# Parameter Estimation and Uncertainty Analysis: A Comparison between Continuous and Event-Based Modeling of Streamflow Based on the Hydrological Simulation Program–Fortran (HSPF) Model

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Study Site and Data Collection

#### 2.2. Description of the HSPF Model

#### 2.3. Multi-Objective Calibration

_{10%}, N

_{50%}, and N

_{90%}represent the fractions of time that streamflow equals or exceeds specific flow rates.

_{n}is the number of hourly time steps in event n, i is the time step index, n is the total number of calibrated events, V is the flow volume of an event, and P is the peak flow rate of an event. For each equation, the performance of a model simulation was assumed to be better when the objective function was closer to zero.

^{2}), which are detailed in the Appendix A. For a better comparison of the parameter estimation between the two scenarios, the best overall performance (BOP) solution and optimized parameter range were both used. The BOP solution was selected by calculating and comparing the Euclidean distances between coordinate origin and Pareto solutions [50]. The optimized range for each parameter was determined as an NSE value more than 0.5.

#### 2.4. Sensitivity and Uncertainty Analysis

#### 2.4.1. Regional Sensitivity Analysis

#### 2.4.2. Generalized Likelihood Uncertainty Estimation

## 3. Results and Discussion

#### 3.1. Comparison of Parameter Calibration

^{2}, NSE, and RMSE) in Table 4 and Table 5. The HSPF model showed good performances compared to literature regarding hydrological modelling using the HSPF model [57,58,59].

#### 3.2. Comparison of Parameter Sensitivity

#### 3.3. Comparison of Uncertainty Quantification

## 4. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A

**Table A1.**Performance variables and criteria of the HSPEXP method [48].

Measure | Criteria (% Error) |
---|---|

Total runoff | ±10% |

Highest 10% flows | ±10% |

Lowest 50% flows | ±15% |

Seasonal volume | ±10% |

Storm peak | ±15% |

Summer storm volume | ±15% |

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**Figure 3.**Water storages and movement represented in the PERLND module. ET: evapotranspiration, AGWS: active ground water storage, LZS: lower zone storage, UPS: upper zone storage, IFWS: interflow storage, SURS: surface detention storage, CEPS: Interception storage.

**Figure 5.**Pareto sets achieved by the multi-objective calibration. The left part is for the continuous modeling, where green, blue, and red dots represent solutions those that achieved four, five, and six criteria of the HSEXP statistics, respectively. The right part is for the event-based modeling, where green dots represent all the Pareto solutions. The original MATLAB figure files (Pareto front-continuous scenario.fig and Pareto front-event-based scenario.fig) are provided as Supplementary Materials.

**Figure 6.**Hyetograph, simulated and observed hydrographs in the calibration and validation periods for the continuous modeling.

**Figure 7.**Hyetograph, and simulated and observed hydrographs in the calibration and validation periods for the 12 rainfall–runoff events.

**Figure 8.**Comparison of the optimized parameter ranges between the continuous and event-based modeling.

**Figure 9.**Comparison of parameter sensitivity between the continuous and event-based modeling Cumulative normalized Nash–Sutcliffe efficiency were plotted against parameter values. Parameter populations from best to worst are ranked and divided into ten bins of equal size.

**Figure 10.**Observations and 95 PPU bounds of streamflow in the continuous modeling. The left and right parts present results of daily and monthly simulation, respectively.

**Figure 11.**Hyetograph, observations, and 95 PPU bounds of streamflow in the representative rainfall–runoff events.

Event Type | 24-h Cumulative Amount | 12-h Cumulative Amount |
---|---|---|

Light | 0.1 mm–9.9 mm | 0.1 mm–4.9 mm |

Medium | 10 mm–24.9 mm | 5.0 mm–14.9 mm |

Heavy | 25.0 mm–49.9 mm | 15.0 mm–29.9 mm |

Very heavy | 50.0 mm–99.9 mm | 30.0 mm–69.9 mm |

Extremely heavy | 100 mm–249.9 mm | 70.0 mm–139.9 mm |

Torrential | ≥250 mm | ≥140 mm |

Event | Start Date | Event Type | Rainfall Amount (mm) | Rainfall Duration (h) | Maximum Intensity (mm/h) |
---|---|---|---|---|---|

1 | 28 August 2013 | Very heavy | 56.5 | 3 | 40.5 |

2 | 21 April 2014 | Medium | 24.0 | 6 | 9.2 |

3 | 20 July 2014 | Heavy | 34.7 | 3 | 28.7 |

4 | 24 July 2014 | Heavy | 33.0 | 5 | 19.6 |

5 | 8 May 2014 | Extremely heavy | 104.7 | 10 | 45.4 |

6 | 9 February 2014 | Heavy | 48.2 | 8 | 10.0 |

8 | 25 June 2013 | Heavy | 31.6 | 6 | 30.3 |

11 | 24 September 2013 | Heavy | 48.1 | 20 | 4.4 |

7 | 15 April 2014 | Medium | 18.2 | 2 | 14.6 |

9 | 25 June 2014 | Heavy | 37.4 | 11 | 7.9 |

10 | 7 August 2014 | Very heavy | 69.2 | 8 | 27.4 |

12 | 23 August 2014 | Heavy | 30.4 | 7 | 11.2 |

Parameter | Description | Unit | Possible Range | Calibrated Values for Continuous | Calibrated Values for Event-Based |
---|---|---|---|---|---|

LZSN | Low zone nominal soil moisture storage | in. | 2.0–15.0 | 5.502 | 3.703 |

INFILT | Index to mean soil infiltration rate | in./h | 0.001–0.50 | 0.079 | 0.067 |

KVARY | Parameter to describe non-linear groundwater recession rate | in^{−1} | 0.0–5.0 | 3.643 | 0.202 |

AGWRC | Groundwater recession rate | day^{−1} | 0.85–0.999 | 0.987 | 0.850 |

DEEPFR | Fraction of infiltrating water which enters deep aquifers | none | 0.0–0.50 | 0.005 | 0.057 |

BASETP | Fraction of potential evapotranspiration which fulfilled only as outflow exists | none | 0.0–0.20 | 0.000 | 0.161 |

AGWETP | Fraction of remaining evapotranspiration that be met from active groundwater storage | none | 0.0–0.20 | 0.000 | 0.166 |

CEPSC | Interception storage capacity | in. | 0.01–0.40 | 0.010 | 0.123 |

UZSN | Nominal upper zone soil moisture storage | in. | 0.05–2.0 | 0.668 | 0.415 |

INTFW | Interflow inflow parameter | none | 1.0–10.0 | 1.000 | 9.999 |

IRC | Interflow recession parameter | day^{−1} | 0.3–0.85 | 0.300 | 0.300 |

LZETP | Index to lower zone evapotranspiration | none | 0.1–0.9 | 0.400 | 0.494 |

Measures | Observed | Simulated | Percent Error % | HSPEXP Performance Criteria |
---|---|---|---|---|

Total runoff (mm) | 543.34 | 534.08 | −1.70 | √ |

Mean of highest 10% flows (m^{3}) | 0.05692 | 0.0535 | −5.97 | √ |

Mean of lowest 50% flows (m^{3}) | 0.0062 | 0.0059 | −4.54 | √ |

Seasonal volume error (mm) | 297.25 | 268.43 | −9.70 | √ |

Mean of storm peaks (m^{3}) | 0.0960 | 0.0940 | −2.06 | √ |

Summer storm volume (mm) | 50.13 | 46.24 | −7.76 | √ |

Coefficient of determination | 0.83 | |||

Nash–Sutcliffe efficiency | 0.82 | |||

Root mean square error | 0.76 |

Event | Start date | NSE | RMSE | R^{2} | PFE ^{a} | SVE ^{b} | PTO ^{c} |
---|---|---|---|---|---|---|---|

1 | 28 August 2013 | 0.91 | 9.72 | 0.92 | 0.16 | 0.05 | 0 |

2 | 21 April 2014 | 0.65 | 1.13 | 0.71 | 0.04 | 0.13 | 0 |

3 | 20 July 2014 | 0.93 | 0.60 | 0.93 | 0.01 | 0.05 | 0 |

4 | 24 July 2014 | 0.90 | 2.11 | 0.93 | 0.20 | 0.10 | 0 |

5 | 5 August 2014 | 0.79 | 27.91 | 0.81 | 0.08 | 0.34 | 0 |

6 | 2 September 2014 | 0.73 | 3.22 | 0.82 | 0.07 | 0.22 | 0 |

7 | 25 June 2013 | 0.83 | 16.17 | 0.98 | 0.38 | 0.29 | 0 |

8 | 24 September 2013 | 0.74 | 0.71 | 0.92 | 0.10 | 0.05 | 5 |

9 | 15 April 2014 | 0.49 | 0.57 | 0.77 | 0.18 | 0.15 | 0 |

10 | 25 June 2014 | 0.83 | 0.48 | 0.84 | 0.07 | 0.04 | 0 |

11 | 7 August 2014 | −0.1172 | 60.81 | 0.12 | 0.21 | 0.28 | 5 |

12 | 23 August 2014 | 0.82 | 0.60 | 0.95 | 0.25 | 0.02 | 0 |

^{a}PFE stands for peak flow error;

^{b}SVE stands for storm volume error;

^{c}PTO stands for peak time offset.

Water Balance | Forest | Tea | Agriculture | Urban | ||||
---|---|---|---|---|---|---|---|---|

Surface flow | 9.23% | 20.9% | 5.76% | 30.13% | 3.44% | 27.53% | 6.34% | 8.44% |

Interflow | 2.69% | 63.03% | 4.34% | 54.92% | 6.11% | 65.61% | 2.75% | 64.33% |

Base flow | 8.45% | 1.31% | 37.6% | 1.95% | 28.46% | 0.03% | 27.44% | 6.06% |

Deep aquifer | 0.52% | 2.65% | 2.26% | 2.92% | 1.72% | 0.08% | 1.65% | 5.72% |

Total ET | 79.1% | 12.11% | 50.04% | 10.08% | 60.26% | 6.74% | 61.82% | 15.45% |

Scenario | Period | p-Factor (%) | r-Factor |
---|---|---|---|

Continuous | 2011–2012 (daily) | 40.25 | 2.58 |

2011–2012 (monthly) | 70.83 | 0.85 | |

Event-based | Event 1 (Very heavy) | 44.44 | 1.20 |

Event 2 (Medium) | 68.75 | 0.52 | |

Event 3 (Heavy) | 33.3 | 1.13 | |

Event 4 (Heavy) | 41.67 | 1.21 | |

Event 5 (Extremely heavy) | 41.17 | 1.79 | |

Event 6 (Heavy) | 65 | 1.17 |

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**MDPI and ACS Style**

Xie, H.; Shen, Z.; Chen, L.; Lai, X.; Qiu, J.; Wei, G.; Dong, J.; Peng, Y.; Chen, X.
Parameter Estimation and Uncertainty Analysis: A Comparison between Continuous and Event-Based Modeling of Streamflow Based on the Hydrological Simulation Program–Fortran (HSPF) Model. *Water* **2019**, *11*, 171.
https://doi.org/10.3390/w11010171

**AMA Style**

Xie H, Shen Z, Chen L, Lai X, Qiu J, Wei G, Dong J, Peng Y, Chen X.
Parameter Estimation and Uncertainty Analysis: A Comparison between Continuous and Event-Based Modeling of Streamflow Based on the Hydrological Simulation Program–Fortran (HSPF) Model. *Water*. 2019; 11(1):171.
https://doi.org/10.3390/w11010171

**Chicago/Turabian Style**

Xie, Hui, Zhenyao Shen, Lei Chen, Xijun Lai, Jiali Qiu, Guoyuan Wei, Jianwei Dong, Yexuan Peng, and Xinquan Chen.
2019. "Parameter Estimation and Uncertainty Analysis: A Comparison between Continuous and Event-Based Modeling of Streamflow Based on the Hydrological Simulation Program–Fortran (HSPF) Model" *Water* 11, no. 1: 171.
https://doi.org/10.3390/w11010171