## 1. Introduction

*M*) with respect to particle size (Nakaya and Terada 1935; Mitchell et al. 1990; Brown and Francis 1995; among others):

*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

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* = 0.012*D*^{1.9} (*D* ≥ 66 μm). Heymsfield et al. (2004, hereinafter H04) derived *M* = 0.0111*D*^{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.

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.

*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

*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.

*μ*m. 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,

*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

*μ*m 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

*μ*m 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} = 1.93*b*_{a} − 1.02 from 1000 simulated 3D ice particles (47 habits classified) and the corresponding projections. A similar relationship, *b*_{m} = 1.46*b*_{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.

*μ*m, 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

*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.

*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:

_{CLH}is the ice water content calculated from the CLH measurements and

*b*

_{a}indicates the exponent of area–dimensional relationships.

**y**, the forward-model vector is written as

*D*) defined in Eq. (4):

*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:

*b*

_{m}is approximately 0.132, which is estimated from the regression.

*P*(

**x**|

**y**) is achieved via minimizing the following cost function:

**x**

_{a}and the associated covariance matrix

**x**before observations. Here

**x**

_{a}and

*a*

_{m}= 0.0128 and

*b*

_{m}= 2.2, is used for

**x**

_{a}. The quantities

^{−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

Equation (7) gives us a clue how measurements (**y**) and prior knowledge (**x**_{a} and

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.

*a*

_{m}and retrieved

*b*

_{m}, respectively. The quantity

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 *b*_{m} and *b*_{a} [Eq. (6)] and remains constant at 0.132 in retrievals. The value of *a*_{m} and *b*_{m} in this case are 58.03% and 7.1%, respectively.

(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

(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

(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

The construction of measurement uncertainty, forward-model uncertainty, Jacobian matrix

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., *A* = 0.23*D*^{1.88} (Mitchell 1996) as a reference.

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

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., *A* = 0.23*D*^{1.88} (Mitchell 1996) as a reference.

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

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., *A* = 0.23*D*^{1.88} (Mitchell 1996) as a reference.

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

In addition, a simple sensitivity test is performed to assess the impact of every influential factor in *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.

(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

(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

(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

## 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.

Summary of microphysical properties and dynamical background.

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

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

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

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., *M*_{300}) as well as *a*_{m}. The maximum *M*_{300} 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.

(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

(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

(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

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 dB*Z*) listed in Mace et al. (2006) derived from long-term ground-based data, the mean of forward-model radar reflectivities for MACPEX (−16.46 dB*Z*; 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 dB*Z* 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.

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 08*D*^{2.26} (bullet rosettes; blue) and *M* = 0.007 39*D*^{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

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 08*D*^{2.26} (bullet rosettes; blue) and *M* = 0.007 39*D*^{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

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 08*D*^{2.26} (bullet rosettes; blue) and *M* = 0.007 39*D*^{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

## 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.

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

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

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

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

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

*a*

_{m}on forward-calculated IWC, we estimate the relative uncertainty in IWC using simple uncertainty propagation as follows:

*a*

_{m}is the Jacobian matrix quantifying the sensitivity of forward-model IWC to

*a*

_{m}and

*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),

*Z*uncertainty is equivalent to a measurement uncertainty in the OE framework.

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

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

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

## 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}, *M*_{300} is presented as well (Fig. 11 and Table 2). Larger *M*_{300} 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 *M*_{300} 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}.

The distribution of retrieved *a*_{m} with respect to forward-calculated radar reflectivities (dB*Z*; *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 dB*Z*. 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

The distribution of retrieved *a*_{m} with respect to forward-calculated radar reflectivities (dB*Z*; *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 dB*Z*. 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

The distribution of retrieved *a*_{m} with respect to forward-calculated radar reflectivities (dB*Z*; *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 dB*Z*. 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

Frequency distributions of *M*_{300} for cases (a) 0416, (b) 0413, (c) 0425, and (d) 0426 (*μ*g). The mean and standard deviation of *M*_{300} are listed in Table 2.

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

Frequency distributions of *M*_{300} for cases (a) 0416, (b) 0413, (c) 0425, and (d) 0426 (*μ*g). The mean and standard deviation of *M*_{300} are listed in Table 2.

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

Frequency distributions of *M*_{300} for cases (a) 0416, (b) 0413, (c) 0425, and (d) 0426 (*μ*g). The mean and standard deviation of *M*_{300} are listed in Table 2.

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

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).

(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*_{m}s 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

(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*_{m}s 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

(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*_{m}s 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

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

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

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

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

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

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

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

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

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

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 *μ*m 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 *μ*m. 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.

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

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

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

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

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

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

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

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.

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

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

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

Similar to the distribution of *a*_{m} with radar reflectivity and temperature, *M*_{300} shows stronger dependence on reflectivity and temperature in cases 0416 and 0426. Meanwhile, we notice such a dependence is more identifiable in *M*_{300} 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 *M*_{300} 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 dB*Z* and ambient temperatures for the whole dataset. W2015 found that reflectivity increased with increasing *a*_{m} but decreased with increasing b*a*_{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 dB*Z*. 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.

The distribution of retrieved (a) *a*_{m} and (b) *b*_{m} with respect to forward-calculated radar reflectivities (dB*Z*; *x* axis) and ambient temperatures (°C; *y* axis). Each bin spans temperature of 5°C and radar reflectivity of 5 dB*Z*. 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

The distribution of retrieved (a) *a*_{m} and (b) *b*_{m} with respect to forward-calculated radar reflectivities (dB*Z*; *x* axis) and ambient temperatures (°C; *y* axis). Each bin spans temperature of 5°C and radar reflectivity of 5 dB*Z*. 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

The distribution of retrieved (a) *a*_{m} and (b) *b*_{m} with respect to forward-calculated radar reflectivities (dB*Z*; *x* axis) and ambient temperatures (°C; *y* axis). Each bin spans temperature of 5°C and radar reflectivity of 5 dB*Z*. 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

As for the temperature dependence, *M*_{300} is likely to increase with increasing temperature in general (Fig. 20). It is evident that the distribution of *M*_{300} weights toward the higher end when the temperature exceeds −50°C. However, we note the dependence of *M*_{300} on temperature could be negligible over a smaller range of temperature. The distribution of *M*_{300} with the temperature regime between −70° and −60°C almost overlaps that of temperatures between −60° and −50°C.

Frequency distributions of *M*_{300} 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

Frequency distributions of *M*_{300} 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

Frequency distributions of *M*_{300} 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

## 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.

## Acknowledgments

Funding for this research was provided by NASA Grants NNX10AM42G, NNX15AK17G, and NNX13A169G.

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