3.4.1. Scattering from the Ocean Surface

The perturbation theoretic approach of Barrick [26], building on the Rice theory for scattering from static rough surfaces [27], has served as the cornerstone of HF radar oceanography for the past five decades. Quite a few generalizations of the Barrick theory have appeared over the years (e.g., [28–30]), as well as a different approach [31] based on the Walsh theory for scattering from static surfaces [32] and extended by Gill and co-workers (e.g., [33–36]). As ocean applications of HF radars dominate, and as bistatic configurations become more widespread, it is hardly surprising to find an emerging literature of papers that apply the fundamental theories to particular circumstances. As a guide, we have tabulated some of these bistatic scatter papers against key parameters: (i) The perturbation order of the approximation, (ii) the scattering geometry, (iii) whether platform motions were taken into account, (iv) the polarization states addressed, and (v) the hydrodynamic dispersion relation employed. This file is available from the author.

The general expression for the scattered field in <sup>→</sup> *k* , ω space, to second order, has the form

$$\begin{split} S(\stackrel{\rightarrow}{k},\omega) &= \int d\stackrel{\rightarrow}{k} \mathcal{\widetilde{L}}(\stackrel{\rightarrow}{k}\_{\text{inc}}) \delta(\stackrel{\rightarrow}{k}\_{\text{scat}} - \stackrel{\rightarrow}{k}\_{\text{inc}} + 2\stackrel{\rightarrow}{k}\_{\text{inc}};\hbar\,\hbar) \delta(\omega - \omega\_{0}) \\ &+ \int d\stackrel{\rightarrow}{\kappa}\_{1} \, F\_{1}(\stackrel{\rightarrow}{k}\_{\text{scat}}, \stackrel{\rightarrow}{k}\_{\text{inc}};\hbar\,\stackrel{\rightarrow}{\chi}\_{1}) \delta(\stackrel{\rightarrow}{k}\_{\text{scat}} - \stackrel{\rightarrow}{k}\_{\text{inc}} + \stackrel{\rightarrow}{\chi}\_{1}) \delta(\omega - \omega\_{0} + \Omega(\stackrel{\rightarrow}{\chi}\_{1})) \\ &+ \int d\stackrel{\rightarrow}{\kappa}\_{1} \, d\stackrel{\rightarrow}{\chi}\_{2} \, F\_{2}(\stackrel{\rightarrow}{k}\_{\text{scat}}, \stackrel{\rightarrow}{k}\_{\text{inc}};\stackrel{\rightarrow}{\chi}\_{1}, \stackrel{\rightarrow}{\chi}\_{2}) \delta(\stackrel{\rightarrow}{k}\_{\text{scat}} - \stackrel{\rightarrow}{k}\_{\text{inc}} + \stackrel{\rightarrow}{\chi}\_{1} + \stackrel{\rightarrow}{\chi}\_{2}) \times \delta(\omega - \omega\_{0} + \Omega(\stackrel{\rightarrow}{\chi}\_{2}) + \Omega(\stackrel{\rightarrow}{\chi}\_{1})) \end{split} (11)$$

In this equation, the dispersion relation appears as Ω → κ*i* ; it is this function that determines the contours of integration that yield the Doppler power spectral density. Recent investigations of HF scatter from sea ice motivated the development of a computational model able to solve for any explicit dispersion relation [11,37]. Expressions for the kernel functions *F*<sup>1</sup> and *F*<sup>2</sup> can be found in the cited literature, while the Fresnel reflection coefficient *R* is a function of the water temperature and salinity.

Viewed diagrammatically in the spatial frequency domain, as shown in Figure 7, an important feature of bistatic Bragg scattering becomes evident, namely the smaller modulus of the resultant wave vector increment, shown in black. At first order, this means that longer ocean waves can be measured directly, while at second order, it means that a broad angular spread of the shorter waves in the participating wave field favors the scattering contributions.

**Figure 7.** Double Bragg scattering processes in <sup>→</sup> κ-space. The red vector represents the incident radiowave, the blue vector the scattered radiowave, and the black vector, the required change in wave vector to be delivered by pairs of ocean waves whose <sup>→</sup> κ -vectors meet at the intersection of the circles of given wavenumbers. Here they are drawn for the case of equal wavenumbers. Tx and Rx indicate the directions of the transmitter and receiver.

It is instructive to see the form of the Doppler spectrum as a function of the bistatic geometry for particular situations and as a function of various parameters. First, we present Figure 8, taken

from [18], which shows, at a most basic level, how a particular seastate modulates the frequency of the scattered signal depending on the scattering geometry: Back scatter, forward scatter, side scatter, or up scatter. In this example, only the VV (vertical in, vertical out) element of the polarization power scattering matrix is presented.

**Figure 8.** An example of the variation of the Doppler spectrum for representative bistatic scattering geometries incident on the same sea state: Backscatter, forward scatter, side scatter, and up scatter, as applicable to different HF radar configurations (reproduced from [18]).

Next, in Figures 9–11, we show the full polarization power scattering matrix for three different radar frequencies, in each case plotting the spectrum for four different bistatic skywave scattering geometries, at a fixed sea state. Here, 180◦ corresponds to backscatter, i.e., monostatic geometry. The vertical angles of incidence and reflection are 40◦ in all cases.

**Figure 9.** Sea clutter Doppler spectrum: ϕ = 45◦(red), 90◦(green), 135◦(blue), 180◦(black); F = 25 MHz.

**Figure 10.** Sea clutter Doppler spectrum: ϕ = 45◦(red), 90◦(green), 135◦(blue), 180◦(black); F = 15 MHz.

**Figure 11.** Sea clutter Doppler spectrum: ϕ = 45◦(red), 90◦(green), 135◦(blue), 180◦(black); F = 5 MHz.

Finally, in Figure 12, we present a range-Doppler map measured with a bistatic HFSWR, along with two modelled range-Doppler maps computed for scattering geometries close to that used for the measurement.

**Figure 12.** Modelled and measured Doppler spectra presented as range-Doppler maps. The variation of Bragg frequency with bistatic angle is clearly seen, especially at low ranges as the bistatic angle approaches its maximum. At near ranges, a decrease in signal power is evident in the measurement—this is due to the transmit antenna gain pattern falling off at angles close to the bearing to the receiver.
