# Vents to events: determining an eruption event record from volcanic vent structures for the Harrat Rahat, Saudi Arabia

## Abstract

Distributed “monogenetic” volcanic eruptions commonly occur in continental settings without obvious structural alignments or rifting/extensional structures. Nevertheless, these may develop as fissures, representing the surface expression of dykes with a range of orientations, especially when stress regimes vary over time and/or older crustal features and faults are exploited by rising magmas. Dykes reaching the surface as fissures can last hours to months and produce groups of closely aligned vents, hiding the true extent of the source fissure. Grouped or aligned vents in a distributed volcanic environment add complexity to hazard modelling where the majority of eruptions are single-vent, point-source features, represented by cones, craters or domes; i.e. vent groups may represent fissure events, or single eruptions coincidently located but erupted hundreds to tens of thousands of years apart. It is common practice in hazard estimation for intraplate monogenetic volcanism to assume that a single eruption cone or crater represents an individual eruptive event, but this could lead to a significant overestimate of temporal recurrence rates if multiple-site and fissure eruptions were common. For accurate recurrence rate estimates and hazard-event scenarios, a fissure eruption, with its multiple cones, must be considered as a single multi-dimensional eruptive event alongside the single-vent eruptions. We present a statistical method to objectively determine eruptive events from visible vents, and illustrate this using the 968 vents of the 10 Ma to 0.6 ka volcanic field of Harrat Rahat, Saudi Arabia. A further method is presented to estimate the number of hidden vents in a thick volcanic pile. By combining these two methods for Harrat Rahat, we determined an updated spatial recurrence rate estimate, and an average temporal recurrence rate of 7.5 × 10^{−5} events/year. This new analysis highlights more concentrated regions of higher temporal hazard in parts of Harrat Rahat, which has significant implications for the city of Al-Madinah and surroundings.

### Keywords

Monogenetic volcanic fields Hazard analysis Spatio-temporal recurrence rate Harrat Rahat## Introduction

Volcanic eruptions in rifting, extensional and volcanic-flank settings can develop as fissures, representing the surface expression of dykes. Post-eruption, a single fissure may be preserved in the geologic and geomorphic record as a group of aligned or sub-aligned vents, most commonly spatter or scoria cones. Examples have been noted in incipient rifts (e.g. Tavenui, Fiji; Cronin et al. 2001), and the large volcanic provinces of the Arabian Peninsula (e.g. Harrat Rahat, Saudi Arabia; Moufti et al. 2010). In many intraplate continental “monogenetic” volcanic fields without strong extension-related structures, aligned or grouped vents also occur e.g. Auckland Volcanic Field (Bebbington and Cronin 2011), Yucca Mountain, Nevada (Connor et al. 2000), Newer Volcanic Region, Victoria, Australia (Lesti et al. 2008) and the Michoacan-Guanajuato Volcanic Field, Mexico (Wadge and Cross 1988). In such settings, where weak or absent extension prevails, two issues persist in considering the hazard of distributed volcanism. First, the hazard associated with a multiple-source or line-source (i.e. open fissure) eruption is often far more widely spread, with less predictable lava flow paths and tephra fall zones, than that of a single point-source eruption (Gudmundsson 1992). Second, the precision of dating methods (usually K-Ar or Ar-Ar for these long-lived fields) is often not adequate to classify neighbouring cones as part of the same eruptive event or not, hence the usual practice is to model the occurrence process of cones (e.g. Wadge and Cross 1988; Lutz and Gutmann 1995; Connor and Hill 1995; Condit and Connor 1996; Conway et al. 1998; Weller 2004; Felpeto et al. 2007; Marti and Felpeto 2010; Kiyosugi et al. 2010). This misidentification of a crater row formed during a single fissure eruption as representing individual asynchronous eruptions results in a significant overestimate of temporal hazard rates. Fissure (or ʻmultiple-vent’) eruptions should thus be considered as multi-dimensional eruptive events alongside the single-vent eruptions in order to obtain both accurate spatial and temporal recurrence rates, and the full range of impacts/magnitudes for event scenarios. Consideration must also be given to hidden or buried eruptive events to quantify the uncertainty involved in using only currently visible vents.

Existing spatio-temporal models for volcanic field hazard analysis have almost invariably assumed that each vent is a discrete, independent eruptive event in both time and space, i.e. a point process (Connor and Hill 1995; Condit and Connor 1996; Conway et al. 1998; Weller 2004; Bebbington and Cronin 2011). This assumption is justified for those volcanic fields where age data show all eruptions to be asynchronous; however, it is no longer valid if there is a suspicion that a large proportion of eruptive events in a field are multiple-vent or fissure eruptions. Consequently for this work, an ‘event’ is defined as a volcanic eruption continuous in both time and space. Independent eruption centres are referred to as single-vent events, and crater rows as multiple-vent events. For recurrence rate estimates, an event is constrained/approximated here with four parameters: eruption age (1), and with ellipsoid dimensions: major semi-axis (2), minor semi-axis (3) and strike (4) (bearing from North of the major semi-axis). We present here a statistical method to account for these eruption types in volcanic field hazard analysis, based on known and estimated eruption parameters and related uncertainties. This approach identifies eruptive events from visible vents in the volcanic field of Harrat Rahat, Saudi Arabia. According to written accounts and geological reconstructions for this field (Moufti et al. 2010; Al-Samhoody 1440–1506 AD), the two most recent historical eruptions involved a series of vents. In addition, while not an active rifting situation, its location outboard from an active rift setting appears to impart strong structural control on the volcanism (Camp and Roobol 1989) and may thus encourage the formation of frequent fissure eruptions. These features, and the excellent exposure of the field, make Harrat Rahat an ideal site from which to develop a method for visible vent to eruptive event determination.

Erosion, burial and the quarrying of scoria cones may all reduce the number of volcanic vents that are counted in a volcanic field. O’Leary et al. (2002) inferred the location of buried vents from high-resolution aeromagnetic data at Yucca Mountain. This method could not be used to identify individual vents in Harrat Rahat due to the low resolution of the available aeromagnetic data (El-Difrawy et al. 2013). Consequently, to obtain an estimate for the number of hidden vents, our approach is based on consideration of the likelihood of a vent being observable today given it occurring at a specific point in space–time (following the purely time-based approach of Guttorp and Thompson 1991). A ‘hidden vent’ is defined here as a volcanic vent which would have been observed after the vent-forming eruption but is not currently visible, due to, for example, burial by the lava flows of more recent eruptions. The application of these two new methods to Harrat Rahat results in updated spatial and temporal recurrence rate estimates that better reflect the multi-dimensional nature of eruptive events within this volcanic field.

## Harrat Rahat volcanism

^{2}covered by localised lava fields (harrats). The harrats show a range of broad spatial patterns, from elongated broad ridges with portions orientated sub-parallel to the active Red Sea rift to the West (e.g. Harrat Rahat), through to those without apparent alignment of orientation (e.g. Harrat Lunayyir) (Camp and Roobol 1989). The youngest Harrat Rahat volcanism is not directly related to the Red Sea rifting, but may instead represent asthenospheric upwelling associated outside the rift margins (Camp and Roobol 1992; Pallister et al. 2010). The Miocene to Recent volcanics overlie a Precambrian basement within the western Arabian Plate (Coleman et al. 1983). The ~20,000 km

^{2}Harrat Rahat is situated immediately SE of the city of Al-Madinah (Fig. 1). Nine hundred sixty-eight volcanic vents (cones, craters and spatter mounds) were identified across the extent of Harrat Rahat using GoogleEarth, SPOT and LandSat images, geologic maps (e.g. Pellaton 1981), and complemented by previous work in the region (Camp and Roobol 1989; Moufti 1985). The volcanic field has produced alkalic and subalkalic basalts as well as a range of evolved compositions erupting predominantly, but not exclusively, through new vents (Moufti 1985).

^{3}of magma from at least seven vents 19 km SE of the city, including a lava flow that terminates at the current city boundaries (Camp et al. 1987; Moufti et al. 2010). The historical eruptions, together with at least six further episodes of known Holocene activity and a possible magma-related earthquake swarm in 1999 AD, means that future volcanism is expected (Moufti et al. 2010).

Harrat Rahat flow unit properties

Stratigraphic unit | Sub-unit | Age range | Number of vents | Dominant alignment angle | Composition |
---|---|---|---|---|---|

Madinah | Qm7 | 1500 BP–present | 11 | −23.56° | Basalt, Hawaiite |

Qm6 | 4500–1500 BP | 45 | Basalt, Hawaiite | ||

Qm5 | 0.3 Ma–4500 BP | 39 | 14.67° | Basalt, Hawaiite, Mugearite, Trachyte | |

Qm4 | 0.6–0.3 Ma | 38 | −42.8° | Basalt, Hawaiite, Mugearite, Benmoreite, Trachyte | |

Qm3 | 0.9–0.6 Ma | 77 | −14.28° | Basalt, Hawaiite, Mugearite, Trachybasalt | |

Qm2 | 1.2–0.9 Ma | 154 | −13.15° | Basalt, Hawaiite, Mugearite | |

Qm1 | 1.7–1.2 Ma | 422 | −10.65° | Basalt, Hawaiite, Mugearite | |

Hammah | Th | 2.5–1.7 Ma | 162 | −10.01° | Basalt |

Shawahit | Tw2 | 8.5–2.5 Ma | 18 | Basalt | |

Tw1 | 10–8.5 Ma | 2 | Basalt |

## Methods—vent to event determination

- a.
Identification of key event characteristics

A volcanic event can be described by various characteristics including: location, age, eruption duration, the number of eruptive centres, eruption style(s), magma composition and volume and geometric features (e.g. length, thickness, bearing, shape, height and diameter of craters/cones). The characteristics selected are dependent on both the scope of the work and the available data. Because very few parameters were available for all 968 volcanic vents in Harrat Rahat, the available data constrained the selection. Four key characteristics were used: age, and ellipsoid dimensions of the structure representing an eruptive event (major and minor semi-axes, and strike—assessed as deviation from the dominant alignment in the harrat).

The ellipsoid dimensions were selected as they could be approximated from visible vent locations and are likely indicative of underlying structures. Individual crater rows may represent a range of controlling factors and a deviation of the strike of any potential multiple-vent event from the dominant alignment of a volcanic field may be informative. In general, groups of vents with strikes orientated close to that of the whole field may be more likely co-genetic than groups orientated off the axis of dominant alignment. This may not be exclusively true, such as if older lineations/faults are exploited by rising magmas. Dominant alignments based on volcanic vent locations within each temporal episode were determined using a squared, asymptotic, mean-squared error method (SAMSE, Wand and Jones 1995, Table 1).

As previously stated, vent age data is poorly known in Harrat Rahat; however, using the mapped temporal episodes (Table 1), it was possible to include a time parameter as an inclusion/exclusion constraint, requiring vents in an event to be in the same episode.

- b.
Estimation of a priori PDFs for each of the key characteristics

The variation in each characteristic across Harrat Rahat was described as a PDF, which allowed incorporation of the uncertainties and facilitated the application of Bayesian techniques. A distribution (PDF) was selected for each characteristic based on the required support or symmetry, and whether the possibility of a positive mode was desirable. The selections were checked against the data using formal tests (Anderson–Darling and Kolmogorov–Smirnov) as appropriate. Two of the characteristics were continuous, with right-skewed distributions (ellipsoid semi-major and semi-minor axes—often referred to here as event length and thickness), thus log-normal or gamma distributions were selected. A log-normal distribution was fitted to San Rafael data by Delaney and Gartner (1997). A gamma distribution was chosen for a further two sets (non-informative and expert elicitation data) as it allows for over-dispersion, i.e. a situation where the majority of the events are single-vent events, as required by the simultaneous modelling of single- and multiple-vent events. In this case, it is possible for an event to have ~zero length and thickness (effectively the vent diameter), hence requiring a distribution that is defined at zero, ruling out the power–law distribution (Gudmundsson and Mohajeri 2013). To be consistent with the SAMSE method used to determine dominant alignments, a normal distribution was selected for the event strike to model deviation from the dominant alignment.

Only one set of a priori characteristic PDFs was required for the application of this method; however, since all distributions are estimates in this Bayesian paradigm, three approaches were applied here to enable a sensitivity analysis and to assess the merits of each of these options: 1. Expert elicitation, 2. Use of an analogue volcanic field and 3. Use of non-informative priors (Fig. 2).- 1.
Expert elicitation (EE) enables information to be obtained from expert judgement using formal, structured and probabilistic procedures (Cooke 1991). Aspinall and Woo (1994) proposed expert elicitation as a tool for volcanic crisis management, and this was successfully implemented at Montserrat from 1995 onwards (Aspinall and Cooke 1998). Expert elicitation methods have also been applied within event tree frameworks both for crisis management (e.g. Exercise Mesimex at Vesuvius; Marzocchi et al. 2008 and Exercise Ruaumoko in New Zealand; Lindsay et al. 2010), and for long-term hazard analyses (Neri et al. 2008; Marti and Felpeto 2010). It was adopted as one of the three approaches here due to this frequency of use, facilitating investigation into its applicability for this vent-to-event method. The Cooke method adopted here entails the following: 1. Select a group of experts, 2. Calibrate them based on level of expertise and 3. Ask them questions of interest and for estimates of related uncertainty levels. This method allows consideration of the fact that not all experts are equal, or free from bias. The number of experts suggested for the Cooke approach is between 8 and 15 (Aspinall 2010). For this work, 15 volcanologists contributed (five geological consultants, ten post-PhD academics), and with experience levels ranging from 5 to 30+ years. The estimates were weighted based on a combination of the number of years of research and published papers on the specific topic. The questionnaire is included as Supplementary material. They were asked eight questions concerning eruptive event and dyke characteristics, including mean value estimates and a ‘credible interval’ for each characteristic (minimum and maximum values representing 99 % certainty levels). To reduce the level of aleatoric uncertainty, to try and avoid “wild guesses”, experts were not required to answer every question. Distributions were obtained via a log-weighted combination of each expert’s answer and fitted using maximum likelihood estimation to obtain a priori PDFs for each of the four key characteristics.

- 2.An analogue volcanic field (AF) can provide comparative information for regions with a scarcity of data spanning past eruptions (Rodado et al. 2011). The inherent assumption is that underlying tectonic and volcanic processes match between regions, the validity of which is considered in the discussion section of this paper. Uniquely, a large data set is available for the highly eroded San Rafael Volcanic Field, Utah, USA (Delaney and Gartner 1997; Kiyosugi et al. 2012), with fitted distributions for dyke thickness, outcrop lengths and strike. The use of this data set assumes similarity between San Rafael and Harrat Rahat, and that eruptive event and dyke dimensions are related. The San Rafael dyke dimensions (e.g. median dyke-segment length ~ 1 km, maximum ~ 9 km) match those from the inferred dyke intrusion in 2009 (e.g. surface fault rupture ~ 8 km, inferred dyke-length ~ 10 km) from the Harrat Rahat neighbouring field, Harrat Lunayyir (Pallister et al. 2010; Mukhopadhyay et al. 2013), and to the historical event dimensions observed for Harrat Rahat (Table 2). Becerril et al. (2013) considered the relationship between the dimensions of eruptive events and dykes on El Hierro Island, Spain. They found that eruptive fissure lengths inferred from the alignments of visible vents were of similar dimensions to those of exposed dykes in the same regions. Galindo and Gudmundsson (2012) assessed dykes and eruptive fissures in both Tenerife and Iceland and looked specifically at basaltic feeder dykes, i.e. dykes which instigated an eruption. They concluded that dykes become thicker and shorter as they reach the surface, but that the eruptive fissure and subsurface dimensions are directly proportional. Thus, it is reasonable to assume that in Harrat Rahat there is also a linear relationship between dyke and eruptive event dimensions.Table 2
Historical eruption event values (Qm7)

Eruptive event

^{a}Event length (m)

Event width (m)

Strike (bearing in degrees from North)

641 AD (4 vents)

820

36

−24.53

1256 AD (7 vents)

1,520

57

−20.74

- 3.
Non-informative (NI) priors were used for the third set of characteristic distribution estimates. These attribute uncertainty to a variable and assume minimal prior information such that the a posteriori distributions are dominated by the measured data (in this case, the dimensions of the historical eruptions). Non-informative prior distributions were selected using plausible values for the mean of each characteristic, but with sufficiently large variance that the prior was effectively uniform (Fig. 2).

- 1.
- c.
Adaptation of a priori PDFs to the volcanic field of interest

For this analysis, the software package WinBUGS (Lunn et al. 2000) was used for the Bayesian analysis and a priori distributions were updated using observed data for the historical events of 641 AD and 1256 AD (Table 2). The model for Bayesian analysis is specified below using the notation of Christensen et al. (2011).

For length and thickness, e.g. length:$$ \begin{array}{l}{l}_1,\dots, \left.{l}_n\right|r,\mu {\sim}_{iid}\mathrm{Gamma}\left(r,\mu \right)\hfill \\ {}\left.r\right|\eta \sim \mathrm{Gamma}\left(\eta, \raisebox{1ex}{${r}_0$}\!\left/ \!\raisebox{-1ex}{$\eta $}\right.\right)\hfill \\ {}\left.\mu \right|\eta \sim \mathrm{Gamma}\left(\eta, \raisebox{1ex}{${\mu}_0$}\!\left/ \!\raisebox{-1ex}{$\eta $}\right.\right)\hfill \\ {}\eta \sim \mathrm{Gamma}\left(0.001,0.001\right)\hfill \end{array} $$i.e. event length variation follows a gamma distribution with shape (

*r*) and location (*μ*) parameters. The shape and location parameters are allowed to vary by following gamma distributions with shape (*η*) and location (\( \raisebox{1ex}{${r}_0$}\!\left/ \!\raisebox{-1ex}{$\eta $}\right.,\raisebox{1ex}{${\mu}_0$}\!\left/ \!\raisebox{-1ex}{$\eta $}\right. \), respectively) parameters where*μ*_{0}and*r*_{0}are the a priori parameters (Fig. 2), and*η*is the hierarchical parameter. This ensures that the mean location and shape parameters are consistent with the prior. The parameter*η*has a reference prior, which should not influence the posterior, thus allowing the data to dominate this part of the model.For strike:$$ \begin{array}{l}{a}_1,\dots, \left.{a}_n\right|\mu, \sigma {\sim}_{iid}\mathrm{Normal}\left(\mu, {\sigma}^2\right)\hfill \\ {}\mu \sim \mathrm{Normal}\left({\mu}_o,{\sigma_o}^2\right)\hfill \\ {}\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{${\sigma}^2$}\right.\sim \mathrm{Gamma}\left(0.001,0.001\right)\hfill \end{array} $$i.e. deviation from dominant strike direction follows a normal distribution with parameters: mean (

*μ*) and standard deviation (*σ*). These parameters are allowed to vary via normal and gamma distributions respectively where*μ*_{0}and*σ*_{0}are the a priori parameters (Fig. 2), achieving the same consistency properties as above. The hierarchical parameter*σ*again has a reference prior. In the absence of greater temporal resolution, this also accounts for variation in the stress field over time.The gamma distribution (PDF) is given by:$$ \begin{array}{cc}\hfill f(x)\sim \mathrm{Gamma}\left(\alpha, \beta \right)=\frac{1}{\varGamma \left(\alpha \right)}{\beta}^x{x}^{\alpha -1}{e}^{-\beta x}\hfill & \hfill \mathrm{where}\kern0.5em \alpha, \upbeta >0\hfill \end{array} $$And the normal distribution (PDF) is given by:$$ \begin{array}{cc}\hfill f(x)\sim \mathrm{Normal}\left(\mu, {\sigma}^2\right)=\frac{1}{\sigma \sqrt{2\pi }}{e}^{-\frac{{\left(x-\mu \right)}^2}{2{\sigma}^2}}\hfill & \hfill \mathrm{where}\kern0.5em {\sigma}^2>0\hfill \end{array} $$The resulting a posteriori distributions are shown in Fig. 3 and Table 3.Table 3Posterior probability density functions (PDFs)

Distribution source

Event length (m)

Event width (m)

Deviation from dominant strike (deg)

PDF

Parameters

PDF

Parameters

PDF

Parameters

Expert elicitation

Gamma

*r*= 2.139*μ*= 1.674 × 10^{−3}Gamma

*r*= 3.044*μ*= 0.07062Normal

\( \overline{x} \) = −1.055

*σ*= 15.26Analogue VF

Log-normal

*μ*= 7.5841*σ*= 0.1444Log-normal

*μ*= 1.0179*σ*= 0.89Normal

\( \overline{x} \) = −0.12

*σ*= 9.3Non-informative

Gamma

*r*= 2.496*μ*= 1.013 × 10^{−3}Gamma

*r*= 3.1313*μ*= 0.04024Normal

\( \overline{x} \) = −0.46

*σ*= 6.75 - d.
Assessment of potential event combinations against a posteriori distributions

Vents were grouped into potential events, and the event dimensions for each of the key characteristics were assessed individually against the respective a posteriori PDFs to obtain the probability of each of the vent groupings being a single event. Assuming each key characteristic was independent, we multiplied the probabilities for each characteristic to obtain the total probability that a vent grouping was a single eruptive event.

For relatively small numbers of vents (< ~10), every potential combination of vent groupings can be assessed. This number is restricted by the enumeration of possible events (Bell numbers). To enable computation of the larger problem, a heuristic approach was taken (Clancey 1985) to obtain a close approximation of the actual event record for the volcanic field, without the need to assess all theoretical combinations. The approach applies a stochastic local search and the movement between candidate event records is based on heuristic values (Appendix A), i.e. the a posteriori PDFs. The vents were randomly grouped into an initial candidate event record by assigning each vent a random number selected from the range 1:

*v*, where*v*is the number of vents. A single potential event was then randomly selected for analysis (e.g. ‘group A’). The*n*nearest neighbours (those groups in the current candidate event record whose centroids were the lowest Euclidean distance from group A) were identified (*n*≥ 2) and the potential combined event dimensions (and consequent probabilities) calculated (‘*join*’). Simultaneously, the probability of breaking group A into single vent events was calculated (‘*break*’). These values were then compared against the probability of group A being an event. A move was then made (*join*or*break*) based on the relative probabilities of the new potential groupings. Locally suboptimal*join*or*break*moves were permitted to allow exploration of wider regions of the global search space. Iterations were then run with an exponentially decaying annealing schedule in place, such that the probability of a less than optimum move tended toward zero.In order to determine the optimal number of nearest neighbours (

*n*), a pre-analysis was conducted for each distribution set, where*n*was varied from two to ten and the results assessed by comparison with the two known historical eruptions augmented with synthetic data representing single and paired vent events. It was found that with only two nearest neighbours, a very slow annealing schedule was required to obtain the known event record. However, for all distributions, this problem was alleviated using*n*≥ 3. Since computational time increased rapidly with the number of nearest neighbours, this optimum value of*n*= 3 was used for the field-wide runs. - e.
Compare potential combinations to obtain the most likely volcanic event record

The potential vent grouping with the highest likelihood across the whole of the volcanic field was assumed to be the most likely volcanic event record.

## Methods—hidden vent estimation

The main premise in our approach to estimating hidden vents is that at each point in time and space, there is a corresponding probability that should an eruptive vent have occurred, the vent would still be visible today.

### Theory

Following the work of Guttorp and Thompson (1991):

**M**_{tA}represents the number of volcanic vents that

*have*occurred in area

*, time*

**A***:*

**t**Where 1[*τ*_{i} ≤ *t*] is a heavy-side function taking the value of 1 if the * i*th eruptive vent occurred at time

**τ**_{i}prior to time

*and zero otherwise, and 1[*

**t***x*

_{i}∈

*A*] takes the value of 1 if an eruptive vent occurred at location

**x**_{i}within area

*and zero otherwise.*

**A**

**Y**_{tA}represents the number of volcanic vents that have occurred and are

*still visible*in area

*, time*

**A***:*

**t**Where * f( )* is a function describing whether a vent is still visible based on when (

**τ**_{i}) and where (

**x**_{i}) it occurred. If all eruptive vents that occurred in area

*, and prior to time*

**A***are still visible,*

**t***would be equal to a heavy-side function as defined above and the number of vents that are visible today (*

**f( )**

**Y**_{tA}) would be equal to the number of vents that have ever erupted (

**M**_{tA}).

*is unknown, the expected value of*

**f( )**

**Y**_{tA}can be estimated from:

Where * ρ*(

**τ**_{i},

**x**_{i}) is the probability that a vent from time

**τ**_{i}, location

**x**_{i}is visible at time

*and*

**t***is a random number generated between 0 and 1, i.e.*

**θ***~*

**θ***(0, 1). Estimates of*

**N**

**M**_{tA}can then be obtained by defining this probability function (or functions) and randomly placing vents in time and space, incrementing

**M**_{tA}by one each time, where

**E[Y**_{tA}

*is incremented by one when*

**]***(*

**ρ**

**τ**_{i},

**x**_{i}) >

*θ*.

### Application

*(*

**p**

**τ**_{i}), with older vents more likely to have eroded than younger vents. Burial of vents due to subsequent eruptions was defined as a function of both space and time,

*(*

**q**

**τ**_{i},

**x**_{i}), as a vent could only be buried in locations with subsequent lava flows. Equation 3 therefore becomes:

i.e. a vent occurring at time **τ**_{i}
* ≤ t*, and location

**x**_{i}within area

*is expected to be visible at time*

**A***if the probabilities of it not having eroded to ‘nothing’:*

**t***(*

**p**

**τ**_{i}) and not having been buried by subsequent flows:

*(*

**q**

**τ**_{i},

**x**_{i}) are greater than random numbers generated between 0 and 1:

**θ**_{p},

**θ**_{q}.

*(*

**p**

**τ**_{i}): It is assumed that the erosion of a vent is proportional to the time since the eruption

*(*

**t − τ**_{i}). The relative degree of erosion with stratigraphic unit is detailed in Camp and Roobol (1991a) with no erosion noted for Qm7 − Qm4, some erosion noted for Qm3 but ‘with distinct craters’, ‘deeply eroded scoria cones’ noted for Qm2 and Qm1, and ‘deeply eroded scoria cones with craters indistinct or removed’ for Th. Assuming no loss of visibility due to erosion for the last 1.2 Ma (post-Qm2), and the complete loss of visibility >10 Ma required for consistency with the observed ages, a linear relationship between time and degree of erosion can be defined as:

Where 10 Ma < * t* < 1.2 Ma, and

*: rate of change of effect of erosion with*

**a***a*= 1.136 × 10

^{−7}year

^{−1}for the assumptions above:

*a*= 1/(10 Ma − 1.2 Ma).

**τ**_{i},

**x**_{i}): It is assumed that the likelihood of a vent to be buried by a subsequent lava flow is proportional to the age (

**τ**_{i}) and location (

**x**_{i}) of the vent, i.e. a younger vent could not be buried by an older flow, whereas an older vent

*could*be buried by a younger flow should that flow be in the same location.

Where * t* is the age of the lava flow. The probability of remaining visible is estimated (for vents older than Qm1) by comparing the diameters of vents which are surrounded by lava with those that are not. The latter are larger on average, indicating that smaller vents are being obscured by lava. Fitting a probability distribution to the observed diameters provides an estimated probability of 50 % that the vents remain visible when inundated by subsequent lava flows. This probability does not differ significantly according to whether the lava is the same age as the vent, or is younger.

With knowledge of the number and temporal episode of currently visible vents, and the location of lava flows also constrained to a temporal episode (Camp and Roobol 1991a), Eqs. 4, 5 and 6 were combined to assess the potential number of hidden vents. Sensitivity analyses were also performed to assess the effect of erosion and burial constants on the estimate of total number of vents.

## Results

### Event determination

Event record results for each distribution set

Episode | Observed vents | Number of events | |||
---|---|---|---|---|---|

Expert elicitation | Analogue VF | Non-informative | Qualitative | ||

Tw1 | 2 | 2 | 2 | 2 | 2 |

Tw2 | 18 | 18 | 18 | 18 | 18 |

Th | 162 | 142 | 125 | 135 | 144 |

Qm1 | 422 | 366 | 336 | 379 | 353 |

Qm2 | 154 | 108 | 77 | 111 | 94 |

Qm3 | 77 | 58 | 50 | 54 | 60 |

Qm4 | 38 | 24 | 20 | 25 | 32 |

Qm5 | 39 | 23 | 22 | 34 | 25 |

Qm6 | 45 | 25 | 19 | 24 | 22 |

Qm7 | 11 | 2 | 4 | 3 | 2 |

Total | 968 | 768 | 673 | 788 | 752 |

Level of agreement between event records

The sensitivity (to the Bayesian prior) analysis presented here (with four independently obtained event records) shows a general agreement value of 51.2 %, and an exact agreement of 38.7 %. The latter of these represents the percentage of vents we can be fairly certain have been assigned to the correct (actual) eruptive event. This in part represents the similarities between the three distribution sets used; increasing the number and detail of these distribution sets would therefore likely result in a decrease in this ‘exact agreement’ value between all event records.

The total number of events identified across Harrat Rahat varies from 673 (AF) to 788 (NI) (Table 4) which is, in part, a reflection of the sensitivity of the method to the distribution set used. The closest match to the actual number of events for the Harrat Rahat is unknown, but upper and lower boundaries for the total event number can be obtained via resampling from individual event determinations. Each vent is grouped into an event by randomly selecting an event record (EE, AF or NI) for it to follow. Should the selected vent grouping be a multiple-vent event, all vents within it are also grouped this way. This results in a new event record comprising events from each of the three distributions. Through extensive repetition and summing the number of events obtained each time, an upper and lower bound for the number of events was obtained. Here, the maximum and minimum values obtained from 100,000 runs were 668 and 804 respectively with a 95 % confidence level that the number of events lies between 748 and 755 for the 968 visible vents.

### Hidden vents

**x**_{i}) within the extent of their and subsequent (Camp and Roobol 1991a) lava flows and randomly assigned an age (

**τ**_{i}) within their temporal episode (Table 1). Equation 6 was applied until

**E[Y**_{ts}

*was equal to the number of visible vents*

**]**

**Y**_{ts}(Table 1). The number of vents randomly placed is an estimate of

**M**_{ts}(the actual number of volcanic vents), thus, for each episode, assuming vents are randomly distributed within the specific flow area, an estimate for the number of hidden vents can be obtained. Using this method, about one third of erupted vents are estimated to still be visible for Harrat Rahat (Table 6). More than half of these hidden vents are ascribed to the oldest temporal episode (Tw1) which spans from 10 Ma to 8.5 Ma. The two observed vents for Tw1 are located just south of the centre of Harrat Rahat, whereas extensive lava flows for this time period are identified across the whole of the southern half of Harrat Rahat (Camp and Roobol 1991a) which supports the idea of multiple hidden vents from this time period.

Hidden vent estimates for Harrat Rahat with temporal episode

Temporal episode | Actual number of visible vents ( | Average estimate of total vents ( | ±1 SD* | Maximum | Minimum |
---|---|---|---|---|---|

Tw1 | 2 | 1,159 | 813 | 7,040 | 10 |

Tw2 | 18 | 96 | 20 | 185 | 40 |

Th | 162 | 422 | 26 | 537 | 333 |

Qm1 | 422 | 677 | 20 | 750 | 599 |

Qm2 | 154 | 198 | 7 | 238 | 175 |

Qm3 | 77 | 92 | 4 | 109 | 79 |

Qm4 | 38 | 42 | 2 | 51 | 38 |

Qm5 | 39 | 40 | 1 | 47 | 39 |

Qm6 | 45 | 46 | 1 | 52 | 45 |

Qm7 | 11 | 11 | 0 | 13 | 11 |

Total | 968 | 2,783 |

Sensitivity analyses were performed to assess the effect of the erosion and burial constants on the estimated number of hidden vents. Varying the erosion constant (±20 %) varies the total estimate of hidden vents between 56 and 1,800 % of the original estimate due to the variation in the number of hidden vents from Tw1 from 25 to 50,000 (likelihood of a vent from Tw1 being visible today ~0). Reducing the burial constant to 0.4 (40 % chance of burial by subsequent flow) resulted in a 13 % decrease in the total number of hidden vents, and increasing it to 0.6 (60 % chance of burial) resulted in a 17 % increase in total number of hidden vents. These analyses suggest that the hidden vent estimates presented here should be updated as further data constraining the age, erosion and the number of flows across the harrat become available.

As is to be expected, the level of uncertainty in the potential number of hidden vents is substantially larger for older time periods than for younger as shown by the minimum and maximum values in Table 6. This is explained mainly by the increase in total area covered by subsequent flows, i.e. the proportion of the total lava flow area covered by younger flows is substantially greater for the oldest sets of vents. The uncertainty is also greater for the longitudinal centre (spine) of Harrat Rahat where the total lava thickness is greatest (up to 380 m, Blank and Sadek 1983) which approximately coincides with the highest regions of observed vent density. This variation in flow thickness could be used to further inform the hidden vent estimation by use of a spatially dependent burial function (vents placed in areas of thicker flow being more likely to be buried). However, to include this, an estimate of the cone height distribution would be necessary as it is possible that cones that form at locations of highest vent density (and greatest lava flow thickness) are larger than those cones that form at the edges of the field whereas estimates of vent size are likely to be biased in the opposite manner (more of a vent is likely to be visible in regions with less lava).

## Discussion

### Temporal recurrence rate implications

*λ*

_{t}) can be estimated via:

*, total number of eruptions;*

**N**

**t**_{0}, age of oldest eruption and

**t**_{y}, age of youngest eruption. The

*vent*temporal recurrence rate estimate (

*λ*

_{t, vent}) for the whole of Harrat Rahat is therefore 9.7 × 10

^{−5}vents/year (or 1 vent produced every ~10,000 years). With a 95 % confidence level that there are 748–755 visible events in Harrat Rahat, the average

*event*temporal recurrence rate estimate (

*λ*

_{t, event}) is 7.5 × 10

^{−5}events/year, or one eruptive event every 13,300 years. This value is ~80 % of the equivalent vent rate. This assumes a constant eruption frequency over the last 10 Ma in Harrat Rahat which does not appear to be the case (Fig. 5). Estimated recurrence rates of basaltic volcanic fields vary by several orders of magnitude, e.g. 5 × 10

^{−4}vents/year for the Eifel Volcanic Field in Germany, 1 × 10

^{−5}vents/year in Pancake, USA and Yucca Mountain (Connor and Conway 2000), and the recurrence rate for Harrat Rahat falls well within but towards the lower frequency end of this range.

By dividing the estimated number of hidden vents (Table 6) by the lowest estimate of average number of vents per event per episode (Fig. 5 inset), an upper bound for temporal recurrence rate can be determined. This results in an estimated 2,553 eruptive events over the last 10 Ma with the upper bound for the average event temporal recurrence rate estimate (*λ*_{t, event}) equal to 2.6 × 10^{−4} events/year, or one eruptive event every 3,900 years.

The temporal variation of the average number of vents per event ranges from ~2 to 1 (Fig. 5 inset). This ratio decreases with age, which suggests either that the older structures were more likely to have eroded or been buried/obscured by subsequent eruptive material, or the number of vents per event is increasing, or both. If the latter possibility is valid, it indicates that the behaviour of the volcanic field is changing over time, with newer eruptions more likely to appear as multiple-vents or fissures than as single-vents. Increasing crustal and asthenospheric tension across this area in the more recent lifespan of the field (Camp and Roobol 1989) could have led to a greater tendency for multiple-vent and fissure eruptions, although further investigation of other volcanic and tectonic data would be needed to confirm this.

Further age data would undoubtedly shed more light on the temporal evolution of Harrat Rahat by reducing the uncertainties in event identification and hidden vent estimation, as well as help identify periods of waxing and waning and any hiatuses in the temporal evolution of Harrat Rahat. Moufti et al. (2013) present new ^{40}Ar/^{39}Ar ages for 25 samples taken from the northern part of Harrat Rahat which suggest that flows as old as 10 Ma (Tw1) extend through this region which would increase hidden vent estimates as the area for these older vents is increased.

### Spatial recurrence rate implications

*, total number of eruptions;*

**N***, kernel bandwidth and*

**h**

**|x**_{i}−

*, distance between*

**x|***th vent at*

**i**

**x**_{i}and location

*. Figure 6 shows the variation in spatial recurrence rates across Harrat Rahat for both vents and the grouped and individual-vent events. The expert elicitation event record was used for this illustration as it shows the highest level of agreement with the other three event records (Table 5). The use of events concentrates the spatial recurrence rate into predominantly smaller, higher density areas with values of approximately an order of magnitude larger than those seen when considering all vents as independent. This results from decreasing the spatial dispersion of probability caused by a large number of vents. The event spatial recurrence rate estimate also identifies an area of higher event density offset slightly to the east of the main spine of Harrat Rahat (Fig. 6d). This area coincides with the occurrence of the most evolved compositions observed in the field (trachytes, benmoreites). These evolved rocks suggest magma is being stored in the crust (Moufti and Moghazi 2012).*

**x**### Quantitative method

*all*eruptive events across the whole of Harrat Rahat. Without knowledge of additional eruptive events, this is impossible to determine directly. However, if a relationship between vent and event dimensions is assumed, the individual volcanic vents of 641 AD and 1256 AD can be compared to field-wide observations. Figure 7 compares the estimated vent diameters for the two historical eruptions (Qm7) with all older (pre-Qm7) vent diameters. The former are smaller than the latter which may suggest that the resulting a posteriori distributions for event length and thickness may be skewed. However, further investigation into other parameters (e.g. eruption type, magma composition) are necessary to assess how typical (or atypical) the two most recent eruptions were of the overall behaviour of Harrat Rahat.

By design, the a posteriori distributions for the non-informative priors were dominated by the dimensions of the historical eruptions. This is especially evident when considering the deviation from dominant strike angle (Fig. 3). For the other two priors, where the prior and additional data align, a more concentrated a posteriori distribution was obtained (Fig. 3), e.g. analogue volcanic field a priori lengths were between 500 and 3,000 m and the historical events lengths fell well within this distribution (820 and 1,520 m), causing the a posteriori distribution range to compress. Where the prior and additional data disagreed, a more distributed a posteriori distribution was obtained, e.g. analogue volcanic field thickness. With future geological work planned in Harrat Rahat it is likely that more vent groups will be definitively identified as multi-dimensional or single-vent events. Increased numbers of known events will greatly benefit this analysis as the actual distribution of eruptive event dimensions becomes better constrained.

### Qualitative versus quantitative event determination

During a qualitative analysis, a large amount of varied information is often considered, e.g. the relative age of vents as interpreted by the colour or degree of weathering, the overlap of deposits and the relative location of vents in relation to a broader region. Therefore, it is likely that the quantitative method uses a subset of the data used for a qualitative analysis since every event characteristic cannot be quantified for every vent. However, in any qualitative analysis, even in rare, well-exposed areas such as Harrat Rahat, there is unknown aleatoric and epistemic uncertainty leading to unquantifiable, subjective and non-repeatable results. The quantitative method presented here is a robust, repeatable way of including these characteristics and uncertainties. The results of the quantitative method may be substantially improved by increasing the number of event characteristics, which could be facilitated by expansion of the method to include incomplete/locally available data. The qualitative analysis identified 752 events for Harrat Rahat (Table 4), well within the 95 % confidence range based on the quantitative event records. Furthermore, general agreement levels of greater than 65 % were found with all of the quantitatively determined event records (Table 5). This suggests that the quantitative method outlined here allows determination of an event record to a relatively good degree of reliability even when using only a few event characteristics, especially with regards to the total number of events, which has significant implications for temporal recurrence rates. The largest variation between quantitative and qualitative event records is with that of the analogue volcanic field. This is likely due to the more densely concentrated probability for the a posteriori distribution for event length which, in turn, refers us back to how well the historical eruptions reflect those of events across the whole of Harrat Rahat. The more concentrated analogue volcanic field distributions resulted in vents in close proximity being grouped more frequently into longer multiple vent events, similar to the 641 AD and 1256 AD eruptions. If it is assumed that the qualitative assessment represents the most likely event record, this would suggest that the two historical events do not reflect event dimensions for Harrat Rahat very well and that the expert elicitation distributions give the most reliable quantitative result.

### Hidden vents

The estimated ratio of hidden to observable vents was approximately 2:1, i.e. only a third of those erupted are still visible today. Accepting this potential high rate of vent “loss”, the quantitative analysis proposed here is thus more robust than a qualitative analysis. It also has the potential to be more accurate under certain conditions; for example, a qualitative assessment is less likely to group two vents separated by a subsequent lava flow (which may or may not be covering a third vent) than the quantitative method which would not be incorrectly biased by this visual aspect. However, this may also suggest that the analogue volcanic field record may be the most accurate as it grouped more vents into multiple-vent events than the qualitative assessment.

## Conclusions

Temporal and spatial recurrence rates are required for almost all aspects of volcanic hazard analyses. However, many difficulties are encountered during the estimation of these recurrence rates which tend to result in the inclusion of numerous assumptions, the validity of which cannot often be ascertained. Aleatoric uncertainties can be reduced by removing some of these assumptions and, by quantifying the epistemic uncertainties, upper and lower bounds for recurrence rates can be obtained.

The two assumptions most often imposed on recurrence rate estimation in volcanic fields have been tackled here: 1. That a volcanic eruption is a point process (discrete in both time and space) and 2. That the number of hidden vents cannot be quantified. The methods demonstrated here are applicable to any volcanic field, especially those with evidence of multiple-vent or fissure eruptions where the assumption of each vent being a discrete, independent eruptive event can no longer be justified and for regions where it is highly likely that some of the eruptive vents are no longer visible. It is of particular use for the majority of volcanic fields where (unlike Harrat Rahat), a qualitative analysis is prevented by vegetation cover, burial or severe erosion.

Further work could expand this statistical method for event determination to include incomplete data sets, the hidden vent method could also be improved with lava thickness to vent height data, and there is the potential to combine the two methods using a Monte Carlo approach to randomly place ‘hidden’ vents into the temporal episode of interest during event determination.

## Notes

### Acknowledgements

This work is part of the VORiSA (Volcanic Risk in Saudi Arabia) collaborative project between King Abdulaziz University and The University of Auckland. We thank W. Aspinall for assistance with the expert elicitation questions. K. Kiyosugi, A. Gudmundsson and an anonymous referee provided valuable feedback on the original manuscript.

## Supplementary material

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