1. Introduction

Clouds play a very important role within the hydrological cycle. The distributions of hydrometeor phase and size within clouds are important factors in the development of precipitation. Thus, the predictability of precipitation in numerical weather forecast models also depends on the correct description of liquid water content of developing clouds. The parameterization schemes used in state-of-the-art weather forecast models have large uncertainties and are not capable of describing actual cloud dynamics on scales that are significant for the formation of convection. One reason for this deficiency is that up to now, no appropriate datasets of microphysical cloud variables have been recorded due to restrictions in measurement capabilities. In the past, sporadic in situ aircraft measurements have been the only reliable method. Ground-based passive microwave measurements have proven to be quite accurate for determining integrated cloud liquid water (LWP; Westwater 1978; Peter and Kämpfer 1992), but retrieving the vertical distribution of cloud liquid water (LWC) from brightness temperature (TB) measurements is very uncertain because the problem is underdetermined. A possibility is to combine active and passive microwave components to infer LWC profile information. In the last 5 to 10 years, active and passive ground-based microwave technologies have been developed for monitoring clouds in a quantitative way (Solheim et al. 1998; Clothiaux et al. 1995). Modern cloud radars have sufficiently high sensitivities to quantify radar reflectivity (Z), Doppler velocity, and polarimetric qualities of clouds. High spectral, spatial, and temporal resolution allow the development of new techniques, which can help to quantify cloud microphysics at finer scales than before. In this study the Forschungszentrum Geesthacht GmbH (GKSS) 95-GHz radar was used together with a radiometer suited to measure the liquid water path with an accuracy of 10%–30%. The combination of both instruments is necessary to constrain the inherent ambiguity of the radar measurements due to the variations of the cloud drop size distribution (DSD).

In the idealized case of Rayleigh backscattering, Z is equal to the sixth moment of the cloud droplet distribution. Since LWC is proportional to the third moment of the DSD, varying DSDs cause large variations in the relationship between Z and LWC. For example, if a volume of 1 m³ contains 0.1 g of liquid water and only droplets 5 μm in radius, the reflectivity will be 8 times smaller than if the volume consists only of droplets 10 μm in radius. A very challenging problem in deriving cloud liquid water is the presence of drizzle drops (50–400 μm in radius), which evaporate on their way to the ground and hence cannot be perceived as precipitation on the ground. These droplets frequently have high contributions to Z but do not necessarily contain considerable amounts of LWC (Fox and Illingworth 1997a).

In past years many attemps have been made to relate cloud radar reflectivity factor measurements to microphysical cloud properties such as liquid water content, effective radius (r_eff), or total number concentration (N_tot). Sauvageot and Omar (1987) developed a universal Z–LWC relationship for non- or very weakly precipitating cumulus clouds. In situ aircraft measurements of drop size distribution were used to calculate Z. Here an upper limit of −15 dBZ was set to the validity of the relationship, because no clear correlation was found for higher reflectivities. This was interpreted as the limit between nonprecipitating and precipitating clouds. Fox and Illingworth (1997b) developed empirical Z–LWC relationships for stratocumulus clouds also based on in situ aircraft measurements. They proposed to distinguish between clouds with and without occasional drizzle according to the form of the vertical Z profile. Errors are estimated to be 50% for a single retrieval of LWC for a nondrizzle cloud. Liao and Sassen (1994) used the output of a one-dimensional diffusional growth cloud model to simulate radar reflectivities from predicted droplet spectra. They found that the Z–LWC relationship was mainly dependent on the activated cloud condensation nuclei (CCN) concentration, which is equal to N_tot in their model. In comparison with empirical based relationships for cumulus and stratocumulus clouds the best agreement was found for N_tot of about 100 cm⁻³. Vivekanandan et al. (1997) used a radar–radiometer combination to derive Z–LWC relationships. They used a gamma function to describe the relationships between N_tot, LWC, Z, and the median diameter D_m. In a first step the radiometer derived LWP was used to determine a constant D_m profile as a first guess. Then equations for Z and LWC were iterated in D_m and N_tot to find an LWC profile corresponding best to the measured LWP. The upper value for D_m was set to 100 μm. Frisch et al. (1995) have developed a combined radar–radiometer LWC profile retrieval algorithm for stratus clouds, where the retrieved LWC is constrained exactly to the observed LWP. Typical DSD parameters for stratus clouds are incorporated.

Here a method is proposed to combine LWP, Z profile, and an LWC model climatology in an optimal way to obtain the LWC profile. If x_i denotes a scalar measurement with variance σ²_i, then m independent measurements can be combined to

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x̂ yielding an optimal estimator for x. This method is called inverse variance weighting or optimal estimation (OE; Rodgers 1976). For profile retrieval, x_i will be a measurement vector and σ²_i the error covariance matrix. Since no direct measurements of LWC can be obtained, the inverse of a forward model function F is used to convert measurements y_i into the space of x. Virtual measurements (e.g., a model climatology) of x can be used as an x_i as long as its variance (covariance matrix) σ²_i is known.

The paper is structured as follows. Section 2 describes the forward model function F and the model climatology. A dynamic cloud model with a resolved liquid cloud droplet spectrum is used. The model is initialized with radiosonde observations to account for realistic atmospheric conditions (sections 2a, b). Then the model output is used to derive both the Z–LWC relations and the LWP retrieval algorithm from simulated passive microwave measurements. The formulation for F is given in section 2e. Attenuation of the radar beam due to hydrometeors is taken into account and a method is proposed to correct for attenuation (sections 3a, b). Section 4 gives an introduction to the theory of optimal estimation (section 4a). The application to the combination radar–radiometer–cloud model is described in detail in section 4b. Here it is shown how the covariance matrices needed for a vector formulation of (1) are constructed. In section 5 the proposed retrieval method is applied to independent model data and compared with a standard retrieval method. The applications of the algorithm to real data is shown in section 6. First, the instruments used to measure a time series of Z profiles and LWP are described (sections 6a, b). Then, the characteristics of the day of measurement (section 6c), and finally the LWC retrieval (section 6d) are discussed. The paper concludes with a summary and a description of further extensions of measurement combination using a highly sophisticated multichannel microwave radiometer.

2. Derivation of the forward model function and model climatology

c. Z–LWC relations

The vertical coordinate used to describe the dependencies of LWC and Z with height is height above cloud base. The typical form relating Z to LWC is a power-law equation of the form

Za^b(3)

(e.g., Sauvageot and Omar 1987; Fox and Illingworth 1997a). The errors when estimating LWC with this equation can be in the range of 1 order of magnitude or more due to variations in the droplet spectrum. To illustrate the sensitivity of (3), the lognormal size distribution, often used to characterize cloud and rain distributions (e.g., Feingold et al. 1997), is used:

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The three parameters describing this DSD are σ_l (logarithmic width), r₀ (modal radius), and N_tot (total number of drops per unit volume). It can be shown that (3) will be exactly satisfied if two of the three DSD describing parameters in (4) are held constant (Erkelens et al. 1998). In the three possible cases, the exponent b will vary between 1 and 4 giving rise to a wide range of variability. LWC derived from a typical cloud reflectivity value of −35 dBZ will vary in one order of magnitude if the exponent b is varied between 1 and 4.

To determine the coefficients a and b from the model simulations (3) is logarithmized to

Zab(5)

with dBZ = 10 log(Z/Z₀), whereby Z₀ is given by 1 mm⁶ m⁻³. Subsequently, in each 250-m height interval above cloud base, a weighted linear regression of (5) is performed. This means that before the regression is performed, each (dBZ, LWC) pair is classified into one of five dBZ classes (A–E). The regression is performed with an equal number of (dBZ, LWC) pairs from each class. Every pair of the class with the minimal number N_min of pairs is taken into the regression, whereas N_min (dBZ, LWC) pairs from the other classes are randomly chosen. To perform a representative regression N_min should at least be equal to 20. This condition could be satisfied up to a height of 1500 m above cloud base for the cloud-model results obtained from the TRAIN dataset. Performing a linear regression with (5) leads to a much stronger weighting of very small values of LWC than larger values due to minimizing the error between dBZ and log(LWC). This results in an almost bias-free determination of LWC in dBZ class A (Fig. 3, Table 1) but increasing absolute biases in classes of higher dBZ. The bias in class E is particularly high. This can be explained by large droplets contributing to high reflectivities but not increasing the mean LWC to values significantly larger than those in the classes C and D. This behavior of LWC over- and underestimations is used to derive a bias corrected Z–LWC relation (Table 1).

Values of the coefficient b derived from the cloud model are in the range of 1.85 to 2.58, whereas a ranges from 0.02 to 0.16. These coefficients are in good agreement with values derived by Sauvageot and Omar (1987) from aircraft measurements of DSDs in a variety of cumulus, stratocumulus and stratus clouds. The coefficients a and b needed for (3) were created using the TRAIN dataset. The coefficients are derived as functions of height above cloud base and cloud thickness. The rms errors when applying the derived coefficients to the independent TEST dataset are used to derive the covariance matrices in section 4b.

e. Formulation of the forward model function

Now two conditions which the forward model function F should meet can be stated. The first l elements of F (corresponding to l height dependent Z–LWC relations) should fulfill

Z_ia_ib_ii(7)

with i denoting the height index (i ∈ [1, l]). Then as a second condition, the last element of F should constrain the retrieved (integrated) LWC_i measurements to the liquid water path as

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where Δz = 250 m is the vertical model resolution. The measurement vector y is defined as (dBZ, LWP), where dBZ denotes the measured radar reflectivity vector and LWP the liquid water path derived from the TB measurements (6). If F in (2) is simplified to a linear problem by means of a Taylor series around a guessed LWC_n vector, then y can be written as

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or, by redefining some of the symbols and omitting higher-order terms,

yy_nK_nLWCLWC_n(10)

with K_n denoting the Jacobi matrix of the problem. Due to the errors in the LWP retrieval, errors in the Z–LWC relation, attenuation errors (section 3), and measurement errors, (10) will never be fulfilled exactly. This error can be minimized, however, by the OE method (section 4).

3. Attenuation

When dBZ profiles at 95 GHz are interpreted to derive cloud microphysical properties, attenuation due to hydrometeors has to be accounted for.

b. Correction for attenuation

If absorption can be described by the Rayleigh approximation, k is related linearly to LWC (Ulaby et al. 1981):

kcT,λ₀(11)

with c a constant depending on the physical temperature T of the cloud, and the radar wavelength λ₀. As before, all clouds are assumed to have a mean temperature of 0°C. The total optical depth τ_cloud of a cloud layer is given by

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The reflectivity factor Z_top, originating from cloud top, attenuated by the water cloud, will then be observed as

Z_atZ_topτ_cloud(13)

at the radar site. With the assumption that Z measured in the lowest cloud level is not attenuated, k can be calculated using (11) and (3)

kz_basecT,λ₀Zz_basea^1/b(14)

Now the optical depth of the lowest layer can be calculated and, consequently, the corrected reflectivity of the second layer can be computed by using an equation of the form of (13). This process can be repeated up to the cloud top in order to provide the complete Z profile corrected for attenuation. To simulate measured radar data, the model results of Z of the TEST dataset are analytically attenuated with an exponential relation as in (13). The term τ_cloud can be calculated using the model output data. Then the attenuated Z profiles are “de”-attenuated by using the method described above. The rms errors of the deattenuated Z values are at maximum about 3 dBZ at cloud-top height for clouds with vertical extension of 1500 m. Most of this error is due to the bias in the region of dBZ values larger than −20 where errors of up to −15 dBZ are possible. In this region of larger optical depths the attenuation correction can become unstable and should be regarded with care. For Z ≤ −20 dBZ the derived errors can then be interpreted as measurement errors and are included in the covariance matrices derived in section 4.

4. Retrieval technique

In this section the retrieval technique for LWC profiles is presented. First, the theory of optimal estimation is introduced following Rodgers (1976). The OE method is often used to retrieve temperature or humidity profiles from multispectral remote sensors (e.g., Ma et al. 1999). Subsequently, it is shown how this technique can be adapted to retrieve LWC profiles from a combination of a model climatology, Z measurements, and LWP measurements.

a. Optimal estimation

One major problem of remote sensing is the inversion of the atmospheric radiative transfer equation. Although forward calculations of atmospheric radiative transfer (i.e., calculation of radiances from the known atmospheric state) can be performed quite accurately, the inversion (even of the linearized ARTE) is complicated because a continuous profile of an atmospheric quantity is desired, whereas only a limited number of often highly correlated observations is available. This leads to an underdetermined problem, meaning that an infinity of solutions exists. As a consequence, profiles which are to be retrieved have to be discretizised and combined with other independent measurements. If additional measurements are not available the measurement vector and its covariance can be combined with a so-called virtual measurement (e.g., the information content contained in the climatology of the desired parameter, here the so-called a priori information). This combination will decrease the number of degrees of freedom of the problem and lead to a solution partially constrained by the climatological mean. If there are m independent measurements y_i with i ∈ [1, m] and x denoting the unknown profile, a linear version of the ARTE can be written as

y_iK_ix(15)

where K_i denotes the linear forward function of radiative transfer. The most likely value of x given the measurements y_i can be calculated by maximizing the conditional probability function P(x | y₁, y₂, … , y_m) with P(x) defined as

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with x₀ denoting the expected value of x, E[x] and S the covariance matrix S = E[(x − x₀)(x − x₀)^T]. Applying Bayes' theorem and maximization leads to the set of the optimal estimation equations yielding the optimal solution

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and its covariance Ŝ

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Here, S_i represents the covariance matrix of the measurement vector y_i. If the error statistics are assumed to be Gaussian, x̂ corresponds both to the maximum likelihood and the expected value solution.

b. Application to combined radar–radiometer measurements

The radar equation valid for meteorological targets (Ulaby 1981) is also a type of ARTE. One can calculate the power backscattered to the radar receiver P_r if transmitted power, antenna gain, wavelength, and the atmospheric composition (i.e., hydrometeors, gases, and temperature) are known:

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with R denoting the range to the scattering volume, τ the opacity, and σ_υ the radar cross section of the scattering volume, the latter two resulting from the atmospheric composition. Here, C is the radar constant and a function of transmitted power, antenna gain, and wavelength. If the backscattering process can be described in terms of Rayleigh scattering, and with the radar reflectivity factor defined as

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with N denoting the number of drops per unit volume and D_i the cloud-drop diameter, σ_υ can be expressed as

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with K a complex quantity dependent on the complex index of refraction of water. Combining (19), (20), and (21), one can write

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with C′ denoting a modified radar constant. The R² dependency is eliminated by multiplying (19) with R², which can be determined by the time difference between the transmitted and received radar pulse. Consequently, P_ra is the so-called range-adjusted power P_rR².

Equation (3) can also be regarded as a simplified ARTE. Together with LWP measurements from the microwave radiometer and an a priori climatology from the cloud model, the OE retrieval method can be applied. In the following the measurement vector y as described in (10) will now be denoted by y₁. Further information is included by taking as a second measurement vector y₂ as the mean model profile climatology of LWC derived from the TEST dataset. The inverse error covariance weighting method will lead to a solution for the LWC profile, which will satisfy the forward function F within its errors as stated in section 2e. Additionally, the LWC retrieval is constrained to the model climatology y₂ taking into account its variance. Thus, the problem consists of two independent measurements that are to be combined. The radiative transfer model MWMOD (Simmer 1994) and the cloud model (Issig 1997) are used to derive the two covariance matrices S₁ and S₂ needed in our case. Covariance matrix S₂ consists of the LWC covariances of the mean model profile derived from the TEST dataset. In this case, K₂ will be the identity. Covariance matrix S₁ is a matrix containing four independet error types. First, the absolute accuracy of the radar (3 dBZ; Danne et al. 1999) is considered by adding Gaussian noise with 3-dBZ standard deviation to the model dBZ. Also, the estimated LWP retrieval error is taken into account (section 2d). Errors due to the correction for hydrometeor attenuation can be interpreted as a measurement error and are also included in S₁. The radar reflectivities calculated from the cloud model are attenuated according to (13). Then the correction for attenuation as described in section 3b is applied to estimate the error. Since the optimal estimation equations do not explicitly include the forward model error due to the assumed Z–LWC relation, this error has been included in S₁ in form of an equivalent error in dBZ. The matrix S₁ is calculated as a full matrix, that is, taking into account the correlations between all described errors. This will also smooth the LWC profile. Separate a priori profiles (Fig. 2) and covariance matrices were calculated for different vertical thicknesses of modeled clouds. This was done to create representative a priori LWC profiles, since the number of thin clouds in the model climatology dominates the number of thick clouds. Due to the nonlinearity of the problem y₁ is written according to (10):

y₁y_1,nK_1,nLWCLWC_1,n(23)

Solving for K_n LWC and taking the model climatology y₂ (equal to LWC₂) into account leads to the following [if (17) is used as an analogon]:

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This equation can now be iterated in n taking as LWC_1,n+1 until covergence is reached. The covergence criterion used here was 0.001 g m⁻³ in all layers. The final product is an LWC profile at all heights, where the measured or modeled radar reflectivity has a significant value above the radar detection limit.

5. Retrieval accuracy

In Fig. 5 the results obtained with the OE algorithm with the inclusion of the a priori profile (OE-ap) and without (OE-pure) are shown. Also an algorithm according to Frisch et al. (1998) is considered for comparison purpose. This algorithm is based on a power-law relation, which adjusts the coefficient a according to the measured LWP weighted with the vertical sum of the Z profile. The coefficient b is held constant at a value of 2 assuming vertically constant droplet concentration and distribution width. This method is independent of radar calibration and DSD, provided that the sixth moment of the DSD can be related to the square of the third moment. The algorithm constrains the retrieved LWC profile exactly to LWP in a linear way.

In the following comparison, the GENE dataset is used to test the different methods. To simulate the real dBZ measurement a Gaussian noise distribution with 3-dBZ standard deviation is added to the dBZ profile (Table 2). The obtained LWP value from the model is subject to noise corresponding to the error resulting from the LWP retrieval (section 2d). In the lowest 250 m above cloud base, both OE methods show a relative retrieval error of about 55% in comparison to 75% for the Frisch method. In the five heights above the lowest layer, the rms errors of the Frisch algorithm and the OE-pure algorithm do not differ significantly, whereas the OE-ap method shows relative errors, which are smaller by about 10% to 15% (except at cloud top). The algorithm bias is negative for both OE methods. This is an artifact of the minimization of logarithmized LWC values (Fig. 3). For the method of Frisch et al., the vertically integrated LWC bias is zero. But, as can be seen in Fig. 5c, the bias varies in a range of 0.06 g m⁻³ with an overestimation in the lower and an underestimation of LWC in the upper part of the cloud. This is due to the fact that total number concentration and distribution width are not constant with height in the modeled clouds. The OE-ap method has a larger negative bias than the OE-pure method because large overestimations of LWC at dBZ values larger than −30 are influenced most strongly by the inclusion of the a priori profile (Figs. 6a,b). For smaller dBZ values the a priori information has a smaller impact because the variance of the a priori profile is quite large. Still, the LWC rms is decreased notably in all dBZ ranges (Fig. 7) using the OE-ap algorithm. The rms error does not change significantly if the bias corrected version of the Z–LWC relation (section 2c) is used in the OE-ap algorithm. Only the bias of the dBZ class A, B, and C is closer to 0. Due to the fact that the OE-ap algorithm is partially constrained to the LWP and the a priori profile, the bias correction is not as effective as in the simple Z–LWC relation.

To test the algorithm sensitivity towards the a priori profiles, which are derived from cloud-model data, the retrieval is performed with modified a priori profiles consisting of the original a priori profiles plus and minus their standard deviations (Fig. 2), respectively. The shift in positive and negative directions, respectively, of the a priori profile leads to a change in LWP bias of ±15 g m⁻² relative to the bias when the original a priori profile is used. The overall performance of the algorithm is not changed significantly due to the described modification.

6. Measurements

On 3 March 1999, combined measurements of the GKSS 95-GHz cloud radar and the 20–30-GHz channel radiometer were made at GKSS in Geesthacht, Germany. Radar and radiometer were located next to each other at a distance of about 5 m. Both instruments were pointing vertically into the atmosphere. This section describes the instruments, the measurement situation, and the results of the new LWC algorithm.

7. Conclusions and outlook

Significant improvements retrieving LWC profiles can be achieved if a cloud radar and a microwave radiometer are combined. Model results indicate that if an a priori climatology is combined with dBZ and LWP measurements and their covariances are taken into account, further improvement of the LWC retrieval can be achieved in comparison to a method constraining the derived LWC profile exactly to the measured LWP. Relative errors in the range of 30% to 60% should be expected when using the described combination of instruments. In comparison, aircraft (in situ) measurements errors of the DSD, can also be in the range of 20% to 40% (French et al. 2000).

An important requirement the OE retrieval method should meet is a correct description of the variability of the modeled atmospheric states. In our case this is indicated by similar values of modeled and empiricaly measured coefficients a and b. It would definitely be of advantage if large and representative climatological data sets of in situ measurements of different cloud DSDs were available. These could be used instead of the model statistics to derive an LWC algorithm.

One limitation to the algorithm is the occurrence of drizzle. Drizzle-sized drops can give the impression of a cloud reaching down close to the ground. This can lead to a false estimate of the liquid water distribution since the larger drizzle-sized drops lead to an overestimation of the LWC. As seen by regarding the Doppler velocities and laser ceilometer measurements, information about the occurrence of drizzle dominating the dBZ signal can be extracted. Methods have been proposed to solely calculate the drizzle liquid water content of a cloud (e.g., Frisch et al. 1995); however, the main problem in future will be not only to distinguish between nondrizzle and drizzle clouds, but to estimate the total liquid water content of both components in one cloud. A promising approach has been proposed by Czekala et al. (1999), who discriminate rain and cloud liquid by taking advantage of the polarization differences at low microwave frequencies due to nonspherical drop shapes.

A further uncertainty of the algorithm is the limitation to pure liquid water clouds. Pure ice cloud cases can be detected in our case since the radiometer frequencies used here do not respond to ice, but this instrument combination does not allow to distinguish between pure liquid and mixed phase clouds. Since very low ice water contents (e.g., 10⁻³ g m⁻³) can give rise to Z values in the range of −30 dBZ, it will be very important to develop methods to discriminate between cloud liquid and ice phase. Here, the utilization of the depolarization ratio of radar measurements or TB measurements at frequencies around 150 GHz could be helpful.

Starting in the summer of 2000, ground-based measurement campaigns will be conducted at different sites of the BALTEX area within the EU project CLIWA-NET. CLIWA-NET (Cloud and Liquid Water-Network) is a network of 12 ground-based stations with microwave and infrared radiometers, cloud radars, laser ceilometers, and standard meteorological observations to determine large-scale variations of LWP fields. These fields will be used to calibrate satellite LWP retrievals and validate numerical weather forecast models. One major site will be at GKSS in Geesthacht, Germany. Here the 95-GHz cloud radar MIRACLE, a multichannel microwave radiometer, a laser ceilometer, and an infrared radiometer will be collocated. The new 22-channel radiometer MICCY (Microwave Radiometer for Cloud Carthography) of the University of Bonn, Germany, will measure radiances in three spectral bands. Besides determining LWP more accurately than a 20–30-GHz combination, temperature and humidity information can be extracted (Crewell et al. 1999). Since microwave emission of liquid water increases with the frequency squared, assumptions can be made about the vertical distribution of LWC with a limited vertical resolution (Solheim et al. 1998). However, in contrast to the radar, the microwave signal is proportional to the third moment of the DSD meaning that even areas “contaminated” with drizzle will contribute with a signal proportional to the LWC, assuming the microwave signal is not saturated. Thus, the instrument combination of cloud radar and MICCY will be a very powerful tool in determining LWC profiles and DSD parameters. Currently an OE algorithm is being developed to combine these instruments. Additionally, aircraft measurements of cloud DSDs are planned during CLIWA-NET for algorithm validation.