The measurement of the physical characteristics of hailstones reaching the ground is usually carried out by means of hailpads, on which the impact of hailstones leaves dents. Hailstone dents provide information about parameters, such as the number N of hailstones, their size M, and their kinetic energy E. In the case of intense hailfalls, however, the dents often overlap and the final measurement may not be totally reliable. This paper presents a computerized simulation with the aim of assessing measurement errors caused by dent overlap. The simulated dents represent several random hailfalls with both exponential size distributions and monodispersed size distributions. The simulated hailpads were measured following the procedure employed in the case of hailpads exposed to authentic hailfalls, and it was thus possible to assess the error due to dent overlap. The results show that dent overlap makes it impossible to measure all the dents, which means that in a real hailfall the number of hailstones registered will often be lower than the number of hailstones that actually hit the ground (up to 25% may go undetected). Consequently, the energy and mass of the hailstones are also underestimated (they may be up to 50% higher than the values registered on a hailpad). The maximum size registered, however, does not depend on the degree of overlapping and neither does the slope parameter λ of the exponential distribution, except when λ takes higher values. Finally, the authors suggest a heuristic correction of the data obtained by real hailpads based on the results of the simulations. An example is provided that applies these corrections to the 228 hailfalls registered by the Italian hailpad network over a period of 10 yr. The results show that, on average, the correction applied because of overlapping increases the number of hailstones in 3.2%, the mass in 1.9%, and the energy in 5.4%. However, there are cases in which these corrections reached much higher values of up to 6.9% in N and M, and up to 25.2% in E. It is therefore advisable to correct dent overlap before carrying out a regional climatic study of hail, since this study would certainly be affected by the errors accumulated by all the hailpads.
Global warming and the subsequent rise in nocturnal temperatures is resulting in an increase in the number of severe storms (Dessens 1995), including hailstorms (Prieto et al. 1999). Hail is one of the hydrometeors that causes the greatest damage to crops (Parker et al. 2005; Sánchez et al. 1996; Segele et al. 2005) and other goods, such as cars and buildings (Hohl et al. 2002a,b).
To estimate the amount of hail that reaches the ground, some type of measurement is required, be it an indirect measurement (inside the cloud with a meteorological radar) or measurements of hailstone parameters on the ground with the help of hailpad networks. This study will focus on the latter.
Hailpads (Long et al. 1980) are simple instruments for measuring hailstone size. They have proved to be extremely useful in studies on hail climatology (Vinet 2001; Palencia et al. 2009). The physical parameters of hailstones are derived from measurements of dents made by the contact of hailstones with the surface of a hailpad, which is made of a deformable material.
With the necessary assumptions (Vento 1976), it is possible to estimate the ice mass that has hit a particular hailpad, the kinetic energy corresponding to the vertical velocity component, the total number of hailstones, and the maximum size registered. This last parameter is linked to the structure of the storm, as pointed out by Cheng et al. (1985).
The main advantage of a hailpad is its simplicity and its low cost, which enables us to install sufficiently dense networks for analyzing the spatial distribution of hailfalls. The main disadvantage of hailpads is the number of assumptions needed to obtain a satisfactory result because this multiplicity often introduces errors in the measurements. The most important assumptions may be summarized as follows: hailstones are assumed to be spherical items; the fall velocity is assumed to be the terminal velocity, depending exclusively on hailstone size and with only a vertical component; and the size of the dent is assumed to be a function of the kinetic energy of the hailstone. In addition, dents tend to have an elliptical shape and only the smaller axis will be used in the calculations (the elliptical shape is supposed to be due to the effect of wind).
Some of these errors may be corrected, at least partially. However, the hail accumulations, which are sometimes observed on the hailpad and may be of several centimeters, cause a measurement error that is difficult to quantify. Recent studies have focused on identifying and quantifying errors in hailpads, as in Palencia et al. (2007), or sampling errors, which will be the focus of future research.
One of the causes of errors in hail measurement is dent overlap on a sensitive hailpad. In the first years of hailpad studies some authors, such as Long et al. (1980) and Rinehart (1983), analyzed a number of possible sources of error. Mezeix and Admirat (1978) also mentioned the saturation of the hailpad surface and the resulting loss of data. However, to our knowledge no other studies have been published as yet on the errors due to dent overlap, even though some authors (Giaiotti et al. 2001) have introduced corrections in the results of their measurements. It must be pointed out that this error will only be noticeable in cases of “overpopulation” of hailstones (large samples). If on one particular hailpad there are only a few dents, then the error will not be relevant. An extreme case may be total overlap: small hailstones hitting the inside of a larger dent, or a large hailstone hitting one or several smaller dents, hiding them completely. In these cases the smaller dents would go unnoticed.
The authors of this study had two main aims when approaching the error in hailstone measurements due to dent overlap: 1) to improve the quality of the data obtained by hailpads in real hailfalls, so that comparative studies on hail climatology may be carried out considering other places and times; and 2) to explain why the hailstone size distribution found does not coincide exactly with the exponential distribution but registers fewer smaller hailstones. Why do we assume that the hailstone size distribution is exponential? The Marshall and Palmer (1948) distribution put forward for raindrop sizes was soon applied to hailstones too (Ulbrich 1974; Federer and Waldvogel 1975; Cheng and English 1983; Cheng et al. 1985). The widespread use of hailpads and data provided by networks of observers (Changnon 1971; Roos 1978) demonstrated that hailstone size distributions were quite similar to the exponential distribution, with a few exceptions in the case of smaller hailstones (Federer and Waldvogel 1978).
It is a well-known fact that one of the causes of the lower number of small hailstones registered when compared with the ones expected by an exponential distribution is the melting of hailstones while falling to the ground (Fraile et al. 2003; Giaiotti and Stel 2006). These missing hailstones are the reason why a number of authors use the exponential distribution only for hailstones larger than 8 or 10 mm (Dessens and Fraile 1994).
Other distributions have been suggested to take into account the deviations with respect to the exponential distribution, for instance, the gamma distribution (Wong et al. 1988; Fraile et al. 1992). However, there is no consensus on the advantages of one over the others (Smith 2003), and it must be admitted that the exponential distribution is the one that is most widely used to compare hailfalls because it is user friendly (Dessens et al. 2001; Fraile et al. 2003). The exponential distribution was therefore chosen in this study as a simple and straightforward way to generate simulations.
The “error due to dent overlap” is a combination of the phenomenon of overlapping and the measurement process. Figure 1 shows a hailpad after its dented surface has been inked black. In the end we will always have an image similar to the one in Fig. 1, which will need interpreting. The measuring process consists of determining the size of each of the dents identified in the image following a pre-established methodology (Palencia et al. 2007). A study on dent overlap, such as this one, must necessarily be related to the process of identifying and measuring dents.
This paper is an approach to dent overlap using a simulation of hailfalls generated using simple hypotheses: the same hypotheses that are used to validate hailpad measurements (spherical hailstones and an exponential size distribution). Then, the results of the measurements carried out on simulated hailpads are compared with those obtained from authentic hailfalls, so that differences can be noticed and corrections suggested that would have to be applied to hail data provided by any hailpad.
2. Material and methods
To test the effect of dent overlap on hailpad measurements, a number of hailfalls were simulated using hailpads of exactly the same size as those used in the hailpad network of León, Spain: a square (30 × 30 cm2) piece of Styrofoam 3 cm thick. The only difference with the hailpad system described in Fraile et al. (1992) was that no section was used for calibration to make the simulation process easier, to increase the sampling area, and to reduce the error due to the edge effect. Thus, the sensitive area is 900 cm2.
The characteristics of the hailfalls had to be decided beforehand. If we assume an exponential hailstone size distribution, then any hailfall may be described using only two parameters: the number N of hailstones and the slope (or rate) parameter λ of the distribution, with the probability density function f(x) = λ exp(−λx). Starting from these data, it is possible to provide a fairly accurate estimation of any other variable, such as ice mass or the kinetic energy due to the vertical velocity component.
As mentioned above, the hailstone size distribution is not exactly exponential because of the loss of the smaller stones due to the melting process while falling to the ground and other causes. Nevertheless, the exponential size distribution will be used as the main hypothesis because 1) it is a simple and straightforward distribution, 2) it fits all hailstone sizes except the smallest ones, and 3) the smallest hailstones are the ones with the smallest influence on dent overlapping on the hailpad, which will be demonstrated throughout this paper.
For the simulated hailfalls, the values of N and λ were selected from the most common ones found in authentic hailfalls, at least in hailpad networks in southern Europe. The relevant literature (Dessens and Fraile 1994; Fraile et al. 1999; Giaiotti et al. 2001) states that hailfalls show little variability in parameter λ and that the value of N varies a lot between hailfalls. In general, the hailfalls recorded in France and Spain show a higher number of hailstones than in Italy. Nevertheless, Lee and Zawadzki (2005) suggest the great variability of these two parameters in the case of hydrometeors.
The distribution of the variables in different hailpads has been analyzed, and the values corresponding to a number of different percentiles were obtained. Table 1 shows the values of N and λ corresponding to the most representative percentiles (multiples of 12.5%). These are the values chosen for the simulations. The percentiles have been determined as follows: the values of N have been represented in ascending order, and the values corresponding to the percentages indicated have been written in columns 2 and 3 in Table 1; next, the values of λ have also been represented in ascending order, and the corresponding values have been included in column 4 in Table 1. That is to say, the characteristic values of N and λ for the eight percentiles selected have been chosen independently.
A first attempt at estimating the effect of dent overlap was made by simulating a number of hailfalls where all the hailstones were of a similar size (monodispersed distribution). The sizes used and the number of simulated hailstones are shown in Table 2. It can be seen that four of the values of N from Table 1 have been taken and that the sizes selected are between 5 and 20 mm, as these are the most frequently registered sizes.
Generating a simulated dented hailpad that may later be measured employing the usual data processing method consists of generating images like the one in Fig. 1: over a black background, a number of dents of different sizes are placed onto different parts of the image. Thus, the images simulated will have the same resolution as scanned images of authentic hailpads. In the method used in León, the resolution is 300 dots per inch (Palencia et al. 2007). A 30 × 30 cm2 area will therefore be represented by a square image with each side consisting of 3500 dots (somewhat less than 12.3 megapixels).
To simulate dents, first the hailstone sizes are organized according to either a monodispersed or an exponential distribution, and then the dents are simulated using experimentally determined relationships between dent sizes and hailstone sizes (calibrations). In other words, the computerized simulations consisted of generating N white circles (dents) of a fixed size (in the case of the monodispersed distribution) or of exponentially distributed sizes (in the case of the exponential distribution). The appendix shows that if the hailstone size distribution is exponential, then so is the dent distribution.
More precisely, to generate these circles, first the sizes of N hailstones had to be generated at random following an exponential or a monodispersed distribution. Once the size of each hailstone is established, we determine the size of the dent it leaves on the hailpad. This is done using a calibration line. Basically, the process used to calibrate hailpads consists of dropping onto a hailpad steel balls of known sizes from a sufficiently high altitude to reach the hailpad with the same kinetic energy as a hailstone of the same size. The relationship between the hailstone sizes and the dents they leave is usually considered to be linear, and it determines the so-called calibration line. Studies on the variability of calibration lines include Vento (1976), Lozowski and Strong (1978), Long et al. (1980), and Palencia et al. (2007).
Calibration lines from a number of hailpad networks in Europe were analyzed, including the ones described in Vento (1976), Giaiotti et al. (2001), and Palencia et al. (2007). The mean of those networks and the network of León was calculated, and the calibration line used in the simulations was
To place the N dents on the simulated hailpad, a system of Cartesian axes was devised coinciding with two sides of the simulated hailpad. The point (x, y) corresponding to the middle of each dent was determined randomly following an independent uniform distribution for both axes.
Once the center and the diameter of each dent were known, white circles were drawn with AutoCAD as shown in Fig. 2. The shading was carried out using CorelDRAW, and the edges were cut so that the images had the required sizes. Figure 3 shows an example of each of the possibilities with the data from Table 2. Similarly, Fig. 4 shows examples of the simulated dented hailpads with exponentially distributed hailstone dents with the data from Table 1.
To determine how often the simulations had to be repeated—that is, how many hailpads had to be hit by one particular hailfall—a study was carried out of the actual situation in real hailfalls in the Mediterranean area. The densest hailpad network—1 hailpad every 4 km2—registered an average of 12.2 hailpads per hailfall in Spain (Fraile et al. 1999). In less dense networks, such as the French network, with 1 hailpad every 48.4 km2 (Dessens and Fraile 1994), the average was 9.9 and 11.7 hailpads hit every day in two different study zones. Thus, it may be estimated that the hailpads that are actually affected by each hailfall may be less than half this figure, as the study zone is considerably vaster. Conversely, the Italian network, which was used as a reference in this paper, has a density of 1 hailpad every 16 km2 (Palencia et al. 2010) and an average of about 7 hailpads per hailfall.
On the grounds of these data, it was considered that in our simulations of hailfalls (with an exponential hailstone size distribution), each hailfall should affect 10 hailpads. The simulations of monodispersed distributions, which will only be used to illustrate the qualitative consequences of overlapping, require only half this number.
So each of the simulations in Fig. 3 was repeated five times and the ones in Fig. 4 were repeated 10 times. In all, 80 simulations were carried out with monodispersed distributions crossing the values of N and D, and another 640 simulations were carried out with exponential distributions crossing the values of N and λ. Summing up, the simulations involved 215 920 hailstones hitting 720 hailpads, and the sizes of the corresponding dents were measured.
Image-Pro Plus was used to identify dents in the images, as it is the software most often employed in processing hailpads (Dessens et al. 2001; Palencia et al. 2007). This program identifies automatically the white circles, counts them, and measures their size. As in the case of a real hailfall, what is measured is the shortest axis of each dent, which the program equals to an ellipse. When the technician detects two or more dents that partly cover each other (i.e., with a certain degree of overlap), the program makes it possible to divide that surface into two. Thus, even though the dents are detected automatically, the technician needs to determine whether the identification was adequately carried out. The technician may modify the selection of the dents to be included in the database. However, a small error is generated in measuring each dent because it is not possible to measure the intersection zone twice. Details about the tasks carried out by the software automatically and the tasks that must be carried out manually by the technician are explained in depth in Palencia et al. (2007).
Hailstone dents are measured by considering the short axis of the ellipse, because even if a hailstone were perfectly spherical, it would still generate an elliptical dent when the hailstone velocity has a horizontal component due to the wind. The short axis is a function of hailstone size, whereas the long axis is a function of wind intensity (Vento 1976). This implies that hailstone size and mass may be accurately calculated from the short axis of the dent. Vento (1978) includes further details on the underestimation of the kinetic energy calculated only with the terminal velocity (vertical).
The measurement process is identical for the simulations and for real hailpads that are scanned after a hailfall: the size of each dent has to be established, as well as the size of the hailstone that caused it. If the sizes are known, then the parameter λ of the hailstone size distribution can be calculated (supposedly an exponential distribution) using the moments method (MM), by establishing the first moment of the distribution (Fraile and García-Ortega 2005). For exponential distributions, the MM and the method of maximum likelihood (MML) coincide (Sneyers 1990), and the MML gives unbiased, though skewed, estimates of the parameter. To avoid the bias in the estimation of other moments of the hailstone size distribution (Smith and Kliche 2005), we have calculated the ice mass and the kinetic energy due to the vertical velocity component.
3. Measured versus simulated hailstone sizes
First, the results of the measurements obtained from the simulations with monodispersed distributions will be presented. It was found that dent overlap causes changes in the dent sizes measured.
It is worth noting the total number of dents measured and the difference with the number of dents actually generated. Figure 5 represents the values of measured N versus simulated ones, the latter being the starting point for each sample. The two numbers do not coincide. In fact, the number of dents found was always lower or equal to the number of generated dents. Henceforth, the subindex m will be used to refer to the variables measured on the hailpad after the simulation (e.g., the hailstone size, mass, or energy), and the subindex s will be used to refer to the sample variables, which are the ones corresponding to the simulated hailstones whose dents are circles randomly drawn onto the hailpad in the simulation using the software described above. Using this notation, it may be said that the Kruskal–Wallis test applied onto the samples in Fig. 5 indicates that Nm and Ns are significantly different for a significance level of α = 0.05.
This disparity between the number Nm of dents measured and the number Ns of simulated balls is found in real hailfalls too and is exclusively due to dent overlap. This simulation with dents of the same size was carried out to make sure that no dents went unnoticed when placing a small hailstone inside a larger one. As a general principle, it may be stated that dent overlap in a hailpad generates a loss in the number of dents measured on it; that is, there will nearly always be more hailstones than the ones we are able to detect and measure. This loss in the number of hailstones depends on the size (the larger the hailstones, the more overlap there is) and on the total number (the more hailstones, the more overlap there is). This difference is quantified later for the case of real hailfalls.
The same dent measurements were carried out in the case of simulated hailfalls with hailstone sizes following an exponential distribution. The number of hailstones was also calculated. With all the data measured on one hailpad, the total number of impacts was determined, as well as the parameter λ of the exponential size distribution. In addition, using the same values as in Fraile et al. (1992) for the physical parameters required, the mass (M) and energy (E) of each hailstone was also calculated. Figure 6 shows the comparison between the values measured on each hailpad (Nm, λm, Mm, and Em) and the sample values (simulated) of Ns, λs, Ms, Es, and the maximum hailstone size (the hailstone size corresponding to the largest simulated dent), which is another commonly used variable in studies on hail climatology (Dessens et al. 2007; Palencia et al. 2010).
Figure 6 shows that the variables Nm, Mm, and Em measured on a hailpad after a hailfall (or in this case, after a simulated hailfall) always present a value lower to the real one because of dent overlap. To verify this result, the Kruskal–Wallis test was applied again, grouping the samples into quartiles. It was found that the masses are significantly different for M > 0.9 kg m−2, the energies for E > 68 J m−2, and the parameter λ of the exponential distribution for values λ > 0.57 mm−1. The number of hailstones N is always significantly different. Parameter λ of the exponential distribution behaves in a different way: its value cannot be expected to be higher or lower when there is overlap, even though, in general, in the case of the higher values, this parameter is lower than simulated. The maximum size is only relatively dependent on dent overlap. In fact, maximum size is less dependent on overlap because it is less likely that the largest dent on a pad will be overlapped by several smaller stones hitting the same area. The authors believe that the maximum size measured depends more on other factors, such as the sampling (Bardsley 1990; Smith and Waldvogel 1989), and future studies will deal with this issue.
The relative error is also different: it is not very high in λ, but it is in the case of the other parameters. In the cases of N, M, and E, the error is directly related to the overlap. A high value of N means that there is a higher likelihood that the dents overlap. The same happens with high values of ice mass or energy: either there is a large number of hailstones or the hailstones precipitating are large. In both cases the dent overlap is considerable.
Figure 6 illustrates that, depending on the tolerance required for our data, it is not always possible to ignore the error because of dent overlap, as it may be substantial in some cases. Nearly 20% of the simulated hailstones may go undetected, for example. The energy measured on a hailpad may represent only 63% of the energy produced by the sample of hailstones that hit a particular hailpad, be it real or simulated. In other words, the energy of a particular sample may be 1.6 times the energy measured on a hailpad.
With respect to the relationship between the error estimated by means of simulations and the error in measuring dented hailpads, it may be said that dent overlap is assumed to be common to both simulated and real hailpads, and the error is likely to be of the same magnitude. However, the complications caused by the shape of dents in authentic hailfalls (hailstones are not perfectly spherical) make the identification of each dent more difficult. Thus, in real hailfalls the error may be considerably higher than the one estimated in the simulation. Moreover, in the case of real hail, elliptical dents are typical and thus the hail area on the hailpad is larger than for simulated dents, because the short axis is the same for both real and simulated dents.
A few additional aspects of parameter λ will be described here. On the one hand, even though it was mentioned above that the difference between the measured value and the simulated value does not depend greatly on dent overlap, a certain trend may be observed: it was noted that when λ is high (e.g., with values of more than 0.5 mm−1), the average value tends to be somewhat lower than simulated (i.e., lower than the value used to generate the dents). This means that when there is a hailfall with an exponential size distribution with a high λ value (i.e., many small hailstones and few large ones), the size distribution measured on the hailpad tends to present a higher proportion between small and large hailstones. One reason for this may be that fewer small hailstones are registered than the ones that actually fall (or are simulated). There are always fewer small hailstones measured than simulated, either because small hailstones can go unnoticed inside a bigger dent, or because when small hailstones overlap they produce a bigger dent. In the case of high lambda values, this is more evident: the formation of bigger dents by overlapping is more significant when the number of big hailstones is low. This is why the value of lambda decreases. This difference between the measured and simulated lambda is higher when the number of hailstones is low. In fact, the dots that lie farther away from the line in Fig. 6 correspond mainly to the lowest values of N.
On the other hand, λ varies somewhat because of dent overlap. This parameter is very important from a statistical point of view, since two different distributions of hailstones may be compared on the basis of the value of λ (Dessens et al. 2001; Fraile et al. 2009). Thus, dent overlap should be taken into account. For example, for the simulations carried out with each of the values of λs, the values of λm found varied in an interval of approximately 7% with respect to the value of λs. Thus, when comparing two or more values of λ measured on different hailpads, it must be taken into account that if they differ in 7%, then this difference may be due to dent overlap only.
Once it has been proved that dent overlap on a hailpad leads to error in the results of the measurement, the following question arises: With the data from dents measured on a hailpad (with dent overlap), is it possible to determine the characteristics of the hailstones that caused those dents?
4. Overlap correction model
a. Setting up the model
From now on we will distinguish between real data (from hailstones that have hit a hailpad) and measured data (calculated from the measurements carried out on that hailpad). For instance, the real size of a hailstone will refer to its actual physical dimensions, whereas from the dent it leaves on a hailpad we may calculate its measured size.
The question posed in the previous section is solved by determining the error due to dent overlap. This error, the difference between the data from the real hailstones and the data from the measurement on the hailpad (affected by dent overlap) is impossible to determine because the information from the real hailstones is never known. Therefore, the only way to estimate the error is to use the simulation data and establish a model that describes the difference between simulated and measured data. The overlap correction model described in this paper may be considered a first step in that direction.
From the measurement of the dents left on a dented hailpad, we will calculate the number of hailstones that have hit the hailpad and their characteristics (size, mass, and energy). The data derived from direct measurement will be subindexed with m, and the results of the correction model will be subindexed with c (corrected). For example, Nm dents will be measured on the hailpad, but the model will estimate that Nc hailstones have fallen.
The process may be described as follows: Nm hailstone dents were measured on a hailpad; the sizes of the hailstones that produced these dents are calculated using a calibration line, and it has been observed that they follow an exponential distribution of parameter λm and that the total kinetic energy is Em. How many hailstones Nc have hit the hailpad? What was the λc parameter of the exponential distribution of those hailstones? What was the kinetic energy Ec of these hailstones that hit the hailpad?
All these questions will be addressed by trying to find simple heuristic relationships. The corrected parameters will be calculated from the variables measured on a hailpad, especially variables Nm, λm, and the variables that need to be corrected (mass or energy). For the relationship between Nc and Nm, the simplest relationships were tested, that is, the relationships that require the lowest number of parameters. A linear relationship would require only two parameters, but the goodness of fit is not particularly high. Another option requiring only two parameters is to build a third-degree polynomial (initially with four parameters) of the type
and then eliminate two parameters as follows: if no hailstone is measured (Nm = 0), then it is obvious that Nc = 0, so D = 0. In addition, if we consider that when there are few hailstones there is no error due to overlap, this means that in those conditions (Nc − Nm → 0),
thus, the relationship between Nc and Nm is
However, there is a difference between the value of N calculated this way and the original value, and this difference seems to be proportional to a power of Nm/λm; thus, it was decided to introduce this new addend—affected by a coefficient—in the previous equation. At the same time, it was settled that the exponents of each variable should be given a better fit, disregarding the condition of being whole numbers. Finally, the expression put forward was of the type
Conversely, a relationship between λc and λm similar to the previous one and also adequate is
In this relationship the change of the exponents of each variable was also tested, but the fit did not improve much when compared with the exponents that equaled 1.
For energy, it was found that the values of Em ≤ 1 J for each hailpad of 900 cm2 (i.e., Em ≤ 11.1 J m−2) are very similar to the values of Ec; thus, no correction is suggested for these values. For higher values an expression was tested that is analogous to the one for the number of hailstones:
With this expression the error in energy lies under 10% in 95% of the hailpads, and if the whole sampling is taken, then the error will always lie under 15%.
Finally, we will present the correction suggested for ice mass. If on a hailpad we measure a mass Mm, then the total ice mass fallen on that hailpad is
It can be noted in Eq. (4) that we have not introduced Nm or λm because the fit does not improve substantially. A simpler relationship was preferred without losing too much accuracy.
Using the information provided by the simulations (Fig. 6), the most adequate parameters for Eqs. (1)–(4) were selected using the least squares method. It was found that the corrections of Nc, λc, Ec, and Mc are explained by the following equations:
where N is expressed in number of hailstones per square meter, λ is in inverse millimeters, E is in joules per meter squared, and M is in kilograms per meter squared.
To apply Eqs. (5)–(8), it must be taken into account that they have been calculated for a particular range of values of the four parameters. The authors have extrapolated these relationships to the simulated hailfalls and have observed that the trend of the corrections stays within the expected limits. However, they recommend their use only for the approximate values included in Table 3. These are merely recommended values because they are the ones that have been used for the simulation; however, in the case of λ, higher values may be used too (except if N is extremely high), since the effect of dent overlap is smaller if the sizes of the hailstones are small too.
As an example, Fig. 7 includes the measured and the corrected values of N, together with Eq. (5) for values of λ of 1 and 0.29 mm−1. It can be seen that the difference between the measured and the corrected values of N increases with the value of N and decreases as λ increases.
The fact that the degree of dent overlap depends on λ and N may be interpreted with the help of Fig. 4. With a low value of lambda, such as 0.3, which represents a broad distribution and more large hailstones, it is obvious that a larger number of corrections will be needed (more small dents overlapped), whereas a lambda of 1 with few large stones implies that a much smaller correction is needed because there is much less overlap. The bottom row of Fig. 4 illustrates this quite well.
Equations (5), (7), and (8) are represented so that the correction due to dent overlap may be calculated directly: it is Nc − Nm, Ec − Em, and Mc − Mm, which corresponds to the second half of each equation, except the values of Nm, Em, and Mm, respectively.
Even though the simulations were carried out on a hailpad of 900 cm2, the results do not depend on the size of the hailpad. This is because the influence of dent overlap depends only on the number N of hailstones per surface unit and on λ, and none of the parameters is dependent on the size of the sample. The relationships in Eqs. (5)–(8) are therefore valid for any sampling surface and for any hailpad size. However, to verify this fact experimentally, new simulations were carried out with a different hailpad size (30 cm × 42 cm) that is commonly used in the Italian network (Giaiotti et al. 2001). The results behave exactly as reflected in Fig. 6.
b. Applying the model
Correction Eqs. (5)–(8) were applied to the hailpads from the Italian network. Only hailpads hit by more than 10 hailstones were considered for hailfalls between 1988 and 1998. In all, there are slightly more than 2000 of these hailpads from 228 hailfalls. A climatic study of this period is summarized in Fig. 8, including the frequencies of the variables N, λ, M, and E in those years. This result corresponds to what has been previously called Nm, λm, Mm, and Em. It was observed that, in general, the hailpads registered a large number of hailstones but not many large hailstones, resulting in energies that were not particularly high. Furthermore, the highest mass and energy categories represent a very small fraction of the total number of hailpads.
Applying the new overlap corrections put forward here, the new frequencies (corrected frequencies fc) were found to be different from those in Fig. 8 (observed or measured frequencies fm). The relative differences in the frequency have been studied, and it was found that in the case of those hailpads that were hit by many hailstones, the relative difference in the frequency in N may be more than 20%. The relative difference in the frequency in λ is not so high but still important, as the value of λ is restricted to smaller ranges. As for the relative differences in the frequency in mass and energy, in general they are higher than in λ and vary greatly when one exceptionally large hailstone appears, even if there are very few of these hailstones. This high relative difference in the frequency in energy ties in with the information provided in Fig. 6.
This global difference between the corrected values and the measured values for each variable is due to the error in each hailpad. Figure 9 summarizes these errors by showing the frequency of various categories of relative differences in the value (as percentages) for each hail variable. For instance, the third panel in Fig. 9 includes few errors in M larger than 9% (four hailpads). These four hailpads register ice masses of more than 0.6 kg m−2; that is, in the most severe hailfalls, the difference between the measured ice mass and the ice mass corrected following the model is higher than in the case of hailfalls with less ice mass. However, severe hailfalls are less frequent than less severe ones. A similar interpretation applies to energy.
Figure 9 shows that, in general, the relative differences in the values of the variables vary up to 8% or even 11% (except in the case of energies, where it may exceed even 25%, though on rare occasions); however, they lead to considerable relative error when carrying out a climatic study including all the hailpads.
The results of applying the overlap correction model presented here to the Italian hailpad network are shown in Table 4. Equations (5)–(8) were applied to each of the Italian hailpads. For each year, Table 4 includes the number of hailfalls on which the overlap correction model has been applied, the value of the average percentage of the correction, the standard deviation, and the maximum and minimum corrections found when applying the model to three variables: the number of hailstones, energy, and mass.
It was observed that, on average, the overlap correction percentages increase the value of the number of hailstones in 3.2%, the value of mass in 1.9%, and the value of energy in 5.4%. However, there have been corrections of up to 6.9% for the values of N and M, and of up to 25.2% for E. These corrections are not as high as the ones obtained for individual hailpads because the corrections in a hailfall are smoothed down by a larger sample of hailstones.
Dent overlap on hailpads is a source of error in determining the characteristics of hail. This error increases, in general, with the number and size of the hailstones registered on the ground. In nonsevere hailfalls this error can be ignored because there is hardly any overlap at all. In contrast, in severe hailfalls (more interesting from a scientific and economic perspective), more hailstones fall than the hailpad is capable of registering and dent overlap occurs. In these cases, the actual mass and energy produced exceed the values measured from the hailpad when dent overlap is ignored. The slope or rate parameter of the exponential distribution is also affected; however, less than the extensive variables, the ones that depend on N: N itself, M, and E (because the extensive variables are those that vary linearly with the size of the sample). This is a trivial consequence of the fact that the higher the number of hailstones hitting a hailpad, the more overlap there is. However, the form of the distribution of the hailstones detected from the overlapped dents is approximately the same as the distribution of the hailstones in the hailfall.
As a consequence, there is a need to modify the data provided by hailpads. The results show that dent overlap makes it impossible to measure all the dents, which means that in a real hailfall, the number of hailstones registered will often be lower than the number of hailstones that have actually hit the ground (up to 25% may go undetected). Consequently, the energy and mass of the hailstones are also underestimated (they may be up to 50% higher than the values registered in a hailpad). However, the maximum size registered does not depend on the degree of overlapping, and neither does the slope parameter λ of the exponential distribution, except when λ takes higher values.
Several hailfalls were simulated, and simple heuristic corrections were developed to reduce the error in the values of N, λ, M, and E from the data provided by the hailpad. Even though the errors of single hailpads may be relatively small (except for kinetic energy), studies on hail, including climatic studies, should only be carried out after applying dent overlap corrections, since this problem has been shown to lead to significant errors in hailfall variables. The application of the overlap correction model onto 228 real hailfalls registered over a period of 10 yr by the Italian hailpad network has shown that, on average, the overlap correction percentages in a hailfall increase the value of the number of hailstones in 3.2%, the value of mass in 1.9%, and the value of energy in 5.4%; however, occasionally there have been corrections of up to 6.9% for the values of N and M, and up to 25.2% for E.
The authors are grateful to Dr. Noelia Ramón for translating the paper into English. Dr. Laura López contributed to this paper by offering interesting suggestions. We thank all the people who have cooperated in the hailpad network in León, Spain. We also thank all the volunteers who contribute to the Italian hailpad network maintenance. We are in debt to Elena Gianesini for her work on hailpad data reduction and to the three anonymous reviewers for their helpful suggestions. This study was supported by the Junta de Castilla y Leon (Grants LE014A07 and LE039A10-2) and by the Spanish Ministerio de Ciencia e Innovación (Grants TEC2007-63216 and TEC2010-19241-C02-01).
On the Exponential Distribution of the Dents
This appendix aims at answering the following question: If the size of the hailstones that hit a particular hailpad follows an exponential distribution of parameter λ, do the dents left on the hailpad also follow an exponential distribution?
Let us consider that the calibration relationship is a line; that is, between the size h of the dent and D, the size of the hailstone, the following relation exists:
If D is distributed exponentially, D − X will also be (X is any constant figure) (Fraile and García-Ortega 2005); thus, the quantity Ah is distributed exponentially with the same λ parameter as D. The expected value of Ah is
which is an exponential distribution whose parameter takes the value λA. As a result, the dents h follow an exponential distribution with that parameter value. Last, it results that