# Hydrometeorological Forecast of a Typical Watershed in an Arid Area Using Ensemble Kalman Filter

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Overview of the Research Area

## 3. Data and Methods

#### 3.1. Data

^{6}m

^{3}), and temperature (°C).

#### 3.2. Algorithm

#### 3.2.1. Stationarity Test

#### 3.2.2. Wavelet Analysis and Wavelet Decomposition

#### 3.2.3. Traditional Framework of EnKF (Ensemble Kalman Filter)

^{a}, at the time of the observation by combining the ensemble model forecast X

^{f}with observations Y. The ensemble methods discussed in this section are based on the Kalman filter (Kalman, 1960) where the updated ensemble mean follows the Kalman update for the state [30], given by:

#### 3.2.4. Preprocessing of Filter Input Data

#### 3.2.5. Accuracy Verification

_{0}is the observed value, Q

_{m}is the predicted value, Q

^{t}represents the observed value at time t, and Q

_{0}represents the total average of the observed values. The closer the value is to 1, the higher the efficiency of the model. The training and test sets were selected, but the test data will not participate in model generation. This selection enables the model to give predictions for each parameter and evaluate the prediction model without subjective bias.

## 4. Results Analysis

#### 4.1. Spatial and Temporal Distribution Characteristics of Parameters

#### 4.2. Stationarity Test and Wavelet Decomposition

#### 4.3. Prediction

#### 4.4. Reducing Data Overreliance and Improving Prediction Ability

#### 4.5. Relationship between Precipitation and Altitude

#### 4.6. Relationship between Evaporation and Temperature

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A

## Appendix B

^{k}convergence to zero matrix while $k\to \infty $. On the other hand, P is a positive definite symmetric matrix. The trace of the left side in Equation (A22) can be constrained by:

_{m}and its limit (if exists) can be calculated as:

_{1}and C

_{2}are constants independent with m. This proves that the input provided by wavelet decomposition will get the asymptotic optimal estimation under EnKF processing, even if the covariance matrix of the measurement matrix may not be convergent. The first inequality of Equation (A26) can be used to provide the confidence interval of hydrometeorological parameter prediction for each time node m.

## Appendix C

_{3}, r

_{4}, r

_{5}, r

_{6}, r

_{7}< 1, r

_{8}= max(r

_{3}, r

_{4}, r

_{5}, r

_{6}, r

_{7}), C

_{3}, …, C

_{8}are constants independent of m, i and each other. It can be seen from Equation (A31) that the robustness of the model varies in different application scenarios. The instability brought by the new data was provided by multiple structures of both the system equation and measurement equation. However, at least in theory, the robustness of the model is excellent.

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**Figure 1.**Sites in the study area, with DEM distribution. The map is from Chinese Standard Map (http://bzdt.ch.mnr.gov.cn/, GS (2019)1822 (accessed on 22 November 2022))

**Figure 3.**Periodic wavelet decomposition based on multi-year moving average. The red boxes represent significant downtrends, which were observed by stationarity test.

**Figure 6.**Relationship between precipitation and altitude. In the figure, the blue line is the regression line which represents the optimal parameters, the gray area is enclosed by the upper and lower confidence bounds under the 0.95 confidence level, and the red points represent the meteorological station.

Site Name | Longitude | Latitude | Altitude (m) |
---|---|---|---|

Karamay | 84°51′ | 45°37′ | 449.5 |

Shawan | 85°37′ | 44°20′ | 522.2 |

Manas | 86°12′ | 44°19′ | 471.4 |

Hutubi | 86°51′ | 44°10′ | 575.1 |

Daxigou | 86°50′ | 43°06′ | 3539.0 |

Bayanbulak | 84°09′ | 43°02′ | 2458.0 |

Baluntai | 86°18′ | 44°19′ | 1739.0 |

Ulan Wusu | 84°62′ | 44°45′ | 480.6 |

Shihezi | 86°02′ | 44°18′ | 493.0 |

Parameter | Test | 2010 | 2011 | 2012 | 2013 | 2014 | 2015 | 2016 | 2017 | 2018 | 2019 | 2020 |
---|---|---|---|---|---|---|---|---|---|---|---|---|

Precipitation | MK | 0.088 | 0.064 | 0.209 | −0.037 | 0.099 | −0.035 | −0.06 | 0.49 | −0.047 | −0.089 | −0.08 |

ACF | 0.032 | 0.064 | 0.238 | −0.063 | 0.089 | 0.024 | −0.384 | −0.28 | −0.097 | −0.03 | −0.606 | |

Evaporation | MK | −0.317 | −0.092 | 0.098 | −0.085 | 0.044 | 0.035 | 0.021 | −0.132 | 0.033 | 0.07 | −0.017 |

ACF | −0.755 | −0.746 | −0.026 | −0.063 | 0.024 | 0.026 | −0.422 | −0.177 | −0.237 | 0.045 | −0.211 | |

Temperature | MK | 0.07 | −0.099 | 0.037 | −0.037 | −0.094 | 0.082 | 0.064 | −0.363 | −0.039 | −0.033 | −0.251 |

ACF | −0.048 | −0.068 | −0.074 | 0.002 | 0.099 | 0.054 | 0.034 | −0.46 | −0.195 | −0.27 | −0.256 | |

Runoff | MK | 0.002 | −0.008 | 0.01 | −0.001 | −0.001 | 0.001 | −0.006 | 0.002 | 0.004 | 0.006 | 0.002 |

ACF | 0.004 | −0.007 | −0.009 | 0 | 0.001 | 0 | 0 | 0.009 | 0.01 | −0.004 | 0.008 |

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**MDPI and ACS Style**

He, G.; Chen, Y.; Fang, G.; Li, Z. Hydrometeorological Forecast of a Typical Watershed in an Arid Area Using Ensemble Kalman Filter. *Water* **2022**, *14*, 3970.
https://doi.org/10.3390/w14233970

**AMA Style**

He G, Chen Y, Fang G, Li Z. Hydrometeorological Forecast of a Typical Watershed in an Arid Area Using Ensemble Kalman Filter. *Water*. 2022; 14(23):3970.
https://doi.org/10.3390/w14233970

**Chicago/Turabian Style**

He, Ganchang, Yaning Chen, Gonghuan Fang, and Zhi Li. 2022. "Hydrometeorological Forecast of a Typical Watershed in an Arid Area Using Ensemble Kalman Filter" *Water* 14, no. 23: 3970.
https://doi.org/10.3390/w14233970