Geoph1 1 J lnt ( 1998) 134, 778 786
Velocity shift in heterogeneous media with anisotropic spatial
correlation
Yann SamuelidesL* and Tapan Mukerjil
1
Centre de Geoltatllttque. Ecole de.1 Mme.\ de Paris. 77305. Fontamehleau, France
cRock Phr11cs Lahorator1. Department of Geophystcs, Stanford Umversity, CA 94305-2215, USA. E-mail: mukerji(alpangea.stanford.edu
Accepted 1998 March 31. Received 1998 March 25; in onginal form 1997 September 2
SUMMARY
Wave propagation in heterogeneous media depends on the relative scales of the wavelength and the spatial correlation length of the heterogeneities. Heterogeneities can give
rise to velocity dispersion due to the fast path effect. Short wavelengths tend to diffract
around the slower inhomogeneities, causing the arrival times to be biased towards lower
values than the arrival times for the average slowness of the medium. The difference
between the high-frequency, short-wavelength velocity obtained from the mean arrival
time and the mean velocity of the heterogeneous medium is known as the velocity shift.
The velocity shift depends on the spatial autocorrelation function of the heterogeneous
random medium. Previous investigations of velocity shift in heterogeneous media have
mostly been limited to media with isotropic spatial correlation functions. We extend the
analysis to media with anisotropic distributions of spatial inhomogeneities. A smallperturbation, asymptotic geometrical optics formulation is used to derive expressions
for the first and second moments of the phase of a wavefield propagating in a random
heterogeneous medium. This allows us to express the velocity shift as a function of
angle between direction of propagation and direction of maximum spatial correlation.
We show how anisotropic spatial heterogeneity can cause a splitting of the isotropic
velocity-shift behaviour. The velocities become much faster along the direction of high
spatial correlation, but are slower along the perpendicular direction.
Key words: anisotropy, heterogeneity, random media, traveltime fluctuations,
velocity shift.
INTRODUCTION
Wave propagation in heterogeneous media is affected by the
spatial distribution of the heterogeneities. In the Earth's crust,
subsurface heterogeneities include variations in lithology,
porosity, permeability, pore-fluid properties,. and conditions of
pore pressure, temperature and stress. They occur over a broad
range of scales, from submillimctre grain and pore scale to
the many-kilometre basin scale, and they arc almost always
spatially anisotropic. Similarly, seismic measurement scales
range from millimetre wavelengths in ultrasonic laboratory
measurements to tens of metres in surface seismic. Therefore,
an important issue for interpreting geophysical images is to
analyse and to understand the impact of spatial heterogeneities
on wave propagation.
• This work was carried out whilst VISitmg the Department of
Mathematics at Stanford University. CA 94305, USA.
778
The measurable arrival times, from which velocities are
determined, of seismic events propagating in heterogeneous
media depend on the scale of the heterogeneity relative to
the seismic wavelength, and the propagation distance. In
general, the velocity inferred from arrival times is slower when
the wavelength is longer than the scale of heterogeneity and
faster when the wavelength is shorter. For normal-incidence
propagation in stratified media, this is the difference between
averaging seismic slownesses, in the short-wavelength limit,
and averaging elastic compliances, in the long-wavelength
limit (e.g. Marion, Mukerji & Mavko 1994). In two and three
dimensions there is also the fast path effect. Shorter wavelengths tend to find faster paths, and so bias the traveltimes
to lower values. In the short-wavelength limit, the slowness
inferred from the average traveltime is smaller than the mean
slowness of the medium. When the propagation distance is
much larger than the scale of the heterogeneity, the path effect
causes the velocity increase from long to short wavelength to be
much larger in two dimensions than in one dimension, and
even larger in three dimensions.
© 1998 RAS
Velocity shift in heterogeneous media
Wave propagation in random media has been studied both
theoretically and numerically by a number of authors (e.g.
Chernov 1960; Keller 1964; Boyse 1986; Frankel & Clayton
1986; Miiller, Roth & Korn 1992; Korn 1993; Shapiro & Kneib
1993; Shapiro, Schwarz & Gold 1996). Typically, the random
medium is characterized by a spatially varying slowness
field, (I I c0 )n(r) =(I I c0 )(l + oJ(r)), where r is the position
vector, 17(r) is a fluctuation with zero mean {(IJ(r)) =0) superposed on the background slowness, I I c0 = s0 , and f « I is a
small perturbation parameter. The spatial structure of the
heterogeneities is described by the spatial autocorrelation
function (fiJ(rt)fiJ(r2 ))=f 2(1] 2)N(r1 -r2 ), where ( ) denotes
the expectation operator, and €2 (17 2 )=~ is the variance
of the relative slowness fluctuations. Wavefields in such a
stochastic medium are themselves random functions with
traveltime and amplitude fluctuations. Statistical moments of
these fluctuations can be related to the moments of the slowness fluctuations of the medium (Ishimaru 1978; Muller eta/.
1992; Korn 1993; Shapiro & Kneib 1993). In one of the most
recent works, Shapiro eta/. (1996) gave detailed accounts of
the velocity shift, both before and after the Fresnel distance,
in media with isotropic correlation functions. We use the
approach of Boyse & Keller (1995) and generalize it to the case
where the correlation function is anisotropic. Anisotropic
correlations were also investigated by van Avendonk & Snieder
(1994) using a different kinematic ray-bending approach to
obtain a similar directional dependence of the velocity shift
to that obtained in this work. However, they considered only
the case of rotationally symmetric correlation functions, with
identical correlation lengths along the horizontal direction.
In this work we consider more general forms of anisotropic
spatial correlations. In addition to the analytical results we
compare the theoretical predictions with finite-difference
computations over multiple stochastic realizations of the
random medium. Gist (1994) used Karal & Keller's (1964)
perturbation expansion to study velocity dispersion in random
media with anisotropic spatial correlation. This velocity dispersion effect, arising from scattering, is inversely proportional
to the wavenumber, k, for a fixed standard deviation of the
slowness fluctuations of the random medium. Hence it becomes
very small at high frequencies (ray-theory limit, k-->oo). In
contrast, the velocity shift we describe in this paper is predominantly a ray-theory, high-frequency phenomenon, caused
by preferential fast paths through the random heterogeneous
medium.
We start by reviewing propagation and velocity shift in
heterogeneous media with an isotropic correlation function.
Next, we consider 2- and 3-D anisotropic autocorrelation
functions where the anisotropic spatial heterogeneity can
cause splitting of the isotropic velocity-shift behaviour. The
velocities become much faster along the direction of high
spatial correlation, but are slower along the perpendicular
direction. Then, we present comparisons with results of
2-D numerical computations, which allows us to study the
saturation of the shift at very large propagation distances, and
the shift in media with large variances in slowness. Finally, we
discuss the implications for laboratory and field measurements
of seismic traveltime.
In seismic measurements we almost never measure elastic
constants or velocities directly. Instead we most often measure
times of wave arrivals, and infer the velocities from the ratio
of distance to traveltime. Therefore, our emphasis here is on
~
1998 RAS, GJ/134, 778-786
779
measurable scale-dependent arrival times of events, as we
would encounter in the laboratory or in the field.
RAY THEORY IN 2- AND 3-D
HETEROGENEOUS MEDIA
We start with monochromatic waves, which obey the
Helmholtz equation:
(I)
where U is the wavefield and k is the wavenumber. In the highfrequency regime, when ka» I (where a is the characteristic
correlation length of the medium), and when the distance of
propagation, X, is small compared to the Fresnel distance
(X «ka 2), so that diffraction effects are negligible, we can use
the geometrical optics approximation
U(r) = A(r) e•k<P(r),
(2)
where A(r) is the amplitude, and the phase </> satisfies the
eikonal equation
(V ¢(r)) 2 = n(r) 2 .
(3)
The phase is related to the traveltime T(r) by T(r) = </>(r)l c0,
I I co being the average slowness in the medium. Our goal is
to compute the first and second moments of the phase, which
will allow us to quantify the velocity shift. What we call shift
here is the difference between the values corresponding to the
average slowness and the values obtained from the mean phase.
Thus, if X is the distance of propagation, the phase shift is
!J.¢= (</>)-X, the traveltime shift is !J.T=(l I c0 )((¢)- X) and
the velocity shift is !J. V = - c0 (!J.<I>I X). In one dimension
(¢)=X, and the velocity shift due to the fast path effect
disappears.
We express ¢ as a power-series in the small perturbation
parameter f:
(4)
Substituting it into the eikonal equation and keeping terms up
to second order gives
V¢o.V¢o= l,
(5)
V</>o.V¢1 =IJ,
(6)
V</>o.V¢2 = 17 2 - (V¢!) 2 .
(7)
Choosing a coordinate basis (£1, £ 2 , £ 3) with initially plane
waves propagating along the £ 1 direction, and boundary
condition </>(X1 =0, X2, X3)=0, the solutions are (Boyse 1986)
</>1(r)=
r
r [J:
¢2(r)=-
(8)
17(~, X2, X3)d~,
VriJ(p, X2, X3)dpr
(9)
d~,
(10)
where VT is the gradient transverse to the X 1 direction,
the direction of wave propagation, and r = (X1, X 2 , X3 ) is the
position vector.
The previous solutions hold under the validity conditions of
geometrical optics. However, to perform analytic calculations
Y Samuelides and T Mukerji
780
~e will need a 'upplementary assumption. namely that the
dtstance of propagation ts much larger than the correlation
dtstance m the direction of propagation. X1 »a 1 • As we are
dealing ~ith plane ~a\e~. and the medtum IS statistically
stationary. ~e can ~ee that the statistical distribution of
cA X1• X~. Xd does not depend on X~ and X1.
For ISOtropic medta, the correlation function N( x,' x2. Xl) =
\'[ v ( Xf +X~+ Xi l] depends only on the lag distance, not on
the directiOn. Boyse ( 19R6) showed that
Var(C>(r))=2X 1 a~ Jf
(II)
i\'(.l)ds.
0
f
.,
,.,
firr~ .\ 1
u
J
-
.\
(.1 )s
I
Since (~)=0 and (¢ 1 )=0, the expression for the mean
phase, keeping terms up to second order, is
f2
2
(rt>(r)) = (~?o(r))-
(¢2(r))
Ix, f~ I~ <-;;--Xa 17(p,,O)-;;--Xa 17(P2·0)~ dp,dfl2·
f2
=X,-~
2
()
• ()
()
(12)
2
(J
2
(18)
Expressing the correlation function in the (e 1, e2) coordinates,
we find
<.
iJ 171Pt, 0) i)X,
iJ 17(P2• 0)
~1 x,
ds,
(!
.
>
I)
where f1 = 0. I /2. I in one, two and three dimensions,
re~pectt\ely. and Var() denote~ the variance.
2-D variance and velocity shift
We now constder the
Gau~stan am~otroptc
XT X~)]
( tlj. , + a~, .
e'l(p [
.\'(r) =
ca~e of a 2-D random medium with a
autocorrelation function given by
( 13)
where a 1 and a~ are the correlation lengths along the x 1and x~-axes. The coordinate basis (e 1, e2) defining N(r) will
tn general be dtfferent from the coordinate basts ( £ 1, £2)
defined by the d1rection of wave propagation. The angle
between the direction of propagation £ 1 and the e 1 axis of the
autocorrelation function will be denoted by :x.
Keepmg terms up to second order gives
Introducing the quantities
I
sin 2 IX
cos 2 IX
Bi
ay
a~
-=--+--
(20)
'
(21)
(22)
where B, is the correlation length in the direction orthogonal
to the propagation, we have
Var( o(r)) = ''Var(otlr))
=t
J J''
2
l,
0
Now. using the le 1 •
I'·
,J I
~-: J'·
-",
II
'
o
(14)
e~) ba~is.
.\'[x,
=(~
we obtam
fl)cos:x. x,=(~- p)sin:x]c/~c/p
II
1
r,
=rr~
(~(~.Ol~(p,O))d~dp.
0
11
(23)
2
[
(cos2 :x <.;-pt+
•
, sin ~-1.;--pt
:x •
')] ch,clfl.
•
exp-a 51
a2
( 15)
Thts mtegral can be analytically calculated when X 1 1 A, 1s
large by replacmg the upper hm1ts on the integrals by -,.. A,
1s the correlatiOn length m the dtrectlon of propagation, and is
gtven by
cos-'
.;fi
tlj'
'
The final
:X
+
sm·' :X
rc~ult
i
~
2
2
(tb(r))=X 1 -tr~X 1
[jiiA.
. (21X)
; + sm
28
2
(
I ) foA~]
I
[)4-
2
C4
-
-- ·
4
(24)
In eq. (24), the term between brackets can be simplified to
jira 1a2
' . ' -- ' ·-,-----.,.1/=2 .
2(aj sm· (:x)+ai cos· (IX))
(25)
(16)
The mean traveltime is given by
is
Varlo(r))=t1~V~tXA,.
and the mean phase (when X1 I A.» I) can be expressed as
(T(r)) = (f/l(r))
(17)
The varian<.-c ts larger in the dtrectiOn of larger correlation
length.
co
(26)
ro 199H RAS. GJ/ 134,778 786
Velocity shift in heterogeneous media
The first term on the right-hand side corresponds to the
average slowness of the medium, while the second term
describes the shift and decrease in mean traveltime caused by
fast path effects. For instance, for propagation along the ei
direction, et = 0, and the shift is
-~
For the validity of geometrical optics, and consequently of
all our results, we normally assume that the quantity v'J[JiK is
small compared to both ai and a2. Actually, considering the
Rytov approximation, the approximation should hold as long
as v'J[JiK is small compared to the correlation distance in the
transverse direction B,:
JXaiXf
~~
(28)
For waves propagating along e2, a=n/2, and then the shift is
-~
~
781
JXa2Xf
2af
This statement can be useful, especially because horizontal
correlation distances are almost always much larger than
vertical ones.
When a I = a 2=a (isotropic media), the shift is
3-D variance and velocity shift
-~
f1tX2
co
2a
We now describe fast path effects for plane waves in 3-D
anisotropic random media. As in two dimensions, to compute explicit formulae we assume a Gaussian correlation
function. Once again, we consider two coordinate bases. In the
(ei, e2, e3 ) basis [coordinates: (xi, x2, x 3 )], the correlation
function can be written as
--~ _V_"_I
It is easy to check that this corresponds to the formula given by
Boyse & Keller (1995).
In Fig. I we plot the quantity
XI- (IP(r))
[
JXaia2
]
(27)
(29)
which is a normalized measure of the anisotropic shift as a
function of angle et. The velocity shift is an increasing function
of the correlation length along the direction of propagation,
and is maximum along the direction of largest spatial correlation. Along the perpendicular direction the velocity shift is
reduced from the isotropic value. Thus for typical geological
heterogeneities where the horizontal correlation is much larger
than the vertical correlation, the velocity shift will be greater
for waves propagating along the horizontal.
Another interesting issue is to compare the shift and the
standard deviation of the phase (or traveltime), which can be
done very easily with our formulae. For instance, in the isotropic case, if (ni/ 4 f2)a~(XI/aJ) 312 ;?: I then the traveltime
shift will be greater than the standard deviation of the traveltimes. This means that numerical computations with specific
realizations of the medium will allow us to actually observe the
shift.
where a I, a 2 and a 3 are the correlation lengths along the
coordinate axes. In the second reference frame (EJ. E2, £3)
[coordinates: (XI, X2, X3)], EI corresponds to the direction of
propagation. For a point with spherical coordinates (r, {}, qJ),
we have the following relations:
a~Xr
=
2<af sin 2 (a)+~ cos2 (a))312
'
+ sin qJ sin {}e2 + cos qJe3 ,
cos qJ cos {}ei - cos qJ sin {}e2 + sin qJe3 ,
sin {}ei + cos {}e2 .
EI = sin qJ cos {}ei
(30)
E2 = -
(31)
£3 = -
(32)
The angles ({}, qJ) are the azimuthal and polar angles in
spherical coordinates defining the direction of plane wave
propagation with respect to the principal axes of the spatial
correlation function. The 3-D calculations are similar to
those in two dimensions. Introducing the quantity A£,, the
correlation distance in the direction EI defined by
I
-,- =
A£,
sin ( qJ ) 2 cos ({}) 2
2
ai
+
sin ( qJ ) 2 sin ({}) 2
a2
10
cos ( qJ ) 2
+ --2-,
(33)
a3
the variance of ¢J can be expressed as
.:::
8
(34)
:E
"'
"0
As in two dimensions, the variance is larger in the direction of
large correlation lengths. The expression for the mean phase is
8
0
N
~
E
z0
4
ai
2
=a2 (isotropic)
0
0
20
40
60
80
angle a (deg.)
Figure I. Normalized anisotropic velocity shift as a function of angle
between direction of propagation and direction of maximum spatial
correlation.
2
0 1998 RAS. GJ/134, TIS-786
We can now work in the (eJ. e2, e3) basis to exploit the
particular form of the correlation function. The calculations
2
Y. Samuelides and T. Muk erji
are tediou but ea y. and th e fina l re ult i
Spherical waves
(37)
where BE, is the correlati on di la nce in the transve rse
direc tion. given by
B~
£,
a~
I
I
I
---,+---=t-,.
a~ a3 A£,
X,
(38)
I
I
a,
. ,
, ·'
,
v ( cos·
,
<p cos·
1
sin 2 U)
IJ( ~ , O . O)d ~,
(41 )
0
02 (r ) = -
/ = - -:I sm· <p cos·
(40)
r:>o (r ) = X 1 ,
o 1(r ) =
and
.
We use the sa me notati on as for pl a ne waves. The perturbati on
so lutions for the phase in the eikonal equati on are
I
X,
0
(I~
0
p
Tl'/(p , O,O) dp)
e
2
d~ .
(42)
The va ria nce calc ul ati ons and the res ult a re th e sa me as for
plane waves . Assuming a Ga ussia n correlati on functi on.
Va r(ct>(r)) = a '~ reX A, .
(43)
By perfo rm ing the ca lcul ati ons in the (e 1, e2, e 3) basis, we
find tha t the shift for spherical waves is a lways one-third of the
shift for pl ane waves. consistent with Roth , Mi.ill er & Snieder
( 1993 ). Thi result is independe nt of the form of the co rrelatio n
function .
(39)
long the three principal di rection of patia l a ni otropy
(e 1. e 2 . e1 )./ = 0. For propaga tion along e3 • <p= O.f = O a nd
the shift i
For propaga tion a long e 1 • <p=rr12. u=O.
·hift i
gai n ( = O and the
The hift for propaga ti on a lo ng e 2 i obtai ned bye changing
the role of a 1 and a2. In i otropi medi a a 1 = a2 = a1.j = 0 a nd
the hift
- a~ ( rrX 2 / a). Thi is twice the 2- D hift a nd
corrre p nd to the formu la give n by Boy e & Kell er ( 1995).
NU M E RICAL MODELLING
We now present the res ults of num eri ca l simul ati ons in
which we explore the velocity shift as a fun cti on of the va ri ance
of slowness fluctua ti ons, a nd the coeffi cient of ani sotro py.
The ray theoretica l res ult s do not all ow us to pred ict the
velocit y shift in heterogeneo us medi a with a la rge va ri ance in
the slowness . T herefore, we usc numerica l modellin g to
ex plo re large variances. 2- D synthetic ra nd om medi a with
Gaussia n a ut ocorrelati on, bot h isot ropic a nd a ni sotropic. we re
ge nerated wi th sta nd ard devia ti ons ra ngin g from 0.5 to 7 per
ce nt in the slowness fluctu ati ons. Fig. 2 show one rea li za ti on
of the rando m med ium (201 x 20 1 pi xels, c0 = 4 km s- 1) with
an a ni otro py ra ti o a 1 /a2 = 5, and it s numeri ca ll y comput ed
correlati on functi n.
A fini te-di ffe rence solve r for the cik onal equati on (Zha ng
1993) was used to c mpute the travcltimes for a n initi all y pl ane
wave propaga ting ac ross the rea li za ti ons oft he random medi um
20
40
0.8
60
0.6
80
0.4
I10o
0.2
>.
120
0
140
- 0.2
100
160
100
180
0
200
100
50
150
200
.Y~
'()(~
x (m)
Figure 2. 2- 0 •ynthctic.: rand m
numcn ally from the 1magc.
~cl
lly model wllh an ani otropy ratio of 5. and its
- 100 - 100
i-
\'3-<;)
\~'\
aussian spati a l a ut oco rrelation function compu ted
a 1998 RA .
Jl 134 ,778 786
Velocity shift in heterogeneous media
from one edge to another. Earlier numerical modelling showed
that the traveltime results from the eikonal solver match very
well the picked travel times for short-wavelength full-waveform
finite-difference synthetic seismograms (Mukerji eta/. 1995).
Fig. 3 shows the numerically calculated arrival times versus
spatial position transverse to the direction of propagation for
different slowness variations (standard deviations of 2, 5 and
7 per cent) but the same realization of an anisotropic random
medium with a 1 I a 2 = 5. The dotted curves are for propagation
perpendicular to the maximum correlation, while the solid
curves are for propagation parallel to the direction of maximum spatial correlation. The shift of traveltimes to lower
values is clearly evident, as is the difference in shift along the
two directions.
We see that the shift is enhanced for propagation along
the direction of larger spatial correlation. Fig. 4 shows the
velocity shift (normalized) as a function of angle between the
direction of propagation and the direction of maximum spatial
correlation. The symbols show results for different realizations
of the random medium, with a u~ of 0.5 and 1.0 per cent, each
with a 1 I a 2 = 2. The corresponding variance in the traveltime
is shown in Fig. 5. While some of the individual realizations
give the right magnitude for the variance, the averaged curve
over all realizations shows a shift. Possible reasons for the
discrepancy are that enough realizations were not averaged or
the fact that the theoretical estimate takes only terms up to
second order into account, and for the variance the higherorder terms may no longer be negligible. In summary, both the
velocity shift and the traveltime variance are greater for
propagation along the maximum spatial correlation.
783
Saturation of the velocity shift
We have shown the existence of a velocity shift due to fast
paths. In two dimensions as well as in three dimensions, the
relative shift ll VI c0 is proportional to ~X1 • Nevertheless, even
when the conditions of validity of geometrical optics hold
(that is the wavelength i. is sufficiently small), it is clear that the
formula cannot be true for any X1 with a given u~. From a
physical point of view, llVIc0 cannot be much larger than u~,
the standard deviation of the velocities in the medium. This is
because most of the velocities lie within [c0 (1-u~), c0 (1 +u~)],
and there cannot be any arrival that is faster than any of
the velocities actually present in the medium. From a mathematical point of view, the formulae come from a power
expansion in u~ and thus are subjected to a limitation in
u~ which depends on X 1• This is because in the expression for
the velocity shift, u~ and X1 appear as a product; therefore,
for larger u~ the expansion is limited to smaller X 1 before
saturation sets in. Mathematically, the problem of large u~
is equivalent to large X1, with a fixed u~. Finally, from
a numerical point of view, the computations seem to show a
saturation of the velocity shift when the ratio X 1 I a increases.
This saturation of the velocity shift, for the isotropic case,
is shown in Fig. 6. In Fig. 6 we see that the numerical curve
for u~ = 2 per cent clearly seems to level off from about
X I a= 13, and falls below the theoretical curve, which continues to increase rapidly. This could show the existence of the
saturation effect. Of course, this does not constitute a rigorous
proof, but is just a plausibility argument based on observations from numerical experiments with multiple stochastic
0.051 . - - - - , - - - - . - - - , - - - , - - - - , - - - . - - - . - - - - . - - - - , - - - - ,
o.~sL--~--~--~-~~--~---~---~---~-~---~
0
20
40
60
80
100
120
140
160
180
200
position (transverse to propagation direction), (m)
Figure 3. Numerically calculated arrival times versus spatial position transverse to the direction of propagation for different slowness variations
(standard deviations of 2. 5 and 7 per cent) for a realization of an anisotropic random medium with a 1 1a~ = 5. Dotted curves: propagation
perpendicular to the maximum correlation <X ra- SO): solid curves: propagation parallel to the direction of maximum spatial correlation (X /a-10).
The total propagation path leng1h was the same in each direction.
C 1998 RAS. GJ/ 134,778-786
784
Y Samuelides and T Mukerji
0 35
...
0.3
...
0.25
~
•
...
0.2
:cen
.......
"0 015
Q)
.!:::!
ro
0.1 ~
i
E
....
*•
' '
' '
' '
'
•
•
...
0.05
... ......
t
...
...
0
•
•
t
0
z
•
t
~
...
1
•
...
...
i
-0.051
!
-0.1
0
10
20
30
40
50
60
70
90
80
angle a (deg.)
Figure 4. Velocity shift (normalized), computed from the fimte-d1fference eikonal traveltime solution as a function of angle between the direction of
propagation and the directiOn of maximum spatial correlation. The symbols show results for different realizations of the random medium with u~ of
0.5 per cent (tnangles) and 1.0 per cent (circles), each with 01 I 02 = 2. Total propagation path= 200 m, OJ = 20 m and 02 = 10 m. Solid curve:
theoretical pred1ct10n (eq. 27); dashed curve. mean over realizations. The velocity sh1ft and traveltime variance (Fig. 5) are greater for propagation
along the maximum spatial correlation.
~--.-----,-----.-----r---~----~-----r-----r----,-----~-,
25
,
Q)
0
c
ro
•
20
·~
I
~1J
•
'
Q)
N
i
·~
I
E
I
010
z
5
,,
,,•
••
•
',,
•
'
., ---r-- -----'
. .....______
'
.•
•
.
•
I
0~~~----~~----~----~----~------~----~----~------~----~~
0
10
20
~
40
50
60
70
80
90
angle a (deg.)
FigureS. Normalized vanance, [Var(tP(r))/(~y'iiX)1 averaged over multiple realizations, corresponding to the traveltimes shown in Fig.4, as a
function of angle between direct1on of propagation and direction of maximum spatial correlation.
© 1998 RAS, GJ/ 134, 778-786
Velocity shift in heterogeneous media
785
cr= 1%
+
+
+
+
0~--~~--~~--~~--~~--~----~----~----~----~----~
10
11
12
13
14
15
16
17
18
19
20
X./a
Figure 6. Slowness shift as a function of X Ia showing saturation of velocity shift for the isotropic case. Solid line: theory (eq. 26 with a1 =a2 );
dotted line: average over five realizations; pluses: individual realizations. The saturation begins for values of X I a of the order of a- 2 / 3 , where a is the
normalized standard devtation of the slowness fluctuations. This is seen in the upper set of curves for u = 2 per cent.
realizations. The two questions to be addressed are (I) when
does the saturation begin, and (2) what is the value of the shift
at the saturation? In an isotropic medium with correlation
length a, we know (White 1984) that at distances of the order
of au;; 2!3 the formation of caustics occurs. Where there are
caustics, the solutions of the eikonal equation become illdefined and we can therefore conjecture that the saturation
takes place in this region. Then, the saturated value of the shift
would not be very different from the value given by the formula
just before the caustic; that is,
.1V
=
co
_
c y'ii[Jcil.u- 213 = C y'iipu413 ,
~
~
~
(44)
where Cis a constant. This seems to be confirmed by numerical
experiments. Fig. 6 shows that the saturation begins for values
of X I a of the order of u;; 2/3. Similarly, the numerical experiments of Roth eta/. (1993) show that for very large values of
X I a, .1 VI c0 = 0.26u~ 33 . For anisotropic media, we think that
the results are similar, the transverse correlation B, playing
once again the role of the correlation distance a in isotropic
media. This means that the small perturbation results will
be valid for longer propagation paths perpendicular to the
maximum correlation compared to propagation parallel to
the maximum correlation, before saturation takes over.
CONCLUSIONS
Heterogeneities affect seismic wave propagation in ways
that both complicate and aid in the interpretation of
seismic images. In this paper we investigated the velocity
shift in random heterogeneous media with an anisotropic
IC 1998 RAS, GJ/134, 778-786
spatial correlation function using asymptotic ray-theoretical
and numerical methods. The velocity shift depends on the
variance of the slowness fluctuations of the medium, and the
spatial anisotropy coefficient. The velocity shift in anisotropic
media increases for propagation along the direction of
maximum spatial correlation. Our main results for the fast
path effects are given by eq. (24) in two dimensions and eq. (37)
in three dimensions. The conditions of validity are described in
Table 1. Numerically, we observed that the relative velocity
shift first increases linearly with the distance of propagation,
as predicted by our formulae, then saturates when there are
caustics.
The traveltime fluctuations have practical implications for
recording data and measuring velocities. When the receiver
spacing is small enough and the array length is long enough to
characterize the fluctuations, then the mean of the traveltimes
will yield the fast ray-theory predictions. Furthermore, the
variance of the fluctuations yields information on the slowness
variance of the medium: the variance is given by eq. (17) in
two dimensions and eq. (34) in three dimensions. Once again,
it is larger in the direction of larger correlation length. The
correlation length of the traveltime fluctuations is a measure of
the spatial correlation length of the medium.
Effects such as velocity shift can impact on reservoir characterization. Therefore, it is important to understand them in
anisotropic random media, because geological heterogeneities
almost always show highly anisotropic spatial correlation. As
an example, consider a case where a 1 = I 00 m, and a 2 = 10 m,
as would be typical in some fluvial-deltaic systems (e.g. Yarus
& Chambers 1994). For a propagation distance of 200m
the fast path effect would introduce an average time shift of
786
Y. Samuelides and T Mukerji
Table l. Cond1t10ns of vahd1ty for the asymptotic ray theory formulae. k is the
wavenumber. a and h are the correlatiOn lengths along the principal directions of
spatial amsotropy. X and X 1 are propagatiOn pathlengths, A, and B, are spatial correlation lengths along the directiOn of propagation. and 11~ IS the normalized standard
deviation of the slowness fluctuations.
Theoretical
conditiOn
Numencal
condition
Meaning
when 1t holds
What happens
if it does not hold
ka »I. kh »I
ka > 2. kh > 2
h1gh frequency
eikonal not valid;
full wave equation required;
effective medium theory
can be used if ka« I
X «kB;
X< 0 3kB;
can neglect diffraction
eikonal not valid;
Rytov can be used
can neglect caustics
saturation of velocity shift
statiStical vanability
of the medium
adequately sampled
results are not
statistically meaningful
XI
-«11
A,
2 1
~
XI
-- < 11
A,
"
XI »I
XI > 10
A,
A,
2 1
about 15 per cent along the direction of maximum correlation,
and negligible time shift in the perpendicular direction. This
apparent amsotropy would have to be taken into account
in additton to amsotropy due to other effects such as thin
layenng and intrinsic anisotropy. The time shift would also be
important when comparing veloctty values found by different
measurements such as cross-well or VSP, which sample the
heterogeneous subsurface along different dtrections. The role
of fast path effects as a possible cause of velocity dispersion,
m addition to fluid-flow-related mechanisms, has been considered both for laboratory measurements in heterogeneously
saturated rocks (Cadoret 1993) and for field conditions (Boirel,
Mukerji & Mavko 1997).
ACKNOWLEDGMENTS
This work was supported by Elf-Aquitaine and Stanford Rock
Physics Project. We acknowledge helpful discussions with
George Papanicolaou, Joe Keller, Pierre Thore and Gary
Mavko. We thank Le-wei Mo and Jerry Harris for providing
Lin Zhang's code. We would also like to acknowledge the
insightful reviews from Sergei Shapiro and Michael Roth,
which helped to improve the paper.
REFERENCES
Boirel. S. Mukerji. T & Mavko. G. 1997. Sonic drift between logs and
vertical se1smic profiles a case study, F.OS. Tran.1 Am. geoph.n
Un . Fall mtng. San FranCISCO, F677
Boyse. WE., 1986. Wave propagation and mvers1on in slightly
mhomogeneous med1a, PhD the.m. Stanford University.
Boyse, W.E. & Keller, J.B .. 1995. Short acoustic, electromagnetic, and
elastic waves m random med1a, J. Opt. Soc. Am . Al2, 380-389.
Cadoret, T., 1993. Effet de Ia saturation eau/gaz sur les proprietes
acoust1ques des roches. Etude aux frequences somHes et ultrasonores. PhD thesis. UniveTSJtc Dems D1derot, Paris VII.
Chernov. L.A., 1960. Wave Propagation in Random Media, McGrawHill, New York.
Frankel. A. & Clayton, R.W., 1986. Finite difference simulations of
seismic scattenng: implications for the propagation of short-period
seismic waves in the crust and models of crustal heterogeneity,
J.geophy.1·. Res., 91,6465 6489.
Gist, G.A., 1994. Seismic attenuation from 3D heterogeneities: a
possible resolution of the VSP attenuation paradox, SEG. 64th Ann.
Mtl(. F.xp. Ahstr. 1042 1045.
Jsh1maru. A .. 1978. Wave Propagation and Scattering in Random
Media, Vol. 2, Academic Press, New York.
Karal, F.C. & Keller, J.B., 1964. Elastic, electromagnetic and other
waves in random media, J. math. Phy.~ .. S, 537-547.
Keller, J.B., 1964. Stochastic equations and wave propagation in
random med1a, Proc. Symp. app/. Math., 16, 145-170.
Korn, M ., 1993. Seismic waves in random media, J. app/. Geophys., 29,
247 269.
Marion, D., Mukerji, T. & Mavko, G., 1994. Scale effects on velocity
dispersion: from ray to effective medium theories in stratified media,
Geophysics, 69, 1613 1619.
Mukerji, T., Mavko, G .. Mujica, D. & Lucet, N .. 1995. Scaledependent seismic velocity in heterogeneous media, Geophysics, 60,
1222 1233.
Muller. G., Roth, M. & Korn, M., 1992. Seismic-wave traveltimes in
random media, Geophy.~. J. Int., 110, 29--41.
Roth, M., Muller, G. & Snieder, R., 1993. Velocity shift in random
media, Geophys. J. Int .. 115, 552-563.
Shapiro, S.A. & Kneib, G., 1993. Seismic attenuation by scattering:
theory and numerical results, Geophys. J. Int., 114, 373 391.
Shapiro, S.A., Schwarz, R. & Gold, N., 1996. The effect of random
isotropic inhomogeneities on the phase velocities of seismic waves,
Geophy.v. J. Int .. 127, 783-794.
van Avendonk, H. & Snieder, R., 1994. A new mechanism for shape
induced seismic anisotropy, Wave Motion. 20,89-98.
White. B.S. 1984. The stochasic caustic, SIAM J. appl. Math., 44,
127 149
Varus, J.M. & Chambers, R.L .. 1994. Stochastic modL•Iing and geo.vtatistic.v: principles. method.v, and case studies, Am. Assoc. Petrol.
Geol.. Tulsa.
Zhang, L., 1993. Imaging by the wavefront propagation method, PhD
thL•si.1·, Stanford.
«'l 1998 RAS, GJ/ 134,778-786