PIERS ONLINE, VOL. 4, NO. 2, 2008 171 Three-scale Radar Backscattering Model of the Ocean Surface Based on Second-order Scattering Ying Yu 1, 2, Xiao-Qing Wang 1, Min-Hui Zhu 1, and Jiang Xiao 1, 1 National Key Laboratory of Microwave Imaging Technology Institute of Electronics, Beijing 100080, China 2 Graduate University of Chinese Academy of Sciences, Beijing 100049, China Abstract Based on the second-order composite surface and stochastic multi-scale models, an ocean surface backscattering model is proposed in this paper, including both the large- /intermediate-/small-scale scattering and the second-order Bragg scattering. Within this frame, we derive a second-order Bragg scattering expression, and develop an analytic solution of hydrodynamic modulation function according to weak hydrodynamic interaction theory. In addition, tilt modulation is simulated through the observation angle transform between nominal and local coordinate systems. The result shows that reasonable agreement between measured and simulated data is obtained, and this model is better than the two models mentioned above. 1. INTRODUCTION In recent years, airborne and spaceborne imaging radars have received considerable attentions in the area of ocean observation. But it is expensive and difficult to obtain accurate data in the measurements due to complicated conditions on the ocean surface. Therefore, a number of radar backscattering models are given, which have been important aroaches in the remote sensing of the ocean. Before 1990s, many physical models of microwave backscatter from the ocean surface have relied on Kirchhoff scattering, Bragg scattering, two-scale and composite surface theories [1 3]. After 1990s, more and more accurate models have been proposed based on the fruit of previous ones. One of them is the improved composite surface model proposed by Romeiser et al., [4], which considers second-order Bragg scattering. And another one is the stochastic multi-scale model proposed by Plant et al., [5], which divides ocean surface waves into large-/intermediate-/smallscale waves continuously. In this paper, the model divides ocean surface into intermediate-small-scale and large-scale spectra. Within this frame, the second-order Bragg scattering expressions are derived and an analytic solution of hydrodynamic function is developed according to weak hydrodynamic interaction theory. In addition, tilt modulation is implemented through the observation angle transform between nominal and local coordinate systems. Finally, a comparison between simulated and measured data is made, which shows well performance of the model. 2. THREE-SCALE MODEL BASED ON SECOND-ORDER SCATTERING 2.1. Large-/Intermediate-small Scale Backscatter Cross Section If only considering single scattering, backscatter cross section for large scale waves is computed by traditional Kirchhoff method [6]: l = exp { 4ε is kz 2 } ke 2 f pq 2 4kz 2 + kx 2 + ky) 2 16kzSS) 4 1/2 exp { Syy kx 2 + S xx ky 2 )} 2S xy k x k y 8 SS k 2 z where k e is transmitted wavenumber, θ i is incidence angle, ε is is mean square of intermediate-small scale wave height, SS is slope variance of large scale waves, S xx, S yy are mean square of large scale slopes in x and y directions, f pq is polarization coefficient [6], p = h s, v s, q = h i, v i denote received and transmitted polarization modes respectively. And backscatter cross section for intermediate-small scale waves is obtained from integral equation: is = k2 e 4π exp [ 4kz 2 ] Γ pq 2 exp [j k B ) x] { exp [ 4kzϕ 2 is x) ] 1 } dx 2) where ϕ is x) is correlation function of intermediate-small scale waves, k B is Bragg wavenumber, Γ pq is polarization coefficient [6]. Because ϕ is x) includes intermediate and small scale waves, is also includes traditional Bragg scattering besides intermediate scale waves scattering. 1)
PIERS ONLINE, VOL. 4, NO. 2, 2008 172 2.2. Second-order Bragg Backscatter Cross Section For a small facet which is tilted with respect to a horizontal reference plane, one obtains: pq0 = 4πke 4 cos θi 2 Γ pq 2 [ψ kb ) + ψ k B )] = T s p, s n ) [ψ k B ) + ψ k B )] 3) where ψk B ) denotes Bragg wave spectrum, s p, s n denote large scale waves slopes parallel and normal to the radar look direction respectively, T s p, s n ) denote tilt modulation term. The Taylor expansion of T s p, s n ) with respect to slope s p, s n ), which only keeps up to second order, reads as follow: T s p, s n ) T 0) + T 1p + T 1n + T 2 + T 2 + T 2pn 4) where T s p = T 1p, 1 2 T 2 s s 2 2 p = T 2, 2 T p s n s p s n = T 2pn. When considering hydrodynamic modulation, wave spectrum turns as: where ψ h = ψ/ψ 0 denotes relatively change rate of spectrum. Do the same thing to ψ as to T s p, s n ), we obtain: ψ = ψ 0 1 + ψ h ) 5) ψ ψ 0) + ψ 1p + ψ 1n + ψ 2 + ψ 2 + ψ 2pn + ψ 0) ψ h + ψ 1p ψ h + ψ 1n ψ h 6) where ψ0 s p = ψ 1p, 1 2 ψ 0 2 s s 2 2 p = ψ 2, 2 ψ 0 p s n s p s n = ψ 2pn. We insert Equation 4) and 6) into 3), and compute expectation values in a cell: pq = pq+ + pq 0) + + 0) + 2) + + 2) 0) + = ψ k B) T 0) 2) + = tt+ + tt+ + pn tt+ + th+ + th+ = ψ kb ) T 2 + T 1p ψ 1p + T 0) ψ 2 th+ = T 0) ψ1p ψ h + ψ 0) ψ h T 1p = ψ kb ) T 2 + T 1n ψ 1n + T 0) ψ 2 th+ = T 0) ψ1n ψ h + ψ 0) ψ h T 1n pn = ψ kb ) T 2pn + T 1n ψ 1p + T 1p ψ 1n + T 0) ψ 2pn 7) Here represents second-order contributions associated with surface slopes parallel to the azimuthal radar look direction, symbol + represents Bragg waves traveling away from the antea, the rest are named by analogy. The second-order hydrodynamic modulation terms are neglected in this paper the same as in literature [4]. Part of second-order scattering terms is derived as follow, the rest can be deduced by analogy: {1 tt+ = 2 ψ k B) 2 T s 2 + T ψ p s p k + ψ ) φ [ + 1 2 T 0) 2 ) ψ kb 2 ) ] k 2 + 2 ψ φb 2 φ 2 +T 0) 2 ψ k φ + 1 2 T 0) ψ 2 k B k s 2 + ψ 2 )} φ B p φ s 2 dkdφk 3 cos 2 φ φ 0 ) ψ k) 8) p T th+ = dkdφψ h k B ) ψk B ) k 2 cos φ φ 0 ) ψk) [ ψ +T 0) dkdφ ψ h k B ) k + ψ )] φ k 2 cos φ φ 0 )ψk) 9) where ψ h k B ) denotes hydrodynamic modulation term of Bragg waves by intermediate scale waves. As we can see, expressions 8) and 9) are consistent with 7).
PIERS ONLINE, VOL. 4, NO. 2, 2008 173 3. HYDRODYNAMIC AND TILT MODULATIONS 3.1. An Analytic Solution of Hydrodynamic Modulation According to weak hydrodynamic interaction theory, the action balance equation reads: dn = dt t + dx dt x + dk ) dt k N = S x, k, t) 10) Nx, k, t) = Ψx, k, t) ρω 0k) 11) k where N is the action spectral density of the wave packet, S is a source function. function in this model is a nonlinear form: ) S k, x, t) = µn 1 NN0 An analytic solution of the action balance equation is derived as follow: δq x, k, t) Q 0 = The source 12) { j [k u K, ωc )] K k Q 0 ) jω c µ + j c g + U 0 ) K exp [j K x ω ct)] + c.c.} dkdω c 13) where Qx, k, t) = 1/Nx, k, t), Q 0 = 1/N 0, µ is relaxation rate, c g is group velocity of the wave packet. 3.2. Tilt Modulation As in Figure 1, according to the transformation between nominal and local coordinate systems, local observation angles are obtained: θ = atg s p ) δ = atg [ s n cos θ] θ i = a cos [cos θ + θ i ) cos δ] ) sin ϕ θi sin δ sin θ + cos θ i sin δ cos θ i = atg sin θ cos θ + cos θ i sin θ θ i represents local incidence angle, ϕ i is local azimuth angle. z z' k e θ i θ δ y x Figure 1: Nominal and local coordinate system. 4. MODEL RESULTS In this section, we compute average cross sections of different parameter sets. And we compare them with the measured data in the literatures cited. The result shows well agreement with the measured data. Figure 2 compares model results with data collected from an airship by Plant et al., [5] as a function of azimuth angle. It shows well agreement between them quantitatively, and suggests this model is not only fit for scattering of intermediate incidence angles but also small incidence angles. The reason lies in that the model considers not only traditional Bragg scattering but also intermediate and large scale scattering.
PIERS ONLINE, VOL. 4, NO. 2, 2008 174 Figure 2: Average cross sections at various azimuthal angles. Frequency: 14 GHz; incidence angle: 10 degree; wind speed: 8 m/s). Figure 3: Average cross sections at various wind speeds. Frequency: 5.3 GHz; VV polarization; incidence angle: 45 degree). Figure 3 shows model predictions and data from Romeiser et al., [4] for cross sections versus wind speed in upwind, downwind and crosswind directions. The fit of all predictions to the data is rather good. As we can see, the magnitudes of backscatter cross sections from the bottom up are in crosswind, downwind and upwind directions. Figure 4: Azimuthally averaged cross sections at various frequencies. Incidence angle: 30 degree; wind speed: 10 m/s). Figure 5: Upwind/crosswind ratio at various frequencies. Parameters are the same as Figure 4). Figure 6: Upwind/downwind ratio at various frequencies. Parameters are the same as Figure 4).
PIERS ONLINE, VOL. 4, NO. 2, 2008 175 Figures 4, 5, 6 compare measured and modeled cross sections, upwind/crosswind ratio and upwind/downwind ratio for the two like-polarizations. The results show fits of the model to the data [7] are reasonably good. And the model predictions are better than those in literature [5]. It is because we consider second-order scattering here which is not included in [5]. 5. CONCLUSIONS The radar backscattering model in this paper is better than those mentioned in [4] and [5]. To explain it theoretically, our model considers not only traditional Bragg scattering but also second-order scattering. Besides Bragg scattering, intermediate and large scale waves scattering is considered as well. Therefore, this model can predict the backscatter characteristics for small to intermediate incidence angles, frequencies from L to Ku band, wind speeds up to 20 m/s. REFERENCES 1. Wright, J. W., A new model for sea clutter, IEEE Trans. Anteas Propag., Vol. 16, 217 223, 1968. 2. Holliday, D., Resolution of a controversy surrounding the Kirchhoff aroach and the small perturbation method in rough surface scattering theory, IEEE Trans. Anteas Propag., Vol. 35, 120 122, 1987. 3. Plant, W. J., A two-scale model of short wind generated waves scatterometry, J. Geophys. Res., Vol. 91, 10,735 10,749, 1986. 4. Romeiser, R. and W. Alpers, An improved composite surface model for the radar backscattering cross section of the ocean surface 1. Theory of the model and optimization/validation by scatterometer data, Journal of Geophysical Research, Vol. 102, No. C11, 25,237 25,250, 1997. 5. Plant, W. J., A stochastic, multiscale model of microwave backscatter from the ocean, Journal of Geophysical Research, Vol. 107, No. C9, 3120, doi:10.1029/2001jc000909, 2002. 6. Fung, A. K., Microwave Scattering and Emission Models and Their Alications, Artech House, Boston London, 1994. 7. Unal, C. M. H., P. Snoeij, et al., The polarization-dependent relation between radar backscatter from the ocean surface and surface wind vector at frequencies between 1 and 18 GHz, IEEE Trans. Geosci. Remote Sens., Vol. 29, No. 4, 621 626, 1991.