HAL Id: hal-02183176
https://hal.archives-ouvertes.fr/hal-02183176
Submitted on 15 Jul 2019
HAL is a multi-disciplinary open access
archive for the deposit and dissemination of sci-
entic research documents, whether they are pub-
lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.
L’archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diusion de documents
scientiques de niveau recherche, publiés ou non,
émanant des établissements d’enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.
Characterisation of the Tri-Modal Discrete Sea Clutter
Model
Luke Rosenberg, Sébastien Angelliaume
To cite this version:
Luke Rosenberg, Sébastien Angelliaume. Characterisation of the Tri-Modal Discrete Sea Clutter
Model. RADAR2018, Aug 2018, BRISBANE, Australia. �hal-02183176�
1
Characterisation of the Tri-Modal Discrete Sea
Clutter Model
Luke Rosenberg
∗
and S
´
ebastien Angelliaume
†
∗
Defence Science and Technology Group, Australia,
†
ONERA, France
Email: Luke.Rosenberg@dst.defence.gov.au
Abstract—Accurately representing the sea clutter amplitude or
intensity distribution is important for achieving a constant false
alarm rate in a detection scheme. This can be difficult as the
backscatter statistics change with the sea surface characteristics,
the geometry of acquisition and the radar parameters. Recently,
a new compound distribution model has been proposed which
models the sea clutter texture discretely, with each component
representing different scattering components. In this paper, the
tri-modal discrete texture (3MD) model is explored using real
aperture data from the Defence Science and Technology Group
Ingara radar and synthetic aperture radar data from the French
Aerospace Laboratory (ONERA) SETHI radar. Both the model
fitting accuracy and the variation of the texture components are
studied to better understand how to relate the model to the
underlying sea-clutter characteristics.
I. INTRODUCTION
The study and analysis of sea clutter is important in many
different applications such as oceanography, maritime surveil-
lance and target classification. The problem of detecting ships
in either real or synthetic aperture radar (SAR) data is typically
approached with constant false alarm rate (CFAR) detection
algorithms using a probability density function (PDF) model
to characterise the target free data. Models for the amplitude
distribution of sea clutter are usually developed empirically
from measurements of real data as it is not currently possible
to accurately predict the PDF of sea clutter under different
conditions using physical models of the sea surface.
There has been a long development of PDF models used
to fit both real aperture radar and synthetic aperture radar.
With coarse range resolution, a reasonable model for the sea-
clutter PDF is the Gaussian distribution, with the envelope
of the returns given by a Rayleigh distribution. As the range
resolution becomes finer, the variation of the sea swell is better
resolved and the effect of breaking waves and other discrete
events (sea-spikes) are more pronounced. These returns have
a larger magnitude which has led to the development of newer
PDF models with longer ‘tails’ such as the log-normal and
Weibull distributions. A popular and widely used framework
for developing PDF models is the compound Gaussian model
which was originally proposed for use in sea-clutter by Ward
[1]. This model includes a temporal or fast varying component
known as speckle which relates to the Bragg scattering, and a
slowly varying component which captures the underlying swell
and models the texture.
Compound distribution models include the K or K+noise
[2], KA [2], KK [3], Pareto or Pareto+noise [4]–[7] and the
K+Rayleigh [8]. While the KA and KK distributions model the
sea clutter very accurately, they are difficult to implement in
practice. Consequently, the Pareto model has become popular
due to its smaller number of parameters and ability to account
for thermal noise. The K+noise distribution has also been
extended to a K+Rayleigh [8] distribution by taking into
account any extra Rayleigh scattering not captured by the
thermal noise. This model has been shown to fit both real
and synthetic aperture radar data very accurately and over a
wide range of geometries [9]. Note that many of these are only
first order compound Gaussian models as they cannot model
the speckle correlation independently of the thermal noise.
The tri-modal discrete texture (3MD) model [10], [11] is
another candidate which has demonstrated great potential for
modelling synthetic aperture radar (SAR) sea-clutter and is
unique in the way it models the sea clutter texture as a combi-
nation of discrete components. The two main contributions in
this paper are to first investigate the suitability of this model
on two different airborne datasets and to study the distribution
modes to determine their relationship to the characteristics
of the underlying sea-clutter. The two data sets include the
Ingara X-band medium grazing angle data set collected by the
Defence Science and Technology (DST) Group in Australia
and both L-band and X-band synthetic aperture radar (SAR)
data collected by the SETHI radar of the French Aerospace
Laboratory (ONERA). This contrast is important as the process
of SAR image formation alters the radar backscatter with the
SAR representation of ocean waves being different from that
of real aperture radar [9].
In Section II, a number of key PDF models are described
with details on how their parameters are estimated and the
model fits assessed. The two datasets are then presented in
Section III with a study of the model fits and modes used for
the 3MD model. Section IV then looks further at the texture
estimates of the 3MD model over a wide parameter space.
II. AMPLITUDE DISTRIBUTION MODELS
To understand the development of the compound distribu-
tion, consider a radar receiving in-phase and quadrature data
from an external clutter source with its amplitude defined
by Gaussian statistics with zero mean and variance, x. In
addition, thermal noise from the radar will add a component
σ
2
n
which is included by offsetting the variance x. In target
2
detection analysis, the envelope of the received pulses is often
converted to power (square law) and the clutter distribution
becomes exponential. This component is known as speckle
in the compound representation. For a frequency agile or
scanning radar with sufficient time between looks, a common
method to improve the detection performance is to sum a
number of looks. If there are M independent exponential
random variables, z =
P
M
m=1
y
m
, then the received power
is described by a gamma PDF,
P (z|x) =
z
M−1
(x + σ
2
n
)
M
Γ(M)
exp
−
z
x + σ
2
n
(1)
where 0 ≤ x ≤ ∞. To include the texture component which
modulates the speckle, we integrate over the speckle mean
power,
P (z) =
Z
∞
0
P (z|x)P (x)dx (2)
where P (x) is the distribution of the texture component. While
there are analytic solutions in many cases, when noise is
included in the model, numerical integration must be used to
evaluate the compound distribution.
A. K+Rayleigh model
The most commonly used PDF model for sea-clutter in
both real and synthetic aperture radar is the K-distribution, or
K+noise when thermal noise is included. However, in many
cases the K-distribution is not able to capture the long tails
present due to high magnitude spiky sea clutter returns. The
K+Rayleigh distribution was formalised in [8] and includes a
further Rayleigh component to better capture these returns. It
is defined by explicitly separating the speckle mean into two
components, x = x
r
+σ
2
r
, where the extra Rayleigh component,
σ
2
r
is modelled in the same fashion as the thermal noise. The
K+Rayleigh model uses a gamma distribution for the texture,
P (x
r
|ν
r
, b
r
) =
b
ν
r
r
Γ(ν
r
)
x
ν
r
−1
r
exp [−b
r
x
r
] , 0 ≤ x
r
≤ ∞ (3)
where ν
r
is the shape and b
r
= ν
r
/σ
2
c
is the scale with the
mean clutter power, σ
2
c
. The influence of the extra Rayleigh
component can be measured by the ratio of the mean of the
Rayleigh component, σ
2
r
to the mean of the gamma distributed
component of the clutter and is defined by k
r
= σ
2
r
/σ
2
c
. For
the Ingara data, it has typical values in the range 0 ≤ k
r
≤ 4.
To calculate the compound distribution in (2), the integration
is then performed with the modified speckle mean level, x
r
instead of the total speckle x. To estimate the distribution
parameters, there are a number of techniques such as the
method of moments, the hz log zi method or a least squares
minimisation between the data and model complementary
cumulative distribution functions (CCDF) [12].
B. Pareto model
The Pareto model is described by only two parameters
(shape and scale), yet can reasonably model the long tails
present in sea-clutter distributions. It was first used for sea-
clutter modelling by Balleri et al. [4] and later by others at
US Naval Research Laboratory (NRL) and DST Group [5]–[7].
For the Pareto distribution, the texture has an inverse gamma
distribution
P (x) =
c
a
Γ(a)
x
−a−1
exp [−c/x] , a > 1, c > 0 (4)
where a is the shape and c = σ
2
c
(a − 1) is the scale. Similarly
to the K+Rayleigh model, the distribution parameters can be
estimated using method of moments, the hz log zi method or
least squares minimisation.
C. 3MD model
The compound models in the literature all assume a con-
tinuous texture distribution function which suggests a small
probability of infinite texture values. This is not physically
sound as it cannot be measured by any real radar system. The
3MD model [10], [11] instead proposes the use of a discrete
texture model that assumes the sea clutter consists of a finite
number of distinct modes or scatterer types, I. This implies
that the scatterers in the observed scene are realisations from
homogeneous clutter random variables with different texture
values. In the original work, it was found that I = 3 modes
were sufficient to model distributions from the spaceborne
SAR imagery, hence the tri-modal in the name. One of the
consequences of this discretisation is that spatial and long-
time correlation cannot be modelled as part of the texture, and
hence the model is less suitable for clutter simulation. The
texture PDF is given by
P (x) =
I
X
n=1
c
n
δ(x − a
n
),
I
X
n=1
c
n
= 1, a
n
, c
n
> 0 (5)
where δ(·) denotes the delta-function, a = [a
1
, . . . , a
I
] are the
discrete texture intensity levels and c = [c
1
, . . . , c
I
] are the
corresponding weightings. The continuous distribution is then
given by
P (z) =
M
M
Γ(M)
z
M−1
I
X
n=1
c
n
exp
−
Mz
ρ
c
a
2
n
+ ρ
n
(ρ
c
a
2
n
+ ρ
n
)
M
(6)
with ρ
c
+ ρ
n
=
σ
2
c
σ
2
c
+ σ
2
n
+
σ
2
n
σ
2
c
+ σ
2
n
= 1.
D. Error metrics
To evaluate how well a model fits a set of observations,
there are many statistical tests and measurement techniques
in the literature including the mean squared error of the
distribution model compared to the data [7], the chi-square
and Kolmogorov-Smirnov tests and the Bhattacharyya distance
[13]. The first metric used in this paper is the Bhattacharyya
distance (BD) which captures the similarity between the actual
PDF, P (·) and the theoretical distribution, Q(·)
D
BD
= − ln
X
x
k
p
P (x
k
)Q(x
k
)
!
(7)
3
where x
k
represents the data samples. The BD ranges from
0 to ∞, where equal distributions have a distance measure of
0. To have more readable results, the BD is reported in dBs.
The second metric is the threshold error which is determined
by first calculating the CCDF for both the empirical data and
the data fit. The threshold error is then the absolute difference
between the two results at a fixed CCDF value. This view of
the data is important due to its relationship with the threshold
in a detection scheme used for distinguishing between targets
and interference. In this context, it is commonly referred to as
the probability of false alarm.
III. DATA SELECTION
A. Ingara real beam data
Ingara is a polarimetric radar system maintained and oper-
ated within the DST Group in Australia [14]. During the ocean
backscatter collections in 2004 and 2006, it was operated at
X-band in a circular spotlight-mode where the aircraft flew
a circular orbit in an anti-clockwise direction (as seen from
above) around a nominated point of interest. Each day the
radar platform performed at least six full orbits around the
same patch of ocean to cover a large portion of grazing
angles between 15
◦
and 45
◦
. The Ingara data has a 200 MHz
bandwidth (0.75 m range resolution) and an azimuth resolution
of approximately 63 m. Alternate pulses transmitted horizontal
(H) and vertical (V) polarisations resulting in a nominal PRF of
300 Hz. For the analysis in this paper, data from a single 2004
flight has been chosen with a Douglas sea state 5 (wind speed
of 10.3 m/s and a wave height of 2.6 m). The data has been
pooled into blocks of 5
◦
azimuth and 3
◦
grazing with each
block containing approximately 10
6
samples. An example of
the data is shown in Fig. 1 for the downwind direction and
30
◦
grazing.
Fig. 1. Ingara data in the downwind direction and 30
◦
grazing.
Fits of the three models from Section II are then shown in
Fig. 2. For each model, the parameters are estimated by a least
squares fit to the data CCDF in the log domain. For the 3MD
model, the fitting process first assumes a single mode (I = 1)
and the threshold error is compared with a threshold value of
0.5 dB. If the error is greater than this threshold, the parameters
for a bi-modal fit (I = 2) are then estimated. Only if this error
is greater than the threshold value, are the parameters estimated
for the full 3MD model (I = 3). The model components,
c
n
, are then ordered from largest to smallest and any modes
where the weightings, a
n
< 10
−3
are removed. Table I shows
the estimated parameters of the three models. In this example,
both the HH and HV polarisations require 3 modes, while VV
only requires 2. The associated error metrics are then reported
in Table II where the threshold error is determined at a CCDF
value of 10
−4
. The BD results focus on the overall fit and
show that the K+Rayleigh model is slightly worse than the
Pareto+noise and 3MD models. However, the threshold errors
which focus on the distribution tail, reveal that the K+Rayleigh
and Pareto+noise models have a similar fitting error, while the
3MD model is significantly lower.
Fig. 2. Ingara CCDF model fits for downwind data and 30
◦
grazing. Blue -
data, red - K+Rayleigh, magenta - Pareto+noise, black - 3MD. Right column
is a zoomed version of the left.
TABLE I. INGARA PARAMETER ESTIMATES FOR EXAMPLE IN FIG. 2.
HH HV VV
CNR (dB) 11.89 6.51 21.29
K+Rayleigh shape 0.10 0.35 8.57
K+Rayleigh k
r
-value 0.70 0.54 0
Pareto shape 3.2 4.83 12.58
3MD mode 1 (a, c) (0.64, 0.76) (0.51, 0.74) (0.75, 0.90)
3MD mode 2 (a, c) (0.35, 1.24) (0.47, 1.16) (0.25, 1.27)
3MD mode 3 (a, c) (0.012, 2.74) (0.021, 2.04) -
TABLE II. INGARA MEASUREMENT ERRORS FOR EXAMPLE IN FIG. 2.
THRESHOLD ERROR IS MEASURED AT 10
−4
.
HH HV VV
K+Rayleigh BM (dB) -30.83 -34.38 -36.21
Pareto BM (dB) -34.97 -36.67 -36.88
3MD BM (dB) -35.94 -36.02 -36.26
K+Rayleigh threshold error 0.25 0.14 0.20
Pareto threshold error 0.24 0.24 0.18
3MD threshold error 0.095 0.096 0.086
B. SETHI Synthetic Aperture Radar data
SETHI is an airborne remote sensing laboratory developed
by ONERA [15] and operates as a pod-based system on a
4
Falcon 20 Dassault aircraft. In 2015, fully polarimetric SAR
data was acquired off the French coast at both X- and L-
bands simultaneously. The SAR imagery has range resolutions
of 0.5 m and 1.0 m for the X and L-bands respectively,
and the imaged area is processed with an azimuth (along-
track) resolution equal to the range resolution. The aircraft
flew at 2743 m (9,000 ft) with an imaged area of 8.8 km in
azimuth and 1.1 km in slant range, covering grazing angles
from 38
◦
-56
◦
. For this paper, the employed SAR data have
been collected over the sea surface in the upwind direction with
a wind speed of 10.8 m/s (sea state 5-6). Example imagery is
shown in Figs. 3 and 4 for the X- and L-band radar systems.
The same PDFs as for the Ingara dataset are now in-
vestigated in Fig. 5 for the dual-frequency SAR data. The
K+Rayleigh, Pareto+noise and 3MD models have been fitted to
the data with the model parameters shown in Tables III and IV
for the X-band and L-band data sets. These results are similar
to the Ingara data with a poor match for the Pareto+noise,
while the K+Rayleigh and 3MD provide better fits. The BD
and threshold errors are then given in Tables V and VI. Overall,
we find a very good correspondence with the BD being lower
than -33 dB for each result, implying a consistently good fit
to the distribution body. When focusing on the tail of the
distribution, low threshold errors are observed for both the
K+Rayleigh and 3MD distributions, while the Pareto+noise
model has a greater mismatch.
Fig. 3. SETHI X-band SAR data in the upwind direction.
TABLE III. SETHI X-BAND PARAMETER ESTIMATES FOR EXAMPLES
IN FIG. 5.
HH HV VV
CNR (dB) 33.66 28.45 36.90
K+Rayleigh shape 0.76 0.43 2.73
K+Rayleigh k
r
-value 0.29 0.53 0.04
Pareto shape 2.95 3.91 4.03
3MD mode 1 (a, c) (0.51, 1.17) (0.49, 0.72) (0.61, 1.04)
3MD mode 2 (a, c) (0.46, 0.63) (0.48, 1.15) (0.30, 0.61)
3MD mode 3 (a, c) (0.018, 2.35) (0.02, 2.23) (0.088, 1.59)
Fig. 4. SETHI L-band SAR data in the upwind direction.
Fig. 5. SETHI SAR CCDF model fits for upwind data and 45
◦
grazing,
X-band (left) and L-band (right). Blue - data, red - K+Rayleigh, magenta -
Pareto+noise, black - 3MD.
TABLE IV. SETHI L-BAND PARAMETER ESTIMATES FOR EXAMPLES
IN FIG. 5.
HH HV VV
CNR (dB) 41.93 36.83 45.22
K+Rayleigh shape 2.06 4.05 4.32
K+Rayleigh k
r
-value 0.27 0.21 0.12
Pareto shape 5.30 7.86 6.75
3MD mode 1 (a, c) (0.72, 0.28) (0.70, 0.86) (0.68, 0.84)
3MD mode 2 (a, c) (0.28, 1.37) (0.30, 1.28) (0.33, 1.28)
3MD mode 3 (a, c) - - -
IV. 3MD TEXTURE ANALYSIS
A. Ingara data set
In order to further study the modes of the 3MD model,
we compare the product of the texture locations a
n
and the
proportion c
n
. Fig. 6 shows this result for each polarisation,
over all azimuth angles and 15
◦
-45
◦
grazing. The dashed lines
indicate either missing data or regions where no mode was