Elastic wave-mode separation for VTI media |

The derivative operators for isotropic and anisotropic media are very
different in both shape and size , and the operators vary with
the strength of anisotropy. In theory, analytic isotropic derivatives
are point operators in the *continuous* limit. If one can do
perfect Fourier transform to
and
(without doing the
approximations to different orders of accuracy as one does
in Figure 2), one gets point derivative operators. This is because
is constant in the
direction (see Figure 6(a)), whose
Fourier transform is delta function; the exact expression of
in the
domain also makes the operator point in the
direction. This makes the isotropic derivative operators point
operators in the
and
direction. And when one applies
approximations to the operators, they are compact in the space domain.

However, even if one does perfect Fourier transformation to and (without doing the approximations for different orders of accuracy) for VTI media, the operators will not be point operators because and are not constants in and directions, respectively (see Figure 6(b)). The domain operators spread out in all directions (Figures 3(b), fig:mop4, fig:mop6, and fig:mop8).

IsoU,AniU
(a)
Isotropic and (b) VTI (
,
) polarization
vectors (Figure ) projected on to the
(left
column) and
directions (right column). The isotropic
polarization vectors components in the
and
directions depend
only on
and
, respectively. In contrast, the anisotropic
polarization vectors components are functions of both
and
.
Figure 6. |
---|

This effect is illustrated by
Figure 3. When the order of
accuracy decreases, the isotropic operators become more compact
(shorter in space), while the anisotropic operators do not get more
compact. No matter how one improves the compactness of isotropic
operators, one does not get compact *anisotropic* operators in the
space domain by the same means.

Because the size of the anisotropic derivative operators is usually large, it is natural that one would truncate the operators to save computation. Figure 7 shows a snapshot of an elastic wavefield and corresponding derivative operators for a VTI medium with and . Figure 8 shows the attempt of separation using truncated operator size of (a) , (b) and (c) out of the full operator size . Figure 8 shows that the truncation causes the wave-modes incompletely separated. This is because the truncation changes the directions of the polarization vectors, thus projecting the wavefield displacements onto wrong directions. Figure 9 presents the P-wave polarization vectors before and after the truncation. For a truncated operator size of , the polarization vectors deviate from the correct ones to a maximum of 10 , but even this difference makes the separation incomplete.

uA,mop5
(a) A snapshot of an
elastic wavefield showing the vertical (left) and horizontal (right)
components for a VTI medium (
and
). (b)
order anisotropic pseudo derivative
operators in
(left) and
(right) direction for this VTI
medium. The boxes show the truncation of the operator to sizes of
,
, and
.
Figure 7. |
---|

pA1,pA3,pA5
separation by
order anisotropic pseudo derivative operators
of different sizes: (a)
, (b)
, (c)
, shown in Figure 7(b). The plot shows the
larger the size of the operators, the better the separation is.
Figure 8. |
---|

truncate1,truncate2,truncate3
The deviation of polarization vectors by truncating the size of the
space-domain operator to (a)
, (b)
, (c)
out of
. The left column shows polarization
vectors from
to
cycles in both
and
directions, and the right column zooms to
to
cycles. The
green vectors are the exact polarization vectors, and the red ones
are the effective polarization vectors after truncation of the
operator in the
domain.
Figure 9. |
---|

Elastic wave-mode separation for VTI media |

2013-08-29