=Paper= {{Paper |id=Vol-1987/paper38 |storemode=property |title=Application of the Particle Swarm Optimization for Determination of Parameters of the Atmosphere over the Sea for the Radar Station |pdfUrl=https://ceur-ws.org/Vol-1987/paper38.pdf |volume=Vol-1987 |authors=Viktor S. Izhutkin,Alexander Zonov }} ==Application of the Particle Swarm Optimization for Determination of Parameters of the Atmosphere over the Sea for the Radar Station== https://ceur-ws.org/Vol-1987/paper38.pdf

Application of the Particle Swarm
Optimization for Determination of
Parameters of the Atmosphere over
the Sea for the Radar Station

Viktor S. Izhutkin Alexander Zonov
Moscow Power Engineering Institute Moscow Power Engineering Institute
GSP-1, Lefortovo District, GSP-1, Lefortovo District,
111020 Moscow, Russia 111020 Moscow, Russia
izhutkin@yandex.ru zonoﬀalexander@gmail.com

Abstract
In a radar-location there is an eﬀect of emergence of a layer of evap-
oration over a surface of the water which begins to inﬂuence distribu-
tion of radio waves there is a distribution trajectory curvature to the
subsequent dispersion of power about a sea surface. The technique of
measurement of height of a layer consists in use of this power. Since the
relationship between the layer parameters and the power is nonlinear,
in the conditions of modern theory it is impossible to get an analytical
solution of the problem. For search of parameters of a layer the method
of “a swarm of particles” and his modiﬁcation by “Levi’s ﬂight” have
been used. As function of optimization the modiﬁed refraction index
is chosen, with his help channels of evaporation of varying complexity
are described. Experiments by determination of parameters of a layer
by two methods, and also their comparison are made.

1 Introduction
The Global methods of optimization are widely used in various problem of radar-location. The features of
such problems are often non-linearity, nondiﬀerentiability, lack of analytical expression, multi-extremality, high
computational complexity of optimized functions and etc.
The global problem in radar-location is the eﬀect of the emergence of an evaporations layer above the water
surface. This atmospheric layer begins to aﬀect the propagation of radio waves - there is a curvature of the
trajectory of propagation. The curvature of the trajectory of the ray is associated with the phenomenon of
refraction. The appearance of so-called waveguide channels entails a change in the operating parameters of the
radar station-a reduction of the maximum range, an increase in the number of sea jammers, and the appearance
of radar holes.
Conditionally, the parameters of the atmosphere can be measured with rocket probes and weather station
predictions. However, these methods are expensive and can not provide real-time channel parameters.

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In: Yu. G. Evtushenko, M. Yu. Khachay, O. V. Khamisov, Yu. A. Kochetov, V.U. Malkova, M.A. Posypkin (eds.): Proceedings of
the OPTIMA-2017 Conference, Petrovac, Montenegro, 02-Oct-2017, published at http://ceur-ws.org

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Evaporation channel usually arises in the case of an abnormal atmospheres state. Such parameters are
temperature, atmospheric pressure, wind speed, and etc. Part of the power is lost, since there is scattering
associated with the state of the sea surface. The technique for measuring the height of a layer is to use this
power. Part of the power is lost, since there is scattering associated with the state of the sea surface. The
technique for measuring the height of a layer is to use this power.

2 Simulation of the Radar Sea Clutter
Articles [Karimian, 2012], [Wang & Wu, 2009], [Zhang et al., 2015] examines the physics of the process and
introduces the appropriate evaporation duct.
A forward simulation of received radar sea clutter power has to be performed, and its accuracy determines the
veracity of evaluating the performances of radar systems and eﬀects the precision of retrieved duct parameters.
Using the classical radar equation, received radar clutter power can be calculated as follows (in dB):

Prc (r, m) = −2L(r, m) + 10log10 (r) + σ 0 (r) + C (1)

where L is one-way propagation loss in ducts and obtained using the split-step fast Fourier transform (FFT)
parabolic equation (PE).r is distance. σ 0 (r) is the normalized sea surface radar cross section (scattering coef-
ﬁcient) at r. C is a constant that includes wavelength, transmitter power, antenna gain, etc. Obviously, σ 0 (r)
and L must be calculated accurately, if we want to get an accurate radar sea clutter power Prc .

2.1 Evaporation Duct Model
Normally, the atmospheric refractivity has a negative slope of the altitude. In this condition, electromagnetic
waves would slowly move away from the surface. If the negative slope is stronger than the curvature of the
earth, the wave will be partially trapped and forced to bend downward, and an atmospheric duct is formed. In
troposphere, refractivity (N ) is usually expressed as:

n = 77.6 4810e
T (p + T ) (2)

where T , p, and e represent absolute temperature (K), atmosphere pressure (hpa) and water vapor pressure
(hpa) of atmosphere, respectively.
The modiﬁed refractivity (M ) is usually introduced in the form of at earth. N and M can be computed as

dh ≤ 0(M − units/km)
dM
(3)

From, (2) and (3), ducts occur when M satisﬁes:

Figure 1: Most common three duct types. (a) Surface-based duct. (b) Elevated duct. (c) Evaporation duct.

Duct models play an important role in retrieving duct parameters, and they eﬀect the nal retrieved results. For
sea environments, there are three major sea ducts frequently encountered (Fig. 1): Surface based ducts, elevated

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ducts, and evaporation ducts. Because evaporation ducts are the main consideration, a single parameter exponent
model for Fig. 2(c) is given by Equation (4)

M (z) = M0 + 0.125z − 0.125dln[ z+z
z0 ]
0
(4)

where z is height; d is the evaporation duct height; M is the evaporation duct strength; z0 = 1.5 ∗ 10−4 . M 0 is
the modiﬁed refractivity at sea surface; its typical value is 370M . This model has been extensively applied and
especially precise to evaporation duct.

3 The Particle Swarm Optimization
The global methods optimization used are considered in [Karpenko, 2014]. The particle swarm optimization
(PSO) is based on the socio-psychological behavioral model of the crowd. The development of the algorithm
was inspired by such tasks as modeling the behavior of birds in the ﬂock and ﬁshes in a jamb. The goal
was to discover the basic principles, thanks to which, for example, the birds in the pack behave surprisingly
synchronously, changing direction of the movement.

3.1 Particle Swarm Optimization
By modern time, the swarm algorithm [Wang & Wu, 2009] has evolved into a highly eﬃcient optimization
algorithm, often competing with other optimization algorithms.
Iterations in the algorithm are
′
Xi = Xi + Vi (5)

Vi = bl Vi + U|X| (0; bc ) ⊕ (Xi∗ − Xi ) + U|X| (0; bs ) ⊕ (Xi∗∗ − Xi ) (6)

where Xi Vi are position and velocity of the each particle. is vector space of the local best values; is vector space
of the global best values; ⊕ is direct sum vector space.

3.2 Particle Swarm Optimization via Levy Flight
There is also a modiﬁcation of the particle swarm optimization via Levy Flight (LPSO) [Zhang et al., 2015].
The Levi ﬂight is a special class of random motion, consisting of a series of short displacements, with long
displacements occurring between them. Many studies have shown that the ﬂight behavior of birds, insects,
herbivores has the typical features of Levy’s ﬂight. This movement can be used as a global search algorithm.
The Levi ﬂight can be successfully applied in a swarm of particles, and the rate of convergence and accuracy
of the method will be higher.
A Levy ﬂight is applied in PSO to generate a new solution from a particle’s position xo using Eq. (8)

Xit+1 = Xit + α ⊕ Levy(λ) (7)

where α is the step size. The product ⊕ means entrywise multiplications. Levy(λ) is a Levy ﬂight, which provides
a random walk while their random steps are drawn from a Levy distribution, Levy(λ): 1 < λ < 3, which has
an inﬁnite variance with an inﬁnite mean. The Levy distribution, named after the French mathematician Paul
Pierre Levy, is a continuous probability distribution for a non-negative random variable, which belongs to the α
stable distribution.
Levy Flight can be described as

Levy(λ) = 0.01 µυ (xti − Xi∗∗ ) (8)

where µ and υ all follow a normal distribution, µ ∼ N (0, σµ2 ), υ ∼ N (0, συ2 ). the value of and can be calculated
by the follow equation:
σµ = ( 2Γ(λ) sin(π(λ−1)/2) 1/λ−1
(λ−2)/2 Γ(λ/2)(λ−1) )
(9)
συ = 1

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3.3 The Steps of PSO and LPSO

PSO algorithm has a good convergence speed in the early calculation period, but it is easily trapped into a local
optimal solution in later calculations. PSO can maintain a high population diversity in the early period. However,
with the increase in iterations, the majority of the groups concentrate in the vicinity of an optimal solution. In
order to deal with this problem, the Levy ﬂight was introduced into standard PSO. That is, when each particle
updates the position, the objective function is not directly calculated, but it further updates individual positions
via Levy ﬂight, and then calculates the value of the objective function. LPSO has a better local search ability,
which can improve the algorithm’s convergence rate and eﬃciency.
The detailed steps for parameter estimation with the LPSO algorithm can be described as follows: Step 1:
Initialize parameters and all the particles.
Step 2: Begin the iteration process.
Step 3: Evaluate the ﬁntes function of each particle and determine the global and local best value.
Step 4: According to the ﬁtness, update position and velocity.
Step 5: If the LPSO algorithm is used, then updating the position using the Levy ﬂight.
Step 6: Go to step 3 and check whether the stopping criterion is met.
Flowchart of the algorithms is shown in Fig 2. Flowchart of the algorithms is shown in Fig 2.

Figure 2: Flowchart of the algorithms.

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4 Experiments
To test the eﬀective of optimization algorithms we employ functions of Rastrigin and Schwefel. After we use
PSO and LPSO for ﬁnd evaporation duct layer.

4.1 Optimization of the Rastrigin Function
Rastrigin function (Fig. 3) is a typical example of non-linear multimodal function. Finding the minimum of this
function is a fairly diﬃcult problem due to existing of large number of local mimina.

Figure 3: Rastrigin function of two variable: (a) In 3D (b) Contour

On an n-dimensional domain it is deﬁned by
∑n
i=1 [xi − 10cos(2πxi )],
2
f (X) = 10n + (10)

where it has a global minima at x = 0 where f (x) = 0.
The result of optimization are shown in Table 1.

Table 1: Results

h, m PSO LPSO
X x1 x2 x1 x2
-0.99 -0.99 0.05 0.03
f(X) 1.98 0.67

PSO was at the local minina, while LPSO was at the global minima.

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4.2 Optimization of the Schwefel’s Function
The Schwefel function (Fig. 4) is complex, with many local minima.

Figure 4: Schwefel’s function of two variable: (a) In 3D (b) Contour

On an n-dimensional domain it is deﬁned by
∑n √
f (X) = i=1 [ |2πxi |], (11)

where it has a global minima at X = 420.9687 where f (x) = −418.9829n.
The result of optimization are shown in Table 2.

Table 2: Results

h, m PSO LPSO
X x1 x2 x1 x2
205.58 401.57 421.04 421.49
f(X) -574.2 -873.93

PSO failed in the optimization task, but LPSO successfully get the global minima.

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4.3 Optimization of the Evaporation Duct Model
Purpose-oriented function has several local minimal (Fig. 5), but the global minima is up to 100 meters.

Figure 5: Modiﬁed refractivity, d = 20

The results of the experiment are shown in Table 3.

Table 3: Results

h, m PSO LPSO
opt z,m % z, m %
20 21.25 6.25 20.065 0.325

If we change altitude Evaporation layer, result of optimization are shown in Table 4 and Fig. 6.

Table 4: Results

h, m PSO LPSO
opt z,m % z, m %
20 20.085 0.425 20.065 0.325
30 30.059 0.197 30.053 0.177
40 40.081 0.17 40.112 0.28

Another important parameter that should be evaluated is the complexity of each method. The amount of
time spent per iteration in each optimization algorithm is diﬀerent.
In both algorithms at each iteration, it is necessary to calculate the value of purpose-oriented function,
update the global and local minima, and update the position. In the modiﬁcation by Levy’s ﬂight, the Levi step
is additionally calculated.
As a result, the total laboriousness of the PSO is

RP SO = τ0 (2N + i) + τ1 N + τ2 N, (12)
where τ0 , τ1 , τ2 are time of the each operations.
Labor intensity LPSO is

RLP SO = RP SO + RLevy
(13)
RLevy = τ3 + 2τ4 + τ2 N

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In the experiment it was discovered that the LPSO ﬁnd more precise solution. Graph of the accuracy of the
solution from convergence is shown in Fig 6.

Figure 6: The result of the eﬀectiveness of methods PSO and LPSO.

For choose an optimization algorithm, it is necessary to take into account not only the accuracy of the solution
and the rate of convergence, but also the laboriousness of the methods. Modiﬁcation of the particle swarm method
leads to an increase in the rate of convergence, with a signiﬁcant increase in accuracy possible. However, the use
of Levy’s method demand many complicated calculations.

5 Conclusion
Based on the results of the experiments it can be concluded that the proposed methods Particle Swarm Opti-
mization (PSO) and Particle Swarm Optimization via Levy Flight (LPSO) successfully cope with the problem.
The accuracy and speed of convergence of the second method is higher. This is primarily due to the fact that the
particles moving by the ﬂight of Levi are able to continue moving even with good accuracy, thereby improving
the result. The convergence rate of the second method is also higher. This is due to the fact that apart from
the usual exchange of information with each other, the particles carry out additional movement by the ﬂight of
Levi, thereby achieving goals faster. If we talk about laboriousness, then Levy’s ﬂight is a more labor-intensive
method, so it is worth choosing between the rate of convergence on one iteration and the computational cost.

References
[Karimian, 2012] Ali Karimian,(2012). Radar Remote Sensing of the Lower Atmosphere, University of California,
San Diego, -118 pages
[Wang & Wu, 2009] B. Wang, Z.-S. Wu, (2009) Retrieving evaporation dust heights from radar sea clutter
using particle swarm optimization (PSO) algorithm Progress In Electromagnetics Research, PIER.
URL:http://www.jpier.org/PIERM/pierm09/08.09090403.pdf

[Zhang et al., 2015] Zhi-Hua Zhang ; Zheng Sheng ; Han-Qing Shi, (2015) Parameter estimation of at-
mospheric refractivity from radar clutter using the particle swarm optimization via Levy ight
PIER.URL:http://remotesensing.spiedigitallibrary.org/article.aspx?articleid=2463
[Karpenko, 2014] Karpenko A. P., (2014) Sovremennye algoritmy poiskovoj optimizacii. Algoritmy, vdohnovlen-
nye prirodoj: uchebnoe posobie / A.P. Karpenko. Moskva : Izdatel’stvo MGTU im. N.EH. Baumana,
446, [2] s. : il. ISBN 978-5-7038-3949-2

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