Velocity continuation by spectral methods |
For the velocity advancement I used an implicit Crank-Nicolson scheme,
which is unconditionally stable independent of the velocity step size.
By writing equation (12) in the matrix form
cheb1
Figure 6. Synthetic seismic data before (left) and after (right) transformation to the Chebyshev grid in squared time. |
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cheb1-inv
Figure 7. The left plots show the reconstruction of the original data after transforming back from the Chebyshev grid to the original grid. The right plots show the difference with the original model. Top: using the original grid size (). Bottom: increasing the grid size by a factor of three. |
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The first advantage of the Chebyshev approach comes from the better conditioning of the grid transform. Figure 6 shows the synthetic data before and after the grid transform. Figure 7 shows a reconstruction of the original data after transforming back from the Chebyshev grid (Gauss-Lobato collocation points). The difference with the original image is negligibly small.
cheb1-fft
Figure 8. Left: Synthetic data after Chebyshev transform. Right: the real part of the Fourier transform in the space coordinate. |
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The second advantage is the compactness of the Chebyshev representation. Figure 8 shows the data after the decomposition into Chebyshev polynomials in and Fourier transform in . We observe a very rapid convergence of the Chebyshev representation: a relatively small number of polynomials suffices to represent the data.
cheb-impl
Figure 9. Impulse responses (Green's functions) of velocity continuation, computed by the Chebyshev- method. Top: without zero padding, bottom: with zero padding on the axis. The left plots correspond to continuation to a larger velocity ( km/sec); the right plots, smaller velocity, ( km/sec). |
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The third advantage is the proper handling of the non-periodic boundary conditions. Figure 9 shows the velocity continuation impulse responses, computed by the Chebyshev method. As expected, no wraparound artifacts occur on the time axis, and the accuracy of the result is noticeably higher than in the case of finite differences (Figure 1).
Velocity continuation by spectral methods |