Abstract

A general drop size distribution (DSD) normalization method is formulated in terms of generalized power series relating any DSD moment to any number and combination of reference moments. This provides a consistent framework for comparing the variability of normalized DSD moments using different sets of reference moments, with no explicit assumptions about the DSD functional form (e.g., gamma). It also provides a method to derive any unknown moment plus an estimate of its uncertainty from one or more known moments, which is relevant to remote sensing retrievals and bulk microphysics schemes in weather and climate models. The approach is applied to a large dataset of disdrometer-observed and bin microphysics-modeled DSDs. As expected, the spread of normalized moments decreases as the number of reference moments is increased, quantified by the logarithmic standard deviation of the normalized moments, σ. Averaging σ for all combinations of reference moments and normalized moments of integer order 0–10, 42.9%, 81.3%, 93.7%, and 96.9% of spread are accounted for applying one-, two-, three-, and four-moment normalizations, respectively. Thus, DSDs can be well characterized overall using three reference moments, whereas adding a fourth reference moment contributes little independent information. The spread of disdrometer-observed DSD moments from uncertainty associated with drop count statistics generally lies between values of σ using two- and three-moment normalizations. However, this uncertainty has little impact on the derived DSD scaling relationships or σ when considered.

1. Introduction

Drop size distributions (DSDs) of rain are critically important for understanding the role of precipitation in the atmosphere. DSDs strongly influence bulk rates of evaporation, collision–coalescence, breakup, and the removal of rain mass in the atmosphere by sedimentation. One approach to characterizing the general features of rain DSDs is the normalization method, in which normalized DSDs are obtained as a function of a nonnormalized DSD and one or more moments of . Here, x is some characteristic measure of individual drops, such as diameter D or mass m. The basic idea is to collapse the variance of DSDs observed over a range of conditions into a single compact DSD representation. The theoretical underpinning for this idea is that consistent DSD forms should evolve from microphysical processes or combinations of processes that act on a population of drops (e.g., Srivastava 1967; List et al. 1987; Hu and Srivastava 1995; McFarquhar 2004; Prat et al. 2012). A well-known example is the nearly exponential equilibrium rain DSD that results from solving the combined problem of drop collision–coalescence and spontaneous breakup from Srivastava (1971), although more detailed breakup formulations that include collisional breakup give deviations from exponential form and often multipeaked DSDs (Valdez and Young 1985; List et al. 1987; Feingold et al. 1988; Hu and Srivastava 1995; McFarquhar 2004; Prat and Barros 2007a,b; Straub et al. 2010). Earlier normalization methods applied to DSD observations assumed exponential or gamma DSD shape and were normalized using the intercept parameter and median volume diameter or DSD slope λ (Sekhon and Srivastava 1971; Willis 1984). These studies proposed power relationships between the third moment (proportional to bulk mass), , and λ or rain rate, in effect reducing the normalizations to a single moment.

Based on these earlier studies, a generalized procedure for normalizing DSDs was developed and applied by Sempere Torres et al. (1994, 1998) that did not make any a priori assumptions about the DSD functional form. By not imposing any DSD form, this approach improved flexibility and generality with regard to remote sensing and other applications. They proposed a DSD expression that describes any as a function of a reference variable, the ith moment of the DSD , and a normalized distribution function :

 
formula

where

 
formula

and α and β are scaling parameters. Because (1) uses a single reference moment, it represents a one-moment normalization. In addition to providing a method to calculate from , this approach clarified the relationship between DSD moments in a general way. However, it did not substantially reduce variability nor collapse normalized DSDs into a compact ; Sempere Torres et al. (1999, 2000) later showed that different are valid when rain is partitioned into stratiform and convective cases. Their results point to the limitations of one-moment normalization and imply that additional degrees of freedom are needed to characterize DSDs across a wide range of conditions.

Testud et al. (2001) developed a two-moment normalization approach that also did not assume any explicit functional form for the DSDs and used and as reference moments. This approach led to a more compact representation of DSDs than using one-moment normalization. Lee et al. (2004, hereafter L04) extended the Sempere Torres et al. (1994, 1998) method and developed a general two-moment normalization utilizing any two reference moments and without imposing any DSD functional form. This is expressed as

 
formula

They showed an explicit relationship between their two-moment scaling and the Testud et al. (2001) method. In addition to using two moments to relate and , L04 derived a general power relationship between DSD moments from this normalization. They showed that several analytic DSD functions (exponential, gamma, generalized gamma) have this scaling property, assuming certain conditions such as a constant gamma shape parameter.

The two-moment normalization of L04 captures variability in the DSD slope and intercept. However, as they state, this method “cannot change the various shapes (especially different curvatures) of DSDs that result from complex physical processes shaping the distribution.” Thus, it cannot collapse DSDs with varying shapes and widths into a single unique normalized DSD. This is illustrated in L04 by the normalized DSD dependence on the shape parameter when applied to gamma DSDs. Berne et al. (2012) compared the one- and two-moment normalizations from Sempere Torres et al. (1994) and L04 and found that, although using two reference moments captured more variability than one, neither captured large DSD variability observed over time scales <10 min. From disdrometer data, Yu et al. (2014) similarly found that the normalization function using two reference moments yielded contrasting shapes during different phases of a rain event; they stated that is “process dependent and not unique as hypothesized in the scaling theory.”

Variability of across a set of DSDs should decrease as the number of reference moments is increased. This is because any distribution function is uniquely defined by an infinite sequence of its moments . This is proven by the Hausdorff moment problem, which has a unique solution and 1:1 mapping between and its moment sequence if the moment sequence is valid (see section 2) and moments are defined by integrating over a bounded interval in x (Hausdorff 1921; Shohat and Tamarkin 1943). It follows that because DSDs are bounded by and , a DSD is uniquely defined by its sequence of moments.

Quantifying the variability of normalized DSDs as a function of the number of reference moments is important for many applications, including understanding how DSDs evolve and developing remote sensing retrievals and instrument simulators. However, this has been difficult to assess directly because of inconsistencies in the various normalization frameworks. For example, some methods have imposed a DSD functional form (Sekhon and Srivastava 1971; Willis 1984; Yu et al. 2014), whereas others have not (Sempere Torres et al. 1994, 1998; Testud et al. 2001, L04). Even among the one- and two-moment frameworks that do not assume a DSD form, the physical dimensions of both the prefactor and argument in differ. As pointed out by Yu et al. (2014), this makes it challenging to directly compare across these frameworks. To address this issue, they unified the one- and two-moment normalizations by assuming a gamma DSD form. They also used this gamma DSD framework to develop a three-moment scaling using , , and as reference moments, and found that it well characterized the observed DSDs, both for individual spectra and time series of spectra. Although this approach allows a more direct comparison of normalized DSDs using one, two, and three reference moments, it is limited by imposing a gamma DSD form, and by prescribing , , and as the reference moments for the three-moment scaling. Moreover, by imposing a three-parameter gamma DSD, the method cannot be extended to use more than three reference moments.

In this paper we extend the basic idea of Yu et al. (2014) in developing a unified normalization framework. In contrast to their method, we do not impose any explicit DSD functional form. Instead, we propose a general normalization framework that has few a priori assumptions for deriving scaling relationships between DSD moments, utilizing any number and combination of reference moments. Specifically, it allows any DSD moment to be expressed as a function of any set of reference moments using generalized power series. In its most general form, such power series can describe a wide assortment of possible functional relationships, including the set of all smooth functions. In contrast to most DSD studies that utilize data from one region or field campaign, we apply the normalization to a large dataset of DSDs collected from locations around the world as well as those modeled using a bin microphysics scheme. Scaling relationships between DSD moments are obtained from applying the normalization to this DSD dataset. In addition to obtaining these relationships, a primary goal is to understand and quantify the reduction of normalized DSD moment variability using different numbers and combinations of reference moments, which is facilitated by this unified framework. After describing the proposed normalization approach, we describe how analytic functional forms commonly assumed for DSDs (two parameter and three-parameter gamma, lognormal) are well characterized by the method. We then apply it to the combined observational and bin-modeled DSD dataset using one through four reference moments. Results are compared to those using the two-moment L04 normalization method. Overall, this work is relevant to remote sensing retrievals of unknown DSD moments from a set of directly observed moments, and for the choice of prognostic variables in multimoment bulk microphysics schemes, particularly those that do not assume an explicit DSD functional form (e.g., Chen and Liu 2004; Szyrmer et al. 2005; Laroche et al. 2005; Kogan and Belochitski 2012).

The rest of the paper is organized as follows. Section 2 gives a description of the approach. Section 3 describes the observations and model providing the DSD dataset. Results applying the normalization to this dataset are presented in section 4. A summary and conclusions are given in section 5.

2. Description of the approach

The proposed general normalization method is centered on characterizing relationships among DSD moments, and thus is different from many previous normalization studies that have focused on the normalized DSD function . Moment scaling relationships are valuable because most physical quantities relevant to bulk microphysics schemes (e.g., bulk process rates) and remote sensing are proportional to DSD moments or sums of moments. Various moments of interest can be derived from a set of known reference moments using this approach without explicitly deriving . Nonetheless, the connection to and previous normalization methods is detailed in the  appendix.

From the Hausdorff moment problem, with moments defined by integrating a real-valued function over a bounded interval of x, a valid sequence of moments uniquely defines (Hausdorff 1921; Shohat and Tamarkin 1943). A moment sequence is valid if and only if it is completely monotonic; that is, it must satisfy for all integers , where is the forward difference operator of order k. Thus, there exists a unique mapping between and the infinite moment sequence that depends on and is given by . The moment sequence can be normalized by any moment to give the normalized moment sequence , with the mapping given by . If is completely monotonic, then the normalized moment sequence must also be completely monotonic and hence itself a valid moment sequence. Normalizing the moment sequence will allow us to conveniently express ratios of moments as a function of other moment ratios below. Because is unique for the normalized moment sequence , then must be well defined from for any real number (in this work we are only concerned with where ). This well-defined relationship between and the normalized moment sequence is expressed by a function .

We expand and write as a multivariate generalized power series of , which assumes it has the following general form:

 
formula

where are coefficients and the exponents are a well-ordered subset of real numbers. Here, we exclude in the product because in this instance . Generalized power series (e.g., Loeb 1991) encompass many other series, including power series when the exponents are integers ≥0 (which includes the family of all polynomials), and Puiseux series when the exponents include fractions. Thus, (4) retains a high degree of generality.1

We use the mathematical identity for and for combined with (4) to write

 
formula

where for and for .

The right-hand side of (5) is approximated by using a finite number J of terms in the sum and by taking a finite number N of known reference moments in the product. The reference moments are , where is a well-ordered subset of integers ≥0 corresponding to the reference moment orders. The moment sequence is normalized by one of the reference moments (i.e., ). For reference moments, this approximation is expressed as

 
formula

where is the “derived” moment and the scaling parameters and are determined by fitting to DSD data. We will refer to (6) as N-moment normalization. To retain generality and flexibility, this normalization does not make any a priori assumptions about the relationship of moments to one another except as required for self-consistency when . This implies an independent set of parameters and for each and combination of reference moments.

For , only is known. Thus, we do not normalize the moment sequence by , but instead approximate as a sum of univariate power functions:

 
formula

This framework has some similarities to previous normalization methods. For example, using and , it gives a power-law relationship between moments that underpins several one-moment normalization methods (e.g., Sempere Torres et al. 1994, 1998; Uijlenhoet et al. 2003). For and , it gives , which is of the same form as the scaling relationships using the two-moment Testud et al. (2001) and L04 methods. However, the exponent parameter b in our general normalization is fit to DSD data, whereas it is determined by the orders of the reference and derived moments in L04, given by . As detailed in the  appendix, the L04 expression for b can be derived from dimensional analysis, and we show that formally this expression is valid when there is no variability in the normalized DSD function , that is, when a set of DSDs collapses to a single, unique nondimensional after normalizing with two reference moments. Similarly, we also use dimensional analysis to obtain an expression relating the exponent parameters to the derived and reference moment orders for three-moment normalization using in (6), valid when there is no variability in after normalizing with three reference moments [see (A8) in the  appendix]. However, in general when normalizing a set of real DSDs some variability will remain in (e.g., Yu et al. 2014). This variability affects statistical relationships between moments and implies that the fit parameters in (6) and (7) will differ from the parameters valid when there is no variability in .

a. Two-moment normalization

If two reference moments and are known, in (6) and the ratio of and is a function of the ratio of and :

 
formula

where for brevity we have omitted the first subscript for b (i.e., ). It can be shown that (8) with conforms exactly to commonly used analytic DSDs with two degrees of freedom, such as gamma functions with a fixed shape parameter and lognormal functions with a fixed variance, with the exponent parameter b given by (A7) in the  appendix. The L04 normalization also conforms exactly to these analytic DSDs because they collapse to a single when normalized by two reference moments (see discussion in the  appendix).

b. Three-moment normalization

If three moments of the DSD are known, we can use in (6) and relate the ratio of and to ratios of , , and . This three-moment normalization is expressed as

 
formula

We next demonstrate that (9) using a single term () is consistent with three-parameter gamma DSDs, which have been used in previous normalization studies (Yu et al. 2014) and three-moment bulk microphysics schemes (e.g., Milbrandt and Yau 2005; Shipway and Hill 2012). The gamma DSD is expressed by , where , μ, and λ are the intercept, shape, and slope parameters. Here, we use , , and as the reference moments, though this analysis could be extended to any three reference moments. By analytically integrating the gamma DSD for , , and , it can be shown that the shape parameter μ is a function of , expressed as ; Milbrandt and Yau (2005) calculated Y as a piecewise second-order polynomial. Calculating and by integrating the gamma DSD allows us to analytically express λ and as functions of , , and μ, and any moment of the gamma DSD is given by

 
formula

Combined with , the analytic expression for is thus

 
formula

We approximate

 
formula

in (10) using

 
formula

If the power-law relationship in (11) was exact, then the three-parameter gamma DSD exactly conforms to the three-moment normalization in (9) using a single term (). This is demonstrated by combining (11) with (10) to give

 
formula

This expression has the same form as the three-moment normalization in (9) using because we can write and . Thus, insofar as X can be represented by a power law using (11), the three-moment normalization (9) using is consistent with a three-parameter gamma DSD. This is tested for a few different values of z. First, parameters c and d are fit to directly calculated values of X following (11) for and using linear regression in log space, with the fit parameters given in Table 1. We then compare the derived and using (12) with the fit values of c and d with and directly calculated from the three-parameter gamma function.

Table 1.

Parameters c and d (unitless) in the power-law relationship (11) fit to directly calculated from analytic three-parameter gamma DSDs as a function of for and 4. The fitted X are used in the three-moment normalization shown in Fig. 1.

Parameters c and d (unitless) in the power-law relationship (11) fit to  directly calculated from analytic three-parameter gamma DSDs as a function of  for  and 4. The fitted X are used in the three-moment normalization shown in Fig. 1.
Parameters c and d (unitless) in the power-law relationship (11) fit to  directly calculated from analytic three-parameter gamma DSDs as a function of  for  and 4. The fitted X are used in the three-moment normalization shown in Fig. 1.

Figure 1 shows the derived and directly calculated and results as a function of the reference moment combination . The plotted values of and are nondimensionalized by scaling with , with z equal to 1 or 4. Differences over a wide range of are small (less than 5%) for both and . Thus, although not exact, analytic three-parameter gamma DSDs are well characterized by the three-moment normalization using (9) with .

Fig. 1.

Nondimensionalized (a) and (b) as a function of calculated directly from the analytic three-parameter gamma DSD (crosses), and estimated from the three-moment normalization (lines). Values of the fit parameters c and d for the normalization are given in Table 1; and are nondimensionalized by scaling with and , respectively. Results are shown over the range of from 1 to 20. This range is consistent with the valid range for the piecewise polynomial fit for from Milbrandt and Yau (2005) used for the gamma DSD calculations herein, corresponding to (and hence covering a wide range of gamma DSD shapes).

Fig. 1.

Nondimensionalized (a) and (b) as a function of calculated directly from the analytic three-parameter gamma DSD (crosses), and estimated from the three-moment normalization (lines). Values of the fit parameters c and d for the normalization are given in Table 1; and are nondimensionalized by scaling with and , respectively. Results are shown over the range of from 1 to 20. This range is consistent with the valid range for the piecewise polynomial fit for from Milbrandt and Yau (2005) used for the gamma DSD calculations herein, corresponding to (and hence covering a wide range of gamma DSD shapes).

3. Methodology and dataset description

In this study, we utilize a large DSD dataset comprising disdrometer observations and bin microphysics model output. The idea is to encompass a wide range of plausible DSDs for applying the normalization method described in section 2. Nonetheless, excluding the bin model DSDs has a limited impact on the results, including the fit normalization parameters (see section 4). We also do not separately analyze different rain regimes, such as stratiform and convective, because the goal is to investigate the general behavior of DSD scaling relationships and variability across regimes.

a. Description of the disdrometer DSDs

Disdrometer data were obtained from the U.S. Department of Energy (DOE) Atmospheric Radiation Measurement (ARM) program observational sites around the world (e.g., Ackerman and Stokes 2003; Mather and Voyles 2013; Gottschalck et al. 2013; Yoneyama et al. 2013; Miller et al. 2016; Sisterson et al. 2016; Long et al. 2016). This includes data from Joss–Waldvogel model RD80 impact disdrometers (JWDs; Joss and Waldvogel 1967) and two-dimensional video disdrometers (2DVDs; Tokay et al. 2001; Kruger and Krajewski 2002). The JWD measures DSDs using 20 irregularly spaced bins with a diameter ranging from approximately 0.3 to 5.5 mm. Although the resolution of the 2DVD is nominally 0.2 mm, smaller drops can still provide enough extinction of light to be counted (Kruger and Krajewski 2002). The ARM data undergo several data quality checks (M. Bartholomew 2017, personal communication). The instruments are calibrated by dropping metal spheres of known sizes into various locations throughout the measurement area. Daily quality-check reports are used to monitor these calibrations. Intercomparisons with nearby rain gauges and other disdrometers are also performed. Data quality reports are available at the DOE ARM data archive. Uncertainty in the disdrometer-observed DSDs and implications are discussed in section 4c.

We use JWD data from the ARM Southern Great Plains (SGP) site from 2006 to 2016 and the tropical western Pacific (TWP) site from 2006 to 2015. The 2DVD data include those from the SGP (2011–16), TWP (2011–15), and eastern North Atlantic (ENA; 2014–16) ARM permanent sites, as well as from field projects in the Indian Ocean [October 2011–February 2012 during Dynamics of the Madden–Julian Oscillation–ARM MJO Investigation Experiment (DYNAMO-AMIE; Yoneyama et al. 2013)] and Finland [February–September 2014 during Biogenic Aerosols—Effects on Clouds and Climate (BAECC; Petaja et al. 2016)]. In total, there are 15.5 million JWD sample periods and 6.1 million 2DVD sample periods. These are 30- or 60-s samples from throughout the data collection periods indicated above and thus contain samples with no rain. To eliminate these “empty” samples, we removed any DSDs that produce moment values <−30 dB. Note that throughout the paper moment values are given in units of dB and thus they are a measure relative to 1 mmz m−3, where z is the moment order; otherwise, values span several orders of magnitude depending on the moment order when expressed directly in units of mmz m−3. Additionally, we removed samples where the sixth moment (nearly the equivalent radar reflectivity factor) dB. This was done to mimic the signal detectability of most precipitation radars (e.g., those operating at S, C, and X bands). Finally, 122 DSD sample periods contained drops that were considered unrealistically large (>8-mm diameter) and were removed. The resulting disdrometer dataset has 671 303 samples (3.11% of the original dataset’s time periods).

b. Description of the bin model DSDs

An ensemble of synthetic DSDs was generated using a one-dimensional explicit warm rain bin-microphysical model (PBT12; Prat and Barros 2007a; Prat et al. 2012). Including the bin model output at various times and heights allows for transient DSDs produced by size sorting (e.g., Milbrandt and Yau 2005; Kumjian and Ryzhkov 2012), which are not well captured by disdrometer observations (e.g., Cao et al. 2008), to be included in our analysis.

We report here the main characteristics and the configurations used for the model runs generated for this work. More details about the PBT12 model can be found in Prat and Barros (2007a, 2009), Prat et al. (2012), and Kumjian and Prat (2014). Briefly, the PBT12 model uses a number and mass conservative scheme solving the stochastic collection–breakup equation using 40 bins. Expressions for the drop fall velocity are taken from Brandes et al. (2002), the gravitational collection kernel is from Pruppacher and Klett (1978), the coalescence efficiency is from Low and List (1982a) and includes the modification for the small drop diameter range proposed by Seifert et al. (2005), and the breakup function is from McFarquhar (2004), which is based on Low and List (1982b). The PBT12 model also accounts for the delineation of the different drop–drop collision outcomes (coalescence, breakup, bounce) within the drop diameter space considered (, ), where and are the diameters of large and small colliding drops, respectively (Testik et al. 2011), and single drop aerodynamical breakup as drops fall through the atmospheric column (Srivastava 1971). For simplicity we neglect the effects of evaporation.

A set of 10 742 normalized gamma DSDs were imposed as initial conditions at the top of the 3-km rainshaft. The normalized gamma DSD is expressed as (e.g., Willis 1984; Testud et al. 2001; Illingworth and Blackman 2002; Bringi et al. 2002)

 
formula

The DSD parameters (, , μ) used as initial conditions cover most of the realistic DSDs observed in nature (Prat and Barros 2009) with conditions imposed as follows:

 
formula

as well as an additional constraint on the nominal rainfall rate to be <500 mm h−1. The PBT12 model was run for a simulated time of 60 min with a time step of 1 s and a 10-m vertical grid spacing. From this set of initial conditions, about 1.99 × 108 individual DSDs were generated from output at each vertical level of the column and every 1 min.

c. Application of the generalized normalization

Applying the same data quality filters to the bin model DSDs as was done for the disdrometer data, we are left with 184 180 279 samples, or 92.48% of the entire dataset. The combined disdrometer and model dataset is thus strongly dominated by the bin simulations. To mitigate any bias this may introduce, we sample in accordance with a climatology of rain rates obtained from 10 years of in situ measurements from the U.S. Climate Reference Network (USCRN), as described below. The USCRN network is composed of about 140 stations in the United States and provides an ensemble of quality-controlled atmospheric parameters at various temporal resolutions (5 min, 1 h, 1 day) (Diamond et al. 2013; Leeper et al. 2015). The 5-min rain-rate probability density function (PDF) and associated statistics are computed for observed rain events when the near-surface air temperature was above 5°C.

The full record of disdrometer and bin model DSDs is reduced to random subsets of equal size, so that neither source dominates the combined sample. From this equally combined dataset, samples are drawn in a way consistent with the rain-rate PDF described above, an approach analogous to Latin hypercube sampling. A total of 2 × 105 individual DSDs are drawn to generate the subsampled dataset for our analysis. This subsampling is robust and drawing different sets of DSDs has little impact on the results. The rain-rate histogram corresponding to this subsampled dataset is shown in Fig. 2a. Given the exponential decrease in the distribution with increasing rain rate, to avoid zero counts a minimum distribution density of 1 mm h−1 is applied for the largest rain rates. Also shown in Fig. 2b are the univariate and joint PDFs of , , and from the disdrometer-observed and modeled DSDs associated with this subsampled dataset. Although there is clearly correlation between and and between and , there is also significant scatter, which indicates the DSDs from this dataset encompass a wide range of characteristics.

Fig. 2.

(a) Histogram of rain rates from which the DSD subsamples are drawn. (b) Joint PDFs of , , and (contour plots) and univariate PDFs for these quantities (line plots aligned diagonally) for the DSD dataset associated with the rain-rate histogram shown in (a). The moment values shown have units of dB and thus are a measure relative to 1 mmz m−3, where z is the moment order.

Fig. 2.

(a) Histogram of rain rates from which the DSD subsamples are drawn. (b) Joint PDFs of , , and (contour plots) and univariate PDFs for these quantities (line plots aligned diagonally) for the DSD dataset associated with the rain-rate histogram shown in (a). The moment values shown have units of dB and thus are a measure relative to 1 mmz m−3, where z is the moment order.

Statistics of the subsampled disdrometer-observed DSDs are shown in Fig. 3a. To obtain these DSDs, we first linearly interpolated the subsampled JWD data to the higher-resolution 2DVD bin grid. For the 2DVD bins smaller than the smallest JWD bin, we simply extended the distribution density in the smallest JWD bin to these smaller 2DVD bins. Then, the subsampled 2DVD and interpolated JWD data were combined and sorted on a bin-by-bin basis to calculate the median, 25th, and 75th percentile DSDs. The median DSD (black line in Fig. 3a) shows some deviation from exponential form, which would appear as a straight line in the plot. The DSD spread quantified by the difference between the 25th and 75th percentiles (red and blue lines in Fig. 3a) is generally an order of magnitude or more, and increases for drop diameters > 1 mm. Normalized DSD functions obtained from applying two- and three-moment normalization to the same dataset are also shown in Figs. 3b and 3c; see the  appendix for details.

Fig. 3.

(a) The median (black line) and 25th (blue line) and 75th (red line) percentiles of the subsampled disdrometer-observed DSDs (nonnormalized) as a function of drop diameter . (b) As in (a), but for the normalized DSDs as a function of normalized drop size , using the two-moment normalization relationship given by (A3). The reference moments for the normalization are and . (c) As in (b), but using the three-moment normalization relationship given by (A4). The reference moments are , , and .

Fig. 3.

(a) The median (black line) and 25th (blue line) and 75th (red line) percentiles of the subsampled disdrometer-observed DSDs (nonnormalized) as a function of drop diameter . (b) As in (a), but for the normalized DSDs as a function of normalized drop size , using the two-moment normalization relationship given by (A3). The reference moments for the normalization are and . (c) As in (b), but using the three-moment normalization relationship given by (A4). The reference moments are , , and .

Normalizations using one through four reference moments are applied to the subsampled DSD dataset. For one-, two-, and three-moment normalizations we apply (7), (8), and (9), respectively. For four-moment normalization we use (6) with to give

 
formula

Only one term in the sum is used for all of the normalizations (i.e., ) for application here; thus, we omit the j subscript from a and b. A number of terms greater than one (i.e., ) could be used in the normalization, but requires rather complicated techniques to obtain best-fit parameters (e.g., more complicated than polynomial regression). Moreover, as shown below and in section 4, using well describes the moment relationships and leads to substantial reduction in variability of normalized compared to nonnormalized moments. We expect parameter fitting is convergent for if the “true” relationship between moments is expressible as a generalized power expression with a number of unique terms greater than or equal to J, provided a sufficient number of data points. A more detailed discussion of parameter fitting for is beyond the scope of this paper.

Using in the normalizations, the simplest and most accurate method to obtain the a and b parameters is to apply linear regression in log space. For the one-moment normalization, the best-fit parameters relating to are determined by directly taking of (7) and applying linear regression. For two-, three-, and four-moment moment normalizations we divide (8), (9), and (15), respectively, by , take , and apply multidimensional linear regression to the moment ratios to determine best-fit parameter values.

The variability of normalized DSD moments is quantified by the logarithmic standard deviation σ. This is calculated by subtracting the derived from the actual in log space for each individual DSD in the subsampled dataset, which yields a set of residual points in log space. Then, we obtain σ by calculating the standard deviation in the usual way from this set of residuals. Because σ is derived from the set of residuals in log space, it is naturally expressed in units of decibels (dB).

Overall, this method provides the following: 1) best-fit parameters to derive any from a set of known reference moments and 2) a set of normalized moments by subtracting the derived from the true for each data point (DSD). In addition, σ quantifies the spread of the normalized and allows us to determine the reduction of DSD variability from applying the various normalizations. For comparison we also apply the L04 two-moment normalization to the subsampled DSD dataset. As with the general normalization, σ is calculated by subtracting the derived from the L04 normalization from the true for each data point and calculating the standard deviation from this set of normalized .

An example applying our two-moment normalization to the subsampled DSD dataset is shown in Fig. 4. The derived moment in this example is and the reference moments are and . The top panel in Fig. 4 shows the actual as a function of for all data points along with the best-fit line, and the bottom panel shows derived values of using the best-fit parameters versus the actual values of . These results clearly illustrate the close relationship between the moments that is well approximated by a linear fit in log space.

Fig. 4.

(a) Joint histogram of moment ratios log vs log from the subsampled DSD dataset, with colored boxes showing log histogram counts indicated by the legend. The blue line indicates the best-fit power law relationship. (b) Scatterplot of derived using the best-fit power relationship shown in the top panel (with reference moments and ) vs the actual from the subsampled DSD dataset. The moment values shown have units of dB and thus are a measure relative to 1 mmz m−3, where z is the moment order.

Fig. 4.

(a) Joint histogram of moment ratios log vs log from the subsampled DSD dataset, with colored boxes showing log histogram counts indicated by the legend. The blue line indicates the best-fit power law relationship. (b) Scatterplot of derived using the best-fit power relationship shown in the top panel (with reference moments and ) vs the actual from the subsampled DSD dataset. The moment values shown have units of dB and thus are a measure relative to 1 mmz m−3, where z is the moment order.

4. Results

a. Application of the general normalization method

Applying the general normalization method from section 2 to the combined disdrometer and bin-modeled subsampled DSD dataset provides best-fit estimates for deriving any moment from a set of reference moments, while uncertainty in these estimates is quantified by the logarithmic standard deviation, σ. Here, all combinations of reference moments from through are utilized for one- through four-moment normalizations. Though not a requirement of the method, we focus on derived/normalized moments that have integer orders.

Results are summarized in Fig. 5. Values of σ for each normalized moment of order 0–10 averaged over all possible combinations of reference moments of orders 0–10 () are shown as a function of the number of reference moments, N. Also shown are values of the minimum σ () for the optimal combination of reference moments. For plotting purposes, and are divided by the logarithmic standard deviation without applying normalization (i.e., zero-moment normalization), . For each normalized moment of order 0–10, Table 2 provides the best-fit normalization parameters and for the optimal combination of reference moments and Table 3 gives values of for zero- through four-moment normalizations.

Fig. 5.

Ratio of logarithmic standard deviation of the normalized moments σ and that with no normalization applied as a function of the number of reference moments. Each plot shows results for normalized moments through . Blue lines show the mean for all possible combinations of reference moments of integer order 0–10, while red lines show the minimum for the optimal combination of integer reference moments. Symbols show for particular combinations of reference moments. Horizontal black lines show the ratio of logarithmic standard deviation associated with uncertainty from disdrometer DSD drop count statistics to .

Fig. 5.

Ratio of logarithmic standard deviation of the normalized moments σ and that with no normalization applied as a function of the number of reference moments. Each plot shows results for normalized moments through . Blue lines show the mean for all possible combinations of reference moments of integer order 0–10, while red lines show the minimum for the optimal combination of integer reference moments. Symbols show for particular combinations of reference moments. Horizontal black lines show the ratio of logarithmic standard deviation associated with uncertainty from disdrometer DSD drop count statistics to .

Table 2.

Summary of results from applying the general normalization method to the subsampled DSD dataset. The derived/normalized moment of order z is shown in the left column. The optimal combination of reference moments that minimizes the logarithmic standard deviation σ for each is shown in the second column for one-, two-, three-, and four-moment normalizations. The third column shows the minimum σ, , in units of dB. The fourth column shows the a parameter fits. The fifth column shows the b parameter fits; for three- and four-moment normalizations the formats are and , respectively. Units of a are for one-moment normalization, for two-moment normalization, for three-moment normalization, and for four-moment normalization, giving derived moments with units of . Here, , , … are the orders of the reference moments.

Summary of results from applying the general normalization method to the subsampled DSD dataset. The derived/normalized moment of order z is shown in the left column. The optimal combination of reference moments that minimizes the logarithmic standard deviation σ for each  is shown in the second column for one-, two-, three-, and four-moment normalizations. The third column shows the minimum σ, , in units of dB. The fourth column shows the a parameter fits. The fifth column shows the b parameter fits; for three- and four-moment normalizations the formats are  and , respectively. Units of a are  for one-moment normalization,  for two-moment normalization,  for three-moment normalization, and  for four-moment normalization, giving derived moments  with units of . Here, , , … are the orders of the reference moments.
Summary of results from applying the general normalization method to the subsampled DSD dataset. The derived/normalized moment of order z is shown in the left column. The optimal combination of reference moments that minimizes the logarithmic standard deviation σ for each  is shown in the second column for one-, two-, three-, and four-moment normalizations. The third column shows the minimum σ, , in units of dB. The fourth column shows the a parameter fits. The fifth column shows the b parameter fits; for three- and four-moment normalizations the formats are  and , respectively. Units of a are  for one-moment normalization,  for two-moment normalization,  for three-moment normalization, and  for four-moment normalization, giving derived moments  with units of . Here, , , … are the orders of the reference moments.
Table 3.

Mean logarithmic standard deviation σ (units of dB) from the subsampled DSD dataset for each derived moment , for all combinations of reference moments through using one-, two-, three-, and four-moment normalizations. The column labeled “zero moments” shows values of σ from the subsampled dataset without any normalization applied.

Mean logarithmic standard deviation σ (units of dB) from the subsampled DSD dataset for each derived moment , for all combinations of reference moments  through  using one-, two-, three-, and four-moment normalizations. The column labeled “zero moments” shows values of σ from the subsampled dataset without any normalization applied.
Mean logarithmic standard deviation σ (units of dB) from the subsampled DSD dataset for each derived moment , for all combinations of reference moments  through  using one-, two-, three-, and four-moment normalizations. The column labeled “zero moments” shows values of σ from the subsampled dataset without any normalization applied.

Differences between the reference and normalized moment orders have a strong influence on σ. Using reference moments with orders near that of the normalized moment gives the smallest σ, which is not surprising. Because we use integer moment orders, this means that using either or as the reference moment for (one-moment normalization) gives for normalized moment . Additional reduction of relative to one-moment normalization occurs using and as the reference moments for (two-moment normalization), especially for z of 2, 3, and 4. For (three-moment normalization), the optimal combination of reference moments giving is , , and . There is a small but notable reduction of for three-moment compared to two-moment normalization for normalized moments , , and , but little reduction for higher-order moments. There is almost no additional reduction of for compared to using three-moment normalization for any normalized moments.

A key result is the asymmetry of σ as a function of the orders of the reference and normalized moments. In general, higher-order reference moments give smaller σ than lower-order reference moments for the same distance in order between the normalized and reference moments. This is illustrated in Fig. 6 by plots of σ for every combination of normalized and reference moments using two-moment normalization. It is clearly seen that σ is smaller using higher-order reference moments compared to lower-order ones for a given distance between the reference and normalized moment orders. For example, using reference moments and gives much smaller σ than using and for normalized moment .

Fig. 6.

Contour plots of logarithmic standard deviation for normalized moments, σ (units of dB), using the general two-moment normalization applied to the subsampled DSD dataset. Each panel shows results for a normalized moment with z of 0–10 for all combinations of reference moments and .

Fig. 6.

Contour plots of logarithmic standard deviation for normalized moments, σ (units of dB), using the general two-moment normalization applied to the subsampled DSD dataset. Each panel shows results for a normalized moment with z of 0–10 for all combinations of reference moments and .

Overall, these results have practical implications for retrieving DSD moments from higher-order observed moments, such as radar reflectivity, which is proportional to . Although there is limited constraint (i.e., large ) using alone to derive lower-order moments (see Fig. 5), it can provide valuable additional constraint when either (proportional to bulk drop surface area) or (proportional to bulk mass) is also known. For example, is 0.47 when normalizing with as the only reference moment, and 0.72 with as the only reference moment. However, using two-moment normalization with the combination of {, } or {, } as reference moments, is reduced to 0.27 or 0.36; thus, adding as a reference moment reduces by about a factor of 2. Even when both and are known, adding as a reference moment constraint for deriving still leads to a reduction of from 0.20 to 0.13. A similar situation holds for deriving any moment of order 0–4. These results highlight the value of having three known reference moments, in particular, for deriving unknown lower-order moments. From remote sensing observations, three reference moments could be obtained from, for example, Raman lidar backscatter (), radar attenuation (close to ), and radar reflectivity ().

These results also have implications for the choice of prognostic moments in bulk microphysics schemes, particularly for calculating bulk process rates and quantities for instrument simulators in schemes that do not assume an underlying DSD functional form (e.g., Chen and Liu 2004; Szyrmer et al. 2005; Laroche et al. 2005; Kogan and Belochitski 2012). Most process rates and parameters of interest in models span (number concentration) through approximately (near the moment proportional to the rain rate). On the other hand, higher-order moments are relevant for some remote sensing applications, in particular but potentially higher moments as well. Most bulk two-moment microphysics schemes prognose and , but it is clear from Fig. 5 that and are poorly constrained by this combination of reference moments (i.e., is large). On the other hand, using and as the prognostic reference moments does not constrain or very well, with of 0.37 for and 0.23 for . Assuming that must be prognosed in order to conserve water mass, our results suggest a good choice of prognostic variables for minimizing the uncertainty of derived moments spanning through is or . Adding a third prognostic moment can further reduce uncertainty in deriving moments. All combinations of any two moments from to plus as the reference moments give limited uncertainty for derived moments through ; the maximum is 0.23, which occurs when normalizing with {, , } as the reference moments. Broadly, these results provide context and support for the continued development and use of three-moment bulk microphysics schemes for modeling rain.

In general, considering all possible combinations of reference moments with orders 0–10, there is a large reduction of as the number of reference moments N is increased from 0 to 1 and from 1 to 2 (see Fig. 5). There is a notable further decrease in when N is increased from 2 to 3, especially for normalized moments , , and . In contrast, there is little decrease of when N is increased from 3 to 4 for any normalized moments. Thus, three-moment normalization can account for the vast majority of DSD variability, in general. Averaging results for all normalized moments of order 0–10, 42.9%, 81.3%, 93.7%, and 96.9% of spread are accounted for with one-, two-, three-, and four-moment normalizations, respectively. These results are consistent with previous studies showing substantial reduction of variability with an increase in the number of reference moments from one to two (L04; Berne et al. 2012). Our result that 81% of the spread of normalized moments is accounted for by two-moment normalization is similar to results from Raupach and Berne (2017). They showed that 85% of the variability for normalized moments of order 0–7 was accounted for by two-moment normalization, based on combined disdrometer and radar analysis.

Including only the subsampled disdrometer-observed DSDs in the analysis does not have a large effect compared to including both the subsampled disdrometer and bin model DSDs. The spread of nonnormalized DSDs is reduced by excluding the bin model DSDs, with decreasing by 15.1% for , 11.3% for , and < 8% for moments and higher. There is a limited impact on the fit normalization parameters. For example, a parameter values for the two-moment normalization differ by a maximum of 45% compared to those using the full dataset, but most (68%) values differ by <10%. A similar result occurs for the fit b parameter values. There is a decrease in the spread of normalized moments by excluding the bin model DSDs, with σ generally about 10%–25% smaller when applying the two-moment normalization.

b. Comparison with the two-moment Lee et al. (2004) normalization

A comparison of the fit b from the general normalization with the calculated b from the L04 normalization [given by (A7) in the  appendix] is shown in Fig. 7a. Overall, the calculated b results are similar to those of the fit b, generally within about 25%. The largest differences occur when the normalized and reference moments orders are far apart, corresponding to relatively large positive and negative b (note that b = 0 or 1 when the normalized moment order is equal to either of the two reference moment orders). For large positive b, the calculated L04 values are larger than the fit values, while for large negative b they are more negative than the fit values. These results are consistent with those presented in L04 (see their Fig. 5). As noted in section 2a, commonly used analytic DSD functions with two degrees of freedom, such as gamma functions with a fixed shape parameter and lognormal functions with a fixed variance, conform to the two-moment normalization with a b exponent parameter identical to that of L04, given by (A7). Thus, differences between the best fit b parameters from the general normalization and the calculated b from the L04 normalization seen in Fig. 7a imply some deviation of the moment relationships from the DSD dataset with those from these analytic DSD functions.

Fig. 7.

(a) A comparison of fit (x axis) and calculated (y axis) values of the exponent parameter b for all combinations of normalized and reference moments using two-moment normalization. Calculated values of b follow from the L04 normalization method [see (A7) in the  appendix] and fit values are from the DSD dataset. (b) A comparison of fit (x axis) and calculated (y axis) values of the exponent parameter for all combinations of normalized and reference moments using three-moment normalization. The calculated follow (A8) in the  appendix. Solid lines indicate a 1:1 relationship.

Fig. 7.

(a) A comparison of fit (x axis) and calculated (y axis) values of the exponent parameter b for all combinations of normalized and reference moments using two-moment normalization. Calculated values of b follow from the L04 normalization method [see (A7) in the  appendix] and fit values are from the DSD dataset. (b) A comparison of fit (x axis) and calculated (y axis) values of the exponent parameter for all combinations of normalized and reference moments using three-moment normalization. The calculated follow (A8) in the  appendix. Solid lines indicate a 1:1 relationship.

Because there is an additional degree of freedom in relating derived moments to the two reference moments using our general approach, it must give σ less than or equal to the σ using the L04 normalization. This is confirmed by the results. Nonetheless, overall differences in σ are fairly small, with a maximum of 13% but generally less than 5%. Differences in σ have similar trends to the differences between the calculated and fit b; differences are larger when the orders of the reference and normalized moments are far apart.

We also compare the fit parameters for three-moment normalization in (9) with the calculated derived using dimensional analysis assuming no variability of the normalized DSD function [see (A8) in the  appendix). As seen in Fig. 7b, the calculated and fit align closely along the 1:1 line with no evidence of bias for large positive and negative values, unlike the calculated and fit b for two-moment normalization.

c. Uncertainty from drop count statistics

There are potentially several sources of uncertainty with the JWD and 2DVD disdrometer measurements, including the filtering of outlier particles with velocities well beyond the range of raindrops, coincident drops in the measuring area, a limited measurable drop size range, and the effects of wind, among others (see, e.g., Loffler-Mang and Joss 2000; Tokay et al. 2001; Kruger and Krajewski 2002; Thurai et al. 2011). There is also uncertainty in the bin model DSDs due to uncertainty in the microphysical process formulations and model numerics. However, this is difficult to quantify and hence we focus on uncertainty associated with the disdrometer measurements.

As in previous DSD normalization studies (Testud et al. 2001, L04), we focus on the uncertainty due to counting statistics, which can be readily quantified. Thus, our uncertainty estimate likely represents a lower bound on the true uncertainty, although we expect counting statistics are a key source of uncertainty over the relatively short sampling times analyzed here (30–60 s). Following Testud et al. (2001) and L04, the probability distribution of the number of particles in each size bin for a sampling time window is described by Poisson statistics, assuming that drops are randomly distributed in time and space. Note that this assumption has been challenged by Larsen and O’Dell (2016), who investigated statistical DSD fluctuations using an observationally constrained Montel Carlo simulation and found that true sampling uncertainty is much larger than that assuming Poisson distributions. Nonetheless, for simplicity we assume Poisson statistics here.

The uncertainty in DSD moments is calculated as follows. For each observed DSD, we first draw random samples from a Poisson distribution with the expected value given by the actual drop count in each bin. The sampled count in each bin is converted to a number density (units of m−4) by dividing the count by the product of a mean sample volume and the bin width. The mean sample volume is estimated simply by the ratio of the actual count and number density, and varies over time and among bins. This approach is used to construct a family of sampled DSDs for each observed DSD. Ten samples are drawn for each bin; redrawing 10 random samples has almost no impact on the results. The sampled DSDs are then integrated to calculate moments. For each observed DSD and the corresponding family of sampled DSDs, the logarithmic standard deviation for each moment is calculated to quantify moment uncertainty (units of dB).

The logarithmic standard deviation for each moment from counting statistics, , as well as the ratio are shown in Table 4. Values of are also indicated by the horizontal black lines in Fig. 5. Because the raw count tends to decrease for larger-sized bins and because higher-order moments are weighted toward larger drop sizes, generally increases with moment order, from less than 0.5 dB for , , and to greater than 2 dB for . Nonetheless, is a fairly small fraction of the total variability, with increasing from 7.0% for to about 15% for . Compared to (see Table 3 and Fig. 5), is smaller for one- through four-moment normalizations for normalized moments and , between values of applying two- and three-moment normalizations for and , and near for two-moment normalization for through .

Table 4.

Logarithmic standard deviation for each moment associated with uncertainty from counting statistics, , and the ratio of and the logarithmic standard deviation with no normalization applied, .

Logarithmic standard deviation for each moment associated with uncertainty from counting statistics, , and the ratio of  and the logarithmic standard deviation with no normalization applied, .
Logarithmic standard deviation for each moment associated with uncertainty from counting statistics, , and the ratio of  and the logarithmic standard deviation with no normalization applied, .

We emphasize, however, that uncertainty due to drop count statistics has little impact on the DSD scaling relationships or σ associated with the normalized moments, even though it leads to uncertainty in the moments themselves. For example, with no normalization, values differ by <5% for the set of Poisson-generated DSDs compared to those calculated directly from the disdrometer dataset without considering the drop count uncertainty. Applying two-moment normalization, σ values from the generated dataset differ by <2% from those calculated directly using the disdrometer dataset. Similarly, there is little impact on values of the fit a and b normalization coefficients. Thus, uncertainty associated with counting statistics has little impact on our main findings regarding DSD scaling relationships. The basic explanation for this is the correlation between DSD moments, particularly when they have orders near one another. In other words, because statistical fluctuations in drop counts affect all DSD moments, often in coherent ways, correlations between moments and thus the DSD scaling relationships are affected less than might be anticipated.

5. Summary and conclusions

We have proposed a DSD normalization framework that relates any DSD moment to a set of N reference moments, for any combination of reference moments. This provides a general method for obtaining DSD moment scaling relationships and facilitates an analysis of normalized DSD moment variability for different numbers and combinations of reference moments. Derived moments were formulated as generalized power series functions of the set of reference moments. Though not the main focus of this study, we also derived general relationships between the DSD function and the normalized DSD function using dimensional analysis, for any number and combination of reference moments. The relationship to previous DSD normalization approaches, including the L04 two-moment normalization, was discussed within this context. We showed the L04 method is exact when the set of DSDs collapses to a single unique after normalizing by two reference moments.

The general normalization method was applied to a large dataset of disdrometer-observed and bin-modeled DSDs spanning a wide range of conditions and geographic locations. The number of reference moments utilized ranged from N = 1 to 4. The spread of normalized moments was quantified by the logarithmic standard deviation, σ. Applying one-, two-, three-, and four-moment normalizations to the DSD dataset, σ was reduced by 42.9%, 81.3%, 93.7%, and 96.9%, respectively, compared to the logarithmic standard deviation with no normalization applied, , averaged for all combinations of normalized and reference moments of integer orders 0–10. Thus, DSDs appear to be well characterized overall using three reference moments, whereas a fourth reference moment adds little independent information.

The difference between the reference and normalized moment orders exerted a strong influence on σ, as expected. The combination of reference moments giving the smallest σ had moment orders near that of the normalized moment. However, even when the minimum distance between the reference and normalized moment orders was unchanged, increasing the number of reference moments often led to significant reductions in σ. For example, values were 0.47, 0.20, and 0.13 when normalizing using only, and , and , , and as the reference moments, respectively. On the other hand, using or as a single reference moment provided little constraint for deriving . These results highlight the value of two- and three-moment normalizations, particularly when lower-order reference moments are combined with higher-order ones such as , when deriving unknown moments from a set of observed reference moments. Overall, these results are relevant to bulk cloud microphysical parameterizations as well as observational retrievals.

Using N = 2 reference moments gave results similar to the two-moment L04 normalization method, giving support to the scaling relationship proposed by L04. Nonetheless, since the exponent parameter b in the moment relationship for the general method is fit to the data rather than diagnosed from the moment orders as in L04, it provided a closer fit to the moment relationships and lower σ, especially when the reference and normalized moment orders were far apart.

We also examined DSD normalization within the context of uncertainty in the disdrometer observations. Although there are numerous potential sources of measurement error, we focused on uncertainty associated with drop count statistics, similar to previous DSD studies (Testud et al. 2001, L04). Uncertainty increased with the moment order, which is expected based on Poisson statistics because higher moments depend more strongly on the distribution of large drops, which generally have a lower count for a given sampling time compared to small drops. The logarithmic standard deviation associated with this measurement uncertainty was generally between values of σ applying two- and three-moment normalization. However, this uncertainty had almost no impact on the derived DSD moment scaling relationships nor σ values associated with the normalized moments because the DSD moments are correlated with one another.

Acknowledgments

We thank S. Collis for providing the processed and combined disdrometer data and C. Williams and M. Bartholomew for guidance on the disdrometer observations. Disdrometer data were obtained from Atmospheric Radiation Measurement (ARM) Climate Research Facility Data Archive in Oak Ridge, Tennessee, compiled and maintained by M. Bartholomew. The USCRN data were obtained from the National Centers for Environmental Information. The U.S. Climate Reference Network program is conducted through a partnership between NOAA’s Air Resources Laboratory/Atmospheric Turbulence and Diffusion Division and NOAA’s NCEI. S. Ellis and W. Wu are thanked for providing comments on earlier versions of the paper. This work was partially funded by U.S. DOE Atmospheric System Research Grant DE-SC0016579. The National Center for Atmospheric Research is sponsored by the National Science Foundation.

APPENDIX

Derivation of General Expressions for the Normalized DSD Function and Relationship to Previous Normalization Methods

Although the focus of this paper is on scaling relationships among DSD moments, here we discuss the connection of the normalization framework proposed in section 2 to the normalized DSD function . Using dimensional analysis, general expressions are derived below for the DSD function in terms of a normalized nondimensional , where is a normalization factor that is a function of one or more moments of . First, we note that any DSD can be characterized as the product of a characteristic number density and nondimensional (e.g., L04; Sempere Torres et al. 1998; Yu et al. 2014) to give

 
formula

for any k. Using drop diameter as the characteristic measure (i.e., ), must have a physical dimension of length in order to render the argument in g dimensionless. Because the dimensions of are for any moment order k, the form of as a function of one or more reference moments can be constrained by dimensional analysis. Using a single reference moment, has dimensions of length if it is proportional to , except when . For reference moments, dimensional consistency of is achieved when it is proportional to for even N, where . For odd N, is dimensionally consistent when it is proportional to , where . Note that for , there is not a unique combination of moments that give dimensional consistency for , but the above expressions are valid for all moment combinations if and .

These relationships for can be combined with , which has dimensions of . Here, we use , but this choice is arbitrary and k could be set to any reference moment. For , must be proportional to to give dimensional consistency. For , dimensional consistency is achieved when is proportional to for even N, and when it is proportional to for odd N.

Using these relations for and with proportionality constants of unity, combined with (A1), gives

 
formula
 
formula
 
formula

Similar expressions can be derived for N ≥ 4.

For N = 1, (A2) is similar to the one-moment normalization of Sempere Torres et al. (1994, 1998), except that the exponent in the normalization is determined from the order of the reference moment to ensure nondimensional and . However, the nondimensional results when using (A2) are highly inconsistent with the best-fit power-law parameters for relating moments using (7) from the general normalization in section 2 and, hence, are not useful in practice. On the other hand, the nondimensional from (A3) and (A4) using two- and three-moment normalizations are more consistent with the best-fit parameters using (6) from the general normalization, though they still differ somewhat as described below. Because , (A3) is identical to the two-moment L04 normalization [see (3) in the introduction]. Thus, dimensional analysis provides an alternative means to derive the L04 normalization, while allowing for straightforward extension to normalizations using more than two reference moments.

Normalized DSD functions obtained from applying two-moment (A3) and three-moment (A4) normalizations to the subsampled disdrometer-observed DSDs (see sections 3a and 3c) are shown in Figs. 3b and 3c. Variability of across the set of normalized DSDs decreases as the number of reference moments is increased, which is seen by comparing the middle and bottom panels in Fig. 3. However, there is still some variability in even after applying three-moment normalization. Thus, normalizing a set of real DSDs does not lead to a single unique , unlike for simple analytic DSDs.

Equations (A2)(A4) can be used to derive DSD scaling relationships. Multiplying from (A1) by [with in (A1)], using the relation on the right-hand side, and integrating the resulting expression gives . Combined with the expression above for using two reference moments [i.e., ], this gives

 
formula

where

 
formula

and

 
formula

Equations (A5)(A7) are identical to the two-moment scaling relationship in L04 {see their Eq. (16), keeping in mind that }. This has the same form as the general two-moment scaling relationship in (8), but with additional constraints on a and b. If the DSDs collapse to a single when normalizing with two reference moments, then must be independent of any other moments and hence dimensionless; (A5)(A7) are exact for any in this instance.

On the other hand, if the DSDs do not collapse to a single when normalizing with two reference moments, then must depend on additional moments. In this case the best-fit b parameter relating particular moments must differ from (A7), and dimensional consistency means that a and hence cannot be dimensionless. Thus, the L04 scaling is exact only when the DSDs collapse to a single when normalizing with two reference moments. As shown in section 4b, the best-fit values of b applying (8) to the subsampled DSD dataset are similar to the calculated b from L04 and (A7) when the normalized and reference moment orders are near one another, but with larger differences when the moment orders are farther apart (Fig. 7a).

We can extend this analysis to the case of . If DSDs collapse to a single using three reference moments in (A4), depends on no other moments and dimensional consistency requires the following relationship between the exponent parameters for the three-moment normalization in (9):

 
formula

Similar expressions can be derived for . The calculated results using (A8) are close to the best-fit applying (9) to the subsampled DSD dataset for all combinations of reference and normalized moments (see section 4b and Fig. 7b).

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Footnotes

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1

Assuming is continuous over a closed interval, from the classical Weierstrass approximation theorem it can be approximated as closely as desired over that interval by a polynomial. Although polynomial approximation is straightforward and efficient, several terms may be needed for reasonable accuracy even when is simple, such as a single-term power function with noninteger exponents; this is indeed the form of for commonly used analytic DSD functions such as gamma or lognormal with a fixed shape/variance. Because generalized power series directly encompass not only all polynomial functions but also all power functions with fractional exponents, equal or greater accuracy is achieved using the same number of terms compared to polynomial approximation. The drawback is that, other than when a single term is used, determining optimal parameter values for generalized power series can be much more challenging.