Geophys. J. R. astr. SOC. (1985) 83,93- 1 10

Gaussian beams, complex rays, and the analytic extension of the Green’s function in smoothly inhomogeneous media

* Ru-S han W U Earth Resources Laboratory, Department of Earth, Atmospheric and Planetary Sciences, Massachusetts Institute of Technology, Cambridge, MA 02139, USA

Acceptcd 1985 Iebruary 3. Received 1985 January 31; in original form 1984 June 15

Summary. Some relations between Gaussian beams, complex rays and the analytic extension of the Green’s function in smoothly inhomogeneous media are shown in this paper. It is found that: (1) a single Gaussian beam is a paraxial approximation of the analytical extension of the ray-approximated Green’s function in smoothly inhomogeneous media by putting the source point into a complex space. The Gaussian beam approximation of the Green’s function has an advantage in computational efficiency and stability and can avoid the singularity problems at caustics. but also introduces a parabolic approximation t o the wavefront and an angle-dependent amplitude damping. Therefore the validity of the Gaussian beam approximation should be checked using other methods. (2) Complex-ray tracing, which does not involve the paraxial approximation. can also avoid the singularity problems, though without the computational efficiency. Therefore, it should be used to verify the Gaussian beam approximation, whenever possible. (3) The decomposition of a plane wave into an ensemble of Gaussian beams is equivalent to approxi- mating the Green’s function (the kernel of the ray-Kirchhoff method) with a single Gaussian beam. This introduces a parabolic approximation to the wave- front and a Gaussian windowing for arrival angles which may cause some problems in modelling wave propagation and scattering and has no advantages over other methods. (4) The representation of a point source field by a super- position of Gaussian beams, on the other hand, is equivalent to approxiniating the Green’s function with a bundle of overlapped Gaussian beams. This representation is similar to a Maslov uniform asymptotic representation. It has n o caustics and has improved accuracies a t the caustics for quasi-plane waves compared to the extended WKBJ method.

1 Introduction

The concepts of complex ray and complex source-points have been introduced and used in electrical science and optics for more than two decades (Keller 1958; Seckler & Keller * Perm anent address: Institute of Geophysical and Geochemical Prospecting, Baiwanzhuang, Peking, China.

94 R. -S. WU

1959a,b; Keller & Streifer 1971; Deschamps 1971; Felsen 1975, 1976; for review see also Kravtsov 1067, Deschamps 1972 and Felsen 1984). On the other hand, not until quite recently has the Gaussian beam approach been proposed and applied to the modelling of seismic wave propagation in smoothly inhomogeneous media(6erveny 198 1 ; Cerveny, Popov & PSenEik 1982: Popov 1982; Cerveny & PSenEik l983a, b; terveny 1983; Nowack & Aki 1984). Here, following Felsen (1984), 1 try t o demonstrate some relations between the analytic extension of the Green’s function, complex rays, and Gaussian beams, through which more insight might be gained on the applicability and limitations of the Gaussian beam method to seismic wave problems.

In an homogeneous medium the analytic extension (or analytic continuation) of a Green’s function to one of a complex source point results in a Gaussian beam in the paraxial region (Felsen 1976). In the case of a smoothly inhomogeneous medium, the analytic extension of the ray-approximated Green’s function has to be done along rays. In the vicinity close to a ray, the analytically extended Green’s function will be shown to be a Gaussian beam. However, unlike the case of homogeneous media. the Gaussian beam will evolve not only by beam spreading (similar to the geometric spreading for spherical waves), but also according t o the velocity variation of the medium. The relation between the initial beam parameters and the corresponding complex location of the source will be found. Therefore, the equivalence of these two representations will be established in this particular case. From this viewpoint, the relations between the method of Gaussian beam summation and the methods of Kirchhoff summation, extended WKBJ, and the Maslov uniform asymp- totic representation will be discussed. I t is found that the two numerical modelling algorithms using Gaussian b e a m for the study of wave propagation in inhomogeneous media, the Gaussian beam summation for plane waves and that for point sources, have quite different features. In the case of plane wave incidence, the procedure of Gaussian beam summation approach is equivalent to a beam-Kirchhoff summation method, which is an approximation t o the ray-Kirchhoff or complex-ray-Kirchhoff summation method and may have some problems in modelling wave propagation and scattering. In the case of a point source, the Gaussian beam summation is similar t o a Maslov uniform asymptotic representation.

2 Homogeneous media b

2.1 P A R A X I A L A P P R O X I M A T I O N T O T H E G R E E N ’ S F U N C T I O N O F C O M P L E X

S O U R C E P O I N T : G A U S S I A N BEAMS

For the purpose of demonstration, only 2-D problems will be discussed. In an homogeneous medium, the Green’s function is defined by the scalar wave equation

( V 2 t k 2 ) G ( r , r ’ ) = - & ( r - r ’ ) . (2.1) and can be written as

which has the asymptotic form

(2 .2)

where r = (x, z ) , r‘ = (x’, z’) are the position vectors for the sensor and source respectively. By putting the source location into a complex space, we can derive the analytic extension of

Gaussian beams in inhomogeneous media 95 the Green’s function, which in the paraxial region will approximately be equal to a Gaussian beam (Deschainps 1971; Felsen 1976, 1984). For the special case r ’ = (0, ib), i.e. x‘= 0, z ’= ib, the analytically extended Green’s function G (r, r‘) can be approximated in the paraxial region as

G ( r , ib) = ! exp ( ikb t i :) 4 rrkD

~ ! ~ ~ e x p ( i a t k b ) e x p [ i k z ~ ~ t XZ

4 nk(z - ib) 3 (z’ t h’)

(2.4)

Here i denotes a complex location, D a complex distance, etc. G denotes the Green’s function with a complex source point. In the derivation the paraxial approximation of distance

b = J(z - . z‘)’ + (x - x’)’ = J(z ~ ib)’ t x’

X’ ZX’ ibx’ = ( z -ib)+-- = ( z - ib) t - ~ + - -

2 (z’ t b2) 3 (z’ t b2) ’ 2 ( z -- ib) (1.5)

has been used (for details see Felsen 1976, 1984), which is based on the assumption that z > x, i.e. only the region close to the z-axis is under consideration. Therefore the paraxial approximation (3.4) of G(r, ib) is a Gaussian beam, which has a Gaussian decay along the perpendicular direction to the axis.

The initial beam at z = 0 has a constant phase front and a Gaussian amplitude profile

Thus 2b equafs the initial beamwidth normalized by the wavelength. The initial phase structure is not necessarily constant. If we assume the complex source point as

i.‘ = (0, - d t ib), (2.7) we obtain

11 XZ exp i - t k b exp i k ( z + d ) I+--- { I 2 [ ( ~ + d ) ~ t b ~ ] 4 ’ r- nk(z t d - ib) ( 1 6 (r, - d t ib) = -

The initial beam at z = 0 has a phase profile

and an amplitude profile

I d I ( z = O ) m e x p

(2.8)

(2.9)

(2.10)

In this case, the initial beam width is 2 (b + dz /b) , and the initial radius of curvature is

96 R. -S. WU

(d + b2/d). Later i t will be shown that J2b/k and d correspond t o the initial beam parameters LM and So in terveny et al. ( 1 982).

The concept of an analytic extension of the Green’s function, introduced by Deschainps ( I97 1 ) and Felsen ( 1 975, 1976), opens a wide range of possible applications. If an initial wavefield. which may result from reflection or scattering by a simple curved interface, can be approximated by an initial beam field, then the solution of that problem would be simply given by (2.8) by substituting in d and b. Therefore, we do not need to solve the problem for that source distribution, we only need to find the corresponding complex source point.

In fact the initial phase profile needs not be symmetric either. If r‘ =(-ia, ib), we will get

- k 2axz + b (x2 - a’) =xi.{:- [ ( z 2 + j 7 y ] } .

- k 2axz + b (x2 - a’) =xi.{:- [ ( z 2 + j 7 y ] } .

At z = 0, the initial phase and amplitude profiles are

p h a s e ( G ) ( z = O ) m e x p - i k - x . 1 11 I G I (z = 01 m exp [ - k (v)]

(2.1 1)

(2.12)

The initial beam has a linear phase front. Another possible application is to use G as the approximate Green’s function wherever

the approximation is allowed, and take the computational advantages offered by C. The analytically extended G is an exact solution of the wave equation, and its paraxial

approximation (2.4) is a solution of the parabolic wave equation. From another point of view, (2.4) may be considered as an analytic extension to the Green’s function of the parabolic wave equation. Equations (2.3) and (2.4) are not necessarily high-frequency approximations (in the 3-D case, the Green’s function itself is in the form of an exponential function). However, in order to be consistent with the case of inhomogeneous media, we take (2.3) as the high-frequency ray approximation to the Green’s function in homogeneous media. Then in the region close to the z-axis (in other words having a small angle with the z-axis), we can take the paraxial approximation

2 (z - - z‘) G,, (I-, r() = ! JT exp ( i i) - exp { ik[(z - z‘) +

4 n k ( z - 2 ‘ ) (2.13)

Gpar is the paraxial approximation of the ray approximation and will satisfy the parabolic wave equation. If we d o the analytic extension to (2.13), i.e. put the source point into a complex space, e.g. x‘ = 0, z’ = ib, we will get Gpar

Gpar has the same form as (2.13), except with a complex curvature (and hence a complex geometric spreading). Therefore, it can be viewed as a bundle of complex rays in the paraxial approxima tion.

From the above analysis, we see that in the case of an homogeneous medium, the

Gaussian beams in inhomogeneous media 97

Gaussian beam can be looked at from three points of view: as the paraxial approximation to the analytic extension of the Green’s function, as the analytic extension to the parabolic approximation of the Green’s function and as a bundle of complex rays from a complex source point (as stated by Deschamps 1971). In smoothly inhomogeneous media, if we first find the approximate Green’s function by high-frequency asymptotic methods, the same procedure of analytic extension can be applied.

2.2 GAUSSIAN BEAMS A N D E X P O N E N T I A L BEAMS

It is interesting to note that, before the concept of the complex source point was introduced, Keller & Streifer (1971) had also treated a Gaussian beam as a bundle of complex rays, not from a source point on the imaginary z-axis but from the initial beam field on the complex transverse plane, which is a more complicated ray tracing system. As shown in Fig. l(a), the complex ray tracing is easy in an homogeneous medium for a complex source point. Since the ray trajectory is a straight line in complex space connecting the source point and sensor point, the complex distance D is simply b = d m w i t h i = z ~ ib. However, in Keller & Streifer’s (1971) procedure, ray tracings start from the points of the initial field on the complex x-axis. Suppose the initial field has a plane phase front and a Gaussian amplitude profile

k ix2 u(z = 0) m exp [; $ I = exp [i--( T)] = exp [ i@l . (2.15)

Considering it as a complex wavefront, at each point the local wavefront will have a different complex direction in the complex space. If we analytically extend (2.15) into a complex line g, then for each point g at z = 0, the complex direction cosine can be found as

(2.16)

Having the complex direction cosine at each point on the g-axis defined by (2.16), the tra- jectory is still a straight line in complex space. Assuming the complex distance is D, for each

( a ) ( b ) Figure 1. (a) A bundle of complex rays of Deschamps from a complex source point, representing the analytically extended Green’s function. The source is located at (0 , ib), the sensors are distributed along the x-axis at z. D is the complex distance from the source to the sensor. (b) A bundle of complex rays of Keller & Streifer, representing a Gaussian beam.

4

98 R. -S. WU

point x at z we can find a corresponding point ray-tracing equations

in the initial complex line at z = 0 by the

(2.17)

Solving equation (2.1 7), we get { b in terms of x, z, after making the assumption that z2 B x 2 , z2 s b2,

& = - ibx/z + 0 ( s - ~ )

xz ibx2 jj t ~- t ~ + o ( ~ - ~ ) .

2z z 2 (2.18)

Therefore Keller & Streifer derived a Gaussian beam as following from a initial Gaussian amplitude function by complex ray tracing:

(2.19)

I t IS also interesting to note that the initial source distribution on the iinaginaryx-axis is x- and z-dependent in Keller & Streifer’s formulation. If we put only one point source on the imaginary x-axis, i.e. put the source at (- iu. 0) in the complex space. we will not get a Gaussiaii beam, but an exponential beam. Since

(2.20)

(2.21)

The beam has an exponential amplitude decay along the real x-axis. For small a, (2.21) is approximately an inhomogeneous plane wave introduced by- Brekhovskikh (1 960), which grows in the negative x-direction while decaying in the positive x-direction. Note that, from Fig. I and (2.2 I ) , we see that Keller & Streifer obtained a Gaussian beam by a superposition of exponential beams.

3 Smoothly inhomogeneous medium

3.1 P A K A X I A L A P P R O X I M A T I O N O F A K A Y F I E L D I N T H E R A Y - C E N T R E D

C 0 0 R D I N A T E S Y S T E M

In an inhomogeneous medium we have to solve the beam propagation problem in order to evaluate the beam variation. There is, however, another approach. We can first find the approximate Green’s function in that medium, and then d o the analytic extension t o obtain the corresponding beam properties. Felsen & Shin (1975) have obtained the solutions for a transversely inhomogeneous 2-D dielectric waveguide excited by a Gaussian beam using this approach. In a 2-D or 3-D smoothly inhomogeneous medium, the approximation to the Green’s function can be obtained by high-frequency asymptotic methods such as the ray series method (Cerveny 1981). The lowest order of the ray series expansion is called the ray

Gaussian beams in inhomogeneous media 99

approximation. By solving the eikonal and transport equations, we can get the ray approxi- mation of the Green's function. In the vicinity of a ray, we can further approximate the ray approximation by the paraxial approximation, by which the value of the Green's function for a point close to a ray can be calculated from the curvature of the local wavefront along the ray. This approximation can be obtained by solving the wave equation in the 'ray-centred coordinate system' (Eel-veny 1981; terveny et al. 1982). In the 2-D case, for a scalar wave (in a smooth medium, the P-wave and the two orthogonal components of the S-wave are uncoupled in the ray-centred coordinate system t o the lowest order approximation), the solution is

(3.1)

where 4 is a complex constant, s is the distance along the ray, u ( s ) is the wave velocity along the ray, n is the transverse distance from the point on the ray, M(s) is the curvature of the local wavefront at s, and q ( s ) is the cross-section of the ray-tube. M ( s ) and q ( s ) can be derived through the dynamic ray tracing system (Popov & P:enEik 1978; terveny et nl. 1982).

3.2 A N A L Y T I C E X T E N S I O N O F T H E P A R A X I A L K A Y F I E L D : A G A U S S I A N B E A M

At this point, (3.1) is the paraxial approximation t o the ray asyniptotics.M(s) and q ( s ) are real. By this approximation, the solution at a point close to a ray can be calculated without doing additional ray tracing once the ray tracing (kinematic and 'dynamic') of the central ray has been done. To extend (3.1) analytically, we put the source point into complex space, then the q (s) and M(s) will become complex, but solution (3.1) will still be valid. Since q ( s ) and M ( s ) are complex, we have to d o both kinematic and dynamic ray-tracing in complex space. Suppose the source point is at (0 , - d t ib) and we start the ray along the z-direction, the complex travel time will be

where d - ib is the complex distance in the negative z-direction from the starting point so. The complex dynamic ray tracing system will be ( terveny er al. 1982)

q , s = up, p , s = - u-2u,..q. (3.2)

4 , S k =&). (3.3)

where

Eerveny et al. (1982) have shown that, since any solution p . 4 of (3.2) can be expressed as a combination of two fundamental solution pairs (pl, ql), and ( p 2 , q 2 ) , the complex system (3.2) can be solved by solving twice the real dynamic ray tracing system using different initial conditions. Therefore, the analytic extension of (3.1) will be

u (s, n , t ) = j , [ u (s)/$ (s)] l i 2 exp [iwuil d i- wui' b ]

(3.4)

100 R.S. WU

which is the same as a Gaussian beam (Cerveny et al. 1982, equations 2.7) except for a complex constant which can be incorporated in &.

The Gaussian beam in the form of (3.4) is not an explicit form as in the case of homo- geneous media (2.8). The parameters of a Gaussian beam (3.4) will change with the variation of the medium property (wave velocity) and have t o be derived through a dynamic ray tracing system; whereas a Gaussian beam in homogeneous media is completely specified by its initial parameters.

3.3 T H E R E L A T I O N BETWEEN T H E I N I T I A L BEAM P A R A M E T E R S A N D T H E

C O M P L E X S O U R C E L O C A T I O N

In order to find the relation between the complex source location and the initial beam parameters, we discuss the case of homogeneous media. In this case, u, nn = 0. From (3.2) we have

$ = P o f constant,

4 = [ upods + Q o = uP0(s - so> + Q0.

Substituting (3.5) into (3.4) with

k 1 q 0 - - - ibvpo = - ib,

= w = V ’

yields

(3.5)

w w 1 ~ ( s , n , r ) = $ [ ~ / q ] ’ / ~ e x p - i w t + i - ( s - s o ) + i -

U 2 [ u (s - so) -ibv]

n 2 }, (3.7) 2 ( z 2 + b 2 )

’” 1- kb

I 1”’ exp { - iwt + ik [ z +

= 2 [ 2

c w ( z - ib)nk 2 ( z 2 + b2)

in which we have combined some complex constant into 5/ and taken z = s - so. Comparing (3.7) with (2.4), we see that (3.7), is really the analytic extension of the Green’s function with the complex source point at (0, - ib). Equations (3.5) and (3.6) give the relation between Po, Q0 and the source location (0, - ib).

The Gaussian beam can also be expressed as ( terveny et a/. 1982, equation 29)

i w

2u - i w [ t - ~ ( s ) ] + - - ( s ) n ~ -

where K ( s ) is the curvature of the local wavefront, L (s) is the effective half-width of the beam. In the case of homogeneous media, compared with (2.8), resulting in

K ( z ) = 1

( z + d ) + b”(2 + d ) ’

(3.9)

We know that L (s) is related to the initial beam parameters Ro(- So of Cerveny’s notation)

Gaussian beams in inhomogeneous media

and L , through (cerveny et al. 1982, equation 3 8 )

101

(3.10)

Therefore we can obtain

(3.1 1 )

This means that the minimum effective beam width is at the source location. and the beam width there is determined by the imaginary distance from the source t o the start point of the beam. In fact, LM is the radius of the Fresnel zone at distance 26 from the source.

The proportional constant of the two fundamental solutions can be determined as

i.e. the complex distance from the complex source point to the start point in real space.

4 Discussion I , plane wave propagation: Gaussian beam summation and the Ray-Kirchhoff summation

Seismologists have introduced the concepts of complex frequency and complex propagation constants for different purposes. Complex propagation constants are good representations for lossy media. However, even for non-lossy media. introduction of a sinall imaginary part into the frequency or wavenumber often gives some improvements in stability or compu- tational efficiency with little loss of information. Some singularity problems in Fourier inte- gration can also be avoided by doing this. Now we can see that by introducing a small imagi- nary part into the spatial coordinates of the source. the same magical effects can happen. Because the source is located in a complex space, rays in a smoothly inhomogeneous media will be complex rays, and the curvatures of the local wavefronts along the rays will also be complex. The complex curvature is an important mechanism in making a Gaussian beam into a useful approximation of the Green’s function in the vicinity of a ray. A complex ray means the local wavefront has a complex direction. However, the concept of complex direction has been introduced by Brekhovskikh ( 1 960) thruugh the ‘complex angle of propagation direc- tion’. This leads t o the concept of an ‘inhomogeneous plane wave’. which has found use in representing surface waves or interface waves (see Aki & Richards 1980). However, this concept has not been found to be of much use in representing local wavefronts, since the inhomogeneous plane waves grow in one direction exponentially while decaying in the opposite direction. On the other hand, the complex curvatures will make the inhomogeneous waves (or evanescent waves) have Gaussian decay in both directions away from the ray, which is the necessary property of a beam.

By introducing the complex frequency, some improvements in stability and computa- tional efficiency have been obtained. The effect of the use of a complex frequency is equiva- lent t o a smoothing of the original spectrum (see Bouchon & Aki 197 1). Similarly, the intro- duction of a complex source point will have a smoothing effect in space of the ray approxi- mation of the Green’s function. From the analysis in Section 3, we can regard [he Gaussian beam (3.4) as a smoothed version or a defocused image of the paraxial approximation of the ray-approximated Green’s function in the space domain (we will discuss the case of the

1 02 R.3. WU

slowness domain in the next section). By introducing an imaginary part into a wave para- meter, we will lose information by the smoothing effects. However, we have gained conzpu- tational efficiency and stability. We save a lot of ray tracing; there is n o need for tinie- consuming two-point ray tracing; and in view of the smoothing effect we can also avoid the singularity problems at caustics. These advantages may be quite remarkable for some problems if we are not concerned about the details that may be lost by using this approach.

One possible application is to use the Gaussian beam as the approximate Green’s function in the Kirchhoff summation method. In a smoothly inhomogeneous medium, the kernel of the Kirchhoff integral may be approximated by ray solutions (Kravtsov & Feizulin 1969; Haddon & Buchen 1981). If both the medium and the initial wavefront are smooth enough, and the propagation distance is not too long, we may use a Gaussian beam t o approximate the Green’s function. In this case the ray-Kirchhoff method becomes a beam-Kirchhoff method. In fact, the procedure for modelling plane wave propagation using Gaussian beams, as described by terveny ( 1 %3), is equivalent to a beam-Kirchhoff method. Since the incident wave is a plane wave, its analytic extension will be approximately a Gaussian beam with the complex source point at (0, - 00 + ib). However, if the plane wave is taken as a superposition of Gaussian patches distributed on the wavefront, this is equivalent t o decomposing the wavefront into a distribution of secondary point sources with complex locations, which is the Huygen’s principle or its analytic extension. By summing up the contributions from all the secondary sources, the resulting wavefield is a Kirchhoff summation. In this way, the Gaussian beam method, which is equivalent t o a beam-Kirchhoff method, can be regarded as an approximation t o the ray-Kirchhoff method. Let us compare these two approaches in more detail.

4.1 R A Y - K I R C H H O F F S U M M A T I O N I N S M O O T H L Y I N H O M O G E N E O U S M E D I A

We start from the Kirchhoff integral (Baker & Copson 1950; Kravtsov & Feizulin 1969)

which is valid in inhomogeneous media. Here So is the plane surface at z = 0, n is the normal of the plane.uo is the initial field on the surface and G is the Green’s function in the inhomo- geneous medium. Since So is a plane surface, we can reduce (4.1) to a Rayleigh integral (Berkhout 1982)

where ro is a point on So. (4.2) can be obtained from (4.1) by introducing a mirror point source (with respect t o the plane So at z = 0) in the reflected medium which is the mirror image in the half-space z < 0 of the inhomogeneous medium in the half-space z > 0.

For a smooth medium, C ( r , ro) can be approximated by a ray solution

G (r, ro) = Gray (r, r01 = A (r, ro) exp [ iw T(r , roll, (4.3)

where A is the amplitude and T is the travel time along the ray. In the ray approximations, we assume A is a slowly varying function of z compared with oT. Therefore

a aT aT ~ G ( r , r o j = i w A - e x p ( i o T ) = i w - Gray(r,ro). a20 az0 az0 (4.4)

Gaussian beams in inhomogeneous media 103

In the paraxial region, we can expand (a/az,) T ( R ) = (a/azo) Z-(r, r o ) into a Taylor series around ro and only keep the first term,

a 1 -- T(r, ro ) = - cos [ 6 ( r , ro)], a Z o UO

(4.5)

where uo is the wave velocity a t ro , and B ( r , ro) is the angle between the z-direction and the direction of the ray at ro . Substituting (4.5), (4.4) into (4.2) yields

This is the ray-Kirchhoff summation in smoothly inhomogeneous media.

4.2 C O M P L E X - R A Y - K I R C H H O F F S U M M A T I O N A N D I T S P A R A X I A L

A P P R O X I M A T I O N : G A U S S I A N BEAM S U M M A T I O N

If we use the analytic extension Gray instead of Gray for the purpose of avoiding the caustics, (4.6) will turn into a complex-ray-Kirchhoff summation. The imaginary part of the starting positions for each ray will give some smoothing effect over the result, but the phase informa- tion will still be correct t o the ray approximation. However, if we use the Gaussian beam as the approximation t o Gray, both phase and amplitude distributions along the wavefront will be distorted. These losses are compensated by the computational efficiency. For each bundle of complex rays starting from one point on the initial plane, only one complex ray tracing (including the dynamic) along the beam axis is needed.

Fig. 2 shows schematically the comparison between the kernels (or weighting functions) of the two summation procedures. We can see that the Gaussian beam kernel is equivalent t o

phose, , G.B.

Figure 2. Phase and amplitude weighting functions of the ray kernel and the Gaussian beam kernel for the Kirchhoff summation method in a smoothly inhomogeneous medium.

104 R.-S. WU a kernel with a parabolic phase front and a Gaussian windowing for arriving angles (dip filtering). From (3.7) i t can be seen that the angular width of the dip filtering is approxi- mately equal to l / b when z s 6 . Therefore, the beam summation method is good only for the small angle forward scattering problem, but may not be appropriate for wide-angle scattering problems because of the phase error and the amplitude decay of the wide-angle scattered signals, when compared with the ray-Kirchhoff summation method.

5 Discussion 11, spherical wave propagation: Gaussian beam summation, EWKBJ and the uniform asymptotic representations of Maslov

5.1 S P H E R I C A L W A V E R A Y T R A C I N G A N D P L A N E W A V E R A Y T R A C I N G

We have discussed the analytic extension of the Green’s function in the space domain. The same procedure can also be done in the slowness domain (or the transform domain of the space domain). In homogeneous media, the Green’s function in the partial transform domain is a plane wave. In a smoothly inhomogeneous medium the ray approximation of the Green’s function in the partial transform domain is a quasi-plane wave or a ‘Snell wave’ as called by Claerbout (1978) (Chapman called it a ‘p-wave’ for the WKBJ method). This quasi-plane wave can be considered as resulting from ray-tracing the initial plane wave in a smooth medium. A more rigorous treatment is to use the Maslov asymptotic theory (Maslov 1971 or Kravtsov 1968; Chapman & Drummond 1982). but the results agree with this intuitive approach. In fact, the ray-tracing method applies t o any locally smooth wavefront. For a ray-approximated wavefield

14 (r) = A (r) exp [ioT(r)J, (5.1)

the eikonal equation

and the transport equation

2UA * V T + A U 2 T = 0

(5.2)

(5.3)

will totally determine the wavefield. We can see that, for the eikonal equation, we only need to know the velocity structure of the medium. For the transport equation, from which the amplitude information can be obtained, we only need to know the gradient and the curvature of the wavefront. Therefore, for an initial spherical wave from a point source, we can first decompse it into a superposition of plane waves (Weyl integral), then ray-trace these plane waves to the receiver and sum up the contributions from all the quasi-plane waves at the receiver (generalized Weyl integral) to obtain the space domain response. Another way to look at the problem is to decompose the wavefield at the receiver into a plane wave super- position. Then ray-trace back these plane waves to the source. The superposition of these quasi-plane waves should match the initial field, so that the coefficients of superposition at the receiver can be determined. The coefficients of the superposition at the receiver can also be determined by comparing the space domain solution with the plane wave superposition after evaluating the superposition integral at regular points by stationary phase method (Frazer & Phinney 1980). This concept of representing the space domain ray solution as a superposition of quasi-plane waves, each of which can be obtained by ray-tracing between the source and receiver, is the heart of WKBJ for laterally homogeneous media (Richards 1973; Chapman 1976, 1978; Dey-Sarkar & Chapman 1978) and EWKBJ (extended WKBJ) for laterally inhomogeneous media (Frazer & Phinney 1980; Sinton & Frazer 1982;

Gaussian beams in inhomogeneous media 105

Chapman & Druniinond 1982). The only difference between the point source ray-tracing and the quasi-plane wave ray-tracing is the initial condition, i.e. the starting curvature of the wavefront. However, this difference gives the two methods different locations of caustics. For the Earth the caustics for quasi-plane waves, which are a t the turning point of the ray, are usually a t depth, so that the ray of the quasi-plane wave is still regular a t the receiver (Chapman & Drummond 1982). The rays can be traced u p to the receiver by the connection formula, in which phase changes on passing through caustics are taken care of by the path index (Kravtsov 1968) (also called KMAH index, Chapman & Drummond 1982).

By summing up the contributions from all the quasi-plane waves at the receiver, we can get the point source field. At regular points the results agree with the point source ray-tracing as can be checked by integrating the superposition integral (slowness integral) using the stationary phase method. However, at caustics for the point source field, where the stationary phase method fails and therefore the point source ray-tracing fails, the results of the quasi- plane wave superposition will still be valid if we evaluate the integral either numerically or by combining it with the inverse Fourier transform into a time domain summation using Chapman’s method (avoiding the oscillatory integral by Fourier transform, see Chapman 1978 and Chapman & Drummond 1982).

5.2 U N I F O R M A S Y M P T O T I C R E P R E S E N T A T I O N O F M A S L O V A N D T H E G A U S S I A N

BEAM S U M M A T I O N

By the method of superposition of quasi-plane waves, the singularities at the caustics for point source ray-tracing have been removed. However, the singularities at the caustics of quasi-plane waves, which are called ‘telescopic points’ by Frazer & Phinney (1980) cause some new problems. On the other hand, these points are regular for the ray-tracing of the point source field. Therefore, a uniform asymptotic representation can be obtained by coni- bining these two methods using the Maslov asymptotic theory (Maslov 1965, 1972, 1981; Kravtsov 1968; Chapman & Drummond 1982). The uniform representation can be written as (Kravtsov 1968; Chapman & Drummond 1982)

(5.4)

where each ui (v) is one type of asymptotic expression in the corresponding subspace of the 6-D (three spatial dimensions and their corresponding transforms) phase space; v is the ray coordinates, e j ( v ) is the weighting function for the corresponding type of asymptotic expres- sion. The weighting function ei (v) is set to nearly one wherever the corresponding type of asymptotic expression is valid and set to zero in the region where the expression fails, and

2 ei(v) = I i For example, we can have

u ( ~ > = e l ( V ) U y l ( V ) + e z ( ~ ) u , p ( V ) , (5.5) where u p l ( v ) is the asymptotic expression using the superposition of quasi-plane waves and Usp (V) is the expression of the ordinary spherical wave ray-tracing. el ( v ) and e2 ( v ) will vary along the ray. Near the caustics for the point source ray-tracing, e2 (v);.. 0 and el (v)= 1 ; near the caustics for the plane wave ray-tracing, on the other hand, e2(v);.. 1, el(v);.. 0. In the intermediate region where both expressions are valid, ez (v) and el (v) vary smoothly. In this way, the uniform asymptotic expression ( 5 . 5 ) is valid everywhere. Note that EWKBJ solutions correspond to the case of e2 0.

106 R.-S. WU Now let us look at the analytic extension of the Green’s function or its paraxial approxi-

can be expressed as combinations of two sets of mation - Gaussian beam (3.4). p and fundamental solutions (Cerveny et al. 1982)

(5.6)

where p 1 and q 1 are the solutions for an initial plane wave, and p 2 , q 2 are that for an initial point source by setting

41 (so) = 1 1 PI (so) = 0

where so is the starting point of the ray, uo is the wave velocity at that point, and S is the complex proportional constant. Therefore the exponential term with the complex curvature in (3.4) can be written as

(5.8)

We see that, at regular points, a Gaussian beam is a combination of the plane wave and the point source field. At the caustics for the point source field, q 2 = 0, p /q = p l / q l + p 2 / i q q l , the curvature will be close to the case of the quasi-plane wave - the bigger S is, the closer the field is to a plane wavefield. On the other hand, at the caustics for the plane wavefield, q1 = 0 and p/Q = p 2 / q 2 + c p l / q 2 , the curvature is close to a point source field, the smaller 5 is, the closer the case is to a point source field. Therefore, a single Gaussian beam is an approximate representation of an asymptotic Green’s function in an intermediate domain between the space domain and its transform domain. By changing i, the domain can be moved close to one or the other. By choosing a big i, the Gaussian beam will represent the approximate Green’s function in the transform domain, a quasi-plane wave. Therefore the superposition of Gaussian beams with large 5 starting from the same point source and radiating to all directions will give similar results to the EWKBJ method. However, since the Gaussian beams have no caustics, and can move closer automatically to the space domain representation at the caustics for quasi-plane waves, the superposition is similar to a uniform asymptotic representation. By varying i, the accuracy near different kinds of singular points can be adjusted. For greater i the accuracies are better at caustics for the point source field, but become worse at caustics for the plane wavefield, and vice versa for smaller 6 .

Recently KlimeS (1984) has derived a uniform asymptotic expression using Maslov theory in a 3-D subspace of the 6-D phase space, which is the same as the superposition of Gaussian beams when the receiver surface is a plane surface. This gives a more rigorous proof about the relation between the representation of a point source field by a superposition of Gaussian beams and the uniform asymptotic representation of Maslov.

5.3 A U N I F O R M R E P R E S E N T A T I O N BY A S U P E R P O S I T I O N O F C O M P L E X R A Y S

The above analysis also applies from the viewpoint of the analytic extension of the Green’s function or complex rays. The analytically extended Green’s function with a complex source point at 2’ = 6 = - d + ib has a different directional pattern from the original one. The amplitude decreases quickly when the receiver moves away from the z-axis; the degree of decay depends on e. Therefore it is not a good approximation of the Green’s function,

Gaussian beams in inhomogeneous media 107 especially in the non-axial direction. To improve this, we can simulate the point source by an equivalent source distribution in a complex space, each of which has a different axial direction. In other words, we superpose an ensemble of an analytically extended Green’s function to represent the original one

where 6, is an analytically extended Green’s function with axis in the 0-direction. For 6,. consider a coordinate system (xl, z I ) with z1 in the direction of its axis (beam axis), where its paraxial approximation can be expressed as a Gaussian beam (2.8) in an homogeneous medium. By a coordinate transformation, it can be written in a common Cartesian coordinate system as

G, (r, 2 ; = - d + ib) = - J-’ ____ exp (kb + i :) 4 n k ( z c o s B + d - i b )

(x cos B - z sin 0)2

2 [(z cos 0 + d + x sin q2 + b2] x e x p I ik[(zcosfJ+xsinf3)+d] I+--------------- ~

1

(x cos e - z sin e )2 2 [z cos 0 + d + x sin 0 j2 + b2]

(5.10)

Putting x = 0, its contribution to the receiver on the z-axis is

d = f J ? - e x p i ~ b + i ~ ) . e x p { i k ~ z c o s e + d ) 4 nk ( z cos 0 + d - ib) z2 sin2 e

2 [(z cose+d)*+b2]

z 2 sin2 8

2 [(zcos 8 + d)’ + b2] x exp{ ~ itb

When d s z, and d B 6,

[ z~;;~~ e)} the phase term = exp ik (z cos 0 + d ) 1 + ~ ,

and the amplitude term

z 2 sin2 e 2d2

= exp { - kb -}. When d = 0, b P z , the phase term

= exp ( ikz cos 6 ( 1 +- z;$0)}7

and the amplitude term

;5: exp {- kT}. z 2 sin2

(5.1 1)

(5.12)

(5.13)

Both cases are approximations of plane waves. Therefore we assume d = 0 in the following. By setting a big b, i.e. a big beamwidth, (5.9) is approximately a plane wave superposition. On the other hand, a small b makes (5.9)only a spatial average of the point source field.

1 ox R. s. w21 From the viewpoint of complex rays. (5.9) represents the summation of complex rays. At

every point, the Green’s function is calculated by summing up all the complex rays from all the complex source points. This will require many more ray-tracings than the ordinary ray- tracing. The Gaussian beam method offers a fast method for calculating 60, making the superposition (5.9) a practical method. On tlie other hand, the viewpoints of analytic extension of Green’s function with complex source points and of complex rays can offer some new insights in the analysis of Gaussian beams. In addition, while the Gaussian beam field is a paraxial approximation, the complex ray field represents the ‘exact’ asymptotic solution of tlie complex point source field. Thus we can check the validity of the paraxial approximation in different problems using complex ray-tracing.

Here we only discussed the case of a smoothly inhomogeneous medium. For other types of heterogeneities. especially for different types of discontinuities, such as corners, edges, curved interfaces, critical reflections, etc., the corresponding canonical problems need to be examined by the complex source point method, as pointed out by Felsen (1984) (see also Choudhary & Felsen 1974 and Wang & Deschamps 1974).

6 Conclusions

( 1 ) In smoothly inhomogeneous media, the paraxial approxiination of the analytical extension of the ray-approximated Green’s function is a Gaussian beam. Therefore, a single Gaussian beam can be regarded as a smoothed version or a defocused image of the paraxial approximation of the ray-approximated Green’s function in the space domain. The Gaussian beam, as the approximation of the Green’s function, has great advantages in both computa- tional efficiency and stability. It can also avoid the singularity problem at caustics because of its smoothing effect. However. in view of its paraxial approximation and angle-dependent amplitude damping, the validity of the Gaussian beam approximation to the Green’s function depends on the particular problem concerned and should be checked using some other method. (2) The plane wave propagation using the Gaussian beam summation method is equivalent

to approximating the ray-Kirchhoff method with a beam-Kirchhoff method. The use of a single Gaussian beam as the approximation of the Green’s function (kernel) in the Kirchhoff summation will introduce a parabolic approximation to the wavefront and a Gaussian windowing for arriving angles, which may cause some problems in modelling wave propaga- tion and scattering. Therefore, tlie representation of a plane wave by a superposition of Gaussian beams should be used with caution and only for small angle forward scattering problems. It has no advantages over other methods.

(3) The Gaussian beam summation procedure for a point source field is equivalent to approxiniating the Green’s function with a bundle of overlapped Gaussian beams, which is sinlilar to a Maslov uniform asymptotic representation of the Green’s function. It has no caustics and has improved accuracies a t the caustics of quasi-plane waves over the EWKBJ method.

(4) For the heterogeneities of ;I discontinuous type, such as block structures, boundaries and interfaces, the responses of the Gaussian beam t o these structures can be studied by applying analytic extension to the Green’s function for the corresponding boundary value problems. In this way, the validity of the Gaussian beam summation method can be checked and the accuracy of the method can also be improved.

(5) Since a Gaussian beam is a paraxial approximation to the analytically extended Green’s function which can be represented by a bundle of complex rays, I propose here that, whenever possible, complex-ray tracing should be done to construct the extended Green’s

Gaussian beams in inhomogeneous media 109 function. The validity of the paraxial approximation can be verified by comparing the results from both Gaussian beam and complex ray methods.

In summary, the use of a single Gaussian beam as the approximation of the Green’s function in the modelling of wave propagation in inhomogeneous media is not robust and should be checked with other methods. On the other hand, representation of the Green’s function by a bundle of overlapped Gaussian beams is similar to a uniform asymptotic representation of the Mzslov type.

Acknowledgments

I a m grateful to Dr I. PknEik for his critical reading of the manuscripts and many interesting discussions during his visit to MIT. I thank also R. Nowack, Professors K. Aki and L. Felsen and Dr V. Cormier for their careful reading of the manuscript and helpful discussions. This research was partly supported by the VSP project of the Earth Resources Laboratory of the Massachusetts Institute of Technology funded by CGG (Compagie GCnCrale de Geophysique).

References

Aki, K . & Richards, P., 1980. Quantitative Seismology, Theory andMethods, vol. 1, W. H. Freeman. San

Baker, B. B. & Copson, E. T., 1950. The Mathematical Theory of Huygens’ Principle, 2nd edn, Oxford

Berkhout, A. J. , 1982. Seismic Migration, Zmaging of Acoustic Energy by Wave Field Extrapolation,

Bouchon, M. & Aki, K . , 1977. Discrete wave-number representation of seismic-source wave fields, Bull.

Frekhovskikh, L. M., 1960. Waves in LayeredMedia, chapter 4, Academic Press, New York. Cerven);, V., 1981. Seismic wave fields in structurally complicated media, Lecture Notes, Rijksuniverssiteit

cerven);, V,, 1983. Syntnetic body wave seismograms, for laterally varying structures by the Gaussian

&men);, V., Popov, M. M. & P;enEik, I . , 1982. Computation of wave fields in inhomogeneous media -

Cerven);, V. & PSenEik, I . , 1983a. Gaussian beams and paraxial ray approximation in three-dimensional

brveny, V. & PSenEIk, I . , 1983b. Gaussian beams in two-dimensional elastic inhomogeneous media,

Chapman, C. H., 1976. A first-motion alternative to geometrical ray theory, Geophys. Res. Lett., 3,

Chapman, C. H., 1978. A new method for computing synthetic seismograms, Geophys. J , 54, 481-518. Chapman, C. H. & Drummond, R., 1982. Body-wave seismograms in inhomogeneous media using Maslov

Choudhary, S. & Felsen, L. B., 1974. Analysis of GaUSSidn beam propagation and diffraction of inhomo-

Claerbout, J. R., 1978. Snell wavcs, Stanford Exploration Project, SEP 15, 57-71. Deschamps, G. A., 1971. Gaussian beams as a bundle of complex rays, Electronics Lett.. 7,684-685. Deschamps, G. A., 1972. Ray techniques in electromagnetics, Proc. ZEEE, 60, 1022- 1035. Dey-Sarkar, S. K. & Chapman, C. H., 1978. A simple method for the computation of body wave seismo-

grams, Bull. seism. SOC. Am., 68, 1577-1593. Felsen, L. B., 1975. Complex rays, Philips Res. Rep., 30, 187-195, special issue in honour of C. .I.

Bouwkamp. Felsen, L. B., 1976. Complex-source-point solutions of the field equations and their relation to the propa-

gation and scattering of Gaussian beams, Symposie Mathematics, Znst. Nationa Aka Matematica, XVIII, 40-56, Academic Press, London.

Francisco.

University Press, London.

A. Theoretical Aspects, 2nd edn, Elsevier, New York.

seism. SOC. Am., 6 1 , 259-277.

Utrecht, Vening-Meinesz Laboratory, Utrecht.

beam method, Geophys. J. R. astr. SOC., 13, 389-426.

Gaussian beam approach, Geophys. J. R. astr. SOC., 10, 109-128.

elastic inhomogeneous media, J Geophys., preprint.

Geophys. J. R. astr. SOC., 12,417-433.

153-156.

asymptotic theory, Bull. seism. SOC. Am., 12, 5277-5311.

geneous wave tracking, Proc. ZEEE, 62, 1530-1541.

110 R.-S. WU I,clscn, L. B.. 1984. Geometrical theory of diffraction. evanescent waves. conlplex rays and Gaussian

I,elsen, L. 13. & Shin, S. Y.. 1975. Rays. beams, and modes pertaining t o the excitation of dielectric wave-

I'razer, L. N. & Phinney, R. A.. 1980. The theory of finite frequency body wave synthetic seismograms in

Haddon. R. A. W. & Ruchcn, P. W., 1981. Use of Kirchhoffs formula for body wave calculations in the

Keller. J . B.. 1958. A geometrical theory o i diffraction, in Calculus of Variatioiis and its Applications.

Keller, J . B . & Karal 1.'. C., 1960.J. appl. Phys., 31, 1039-1042. Keller, J . B. & Streiier, W., 197 1. Complex rays with an application t o Gaussian beams, J. opt. Soc. A m . .

Kliine'z. L.. 1984. The relation between Gaussian beams and Maqlov asymptotic theory, preprint. Kravtsov. Yu. A.. 1967. Complex rays and complex caustics, Radiophys. Quantum Elecrronics. i0,

Kravtsov, Y u . A,, 1968. Two ne\v asymptotic methods in the theory of wave propagation in inhomo-

Kravtsov, Y u . A. & P'eizulin. Z. I., 1969. Some consequences of the Huygens-Kirchhoff principle for a

Maslov, V. P., 1965. Theory of Perturbations and Asymptotic Merhods (in Russian). Izd. MGV. Moscow. Maslov. V. P., 1972. Tlieorie des Perturbations el Methodes Asymptofiques. Dunod, Paris. Maslov. V. P.. 198 I . Semi-classical Approximation in Quantum Mechanics (trans. from Russian by Niederle

Nowack, R. & Aki, K., 1984. The 2-D Gaussian beam synthetic method: testing and application. J.

Popov, bl. M . , 1982. A new method of computation of wave fields using Gaussian beams. Wavehfofion.

Popov, M. M. & P'zcnPik. I . , 1978. Computation of ray amplitudes in inhomogeneous media with curved

Richards. P. G., 1973. Calculation of body waves for caustics and tunnelling in core phaces. Geoplzys.

Seckler, B. D. & Kellcr. J . B.. 1959a. Geonietrical theory o f diffraction in inhomogcncous media, J. acousf.

Seckler. 13. D. & Keller, J. B . , 1959b. Asymptotic theory of diffraction in inhomogeneous rnedia,J. acousf.

Sinton, J . B . & k'razcr, L. N., 1982. A method for thc computation of finite frequency body wave

Wang, W. Y . 0. L Deschamps, G. A., 1Y74. Application of complex ray tracing to scattering problems,

Ixams, Geopliys. J. R. astr. Soc.. 79, 17 - 88.

Fuides, IEEE Trans. Microwave fheor. Tech.. MTT-23, 150-161.

inhomogeneous elastic media, Geophys. J . , 63, 691-7 13.

earth. G e o p h j x J. R. asrr. Soc.. 67, 587 - 598.

Doc. Symp. appl. Math., 8, McGraw-Hill, New York.

61, 40 43.

1283 1304.

geneous media (review). Soviet Pliys. Acoust., 14, I - 17.

sniootlily nonuniform nicdiurn. Izv. VUZ. Radiofiz., 12, 886-893.

and Tolar), Reidel, Boston.

~ e o p h y s . Res., 89, 7797-7819.

4 ,85-97.

interfaces, Studia geophys. geod., 22, 248- 258.

J. R. astr. Soc., 35, 243 -264.

Soc. Am., 31, 192-205.

Soc. Am., 31, 2 0 6 - 316.

synthetic seismograms in laterally varying media, Gcophys. J. , 71, 37-55.

BOC. IEEE. 62,1541-1551.