**1. Introduction**

Coastal high frequency or HF radar (3–30 MHz) is a tool that has enabled users to remotely measure ocean currents, winds and waves, in real-time, since the initial observation of Crombie [1] in 1955, who realised the relationship between ocean wave and HF radar Doppler spectra (such as that shown in Figure 1). The measurements are important in a number of coastal engineering topics, including testing of and assimilation in operational wave models, sea vessel navigation, land/beach erosion, designing offshore structures, and in supporting marine activities. Additionally, collecting ocean data over a long period of time can be useful in climate changes studies; the same data can also be used to assess the potential of a coastal region to become a wave/wind farm as shown by Wyatt [2]. With such important applications, the accuracy of the measurements is imperative.

The majority of the existing theory is for monostatic radar, where the transmitter and receiver are co-located. Ocean surface current measurements from such a radar are robust, and hundreds of radars provide real-time current measurements all over the world; see the work of Paduan and Washburn [3] for an introduction to the subject. Methods for measuring wind direction and variability are also reliable, such as the method of Wyatt et al. [4], where the maximum likelihood method is used fit the wind distribution to the available radar data. Ocean wave measurements are also possible, however, they can be less robust as they are more vulnerable to noise (for limitations see

Wyatt et al. [5]). Notable methods of Lipa [6,7], Howell and Walsh [8], and Hisaki [9] are all significant in the history of measuring the wave spectrum. Another key method is the Seaview method, presented by Wyatt [10], and Green and Wyatt [11], which will be explained in Section 3.

**Figure 1.** Example of a radar Doppler spectrum measured by a bistatic HF radar on the south coast of France on 09/07/2014 00:01. Radar data provided by Celine Quentin, University of Toulon.

To obtain the ocean measurements using radar data, many researchers use the *radar cross section* of the ocean surface, which models what the radar output will be for a given ocean state and radar frequency. For monostatic radars, the most commonly used radar cross section is that of Barrick [12,13], derived in 1972, based on the perturbation method of Rice [14]. The expression is split into its first and second order components; the first order radar cross section, *σ*(1)(*ω*), is due to resonance between the emitted radio waves and ocean waves of a particular length and direction, and the second order radar cross section, *σ*(2)(*ω*), is due to double scattering of the emitted radio waves from two ocean waves and the non linear combination of the same two ocean waves. In full, for radar wavenumber *k*<sup>0</sup> and ocean spectrum *S*( *k*),

$$
\sigma^{(1)}(\omega) = 2^6 \pi k\_0^4 \sum\_{m=\pm 1} S(mk\_B^\star) \delta(\omega - m\omega\_B),
\tag{1}
$$

where *kB* = 2*k*0, is known as the *Bragg wavenumber* and

$$
\omega\_B = \sqrt{2gk\_0 \tanh(2k\_0 d)}\tag{2}
$$

is known as the *Bragg frequency*, for ocean depth *d* and gravity *g*. The second order term is given by

$$\sigma^{(2)}(\omega) = 2^6 \pi k\_0^4 \sum\_{m, m' = \pm 1} \iint\_{-\infty}^{\infty} |\Gamma\_T|^2 \mathcal{S}(m\vec{k}\_1) \mathcal{S}(m'\vec{k}\_2) \delta(\omega - m\omega\_1 - m'\omega\_2) \, dp \, dq,\tag{3}$$

where *k* -<sup>1</sup> and *k* -2, with respective angular frequencies *ω*<sup>1</sup> and *ω*2, are the two contributing wave vectors, defined by the relationship *k* -<sup>1</sup> + *k* -<sup>2</sup> = *k* - *<sup>B</sup>*. The |Γ*T*| <sup>2</sup> term is known as the *coupling coefficient* and contains the mathematics of the nonlinear combinations of the waves and, as such, is a function of *k* -<sup>1</sup> and *k* -2 . More detail on the monostatic coupling coefficient is given by Lipa and Barrick [15].

Numerical methods like that of Holden and Wyatt [16] can be used to simulate monostatic Doppler spectra for given ocean conditions and radar settings, using Equations (24) and (28); a comparison of a radar measured Doppler spectrum and a simulated Doppler spectrum, using co-spatial and co-temporal wave buoy data as input to the simulation, is shown in Figure 2 where good agreement is shown.

**Figure 2.** Comparison of measured and simulated monostatic Doppler spectra. The simulated Doppler spectrum has been generated using wave buoy data, measured at the same time and place as the radar Doppler spectrum. Radar and buoy data provided by Daniel Conley, Univeresity of Plymouth

Recently, bistatic radar—where the transmitter and receiver are separated by a notable distance—is on the ascendancy. Therefore, conversely to monostatic radar which receives backscatter, the detected radio waves in a bistatic radar have been scattered at a non-zero angle. A traditional coastal HF radar site consists of two monostatic radars, each providing data from ocean backscatter. However, a third dataset can be obtained at no additional cost if one of the receivers also receives bistatic scatter from the other transmitter. In this case, the radar site is called multistatic. Each different radar setup is shown in Figure 3. The advantages of employing a bistatic/multistatic radar setup are that (1) it can reduce the cost of setting up/maintaining a HF radar and (2) it can increase spatial coverage and data quality as shown by Whelan & Hubbard [17].

**Figure 3.** Comparison of (**a**) monostatic (receiver and transmitter colocated), (**b**) bistatic (receiver and transmitter separated) and (**c**) multistatic (**b** with extra receiver) radar geometries. In each case, the transmitter is shown by the blue cross and the receiver is shown by the blue circle (in the monostatic case, is also at the same location as the transmitter). An example scatter point is shown by the red star and the path the signal takes is shown by the solid black line. The line of constant range for each particular range is shown by the dashed black line and the angle marker shown represents the bistatic angle.

In this paper, the aim is to obtain directional wave measurements from bistatic radar data. Previously, for bistatic radar, Zhang and Gill [18] developed an inversion algorithm to obtain the nondirectional wave spectrum from bistatic radar data and, when tested on simulated data, they obtained good results. Silva [19] has also presented results of simulated bistatic HF radar data

inversion where the directional wave spectrum was estimated using Tikhonov regularization. They achieved good results for simulated data, however, the method is limited as a model is assumed for the direction of the spectrum, and this assumption may not always be appropriate.

In the existing numerical methods for extracting wave measurements from monostatic radar data, the aim is to invert Equation (3), to obtain the ocean spectrum *S*( *k*) in terms of the measured *σ*(*ω*). Anderson [20] stated that the existing algorithms should work for a bistatic system if the inverted radar cross section is changed to the bistatic expression. In this work, we test this hypothesis and modify the Seaview method to measure the directional wave spectrum from bistatic HF radar data. Therefore, to do this, the bistatic radar cross section must be known.

In 1975, Johnstone [21] presented a bistatic radar cross section of the ocean surface and then, in 2001, Gill and Walsh [22] presented an alternative expression. However, under monostatic conditions, neither of the expressions reduce exactly to the monostatic term of Barrick [12,13] (which the Seaview method depends on). The derivation of Johnstone appears to have an error which causes the difference in the resulting expressions; Gill and Walsh followed a more complicated method, however, it has been shown that the monostatic form of their radar cross section is similar to Barrick's and it is, therefore, unnecessary to change the existing operational inversion programs to use theirs instead. Another recent derivation is given in Chen et al. [23].

A bistatic radar cross section that reduces exactly to the monostatic term of Barrick [12,13] would be beneficial to systems based on Barrick's expression (such as the Seaview inversion) as, in the radar coverage area, the bistatic angle can vary between 0° and 90°, so the discontinuity between the monostatic and bistatic radar cross section expressions would cause a discontinuity in the inversion program used and perhaps, then, the results. Therefore, in this work, we follow the method of Barrick, whilst retaining the bistatic angle, to derive the bistatic radar cross section of the ocean surface. A reviewer of this paper has drawn our attention to similar work by Hisaki and Tokuda [24] who allowed for a finite scattering area and showed that their equations reduced to those of Barrick for an infinite scattering area and a monostatic geometry.

We begin with an overview of the derivation of the bistatic radar cross section in Section 2.1, before presenting the numerical solution of the resulting expression in Section 2.2. Details of the Seaview inversion method (for which details of the cross-section equations and numerical simulations are a pre-requisite) are then given in Section 3. The results of the modified Seaview inversion, when tested on simulated bistatic data, are given in Section 4 and these are discussed in Section 5 which also includes some concluding remarks.
