Geophys. J. Int. (1999) 136, 205^217
Computation of second-order traveltime perturbation by
Hamiltonian ray theory
Vëronique Farra
Dëpartement de Sismologie, Institut de Physique du Globe de Paris, 4 Place Jussieu, 75252 Paris cedex 05, France
Accepted 1998 August 12. Received 1998 July 9; in original form 1998 March 11
S U M M A RY
Ray perturbation theory may be used to compute changes in ray paths and physical
attributes (traveltime, polarization, amplitude) due to changes in the medium or in the
boundary conditions of the rays. The theory developed in the Hamiltonian approach
is valid for both isotropic and anisotropic media, including models with structural
interfaces. First-order traveltime perturbation is given by the integral of the ¢rstorder term of the Hamiltonian perturbation. Computation of the second-order traveltime perturbation needs the computation of the ¢rst-order ray perturbation and the
expression of the second-order term of the Hamiltonian. Endpoint boundary conditions
are easy to introduce in this formulation using the paraxial propagator matrix.
Examples computed in transverse isotropic media show that P-wave traveltimes may be
well approximated by using the second-order expression even for strong anisotropy. A
reference medium with elliptical anisotropy seems to be a better choice to develop the
perturbation approach than an isotropic medium.
Key words: anisotropy, ray perturbation theory, traveltime tomography.
1
IN T ROD U C T I O N
The accurate determination of traveltimes and other observed
attributes (polarization, amplitude) is extremely important
in seismology. In many applications, rays are known in a
model and perturbation theory may be used to estimate the ray
positions when the elastic parameters of this model are perturbed. For example, a perturbation approach is very attractive
for the determination of rays in anisotropic structures because
in such structures the computation is more complicated and
time consuming than in isotropic structures.
In tomographic inversion, it is crucial to have accurate
estimates of traveltimes. Most recent tomographic inversion
methods use ¢rst-order expression of the traveltime perturbation with higher-order terms included in the inversion
via iteration (see e.g. Chapman & Pratt 1992; Jech & Psencik
1992; Le Bëgat & Farra 1997). The expression of the ¢rst-order
traveltime perturbation in smooth anisotropic media has
been obtained by Cerveny (1982) and Cerveny & Jech (1982)
(see also Hanyga 1982; Jech & Psencik 1989; Nowack &
Psencik 1991). An expression for the second-order traveltime
perturbation was derived for smooth isotropic media by
Snieder & Sambridge (1992, 1993). They used a Lagrangian
approach to the problem and showed that the second-order
traveltime perturbations could be obtained from ¢rst-order ray
perturbation computation. As shown by Farra et al. (1994),
Hamiltonian and Lagrangian approaches lead to the same
ray perturbations. However, treating ray perturbation theory
ß 1999 RAS
with a Hamiltonian approach has the advantage that the
equations are simple to derive in heterogeneous isotropic and
anisotropic media. Moreover, in the Hamiltonian approach,
discontinuities may be introduced easily. The second-order
traveltime perturbation may be very useful when applying
non-linear tomography without ray tracing at each iteration.
For anisotropic media, this may be crucial if one wants to
use rays obtained in an isotropic medium in order to obtain
accurate computations of traveltimes and Frechet derivatives
in anisotropic media.
2 H A MI LTO N I A N FORM U L ATI O N OF R AY
T R AC I NG
In the ¢rst part of this paper, we recall the Hamiltonian
formulation used for ray tracing in isotropic and anisotropic media (see Farra & Le Bëgat 1995). Ray and paraxial
ray tracing are performed in Cartesian coordinates (see
e.g. Cerveny 1972). Boundary equations at interfaces can be
introduced very easily in this formulation.
2.1 Ray tracing equations
In the high-frequency approximation, the elastodynamic
equation yields a non-linear ¢rst-order partial di¡erential
equation for the traveltime (the eikonal equation):
H(x, p)~0 .
(1)
205
206
V. Farra
The function H is called the Hamiltonian, p~=T is the slowness vector and T is the traveltime. Many suitable forms of the
Hamiltonian can be used [see Cerveny (1989) and Farra (1993)
for a discussion]. For example, in isotropic media a useful form
of the Hamiltonian was proposed by Burridge (1976):
1
H(x, p)~ [p2 {u2 (x)] ,
2
(2)
where the slowness u(x) is the reciprocal of the velocity. The
theory developed in this paper is independent of the chosen
form of the Hamiltonian and is valid in isotropic as well as
anisotropic media. The most common way of solving eq. (1) is
to use the ray tracing method. A ray is de¢ned by its canonical
vector
"
#
x(q)
,
y(q)~
p(q)
where x(q) is the position along the ray, p(q) is the slowness
vector of the wavefront at position x(q) and q is a sampling
parameter that depends on the chosen form of the Hamiltonian
(Cerveny 1989). The canonical vector of the rays satis¢es
Hamilton's equations:
xç ~=p H ,
(3)
pç ~{=x H ,
where =x and =p denote the gradients with respect to the
vectors x (position vector) and p (slowness vector), respectively,
and dots indicate derivatives with respect to the sampling
parameter q. The six equations in (3) are not independent since
at least one of them may be eliminated by using the fact that
the slowness vector p should satisfy the eikonal equation (1).
The traveltime is obtained by simple integration along the
ray path:
… qr
T (qs , qr )~
p . xç dq ,
(4)
The presence of these interfaces requires the introduction of
appropriate boundary conditions for ray tracing and paraxial
ray tracing. Let the interface location be de¢ned by the general
relation f (x)~0. Let us consider a reference ray whose
canonical vector is
"
#
x(q)
.
y(q)~
p(q)
This reference ray hits the interface at x(q{
i ) with a local
slowness vector p(q{
i ). Quantities measured on the re£ected/
z
{
transmitted ray at the interface will be written at qz
i , qi ~qi
{
(qi is used for quantities measured on the incident ray at
the interface). At the interface, the ray satis¢es the following
equations (Farra et al. 1989; Farra 1989):
{
x(qz
i )~x(qi ) ,
{
p(qz
i )|=f ~p(qi )|=f ,
(7)
z
{
{
H(x(qz
i ), p(qi ))~H(x(qi ), p(qi ))~0 ,
where the cross-product has been denoted by | and =f is
the gradient of f computed at x(q{
i ).
We now consider the expression for the paraxial propagator
in a layered medium. Transformation matrices at the interfaces
crossed by the ray have to be introduced. The generalized
propagator is given by the expression:
P(q, q0 )~P(q, qz
N)
N
Y
i~1
z
Mi P(q{
i , qi{1 )
(8)
(see Farra & Le Bëgat 1995), where N is the number of interz
faces crossed by the reference ray between q0 and q, q{
i ~qi
is the value of the q parameter of the central ray at the ith
intersection point with an interface and Mi is the interface
transformation matrix (see Farra & Le Bëgat 1995).
qs
where we denote by T (qs , qr ) the traveltime between x(qs ) and
x(qr ) and by a . b the scalar product of vectors a and b.
Let us assume that we have traced a ray in the medium.
Along this reference ray, we can compute the paraxial
propagator. The paraxial propagator is the propagator matrix
of the paraxial system
P_ ~A(q)P ,
where
"
A(q)~
(5)
=x =p H
=p =p H
{=x =x H
{=p =x H
#
(6)
is a 6|6 matrix computed on the reference ray at q. The
paraxial propagator matrix has many applications, for
example the computation of the so-called paraxial rays of the
reference ray, the second-order derivatives of the traveltime,
the ray theoretical amplitude, as well as rays in perturbed
media (see e.g. Cerveny et al. 1988; Cerveny 1989; Farra &
Madariaga 1987; Farra 1993).
2.2
Transformation across an interface
The model can be divided into individual layers separated
by interfaces of zero-order (elastic parameters discontinuity).
2.3
Ray perturbation theory
Let us assume that we have a reference model characterized
by the Hamiltonian H0 . Let us consider a perturbation of
the model such that the Hamiltonian is changed from H0 to
H~H0 z*H, where H0 is the Hamiltonian for the reference
medium. The procedure of perturbation theory is to identify
a small parameter, denoted , such that when ~0 the problem
is solvable. The perturbation *H can be expanded in a
perturbation series in powers of :
*H~H1 z2 H2 z . . . .
(9)
The parameter facilitates a systematic perturbation approach
but, at the end of the procedure, one can set ~1 to recover the
original problem.
We assume that a ray has been de¢ned in the reference model
by its canonical vector
"
#
x0 (q)
y0 (q)~
.
p0 (q)
Using perturbation theory, it is possible to obtain rays in
the perturbed medium that deviate slightly from the reference
ray (Farra & Madariaga 1987). We introduce the perturbed
canonical vector y(q)~y0 (q)z*y(q) of these rays, where the
ß 1999 RAS, GJI 136, 205^217
Second-order traveltime perturbation
perturbation
"
#
*x
*y~
*p
We introduce brie£y the equations corresponding to the
second-order perturbation
" #
x2
.
y2 ~
p2
can be written as a perturbation series in powers of :
*x(q)~x1 (q)z2 x2 (q)z . . . ,
(10)
*p(q)~p1 (q)z2 p2 (q)z . . . .
Developing the ray equations (3) to second order in ,
one obtains a linear system similar to (11). Given the initial
perturbation y2 (qs ), the subsequent evolution of the perturbed
canonical vector y2 (q) is given by
We write the expressions of the ¢rst-order perturbation of
the rays. Inserting the perturbed canonical vector in (3), with
(10) taken into account, and developing to ¢rst order, one
obtains the following linear system:
y2 (q)~P0 (q, qs )y2 (qs )zyb2 (q) ,
xç 1 ~=x =p H0 x1 z=p =p H0 p1 z=p H1 ,
yb2 (q)~
pç 1 ~{=x =x H0 x1 {=p =x H0 p1 {=x H1 .
(11)
The perturbed rays (Farra 1989) are given to ¢rst order by
…q
y1 (q)~P0 (q, qs )y1 (qs )z
P0 (q, q')B1 (q') dq' ,
(12)
qs
where
"
y1 ~
x1
#
is the ¢rst-order perturbation of the ray canonical vector, y1 (qs )
is its initial perturbation, P0 (q, qs ) is the paraxial propagator
computed along the reference ray in the reference medium and
"
#
=p H1
.
B1 ~
{=x H1
Moreover, the perturbation y1 (q) satis¢es a condition derived
from the eikonal equation (1):
=x H0 . x1 (q)z=p H0 . p1 (q)zH1 ~0 ,
(13)
where the Hamiltonian H0 , its partial derivatives and H1 are
computed at y0 (q).
We now consider a reference medium containing discontinuities. Perturbations of these discontinuities may be
considered as shown in Farra & Le Bëgat (1995). Given
the initial perturbation y1 (qs ), the subsequent evolution of the
perturbed canonical vector y1 (q) is given by
y1 (q)~P0 (q, qs )y1 (qs )zyb1 (q) ,
(14)
where P0 (q, qs ) is the generalized propagator (see eq. 8) computed along the reference ray in the reference medium and
yb1 (q), given by
…q
qs
(16)
where yb2 (q) is given by an expression similar to (15):
…q
qs
P0 (q, q')B2 (q') dq'z
N
X
i~1
I
P0 (q, qz
i )y2 .
(17)
The term B2 , which is related to the perturbation of the
elastic parameters, contains products of partial derivatives of
Hamiltonian terms H0 , H1 and H2 and ¢rst-order perturbation
terms x1 and p1 . The perturbation yI2 contains terms related to
the ith incidence of the reference ray at an interface.
Moreover, the perturbation y2 (q) satis¢es a condition
derived from the eikonal equation (1):
=x H0 .x2 z=p H0 . p2 z=x H1 . x1 z=p H1 . p1
p1
yb1 (q)~
207
P0 (q, q')B1 (q') dq'z
N
X
i~1
I
P0 (q, qz
i )y1 ,
(15)
models the bending e¡ect due to the perturbation of the
structure. The perturbation yI1 is related to the ith incidence
of the reference ray at an interface and is given by eq. (B6)
in Farra & Le Bëgat (1995). We note that when there is no
perturbation of the medium and the perturbation y1 (qs ) is
non-zero, expression (14) reduces to the paraxial ray equations.
ß 1999 RAS, GJI 136, 205^217
1
z (xT1 =x =x H0 x1 z2xT1 =x =p H0 p1 zpT1 =p =p H0 p1 )zH2 ~0 ,
2
(18)
where H0 , H1 , H2 and their partial derivatives are computed
at y0 (q) and the superscript T means transpose.
The perturbation solutions (14) and (16) may be used to solve
a number of initial and boundary value problems. Boundary
conditions may be introduced very easily for the ¢rst-order ray
perturbation by using a projection matrix (see e.g. Farra et al.
1989; Farra & Le Bëgat 1995).
3 SE C O N D - OR D E R T R AV E LT I M E
PE RT U R BAT IO N
The use of ¢rst-order perturbation is very useful in traveltime
tomography. Most recent tomographic inversion methods
use a ¢rst-order expression of the traveltime with higher-order
terms included in the inversion via iteration. In this section,
the second-order perturbation of traveltime is derived for any
form of the Hamiltonian. The medium is assumed to have
smooth properties, and smooth perturbations of the elastic
parameters of the medium are considered.
3.1 Second-order extrapolation of the traveltime
Suppose a ray has been traced in the unperturbed medium
characterized by the Hamiltonian H0 . Let
"
#
x0 (q)
y0 (q)~
p0 (q)
be the canonical vector of this reference ray. We consider a
ray in the perturbed medium that deviates slightly from the
208
V. Farra
reference ray. The canonical vector of this ray is
"
#
x(q)
y(q)~
.
p(q)
Using perturbation series (10), we can write to second order in p . xç ~p0 . xç 0 z(p0 . xç 1 zp1 . xç 0 )z2 (p0 . xç 2 zp1 . xç 1 zp2 . xç 0 ) .
Let us assume that a ray with canonical vector y0 (q) has been
de¢ned in the reference medium between qs and qr . In the
vicinity of this reference ray, rays may be traced in the perturbed medium between q's and q'.r The value of the sampling
parameters q's and q'r may be di¡erent from the values qs and qr
but *qs ~q'{q
s
s and *qr ~q'{q
r
r are perturbations that can be
written as series:
(19)
*qs ~qs1 z2 qs2 z . . . ,
(24a)
From expressions (4) and (19), we can write the second-order
expression of the traveltime along the perturbed ray,
*qr ~qr1 z2 qr2 z . . . .
(24b)
T (qs , qr )~T0 (qs , qr )
… qr 1
1
2
.
.
H1 z H2 z =x H1 x1 z =p H1 p1 dq
{
2
2
qs
1
z[p0 . x1 ]qqrs z2 [p0 . x2 ]qqrs z [p1 . x1 ]qqrs ,
(20)
2
The Taylor expansion of the traveltime T (qs z*qs , qr z*qr )
may be written from (4),
where we use the ray equations (3) and (11) and the relations
(13) and (18) and we„ apply integration by parts. In (20), the
q
quantity T0 (qs , qr )~ qsr p0 . xç 0 dq is the traveltime along the
reference ray and we use the notation [g]qqrs ~g(qr ){g(qs ). In
the integral, H2 , H1 and its partial derivatives are computed at
" #
x0
.
y0 ~
p0
Until now, the perturbation of the ray canonical vector and
traveltime was de¢ned with respect to reference expressions
computed at the same value of the q-parameter. Let us now
develop the perturbations between quantities obtained at
slightly di¡erent q-values. The perturbation of the canonical
vector y(q') may be written with respect to the reference
vector y0 (q), assuming that *q~q'{q is a perturbation.
Denoting dx~x(q'){x0 (q) and dp~p(q'){p0 (q), we introduce
the perturbation series up to second-order in :
T (qs z*qs , qr z*qr )
~T (qs , qr )z [p0 . xç 0 q1 ]qqrs
qr
1 d
z2 (p0 . xç 1 zp1 . xç 0 )q1 zp0 . xç 0 q2 z
[p0 . xç 0 ]q21 . (25)
2 dq
qs
Using eqs (3), (20), (22) and (23) written in the vicinity of y0 (qs )
and y0 (qr ), one gets from (25)
T (qs z*qs , qr z*qr )
~T0 (qs , qr ){
(21)
dp~dp1 z2 dp2 .
Introducing the perturbation series (10) and (21) into the
Taylor expansion of y(q') to second order in *q,
x(q')~x(q)zxç (q)*qz
1
x« (q)*q2 ,
2
p(q')~p(q)zpç (q)*qz
1
p« (q)*q2 ,
2
dx1 ~x1 (q)zq1 xç 0 (q) ,
(22a)
dp1 ~p1 (q)zq1 pç 0 (q) ,
(22b)
1
dx2 ~x2 (q)zq1 xç 1 (q)zq2 xç 0 (q)z q21 x« 0 (q) .
2
(22c)
Moreover, expanding H(x(q'), p(q'))~0 to ¢rst order, one
obtains
=x H0 . dx1 z=p H0 . dp1 zH1 ~0 ,
(23)
where H0 , H1 and the partial derivatives of H0 are computed
at y0 (q).
(26)
where superscripts s and r were added in order to show that
the perturbations are calculated with respect to y0 (qs ) and
y0 (qr ). Expression (26) gives the traveltime between two points
situated on the perturbed ray with q-values in the vicinity of qs
and qr , respectively.
3.2
one can write the following relations:
qs
1
H1 z2 H2 z =x H1 . x1
2
1
z =p H1 . p1 dqzp0 (qr ) . dxr1
2
1
1
z2 p0 (qr ) . dxr2 z dpr1 . dxr1 { qr1 H1 (qr , y0 )
2
2
1
{ p0 (qs ) . dxs1 z2 p0 (qs ) . dxs2 z dps1 . dxs1
2
1
{ qs1 H1 (qs , y0 ) ,
2
*q~q1 z2 q2 ,
dx~dx1 z2 dx2 ,
… qr Boundary conditions of the perturbed ray
Let us assume that a ray has been de¢ned in the reference
medium such that the points xs and xr correspond to x0 (qs )
and x0 (qr ), respectively. First-order perturbation theory can
be used in order to ¢nd, in the vicinity of this reference ray,
the ray of the perturbed medium that connects the points
x'~x
s
s zdxs and x'~x
r
r zdxr (the expressions corresponding
to rays normal to an interface may be found in Appendix A).
This ray can be de¢ned in the phase space by its canonical
vector
"
#
x(q)
y(q)~
,
p(q)
ß 1999 RAS, GJI 136, 205^217
Second-order traveltime perturbation
with
x(qs )~x'~x
s
s zdxs ,
(27)
x(q'r )~x'~x
r
r zdxr .
The sampling parameter q'r is generally di¡erent from the
parameter qr but the quantity *qr ~q'{q
r
r can be expanded as a
perturbation series (24b). We assume that q's is equal to qs, so
that qs1 and qs2 are equal to zero. The boundary conditions (27)
of the perturbed ray may be written as
dxs1 ~x1 (qs )~dxs ,
dxr1 ~dxr ,
dxs2 ~x2 (qs )~0 ,
dxr2 ~0 ,
(28)
where we use the perturbation series developed around y0 (qs )
and y0 (qr ). Let us introduce a vector n such that n . dxr ~0. In
the case dxr ~0, this vector is chosen as n~p0 (qr ). By taking the
scalar product of (22a) with n, we obtain
n . x1 (qr )
qr1 ~{ .
.
(29)
n xç 0 (qr )
Introducing (29) into (22a) and (22b), one ¢nds that, to ¢rstorder, a linear relation exists between dyr ~y(qr'){y0 (qr ) and
y1 (qr ):
dxr1 ~nr1 x1 (qr ) ,
(30)
dpr1 ~nr2 x1 (qr )zp1 (qr ) ,
where dpr1 is the ¢rst-order expression of the quantity
dpr ~p(qr'){p0 (qr ). The 3|3 matrices nr1 and nr2 are given by
nr1 ~I{
nr2 ~{
=p H0 n
xç 0 (qr ) n
,
~I{
xç 0 (qr ) . n
=p H0 . n
T
where T0 (xs , xr ) is the traveltime computed in the reference
medium, qr1 is given by (29) and dpr1 by (30). In the integral, the
terms H2 , H1 and its partial derivatives are computed on the
reference ray y0 ; the ¢rst-order canonical vector
" #
x1
y1 ~
p1
of the perturbed ray is computed from (12) with initial
condition y1 (qs ) corresponding to boundary conditions (28).
Expression (32) of the second-order traveltime does not
depend on the second-order ray perturbation
" #
x2
.
y2 ~
p2
This means that in order to obtain expressions of the traveltime
perturbation that are valid to second order, it su¤ces to know
the ray perturbation to ¢rst order [in agreement with the results
obtained by Snieder & Sambridge (1992) in isotropic media].
Expression (32) is independent of the chosen form of the
Hamiltonian and may be used for any sampling parameter q
along the reference ray.
We note that when there is no perturbation of the medium,
expression (32) reduces to
T (xs zdxs , xr zdxr )~T0 (xs , xr )zp0 (qr ) . dxr
1
z 2 dpr1 . dxr {p0 (qs ) . dxs
2
T
T
T
pç 0 (qr ) n
=x H0 n
,
~
xç 0 (qr ) . n =p H0 . n
T (xs zdxs , xr zdxr )
… qr 1
H1 z2 H2 z =x H1 . x1
~T0 (xs , xr ){
2
qs
1
z =p H1 . p1 dqzp0 (qr ) . dxr
2
1
z 2 [dpr1 . dxr {H1 (qr , y0 )qr1 ]
2
1
{p0 (qs ) . dxs { 2 p1 (qs ) . dxs ,
2
ß 1999 RAS, GJI 136, 205^217
1
{ 2 p1 (qs ) . dxs ,
2
(31)
where I denotes the 3|3 identity matrix and the superscript T
in nT means transpose. We emphasize that the quantities
=p H0 nT and =x H0 nT represent dyadic products. nr1 and nr2 are
submatrices of the projection matrix %r used to extrapolate
the perturbation canonical vector y1 (qr ) on the plane going
through xr ~x0 (qr ) and perpendicular to vector n (see Farra &
Le Bëgat 1995). The ¢rst equation of (30) is used together with
(28), (12) and (13) written at qs to obtain the initial condition
y1 (qs ) of the ¢rst-order canonical vector of the perturbed ray
satisfying the boundary conditions (27) (see Farra & Le Bëgat
1995). Inserting the initial condition y1 (qs ) in (12), one obtains
the ¢rst-order canonical vector y1 of the perturbed ray at every
value of q.
Introducing the boundary conditions (28) into (26), we
obtain the traveltime between x's and x'r in the perturbed
medium up to second-order:
(32)
209
which corresponds to the traveltime along paraxial rays in the
reference medium.
The traveltime between ¢xed endpoints xs and xr is given by
T (xs , xr )~T0 (xs , xr )zT1 (xs , xr )z2 T2 (xs , xr ) ,
where
T1 (xs , xr )~{
T2 (xs , xr )~{
… qr
qs
H1 dq ,
… qr 1
1
H2 z =x H1 . x1 z =p H1 . p1 dq
2
2
qs
(33)
1
{ H1 (qr , y0 )qr1 ,
2
where qs and qr are the sampling parameters of the reference
ray at positions xs and xr , respectively, and qr1 is given by
qr1 ~{
p0 (qr ) . x1 (qr )
.
p0 (qr ) . xç 0 (qr )
(34)
In (33), all terms are computed along the reference ray. The
¢rst part of expression (33), which is valid in anisotropic media,
is the well-known ¢rst-order traveltime perturbation. The ¢rstorder term of the traveltime perturbation is the integral of the
¢rst-order term of the Hamiltonian. The second-order term of
the traveltime is the integral of the second-order term of the
Hamiltonian and of the variation of the ¢rst-order term of
the Hamiltonian due to the perturbation of the reference ray;
moreover, there is a term due to the change qr1 of the q-length of
the ray.
210
4
V. Farra
E XA M PL E : IS OT RO P IC M E D I U M
As an example, let us assume that the reference and the
perturbed media are isotropic. For isotropic media, we will
adopt the Hamiltonian (2) proposed by Burridge (1976). The
reference medium is de¢ned by the squared slowness u20 (x). In
the perturbed medium, the squared slowness is given by
u2 (x)~u20 (x)zw1 (x) ,
(35)
where w1 is the perturbation of the squared slowness.
The Hamiltonian H0 in the reference medium and the
perturbations H1 and H2 are given by
1
H0 (x, p)~ [p2 {u20 (x)] ,
2
1
H1 (x, p)~{ w1 (x) ,
2
(36)
H2 (x, p)~0 .
First-order equations of perturbed rays are obtained from (11)
and given by
" # "
#" # 2 0 3
xç 1
0 I
x1
5
~
z4 1
=w1
U0 0
pç 1
p1
2
(see Farra et al. 1989), where U0 is the 3|3 matrix de¢ned
by U0ij ~[L2 u20 /Lxi Lxj ]/2. The reference traveltime and the
¢rst- and second-order traveltime perturbations corresponding
to the perturbed ray passing through the endpoints xs and xr
are given by
… qr
… qr
T0 (xs , xr )~
p0 . xç 0 dq~
u20 dq ,
qs
1
T1 (xs , xr )~
2
T2 (xs , xr )~
1
4
… qr
qs
… qr
qs
qs
w1 dq ,
=w1 . x1 dq{
(37)
1 w1 (xr )
p0 (qr ) . x1 (qr ) .
4 u20 (xr )
Another form of the Hamiltonian, which is quite frequently
used, is the one for which the sampling parameter is the
arclength s. The Hamiltonian is given by
1
H(x, p)~ u{1 (x)[p2 {u2 (x)] .
2
(38)
Let us assume that the reference medium is de¢ned by the
slowness u0 (x). In the perturbed medium, the slowness is
given by
u(x)~u0 (x)zu1 (x) ,
(39)
where u1 is the perturbation of the slowness. The Hamiltonian
H0 in the reference medium and the perturbations H1 and H2
are given by
1
H0 (x, p)~ u{1
[p2 {u20 ] ,
2 0
1
H1 (x, p)~{ u{2
u1 [p2 {u20 ]{u1 ,
2 0
(40)
1
1
H2 (x, p)~ u{3
u2 [p2 {u20 ]z u{1
u2 .
2 0 1
2 0 1
First-order expressions of the perturbed rays may be
found in Farra et al. (1994). The reference traveltime and the
¢rst- and second-order traveltime perturbations corresponding
to the perturbed ray passing through the endpoints xs and xr
are given by
… Sr
… Sr
T0 (xs , xr )~
p0 . xç 0 ds~
u0 ds ,
Ss
T1 (xs , xr )~
T2 (xs , xr )~
… Sr
Ss
1
2
Ss
(41)
u1 ds,
… Sr
Ss
=u1 . x1 ds{
1 u1 (xr )
p0 (Sr ) . x1 (Sr ) ,
2 u0 (xr )
where Ss and Sr are the arc lengths of the reference ray at xs
and xr , respectively. The ¢rst-order traveltime perturbation
T1 contains the integral of the slowness perturbation along
the reference ray. This expression is used in linear traveltime
tomography. We note that, for the Hamiltonian proposed by
Farra et al. (1994) in order to introduce a coordinate stretching
along the perturbed ray so that the perturbation qr1 is forced to
be zero, relations (33) give identical expressions to those found
by Snieder & Sambridge (1993).
5 SE C O N D - OR D E R T R AV E LT I M E
PE RT U R BAT I O N I N A L AY E R E D M E D I U M
Until now, the reference model and the perturbed model were
assumed to have continuous properties. Let us now consider a
reference medium with one interface. We consider a smooth
perturbation of the shape of this interface. The reference
interface is de¢ned by the equation f0 (x)~0 and the perturbed
interface by f (x)~f0 (x)z f1 (x)~0. Let us consider a reference
ray with position vector x0 (q) and a ray in the perturbed
medium that is situated in the neighbourhood of this reference
ray. The incident reference ray intersects the reference interface at x0 (q{
i ) and the incident perturbed ray intersects the
perturbed interface at x(qi'{ ). qi'{ is generally di¡erent from q{
i
and the quantity *qi ~qi'{ {q{
i is a perturbation. Quantities
measured on the perturbed re£ected/transmitted ray at the
interface will be written at qi'z , qi'z ~qi'{ . The reference
ray satis¢es relations (7) at qi in the reference medium. The
perturbed ray satis¢es relations (7) at qi' in the perturbed
medium.
Let us develop the equations in the medium in which the
incident wave propagates. Denoting dx{ ~x(qi'{ ){x0 (q{
i )
{
{
and dp{ ~p(qi'{ ){p0 (q{
and *qi in
i ), developing dx , dp
2
{
{
{
2
{
perturbation series, dx{~ dx{
1 z dx2 , dp ~ dp1 z dp2
and *qi ~qI1 z2 qI2 , one obtains relations similar to (22).
Moreover, we can write the Taylor expansion to second
order of the function f (x) at x0 (q{
i ):
1
f (x0 zdx{ )~f (x0 )z=f . dx{ z dx{T ==fdx{ ,
2
(42)
where the vector =f and the matrix ==f are computed at
{
{
x0 (q{
i ). Taking into account the fact that x0 (qi ) and x(qi' )
belong to the reference interface and the perturbed interface,
respectively, so that f0 (x0 )~0 and f (x0 zdx{ )~0, and using
perturbation series in of dx{ and f (x) in equation (42), one
obtains the following equations:
=f0 . dx{
1 zf1 ~0 ,
1 {T
. {
=f0 . dx{
==f0 dx{
1 z=f1 dx1 ~0 ,
2 z dx1
2
(43)
ß 1999 RAS, GJI 136, 205^217
Second-order traveltime perturbation
where f0, f1 and their derivatives are computed at x0 (q{
i ).
Inserting (22a) written at q{
i into (43), one obtains
qI1 ~{
=f0 . x1 (q{
i )zf1
.
=f0 . xç 0 (q{
i )
(44)
Introducing (44) into (22a) and (22b), one obtains a linear
relation between
"
#
x1 (q{
i )
)~
y1 (q{
i
p1 (q{
i )
and
"
dy{
1 ~
dx{
1
dp{
1
5
f1
xç 0 (q{
i ),
=f0 . xç 0 (q{
i )
I
{
{
dp{
1 ~n2 x1 (qi )zp1 (qi ){
f1
pç 0 (q{
i ),
=f0 . xç 0 (q{
)
i
(45)
on the plane tangent to the reference interface at x0 (q{
i ).
z
z
z
Denoting dxz ~x(qi'z ){x0 (qz
i ) and dp ~p(qi' ){p0 (qi )
z
z
and developing dx and dp in perturbation series, similar
z
relations to (45) can be obtained between y1 (qz
i ) and dy1 .
Using the continuity of the reference and the perturbed ray
z
paths at the interface, dx{ ~dxz , so that dx{
1 ~dx1 ,
z
{
dx2 ~dx2 , as well as the perturbation of Snell's law (7), a
linear relation is obtained between
"
#
x1 (qz
i )
z
y1 (qi )~
p1 (qz
i )
y1 (q{
i )~
x1 (q{
i )
p1 (q{
i )
5
(46)
where Mi is the transformation matrix of the paraxial
propagator at the interface and yI1 is given by eq. (B6) of Farra
& Le Bëgat (1995). The perturbation of the ray canonical
vector is given to ¢rst order by (14)^(15), expressions that take
into account all interfaces encountered by the reference ray.
Endpoint boundary conditions may be introduced as explained
in Section 3.2.
Let us now compute the traveltime along the perturbed ray.
The traveltime T (qs', qi') may be developed to second order with
respect to T0 (qs , qi ) as in eq. (26). The perturbations terms
related to the interface are given by the boundary term
. {
p0 (q{
i ) dx1
z
(48a)
1
z .
{
z .
T2I ~[p0 (q{
i ){p0 (qi )] dx2 z [dp1 {dp1 ] dx1
2
1
z
{ qI1 [H1 (q{
i , y0 ){H1 (qi , y0 )] .
2
(48b)
{ .
[p0 (qz
i ){p0 (qi )] =f0
f1 ,
(49a)
=f0 . =f0
{ .
[p0 (qz
T
i ){p0 (qi )] =f0 1
. dx1
==f
dx
z=f
dx
T2I ~
0
1
1
=f0 . =f0
2 1
1
1 I
z .
{
z
z [dp{
1 {dp1 ] dx1 { q1 [H1 (qi , y0 ){H1 (qi , y0 )] ,
2
2
(49b)
where qI1 is given by (44) and f0 , f1 and their derivatives are
computed at x0 (q{
i ).
When there is no interface perturbation, f1 ~0, the
perturbation terms T1I and T2I are given by
T1I ~0 ,
T2I ~
1 I
{
q [H1 (qz
i , y0 ){H1 (qi , y0 )]
2 1
(see Appendix B), where
qI1 ~{
=f0 . x1 (q{
i )
.
=f0 . xç 0 (q{
i )
Therefore, the traveltime between ¢xed endpoints xs and xr is
given by
T (xs , xr )~T0 (xs , xr )zT1 (xs , xr )z2 T2 (xs , xr ) ,
#
{
I
y1 (qz
i )~Mi y1 (qi )zy1 ,
2
z .
T1I ~[p0 (q{
i ){p0 (qi )] dx1 ,
T1I ~
where nI1 and nI2 are given by (31) with n~=f0 . nI1 and nI2 are
submatrices of the projection matrix %I used to extrapolate the
perturbation canonical vector
"
#
x1 (q{
i )
)~
y1 (q{
i
p1 (q{
i )
"
where qI1 is given by (44). A similar term may be obtained
for the traveltime T (qi', q'r ). We introduce the notation
z
z
dx1 ~dx{
and dx2 ~dx{
1 ~dx1
2 ~dx2 . The terms of the
traveltime perturbation related to the interface can be written
as T1I z2 T2I with
Because of Snell's law, (7) written for the reference ray
z
p0 (q{
i ){p0 (qi ) is along the vector =f0 . Therefore, using
relations (43) one obtains
#
I
{
dx{
1 ~n1 x1 (qi ){
and
211
. { 1
p0 (q{
i ) dx2 z
2
dp{
1
ß 1999 RAS, GJI 136, 205^217
1 I
{
. dx{
1 { q1 H1 (qi , y0 ) ,
2
(47)
where T0 (xs , xr ) is the traveltime computed in the reference
medium and
… qr
N
X
H1 dqz
T1I ,
T1 (xs , xr )~{
qs
T2 (xs , xr )~{
… qr qs
i~1
1
1
H2 z =x H1 . x1 z =p H1 . p1 dq
2
2
(50)
N
X
1
{ H1 (qr , y0 ) qr1 z
T2I ,
2
i~1
where qs and qr are the sampling parameters of the reference
ray at positions xs and xr , respectively, N is the number of
interfaces crossed by the reference ray between qs and qr , qr1 is
given by (34) and T1I and T2I are given by (49a) and (49b).
Moreover, if we consider source and receiver perturbations,
the traveltime perturbation is given by an expression similar
to (32) where one has to add perturbation terms T1I and T2I
computed at each interface encountered by the reference ray.
In the integral, the terms H2 , H1 and its partial derivatives are
212
V. Farra
computed on the reference ray y0 ; the ¢rst-order canonical
vector y1 of the perturbed ray is computed from (14) with initial
condition y1 (qs ) corresponding to the boundary conditions
(28). We note that the second-order term of the traveltime
perturbation does not depend on the second-order term y2
of the ray perturbation when interface perturbations are
considered.
6
E XA M PL E : T I M E DI U M
The expressions obtained in the preceding sections are valid for
any heterogeneous isotropic or anisotropic media with interfaces. The ¢rst-order expressions of perturbed rays in a slightly
anisotropic medium without interfaces have been given by
Nowack & Psencik (1991) (see also Jech & Psencik 1989).
The second-order term of the Hamiltonian may be obtained
following the procedure used by Nowack & Psencik (1991). We
consider here a medium with transverse isotropy (TI medium)
and a vertical symmetry axis. For such a medium, we use
a parameter set, similar to that introduced by Thomsen
(1986), which simpli¢es the Hamiltonian expression (Farra
1989, 1990). Only qP waves are considered here. The parameters describing the qP-wave propagation are u2P (the square
of qP-wave phase slowness along the symmetry axis),
l2 ~u2P /u2S (the square of the ratio of qP to qS phase slowness
along the symmetry axis), a and $a ~da {a (non-dimensional
parameters that describe the amount of anisotropy and the
anellipticity of the slowness sheets, respectively). Following
Appendix C, the qP-wave Hamiltonian for small values of $a
may be obtained to second order as H~H0 zH1 zH2 , where
$a ~0. Let us introduce the normalized traveltime residual
*T /Ti ~(T {Ti )/Ti with respect to the traveltime Ti calculated in the isotropic medium with P velocity oP ~3 km s{1 .
We denote Tt the exact traveltime computed in the TI medium
with an exact ray tracing, T0 the traveltime computed in the
reference medium with elliptical anisotropy and T1 and T2 the
¢rst- and second-order approximations of the traveltime in
the TI medium. In Fig. 1, the four quantities (Tt {Ti )/Ti ,
(T0 {Ti )/Ti , (T1 {Ti )/Ti and (T2 {Ti )/Ti are plotted as
functions of the angle h of the ray with the vertical axis. In
homogeneous media, these quantities are independent of the
distance between the station and the source. The variability
of (Tt {Ti )/Ti with h is related to the anisotropy. One can see
that the maximum traveltime residual (around 15 per cent)
with respect to the isotropic traveltime is for the horizontal
incidence. The traveltime obtained in the elliptical reference
medium is closer to the traveltime obtained in the TI medium
than the isotropic traveltime. The di¡erence between ¢rstorder and exact traveltimes is around 0:6 per cent. The secondorder traveltime approximates very well the exact traveltime
with an error of less than 0:1 per cent (the two curves are
almost indistinguishable). The analysis of the terms contributing to the traveltime perturbation shows the important
e¡ect of the H2 term as well as the ¢rst-order ray perturbation
in the value of the second-order traveltime perturbation for
strong anisotropy.
1
H0 (x, p)~ [(1za )( p2x zp2y )zp2z {u2P (x)] ,
2
( p2x zp2y )p2z
1
H1 (x, p)~ $a
,
(1zma )( p2x zp2y )zp2z
2
H2 (x, p)~{
(51)
[( p2x zp2y )p2z ]2
1 $2a
.
2 1{l2 [(1zma )( p2x zp2y )zp2z ]3
In (51), ma ~a (1{l2 ){1 is a parameter proportional to a
and ( px , py , pz ) are the slowness vector components in the
Cartesian coordinate system. The reference Hamiltonian H0
corresponds to a medium with elliptical anisotropy characterized by the parameters a and da such that $a ~0. In such
a medium, ray tracing is not more di¤cult than in an isotropic
medium (see Farra 1989). We note that in this formulation
the parameter a is not assumed to be small. First-order
expressions of perturbed rays in TI media are given by
Farra (1989, 1990) for the perturbation series (51) of the
Hamiltonian H.
In order to illustrate the perturbation technique, we con{1
sider a simple homogeneous TI medium with u{1
P ~3 km s
and l~0:333; the anisotropic parameters are de¢ned by
a ~0:4 and da ~0:0 ($a ~{0:4) so that the medium has
a strong anisotropy. This anisotropy is around 15 per cent
[remembering that the anisotropic parameters used by Farra
(1989, 1990) are multiplied by 2 with respect to those de¢ned by
Thomsen (1986)]. In a homogeneous anisotropic medium, ray
curves are straight lines; however, the slowness vector direction
di¡ers from the ray direction and the phase slowness depends
on the ray direction. The reference medium is the homogeneous
medium with elliptical anisotropy de¢ned by a ~0:4 and
Figure 1. Normalized traveltime residual as a function of the ray
angle h with the vertical axis. The normalized traveltime residual
*T /Ti ~(T {Ti )/Ti is computed with respect to the traveltime Ti
obtained in the isotropic medium with P velocity op ~3 km s{1 .
Tt is the exact traveltime calculated in the TI medium de¢ned by
{1 2
, l ~0:333, a ~0:4, $a ~{0:4. T0 is the traveltime
u{1
P ~3 km s
computed in the reference medium (homogeneous medium with
{1
, a ~0:4, $a ~0). T1
elliptical anisotropy de¢ned by u{1
P ~3 km s
and T2 are the ¢rst- and second-order approximations of the traveltime
computed from relations (33) with Hamiltonian terms given by (51).
ß 1999 RAS, GJI 136, 205^217
Second-order traveltime perturbation
{1
Let us now consider the TI medium with u{1
,
P ~3 km s
l~0:333, a ~0:66 and da ~1:1 ($a ~0:44). This medium has a
strong anisotropy of around 22 per cent. The reference medium
has an elliptical anisotropy de¢ned by a ~0:66 and $a ~0.
Fig. 2 shows the normalized traveltime residuals (Tt {Ti )/Ti ,
(T0 {Ti )/Ti , (T1 {Ti )/Ti and (T2 {Ti )/Ti as functions of the
ray angle h. One can see that for this medium the ¢rst-order
traveltime is itself a good approximation of the exact traveltime. The di¡erence between the ¢rst-order and exact
traveltimes is less than 0:3 per cent. The second-order traveltime approximates very well the exact traveltime with an
error of the order of 0:05 per cent (the two curves are almost
indistinguishable). One can show that, for positive $a , the ¢rstorder traveltime approximation underestimates the traveltime
and the second-order traveltime approximation overestimates
it, so that T1 < Tt < T2 < T0 . For negative $a , as in the
example shown in Fig. 1, the ¢rst- and second-order traveltime approximations underestimate the traveltime, so that
T0 < T1 < T2 < Tt . This phenomenon is related to the form
of the series (C3), which has terms with alternate signs for
positive $a and negative terms for negative $a .
In weakly anisotropic media, an isotropic medium is
generally used as the reference medium. Di¡erent choices of
the reference medium as well as the anisotropic parameters can
be found in the literature (Thomsen 1986; Sayers 1994; Mensch
& Rasolofosaon 1997; Tsvankin 1997, Psencik & Gajewski
1998). We choose as the reference medium the isotropic
medium whose velocity is the vertical P-wave phase velocity
of the TI medium. For such a reference medium, the qP-wave
Figure 2. Same as Fig. 1 for the homogeneous TI medium de¢ned by
{1 2
, l ~0:333, a ~0:66, $a ~0:44. The reference medium
u{1
P ~3 km s
is the homogeneous medium with elliptical anisotropy de¢ned by
{1
, a ~0:66, $a ~0. T1 and T2 are the ¢rst- and secondu{1
P ~3 km s
order approximations of the traveltime computed from relations (33)
with Hamiltonian terms given by (51).
ß 1999 RAS, GJI 136, 205^217
213
Hamiltonian for small values of a and $a may be obtained to
second order as H~H0 zH1 zH2 , where
1
H0 (x, p)~ [p2 {u2P (x)] ,
2
( p2x zp2y )p2z
1
1
H1 (x, p)~ a ( p2x zp2y )z $a
,
(1zma )( p2x zp2y )zp2z
2
2
H2 (x, p)~{
(52)
[( p2x zp2y )p2z ]2
1 $2a
.
2 1{l2 [(1zma )( p2x zp2y )zp2z ]3
Using expression (33) with the Hamiltonian terms (52), we
compute the ¢rst- and second-order traveltimes for the same
TI medium as used in Fig. 1. Fig. 3 shows the normalized
traveltime residuals (Tt {Ti )/Ti , (T0 {Ti )/Ti , (T1 {Ti )/Ti and
(T2 {Ti )/Ti as functions of the ray angle h. Of course,
T0 {Ti ~0 because the isotropic medium corresponds to the
reference medium. The ¢rst-order traveltime is a good
approximation for h less than 150. The second-order traveltime
approximates the exact traveltime very well for angle h less
than 250 but is not very good for large angles of incidence. We
note that the ¢rst-order traveltime computed with expression
(52) is a little closer to the exact traveltime in the near-vertical
incidence than the ¢rst-order traveltime computed with
expression (51) (see Fig. 1). The use of an isotropic reference
medium as the reference medium should be interesting for
smaller anisotropic parameters a and $a . Fig. 4 shows the
normalized traveltime residuals obtained in the TI medium
{1
with u{1
, l~0:333, a ~0:2 and da ~0:4 ($a ~0:2).
P ~3 km s
The reference medium is the isotropic medium with P velocity
Figure 3. Same as Fig. 1 for the homogeneous TI medium de¢ned by
{1 2
, l ~0:333, a ~0:4, $a ~{0:4. The reference medium
u{1
P ~3 km s
is the isotropic medium with P velocity op ~3 km s{1 . T1 and T2 are
the ¢rst- and second-order approximations of the traveltime computed
from relations (33) with Hamiltonian terms given by (52).
214
V. Farra
One can see that in this case the ¢rst-order approximation is
already very good. To be e¤cient, perturbation algorithms
require a good parametrization of the medium. The examples
presented here show that for TI media the utilization of an
elliptically anisotropic reference medium and the parameter $a
as the small perturbation parameter is well adapted for strong
anisotropy.
7
Figure 4. Same as Fig. 1 for the homogeneous TI medium de¢ned by
{1 2
, l ~0:333, a ~0:2, $a ~0:2. The reference medium is
u{1
P ~3 km s
the isotropic medium with P velocity op ~3 km s{1 . T1 and T2 are the
¢rst- and second-order approximations of the traveltime computed
from relations (33) with Hamiltonian terms given by (52).
oP ~3 km s{1 . One can see that the second-order approximation is a good estimation of the traveltime for all ray
directions. The di¡erence between the exact and second-order
traveltimes is around 0:2 per cent. Fig. 5 shows the normalized
traveltime residuals computed for the same TI model with an
elliptically anisotropic reference medium de¢ned by a ~0:2.
CO NC LUSI O N S
Very simple expressions of the second-order traveltime perturbations are obtained in the framework of the Hamiltonian
approach. The formulation is independent of the chosen form
of the Hamiltonian and, consequently, may be used for any
sampling parameter q along the reference ray. The theory is
valid for both isotropic and anisotropic media, including
models with structural discontinuities. Perturbations of the
elastic parameters and the interface shape are considered here.
The ¢rst-order traveltime perturbation is given by the integral
of the ¢rst-order term of the Hamiltonian perturbation. The
computation of the second-order traveltime needs to take into
account the ¢rst-order perturbation of the ray as well as the
second-order term of the Hamiltonian. Moreover, the ¢rstand second-order traveltime perturbations include the contribution of boundary terms at the discontinuities encountered by
the reference ray. Endpoint boundary conditions are easy to
introduce in this formulation using the paraxial propagator.
The computation of traveltime in weakly anisotropic media
is an interesting application of perturbation theory. Examples
obtained in transverse isotropic media show that traveltimes
may be estimated well using the second-order approximation,
even for strong anisotropy. An elliptically anisotropic medium
seems to be a good choice for a reference medium and the
Thomsen parameter da {a the natural parameter to develop
the perturbation approach. In anisotropic media of arbitrary
symmetry, the second-order term of the Hamiltonian may be
obtained easily following the procedure used by Nowack &
Psencik (1991).
AC K NOW L E D G ME N T S
This work was partly funded by the European Commission
in the framework of the Joule project `Reservoir-oriented
delineation technology'. I thank Ivan Psencik and Robert
Nowack for critical comments on the manuscript and
Professor Cerveny for his very stimulating review of the paper.
This is IPG contribution No. 1566.
R E F ER E NC E S
Figure 5. Same as Fig. 1 for the homogeneous TI medium de¢ned by
{1 2
, l ~0:333, a ~0:2, $a ~0:2. The reference medium
u{1
P ~3 km s
is the homogeneous medium with elliptical anisotropy de¢ned by
{1
, a ~0:2, $a ~0. T1 and T2 are the ¢rst- and secondu{1
P ~3 km s
order approximations of the traveltime computed from relations (33)
with Hamiltonian terms given by (51).
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Nowack, R.L. & Psencik, I., 1991. Perturbation from isotropic to
anisotropic heterogeneous media in the ray approximation,
Geophys. J. Int., 106, 1^10.
Psencik, I. & Gajewski, D., 1998. Polarization, phase velocity and
NMO velocity of qP waves in arbitrary weakly anisotropic media,
Geophysics, in press.
Sayers, C.M., 1994. P-wave propagation in weakly anisotropic media,
Geophys. J. Int., 116, 799^805.
Snieder, R. & Sambridge, M., 1992. Ray perturbation theory for
traveltimes and ray paths in 3-D heterogeneous media, Geophys. J.
Int., 109, 294^322.
Snieder, R. & Sambridge, M., 1993. The ambiguity in ray perturbation
theory, J. geophys. Res., 98, 22 021^22 034.
Thomsen, L.A., 1986. Weak elastic anisotropy, Geophysics, 51,
1954^1966.
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orthorhombic media, Geophysics, 62, 1292^1309.
A PPE N D I X A : B O U N DA RY CO N D I T I O N S
C OR R E S P O N DI NG TO R AYS NORM A L TO
A N I N T E R FAC E
Let the interface location be de¢ned by the general relation
f (x)~0, =f de¢nes the local normal to the interface. We
assume that a ray with canonical vector y0 (q) has been de¢ned
in the reference medium with the following boundary conditions: one endpoint is x0 (qs )~xs and at the other endpoint,
x0 (qr ), the slowness vector p0 (qr ) is perpendicular to the
interface. The corresponding boundary conditions are
x0 (qs )~xs ,
f (x0 (qr ))~0 ,
ß 1999 RAS, GJI 136, 205^217
p0 (qr )|=f0 ~0 ,
(A1)
215
where =f0 is computed at x0 (qr ). In the vicinity of this reference
ray, one can ¢nd rays in the perturbed medium with the
following boundary conditions:
x(qs )~x'~x
s
s zdxs ,
f (x(q'r ))~0 ,
p(q'r )|=f ~0 .
(A2)
In (A2), q'r can be di¡erent from qr but *qr ~q'{q
r
r may be
developed in a perturbation series: *qr ~q1 z2 q2 . We denote
dx~x(q'r ){x0 (qr ) and dp~p(q'r ){p0 (qr ). Using perturbation
series (10) and (21), one obtains similar relations to (22).
Moreover, we can write the Taylor expansion to second order
of the function f (x) at x0 (qr ):
1
f (x0 zdx)~f (x0 )z=f0 . dxz dxT ==f0 dx ,
2
(A3)
where the vector =f0 and the matrix ==f0 are computed
at x0 (qr ). The matrix ==f is the matrix of second-order
derivatives of the function f (x). Because x0 (qr ) and x(q'r ) belong
to the interface f (x)~0, one obtains
=f0 . dx1 ~0 ,
=f0 . dx2 ~{
1 T
dx ==f0 dx1 ,
2 1
(A4)
where perturbation series (21) were used for dx in (A3).
Inserting (22a) into (A4), one obtains
q1 ~{
=f0 . x1 (qr )
.
=f0 . xç 0 (qr )
(A5)
Introducing (A5) into (22a) and (22b), one obtains:
dx1 ~nr1 x1 (qr ) ,
(A6)
dp1 ~nr2 x1 (qr )zp1 (qr ) ,
where dp1 is the ¢rst-order expression of the quantity dp and nr1
and nr2 are given by (31) with n~=f0 . Moreover, developing the
third equation of (A2) to ¢rst order, one obtains
=f0 |dp1 ~p0 (qr )|(==f0 dx1 ) .
(A7)
Eq. (A7) may be written in the following form:
M1 dx1 zM2 dp1 ~0 ,
(A8)
where M1 and M2 are matrices of rank 2. Using relations (A6)
in (A8), one obtains
[M1 nr1 zM2 nr2 ]x1 (qr )zM2 p1 (qr )~0 .
(A9)
Moreover, the perturbed ray should satisfy the following
boundary conditions:
dxs1 ~x1 (qs )~dxs ,
dxs2 ~x2 (qs )~0 .
(A10)
Six independent boundary conditions are needed to solve
the boundary problem associated with the ray perturbation
equations (11). Eqs (A9)^(A10) correspond to ¢ve independent
boundary equations. The last equation is the eikonal equation
(13) written at qs . Inserting (14) into eqs (A9) and using (A10)
and (13), one obtains three independent equations for p1 (qs ).
Using y1 (qs ) in (14), one can compute the ¢rst-order perturbed
ray satisfying boundary conditions (A2).
216
V. Farra
The traveltime between x's and x'~x(q
'r ) is given to
r
second-order by (26):
… qr 1
H1 z2 H2 z =x H1 . x1
T (x's , x'r )~T0 (xs , xr ){
2
qs
1
z =p H1 . p1 dqzp0 (qr ) . dx1
2
1
1
z2 p0 (qr ) . dx2 z dp1 . dx1 { qr1 H1 (qr , y0 )
2
2
{p0 (qs ) . dxs {
2
p1 (qs ) . dxs ,
2
is given by (A5) and dp1 by (A6). Because the slowness
vector p0 (qr ) is normal to the interface at x0 (qr ), from (A4) one
can write p0 (qr ) . dx1 ~0. Moreover, we express dp1 as
(=f0 . dp1 )=f0 z(=f0 |dp1 )|=f0
.
=f0 . =f0
{
z
{
=f0 |dpz
1 ~=f0 |dp1 z[p0 (qi ){p0 (qi )]|(==f0 dx1 ) .
(B3)
z
We express dp{
1 and dp1 as
{
(=f0 . dp{
1 )=f0 z(=f0 |dp1 )|=f0
,
.
=f0 =f0
z
(=f0 . dpz
1 )=f0 z(=f0 |dp1 )|=f0
dpz
.
1 ~
.
=f0 =f0
dp{
1 ~
(B4)
Taking into account (B3), (B4) and (43), one obtains
(A11)
where qr1
dp1 ~
and (B2), we obtain the perturbed Snell's law to ¢rst order:
(A12)
Taking into account (A4) and (A7), one can write:
1
1 p0 (qr ) . =f0
dxT1 ==f0 dx1 ~{p0 (qr ) . dx2 .
dp1 . dx1 ~
2
2 =f0 . =f0
{ .
1
1 [p0 (qz
z .
i ){p0 (qi )] =f0
dx1 T ==f0 dx1 ,
(dp{
1 {dp1 ) dx1 ~{
.
=f0 =f0
2
2
(B5)
which can be written as
{ .
1
[p0 (qz
z .
i ){p0 (qi )] =f0
=f0 . dx2
(dp{
1 {dp1 ) dx1 ~
2
=f0 . =f0
{ .
~[p0 (qz
i ){p0 (qi )] dx2 ,
(B6)
where we used eqs (7) and (43).
(A13)
Therefore, the traveltime along the perturbed ray is given by
… qr 1
H1 z2 H2 z =x H1 . x1
T (x's , x'r )~T0 (xs , xr ){
2
qs
1
z =p H1 . p1 dq{p0 (qs ) . dxs
2
1
{ 2 fp1 (qs ) . dxs zqr1 H1 (qr , y0 )g ,
2
(A14)
where qr1 is given by (A5), the terms H2 , H1 and its partial
derivatives are computed on the reference ray y0, and the ¢rstorder canonical vector y1 of the perturbed ray is computed
from (14) with initial condition y1 (qs ) corresponding to the
boundary conditions (A2). Note that expression (A14) of
the second-order traveltime perturbation does not depend
on the second-order ray perturbation y2 (q).
In this section, we assume that there is no perturbation of
the interface. The perturbed ray satis¢es Snell's law (7) in the
perturbed medium:
=f|p(qi'z )~=f|p(qi'{ ) ,
(B1)
where =f is computed at x(qi'{ )~x(qi'z ). The reference ray
satis¢es Snell's law in the reference medium. Developing the
¢rst equation of (B1) to ¢rst order in and using the notation
z
introduced in the text, one obtains dx{
1 ~dx1 . Moreover, the
¢rst-order Taylor expansion of the vector =f may be written as
=f~=f0 z==f0 dx1 ,
z
dx1 ~dx{
1 ~dx1 ,
Farra (1989) introduced the Hamiltonian for an anisotropic
medium as
1
H(x, p)~ u2P (x)[GP (x, p){1] ,
2
(B2)
and the vector =f0 and
where we denote
z
the matrix ==f0 are computed at x0 (q{
i )~x0 (qi ). From (B1)
(C1)
where GP (x, p) is the qP eigenvalue of the Christo¡el matrix
and uP is the vertical phase slowness of the qP wave. We consider here a medium with transverse isotropy (TI medium) and
a vertical symmetry axis. The TI medium can be de¢ned by
¢ve independent density-normalized elastic parameters Aij . We
introduce the following parameters (Farra 1989):
2
u2P ~A{1
33 , l ~
ca ~
A PPE N D I X B : R E L AT I O N S B ET W E E N
S E C O N D - A N D F I R ST- OR D E R
PE RT U R BAT I O N S AT A N IN T E R FAC E
x(qi'{ )~x(qi'z ) ,
A PPE N D I X C : C OMP U TAT IO N OF T H E
qP - WAVE H A M I LTO N I A N I N A T I ME D I U M
A44
A11 {A33
, a ~
,
A33
A33
A66 {A44
(A13 zA44 )2 {(A33 {A44 )2
,
, da ~
A44
A33 (A33 {A44 )
where uP is the quasi-P-wave (qP) phase slowness (velocity
reciprocal) along the symmetry axis and l is the ratio of quasiP-wave to quasi-S-wave phase slowness along the symmetry
axis. The parameters a /2 and ca /2 are the conventional
measures of P- and S-wave velocity anisotropy (Thomsen
1986). The parameter $a ~da {a describes the anellipticity
of the qP-slowness sheet. The three parameters (a , $a , ca )
are dimensionless and zero in isotropic media. Note that $a
reduces to zero in elliptically anisotropic media (Thomsen
1986; Farra 1989).
In a TI medium, one can write (see Farra 1989)
1
1
u2P GP ~ [(1zl2 )p2 za p2r ]z [(1{l2 )p2 za p2r ]
2
2
s
(1{l2 )p2r p2z
,
| 1z4$a
[(1{l2 )p2 za p2r ]2
(C2)
ß 1999 RAS, GJI 136, 205^217
Second-order traveltime perturbation
where pr 2 ~p2x zp2y . Using Taylor's series expansion in $a , one
obtains
u2P (x)GP (x, p)~p2 za p2r z$a
{$2a
(1{l2 )p2r p2z
(1{l2 )p2 za p2r
(1{l2 )2 p4r p4z
z... .
[(1{l2 )p2 za p2r ]3
Introducing the parameter ma ~a (1{l2 ){1 , one can write
the Hamiltonian (C1) as a perturbation series in $a ,
ß 1999 RAS, GJI 136, 205^217
H~H0 zH1 zH2 z . . . , where the ¢rst terms are given by
1
H0 (x, p)~ [(1za )p2r zp2z {u2P (x)] ,
2
1
p2r p2z
,
H1 (x, p)~ $a
(1zma )p2r zp2z
2
(C3)
217
H2 (x, p)~{
(C4)
1 $2a
p4r p4z
.
2 1{l2 [(1zma )p2r zp2z ]3
H0 is the Hamiltonian for an elliptically anisotropic medium
with parameter a (remembering that elliptical anisotropy
corresponds to transverse isotropy with $a ~0).