A hexagonal finite difference mesh for 2D TTI RTM
Cen Ong, Damir Pasalic and Ray McGarry, Acceleware Corp.
Summary
Theory
A common finite difference implementation of reverse time
migration with tilted transverse isotropy (TTI) follows the
formulation of Alkhalifah (2000), Fletcher et. al. (2008)
and Zhou et. al. (2006). The finite difference
implementation of these equations in Cartesian coordinates
necessitates the computation of mixed partial derivatives of
the acoustic pressure in the spatial domain. Different
methods exist for the computation of these mixed
derivatives including sequentially computing centered first
derivatives in each direction, computing staggered first
derivatives with interpolation or the pseudospectral
method. The computation of centered first derivatives
causes ringing in the output but using staggered derivatives
requires interpolation of the staggered points back to the
original grid. The pseudospectral method necessitates the
computation of a 2D FFT in a 2D simulation. In this paper,
we propose a hexagonal mesh for the finite difference
implementation of 2D TTI RTM. The implementation
eliminates the need for mixed partial derivatives and
reduces grid dispersion in wave simulations.
Introduction
Reverse time migration is gaining popularity with
technological improvements in computing infrastructure
and the need for accurate imaging in ever more challenging
geologies. The algorithm is robust and does not suffer any
dip limitations because it is based on two-way wave
propagation. The derivation of a two-way scalar acoustic
wave equation for isotropic media is straightforward. In
many exploration areas however, isotropic imaging is not
sufficiently accurate, making anisotropic imaging
necessary. The two-way wave equations which incorporate
tilted transverse isotropy (TTI) (Alkhalifah, 2000; Fletcher
et. al., 2008; Zhou et. al., 2006) are robust for modeling and
migrating most seismic anisotropy.
Finite difference anisotropic migrations can be memory and
computationally intensive. The anisotropic wave equations
not only require homogeneous partial second derivatives
but also mixed derivatives. Spatial discretization must be
sufficiently fine to reduce grid dispersion effects. In an
explicit implementation, the temporal discretization is
limited to maintain stability.
This work proposes utilizing hexagonal discretizations for
the finite difference implementation of 2D TTI RTM.
Hexagonal discretizations eliminate the need for mixed
derivatives and reduce overall spatial dispersion.
In Alkhalifah (2000), a pseudo-acoustic approximation for
vertical transversely isotropic media was introduced. The
approximation was further developed for computational
efficiency by Fletcher et. al. (2008) and Zhou et. al. (2006).
We start with the formula for TTI RTM, similar to that of
Fletcher et. al. (2008),
where
(1)
(2)
sin2θ
" !
(3)
(4)
1 2$
1 2%
Now, we reformulate this equation to hexagonal mesh as
follows. Letting &' and (̂ represent unit vectors in x- and zaxes in Cartesian coordinates, unit vectors in hexagonal
coordinates can be written as,
' ,
-
√+
√+
(5)
(6)
&' (̂
&' " (̂
while preserving the unit vector in z-axis. The orientation
of v- and w-axes are shown in Figure 1. The first
derivatives can then be written as,
.
2
/
/
&' &' (̂ 0 · ' √+ (̂ 0 · ,
-
√+ "
(7)
(8)
and the second derivatives are derived by further
differentiating equations (7) and (8), to obtain:
. 2
+ 3 + 3 "
√+ 3 √+ 3 3 (9)
3 (10)
Hexagonal FD mesh for 2D TTI RTM
We transform equations (3) and (4) to hexagonal axes using
the following substitutions,
Derivatives in the spatial domain are typically computed
using a higher order approximation, and can be represented
as:
4
2 4
2 4
1 4
"
4& 3 4 3 4, 3 4( N
89
=
HI 9 K IL @ABLMF ∆FP
8F9 ∆F9 J
4
1 4
1 4
"
4&4( √3 4
√3 4,
LO=
which results in:
√+
6 +
√+
sin2θ7
. 6 "
+
sin2θ7 / " 0 2
+
/
+ . 2 where > is the temporal discretization and :E is the
frequency domain representation of P.
0 " where ∆F is the spatial discretization in v-axis, QIJ , S , IN T
are Taylor spatial derivative coefficients, M is the order of
approximation and MF is the wavenumber in the v-axis.
Similar equations can be written for the spatial derivatives
in w- and z-axes.
(11)
(12)
From equations (11) and (12), we see that transforming the
TTI equations from Cartesian axes to hexagonal axes
eliminates the need to compute finite difference mixedderivatives using field values at co-located gridpoints.
Substituting the frequency and wavenumber representations
into equations (1), (2), (11) and (12), and eliminating P and
R, we obtain
16 sin3 /
W3
VW
VW
0 4 sin / 0
2 2 ?Y Y D Y Y " Z Z 0
W
where Y , Y , Z , Z are functions of the form,
\] , $, %, K
_bQ.,2,T
^_ ] , $, %, `
f
1
Ha
∆ ]
2 K ac de_ ∆Pg
cO
Figure 1 Transforming rectangular to hexagonal grid finite
difference
Dispersion Analysis
Grid dispersion is a significant problem limiting the
computational and memory efficiency of point-wise
discretization schemes for the wave equation (Dablain,
1986). Grid dispersion analysis is performed on the
hexagonal mesh formulation using the approach presented
by Liu and Sen (2010). Assuming a second order finite
difference Taylor approximation, the temporal derivative
can be represented in the frequency domain as follows:
89 :
=
9 ?"9 9@ABC>D:E
9
8;
>
and ^_ ] , $, %, are coefficients that are dependent on
spatially varying velocity, Thomsen parameters and dip.
Solving the equation allows us to determine the numerical
temporal frequency V
h, which we can then use to calculate
h
k
the numerical phase velocity given by ij where k is
l
the wavenumber. The extent of numerical dispersion can
then be calculated as follows,
m
ij
"1
nnop_qno
where the analytical velocity is given by,
nnop_qno
1
] r $ sin stt
2
1
uv1 2$ sin stt w " 8$ " % sin stt cos stt {
2
and stt is the difference between the angle of the k vector
of the wave and the dip angle of the medium.
Hexagonal FD mesh for 2D TTI RTM
different dip angles. An 8th-order finite difference
approximation is used for the spatial derivatives. The
dispersion contour curves versus wavenumber and wave
angle for three different dip angles are plotted in Figures 2,
3 and 4.
Results
Grid dispersion analysis is performed on a constant velocity
c
model with ] 3000 , $ 0.3, % 0.1 for three
|
(a)
(b)
(a)
(b)
Figure 2 Contour plots of d, deviation of finite difference velocity from analytical for 0° dip using (a) hexagonal mesh (b) rectangular mesh
Figure 3 Contour plots of d, deviation of finite difference velocity from analytical for 45° dip using (a) hexagonal mesh (b) rectangular mesh
Hexagonal FD mesh for 2D TTI RTM
(a)
(b)
Figure 4 Contour plots of d, deviation of finite difference velocity from analytical for 60° dip using (a) hexagonal mesh (b) rectangular mesh
In Figures 2-4, hexagonal mesh dispersion contour curves
are compared against those on an equivalently dense
rectangular mesh dispersion contour curves on the same
scale. It can be seen that hexagonal mesh introduces less
phase velocity error at 0, 45 and 60 degree dips,
particularly at high wavenumbers.
The results for wave propagation in constant velocity media
show less dispersion in the hexagonal mesh when
compared to that in the rectangular mesh. This is evident in
the presence of wave ringing around the shear wave in
Figure 5 compared to Figure 6.
Figure 6 Hexagonal grid wave field
Conclusions
The finite difference implementation of reverse time
migration with tilted transverse isotropy on hexagonal
mesh is more computationally efficient than an equivalent
rectangular mesh because the approach eliminates the need
for computing mixed partial derivatives. Hexagonal mesh
also introduces less grid dispersion in wave simulations.
Figure 5 Rectangular grid wave field