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A First Course in Wavelets with Fourier Analysis, 2nd Edition by Francis J. Narcowich, Albert Boggess

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4

HAAR WAVELET ANALYSIS

4.1 WHY WAVELETS?

Wavelets were first applied in geophysics to analyze data from seismic surveys, which are used in oil and mineral exploration to get “pictures” of layering in subsurface rock. In fact, geophysicists rediscovered them; mathematicians had developed them to solve abstract problems some 20 years earlier, but had not anticipated their applications in signal processing.1

Seismic surveys are made up of many two-dimensional pictures or slices. These are sewn together to give a three-dimensional image of the structure of rock below the surface. Each slice is obtained by placing geophones–seismic “microphones”–at equally spaced intervals along a line, the seismic line. Dynamite is set off at one end of the line to create a seismic wave in the ground. Every geophone along the line records the movement of the earth due to the blast, from start to finish; this record is its seismic trace (see Figure 4.1).

The first wave recorded by the geophones is the direct wave, which travels along the surface. This is usually not important. Subsequent waves are reflected off rock layers below ground. These are the important ones. Knowledge of the time that the wave hits a geophone gives information about where the layer that reflected it is located. The “wiggles” that the wave produces tell something about the fine details of the layer. The traces from the all the geophones on a line can be combined to give the slice for the ground directly beneath the line.

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