a. Motivation

The representation of ice particles, especially the parameterization of their microphysical properties, is an important part in atmospheric models and spaceborne sensors retrieval algorithms. It has long been acknowledged that ice particle mass, size, and related quantities such as density are key modules in ice microphysical schemes (Schmitt and Heymsfield 2009; Erfani and Mitchell 2016; Cotton et al. 2013), hence the acute need for improved representation of ice particles. Furthermore, interest in particle microphysical properties has continued to grow since high-altitude ice crystals were identified as a threat to commercial aviation, causing aircraft engine and air data probe icing. This led aircraft industry leaders, aviation regulatory authorities, and research facilities to join forces to launch the High Altitude Ice Crystal–High Ice Water Content (HAIC–HIWC) projects with the overall objective to enhance flight safety when flying in mixed and glaciated icing conditions. Technical developments of detection and awareness technologies and modeling of ice accretion mechanisms within engine modules and air data probes require better characterization of ice particle mass, size, and concentration. These are the main rationale for the observation of ice particle physical properties and for the development of an empirical model that accurately describes particle mass as a function of size, temperature, and any other relevant physical parameters.

b. State of the art

To address the abovementioned research topics, many studies have been conducted in the past with the primary goal to characterize cloud microphysical properties and to derive expressions for ice particle mass. In the following subsections, some state-of-the-art studies providing mass–size relationships from particle measurements are reviewed, with emphasis placed on the hypotheses and analytical tools they use.

c. Mass–size relationships: Limitation and opportunities for improvement

In conclusion of this partial literature review, there is a consensus in using power laws to describe the dependence of ice particle mass on size with a few exceptions, such as the Erfani and Mitchell (2016) scheme and the Jackson et al. (2012) habit-dependent approach. In practice, the power-law assumption 1) reduces the search of to the computation of two scalars, a and b; 2) produces curves that are globally matching theoretical expectations (in terms of magnitude and tendency for ice particle density to decrease with size); and 3) enables the calculation of particle ensemble properties (IWC or for instance) that agree reasonably well with observations. However, most of the published relationships based on this power-law assumption are subject to a fundamental weakness, as acknowledged by recent authors (Schmitt and Heymsfield 2010; Erfani and Mitchell 2016): they might not accurately reflect the contribution of the different growth processes to the ice crystals’ masses. Especially, they cannot capture potential irregular variations of the mass with size for crystals ranging from a few microns to a few centimeters in size. Because of the preponderance of different ice growth mechanisms within distinct size intervals, the smaller size classes up to a few hundred micrometers are generally populated by small pristine ice crystals formed by vapor deposition, whereas crystal aggregation dominates in the largest particles. It is well known that the properties of dense pristine ice and fluffy aggregates may be significantly different (Pruppacher and Klett 2010). The effect of riming could also change the mass properties in some intermediate bins so that the resulting curve may significantly deviate from a regular power-law relationship. The results reported in Jackson et al. (2012) support these considerations, which suggest that accounting for the size dependence of particles’ properties, such as crystal habit distribution, could improve the mass retrieval. The situation is even worse when only one pair is derived from an entire dataset. In this case the parameterization fails to capture the spatiotemporal variability in the microphysical properties of ice particles within clouds.

Since earlier methods based on individual crystal mass measurements are impractical for use in large datasets of mixed-habit ice particles, there is still a strong need for the development of alternative concepts for retrieval methods.

This study presents a nonstandard approach that leverages the optimal use of common in situ measurements—namely, PSD and IWC—to waive the power-law constraint and therefore better capture the dependence of particle microphysical properties on size. Section 2 introduces the assumptions made to set up a simple forward model that relates particle-size-dependent mass to PSD and bulk IWC values, as well as the mathematical formalism and tools selected to retrieve the particles’ mass by solving the inverse problem. The method is then applied to simulated crystal populations (section 3) and to real cloud in situ measurements extracted from the HAIC–HIWC dataset (section 4). In this article, the emphasis is placed on the problem statement and technical aspects of this new mass retrieval process rather than on the implications of the preliminary results presented in section 4 for modeling and remote sensing retrieval algorithms.

a. Forward model

2) Physical considerations and approximations

The IWC value is by definition the sum of the individual crystal masses within a sampled cloud volume. The mass M of an ice particle is mostly determined by the air and ice fractions and the respective densities of these two components. Actually, it is known from theoretical considerations that the particle mass is not evenly distributed over the particle volume as a result of complex interactions of several growth processes, such as water vapor diffusion, aggregation, and riming (Pruppacher and Klett 2010). An accurate computation of the mass of ice particles would require knowledge about how these different mechanisms contributed to the particle’s growth throughout its life cycle. In this Lagrangian-like approach, in situ measurements would follow the particle transportation from the nucleation site across different cloud parcels. However, the complex transport of ice particles within a cloud—especially within deep convection, where turbulent mixing processes occur over several kilometers in height—makes such data almost impossible to collect. Therefore, the proposed model uses the Eulerian approach, which is prevalent in literature, whereby the ice particles are characterized at a given location and time by collocated and coincident in situ measurements of the cloud environment. It is postulated that a series of measured parameters can be identified and used to determine the mass of ice particles. Among others, one can think of environmental parameters such as static air temperature (ambient temperature) and cloud type (the dependence of particle properties on cloud type is reported in several studies quoted in section 1), and morphological parameters derived from the analysis of OAP images of individual particles (e.g., dominance of some crystal habits, presence of riming) as potentially characteristic parameters. In the following, these parameters are denoted as . In summary, the mass of a particle depends on its volume and on some other characteristic parameters, as formally written in Eq. (1):

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Two simplifying assumptions are made to establish the forward model.

(i) Characteristic parameters

Time invariance of the parameters is assumed for each 5-s average data point: all particles sampled during a 5-s time interval are characterized by the same parameters (e.g., the time-averaged value computed from the 1-Hz original measurements). Subsequently, datasets can be sorted into subsets of consistent data points using these parameters as sorting criteria. In such a subset where the characteristic parameters are invariant, the mass depends only on volume. Equation (1) becomes

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(ii) Volume

In the ith bin of the PSD, the variability in the particle’s size, , depends on the bin width . As is generally small, the variability in is small. It is further postulated that the spread in volume is also reasonably low within a bin (discussed in more detail in section 3b). This assumption implies that PSD bins are natural subensembles, where the crystals’ mass is uniform, provided data points have been sorted as per assumption i. As a consequence, a statistical “reference mass” can be defined for each bin: . We note as the reference mass of the ith bin in the PSD, such that and are the reference masses of the first bin (the smallest particles with ) and the last bin (largest particles with , where N is the number of bins in the PSD), respectively. There are as many reference masses as bins in the PSD, and in the following we note the vector of reference masses.

With the two abovementioned assumptions, we now develop the mathematical equations that relate bin reference masses to PSD and IWC measurements. For a data point measured at , the ice water content is calculated by summation over the PSD bins of the number of crystals in each bin [] times the bins’ reference mass (m_i):

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Equation (3) can be written as many times as there are data points in a subset satisfying assumption i, say P points in this explanation. This results in a system of linear equations with N unknowns and P equations:

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For the sets of synthetic data (section 3) and experimental data (section 4) presented in the next sections, and 240 and and 1051, respectively. This system of linear equations can be written as a matrix equation:

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where

• is a matrix in which are vertically concatenated,

• is a vector of P IWC values, and

• is the vector of N unknown reference masses.

Thus, the forward model is formulated as a matrix equation that relates in situ measurements to the unknown bin reference masses. It is worth mentioning that the model does not require prior knowledge of the descriptors . Sensitivity studies in which reference masses are retrieved from two distinctive subsets will reveal whether the parameter that differentiates between the two subsets influences the mass of ice particles (see section 4 for an example with sampling temperature as a descriptor).

b. Solving the inverse problem

1) Solving an ill-conditioned inverse problem

The mass retrieval process now consists of solving the inverse problem. Estimates of the problem condition number—defined as , where and are the largest and smallest singular values of the matrix, respectively—indicate that measurement uncertainties and noise inherent to the measured data significantly influence the solution: the problem written with synthetic data (case presented in section 3) is severely ill-conditioned () and the problem using HAIC–HIWC real data (case presented in section 4) is ill-posed (). To circumvent the difficulty related to numerical instabilities, we follow the regularization approach presented in Idier (2008) and search for a solution that minimizes a composite criterion, noted as :

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where is twofold:

• The first term is referred to as the least squares term. Minimizing that term forces the solution vector to fit the measurements and to satisfy Eq. (5).

• The second term, referred to as regularization term, allows the introduction of some prior information that the solution should fulfill in order to qualify from a physical point of view and helps in reducing the noise propagation on the solution.

The retrieved solution will be a trade-off between fidelity to the measured data and compliance with prior information embedded in the regularization term. The compromise between the two sources of information is adjusted via the value of the regularization parameter λ. Since the retrieval process relies on a series of measurements, a measurement error covariance matrix is added to weight the contribution of each data point according to their respective uncertainty, as presented in Borsdorff et al. (2014). The cost function J becomes

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In the synthetic case, data points are generated from populations of synthetic crystals, so there is no measurement error associated with the data: they are all equally uncertain and the covariance matrix is an identity matrix. The real data are impacted by measurement errors that may vary from one data point to the other: is a diagonal matrix (measurement errors are assumed uncorrelated between data points) and its coefficients are specified in section 4.

3) Other algebraic operations

The particle mass is proportional to volume and increases superlinearly with the size parameter. When the size range of interest spans several orders of magnitude, the reference masses are also expected to span several orders of magnitude, which makes the problem poorly scaled. A diagonal scaling may be applied to improve the problem formulation and to decrease the condition number. The optimization problem is solved in terms of the new variable m′, defined as , where is a diagonal matrix as defined below:

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This operation is applied to HAIC–HIWC data, where the size range of interest is . It is not applied to synthetic data, where the size range of interest is limited to .

4) Optimization algorithms

All algorithms used in this paper are sourced from Nocedal and Wright (2006). The problem is considered as an unconstrained optimization problem and a line search strategy is adopted to identify the minimizer of the objective function. Briefly, the solution is found by creating a convergent sequence of iterates such that decreases. At each iteration, the new iterate is calculated as per Eq. (9):

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where is the search direction and is the step length.

At the kth iteration, Newton’s method is used to compute from the gradient and the Hessian matrix of the objective function [Eq. (10)]. Since the computation of the Hessian matrix is not cumbersome for small size problems, this method was chosen for its fast rate of convergence and because it is relatively insensitive to poor scaling.

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The step length is computed either iteratively with a backtracking line search algorithm given in Nocedal and Wright (2006) (applied to the HAIC–HIWC dataset in section 4) or explicitly from Eq. (11) (applied to synthetic data in section 3):

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The new iterate is computed from Eq. (9), and this calculation is repeated until convergence of the sequence is reached, with as the stopping criterion.

The regularization parameter λ controls the degree of regularization applied to the problem. Several techniques are available to identify the optimal λ, that is, the value yielding the best compromise between the two terms of the objective function. The L-curve technique, a simple graphical way of determining from plots of the least squares term against the regularization term, is used here (Hansen 1992). An example of such a curve is given in section 3c.

a. Generation of the synthetic dataset

The generation of a synthetic dataset starts with the simulation of ice particle models using an Interactive Data Language (IDL) program (Fontaine 2014; Fontaine et al. 2014). A model is characterized by its shape and dimensional properties, which are defined in Fig. 1 and summarized in Table 1. Each model is randomly rotated in space and projected to produce a 2D image. This process reproduces the working principles of OAPs and how 2D images are produced from real 3D crystals. The rotation/projection step is repeated 100 times, such that 100 2D images are created from each 3D model. The images are then processed to derive using the same image processing software as for OAP images. The resulting size interval of available for each crystal habit is indicated in the last column of Table 1.

Simulated crystals: habit and size definitions.

Citation: Journal of Atmospheric and Oceanic Technology 34, 11; 10.1175/JTECH-D-17-0013.1

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Simulated crystals: habit and size definitions.

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Simulated crystals: habit and size definitions.

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Table 1.

Synthetic dataset: habits and sizes of simulated crystal models.

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Mass is calculated from the 3D volume by prescribing a constant ice density g cm⁻³ for all particles: it has the effect of fixing all physical parameters in the forward model (assumption i is satisfied), hence the particle mass depends solely on volume. For each habit, a lookup table is created where a single theoretical mass is associated with 100 resulting from the 100 respective projections.

The next step is the creation of a synthetic data point that replicates a 5-s average in situ measurement data point. It consists of forming a population of thousands of simulated ice particles described by one PSD and its corresponding theoretical IWC value, noted . Two input requirements (user’s input) control the generation of a simulated crystal population:

• PSD shape: The size range considered in the synthetic test cases is . Simulated crystals are sized into 93 equally spaced 10-μm-wide bins, and the number of particles per bin is prescribed by a normal probability density function of the bin midpoint multiplied by a prefactor so that there are approximately 10 000 elements in each population (or data point). The normal distribution parameters—namely, its mean μ and standard deviation σ—are changed from one data point to the other when generating the synthetic dataset, as detailed below.

• Habit mixing: A mixing law is defined to describe the fraction of each crystal habit within each bin, as in Table 2, for example.

Table 2.

Mixed-habit population: Mixing laws for different size intervals.

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PSD bins are then populated by randomly picking up crystals based on their from habit tables with respect to the number and habits prescribed by the user’s input. For each bin the theoretical bin reference mass, for the bin, is calculated as the average of the theoretical masses of the binned particles. This value is stored for later comparison with the bin reference mass retrieved by solving the inverse problem. is calculated by summing the actual mass of all the elements in the population.

Finally, a synthetic dataset is composed of P different data points that are generated by randomly varying μ () and σ (). To test the retrieval method, three synthetic datasets are used:

• One optimization dataset, noted as DS1: this is the dataset to which the optimization process is applied. Twice as many PSDs as unknowns are generated () to ensure that the matrix in Eq. (5) has a full rank. In DS1, . The unknown mass vector m of Eq. (5) (whose elements are the sought bin reference masses) is found by minimizing Eq. (6) written with and quantities.

• Two validation datasets, noted DS2 and DS3, are used to test the mass vector retrieved from DS1 on populations dominated by small () and large () particles, respectively. IWC values, noted and , are calculated from the retrieved mass vector m and and matrices, respectively. These calculated values are then compared to the corresponding for each data point.

Furthermore, some statistical reference vectors can be defined, since the masses of all individual particles in the population are known:

• Lower and upper bounds: vectors whose elements are the minimum and maximum theoretical bin reference masses, respectively, found across the dataset

• Average bin reference mass []: vector whose elements are the average of the theoretical bin reference mass () calculated over the dataset

Some quantitative criteria are calculated to estimate the quality of retrieved values:

• Mass percent error (PE) is computed for each bin from the retrieved mass value (, optimization output) and :

  
• Mass mean absolute percentage error:

  

  where N is the number of bins in the PSD, indicates the accuracy of the mass retrieval process

• IWC mean absolute percentage error:

  

  where P is the number of data points in the set, indicates the accuracy of the IWC retrieval process

b. Study of input noise in PSD data

The mass retrieval process uses PSD as input data. In addition to the unavoidable measurement uncertainties, approximations made in the forward model also make the PSD a noisy input for inversion. In the frame of the proposed inverse problem approach, the synthetic dataset is first used as an opportunity to study the noise contained in PSDs based on OAP images.

The concept of bin reference mass relies on the assumption that PSD bins are natural subensembles where the crystals’ mass is uniform, which in turn holds if: 1) PSD bins are narrow and 2) the size descriptor used to generate PSD accurately describes the particles’ volume. These conditions are hardly met in reality. On the one hand, the width of a PSD size class is driven by the OAP specifications: it cannot be smaller than the probe resolution, which is nonnegligible (e.g., 10–100 μm, as detailed later in section 4). On the other hand, any size descriptor derived from 2D projected images will fail to accurately describe particle volume: with exception of spherical particles, whose volume is precisely described by a single size parameter (e.g., its diameter); an accurate volume description of real ice particles with complex geometry (e.g., plates, columns, bullet rosettes, capped columns, or even aggregates of those) would require several size parameters. In addition, the projection of 3D objects onto a 2D plane inevitably distorts the volume’s dimensional properties so that the arbitrary projected 2D image scarcely represents the original 3D volume information. This results in a large variability of particle masses when sized into bins based on or similar size parameters derived from 2D projected images.

Figure 2a illustrates this variability for a population of simulated hexagonal plates (, code = C1g in Table 1). The particles are sized into equally spaced 10-μm-wide PSD bins using and the minimum, mean, and maximum of particle masses in each bin are plotted against bin centers. For that specific habit, the particles’ minimum and maximum mass differ by at least one order of magnitude in the 15–465-μm size interval. Above 465 μm, the maximum mass curve plateau observed up to 935 μm indicates that the largest available simulated particle in this population (; ) can end up in 48 different bins depending upon its spatial orientation before projection.

(a) Mass variability in PSD bins for C1g population; (b) spread in mass in one bin () for all simulated habits.

Citation: Journal of Atmospheric and Oceanic Technology 34, 11; 10.1175/JTECH-D-17-0013.1

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(a) Mass variability in PSD bins for C1g population; (b) spread in mass in one bin () for all simulated habits.

Citation: Journal of Atmospheric and Oceanic Technology 34, 11; 10.1175/JTECH-D-17-0013.1

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(a) Mass variability in PSD bins for C1g population; (b) spread in mass in one bin () for all simulated habits.

Citation: Journal of Atmospheric and Oceanic Technology 34, 11; 10.1175/JTECH-D-17-0013.1

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Similarly, the box plot given in Fig. 2b illustrates the mass spread for different crystal shapes ending up in the 105 ± 5-μm size bin, with habit codes as indicated in Table 1. Each box plot gives the median, 25th, and 75th percentiles, as well as extreme values. One can see that generally the spread increases as the particle shape departs from the sphere. It is worth mentioning that the process of sizing multidimensional 3D particles with a size parameter derived from 2D projected images is not the only source of variability: a nonnegligible mass spread is also observed for spherical particles (code = Sph) as a result of the bin width, that is, sorting particles whose diameters range from 100 to 109 μm into the same 10-μm-wide bin. A preliminary study where the population of simulated hexagonal plates (; code = C1g in Table 1) is sized using different size parameters reveals that using the maximum dimension as a size parameter decreases the spread of mass values within bins, compared to using .

This spread in the particles’ mass within a bin is expected to be even more pronounced in real data: OAP images reveal that several particle habits are mixed within the ice cloud particle populations sampled in airborne field campaigns. These considerations give an insight into the level of noise included in PSD data, which is detrimental to the accuracy of the mass retrieval process. It also underlines the need for regularization when attempting to inverse the forward model presented in section 2a. With this in mind, we proceed with OAP measurements for lack of better inputs.

c. Application and validation of the mass retrieval process to synthetic data

2) Results and discussion

Figure 3 presents optimization results for the mixed-habit population. Figures 3a and 3b show the mass vectors retrieved for the nonregularized () and the optimally regularized () least squares problem, respectively. It highlights the importance of the regularization to extract meaningful information from noisy data.

Mass retrieval from synthetic data: (a),(b) plots for and , respectively; (c) L curve; (d) mass PE between calculated and theoretical bin reference masses; (e)–(g) comparison between retrieved and theoretical IWC for DS1–DS3, respectively.

Citation: Journal of Atmospheric and Oceanic Technology 34, 11; 10.1175/JTECH-D-17-0013.1

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Mass retrieval from synthetic data: (a),(b) plots for and , respectively; (c) L curve; (d) mass PE between calculated and theoretical bin reference masses; (e)–(g) comparison between retrieved and theoretical IWC for DS1–DS3, respectively.

Citation: Journal of Atmospheric and Oceanic Technology 34, 11; 10.1175/JTECH-D-17-0013.1

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Mass retrieval from synthetic data: (a),(b) plots for and , respectively; (c) L curve; (d) mass PE between calculated and theoretical bin reference masses; (e)–(g) comparison between retrieved and theoretical IWC for DS1–DS3, respectively.

Citation: Journal of Atmospheric and Oceanic Technology 34, 11; 10.1175/JTECH-D-17-0013.1

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Qualitatively, one can see that the mass vector retrieved for matches expectations: masses are all positive and increase with size in close relation with the average bin reference masses, computed from actual masses of simulated particles (Fig. 3b). The regularization has an obvious effect of smoothing sharp discontinuities that are present in the synthetic data: the pronounced discontinuity between 795 and 805 μm, where bin mean mass is approximately multiplied by 4 due to the introduction of spherical particles, is spread between 695 and 865 μm. Smaller discontinuities, like the one between 205 and 215 μm, are totally smoothed out. Figure 3c illustrates how is selected: the curve, referred to as the L curve, of the least squares term against the regularization term has a characteristic L shape, hence its name, and the value of λ that yields the best compromise in amplitude between the least squares term and the regularization term can be found at the L-shaped “corner.” Finally Figs. 3d–g give a quantitative assessment of the solution accuracy:

• The mass PE is plotted in Fig. 3d for each bin. If large disagreements are found in boundary bins where particle habits change abruptly ( and 795 μm, with PE in excess of 50%), the resulting mass MAPE of only 18.1% indicates that bin reference masses computed by optimization algorithms are generally in good agreement with theoretical values.

• Figures 3e–g illustrate how the calculated values compare to the theoretical values for DS1–DS3 [scatterplots, where the red curve underneath the data points (blue symbols) is the x = y line]. The mass vector retrieved from DS1 produces excellent estimates of the IWC in the validation sets: as shown in Table 3, IWC MAPE values calculated for DS2 and DS3 are less than 1.6% and 0.6%, respectively.

Table 3.

Optimization results for the synthetic dataset.

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The effective density , defined in this study as the bin reference mass divided by the volume of a sphere of diameter equal to the bin center, is plotted against size in Fig. 4. The retrieved effective density varies between 0.115 and 0.928 g cm⁻³ following a pattern controlled by the crystal habits. The retrieved effective density values (blue cross symbols) generally compare well with the theoretical values

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(black dots in the figure) especially for particle sizes above 250 μm, where the plateau values are well reproduced notwithstanding the smoothing effect of the regularization. The results are not as good for smaller particles: this is due to the low contribution of these particles to the total ice mass resulting from 1) the small individual weight of these particles and 2) the relatively small number of particles in these bins as a result of the choice of normal distribution centered around 300 and 400 μm. This demonstrates the potential of this method to capture size-dependent features much better than any power-law-based method, producing a regular polynomial curve by construction:

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Test case 3: Effective density against size.

Citation: Journal of Atmospheric and Oceanic Technology 34, 11; 10.1175/JTECH-D-17-0013.1

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Test case 3: Effective density against size.

Citation: Journal of Atmospheric and Oceanic Technology 34, 11; 10.1175/JTECH-D-17-0013.1

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Test case 3: Effective density against size.

Citation: Journal of Atmospheric and Oceanic Technology 34, 11; 10.1175/JTECH-D-17-0013.1

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a. HAIC–HIWC dataset

In the frame of HAIC–HIWC projects (Dezitter et al. 2013; Strapp et al. 2016a), two airborne field campaigns were held in Darwin, Australia, in 2014 and out of Cayenne, French Guyana, in 2015. With the overarching goal of characterizing ice particles in mesoscale convective systems (MCSs), these campaigns resulted in nearly 15 000 data points each, sampled at different stages of cloud life cycle in both oceanic and continental convection. This makes the HAIC–HIWC dataset one of the most comprehensive datasets on MCSs from the standpoint of the microphysical properties of hydrometeors.

In this section we use data taken from three flights, as reported in Table 4. The optimization is conducted on the data of Darwin flight 16 (D16): all data points were sampled in the same oceanic MCS at a single temperature level (236.1 K) such that the dataset is consistent with respect to two parameters (cloud temperature and cloud type) potentially influencing the particles’ mass. The other reasons for choosing D16 as the optimization dataset is the high number of data points (1051 data points) and the large size range covered by the sampled particles.

Table 4.

HAIC–HIWC dataset: Overview of the data from selected flights.

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The validation is carried out on data from two arbitrarily chosen flights:

• Darwin flight 19 (D19) was conducted in an oceanic MCS 2 days after D16. Two temperature levels (237 and 226 K) were sampled but only the 212 data points sampled at 237 K (consistent with the temperature range of D16) are used in this study.

• Cayenne flight 26 (C26) was performed one year later in coastal convection. Three temperature levels (264, 244.2, and 229.4 K) were sampled, and all data points (1298 in total) are conserved so that the potential effects of temperature on the retrieved particle mass can be assessed.

The analysis considers only data points measured during level flight periods in areas where IWC values are larger than 0.1 g m⁻³. As documented in Davison et al. (2016), the uncertainty in IWC measurements mostly depends on the temperature and the IWC value: although relative uncertainty remains below a few percent in most of the operating conditions, it can reach 50% for warm temperatures (263.15 K) and small IWC values (0.1 g m⁻³), hence the need to consider measurement errors in the retrieval process. Given the conditions of D16, the coefficients of the matrix are derived from IWC relative uncertainty values , calculated using Eq. (12) (adapted from Fig. 1 in Davison et al. 2016):

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The in situ measured PSD are derived from 2D projected images of the 2D-S probe (Lawson et al. 2006; 10–1280-μm size range; 10-μm resolution) and the PIP (Baumgardner et al. 2011; 100–6400-μm size range; 100-μm resolution). The particles are sorted into size classes according to their , and a composite PSD with bin centers ranging from 15 to 12 800 μm is computed by combining data from the two imaging probes. The composition technique is adapted from Leroy et al. (2016) with two minor deviations: 1) the overlap limits are 800–1250 μm; and 2) the width of the PSD bins is kept as per the OAPs’ original resolution except in the overlap , where the width of PIP bins is numerically decreased from 100 to 10 μm by linear interpolation to ease merging with 2D-S data. As a result, composite PSDs have 240 bins with a bin width of 10 μm from 15 to 1245 μm and a bin width of 100 μm from 1300 to 12 800 μm. For D16, the PSD matrix has a rank of 231 because 9 bins out of 240 are empty throughout the flight. These bins are at the largest end of PSDs (bin centers are 1.17, 1.19, 1.2, and 1.23–1.28 cm) where the concentrations are almost insignificant if any. The uncertainty in concentration measurements is quite hard to establish because of the paucity of data published in literature for these two probes. In this study, estimates provided by Baumgardner et al. (2017) are used with minor deviations:

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The impact of concentration uncertainties on the retrieved masses is evaluated by solving the Eq. (7) for two extreme conditions: once with concentrations systematically underestimated (, with negative values of u_PSD) and once with concentrations overestimated (, with positive values of u_PSD) which gives the upper and lower mass bounds in Fig. 5, respectively.

Mass retrieval from real HAIC–HIWC data: (a) plot for and (b) effective density. (c)–(e) Comparison between and for D16, D19, and C26 datasets, respectively.

Citation: Journal of Atmospheric and Oceanic Technology 34, 11; 10.1175/JTECH-D-17-0013.1

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Mass retrieval from real HAIC–HIWC data: (a) plot for and (b) effective density. (c)–(e) Comparison between and for D16, D19, and C26 datasets, respectively.

Citation: Journal of Atmospheric and Oceanic Technology 34, 11; 10.1175/JTECH-D-17-0013.1

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Mass retrieval from real HAIC–HIWC data: (a) plot for and (b) effective density. (c)–(e) Comparison between and for D16, D19, and C26 datasets, respectively.

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b. Mass retrieval from in situ measurements

The retrieved relationship is plotted in Fig. 5a (blue cross symbols) along with the upper and lower mass bounds (shaded area) computed from the underestimated and overestimated PSDs. The red and yellow dashed lines represent the masses of spherical ice particles () and spherical air particles (), respectively. Qualitatively, the difference between the retrieved curve and the straight line that any power-law based method would have produced is noticeable. Masses are found to globally increase over the size interval although the function is not monotone, especially in the subinterval. The plot of effective density (Fig. 5b), calculated from this mass vector, confirms the tendency of the effective density to decrease with size, as commonly reported in literature (e.g., Pruppacher and Klett 2010), albeit for other density definitions. Qualitatively, the shape of the curve reveals quite irregular variations of the effective density as a function of particle size and two distinctive regimes for small () and large () particles: the effective density of small particles, lying between 0.66 and as the size increases from 15 to , is one order of magnitude larger than the effective density found for particles above , where the values lie below . A preliminary assessment of the impact of the choice of the regularization term (different weighting matrices were tested) on the effective density shows that the effective density calculated for the bins of the PSD ends (for particle sizes below 200 μm and for sizes above 3000 μm to a lesser extent) can be significantly influenced by the definition of the regularization term. In contrast, effective density values are more consistent for intermediate size bins. In the size range, the effective density decreases from 0.3 to approximately, regardless of the definition of the tested weighting functions (the results of the preliminary sensitivity study are not presented here). The unusually low values retrieved for particles larger than 2000 μm, especially the trough between 2700 and 7700 μm, are subject to discussion. In this size range, the larger bins are populated in only a few data points [90% of the bins above 2000 μm are empty in more than 50% of the data points (529 out of 1051) and 1022 data points have more than 90% of the bins empty above 4000 μm], and for these data points the measured concentrations are extremely small. Therefore, the contribution of largest particles to the total ice mass is negligible. As a consequence of the small significance of the total mass of the smallest and largest particles in the least squares term, it could be argued that the masses of these particles are neglected by the minimization algorithms. This limitation could probably be alleviated with better scaling of the data (here, the applied scaling focuses on balancing the reference masses’ contributions with respect to a supposed superlinear increase with size). Alternatively, truncated PSDs could be used as input data. However, in this study, the span of PSDs is purposely kept as defined in other HAIC–HIWC studies [i.e., up to 1.28 cm as in Leroy et al. (2016)] in order to demonstrate the method’s ability to handle a large number of unknowns.

Since the particle’s true masses are unknown, the quality of the mass retrieval is indirectly estimated by comparing the IWC values calculated from this mass vector to the reference IWC values measured by the IKP-2 instrument. A comparison between the calculated and measured IWC values is presented in Figs. 5c–e for the three datasets, and the quantitative results are summarized in Table 5. Unsurprisingly, the best agreement () is found for D16, which supports the minimization process. The IWC prediction is equally satisfying for D19 (), where efforts have been made to sort data points into a coherent subset with respect to sampling temperature. Perhaps most interesting are the results of the IWC retrieval from C26 data. In Fig. 5c, points are color coded with temperature and three distinct groups are clearly visible (dark blue for the 229.4 K level, light blue for 244.2 K, and yellow for 264 K). Data points measured at a sampling temperature close to that of D16 yield the most accurate IWC predictions, as they are well distributed along the line. In the other two temperature subsets, the accuracy of IWC prediction decreases as the sampling temperature departs from 236.1 K. At 244.2 K, compares well with , although the method produces underestimated IWC values. Finally, values are generally underestimated by a factor of 1.5–4 for the points measured at 264 K. The good overall agreement found for C26 () contrasts with the noticeable differences observed between the predicted and measured IWC values for the two warmest temperature levels: it can be explained by the dominance of data points sampled at 229.4 K in the C26 dataset (936 out of 1298; see Table 4). These results suggest that 1) ice particles observed in the D16, D19, and C26 MCSs at comparable temperature levels may have similar mass properties and 2) in the same cloud, particles sampled at different temperature levels may have different mass properties.

Table 5.

Optimization results for HAIC–HIWC partial dataset.

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Since the good agreement between calculated and measured IWC values observed for D19-244.2 K and C26-229.4 K may hide compensating errors, the next step will be to apply the method on subsets sampled at the same temperature level in different clouds and compare the retrieved quantities. Subsequently, the method will be applied to subsets sampled at different temperature levels in order to evaluate whether a first-order approximation model describing the mass of ice particles as a function of their size and the cloud temperature produces consistent results. If not, we will expand the study to a multiparameter approach and search for potential correlations between the crystal mass and some other yet undefined parameters by applying the method to different subsets with specific environmental conditions. This, as well as any specific findings in ice particle microphysical properties and the implications for modeling and remote sensing retrieval algorithms, is well beyond the scope of this technical paper, which is focusing on the development of the technique and the numerical tool.

Last but not least, the convergence of the optimization algorithms is reached after a very small number of iterations. Practically, the mass vector plotted in Fig. 5 is computed within few seconds on an ordinary computer. It underlines the efficiency of the present method in retrieving particle mass from large datasets of collocated and simultaneous PSD and IWC in situ measurements.