sensitivity of multiple-scattering inverse transport methods to measurement errors

1972 J. Opt. Soc. Am. A/Vol. 2, No. 11/November 1985

Sensitivity of multiple-scattering inverse transport methodsto measurement errors

J. C. Oelund and N. J. McCormick

Department of Nuclear Engineering, University of Washington, Seattle, Washington 98195

Received January 7, 1985; accepted April 2, 1985

Inverse multiple-scattering transport methods, derived from the linear Boltzmann equation, are potentially suit-able for remote-sensing studies because they do not require measurements within the scattering medium. Themethods investigated yield the single-frequency angular redistribution function for a homogeneous one-dimension-al slab medium that is uniformly illuminated. Illustrative numerical results for a monodirectional incident beamshow that the methods are ill conditioned with respect to simulated measurement errors, with the severity depend-ing strongly on the direction of the incident beam.

1. INTRODUCTION

Earlier methods based on the radiative transfer equation forsolving multiple-scattering inverse transport problems re-quired in situ measurements of the homogeneous medium'or were limited to a scattering anisotropy of at most secondorder in the cosine of the scattering angle.2.3 Once a sufficientnumber of equations became available to enable the entireangular redistribution function for any degree of scatteringanisotropy to be determined from measurements external toa homogeneous slab medium, 4 the methods became poten-tially suitable for use in remote-sensing studies. More re-cently, methods have been developed that apply to moregeneral geometries. 5' 6

For a homogeneous one-dimensional slab medium withuniform spatial illumination at the boundaries, there are ac-tually an infinite number of equations sets that can be em-ployed7; however, only two are linear and can be easily solved.This study investigates these two linear inverse methods, withthe further restriction that the external incident illuminationbe monodirectional. The physical situations to which themodel applies encompass passive illumination of an atmo-sphere by the sun or active illumination of a dense target ofscatterers by a beam-expanded laser. Although multiple-scattering inverse methods exist that include the effects ofpolarization, 8 '9 such effects are neglected here.

Application of these inverse methods requires the evalua-tion at both boundaries of the angular moments of the specificintensity with respect to the set of spherical harmonics, asdiscussed in Section 2. Tests of the two linear methodssuggest that they provide an exact solution when the intensityis known exactly and continuously as a function of directionon both boundaries'0 ; this does not guarantee, however, thatthe angular moments can be accurately determined from onlya finite number of data points or that the methods are wellbehaved with respect to experimental errors. Inversion usingonly a finite number of exact data points has been studiedpreviously" and is addressed again in Section 3 with signifi-cant improvements in the numerical techniques used forcalculation of the angular moments.

The main purpose of this study is to determine the sensi-tivity of the inversion methods to measurement errors gen-erated by a realistic modeling of an experiment based on usingthe FN method to calculate the intensities at the bounda-ries.1 2-14 A previous sensitivity study has been made for aninverse method involving nonlinear equations applicable toa problem with anisotropic scattering of at most second orderin the cosine of the scattering angle.15 In this paper a sensi-tivity analysis of the two linear systems for general anisotropyis done by numerically simulating experimental errors at afinite number of measurement positions.

The effects of experimental errors on inversion are dis-cussed in Section 4, where the errors are those incurred inmeasuring the direction of the incident monodirectional il-lumination and the magnitude of the uncollided transmittedbeam and random errors, which include variations not onlyin the magnitude of the specific intensity for a given directionbut also in measuring the direction itself.

An analytical investigation in Section 5 of the numericalresults from Sections 3 and 4 shows that, because of the fun-damental structure of the inverse equations, the methods asapplied here are ill conditioned with respect to measurementerror.

Concluding statements about the multiple-scattering in-verse techniques are given in Section 6.

2. INVERSE EQUATIONS AND NOTATION

The inverse methods provide for determination of the angularredistribution function expressed as an expansion over the setof Legendre polynomials

Nf(cv) = (47r)-l (2n + )f&Pn(w),

n=O

where is the cosine of the angle between the directions beforeand after scattering, the fn coefficients are the unknowns tobe solved for, and N is the degree of the scattering anisot-ropy.

In order to make comparisons among numerical test casesand provide an estimate for the degree of error inherent in the

0740-3232/85/111972-07$02.00 © 1985 Optical Society of America


Vol. 2, No. 11/November 1985/J. Opt. Soc. Am. A 1973

scattering function, errors are expressed in terms of the L2

metric

Itf(,w) - I(,w) 4w J (f(w) - ?(W)) 2dw

N=- (2n + 1)(fn-I)2,2 n=O

where the fractional error of an estimated I with respect to fis defined to be I1f - I1llf II.

For the test cases discussed in this paper, a binomial modelhas been used for the scattering redistribution function

f(w) = (47r)'lfo(1 + a)2-(1 + W)a, a > 0.

This model has the advantage that highly anisotropic scat-tering can be generated, but for integer values of a the scat-tering is of finite order, allowing sample calculations to bemade without truncation error. In any event, the principalconclusions of this study are independent of the model se-lected for the angular redistribution function.

The methods require that the specific intensity I(r, Au, 4)be known on both boundaries r = 0 and r = r,,, where r is thedimensionless optical depth, g is the cosine of the polar angle

with respect to the positive axis, and 4 is the azimuthal angle.Although it is not a necessary requirement for inversion, theexternal illumination is taken to be monodirectional and atthe left boundary

Fn = /(1 -f.) (4b)

and Amnk and Snk are evaluated as differences of momentsof the specific intensities on the slab boundaries

Amnk = (_)n-mamn[J d' f dAPnfm(A)

(5a)

Smk~~1 =4 ,2kI (T. -u)I-(T. AW To. (5b)

The moments in Amnk are projections of AgkI(,, 4') over thespherical harmonics Yn(u, ) = Pm(,g)cos(mo), withPn m() being the associated Legendre functions, and

camn = (2n + 1)(n - m)!/(n + m)!.

For monodirectional illumination the elements Sn k simplifyto

Smk = -4Au 2kIC m(O, -A.).

Thus the elements of the source vectors S in Eq. (3a) are de-termined from the Fourier moments of 1(0, -O, 4'), whereasthe elements of A come from moments taken over the entireangular surface. (Here and in what follows, the superscriptk will be suppressed, with the equations to be understood asreferring to the particular set under consideration, k = 0 or 1.)

0< A, AO < 1,

-1 CA <0.

The following discussion is facilitated by splitting the in-tensity into two contributions:

I(r, Ai, 4) = IS (r, A, (A) + I. (T. , 4').

Here, I, is the collided intensity and I, is the uncollided sin-gular contribution to the intensity arising from the monodi-rectional incident illumination:

(1)

I,(T, At, 4') = 6(U - A.)6(4')exp(-T/xA), (2)

where Im is the mth Fourier moment of I over 4 at ,

In = _ 2'r cos(mO)I(-r, A, O)do.

2r JO

Note that the collided intensity is symmetric in 0 because ofthe boundary conditions and that the Fourier cosine expan-sion is of the same order as the degree of scattering anisotropyfor the angular redistribution function.

The basic inverse equation for both methods is"ll6

AkFk = Sk (3a)

or, in component form,

NA i Amn kFn k = S k,

n=mm = OtoN

for k = 0 or 1, corresponding to method 0 and method 1.the unknowns are

Fno = Ans

3. INVERSION AT A FINITE NUMBER OFDATA POINTS

The solution of Eqs. (3) requires assuming a degree of scat-tering anisotropy N* so that

NE AmnFn = Sm.

n=m

In practice, it is the convergence of F with respect to N* thatis important. From the upper triangularity of A, if A-1 is theinverse for the true order of anisotropy N, then

ANFn = E A S.

m=n

so that the elements Anl for n, m < N* do not change withincreasing N*. Thus the change in F with ascending valuesof N* is equivalent to merely increasing the dimension of S,with no error being introduced by the inversion of A.

Calculation of the angular moments for the matrix and thesource vector from a finite number of data points requires theselection of an angular collocation set. Previously the set usedwas rectangular in g and X, where the moments for both thesource vector and the matrix were calculated by using a seg-mented Gauss-Legendre quadrature formula, requiring a localtwo-dimensional Lagrange interpolation to points not on thecollocation mesh."

Preliminary calculations showed that the inverse methodsare sensitive to the values of Sm for large m and that the pre-viously used two-dimensional interpolation from points onthe rectangular grid to points at -/O leads to unpredictableerrors in the Fourier moments for Sm. Thus, to minimize theerror (and permit a consistent error analysis), measurementsare now required to be made at -yO. This means the grid onthe incident face is effectively split into two parts: one part

I(0, A, (I) = (g - 0.M)

while at the right boundary

I(-T-, /U, 0) = 0,

NIC(r, gu, 4) = E (2- 6io)Icin(T,A)cos(mO),

m=0


1217%X cos(M0)AkJ(_r, A, 0)

0


obtain better results for inversion at a finite number of datapoints.

An improvement has been made also in the selection of thegrid points to be used for calculation of the angular momentsrequired for the matrix elements in A, since a rectangular meshin y and 0 is inefficient because the nodes at the top of thehemisphere are too tightly packed. The intent has been notto select the best mesh from a purely mathematical standpointbut rather to select a mesh for which the angular moments canbe rapidly evaluated and which may be representative of anactual experimental data set. The collocation points foremerging polar directions 0 < 0 Omax < w/2 and for 0 < 0 <7r are

O = (i - )Omax/(No - 1),

Oj = ( - 1)7/(N. - 1),

Fig. 1. Split-mesh collocation scheme for No = 5 with seven pointsat -A,,. ( denotes a node for calculation of A, and denotes a nodefor calculation of S at -.. )

for the determination of the Fourier moments at -,u., for Sand one part for the calculation of the surface moments forthe elements of A.

For calculation of S, the points at -o are taken to beequally spaced in to allow direct computation of the mo-ments I(,r-,u ) on the half-range 0 < 0 7r using'7

MIm(r, -. ) = (M - 1)-1 E (1- k1/2)

k=1

X (1 -bkM/2)I(T, 0 Sk)cos(m'k), (6)

where k = (k - 1)w(M - 1), k = Ito M, and M is thenumber of points. This approach has been chosen becauseit satisfies the limiting case in which, if M exceeds N, thenfrom Eq. (1) there will be no error in S given exact data. Itis primarily this improvement that has made it possible to

i = 1 to No,

i = 1toNi,where No is the number of nodes in 0, and Ni is the numberof nodes in 0 at the ith node in 0. This collocation scheme isshown in Fig. 1. Here the constant spacing in 0 and 0 has beenchosen to allow rapid calculation of the angular moments toprevent the Monte Carlo calculations in Section 4 from be-coming prohibitively expensive. The restriction of the do-main of 0 is necessary to account for geometric or optical sit-uations in which measurements cannot be made for 0 > ma_For the ith node in 0, the number of nodes in 0 has been se-lected to provide a uniform distribution of points on thequarter-sphere using

Ni = 1 + integer[3 sin(Oi)/sin(0 2 ) + 1/2].

For example, for No = 5 and ma = 90°, the values of Ni fori = I to 5 are 1, 4, 7, 8, and 9, giving a mesh containing 29points.

Calculation of the angular moments for A is done by eval-uation of the Fourier moments over at each node in [Eq.(6)] and then by direct numerical integration overu = cos 0[Eq. (5a)] using a segmented Gauss-Legendre integrationscheme with 10 equal-length segments and 5 nodes per seg-ment.

Cubic splines were chosen for interpolation to the Gauss-Legendre integration node points because they possess severalideal properties.' 7 The use of splines requires the selection

Table 1. Scattering Coefficients and Percent Error in Scattering Functiona

Percent Errorin Scattering

N* fo fi f2 f3 /4 FunctionRectangular collocation mesh (from Ref. 11)

3 9.312(-1) 5.063(-1) 1.808(-1) 2.774(-1) - 40.04 9.385(-l) 5.589(-1) 2.581(-l) 6.745(-2) 7.547(-3) 6.55 9.412(-1) 5.791(-1) 2.917(-1) 9.065(-2) 1.058(-2) 5.46 9.451(-1) 6.080(-1) 3.435(-1) 1.338(-1) 3.387(-2) 8.97 9.885(-1) 9.205(-1) 8.791(-1) 7.920(-1) 6.121(-1) 98.0

Split collocation mesh (this paper)3 9.415(-1) 5.692(-l) 1.871(-1) 2.748(-2) - 15.54 9.498(-1) 6.302(-1) 2.699(-1) 6.776(-1) 7.539(-3) 0.405 9.498(-1) 6.302(-1) 2.699(-1) 6.776(-1) 7.539(-3) 0.406 9.498(-1) 6.302(-1) 2.699(-l) 6.776(-1) 7.539(-3) 0.407 9.498(-1) 6.302(-1) 2.699(-1) 6.776(-1) 7.539(-3) 0.40

Exact 9.500(-1) 6.333(-1) 2.714(-1) 6.786(-2) 7.340(-3)

a Binomial scattering function has a = 4 and o = 0.95, while ,, = 0.8 and ,, = 5; split mesh has No = 5, Omax = 90° and six points at -,,.

e5



Table 2. Percent Error in Scattering Function forMethods 0 and 1 versus Omaxa

MethodOmax 0 1

90 0.63 0.03

80 0.21 0.00

70 1.32 0.03

60 4.48 0.03

a Binomial scattering function has a = 4 and fo = 0.95, while Ma = 0.8 andTo = 5.

of suitable boundary conditions; at A = 1 the derivatives of

I, (r, A, 4) with respect to /.t were made continuous and at / =

cos 0imax, an open boundary, the second derivative of Ii(-r, u)

was taken to be the same at both Omax and the next lower 0node.

A comparison of the results for exact data points betweenthe new collocation scheme and the rectangular mesh is shownin Table 1. For the previously used rectangular mesh, thereis no marked convergence of the values for f,, with increasing

values of N*, but the improved technique is stable and pro-vides (as should be the case) the same result for orders N* >

N = 5, the exact order of scattering. This stability is due tothe calculation of the source vector using the separate grid at,g = -A. Also note that the percent error (fractional errorX 100%) in the scattering function with the new mesh is sig-

nificantly lower for all N*, although the total number of points

per face for this mesh is only 29 (plus six points at -lo for the

face at T = 0), whereas that for the rectangular mesh is 150,consisting of 10 nodes in A and 15 nodes in 0.

The percent error introduced in the scattering function byrestricting the domain of measurements to 0 < Omax is shownin Table 2. This restriction introduces little error, i.e., at Omax

= 600, the error for method 0 is -5%, whereas for method 1

it is -0.03%. Method 1 provides a better estimate for smaller

values of Omax because of the weighting of the angular mo-ments of the collided intensities [Eq. (5a)] by g, and, as is

shown in Table 2, the error is almost constant for Omax 2 60°.The errors in the scattering function with such a restrictionin Omax are negligibly small in comparison with the errorsarising from inexact data, as will be seen in Section 4.

4. EFFECT OF, MEASUREMENT ERRORS ONINVERSION

There are a large number of variables that are potentiallyimportant when studying the effect of measurement errors.Besides measurement errors, the error in inversion is also

dependent on the scattering function studied, the angle ofincidence of the external monodirectional beam, the thickness

of the scattering medium, and the number and the distribu-tion of the collocation-mesh points. Fortunately, the resultsof this section and the analysis in Section 5 show that thebehavior of the inverse methods can be explained in general

terms that are valid except when the slab thickness or thecosine of the incident angle for the monodirectional beambecomes small.

The modeling must include errors not only in the magnitudeof the intensity at each collocation point but also errors made

in measuring the values 4 and 0 at each point, which specifythe direction of the intensity. A Monte Carlo sampling

scheme is used to simulate these errors by introducing per-turbations in the directions or in the intensities as calculatedat each node point using the FN direct transport method. Theinverse method is applied for each case history, after whichthe average values and standard deviations of the f,& coeffi-cients are updated.

The effect of fixed errors in the direction of the incidentbeam is shown in Table 3. As can be seen,-the effect of anerror in j,, i.e., 6,u, is more severe than an error in qO,, 64'. Thisappears to be true in general and is due to the fact that avariation in ,, causes a greater change in the values for S, theleading cause of error, than does a change in 4s. As expected

from Eq. (2) and as has been verified by numerical calcula-tions, the effect of error in the magnitude of the emergingattenuated monodirectional beam is negligible for thick slabsand is not an important source of error.

Errors in the magnitude of the intensity are introduced ona fractional (percentage) basis that, for cases in which theintensity varies by a large factor, provides a better represen-tation than a fixed error. The fractional error X is defined interms of the standard deviation a, so that cr = XI. Here acosine-squared probability-density function has been chosenfor the sampling,

p(x) = cos2(7rx/2), IxI < 1,

where the variation for the intensity 61 is given in terms of therandom variable x as

5I = (1/3 - 2/r 2)-l/2¢x.

This distribution was selected as representative of what canbe expected experimentally. In any case, because of thestructure of the inverse equations in which sums of randomvariables are taken, the exact form for the probability-densityfunction is not important.

The effect of random errors in the magnitude of the inten-sity, as dependent on the number of points used in the splitcollocation mesh, is shown in Table 4. For both methods 0and 1, the error is almost totally independent of the numberof points used in the collocation scheme, and for method 0 theerror in the scattering function is seen to be linear with respectto the error in the intensity. This linearity is due to the factthat the error in inversion is mostly dependent on variationsin S, whereas the variations in the moments for calculationof A are of secondary importance. Thus, from Eqs. (4a) and(3b), for method 0 the error in the scattering function will be

approximately linear in S.

Table 3. Percent Error in Scattering Function forMethods 0 and 1 versus Different Combinations ofErrors in 0 and 0 for the Incident Illuminationa

Method60 6 0 1

0.0 0.0 0.63 0.030.0 1.0 0.81 0.18

0.5 0.0 3.95 2.651.0 0.0 7.87 5.13

1.5 0.0 11.5 7.47

a Binomial scattering function has a = 4 with fo = 0.95, while Ma = 0.8 andTo = 5; split mesh has No = 5, Om_ = 900, and six points at -,uo.


1976 J. Opt. Soc. Am. A/Vol. 2, No. 11/November 1985 J. C. Oelund and N. J. McCormick

Table 4. Percent Error in Scattering Function for Different Split-Mesh Collocation Schemes versus PercentRandom Error in Intensitiesa

Number of PointsPercent Error 29 (NO = 5) _57 (NO = 7) .117 (NO = 10)in Intensity Method 0 Method 1 Method 0 Method 1 Method 0 Method 1

0.00 0.63 0.03 0.28 0.01 0.12 0.000.03 9.16 7.34 8.59 7.10 8.36 6.840.06 18.7 15.0 17.2 15.2 16.7 14.50.12 37.4 33.3 34.4 44.6 33.4 36.80.25 78.0 720. 71.7 235. 69.7 1111.

a Binomial scattering function has a = 4 and o = 0.95, while Mo = 0.8 and T 0 = 5; split mesh has Om = 90° and six points at -,M.

510

4

10

RI 2

10

10

010

-110

1 2 3 4 5 6 7 8

NFig. 2. Ratio R1 of the percent error in scattering function to percenterror in the intensity versus N*. (Binomial scattering function hasa = 8 and to = 0.99 with -r = 5.)

For method 1, however, the error in the scattering functionis nonlinear, as is shown in Table 4. Analytically this followsfrom the fact that the transformation from F to [Eq. (5b)]is hyperbolic. In fact, from the theory of functions of onerandom variable, for F (x), the probability density functionof F,,, the standard deviation for f,& is given by18

a 2 = [X/(1 + x)] 2F (x)dx.

Thus, if F (x) is nonvanishing in the neighborhood of x =-1,then u, becomes unbounded. This has been observed indi-rectly in results from Monte Carlo studies in which the stan-dard deviation was erratic and did not converge with thenumber of case histories. For this reason, method 1 is fun-damentally ill conditioned with respect to measurementerror.

From the strong linear response of method 0 to errors in theintensity and the insensitivity of the results to errors in themoments required for A, it is necessary to study only the ratioRI of the fractional error in the scattering function to thefractional error in the intensity. The Monte Carlo approachwas not used here because, with S the only source of error, it

was possible to calculate analytically the error in the scatteringfunction by using statistics; these errors, however, do not in-clude errors resulting from truncating S at dimension N*.

The behavior of the ratio RI with respect to N* is shown inFig. 2, where eight points at -A, were used for the calculationof S. We see that RI increases exponentially with respect tolarge N*. This is a general result, as verified by other testcases and as shown from the analysis presented in Section 5.Thus method 0 is seen to be ill conditioned with respect toincreasing N*, with the degree of ill conditioning stronglydependent on /I, and decreasing as ,, approaches zero.

Errors in direction are introduced by using the cosine-squared distribution to sample for , the arccosine of -a',the solid angle between the perturbed and the unperturbeddirections. A local azimuthal angle 7 is determined by sam-pling from a uniform distribution, and then the local coordi-nates Land -q are mapped into the perturbed values 0' and 'of the coordinates 0 and . The error in direction is taken tobe the standard deviation for sampling .

The effect of random errors in direction on inversion usingmethod 0 is shown in Fig. 3, where the standard deviation in

4I0

310

210

R

a 1I

010

-I10

-210

1 2 3 4 5 6 7 8*

NFig. 3. Ratio RR of the percent error in scattering function to theerror in direction in degrees. (Binomial scattering function has a =8 and fo = 0.99 with -r = 5.)


the direction is 0.10. (Here the modeling parameters are the

same as those used for Fig. 2.) The error in the scatteringfunction is expressed in terms of Rq, the ratio of the percenterror in the scattering function to the error in direction indegrees. As should be expected, the behavior of R0 with re-spect to N* is similar to the behavior of RI with respect to N*,

and in fact the rate of change of R9 with respect to N* is thesame as for RI, as shown by a comparison between Figs. 2 and

3. This is a consequence of the structure of inverse method

0 and follows from the analysis given in Section 5.

5. ANALYSIS OF ILL CONDITIONING

The numerical calculations have suggested that the methodsare most sensitive to errors incurred in the calculation of S,whereas the errors introduced by the matrix elements of A areof secondary importance. This is due to two features asso-ciated with the structure of A. First, the matrix coefficientsAm,, for high values of m are, to a strong degree, independent

of the error in the collided intensity. This holds because I, (0,

A, 4) is a smooth function of A and 4, so that its projection over

the basis set of spherical harmonics makes a small contribu-tion to the matrix elements for large m relative to the singularcontribution from the incident monodirectional beam at T =

0. Thus, for thick slabs, where the matrix elements will beprimarily dependent on the contributions from the incidentface, the matrix elements for large m will be principally de-

pendent on the incident beam, so that from Eq. (6)

Am, (-1)n-mammnPnm(_/o)2 for large m. (7)

The second feature about the structure of A can be seen by

studying the condition number1 6

CK = I|AIIIIA-111,

which depends on K, a particular value of N*. The condition

number provides an upper bound on the relative error formatrix inversion, so that

ll1601110l < CKOISIM/ISII, (8)

where f and bf are the scattering function and its variation,as expressed over the basis set of Legendre polynomials, and6S is the variance in S. For XK, the largest eigenvalue of A-1 ,

it is true that

IIA-1 || > I XK|,

so that

CK > 11 A111 K1 (9)

Now, from the upper triangularity of A, for large K the largesteigenvalue for the inverse matrix is the reciprocal of thesmallest eigenvalue for the direct matrix AKK, so that fromexpression (7)

1/XK = AKK = aKKPKK(ttO)2 .

Also,K-1

PKK(Ao) = (1 - A2)K/2 f (2k + 1).k=O

Thus the condition number depends on K as

CK/CK+1 IXK/XK+lI = [(2K + 3)/(2K + 1)](1 - ,12),

so that for large K

CK+1 CK(1 - A12)-1. (10)

This estimate succinctly explains the exponential behaviorof the relative error in the scattering function with respect toN* and the dependence on /1o, as was seen in Fig. 2. The ill

conditioning is strongly dependent on ,u4, where, in the limitas ,, goes to zero, the ill-conditioned behavior is seen tovanish.

6. SUMMARY AND CONCLUSIONS

The determination of the scattering properties of a homoge-neous medium with a large number of multiple scatterings isa difficult task. This is because the multiple scatterings serve

to smooth out the angular distributions of the intensity at theboundaries. Nonetheless, in Section 2 it has been shown thatboth methods can yield good estimates for the scatteringfunction using only data at a finite number of directions-provided that the data are exact.

When the measurements made at a finite number of pointsare inexact, however, method 1 is seen to be fundamentallyill conditioned, where the error can become unbounded.Method 0 is better behaved in that it is approximately linearin response to input errors, but the implicit error increasesexponentially with the assumed degree of scattering anisot-ropy so that it too is poorly conditioned. The rate of expo-nential increase of the error in the scattering function torandom errors in the intensity and the measurement directionhas been shown to be strongly dependent on the cosine of the

angle of the incident beam A10, although relatively good esti-mates of the scattering function may be obtainable if ,uo ap-

proaches zero.The ill conditioning actually stems from the properties of

A-', whose eigenvalues are the reciprocal of A,,,. These

values for large n have been shown to be increasing expo-

nentially with respect to n. This behavior for the inversematrix is characteristic of ill-posed linear inversion prob-lems.'9 For such methods, the remedy lies not in choosingbetter numerical techniques for inversion (although poortechniques can make matters worse) but in finding andapplying reasonable external constraints that are not inherentin the inversion equations themselves. Thus, with respect topotential future applications, it must be realized that thisstudy is only an initial step in assessing the potential of theinverse methods investigated here.

ACKNOWLEDGMENT

This research was supported in part by National ScienceFoundation grant CPE-8209908.

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19. S. Twomey, Introduction to the Mathematics of Inversion inRemote Sensing and Indirect Measurements (Elsevier, NewYork, 1977).

sensitivity of multiple-scattering inverse transport methods to measurement errors

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