An introduction to combined Fourier–wavelet transform and its application to convectively coupled equatorial waves

Abstract

Convectively coupled equatorial waves (CCEWs) are major sources of tropical day-to-day variability. The majority of CCEWs-related studies for the past decade or so have based their analyses, in one form or another, on the Fourier-based space–time spectral analysis method developed by Wheeler and Kiladis (WK). Like other atmospheric and oceanic phenomena, however, CCEWs exhibit pronounced nonstationarity, which the conventional Fourier-based method has difficulty elucidating. The purpose of this study is to introduce an analysis method that is able to describe the time-varying spectral features of CCEWs. The method is based on a transform, referred to as the combined Fourier–wavelet transform (CFWT), defined as a combination of the Fourier transform in space (longitude) and wavelet transform in time, providing an instantaneous space–time spectrum at any given time. The elaboration made on how to display the CFWT spectrum in a manner analogous to the conventional method (i.e., as a function of zonal wavenumber and frequency) and how to estimate the background noise spectrum renders the method more practically feasible. As a practical example, this study analyzes 3-hourly cloud archive user service (CLAUS) cloudiness data for 23 years. The CFWT and WK methods exhibit a remarkable level of agreement in the distributions of climatological-mean space–time spectra over a wide range of space–time scales ranging in time from several hours to several tens of days, indicating the instantaneous CFWT spectrum provides a reasonable snapshot. The usefulness of the capability to localize space–time spectra in time is demonstrated through examinations of the annual cycle, interannual variability, and a case study.

Appendix: Representation of the CFWT spectrum and estimation of a red-noise background spectrum

Representation of the WT spectrum sometimes gives rise to confusion. The WT spectrum is usually represented as a function of wavelet scale and the wavelet scale is usually associated with Fourier frequency for analysis (e.g., Meyers et al. 1993). However, the WT spectrum in that representation is biased in magnitude toward higher frequencies (e.g., Telfer and Szu 1992; Liu et al. 2007), making direct comparison with the FT spectrum difficult. Figure 10 illustrates the situation for the case of the CFWT: the WT spectra of both the symmetric and antisymmetric components at a particular zonal wavenumber (top panel) are blue (i.e., the spectrum increases with increasing frequency), while the FT counterparts are red (bottom panel).

Fig. 10

Power spectrum estimates of T_b at zonal wavenumber −5. a Wavelet scale-dependent representation of the CFWT spectrum, b frequency-dependent representation of the CFWT spectrum, and c FT spectrum. Symmetric and antisymmetric components are denoted by solid and dashed curves, respectively. AR(1) red noise power spectra and their 1.3 times values are denoted by gray solid and dashed curves, respectively. Each spectrum is suitably scaled so that the maximum value is ~1

In order to bridge the gap, we developed a method to convert the scale-dependent representation of the CFWT spectrum into a frequency-dependent representation. In light of energy conservation (8), the frequency-dependent spectrum $P\left( {k,f_{n} } \right)$ at discrete frequency f _n may be defined, making use of (5) and (9), as

$$\left| {P\left( {k,f_{n} ,\tau } \right)} \right|^{2} = \frac{1}{{C_{\psi } }} \int \limits_{{s_{1} }}^{{s_{2} }} \left| {\left( {\mathcal{T}g} \right)\left( {k,s,\tau } \right)} \right|^{2} \frac{ds}{{s^{2} }}$$

where ${{s_{1} = \left[ {\omega_{0} + \left( {\omega_{0}^{2} + 2} \right)^{1/2} } \right]} \mathord{\left/ {\vphantom {{s_{1} = \left[ {\omega_{0} + \left( {\omega_{0}^{2} + 2} \right)^{1/2} } \right]} {\left[ {2\pi \left( {f_{n - 1} + f_{n} } \right)} \right]}}} \right. \kern-0pt} {\left[ {2\pi \left( {f_{n - 1} + f_{n} } \right)} \right]}}$, ${{{\text{s}}_{2} = \left[ {\omega_{0} + \left( {\omega_{0}^{2} + 2} \right)^{1/2} } \right]} \mathord{\left/ {\vphantom {{{\text{s}}_{2} = \left[ {\omega_{0} + \left( {\omega_{0}^{2} + 2} \right)^{1/2} } \right]} {[2\pi (f_{n} + f_{n + 1} )]}}} \right. \kern-0pt} {[2\pi (f_{n} + f_{n + 1} )]}}$. The resulting frequency-dependent CFWT power spectra (middle panel) are red and are in good agreement with the FT counterparts (bottom panel), yet they are somewhat smeared, resulting in more moderate peaks at the diurnal and semidiurnal frequencies.

Now, because of the similarity in appearance, we can anticipate that a technique that has been developed to assess FT space–time spectrum peaks is also applicable to the frequency-dependent CFWT spectrum. To determine the level of significance, we need to estimate the background noise spectrum and DOF. Because we discussed how to estimate the DOF for the case of the CFWT in Sect. 2.2, we here focus on how to estimate the background spectrum. In contrast to the case of a one-dimensional time series, whose background noise is usually assumed to be generated by a process represented by a simple mathematical model (by the first-order autoregression AR(1) model for most geophysical time series), there is no consensus on how to estimate the background noise spectrum in a space–time series at this point. In their pioneering work, Wheeler and Kiladis (1999) defined the background spectrum by smoothing the total (i.e., sum of the symmetric and antisymmetric parts) raw spectrum in zonal wavenumber and in frequency many times. Recently, Masunaga et al. (2006) and Hendon and Wheeler (2008) developed a different approach by considering the fitting of the AR(1) model at each zonal wavenumber. This approach is arguably more objective than that of Wheeler and Kiladis (1999).

We took a similar approach but with more elaboration. To check the validity of model assumption, we considered three mathematical models including the power law distribution and bending power law distribution as well as the AR(1) model (for exact formulae see Vaughan et al. 2011). They together can represent a wide range of realistic noise spectra. At each zonal wavenumber the degree to which each model can fit to the total raw spectrum and the parameters of each model that provide best fit were determined by means of the maximum likelihood method (e.g., Stella et al. 1994). It was found that the AR(1) model fits best at nearly all the zonal wavenumbers within ~160 for both the FT and CFWT global spectra (not shown).

Figure 11 shows the resulting background spectrum obtained based on the AR(1) model. Prior to the fitting of the AR(1) model, the power in the MJO spectral range (positive zonal wavenumbers 1–3 and frequencies up to 1/30 cpd) was halved in the total raw spectrum as with Hendon and Wheeler (2008). This treatment little affects the overall shape of the background spectrum. Clearly, both the CFWT and FT background spectra are to a large extent symmetric about zonal wavenumber 0 and highly red in both wavenumber and frequency.

Fig. 11

Zonal wavenumber-frequency spectra of the base-10 logarithm of the background red noise power estimated by fitting the AR(1) model to the total raw spectrum for a CFWT and b FT