1. Introduction
Over the past 15 years, scattering of seismic waves in random media in two dimensions has
Ž been studied numerically Frankel and Clayton,
1984, 1986; Gibson and Levander, 1988; Ikelle .
et al., 1993; Roth and Korn, 1993 . The random media generally consist of a deterministic back-
ground velocity plus a random component de- fined by its wave number spectra. Gaussian,
exponential and self-similar random media are often used in these scattering studies since these
are easily defined by their respective wave num- ber spectra. Self-similar media, however, appear
to best describe the spectral content of velocity
Ž logs from boreholes Holliger, 1997; Dolan and
. Bean, 1997; Leary, 1997 . Isotropic self-similar
media are completely described by the standard deviation of the velocity fluctuations, correla-
tion distance and Hurst number.
Analytical expressions for the amplitude de- cay of a forward propagating wave due to scat-
tering may be estimated using single scattering Ž
. theory Chernov, 1960 . An important parame-
ter in the analytical expressions for scattering Q is u
, the minimum angle that energy must be
min
scattered so as not to be considered as contribut- ing to the forward propagating wave. At high
wave numbers, 1rQ can vary by several orders of magnitude depending upon the choice u
.
min
In order to estimate scattering Q from borehole velocity logs it is, therefore, important to know
what u to use in the analytical expressions.
min
Comparison of numerical simulations with the analytical functions is one method for determin-
ing u . For 2-D media, Frankel and Clayton
min
Ž .
Ž .
1986 and Roth and Korn 1993 estimated u
min
to be about 308 for Gaussian and self-similar Ž
. media. Sato 1982 derived a value of 298 from
a theoretical viewpoint. In this paper, we perform a similar compari-
son in 2-D as in the previous studies, but with significantly larger models and more synthetic
data, which give better determined estimates. We also perform a 3-D comparison, which has
not been done before, to investigate if a differ- ence between 2-D and 3-D scattering simula-
tions can be observed, or if 2-D is enough when studying scattering.
In the first part of this paper, we discuss the properties of random media and how they are
generated. We will in detail investigate the dif- ferences between continuous and discrete media
and point at the importance of this difference. In the second part, we generate synthetic seismo-
Ž .
grams for vertical seismic profiling VSP ac- quisition geometries in Gaussian and self-simi-
lar 2-D and 3-D acoustic random media. The Hurst number and standard deviation of the
velocity fluctuations of the random media are systematically varied. From the seismograms,
we calculate the scattering attenuation observed in the finite difference seismograms and com-
pare these results with analytical formulas ŽChernov, 1960; Wu, 1982; Frankel and Clay-
. ton, 1986 .
2. Random media
The random medium most often used to de- scribe heterogeneities in the crust is a spatially
isotropic self-similar medium with a Gaussian probability density function, an autocovariance
Ž . function N r
and corresponding power spec- Ž
. Ž
trum P k given by e.g., Frankel and Clayton,
r
. 1986
2 n
s r
r N r s
K 1
Ž . Ž .
n ny1
ž ž
2 G n
a a
Ž .
E 2
s 2 p a
G n q Er2
Ž .
Ž .
P k s 2
Ž . Ž .
nqEr2 2
2
G n 1 q k a
Ž . Ž .
where r is the spatial lag, s the standard deviation of the velocity perturbations, a the
correlation distance, n the Hurst number, G the Gamma function, k the wave number and E the
Euclidean dimension. It has fractal properties with similar velocity fluctuations on all length
scales less than the correlation distance. For
n s 0.5, Eq. 1 corresponds to the well-known exponential autocovariance function.
We generate our media in the wave number domain by creating a spectrum with a uniform,
random phase between yp and p , and an amplitude equal to the square root of the desired
power spectrum, according to the autocorrela-
Ž . tion theorem. Note that P 0 should be set to
zero for a zero mean distribution. The final random medium is then obtained by inversely
Fourier transforming the wave number spectrum to the spatial domain.
When generating the random medium, we believe it is common to scale the medium to the
desired standard deviation as the last step. This can introduce an error since the standard devia-
tion of a discrete medium differs from that of a continuous one, in some cases, quite signifi-
Ž .
cantly. Frankel and Clayton 1986 discuss this aspect for 2-D Gaussian and exponential media
and the special case of 2-D self-similar media with a zero Hurst number. Here, we will gener-
alise this concept for self-similar and Gaussian media in both two and three dimensions.
The standard deviation, s , can be calculated in the wave number domain in 2-D and 3-D by
` `
1
2
s s
P k , k d k d k
Ž .
H H
2- D x
z x
z 2
4p
y` y` `
1 s
P k k d k
3
Ž . Ž .
H
r r
r
2p s
2 3- D
` `
`
1 s
P k , k , k d k d k d k
Ž .
H H H
x y
z x
y z
3
8p
y` y` y` `
1
2
s P k
k d k 4
Ž . Ž .
H
r r
r 2
2p
2 2
2 2
2
where k s k q k and
k q k q k , re-
r x
z z
y z
spectively. In the discrete case, these integrals, or now rather the sums, will be truncated at the
Nyquist frequency, causing the discrete standard deviation, s , to be less than the corresponding
d
continuous one. Further, since the size of the model is limited, the low wave numbers will not
be present, further reducing s . That is, Eqs. 3
d
and 4 take the form 1
2
s s
ÝÝP k ,k D k D k 5
Ž .
Ž .
d ,2 - D x
z x
z 2
p 1
2
s s
ÝÝÝP k ,k ,k D k D k D k
Ž .
d ,3- D x
y z
x y
z 3
p 6
Ž .
where D k s 2prND and 0 F k F k s
x x
Nyq, x
Ž .
prD the same for y and z , where D is the grid spacing. The sums over P are calculated by
taking the average of neighbouring nodes. Fig. 1 shows how s varies with D in a model with a
d
fixed number of nodes. The decrease in s for
d
Ž .
larger D D 4 m in Fig. 1 is caused by the truncation of the sum at the Nyquist wave num-
Ž ber. The decrease in s
for smaller D D - 4
d
. m is caused by the limited size of the model,
i.e., the loss of low wave numbers. The reason for the reduction of s at the low
d
and high ends of the spectrum are of a com- pletely different nature. The ‘‘Nyquist decrease’’
is not an error, it is only the effect of going from the continuous to the discrete case. How-
Fig. 1. The discrete standard deviation variation vs. grid Ž
.
3
spacing as predicted by Eq. 6 solid line in a model 128 nodes. Standard deviations observed in our random models
Ž .
with Ds 0.5, 1.0, 2.0, 3.0, 4.0, 6.0 and 8.0 m are plotted by stars.
ever, it is important to be aware of this differ- ence in order to obtain a discrete medium with
the corresponding continuous properties. If the random medium is scaled at the end to the
desired continuous standard deviation, this will result in a medium with the wrong character-
istics. If the generation of the wave number spectrum and the inverse transform are done
correctly, the standard deviations predicted by expressions 5 and 6 will fall out naturally, and
should not be corrected for. The stars plotted in Fig. 1 show the standard deviations calculated
from our random models without any scaling. There is a good agreement between the pre-
dicted standard deviation and the observed stan- dard deviation in the model.
On the other hand, the decrease caused by the size of the model will result in a model with
completely wrong characteristics. If the model is too small, the low wave numbers will be
missing, along with the information about the correlation length. We calculated the power
spectra and the autocorrelations of our models to check that we really had the characteristics
that we wanted. In Fig. 2, the power spectra calculated from two models with the same pa-
rameters and grid spacing but different sizes are
Ž
3
. shown. The larger model 128
nodes is 12 times larger than the correlation length and the
3
Ž Fig. 2. Power spectra for a model with 32
nodes black .
3
Ž .
line and a model with 128 nodes grey, thick line . Fig. 3. Autocorrelations of the same models as in Fig. 2
Ž
3 3
dotted line for the 32 model, dashed line for the 128
. model , in comparison with the theoretical correlation
Ž .
function solid line .
power spectrum shows both the flat part and the sloping fractal part, where the transition be-
tween the two is determined by the correlation length, a, i.e., where ka s 1. On the other hand,
Ž
3
. for the smaller model 32
nodes , which is only three times larger than the correlation
length, the power spectrum only contains infor- mation about the fractal part and does not give
any information about the correlation length. In Fig. 3, the theoretical correlation function given
by Eq. 1 is plotted in comparison with the observed autocorrelations of the two models.
These observed correlations are calculated by extracting several synthetic 1-D velocity logs
from the models, calculating the autocorrela- tions for each log and then taking the average.
The correlation curve from the smaller model deviates severely from the theoretical curve
showing that this model does not have the de- sired characteristics. Our conclusions are that
the size of the model must be large enough so that the minimum wave number, k
s 2prND,
min
is smaller than the corner wave number, k s
c
1ra, i.e., ND 2p a
7
Ž .
where ND is the length of the model.
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
Kategori