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
The deviation from the 10–208 u curves
min
for high ka in Figs. 8 and 9 for Gaussian and exponential media could be caused by multi-
pathing or multiple scattering, explanations that have been mentioned earlier by both Frankel
Ž .
Ž .
and Clayton 1986 and Roth and Korn 1993 . The authors suggested that larger models might
improve the results. We tried using larger mod- els, but our results were the same in the high ka
range. The only difference was that the standard deviation decreased, the mean value remained
the same. We believe that the deviation is caused by the fact that single scattering is not valid in
the high ka range for Gaussian and exponential
Ž .
media. Levander et al. 1994 show a scattering Ž
. regime diagram Fig. 5 in their paper where
Ž .
Ž .
log ka y log Lra space, L is path length, is divided into different regions with different
types of scattering. All our simulations fall into the weak scattering region, i.e., single scattering
should dominate. The simulations with low ka fall into the subpart of the weak scattering
region that can be explained by classical scatter- ing, whereas the simulations with high ka ap-
proach the geometrical optics part where scatter- ing is not relevant. We did some simulations in
2-D for the different models with very large correlation lengths, a s 640 m on a 10 = 6-km
model, and a s 1280 m on a 12 = 12-km model. These simulations are well within the geometri-
cal optics part. The results are shown in Fig. 12 together with theoretical curves for u
s 108.
min
For the self-similar medium, with n s 0.1, the simulations still agree with single scattering the-
ory. However, for the self-similar media with higher Hurst numbers, as well as for the Gauss-
ian medium, simulations and theory diverge and single scattering theory cannot explain the atten-
uation observed in these cases.
In Gaussian and exponential media, the corre- lation length corresponds to the characteristic
scale of the heterogeneities, or ‘‘blobbs’’, in the model. A large value of a means large blobbs,
and the energy will be mainly reflected at the blobbs, not scattered. This means that the wave-
field, and, thereby, Q, will be sensitive only to the velocity contrast in the model, and not what
type of medium it is. However, for self-similar media with low Hurst numbers, scattering is
relevant in the high ka range as well. This type of medium does not have a characteristic scale,
and small-scale heterogeneities, causing scatter- ing, will be present in the model, independently
of a. A low Hurst number will also mean a high content of small-scale heterogeneities. We did
some simulations in models with a low Hurst number and infinite, or absence of, correlation
length, i.e., media that are fractal on all scales, and the estimated 1rQ still agreed well with
single scattering theory for a u
of 10–208.
min
The finite difference modelling in this study shows that the observed Q is dependent upon
the choice of all three parameters, s , n and a. Analyses of the amplitude decay of first arrivals
in VSP data could provide information on these parameters. However, there are a couple of
problems that must be dealt with prior to any ‘‘inversion’’ attempt. Firstly, the physical sig-
nificance of a in real rocks must be further
Ž .
investigated. Holliger 1996 gives various esti- mates of a, but these have been made after the
data have been detrended. Analyses on velocity log data without any detrending indicate a to be
Ž .
Ž .
Fig. 12. 1rQ estimates from 2-D numerical simulations in self-similar models with n s 0.1 squares , n s 0.3 stars , Ž
. Ž
. n s 0.5 circles and Gaussian models diamonds , with correlation lengths of 640 and 1280 m. For visualization, the
estimates are spread out slightly sideways around the dotted lines, which indicate the true position of the estimates. Theoretical curves for u
s 108 are also plotted.
min
Ž .
infinite Dolan and Bean, 1997; Leary, 1997 , i.e., the variations in velocity are scale indepen-
dent. The practical limit for a may be the Ž
. thickness of the crust Dolan and Bean, 1997 .
However, our study shows that in the case of a self-similar medium with a low Hurst number,
single scattering theory can successfully predict the scattering attenuation observed, indepen-
dently of a, so, from the simulation point of view, an infinite a need not be a problem.
Secondly, the observed amplitude decay is highly dependent on the location of the borehole
in the random media. Some locations will give a
Ž .
strong amplitude decay low Q , while others Ž
may even give an amplitude increase negative .
Q . Since generally only one borehole, or at best a few boreholes, are available in a given
area, caution should be used when interpreting scattering Q from VSP data.
Finally, since single scattering theory cannot predict the 1rQ estimates obtained for high ka
in our simulations, and sometimes quite similar values can be obtained for the different models,
it is not possible to discriminate between the different types of random media in this range.
To be able to discriminate a self-similar medium from an exponential or a Gaussian one, ka must
be less than ; 5. Unfortunately, the difference between predicted 1rQ values for the different
types of media is small in the low ka range, indicating the difficulty in extracting informa-
tion about the type of media from only seismic data.
We conclude that, even in the simple case of synthetic data, with pure scattering, it is difficult
to characterize the scattering medium, and to estimate the scattering attenuation. Therefore,
any attempt to separate scattering and intrinsic
attenuation from real data will most likely be very difficult.
8. Conclusions
