Generalization of the Bremmer coupling series

Generalization of the Bremmer coupling series
Maarten V. de Hoopa)
Schlumberger Cambridge Research, High Cross, Madingley Road,
Cambridge CB3 OEL, England
~Received 20 September 1995; accepted for publication 25 March 1996!
An operator formalism is developed to expand the acoustic wave field in a multidimensionally smoothly varying medium, generated by a source localized in space
and time, into a sum of constituents each of which can be interpreted as a wave that
has traveled up and down with respect to a direction of preference a definite number of times. This expansion is a generalization of the Bremmer coupling series.
The condition of smoothness of the medium relates to the width of the signature of
the source in the configuration. Both the existence and the convergence ~in the
weak sense! of the expansion are discussed. The operator calculus involved leads to
a natural generalization of the concept of slowness surface to multi-dimensionally
smoothly varying media. The operator associated with the corresponding generalized vertical slowness induces the full one-way wave operator in the type of media
under consideration. In addition, a wavefield decomposition operator as well as an
interaction operator that couples the decomposed constituents, are derived.
© 1996 American Institute of Physics. @S0022-2488~96!01407-7#
I. INTRODUCTION
In recent years, there has been an increasing interest in the use of one-way ‘‘parabolic’’
approximations to the wave operator in the application of seismic modeling and migrationinversion techniques,1–3 and in the application of long-range waveguiding problems in ocean
acoustics4–6 and integrated optics.7,8 The parabolic approximation arises in the decomposition ~or
‘‘splitting’’! of the acoustic wave field into constituents that travel ‘‘up’’ and ‘‘down’’ with
respect to a given direction of preference, such that the two constituents satisfy coupled partial
differential equations of a specific type. In this paper, we shall discuss the mathematical theory
underlying this decomposition technique. The theory builds on the work of Seeley,9,10
Hörmander,11 and Duistermaat and Guilleman,12 and is based on the calculus of pseudodifferential operators. The use of such operators, in particular in the field of underwater acoustics
where it yields the factorization of the Helmholtz operator, has been noticed by Fishman and
McCoy,13–17 Fishman,18,19 Fishman and Wales,20 McCoy and Frazer,21 and Weston.22 The interaction of up- and downgoing constituents has been discussed by McCoy, Fishman, and Frazer.23
Within the parabolic approximations, Corones24 has put the interaction in the context of the
Bremmer series.
The direction of preference, which is assigned to the ‘‘vertical’’ direction, arises from the
medium’s variations. In its exact form, the decomposition procedure transforms the scattering
problem in n dimensions into a continuous family of ~n21!-dimensional problems, such that the
remaining scattering phenomenon can be solved with the aid of a Neumann series in the relative
vertical changes in the medium parameters. This series is a generalization to multi-dimensionally
varying media of the Bremmer coupling series that has been used in one-dimensional scattering
problems ~for an example, see Ref. 25!.
a!
Present address: Center for Wave Phenomena, Colorado School of Mines, Golden, Colorado 80401–1887.
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J. Math. Phys. 37 (7), July 1996
0022-2488/96/37(7)/3246/37/$10.00
© 1996 American Institute of Physics
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Maarten V. de Hoop: Generalized Bremmer coupling series
3247
The decomposition introduced applies to media which vary smoothly on the scale of the
irradiating pulse, and generalizes the standard decomposition to media that are no longer translationally invariant in the ‘‘horizontal’’ directions. Owing to this generalization, more refined mathematical tools are required, such as the calculus of pseudo-differential operators. The up/down
decomposition does not allow any further decomposition into horizontal, right/left, directions.
Media, in which discontinuities in their physical properties occur, should be smoothed on the scale
of the irradiating pulse with the aid of equivalent medium averaging prior to the decomposition.
With regard to media with discontinuities, we mention the alternative, quasi-decomposition into
up- and downgoing waves in the neighborhood of a ‘‘rough’’ interface separating two homogeneous half-spaces with the aid of the modified Rayleigh hypothesis.26,27
The key applications of the Bremmer series are ~i! an efficient way of numerically solving a
direct scattering problem, ~ii! identification of multiple scattered wave constituents, and ~iii! formulations of various inverse scattering procedures. Fast numerical schemes require sparse matrix
representations of the kernel associated with the relevant integral or pseudo-differential part of the
one-way wave operator in space domain. The properties of the kernel, however, are such that
generic bases in which its representation becomes sparse do not exist. Parabolic-type approximations of the kernel’s symbol, on the other hand, lead to possible sparsifications. The validity of
such approximations has been discussed in previous papers.2,3 They typically capture the precritical angle phenomena in the wave propagation. Beyond this regime, matrix representations for the
exact cokernel, acting in horizontal slowness space, have to be considered. A list of references to
the development and applications of parabolic theories can be found in Ref. 2. Approximations of
a different nature and with a different range of validity result from the method of phase screens.28
Other numerical procedures are based on constructing a spectral representation of the pseudodifferential part of the one-way wave operator and relate to normal-mode summation.
The solution of the direct scattering problem in the form of a Bremmer series allows one to
identify or predict multiple scattered constituents in the configuration. Applying this process to
physical measurements, however, requires some knowledge about the medium in which the experiment has been carried out. In fact, the Bremmer series yields an expansion of the acoustic
wavefield in terms of the spatial derivatives of the medium properties, as opposed to an expansion
in the medium’s contrast with respect to a given embedding through a contrast-source integral
representation. The leading term in the former expansion is a high-frequency ~Rytov-like! approximation to the wavefield; in the latter expansion, the leading term is the ~distorted! Born approximation in the embedding. It is noted that, once the former procedure has led to a construction of
the Green’s function in the embedding, the latter procedure can be applied to the contrast ~possibly
with discontinuities!.
The Bremmer coupling series essentially recomposes the solutions of the system of coupled
one-way wave equations into a two-way solution. As such, it connects the one-way wave formulation of scattering to the Dirichlet-to-Neumann map formulation ~see also Refs. 29 and 30!, and
also yields a solution of the associated invariant imbedding equations. We note that the decomposition of the direct scattering problem is an integral part of the layer stripping approach to the
inverse scattering problem ~see, for example, Ref. 31 for the one-dimensional formulation and
Ref. 32, for a multi-dimensional formulation!. In fact, the Bremmer series representation allows
one to link the asymptotic single-scattering approach ~see, for example, Ref. 33 and 34! with the
mentioned multiple-scattering approach.
The Bremmer coupling series becomes a powerful tool in those configurations in which the
complexity of the medium is such that ray-theoretic approaches become intractable or the approximation by homogeneous horizontal layers breaks down.
The remainder of this paper is organized as follows. In Sec. II, the principle of directional
decomposition is explained. In Sec. III, the decomposition problem is related to the solution of an
elliptic problem in one dimension less than the original scattering problem. In Sec. IV, the original
system of two-way wave equations is transformed into a system of coupled one-way wave equa-
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Maarten V. de Hoop: Generalized Bremmer coupling series
tions. The one-way wave equations define a generalization of the concept of slowness surface,
which is discussed in Sec. V. In Sec. VI, the fundamental properties of the Green’s functions of
the one-way equations are derived. These functions are used in Sec. VII to transform the system
of one-way integro-differential equations into a system of integral equations, which is then solved
in terms of a Neumann series expansion. Finally, presented in Sec. VIII is a series expansion for
the generalized slowness surface that yields an explicit solution of the elliptic problem posed in
Sec. III. Section IX concludes the paper.
We note that our analysis differs slightly from the standard mathematical treatment of factorizing differential operators, since in our case the ~n21!-dimensional ~‘‘horizontal’’! space is not
assumed to be compact. However, with regard to the numerical implementation of the theory,
periodic boundary conditions may be imposed in the horizontal directions. The causal acoustic
waves are well defined on T n21 3R, where T n21 denotes the ~n21!-dimensional torus, for a
finite time window.
II. DIRECTIONAL DECOMPOSITION OF THE ACOUSTIC SCATTERING PROCESS
In each subdomain of the configuration where the acoustic properties vary continuously with
position, the acoustic wavefield satisfies the hyperbolic system of partial differential equations
] k p1 r] i v k 5 f k ,
~II.1!
k ] t p1 ] r v r 5q,
~II.2!
where p5acoustic pressure ~Pa!, v r 5particle velocity ~m/s!, r5volume density of mass ~kg/m3!,
k5compressibility ~Pa21!, q5volume source density of injection rate ~s21!, f k 5volume source
density of force ~N/m3!, and $ x 1 ,x 2 ,x 3 % are the right-handed, orthogonal, Cartesian coordinates, t
is the time, and the subscript notation and the summation convention for Cartesian tensors are
employed. We assume that the coefficients r and k are smooth, i.e., infinitely differentiable
functions of position, and time independent. Furthermore, we assume that these functions are
constant outside a sphere of finite radius. This provision enables us to formulate the acoustic wave
propagation, when necessary, as a scattering problem in a homogeneous embedding. The smoothness entails that the singularities of the wavefield ~in particular the ones on the wavefront! arise
from the ones in the signatures of the source distributions. Further, causality of the wave motion
is enforced. This implies that if the sources that generate the wavefield are switched on at the
instant t50, the wavefield quantities satisfy the initial conditions
p ~ x m ,t ! 50
for t,0
and all
xm ,
~II.3!
v r ~ x m ,t ! 50
for t,0
and all
xm .
~II.4!
Due to the time invariance of the medium, the causality of the wave motion can also be taken into
account by carrying out a one-sided Laplace transformation with respect to time and requiring that
the transform-domain wave quantities are bounded functions of position in all space when the time
Laplace-transform parameter s, which is in general complex, lies in the right half Re$ s % .0 of the
complex s plane. The limiting case of sinusoidal oscillations of angular frequency vPR is covered
by considering the limiting case s→iv , in which i is the imaginary unit, the limit being taken via
Re$ s % .0. In view of Lerch’s theorem,35 however, it is sufficient to consider values with Im$ s % 50
and s>s 0.0; s 0 will be specified at several stages in the analysis.
To show the notation, we give the expression for the acoustic pressure,
p̂ ~ x m ,s ! 5
E
`
t50
exp~ 2st ! p ~ x m ,t ! dt.
~II.5!
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Maarten V. de Hoop: Generalized Bremmer coupling series
3249
Under this transformation, assuming zero initial conditions, we have ]t →s. The transformed
system of first-order equations follows from Eqs. ~II.1! and ~II.2! as
] k p̂1s r v̂ k 5 f̂ k ,
~II.6!
s k p̂1 ] r v̂ r 5q̂.
~II.7!
The change of the wavefield in space along a direction of preference can now be expressed in
terms of the changes of the wavefield in the plane perpendicular to it. The direction of preference
is taken along the x 3 axis ~or ‘‘vertical’’ axis! and the remaining ~‘‘horizontal’’! coordinates are
denoted by x m , m51,2. The procedure requires a separate handling of the horizontal components
of the particle velocity. From Eqs. ~II.6! and ~II.7! we obtain
v̂ k 52 r 21 s 21 ~ ] k p̂2 f̂ k ! ,
~II.8!
leaving, upon substitution, the matrix differential equation
~ ] 3 d I,J 1sÂ I,J ! F̂ J 5N̂ I ,
I,JP $ 1,2% ,
~II.9!
in which the elements of the acoustic field matrix are given by @in Eq. ~II.7! r5n,3#
F̂ 1 5 p̂,
~II.10!
F̂ 2 5 v̂ 3 ,
~II.11!
the elements of the acoustic system’s operator matrix are given by
Â 1,15Â 2,250,
~II.12!
Â 1,25 r ,
~II.13!
Â 2,152s 21 ] n ~ r 21 s 21 ] n ! 1 k ,
~II.14!
and the elements of the notional source matrix by
N̂ 1 5 f̂ 3 ,
~II.15!
N̂ 2 52s 21 ] n ~ r 21 f̂ n ! 1q̂.
~II.16!
It is observed that the right-hand side of Eq. ~II.8! and Â I,J contain spatial derivatives with respect
to the horizontal coordinates only. Further, it is noted that Â 1,2 is a multiplicative operator,
whereas Â 2,1 is a partial differential operator. Equation ~II.9! is sometimes called the two-way
wave equation ~Ref. 36!.
To be able to solve the scattering process along the vertical direction separately from the
scattering process in the ~family of! planes perpendicular to it, we decouple the two operators on
the left-hand side of Eq. ~II.9!. This procedure will possibly lead to an additional source term on
the right-hand side that accounts for the coupling. To achieve this, we shall construct an appropriate linear operator L̂ I,J with
F̂ I 5L̂ I,J Ŵ J
~II.17!
that, with the aid of the commutation relation
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Maarten V. de Hoop: Generalized Bremmer coupling series
FIG. 1. Directional decomposition.
~ ] 3 L̂ I,J ! 5 @ ] 3 ,L̂ I,J #
~II.18!
~@.,.# denotes the commutator!, transforms Eq. ~II.9! into
L̂ I,J ~ ] 3 d J,M 1sL̂J,M ! Ŵ M 52 ~ ] 3 L̂ I,J ! Ŵ J 1N̂ I ,
~II.19!
so as to make L̂J,M , satisfying
Â I,J L̂ J,M 5L̂ I,J L̂J,M ,
~II.20!
a diagonal matrix of operators. We denote L̂ I,J as the composition operator and Ŵ M as the wave
matrix. The elements of the wave matrix represent the local weights of the down- and upgoing
constituents ~see also Fig. 1!. The expression in parentheses on the left-hand side of Eq. ~II.19!
represents the two so-called one-way wave operators ~Ref. 36!. The first term on the right-hand
side of Eq. ~II.19! is representative for the scattering due to variations of the medium properties in
the vertical direction. The scattering due to variations of the medium properties in the horizontal
directions is contained in L̂J,M and, implicitly, in L̂ I,J .
To investigate whether solutions (L̂ I,J ,L̂J,M! of Eq. ~II.20! exist, we introduce the column
matrix, or generalized eigenvector, operators L̂ (6)
according to
I
L̂ ~I 1 ! 5L̂ I,1 ,
~II.21!
L̂ ~I 2 ! 5L̂ I,2 .
~II.22!
Upon writing the diagonal elements of L̂J,M as
L̂1,15Ĝ~ 1 ! ,
~II.23!
L̂2,25Ĝ~ 2 ! ,
~II.24!
Eq. ~II.20! decomposes into the two systems of equations
Â I,J L̂ ~J1 ! 5L̂ ~I 1 ! Ĝ~ 1 ! ,
~II.25!
Â I,J L̂ ~J2 ! 5L̂ ~I 2 ! Ĝ~ 2 ! .
~II.26!
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Maarten V. de Hoop: Generalized Bremmer coupling series
3251
By analogy with the case where the medium is translationally invariant in the horizontal directions, we shall denote Ĝ~6! as the vertical slowness operators. Notice that the operators L̂ ~6!
1
compose the acoustic pressure and that the operators L̂ ~6!
2 compose the vertical particle velocity,
whereas the elements of Ŵ M may be physically ‘‘nonobservable.’’
~6!
Through mutual elimination, the equations for L̂ ~6!
1 and L̂ 2 can be decoupled as follows:
Â 1,2Â 2,1L̂ ~16 ! 5L̂ ~16 ! Ĝ~ 6 ! Ĝ~ 6 ! ,
~II.27!
Â 2,1Â 1,2L̂ ~26 ! 5L̂ ~26 ! Ĝ~ 6 ! Ĝ~ 6 ! .
~II.28!
The partial differential operators on the left-hand sides, which are given by
Â 2,1Â 1,252s 21 ] n ~ r 21 s 21 ] n ~ r ! …1 k r ,
~II.29!
Â 1,2Â 2,152 r s 21 ] n ~ r 21 s 21 ] n ! 1 r k ,
~II.30!
are strongly elliptic in the horizontal plane R2 for each value of the vertical coordinate x 3PR and
all frequencies s under consideration; they differ from one another in case the volume density of
mass does vary in the horizontal directions. To ensure that nontrivial solutions of Eqs. ~II.27! and
~II.28! exist, one equation must imply the other. To construct a formal solution, an Ansatz is
that restricts the
introduced in the form of a commutation relation for one of the components L̂ (6)
J
freedom in the choice for the other component. Three choices will be considered.
A. Acoustic-pressure normalization analog
Our first Ansatz assumes that L̂ ~6!
2 can be chosen such that
@ L̂ ~26 ! ,Â 2,1Â 1,2# 50.
~II.31!
In view of Eq. ~II.28!, the Ĝ~6! must then satisfy
Â 2,1Â 1,22Ĝ~ 6 ! Ĝ~ 6 ! 50.
~II.32!
(6)
The commutation relation for L̂ ~6!
follows as [Â 21
1
1,2 L̂ 1 ,Â 2,1Â 1,2#50 and a possible solution of
Eqs. ~II.25! and ~II.26! is
L̂ ~26 ! 5Ĝ~ 6 ! ,
L̂ ~16 ! 5Â 1,2 .
~II.33!
as given by Eq. ~II.33! satisfies Eq. ~II.31!, the Ansatz is justified. In view of the
Since L̂ ~6!
2
up/down symmetry, the solutions of Eq. ~II.32! are written as
Ĝ~ 1 ! 52Ĝ~ 2 ! 5Ĝ5Â1/2,
~II.34!
where Â[Â 2,1Â 1,2. Thus, the composition operator becomes
L̂5
S
Â 1,2 Â 1,2
Ĝ
2Ĝ
D
~II.35!
.
In terms of the inverse vertical slowness operator, Ĝ215Â21/2, the decomposition operator then
follows as
L̂ 21 5
1
2
S
Â 21
1,2
Ĝ21
Â 21
1,2
2Ĝ21
D
.
~II.36!
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Maarten V. de Hoop: Generalized Bremmer coupling series
In this normalization, the elements of the wave matrix correspond with pressures up to the action
of Â 1,2; pressures are typically measured with hydrophones.
B. Vertical-particle-velocity normalization analog
A second Ansatz assumes that L̂ ~6!
1 can be chosen such that
@ L̂ ~16 ! ,Â 1,2Â 2,1# 50.
~II.37!
Then, Ĝ~6! must satisfy @cf. Eq. ~II.27!#
Â 1,2Â 2,12Ĝ~ 6 ! Ĝ~ 6 ! 50,
~II.38!
and a possible solution of Eqs. ~II.25! and ~II.26! is
L̂ ~16 ! 5Ĝ~ 6 ! ,
L̂ ~26 ! 5Â 2,1 ,
~II.39!
which satisfies the second Ansatz. The solutions of Eq. ~II.38! are written as
Ĝ~ 1 ! 52Ĝ~ 2 ! 5Ĝ5Â1/2,
~II.40!
where Â[Â 1,2Â 2,1. Thus, the composition operator is given by
L̂5
S
Ĝ
2Ĝ
Â 2,1 Â 2,1
D
~II.41!
.
Now, the decomposition operator becomes
L̂ 21 5
1
2
S
Ĝ21
Â21 Â 1,2
2Ĝ21
Â21 Â 1,2
D
.
~II.42!
In this normalization, Â 2,1 acting on the elements of the wave matrix results in vertical particle
velocities; particle velocities are typically measured with geophones.
C. Vertical-acoustic-power-flux normalization analog
It will appear to be advantageous to consider a third Ansatz, viz., the one arising from the
acoustic-power-flux normalization. For this, the commutation relation
1/2
1/2
~6!
@ Â 21/2
1,2 L̂ 1 , Â 1,2Â 2,1Â 1,2 # 50
~II.43!
is imposed on L̂ ~6!
1 . Then the vertical slowness operators must satisfy the equation
1/2
~6! ~6!
Ĝ 50.
Â 1/2
1,2Â 2,1Â 1,22Ĝ
~II.44!
1/2
1/2 21
] n „r 21 s 21 ] n ~ r 1/2 ! …1 k r ,
Â[Â 1/2
1,2Â 2,1Â 1,252 r s
~II.45!
Note that the operator Â,
is self-adjoint with respect to the standard real L 2 inner product in ~almost all of! L 2. A possible
solution of Eqs. ~II.25! and ~II.26! is now given by
L̂ ~16 ! 5 ~ Â 1,2/2! 1/2~ Ĝ~ 1 ! ! 21/2,
L̂ ~26 ! 56 ~ 2Â 1,2! 21/2~ Ĝ~ 1 ! ! 1/2.
~II.46!
The solutions of Eq. ~II.44! are written as
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Maarten V. de Hoop: Generalized Bremmer coupling series
Ĝ~ 1 ! 52Ĝ~ 2 ! 5Ĝ5Â1/2.
3253
~II.47!
Thus, the composition operator is given by
L̂5
1
&
S
21/2
Â 1/2
1,2Ĝ
21/2
Â 1/2
1,2Ĝ
1/2
1/2
Â 21/2
2Â 21/2
1,2 Ĝ
1,2 Ĝ
D
~II.48!
.
This composition operator L̂: (L 2 ) 2 →(L 2 ) 2 is normalized in the sense that @cf. Eq. ~II.46!#
L̂ T JL̂5
S
I
0
0
2I
D
~II.49!
,
with
J5
S D
0
I
I
0
~II.50!
.
This normalization establishes the connection with asymptotic ray theory in the vicinity of the
wavefronts.
Using Eq. ~II.48!, we can map the pressure to the vertical particle velocity, viz.,
F̂ 2 5Ŷ~ 1 ! F̂ 1
if Ŵ 2 50,
~II.51!
F̂ 2 5Ŷ~ 2 ! F̂ 1
if Ŵ 1 50,
~II.52!
21/2
6Ŷ~ 6 ! 5Ŷ5Â 21/2
1,2 ĜÂ 1,2
~II.53!
where
has the interpretation of admittance operator. The latter operator discriminates the decomposed
constituents. Note that F̂ T1 F̂ 2 represents the vertical component of the Poynting vector.
The decomposition operator becomes
L̂ 21 5
1
&
S
Ĝ1/2Â 21/2
1,2
Ĝ21/2Â 1/2
1,2
Ĝ1/2Â 21/2
2Ĝ21/2Â 1/2
1,2
1,2
D
.
~II.54!
It is observed that all the operators involved can be directly constructed from Â21/4, viz.,
Ĝ21/25Â21/4, Ĝ1/25Â~Â21/4!3, and Ĝ5Â~Â21/4!2. All these powers of Â are self-adjoint in ~almost
all of! L 2 as well as positive definite, since it has been assumed that Im$ s % 50 and Re$s%.0.
Apparently a whole class of composition operators L̂ I,J , all leading to different representations of the scattering process in the horizontal space, exists. The final results for the acoustic
pressure and the vertical particle velocity, however, will not depend on a particular choice: in
terms of observables the decomposition yields
F̂ ~I 1 ! 5L̂ ~I 1 ! ~ L̂ 21 ! 1,J F̂ J ,
~II.55!
F̂ ~I 2 ! 5L̂ ~I 2 ! ~ L̂ 21 ! 2,J F̂ J .
~II.56!
For practical applications, one adjusts the normalization to the sensors being used; it is quite
common that only one of the two relevant field components is being measured.
At this point, it is emphasized that Ĝ~6! are still unknown. It is noted that the key property we
have used so far is that the diagonal of the system’s operator matrix Â vanishes. For the evaluation
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Maarten V. de Hoop: Generalized Bremmer coupling series
of Ĝ~6!, and L̂ (6)
J , we need to introduce ~fractional! powers of an elliptic partial differential
operator in R2 for each value of x 3PR @cf. Eq. ~II.32!, ~II.38!, or ~II.44!#. How this can be done
will be discussed in Sec. III. In this respect it is noted that only a proper definition of a negative
~fractional! power is needed, since positive fractional powers are constructed from the negative
ones through the application of the operator itself an appropriate number of times. For the time
being, the vertical coordinate will play the role of parameter, which will be indicated by writing
R35R23R. In this framework, the wavefield is viewed as a map of x 3 with values in a function
space on R2.
III. THE DIMENSIONALLY REDUCED SCATTERING PROBLEM
In this section we will consider the acoustic-pressure normalization analog. All the other
normalizations lead to similar results. Thus, consider the partial differential operator Â5Â2,1Â1,2
on R2 @cf. Eq. ~II.29!# which is elliptic with a parameter yet to be specified. The dependence on
x 3 will be suppressed in this section.
In the following analysis it is assumed that the field matrix F̂ I and the wave matrix Ŵ M are
contained in proper spaces, which, in view of the smoothness of the medium, is controlled by the
source distributions @cf. Eqs. ~II.6! and ~II.7!#. The elliptic operator is clearly well-defined on the
space C `0 of smooth functions with compact support in R2. It can be extended as a bounded
operator Â: H r →H r22 for any real r ~Ref. 37, Theorem 8.9!. Here, H r is a reserved symbol for
the Sobolev spaces. @Whenever we write H r , we mean H r ~R2!; otherwise, the underlying space
will be specified.# The norm on H r will be denoted as i.ir , and the norm of an operator
H r →H r 8 as i . i r,r 8 . The norms are implicitly scaled with the time Laplace-transform parameter s;
we postpone the discussion of this aspect to Sec. V. In particular, when r50, we have an operator
Â: L2→H22, which is bounded. On the other hand, note that Â: L 2 →L 2 is unbounded in general.
First, we shall discuss the existence and integral representations of powers of Â. To this end,
we need to analyze the properties of its resolvent. The construction of the resolvent is, essentially,
the solution to the reduced scattering problem and will be dealt with in Sec. VIII.
A. Properties of the resolvent
Let l be a complex variable. The resolvent R̂ l of Â is defined as
R̂ l 5 ~ Â2lI ! 21 : H r →H r1l ,
0<l<2.
~III.1!
It exists for l¹s~Â!, which defines the spectrum s~Â! of Â. We refer to this spectrum as the
horizontal spectrum. Whenever confusion would arise, the resolvent will be denoted as R̂ ~A!
l rather
than R̂ l to explicitly show its relation to the operator Â.
First, to analyze the spectrum of Â, we consider the case r5l50. We have Â: L 2 →L 2 . In
accordance with the structure of our horizontal partial differential operators, we introduce a family
of inner products on L 2 with respect to û5û(x m ,s) and v̂ 5 v̂ (x m ,s) as
^ û, v̂ & 0 p 5
E
x m PR
û * v̂ r p dx 1 dx 2 ,
~III.2!
where * denotes complex conjugate ~note that for s real all the quantities are, however, real! and
corresponding L 2-norms as
i û i 20 5 ^ û,û & 0 p
p
~III.3!
with 21<p<1. For the acoustic-pressure normalization analog we take p51. ~For the verticalparticle-velocity normalization analog we take p521 and for the vertical-acoustic-power-flux
normalization analog we take p50. When p50 the subscript will be omitted.! Using
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Maarten V. de Hoop: Generalized Bremmer coupling series
E
x m PR
@ 2s 21 ] n „r 21 s 21 ] n ~ r û ! …# * v̂ r dx 1 dx 2
52
5
3255
E
E
x m PR
x m PR
r 21 [2s 21 ] n ~ r û ! ] * ~ s * ! 21 ] n ~ r v̂ ! dx 1 dx 2
û * @ 2 ~ s * ! 21 ] n „r 21 ~ s * ! 21 ] n ~ r v̂ ! …# r dx 1 dx 2 ,
~III.4!
it is found that Â is self-adjoint in L 2, i.e.,
^ Âû, v̂ & 0 1 5 ^ û,Âv̂ & 0 1 .
~III.5!
In the derivation of Eq. ~III.4! it was used that the sum of contributions from the boundaries in the
horizontal plane at infinity vanishes. In fact, the proof is obvious on C `0 ; subsequently, use that C `0
is dense in L 2. Note that the self-adjointness @cf. Eq. ~III.5!# in combination with the unboundedness is not in contradiction with the Hellinger–Toeplitz theorem,38 since it only holds for functions that satisfy boundary conditions associated with causal solutions to the spectral-domain
acoustic equations.
From Eq. ~III.4! it also follows that ~here, we need the condition Im$ s % 50!
^ Âû,û & 0 1 5
E
x m PR
r 21 u s 21 ] n ~ r û ! u 2 dx 1 dx 2 1
E
x m PR
k r u û u 2 r dx 1 dx 2 ,
~III.6!
so that
^ Âû,û & 0 1 > ^ c 22 &^ û,û & 0 1 ,
~III.7!
^ c 22 & 5 inf $ k r % .0.
~III.8!
in which
x m PR
This shows that Â is positive and semi-bounded from below in i . i 0 1 . Again, these properties
trivially hold, e.g., on C `0 . Since by Cauchy–Schwarz’ inequality
i~ Â2lI ! û i 0 1 i û i 0 1 > u ^ ~ Â2lI ! û,û & 0 1 u ,
~III.9!
while @cf. Eq. ~III.7!#
u ^ Âû,û & 0 1 2l * ^ û,û & 0 1 u > @~ ^ c 22 & 2Re$ l % ! 2 1 ~ Im$ l % ! 2 # 1/2^ û,û & 0 1 5 u ^ c 22 & 2l u ^ û,û & 0 1
~III.10!
if Re$l%<^c 22&, we obtain
i~ Â2lI ! û i 0 1 i û i 0 1 > u ^ c 22 & 2l u i û i 20 .
1
~III.11!
Hence, when l¹s~Â! and Re$l%<^c 22&, we arrive at the estimate
i R̂ l i 0 1 ,01 < u ^ c 22 & 2l u 21
~III.12!
for the operator norm of the resolvent as an operator L 2 →L 2 . When l is large enough, this implies
that there exists a constant C 0,0 such that
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3256
Maarten V. de Hoop: Generalized Bremmer coupling series
i R̂ l i 0 1 ,01 <C 0,0 / u l u .
~III.13!
Now, let the inner product on H r using Parseval’s formula be defined through
with Ĝ0 5 @ 2s 22 ] s ] s 1 ^ c 22 & # 1/2.
^ .,. & r p 5 ^ ~ Ĝ0 ! r ., ~ Ĝ0 ! r . & 0 p
Thus, the inner product and the corresponding norm contain the parameter s. The estimate for the
norm of R̂ l can be generalized following Ref. 9, Theorem 1, Corollary 1, and is given in Sec. VIII:
i R̂ l i r,r1l <C r,l / u l u 12l/2,
0<l<2,
~III.14!
when l is large enough and lies in the sector L5L0øLsp of the complex plane, where L0 is
defined as p/2,uarg~l!u<p and Lsp is defined as 0,uarg~l!u<p/2.
From Eq. ~III.7! ~r50! it also follows that the spectrum must be real and positive and bounded
from below, i.e., when lPs~Â!,
l> ^ c 22 & ;
~III.15!
the ‘‘smoothness’’ of the possible eigenfunctions is estimated in their appropriate Sobolev space
H r . The property that the spectrum is semibounded from below extends to Â as an unbounded
operator H r →H r also for rÞ0 ~see also Ref. 37, Theorem 13.31!. We have to ensure that the
spectrum is strictly positive in H r for rÞ0. Since the multiplication operator, arising from the
multiplicative part, say f̂, of the elliptic operator Â satisfies (Ĝ0!rf̂~Ĝ0!2r 5 f̂ 2[ f̂ , (Ĝ0!r#~Ĝ0!2r
while [ f̂ ,Ĝ0#5O~s21! as s→ `, it is found that ^ f̂ û,û & r p > 0 if the medium is sufficiently smooth
or s is large enough and f̂>0. Anyway, the spectrum can be controlled by imposing constraints on
the compressibility or on the topology of the underlying horizontal space. In general, the spectrum
will consist of absolute continuous ~branch cut!, pure point and possibly singular continuous
contributions. By requiring that k→ ` as uxmu→ `, or by applying periodic boundary conditions in
the horizontal directions, the operator Â becomes compact, and its spectrum becomes discrete.
It is observed that the estimate in Eq. ~III.14! and the properties of Â hold at each depth level
x 3 , provided that ^c 22& is a positive and bounded function of x 3 . How to obtain, via a parametrix,
the resolvent, which is a two-dimensional problem, will be discussed at the end of this paper.
Given the resolvent R̂ l for l¹ s~Â!, we then construct general powers of the differential
operator Â, following a standard procedure from functional analysis. This will be discussed in the
next subsections.
B. Negative fractional powers of the elliptic operator
Let the power lz of a complex variable l with zPR be defined as
l z 5 u l u z exp@ iz arg~ l !# ,
~III.16!
with arg~l!P~2p,p!. With this definition, the branch cut of lz is along the negative real axis. Let
B be a contour of integration in the l plane around the branch cut, counter-clockwise oriented,
staying away a small but finite distance from the origin ~the branch point!, not intersecting the
spectrum s~Â!, and going to infinity in the sector L0 . Then, for zPR,0, the Dunford integral
Âz 5
1
2pi
E
lPB
l z R̂ l dl
~III.17!
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Maarten V. de Hoop: Generalized Bremmer coupling series
3257
FIG. 2. Contours for the Dunford integral ~Seeley’s rays of minimal growth!.
converges in the operator norm i•i r,r22z on H r ~the proof relies on the symbol calculus of Secs.
V and VIII for the parametrix of Â2lI for lPL0 large, in combination with the knowledge about
the spectrum s~Â! near the origin!. The integral satisfies the composition equation
Âz Âw 5Âz1w
~III.18!
for z, wPR,0. To show this, consider another contour B 8 around the branch cut such that B is in
between B 8 and the branch cut. The integral in Eq. ~III.17! remains the same when the contour B
is deformed into B 8 ~see Fig. 2!, since the contributions from the arcs connecting B and B 8 at
infinity vanish. Using B 8 to evaluate Âz and B to evaluate Âw , we get
Âz Âw 52
1
4p2
E E
lPB 8
m PB
R̂ l R̂ m l z m w dld m 52
1
4p2
E E
lPB 8
m PB
l zm w
~ R̂ l 2R̂ m ! dld m ,
l2 m
~III.19!
in view of the Hilbert identity. Since for the first term we have owing to the theorem of residues
E
lPB 8
FE
m PB
G
mw
d m l z R̂ l dl52 p i
l2 m
E
lPB 8
l w1z R̂ l dl
~III.20!
and, upon changing the order of integration, for the second term
E FE
m PB
G
lz
dl m w R̂ m d m 50,
lPB 8 l2 m
~III.21!
in view of Cauchy’s theorem, Eq. ~III.19! reduces to
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3258
Maarten V. de Hoop: Generalized Bremmer coupling series
Âz Âw 5
E
1
2pi
l z1w R̂ l dl,
~III.22!
l 21 R̂ l dl,
~III.23!
lPB 8
from which Eq. ~III.18! is found.
C. The inverse operator
In the case z521, Eq. ~III.17! implies
Â21 5
1
2pi
E
lPS
where S encircles the origin counter-clockwise ~note that the contributions from the branch cut
cancel!. The spectrum of Â lies outside S in the l plane. The change of variables m5l21 leads to
Â21 5
E
1
2pi
m PS 8
m R̂ m 21 m 22 d m ,
~III.24!
where S 8 denotes the contour in the m plane corresponding to S in the l plane, but also
encircling the origin counter-clockwise. Notice that the spectrum of Â lies inside S 8 in the m
plane, hence R̂ m21 is well defined on S 8. Since
R̂ m 21 5 m Â21 ~ m I2Â21 ! 21 ,
~III.25!
certainly when mPS 8, substitution in Eq. ~III.24! yields
Â21 5
1
Â21
2pi
E
m PS 8
m 21 ~ I2 m 21 Â 21 ! 21 d m .
~III.26!
Since the operator Â21 must be bounded, ~I2 m 21 Â21!21 can be expanded in the Neumann series
(I2 m 21 Â21!215( `n50 m 2n Â2n. In view of Cauchy’s theorem, only the term n50 in Eq. ~III.26!
contributes, from which it follows that
Â215Â21
~III.27!
~see, e.g., Ref. 39, III Theorem 6.15!. Equations ~III.18! and ~III.27! show that the operators Âz
behave like ordinary powers for negative values of z.
D. Non-negative fractional powers of the elliptic operator
With the aid of Eq. ~III.17! a non-negative fractional power of Â can be readily introduced
through
Âz 5Âk Âz2k ,
~III.28!
where k is an integer such that k.z. The resulting operators behave, again, like ordinary powers,
i.e.,
Âz Âw 5Âz1w
~III.29!
~note that Â and its resolvent commute!.
In view of Eqs. ~III.17!, and ~III.28! and ~III.14!, which are based on the results of Sec. VIII,
it follows that Âz : H r →H r22z is bounded for general r and for all zPR.
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Maarten V. de Hoop: Generalized Bremmer coupling series
3259
E. The Schrödinger problem
If Â is compact and self-adjoint, its eigenfunctions c and spectrum can be used to evaluate its
negative fractional powers, viz., upon choosing the contour B in the Dunford integral equation
~III.17! in Lsp around s~Â!. Then, applying the theorem of residues leads to
Âz c @ N # 5
1
2pi
E
lPB
l z R̂ l c @ N # dl5
1
c
2pi @N#
E
lPB
l z ~ l @ N # 2l ! 21 dl5l @zN # c @ N # .
~III.30!
Expanding the components of the wave matrix into the eigenfunctions of Â leads to a diagonal
representation of Âz , viz.,
Âz û5
l @zN # û @ N # c @ N #
(
@N#
with
û5
û @ N # c @ N # .
(
@N#
~III.31!
Finding the horizontal spectrum and eigenfunctions, Âc 5lc, is a Schrödinger problem in two
dimensions; note that this spectrum may vary with x 3 . The Dunford integral around the spectrum
links the current methodology to the theory of waveguides in two dimensions, since s 2 Â corresponds with a dimensionally reduced wave equation.
F. Vertical derivatives of powers of the elliptic operator
Consider, again, Eq. ~II.19!. We are now in a position to show that ~] 3 L̂ I,J ! exists. As before,
it is sufficient to prove that the expression for ~] 3 Âz! converges in operator norm when z,0. To
this end, we consider the integral
1
2pi
E
lPB
l z ] 3 R̂ l dl,
in which
] 3 R̂ l 52R̂ l ~ ] 3 Â! R̂ l ,
~III.32!
where ]3 in ~]3Â! acts on the coefficients of Â only. In fact, ~]3Â!: H r →H r is a multiplication
operator. Note that this operator is bounded, since the derivatives of the medium parameters are
assumed to be continuous. Further, ]3R̂ l vanishes in those regions where the medium properties
are independent of x 3 . The norm of ]3R̂ l satisfies the estimate @cf. Eq. ~III.14!#
i ] 3 R̂ l i r,r1l1m < i R̂ l i r1l,r1l1m i~ ] 3 Â!i r1l,r1l i R̂ l i r,r1l <C r,l,m / u l u 22 ~ l1m ! /2
~III.33!
for 0<l, m<2, and lPL0 large. It then also follows that ~]3Âz!: H r →H r22(z21) exists and is
bounded for Re$z%,0 ~the proof relies on the symbol expansions to be discussed in Sec. VIII!.
Now, use the relation
k
~ ] 3 Âz ! 5 ~ ] 3 Âk Â z2k ! 5
(
q50
Âq ~ ] 3 Â!Âk2q Âz2k 1Âk ~ ] 3 Âz2k !
to extend the result from z,0 to z>0.
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3260
Maarten V. de Hoop: Generalized Bremmer coupling series
IV. THE SYSTEM OF ONE-WAY WAVE EQUATIONS IN THE
TIME-LAPLACE-TRANSFORM DOMAIN
Next, we complete the directional decomposition procedure. Using any of the decomposition
schemes, Eq. ~II.19! transforms into
~ ] 3 d I,M 1sL̂I,M ! Ŵ M 52 ~ L̂ 21 ! I,M ~ ] 3 L̂ M ,K ! Ŵ K 1 ~ L̂ 21 ! I,M N̂ M ,
~IV.1!
which can be interpreted as a coupled system of one-way wave equations. The coupling between
the components of Ŵ M is apparent in the first sourcelike term on the right-hand side. In particular,
we shall further investigate the decomposition operator associated with the vertical-power-flux
normalization and given by Eq. ~II.48!. For this normalization, the coupling operator becomes
2L̂ 21 ~ ] 3 L̂ ! 5
S D
T̂
R̂
R̂
T̂
,
~IV.2!
in which
21/2 1
2 2 @ Ĝ1/2, ~ ] 3 Ĝ21/2!# ,
T̂52 41 Ĝ21/2@ Ĝ, Â 21
1,2 ~ ] 3 Â 1,2 !# Ĝ
~IV.3!
where we have used the property that Ĝ1/2~]3Ĝ21/2!52~]3Ĝ1/2!Ĝ21/2, is the transmission operator
that consists of commutators only, and
21
21/2
R̂52 41 Ĝ21/2„@ Ĝ, Â 21
1,2 ~ ] 3 Â 1,2 !# 12Â 1,2 ~ ] 3 Â 1,2 ! Ĝ22 ~ ] 3 Ĝ ! …Ĝ
21/2 1
52 41 Ĝ21/2$ Ĝ,Â 21
2 2 $ Ĝ1/2, ~ ] 3 Ĝ21/2! % ,
1,2 ~ ] 3 Â 1,2 ! % Ĝ
~IV.4!
where $.,.% denotes the anticommutator, is the reflection operator. In the limit of a horizontally
homogeneous medium ~or as s→ `!, the physical interpretation of Eq. ~IV.1! simplifies since then
L̂ 21 ( ] 3 L̂) becomes purely off-diagonal. In this case, therefore only counter-propagating constituents interact. This property reveals the consistency of the decomposition method with asymptotic
ray theory.
The reflection and transmission operators are bounded and vanish, due to our initial assumption of a homogeneous, isotropic embedding, outside a closed interval along the x 3-direction. To
show the boundedness, note that the multiplication operator,
s
s
Â 21
1,2 ~ ] 3 Â 1,2 ! : H →H
`
with Â 21
1,2 ~ ] 3 Â 1,2 ! PC 0 ,
~IV.5!
is bounded for all H s . Further, Ĝ1/2~]3Ĝ21/2!52~]3Ĝ1/2!Ĝ21/2: H r →H r1l with 0<l<2 are
bounded. Thus, the norms of the reflection and transmission operators in R2 can be estimated as
1/2
i T̂ i r,r <C Tr,1 @ i Ĝ21/2i r21/2,r i Â 21
1,2 ~ ] 3 Â 1,2 !i r21/2,r21/2i Ĝ i r,r21/2
21/2
i r,r11/2# 1C Tr,2 i@Ĝ1/2, ~ ] 3 Ĝ21/2!#i r,r
1 i Ĝ1/2i r11/2,r i Â 21
1,2 ~ ] 3 Â 1,2 !i r11/2,r11/2i Ĝ
~IV.6!
and
1/2
i R̂ i r,r <C Rr,1 @ i Ĝ21/2i r21/2,r i Â 21
1,2 ~ ] 3 Â 1,2 !i r21/2,r21/2i Ĝ i r,r21/2
21/2
i r,r11/2# 1C Rr,2 i $ Ĝ1/2, ~ ] 3 Ĝ21/2! % i r,r
1 i Ĝ1/2i r11/2,r i Â 21
1,2 ~ ] 3 Â 1,2 !i r11/2,r11/2i Ĝ
~IV.7!
uniformly in s>s 0 .0.
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Maarten V. de Hoop: Generalized Bremmer coupling series
3261
So far we have replaced Eq. ~II.9! by Eqs. ~IV.1! and ~II.17!. Equation ~IV.1! shows that at
this stage the vertical derivative operator needs further analysis. To illustrate how the source
distributions control the space to which the wave matrix belongs, suppose that f̂ k PH r and
q̂PH r21 . Then N̂ 1 PH r and N̂ 2 PH r21 @cf. Eqs. ~II.15! and ~II.16!#. In accordance with the matrix
operator in Eq. ~II.9!, F̂ 1 PH r11 and F̂ 2 PH r . Further, we have Â 1,2: H r(11) →H r(11) , whereas
Â 2,1: H r11 →H r21 @cf. Eqs. ~II.12!–~II.14!#. In view of Eq. ~II.48! we then arrive at Ŵ M PH r11/2.
In practice, we set r52 21.
A. Factorization of the ‘‘Helmholtz’’ operator
If we constrain our configuration to a vertically homogeneous ~thin! slab, the directional
decomposition implies a factorization of the Laplace-domain analog of the second-order wave
equation. Using Eq. ~II.40!, we find that
~ ] 3 1sĜ~ 1 ! !~ ] 3 1sĜ~ 2 ! ! 5 ] 23 2s 2 Â.
~IV.8!
Indeed, the pressure satisfies the equation @cf. Eq. ~II.9!#
~ ] 23 2s 2 Â 1,2Â 2,1! F̂ 1 5 ] 3 N̂ 1 2Â 1,2 sN̂ 2 .
~IV.9!
However, it is emphasized that the factorization does not hold in this form for vertically heterogeneous media.
V. THE GENERALIZED VERTICAL SLOWNESS
For the proofs of the basic results in Sec. III, for the evaluation of the resolvent R̂ ~A!
l and hence
of Ĝ and L̂ I,J , as well as in preparation of the evaluation ~and the associated numerical implementation with respect to a Fourier basis! of the Green’s functions belonging to the left-hand side
of Eq. ~IV.1!, the calculus of pseudo-differential operators is employed. An overview of the
pseudo-differential-operator calculus can be found in several textbooks.11,40–42
It is obvious that Â: H r →H r22 can be interpreted as a pseudo-differential operator of order
2. The existence of the resolvent, via a parametrix, as a pseudo-differential operator has been
shown by Seeley,9 Section 6. As a consequence of this, the vertical slowness operator
Ĝ5Â1/2: H r →H r21 can be represented by a pseudo-differential operator of order 1.
A. General considerations
First, we present some rules for a general pseudo-differential operator Ĝ: H r →H r2d of order
d; later on, we will focus our attention on the particular case of the vertical slowness operator for
which d51.
The Fourier transformation in the horizontal plane is defined as
ũ ~ a m ,x 3 ,s ! 5
E
x m PR
û ~ x m ,s ! exp~ is a m x m ! dx 1 dx 2 .
~V.1!
Here, i a m are identified as the horizontal slownesses. Now, the Sobolev norm ~with parameter s!
on H r is written as ~in view of Plancherel’s theorem!
i û i 2r 5
S DE
s
2p
2
a m PR
†u ũ ~ a m ,x 3 ,s ! u @~ ^ c 22 & 1 a s a s ! 1/2# r ‡2 d a 1 d a 2 .
~V.2!
In the space domain, ia l corresponds to the horizontal slowness operator
D l 52
1
] .
s l
~V.3!
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3262
Maarten V. de Hoop: Generalized Bremmer coupling series
Note that D l and the multiplication by x k do not commute, since
@ xk , Dl#5
1
d .
s k ,l
~V.4!
However, in the limit s→` the commutator vanishes. By letting the operator Ĝ act on a Fourier
component exp~2is a m x m ! we introduce its left symbol ĝ (x k , a l ),
Ĝ~ x k ,D l ! exp~ 2is a m x m ! 5 ĝ ~ x k , a l ! exp~ 2is a m x m ! .
~V.5!
For a general test function û this implies
E
Ĉ ~ x m ,x n8 ! û ~ x n8 ! dx 18 dx 28 ,
~V.6!
ĝ ~ x m , a n ! exp@ is a n ~ x n8 2x n !# d a 1 d a 2 .
~V.7!
„Ĝ~ x k ,D l ! û…~ x m ! 5
x n8 PR
in which, with the use of Eq. ~V.1!,
Ĉ ~ x m ,x n8 ! 5
S DE
2
s
2p
a n PR
Here, Ĉ is called the Schwartz kernel of the pseudo-differential operator Ĝ. The left symbol and
the Schwartz kernel are related through the Fourier transformation @cf. Eq. ~V.7!#
ĝ ~ x m , a l ! 5
E
x n8 PR
Ĉ ~ x m ,x n8 ! exp@ is ~ x l 2x l8 ! a l # dx 18 dx 28 .
~V.8!
In the horizontal space Fourier-transform domain Eq. ~V.6! becomes
~ Ĝũ !~ a m ! 5
S DE
2
s
2p
g̃ ~ a m 2 a m8 , a n8 ! ũ ~ a n8 ! d a 18 d a 28 ,
~V.9!
exp~ is a m x m ! Ĝ~ x k ,D l ! û ~ x n ! dx 1 dx 2
~V.10!
exp~ is a m x m ! ĝ ~ x m , a n8 ! dx 1 dx 2 .
~V.11!
a n8 PR
where G̃ is defined as
~ Ĝũ !~ a m ! 5
E
x n PR
and g̃ as
g̃ ~ a m , a n8 ! 5
E
x n PR
Equation ~V.9! explicitly shows the interaction between the different Fourier components ~see also
Refs. 43–45!. The quantity g̃ is denoted as the cokernel42 of Ĝ. Its representation is useful for
numerical computations.
The notation in Eq. ~V.6! is justified by the fact that if ĝ would be a polynomial in al , as is
the case when Ĝ is a partial differential operator, then Ĝ would be obtained from ĝ by replacing
ia l by D l put to the right of the coefficients. Still, we omit the dependencies of ĝ and û on x 3 and
s for the time being. The integral in Eq. ~V.6! converges with Eq. ~V.7! even when ĝ becomes
large, as long as ĝ oscillates more slowly than the exponential. The Schwartz kernel @cf. Eq. ~V.7!#
is a so-called oscillatory integral. To guarantee that the right-hand side of Eq. ~V.7! exists as a
distribution, the symbol ĝ must lie in a space S d ~R23R2!, d being a real number, which means
that for all m 1 ,m 2 ,n 1 ,n 2 there exists a constant C m 1 ,m 2 ,n 1 ,n 2 such that ~Ref. 11, Definition 18.1.1!
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Maarten V. de Hoop: Generalized Bremmer coupling series
u ~ ] a 1 ] a 2 ] x 1 ] x 2 ĝ !~ x k , a l ! u <C m 1 ,m 2 ,n 1 ,n 2 @~ ^ c 22 & 1 a s a s ! 1/2# d2m 1 2m 2
m
1
m
n
2
n
1
2
3263
~V.12!
for all x kPR, alPR. The constant C m 1 ,m 2 ,n 1 ,n 2 may depend on s but is O~1! as s→ `. The
number d is called the order of the space S d . We write S 2` 5 ù dPRS d . Under the condition Eq.
~V.12! it follows that Ĉ is a distribution of order <k with k.d12 ~Ref. 46, Theorem 7.8.2 and
Appendix A!, while Ĝ: H r →H r2d is continuous ~Ref. 11, Theorem 18.1.13! and d is the so-called
order of the operator; the corresponding operator norm associated with the Sobolev norm with
parameter is O~1! as s→ ` if the symbol is O~1!. Then the kernel is smooth outside the diagonal
in R23R2. The space of pseudo-differential operators of which the left symbols are in S d is
denoted by Op S d . It is observed that Op S 2` is the space of operators the Schwartz kernels of
which are in C `~R23R2!. The expansions of symbols to be considered later on will all be
mod S 2`.
B. The equation for the slowness surface
The left symbol â5â(x m , a n ) of the normalized elliptic differential operator given in Eq.
~II.45!,
Â52 r 1/2s 21 ] n „r 21 s 21 ] n ~ r 1/2 ! …1 k r
52s 22 ] n ] n 1 k r 1 43 r 22 s 22 ~ ] n r ! 2 2 21 r 21 s 22 ~ ] n ] n r ! ,
~V.13!
using Eq. ~V.5!, is obtained as
â5 a n a n 1 k r 1 43 r 22 s 22 ~ ] n r ! 2 2 21 r 21 s 22 ~ ] n ] n r ! .
~V.14!
The latter expression is real valued, while terms O(s 21 ) do not occur. The symbol lies in S 2 ~note
that the third- and higher-order derivatives with respect to an vanish and that the volume density
of mass and the compressibility together with their derivatives are bounded functions of position
in space!. The corresponding Schwartz kernel is given by
Â~ x m ,x m8 ! 52s 22 ] n ] n d ~ x m 2x m8 !
1 @ k r 1 21 r 21 s 22 $ 23 r 21 ~ ] n r ! 2 2 ~ ] n ] n r ! % #~ x m ! d ~ x m 2x m8 ! ,
~V.15!
with as its support x m 5 x m8 , i.e., the diagonal in R23R2.
To transform the operator equation ~II.44! into an equation for the corresponding left symbols,
we consider the composition of two pseudo-differential operators. Representing the operators as in
Eqs. ~V.6! and ~V.7!, the composition rule for the respective left symbols is found ~see the
Appendix!. Application of this rule yields the definition of the generalized slowness surface as the
solutions ĝ~6!PS 1 of @cf. Eq. ~V.14!#
2
S DE E
s
2p
2
x 8n PR
a 8n PR
ĝ ~ x m , a l8 ! exp@ is ~ x s 2x s8 !~ a s 2 a s8 !# ĝ ~ x l8 , a n ! d a 81 d a 82 dx 81 dx 82
1 a n a n 1 k r 1 43 r 22 s 22 ~ ] n r ! 2 2 21 r 21 s 22 ~ ] n ] n r ! 50.
~V.16!
The branches are ĝ (6) (x k , a l ) such that
Re$ ĝ ~ 1 ! ~ x m , a n ! % >0
and
Re$ ĝ ~ 2 ! ~ x m , a n ! % <0.
Due to the isotropy ~up/down symmetry! of the medium we have ĝ (1) 52 ĝ (2) . Further, note that
as s→ ` the composition of symbols tends to an ordinary multiplication. The solution of the
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3264
Maarten V. de Hoop: Generalized Bremmer coupling series
associated equation for the slowness surface yields the principal vertical slowness, which coincides pointwise with the vertical gradient of travel time along a characteristic.
So far, we had to assume that the medium properties ~i.e., the coefficients in the elliptic
operator! were smooth. This condition can be somewhat relaxed. Media, in which discontinuities
in their physical properties occur, should be smoothed on the scale of the irradiating pulse width
with the aid of equivalent medium averaging. To allow singularities in the medium and the volume
source densities to coexist, however, requires a novel analysis of pseudo-differential operators.47
Further, our analysis in the horizontal plane builds on the one on the torus; thus, we have chosen
to use left symbols rather than Weyl symbols ~Ref. 11, Sections 18.4 and 18.5! in this paper.
VI. THE GREEN’S FUNCTIONS OF THE ONE-WAY WAVE OPERATORS
We now subject the left-hand side of Eq. ~IV.1! to a further investigation. In it, we recognize
the operators
] 3 1sĜ~ 6 ! : L„R6 ,H r ~ R2 ! …→L„R6 ,H r21 ~ R2 ! …,
~VI.1!
where L„R6 ,H r ~R2!…, denotes a Banach space of maps R6→H r ~R2!. The operators in Eq. ~VI.1!
are the full one-way wave operators. A technical complication arises because the operators in Eq.
~VI.1! cannot be identified as pseudo-differential operators H r ~R23R!→ H r21 ~R23R! ~see also
Ref. 41!.
To arrive at the coupled system of integral equations that is equivalent to Eq. ~IV.1! and that
can be solved in terms of a Neumann expansion, we have to invert the operator occurring on the
left-hand side. The one-sided elementary kernels Ĝ (6) (x m ,x 3 ;x 8n ,x 83 ) associated with the operators
Ĝ~ 6 ! 5 ~ ] 3 1sĜ~ 6 ! ! 21 : L„R6 ,H r ~ R2 ! …→L„R6 ,H r ~ R2 ! …
in three-dimensional space are the so-called Green’s functions. They satisfy the equations
] 3 Ĝ
~6!
1sĜ~ 6 ! Ĝ
~6!
together with the condition of causality.
We will consider the case Ĝ5Ĝ~1!, Ĝ =Ĝ
test function û as
~ Ĝû !~ x m ,x 3 ! 5
E E
z PR x 8n PR
5 d ~ x n 2x 8n ! d ~ x 3 2x 83 ! ,
~VI.2!
~1!
, and Ĝ5Ĝ~1! in detail. The operator Ĝ acts on a
Ĝ ~ x m ,x 3 ;x 8n , z ! û ~ x 8n , z ! dx 81 dx 82 d z .
~VI.3!
Let us define the initial-value problem of determining the function Û(x m ,x 3 ; z ) satisfying
~ ] 3 1sĜ! Û50
for x 3 > z ,
Û ~ x m , z ; z ! 5û ~ x m , z ! .
~VI.4!
Û ~ x m ,x 3 ; z ! d z .
~VI.5!
Then it is observed that
~ Ĝû !~ x m ,x 3 ! 5
E
x3
z 52`
A. Properties of the inverse one-way wave operator
Now, to estimate in a proper norm the operator Ĝ, let
Ĝ5L̂1Ê,
~VI.6!
where
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Maarten V. de Hoop: Generalized Bremmer coupling series
1/2
L̂5 @ 2s 22 ] s ] s 1c 22
L #
3265
~VI.7!
is an elliptic operator independent of x 3 ~which can be identified as the vertical slowness operator
of order 1 in a homogeneous medium with slowness c 21
L ! and where
~VI.8!
Ê5Ĝ2L̂
is a pseudo-differential operator of order <0. Note that @cf. Eq. ~V.1! and below#
i L̂û i r >c 21
L i û i r
~VI.9!
uniformly in s ~and x 3!. In this framework, our initial-value problem is written as
~ ] 3 1sL̂! Û52sÊÛ
for x 3 >x 83 ,
Û ~ x m ,x 83 ;x 83 ! 5û ~ x m ,x 83 ! .
~VI.10!
Thus, the causal or one-sided propagator Û satisfies @cf. Eq. ~VI.10!#
Û ~ x 3 ;x 83 ! 5exp@ 2s ~ x 3 2x 83 ! L̂# Û ~ x 83 ;x 83 ! 2s
E
x3
z 5x 83
exp@ 2s ~ x 3 2 z ! L̂# Ê~ z ! Û ~ z ;x 83 ! d z .
~VI.11!
Taking Sobolev norms on both sides yields on account of Eq. ~VI.9!
i Û ~ x 3 ;x 83 !i r <exp@ 2s ~ x 3 2x 83 ! c 21
L #i Û ~ x 8
3 ;x 8
3 !i r
1s
E
x3
z 5x 83
exp@ 2s ~ x 3 2 z ! c 21
3 !i r d z .
L #i Ê~ z !i r,r i Û ~ z ;x 8
~VI.12!
Now, let
w ~ x 3 ! 5exp~ sx 3 c 21
L !i Û ~ x 3 ;x 8
3 !i r ,
~VI.13!
then Eq. ~VI.12! leads to
w ~ x 3 ! <w ~ x 83 ! 1s h
E
x3
z 5x 83
w~ z !dz,
~VI.14!
where ~note that Ê must be bounded!
h ~ c 21
L ! 5 sup i Ê~ x 3 !i r,r .
x 3 PR
~VI.15!
~In view of the structure of Ĝ note that h depends on s but that an estimate can be given uniformly
in s for values away from zero!. Application of Gronwall’s theorem ~Ref. 48, p. 37! to Eq. ~VI.14!
yields
w ~ x 3 ! <w ~ x 83 ! exp@ s h ~ x 3 2x 83 !# ,
~VI.16!
for x 3 > x 83 , so that upon using Eq. ~VI.13! we have
i Û ~ x 3 ;x 83 !i r < i Û ~ x 83 ;x 83 !i r exp@ 2s ~ x 3 2x 83 !~ c 21
L 2 h !#
~VI.17!
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3266
Maarten V. de Hoop: Generalized Bremmer coupling series
for x 3 > x 83 . To find a useful estimate for the norm of Ĝ, c 21
must be chosen such that
L
21
c 21
.
h
(c
)
uniformly
in
s
for
s>s
.
Let
ê
denote
the
left
symbol
of
Ê. An expansion for this
L
L
0
symbol follows from Eq. ~V.16! and the results of Sec. VIII:
1/2
~0!
,
ê5 ~ a n a n 1c 22 ! 1/22 ~ a s a s 1c 22
L ! 1C
~VI.18!
where c 225kr, and C ~0! is in S d , d<0 independent of c L . Let
m5 sup u c 21 2c 21
L u.
~VI.19!
x m PR3
In a realistic medium, we can arrange the parameters such that there exists an estimate m<m0
with m0 independent of c L . We have
1/2
u ~ a n a n 1c 22 ! 1/22 ~ a s a s 1c 22
L ! u <m.
Since C ~0! is continuous, we find the estimate
u ê ~ x m , a l ! u <m1c ~ 0 ! .
~VI.20!
Further, we obtain
] a m ê5 a m
1/2
22 1/2
!
~ a s a s 1c 22
L ! 2 ~ a n a n 1c
1/2
~ a n a n 1c 22 ! 1/2~ a s a s 1c 22
L !
1C ~ 21 ! ,
C ~ 21 ! 5 ] a m C ~ 0 ! .
~VI.21!
Since C (21) PS d , d<21, we find the estimate
u ] a m ê ~ x m , a l ! u < ~ m1c ~ 21 ! !~ ^ c 22 & 1 a s a s ! 21/2
~VI.22!
with c ~21! independent of c L . This way, we can continue to analyze estimates like Eq. ~V.12! for
ê up to any order of differentiation. Let c 0 denote the supremum of all c (d) s; the calculus of
symbols ~and the proof of continuity of pseudo-differential operators11! then implies the estimate
sup i Ê~ x 3 !i r,r < b m1C 0 ,
x 3 PR
C 05 b c 0 ,
b .1,
~VI.23!
where C 0 is independent of c L . Now, choose c L so that
21
c 21
L . b m1C 0 > h ~ c L ! 5 sup i Ê~ x 3 !i r,r .
x 3 PR
~VI.24!
To be able to find a c 21
L , C 0 must satisfy the inequalities @cf. Eq. ~VI.24!#
0,2C 0 ,2 ~ b 21 ! sup c 21 1 ~ b 11 ! inf c 21 ,
x m PR3
x m PR3
from which it follows that b.1 must be chosen in accordance with the conditions
1<
supx m PR3 c 21 b 11
.
,
infx m PR3 c 21 b 21
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Maarten V. de Hoop: Generalized Bremmer coupling series
3267
Let b1 satisfy the conditions, and let b5b11b2 ; then b m<b1m1b2m0 , and b m0 can be absorbed in C 0 to recover an estimate of the type Eq. ~VI.24!. @In a horizontally homogeneous
21
21
2 c 21
medium we find m(c 21
L ) 5 supx 3 PRu c
L u , hence c L must be chosen in accordance with
21
supx 3 PRc 21 .#
c 21
L . b (11 b )
Now, take Sobolev norms on both sides of Eq. ~VI.5!:
i~ Ĝû !~ x 3 !i r <
<
5
E
E
E
x3
z 52`
x3
z 52`
x3
z 52`
i Û ~ x 3 ; z !i r d z
exp@ 2s ~ x 3 2 z !~ c 21
L 2 h !# i Û ~ z ; z !i r d z
exp@ 2s ~ x 3 2 z !~ c 21
L 2 h !# i û ~ z !i r d z .
~VI.25!
Apparently, a useful norm on L„R6 , H r ~R2!… for the wavefield in three-dimensional space is
given by
i • i r;3 5 sup i • i r .
~VI.26!
x 3 PR
Then, from Eq. ~VI.25! it follows that
i Ĝi r;3,r;3 <
1
s ~ c 21
L 2h!
,
~VI.27!
s>s 0 .
This estimate has been made explicit for r50 @cf. below Eq. ~VI.17!#. Similar steps can be carried
out upon replacing ~1! by ~2!.
B. Path integral representations
With the vertical slowness symbols following from the resolvent, which represents the scattering process in the horizontal directions, we can construct the Green’s functions Ĝ ~6! using a
Hamiltonian path integral representation.49–51
First, it is observed that the vertical slowness operators at different levels of x 3 do not
necessarily commute with one another due to the heterogeneity of the medium. Thus we arrive at
a ‘‘time’’-ordered product integral representation ~see, e.g., Ref. 50! of the one-sided propagators
@cf. Eq. ~VI.4!# associated with the one-way wave equations, where ‘‘time’’ refers to the vertical
coordinate x 3 ,
H)
J
x3
Û ~ 6 ! ~ .,x 3 ;x 83 ! 56H ~ 7 @ x 83 2x 3 # !
exp@ 2sĜ~ 6 ! ~ ., z ! d z # û ~ .,x 83 ! .
z 5x 83
~VI.28!
In this expression, the operator ordering is initiated by exp@ 2sĜ(.,x 38 )d z # acting on û(.,x 38 )
followed by applying exp@2sĜ~.,z!dz# to the result, successively for increasing z.
If the medium in the interval @ x 38 , x 3 # were weakly varying in the vertical direction, the
Trotter product formula can be applied to the product integral in Eq. ~VI.28!. This results in the
Hamiltonian path integral representations for the Green’s functions,
Ĝ
~6!
~ x n ,x 3 ;x m8 ,x 83 ! 56H ~ 7 @ x 83 2x 3 # !
E
P
D ~ x 9n , a 9n !
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3268
Maarten V. de Hoop: Generalized Bremmer coupling series
F
3exp 2s
E
x3
z 5x 38
G
d z $ i a s9 ~ d z x s9 ! 1 ĝ ~ 6 ! ~ x m9 , z , a n9 ,s ! % ,
~VI.29!
P being a set of paths „x m9 ( z ), a n9 ( z )… in ~horizontal! phase space satisfying
x n9 ~ z 5x 38 ! 5x n8 ,
x n9 ~ z 5x 3 ! 5x n .
~VI.30!
Omitting the Heaviside function in the expression for the Ĝ ~6! yields the kernel ĝ ~6! of the
so-called phase shift operator ~Ref. 52!. A perturbative approximation of the latter operator based
on the split-step Fourier transform is discussed in Ref. 53. In Eq. ~VI.29! we have restricted
ourselves to causal solutions, since the conditions Re$ĝ ~1!%>0 and Re$ĝ ~2!%<0 imply that Ĝ ~6!
remain bounded as ux 3u→ `.
The path integral in Eq. ~VI.29! is to be interpreted as the lattice multiple integral
Ĝ
~6!
~ x n ,x 3 ;x m8 ,x 38 !
E) S
N
56H ~ 7 @ x 83 2x 3 # ! lim
N→`
F
i51
s
2p
D
N21
2
d 2 a ~ni !
)
j51
d 2 x ~nj !
N
3exp 2s
(
k51
$ i a ~sk ! ~ x ~sk ! 2x ~sk21 ! ! 1 ĝ ~ 6 ! ~ x ~mk ! , z k 2 21 N 21 Dx 3 , a ~nk ! ,s ! N 21 Dx 3 %
G
~VI.31!
with
x ~n0 ! 5x n8 ,
x ~nN ! 5x n ,
~VI.32!
and
Dx 3 5x 3 2x 38 .
~VI.33!
Note that the function
N
t ~ x n ,x 8n ! 5 ( $ i a ~sk ! ~ x ~sk ! 2x ~sk21 ! ! 1 ĝ ~ 6 ! ~ x ~mk ! , z k 2 21 N 21 Dx 3 , a ~nk ! ,s ! N 21 Dx 3 %
k51
can be associated with travel time along a path. All the integrations are taken over the interval
~2`,`!, N 21 Dx 3 is the step size in z, and (x m( j) , a (nj) ) are the coordinates of a path at the discrete
values z j of z as j51,...,N. If Dx 3 is sufficiently small, the path integral reduces to
Ĝ
~6!
~ x m ,x 3 ;x n8 ,x 38 ! .6H ~ 7 @ x 38 2x 3 # !
E S pD
s
2
2
d a 19 d a 29
3exp@ 2s $ i a s9 ~ x s 2x s8 ! 1 ĝ ~ 6 ! ~ x m ,x 3 2 21 Dx 3 , a 9n ,s ! Dx 3 % # .
~VI.34!
In the analysis ĝ ~6! may be interpreted as ~nonstandard square-root! Hamiltonians.
If the medium varies strongly in the vertical direction, the interval @ x 83 , x 3 # is divided up into
thinner slabs, and the product integral is used to arrive at a composition of one-sided propagators
through these slabs, for which the lattice multiple integrals are then substituted. The resulting
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Maarten V. de Hoop: Generalized Bremmer coupling series
3269
multiple integral is similar to the one in Eq. ~VI.31!. It is conjectured that the stationary-phase
approximation of the path integral in Eq. ~VI.29! leads to the leading term of the asymptotic ray
expansion54,55 including the KMAH index,55 in the presence of caustics.
The expression in Eq. ~VI.34! serves as the basis for numerical computations based on Fourier
transformations. Rather than using this thin-slab propagator, quasi-Monte Carlo methods can be
applied to numerically calculate the propagator over larger vertical distances. Also techniques
from the theory of symplectic integrators ~Ref. 56! may prove to be useful in the propagation over
long distances.
C. The Schwartz kernel
The one-sided Green’s function is directly related to the Schwartz kernel associated with the
vertical slowness operator. Since @cf. Eqs. ~VI.3! and ~VI.29!#
] 3 ~ Ĝû !~ x m ,x 3 ! 2û ~ x m ,x 3 ! 5
E E
x3
x n8 PR
z 52`
] 3 Ĝ ~ x m ,x 3 ;x n8 , z ! û ~ x n8 , z ! dx 18 dx 28 d z ,
~VI.35!
while @cf. Eq. ~VI.5!#
] 3 ~ Ĝû !~ x m ,x 3 ! 2û ~ x m ,x 3 ! 5
E
x3
z 52`
] 3 Û ~ x m ,x 3 ; z ! d z
for all x 3 , we have @cf. Eq. ~VI.4!#
2sĜ~ .,.;x 3 ! Û ~ .,x 3 ; z ! 5
E
x 8n PR
] 3 Ĝ ~ .,x 3 ;x 8n , z ! û ~ x 8n , z ! dx 81 dx 82 .
~VI.36!
Upon taking the limit z↑x 3 , we thus obtain
2s„Ĝ~ .,.;x 3 ! û…~ x m ,x 3 ! 5
E
lim ] 3 Ĝ ~ x m ,x 3 ;x 8n , z ! û ~ x 8n , z ! dx 81 dx 82
x 8n PR z ↑x
3
~VI.37!
so that
Ĉ ~ x m ,x 8n ;x 3 ! 5 lim 2
z ↑x 3
1
] Ĝ ~ x m ,x 3 ;x 8n , z ! .
s 3
~VI.38!
This expression implies that, in the special case of a homogeneous medium, the Schwartz kernel
reduces to the vertical particle velocity ~F̂ 2! response due to a vertical point-force source @ f̂ 3
5 d (x m 2 x m8 ) d (x 3 2 x 83 ) # at zero vertical offset.
VII. THE BREMMER COUPLING SERIES
The resolvents R̂ ~A!
fully describe the scattering in the level surfaces of x 3 . From these
l
resolvents the left vertical slowness symbols have been derived, which in their turn are used in
constructing the Green’s functions introduced in Sec. VI. Employing the Green’s functions of the
one-way operators, we are now able to formulate the scattering process along the vertical direction
in terms of a coupled system of integral equations.
To simplify the notation, we set
X̂ 1 5 ~ L̂ 21 ! 1,M N̂ M ,
X̂ 2 5 ~ L̂ 21 ! 2,M N̂ M .
~VII.1!
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3270
Maarten V. de Hoop: Generalized Bremmer coupling series
Using Eq. ~VII.1!, we rewrite Eq. ~IV.1! as @cf. Eqs. ~IV.3! and ~IV.4!#
] 3 Ŵ 1 1sĜ~ 1 ! Ŵ 1 5X̂ 1 1T̂Ŵ 1 1R̂Ŵ 2 ,
~VII.2!
] 3 Ŵ 2 1sĜ~ 2 ! Ŵ 2 5X̂ 2 1R̂Ŵ 1 1T̂Ŵ 2 .
~VII.3!
To derive an integral representation for Ŵ 1,2, we introduce the adjoint Green’s functions Ĝ
satisfying
] 3 Ĝ
~6!
~6! T
# Ĝ ~a6 ! 5 d ~ x n 2x 8n ! d ~ x 3 2x 83 ! ,
a 2s @ Ĝ
with @ Ĝ~ 6 ! # T 5Ĝ~ 6 !
~6!
a
~VII.4!
since Ĝ~6! is self-adjoint in L 2. Note that
~6!
a ~x8
n
Ĝ
~6!
, z ;x m ,x 3 ! 52Ĝ
~ x m ,x 3 ;x 8n , z ! .
~VII.5!
In fact, in view of the up/down symmetry of the medium, we also have
Ĝ
~6!
a ~x8
n
, z ;x m ,x 3 ! 5Ĝ
~7!
~ x 8n , z ;x m ,x 3 ! .
~VII.6!
@Equations ~VII.5! and ~VII.6! constitute reciprocity relations.# Combining Eq. ~VII.4! for the
adjoint Green’s functions with ~VII.2! and ~VII.3!, it is found that
] 3 ^ Ĝ
~1!
a ,Ŵ 1 0 5
^ Ĝ
~1!
8
a ,X̂ 1 1T̂Ŵ 1 1R̂Ŵ 2 0 1Ŵ 1 ~ x n
,x 38 ! d ~ x 3 2x 38 ! ,
~VII.7!
] 3 ^ Ĝ
~2!
a ,Ŵ 2 0 5
^ Ĝ
~2!
8
a ,X̂ 2 1R̂Ŵ 1 1T̂Ŵ 2 0 1Ŵ 2 ~ x n
,x 38 ! d ~ x 3 2x 38 ! .
~VII.8!
&
&
&
&
Now, we have
^ Ĝ
~1!
8
a ~ .,x 3 ;x m
,x 83 ! ,Ŵ 1 ~ .,x 3 ! & 0 50
~VII.9!
as x 3→ ` since Ĝ (1)
a 50 when x 3 . x 8
3 while in view of the assumption that in some upper
half-space the fluid is homogeneous, Ŵ 150 as x 3→ 2` on the basis of causality. A similar
reasoning leads to
^ Ĝ
~2!
8
a ~ .,x 3 ;x m
,x 83 ! ,Ŵ 2 ~ .,x 3 ! & 0 50
~VII.10!
as x 3→6`. Integration of Eqs. ~VII.7! and ~VII.8! over all x 3 then yields a coupled system of
integral equations which can be written in operator form as @cf. Eq. ~VII.5!#
~ d I,J 2K̂I,J ! Ŵ J 5Ŵ 0I ,
~VII.11!
in which
S DS
Ĝ~ 1 !
Ŵ 01
0 5
Ŵ 2
0
0
~2!
Ĝ
DS D
X̂ 1
,
X̂ 2
~VII.12!
i.e.,
Ŵ 01 ~ x m ,x 3 ! 5
E E
Ĝ
E E
Ĝ
Ŵ 02 ~ x m ,x 3 ! 5
x3
z 52`
x n8 PR
`
z 5x 3
x 8n PR
~1!
~2!
~ x m ,x 3 ;x n8 , z ! X̂ 1 ~ x n8 , z ! dx 18 dx 28 d z ,
~VII.13!
~ x m ,x 3 ;x 8n , z ! X̂ 2 ~ x 8n , z ! dx 81 dx 82 d z
~VII.14!
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Maarten V. de Hoop: Generalized Bremmer coupling series
3271
denote the directly transmitted waves, and
K̂5
S
Ĝ~ 1 !
0
0
Ĝ~ 2 !
DS D
T̂
R̂
R̂
T̂
~VII.15!
~note that the second matrix operator acts in the horizontal directions only, whereas the first matrix
operator acts in the full space!, i.e.,
~ K̂1,1Ŵ 1 !~ x m ,x 3 ! 5
E E
Ĝ
~ K̂1,2Ŵ 2 !~ x m ,x 3 ! 5
E E
Ĝ
E E
Ĝ
x3
x 8n PR
z 52`
x3
x 8n PR
z 52`
~ K̂2,1Ŵ 1 !~ x m ,x 3 ! 5
~ K̂2,2Ŵ 2 !~ x m ,x 3 ! 5
`
x 8n PR
z 5x 3
E E
`
x n8 PR
z 5x 3
Ĝ
~1!
~ x m ,x 3 ;x 8n , z !~ T̂Ŵ 1 !~ x 8n , z ! dx 81 dx 82 d z , ~VII.16!
~1!
~ x m ,x 3 ;x 8n , z !~ R̂Ŵ 2 !~ x 8n , z ! dx 81 dx 82 d z , ~VII.17!
~2!
~ x m ,x 3 ;x 8n , z !~ R̂Ŵ 1 !~ x 8n , z ! dx 81 dx 82 d z ,
~2!
~ x m ,x 3 ;x n8 , z !~ T̂Ŵ 2 !~ x n8 , z ! dx 18 dx 28 d z
~VII.18!
~VII.19!
are representative for the multiple scattering formalism. Now, consider the operators K̂I,J :
L„R6 , H 0~R2!…→L„R6 , H 0~R2!…. In the space of wave matrices we introduce the norm @cf. Eq.
~VI.26!#
S(
2
i Ŵ i 5
J51
i Ŵ J i 20;3
D
1/2
.
~VII.20!
2
i K̂I,J i 20;3,0;3 . Using the norm estimates of the preceding sections, it is found
Hence, i K̂i 2 < ( I,J51
21
that iK̂i5O(s ) as s→ `, which implies that the norm of K̂ is less than 1 when s>s 0 , for s 0
sufficiently large. In that case a convergent Neumann expansion of Eq. ~VII.11! yields its solution.
Thus, the solution of Eq. ~VII.11!,
Ŵ5R̂ ~ K! Ŵ 0 ,
~VII.21!
is found in the form of a sum of generalized-ray-like constituents, the Bremmer series,57 upon
employing the Neumann expansion for the resolvent of K̂:
`
R̂
~ K!
5 ~ I2K̂!
21
5
(
K̂n .
n50
~VII.22!
To emphasize that we have found the solution of the direct scattering problem as a summation
over multiple scattered constituents, we write
`
Ŵ5
(
n50
Ŵ ~ n !
with Ŵ ~ n ! 5K̂Ŵ ~ n21 ! .
~VII.23!
Figure 3 illustrates the decomposition procedure and the interrelation between the different resolvents. The analog of the series in a horizontally shift invariant medium can be found, e.g., in Refs.
58 and 59.
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3272
Maarten V. de Hoop: Generalized Bremmer coupling series
FIG. 3. The decomposition of the scattering process ~the solid rays refer to one term in the series!.
Upon substituting in K̂ the path integral representations for the Green’s functions of Sec. VI,
the recursion formula ~VII.23! leads to path integral representations for all the constituents in the
coupling series. Upon substituting for the transmission/reflection operators their Schwartz kernel
representations, Eq. ~VII.23! essentially composes path integrals at any level where interaction
takes place. In particular, one finds a path integral representation for the leading order backscattered field, Ŵ ~1!.
Finally, Eq. ~II.17! must be employed to compose the acoustic field matrix per constituent @cf.
Eq. ~VII.21!#, i.e., to obtain the observables. The uniqueness of the time-domain counterpart of
this result for s>s 0 is guaranteed by Lerch’s theorem ~see Ref. 35!. Note that the convergence of
the series is guaranteed essentially in the time domain; the convergence criterium in the frequency
or complex Laplace domain as described by Wing,60 and earlier by Atkinson,61 is different from
ours. From the final representation of the acoustic field matrix, the associated representation for
the Dirichlet-to-Neumann map can be obtained.29
VIII. ANALYSIS OF THE SYMBOLS
The scattering process in horizontal space is governed by a composition equation for the
~unknown! resolvent of a ~known! elliptic operator. Here, we shall discuss an asymptotic expansion for the left symbol of the resolvent belonging to Â introduced in Sec. III as the slowness
vector becomes large to find the solution of the composition equation ~V.16! as well as the other
powers needed to transform Eq. ~II.9! into Eq. ~IV.1!. Using the first few terms of the asymptotic
expansion, a Neumann series is derived for the resolvent. The latter expansion is the counterpart
in horizontal space of the Neumann expansion introduced in Sec. VII.
A natural decomposition of the left symbol of the partial differential operator Â is @cf. Eq.
~V.14!#
â5â ~ ` ! 1â ~ 22 ! ,
~VIII.1!
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Maarten V. de Hoop: Generalized Bremmer coupling series
3273
the first term being O~1! and the second term being O(s 22 ) as s→ `. We have
â ~ ` ! 5 a n a n 1č 22 c 22
` ,
â ~ 22 ! 5 43 r 22 s 22 ~ ] n r ! 2 2 21 r 21 s 22 ~ ] n ] n r ! ,
~VIII.2!
where
č 21 5 ~ c/c ` ! 21
~VIII.3!
c 22 5 k r ,
~VIII.4!
and
in which c 21
` is an appropriate parameter, introduced to enforce the correct asymptotic behavior.
The differential equation for the symbol r̂ l of the resolvent follows from the equation
~Â2lI!R̂ l 5I as @cf. Eq. ~VIII.2!#
~ is 21 ] x s 1 a s ! 2 r̂ l 1 ~ c 22 1â ~ 22 ! 2l ! r̂ l 2150.
~VIII.5!
This equation must be solved for r̂ l PS 22. The left symbols of the negative real powers then
follow from @cf. Eq. ~III.17!#
â z 5
1
2pi
E
B
l z r̂ l dl.
~VIII.6!
The symbols of the positive real powers are obtained using the composition equation for left
symbols repeatedly ~see the Appendix!.
A. The parametrix: asymptotic analysis
To carry out the asymptotic analysis, the symbol of the operator Â2lI with parameter is
written as
â l 5â l,21â l,0 ,
~VIII.7!
where
â l,25â ~ ` ! 2l,
â l,05â ~ 22 ! .
~VIII.8!
The correct behavior of the symbol of the resolvent as l and c 21
` become large is achieved by
thinking of l and c 22
as
the
squares
of
the
Fourier
domain
counterparts
of two new independent
`
variables. Actually, it is natural to treat the slowness of the medium as if it were a component of
the slowness vector. This way, the term 2l and the one linear in c 22
` are absorbed in the principal
part of the symbol. It is noticed that â l,2 is homogeneous of degree 2 in ~am ,l1/2,c 21
` !, i.e.,
21
2
â t 2 l,2~ x m ,t a n ,tc 21
` ! 5t â l,2~ x m , a n ,c ` !
~VIII.9!
for t.0 such that t 2lPL, while â l,0 is homogeneous of degree 0 in the same sense. Further, it
follows that â l,2Þ0 for lPL and ~asas!1/21ulu1/21uc 21
` uÞ0, where L is, again, the sector in the
complex l plane defined by 0,uarg~l!u<p. Hence, the operator associated with â l is ‘‘elliptic
2
2
2
with parameters l and c 21
` ,’’ whereas the symbol itself is in S L~R 3R ,R!. The extension of this
concept to anisotropic elastic media has been given by de Hoop and de Hoop.62 In the following
it is crucial to restrict l to the sector L in the complex plane.
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3274
Maarten V. de Hoop: Generalized Bremmer coupling series
Consider the parametrix B̂ l , which is an approximation to the resolvent R̂ l in the following
sense. Let the symbol of the resolvent, too, be expanded in a sum of symbols b̂ l,222 j which are
homogeneous of degree 222j in ~am ,l1/2,c 21
` !; then this sum defines the parametrix, which
resembles the resolvent up to an integral operator in Op S 2` with an infinitely differentiable
kernel ~Ref. 63, p. 20!. The successive terms in the series have increasingly smooth kernels. This
way, a parametrix is constructed with the correct behavior as ulu→ ` or c 21
` → ` ~the latter
corresponds to s→ `!. Thus, the symbol of the parametrix is written as
`
b̂ l 5
(
j50
~VIII.10!
b̂ l,222 j .
The terms b̂ l,222 j , j50,1,..., are determined as follows. Substitute the expansion Eq. ~VIII.10!
into Eq. ~VIII.5! and collect terms of equal degrees. Then we arrive at
â l,2b̂ l,22 51,
â l,2b̂ l,23 12is 21 a m ] x m b̂ l,22 50,
â l,2b̂ l,222 j 12is 21 a m ] x m b̂ l,212 j 1 @ â l,02s 22 ] x m ] x m # b̂ l,2 j 50,
~VIII.11!
j52,3,... .
It can be shown that the solutions must satisfy ~following Ref. 9!
1/2 22
u ~ ] a 1 ] a 2 ] x 1 ] x 2 b̂ l,222 j !~ x k , a m ! u <s 2 j C m 1 ,m 2 ,n 1 ,n 2 @~ ^ c 22 & 1 a n a n 1c 22
` 1ulu ! #
m
1
m
2
n
n
1
2
3 @~ ^ c 22 & 1 a s a s ! 1/2# 2 j2m 1 2m 2 .
~VIII.12!
For the asymptotic sum as following from Eq. ~VIII.11!, we have the estimate
K21
b̂ l 2
(
j50
b̂ l,222 j 5O ~ u a u 222K !
as u a u → `
~VIII.13!
for K51,2,3,... . Let B̂ l,222 j be the operator that corresponds to the symbol b̂ l,222 j , and let
K21
B̂ ~lK ! 5
(
j50
B̂ l,222 j .
~VIII.14!
From Eq. ~VIII.12!, using that for l50,1,2 we have
1/2 22
1/2 2l
< u l u 211l/2 @~ ^ c 22 & 1 a n a n 1c 22
@~ ^ c 22 & 1 a n a n 1c 22
` 1ulu ! #
` ! # ,
~VIII.15!
we obtain the estimate for B̂ (K)
l :
9 / u l u 12l/2
i B̂ ~lK ! i r,r1l <C r,l,K
~VIII.16!
there is associated the truncated expansion of Â, viz., Â(K)
with l50,1,2 and lPL. With B̂ (K)
l
K21
(`)
5 ( j 8 50 Â22 j 8 ~set â 2 5â , â 0 5â (22) , â j 8 50 otherwise!. In general, if K>2 we have
Â2Â(K) POp S 0, so that the latter difference is bounded and continuous as an operator H r →H r ;
the same holds for Â2Â(K) : H r →H r1K22 . Using this and Eq. ~VIII.16!, it follows that for K>2
8 / u l u 12l/2.
i~ Â2lI ! B̂ ~lK ! 2 ~ Â~ K ! 2lI ! B̂ ~lK ! i r,r1l1K22 <C r,l,K
~VIII.17!
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Maarten V. de Hoop: Generalized Bremmer coupling series
3275
Using Eqs. ~A15!, ~VIII.11!, and ~VIII.12!, after some manipulations as in Refs. 9 and 10, we
arrive at
i I2 ~ Â2lI ! B̂ ~lK ! i r,r1l1K22 <C r,l,K / u l u 12l/2
~VIII.18!
with l50,1,2 and lPL, whereas C r,l,K 5O(s 2K ) as s→ `. Hence, for K52, setting
Ĉ l 5I2 ~ Â2lI ! B̂ ~l2 ! ,
~VIII.19!
we get for sufficiently large l @cf. Eq. ~VIII.18!#
i Ĉ l i r,r < 21 .
~VIII.20!
Thus, the resolvent follows as the convergent Neumann series
S( D
`
R̂ l 5B̂ ~l2 !
n50
Ĉ nl
for lPL large.
~VIII.21!
Now, using that
`
(
n50
i Ĉ nl i r1l,r1l <2,
~VIII.22!
in combination with Eq. ~VIII.16!, finally leads to the estimate in Eq. ~III.14!. Following Ref. 9
~Theorem 2!, through the explicit evaluation of the symbols @cf. Eq. ~VIII.6!# it can be shown that
the integral in Eq. ~III.17! defines a pseudo-differential operator of order 2z.
Solving the system of equations ~VIII.11! yields
b̂ l,22 5â 21
l,2 ,
~VIII.23!
b̂ l,23 52 a m ~ is 21 ] x m â l,2! â 23
l,2 ,
~VIII.24!
while
22
22
b̂ l,24 52â l,0 â 22
~ ] x m ] x m â l,2! â 23
~ ] x m â l,2!~ ] x m â l,2!
l,2 2s
l,2 1 @ 2s
21
14 a m s 21 a n s 21 ~ ] x m ] x n â l,2!# â 24
] x m â l,2!
l,2 212~ a m s
3 ~ a n s 21 ] x n â l,2! â 25
l,2
~VIII.25!
and so on.
It is observed that the Neumann series for the vertical scattering gives rise to a decomposition
into constituents that have traveled up and down a definite number of times, while the Neumann
series in Eq. ~VIII.21! clearly does not separate the wavefield into constituents that travel from
right to left or vice versa.
B. The vertical slowness
From a physical point of view, it is interesting to compare the contributions to the generalized
slowness surface from the successive terms of the parametrix. For this, the integration over l has
to be carried out and the original elliptic operator has to be applied to the result. Using Eqs.
~VIII.6!, ~VIII.23!, and ~VIII.24!, we have
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3276
Maarten V. de Hoop: Generalized Bremmer coupling series
â 21/2~ x m , a n ! 5 ~ c 22 1 a s a s ! 21/2 @ 11 43 ~ c 22 1 a n a n ! 22 i a m s 21 ~ ] x m c 22 ! 1••• # .
With this the left symbol for the vertical slowness becomes
ĝ ~ x m , a n ! 5 ~ is 21 ] x s 1 a s ! 2 â 21/21 ~ c 22 1â ~ 22 ! ! â 21/2
5 ~ c 22 1 a s a s ! 1/2 @ 11 ~ c 22 1 a n a n ! 21 â ~ 22 ! 1 21 ~ c 22 1 a n a n ! 22
3 $ 2 21 i a m s 21 ~ ] x m c 22 ! 1s 22 ~ ] x m ] x m c 22 ! % 1••• # .
~VIII.26!
Note that this expansion is valid for real-valued am; it is, however, nonuniform. In the complex
radial horizontal slowness plane, a set of branch points, where the argument of the square root
vanishes, has been introduced. Near the branch points the polyhomogeneous expansion does not
behave properly, and a uniform expansion must be found. It is an open issue whether a parallel
analytic continuation of the symbols into the complex radial horizontal slowness and complex
Laplace planes exists and would be stable. However, in the angular frequency ~v! domain with
s5i v and an52i a ~nv! , vPR and a~nv!PR, a uniform expansion has been found by Fishman and
Gautesen.30
Spectral theory ~Sec. III! can also be employed to construct a convergent expansion for â z ,
â z ~ x m , a n ! 5
(
@N#
l @zN #
N@N#
c @ N #~ x m !
E
x 8n PR
c @ N # ~ x 8n ! exp@ is ~ x s 2x s8 ! a s # dx 81 dx 82
~VIII.27!
if
E
x m PR
c @ N # ~ x m ! c @ M # ~ x m ! dx 1 dx 2 5N @ N # d @ N # , @ M # .
~VIII.28!
Upon taking z52 21 and composing the result with â, the vertical slowness symbol is found, as
before. The latter construction implies an explicit regularization of the vertical slowness operator.
Numerical algorithms associated with a construction of this kind can be found in the literature on
the Mode Expansion Method64 ~see also Ref. 65!.
IX. DISCUSSION OF THE RESULTS
In this paper, we have generalized the Bremmer coupling series to configurations with multidimensionally varying media with properties that are up/down symmetric. The setup of the series
required the introduction of the directional wavefield decomposition into, the one-way wave
equations for, and the interaction of up/down constituents. The decomposition into ‘‘up’’ and
‘‘down’’ no longer permits a separation into ‘‘left’’ and ‘‘right.’’ The convergence of the series in
space–time has been proved.
The solution of the direct scattering problem in smoothly varying media has been given in
terms of two nested series expansions. Both expansions represent resolvents, one associated with
the coupling of counter-propagating constituents, and the other associated with the evaluation of
the generalized slowness surface and the ~de!composition operators. For practical purposes, one
hopes that just a few terms of both series suffice to describe the scattering phenomenon under
investigation; particular numerical advantage is achieved when only a few frequencies are of
physical importance. Smoothness of the medium is understood relative to the pulse width associated with the irradiating source.
The derivation of the generalized Bremmer coupling series implies two basically alternative
numerical approaches: a spectral approach based upon the eigenfunctions of the elliptic operator,
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Maarten V. de Hoop: Generalized Bremmer coupling series
3277
and a phase space approach making direct use of the left symbols. The spectral approach is
rigorous, but not as efficient as the phase space approach. The most straightforward way to
develop a propagation algorithm of the first kind is based on a matrix representation of the elliptic
operator on the torus in terms of a basis of pyramid-type functions ~rather than a basis of eigenfunctions!; this leads to a finite difference approximation of the partial differential operator. The
matrix is then diagonalized with the aid of the Lanczos method66 in which only the relevant
eigenpairs are calculated and propagated. The remaining calculations make use of the diagonal
form thus obtained.
The phase space approach lends itself for various different approximations to enhance its
computational efficiency. Among those are the ~rational! parabolic approximations and the phasescreen reduction of the vertical slowness symbol. The phase-screen approximation is only valid in
relatively weakly heterogeneous media. In the rational approximation method, special care has to
be taken to keep the associated, approximate vertical slowness operator self-adjoint; inherently,
the distinction between the principal part of and the higher-order contributions to the vertical
slowness symbol becomes obscure ~see Ref. 67!. The uniform expansion of Fishman and
Gautesen30 lends itself to a competing algorithm, and includes critical scattering-angle phenomena, unlike the rational approximation approach. We note that the generalized Bremmer coupling
series as presented in this paper lends itself to understanding the limits of approximate one-way
wave theories.
Several approaches exist for the transformation back to the time domain. A numerical inverse
Laplace transform can be used under the assumption that we restrict our scattered field to a finite
time window. It is emphasized that causility in this approach throughout the calculations is preserved. For a review of various algorithms we refer the reader to Ref. 68; pioneering work was
carried out by Papoulis.69 As a candidate, we mention the Stehfest algorithm.
ACKNOWLEDGMENTS
The author would like to thank Professor J. J. Duistermaat and Professor A. T. de Hoop for
their interest in this work and their many valuable comments.
This research was supported in part by the Institute for Mathematics and its Applications,
Minneapolis, MN, with funds provided by the National Science Foundation. Part of the research
was carried out at the Koninklijke/Shell Exploratie en Produktie Laboratorium, Rijswijk, the
Netherlands.
APPENDIX: THE CALCULUS OF PSEUDO-DIFFERENTIAL OPERATORS
1. The composition equation
In this subsection we consider the composition of two pseudo-differential operators,
B̂35B̂1B̂2 , say. Representing the operators as in Eqs. ~V.6! and ~V.7!,
E
B̂ ~ x m ,x 8n ! û ~ x 8n ! dx 81 dx 82 ,
~A1!
b̂ ~ x m , a n ! exp@ is a n ~ x n8 2x n !# d a 1 d a 2 ,
~A2!
„B̂~ x k ,D l ! û…~ x m ! 5
x 8n PR
in which
B̂ ~ x m ,x n8 ! 5
S DE
s
2p
2
a n PR
a composition rule for the corresponding left symbols, b̂ 1 , b̂ 2 , and b̂ 3 , is found. To begin with,
the Schwartz kernels must satisfy the composition rule
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3278
Maarten V. de Hoop: Generalized Bremmer coupling series
E
B̂ 3 ~ x m ,x n8 ! 5
x n9 PR
B̂ 1 ~ x m ,x n9 ! B̂ 2 ~ x n9 ,x n8 ! dx 19 dx 29 .
~A3!
Hence,
S DE
s
2p
2
a n PR
b̂ 3 ~ x m , a n ! exp@ is a s ~ x s8 2x s !# d a 1 d a 2 5
E
x 9n PR
B̂ 1 ~ x m ,x 9n ! B̂ 2 ~ x 9n ,x 8n ! dx 91 dx 92 .
~A4!
Substituting in Eq. ~A4! u n 5x n and u n 5 2(x 8n 2 x n ), we arrive at
S DE
s
2p
2
a n PR
b̂ 3 ~ u m , a n ! exp~ 2is a s v s ! d a 1 d a 2 5
E
x n9 PR
B̂ 1 ~ u m ,x n9 ! B 2 ~ x n9 ,u n 2 v n ! dx 19 dx 29 .
~A5!
By inverse Fourier transformation it now follows that
b̂ 3 ~ u m , a n ! 5
E E
x 9n PR
v n PR
B̂ 1 ~ u m ,x 9n ! B̂ 2 ~ x 9n ,u n 2 v n ! exp~ is a s v s ! dx 91 dx 92 d v 1 d v 2 .
~A6!
Substituting Eq. ~A2! twice yields
b̂ 3 ~ u m , a n ! 5
S DE E E E
4
s
2p
x 9n PR
a 8n PR a 9n PR v n PR
b̂ 1 ~ u m , a 8n ! b̂ 2 ~ x 9n , a 9n !
3exp@ is $ ~ a s 2 a s9 !v s 1 ~ u s 2x s9 !~ a s9 2 a s8 ! % #
3d v 1 d v 2 d a 19 d a 92 d a 81 d a 82 dx 91 dx 92 .
~A7!
Upon performing four of the integrations, we arrive at (u m 5x m )
b̂ 3 ~ x k , a l ! 5
S DE E
s
2p
2
x n9 PR
a n8 PR
b̂ 1 ~ x m , a n8 ! b̂ 2 ~ x n9 , a n !
3exp@ is ~ x s 2x s9 !~ a s 2 a s8 !# d a 18 d a 28 dx 19 dx 29 .
~A8!
This equation can also be written as a differential equation. To this end, we introduce the fourdimensional Fourier transformation in phase space
b5 ~ j m , h n ! 5
S DE E
s
2p
2
x m PR
a n PR
b̂ ~ x m , a n ! exp@ is ~ j k x k 1 h l a l !# d a 1 d a 2 dx 1 dx 2
~A9!
and its inverse
b̂ ~ x m , a n ! 5
S DE E
s
2p
2
j m PR h n PR
b5 ~ j m , h n ! exp@ 2is ~ j k x k 1 h l a l !# d h 1 d h 2 d j 1 d j 2 .
~A10!
Using Eq. ~A10!, we have
S DE E
s
2p
2
x k PR
a l PR
b̂ ~ x k , a l ! exp~ isx s a s ! d a 1 d a 2 dx 1 dx 2
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Maarten V. de Hoop: Generalized Bremmer coupling series
5
S DE E E E
s
2p
4
a l PR x k PR j k PR h l PR
3279
b5 ~ j k , h l ! exp@ 2is ~ j s x s 1 h s a s
2x s a s )]d h 1 d h 2 d j 1 d j 2 dx 1 dx 2 d a 1 d a 2
5
S DE E
F G
s
2p
5exp
2
j k PR h l PR
b5 ~ j k , h l ! exp~ 2is j s h s ! d h 1 d h 2 d j 1 d j 2
i
] ] b̂ ~ x k , a l ! u ~ x k , a l ! 5 ~ 0,0! .
s as xs
~A11!
Using this equality in Eq. ~A8!, it is found that
F
b̂ 3 ~ x k , a l ! 5exp
G
i
] ] b̂ ~ x , a 8 ! b̂ ~ x 8 , a !
s a s8 x s8 1 k n 2 m l
U
.
~ x k8 , a l8 ! 5 ~ x k , a l !
~A12!
The interpretation of the exponential operator follows upon analyzing
F
B̂ 3 ~ x k , a l ,x m8 , a 8n ! 5exp
G
i
] ] b̂ ~ x , a 8 ! b̂ ~ x 8 , a !
s a s8 x s8 1 k n 2 m l
~A13!
introducing
M 21
r̂ M ~ x k , a l ,x m8 , a 8n ! 5B̂ 3 ~ x k , a l ,x m8 , a 8n ! 2
(
m50
SD
1 i
m! s
m
~ ] a 8 ] x 8 ! m b̂ 1 ~ x k , a 8n ! b̂ 2 ~ x m8 , a l ! .
s
s
~A14!
Note that b̂ 3 (x k , a l )5B̂ 3 (x k , a l ,x k , a l ). Suppose that b̂ 1 lies in a space S s 1 and that b̂ 2 lies in
a space S s 2 . Then the following estimate holds ~see the proof of Theorem 18.1.8 in Ref. 11!
m
m
n
n
m 81 m 82 n 81 n 82
] a ] x ] x r̂ M !~ x k , a l ,x m8 , a 8n ! u
a
2
18
28
18
28
u~ ] a 1] a 2] x 1] x 2]
1
2
1
<C M ,m 1 ,m 2 ,n 1 ,n 2 ,m 8 ,m 8 ,n 8 ,n 8 @~ 11 a 8r a 8r ! 1/2# s 1 2M 2m 81 2m 82
1
2
1
2
3 @~ 11 a s a s ! 1/2# s 2 2m 1 2m 2 ,
~A15!
which implies that r̂ M (x m , a n ,x m , a n ) P S s 1 1s 2 2M .
2. Continuity
We will review the proof of continuity of a pseudo-differential operator B̂: H r →H r2d , given
that its symbol b̂ is contained in S d . Let Ĝ be the pseudo-differential operator of order 1,
Ĝ0 5 @ 2D s D s 1 ^ c 22 & # 1/2,
~A16!
which, by Fourier analysis, is trivially continuous as an operator H r →H r21 . Then also
Ĝd0 : H r →H r2d
continuous for dPR.
~A17!
r
Now, let ûPH r . Then û 0 5Ĝro û P L 2 ; using the parametrix Ĝ2r
o of Ĝo , we can write
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3280
Maarten V. de Hoop: Generalized Bremmer coupling series
û5Ĝ2r
0 û 0 1 v̂ 0
with û 0 PL 2 and v̂ 0 PC ` .
The contribution from v̂ 0 is trivially dealt with. We obtain
r2d
2r
Ĝr2d
0 B̂û5Ĝ0 B̂Ĝ0 û 0 .
~A18!
2r
2
2
B̂0 [Ĝr2d
continuous.
0 B̂Ĝ0 : L →L
~A19!
Hence, continuity of B̂ is proved, if
From the calculus of symbols, discussed in the preceding subsection, we find that the symbol b̂ 0
of the latter operator is contained in S 0.
Step 1: b̂ PS 2n21 . Let m51,...,n, and b̂ PS 2n21 . Then
u B̂ ~ x m ,x 8n ! u <
S DE
s
2p
2
a n PR
u b̂ ~ x m , a n ! u 2 d a 1 •••d a n <C.
Note that
~ x m 2x m8 ! l m B̂ ~ x m ,x n8 ! corresponding with i l m ] am b̂ ~ x m , a n !
l
m
must be bounded as well, hence
~ 11 @~ x m 2x m8 ! 2 # 1/2! n11 u B̂ ~ x m ,x 8n ! u <C.
Schur’s lemma states that if B̂(x m ,x n8 ) is continuous and
sup
x 8n
E
x m PR
u B̂ ~ x m ,x 8n ! u dx 1 •••dx n <C
and
sup
xm
E
x 8n PR
u B̂ ~ x m ,x 8n ! u dx 81 •••dx 8n <C, ~A20!
that then B̂:L 2 →L 2 is bounded with norm <C. ~This is a consequence of Cauchy–Schwarz
inequality.! Conditions ~A20! are satisfied for b̂ PS 2n21 .
Step 2: b̂ PS m , m<21. Let b̂ PS m , m<21. Let b̂ * PS m be the symbol of the adjoint
operator B̂T. Set Ĉ[B̂T B̂. Then
i B̂û i 2 < i Ĉû i i û i .
Hence, if Ĉ is continuous, then B̂ must be continuous. Let the symbol of Ĉ be contained in S 2m .
By induction, we find continuity for
m<2
n11
,
2
m<2
1 n11
,
2 2
...,
m<21.
Step 3: b̂ PS 0. Let b̂ PS 0. Then there is an estimate
M .2 sup u b̂ ~ x m , a n ! u 2 .
Set
d̂ ~ x m , a n ! [ @ M 2 u b̂ ~ x m , a n ! u 2 # 1/2PS 0 .
Since
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Maarten V. de Hoop: Generalized Bremmer coupling series
3281
M 2 u b̂ ~ x m , a n ! u 2 >M /2,
the symbol d̂ is well defined. Using the calculus of the previous subsection, form the operator
D̂T D̂; then
D̂T D̂5M 2B̂T B̂1Ê,
êPS 21 .
From this operator equality, we obtain
i B̂û i 2 < i û i 2 M 1 ^ Êû,û & .
In the previous step we have shown that Ê must be continuous ~êPS 21!; hence B̂ must be
continuous.
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