文档介绍：Gravity, Topography and the
Continuous Wavelet Transform
Jon Kirby
Curtin University of Technology
Perth, Western Australia
Continuous wavelet transform
• Developed to analyse signals in real space, rather
than in the Fourier domain
• Can do this because the basis pact in both real
space (“x-space”) and wavenumber space (“k-
space”)
• Contrast with the Fourier transform which uses
infinitely-plex exponentials, and has no
x-space localisation
CWT (cont…)
• The two-dimensional (2D) CWT may be performed in x-
space as the convolution of the signal with the wavelet:
~ ∗
f ()x, s = f (x)∗ψ s (x)
•orin k-space as the product of Fourier transforms:
~
f ()x,s = F−1[fˆ(k)ψˆ∗(sk)]
Harmonic functions
•If g(x) is a harmonic function, then it satisfies Laplace’s
equation:
∇2 g = 0
• Using the derivative identity of convolution,
∇2 ()f ∗ g = (∇2 f )∗ g = f ∗(∇2 g)
• We see that
∇2 ()g~ = ∇2 ()g ∗ψ∗= (∇2 g)∗ψ∗= 0
• Therefore, the CWT of a harmonic function is itself
harmonic
• This enables us to interpret the CWT itself as the field due
to a distribution of sources
Non-orthogonal 2D wavelets
• Consider 3 wavelets:
– Poisson wavelet
–Halo wavelet
–Marr wavelet
• These are all real wavelets, therefore contain no
phase information
• They are all isotropic wavelets (functions of |k|)
– No directional preference, which is essential when analysing
randomly-anisotropic 2D signals
– In theory, it is probably impossible to construct plex,
isotropic wavelet (Farge, 1992)
Poisson wavelet
A “true” harmonic wavelet – derived from the upward
continuation operator of potential theory
2s
ψˆ()s k = k e−s k
3π
λ=F π s 2
Halo wavelet
2D analogue of Morlet wavelet (Dallard & Spedding, 1993)
Take k0 = 6 to ensure admissibility
s 2
ψˆ()s k = e−()k − k0 / 2
2 k0 ππ
4π s 2
λF =
k0 + 4 + k0
Marr wavelet
2nd derivative of Gaussian (m = 2)
− s m 2 2
ψˆ()s k = ()is k e−s k / 2
πm!
2π s 2
λ