TAILIEUCHUNG - Mathematics - Fourier Transforms and Waves

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 frequency). In seismology, the earth generally changes rather strongly with depth, so we cannot usefully Fourier transform the depth axis and we are stuck with differential equations in . On the other hand, we can model a layered earth where each layer has material properties that are. | FOURIER TRANSFORMS AND WAVES in four lectures JonF. Cl rbout Cecil and Ida Green Professor of Geophysics Stanford University January 18 1999 Contents 1 Convolution and Spectra 1 SAMPLED DATA AND Z-TRANSFORMS. 1 FOURIER SUMS. 5 FOURIER AND Z-TRANSFORM. 8 CORRELATION AND SPECTRA. 11 2 Discrete Fourier transform 17 FT AS AN INVERTIBLE MATRIX. 17 INVERTIBLE SLOW FT PROGRAM. 20 SYMMETRIES. 21 TWO-DIMENSIONAL FT. 23 3 Downward continuation of waves 29 DIPPING WAVES . 29 DOWNWARD CONTINUATION. 32 A matlab program for downward continuation. 36 . 38 . 38 . 38 . 38 . 38 . 38 . .

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