# Relationship between Sunspot Numbers and Mean Annual Precipitation: Application of Cross-Wavelet Transform—A Case Study

^{*}

## Abstract

**:**

## 1. Introduction

#### 1.1. Previous Studies

#### 1.2. Objectives

## 2. Materials and Methods

#### 2.1. Case Study and Data

#### 2.2. Wavelet Analysis

_{x}(s,τ), which describe the contribution of the scales s to the time series x(t) at different time positions τ [37,39]. Here, $\sqrt{\frac{\delta t}{s}}$ is a parameter used to normalize the Morlet wavelet function to unit variance, in order to allow direct comparisons of the wavelet coefficients W

_{x}(s,τ) across the different scales s and time positions τ [37,38]. These wavelet coefficients can be used to compute the bias-corrected local wavelet power, which describes how the contribution of each frequency or period in the time series varies in time [37,39,40]:

^{s}is the bias correction factor [40]. The local wavelet power spectrum can then be visualized via contour plots [38,39]. The scale s of the Morlet wavelet is related to Fourier frequency f [39,41]:

#### 2.2.1. Zero-Padding and the Cone of Influence

^{−2}[37]. Hence, any region falling below the COI is susceptible to edge effects.

#### 2.2.2. Statistical Significance Testing

^{2}represents the variance of the time series, “→” means “is distributed as,” and ${x}_{2}^{2}$ represents the chi-square distribution with two degrees of freedom. The value of P(k) is the mean wavelet power spectrum at frequency k that corresponds to the wavelet scale s [37]. Using this equation, one can construct 95% confidence contour lines at each scale using the 95th percentile of the chi-square distribution ${x}_{2}^{2}$ [37].

#### 2.2.3. Wavelet Coherence

_{x}(s,τ) and W

_{y}(s,τ). Then, the cross-wavelet transform is computed via [37,38]:

_{x}(s,τ) and W

_{y}(s,τ), and cross-wavelet spectrum W

_{x,y}(s,τ), is then computed as [38,39,45]:

## 3. Results and Discussion

## 4. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**(

**a**,

**c**): the Sun during solar maximum (19 July 2000), with high sunspot numbers (SSN); (

**b**,

**d**): the Sun during solar minimum (18 March 2009), with low SSN [3].

**Figure 2.**Selected stations in the study area [35].

**Figure 5.**(

**a**) SSN and MAP of all stations combined, and (

**b**) the XWT analysis result of the two time series.

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**MDPI and ACS Style**

Nazari-Sharabian, M.; Karakouzian, M.
Relationship between Sunspot Numbers and Mean Annual Precipitation: Application of Cross-Wavelet Transform—A Case Study. *J* **2020**, *3*, 67-78.
https://doi.org/10.3390/j3010007

**AMA Style**

Nazari-Sharabian M, Karakouzian M.
Relationship between Sunspot Numbers and Mean Annual Precipitation: Application of Cross-Wavelet Transform—A Case Study. *J*. 2020; 3(1):67-78.
https://doi.org/10.3390/j3010007

**Chicago/Turabian Style**

Nazari-Sharabian, Mohammad, and Moses Karakouzian.
2020. "Relationship between Sunspot Numbers and Mean Annual Precipitation: Application of Cross-Wavelet Transform—A Case Study" *J* 3, no. 1: 67-78.
https://doi.org/10.3390/j3010007