Journal of Electromagnetic Analysis and Applications, 2012, 4, 492-496
http://dx.doi.org/10.4236/jemaa.2012.412069 Published Online December 2012 (http://www.SciRP.org/journal/jemaa)
Reconstruction Parameters of Local Scattering Sources of
a Metallic Strip from the Backscattering Pattern
Stanislav N. Kutishchev
Department of Physics and Chemistry, Voronezh State University of Architecture and Civil Engineering, Voronezh, Russia.
Received October 11th, 2012; revised November 13th, 2012; accepted November 23rd, 2012
ABSTRACT
In this paper, it is proved the ability of quantity reconstruction, amplitudes and coordinates of metallic strip local scat-
tering sources from the backscattering pattern. They are performed as the results of numerical solution for the infinite
perfect conducting strip in case of E-polarization of the incident plane electromagnetic wave. In this case it is necessary
to fulfill the following conditions. The local sources amplitudes should be the same order, in transverse and longitudinal
directions the local sources should be separated into distances more than apparatus resolution, and the object maximum
size does not have to be more than approximately 50λ. It was shown the limit and ability of the further development of
the offered method.
Keywords: Backscattering Pattern; Local Scattering Sources; Radar Image; Electromagnetic Wave; Method; Metallic
Strip
1. Introduction
Nowadays the researches of creating of objects scattering
structure reconstruction methods in scattering field in
radiolocation [1-7], antenna theory [8], radio astronomy
[9], optics [10] and other scientific regions are being done.
Urgency of radiolocation work in this region is brought
about, for example, the essential of getting rather full in-
formation about space structure of local scattering sources
[11] on the surface of complex shape objects with the pur-
pose of improving means and methods of the objects radar
visibility decreasing.
Doing radiolocation characteristic research of objects at
ranges and in laboratories the scattering pattern is built as
a result of the object turning. So reconstruction problem of
local scattering sources of this object appears from the
backscattering pattern. For the object model [12] as the
system of isotropic rigidly tied and electrodynamically
independent scatters in work [7] the method of numerical
solution of the problems was offered. It is interesting to
use this method for the solution of analyzing problem in
case of the infinite (along z axis) perfect conducting strip
(Figure 1). The strip is believed to be infinitely thin.
Work aim is the researching of the reconstruction abi-
lity of parameters (quantity, amplitudes and coordinates)
of metallic strip scattering sources from the backscattering
pattern.
The reconstruction ability of the parameters of the me-
tallic cylindrical object (metallic strip) local scattering
sources has been first investigated in the paper. The solu-
tion of the inverse electrodynamical problem is based on
apriori assumption about discretion of local sources the
cylindrical object (the model of isotropic rigidly tied and
electrodynamically independent scatters). The dependences
of the reconstructed amplitudes and transverse coordi-
nates of two local sources of the metallic strip on the
aspect angle (the case of a monostatic scattering) have
been first numerically obtained. As the result of analysis
of the obtained numerical data it was found that the ab-
solute error of the reconstruction of the transverse coor-
dinates of the local sources does not exceed approxi-
mately 0.03λ. The limits of the offered method were esti-
mated.
2. Method of Reconstruction
Let’s look at the case of a monostatic scattering of the
plane electromagnetic wave from the system of N iso-
tropic rigidly tied and electrodynamically independent
scatters (Figure 2) which is the electrodynamical model
[12] of the object. Receiving narrow-band reflected signal
in the far zone of the object and transmit-receive antenna
the backscattering pattern including the results of work [2]
and the problem geometry (believing that measurements
are done in xy-plane) can be performed as:

 
0
1
exp 2cossin
N
iii
i
EEjkxy




 


, (1)

Reconstruction Parameters of Local Scattering Sources of a
Metallic Strip from the Backscattering Pattern
493
Figure 1. Metallic strip.
Figure 2. Object model.
where xm, ym are the coordinates of m-scatter; m is the
amplitude of the signal scattering from m-scatter; φ is the
observation angle calculated from x-axis directed along
bisector of angles Δφ;
E
2πk

 is the wave number; λ is
the wavelength. Formula (1) is right for every observation
angles φ.
When 21

= formula (1) can be linearized and
changed as:

  
0
1
exp 2,sin2
N
ii
i
EuE jyuukk 


%=, (2)
where ,

exp 2
nn n
EE jkx
%

sinuk

—spatial fre-
quency.
So this problem can be expressed as: for observing ob-
ject model is necessary to find the number of local scat-
tering sources N, their amplitudes iand their transversal
and longitudinal coordinates i and i
E
y
x
according to the
known from experiment backscattering pattern ,
where


0
Eu

sin 2uk k  =.
The method of the solution of the considered problem
consists of some stages [7].
Stage 1. From fragment


0
E

, which is known at
21   =, quantity of local sources N, their trans-
verse coordinates iand amplitudes are found. Namely,
spatial frequency spectrum
y


,Eu



is found out from
the fragment known from measurements backscattering
pattern of the object

(0)
E

(1):


 

0,sin2
,0,sin2 .
Eu ukk
Eu uk








,
(3)
Writing the spatial frequency spectrum


,Eu


in
(3) a rectangular window [13] is used, legal using of
which is connected with the small value of observation
angles sector Δφ.
Using spatial frequency spectrum

,Eu


 one-di-
mensional radar image


,Jy

 of the object [3-7,14] is
found:
 


1
1
,,exp
2π
sin 2
2,
2π2
2
i
N
i
i
i
d
J
yEujyu
kyy
kEkyy







 








u
(4)
From this formula it is known that the number of main
maxima equals to the number of local sources N. The
transversal coordinates of the local sources equal a half of
the transverse ones going with main maxima. The values
of the main maxima of the one-dimensional radar image


,Jy


(4) correspond to the amplitudes of the
object local sources.
i
E
Stage 2. Using the fragment


0
E

which is known
at 021
   =, with algorithm help of the 1st
stage the transverse coordinates i of the local sources
in x'y' coordinates system turned to the angle 0
y

with
respect to xy coordinates system (Figure 2) are calculated.
Let us look at the backscattering pattern


0
E

in
the aspect angles sector 021
    (Figure 2).
Using the well-known relations [15] for the coordinate
transformation of the local scattering sources let us write
it in x'y' coordinates system as [4]:




  
0
00
1
0
exp 2cossin,
21.
N
ii i
i
E
Ejkx y

 
 


 






(5)
In this situation 0

can take any values.
From the match up (5) and (1) it is summed that the
algorithm of calculation of the transverse coordinates i
y

of the second stage is analogous to the algorithm of cal-
culation of the transverse coordinates of the first
stage.
i
y
From the resulting local sources transverse coordinates
,
ii
yy

their longitudinal ones are calculated (Figure
2) [4]:
i
y


0
ctg1, ,
ii
x
yi  N. (6)
From formula (6) it is known that if we increase 0

, an
error of definition of the local sources longitudinal co-

Reconstruction Parameters of Local Scattering Sources of a
Metallic Strip from the Backscattering Pattern
494
ordinates decreases. For
090
 ο
ii
x
y

 . (7)
In this case the error of the reconstruction of the longi-
tudinal coordinates in xy system equals to the error of the
reconstruction of the transverse coordinates in x'y' system.
So at the second stage it is better to choose 090
 
ο
.
3. Numerical Results and Discussion
Later there are the results of the solution of the observing
problem for the metallic strip with the lower edge co-
ordinates (0, −5λ) and upper edge coordinates (0, 5λ)
(Figure 1). The backscattering pattern of the strip was
calculated by the rigorous method of the integral equa-
tions [16] in case of E-polarization (E is directed along z
axis) of the incident plane monochromatic electromag-
netic wave with the amplitude equals 30.
In Figure 3 the dependences of the reconstructed trans-
verse coordinates of two local sources of the strip on the
aspect angle φ are shown. The value of the aspect angles
sector ∆φ = 12˚. In this case two local sources starting
with φ0 = 6˚ are reconstructed when the normal angle to
the strip surface is not thrown into the aspect angles sector
(φ0 = 0˚). Curves 1 and 2, Figure 3 practically equal, so
the first local source is located on the lower edge of the
strip. For the aspect angles φ0 > 48˚ the amplitude of the
first local source is extremely small (Figure 4) and this
local source is not reconstructed. Curves 3 and 4, Figure 3
practically match, so the second local source is located on
the strip upper edge for the aspect angles (6˚, 90˚). The
local scattering sources of the strip are located on its sur-
face, so the reconstruction of its longitudinal coordinates
is obvious.
In Figure 4 the dependences of reconstructed ampli-
tudes of two local sources of the strip are shown on the
Figure 3. The dependence of the reconstructed transverse
coordinates of two local sources of the strip on the aspect
angle φ0 (1) The reconstructed coordinates of the first local
source; (2) Coordinates of the strip lower edge; (3) The re-
constructed coordinates of the second local source; (4) Co-
ordinates of the strip upper edge.
Figure 4. The dependence of the reconstructed amplitudes
of two local sources of the strip on the aspect angle φ0. (1)
The first local source; (2) The second local source.
aspect angle φ0. Their amplitudes are practically equal for
small aspect angles. Increasing the observation angle φ0
the amplitude of the second local source starts over the
amplitude of the first local source. For the aspect angles φ0
> 48˚ the amplitude of the first local source becomes
negligibly small, as the amplitude of the second one at φ0
= 90˚ equals 4.67.
Further as an illustration the results of reconstruction of
the local sources of this metallic strip for case ∆φ = 12˚
and φ0 = 28˚ are shown.
The curve Figure 5 is a fragment of the amplitude
backscattering pattern


Eu of the strip in spatial fre-
quency sector u

[k sin(22˚); k sin(34˚)], (∆φ = 12˚, φ0
= 28˚).
The curve Figure 6 is a fragment of the phase back-
scattering pattern


arg u (let us notice that








exEu Euup argj) of the strip in spatial fre-
quency sector u

[k sin(22˚); k sin(34˚)] (∆φ = 12˚, φ0 =
28˚).
In Figure 7 the modulus of the one-dimensional image
of the strip
J
for u

[k sin(22˚); k sin(34˚)] (∆φ = 12˚,
φ0 = 28˚) is shown. Two main maxima correspond to two
local sources (N = 2). The reconstructed transverse coor-
dinates of the local sources r (Table 1) equal to a half
of transverse coordinates of the corresponding main maxi-
ma. The transverse coordinates y' (Table 1) correspond to
the transverse coordinates of the strip edges. The modulus
values of the main maxima of the one-dimensional image
y


,Jy


correspond to the modulus of the recon-
structed amplitudes of the local sources r E (Table 1).
As the result of analysis of the obtained numerical data
(Figure 3) it was found that the absolute error of the re-
construction of the transverse coordinates of the local
sources does not exceed approximately 0.03 λ. The analy-
sis of numerical results (Table 1) affords us to summarize
that in the considered example (∆φ = 12˚, φ0 = 28˚) the
absolute error of the reconstruction of the transverse co-
ordinates of the local sources 0.03y

 .

Reconstruction Parameters of Local Scattering Sources of a
Metallic Strip from the Backscattering Pattern
495
Figure 5. The amplitude backscattering pattern of the strip
in spatial frequencies sector u

[k sin(22˚); k sin(34˚)] (∆φ
= 12˚, φ0 = 28˚).
Figure 6. The phase backscattering pattern of the strip in
spatial frequencies sector u

[k sin(22˚); k sin(34˚)] (∆φ =
12˚, φ0 = 28˚).
Figure 7. The modulus of the one-dimensional image J of
the strip for u

[k sin(22˚); k sin(34˚)] (∆φ = 12˚, φ0 = 28˚).
Table 1. Data of the strip local scattering sources.
i y' r
y r E
1 −4.42λ −4.45λ 2.74
2 4.42λ 4.45λ 7.35
4. Conclusions
As a result, this method allows us to reconstruct from
backscattering pattern the parameters (quantity, ampli-
tudes and coordinates) of isotropic stiff-tied and electro-
dynamically independent local scattering sources of a me-
tallic strip. In this case it is necessary to fulfill the fol-
lowing conditions. The local sources amplitudes should be
the same order, in transverse and longitudinal directions
the local sources should be separated into distances more
than resolution


2



, and the object maximum
size does not have to be more than approximately 50λ.
Further it is planned to use this method for the recon-
struction of the local scattering sources of two-dimen-
sional cavities of a complex shape.
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Metallic Strip from the Backscattering Pattern
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