# Nonlinear Dynamics of Sea Clutter

Authors: Timothy R. Field; Simon Haykin
Journal: International Journal of Navigation and Observation
 (2008)

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## Abstract

We review experimental evidence for the nonlinearity of sea clutter
and the role of thez-parameter or Mann-Whitney rank-sum statistic in
quantifying this nonlinear behavior in the context of a hybrid AM/FM model
for sea clutter, viewed as a cyclostationary process. An independent theoretical
derivation of the stochastic dynamics of radar scattering in a sea clutter
environment, in terms of a pair of coupled stochastic differential equations for
the received envelope and radar cross-section (RCS), enables the identification
of nonlinearity in terms of the shape parameter for the RCS. We are led
to conclude that, from both experimental and theoretical points of view, the
dynamics of sea clutter are nonlinear with a consistent degree of nonlinearity
that is determined by the sea state.


---

## Body

## 1. Introduction

Haykin et al. [1] advocated a state-space
formalism for the processing of radar signals in the presence of sea clutter (i.e.,
radar backscatter from an ocean surface). Such a model not only accounts for
the temporal dimension of sea clutter in an explicit manner but also for its
statistical characterization. Basic to this formalism is whether the underlying
dynamics of sea clutter are linear or nonlinear.In the detailed experimental study reported in Haykin et al., [1] it was also demonstrated that sea
clutter is a nonlinear dynamic process, with the
degree of nonlinearity increasing as the “sea state” becomes higher. The
conclusion reached on the nonlinearity of sea clutter was based on two
premises, using real-life data collected with an instrument-quality coherent
radar system.(1) The characterization of sea clutter embodies two
forms of continuous-wave modulation:(i)
amplitude modulation (AM), which is linear, and(ii)
frequency modulation (FM), which is nonlinear.The latter phenomenon is responsible for the nonlinearity of sea clutter.(2) Thez-parameter, denoting the Mann-Whitney
rank-sum statistic, is less than the special value −3, which is a strong indicator of nonlinearity.With regards to point 1, it is also noteworthy that in
another study that focussed on the spectral characterization of sea clutter
using the Loève transform [2], it was
discovered for the first time that sea clutter is a cyclostationary process.
Cyclostationarity is ordinarily associated with modulation. But knowing that
sea clutter is cyclostationary, it does not tell us the type of modulation
involved in the characterization of its waveform.In this paper, we expand on the characterization of
sea clutter as a nonlinear dynamic process, using a principled theoretical
approach. In particular, the approach is rooted in stochastic differential
equation (SDE) theory. The issue of the dynamics of radar scattering in a sea
clutter environment has been addressed in the literature independently from
both theoretical and experimental points of view. Perhaps most notably in the
former case, Field & Tough [3, 4] develop a theoretical basis for the dynamics
which is demonstrated to agree with experimental data to a remarkable degree of
accuracy. In the latter case, Haykin et al. [1] study experimental data to motivate
a line of argument leading to the conclusion that sea clutter is inherently
nonlinear (and indeed possibly chaotic). In the current paper, we bring these
two independent lines of development together in a consistent way in order to
establish the nonlinear nature of sea clutter from both physical and
mathematical viewpoints. More precisely, the scattering dynamics can be derived
from first principles in terms of a pair of stochastic differential equations
(SDEs) for the received envelope and the radar cross-section (RCS) that feature
a nonlinear coupling and encode the statistical character of the sea state in
terms of a certain “shape parameter.” Examination of the differentiable parts
in this system of SDEs reveals a corresponding “noise-free skeleton,” that is,
a nonlinear vector process, with a degree of nonlinearity dependent on the
shape parameter in a manner consistent with that shown experimentally by Haykin
and coworkers. This significant development affirms the case for the nonlinear
character of radar sea clutter.The paper is organized as follows. Section2 provides a
summary of the experimental study  that led to the formulation of a hybrid AM/FM model, and the conclusion that sea clutter is a nonlinear dynamic process. Section 3 summarizes the essential ingredients of
SDE theory necessary for the basic interpretation of the SDE dynamics of radar
sea clutter. In Section 4 we apply this
formalism to establish the nonlinear character of the stochastic dynamics of the vector process consisting of the radar cross-section (RCS) and resultant back-scattered amplitude or “received envelope.” This is achieved from first principles via an extended random walk model. The extent of the nonlinearity in the resulting SDE description is quantified in terms of a certain
“shape parameter” (the relative variance in the RCS, minus one) that encodes the sea
state. We conclude in Section 5 with a
discussion of the interplay between the two independent lines of enquiry that lead to the common conclusions concerning the nonlinear character of radar sea clutter. We also indicate how our results
may suggest which types of experiments to perform to further substantiate and
enhance the theoretical framework, and discuss future prospects for the
investigation of chaotic dynamics.We refer the reader also to the recent book by
Haykin [5], where the experimental results of the current paper are mentioned
in the broader context of adaptive radar signal processing.


## 2. The Hybrid AM/FM Model of Sea Clutter

In an independent study reported in Gini & Greco [6], sea clutter was viewed as a fast “speckle” process multiplied by a “texture” component that represents
the slowly varying power level of the sea clutter signal; such a model is
perceptually satisfying. This is known as the K-distribution model and is widely used in the
literature. It is the model that we will be concerned with in our dynamical
description of sea clutter throughout the paper. The slow variation of the sea
clutter power level was attributed to the large ocean waves passing through the
observed ocean patch. The speckle was modeled as a stationary compound complex
Gaussian process, and the texture was modeled as a harmonic process.Inspired by the Gino-Greco model of sea clutter,
Haykin and coworkers carried out an extensive physical study of sea clutter
collected by the instrument quality coherent IPIX radar, where the radar data
were recorded on the East Coast of Canada [1]. In that paper, it was
demonstrated that amplitude modulation and frequency modulation play important
roles in the waveform description of sea clutter. The hybrid AM/FM model of sea
clutter has been substantiated further in Greco & Gini
[7].To explain the physical presence of modulation in sea
clutter, we observe that when a large wave passes across a patch of the ocean
surface, it will first accelerate and then decelerate the water's motion on the
ocean surface. The continuous tilting of the ocean surface by the waves gives
rise to amplitude modulation.Moreover, the ocean wave will cause a cyclic motion of
the instantaneous velocity of scatterers on the ocean surface, thereby giving
rise to frequency modulation as another characteristic of the sea clutter
waveform. When the mean velocity of the scatterers is high at a given instant
of time, then the spectral spread (i.e., the bandwidth occupied by the
frequency modulation) around that mean is correspondingly high, which is in
perfect accord with modulation theory.It is well known that, unlike amplitude modulation,
frequency modulation is a nonlinear process [8]. Therefore, the
presence of frequency modulation in the physical behavior of the sea clutter
waveform leads us to hypothesize that sea clutter is a nonlinear dynamic
process. To validate this hypothesis, Haykin et al. use 78 different coherent radar
data sets to compute the z-parameter, which denotes the Mann-Whitney
rank-sum statistic [9]. The results of this test are reproduced in
Figure 1, where the z-parameter is plotted against the spectral width modulation.Figure 1
Z value versus NMAD (ϕ˙), computed for 78 data sets measured by the
IPIX radar at various experimental conditions.A value ofz less than −3 is considered to be a strong reason for
rejecting the null hypothesis that the sea clutter data under test can be
described by linearly correlated noise. In Figure 1, we clearly see that the large majority of the experimental points lie below z=−3. Those points were in actual fact representative of high sea states. Based on
these experimental results, Haykin et al. concluded that sea clutter is indeed a nonlinear dynamic process, with the degree of nonlinearity increasing with increasing sea state.


## 3. Elements of SDE Theory

Stochastic differential equation (SDE) theory has significant implications for statistical
signal processing. It has recently proven successful in this context in the
application to radar sea clutter [3, 4, 10]. In a more general physical context, including optical propagation, the
stochastic calculus has led to substantial new theoretical developments in the
subject of electromagnetic scattering from random media [11–13].More recently, SDE techniques have been applied to
wireless channel modelling [14]
to include the effects of phase fluctuations in multipath reception. The fact that similar techniques are applicable to both the radar
backscattering and wireless propagation problems stems from the fact that each
is multipath in nature, with the only essential difference being that for radar
the receiver and transmitter are colocated. This latter feature, however, does
not affect the structure of the mathematical model used to describe the
resulting amplitude signal.In this paper, we will consider the RCS and received
envelope processes to evolve according to the dynamics governed by a stochastic
differential equation (SDE). In the context of the radar cross-section, such
dynamics arise from taking the continuum limit of a generic population dynamic
model for the (discrete) number of component scatterers. For the scattered
radiation, the origin of the SDE dynamics lies in the behavior of the component
phases which are taken to evolve in time according to a Wiener processWt [15] on a suitable (Rayleigh)
timescale. Thus, as we will see explicitly in Section 3, we are able to represent the essential ingredients of the radar
back-scatter temporally, in the form of a set of continuous time SDEs, the
basic mathematics of which we now introduce. Consider an arbitrary continuous
time stochastic process, say qt, which evolves in time according to (1)dqt=btdt+σtdWt. Herein, bt is a random process referred to as the
“drift,” and represents the ordinary time derivative of the process qt in the case that σ vanishes. The quantity σt, on the other hand, is the amplitude of the noise or fluctuating part of qt, in general a random process, and referred to as the “stochastic volatility” of qt. In the cases we study, it will become apparent that bt=b(t,qt) and σt=σ(t,qt) for some specific functions b, σ, and accordingly the process qt is called a “diffusion.”In contrast to the part ofdqt containing bt, the σt term contributes an essential part to qt that is not differentiable, in the ordinary
sense that ddt is well-defined. Nevertheless, the (Ito)
stochastic differential of qt can be well defined.In the engineering physics literature, one is perhaps
more familiar with the “Langevin” equation for the time
derivative(2)dqdt=bt+σtΓt in which Γt is the familiar white noise process and Γt has the autocorrelation property 〈ΓtΓt′=δ(t−t′)〉. For our purposes, it will be sufficient to understand and interpret from the
dynamical equations for the RCS and the received radar amplitude that, in a
discrete-time setting, (3)δqti=btiδt+σtintiδt1/2, where {ti} is a discrete set of observation times, δt=ti+1−ti, and {nti} are a collection of independent 𝒩(0,1) random variables. Thus, in terms of the Wiener
process, we make the discrete time identification δWti=ntiδt1/2. Moving from
(2) to
(3), the same drift and volatility coefficients become sampled at this discrete set of times.Then the above properties ofq and its time derivative are evident. The
essential distinguishing feature of the Ito stochastic differential is that it
refers to an integration of
(3) in which the volatility is to be evaluated at the left most point of each time subinterval (see [15]
for a detailed rigourous account).The essence of the approach taken is therefore to
postulate the exact dynamics in continuous time, and then sample at a discrete
set of times corresponding to the physical measurements. This procedure is
inevitably more precise than an attempt at a model that is fundamentally
discrete time in nature, since the physical observables are not quantized in
time.


## 4. Nonlinear Dynamics from SDE Theory

We will assume the (dynamical extension of the) random walk model for the resultant
back-scattered amplitude or “received envelope”(4)ℰt(N)=∑j=1Najexp[iφt(j)]︷s(j) with (fluctuating) population size N, random phasor step s(j), “form factors” aj, and component phases φ(j), wherein the collection {N,aj,φ(j)} is assumed to be mutually independent. Our
basic dynamical assumption is that the component phases φt(j) evolve according to a Wiener process on a
suitable (Rayleigh) timescale, that is, that dφt(j)=ℬdWt(j), which relation serves to define the constant ℬ.The key result of relevance to our discussion is obtained by taking the (Ito) stochastic differential of (4), in the limit that N, the number of component scatterers becomes large. Accordingly we introduce the
normalized amplitude process Ψt=ℰt(N)/N1/2, and a continuous valued RCS xt via N=N¯x, where N¯ denotes the mean of the discrete scattering
population size. In terms of these quantities, we now provide the following
coupled stochastic dynamics of the RCS and scattered amplitude/received
envelope. (We can express Ψt=It+jQt (j=−1), the familiar sum of its “in-phase” and
“quadrature phase” components.)Proposition 1.
The dynamics of the RCS and received
envelope for radar sea clutter, with shape parameterν=α−1, are given by the following set of nonlinearly coupled SDEs:
(5)dxt=𝒜(α−xt)dt+(2𝒜xt)1/2dWt(x),(6)dΨtΨt=[𝒜(2(α−xt)−14xt)−12ℬ]dt+(𝒜2xt)1/2dWt(x)+ℬ1/2γtdξt,in which γt is a unit power Rayleigh process, whose
dynamics are obtained by setting xt equal to a constant of unity and 𝒜=0 in the above system.This result pertains to (the simulation of) sea clutter from a generic radar system.Thus, in terms of the familiarK-distribution model for sea clutter, the
fast-speckle component is represented by γt (or its modulus squared) which is multiplied by a “texture” component, the RCS xt, according to the product representation Ψt=xt1/2γt. Incidentally, the separation of the radar scattering process into the RCS and
received amplitude (or intensity) components in this manner is introduced in a
statistical context in Jakeman [16], Jakeman & Tough [17] and developed
in a stochastic dynamical context in Field & Tough [3, 4]. (The
original proof of this result appears as [4, Proposition 2.1], and we will omit the details of this mathematical derivation which are outside
the scope of this paper.)It is beneficial at this point, in relation to the
above proposition, to explain the roles of the various quantities that occur in
more familiar radar terminology. The shape parameterν used in the SDE model is the same as that
familiar from the standard K-distribution statistical model of sea clutter. The quantity Ψ is the total radar backscattered amplitude, or received envelope,
incorporating both speckle and texture components; its modulus squared is equal
to the total backscattered intensity, that is taken to be K-distributed. The RCS or texture component is
represented by the correlated process xt.Thus, the nonlinear SDE forΨt is derived theoretically from first principles
beginning with the random walk model for the scattered electric field under the
assumption of a uniform phase distribution. (The
assumption of a uniform phase distribution can be relaxed, and a corresponding
detailed dynamical description in terms of SDEs has been given in [11].) An immediate consequence of this
dynamical equation is the “noise-free skeleton”, obtained by setting the
volatility coefficients of the fluctuating Wiener terms, that is, those
containing Wt, equal to zero. Accordingly, the randomness of the process is eliminated and the
residual dynamics are deterministic and differentiable. Physically, this
corresponds to an evolution conditioned on the current state of the system and
then averaged over an ensemble. (In other words, for
an Ito process qt with SDE dqt=btdt+σtdWt, the ensemble average evolution is determined by E|t[dqt]=btdt, where E|t denotes the expectation conditional on
information at time t.)
The concept of the residual noise-free part is explored further below.This set of coupled stochastic dynamical equations is
manifestlynonlinear by virtue of
the reciprocal term in xt appearing in the  amplitude equation, and only
reduces to linear dynamics in the special case that 𝒜 vanishes, that is, the scattering
cross-section is constant  (Rayleigh scattering). It turns out that a natural
quantifier of this nonlinearity, in the context of the SDE model for K-distributed noise, is the parameter α appearing in the coupled system of Proposition 1, as discussed below and illustrated in Figure 2.Figure 2
Normalized RCS time-series for low/moderate/high values of the shape parameter;
simulated data with𝒜δt=0.001, α=1,10,100.


### 4.1. Radar Parameters

In the present context, it is worth remarking on some of the key salient features of the SDE
theory, in relation to the sensitivity analysis of sea clutter to certain radar
parameters. Most notably, this kind of description is illuminating in respect
of the following issues.


#### 4.1.1. Correlation

The constants𝒜,ℬ in (5),
(6) have the physical dimension
of frequency, so that their reciprocals represent correlation timescales for
the RCS modulation (texture) and unit power Rayleigh (speckle) components,
respectively. The constant ℬ is electromagnetic in origin with a value ℬ∼c|k|, where k is the wave vector of the carrier and c is the speed of light. In radar situations,
the illuminating radiation is such that 𝒜≪ℬ, with the value of 𝒜 being determined as an intrinsic property of
the statistics of the scattering surface, independent of the electromagnetic
wave. Accordingly, in radar, the correlation time for the RCS is much longer
than that of the Rayleigh speckle (cf. also the discussion of amplitude and
frequency modulation in Section 5). The pulse frequency of the radar is the reciprocal of δt in the discrete implementation of the coupled
system of Proposition 1, and is assumed small compared to the Rayleigh correlation timescale ℬ−1, which amounts to the dimensionless criterion ℬδt≪1.


#### 4.1.2. Superposition

In light of the SDE theory, we may argue that the SDE of sea clutter is independent of the
amplitude profile of a transmitted pulse, provided the transmit energy is
maintained constant. This property, which is derived explicitly in
Field [12], is related to the fact that the form factors (i.e., the amplitude
weightings) in (4) may be taken as a unity for an asymptotically large population (cf. also [17] where the emergent statistical properties are independent of the choice of form factors).For a radar pulse of constant amplitude, suppose that
the two halves of the pulse have transmit frequenciesω1 and ω2. Then we may consider the correlation between the SDE of sea clutter for the two
portions as frequency ω1 increases relative to ω2. The transmit frequencies are proportional to the Rayleigh constant ℬ appearing in (6) (ℬ∼c|k|,
as explained above), and the relationship between the two SDEs, for the two
different transmit frequencies, is through (6). The two terms involving the constant 𝒜 are the same for both SDEs. On the other hand,
the terms involving the Rayleigh constant have different ℬ values corresponding to the two transmit
frequencies. Nevertheless, on physical grounds, the two complex Wiener
processes ξt for each transmit frequency should be
considered perfectly correlated. The reason for this correlation is that the
physical origin of the component phase fluctuations ϕ(j) is (microscopic) Doppler—the Doppler
frequency ratio ω1/ω2 is a function of the radial velocity of the jth member of the population, so the
micro-Doppler phase shift scales with the transmit
frequency; the ξt process is the same for any transmit frequency
(assuming that these are transmitted simultaneously) as this depends only on
the behavior of the component scatterer.In a similar fashion, consider the simultaneous
transmission of two pulses of constant amplitude, with two different
frequencies as above, and the resulting SDE of sea clutter received by a common
antenna. Since Maxwell's equations of electromagnetism are linear, the
resultingΨ is a linear superposition Ψ=κ11/2Ψ(1)+κ21/2Ψ(2), where κ1, κ2 are the relative intensities of the two
transmit waveforms, normalized so that κ1+κ2=1, and Ψ(i) are the constituent complex amplitude
processes, both satisfying the SDE (6),
with different Rayleigh constants ℬ corresponding to the two transmit frequencies.
Since the beams are simultaneous, the ξ processes are perfectly correlated, with the
remaining parts of (6) involving the constant 𝒜, the same for both transmit frequencies. Thus, the nonlinear dynamics do not
infringe the principle of superposition inherent in Maxwell's equations.
(It is necessary to assume here that the scattering
populations N(1) and N(2) pertaining to the different transmit
frequencies are equal.)


#### 4.1.3. Sea State and Polarization

Next, consider the two different copolarizations “HH” and “VV.” The SDE theory conveniently
represents the spikiness in the RCS of sea clutter due to “HH,” versus the
noise-like character due to “VV,” as follows.The cross-section SDE (5) emerges as a large N limit of an underlying
discrete-valued model for the scattering
population, the so-called birth-death-immigration (BDI) model
[18], in which α occurs as the ratio of the immigration and birth
rates. A property of the continuum limit of this population model, as
represented by the SDE (5), is that the distribution of xt is (univariate) gamma, with parameter α. As a consequence, since the distribution of the modulus amplitude for a given
value of the RCS is Rayleigh (as follows from (4) for fixed N), the intensity emerges as being K-distributed (also parameterized by α). Thus, the BDI population model is
appropriate to an RCS that generates K-distributed data. Now, for this gamma
distribution, we have Var[x]=E[x]=α. So the absolute magnitude of fluctuations in the RCS, that give rise to the K-distribution for the intensity (as opposed to
the Rayleigh “noise-like” distribution), becomes more appreciable as α increases. However, the appropriate theoretical
measure of “spikiness” is the relative variance R given by (7)R=Var[x](E[x])2,(E denotes the expectation functional) which is
the physical parameter of interest  since it is dimensionless and invariant
under rescaling of the RCS. In the case of K-scattering that we consider, R is equal to 1/α, and therefore the horizontal copolarization “HH” has small α, with larger α for vertical copolarization “VV.” The SDE
theory explains that if the ratio α of the immigration to birth rates is small,
then the sea clutter possesses spikes. It is therefore a natural mathematical,
as opposed to a detailed phenomenological, way of encoding this physical
property of the sea surface. (However, the SDE
theory does not explain why for “HH” polarization one should expect the
population to behave this way, the phenomenological reasons for which we do not
describe here.) Correspondingly, there are two different K-distributions for the intensity, indexed by
different values of the shape parameter ν=α−1, for the respective polarizations, where α is the SDE parameter appearing in Proposition 1.The situation as regards the extent of the temporal
fluctuations in the RCS for low/moderate/high values of the shape parameter is
illustrated in Figure2, which has been generated independently via a direct numerical integration of (5) according to (3), for various values of
the shape encoding parameter α. The figure
demonstrates the extreme departures from the mean value for large R,
which represents in physical terms sea spikes or glints in the scattering
surface. As the sea state settles down to a low value, the (normalized) RCS has
small fluctuations away from its mean (unity), so that there is no significant
modulation of the Rayleigh scattering time series—in other words, the scattering
is of constant local power. We remark also that spikiness should also be more
apparent at low grazing angles, represented by corresponding small values of
the shape parameter.As the sea state diminishes, correspondingly in terms
of the SDE dynamics, the parameterα→∞ and the relative variance in the RCS tend to
zero. Thus in Figure 2 the nonlinear term becomes less
pronounced. Accordingly, as we have seen in Section 2, so does the degree of nonlinearity as measured by the z-parameter, which further substantiates the
experimental findings reported in Haykin et al. [1].Our analysis therefore establishes the precise
relationship between the radar shape parameter, its statistical interpretation,
and the dynamical SDE theory, via the explicit appearance ofα in the coupled system of Proposition 1.


## 4.1. Radar Parameters

In the present context, it is worth remarking on some of the key salient features of the SDE
theory, in relation to the sensitivity analysis of sea clutter to certain radar
parameters. Most notably, this kind of description is illuminating in respect
of the following issues.


### 4.1.1. Correlation

The constants𝒜,ℬ in (5),
(6) have the physical dimension
of frequency, so that their reciprocals represent correlation timescales for
the RCS modulation (texture) and unit power Rayleigh (speckle) components,
respectively. The constant ℬ is electromagnetic in origin with a value ℬ∼c|k|, where k is the wave vector of the carrier and c is the speed of light. In radar situations,
the illuminating radiation is such that 𝒜≪ℬ, with the value of 𝒜 being determined as an intrinsic property of
the statistics of the scattering surface, independent of the electromagnetic
wave. Accordingly, in radar, the correlation time for the RCS is much longer
than that of the Rayleigh speckle (cf. also the discussion of amplitude and
frequency modulation in Section 5). The pulse frequency of the radar is the reciprocal of δt in the discrete implementation of the coupled
system of Proposition 1, and is assumed small compared to the Rayleigh correlation timescale ℬ−1, which amounts to the dimensionless criterion ℬδt≪1.


### 4.1.2. Superposition

In light of the SDE theory, we may argue that the SDE of sea clutter is independent of the
amplitude profile of a transmitted pulse, provided the transmit energy is
maintained constant. This property, which is derived explicitly in
Field [12], is related to the fact that the form factors (i.e., the amplitude
weightings) in (4) may be taken as a unity for an asymptotically large population (cf. also [17] where the emergent statistical properties are independent of the choice of form factors).For a radar pulse of constant amplitude, suppose that
the two halves of the pulse have transmit frequenciesω1 and ω2. Then we may consider the correlation between the SDE of sea clutter for the two
portions as frequency ω1 increases relative to ω2. The transmit frequencies are proportional to the Rayleigh constant ℬ appearing in (6) (ℬ∼c|k|,
as explained above), and the relationship between the two SDEs, for the two
different transmit frequencies, is through (6). The two terms involving the constant 𝒜 are the same for both SDEs. On the other hand,
the terms involving the Rayleigh constant have different ℬ values corresponding to the two transmit
frequencies. Nevertheless, on physical grounds, the two complex Wiener
processes ξt for each transmit frequency should be
considered perfectly correlated. The reason for this correlation is that the
physical origin of the component phase fluctuations ϕ(j) is (microscopic) Doppler—the Doppler
frequency ratio ω1/ω2 is a function of the radial velocity of the jth member of the population, so the
micro-Doppler phase shift scales with the transmit
frequency; the ξt process is the same for any transmit frequency
(assuming that these are transmitted simultaneously) as this depends only on
the behavior of the component scatterer.In a similar fashion, consider the simultaneous
transmission of two pulses of constant amplitude, with two different
frequencies as above, and the resulting SDE of sea clutter received by a common
antenna. Since Maxwell's equations of electromagnetism are linear, the
resultingΨ is a linear superposition Ψ=κ11/2Ψ(1)+κ21/2Ψ(2), where κ1, κ2 are the relative intensities of the two
transmit waveforms, normalized so that κ1+κ2=1, and Ψ(i) are the constituent complex amplitude
processes, both satisfying the SDE (6),
with different Rayleigh constants ℬ corresponding to the two transmit frequencies.
Since the beams are simultaneous, the ξ processes are perfectly correlated, with the
remaining parts of (6) involving the constant 𝒜, the same for both transmit frequencies. Thus, the nonlinear dynamics do not
infringe the principle of superposition inherent in Maxwell's equations.
(It is necessary to assume here that the scattering
populations N(1) and N(2) pertaining to the different transmit
frequencies are equal.)


### 4.1.3. Sea State and Polarization

Next, consider the two different copolarizations “HH” and “VV.” The SDE theory conveniently
represents the spikiness in the RCS of sea clutter due to “HH,” versus the
noise-like character due to “VV,” as follows.The cross-section SDE (5) emerges as a large N limit of an underlying
discrete-valued model for the scattering
population, the so-called birth-death-immigration (BDI) model
[18], in which α occurs as the ratio of the immigration and birth
rates. A property of the continuum limit of this population model, as
represented by the SDE (5), is that the distribution of xt is (univariate) gamma, with parameter α. As a consequence, since the distribution of the modulus amplitude for a given
value of the RCS is Rayleigh (as follows from (4) for fixed N), the intensity emerges as being K-distributed (also parameterized by α). Thus, the BDI population model is
appropriate to an RCS that generates K-distributed data. Now, for this gamma
distribution, we have Var[x]=E[x]=α. So the absolute magnitude of fluctuations in the RCS, that give rise to the K-distribution for the intensity (as opposed to
the Rayleigh “noise-like” distribution), becomes more appreciable as α increases. However, the appropriate theoretical
measure of “spikiness” is the relative variance R given by (7)R=Var[x](E[x])2,(E denotes the expectation functional) which is
the physical parameter of interest  since it is dimensionless and invariant
under rescaling of the RCS. In the case of K-scattering that we consider, R is equal to 1/α, and therefore the horizontal copolarization “HH” has small α, with larger α for vertical copolarization “VV.” The SDE
theory explains that if the ratio α of the immigration to birth rates is small,
then the sea clutter possesses spikes. It is therefore a natural mathematical,
as opposed to a detailed phenomenological, way of encoding this physical
property of the sea surface. (However, the SDE
theory does not explain why for “HH” polarization one should expect the
population to behave this way, the phenomenological reasons for which we do not
describe here.) Correspondingly, there are two different K-distributions for the intensity, indexed by
different values of the shape parameter ν=α−1, for the respective polarizations, where α is the SDE parameter appearing in Proposition 1.The situation as regards the extent of the temporal
fluctuations in the RCS for low/moderate/high values of the shape parameter is
illustrated in Figure2, which has been generated independently via a direct numerical integration of (5) according to (3), for various values of
the shape encoding parameter α. The figure
demonstrates the extreme departures from the mean value for large R,
which represents in physical terms sea spikes or glints in the scattering
surface. As the sea state settles down to a low value, the (normalized) RCS has
small fluctuations away from its mean (unity), so that there is no significant
modulation of the Rayleigh scattering time series—in other words, the scattering
is of constant local power. We remark also that spikiness should also be more
apparent at low grazing angles, represented by corresponding small values of
the shape parameter.As the sea state diminishes, correspondingly in terms
of the SDE dynamics, the parameterα→∞ and the relative variance in the RCS tend to
zero. Thus in Figure 2 the nonlinear term becomes less
pronounced. Accordingly, as we have seen in Section 2, so does the degree of nonlinearity as measured by the z-parameter, which further substantiates the
experimental findings reported in Haykin et al. [1].Our analysis therefore establishes the precise
relationship between the radar shape parameter, its statistical interpretation,
and the dynamical SDE theory, via the explicit appearance ofα in the coupled system of Proposition 1.


## 4.1.1. Correlation

The constants𝒜,ℬ in (5),
(6) have the physical dimension
of frequency, so that their reciprocals represent correlation timescales for
the RCS modulation (texture) and unit power Rayleigh (speckle) components,
respectively. The constant ℬ is electromagnetic in origin with a value ℬ∼c|k|, where k is the wave vector of the carrier and c is the speed of light. In radar situations,
the illuminating radiation is such that 𝒜≪ℬ, with the value of 𝒜 being determined as an intrinsic property of
the statistics of the scattering surface, independent of the electromagnetic
wave. Accordingly, in radar, the correlation time for the RCS is much longer
than that of the Rayleigh speckle (cf. also the discussion of amplitude and
frequency modulation in Section 5). The pulse frequency of the radar is the reciprocal of δt in the discrete implementation of the coupled
system of Proposition 1, and is assumed small compared to the Rayleigh correlation timescale ℬ−1, which amounts to the dimensionless criterion ℬδt≪1.


## 4.1.2. Superposition

In light of the SDE theory, we may argue that the SDE of sea clutter is independent of the
amplitude profile of a transmitted pulse, provided the transmit energy is
maintained constant. This property, which is derived explicitly in
Field [12], is related to the fact that the form factors (i.e., the amplitude
weightings) in (4) may be taken as a unity for an asymptotically large population (cf. also [17] where the emergent statistical properties are independent of the choice of form factors).For a radar pulse of constant amplitude, suppose that
the two halves of the pulse have transmit frequenciesω1 and ω2. Then we may consider the correlation between the SDE of sea clutter for the two
portions as frequency ω1 increases relative to ω2. The transmit frequencies are proportional to the Rayleigh constant ℬ appearing in (6) (ℬ∼c|k|,
as explained above), and the relationship between the two SDEs, for the two
different transmit frequencies, is through (6). The two terms involving the constant 𝒜 are the same for both SDEs. On the other hand,
the terms involving the Rayleigh constant have different ℬ values corresponding to the two transmit
frequencies. Nevertheless, on physical grounds, the two complex Wiener
processes ξt for each transmit frequency should be
considered perfectly correlated. The reason for this correlation is that the
physical origin of the component phase fluctuations ϕ(j) is (microscopic) Doppler—the Doppler
frequency ratio ω1/ω2 is a function of the radial velocity of the jth member of the population, so the
micro-Doppler phase shift scales with the transmit
frequency; the ξt process is the same for any transmit frequency
(assuming that these are transmitted simultaneously) as this depends only on
the behavior of the component scatterer.In a similar fashion, consider the simultaneous
transmission of two pulses of constant amplitude, with two different
frequencies as above, and the resulting SDE of sea clutter received by a common
antenna. Since Maxwell's equations of electromagnetism are linear, the
resultingΨ is a linear superposition Ψ=κ11/2Ψ(1)+κ21/2Ψ(2), where κ1, κ2 are the relative intensities of the two
transmit waveforms, normalized so that κ1+κ2=1, and Ψ(i) are the constituent complex amplitude
processes, both satisfying the SDE (6),
with different Rayleigh constants ℬ corresponding to the two transmit frequencies.
Since the beams are simultaneous, the ξ processes are perfectly correlated, with the
remaining parts of (6) involving the constant 𝒜, the same for both transmit frequencies. Thus, the nonlinear dynamics do not
infringe the principle of superposition inherent in Maxwell's equations.
(It is necessary to assume here that the scattering
populations N(1) and N(2) pertaining to the different transmit
frequencies are equal.)


## 4.1.3. Sea State and Polarization

Next, consider the two different copolarizations “HH” and “VV.” The SDE theory conveniently
represents the spikiness in the RCS of sea clutter due to “HH,” versus the
noise-like character due to “VV,” as follows.The cross-section SDE (5) emerges as a large N limit of an underlying
discrete-valued model for the scattering
population, the so-called birth-death-immigration (BDI) model
[18], in which α occurs as the ratio of the immigration and birth
rates. A property of the continuum limit of this population model, as
represented by the SDE (5), is that the distribution of xt is (univariate) gamma, with parameter α. As a consequence, since the distribution of the modulus amplitude for a given
value of the RCS is Rayleigh (as follows from (4) for fixed N), the intensity emerges as being K-distributed (also parameterized by α). Thus, the BDI population model is
appropriate to an RCS that generates K-distributed data. Now, for this gamma
distribution, we have Var[x]=E[x]=α. So the absolute magnitude of fluctuations in the RCS, that give rise to the K-distribution for the intensity (as opposed to
the Rayleigh “noise-like” distribution), becomes more appreciable as α increases. However, the appropriate theoretical
measure of “spikiness” is the relative variance R given by (7)R=Var[x](E[x])2,(E denotes the expectation functional) which is
the physical parameter of interest  since it is dimensionless and invariant
under rescaling of the RCS. In the case of K-scattering that we consider, R is equal to 1/α, and therefore the horizontal copolarization “HH” has small α, with larger α for vertical copolarization “VV.” The SDE
theory explains that if the ratio α of the immigration to birth rates is small,
then the sea clutter possesses spikes. It is therefore a natural mathematical,
as opposed to a detailed phenomenological, way of encoding this physical
property of the sea surface. (However, the SDE
theory does not explain why for “HH” polarization one should expect the
population to behave this way, the phenomenological reasons for which we do not
describe here.) Correspondingly, there are two different K-distributions for the intensity, indexed by
different values of the shape parameter ν=α−1, for the respective polarizations, where α is the SDE parameter appearing in Proposition 1.The situation as regards the extent of the temporal
fluctuations in the RCS for low/moderate/high values of the shape parameter is
illustrated in Figure2, which has been generated independently via a direct numerical integration of (5) according to (3), for various values of
the shape encoding parameter α. The figure
demonstrates the extreme departures from the mean value for large R,
which represents in physical terms sea spikes or glints in the scattering
surface. As the sea state settles down to a low value, the (normalized) RCS has
small fluctuations away from its mean (unity), so that there is no significant
modulation of the Rayleigh scattering time series—in other words, the scattering
is of constant local power. We remark also that spikiness should also be more
apparent at low grazing angles, represented by corresponding small values of
the shape parameter.As the sea state diminishes, correspondingly in terms
of the SDE dynamics, the parameterα→∞ and the relative variance in the RCS tend to
zero. Thus in Figure 2 the nonlinear term becomes less
pronounced. Accordingly, as we have seen in Section 2, so does the degree of nonlinearity as measured by the z-parameter, which further substantiates the
experimental findings reported in Haykin et al. [1].Our analysis therefore establishes the precise
relationship between the radar shape parameter, its statistical interpretation,
and the dynamical SDE theory, via the explicit appearance ofα in the coupled system of Proposition 1.


## 5. Discussion

We have described a detailed analysis of radar sea clutter data, whose primary purpose
is to address the presence of nonlinearity, from real experimental data. A
natural quantifier for this nonlinearity is thez-parameter or Mann-Whitney rank-sum
statistic, which has been successfully applied in the context of a hybrid AM/FM
model for sea clutter. The SDE dynamical model of radar sea clutter has also
been verified previously to a remarkable degree of accuracy, in terms of real
experimental data (see [3, Section 4(b)]). Moreover, an independent
theoretical account for such a model was provided in Field & Tough [4], and has served as the basis for other significant developments
[11, 12]. As we have seen in Section 4, this stochastic dynamic behavior is
inherently nonlinear, due to the broader timescale fluctuations in the RCS. The
extent of nonlinearity arises naturally in the SDE description through the
relative variance or shape parameter, which encodes the sea state. Thus, from
an SDE dynamical perspective, the nonlinear character of radar sea clutter is
firmly established, both theoretically and experimentally.Calculation of thez-parameter is from real data containing noise, the latter being akin to
the stochastic fluctuating terms present in (5), (6). However, z has the stochastic element removed, that is,
it is not a random variable. Accordingly, some ensemble averaging takes place
in the calculation of z, and for this purpose, the statistical properties of ergodicity and stationarity
are assumed, legitimate over realistic short timescales. In terms of the
parameter 𝒜 of (5), such timescales are short enough that
the assumption of constant 𝒜 is valid. Nevertheless, they should be long
enough (of the order of 𝒜−1) for the fluctuations in the RCS (or
equivalently, as we elucidate below, the frequency modulation effect) to be
appreciable so that nonlinearity can indeed be detected.From an engineering physics perspective, the dynamics
of sea clutter are perhaps more naturally viewed in terms of amplitude (AM) and
frequency modulation (FM). Studies have indicated that the degree of
nonlinearity is governed by the extent of FM which, in turn, is more noticeable
for higher sea states (i.e., the shape parameterν is large). To relate this further to the SDE
description of Section 4, it is convenient to view the resultant amplitude process Ψt in the product representation Ψ=x1/2γ, in which x is the RCS and γ is a unit power Rayleigh process. Then, the AM
consists of the fluctuations of γt (Rayleigh “speckle”) which is “frequency”
modulated by the RCS process xt over a much broader timescale. The FM/AM
contributions therefore have characteristic frequencies determined by 𝒜, ℬ, respectively. With zero FM, that is, xt constant, the dynamics of the resultant
amplitude are rendered linear, according to Proposition 1.It is worth emphasizing that the roles ofα and 𝒜, ℬ are essentially different, both theoretically
and in terms of their radar significance. The parameter α determines the associated gamma distribution
for the RCS and corresponding intensity K-distribution, and provides a scale invariant
measure of the spikiness of the backscatter. On the other hand, the frequency
constants 𝒜 and ℬ set the fluctuation timescales of the
respective texture and speckle processes, and thus leave the (asymptotic)
statistics invariant. In terms of Figure 2, the adjustment of 𝒜 can be considered as an amplitude preserving
dilation of the time series along the temporal axis, with smaller values of 𝒜 yielding longer duration between peaks in the
texture component.Observe that, whereas the FM/AM characteristics of the
received envelope do not map to unique stochastic dynamics, conversely the SDE
description allows for explicit extraction of both the FM/AM constituents
(see [12]), and therefore the SDE description is more fundamental than the
spectral one. Indeed, given the SDE dynamics, we are able to extract all higher
order statistical information through the propagators obtained as solutions of
the associated Fokker-Planck equations
[4, 19]. In this way, the SDE description of sea clutter should be viewed as the
most complete dynamical description, which preserves the inherent randomness in
the physical processes involved.We have seen that independent lines of enquiry, from
theoretical and experimental perspectives, lead to the common conclusion that
radar sea clutter is nonlinear over appreciable timescales such that the
temporal variation in the RCS is significant. The degree of nonlinearity is
determined by the sea state, which is represented by a certain “shape
parameter”ν that features in the SDE for the RCS.
(More precisely, ν=α−1,
where α is the parameter in the SDE for the RCS, and α≥0 arises from the parameters in the scattering
population model.) Consistently, the nonlinearity
is also determined by the extent of frequency modulation which, in terms of
real experimental data, has been quantified in terms of a certain z-parameter representing the Mann-Whitney
rank-sum statistic.From a theoretical point of view, the deterministic
part of the stochastic dynamics (5), (6)
is nonlinear, and is augmented with the addition of fluctuating Wiener terms in
the description of real experimental data, which is inherently noisy (cf. the
discussion of chaos surrounding
[20, Figure 2], and
also [21, 22].) We recommend that further studies be
made on the noise-free skeleton of the coupled system (5),
(6), which is manifestly nonlinear, to establish the existence or otherwise of an underlying deterministic chaotic behavior. If
chaos is present, then this system of nonlinearly coupled SDEs is an instance
of “stochastic chaos.” We remark in this respect that the presence of the
Wiener fluctuating terms in the system has the effect of stabilizing the
system, so that any chaotic behavior may no longer be observable
experimentally. These issues will be pursued in a subsequent paper.It is worth emphasizing again that the SDE theory of
sea clutter is experimentally valid, in its own terms
(see [3, Section 4(b)]),
and has also succeeded in practical applications, such as radar anomaly
detection, to a remarkable degree of accuracy. The theory also provides a way
of generating synthetic data, over which we have direct control, in terms of
its dependence on the sea state. Thus, in principle, we could measure the z-parameter for a data set simulated using
SDEs, for which the shape parameter is known, and thereby develop the precise
relationship between the z-parameter and shape parameter quantifiers of
nonlinearity. It may, indeed, also be possible to relate the two parameters in
purely theoretical terms. We can also generate data with and without noise,
which forms the basis for further experiment. We suggest that these two lines
of enquiry could form the basis of future developments in the investigation of
the nonlinear properties of radar scattering dynamics.The novelty of the current paper can be summarized as follows. In Haykin's paper, it was experimentally demonstrated that sea clutter becomes increasingly nonlinear as the sea state
increases. In this new paper, for the first time, theoretical justification of
this important result has been presented based on the earlier results of Field.



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*Source: 863129-2008-11-10.xml*