1. Introduction
It is a common practice to fit observed hydrometeor particle size distributions (PSDs) into a known distribution form. A typical choice is the gamma distribution (Ulbrich 1983; Willis 1984). One may even hope that by fitting an observed PSD into a known function form, the PSD may be extrapolated beyond the observed range, if the chosen function form is physically based in a robust manner. If such a procedure is possible, the various characteristics for PSDs such as the total particle number and the mean particle mass would also be better estimated as a result. Such a possibility has led to consideration for a new methodology for determining the hydrometeor PSDs, inspired from the maximum entropy principle.
The maximum entropy principle defines the statistically most likely distribution of a variable under a given set of physical integral constraints (Jaynes 2003). This procedure consists of the two major steps. First, a general form of the distribution is derived under a given set of constraints. Second, free parameters of the distribution are determined from the given physical values for these constraints.
A basic starting point of the present study is to note that the above second step can be taken out from the whole procedure and be used in a standalone manner. Here, in general, the system may not follow the most likely distribution as defined from the maximum entropy principle. However, whatever way it may be defined, if an assumed general form of the distribution is physically based, to that extent, a finite set of physical integral constraints would accurately determine the distribution even without referring to an observed distribution directly. The present paper applies this idea to the ice hydrometeor PSDs by taking these integral physical constraints to be the total particle concentration NT, the mean particle mass 
If such a procedure is possible in general, there would no longer be a need for measuring the PSD except for the total particle concentration. Instead, we just need to focus on the less difficult measurements of the bulk quantities of the hydrometeors such as the total mass per volume. The present paper explores this possibility.
The paper is organized as follows. The hydrometeor data to be analyzed in the study is described in the next section. Some basic characteristics of the hydrometeor measurements are also discussed therein. Some of the measurement uncertainties described therein provide the further motivations for a new PSD fitting method. The traditional methodology for determining the PSD is reviewed in section 3, and a new methodology is introduced in section 4. PSDs are analyzed in section 5 based on both traditional and new methods, and the results are compared. The paper concludes with a summary and further discussion in section 6.
2. Data
a. Study case
Data from the Tropical Composition, Cloud and Climate Coupling Experiment (TC4; Toon et al. 2010) are taken for the study. TC4 was conducted over Costa Rica and the Caribbean during July and August 2007 with a focus on convective clouds. A flight leg on 8 August for ice particle measurements is used for the present study. This particular leg was chosen since it was the longest continuous in-cloud leg during the project with cold temperatures in a relatively dense anvil cloud. The analysis is limited to the period of 1230–1530 UTC in order to ensure that only the full ice phase is considered in the temperature range from −35° to −50°C. The data are averaged over 5-s intervals corresponding approximately to a 1-km flight segment (Fig. 1).
Time series of (a) the observed total particle number density NT and (b) the IWC during the flight lag adopted for the present study. Note that the analysis period spans 1230–1530 UTC.
Citation: Journal of the Atmospheric Sciences 75, 3; 10.1175/JAS-D-17-0145.1
b. Instruments and measurements
For measuring properties of the ice particles, the cloud imaging probe (CIP) is used for the range of 25 μm–1 mm, and the precipitation imaging probe (PIP) for the range of 1–30 mm. These particle probes measure the size of each observed particle. Additionally, the counterflow virtual impactor (CVI) measures the total hydrometeor mass per volume [ice water content (IWC)] by vaporizing the ice particles within the zero-humidity air (Ogren et al. 1985; Noone et al. 1988; Twohy et al. 1997).
Arguably, a major function of the particle probes is to count the total number of hydrometeor particles. However, these instruments do not “count” the particles in literal sense. First of all, these two particle probes do not have the same sample volume: the CIP has a quarter of the sample volume of the PIP. Furthermore, the effective sample volume depends on the particle size owing to the optics of the instrument and the way that partially imaged particles are handled, especially those that touch an edge of the image area. Thus, each particle count contributes to the total particle number with a weight inversely proportional to its effective sample volume. In this manner, the uncertainty of the particle size estimate also affects the estimate of the particle concentration. For further basic technical issues of the particle probes, see Heymsfield and Parrish (1978).
The particle size range recorded by the probes is typically from 25 μm to several centimeters in size. However, the probes do not measure all particle sizes with equal accuracy. Measurement uncertainties are higher for smaller particles, and measurements below 100 μm in size are particularly unreliable owing to the effects of shattering (Korolev et al. 2013) and optical limitations. Reported concentrations of smaller particles are often unrealistically large for these reasons. Larger particles are often only partially seen in the image area, and those above a certain size need to be excluded.
With these considerations, all the measurements below 100 μm are excluded from the present analysis. The remaining measurements from both probes are tabulated by dividing these ranges into 34 bins. The first three bins have the size ranges of [100 μm, 150 μm], [150 μm, 200 μm], [200 μm, 250 μm], and the last bin considered is [16 mm, 18 mm]. The bin sizes increases roughly logarithmically with the increasing size. Furthermore, in the analysis of section 6, the three particle size ranges are considered: (i) the full range, 100 μm–18 mm; (ii) the medium range, 150 μm–10 mm, which excludes least-reliable measurements; and (iii) the narrow range (550 μm–5 mm), in which only the most reliable measurements are retained. In the small-size limit, depth of field uncertainty is reduced beyond 150 μm and shattering uncertainties are reduced beyond 550 μm, since most shattered particles are smaller than this. In the large-size limit, a 5-mm limit stays within the array width of the PIP (6.4 mm), and 10 mm includes particles where a reasonable size may be estimated from a partial image.
The basic principle of the CVI is to sublimate or vaporize all the condensed water within a zero-relative-humidity environment, and thus the total condensed water content (IWC for the present study) is obtained. However, not all the hydrometeors are captured into the inlet. First, the inlet diameter is 5.8 mm; thus, any particles larger than this size would not fully enter the inlet, and large particles of comparable sizes are not sampled effectively, either. We, rather arbitrarily, assume the maximum size that enters the inlet to be 2 mm in the following analysis. Particles must also have enough inertia to penetrate the counterflow at the probe inlet, preventing very small particles from entering the instrument (Ogren et al. 1985). We assume the minimum size to be 7 μm. By comparison with the particle imaging probes, the IWC values less than 0.01 g m−3 were removed from the analysis so that the errors from residual vapor/condensate in the CVI plumbing after exiting cloud are much reduced. Furthermore, CVI measurements saturate above the upper limit of the instrument’s capability to fully evaporate the liquid water, at about 2 g m−3 (Heymsfield et al. 2009). By limiting the data in this manner, the total of 1427 samples is considered in the present study.
c. Basic characteristics of the measurements
A major issue with the particle probes is that the particle count is not very reliable right at the edge of the resolution of the probe (25 μm in the present case). Uncertainties are also introduced into mass estimation owing to ambiguities in estimating the particle size at the resolution limit and with out-of-focus ice particle images. The cutoff at larger sizes makes the data more reliable. However, a deeper cutoff leads to a further “undercounting” of both particle concentration and the total mass (IWC). Here, note that the IWC is defined by an integral over a finite size range [Eq. (4.6) below]. Examining this effect is important for understanding the uncertainties in estimating PSDs.
Figure 2a plots the frequency distribution of the fractional particle number counted when the cutoff sizes are increased from 100 to 150 (solid) and to 200 μm (long dashed). It is most likely that the total particle count decreases by about 30% by increasing the cutoff size to 150 μm. A substantially long tail in the distribution suggests that undercounting could be substantially worse. A further increase to 200 μm clearly loses more counts and the most likely fractional number retained is about 50% with a fairly wide distribution.
Various characterizations of the observations during the flight leg adopted for the present study. (a) The frequency distribution of the fractional particle concentration relative to the full data with the cutoff sizes 150 (solid) and 200 μm (long dashed). (b) The frequency distribution of the IWC ratio: IWCp/IWCm. Here, IWCp is IWC based on the particle probe measurements, and IWCm is IWC based on CVI. The three curves are for IWCp with the cutoff size: 100 (solid), 150 (long dashed), and 200 μm (short dashed). (c) As in (b), but for the number of cases against the total number of the counted bins. (d) As in (c), but for the mean total particle concentration NT. (e) As in (c), but for the mean IWC. (f) Number of cases as a function of the maximum size measured.
Citation: Journal of the Atmospheric Sciences 75, 3; 10.1175/JAS-D-17-0145.1
Figure 2b shows the ratio of the IWC estimated by the particle probes to that measured by CVI [see also Fig. 3 of Heymsfield et al. (2013)]. Recall that the full particle probe data are tuned to provide an average IWC identical to that from CVI. This fact is confirmed by the maximum frequency corresponding to the ratio of unity when the full data are considered (solid curve). A long tail toward the small ratio, IWCp/IWCm, may also be worthwhile to note. This long tail may partially be explained by the tendency of the CVI measurements to smooth out the time series due to the hysteresis, as well as the saturation of the instrument. Here, coincidentally, the three IWC ratio curves corresponding to the three cutoff sizes match the other relatively well, though the curves shift leftward with increasing cutoff sizes (as they should).
To perform a reliable PSD estimate, the available number of particle bins must be large enough. In Fig. 2c, the solid curve shows a number distribution of the counted number of particle bins during the flight leg. The most frequently found bin number is 10, with the range of 8–18 fairly frequent. A few flight segments contain only one particle bin measurement, whereas the maximum number of bins counted is 30. No flight segment counted all the possible particle bins. When the size range is slightly limited to [150 μm, 10mm] (long dashed), the distribution simply shifts to the left by about the two bins. However, when the size range is further limited to [0.55mm, 5mm] (short dashed), the frequency simply decreases with the increasing total bin numbers (with an exception of a minor peak at 17 bins). See Hallett (2003) for discussions on the errors associated with a particle number counting per bin.
To see a possible reason for missing measurements at given bins, the mean for total counted particle number NT (cm−3) is plotted as a function of the total measured bin numbers for the full (solid), the medium (150 μm–10 mm: long dashed), and the narrow ranges (0.55 μm–5 mm: short dashed) in Fig. 2d. Regardless of the cutoff sizes, the total particle concentration presents a clear trend of increasing with increasing available bin numbers. This leads to the natural interpretation that a low number of counted bins typically occurs when the total particle concentration is also low. Under such a situation, a particle is counted at a certain bin by chance, and not in the others by chance, because the turbulent flow within the cloud stirs the particles intermittently in a random-looking manner (Ottino 1989). As a result, interpolating the counted bin values may even not be a good idea. Note that the number of missing bins can be reduced by taking a longer averaging time for sampling, but at the expense of reduced spatial resolution. Figure 2e shows the same, but for the IWC.
Similar statistics can be taken as functions of the maximum particle size Dmax measured. Figure 2f shows that most of the flight segments correspond to the maximum size of the range from 0.2–0.3 to 2–3 mm. Recall that the maximum ice particle available for a measurement is 18 mm. We further find that the available fraction roughly monotonically decreases with the increasing maximum measured size. When the maximum size is larger than 3 mm, the missing fraction may be larger than 10%.
3. Traditional PSD estimate method
See Smith and Kliche (2005), Kliche et al. (2008), Mallet and Barthes (2009), Smith et al. (2009), and Handwerker and Straub (2011) for discussions on the sampling errors associated with the standard estimate method.
4. New PSD estimate method
To some extent, a set of the constraints (4.1) may be considered a generalization of the moment constraints given by Eq. (3.2) in the last section. However, this generalization is associated with two major implications. First, the constraint itself can be generalized in any possible manner. The exponent in the moment may not be necessarily constrained to integers, and a fractional power may be considered. The total mass constraint introduced below is such a particular example. Though we do not consider any further generalizations in the present paper, we should also emphasize that the function Fl for the defining a constraint can take any form, even a one not at all related to a moment in a remote sense, as further discussed in section 6d.
The second implication stems from the maximum entropy principle (Jaynes 2003): it is desirable to constrain a distribution in a physically meaningful manner, rather than just considering it as a simple curve fitting problem. In other words, the choice of the constraint function Fl should better be based on physical considerations, but mathematically in a more general manner, rather than a purely statistical consideration that may lead to a choice of a type of moments. A specific choice of the constraints below is also based on this consideration.
From a point of view of observational measurements, we may further insist that a distribution is best constrained by an available set of measurements. Use of the CVI mass measurement is a particular example considered here. It further leads to a perspective that in order to estimate a PSD, no information on the PSD itself may be necessary, but only the observational information on the bulk integrated constraints Cl (l = 1, …, N) is required, as long as the assumed PSD Eq. (4.2) is well physically based so that a certain universality can be expected.
To perform the integral (4.5b), the particle mass m(D) must also be specified as a function of the size D. In practice, the mass–size relationship differs from one particle habit to another, and estimating the optimized relationship for a given dataset is an important subject in its own right. In the present study, however, we simply follow the Eq. (2.1) introduced by Heymsfield et al. (2002), as discussed in section 2, but with different coefficient values introduced earlier. In the next section, we will consider two cases: the first fully considers the two constraints (4.5a) and (4.5b), and the second only the single constraint (4.5b). The assumed PDF forms for each case are discussed in the reminder of this section.
a. Under two constraints
b. Under a single constraint
An ultimate goal of this study is to seek the possibility of estimating a PSD based only on bulk characteristics of the hydrometeors, which are measured by other instruments independent of the particle probes. The IWC measured by CVI is an example. The mean size 
5. Analysis
In presenting the results in this section, we show the number density 
Note that the error measure (5.1) is designed to directly quantify the degree of the fit under the double-logarithmic plot (as shown by Figs. 6, 7, 11, and 12 below). As pointed out by McFarquhar and Heymsfield (1997), different error measures provide qualitatively different results. Recall that we minimize Eq.(4.4) for obtaining the PSD 
a. Under the two constraints
Here, the PSD forms are optimized under the two constraints: the mean size 
The mean success rate of the minimization of Eq. (4.4) as a function of the total bins assuming the PSD forms (a) 
Citation: Journal of the Atmospheric Sciences 75, 3; 10.1175/JAS-D-17-0145.1
Figure 3d summarizes the performance of the fit to these three PSD forms by choosing the case with the full range with the IWC based on the particle probes. The success rate for the traditional approach is also included with a green curve: it is almost perfect for the total bin numbers above 5. The fits to the PSD forms [Eqs. (4.7a) and (4.7b)] defined under the maximum entropy principle succeed more than 95% of times for the range of the total bin numbers 2–22. The fit to the gamma distribution only has a success rate about 70%.
The fitting error (5.1) under the traditional method is shown by the green curve in all the frames of Fig. 4 as a reference: the mean error is always less than 0.5 for all the total bin numbers under the traditional method. There is a great deal of contrast with the errors associated with estimates based on the proposed new method as shown in Figs. 4a–c by the other curves. In all cases, the error increases rather dramatically above about 16 total bins, with the increasing bin numbers. We will further discuss the implications in section 6c.
As in Fig. 3, but for the fitting errors. Furthermore, the errors under the traditional method are added as a green curve in all the frames.
Citation: Journal of the Atmospheric Sciences 75, 3; 10.1175/JAS-D-17-0145.1
As for the success rates (Fig. 3), the errors are clustered by the choice of the mass measurement methods: IWC based on the particle probe systematically provides (red curves) better fit than those based on CVI (blue curves). This is rather expected, because CVI-based IWC is not consistent with the PSD data provided by the particle probes. Nevertheless, the relative success with the CVI-based fit may be considered encouraging. We also note that the change of the fitting range does not represent any consistent tendency for deteriorating the fitting error, although the fitting error itself is defined for the whole size range.
A summary of these results is given in Fig. 4d: the traditional fitting method (green curve) always performs well independent of the total bin number. Below total bin number 12, the proposed new method also works equally well, but above the total bin numbers 12, 16, and 20, respectively, the error suddenly increases for the forms (4.7b), (4.7a), and (3.1). The gamma distribution also fits the best to data even under the new method.
Figure 5 is as in Fig. 4, but for functions of the maximum size. Though the correspondence is not one to one, we see qualitatively the same results, because the mean maximum size generally increases with increasing available bin numbers. In particular, the error suddenly begins to increase above the 2-mm maximum size with the new fitting method.
As in Fig. 4, but as functions of the maximum size.
Citation: Journal of the Atmospheric Sciences 75, 3; 10.1175/JAS-D-17-0145.1
Finally, two examples of the actual curve fit results when the full range is used are shown with the particle probes and CVI for IWC, respectively, in Figs. 6 and 7. Here, we have chosen particular cases that the four curve fits diverge well enough so that we can see differences. In many cases, those four curves fit together rather well, as the statistics in Figs. 4 and 5 above suggest.
Two examples of fit of PSD measurements (solid) into prescribed forms: 
Citation: Journal of the Atmospheric Sciences 75, 3; 10.1175/JAS-D-17-0145.1
As in Fig. 6, but using the CVI-based IWC.
Citation: Journal of the Atmospheric Sciences 75, 3; 10.1175/JAS-D-17-0145.1
The first example (Figs. 6a and 7a) is with a relatively narrow size range with a maximum size of 2 mm. This is a case where the traditional gamma fit (double-chain dashed) substantially underestimates (overestimates) the PSD in the small (large) size limit compared to the observation (solid). In this case, the new method provides a much better fit overall, with the exception of the gamma distribution fit with the IWC constrained by CVI (long dashed in Fig. 7a). The second (Figs. 6b and 7b) is a typical example when a large number of bin measurements is available: the gamma distribution can fit to the data (solid curve) well based on traditional (double chain) method. The fit based on the new method works less well for all the three distributions forms considered. The performance with the all three forms is also fairly comparable when the IWC is constrained by CVI (Fig. 7b). On the other hand, the gamma distribution fit performs clearly the best, and the distribution with Eq. (4.7a) the second, when the IWC is constrained by the particle probes (Fig. 6b).
b. Under a single constraint
Here, the distribution forms (4.8a), (4.8b), and (4.8c) are optimized solely based on the constraint of the total mass (IWC) M after a normalization in terms of the total particle concentration NT. Arguably, an estimation of the mean size 
Figures 8–10, respectively, show the corresponding results as for Figs. 3–5. A reduction of a number of parameters clearly facilitates a procedure for identifying the minimum, and as a result, the success rate overall increases: all the cases have almost 100% success rate above the total bin number 8 and Dmax = 0.1 mm. In spite of the decrease of the number of free parameters, the deterioration of the fit to the observations is not dramatic: the deterioration is the least with the gamma distribution (Figs. 9a and 10a), but the deterioration becomes gradually more apparent to the order of the fits to Eqs. (4.8b) and (4.8c).
As in Fig. 3, but for the cases under a single constraint.
Citation: Journal of the Atmospheric Sciences 75, 3; 10.1175/JAS-D-17-0145.1
As in Fig. 4, but for the cases under a single constraint.
Citation: Journal of the Atmospheric Sciences 75, 3; 10.1175/JAS-D-17-0145.1
As in Fig. 9, but as functions of the maximum size.
Citation: Journal of the Atmospheric Sciences 75, 3; 10.1175/JAS-D-17-0145.1
Examples of a fit are shown in Figs. 11 and 12: the same cases as for Figs. 6 and 7 are shown. Comparison of the two pairs suggests that a reduction of fit performance due to a reduction of the number of the parameters is relatively small as suggested above.
Examples of fit of SPD measurements (solid) into prescribed forms: 
Citation: Journal of the Atmospheric Sciences 75, 3; 10.1175/JAS-D-17-0145.1
As in Fig. 11, but using the CVI-based IWC.
Citation: Journal of the Atmospheric Sciences 75, 3; 10.1175/JAS-D-17-0145.1
6. Discussion and conclusions
a. Proposed basic principle
The present study has proposed the determination of the PSD as a physical problem rather than as a mere curve-fitting procedure. The study has been inspired by the maximum entropy principle (Yano et al. 2016), which can define the most likely distribution of a variable when a set of physical integral constraints are known. The constants (parameters) of the derived distribution are determined from an imposed set of integral constraints. The main starting point of the present study has been to realize that this last procedure can be used in its standalone manner. A general distribution form must be prescribed in a certain manner but not necessarily based on the maximum entropy principle. The present study provides a proof of this concept by analyzing the ice particle measurements from the TC4 campaign (Toon et al. 2010).
The conventional wisdom would be that in order to determine the hydrometeor particle size distribution (PSD) observationally, the particle sizes must be directly measured, say, with optical probes. Unfortunately, collecting in situ particle size distribution data is expensive. In this respect, the proposed methodology is potentially powerful, because it suggests estimating the PSD solely based on the integrated physical characterizations of the hydrometeors instead of counting every hydrometeor. As a result, the task of the particle probes reduces to that of carefully counting the total particle concentration.
Integrated constraint can be alternatively provided, for example, by the total mass measured by the counterflow virtual impactor (CVI). To see this possibility explicitly, the CVI data are also used in the present study as an alternative source for the total mass constraint. The overall fit error increases with use of the CVI data, owing to the discrepancy between the CVI and particle probe data for the IWC, but only by a relatively minor degree. Arguably, this method may even self-correct the PSD form by providing a more reliable IWC measurement.
We may further argue that bulk quantities such as mean size and total mass are likely to provide more reliable information than the individual counting of hydrometeors. The new methodology proposes to exploit this bulk information more fully. This general approach also permits us to focus on a narrower range of the particle sizes where the measurements are more reliable. If the prescribed general form is ever “physically based,” a proper fitting procedure would allow us to extrapolate the result.
The bulk information that can be adopted under this methodology is completely general so long as they are expressed in terms of a general integral form (4.1), an integral of the PSD weighted by a certain function.
b. Test study
In this study, the total particle concentration, total mass (IWC), and the mean particle size are adopted as the integrated physical constraints of the problem. The gamma distribution and the two distributions obtained under the maximum entropy principle by taking the size and the particle mass, respectively, as a restriction variable are adopted as distribution functions to fit. The fitting errors [Eq. (5.1)] of the PSD for these prescribed distributions with the proposed method [Eq. (4.4)] are comparable to those based on the measurements, especially when the observed maximum particle size is less than 2 mm. The same method is further applied to the case when distributions are constrained only by the total particle number concentration and the mass (IWC). In spite of a further decrease of free parameters for the distribution form, it is found that the proposed method still works relatively well.
Among the three possible distribution forms adopted for fit, the gamma distribution consistently leads to better results than the other two distributions derived under the maximum entropy principle. Though a physical reason for a superior fit of the gamma distribution is yet unclear, the result suggests an inherent limitation of the maximum entropy principle for estimating the hydrometeor PSDs.
An obvious issue left out of the present study is that of evaluating both the data and the estimation reliabilities as well as associated errors in a more objective manner. In general, when narrowing down the analysis range for the size of hydrometeors, data reliability increases, but the uncertainties associated with the estimation of the PSD form also increases. The optimal analysis range should be ultimately decided based on a compromise between these two opposite tendencies.
c. The tail issue
The test with the new proposed methodology has encountered an inherent difficulty of obtaining a good fit to the tail part of the PSD.
The highest moment that we could identify under this principle was the total particle mass, which only roughly corresponds to the second moment, or the third moment when the particles are spherical. As a result, the proposed fitting method faces an inherent difficulty in accurately fitting the PSD to data in the tail part compared to the traditional method, in which the PSD is constrained by the sixth moment. As a result, the fitting error inevitably increases as a measured size range increases, and rather dramatically when the observed maximum particle size exceeds 2 mm. This difficulty may be overcome if we adopt radar reflectivity as an additional physical constraint, for example. Possibilities of exploiting different types of measurements should be further pursued.
On the other hand, as the most basic premise of the present study, if an assumed PSD form is physically based in robust manner, a given set of integral constraints must be sufficient for estimating the PSD accurately, to the extent that the observed integral constraints are accurate enough. From this perspective, an alternative possibility is that the introduced PSD forms from the maximum entropy principle do not describe the tail part of the observed PSD well, and that is the real reason why we have inherent difficulties for an accurate fit in the tail part. One specific possibility is that the tail part distribution is strongly controlled by transient sedimenting ice particles, which are not properly described in the maximum entropy principle due to its transient tendency. This alternative possibility would worth further investigations.
d. Further perspectives
The idea proposed here may appear to be not fundamentally original in a practical sense: the traditional methodology (e.g., Kozu and Nakamura 1991; Heymsfield et al. 2002; Zhang et al. 2003; Schmitt and Heymsfield 2009) also fits the observed PSD into the gamma distribution solely based on certain integral constraints called moments. However, the original rationale for using moments for fitting the PSD is that the former well quantifies the latter from a statistical point of view.
On the other hand, the present study considers these constraints from the physical basis and also emphasizes an importance of directly adopting the quantities that can be measured, preferably separately from the particle probe measurements. The IWC obtained from the CVI, as considered in the present study, is a particular example. In addition to the in situ cloud measurements, potentially, remote sensing measurements such as those from radar as well as satellite can be adopted as physical constraints. These possibilities should be pursued further.
There are already many attempts for estimating the PSD from the radar measurements (e.g., Seliga and Bringi 1976; Gorgucci et al. 2002; Yoshikawa et al. 2014), but from very different perspectives. The proposed new methodology, on the other hand, establishes a clear link of these attempts to the standard PSD estimate based on the moments, by generalizing the latter. Characteristics obtained from radar measurements, for example, as defined by Eqs. (4)–(6) in Yoshikawa et al. (2014), are considered particular examples of those general physical constraints. Unlike the earlier radar-based PSD retrievals, under this new approach, the only preliminary requirement for the radar data processing is to remove the attenuation effect of the radar signal. Nonattenuated radar parameters can directly constrain the PSD form under the new methodology proposed. This is a particular possibility that is worthwhile to further pursue.
An important ingredient for the success of this approach is that the adopted general PSD form must be physically based to a good extent so that its universality can be anticipated. The present study has not considered the issue of the “physically based” distribution in depth but merely adopted distributions obtained by straight applications of the maximum entropy principle. As it turns out, those distributions somehow never overperform the standard gamma distribution. The reason for this tendency as well as applicability and the limitations of the maximum entropy principle are issues that warrant further investigation. These investigations are especially important for establishing a robust physical theory for the PSD so that the observed PSDs are interpreted under a more solid physical basis and more readily applied for improving the microphysical modeling.
APPENDIX
Numerical Procedure for the Minimization
In this appendix, in presenting the technicalities of the minimization procedure, we designate these parameters as 
The minimization is performed on the plane consisting of 
Thus, the basic sequence of the procedure is as follows:
As the initialization, choose the three points
The minimum points in x direction is sought by an iteration taking
In a similar manner, the minimum points in y direction is sought by setting
More precisely, the iteration procedure in x direction is performed in the following manner: After evaluating the error measure ϵ [Eq. (4.4)] for the given three points, we set the points giving the first two minimums for ϵ as 
The iteration in x direction is repeated until one of the following conditions is satisfied:
- with
- and the number
the number
A similar set of conditions is also introduced in y direction for terminating the whole iteration procedure, but in a slightly different manner:
- with
- and
- with
- with
The last case is considered an abnormal convergence and returned with a flag “failure.” Finally, the maximum number of total iterations (i.e., accumulated number of iterations in x direction) is set to 105, and the iteration is terminated at that point returning a flag failure.
For a certain value of y, no minimum in the x direction may be found. In this case, the testing points in y must be reset. Assuming 
The minimization is also performed in the analogous manner with the case with the distributions (4.8a)–(4.8c). In this case, in order to ensure finding the real minimum, the two points with the smallest values are chosen from the 100 points obtained by dividing the range [1, 103] for λ by equal interval. These two points are used as 
Note that the current version of the minimization code is not optimized.
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