# Analytic Expressions for Radar Sea Clutter WSSUS Scattering Functions +
## 2 Results
### Experimental Validation
#### 2.8.2 Validation of WSSUS Assumption  
Another validation test that was performed using the Ingara dataset was to determine the validity of the WSSUS assumption. If the clutter impulse response \(h(\tau,t)\) is non-WSS in \(t\), then the autocorrelation \(A_{h}(\tau,\Delta t)\) would not just depend on the time-difference \(\Delta t\), but also the absolute time \(t\).  
To characterize the change in the autocorrelation over time, we computed the autocorrelation of the data over a sliding window of width 1 s over the entire record length of 305 s in range bin 1. The start time of each window was advanced by 10 ms, leading to a 99% overlap between windows. We will denote this function \(A_{t}(\Delta t)\). To characterize the autocorrelation at each time instant \(t\), we measure the decorrelation time \(T_{c}\), defined implicitly by:  
\[\frac{A_{t}(T_{c})}{A_{t}(0)}=e^{-1}, \tag{45}\]  
i.e., \(T_{c}\) is the time it takes for the autocorrelation centered at time \(t\) to decay to \(1/e\) from its peak value. The measured values of \(T_{c}\) versus time are plotted in Figure 10.  
We will use the stability of \(T_{c}\) over time as a proxy to for the stationarity properties of the clutter. It is clear that the clutter is nonstationary over long periods of time, but for short instants of time it may  
Figure 10: Clutter decorrelation times measured over time in range bin 1. Note that the decorrelation time remains roughly constant over several intervals much longer than a CPI. This provides support for the assumption that the clutter statistics are locally WSS for sufficiently long instants in time for the assumptions in Section 2.4 to apply.  
be approximated as being locally WSS. Ironically, even researchers that study the nonstationary nature of clutter spectra must implicitly assume the clutter signal is locally WSS for them to be able to apply the Wiener-Khinchin theorem to estimate the clutter power spectrum in the first place! So the important question is not whether or not clutter is stationary, but how long does it remain locally stationary? Our modeling so far has assumed that the stationarity duration is larger than a CPI.  
To test this assumption, we created a statistic called the \"stability duration\" of \(T_{c}\), which is defined as the region of time over which \(T_{c}\) changes less than \(\mathrm{X}\%\). (We used 10 percent, but this is admittedly arbitrary.) Because \(T_{c}=T_{c}(t)\) is itself a function of time, the stability duration was computed relative to the value of \(T_{c}(t)\) at each timestep. A histogram of the stability times over the entire record length is shown in Figure 11. It can be seen that the median stability time is about \(1.5\) s, with the mean being even higher. Thus for any waveform with a PRI on the order of one millisecond or less (i.e., \(\mathrm{PRF}>1\) kHz, which is very common) and integrating less than 1000 pulses, this is a perfectly reasonable assumption. This assumption may be violated for low-PRF waveforms with long integration times, such as those used in surveillance radars.