1. Introduction

Hail is a risk to human life and property. The damage it causes depends on the size, mass density, and hardness of hailstones hitting the object (Brown et al. 2015). Objects get damaged if they are hit by hailstones larger than some object-specific threshold. For cars and property this threshold is often assumed to be a hailstone diameter of 2 cm (see, e.g., GdV 2014). This paper introduces a model to estimate how many hailstones larger than a given threshold hit an object of a given size, where the diameter of the largest hailstone is known or modeled.

According to Gunturi and Tippett (2017) large hail is the greatest contributor to insured losses from thunderstorms. Individual hail events can cause losses well above $1 billion (Sander et al. 2013). Hail losses are especially high if urban areas are affected. In 2016, two hailstorms in Texas generated a combined loss of $4.7 billion as a result of hail (Swiss RE 2017). The hail events on 27 and 28 July 2013 in Germany caused insured losses of $3.9 billion (Heymsfield et al. 2014). Before these events, the Munich hailstorm from 12 July 1984 was the costliest hailstorm in Europe, with $2.1 billion loss. This event—which damaged approximately 70 000 properties, 200 000 cars, and even airplanes waiting at the airport terminal (Allianz 2009)—highlights how auto losses can approach or outweigh property losses. Quantifying the risk from damaging hail is of great importance for the insurance industry. For example, Gunturi and Tippett (2017) showed that in the United States average annual losses resulting from severe convective storms have reached those of hurricanes, from data covering the period 2003–15.

Databases with hail reports are available for the United States (Schaefer and Edwards 1999) and Europe (Dotzek et al. 2009). However, these databases suffer from observational issues like clustering of hail reports toward highly populated areas (Allen and Tippett 2015) and the fact that they consist of observer supplied estimates of hail size rather than actual measurements (Ortega 2018). Another disadvantage of observed maximum diameters of the largest hailstone is that the largest hailstone of an event is hardly observed. Blair and Leighton (2012) and Blair et al. (2017) compared high-resolution hail observations based on social media and poststorm ground surveys to the respective entries in the National Centers for Environmental Information (NCEI) database. In many cases, the maximum hail size of the high-resolution data was much larger, likely because of the increase in sampling area. The difference in maximum hail size reached several inches (1 in. = 2.54 cm). However, databases of observed maximum hail size are used to estimate return periods of large hail (Allen et al. 2017).

In the United States, the NCEI has collected reports of hail (Schaefer and Edwards 1999) dating back to 1955. The number of reported hail days per year increased significantly between 1955 and 2000, but it has been stationary since then with an average of 226.75 days per year with hail of at least 2 cm in diameter. On average, the largest hailstone observed per year in the United States has a diameter of 14.13 cm. The largest hailstone observed in the United States in the modern era had a maximum diameter of 20.3 cm and was collected in Vivian, South Dakota, on 23 July 2010 (Monfredo 2011). According to the European Severe Weather Database (ESWD; Dotzek et al. 2009), the largest hailstone observed in Europe in the modern era had a diameter of 14.1 cm and was observed in Germany on 6 August 2013. The ESWD also contains an entry of an estimated maximum hailstone diameter of 20 cm reported in the year 1403 in Bavaria, Germany. For the period 2010–15, the ESWD contains 8 days with largest observed hailstone diameters of at least 10 cm. During the same period, 501 days with hailstones of at least 2 cm diameter were observed, which corresponds to 83.5 days per year on average. Hailstones with sizes that can damage cars and property are common in Europe and the United States.

Historical hailstone size data can be used to study the probability of such occurrences. For example, Allen et al. (2017) used an extreme value model to estimate local return periods for the maximum diameter of the largest hailstone for the United States on the basis of the NCEI data. They used the observed largest hailstone diameter within 3-h intervals on a 1° × 1° spatial grid. Similarly, Fraile et al. (2003) used the size of the largest hailstone observed by hailpads per year in southwestern France to estimate regional return periods of largest hailstones by fitting a Gumbel distribution.

Knowing the return period of the largest hailstone per hail event is very valuable for predicting the full range of possible future outcomes in a catastrophe risk model. However, we need more information to translate the largest hailstone size into expected damage. To understand damage, we need to estimate how many hailstones larger than a damaging threshold hit an object of a given size during a hail event. This paper aims to translate a local maximum diameter of the largest hailstone D_L into the number of hailstones above a threshold diameter that fall in an area during an event. To achieve this aim, we need to know the hail particle size distribution (PSD). We therefore discuss the role of the PSD in section 2.

Other metrics than the maximum diameter of the largest hailstone characterize the intensity of hail events. For example, weather radar observations provide spatiotemporal patterns and are widely used in weather nowcasting. In the simplest case, reflectivities above a threshold are taken to indicate hail (Mason 1971). But this can be misleading since severe rain can generate radar reflectivities of up to 60 dBZ while reflectivities due to hail are usually above 45 dBZ (Aydin et al. 1986; Vivekanandan et al. 1999). Modern dual-polarization radar observations can distinguish between different hydrometeor classes and general categories of hail size (Vivekanandan et al. 1999; Ryzhkov et al. 2013) and therefore overcome the limitations of traditional radar observations. Nevertheless, to the authors’ knowledge, a unique and reliable translation from radar observations into maximum hailstone size is not available. Several radar-based hail parameters exist, like the probability of severe hail and the maximum expected size of hail (MESH) (see Witt et al. 1998). However, after comparing nine such radar-based hail indicators with ground observations from the Severe Hazards Analysis and Verification Experiment (SHAVE), Ortega (2018) found that, although it was the best-performing parameter, MESH still showed weak skills. He concluded that “MESH should not be used as a direct proxy for the actual hail size that fell.” Similarly, Brown et al. (2015) compared losses from the hail event that hit the Dallas–Fort Worth (Texas) area on 24 May 2011 with radar data. They found that areas of maximum loss ratios were shifted away from areas with high MESH and that some areas with high loss ratios did not correspond to any high MESH values.

A further way to express hail intensity is the instantaneous hail kinetic energy flux (W m⁻²). As shown by Waldvogel et al. (1978) hail kinetic energy flux can be deduced from radar reflectivity. Including assumptions about the duration of hail, a local total hail kinetic energy (HKE; J m⁻²) can be estimated. Hohl et al. (2002) were the first to use radar-based hail kinetic energy in a catastrophe risk model. However, as an aggregated variable it does not contain any information about the fraction of that energy that is contributed by various hailstone sizes.

Many more hail intensity metrics are imaginable, like the total hail volume per square meter on the ground, the mass, and the number of stones above a threshold size. While they might be of interest from a hail risk point of view, they might be difficult to observe. But what all these characteristics share is that they are integrals over the hail PSD. The PSD is the key that links all integrated variables like hail kinetic energy and radar reflectivity.

The goal of this paper is modeling the hail PSD at the ground as function of the maximum hailstone size since the latter is often observed and registered or modeled while the first is important for damage assessments. As many authors before us did—for example, Field et al. (2019), we assume an exponential PSD aloft leading to a gamma-distributed PSD at the ground. We use observations of the size of the smallest, average, and largest hailstone; the duration of hailfall; and the number of hailpad hits from the Community Collaborative Rain, Hail and Snow Network (CoCoRaHS) to calibrate the model (see section 4). In section 5 results are discussed and are compared with those of previous studies. Section 6 contains a comparison with independent observations from France. Some example applications are shown in section 7. A final summary, conclusions, and outlook can be found in section 8.

2. Hailstone size distribution and integrated variables

The range of sizes of hailstones within an air parcel can be expressed by a hailstone particle size distribution n(D). It provides the number of hailstones in the diameter interval from D to D + dD per volume of air. Therefore, it can be expressed in units of mm⁻¹ m⁻³. If n(D) is known, a variety of integrated variables can be calculated. We list the most important ones in Table 1. In this table, the equation for radar reflectivity is based on Rayleigh scattering. It is based on the assumption that hailstones are spherical, dry, and much smaller than the radar wavelength. However, these conditions are hardly fulfilled by real hail and this equation has to be used with care. More realistic calculations of radar reflectivity need electromagnetic scattering calculations, for example, the widely used T-matrix formulation (Waterman 1969; Mishchenko 2000). Furthermore, real radar observations are affected by attenuation. Several correction methods have been tested and used, for example, by Snyder et al. (2010) and Kalina et al. (2014). A comparison of scattering properties of real hailstones and spheroids can be found in Jiang et al. (2019).

Table 1.

Some definitions of hail-related variables as function of the hailstone size distribution and results for an exponential hailstone size distribution. Hailstones are regarded as perfect spheres for V, H, R, E, and $\dot{E}$; γ(,) is the incomplete gamma function. Note that the results for D_S = 0 and D_L = ∞ simplify because the content of the parentheses reduces to 1.

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Some of the integrated variables in Table 1 are air-volume specific, such as number density, average size (aloft), kinetic energy, and radar reflectivity. Others are ground specific, such as hit rate, average size (ground), and kinetic energy flux. The spectral density of hailstones reaching the ground per time interval dt is n_g(D) = υ(D)n(D) expressed, for example, in units of per millimeter per meter squared per second. The falling speeds of hailstones υ(D) is the sum of the terminal velocity υ_T(D) that results from an equilibrium between drag and gravity forces and the vertical speed of the air through which the hailstones fall. Here we neglect the latter and approximate the falling speed by the terminal velocity.

If we know the hail PSD and the sizes of the smallest and largest hailstone, D_S and D_L, we can calculate all integrated variables of Table 1. Additionally, we can estimate how many stones of a given size hit an object per time interval during a hail event. This approach goes beyond the integrated variables like radar reflectivity, kinetic energy, or mass per square meter. It lets us calculate the probability that an object of a given size is hit by a damaging hailstone during a hail event. Such objects could be a whole roof of a house, a car, a solar panel, an individual shingle, or a hailpad. The framework can be used to calculate these probabilities and thereby pave the way toward a physically meaningful calibration of damage ratios given the diameter of the largest hailstone.

We can disregard the impact of horizontal wind speed on the total kinetic energy of hailstones here, knowing that horizontal speeds can just be vector added to the vertical velocity to get total speeds and from that total hail kinetic energies of individual hailstones and impact angles.

In the following section we discuss a widely used two-parameter hail PSD that can be used to solve the integrals of Table 1. We show how the results can be used to estimate these parameters from the observation of the diameters of the smallest, average, and largest hailstone (D_S, D_A, and D_L) and the hit rate Hr.

3. Theoretical hailstone size distribution

By following two steps—first inserting a specific hail PSD into the definitions in Table 1, and second approximating the vertical velocity as the terminal velocity—one can integrate these equations. We inserted the exponential distribution to obtain the results shown in Table 1.

The size distribution of hailstones hitting the ground follows a gamma distribution if the hail PSD is an exponential distribution and we approximate the falling speed as the terminal velocity as expressed in Eq. (2) or Eq. (3). The difference in falling speeds of hailstones of different sizes leads to size separation as discussed, for example, in Milbrandt and Yau (2005), Kumjian and Ryzhkov (2012), or Loftus et al. (2014).

To apply the equations of Table 1, one needs to know the two parameters of the exponential distribution (N₀ and λ) as well as the size of the smallest and largest hailstones (D_S and D_L). With this information, one can calculate all the integrated variables and convert them into each other. To the authors’ knowledge, Ulbrich and Atlas (1982) first did this based on hailpad observations, using the size of the largest hailstone observed on hailpads and zero as the size of the smallest hailstone. Other authors parameterized hail size distributions at the ground as exponential functions based on ground observations, for example, Cheng and English (1983) and Cheng et al. (1985), but without applying size limits. Federer and Waldvogel (1975) used hail spectrometer data at the ground to fit an exponential hail PSD but again disregarded the sizes of the smallest and largest hailstones. Other studies fit hail PSDs to in-cloud observations of hailstones to improve the microphysical parameterizations of cloud models (Spahn and Smith 1976; Musil et al. 1991; Peterson et al. 1991; Field et al. 2019). To the authors’ knowledge, our study is the first to fit a hail PSD including lower and upper limits to ground-based observations.

In the next section we discuss how we use ground-based observations of D_S, D_A, and D_L as well as the duration and number of stones observed with hailpads to estimate λ and N₀ as functions of D_L.

4. Fit to observed hail characteristics

Several hail observing networks provide hail data. NOAA’s Storm Data (Schaefer and Edwards 1999), SHAVE (Ortega et al. 2009), the Hail Spatial and Temporal Observing Network Effort (HAILSTONE; Blair et al. 2017), and the European Severe Weather Database (Dotzek et al. 2009) provide the size of the largest locally registered hailstone. For our application, however, we need to know the size of the smallest and average hailstone and the duration of the hail event as well. To our knowledge, only CoCoRaHS (https://www.cocorahs.org) provides all four variables. CoCoRaHS is a volunteer-based network of weather observations in the United States (Doesken and Reges 2011; Reges et al. 2016). Among other equipment, volunteers get hailpads and hailcards, which, if a hail event occurs, they send to the CoCoRaHS headquarters at the Colorado Climate Center at Colorado State University (CSU). There, the hailpads and the information from the hailcards get analyzed, quality controlled, and inserted into a database. Five variables are necessary for the goals of this paper. Those are the duration of the hail event; the maximum diameter of the smallest, average, and largest hailstone; and the number of hailstones on the hailpad. CoCoRaHS kindly provided us with this information for the period from May 1999 until March 2019 (37 726 records). CoCoRaHS rain and snow observations have been used previously within the scientific literature—for example, by Cocks et al. (2017). A comprehensive publication list is on the CoCoRaHS website (https://www.cocorahs.org/Content.aspx?page=publications_data_usage). The hail data were used as ground truth by the CSU-CHILL National Radar Facility to validate their radar-based hail detection (Kennedy and Junyent 2017; Kennedy 2018). NASA uses CoCoRaHS hailpads at Kennedy Space Center together with their hail transducers: first to register hailstones smaller than the cutoff size of the hail transducers (approximately 9 mm), and second to correct observations of the hail transducers for observations above that cutoff size (Lane et al. 2012). For more information on the calibration of hailpads see Long et al. (1980).

Since this paper represents the first use of the CoCoRaHS hail data in the scientific literature, we briefly discuss the data here. The volunteer measures the local duration of a hail event together with an uncertainty. The volunteer also estimates the sizes of the smallest, average, and largest hailstone and provides them as binned sizes. For hailstones up to 1 in., the bins are n/8 of an inch with n = 1, 2, …, 8. Between 1 and 2 in. the accuracy is 0.25 in. Further possible values for entries on the hailcard are 2.5, 3, 4, or 4.5 in. For larger sizes, volunteers can directly provide their observations. Volunteers are encouraged to send photos of their observations. Hailpads are analyzed at the Colorado Climate Center at Colorado State University. The number of hailstone hits on hailpads is counted by student interns. However, in many cases the count of hailstones on the hailpad is missing since either the volunteer had no hailpad installed or reliable counting was not possible. The information we needed was only complete for 4632 of 37 726 cases. After excluding cases with an uncertainty in hail duration of more than 3 min, we were left with 4192 cases. Since only discrete hailstone sizes are reported in the CoCoRaHS data, it can happen that the reported smallest and average hailstone size or average and largest hailstone size could not be distinguished. However, to fit a hailstone size distribution the largest observed hailstone has to be larger and the smallest hailstone has to be smaller than the average hailstone. The first of these conditions reduces the number of cases to 2899; the second reduces it further to 1579 cases. In some cases, hail durations of several hours are registered. Such durations can originate either from observers including accompanying rain into the duration or from the passage of several hail cells. We therefore exclude all observations with hail durations larger than 40 min, which further reduces the number of cases to 1460. We also excluded observations with too few hits. To calculate the hit rate (m⁻² s⁻¹) we need the number of hailpad hits. The hailpads have a size of 12 in. × 12 in., which corresponds to 0.0929 m². Cases with a low number of hailpad hits would introduce a large uncertainty into the estimated hit rate. We therefore exclude all cases in which fewer than five hailstones were registered on the hailpad, leaving 1421 cases.

Figure 1 shows the number of observations, the expected diameter of the smallest and average hailstone and the expected duration and hit rate per equivalent diameter of the largest hailstone for the 1421 cases together with their standard errors (except for the number of observations where a sampling error is not applicable). Best-fit functions are drawn in Fig. 1 as well. They fit the observations very well and can therefore be used to interpolate between observations. The parameters of all best-fit functions of this paper are provided in Table 2. However, while all functions are fit to equivalent diameters, Table 2 shows the parameters for maximum diameters. The number of observations decreases exponentially with the equivalent diameter of the largest hailstone. In this case, the translation from equivalent to maximum diameter changes the function from exponential to Weibull. Expected smallest and average hailstone diameters, duration and hit rate can be modeled well as power functions of the equivalent and the maximum diameter of the largest hailstone.

Some statistical features of the CoCoRaHS data as function of the equivalent diameter of the largest hailstone. Dots are sample averages, lines are best-fit functions, and vertical bars are standard errors. Shown are the (top left) number of observations, (top right) average (blue) and smallest (red) hailstone diameter, (bottom left) duration, and (bottom right) hit rate. Maximum diameters of the largest hailstone are provided at the upper axes.

Citation: Journal of Applied Meteorology and Climatology 58, 10; 10.1175/JAMC-D-18-0334.1

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Some statistical features of the CoCoRaHS data as function of the equivalent diameter of the largest hailstone. Dots are sample averages, lines are best-fit functions, and vertical bars are standard errors. Shown are the (top left) number of observations, (top right) average (blue) and smallest (red) hailstone diameter, (bottom left) duration, and (bottom right) hit rate. Maximum diameters of the largest hailstone are provided at the upper axes.

Citation: Journal of Applied Meteorology and Climatology 58, 10; 10.1175/JAMC-D-18-0334.1

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Some statistical features of the CoCoRaHS data as function of the equivalent diameter of the largest hailstone. Dots are sample averages, lines are best-fit functions, and vertical bars are standard errors. Shown are the (top left) number of observations, (top right) average (blue) and smallest (red) hailstone diameter, (bottom left) duration, and (bottom right) hit rate. Maximum diameters of the largest hailstone are provided at the upper axes.

Citation: Journal of Applied Meteorology and Climatology 58, 10; 10.1175/JAMC-D-18-0334.1

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Table 2.

Some functions of this work. Here, $\widehat{a}$ and $\widehat{b}$ are best estimates of parameter values, σ_a and σ_b are the standard errors of the parameters (multiplicative in the case of a; additive in the case of b), and R² is the variance explained. The D_S, D_A, and D_L are the maximum diameters of the smallest, average, and largest hailstone in millimeters; nObs is the number of observed hailstones on a hailpad; T is the duration of the hail event; Hr is the hit rate; and λ and N₀ are the parameters of the exponential hailstone size distribution aloft. The N, M, Z, and HKE are the number of stones, their mass, radar reflectivity, and hail kinetic energy. Index c refers to the case of constant D_L during the hail event; index q refers to the case of parabolically varying D_L.

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We can now apply Eqs. (6) and (7) in several ways to calculate λ and N₀. First, we can calculate values for λ and N₀ for each of the 1421 observations. Second, we can apply the equations to the averaged observations per largest observed hailstone. The third option is using the best-fit functions expressing D_S, D_A, and the hit rate Hr as function of D_L.

Using a time-dependent diameter of the largest hailstone leads to time-dependent values for λ and N₀. Using Eq. (6) provides the time-dependent values of λ. Since we have no knowledge of the time-dependent hit rate, we cannot use Eq. (7) to calculate a time-dependent N₀. We therefore assumed the relation N₀(λ) = Aλ^b following, for example, Cheng and English (1983), Cheng et al. (1985), Ferrier (1994), and Field et al. (2019). The observed number of hailstones was reproduced well for $\widehat{A}=60$ and $\widehat{b}=3.9$.

Data accuracy affects the results in several ways. For small hailstone sizes, the resolution of the CoCoRaHS data is 1/8 in. (3.175 mm). This means that all samples with a largest hailstone of 3/8 in. (9.375 mm) have an attributed smallest hailstone size of 1/8 in. and an average hailstone size of 2/8 in. (6.35 mm). Therefore, Eq. (6) attributes the same value of λ to all observations with a largest hailstone diameter of 3/8 in. In this case, λ is fully determined by the resolution of the observations and the spread of estimates of N₀ depends solely on the hit rate. For a largest hailstone size of 4/8 and 5/8 in. (12.7 and 15.875 mm), we get three and six possible combinations of the smallest and average hailstone size, respectively. Given the resolution of the observations, as maximum hailstone size increases, so does the number of possible combinations of smallest and average hailstone size, but at the same time the number of observations decreases. We chose a largest hailstone size of at least 6/8 in. (19.05 mm), which leaves 381 observations, which we consider a large enough sample size. It allows for at least 10 combinations of smallest and average hailstone size, which we consider to be enough degrees of freedom.

5. Results and comparison with previous studies

We used ground-based observations of the diameters of the smallest, average, and largest hailstones as well as number of hailstones and duration of hail to fit an exponential distribution for the hail PSD aloft. This section compares the results for the two parameters (λ and N₀) of the exponential distribution for the four different options of this paper to previous estimates. Figure 2 shows our results for the 1421 observations as dots. Gray dots represent the cases where the largest hailstone has a diameter of 3/8 in. Given the resolution of the observations, in this case the smallest hailstone has a diameter of 1/8 in. and the average hailstone has a diameter of 2/8 in. According to Eq. (6) all of these cases lead to the same estimate of λ while N₀ varies over a wide range that is uniquely linked to the different number of hailstones and hail durations observed. All these cases lead to values outside the range calculated by Field et al. (2019) from in-cloud observations indicated as a dash–dotted line in Fig. 2. For a largest hailstone diameter of 4/8 in. three possible values of λ could be realized of which two actually are. Only one of them is in the range calculated by Field et al. (2019). They are depicted in Fig. 2 as yellow dots. For a largest hailstone diameter of 5/8 in., four of the six possible values of λ are realized (green dots in Fig. 2). We exclude all of these cases from further analysis since they are affected by the resolution of the observations. For largest hailstones with a diameter of 6/8 in., 10 combinations of smallest, average, and largest hailstone diameters can be resolved. Five of them are realized by the 111 observations with this largest hailstone diameter. The number of possible combinations grows quickly with the size of the largest hailstone, and estimates of λ become increasingly accurate. We use all 327 cases for which the largest hailstone has a diameter of at least 6/8 in. (red dots in Fig. 2). The range we obtained from the CoCoRaHS data overlaps substantially with the range obtained by Field et al. (2019) from aircraft measurements. However, some of the estimated values of λ from this paper are smaller than those by Field et al. (2019). Field et al. (2019) observe values of λ up to 900 m⁻¹, whereas all of our values are smaller than 450 m⁻¹. The reason for this difference is that, according to Eq. (6), λ is inversely proportional to the average hailstone diameter. If we assume a smallest hailstone of size zero and a largest hailstone of infinite diameter, we would get $\lambda =D_A^{-1}$. Since we exclude small hailstones from our analysis, we cannot observe large values of λ. On the other hand, Field et al. (2019) did not observe hailstones as large as the largest ones observed by CoCoRaHS.

Coefficients λ and N₀ of an exponential hail particle size distribution from various sources. Dots represent fits to individual CoCoRaHS data (gray for largest hailstone diameter D_L = 0.375 in. (9.375 mm), yellow for D_L = 0.5 in. (12.7 mm), green for D_L = 0.625 in. (15.875 mm), and red for D_L = 0.75 in. (19.05 mm) or larger). Orange triangles represent values in the case in which averages for D_S, D_A, and hit rate are used per D_L. Blue squares are results if the best-fit relations of D_S(D_L), D_A(D_L), and Hr(D_L) are used. The thick red line is for the assumption of a constant hit rate. The blue line represents the relation between λ and N₀ that reproduced the observed number of hailstones best in the case of a time-dependent hit rate. The black lines represent results from Cheng and English (1983) and Cheng et al. (1985). The black square, triangle, and diamond depict the results of Douglas (1963), Federer and Waldvogel (1975), and Seino (1980), respectively. The dash-bordered rectangle indicates the range provided by Federer and Waldvogel (1978), and the dot–dash-bordered rectangle depicts the range that Field et al. (2019) deduced from in-cloud observations.

Citation: Journal of Applied Meteorology and Climatology 58, 10; 10.1175/JAMC-D-18-0334.1

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Coefficients λ and N₀ of an exponential hail particle size distribution from various sources. Dots represent fits to individual CoCoRaHS data (gray for largest hailstone diameter D_L = 0.375 in. (9.375 mm), yellow for D_L = 0.5 in. (12.7 mm), green for D_L = 0.625 in. (15.875 mm), and red for D_L = 0.75 in. (19.05 mm) or larger). Orange triangles represent values in the case in which averages for D_S, D_A, and hit rate are used per D_L. Blue squares are results if the best-fit relations of D_S(D_L), D_A(D_L), and Hr(D_L) are used. The thick red line is for the assumption of a constant hit rate. The blue line represents the relation between λ and N₀ that reproduced the observed number of hailstones best in the case of a time-dependent hit rate. The black lines represent results from Cheng and English (1983) and Cheng et al. (1985). The black square, triangle, and diamond depict the results of Douglas (1963), Federer and Waldvogel (1975), and Seino (1980), respectively. The dash-bordered rectangle indicates the range provided by Federer and Waldvogel (1978), and the dot–dash-bordered rectangle depicts the range that Field et al. (2019) deduced from in-cloud observations.

Citation: Journal of Applied Meteorology and Climatology 58, 10; 10.1175/JAMC-D-18-0334.1

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Coefficients λ and N₀ of an exponential hail particle size distribution from various sources. Dots represent fits to individual CoCoRaHS data (gray for largest hailstone diameter D_L = 0.375 in. (9.375 mm), yellow for D_L = 0.5 in. (12.7 mm), green for D_L = 0.625 in. (15.875 mm), and red for D_L = 0.75 in. (19.05 mm) or larger). Orange triangles represent values in the case in which averages for D_S, D_A, and hit rate are used per D_L. Blue squares are results if the best-fit relations of D_S(D_L), D_A(D_L), and Hr(D_L) are used. The thick red line is for the assumption of a constant hit rate. The blue line represents the relation between λ and N₀ that reproduced the observed number of hailstones best in the case of a time-dependent hit rate. The black lines represent results from Cheng and English (1983) and Cheng et al. (1985). The black square, triangle, and diamond depict the results of Douglas (1963), Federer and Waldvogel (1975), and Seino (1980), respectively. The dash-bordered rectangle indicates the range provided by Federer and Waldvogel (1978), and the dot–dash-bordered rectangle depicts the range that Field et al. (2019) deduced from in-cloud observations.

Citation: Journal of Applied Meteorology and Climatology 58, 10; 10.1175/JAMC-D-18-0334.1

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We also calculated pairs of λ and N₀ from sample averages of D_S, D_A, and Hr per diameter of the largest hailstone. They are shown as orange triangles in Fig. 2 for the nine different largest hailstone sizes used. The third option was using the best-fit functions for D_S(D_L), D_A(D_L), and Hr(D_L) instead of the observed averages per largest hailstone diameter. The resulting pairs of λ and N₀ are drawn as blue squares in Fig. 2. Inserting the best-fit functions for D_S, D_A, and Hr into Eqs. (6) and (7) allows us to calculate λ–N₀ pairs for arbitrary values of D_L. They are depicted as a red line in Fig. 2. They fall well among the λ–N₀ relations Cheng and English (1983) and Cheng et al. (1985) found from ground observations, drawn as solid black lines in Fig. 2.

In all of these cases we assumed that the instantaneous values of the largest hailstone diameter and the hit rate are constant during the hail event. As an alternative, we assumed a parabolic function for D_L. The power function for N₀(λ) that reproduced the number of observed stones best is drawn as a blue line in Fig. 2.

Early work in the literature estimated numerical values for the parameters N₀ and λ of the exponential distribution. Three examples (Douglas 1963; Federer and Waldvogel 1975; Seino 1980) are depicted in Fig. 2 as a black square, black triangle and black diamond, respectively. Assuming that all hailstone sizes between zero and ∞ are possible, the integration over the exponential distribution leads to the upper limit of the number of hailstones per cubic meter as N₀/λ. For the three examples these numbers are 10, 28.6, and 20.3, respectively. The maximum number of hailstones larger than 1 cm is N₀/λe^−λ1cm and in all three cases about one every 2 m³. This result follows directly from the assumption of constant values for λ and N₀. A variable hailstone number density is only possible if at least one of the parameters is variable.

Cheng and English (1983) and Cheng et al. (1985) have shown that observations of hail size spectra suggest that both N₀ and λ are variable and not independent. They fit the relation N₀ = Aλ^b with positive constants A and b. This relation reflects that intense hail can either mean many small hailstones or few large ones. Hail events that create thick layers of large stones would contradict this assumption but are not observed. Their three functions are drawn in Fig. 2 as well (black curves) and cover our best fit (red line). For large values of λ our results for the time-dependent hit rate (blue line) are slightly outside the range they cover. They are however, well in the range Field et al. (2019) deduced from in-cloud observations.

We can calculate all aggregated variables of Table 1 with the functions D_S(D_L), D_A(D_L), and Hr(D_L) and the assumption of either a constant D_L or parabolically varying D_L during the hail event. Figure 3 shows how the number of hailstones at the ground, their total mass, radar reflectivity, and total HKE (J m⁻²) depends on the diameter of the largest hailstone. Results for constant D_L are drawn in red. Those for parabolically varying D_L are shown in blue. Dashed lines show the contribution by hailstones with equivalent diameters larger than 2 cm. Those are the ones that can potentially cause damage to cars and buildings. It can clearly be seen from Fig. 3 that while the total number of hailstones is approximately the same in both models, the number of hailstones larger than 2-cm diameter differs considerably. In the case of the parabolically varying D_L the number of hailstones larger than 2 cm never exceeds 169.1 hailstones per square meter, whereas it reaches 741.5 in the case of a constant D_L. The total hail mass at the ground (kg m⁻²) increases considerably more steeply in the case of constant D_L relative to parabolically varying D_L. The radar reflectivity in the case of parabolically varying D_L is time dependent. Figure 3 shows the instantaneous maximum radar reflectivities near the ground. They are above 50 dBZ for maximum hailstone diameters larger than 6 mm in the case of constant D_L. For a largest hailstone diameter of 25 mm the calculated radar reflectivity is 63.2 dBZ in the case of constant D_L and 59.98 dBZ in the case of parabolically varying D_L. A comparison with real radar observations is difficult since radar signals are averages over space, taken at discrete times and for the reasons we already discussed in section 1. In the case of constant D_L, hail kinetic energy reaches 1000 J m⁻² for D_L = 3.9 cm and exceeds 5000 J m⁻² at D_L = 9.2 cm. In the case of parabolically varying D_L, a hail kinetic energy of 1000 J m⁻² is exceeded for D_L = 8.3 cm, 2000 J m⁻² is exceeded for D_L = 12.5 cm, and 5000 J m⁻² is not reached for D_L < 20 cm.

Some aggregated hail characteristics as function of the equivalent diameter of the largest hailstone assuming a constant hail rate (blue lines) and a time-dependent hail rate (red lines). The dashed lines represent the contribution from hailstones with a maximum diameter larger than 2 cm. Shown are (top left) the number of hailstones per square meter, (top right) hail mass per square meter, (bottom left) radar reflectivity, and (bottom right) aggregated hail kinetic energy.

Citation: Journal of Applied Meteorology and Climatology 58, 10; 10.1175/JAMC-D-18-0334.1

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Some aggregated hail characteristics as function of the equivalent diameter of the largest hailstone assuming a constant hail rate (blue lines) and a time-dependent hail rate (red lines). The dashed lines represent the contribution from hailstones with a maximum diameter larger than 2 cm. Shown are (top left) the number of hailstones per square meter, (top right) hail mass per square meter, (bottom left) radar reflectivity, and (bottom right) aggregated hail kinetic energy.

Citation: Journal of Applied Meteorology and Climatology 58, 10; 10.1175/JAMC-D-18-0334.1

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Some aggregated hail characteristics as function of the equivalent diameter of the largest hailstone assuming a constant hail rate (blue lines) and a time-dependent hail rate (red lines). The dashed lines represent the contribution from hailstones with a maximum diameter larger than 2 cm. Shown are (top left) the number of hailstones per square meter, (top right) hail mass per square meter, (bottom left) radar reflectivity, and (bottom right) aggregated hail kinetic energy.

Citation: Journal of Applied Meteorology and Climatology 58, 10; 10.1175/JAMC-D-18-0334.1

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Contrary to radar reflectivity, calculated hail kinetic energy depends on the parameterization of the terminal velocity and hence our results might be highly affected by the parameterization we chose. Therefore, in the following section, we compare the hail kinetic energy calculated here to independent observations.

6. Comparison with independent observations

While hailpads can be used to reliably estimate hail kinetic energy (driven by all stones collected) the observed maximum hailstone size is usually an underestimation due to the small sampling area. Smith and Waldvogel (1989) published a table comparing 55 human-observed maximum hailstone sizes D_L in the vicinity of hailpads with the maximum hailstone sizes observed by hailpads D_Lp. They found the regression line

$\widehat{D_{\mathrm{Lp}}}=5.06\hspace{0.17em}\mathrm{mm}+{\displaystyle \frac{0.493}{\mathrm{mm}}}{D}_L.$(12)

Figure 4 shows the expected maximum hailstone diameter following from the hailpad observations and inverting Eq. (12) and the hail kinetic energy observed by hailpads as black symbols. In this case the best-fit potential function is

$\mathrm{HKE}\left(D_L\right)=0.3241{\displaystyle \frac{\text{J}}{{\mathrm{m}}^2}}{\left({\displaystyle \frac{D_L}{1\hspace{0.17em}\mathrm{mm}}}\right)}^{1.843}.$(13)

This best-fit power function and its 95% confidence interval are shown in Fig. 4 as well. The exponent of the best-fit power function is smaller than 2. This value is far lower than the exponent of 3.54 that Heymsfield et al. (2018) calculated for monodisperse hailstones based on observed terminal velocities. It is even lower than the exponent 2.1 that follows from Pareto-distributed hailstone sizes with the coefficients that Auer (1972) suggested (see section 2).

Observed and modeled hail kinetic energy as function of maximum hailstone size for hailpad observations in nine counties in southwestern France. Observations are shown as black times signs. A best-fit power function to the observations is drawn as a black line. The 95% uncertainty interval is shown as the shaded area. The red line depicts the model results in the case of constant D_L; the blue line is for parabolically varying D_L.

Citation: Journal of Applied Meteorology and Climatology 58, 10; 10.1175/JAMC-D-18-0334.1

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Observed and modeled hail kinetic energy as function of maximum hailstone size for hailpad observations in nine counties in southwestern France. Observations are shown as black times signs. A best-fit power function to the observations is drawn as a black line. The 95% uncertainty interval is shown as the shaded area. The red line depicts the model results in the case of constant D_L; the blue line is for parabolically varying D_L.

Citation: Journal of Applied Meteorology and Climatology 58, 10; 10.1175/JAMC-D-18-0334.1

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Observed and modeled hail kinetic energy as function of maximum hailstone size for hailpad observations in nine counties in southwestern France. Observations are shown as black times signs. A best-fit power function to the observations is drawn as a black line. The 95% uncertainty interval is shown as the shaded area. The red line depicts the model results in the case of constant D_L; the blue line is for parabolically varying D_L.

Citation: Journal of Applied Meteorology and Climatology 58, 10; 10.1175/JAMC-D-18-0334.1

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The calculated hail kinetic energies in the case of constant D_L and parabolically varying D_L are drawn into Fig. 4 for comparison as red and blue lines, respectively. For constant D_L the calculated hail kinetic energy is HKE_c = 1.255 J m⁻² (D_L/1 mm)^1.831 (see Table 2). While the exponent is in line with the observations, the factor is nearly 4 times as large. Hail kinetic energy is overestimated by the model with constant D_L. Using a parabolically varying D_L leads to HKE_c = 0.633 J m⁻² (D_L/1 mm)^1.669. Modeled hail kinetic energies are closer to observations. Nevertheless, for large D_L, the model underestimates hail kinetic energies. Many factors might contribute to the differences between modeled and observed hail kinetic energies. The use of a constant D_L overestimates the fraction of large hailstones and thus overestimates the hail kinetic energy independent of D_L. The one alternative we study in this paper (the parabolically varying D_L) might underestimate the fraction of large hailstones. For D_L just above 5 mm both alternatives need to provide the same results. They are slightly higher than the best fit to the observations of Fraile et al. (2003). Therefore, part of the difference between the modeled and observed hail kinetic energy must originate from other reasons than the assumption of constant or parabolically varying D_L. For example, our assumption of ρ_a = 1.2 kg m⁻³ is representative for sea level, but some of the hailpads in southwestern France are in mountainous regions—and in fact, some of the largest hailstones observed in the dataset published by Fraile et al. (2003) are from mountainous counties.

Other reasons for the differences might be our use of a constant drag coefficient of C_D = 0.55, a constant hailstone density of ρ_h = 910 kg m⁻³, and the use of Eq. (2) instead of the more general form of Eq. (3).

The hail kinetic energy calculated for parabolically varying D_L is well within the 95% uncertainty interval of a power function fit to the observations provided by Fraile et al. (2003). We think this justifies using this model to calculate probabilities for an object being hit by a hailstone larger than an arbitrary threshold size, where the diameter of the largest hailstone is known.

7. Example applications

The model attributes hailstone size distributions to locations where the largest hailstone diameter D_L is known. In this sense, a location is an area much smaller than the spatial variability of hail. This condition is usually fulfilled for automobiles and for individual components of buildings like shingles, tiles, or solar panels. In this case we can assume that a completely random number of hailstones within predefined hailstone size bins hits an individual component. This can be modeled as a Poisson distribution. The rate of the Poisson distribution is the expected number of hits per square meter of stones within the corresponding size bin. For a maximum hailstone size of 50 mm and size bins of 5 mm in width we get the rates provided in Table 3. According to these rates, we expect on average about 3 hailstones larger than 45 mm, 10 hailstones larger than 40 mm, and 23 hailstones larger than 35 mm m⁻².

Table 3.

Expected number of stones (rates) that fall per square meter and on a hailpad of 1/16 m² if the maximum hailstone size is 50 mm.

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Since the sum of two Poisson-distributed random variables is again a Poisson-distributed random variable, and since we assume horizontal homogeneity of hail, we can apply Eq. (15) also to subjects at risk that are not of the size of 1 m². We only need to adjust the rates according to the size of the subject.

As an example, we assume a hailpad of the size of 1/16 m². Then the rates are lower by a factor of 16 than for 1 m² and we can calculate the probability mass function (PMF) for the largest hailpad-observed hailstone diameter given the largest occurring hailstone. For a wide range of D_L the PMF is depicted in Fig. 5. Also shown in Fig. 5 is the expected diameter of the largest hailpad-observed hailstone given the largest hailstone (white line) and a best fit of the form

$\widehat{D_p}=a\left[1-e^{-c\left(D_L-b\right)}\right],$(16)

with the best-fit parameters a = 82.16 mm, b = 1.123 mm, and c = 0.013 19 mm⁻¹. This fit matches the modeled results extremely well (blue dashed line in Fig. 5). We could not find a power function that fit the data that well. Equation (16) converges against an upper threshold for $\widehat{D_p}$ of about 82 mm. This reflects the fact that the larger the largest hailstone of an event is, the fewer of those large stones exist. For comparison, also the linear equation by Smith and Waldvogel (1989) is drawn in Fig. 5. This fit to observations did not include giant hailstones. We expect it to overestimate the expected largest hailpad-observed hailstone diameter in the case of giant hailstones.

Probability mass function of the largest hailpad-observed hailstone diameter as a function of the largest hailstone diameter. The white line is the expected largest hailpad-observed hailstone diameter according to our model. The blue dashed line is a best fit to it. The straight black line is the linear best fit to observations by Smith and Waldvogel (1989).

Citation: Journal of Applied Meteorology and Climatology 58, 10; 10.1175/JAMC-D-18-0334.1

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Probability mass function of the largest hailpad-observed hailstone diameter as a function of the largest hailstone diameter. The white line is the expected largest hailpad-observed hailstone diameter according to our model. The blue dashed line is a best fit to it. The straight black line is the linear best fit to observations by Smith and Waldvogel (1989).

Citation: Journal of Applied Meteorology and Climatology 58, 10; 10.1175/JAMC-D-18-0334.1

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Probability mass function of the largest hailpad-observed hailstone diameter as a function of the largest hailstone diameter. The white line is the expected largest hailpad-observed hailstone diameter according to our model. The blue dashed line is a best fit to it. The straight black line is the linear best fit to observations by Smith and Waldvogel (1989).

Citation: Journal of Applied Meteorology and Climatology 58, 10; 10.1175/JAMC-D-18-0334.1

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We can use the Poisson assumption to get the exceedance probability function EP_hp for the largest hailstone hitting a hailpad as function of the largest hailstone occurring. For the case of D_L = 50 mm it is shown in Table 3.

During the hail event of 6 August 2013 in Germany a hailstone of 141 mm in diameter was collected. To demonstrate how unlikely it is to observe giant hailstones with hailpads, we assume that in the area of homogeneous hailfall in which this hailstone fell, a hailpad of the size of 1/16 m² was installed. Figure 6 provides the exceedance-probability curve for such a hailpad in this event according to our model. In more than 60% of the cases, the maximum hailpad-observed hailstone size is smaller 75 mm.

Exceedance probability function of the largest hailpad-observed hailstone diameter in the case of a largest hailstone diameter of 141 mm.

Citation: Journal of Applied Meteorology and Climatology 58, 10; 10.1175/JAMC-D-18-0334.1

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Exceedance probability function of the largest hailpad-observed hailstone diameter in the case of a largest hailstone diameter of 141 mm.

Citation: Journal of Applied Meteorology and Climatology 58, 10; 10.1175/JAMC-D-18-0334.1

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Exceedance probability function of the largest hailpad-observed hailstone diameter in the case of a largest hailstone diameter of 141 mm.

Citation: Journal of Applied Meteorology and Climatology 58, 10; 10.1175/JAMC-D-18-0334.1

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8. Summary, conclusions, and outlook

The study presents a climatological study of CoCoRaHS hailpad observations and uses these observations to show how aggregated hail intensity characteristics are linked to each other by the hailstone size distribution. Solutions for the widely used exponential hailstone size distribution are shown.

We used the observations of the diameters of the smallest, average, and largest hailstones as well as duration and number density of stones at the ground as observed by CoCoRaHS to show that the largest observed hailstone determines the expectation of the diameters of the smallest and average hailstones and hit rate. Assuming an exponential hailstone size distribution (aloft), this allows us to estimate the parameters N₀ and λ. The estimated values are in line with the results of previous studies.

In a next step two assumptions are compared, a constant diameter of the largest hailstone during the duration of hailfall and a parabolically varying diameter. We then calculated various aggregated measures of hail intensity like total hail mass and hail kinetic energy. While both assumptions can reproduce the total number of hailstones counted at the ground, the assumption of parabolically varying diameter of the largest hailstone leads to a far smaller number of large hailstones as well as lower mass of hail at the ground, lower radar reflectivity and lower hail kinetic energy.

As a validation experiment, we used French observations of maximum hailstone size and according hail kinetic energies as published by Fraile et al. (2003). The data are very noisy because only the results of one hailpad (the one hit by the largest hailstone) per year and per county for nine counties and 15 years are published. The modeled hail kinetic energies in the case of a parabolically varying diameter of the largest hailstone fit the observed HKE very well given the simplicity of the model. The HKE of our model is proportional to $D_L^{1.67}$, whereas the observations by Fraile et al. (2003) suggest $\mathrm{HKE}\propto D_L^{1.84}$ (excluding the five mountainous counties leads to $\mathrm{HKE}\propto D_L^{1.75}$). Although our results are within the 95% confidence interval of the best fit to the observations, part of the difference might be explained by the fact that we used a constant air density that is representative for sea level, whereas the data Fraile et al. (2003) used were taken from hailpads above sea level, and larger hailstones were observed in higher areas. The observations by Fraile et al. (2003) and our results both show a weaker dependency of HKE on D_L than the Pareto distribution that Auer (1972) used, which leads to $HKE\propto D_L^{2.1}$ in the case of ${\upsilon }_T\propto D_L^{1/2}$.

The validation experiment confirms that the model can reproduce hail kinetic energies in the case of a parabolically varying D_L. Other time dependencies might be considered in the future as well. The authors are convinced that measurements of the full hail size spectrum at the ground can improve the understanding of hail risk considerably, and by that lead to better risk models. In particular, the assumption that the first and last hailstones that fall locally are of the size of 5 mm is not justified by observations. However, because we disregard hailstones with a diameter of less than 2 cm, our results seem not to be affected by this assumption.

The use of a hailstone size distribution gives concrete meaning to a maximum hailstone size. We showed some examples of the wealth of applications that the hail model can have. It allows us to calculate the probability that an object is hit by a certain number of hailstones within a certain size range given the size of the object and the maximum hailstone size. We are aware that the horizontal wind speed needs to be taken into account when calculating the hail-effective size of the object at risk. The velocity vector of a hailstone is the vector sum of its vertical velocity and horizontal wind speed. The hail-effective size of a surface at risk follows directly from the scalar product of its normal vector and the velocity vector of the hailstones.

When applying the model, we used the best-fit functions for D_S(D_L), D_A(D_L), and Hr(D_L) and integrated over the duration T(D_L). This means that we attribute one representative hailstone size distribution to a location as function of the largest local hailstone. However, in reality hail cells move with different forward speeds, and therefore the duration of hail is itself a random variable and should be treated as such. This can be included into the hail model. Instead of using the best-fit function and therefore the expected duration as function of D_L, one could sample from the distribution of the CoCoRaHS observations. Apart from the impact of vertical and horizontal wind and the elevation of hailpads, this may explain part of the scatter observed in Fig. 4.