Why is the phase of the input wavelet rotated +- 45º when 2D finite difference modeling is performed for seismic wave propagation?
By Dablain (1986) and others I know that this does not happen for 1D and 3D modeling, only for 2D. Could someone provide me also some reference?
The phase shift is associated with the properties of waves on a 2-D membrane. Morse and Feshbach, 1953 (Methods of Theoretical Physics, Part I, Section 7.3, and especially Figure 7.11 and the summary on page 893-894) give the Green's functions for 1-D, 2-D and 3-D media. In the time domain in 2-D the pulse shape has a "tail" that is not present in the 1-D and 3-D solutions. This tail corresponds to a Hankel function in the frequency domain (page 891) which has a frequency independent phase shift of 90degrees (-i). Also see Chapman, 1978, GJRAS v54, 481-518.
Mr Stephen, thanks for your answer! What a coincidence, yesterday I was reading your paper from 1988 about the review of FD methods. This is cool! Really thanks! I was looking for this answer since I started the FD modeling and was not successful up to now.
2D wave propagation in the x-z plane (for y=0) can be modeled by having a line of sources in the y-direction. Contributions from this spatially extended source from outside the y=0 plane have a time delay with respect to contributions from the part of the source that is at y=0. These 'out of plane' contributions lead to a tailing of amplitudes after the onset. Integration over a line of sources extending to infinity (all time delays) leads to a pi/4 phase shift of the wavefield with respect to the wavefield that would be modeled if there were only a point source at (x,y,z)=(0,0,0).
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