1. Introduction

An understanding of the microphysical properties of ice, such as the relationship between particle mass and maximum dimension, is an important consideration in the representation of ice cloud and precipitation in models of all scales including global climate models (GCMs; Thompson et al. 2008; Morrison and Milbrandt 2015) as well as for retrievals of ice microphysics (e.g., ice water content and effective radius) from active and passive remote sensors. The distribution of ice mass with size, a key descriptor of the ice particle habit or shape, is a piece of information essential in these studies, yet this quantity has not been unambiguously derived from in situ data even when bulk mass and particle size distributions (PSD) are measured independently (Heymsfield et al. 2010). Moreover, ice crystal habits are extremely varied, and irregular shapes are predominant in natural clouds (Korolev et al. 1999; Bailey and Hallett 2009). This situation renders it impossible either to approximate them as spheres, like the treatrment for liquid droplets, or to classify them as discrete habits such as has been common in most studies. Correspondingly, either a discrete relationship for the effective ice density (e.g., Heymsfield et al. 2004) or a form of power law is commonly assumed to represent the variation of ice particle mass (M) with respect to particle size (Nakaya and Terada 1935; Mitchell et al. 1990; Brown and Francis 1995; among others):

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where D is the maximum dimension of a nonspherical ice crystal. Hereinafter, Eq. (1) will be referred to as a mass–dimensional (M–D) relationship. Accordingly, a_m and b_m will be referred to as M–D parameters or the M–D prefactor and exponent, respectively. For consistency, cgs units apply to all M–D parameters mentioned in this paper. Bulk ice water content (IWC) is expressed as the integral of the mass of individual ice particles in a volume across the PSD.

M–D power-law relationships have been widely assumed in the parameterization of microphysical processes in predictive models (e.g., Lin and Colle 2011), retrievals of ice microphysical properties, such as IWCs and snowfall rates, from various remote sensing measurements (Zhang and Mace 2006; Deng et al. 2010; Matrosov and Heymsfield 2008; Delanoë and Hogan 2008, 2010; among others), and calculations of ice particle terminal velocities as well (e.g., Mitchell and Heymsfield 2005). Traditionally, modelers and retrieval algorithm developers choose discrete habits that fix a_m and b_m for certain habits within specific size ranges or for composite particle populations, while sensitivity tests demonstrate that the variability in M–D relationships could cause one order of magnitude variability in the calculated radar backscatter cross sections (Matrosov et al. 2009; Hammonds et al. 2014). It is becoming increasingly apparent that choosing specific power-law parameters tends to drive uncertainty in both modeling and retrieval algorithm applications (Mace and Benson 2017), necessitating a more rigorous quantification of uncertainties associated with M–D parameters but also uncertainties resulting from the assumption of M–D relationships.

Before data from research aircraft became more reliable and extensive, M–D relationships were mainly derived by collecting single ice particles at the surface level (often at high-elevation sites), which were photographed to document their size and shape, and then melted to determine their mass (e.g., Locatelli and Hobbs 1974). Via this method, habit-dependent M–D relationships were characterized with the applicable range of the particle size specified, although the statistical representativeness of such samples were always in question. Mitchell (1996) summarized a list of important work for various habits from dendrites, columns to various aggregates, and extremely rimed particles (black asterisks in Fig. 1) that shows that a_m varies over a factor of 100 and b_m varies in a manner strongly correlated with a_m within a smaller range from approximately 1.7 to 2.8, though the latter’s influence could be substantial as an exponent. Another typical approach to infer M–D relationships is pursuing a good fit to airborne IWC measurements, given the coincidently observed PSD. The increasing availability of aircraft data largely expands the size and climatological variability of sampling as well as allowing M–D relationships for ice particle ensembles to be developed. Brown and Francis (1995, hereinafter BF95) utilized the data from just two flights within warm cirrus layers using a P3 aircraft and demonstrated the M–D relationship appropriate for aggregates of unrimed bullets, columns, and side planes, among those derived by Locatelli and Hobbs (1974), generated the best estimate of IWC. The particle size was originally defined in BF95 as the mean of the maximum dimensions measured parallel and perpendicular to the photodiode array. Hogan et al. (2012) provided a transformation of BF95 in terms of the maximum dimension, which is M = 0.012D^1.9 (D ≥ 66 μm). Heymsfield et al. (2004, hereinafter H04) derived M = 0.0111D^2.4, for synoptically generated ice clouds, from condensed water contents acquired by a counterflow virtual impactor (CVI) and PSDs collected by a particle spectrometer. Note how different the exponent of BF95 is from H04, though both M–D relationships were estimated from aircraft observations focusing on ice cloud layers primarily formed through large-scale lifting. Recently, Maahn et al. (2015) introduced a new method to obtain M–D relationships from the derived power-law relationship between radar reflectivity and Doppler velocity measurements. Similar to H04 and BF95, one pair of M–D parameters was estimated for the entire dataset. Also, they found assuming temperature-dependent M–D parameters could improve the agreement between modeled moments of Doppler spectrum and observations.

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The black asterisks indicate the empirical M–D relationships for various habits derived from the ice crystals collected at the ground, summarized in Table 1 in Mitchell (1996). The red solid line is the fit derived from all M–D parameters in his Table 1, [b_m = 3.457 + 0.51 log₁₀(a_m)]. The orange line is also derived from the M–D parameters in his Table 1, with two points excluded (b_m = 2.8 and b_m = 2.91). The blue dashed line [b_m = 4.088 + 0.837 log₁₀(a_m)] and the blue solid line [b_m = 3.7038 + 075 log₁₀(a_m)] are the fits from the retrievals in Heymsfield et al. (2010) for stratiform and convective clouds, respectively. The blue triangles are their retrieved composite M–D relationships (M = 0.005 28D^2.1 and M = 0.0078D^2.2).

Citation: Journal of Applied Meteorology and Climatology 56, 3; 10.1175/JAMC-D-16-0222.1

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Composite M–D relationships have been widely utilized, since the coexistence of various habits within natural clouds is the usual case and assigning a habit within a specific size range is mostly infeasible, especially when aggregation and riming rather than the diffusional growth dominate the PSD evolution. However, M–D properties can vary to a great extent, depending on the environmental history of the ice crystal population. During the last decade, M–D relationships were developed from distinct datasets focusing on ice clouds generated under different conditions and increasing attention has been devoted to exploring how they vary with cloud conditions and temperatures (e.g., Heymsfield et al. 2007a, 2013). The commonly applied BF95 relationship was evaluated with data from six field campaigns that recorded ice cloud properties of various cloud types and temperatures (Heymsfield et al. 2010, hereinafter H2010). They found BF95 could provide IWC matching the measurements in a mean sense, while the uncertainty of inferred IWC could be as large as a factor of 5 for ice particles of certain habits within some specific size ranges. H2010 pointed out that dependencies of ice mass on temperature and crystal size were not well represented by BF95. These dependencies reflect the complex microphysical processes operating within cold clouds that, when not accurately represented, could result in considerable error in retrievals of other PSD moments such as precipitation rates (Hammonds et al. 2014; Mace and Benson 2017).

Jackson et al. (2012) applied an automated habit identification scheme to in situ 2D imagery at 60-s resolution, depending on which M–D parameters were selected from a list of 9 habits: spheres, dendrites, stellars, columns, plates, rosettes and budding rosettes, small irregulars, and big irregulars. They found the mean difference between IWC measurements and habit-dependent derived IWC was approximately 50%. These results among others strongly suggest that assumptions of M–D relationships introduce significant uncertainties. In response, the modeling communities have been attempting to simulate the evolution of ice particles more realistically by allowing M–D parameters to vary with the degree of riming (Morrison and Grabowski 2008; Morrison and Milbrandt 2015) and ambient temperature (Lin and Colle 2011) rather than employing constant values empirically derived in observational studies. It is necessary that the observational community provide sufficient constraints on this innovative modeling work.

It was demonstrated that a_m could always be found to fit the IWC measurements, given a b_m within the reasonable range (Heymsfield et al. 2007a), while the derived higher moments of the PSD (e.g., radar reflectivities) with such M–D parameters were subject to significant uncertainty (Heymsfield et al. 2007b). This suggests that the problem is ill posed, when a_m and b_m are concurrently inferred from only the measurement of IWC and the PSD. Furthermore, a strikingly strong correlation was found between a_m and b_m (Wood et al. 2015), which we take advantage of later to help constrain this ill-posed problem. From the M–D parameters inferred for a variety of habits (Mitchell 1996, hereinafter M1996), a fit was acquired and formulated in log base as

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which indicates b_m is very likely to increase with a_m, while the processes of aggregation and riming are increasingly involved. Based on early single-particle samplings as well (Locatelli and Hobbs 1974; Mitchell et al. 1990; Zikmunda and Vali 1972; among others), Wood et al. (2014) reported the correlation between ln(a_m) and b_m to be as high as 0.753. Meanwhile, similar correlations were also noted between a_m and b_m retrieved from aircraft measurements. The retrieved M–D parameters were respectively fitted for stratiform cases (blue dashed line in Fig. 1) and convective cases (blue solid line in Fig. 1). When we exclude two data points (a_m = 0.049, b_m = 2.8, graupel and a_m = 0.001 66, b_m = 2.91, hexagonal columns) with the exponents obviously beyond the common range, we obtain a fit (orange line in Fig. 1) whose slope agrees well with those of the blue lines from H2010, which suggests a_m and b_m covary in such a way as to capture the variation of ice mass with particle size. The different intercepts are in accordance with the fact that M–D properties are subject to the specific processes that take place in clouds of various types, and they tend to also depend on the environmental history of the ice crystal population. Comparing the M–D parameters for different clouds, Fig. 1 shows that larger a_m is derived for convective clouds with the same b_m given, which can be attributed to the vertical mixing of habits and PSDs within the convective clouds (Heymsfield et al. 2007a; H2010). During the sedimentation of ice crystals through natural clouds, environmental conditions change toward those more conducive to the growth of ice crystals by riming and aggregation, in agreement with which the fit through the ground-based samplings in Fig. 1 shifts toward larger b_m from the fits based on airborne measurements.

In this study, an optimal estimation (OE) methodology is employed to derive uncertainties in M–D relationships using airborne in situ measurements. In addition to IWCs, time-varying ice particle area–dimensional (A–D) relationships are inferred from optical array imaging probes, which allows the information regarding ice crystal projected areas to contribute to this ill-constrained problem. Section 2 introduces datasets and details the retrieval implementation. Section 3 contains overall results of retrieved A–D and M–D relationships. Section 4 evaluates the retrieval algorithm and discusses the influence of M–D uncertainties. Section 5 presents case study comparisons with concurrent satellite observations and analysis of large-scale dynamics and retrieval uncertainties of M–D relationships. The major conclusions are summarized in section 6.

2. Algorithm development

a. Data

The Midlatitude Airborne Cirrus Properties Experiment (MACPEX; Jensen et al. 2013; Rollins et al. 2014) was undertaken in March and April 2011, designed to emphasize sampling of in situ cirrus. The WB-57 research aircraft flew 14 science flights mainly over the south-central United States, with the payload including a number of cloud microphysics probes and instruments that measured state parameters and water vapor concentration. Data from 10 flights are analyzed in this study, 6 of which targeted synoptic cirrus and the rest sampled anvil cirrus. The measurements used in this study are provided by the two-dimensional stereo probe (2D-S; Lawson et al. 2006), the high-volume precipitation spectrometer (HVPS), the first-generation Colorado closed-path tunable-diode laser hygrometer (CLH; Davis et al. 2007b), and the Meteorological Measurement Systems (MMS). To focus on cold midlatitude cirrus, which characterizes MACPEX, we employ a few state parameters as criteria and select data points when ambient temperatures are colder than −30°C and IWC as measured by the CLH greater than 0.005 g m⁻³.

The CLH probe measures the enhanced total water content that is the sum of ambient water vapor and IWC multiplied by the size-dependent enhancement factor (eTW = IWC × EF + W; Davis et al. 2007b). During MACPEX, JPL Laser Hygrometer (JLH) and JPL Unmanned Aircraft System Laser Hygrometer (ULH) provided water vapor measurements, which would be replaced by the CLH background value in clear air before and after the cloud penetration, when in situ measurements were unavailable. The IWC in units of density was calculated using the enhancement factor as well as the temperature and pressure obtained by MMS. The uncertainty of calculated IWC was estimated to be approximately 27% at two-sigma confidence, but could approach 50%, when IWC values were less than 0.005 g m⁻³ (Davis et al. 2007a,b). Correspondingly, an IWC threshold of 0.005 g m⁻³ was used in this paper. The first piece of information needed for inference of the M–D relationship are 10-s-averaged IWCs from CLH.

The 2D-S is an optical array imaging probe manufactured by the Stratton Park Engineering Corporation (SPEC) that records the projected areas of three-dimensional ice particles as well as the PSDs over the particle size ranging from 10 to 1280 μm. To mitigate the ice crystal shattering problem, the probe tips were modified and the arrival time algorithm was applied to the collected sampling to remove the artifacts (Lawson 2011). We use linear regression to infer an area–dimensional relationship from the size-resolved projected area normalized by total number of particles in each size bin for each 10-s interval,

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where a_a and b_a are area–dimensional parameters in cgs units. The quantities A and D indicate the ice particle projected area and its maximum dimension, respectively. Particles smaller than 100 μm are not taken into account in deriving the fit, since the averaged projected area is subject to uncertainties due to the considerable number of small particles, and it was found that the microphysical and radiative contributions of particles with size below 100 μm are relatively minor (Lawson et al. 2010). Only a_a and b_a within a physically reasonable range are accepted, which requires that a_a is less than π/4 and b_a falls between 1.4 and 2. To ensure a significant performance of the linear fit data points are excluded if the standard deviation of derived b_a exceeds 0.15, which agrees with the standard deviation of b_a estimated in Erfani and Mitchell (2016, hereinafter EM2016) in general.

Baker and Lawson (2006) found better-constrained M–D relationships could be expected with information regarding ice particle projected area involved. Also, the relationship between b_a and b_m has been discussed in previous studies: 1) Schmitt and Heymsfield (2010) determined the scaling factor (i.e., S = b_m/b_a) from fractal geometry and found S varies from 1.23 to 1.33 depending on aspect ratio. 2) Fontaine et al. (2014) fitted b_m = 1.93b_a − 1.02 from 1000 simulated 3D ice particles (47 habits classified) and the corresponding projections. A similar relationship, b_m = 1.46b_a − 0.354, could be derived from M1996 as well. Therefore, in addition to CLH IWC, information provided by projected areas recorded by 2D-S and the correlation between b_a and b_m will be used in this retrieval.

We created 10-s-average PSDs via merging the in situ data collected by 2D-S and HVPS. The HVPS data cover the range where particle size exceeds 1000 μm, below which 2D-S data are used. A method of moments approach was used to fit the merged PSDs to two modified gamma functions for the small (s) and large (l) particle modes (Zhao et al. 2011), which is formulated as

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where i stands for the particle mode and D is the maximum dimension of the ice particle projected area; N_x (cm⁻⁴) is a proportionality term related to the total number concentration; D_x (cm) and α represent the size parameter and the shape parameter, respectively. It was demonstrated that use of a gamma distribution estimated from a moment preserving approach could generate radar reflectivities most consistent with corresponding observations (Maahn et al. 2015), although a quantitative estimate of the uncertainties due to the chosen parameterization of PSD needs further investigation. In addition, Mascio and Mace (2017) find a several-decibel high bias in forward-calculated radar reflectivities relative to CloudSat measurements when the binned data instead of gamma-fitted PSDs are used. Furthermore, the use of such a distribution function implicitly smooths the uncertain binned data as well as extrapolates to sizes that are poorly sampled by the probes both on the large and small ends of the size spectrum. Essentially we make the assumption that the PSD is smooth and continuous. Therefore, we chose a combined gamma PSD fit over the original 2DS and HVPS bin data and utilize Eq. (4) in the calculation of forward-model IWCs as well as forward-model radar reflectivities.

b. Optimal estimation

Based on Bayes’s theorem, the OE method (Rodgers 2000) generates an optimal solution of the state vector x for a given set of measurements y and prior knowledge of the state vector x_a, with a forward model F(x) employed to simulate the measurement space from the candidate x. Note neither the inputs (y and x_a) nor the forward model are considered perfect, in response to which each piece of information and the relationships among the pieces of information are assumed to follow Gaussian statistics. The optimal estimate is defined at the point where the posteriori probability distribution function (PDF), P(x | y) = P(y | x)P(x)/P(y), reaches the maximum. Here, P(y) indicates the probability space of the measurements, while P(x) is the PDF of the quantities to be retrieved, and P(y | x) is the conditional PDF that maps the information in y to the state x. The strength of OE is to track the sources of uncertainty with a relatively simple implementation and acceptable computational cost, despite its limitations of assuming Gaussian statistics.

Here we apply the OE approach to estimate a_m, b_m power-law pairs as well as their associated uncertainties from the bulk IWC and the exponent of area–dimensional relationships, with given PSDs. Rewriting M–D relationships in the form of a natural logarithm as ln(M) = ln(a_m) + b_m ln(D), the state vector x and the measurement vector y can be constructed as follows:

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IWC_CLH is the ice water content calculated from the CLH measurements and b_a indicates the exponent of area–dimensional relationships.

With a similar structure as y, the forward-model vector is written as

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where the subscript FM stands for forward model. The forward-calculated ice water content (IWCFM) is obtained by integrating the individual ice particle masses represented by the M–D power-law relationship across N(D) defined in Eq. (4):

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The A–D exponent in the forward model (b_a,FM) is calculated by taking advantage of a linear relationship between the A–D exponents and M–D exponents fitted from M1996 with all the habits in his Table 1 included:

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The standard deviation of b_m is approximately 0.132, which is estimated from the regression.

In the specific implementation, maximizing P(x | y) is achieved via minimizing the following cost function:

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where is the covariance matrix of total uncertainties, defined as the sum of observational uncertainties () and forward-model uncertainties. The a priori state x_a and the associated covariance matrix represent our best knowledge of x before observations. Here x_a and are extracted from a series of habit-dependent M–D relationships reported in M1996, excluding which are assigned for hail and graupel. A priori mean, a_m = 0.0128 and b_m = 2.2, is used for x_a. The quantities and are two covariance matrices that characterize the uncertainties of prior information and measurements, respectively:

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The diagonal elements of the above matrices are the variances of the corresponding variables in the natural logarithm space, while the off-diagonal elements indicate the covariance of these variables. The is inferred from the M–D relationships summarized in M1996, which covered a variety of habits from dendrites to rimed aggregates. As for measurement uncertainties, 1-sigma standard deviation of IWC calculated from CLH is estimated to be approximately 10% (Davis et al. 2007b) when IWC is greater than 0.005 g m⁻³. The uncertainty of b_a is returned by the linear regression applied to 2D-S data. We assume the measurements of IWC and the A–D exponent are independent, which renders the covariance in zero.

Equation (7) gives us a clue how measurements (y) and prior knowledge (x_a and ) collaboratively constrain the retrieval problem. Whether an optimal solution can be returned or how efficiently it is returned depends on the accuracy of the measurements, how well the prior knowledge characterizes the geophysical quantities to be inferred, and how well the forward model replicates reality.

In this case, a source of forward-model uncertainty arises from the use of 10-s-averaged PSDs, whose values depend on to what degree PSDs vary within the 10-s interval. For example, a dramatic change of PSDs might be observed when the aircraft passes the cloud edge causing large forward-model uncertainties. The 10-s-averaged PSD employed in our retrieval is created by averaging ten 1-s PSDs for each 10-s interval. To estimate the corresponding forward-model uncertainties, we create 10 random PSDs, each of which is created by averaging 6 randomly chosen PSDs within the 10-s period. By choosing 6 of 10 PSDs, we expect any gross variability would be captured, while there is enough information to smooth out random variations in the 1-s data. Then 10 forward-calculated IWCs and their variances are obtained, with Eq. (4) with fixed M–D parameters. The other source of forward-model uncertainty, the variance of b_a,FM, is determined by the linear fit based on M1996, which does not vary with time in this retrieval.

With the forward model being moderately linear, Newtonian iteration is executed to minimize the cost function. When the convergence criteria is met, the solution of the state vector along with the associated uncertainty matrix are determined statistically:

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where is a Jacobian matrix and indicates the sensitivity of forward-modeled measurements to the state vector. In our retrieval, is a 2 × 2 matrix that is constructed as

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where and are the first standard deviations of retrieved a_m and retrieved b_m, respectively. The quantity is their correlation.

To shed light on the sources of uncertainties, we present the following example. At 1621 UTC 7 April 2011, the WB-57 was flying near the cirrus cloud base, where IWC measured by CLH was 0.0374 g m⁻³ and the ambient temperature was −36.7°C. As shown in Fig. 2, one source of forward-model uncertainties is rooted in the process of creating the 10-s-averaged PSD. Ten PSDs created by randomly sampling 6 of the 10 1-s PSDs vary little from the 10-s-averaged PSD within the particle size ranging from approximately 300 to 800 μm, while large discrepancies are present when the size is less than 100 μm or greater than 1200 μm . As a result, employing the 10-s-averaged PSD would cause uncertainty in the forward-model IWC, to account for which we calculate IWCs with the 10 random PSDs and fill its one standard deviation in the matrix of that describes the forward-model uncertainty. In this example, the uncertainty in forward-model IWC is 0.006 24, which is approximately a factor of 1.7 of (Table 1). The other source of forward-model uncertainty arises from taking advantage of the correlation between b_m and b_a [Eq. (6)] and remains constant at 0.132 in retrievals. The value of is determined by the linear fit (Fig. 3) between two-dimensional projected area and particle size and is returned in this retrieval as 0.037, which is a typical value in MACPEX. Following Eq. (8) which is composed of uncertainties listed above, maps to with given and . The derived relative uncertainties of a_m and b_m in this case are 58.03% and 7.1%, respectively.

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(a) Modified gamma fit of particle size distributions and (b) binned data combined from 2D-S and HVPS at 1621 UTC 7 Apr 2011. The black solid line indicates the 10-s-averaged PSD, and dashed lines in color indicate 10 random PSDs during the 10-s period.

Citation: Journal of Applied Meteorology and Climatology 56, 3; 10.1175/JAMC-D-16-0222.1

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Table 1.

The construction of measurement uncertainty, forward-model uncertainty, Jacobian matrix , and their values at 1621 UTC 7 Apr 2011.

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The linear fit between 2D-S projected area and particle size at 1621 UTC 7 Apr 2011. Black dots denote 2D-S size-resolved total cross-sectional area averaged by the number of particles in each size bin. The black dashed line indicates the derived A–D relationship (i.e., ). The blue asterisk-dashed line presents A = 0.23D^1.88 (Mitchell 1996) as a reference.

Citation: Journal of Applied Meteorology and Climatology 56, 3; 10.1175/JAMC-D-16-0222.1

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In addition, a simple sensitivity test is performed to assess the impact of every influential factor in . We perturb each source of uncertainty in separately and observe how retrieved relative uncertainties of M–D parameters respond to such perturbations. The retrieved relative uncertainty of a_m presents more significant sensitivity to the perturbations than b_m does, which is consistent with the fact that the natural variance of a_m is much larger than that of b_m. We find the uncertainties of a_m are dominated by the uncertainty in IWC measurements as well as that of the forward-model IWC, and contributions from these two sources are at a comparable level. Meanwhile, the uncertainties of b_m are mainly determined by the uncertainties in measured b_a and forward-model b_a. The trend shown in Fig. 4 is as expected, since IWCs constrain both a_m and b_m and a linear relationship between b_a and b_m without a_m involved serves as an additional constraint in our retrieval.

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(a) The impact of CLH measurement uncertainty on relative uncertainty of (left) a_m and (right) b_m. The x axis is the CLH IWC uncertainty scaled by the value of CLH IWC. (b) As in (a), but for the forward-model uncertainty that resulted from 10-s-averaged PSDs. (c),(d) As in (a), but for the uncertainty in measured b_a and forward-model b_a, respectively.

Citation: Journal of Applied Meteorology and Climatology 56, 3; 10.1175/JAMC-D-16-0222.1

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3. Overall results

We focus on cirrus data collected during 10 MACPEX flights that were conducted in meteorological situations conducive to synoptically and convectively generated cirrus. As summarized in Table 2, jet stream and frontal cirrus dominate the synoptic cases. Since MACPEX was a short-term field campaign conducted in the month of April 2011, more cases than might be considered normal were affected by convective activity. Among the 10 cases, there are 4 that sampled primarily anvil cirrus of various ages. The number of data points varies considerably with case, which is basically determined by the operational condition of the CLH and 2D-S probes. Ambient temperatures span from −70° to −30°C with an uneven distribution over this broad range (Fig. 5a). Approximately half the data fall within the range from −45° to −33°C, with a minor peak located around −55°C as well as a spike right below −60°C. There are usually one or two dominant temperature ranges in each case determined by the flight levels of the WB57. In terms of IWC, the values are larger than the IWC of typical midlatitude cirrus (Mace et al. 2006) in general, except for 16 April, when the aircraft was sampling a thin, patchy band of subtropical jet stream cirrus completely independent from convection. In Fig. 5b, there is a long tail that extends to some IWC values far beyond the normal range of midlatitude cirrus, which is exclusively contributed by an outlier (case 0426). The mean IWC of case 0426 is an order of magnitude higher than that of case 0416 (Table 2). Data studied in the former case were collected when the aircraft flew at an altitude between 7 and 8 km, in clouds with ambient temperatures mostly warmer than −40°C and reaching −30°C. In addition, as discussed in section 2, data were screened to ensure IWC greater than 0.005 g m⁻³ because of the larger uncertainties of CLH measurements at low IWC, which might also bias the relatively high mean IWC shown here.

Table 2.

Summary of microphysical properties and dynamical background.

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Distributions of (a) temperature (°C) and (b) IWC measured by CLH in log base (g m⁻³) for all of the samplings.

Citation: Journal of Applied Meteorology and Climatology 56, 3; 10.1175/JAMC-D-16-0222.1

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The range of derived area–dimensional relationships and mass–dimensional relationships (Fig. 6) are generally consistent with previous studies (e.g., Mitchell 1996). The A–D power-law exponent, b_a, falls between 1.6 and 1.9, while the b_m distribution falls within the range from 2.1 to 2.5. The values of both pairs of parameters exhibit significant variability. For the purpose of illustration and because increases in a b_m can be countered by decreases in b_m, we compare cases by examining the mass of an ice particle with a maximum dimension of 300 μm (i.e., denoted as M₃₀₀) as well as a_m. The maximum M₃₀₀ appeared in the case of aged anvils (i.e., 0411; Table 2), and the minimum appeared in case 0425, which focused on fresh anvils, as compared with synoptic cases that show less variability. Based on 2D-S images, habits recorded include rosette shapes (mostly observed in cases 0413, 0414, and 0416), complex platelike crystals, and occasionally columns and aggregates; there was also a considerable occurrence of irregular shapes that cannot be identified resembling one specific habit. In addition to the relationship between a_m and b_m that has been discussed in detail, a nontrivial correlation between a_m and b_a is observed and such a correlation is not always apparent in literature, as Wood et al. (2015, hereinafter W2015) also pointed out. How solid these relationships are is a problem worth further studying with broader datasets, since correlations between M–D and A–D parameters could provide additional constraints on retrievals of some crucial quantities (e.g., precipitation rates), which are usually ill posed.

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(a) The distributions of retrieved a_a (x axis) and b_a (y axis) for all samplings included in this study (cgs units). The color bar indicates the frequency in logarithm base 10. (b) As in (a), but shown for a_m and b_m.

Citation: Journal of Applied Meteorology and Climatology 56, 3; 10.1175/JAMC-D-16-0222.1

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Forward-model radar reflectivities are calculated with the retrieved a_m and b_m pairs, which are temporally varied. Within the radar forward model, backscatter cross sections are calculated for the prescribed M–D relationships by scaling T-matrix derived backscatter cross sections (Matrosov 2007) with the Clausius–Mossotti factor for ice-phase hydrometeors size ranging from 1 μm to 1.9 cm, assuming oblate spheroids with the aspect ratio of 0.6 (Hogan et al. 2012). Then radar reflectivities are derived by integrating backscatter cross sections across PSDs (Posselt and Mace 2014).

In comparison with the annual average values (−24.84 ± 12.18 dBZ) listed in Mace et al. (2006) derived from long-term ground-based data, the mean of forward-model radar reflectivities for MACPEX (−16.46 dBZ; Fig. 7) leans toward the higher end but is still within one standard deviation, which is consistent the relatively large IWCs and temperatures in our dataset. The presence of radar reflectivities greater than −5 dBZ corresponds to the aggregates as large as a few thousand micrometers that were observed during some flight segments on 0421 and 0426, which is atypical for midlatitude cirrus. Moreover, radar reflectivities are also calculated with the constant M–D relationships assuming the habit is bullet rosettes and assemblages of planar polycrystals in cirrus, respectively. The habits and values of the selected M–D parameters are expected to somehow represent the data we study. However, it is shown in Fig. 7 that assuming bullet rosettes yields underestimated radar reflectivities, and the other assumption results in overestimated reflectivities.

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The distribution of forward-model radar reflectivities calculated with the retrieved M–D relationships (black). The mean and standard deviation of forward-model radar reflectivities calculated with M = 0.003 08D^2.26 (bullet rosettes; blue) and M = 0.007 39D^2.45 (assemblages of planar polycrystals in cirrus; red) are shown for reference. The dashed lines and arrows represent mean and standard deviation, respectively. The constant M–D relationships with habits assigned are from M1996.

Citation: Journal of Applied Meteorology and Climatology 56, 3; 10.1175/JAMC-D-16-0222.1

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4. Uncertainty analysis

Along with values of a_m and b_m, associated uncertainties were obtained as well from the OE framework. Figure 8 presents the relative uncertainties of M–D parameters, which are defined as the standard deviations divided by the means of retrieved Gaussian PDFs. The uncertainties of a_m are mostly (96.1% of the data) between 50% and 80%, and the main range of b_m uncertainties (96.2% of the data) is approximately from 6% to 9.5%. Mean relative uncertainties of each case are summarized in Table 3. We find relative uncertainties do not vary with cases as substantially as M–D parameters. To offer a context to evaluate our retrieval algorithm and the level of uncertainties, we list the findings of a few recent studies focusing on ice particle M–D relationships (Table 4). H2010 assumed a range of b_m with values from 1.5 to 2.5 in increments of 0.1 and derived the corresponding a_m from measurements of IWCs and PSDs. Based on their Fig. 5, 100% is likely to be a good approximation of their relative uncertainty of a_m, which was also assumed in Hammonds et al. (2014). Extracting M–D relationships from IWCs is a traditional method and has been widely used (Heymsfield et al. 2007a), which could act as a reliable reference point here. Another recent study by Wood et al. (2015) applied OE methodology to retrieve M–D and A–D relationships of snow from radar reflectivities, precipitation rates and fall speeds. They reported the derived uncertainty of a_m was about 120% and uncertainty of b_m was approximately 16.6% (their Table 4). EM2016 developed M–D relationships by combining measurements of ice particle area from 2D-S and a cloud particle imager and deploying empirical relationships between ice particle mass and area (Baker and Lawson 2006). Uncertainties of b_m were estimated as 6.715% for anvil cirrus and 9.031% for synoptic cirrus, which are much lower than the 16.6% reported by Wood et al. (2015) and agree well with our findings. Also, our uncertainties of a_m are smaller than 100% in general. On the one hand, the fact that M–D properties are better constrained in both EM2016 and this study indicates that measurements of projected area provide valuable information in the retrieval of M–D relationships. On the other hand, results of EM2016 are based on dataset from the Small Ice Particles in Cirrus Experiment (SPARTICUS; January–June 2010), which was conducted over the continental United States with a focus on midlatitude cirrus and therefore shares considerable similarities with our dataset. It requires further validation to judge to what degree the information regarding ice particle area contributes to constraining M–D properties of ice clouds or snow in general.

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As in Fig. 6, but for relative uncertainties of a_m (x axis) and b_m (y axis).

Citation: Journal of Applied Meteorology and Climatology 56, 3; 10.1175/JAMC-D-16-0222.1

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Table 3.

Mean values of M–D parameter uncertainties in percentage for each case.

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Table 4.

A priori relative uncertainties of M–D relationships calculated from M1996 and relative uncertainties of retrieved M–D relationships reported in literature.

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To evaluate the influence of the variance of a_m on forward-calculated IWC, we estimate the relative uncertainty in IWC using simple uncertainty propagation as follows:

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where ∂ lnIWC/∂ lna_m is the Jacobian matrix quantifying the sensitivity of forward-model IWC to a_m and represents relative uncertainties. Both terms could be easily obtained during the implementation of OE. Were we to assume that is 100% (Hammonds et al. 2014), which should represent the level of uncertainty in a_m retrieved from only IWCs and PSDs, the incurred mean relative uncertainty in IWC would be around a factor of 2.83 (Fig. 9). Similarly applying Eq. (9), derived in this study results in approximately a factor-of-2.01 relative uncertainty in forward-model IWC. If translated into radar reflectivities, uncertainties of a factor of 2.83 and 2.01 in IWC would cause uncertainties of reflectivities on the order of 3.5 and 2.5 dB, respectively, according to Hammonds et al. (2014). While this seems to be some improvement, we note that a radar whose calibration is not known to within 2.5 dB would be considered a very poor radar indeed. However, a forward-model Z uncertainty is equivalent to a measurement uncertainty in the OE framework.

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The frequency distribution of relative uncertainties in the forward-calculated IWC, assuming the relative uncertainties of a_m is 100% (blue) and retrieved values (black), respectively.

Citation: Journal of Applied Meteorology and Climatology 56, 3; 10.1175/JAMC-D-16-0222.1

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5. Case studies

In our retrieval algorithm, each pair of retrieved M–D parameters represents the composite particle population at a certain time point, whose values are determined by the measured IWC, the 2D projected area distribution, and the PSD. The evolution of these microphysical properties is never independent, but is rather an extremely complex process involved with atmospheric dynamics and physical processes over a wide range of scales from the small-scale turbulence to the synoptic scale (Comstock et al. 2008; Berry and Mace 2013). Even though MACPEX was designed to mainly focus on midlatitude synoptic cirrus, striking differences regarding M–D properties are found among cases (Fig. 10). Four representative cases are selected for detailed discussion in terms of large-scale dynamics, microphysics as well as environmental conditions, three of which are synoptic cirrus (cases 0413, 0416, and 0426) and one anvil cirrus flight (case 0425). Cases 0416 and 0413 shared two temperature regimes (−54° ≤ T ≤ −49°C and −44° ≤ T ≤ −39°C), while the latter has a broader range. Case 0425 is a “cold” (T ≤ −44°C) case and 0426 is a “warm” (T ≥ −44°C) case. In Fig. 10, all cases somehow show the positive correlation between a_m and reflectivities within each temperature range, but such correlation is much more apparent in cases 0416 and 0426 and least evident in case 0425. Similarly, the values of a_m of cases 0416 and 0426 are more likely to increase with temperature, suggesting that an active aggregation or other microphysical process was actively occurring while the aircraft was sampling those clouds. For a comprehensive grasp of the variance of a_m and b_m, M₃₀₀ is presented as well (Fig. 11 and Table 2). Larger M₃₀₀ tends to be present in the cases with higher values of a_m because of the positive correlation between a_m and b_m. For example, the difference regarding M₃₀₀ between cases 0425 and 0426 is as evident as their difference of a_m. Nonetheless, it should be noted that b_m does not necessarily increase with a_m.

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The distribution of retrieved a_m with respect to forward-calculated radar reflectivities (dBZ; x axis) and ambient temperatures (°C; y axis) for cases (a) 0416, (b) 0413, (c) 0425, and (d) 0426. Each bin spans a temperature of 5°C and radar reflectivity of 5 dBZ. Bins that contain <5 samplings are not shown. The color bar indicates the mean value of a_m for each bin in logarithm base 10.

Citation: Journal of Applied Meteorology and Climatology 56, 3; 10.1175/JAMC-D-16-0222.1

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Frequency distributions of M₃₀₀ for cases (a) 0416, (b) 0413, (c) 0425, and (d) 0426 (μg). The mean and standard deviation of M₃₀₀ are listed in Table 2.

Citation: Journal of Applied Meteorology and Climatology 56, 3; 10.1175/JAMC-D-16-0222.1

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As shown in Fig. 10, the values of retrieved a_m in case 0416 are much lower than those in 0413, even when the temperatures are similar. In terms of synoptic background, both cases could be classified as jet stream cirrus, while they tend to be different in many respects. On 16 April, an optically thin, patchy band of cirrus (Figs. 12 a,b) extended along the southwesterly flow, near the axis of an upper-level trough. There was no obvious large-scale uplift or advection of moisture shown within the vicinity of sampling. On the other hand, case 0413 (Fig. 13) was associated with a moist subtropical jet stream over southwestern Texas, which was likely enhanced by gravity wave activity from the Sierra Madre range in northern Mexico (Fig. 14a). The event occurred downstream of an upper-tropospheric trough, within the southwesterly divergent flow. When compared with case 0416, a significant difference is that a high center of relative humidity was situated upstream, continuously streaming moisture into the study area. Correspondingly, the mean of IWC measurement during this flight is approximately a factor of 2 of what was recorded in case 0416 (Table 2). As a signature habit of cold in situ cirrus, a fair amount of rosette shapes could be identified on the 2D-S images for both cases, while shapes in case 0413 are much more variable and irregular, which is as expected, owing to the stable supply of moisture and variability of temperature and ice supersaturation enhanced by gravity wave activity. Averaged PSDs of each case are shown in Fig. 15, with the averaged PSD of the whole dataset superimposed as a reference. We notice considerable numbers of small crystals and lower concentration of large particles were sampled in case 0413 as compared with case 0416. The reasons could be twofold. First, cirrus in these two cases was at a very different stage of cloud life cycle, which is also qualitatively confirmed by satellite images (Figs. 12b and 14b). Small particles in case 0416 either sublimate or grow to larger size. Second, approximately 28% of the data of case 0413 were collected around −58°C, while temperatures of case 0416 are all greater than −54°C. This difference of PSDs agrees with the previous finding that maximum D increases with temperature for both convective and synoptic cirrus (Jackson et al. 2015).

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(a) Visible (VIS) and (b) IR satellite images at 1945 UTC 16 Apr 2011 with aircraft track overlay (solid lines in color). The images were provided through the courtesy of P. Minnis and L. Nguyen, NASA Langley Research Center. (c) Retrieved a_ms in case 0416 overlaid on map. Colors indicate the values of a_m in logarithm base 10.

Citation: Journal of Applied Meteorology and Climatology 56, 3; 10.1175/JAMC-D-16-0222.1

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NCEP–NCAR weather analysis at 1800 UTC 13 Apr 2011. (top) The 300-hPa wind barbs, wind speed (color shading), and geopotential height contours. (bottom) The 300-hPa vertical velocity (dashed contours) and relative humidity (color shading).The red bar marks the track of the flight.

Citation: Journal of Applied Meteorology and Climatology 56, 3; 10.1175/JAMC-D-16-0222.1

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As in Fig. 12, but for (a),(b) 1945 UTC 13 Apr 2011 and (c) case 0413.

Citation: Journal of Applied Meteorology and Climatology 56, 3; 10.1175/JAMC-D-16-0222.1

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The average combined PSDs (number concentrations: cm⁻¹; black open squares) and relative uncertainties of the averaged PSD (%) for cases (a) 0416, (b) 0413, (c) 0425, and (d) 0426. The blue asterisks indicate the averaged PSD (number concentration: cm⁻¹) of the whole dataset.

Citation: Journal of Applied Meteorology and Climatology 56, 3; 10.1175/JAMC-D-16-0222.1

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Another synoptic case (0426) presented here occurred on the last day of MACPEX, 26 April. This cirrus event was generated by large-scale uplift associated with the activity of a frontal system. Data included in this study were collected approximately from 1900 to 2100 UTC, during which the trough at 400 hPa extended toward the southeast, along with the high center of upper-tropospheric ascent building in the vicinity of the sampling location (Figs. 16 and 17). As a deep synoptic system, the deepening of the trough at lower levels and further development of surface frontal system were also observed (not shown). The WB-57 flew at or below 8 km, where temperatures were mostly between −40° and −30°C. The mean IWC was approximately 0.12 g m⁻³, which is an order of magnitude larger than that of case 0416. Among large amounts of irregular shapes, some rosettes and columns were present, when temperatures were at the lower end. As the aircraft descended down to 7 km, the 2D-S recorded increasing number of large ice particles with size larger than 500 μm and aggregates that could be as large as 2 mm, indicating conditions were becoming more favorable for depositional growth and aggregation as temperature increases. The retrieved M–D relationships where IWC approaches 1 g m⁻³ (Table 5) agree well with those specified for assemblages of planar polycrystals in cirrus (a_m = 0.007 39 and b_m = 2.45) in M1996. The amount of large particles contained in this case is far beyond the average of the whole dataset (Fig. 15d). Bimodality in the PSD is significant, with the boundary located around 100 μm. It is very likely sampling occurred near the lower part of the cirrus layer. Despite the large-scale dynamics, 0426 is not a typical synoptic cirrus case in terms of microphysical properties and features of the ice crystal habit, which is in accordance with the high values of retrieved M–D parameters. The mean a_m is 0.007 35, and the mean b_m is 2.29 (Table 2). However, this case may be representative of cirrus evolving into altostratus, which is very typical of processes that occur along the warm conveyor belt of midlatitude cyclones.

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As in Fig. 13, but for 1800 UTC 26 Apr 2011 at 400 hPa.

Citation: Journal of Applied Meteorology and Climatology 56, 3; 10.1175/JAMC-D-16-0222.1

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As in Fig. 12, but for (a),(b) 2003 UTC 26 Apr 2011 and (c) case 0426.

Citation: Journal of Applied Meteorology and Climatology 56, 3; 10.1175/JAMC-D-16-0222.1

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Table 5.

Retrieved M–D parameters and median mass diameters where IWC approaches 1 g m⁻³ on 26 Apr.

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In addition to the three synoptic cases discussed above, a cold anvil case with distinctively low values of M–D parameters is presented (Fig. 10c). Case 0425 sampled mostly fresh anvils attached with ongoing convection, at the edge of which the WB-57 flew a racetrack pattern (Fig. 18) where the temperatures ranged from −68° to −53°C. In contrast with synoptic cirrus, rosette shapes were mostly absent, while there were a fair amount of irregular aggregates evident in the 2D-S imagery. Also, the distribution of particle size is more uniform and less likely to appear as bimodal (Fig. 15c). Because of the low temperatures, there are low concentrations of large aggregates in the anvil case. In Fig. 10c, we find a_m shows almost no tendency to vary with temperature and its positive correlation with reflectivities is apparently weaker than other cases, which may be attributed to the homogeneous effect of vertical mixing within convectively generated clouds (Heymsfield et al. 2007a). Although this case is totally different from case 0416 in terms of generation mechanism, their retrieved a_m values are comparable, which suggests large-scale dynamics alone could be a poor predictor for M–D properties.

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As in Fig. 12, but for (a),(b) 2115 UTC 25 Apr 2011 and (c) case 0425.

Citation: Journal of Applied Meteorology and Climatology 56, 3; 10.1175/JAMC-D-16-0222.1

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Similar to the distribution of a_m with radar reflectivity and temperature, M₃₀₀ shows stronger dependence on reflectivity and temperature in cases 0416 and 0426. Meanwhile, we notice such a dependence is more identifiable in M₃₀₀ than in a_m for case 0425, which indicates a less substantial correlation between a_m and b_m in this case. Under such circumstance, considering only one of the M–D parameters could lead to misinterpretation. Interestingly, both a_m and M₃₀₀ of case 0413 exhibit weak dependence on the ambient temperature and yet the reasons are not completely clear and may be attributed to the active mesoscale activity in this case. H2010 reported M–D relationships for convectively generated ice clouds that yielded mass approximately twice as large as that for synoptically generated ice clouds. A similar trend is not found here, which is not surprising, since only a single anvil case is evaluated and the variability of genre is lacking.

Ice supersaturation and temperatures are two dominant factors shaping ice crystals and the shapes of ice crystals are closely correlated with M–D parameters. Instead of using one pair of M–D parameters for all conditions, increasing effort has been devoted to studying how to determine these parameters using physically based arguments. As a quantity that is important yet that can be obtained relatively easily, the dependence of M–D relationships on temperatures has been explored in many studies (e.g., H2010; EM2016; Lin and Colle 2011). On the other hand, how the choice of a_m and b_m affects forward-model radar reflectivities is another problem of common concern. Figure 19 presents the distributions of M–D parameters with respect to the forward-calculated radar reflectivities in dBZ and ambient temperatures for the whole dataset. W2015 found that reflectivity increased with increasing a_m but decreased with increasing ba_m. However, Fig. 19 shows reflectivities tend to positively correlate with a_m, while the correlation between reflectivities and b_m is barely observed, except in the regime when reflectivities increase from −8 to 8 dBZ. This discrepancy may be due to the particle size range of interest and the genre of clouds considered, since W2015 focus on precipitating mixed-phase snow, in which case aggregation and riming process that predominates at warmer temperatures in the presence of liquid water are likely to result in very different behavior microphysically.

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The distribution of retrieved (a) a_m and (b) b_m with respect to forward-calculated radar reflectivities (dBZ; x axis) and ambient temperatures (°C; y axis). Each bin spans temperature of 5°C and radar reflectivity of 5 dBZ. For clarity, bins that contain <10 samplings are omitted. The color bars indicate the mean value of a_m for each bin in logarithm base 10 and the mean value of b_m of each bin, respectively.

Citation: Journal of Applied Meteorology and Climatology 56, 3; 10.1175/JAMC-D-16-0222.1

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As for the temperature dependence, M₃₀₀ is likely to increase with increasing temperature in general (Fig. 20). It is evident that the distribution of M₃₀₀ weights toward the higher end when the temperature exceeds −50°C. However, we note the dependence of M₃₀₀ on temperature could be negligible over a smaller range of temperature. The distribution of M₃₀₀ with the temperature regime between −70° and −60°C almost overlaps that of temperatures between −60° and −50°C.

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Frequency distributions of M₃₀₀ within four temperature regimes. Black lines indicate that the temperature is between −70° and −60°C. Blue lines indicate that the temperature is between −60° and −50°C. Green lines indicate that the temperature is between −50° and −40°C. Red lines indicate that the temperature is between −40° and −30°C.

Citation: Journal of Applied Meteorology and Climatology 56, 3; 10.1175/JAMC-D-16-0222.1

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6. Summary

We apply optimal estimation methodology to infer ice particle mass–dimensional relationships from ice particle size distributions and bulk water contents independently measured during MACPEX. We also utilize ice particle projected area recorded by the 2D-S imaging probe, offering a further constraint to this ill-posed problem. M–D relationships are retrieved using samples collected during 10 flights, with a total of 2690 10-s-averaged data points. Several case studies are presented and consist of two jet stream cirrus cases: one frontal cirrus and one anvil cirrus case. Substantial variability of M–D properties is found to be present among cases. The values of M–D parameters tend to increase with temperature in a general sense, but such dependence could be almost negligible within a smaller range of temperature. It is suggested that M–D properties not only depend on large-scale dynamics and environmental conditions in a convolved way, but also are subject to the variations of temperature and ice supersaturation associated with mesoscale or even turbulent-scale dynamics. Determining M–D relationships simply based on temperature or general cloud type may cause significant uncertainties, not to mention applying one M–D relationship to all conditions.

In addition to the values of retrieved mass–dimensional parameters, the associated uncertainties are conveniently acquired in the OE framework, within the limitations of assumed Gaussian statistics. We find, given the constraints provided by the bulk water measurement and ice particle areas, that the relative uncertainty of mass–dimensional power-law prefactor (a_m) is approximately 70%, and the relative uncertainty of exponent (b_m) is 6%–9.5%. It shows taking into account the self-consistent A–D relationships significantly contributes to constraining the uncertainties of M–D parameters. With this level of uncertainty, the forward-model uncertainty in radar reflectivity would be on the order of 2.5 dB, or a factor of approximately 2 in ice water content. The implications of this finding are that inferences of bulk water from either in situ measurements of particle spectra cannot be more certain than this when the mass–dimensional relationships are not known a priori, which is almost never the case.

Since MACPEX targeted midlatitude synoptic and convective cirrus, findings in this study may have potential limitations. We plan to further explore this topic by applying the retrieval algorithm to broader datasets that contain a variety of cloud types in future work.