# Constraining Flood Forecasting Uncertainties through Streamflow Data Assimilation in the Tropical Andes of Peru: Case of the Vilcanota River Basin

^{1}

^{2}

^{3}

^{4}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Study Area

^{2}, spanning from 2959 m.a.s.l to 6268 m.a.s.l. Also, we defined a regular domain between latitudes 12.9° S–14.8° S and longitudes 70.6° W–72.7° W to download and merge satellite-based precipitation and pluviometric station observations in the study area.

^{3}/s to 140 m

^{3}/s, with peak flows often occurring in January and February.

#### 2.2. Parsimonious Sub-Daily Hydrological Modeling

#### 2.3. Design of Streamflow Data Assimilation Experiments

#### 2.4. Quantification of Model Errors

_{p}applied at each hourly time step, according to the methodology proposed in [54] and presented in Equations (2) and (3). Also, as in [30], random perturbations were provided by a first-order autoregressive model to guarantee temporal correlation of the time-variant forcings. The fractional error parameter was set to 0.65 and temporal decorrelations lengths were defined as 5 h for rainfall and 3 h for potential evapotranspiration based on an autocorrelation analysis.

_{p}is the error parameter for precipitation, and u

_{p}is a uniform random number, such that φ

_{p}is a realization from a uniform distribution ranging from 1 − ε

_{p}to 1 + ε

_{p}.

_{obs}

^{2}) for describing the measurement noise and parameterizing it as a function of the discharge observation (Q

_{obs}), as presented in Equation (6).

_{obs}was set to 0.1, the quantile 10 (Q

_{10}) was used as the minimum threshold to prevent underestimated error variances in the case of low discharges, and the variance was evaluated proportionally to Q

_{10}

^{2}for values below Q

_{10}.

#### 2.5. Evaluation of Model Forecast

## 3. Results

#### 3.1. Model Calibration and Validation

#### 3.2. Estimation of Model Uncertainties in Streamflow Data Assimilation

^{3}/s) simulations performed better than GSMaP-NRT’ (71.76 m

^{3}/s). This is notable when comparing the observed and OL series. For instance, IMERG-E’ simulations usually overestimated discharge, while for the same period, GSMaP-NRT’ tended to underestimate high flow.

^{3}/s).

#### 3.3. Forecasting Performance Assessment

^{3}/s) and 24 h (302 m

^{3}/s) hugely subestimated the observed value (465 m

^{3}/s). In contrast, high peak flow forecasts given by the most recent prediction (e.g., 1–6 h) were considered to be the best estimate, as shown in Figure 8a–c. Nevertheless, the ensemble means forecasted discharges were still below the observed values, but at much less magnitude than the OL runs and during model calibration-validation.

## 4. Discussion

#### Limitations and Potential of Streamflow Data Assimilation in the Vilcanota River Basin

^{2}, especially in terms of the impact of local variability on the basin’s hydrology. Hence, we highlight the benefits of streamflow DA in an Andean basin of Peru to improve forecasting accuracy using real-time discharges at the basin outlet. Future work will assess DA techniques in the semi-distributed Vilcanota systems presented in [43] to prove if incorporating forcing spatialization, river routing, and soil moisture sub-basin spatialization in conceptual models, such as in [25,26], is more appropriate for Andean basins with sparse data availability. For instance, ref. [24] suggests that hydrologic river routing, such as the Muskingum method, is subject to potentially significant errors from structural and parametric uncertainties.

## 5. Conclusions

^{3}/s ≤ BIAS ≤ −10 m

^{3}/s, MRMSE ≤ 70 m

^{3}/s and CRPSS ≥ 0.75) from the four experiments.

## Author Contributions

## Funding

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

BIAS | Error in observations and/or simulations. |

BoxNSE | Nash–Sutcliffe Efficiency criterion with Box–Cox transformed values. |

CRPS | Continuous Ranked Probability Score. |

CRPSS | Continuous Ranked Probability Skill Score. |

DA | Data Assimilation. |

EnKF | Ensemble Kalman Filter. |

GR4H | Génie Rural à 4 paramètres Horaire. |

GSMaP-NRT | Global Satellite Mapping of Precipitation Near Real-Time product. |

GSMaP-NRT’ | GSMaP-NRT product merged with pluviometric stations. |

GSMaP-NRT’+EnKF | EnKF experiment applied to the hydrological model forced with GSMaP-NRT’. |

GSMaP-NRT’+OL | Open Loop for the hydrological model forced with GSMaP-NRT’. |

GSMaP-NRT’+PF | PF experiment applied to the hydrological model forced with GSMaP-NRT’. |

IMERG-E | Integrated Multi-satellitE Retrievals for GPM Early Runs product. |

IMERG-E’ | IMERG-E product merged with pluviometric stations. |

IMERG-E’+EnKF | EnKF experiment applied to the hydrological model forced with IMERG-E’. |

IMERG-E’+OL | Open Loop for the hydrological model forced with IMERG-E’. |

IMERG-E’+PF | PF experiment applied to the hydrological model forced with IMERG-E’. |

KGE | Kling-Gupta Efficiency criterion. |

logNSE | Nash–Sutcliffe Efficiency criterion with logarithmic values. |

MRMSE | Mean of Ensemble Root Mean Squared Error |

NSE | Nash–Sutcliffe Efficiency criterion. |

OL | Open Loop. |

PF | Particle Filter. |

RMSE | Root Mean square Error. |

SM | Soil Moisture. |

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**Figure 1.**The Pisac gauge station in the Vilcanota River Basin, located in the southern Peruvian Tropical Andes. The available pluviometric network in the study domain is shown as orange triangles.

**Figure 4.**(

**a**,

**b**) Comparison of simulated (IMERG-E’ and GSMaP-NRT’) and observed hourly flow series at the Pisac stream gauge station during 2017–2022. The red dashed box represents the period February–March 2022 selected for testing streamflow DA techniques. (

**c**,

**d**) Scatter plots between observed and simulated discharges for the period February–March 2022.

**Figure 5.**Comparison between observed and simulated hourly flow series during February and March 2022 at the Pisac stream gauge station for (

**a**) IMERG-E’+EnKF, (

**b**) IMERG-E’+KF, (

**c**) GSMaP-NRT’+EnKF, and (

**d**) GSMaP-NRT’+PF experiments.

**Figure 6.**Uncertainty in simulated soil moisture (S) from the GR4H model in the Vilcanota Basin during February and March 2022 for (

**a**) IMERG-E’+EnKF, (

**b**) IMERG-E’+KF, (

**c**) GSMaP-NRT’+EnKF, and (

**d**) GSMaP-NRT’+PF experiments.

**Figure 7.**Performance skills of hourly forecasted discharges from 1 to 24 h of lead times at the Pisac stream gauge station from February to March 2022, based on (

**a**) NSE, (

**b**) BIAS, (

**c**) MRMSE and (

**d**) CRPSS coefficients.

**Figure 8.**Comparison of observed and forecasted hourly flow series at the Pisac station from February to March 2023, for lead times of (

**a**) 1, (

**b**) 3, (

**c**) 6, (

**d**) 12, (

**e**) 18, and (

**f**) 24 h in the GSMaP-NRT’+EnKF experiment.

Type | Station | Abrev. | Longitude [°W] | Latitude [°S] | Elevation [m.a.s.l.] |
---|---|---|---|---|---|

Fluviometric | Pisac | PIS | 71.84 | 13.43 | 2791.65 |

Pluviometric | Acjanaco Gore | AGR | 71.62 | 13.20 | 3466.11 |

Calca | CAL | 71.96 | 13.33 | 2921.24 | |

Casaccancha | CAS | 72.30 | 13.99 | 4033.16 | |

Huayllabamba | HUA | 72.45 | 13.27 | 2976.55 | |

Intihuatana M | INM | 72.56 | 13.17 | 1778.23 | |

Machupicchu | MAC | 72.55 | 13.18 | 2399.80 | |

Marcapata Gore | MAR | 70.90 | 13.50 | 1792.76 | |

Qorihuayrachina | QOR | 72.43 | 13.22 | 2517.25 | |

Salcca | SAL | 71.23 | 14.17 | 3920.10 | |

San Pablo | SPB | 72.62 | 13.03 | 1228.11 | |

Santo Tomas | STM | 72.10 | 14.45 | 3665.48 | |

Sicuani | SIC | 71.24 | 14.24 | 3534.95 |

**Table 2.**Statistic al metrics and their corresponding equations used for evaluating the hydrological performance of the GR4H model.

Statistical Metric | Equation | Min, Max, Optimal | Emphasis |
---|---|---|---|

Logarithmic Nash–Sutcliffe Efficiency (logNSE) [-] | $logNSE=1-\frac{{\sum}_{t=1}^{n}{\left({\mathrm{l}\mathrm{o}\mathrm{g}(Qsim}_{t})-{\mathrm{l}\mathrm{o}\mathrm{g}(Qobs}_{t})\right)}^{2}}{{\sum}_{t=1}^{n}{\left({log(Qobs}_{t})-\overline{log\left(Q\right)}obs\right)}^{2}}$ | −∞,1,1 | Low flows [51] |

Nash–Sutcliffe Efficiency with Box–Cox transformation (BoxNSE) [-] | $\mathrm{B}\mathrm{o}\mathrm{x}\mathrm{N}\mathrm{S}\mathrm{E}=1-\frac{{\sum}_{t=1}^{n}{\left({Qsim\prime}_{t}-{Qobs\prime}_{t}\right)}^{2}}{{\sum}_{t=1}^{n}{\left({Qobs\prime}_{t}-\overline{Q\prime obs}\right)}^{2}}$ ${Q}^{\prime}=\frac{{(Q+1)}^{\gamma}-1}{\gamma}$ | −∞,1,1 | Middle flows [26] |

Kling–Gupta Efficiency (KGE) [-] | $KGE=1-\sqrt{{(r-1)}^{2}+{(\alpha -1)}^{2}+{(\beta -1)}^{2}}$ $r=\frac{{\sum}_{t=1}^{n}\left[\left({Qsim}_{t}-\overline{Q}sim\right)\left({Qobs}_{t}-\overline{Q}obs\right)\right]}{\sqrt{{\sum}_{t=1}^{n}{\left({Qsim}_{t}-\overline{Q}sim\right)}^{2}}\sqrt{{\sum}_{t=1}^{n}{\left({Qobs}_{t}-\overline{Q}obs\right)}^{2}}}$ $\alpha =\frac{{\sigma}_{sim}}{{\sigma}_{obs}};\beta =\frac{{\mu}_{sim}}{{\mu}_{obs}}$ | −∞,1,1 | Variance and high flows [52] |

Bias (BIAS) [%] | $BIAS={\left[\mathrm{max}\left(\frac{\overline{\mathrm{Q}}sim}{\overline{\mathrm{Q}}obs},\frac{\overline{\mathrm{Q}}obs}{\overline{\mathrm{Q}}sim}\right)-1\right]}^{2}$ | 0,+∞,0 | Average trend of simulated flows [26] |

Statistical Metric | Equation |
---|---|

Nash–Sutcliffe Efficiency (NSE) [-] | $\mathrm{N}\mathrm{S}\mathrm{E}=1-\frac{{\sum}_{t=1}^{n}{\left({Qfcst}_{t}-{Qobs}_{t}\right)}^{2}}{{\sum}_{t=1}^{n}{\left({Qobs}_{t}-\overline{Q}obs\right)}^{2}}$ |

Bias (BIAS) [m ^{3}/s] | $\mathrm{B}\mathrm{I}\mathrm{A}\mathrm{S}=\frac{1}{n}{\sum}_{t=1}^{n}\left({Qfcst}_{t}-{Qobs}_{t}\right)$ |

Mean of Ensemble Root Mean Squared Error (MRMSE) [m ^{3}/s] | $MRMSE=\frac{1}{n}{\sum}_{t=1}^{n}\sqrt{\frac{1}{F}\sum _{t=1}^{F}{\left({{Q}^{j}fcst}_{t}-{Qobs}_{t}\right)}^{2}}$ |

Continuous Ranked Probability Skill Score (CRPSS) [-] | $CRPS=\frac{1}{n}{\sum}_{t=1}^{n}\underset{-\infty}{\overset{\infty}{\int}}{\left({Ffcst}_{t}\left(Q\right)-{Fobs}_{t}\left(Q\right)\right)}^{2}dQ$ ${Fobs}_{t}\left(Q\right)=\left\{\right)separators="|">\begin{array}{c}0\hspace{1em}(Q{Qobs}_{t})\\ 1\hspace{1em}(Q\ge {Qobs}_{t})\end{array}$ $CRPSS=1-\frac{CRPS}{{CRPS}_{OL}}$ |

Statistic Metric | Calibration | Validation | ||
---|---|---|---|---|

IMERG-E’ | GSMaP-NRT’ | IMERG-E’ | GSMaP-NRT’ | |

logNSE | 0.875 | 0.792 | 0.878 | 0.786 |

BoxNSE | 0.883 | 0.831 | 0.878 | 0.819 |

KGE | 0.912 | 0.871 | 0.869 | 0.789 |

BIAS | 0.003 | 0.003 | 0.016 | 0.029 |

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## Share and Cite

**MDPI and ACS Style**

Llauca, H.; Arestegui, M.; Lavado-Casimiro, W.
Constraining Flood Forecasting Uncertainties through Streamflow Data Assimilation in the Tropical Andes of Peru: Case of the Vilcanota River Basin. *Water* **2023**, *15*, 3944.
https://doi.org/10.3390/w15223944

**AMA Style**

Llauca H, Arestegui M, Lavado-Casimiro W.
Constraining Flood Forecasting Uncertainties through Streamflow Data Assimilation in the Tropical Andes of Peru: Case of the Vilcanota River Basin. *Water*. 2023; 15(22):3944.
https://doi.org/10.3390/w15223944

**Chicago/Turabian Style**

Llauca, Harold, Miguel Arestegui, and Waldo Lavado-Casimiro.
2023. "Constraining Flood Forecasting Uncertainties through Streamflow Data Assimilation in the Tropical Andes of Peru: Case of the Vilcanota River Basin" *Water* 15, no. 22: 3944.
https://doi.org/10.3390/w15223944