# Global Atmospheric Dynamics Investigated by Using Hilbert Frequency Analysis

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## Abstract

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## 1. Introduction

## 2. Dataset and Method

## 3. Results

## 4. Discussion

## 5. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**Figure 1.**Calculation of the instantaneous frequency from surface air temperature (SAT) daily data, at a site which is located in east Asia and is indicated with a circle in Figure 2. (

**a**) Original SAT series; (

**b**) Transformed series; (

**c**) Trajectory; (

**d**) Phase series; (

**e**) Frequency series; (

**f**) Probability distribution function (PDF) of instantaneous frequency.

**Figure 2.**(

**a**) Map of time-averaged frequency displayed in colour coded units of rad/day. The colour scale is adjusted such that the instantaneous frequency corresponding to the annual cycle is displayed in white. The circle, triangle, and square indicate the geographical locations considered in Figure 1, Figure 3, and Figure 4, respectively; (

**b**) Distribution of time-averaged frequency, weighted according to the area of each site. The blue vertical dashed line indicates the value of frequency that corresponds to the annual cycle (i.e., $0.017\mathrm{rad}/\mathrm{day}$).

**Figure 3.**Analysis of a site that is located in the west Pacific Ocean (indicated in Figure 2a with a triangle). In this region, the average frequency is higher than the expected value. (

**a**) Trajectory; (

**b**) Distribution of instantaneous frequency; (

**c**) Time series of instantaneous frequency.

**Figure 4.**Analysis of a site that is located in the central Pacific Ocean, at the border of the area of high frequency (indicated in Figure 2a with a square). (

**a**) Trajectory; (

**b**) Distribution of instantaneous frequency; (

**c**) Time series of instantaneous frequency.

**Figure 5.**(

**a**) Map of standard deviation of frequency fluctuations, where the colour scale is in units of rad/day; (

**b**) Map of annual mean precipitation in units of mm/day. (We reproduced this figure using data from The Version 2 Global Precipitation Climatology Project (GPCP) [15].)

**Figure 6.**Hilbert frequency calculated as the linear trend of the phase in a window of one month. (

**a**) Map of average frequency. (

**b**) PDF of averaged frequency, weighted according to the area of each site. The vertical line indicates the value of frequency that corresponds to the annual cycle (i.e., $0.017\mathrm{rad}/\mathrm{day}$). (

**c**) Map of standard deviation of the frequency. (

**d**–

**f**) PDF of instantaneous frequency in the regions indicated with a circle, a triangle, and a square, respectively, in Figure 2a.

**Figure 7.**Results obtained by selecting summer or winter in the frequency series. The four colour scales are in units of rad/day. (

**a**) Average frequency in summer; (

**b**) Average frequency in winter; (

**c**) Standard deviation of frequency in summer; (

**d**) Standard deviation of frequency in winter.

© 2016 by the authors; licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC-BY) license (http://creativecommons.org/licenses/by/4.0/).

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**MDPI and ACS Style**

Zappalà, D.A.; Barreiro, M.; Masoller, C.
Global Atmospheric Dynamics Investigated by Using Hilbert Frequency Analysis. *Entropy* **2016**, *18*, 408.
https://doi.org/10.3390/e18110408

**AMA Style**

Zappalà DA, Barreiro M, Masoller C.
Global Atmospheric Dynamics Investigated by Using Hilbert Frequency Analysis. *Entropy*. 2016; 18(11):408.
https://doi.org/10.3390/e18110408

**Chicago/Turabian Style**

Zappalà, Dario A., Marcelo Barreiro, and Cristina Masoller.
2016. "Global Atmospheric Dynamics Investigated by Using Hilbert Frequency Analysis" *Entropy* 18, no. 11: 408.
https://doi.org/10.3390/e18110408