Mathematics - Fourier Transforms And Waves

Mathematics - Fourier Transforms And Waves - FOURIER...

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Unformatted text preview: FOURIER TRANSFORMS AND WAVES: in four lectures Jon F. Claerbout Cecil and Ida Green Professor of Geophysics Stanford University c January 18, 1999 Contents 1 Convolution and Spectra 1 1.1 SAMPLED DATA AND Z-TRANSFORMS . . . . . . . . . . . . . . . . . 1 1.2 FOURIER SUMS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.3 FOURIER AND Z-TRANSFORM . . . . . . . . . . . . . . . . . . . . . . 8 1.4 CORRELATION AND SPECTRA . . . . . . . . . . . . . . . . . . . . . . 11 2 Discrete Fourier transform 17 2.1 FT AS AN INVERTIBLE MATRIX . . . . . . . . . . . . . . . . . . . . . 17 2.2 INVERTIBLE SLOW FT PROGRAM . . . . . . . . . . . . . . . . . . . . 20 2.3 SYMMETRIES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 2.4 TWO-DIMENSIONAL FT . . . . . . . . . . . . . . . . . . . . . . . . . . 23 3 Downward continuation of waves 29 3.1 DIPPING WAVES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 3.2 DOWNWARD CONTINUATION . . . . . . . . . . . . . . . . . . . . . . 32 3.3 A matlab program for downward continuation . . . . . . . . . . . . . . . . 36 3.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 3.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 3.6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 3.7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 3.8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 3.9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 3.10 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 CONTENTS 3.11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 3.12 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 3.13 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 3.14 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 3.15 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 3.16 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 3.17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 3.18 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 Index 39 Why Geophysics uses Fourier Analysis When earth material properties are constant in any of the cartesian variables then it is useful to Fourier transform (FT) that variable. In seismology, the earth does not change with time (the ocean does!) so for the earth, we can generally gain by Fourier transforming the time axis thereby converting time-dependent differential equations (hard) to algebraic equations (easier) in frequency (temporal fre- quency)....
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This note was uploaded on 03/09/2010 for the course MECHANICAL ME9802701 taught by Professor Prof.william during the Spring '10 term at Institut Teknologi Bandung.

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Mathematics - Fourier Transforms And Waves - FOURIER...

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