zotero/storage/UNFK2QPE/.zotero-ft-cache

Geophys. J . Inr. (1992) 108, 364-371
RESEARCH NOTE
Ray-theory Green’s function reciprocity and ray-centred coordinates in anisotropic media
J-M. Kendall, W. S. Guest and C. J. Thomson
Department of Geological Sciences, Queen’s University, Kingston, Ontario, Canada K7L 3N6
Accepted 1991 June 12. Received 1991 June 11; in original form 1990 December 10
SUMMARY Lighthill and others have expressed the ray-theory limit of Green’s function for a point source in a homogeneous anisotropic medium in terms of the slowness-surface Gaussian curvature. Using this form we are able to match with ray theory for inhomogeneous media so that the final solution does not depend on arbitrarily chosen ‘ray coordinates’ or ‘ray parameters’ (e.g. take-off angles at the source). The reciprocity property is clearly displayed by this ‘ray-coordinate-free’ solution. The matching can be performed straightforwardly using global Cartesian coordinates. However, the ‘ray-centred’ coordinate system (not to be confused with ‘ray coordinates’) is useful in analysing diffraction problems because it involves 2 X 2 matrices not 3 x 3 matrices. We explore ray-centred coordinates in anisotropic media and show how the usual six characteristic equations for three dimensions can be reduced to four, which in turn can be derived from a new Hamiltonian. The corresponding form of the ray-theory Green’s function is obtained. This form is applied in a companion paper. Key words: body-wave seismology, geometrical spreading, point sources, reciprocity, ray-centred coordinates.
1 INTRODUCTION
In scattering or diffractionproblems, it is sometimes necessary to know the zeroth-order ray approximation to Green’s function and to exploit the reciprocity property of this approximation. See, for example, Coates 8c Chapman (1990) and Thomson, Kendall & Guest (1992, this issue). The reciprocity of point-source geometrical spreading functions is at the heart of the issue and Coates & Chapman (1990) refer to Richards (1971, section 3.5) for the proof. The discussion in Richards (1971) applies to multilayered media, but anisotropy and the relevance of symmetry in the geometrical spreading equations are not considered. Here, the approximate Green’s function is found by matching zeroth-order ray theory with the exact point-source solution for a homogenous anisotropic medium obtained by Lighthill (1960), Buchwald (1959) and Burridge (1967). This point-source solution is expressed in terms of the Gaussian curvature of the slowness surface at the source and it matches elegantly with the geometrical spreading function in global Cartesian coordinates. Arbitrarily chosen ray coordinates or parameters at the source do not appear in the resulting Green’s function and the reciprocity is evident in this ‘coordinate-free’solution.
Ray-centred coordinates (as opposed to ray coordinates or parameters) are useful for local analysis problems such as that in Thomson et af. (1991). This is partly because the 6 X 6 system of geometrical spreading equations in Cartesian coordinates reduces to a 4 x 4 system in ray-centred coordinates. Popov& PSenEik (1976) have described the latter for isotropic media and deduced the corresponding ray and geometrical spreading equations from a Lagrangian argument involving stationarity of ray traveltime. Here we obtain the ray and geometrical spreading equations for anisotropic media by starting from the eikonal or phase equation expressed in ray-centred coordinates and applying the method of characteristics. This is quite analogous to the procedure used with Cartesian coordinates and six ray equations are still obtained for a 3-D space. However, the properties of ray-centred coordinates allow this system to be reduced to four equations. We find these reduced equations may be considered to come from a Hamiltonian (other than the eikonal equation). This Hamiltonian turns out to be related to one component of
364

Reciprocity and ray -centred coordinates 365
slowness in the ray-centred system, as noted by Farra & Madariaga (1987) for the isotropic equations of Popov & PSenfik (1976). As the reduced system of ray equations may be derived from a Hamiltonian, the corresponding 4 x 4 system of geometrical spreading equations has the symplectic property. Point-source reciprocity and other symmetries follow straight away.
The forms of the eikonal and other equations in ray-centred coordinates are generally quite complicated. Fortunately though, explicit forms are not needed to derive the results being sought. As a rule it seems ray-centred coordinates are useful for proving local analytical results, as in Thomson et al. (1992), but the Cartesian equations are simpler for numerical implementations. We should note that the theory of narrow beams in inhomogeneous anisotropic media relies on ray-centred coordinates and that Hanyga (1982,1986) has also considered the geometrical spreading equations in this system. It is hoped that the discussion of these coordinates, reciprocity and other properties given here, coupled with the companion paper Thomson et al. (1992), will be helpful in realizing their wider potential in seismological scattering problems.
2 MATCHING WITH A POINT SOURCE A N D RECIPROCITY IN ANISOTROPIC RAY THEORY
2.1 Point source in a uniform anisotropic medium
The field due to a point force in a uniform anisotropic medium has been considered by, among others, Buchwald (1959), Lighthill (1960), Duff (1960) and Burridge (1967).
In our notation, the appropriate form of the Green’s function given by the latter author is
(1)
[Burridge (1967), equation (6.8)]. This is the p-component of displacement due to the q-component of a unit body force per unit volume at the origin [not per unit mass as in Burridge (1967), note]. The displacement eigenvectors g, are defined in the usual way [Kendall & Thomson (1989), equation (4)] and are normalized to one. Vector v is the total/group/ray velocity and x is the receiver position corresponding to slowness po at the origin. The quantity K(p,) is the Gaussian curvature of the slowness surface at p,,. Lighthill (1960) and Buchwald (1959) give the following expression for this curvature, adapted to our notation,

where the ray theory Hamiltonian H(p,x)= H(p) in the uniform medium, H = O defines the slowness surface and Hi= a H / a p i = ui.The summation in (2a) is wrt cyclical permutation of i, j and k. The following alternative form of the curvature is useful

where the ijk summation is still cyclical and lmn are summed in the usual manner. This form makes the relationship to the wavefront vector-area easier to recognize.
For S-waves, it is possible that the slowness surface is not convex in the direction of po and a form other than (1) should be used. In some directions the usual ray approximation does not apply at all for S-waves. Complications such as these and the affects of caustics far from the source will not be considered in the present work.
The ray theory solution in an inhomogeneous medium may be written

- - [eerveng (1972), following equation (36)]. The function qois to be found by matching (3) with (1) as the source is approached.
Matching the phase ,is simple since t ( x ) + Ixl/lvl = po v Ixl/lvl -- po * x, where we have used the fact that po v = 1. The

geometrical spreading in (3) is defined by [Cervenf (1972), equation (27)]

- v,J = u, p q , x x aqzxl= Iv (d,,x x aq2x)1,

(4)

where u, is the normal velocity. The parameters q 1 and q2 uniquely define a ray leaving the source. The derivatives in (4) are at constant t and so the vector product defines an area in the wavefront. There is some freedom in the choice of the ray parameters or coordinates. The final solution (3) cannot depend on this choice and so the unknown y omust be a function of q 1 and q2 in such a way as to remove the dependence on how these two are chosen. It should be possible to rewrite the ray solution (3) in terms of unique quantities, such as p and v at the source, rather than arbitrary parameters on which these quantities have been made to depend simply for convenience. This ‘ray-coordinate free’ form of the solution will also make the reciprocity relation apparent.

366 J - M . Kendall, W . S. Guest and C.J. Thomson
2.2 Geometrical spreading function near the source It is appropriate to consider the geometrical spreading equations in the Cartesian form used by Thomson & Chapman (1985).
The geometrical spreading equations are
%Ay,
dt where

For present purposes, the elements of vector y may be considered to be (d,,xT, d , , ~ ~ w) h~er,e q, = q , or q 2 . At the point source, only the latter three elements of y are non-zero and so we have for small t

aqTi(T) = B;j(O)(aq,pj)r+ O(t').

(6)

Substituting this into (4) yields the leading-order form

v,J = E j i k v i B j , B k m g ~ ' ) q ~ ) tw2h, ere qjl' = a4,pi

(7)

at the source. In order to bring this into a form such as (2b), first note the equivalent form

v,J = - € i l k V ; B j l B k , q ~ ' q ~ 2 ' t 2 ,

(8)

from the properties of E;,k. Adding (7) and (8) leads to

l r 2 . u , J =

1,
2 ijk

i

B j

l

B

k

m

(41(1)

q

m( 2

)

-

q

m(l)

qI(

2

)

(9)

At this stage the summation in (9) takes place over all 1 and all rn. However, the case 1 = m clearly gives zero contribution. Moreover, the contributions for the pair lm and the pair ml are equal. This follows from noting that when 1 and m are interchanged (i) the bracketed ( ) term changes sign and (ii) the effect on the B , terms is effectively to interchange j and k and hence the sign of E+. Lastly we note that the term in brackets is parallel to the p-component of v (p # 1, p # m ) . Thus we have the leading approximation near the source

v,J = cot2 EijkviBjlBkmvp,
plm

with cyclical summation on p , 1 and rn and

at the source. Comparing (2b) and (10) shows that the matching is now straightforward and one obtains for inhomogeneous media the
leading approximation

where po is the density at the source.
2.3 Reciprocity and geometrical spreading
Now consider the propagator matrix W(t, 0) of the system of geometrical spreading equations (5). The ray path along which this system is integrated i s written x = x(%, po, r ) , p = p(%, po, t) and the elements of the propagator are the same as the elements of the determinant

It is important to note that the xo, and p,, are considered independent for the partial derivatives in W. The vector solution to a
specific problem, y in ( 9 ,is given by y = Wc, where c is an initial vector that imparts the interdependences of the xo, and the
po, for the problem at hand. Thomson & Chapman (1985) show that the propagator satisfies

Reciprocity and ray -centred coordinates 367
where I is the 3 x 3 identity matrix. The matrix W is said to be symplectic, a property which follows from the structure of the matrix of Hamiltonian derivatives A (5).
The propagator is composed of its 3 X 3 blocks according to

and for point-source problems the block R , = %xi/i5po,is important ( ‘ R for Reciprocity). It follows from (13) and the definition of a propagator (Gilbert & Backus 1966) that

R(t, O ) = -RT(O, t).

(14b)

The generalization of equation (7) to large t is

V,J = E i , k V i R j / R k m q i l ’ q E ) ,

#’ = 34/P(,’

(15)

where we now use subscript 0 to distinguish values at the source. Steps analogous to those leading to (10) now lead to its generalization

where C , is still given by (11). Equation (16) is almost the result we seek. Note that it allows us to replace v,J/C, in the
matched ray theory result (12) with the sum on the rhs of (16). This eliminates the dependence of the solution (12) on the arbitrary parameters ql,*.
In order to establish reciprocity we must replace the cyclical sum in (16) with a complete sum as follows. For each cyclic pair lm in the sum
2 = EijkViRjIRkmvOp
plm
interchanging 1 and m is equivalent to interchanging j and k. Hence, from the properties of eijk,
% = - EiikViRjmRklVOp
plm
and

where now the summation of plrn is in the conventional manner repeated subscripts. Now the ray theory Green’s function may be written

with R given by (17). Once again we note that modifications due to the presence of caustics are not considered and hence R is assumed to be positive.
Reciprocity in the geometrical spreading contribution to (18) is recognized by noting that interchanging the source and
receiver in (17) is equivalent to replacing (R)jl with (FIT),,. The sum 99 is left unchanged, though, because of (14b). Lastly, it turns out to be useful for Thomson et al. (1992) if we note that the adjugate (sometimes ‘adjoint’) of the matrix R
can be written

aps (adj R)si = det IRI (8-‘)s=j EijkRjrRkrn plm
and hence an alternative form of (17) is

R = v: adj Rv = det (RIv: R-’v.

(19)

3 RAY-CENTRED COORDINATES IN THE ANISOTROPIC CASE
The ray or Hamilton equations are usually expressed in the 3-D Cartesian coordinates used for the wave and eikonal equations (e.g. Eervenf 1972; Thomson & Chapman 1985; Kendall & Thomson 1989). /This leads to a sixth-order system for the equations of variation or the geometrical spreading or the dynamic ray-tracing equations. Sometimes it is convenient to use ray-centred coordinates, as for the proof in Appendix B of Thomson et al. (1992), because these lead to a lower order system. POPOV& PSenEik (1976) describe ray-centred coordinates for isotropic media and motivate an appropriate form of the

368 J - M . Kendall, W . S . Guest a n d C. J . Thomson
Hamiltonian by considering stationanty of the traveltime along a ray. Farra & Madariaga (1987) and Coates & Chapman (1990) use the isotropic Hamiltonian of Popov & PSenEik (1976) as a starting point, but they do not indicate how it is motivated.
Here we take as our starting point the anisotropic eikonal equation in 3-D ray-centred coordinates and consider how the corresponding characteristic equations (Courant & Hilbert 1962, p. 78) may be reduced in number. It is found that these reduced ray equations may be considered to have come from a certain Hamiltonian [which is essentially that of Popov & PSenEik (1976) in isotropic media]. The ray-centred coordinates themselves have been considered for the anisotropic case by Hanyga (1982, 1986), where a novel alternative approach to the geometrical spreading problem, not involving Hamiltonians, may be found. We use the same notation for the coordinates. They are not necessarily orthogonal and there is some arbitrariness in the way they are chosen (see Section 3.2).

3.1 General theory

The ray-centred coordinates ( t , y ,,y,) are defined relative to the reference ray R(t).Coordinate t is time along the reference ray and y , , y, lie in the plane tangential to the wavefront at xo(t). A general position is written

x,(t,y , , Y 2 ) = xo,(t)+ K A t ) Y , , a = 1, 2,

(20)

where yi,(t)depends on precisely how the y, are chosen. See Hanyga [1982, equations (21)-(23)] and Section 3.2 for examples.

The eikonal or phase function is written t ( x ) = f ( t , y , , y,) and by the chain rule

In equations (21), % x i / & and axi/dya are found from (20). We note that on the reference ray itself the a f / 3 y a= 0 and
a t / & = 1, though on a general ray these vary. The role of ‘momenta’ in the ray-centred coordinates will be played by

and the three equations (21) may be written

Pb = Th&, y , I Y,)Pi, pi = %At, Y , . Y Z P h * b = t ? 1, 2.

(22b)

The relations (22b) may be inverted to give the Cartesian components of slowness p , in terms of the new momenta and as a

result of these transformations the original eikonal equation H(x,p) = 0 is transformed into the new equation

A(t,y , ,y,, E , PI, P,) = 0.

(23)

The method of characteristics applied to (23) yields the system of equations

-d=t A,, -d=Y,f i e ,

dv

dv

dv

_ dPt - - f i t ,
dv

- dP, = - H-y I ,
dv

_ dP2 - - H M ,
dv

_d f -- PtH, + PIHpl+ P 2 f i f i , (24)
dv

where f i e = a f i / a P , , etc. (Courant & Hilbert 1962, p. 78). The variable v may be, for example, time or arclength along the ray through (t,y , ,y,). On the reference ray itself we expect that d t / d v # 0 (since the wavespeed of the medium is finite). Hence in some vicinity of the reference ray we may divide the last six of equations (24) by the first and take t to be the independent variable. These six new equations may be integrated to yield y , , y 2 , PI, P I , P, and f(t, y , , y z ) at a given time t along the reference ray. However, of these new equations the one for PI is redundant since for given t , y , , y,, PI and P2 one may in principle obtain P, from (23) (one of the characteristic equations is always redundant in view of the original eikonal equation). Therefore instead of the first six of equations (24) it is sufficient to consider only the four

plus the equation for 5. Equations (25) are in Hamilton form for the time-dependent Hamiltonian
recognized by differentiating (23) at constant t to obtain, for example,

= -Pt(t, y , , y,, PI,P,). This may be

which is just the negative rhs of the first of equations (25). Thus (25) are in the Hamilton form

Reciprocity and ray -centred coordinates 369
The Hamiltonian of Popov & PSenEik [1976, equation (3.14)] is essentially F for the isotropic case, as was noted by Farra &
Madariaga (1987). The only difference is that they use arclength rather than time along the reference ray as one of the coordinates.
+ The geometrical spreading equations may be obtained by considering the two rays through (t,y , , y z ) and ( t , y , by,, y , + + Sy,). On these two rays the ray-centred coordinate momenta are (Pl, P2) and (PI+ bP,, P2 bP2). The leading-order equations
for y = ( b y l , by,, 6 P , , UP,) are obtained as usual by differentiating the system (27). As (27) is a Hamiltonian system the resulting geometrical spreading equations display the symplectic property, (Thomson & Chapman 1985). This fourth-order geometrical spreading system is written
!LAy
dt where
Equations (28) describe the geometrical spreading along a general ray, provided it is close enough to the reference ray for the ray-centred coordinates to be useful. The initial conditions may be of point-source or plane-wave type and the propagator-matrix solution of (28), W(r, 0), has the same elements as the Jacobian
a(Y1, Y Z , PI1 PZ) d(Y10, Y20, ploy PZO) ’ where subscript 0 indicates values at t = 0. For use in Section 3.3 and Thomson et a[. (1992), we define the 2 x 2 blocks of this propagator according to

[cf. ( 1 4 4 1 . The solution of the transport equation of ray theory involves the Jacobian a ( x , , xz, n,)/a(t, q l , q2), where gl,z are two
parameters which define a single ray. From the chain rule for Jacobians

The first Jacobian on the rhs is that corresponding to the transformation to ray-centred coordinates [see matrix T in (22b)l and the second-order Jacobian is obtained from the solution of the geometrical spreading equations (28). Note that in (31) we are using the time t along the reference ray on the Ihs, as opposed to the time or arclength along the actual ray in question. The latter is perhaps more usual in geometrical spreading considerations and if it were used instead, a slightly longer argument is needed in order to reduce the results to a second-order Jacobian on the right. See Popov & PSenEik [1976, equations (3.8) to (3.11), which are essentially unchanged in the anisotropic case]. In practice, it is the spreading along the reference ray itself which is of main interest.

3.2 Particular ray-centred coordinates

The slowness po and total velocity vo on the reference ray may be used to define two orthogonal vectors m and n in the wavefront according to

mk = Ek[mpOlHp,,,9

- n~= ErikPOjmk = (‘1) POjHp,)POj = p f ( t ) p O ] ~

(32)

where the identity crlkcklm= b,,b,, - 6, b,, has been used. These definitions break down in the isotropic case, when po and vo are parallel. However, we note that the operator P: on the rhs of (32) has the property p , P f = p o , -po,(po,vo,)= 0, since
po,uo,= 1 by definition. Thus, for fixed j , P; is a vector in the wavefront in either the isotropic or anisotropic case. The
operators P: were introduced by Hanyga (1982) (hence we use the same symbol). There are three such operators P’,, P i and P,,two of which suffice to define two directions in the wavefront. Hanyga [1982, equation (23)] has chosen the defining

relation for the ray-centred coordinates as

+ xJt) =x&) Pb(t)y,, a = 1, 2,

(33)

and then we have the Jacobian

370 J - M . Kendall, W. S. Guest and C . J . Thomson
[Hanyga (1982), equation (24)]. If H,,, = O another choice of two from the Pi must be used to obtain a well-defined transformation.
Evidently a choice yielding a non-zero value for the transformation Jacobian on the Ihs of (34) at one point on the ray may not do so at another point on the ray. It is of interest to enquire if a linear combination of the P; can be taken so that the Jacobian is never zero. However, such a combination has not been found.
Directions in the wavefront can be defined in other ways. For example, by taking the vector product of po with a unit vector along a Cartesian coordinate axis. The three wavefront vectors obtained this way have much the same properties as the Pi. A third alternative is to start from the normal and binormal for the ray (Mathews & Walker 1964, p. 409; Popov & PSenEik 1976). The projections of these two vectors in the wavefront can be constructed easily and intuitively it seems unlikely that these projections could be colinear in any reasonable situation.
The choice of ray-centred coordinates will not be considered further at this time and we return to the point source problem.
3.3 Matching with a point source in ray-centred coordinates
Two possible approaches to this problem will be described. It is understood that from now on all quantities are evaluated on the reference ray itself and that subscript zero indicates values taken at the point t = 0 on the reference ray. We take pOl and Po, as the parameters defining the rays leaving the source point [i.e. as q1 and q2 in (31) above] and introduce the notation [see
(31), (22b) and (30)j
Note that the matrix R is the 2 x 2 counterpart of the 3 X 3 submatrix R of equation (14a), and that R also has the reciprocity
property (14b).
Method I
Here the aim is to exploit equation (18) of Section 2.3, the result of matching in Cartesian coordinates. The task reduces to finding the relationship between the elements of the 3 x 3 Cartesian spreading matrix R,J= dx,/dp,,, and the elements of the
Jacobian 3 (35). From the chain rule and transformation (22b) at the source point

and the summation over pmn is cyclic over the three Cartesian components. The vector to,, has a simple interpretation obtained as follows: by taking the cross product of the second two columns of the defining relationship I = TOToone finds

and hence

top= det ITo[-' TolP= det ITo/-' uOP,

since the first row of TOis just vo (20,22b). As a result (37) and (35) lead to

det lTolf = det [Totdet IT[det lRl = E ~ ~ ~ = 9~ , ~ R ~ ~ R ~ ~ u ~ ~

(39)

Plm

where 3 was previously defined in equation (17a). Thus, f and R may be inserted in equation (18) of Section 2.3, giving the ray

solution at large distances in the ray-centred coordinates. Note that reciprocity is apparent in the second part of equation (39),

in view of the way To and T appear and the properties of R.

Method II
In this method the matching with the point source solution [Section 2.1, equation (l)] is performed directly in the ray-centred coordinates. The ray theory solution at a general point is written [see (3)]

and the aim is to determine I&

Reciprocity and ray-centred coordinates 371

As the source is approached one obtains from (29) that

+ R , = (a, a,P)t ojt'), i, j = 1, 2,

(41)

where the derivatives of Hamiltonian are evaluated at the source. The three second derivatives of P are found in terms of the
six derivatives of H (23) by differentiation of (26). This yields

where a subscript notation Hi, i = t, 1, 2, is used for derivatives. With (41) and (42) the Jacobian det IR( may be written in terms of derivatives of H at the source. After some cancellations and rearrangement one finds that

where the summation on plrn is cyclical over t , 1 and 2 and H represents the matrix of second derivatives of H. In order to relate (43) to the Gaussian curvature of the slowness surface at the source (equations 1, 2) we must pull back from the P, to the piusing (22b). From that equation at the source
Hi = H,,,Tnmi, Hij = H,,,,TomiTnnJ, i.e. T$-lTo,
and then from the last form of (43) it is apparent that

The denominator here may be shown to be just unity by forming the product of (38b) with Top,. Hence,
( det (R(= det lTol-2 pmn Hi€ijk?f,,,,Hk,Hp)t+2 o(t3)),

(Mb)

Hi, etc. evaluated at the source. Combining (44b) with (35) and then matching (40) with the point source solution (1) shows
q: that is just (4n)-2p;' det IT&' and we recover the same results as method I.

ACKNOWLEDGMENTS
This work was supported by an NSERC Operating Grant, an Energy, Mines & Resources Research Agreement and a University Research grant from Imperial Oil/Esso Resources Canada. JMK is supported by an Amoco Canada Graduate Scholarship and WSG is supported by an NSERC Graduate Studentship. The authors gratefully acknowledge careful and constructive reviews from Richard Coates.

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