2015 IEEE International Conference on Advanced Intelligent Mechatronics (AIM)
July 7-11, 2015. Busan, Korea

Altitude Estimation Using Particle Filter with Monopulse Radars
in a Multipath Environment
Yuki Takabayashi, Yasushi Obata, and Ryo Kurazume
method as stated above, our proposed method is able toestimate the altitude but the preset ones by using particle filter
theorem. Therefore, the accuracy of altitude estimation ismuch improved by the proposed method.

Abstract— The problem of multipath propagation in the
tracking low-altitude targets with a radar is addressed. It is wellknown that multipath fading causes bias errors to the target
altitude over the sea. Since the bias error caused by multipath
propagation depends on a large number of parameters such as
the frequency of the radar waveform, the actual target altitude,
and range, it is difficult to estimate the bias errors. In this paper,
we propose an altitude estimation method using particle filter
and multipath propagation model. The performance of the
proposed method is verified through computer simulations.

This paper is organized as follows. MP model is describedin Section Ⅱ. Proposed method is described in Section Ⅲ.
Simulation results for 3 simulation scenarios are presented in
Section Ⅳ. Finally, a summary of the results is given in
Section Ⅴ.

I. INTRODUCTION

II. MULTIPATH PROPAGATION MODEL

It is well known that the problem of multipath propagation(MP) arises in the situation of tracking low-altitude targets
with a radar and causes large bias errors to the real altitude of
the target. In the presence of multipath propagation,interference among the direct path and surface-reflected ones
is caused. The interference affects bias errors of the target
angle. Estimation of the bias errors is normally a difficult
problem due to the dependence on a large number ofparameters such as the frequency of the radar waveform, radar
altitude above the sea, the actual target altitude, and the range
from a radar to the target and the surface reflectivity[1].

The geometry of multipath phenomenon is shown as Fig.1.
There are two separate paths between the target and the radar:the direct path and the indirect path via reflection with grazing
angle ψ g from the sea (or ground) surface.
The surface reflected signal consists of two components,that is, specular and diffuse. The specular reflection is caused
by a smooth (mirror-like) surface and the diffuse reflection iscaused by the surface irregularities. While the specularreflection coefficient is a deterministic number which depends
on several parameters, the diffuse reflection has a random
nature. Therefore, in this paper, we do not take into accountthe diffuse reflection because a proposed tracking method
using a particle filter is able to remove random errors by the
diffuse reflection.

Tracking algorithm for MP problem have been widely
studied[2]-[7]. A conventional tracking method for MP
problem is able to eliminate large peak errors (spike errors)
but is not able to eliminate constant bias errors to the real
altitude of the target over the sea. On the other hand, we
proposed altitude estimation algorithm which calculates thereliability of assumed altitude hypothesis, which is chosenfrom several preset altitude hypothesis, using bias error basedon multipath propagation model[8]. However, there is aproblem that the altitude estimation method mentioned above
is able to estimate only the preset altitude. Therefore, theaccuracy of the altitude estimation method should be degradedwhen the altitude except the preset ones is estimated.

Under precondition as above, signal-to-noise ratio (SNR)
and the constant bias errors to the target altitude is depended
on specular reflection at the surface of indirect path andrelative phase between direct and indirect paths[1].
In this Section, we present the mathematical expressions
for the elevation angle error, SNR, and the probability ofdetection by specular reflection.
TargetIn this paper, we propose an altitude estimation method
using a particle filter based on MP model. Our method cancompensate the constant bias errors to the real altitude of thetarget through tracking. Compared with conventional tracking
method which regards bias errors as random errors, proposed
method calculates the importance weight of assumed altitudes,
which corresponds to particles in particle filter, incorporating
bias error based on multipath propagation model to estimate
target altitude. Moreover, compared with our conventional

Radar
θ0
ψg
Figure 1.

Yuki Takabayashi and Yasushi Obata are with Information Technology
R&D center, Mitsubishi Electric Corporation, 5-1-1, Ofuna Kamakura,
Kanagawa, 247-8501, Japan(e-mail: Takabayashi.Yuki@ dp.Mitsubishi
Electric.co.jp, Obata.Yasushi@ dr.MitsubishiElectric.co.jp).
Ryo Kurazume is with Graduate School of Information Science andElectrical Engineering, Kyushu University, 744, Motooka, Nishi-ku,
Fukuoka, 819-0395, Japan(e-mail: kurazume@ ait.kyushu-u.ac.jp).

978-1-4673-9106-1/15/$31.00 ©2015 IEEEψg

Surface

Geometry of multipath propagation.

A. The elevation angle error
The elevation angle error with respect to the target is given
by[7]

202

ΔE = −

ρ ( ρ + cos ΔΦ )
θ0
1 + 2 ρ cos ΔΦ + ρ 2

where r1 and r2 are the ground distances from the reflection
point to the radar normal projection and the target normal
projection points, respectively, r = r1 + r2 ,and re is Earthradius.

(1)where ρ is the specular reflection coefficient, ΔΦ is therelative phase between the direct path and the indirect path andθ 0 is the angle difference between the two paths in Fig.1. The
specular reflection coefficient ρ in (1) is composed of threefactors: Fresnel reflection coefficient Γ , specular scatteringfactor ρ s and divergence factor D and is given by[7]ρ = Γ ρs D

Fig.2 shows an example of the measured target altitude
including ΔE . The horizontal axis is the range from the radarto the target, the vertical axis is the target altitude, solid line
represents the true altitude and solid line with cross sign
represents the measured target altitude. As can be seen from
Fig.2, the elevation angle error ΔE in (1) depends on thetarget position.

(2)

The Fresnel reflection coefficient for a smooth surface isdetermined by the electromagnetic properties of the surface.
The Fresnel reflection coefficient for vertical polarization is
given by[7][9]Γ=

ε c sinψ g − ε c − cos 2 ψ g

Spike error(3)

ε c sinψ g + ε c − cos 2 ψ gConstant bias error

In the above equations, ψ g is the grazing angle in Fig.1 andε c is the complex dielectric constant, given by[9]ε c = ε r − jε i
εr =

εi =

(4)

εs − ε0
1 + ( 2π f τ )

2

+ ε0

2π f τ ( ε s − ε 0 )
1 + ( 2π f τ )

2

+

(5)
2σ i
f

Figure 2. True and measured altitude.

(6)

B. SNR( signal-to-noise ratio)
The SNR on MP model is given by[1]where f is radar frequency, ε 0 is constant( ε 0 =4.9), ε s is static
dielectric constant, τ is attenuation coefficient, and σ i is ion
conductivity. In the following simulation, ε s , τ and σ i is set to
be 69.1, 9.2×10-12, 4.7×10-10, respectively[9].

 R 
SNRmlt = SNR0 + 40 log10  F ⋅ d 0 R 

where SNR0 is SNR at the reference range Rd 0 , R is therange from the radar to the target and F is the patternpropagation factor. In addition, F is given by[1][7]The specular scattering factor ρ s in (2) is given by [7] 2πσ h sinψ g  
 
λ

 
2

ρ s = exp  −2 (7)

F = 1 + ρ e j ΔΦ = 1 + 2 ρ cos ΔΦ + ρ 2

where the grazing angle is the angle between indirect path andsurface which depends on target altitude, and λ is the
wavelength of the waveform. σ h is the rms(root-mean-square)of the waveheight.

−

(10)

C. Probability of Detection
In this paper, assuming that the target fluctuates according
to a Swerling case І[10], detection probability Pd is given by

Divergence factor is another quantity that affects thereflection coefficient. This factor is considered due to thecurvature of the Earth. The reflected waves from a surface
diverge and this divergence causes an attenuation in the power
density of the waves. An approximate value for the divergencefactor is given by[1][7] log ( Pfa ) 
Pd = f d ( R ) = exp  (1 + G ) 



(11)

where Pfa is false-alarm probability and G is antilogarithm of
SNR in (9).

1

2r1r2  2
D = 1 + re r sinψ g 



(9)

III. PROPOSED METHOD(8)

MP errors depend on the real altitude of the target and therange from the radar to the target which are unknown.
Therefore, we propose a method which substitutes assumed
altitudes for the actual altitude utilizing particle filter, where203removed, and new particles with assumed altitudes
are generated according to uniform random numbers.

the assumed altitude is regarded as particle, to estimate thetarget altitude. Our proposed method calculates theimportance weight wm of an assumed altitude za,m which is
calculated by using estimated horizontal state vector, thealtitude hmlt,m including the constant bias errors based on the
MP model(where hmlt,m is called “multipath altitude” in thispaper, in addition, the subscript “m” means the model number
from 1 to M which identify an assumed altitude, M means the
maximum number of assumed altitudes and the subscript “a”
means the assumed value). And then, proposed method
estimates the altitude of the target using assumed altitudes andthe importance weight of those. Fig.3 shows the block diagramof our proposed method. As shown in Fig.3, our proposed
method is composed of 6 blocks (which are “Initialization”,“Tracking”, “Multipath propagation generator”, “Importancecalculator” , “Resampling” and “Altitude Estimator”). The
processes of our method are shown as the following.
a)f)“Altitude Estimator” calculates weighted average
altitude using the importance weight of the assumed
altitudes which are the top Pu percent in theimportance order, and outputs the weighted average
one as the estimated altitude of the target.
The details of the above processes from a) to f) aredescribed as follows. In addition, Fig.4 shows the flowchart ofproposed method.
Radar
Measurements
Tracking
Radar frequency

Initialization“Initialization” process sets the assumed altitudes inparticles within a predetermined range at equalintervals (See table Ⅱ).

b)

Initialization

Multipath h
mlt
propagation
generator

Importance w, zaAltitudeweightEstimation
calculator

Estimated
altitudeResampling

STARTMultipath propagation generator (Section B)“Multipath propagation generator” calculates themultipath altitude hmlt,m for each assumed altitudeza,m(which is set in advance) using track estimates xs,
ys, and radar frequency f.

Initialization

za means assumed altitude set, hmlt means multipath
altitude set as follows.

Tracking phase?

Tracking
No

Yes
In gate?

NoYeshmlt = ( hmlt ,1 , hmlt ,2 , , hmlt , M )Multipath Propagation Generator

Importance weight calculator (Section C)

Importance Weight Calculator“Importance weight calculator” caluculates theimportance weights for all assumed altitudes using
measured altitude zo, the altitude error variance σ z2and the multipath altitude, which are from “Multipath
propagation generator”.

Resampling
Altitude Estimation
END

w means importance weight set as follows.
w = ( w1 , w2 , , wM )e)Particle Filter

Figure 3. Block diagram of proposed methodza = ( za ,1 , za ,2 , , za , M )d)zaMeasured altitude zo,
Altitude error variance σ z

Track position
xs, ysza

Tracking (Section A)“Tracking” process calculates target state estimates
and their covariance matrix using tracking filter. This
process also outputs correlation result (e.g., radarmeasurements are in/out of tracking gate).

c)

Altitude Estimator (Section E)

Figure 4. Flowchart of proposed method

Resampling (Section D)“Resampling” process resample particles withassumed altitudes based on the importance weights
according to the particle filter theorem. The particles
with assumed altitudes which are the top Pu percent in
the importance order are resampled according to theimportance weights, and random noise is added to theassumed altitudes. The rest of the particles are

A. Tracking
Tracking process uses two coordinate systems which areNorth-East-Up (NEU) coordinates and Polar (POL)
coordinates (see Fig.5).

204spike errors and constant bias errors. “Tracking” in our
proposed method has removed the spike errors before thisprocess “Multipath propagation generator”, so that the spikeerrors are not needed to be modeled.

Z (Up)z

TargetC. Importance weight calculator
In case of that measured altitude zok is given at time tk , theimportance weight wm ( m = 1, 2,, M ) which corresponds tothe probability of the measured altitude under the assumed
altitude za,m[12], is given by

R

Radar
( xr , y r , z r )

E
Az

y

0(origin)x

Y (North)γ

wm = M m

X（East）

m =1

Figure 5. NEU and POL coordinates

where γ m is defined as

In Fig.4, the radar position vector is definedT
as [ xr yr zr ] , so that the relationship between NEU and
POL is defined as



2
2
2 
x
−
x
+
y
−
y
+
z
−
z
(
)
(
)
(
)
r
r
r



R 



z − zr
  
−1 


 E  =  tan 22 x − xr ) + ( y − yr )   Az  
(

  



−1  x − xr 
tan 



y
y
−
r 




γ m = P ( zok | za , m )2 (17)
1 ( zok − hmlt , m ( k ) ) = g ( zok ; hmlt , m ( k ) ,σ ) =
exp −
2
 2

σ zk
2πσ zk22zkνkT Sk−1νk ≤ d(12)za , j ( k +1) = za , j ( k ) +h(13)

Moreover, proposed method deletes the rest of assumed
altitudes (which correspond to 100-Pu percent of all assumed
altitudes), and generates new assumed altitudes the same
number as the deleted ones. Note that new assumed altitudes
are generated from uniform distribution in the interval [a,b],that is, the probability density function is given by

In addition, tracking begins after at least 3 measurements
are observed through all time. We call the time after tracking
started “tracking phase”.

 1
p ( z) =  b − a 0

B. Multipath propagation generator
The multipath altitude hmlt,m of an assumed altitude za,m attime tk is defined as follows.
2 + ΔE (15)k22 ( xsk − xr ) + ( ysk − yr ) za , m − zr

ysk

(19)

elsewhere

E. Altitude Estimator
We calculate weighted average altitude using theimportance weight of the assumed altitudes which are the top
Pu percent in the importance order. The weighted average
altitude is given by

where the elevation angle error ΔEk is calculated by using (1)and target position vector =  xskz ∈ [ a, b ]where z is new assumed altitude for next sampling time, a isthe same setting value as the lowest limit, b is the samesetting value as the upper limit of the preset assumed altitude
in “Initialization” process (See table Ⅱ).

( xsk − xr ) + ( ysk − yr ) ⋅ tan {Eok } + zr (14)
Eok = tan −1 



(18)

where h is Gaussian random variable with mean 0 andvariance σ h .

where vk is measurement residual (the difference between theposition vector of a measurement and predicted position
vector from tracking), Sk is residual covariance, and d is gate
size which is determined chi-square distributed with 3 degrees
of freedom.

2

1

D. Resampling
The resampling takes place with probabilities proportional
to the importance weights of the top Pu percent in theimportance order. Where Pu is one of the parameters inproposed method and only Pu percent of all assumed altitudes
are resampled. In addition, our proposed method adds thevalue drawn from Gaussian random distribution with standard
deviation σ h (which is so called proposed distribution inparticle filter[12]) to the resampled assumed altitude, forinstance, which is j-th assumed altitude for next sampling time
as follows.

Our proposed method uses tracking filter to improve theposition accuracy in XY coordinates and to remove abnormal
measurements which have spike errors. Tracking gate canremove the measurements which have spike errors. The
tracking gate regards the measurements which satisfy (13) as“in” gate, otherwise as “out” of gate[11].

hmlt , m ( k ) =

(16)

γ m

T

za , m  . Note that, as

can be seen from Fig.2, the elevation angle error ΔEk contains
205zˆ=  wm za , m

(20)

m∈M t

where M t is the index set which corresponds to the top Pu
percent in the importance order.
IV. SIMULATION RESULTS
The validity of our proposed method has been examinedthrough computer simulations as follows.
A. Simulation Scenario
Fig.6 shows the geometry in simulation scenario. In all
scenarios, a target starts from the point A (40km distance fromthe radar) to the point B with constant horizontal speed 250m/s.
The target has a constant altitude of ht (100m, 250m and700m) above the sea surface. The radar is located at the point
C which has altitude of 4m.

In addition, we compare the proposed method with theconventional method[8]. The conventional method calculates
the reliability of assumed altitudes which are predetermined
and fixed to output the one with highest reliability as
estimated altitude. The conventional method parameters are
summarized in table Ⅲ . The assumed altitudes of aconventional method are in the range of from 100m to 600m
at intervals of 20m and the number of those is 26. In table Ⅲ,
ThSNR is threshold for deleting multipath altitudes whichcontain spike error as shown in Fig.2, Thβ and ThCnt are
thresholds for selecting a sequential assumed altitude withhigher reliability than specified value as estimated altitude.
Thβ is the specified value stated above for judging thereliability, ThCnt is threshold for judging the continuity[8].
TABLE II.

Z
Assumedaltitudes

TargetB
250m/s

Radar

A
ht

O

X

39.5[km]
40[km]TABLE III.

Figure 6. Simulation scenario

B. Simulation Parameters
The radar parameters are summarized in table I.
The parameters of the tracking filter are set to be as follows.
The range, the azimuth and the elevation measurement noise
standard deviation are 50m, 10mrad, and 10mrad respectively.
The process noise standard deviation at axis x, y and z are
1m/s2, 1m/s2, and 0.01m/s2 respectively. The gate size d isdetermined by chi-square distribution with significance level
0.01% and 3 degrees of freedom.
TABLE I.

Value
1001
100
600
0.5
99
10

CONVENTIONAL METHOD PARAMETERS

Parameter
The number M
The lowest limit (m)Assumedaltitudes
The upper limit (m)Interval (m)False-alarm probability
ThSNR (dB)
Thβ
ThCnt

Value
26
100
600
20
10-6
3
0.95
2

C. Results
The performance of our proposed method is evaluated by
the RMSE (Root Mean Square Error) of the target position
estimate in 50 Mote Carlo runs. Fig.7, 8 and 9 show the RMSE
of target position estimate where the true altitude of the target
is respectively 100m, 250m and 700m. In the figures, thehorizontal axis represents the simulation time, the vertical axis
represents the RMSE, the solid line with triangle sign
represents proposed method, and the dashed line with plus
sign represents conventional method.

RADAR PARAMETERS

Parameter
Range error standard deviation (m)Azimuth error standard deviation (mrad)
Elevation error standard deviation (mrad)
Sampling rate(s)False-alarm probability

Parameter
The number M
The lowest limit (m)The upper limit (m)Interval (m)Pu (%)
Standard deviation σ h (m)

C

4[m]

PROPOSED METHOD PARAMETERS

Value
50
10
10
2
10-6

The parameters of the proposed method are summarized intable Ⅱ. In table Ⅱ, the assumed altitudes of a proposed
method are in the range of from 100m to 600m at intervals of
0.5m and the number of those is 1001. Note that thoseparameters are set in “Initialization” process. Also, Pu
represents the percentage of the assumed altitudes resampledfrom all the ones.

As results, the RMSE of proposed method is about a halfof that of conventional method at the time 80s in Fig.8, at thetime 140s in Fig.9. However, in Fig.7, the RMSE of proposed
method is almost same as that of conventional method because
the true altitude of 100m is one of the preset assumed altitudes
in conventional method.

206

From these simulation results, it is verified that theaccuracy of altitude estimation is improved by the proposed
method compared with the conventional method.

V. CONCLUSION
In this paper, we proposed a new altitude estimation
method using a particle filter for the problem of multipath
propagation over the sea. A conventional method whichcalculates the reliability of the preset assumed altitude was notable to estimate the altitudes but the preset target altitudes.
Therefore, the altitude accuracy deteriorates.
On the other hand, our proposed method calculated theimportance weight of assumed altitudes, which corresponds toparticles in particle filter, based on multipath propagation
model to estimate target altitude. Compared with conventional
method, our proposed method has the advantage of being able
to estimate the altitude but the preset ones by using particle
filter theorem. Simulation results showed that our proposed
method reduced altitude estimation error to a half of that ofconventional method, particularly in the far distance from aradar.
Through computer simulation trials, our proposed method
showed the high track accuracy compared with conventional
method.

Figure 7. RMSE of position estimate (target altitude=100m).

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Figure 8. RMSE of position estimate (target altitude=250m).

Figure 9. RMSE of position estimate (target altitude=700m).

207