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. DevaneyA. J. Devaney Associates, 355 Boyiston Street Boston, MA 02116

andNortheastern University, Boston MA

Abstract

This paper presents a brief review of the various integral equation formuiations that have been employedfor the inverse source problem for the inhomogeneous scalar Heimhoitz equation. It is shown that theseformulations apply only in cases where either the data are prescribed on a closed surface surrounding theunknown 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 sourcescan exist on both sides of an open measurement surface. This latter problem arises in geophysical remotesensing and the derived integral equation offers an approach to this class of problems not offered bycurrently employed techniques.

1. Introduction

R.P. Porter [1,2,3} and N. Bojarski [4] independently derived an integral equation that relates an unknownsource p to the inhomogeneous Helmholtz equation to an image of this source generated from field dataspecified over a closed surface surrounding the source. This integral equation, and certain generalizationsknown 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 canalso be shown to be closely tied to the underlying structure of the reconstruction algorithms of diffractiontomography [8].

In this paper we present a unified treatment of the Porter-Bojarski integral equations and derive a newequation of this general type that applies to the case where data are prescribed on an open measurementsurface and unknown sources are located external to the half-space in which the radiating source of interestis located. The external sources can be either primary or secondary sources and, in particular, can arise fromscattering from unknown structures (scatterers) located exterior to the source region. The derived equationhas application in a number of problems in geophysical imaging which include wavelet estimation in thepresence of reflecting boundaries [9] and acoustic and electromagnetic tomography and offset VSP [10].

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2. Review

We consider a scalar source p(r) that radiates a wavefield b(r) according to the inhomogeneous Helmholtzequation

[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 ineither R2 or R3. In addition to the requirement that satisfies Eq.(1) we also impose the condition thatit 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 thescalar source p(r) from measurements of the radiated field (r) performed external to the support of thesource. It follows immediately from Eqs.(1) and (2) that the radiated field ib and the source p are relatedaccording 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 toeiklrr'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 integralequation to be solved for p.

Formulating the inverse source problem in terms of Eq.(3) has several drawbacks, the most serious ofwhich is that domain over which the data is available is disjoint from the support of the unknown sourcep. Because of this it is not possible to employ methods such as Fourier analysis that require an equalityover all of space. A second objection to this equation is that it can be employed only if p is the only sourcecontributing 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 itdoes 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 solutionor 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 forthe inhomogeneous Helmholtz equation into an integral equation form that avoids many of the limitationsassociated with Eq.(3). They considered the case where an unknown source p is contained within some finiteregion V (the "source region") bounded by a closed surface E. Their approach is based on a straightforwardapplication of Green's theorem applied to the Helmholtz equation (1) and the associated equation satisfiedby any free space Green function . By "free space" Green function we mean a Green function whose onlysingularity 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 toeither 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)SPIE Vol. 1767 Inverse Problems in Scattering and Imaging (1992)! 171

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where E is a closed surface completely surrounding the source region, O/9n' is the derivative along theoutward 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 andsubtracting the expression for the radiated field given by this equation from the expression for givenby 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 everywherewithin 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 sourcegenerated 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 theHelmholtz 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 operationof generating this image from the field data is a form of txickpmpagation and is commonly employed in anumber of applications that range from optical imaging and diffraction [11,12] to inverse scattering anddiffraction tomography [7,8].

Eq.(6) has the advantage over Eq.(3) that the domains of the "data" (the right-hand side of thisequation) and the unknown source p overlap. It still suffers from the fact that Eq.(3) was employed in itsderivation so that it is not valid if sources exterior to V contribute to the field b and that it holds onlyover the finite region V. Both of these drawbacks are removed in a modified form of the Porter-Bojarskiintegral equation which seems to have been first derived by Bleistein and Cohen [5]. This latter form of theequation is obtained directly from Eqs.(5) by first taking = G and then taking = and subtractingthe 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 overall 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/'/n' be specifiedover the boundary E. As is well known these two quantities can not be specified independently over aclosed surface so that the surface terms in the above integral equations are, in fact, over specified. Amodified 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 tohold even if we require to satisfy homogeneous Dirichlet or Neumann conditions on . Thus, one canemploy the same steps as were used in deriving Eq.(6) to obtain an integral equation of the same formas Eq.(6) but where G is replaced by G and 2iG by G G, with GE being the Green function thatsatisfies 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 wellbehaved and non-singular.

Unfortunately, the generalization of Eq.(7) to Green functions that satisfy homogeneous conditions onthe surface results in the trivial identity 0 = 0. In particular, although Eq.(7) can be shown to be valid ifG is replaced by the Green function GE, the imaginary part of this latter quantity must vanish identicallydue to the fact that (i) it satisfies the homogeneous Helmholtz equation throughout V and (ii) must satisfyhomogeneous conditions on . Thus, a Porter Bojarski type integral equation for the case of Dirichlet or

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Neumann conditions on exists only in the form of Eq.(6). Moreover, this latter equation is valid only ifthere 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 opensurface E0 and field measurements are only available over the open surface Eo. To obtain Porter's resultwe 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 interms of the source p by substituting for from Eq.(3) and we obtain

(r) = J d3r' 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 byany Green function and, in particular, can employ the complex conjugate of a Green function thatsatisfies homogeneous Dirichiet or Neumann conditions on o and that is outgoing at infinity. Thus, aswith Eq.(6), this integral equation comes in a form that requires only the value of the field or its normalderivative on the measurement boundary Again, however, as with Eq.(6), these integral equations donot hold over all of space and, in addition, since Eq.(3) is required in their derivation they hold only if p isthe 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 equationstogether 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 boundedby an open surface E0. Here, however, we will allow additional sources to exist exterior to the half-spacecontaining the source of interest p. None of the various forms of the Porter-Bojarski integral equationderived in the preceding section apply to this case and our main goal in this section is to derive such anintegral equation.

We begin our discussion by recalling that Eq.(3) plays a key role in the derivation of both Eq.(6) andEq.(9). Although this equation ceases to hold if sources are present exterior to V, it is possible to define anauxiliary field (r), constructed from boundary values of and its normal derivative on E, that effectivelygeneralizes Eq.(3) to such cases. In particular, if in Eq.(5) we take to be the outgoing wave Greenfunction G, we obtain the identity

(r) =_ f dS' [G(r -r) -(r'), r') (11)

= j dr'p(r')G(r r'), (12)

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Porter-Bojarski type integral equations

Data Domain Source Region Integral Equation

.

Cauchy conditions on closedsurface E

Interior of E Interior of E Eq.(6)

All Space Arbitrary Eq.(7)

Dirichiet or Neumannconditions on closed surface E

Interior of E Interior of E Same form as Eq.(6)

Cauchy conditions on opensurface E0

Half-space Half-space Eq.(9)

Dirichiet or Neumannconditions 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 exteriorto this region. Moreover, this equation also holds for the case where the region V is the half-space boundedby the infinite surface Eo. In this case the integration in Eq.(11) is performed only over Eo since satisfiesthe Sommerfeld radiation condition.

The auxiliary field x defined in Eqs.( 1 1) is not , in general, the total field radiated by all sources, butrather is that portion of the total field due only to sources contained in the region V. Thus, the surfaceintegral in Eq.(1 1) filters out the portion of the total field due to sources lying outside the region V andisolates only the contribution to the total field due to sources lying within V. For this reason this equalityand, in general, the lower equality in Eqs.(5), is sometimes referred to as the "extinction theorem". Inthe absence of external sources the auxiliary field x obviously reduces to the total field; i.e., x = L' if nosources exist outside of V.

An integral equation of the Porter-Bojarski type can be derived by making use of Eq.(8) with thesurface replaced by E(+) which lies an epsilon distance outside of o and where G is replaced by theoutgoing wave Green function satisfying homogeneous Dirichlet or Neumann conditions on the surface Eand where the boundary value field L' is replaced by the auxiliary field x. For the case of homogeneousDirichiet conditions on we obtain

(r) =

dS" X(r")GD(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 homogeneousNeumann conditions on

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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(rr')= I dSIFG(rl_rF) (14)4iifor the case of homogeneous Dirichiet conditions on the plane E and by a similar expression for thecase of homogeneous Neumann conditions on this plane. Moreover, by substituting for in Eq.(13) fromEq.(11) we can express the field b directly in terms of the boundary value of and its normal derivativeon the plane E0. In particular, we obtain

b(r) =- lEo dS' {H(r - rI)8 _ (15)

where H is defined in Eq.(14) for the case of homogeneous Dirichlet conditions.

4. SummaryThe integral equations of the Porter Bojarski type relate an unknown radiating source p(r) to an imageof this source generated from boundary values of the field and/or its normal derivative O/8n specifiedover closed or open measurement surfaces that bound the source region V. These integral equationshave 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 diffractiontomography [8]. However, as indicated in the paper, these integral equations are inapplicable for openmeasurement boundaries if unknown sources are present that are external to the source region V. A newintegral equation was derived in the paper does not suffer this restriction and, in particular, can be employedindependent of whether or not sources are present external to the source region V. These external sourcescan be either primary or secondary sources and, in particular, can arise from the scattering of the radiationgenerated by sources within the source region from unknown structures (scatterers) located exterior to thisregion. This latter situation arises in a number of applications [9,1OJ and it is hoped that the derivedintegral 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 ofthe 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, 18:19420 1, 1977.

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[6] R. P. Porter and A.J. Devaney. Holography and the inverse source problem. J. Opt. Soc.Am., 72:329,1982.

[7] K.J. Langenberg. Applied inverse problems for acoustic, electromagnetic and elastic scattering. InP.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 Sonicsand Ultrasonics, SU-30:355, 1983.

[9] A.B. Weglein and B.G. Secrest. Wavelet estimation for a multidimensional acoustic or elastic earth.Geophysics, 55:902913, 1990.

[10] A.J. Devaney. Geophysical diffraction tomography. IEEE Transactions on Geoscience and RemoteSensing, GE-22:3, 1984.

[11] J.R. Shewell and E. Wolf. Inverse diffraction and a new reciprocity theorem. J. Opt. Soc. Am.,58:15961603, 1968.

[12] G.C. Sherman. Diffracted wave fields expressible by plane-wave expansions containing only homoge-neous waves. J. Opt. Soc. Am., 59:697711, 1969.

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