1. Introduction
Detailed knowledge of cloud and precipitation microphysical properties is critical for determining many key aspects of the earth’s climate system. This is especially true as numerical models of the climate approach resolutions that can begin to resolve cloud processes. While in situ measurements can be used to determine particle size distribution (PSD) characteristics, in situ data often lack the whole-cloud context that is needed to understand clouds at a process level. Remote sensors provide this context, but at the expense of having to infer geophysical properties via inversion algorithms. Several recent studies have documented success characterizing liquid and ice properties using a combination of Doppler radar and microwave radiometer brightness temperature (Tb) observations (Löhnert et al. 2001, 2003; McFarlane et al. 2002; Delanoë and Hogan 2008; Matrosov et al. 2008; Ebell et al. 2010; Wood 2011; Rambukkange et al. 2011). Several of these studies cast the retrieval problem in a probabilistic framework, in which uncertainties in observations and prior knowledge are each assigned a probability distribution with width proportional to the magnitude of the uncertainty (Delanoë and Hogan 2008; Ebell et al. 2010; Wood 2011). A solution is produced by combining all sources of information in a Bayesian context. The problem is made tractable by assuming all probability distributions are Gaussian, and solving the retrieval in a least squares optimal estimation (OE; Rodgers 2000) framework.
The strength of Bayesian inference in cloud and precipitation property retrieval algorithms is the ease with which multiple observational constraints can be applied to a problem. In liquid-phase clouds, Frisch et al. (1998) exploited the integral constraint that is provided by microwave radiometer-derived liquid water path and showed that liquid water content profiles can be inferred from profiles of radar reflectivity raised to an appropriate exponent. McFarlane et al. (2002) and Löhnert et al. (2003) extended these ideas to nonprecipitating cumulus and stratocumulus, respectively, using in situ aircraft statistics and cloud-resolving model output in an OE framework. Ebell et al. (2010) show that the uncertainties in these estimates depend fundamentally on the validity of a priori assumptions that must be derived from in situ aircraft data. In thin ice clouds, an integral constraint equivalent to the liquid water path in warm clouds is provided by the downwelling infrared radiation (Matrosov et al. 1994; Mace et al. 1998; Zhang and Mace 2006). These ideas were extended to the use of radar Doppler moments by Deng and Mace (2006) and to constraints provided by both lidar and radar by Donovan (2003) and Zhao et al. (2011). Delanoë and Hogan (2008) combined active and passive measurements in an innovative OE algorithm applicable to cirrus. All of these ice cloud studies discussed limitations to accuracy that arose because of assumptions regarding the microphysical properties of ice crystals—specifically the mass– and area–dimensional relationships that were assumed. There have been few applications of Bayesian inference in mixed-phase clouds using active and passive sensors.
Bayesian OE has great utility in that it explicitly represents each piece of information and the associated uncertainty, and, if properly implemented, can produce a quantitative estimate of the retrieval error. Even so, the assumption of Gaussian uncertainty and a linear least squares framework is a limitation that can lead to misinterpretation of retrieval results (Posselt et al. 2008). If the relationship between retrieved quantities and observations is nonunique, the OE algorithm may not produce the optimal set of retrieved parameter values. In addition, Gaussian probability distributions are not suitable representations of uncertainty in cases for which parameters are constrained to be positive definite (Posselt et al. 2014).
Markov chain Monte Carlo (MCMC) algorithms (Metropolis et al. 1953; Hastings 1970; Posselt and Vukicevic 2010, hereinafter PV10; Posselt and Bishop 2012, hereinafter PB12; Posselt 2013) have recently been used to examine the information content of remote sensing retrievals (Tamminen and Kyrölä 2001; Posselt et al. 2008) and in rare cases to perform the retrieval itself (Tamminen 2004). MCMC methods produce a Bayesian estimate of the probability distribution of a set of parameters given a set of observations, prior knowledge, and a model that relates parameters and observations. There is no requirement that the probability distributions be Gaussian, and additional constraints (e.g., positive definite PSD parameters) can be easily and flexibly accommodated. As such, MCMC algorithms can be used to obtain an accurate assessment of observation information content, to assess the effect of uncertainty in the forward model, and determine which observations might be used to constrain the retrieval should the solution prove to be nonunique. The cost of this flexibility is the computational burden required to implement MCMC algorithms, as will become clear below.
In this paper, a Markov chain Monte Carlo algorithm is used to retrieve cloud and precipitation properties in a case of orographic snowfall for the purpose of illustrating the sensitivity of such retrievals to assumptions and prior information. The results 1) indicate whether there is sufficient information in a combined microwave radiometer and radar retrieval to constrain a bimodal (liquid and ice) cloud particle size distribution, and 2) determine, via computation of the probability distribution of retrieved liquid and ice, the error characteristics of the retrieval and the dominant sources of uncertainty. It should be noted at the outset that errors in any retrieval of cloud PSD properties tend to be strongly regime dependent, and retrieval outcomes in general are sensitive to the specific forward model(s) used and measurement errors assumed. Our experiments highlight the uncertainty in mixed-phase PSD retrievals from surface-based measurements: W-band cloud radar and low-frequency microwave radiometer observations. The results are specific to our combination of forward models and were obtained from a single orographic snowfall case. As such, though the MCMC methodology is robust, and our forward models well tested, our results should be viewed as applicable to a relatively narrow range of conditions. We will return to this point in the conclusions, and offer a preview of how we are currently increasing the generality of our experiments.
The remainder of this paper is organized as follows. The case study and forward models for radar and microwave radiometer are described briefly in section 2 and in more detail in appendixes A and B. Section 3 contains an analysis of how forward-modeled measurements change in response to changes in cloud PSD—the cloud property response functions. Results from idealized and real-data retrievals using MCMC are presented in section 4, and our major conclusions are summarized in section 5.
2. Case study, forward models, and retrieval algorithms
a. StormVEx experiment
During the winter of 2010/11, the second Atmospheric Radiation Measurement Program (Mather and Voyles 2013) Mobile Facility (AMF2) conducted its maiden deployment to the Park Range Mountains near Steamboat Springs, Colorado, as the principal component of the Storm Peak Laboratory Cloud Property Validation Experiment (StormVEx; Mace et al. 2010). The purpose of StormVEx was to document the properties of orographic mixed-phase snow clouds in conjunction with the collection of correlative data at Storm Peak Laboratory (SPL) located at the summit of Mount Werner (Borys and Wetzel 1997; Hallar et al. 2011a,b). The remote sensors used for this study were located near the Thunderhead Lodge at an elevation of 2759 m approximately 2.4 km to the west and 400 m below SPL and collected data continuously from late November 2010 until early April 2011. The Scanning W-Band Cloud Radar (SWACR) and a 2-channel (23 and 31 GHz) microwave radiometer, as well as ancillary data from a laser ceilometer, are the primary instruments used in this study.
The SWACR collected data during ~30-min sequences in zenith-pointing, RHI, and plan position indicator (PPI) scan modes. During the zenith periods, which occupied 18 min of every 30-min sequence, Doppler spectra were collected and archived, and Doppler moments were calculated from those spectra. The campaign benefitted from one of the snowiest winters on record in the northern Colorado Rocky Mountains with cloud cover exceeding 60%. Measureable snow was recorded on more than 75 days of the campaign. Major snowfall exceeding 10 cm day−1 was recorded on approximately 40 of those days. Total measureable snowfall exceeded 800 cm over the entire winter season.
The case we examine in this paper was recorded on 4 February 2011 when a quiescent northerly flow existed over the Park Range. This case was selected because both liquid and ice particles were known to exist within the cloud, and because snowfall was light and nearly continuous over a period of several hours. Snow began to fall during the early local afternoon from a mixed-phase cloud layer with bases (determined from ceilometer data) that varied from near to just below the SPL level where temperatures were in the −10°C range. Cloud tops from the SWACR were observed to be approximately 2 km above the radar where temperatures from an afternoon radiosonde were −15°C. Operators at SPL reported rime buildup on the instruments during the afternoon with the light snowfall that consisted primarily of rimed dendritic ice crystals. Snowfall was light enough that automated snowfall instruments recorded no measureable accumulation. Figure 1 shows the radar Doppler moments that were collected during the 1-h period centered on the profile that we evaluate in this study. It is notable that a bimodal (and in select layers, trimodal) spectrum is evident in the observations of Doppler spectral width. It is not uncommon to observe a population of precipitating ice hydrometeors collocated with smaller ice particles falling more slowly, and it is likely that this was the case during a portion of the observed time period. On this particular afternoon, the SWACR did not conduct routine RHI and PPI scans, and only zenith-pointing data were collected.
SWACR data collected on 4 Feb 2011. (a) Radar reflectivity height–time cross section for 2100–2200 UTC. (b) Doppler velocity for 2100–2200 UTC (positive is toward the radar, or downward motion). (c) Doppler spectra with height at 2117 UTC—the time used in the bulk of the retrieval analyses.
Citation: Journal of Applied Meteorology and Climatology 53, 8; 10.1175/JAMC-D-13-0237.1
b. Forward models
To calculate radar measurables, we integrate across the bimodal PSDs to determine the radar backscattered power and attenuation through the cloudy column using scattering and extinction efficiencies as a function of particle size. To accomplish this, we neglect multiple scattering and derive the backscatter cross section as a function of size using Mie theory for liquid (Bohren and Huffman 1983), with refractive indices after Hale and Querry (1973) and T-matrix-derived cross sections for ice-phase hydrometeors provided by Matrosov et al. (2008) in size intervals ranging from 100 μm to 2 cm in 100-μm increments. The T-matrix cross sections were calculated assuming oblate spheroids with an aspect ratio of 0.6 and a mass–diameter relationship of 
We have collected and adapted a set of published forward models that relate hydrometeor properties to measurements. With the exception of the radar forward model, we use published radiative transfer codes, including the microwave radiative transfer algorithm based on the Eddington approximation described by Kummerow et al. (1996) and modified by Lebsock et al. (2011). As the microwave radiometer of interest is vertically pointing (upward looking), the forward-modeled Tbs are nearly insensitive to the value of surface emissivity. We have used a value of 0.8 to strike a balance between the relatively low emissivity of fresh deep dry snow and high emissivity of aged wet snow (Hewison and English 1999).
c. Markov chain Monte Carlo algorithm
Note that the form of likelihood and prior are completely general—any distribution shape may be assumed. The value of the acceptance ratio 
The sample of the posterior PDF generated by an MCMC algorithm represents the most complete characterization of the retrieval solution for a given forward model, prior, and set of measurements. Sources of uncertainty can be easily identified, observation information content can be quantified, and the impact of changes to observation uncertainty as well as the introduction of new observations can be tested in a straightforward manner. The implementation of the MCMC algorithm in this paper is based on the algorithm described in Posselt et al. (2008), and PB12, and illustrated in the flowchart presented in Fig. 2. In each of the aforementioned papers, the set of parameter values was fixed in time and space.
Schematic depiction of the order of operations in each MCMC iteration (adapted from PV10).
Citation: Journal of Applied Meteorology and Climatology 53, 8; 10.1175/JAMC-D-13-0237.1
It should be noted that the implementation of MCMC used in this paper has been modified to account for the fact that, as opposed to a set of scalar quantities, vertical profiles of cloud properties are estimated. There are a number of ways to estimate vertical profile variables in an MCMC algorithm. One can assume parameters are spatially uncorrelated, and randomly generate proposed parameter profiles by perturbing a single layer at a time or perturbing all layers simultaneously and randomly. In practice, the former option leads to very slow convergence, while the latter introduces a significant amount of noise in the profiles. An alternative option is to smooth perturbations in the vertical, either by perturbing a single layer at a time and spreading the influence using a correlation function with specified vertical decorrelation length, or by perturbing all layers simultaneously, then adjusting the proposed profile using a specified smoothing function. Tests with each option indicate the former technique is most effective in producing a robust estimate of the posterior PDF in the most computationally efficient manner. The vertical decorrelation length scale was estimated from the vertical decorrelation of the radar reflectivity profiles, and was determined to be approximately 400 m. The results were relatively insensitive to the choice of decorrelation length, though of course large decorrelation length essentially causes the entire profile to be perturbed simultaneously, unrealistically constraining the vertical variability in the profile. As mentioned above, very small lengths lead to effectively independent perturbation of each layer. This is not a problem per se, but it leads to much slower convergence in the MCMC algorithm.
As in PV10 and PB12, the prior probability density function for all control variables is assumed to be uniform with minimum and maximum bounds set to physically realistic values of each of the PSD parameters (Table 1). The mass– and area–dimensional relationships are estimated by summarizing values reported in Mitchell (1996), Szyrmer and Zawadzki (2010), and Heymsfield et al. (2010) and are appropriate for the aggregates and dendritic particles observed on this case day. The PSD parameters were determined using data collected in situ during StormVEx from the Colorado Airborne Mixed Phase Study (CAMPS; Dorsi 2013) that was funded by the National Science Foundation. We perform two sets of experiments: one in which the true values of each PSD parameter are predetermined and used to generate simulated radar and microwave radiometer observations, and another in which MCMC is used to retrieve PSD properties from observations taken from the 4 February 2011 StormVEx case. The idealized experiment is used to examine the functional relationships between changes to PSD parameters and forward-model output (the forward-model response function). Following idealized tests, MCMC experiments are used to retrieve the vertical profile of liquid and ice PSD parameters for a single representative cloud profile. The information content of the observations is assessed and the effect of uncertainty in the cloud PSD is examined.
List of retrieved parameters, along with their ranges and values used in idealized and real-data cases. Note that the real-data values for α, am, and bm correspond to the settings used in the experiments for which these parameters were set fixed. Liquid and ice modal diameter and number are allowed to vary in all real-observation cases.
3. Idealized retrieval experiments
Prior to conducting MCMC experiments, we explore the performance of the combined radar and microwave radiometer retrieval for an idealized case. We first analyze the response of the forward-modeled observations to changes in particle size distribution parameters with the intent of 1) exploring the inherent sensitivity in the retrieval and 2) characterizing the (univariate) parameter–observation functional relationship. We then specify a (vertically homogeneous) vertical profile of liquid and ice content based on the specified default parameters listed in Table 1, and generate simulated observations by using these “true” profiles in the radar and microwave radiometer forward models. The resulting simulated observations are then used to retrieve the cloud particle size distribution parameters, using observation uncertainty consistent with the real-world StormVEx observations. The results provide information as to whether the observations provide sufficient constraint on the retrieval.
a. Response functions
While it is ultimately the multivariate relationships that are of interest in the retrieval, it is useful to examine the univariate retrieval sensitivity. Such an analysis yields information as to the functional dependence of forward-model output on changes to control parameters, and can provide an early indication of nonlinearity and/or nonuniqueness in the retrieval. In the following analysis, radar and microwave radiometer forward models are run for a range of values of each input parameter (Table 1) one at a time, fixing the remaining parameter values at their default values. Forward-model sensitivities are presented in terms of the modal diameter and particle number, as these are the fundamental retrieved variables (Deng and Mace 2006). In the retrieval framework, the maximum likelihood estimate and associated PDF are converted to liquid and ice water content and total number via equations in appendix A.
Examination of the results (Fig. 3) reveals the following. W-band radar reflectivity (Fig. 3a) is insensitive to changes in liquid particle size up to modal diameter of approximately 10 μm at which point liquid drops become large enough to produce changes to W-band reflectivity. However, the changes are less than a few decibels for a factor-of-3 change in model drop size, suggesting that realistic changes to drop size would be difficult to discern in reality. Doppler velocity (Fig. 3b) decreases as the liquid size increases. This counterintuitive result arises because the retrieval accounts for the effects of the snow as well as the liquid cloud mode. Recall that the Doppler velocity is a Z-weighted quantity, that is, 
Response of (a),(e),(i),(m) forward-modeled reflectivity; (b),(f),(j),(n) Doppler velocity; (c),(g),(k),(o) 23-GHz microwave radiometer Tb; and (d),(h),(l),(p) 31-GHz microwave radiometer Tb to changes in liquid modal (top) diameter and (top middle) number and snow modal (bottom middle) diameter and (bottom) number.
Citation: Journal of Applied Meteorology and Climatology 53, 8; 10.1175/JAMC-D-13-0237.1
Forward-modeled low-frequency microwave radiometer Tbs (Figs. 3c,d) display the well-known sensitivity to liquid water mass, while reflectivity changes little with liquid particle number (Fig. 3e), primarily because the liquid particles are assumed to be small (default modal diameter set equal to 5 μm). As expected, liquid particle fall velocity, as viewed through the lens of mean Doppler velocity, is insensitive to changes in particle number (Fig. 3f). Liquid water content is linearly dependent on particle number (see appendix A for details), and this dependence is reflected in the microwave radiometer Tb (Figs. 3g,h). Reflectivity and Doppler velocity (Figs. 3i,j) are highly sensitive to changes in snow modal diameter, but, in contrast to the liquid particles, Doppler velocity sensitivity begins to saturate as particle size increases because of hydrodynamic drag. In comparison with the liquid diameter–Tb response function plots (Figs. 3c,d), low-frequency microwave radiometer Tbs are less sensitive to changes in ice diameter (Figs. 3k,l) for equivalent changes in ice particle size, although this sensitivity increases dramatically at large sizes because of scattering effects at these weakly absorbing wavelengths. Radar reflectivity is significantly sensitive to changes in the ice number (Fig. 3m), because the modal ice diameter is set by default to a relatively large 200 μm (Table 1). As with liquid particle number, Doppler velocity demonstrates a very weak dependency on the ice particle number density (Fig. 3n). Microwave radiometer Tbs are linearly related to changes in ice modal number, but the sensitivity is small because of the far smaller absorption of microwave energy at gigahertz frequencies in ice.
Considered as a whole, response functions indicate W-band radar reflectivity and Doppler velocity observations should contain enough information to constrain the ice particle size and ice particle number. Resolving the liquid drop size in the presence of snowfall would be possible only under certain conditions, such as low turbulence and large drop sizes, as recently shown by Luke and Kollias (2013). Addition of low-frequency microwave radiometer Tb observations provides constraint on the liquid number density and size, while penalizing large values of ice diameter. It cannot be overstated that the results presented above are only applicable to the specific combination of observations and forward models considered here (W-band radar and low-frequency passive microwave radiometer). The conclusions would almost surely differ for other radar wavelengths (and for multifrequency radar) and high-frequency microwave radiometer observations.
b. Idealized retrieval
Observed profile of radar reflectivity and Doppler velocity, along with microwave radiometer Tbs and their associated observation uncertainties from 2114 UTC 4 Feb 2011.
The results reflect the sensitivity of the forward-model output to changes in the PSD parameters shown in the response function plots (Fig. 3), and by extension the information content of the simulated observations. Observations of W-band reflectivity alone are insufficient to constrain either liquid or ice particle size distribution, though large liquid and ice diameter are precluded (Figs. 4a,d). The addition of Doppler velocity renders a unique solution for the ice PSD parameters (Fig. 4e), and results in elimination of liquid modal diameters larger than about 12 μm (Fig. 4b). Addition of low-frequency microwave radiometer Tb observations places additional constraint on the liquid PSD (Fig. 4c), but it is notable that the solution is still nonunique. There exist a near infinite number of combinations of modal diameter and number that will produce the identical set of forward observations. Examination of the response functions provides the explanation. Tb observations prevent large values of liquid diameter, but within a reasonable range of diameter and particle number, increases in diameter can be compensated by decreases in number. It is worth pointing out that this retrieval is perhaps the most idealized one can expect for the given set of observations, as the cloud fields were set to be vertically homogeneous, the true values were known, and the gamma width parameter and mass–dimensional parameters are specified. In the next section we examine the results of a retrieval in which real-world observations are used to constrain vertically varying profiles of liquid and ice content and number, and in which gamma distribution width parameter and mass–dimensional relationships are allowed to realistically vary.
PDFs of (top) liquid and (bottom) ice cloud properties obtained from an idealized PDF-based retrieval using observations of (a),(d) dBZ only; (b),(e) dBZ and Vd; and (c),(f) dBZ, Vd, and Tb. Red solid lines indicate the position of the specified (true) value of each parameter, while color-filled contours depict the normalized likelihood P(y | x).
Citation: Journal of Applied Meteorology and Climatology 53, 8; 10.1175/JAMC-D-13-0237.1
4. Retrievals of cloud PSD from StormVEx observations
We now use W-band radar and low-frequency microwave radiometer observations obtained during the StormVEx field campaign to retrieve vertical profiles of liquid and ice using MCMC. As our primary goal is an exploration of the sensitivity of the retrieval, we do not need to retain the full vertical resolution in the observations or estimated parameters. Estimation of cloud PSD properties for every radar range bin would also add significantly to the computational expense of implementing MCMC. For this reason, we significantly reduce the vertical resolution of the SWACR data from 42-m range bins to just three layers. These layers are evenly divided in the vertical, where the lowest layer (142–741 m above the surface) captures primarily the subcloud layer where mostly precipitation is found, the midlayer (741–1341 m above the surface) is the mixed-phase region where snow is falling through liquid cloud, and the upper layer (1341–1983 m above the surface) is also mixed phase but the ice particles are smaller and the liquid phase contributes more significantly to the radar observables. The vertical averaging is accomplished by summing the radar reflectivity Z over each layer and dividing by the number of range bins in the layer, while the Doppler velocity is computed as a Z-weighted average in the layer. A single representative profile is chosen for analysis (Table 2), and two different experiments are conducted; one in which the ice mass–dimensional parameters and liquid and ice gamma distribution width parameters are set equal to default values, and one in which they are allowed to vary. In each experiment, observations of reflectivity, Doppler velocity, and microwave radiometer Tb are used to constrain the retrieved particle size distribution. The vertical column is perturbed one layer at a time, with perturbations spread in the vertical via a Gaussian function with 400-m decorrelation length. A single Markov chain is used, and the MCMC algorithm is run for increasing numbers of iterations of the forward model as the dimensionality of the problem is increased (Table 3). Convergence of the Markov chains is assessed via intercomparison of multiple randomly drawn subsamples of 105 iterations, as well as via visual inspection of the chains themselves (not shown in the interest of brevity). In each case, the integrated absolute differences between the subsampled PDFs were less than 2% and the chains were observed to be mixing readily in the solution space.
List of parameters varied and the number of MCMC iterations used in each experiment.
a. Fixed gamma distribution width and mass–dimensional relationships
We first examine the constraint on the retrieval imposed by adding successively greater amounts of information to the system. In each of the experiments that follow, we begin with a reflectivity-based retrieval, then add information first from Doppler velocity then microwave radiometer Tb. Retrieval information content can be estimated via examination of the Bayesian posterior PDF produced by MCMC. By definition, the optimal fit to the available observations is the maximum likelihood point in the PDF, the point at which the posterior PDF has a maximum in value. If there are multiple maxima, this reflects the fact that multiple different sets of parameter values produce forward-modeled values that are consistent with the real observations. If there is no unique maximum likelihood point, or the probability maximum is broad in the vicinity of such a point, then by definition the retrieval is not well constrained.
Examination of the posterior PDF of the liquid number and content (Fig. 5) indicates, as expected, W-band reflectivity alone is not capable of producing a unique solution for the cloud of interest (Figs. 5a,d,g). While the liquid water content is restricted to relatively low values, the maximum likelihood region in the PDF is spread across a large range of values of number density. The addition of observations of Doppler velocity results in a strong constraint on the liquid PSD in the top layer (Fig. 5b) in accordance with our earlier discussion of Fig. 3b. Whern the ice and liquid modes both contribute significantly to the radar measurables, the reflectivity-weighted velocity decreases as the liquid phase increases in mean size. Whereas the combined reflectivity changes in the same sense with increases or decreases in the ice or liquid PSD modes, the negative tendency in Doppler velocity for increases in the liquid mode provides a unique constraint on the microphysical properties. The solution is clearly bimodal in the lower and midlayers (Figs. 5e,h), especially for the liquid-phase clouds. A solution that consists of high number and low liquid water content is nearly as likely as lower droplet number but marginally higher water content. Note that the ice content in the top layer is low (Figs. 6a–c), and as such the amounts of liquid and ice are comparable.
Retrieved PDF of liquid PSD in (a)–(c) top, (d)–(f) middle, and (g)–(i) lower layers using observations of (left) radar reflectivity alone, (center) reflectivity and Doppler velocity, and (right) reflectivity, Doppler velocity, and microwave radiometer Tb. White dot–dashed, dashed, solid, and dotted lines correspond to the 99.7%, 95%, 68%, and 38% probability contours, respectively. Color-filled contours correspond to the probability density normalized to 1.0 over 51 bins in each parameter. Note that the number density is in units of per centimeter cubed while liquid water content has units of grams per meter cubed.
Citation: Journal of Applied Meteorology and Climatology 53, 8; 10.1175/JAMC-D-13-0237.1
As in Fig. 5, but for the retrieved PDF of ice PSD. Note that in these plots, the number density has units of per liter while ice water content has units of grams per meter cubed.
Citation: Journal of Applied Meteorology and Climatology 53, 8; 10.1175/JAMC-D-13-0237.1
Addition of low-frequency microwave radiometer Tb observations produces a preferred mode in the solution, but the secondary mode is still evident, particularly at low and midlayers (Figs. 5f,i). In retrieval algorithms that seek a solution via iterative convergence to a local maximum in the PDF (e.g., optimal estimation), it is possible that the algorithm might select the incorrect (less likely) solution. In all cases it is clear that the posterior PDF departs significantly from a Gaussian shape. In select cases (e.g., observation of reflectivity alone, or the PDFs in the top layer (Figs. 5b,c), the PDF is unimodal and skewed, but in all other cases there are multiple possible solutions (multiple modes in the PDF). In each case, simply the physical requirement that PDFs be truncated at zero liquid content and number can present problems for methods that assume Gaussian PDFs.
In contrast to the liquid particle size distribution, use of reflectivity observations almost immediately produces a more well constrained solution for the ice water content and number density in all layers (Figs. 6a,d,g), though a bimodal PDF is still clearly evident. Addition of Doppler velocity observations serves to produce a unique solution in the upper layer and leads to a preferred mode in the PDF in the lower and midlayers (Figs. 6e,h). At first glance, addition of microwave radiometer Tb observations appears to introduce greater uncertainty into the retrieval of ice content and number in the upper and midlayers (a greater number of possible solution states is included in the posterior PDF). In the absence of Tb observations, the large liquid water content solution is coupled with the low ice content solution so that the forward-modeled reflectivity is consistent with the observations. Upon the addition of microwave radiometer observations, the relatively high LWC and low IWC solution emerges as preferred. As with the liquid PDFs, none of the posterior distributions for ice content and number can be said to be Gaussian in form, and most cannot even be claimed to be approximated by a Gaussian distribution, much less a distribution with a single mode.
In all of the results presented in this section, the gamma distribution width parameter and snow mass–dimensional relationships were assumed known. In reality, these may vary over a potentially large range of values introducing additional uncertainty into the retrieval. To be sure, in most real-world applications, the most appropriate values of these empirical parameters are usually not known, much less their variability. The effect of this additional uncertainty is explored in the next section.
b. Variable gamma width and mass–dimensional relationships
Prior to conducting MCMC experiments in which the width and mass–dimensional parameters are allowed to vary, it is first useful to explore the response of the forward-modeled radar and microwave radiometer observations to changes in these variables. The results of similar response function experiments to those described in section 3 above are presented in Fig. 7. From Eqs. (A1) and (A3) it can be seen that the width parameter affects both the water content and total number via its presence in the gamma function, while the coefficient and exponent in the mass–dimensional relationship affect only the water content. Because the terminal velocity of particles depends on the ratio of particle mass to area, the mass–dimensional relationship parameters influence the forward-modeled Doppler velocity [see Eqs. (A5) and (B4)]. It is also clear that changes in the width parameter for liquid have negligible effect on W-band radar reflectivity and Doppler velocity (Figs. 7a,b) and a very small effect on 23- and 31-GHz microwave radiometer Tbs (Figs. 7c,d). In contrast, variability in the width parameter for ice has a relatively large effect on the radar forward observations (Figs. 7e,f), and on the microwave radiometer Tb (Figs. 7g,h), especially at large gamma distribution widths (larger numbers of large particles).
Response of (a),(e),(i),(m) forward-modeled reflectivity; (b),(f),(j),(n) Doppler velocity; (c),(g),(k),(o) 23-GHz microwave radiometer Tb; and (d),(h),(l),(p) 31-GHz microwave radiometer Tb to changes in (top) liquid and (top middle) ice gamma width parameter and the (bottom middle) coefficient and (bottom) exponent in the mass–dimensional relationship for snow.
Citation: Journal of Applied Meteorology and Climatology 53, 8; 10.1175/JAMC-D-13-0237.1
Variability in the mass–dimensional relationship is only relevant to the ice phase, as liquid particles in this particular case are small and can be assumed to be spherical. Note that in the current version of the radar forward algorithm, reflectivity is rendered insensitive to changes in the mass–dimensional relationships by design. We are well aware that this is a limitation of the current forward model, and an improved version of the model, in which the radar backscatter cross sections are functionally related to the mass–dimensional parameters, is currently in development (Hammonds et al. 2014). Forward-modeled Doppler velocity is very sensitive to changes in both the mass–dimensional parameters, while microwave radiometer Tb exhibits little sensitivity (Figs. 7k,l,o,p). Note that there is an inverse functional relationship between the Doppler velocity and the coefficient and exponent in the mass–dimensional relationship, indicating that increases in the mass coefficient can be effectively compensated by decreases in the mass exponent (and vice versa).
We interpret the sensitivity in the microwave Tbs to the assumed ice parameters to be primarily a scattering phenomenon. We note that the effects are larger at 31 GHz than at 23 GHz even though the complex index of refraction taken from Sadiku (1985) is smaller at 31 GHz (1.0 × 10−3) than at 23 GHz (1.4 × 10−3). The scattering cross section varies approximately as the inverse wavelength raised to the −4 power and directly as the particle size raised to the exponent 6 in the Rayleigh regime. Therefore, the stronger response at 31 GHz may be ascribed to the inverse wavelength dependence. As the shape parameter increases (Figs. 7g,h) the ice size distribution broadens and ice mass is increasingly moved to large particle sizes where the D6 relationship results in increased scattering cross sections. For the mass–dimensional relationship parameters 
Comparison of the retrieved joint PDFs of liquid water content and number density (Fig. 5) with those retrieved in the presence of fixed α, 
Retrieved PDF of liquid PSD using observations of radar reflectivity, Doppler velocity, and microwave radiometer Tb and allowing variability in (a),(d),(g) ice and liquid gamma width parameters; (b),(e),(h) mass–dimensional parameters for ice; and (c),(f),(i) both. As in Figs. 5 and 6, white dot–dashed, dashed, solid, and dotted lines correspond to the 99.7%, 95%, 68%, and 38% probability contours, respectively.
Citation: Journal of Applied Meteorology and Climatology 53, 8; 10.1175/JAMC-D-13-0237.1
As with the liquid cloud properties, the shape of the retrieved ice PDF (Fig. 9) is consistent across all experiments; IWC and ice number are positively correlated and the magnitude of the correlation is similar. However, in contrast to the liquid experiments, the location of the mode in the PDF does not change as the gamma distribution width and ice mass parameters are allowed to vary. Variability in α (Figs. 9a,d,g), as in the liquid cloud case, changes the PDF structure, leading to removal of the mode in the PDF associated with very low ice water content and large ice number concentration. When the mass parameters are allowed to vary (Figs. 9b,e,h), the structure of the solution space changes little, but the width of the posterior PDF increases significantly, especially in the lowest layer. Solutions are now relatively likely (e.g., large ice water content and number density) that were prohibited when 
As in Fig. 8, but for the ice water content and number density.
Citation: Journal of Applied Meteorology and Climatology 53, 8; 10.1175/JAMC-D-13-0237.1
The absence of the mode in the midlayer liquid and ice PSD PDFs can be explained via examination of the PDF of the modal diameter (D0) and number (N0) of liquid and ice (Fig. 10). Recall that these are the parameters that are, in actuality, varied in the retrieval process. The mode in the PDF of LWC and N0liq (Fig. 5f) at large LWC and small N0liq is associated with the mode in Fig. 10a at relatively large modal diameter (approximately 30 μm) and small number (by inspection N0 > 1200 cm−4). When the gamma distribution width is specified to be zero, it is possible to fit the available radar and microwave radiometer Tb observations with a small number of relatively large cloud droplets. The fit is only reasonable in a very small range; however, because of the sensitivity of the W-band radar reflectivity to small changes in particle diameter once the particles are large. Inspection of the joint PDF of modal liquid diameter and ice number (Fig. 10b) reveals there is a large range of ice numbers compatible with the large liquid modal diameter values, but these are necessarily associated with a small ice diameter (Fig. 10c). This explains the secondary mode in the IWC/N0ice PDF (Fig. 6f) centered at small IWC values. The liquid particles are providing the major portion of the radar signal and necessitate the retrieval of small ice particles. Since the particles are small, a relatively large range of sizes will produce a reasonable fit to the observations. The mode in the joint PDFs of liquid and ice exists only because the liquid PSD width is specified and set equal to zero. Once the width parameter is allowed to vary, there is no preference for a specific set of values of liquid modal diameter and number. A large number of combinations of gamma distribution width, modal number, and diameter will fit the observations and as such there will be no localized mode in the PDF.
The 2D PDFs of (a) liquid modal diameter and number, (b) liquid modal diameter and ice modal number, and (c) ice modal diameter and liquid modal number. Color-filled and solid contours are as in Figs. 5, 6, 8, and 9.
Citation: Journal of Applied Meteorology and Climatology 53, 8; 10.1175/JAMC-D-13-0237.1
Note that the localized mode is also eliminated when the mass–dimensional parameters are varied (Fig. 9e). Changing 
The fact that a localized mode (and hence a nonunique solution) is produced when the system is placed under tighter constraints is a result that might not have been expected a priori, and it is one of the reasons a detailed examination of the retrieval probability structure is useful.
5. Summary and conclusions
This paper provides an analysis of the probability structure associated with a combined surface-based radar and passive microwave radiometer retrieval of a bimodal cloud particle size distribution for a case of snowfall from a mixed-phase orographic cloud. While optimal estimation type retrievals have been used to retrieve the PSD parameters of interest for combined active–passive observing systems, the true nature of the solution space has remained unknown. The nature and degree of constraint of snowfall retrievals by radar and/or microwave radiometer observations has also been unclear. Characterization of the full solution space allows an evaluation of the robustness (or lack thereof) of optimal estimation retrievals, and evaluation of the sources of uncertainty and their effect on the resulting retrieval. Markov chain Monte Carlo methods produce a Bayesian sample of the retrieval solution space, and are more efficient than multivariate brute force perturbation and more complete than one-at-a-time response function analysis.
This paper focused specifically on an MCMC-based exploration of the solution space associated with a surface-based combined W-band radar–low-frequency microwave radiometer brightness temperature retrieval of cloud properties for a case of orographic snowfall observed during the StormVEx field campaign. Three-layer liquid and snow water content and number were retrieved from 94-GHz radar and 23- and 31-GHz microwave radiometer Tb observations, and the successive addition of observational constraints provided information as to whether a radar-only retrieval is capable of constraining the liquid and ice particle size distributions. The effect of variability in PSD parameters and in the ice crystal shape on the retrieval solution was also assessed.
The major conclusions of this study are the following:
W-band radar reflectivity alone is insufficient to constrain the ice PSD; at minimum Doppler velocity is needed if a unique solution is to be obtained.
If Doppler velocity is included as an observable, radar-only ice retrievals may return a robust solution, provided the mass–dimensional relationships are well constrained. Retrieval of liquid cloud properties requires observations of microwave radiometer Tb, but in the case of low-frequency microwave radiometer observations there remains a nonunique relationship between number and liquid water content.
When the constraint on the assumed gamma distribution width parameter is relaxed, the retrieval may still be able to produce a unique solution, but the solution mode will differ from that obtained with a specified value.
When assumed mass–dimensional parameters are allowed to vary, the retrieved liquid and ice water path and number become far less certain, and retrieval of a single unique value is rendered more difficult.
In select cases, specification of the mass–dimensional parameters and the PSD width produces a nonunique solution that does not exist when these parameters are allowed to vary. Specifically, when the gamma distribution width is set equal to zero and the mass–dimensional parameters are fixed, a combination of a small number of large liquid cloud droplets with generally small ice particles provide a fit to the combined radar and microwave radiometer observations over a very narrow range. The large sensitivity of the observations to small changes in the liquid particle size distribution in this region of the solution space leads to the production of a highly localized solution mode that disappears when gamma distribution width and ice crystal shape are allowed to vary.
This study illustrates the importance of a rigorous exploration of the retrieval solution space, and indicates that care is needed when interpreting the results of radar-only and combined radar-passive cloud property retrievals. This is especially true when the empirical parameters describing the ice crystal habit are specified, if a priori information is not available to actually constrain those parameters. The results also illustrate the important role of prior knowledge in optimal estimation-type retrievals. In those cases for which MCMC returned a multimode (nonunique) solution, prior knowledge could serve to restrict the solution space to a unique set of PSD parameters. However, global specification of prior information is a difficult exercise at best, and at worst may lead to the retrieval of the wrong set of PSD parameters when the solution space is by nature nonunique.
In closing, it is appropriate to point out a number of limitations to the current study. First, the temperature profile was obtained from a nearby sounding and was not allowed to vary as part of the retrieval. Undoubtedly, variability in temperature would have had an effect on the microwave radiometer Tbs as well as a (small) effect on the retrieved Doppler velocity, which exhibits a weak dependence on the air density. Averaging of the observations into three broad layers had an unknown effect on the retrieval error characteristics. While it may be reasonable to expect that averaging (as a smoothing operator) leads to smaller noise in the solution, this does not necessarily mean that the uncertainty is reduced. Additional retrieval error is also present because radar attenuation was not represented in the radar forward model, and changes in the ice particle shape were not functionally related to changes in the radar reflectivity. While many in situ analyses of hydrometeor populations indicate a gamma function provides a close match to observed particle size distributions, the assumption that PSDs adhere to any particular functional form introduces additional (and unknown) uncertainties into the retrieval. It should also be emphasized that the results reported here are preliminary, as the analysis has only been applied to a single vertical profile on a single case day. In the near future, we plan to explore the characteristics of the solution space for a time series of profiles, as well as for other case days observed during the StormVEx field campaign.
Last, and perhaps most significant, it should be emphasized that the results and conclusions presented in this study are only applicable to the particular observations, forward models, and physical environment we have chosen. Much of the nonuniqueness observed in the posterior PDFs may not exist for other cloud system types; and in particular for bimodal liquid PSDs. In addition, it is expected that high-frequency microwave radiometer observations (far more sensitive to scattering from ice crystals) may provide enough unique information to constrain aspects of the snow PSD. The use of multifrequency radar observations and use of observations of Doppler spectral width would also likely render a unique solution for many of the parameters studied here. Of course, the converse is also true; in other types of ice cloud, W-band reflectivity and Doppler velocity may not be sufficient to uniquely constrain the retrieval of ice content and number. While we view the methodology and results to be robust for our chosen case, care must be taken not to assume these results apply globally to all cloud systems. We are currently using the framework described above to explore retrieval sensitivity to forward-model assumptions, potential increase in information provided by other sensors, and the error characteristics of retrieved PSDs for other cloud system types.
Acknowledgments
This research was supported primarily by the U.S. Department of Energy’s Atmospheric System Research, an Office of Science, Office of Biological and Environmental Research program, under Grants DE-SC0007059 (DP), ER65324-1038673-0017514 (GM), and ER65235-1038449-0017520 (GM). Data were obtained from the Atmospheric Radiation Measurements Program sponsored by the U. S. Department of Energy Office of Science, Office of Biological and Environmental Research, Environmental Science Division. We also thank members of StormVEx science team for their tireless work in making the program a success: Gannet Hallar, Ian McCubbin, Matt Shupe, Roger Marchand, Sergey Matrosov, and Chuck Long. Sergey Matrosov provided T-matrix calculations of the W-band backscatter cross sections. Stephanie Avey and Qiuqing Zhang at the University of Utah provided programming assistance.
APPENDIX A
Microphysical Parameter Equations
Calculating microphysical properties from the modified gamma PSD functions generally entails integrating the PSDs after multiplying by an empirical relationship that describes cross-sectional area 
Using the notation described above, where subscript i represents either small (s: typically representing cloud) or large (l: typically representing precipitation) particle modes, the microphysical parameters of interest are
- the total particle number concentration with cgs units of per centimeter cubed:
- the volume extinction coefficient taken to be 2 times the total cross-sectional area with cgs units of per centimeter:
- the water content with cgs units of grams per centimeter cubed:and 4) the effective radius is defined as the area-weighted mean size (Hansen and Travis 1974) with cgs units of centimeters:
APPENDIX B
The Radar Forward-Model Equations
Doppler spectrum
Because, in most applications of the radar forward model, we are interested in multimodal PSDs existing within turbulent air motions that include mean ascent or descent of air within the radar sample volume, it is often more convenient to deal directly with simulated Doppler spectra. While mean air motions cause a quiet-air spectrum to be translated in velocity, turbulence causes the spectrum to broaden and must be calculated by formally convolving the quiet-air Doppler spectrum with the turbulent PDF (Gossard 1994). We approach this problem in a manner identical to the development in Deng and Mace (2006), except that we adopt the bimodal PSD described by the modified gamma function above, and we allow for the radar backscatter cross sections and fall speed power laws to vary as a function of size. With these adaptations, Eqs. (8) and (9) of Deng and Mace (2006) are solved numerically for each PSD mode, and the backscattered power in each velocity bin is given by the sum of the backscattered power from each PSD mode.
REFERENCES
Bohren, C. F., and D. R. Huffman, 1983: Absorption and Scattering of Light by Small Particles. Wiley-Interscience, 530 pp.
Borys, R. D., and M. A. Wetzel, 1997: Storm Peak Laboratory: A research, teaching and service facility for the atmospheric sciences. Bull. Amer. Meteor. Soc., 78, 2115–2123, doi:10.1175/1520-0477(1997)078<2115:SPLART>2.0.CO;2.
Delanoë, J., and R. J. Hogan, 2008: A variational scheme for retrieving ice cloud properties from combined radar, lidar, and infrared radiometer. J. Geophys. Res., 113, D07204, doi:10.1029/2007JD009000.
Deng, M., and G. Mace, 2006: Cirrus microphysical properties and air motion statistics using cloud radar Doppler moments. Part I: Algorithm description. J. Appl. Meteor. Climatol., 45, 1690–1709, doi:10.1175/JAM2433.1.
Donovan, D. P., 2003: Ice-cloud effective particle size parameterization based on combined lidar, radar reflectivity, and mean Doppler velocity measurements. J. Geophys. Res., 108, 4573, doi:10.1029/2003JD003469.
Dorsi, S. W., 2013: Airborne and surface-level in situ observations of wintertime clouds in the southern Rockies. Ph.D. dissertation, University of Colorado, 204 pp.
Ebell, K., U. Löhnert, S. Crewell, and D. D. Turner, 2010: On characterizing the error in a remotely sensed liquid water content profile. Atmos. Res., 98, 57–68, doi:10.1016/j.atmosres.2010.06.002.
Frisch, A. S., G. Feingold, C. W. Fairall, T. Uttal, and J. B. Snider, 1998: On cloud radar and microwave radiometer measurements of stratus cloud liquid water profiles. J. Geophys. Res., 103, 23 195–23 197, doi:10.1029/98JD01827.
Gossard, E. E., 1994: Measurement of cloud droplet size spectra by Doppler radar. J. Atmos. Oceanic Technol., 11, 712–726, doi:10.1175/1520-0426(1994)011<0712:MOCDSS>2.0.CO;2.
Haario, H., E. Saksman, and J. Tamminen, 1999: Adaptive proposal distribution for random walk Metropolis algorithm. Comput. Stat., 14, 375–395, doi:10.1007/s001800050022.
Hale, G. M., and M. R. Querry, 1973: Optical constants of water in the 200 nm to 200 μm wavelength region. Appl. Opt., 12, 555–563, doi:10.1364/AO.12.000555.
Hallar, A. G., G. Chirokova, I. B. McCubbin, T. H. Painter, C. Wiedinmyer, and C. Dodson, 2011a: Atmospheric bioaerosols transported via dust storms in western United States. Geophys. Res. Lett., 38, L17801, doi:10.1029/2011GL048166.
Hallar, A. G., D. H. Lowenthal, G. Chirokova, C. Wiedinmyer, and R. D. Borys, 2011b: Persistent daily new particle formation at a mountain-top location. Atmos. Environ., 45, 4111–4115, doi:10.1016/j.atmosenv.2011.04.044.
Hammonds, K. D., G. G. Mace, and S. Y. Matrosov, 2014: Characterizing the radar backscatter cross-sectional sensitivities of ice-phase hydrometeor size distributions via a simple scaling of the Clausius–Mossotti factor. J. Appl. Meteor. Climatol., doi:10.1175/JAMC-D-13-0280.1, in press.
Hansen, J. E., and L. D. Travis, 1974: Light scattering in planetary atmospheres. Space Sci. Rev., 16, 527–610, doi:10.1007/BF00168069.
Hastings, W. K., 1970: Monte Carlo sampling methods using Markov chains and their applications. Biometrika, 57, 97–109, doi:10.1093/biomet/57.1.97.
Hewison, T. J., and S. J. English, 1999: Airborne retrievals of snow and ice surface emissivity at millimeter wavelengths. IEEE Trans. Geosci. Remote Sens., 37, 1871–1879, doi:10.1109/36.774700.
Heymsfield, A. J., C. Schmitt, A. Bansemer, and C. H. Twohy, 2010: Improved representation of ice particle masses based on observations in natural clouds. J. Atmos. Sci., 67, 3303–3318, doi:10.1175/2010JAS3507.1.
Kummerow, C., W. S. Olson, and L. Giglio, 1996: A simplified scheme for obtaining precipitation and vertical hydrometeor profiles from passive microwave sensors. IEEE Trans. Geosci. Remote Sens., 34, 1213–1232, doi:10.1109/36.536538.
Lebsock, M. D., T. S. L’Ecuyer, and G. L. Stephens, 2011: Detecting the ratio of rain and cloud water in low-latitude shallow marine clouds. J. Appl. Meteor. Climatol., 50, 419–432, doi:10.1175/2010JAMC2494.1.
Locatelli, J. D., and P. V. Hobbs, 1974: Fall speeds and masses of solid precipitation particles. J. Geophys. Res., 79, 2185–2197, doi:10.1029/JC079i015p02185.
Löhnert, U., S. Crewell, C. Simmer, and A. Macke, 2001: Profiling cloud liquid water by combining active and passive microwave measurements with cloud model statistics. J. Atmos. Oceanic Technol., 18, 1354–1366, doi:10.1175/1520-0426(2001)018<1354:PCLWBC>2.0.CO;2.
Löhnert, U., G. Feingold, T. Uttal, A. S. Frisch, and M. D. Shupe, 2003: Analysis of two independent methods for retrieving liquid water profiles in spring and summer Arctic boundary clouds. J. Geophys. Res., 108, 4219, doi:10.1029/2002JD002861.
Luke, E. P., and P. Kollias, 2013: Separating cloud and drizzle radar moments during precipitation onset using Doppler spectra. J. Atmos. Oceanic Technol., 30, 1656–1671, doi:10.1175/JTECH-D-11-00195.1.
Mace, G. G., T. P. Ackerman, P. Minnis and D. F. Young, 1998: Cirrus layer microphysical properties derived from surface-based millimeter radar and infrared interferometer data. J. Geophys. Res.,103, 23 207–23 216, doi:10.1029/98JD02117.
Mace, J., and Coauthors, 2010: STORMVEX: The Storm Peak Lab Cloud Property Validation Experiment science and operations plan. U.S. Department of Energy Tech Rep. DOE/SC-ARM-10-021, 45 pp.
Mather, J. H., and J. W. Voyles, 2013: The ARM Climate Research Facility: A review of structure and capabilities. Bull. Amer. Meteor. Soc., 94, 377–392, doi:10.1175/BAMS-D-11-00218.1.
Matrosov, S. Y., B. W. Orr, R. A. Kropfli, and J. B. Snider, 1994: Retrieval of vertical profiles of cirrus cloud microphysical parameters from Doppler radar and infrared radiometer measurements. J. Appl. Meteor., 33, 617–626, doi:10.1175/1520-0450(1994)033<0617:ROVPOC>2.0.CO;2.
Matrosov, S. Y., M. D. Shupe, and I. V. Djalalova, 2008: Snowfall retrievals using millimeter-wavelength cloud radars. J. Appl. Meteor. Climatol., 47, 769–777, doi:10.1175/2007JAMC1768.1.
McFarlane, S. A., K. F. Evans, and A. S. Ackerman, 2002: A Bayesian algorithm for the retrieval of liquid water cloud properties from microwave radiometer and millimeter radar data. J. Geophys. Res., 107, 4317, doi:10.1029/2001JD001011.
Metropolis, N., A. W. Rosenbluth, M. N. Rosenbluth, A. H. Teller, and E. Teller, 1953: Equation of state calculations by fast computing machines. J. Chem. Phys., 21, 1087–1092, doi:10.1063/1.1699114.
Mitchell, D. L., 1996: Use of mass- and area-dimensional power laws for determining precipitation particle terminal velocities. J. Atmos. Sci., 53, 1710–1723, doi:10.1175/1520-0469(1996)053<1710:UOMAAD>2.0.CO;2.
Mitchell, D. L., and A. J. Heymsfield, 2005: Refinements in the treatment of ice particle terminal velocities, highlighting aggregates. J. Atmos. Sci., 62, 1637–1644, doi:10.1175/JAS3413.1.
Posselt, D. J., 2013: Markov chain Monte Carlo methods: Theory and applications. Data Assimilation for Atmospheric, Oceanic and Hydrologic Applications, 2nd ed. S. K. Park and L. Xu, Eds., Springer 59–87.
Posselt, D. J., and T. Vukicevic, 2010: Robust characterization of model physics uncertainty for simulations of deep moist convection. Mon. Wea. Rev., 138, 1513–1535, doi:10.1175/2009MWR3094.1.
Posselt, D. J., and C. H. Bishop, 2012: Nonlinear parameter estimation: Comparison of an ensemble Kalman smoother with a Markov chain Monte Carlo algorithm. Mon. Wea. Rev., 140, 1957–1974, doi:10.1175/MWR-D-11-00242.1.
Posselt, D. J., T. S. L’Ecuyer, and G. L. Stephens, 2008: Exploring the error characteristics of thin ice cloud property retrievals using a Markov chain Monte Carlo algorithm. J. Geophys. Res., 113, D24206, doi:10.1029/2008JD010832.
Posselt, D. J., D. Hodyss, and C. H. Bishop, 2014: Errors in ensemble Kalman smoother estimates of cloud microphysical parameters. Mon. Wea. Rev., 142, 1631–1654, doi:10.1175/MWR-D-13-00290.1.
Pruppacher, H. R., and J. D. Klett, 1978: Microphysics of Clouds and Precipitation. Reidel, 714 pp.
Rambukkange, M. P., J. Verlinde, E. W. Eloranta, C. J. Flynn, and E. E. Clothiaux, 2011: Using Doppler spectra to separate hydrometeor populations and analyze ice precipitation in multilayered mixed-phase clouds. IEEE Geosci. Remote Sens. Lett., 8, 108–112, doi:10.1109/LGRS.2010.2052781.
Rodgers, C. D., 2000: Inverse Methods for Atmospheric Sounding, Theory and Practice. Series on Atmospheric, Oceanic and Planetary Physics, Vol. 2, World Scientific, 240 pp.
Sadiku, M. N. O., 1985: Refractive index of snow at microwave frequencies. Appl. Opt., 24, 572–575, doi:10.1364/AO.24.000572.
Schmitt, C. G., and A. J. Heymsfield, 2010: The dimensional characteristics of ice crystal aggregates from fractal geometry. J. Atmos. Sci., 67, 1605–1616, doi:10.1175/2009JAS3187.1.
Szyrmer, W., and I. Zawadzki, 2010: Snow studies. Part II: Average relationship between mass of snowflakes and their terminal velocity. J. Atmos. Sci., 67, 3319–3335, doi:10.1175/2010JAS3390.1.
Tamminen, J., 2004: Validation of nonlinear inverse algorithms with Markov chain Monte Carlo method. J. Geophys. Res., 109, D19303, doi:10.1029/2004JD004927.
Tamminen, J., and E. Kyrölä, 2001: Bayesian solution for nonlinear and non-Gaussian inverse problems by Markov chain Monte Carlo method. J. Geophys. Res., 106, 14 377–14 390, doi:10.1029/2001JD900007.
Tarantola, A., 2005: Inverse Problem Theory and Methods for Model Parameter Estimation. SIAM, 342 pp.
Wood, N. B., 2011: Estimation of snow microphysical properties with application to millimeter-wavelength radar retrievals for snowfall rate. Ph.D. dissertation, Colorado State University, 231 pp. [Available online at http://hdl.handle.net/10217/48170.]
Zhang, Y., and G. G. Mace, 2006: Retrieval of cirrus microphysical properties with a suite of algorithms for airborne and spaceborne lidar, radar, and radiometer data. J. Appl. Meteor. Climatol., 45, 1665–1689, doi:10.1175/JAM2427.1.
Zhao, Y., G. G. Mace, and J. Comstock, 2011: The occurrence of particle size distribution bimodality in midlatitude cirrus as inferred from ground-based remote sensing data. J. Atmos. Sci., 68, 1162–1179, doi:10.1175/2010JAS3354.1.
