a. Two-scale quasi-specular model

Theoretical models of the interaction between electromagnetic waves and random surfaces have been developed in order to help interpret the observations. For near-nadir observations, the quasi-specular approach (Barrick 1968) relates σ° to the slope probability density function (pdf) via the Fresnel reflection coefficient. It was further recognized that a two-scale approach was necessary, as reviewed by Valenzuela (1978). The two-scale approach requires the introduction of a cutoff wavenumber k_d. Then, the near-nadir σ° is expressed according to

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where θ is the incidence angle and R′(0) is the Fresnel reflection coefficient for normal incidence, corrected in order to account for diffraction effects due to the small-scale roughness. The function p(ζ_x, ζ_y) is the sea surface slope pdf of ocean waves whose wavenumbers are smaller than k_d, and direction x is oriented toward the radar-viewing direction, while direction y is perpendicular to x.

Assuming for instance an isotropic rough surface of Gaussian slope pdf with an effective mean square slope s² (obtained by integrating the slope spectrum over the domain of wavenumbers smaller than k_d), expression (1) can be rewritten as

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Although the two-scale quasi-specular model provides a very simple formulation of the near-nadir backscatter, it is unable by itself to yield the appropriate values of k_d and R′(0). It turns out, however, that for near-nadir incidences the modeled σ° is substantially affected by the choice of k_d. Jackson et al. (1992) proposed a ratio of the electromagnetic wavenumber k_o to k_d of the order of 3–6, on the basis of a comparison with the physical optics approach in the infinite conductivity case.

b. Diffraction-modified Fresnel coefficient

In the framework of polarimetric passive remote sensing of the sea surface, Yueh et al. (1994) and Yueh (1997) have developed the small perturbation method (SPM) to second order for randomly rough surfaces. The scattered field is decomposed into coherent and incoherent components. The zero-order scattered field is a coherent field propagating in the specular direction, and its amplitude is characterized by the Fresnel reflection coefficients R⁽⁰⁾_HH and R⁽⁰⁾_VV for horizontal and vertical polarizations, respectively. While the first-order scattered field is incoherent, the second-order scattered field is coherent and gives a correction to the Fresnel coefficient. For HH and VV polarizations the coherently scattered field should be largely dominant in the vicinity of the specular direction, and since we are dealing with near-nadir observations, the incoherent component will be ignored in our analysis. To assess the range of validity of this approximation, one may refer to the work by Jackson et al. (1992), who compared the two-scale model approximation with the solution of the physical optics (PO) integral performed in the perfect conductivity case. They obtained that the two-scale quasi-specular model approximates the PO integral to within a few percent over the angular range θ ≤ arctan (3^1/2s), where s² is a nominal value of the mean square slope (mss). The limitation of our analysis to θ ≤ 10° is compatible with this value over the range of winds tested (u ≥ 5 m s⁻¹).

Instead of a diagonal matrix involving both Fresnel reflection coefficients R⁽⁰⁾_HH and R⁽⁰⁾_VV, the coherent reflection matrix including the zero- and second-order fields is obtained by Yueh et al. (1994):

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where R⁽²⁾_αβ, with α and β being V or H, are the second-order corrections of the specular reflection coefficients caused by small surface perturbation. They are given by Yueh (1997) as a function of incidence and azimuth angles θ_i and ϕ_i as

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where k_ρi = k_osinθ_i and F_s(k, ϕ) [written as a function of Cartesian coordinates in Eq. (4)] is the small-scale sea wave spectrum, defined as equal to the sea wave spectrum F(k, ϕ) for k higher than the cutoff wavenumber k_d, and equal to zero otherwise. The sea wave spectrum F(k, ϕ) is normalized so that h² = ∫^2π₀ ∫^∞₀ F(k, ϕ)k dk dϕ, where h² is the mean square height of the waves. The second-order scattering coefficients g⁽²⁾_αβ are functions of permittivity and geometry, and given by Yueh (1997). The complex permittivity is taken from Ellison et al. (1998), assuming a sea surface temperature (SST) of 10°C and a sea surface salinity (SSS) of 35 practical salinity units (psu). Increasing SST to 15°C, or SSS to 36 psu, would increase |R′(0)|² by 0.02 dB and 2 × 10⁻⁴ dB, respectively, which would not lead to significant modifications of the results of this study. In practice, the precise operating frequency of a Ku-band radar, although close to 14 GHz, depends upon the instrument. The nominal runs of the model have been performed here for a frequency f_o = 14 GHz, but in order to explore the sensitivity to the choice of the operating frequency, other tests were performed at slightly different frequencies. The effect of this modification is weak, as will be discussed in section 4.

In this paper, the diffraction-modified reflection coefficient R′(0) introduced in the two-scale quasi-specular model [Eq. (1)] is identified with the specular reflection coefficient including corrections up to the second order. Thus, |R′(0)|² is expressed as

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For the anisotropic sea spectrum, |R′(0)|² will be slightly dependent on ϕ and on polarization α. Since only azimuthally averaged values of σ^o are included in our analysis, we take the mean of |R′(0)|² over azimuth ϕ. The result is independent upon whether α = H or V.

a. Nadir model functions σ°(θ = 0)

Providing accurate absolute calibration of σ° is a difficult task. Empirical model functions for σ° at nadir were first developed for the Seasat altimeter (Chelton and McCabe 1985; Chelton and Wentz 1986). Chelton and Wentz (1986) also reported a general bias of −0.5 dB of σ° estimates from Seasat altimeter (ALT) data as compared to scatterometer (SASS) data at nadir, for situations where σ° is greater than about 10.5 dB. This bias was shown by Chelton et al. (1989) to be due to the use of a flat-earth approximation, which had been used to determine the altimeter footprint area.

A further study involving cross calibration between Geosat and Seasat altimeter estimates of σ° was performed by Witter and Chelton (1991). They reached the conclusion that the Seasat altimeter estimates of σ° were miscalibrated for values of σ° ≤ 11 dB and proposed a modification to the altimeter model function previously proposed by Chelton and Wentz (1986). This was called by Witter and Chelton the modified Chelton and Wentz (MCW) model function.

Comparing TOPEX/Poseidon and Geosat altimeter measurements of σ°, Callahan et al. (1994) identified an offset between both instruments. Callahan et al. mentioned that TOPEX used a round earth correction, which was not used for Geosat. If this correction were applied, the Geosat σ° would be 0.5 dB higher. Also, unlike Geosat, the TOPEX values have an additional correction for atmospheric absorption applied. The minimum absorption correction is about 0.2 dB at Ku band. To minimize the rms difference between the TOPEX and Geosat data, Callahan et al. indeed found that an offset of 0.7 dB between both datasets should be introduced. Callahan et al.’s (1994) conclusion is thus that the use of the MCW model function, with a + 0.7 dB offset added in order to account for a round earth and atmospheric corrections, should provide an accurate estimate of σ° at nadir at Ku band as a function of wind speed.

From an analysis performed on the basis of 1 yr of σ° measurements from the TRMM PR radar, Freilich and Vanhoff (2003) obtained a new model function for σ°(θ = 0). Following Freilich and Challenor (1994), they expressed their new model function by means of a four-parameter expression. Performing a nonlinear least squares fit of the coefficients to the TRMM PR data, they obtained

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where u is the neutral stability wind at 10-m height, expressed in meters per second.

The model function from Freilich and Vanhoff (2003), as well as the MCW model function, are shown in Fig. 1. As was mentioned by Freilich and Vanhoff, it turns out that their model is very similar to the MCW model, except that it exhibits a +1.9 dB offset compared to MCW. Callahan et al.’s (1994) model is also plotted in Fig. 1. As mentioned above, Callahan et al.’s model is identical to the MCW model, except for a +0.7 dB offset to account for a round earth and atmospheric corrections.

It should be mentioned that a two-parameter nadir σ° model function, including the effects of both wind speed and wave height, has been developed by Gourrion et al. (2002) from the TOPEX altimeter observations. Such a refined two-parameter model function, elaborated for the TOPEX altimeter, is unfortunately not available for the other satellites (Geosat or TRMM PR). In the present paper, where we are concerned with the comparison between model functions obtained from various satellites, such effects of wave conditions (including either fetch-limited situations or situations with swell) are therefore not studied. We limit ourselves to using the one-parameter model functions σ°(u) built from satellite observations, and we assume that they may be considered to be representative of fully developed situations.

b. Shape of the σ°(θ) profile

In the limit of a Gaussian isotropic sea surface slope distribution, the quasi-specular radar cross section from the sea surface is given by Eq. (2) and rewritten here as follows:

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where the effective mean square slope s² accounts for ocean waves whose wavenumbers are smaller than the scale separation wavenumber k_d. Jackson et al. (1992) obtained empirical determinations of s² by fitting the shape of the profile given by Eq. (7) to the observations. Similarly, Freilich and Vanhoff (2003) were able to infer effective mean square slopes from fits to PR data and gave an accurate parameterization of s for 5 ≤ u ≤ 17 m s⁻¹ according to

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Beyond u = 17 m s⁻¹, the simple expression (8), however, becomes less accurate (see Freilich and Vanhoff’s Fig. 7), and therefore for the highest wind speed studied here (u = 20 m s⁻¹), s²(u) was taken as 0.046 to be consistent with Freilich and Vanhoff’s Fig. 7 [instead of 0.048 as would result from Eq. (8)].

It is to be noted that, in order to determine s², Freilich and Vanhoff’s fits were performed using σ° in natural, not decibel, units. Owing to the rapid monotonic decrease of σ° with θ, the fit is therefore most sensitive to small incidence angles, where σ° is large. By this way, the authors show that the quality of the fits is excellent for θ ≤ 10° at all wind speeds, even though the fit was formally performed over a wider range of incidence angles.

In this paper, we use Freilich and Vanhoff’s (2003) empirical model function as a reference for radar cross section, obtained by combining Eqs. (6)–(8).

a. Standard method

For every wind speed, we compute the theoretical function σ°(θ) from the electromagnetic model described in section 2. For this purpose, it is necessary to determine the diffraction-modified Fresnel coefficient |R′(0)|², as well as the slope variance s² of the low-pass-filtered surface. The former may be deduced from Eqs. (4) and (5), provided that the sea wave spectrum F(k) is known for k ≥ k_d. As for the slope variance s², it may be readily computed by integration of k²F(k) over the range k ≤ k_d. Thus the knowledge of the whole sea wave spectrum F(k) is required in order to determine σ°(θ).

Several spectral models of the sea surface are available in the literature (Donelan and Pierson 1987; Apel 1994; Elfouhaily et al. 1997; Lemaire et al. 1999; Kudryavtsev et al. 1999). In this paper, fully developed situations are assumed, and Elfouhaily et al.’s (1997) model is used. The reason for this choice, as well as trials performed with other spectral models, will be discussed in section 4b below.

To avoid an unphysical negative value of Elfouhaily et al.’s Phillips–Kitaigorodskii equilibrium range parameter for short waves α_m at low wind, we have followed Freilich and Vanhoff’s approach, which consists of replacing Elfouhaily et al.’s two-regime logarithmic law for α_m with

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with c_m = 0.23 m s⁻¹ corresponding to the capillary–gravity phase speed minimum.

Also, in order to determine the friction velocity u* for a given u, the parameterized polynomial form of the 10-m neutral drag coefficient developed by Taylor and Yelland (2001) was used.

Once the sea wave spectrum is determined, the two-scale electromagnetic model can be run, provided that a value of the scale-separation wavenumber k_d is given. We have chosen to give no a priori information on k_d and, therefore, we leave it as a free parameter. The value of k_d is adjusted so as to minimize the rms error E between the output of the electromagnetic model and the empirical model function of σ°.

An example of the fit obtained for u = 10 m s⁻¹ is given in Fig. 2. To see how much the modeled σ° is sensitive to the choice made for the ratio r = k_o/k_d, the modeled σ° is plotted for different assumptions from r = 2 to r = 6. Here the best-fitted value of r is found to be r = 2.95, and the rms error of the fit is then E = 0.19 dB for σ°. As a comparison, σ° obtained with r = 2.95, but using the nominal (uncorrected) Fresnel coefficient, is also plotted as the dotted line in Fig. 2. Note that the correction to the Fresnel coefficient introduced by the second-order term in Eq. (5) reduces σ° by 0.75 dB in that case. This effect increases as the ratio r = k_o/k_d increases. Thus, for the same wind speed u = 10 m s⁻¹, if a ratio r = 6 were taken, the correction to the Fresnel coefficient due to the second-order term would become as high as 2.6 dB.

For other wind speeds between 5 and 20 m s⁻¹, similar fits are obtained, with rms errors ranging from E = 0.12 to E = 0.23 dB. Freilich and Vanhoff mention that TRMM Microwave Imager (TMI) measurements, which they used to estimate wind speeds, underpredict wind speed below 3 m s⁻¹, and therefore their inferred effective mean square slope s²(u) is probably inaccurate below 3 m s⁻¹. Additional effects such as non-Gaussian peakedness (Chapron et al. 2000), which have not been considered here, may also become nonnegligible, and therefore such low wind speeds were not considered here.

Due to the shape of the empirical model function as a function of θ [Eq. (7)], which is identical to the theoretical electromagnetic model function, E should always be exactly equal to 0. The example of Fig. 2 shows, however, that E is generally not equal to 0, because the fit must also ensure that the computed absolute level of σ° coincides with the level of the empirical model function. Thus, for example in Fig. 2, the computed σ° (solid line) is slightly too low for small values of θ and slightly too high for θ approaching 10°. Taking the model function by Freilich and Vanhoff (2003) as a reference, we have therefore performed the fit of the electromagnetic model to the empirical model function, introducing an offset to the empirical model function, and the residual error E was computed as a function of the offset. The results are displayed in Fig. 3. For every wind speed, it can be seen that E tends to decrease when a negative offset is introduced. Depending on the wind speed, a minimum E = 0 is attained for an offset between −0.8 and −1.4 dB, although no clear systematic dependence of this optimum offset with wind speed is noticeable (extreme values of the optimal offset are obtained for u = 5 or 15 m s⁻¹, while intermediate values are obtained for u = 10 or 20 m s⁻¹). Denoting as 〈E〉 the quadratic mean of E over the range 5 ≤ u ≤ 20 m s⁻¹ (solid line in Fig. 3), 〈E〉 is of the order of 0.17 dB for zero offset, while the minimum of 〈E〉 is attained for an overall offset of −1.2 dB.

As mentioned by Freilich and Vanhoff (2003), their model is virtually 1.9 dB above the MCW model, while Callahan et al.’s (1994) model is 0.7 dB above MCW (see section 3a). The offsets corresponding to each of the three models discussed here are indicated as straight vertical lines in Fig. 3. It turns out that Callahan et al.’s (1994) model coincides exactly with the minimum of the quadratic mean error 〈E〉, while the MCW model is too low. It may be concluded that, under the standard assumptions made here, there is virtually perfect consistency between the absolute calibration proposed by Callahan et al. (1994) and standard electromagnetic modeling.

It should be recalled that the computations were performed here for a radar frequency taken as f_o = 14 GHz. Shifting the frequency to f_o = 13.8 GHz (the one of the TRMM PR) would lead to a slight shift of the optimal offset to −1.1 dB (instead of −1.2 dB), but would not change qualitatively the results obtained here.

The values obtained for the ratio r = k_o/k_d between the electromagnetic wavenumber and the scale-separation wavenumber, when an offset of −1.1 dB is introduced, are displayed in Fig. 4. It is of the order of 7.4 for u = 5 m s⁻¹, tends to decrease with increasing wind speed, and reaches about 4.1 for u = 20 m s⁻¹. Such values are in basic agreement with, although slightly higher than, the values estimated by Jackson et al. (1992), who found r to be of the order of 3–6 from a Ku-band airborne radar altimeter.

It may be noted that for the SPM method discussed in section 2b to be applicable, the short-scale rms height h_s = [∫^2π₀ ∫^∞_kd F_s(k, θ)k dk dϕ]^1/2 should verify the criterion (e.g., Barrick and Peake 1968)

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With increasing wind speed, the sea surface roughness increases. Since h_s is obtained by integrating the sea wave spectrum over wavenumbers k ≥ k_d, the separation wavenumber k_d ought to increase (and therefore r = k_o/k_d should decrease) to maintain h_s at a moderate value in such a way that the criterion (10) continues to be verified. The variation of r with wind speed exhibited in Fig. 4 thus follows the same trend as the variation of the limit of validity of the SPM method with wind speed. We may mention, too, that computations of h_s performed with the values of r found here give the inequality (10), which is well verified over the whole range of winds studied.

Finally, as a comparison, the ratio r for an offset of 0 dB, also indicated in Fig. 4, would be lower by a factor of 1.4–1.6.

b. Role of the wave spectral model

As mentioned above, the sea wave spectral model used in this study is Elfouhaily et al.’s (1997) model. It was chosen here both because of its simplicity and for the fact that it is consistent with a variety of high-quality nonradar measurements, including Jähne and Riemer’s (1990) laser slope gauge data and Cox and Munk’s (1954) measurements of mean square slope (see Elfouhaily et al.’s Fig. 7b), which is an essential parameter as far as radar altimetry is concerned. As a comparison, the mean square slopes of Donelan and Pierson’s (1987) and Apel’s (1994) models at 10 m s⁻¹ are, respectively, 2.4 and 1.4 times the ones of Cox and Munk’s clean sea model. Although some artifacts of Elfouhaily’s model were identified for L-band radiometric models (Dinnat et al. 2003), Anderson et al. (2002) mentioned that the qualitative agreement of the altimeter cross section obtained with different wave spectra (including Elfouhaily et al.’s model) could not be distinguished at Ku band.

To assess the role of the shape of the wave spectral model used in this study, we performed trials using Apel’s (1994) model. However, in order to be consistent with Cox and Munk’s (1954) mean square slope measurements, Apel’s (1994) model was rescaled by a proportionality factor (depending upon wind speed). Compared to Fig. 3, the results (not shown) then give a much larger rms residual error, with a minimum attained for an offset that is highly dependent upon wind speed. When averaged over wind speed, the residual rms error reaches a minimum of 0.95 dB (to be compared to 0.02 dB obtained with Elfouhaily et al.’s model, as seen in Fig. 3), even though the offset corresponding to this minimum is only slightly modified (−1.0 instead of −1.2 dB). Clearly these results cannot be interpreted in terms of a mere calibration offset. When Donelan et al.’s model, rescaled by the same method, is used, a strong variability of the rms residual error is again obtained and, when averaged over wind speeds, an unrealistic optimal offset of less than −2.2 dB is found. Thus, the shape of the spectral model plays a significant role, and an accurate knowledge of the spectral shape of the sea surface, indeed constitutes an issue to be addressed in order for this method to become fully reliable. However, the fact that virtually the same optimal offset was found for the various wind speeds tested (from 5 to 20 m s⁻¹), as displayed in Fig. 3, and the fact that the optimal residual rms error was extremely low (0.02 dB) give some good confidence for the choice made of Elfouhaily et al.’s model.

c. Role of peakedness of the sea surface

Chapron et al. (2000) showed that the peakedness of the sea surface, caused by a fourth-order correction to the sea surface slope pdf p(ζ, 0), affects the absolute level of σ° at nadir, as well as the σ°(θ) profile. Those authors interpret non-Gaussian peakedness by considering the sea surface as being composed of patches where the sea surface is locally Gaussian, but assuming that the slope variance varies from patch to patch. They show that, because of those fluctuations in s², our Eq. (2) should be replaced by the following more exact quasi-specular expression for σ°:

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where Δ is the variance of s². Also, Chapron et al. use the zero-order Fresnel coefficient |R⁽⁰⁾(0)|² instead of the diffraction-modified Fresnel coefficient |R′(0)|² used in Eq. (11).

Thus two different explanations of the behavior of σ° at low incidence angles are proposed. In the present paper, the effective reflection coefficient is explained by second-order scattering due to subfacet roughness while Chapron et al. explain it by peakedness effects.

On the basis of the analysis of Cox and Munk (1954) and Cox and Munk (1956), Chapron et al. suggest a value of Δ independent of wind and estimated as Δ = 0.23. The electromagnetic model was thus run, keeping Elfouhaily et al.’s (1997) spectral model but now using Eq. (11) with Δ = 0.23. As compared to the overall optimal fit of −1.2 dB obtained in Fig. 3, the result (not shown) then gives an optimal fit for an offset that depends slightly on wind speed (between −0.3 and +0.1 dB), with an average of −0.2 dB. The corresponding ratio r is between 4.3 and 2.3 depending upon wind speed. In that case the model function giving the best consistency with the electromagnetic modeling would be that of Freilich and Vanhoff.

It should be noted, however, that the fourth-order coefficients of the Gram–Charlier development of the slope pdf (from which Δ can be evaluated) were obtained by Cox and Munk (1954) with a large uncertainty (c₄₀ = 0.4 ± 0.2, c₂₂ = 0.1 ± 0.05, c₀₄ = 0.2 ± 0.4). Cox and Munk also mention that it is possible that systematic errors in the correction for background light may account for much of the observed peakedness. In view of Cox and Munk’s inability to measure the steep slopes, Chapron et al. suggest that alternate measurements to clarify variability and magnitude of Δ would be beneficial. A complete treatment of the problem including both the diffraction-modified reflection coefficient and peakedness effects would indeed require an accurate evaluation of Δ as a function of u, which is unfortunately not available at this time. The consequence is that we cannot exclude the possibility that, due to peakedness effects, the offset required to match the Freilich et al. model would be smaller (in absolute value) than the −1.1 dB that we found in section 4a in a context where peakedness effects were assumed negligible.