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Graham Heinson, Jon Kirby, Kent Inverarity, Katherine Stoate, Tania DhuFrome AEM Survey Data Interpretation WorkshopWednesday 30th November 2011Mawson Laboratories, University of Adelaide, Adelaide SA 5001
The Fractal Dimension of the Frome AEM Data
40-60 m Conductivity slice
“We show here that electromagnetic responses are fractal signals, reflecting a very rough distribution of electrical conductivity. Apparent conductivity profiles across a floodplain and a fractured sandstone aquifer both show that the fractal properties of the surface response depend on the complexity of the causative geological structure.”
Sandstones (heterogeneous) have greater slopes of power spectra than (well sorted) alluvial sediments
Many physical and geophysical phenomena arefractal in nature, such as topography and gravity field. These data have spectra with a power-law decay |k|-, where |k| = kx
2+ ky2
The slope of the power spectra is related to thefractal dimension by = 8 - 2FD (or FD = 4-/2)
Wavelet Transform
Wavelet Transform
Wavelet transform enables a ‘‘power spectrum’’ to be calculated at each and every location of a specified signal, i.e., power as a function of space and frequency
Fourier power spectrum, yields power as a function only of frequency - spatial information has been lost
Wavelets use localised basis functions, rather thaninfinitely repeating sines and cosines.
Wavelet transform of a signal,g(x,y) is computed from the convolution of the signal with the complex conjugate of a wavelet, (x,y)
Australian DEM (a), and its scalograms at 21 km scale, for (b) DoG, (c) Fan, (d) Halo, (e) Morlet, (f) Paul, (g) Perrier, and (h) Poisson wavelets. Grey-scale for scalograms is log10(power), from high (light) to low (dark) power.
What information is contained in the Frome AEM data set?•Preliminary study looked at the 40-60 m conductivity slice•Gridded at 0.625 km squaresCan wavelet analysis indicate variability in power and fractal scaling?
DoG Wavelet ScalogramPower contained in
wavelet
40-60 m Conductivity Slice DoG Wavelet at 10 km scale
DoG Fan Halo
Morlet Perrier Poisson
40-60 m Conductivity Slice
Fractal dimension indicates spatial correlation
Conclusion
• Why might is be worth this analysis?– Power-spectra indicate localised change, edges– Differences between homogenous and
heterogeneous Earth– Fractal dimension indicates spatial self similarity
over different scale lengths (correlation with wavelengths)
– Potentially link fractal dimension with geology (fractures, lithology, stratigraphy, groundwater, etc)