# A Dynamic Study of a Karst Spring Based on Wavelet Analysis and the Mann-Kendall Trend Test

^{1}

^{2}

^{3}

^{4}

^{*}

## Abstract

**:**

## 1. Introduction

^{3}/d to 100,000 m

^{3}/d, and 90% of the urban water supply was instead taken from the Yellow River (this water being classed as inferior to the spring water). Although groundwater exploitation has declined drastically in recent years, there is still a threat of spring water discontinuity during the dry season. With issues relating to flow levels in the spring not being resolved with the substantial reduction in exploitation, we can infer that influencing factors on spring dynamics due to anthropogenic activities have changed.

## 2. Materials and Methods

#### 2.1. The Background Conditions of the Study Area

#### 2.2. Data

#### 2.3. Periodic Inspection Method

_{f}(a, b) is the wavelet transform coefficient; f(t) is a signal or square-integrable function; a is the scaling scale; b is the translation parameter; and $\overline{\psi}(\frac{x-b}{a})$ is the complex conjugate function of $\psi (\frac{x-b}{a})$. Most of the time-series data observed in geosciences are discrete, and the function is f(kΔt) (k = 1, 2, ..., N); Δt is the sampling interval. The discrete wavelet transform of Equation (1) was:

#### 2.4. Trend Test Method

_{1}, x

_{2}, …, x

_{n}) and is independent and identically distributed with no trend. Suppose that H1 (bilateral test) representing the data distribution of the time series is different, and the statistical variable S to be tested is calculated as follows:

_{c}converges to a normal distribution and is calculated by:

_{c}| > Z1 − α/2 is rejected at the given α confidence level, the original hypothesis H0 is rejected, i.e., there is a clear upward or downward trend of the time series data at the α confidence level. ±Z1 − α/2 is the standard normal distribution (1 − α/2) quantile, and α is the test’s confidence level. The size of the trend can be expressed using the Kendall inclination β, which is calculated as follows:

#### 2.5. Mutation Test Method

_{k}is the cumulative number of times the value of i at the moment i is greater than the number of values at time j. Under the assumption of random independence of time series, define statistics:

_{k}) and Var(S

_{k}) are the mean and variance of the cumulative number S

_{k}, respectively. This value is calculated when x

_{1}, x

_{2}, …, x

_{n}are independent and have the same continuous distribution as:

_{i}is a standard normal distribution, which is a sequence calculated according to time series x order x

_{1}, x

_{2}, …, x

_{n}. Given a significance level a, in comparison with the data in the known normal distribution table, and if UF

_{i}> Ua, then significant changes exist in the trend. This method can also be applied to the inverse sequence of the time series, and the above procedure can be repeated by x

_{n}, x

_{n}

_{−1}, …, x

_{1}, thus making UF

_{k}= −UB

_{k}, k = n, n − 1, …, UB = 0. Given the significance level α, the two curves of UF

_{k}and UB

_{k}and the significant horizontal line are plotted on the same graph. If the values of UF

_{k}and UB

_{k}are greater than 0, then the sequence shows an upward trend; values below 0 indicate a downward trend. When the value exceeds the critical line, this indicates that the rising or falling trend is significant. The range beyond the critical line is defined as the time zone of mutation. If the UF

_{k}and UB

_{k}curves appear on an intersection point, and the intersection point is between the critical line, then the intersection point corresponds to the time the mutation begins. More detailed descriptions of this method are introduced in Reference [51].

## 3. Results

#### 3.1. General Characteristics of Atmospheric Precipitation and Spring Water Level

#### 3.2. Atmospheric Precipitation and Spring Water Level Cyclical Changes

#### 3.3. Differences between the Trend of Atmospheric Precipitation and Spring Water Level Change

#### 3.4. Atmospheric Rainfall and Detection of Spring Water Level Changes

_{1}− 0.278X

_{2}+ 0.002X

_{3}+ 0.0004X

_{4}

_{1}− 0.002X

_{2}− 0.05X

_{3}+ 0.001X

_{4}

_{1}is current precipitation; X

_{2}is urban exploitation; X

_{3}is external extraction; and X

_{4}is precipitation of the previous year.

## 4. Conclusions

## Author Contributions

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 4.**Wavelet coefficient contour map. (

**a**) For atmospheric precipitation; (

**b**) for spring water level.

**Figure 5.**Wavelet coefficient model contour map. (

**a**) For atmospheric precipitation; (

**b**) for spring water level.

**Figure 6.**Contour map of the wavelet coefficients modulus square. (

**a**) For atmospheric precipitation; (

**b**) for spring water level.

**Figure 8.**Wavelet coefficient main period chart. (

**a**) 15-year scale for atmospheric precipitation; (

**b**) 27-year scale for atmospheric precipitation; (

**c**) 15-year scale for spring water level; (

**d**) 27-year scale for spring water level.

Project | U | β |
---|---|---|

Annual groundwater level | 0.9996 | −0.065 |

Annual precipitation | 0.2047 | 1.2647 |

Influencing Factors | Year | |
---|---|---|

1960–1967 | 1968–1989 | |

Current precipitation and groundwater level | 0.959 | 0.285 |

Precipitation and groundwater level in the previous year | 0.95 | 0.094 |

The first two years of precipitation and groundwater level | 0.479 | |

Urban exploitation and groundwater level | −0.873 | −0.017 |

Periphery exploitation and groundwater level | −0.572 |

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**MDPI and ACS Style**

Xing, L.; Huang, L.; Chi, G.; Yang, L.; Li, C.; Hou, X.
A Dynamic Study of a Karst Spring Based on Wavelet Analysis and the Mann-Kendall Trend Test. *Water* **2018**, *10*, 698.
https://doi.org/10.3390/w10060698

**AMA Style**

Xing L, Huang L, Chi G, Yang L, Li C, Hou X.
A Dynamic Study of a Karst Spring Based on Wavelet Analysis and the Mann-Kendall Trend Test. *Water*. 2018; 10(6):698.
https://doi.org/10.3390/w10060698

**Chicago/Turabian Style**

Xing, Liting, Linxian Huang, Guangyao Chi, Lizhi Yang, Changsuo Li, and Xinyu Hou.
2018. "A Dynamic Study of a Karst Spring Based on Wavelet Analysis and the Mann-Kendall Trend Test" *Water* 10, no. 6: 698.
https://doi.org/10.3390/w10060698