## 1. Introduction

The technique of wavelet analysis has increasingly become a common tool for time series examinations. It not only helps overcome the limitations of Fourier transform in analyzing nonstationary time series, but also tackles the problem of constant window width and thus the time-resolution issue over frequencies (Daubechies 1992). This is done by decomposing the time series into a time–frequency space with multi-time-resolution, which allows one to determine the dominant modes of variability and the change of those modes with time (Torrence and Compo 1998). It is very useful in the field of geophysics, in which trends and periodicities of oscillations with various frequencies are often studied.

Many studies have been carried out previously on the application of wavelet analysis to geophysics and some free toolkits have also been provided. For example, Torrence and Compo (1998) gave an excellent guide with a user-friendly toolkit for wavelet transform. Maraun and Kurths (2004) presented the evolution of cross-wavelet transform, developed a statistical test for wavelet coherence (WTC),^{1} and discussed some pitfalls in wavelet applications. Extending the toolkit of Torrence and Compo (1998), Grinsted et al. (2004) provided a software package for cross-wavelet transform and WTC.

Recently, Mihanović et al. (2009) introduced two additional wavelet analysis techniques—partial wavelet coherence (PWC) and multiple wavelet coherence (MWC)—to the field of marine science for calculating the resulting WTC of two effects after eliminating their common dependent factor and the proportion of WTC that can be explained by multiple independent variables on one dependent, respectively. Ng and Chan (2012) applied PWC to remove the El Niño–Southern Oscillation (ENSO) effect, revealing the unlikely dependence of tropical cyclone intensity on local sea surface temperature in the Bay of Bengal.

While the PWC and MWC techniques should have wide applications in the study of geophysical systems, Mihanović et al. (2009) applied them only to extract oscillations with a particular frequency as a time series. The direct usage of these techniques in the frequency–time space, which is believed to be as valuable, has not yet been well documented. The present paper therefore aims to contribute toward such documentation by demonstrating the use of PWC and MWC in frequency–time space. Moreover, a Matlab software package for these techniques is developed, which also includes modified software [based on the program of Torrence and Compo (1998) and Grinsted et al. (2004)] that rectified the bias in the wavelet power spectrum (WPS) and wavelet cross-spectrum (Liu et al. 2007; Veleda et al. 2012).

Although several studies have suggested that the rise of sea surface temperature due to global warming may induce increases in tropical cyclone frequency and intensity (e.g., Knutson and Tuleya 1999; Emanuel 2005; Webster et al. 2005), Chan and Liu (2004) found that tropical cyclone activity over the western North Pacific (WNP) does not seem to depend on the rising sea surface temperature, though a large percentage of its variation is forced by large-scale atmospheric factors that are found to be associated with ENSO. The result is confirmed by Chan (2009). In this paper, an investigation using wavelet analysis of the ENSO-related impact of these large-scale atmospheric factors on tropical cyclone activity over the western North Pacific is conducted.

Section 2 contains the description of the ENSO example, data, and methodology used in this paper. Section 3 gives a summary review of the basics of wavelet analysis, including the continuous wavelet transform, cross-wavelet transform, phase angle, and WTC, which are necessary for a better understanding of the PWC and MWC methods. The PWC and MWC are then respectively introduced in sections 4 and 5, with their applications demonstrated. This paper is summarized with some discussions on wavelet applications in section 6.

## 2. Data and methodology

### a. Tropical cyclone data

The tropical cyclone data of the International Best Track Archive for Climate Stewardship (IBTrACS; Knapp et al. 2010; Kruk et al. 2010) Project over the WNP during 1960–2008 are employed. The ratio of the number of typhoons to the number of tropical cyclones (RTY) is investigated in this study. Tropical cyclones and typhoons are only counted for maximum sustained wind speed when >34 and >64 kt, respectively, is reached in their lifetime.

### b. Atmospheric data

The wind, temperature, geopotential height, and specific humidity data are from the reanalysis of the National Centers for Environmental Prediction–National Center for Atmospheric Research (Kalnay et al. 1996; see online at http://www.esrl.noaa.gov/psd/), which is of 2.5° latitude × 2.5° longitude horizontal resolution. Similar to Chan and Liu (2004), an empirical orthogonal function (EOF) is first performed via correlation matrices on the thermodynamic and dynamic parameters suggested by Gray (1979) that govern tropical cyclone genesis and development. The domain of 5°–30°N, 120°E–180° is selected, as tropical cyclones over the WNP mostly reach their maximum intensity south of 30°N and east of 120°E (Xue and Neumann 1984). Because over 90% of tropical cyclones in this region are recorded during May–November, only data from these months are used. The time series of the principal components (PCs) that are significantly correlated with that of RTY at a 95% confidence level and related to ENSO are then analyzed. The sign of the EOFs is chosen to give a positive correlation with ENSO.

### c. The climatic oscillation indexes

The Niño-3.4 and Niño-3 indexes are extracted from the National Oceanic and Atmospheric Administration (NOAA) Earth System Research Laboratory website (http://www.esrl.noaa.gov/psd/data/climateindices/list). Both the Niño-3.4 and Niño-3 indexes are often used to detect the variability of the canonical ENSO.

The Indian Ocean dipole mode index (DMI) is extracted from the website of the Japan Agency for Marine-Earth Science and Technology (http://www.jamstec.go.jp/frcgc/research/d1/iod/).

## 3. Basic applications of wavelet analysis

### a. Continuous wavelet transform

Wavelet transform is a tool often used for analyzing a nonstationary time series with different power at different frequencies. There are two types of wavelet transform—the continuous wavelet transform and discrete wavelet transform. Only the latter suits orthogonal wavelet bases, while both can be used for nonorthogonal wavelet functions. Continuous wavelet transform is discussed in this paper.

*s*is the set of scales used, and

*N*times the convolution given in (2) for each scale (Torrence and Compo 1998). WPS is defined as

In general, either the white- or red-noise background spectrum is suitable for geophysical processes and hence used to define the null hypothesis for the significance test for peaks in WPS. Those phenomena of red-noise characteristics, like our example, can be well modeled by the lag-1 autoregressive process (Gilman et al. 1963; Allen and Smith 1996). With an appropriate background spectrum and desired confidence level, significant regions can be easily found.

While the WPSs of both the *monthly* Niño-3 and Niño-3.4 indexes (Fig. 1) are dominated at the period around the 2–7-yr band and do not appear to be very different, a slightly greater areal extent of significant regions is shown for the WPS of the Niño-3.4 index, particularly since 1980, when the ENSO Modoki started occurring more frequently. This could be because the Niño-3.4 index takes up signals from both the canonical ENSO and the ENSO Modoki given its geographical location, which has been suggested previously (Ashok et al. 2007; Chen and Tam 2010). We therefore use the Niño-3 index in this study to represent the canonical ENSO. Note that regions inside the cone of influence (COI; Torrence and Compo 1998), where discontinuities at end points occurred because of padding with zeros that may have distorted the results, are not considered.

### b. Cross-wavelet transform and phase angle

### c. WTC

Wavelet coherence (WTC) is a tool for identifying possible relationships between two processes by searching frequency bands and time intervals during which they covary. In other words, WTC may enhance linear correlation analyses that help reveal intermittent correlations between two phenomena (Gurley and Kareem 1999; Gurley et al. 2003), and their significant linear correlation relationship, if it is real, should thus be shown in WTC analyses as well. Unlike cross-wavelet transform, it is always helpful to implement the WTC analysis for relationship studies, even at intervals where high coherence exists but only minimal power is shown in the WPS of the two processes.

*W*operator is the continuous wavelet transform when it has one argument and cross-wavelet transform when it has two, and

*S*is the smoothing operator that helps strike a balance between resolution and significance. Because of the normalizing nature of WTC, the bias problem exists in WPS and wavelet cross spectrum does not occur in WTC. The same applies for PWC and MWC. The term “coherence” usually stands for the WTC squared, which ranges from 0 to 1 (1 being the highest coherence), given by smoothing operators. Monte Carlo methods are used to determine the statistical significance level of WTC.

The time series of the first PC of 850-hPa relative vorticity (RV1), that of 200–850-hPa vertical wind shear of zonal wind (VWS1), and the second PC of moist static energy (MSE2) are highly correlated with RTY and the Niño-3 index at a >95% confidence level with correlation coefficients of >0.37 and >0.57, respectively. These time series provide a demonstration of the application of WTC (Fig. 2). For example, the WTC of RTY and RV1 suggests that the two series have apparent coherence throughout the study period around the 3–16-yr band, where the mean phase of oscillations within the significant regions and outside COI is ~32°. The WTC of RTY and VWS1 and of RTY and MSE2 implies that their relationships are dominant only after the mid-1980s around the 4–16-yr band, in which the corresponding mean phase angles

## 4. PWC

Since RTY appears to be related to ENSO at around the 2–8-yr band (Fig. 3) and the time series of RV1, VWS1, and MSE2 are all significantly correlated with the Niño-3 index, the “stand-alone” relationship between RTY and the PCs should be studied by removing the effect of ENSO. Partial correlation is one of the methods that can be used in a simple correlation concept. In wavelet applications, we can perform this with the help of PWC.

In this case, a low PWC squared shown at where a high wavelet coherence squared was found implies that the time series

The squared PWCs of RTY and RV1, RTY and VWS1, and RTY and MSE2 over the WNP (after the removal of the ENSO effect) are shown in Fig. 5. The stand-alone coherence relationships between them can be reanalyzed with these squared PWCs. Substantial reductions in the amplitude and areal extent of the previously found WTC squared (Fig. 2), especially for interannual variations, are observed. Except for the significant region present in the squared PWC of RTY and RV1 around the 8–10-yr band, where RTY does not seem to be related to ENSO (Fig. 3), almost all the significant regions have disappeared. The reductions imply that there is a significant contribution of the ENSO effect to the relationship between RTY and the large-scale parameters. This is evident that the variability of tropical cyclone activity is forced by ENSO via the alteration of the atmospheric circulation, as suggested by Chan and Liu (2004).

## 5. MWC

Other than the canonical ENSO, the ENSO Modoki is also found to possess obvious relationships with RV1, VWS1, and MSE2, with correlation coefficients >0.42 (>99% confidence level). The WTCs of the EMI and the PCs show significant dominant oscillations, mostly around the 8–16-yr band (not shown), which suggests that the ENSO Modoki may have a significant impact on the interdecadal variability of tropical cyclone activity. The WTC of the monthly Niño-3 index and EMI show only a few significant regions (Fig. 7), indicating that the canonical ENSO and Modoki ENSO are largely independent, which confirms the suggestion of Ashok et al. (2007) further *to all scales*. The application of MWC is therefore illustrated by investigating the composite effect of the canonical ENSO and ENSO Modoki on the variability of the PCs, respectively.

The combined impact of the canonical ENSO and ENSO Modoki on RV1 can be studied with MWC squared by putting RV1 as

With MWC, by comparing different combinations of the independent variables, the combination that best represents the dependent one may also be found. For example, as a significant relationship exists between VWS1 and the DMI, with a correlation coefficient of 0.40 (>95% confidence level), by comparing the squared MWC of VWS1, the DMI, and the Niño-3 index (Fig. 9) with that of VWS1, the EMI, and the Niño-3 index (Fig. 8b), the areal extent and amplitude of significant regions of the former are not comparable to those of the latter. This infers that EMI and the Niño-3 index together provide a better explanation of the variability of VWS1 than using the DMI and the Niño-3 index. Even worse results are found for the combination of the DMI and the EMI (not shown). This example demonstrates the usefulness of MWC in model simulations in helping to find the best group of predictors that explain the predictand *on all scales*.

## 6. Discussion and summary

The present study demonstrates the applications of PWC and MWC. The possible ENSO-related impact of the large-scale atmospheric factors leading the variability of tropical cyclone activity over the WNP is used as an example. PWC is a technique similar to partial correlation that helps find the resulting WTC between two time series after eliminating the influence of their common dependence, while MWC, working like multiple correlation, is useful in seeking the resulting WTC of multiple independents on a dependent. Given the similar working principle to that of traditional correlation coefficients, WTC can be seen as a localized correlation in a time–frequency space (Grinsted et al. 2004). PWC and MWC can therefore be defined easily with the concepts of partial and multiple correlation, as suggested by Mihanović et al. (2009).

With wavelet analysis, the problem of studying nonstationary time series has been much improved compared to Fourier transform, and one may now be able to study the variations of phenomena in a time–frequency space. However, imperfections of this tool cannot be neglected. First, because wavelet analysis is an expansion of a one-dimension time series, like simple correlation, the domain selection for analysis is very critical. In other words, the existence of noise is almost inevitable. This error may even be magnified when undergoing a time–frequency expansion. For example, the domain for large-scale parameters chosen in this study may not have consistent relationships with RTY and averaging across the domain could disrupt the relationships. Second, as wavelet analysis is developed mainly for studying periodicity, even though phase angle may help validate relationships, conclusions should not be drawn about relationships without any expectation of further investigation. Significant coherence could result even for random signals that happen to oscillate at the same frequency at particular intervals simultaneously. On the other hand, although angular standard deviation^{2} measures the scattering of phase angle and may be able to test the credibility of angle mean, a large circular standard deviation does not necessarily imply doubts to that significant region and thus to the possible relationship between those phenomena. As long as the phase angles do not have sudden changes in the significant regions, as mentioned in section 3b, the possible physical relationship revealed may still be applicable.

To conclude, PWC and MWC further expand the applications of wavelet analysis, which helps diagnose the time series in a time–frequency space. They should have wide applications in the study of geophysical systems, for both observational and modeling ones, where trends and periodicities of oscillations with various frequencies are often investigated.

## Acknowledgments

Special thanks to T. W. Kwok for his help in developing the software package. The wavelet software was provided by C. Torrence and G. Compo (see online at http://paos.colorado.edu/research/wavelets/), while the cross-wavelet and wavelet coherence software, from which some part of the code is used in ours, were provided by A. Grinsted (see online at http://www.pol.ac.uk/home/research/waveletcoherence).

## APPENDIX

### List of Symbols and Acronyms

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