This document presents a review and derivation of integral equation formulations for solving the inverse source problem for the inhomogeneous scalar Helmholtz equation. It summarizes existing formulations, including the Porter-Bojarski integral equations, and derives a new generalized integral equation that applies when unknown sources exist on both sides of an open measurement surface. Specifically: 1) It reviews existing integral equation formulations including the Porter-Bojarski equations and their limitations when external sources are present. 2) It derives a new generalized integral equation relating the unknown source to boundary measurements of an auxiliary field, which isolates the contribution from sources within the measurement region. 3) This new equation applies to problems with open surfaces and external

1. The Inverse Source Problem in The Presence of External Sources
A. B. Weglein
ARCO Oil and Gas Company, 2300 West Piano Pkway, Piano, TX 75075
and
A.J. Devaney
A. J. Devaney Associates, 355 Boyiston Street Boston, MA 02116
and
Northeastern University, Boston MA
Abstract
This paper presents a brief review of the various integral equation formuiations that have been employed
for the inverse source problem for the inhomogeneous scalar Heimhoitz equation. It is shown that these
formulations apply only in cases where either the data are prescribed on a closed surface surrounding the
unknown source or where the unknown source lies entirely on one side of an open measurement surface.
A generalized integral equation is derived that applies to the more general case where unknown sources
can exist on both sides of an open measurement surface. This latter problem arises in geophysical remote
sensing and the derived integral equation offers an approach to this class of problems not offered by
currently employed techniques.
1. Introduction
R.P. Porter [1,2,3} and N. Bojarski [4] independently derived an integral equation that relates an unknown
source p to the inhomogeneous Helmholtz equation to an image of this source generated from field data
specified over a closed surface surrounding the source. This integral equation, and certain generalizations
known collectively as the "Porter-Bojarski integral equations", have formed the basis for a number of appli-
cations in various inverse problems related to the Helmholtz equation [5,6,7] . These integral equations can
also be shown to be closely tied to the underlying structure of the reconstruction algorithms of diffraction
tomography [8].
In this paper we present a unified treatment of the Porter-Bojarski integral equations and derive a new
equation of this general type that applies to the case where data are prescribed on an open measurement
surface and unknown sources are located external to the half-space in which the radiating source of interest
is located. The external sources can be either primary or secondary sources and, in particular, can arise from
scattering from unknown structures (scatterers) located exterior to the source region. The derived equation
has application in a number of problems in geophysical imaging which include wavelet estimation in the
presence of reflecting boundaries [9] and acoustic and electromagnetic tomography and offset VSP [10].
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2. 2. Review
We consider a scalar source p(r) that radiates a wavefield b(r) according to the inhomogeneous Helmholtz
equation
[V2 + k2](r) = —4irp(r) (1)
where k is a real scalar constant called the wavenumber of the field L' and V2 is the Laplacian operator in
either R2 or R3. In addition to the requirement that satisfies Eq.(1) we also impose the condition that
it satisfy the Sommerfeld radiation condition; e.g., in R3
(rs) f(s)ezkn/r (2)
as kr — 00 along the direction s and where f(s) is the radiation pattern of the source p.
The inverse source problem for the inhomogeneous scalar Helmholtz equation consists of estimating the
scalar source p(r) from measurements of the radiated field (r) performed external to the support of the
source. It follows immediately from Eqs.(1) and (2) that the radiated field ib and the source p are related
according to the linear transformation
(r) = Jd3r'p(r')G(r_ r') (3)
where G is the free space Green function that satisfies the radiation condition and, for example, is equal to
eiklr—r'I/Ir r'I in R3. Equation (3) then provides one possible formulation for the inverse source problem:
b(r) is specified over some region external to the support of p and Eq.(3) is regarded as a Fredholm integral
equation to be solved for p.
Formulating the inverse source problem in terms of Eq.(3) has several drawbacks, the most serious of
which is that domain over which the data is available is disjoint from the support of the unknown source
p. Because of this it is not possible to employ methods such as Fourier analysis that require an equality
over all of space. A second objection to this equation is that it can be employed only if p is the only source
contributing to the field i/'. In some applications other, extraneous sources, both primary and secondary
(e.g., scatterers) may be present so that Eq.(3) is not valid. A final objection to this equation is that it
does not provide much insight into the underlying mathematical structure of the inverse source problem.
For example, it is not clear from this equation whether the inverse source problem has a unique solution
or if such a solution exists what minimal data is required to attain this solution.
R.P. Porter [1,2,3] and N. Bojarski [4] appear to have been the first to cast the inverse source problem for
the inhomogeneous Helmholtz equation into an integral equation form that avoids many of the limitations
associated with Eq.(3). They considered the case where an unknown source p is contained within some finite
region V (the "source region") bounded by a closed surface E. Their approach is based on a straightforward
application of Green's theorem applied to the Helmholtz equation (1) and the associated equation satisfied
by any free space Green function . By "free space" Green function we mean a Green function whose only
singularity in the finite domain is at r = r'. In particular, these Green functions satisfy the equation
[V2 + k2]c(r — r') = —4ir(r —
r') (4)
where 5 is the Dirac delta function in B" (n = 2 or 3) and where the Laplacian is taken with respect to
either r or r'. It is readily concluded from Eqs.(1) and (4) that
d3r' p(r')c(r - r') + J dS' [c(r -r')-(r'),r) =
{
(r) if rEV
(5)
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3. where E is a closed surface completely surrounding the source region, O/9n' is the derivative along the
outward pointing normal to this surface and V is the interior of this surface; i.e., the source region.
By choosing g in Eqs.(5) to be the complex conjugate of the outgoing wave Green function G and
subtracting the expression for the radiated field given by this equation from the expression for given
by Eq.(3) we obtain one form of the so-called Porter-Bojarski integral equation
Jd3n'P(n')(r - r') =
jdS'[(r- rF)0 (r'), r') (6)
where G is the imaginary part of the outgoing wave Green function G and where Eq.(6) holds everywhere
within the interior of the source region V.
The Porter-Bojarski integral equation is seen to relate the unknown source p to an image of this source
generated from the field data specified on the surface E. The image is generated by evolving or "migrating"
these boundary values via the complex conjugate of the outgoing wave, free space Green function to the
Helmholtz equation. This image, as given by the right-hand side of Eq. (6), is actually a wavefield itself and,
in particular, satisfies the homogeneous Helmholtz equation throughout the source region V. The operation
of generating this image from the field data is a form of txickpmpagation and is commonly employed in a
number of applications that range from optical imaging and diffraction [11,12] to inverse scattering and
diffraction tomography [7,8].
Eq.(6) has the advantage over Eq.(3) that the domains of the "data" (the right-hand side of this
equation) and the unknown source p overlap. It still suffers from the fact that Eq.(3) was employed in its
derivation so that it is not valid if sources exterior to V contribute to the field b and that it holds only
over the finite region V. Both of these drawbacks are removed in a modified form of the Porter-Bojarski
integral equation which seems to have been first derived by Bleistein and Cohen [5]. This latter form of the
equation is obtained directly from Eqs.(5) by first taking = G and then taking = and subtracting
the two equations that result. We obtain
Iv d3r'p(r')G(r —
r') =
— j dS' [G(r —
r')—
(r'), r') (7)
with Eq.(7) holding over all of space. This latter form of the Porter Bojarski integral equation holds over
all of space even in the presence of external sources that lie outside the region V.
Eqs.(6) and (7) require that both the value of the field b and its normal derivative th/'/ôn' be specified
over the boundary E. As is well known these two quantities can not be specified independently over a
closed surface so that the surface terms in the above integral equations are, in fact, over specified. A
modified form of Eq.(6) can be derived that requires only one of these boundary conditions and that,
hence, avoids the problem of inconsistent data. In particular, the upper equality in Eq.(5) continues to
hold even if we require to satisfy homogeneous Dirichlet or Neumann conditions on . Thus, one can
employ the same steps as were used in deriving Eq.(6) to obtain an integral equation of the same form
as Eq.(6) but where G is replaced by G and 2iG by G — G, with GE being the Green function that
satisfies either homogeneous Dirichlet or Neumann conditions on E. It is important to note tkat G —
has no singularities within the source region V so that kernel in the associated integral equation is still well
behaved and non-singular.
Unfortunately, the generalization of Eq.(7) to Green functions that satisfy homogeneous conditions on
the surface results in the trivial identity 0 = 0. In particular, although Eq.(7) can be shown to be valid if
G is replaced by the Green function GE, the imaginary part of this latter quantity must vanish identically
due to the fact that (i) it satisfies the homogeneous Helmholtz equation throughout V and (ii) must satisfy
homogeneous conditions on . Thus, a Porter Bojarski type integral equation for the case of Dirichlet or
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4. Neumann conditions on exists only in the form of Eq.(6). Moreover, this latter equation is valid only if
there are no sources exterior to the source region.
Porter [1] investigated the case where the source p lies in a half-space bounded by an infinite open
surface E0 and field measurements are only available over the open surface Eo. To obtain Porter's result
we consider the quantity
(r) = - lEo dS" [(r - r") - (r"),,r")
(8)
with r lying in the half-space occupied by the source p. The above expression can be directly evaluated in
terms of the source p by substituting for from Eq.(3) and we obtain
(r) = Jd3r' p(r')H(r — r') (9)
where
H(r — r') = —
JE0
dS" [(r — r") 11
r') G(r" — r') II
r")
(10)
Eq.(9) is an integral equation relating the source p to the boundary value of the field and its normal
derivative over E0. Moreover, it is clear from the derivation of this equation that we can replace by
any Green function and, in particular, can employ the complex conjugate of a Green function that
satisfies homogeneous Dirichiet or Neumann conditions on o and that is outgoing at infinity. Thus, as
with Eq.(6), this integral equation comes in a form that requires only the value of the field or its normal
derivative on the measurement boundary Again, however, as with Eq.(6), these integral equations do
not hold over all of space and, in addition, since Eq.(3) is required in their derivation they hold only if p is
the only source contributing to the field b. Finally, we note that the kernel H in the integral equation (9)
is well-behaved and non-singular as long as the measurement surface lies outside the source region V.
We present in the table below a summary of the various forms of the Porter-Bojarski integral equations
together with conditions required for their validity.
3. External Sources
In this section we again consider the case of an unknown source located within a half-space V bounded
by an open surface E0. Here, however, we will allow additional sources to exist exterior to the half-space
containing the source of interest p. None of the various forms of the Porter-Bojarski integral equation
derived in the preceding section apply to this case and our main goal in this section is to derive such an
integral equation.
We begin our discussion by recalling that Eq.(3) plays a key role in the derivation of both Eq.(6) and
Eq.(9). Although this equation ceases to hold if sources are present exterior to V, it is possible to define an
auxiliary field (r), constructed from boundary values of and its normal derivative on E, that effectively
generalizes Eq.(3) to such cases. In particular, if in Eq.(5) we take to be the outgoing wave Green
function G, we obtain the identity
(r) = _ f dS' [G(r -
r)ô-(r'), r') (11)
= j dr'p(r')G(r
—
r'), (12)
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5. Porter-Bojarski type integral equations
Data Domain Source Region Integral Equation
.
Cauchy conditions on closed
surface E
Interior of E Interior of E Eq.(6)
All Space Arbitrary Eq.(7)
Dirichiet or Neumann
conditions on closed surface E
Interior of E Interior of E Same form as Eq.(6)
Cauchy conditions on open
surface E0
Half-space Half-space Eq.(9)
Dirichiet or Neumann
conditions on open surface E0
Half-space Half-space Same form as Eq.(9)
which holds at ali points exterior to the source region V regardless of whether or not sources exist exterior
to this region. Moreover, this equation also holds for the case where the region V is the half-space bounded
by the infinite surface Eo. In this case the integration in Eq.(11) is performed only over Eo since satisfies
the Sommerfeld radiation condition.
The auxiliary field x defined in Eqs.( 1 1) is not , in general, the total field radiated by all sources, but
rather is that portion of the total field due only to sources contained in the region V. Thus, the surface
integral in Eq.(1 1) filters out the portion of the total field due to sources lying outside the region V and
isolates only the contribution to the total field due to sources lying within V. For this reason this equality
and, in general, the lower equality in Eqs.(5), is sometimes referred to as the "extinction theorem". In
the absence of external sources the auxiliary field x obviously reduces to the total field; i.e., x = L' if no
sources exist outside of V.
An integral equation of the Porter-Bojarski type can be derived by making use of Eq.(8) with the
surface replaced by E(+) which lies an epsilon distance outside of o and where G is replaced by the
outgoing wave Green function satisfying homogeneous Dirichlet or Neumann conditions on the surface E
and where the boundary value field L' is replaced by the auxiliary field x. For the case of homogeneous
Dirichiet conditions on we obtain
(r) = —
dS" X(r")
ÔGD(r r")
(13)
where GD is the outgoing wave Green function satisfying homogeneous Dirichlet conditions on the plane
A similar equation holds if GD is replaced by GN, the Green function satisfying homogeneous
Neumann conditions on
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6. We conclude from Eq.(12) that the field 4 as defined in Eq.(13) is still related to the source p via Eq.(9)
where, however, the kernel H is given by
H(r—r')= I dSIFG(rl_rF)° (14)4ii•
for the case of homogeneous Dirichiet conditions on the plane E and by a similar expression for the
case of homogeneous Neumann conditions on this plane. Moreover, by substituting for in Eq.(13) from
Eq.(11) we can express the field çb directly in terms of the boundary value of and its normal derivative
on the plane E0. In particular, we obtain
çb(r) =
- lEo dS'{H(r -
rI)8_ (15)
where H is defined in Eq.(14) for the case of homogeneous Dirichlet conditions.
4. Summary
The integral equations of the Porter Bojarski type relate an unknown radiating source p(r) to an image
of this source generated from boundary values of the field and/or its normal derivative O/8n specified
over closed or open measurement surfaces that bound the source region V. These integral equations
have found a number of applications in various inverse problems related to the Helmholtz equation [5,
6,7] and are also closely tied to the underlying structure of the reconstruction algorithms of diffraction
tomography [8]. However, as indicated in the paper, these integral equations are inapplicable for open
measurement boundaries if unknown sources are present that are external to the source region V. A new
integral equation was derived in the paper does not suffer this restriction and, in particular, can be employed
independent of whether or not sources are present external to the source region V. These external sources
can be either primary or secondary sources and, in particular, can arise from the scattering of the radiation
generated by sources within the source region from unknown structures (scatterers) located exterior to this
region. This latter situation arises in a number of applications [9,1OJ and it is hoped that the derived
integral equation can form the basis for reconstruction algorithms of use in such situations.
References
[1] R.P. Porter and W.C. Schwab. Optimum imaging, closed holograms and optical channel capacity.
Journal of the Optical Society of America, 61:789, 1971.
[2] R.P. Porter. Diffraction limited scalar image formation with holograms of arbitrary shape. Journal of
the Optical Society of America, 60:105 1, 1970.
[3] R.P. Porter. Image formation with arbitrary holographic type surfaces. Physics Letters, 29A:193,
1969.
[4] N.N. Bojarski. Inverse scattering. Project Report N00019-73-C-0312, Naval Air Systems Command,
1973.
[5] N. Bleistein and J.Cohen. Nonuniqueness in the inverse source problem in acoustics and electromag-
netics. Journal of Mathematical Physics, 18:194—20 1, 1977.
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7. [6] R. P. Porter and A.J. Devaney. Holography and the inverse source problem. J. Opt. Soc.Am., 72:329,
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[7] K.J. Langenberg. Applied inverse problems for acoustic, electromagnetic and elastic scattering. In
P.C. Sabatier, editor, Basic Methods of Tomography and Inverse Problems, Adam Hilger, 1987.
[8] A.J. Devaney. Inverse source and scattering problems in ultrasonics. IEEE Transactions on Sonics
and Ultrasonics, SU-30:355, 1983.
[9] A.B. Weglein and B.G. Secrest. Wavelet estimation for a multidimensional acoustic or elastic earth.
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[10] A.J. Devaney. Geophysical diffraction tomography. IEEE Transactions on Geoscience and Remote
Sensing, GE-22:3, 1984.
[11] J.R. Shewell and E. Wolf. Inverse diffraction and a new reciprocity theorem. J. Opt. Soc. Am.,
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[12] G.C. Sherman. Diffracted wave fields expressible by plane-wave expansions containing only homoge-
neous waves. J. Opt. Soc. Am., 59:697—711, 1969.
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