With the same argument as above, if the grid spacing is too large, no information about the
sloping part of the power spectrum, and, thereby, the correlation length and the Hurst number will
be present in the model. The grid spacing should be chosen at least so that the Nyquist wave
number, k
s prD is greater than k , i.e.,
Nyq c
D -p a. To get further constraints on how small D should be, we performed scattering simula-
tions in both self-similar and Gaussian media with varying D. The analysis of the simulations,
Ž .
which is presented later Comparison section , lead to the conclusion that one grid point per
correlation length is enough for the medium to support the desired characteristics, i.e.,
D F a 8
Ž .
However, the grid spacing is normally deter- mined by the velocity of the model and the
frequency of the source to satisfy the dispersion requirements, so D will be a.
3. Theoretical Q expressions
The theoretical expressions for the frequency dependence of Q are derived from single scat-
Ž .
tering theory, first presented by Chernov 1960 . Details of the derivation may be found in Cher-
Ž .
Ž .
nov 1960 , Wu 1982 and Frankel and Clayton Ž
. 1986 . The relationship between inverse Q and
the power spectrum of the random medium is,
2 p
k u
y1
Q k s
P 2 ksin du
9
Ž . Ž .
H
2- D 2
ž
p 2
u
min
u
min y1
Q k s k P 2 ksin
yP 2 k 10
Ž . Ž
. Ž .
3- D 1
1
ž
ž
2 Ž .
Ž . where P k is the 1-D power spectrum, P k
1 2
is the 2-D power spectrum of the random medium and u
is the minimum scattering
min
angle, i.e., the minimum angle that energy must be scattered to be regarded as not contributing
to the propagating wave. Eq. 9 must be solved numerically.
For self-similar media, Eqs. 9 and 10 become Q
y1
k
Ž .
2- D p
1
2 2
2
s 4 k a s n du
H
nq1 2
u
min
u 1 q 2 kasin
ž
ž
2 11
Ž .
2
2 p s kaG n q 1r2
Ž .
y1
Q k s
Ž .
3- D
G n
Ž .
= 1
nq1r2
u
min 2
2 2
1 q 4 k a sin
ž
2 1
y 12
Ž .
nq1r2 2
2
1 q 4 k a
Ž .
The choice of u is very important, and it is
min
Ž .
not clear what value to use. Sato 1982 derived a value of 298 from a theoretical viewpoint. In
previous comparisons with numerical simula- Ž
tions in 2-D Frankel and Clayton, 1986; Roth .
and Korn, 1993 , a u of around 308 was
min
suggested. In this paper, we will make a similar comparison, both for the 2-D and 3-D case.
4. Synthetic data
We have generated synthetic VSP seismo- grams using a finite difference operator that is
second order accurate in time and fourth order Ž
. accurate in space
Alford et al., 1974 . The source was a plane wave with a dominant fre-
quency of 80 Hz, inserted at the top of the model and allowed to propagate downwards.
We used a Ricker wavelet, i.e., the second derivative of a Gaussian function, which has
smooth derivatives and a smooth frequency
spectrum. Symmetric boundary conditions were implemented at all edges except at the top where
a free surface was used. Since theoretical scat- Ž
. tering Q is a function of ka Eqs. 11 and 12 ,
we kept the frequency content of the source constant and instead varied the correlation dis-
tance. Thereby, we could use the same size for our models without changing the number of
wavelengths the wave could propagate.
The sizes of our 2-D and 3-D models were 1666 = 2000 and 167 = 167 = 1333 grid points,
respectively. The standard deviation of the ran- dom velocity fluctuations was 400 mrs super-
imposed on a constant background velocity of 5000 mrs. We varied the correlation length
from 5 to 160 m, and the Hurst number from 0.1 to 0.5 for the self-similar media. With a grid
spacing of 3 m both expressions 7 and 8 are satisfied, and with a time step of 0.2 ms in 2-D
and 0.15 ms in 3-D, the conditions for stability and numerical dispersion are satisfied.
Fig. 4. One of the synthetic VSP seismograms recorded in a 3-D model with s s8, n s 0.1 and as 40 m. The
source is a plane Ricker wave with a dominant frequency of 80 Hz. The background velocity is constant at 5000
mrs. Fig. 5. Peak amplitude vs. traveltime in synthetic VSP data
Ž .
Ž .
from acoustic solid line and elastic dotted line model- ing.
About 100 evenly distributed VSP seismo- grams were recorded for each model, 48 m
apart in the horizontal direction, in both 2-D and 3-D. No seismograms were recorded close
to the edges to avoid possible edge effects. The downgoing wave was recorded every 33 m for
each VSP. One example of a synthetic seismo- gram is plotted in Fig. 4. It shows a clear first
arrival from the downgoing wave followed by scattered energy from the heterogeneities.
The modelling was performed acoustically, since it is computationally less expensive than
elastic modelling, and we wanted to do a large number of simulations. In our case, with a plane
wave and vertical incidence, the difference be- tween acoustic and elastic modelling should be
minor. We carried out a 2-D elastic test run Ž
. Levander, 1988 for comparison, and, as ex-
pected, could not observe any major differences in the amplitude decay of the P-wave for the
Ž .
two types of modelling Fig. 5 . We believe that the results obtained in this paper hold for the
elastic case as well.
5. Q value estimation
