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

- 519 Downloads
- 6 Citations

## 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.

## Keywords

Convectively coupled equatorial waves Tropical convection Wavelet Annual cycle Interannual variability## Notes

### Acknowledgments

This research was supported by NSF Grant AGS-1005599 and by the Global Research Laboratory (GRL) Program from the Ministry of Education, Science, and Technology (MEST), Korea. Additional support was provided by the JAMSTEC through its sponsorship of research activities at the IPRC. These results were obtained using the CLAUS archive held at the British Atmospheric Data Centre, produced using ISCCP source data distributed by the NASA Langley Data Center. The author acknowledges the use of the 1D WT program provided by C. Torrence and G. Compo, which is available at URL: http://atoc.colorado.edu/research/wavelets/ to develop the CFWT code and of a package provided by CCSM AMWG to compute Fourier-based zonal wavenumber-frequency power spectrum. The Niño3.4 index, based on OISST.v2 product, was obtained from the U.S. Weather Service’s Climate Prediction Center Web site at http://www.cpc.ncep.noaa.gov/data/indices/. NOAA_OI_SST_V2 data provided by the NOAA/OAR/ESRL PSD, Boulder, Colorado, USA, from their Web site at http://www.esrl.noaa.gov/psd/. The author thanks Dr. Yasunaga for his comments on how to estimate the background power spectrum, Dr. George N. Kiladis for helpful comments on an earlier version of the manuscript, and anonymous reviewers for their suggestions and comments.

## References

- Addison PS (2002) The illustrated wavelet transform handbook: introductory theory and applications in science, engineering, medicine and finance, 1st edn. Taylor & Francis, UKCrossRefGoogle Scholar
- Baldwin MP, Gray LJ, Dunkerton TJ, Hamilton K, Haynes PH, Randel WJ, Holton JR, Alexander MJ, Hirota I, Horinouchi T, Jones DBA, Kinnersley JS, Marquardt C, Sato K, Takahashi M (2001) The quasi-biennial oscillation. Rev Geophys 39:179–229CrossRefGoogle Scholar
- Chen SS, Houze RA Jr (1997) Diurnal variation and life-cycle of deep convective systems over the tropical Pacific warm pool. Q J R Meteorol Soc 123:357–388CrossRefGoogle Scholar
- Chia HH, Ropelewski CF (2002) The interannual variability in the genesis location of tropical cyclones in the northwest Pacific. J Clim 15:2934–2944CrossRefGoogle Scholar
- Cho HK, Bowman KP, North GR (2004) Equatorial waves including the Madden–Julian oscillation in TRMM rainfall and OLR data. J Clim 17:4387–4406CrossRefGoogle Scholar
- Davis RE (1976) Predictability of sea surface temperature and sea level pressure anomalies over the north Pacific ocean. J Phys Oceanogr 6:249–266CrossRefGoogle Scholar
- Dias J, Leroux S, Tulich SN, Kiladis GN (2013) How systematic is organized tropical convection within the MJO? Geophys Res Lett 40:1420–1425. doi: 10.1002/grl.50308 CrossRefGoogle Scholar
- Dickinson M, Molinari J (2002) Mixed Rossby-gravity waves and Western Pacific tropical cyclogenesis. Part I: synoptic evolution. J Atmos Sci 59:2183–2196CrossRefGoogle Scholar
- Dunkerton TJ (1990) Annual variation of deseasonalized mean flow acceleration in the equatorial lower stratosphere. J Meteorol Soc Jpn 68:499–508Google Scholar
- Dunkerton TJ, Baldwin MP (1995) Observation of 3–6-day meridional wind oscillations over the tropical Pacific, 1973–1992: horizontal structure and propagation. J Atmos Sci 52:1585–1601CrossRefGoogle Scholar
- Duval-Destin M, Murenzi R (1993) Spatio-temporal wavelets: application to the analysis of moving patterns. In: Meyer Y, Roques S (eds) Progress in wavelet analysis and applications (Proc. Toulouse 1992). Frontières, Gif-suf-Yvette, pp 399–408Google Scholar
- Farge M (1992) Wavelet transforms and their applications to turbulence. Annu Rev Fluid Mech 24:395–457CrossRefGoogle Scholar
- Frank WM, Roundy PE (2006) The role of tropical waves in tropical cyclogenesis. Mon Weather Rev 134:2397–2417CrossRefGoogle Scholar
- Gabis IP, Troshichev OA (2011) The quasi-biennial oscillation in the equatorial stratosphere: seasonal regularity in zonal wind changes, discrete QBO-cycle period and prediction of QBO-cycle duration. Geomag Aeron 51:501–512CrossRefGoogle Scholar
- Gu GJ, Zhang CD (2001) A spectrum analysis of synoptic-scale disturbances in the ITCZ. J Clim 14:2725–2739CrossRefGoogle Scholar
- Hendon HH, Wheeler MC (2008) Some space-time spectral analyses of tropical convection and planetary-scale waves. J Atmos Sci 65:2936–2948CrossRefGoogle Scholar
- Hendon HH, Wheeler MC, Zhang CD (2007) Seasonal dependence of the MJO-ENSO relationship. J Clim 20:531–543CrossRefGoogle Scholar
- Hodges KI, Chappell DW, Robinson GJ, Yang G (2000) An improved algorithm for generating global window brightness temperatures from multiple satellite infrared imagery. J Atmos Ocean Technol 17:1296–1312CrossRefGoogle Scholar
- Huang P, Huang RH (2011) Climatology and interannual variability of convectively coupled equatorial waves activity. J Clim 24:4451–4465CrossRefGoogle Scholar
- Hudgins L, Friehe CA, Mayer ME (1993) Wavelet transforms and atmospheric turbulence. Phys Rev Lett 71:3279–3282CrossRefGoogle Scholar
- Kawatani Y, Sato K, Dunkerton TJ, Watanabe S, Miyahara S, Takahashi M (2010) The roles of equatorial trapped waves and internal inertia-gravity waves in diving the quasi-biennial oscillation. Part I: zonal mean wave forcing. J Atmos Sci 67:963–980CrossRefGoogle Scholar
- Kikuchi K, Wang B (2010) Spatiotemporal wavelet transform and the multiscale behavior of the Madden–Julian oscillation. J Clim 23:3814–3834CrossRefGoogle Scholar
- Kiladis GN, Thorncroft CD, Hall NMJ (2006) Three-dimensional structure and dynamics of African easterly waves. Part I: observations. J Atmos Sci 63:2212–2230CrossRefGoogle Scholar
- Kiladis GN, Wheeler MC, Haertel PT, Straub KH, Roundy PE (2009) Convectively coupled equatorial waves. Rev Geophys 47:RG2003. doi: 10.1029/2008RG000266
- Kumar P, FoufoulaGeorgiou E (1997) Wavelet analysis for geophysical applications. Rev Geophys 35:385–412CrossRefGoogle Scholar
- Labat D (2005) Recent advances in wavelet analyses: Part I. A review of concepts. J Hydrol 314:275–288CrossRefGoogle Scholar
- Lau KM, Weng H (1995) Climate signal detection using wavelet transform: how to make a time series sing. Bull Am Meteorol Soc 76:2391–2402CrossRefGoogle Scholar
- Liebmann B, Hendon HH (1990) Synoptic-scale disturbances near the equator. J Atmos Sci 47:1463–1479CrossRefGoogle Scholar
- Liu YG, Liang XS, Weisberg RH (2007) Rectification of the bias in the wavelet power spectrum. J Atmos Ocean Technol 24:2093–2102CrossRefGoogle Scholar
- Marrat S (2008) A wavelet tour of signal processing, 3rd edn. Academic Press, WalthamGoogle Scholar
- Masunaga H (2007) Seasonality and regionality of the Madden–Julian oscillation, Kelvin wave, and equatorial Rossby wave. J Atmos Sci 64:4400–4416CrossRefGoogle Scholar
- Masunaga H, L’Ecuyer TS, Kummerow CD (2006) The Madden–Julian oscillation recorded in early observations from the Tropical Rainfall Measuring Mission (TRMM). J Atmos Sci 63:2777–2794CrossRefGoogle Scholar
- Matsuno T (1966) Quasi-geostrophic motions in the equatorial area. J Meteorol Soc Jpn 44:25–43Google Scholar
- McPhaden MJ (2008) Evolution of the 2006–2007 El Niño: the role of intraseasonal to interannual time scale dynamics. Adv Geosci 14:219–230CrossRefGoogle Scholar
- Meyers SD, Kelly BG, Obrien JJ (1993) An introduction to wavelet analysis in oceanography and meteorology: with application to the dispersion of Yanai waves. Mon Weather Rev 121:2858–2866CrossRefGoogle Scholar
- Mitchell TP, Wallace JM (1992) The annual cycle in equatorial convection and sea-surface temperature. J Clim 5:1140–1156CrossRefGoogle Scholar
- Moncrieff MW, Waliser DE, Miller MJ, Shapiro MA, Asrar GR, Caughey J (2012) Multiscale convective organization and the YOTC virtual global field campaign. Bull Am Meteorol Soc 93:1171–1187CrossRefGoogle Scholar
- Mounier F, Kiladis GN, Janicot S (2007) Analysis of the dominant mode of convectively coupled Kelvin waves in the West African monsoon. J Clim 20:1487–1503CrossRefGoogle Scholar
- Nakazawa T (1988) Tropical super clusters within intraseasonal variations over the western Pacific. J Meteorol Soc Jpn 66:823–839Google Scholar
- Okumura YM, Deser C (2010) Asymmetry in the duration of El Nino and La Nina. J Clim 23:5826–5843CrossRefGoogle Scholar
- Reynolds RW, Rayner NA, Smith TM, Stokes DC, Wang WQ (2002) An improved in situ and satellite SST analysis for climate. J Clim 15:1609–1625CrossRefGoogle Scholar
- Roundy PE (2012) The spectrum of convectively coupled Kelvin waves and the Madden–Julian Oscillation in regions of low-level easterly and westerly background flow. J Atmos Sci 69:2107–2111CrossRefGoogle Scholar
- Roundy PE, Frank WM (2004) A climatology of waves in the equatorial region. J Atmos Sci 61:2105–2132CrossRefGoogle Scholar
- Roundy PE, Janiga MA (2012) Analysis of vertically propagating convectively coupled equatorial waves using observations and a non-hydrostatic Boussinesq model on the equatorial beta-plane. Q J R Meteorol Soc 138:1004–1017CrossRefGoogle Scholar
- Schreck CJ III, Molinari J, Aiyyer A (2012) A global view of equatorial waves and tropical cyclogenesis. Mon Weather Rev 140:774–788CrossRefGoogle Scholar
- Seiki A, Takayabu YN (2007) Westerly wind bursts and their relationship with intraseasonal variations and ENSO. Part I: statistics. Mon Weather Rev 135:3325–3345CrossRefGoogle Scholar
- Serra YL, Houze RA (2002) Observations of variability on synoptic timescales in the East Pacific ITCZ. J Atmos Sci 59:1723–1743CrossRefGoogle Scholar
- Serra YL, Kiladis GN, Cronin MF (2008) Horizontal and vertical structure of easterly waves in the Pacific ITCZ. J Atmos Sci 65:1266–1284CrossRefGoogle Scholar
- Stella L, Arlandi G, Tagliaferri G, Israel GL (1994) Continuum power spectrum components in x-ray sources: detailed modelling and search for coherent periodicities. In: Rao TS, Priestley MB, Lessi O (eds) Applications of time series analysis in astronomy and meteorology [arXiv:astro-ph/9411050v1]Google Scholar
- Straub KH, Kiladis GN (2002) Observations of a convectively coupled Kelvin wave in the eastern Pacific ITCZ. J Atmos Sci 59:30–53CrossRefGoogle Scholar
- Takayabu YN (1994) Large-scale cloud disturbances associated with equatorial waves. Part I: spectral features of the cloud disturbances. J Meteorol Soc Jpn 72:433–449Google Scholar
- Takayabu YN, Nitta T (1993) 3-5 day-period disturbances coupled with convection over the tropical Pacific Ocean. J Meteorol Soc Jpn 71:221–246Google Scholar
- Telfer B, Szu HH (1992) New wavelet transform normalization to remove frequency bias. Opt Eng 31:1830–1834CrossRefGoogle Scholar
- Tian BJ, Waliser DE, Fetzer EJ (2006) Modulation of the diurnal cycle of tropical deep convective clouds by the MJO. Geophys Res Lett 33:L20704. doi: 10.1029/2006GL027752 CrossRefGoogle Scholar
- Torrence C, Compo GP (1998) A practical guide to wavelet analysis. Bull Am Meteorol Soc 79:61–78CrossRefGoogle Scholar
- Tulich SN, Kiladis GN (2012) Squall lines and convectively coupled gravity waves in the tropics: why do most cloud systems propagate westward? J Atmos Sci 69:2995–3012CrossRefGoogle Scholar
- Vaughan S, Bailey RJ, Smith DG (2011) Detecting cycles in stratigraphic data: spectral analysis in the presence of red noise. Paleoceanography 26:PA4211. doi: 10.1029/2011PA002195
- Vecchi GA, Harrison DE (2000) Tropical Pacific sea surface temperature anomalies, El Nino, and equatorial westerly wind events. J Clim 13:1814–1830CrossRefGoogle Scholar
- von Storch H, Zwiers FW (1999) Statistical anlysis in climate research. Cambridge University Press, CambridgeGoogle Scholar
- Waliser DE, Gautier C (1993) A satellite-derived climatology of the ITCZ. J Clim 6:2162–2174CrossRefGoogle Scholar
- Waliser DE, Moncrieff MW, Burridge D, Fink AH, Gochis D, Goswami BN, Guan B, Harr P, Heming J, Hsu H–H, Jakob C, Janiga M, Johnson R, Jones S, Knippertz P, Marengo J, Hanh N, Pope M, Serra Y, Thorncroft C, Wheeler M, Wood R, Yuter S (2012) The “year” of tropical convection (May 2008–April 2010) climate variability and weather highlights. Bull Am Meteorol Soc 93:1189–1218CrossRefGoogle Scholar
- Wang B, Chan JCL (2002) How strong ENSO events affect tropical storm activity over the Western North Pacific. J Clim 15:1643–1658CrossRefGoogle Scholar
- Wang B, Xie X (1996) Low-frequency equatorial waves in vertically sheared zonal flow. Part I: stable waves. J Atmos Sci 53:449–467CrossRefGoogle Scholar
- Wheeler M, Kiladis GN (1999) Convectively coupled equatorial waves: analysis of clouds and temperature in the wavenumber-frequency domain. J Atmos Sci 56:374–399CrossRefGoogle Scholar
- Wong MLM (2009) Wavelet analysis of the convectively coupled equatorial waves in the wavenumber-frequency domain. J Atmos Sci 66:209–212CrossRefGoogle Scholar
- Wu B, Zhou T, Li T (2009) Seasonally evolving dominant interannual variability modes of east Asian climate. J Clim 22:2992–3005CrossRefGoogle Scholar
- Xie XS, Wang B (1996) Low-frequency equatorial waves in vertically sheared zonal flow. Part II: unstable waves. J Atmos Sci 53:3589–3605CrossRefGoogle Scholar
- Yang D, Ingersoll AP (2011) Testing the hypothesis that the MJO is a mixed rossby-gravity wave packet. J Atmos Sci 68:226–239CrossRefGoogle Scholar
- Yano JI, Moncrieff MW, Wu XQ, Yamada M (2001) Wavelet analysis of simulated tropical convective cloud systems. Part I: basic analysis. J Atmos Sci 58:850–867CrossRefGoogle Scholar
- Yasunaga K, Mapes B (2012) Differences between more divergent and more rotational types of convectively coupled equatorial waves. Part I: space-time spectral analyses. J Atmos Sci 69:3–16CrossRefGoogle Scholar