• The area reflectivity (normalized radar cross section) for spatially distributed surface clutter.

• The volume reflectivity, , for volume distributed clutter, such as precipitation and chaff.

• The amplitude distribution of clutter returns.

• The Doppler spectrum of returns.

• The spatial variation of clutter characteristics.

• Polarization characteristics (the polarization scattering matrix).

• Discrete features (sea clutter spikes, discrete land clutter features, etc.).

2.11.2.1.1 Normalized clutter reflectivity,

A perfectly smooth and flat conducting surface will act as a mirror, producing a coherent forward reflection, with the angle of incidence equal to the angle of reflection. However, if the surface has some roughness, the forward scatter component (called coherent or specular reflection) is reduced by diffuse, non-coherent scattering in other directions. For monostatic radar, clutter is the diffuse backscatter in the direction towards the radar. This is illustrated in Figure 11.1.

The magnitude of the backscattered signal is characterized by the normalized radar cross-section . At some instance during the propagation of the pulse, a pulsed radar will illuminate a patch on the surface, defined (for low grazing angles) to a first order by the pulse length, the antenna azimuth beamwidth and the local grazing angle. The backscatter from land or sea is then modeled assuming multiple scatterers distributed spatially uniformly over this clutter patch. This is illustrated in Figure 11.2.

The normalized clutter reflectivity, , is defined as the total RCS, , of the scatterers in the illuminated patch, normalized by the area, , of the patch:

(11.1)

is usually defined in units of (dB relative to radar cross-section, per of area). Referring to Figure 11.2, the area of the clutter patch is given by

(11.2)

where is the antenna azimuth beamwidth and is the local grazing angle. The range resolution, , is related to the radar pulse bandwidth, B, by , where c is the velocity of light. The factor accounts for the actual compressed pulse shape and the azimuth beamshape, including the range and azimuth sidelobes. For a rectangular shaped pulse and beamshape, , while for a Gaussian-shaped beam and rectangular pulse, .

The grazing angle is defined in terms of the average local surface. For a nominally average flat surface, such as the sea, the grazing angle can be defined in terms of the radar height and propagation over a curved earth. In this case, the grazing angle, , can be written as:

(11.3)

where h is the height (altitude) of the radar, is the effective Earth’s radius and R is the slant range. For routine calculations at sea level, , where r is the true earth radius. This allows for the effects of atmospheric refraction for a typical refractive index profile. The actual effective grazing angle will depend on the local propagation and may be greatly changed under conditions of anomalous propagation, such as a surface ducts [2]. Over land, the local grazing angle will often be dominated by the terrain elevation and slope, which must be determined for specific cases.

The above expressions apply at low grazing angles, when the illuminated patch is defined by the azimuth beamwidth and pulse length. At high grazing angles, or for low bandwidth radars, the illuminated patch may be defined by the antenna azimuth and elevation beamwidths. Care should be taken if the grazing angle varies significantly over the illuminated patch, as will not be constant.

2.11.2.1.2 Clutter volume reflectivity

A similar approach is taken for volume scattering. This is defined in terms of the volume reflectivity, :

(11.4)

with units of .

The illuminated volume, , is given approximately by:

(11.5)

where is the one-way 3 dB elevation beamwidth. This is illustrated in Figure 11.3.

Clearly, this expression for the illuminated volume assumes that the volume scatterers fully fill the antenna beam and pulse length at a given range. If this is not the case, appropriate corrections must be made. For example, a volume search radar may have a narrow azimuth beam and a broad elevation beam. When illuminating rain, the rain ceiling may subtend a smaller angle than the upper edge of the beam.

2.11.2.1.3 Amplitude statistics

As illustrated in Figures 11.2 and 11.3, the return from clutter is usually assumed to comprise the backscatter from multiple scatterers, uniformly spatially distributed over an area or volume.

The scattered field, y, can be written as the vector sum from N random scatterers:

(11.6)

where is the radar cross-section of a single scatterer, is a phase term (related to the reflection coefficient and the relative range of each scatterer from the radar). Provided , the probability density function of the real and imaginary parts of y will be Gaussian, through the application of the central limit theorem (CLT). The corresponding PDFs for the envelope and intensity of the signal will be:

(11.7)

where r is , , .

This representation of clutter is applicable to spatially uniform clutter observed with low-resolution radars (i.e., with a large clutter patch, so ). This may apply to sea clutter, observed at high grazing angles and low resolution, or to, say, large flat areas of monoculture on land, such as woods or fields.

In many cases this representation is too simplistic. For example, the clutter mean intensity, above, may vary from one clutter cell to another, even though each cell still comprises multiple scatterers as in (11.6). Under these circumstances, the overall PDF will no longer have simple Gaussian statistics. Figure 11.4 illustrates the difference between returns with a Rayleigh PDF compared with clutter with strongly non-Gaussian statistics, modeled by a K distribution PDF, which is discussed below. A variation of the underlying mean intensity can be incorporated into the overall amplitude statistics by extending the simple model for PDF to include a dependence on the local mean intensity, which we shall call . Now:

(11.8)

The mean level is itself a random variable with PDF , so that the overall PDF can be written

(11.9)

This way we obtain the so-called compound-Gaussian model. The use of this type of model to represent radar sea clutter was originally described in [6–8]. The compound K distribution form of the model, discussed below, was originally formulated by Ward et al. [9,10]. There is further discussion in [9–21], and references therein. According to this model, each sample of the complex envelope of the sea clutter process is the product of two random variables: the texture and the speckle and can be represented as . The term represents a stationary complex Gaussian process, called speckle, which accounts for local backscattering; e are the in-phase and quadrature components of the speckle complex envelope . They satisfy the property and , so that , i.e., the speckle complex samples have unit power. The factor is a non-negative real random process, called texture that, as said, models the local clutter power.

The compound-Gaussian model can be derived also as an extension of the CLT, allowing the number of scatterers N in Eq. (11.6) to be a random variable [16,22]. In the particular case in which the number of scatterers is distributed following a negative binomial PDF, the texture can be shown to be a Gamma-distributed random variable (r.v.), and the amplitude a K-distributed r.v. (see Eq. (11.10)).

The compound-Gaussian model counts among its particular cases some families of distribution that are very popular in clutter modeling. The analytical expressions for these PDFs and their moments are reported below, where denotes the clutter amplitude.

K-model (K):

Replacing in Eq. (11.9) the generic with the Gamma PDF , we obtain

(11.10)

(11.11)

where is the gamma function, is the modified Bessel function of the second kind, of order is the shape parameter, and is the mean.

Generalized K model with lognormal texture (LNT):

(11.12)

(11.13)

where is the shape parameter, and m is the scale parameter. This model cal be obtained with the lognormal texture PDF .

Generalized K model with generalized Gamma texture (GK):

Putting in Eq. (11.9) the generalized Gamma texture PDF we obtain

(11.14)

(11.15)

Weibull model (W):

(11.16)

(11.17)

where c is the shape parameter and b is the scale parameter. The Rayleigh PDF is a particular case of the Weibull PDF for c = 2 [23]. Unfortunately, for the Weibull distribution, the PDF of the texture does not have a closed form and it is a compound-Gaussian model only for c < 2 [12].

Other sources of variability in clutter give rise to non-Gaussian statistics. While compound distributions can give some insight into the physical model underlying the non-Gaussian behavior, it is often sufficient to find an empirical fit of the overall amplitude statistics to a generalized PDF. Other popular distributions include the lognormal model, which does not belong to the compound-Gaussian family. The expressions of PDF and the moments are given below.

Lognormal model (LN):

(11.18)

(11.19)

where is the shape parameter, and is the scale parameter and the unit step function. Unfortunately, the LN model does not satisfies any of the compound-Gaussian properties [12].

The non-Gaussian PDFs K, W, and LN each have two parameters, a shape and a scale parameter, that can be adjusted to fit the observed data. The GK, conversely, has three parameters. Figures 11.5–11.7 illustrate examples of PDFs from different families.

The K and Weibull distributions are very similar, both including the Rayleigh distribution as part of their family ( in the Weibull distribution and in the K distribution) and they are often used for sea clutter modeling. The lognormal distribution gives PDFs with much longer tails (i.e., a higher probability of achieving larger amplitudes relative to the mean amplitude). This is often used for modeling land clutter, when the presence of large discrete scatterers can give rise to long tails if they are included in distributed clutter. A model that explicitly includes discrete spikes is the KA distribution, which is described in detail in [24,25].

2.11.2.1.4 Doppler spectrum

The simple model for clutter scattering given by (11.6), does not include any variation over time. If the individual clutter scatterers are moving radially with respect to the radar, the phase will vary with time, so that

(11.20)

Assuming random motion of the scatterers, the temporal variations of the return must be described in terms of the autocorrelation function, ACF:

(11.21)

The power spectrum of the returns can be related to the ACF from its Fourier transform (Weiner Khintchine theorem):

(11.22)

where is the Doppler radian frequency and , where f is the Doppler frequency.

The Doppler power spectrum (power spectral density PSD) is often modeled as having a Gaussian shape:

(11.23)

This is usually a mathematical convenience rather than any attempt at realism. Often the Doppler spectrum will be strongly asymmetric and the mean Doppler shift, , may not be zero. Clearly for land clutter is usually zero, but for rain and sea clutter in general and will be dependent on the wind speed and direction. Moreover, for sea clutter, as first reported by Pidgeon at C-band [26], and in X-band [27], the sea spectrum exhibits different peaks in the HH and VV polarizations [28,25,10]. Generally the spectral component corresponding to the lower frequency peak relative to the VV polarization, associated with the Bragg scattering component, is well described by the Gaussian function (11.23). On the contrary, the HH polarization is characterized by a higher frequency peak in the spectrum, maybe owing to the scattering from fast moving (faster than Bragg scatterers) and short-life scatterers. In this polarization the clutter PSD is well described by the Lorentzian function (autoregressive model of order 1)

(11.24)

where the constant k depends on the mean lifetime of scatterers.

In some practical situations the above-mentioned two models do not suffice to fit the real spectra shape, especially for the HH polarization. In [28] the authors consider first the scattering from fast-to-intermediate scatterers (e.g., bound-Bragg waves, etc.) whose PSD is characterized by a convolution of the Gaussian and Lorentzian profiles, resulting in the Voigtian function, and then a linear combination of these three models to improve the fitting.

Another simple model often used for both sea and land clutter PSD is the autoregressive (AR) one. The rationale for adopting AR models for the radar echoes is to have a highly parameterized model with a minimum number of parameters that can be easily estimated. Some examples can be found in [29,30].

For moving platforms, the antenna motion with respect to the clutter will also modify the Doppler spectrum. This is illustrated in Figure 11.8 for a side looking antenna, with one-way 3 dB beamwidth and platform velocity v. If the antenna has a Gaussian shape, the combined Doppler spectrum due to platform motion and internal clutter motion will be approximately:

(11.25)

Figure 11.9 shows an example of the Doppler spectrum of sea clutter as function of time, derived from radar data collected by CSIR [31]. The radar was vertically polarized with a frequency of 9 GHz, a pulse repetition frequency, , of 5 kHz and a range resolution of 15 m. The local grazing angle for the data collected was approximately . The radar look direction was , with wind of 15 kts from and a wave direction of , significant wave height 2.2 m. The raw data was then processed over bursts of L = 512 samples with an FFT, using a −55 dB Dolph-Chebyshev weighting function in the time domain. It can be seen that the clutter intensity varies in time in a periodic manner. The spectrum width also varies with time, with occasional extreme Doppler excursions, such as seen around 27 s into the time record, perhaps as the result of local wind gusting. Finally, the spectrum appears asymmetric in shape, with a non-zero mean Doppler shift. These features highlight the complexity of the relationship between the intensity modulation and the form of the spectrum, the former being dominated by the swell structure in the sea surface and the latter being additionally affected by the local gusting of the wind and the detailed scattering mechanism. This non-stationary behavior of sea clutter will be addressed with more detail in Section 2.11.3.1.7. However, despite this complexity it should be noted that the compound modulated Gaussian process is still applicable in the spectral domain and will affect the performance of both coherent and non-coherent radars.

2.11.2.1.5 Polarization characteristics

It is observed that most clutter characteristics are very dependent on the polarization of the radar signal and so an understanding of polarization is important.

A wave is said to be polarized if the direction of the electric, E, and magnetic, H, fields reside in a fixed plane. The plane in which the E vector moves is called the plane of polarization. The polarization scattering matrix, S, describes the amplitude and relative phase of returns from different combinations of polarizations on transmit and receive:

(11.26)

where:

For monostatic backscatter and .

The discussion above is for linear polarization and some examples of polarization scattering matrices for different targets types are shown in Table 11.1. Similar matrices can be used to describe the relationship for circular polarized signals (right or left handed) or any orthogonal coordinate system.

If the V and H components of the electric field are in phase, linear polarization is obtained. In general, an arbitrary phase between the V and H fields produces an elliptical polarization. The special case of a phase shift gives circular polarization.

For circular polarization, left-hand polarization is defined when on transmit; right-hand polarization is defined with .

The circular-polarization scattering matrix is defined as

(11.27)

while the linear-polarization has a scattering matrix given by

(11.28)

The circular polarization RCS terms can then be related to the linear polarization RCS terms by:

(11.29)

Using the definitions above, it can be seen that for a sphere, where and :

(11.30)

For this reason circular polarization is often used to reduce the return from rain clutter. Odd-bounce scatterers such as spheres or trihedrals will reverse the hand of polarization on reflection and a perfect sphere will have . Unfortunately, raindrops are not perfectly spherical but, even so, the reflectivity of rain may be reduced by about 15 dB to as much as 30 dB, dependent on conditions.

Target signatures, such as high-resolution range profiles, may be quite different according to polarization. For example using circular polarization, will show odd bounce scatterers while will show even bounce scatterers.

2.11.2.1.6 Spatial correlation

The returns from spatially uniform clutter will have Gaussian amplitude statistics, as described in Section 2.11.2.1.3. The magnitude of the return will change as the viewing geometry changes, such as when the antenna beam scans in azimuth or the range from the radar is changed. For a square beam and pulse shape (see Section 2.11.2.1.2), the returns from clutter patches spaced by more than one beamwidth or one pulse length will be independent and uncorrelated. However, if the successive clutter patches overlap spatially, then the returns will be correlated. A convenient measure of the spatial correlation of a sequence of intensity samples, is given by the estimation of the correlation coefficient:

(11.31)

where is an estimate of the mean intensity and k is the correlation lag . For spatially uniform clutter with Gaussian statistics and a fractional overlap of the beam or pulses between successive samples of , then the correlation coefficient of the clutter intensity is .

Spatial correlation of the clutter returns may also be observed if the local normalized reflectivity is changing in a systematic way. One example of this is observed over the short term in sea clutter, when the spatial variations caused by the sea swell or waves causes a related variation in the local mean reflectivity. This is illustrated in Section 2.11.3.1. Some examples of the range profiles of the local mean intensity of sea clutter and their corresponding range correlation coefficients are shown in Figure 11.10.

2.11.2.1.7 Discrete scatterers

The models for spatially distributed clutter are very useful for representing the returns from rain and often from land and sea, especially at high grazing angles. However, under some conditions the underlying assumptions of spatially uniform scatterers is no longer valid. For example, at low grazing angles terrain scattering may become very patchy and spiky, and is dominated by local high structures [32]. At higher grazing angles a distributed clutter model for terrain becomes more useful, but there will often also be a number of very large discrete scatterers in any scene, due to natural and man-made features. Barton [33] has analyzed a number of results from the literature and suggests that discrete clutter echoes of RCS might have a typical density of RCS a density of ; and RCS a density of . Discrete scatterers as large as RCS may be found. Long [34] suggest a density of about for RCS and for RCS.

Sea clutter may also include discrete spikes that have distinctly different properties from the surrounding clutter. In particular, specular scattering with HH polarization from the crests of incipient breaking waves may give rise to very localized returns having a large RCS [25]. The KA distribution [24,25] can be used to model discrete spikes that are added to the standard compound K distribution model. The amplitude distribution of clutter spikes has also been investigated in [35] who used the KK distribution to achieve a good fit to the tail of the distribution of clutter-plus-noise data recorded at medium grazing angles.

2.11.2.2.1 Theoretical and empirical models for sea clutter reflectivity

The radar backscatter from the sea is derived from a complex interaction between the incident electromagnetic waves and the sea surface. There are many theoretical models for backscatter, based on the physics of scattering from rough surfaces and approximations to scattering mechanisms. The simplest models attempt to represent the surface as many small segments, called facets, with orientations modulated by the waves. Scattering from wind-driven ripples may be approximated by Bragg scattering. The tilting of the ripples by longer sea waves changes the scattered power. This type of model, introduced by Wright [36] and Bass et al. [37], is discussed in detail in [25] and can give good results at medium to high grazing angles. However, at low grazing angles and high sea states the electromagnetic scattering becomes much more complex, with multiple reflection paths and shadowing from adjacent waves. There will also be breaking waves that can contribute considerably to the backscatter and are not modeled by simple modulated Bragg scattering. Some progress has been made recently in the understanding of electromagnetic scattering at low grazing angles [25]. However, the practical development of sea clutter models still mainly relies on empirical measurements.

Figure 11.11 shows a typical plot of normalized clutter reflectivity, , for sea clutter as a function of grazing angle, for VV and HH polarizations. At near vertical incidence, the backscatter is quasi-specular. In this region, the backscatter varies inversely with surface roughness with maximum backscatter at vertical incidence for a perfectly smooth surface. At medium grazing angles the reflectivity shows a lower dependence on grazing angle. This is often called the plateau region. Here the reflectivity is well modeled by the composite model. Below some critical angle (typically around grazing angle, dependent on the roughness) it is found that the reflectivity reduces much more rapidly with smaller grazing angles. This is known as the interference region, where propagation is strongly affected by multipath scattering and shadowing. Also shown in Figure 11.11 is the dependence of the reflectivity on radar polarization. In the plateau region, the backscatter for HH polarization is significantly lower than for VV polarization. This is evident of the significantly different scattering mechanisms for VV and HH polarizations.

In addition to a dependence on polarization and grazing angle, it is found that the reflectivity is strongly dependent on wind speed, which creates local surface roughness. This is often associated with the sea state but it should be noted that a strong sea swell in the absence of local wind may have a low reflectivity, while a strong wind may create a high reflectivity from a comparatively flat sea. The reflectivity will also depend to some extent on radar frequency. Another important consideration is propagation effects, such as ducting, which can change the local grazing angle of the signal incident on the sea surface. Indeed, observation of variation of surface reflectivity may be used to infer the presence of ducts [38].

There are various empirical models for the normalized clutter reflectivity that are used by radar designers. Tables of for different radar frequencies, grazing angles and sea states and for V and H polarizations are given in [1]. These values are the result of averaging measurements from many experiments. It may be noted that they do not model the variation of reflectivity with wind direction.

Another useful model, known as the GIT model [39], was developed by workers at the Georgia Institute of Technology in the 1970s. This model covers radar frequencies from 1 to 100 GHz and grazing angles from 0.1 to . It is based on an underlying multipath model as well as more general trends observed in experimental data sets. The normalized reflectivities modeled for H and V polarizations, (H) and (V), respectively, are given by the following expressions:

Radar frequency 1–10 GHz

(11.32)

The adjustment factors are

Radar frequency 10–100 GHz

(11.33)

and the adjustment factors are

The units and symbols used here are:

reflectivity for H and V polarizations, dB

average wave height, m

radar wavelength, m

wind velocity,

grazing angle, rad (0.1 )

look direction relative to wind direction, rad

Radar performance is often specified in terms of sea state and a useful relationship between sea state, s, and wind speed for a fully developed sea is:

(11.34)

Figures 11.12 and 11.13 show examples, using the GIT model, of and as a function of sea state and grazing angle, for the radar looking cross-wind and radar frequency 10 GHz (using Eq. (11.32)).

It may be noted that the values of normalized reflectivity given by the GIT model are significantly different from some of the values given in [1], especially at low grazing angles. This is not surprising, given that the models were derived from different data sets, and reflects the wide variation of values that may be encountered for nominally similar conditions.

2.11.2.2.2 Sea clutter amplitude statistics

Observation of the returns from the sea surface has identified two distinct components of the amplitude fluctuations. The first is a spatial variation, the texture, often associated with the sea swell. This represents a spatial and longer-term temporal variation of the local mean clutter level. This component de-correlates only slowly with time and is not affected by radar frequency agility. The second is a speckle component, associated with scattering from multiple scatterers in a given range cell. The speckle component de-correlates with time due to relative motion of the scatterers or due to changes in radar frequency.

These characteristics are shown in Figures 11.14 and 11.15 [25], which show recordings of radar sea clutter data, with a radar operating in X band (9 GHz), with a antenna beamwidth and 4.2 m range resolution.

Figure 11.14 shows range-time intensity plots of returns for a range interval of 800 m and a period of 125 ms, with a PRF of 1 kHz. The radar range was 5 km and the grazing angle was . The upper part of the figure shows fixed frequency returns. At a given range the returns exhibit a correlation time of . The underlying swell pattern is clearly visible. The lower figure shows frequency agile returns. Now returns at each range are decorrelated from pulse to pulse but the swell pattern is not affected.

In Figure 11.15, the fluctuating component (speckle) has been reduced by adding successive pulses. After 60 s the polarization changes from VV to HH. The VV POL returns show a clear swell-like component while the HH POL returns show short lived clutter “spikes” which still appear to be associated with the swell peaks. It may also be noted that the overall mean level of the HH returns appears to be lower than that of the VV returns, evidence of a lower reflectivity, .

Using data of the type shown in Figure 11.15, it has been shown [10] that the distribution of the local mean power fits well to the Gamma distribution. The local scattering (at a given range in Figure 11.14) has Gaussian statistics, resulting in the envelope of the overall return having the compound form of the K distribution (11.10). An empirical model for the dependence of the shape parameter on radar, environmental and geometric parameters has been developed through the analysis of experimental data at X-band (9–10 GHz). The model is [25]:

(11.35)

where

(The last term is omitted if there is no swell).

There are no known comparable models for the shape parameters of the Weibull and Lognormal distribution models.

2.11.2.2.3 Sea clutter Doppler spectrum

The Doppler spectrum of sea clutter varies with environmental condition, just as do the values of and the amplitude statistics. A useful simple low grazing angle model [40] for the shape of the average clutter velocity spectrum upwind at a given wind speed is a Gaussian shape with a mean velocity of and standard deviation of given by (in

(11.36)

where (VV) and (HH) denote the values for vertical and horizontal polarizations, respectively. This model can be applied to Eq. (11.23) with and .

There are many observations from data that can be used to improve this simple model. As the radar scans away from upwind, the mean velocity changes approximately proportionately with the view direction component of the wind vector, while the standard deviation remains approximately unchanged. The shape of the spectrum is skewed somewhat in the direction of the wind, and the shape varies in a complicated manner as the local mean power in the compound K distribution changes [41,42]. This is dependent on polarization and sea conditions. It causes the shape parameter of the magnitude distributions in individual frequency filters (after Doppler radar processing) to vary with frequency [10,42,43], thus causing difficulties for false alarm rate control in Pulsed Doppler and MTI radar systems. More research is needed to develop quantitative models to characterize these effects.

2.11.2.3.1 Empirical models of for land clutter

2.11.2.3.2 Empirical models of Doppler spectra for land clutter

Billingsley [32] has developed an empirical model for the Doppler spectrum of ground clutter. This is discussed in detail in Section 2.11.3.2.

2.11.2.4.1 Atmospheric attenuation

A detailed description of the attenuation of radar signals by the atmosphere has been provided by Blake [50]. Atmospheric attenuation varies with altitude, radar frequency and humidity. The references give typical figures for attenuation through the whole troposphere, attenuation over paths at sea level and so on. A simple rule of thumb for two-way atmospheric attenuation, , at sea level for a frequency f GHz is given by:

(11.39)

2.11.2.4.2 Rain attenuation

Rainfall is usually specified in terms of a rainfall rate, r, with the values for different levels of precipitation are given by Nathanson [1]:

Drizzle r = 0.25 mm/h

Light Rain r = 1.0 mm/h

Moderate Rain r = 4.0 mm/h

Heavy Rain r = 16.0 mm/h

Excessive rain r = 40.0 mm/h

The attenuation through rain is difficult to calculate theoretically, not least because conditions (humidity, drop size distribution and so on) can vary considerably for a nominal rainfall rate. Rainfall rates also vary with altitude and spatially across a given rainstorm. Again Nathanson [1] gives useful rules of thumb. The diameter, D, of a rainstorm can be represented by

(11.40)

and if the rainfall rate at ground level is , the rate at height h km, is given by:

(11.41)

The attenuation of two-way radar signals through rain is approximated by:

(11.42)

2.11.2.4.3 Theoretical and empirical model for rain reflectivity

Backscatter from rain can be modeled as scattering from multiple spheres. This is generally valid for small raindrops and theoretical models can be used to predict reflectivity, within their limits of validity. For higher rainfall rates (larger drops) these assumptions may no longer be valid and again the radar designer will resort to empirical models. For Rayleigh scattering from spherical drops , the volume reflectivity can be expressed as [2]:

(11.43)

where D is the drop diameter, N is the number of drops, is the radar wavelength and , dependent on the dielectric constant and . It is found that

(11.44)

where r is the rainfall rate in mm/h.

Empirical values for rain reflectivity are given by Nathanson [1], Currie [48], and Skolnik [2]. As an example, Table 11.6 shows some typical values of reflectivity given in [1]. Also shown in this table are values for the probability of different rainfall rates occurring, which is useful information when designing a radar for a particular application. The probabilities given in Table 11.6 are for the UK and values for other areas of the world can be found in [1].

2.11.2.4.4 Rain Doppler spectrum

The spectrum of wind-driven rain is of particular interest to radar designers. For radar meteorologists it provides detailed information on weather patterns. For weather avoidance radar on aircraft, Doppler provides important information on wind shear and other dangerous conditions.

A good discussion of the Doppler spectrum of rain is given in [1] and this is summarized here. The velocity spectrum and equivalent Doppler spectrum of rain clutter are conveniently modeled as having a Gaussian shape, with a standard deviations and , respectively. The spectrum of rain clutter is found to derive from various physical mechanisms, of which the principle ones are:

On the assumption that these spectral components are independent, the total velocity spectrum width can be given as:

(11.45)

Figure 11.16 illustrates how wind shear contributes to variations in Doppler shift across the radar beam. Wind shear causes a change in wind velocity with height, which can be approximated as a constant velocity gradient with zero velocity at ground level. Across the elevation beam of a ground-based antenna there will be a linear change of radial velocity, , which for low elevation angles will equal the difference in horizontal wind speeds. For a Gaussian shaped beam:

(11.46)

where k is the shear gradient (typical value , R is the range to the radar and is the one-way 3 dB elevation beamwidth.

For altitudes up to about 3 km, . Beam broadening is due to tangential velocity variation across the antenna in azimuth and

(11.47)

where is the tangential wind velocity at the beam centre, is the one-way antenna azimuth 3 dB beamwidth, and is the azimuth angle relative to the wind direction at the beam centre. is usually only a small component of the spectrum.

Finally, the distribution of vertical velocity amongst the raindrops will cause a spread in the velocity spectrum. A typical spread of vertical velocities is about so that at an elevation angle we have

(11.48)

2.11.3.1.1 Statistical models of clutter amplitude

As already said, many distributions are described in the literature to model the amplitude probability density function (PDF) of high-resolution non-Gaussian clutter. Here we compare the empirical PDF with lognormal (LN), Weibull (W), K, and Generalized K (GK) and Generalized K with lognormal texture (LNT) PDFs whose expressions are given in Section 2.11.2.1.3.

The characteristic parameters of the theoretical PDFs can be estimated by the classical method of moments (MoM) [53], which consists of equating experimental moments with the corresponding theoretical moments. The estimated moments are given by:

(11.49)

For the data at hand, samples have been processed for each range cell.

2.11.3.1.2 Cumulant domain analysis

To perform additional analysis of the compound-Gaussian model and to investigate whether the deviation from the theoretical models in the highest two resolutions, i.e., 9 m and 3 m, may be due to the presence of non-negligible thermal noise, we can apply the theory of cumulants² [56]. It is widely known in the literature that cumulants of order greater than two for a Gaussian process are identically zero [57,56]. Thus, if we consider the clutter process , where is a Gaussian process and is a non-Gaussian process, independent of , we have:

(11.50)

so the cumulants of can be derived from the cumulants of , that is, the only contribution in the cumulants of the overall process is that of the non-Gaussian component. In our case the in-phase (I) and quadrature (Q) components of the thermal noise are zero-mean Gaussian processes, then only non-Gaussian clutter contributes to the cumulants of order k > 2 of the observed complex data.

In [51] the authors estimated from the data the second, third, and fifth order normalized cumulants at zero-lags, i.e., for whose definition is:

(11.51)

where superscripts I and Q refer to the in-phase and quadrature components, i.e., the real and imaginary parts of the complex data. Then the estimates have been compared with the (normalized) theoretical cumulants of the compound-Gaussian model calculated at zero-lags. All the theoretical cumulants of the compound-Gaussian model of odd order calculated at the origin are equal to zero since the PDF of a complex compound-Gaussian process is symmetric with respect to the mean value (zero, in our case).

In Figures 11.20 and 11.21 we show the normalized cumulants and versus the second order cumulant for the resolutions of 9 m and 3 m, respectively. The results show that at a range resolution of 9 m the compound-Gaussian model is still accurate for VV and VH data because and are close to zero. Conversely, the fifth order cumulant for HH data shows a large deviation from zero in most cells. At a range resolution of 3 m, for most cells and all polarizations, the estimated cumulants deviate significantly from zero. This is an indication that the thermal noise, whose contribution is null in the cumulants of order higher than two, in these cases is not the cause of the deviation from the compound-Gaussian family.

2.11.3.1.3 Correlation analysis and power spectrum estimation

As said, the compound-Gaussian clutter model assumes the presence of two components, speckle and texture, with very different correlation times (some milliseconds for the first component and some seconds for the second one, in X-band). If the two components are statistically independent the overall autocorrelation function is the product of the autocorrelation functions of the two components [15,58]:

(11.52)

(if the speckle is a complex-valued stationary circular process, then . In practice, the decorrelation time of the coherent signal is equal to that of the faster component [15].

2.11.3.1.4 Estimation of the speckle autocorrelation and cross-correlation sequences

Since the texture can be considered constant over short time intervals, we can estimate the speckle autocorrelation functions and by using coherent signal samples from such short intervals with or without overlapping.

(11.53)

(11.54)

where is the number of data bursts and is the estimated value of the texture in the kth burst where .

Figure 11.22 shows two plots of the real and the imaginary parts of the speckle autocorrelation function for 60 m and 3 m estimated with . This result provides a clear indication that for all resolutions and all polarizations, the speckle correlation time is about 10 ms and the behavior is oscillatory.

2.11.3.1.5 Estimation of the texture autocorrelation sequence

To check the validity of the hypothesis made on the correlation times of the two components, we can estimate the texture autocorrelation sequence with the formula:

(11.55)

and the texture covariance as

(11.56)

It is useful to observe that with a 50% of overlap, we can estimate the texture correlation and covariance every lags. In the figures we plot the texture correlation coefficient, that is , with N = 128.

Figures 11.23 and 11.24 show the texture correlation coefficient. At the same resolutions and without differences in the polarizations, the texture correlation time is on the order of seconds. Furthermore, the texture presents periodicities with a period of 8 s at a range resolution of 60 m and of 3 s at a range resolution of 30 m. The periodicity is particularly evident in the VV polarized data for the resolution of 60 m in all the analyzed files. These results show as well that, with increasing resolution, the texture correlation time decreases, but still in the order of few seconds and the periodicities tend to disappear, due to the strong contribution of the thermal noise (see Figure 11.24).

Figure 11.25 reports two examples of the average spectrogram in semi-logarithm scale calculated as

(11.57)

where is the number of sequences in which the received vector for each cell has been divided, is the number of samples per sequence, k is the normalized frequency and is the kth sample of the periodogram of the rth sequence.

In these results, the periodogram shows, for all the polarizations, for all the range resolutions, a peak located around 150 Hz. Moreover, with a resolution of 60 m and 30 m, most of the analyzed cells show a bimodal spectrum, particularly evident in HH polarization, and then a second peak near −150 Hz; the power of the second peak is much lower than the power of the main one (see upper figure). From the resolution of 15 m, the IPIX radar seems to add a frequency interfering line in the spectrum at about −220 Hz (see lower figure). The line at 0 Hz is due to a residual of the continuous component. It is evident that, as resolution increases, the thermal noise effect becomes very important. In fact, the clutter-to noise ratio (CNR) decreases from for a resolution of 60 m and VV data to less than −5 dB for a resolution of 3 m and VH data. The CNR has been roughly estimated from the spectrum figures reading the value of the noise floor from each figure and calculating the clutter power as the difference between the overall disturbance power and the noise power. The values calculated for each cell is reported in the figure captions.

2.11.3.1.6 Mean range texture autocovariance sequence

To conclude the correlation analysis, in order to highlight further differences due to the resolution, we can also calculate the average range autocovariance function of the texture given by

(11.58)

where is the estimate of the texture on the mth burst of the ith cell, is the number of illuminated cells, is the number of bursts and is the texture average value in the mth burst .

Since at 30 m range resolution the illuminated zone is different, we compare here only the results found at 15 m, 9 m, and 3 m range resolutions.

Figure 11.26 shows the results obtained for VV and HH polarizations. From the figure it is evident that with a resolution of 3 m, it is possible to highlight and resolve shorter-range periodicities that are not visible in the other resolutions in all the polarizations.

2.11.3.1.7 Sea clutter non-stationarity: Bragg scattering and long waves

So far we have considered the sea clutter process as a stationary stochastic process. Actually this is not true, and the behavior we have proved in the previous paragraphs holds true only “in average.” The non-stationarity of the sea clutter is due to its nature and some explanation is needed. Sea clutter is backscattered by a moving rough surface [59,60] characterized in terms of two fundamental types of waves. The first type is represented by capillary waves with wavelengths on the order of centimeters or less, the second by the longer gravity waves (sea or swell) with wavelengths ranging from a few hundred meters to less than a meter [59]. In deep water, for the capillary waves we have , whereas for the gravity waves the wavelength is . Capillary waves are usually generated by turbulent gusts of near surface wind and their restoring force is the surface tension. On the contrary, swells are produced by stable winds and their restoring force is the gravity. Then, at any point on the surface the waves are complex summations of the locally generated wind waves and waves that have propagated in from other areas and different directions, resulting in a complex interaction [61].

To take into account the presence of different scales of roughness in the sea surface, Wright [36] and Bass et al. [37] developed a two-scale model of the sea surface scattering in which the surface height is partitioned into a large-scale displacement and a small-scale displacement. For this model, it is assumed that over any patch of the surface that is large compared with small-scale lengths, but small compared with large-scale lengths, the scattering can be modeled as first-order Bragg scattering from the small-scale structure. Thus, the effect of the large-scale structure is to change the distance between the antenna and each point of the considered patch, by tilting the surface and advecting the small-scale structure both vertically and horizontally. The effect of large-scale surface tilt is to introduce an effective amplitude modulation of the small-scale scattering [62]. Conversely, the effect of the advection is to influence the frequency content of the overall scattering.

The Bragg scattering is based on the principle that the return signals from scatterers that are half a radar wavelength apart, measured along the line of sight from the radar, reinforce each other since they are in phase [59]. The Bragg resonant length is where is the wavelength of the radar signal and is the grazing angle. At microwave frequencies, the Bragg scattering is from capillary waves and , where g is the acceleration of free fall. As a consequence, the capillary waves approaching and receding in the radar line-of-sight direction give rise to two Bragg spectral lines located at , at least in absence of other scattering phenomena and of the long waves. The magnitude of these lines depends on the azimuth look direction of the radar relative to the wind direction [41]. In a real scenario, the Bragg scatterers are advected by the orbital velocity of the intermediate waves (waves with wavelengths longer than the Bragg wavelength but shorter than the radar resolution cell) and of the long waves (waves with wavelengths longer than the radar resolution cell). The sum of the orbital velocities of the unresolved intermediate scale waves causes a spectral broadening around the Bragg lines. For many ocean conditions, these orbital motions broaden the Bragg lines by more than their separation, causing the lines to be unresolved, and generating only one Doppler peak. At X-band frequencies, this is generally the case [61,63]. Therefore, according to the Bragg theory, long waves that are resolved by high-resolution radars may be assumed to be constant over each illuminated cell. Consequently, their effect on the Doppler spectrum is to shift the Doppler peak according to the long wave orbital velocity. The orbital velocities are given by the simple harmonic motion , where f is the frequency of the long gravity wave of period T and H is its height from crest to trough. The contribution of the orbital velocity to the motion of the Bragg scatterers is the horizontal component, that is , where K is the wave number and x the spatial position [64]. The velocities of the scatterers in the nodes, crests and troughs of the waves are very different. Finally, an additional Doppler shift results from any surface currents present, including wind drift. A formula often used to represent this drift is , where is the wind speed. Therefore, by considering the contribution of orbital velocity, current velocity and wind drift, we can calculate the time-varying instantaneous Doppler shift as (see the picture in Figure 11.27) where is the intrinsic phase speed of the Bragg wave given by the wave dispersion relation. Due to the periodicity of should be periodic as well. However, the Bragg scattering is not the only phenomenon determining the clutter return, particularly if breaking waves are present on the sea surface as clearly showed in [65,66] but it is generally dominant in down-wind VV data. Therefore, to see clearly the effect of the long waves on the sea return, we show here in detail some results of the analyses carried out on down-wind VV data [41].

It is worth observing that in the modern statistical sea clutter literature, as already written, the small-scale structure scattering of the two-scale model corresponds to the speckle ([11,12,15,55]). The variations of the local power, due to the amplitude modulation of the speckle introduced by the tilting of the small-scale structure correspond to the texture. According to the compound-Gaussian model the texture and the speckle are two independent processes, the speckle is stationary and the texture amplitude modulates the speckle. However, as suggested by Haykin et al. [59], and as provided for by the two-scale model, the relationship between slowly and rapidly time-varying processes is much more complicated: the slowly varying swell motion does not only modulate the amplitude of the speckle, but also its mean frequency (Doppler centroid) and its bandwidth [41].

In this section of this tutorial, evidence of the modulating effect of long waves on speckle backscattering is verified through the analysis of experimental sea clutter data, collected in Dartmouth, Nova Scotia, at Osborne Head Gunnery Range (OHGR), with the IPIX radar.

The effect of the long waves on the overall scattering has been already considered in literature, but generally the analyses are theoretical or, when experimental, they consider only the average spectrum, that is, the spectrum calculated on the entire data set [27,28,61,67]. This kind of analysis can be exhaustive only when the radar resolution is low. In this case in fact, as explained in [67], the low-resolution radar performs a spatial averaging over many waves and the radar cannot see the different features of the waves passing through the resolution cell during the recording. Some evidence of the long wave and breaking wave amplitude and frequency modulation on the time-varying spectrum of real sea clutter is presented, for instance, in [59,64,68].

The sea clutter data shown here were collected at Osborne Head Gunnery Range (OHGR) with IPIX radar during the Dartmouth campaign. The characteristics and acquisition conditions of the analyzed file are summarized in Table 11.15. Data from all polarizations were processed, however, in this section we describe in more details the experimental results relative to the VV polarized data of the file Starea4 recorded on the 7 November 1993 at 11:23 pm. This dataset was recorded in conditions particularly apt to reveal the long wave modulating effects (that is our main interest here), both for the fully developed sea state and for the radar downwind direction. During the time the data was collected, the Canadian Forecast Service reported that the significant wave height was of 2.23 m with an average period of 8.3 s. The wind was blowing with a speed of 4–15 km/h since 8 pm of 6 November, coming to a relative calm state (1–3 in the Beaufort scale) after it has been blowing with strong velocity (35–45 km/h) for the previous 24 h. The wind direction was from the North since 10 am, and the azimuth angle was fixed at from the North ( of difference, approximately down-wind). Due to these conditions in the VV data the Bragg scattering was dominant.

A statistical analysis performed on these data as described in the previous paragraphs, showed that this clutter is GK distributed [41].

As already stated, the selection of the vertical polarization is justified because Bragg scattering was dominant (in the recording conditions of file Starea4) and the effect of long-waves should be clear and evident. Figure 11.28 shows the time evolution of the texture, estimated as , where , is the nth sample of the complex envelope of the data, is the number of bursts in which the entire sequence of the range cell data has been divided, is the number of samples in the lth burst (adjacent bursts have an overlap of 50%). It has been assumed (and verified) that the texture is constant within each short burst of 0.128 s. The almost-periodic behavior of the texture is evident for both polarizations.

In the frequency domain, the presence of periodicity in the power, and then of an amplitude modulating effect of the swells, is evidenced by the time-varying spectrogram in Figure 11.29a, where the spectrogram is plotted in a linear colormap from blue to red in the frequency range −200 to 200 Hz, where we have most of the sea clutter contribution (the variability of the sea spectrum was also visible in Figure 11.9. The plots in this figure were obtained by computing the FFT over a sliding window of 512 samples (0.512 s), weighed by a Hanning window, and with an overlap of 50%. The 512-points FFT provides the spectrogram in the frequency range −500 to 500 Hz. It is apparent that the mean texture and spectrogram illustrate a common periodic trend with a period of about 10 s that is comparable with those of the swells after projection on the radar line of sight. However, the effect of long waves is not limited to an amplitude modulation. It involves also a periodic change in the shape, bandwidth and maximum of the time-varying spectrum. This aspect is better revealed by the normalized spectrogram reported in Figure 11.29b (the spectrogram in Figure 11.29a was not normalized). The entire sequence of data has been again divided into bursts of samples each, with an overlap between adjacent bursts of 50%. Individual spectra have been calculated using a 512-point FFT and Hanning windowing, and then normalized with respect to the local clutter power. The non-stationarity of the sea clutter related to the periodic spectral variations is evident. These variations correspond to the changes of the frequency extent and of the Doppler peak due to the frequency modulation induced by the long waves on the Bragg-wave scattering.

In order to measure the temporal variation of the backscattered Doppler spectra, we can measure the Doppler centroid of the clutter spectrum and the rms (root mean square) bandwidth, defined respectively as

(11.59)

where f is the frequency and is the clutter PSD. The behavior of the Doppler centroid is investigated instead of that of the PSD peak because it is generally more representative of the short-time spectrum change, especially in the case of non-symmetric spectral shape. To estimate the time-varying centroid and bandwidth we can calculate:

(11.60)

where is the digital frequency, is the periodogram of the lth data burst (composed of 512 samples) corresponding to f (n), computed on points and .

Figure 11.30 shows the time evolution of the texture (on arbitrary scale and units, for ease of representation), Doppler centroid, and bandwidth, obtained by processing the data from the third range cell. They exhibit a common periodic trend with their own reciprocal delay. The Doppler centroid varies between −50 Hz and 5 Hz, the bandwidth between 25 Hz and 225 Hz. We observe that the maxima in the texture values are coincident with the minima of the bandwidth and vice versa. The spectral enlargement could be due to the modulation induced by the long waves or the presence of thermal noise. The effect of this noise in the spectral and temporal features of the received echo may be quite different depending on the clutter-to-noise power ratio (CNR). In the analyzed file, the CNR ranges from about 80 dB when the texture is maximum to about −5 dB when the texture is minimum [41]. In the latter case, the contribution of the noise to the echo is strong. The Doppler centroid and the bandwidth follow the texture behavior but with small delays of 2–3 s and 5 s respectively, i.e., about 1/4 and 1/2 of the swell period.

In order to quantify the relationship between texture, centroid and bandwidth, we can use their cross-covariance functions (CCF). The CCF between the time-varying texture and the Doppler centroid signals can be calculated as

(11.61)

and for texture and bandwidth:

(11.62)

where, and are the estimated texture, Doppler centroid and bandwidth, and

(11.63)

are their respective sample means. Figure 11.31a and b shows the plots of and , respectively. They show a behavior very similar to the cross-covariance functions between two truncated sinusoids with a common period of about 10 s separated by a lag of about 2.5 s and 5 s, respectively.

In order to retrieve the actual frequency values of the common sinusoidal components, it is useful to calculated the absolute value of the mutual PSDs, i.e., the absolute value of the Fourier transform of and . The results of this calculation are shown in Figure 11.32a and b. The information on the frequency “similarity” of texture and centroid provided by these mutual PSDs is analogous to that contained in the modulation transfer function (MTF) known to the geophysics community (see, for instance, [62] and references therein). The peaks in the mutual PSD correspond to the frequencies common to the two analyzed signals. This means, for instance, that the frequency component at is present in both texture and Doppler centroid signals and in both texture and bandwidth signals. This fact is confirmed by the normalized Fourier spectrum of texture, Doppler centroid and bandwidth (calculated after sample mean removal). The curves are in Figure 11.33a–c. They confirm the hypothesis of a near line-shape Doppler spectrum with components located in correspondence with the long wave frequency . Moreover, secondary components are present close to and, especially for texture and bandwidth spectra, at . These results suggest that, at least when Bragg scattering is dominant, the long waves modulate the speckle scattering in amplitude, causing the variation of the local power, and in frequency, causing a periodic variation of the spectrum centroid and bandwidth.

2.11.3.1.8 AR model

As apparent, there are two goals related to clutter modeling. The first is to provide further insight into the physical and electromagnetic factors that play a role in forming the clutter signal. The second is to produce a mathematical model, physically grounded, with which the clutter signal can be generated and processed to test detection algorithms. In this section, we suggest the use of AR modeling to describe the physical phenomenon of the long wave modulation analyzed in the previous section. As known, the autoregressive models can fit both Gaussian and non-Gaussian (as in our case) processes [70, Chapter 4].

The model’s ability to describe a physical phenomenon is not the main constraint, its mathematical tractability is also very important. In [41] the authors found that an AR model of the third order, AR(3), allows these constrictions. As regards the tractability, the AR(3) model is defined by seven real parameters, the complex value of its three poles and the final prediction error. Moreover, an AR process can be simply generated by properly filtering white noise.

The normalized periodogram of the data was obtained by averaging the individual normalized Fourier spectra related to single sliding bursts. The entire sequence of data has been divided into bursts of samples each, with an overlap between adjacent bursts of 50%. Next, individual Fourier spectra were calculated using a 512-point FFT and a Hanning window and then normalized with respect to the local clutter power. As regards the AR spectrum, it was obtained by processing all the range cell samples via Yule-Walker’s method [71]. In Figure 11.34 we report some results for an AR(3) process. It seems that the AR(3) can describe the basic shape features such as Doppler centroid and bandwidth.

This is confirmed by Figure 11.35a and b that represent the time evolution of these quantities, estimated both by the spectrogram and by the AR method. In particular, the estimated AR centroid is given by

(11.64)

while the estimated AR bandwidth is

(11.65)

where is the digital frequency, is the AR spectrum computed by processing the 512 samples of the lth data burst, via Yule-Walker’s method on points, and . Figure 11.36a and b shows the Fourier spectrum of the AR(3)-centroid and AR(3)-bandwidth. AR centroid and AR bandwidth still preserve the spectral components of the corresponding quantities obtained by the Fourier analysis. Also, the time delay with respect to texture is preserved, as shown by the cross-covariance (Figure 11.37a and b). Therefore, this analysis suggests that AR(3) model is able to model the basic effects of long waves modulation on the capillary backscattering.

It is also of interest to investigate how its parameters vary. Figures 11.36a–c show the time evolution of the two dominant AR poles and final prediction error, together with texture. The elements of correlation are evident. The dominant pole, with almost constant unitary modulus, has an instantaneous frequency that varies with the same period of the texture. Also, the second pole follows the same periodic trend, being almost constant when the signal is strongly reflected by the crest of a wave and noisy otherwise. The third pole is dominated by noise, while the final prediction error follows the texture trend.

From the results shown in the previous pages, it is apparent that there is not a unique optimum amplitude and spectral model for the sea clutter in all the resolutions. The statistical parameters change with time, range resolution, polarization, sea conditions. Generally the compound-Gaussian model provides a good fitting to the data but it shows some limits at the very high resolutions particularly for HH data, which are almost always spikier than VV data. Moreover, the sea clutter cannot be considered stationary on long periods of time. As particularly evident in X-band and VV pol, the long waves modulate the sea scattering in amplitude and frequency, determining the temporal and spatial periodical changes of the texture and of the bandwidth and Doppler centroid of the speckle spectrum.

2.11.3.2.1 Modified Kolmogorov-Smirnoff statistical test

The statistical analysis of data amplitude was concluded by applying a statistical hypothesis test as for the sea clutter data. The KS goodness-of-fit test has been largely used to determine which distribution (Rayleigh, log-normal, Weibull, or K in our case) best fits the data. Unfortunately, as already said for sea clutter, in some cases it is not useful, because it places an equal importance on all regions in the probability space. In practical radar applications a good fit is important in the tail regions of the PDFs. The tails, in fact, contain the strong values (i.e., the spikes) that, considered as target returns by a detector, can increase the FAR. Since good fitting in the tails is mandatory for correct design of CFAR processors, especially when low values are required, the KS test is of limited use for clutter data (as recognized also in [32]). To overcome this problem we can use a modified Kolmogorov-Smirnoff (MKS) goodness-of-fit test [29].

The idea is simple: apply the standard KS test by taking into account only the tail regions, i.e., by considering only the data above a given threshold AMKS and the modified theoretical PDFs where is the unit step function. The standard two-sample KS test verifies whether the recorded data are distributed in accordance with a hypothetical PDF, as already said for the sea clutter data. If the probability of type I is very low, for instance <1%, then this hypothesis should be rejected [72]. By applying the standard two-sample test, which considers the entire definition range of the random variable Z under investigation, the obtained probability of Type I error is always <1%, for all the distributions (Rayleigh, log-normal, Weibull, and K); thus, in the classical formulation, the KS test is not useful with these data. This is due to the differences exhibited by the distributions in the region of low values of Z. On the contrary, if we apply the MKS test setting the threshold AMKS in the region (0.01, 0.03), where the distributions are very similar (except for the Rayleigh), the result for the probability of type I is always 100%. This means that, in this central region, the KS test cannot distinguish between the different proposed models. But, as written above, we are mainly interested in the tails of the distribution, so we can applied the MKS test with, for instance, . In this way we leave out of the analysis the central region of the PDF, corresponding to high values of (say, . The results are reported in Table 11.17.

Generally, the value of probability of error of type I, , for Weibull PDF is higher than for the other PDFs, so the good fit of these ground clutter data to the Weibull model is confirmed.

2.11.3.2.2 Windblown trees data

The same analysis of farmland data was performed in [73] on a different Phase One X-band file, namely H067032.2, for which the data were measured from range cells containing windblown trees in contrast to open farmland. The clutter data shown here were recorded at Katahdin Hill site by Lincoln Laboratory personnel on 17 April 1985 [32]. The analyzed data set contains 30,720 samples per range cell, of HH-polarization, with pulse repetition frequency (PRF) equal to 500 Hz, and therefore pulse repetition interval (PRI) of T = 1/PRF = 2 ms. The return from each pulse is provided in in-phase and quadrature format. Data were recorded from 76 contiguous range gates utilizing the Phase One X-band stationary antenna in a fixed azimuth position (). These 76 range cells were located from 2.0 km to 3.1 km. The radar depression angle was about and the azimuth beamwidth of the antenna was 0.018 rad. The radar range resolution was 15 m. The illuminated area was tree covered, primarily with mixed deciduous trees, but also with occasional pine and cedar. At the time of the experiments, the deciduous trees did not yet have their leaves. During this experiment the wind was quite strong: 15–20 knots. Details about how the data were collected can be found in [32]. The 3D power map of the clutter is shown in Figure 11.42.

In this figure the presence of two regions, with different power values, is evident. The curves representing the received power versus range exhibit a stepwise behavior. This is due to the transition between agricultural fields/wetland (first region) and windblown trees (second region). In our analysis here we consider only single range cells from the second region. In Figure 11.43 the histogram of the data relative only to the range cells 34–36 is compared with the Rayleigh PDF with the same variance.

The results obtained for the open farmland data (two data files N007001.34 and N007001.35) are very different from these windblown trees data. This is partly due to the different land covers of the illuminated areas, but also importantly, partly due to the forest data embodying temporal variations, not spatial. The area relative to the H067032.3 data file was homogeneously tree covered, primarily with mixed deciduous trees and with occasional pine and cedar. In the other two analyzed data files, the returns came from a large spatial population of fixed discrete sources on open farmland. This heterogeneity introduces a considerable spread in the distributions as already noted in [32]. The differences are also due to the way of recording the data: the windblown tree data are temporal statistics (variations in time on a given range cell, or on few cells) recorded with fixed antenna; the Wolseley farmland data are spatial statistics, recorded with a scanning antenna on many range cells.

2.11.3.2.3 The experimentally-measured Doppler spectrum of ground clutter

The Katahdin Hill data are particularly interesting for the shape of the PSD.

A suitable analytic expression of the PSD of ground clutter, , back-scattered from regions containing windblown vegetation [32], is

(11.67)

where f is the Doppler frequency in Hz, r is the ratio of dc power to ac power in the spectrum, is the Dirac delta function which represents the shape of the dc component in the spectrum and represents the shape of the ac component of the spectrum, normalized such that . A phenomenological interpretation of this spectral shape was proposed in [74], where the exponential spectrum is obtained as the limit power spectrum in the presence of a random number of fluctuation scales in the Doppler components of windblown vegetation. From the detailed analyses of the MIT-LL measurements, it is apparent that the value of the dc/ac ratio r in Eq. (11.67) is strongly dependent on both wind speed and radar frequency and generally follows the empirically-derived analytical expression where w is wind speed in miles/hour, and is the radar carrier frequency in MHz. As described in [32], the MIT-LL measurements show a good fit to the model of Eq. (11.67) when the ac component of the spectrum is characterized by the two-sided exponential spectral shape

(11.68)

where is the radar transmission wavelength and is the exponential shape parameter. The value of is a function of the wind conditions such that the spectral width increases with increasing wind speed (see Table 11.18); however, is largely independent of radar carrier frequency over the range from VHF to X-Band. An algebraic expression for that incorporates the linear dependency of spectral width on the logarithm of the wind speed w (mph) as observed in the data is .

The described equations and Table 11.18 constitute a simple but complete model for characterizing the complex physical phenomenon of windblown clutter spectra over spectral dynamic ranges reaching 60–80 dB below zero-Doppler peaks, applicable over the range from VHF to X-band [32].

The Gaussian model for the windblown clutter spectrum, which was supported by early measurements with very limited dynamic range, is given by

(11.69)

where is the standard deviation of the Gaussian spectrum. The exponential model has wider tails than the Gaussian model. For use with somewhat increased dynamic range, the other alternative popular spectral shape function has been the power-law spectrum, which is characterized by two parameters, the shape parameter n, and the break-point Doppler frequency that determines where the shape function is 3 dB below its peak zero-Doppler level:

(11.70)

The MIT-LL measurements of much increased dynamic range clearly indicate that clutter spectral shapes are indeed wider than Gaussian; but that observed rates of decay modeled as power-law at upper levels of spectral power do not continue as power-law to lower levels of spectral power, but fall off much faster at the lower levels. The tails of the exponential model show an intermediate rate of decay—between the fast decay of the Gaussian model and the slow decay of the power-law model—that yields a good fit to the measured data.

For analyzing the PSD of windblown tree clutter, many data from various windblown-tree range cells were processed. Here we report only the results from range cell #35 (second region) as representative of results achieved by processing other range cells in the same region. The PSD has been first estimated non-parametrically, by using the modified periodogram method (Welch method) with the four-sample Blackman-Harris window [73]:

(11.71)

with coefficients , and . The set of 30,720 data samples has been divided in L = 15 subsets , with , of N = 2048 samples each and then the windowed periodogram calculated from each subset as follows

(11.72)

where , for , and is the nth complex sample relative to the kth subset. To obtain the final spectrum estimate, the K estimates are averaged as . Finally, the estimate is normalized such that the area is unity, i.e., divided by the estimate of the total disturbance power . The disturbance power is estimated to be . The resulting curve is shown in Figure 11.44, labeled as “WP,” which stands for weighted periodogram. Note that the PSD is plotted from −250 Hz to 250 Hz because the PRF is 500 Hz. It is quite evident in these figures that, for , thermal noise predominates over clutter; thus we fit the clutter PSD only over the frequency range . In [73] the authors used the frequency bins in the range to estimate the clutter-to-noise power ratio (CNR), which is obtained by solving the equation with respect to CNR, where is the thermal noise power, denotes the normalized PSD estimate and the sum is over all i’s such that . For the 35th range cell it results that .

To estimate the clutter PSD parameters, nonlinear least squares (NLLS) method [15] has been applied. The NLLS method can be applied directly to either or to its logarithm. In the second case, the estimates of the model parameters are derived by solving the following minimization problem:

(11.73)

where is the vector of model parameters that have to be estimated. We refer to the method in (64) as the NLLS method. For the Power Law model in Eq. (11.61), we consider the two cases n = 2 (PL2) and n = 3 (PL3), and then we use the NLLS method to estimate the PSD parameters. In Figure 11.44 the various models obtained by the NLLS method are compared with the weighted periodogram. In Table 11.19 we report the estimates of , and obtained by the NLLS method. The value of r is always lower than , so the dc component in cell #35 is negligible. The best fit is provided by the exponential model.

The analysis of the land clutter shows that the clutter statistics are heavily site-dependent. As a matter of fact the amplitude of the clutter at X- and L-band is Gaussian distributed for homogeneous clutter as that of windblown trees, non-Gaussian distributed if backscattered by discrete sources as pylons, fences, etc. Even the PSD depends on the sites, and, often, on the dynamic range of the radar as well.

2.11.5.1.1 Signal processing and design of detectors

The radar equation provides a means of calculating the signal-to-clutter-plus-noise ratio, SCNR, and the clutter-to-noise ratio, CNR, for a single pulse return. For many radar systems, the returns from several successive pulses are integrated, to benefit from more radar power on the target. These pulses may sometimes be at different frequencies (frequency agility). In a typical example, during a dwell on a target, a pulse Doppler radar may transmit a burst of pulses with a fixed PRF and frequency, followed by successive bursts at different PRFs and frequencies. Returns from trains of pulses are used in subsequent signal processing to improve target detection and discrimination against clutter and noise.

To calculate and following detection processing in the radar requires knowledge of the effect of the processing on the amplitude statistics immediately prior to the final detection threshold or decision. For coherent processing with Doppler filter banks (typically achieved with an FFT over a fixed frequency burst of pulses), the Doppler spectra of target and clutter must be known to calculate the subsequent SCNR and CNR in each Doppler filter. The outputs from each Doppler filter may then be detected and then subsequent detections from successive bursts may be non-coherently integrated. Again the PDF of the integrated signals must be calculated. In the case of clutter, it is very important to understand any correlation between the samples that are being integrated. Thermal noise will always be independent from pulse to pulse. For clutter, the situation is more complex, with different components fluctuating at different rates. For example, pulse-to-pulse frequency agility can decorrelate the speckle component of sea clutter [10], but the underlying mean level component is not affected and may effectively be constant over a dwell. The target returns may also fluctuate at different rates over a dwell. This type of behavior is usually approximated by estimating the effective number of independent samples observed over a dwell period. Again, for the compound K distribution model it is possible to model the effects of pulse-to-pulse integration taking into account the effective number of independent samples of target, thermal noise and the two clutter components, each of which may be different [25].

The design of optimum and suboptimum, adaptive and non-adaptive, coherent and non-coherent detectors for detecting point-targets embedded in Gaussian disturbance is a mature and well understood topic. There are many excellent books and papers addressing different aspect of the problem. Some references are listed in [81]. Conversely, the detection of targets embedded in non-Gaussian clutter is a less mature topic, even though there are many references published in the last 20 years (see for instance [19,22,25,82–92] and references therein). The main problems that must be taken into account are the estimation of the disturbance covariance matrix, the estimation of the parameters of the clutter distribution [17,25,93–95], the non-stationarity of the clutter in the case of sea scattering [96] and the site-dependence of the clutter in the case of ground scattering [97,98]. New radar systems must be able to rapidly adapt to changing conditions. Knowledge or estimation of clutter conditions can assist with the selection of both transmitted waveforms and detection algorithms.

2.11.5.2.1 Clutter models for performance acceptance and trials analysis

An increasingly important role for models is for the analysis of radar trials and the acceptance by a customer of a new radar into service.

Acceptance trials are an integral part of any radar development program. These are intended to demonstrate to the customer that the radar provides the required functionality and is “fit for purpose.” It is also usually desired that trials be used to quantify the key performance parameters of the radar (e.g., detection range against a given target type). However, in many cases the reliable measurement of performance on trials is very difficult and may at the very least require many hours of trials. This is particularly true for radars required to detect very small targets in difficult clutter conditions or for radars that are continuously adapting their parameters (such as signal processing, waveforms, antenna beamshape, etc.) in response to changing environmental conditions. The customer may specify that a target of given RCS must be detected at a certain range, in particular environmental conditions (rainfall rate, sea state, etc.). Given an agreed set of models, such a situation could be modeled and provide reasonably accurate estimates of performance, within the limitations of the models used. However, measurement of these parameters in a trial can be very difficult. Even the apparently simple task of measuring the noise-limited detection range of a moving target is not straightforward [102]. If the original requirement was, say, to detect a target in sea state 3, in 1mm/h of rain, the radar designer would have to put a very large margin on expected trials results to allow for uncertainty in assessing whether sea state three is present, what the rainfall rate was (at the target and over the intervening path), combined with the large range of clutter characteristics, such as associated with a given sea state. Although the radar designer would be able to model performance for given clutter model parameters, it would not usually be possible to relate these to the conditions actually prevailing on a trial.

One solution is to combine trials measurements with modeling and factory measurements. Factory measurements can determine the basic radar parameters that determine noise-limited performance in, say, clear air. Modeling can predict the expected mean performance for a wide range of target sizes and clutter characteristics, which could never be tested in practical trials of reasonable duration. The empirical models that are used in radar design have themselves been developed from many hours of trials data, representing an average expected conditions that do not then have to be reproduced in acceptance trials. Then on a trial, detection performance can be measured (detection range, for example) for the target and environmental conditions applying on the trial. At the same time, the raw radar returns can be recorded for subsequent analysis. This analysis can yield the actual target and clutter characteristics that were present, provided that the radar noise-limited performance is suitably calibrated. These observed characteristics can then be input to the performance prediction model with the results compared with those actually achieved in the trial.

If raw radar data is recorded on a trial, it should also be possible to replay it through the radar signal processing, to assess the effects of different detection algorithms. In addition, it may be possible to inject synthetic targets, to further investigate detection performance.

This type of approach, combining trials with performance prediction modeling, can allow quantitative assessment of performance that would be effectively impossible from trials alone (due to uncertainty over the prevailing conditions and the time and cost of extensive trials). Of course, it is far from straightforward to achieve. It implies the measurement of detection ranges and false alarms rates from trials observations, and the estimation of clutter and target characteristics from recorded data. Each of these measurements is subject to its own measurement errors and the combined results would have to be analyzed in detail to assess the buyer and seller risks [25] associated with eventual equipment acceptance or rejection. The analysis methods and also the underlying performance prediction model must be agreed in detail between customer and supplier (preferably well in advance of any acceptance trials actually take place).