MATCHING SWERLING MODELS WITH RADAR CROSS SECTION
PROJECTION INSTRUMENT ON MILITARY PLATFORMS
Abstract:
Although the use of “Swerling Target Models” is a well-known statistical approach found in
the 1950s, today it is still extensively used especially for target RCS production at radar simulators
due to its fast and simple solution performance. While it has lots of application areas, to the best
knowledge of the authors, no research on the accuracy of Swerling Target Models has been
reported which uses well-developed Radar Cross Section (RCS) prediction tools. This is in spite of
the fact that powerful RCS prediction methods and tools have been around during the last two
decades.
In this paper, the Swerling Target Models of selected targets are compared with the RCS
Probability Density Functions (PDFs) obtained with a Shooting and Bouncing Ray (SBR) based
RCS computing program (RKAY RCS Prediction Program). This code has been developed inhouse and has been validated against results found in the literature. As a result of comparison it is
seen that the Swerling Target Models are generally coherent with the RCS results of the prediction
tool but since they model targets too generally, better stochastic RCS models need to be
developed.
Probability Density Functions of RCS of several (simple and complex) targets generated with
the prediction tool will be presented at the conference. Convergence of these PDFs with respect to
angular sampling rate will be discussed. Comparison of these PDFs with appropriate Swerling
models will be made and discrepancies will be highlighted. Possible directions for developing new
stochastic target models will be suggested.
Key Words: Swerling Target Models, Radar Cross Section, Probability Density Functions, RKAY
RCS Prediction Program.
1.
Introduction:
The first studies were done to solve the problem of detection of the signal in noise after
WWII. However, these studies were not published before 1960s. After that, these studies
continued in two ways. In one side Chi-Square Model, Log-Normal Model and Rice Model were
suggested and on the other, parallel to the development of computational techniques calculation
algorithms developed for detection.
Many of today’s radar targets have complex architecture. Targets could be modeled with
different statistical models when viewed from different perspectives or frequencies. This
necessitates lots of mathematical models to characterize target Radar Cross Section (RCS).
In 1960, Marcum J.I. published his study on Determination of Detection Probability of the
Stationary RCS Targets. Then, mathematical models developed for determining detection
probability of fluctuating RCS targets were published by American mathematician Peter Swerling.
Swerling Target Models (STMs) are used in radar simulators due to its fast computation
speed. Contrary to RCS prediction tools, STMs generate quick solutions by a simple algorithm.
Since it’s based on statistical computation method, it is independent of the detection of the
target.
For STM, it’s enough to know of Freedom Level (m) and Average RCS Value (sigma ort) to
obtain the statistical RCS outputs. Even though STM has a simple algorithm and statistical
computational method it produces satisfactory results for certain targets.
Modern RCS Prediction Tools, unlike STM make RCS calculations with more sophisticated
algorithm. With these tools, targets’ RCS are calculated in accordance with physical laws. Thus,
RCS results are directly related with the targets’ geometrical form. It also increases the
computation time due to the sophisticated algorithm of the Modern RCS Prediction Tools. It’s
widely used in especially while designing low RCS targets.
In Modern RCS Prediction Tools, frequency, polarization, measurement technique,
computation numbers and measurement of several parameters such as angle is required to be
entered into the program. Even though it will bring extra load to take part in so many variables, it
contributes to get more detailed and precise results.
2.
Paper Contents and Requirements:
Among the studies related to the RCS computation methods there aren’t any study that
shows the performance of STMs by comparing it with a Modern RCS Prediction Tools. (to the best
knowledge of the author).In this paper the author aims to analyze the performance of the Swerling
Target Models. In this context, probability density functions (PDFs) has been used as a
comparison tool. With the STMs it can be obtained only PDF as an output of the target RCS
computation. In the study, it is used “RKAY RCS Prediction Program” which is based on shooting
and bouncing ray (SBR) method as a Modern RCS Prediction Tool. After a period of process done
to RKAY RCS Prediction Program, it’s gained the PDF graphs. The final results are obtained by
making comparisons of STM’s and RKAY RCS Prediction Program’s PDF results.
3.
Swerling Target Models:
Even though it’s accepted that the signals returning from the target are fixed, if the target is
moving they are not so. The RCS of the sophisticated structure targets are directly related with its
surface, polarization and frequency. That's why returning signal changes when the surface of the
target changes according to the radar position.
It should be especially known that the PDF of the target and the time dependent correlation
specifications in order to express the fluctuations due to the movement of a target RCS correctly.
However, generally it’s not practical for many radars to make computations providing these
experimental information. Swerling determined considerable statistical models for RCS fluctuations
and he tried to get a method to compute the target’s RCS computations more economically by
analyzing them mathematically. For these computations he used Chi-Square Distribution to model
the statistical fluctuations. He calculated probability of detection for 5 different RCS models. RCS
fluctuations models known as Swerling Conditions are shown in Table 1. Chi-Square Distribution is
mathematical model which shows statistical fluctuations.
(1)
(2)
Swerling Condition 1-2 characterizes the behavior of a lot of independent and similar size
objects. This is suitable for many flying targets. Condition 3-4 is suitable for the targets which have
big body with small particles on it. This is suitable mostly for “Ship” and “Guided Missile (G/M)”
targets. In Swerling Condition 5 also named as Swerling Condition Zero there is no fluctuation in
the target RCS. It’s stable and used for comparison. It’s analyzed widely by Marcum. The Sphere
could be exemplified for Swerling Condition 5 since it has the same RCS in every respect.
(3)
Table 1-The Specifications of the Swerling Conditions
Target
Model
Target RCS’s
PDF (Freedom
Level)
1
Target RCS’s Fluctuation Rate
Target
Geometry
Sample
Objective
s
Rayleigh (2)
Slow (fluctuation scan to scan)
2
Rayleigh (2)
Fast (fluctuation pulse to pulse)
Numerous
independent,
equivalent
RCS
Airplane,
Battle
Tank
3
Chi-square (4)
Slow (fluctuation scan to scan)
Ship, G/M
4
Chi-square (4)
Fast (fluctuation pulse to pulse)
Small sections
around a large
RCS
5
Fixed (∞)
No fluctuation
Stationary
RCS targets
Sphere
4.
Techniques and Methods Used in RCS Analysis:
The techniques and methods used in the context of the paper are shown below.
a.
Designing targets for RCS analysis by means of Rhinoceros 3D Design Program,
b.
Computing RCSs of designed targets via RKAY RCS Prediction Program,
c.
Obtaining PDF graphs according to RCS results,
d.
Inputting m and σort into STM and gaining PDF graph with a MATLAB program,
e.
Comparing RKAY RCS Prediction Program outputs with Swerling PDF outputs for “2
norms” and “∞ norms”.
5.
Designing Targets:
The targets which are shown below are designed by Rhinoceros 3D Design Program. In the
context of the paper it’s assumed that these targets are perfectly conductor.
First of all, it is created 3 main targets and then created 3 more targets for each of them. The
later targets are derived from those produced first. (Totally 12 targets are produced.) Main targets
are G/M, Airplane and Ship. First of the subordinate targets are comprised of the Ellipsoid, second
of the subordinate targets are comprised of the Polygon, third of the subordinate targets are
comprised of the mixture of the both Ellipsoid and Polygon targets and the last subordinate targets
are the real platforms.
Figure 1-G/M Ellipsoid, G/M Polygon, G/M Ellipsoid with Small Parts and Penguin Models.
Figure 2-Airplane Ellipsoid, Airplane Polygon, Airplane Ellipsoid with Small Parts and F-18
Models.
Figure 3-Ship Ellipsoid, Ship Polygon, Ship Ellipsoid with Small Parts and Ship Models.
The targets’ RCSs are computed via RKAY RCS Prediction Program for the conditions
shown in Table 2.
Table 2-Information on the Analysis Conducted in the Paper
Target
Lengt
h
Horizontal
Angle
Scanning
Ranges
Vertical
Angle
Polarization
Analysis
Frequency
G/M
3m
360o
1o
45o(fixed)
Vertical-Vertical
3/10/15
GHz
Airplane
30 m
360o
2o
45o(fixed)
Vertical-Vertical
3/10/15
GHz
Ship
100 m
360o
2o
45o(fixed)
Vertical-Vertical
3/10 GHz
6.
RCS Computations of Targets via RKAY RCS Prediction Program:
After creation of targets with Rhinoceros 3D Design Program, their RCSs are calculated with
RKAY RCS Prediction Program. The outputs of the computations are processed with a MATLAB
program.
G/M Ellipsoid Elipsoid (3 GHz, 1o Horizontal)
missile elips 3GHz 1lik
35
RKAY Olasılık Yoğunluk Fonksiyonu
RKAY PDF
Probability
Density
Function
(PDF)
Olasılık Yoğunluk Fonksiyonu
30
25
20
15
10
5
0
0
0.05
0.1
0.15
0.2
0.25
0.3
Radar Kesit Alanı
0.35
0.4
0.45
0.5
Radar Cross Section (RCS)
Figure 4-The PDF Graph of G/M Ellipsoid Model Obtained From RKAY RCS Prediction
Program at 3 GHz Scanned with 1o Steps.
7.
Creating Swerling PDF Graphs:
STMs are comprised of “σort” Chi-Square Distribution. While for Swerling Condition
1-2 Equation 1 is used, for Swerling 3-4 Equation 2 is used.
It’s created a MATLAB program to monitor the fluctuations of the STMs’ PDFs by means of
the parameter “m” and “σort.”
8.
Comparing the PDF Graphs of RKAY RCS Prediction Programs and Swerling Target
Models:
All the “m” and “σort ” are tried to discover the most compatible Swerling PDF graph with the
RKAY Prediction Program generated PDF graph. The PDF graphs are compared according to “2
norm” and “∞ norm”.
Target “G/M Ellipsoid” is scanned at 3 Ghz frequency, with horizontal 1 degree steps which is
shown in Figure 5-6.
G/M Ellipsoid Modelmissile
(3 GHz,
1o Horizontal, σort=0.04, m=1.7, norm 2)
elips 3ghz 1lik-norm *** Ortalama Radar Kesit Alanı=0.04 ve m=1.7
35
RKAY Olasılık Yoğunluk Fonksiyonu
RKAY
Swerling PDF
Olasılık Yoğunluk Fonksiyonu
Swerling PDF
Probability
Density
Function
(PDF)
Olasılık Yoğunluk Fonksiyonu
30
25
20
15
10
5
0
0
0.05
0.1
0.15
0.2
0.25
0.3
Radar Kesit Alanı
0.35
0.4
0.45
0.5
Radar Cross Section (RCS)
Figure 5-Comparison of Swerling Target Models and RCS Prediction Program’s PDFs’
Graphs of G/M Ellipse Model at 3 GHz Frequency with Horizontal 1o Steps in Norm 2.
In figure 5-6, while red stripe shows the STM’s PDF graph, blue sticks show the RKAY RCS
Prediction Program’s PDF graph.
G/M Ellipsoid Model
(3 GHz, 1o Horizontal, σort=0.045, m=1.7, norm ∞)
missile elips 3ghz 1lik-MAX *** Ortalama Radar Kesit Alanı=0.045 ve m=1.7
35
RKAY Olasılık Yoğunluk Fonksiyonu
RKAY
PDF
Swerling Olasılık Yoğunluk Fonksiyonu
Swerling PDF
Probability
Density
Function
(PDF)
Olasılık Yoğunluk Fonksiyonu
30
25
20
15
10
5
0
0
0.05
0.1
0.15
0.2
0.25
0.3
Radar Kesit Alanı
0.35
0.4
0.45
0.5
Cross
Section (RCS)
Radar Radar
Kesit Alanı
(metrekare)
Figure 6- Comparison of Swerling Target Models and RCS Prediction Program’s PDFs’
Graphs of G/M Ellipse Model at 3 GHz Frequency with Horizontal 1o Steps in Norm ∞.
9.
The Results of the Target “G/M Polygon” at 15 GHz:
G/M Polygon target model comprises of polygons and hexagonal cylinder body with wings.
(Figure 1) In Figure 7, it’s shown the PDF graph of the target “G/M Polygon” at 15 Ghz in RKAY
RCS Prediction Programs.
It’s discovered the STM’s most compatible Chi-Square distribution with RKAY PDF graph for
norm 2 and norm ∞.
In this condition, norm 2 is used as well. (Figure 7) It’s discovered σort = 0.03 and
m= 1 the average with the result of the graph.
It is stated that G/M and Ship model are categorized as Swerling Condition 3-4 and the
Airplane is categorized as Swerling Condition 1-2. Here, G/M Polygon model, Ellipsoid Small Parts
model and Penguin model are categorized as Swerling Condition 1-2 due to their relatively big
wings and fringes. The Swerling graph confirms this.
G/M Polygon Güdümlü
Model
(15 GHz, 1o Horizontal, σort=0.03, m=1, norm 2)
Mermi Poligon Modeli-15Ghz-1lik-norm *** Ortalama RKA=0.030 ve m=1 (aynı)
50
RKAY Olasılık Yoğunluk Fonksiyonu
RKAY
PDF Yoğunluk Fonksiyonu
Swerling Olasılık
Swerling PDF
45
Probability
Density
Function
(PDF)
Olasılık Yoğunluk Fonksiyonu
40
35
30
25
20
15
10
5
0
0
0.05
0.1
0.15
0.2
0.25
0.3
Radar Kesit Alanı
0.35
0.4
0.45
0.5
Radar Cross Section (RCS)
Figure 7- Analysis results of G/M Polygon model of RKAY RCS Prediction Programs at 15
GHz and Swerling Target Models in norm 2
G/M Polygon Model (15 GHz, 1o Horizontal, σort=0.03, m=1, norm ∞)
Güdümlü Mermi Poligon Modeli-15Ghz-1lik-MAX *** Ortalama RKA=0.030 ve m=1 (aynı)
50
RKAY Olasılık Yoğunluk Fonksiyonu
45
RKAYOlasılık
PDF Yoğunluk Fonksiyonu
Swerling
Swerling PDF
Probability
Density
Function
(PDF)
Olasılık Yoğunluk Fonksiyonu
40
35
30
25
20
15
10
5
0
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
Kesit Alanı
Radar CrossRadar
Section
(RCS)
Figure 8-Analysis Results of G/M Polygon Model of RKAY RCS Prediction Programs at 15
GHz and Swerling Target Models in Norm ∞.
It’s created totally 12 models and each models scanned at 3 GHz, 10 GHz and 15 GHz
frequencies respectively in RKAY RCS Prediction Program in the context of the paper. Then, their
PDF graphs are composed in the same program. Finally Swerling Statistical Models are adapted
according to this graph. Final Results of the RCS computations are shown in Table 3.
In Table 3, the blue numbers show the results of the STM for G/M and Airplane Models.
According to the results, it is seen that STM’s outputs are compatible with the RKAY RCS
Prediction Program’s outputs. However, with the red numbers, it is seen that there are
discrepancies between the outputs of the two computation programs.
According to the results, while G/M models were supposed to be categorized in Swerling
Condition 3-4, it is seen that G/M Ellipsoid, Ship Ellipsoid and Ship Polygon models do not
correspond with the Swerling Condition 3-4. It’s because there are not any fringes or parts on
targets’ body. It is also seen that Ship Ellipsoid and Ship Polygon models correspond to Swerling
Condition 1-2. Similar to the previous situation, while G/M Polygon, G/M Ellipsoid Small Parts and
Penguin models are expected to be correspond with the Swerling Condition 3-4 it is seen that they
correspond with the Swerling Condition 1-2 instead. It’s because there are parts and fringes which
are as big as their body on targets’ body.
According to the analysis results it is seen that all of the airplane models correspond with
Swerling Condition 1-2.
Lastly, while expecting Ship Ellipsoid Small Parts and Ship models to be in Swerling
Condition 3-4, it’s seen that they correspond to Swerling Condition 1-2 instead. As far as the
author’s assessment, this situation derives from the parts on the superstructure of the ship. These
parts cause hot spots and affect the overall RCS.
Table 3-Final Results of Radar Cross Section Computations
G/M (3 m)
3 GHz
σort
m
Ellipsoid 1.7
10 GHz
m
σort
Airplane (30 m)
Ship (100 m)
15 GHz
3 GHz
10 GHz
σort
m σort
m
σort
m
σort
m
σort
m
σort
1
14
1
12
1
13
1.05
26
1.1
25.5
m
11.1 0.03 5.7 0.03
15 GHz
3 GHz
10 GHz
Polygon
1
0.02
1
0.04
1
0.03
1
2
1
2
1
2
1
3
1
2.4
Ellipsoid
Small
Parts
1
0.8
1
1.5
1
1.7
1
19
1
19
1
18
1
1200
1
3000
Real
Target
1
0.29
1
0.32
1
0.26
1
2.5
1
3.25
1
3.35
1
2500
1
4000
10.
Conclusion:
The assessment and conclusion of the paper which aims to analyze the STM’s RCS
computation performance by comparing with RKAY RCS Prediction Program is stated below.
In the context of the study 12 targets are designed in Rhinoceros 3D Design Program and
these targets’ RCSs are calculated by STM and RKAY RCS Prediction Program at 3, 10, 15 GHz
frequencies by means of PDFs. The PDF plot produced from RKAY RCS prediction Program
scanned from 360 points for G/M target and scanned from 180 points for airplane and ship targets.
Within the study all of the target RCS computations are done both with STM and RKAY RCS
Prediction Program. However, in this paper only a few examples which reflect the bottom line of
the study are shown.
According to the study it is observed that:
a.
All of the Airplane targets’ RKAY RCS outputs correspond to STM,
b.
G/M Polygon, G/M Ellipsoid Small Parts and Penguin targets’ RKAY RCS outputs
correspond to STM. However, G/M Ellipsoid target doesn’t correspond to STM due lack of parts or
fringes on it,
c.
None of the Ship targets correspond to STM. It’s mostly because of the existing
RCS hot spot parts on the ship body.
According to the study it is seen that Swerling Target Model’s RCS computation performance
is considerably good for both G/M and Airplane target. With its quick RCS computation speed
Swerling Target Model is superior to RKAY RCS prediction Program. However, Swerling Target
Model’s performance is not satisfactory for the Ship target. The reasons of the discrepancies
between the two RCS computation tools have been stated above.
For better RCS computation performance the Chi-Square distribution of the Swerling Target
Models may be changed with the Log-normal or Rice distribution or that kind of tools.
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