criteria obtained for the different functions. The intervals represent the range of the metric for the estimated years in each validation step, and 2005 and 2014 as example for extreme events.

Function | Kendall’s τ (–) | Linear R^{2} (–) | Max Underestimation/Max Overestimation (mg·L^{−1}) | PBIAS (%) | NSE (–) |
---|---|---|---|---|---|

$\mathrm{SSSC}={\mathrm{aQ}}^{\mathrm{b}}+\mathsf{\epsilon}$ | (0.38,0.85) | (0.34,0.95) | −3620/3435 | (−77,49) | (0.14,0.93) |

2005 | 0.85 | 0.86 | −370/3130 | 46 | 0.86 |

2014 | 0.38 | 0.61 | −2900/430 | −77 | 0.61 |

$\mathrm{SSSC}={\mathrm{aQ}}_{\mathrm{s}}+{\mathrm{bQ}}_{\mathrm{b}}+\text{}\mathsf{\epsilon}$ | (0.64,0.97) | (0.44,0.97) | −2435/3340 | (−73,29) | (0.35,0.94) |

2005 | 0.91 | 0.93 | −440/2380 | 29 | 0.68 |

2014 | 0.83 | 0.85 | −1575/214 | −27 | 0.79 |

$\mathrm{SSSC}=\mathrm{Bivariate}\text{}\mathrm{Copula}\text{}+\mathsf{\epsilon}$ | |||||

$\mathrm{SSSC}\_1=\left(\mathrm{R}-\mathrm{SSSC}\right)$ | (0.53,0.91) | (0.78,0.97) | −1495/2640 | (−41,28) | (0.43,0.97) |

2005 | 0.91 | 0.97 | −380/2050 | 28 | 0.77 |

2014 | 0.53 | 0.90 | −1000/610 | −7 | 0.90 |

$\mathrm{SSSC}\_1=\left(\mathrm{Q}-\mathrm{SSSC}\right)$ | (0.52,0.85) | (0.64,0.97) | −1630/2,410 | (–48,31) | (0.4,0.84) |

2005 | 0.85 | 0.90 | −640/1910 | 22 | 0.75 |

2014 | 0.64 | 0.78 | −1500/920 | −10 | 0.79 |

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

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**MDPI and ACS Style**

Ayes Rivera, I.; Callau Poduje, A.C.; Molina-Carpio, J.; Ayala, J.M.; Armijos Cardenas, E.; Espinoza-Villar, R.; Espinoza, J.C.; Gutierrez-Cori, O.; Filizola, N. On the Relationship between Suspended Sediment Concentration, Rainfall Variability and Groundwater: An Empirical and Probabilistic Analysis for the Andean Beni River, Bolivia (2003–2016). *Water* **2019**, *11*, 2497.
https://doi.org/10.3390/w11122497

**AMA Style**

Ayes Rivera I, Callau Poduje AC, Molina-Carpio J, Ayala JM, Armijos Cardenas E, Espinoza-Villar R, Espinoza JC, Gutierrez-Cori O, Filizola N. On the Relationship between Suspended Sediment Concentration, Rainfall Variability and Groundwater: An Empirical and Probabilistic Analysis for the Andean Beni River, Bolivia (2003–2016). *Water*. 2019; 11(12):2497.
https://doi.org/10.3390/w11122497

**Chicago/Turabian Style**

Ayes Rivera, Irma, Ana Claudia Callau Poduje, Jorge Molina-Carpio, José Max Ayala, Elisa Armijos Cardenas, Raúl Espinoza-Villar, Jhan Carlo Espinoza, Omar Gutierrez-Cori, and Naziano Filizola. 2019. "On the Relationship between Suspended Sediment Concentration, Rainfall Variability and Groundwater: An Empirical and Probabilistic Analysis for the Andean Beni River, Bolivia (2003–2016)" *Water* 11, no. 12: 2497.
https://doi.org/10.3390/w11122497