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Investigation on the scattering from one-dimensional nonlinear fractal sea surface by second-order small-slope approximation.

1. INTRODUCTION

The study of electromagnetic (EM) scattering from sea surfaces is an old but vigorous research realm. Stochastic waves can be used to form a rough sea surface, and has been widely applied to the research areas such as oceanic surveillance, target detection [1-8], remote sensing and so on [9-15]. But the fractal sea surface is also a very good tool for characterizing the natural sea surface as a result of its semi-stochastic and semi-periodic features [16,17]. So far, a lot of fractal sea surface models have been applied to the study of EM scattering from rough surfaces [18-25]. Most of the models are linear sea surfaces, and the Kirchhoff approximation (KA) is usually employed to analyze the EM scattering problems.

However, the real surface is usually nonlinear and the EM scattering of the nonlinear sea surface might have a big difference from the linear surface, especially for Doppler spectrum. A nonlinear fractal sea surface is developed in [24,26] by applying the two-scale method of Longuet-Higgins and Stewart and the backscattering coefficients discrepancies between linear and nonlinear models are observed. Yet, there is no more information in [24, 26] about the discrepancies of the time-varying signals, and the nonlinearity of sea surface might have great effect on the time-dependent signals such as Doppler spectrum according to some recently published researches about the scattering from nonlinear sea surfaces [27-29], so it is necessary to establish another nonlinear fractal sea model, which should be able to carry more signals scattering from rough surfaces.

A one-dimensional nonlinear fractal sea surface model has been established based on the narrow-band Lagrange model in this paper, in which the vertical and horizontal skewnesses are both considered [30-32]. In fact, this model has been used to study the Doppler spectrum of sea surface by Wang et al. in [32], but the method used in [32] is two-scale method, which can not calculate the scattering field directly from linear and nonlinear sea surfaces. Accounting for this problem and the shortages of KA, the method of SSA-II is employed to calculate the scattering field of fractal sea surface in this paper. The SSA-II method is developed by Voronovich [33,34] and has been widely applied to analyze the EM scattering from sea surfaces [27, 35], which can reach an enough accurate electromagnetic description.

We would like to exhibit the influence of nonlinearity on the EM scattering from fractal sea surface in this paper. In Section 2, the one-dimensional nonlinear fractal sea surface is established by using the model of narrow-band Lagrange. Section 3 presents the model of SSA-II scattering from one-dimensional fractal sea surface detailed, and the results of backscattering, bistatic scattering and Doppler spectrum of linear and nonlinear fractal sea surface are displayed in Section 4. Some conclusions are presented in the final section.

2. NONLINEAR FRACTAL SEA SURFACE MODEL

A linear fractal sea surface model is represented as follows [23,25, 36]:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1)

where [sigma] is the standard deviation of the sea wave amplitude, C the normalization constant, s the fractal dimension (1 [less than or equal to] s [less than or equal to] 2), [K.sub.0] the fundamental spatial wavelength, b (b > 1) the scale factor, V the observer platform velocity, [[omega].sub.n] the angular frequency, and [N.sub.f] the number of sinusoidal components.

The phase [[phi].sub.n](t) are stochastic and are set according to the following relationship:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2)

where [[phi].sub.n], [[psi].sub.n](t) are independent random variables, which are uniformly distributed between [-[pi], [pi]]. [[psi].sub.n](t) are time-varying. In order to ensure spatial correlation of fractal sea surface varying with time and calculate the Doppler spectrum, [[psi].sub.n](t) are set time invariant in a short time interval.

The linear fractal sea surface can be rewritten as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3)

where [A.sub.n] = [sigma]C[b.sup.(s-2)n], [K.sub.n] = [K.sub.0][b.sup.n], [W.sub.n] = [[omega].sub.n] - [K.sub.n]V, [[chi].sub.n] = [[phi].sub.n] - [pi]/2. Then we can apply the Lagrange model to establish the nonlinear fractal sea surface conveniently. For deep water, the nonlinear fractal sea surface can be written as [30-32]

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4)

where [[alpha].sub.m] = [xi]/[W.sup.2.sub.m] and [xi] is the parameter that donates the relation between the horizontal acceleration of water particles and vertical displacement and more detailed description can be find in [31,32]. [K.sub.ave] is the average wavenumber defined as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (5)

And also, the horizontal displacement of water particles can be expressed as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (6)

The nonlinear fractal sea surface can be formed from (4) and (6). Figure 1 shows the linear and nonlinear fractal sea surface of wind speed 5m/s. Figure 1(a) shows the comparison of linear and nonlinear sea surface, and Figure 1(b) shows the horizontal and vertical skewness of nonlinear sea surface. For simplicity, the platform velocity is set V = 0, and [xi] = 0.4 is the same as in [32]. The fractal dimension s = 1.1, scale factor b = 1.2 and [N.sub.f] = 30. K = [0.877.sup.2]g/u and C = 0.124[u.sup.2]/4 x 1.62. From Figure 1, we can clearly see that the nonlinear sea surface will become steeper at wave crests and will become more flat at wave troughs.

3. SSA-II MODEL SCATTERING FROM ONE-DIMENSIONAL FRACTAL SEA SURFACE

In this section, the SSA-II model is employed to study the scattering of the fractal sea surface. The geometry of the scattering process is illustrated in Figure 2. [[theta].sub.i] and [[theta].sub.s] denote the incident angle and scattered angle. [k.sub.i] is the incident wave vector and [k.sub.s] is the scattering wave vector. They can be decomposed into horizontal and vertical components respectively.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (7)

Suppose a tapered wave is incident on the sea surface and can be expressed as [37]

[E.sub.i](x, f) = G(x, f, [[theta].sub.i]) exp[-jk(x sin [[theta].sub.i] - f cos [[theta].sub.i])] (8)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (9)

where k donates the wavenumber of incidence wave, G(x, f) is the taper function and v is the parameter that controls the beam waist.

The scattering amplitude of SSA-II can be expressed as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (10)

where [P.sub.inc] is the incident wave power, it can be expressed as

[P.sub.inc] = [integral] [[absolute value of [E.sub.i](x, 0)].sup.2] dx (11)

B and M can be found in [35], are dependent on configuration angles, the polarization, and the complex permittivity of sea surface. For simplicity, the expressions of B and M are given in Appendix A. F([zeta], t) is the Fourier transform of the sea surface elevation

F ([zeta],t) = [1/(2[pi])] [integral] f (x, t) exp (-j[zeta]x)dx (12)

For the nonlinear fractal sea surface, the integral variables x in (10) should be replaced by X = x + h(x, t) due to the horizontal displacement. Then the Jacobian J of the transformation from x to X is utilized to accomplish this change of integral variables. So Equation (10) should be rewritten as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (13)

where J(x,t) = 1 + [partial derivative]h(x,t)/[partial derivative]x. Then the average NRCS can be expressed as

[[sigma].sub.SSA-II] = 4[pi][q.sub.i][q.sub.s]<[[absolute value of S([k.sub.i], [k.sub.s]; t)].sup.2] (14)

The nonlinearity affects not only the NRCS of sea surface, but also the Doppler spectrum of the sea clutter. And also, the Doppler analysis is a much more precise and sensitive tool for evaluating the validity of the scattering model than the NRCS, so the Doppler spectrum of sea clutter is of great interest for the remote sensing from the sea surface.

The Doppler spectrum is typically defined as the power spectral density of the time-evolving scattering amplitude and its expression is [38]

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (15)

The angle bracket denotes the ensemble average over much surface realizations. T is the duration of the sea surface evolution.

4. NUMERICAL RESULTS

4.1. Comparison of NRCS

In this section, the NRCS given by Equation (14) of wind speed 5m/s is simulated, the frequency is 1.2 GHz, and the complex relative permittivity is [epsilon] = 73.5 + j61.0. The size of the sea surface is [L.sub.x] = 64 m, the sample points is 2048, and the sample interval is [lambda]/8. The parameter v is chosen to be [L.sub.x]/6. Each NRCS is obtained over 100 samples of sea surfaces.

The average backscattering NRCS versus incidence angles from linear and nonlinear surfaces is shown in Figure 3. It is seen that, the average NRCS of nonlinear surfaces is larger than its linear surface. At small incidence angles, the discrepancies are not large, and the discrepancies become larger with the increase of incidence angles, which indicates the nonlinearity is very important for sea surface scattering at low grazing angles. The differences value (DV) of NRCS between linear and nonlinear surfaces at incident angles of 9[degrees] and 81[degrees] are calculated, and the corresponding absolute percentage error (APE) are also included in Table 1. For bistatic case, the average NRCS versus scattering angles for incidence angles of 0[degrees] and 45[degrees] are shown in Figure 4. It can be seen from Figure 4, the discrepancies between linear and nonlinear surface are not large near the specular direction and the discrepancies become larger as the scattering angles departing from the specular direction. The corresponding DV and APE of NRCS for different scattering angles are listed in Table 2.

4.2. Doppler Analysis

Compared with NRCS, the Doppler spectrum of sea surface backscattered signals is a more precise and sensitive tool for detecting to changes in the fluid motion. For deep water, the phase velocity of Bragg scattering can be written as

[f.sub.B] ([[theta].sub.i]) = [1/2[pi]] [square root of (2[g.sub.0] k sin [[theta].sub.i])] (16)

where [g.sub.0] is the gravity acceleration. It is obvious that [f.sub.B] ([[theta].sub.i]) is determined by the incidence angle and incidence frequency. However, the frequency shift and Doppler spectrum bandwidth of the measured Doppler spectrum is usually larger than the simulations from linear sea surface, which is different from the Bragg theory. So it is necessary to consider the nonlinearity of sea surface.

In order to measure the characters of the Doppler spectrum, the Doppler shift [f.sub.s] and the bandwidth of the Doppler spectrum [f.sub.w] are employed to indicate the features of the linear and nonlinear fractal sea surfaces. Their expressions are given in [39]

[f.sub.s] = [[integral]fS[D.sub.op] (f)df]/[[integral][S.sub.Dop] (f)df], [f.sup.2.sub.w] = [[integral] [(f - [f.sub.s]).sup.2] [S.sub.Dop] (f) df]/[[integral][S.sub.Dop] (f) df], (17)

Figure 5 shows the Doppler spectrum for different incidence angles of a wind speed of 5 m/s, and Figure 7 shows the Doppler spectra of a larger fractal dimension s = 1.4 for 5m/s and 7m/s at [[theta].sub.i] = 80[degrees]. For 5 m/s, the size of the sea surface is the same as above [L.sub.x] = 64 m, and the size of sea surface is [L.sub.x] = 128 m for 7 m/s. The time step is 0.02 s, and each spectral realization is performed on 128 samples. The average spectrum is obtained from 100 realizations. It can be seen from Figure 5, the Doppler spectra for linear and nonlinear fractal sea surfaces coincide with each other at small incident angles. When the incident angle increases, the bandwidths of Doppler spectra firstly increase and then decrease both for linear and nonlinear surfaces. This effect is clearly observed in Figure 6 where the corresponding Doppler spectra bandwidths are shown. These are similar with the results of stochastic wave sea surfaces in [27, 40]. In Figure 6, if the largest incident angle, for example [[theta].sub.i] = 80[degrees], is considered, it can be seen that the bandwidths of nonlinear surfaces can reach as large as 2.4 Hz while the bandwidths of linear surfaces are just about 1.0Hz. This is due to the fact that the nonlinear-wave components propagate faster than the linear-wave components.

Compared with the result s = 1.1, the Doppler spectrum has larger bandwidth in Figures 7(a) and (b) for wind speed 5 m/s, s = 1.4 at [[theta].sub.i] = 80[degrees], which is caused by the increased roughness of fractal sea surface. The Doppler shifts and bandwidths corresponding Figure 7 are listed in Table 3. From Table 3, we can see that the nonlinearity has a large influence on the bandwidths, while the influence on the Doppler shifts is little for wind speed 5 m/s. And the nonlinearity not only has a large influence on the bandwidths, but also has a great influence on the Doppler shifts under wind speed 7 m/s. And also, he peak frequencies in Figures 7(c) and (d) for 7 m/s can reach relative large values (HH -7.2 Hz, VV -7.2 Hz), while the values (HH -3.3 Hz, VV -3.0Hz) in Figures 7(a) and (b) for 5 m/s is smaller. This is attributed to the fact that the nonlinear fractal sea surface corrects the phase velocities by adding the horizontal and vertical skewness, and the influence of nonlinearity increases with the increasing of wind speed.

5. CONCLUSION

In this paper, a one-dimensional nonlinear fractal sea surface model has been established based on the narrow-band Lagrange model. Analyzed by SSA-II, the NRCS and Doppler spectrum of linear and nonlinear fractal sea surface is calculated. The NRCS of nonlinear sea surface is larger than the linear sea surface for backscattering, especially for large incidence angles. And for bistatic case, the result of nonlinear sea surface is also larger than the linear fractal sea surface, which is characterized as the discrepancies being small near specular direction, while the discrepancies becoming larger as the scattering angles departing from the specular direction. These indicate the nonlinearity of sea surface has important influence on the scattering echoes of sea surface, especially for large incidence angles and large scattering angles. For Doppler spectrum, at small incidence angles, the differences between nonlinear and linear surface is not obvious. As the increase of incidence angles, the nonlinearity of sea surface effects greatly enhances the Doppler shift and the Doppler spectrum bandwidth. These are attributed the fact that the nonlinear-wave components propagate faster than the linear-wave components and the nonlinear fractal sea surface corrects the phase velocities by adding the horizontal and vertical skewness. All the result shows the validity of this nonlinear model.

The scattering results in this paper from the nonlinear fractal sea surface model can help to better understand the scattering from fractal sea surface, especially for the time-dependent signals, though the fractal sea surface is limited to one-dimension. And also, this nonlinear fractal sea surface model might be applicable to establish a two-dimensional nonlinear fractal sea surface, which can potentially provide a higher precision RCS and Doppler spectrum in studies of electromagnetic scattering from the sea surfaces.

ACKNOWLEDGMENT

The authors would like to thank the Fundamental Research Funds for the Central Universities, the National Natural Science Foundation of China under Grant No. 60871070, and the Foundation of the Science and Technology on Electromagnetic Scattering Laboratory to support this kind of research.

APPENDIX A.

The general expressions for kernel functions B and M based on vertically and horizontally polarized waves are given in [35]. The first order B can be expressed as:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (A1)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (A2)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (A3)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (A4)

The second order M can be expressed as:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (A5)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (A6)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (A7)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (A8)

where [??] and [[??].sub.0] donate the horizontal wavevector of scattered wave and incident wave, respectively. In this paper, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] just need to be replaced by [k.sub.s], [k.sub.i], and [zeta]. [epsilon] is the complex relative permittivity of sea water, and c is the speed of light. [q.sup.(1,2).sub.k] and [q.sup.(1,2).sub.0] are the vertical components of the wavevectors, and can be expressed as:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (A9)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (A10)

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G. Luo and M. Zhang *

School of Science, Xidian University, Xi'an 710071, China

* Corresponding author: Min Zhang (mzhang@mail.xidian.edu.cn).

Received 27 August 2012, Accepted 25 October 2012, Scheduled 1 November 2012

Table 1. DV and APE of NRCS at incident angles of
9[degrees] and 81[degrees] in Figure 3.

Incident       9[degrees]          81[degrees]
angles

           DV (dB)   APE (%)   DV (dB)   APE (%)

HH         0.6059     3.999    12.0276   30.0538
VV         0.6092     4.02     12.0237   129.3127

Table 2. DV and APE of NRCS at different scattering angles for
0[degrees] and 45[degrees] incident angle in Figure 4.

0[degrees]    Components   Scattering angle    Scattering angle
incident                    -89[degrees]        -1[degrees]
angle

                           DV (dB)   APE (%)   DV (dB)   APE (%)

                  HH       2.6215     6.463    0.0006     0.004
                  VV       2.3813    10.944    0.0021     0.014

45[degrees]   Components   Scattering angle    Scattering angle
incident                   -89[degrees]        45[degrees]
angle

                           DV (dB)   APE (%)   DV (dB)   APE (%)

                  HH       5.9039    11.993    0.3369     2.342
                  VV       5.5022    23.089     0.334     2.532

Table 3. The bandwidths of the Doppler spectra and Doppler
shifts in Figure 7.

Bandwidths      S = 1.4, 5 m/s       S = 1.4, 7m/s

             Linear    Nonlinear   Linear    Nonlinear

HH (Hz)      1.1101     3.6702     1.8056     4.1497
VV (Hz)      0.9481     3.1991     1.4822     4.3045
Shifts       Linear    Nonlinear   Linear    Nonlinear
HH (Hz)     -3.4497    -3.107     -4.7148    -7.8185
VV (Hz)     -3.3232    -3.2652    -5.2797    -8.0025
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