# Analytic Expressions for Radar Sea Clutter WSSUS Scattering Functions +
## 2 Results
### Experimental Validation
#### 2.8.1 Doppler Spectrum Modeling
We will use the MaxEnt procedure of Section 2.7.1 to estimate the ICM prior. For these models, it is necessary to predict mean and variance. Because the spectrum is periodic, we estimated the mean \(\mu_{\rho}\) Doppler shift using the circular mean [45]:  
\[\mu_{\rho}\approx\frac{F_{r}}{2\pi}\arg\left(\mathbb{E}_{P}\left[e^{j2\pi\rho/F _{r}}\right]\right), \tag{41}\]  
where \(\mathbb{E}_{P}\) denotes expectation is taken with respect to the empirically observed spectrum \(P(\rho)\), normalized to unit area. From the data in Figure 9, \(\mu_{\rho}=-47.9\) Hz, which corresponds to a radial wave speed of \(-0.71\) m/s. We will assume that \(\mu_{\rho}\) accounts for both the mean ICM Doppler shift as well as antenna pointing errors. For estimating the variance \(\sigma_{\rho}^{2}\) of the prior, we need to account for the fact that, according to Equations (27) and (35), the spectral width is due to the convolution of the prior with the antenna with the waveform ambiguity, therefore each step \"broadens\" the spectrum. In the elementary case when all three functions are Gaussian, the observed spectral variance \(\sigma_{\text{obs}}^{2}\) is the sum of the component variances:  
\[\sigma_{\text{obs}}^{2}=\sigma_{\rho}^{2}+\sigma_{a}^{2}+\sigma_{w}^{2}, \tag{42}\]  
where \(\sigma_{a}\) and \(\sigma_{w}\) are the RMS Doppler widths of the antenna pattern and waveform, respectively. This can be used to estimate the variance of the clutter prior:  
\[\sigma_{\rho}\approx\sqrt{\sigma_{\text{obs}}^{2}-\sigma_{a}^{2}-\sigma_{w}^{ 2}}. \tag{43}\]  
However, because the observed distribution \(P(\rho)\) and the waveform ambiguity cut \(|\chi_{NT}(0,\rho)|^{2}\) are both periodic, the appropriate way to measure \(\sigma_{\text{obs}}\) and \(\sigma_{w}\) are using the circular standard deviation [45]:  
\[\sigma_{\text{obs}} =\sqrt{-2\ln\left(\left|\mathbb{E}_{P}\left[e^{j2\pi\rho/F_{r}} \right]\right|\right)} \tag{44}\] Figure 8.0  
where \(\mathbb{E}_{|\chi|^{2}}\) is expectation with respect to the normalized ambiguity squared. The predicted spectrum using the WSSUS framework with the MaxEnt priors, labeled \"WSSUS u.v.\" for the unknown variance case, and \"WSSUS k.v.\" for the known variance case,  
Figure 8.1 Empirical ensemble average spectrum plotted versus WSSUS predicted spectra for the unknown variance (u.v.) and known variance (k.v.) cases. As well, the “standard” Gaussian clutter spectrum when the variance is known is plotted for reference. It can be seen that the Gaussian spectrum model vastly underestimates the clutter floor caused by antenna and waveform sidelobes. as well a plot of the \"standard\" Gaussian Doppler spectrum model for \(P(\rho)\), are shown in Figure 8.2 It can be seen that both predicted spectra capture the shape of the empirical spectrum; the specification of the variance produces a spectrum that is within approximately 1 dB of the empirical spectrum at all frequencies. The Gaussian model for \(P(\rho)\) produces excelled agreement for in the main lobe but does not predict the presence of the sidelobe clutter floor of approximately -29 dB. This could lead to excessively optimistic predictions of radar performance in the sidelobe clutter region. It is expected that for a side-looking radar in level flight the clutter Doppler spectrum will be centered at 0 Hz. The negative frequency bias of the observed Doppler spectrum in Figure 9 could be due to:  
* Wave motion away from the radar, as described in Section 2.7.2, of approximately 0.71 m/s,
* Aircraft crabbing (i.e., translational motion in the \(Y\)-direction),
* Antenna pointing errors; using Equation (23) yields a pointing error of approximately 0.4 degrees,
* Some combination of the above effects.