Introduction

Improving hazard assessments with accurate probabilistic forecasts is an ongoing task, driven by high and increasing population densities around volcanoes across the world (Small and Naumann 2001). An unexpected large-scale eruption may trigger volcanic disasters (e.g. Krakatau 1882; Self and Rampino 1981; Mt. Pelee 1902; Tanguy 1994; Mt. Vesuvius AD79; Sheridan et al. 1981), but even ‘small’ eruptions (< 0.01 km3 of pyroclastics), like that of Ruapehu (New Zealand) in 1995–1996, impact greatly on our increasingly more technological societies (Johnston et al. 2000). Long-term dormant volcanoes pose a significant compounding risk, because the risk is often underestimated by their inhabitants. For example, Pinatubo was dormant for 650 years prior to its devastating 1991 eruption (Dartevelle et al. 2002; Gaillard 2006), misleading the local population. Even with an increasing understanding of volcanic systems, many mafic to intermediate volcanoes with irregular reawakening eruption patterns remain inexplicably diverse in behaviour and poorly predictable (e.g. Miyaji 1988; Scandone et al. 1993; Melnik and Sparks 1999; Andreastuti et al. 2000). Robust eruption forecasts at many volcanoes are hindered by a lack of well-documented volcanic histories. Event records that are long and detailed enough to enable identification of fundamental eruption recurrence patterns are rare.

In this study, a new and highly detailed eruption record extending back to 30,000 cal year BP was examined for Mt. Taranaki in order to test methods of probabilistic eruption forecasting. The most important feature of this new record was that it spans a wide dispersal arc around this volcano for the first time (Fig. 1). Previous studies (Turner et al. 2008, 2009) used one or two stratigraphic records to develop probabilistic forecasts, but this base data has been superseded in recent studies by Damaschke et al. (2017a), who added four new lake and peat tephra records. The distribution of these sampling sites around a ~ 120° northeast-southeast arc downwind of the volcano provides the opportunity to track the dominant tephra dispersal directions and also evaluate tephra preservation biases and variations between sites.

Fig. 1
figure 1

Location map of the tephra deposition sites (pv = previous studied sites by Turner et al. 2008, 2009; highlighted in yellow), Cape Egmont, western North Island, New Zealand. Stars indicate specific coring locations. TVZ Taupo Volcanic Zone

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A well-constrained multiple-site event stratigraphy makes the overall eruption history record more representative and inclusive of the actual eruptive behaviour, providing a more robust basis for improved probabilistic hazard forecasts. Nevertheless, combining sequences from more than one site must be conducted with care, to avoid over- or under-estimating the true eruption frequency. This is in part due to self-similar chemical compositions making individual andesitic tephra bed correlation challenging (Cronin et al. 1996; Donoghue et al. 2007; Damaschke et al. 2017a, b). Statistical methods for correlating multiple records using radiocarbon age and stratigraphy are well demonstrated (Turner et al. 2009; Green et al. 2014), but the quality of the result depends on the accuracy of the age determinations. Particularly difficult is the matching of near-contemporaneous tephras, as well as situations where the sedimentation regime trapping tephra deposits is discontinuous, or highly variable between sites. These issues are addressed in this study and a new probabilistic eruption hazard forecast is presented for Mt. Taranaki. Further, this paper outlines how to avoid pitfalls during stratigraphic merging procedures in order to derive more reliable probabilistic eruption forecasts for periodically active volcanoes.

Eruption history of Mt. Taranaki

Mt. Taranaki is an andesitic stratovolcano located in the western North Island of New Zealand (Fig. 1). Its current period of dormancy has lasted for ~ 200 years (Platz et al. 2012). Volcanic activity is known from this centre for over 170,000 years, based on mass flow and tephra deposits on its lower flanks and ring-plain (Alloway et al. 1995, 2005; Turner et al. 2008, 2009, 2011a; Zernack et al. 2009, 2011). Alloway et al. (1995) estimated a sub-Plinian eruption frequency of one event per 330 years, based on post-32-ky tephra deposits, but implied that smaller-scaled eruptions were likely far more frequent. Taranaki-sourced tephras within distal lacustrine sediments have also been identified to the N-NE of the volcano, with one fall per 1.1 kyr identified in Waikato lakes (Lowe 1988), and one event per 1.4 kyr in Auckland palaeolakes (Shane 2005; Molloy et al. 2009). These also highly underestimate the underlying volcanic activity, because they only include the larger magnitude events that disperse ash up to 270 km and in a narrow specific wind direction to the NNE. The first temporal high-resolution tephrochronological studies at this volcano (Turner et al. 2008, 2009) identified a clear time-variability in eruptions, with semi-regular ~ 1500-year period cycles in eruption frequency noted over the past ~ 10 ky BP. Between the extremes of these cycles, there was a fivefold difference in the annual eruptive frequency of Mt. Taranaki. Based on a tephra record from sediments below Lake Umutekai, north-north-east of the volcano (Fig. 1), Turner et al. (2008) estimated a 37–48% chance of an eruption over the next 50 years. After combining these data with a second tephra record from a location southeast of the volcano (Lake Rotokare), some apparent gaps in the original eruption record were ‘filled’, leading Turner et al. (2009) to raise the 50-year probability for an eruption at Mt. Taranaki to 52–59%.

Approach and methods

Dataset—a new Mt. Taranaki tephrostratigraphy

A complete eruptive activity record is not achievable from a single stratigraphic section or core, due to the variability of preservation and wind dispersal of tephra from volcanoes. Here, we use a multiple-source temporal high-resolution eruption record recently obtained by Damaschke et al. (2017a), merged first by manual correlation and subsequently by a statistical matching approach. Damaschke et al. (2017a) recovered nine cores from four new deposition sites, including Lake Richmond, Eltham Swamp, Ngaere Swamp and Tariki Swamp, as well as combined those with two existing cores from Lake Umutekai and Lake Rotokare collected in earlier studies by Turner et al. (2008, 2009) (Fig. 1, Table 1).

Table 1 Applied dataset obtained in previous studies by Damaschke et al. (2017a)
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Mineral layers within these sequences were sampled and primary tephras identified by optical microscopy and geochemical analyses. A small number of mineral layers were not classified as tephras, because they were dispersed within host sediment and/or showed diverse physical and geochemical characteristics (Damaschke et al. 2017a). Tephras in multiple cores from single sites were previously combined using chemical groupings, age-depth models and mineral/physical characteristics to produce six site-composite tephra records.

Individual tephra ages were estimated using vertically well-spaced radiocarbon dates in each core (Damaschke et al. 2017a) and modelling the age between these using a piecewise cubic Hermitian interpolating polynomial (Fritsch and Carlson 1980). Calibration was performed using OxCal Version 4.2 (Bronk Ramsey 2013) and the Southern Hemisphere SHCal13 atmospheric calibration curve (Hogg et al. 2013). A summary of the dataset used as the basis of this study is given in Table 1.

Temporal models for eruption occurrence

Several statistical models have been applied to model eruption histories. The simplest approach, used mainly with sparse datasets is a Poisson process (Wickman 1966; De la Cruz-Reyna 1991), where repose intervals are described by an exponential (memory-less) distribution. Hence, the rate of volcanic eruptions λ(t) is a constant λ independent of time t. The probability of an eruption onset occurring in a short time interval (t, t + Δ) is approximately λΔ, and thus λ can be thought of as the instantaneous rate of occurrence.

A memory-less repose assumption does not fit when repose intervals are more complex and variable over time (Wickman 1966; Bebbington and Lai 1996a). Hence, Bebbington and Lai (1996a) introduced the Weibull renewal model. A renewal process depends on the inter-event time τ, rather than on the absolute time t. The inter-event times are independent and identically distributed according to a known probability density function ƒ(τ). Hence, the time to the next event is dependent on the time since the last event (e.g. Reyment 1969; Bebbington and Lai 1996a; Bebbington 2010). The occurrence rate of a renewal process is given by:

$$ \uplambda (t)=\frac{\mathrm{f}\left(t\hbox{--} s\right)}{1\hbox{--} F\left(t\hbox{--} s\right)}, $$
(1)

with t − s = τ as the elapsed repose time, hence,

$$ \uplambda \left(\uptau \right)=\frac{\mathrm{f}\left(\uptau \right)}{1\hbox{--} F\left(\uptau \right)}, $$
(2)

where the most recent event occurred at time s < t, and ƒ(τ) = F(τ)’ is the renewal density function. Marzocchi and Bebbington (2012) summarised a large number of renewal distribution functions used in volcanology, but that used here is the Weibull distribution, where:

$$ \mathrm{f}\left(\uptau \right)=\alpha {\left(\upbeta \uptau \right)}^{\upalpha -1}\ \exp \left[-{\left(\upbeta \uptau \right)}^{\upalpha}\right],\alpha, \beta >0, $$
(3)

with α and β being shape and scale parameters, respectively.

If α = 1, the Weibull distribution reduces to an exponential distribution; if α < 1, the model is ‘over-dispersed’, or shows clustering behaviour; and if α > 1, the result is a semi-periodic distribution with a mode at τ = (1/β)(1 − 1/α)1/α. The Weibull renewal process has been successfully fitted to historical eruption data from a large number of volcanoes (e.g. Bebbington and Lai 1996a, b; Cronin et al. 2001; Watt et al. 2007; Dzierma and Wehrmann 2010; Garcia-Aristizabal et al. 2012).

However, when the distribution of inter-event times is bimodal, a mixture of Weibull distributions may be more appropriate (Jiang and Murthy 1995, 1998; Turner et al. 2008):

$$ f\left(\tau \right)=p{\alpha}_1{\beta}_1^{\alpha_1}{\tau}^{\alpha_1-1}\exp \left[-{\left({\beta}_1\tau \right)}^{\alpha_1}\right]+\left(1-p\right){\alpha}_2{\beta}_2^{\alpha_2}{\tau}^{\alpha_2-1}\exp \left[-{\left({\beta}_2\tau \right)}^{\alpha_2}\right], $$
(4)

where 0 < p < 1 is the mixing proportion, and the modes are at \( \tau ={\beta}_1^{-1}{\left(1-1/{\alpha}_1\right)}^{1/{\alpha}_1} \) and \( \tau ={\beta}_2^{-1}{\left(1-1/{\alpha}_2\right)}^{1/{\alpha}_2} \). Given any density function ƒ(τ) and underlying inter-event times τ, the parameters can be estimated by maximising the log-likelihood:

$$ \mathrm{logL}\left(\uptau \right)=\left[1\hbox{--} F\left({\uptau}^{\ast}\right)\right]{\prod}_{i=1}^{\mathrm{n}}f\left({\uptau}_i\right), $$
(5)

where τ* is the elapsed repose time since the last event and τ1,…, τn are the inter-event times. Turner et al. (2008) used Monte Carlo simulation to model uncertainties associated with event ages and hence inter-event times. To determine the start of the observation period is effectively impossible; hence, observation is normally assumed to start at the first recorded eruption.

The method of moments or least squares are alternative techniques to estimate the parameters of a given distribution (Press et al. 1992; Bebbington and Lai 1996b; Boudt et al. 2011). A more serious issue than the calculation of parameter values is that of selecting the distribution that should be used. Distributions with more parameters, if they include a nested simpler model (for example, the exponential distribution contained in the Weibull), will always fit better. However, the more parameters used in any model, the higher the likelihood of overfitting. To evaluate which model best balances this bias-variance trade-off, the Akaike Information Criterion (AIC; Akaike 1977) can be applied, which measures the goodness of fit using:

$$ \mathrm{AIC}=\hbox{--} 2\mathrm{logL}+2k, $$
(6)

where k is the number of parameters and logL is the log-likelihood of the model. The model with the lower AIC value is favoured. Given any two estimated models, a difference of 1.5–2 in the AIC value is usually considered as significant (Akaike 1977).

Results

Eruption frequency records

Tephras recovered in lake and peat sediment cores surrounding Mt. Taranaki are our basis for understanding eruption rates. The time frame encapsulated by the records and the number of tephra layers preserved is not uniform across the cores (Fig. 2, Table 1). Estimates were thus made of the time-varying annual eruption rates over time at each single site with the use of a Gaussian kernel smoother (Silverman 1984, 1986; Wand and Jones 1994) with a bandwidth of 100 years, (Fig. 2a). However, integration across all sites is important, because one or two sites are likely to have little chance of recording all tephra falls due to variable wind and tephra dispersal patterns (Carey and Sparks 1986) and/or variations in eruption style and magnitude (Woods 1995; Bursik 1998).

Fig. 2
figure 2

a The rates of deposition of tephras within lake and peat sequences recovered from six different localities at Mt. Taranaki (age data from Damaschke et al. 2017a) presented as histograms of tephras deposited over 500-year intervals (left axis, grey columns), and annual tephra deposition rates generated using a Gaussian kernel smoother (Silverman 1984, 1986; Wand and Jones 1994) with a 100-year bandwidth (right axis, red line). Numbers indicate intervals during which particularly high rates of tephra deposition occurred. b Cumulative deposition of the same tephra units. Letters indicate periods of low rates of tephra deposition and/or no deposition. Yellow coloured fields represent disturbed or oxidised/dried sediment horizons

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At Lake Richmond, 129 Taranaki eruptions were recorded with an average interval of 113 years, similar to that found at Lake Umutekai (111 years). Both lakes show similar cyclical patterns of ~ 1000–1500-year-long periods with particularly high rates of tephra deposition (HRTD) centred at ~ 0.5, 2, 3.5, 5, 7, 8.5, 10.5 and 12.5 cal ky BP (indicated by numbers 1–8 in Fig. 2a), separated by intervals of low rates of tephra deposition (LRTD) and/or a lack of tephra deposition (both indicated by letters A-H in Fig. 2b). Older variations in the rate of tephra deposition are documented only within the Tariki Swamp record (HRTD intervals-9 and -10 at ~ 15 and 17 cal ky BP, respectively, separated by LRTD interval-I at ~ 16.5 cal ky BP; Fig. 2a, b).

The HRTD interval-1 (~ 0.5 cal ky BP; Fig. 2a) is only recognised in the Lake Richmond and Lake Rotokare cores, because most of the Taranaki peatlands have been drained and developed into pasture, causing oxidation, mixing and loss of the upper ~ 1 m of sediment. In Lake Umutekai, uncompacted suspended material at the top of the sediment core was lost during extraction (Turner 2008). HRTD interval-1 is preceded by a distinct period of ~ 700 years where no tephra deposition is observed (A; Fig. 2b).

The HRTD interval-2 (~ 2 cal ky BP) is recognised in all sediment cores, but is most prominent within the Lake Richmond record (0.028 units per year; Fig. 2a). HRTD interval-2 was generated by eruptions from two vents, mafic eruptions from Fanthams Peak (1966 m elevation on the southern flanks of the volcano) producing the Manganui tephras, which intercalate with more silicic units (e.g. Maketawa tephras) from the summit vent of the 2518 m Mt. Taranaki (Damaschke et al. 2017b; Torres-Orozco et al. 2017). The preceding HRTD interval-3 (~ 3.5 cal ky BP) only encompasses tephras from the summit vent, including the prominent and widespread Inglewood tephra unit. It is noteworthy that the HRTD intervals-2 and -3 within the Lake Rotokare record contain tephras that are ~ 300–500 years older than their apparent age determinations of their tephra correlatives (Fig. 2a).

The HRTD interval-4 (~ 5 cal ky BP; Fig. 2a) is best observed within the Lake Richmond and Lake Umutekai cores, because during this time interval discontinuous peat accumulation hindered complete tephra preservation within the peat records. The HRTD interval-5 (~ 7 cal ky BP; Fig. 2a), however, is also prominent within the Ngaere Swamp, and so expands the length of this phase of high rates of tephra deposition to ~ 1000 years. The preceding HRTD interval-6 (~ 8.5 cal ky BP; Fig. 2a) is most prominent within Lake Umutekai and preserves tephra layers not recognised/preserved within the rest of the tephra records (i.e. U-78 to U-87, ~ 9–9.4 cal ky BP; Damaschke et al. 2017a).

The HRTD interval-7 (~ 10.5 cal ky BP; Fig. 2a) is noted within most records, except Tariki Swamp (section of disturbed sediment), and it is too old for Lake Rotokare. A tephra deposition rate of up to 0.02 units per year is recorded for this interval. The HRTD interval-8 (~12.5 cal ky BP; Fig. 2a) appears to be time-transgressive (e.g. Tariki Swamp vs. Ngaere Swamp). This probably results from large age errors, because a distinctive geochemical fingerprint of this tephra sequence allows its correlation between sites. The HRTD interval-8 is separated from HRTD interval-9 (~ 15 cal ky BP) by a period of non-tephra deposition lasting > 500 years (H; Fig. 2b). The HRTD intervals-9 and -10 (~ 17 cal ky BP) are only documented within the Tariki Swamp tephra record, because none of the other sequences contained recognisable tephras in this time interval.

It is clear from these data that multiple tephra records can help to construct a more robust and complete eruption record from a volcano. Site-specific gaps, overlaps and/or offsets in the rate of tephra deposition can be recognised between locations. However, it must be remembered that, although this NE-SE sampling arc is downwind for the prevailing winds, it covers only about 120° around the eastern side of the volcano and is likely missing eruptive events dispersed to the unfavourable upwind direction to the south and west, as well as effusive eruptions with minor or no associated falls. Hence, the frequency data remain an underrepresentation of the true volcanism and any synthesis of them is only a minimum-estimate of reality. Methods of combining multiple eruption deposits and sequences into a single record of eruption events are described below.

Merging tephra records

A first-order composite stratigraphic record was created by conventional manual merging of core segments based on distinctive appearance or properties of individual tephras or sequences of them, reinforced by distinct geochemical tephra sequences (Damaschke et al. 2017a). This ‘composite manual’ stratigraphy includes ~ 228 tephras, spanning the last 30 cal ky BP. The quality of this record varies, with the older part contained within only one core and being compromised by a large break in deposition. As a first-order estimate of eruption frequency, this composite record is most reliable only in the post-17.5 cal ky BP part. This partial record includes ~ 189 tephra layers with an average interval between eruptions of 92 years (0.011 tephra deposition events per year, Fig. 3a).

Fig. 3
figure 3

Variation in the deposition rate of tephra layers across all sites analysed presented as histograms of tephras deposited over 500-year intervals (left axis, columns), and annual tephra deposition rates generated using a Gaussian kernel smoother (Silverman 1984, 1986; Wand and Jones 1994) with a 100-year bandwidth (right axis, red line). Cumulative deposition rates are also shown. Two different matching models are presented: a A manually merged composite record with conservative traditional stratigraphic matching, based on tephra appearance (individual and patterns) and geochemical matches; and b a statistically combined record developed using a matching algorithm (following the approach of Green et al. 2014). Numbers indicate periods of high rates of tephra deposition and letters indicate periods where tephra deposition rates were low including two quiescence periods (A and H; marked as red-shaded fields) (refer to text). Note: HRTD intervals-3 and -4 may indicate one long-lasting high-frequency interval, since no repose times > 200 years are recorded within this particular period (dotted line). LRk Lake Rotokare, LRi Lake Richmond, LU Lake Umutekai, NS Ngaere Swamp, ES Eltham Swamp, TS Tariki Swamp

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A second means of merging tephra sequences applied the Green et al. (2014) matching algorithm, based on interpolated age determinations of tephra layers at each site, with the stratigraphic-order at each site fixed, and geochemical compositions used to force (or exclude) specific tephra matches. Due to the stratigraphic constraints of the process, well-matched tephras produce flow-on consequences to correlations of units stratigraphically above or below them. For this dataset, the resulting statistically combined record comprises ~ 272 tephra layers spanning the last 30 cal ky BP, including ~ 231 tephra layers within the continuous sequences younger than 17.5 cal ky BP (Fig. 3b). The latter implies an average ashfall frequency of one event per 75 years (0.013 tephra deposition events per year).

The composite manual approach provided an underestimate of event frequency, because it cobbles-together blocks of tephra sequences at individual sites, and this mostly avoids any possibility of doubling-up closely timed events. Where strong winds prevail (common in New Zealand), tephras may be dispersed in very narrow lobes (e.g. Ruapehu 1995 and 1996 eruptions, Cronin et al. 2003; Te Maari eruption 2012; Turner et al. 2014). Hence, a single site will never capture a record of all eruptions. A good example of this is at ~ 4.3 cal ky BP, where the ‘best-core’ record is a segment of the Lake Umutekai sequence that misses tephras seen at Lake Richmond (C; Fig. 3a). This gap is ameliorated within the statistically combined record (Fig. 3b).

Using a piecewise cubic Hermitian interpolating polynomial (Fritsch and Carlson 1980) to develop an age model for each sediment record, uncertainties vary down-core depending on the intervals between dates and the curvature of the fitted polynomial. Thus, sudden changes in sediment accumulation rate and unconformities may blurred in the resulting age-depth relationships (Telford et al. 2004; Blaauw 2010). Comparing cores, there is an apparent temporal offset of HRTD intervals-2 and -3 in the Lake Rotokare record, compared to the rest of the frequency records (Figs. 2 and 4). Geochemical compositions of the glass, titanomagnetite grains and whole-lapilli samples show however that these horizons should be correlated (Damaschke et al. 2017a). Thus, the depositional ages in the Lake Rotokare record are inaccurate. At least one of the 14C determinations for this site may have been contaminated by an old carbon effect (e.g. Keaveney and Reimer 2012; Philippsen 2013; Hart 2014). Lake Rotokare is unique amongst the sites studied here in being a landslide-dammed lake within a steep valley set in strongly eroding Tertiary-aged calcareous mudstone and sandstone. Turner (2008) noted frequent landslide deposition into this lake, and the tephra layers are intercalated with the mud-rich landslide debris. The age inconsistencies meant that the Lake Rotokare tephra ages could not be used directly within the statistical matching process, without detailed in-core correlation adjustments. Due to the multiple age inconsistencies, such corrections would have needed additional overlapping core coverage to avoid potentially circular logic in the adjustments. We thus omitted a potentially complex site-correction in this study, and subsequently excluded the Lake Rotokare record from further statistical matching procedures.

Fig. 4
figure 4

Annual tephra deposition rates of each single record in comparison with the statistically combined record. Rates are generated using a Gaussian kernel smoother (Silverman 1984, 1986; Wand and Jones 1994) with a 100-year bandwidth. Note the offset of the Lake Rotokare events

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Highly variable background sedimentation regimes may also disrupt statistical merging procedures, as observed for certain horizons within the Tariki Swamp record (Figs. 2 and 4). In parts of this sequence, thick silt loam horizons occur, in which tephra deposits are diffuse and possibly reworked (Damaschke et al. 2017a). Consequently, age determinations may be biased, and have large error ranges that add disruptive ‘noise’ to the matching procedure. For this reason, the affected part of the Tariki Swamp record (e.g. 3–12 cal ky BP) was also excluded from further statistical matching.

Past studies have shown that additional geochemical data can provide valuable constraints within a statistical matching procedure (Turner et al. 2009; Green et al. 2014). These data can be used to prevent matches with incompatible compositions, and also force some that are obviously distinctive (Damaschke et al. 2017a; Table 2), over-riding and correcting temporal positioning. The composition of mineral, glass and whole-rock samples from many andesitic volcanoes rarely allow unique correlations (e.g. Cronin et al. 1996). Therefore, only a few geochemically fixed points (Table 2) were applied. Using these, we derived the most robust statistically combined record yet for Mt. Taranaki eruptions (Figs. 3b and 4).

Table 2 Correlation pairs used in the merging algorithm (c.f., Green et al. 2014). They are based on the physical and geochemical characteristics of the tephras, as well as, on their stratigraphic position and age constraints (implemented by Damaschke et al. 2017a)
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Irrespective of the merging method, all records of eruption here and in earlier studies (Turner et al. 2008, 2009, 2011b) show a similar pattern of cyclic variation in eruption frequency over time at Mt. Taranaki (Fig. 3a, b). Our new data defines ten periods of high tephra deposition/eruption frequency, each ~ 1000–1500 years in duration. These are separated by eight periods of low eruption frequency, including two intervals of > 500 years with no tephra deposition (A and H, centred on ~ 1 and ~ 14 cal ky BP, respectively; Fig. 3a, b), which may indicate two significant quiescence periods at Mt. Taranaki. On the other hand, two of the HRTD intervals-3 and -4 (from 6 to 3 cal ky BP) are so close together that they could be considered a single continuous high-activity period, since no > 200-year. repose intervals occur (Fig. 3b). Over this 3000-year period, the average interval between ashfalls is 52 years (0.019 events per year).

Probabilistic modelling

Forecasting of a future tephra-producing eruption was carried out using the final statistically combined tephra dataset for < 14.5 cal ky BP events using: (1) the exponential distribution (Poisson process), (2) the Weibull renewal and (3) the mixture of Weibull renewal models. The Tahurangi eruption, dated by dendrochronology at AD1755, is the last recorded significant tephra fall (Druce 1966). However, the summit dome was emplaced and partly collapsed after this in the Sisters eruption. In the absence of reliable radiocarbon dating, Platz et al. (2012) estimated this last eruption of Mt. Taranaki occurred between AD1785 and AD1820. This age range was incorporated as the last event in the forecasting model.

The estimated parameters and average log likelihoods for each model are shown in Table 3. In total, 200 events span the last ~ 14.5 cal ky BP in our final record. A Monte Carlo simulation was run with n = 100 (i.e. 19,900 sampled inter-event times) to derive the fitted densities shown in Fig. 5. In an update from previous procedures (Turner et al. 2008, 2009), this study used calibrated calendar ages applying OxCal Version 4.2 (Bronk Ramsey 2013) and the Southern Hemisphere SHCal13 atmospheric calibration curve (Hogg et al. 2013).

Table 3 Fitted parameters and average log likelihoods for the models tested for both of the last explosive eruptions of Mt. Taranaki at AD1785 and AD1820
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Fig. 5
figure 5

Histogram of 19,900 sampled inter-event times based on Monte Carlo simulations of the new statistically combined Mt. Taranaki eruption record. Curves show the different densities fitted for this dataset with AD1785 (red) and AD1820 (blue) as last volcanic activity events

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The best fit for the distribution of modelled inter-event periods was achieved by the mixture of the Weibull model according to the AIC statistic (Table 3). The Poisson process is less well suited, because event times are systematically variable. The Weibull renewal distribution has α < 1 implying clustering behaviour. In this case, α is almost = 1; thus, the result is very similar to a simple Poisson process (indicated by the dotted and dashed lines almost overlying (Figs. 5 and 6). The mixtures distribution includes a strong primary mode at inter-event times of ~ 9 years and a secondary mode at inter-event times of ~580 years (Table 3). The overall pattern is similar in form to that in an earlier work (Turner et al. 2008, 2009), but with a much more refined outcome, indicated by increased α and p values (with p = 1 corresponding to a simple unimodal Weibull distribution; Table 3). The primary short-repose mode (mean repose ~ 65 years) now accounts for 98% of interval times, while the secondary long repose mode reduces to only 2% compared to 25–4% in previous studies. The long repose mode is also longer at ~ 580 years compared to ~ 165 and ~ 210 years in a previous work (Turner et al. 2008, 2009; Fig. 7b). This implies that activity at Mt. Taranaki is overall more frequent than previously detected, but there are better defined long breaks where tephra-producing eruptions are not occurring.

Fig. 6
figure 6

a Probabilities of no eruption of Mt. Taranaki occurring over future time periods, based on three models of inter-event distributions with AD1785 (red) and AD1820 (blue) as last volcanic activity events. b Annual eruption probabilities estimated for Mt. Taranaki, assuming the last volcanic activity event was at AD1785 (red) and AD1820 (blue). The Weibull renewal distribution is similar to a simple Poisson process

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Fig. 7
figure 7

a Probabilities of no eruption at Mt. Taranaki occurring over future time periods, based on the inter-event distribution constructed in each previous paper and for the statistically combined record built in this study, with AD1800 (compromise date between AD1785 and AD1820) as the last volcanic event. b Annual eruption probabilities of Mt. Taranaki estimated for different records proposed in previous studies and for the statistically combined record built in this study, assuming the last volcanic event was at AD1800

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The probability of no eruption in the next t years, given that the most recent event occurred at T years ago, is:

$$ \Pr \left(\uptau >T+t|\uptau >T\right)=\frac{\int_{T+t}^{\infty }f\left(\uptau \right)d\uptau}{\int_T^{\infty }f\left(\uptau \right)d\uptau}. $$
(7)

Assuming that the last eruption event occurred between AD1785 and AD1820, the probability of an eruption within the next 50 years is 33–42%, based on the best fitting mixture of the Weibull model (Fig. 6a). This is significantly lower than previous probability models using only two records, which suggested a 52–59% chance of eruption within the next 50 years (Turner et al. 2009; Fig. 7a). Plotting the annual eruption probability progression (Fig. 6b), the preferred mixture of the Weibull model indicates that a high-probability peak occurred at approximately AD1900 (annual eruption probability of ~ 0.016 or 1.6%) and this will decline to a trough at ~ AD2200 (annual eruption probability of ~ 0.003 or 0.3%). The current annual eruption probability is estimated at 0.01 to 0.013 (1 to 1.3%; Fig. 6b). This indicates that the likelihood of a long repose is steadily increasing with a steady decrease in the annual probability of a future eruption within the short to medium term future.

These results contrast strongly with previous estimates made by Turner et al. (2009) (Fig. 7), who estimated a current annual eruption rate of 0.016 (1.6%) with the trough occurring earlier (i.e. after 210 years in the two-core merged record). This subtle difference in the secondary mode of inter-event times strongly affects the eruption probability progression of Turner et al. (2008, 2009) compared to our update (Fig. 7b). Part of the reason for this is that Turner et al. (2009) used the Lake Rotokare record to fill potential gaps within the Lake Umutekai record. However, as revealed in here, the Lake Rotokare record is strongly, but non-systematically, biased in its age estimates. This led to an apparent infill of long intervals, and possibly an inflation of the true tephra fall frequency, and the differences in forecast results described above.

Discussion

Long and complete chronological eruption records from volcanoes are fundamental to robust probabilistic hazard models (e.g. Scandone et al. 1993, 2008; Behncke and Neri 2003; Turner et al. 2008, 2009). In this case, tephra deposition in a series of lake and peat bog sequences in a 120° downwind arc was used as a proxy for eruptions. It is clear that this sampling net will miss tephra dispersed in other, less-common wind directions, small-scale eruptions with only proximal deposits and effusive-only events. Putting this limitation aside, tephra-forming eruptions are hazardous to surrounding populations and tracking at least this type and scale of volcanism is a good way of understanding overall volcanic behaviour.

To build robust tephra records at any volcano, methods to extend from single-site to multiple-location stratigraphy are needed to capture a range of possible tephra dispersal directions, variations in eruption sizes/styles, and account for variable tephra preservation issues. Good age determinations for tephras rely strongly on a stable sedimentation background regime. In this study, one of the sites (Lake Rotokare) was discovered to have a distinctive old-carbon effect within its radiocarbon ages. It was not possible to isolate this in previous two-record combinations (Turner et al. 2009), because cross-validation required the information from the several new sites reported here. In other sites, parts of the records are disturbed, but with enough sampling sites to bridge these gaps, a good overall coverage can be achieved through subsequent merging procedures.

It has also been confirmed in this work that incorporating geochemical constraints is far more reliable for indicating incompatible matches, rather than forcing direct match-links, although the latter enormously decrease computation time (for this exercise, on the order of 2–3 days on an Intel i7 3.4 GHz desktop). Hence, a balanced approach in the use of geochemical covariates within the statistical merging procedures is suggested. Distinct titanomagnetite compositional groups for packages of tephras identified in earlier studies (Damaschke et al. 2017a) were useful to inform some correlations, but currently the matching algorithm does not include patterns of compositions, only single tephra compositions. Additional independent tools are needed to improve the correlation of tephras between multiple sites.

Probabilistic assessments of volcanic recurrence at other volcanic sites have been carried out, but due to the greater difficulties of assembling a complete record at polygenetic volcanoes, many studies have focussed on monogenetic volcanic fields and caldera or rift-zone volcanoes (e.g. Yucca Mountain region, Connor and Hill 1995; Ho and Smith 1997, 1998; Connor et al. 2000; Springerville volcanic field, Arizona, Condit and Connor 1996; Taveuni volcano, Fiji, Cronin et al. 2001; Auckland volcanic field, Molloy et al. 2009; Bebbington and Cronin 2011; Campi Flegrei caldera, Orsi et al. 2009, Bevilacqua et al. 2016). These tend to have a far more erratic temporal variation compared to long-lived subduction-related stratovolcanoes. However, Wadge (1982) conjectured that stratovolcanoes may exhibit periods of high-frequency activity interspersed by longer periods of quiescence.

Many studies (e.g. Klein 1982; De la Cruz- Reyna 1993; Behncke and Neri 2003; Cappello et al. 2013) of eruption forecasting have been made using historical data, which often means short (c. 200–400 year) eruption records at individual volcanoes. A number of these cases were shown to exhibit cyclic patterns on the order of decades to centuries (Bebbington 2010). Another longer, relatively complete, record is Mt. Vesuvius (Scandone et al. 1993, 2008), which shows a well-defined eruption cycle, where each cycle is initiated (or ended) by a large (sub-Plinian to Plinian) eruption, similar to the behaviour of Mt. Taranaki postulated by Turner et al. (2011b).

According to the present study, Mt. Taranaki exhibits strong cyclic patterns in event frequency, with an apparent long secondary mode in the repose time distribution, which may have implications for the details of its magmatic system. Those longer inter-event intervals may correspond to less-frequent feeding pulses of magma replenishment to lower-mid crustal storage systems between bursts of rapid supply/replenishment rates (Turner et al. 2011b). The cycles may also be linked with distinct tectonic regime processes, such as changes in the regional crustal stress, enabling magma movement and accommodation in the crust (cf., Walter et al. 2007; Watt et al. 2009; Caricchi et al. 2014). Regular gravitational loading and unloading of the volcano due to regular collapse and re-growth cycles as seen at Mt. Taranaki (Zernack et al. 2012) may also modify the stress-field enough to suppress or enhance magma ascent rates and/or dyke propagation (e.g. Pinel and Jaupart 2000; Muller et al. 2001; McGovern et al. 2013). Suppression phases could lead to periods of low-activity and be terminated due to unloading of the volcano by large debris avalanche events (Manconi et al. 2009; Pinel and Albino 2013). At Mt. Taranaki two major debris avalanches occurred during the time frame investigated in this study, the ~ 14 cal ky BP Motumate debris avalanche, and the ~ 7.5 cal ky BP Opua debris avalanche (Zernack et al. 2011). It is unclear if the earlier collapse had any influence, because it is at the beginning of the robust combined record. The window of the 7.5 cal ky BP terminating growth/collapse cycle is, however, captured in the new record, but clear pre/post-collapse variability is not immediately obvious, because shorter-period variability dominates this time frame.

Conclusions

This study has shown how important it is to assemble as large a number of independent records of eruption frequency as possible, in this case from high-sensitivity tephra deposition sites. The large number of sites allow potentially anomalous records to be identified (e.g. when dating is erroneous or deposition rates are too irregular). Adding additional stratigraphic records to such studies comes with the added complexity, however, of the need to find reliable means to combine and merge individual sequences to create a robust overall eruption record for modelling. Here, we conclude that applying statistical methods that compare age-error models between deposition sites, within a rigid stratigraphic requirement, is a first step. Adding geochemical constraints to this, in terms of direct layer correlations or by enabling exclusion of potential matches based on composition, can markedly improve the reliability of the process. Future work is needed to incorporate less well-defined geochemical criteria, such as the identification of temporal chemical groupings, as is seen at Mt. Taranaki. In addition, incorporation of proximal data and understanding missing elements of individual eruption records are needed.

Based on the best model yet for Mt. Taranaki, we estimate there is a 33–42% probability of an eruption occurring within the next 50 years. This is a reduction in likelihood compared to a previous work. The new record and model provide strong evidence that Mt. Taranaki is currently within a ‘long’ repose. We confirm that there is a cyclicity in eruption frequency over time at Mt. Taranaki, which was proposed in previous studies. There is one unusual double-period, however, between 3 to 6 cal ky BP, where there is highly frequent volcanism with no repose periods longer than 200 years. Understandings of the drivers behind this remain unclear, although it may relate to the aftermath of the latest debris avalanche from this volcano.

The principles applied here can be implemented at any similar ‘re-awakening’ volcano. It is thus important to build such high-resolution records at other sites in order to test whether cyclic variations in eruption frequency and bimodal distributions of inter-event times are a characteristic feature of andesitic stratovolcanoes.