Angular Composite Facet Model with Shadowing Treatment for Backscattering from Two-Dimensional Rough Sea Surface

Abstract

Recent diverse applications on ocean wave synthetic aperture radar (SAR) images have a potential demand for individual facet scattering contribution rather than the total average radar cross-section (RCS) by statistical scattering models. However, only a few facet methods have been proposed and fewer have considered shadowing effects, which are thought to be non-negligible at large incident angle. In view of this, we proposed an angular composite facet model (ACFM) with shadowing treatment to investigate the backscattering from two-dimensional rough sea surface. First, a region division formula for ACFM without shadowing treatment is proposed to classified sea surface facets into specular facets and diffusion facets based on which kind of scattering is dominate from each facet, and the corresponding scattering contribution is calculated either by Kirchhoff approximation (KA) or by a small perturbation method (SPM) with a tilting process. Second, an electromagnetic shadowing algorithm based on facets grouping is adopted to handle the shadowing effects in a geometric manner with moderate computation complexity gained. Finally, comparisons between numerical backscattering evaluations and experimental data demonstrate that this new ACFM can attain accurate numerical results and the geometric facets grouping method is a practical way to tackle shadowing effects despite a certain acceptable gap. Therefore, the whole ACFM can simulate ocean wave SAR imaging, especially for those electrically large surfaces and to evaluate the scattering from sea surface with different local nature, such as sea spikes, foams, spilled oil, swells, and ship wake.

1. Introduction

In the past decades, electromagnetic (EM) scattering has attracted many researchers as well as government departments. Among these individuals and departments, the European Space Agency alone has developed many types of radar for airborne and spaceborne platforms, including advanced SAR on the satellite ENVISAT-1 and Sentinel-1. Other countries have also launched their satellites with carried SARs for different purposes. Canada, for example, launched Radarsat-1 and Radarsat-2 in 1995 and 2007, respectively. These SARs provided valuable radar image data for various applications, such as environment monitoring, natural resources managing, and coastal surveillance. However, those ocean image data acquired from SAR are often limited by continuous variation in ocean conditions, observation geometry and radar parameters. Hence, as a very efficient and cost-effective approach, the EM scattering estimation of marine scene becomes more valuable and has been proven to be useful in many fields, e.g., sea surface salinity or wind speed retrieval, ship wake detection, and oil spill monitoring [1,2,3,4,5,6].

Due to its vital importance, developing accurate approaches to evaluate the scattering of marine scene is crucial. However, when comes to rough sea surfaces, EM scattering simulation faces so many challenges that researchers have investigated a variety of approaches to tackle them. Compared with approximate approaches, numerical approaches [7,8], even their acceleration algorithms [9,10], have their inherent disadvantages—they are computationally demanding and time-consuming—so that they are often applied to one-dimensional sea surface and simple three-dimensional targets and are rarely applied to two-dimensional (2D) sea surface. Composite surface methods (CSM) or TSM [11,12] is an outstanding approximate approach to obtain accurate and acceptable results for 2D sea surface, which is developed from two basic approximate approaches—KA [13] and SPM [14]—and can extend the domain of validity of the two. Another effective method—small slope approximation (SSA) [15,16,17]—can be degenerated into the two while with similar accuracy to method of moment (MoM), which allows it to be the most common method to make comparison with, even with relatively higher computation complexity than other approximate approaches.

However, both CSM and SSA only provide the total average RCS, which cannot provide specific facet scattering contribution to meet the demand of diverse applications of ocean wave SAR image. Even though similar applications are likely to be much more common as the spatial resolution and temporal resolution of SAR constantly enhance, only a few facet-based methods have been proposed to deal with it. Wang and Tong [18] adopted a Facet-Based GO-SSA Model to calculate the normalized RCS from 2D sea surface with visibility factor evaluated by shadow functions [19]. Rochidi [20] applied physical optics and the method of equivalent currents to calculate the facet scattering along with back-face culling algorithm and painter’s algorithm evaluating the self-shadowing and mutual shadowing effect, respectively, and then used multiple-bounce scattering mechanisms to calculate the bistatic scattering for complex objects over sea surface. The semi-deterministic two-scales method (SD-TSM), proposed by Arnold-Bos [21], is the prototype for scattering evaluation of specific facet based on composite model, which provides a vision to clarify the RCS fluctuation with local features considered, while macroscopic visibility factor is calculated by ray tracing or Z-buffer procedure with facets sizes compared to incident wavelength. Nie and Zhang [22] proposed an angular cutoff composite model (ACCSM) with local scattering angle as its angular division criterion for backscattering evaluation, whereas they merely take the self-shadowing effect into consideration. Li and Zhang [23] advanced complete facet scattering calculation with breaking waves based on their previous work on capillary wave modification facet scattering model. In their model, an additional phase factor is calculated by Fuks’ formula and the individual facet scattering and shadowing effect is considered using computer graphics and Z-buffer algorithm.

In most cases for KA and methods combined KA, shadowing effects are thought to be non-negligible, especially at a large incident or scattering angle. However, the traditional statistical shadow functions discussed by Bourlier [19] can only provide average shadowing factors so that they are not able to quantitatively analyze the shadowing effects on the scattering contribution of facets, especially for facets with local nature, such as foams, spilled oil, swells, and ship wake. In addition, the shadowing effects treated in a geometric manner for facet-based methods are often evaluated with smaller facets sizes compared to incident electromagnetic wavelength, which bring about a huge computation burden. With aforementioned facet-based methods, scattering contribution from facets can be calculated efficiently and the results are accurate or acceptable. Unfortunately, most of these above approaches either did not take shadowing effects into consideration in a more practical way like shadow functions used by Wang [18] and only self-shadowing factor adopted by Nie [22], or brought about an increase in computational complexity using finer facets by Arnold-Bos [21,24].

Therefore, an angular composite facet model with shadowing treatment is proposed to investigate the backscattering from 2D sea surface in this paper. First, the scattering contribution of facets without shadowing effects using the angular composite model with definite region division expressions is described as in our previous work [25]. Then, we present how two kinds of shadowing effects affect the scattering estimation of every single facet, where the self-shadowing factor is calculated based on incident and scattered wave vectors and normal vector of every facet and Ji’s theory of electromagnetic shadowing algorithm based on facets grouping for 3D objects [26] is modified to evaluated the mutual shadowing factor. Finally, the scattering contributions from every single facet are added together to get the average RCS to compare with others’ work and experimental data, and comparisons on shadowing effects have also been made to prove that the mutual shadowing effect is more significant than self-shadowing at very large angles, especially at high wind speed and the geometric facets grouping method is a practical way to tackle shadowing effects. The rest parts are organized as follows, the sea spectrum, KA component and SPM component in ACFM, domain of validation for each component and self-shadowing effect; the mutual shadowing effect and its complexity evaluation are detailedly described in Section 2. In Section 3, analysis of components, angular distribution in Ku-band, angular distribution in different bands, wind speed variation, shadowing factor, angular distribution with shadowing effects, and modulus distribution are analyzed. In Section 4, the final conclusion is given.

2. Proposed Methods

2.1. Sea Spectrum

To precisely evaluate the EM scattering in an approximate facet model, the choice of reliable sea spectrum is of vital importance. Gravity and capillary-gravity waves are often treated as two different kinds of roughness in microwave frequency band and more attention should be paid to small-scale waves for their dominant effects at middle and large incident angles. Of all the spectral models, the PM spectrum and JONSWAP spectrum are only able to describe large-scale waves, which means they fail to fully express the small-scale roughness [27]. Therefore, we chose the Fung spectrum [28] to generate the sea surface height map and to be employed in EM computations later.

$$S_{1d}\left(K\right)=\{\begin{array}{l}\frac{1.4\times {10}^{-3}}{K^3}exp\left[-\frac{0.74g_0^2}{K^2{U}_{19.5}^4}\right]ifK<0.04\mathrm{rad}/\mathrm{cm}\\ a_0\left(1+3K^2/k_m^2\right){[K+\frac{K^3}{k_m^2}]}^{-\frac{p+1}{2}}ifK>0.04\mathrm{rad}/\mathrm{cm}\end{array},$$

(1)

where $g_0=981\mathrm{cm}/\mathrm{s}$, $U_{19.5}$ is wind spend at 19.5 m above the mean sea level (in cm/s), $a_0=0.85{(2\pi )}^{\mathrm{p}-1}{g}_0{}_{}^{\left(1-\mathrm{p}\right)/2}$, $p=5-{log}_{10}\left(U_c\right)$, $k_m^{}=3.63\mathrm{rad}/\mathrm{cm}$ and $U_c$ is the friction wind speed (in cm/s). The relationship between the wind speed at different altitude and friction wind speed is described by Fung [28].

To better describe a two-dimensional sea surface, a number of different angular distribution models have been proposed as different functions of wave propagation wave direction, wind direction, and wave number. The spread function proposed by Fung is a very practical one that links the spread function with wave number [28,29].

$$S_{dis}\left(K,\mathsf{\varphi }\right)=a_1+a_2\left(1-e^{-b{K}^2}\right)cos\left(2\left({\mathsf{\varphi }}_{obs}-{\mathsf{\varphi }}_{wind}\right)\right),$$

(2)

where $a_1=1/\left(2\pi \right)$, $a_2=\frac{\left(1-R\right)/\left(1+R\right)}{\pi \left(1-\upsilon \right)}$, $b=1.5{\mathrm{cm}}^2$, $R$, and $\upsilon$ are variables related to slope variance, ${\mathsf{\varphi }}_{obs}$ is the angle of observation, and ${\mathsf{\varphi }}_{wind}$ is the angle of wind direction. The two-dimensional sea spectrum is then described as

$$S_{2d}\left(K,\mathsf{\varphi }\right)=S_{1d}\left(K\right)*S_{dis}\left(K,\mathsf{\varphi }\right)/K.$$

(3)

Figure 1 shows a two-dimensional sea surface geometric model constructed by a linear filtering method with Fung spectrum under the condition of 5 m/s wind speed. The total surface size is 64 m × 64 m and the facet size 1 m × 1 m (m for meter).

2.2. KA Component

We used the assumption that a facet can be considered as either a diffusion facet or a specular facet depending on which kind of scattering dominates [30], then, a diffusion facet means that the scattering contribution made by this facet can be mainly expressed as diffusion scattering and can be evaluated by SPM due to its outstanding performance in diffusion region. In contrast, a specular facet means that the scattering that comes from this facet is mainly specular scattering where KA is more appropriate to be applied to calculate the scattering coefficient. To illustrate the facet division clearly, the geometry of sea surface scattering is shown in Figure 2. A facet whose propagation direction of scattered waves is in the inner region of the cone within a $20°$ semi-cone angle is a specular facet; otherwise, it is a diffusion facet. Stationary-phase approximation is adopted here to describe the KA component or scattering contribution from specular facets in ACFM within the physical optics framework. Thus, the first step is to figure out the relationship between two coordinates—the local coordinate and the global coordinate—where the three axes with no subscript are set to be the global framework and the three axes with subscript l to be the local framework.

Let $z\left(x,y\right)$ be the height of a two-dimensional sea surface and $Zx$ and $Zy$ donate its derivatives along the x axis and y axis, respectively, so for every single facet, the theorical unit normal vector in the global framework can expressed as

$$n_l=-xZx-yZy+z/\sqrt{1+Z{x}^2+Z{y}^2},$$

(4)

and the propagation directions of incident and scattering wave in the global framework are, respectively, equal to

$$n_i=xsin{\theta }_{i}cos{\mathsf{\varphi }}_i+ysin{\theta }_{i}sin{\mathsf{\varphi }}_i-zcos{\theta }_i,$$

(5)

$$n_s=xsin{\theta }_{s}cos{\mathsf{\varphi }}_s+ysin{\theta }_{s}sin{\mathsf{\varphi }}_i+zcos{\theta }_s.$$

(6)

Accordingly, the three local axes in the local framework are defined as below

$$z_l=n_l,$$

(7)

$$y_l=\frac{\left(n_i\times n\right)}{\left|n_i\times n\right|},$$

(8)

$$x_l=y_l\times z_l.$$

(9)

Then the local angle configuration $\left({\theta }_{il},{\theta }_{sl},{\mathsf{\varphi }}_{il},{\mathsf{\varphi }}_{sl}\right)$ can be calculated by the use of the global angle configuration $\left({\theta }_i,{\theta }_s,{\mathsf{\varphi }}_i,{\mathsf{\varphi }}_s\right)$:

$$cos{\theta }_{il}=-\left(n_l\mathbf{·}{n}_i\right)=\frac{\left(cos{\theta }_i+Zxsin{\theta }_{i}cos{\mathsf{\varphi }}_i+Zysin{\theta }_{i}sin{\mathsf{\varphi }}_i\right)}{\sqrt{1+Z{x}^2+Z{y}^2}},$$

(10)

$$cos{\theta }_{sl}=n_l\mathbf{·}{n}_s=\frac{\left(cos{\theta }_s+Zxsin{\theta }_{s}cos{\mathsf{\varphi }}_s+Zysin{\theta }_{s}sin{\mathsf{\varphi }}_s\right)}{\sqrt{1+Z{x}^2+Z{y}^2}},$$

(11)

$$cos{\mathsf{\varphi }}_{il}=\{\begin{array}{ll}\frac{n_i\mathbf{·}{x}_l}{sin{\theta }_{il}},& if{\theta }_{il}\ne 0\\ 1,& if{\theta }_{il}=0,\end{array}$$

(12)

$$cos{\mathsf{\varphi }}_{sl}=\{\begin{array}{ll}\frac{n_s\mathbf{·}{x}_l}{sin{\theta }_{il}},& if{\theta }_{is}\ne 0\\ 1,& if{\theta }_{is}=0,\end{array}$$

(13)

The angle $\chi$ between the bisector angle of XOR and local unit normal vector is the region division angle, and its cosine is enough to figure out whether a facet is a specular facet or a diffusion facet.

$$cos\chi =\frac{\left(n_{sl}-n_{il}\right)\mathbf{·}{z}_l}{\left|n_{sl}-n_{il}\right|}.$$

(14)

Under the backscattering situation ${\theta }_i={\theta }_s,{\mathsf{\varphi }}_i={\mathsf{\varphi }}_s+\pi$, the region division angle $\chi ={\theta }_{il}={\theta }_{sl}$ turns out to be the same as in backscattering configuration in ACCSM. The scattering coefficient of a specular facet can be expressed as

$$\{\begin{array}{c}cos\chi \ge cos\left(20°\right)\\ {\sigma }_{mn,\alpha \alpha }=\frac{\pi k_i{\left|q\right|}^2}{q_z^4}\left|U_{\alpha \alpha }\right|Pr(Zx,Zy)\end{array},$$

(15)

where $\alpha =HorV$; the second $\alpha$ denotes polarization of scattered wave, the first $\alpha$ expresses the polarization of incident wave, H and V for horizontal and vertical polarization, respectively, m, n denotes the number of discretized points, namely discretized facet, and $Pr(Zx,Zy)$ is the probability density function [31]. Other variables except for polarimetric parameter $\left|U_{\alpha \alpha }\right|$ are expressed as follows

$$q=k_i\left|n_s-n_i\right|,$$

(16)

$$q_z=k_i\left(cos{\theta }_s+cos{\theta }_i\right),$$

(17)

where $k_i$ is the incident wave number. It is important to note that the polarimetric parameter $\left|U_{\alpha \alpha }\right|$ is related to polarization vectors of incident and scattered waves $\left(h_i,v_i,h_s,v_s\right)$ and local Fresnel reflection coefficients, where $R_{HH}$ and $R_{VV}$ are for horizontal and vertical polarizations, respectively.

$$U_{HH}=\frac{q\left|q_z\right|\left[R_{VV}\left(h_s\mathbf{·}{n}_i\right)\left(h_i\mathbf{·}{n}_s\right)+R_{\mathrm{HH}}\left(v_s\mathbf{·}{n}_i\right)\left(v_i\mathbf{·}{n}_s\right)\right]}{\left[{\left(h_s\mathbf{·}{n}_i\right)}^2+{\left(v_s\mathbf{·}{n}_i\right)}^2\right]k_i{q}_z},$$

(18)

$$U_{VV}=\frac{q\left|q_z\right|\left[R_{HH}\left(h_s\mathbf{·}{n}_i\right)\left(h_i\mathbf{·}{n}_s\right)+R_{VV}\left(v_s\mathbf{·}{n}_i\right)\left(v_i\mathbf{·}{n}_s\right)\right]}{\left[{\left(h_s\mathbf{·}{n}_i\right)}^2+{\left(v_s\mathbf{·}{n}_i\right)}^2\right]k_i{q}_z},$$

(19)

$$U_{HV}=\frac{q\left|q_z\right|\left[R_{VV}\left(h_s\mathbf{·}{n}_i\right)\left(v_i\mathbf{·}{n}_s\right)-R_{\mathrm{HH}}\left(v_s\mathbf{·}{n}_i\right)\left(h_i\mathbf{·}{n}_s\right)\right]}{\left[{\left(h_s\mathbf{·}{n}_i\right)}^2+{\left(v_s\mathbf{·}{n}_i\right)}^2\right]k_i{q}_z},$$

(20)

$$U_{HV}=\frac{q\left|q_z\right|\left[R_{VV}\left(v_s\mathbf{·}{n}_i\right)\left(h_i\mathbf{·}{n}_s\right)-R_{\mathrm{HH}}\left(h_s\mathbf{·}{n}_i\right)\left(v_i\mathbf{·}{n}_s\right)\right]}{\left[{\left(h_s\mathbf{·}{n}_i\right)}^2+{\left(v_s\mathbf{·}{n}_i\right)}^2\right]k_i{q}_z},$$

(21)

$$R_{HH}=\frac{cos{\theta }_{il}-\sqrt{1-{sin}^2{\theta }_{il}}}{cos{\theta }_{il}+\sqrt{1-{sin}^2{\theta }_{il}}},$$

(22)

$$R_{VV}=\frac{{\epsilon }_{r}cos{\theta }_{il}-\sqrt{1-{sin}^2{\theta }_{il}}}{{\epsilon }_{r}cos{\theta }_{il}+\sqrt{1-{sin}^2{\theta }_{il}}},$$

(23)

where the most confusing terms are the relationships between polarization vectors and the incident and scattered vectors, so the final result is given out directly here. The detailed derivation refers to work of Ulaby [32].

$$h_s\mathbf{·}{n}_i=-sin{\theta }_{i}sin({\mathsf{\varphi }}_s-{\mathsf{\varphi }}_i),$$

(24)

$$h_i\mathbf{·}{n}_s=sin{\theta }_{s}sin({\mathsf{\varphi }}_s-{\mathsf{\varphi }}_i),$$

(25)

$$v_s\mathbf{·}{n}_i=sin{\theta }_{i}cos{\theta }_{s}cos({\mathsf{\varphi }}_s-{\mathsf{\varphi }}_i)+cos{\theta }_{i}sin{\theta }_s,$$

(26)

$$v_i\mathbf{·}{n}_s=cos{\theta }_{i}sin{\theta }_{s}cos({\mathsf{\varphi }}_s-{\mathsf{\varphi }}_i)+sin{\theta }_{i}cos{\theta }_s,$$

(27)

2.3. SPM Component

For the nonspecular region, the diffusion scattering contribution of a facet can be evaluated by SPM in the local framework but with a tilting process (or CSM) to adapt to the global configuration since the SPM in the global configuration could not give credible results at large incident angles.

$${\sigma }_{mn,\alpha \alpha }^0=8k_i{}^4{cos}^2{\theta }_{il}{cos}^2{\theta }_{sl}{\left|{\beta }_{\alpha \alpha }\right|}^2{S}_{2d}\left(K,\mathsf{\varphi }\right),$$

(28)

$$\{\begin{array}{c}cos\chi \le cos\left(20°\right)\\ {\sigma }_{mn,hh}=\left(h_i\mathbf{·}{h}_{il}\right){\sigma }_{mn,\alpha \alpha }^0\left(1+Zxtan{\theta }_i\right)Pr(Zx,Zy)\\ {\sigma }_{mn,vv}=\left(v_i\mathbf{·}{v}_{il}\right){\sigma }_{mn,\alpha \alpha }^0\left(1+Zxtan{\theta }_i\right)Pr(Zx,Zy)\end{array},$$

(29)

where $h_{il}$ is the corresponding local horizontal polarization vector, $v_{il}$ is the vertical counterpart, and ${\beta }_{\alpha \alpha }$ is the polarimetric coefficient in terms of local angles and sea water permittivity ${\epsilon }_r$.

$${\beta }_{HH}=-\frac{\left({\epsilon }_r-1\right)cos{\mathsf{\varphi }}_{sl}}{\left[cos{\theta }_{il}+\sqrt{\left({\epsilon }_r-{sin}^2{\theta }_{il}\right)}\right]\left[cos{\theta }_{sl}+\sqrt{\left({\epsilon }_r-{sin}^2{\theta }_{sl}\right)}\right]},$$

(30)

$${\beta }_{VV}=\frac{\left({\epsilon }_r-1\right)\left[{\epsilon }_{r}sin{\theta }_{il}sin{\theta }_{sl}-\sqrt{\left({\epsilon }_r-{sin}^2{\theta }_{il}\right)\left({\epsilon }_r-{sin}^2{\theta }_{sl}\right)}cos{\mathsf{\varphi }}_{sl}\right]}{\left[{\epsilon }_{r}cos{\theta }_{il}+\sqrt{\left({\epsilon }_r-{sin}^2{\theta }_{il}\right)}\right]\left[{\epsilon }_{r}cos{\theta }_{sl}+\sqrt{\left({\epsilon }_r-{sin}^2{\theta }_{sl}\right)}\right]},$$

(31)

$${\beta }_{HV}=\frac{\left({\epsilon }_r-1\right)\sqrt{\left({\epsilon }_r-{sin}^2{\theta }_{il}\right)}sin{\mathsf{\varphi }}_{sl}}{\left[{\epsilon }_{r}cos{\theta }_{il}+\sqrt{\left({\epsilon }_r-{sin}^2{\theta }_{il}\right)}\right]\left[cos{\theta }_{sl}+\sqrt{\left({\epsilon }_r-{sin}^2{\theta }_{sl}\right)}\right]},$$

(32)

$${\beta }_{VH}=\frac{-\left({\epsilon }_r-1\right)\sqrt{\left({\epsilon }_r-{sin}^2{\theta }_{il}\right)}sin{\mathsf{\varphi }}_{sl}}{\left[cos{\theta }_{il}+\sqrt{\left({\epsilon }_r-{sin}^2{\theta }_{il}\right)}\right]\left[{\epsilon }_{r}cos{\theta }_{sl}+\sqrt{\left({\epsilon }_r-{sin}^2{\theta }_{sl}\right)}\right]}.$$

(33)

2.4. Angular Composite Facet Model

Combining the two components together, the average scattering coefficient of ACFM without shadowing effect is given by

$$\langle \sigma \rangle =\frac{1}{M}\frac{1}{N}{\displaystyle \sum _{m=1}^M{\displaystyle \sum _{n=1}^N{\sigma }_{mn,\alpha \alpha }}}.$$

(34)

Since the shadowing effect is non-negligible at large incident or scattering angles, a visibility factor $\Delta vis$ is used to indicate its influence, and then the average scattering coefficient of ACFM with shadowing treatment can be expressed by

$$\langle \sigma \rangle =\frac{1}{M}\frac{1}{N}{\displaystyle \sum _{m=1}^M{\displaystyle \sum _{n=1}^N{\Delta vis}_{mn}{\sigma }_{mn,\alpha \alpha }}}.$$

(35)

Note that the visibility factor $\Delta vis=1-\Delta shw$, where $\Delta shw$ is the shadowing factor tied to the mutual shadowing factor ${\Delta mut}_{shw}$ and self-shadowing factor ${\Delta self}_{shw}$. When ${\Delta mut}_{shw}=1$ or ${\Delta self}_{shw}=1$, $\Delta vis=0$.

2.5. Domain of Validation

As we mentioned before, KA is only valid when close to the specular region since it underrates the diffuse component when the scattered wave vectors are not inside the cone. However, it is adequate to compute the specular component from large sea surface with incident frequency above 1 GHz. The validity conditions for KA and SPM proposed by researchers differ, in this paper the validity conditions quoted from Nie [30] are adopted.

$$k_i\sigma >\frac{\sqrt{10}}{\left|cos{\theta }_i+cos{\theta }_s\right|}{k}_{i}l>6R_{cur}>{\lambda }_{i}forKA,$$

(36)

$$k_i\sigma <0.3k_i\sigma <\frac{0.3}{\sqrt{2}}{k}_{i}lforSPM,$$

(37)

where $\sigma$ is the standard deviation of the sea surface, $l$ is the surface correlation length, and $R_{cur}$ is the average radius of curvature.

2.6. Shadowing Effect

For Kirchhoff approximation (KA), or methods combined with KA, shadowing effects are thought to be non-negligible at large incident angle. Since traditional statistical shadow functions can only provide average shadowing factors, they are not able to quantitatively analyze the shadowing effects on the scattering coefficient of facets with local nature, such as foams, spilled oil, swells, and ship wake. Even though ACFM does not consider multiple scattering and breaking wave multipath and volume scattering at large scattered angle, it is still necessary to take shadowing effects into consideration to improve it in a facet-based way. Figure 3 shows the process of a classical whole shadowing judgement.

The self-shadowing factor is easy to understand and describe. It represents a local nature relevant to the local normal vector and it is not affected by shadowing caused by other facets. For a given facet, it is impossible the ray which comes from the transmitter or goes to the receiver crosses the air-water interface. Therefore, the facet satisfies the following condition gives no contribution at all.

$${\Delta self}_{shw}=1,if{n}_i\mathbf{·}{z}_l>0or{n}_s\mathbf{·}{z}_l<0.$$

(38)

2.7. Mutual Shadowing Effect

When ${\theta }_i$ or ${\theta }_s$ is less than $65°$, both the self-shadowing effect and mutual shadowing effect can be safely omitted [21], which means that $\Delta vis=1$ and the average scattering coefficient of ACFM can be calculated by equation 34. Conversely, when ${\theta }_i$ and ${\theta }_s$ are from $65°$ to near $90°$, some facets will be in the shadow of other facets, thus the mutual shadowing effect between them needs to be clarified. Here, an electromagnetic shadowing algorithm based on facets grouping (FGSA) is used to consider it. In a bistatic scattering configuration, the process of shadowing judgments must be performed twice: once for the incident wave and once for the scattered wave. However, the process of mutual shadowing judgment is only performed once for the deterministic sea surface in backscattering configuration due to the path symmetry of electromagnetic wave propagation shown in Figure 4.

2.7.1. Triangulation and Projection Transformation

Before the process of projection transformation, the number of the facets is increased by one interpolation along the x and y axes of the sea elevation map. Thus, every facet is divided into two triangular facets so that every facet in the sea elevation map is divided into eight triangular facets. This triangulating process ensures that the number of triangular facets used to judge the mutual shadowing effect is much larger than the number of facets divided by the composite model to achieve a certain precision. It is worth to note that the facet size used here is almost 10 to 20 times that of the electromagnetic wave, which has been proven to be able to obtain relatively accurate results in the microwave band, since the short-wave contribution has been included in facet-based models. This allows a huge reduction of computation complexity in the shadowing effect, because in most cases the size of facets is often much smaller than electromagnetic wavelength. Certainly, as the wavelength of electromagnetic wave is small, it is wise to reduce the size of the facets properly.

For a given direction of observation, a new reference of observation ${\mathrm{x}}_{\mathrm{v}}{\mathrm{y}}_{\mathrm{v}}{\mathrm{z}}_{\mathrm{v}}$ is defined, where ${\mathrm{z}}_{\mathrm{v}}$ points to the observer and ${\mathrm{x}}_{\mathrm{v}}{\mathrm{o}}_{\mathrm{v}}{\mathrm{y}}_{\mathrm{v}}$ is the projection plane. Through the transformation of two coordinate systems, the triangulated sea surface can be dealt with in the projection plane, where a small triangular facet becomes a plane triangle or degenerated triangle, that is, when three points are on the same line. In this way, the shadowing judgement of facets changes to the overlapping judgement of triangles.

2.7.2. Facets Grouping and Overlapping Judgement

The main idea of the electromagnetic shadowing algorithm is to group the triangular facets in the projection plane and to record the height z of each vertex of all discrete facets, then to judge whether the triangular facets in the group are intersected or overlapped. The facet grouping steps: (1) determine a rectangle on the projection plane so that all the facets in the projection plane are included in the area S; (2) divide the rectangle into a number of square with side length L, so the area is S = L² and the total number of facets is K = S/L²; and (3) consider that all the valid facets after the self-shadowing judgement in the same square are in the same group. Since a triangle may be intersected with different square, so a facet can belong to different groups.

If any two plane triangles in the same group are overlapped, then the facet depth d is calculated from the height of the three vertexes (z1, z2, z3) using d = (z1 + z2 + z3)/3. For a given triangular facet relative to the incident field, a larger facet depth means that the facet is farther away from the observer and is considered to be obscured by other facets; otherwise it is not obscured. This method makes full use of the real sea elevation map and have no limits on sea state, so it is more realistic than the empirical formula.

Two plane triangles overlap, provided that there are intersecting edges or there is a vertex or vertices of one triangle inside the other triangle. Assuming that the endpoint coordinates of line segments AB and CD are {A1, A2}, {B1, B2}, {C1, C2}, and {D1, D2}, respectively. Two conditions should be satisfied at the same time: Condition 1, Point A and point B are on the different sides of straight line CD and Condition 2, point C and Point D are on the different sides of line AB, which is the necessary and sufficient condition for the intersection of two line segments. Let the line equation of CD be $F_{CD}\left(X,Y\right)=0$, then Condition 1 is equivalent to $F_{CD}\left(A1,A2\right)F_{CD}\left(B1,B2\right)\le 0$. The same method can be used to judge the Condition 2.

2.7.3. Complexity Evaluation

To evaluate how many judgements are needed for the whole surface, all triangles in the projection plane need an ergodic judgement in the same group. All triangles are considered to be in the same group in the traditional geometric method, so the computational complexity is O(N²). If all the facets of the sea surface are judged by this method, the computational complexity is quite large. In our case, the sea surface is divided into 64 × 64 × 8 facets, and there are no more than 64 × 64 × 8 valid facets, so the number of shadowing judgments is approximately 32,768 × 32,767/2 = 536,854,528 times. For a specific sea surface, the number of facets overlapping with a facet is generally no more than 10, and a large amount of calculation time is used for judgment between two facets that are far apart. The idea of facet grouping is to avoid judgments between facets far away in the projection plane, but only for facets that are closer together.

The mutual shadowing process is only used to judge whether a facet is obscured by other facets in this group. Assuming that the facets are divided into 2048 groups and each group has 16 facets, since there are no more than 64 × 64 × 8 valid facets, the number of shadowing judgments is approximately 16 × 15/2 × 2048 = 245,760 times, which is 0.045% that of without grouping. The mutual shadowing factor is then calculated by assemble average of the mutual shadowing factor of the eight triangles belonging to the same facets before the triangulating process. The shadowing effect of a sea surface at 5 m/s wind speed with incident angle $88°$ is shown in Figure 5, where the white facet means a visible facet, black facet means a nonvisible facet, and gray facet means that not all eight triangles within a facet are visible. The self-shadowing factor in Figure 5b is 0.2292, which is less than mutual shadowing factor 0.3982 in Figure 5c. The mutual shadowing is 73.7% larger than self-shadowing under this situation, which demonstrates that at large incident angles the mutual shadowing effect is greater than self-shadowing and worth careful treatment.

3. Results

3.1. Analysis of Components

Except for the special declaration, in the following calculations in ACFM, the incident frequency f is 14.0 GHz, and $U_{10}$ is the wind speed at 10 m above the sea level. The permittivity of the seawater calculated in Kelvin model is 48.396 − 36.603i, where i is the imaginary unit.

Unlike other facet-based methods, where the scattering contribution added together the specular contribution computed by KA with the diffusion contribution by SPM or CSM for every facet, the final backscattering coefficients are composed of scattering contributions of each individual facet on the sea surface, either given by the KA if the region division judgement in the local vectors is satisfied or calculated by the SPM.

Table 1. The number of facets processed by Kirchhoff Approximation (KA) and mall perturbation method (SPM) with $U_{10}=5\mathrm{m}/\mathrm{s}$ in total 64 × 64 facets.

Incident Angle	14°	16°	18°	19°	20°	21°	22°	24°	26°
KA	3892	3552	2871	2408	1891	1443	1004	439	131
SPM	204	544	1225	1688	2205	2653	3092	3657	3965

illustrates the number of facets processed by KA and SPM in total 64 × 64 facets on the two-dimensional sea surface with $U_{10}=5\mathrm{m}/\mathrm{s}$. When the incident angle in the global framework is less than $20°$, the number of facets processed by KA is more than that of SPM for that most of the facets is considered to be in the specular region. Contrarily, when the incident angle in the global framework is more than $20°$, the number of facets processed by SPM is more than that of KA since more facets are considered to be in the diffusion region. However, the number of facets processed by SPM is more than KA when the incident angle is equal to 20°. The tilting effect of gravity wave has led to the fact that the local scattered angle of every single facet is not equal to $20°$ exactly.

3.2. Angular Distribution in Ku-Band

Backscattering is obviously of most importance in satellite SAR images and classic radars; here we make comparison of numerical backscattering coefficients calculated from various approaches in the Ku Band. In the backscattering configuration, the incident angle is equal to the scattered angle and the azimuth difference is equal to π. Figure 6 shows the incident angular distribution of the backscattering coefficients with different models at wind speed 5 m/s and 15 m/s. In ACFM and the SPM, backscattering coefficients are evaluated using the Fung sea spectrum. Keeping the consistency of all the approaches, we compare the theoretically predicted horizontal and vertical polarized scattering in the Ku Band with SASS-II model which has been confirmed by airborne measurements [33]. Several important deductions can be made from Figure 6. First, a good agreement is gained between the ocean measurements and the calculated backscattering coefficients in ACFM and SSA. In the specular region (near the normal), where the incident angle is less than $20°$, the calculation results of KA and ACFM are fitted to the experimental data. This observation makes sense for the KA hypotheses and the assumption of specular region. Second, in the median region (incident angle from $25°$ to $60°$), SPM and SSA have great performance compared to the measurements. ACFM gives out similar performance but with a slightly discrepancy. The experimental data in VV polarization is slightly smaller than the backscattering coefficient, which may be due to the stochastic characteristic of the sea surface. Third, at the large angles (incident angle more than 60°), neither KA nor SPM are accurate enough to evaluate the scattering coefficients, but ACFM can give credible results. Specifically, SSA raises great numerical problems in this region so that it cannot provide reliable results [17]. SPM is indeed applicable in the local framework with respect for its theory hypothesis, so the averaging backscattering coefficient given by SPM with tilting process (induced by slope distribution) can adapt the results to the global framework. Moreover, it is the fact that the region determination is based on local angle and final averaging scattering coefficient is calculated from that of each single facet ensures smooth transition in the median region in the ACFM. Lastly, the whole simulation shades light on the wind speed effect on the EM scattering. The EM scattered energy will be primarily in the specular direction as wind speed decreases, and conversely the EM scattered energy is significantly attenuated by several dBs as the wind speed increases.

3.3. Angular Distribution in Different Bands

To further examine the effectiveness of ACFM, we also give the numerical results in other frequency bands and compared with existed experimental data at different wind speeds. From Figure 7, there is an overall agreement with RRL (Radio Research Laboratory) experimental data [34] in the X band (frequency 10GHz) and similar agreement with NRL-4FR (US Naval Research Laboratory 4-band Radar) experimental data [35] in the C band (4.455GHz). In the region with an incident angle above 60°, there is an acceptable gap between the simulation results and the measured data. Moreover, it can be seen that as wind speed increases, the scattering coefficient in VV polarization and HH polarization both increase.

In Figure 8, a similar trend with wind speed can also be observed in L band. Even though there is a certain gap between the VV polarized scattering coefficient and the measured data at large incident angles, the NRL-4FR measured data and simulated HH polarized scattering coefficient agree well. As a probable result of the stochastic characteristic of the simulated sea surfaces slopes, the gap between the VV polarized scattering coefficient and the measured data is slightly larger than HH scattering coefficient, which means that the stochastic characteristic of sea surfaces slopes affects the accuracy of the numerical results. Ulaby has reminded us that vertical polarization is more sensitive to surface slopes, so the agreement between numerical results and experimental data is consequently poorer [32]. Therefore, in ACFM the gap may due to the higher sensitivity of VV polarization on statistical characteristic of sea surface slopes than that of HH polarization.

3.4. Wind Speed Variation

Figure 9 shows the influence of wind speed on the backscattering from sea surface for three different incident angles. Numerical results are also compared with SDFSM simulations [36]. The results of ACFM are in good accordance with that of SDFSM with 3-5dB difference. Based on the above comparisons, it can be concluded that ACFM is accurate enough for estimation of backscattering from two-dimensional rough sea surfaces.

3.5. Shadowing Factor

To further illustrate the shadowing effect in the ACFM, the shadowing factor with its components at 5 m/s wind speed and visibility factor at 5 m/s and 15 m/s wind speed calculated by FGSA (this paper), Z-buffer in a computer graphics way [36] and traditional Smith shadowing function are shown in Figure 10. Note that the mutual shadowing factor here is defined as the difference between the whole factor simulated by FGSA and self-shadowing effector calculated by Equation (38), thus the whole process for shadowing judgement tells us that mutual shadowing factor is judged after the self-shadowing process. Several important deductions can be made from Figure 10. First, both self-shadowing and mutual shadowing factor increase with the incident angle except for at $90°$, which means shadowing effects become more significant as incident angle increases. The exception at 90° is probably due to the definition of mutual shadowing factor, which means facets obscured by themselves are not counted in the mutual shadowing factor and these facets are more than facets obscured only by other facets. Second, the fact that mutual shadowing factor is larger than the self-shadowing factor means that taking the self-shadowing effect into consideration is not enough since the mutual shadowing factor can exert a larger impact on scattering coefficients than the self-shadowing factor. Third, the visibility factors calculated by all three methods have the same trend: they decrease as incident angle gets larger and it decreases as wind speed increases. Lastly, the largest visibility difference between traditional Smith shadowing function and FGSA is less than 0.1 and it is also true for Z-buffer, which is an acceptable difference. The difference for Z-buffer is probably due to the color and size recognition error of facets and the difference for FGSA is mainly due to the stochastic characteristics of surface slope.

3.6. Angular Distribution with Shadowing Effects

Figure 11 shows the backscattering changes when taking shadowing effects into consideration and comparisons are made with results from SDFSM and GOSSA shown in [18,36]. With consideration of shadowing effect, both VV and HH polarized scattering decrease at very large incident angles above $80°$. The comparison with GOSSA and SDFSM shows the difference gap at large incident angles is within 10 dB, where GOSSA is cooperated with Smith shadowing function and SDFSM is cooperated with Z-buffer algorithm. As Arnold [21] reminded us that an ‘average’ radar cross-section can change as much as 10 dB in a 1-minute interval, this is also an acceptable difference with other simulated methods.

3.7. Modulus Distribution

At last, in Figure 12, we show the backscattering modulus distribution in VV and HH at 5 m/s windspeed in 14 GHz at $88°$. It can be seen that it can provide local facet scattering and is probably to simulate ocean SAR imaging. Therefore, ACFM can provide accurate evaluation on backscattering from two-dimensional sea surface and can probably be used in ocean SAR imaging and in other applications.

4. Conclusions

In this article, an angular composite facet model (ACFM) with shadowing treatment was proposed to investigate the backscattering from a two-dimensional sea surface. First, a region division formula for ACFM without shadowing treatment is proposed to classified sea surface facets into specular facets and diffusion facets based on which kind of scattering is dominate from each facet, and the corresponding scattering contribution is calculated either by Kirchhoff approximation (KA) or by small perturbation method (SPM) with a tilting process. Second, an electromagnetic shadowing algorithm based on facets grouping is adopted to handle the shadowing effects in a geometric manner with moderate computation complexity gained. Finally, the backscattering coefficients in different frequency bands, at different wind speeds, and with two different common polarization states are calculated and analyzed. There is a better agreement between ACFM and the experimental data except for some acceptable difference in the L band with VV polarization at large angles. Divided into self-shadowing effect and mutual shadowing effect, shadowing effects on backscattering has been analyzed and compared and it shows that the mutual shadowing effect is more significant than self-shadowing and non-negligible at very large angles, especially at high wind speed. All in all, this new ACFM can attain accurate numerical results and is possible to simulate ocean wave SAR imaging, especially for those electrically large rough sea surfaces and FGSA is suitable for shadowing effects estimation. But there are also some aspects we could do to improve our results further. First, the SPM results employed in this paper were derived from a the first-order solution including no multipath effect. Second, foams, spikes, and ship wakes are common natural features on the sea surface, which will change the sea surface into a layered medium. With these features, refractions and volume scattering should be considered. Lastly, even though it is possible to simulate ocean SAR image, more efforts could be made to gain more accurate scattering results and to study the Doppler effects.