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
Figs. 8 and 9 show the comparison between theoretical Q values and the ones estimated
from the numerical 2-D and 3-D simulations, respectively. In all cases, the Q estimates for
a s 5, 10, 20, 40, 80 and 160 m are plotted. Also plotted are the corresponding theoretical
curves for various minimum scattering angles.
Fig. 8. Comparison between 1r Q estimates from 2-D numerical simulations with error bars representing one
Ž .
standard deviation and theoretical 1r Q values thin lines . Ž
. The models are a self-similar medium top an exponential
Ž .
Ž .
medium middle and a Gussian medium bottom .
Fig. 9. Same type of plot as Fig. 8 but for the 3-D simulations.
In all cases, the simulations fall on the 10–208 u
curves for ka - 5. For ka 5, the simula-
min
tions deviate from the 10–208 curves for the Gaussian and exponential medium, where a too
high 1rQ is obtained compared to what is predicted by theory. 1rQ remains almost con-
stant for ka 5. However, for the self-similar medium with n s 0.1 the simulations still fall
on the 10–208 u
curves. This is a somewhat
min
lower value for u than the 298 predicted
min
Ž .
theoretically by Sato 1982 and the 308 sug- gested
by previous
numerical simulations
ŽFrankel and Clayton, 1986, Roth and Korn, .
1993 . We believe the difference between our results and the previous numerical simulations
has two reasons. Firstly, as we stated earlier, the final Q estimate is highly sensitive to how the
Ž .
Q analysis is done. Frankel and Clayton 1986 used the peak amplitude method in the time
domain after bandpass filtering the data. This is similar to not applying a window in our analysis
Ž
. Ž
. Fig. 7 . Sato 1982 did not include a window
at all in his theoretical argument. Roth and Korn Ž
. 1993 applied a 90-ms window to their data
and did the analysis for each frequency between 10 and 80 Hz. This means that the 10-Hz
component will ‘‘see’’ a shorter window than the 80-Hz component and, according to Fig. 7, a
lower 1rQ will be observed. To make a fair comparison for different frequencies, different
window lengths should be applied. We used a window corresponding to 1.5 times the source
wavelet length, a 40-ms window. Using a larger window, or no window, would have given us a
larger u
. Secondly, the model size is of im-
min
Ž .
portance. Frankel and Clayton 1986 report that a random model of 17 wavelengths is enough to
obtain accurate estimates of Q. Roth and Korn Ž
. 1993 used only ; 10 wavelengths. We be-
lieve this is too small. With such a small model, varying the random seed of the random number
generator can change the results significantly. In our simulations, we used a 6-km-deep model in
2-D and a 4-km-deep one in 3-D, and did the analysis over a depth range corresponding to 90
and 55 wavelengths, respectively. With these model sizes, we obtained stable Q estimates.
We also tried varying s , the velocity stan- dard deviation of the random models, to see if
this influenced the choice of the minimum scat- tering angle. We did the simulation in 2-D with
a s 40 m, n s 0.1, and varied s between 2 and 20. Since the models with high s could
contain extremely low velocities we used a source with a lower dominant frequency, 50 Hz,
to avoid numerical dispersion. Fig. 10 shows the result, where all estimates are normalized with
Fig. 10. 1rQ estimates from 2-D numerical simulations with a s 40 m, n s 0.1 and varying standard deviation, s . The source is a plane Ricker wave with a dominant frequency of 50 Hz.
s
2
to allow a direct comparison. It is clear that the choice of u
depends strongly on s .
min
However, the smaller s , the larger the error bars since a medium close to homogeneous will
only slightly attenuate the wave and, therefore, the Q estimate is quite unstable. This s depen-
dence might also explain the difference between our results and those that Frankel and Clayton
Ž
. Ž
. 1986 obtained. They used a higher s 10 in
their models than we did. When comparing Figs. 8 and 9, no major
differences between the 2-D and 3-D simula- tions can be observed. However, we noticed that
the 2-D simulations had larger standard devia- tions than the 3-D simulations when the models
had the same depths. That is, 3-D simulations appear to be more stable than 2-D. Using a
6-km-deep 2-D model and a 4-km-deep 3-D model resulted in roughly the same standard
deviations.
As mentioned in the Random media section, we also did some simulations to obtain con-
straints on the choice of the grid spacing, D. We did the simulations in 2-D for both self-similar
and Gaussian media, with a s 5 m, s s 8,
Fig. 11. 1r Q estimates in media with varying grid spac- ing, D. as 5 m and D varies between 1 and 10 m.
and varied D between 1 and 10 m. The source was a 80-Hz Ricker wavelet. The result is plot-
ted in Fig. 11. Roughly the same 1rQ estimate is obtained for D - 1.5a, but for D 1.5a, the
1rQ estimates starts deviating. We conclude that with D F a, the random model will support
the desired characteristics, such as correlation length and Hurst number. It is somewhat sur-
prising that the grid spacing can be on the same order as a, and a stable estimate of 1rQ can be
obtained. This may be due to that the dominant wavelength used in these simulations was ap-
proximately 60 m, and, therefore, was insensi- tive to variations on the order of 5 m.
7. Discussion
