Lowrank finite-differences and lowrank Fourier finite-differences for seismic wave extrapolation in the acoustic approximation |

In a matrix notation, the lowrank decomposition problem takes the following form:

Note that is a matrix related only to wavenumber .
We propose to further decompose it as follows:

where is an matrix,

and

According to the shift property of FFTs, we finally obtain an expression in the space-domain

where , and .

Equation 14 indicates a procedure of finite differences for wave extrapolation: the integer vector,
provides the stencil information, and
stores the corresponding coefficients.
We call this method *lowrank finite differences* (LFD)
because the finite-difference coefficients are derived from a lowrank approximation of the mixed-domain propagator matrix.
We expect the derived LFD scheme to accurately propagate seismic-wave components within a wide range of wavenumbers,
which has advantages over conventional finite differences that focus mainly on small wavenumbers.
In comparison with the Fourier-domain approach, the cost is reduced to ,
where , as the row size of matrix , is related to the order of the scheme.
can be used to characterize the number of FD coefficients in the LFD scheme, shown in equation 14.
Take the 1-D 10th order LFD as an example, there are 1 center point, 5 left points () and 5 right ones ().
So
, and
.
Thanks to the symmetry of the scheme,
coefficients of and are the same, as indicated by equation 14.
As a result, one only needs 6 coefficients: .

Mexact,Mlrerr,Mapperr,Mfd10err
(a) Wavefield extrapolation matrix for 1-D linearly increasing velocity model. Error of wavefield extrapolation matrix by:(b) lowrank approximation, (c) the 10th-order lowrank FD (d) the 10th-order FD.
Figure 1. |
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We use a one-dimentional example shown in Figure 1 to demonstrate the accuracy of the proposed LFD method. The velocity linearly increases from 1000 to 2275 m/s. The rank is 3 () for lowrank decomposition for this model with 1 ms time step. The propagator matrix is shown in Figure 1a. Figure 1b-Figure 1d display the errors corresponding to different approximations. The error by the 10th-order lowrank finite differences (Figure 1c) appears significantly smaller than that of the 10th-order finite difference (Figure 1d). Figure 2 displays the middle column of the error matrix. Note that the error of the LFD is significantly closer to zero than that of the FD method.

slicel
Middle column of the error matrix. Solid line: the 10th-order LFD. Dash line: the 10th-order FD.
Figure 2. | |
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To analyze the accuracy, we let

For 1-D 10th order LFD, , and . With equation 16, we can calculate phase-velocities () by 1-D 10th order LFD with different velocities ( ), and we use the ratio to describe the dispersion of FD methods. Figure 3a displays 1D dispersion curves by 1-D 10th order LFD, and Figure 3b shows those by conventional FD method. Note that compared with the conventional FD method, LFD is accurate in a wider range of wavenumbers (up to 70% of the Nyquist frequency).

app,fd10
Plot of 1-D dispersion curves for different velocities, (red), (pink), (green), (blue) , ms, m by: (a) the 10th-order LFD (b) the 10th-order conventional FD.
Figure 3. |
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Lowrank finite-differences and lowrank Fourier finite-differences for seismic wave extrapolation in the acoustic approximation |

2013-07-26