Wave spectrum modulated by sea surface currents

This section introduces the time-evolving deterministic Creamer-II nonlinear sea surface based on Elfouhaily spectrum. The Elfouhaily omnidirectional spectrum expresses as a sum of two spectrums

(1)$$S\lpar k\rpar = k^{{-}3}\lpar B_l + B_h\rpar \comma \;$$

here B_l and B_h are long-wave curvature spectrum and short-wave curvature spectrum, respectively. The presentations of B_l and B_h are shown in [Reference Elfouhaily, Chapron, Katsaros and Vandemark5] in detail. The wave spectrum consists of an omnidirectional spectrum and a directional spectrum.

(2)$$W\lpar k\comma \;\varphi \rpar = S\lpar k\rpar \Phi \lpar k\comma \;\varphi \rpar . $$

In this paper, we use a unilateral cosine direction spectrum proposed by Longuet-Higgins et al.,

(3)$$\Phi \lpar k\comma \;\varphi \rpar = G\lpar s\rpar \left\vert {\cos \left({\displaystyle{{\varphi -\varphi_w} \over 2}} \right)} \right\vert ^{2s}\comma \;$$

where φ _w is the wind direction angle,

(4)$$\eqalign{& s = 1-\displaystyle{1 \over {\ln 2}}\ln \left[{\displaystyle{{1-\Delta \lpar k \rpar } \over {1 + \Delta \lpar k \rpar }}} \right]\comma \;\cr & G\lpar s \rpar = \displaystyle{{2^{2s-1}} \over {\rm \pi }}\displaystyle{{\Gamma ^2\lpar {s + 1} \rpar } \over {\Gamma \lpar {2s + 1} \rpar }}.} $$

The Elfouhaily spectrum describes the form of the wind-driven waves, but the actual ocean movement is much more complicated. The currents in spatial variation such as tidal currents and internal waves will modulate the wave spectrum and change the roughness of the sea surface, which is characterized by hydrodynamic modulation in SAR imaging.

According to the “weak hydrodynamic interaction” theory, wave packets travelling can be described by the action balance equation [Reference Romeiser and Alpers10]

(5)$$\displaystyle{{{\rm d}N} \over {{\rm d}t}} = \left({\displaystyle{\partial \over {\partial t}} + \displaystyle{{{\rm d}{\bi x}} \over {{\rm d}t}}\displaystyle{\partial \over {\partial {\bi x}}} + \displaystyle{{{\rm d}{\bi k}} \over {{\rm d}t}}\displaystyle{\partial \over {\partial {\bi k}}}} \right)N = Q\lpar {\bi k}\comma \;{\bi x}\comma \;t\rpar \comma \;$$

here N denotes the action density of a surface wave packet. The source function Q describes the combined effect of wind input, wave–wave interaction, and dissipation. The action spectrum $N\lpar {\bi k}\comma \;{\bi x}\semicolon \;t\rpar$ and wave spectrum $W_{st}\lpar {\bi k}\comma \;{\bi x}\comma \;t\rpar$ are as follows

(6)$$N\lpar {\bi k}\comma \;{\bi x}\comma \;t\rpar = W_{st}\lpar {\bi k}\comma \;{\bi x}\comma \;t\rpar \displaystyle{{\rho \omega _0\lpar k\rpar } \over k}\comma \;$$

where $\omega _0 = \lpar gk + \tau k^3/\rho \rpar ^{1/2}\comma \;g = 9.81\,{\kern 1pt} {\rm m/}{\rm s}^2\comma \;\tau = 0.079\,{\rm N/m\comma } $, $\rho = 1025\,{\rm kg/}{\rm m}^3$. In this paper, a numerical integration algorithm developed by Romeiser [Reference Romeiser and Alpers10] is used to calculate the action spectrum $N\lpar {\bi k}\comma \;{\bi x}\semicolon \;t\rpar$, which is obtained by tracing the trajectories from an interesting position. Then the new spatio-temporal variation wave spectrum $W_{st}\lpar {\bi k}\comma \;{\bi x}\comma \;t\rpar$ can be obtained.

Model of the deterministic nonlinear sea surface

The fast Fourier transform (FFT) is used to generate time-evolving deterministic nonlinear sea surface. The spatial Fourier components of the surface at any time t is

(7)$$\eqalign{F\lpar k_{m_k}\comma \;k_{n_k}\semicolon \;t\rpar = & \gamma _n{\rm \pi }\sqrt {2W\lpar k_{m_k}\comma \;k_{n_k}\rpar /\lpar L_xL_y\rpar } {\rm exp\lpar }-{\rm i}\omega \lpar {\bi k}\rpar t\rpar \cr & + \gamma _{{-}n}^\ast {\rm \pi }\sqrt {2W\lpar {-}k_{m_k}\comma \;-k_{n_k}\rpar /\lpar L_xL_y\rpar } {\rm exp\lpar i}\omega \lpar -{\bi k}\rpar t\rpar .} $$

$W\lpar k_{m_k}\comma \;k_{n_k}\rpar$ in (5) and W(k, φ) in (2) are the expressions of the wave spectrum in Cartesian coordinates and polar coordinates, respectively. ${\lcub }\gamma _n{\rm \rcub}$ is a set of independent complex random numbers with a Gaussian probability density function and unit variance. L _x and L _y are the length of the sea surface. $\omega \lpar {\bi k}\rpar$ is the dispersion relationship of seawater.

(8)$$\omega \lpar {\bi k}\rpar ^2 = gk\lsqb {1 + {\lpar {k/k_m} \rpar }^2} \rsqb \tanh \lpar {kh} \rpar + {\bi k}\cdot {\bf u}_{\bf c}.$$

g is gravity acceleration, h is the depth of seawater, k_m equals 370 rad/m, ${\bi u}_c$ is the speed of the sea surface current. Then the linear sea surface profile is obtained by taking the inverse Fourier transform of (5)

(9)$$f\lpar x_m\comma \;y_n\rpar = \sum\limits_{m_k = {-}M/2}^{M/2} {\sum\limits_{n_k = {-}N/2}^{N/2} {F\lpar k_{m_k}\comma \;k_{n_k}\semicolon \;t\rpar \exp \left[ {{\rm i}\lpar {k_{m_k}x_m + k_{n_k}y_n} \rpar } \right] } } .$$

Creamer introduced Hilbert transform to obtain nonlinear sea surface based on the linear sea surface. The Creamer nonlinear Fourier components are approximately as

(10)$$A_{NL}^{\lpar n\rpar } \lpar {\bi K}\comma \;t\rpar = \displaystyle{1 \over N}\sum\limits_{\bf r} {\displaystyle{{{\lsqb {\rm i}{\bi K}\cdot {\bf \varsigma }\lpar {\bf r}\comma \;t\rpar \rsqb }^n} \over {n!K}}\exp \lpar\! -{\rm i}{\bi K}\cdot {\bi r}\rpar } .$$

The first order of this series $A_{NL}^{\lpar 1\rpar }$ identifies with F(k _m, k _n;t) in (5), then the second-order is given by

(11)$$\eqalign{A_{NL}^{\lpar 2\rpar } \lpar K_x\comma \;K_y\semicolon \;t\rpar = & -\displaystyle{{K_x^2 } \over {2K}}F^D\lpar f_{0x}^2 \rpar \cr & \quad -\displaystyle{{K_x^{} K_y} \over K}F^D\lpar f_{0x}^{} f_{0y}^{} \rpar -\displaystyle{{K_y^2 } \over {2K}}F^D\lpar f_{0y}^2 \rpar .} $$

F ^D( ⋅ ) represents a two-dimensional Fourier transform. Therefore, a second-order Creamer nonlinear sea surface can be written as

(12)$$f_{NL}\lpar {\bi r}\comma \;t\rpar = \sum\limits_{\bi K} {\lsqb F_L\lpar {\bi K}\comma \;t\rpar + A_{NL}^{\lpar 2\rpar } \lpar {\bi K}\comma \;t\rpar \rsqb \exp \lpar {\rm i}{\bi K}\cdot {\bi r}\rpar } .$$

Electromagnetic scattering model

This section describes CWMFSM, which is an improved TSM. According to the first-order small perturbation theory proposed by Fuks, the scattering field of an arbitrary tilted small patch can be expressed as

(13)$${\bi E}_{PQ}^{{\rm facet}} \lpar \hat{{\bi k}}_i{\rm \comma \;}\hat{{\bi k}}_s\rpar = \lsqb \hat{{\bi H}}_s\comma \;\hat{{\bi V}}_s\rsqb 2{\rm \pi }\displaystyle{{{\rm e}^{{\rm i}kR_0}} \over {{\rm i}R_0}}{\bf {\bf S}}_{PQ}\lpar \hat{{\bi k}}_i{\rm \comma \;}\hat{{\bi k}}_s{\rm \rpar \comma \;}$$

where $\hat{{\bi k}}_i$ and $\hat{{\bi k}}_s$ are incident and scattering electromagnetic wave vectors, respectively (Fig. 1), S_PQ is scattering amplitude matrix

(14)$${\bf { S}}_{PQ}{\rm (}{ \hat{\bi k}}_i{\rm ,}{ \hat{\bi k}}_s{\rm )} = \displaystyle{{k^2(1-\varepsilon )} \over {8{\rm \pi }^2}}{ \tilde{{\bf F}}}_{PQ}\int\!\!\!\int {\zeta ({\bi r}) {\rm e}^{-{\rm i}{\bi q}\cdot {\bi r}} } {\rm d} {\bi \rho }{\rm .}$$

here ${\bi q} = {\bi k}_s-{\bi k}_i\comma \;\tilde{{\bf {F}}}_{PQ}\comma \;$ is scattering factor matrix and its expression is in [Reference Zhang, Chen and Yin11].

Fig. 1.

The coordinate system of electromagnetic scattering model CWMFSM.

The capillary wave component, which satisfies the Bragg resonance condition along the radar observation direction, dominates the intensity of scattering electromagnetic wave from the sea surface. In CWMFSM, this component can be considered a set of monochromatic cosine waves.

(15)$$\zeta({\bi \rho}_c\comma\; t)=B({\bi k}_c\comma \;{\bi \rho}_c\comma \; t)cos({\bi k}_c\cdot{\bi \rho}_c-\omega_ct)$$

where ${\bi k}_c$ is sea surface Bragg wave number, ω _c is angular frequency corresponding to wave number ${\bi k}_c$. Because of the introducing spatial surface current velocity, the dispersion relation of the model in (8) is different from that of the traditional CWMFSM [Reference Zhang, Chen and Yin11]. Furthermore, the amplitude of the cosine waves $B\lpar {\bi k}_c\comma \;{\bi \rho }_c\comma \;\;t\rpar$, which is also different from the traditional CWMFSM [Reference Zhang, Chen and Yin11], is spatio-temporal variable determined by the wave spectrum, thus

(16)$$B\lpar {\bi k}_c\comma \;{\bi \rho }_c\comma \;\;t\rpar = 2{\rm \pi }\sqrt{W_{st} \lpar {\bi k}_c\comma \;{\bi \rho }_c\comma \;\;t\rpar /\Delta x\Delta y} \comma \;$$

where the spatio-temporal wave spectrum $W_{st}\lpar {\bi k}\comma \;{\bi \rho }_c\comma \;\;t\rpar$ in (6), Δx and Δy are the facet length in x and y direction, respectively.

Therefore, the integration in (14) can be simplified into

(17)$$\eqalign{ & \int\limits_{\Delta S} {\zeta \lpar {\bf r}\comma \;t\rpar \exp \lpar {-{\rm i}{\bi q}\cdot {\bi r}} \rpar {\rm d}{\bi \rho }} = \displaystyle{{B\lpar {\bi k}_c^{} \comma \;{\bi \rho }_c\comma \;t\rpar \Delta S} \over {2n_z}}{\rm e}^{-{\rm i}{\bi q}\cdot {\bi r}_0} \cr & \sum\limits_{n = {-}\infty }^\infty {{\lpar -{\rm i}\rpar }^n} J_n\lsqb q_zB\lpar {\bi k}_c\comma \;{\bi \rho }_c\comma \;t\rpar \rsqb {\bf I}_0\lpar {\bi k}_c\rpar .}$$

J _n is n-order Bessel function, and ΔS is the area of the small patch.

(18)$$\eqalign{{\bf I}_0\lpar {\bi k}_c\rpar = & \exp \lsqb {-i\lpar n + 1\rpar \omega_ct} \rsqb {\rm sinc}\left\{{\displaystyle{{\Delta x} \over 2}\lsqb {\lpar n + 1\rpar k_{cx}-q_x-q_zZ_x} \rsqb } \right\}\cr & \cdot {\rm sinc}\left\{{\displaystyle{{\Delta y} \over 2}\lsqb {\lpar n + 1\rpar k_{cy}-q_y-q_zZ_y} \rsqb } \right\}\cr & + \exp \lsqb {i\lpar 1-n\rpar \omega_ct} \rsqb {\rm sinc}\left\{{\displaystyle{{\Delta x} \over 2}\lsqb {\lpar 1-n\rpar k_{cx} + q_x + q_zZ_x} \rsqb } \right\}\cr & \cdot {\rm sinc}\left\{{\displaystyle{{\Delta y} \over 2}\lsqb {\lpar 1-n\rpar k_{cy} + q_y + q_zZ_y} \rsqb } \right\}.} $$

The two terms of ${\bi I}_0\lpar {\bi k}_c\rpar$, respectively, represent the contribution of the Bragg wave propagating towards and away from the radar incident direction, respectively. Z _x and Z _y are the derivatives of the second-order Creamer nonlinear sea surface height $f_{NL}\lpar {\bi r}\comma \;t\rpar$ in x and y direction, respectively, representing the contribution of wave–wave interaction to the scattering field.

The total scattering field can be obtained by summing scattering field in (13) corresponding to small facet in (12).

(19)$${\bi E}_{PQ}^s \lpar \hat{{\bi k}}_i{\rm \comma \;}\hat{{\bi k}}_s\comma \;t\rpar = \sum\limits_{m = 1}^M {\sum\limits_{n = 1}^N {{\bi E}_{PQ\comma mn}^{\,facet} \lpar {\hat{{\bi k}}}_i{\rm \comma \;}{\hat{{\bi k}}}_s\rpar } } .$$

Doppler spectrum and SAR images

The Doppler spectrum of time-varying sea surface can be obtained by the following formula [Reference Toporkov and Brown13]

(20)$$S_0\lpar f \rpar = \displaystyle{1 \over T}\left\vert {\int_0^T {{\bi E}_{PQ}^{{\rm scatt}} \lpar {\hat{{\bi k}}}_i{\rm \comma \;}{\hat{{\bi k}}}_s\comma \;t\rpar \exp \lpar {-{\rm i}2{\rm \pi }f\,{\kern 1pt} t} \rpar {\rm d}t} } \right\vert ^2.$$

The detailed realization process of Doppler spectrum is as follows:

1. (1) Input a set of independent complex random numbers ${\rm \lcub}\gamma _n{\rm \rcub}$, and the nonlinear sea surface at a certain time t is generated according to the method described in the section “Methods–Model of the deterministic nonlinear sea surface”;

2. (2) The scattering field of each nonlinear sea surface is obtained by CWMFSM;

3. (3) Generate the nonlinear sea surface at the next time, and repeat step (2), until the time series of sea surface scattering field is obtained within the required integration time;

4. (4) The Doppler spectrum of each sea surface sample is obtained through the Doppler spectrum formula (20);

5. (5) Input another set of independent complex random numbers ${\rm \lcub }{\gamma }{\prime}_n{\rm \rcub }$, repeat steps (1)–(4), then the Doppler spectrum under the required number of samples is obtained;

6. (6) The Doppler spectrum of each sample is statistically averaged.

Since we obtained the scattering field for each discrete facet of the nonlinear sea surface, it is easy to simulate the ensemble averaged image intensity through the velocity bunching (VB) model proposed by Alpers [Reference Alpers, Duncan and Clifford14] in the following translation:

(21)$$\eqalign{& I\lpar x\comma \;y\rpar = \int_{{-}L_x/2}^{L_x/2} {\int_{{-}L_y/2}^{L_y/2} {\sigma _I\lpar x_g\comma \;y_g\rpar } } f_y\lpar x-x_g\rpar \cdot \lpar \rho _{aN}/\rho _{aN}^{\rm {\prime}} \rpar \cr & \exp \lcub {-{\lsqb {\pi \lpar y-y_g-R\cdot u_r\rpar /\lpar \rho^{\prime}_{aN}V\rpar } \rsqb }^2} \rcub {\rm d}x_g{\rm d}y_g\comma \;} $$

where σ _I(x _g, y _g) = σ(x _g, y _g)(1 + f _SAR(x _g, y _g)), denotes the backscattering coefficients (BSC) of the nonlinear sea surface that takes the hydrodynamic modulation and velocity bunching into account. f _SAR(x _g, y _g) is the SAR modulation function. σ(x _g, y _g) is the NRCS based on this paper's proposed ocean scattering model. u _r is the orbital velocity in the range direction. Further, f _y(x − x _g) is the range resolution function, ρ _aN = N _lρ _a denotes the stationary target azimuth resolution for N _l incoherent looks, ρ _a = λR/2VT is the azimuth resolution, λ is the electromagnetic wavelength, V is the platform velocity, T is the synthetic aperture duration, and ${\rho }^{\prime}_{aN}$ is the degraded azimuth resolution due to the target acceleration and finite scene coherence time

(22)$$\rho _{aN}^{\rm {\prime}} = \rho _{aN}\left\{{1 + \displaystyle{1 \over {N_l}}\left[{{\left({\displaystyle{{\pi T^2} \over \lambda }a_r\lpar x^\prime\comma \;{y}^{\prime}\rpar } \right)}^2 + {\left({\displaystyle{T \over {\tau_s}}} \right)}^2} \right]} \right\}^{1/2}\comma \;$$

where a _r is the orbital acceleration in the range direction, and τ _s is the scene coherence time.