1 Introduction

Active sinkholes cause risk situations worldwide with important economic and societal implications. Gutiérrez (2016) listed some of the most damaging sinkhole events reported in the literature, occurred in countries such as Spain, South Africa, Kuwait, United States, China, or Russia. As illustrated by these examples, sinkholes can cause economic losses amounting millions of euros, the evacuation of population, the demolition of residential and industrial buildings, service interruptions in major transportation infrastructure, and even fatalities. Population and economic growth have resulted in an increase of the vulnerable elements exposed to natural damaging processes. Moreover, very often sinkhole activity is triggered or accelerated by human activities that modify karst environments (Cooper 2002; Beck 2012; Song et al. 2012; Parise et al. 2015; Parise 2019; De Waele and Gutierrez 2022). Therefore, there is an increasing need for developing sinkhole hazard assessments, but the underlying concepts and methodologies have been scarcely studied, in contrast to other ground-instability hazardous processes such as landslides.

Waltham et al. (2005) defined sinkhole hazard as the probability of a sinkhole of a particular magnitude occurring in a given area and within a specific period of time. That is, the spatiotemporal probability of occurrence of an event with a specific magnitude (e.g., Cruden and Fell 2018). Nevertheless, there are multiple approaches for conducting hazard assessments. The vast majority consider two elements: (1) the size of the event; and (2) its frequency (or probability of occurrence), separately or in combination. For example, there are methods for hazard estimation that take into account the frequency of occurrence of events regardless of their size (e.g., Remondo et al. 2005), some consider the annual increment of the area affected by the process (e.g., Hufschmidt and Crozier 2008), others estimate an average size for the process and apply theoretical probability functions to forecast its size distribution (e.g., Tolmachev and Leonenko 2011), or evaluate the frequency-size relationships (e.g., Galve et al. 2011; Taheri et al. 2015). In this regard, frequency-size relationships are commonly applied in hazard studies to model spatiotemporal distribution functions that describe the empirical data and serve to establish future projections for risk management (Malamud 2004; Corral and González 2019; Bernatek-Jakiel et al. 2019). They were firstly applied by Gutenberg and Ritcher (1954) to describe the empirical relation between the magnitude of an earthquake and its cumulative frequency. Afterwards, they have continued to be used to further investigate earthquakes (e.g., Pacheco et al. 1992; Coppersmith et al. 2009; Bayrak et al. 2015) and other hazardous geological processes such as landslides (e.g., Dai and Lee 2001; Guzzetti et al. 2002; Malamud et al. 2004; Remondo et al. 2005; Casas et al. 2016; Corominas et al. 2018), floods (e.g., Machado et al. 2015; Cloete et al. 2018; Chiverrell et al. 2019), piping collapses (Bernatek-Jakiel et al. 2019), and sinkholes (Galve et al. 2009, 2011; Tolmachev and Leonenko 2011; Taheri et al. 2015; Gutiérrez and Lizaga 2016).

Despite the growing amount of research addressing different aspects of sinkholes (see review in Gutiérrez et al. 2014), there is a notable lack of quantitative hazard assessments. This scarcity is generally caused by: (1) the incompleteness of sinkhole inventories; (2) the common low rate of sinkhole occurrence and thus the limited chronological data; and (3) the lack of data on the sinkhole size at the time of occurrence. In the Ebro Valley, Spain, Galve et al., (2009; 2011) presented quantitative sinkhole hazard models including frequency-size relationships, but based on size data with limited accuracy. In Hamedan, Iran, Taheri et al. (2015) produced frequency-size relationships, but based on a limited number of events.

Sinkhole hazard analyses are often based on incomplete sinkhole inventories providing minimum estimates (Gutiérrez 2016). Moreover, in most cases it is implicitly considered that sinkhole hazard remains constant over time, which is an uncertain assumption that may lead to significant hazard under- or over-estimates. Cruden and Hu (1993) introduced for the first time a non-steady statistical approach to model the temporal distribution of rock slides, called exhaustion model, whereby, given a limited number of potential sliding sites, the probability of occurrence would decrease after slope movement and landform stabilization. Subsequent works on hazard assessment evaluated the potential variability of frequency and size of hazardous processes related to the geomorphic setting (e.g., Hufschmidt and Crozier 2008), geological constraints (e.g., Corominas et al. 2018), or external factors such as urban development, land-use variation, or climate change (e.g., van Westen et al. 2006; Bonachea et al. 2009; Crozier 2010; Kundzewicz et al. 2017; Hounkpè et al. 2019; Bates et al. 2021). The changes in hazardous systems have crucial consequences on the accuracy and reliability of hazard and risk predictions and could reduce the effectiveness of mitigation measures.

To our knowledge, sinkhole hazard assessments published in the literature have never evaluated the temporal variability of frequency-size relationships. Nevertheless, the highly active salt karst of the Dead Sea, and the technical possibility of obtaining data on sinkhole occurrence with high temporal and spatial resolution, offer an exceptional opportunity to comprehensively assess hazard and its temporal evolution. Here, thousands of human-induced sinkholes have formed since 1980s in parallel with a persistent lake level decline mainly because of the use of water from its catchment and the lake itself (Abelson et al. 2006). The extraordinary high rate of sinkhole occurrence allows assessing, within a relatively short investigation period, sinkhole hazard from a dynamic perspective. The analysis, based on complete sinkhole inventories including chronological and size data of each sinkholes, illustrates the temporal variability of sinkhole hazard. Despite sinkholes occurring on the western and eastern coast of the Dead Sea have been extensively studied in recent years (Frumkin and Raz 2001; Baer et al. 2002, 2018; Abelson et al. 2006; Closson and Karaki 2009; Ezersky et al. 2009; Filin et al. 2011; Avni et al. 2016; Yechieli et al. 2016; Al-Halbouni et al. 2017, 2019, 2021; Yizhaq et al. 2017; Nof et al. 2019; Watson et al. 2019; Arav et al. 2020; Ezersky and Frumkin 2020, 2021; Sevil and Gutiérrez 2023) and constitute a highly damaging process, no quantitative hazard analyses have been conducted so far.

2 Geological setting

The Dead Sea is a hyper-saline endorheic lake located at the lowest elevation on Earth (~ 437 m below sea level in 2021 according to the geodetic data gathered by Sevil and Gutiérrez 2023). It lies at the bottom of the Dead Sea pull-apart basin, associated with the left-stepping Arava and Jericho strike-slip faults. These NNE-trending structures occur in the southern sector of the left-lateral Dead Sea transform fault system, between the Arabian plate and the Sinai sub-plate (Garfunkel and Ben-Avraham 1996) (Fig. 1a). The study area, covering 2 km2, is located on the western coast of the Dead Sea, in the piedmont of the Western Boundary Fault escarpment, extending between the Nahal Hazezon alluvial fan and 100 m north of the Ein Kedem North thermal spring. The Western Boundary Fault juxtaposes Holocene Ze’elim lake deposits in the downthrown block against the Late Pleistocene Samra and Lisan lacustrine formations, deposited in much larger paleolakes. The underlying bedrock consists of carbonate units of the Judean Mountains (Mor and Burg 2000; Neugebauer et al. 2014; Coianiz et al. 2019, 2020) (Fig. 1b). From west to east, the area comprises three morpho-sedimentary environments: (1) a strip of steep slopes underlain by colluvial deposits shed from the fault escarpment; (2) alluvial fans consisting of slightly cemented gravels; and (3) a mudflat on soft, laminated lake deposits exposed by the human-induced decline of the lake level (Fig. 1c–d).

Fig. 1
figure 1

Geological setting of the study area. a. Satellite image of the Dead Sea located within the pull-apart basin developed between the Arabian plate and the Sinai sub-plate. b. Geological sketch including the studied sector in the western coast of the Dead Sea, south of the Nahal Hazezon alluvial fan (modified from Mor and Burg 2000). c. Geomorphological map of the study area indicating the main morpho-sedimentary units and the sinkholes mapped in 2021 (modified from Sevil and Gutiérrez 2023). Coordinate system in b and c EPSG:2039. d. Image of the study area taken from the southwestern escarpment

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The studied sinkholes mostly occur in the mudflat, and to a much lesser extent in the southern sector of the Nahal Hazezon alluvial fan. At the boundary between these two geomorphic units sinkhole activity caused the destruction and abandonment of the Mineral Beach touristic resort (Fig. 1c). Here, there is thermal deep groundwater rising at 30 °C and unsaturated with respect to halite, with a salinity of ca. 120 g/L (halite solubility 356 g/L, per liter of solvent) (Abelson et al. 2006; Yechieli et al. 2006). In the analyzed area, the Ze’elim Fm. consists of interdigitated gravelly alluvial fan and lacustrine muds, and a basal porous and poorly lithified (i.e., eogenic) halite unit 13–18 m thick at around 20–30 m depth that pinches out to the west. Borehole data indicate a significant lakeward elevation drop of the top and bottom of the salt in short distances (10 m in 200 m distance) (Frumkin et al. 2011; Ezersky and Frumkin 2013), suggesting that the unit is vertically offset by a concealed down-to-the-east fault, as supported by seismic reflection profiles (Abelson et al. 2003) and the linear distribution of sinkholes. The displacement of the brackish-saline groundwater interface induced by the lake-level drop, and the accompanying dissolution of the salt unit, has resulted in the generation of the more than 6000 sinkholes since 1980 on the western coast of the Dead Sea (Abelson et al. 2006; Yechieli et al. 2006), damaging transportation infrastructure and tourist resorts, and compromising human safety (e.g., Frumkin et al. 2011; Abou Karaki et al. 2016; Salameh et al. 2019; Vey et al. 2021). Three main compatible models have been proposed to explain the dissolution of the salt unit: (1) deep groundwater from bedrock aquifers rising along concealed faults (hypogene karst) (Abelson et al. 2003; Closson 2005; Shalev et al. 2006; Frumkin et al. 2011; Closson et al. 2013; Charrach 2019); (2) shallow groundwater from the marginal alluvial fan gravels that dissolves the salt unit along its edge (e.g., Ezersky et al. 2009; Frumkin et al. 2011); and (3) surface water that percolates via sinkholes acting as swallow holes during flash floods (Avni et al. 2016; Arav et al. 2020).

3 Methodology

In order to analyze the frequency-size relationships of sinkholes (i.e., sinkhole hazard) and their temporal variability, we produced six multi-temporal sinkhole maps with a time lapse of three years, ranging from 2.8 to 3.9 years and spanning the period 2005–2021. Sinkhole mapping was conducted using the following remote-sensed imagery: (1) aerial photographs from 2005 to 2011 (100–66 cm/pix; Geological Survey of Israel); (2) satellite images from 2015 (25 cm/pix; Geological Survey of Israel); and (3) high-resolution orthomosaics (4 cm/pix) and digital surface models (14–17 cm/pix) that we produced by Structure from Motion (SfM) Photogrammetry with drone images taken by Terrascan-Labs LTD during two fieldwork campaigns conducted in 2018 and 2021. Despite the lower resolution of the imagery covering the older periods (2005–2011), the higher quality of the subsequent data, together with the barren and relatively flat nature of the area (Fig. 1d), allowed the precise mapping of the sinkhole edges in a GIS environment, producing detailed and complete time-lapse sinkhole inventories (Sevil and Gutiérrez 2023). The multi-temporal sinkhole maps capture four important processes involved in the evolution of the sinkhole field: (1) new sinkhole occurrence; (2) sinkhole reactivation; (3) sinkhole expansion by mass wasting processes; and (4) sinkhole coalescence related to the lateral growth of adjoining sinkholes or the formation of new sinkholes intersecting pre-existing ones, to form compound sinkholes.

In contrast to the previous analysis on the morphometric and spatiotemporal evolution of the sinkholes (Sevil and Gutiérrez 2023), in this hazard-oriented work we mainly focus our analysis on the new single sinkholes occurred over each time interval (2005–2008, 2008–2011, 2011–2015, 2015–2018, 2018–2021). The new sinkholes were classified into non-redundant and redundant, depending on whether they occurred in areas previously unaffected by sinkholes, or if they were nested into or intersecting pre-existing ones (Fig. 2). The underlying concept behind the differentiation of redundant sinkholes is that they may not involve any increase in the area affected by subsidence. The mapped sinkholes can be classified as cover collapse sinkholes and cover collapse and sagging sinkholes. Since the collapse mechanism is the dominant process in all the mapped sinkholes, they were analysed jointly. Two morphometric parameters were automatically extracted for each new sinkhole: length (distance between antipodal points) and area. These values can be considered as a good approximation to the size of the sinkholes at the time of occurrence given the temporal resolution of the data, and consequently adequate for conducting hazard assessments. The data was processed with the assistance of R (R Core Team 2022) and the packages tidyverse (Wickham et al. 2019) and sf (Pebesma 2018).

Fig. 2
figure 2

Examples illustrating the difference between new non-redundant sinkholes and new redundant sinkholes, occurring in areas previously unaffected or affected by sinkholes, respectively. The orthoimages show changes occurred over the time intervals 2015–2018 (a), and 2018–2021 (b). Note that new sinkholes can be fully redundant or partially redundant depending on whether they are nested within pre-existing sinkholes or intersect previous sinkholes, respectively. The former does not involve any increase in the area affected by sinkholes

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The frequency-size relationships of the new sinkholes occurred in different time intervals were explored considering their length. Very similar results would be obtained with the area, given the high circularity of the sinkholes, with a mean circularity ratio of 0.93 (ratio between sinkhole area and area of a circle with the same perimeter). We plotted sinkhole length (L) in logarithmic scale against the number of sinkholes equal or larger than a given size by unit area and time (i.e., cumulative spatiotemporal frequency; number of sinkholes ≥ L per km2 and per year).

4 Results

The sinkholes mapped in the study area are confined within a narrow N–S trending strip including alignments and tightly packed clusters (Sevil and Gutiérrez 2023). Sinkhole coalescence played an important role in the evolution of the sinkhole field because of three main factors: (1) highly clustered distribution; (2) high rate of new sinkhole occurrence; and (3) rapid expansion of the steep-sided sinkholes by mass wasting. Overall, new sinkholes expanded the sinkhole belt towards the lake and progressively filled the space between the sinkhole clusters (Fig. 3). Sinkholes at the time of formation had a mean length of 5.6 m (0.6 – 35.0 m range) and an average area of 30.6 m2 (0.2 – 658.5 m2 range). The number of new sinkholes (197, 119, 151, 128, and 72) and the spatiotemporal frequency of sinkhole occurrence, expressed as number of sinkholes per km2 and year (24.9, 21.3, 19.3, 19.6, 12.0) shows a general decrease throughout the analyzed time intervals (2005–2008, 2008–2011, 2011–2015, 2015–2018, 2018–2021) (Fig. 4a). The frequency of new sinkholes in the interval 2018–2021 was around half of that in 2005–2008 (24.9 vs. 12.0 sinkholes/km2 yr).

Fig. 3
figure 3

Evolution of the spatiotemporal distribution of sinkholes between 2005 and 2021 (modified from Sevil and Gutiérrez 2023). The maps of the study area differentiate new sinkholes and pre-existing ones. The sequence records the lake-level decline and the consequent expansion of the mudflat. The maps also indicate the location of the Mineral Beach touristic resort that was abandoned in 2015 because of sinkhole damage and the anthropogenic fill dumped between 2008 and 2011 to mitigate sinkhole risk. Coordinate system EPSG:2039

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Fig. 4
figure 4

Evolution of the new single sinkholes mapped in the study area over each year interval from 2005 to 2021. a. Spatiotemporal frequency differentiating redundant and non-redundant sinkholes. b. Comparison between the cumulative area of new and all non-redundant single sinkholes. c. Lengths of all the new non-redundant single sinkholes mapped in each year interval. The boxplots include in red the mean length

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As expected for a sinkhole field with high clustering and areal density, the proportion of redundant sinkholes increased substantially through time. The percentage of redundant sinkholes nested in or intersecting pre-existing ones escalated from 3% in 2005–2008 to 53% in 2018–2021. For the same time intervals, the spatiotemporal frequency of new redundant sinkholes increased from 0.6 to 6.4 sinkholes km−2 yr−1 (Fig. 4a). The progressive rise in spatial redundancy reflects the reduction in the sinkhole-free area and the densification of the sinkhole field (i.e., competition for space; De Waele and Gutierrez 2022).

Despite the decrease in sinkhole frequency and the growing spatial redundancy, the area affected by single non-redundant sinkholes showed a steady increase, indicating relatively constant areal increments (Fig. 4b). This apparent inconsistency is explained by the general increase in the size of the new sinkholes (Fig. 4c). The average length of the new non-redundant single sinkholes increased from 5.0 in 2005–2008, to 8.4 m in 2018–2021. Overall, the study area has been affected by progressively lower numbers of new sinkholes with higher spatial redundancy, but because of their larger dimensions, the sinkhole-affected area has increased steadily.

Figure 5 shows cumulative frequency-size relationships of all new single sinkholes and non-redundant new single sinkholes for different time intervals (i.e., hazard curves). The cumulative spatiotemporal frequency is expressed as the number of new sinkholes equal or larger than L (length) per km2 and per year. The comparison between the “all new” and “only non-redundant” curves allows assessing the deviation between hazard curves generated considering and obviating spatial redundancy. The empirical data can be satisfactorily modelled over most of the size distribution by logarithmic functions with high goodness of fit (R2adj 0.94–0.98). Interestingly, most large new sinkholes are non-redundant (Fig. 5a). This is explained by the fact that new large sinkholes essentially occur on the east side of the sinkhole belt, away from pre-existing sinkholes, as indicated by the multi-temporal mapping (Fig. 3). The frequency-size relationships of new sinkholes formed during the different time intervals show a substantial deviation between the “all new” and “only non-redundant” curves from 2015 onwards, indicating the time at which spatial redundancy starts to have a significant impact in the decrease of the effective hazard (i.e., subsidence in previously unaffected areas). The frequency-size relationships representing new sinkholes for three successive time periods (2005–2011, 2011–2018, 2018–2021) show a progressive slope decline and an intersection point at ca. 10 m length (Fig. 5c). This indicates a significant decrease in the frequency of sinkholes smaller than 10 m and an increase in the frequency of larger sinkholes.

Fig. 5
figure 5

Cumulative frequency-size relationships of all new single sinkholes and new non-redundant single ones for different time intervals. The plots include the logarithmic functions that better fit the empirical distributions (function and adjusted R2 (R.2adj)). a. Frequency-size relationships of new single sinkholes at any time from 2005 to 2021. b. Frequency-size relationships at the time of occurrence. c. Frequency-size relationships for three successive time periods (2005–2011, 2011–2018, 2018–2021)

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5 Discussion

The production of complete multi-temporal cartographic sinkhole inventories with morphometric data in a salt karst area characterized by extraordinarily high frequency of sinkhole occurrence and high clustering has allowed us to explore: (1) the potential impact of the commonly disregarded spatial redundancy on hazard assessments; (2) the advantages and disadvantages of different hazard assessment approaches; and (3) the temporal variability of sinkhole hazard.

Regarding spatial redundancy, the underlying concept is that redundant sinkholes do not necessarily involve an increase in the area affected by sinkholes, and consequently their consideration may lead to hazard overestimates (e.g., new sinkholes nested in pre-existing ones). In the analyzed zone, the available sinkhole-free area is rapidly exhausted because of the high rate of sinkhole occurrence, tight sinkhole clustering, and rapid expansion of the sinkholes by mass wasting. This explains the important morphogenetic role played by sinkhole coalescence and the high competition for space leading to significant spatial redundancy. Moreover, the rapid areal growth of the sinkholes has a significant impact on the establishment of set-back distances and the delineation of standoff zones. Sinkhole hazard can be markedly different in theoretical sinkhole fields with the same frequency of sinkhole occurrence, but with different clustering and sinkhole density. Spatial redundancy is higher in sinkhole fields with tight clustering, and it increases through time in specific fields as the areal density of sinkholes increases. In our particular case, spatial redundancy shows a substantial increase from 2015 onwards. The proportion of new redundant sinkholes in the interval 2005–2008 was 2%, whereas in the intervals 2015–2018 and 2018–2021 raised to 27% and 53%, respectively (Fig. 4a). Consistently, the frequency-size relationships (i.e., hazard curves) for “all new” and “only non-redundant” sinkholes start to show a significant deviation from 2015 (Fig. 5b). Hazard curves generated with all the new sinkholes suggest a significantly higher subsidence hazard than the more reasonable “effective hazard” curves obviating the new redundant sinkholes (Fig. 5).

Sinkhole hazard can be assessed using various parameters and approaches. A widely accepted parameter is the spatiotemporal frequency of sinkhole occurrence regardless of sinkhole size, expressed as number of sinkholes per unit area and time. Abelson et al. (2006) reported an acceleration trend in the occurrence rate of sinkholes in the Dead Sea from the 1980s to 2002. In our study area, we have observed a substantial decrease in the frequency of sinkhole occurrence dropping from 24.9 to 12.0 sinkholes km−2 yr−1, suggestive of a significant hazard decrease. The sinkhole frequency decline is even more important when considering only non-redundant sinkholes for the same intervals (from 24.2 to 5.7). Sinkhole hazard can also be assessed computing the area affected by new sinkholes per unit time, implicitly integrating sinkhole frequency and size. In our study area, the areal increments remain relatively constant through time, suggesting a relatively unchanged hazard level. The apparent contradiction with the hazard assessment based on the number of new sinkholes is related to the occurrence of progressively larger sinkholes (Fig. 4).

Sinkhole hazard can be appraised more comprehensively generating empirical frequency-size relationships. The hazard curves generated for different time intervals (Fig. 5c) show a progressive flattening, apparently suggesting an overall declining hazard trend. The distribution of the curves indicates a substantial decrease in the frequency of small sinkholes and a slight increase in the frequency of large sinkholes. For instance, for the intervals 2005–2011 and 2018–2021, the frequency of new non-redundant sinkholes ≥ 2 m long dropped from 20.6 to 4.8 sinkholes per km2 and year, whereas that of new non-redundant sinkholes ≥ 20 m long increased from 0.1 to 0.8 sinkholes per km2 and year. The latter frequency increase is comparatively small but has a critical contribution to the hazard because of three main reasons. First, the area of a circular sinkhole 20 m long is equivalent to the area of 100 circular sinkholes 2 m long. Second, the capability of collapse sinkholes to cause damage (i.e., severity) largely depends on the sinkhole size at the time of occurrence. Third, the maximum observed sinkhole size is frequently used as the design parameter for sinkhole-resistant structures. This value has increased for the cover collapse sinkholes of the study area from 23 to 35 m between 2005–2011 and 2018–2021.

An interesting spatiotemporal pattern observed in the study area is the occurrence of larger sinkholes on the western sector of the subsidence belt, at some distance from the pre-existing sinkholes (non-redundant). A probable explanation is that dissolution is migrating toward the downthrown side of the concealed fault inferred by seismic data that has controlled the sinkhole strip (Abelson et al. 2006), and that has offset vertically the salt layer, as supported by borehole data (Ezersky and Frumkin 2013). This interpretation whereby dissolution is advancing toward areas with thicker overburden is compatible with the decrease in sinkhole frequency and the size increase of the sinkholes, although it remains as an untested hypothesis due to lack of sufficient subsurface data (e.g., boreholes).

6 Conclusions

In this research paper we show that sinkhole hazard can significantly change over time. The non-steadiness of the frequency-size relationships has important consequences inasmuch as it restricts the reliability and temporal validity of the hazard assessments.

In the study area of the western shore of the Dead Sea, the frequency-size relationships have displayed contrasting temporal changes in different sinkhole hazard components. Regarding the frequency of generation of new sinkholes, regardless of their size, sinkhole hazard showed a decreasing pattern. By contrast, the increment of the area affected by new non-redundant sinkholes displayed little variation suggesting rather constant hazard levels. However, the mean and maximum size of new sinkholes increased in time together with their relative frequency revealing a hazard and severity rise.

Moreover, the spatial redundancy of sinkholes, controlled by clustering and coalescence related to sinkhole interception or post-collapse expansion by mass wasting processes, can change the susceptibility status of the system and reduce the effectiveness of the damaging process. This characteristic is commonly disregarded on sinkhole hazard assessments and may cause their overestimation.

The presented results are applicable to other sinkhole-prone areas worldwide and illustrate that comprehensive hazard assessments require the construction of complete cartographic sinkhole inventories with chronological and morphometric data. Moreover, it is advisable to explore the temporal variability of frequency-size relationships to produce more reliable hazard estimates and better understand the controlling factors.