Clutter Simulation in Maritime Environments Peter E. Lawrence & Anthony P. Szabo Weapons Systems Division, Defence Science & Technology Organisation, PO Box 1500, Edinburgh SA 5 1 1 1. Email:
[email protected] &
[email protected]
Absmrcf-Knowledge of clutter characteristics are an important factor in determining radar system performance, especially for target detection. Clutter may be modelled as a random process, with the clutter characteristics embodied in the statistics of the process. Most modern pulsed-Doppler radars utilize rangeDoppler maps in their detection schemes, and so it is crucial to understand the statistical properlies of clutter in these maps in order to develop effective target detection algorithms. In this paper, we discuss the simulation of the sea clutter seen on the rangeDoppler map of an X-band pulseDoppler radar operating in a maritime envimnment. We shall implement the compound K-distribution model far maritime clutter returns and incorporate temporal and spatial correlations in clutter map realizations.
c
ronments for low grazing angles and when using high resolution radars [3], [7], [8]. In these cases, the empirical distributions exhibit both higher tails and larger standard4eviation to mean ratios than the Rayleigh distribution and so appear more “spiky” than at lower resolution. It has also been well known since the 1950’s that amplitude fluctuations of sea echoes in high resolution radar were caused by at least two different phenomena [9]; the rapid fluctuations caused by the presence of capillary waves and wind ripples, and the slower amplitude fluctuations caused by the passage of sea waves through a fixed spatial location. Hence different clutter models are required for the simulation of clutter in the maritime environment corresponding to different radar resolutions.
I. IXTRODUCTION
LUlTER characteristics play an important role in de- A. Clutter Measurements and Models Several models have been proposed in the literature for sea termining radar systems performance in many applications. For example in the maritime environment, sea clutter clutter. For example, probabilistic models such as Rayleigb, may limit the detection performance of radars when searching Log-Normal, Weibull and K-distributed clutter amplitude for small targets, such as periscopes [I]. Even when search- fluctuations [3], [IO], [ I l l have all been used to describe sea ing for larger targets, sea clutter may impact on the detection clutter. Jakeman and Pusey 131, 1121 first introduced a comefficiency of the target detection algorithms used [2]. Hence, pound sea-clutter model fitting data to the K-distribution. when designing an efficient target detection algorithm it is im- However, they did not emphasize the significance of this portant to have a good knowledge of the sea clutter statistics, model in jointly describing both the temporal and spatial amas well as the capability to simulate the clutter seen in the radar plitude statistics of sea clutter. Ward [I31 first demonstrated the significance of this model in jointly describing both the to test the algorithm. The realistic modelling of clutter is far from a new problem temporal and spatial amplitude statistics of sea clutter. In this and many authors have presented the results of theoretical and formulation the clutter return in each range cell consists of two experimental investigations of surface clutter, see e.g. [3], [4], components: a slowly varying (mean level) component gener(51. Here we will develop the method of Szabo [6] for simu- ated by the ChiAistribution, and a fast (speckle) component lating the clutter seen by an airbome pulse-Doppler radar. In generated by the Rayleigh-distribution. Ward [7] summarizes this approach, surface clutter is modelled as an uncorrelated the comprehensive evidence in support of the K-distribution, twdimensional Gaussian random field characterized by the including the effects of various geometrical and electromagmean clutter power in each range-Doppler bin. The impact netic factors on the clutter statistics. of the waveform and receiver processing on the clutter signal is modelled by “folding” the clutter signal onto an unambigu- E. K-disrriDured Clurrer Model ous range-Doppler interval and convolving with the ambiguA realistic model of maritime clutter returns needs to repity function of the waveform. Here this method is expanded resent the clutter amplitude fluctuations as functions of both to incorporate spatially-correlated K-distributed clutter, as re- time (from a given location) and position (at a given time). quired for modelling the clutter returns in a maritime environ- These fluctuations may be described separately by temporal ment. and spatial clutter distributions. The performance of signal This paper is organized as follows. An overview of mar- processing techniques such as target detection depend heavily itime clutter statistics and models is presented in Section I1 on on the correlation properties of the underlying noise proalong with the merits of a K-distribution clutter model for sea cesses in a range-Doppler map. In this way a higher emphasis clutter. The simulation of temporally and spatially correlated should he put on the correct encoding of correlations than on K-distributed clutter is discussed in Section III,and an exanthe choice of statistical model when simulating clutter returns ple is used to illustrate the key features of the method. for the purpose of assessing signal processing performance . Many methods are proposed in the literature for the incor11. MARITIMECLUTTERSTATISTICS poration of correlation into a given probability model. Here Experimental measurements indicate that large deviations the correlations are included by the modelling the clutter as from Rayleigh clutter statistics are observed in maritime envi- a K-distributed spherically invariant random process (SIRP) ”
0-7803-7871-7/03/$17.00 0 2003 IEEE
619
Radar 2003
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[141, [15]. Ingeneral SIRPsareexpressibleastheproductofa zerwnean complex Gaussian process and a non-negative random variate 1161. The temporal and spatial correlation properties of the clutter process are thus encoded into the respective random variables. One must note that in general a realization of a SIRP does not represent a realization of a clutter process. The ensemble statistical properties of a SIRP represent certain clutter characteristics, and in general time averages of a SIRP do not coincide with corresponding ensemble averages. A single realization of a SIRP can, however, represent the radar return from a portion of an illuminated surface with constant reflectivity,and this will be the case if the time-space resolution of the sea surface is small compared with the large-scale surface features; namely, those attributed to gravity waves [14]. The K-distribution is a popular SIRP model for sea clutter, being the result of analysis of random-walk models for bunched scatterers [4]. For grazing angles 0.1’ 5 &, 5 loo and range resolutions of the order of the small scale features of the sea surface, sea clutter returns in X-band (9.5-10 GHz) are accurately described in amplitude by the distribution [3]
where A is the random variable describing the clutter amplitude, K,( . ) is the modified Bessel function of second kind and of order U ,and r(.) is the Gamma function. This probability distribution function is characterized by the shape and scale parameters U and C, respectively, which are related to environmental attributes such as swell direction, wind speed, polarization and average clutter power [7]. The density function ( I ) reduces to a Rayleigh distribution in the limit U i 00 and can be considered effectively Rayleigh for shape parameters U > 20. Formulae for the estimation of these parameters will be introduced in the following section when constructing appropriate K-distributions for each range-Doppler map cell. As discussed above, in its compound form, the Kdistributed random variate A is the product of a generalizedChi variate of order v (the texture) and a statistically independent Rayleigh distributed random variable (the speckle). The speckle component has a relatively short de-correlation period on the order of tens of milliseconds, and can be de-correlated pulse-to-pulse by using a frequency agile waveform. The texture component has a long de-correlation period and is not affected by frequency agility. Therefore for a fixed frequency waveform with a relatively short processing interval the mean level can be assumed constant. For a given realization of a large scale structure, the small scale features of different patches are usually uncorrelated, therefore the speckle is assumed to be entirely de-correlated from one range-Doppler cell to another [17]. Throughout the course of this paper we shall adopt this compound K-distribution model for sea-clutter retums, provided we remain within suitable bounds on environmental and system parameters such as those mentioned above. 111. CLUTTERMAP SIMULATION The method of Szabo [6] has been expanded to generate realizations of the clutter signal seen on the range-Doppler map of a pulse-Doppler radar in a maritime environment. This approach may be summarized as follows. The mean clutter power is calculated as a function of range and Doppler shift
relative to the platform carrying the radar. This calculation includes detailed models of both the antenna pattern of the radar and the reflectivity of the surface. The folded mean clutter power, i.e. the mean clutter power seen on the unambiguous range-Doppler interval of the radar (the range-Doppler map), is then calculated and used to sample a realization of the clutter signal on the range-Doppler map. The resulting signal is then convolved with the ambiguity function of the waveform to model the resolution and correlation characteristics of the receiver. See [6] for a full description of the method. Here this method has been modified as follows. The mean clutter power is in this case a function of range, Doppler shift and the sea conditions such as wave height and wind speed. This dependence of sea conditions is present through the mean reflectivity function of the maritime environment. This mean clutter power is used together with an estimate of the shape parameter for each range-Doppler bin to obtain a K-distributed clutter field. This was not required in the original work of Szaho as the clutter was Gaussian distributed in its in-phase and quadrature (I and Q) components and hence the folded clutter signal was also I and Q Gaussian distributed. Szabo also modelled the ground clutter as a random field with uncorrelated increments. This was appropriate for ground clutter returns where the features on the surface are typically smaller than the range bins used by the radar. However, retums from the sea surface are typically correlated on length scales larger than a range bin and hence this effect needs to he incorporated when sampling a realization of the clutter signal. The method developed for doing this is described in detail below.
A. Correlated K-distributed Clutter Map Realizations Given a bandlimited real signal s ( t ) represented by the ] H [. ] complex analytic signal &(t) = s ( t ) i H [ s ( t ) where is the Hilbert transform, the received signal &(t), due to the illumination of surface of electromagnetic scatterers, at a fixed time t can he expressed as a stochastic integral of the form
+
&(t)
~
dz
/ / [ G S ( t - T ) ~ z ~ a ( u + u ~ ( ~ . u ; t ) ) ( ~ -iT( 7) : U ; t - T )
+ q,.(t - .)e2=z(”+u:,(i,”;t))(t--i)
dzZ ( T > U ;t - T I ]
(2)
where dZj(r;U ;t ) is the random complex modulation at time t due to an incremental scattering area centered at range R = m/2, with c the speed of light, and azimuth determined by the is-Doppler locus with Doppler frequency shift U ; the subscript j = 1 , 2 refers to the hemisphere the contributing clutter patch is located in, see Figure I . The function uA(r:v;t ) denotes the average Doppler frequency of the maritime surface within the incremental area of illumination. This maritime Doppler consists of a number of components attributed to different scattering mechanisms in a clutter patch [IS], however for simplicity we shall take U ;t ) to be the Doppler shift due to radial sea swell wave-speed within the given clutter patch. Note that we have ignored the presence of additive receiver noise in (2) for expedience in our discussion of clutter realizations. To be precise, a complex stochastic process n(t) should be added to (2) whose in-phase and quadrature components are uncorrelated strict-sense stationary white-noise N ( 0 , kTB) Gaussian processes, where k is Boltemann’s constant, T the receiver’s temperature in degrees Kelvin and B the effective bandwidth of the signal s ( t ) . The
.A(.,
620
mitted signal ei.(t) is given by
analyses that are to follow can he carried out in precisely the same manner if receiver noise was to be added to the received signal, and this would result in a more accurate range-Doppler map realization, especially in regions where the clutter-tw noise ratio (CNR)was low. The integration domain in (2) is [-U,,, ] , U x [O:r,,,] where ,U = 2V/X with V the radar platform speed, X the RF carrier wavelength, and r,, the delay associated with a return from the radar horizon.
m
A d Y , ( r : U ) (4)
where the overbar denotes complex conjugation and ~ ‘ . , ~ , “ ( t ) is the signal with analytic representation es;,7,w(t) = Wj(r,U ;t)d,(t). Given the unambiguous delay ru = T, (the PRI) and unambiguous Doppler uU = 1/T, (the PRF), the receiver response can he decomposed as X d . ’ : U’) =
+ 4 ( 7 >U ) ) ) d Y , ( r ; U ) where the summation limits l,,, kmi, and k,, are dependent on the dive angle of the platform, the radar horizon range, and other radar parameters (see [61). Here we have “absorbed” the unit magnitude phasor e-2rriv’r into the random field dY,. Under the change of variables (r,U ) H ( r lru:u ku,) and by Fubini’s theorem we simplify expression to
+
Fig. 1. Clutter cells defined by iio-rage and iso-Doppler loci. Contnbutionr to clutter are made by cells either side of the heavy mid-line on the sphere, which lies under the radar platform vector. It in 10 be understood that only one of the integrals in equation ( 2 ) is coinpired of the stochastic increments dZi which intersect this ruid-he.
We assume that the increments of the stochastic fields in (2) are K-distrihuted and thus may be factorized to where d Y , is a give dZ,(r,v;t) = Wj(~,u;t)dYj(r;u) generalizedchi variate and Wj(7>u ; t ) is a zero-mean complex Gaussian process. We have assumed that the processing interval is short compared to the de-correlation time of the modulating process dY, ( r ,U ;t ) so that this field can be regarded as a random constant during this period. The spatial correlation of the clutter process is inherent to this.component. The process ”r;(7; U ;t ) is uncorrelated between different (non-overlapping) clutter patches [17]. For a given clutter patch the speckle component Wj is correlated according to a temporal autocorrelation function (ACF) related, by the Fourier transform, to the power spectral density attributed to different microwave scattering mechanisms on the sea surface. There is evidence to suggest thal at low grazing angles (about loo) the sea surface scatterers arc lifetime4ominated [ I 81. To this extent if we assume that the sizes of the scatterers to be the same and that all are moving at the same speed, the temporal ACF of the speckle component for a given clutter patch is found to be
where t k is the time step difference, U is the scatterer Doppler frequency and t, is the average lifetime of the scatterer. For simplicity we shall assume this model. The response of a matched receiver matched to the trans-
(U
+
U&(.;
U)))dq(T
+
+
/Tu;U
+ kU,).
If the peaks of the ambiguity function ~ ~ ~ ~ ; , , are , ~ ( r ~ u ) ~ ~ approximately equal in height over the region [O; lm,ru] x [k,i,u,, k,,,u,] then the response of the receiver is given approximately by
where
is the folded incremental mean level field due to the ambiguities of the waveform in range and Doppler-shift. This approximation reduces the calculation of the receiver response to integration over a single unambiguous zone (0:r,,] x [0,U,]. Evaluation of the integral ( 5 ) is performed as a finite sum approximation in range- and Doppler-resolution increments over the domain of integration. The precise analysis and computation of stochastic integrals is beyond the scope of the current discussion; the interested reader is referred to 83.2 [19].
The shape and scale parameters associated with the random field dZj(r;U) are derivable as follows. The scale (or power) parameter C in Equation (1) for a given range-Doppler cell is related to the shape parameter v and the mean clutter power
62 1
~
ofthe returns in that cell. Similarly to Szabo [6], assuming a monostatic radar the mean clutter power attributed to an incremental scatterer, denoted by dc(r,U), is given by
where PT is the peak transmitted RF power, F the pattem propagation factor accounting for multipath propagation, diffraction, refraction and the antenna pattem, G the one-way power gain of a directive antenna, the (temporally) average reflectivity of the incremental scatterer with 19 a collection of radar system and maritime environment parameters used to construct a reflectivity model, R = cr/2 the range to the scatterer and dA(r,U) the area of the incremental scatterer patch whose center is located at the intersection of the is-range R-locus and the is-Doppler U-locus in the appropriate hemisphere. The shape parameter is then given by [I71
The shape parameter U depends on the sea swell as well as on other radar system parameters and so, at best, is estimated via empirical formulae. Such formulae have been developed for X-band radar frequencies, such as the following for waveforms of pulse length of 30ns (a range resolution of 4.2m) has been developed by Ward 171 from data collected at X-band frequencies of 9.5-10 GHz
where qig t [0.1, 101 is the grazing angle in degrees, Ap is the cross-range resolution, p = 1 for VV polarization or p = 1.7 for HH polarization, and U = -cos(28)/3 with 6 the angle between the direction of swell propagation and the antenna boresight axis. For pulse lengths longer than 30ns Ryan and Johnson [201 devised an estimate for the shape parameter which involves the addition of the term logl,(rp/30)loglo(50/$,) loglo(S.54Q)o-8 to the right-hand side of equation (91, where rp is the pulse length in nanoseconds. There exist other shape parameter estimations, some of which take into account the sea swell wavelength 181. As a rough guide to the values of the shape parameter we shall use the formula (9) or its modification due to Ryan according our waveform's pulse length. One must be aware that these formulae apply to clutter patches in the mainbeam footprint. To extend their validity to sidelobe clutter patches we take 0 to be the angle between the platform line-of-sight (LOS) to the center of the clutter patch and the swell direction and, similarly to [211, replace the cross-range resolution with the cross-range separation of the antenna pattem nulls between which the given clutter patch is located within or closest to. Having calculated the shape and scale parameters, we take for each time t the I- and Q-components of the speckle Wj (r:U,t) to have respective variances 2v/C2 in a given clutter cell, where upon the variate d l j ( r , U) is generalized-Chi distributed with degree U ; namely, with pdf
We now discuss the incorporation of spatial correlation into the field dYj(r,U). It is known that a generalized-Chi variate can be expressed as the square-root of a gamma variate. " I s d Y , (r,U) = fi where y is a gamma distributed rmdom variate with parameters ( U : U ) for v the shape parameter associated with the range-Doppler coordinate (r:U). When U E $W a semi-integer, y is a Chi-square variate and so is exg: for iid zero-mean Gaussian variates pressible as y = g, each with variance 1/2v. In order to correlate two gamma variates y1 and yz with the same semi-integer shape parameters ( U , v) we correlate their underlying Gaussian variates g j i in the expressions yj = gi, for j = I, 2. BY generating correlated pairs {g,,, g2i} such that their correlation coefficient pg,i,wi is equal to p for each i = 1,. . . 2v, it is found that [22]
Consider now the case of two gamma variates yi and y z with different semi-integer parameters (vi,vi) and (u2:v 2 ) respectively. Assuming that ul < u2. with iid Gaussian decompositions yi = g$ such that the pairs ,:g and 72 = { g l i ; g z i } for 1 5 i 5 2vl are correlated with correlation coefficient p9,i,92, = p and with the g ~ for , 2vl < i 5 2u2 being independent of all the other gil for 1 5 i 5 2ul. Using the results of [24] it can be shown that
Finally, for the case of gamma variates with non semi-integer parameters we use a non-linear transformation to convert these variates to semi-integer ones which can then be manipulated in the preceding fashion. This can be done as follows. Given a Chi-squared variate X = 9: 922 of degree 2 whose two constituent Gaussian variates 91 and 92 are iid N(O:l), upon solving the following equation for the variate y we obtain a gamma variate with non semi-integer parameters (U; v) [221
+
where ri( .: .) is the incomplete gamma function and r(.) is the Eulerian gamma function. Thus upon encoding the required correlations into the two constituent Gaussian variates that make up the Chi-squared variate X ,we can construct a gamma variate of non semi-integer parameters with the required correlation properties. As there is no closed form solution for the inverse of the incomplete gamma function, solution of equation (13) for y must be performed numerically. As stated in 1221, the preceding relations ( I I ) and (1 2) still hold in the case of gamma variates with non semi-integer parameters. In this way a given spatial autocorrelation function can be encoded into the clutter process by transforming the associated correlation coefficients to those for the constituent Gaussian variates as above. The correlated Gaussian variates are easily generated by standard linear filtering techniques [17]. Note that a spatially uncorrelated field dYj(r:U) may be generated easily by methods such as those described in 1231.
B. Clutter Map Examples We consider a radar platform flying straight and level at a height of h = lOOm with a speed of V = 200 knots 622
The following figures depict the results of maritime clutter simulation using equation ( 5 ) . All plots of receiver response power have been normalized by the transmitter power and receiver gain. Figure 2 is a realization of the folded receiver response. The mainlobe returns are observed to constitute an elongated bright patch at the bottom of the plot, with horizontal patterns caused by the antenna sidelobe structure and vertical structures due to the waveform's ambiguity function Doppler sidelobes. Here a PRF of 5.2kHz has been used. Figure 3 shows a realization of the associated texture field dY, (R?U) over a small region of the range-Doppler (RD) map in Figure 2 consisting of spatially correlated increments giving rise to localized regions of homogeneity consistent with wave-like structures. Figure 4 is a magnification of Figure 2 over the same RI-region in Figure 3. This receiver response power results in applying (5) to the spatially correlated texture field depicted in Figure 3.
(- 102.9ms-'). On each processing interval the radar will be emitting a train of 256 pulses each being phase
