South Africa's topography consists of a large central plateau with extensive grasslands; a continuous escarpment of mountain ranges surrounding the plateau on the west, south, and east; and a narrow strip of low-lying land near the coastline. The central plateau is bounded by the Great Escarpment (see Fig. a), a major topographical feature in Africa consisting of steep slopes that drop from the high central plateau toward the oceans that surround southern Africa on three sides. The eastern part of the Great Escarpment in the border region of South Africa and Lesotho is the Drakensberg Escarpment. It is the highest mountain range in southern Africa, reaching an elevation of 3482 a.m.s.l. (above mean sea level).

South Africa's climate is determined by the subtropical zone of the Southern Hemisphere, the location between the Atlantic and the Indian oceans, and its orographic characteristics. Climate conditions, influenced by the ocean along the east and west coasts and the interior plateaus, vary between subtropical and temperate: subtropical in the northeast, Mediterranean in the southwest, and a warm dry desert environment in the central west and northwest.

Intense thunderstorms are routinely observed by visible and infrared imagery from geostationary satellites for forecasting and warning purposes . In particular in the infrared channel, deep convective cloud tops atop updrafts appear as cold spots growing near to or above the tropopause level, surrounded by a warmer anvil . Cloudy air masses are propelled upwards in the storm's core before rebounding or dissolving again on timescales of a few minutes.

Detection of these OTs hereinafter referred to as OT detections has been automated by , revealing the climatological distribution in North America as well as in Europe and Australia . An advanced version of the OT detection algorithm described by delivers a 3 km gridded probabilistic estimate of OT likelihood based on a statistical combination of tropopause-relative infrared (IR) brightness temperature, prominence of an OT relative to the surrounding anvil, and the area and spatial uniformity of the anvil cloud surrounding an OT candidate region. OTs detected with a probability > 50 % and with a surrounding anvil cloud (green and yellow colors in Fig. b) are used in this work . The method was validated by using OT identifications from gridded weather radar observations and by using human OT identifications over the United States (US). A relation between hail size estimated from radar and OT intensity has been suggested in several studies e.g.,.

In the study, the human analysts identified OTs with two confidence levels, resulting in a conservative mask with only the most confident OTs and a liberal mask that also included less confident OT identifications that did not appear as prominently in the imagery as those in the conservative mask. For the Geostationary Operational Environmental Satellite GOES-16, probability of detection (POD) at an OT probability > 0.5 ranged from 0.51 to 0.95 and false alarm ratio (FAR) ranged from 0.04 to 0.24, with the highest POD for the conservative mask and the lowest FAR for the liberal mask. For GOES-13, POD decreased to 0.8 for the conservative mask but remained nearly the same as GOES-16 for the liberal mask. FAR increased by 0.10 for the liberal mask. In , an OT probability of 0.5 corresponded to a median 20 dBZ precipitation echo top near the tropopause and a FAR ranging from ∼0.1 to 0.5 depending on the reflectivity level used to define the precipitation echo top (e.g., 10 or 20 dBZ), the height of the echo, and the satellite data used as input (e.g., GOES-13 vs. GOES-16). POD based on these comparisons with echo tops ranged from 0.35 to 0.75. In summary, even the most prominent OTs are less evident and harder to detect in the GOES-13 data. Given the reduced prominence from coarser resolution, OT detection algorithm sensitivity settings must allow detection of smaller temperature gradients within anvils, which results in increased FAR.

False OT detections in very cold outflow near to actual OT regions is the most common source of error. Despite these false alarms, which in our opinion are impossible to completely eliminate, the OT detection method improves upon the version used in previous hailstorm climatology studies and will represent the convection climatology across South Africa quite well.

Imagery of the Meteosat Second Generation (MSG) Spinning Enhanced Visible and Infrared Imager (SEVIRI) instrument between January 2005 and December 2018 is scanned for OTs at a temporal resolution of 15 min. Since South Africa is not continuously covered by the high-resolution visible imagery product of MSG, only the IR channel data (given as 10.8 µm brightness temperature) are provided to the detection algorithm. An example is shown in Fig. b, where brightness temperature during a very strong hail event on 28 November 2013 and detected OTs are displayed.

The spatial distribution of OT pixel detections across the entire study domain and 14-year study duration is depicted in Fig. a. Clearly, convective storms are most common in the prevailing moist subtropical climate in east South Africa, along the Great Escarpment, including the southeastern flanks of the Drakensberg and stretching north through the Mpumalanga province (see Fig. a and Sect. ). Here the complex terrain with a height of more than 2300 m a.s.l. induces uplift to serve as a trigger for convection initiation. By contrast, OT frequency decreases towards the west and towards the coast of the Indian Ocean, where the climate is mainly semi-arid to desert.

Compared to , we note the absence of an OT frequency maximum over the country of Lesotho, even in the unfiltered OT data, and higher values to the north and southeast. We attribute these differences to the coarse spatial resolution of the ERA-Interim reanalysis used in the abovementioned study, which is likely insufficient to resolve local orography. The high altitudes in southern Lesotho (mostly 2500–3500 m a.s.l.; see Fig.a) seem to suppress deep convection to some degree, similar to the situation in the interior of the Alps in Europe .

Reports of hail observations including estimates of hail sizes are registered in several continental-scale, centralized databases for North America, Europe, and Australia. These reports are very helpful for validating the severity of the storms with detected OTs. In addition, derived hail size spectra are required as a measure of intensity in hail risk models. For South Africa no such database of comparable extent is available. However, comparing studies that estimated hail size spectra e.g.,, one can conclude that hail size spectra do not tend to vary greatly among regions and even continents. For this reason, we computed hail size spectra for the stochastic modeling component of this work from 26 884 hail reports archived by the European Severe Weather Database ESWD; for the period 2005–2019 and from 3764 reports provided by the Severe Storms Archive of Australia's Bureau of Meteorology for the period 1950–2019. Reporting policies meant that events of a hail diameter of 2 cm or more are covered, but in some cases reports with smaller stones accumulating to thick layers are included. A uniformly distributed random value between -0.5 and +0.5 cm was added to each reported hail diameter to compensate for rounding in the hailstone measurement process and to obtain a smooth distribution.

In addition, 1423 hail damage claims between 1984 and 2017 including a reference to a location were obtained from several insurance companies of South Africa (see Fig. ). Data did not include information on the size of hail, hour of occurrence, or type of asset affected. Claims data tend to be biased towards population centers and – in this case – towards major hail days but still has the advantage of providing direct evidence for hail occurrences.

ERA5 is the fifth-generation reanalysis of the European Centre for Medium-Range Weather Forecasts (ECMWF). It is a global observation-guided model representation of past weather. Data were obtained for the period 2005–2018 at a spatial resolution of 0.25∘ and hourly resolution. While CAPE and the height of the freezing level are provided as output variables, bulk wind difference between the near surface (10 m) and 6 km above the ground (0–6 km wind difference) was computed based on pressure-level data using linear interpolation.

Scattering of surface-emitted microwave radiation by hailstones is an alternative method of hailstorm detection by satellites . The measurement principle has the advantage that in contrast to OTs the signal is directly caused by hailstones but at the cost of spatial and temporal coverage, commonly limited to two overpasses per day, and at the risk of nonuniform beam filling of the field of view. Beam sizes of current-generation sensors e.g., are of the same order of magnitude as convective storm core diameters, meaning that the sensor and storm need to be aligned for successful detection.

In this work, such detections from the Tropical Rainfall Measuring Mission (TRMM) and the Global Precipitation Measurement Mission (GPM) satellites over the South African domain were evaluated for the period 1998–2019. Cases with a hail probability greater than 10 % over South Africa and Lesotho were retained. The threshold here is lower than in the studies cited above (threshold of 20 %) because our intention is to consider also small hail diameters and not just significant hail. Along with the damage claims, they are used to compare hail-prone environments in South Africa to those in Europe and Australia and to constrain OT detections to a certain range of convection-related parameters from ERA5 reanalysis (see next section).

Before clustering the OT detections, a filter is applied based on surrounding atmospheric conditions in terms of wind shear and CAPE obtained from ERA5 for days and locations with microwave hail detections and damage reports (Fig. ). Since insurance data are only available for South Africa (see Fig. ), the filter criteria are also only determined for the territory of South Africa and Lesotho. Note that the purpose of this filter is distinct from other studies aiming to identify hail-prone conditions from reanalysis e.g.,, which tend to suggest much stricter criteria.

The filter design used for Europe and Australia was retained in this work. For the filtering, thresholds for the different ambient parameters were defined based on the OT distribution for insurance loss data and microwave hail detections. The thresholds are then determined from the 2nd and 98th percentiles of the respective distribution functions. OT detections that occur outside that range are then removed to obtain the filtered OT dataset.

Ambient conditions near OT detections are interpolated spatially from the much higher resolved ERA5 reanalysis instead of ERA-Interim (25 km rather than 80 km). In contrast to the latter, ERA5 has hourly rather than 6-hourly fields, so values at the full hour are used for OT or microwave detections in the following 60 min. This reduces false filtering due to model uncertainty to resolve, for example, the rapidly evolving CAPE field with a strong diurnal cycle. For insurance loss data without a time of occurrence, we have chosen 12:00 UTC in ERA5 by default.

The bulk wind difference (near surface to 6 km) and freezing level for both microwave hail detections and insurance claims in the vicinity of OT detections are shown in Fig.  for South Africa and Lesotho. OTs occur at a somewhat lower 0–6 km wind difference and higher freezing level compared to microwave detections, confirming the filter choice in . Note that microwave hail and OT detections occur most frequently at a 0–6 km wind difference between 10 and 20 m s-1, which represents the lower limit for organized convective storms such as multicells, supercells, or mesoscale convective storms (MCSs) to occur e.g.,. Using a higher threshold would exclude a relevant fraction of situations where hail is likely based on the microwave algorithm. Damage reports often occur at 0–6 km wind difference between 20 and 30 m s-1, indicating a bias towards the more organized storms producing more damaging hail. But given the high concentration of these claims in populated regions, we refrain from using a more restrictive threshold based on this data alone. The somewhat odd distributions of damage reports shown in Fig. a and b are due to heavily population-biased sampling locations. As 9.5 % of the OTs but only 3.5 % of the microwave hail detections and 2.5 % of the claims occur at a melting level of fewer than 2400 m, this altitude was used for the lower threshold with this parameter.

OT detections are thus retained if the surroundings of a given OT fulfill minimum conditions of convective instability (CAPE > 100 J kg-1), a 0–6 km wind difference >1.5 m s-1 and a height of the melting level >2400 m (<4845 m a.g.l.). As can be seen in the spatial distribution of the filtered OTs in Fig. b, the filter removes OT detections in particular over the ocean, central Mozambique, and the Lesotho Highlands. The latter feature is due to the minimum freezing-level condition and remains to be confirmed by independent observations, for example, by hailpad or hail sensor data from the region.

A similar pattern of hail events shown in Fig. b is found in the global hail study by Fig. 11 as well as in passive microwave data by Fig. 7; see also the discussion in the Appendix. A final judgment on the actual occurrence of significant hail on the ground would require surface observation data, such as hailpads or sensors covering multiple regions of South Africa. Such direct observation data, however, are not sufficiently available for the entire South Africa.

All subsequent analyses presented in the next sections are based on the filtered OT detections.

Figure illustrates the definition of historic potential events based on observed and filtered OT detections. These events are formed by computing the spatial and temporal distance between individual OT detections. OTs are assigned to the same event if they are separated by fewer than 1 h and fewer than 30 km. This simple approach can detect both single cells and other more organized forms of convection including MCSs or squall lines. Event centroids are defined as the mean latitude and longitude of the event, or grouped, OTs. An event is approximated by an ellipse and characterized by its length, width, and orientation relative to the meridian, as well as the fraction filled with OTs. In addition, we also considered the highest OT–anvil mean temperature difference among the OTs assigned to an event as a criterion for storm severity (see Sect. ). Lifetime and propagation speed are estimated based on the initial and final OT detections within an event. Events can overlap when several storms pass over the same region on a given day. This actually happened in the example on 28 November 2013, shown in Fig. . The three events overlap, but we can assume that the activity of the smaller ones is not related to the main event, since the convection occurred several hours later when the main storm had already moved away and also was much weaker.

The event definition criteria are more restrictive compared to , so the events are better constrained to zones of possible hail activity. Events made up by OT detections only at a single time step, hence lasting for fewer than 30 min, are neglected. In total, 33 820 events were identified from the total of 8 272 509 filtered OT detections for the entire 14-year study period. This means one event on average contains 245 individual OT detections.

Histograms of duration and propagation speed of SCS events – not considered in – are shown in Fig. . The distribution of duration is exponential in shape, in line with radar-based studies . While duration is roughly proportional to length, it becomes clear that propagation speed varies over a wide range. Most frequently, the speed ranges around 30–35 km h-1 (≈8–10 m s-1), slightly lower than the range of 10–30 kn (≈18–55 km h-1) found by . Very high values beyond 100 km h-1 are explained by cases where convection is triggered simultaneously over a larger area and OTs from several storms are unintendedly grouped to an event. Such unrealistic events can be simply excluded in the event set.

Both length l and width w distributions, determined from the observed OT events as described in Sect. , decay rapidly with increasing values (histograms shown in Fig. c and d). In contrast to previous model versions, we choose to approximate both properties with generalized extreme value (GEV) distributions rather than exponential distributions. This improves their overall fit but particularly at the lower end of the range. The good match represents a use case for the GEV function family beyond extreme value theory. Positive shape parameters κ for length, width, and event-to-storm-area ratio (Table ) indicate that the GEV converges to a Fréchet distribution, also known as a Fisher–Tippett Type II distribution. These distributions exhibit what are called heavy tails, meaning that the probability density function decreases rather slowly for large values of x . Consequently, the GEV tends to give too large values, which is why length and width were truncated at 1.5 times the largest observed values at which events effectively cover the entire country (1445 km × 677 km). In addition, low widths are somewhat overrepresented, which can be attributed to the design of the event definition procedure for historic potential hail events.

The fraction f of the event area – the area of the ellipse spanned by major and minor axes of lengths l and w, π/4lw – covered by OT events is also modeled using the GEV (histogram for the logarithm of the effective event area shown in Fig. e). Here we fitted the GEV to the inverse of that fraction, the event-to-storm-area ratio. This function was found to match observations well as the inverse has no upper bound. Furthermore, very OT-sparse events are extremely rare in the event catalogue. For the highest class (>105 km2), the fraction in the stochastic set is significantly higher, while the match is otherwise satisfactory. Table  lists the distributions and parameters for these event properties.

The orientation, or the direction of the major axis of an event, generally aligns with the direction of propagation. We find that most frequently events have an orientation of around 100∘, i.e., propagate eastward to southeastward (Fig. f) most frequently. This, however, applies only for the whole country but not for all regions such as the high hailfall region Gauteng, where storms preferably propagate in northeasterly directions. Stochastic modeling was performed using a von Mises distribution function .

For the sake of completeness, Fig. g shows the distribution of maximum hail diameters dmax modeled with a gamma distribution function. As can be clearly seen, the distribution has an exponential decay for sizes larger than 2 cm. The decrease in hail counts for smaller diameters is only due to reporting practices and does not reflect reality. However, small hail is not relevant to hail damage and risk.

Because event properties vary across the South Africa domain, a box-window average over a 2.5∘ window is used to estimate these variations and to scale event properties from the historic event set. Figure a shows the spatial variation of event length, estimated using maximum likelihood estimation. Here the objective function is the negative logarithm of the product of the probabilities of each sample length given the assumed (GEV) distribution of lengths. The distribution parameters were then varied to find the maximum of the objective function.

Figure b represents the most common orientation, obtained by fitting a von Mises distribution to the regional events on a national scale. Events occurring offshore, where MCSs occur more frequently than over land , tend to run longer, whereas events are shorter in the western part of the domain, hinting towards less organized forms of convection. Orientation varies from southeastward in the southwest to northeastward in the northeast and is most aligned over the Eastern Cape region. found winds from the northwesterly sector to prevail at 850 hPa on convective storm days in northeastern South Africa but from a much smaller sample. The larger spread in orientation towards the north and west can be explained by (i) the prevalence of storm systems affecting larger regions at the same time, hence grouping multiple parallel storms, as well as by (ii) small, quasi-stationary events whose orientation does not reflect an influence of storm propagation.

At present, SCS intensity cannot be estimated accurately from geostationary satellite measurements alone, even if suggest that storms' tornadic intensity is related to the OT area. In a recent study, showed that the difference between the OT temperature and the tropopause temperature and the OT temperature and surrounding anvil in general is larger for significant severe (≥5 cm) hailstorms than storms with smaller hail reports. But, because of the large uncertainty involved in these relationships, hail sizes estimated from the temperature differences are not further considered in our stochastic modeling approach. To at least separate days with strong convection from fewer convective days, we determined a severity indicator based on the temperature difference between the overshooting top and surrounding anvil. This extent can be somehow related to the strength of the updraft supporting hailstone growth . To quantify the hail severity indicator, first the temperature difference between the OT and the surrounding anvil is computed. The severity index is then assigned to the highest such temperature difference of all OTs comprised within an event (see also discussion in the Appendix).

For the estimation of the damage and the risk, the hail model needs as intensity metric the hail size. Because OT data do not allow us to estimate reliable hail sizes (see previous Sect. ), this quantity is only considered and implemented in the stochastic part of the model based on the hail size distributions obtained from severe weather reports. Since hail size observations in South Africa are very rare, we assume that the maximum hail diameters follow the same distribution as large hail diameters recorded in the databases of ESWD and Severe Storms Archive (see Sect. ). Hail sizes for all hailswaths within an event are derived from the attributed largest hail diameter. This means that hailswaths with small hail sizes are more frequent in the stochastic model than the distribution derived from the database of maximum diameters would suggest. That is also the case in reality, as hail with a small diameter is less likely to be recorded in a database than large hail. The hailstone size distribution used in the stochastic model is assumed to be exponential as suggested by various authors using hailpad data . As we do not know the hail spectra for South Africa, this uncertainty remains in the model. However, when the final hail risk model is calibrated using historical loss events, this uncertainty in the result can be reduced to a large degree.

An important feature of the stochastic hail model is that it conserves the relation of different properties of the historic potential hail events. The most relevant properties that are correlated among each other are event length l, event width w, event severity, and fraction f of event area covered by potential hail streaks see also.

The scatterplot of length and width (Fig. a) confirms that correlations between these event properties are conserved in the model. As in , the correlations between length and width to the storm severity indicator – represented by the minimum OT temperature difference for historic events as described in Section – are likewise conserved here. By the same method, the fraction of the event area affected by potential hail events (“effective track area” determined from the ellipse area spanned by length l and width w; see also Sect. ) was also taken into account. This is achieved by first drawing correlated sets of random numbers for each property from a uniform distribution and determined ranks. Then, for each property, we draw values from the actual distribution, sort them, and attribute them to events using the pre-determined ranks .

Figure b shows the relation of the event length l and fraction f area, indicating that in large events, a lower fraction of the area is affected. Table  summarizes the Spearman rank correlation coefficients of the four variables considered. Accordingly, longer hail events tend to be wider, have a higher storm intensity, and have a lower fraction of the event area covered by hail streaks.

Hailstorm frequency shows considerable year-to-year variation in both the annual number of hail events and hail days (Fig. ). Even if there is a strong correlation between all regions shown in the figure (i.e., GAU, HVD, and KZN), smaller regions because of the short temporal and small spatial scales of potential hail events tend to experience relatively higher variability. This information helps to better understand the year-to-year variability of hail hazard and resulting losses. Note, however, that the year 2013/2014 had the second-lowest event count in the Gauteng region, despite the large damage from the 28 November 2013 event. Figure  shows patterns for the first 4 years of the historic event set (left), along with 5 years of the stochastic event set for a visual impression of spatiotemporal variability. Overall, the variations in location and intensity of the annual maximum look realistic. The model has fewer events in the north compared to the observed event count as it is based on the scaled OT frequency instead of events, and events in this region contain fewer OTs. The modeled distribution appears overly smoothed at lower rates, but this is unlikely to be a concern.

Quite relevant in practice is the representation of multiple events occurring on a single day, not considered by . In fact, it turns out that only 15 % of the potential hail event days have just one event, whereas on a few occasions, more than 30 events were detected on a single day. Figure  shows the number of days with a given number of observed events per day for the South Africa domain (KwaZulu-Natal – KZN, Highveld – HVD, and Gauteng – GAU regions as displayed in Fig. ). Naturally, the smaller the study area, the lower the respective counts. Over the Highveld, the events are concentrated on a smaller number of days compared to the similar-sized KZN region.

The panels in Fig.  show the same regional frequency spectra, comparing the historic events to six equivalent subsets of the stochastic set. For example, the Highveld region has 2530 OT events over 813 d. This corresponds to around 180 hail events on 58 d each year. In an equivalent sample of subsets from the stochastic event set, the event count ranges from 2281 to 2942 (2523±232) events on 805 to 893 d (838±32). Table  summarizes the annual OT and model event statistics for the four regions.

The result is satisfactory for the South Africa and Highveld regions, whereas for the KwaZulu-Natal region, events concentrate on slightly fewer days. Given the absence of multiple-event treatment in previous model versions, this new event approach represents an improvement for the estimation of damage on individual days.

In the Highveld region, there were 74 d yr-1 with OT detections, 58 d with OT events, and 60±2 events in the model event set, whereas in the Gauteng region, these numbers were 32 and 23, respectively (Table ).

In contrast, found 69 hail days per year from hail reporting from a network of voluntary observers by mail for a portion of Gauteng (2800 km2). The observers reported three size categories: 3–10, 11–30, and >31 mm; information about the number of days for the different classes, however, is missing. Assuming an exponential hail size distribution, one can assume that the given number of hail days is dominated by events with small hail sizes. These are underrepresented in our model, as they are not relevant for hail damage. This hypothesis is supported by the fact that reported about severe hail (>3 cm in diameter) on an average of only 3.3 d yr-1.

When considering only hail of this size and assuming the hail severity indicator of the stochastic event set would correspond to an actual diameter, the stochastic event set has 14.5 hail days per year in the Gauteng region. Clearly, the frequency of severe events may have changed since the time of Smith's observations, through natural variability or climatic change, but the effect is likely small compared to the uncertainty of both past and present estimates.

When applying the stochastic hail risk model to an insurer's portfolio, going through millions of events for – potentially – millions of assets is a time-consuming process. While the complete event set is optimal for describing the hail hazard, an intermediate step, called importance sampling (IS), is introduced to make risk calculations more efficient, reducing the event count by a factor of approximately 10. However, the most important events in terms of damage potential are overrepresented to allow for adequate statistics (notably, computing damage at higher return periods reliably).

The newly introduced explicit modeling of the date requires that all events occurring on the same day need to be considered together. Consequently, daily aggregated damage potential (here: ellipse area × hail severity indicator) is the relevant quantity to rank event days by overall aggregated severity. The class thresholds correspond to the 50th, 80th, 95th, and 99th percentiles of aggregated severity, splitting the event set into five classes, of which 2.5 %, 7.5 %, 5 %, 10 %, and 100 % were retained. To compensate, the retained events are attributed a higher frequency (default is once per event set period, e.g., 25 000 years), so the total damage potential is conserved. These frequency weights are hence the inverse of the retained fraction per class, i.e., 40, 13.3, 20, 10, and 1 for classes 1 to 5.

The large differences in event frequency across South Africa mean that local statistics on, for example, hail damage or probable maximum loss for a 200-year return period, can rely on a much bigger sample in the hail hotspots of the country compared to the less hail-prone regions in the west. It is hence important to retain a minimum number of events in those low-hail regions, in practice at least one per 250-year batch (if one is present in the first place).

Figure  illustrates this point: Fig. a shows the distribution of events retained in the importance sampling; some areas in the west and far north have fewer than 50 events in a 0.5∘×0.3∘ box (≈30 km in extent). As to be expected, when frequency weights are applied (Fig. b and c), the distribution corresponds very well to that of the full event set (Fig. ).

Finally, the areas affected by hail with corresponding hail sizes need to be determined for the stochastic events, forming a hail “footprint”. Given the arbitrary paths of thunderstorms observed for the historic events, we chose to achieve this by a randomized process of allocating ellipsoid hail streaks within the event area. These streaks are aggregated across all events of a day to form the daily hail footprint, which is applied to portfolios in the further stages of the risk model. The footprint catalog gives local information on hail occurrence and a severity indicator for each day on a 2×2 km2 grid covering continental South Africa.

The footprint generation algorithm attempts to mimic observed patterns of hailfall in an empirical way. A first streak is located in the center of the event, and its length, orientation, and severity indicator match those attributed to the event. Streak widths were chosen to approximate hail streaks in ground- and radar-based studies , without strictly following observed distributions. An exponential distribution is assumed, with a mean width of 6 km and a maximum of 20 km. Hail severity decreases towards the streak's edges in a parabolic way, as proposed by . This is deemed acceptable, since actual hail patterns on the ground are largely uncertain across the world. In an iterative process, further streaks are added until the combined streak area covers the prescribed fraction of the event area. They are located randomly within the event area, and the possible event length decreases with each new streak. Streak orientation is varied by ±10∘ around the event orientation to account for both the uncertainty in the tracking of OTs and new cell formations preferably at the downshear flank.

Accumulating events over a time equivalent of 2500 years, Fig. a and b show the number of events hitting each cell for a 10 % sample of the importance-sampled event set, for the least and most severe class (1 and 5, respectively). Clearly, the footprint frequency over the Highveld and KwaZulu-Natal increases from class 1 to 5, whereas it decreases in the western half of the country. Consequently, hailstorms are relatively more often severe over the Highveld than other parts of the country, with important consequences for the financial risks associated with the peril. Figure c shows the accumulated, frequency-weighed annual sum of hail occurrences in the model. We note that the local frequency-weighed hail count per year is about 2 in the KwaZulu-Natal maximum and around 1 in the Highveld and Gauteng regions, roughly in line with 0.81 normalized hail days per year in . For a maximum hail severity indicator greater than 2 cm, the overall number of events decreases by about 75 %, while the overall spatial distribution remains almost the same (Fig. d).

Based on the local hail count and event set length, return periods, i.e., inverse frequencies, can be estimated for given hail severity thresholds. Hail severity for a fixed return period was calculated by linear interpolation between such thresholds fixed at constant intervals. Summarizing the information contained in the event set, and assuming hail severity estimates were corresponding to actual hail sizes, Fig.  shows the maximum hail severity that can be expected once per decade at a given location: in the most affected parts in the east, near Newcastle and Ladysmith, hailstones of around 4 cm were to be expected, followed by 3.5 cm in Pietermaritzburg, Mthatha, or Nelspruit (officially known as Mbombela); 3.1 cm in Durban; and 2.8 cm in the major cities of Gauteng but fewer than 2 cm in the western third of the country. Despite the uncertainty regarding the exact hail-size–OT relation, this information has clear implications for the need for mitigation measures to reduce hail risk, such as roof-cover robustness or covered parking of vehicles.