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
The scattering attenuation in the synthetic data is estimated by the peak amplitude method
Fig. 6. The distribution of Q values calculated from the synthetic data described in Fig. 4.
giving an apparent Q for the model. Briefly, the method consists of comparing the peak ampli-
tude of the downgoing wave at different depths. The amplitude of a plane wave can be described
Ž .
by Aki and Richards, 1980
p f z y
A z s A e 13
Ž . Ž .
Qc
where Q is the number of wavelengths before the amplitude has been attenuated to e
yp
, or about 4 of its original value, f is the fre-
quency of the wave, z is distance and c is wave velocity. Comparing the amplitudes at distance
z and z , assuming approximately constant
1 2
velocity and taking the natural log, gives p f
ln A z y ln A z
s y DT
14
Ž . Ž .
Ž .
Ž .
Ž .
2 1
Q where DT is the traveltime from z
to z . A
1 2
Ž . plot of ln A vs. DT will yield a straight line,
where Q can be calculated from the slope. In
Ž .
Fig. 7. 1rQ estimates squares from the same model as in Fig. 4 with error bars representing one standard deviation. Solid Ž
. lines show values predicted by single scattering theory. The circle at zero window length shows the 1rQ estimate obtained
if no window is used, which corresponds to 1rQ being estimated in the time domain. The arrow indicates the 40-ms window used in our study.
practice, the slope is determined by least squares. We did the Q estimation in the frequency do-
main by applying a window around the first arrival and Fourier transforming the data. The
apparent attenuation is then calculated with the peak amplitude method for the 80-Hz compo-
nent, the dominant frequency of our source.
The calculation of Q is quite unstable, and, in many cases, several synthetic seismograms
are generated and stacked prior to the analysis Ž
. e.g., Korn, 1993 . However, stacking and esti-
mating mean field attenuation can give the Ž
. wrong result Sato, 1982; Wu, 1982 , and, in
this study, we analysed each VSP seismogram individually, which resulted in a distribution of
Ž .
Q values for each model Fig. 6 . From this distribution, we then calculated a mean value
and a standard deviation, and compared it with what is predicted by single scattering theory.
The Q estimate is highly dependent on where in the model the seismogram is recorded. Gen-
erally, the amplitude decay does not plot as a straight line but more chaotically. In some cases,
the amplitude even increases with distance, re- sulting in a negative Q estimate. Further, quite
different results can be obtained depending on how the analysis is done, as has been discussed
Ž by several authors
Sato, 1982; Wu, 1982; .
Shapiro and Kneib, 1993 . The length of the window around the first arrival greatly affects
Ž .
the final Q estimate Shapiro and Kneib, 1993 . Fig. 7 shows the different results that can be
obtained when varying the length of the win- dow. 1rQ is plotted vs. the window length
together with the predicted theoretical 1rQ val- ues for some minimum scattering angles. In-
creasing the window length results in that more scattered energy is included in the analysis and
a lower 1rQ, i.e., a lower damping is observed. In the following we have used a window length
of 40 ms, 1.5 times the source wavelet length, to exclude the scattered coda and only account
for the amplitude change in the first arrival. We also tried varying the tapering length of the
window, but this did not have any influence on the final estimate.
6. Comparison
