*2.1. Bistatic Radar Cross Section of the Ocean Surface*

To derive the bistatic radar cross section of the ocean surface, we follow the method of Barrick [12], where the equivalent monostatic radar cross section was derived. In his work, Barrick used the perturbation analysis of Rice [14] where, by assuming small waveheights and slopes, the electric field scattered from the ocean surface, *Es*, was calculated. They key points of the derivation follow, however, more details can be found in the work of Hardman [25].

The value of *E<sup>s</sup>* depends on both the incident radio waves and the properties of the scattering surface. Firstly, the incident waves will propagate as vertically polarised ground waves. Secondly, as the ocean varies in both time and space, by assuming that these variations are periodic and that the surface is of infinite extent, we can define the surface, *f*(*x*, *y*, *t*), as a Fourier series expansion, such that

$$z = f(x, y, t) = \sum\_{mnl = -\infty}^{\infty} P(m, n, l) \mathbf{e}^{ia(mx + ny) - i nlt},\tag{4}$$

for wavenumber *a* = 2*π*/*L*, angular frequency *w* = 2*π*/*T* and Fourier coefficients *P*(*m*, *n*, *l*) which are dependent on the integers *m*, *n* and *l*.

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The electric field scattered from this surface, will also be periodic with the same fundamental spatial and temporal periods and hence Rice defined - *Es* as a Fourier series. For a perfectly conducting flat surface, the exact solution can be found. Therefore, we perturb the solution around the flat surface, with ordering parameter *k*<sup>0</sup> *f* (for radar wavenumber *k*0), using Maxwell's equations and the tangential boundary condition to obtain the first and second order Fourier coefficients. Details of the calculations for vertically polarised waves can be found in the work of Hardman [25]; for horizontally polarised waves, the details are provided by Rice [14]. The resulting scattered electric field has components

$$\begin{split} E\_x &= 2\sum\_{mml} E(m, n, z, l) e^{-i\omega\_0 t} \left[ i(k\_0 - am)P(m - \nu, n, l) \right] \\ &+ \sum\_{qrs} \left\{ a^2 (m - q)(\nu - q) k\_0 + (k\_0 - am)b^2(q, r) \right\} Q(m, n, l, q, r, s) \right], \\ E\_y &= 2a \sum\_{mml} E(m, n, z, l) e^{-i\omega\_0 t} \left[ -inP(m - \nu, n, l) \right. \tag{6} \\ &+ \sum\_{qrs} \left\{ a(n - r)(\nu - q) k\_0 - nb^2(q, r) \right\} Q(m, n, l, q, r, s) \Bigg] \\ E\_z &= 2a^2 \theta^{\mathrm{an}} e^{-i\omega\_0 t} + 2 \sum\_{mml} \frac{E(m, n, z, l)}{b(m, n, l)} e^{-i\omega\_0 t} \left[ \left( -i(a(m - \nu) k\_0 + b^2(m, n)) \right) P(m - \nu, n, l) \right. \\ &\left. + \sum\_{qrs} \left\{ \left( a^3 (q - \nu) (m^2 + n^2 - qm - rn) k\_0 \right) \right. \right. \tag{7} \\ &\left. + \left. a(a(m^2 + n^2) - mk\_0) b^2(q, r) \right] Q(m, n, l, q, r, s) \right], \end{split} \tag{7}$$

where *ω*<sup>0</sup> is the angular frequency of the emitted radio waves,

$$E(m, n, z, l) = \mathbf{e}^{i(a(mx+ny) + b(m,n)z)} \mathbf{e}^{-iwlt}$$

in which

$$b(m,n) = \begin{cases} (k\_0^2 - a^2m^2 - a^2n^2)^{1/2} & \text{if } m^2 + n^2 < k\_0^2/a^2\\\ i(a^2m^2 + a^2n^2 - k\_0^2)^{1/2} & \text{if } m^2 + n^2 > k\_0^2/a^2 \end{cases},$$

and

$$Q(m,n,l,q,r,s) = \frac{P(q-\nu,r,s)P(m-q,n-r,l-s)}{b(q,r)}.$$

By definition, the electric field in Equations (5)–(7) corresponds to the scattering of infinite plane waves from a surface of infinite extent. To transform the fields to finitely scattered fields, which is necessary as only a portion of the whole ocean will be illuminated by the radar, we can use the equation presented by Johnstone [21] who followed the work of Stratton [26]. He showed that if the electric field is known on a finite section of an infinitely large volume such as the surface *S*<sup>1</sup> on the hemisphere shown in Figure 4, then the scattered electric field at a point (*x* , *y* , *z* ) inside of the volume can be calculated.

The equation is given by

$$\begin{split} \mathcal{E}(x', y', z') &= \frac{\mathbf{e}^{ik\_0 R}}{4\pi R} \int\_{S\_1} \left[ \left( \left( \frac{\partial E\_x}{\partial z} - \frac{\partial E\_z}{\partial x} \right) \vec{d}\_x + \left( \frac{\partial E\_y}{\partial z} - \frac{\partial E\_z}{\partial y} \right) \vec{d}\_y \right)\_{z=0} \right. \\ &\left. + ik\_0 \left[ E\_x \cos\theta \vec{d}\_x + E\_y \cos\theta \vec{d}\_y - \left( E\_x \sin\theta \cos\varphi + E\_y \sin\theta \sin\varphi \right) \vec{d}\_z \right]\_{z=0} \\ &\quad + \left[ \left( \frac{\partial E\_z}{\partial x} - \frac{\partial E\_x}{\partial z} \right) \sin\theta \cos\varphi + \left( \frac{\partial E\_z}{\partial y} - \frac{\partial E\_y}{\partial z} \right) \sin\theta \sin\varphi \right]\_{z=0} \\ &\quad \cdot \left[ \sin\theta \cos\varphi \vec{d}\_x + \sin\theta \sin\varphi \vec{d}\_y + \cos\theta \vec{d}\_z \right] \right] \mathbf{e}^{-i\vec{k}\_0 \cdot \rho} dS\_{1,z} \end{split} \tag{8}$$

where *ρ* is the vector (*x*, *y*, *z*) on *S*1, (*R*, *θ*, *ϕ*) are the radius, polar angle and azimuthal angle measured from the origin to (*x* , *y* , *z* ) and *k* -<sup>0</sup> is the radar wavevector given by

**Figure 4.** The finite scattering surface, *S*1, with boundary *C*, as part of a hemispherical surface. The vector *ρ* denotes the position (*x*, *y*, *z*) on *S*1; the vector*rr* is the vector from (*x*, *y*, *z*) to some distant point (*x* , *y* , *z* ) where the scattered electric field is desired. The vector *k* -<sup>0</sup> is the radar wavevector in the direction of the scattered radio wave, - *R*.

Note that the integral can be evaluated on any plane and *z* = 0 is used for convenience. However, as pointed out by Hisaki and Tokuda [24], this choice is in fact the infinite scattering surface limit. Evaluating the integral on *z* = *f* as they did changes the resulting power spectrum of the scatter but differences are very small at the Doppler frequencies used for inversion so *z* = 0 is sufficient for this work. To find the scattered field at a point (*x* , *y* , *z* ), the values for *Ex*, *Ey* and *Ez* from Equations (5)–(7) are substituted into Equation (8) and the integral is calculated. The vertically polarised component, *E<sup>θ</sup>* (*t*), is then identified in the resulting expression (as these are the radio waves that the receiver will detect) such that

$$\begin{split} E\_{\theta}(t) &= \frac{i e^{i\mathbf{k}\_{0} \cdot \mathbf{R}}}{2\pi R} L^{2} \sum\_{nml} \left\{ B(t) \left[ -i\mathbf{x}\_{1} P(m-\nu,n,l) + \sum\_{qrs} x\_{2} Q \right] + \mathcal{C}(t) \left[ -i\mathbf{y}\_{1} P(m-\nu,n,l) + \sum\_{qrs} y\_{2} Q \right] \\ &+ D(t) \left[ -i\mathbf{z}\_{1} P(m-\nu,n,l) + \sum\_{qrs} z\_{2} Q \right] \right\}, \end{split} \tag{9}$$

where,

$$\begin{aligned} x\_1 &= am - k\_0; & x\_2 &= a^2(m - q)(\nu - q)k\_0 + (k\_0 - am)b^2(q, r) \\ y\_1 &= an; & y\_2 &= a^2(n - r)(\nu - q)k\_0 - anb^2(q, r) \\ z\_1 &= a(m - \nu)k\_0 + b^2(m, n); & z\_2 &= \left[a^3(q - \nu)(m^2 + n^2 - qm - rn)k\_0 + a\left(a(m^2 + n^2) - mk\_0\right)b^2(q, r)\right]. \end{aligned}$$

and,

$$B(t) = \sum\_{mnl} \cos \varphi (k\_0 + b(m, n) \cos \theta) \sin(XR) \sin(YR) e^{-i(wl + \omega\_0)t} \tag{10}$$

$$\mathbf{C}(t) = \sum\_{mnl} \sin \varrho (k\_0 + b(m, n) \cos \theta) \sin \mathbf{c}(XR) \sin \mathbf{c}(YR) \mathbf{e}^{-i(wl + \omega y)t} \tag{11}$$

$$D(t) = \sum\_{mnl} -\cos\theta (am\cos\varphi + an\sin\varphi) \frac{\text{sinc}(XR)\,\text{sinc}(YR)}{b(m,n)} \mathbf{e}^{-i(wl+\omega\_0)t},\tag{12}$$

for *XR* <sup>=</sup> *<sup>L</sup>* <sup>2</sup> (*am* <sup>−</sup> *<sup>k</sup>*<sup>0</sup> sin *<sup>θ</sup>* cos *<sup>ϕ</sup>*) and *YR* <sup>=</sup> *<sup>L</sup>* <sup>2</sup> (*an* <sup>−</sup> *<sup>k</sup>*<sup>0</sup> sin *<sup>θ</sup>* sin *<sup>ϕ</sup>*).

In the scattered electric field in Equation (9), the first order components (namely, the terms including a single *P* term) represent the single scattering of one electromagnetic wave, to the receiver, from one ocean wave. The second order components (which include a factor of *Q*) represent doubly scattered electromagnetic waves, to the receiver, from two single ocean waves. The order of the ocean wave, currently denoted by *P*(*m* − *ν*, *n*, *l*), has not yet been considered and is assumed to be first order. However, in making such an assumption, a second order contribution from first order scattering from second order oceans waves is missed, where a second order ocean wave is the result of the nonlinear interaction between two first order ocean waves.

To allow for the second order hydrodynamic effects in shallow water, Barrick and Lipa [27] used a perturbation method to relate the second order coefficients *P*(2)( *k*, *ω*), of a surface defined by *z* = ∑*<sup>k</sup>*,*<sup>ω</sup> P*( *k*, *ω*)e*<sup>i</sup> k*·*r*−*iωt* , to the first order coefficients *P*(1)( *k*, *ω*). Their method involved expanding the surface height Fourier coefficients around the flat surface, i.e.,

$$P(\vec{k},\omega) = P^{(1)}(\vec{k},\omega) + P^{(2)}(\vec{k},\omega) + \dots,$$

alongside boundary conditions from the equations of motion, also expanded to second order. They showed that for ocean waves with wavevectors *k* -<sup>1</sup> and *k* -2, with corresponding angular frequencies *ω*<sup>1</sup> and *<sup>ω</sup>*<sup>2</sup> (related by the dispersion relation of ocean waves given by *<sup>ω</sup>* = *gk* tanh(*kd*)),

$$P^{(2)}(\vec{k}^{\prime\prime},\omega^{\prime\prime}) = \sum\_{\vec{k\_1}\vec{k\_2}} \sum\_{\omega\gamma\cup\omega\_2} \Gamma\_H(\vec{k\_1},\omega\_1,\vec{k\_2},\omega\_2) P^{(1)}(\vec{k\_1},\omega\_1) P^{(1)}(\vec{k\_2},\omega\_2),\tag{13}$$

where *k* - = *k* -<sup>1</sup> + *k* -2, *ω* = *ω*<sup>1</sup> + *ω*2, and

$$\begin{split} \Gamma\_{H} &= \frac{1}{2} \left\{ k\_{1} \tanh(k\_{1}d) + k\_{2} \tanh(k\_{2}d) + \frac{\omega \prime \prime}{g} \frac{(\omega\_{1}^{3} \operatorname{csch}^{2}(k\_{1}d) + \omega\_{2}^{3} \operatorname{csch}^{2}(k\_{2}d))}{(\omega \prime \prime - gk \prime \tanh(k^{\prime}d))} \right. \\ &\left. + \frac{(k\_{1}k\_{2} \tanh(k\_{1}d) \tanh(k\_{2}d) - \vec{k}\_{1} \cdot \vec{k}\_{2})}{\sqrt{k\_{1}k\_{2} \tanh(k\_{1}d) \tanh(k\_{2}d)}} \left( \frac{g k^{\prime\prime} \tanh(k^{\prime}d) + \omega^{\prime\prime 2}}{g k^{\prime\prime} \tanh(k^{\prime}d) - \omega^{\prime\prime 2}} \right) \right\}, \end{split} \tag{14}$$

is called the *hydrodynamic coupling coefficient*.

To include the second order hydrodynamic effects in the scattered electric field, we expand *<sup>P</sup>*(*<sup>m</sup>* <sup>−</sup> *<sup>ν</sup>*, *<sup>n</sup>*, *<sup>l</sup>*) into *<sup>P</sup>*(1)(*<sup>m</sup>* <sup>−</sup> *<sup>ν</sup>*, *<sup>n</sup>*, *<sup>l</sup>*) + *<sup>P</sup>*(2)(*<sup>m</sup>* <sup>−</sup> *<sup>ν</sup>*, *<sup>n</sup>*, *<sup>l</sup>*) and then substitute in the value of *<sup>P</sup>*(2)(*<sup>m</sup>* <sup>−</sup> *<sup>ν</sup>*, *<sup>n</sup>*, *<sup>l</sup>*) using Equation (13) (by letting *k* -= (*m* − *ν*, *n*) and *ω* = *l*), and so Equation (9) becomes

$$E\_{\theta}(t) = \frac{i e^{ik\_{0}R}}{2\pi R} L^{2} \sum\_{mml} \left\{ -i l\_{\text{s}}^{\prime\prime}(t) P^{(1)}(m - \nu, n, l) + \sum\_{qrs} [-i l\_{\text{s}}^{\prime\prime}(t) \Gamma\_{l} b(p, q) + \overline{\xi}(t)] \right. \\ \left. Q(m, n, l, q, r, s) \right\}, \tag{15}$$

where for brevity

$$\mathcal{Z}(t) = B(t)\mathbf{x}\_1 + \mathcal{C}(t)y\_1 + D(t)z\_1$$

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and

$$\mathcal{G}(t) = B(t)\mathbf{x}\_2 + \mathcal{C}(t)\mathbf{y}\_2 + D(t)z\_2.$$

Following Johnstone [21], we finally calculate the radar cross section by substituting Equation (15) into

$$\sigma(\omega) = \lim\_{R \to \infty} 4\pi R^2 \frac{\mathcal{F}\left[ \langle E\_{\theta}(t\_1) E\_{\theta}^\*(t\_2) \rangle \right]}{L^2},\tag{16}$$

where *R* is the distance from the scatter patch to the receiver and F denotes a Fourier transform, with definition

$$\mathcal{F}[f(t)] = \frac{1}{2\pi} \int\_{\infty}^{\infty} f(t)e^{-i\omega t} \, dt. \tag{17}$$

To calculate Equation (16), the following properties of the surface height Fourier coefficients are used:

• The Fourier coefficients are normally distributed about zero; hence

$$
\langle P(m, n, l) \rangle = 0.\tag{18}
$$

• As the surface is real, *f*(*x*, *y*, *t*) is equal to *f* ∗(*x*, *y*, *t*), which is true when

$$P(-m, -n, -l) = P^\*(m, n, l). \tag{19}$$

• From Thomas [28],

$$
\langle P\_1 P\_2 P\_3 \rangle = 0 \tag{20}
$$

and

$$
\langle P\_1 P\_2 P\_3 P\_4 \rangle = \langle P\_1 P\_2 \rangle \langle P\_3 P\_4 \rangle + \langle P\_1 P\_3 \rangle \langle P\_2 P\_4 \rangle + \langle P\_1 P\_4 \rangle \langle P\_2 P\_3 \rangle. \tag{21}
$$

• The surface roughness spectrum *S*(*p*, *q*, *wl*), found by utilising the Wiener–Khinchin theorem, is related to the surface height Fourier coefficients by

$$P(P(m,n,l)P(q,r,s)) = \begin{cases} \frac{(2\pi)^3 S(p\_\prime q\_\prime, wl)}{L^2 T} & \text{if } q\_\prime r\_\prime s = -m\_\prime - n\_\prime - l\\ 0 & \text{if else} \end{cases} \tag{22}$$

where *p* = *am* and *q* = *an*.

Then, substituting *E<sup>θ</sup>* (*t*) from Equation (15) into Equation (16) and using Equation (20) leads to

$$\begin{split} \sigma(\omega) &= \frac{1}{\pi} \mathcal{F} \left[ \mathbf{L}^{2} \sum\_{\begin{subarray}{c} \text{null} \\ \mathbf{m}' \mathbf{n}' \end{subarray}} \left\{ \mathbb{J}(t\_{1}) \mathbb{J}^{\prime \*}\_{\mathbf{s}}(t\_{2}) \left\langle P^{(1)}(m-\nu,n,l) P^{(1)^{\prime}\*}(m'-\nu,n',l') \right\rangle \right. \\\\ &+ \sum\_{\begin{subarray}{c} \text{\$q\$rr\$} \\ \mathbf{q}' \mathbf{r} \text{s}' \end{subarray}} \left\{ \left[ -\mathbf{i}\_{\mathbf{s}}^{\prime}(t\_{1}) b(\mathbf{q},r) \Gamma\_{\mathrm{H}} + \mathbf{j}\_{\mathbf{s}}^{\prime}(t\_{1}) \left[ \left[ \mathbf{i}\_{\mathbf{s}}^{\prime \prime \*}(t\_{2}) b^{\ast}(\mathbf{q}^{\prime},r^{\prime}) \Gamma\_{\mathrm{H}}^{\prime} + \mathbf{j}\_{\mathbf{s}}^{\prime \*}(t\_{2}) \right] \left\langle Q Q^{\prime \*} \right\rangle \right] \right\} \right], \end{split} \tag{23}$$

where the arguments of *Q* are implied. The calculation of Equation (23) can be separated into its first and second order terms, such that

$$
\sigma(\omega) = \sigma^{(1)}(\omega) + \sigma^{(2)}(\omega),
$$

where the first order radar cross section, *σ*(1)(*ω*), is defined by the term including the average ) *<sup>P</sup>*(1)(*<sup>m</sup>* <sup>−</sup> *<sup>ν</sup>*, *<sup>n</sup>*, *<sup>l</sup>*)*P*(1) <sup>∗</sup>(*m* − *ν*, *n* , *l* ) \* , and the second order, *<sup>σ</sup>*(2)(*ω*), including *QQ*∗. To calculate each of *σ*(1)(*ω*) and *σ*(2)(*ω*), the properties in Equations (18)–(22) are used to enforce restrictions on the Fourier coefficients and to introduce the roughness spectrum *S*( *k*). The mathematical details are spared here, but can be found in the work of Hardman [25].
