## A Practical Guide to Wavelet Analysis (1998)

Citations: | 230 - 2 self |

### BibTeX

@MISC{Torrence98apractical,

author = {Christopher Torrence and Gilbert P. Compo},

title = {A Practical Guide to Wavelet Analysis},

year = {1998}

}

### Years of Citing Articles

### OpenURL

### Abstract

A practical step-by-step guide to wavelet analysis is given, with examples taken from time series of the El Nio-- Southern Oscillation (ENSO). The guide includes a comparison to the windowed Fourier transform, the choice of an appropriate wavelet basis function, edge effects due to finite-length time series, and the relationship between wavelet scale and Fourier frequency. New statistical significance tests for wavelet power spectra are developed by deriving theoretical wavelet spectra for white and red noise processes and using these to establish significance levels and confidence intervals. It is shown that smoothing in time or scale can be used to increase the confidence of the wavelet spectrum. Empirical formulas are given for the effect of smoothing on significance levels and confidence intervals. Extensions to wavelet analysis such as filtering, the power Hovmller, cross-wavelet spectra, and coherence are described. The statistical significance tests are used to give a qu...

### Citations

1727 | Ten Lectures on Wavelets - Daubechies |

854 | Ideal spatial adaptation by wavelet shrinkage - Donoho, Johnstone - 1994 |

514 |
The wavelet transform, time-frequency localization and signal analysis
- Daubechies
- 1990
(Show Context)
Citation Context ...dependent, such as wavelet analysis, should be employed. b. Wavelet Transform The wavelet transform can be used to analyze time series that contain non-stationary power at many different frequencies (=-=Daubechies 1990). Assume -=-that one has a time series, x n , with equal time spacing ffit and n = 0 : : : N\Gamma1. Also assume that one has a wavelet basis function, / 0 (j), that depends on a non-dimensional "time" ... |

121 |
A friendly guide to wavelets
- Kaiser
- 1994
(Show Context)
Citation Context ...h N ffit, thus returning frequencies from T \Gamma1 to (2ffit) \Gamma1 at each time step. The segments can be windowed with an arbitrary function such as a boxcar (no smoothing) or a Gaussian window (=-=Kaiser 1994). As disc-=-ussedby Kaiser (1994), the WFT represents an inaccurate and inefficient method of time-frequency localization, as it imposes a scale or "response interval" T into the analysis. The inaccurac... |

114 |
Spectral analysis and its applications
- Jenkins, Watts
- 1968
(Show Context)
Citation Context ...ex, and hence one can define the cross-wavelet power as jW XY n (s)j. Confidence levels for the cross-wavelet power can be derived from the square-root of the product of two chi-square distributions (=-=Jenkins and Watts 1968). A-=-ssuming both wavelet spectra are �� 2 distributed withsDOF, the probability distribution becomes f(z) = 2 2\Gamma \Gamma 2 \Gammas2 \Delta zs\Gamma1 K 0 (z); (30) where z is the random variable, \... |

109 |
Wavelet transforms and their applications to turbulence
- Farge
- 1992
(Show Context)
Citation Context ... Blumen 1993), central England temperature (Baliunas et al. 1997), the dispersion of ocean waves (Meyers et al. 1993), wave growth and breaking (Liu 1994), and coherent structures in turbulent flows (=-=Farge 1992-=-). A complete description of geophysical applications can be found in Foufoula-Georgiou and Kumar (1995), while a theoretical treatment of wavelet analysis is given in Daubechies (1992). Unfortunately... |

33 | On estimation of the wavelet variance - Percival - 1995 |

32 |
Interdecadal Changes in El Niño onset in the last four decades
- Wang
- 1995
(Show Context)
Citation Context ... regions return in the 1950s, with the strongest amplitudes after 1970. The diminished power after 1990 is within the COI, yet may reflect the changes in ENSO structure and evolution in recent years (=-=Wang 1995-=-). The zonal average of the power hovmoller (Fig. 10b), gives a measure of global 2--8 year variance in this latitude band. Comparing this to Fig. 9, one can see that the peaks in zonal-average power ... |

26 |
Secular changes of annual and interannual variability in the tropics during the past century
- Gu, Philander
- 1995
(Show Context)
Citation Context ...iability and how those modes vary in time. The wavelet transform has been used for numerous studies in geophysics, including tropical convection (Weng and Lau 1994), the El Nino-Southern Oscillation (=-=Gu and Philander 1995-=-; Wang and Wang 1996), atmospheric cold fronts (Gamage and Blumen 1993), central England temperature (Baliunas et al. 1997), the dispersion of ocean waves (Meyers et al. 1993), wave growth and breakin... |

21 |
Climate signal detection using wavelet transformation: How to make a time series sing. 76:2391–2402
- Lau, Weng
- 1995
(Show Context)
Citation Context ...t power spectrum, to compare against either the mean background or against other peaks. e. Stationarity It has been argued that wavelet analysis requires the use of non-stationary significance tests (=-=Lau and Weng 1995-=-). In defense of the use of stationary tests such as those given above, the following points are noted: 1. A non-arbitrary test is needed. The assumption of stationary statistics provides a standard b... |

21 |
Temporal structure of the Southern Oscillation as revealed by waveform and wavelet analysis
- Wang, Wang
- 1996
(Show Context)
Citation Context ...modes vary in time. The wavelet transform has been used for numerous studies in geophysics, including tropical convection (Weng and Lau 1994), the El Nino-Southern Oscillation (Gu and Philander 1995; =-=Wang and Wang 1996-=-), atmospheric cold fronts (Gamage and Blumen 1993), central England temperature (Baliunas et al. 1997), the dispersion of ocean waves (Meyers et al. 1993), wave growth and breaking (Liu 1994), and co... |

20 | Improved predictability of two-dimensional turbulent flows using wavelet packet compression. Fluid Dyn - Farge, Goirand, et al. - 1992 |

19 |
An introduction to wavelet analysis in oceanography and meteorology: With applications to the dispersion of Yanai waves
- Meyers, Kelly, et al.
- 1993
(Show Context)
Citation Context ...-Southern Oscillation (Gu and Philander 1995; Wang and Wang 1996), atmospheric cold fronts (Gamage and Blumen 1993), central England temperature (Baliunas et al. 1997), the dispersion of ocean waves (=-=Meyers et al. 1993-=-), wave growth and breaking (Liu 1994), and coherent structures in turbulent flows (Farge 1992). A complete description of geophysical applications can be found in Foufoula-Georgiou and Kumar (1995), ... |

19 |
The annual cycle of persistence in the El Nino southern oscillation
- Torrence, Webster
- 1985
(Show Context)
Citation Context ...ions in the frequency of occurrence and amplitude of El Nino. During 1875--1920and 1960--1990 there were many El Nino events of large amplitude, while during 1920--1960 there were few El Nino events (=-=Torrence and Webster 1997-=-). From 1875--1910, there was a slight shift from a period near four years to a period closer to two years, while from 1960--1990 the shift is from shorter to longer periods. These results are similar... |

17 | Wavelets in Geophysics - Foufoula-Georgiou, Kumar - 1994 |

16 | Wavelet Transforms and Atmospheric Turbulence - Hudgings, Friehe, et al. - 1993 |

16 | Version 2.2 of the global sea-ice and sea surface temperature data set, 1903–1994. Climate Research Tech. Note 74, CRTN74, Hadley Centre for Climate Prediction and Research, 25 pp. [Available from Hadley Centre for Climate Prediction and Research, Meteoro - Hackett - 1996 |

14 |
Version 2.2 of the global sea-ice and sea surface temperature data set, 1903–1994. Climate Tech. Note 74, Hadley Centre for Climate Prediction and Research
- Rayner, Horton, et al.
- 1996
(Show Context)
Citation Context ...index is defined as the seasonal SST averaged over the central Pacific (5 ffi S--5 ffi N, 90 ffi W-- 150 ffi W). Data for 1871--1996 are from an area-average of the UK Meteorological Office GISST2.3 (=-=Rayner et al. 1996-=-), while data for January--June 1997 are from the Climate Prediction Center (CPC) optimallyinterpolated NINO3 SST index (courtesy D. Garrett at CPC, NOAA, USA). The seasonal means for the entire recor... |

13 | The discrete wavelet transform and the scale analysis of the surface properties of sea ice - Lindsay, Percival, et al. - 1996 |

11 | Wavelet Spectra Compared to Fourier Spectra, Publication of ENS - Perrier, Philipovitch, et al. - 1999 |

10 |
Comparative analysis of low-level cold fronts: Wavelets, Fourier, and empirical orthogonal function decomposition
- Gamage, Blumen
- 1993
(Show Context)
Citation Context ...been used for numerous studies in geophysics, including tropical convection (Weng and Lau 1994), the El Nino-Southern Oscillation (Gu and Philander 1995; Wang and Wang 1996), atmospheric cold fronts (=-=Gamage and Blumen 1993-=-), central England temperature (Baliunas et al. 1997), the dispersion of ocean waves (Meyers et al. 1993), wave growth and breaking (Liu 1994), and coherent structures in turbulent flows (Farge 1992).... |

10 |
Wavelet spectrum analysis and ocean wind waves. In
- Liu
- 1994
(Show Context)
Citation Context ...ang and Wang 1996), atmospheric cold fronts (Gamage and Blumen 1993), central England temperature (Baliunas et al. 1997), the dispersion of ocean waves (Meyers et al. 1993), wave growth and breaking (=-=Liu 1994-=-), and coherent structures in turbulent flows (Farge 1992). A complete description of geophysical applications can be found in Foufoula-Georgiou and Kumar (1995), while a theoretical treatment of wave... |

9 |
Schaum's Outline of Theory and problems of Probability and Statistics. pp 372
- Spiegel
- 1975
(Show Context)
Citation Context ... Fourier spectrum (Fig. 3), the power at each frequency is independent of the others, and the average of the power at M frequencies, each with two DOF, is �� 2 distributed with 2M degrees of freed=-=om (Spiegel 1975). F-=-or the time-averaged wavelet spectrum, one is also averaging points that are �� 2 2 distributed, yet Figs. 1b and 4 show that these points are no longer independent, but are correlated in both tim... |

8 |
Time scales and trends in the Central England Temperature data (1659–1990): A wavelet analysis, Geophys
- Baliunas, Frick, et al.
- 1997
(Show Context)
Citation Context ...g tropical convection (Weng and Lau 1994), the El Nino-Southern Oscillation (Gu and Philander 1995; Wang and Wang 1996), atmospheric cold fronts (Gamage and Blumen 1993), central England temperature (=-=Baliunas et al. 1997-=-), the dispersion of ocean waves (Meyers et al. 1993), wave growth and breaking (Liu 1994), and coherent structures in turbulent flows (Farge 1992). A complete description of geophysical applications ... |

7 |
The Analysis of Time Series: An Introduction, 4th ed
- Chatfield
- 1989
(Show Context)
Citation Context ... x k j 2 =oe 2 , where N is the number of points, b x k is from (3), and oe 2 is the variance of the time series. If x n is a normally distributed random variable, then b x k is normally distributed (=-=Chatfield 1989). S-=-ince b x k is complex, then jb x k j 2 is chi-square distributed with two degrees of freedom (DOF), denoted by �� 2 2 . To determine the 95% confidence level (significant at 5%), one multiplies th... |

7 |
Spatial and temporal variations of the Southern Oscillation
- Trenberth
- 1976
(Show Context)
Citation Context ...oise. In Fig. 1b most of the power is concentrated within the El Nino band of 2--8 years, although there is appreciable power at longer periods. The 2--8 year band for ENSO agrees with other studies (=-=Trenberth 1976-=-), and is also seen in the Fourier spectrum in Fig. 3. With wavelet analysis, one can see variations in the frequency of occurrence and amplitude of El Nino. During 1875--1920and 1960--1990 there were... |

7 | Worldwide marine temperature fluctuations 1856–1981 - Folland, Parker, et al. - 1984 |

7 | Orthogonal wavelet analysis: Interannual variability in the sea surface temperature - Mak - 1995 |

6 | Time frequency variability of ENSO and stochastic simulations - Kestin, Karoly, et al. - 1998 |

3 | Wavelet spectrogram of noisy signals - Qiu, Er - 1995 |

2 | Available from Meteorological Office, London Road, Bracknell, Berkshire RG12 2SY, United Kingdom - Spiegel - 1975 |

1 |
On the power spectrum of "red noise
- Gilman, Fuglister, et al.
- 1963
(Show Context)
Citation Context ... with two degrees of freedom (DOF), denoted by �� 2 2 . To determine the 95% confidence level (significant at 5%), one multiplies the background spectrum (16) by the 95th percentile value for ��=-=�� 2 2 (Gilman et al. 1963-=-). The 95% Fourier confidence spectrum for the NINO3 SST is the upper dashed curve in Fig. 3. Note that only a few frequencies now have power above the 95% line. In the previous section, it was shown ... |

1 | The annual cycle of persistence in the El Nio--Southern Oscillation - Torrence, Webster - 1998 |

1 | Interdecadal changes in El Nio onset in the last four decades - Wang - 1995 |