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Figure 17 The wind tunnel turbulence data, with the Hilbert, singularity, and marginal spectra, respectively. Annu Rev. Fluid. Mech. 1999.31:417-457. Downloaded from arjoumals.annualreviews.org by CALIFORNIA INSTITUTE OF TECHNOLOGY on 09/08/05. For personal use only.

Figure 17 The wind tunnel turbulence data, with the Hilbert, singularity, and marginal spectra, respectively. Annu Rev. Fluid. Mech. 1999.31:417-457. Downloaded from arjoumals.annualreviews.org by CALIFORNIA INSTITUTE OF TECHNOLOGY on 09/08/05. For personal use only.

Related Figures (17)

Figure 1 Comparisons of Wavelet and Hilbert representation for simple symmetric data. The Hilbert transfonm can be applied to these types of data to give better time-frequency resolution without difficulty. Annu. Rev. Fluid. Mech. 1999.31:417-457. Downloaded from arjoumals.annualreviews.org by CALIFORNIA INSTITUTE OF TECHNOLOGY on 09/08/05. For personal use only.
Figure 1 Comparisons of Wavelet and Hilbert representation for simple symmetric data. The Hilbert transfonm can be applied to these types of data to give better time-frequency resolution without difficulty. Annu. Rev. Fluid. Mech. 1999.31:417-457. Downloaded from arjoumals.annualreviews.org by CALIFORNIA INSTITUTE OF TECHNOLOGY on 09/08/05. For personal use only.
Figure 2. Demonstration of the empirical mode decomposition method from length-of-the-day deviation data; the unit is mi: croseconds.
Figure 2. Demonstration of the empirical mode decomposition method from length-of-the-day deviation data; the unit is mi: croseconds.
Figure 3 Reconstruction of the original data from the IMF components. The difference between the reconstructed and the original data is only 10-° microsecond. Intermittency Test
Figure 3 Reconstruction of the original data from the IMF components. The difference between the reconstructed and the original data is only 10-° microsecond. Intermittency Test
Figure 4 Effect of intermittency criterion in the EMD sifting process invoked to eliminate mode mixing. Sequence b-d with intermittency check; sequence e-h without intermittency check.
Figure 4 Effect of intermittency criterion in the EMD sifting process invoked to eliminate mode mixing. Sequence b-d with intermittency check; sequence e-h without intermittency check.
Figure 5 The comparison of Hilbert and Fourier spactral analysis for highly transient data. The Fourie method cannot reveal of the variation of the signal with time. Annu Rev. Fluid. Mech. 1999.31:417-457. Downloaded from arjoumals.annualreviews.org by CALIFORNIA INSTITUTE OF TECHNOLOGY on 09/08/05. For personal use only.
Figure 5 The comparison of Hilbert and Fourier spactral analysis for highly transient data. The Fourie method cannot reveal of the variation of the signal with time. Annu Rev. Fluid. Mech. 1999.31:417-457. Downloaded from arjoumals.annualreviews.org by CALIFORNIA INSTITUTE OF TECHNOLOGY on 09/08/05. For personal use only.
Figure 6 The Hilbert and marginal spectra for the length of the day data. It gives detailed time variation of the frequency with time. Annu Rev. Fluid. Mech. 1999.31:417-457. Downloaded from arjoumals.annualreviews.org by CALIFORNIA INSTITUTE OF TECHNOLOGY on 09/08/05. For personal use only.
Figure 6 The Hilbert and marginal spectra for the length of the day data. It gives detailed time variation of the frequency with time. Annu Rev. Fluid. Mech. 1999.31:417-457. Downloaded from arjoumals.annualreviews.org by CALIFORNIA INSTITUTE OF TECHNOLOGY on 09/08/05. For personal use only.
Figure 7 Camparison of Hilbert and Wavelet spectra for the intta-wave modulation case The Hilbert view avoids the harmonics, and itgives the real intra-wave frequency modulations Annu. Rev. Fluid. Mech. 1999.31:417-457. Downloaded from arjoumals.annualreviews.org by CALIFORNIA INSTITUTE OF TECHNOLOGY on 09/08/05. For personal use only.
Figure 7 Camparison of Hilbert and Wavelet spectra for the intta-wave modulation case The Hilbert view avoids the harmonics, and itgives the real intra-wave frequency modulations Annu. Rev. Fluid. Mech. 1999.31:417-457. Downloaded from arjoumals.annualreviews.org by CALIFORNIA INSTITUTE OF TECHNOLOGY on 09/08/05. For personal use only.
Figure 9 The Hilbert and Marginal spectra for the Duffing equation solution. The Hilbert reveals the detailed frequency modulation of the data. Annu. Rev. Fluid. Mech. 1999.31:417-457. Downloaded from arjoumals.annualreviews.org by CALIFORNIA INSTITUTE OF TECHNOLOGY on 09/08/05. For personal use only.
Figure 9 The Hilbert and Marginal spectra for the Duffing equation solution. The Hilbert reveals the detailed frequency modulation of the data. Annu. Rev. Fluid. Mech. 1999.31:417-457. Downloaded from arjoumals.annualreviews.org by CALIFORNIA INSTITUTE OF TECHNOLOGY on 09/08/05. For personal use only.
Figure 8 Data, phase diagram, and IMF components for the solution of the Duffing equation. Trajectories of motion show clear bunching at the preferred paths, indicating additional time scales.
Figure 8 Data, phase diagram, and IMF components for the solution of the Duffing equation. Trajectories of motion show clear bunching at the preferred paths, indicating additional time scales.
Figure 10 Data, phase diagram, and IMF components for the solution of the Rossler equation. This is the case of period doubling, which indicates two time scales. in which yw is a constant parameter. The numerical value of x computed with je = 3.5 is given in Figure 10a with its phase diagram in Figure 10b. Thisis the
Figure 10 Data, phase diagram, and IMF components for the solution of the Rossler equation. This is the case of period doubling, which indicates two time scales. in which yw is a constant parameter. The numerical value of x computed with je = 3.5 is given in Figure 10a with its phase diagram in Figure 10b. Thisis the
Figure 11 The Hilbert and Marginal spectra for the R6ssler equation solution. The peak of the Fourier spectum is not any real scale of the oscillation, but a weighted mean. Annu. Rev. Fluid. Mech. 1999.31:417-457. Downloaded from arjoumals.annualreviews.org by CALIFORNIA INSTITUTE OF TECHNOLOGY on 09/08/05. For personal use only.
Figure 11 The Hilbert and Marginal spectra for the R6ssler equation solution. The peak of the Fourier spectum is not any real scale of the oscillation, but a weighted mean. Annu. Rev. Fluid. Mech. 1999.31:417-457. Downloaded from arjoumals.annualreviews.org by CALIFORNIA INSTITUTE OF TECHNOLOGY on 09/08/05. For personal use only.
Figure 12 The selected Hilbert, Wavelet and their marginal spectra for the laboratory data of the Stokes waves. Both Wavelet and Hilbert spectral analysis reveal the front-and-back-asymmetry. The Hilbert spectra offers better frequency resolution. Annu. Rev. Fluid. Mech. 1999.31:417-457. Downloaded from arjoumals.annualreviews.org by CALIFORNIA INSTITUTE OF TECHNOLOGY on 09/08/05. For personal use only.
Figure 12 The selected Hilbert, Wavelet and their marginal spectra for the laboratory data of the Stokes waves. Both Wavelet and Hilbert spectral analysis reveal the front-and-back-asymmetry. The Hilbert spectra offers better frequency resolution. Annu. Rev. Fluid. Mech. 1999.31:417-457. Downloaded from arjoumals.annualreviews.org by CALIFORNIA INSTITUTE OF TECHNOLOGY on 09/08/05. For personal use only.
Figure 13 Fourier spectra for two stations with different frequency resolutions, to demonstrate that downshift of the spectral peak depends on frequency resolution.
Figure 13 Fourier spectra for two stations with different frequency resolutions, to demonstrate that downshift of the spectral peak depends on frequency resolution.
Figure 14 Joint distribution of phase and amplitude and waves at different stations all relative to the first station. Phase variation is discrete. Huang et al (1996) opted for the use of Hilbert analysis, with which variation of frequency can be defined much more precisely and locally. Phase variation can be presented in two ways: first, the total phase value changes with respect to the reference station, e.g. station 1. This revealed that the decrease of the total phase values was an integer multiple of 27. Secondly, the relative phase variation can be presented in ajoint distribution with the elevation. The selected results are shown in Figure 14a-d. The discrete location of the phase variation is
Figure 14 Joint distribution of phase and amplitude and waves at different stations all relative to the first station. Phase variation is discrete. Huang et al (1996) opted for the use of Hilbert analysis, with which variation of frequency can be defined much more precisely and locally. Phase variation can be presented in two ways: first, the total phase value changes with respect to the reference station, e.g. station 1. This revealed that the decrease of the total phase values was an integer multiple of 27. Secondly, the relative phase variation can be presented in ajoint distribution with the elevation. The selected results are shown in Figure 14a-d. The discrete location of the phase variation is
Figure 15 Fusion of two waves into one also coincides with a phase jump. Magnitude of phase jump at 49 seconds is exactly 27.
Figure 15 Fusion of two waves into one also coincides with a phase jump. Magnitude of phase jump at 49 seconds is exactly 27.
Figure 16 A selected section of the Hilbert and Wavelet spectra together with the wave profile. The Hilbert spectrum can track the frequency variation closely. ‘The marginal spectra show that the Wavelet spectrum has very poor frequency resolution. Annu Rev. Fluid. Mech. 1999.31:417-457. Downloaded from arjoumals.annualreviews.org by CALIFORNIA INSTITUTE OF TECHNOLOGY on 09/08/05. For personal use only.
Figure 16 A selected section of the Hilbert and Wavelet spectra together with the wave profile. The Hilbert spectrum can track the frequency variation closely. ‘The marginal spectra show that the Wavelet spectrum has very poor frequency resolution. Annu Rev. Fluid. Mech. 1999.31:417-457. Downloaded from arjoumals.annualreviews.org by CALIFORNIA INSTITUTE OF TECHNOLOGY on 09/08/05. For personal use only.
Figure 18 Selected sections of the intrinsic mode function expansion of the turbulence data Events A and B show the conflict with the fractal hypothesis.
Figure 18 Selected sections of the intrinsic mode function expansion of the turbulence data Events A and B show the conflict with the fractal hypothesis.
Related topics:
EngineeringEarth SciencesTime-Frequency AnalysisNonlinear WavesWavelet AnalysisEmpirical mode decomposition
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