Introduction

Latent heat released by precipitation accounts for 75% of the heat transfered from the surface to the atmosphere (Simpson et al. 1996), and precipitation is intimately connected with both large- and small-scale circulations in the atmospheres and oceans. Of all environmental parameters, rainfall plays one of the most significant roles in global heat and water budgets, yet a lack of accurate rainfall information persists over vast areas of the globe. The presence of large oceanic regions, deserts, and jungle, particularly in the Tropics where two-thirds of all precipitation falls, precludes the use of rain gauge and other ground-based rainfall measurements in these regions. As a result, spaceborne instruments provide the only means of obtaining global measurements of rainfall (Simpson et al. 1988).

Numerous techniques based on satellite measurements of microwave emission have achieved some success in mapping global rainfall [see, e.g., Wilheit et al. (1977); Spencer et al. (1989); Kummerow and Giglio (1994); Smith et al. (1994); Petty (1994a,b)]. One drawback to these methods, however, is that few are capable of accurately determining the vertical structure of precipitation at the resolution required in global weather and climate models (e.g., Cess et al. 1993; Wang et al. 1996). In addition, such methods suffer from nonuniqueness resulting from the fact that substantially different profiles of precipitating liquid and ice hydrometeors in the atmospheric column yield similar microwave radiance signatures introducing a substantial element of uncertainty in their surface rain-rate estimates.

The profiling capabilities of spaceborne radars have the potential to add vertical structure information to complement techniques based on passive measurements. As a result, the Tropical Rainfall Measuring Mission (TRMM) carried the first radar designed for precipitation measurements flown in space. Given the success of the 14-GHz precipitation radar (PR), which has been operational for more than 3 yr, future spaceborne radar missions are being discussed. The potential for extensive spatial coverage offered by satellite-based radars has also been recognized by the CloudSat science team (Stephens et al. 2000, manuscript submitted to Bull. Amer. Meteor. Soc.), and a 94-GHz cloud profiling radar (CPR) will be flown on that satellite, which may be capable of measuring light rainfall in addition to clouds. As a result of the logistics of flying a radar in space, most notably size, weight, and power restrictions, only frequencies greater than 10 GHz are practical for a spaceborne radar system (Fujita and Satake 1997). Consequently, all spaceborne radars are subject to significant attenuation by the rain and clouds they are measuring, necessitating new radar-rainfall retrieval algorithms that explicitly account for attenuation effects. Hitschfeld and Borden (1954) were the first to propose a method for directly correcting measured reflectivities for attenuation. Since then, numerous retrieval schemes have been developed that incorporate aspects of this early work along with the more recent technique of constraining retrievals using path-integrated attenuation (PIA) derived from surface return echoes. Iguchi and Meneghini (1994), Marzoug and Amayenc (1994), and Amayenc et al. (1996) describe many such algorithms and evaluate them in the context of airborne and spaceborne operation.

Many ground-based radar retrieval algorithms are based on the Rayleigh approximation that the rain drops are small with respect to the radar wavelength and assume some drop size distribution (DSD) to derive a simple relationship between radar reflectivity and rain rate. From an operational standpoint these algorithms are computationally efficient, relatively simple to implement, and produce useful results but they can incur uncertainties that are difficult to estimate as a result. Errors associated with the assumption of an idealized size distribution were pointed out by Wexler (1948); Gunn and East (1954) showed that the Rayleigh approximation is not valid for wavelengths less than 3 cm (or frequencies greater than 10 GHz), emphasizing the need for an algorithm that is not restricted by such assumptions in spaceborne applications. This has been recognized by the designers of the TRMM PR rainfall profile algorithm who select a DSD based on rain type and the presence or absence of a bright band and correct the DSD based on a PIA estimate (Iguchi et al. 2000).

Another shortcoming of many algorithms is their inability to adapt measurements from other instruments except in a limited capacity. With the growing trend toward multisensor platforms as exemplified by the TRMM satellite, it is important to develop algorithms better suited to combining information from a variety of different sources simultaneously in rainfall retrievals. Again, this has been recognized by the TRMM science team and a first-generation combined PR–TRMM Microwave Imager (TMI) rainfall algorithm has been developed, which employs a Bayesian approach to retrieve DSD parameters from PR reflectivities with a PIA estimate derived from the 10.6-GHz channel of the TMI (Haddad et al. 1997).

Rigorous estimates of the uncertainty in the retrieved rainfall profiles are required for model validation and data assimilation applications, but many algorithms are limited to a rough estimate of their accuracy. For this reason, all operational TRMM algorithms, including the two PR algorithms described above, are required to provide estimates of their accuracy, and considerable emphasis will be placed on error diagnostics for the algorithm presented below.

This paper presents a complete algorithm for retrieving profiles of rainfall from spaceborne radar measurements that seeks to address these issues and provide an alternate method of inversion to either of those currently employed in the TRMM algorithms. This new algorithm differs from those of many previous works in that it can be adapted to use any DSD, requires no simplifying assumptions regarding particle shape provided that backscatter information can be computed, and is readily modified to include any additional measurements that are available. Section 2 introduces a general model for computing radar reflectivities for any profile of atmospheric particles. A detailed description of the inversion method is presented in section 3, and results of some synthetic retrievals are discussed in section 4. The utility of adding a simple total liquid water path constraint is demonstrated in section 5.

Attenuating radar forward model

In general, the return power P_r received by a radar, which transmits a power P_t at wavelength λ is given by

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where G is the antenna gain, θ and ϕ characterize the half-power beamwidth of the pulse, h is the length of the emitted pulse, and r is the range to the target. With k_ext defined as the attenuation coefficient, the exponential factor accounts for the two-way attenuation due to all atmospheric constituents along the slant path s of the radar beam. Last, η is the radar reflectivity per unit volume V given by

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where the σ_b are the backscatter cross sections (per unit volume) of the particles in the target volume.

Many ground-based radar models assume that the particles in the target volume are much smaller than the wavelength of the incident radiation πD/λ ≪ 1 and use the Rayleigh approximation for the backscatter cross-section yielding

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where Z = ∫ D⁶N(D) dD is the Rayleigh reflectivity factor and K = (m² − 1)/(m² + 2) is the dielectric factor of the scattering target. Here N(D) dD is used to denote the number of particles with diameters between D and D + dD in the target volume, and m = n − ik is the wavelength and temperature-dependent complex refractive index of the scatterers. At the frequencies used in most spaceborne applications, the Rayleigh approximation breaks down in rain. Figure 1a illustrates the ratio of Mie to Rayleigh reflectivity factors at 14, 35, and 94 GHz as a function of size parameter χ = πD/λ, assuming spherical liquid raindrops distributed according to the DSD of Marshall and Palmer (1948, hereinafter referred to as MP). Plotted in this way, the frequency dependence of the ratio can be attributed solely to differences in the real part of the refractive index, which is greatest at 14 GHz and lowest for 94 GHz. The results show that using the Rayleigh approximation leads to large errors for size parameters greater than about 0.15 when integrating over a size distribution. Cast as a function of rainfall rate in Fig. 1b, the Rayleigh approximation is only suitable for rain rates less than 2.5 mm h⁻¹ at 14 Ghz and is not applicable in any appreciable rainfall at 35 or 94 GHz.

Based on these results, a more general form for the reflectivity η is adopted:

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where P(Θ = 180) is the scattering phase function evaluated in the backscatter direction, Q_sca is the scattering efficiency equal to the ratio of the particle's scattering cross section to its geometric cross section, and N(D) is an arbitrary number distribution of particles as a function of their effective diameter D. A series of tables of P(Θ = 180) and Q_sca are created as a function of particle size using Mie theory for spherical particles or some alternate computational method for more complicated particle shapes. This allows flexibility with regard to particle size, shape, and number concentration, all of which can be adapted to any given problem as needed.

Combining Eqs. (1), (3), and (4), a model to simulate an effective radar reflectivity factor for any atmospheric profile is constructed:

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Unattenuated reflectivities are computed at each level, along with an attenuation correction due to the hydrometeors in that level. Each reflectivity is then corrected in turn for the attenuation through the levels above it.

Optimal estimation approach to inversion

Many radar algorithms assume a size distribution and Rayleigh scattering to obtain relations of the form Z = aR^b and k = αR^β, which are directly invertible to obtain rain rate given radar reflectivity measurements. These methods have the advantage that they cast the problem into a simple analytical form, which results in a computationally quick and relatively simple retrieval algorithm. These assumptions also lead to two potential sources of uncertainty: 1) the assumption of a DSD strongly influences the values of a, b, α, and β, and 2) the Rayleigh assumption is not valid for radars with frequencies in excess of 10 GHz, a characteristic of both the TRMM PR and CloudSat CPR. In the present work, a significantly different technique is applied to invert Eq. (5) based on the works of Rodgers (1976, 1990) and Marks and Rodgers (1993) and applications of their work by Engelen and Stephens (1999) and Miller et al. (2000). Within the constructs of this approach a high degree of generality is maintained, which allows one to vary assumed particle size and shape and facilitates the addition of complementary information from different sensors if so desired.

Basic theory

Denoting the forward model in the previous section by F, a general profile of radar reflectivities Z can be expressed as

ZFRbϵ_Z(6)

where ϵ_Z is a vector of measurement uncertainties. For the purpose of illustrating the method, we cast the retrieval in terms of the vertical profile of rain rate R (mm h⁻¹), representing all other parameters in the radar forward model by b. This choice is purely arbitrary and the algorithm could, equivalently, be developed in terms of any other set of parameters that completely describe the DSD. An alternate formulation may be particularly useful when other observations are available to supplement the reflectivities to constrain the shape of the size distribution. In either case, an appropriate F to be used in the retrieval can be selected from criteria such as rain type, freezing level, or the presence of a bright band and uncertainties in the parameter vector b must be explicitly accounted for in the inversion approach.

Our problem is to estimate the rain-rate profile that most likely produced a given observed profile of reflectivity. Following Olson et al. (1996), the optimal rain-rate profile is given by

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where pdf(R) is a probability density function proportional to the conditional probability that R is the true atmospheric rain-rate profile given the observed reflectivity profile P(R = R_true | Z_true = Z). Invoking Bayes's theorem, the probability density function can be recast as a product of the probability of observing a reflectivity profile Z given a simulated reflectivity profile, F(R), and the a priori probability that R is the true rain-rate profile,

RPZFRP_aR(8)

Jaynes (2001) argues that, according to the principle of maximum entropy, the Gaussian distribution is the “most honest” representation of errors if only the mean and variance of a probability distribution are known. Alternate distributions, unless rigorously justifiable, add spurious information to the retrieval, therefore biasing the results. For this reason, uncorrelated and unbiased Gaussian statistics will be adopted to ensure that only distribution mean and variance information are introduced in the retrieval. Under this assumption,

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where 𝗦_Z is the total error covariance matrix representing the sum of the measurement and model errors. Similarly, if 𝗥_a represents some a priori guess at a rain-rate profile, the probability of finding the true atmospheric rain-rate profile R will be expressed as

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where 𝗦_a is the error covariance matrix associated with the a priori guess. Given the symmetry of the Gaussian distribution, the “best” estimate of a rainfall profile from Eq. (7) is that which maximizes the joint probability

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or, alternatively, minimizes the scalar cost function

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with respect to the rain-rate profile R. The resulting rain-rate profile is

R̂R_aa^T−1_ZZFR̂(13)

Here 𝗞= ∂F/∂R is the Kernel or weighting function representing the sensitivity of the model to the parameter being retrieved. From Eq. (13) it is clear that a model that is very sensitive to the rainfall-rate profile is desirable since the Kernel functions weight the measurement portion of the retrieval. It is worth noting that, under the present assumptions of uncorrelated, unbiased, Gaussian error statistics, the maximum-likelihood estimate computed here is identical to the minimum variance solution (Daley 1991), which guarantees that R estimated using Eq. (13) will be the best possible fit to the observations given our uncertainty estimates.

Equation (13) is implemented numerically through a Newtonian iteration scheme,

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where 𝗦_R̂ = (𝗦⁻¹_a + 𝗞^T_i𝗦⁻¹_Z𝗞_i)⁻¹ is the covariance matrix of the retrieved rainfall profile. The iteration proceeds until such time as the covariance-weighted square difference between successive estimates is much less than the total number of independent variables being retrieved, in this case the number of layers in the atmosphere, N_l:

R̂_i+1R̂_i^T−1_R̂R̂_i+1R̂_iN_l(15)

Error diagnostics

The preceding optimal estimation formulation of the inversion problem furnishes a number of useful diagnostics to measure the quality of the results. The most important of these is the covariance matrix of the retrieved parameters:

_R̂⁻¹_a^T_i⁻¹_Zi⁻¹(16)

Equation (16) provides a direct estimate of the uncertainty in the retrieved rain-rate profile due to uncertainties associated with the a priori profile, forward model, and measurements themselves. The diagonal elements of 𝗦_R̂ are the variances in the retrieved rain rate at each level while the off-diagonal elements indicate correlations between uncertainties at different levels. In regard to the details of the forward model F, for example, uncertainties in the parameter vector b, such as those associated with assumptions regarding the highly variable DSD, must be accounted for in the measurement and model covariance matrix 𝗦_Z. An appropriate 𝗦_Z dictates the reliability of the observations and forward model in the retrieval and ensures a reliable estimate of retrieval accuracy through Eq. (16).

Defining the matrix

_R̂^T_i⁻¹_Zi(17)

Eq. (14) can be written,

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In this way, the contributions to the retrieved profile from the measurements and from the a priori profile are isolated. In an ideal retrieval, the so-called a priori matrix 𝗦 will be an identity indicating that the retrieval is based exclusively on the measurements. Departures from the identity represent a combination of influences from the a priori data and reflectivity measurements from different atmospheric levels on the rain rate retrieved at a given level.

One can investigate the validity of the assumption of Gaussian error statistics using the χ² test. If the assumption is valid,

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will be approximately equal to the number of independent parameters in the retrieval, in this case N_l. In cases where our physical assumptions, particularly those relating to the DSD, are grossly different from those being observed or in the event that the Gaussian distribution and assumed variances do not properly characterize uncertainties in the measurements and model, χ² will be much greater than N_l. In global applications, a significant number of retrievals characterized by large χ² is indicative of overoptimistic error covariance assumptions in the algorithm, such as a misrepresentation of the DSD variability encountered by the satellite. On the other hand, if χ² is much less than N_l, the measurement, model, and/or a priori covariances may have been overestimated and 𝗦_R̂ will correspondingly overestimate the retrieval uncertainty.

Synthetic retrievals

The algorithm was tested using the cloud database used in the Goddard Profiling Algorithm (GPROF), which is currently employed in rainfall retrievals from the TRMM Microwave Imager [see Kummerow and Giglio (1994) and Olson et al. (1996) for details regarding the cloud database and the GPROF algorithm]. The database consists of approximately 10 000 cloud and rainfall profiles varying from nonprecipitating cirrus and stratus clouds to heavily precipitating cumulonimbus cloud complexes and represents a wide range of conditions with which to test the algorithm. In order to illustrate the utility of the model, reflectivity profiles with 0.5-km vertical resolution were computed for all cloud and rain-rate profiles in the database assuming an MP distribution of spherical raindrops and cloud droplets of diameter D, N(D) = N₀e^−ΛD. The distribution is entirely described by the rain rate R assuming the parameters N₀ = 8 × 10⁶ m⁻⁴ and Λ = 4.100R^−0.21 mm⁻¹. These reflectivity profiles were then perturbed by adding a randomly distributed noise component and run through the retrieval algorithm to recover the original profile. The noise was set to 1 dB (or 23%) for profiles with surface rain rates less than 20 mm h⁻¹ and 2 dB (or 46%) for higher rain rates to reflect the fact that errors due to particle shape are more significant for larger raindrops than smaller ones (Green 1975). In the retrieval, the measurement covariance matrix is assumed to be diagonal with elements equal to 1.0 dB² (or 4.0, where applicable). No attempt is made at this time to account for uncertainties in DSD parameters introduced by their variability in nature but it is acknowledged that this contribution will be extremely important in operational applications using this technique. A variance of 25.0 mm² h⁻² is assumed in all a priori rain rates and all correlations between errors at different levels are neglected. In this way, the retrieval is based far more heavily on the measurements and the a priori merely serves as an initial guess and an extremely weak positivity constraint on the retrieval. In addition, the assumption of high a priori variances minimizes errors incurred as a result of neglecting correlations between uncertainties at different levels, which are, in practice, extremely difficult to determine.

A scatterplot of retrieved surface rain rate against the GPROF input at 14 GHz is presented in Fig. 2. The results show excellent agreement for surface rain rates below 40 mm h⁻¹. Significant attenuation by large raindrops at higher rain rates, however, degrades the quality of the retrievals. Figure 3 shows similar results obtained from a retrieval using reflectivities at 94 GHz. Attenuation is severe for all rainfall at 94 GHz and only rain rates less than 1.5 mm h⁻¹ are retrieved accurately in the absence of additional information.

Figures 4 and 5 present the estimated error in the retrieved surface rainfall rate and the corresponding component of the a priori matrix at 14 GHz. Together, the two figures demonstrate that, under the present assumptions, the uncertainty in retrieved rainfall is below 30% for all profiles with surface rain rates between 0 and 40 mm h⁻¹. In heavier rain, the retrieval relies heavily on the a priori guess, so, although the retrieval errors appear to be acceptable, the results do not reflect the information contained in the measurements and must be discarded.

At 94 GHz, the enormous retrieval uncertainties illustrated in Fig. 6 demonstrate the fact that reflectivity measurements do not provide sufficient information to accurately determine surface rainfall rate. The fractional error only remains within tolerable limits up to about 1.0 mm h⁻¹. In heavier rain, uncertainties become large and differences in the vertical distribution of rainfall above the surface give rise to scatter in the results. Many of the 94-GHz retrievals also yield extremely high values of χ² (not shown) indicating that the retrieved profile of rainfall seldom attains satisfactory agreement with the observations.

These results may be attributed to the method adopted for applying the attenuation correction. Since the procedure, which is analogous to the Hitschfeld and Borden (1954) method, first corrects the layers closest to the radar and then adjusts lower layers according to rain rates derived from these corrections, it leads to the propagation and magnification of errors as the algorithm proceeds to lower levels within the cloud. Figure 7, which presents retrieved rain rates at 4.0 km, near the top of the liquid rain column, supports this hypothesis. Although the retrievals exhibit some deviation from the input at high rain rate, the agreement is clearly far superior than that at the surface. Figure 8 shows that uncertainties in retrieved rain rates are better constrained at 4.0 km.

Precipitation water path as a constraint

Because of the severity of attenuation by rainfall at high frequencies, it is not sufficient to rely exclusively on a profile of radar reflectivity to make estimates of rainfall profiles. Fortunately, many spaceborne platforms such as TRMM, offer additional sources of information from other instruments. Microwave radiometers can be used to obtain an estimate of the total precipitation water path [PWP; see, e.g., Fujita et al. (1985); Alishouse et al. (1990)], which is defined as the column-integrated liquid water that is in the form of precipitation. This estimate complements the radar reflectivity measurements in that it imposes a constraint on the total attenuation through the rain column.

Theory

In the presence of such a constraint, the cost function can be expressed as

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where PWP_obs is the observed PWP with standard deviation σ_PWP, and PWP_sim is the liquid water path obtained by integrating the rain-rate profile in height. Minimizing with respect to R, we obtain

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with 𝗦_R̂ = (𝗦⁻¹_a + 𝗞^T_i𝗦⁻¹_Z𝗞_i + (Δz²/σ²_PWP)L_iL^T_i)⁻¹. The vector ΔzL consists of the derivatives of PWP with respect to the rain rate at each level.

One measure of the strength of the constraint is to determine the relative contributions of the measurements, a priori guess, and the constraint to the retrieval error covariance matrix 𝗦_R̂. Defining

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the total retrieval error covariance matrix can be expressed as

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where 𝗦_PWP is a diagonal matrix with σ²_PWP on the diagonal. Diagonal elements of each term represent the contributions of the measurements, a priori guess, and the constraint to the retrieval uncertainty making it possible to assess the relative strengths of the three components in determining the rain-rate profile. In addition, the χ² test can, again, be employed to test the assumption of Gaussian statistics and the values assumed for the variances:

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Conclusions

With the growing trend toward space-based observation platforms that carry high-frequency cloud and rain profiling radars, the need for accurate, flexible algorithms for retrieving rainfall at attenuating frequencies is paramount. In this paper, we have presented an algorithm that is not, in principle, restricted by assumptions of size distribution or particle shape and accounts for attenuation through direct integration of two-way extinction due to all rain along the path to each target layer. The algorithm maximizes a posterior probability density function derived from an assumption of uncorrelated, unbiased errors following a Gaussian distribution. The Gaussian distribution, by virtue of its symmetry, adds the least bias of any assumed error distribution in the absence of additional information and ensures equivalence between the location of its maximum and its mean, a necessary condition for the solution to be optimal.

Preliminary results, using simplifying assumptions, illustrate the utility of the algorithm, particularly at low frequencies. At 14 GHz, synthetic retrievals of rainfall profiles using the GPROF cloud database accurately reproduce the input for rain rates as high as 40 mm h⁻¹. At 94 GHz, however, the algorithm grossly over- or underestimates surface rainfall at all rain rates above 1.5 mm h⁻¹ because of instability inherent in the method employed to correct for attenuation. The PIA approach to correct for attenuation, which makes use of the surface-return echo to estimate the total attenuation through the rain column, is stable and yields much more accurate results when attenuation is severe (Iguchi and Meneghini 1994). This is confirmed by the remarkable improvement in the retrieval when a precipitation water path constraint is imposed on the retrieval. Since the PIA derives from the precipitation water path, both constraints perform essentially the same function. Either method (or both) can be used in the present algorithm with very little modification. It is also worth noting that the application of this algorithm to the CloudSat radar is explicitly for the purpose of retrieving very light rain that may not be detectable at lower frequencies. In this light, the results presented here are encouraging.

While the assumption of an exponential DSD and spherical particles oversimplifies the problem, the early results presented above suggest that it is worth pursuing extensions of the model to include more direct estimation of size distribution parameters making more complete use of the other observations available to both the TRMM and CloudSat missions. In a slightly more general form, Eq. (12) can be written as

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where x is a more general vector of retrieval parameters, perhaps consisting of DSD parameters at each level, y is a general vector of satellite-based measurements, and F is an appropriate combination of forward models that map the retrieval parameters into the basis of these measurements. This equation can be solved in exactly the same manner as above, yielding an iterative solution to determine the best estimate of rain rate based on all available observations simultaneously. As the TRMM PR algorithms have demonstrated, directly accounting for DSD variability is of paramount importance for the next generation of satellite radar retrievals. Studies are currently under way to recast the algorithm in terms of a more general DSD and a parallel effort is being undertaken to directly incorporate radiances from the TMI to provide the more rigorous constraint required to estimate a more flexible DSD.

In the near future, the CloudSat CPR in formation with the Advanced Microwave Scanning Radiometer (AMSR-E) on board the Earth Observing System (EOS) Aqua satellite will provide an unprecedented opportunity to examine the possibility of making quantitative rainfall estimates from a combination of a microwave radiometer and a 94-GHz cloud radar in space. The algorithm presented here is ideally suited for this purpose.