# Analytic Expressions for Radar Sea Clutter WSSUS Scattering Functions +
## 2 Results
### Experimental Validation  
#### 2.8.1 Doppler Spectrum Modeling  
To validate our framework, we will compare the modeled clutter spectra to empirically measured spectra. For our comparison, we will use the Ingara dataset, which is a medium grazing angle dataset containing measured returns from an airborne radar flying in a circular path at a speed of approximately 200 knots, operating in spotlight mode (i.e., illuminating the patch of sea at the center of the circle.) Data were collected at grazing angles from 15 to 45 degrees. The radar operates at a center frequency of 10.1 GHz using a linear frequency modulated (LFM) waveform with a pulse width of 20 us and a bandwidth 200 MHz, leading to a range resolution of about 0.75 m.  
The data in the flight that we will use for comparison was collected from an altitude of 0.5 nm with the plane flying in a circular path of radius 1.9 nmi, leading to a grazing angle of approximately 15 degrees. This run contains the measured radar IQ returns over a 305 second interval, or about 1.4 revolutions around the circular path. A plot of the downrange video at each slow time is shown in Figure 7. Close inspection of this picture shows that the individual wave crests and troughs are clearly visible. To test the ability of the WSSUS framework to predict Doppler spectra, we will compare the predicted spectra to the Doppler spectra observed while the radar is looking downwind. The observed range-Doppler map seen when the radar is looking downwind is shown in Figure 8. It can be seen  
Figure 5: Maximum entropy (MaxEnt) Doppler priors for unknown variance (u.v.) and known variance (k.v.) cases. For the u.v. case, the MaxEnt prior is an exponential distribution, whereas for the k.v. case, the MaxEnt prior is a Gaussian. Note that additional knowledge of the variance significantly reduces the predicted spectrum width. Figure 6: Output power distributions for unknown variance (u.v.) and known variance (k.v.) cases. Note that in this case the predicted spectrum when the variance is known becomes waveform-limited because \(\sigma_{\rho}\ll F_{r}/N\). that the spectrum is biased in the negative Doppler direction, as expected. Significant variation in the mean Doppler frequency in each range bin can be seen; this is likely due to the high range resolution of the radar. Because these minute fluctuations are essentially random and difficult to predict, we will take the ensemble average of the normalized spectra over \(\tau\). In this dataset, the elevation beampattern was removed as a preprocessing step, so it is expected that the Doppler spectra in each range bin will have similar statistics with the exception of path loss-induced amplitude decay. The ensemble average spectrum is shown in Figure 9, where it is labeled \"empirical\". Because the Ingara radar is side-looking and is flying at a constant altitude, we will use the simplifications of the WSSUS model in Section 2.5.2 to compute the scattering function \(S_{ht}\), which we will convolve with the ambiguity to obtain the delay-Doppler map \(P(\tau,\rho)\). The azimuth antenna pattern and waveform parameterization were supplied with the dataset, so the only unknown in our prediction efforts is the ICM Doppler prior \(\underline{b}(\rho|\tau)\). Figure 8: Range-Doppler map when the radar is looking in the downwind direction, obtained over a coherent processing interval (CPI) of 128 pulses (0.52 s). The wind speed was 8.5 m/s. Note that the Doppler shifts are biased towards negative frequencies. This bias could be due to wave motion away from the radar as well as antenna pointing errors. Figure 7: Ingara data from trial SCT04, flight F42, run 34,877. The Ingara radar contains 1024 range gates covering a range span of approximately 767 m, captured from an airborne radar operating in spotlight mode flying in a circular path at an altitude of 0.5 nmi and observing the ocean at a grazing angle of 15\({}^{\circ}\). In this run, the plane completes approximately 1.4 revolutions of its circular flight path.