**4. Numerical analysis of detectors**

#### **4.1 The lngara data**

Before proceeding with an analysis of the optimal and suboptimal detectors, the Ingara data is discussed briefly. This sea clutter set was collected by DSTO, using their radar testbed called Ingara. This is an X-Band fully polarised radar. The 2004 trial used to collect the data was located in the Southern Ocean, roughly 100km. south of Port Lincoln in South Australia. Details of this trial, the Ingara radar, and data analysis of the sea clutter can be found in Crisp et al (2009); Dong (2006); Stacy et al (2003; 2005).

The radar operated in a circular spotlight mode, so that the same patch of sea surface was viewed at different azimuth angles. The radar used a centre frequency of 10.1 GHz, with 20 µs pulse width. Additionally, the radar operated at an altitude of 2314m for a nominal incidence angle of 50°, and at 1353m for 70° incidence angle. The trial collected data at incidence angles varying from 40° to 80°, on 8 different days over an 18 day period. As in Dong (2006), we focus on data from two particular flight test runs. These correspond to run34683 and run34690, which were collected on 16 August 2004 between 10:52am and 11:27am local time Dong (2006). Dataset run34683 was obtained at an incidence angle of 51.5°, while run34690 was at 67.2°. In terms of grazing angles, these correspond to 38.7° and 22.8° respectively. Each of these datasets were also processed in blocks to cover azimuth angle spans of 5° over the full 360° range. Roughly 900 pulses were used, and 1024 range compressed samples for each pulse were produced, at a range resolution of 0.75m. In Dong (2006) parameter estimates for the data sets run34683 and run34690 are given, enabling the fitting of the K- and KK-Distributions to this data. This will be employed in the numerical analysis to follow.

As reported in Dong (2006), the radar is facing upwind at approximately 227° azimuth, which is the point of strongest clutter. Downwind is at approximately 47°, which is the point where the clutter is weakest. Crosswind directions are encountered at 137° and 317° approximately.

#### **4.2 Target model and SCR**

Since the Ingara clutter does not contain a target, an artificial model is used to produce the receiver operating characteristics (ROC) curves. These curves plot the probability of detection against the signal to clutter ratio (SCR). It hence indicates the performance of a detector relative to varying signal strengths in the clutter model.

Throughout, a Swerling 1 target model is used Levanon (1988). This is equivalent to assuming that the in-phase and quadrature components of the signal return are complex Gaussian in distribution. For the problem under investigation, the SCR is given by

$$SCR = \frac{\mathbb{E}(|R|^2) \, |\, \mathfrak{u} \, |^2}{\mathbb{E}(S^2)N},\tag{36}$$

and it is not difficult to show that

$$\mathbb{E}(S^2) = 4\nu \left[ \frac{1-k}{c\_1^2} + \frac{k}{c\_2^2} \right]. \tag{37}$$

Hence, assuming for the Swerling 1 target model, EIRl2 = 2<T2, an application of this to (36) yields

$$SCR = \frac{\sigma^2 \|\boldsymbol{\mu}\|^2}{2N\nu \left[\frac{1-k}{c\_1^2} + \frac{k}{c\_2^2}\right]}.\tag{38}$$

The ROC curves to follow are produced by determining the parameter <T2, for a given SCR, using (38).

#### **4.3 Receiver operating characteristics curves**

Four examples of detector performance are provided. In all cases, the ROC curves have been produced using Monte Carlo simulations, with approximately 105 runs. Each simulation is for the case of a false alarm probability of 10- 6• The SCR is varied from -10 to 30 dB. Each ROC curve shows the performance of the optimal decision rule (33), the GLRT decision rule using (35) to estimate the target strength and the performance of the whitening matched filter. The latter is the optimal detector for targets in Gaussian distributed clutter, and can be used as a suboptimal decision rule. For a return **r,** it is given by

$$M(\mathbf{r}) = \left| \mathfrak{u}^H \mathbf{r} \right|^L,\tag{39}$$

where uH is the Hermitian transpose. Although the decision rule (33) is dependent on the target parameters, and so is not useful in practice, it is used here to gauge the performance of the two suboptimal decision rules.

Each simulation uses parameters estimated from a specific Ingara data set. The clutter used for each ROC curve is produced by simulating a zero mean complex Gaussian process whose covariance matrix is specified, and so is not regressed from real data. This is then multiplied by a simulation of *S* with density (28) to produce a simulation of the SIRP.

#### **4.3.1 Examples for horizontal polarisation**

Two cases are considered for the horizontally polarised case. The first is for the scenario where the KK-Distributed clutter takes parameters c1 = 1, c2 = 3.27, *v* = 4.158, and *k* = 0.01. These parameters have been selected based upon the estimates for the clutter set run34683 described above, and with parameters estimated based upon the results in Dong (2006). For this case, the azimuth angle is 225°, which is nearly upwind. The number of looks is *N* = 30, and the clutter covariance matrix has been produced using simulated variables. Additionally, for this example, the normalised Doppler frequency is generated from *fv* = 1 and *Ts* = 0.5. The corresponding ROC curve can be found in Figure 2 (left subplot). This shows the optimal detector and GLRT detector matching very closely. Increasing the number of Monte Carlo samples improves the plot, but takes long periods to generate. The main feature one can observe from this plot is that the GLRT performs very well in this case, and certainly outperfoms the WMF.

The second example considered is illustrated also in Figure 2 (right subplot). This is for the case where the KK-Distribution has parameters c1 = 8, c2 = 46.16, *v* = 4.684, *k* = 0.01, *fv* = 0.8, *Ts* = 0.5, with the number of looks *N* = 20. The clutter parameters have been estimated from data set run34683 with azimuth angle 190°. The clutter covariance matrix was generated with random Gaussian numbers. The plot shows the same phenomenon as for the previous example.

Fig. 2. ROC curves, Horizontal polarisation, clutter set run34683, azimuth angles 225° (left plot) and 190° (right plot). OPT corresponds to (33), GLRT to the suboptimal rule using (35) and WMF is (39).

#### **4.3.2 Examples for vertical polarisation**

The vertically polarised case is illustrated in Figure 3. The first subplot is for the scenario where c1 = 25, c2 = 26.5, *v* = 8.315, *k* = 0.01, */d* = 100 and *Ts* = 0.5. *N* = 10 looks have been used . These correspond to run34690, with azimuth angle 45°, which is almost down wind in direction . For this case, it is interesting to note that the GLRT and WMF match very closely. It is only under increased magnification that it can be shown the GLRT is marginall y better. The same phenomenon is also demonstrated for the case where c1 = c2 = 35, *v* = 12.394, *Jo* = 10 and *Ts* = 0.5, with *N* = 5 looks. This is illustrated in Figure 3 (right subplot). This example is based upon the clutter set run34690, at an azimuth angle of 315°, which is approximately crosswind.

Fig. 3. ROC curves corresponding to the vertically polarised case, for clutter set run34690, azimuth angles 45° (left plot) and 315° (right plot). Here, the GLRT and WMF match almost exactly, suggesting the WMF is a suitable suboptimal decision rule .

#### **4.3.3 Analysis of detectors**

Examination of ROC curves for other Ingara clutter sets showed roughly the same results. For the vertically polarised channel, the WMF was a valid approximation to the GLRT, and is preferable because it does not require knowledge of clutter parameters. This result can be explained from the fact that the vertically polarised clutter is not as spiky as the horizontally polarised case, and the Gaussian distribution is the limit of a K-Distribution as the shape parameter increases. The latter results in less spiky clutter. For the horizontally polarised case, the clutter is spikier and so the GLRT is much better.
