Statistical Analysis of Sea-Clutter using K-Pareto, K-CGIG, and
Pareto-CGIG Combination Models with Noise
HOUCINE OUDIRA1*, MEZACHE AMAR2, AMEL GOURI3
1LGE Laboratory university of M'Sila,
Department of Electronics,
University Mohamed Boudiaf of M'Sila,
ALGERIA
2SISCOM Laboratory university of M'Sila,
Department of Electronics,
University Mohamed Boudiaf of M'Sila,
ALGERIA
3LASS Laboratory university of M'Sila
Department of Electronics,
University Mohamed Boudiaf of M'Sila,
ALGERIA
*Corresponding Author
Abstract: - In this paper, the combinations of two compound Gaussian distributions plus thermal noise for
modeling measured polarimetric clutter data are proposed. The speckle components of the proposed models are
formed by the exponential distribution, while the texture components are mainly modeled using three different
distributions. For this purpose, the gamma, the inverse gamma, and the inverse Gaussian distributions are
considered to describe these modulation components. The study involves the analysis of underlying mixture
models at X-band sea clutter data, and the parameters of the combination models are estimated using the non-
linear least squares curve fitting method. Compared to existing K, Pareto type II, and KK clutter plus noise
distributions, experimental results show that the proposed mixture models are well matched for fitting sea
reverberation data across various range resolutions.
Key-Words: - Combination models, Nelder-Mead algorithm, Sea clutter, Random textures, speckle, Noise.
Received: July 6, 2022. Revised: October 16, 2023. Accepted: November 19, 2023. Published: December 27, 2023.
1 Introduction
In radar systems with low resolution capabilities, the
intensity statistics of the sea echoes have been found
to be described by the exponential (i.e., Gaussian
clutter case) probability density function (PDF).
With the development of radar technologies
operating at low grazing angles, the resolutions of
sea clutter statistics have been greatly reduced and
have been observed to deviate from Gaussianity,
[1]. Consequently, these deviations occur when the
Central Limit Theorem does not apply in evaluating
the strength of the background electromagnetic
energy (i.e., when independent random variables are
added, their sum does not tends toward a normal
distribution). Nowadays, compound Gaussian (CG)
distributions are commonly used to fit high-
resolution sea reverberation data and are the basis of
the construction of most target detection schemes
with CFAR (Constant False Alarm Rate) behavior,
[2]. The CG models are formed by means of two
components; the speckle component and the texture
component that is also termed by the modulation
component. In the sense of tail fitting improvements
to high-resolution real data, some distributions
related to the texture component have been
proposed in the open literature, [3], [4], [5], [6], [7].
The inverse gamma and the inverse Gaussian
distributed texture components were used to obtain
the generalized Pareto (GP) and the compound
Gaussian inverse Gaussian (CGIG) PDFs
respectively, [3], [4]. These models have been
effectively tested to fit the McMaster Intelligent
Pixel Processing (IPIX) radar lake-clutter
measurements and are extended to incorporate the
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thermal (system) noise in order to attaint better
goodness-of-fit against the Weibull, Log-normal and
K distributions, [5], [6]. Compared to the standard
Pareto model, the fractional order Pareto
distribution was shown to produce excellent fits to
the Defence Science Technology Organization
(DSTO) Ingara data collected by X-band maritime
surveillance radar, [7]. Another way to obtain the
best fitting to empirical data is to use the mixture
models, i.e., the mixture of two or more
distributions. In this context, the KK distribution is
applied for the analysis of the Ingara data collected
at medium to high grazing angles, [8]. The model is
generalized to account the addition of multiple looks
and a thermal noise component to produce greater
accuracy of the underlying shape of the fitted PDF.
The required detection threshold to achieve a
constant false alarm rate was also studied and
compared with the K-distribution. [9], proposed a
mixture of a Rayleigh and a K PDFs for
representing active sonar data comprising clutter
sparsely observed in a Rayleigh-distributed
background. The K-Rayleigh mixture was seen to
provide improved PDF fits and inference on the
clutter statistics. A parameter estimation technique
based on the expectation maximization (EM)
algorithm is proposed and shown to perform
adequately, [9]. The reference, [10], proposed an
alternative statistical model, which is a mixture of
K-distribution and log-normal distribution for
modeling the SAR (synthetic aperture radar) data.
This mixture model is able to model the clutter data,
the target data, or the mixed data of clutter and
target. The flowchart of the maximum likelihood
(ML) method using the EM approach was presented
for estimating the respective parameters of the
proposed mixture model.
In this paper, the modelling of sea radar clutter
using a mixture of two compound Gaussian
distributions plus thermal noise is presented. The
speckle component of the proposed models is
formed by an exponential distribution where the
texture components are particularly modelled by
means of two different distributions, [11]. The work
presented in, [11], is extended in this paper to
account three mixture models which are compared
to the K, GP and KK clutter plus noise models. To
do this, the gamma, the inverse gamma and the
inverse Gaussian distributions are considered to
describe the modulation components. The proposed
models are analysed and the non-linear least squares
curve fitting technique based on the Nelder-Mead
algorithm, [12], is employed to obtain the optimal
parameter estimation. Compared to the existing KK,
K, and Pareto clutter plus noise distributions,
experimental results show that the proposed mixture
models with different random textures are well
suited to fit high resolution sea clutter data in most
cases. This paper is structured in the following
manner. In section 2, we briefly recall the
expressions of the K, the Pareto and the CIG
(compound inverse Gaussian) clutter plus noise
PDFs. Section 3 describes the proposed mixture
models where the flowchart of the the N-M
algorithm is presented. Section 4 investigates
modeling comparisons using IPIX data of the
proposed mixture models against the existing K, GP
and KK distributions plus noise. Finally, main
concluding remarks are listed in section 5.
2 Review of K, Pareto and CIG plus
Noise Distribution
This section introduces compound Gaussian
processes for characterizing sea-clutter returns,
which are composed of a rapidly fluctuating speckle
component influenced by a slowly fluctuating
texture component. Assuming independent and
identically distributed (iid) single look data, the
combined CG distribution of the random variable X
is described as per reference, [1].
0
)()( dyypyxpxp Y
YX
X
(1)
If a square law detector is used and the thermal
noise power denoted by
2
2
n
p
is incorporated,
the speckle component (namely the conditional PDF
of x given y) follows the exponential distribution
(i.e., single pulse case) given by:
nn
YX py
x
py
yxp exp
1
(2)
The K-distribution plus noise is obtained if the
texture component fluctuates according to a gamma
PDF, [2].
(3)
where
is the shape parameter which governs the
spikness of the clutter, b is the scale parameter and
(.)
is the gamma function. Substituting (2) and (3)
into (1), the overall K plus noise PDF is given in
integral form, [13], [14].
dyby
py
x
py
yb
xp
nn
X
expexp
0
1
(4)
The complementary cumulative distributed function
(CCDF) related to (4) is:
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dyby
yp
T
y
b
CCDF
n
expexp
0
1
(5)
where T denotes the normalized detection threshold.
It is shown in, [14], that the moment formula of
order r>0 can be expressed as
n
r
n
r
bp
rFrpx 1
;.;,1 0
2
(6)
where
.,.;.;.
02 F
is the generalized hyergeometric
function.
When the modulation component is an inverse
gamma PDF, the Pareto plus noise distribution is
constructed, [3], [15].
y
y
ypY
exp
)(
)(
1
(7)
Where
is the shape parameter which depends
on heavy tailed clutter and b is the scale parameter.
Substituting (2) and (7) into (1), the Pareto plus
noise PDF is obtained, [16], [17].
dy
y
yp
x
yp
y
xp
nn
X
expexp
0
1
(8)
Integrating (8) from T to +
, the
corresponding CCDF is also given in an integral
form:
dy
y
yp
T
yCDFC
n
expexp
0
1
(9)
Using (8), the expression of moments of order
r
is derived in, [17], to be:
n
r
rp
rrF
rr
x;;,
)(
)1(
02
(10)
If the inverse Gaussian law is used to describe
the modulation component in (1), the CIG plus
noise PDF is obtained. The underlying inverse
Gaussian distribution is presented in, [18], [19].
y
y
y
ypY2
2
2/3
2/1
2
)(
exp
2
)(
(11)
Where
is the shape parameter and
is the
mean. Note that,
relies upon sea conditions and
radar parameters. Spiky clutter corresponds to
values of
10
and the Exponential distribution
or Gaussian clutter is attained for
→∞.
Substituting (2) and (11) into (1), the CIG PDF plus
noise is expressed by:
dy
y
y
py
x
py
y
xp
nn
X
2
2
0
2/32/1
2
)(
expexp
2
(12)
The corresponding CCDF of (12) is determined
to be:
dy
y
y
yp
T
yCCDF
n
2
2
0
2/3
2/1
2
)(
expexp
2
(13)
Contrary of (6) and (10), it is difficult to solve
the integral of moment’s expression from (12) of
order r. Numerical integration is used to evaluate
the following non-integer order moments
0
2
2/3/
22
exp
2
1dy
y
ypyyerx r
n
r
(14)
3 Proposed Combination of CG
Models
In this section, mixtures CG models are presented
with different random textures for the best tail
fitting to real data. Three texture components are
considered as devoted in Section 2. To this end, we
resort to combine two CG distributions with an
appropriate weighting factor k (0<k<1) given by,
[10].
)()1()( 2211
xpkxkpxp
(15)
where
21 ,,
k
is a vector of unknown
parameters to be estimated at each estimation task.
The two CG distributions p1(x) and p2(x) have the
same exponential distribution for the speckle
component given by (2) and two different texture
components. For instance, if we choose the CIG and
the Pareto plus noise models to describe p1(x) and
p2(x) respectively, (15) becomes (after substitution
(8) and (12) into (15)).
dyy
yp
x
yp
yk
dy
y
y
py
x
py
y
kxp
nn
nn
X
/expexp
)1(
2
)(
expexp
2
0
1
2
2
0
2/3
2/1
(16)
Note that (16) spans from CIG plus noise PDF
to Pareto plus noise PDF. With k=1 and k=0
corresponding to the purely CIG plus noise
distribution and the purely Pareto plus noise
distribution respectively. If 0<k<1, (16) is a mixture
of the CIG and the Pareto plus noise PDFs.
Consequently, several combinations between
K+noise, Pareto+noise and CIG+noise PDFs can be
used in (15). However, it is shown in, [12], that the
best estimation of the parameters of K-clutter plus
noise model can be achieved when the
corresponding CCDF is used in the objective or
fitness function of the N-M algorithm. To this
effect, we apply in this work the PCFE (parametric
curve fitting estimation) method described in, [11],
[12], to optimize the parameters of the following
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CCDFs instead of mixture models used in (15). In
the case of the CCDF obtained from a mixture of the
CIG and Pareto plus noise models, it is easy to
obtain:
dyy
yp
T
y
k
dy
y
y
py
T
ykCCDF
n
n
CIG
Pareto
/expexp
)1(
2
)(
expexp
2
0
1
2
2
0
2/3
2/1
(17)
where ,
n
pbk ,,,,,
. The CCDF obtained
from a mixture of CIG and K plus noise models is
dyby
py
T
y
bk
dy
y
y
py
T
ykCCDF
n
n
CIG
K
expexp
)1(
2
)(
expexp
2
0
1
2
2
0
2/3
2/1
(18)
In this case,
n
pbk ,,,,,
. Also, the
resulting CCDF from a mixture of K and Pareto plus
noise models is given by
dyy
yp
T
y
k
dyby
py
T
y
b
kCCDF
n
n
K
Pareto
/expexp
)1(
expexp
0
1
0
1
(19)
where
n
pcbk ,,,,,
. In (17)-(19), we have six
unknown parameters to be optimized. Due to this
complexity of parameter estimation, the PCFE
based on the N-M simplex search algorithm is used.
The best fit is simply achieved by a direct
comparison of the experimentally measured CCDF
with a set of curves derived from the theoretical
CCDF given by (17)-(19). Thus, the residual can be
formulated, in our case, as the difference between
the experimentally measured CCDF of the recorded
data and the theoretical model. An adequate tail
fitting regions of the theoretical CCDFs with real
data are optimized by means of the N-M algorithm
which is the best known algorithm for
multidimensional unconstrained optimization
without derivatives. After a sufficient number of
iterations, the algorithm converges to the global
minimum and outputs the numerical estimates of the
parameters. From the flowchart of Figure 1, the
following basic steps of the N-M algorithm are
given below with fixed parameters,
, 2/12,1
and
2/1
, [11].
Step 1: Estimate the real CCDF of the recorded
data.
Step 2: Initialize the method.
Step 3: Calculate the initial working N–M simplex
from the initial point given above.
Step 4: Evaluate the summed square of residuals
between theoretical CCDFs with real data at each
point (vertex) of the working N–M simplex.
Step 5: Repeat the following steps until the
termination test is satisfied.
1 Calculate the termination test information.
2 If the termination test is satisfied then,
accept the best vertex of the working N–M
simplex and go to step 6, otherwise
transform the working N–M simplex and go
to step 4.
Step 6: Return the best point (vertex) of the working
simplex ∆ and the associated function value.
In the following section, the radar data that is used
for sea clutter modeling is described. After that, the
procedure to be followed for data analysis is
provided.
4 Modeling Assessment using IPIX
Data
The capabilities of the proposed mixture models
given by (15) to fit the real PDFs and CCDFs given
by (17-19) for various sets are investigated in this
section. This modeling performance is assessed
using real-world IPIX lake clutter. The lake-clutter
data we processed were collected at Grimsby,
Ontario, with the McMaster University IPIX radar.
IPIX is an experimental X-band search radar,
capable of dual polarized and frequency agile
operation, [20]. As in reference, [4], we focus our
analysis on the datasets 84, 85 and 86 which
correspond to the range resolutions 30m, 15m and
3m respectively. The radar site was located at east
of the “Place Polonaise” at Grimsby, Ontario
(Latitude 43:2114±N, Longitude 79:5985±W),
looking at lake Ontario from a height of 20 meter
(m). The nearest shore on the far side of the lake is
more than 20 Km away. The data of the Grimsby
database are stored in 222 files, as 10 bits integers.
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Fig. 1: Flowchart of the Nelder-Mead method
There are co-polarizations, HH and VV (Lpol),
and cross-polarizations, HV and VH (Xpol),
coherent reception, leading to a quadruplet of I and
Q values for co-pol and cross-pol. Here, the
experimental modeling analysis is carried out for
HH and VV antenna polarizations, 3 m, 15 m and 30
m range cell resolutions. During the recordings, the
radar was transmitting with a pulse-repetition
frequency (PRF) of 1000 Hz and a pulse length of
0.06
s
. The received IPIX data is treated by the
arrival order and registered in a (60000x34) matrix
where 34 denotes the number of range cells and
60000 is the number of pulses. Because parameter
estimation of compound Gaussian models cannot be
obtained using low sample sizes, the proposed
mixture models were validated using 60000
recorders rather than 34 range samples. High
resolution sea clutter depends on both azimuth
resolution which is related to the beam width and
range resolution, ie.,
2/
cd
. In cases of d=3m,
d=15m and d=30m, the pulse duration (i.e.,
sampling time) have three different values where the
grazing angle is fixed at a low value. Thus, these
data does not have a connection with the grazing
angle.
In order to investigate the statistical properties
of the data, we compare the empirical PDF and
CCDF of the data with their theoretical mixture
models in the case of single look data. Fitting with
multilook data using for example 10 to 20 cells is
not possible using the proposed theoretical
distributions.
The tail fitting to real data is important in radar
detection applications. Thus, 1000 independent
samples are not sufficient to fit the tail of the
corresponding PDFs and CCDFs. Usually, 105
samples are needed for a desired CCDF value of 10-
3. Each range cell, 60 000 measurements are
therefore necessary to compare the tail fitting of the
different models.
The following experimental procedure focuses
firstly on the parameter estimation found by the N-
M algorithm and secondly on the validation of the
mixture models using the real data described
previously. The MSE (Mea Square Error) values are
calculated from the fitted and empirical CCDFs
curves. According to these values which are
obtained from specific range of the CCDFs between
10-3 and 10-2, the proposed mixture models give
lower values allowing best tail fitting as several
scenes will show. The optimal estimates of the
various models parameters as well as the MSE
values are illustrated in Table 1 and Table 2. For
HH polarization, resolution of 30m and 19th range
cell, the PDFs and the CCDFs curves are depicted in
Figure 2. It is clearly seen from this experiment that
a mixture model constructed by the CIG plus K
distributions provides the smallest value of MSE
which means the best fit to empirical data. Now if
the case of a resolution of 15m, a VV polarization is
considered with 32th range cell, the theoretical
mixture models CIG plus GP and K plus GP have
quasi-similar results with the empirical CCDFs as
shown in Figure 3. Next, we conduct the same test
for a resolution of 3m, HH polarization, 17th range
cell; better modeling performance is obtained by the
K plus GP and CIG plus GP CCDFs as depicted in
Figure 4. If another study based on the use of the
same resolution with VV polarization and 9th range
cell, Figure 5 illustrates the different PDFs and
CCDFs for all considered models. In this
experiment, it can also be seen that the tail of the
proposed mixture models (CIG plus K and CIG plus
GP) leads to the best fit. From these modelling
experiments, it is pinpointed out that a mixture
model constructed by the sum of the CIG, K, and
GP distributions with noise is mostly an accurate
statistical model of IPIX data, but it requires more
computational time due to the number of estimated
parameters.
Finally, the results obtained by the proposed
mixture models are also assessed against those
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obtained by the existing KK model with thermal
noise, [8]. To this effect, we used the same PCFE
based method given in Figure 1 to compute the KK
clutter plus noise parameters. For this, the
corresponding CCDF of the KK model with thermal
noise is given by
dyby
py
T
y
b
k
dyby
py
T
y
b
kCCDF
n
n
K
K
expexp)1(
expexp
0
1
1
0
1
1
1
(20)
where
n
pbbk ,,,,, 11
.
Table 3 illustrates the MSE values as well as the
optimal estimates of the various models parameters.
For VV polarization, resolution of 30m and 19th
range cell, the PDFs and the CCDFs curves are
depicted in Figure 6. It is clearly seen from this
experiment that a mixture model constructed by the
CIG plus GP distributions provides the smallest
value of MSE which means the best fit to empirical
data.
Fig. 2: Fitted PDFs and fitted CCDFs of mixture
models for HH polarization, resolution of 30m and
19th range cell, dataset 84
Fig. 3: Fitted PDFs and fitted CCDFs of mixture
models for VV polarization, resolution of 15m and
32th range cell, dataset 85
Fig .4: Fitted PDFs and fitted CCDFs of mixture
models for HH polarization, resolution of 3m and
17th range cell, dataset 86
0 5 10 15 20 25 30
10-4
10-2
100
Intensity, x
PDFs
Emperical data
K with noise
GP with noise
CIG+K with noise
CIG+GP with noise
K+GP with noise
-5 0 5 10 15 20 25
10-3
10-2
10-1
100
Normalized threshold, T(dB)
CCDFs
Emperical data
K with noise
GP with noise
CIG+K with noise
CIG+GP with noise
K+GP with noise
0 5 10 15 20 25
10-4
10-2
100
Intensity, x
PDFs
-5 0 5 10 15 20
10-3
10-2
10-1
100
Normalized threshold, T(dB)
CCDFs
Emperical data
K with noise
GP with noise
CIG+K with noise
CIG+GP with noise
K+GP with noise
Emperical data
K with noise
GP with noise
CIG+K with noise
CIG+GP with noise
K+GP with noise
0 5 10 15 20 25 30
10-4
10-2
100
Intensity, x
PDFs
Emperical data
K with noise
GP with noise
CIG+K with noise
CIG+GP with noise
K+GP with noise
-5 0 5 10 15 20 25
10-3
10-2
10-1
100
Normalized threshold, T(dB)
CCDFs
Emperical data
K with noise
GP with noise
CIG+K with noise
CIG+GP with noise
K+GP with noise
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Fig. 5: Fitted PDFs and fitted CCDFs of mixture
models for VV polarization, resolution of 3m and 9th
range cell, dataset 86
Fig. 6: Fitted PDFs and fitted CCDFs of mixture
models for VV polarization, resolution of 30m and
19th range cell, dataset 84
Fig. 7: Fitted PDFs and fitted CCDFs of mixture
models for HH polarization, resolution of 15m and
10th range cell, dataset 85
Fig. 8: Fitted PDFs and fitted CCDFs of mixture
models for HH polarization, resolution of 3m and
17th range cell, dataset 86
If the HH polarization is used with resolution of
15m and the 10th range cell, all proposed models
with noise as shown in Figure 7 overcome the KK
model and achieve almost the same tail fitting to
0 5 10 15 20
10-4
10-2
100
Intensity, x
PDFs
Emperical data
K with noise
GP with noise
CIG+K with noise
CIG+GP with noise
K+GP with noise
-5 0 5 10 15 20
10-3
10-2
10-1
100
Normalized threshold, T(dB)
CCDFs
Emperical data
K with noise
GP with noise
CIG+K with noise
CIG+GP with noise
K+GP with noise
0 5 10 15 20 25 30
100
Intensity, x
PDFs
-5 0 5 10 15 20
10-2
100
Normalized threshold, T(dB)
CCDFs
Emperical data
kk with noise
CIG+K with noise
CIG+GP with noise
K+GP with noise
Emperical data
kk with noise
CIG+K with noise
CIG+GP with noise
K+GP with noise
0 5 10 15 20 25 30
100
Intensity, x
PDFs
Emperical data
kk with noise
CIG+K with noise
CIG+GP with noise
K+GP with noise
-5 0 5 10 15 20
10-2
100
Normalized threshold, T(dB)
CCDFs
Emperical data
kk with noise
CIG+K with noise
CIG+GP with noise
K+GP with noise
0 5 10 15 20 25 30
10-4
10-2
100
Intensity, x
PDFs
-5 0 5 10 15 20
10-3
10-2
10-1
100
Normalized threshold, T(dB)
CCDFs
Emperical data
CIG+K with noise
CIG+GP with noise
K+GP with noise
kk with noise
Emperical data
kk with noise
CIG+K with noise
CIG+GP with noise
K+GP with noise
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real data with a slight superiority of the CIG plus K
that is illustrated by the smallest value of the MSE
(Table 3). Now, in the case of a resolution of 3m,
HH polarization and 17th range cell, the estimated
CIG plus GP CCDF as presented in Figure 8 offers a
goodness of fit compared to the obtained KK curve.
Table 1. Estimated parameters of K and Pareto plus
noise models for HH and VV polarizations
Table 2. Estimated parameters of mixture models for
HH and VV polarizations
Table 3. Estimated parameters of mixture models
for HH and VV polarizations
5 Conclusion
The modeling of high resolution sea clutter has been
discussed and the mixture distribution with different
random textures has been proposed. The additive
thermal noise has been incorporated to provide an
appropriate model for sea clutter statistics collected
by IPIX X-band radar. The proposed model is based
on the sum/mixture of two different compound
Gaussian distributions plus noise. First, the CIG the
K and the Pareto plus noise distributions were
combined to achieve better tail fitting. Then,
unknown parameters were acquired by means of the
PCFE method based on the N-M algorithm. Using
experimental data, the proposed models can quickly
produce satisfactory curve fitting results in most
cases compared with standard K plus noise, Pareto
plus noise, and KK plus noise models. The only
drawback of the new mixture models lies in the
computational requirements due to the numerical
computation of unknown parameters.
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WSEAS TRANSACTIONS on SIGNAL PROCESSING
DOI: 10.37394/232014.2023.19.17
Houcine Oudira, Mezache Amar, Amel Gouri
E-ISSN: 2224-3488
166
Volume 19, 2023
Contribution of Individual Authors to the
Creation of a Scientific Article (Ghostwriting
Policy)
- The research project was led by Houcine Oudira,
who supervised the simulation, algorithm
implementation, paper preparation and review, and
paper editing.
- Amar Mezache and Amel Gouri carried out the
writing, reviewing, and editing of the paper.
Sources of Funding for Research Presented
in a Scientific Article or Scientific Article
Itself
No funding was received for conducting this study.
Conflict of Interest
The authors have no conflicts of interest to declare.
Creative Commons Attribution License 4.0
(Attribution 4.0 International, CC BY 4.0)
This article is published under the terms of the
Creative Commons Attribution License 4.0
https://creativecommons.org/licenses/by/4.0/deed.en
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WSEAS TRANSACTIONS on SIGNAL PROCESSING
DOI: 10.37394/232014.2023.19.17
Houcine Oudira, Mezache Amar, Amel Gouri
E-ISSN: 2224-3488
167
Volume 19, 2023