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Comptes Rendus Géoscience
Volume 344, n° 10
pages 483-497 (octobre 2012)
Doi : 10.1016/j.crte.2012.09.007
Received : 6 December 2011 ;  accepted : 28 September 2012
The improvement of the Morlet wavelet for multi-period analysis of climate data
Perfectionnement de la méthode de l’ondelette de Morlet pour l’analyse multipériodes des données climatiques

Hua Yi a, b, , Hong Shu c
a Department of Mathematics, Jinggangshan University, Ji’an, Jiangxi Province, 343009, People’s Republic of China 
b School of Mathematics and Statistics, Wuhan University, Wuhan, People’s Republic of China 
c The State Key Laboratory for Information Engineering in Surveying, Mapping and Remote Sensing, Wuhan University, Wuhan, People’s Republic of China 

Corresponding author. Department of Mathematics, Jinggangshan University, Ji’an, Jiangxi Province, 343009, People’s Republic of China

The multi-level dynamics of an atmosphere system exhibits temporal structures in different types of climate data. This article addresses two issues in multi-period analysis of climate data. Firstly, the advantages of the modified Morlet wavelet transform (MMWT) for analyzing multi-period structure of time series over Morlet wavelet transform (MWT) are emphasized. Secondly, the multi-period issues of temperature data are studied with MMWT through four steps: the four dominant periods of 60 year temperature data are determined with the wavelet variance; by analyzing the real part of MMWT, the warm and cold stages of the temperature data at different scales are determined, and the time intervals of the warm and cold interchange are singled out; the amplitude of each periodic component is quantitatively characterized by the amplitude of wavelet coefficients; the most intensive oscillation time intervals are computed by the squared modulus of the MMWT (MMPS).

The full text of this article is available in PDF format.

La dynamique multiniveaux d’un système atmosphérique indique des structures temporelles dans différents types de données climatiques. Cet article concerne deux aspects relatifs à l’analyse multipériodes des données climatiques. D’abord, les avantages de la transformée modifiée d’ondelette de Morlet (MMWT) pour analyser la structure multipériodes des séries temporelles sur transformée d’ondelette de Morlet (MWT) sont mis en évidence. Ensuite, les aspects multipériodes des données sur la température sont étudiés par la méthode MMWT en quatre étapes : les quatre périodes dominantes sur les mesures de température annuelle sont déterminées par la variance d’ondelette ; par l’analyse de la part réelle de MMWT, les phases de température chaudes et froides à différentes échelles sont déterminées, et les intervalles de temps dans l’échange chaud–froid distingués ; l’amplitude de chaque composant périodique est caractérisée par l’amplitude des coefficients d’ondelette ; les intervalles de temps d’oscillation les plus intenses sont calculés par le module carré de MMWT (MMPS).

The full text of this article is available in PDF format.

Keywords : Morlet wavelet, Gaussian window, Scale-to-frequency formula, Wavelet variance, Multi-period

Mots clés : Ondelette de Morlet, Fenêtre gaussienne, Formule échelle-fréquence, Variance d’ondelette, Multipériodes

 This work is jointly supported by the National Natural Science Foundation of China (No. 41171313) and the National Basic Research Program of China (973 Program) (No. 2011CB707103).

1  If we modify the wavelet defined in as:  So this wavelet is admissible by calculation that  . Furthermore, the theoretical analyses in the following part of this article for two definitions of mother wavelet are almost the same.
2  The introduction of the sampling frequency is necessary when we analyze the discrete signal. In fact, if we neglect the sampling frequency dt in Theorem 1 or in the equation (27), some errors may be produced. For example, in Table 1, the quantitative values of the row “the amplitude w.r.t MWT” is calculated from Eq. (27), and in this calculation, dt , the sampling time of the signal in Fig. 1 of this paper, is  . It is obvious that if the sampling time in Eq. (27) is not introduced, the corresponding values in Table 1 will be wrong, and will contradict with Fig. 2.
3  It is not necessary to introduce the sampling time dt in Theorem 2 while it is necessary in Theorem 1, which may be another merit of MMWT compared with MWT.
4  Wavelet coefficients are endowed with the unit of temperature due to the fact that the quantitative depict of each period of data is given by taking advantage of MMWT.

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