#
Analytic Expressions for Radar Sea Clutter WSSUS Scattering Functions^{ †}

^{†}

^{‡}

## Abstract

**:**

## 1. Introduction

## 2. Results

#### 2.1. Mathematical Background

- The LTV impulse response $h(\tau ,t)$ is only valid over a single coherent processing interval (CPI) of N transmitted pulses
- Over a single CPI, the range to target(s) is roughly constant
- Relative motion produces a Doppler shift on the carrier only and does not introduce time dilation of the pulse. The conditions for this assumption to be valid are that $BT<0.1c/\dot{r}$, where $BT$ is the time-bandwidth product of the waveform and $\dot{r}$ is the maximum range rate [47] (p. 382).

#### 2.2. WSSUS Processes

#### 2.3. Simulation Geometry

- ${P}_{T}$ = transmit power
- $d{P}_{R}$ = incremental received power from isorange ring at delay $\tau $
- $\varphi ,\theta $ = azimuth and depression angles relative to platform
- $\alpha $ = grazing angle, which equals $\theta $ in a flat earth model$\Rightarrow \alpha =\theta ={sin}^{-1}(H/\left(\frac{1}{2}c\tau \right))$
- $G(\varphi ,\theta )$ = one-way antenna power gain
- $\lambda $ = carrier wavelength
- ${\sigma}^{0}={\sigma}^{0}(\alpha ,\lambda ,\mathrm{sea}\phantom{\rule{4.pt}{0ex}}\mathrm{state},\dots )=$ surface normalized radar cross section (NRCS)
- $dA$ = $\frac{\pi}{2}{c}^{2}\tau d\tau $ = incremental area of isorange ring at delay $\tau $.

#### 2.4. WSSUS System Function Derivations

#### 2.5. Important Special Cases

#### 2.5.1. No ICM

#### 2.5.2. Side-Looking Antenna, Level Flight Path

#### 2.5.3. Arbitrary Orientation

#### 2.6. Output Time-Frequency Power Distribution

#### 2.7. ICM Spectrum Characterization

#### 2.7.1. Maximum Entropy Prior

#### 2.7.2. Known Mean, Unknown Variance

- The mean wave speed (and hence mean Doppler shift) is proportional to the wind speed. For example, it is commonly assumed that the wave speed is 1/8th the wind speed [57], and
- The wind velocity vector has no Z component, and
- The waves move in the same direction as the wind,

- ${\mu}_{\rho}\propto {v}_{w}cos\theta cos(\varphi -{\varphi}_{w})$,
- $\rho sgn\left({\mu}_{\rho}\right)\in [0,\infty )$.

#### 2.7.3. Known Mean and Variance

#### 2.7.4. Distribution Comparison

- ${F}_{r}=2$ kHz
- $N=32$ pulses
- Dolph-Chebyshev window with −55 dB sidelobes
- ${\mu}_{\rho}=62.5$ Hz
- ${\sigma}_{\rho}^{2}=20$ Hz
^{2} - Overall clutter to noise ratio (CNR) = 20 dB

#### 2.8. Experimental Validation

#### 2.8.1. Doppler Spectrum Modeling

- 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.

#### 2.8.2. Validation of WSSUS Assumption

## 3. Discussion and Conclusions

## 4. Materials and Methods

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Wide-sense stationary uncorrelated scattering (WSSUS) system function relationships. Knowledge of any one of these functions grants complete knowledge of the second-order statistics of the system.

**Figure 2.**Flat earth geometry in the X-Z plane. Because the goal is to ultimately produce a range-Doppler map, we need to express all quantities in terms of downrange delay $\tau $ and absolute time t.

**Figure 3.**Return from a single infinitesimal isorange ring on the sea surface. The autocorrelation function of the impulse response of this ring is computed at delay $\tilde{\tau}$, then the result is integrated over all $\tilde{\tau}$ to get the total autocorrelation as a function of delay $\tau $ and time offset $\Delta t$.

**Figure 4.**Constant-$\tau $ cut of the no-internal clutter motion (ICM) scattering function Equation (21) with the antenna pattern removed (i.e., $G=1$) and no vertical motion (${\rho}_{Z}^{\prime}=0$). The true scattering function will be windowed by the antenna pattern to focus on a specific region of $\rho $—a side-looking antenna will be focused near $\rho =0$, whereas a nose-aspect antenna will be focused near $\rho ={\rho}_{X}^{\prime}$.

**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 ${\sigma}_{\rho}\ll {F}_{r}/N$.

**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°. In this run, the plane completes approximately 1.4 revolutions of its circular flight path.

**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 9.**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.

**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.

**Figure 11.**Histogram of stability durations. It can be seen that the decorrelation time ${T}_{c}$ is roughly constant over intervals much greater than one CPI for medium- and high-PRF waveforms.

© 2019 by the author. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

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Cooke, C. Analytic Expressions for Radar Sea Clutter WSSUS Scattering Functions. *Entropy* **2019**, *21*, 915.
https://doi.org/10.3390/e21090915

**AMA Style**

Cooke C. Analytic Expressions for Radar Sea Clutter WSSUS Scattering Functions. *Entropy*. 2019; 21(9):915.
https://doi.org/10.3390/e21090915

**Chicago/Turabian Style**

Cooke, Corey. 2019. "Analytic Expressions for Radar Sea Clutter WSSUS Scattering Functions" *Entropy* 21, no. 9: 915.
https://doi.org/10.3390/e21090915