We analyze vertically and horizontally polarized brightness temperatures at 85.5 GHz that were measured by F5 Special Sensor Microwave Imager (SSM/I) over the world oceans. The columnar liquid cloud water content is restricted to below 0.04 mm. We develop a radiative transfer model function in this frequency region that provides a functional relationship between the measured microwave brightness temperatures and the essential geophysical parameters, which are the atmospheric temperature, ocean surface temperature, and wind speed. This will provide the basis for including the higher SSM/I, Tropical Rainfall Measurement Mission Microwave Imager (TMI), and Advanced Microwave Scanning Radiometer frequency channels into oceanic and atmospheric retrieval algorithms, which rely on an accurate forward model.
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 Because of the lack of an accurate, consistent geophysical model function for the measured brightness temperatures of the high-frequency channels of airborne passive microwave sensors (85.5 GHz at Special Sensor Microwave Imager (SSM/I) and Tropical Rainfall Measurement Mission Microwave Imager (TMI) and 89 GHz at Advanced Microwave Scanning Radiometer (AMSR)), these channels have so far not been included into the Remote Sensing Systems well-calibrated retrieval algorithms [Wentz, 1997], which rely on an accurate geophysical forward model. The 85.5 GHz SSM/I has been used in data assimilation systems for numerical weather predictions [Phalippou, 1996; Prigent et al., 1997], which adopt a variational approach and are therefore able to tolerate larger forward model errors at 85.5 GHz without completely dropping the channel.
 In the absence of rain the brightness temperature TB that is received by the satellite depends on the following: (1) the sea surface temperature (SST), TS, (2) the atmospheric temperature profile, T(z), (3) the atmospheric air pressure profile, p(z), (4) the atmospheric profile of water vapor density, ρV(z), (5) the atmospheric profile of liquid cloud water density, ρL(z), (6) the ocean surface wind speed, W, and (7) the ocean surface wind direction (relative to the looking azimuth), φ. It is the aim of our study to derive a model function depending on these geophysical parameters for vertical (v) and horizontal (h) polarization and Earth incidence angles of ∼53° from 85.5 GHz SSM/I observations.
 The 85.5 GHz channel is very sensitive to cloud absorption. In order to not swamp the signals for vapor, SST, and wind, we consider only cases with low columnar cloud liquid water content, i.e., L ≤ 0.04 mm. Furthermore, we will consider only the isotropic wind response of the ocean surface for the moment and neglect the dependence on wind direction.
2. Study Data Set
 For our analysis we are using brightness temperatures that were measured by F15 SSM/I during 2002. SSM/I also provides values for the columnar water vapor
the columnar liquid cloud water
and the surface wind speed W. H denotes the satellite altitude. All observations are binned into 0.25° latitude (LAT)-longitude (LON) maps. Because of the restriction L ≤ 0.04 mm, they are automatically filtered for rain. V, L, and W are retrieved using only the 19, 22, and 37 GHz channels of the SSM/I instrument [Wentz, 1997].
 The atmospheric profiles of T(z), p(z), ρV(z), and ρL(z) as well as the SST TS are obtained from the National Centers for Environmental Prediction (NCEP) final analysis, which is available four times daily on a 1° LAT-LON grid. The NCEP events are trilinearly interpolated to the SSM/I events in LAT, LON, and time. The numerical weather prediction moisture analyses are accurate to 10% at best. We therefore scale the NCEP water vapor density ρV(z) so that its columnar integral equals the SSM/I value.
3. Radiative Transfer Equation
 In general, the brightness temperature that is received by the satellite is given by the following radiative transfer equation:
where E is the sea surface emissivity and depends on TS, W, and φ. The total atmospheric transmittance is denoted by τ and is given by
where the AI is the columnar integral over the atmospheric absorption coefficients α1 for I = O (oxygen), V (water vapor), and L (liquid cloud water) as follows:
TBU is the upwelling atmospheric brightness temperature, and TBD is the downwelling atmospheric brightness temperature that is reflected at the sea surface. Both quantities are given as weighted integrals of the atmospheric temperature profiles
and τ = τ(0,H,θ). The scatter term Ωscat correction accounts for the fact that, for a wind-roughened sea surface, radiation can be reflected from directions of the sky other than the specular direction. TC = 2.7 K is the cold space radiation.
4. Dielectric Constant of Seawater
 A major input into the radiative transfer model is the dielectric constant of seawater ε. It determines the specular sea surface emissivity E0 for both v and h polarization on the basis of the Fresnel equations
Guillou et al.  have recently provided laboratory measurements of this quantity at 85.5 and 89 GHz. Alternatively, one can predict ε for a frequency ν from the Debye dipole relaxation theory [Debye, 1929]
 The value ε0 denotes the value for the static dielectric constant, ε∞ denotes its value at infinite frequency, σ denotes the seawater conductivity, η denotes the Cole-Cole spread factor [Cole and Cole, 1941], νR denotes the Debye relaxation frequency, and . The values ε∞, ε0, σ, and νR depend on SST. In addition, ε0, σ, and νR depend on the salinity of seawater. The value η is an empirical constant, which describes the spread of the relaxation wavelengths. A model for these parameters was provided by Wentz and Meissner  which updates an earlier work by Klein and Swift . The analysis is based on measurements at low frequencies and has proven to describe well the frequency dependence of ε for frequencies below 37 GHz. At 85.5 GHz, both methods are consistent in warm water TS > 20°C, whereas in cold water there are substantial differences. For example, the specular surface emissions E0TS for both v and h polarization are ∼3.5 K lower in the case of Guillou et al.  than in case of the Debye model. A more elaborate Debye model containing two different relaxation wavelengths has been developed by Liebe et al.  for fresh water and by Stogryn et al.  for freshwater and seawater. Recent airborne measurements by the Millimeter-Wave Imaging Radiometer at 89 GHz [Wang, 2002] favor the model of Stogryn et al. . It is one aim of our study to provide a validation for ε at 85.5 GHz from SSM/I measurements.
5. Atmospheric Absorption
 The atmospheric absorption models relate the absorption coefficients α1, I = O, V, and L to the atmospheric profiles for temperature, pressure, water vapor, and liquid cloud water.
5.1. Oxygen Absorption
 We use the oxygen absorption coefficients provided by P. W. Rosenkranz (personal communication, 1998). These coefficients are based on the work by Liebe et al.  and Schwartz .
5.2. Water Vapor Absorption
 The 85.5 GHz channel lies in the water vapor window and is therefore very sensitive to the water vapor continuum absorption. An updated and comprehensive analysis has been performed by Rosenkranz , who uses a combination of the millimeter wave propagation models (MPM) created by Liebe and Layton  (MPM 87) for the foreign-broadened continuum and by Liebe et al.  (MPM 93) for the self-broadened continuum. This combination has been validated against numerous experimental observations [Westwater et al., 1990]. An exception is found in upward looking combined aircraft-radiosonde measurements by English et al. . In a moist atmosphere (V = 45 mm) their measurements show a larger absorption than the model of Rosenkranz .
 Very recently, an atmospheric radiative transfer model, called MonoRTM, has been developed by Atmospheric and Environmental Research, Inc. (AER) [Boukabara et al., 2002]. It is the aim of this paper to compare the two water vapor models by Rosenkranz  and Boukabara et al.  using the SSM/I data.
5.3. Liquid Cloud Water Absorption
 For the liquid cloud water absorption, we use the Rayleigh approximation
where λ is the radiation wavelength (in cm), ρL is the liquid cloud water density (in units of g/cm3), and ε is the dielectric constant of cloud (pure) water. For ε at 85.5 GHz we have used the model of Liebe et al. . The liquid cloud water absorption for the low liquid cloud water content L ≤ 0.04 mm is rather insensitive to ε even if supercooled clouds are included. We have found no noticeable difference when using the dielectric model of Stogryn et al. .
6. Special Sensor Microwave Imager (SSM/I) Analysis for Specular Sea Surface
 For low wind speeds W ≤ 5 m/s the 85.5 GHz v polarization brightness temperature changes very little with wind speed and is therefore independent on the sea surface roughness. This provides a convenient way to measure the properties of the specular emission and the atmospheric transmittance at this frequency.
6.1. Validation of the Dielectric Properties of Seawater
 In order to analyze the temperature dependence of the dielectric constant of seawater we need to avoid possible errors that can arise owing to deficiencies of the water vapor absorption model because of the strong geographic correlation between SST and water vapor. We first confine ourselves to dry atmospheres where the total vapor content is <15 mm. Figure 1 shows the difference TB − F between the measured brightness temperatures TB and the computed model function F. We have used the values of the dielectric constant from Guillou et al.  (model 1; top panel of Figure 1), the prediction by the Debye theory with the parameters of Wentz and Meissner  (model 2; middle panel of Figure 1), and the double Debye relaxation model of Stogryn et al.  (model 3; bottom panel of Figure 1). The results are binned with respect to SST in the interval between 0° and 25°C. The bin population is ∼30,000 events for the lowest SST bin and is decreasing to ∼1200 events in the highest SST bin. In computing the model function F we have used the water vapor absorption model of Rosenkranz . As we will show in the next section, the difference to the MonoRTM [Boukabara et al., 2002] is very small in dry atmospheres.
 For all three models in Figure 1 we observe small overall biases (−1.3 K for model 1, −2.1 K for model 2, and −2.2 K for model 3). These brightness temperature offsets arise because of deficiencies in the dielectric constant model itself, instrument calibration errors, and deficiencies in the atmospheric absorption model, especially the oxygen absorption, which is nearly constant over the world oceans. At this point it is not possible to separate these effects. In order to assess which of the three models fits the data best, we need to look at the average quadratic deviation of each bin from a constant line, which is given by the overall biases mentioned above. Quantitatively, this is given by the expression . The total number of bins is denoted by n, which is 13 in this case, and where yi is the average of TB − F in each bin, is the overall bias of TB − F and the wi are appropriate weights for each bin. We use wi = 1. The results are 0.48 K (model 1), 1.45 K (model 2) and 0.83 K (model 3). Furthermore, we observe that for model 1 the deviations from the overall bias are basically constant over the whole SST range. For model 2, TB − F increases monotonically by ∼3.0 K over the SST range. For model 3 the increase is much less, ∼1.5 K. These values are to be compared with the values of the error bars σ in each bin, which are between 1.1 and 1.7 K. This supports the measurements of Guillou et al.  for dielectric constant of seawater at 85.5 GHz. It also shows that the double Debye relaxation model fits the data noticeably better than the single Debye relaxation model.
6.2. Validation of the Water Vapor Absorption Models
 Having validated the measurement for the dielectric constant of seawater of Guillou et al. , we will use their value from now on and extend the analysis to the whole water vapor region between 2 and 60 mm and test the vapor absorption models by Rosenkranz  and the AER MonoRTM [Boukabara et al., 2002]. The bin size is 2 mm. The population of the lowest vapor bin is ∼500 events, increasing to just above 60,000 events for vapor bins above 15 mm. Figure 2 displays the difference between measured and computed brightness temperatures. In the upper panel of Figure 2 we have used the columnar water vapor V from SSM/I and, as mentioned earlier, scaled the NCEP profiles so that their value of total vapor content agrees with the SSM/I value. Both models in Figure 2 give very similar results: There are basically no differences for V < 10 mm. For moist atmospheres (V > 40 mm) the MonoRTM gives a slightly larger absorption amounting to a computed brightness temperature that is smaller than the Rosenkranz  model gives. The values for the overall biases are +0.4 K (for the Rosenkranz  model) and −0.2 K (for the MonoRTM). The values for are 0.75 K and 0.73 K, respectively, and are to be compared with typical values of 3.3 K for the error bars in each vapor bin. This means that the MonoRTM vapor model fits the data slightly better than the Rosenkranz  model. We cannot see the observed deficiency of the model absorption reported by Rosenkranz  when comparing his model with the measurements by English et al.  (compare section 5.2).
 For comparison, the bottom panel of Figure 2 shows the results if the NCEP profiles are not scaled and the columnar water vapor from NCEP instead of the SSM/I value is used. The fluctuations of TB − F around the mean increase. Therefore the values for also increase to 1.21 and 0.90, respectively, but the basic pattern and conclusions remain the same.
7. SSM/I Analysis of the Isotropic Wind-Induced Emissivity
 The emissivity of the wind-roughened ocean surface differs from the specular emissivity owing to the following:
The presence of large gravity waves. This is commonly treated in the Geometric Optics (GO) model (Kirchhoff Approximation) using an ensemble of tilted facets, each acting as a specular tilted surface [Stogryn, 1967]. The slope variance of the tilted facets increases proportionally with wind speed [Cox and Munk, 1954].
The presence of small capillary waves resulting in Bragg Scattering.
The appearance of sea foam at larger wind speeds W ≥ 7 m/s.
 For the lower-frequency channels, it has turned out that a satisfactory theoretical modeling of all these effects is very difficult [Wentz and Meissner, 1999; Wentz, 1997]. We do not attempt to do this but derive the wind induced isotropic sea surface emissivity empirically from SSM/I observations. The only exception is the scatter term Ωscat in equation (1), which has to be removed from the measured TB in order to extract the sea surface emissivity. We calculate this term using the GO model [Wentz and Meissner, 1999; Wentz, 1997] at 85.5 GHz.
Figure 3 shows the results for the wind-induced emissivities at 85.5 and 37 GHz, binned with respect to wind speed between 0 and 20 m/s. The bin size is 1 m/s. The bin population ranges from ∼3000 at 20 m/s to over 300,000 at 7 m/s. For the computation we have used the dielectric constant of [Guillou et al., 1998] and the water vapor absorption model of MonoRTM, both of which we validated in section 6. The range of the water vapor comprises the interval between 2 and 40 mm. For comparison, we have also included the results of the GO model with the Cox-Munk slope distribution [Cox and Munk, 1954] as specified by Wentz and Meissner . We note that the 85.5 GHz v polarization wind speed signal is basically zero over the whole wind speed range. For a surface wind speed of 15 m/s the h polarization wind speed signal increases by ∼25% from 37 to 85.5 GHz. A similar observation was made by Rosenkranz .
 Interesting also is the deviation between the observed emissivities and the GO model, which is noticeably larger at 85.5 GHz than at 37 GHz. At higher wind speeds this is likely due to the larger emissivity of sea foam at the higher frequency. At low wind speeds the observed difference can arise from the presence of capillary waves, as mentioned above, or it could point to an insufficiency of the Cox-Munk slope distribution, which has been used in the GO calculation.
8. Summary and Conclusions
 Our study of the 85.5 GHz SSM/I v and h polarization channels for low cloud cases validates the measurements for the dielectric of seawater by Guillou et al.  and the water vapor continuum absorption models by Rosenkranz  and the AER MonoRTM [Boukabara et al., 2002]. We also provide a prediction for the isotropic wind-induced sea surface emissivity.
 In order to complete this analysis, we will include cases for larger liquid cloud water and study the liquid cloud water absorption at 85.5 GHz in more detail. We also plan to investigate the size of the wind direction signal at this frequency.
 This research was funded by NASA contract NAS5-32594 (AMSR). We are thankful to P. W. Rosenkranz for the FORTRAN codes for calculating the water vapor and oxygen absorption coefficients, to S. Boukabara (AER) for the FORTRAN code of the MonoRTM, and to P. Wang for the FORTRAN codes for the double Debye model of Stogryn et al. .