1 INTRODUCTION

The technique of measuring the directional height spectrum of ocean wind waves from real-aperture radar data, recorded by relatively simple, scanningbeam microwave radar systems, was first presented by Jackson (1981). The basic principle of the ocean wave spectrometer is that, at low incidence angles, the change in the backscattering cross section of the ocean surface linearly varies with the slope of long ocean waves. The principle was validated by airborne radar experiments, namely ROWS (Jackson, 1985), RESSAC (Hauser, 1992), and STORM (Mouche et al., 2005). The CFOSAT (China-France Oceanography Satellite) mission, whose launch is planned for 2018, will carry two radar payloads that will monitor wind and waves over the oceans. One of these two radar instruments, designed and developed by the French Space Agency CNES, is called SWIM (Surface Waves Investigation and Monitoring) and was designed to measure directional ocean wave spectra at a scale of about 70×70 km² along a 180-km-wide swath. The instrument consists of a rotating antenna that will transmit and receive signals in six distinct elevation directions, namely the nadir and incidence angles of 2°, 4°, 6°, 8° and 10°. The selected sun synchronous orbit (characterized by an altitude of 514 km and a 13-day cycle) ensures nearly full coverage of the world’s oceans.

According to Jackson’s principle of measuring the directional height spectrum of the sea surface with near vertical incidence (incidence angle less than 15°), microwave backscatter from the sea surface occurs through quasi-specular reflections from wave facets oriented normal to the radar system’s line of sight. Therefore, the radar cross-section is linearly related with the slope probability density function (pdf) of the wave components that are longer than the diffraction limit (about three times the electromagnetic wavelength (Brown, 1978)). Neglecting the effects of hydrodynamic forcing and wave-wave interactions, which are considered of secondary importance, the sea surface is then treated as a free wave superposition possessing Gaussian statistics. If the large-wave slopes are then assumed to be small compared with the incidence angle and the total root-mean-square surface slope, the modulation can be modeled using a linear “tilt model”. With the tilt model, it is found that the change in the radar backscattering cross-section of the ocean surface linearly varies with the slope of long ocean waves.

An end-to-end simulator (Tison et al., 2009; Enjolras et al., 2009) has been developed to validate the wave spectrum inversion performance of SWIM. A similar simulation tool for SWIM has also been developed in China (Lin, 2010).

Generally, in simulation studies, Gaussian statistics are assumed to describe the sea surface slope and to calculate the normalized radar cross section. However, the works of Cox and Munk (1954) and Longuet-Higgins (1963) have revealed differences from these Gaussian statistics for the sea surface. It is not known whether the non-Gaussian properties of the sea surface will affect the inversion performance of the wave spectrum if following the existing inversion steps and methods.

In this paper, to describe the non-Gaussian properties of the sea surface, the pdf of the sea surface slope is expressed as a Gram-Charlier fourthorder expansion, which is quasi-Gaussian. The modulation transfer function (MTF) is derived for the non-Gaussian slope pdf. The effects of non-Gaussian properties of the sea surface slope on the inversion process and result are then studied using a simulation set for the SWIM. The simulation includes both a direct simulation and a wave spectrum inversion simulation. In the direct simulation, the non-Gaussian properties of the sea surface are taken into account using the quasi-Gaussian sea surface slope pdf. In the inversion simulation, considering and ignoring these non-Gaussian properties of the sea surface, three methods of obtaining the MTF are studied. Furthermore, the inversion performances achieved using the different MTFs are estimated.

The remainder of the paper is organized as follows. Section 2 describes Jackson’s principle of the wave spectrum measurement for the sake of completeness. In Section 3, the MTF for the non-Gaussian sea surface slope is derived. For the wave spectrometer SWIM, forward and inversion simulation methods are described and simulation results are presented in Section 4. A conclusion is presented in Section 5.

2 JACKSON’S PRINCIPLE OF A WAVE SPECTRUM MEASUREMENT

For near vertical incidence (i.e., an incidence angle less than 11° for SWIM), microwave backscatter is dominated by quasi-specular reflections:

where θ is the incidence angle from the nadir, φ is the azimuthal angle, and ρ is the diffraction-modified normal incidence Fresnel reflectivity; for the Ku band of SWIM, ρ = 0.61. P(tanθ, 0) is the slope pdf of wave components that are longer than the diffraction limit, satisfying the specular condition for backscatter, ∂ζ/∂X = tanθ, ∂ζ/∂Y = 0, and X and Y are in the range and azimuth directions, respectively.

The backscatter cross section of a small patch of the sea surface having area A is given by σ = σ⁰A, where the normalized radar cross section σ⁰ is assumed to be the average σ⁰ of the sea surface in a tilted reference frame. If the tilt ε due to the long wave is small, then the fractional variation of the radar crosssection due to the tilt ε is δ σ/σ = α·∂ζ/∂X, where ∂ζ/∂X is the slope of the large-scale wave. Without any assumption of the backscattering mechanism or the surface slope pdf, α(θ) is called the MTF and is expressed as

For the specular point backscattering mechanism,

The fractional range-reflectivity modulation seen by the radar is δσ/σ averaged laterally across the beam:

where G is assumed to be a Gaussian-shaped azimuth gain pattern. The signal fluctuation spectrum with noise is defined by where the angle brackets denote the ensemble average. If we consider the case of very large footprints (e.g., 18×18 km² for the SWIM 10° beam), then the directional modulation spectrum with the noise removed, P_m, can be subtracted. P_m is proportional to the directional slope spectrum: where F is the two-sided, polar-symmetric height spectrum.

To invert F from P_m, it is important to determine the coefficient α(θ) in Eq.6. Jackson et al. (1985), by assuming the isotropic Gaussian slope pdf of the sea surface P(tanθ, 0) = 1/(Πv)·exp(-tan²θ/v) used Eq.2 to derive α(θ) as

where v is the slope variance. If the wind speed U is known in advance, v can be obtained from the empirical relation between U and v. Here, because the isotropic Gaussian slope pdf is assumed, the values of α(θ) are the same for any azimuthal angle φ.

3 MODIFIED EXPRESSION OF α(θ, φ)

Under the assumption of a Gaussian slope pdf, a sea surface is represented as the sum of independent components. However, the phase between these different wave elevation and slope components may locally exhibit correlations due to local strong nonlinear interactions. This modifies the Gaussianity of the distribution. A slope pdf that is close to Gaussian but with a non-Gaussian property is usually expressed as a Gram-Charlier fourth-order expansion. Cox and Munk (1954) estimated the parameters (hereinafter referred to as CM parameters) in the Gram-Charlier expansion from the glitter pattern of reflected sunlight in airborne photographs, whereas Breon and Henriot (2006) estimated the parameters (hereinafter referred to as BH parameters) from spaceborne observations of ocean glint reflectance. The non-Gaussian slope pdf is consistent with the analysis of σ⁰ values at nadir as measured by satellite altimeters using specular point (geometrical optics) theory (Chapron et al., 2000).

Here, a Gram-Charlier fourth-order expansion is used to express the non-Gaussian slope pdf:

where Γ_{u, c} = ζ_{u, c}/σ_{u, c} denotes the slopes of upwind and crosswind ζ_u and ζ_c normalized by the root-meansquare slope (σ_u and σ_c). Here, ζ_u = ζ_xcosφ₀ + ζ_ysinφ₀ and ζ_c = -ζ_xsinφ₀ + ζ_ycosφ₀, where φ₀ is the azimuth angle of the upwind direction, and ζ_x = tanθcosφ and ζ_y = tanθsinφ, where ζ_x and ζ_y are the slopes of the specular point of the radar.

The parameters c₁₂ and c₃₀ are skewness factors, whereas c₀₄, c₂₂ and c₄₀ are peakedness factors.

If the azimuthal reference axis φ is set in the upwind direction, then φ₀ is equal to 0°. We can also express Eq.8 with θ and φ as

If ignoring the non-Gaussian terms, then Eq.8 becomes the Gaussian distribution

Substituting into Eq.3 the derivative of Eq.9, an alternative expression of α(θ, φ) is obtained for the non-Gaussian slope pdf:

In the Gaussian case, Eq.11 becomes

4 SIMULATION METHOD AND RESULT

Our simulation assesses the effects of non-Gaussian statistics of the sea surface slope on σ⁰ received by the spectrometer, and the MTF α(θ, φ) and the inversion performance of the wave spectrum when non-Gaussian features are neglected. The simulation configuration refers to the SWIM instrument as presented in Table 1.

The simulation comprises a direct simulation step and an inversion step. Direct simulation includes a simulation of the sea surface, computation of the radar signal integrated over several samples, and the addition of speckle and thermal noise perturbations. Inversion simulation includes calculation of the signal modulation and spectrum, correction for perturbing effects (mean level of thermal noise and modulations due to speckling), and averaging of modulation spectra over an azimuth angle of 15°.

4.1 Direct simulation

The simulated surfaces cover an area of 18 km×18 km, which is discretized by 1 024×1 024 points. Here, satellite movement and the displacement of the radar footprint over the integration time are ignored. The horizontal sampling ΔX = 17.5 m, which is about six times the radar’s horizontal resolution. The corresponding maximum wavenumber is k_max = 0.18 rad/m and the wavenumber resolution dk = 2π/Lx = 1.745×10^-4 rad/m. The slopes of the largescale waves are efficiently generated using the fast Fourier transform (Yang et al., 2002).

The sea-state spectra used in our simulation correspond to three cases, namely, a fully developed sea, developing sea and swell state. Table 2 gives the analytical forms of sea wave height spectra and conditions chosen in each case.

In the direct simulation step, α(θ, φ) of Eq.11 is substituted into Eq.4 to generate σ⁰ modulated by the large-scale waves. In Eq.11, the parameters σ_u, σ_c, c₁₂, c₃₀, c₀₄, c₄₀ and c₂₂ are the same as those in Eq.8, which is a non-Gaussian slope pdf. Thus, the non-Gaussian nature of the sea surface is included in the mean trend of simulated σ⁰.

We now discuss how to set the above non-Gaussian parameters in the direct simulation. So far, there are three sets of parameters of non-Gaussian slope pdfs for the sea surface. Two sets were obtained from airborne and spaceborne optical data, namely the CM and BH parameters. The third set of parameters (Chu et al., 2012, hereinafter referred to as Chu parameters) were obtained, employing the same method used by Cox and Munk (1954), from microwave data of precipitation radar by the Tropical Rainfall Mapping project. However, the Chu parameters only include σ_u, σ_c, c₁₂, and c₃₀. For SWIM observing in the Ku band, we use the third set of parameters for σ_u, σ_c, c₁₂, and c₃₀, presented by Chu et al. (2012, Fig. 12). To express them more clearly, we describe Chu’s parameters as linear functions of wind speeds, as shown in Table 3, employing the linear fitting method and using the curves in Chu’s Fig. 12.

Peakedness coefficients in Table 3 are taken directly from CM parameters of a slick sea, because the sea slope parameters obtained from radar measurements are more similar to those of the slick sea slope measured by optical instruments (Chu et al., 2012).

σ⁰ modulated by long waves is generated by calculating m(X, φ) in Eq.4. Speckle noise and thermal noise are added to σ⁰ employing the method used by Hauser et al. (2001). To simulate the radar signal perturbed by speckle noise, we made a random selection from samples of a gamma function of six looks with a mean of m(X, φ)+1 calculated using Eq.4.

4.2 Inversion simulation

In the inversion simulation step, to get the modulation function m(X, φ) from the radar received signal, the mean trend of σ⁰ varying with X is needed. Here, the mean trend of σ⁰ is obtained simply by averaging the received σ⁰ directly. For the range direction of 18 km, which corresponds to 1 024 points, the values of σ⁰ are averaged over 32 adjacent points.

The signal fluctuation spectrum P(k, φ) is then calculated using Eq.5. It is noted that here P(k, φ) includes a noise spectrum. Following the approach of Enjolras et al. (2009), P(k, φ) is expressed as

where P_m(k, φ) is the modulation spectrum, P_th(k) is the thermal noise spectrum, R(k) is the impulse response, approximated as 1 here, P_sp(k) is the speckle spectrum, Δx is the horizontal resolution, N_imp is the number of independent samples, SNR is the signal-to-noise ratio, and L_dis is the number of range bins. When the incidence angle is 10°, we set N_imp = 237, SNR = 2.3 dB, and L_dis = 6.

P_m(k, φ) is then calculated using Eq.5. Furthermore, to get F(k, φ) from P_m(k, φ), the MTF α(θ, φ) needs to be estimated. There are three methods of obtaining the MFT. The first method (method 1) is based on Eq.2, where the MTF is simply obtained from the derivative of the mean trend of σ⁰ versus the incidence angle. In the second and third methods (methods 2 and 3), it is assumed that the wind speed is known in advance. The MFT α(θ, φ) in method 2 is calculated using Eq.12, where the sea surface slope pdf is considered as Gaussian, and the two parameters σ_u and σ_c are taken from Table 3. The MFT α(θ, φ) in method 3 is calculated using Eq.11, where the sea surface slope pdf is considered as non-Gaussian, and all seven parameters, σ_u, σ_c, c₂₁, c₀₃, c₀₄, c₄₀ and c₂₂, are again taken from Table 3.

4.3 Mean trend of σ⁰ used in the inversion simulation

Figure 1a and b show σ⁰ simulated in the upwind and crosswind directions (green thin lines) as functions of the incidence angle ranging 5°-11°, where the JONSWAP wave spectrum is used with a wind fetch of 90 km and wind speed of 13 m/s. The slope variances and skewness coefficients are set according to Table 3, and peakedness coefficients are set as c₄₀ = 0.26, c₀₄ = 0.36, and c₂₂ = 0.1. The rapid fluctuations of σ⁰ received are due to the modulation of the large-scale waves and speckle effects. In the same figure, the black lines are the mean trend of σ⁰ obtained by averaging σ⁰ over 32 points in the range direction, and the blue lines are σ⁰ calculated using Eq.1, where the sea surface slope pdf is assumed Gaussian, as expressed by Eq.10. The red lines are σ⁰ calculated using Eq.1, where the sea surface slope pdf is assumed non-Gaussian, as expressed by Eq.9.

Although each black line is obtained by averaging simulated σ⁰ over some points, it is still relatively smooth because of the high resolution (18 km, 1 024 points) of σ⁰ in the range direction. The mean trend of σ⁰ shown by a black line is used to calculate the MTF in the inversion simulation with method 1.

Figure 1 reveals that the mean trend of simulated σ⁰, shown by the black line, is almost the same as σ⁰ calculated under the non-Gaussian assumption, shown by the red line. This is because, as shown by Eq.1, the mean trend of simulated σ⁰ is proportional to the sea surface slope pdf, which has been used in the direct simulation. As described in Section 4.1, the non-Gaussian sea surface slope pdf is used in the direct simulation. Therefore, the mean trend of simulated σ⁰ should agree with σ⁰ calculated using Eqs.1 and 9 in the case of no noise. As mentioned in Section 4.1, speckle and thermal noise have been added in the forward simulation. However, owing to the averaging of σ⁰ over 32 points, the speckle noise has little effect on the mean trend of simulated σ⁰. Figure 1a and b show that the mean trend of simulated σ⁰ is very close to the theoretical values obtained using Eqs.1 and 9.

The same figure shows that the difference between black and blue lines, which is calculated for the case of the Gaussian assumption, is not negligible. We therefore conclude from Fig. 1 that the mean trend of simulated σ⁰ is proportional to the corresponding sea surface slope pdf.

Figure 2 shows σ⁰ as a function of the azimuth angle for a wind speed of 13 m/s and an incidence angle of 10°. The simulated σ⁰, the mean trend of σ⁰ obtained by averaging over the azimuthal angle range of 3°, σ⁰ calculated with the Gaussian assumption and σ⁰ calculated with the non-Gaussian assumption are plotted with green, black, blue and red lines, respectively. It is seen that for the black line, there is a difference between the upwind and downwind directions of about 0.3 dB, which is almost the same as the case for the red line. Therefore, the mean trend of σ⁰ indeed includes the upwind/downwind asymmetry of the sea surface.

4.4 Effects of non-Gaussian parameters on the mean of σ⁰

In this section, we discuss the effects of non-Gaussian parameters of the sea surface slope pdf, peakedness coefficients and skewness coefficients, on the mean of σ⁰. As shown in Fig. 1, the mean of σ⁰ is almost the same as σ⁰ calculated using Eqs.1 and 8. Therefore, we here obtain σ⁰ using the calculation method rather than through full simulation. Because we only focus on non-Gaussian parameters, for the sake of comparison, the values of σ_u and σ_c are set according to Chu’s parameters in Table 3, which are suitable for the SWIM frequency.

The peakedness coefficients are not included in Chu parameters. For both CM and BH parameters obtained employing the optical method, the peakedness coefficients are modeled independently of the wind speed. Because the sea slope parameters obtained from radar measurements are more similar to those of the slick sea slope obtained using optical instruments, we here only consider peakedness coefficients given as CM parameters of the slick case. The ranges of c₄₀, c₀₄, and c₂₂ are respectively 0.26±0.31, 0.36±0.24, and 0.1±0.05 for the CM slick case. To study the effect of the separate peakedness coefficient on the mean value of σ⁰, we here fi x two of the three peakedness coefficients and set the third parameter as the maximal and minimal values according to the change range of that parameter in the CM model for the slick case.

According to specular point theory, as revealed by Eq.9, in the down/upwind direction, the variation of σ⁰ with incidence angle is related to the peakedness coefficients c₄₀ and c₂₂ and the skewness coefficients c₁₂ and c₃₀ but is independent of the peakedness coefficient c₀₄. Meanwhile, in the crosswind direction, the shape of the mean of σ⁰ is dependent on c₀₄ and c₂₂ and independent of the skewness coefficients. The mean of σ⁰ as a function of the incidence angle is shown in Fig. 3. The effect of c₄₀ on σ⁰ in the upwind direction is shown in Fig. 3a, where c₂₂ = 0.11 and the skewness coefficients c₁₂ and c₃₀ are set using Chu’s parameters for different wind speeds. The blue curve corresponds to c₄₀ = -0.05 and the red curve to c₄₀ = 0.57, while the solid, dotted and dashed lines correspond to wind speeds of 5, 9 and 15 m/s, respectively. The effect of c₂₂ on the average value of σ⁰ in the upwind direction is shown in Fig. 3b, where c₄₀ = 0.26, and c₂₂ = 0.05 and 0.15. The effect of c₀₄ on the average value of σ⁰ in the crosswind direction is shown in Fig. 3c, where c₂₂ = 0.11 and c₀₄ = 0.12 and 0.6.

Figure 3a reveals that a larger value of c₄₀ leads to a lower average σ⁰ for an incidence angle more than approximately 6°. The difference between two curves with different values of c₄₀ increases as the incidence angle increases from 6° to 11°. The sensitivity of mean σ⁰ to c₄₀ thus increases with the incidence angle increasing from 6° to 11°. Figure 3a also reveals that the sensitivity of σ⁰ to wind speed strengthens with the incidence angle increasing from 5° to 11°. Therefore, the change in σ⁰ due to the wind speed increasing from 5 to 15 m/s is greater than the change in σ⁰ due to c₄₀ increasing from -0.05 to 0.57 when the incidence angle is between 5° and approximately 7°, while the change in σ⁰ due to the wind speed is less than the change in σ⁰ due to c₄₀ when the incidence angle is about 11°.

Figure 3b shows that the change in σ⁰ due to the change in c₂₂ is very weak because the possible range of variation in c₂₂ is small. Figure 3c reveals that the effects of c₀₄ on the curve of σ⁰ versus the incidence angle are similar to those of c₄₀ because the range of c₀₄ is similar to that of c₄₀. With increasing c₀₄, the shape of the curve of σ⁰ versus the incidence angle becomes steeper, while σ⁰ at an incidence angle of 5° remains almost constant.

Again according to specular point theory, the difference in mean σ⁰ between the downwind and upwind directions is related to the skewness coefficients c₁₂ and c₃₀ but is independent of the peakedness coefficients. We choose the skewness parameters given in Table 3. The peakedness coefficients are set the same as those in Fig. 1; i.e., c₀₄ = 0.36, c₄₀ = 0.26, and c₂₂ = 0.1.

Figure 4 presents σ⁰ as a function of the azimuthal angle, where the red curve is for the wind speed of 5 m/s and the black curve is for 15 m/s. According to the Chu model, c₁₂ = 0.001 9 and c₃₀ = 0.010 1 for the wind speed of 5 m/s and c₁₂ = 0.023 5 and c₃₀ = 0.096 1 for the wind speed of 15 m/s. The solid curves in the upper part of the figure are for an incidence angle of 6°, whereas the dotted curves in the lower part are for an incidence angle of 10°.

Figure 4 reveals that, for wind speed of 5 m/s, there is a weak upwind/downwind asymmetry at an incidence angle of 6°, whereas the asymmetry is negligible at an incidence angle of 10°; and for wind speed of 15 m/s, there is apparent upwind/downwind asymmetry in σ⁰ values. It is thus clear that the upwind/downwind asymmetry strengthens as the wind speed increases. It is also shown that the upwind/downwind asymmetry of about 0.3 dB at incidence of 6° is slightly weaker than that of about 0.4 dB at incidence of 10°.

4.5 Comparison of α(θ, φ) obtained using the three methods

As described in Section 4.2, there are three methods of obtaining the MFT α(θ, φ). Method 1 based on Eq.2 obtains MTFs simply from the derivative of the mean trend of σ⁰ versus the incidence angle. Method 2 calculates the MFTs using Eq.12, where the sea surface slope pdf is considered as Gaussian. Method 3 calculates the MFTs using Eq.11, where the sea surface slope pdf is considered as non-Gaussian (Eq.9). It is clear that the MTFs obtained using method 3 in the inversion are as the same as those used in the direct simulation.

Figure 1 shows that, for different azimuthal angles, the mean trend of σ⁰ versus the incidence angle is the same as that of σ⁰ calculated using Eqs.1 and 9, where the sea surface slope pdf is assumed non-Gaussian. Therefore, for method 1, the MFTs obtained from the derivative of the mean trend of σ⁰ versus the incidence angle will be suffi ciently accurate, and the same as those calculated using method 3, where MTFs are derived from Eqs.9, 1 and 2. In fact, the MTF obtained using method 1 depends on the type of slope pdf used in the direct simulation, and there is no need for any assumption to be made for the slope pdf. Our direct simulation uses a non-Gaussian slope pdf. The non-Gaussian statistical characteristics of the sea surface will therefore be automatically included in the MTF obtained using method 1.

The following figures present only the MFTs obtained using methods 2 and 3 because MTFs obtained using methods 1 and 3 are almost the same.

For methods 2 and 3, it is assumed that the wind speed is known in advance. Here, the parameters in Table 3 are used in methods 2 and method 3. The slope variances and skewness coefficients vary with the wind speed and the peakedness coefficients are set as c₄₀ = 0.26, c₀₄ = 0.36, and c₂₂ = 0.1. α(θ) in the crosswind direction and α(φ) for an incidence angle of 10° are shown in Fig. 5a and b, respectively. The values of α(θ) and α(φ) obtained using methods 2 and 3 are plotted in red and blue, respectively. The results for wind speeds of 5, 9 and 15 m/s are shown by solid, dotted and dashed curves, respectively.

Figure 5a shows the difference in the shapes and values of α(θ) obtained using methods 2 and 3, in the crosswind direction. The difference results from the peakedness coefficients c₀₄ and c₂₂. Figure 5b shows that values of α(φ) in the downwind direction (φ = 180°) obtained using method 2 under the Gaussian assumption are the same as those in the upwind direction (φ = 0°), whereas there is a difference between the values of α(φ) in downwind and upwind directions obtained using method 3 under the non-Gaussian assumption. The difference, resulting from the skewness coefficients, increases with wind speed. For the same wind speed and incidence angle, the difference in α(φ) between the Gaussian case and non-Gaussian case is greatest in the crosswind direction.

4.6 Comparison of one-dimensional (1D) slope spectrum inversion

As shown in Section 4.5, the three methods give different MTF α(θ, φ) results. Because the MTFs obtained using methods 1 and 3 are almost the same, so only two kinds MTF—one for the case of the Gaussian assumption using method 2 (case 1) and the other for the case of the non-Gaussian assumption using method 3 (case 2)—are used to calculate the 1D slope spectrum S(k, φ)= k²F(k, φ) from P_m(k, φ). For case 1, the non-Gaussian statistics of the sea surface slope are ignored in the MTF of the inversion simulation. For case 2, the non-Gaussian statistics are included in the MTF. S(k, φ) calculated with the MTF obtained using method 1 is considered the same as that calculated for case 2. Here P_m(k, φ) is obtained using a 10° incident beam.

The 1D slope spectra S(k, φ) in the upwind direction calculated for cases 1 and 2 are compared for a developing sea, fully developed sea and swell in Figs.6-8, respectively. The inversion performances for the 1D slope spectrum are assessed using two indexes: the correlation coefficient C of the reference wave slope spectrum and the inverted, and the integrated energy error where k_min = 0.012 6 rad/m, corresponding to a wavelength of 500 m, and k_max =0.120 8 rad/m, corresponding to a wavelength of 52 m. Figures 6-8 are for viewing directions aligned with the wave propagation direction and with 15° azimuthal averages; i.e., within ± 7.5° along the wave direction.

The inversion performances for 1D wave slope spectra of JONSWAP in the wave direction are presented in Fig. 6a-d for wind speeds of 8, 10, 13, and 16 m/s, respectively. Figure 6 shows that the correlation coefficients obtained via the two inversions are the same. When the wind speed is 8, 10, 13 or 16 m/s, the energy error in case 2 is smaller than that in case 1, by 0.96%, 1.50%, 4.30% and 6.07% respectively. The worse inversion performance in case 1 is due to the inaccuracy of the MTF α(θ, φ) in case 1. As shown in Fig. 4, the MTF (blue curves) obtained using method 2 in case 1 deviates from the correct MTF (red curves) used in case 2. Therefore, the non-Gaussian property of the sea surface should be taken into account in 1D wave spectrum inversion.

It is noted that, although MTFs obtained with method 3 and used for case 2 can take the non-Gaussian statistics of the sea surface slope into account, there is a need to know in advance the wind speed and all seven parameters, σ_u, σ_c, c₄₀, c₀₄, c₂₂, c₃₀, c₁₂, in the Gram-Charlier fourth-order expansion of (8). However, these non-Gaussian parameters are not well known for microwave bands. Therefore, crude assumptions have to be made for these parameters. Errors introduced by these assumptions and inaccurate wind speeds may be as important as assuming Gaussian conditions.

For a fully developed sea (using Pierson-Moscowitz spectra) and swell (using Durden and Vesecky spectra), similar results are obtained, as shown in Figs.7 and 8. Figure 7 shows that the inaccuracy arising from not considering the non-Gaussian property of the sea surface increases with the wind speed. Figure 8 shows that, for a swell, the energy error in both two cases, for wind speed U = 5 m/s and signifi cant height H_s = 1 m, is greater than 20%. It is noted that the mission requirement for SWIM is the energy error<20%. It is also shown the imprecision due to not considering the non-Gaussian property of the sea surface increases as Hs decreases.

4.7 Comparison of the inverted two-dimensional (2D) wave spectrum

The comparison of the inverted 2D wave spectrum is shown in Fig. 9. Figure 9a presents the reference JONSWAP wave spectrum with wind speed of 13 m/s, corresponding peak wavelength λ_p = 99.6 m, and main wave direction φ_p =0°, Fig. 9b presents the directional wave spectrum of inversion in case 1, and Fig. 9c presents the directional wave spectrum of inversion in case 2. The difference in results obtained via the two inversions is not easily seen in the 2D wave spectra.

To assess the inversion performance for 2D wave spectra, the signifi cant height H_s is calculated as , and the errors of ΔH_s for case 1 (Gaussian assumption) and case 2 (non-Gaussian assumption) are presented in Fig. 9b and c.

ΔH_s is also calculated for different wind speeds and different sea states, as shown in Figs.6-8. For all considered wind speeds and sea states, the error in H_s when considering the non-Gaussian statistics of the sea surface slope in the inversion is always smaller than that when ignoring the non-Gaussian statistics in the inversion.

However, Fig. 9 shows that the peak wavelength and wave direction are the same in the two inversions of cases 1 and 2. Therefore, ignoring the non-Gaussian statistics of the sea surface slope in the inversion has no effect on the inverted peak wavelength and wave direction.

5 CONCLUSION

The present study investigated the effect of ignoring non-Gaussian features of the sea surface pdf on the inversion of wave spectra by a low-incidenceangle spectrometer.

We first studied the effect of the Gaussian or non-Gaussian nature of the surface on σ⁰ at a low incidence angle in the presence of long waves, and on the tilt MTF that relates long wave slopes to signal modulations. In the non-Gaussian case, the simulations use a Gram-Charlier fourth-order expansion of the slope pdf. Results show that the mean trend of received σ⁰ versus the incident and azimuthal angle is proportional to the slope pdf of the sea surface. The speckle noise has little effect on the mean trend of the received σ⁰ if averaging σ⁰ over tens of points in both incident and azimuthal directions. For the non-Gaussian slope pdf, the sensitivity of σ⁰ to the peakedness coefficient increases with an incidence angle increasing from 6° to 11°, the skewness coefficients lead to the upwind/downwind asymmetry of σ⁰, and the asymmetry strengthens with increasing wind speed and incidence angle. Meanwhile, the mean trend of received σ⁰ may be used to derive the parameters of the sea surface slope pdf, especially non-Gaussian parameters such as the peakedness and skewness coefficients, which are not well known for the microwave band.

Then, for a case simulated with the non-Gaussian assumption, combined with simulation of the SWIM instrumental configuration, we inverted wave spectra using three methods to estimate the tilt MTF. For the first method, where the experimental profi le of σ⁰ versus the incidence angle is used to estimate the tilt MTF, the inverted wave spectra are insensitive to the pdf assumption in the forward simulation. For the second and third methods, where analytical expressions based on Gaussian and non-Gaussian pdfs are used to estimate the tilt MTF respectively, the energy errors of 1D wave spectra and H_s errors of 2D wave spectra inverted using method 2 are larger than those inverted using method 3, for wind waves and swell. This is because method 2 ignores the non-Gaussian features of the sea surface applied in the forward simulation, whereas method 3 includes them in the inversion. The simulation results of 2D wave spectra also show that ignoring the non-Gaussian property of the sea surface in the inversion has no effect on the inverted peak wavelength and wave direction.

It is noted that, although MTFs obtained using method 3 can take non-Gaussian statistics of the sea surface slope into account, there is a need to know in advance the wind speed and all seven parameters in the Gram-Charlier fourth-order expansion of (8). However, these parameters are not well known for microwave bands. Therefore, crude assumptions have to be made for these parameters. Errors introduced by these assumptions and inaccurate wind speeds may be as important as assuming Gaussian conditions. Fortunately, owing to the high resolution of σ⁰ in range and azimuth directions, MTFs obtained using method 1 are as accurate as those obtained using method 3, and the same inversion performance using the MTFs of method 3 can be achieved using the MTFs of method 1. Furthermore, method 1 need not know in advance the wind speed and all seven parameters as input ancillary information. It is thus recommended to use method 1 to obtain MTFs in the application of wave spectrum inversion.