# PISCO_HyM_GR2M: A Model of Monthly Water Balance in Peru (1981–2020)

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## Abstract

**:**

_{sqrt}), and water balance error (WBE) metrics. The results show a very well representation of monthly discharges for a large portion of Peruvian sub-basins (KGE ≥ 0.75, NSE

_{sqrt}≥ 0.65, and −0.29 < WBE < 0.23). Finally, this study introduces a product of continuous monthly discharge rates in Peru, named PISCO_HyM_GR2M, to understand surface water balance in data-scarce sub-basins.

## 1. Introduction

## 2. Study Area

^{2}and a population of approximately 32.5 million people. It borders on the west with the Pacific Ocean, on the north with Ecuador and Colombia, and on the southeast with Brazil, Bolivia, and Chile. The Andes mountain range creates a complex topography and introduces hydroclimatic variability along its three hydrographic regions: Pacific, Atlantic, and Titicaca. This natural orographic barrier traps atmospheric moisture from the Atlantic, producing high rainfall over the Andean–Amazon region and Amazon lowlands (eastern side) and low rainfall on the coast (western side) [40], leading to the great contrast of water resources in the country, characterized by a much larger water supply on the Atlantic slope than on the Titicaca and Pacific slopes [1]. Rainfall is highly variable in both space and time [49]. Maximal rainfall rates occur between November and March. Arid conditions with low rainfall rates characterize coastal areas in the Pacific slope (<~150 mm/year) and semi-arid conditions (<~400 mm/year) in the western flank of the Andes [18]. The Atlantic and Titicaca slopes have humid conditions with high rainfall rates in the eastern flank of the Andes (~1100 mm/year), the Andes–Amazon transition (~3200 mm/year), and the Amazon lowland (~2550 mm/year) [11]. Mean annual temperature fluctuations over the country appear indirectly related to elevation (lower altitude, more temperature).

^{2}(Figure 1). Moreover, 3594 river streams and sub-basins with a median area of 300 km

^{2}(with extremes values of 40 km

^{2}and 2500 km

^{2}) were delimited to obtain fine streamflow spatialization according to meteorological inputs resolution (~10 × ~10 km) and considering a unique river stream by sub-basin to compute flow accumulation. In Figure 1, gauged areas correspond to drainage areas covered by a hydrometric station, while ungauged areas correspond to areas without any hydrometric control downstream.

## 3. Data and Methods

#### 3.1. Hydrometeorological Data

#### 3.1.1. The PISCO Dataset

#### 3.1.2. Discharge Data

#### 3.2. Semi-Distributed GR2M Model

#### 3.3. Sensitivity Analysis

#### 3.4. Calibration Regions and Sub-Regions

#### 3.5. GR2M Calibration and Validation Strategy

_{sqrt}) [62] is used to evaluate model performance in general flows, and the Water Balance Error (WBE) [63] was used to assess the model bias. The summary of the statistical metrics used and their corresponding equations are shown in Table 3. The validation process consisted of evaluating the model’s outputs based on the previously calibrated parameters using the remaining 30–40% of the available streamflow records.

#### 3.6. Discharge Simulation at a National Level

## 4. Results

#### 4.1. Sensitivity Analysis and Calibration Regions

#### 4.2. Model Performance Assessment

_{sqrt}, the model performs well during the calibration period, with values above 0.75 and 0.65, respectively, at stations on the Pacific and the Andes–Amazon transition; however, low values of KGE and NSE

_{sqrt}(<0.50) predominate at stations on the Amazon lowlands. Performance remains the same during validation but with a slight decrease in the KGE and NSE

_{sqrt}metrics at stations belonging to the Andes–Amazon transition. Regarding the WBE, balance errors close to zero during the calibration period are observed at stations with high KGE and NSE

_{sqrt}values, except for positive balance errors not greater than 0.25 at stations in the Amazon lowlands. Negative balance errors increase in the validation period, up to −0.38 on the Pacific coast and the Andes–Amazon transition.

_{sqrt}values higher than 0.65 for 70% of the stations demonstrate a good representation of the sub-basin’s general flows. In terms of the WBE, negative values of not less than −0.20 are evident at most of the stations with high KGE and NSE

_{sqrt}values, and positive values of not more than 0.23 are evident for the Amazon lowlands. This behavior indicates that stations with a good fit in terms of KGE and NSE

_{sqrt}tend to slightly overestimate the total runoff, while on the Amazon lowlands, it tends to be underestimated.

^{3}/s average annual flow (Cañete basin) up to basins with 9000 m

^{3}/s (Ucayali basin). The simulated series fit very well to the observations at most of the stations evaluated, except for SOC, where the wet season’s streamflows (December–March) were slightly overestimated. At an annual scale, the model can represent very dry (e.g., 1991 and 1992) to very wet years (e.g., 1997) in sub-basins of the three Peruvian slopes. The seasonal variation curves adequately represent the peak flow month, except for the PUC station, where there is one lag month (Figure 7d). This adequate seasonal and interannual representation is repeated in the remaining hydrometric stations (not shown), except for those located in the Amazon lowlands, where the monthly model performs poorly in terms of NSE

_{sqrt}(Figure 6b).

_{sqrt}(Figure 6a,b). In contrast, the regions with smaller variations in both X1 and X2 parameter values correspond to stations with a good model fit.

_{sqrt}, the model performance using the regional approach (Figure 8c,d) declines mainly in sub-basins of the A–D regions (see Figure 5b) and is relatively stable (except for region M) in south-central regions. Since ungauged areas are located predominantly in regions F, L, and N, the regional approach of parameters in these sub-basins is suitable for estimating monthly discharges.

#### 4.3. Product of Simulated Monthly Discharges at a National Level

_{sqrt}[64] performance categories for gauged areas are shown in Figure 9a,b, respectively. Both metrics agree that simulated monthly discharge in the central and southern of the study area are well represented, while those for the northeast (Amazon lowlands) should be interpreted with caution. The latter varies depending on the hydrograph assessment approach because the KGE metric emphasizes high flows [65], while NSE

_{sqrt}reduces this effect and emphasizes the general representation of streamflows [64].

## 5. Discussion

#### 5.1. Sensitivity Analysis and Calibration Regions

#### 5.2. Model Simulations at a National Level

_{P}sub-product biases probably because of the lack of adequate rainfall estimates in equatorial regions [11]. This lower model performance is similar to that were obtained in [67,68] using different hydrological models in a daily timestep and different sets of satellite precipitation products. Thus, rainfall uncertainties propagate to model outputs and reduce the model’s predictable capacity [26]. Additionally, PET climatology used in this paper for operational purposes might not be reflecting actual evapotranspiration in the Amazon plain. Future works will incorporate a robust assessment of evapotranspiration in the hydrological modeling with a data scares scenario and its impacts on the water balance.

^{2}) such as in the Amazon plain. Future works will incorporate routing models such as the Routing Application for Parallel computatIon of Discharge (RAPID) [72] to improve flow routing throughout the national drainage network, especially in the Atlantic slope.

## 6. Conclusions

- (a)
- The hydrological performance of the GR2M model in Peru performed well in sub-basins of the Pacific slope and the Andes–Amazon transition (part of the Titicaca and the Atlantic slopes). The model adequately represents the seasonality and interannual variability of the streamflows, except for the Amazon lowlands, where only high flows are well-represented.
- (b)
- Through the monthly meteorological PISCO sub-products, it is possible to simulate the runoff volume over most of Peru adequately. However, the uncertainties associated with these sub-products are more significant towards the north of the country where there are not enough meteorological stations, so this error propagates towards the hydrological model outputs for the Amazon lowlands.
- (c)
- The proposed methodology to define the calibration regions based on the spatial patterns of two hydroclimatic indices’ relative sensitivities proved to be an appropriate technique for calibrating and validating the GR2M model and estimating monthly discharge in ungauged sub-basins.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**(

**a**) Study area and hydrometric stations in the Pacific, Titicaca, and Atlantic slopes. Detail of (

**b**) sub-basins and (

**c**) river streams used for the semi-distributed hydrological model at a national level.

**Figure 2.**Methodological framework. Where: P = precipitation, PET = Potential Evapotranspiration, Q = Discharge, WFAC = Weighted Flow Accumulation, FAST = Fourier Amplitude Test, RR = Rainfall-Runoff Index, RV = Runoff Variability Index, and SCE-UA = Shuffle Complex Evolutionary Algorithm.

**Figure 3.**Framework to calculate accumulated discharges from sub-basins to river streams in a semi-distributed GR2M adaptation. Where: q = runoff in mm, CONDEM = Hydrologically Conditioned Digital Elevation model, FlowDir = Flow Direction, WFAC = Weighted Flow Accumulation, FlowAcum = Flow Accumulation, and Q = discharges in m

^{3}/s.

**Figure 4.**Strategy for the GR2M model calibration at a national level. Steps are based on the stream network configuration and location of the hydrometric stations in gauged areas. Each basin might contain more than one set of X1 and X2 parameters. Gray areas correspond to ungauged areas.

**Figure 5.**(

**a**) Relative sensitivities of RR and RV indices obtained from FAST. (

**b**) Delimited calibration regions and sub-regions for the GR2M model at a national level, each region contains at least one hydrometric station.

**Figure 6.**(

**a**–

**c**) Statistical metrics for evaluating the GR2M model performance at a national level during the calibration, the validation, and the total period. In terms of the KGE and NSE

_{sqrt}metrics, cold colors represent good model performance while warm colors represent inadequate model performance. In WBE, blue colors are associated with the underestimation of the total volume of surface runoff and red colors indicate the overestimation.

**Figure 7.**Observed and simulated monthly and annual discharges (Qm), and seasonal variation curves, for six representatives’ hydrometric stations in the (

**a**,

**b**) Pacific, (

**c**) the Titicaca, and (

**d**–

**f**) the Atlantic slopes, with more extended data availability from January 1981 to March 2020.

**Figure 8.**Boxplot of calibrated (

**a**) X1 and (

**b**) X2 parameters for each calibration region. (

**c**–

**e**) Scatterplot of statistical metrics used for evaluating model performance changes using calibrated GR2M model parameters by sub-regions (Sub) and applying the median of X1 and X2 for each calibration region (Reg).

**Figure 9.**Qualitative ratings of streamflow simulation result at 43 hydrometric stations across the study area, based on (

**a**) KGE and (

**b**) NSE

_{sqrt}values.

**Table 1.**Hydrometric stations selected for hydrological modeling at a national level. Coverage [%] is considering from January 1981 to March 2020.

Slope | Station | Abrev. | Latitude [°S] | Longitude [°W] | Watershed | Source | Coverage [%] |
---|---|---|---|---|---|---|---|

Pacific | Huatiapa | HUA | −16.008 | −72.484 | Camaná | SENAMHI | 45.3 |

Socsi | SOC | −13.029 | −76.195 | Cañete | SENAMHI | 92.9 | |

Santo Domingo | SDO | −11.384 | −77.050 | Chancay-Huaral | SENAMHI | 66.0 | |

Racarumi | RRI | −6.633 | −79.317 | Lambayeque | SENAMHI | 99.8 | |

Salinar | SAL | −7.661 | −78.961 | Chicama | SENAMHI | 92.1 | |

Obrajillo | OBR | −11.452 | −76.622 | Chillón | SENAMHI | 58.8 | |

El Ciruelo | ECI | −4.300 | −80.150 | Chira | SENAMHI | 97.6 | |

Malvados | MAL | −10.340 | −77.630 | Fortaleza | JU FORTALEZA | 31.2 | |

Pte. Ocoña | POC | −16.422 | −73.115 | Ocoña | SENAMHI | 33.3 | |

Letrayoc | LET | −13.640 | −75.720 | Pisco | SENAMHI | 89.5 | |

Pte. Sánchez Cerro | PSC | −5.194 | −80.623 | Piura | PE CHIRA PIURA | 49.1 | |

Condorcerro | CCO | −8.658 | −78.262 | Santa | PE CHAVIMOCHIC | 98.1 | |

Pte. Santa Rosa | PSR | −17.030 | −71.690 | Tambo | JU TAMBO | 92.9 | |

El Tigre | ETI | −3.769 | −80.457 | Tumbes | SENAMHI | 97.4 | |

Chosica | CHO | −11.930 | −76.690 | Rímac | SENAMHI | 54.7 | |

Titicaca | Pte. Huancané | HNE | −15.216 | −69.793 | Huancané | SENAMHI | 79.7 |

Pte. Ramis | RAM | −15.255 | −69.874 | Ramis | SENAMHI | 72.4 | |

Pte. Unocolla | COA | −15.451 | −70.192 | Coata | SENAMHI | 74.6 | |

Pte. Ilave | ILA | −16.088 | −69.626 | Ilave | SENAMHI | 72.6 | |

Atlantic | Egemsa Km105 | EKM | −13.183 | −72.533 | Urubamba | SENAMHI | 86.1 |

Borja | BOR | −4.470 | −77.548 | Marañón | HYBAM | 86.5 | |

Jesús Tunel | JTU | −7.221 | −78.404 | Crisnejas | SENAMHI | 97.9 | |

Cumba | CUM | −5.944 | −78.661 | Marañón | SENAMHI | 13.7 | |

Los Naranjos | LNA | −5.756 | −78.432 | Marañón | SENAMHI | 17.5 | |

Puente Tocache | TOC | −8.181 | −76.506 | Huallaga | SENAMHI | 58.3 | |

Tingo María | TMA | −9.290 | −76.003 | Huallaga | SENAMHI | 51.9 | |

Picota | PIC | −6.949 | −76.325 | Huallaga | SENAMHI | 42.3 | |

Chazuta | CHA | −6.570 | −76.119 | Huallaga | HYBAM | 41.7 | |

Paucartambo | PAU | −13.321 | −71.594 | Urubamba | SENAMHI | 28.8 | |

Pisac | PIS | −13.422 | −71.855 | Vilcanota | SENAMHI | 66.5 | |

Pte. Cunyac | PCU | −13.560 | −72.574 | Apurímac | SENAMHI | 26.5 | |

Pte. Stuart | PST | −11.802 | −75.490 | Mantaro | ELECTROPERU | 68.8 | |

Puerto Inca | PUI | −9.384 | −74.968 | Pachitea | HYBAM | 43.8 | |

Tamshiyacu | TAM | −4.003 | −73.162 | Amazonas | HYBAM | 90.6 | |

Requena | REQ | −5.030 | −73.830 | Ucayali | HYBAM | 58.5 | |

Lagarto | LAG | −10.607 | −73.871 | Ucayali | HYBAM | 23.1 | |

Bellavista | BEL | −3.482 | −73.073 | Napo | HYBAM | 71.2 | |

Pte. Corral Quemado | PCQ | −5.755 | −78.692 | Marañón | SENAMHI | 13.7 | |

La Pastora | LPA | −12.584 | −69.214 | Madre de Dios | HYBAM | 24.4 | |

Napo | NAP | −0.917 | −75.396 | Napo | HYBAM | 40.6 | |

Tabatinga | TAB | −4.250 | −69.950 | Amazonas | HYBAM | 87.6 | |

Pucallpa | PUC | −8.390 | −74.530 | Ucayali | HYBAM | 43.6 | |

San Regis | SRE | −4.513 | −73.907 | Marañón | HYBAM | 53.0 |

Index | Unit | Equation | Description |
---|---|---|---|

Runoff Ratio (RR) | - | $RR=\frac{X}{P}$ | The ratio of simulated runoff to precipitation |

Runoff Variability (RV) | - | $RV=\frac{{\sigma}_{X}}{{\sigma}_{P}}$ | The standard deviation of simulated runoff to the standard deviation of precipitation |

_{X,P,}standard deviation of simulated runoff and rainfall.

**Table 3.**Statistical metrics and their corresponding equations used for evaluating the hydrological performance of GR2M model.

Statistical Metric | Unit | Equation | Optimal Value |
---|---|---|---|

Kling–Gupta efficiency (KGE) | - | $KGE=1-\sqrt{{\left(r-1\right)}^{2}+{\left(\alpha -1\right)}^{2}+{\left(\beta -1\right)}^{2}}$ $r=\frac{{{\displaystyle \sum}}_{i=1}^{n}\left[\left({X}_{i}-\overline{X}\right)\left({O}_{i}-\overline{O}\right)\right]}{\sqrt{{{\displaystyle \sum}}_{i=1}^{n}{\left({X}_{i}-\overline{X}\right)}^{2}}\sqrt{{{\displaystyle \sum}}_{i=1}^{n}{\left({O}_{i}-\overline{O}\right)}^{2}}}$ $\alpha =\frac{{\sigma}_{X}}{{\sigma}_{O}};\beta =\frac{{\mu}_{X}}{{\mu}_{O}}$ | 1 |

Nash–Sutcliffe Squared (NSE_{sqrt}) | - | $NS{E}_{sqrt}=1-\frac{{{\displaystyle \sum}}_{i=1}^{n}{\left(\sqrt{{O}_{i}}-\sqrt{{X}_{i}}\right)}^{2}}{{{\displaystyle \sum}}_{i=1}^{n}{\left(\sqrt{{O}_{i}}-\sqrt{\overline{O}}\right)}^{2}}$ | 1 |

Water Balance Error (WBE) | - | $WBE=\frac{{{\displaystyle \sum}}_{i=1}^{n}\left({O}_{i}-{X}_{i}\right)}{{{\displaystyle \sum}}_{i=1}^{n}{O}_{i}}$ | 0 |

**Table 4.**Resume of calibration regions delimited for the study area, including gauged and ungauged areas.

Calibration Region | Number of Sub-Regions | Number of Sub-Basins | Number of Hydrometric Stations |
---|---|---|---|

A | 8 | 380 | 2 |

B | 6 | 346 | 1 |

C | 13 | 239 | 6 |

D | 6 | 385 | 5 |

E | 2 | 229 | 1 |

F | 6 | 158 | 5 |

G | 10 | 190 | 2 |

H | 7 | 269 | 2 |

I | 6 | 316 | 3 |

J | 6 | 220 | 2 |

K | 2 | 258 | 1 |

L | 8 | 159 | 4 |

M | 12 | 324 | 7 |

N | 4 | 121 | 2 |

Total | 96 (100.0%) | 3594 (100.0%) | 43 (100.0%) |

Gauged area | 84 (87.5%) | 2605 (72.5%) | 43 (100.0%) |

Ungauged area | 12 (12.5%) | 989 (27.5%) | 0 (0.0%) |

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**MDPI and ACS Style**

Llauca, H.; Lavado-Casimiro, W.; Montesinos, C.; Santini, W.; Rau, P.
PISCO_HyM_GR2M: A Model of Monthly Water Balance in Peru (1981–2020). *Water* **2021**, *13*, 1048.
https://doi.org/10.3390/w13081048

**AMA Style**

Llauca H, Lavado-Casimiro W, Montesinos C, Santini W, Rau P.
PISCO_HyM_GR2M: A Model of Monthly Water Balance in Peru (1981–2020). *Water*. 2021; 13(8):1048.
https://doi.org/10.3390/w13081048

**Chicago/Turabian Style**

Llauca, Harold, Waldo Lavado-Casimiro, Cristian Montesinos, William Santini, and Pedro Rau.
2021. "PISCO_HyM_GR2M: A Model of Monthly Water Balance in Peru (1981–2020)" *Water* 13, no. 8: 1048.
https://doi.org/10.3390/w13081048