# Mechanism Analysis and Demonstration of Effective Information Extraction in the System Differential Response Inversion Estimation Method

^{1}

^{2}

^{3}

^{4}

^{5}

^{*}

## Abstract

**:**

## 1. Problems of the Optimization Based on Objective Functions

_{1}(a)/da = 0 and dF

_{2}(a)/da = 0, Equation (5) is obtained

## 2. The Inverse Estimation Method of System Differential Response and Theoretical Demonstration of Its Effective Information Extraction

#### 2.1. Basic Method

#### 2.2. The Relation Degree Coefficient and the Information Extraction Mechanism

## 3. Analysis and Demonstration of Information Extraction Effect

#### 3.1. Synthetic Case

#### 3.2. Single-Factor Information Extraction under the Influence of Noise Intensity

#### 3.3. Multi-Factor Information Extraction Affected by Noise Intensity

#### 3.3.1. Feature Analysis of Distinguishing and Extracting Information for Two-Factor Inversion Estimation

#### 3.3.2. Demonstration of Multi-Factor Information Extraction Effect

^{2}is the coefficient of determination, and RE is the relative effective coefficient. An R

^{2}value of 1 indicates a perfect match of the model output to the observed data, while an R

^{2}value of 0 means that the performance of the model is only as accurate as the mean of the observations. RE represents the degree of reduction of the model calculation error using the inverse estimation method of system differential response compared to the original simulations. P, E, W, and R in brackets after RE are the estimated rainfall, evaporation, initial soil water, and runoff, respectively. The R

^{2}and RE calculation formulas are shown in Equations (15) and (16), respectively.

^{2}underscore the significant reduction in forecasting errors achieved by implementing distinct inversion estimates for each flood event. The average REs of 32 floods are 0.832, 0.726, 0.695, 0.904, and 0.835 for single-factor P, e, W, R, and multi-factor PEW, respectively. The most effective among them is the inversion estimation of runoff, followed by the simultaneous inversion estimation of P, E, and W, followed by the single-factor estimation of P, E, and W. The reason is that most of the error factors, including the errors from rainfall, evaporation, initial soil moisture, and the structure of the runoff model, are embedded in runoff errors and thus are considered indirectly when we estimate the runoff errors. However, the simultaneous estimation of P, E, and W only considers the errors of P, E, and W without considering those of the model structure. The single-factor estimation considers a single error factor, resulting in a lower RE. The results show that the more error factors there are involved in the inversion estimation, the more effective information contained in the flow hydrograph can be extracted by the system differential response, which makes the inversion estimation better.

## 4. Conclusions

- (1)
- The inversion estimation method of system differential response can selectively extract effective information, and its mechanism is based on the correlation between the system differential response curve of the factors to be estimated and the information contained in the flow hydrograph.
- (2)
- The more error factors considered in the inversion estimation, the more effective information contained in the flow hydrograph can be extracted by the system differential response, which makes the inversion estimation better.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**The unit curves of flow routing. Us is the unit hydrograph of surface runoff, and Ug is the unit hydrograph of underground runoff.

**Figure 2.**Model-calculated flow deviations and error series. DQ represents the total residual information process of the flood discharge, and e represents the error series.

**Figure 3.**The process line of extracted effective information. IRS and IRG represent the effective information-extracting process of surface runoff and underground runoff extracted by the system differential response inversion estimation method.

**Table 1.**Results of the estimated runoff obtained by the system differential response method. V1 and V2 are the mean square errors of estimated runoff. δ1 and δ2 are the relative errors of the estimated runoff yield. α represents the proportional coefficient between the maximum possible error and the average discharge.

α | V1 | V2 | δ1 | δ2 |
---|---|---|---|---|

0 | 0 | 0 | 0 | 0 |

0.01 | 0.002 | 0.002 | 0.002 | 0.001 |

0.02 | 0.004 | 0.004 | 0.004 | 0.003 |

0.03 | 0.007 | 0.006 | 0.006 | 0.005 |

0.04 | 0.010 | 0.008 | 0.008 | 0.006 |

0.05 | 0.013 | 0.010 | 0.010 | 0.008 |

0.06 | 0.015 | 0.013 | 0.012 | 0.010 |

0.07 | 0.017 | 0.014 | 0.014 | 0.011 |

0.08 | 0.020 | 0.017 | 0.016 | 0.013 |

0.09 | 0.023 | 0.019 | 0.018 | 0.015 |

0.10 | 0.024 | 0.020 | 0.020 | 0.016 |

0.11 | 0.027 | 0.023 | 0.022 | 0.018 |

0.12 | 0.031 | 0.026 | 0.025 | 0.021 |

0.13 | 0.032 | 0.027 | 0.026 | 0.021 |

0.14 | 0.036 | 0.030 | 0.028 | 0.024 |

0.15 | 0.039 | 0.033 | 0.031 | 0.026 |

0.16 | 0.040 | 0.034 | 0.032 | 0.027 |

0.17 | 0.042 | 0.035 | 0.033 | 0.028 |

0.18 | 0.045 | 0.038 | 0.036 | 0.030 |

0.19 | 0.049 | 0.041 | 0.039 | 0.032 |

0.20 | 0.048 | 0.040 | 0.038 | 0.032 |

**Table 2.**Multi-factor inversion estimation performance. Flood code represents a unique identifier assigned to each analyzed flood event. R

^{2}represents the coefficient of determination indicating the reliability of the model fit. RE(P), RE(E), and RE(W) represent the relative error attributed to discrepancies in precipitation estimation, evaporation estimation, and initial soil water content estimation, respectively. RE(PEW) represents the combined relative error for factors P, E, and W, showing their compounded impact.

Flood Code | R^{2} | RE(P) | RE(E) | RE(W) | RE(PEW) | RE(R) |
---|---|---|---|---|---|---|

31060517 | 0.994 | 0.420 | 0.000 | 0.271 | 0.581 | 0.736 |

31070613 | 0.963 | 0.907 | 0.899 | 0.901 | 0.938 | 0.968 |

31070917 | 0.986 | 0.786 | 0.541 | 0.518 | 0.827 | 0.891 |

31071007 | 0.951 | 0.932 | 0.841 | 0.828 | 0.936 | 0.96 |

31080609 | 0.985 | 0.919 | 0.878 | 0.853 | 0.935 | 0.963 |

31080610 | 0.985 | 0.612 | 0.625 | 0.629 | 0.660 | 0.710 |

31090808 | 0.968 | 0.953 | 0.893 | 0.960 | 0.920 | 0.967 |

31090812 | 0.990 | 0.64 | 0.554 | 0.460 | 0.790 | 0.797 |

31100410 | 0.979 | 0.852 | 0.811 | 0.854 | 0.502 | 0.899 |

31110603 | 0.950 | 0.931 | 0.288 | 0.627 | 0.850 | 0.952 |

31110611 | 0.970 | 0.852 | 0.684 | 0.548 | 0.892 | 0.896 |

31110617 | 0.973 | 0.848 | 0.737 | 0.729 | 0.872 | 0.944 |

31110806 | 0.944 | 0.827 | 0.939 | 0.937 | 0.929 | 0.991 |

31110825 | 0.970 | 0.89 | 0.855 | 0.909 | 0.667 | 0.964 |

31120617 | 0.983 | 0.782 | 0.781 | 0.679 | 0.694 | 0.839 |

31120714 | 0.968 | 0.831 | 0.552 | 0.769 | 0.836 | 0.922 |

31120802 | 0.954 | 0.896 | 0.737 | 0.858 | 0.833 | 0.945 |

31120807 | 0.973 | 0.936 | 0.916 | 0.690 | 0.907 | 0.952 |

31130429 | 0.967 | 0.84 | 0.759 | 0.739 | 0.932 | 0.931 |

31130606 | 0.984 | 0.723 | 0.64 | 0.557 | 0.368 | 0.857 |

31130626 | 0.974 | 0.897 | 0.798 | 0.828 | 0.920 | 0.939 |

31131005 | 0.971 | 0.928 | 0.899 | 0.830 | 0.930 | 0.947 |

31140525 | 0.987 | 0.629 | 0.663 | 0.597 | 0.778 | 0.633 |

31140620 | 0.962 | 0.877 | 0.790 | 0.761 | 0.910 | 0.914 |

31140818 | 0.986 | 0.873 | 0.853 | 0.242 | 0.895 | 0.933 |

31140918 | 0.976 | 0.854 | 0.745 | 0.575 | 0.908 | 0.909 |

31150704 | 0.981 | 0.842 | 0.775 | 0.652 | 0.876 | 0.887 |

31150710 | 0.970 | 0.818 | 0.653 | 0.492 | 0.880 | 0.888 |

31150928 | 0.975 | 0.824 | 0.517 | 0.788 | 0.925 | 0.941 |

31160527 | 0.966 | 0.875 | 0.850 | 0.401 | 0.955 | 0.962 |

31160628 | 0.939 | 0.936 | 0.916 | 0.921 | 0.945 | 0.957 |

31170611 | 0.962 | 0.886 | 0.843 | 0.828 | 0.93 | 0.947 |

Mean | 0.972 | 0.832 | 0.726 | 0.695 | 0.835 | 0.904 |

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**MDPI and ACS Style**

Chen, Y.; Liu, K.; Jiang, S.; Sun, Y.; Chen, H.
Mechanism Analysis and Demonstration of Effective Information Extraction in the System Differential Response Inversion Estimation Method. *Water* **2023**, *15*, 4016.
https://doi.org/10.3390/w15224016

**AMA Style**

Chen Y, Liu K, Jiang S, Sun Y, Chen H.
Mechanism Analysis and Demonstration of Effective Information Extraction in the System Differential Response Inversion Estimation Method. *Water*. 2023; 15(22):4016.
https://doi.org/10.3390/w15224016

**Chicago/Turabian Style**

Chen, Yang, Kexin Liu, Sijun Jiang, Yiqun Sun, and Hui Chen.
2023. "Mechanism Analysis and Demonstration of Effective Information Extraction in the System Differential Response Inversion Estimation Method" *Water* 15, no. 22: 4016.
https://doi.org/10.3390/w15224016