Lowrank seismic wave extrapolation on a staggered grid

A simple illustration of the accuracy

We first use a simple 1-D example shown in Figure 4 to demonstrate the accuracy of the SGL method and SGLFD method when they are used to calculate the partial derivatives in equation 9. The velocity increases linearly from 1000 to 2275 m/s. The rank is 2 for lowrank decomposition, assuming 1 ms time step. The exact $kx$ -space operator $\partial /\partial ^+x$ in equation 9 is shown in Figure 4a. Figures 4b, 4c and 4d display errors of approximation operators of SGL, SGLFD and conventional staggered grid finite-difference (SGFD), respectively. Figure 5 shows the middle column of the error matrix. The errors of SGL and SGLFD are significantly smaller than that of SGFD.

Mexact,Mlrerr,Mapperr,Mfd10err
Figure 4. (a) $kx$ -space operator $\partial /\partial ^+x$ for 1-D linearly increasing velocity model. (b) Error of SGL operator. (c) Error of 8th order SGLFD operator. (d) Error of 8th order SGFD operator.

slicel
Figure 5. Middle column of the error matrix. Blue dashed line: SGL operator. Green dotted line: the 8th order SGLFD operator. Red solid line: the 8th order SGFD operator.

Lowrank seismic wave extrapolation on a staggered grid