The study of the tropical water cycle and of its evolution under climate modifications strongly relies on a documentation of the horizontal and vertical distribution of the water vapour field and on the understanding of the different processes that interact with it at all scales (Roca et al., 2010). The observation from space of the free tropospheric humidity began in the early 1970s with the launch of the Nimbus-4 satellite, which carried the Temperature–Humidity Infrared Radiometer (THIR), with a specific channel in the 6.3 µm strong absorption band (Allison et al., 1972). A continuous monitoring of the atmospheric water vapour has been provided since 1983 by the series of METEOSAT geostationary satellites and analyses of this database have revealed specific features of the tropical tropospheric water vapour, be it at the scale of the convective event (e.g. Roca et al., 2005) or at the climatological scale (e.g. Brogniez et al., 2009). Moreover, a global view of the atmospheric humidity content is brought by the succeeding NOAA polar orbiting satellites carrying the HIRS radiometer since 1979, allowing trend studies of the upper tropospheric humidity (e.g. Bates and Jackson, 2001; McCarthy and Toumi, 2004). However, such measurements are limited to clear-sky areas and scenes of low cloud cover (e.g. Brogniez et al., 2006), which limits the comprehension of the processes at play in the atmospheric water cycle.
Microwave techniques give the opportunity to observe the Earth's surface and its atmosphere from space even in the presence of clouds, which are largely transparent at frequencies below 100 GHz. Indeed, considering the mean size of water droplets and ice crystals (∼10–100 µm) with respect to the wavelength at microwave frequencies (∼1 cm, for ν > 60 GHz), the effect of scattering by non precipitating particles can be reasonably well represented by the Rayleigh approximation in the radiative transfer. Recent efforts towards the exploitation of 183.31 GHz observations have been made, dedicated to retrieval algorithms (Sohn et al., 2003; Houshangpour et al., 2005), to process studies (e.g. Brogniez and Pierrehumbert, 2006; Eymard et al., 2010) or to the elaboration of intercalibrated long-term datasets (Buehler et al., 2008). Indeed, the use of microwave radiometer allows for the study of atmospheric humidity close to the deep convective core, the spatial resolution still being a challenge for the processes at play in between the rainy convective cells: the current space-borne microwave sounders dedicated to water vapour observations have a footprint of 16 km at nadir (Microwave Humidity Sounder (MHS) onboard the MetOp platforms and Advanced Microwave Sounding Unit-B (AMSU-B) onboard the NOAA satellites), while the infrared radiometers offer a view of the water vapour field with a sub-satellite point of up to 3 km (example of the SEVIRI imager onboard the METEOSAT Second Generation series).
The Indian–French Megha-Tropiques mission (Roca et al., 2012, this issue) will sample the tropical belt within ±30° in latitude. The payload includes two microwave radiometers dedicated to the atmospheric hydrological cycle: MADRAS (Microwave Analysis and Detection of Rain and Atmospheric Structures) is a conical scanning imaging radiometer with channels ranging from 18 to 157 GHz; SAPHIR (Sondeur Atmosphérique du Profil d'Humidité Intertropicale par Radiométrie) is a cross-track scanning sounder operating in the 183.31 GHz water vapour band, with a footprint at nadir of 10 km. The channels' central frequencies and bandwidths of SAPHIR and MADRAS are listed in Table 1, along with those from AMSU-B and MHS.
Table 1. Channel central frequencies and bandwidths for AMSU-B, MHS, SAPHIR (all channels are vertically polarized at nadir) and for MADRAS. The ‘–’ symbol indicates that the same channel is present.
Central frequency (GHz)
183.31 ± 1.0
183.31 ± 3.0
183.31 ± 7.0
183.31 ± 0.2
183.31 ± 1.1
183.31 ± 2.8
183.31 ± 4.2
183.31 ± 6.6
183.31 ± 11.0
18.7 (H & V)
36.5 (H and V)
89.0 (H and V)
157.0 (H and V)
In this paper we present the expected improvements in the estimation of the relative humidity profile brought by the combination of those two instruments with respect to the current operational platforms carrying similar radiometers. For this purpose, the method of restitution of the profiles is based on a generalized additive model that addresses the strong nonlinearity of the problem with additive cubic smoothing splines, and the training of the model is accomplished with a 10-year set of radiosoundings limited to oceanic and purely non-diffusive profiles. This exercise finally discusses the interest of estimating the whole distribution of the possible profiles given a set of explanatory variables, such as the brightness temperatures (TB), and not only its expectation. Such a retrieval could be a guide for a new type of operational product.
The rest of this paper is organized as followed. Section 2 briefly recalls the information given by observations in the 183.31 GHz water vapour absorption band, section 3 is dedicated to a description of the data and tools used for the study, while section 4 provides a short discussion on the choice of training dataset for the specific problem of relative humidity profile retrieval. Section 5 discusses the statistical model used to perform the retrieval and its results in a comparative study on the number of inputs. Finally, a conclusion is drawn in section 6.
2. Information content of the 183.31 GHz band
The use of microwave radiometry to estimate the Earth's tropospheric water vapour has spread since the launch of the Nimbus-E microwave spectrometer in the 1970s (Staelin et al., 1976). This satellite was designed for the study of atmospheric water vapour and liquid abundances from the 22.235 GHz water vapour rotational transition, using the open ocean as a cold background. Although this line is very well adapted for the measurement of the total water vapour burden, measurements at several frequencies around the stronger rotational transition line at 183.31 GHz (see Figure 1) are better suited to estimate the vertical relative humidity profile over land and oceans (Schaerer and Wilheit, 1979; Rosenkranz etal., 1982; Kakar, 1983; Wang et al., 1983). One also sees in Figure 1 the absorption due to the water vapour continuum, which increases significantly with frequency.
The reference study by Schaerer and Wilheit (1979) on the use of selected frequencies in a 18 GHz bandwidth around the 183.31 GHz water vapour line clearly showed the theoretical feasibility of atmospheric profiling of relative humidity from a passive radiometer over both land and oceanic surfaces. This was confirmed with airborne measurements by the Advanced Microwave Moisture Sounder (AMMS) during the CCOPE campaign (Cooperative Convective Precipitation Experiment; Wang et al., 1983), with three channels in the 183.31 GHz band (±2.25, ±5.0 and ±8.75 GHz; use of the double sideband mode for a better radiometric sensitivity). This experiment helped in designing the current configuration of the space-borne water vapour sounders (e.g. AMSU-B; Saunders et al., 1995; Table 1) with channels at 183.31 ± 1.0, 183.31 ± 3.0 and 183.31 ± 7.0 GHz, which were shown to be a more favourable distribution of the weighting functions for a retrieval at relatively high altitudes (∼12 km; e.g. Wang and Chang, 1990).
Several recent humidity sounding radiometers share the same spectral characteristics, such as the Special Sensor Microwave/Temperature-2 (SSM/T-2) instrument onboard the DMSP satellites or the recent Microwave Humidity Sounder (MHS) onboard the MetOp platform. SAPHIR slightly differs from these sensors since it has six channels ranging from 183.31 ± 0.2 to 183.31 ± 11.0 GHz (see Table 1 for the characteristics of both AMSU-B and SAPHIR). A detailed description regarding the instrument is found in Roca et al.(2012, this issue) and in a preliminary study by Eymard et al.(2002). The first channel, close to the centre of the 183.31 GHz band, is aimed at reaching higher layers of the atmosphere compared to AMSU-B, and the use of a large bandwidth of 11.0 GHz around the central frequency locates the sixth channel on the wings of the absorbing line for a deeper sounding of the atmosphere. The usual observations in the 150 GHz window channel generally used to remove the surface contribution (present on AMSU-B; a 157 GHz channel is on MHS; see Table 1) are provided by the MADRAS instrument, with two 157 GHz channels measuring the two polarizations.
Figure 2 shows the sensitivity functions of the different observing channels (∂TB/∂RH, i.e. the relative humidity Jacobians) for a standard tropical relative humidity profile for AMSU-B-like sounders and for the SAPHIR sounder. Sensitivity functions of channels located near the centre of the 183.31 GHz water vapour line peak higher in the atmosphere than those of channels that are farther from the line centre. The shape and behaviour of the sensitivity functions reveal the nonlinearity of the retrieval problem: as an absorbing component is added to the atmosphere, brightness temperatures decrease and the peaks of all channels shift upward, whereas for a drier atmosphere the functions shift downward (e.g. Schaerer and Wilheit, 1979; Blankenship et al., 2000). In that spectral area the Rayleigh–Jeans approximation directly links the measured brightness temperature to the vertical profile of the atmospheric opacity (i.e. the absorber amount) and to the thermal structure. Without any a priori information on the temperature profile, the distinction between the two contributions is delicate, yielding to estimate directly the relative humidity profile, although in the Tropics the variability of the temperature field plays a minor role in the relative humidity variability (Peixoto and Oort, 1996).
Both multivariate regression methods and neural network algorithms are built on a training dataset that is assumed to be representative of the atmosphere. The choice of training dataset is thus essential since it holds all the physics of the problem. Iterative schemes adjust the estimated relative humidity profile using successive perturbations on an a priori state of the atmosphere (selected, for example, using a Bayesian approach; Rieder and Kirchengast, 1999; Rosenkranz, 2001) and the physical constraint comes from radiative transfer computations, the best profile minimizing a cost function (e.g. Wilheit and Al-Khalaf, 1994).
For the Megha-Tropiques mission, the operational algorithm is a neural network algorithm (a multilayered perceptron) that combines the SAPHIR and MADRAS measurements to retrieve a six-layer relative humidity profile (for details of the method see Aires et al., 2012, this issue). To physically constrain the retrieval, a priori information from ECMWF analyses (collocated temperature profile and surface temperature over land), and a climatology of microwave land surface emissivities derived from Special Sensor Microwave/Imager (SSM/I) observations complete the input vector. In the present study, we decided to use an alternative approach (see details in section 5.2) to this operational algorithm in order to suggest new scientific possibilities, which will be discussed.
As stated earlier, this study aims at showing the expected gain of the SAPHIR and MADRAS combination and therefore focuses on the Tropics, limited to ±30° around the Equator. For this purpose, we consider only the oceanic situations (emissivity around 0.4, corresponding to a cold background) in order to avoid the problem of a strong contribution of the land surfaces to the upwelling radiation at these frequencies (emissivity typically between 0.8 and 1) (e.g. Bennartz and Bauer, 2003; Karbou et al., 2010).
Seeing through clouds is the big advantage of microwave soundings over infrared soundings. Contamination by clouds is largely negligible in the 183.31 GHz band, although a significant depression occurs from scattering of the upwelling radiation back to the surface by precipitating particles of deep convective clouds and cirrus anvil clouds (e.g. Rosenkranz et al., 1982; Burns et al., 1997; Greenwald and Christopher, 2002). Methods that filter out the diffusive cases exist and allow interpretation of the 183.31 GHz TB in terms of relative humidity only (e.g. Hong et al., 2005; Buehler et al., 2007). However, in the present case where no observed TBs are yet available, the development of a realistic training database requires association of a synthetic TB to each thermodynamic profile, which is done using a radiative transfer model (see section 3.4).
To our knowledge, there is no database of thermodynamic profiles that also includes in situ measurements of the cloud profile (i.e. the liquid and ice water contents). To overcome this problem, cloud profiles could be extracted from the ECMWF archive, but a recent evaluation study (Delanoe et al., 2011) shows that, with respect to the ice water path (IWP) derived from joint observations by CloudSat and CALIPSO, the ECMWF model overestimates very thin clouds and underestimates the global IWP. The depression in the TB induced by ice particles being a strong function of microphysics (Liu and Curry, 1996), it would be unwise to use the ECMWF cloud profiles to produce all-sky TBs.
For this reason, we decided to concentrate on the case of non-diffusive atmospheres in order to avoid any bias in the computation of synthetic TBs that could inevitably translate into a bias of retrieval, not due to the retrieval algorithm itself but to the inputs of the algorithm.
3.2. Radiosounding measurements
Radiosondes are extracted from the ECMWF operational sounding archive used during the assimilation process in the ECMWF reanalyses. The data (pressure, temperature and moisture) have been analysed and underwent quality control in order to discard incomplete profiles (threshold of 30 hPa for temperature and 350 hPa for humidity), and a vertical extrapolation was applied up to the top of the atmosphere (2.10−3 hPa) using climatology. Finally, the measurements are interpolated on a fixed-pressure grid (the ARSA database, developed by the Laboratoire de Météorologie Dynamique; see http://ara.abct.lmd.polytechnique.fr/index.php?page=arsa). The selected clear and oceanic cases cover the 1996–2006 period and form a base of about 1400 profiles, limited for the present exercise to 21 levels in the troposphere (from the surface up to 100 hPa).
3.3. ECMWF reanalyses
ERA-Interim is the latest global atmospheric reanalysis produced by the ECMWF, covering the period 1989 onward (Berrisford et al., 2009) and provides a bridge between the former reanalysis ERA-40 (1957–2002) and the next-generation reanalyses planned at ECMWF. The assimilating model represents the basic dynamical fields on 60 vertical hybrid levels from the surface up to 0.1 hPa. Its main improvements concern many aspects, among them the use of variational bias correction for satellite data and a revised humidity analysis. In the present study, we selected clear-sky and oceanic temperature and humidity profiles from the reduced resolution grid (1.125°×1.125°) for the months of July and December 2001. This sample represents about 40 000 profiles, with equal weight for both months. Finally, in order to easily compare these profiles to the radiosounding profiles, they have been interpolated from the hybrid model levels to the fixed-pressure grid of the latter.
3.4. The RTTOV radiative transfer model
The Radiative Transfer for Television and Infrared Observation Satellite Operational Vertical Sounder (TIROS-OVS) model (RTTOV version 9.3, supported by ECMWF NWP SAF Saunders et al., 1999; Matricardi et al., 2004) is used to simulate AMSU-B, SAPHIR and MADRAS instruments. This model is based on a regression scheme of the transmittances and thus allows for rapid simulations of brightness temperatures for satellite infrared and microwave radiometers for a given atmospheric state. The oceanic emissivities are computed by the FASTEM-3 surface model (Deblonde and English, 2001). For the present study, AMSU-B and SAPHIR brightness temperatures are simulated at nadir viewing angle, while for MADRAS the simulations are performed at the viewing angle of 53° (conical scan).
4. Discussion on the training dataset
The use of reanalysis profiles for training purposes in a retrieval problem is more and more widespread since they offer a larger spatial and temporal coverage of the system, thus forming statistically representative training databases of various atmospheric situations. However, while such databases are very well suited for the study of integrated contents of water vapour (Brogniez et al., 2009; Obligis et al., 2009), their use for the purpose of the vertical distribution of relative humidity is more delicate. Indeed, recent comparisons between satellite observations of single-layer upper tropospheric humidity (∼250 hPa, estimated from HIRS and AIRS) and reanalyses (ERA-40 and NCEP) have highlighted discrepancies at the interannual scale (Huang et al., 2005; Chuang et al., 2010) that are reduced in the ERA-Interim model (Dee and Uppala, 2009). However, on the vertical the ERA-Interim humidity field still exhibits noticeable inconsistencies in the tropical water and energy budget (Chiodo and Haimberger, 2010; Eymard et al., 2010), indicating the need for special care when using profiles of moisture from reanalyses.
The vertical differences between profiles from reanalyses and profiles from in situ observations are illustrated in Figure 3, which shows the vertical correlations between the various levels of relative humidity provided by ERA-Interim and by the radiosounding record. The structures of the correlation matrices reveal noticeable differences between the two sets of profiles: the relative humidity profiles of the radiosoundings show a strong correlation between adjacent levels of the upper troposphere (∼400–100 hPa), which is not present in the ERA-Interim set, while the latter show a weak pattern of correlation throughout the free troposphere. The differences of patterns in the correlation matrices suggest that some information on the mechanisms that regulate the upper tropospheric humidity (roughly 350–100 hPa) is missing in the ERA-Interim profiles. Indeed, at these levels, the upper tropospheric moisture is mainly influenced by the level of maximum outflow of saturated air by deep convective clouds (see Folkins etal., 2002; among others), together with drying effects induced by mixing or extratropical air intrusion (e.g. Pierrehumbert and Roca, 1998). Thus, although in ERA-Interim the moist physics, in particular, has been improved (Chuang etal., 2010) the differences on the vertical can bias training and thus affect retrieval of the relative humidity profiles. Based on these results, we decided to use radiosoundings instead of re-analysed profiles for elaboration of the training database. However, this paper is not aimed at studying thoroughly these differences and this is left for a future work in the framework of profile retrieval.
5. Comparative study: three versus six channels around 183.31 GHz
5.1. Characteristics of the training dataset
Figure 4 summarizes the characteristics of the sampled database. The total column water vapour (TCWV) evolves with the sea-surface temperature following a well-known pattern of the tropical atmosphere, with an increase in the TCWV as the surface temperature increases (see Stephens, 1990; among others). Synthetic brightness temperatures are associated with each thermodynamic profile with the RTTOV model (see section 3.4). Then, for the present exercise, the 21 levels of relative humidity are averaged into the six layers, delimited in Figure 3: 100–200, 200–325, 325–450, 450–700, 700–900 and 900 hPa–surface. These limits have been defined to frame the structure of the correlation matrix. This database of six-layer relative humidity profiles is finally randomly split in two: a training base (50%) and a validation base (50%).
5.2. Description of the retrieval model
To illustrate the interest of the combination of SAPHIR and MADRAS observations, a generalized additive model (GAM; Hastie and Tibshirani, 1987) is developed for estimation of the tropospheric relative humidity at a given level k, namely the response (or explained) variable Yk, from brightness temperatures only, namely the explanatory variables Xj. Such a model allows one to infer nonlinear relationships between the response variable and the different chosen explanatory variables. This model has the form
and expresses the expectation of Yk (here the relative humidity of level k) conditionally on a p-dimensional covariate X (X1,…,Xp) (here the vector of TBs), as a sum of fj(Xj). fj(·) are smooth non-parametric functions of the explanatory variables, and is the residual that follows a Gaussian distribution. In the present case, the basis of regression cubic smoothing splines (sj(·); third-order polynomial functions) are used to represent the nonlinear variations fj(·) of the TBs. Such functions offer a nice trade-off between over- and under-smoothing. In fact, a GAM is a conditional density function. It consists of two components: a parametric conditional normal density function, and a functional relationship between the normal parameters and X.
Figure 5 shows the estimated spline functions sj(·) for determination of the 325–450 hPa relative humidity layer with the six TBs of SAPHIR as explanatory variables, using the training dataset described above. In this figure, the x-axes correspond to each explanatory variable Xj and the y-axes show the contribution of Xj, i.e. sj(Xj), to explain Y325−450. Hence, according to the weighting functions presented in Figure 2, the 325–450 hPa layer is mainly documented by channels ±2.8 GHz (X4), ±1.1 GHz (X5) and ±0.2 GHz (X6), with a major contribution of the ±1.1 GHz channel (X5). Figure 5 shows indeed that this channel has the larger dynamic of variation ([−1;1]), with an increase of TB as the 325–450 hPa relative humidity decreases, while the X4 and X6TBs have a smaller contribution to the signal ([−0.3;0.8] and [−0.7;0.5] respectively). Finally, the estimated contributions from each of the six TBs to the variation of the 325–450 hPa relative humidity are nonlinear, the nonlinearity being higher for the 183.31 ± 4.2 GHz TB (X3) and the 183.31 ± 2.8 GHz TB (X4) than for the other TBs. This highlights the flexibility of a GAM to represent more or less strong nonlinear relationships.
A GAM was developed for each of the six layers of relative humidity. In order to quantify the gain induced by the six channels of SAPHIR with respect to the three usual channels of a AMSU-B-like radiometer (one near the centre and two more), a GAM has been applied for these two configurations. The statistical characteristics of each model are summarized in Table 2, with the percentage deviance of the data explained by the model, the mean and the variance of the error for each layer (estimated − observed relative humidity), and the Pearson correlation coefficient in order to quantify the scatter induced by the model. The deviance measures how much fit is lost (in terms of likelihood) by the modelling compared to a perfect (or saturated) model: D = −2[lnLc − lnLs], with Lc the likelihood of the current model and Ls the likelihood of the saturated model.
Table 2. Statistics of the different GAM-based estimations of the six-layer relative humidity profile (radiosoundings from the 1996–2006 period) using a SAPHIR-only configuration compared to an AMSU-B-like configuration (values in parentheses). For the model itself the deviance explained (%) is provided. For the validation step: the mean (%) and variance (%2) of the residuals (estimated − observed relative humidity) as well as the Pearson correlation coefficient are given.
SAPHIR-only (AMSU-B like)
deviance explained (%)
Mean of residuals (%)
Variance of residuals (%2)
The addition of three channels clearly improves the restitution of the relative humidity throughout the entire troposphere, with a more pronounced improvement for the upper layer, and an increase of the explained deviance in the adjustment step, from 55% with a AMSU-B radiometer to 87.1% with a SAPHIR radiometer, and a noticeable reduction of the scatter, with the correlation coefficient reaching 0.91 with SAPHIR data (0.76 with AMSU-B). Similar observations are made for the other layers, and more specifically for the lower layers. Figures 6 and 7 support these numbers, with the use of box-and-whiskers diagrams to represent the variability (median, first and third quartiles, and upper and lower limits of the distribution) of the observed and estimated data at each layer. While the mean profile does not show strong differences when the GAM is elaborated with three channels or with six channels (Table 2), the addition of three channels in the 183.31 GHz band allows for a better restitution of the distribution of the data. Indeed, from the AMSU-B configuration (Figure 6(a)) to the SAPHIR configuration (Figure 6(b)) the boxes representing the majority of the data (between 25% and 75% percentiles) are closer to each other, as well as the whiskers. This is particularly convincing for the two upper layers and for the lowest layer.
5.4. Contribution of the MADRAS channels
Three channels of the MADRAS radiometer are sensitive to the water vapour content of the atmosphere: the 23.8 GHz channel and the two 157 GHz channels (V and H polarizations). While the 23.8 GHz channel is often used to retrieve the integrated water vapour content of the atmosphere, measurements at 157 GHz are useful for the estimation of the water vapour continuum (e.g. English et al., 1994). Thus, although the prime role of this channel is to improve the description of high-level ice clouds associated with the convective systems, it can also be used as an additional channel for observation of the water vapour in the low-level atmosphere. For this purpose, a 150 GHz channel is present on the AMSU-B/NOAA radiometers, whereas there is a 157 GHz channel on the MHS/MetOp radiometers. To test their contributions, a GAM was developed adding these three channels of MADRAS to the six channels of SAPHIR. The statistics of this model are provided in Table 3 and a quick comparison with Table 2 clearly shows the interest in these channels for the restitution of the atmospheric water vapour: the model adjustment is improved for the whole profile, and for each layer the scatter is reduced, with a greater reduction for the lower layers. Again, the box-and-whiskers diagrams of Figures 6(c) and 7(c) support the interest of the combination of SAPHIR and MADRAS for the restitution of relative humidity profiles, since the distributions of the estimated values are much closer to the observed ones when information on the 23.8 GHz channel and of the two 157 GHz channels are added to the model.
Table 3. The same as Table 2 but for the Megha-Tropiques payload, i.e. including the 23.8 GHz and 157 V and H GHz channels of MADRAS.
SAPHIR and MADRAS
deviance explained (%)
Mean of residuals (%)
Variance of residuals (%2)
The purpose of this article is to present the expected gain in the retrieval of the relative humidity profile by the microwave payload of the Megha-Tropiques platform. Synthetic brightness temperatures of the two microwave radiometers are computed using the RTTOV-v9.3 radiative code and a 10-year radiosounding dataset covering the tropical belt. Our results have revealed noticeable differences of patterns in the correlation matrices of the relative humidity profiles between this source of profiles and ERA-Interim profiles. Note that the impact of these differences in the retrieval of ∼200 hPa wide layers of relative humidity has not been evaluated and should be performed in a dedicated work.
The question of all-sky profiles should be also addressed: a combination of thermodynamic profiles measured by radiosoundings and of profiles of cloud liquid and ice-water content derived from CloudSat-CALIPSO observations could be an interesting approach that should help in simulating realistic cloudy TBs. Such an approach needs to be fully evaluated against the use of ECMWF profiles, which offer the advantage of consistent all-sky thermodynamics.
The retrieval algorithm uses a generalized additive model that allows modelling of the strong nonlinearities of the problem with additive cubic smoothing splines. The use of AMSU-B-like information versus the use of SAPHIR observations as inputs of the retrieval algorithm has shown a clear reduction of the error of retrieval, the highest improvement concerning the upper and lower layers of the profile, with a variance of the residuals divided by 2 for these layers and correlation coefficients reaching at least a value of 0.8. The addition of MADRAS measurements, at 23.8 GHz and 157 GHz, improves even further the retrieval: throughout the troposphere, the mean residual is less than 1% and the correlation coefficients exceed 0.89. Although the present exercise is restricted to the oceanic and clear-sky situations for simplification purposes, similar improvements should be expected when dealing with cloudy atmosphere and continental surfaces. For those cases, however, it is likely that the other channels of MADRAS (at 18.7 and 36.5 GHz) will bring valuable information to better constrain the retrieval of the lowest layer. Moreover, the use of other explanatory variables from climatologies such as the sea-surface temperature or vertical stratification of the atmosphere (e.g. Obligis et al., 2009), and the correlations between the successive levels, should also provide additional constraints.
Currently, numerous methods of retrieval exist, among them the neural network approach (e.g. Cabrera-Mercadier and Staelin, 1995) or the use of iterative schemes mixing physical constraints and statistics (e.g. Kuo et al., 1994): all of them provide a conditional estimate of the expectation of relative humidity given a set of inputs. A future study will consider the use of the generalized additive model presented here to estimate the whole distribution and not only its expectation. In other words, a GAM can be applied to generate ensembles of probable relative humidity profiles conditioned on given TBs (and other explanatory variables). Such a conditional distribution of the relative humidity profile should be more advantageous to users than deterministic products since the distribution will enable access to the whole variability of the atmospheric moisture. It will also be useful to address the issue of the possible non-Gaussian or even non-symmetric distribution of the relative humidity at each layer to infer a more appropriate model. A generalized additive model for location, scale and shape (GAMLSS; Rigby and Stasinopoulos, 2005) may be used for such a purpose.
The authors acknowledge the LMD/ABC(t)/ARA group for producing and making available to the community their radiosounding database ARSA (http://ara.abct.lmd.polytechnique.fr/index.php?page=arsa). Thanks also to Rémy Roca and to Nicolas Viltard for fruitful discussions on the radiative transfer in the microwave spectrum. The first author thanks Mathieu Vrac for his advice on the use of generalized additive models and for proofreading the manuscript. Finally, the authors thank the two anonymous reviewers for their comments, which greatly enhanced the quality of the manuscript.