# Residual-Oriented Optimization of Antecedent Precipitation Index and Its Impact on Flood Prediction Uncertainty

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## Abstract

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## 1. Introduction

## 2. Materials and Methods

#### 2.1. Study Area and Data

^{2}. Due to the subtropical monsoon climate, the study area is warm and humid, producing an average annual flow of $3.75\times {10}^{9}$ m

^{3}at the outlet of the basin (Changshangang Station). The average annual precipitation of the study area is about 1500 to 2300 mm, while 80% of the precipitation falls between April and September. The uneven temporal distribution of the precipitation causes frequent flood disasters along the river network during rainy periods.

^{3}/s to 5278.3 m

^{3}/s. Hourly precipitation series of 12 rain gages in the Changshangang River basin are collected and then averaged by the Thiessen polygon method. For the same period, hourly stream flow and potential evapotranspiration data are collected from the Changshangang Station. Note that an antecedent 240 h (ten days) data series of warmup periods is included in each flood event. All data are obtained from the Taihu Basin Authority of the Ministry of Water Resources.

#### 2.2. The Xinanjiang Model

_{m}, L

_{m}, and D

_{m}, respectively; the impervious area of the sub-basin I

_{m}; the areal mean free water storage capacity S

_{m}; the ratio of potential evapotranspiration to pan evaporation K; the exponent of the tension water capacity curve B; the coefficient of deep evapotranspiration C; the runoff-producing area F

_{r}; as well as ten runoff routing component parameters: outflow coefficients of the free water storage to groundwater and interflow K

_{g}and K

_{i}, respectively; recession constants of groundwater, interflow, and channel system C

_{g}, C

_{i}, and C

_{s}, respectively; Muskingum parameters K

_{e}and X

_{e}; the lag time of routing L. For a detailed explanation of the parameters, readers should refer to [20].

#### 2.3. Kernel-Based Residual Error (KRE) Model

_{m}is called the sensitivity parameter, which satisfies 0 ≤ α

_{m}≤ 1. The adaptive bandwidth selector reduces to the fixed bandwidth when α

_{m}= 0 and equals to the nearest neighbor estimator when α

_{m}= 1.

_{σ}according to Equation (6).

#### 2.4. Residual-Oriented Antecedent Precipitation Index (RAPI)

_{t}is the precipitation in the tth antecedent hour, M is the statistical number of antecedent hours involving the estimation of RAPI, and k is the decay constant. The parameters of the RAPI model are then ${\theta}_{A}=\left\{k,M\right\}$.

#### 2.5. Calibration Method

#### 2.5.1. The MI-LXPM Algorithm

#### 2.5.2. Two-Stage Calibration Procedure

#### 2.6. Probabilistic Predictions

- (1)
- Sample innovations from the inverse of the estimated innovation distribution in Equation (8):$${\epsilon}_{t}^{\left(r\right)}\leftarrow {F}^{-1}(\epsilon |b)$$
- (2)
- Model temporal structure with Equation (7):$${\delta}_{t}^{\left(r\right)}=\widehat{\phi}{\delta}_{t-1}^{\left(r\right)}+{\epsilon}_{t}^{\left(r\right)}$$
- (3)
- Calculate the value of regression function ${\widehat{m}}_{n}(\xb7)$ and CV function ${\widehat{\sigma}}_{n}(\xb7)$ by substituting the corresponding regressor ${R}_{t}$ into Equations (4) and (6) sequentially.
- (4)
- Generate samples of ${e}_{t}^{\left(r\right)}$ using Equation (3):$${e}_{t}^{\left(r\right)}={\widehat{m}}_{n}\left({R}_{t}\right)+{\widehat{\sigma}}_{n}\left({R}_{t}\right){\delta}_{t}^{\left(r\right)}$$
- (5)
- Combine with the deterministic model output ${\widehat{y}}_{t}$$${y}_{t}^{\left(r\right)}={\widehat{y}}_{t}+{e}_{t}^{\left(r\right)}$$

#### 2.7. Probabilistic Prediction Performance Metrics

#### 2.7.1. Reliability Metric

#### 2.7.2. Precision Metric

## 3. Results and Discussion

#### 3.1. Flood Prediciton Residuals and RAPI Estimates

^{3}/s at 23:00 24 June 2017) and absolute residual value (1746 m

^{3}/s at 2:00 25 June 2017). Figure 6 shows that lower values of the optimal RAPI correspond to stable residuals near zero, while a larger RAPI follows with strong fluctuations in residuals. This phenomenon suggests that the optimal RAPI can capture the heteroscedasticity of the residual.

#### 3.2. Impact of the Optimal RAPI on Flood Residual

^{3}/s corresponds to the lowest RAPI of 0 mm, while a maximum CV value of 782.5 m

^{3}/s corresponds to the largest RAPI of 149.82 mm. The increasing trend in the CV function of the residual suggests that the predictive uncertainty magnifies almost 20 times between the minimal and maximum optimal RAPI value. A relatively high R-squared value of 0.47 suggests that RAPI provides an adequate representation of changes in the CV function of flood residuals.

#### 3.3. Stochastic Predictive Performance

#### 3.4. Impact of Soil Moisture on Predictive Performance of Flood

_{S}illustrate statistically significant linear trends. According to the definition of the two metrics, although given opposite trends, the result suggests improved predictive precision and unbiasedness. In the comparison of Figure 11a,b, the unbiasedness of flood prediction is found to be more sensitive to the variation in soil moisture. At the same time, the predictive precision is more sensitive to the volume of soil moisture.

#### 3.5. Comparison of the Regressor

#### 3.6. Limitations and Future Work

## 4. Conclusions

- For hourly flood predictions, the optimal RAPI can be the weighted average of hourly precipitation falls in the antecedent days with a mild decay. The distribution of the optimal RAPI is found to be highly peaked with positive skewness.
- The optimal RAPI influences the residual conditional volatility more than the conditional mean. As a result, a poor bias-correction ability can be found when making probabilistic flood predictions with RAPI.
- The reliability of probabilistic flood prediction is almost independent of the RAPI value. On the contrary, prediction precision and unbiasedness are found to improve with increasing value and variability of the RAPI.
- As a regressor, the RAPI produces better probabilistic flood predictions for small to median flood events than the deterministic model output $\widehat{\mathit{y}}$. On the contrary, $\widehat{\mathit{y}}$ provides better predictions of extreme flood events.

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Map of the Changshangang River Basin. Rain gages are shown with green circles, and runoff observations are collected from the Changshangang station (red circle).

**Figure 3.**Scatter plot of hourly flow observations and the discharge simulated by the calibrated XAJ model for all 22 flood events. The fitted linear trend (red dashed line) and the 1:1 line (black solid line) are also shown.

**Figure 5.**Boxplots for the optimal RAPI samples in different months. On each box, the central mark indicates the median, and the bottom and top edges of the box indicate the 25th and 75th percentiles, respectively. Outliers are plotted using solid points.

**Figure 6.**Model residual and RAPI time series of the flood event with the largest peak flow. The optimal RAPI estimates are plotted using the blue solid line, and residuals are plotted using black triangle markers.

**Figure 7.**Illustration of the impact of the optimal RAPI on flood simulation residuals: (

**a**) Optimal kernel regression function and (

**b**) conditional volatility function of the residuals given RAPI as the regressor. The R-squared values are also shown.

**Figure 8.**Statistics for the innovation of the KA scenario: (

**a**) autocorrelation (black dots) for different lags with 95% significance levels (red dash line); (

**b**) histogram and kernel estimate of the probability density function of the innovation. The kernel density is estimated using Equation (8) with an optimal bandwidth b = 0.08.

**Figure 9.**Typical examples of probabilistic streamflow predictions in the Changshangang River Basin for the ‘KA’ scenario: (

**a**) mean and the 95% PUBs for the flood prediction with deterministic model output and flood observations; (

**b**) PQQ plot for all simulated flood events.

**Figure 10.**Typical examples of probabilistic streamflow predictions in the Changshangang River Basin for the ‘KF’ scenario: (

**a**) mean and the 95% PUBs for the flood prediction with deterministic model output and flood observations; (

**b**) PQQ plot for all simulated flood events.

**Figure 11.**Scatter plot of normalized predictive performance metric values: normalized metric values as a function of (

**a**) the natural logarithm of the mean optimal RAPI and (

**b**) the standard deviation of optimal RAPI for each flood event. Linear trends and corresponding R-squared value.

**Figure 12.**Comparison of the predictive performance of the ‘KA’ and ‘KF’ scenarios under different total flood volume: (

**a**) scatter plot of the reliability metric; (

**b**) scatter plot of the precision metric; (

**c**) scatter plot of the NSE

_{S}. Fitted linear trends and corresponding R-squared values are also shown. Blue markers and lines represent the ‘KA’ scenario, while red markers and lines represent the ‘KF’ scenario. The natural logarithm is implemented to the flood volume to strengthen the linear trends.

Parameter | K * | B * | W_{m} * | I_{m} | F_{r} | U_{m} | L_{m} | C | D_{m} | S_{m} * |
---|---|---|---|---|---|---|---|---|---|---|

Upper limit | 0.7 | 0.1 | 100 | - | - | - | - | - | - | 5 |

Estimate | 1.190 | 0.662 | 136.483 | 0.010 | 0 | 20.000 | 60.000 | 0.180 | 56.483 | 10.453 |

Lower limit | 1.3 | 0.8 | 150 | - | - | - | - | - | - | 50 |

Parameter | E_{x} | K_{i} | K_{g} * | C_{i} * | C_{g} * | C_{s} * | L * | X_{e} * | K_{e} * | |

Upper limit | - | - | 0.01 | 0.8 | 0.93 | 0 | 0 | −0.5 | 1 | |

Estimate | 1.500 | 0.359 | 0.341 | 0.892 | 0.995 | 0.865 | 6 | −0.195 | 1.500 | |

Lower limit | - | - | 0.69 | 0.95 | 0.995 | 1 | 20 | 0.5 | 2.5 |

Parameter | ${\mathit{\alpha}}_{\mathit{m}}$ | ${\mathit{\alpha}}_{\mathit{\sigma}}$ | $\mathit{\phi}$ | $\mathit{b}$ | $\mathit{k}$ | $\mathit{M}$ |
---|---|---|---|---|---|---|

Upper limit | 0.01 | 0.01 | 0.01 | 0.10 | 0.10 | 1 |

Estimates for KA | 0.92 | 0.10 | 0.07 | 0.93 | 0.97 | 51 |

Estimates for KF | 0.59 | 0.22 | 0.12 | 0.95 | - | - |

Lower limit | 1.00 | 1.00 | 10.00 | 0.99 | 0.99 | 240 |

Scenario | Reliability Metric | Precision Metric | $\overline{\mathbf{N}\mathbf{S}{\mathbf{E}}_{\mathit{S}}}$^{1} | −log(L_{k}) |
---|---|---|---|---|

KA | 0.04 | 0.33 | 0.82 | 30077 |

KF | 0.04 | 0.35 | 0.84 | 30221 |

^{1}${\mathrm{NSE}}_{\mathrm{S}}$ represents the NSE value comparing flood observations and the mean of probabilistic predictions. $\overline{{\mathrm{NSE}}_{\mathrm{S}}}$ then evaluates the average bias-correction ability for each modeling scenario.

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## Share and Cite

**MDPI and ACS Style**

Liang, J.; Hu, Z.; Liu, S.; Zhong, G.; Zhen, Y.; Makhinov, A.N.; Araruna, J.T.
Residual-Oriented Optimization of Antecedent Precipitation Index and Its Impact on Flood Prediction Uncertainty. *Water* **2022**, *14*, 3222.
https://doi.org/10.3390/w14203222

**AMA Style**

Liang J, Hu Z, Liu S, Zhong G, Zhen Y, Makhinov AN, Araruna JT.
Residual-Oriented Optimization of Antecedent Precipitation Index and Its Impact on Flood Prediction Uncertainty. *Water*. 2022; 14(20):3222.
https://doi.org/10.3390/w14203222

**Chicago/Turabian Style**

Liang, Jiyu, Zichen Hu, Shuguang Liu, Guihui Zhong, Yiwei Zhen, Aleksei Nikolavich Makhinov, and José Tavares Araruna.
2022. "Residual-Oriented Optimization of Antecedent Precipitation Index and Its Impact on Flood Prediction Uncertainty" *Water* 14, no. 20: 3222.
https://doi.org/10.3390/w14203222