An approximate method to determine the index of refraction of a dielectric obstacle is presented. For simplicity one dimensional models of electromagnetic scattering are treated. The governing equations yield a second order boundary value problem, in which the index of refraction appears as a functional parameter. The availability of reflection coefficients yield two additional boundary conditions. The index of refraction by a k-th order spline which can be written as a linear combination of B-splines is approximated. For N distinct reflection coefficients, the resulting N boundary value problems yield a system of N nonlinear equations in N unknowns which are the coefficients of the B-splines

Dunn, M. H.

Hariharan, S. I.

English

NASA Technical Reports Server

NASA Contractor Report 166114 
leASE 
NASA-CR-166114 
19830016495 
I \--------------------------
NUMERICAL COMPUTATIONS ON ONE-DIMENSIONAL 
INVERSE SCATTERING PROBLEMS 
Mark H. Dunn 
and 
S. 1. Hariharan 
Contract Nos. NASl-16394, NASl-17070, NASl-17130 
April 1983 
INSTITUTE FOR COMPUTER APPLICATIONS IN SCIENCE AND ENGINEERING 
NASA Langley Research Center, Hampton, Virginia 23665 
Operated by the Universitite Space Research Association 
NI\SI\ 
National AeronautiCS and 
Space Administration 
Langley Research Center 
Hampton. Virginia 23665 
~(BRAR,{ COpy 
I. i IJ:~' ? 6 1983 
lJ.~GLEY RESr:ARI' ~ i'~ENTER 
L1BPARY NASA 
HA\~?TON, VIRGINIA 
i 1111111111111 11111111111111111111111111111111 
, NF02448 
https://ntrs.nasa.gov/search.jsp?R=19830016495 2020-03-21T04:13:51+00:00Z
NUMERICAL COMPUTATIONS ON ONE-DIMENSIONAL 
INVERSE SCATTERING PROBLEMS 
Mark H. Dunn 
Institute for Computer Applications in Science and Engineering 
S. I. Hariharan 
Institute for Computer Applications in Science and Engineering 
ABSTRACT 
In this note we present an approximate method to determine the index of 
refraction of a dielectric obstacle. For simplicity we treat one-dimensional 
models of electromagnetic scattering. The governing equations yield a second 
order boundary value problem, in which the index of refraction appears as a 
functional parameter. The availability of reflection coefficients yield two 
additional boundary conditons. ~"e approximate the index of refraction by a 
k-th order spline which can be written as a linear combination of B-splines. 
For N distinct reflection coefficients, the resulting N boundary value 
problems yield a system of N non-linear equations in N unknowns which are 
the coefficients of the B-splines. 
Research reported in this paper was supported by the National Aeronautics 
and Space Administration under NASA Contract Nos. NASl-16394, NASl-17070 and 
NASl-17130 for the first author and NASl-17070 for the second author while 
they were in residence at ICASE, NASA Langley Research Center, Hampton, VA 
23665. 
i 
Introduction 
In this note we treat a class of inverse scattering problems in one 
dimension. Similar problems hae been studied and motivated by Hagin [1] and 
Gray and Hagin [2]. Our goal, like theirs is to reconstruct an unknown 
function, which we refer to as the index of refraction, to moderate accuracy 
and by simple means. 
The problem we solve is as follows: 
We seek a function n(x) 
problem 
such that the solution u of the boundary value 
o xe(O,1) 
(P) 
satisfies u(O) = few) and u'(O) = g(w), where f and g are known 
functions of the real parameter w. 
This class of problems arises in several situations. In order to 
motivate the above problem we shall describe one such situation arising from 
electromagnetic wave propagation. 
For a general graded index medium, whose refractive index is n(x), the 
time harmonic Maxwell's equation, with single angular frequency k, can be 
written as [3] 
2 
2 
curl H = ikn (~)E~ 
curl E = -ik~~ (M) 
div H = 0 
where Hand E are the electric and magnetic fields and EO and ~O are 
the permittivity and magnetic permeability. 
thickness L (Figure 1) parallel to the y-axis 
air 
n 
o 
o 
y 
n(x) 
Figure 1 
Consider a dielectric slab of 
air 
L 
n 
o 
x 
We assume the magnetic permeability of the slab is the same as for the ambient 
medium (air). Suppose there is a transverse electromagnetic wave 
present. That is to say the electric field ~ and the magnetic field ~ are 
at right angles to each other and to the direction of propagation which is 
3 
taken as the x-axis. E = (O,u(x),O)T satisfies (M)3. H = (o,o,iU~~X))T 
o 
satisfies (M)2 and (M)4. Demanding Hand E to satisfy (M)l gives 
Upon nondimensionalization, in particular 
replacing x with x/L, the above becomes 
o (1.1 ) 
where The bondary conditions are that the tangential 
components of E and H are continuous across the interfaces of the slab. 
This implies that u and u' are continuous at x = 0 and x = 1. The 
problem that we consider is that the index of refraction of the ambient medium 
is a constant nO. 
coming from x = -~ 
The process of wave propagation is initiated by a wave 
inOwx 
of the form e Then the solution of (1.1) for 
x < 0 can be written as 
u(x) 
inowx -inOwx 
e + R(w)e , (1. 2) 
where R(w) is called the reflection coefficient. For x > L part of this 
wave is transmitted and the solution has the form 
u(x) (1.3) 
where T(w) is called the transmission coefficient. Since u and u' are 
continuous at x = 0 and x = 1, then (1.2) and (1.3) imply that 
(1. 4) 
and 
4 
o (1.5) 
In practice the reflection coefficient R(w) can be measured for any desired 
value of w. The continuity of u and u' at x = 0 together with (1.2) 
imply that 
u(O) 1 + R(w) 
and (1.6) 
u'(O) inOw(1-R(W» 
Thus u(O) and u'(O) are known functions of w. 
2. Solution Procedure 
Our solution procedure consists of approximating n(x) with a k-th order 
spline n(x). This choice of approximation is motivated by the need to solve 
the problem (P) quickly and accurately. We will expound on this statement 
after the description of the scheme. 
We begin with a brief description of the spline spaces Sk (.!). Let 
x = { }N+1 xi i=1 be a partition of [0,1], and let Sk(~) denote the space of k-th 
order splines with knots at each xi· seSk(~) implies that in each interval 
[Xi ,xi +1] s is a polynomial of degree at most k-1 and sec (k-2) [0,1]. Let 
{ }N+k-1 Bi,k(x) i=1 denote the B-spline basis for The general properties 
of B-splines are well known and can be found in [4] for example. A property 
for which we will have occassion to utilize is 
xe[xi_k+1,Xi+1]. As a consequence, if f(x) 
xe[x
m
,xm+1] then f(x) is simply given by 
that Bi k(x) = 0 if 
N+K-1 ' 2 aiBi k(x) and if 
i=l ' 
5 
(2.1) 
The approximation of n proceeds as follows: 
For given Nand k we seek an approximation 
n (x) 
(N) 
(2.2) 
where N = N + k - 1, such that, given the distinct pairs (Wj,R(Wj )) 
j = 1,···,t, where t = rN/21, the A(N),s are chosen such that the solutions i 
u. of the initial value problems 
J 
~" + wj2u2 (x)u. 0 j (N) J 
u. (0) 
J 
(I) 
1,2,···,t. This is equivalent to solving 
where A = (A(N) ••• A(N))T 
- l' , N 
F(A) = 0 
and E(~) = (fl(~),···,fN(~))T and where 
and f 2j _1(l) = Imag((uj-iWjUj ) (1») 
(2.3) 
j = 1,2,···,t. Before proceeding to the motivation we note that the system 
(2.3) is highly nonlinear. 
In order to solve (2.3) one must solve (I) many times. In general this 
requires the numerical solution of these problems which leads to a large cost 
6 
in computer time. Our choice of approximation reduces this time to a great 
extent. For examp Ie, when k lone can obtain the solution of (I) in 
closed form, and when k ~ 2 one can easily obtain truncated Taylor 
expansions, whose accuracy is comparable to the numerical integrations of 
(I), but the time taken to compute the expansions is, in general, much less 
than the time taken to integrate (I) nUMerically. The Taylor expansions are 
computed as follows: For is a polynomial of degree at 
most k - 1 and is given by 
n (x) (i) 
m+k-1 (N") 
L \ Bi k(x). 
i=m ' 
This follows from (2.1). Let and l\n(x) denote two linearly 
independent Taylor series solutions, expanded about xm' of (1)1. Then for 
is given by ~.(x) = a S (x) + b H (x), J m m m m (I) the solution of 
where the am's and bm's are chosen so that u j and uj are continuous at 
each xm and also to satisfy the initial conditions. 
It is known [4], that if fEC(j) [0,1] for j = 0,1,···,k-1, then 
inf IIf-sll 
00 
SESk(~) 
where h max (xi+Cxi)' Ck . is a constant which depends only on k and l~i~N ,J 
j, and 
sup{lf{j)(x)-f(j)(y)l: x,yE[O,l], Ix-yl(h} 
is the modulus of continuity of f(j) at h. It is also known that the above 
estimate cannot be improved. Thus, if nEC(k) [0,1], then 
7 
lln-~ II w(n(k-l;)h) C hlln(k)lloo 
o (hk ). As will be seen, in 
and the best that we can hope for is (N') 00 
several numerical examples tested the optimal 
convergence rate was attained. 
3. Numerical Results 
In this section we present some numerical examples for k = 1,2 and 3 
with knot sequence { i-l}N+l N i=l· In all examples the Levenberg-l1arquardt 
algorithm [5] was used to solve (Z.3). The R( w.), s were computed by 
J 
inputting the exact solution n(x) into (P). 
To solve a large non-linear system of equations it is usually necessary 
to have a good initial approximation to the solution. For fixed Nand k 
our solution was built up according to the following algorithm: 
(i) Select a sequence of integers ~ = {mi}~=1 with ml 
and 
(ii) Compute by solving (Z.3) with an initial guess 
1. 
k, IIJ.t N 
(iii) For j = 1,000 ,M-l compute n( ) by solving (Z.3) with the 
_ mj +1 
initial guess chosen so that
m
.:Im
j
+
1
) interpolates TI(m
j
) at the 
mj +1 distinct points {Y.t} .t:l. This gives a linear system for (m.+l) 
the initial Ai J 's that is invertible if and only if 
.t = 1,000 ,mj+l [4]. 
We found that the best results were obtained by setting m 
for k = 1 and ~ = {Zi}~=1 for k = Z and 3. 
{Zi}M i=1 
8 
The maximum attainable N is limited by considerations of computer time 
and accuracy. In the examples tried we found that satisfactory results were 
obtained if N = 32 for k = 1 and N varied between 10 and 14 for k = 
2 and 3. 
Example 1. (Figures 2, 6, 10) 
n(x) = 1 + x2 
k 
N 
IIn-u(N) II"" 
computational time (cpu sec.) 
estimated convergence rate 
Example 2. (Figures 3, 7, 11) 
xe[ 0 ,-!)U( i, 1] 
+ sin221T(x- t) 
Table I 
1 2 3 
32 12 10 
.036 .0064 2.4 x 10-6 
123 31.2 8.30 
1.0 2.1 
k 
N 
IIn-n(N) 1100 
computational time (cpu sec.) 
estimated convergence rate 
Example 3. (Figures 4, 8, 12) 
n(xl = {: 
k 
IIn-n(N) 1100 
xe[ O,i) 
xe[i, 1] 
computational time (cpu sec.) 
estimated convergence rate 
Table II 
1 
32 
.16 
195 
1.1 
Table III 
1 
16 
.50 
25.6 
9 
2 3(a) 
12 10 
.058 .021 
101 35.7 
.96 
2 3 
8 10 
.46 .47 
20.2 104 
(a)These results were obtained by demanding the initial approximation to 
interpolate the final approximation for k = 2. 
10 
Example 4. (Figures 5, 9, 13) 
n(x) ={ t; 2x xe[ o,i) xe[i ' i] 
xe( i ,1] 
k 
N 
IIn-o(N) 1100 
computational time (cpu sec.) 
estimated convergence rate 
4. Discussion 
Table IV 
1 2 3 
32 14 10 
.050 .028 .025 
09.2 117 51.1 
1.1 1.1 
In all examples with k = 1 and 2, except for example 3, the optimal 
convergence rate was attained. 
It is obvious from the results listed in Tables 1 - 4 that if a desired 
accuracy is specified, then the minimum computational time is achieved by 
increasing the order of the splines rather than increasing the number of 
unknowns for a fixed order. 
In practice the measurement of the reflection coefficients is subject to 
experimental error. To simulate this situation we introduced Gaussian type 
noise in the values of R(w). Stable results were obtained if the amplitude 
of the noise was less than 10-6• 
11 
Our results are comparable in accuracy and computational time to those 
obtained in [1] and [2]. However, our method is applicable to any boundary 
value problem in which the unknown appears as a parameter and sufficient 
additional conditions are known. 
12 
REFERENCES 
1 F. HAGIN, Some numerical approaches to solving one-dimensional inverse 
problems, J. Compo Phys., Vol. 43, No.1, (1981), pp. 16-30. 
2 S. GRAY AND F. HAGIN, Toward precise solution of one-dimensional 
velocity inverse problems, SIAM J. Appl. Math., Vol. 42, No.2, 
(1982), pp. 346-355. 
3 D. MARCUSE, "Light Transmission Optics," Van Nostrand Reinhold, New 
York, 1972. 
4 C. deBOOR, "A Practical Guide to Splines," Springer-Verlag, New York, 
1978. 
5 R. FLETCHER, "Practical Methods of Optimization," Vol. 1, John Wiley & 
Sons, New York, 1980. 
2.5 
2.0 
R 
'-' 1. 5 
z 
1.0 ................ -
K-l 
• 
• 5 L--___ '--___ J...-___ --'-___ ---1 
Figure 2 
2.5 .-
• 2.0 fo- • 
-
• 
• 
'""' ~ 1.5 fo- • '-' z 
• 
1.0 • • • • 
• 
• • 
.5 I I I I 
a .25 .50 .75 1.00 a 
X 
Figure 4 
exact solution 
• computed solution 
• • • 
.25 
13 
Figure 3 
.50 
X 
Figure 5 
• • 
••• 
.75 1.00 
14 
K-2 
2.5 
2.0 
R 
....., 1.5 
z 
1.0 L-_ ....... ,...... 
.5L-____ -L ______ ~ ______ ~ ____ ~ 
Figure 6 
2.5 r-
• 2.0 I- • a 
-. -
• 
......... 
:xl Z 1.5-
• 
1.0~~~a~a~.~. ___ • __ J 
.5L-__ ~1 ____ ~1 ____ ~1 ____ ~1 
a .25 .50 .75 1.00 a 
X 
Figure 8 
exact solution 
• computed solution 
.25 
Figure 7 
.50 
X 
Figure 9 
.75 1.00 
K-3 
205 
2.0 
R Z 1.5 
1. 0 .L..-__ "'--
05~ ______ ~ ______ ~ ______ -L ______ ~ 
Figure 10 
2.5 r-
• • • • 2 0 0 :-
- .. 
• 
1.5-
• 
• 1.0~~-~·~~--~--~ 
••• 
05~ ____ ~1 ____ ~1 ____ ~1~ __ ~1 
o .25 .50 .75 1.00 
X 
Figure 12 
o 
exact solution 
• computed solution 
.25 
15 
Figure 11 
.50 
X 
Figure 13 
.75 
1 Report No I 2 Government Accession No 3 Recipient's Catalog No 
NASA CR-166114 
4 Title and Subtitle 5 Report Date 
NUMERICAL COMPUTATIONS ON ONE-DIMENSIONAL INVERSE Apnl 1983 
SCATTERING PROBLEMS 6 Performing OrgaOlzatlon Code 
7 Author(s) 8 Performing Organization Report No 
Mark H. Dunn and S. 1. Har1.haran 83-5 
10 Work Unit No 
9 Performing Organization Name and Address 
Institute for Computer Applicat1.ons in Science & Engineer1. 
:fl, Contract Zr Grant No Ma1.l Stop l32C, NASA Langley Research Center NASl-1639 , NASl-17070 
Hampton, VA 23665 NASl-17130 
13 Type of Report and Period Covered 
12 Sponsoring Agency Name and Address 
Contractor Report Nat1.onal Aeronautics and Space Adm1.n1.strat1.on 
Wash1.ngton, DC 20546 14 Sponsoring Agency Code 
15 Supplementary Notes 
Langley Techn1.cal Mon1.tor: Robert H. Tolson 
Final Report 
16 Abstract 
In th1.s note we present an approx1.mate method to determ1.ne the 1.ndex of refract1.on 
of a d1.electr1.C obstacle. For s1.mpl1.c1.ty we treat one-d1.mensional models of electro-
magnet1.c scatter1.ng. The govern1.ng equat1.ons Y1.eld a second order boundary value 
problem, 1.n wh1.ch the 1.ndex of refraction appears as a funct1.onal parameter. The 
ava1.lab1.l1.ty of reflect1.on coeff1.c1.ents Y1.eld two add1.t1.onal boundary cond1.t1.ons. 
We approx1.mate the 1.ndex of refract1.on by a k-th order spline wh1.ch can be wr1.tten as 
a l1.near comb1.nat1.on of B-spl1.nes. For N dist1.nct reflect1.on coeff1.c1.ents, the 
result1.ng N boundary value problems Y1.eld a system of N non-linear equat1.ons 1.n 
N unknowns which are the coeff1.c1.ents of the B-spl1.nes. 
17 Key Words (Suggested by Author(s)) 1 B Distribution Statement 
Electromagnet1.c scattering, B-spline 
reflection coeff1.cients, index of Unclass1.f1.ed-Unl1.m1.ted refract1.on 
Subject Category 33 
19 Security aasslf (of thiS report) 20 Security Classlf (of thiS page) 21 No of Pages 22 Price 
Unclass1.f1.ed Unclass1.f1.ed 17 A02 
For sale by the National Technical InformatIOn Service, Springfield, Virginia 22161 
NASA-La n9 1 ey. 1983 
End of Document 


Numerical computations on one-dimensional inverse scattering problems

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