A Physically-based Model of Ocean Backscatter for Wind Speed Retrieval from Sar, Scatterometer and Altimeter

A comprehensive theoretical model of radar backscatter from the ocean surface is discussed. It combines alternative models of the interaction between the wind and the sea surface, and different descriptions of the ocean waveheight spectrum of wind seas, with two alternative scattering theories, namely the composite-surface model and the integral equation method. The effects of swell, limited fetch, and perturbations due to rain, surface slicks and currents may be included. The model is able to show the expected sensitivity of radar backscatter to all these factors, and some results for the dependence on swell are shown. Hence it indicates possible improvements to current empirical models as well as implications for future missions such as Envisat. The model predictions are also compared against Topex altimeter data which are collocated with buoy measurements, and against an empirical scatterometer model function. The results at nadir agree well with the empirically observed dependence on wind speed, while the dependence on swell is consistent with recent studies of the influence of swell on the relation between wind stress and wind speed. Results at off-nadir incidence angles are more sensitive to the choice of ocean waveheight spectrum. In particular, the directional spread of waves is not well understood; this leads to a systematic discrepancy between dependences of backscatter on wind direction predicted by the empirical and theoretical models. INTRODUCTION The information on wind and wave fields which is obtainable from spaceborne radar is important for a wide range of applications, including maritime operations, weather forecasting and climate modelling. Empirical models of ocean radar backscatter, such as the CMOD family of C-band models for the ERS Wind Scatterometer [1], are widely used to provide quantitative information about wind fields. However, these models are limited because they do not include all the relevant physical effects, e.g. swell, rain, surface slicks and currents. Recent theoretical developments address these aspects, and here we attempt to construct a ‘unified’ theoretical model, that is, one with a common set of parameters applicable to synthetic-aperture radar (SAR), scatterometer and altimeter. Below, we discuss the construction of a relevant theoretical model, based on the present understanding of three aspects, namely the interaction between the wind and the sea surface, the description of the sea surface, and the interaction of electromagnetic radiation with this surface. Next, we outline the predicted sensitivity of the model to swell, which is one of the key environmental factors of interest. We then compare the model predictions with observed backscatter crosssections from satellite altimeter data which are collocated with in situ buoy measurements, and with an empirical scatterometer model function. Finally, we draw conclusions on the model successes and deficiencies. THEORETICAL MODEL We discuss in turn the formulations used in our model for the three aspects mentioned above. _______________________________________________________________________________________________ This work was funded by the European Space Agency under ESTEC Contract No. 12934/98/NL/GD. Interaction Between the Wind and the Sea Surface It is widely accepted that the transfer of momentum from the atmosphere to the ocean occurs through the influence of surface waves of all wavelengths. However, the quantitative description of the influence of the sea state has been the subject of much debate; [2] and [3] summarise the key studies in recent years. Here, we consider the dependences derived from three studies, namely: (i) the HEXOS experiment [4], which fitted a dependence of the roughness length of the sea surface, z0, on the wave age, cp/u*, where cp is the phase speed at the peak of the wave spectrum and u* is the friction velocity (defined as the square root of the ratio of wind stress to air density); (ii) the dependence fitted by [5] to the neutral drag coefficient, defined as (u*/U10), where U10 is the wind speed at 10-m altitude; and, (iii) an explicit dependence on sea state, using the description of z0/Hs as a power-law dependence on Hs/Lp fitted by [2], where Hs is the significant wave height and Lp is the peak wavelength in the waveheight spectrum. Model (ii) describes the neutral drag coefficient as a linear function of U10 above U10 = 6 m/s, with a more complicated function at lower wind speeds. This model is consistent with the conclusion of [3] that very low wind speeds are always associated with high drag coefficients. Model (iii) has been shown to fit a wide range of deep-water observations. Strictly there is a discontinuity when it is applied to combined wind seas and swell, because Lp will shift from the peak of the wind sea to the peak of the swell if the swell height increases at fixed wind speed. In practice, however, this discontinuity does not have a significant impact on our results. For all the models, we represent the wind speed at height z by the profile:       Ψ −     = 0 ln 4 . 0 * ) ( z z u z U (1) where Ψ is a stability function which describes the effects of differences between the air and sea temperatures. Description of the Sea Surface A description of the sea surface is required over a broad range of scales, from centimetre-scales commensurate with the radar wavelength out to several hundred metres to represent ocean swell. This has been a key area of research for many years and there is still a considerable lack of consensus amongst the alternative, published models. In recent years, a number of attempts have been made to develop realistic descriptions of the ocean waveheight spectra of wind seas, without using radar data as a constraint. (Otherwise this would lead to a circular argument in the testing of the combined surface and backscatter model against radar data.) Here we consider the descriptions of Apel [6], Elfouhaily et al. [7] and Lemaire et al. [8]. The Apel model gives the two-dimensional surface height spectrum as a function of wind speed and direction, but only represents fully-developed wind seas. The other two models [7,8] also include the effects of fetch and friction velocity. [7] in particular is consistent with a wide set of observational data, as well as being analytically continuous and with no aphysically tuned parameters. The Lemaire et al. spectrum requires a free parameter to be set, namely the significant slope, which is defined as the r.m.s. surface height divided by the wavelength of the peak in the spectrum. Lemaire et al. estimate it by fitting to radar data. Operationally, we would prefer to derive this parameter from a wave model such as WAM. However, WAM results for idealised runs at different wind speeds give different values of significant slope to the values fitted by Lemaire et al. (Fig. 1). This may be because the fitting procedure of Lemaire et al. is affected by the presence of swell. Our implementation allows the inclusion of a two-dimensional gaussian spectrum in wavenumber space to represent swell. We also incorporate published models of the perturbations in the wave spectrum due to rain [9] and surface slicks [10]. We use our own implementation for currents, but the results are similar to those of [11]. Fig. 1. Significant slopes calculated from different spectra, compared with the best fitting value of 0.70 % from [8]. Description of Backscatter from the Sea Surface The scattering model must include the contributions from surface scales commensurate with the radar wavelength as well as from longer scales. The relative contributions from these scales depend on the incidence angle and affect the dependence on radar polarisation. The most relevant scattering models are found to be composite-surface model (which considers the surface to consist of patches of small scale roughness tilted and modulated by longer wavelengths [12]) and the Integral Equation Method (IEM, which, in theory, treats all wavelengths in a consistent manner [13]). The split into two scales is an artifice of the composite-surface model. The calculated cross-section is insensitive to the wavenumber of this split, provided that wavenumber is about 0.3 times the radar wavenumber. In practice, the IEM requires two dividing wavenumbers to be set for its numerical implementation. A physically based wavenumber must be chosen to mark out the wavelengths that contribute to specular scattering when the incidence angle tends towards zero, and a computationally based wavenumber, related to the grid size used in the Fourier transform, must also be set. Fig. 2 shows the predicted dependences of the polarimetric backscatter for the composite-scale and IEM backscatter models at 20° and 50° incidences. HH and VV show similar sensitivity to wind speed whereas their ratio only varies by around 1.5 dB over the wind speed range. At large incidence angles the backscatter in HH polarisation is around 10 dB less than VV which, depending on the relative noise level, may give VV polarisation an advantage for the estimation of wind speed. Note that, although the two-scale and IEM models give similar results in VV polarisation, they differ by up to 2 dB in HH. This corresponds to a difference of around 2 m s in retrieved wind speed. Fig. 2. Predicted dependence of C-band backscatter on wind speed using Elfouhaily et al. wave spectrum [7] and HEXOS model of roughness length [4]. ‘comp’ and ‘IEM’ refer to the composite-surface and IEM models, respectively. SENSITIVITY TO SWELL The model is able to predict the sensitivity of the radar backscatter to environmental factors which are not represented in the empirical models such as CMOD. One key issue is the effect of the presence of swell. Fig. 3. shows the predicted behaviour for C band at nadir using the Elfouhaily et al. spectrum [7] and the composite-surface backscatter model, with dependences (ii) ([5], left) and (iii) ([2], right) for the interaction between the wind and the sea surface. It can be seen that the choice of dependence for this interaction has a significant effect on the predicted influence of swell, with dependence (iii) showing the strongest effect. However, we also find that the predicted influence of swell is small at an incidence angle of 50°. Hence swell is predicted to have a significant effect on the mean backscatter only for the lower end of the range of incidence angles accessible to the Envisat SAR. Fig. 3. Nadir backscatter at C band predicted from composite-surface model, as a function of swell height for 200 m swell with wind sea modelled by spectrum of Elfouhaily et al. [7], and descriptions of air-sea interaction from [5] (left) and [2] (right). The discontinuities in the right-hand plot result from the influence of swell height on the location of the peak wavelength in the combined spectrum of wind sea and swell. COMPARISONS OF MODEL PREDICTIONS AGAINST OBSERVATIONS Behaviour at Nadir Incidence Fig. 4 shows the predictions from the composite-surface backscatter model at C band, compared against an extensive set of Topex data collocated with buoy measurements at 41 sites over the period 1992-1997. Note the variation in backscatter between the different ocean spectra, indicating the errors which may be introduced due to uncertainties in the sea surface description. Also note that the absolute calibration of the altimeter is poorly known. Hence it is not possible to decide which of the wave spectrum descriptions fits best; they all reproduce the observed rate of change of backscatter with wind speed very well. A more detailed analysis of the Topex data-set reveals a dependence on sea state, which may be parameterised in terms of the significant wave height Hs. The observed sensitivity corresponds to reductions of about 1.2 dB and 0.4 dB in Kuband backscatter for a 1 m increase in r.m.s. swell height, for 200-m swell and a wind speed of 3 m/s and 18 m/s, respectively. These sensitivities agree well with the model predictions at Ku band using the composite-surface model with the Elfouhaily et al. spectrum [7] and [2] for the description of the air-sea interaction. (These predictions are not shown here, but they are similar to the C-band results in Fig. 2, apart from a difference in the absolute backscatter.) Hence we conclude that the collocated Topex-buoy data set is able to provide a test of our understanding of the influence of sea state on the interaction between the wind and the sea surface. 8 10 12 14 16 18 20 22 24 26 0 5 10 15 20 25 C b an d cr os sse ct io n (d B ) wind speed (m/s) Apel Elfouhaily Lemaire sig. slope = 0.0065 Lemaire sig. slope = 0.0080 Fig. 4. Topex altimeter and composite-surface model cross-sections as a function of wind speed for different assumed wave spectra. Behaviour at Off-Nadir Incidence Angles This model is also tested against the IFREMER-2 empirical scatterometer model for C band, VV polarisation [14]. Fig. 5 compares the behaviour of the composite surface scattering model with the Apel [6], Elfouhaily et al. [7] and Lemaire et al. [8] spectra at C band. The behaviour at Ku band is similar. The differences are greater for low incidence angles. At 50° incidence all the theoretical predictions are within 1-2 dB for VV polarisation, but they diverge from the empirical IFREMER-2 model [14] above wind speeds of about 10 m/s. In HH polarisation there is more sensitivity to the significant slope in the Lemaire et al. spectrum. The differences between the models are greater at 25° incidence, with changes of up to 4 dB on changing the significant slope from 0.007 to 0.006 or 0.008 in the Lemaire et al. spectrum. Fig. 5. Comparison of models of ocean backscatter at C band. The accuracy of the theoretical model off-nadir is limited by the present understanding of the description of ocean wave spectra. In particular, the directional spread of waves is not well understood. This leads to a systematic discrepancy between dependences of backscatter on wind direction predicted by the empirical and theoretical models which is apparent in the poorer agreement for the crosswind case shown in Fig. 6. Fig. 6. Comparisons of theoretical models with empirical model IFREMER-2 for C band, VV polarisation, upwind (left) & crosswind (right). ‘comp’ and ‘IEM’ refer to the composite-surface and IEM backscatter models, respectively. CONCLUSIONS A theoretical model of radar backscatter from the ocean surface which attempts to unify the predictions for nadir and off-nadir incidence angles has been discussed. The dependence of radar backscatter on swell may be included in this model. This represents an advance upon present empirical models for the off-nadir case. The predicted dependence on swell was shown to be sensitive to the relationship which is assumed for the interaction between the wind and the sea surface. The best agreement with a collocated altimeter-buoy data set was obtained by using a new relationship [2] which incorporates the effect of sea state on the roughness length of the sea surface. The predictions at nadir would be constrained further if the altimeter data were calibrated in terms of absolute cross-section. This aspect should be testable with the Envisat RA-2 altimeter, where the plans to achieve such calibration are already well advanced. Predictions from two alternative scattering theories, namely the composite-surface model and the integral equation method, were compared for different descriptions of the ocean waveheight spectrum of wind seas. These two scattering theories showed small differences, ~ 2 dB in the predicted ratio of HH/VV backscatter at C band. Hence observations from the alternating polarisation mode of the Envisat should be accurate enough to discriminate between these alternative scattering theories. 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