A Review of Statistical Methods Used to Assess Probabilistic Hazard in Monogenetic

Basaltic Fields Elaine Smid, Jan Lindsay, and Gill Jolly

Report 1-2009.04 | June 2009

ISBN: [Print] 978-0-473-16142-2 [PDF] 978-0-473-16143-9

A Review of Statistical Methods Used to Assess Probabilistic Hazard in Monogenetic

Basaltic Fields Elaine Smid1*, Jan Lindsay1,2, and Gill Jolly3

1Institute of Earth Science and Engineering, The University of Auckland, Private Bag 92019, Auckland 2School of Environment, The University of Auckland, Private Bag 92019, Auckland

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*e.smid@auckland.ac.nz GNS Science, Private Bag 2000, Taupo

IESE Report 1-2009.04 | June 2009

ISBN: [Print] 978-0-473-16142-2 [PDF] 978-0-473-16143-9

Disclaimer: While the information contained in this report is believed to be correct at the time of publication, the Institute of Earth Science and Engineering, The University of Auckland, and its working parties and agents involved in preparation and publication, do not accept any liability for its contents or for any consequences arising from its use.

This report was prepared by IESE as part of the DEVORA Project, Theme 2, Objective 2.

Copyright: This work is copyright of the Institute of Earth Science and Engineering. The content may be used with acknowledgement to the Institute of Earth Science and Engineering and the appropriate citation.

TABLE OF CONTENTS Abstract ...................................................................................................................................... 1 Introduction ................................................................................................................................ 2 Temporal Trends ........................................................................................................................ 2

Recurrence Rates ................................................................................................................... 2 Average Recurrence Rate/Simple Poisson Models ............................................................. 3 Linear Regression Models .................................................................................................. 4 Non-Homogeneous Poisson Models ................................................................................... 6

Spatial Heterogeneity ................................................................................................................. 6 Univariate Statistical Methods................................................................................................. 6

Vent Density ....................................................................................................................... 6 Spatial Cluster Analysis ...................................................................................................... 9 Vent Alignments ............................................................................................................... 11 Shifts in Locus of Activity .................................................................................................. 13

Spatio-Temporal Models .......................................................................................................... 14 Spatio-temporal Nearest Neighbour Recurrence Rate Model ............................................... 14 Kernel Technique ................................................................................................................. 14 Alternative Methods .............................................................................................................. 16

Using GIS to assess point distribution .............................................................................. 16 Bayesian Inference for Long-Term Eruption Forecasting .................................................. 16

Studies on the Auckland Volcanic Field .................................................................................... 18 Eruption Magnitude and Frequency ...................................................................................... 18 Spatial Heterogeneity ........................................................................................................... 19 Analogue Fields .................................................................................................................... 20

Considerations ......................................................................................................................... 20 Cryptotephra & Geochronology ............................................................................................ 21 Petrology .............................................................................................................................. 21 Structure of the Crust ........................................................................................................... 21 Analogue/Miscellaneous ....................................................................................................... 22

Recommendations ................................................................................................................... 22 References ............................................................................................................................... 24

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ABSTRACT One main aim of the Determining Volcanic Risk in Auckland (DEVORA) project is to determine when and where future eruptions are likely to occur in the Auckland Volcanic Field (AVF). This literature review describes the statistical methods that have previously been used to represent spatial and temporal patterns in monogenetic fields in order to design an appropriate model for the AVF. Two end-member volcanic fields have been recognized in the literature—one with predictable volume of eruption, and another that is time-predictable. The activity in most volcanic fields, however, waxes and wanes with time and does not follow either of these patterns. Therefore, simple models, or ‘homogeneous Poisson models,’ cannot accurately represent the irregular timing of eruptions in a volcanic field--these methods are useful only in the sense that they offer a means to compare fields. A model that incorporates the waxing and waning of activity is necessary—most authors use and recommend a variant of the non-homogeneous Poisson model to mathematically represent temporal heterogeneity in monogenetic volcanic fields. Many factors may influence the location of eruptions: faults, variations in the strength and structure of the crust, magma flux, source plume migration, and plate motion, among others. Similar to the waxing and waning activity found in monogenetic fields over time, vent locations are usually irregularly spaced. The past pattern of vent locations in fields can be tested to determine if the field is random with respect to vent locations using the Hopkins F, Clark-Evans, or K function test. Several other tests have been designed to detect vent alignments and clusters of vents, which are common within volcanic fields. Authors recommend applying cluster analyses, such as a weighted-centroid cluster analysis or uniform kernel density fusion cluster analysis, before vent alignment detection, due to the way clusters may affect vent alignments. Several statistical methods can be used to determine the presence, location, and orientation of vent alignments within volcanic fields: the two-point azimuth, Hough transform, strip methods, and the two-dimensional Fourier transform methods. The most advanced of these methods is the strip method, which does not assume vent homogeneity, and comes in many forms. The spatial and temporal methods described above may be integrated into a spatio-temporal model, and used to create probability estimate maps. There are two types of this model: the kernel estimation technique and the spatio-temporal nearest-neighbour model. Ages do not need to be known for the kernel estimation technique, which gives it a distinct advantage over the spatio-temporal model. A combination of these two methods is also sometimes used. Additional data sets which may influence timing and location of future eruptions, such as geophysical structure, geochemistry, etc., may be incorporated into a spatio-temporal model by using Bayesian inference. Bayesian inference allows for a more accurate representation of the field as a whole. Any spatial data set may be turned into a likelihood function and incorporated into the model. As new information is discovered about the field, the model will become increasingly refined. Several statistical tests have been performed on the AVF, with sometimes contradictory results. Given that the activity of the AVF waxes and wanes (with possible ‘flare ups’ of activity), the irregular volumetric output and non-typical behaviour exhibited by the latest eruption at Rangitoto, and the evidence for multiple eruptions from a single vent, a non-homogeneous, complex model will most certainly be necessary to assess future eruption timing and location probabilities. This report recommends an integrated approach of many of the methods mentioned above, including Bayesian inference, to construct a probability estimate map for future eruptions in Auckland. The South Auckland Volcanic Field could potentially be used as an analogue to the AVF.

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INTRODUCTION One of the main objectives of the DEVORA project is to assess the volcanic hazard and risk of future eruptions in the AVF. The first step in analysing the probability of future events is to get a clear picture of the timing of activity and the location of previous events, as hazard probability models rely on temporal and spatial recurrence rates to construct hazard maps. This literature review outlines the various methodologies used to assess the temporal and spatial patterns found in other monogenetic basaltic volcanic fields in order to select and design an appropriate long-term hazard probability model to apply in the AVF. The first section of this report briefly describes each of the statistical methods that have been used to assess the temporal and spatial trends in monogenetic basaltic volcanic fields, outlines their advantages and disadvantages, data required (input), assumptions, and lists the studies in which they were used, with an emphasis on how each technique may or may not be applicable in the AVF. The last section consists of a review of the magnitude-frequency studies focusing on the AVF, and also describes the characteristics of the AVF that must be considered and the key questions that need to be answered before designing a model that adequately reflects behaviour seen in the AVF. This section concludes the report with recommendations for future work in the AVF based on the literature.

TEMPORAL TRENDS

Recurrence Rates Recurrence rates are used to assess the probability of an eruption occurring within some specified time period. Some questions that arise when calculating the recurrence rates in volcanic fields include:

• What is an ‘event?’ Is it satisfactory to count each vent as an event? How should one count polygenetic behaviour or ‘flare ups’ of activity?

• Is each event accurately dated?

• Could there have been events that now have no surface expression, and therefore, have

not been recognized? These questions are of particular importance in the AVF due to recent findings indicating that Rangitoto Volcano may have had two eruptive episodes, one 525 Cal yrs BP and a separate eruption 604 Cal yrs BP, rendering it a ‘compound monogenetic volcano’ as defined by Martin et al. (2003) (Needham, 2009). Moreover, there is evidence that eruptions from multiple vents (e.g., Crater Hill, Puketutu, and Wiri Mountain) may have occurred almost simultaneously at several times in the field’s history (Cassidy et al., 2007; Cassata et al., 2008). This behaviour is not uncommon in basaltic volcanic fields (Connor and Conway, 2000). A majority of volcanic centres in the AVF have not been dated reliably and many have not been dated at all (Lindsay and Leonard, 2009). Accurate ages for each volcanic centre or vent are critical to developing a recurrence rate model in volcanic fields. Moreover, the intense urbanization of the Auckland area could mean that there were more events than are currently recognized. Geophysical and cryptotephra studies may reveal other eruptive events (Molloy et al., 2009); cryptotephra studies are currently underway as part of the DEVORA project.

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AVERAGE RECURRENCE RATE/SIMPLE POISSON MODELS In general, long-lived monogenetic basaltic fields exhibit average recurrence rates on the order of 10-4 to 10-5 events per year (Table 1) (Connor and Conway, 2000). While average recurrence rates are useful when comparing overall trends in volcanic fields, the results from these calculations and simple Poisson models (chi-square goodness of fit; Kolmorgorov-Smirnov tests) do not reflect the temporal clustering of activity that is characteristic of monogenetic basaltic volcanic fields, and therefore, they should not be used to predict future events (Connor and Conway, 2000; Condit and Connor, 1996; Ho, 1991). In the AVF, there is evidence of at least one ‘flare up’ of activity, though the average recurrence rate for the AVF over the last 80 ky falls within the expected average recurrence rate for volcanic fields at 2.9 x 10-4 events per year (Table 1) (Molloy et al., 2009). Conway et al. (1998) used a simple temporal Poisson model to examine probabilities of future eruption timing in the SP cluster of the San Francisco Volcanic Field in Arizona. They found an average recurrence rate of 1 event per 15 ky for the past 780 ky. Several other studies have described the average recurrence rates at various intervals in volcanic fields (Table 1).

Table 1. Average recurrence rates for various monogenetic fields around the world. Table adapted from Connor and Conway (2000) and information compiled by Lucy McGee.

Volcanic field Average Recurrence Rate (events/year)

Reference

Eifel Volcanic Field 5 x 10-4 Condit and Connor, 1996; Schminke,

et al., 1983; Valentine and Perry, 2007

Michoacan-Gunajuato Volcanic Field

3 x 10-4 Hasenaka and Carmichael, 1987

Auckland Volcanic Field 2.9 x 10-4 Molloy et al., 2009

Springerville Volcanic Field 2 x 10-4 Condit and Connor, 1996; Molloy et al., 2009

Kluycheskoy Group 2 x 10-4 Condit and Connor, 1996; Fedotov et al., 1991

Higashi-Izu Volcanic Field 1.3 x 10-4 Martin et al., 2003

Yucca Mountain, Nevada 1 x 10-4 Connor and Hill, 1995

Sierra del Chichinautzin 1 x 10-4 Siebe, 2004

San Francisco Volcanic Field 1 x 10-4 Conway et al., 1998

Camargo 1 x 10-4 Condit and Connor, 1996; Luhr et al., 1997

Cima Volcanic Field 8 x 10-5 Dohrenwend et al., 1986; Turrin et al., 1985

San Francisco Volcanic Field 7 x 10-5 Tanaka et al., 1986; Conway et al., 1998; Conway et al., 1998

Coso, California 3 x 10-5 Duffield et al., 1980; Bacon, 1982

Big Pine Volcanic Field 2 x 10-5 Ormerod et al., 1991; Turrin and Gillespie, 1986

Pancake 1 x 10-5 Foland and Bergman, 1992; Scott and Trask, 1971

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LINEAR REGRESSION MODELS Gaining in popularity are linear regression models, which examine the relationship between eruption volumes (or area covered by eruptive products) and the timing of each event. If volcanism is volume- or time-predictable, plotting cumulative erupted volume versus time will reveal a linear relationship, the slope of which is the eruptive volume flux (Valentine and Perry, 2007). Volumetric steady-state volcanism patterns have been detected in several volcanic fields (Connor and Conway, 2000; Ho, 1991). In the Coso Volcanic Field in California, researchers found a link between the volume of the last eruption and the repose period for the past 0.5 My, allowing the prediction of future eruption timing if rates remain steady. They hypothesized that the extensional strain rate was controlling the pattern of volcanism (Bacon, 1982). A study on the Southwestern Nevada Volcanic Field revealed a time-predictable relationship between eruptive volume (including tephra fall) and repose time in the past 3 My (Valentine and Perry, 2007). Similarly, in a study of the Springerville Volcanic Field in Arizona, the area covered by lava flows was used as a proxy for erupted volume; researchers found that the cumulative area impacted by eruptions was consistent through time (Connor and Conway, 2000; Condit and Connor, 1996). Condit and Connor (1996) analysed the number of events through time to detect waxing and waning phases and to calculate the maximum and minimum recurrence rates for the Springerville Volcanic Field. For events where the exact ages of events were unclear, both the maximum and minimum ages were plotted. They then plotted the cumulative area and cumulative number of events with time and demonstrated that the magma output rate through time was somewhat constant, while vent formation increased with time. Along with the increased vent formation, there was an increase in more evolved basaltic, highly alkalic eruptions during a time period when tholeiitic basaltic eruptions declined. Similar analyses can be used to gain insight into the chemistry and plumbing system of monogenetic volcanoes and, conversely, illustrates how geochemistry may contribute to predictions of future activity. After examining the field for spatial trends, they then plugged all of this information into an appropriate spatio-temporal recurrence rate model. Valentine and Perry (2007) identified two end-member types of volcanic fields: tectonically controlled and magmatically controlled (Figure 1). Tectonically controlled fields, such as the Coso Volcanic Field, are more likely to exhibit time-predictable behaviour and exhibit low eruptive fluxes, while magmatically controlled fields may be volume-predictable and have high eruptive rates. Even in volcanic fields that exhibit predictable behaviour, however, the eruptive output rates may wax and wane over various time scales as the phenomena (e,g., extension rates, source depths, partial melting rates or temperatures) that is controlling them evolves, disrupting the pattern. For example, in the SP cluster of the San Francisco Volcanic Field in Arizona over the last 780 ky, the recurrence rate increased seven-fold but the eruptive volume rate only doubled, as compared to pre-780 ky recurrence and volume output rates.

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Figure 1. Two types of predictable behaviour in volcanic fields: a) volume-predictable behaviour (magmatically controlled) and b) time-predictable behaviour (tectonically controlled). In a), circles represent the age of the most recent eruption and the volume of all eruptions to that point, while crosses represent the age of the next eruption and the volume of all eruptions excluding the next eruption—the new cumulative eruptive volume can be predicted from where a vertical line running through the cross intersects the slope of the volume-predictable line. In b), circles represent the age of the most recent eruption and the cumulative volume of all eruptions, while crosses represent the cumulative volume of all eruptions excluding the most recent—the age of the next eruption can be predicted from where a horizontal line running through the circle intersects the slope of the time-predictable line. The slope of each line is the eruptive volume flux (Valentine and Perry, 2007). These authors have had success correlating eruption volumes (or areas) with time, and in some cases have been able to predict the timing or eruptive volume of subsequent eruptions. However, in more studies than not, such linear relationships are not found. Given the complex nature of the tectonic setting, tectonic history, small vent population, field age, and varied volume and vent distribution of the AVF, it does not seem likely that such simple correlations will be detected. To further support this conclusion, in the AVF, an 80 ky tephra record shows that volumetric output changes on time scales of 103 to 104 years; the most recent eruption at Rangitoto Volcano produced the largest known volume of material than any other event in the field’s history, after a relatively quiet period; and finally, Molloy et al. (2009) found no correlation between the age and repose period of eruptions in the AVF. As volcanism in the AVF has a history of waxing and waning, variable magma output rates, and evidence for multiple eruptions from a single vent, simple Poisson models should not be used to represent time trends in this field (Allen and Smith, 1994).

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NON-HOMOGENEOUS POISSON MODELS Ho (1991) questions the idea that magma output rates (via volume or area calculations) and repose periods can be related by simple models like those used in Bacon (1982), Valentine and Perry (2007), and Condit and Connor (1996). In the Yucca Mountain test site in Nevada, which Ho focuses on, there is no such correlation between magma production and repose period. Ho also points out that a field with waning eruptive activity can also have a trend of increasing volume (as found in the AVF) and that erosion can affect the volume estimates dramatically. His position is that simple Poisson models should not be used to accurately predict future eruption timing in these fields. Instead, Ho (1991) is a proponent of the ‘Weibull intensity with nonhomogeneous Poisson model’ to determine the instantaneous recurrence rate in the Yucca Mountain, Nevada region. The Weibull intensity is specifically used for waxing/waning activity. He then used a homogeneous Poisson process to predict future activity, because he hypothesized that random influences on the timing of future eruptions would not be taken into account by a non-homogeneous Poisson model. From his calculations, he estimates that the instantaneous recurrence rate of the Yucca Mountain volcanic region is 5.5 x 10-6 years. He cautions that stating this figure without acknowledging the time period in which nuclear waste is susceptible to volcanic activity is misleading; it is better to state the cumulative risk over each time period (i.e., 5% over 104 years; 42% over 105 years) to get a clear picture of the true risk.

SPATIAL HETEROGENEITY Spatial heterogeneity of vents, in the form of clusters and/or local- and regional-scale lineaments, is a common feature of basaltic volcanic fields. The main causes of these lineaments and clusters are unknown, however, crustal extension and compression may influence vent alignment and spatial heterogeneity (Martin et al., 2004). Structural weaknesses such as faults within the crust hypothetically allow vents to more easily form along those planes, and vent alignments are sometimes correlated to these faults. However, vent clusters do not always form near faults. In addition to structural anisotropies within the crust, vent distribution may also be influenced by plate motion or shifts in the locus of the magma supply (Connor and Conway, 2000).

Univariate Statistical Methods Similar to the average recurrence rate, which does not reflect temporal variations of eruptions, univariate statistical methods that calculate the number of vents per unit area are only useful in the overall vent density comparison amongst volcanic fields, as they cannot describe spatial variation such as vent alignments (Connor, 1990; Connor et al., 1992). Therefore a non-homogeneous Poisson model must be used in most point analyses of vent distributions (Martin et al, 2004; Ho, 1991). Density mapping or contouring has also been used, but is severely crippled by the assumptions one must make, such as grid spacing or search radius (Connor, 1990).

VENT DENSITY Statistical tests may be performed using the data to determine how likely it is that the vent locations were determined by chance. Volcano intensity (number of expected events per unit area) can be calculated in two ways: a random point to volcano distance, and a volcano to volcano distance (Connor and Hill, 1995). A statistical examination of a volcanic field’s vent locations may reveal clusters or lineaments along which or around future eruptions are likely to occur.

Simple Statistical Tests The Clark-Evans, Hopkins F, and K function tests determine whether a set of points is random, clustered, or can be represented by a Poisson distribution. Once a determination of the type of

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vent distribution pattern (random, clustered, homogeneous Poisson) is made, one can go about defining the characteristics of these patterns in a spatial-temporal model. For non-random events, a non-homogeneous Poisson distribution is appropriate for modelling the temporal-spatial distribution of vents; the nearest-neighbour and kernel estimation models are the most popular methods. Martin et al. (2004) prefer the kernel technique because it does not require exact ages as inputs as the nearest neighbour models do. The Clark-Evans test was used in Martin et al. (2004) to show that the distribution of vents in the Tohoku volcanic arc is not random. Using the Clark-Evans dispersion index, they found that the distribution is clustered. Connor and Hill (1995) used the Clark-Evans, Hopkins F, and the K function tests to determine that vents were not randomly scattered across the Yucca Mountain region. Connor et al. (1992) contoured vent density distribution in the Springerville Volcanic Field using a minimum curvature contouring algorithm. They used a grid spacing of 10 km and a search radius of 7.5 km, and were able to identify where the highest concentration of vents was located, as well as a possible alignment within the field using this method (Figure 2).

Figure 2. Vent density contours in the Springerville Volcanic Field, Arizona, USA were created using a minimum curvature contouring algorithm. Contouring vent density is a simple way to examine vent concentrations within a field, as well as locate any vent clusters or alignments (Connor et al., 1992).

Kernel Estimation Technique to Examine Vent Density The kernel function technique defines the intensity of volcanic events at a point in the field using a smoothing constant, h, and a distribution function to describe the distance to nearby volcanoes. The Epanechnikov, Cauchy, and Gaussian kernel functions are the most used functions, the choice of which depends on the characteristics of the volcanic field in question (Martin et al., 2003). The Gaussian function should be used with most small volcanic fields like the AVF (Martin et al., 2004). The Gaussian kernel function bases its predictions on the idea that subsequent eruptions will occur close to existing volcanoes. Although the function used is important, the smoothing coefficient, h, exerts the most control over the results of this technique and if not chosen carefully, can affect the results dramatically (Martin et al., 2003; Martin et al., 2004). The smoothing coefficient, h, controls how local intensity rate varies with distance from existing volcanoes. The smoothing coefficient should be determined by the size of the volcanic field,

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size and degree of clustering, and how confident one is about the data (i.e., vent locations). A range of smoothing constants should be tested to obtain the best results (Martin et al., 2003). Martin et al. (2004) optimized determinations of h values by plotting cumulative probability density functions (Cauchy and Gaussian) versus vent distances and fraction of events. For this test, edge correction is also important. A sporadic distribution of vents and the nature of defining boundaries around volcanic fields lead to lack of data points around the edge of the field. Martin et al. (2003) maintains that the best guide for the choice of the edge correction factor is the number of volcanic events in the field. In Condit and Connor’s (1996) analysis of trends in the Springerville Volcanic Field, the kernel estimation technique was used to examine vent density. After first examining the field using recurrence rate and linear regression models, they studied vent density using an Epanechnikov kernel technique. This technique estimates vent density per some specified time interval, in this case 0.25 My intervals, because the number of robust age determinations was approximately equal for each time interval (Figure 3). It is a simple way to examine how vent density changes in a field through time. The Epanechnikov probability distribution function was chosen because it is symmetrical about each vent. They used a large enough area compared to the size of the field to allow the edge correction to be 1. Small smoothing constants maximize the variations in vent density while large constants will gloss over changes in density. Connor and Hill (1995) based h on vent spacing and cluster analysis, while Condit and Connor (1996) chose 5 km for h because most of the vent-vent distances in the Springerville Volcanic Field are less than this distance during each 0.25 My time interval. This method also shows how volcanic activity moved through the field with time. Vent density maps for each 0.25 My time period were constructed. In conjunction with temporal recurrence rate information, these results were used to adapt the spatio-temporal near-neighbour model to the Springerville volcanic field. Probability maps were then created from the recurrence rate values using a Poisson distribution.

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Figure 3. Vent density in the Springerville volcanic field, Arizona, using the Epanechnikov kernel method with a contour interval of 0.05 volcanic events per square kilometer (Condit and Connor, 1996).

SPATIAL CLUSTER ANALYSIS Clusters of vents are common in monogenetic basaltic fields. Temporal or geochemical patterns may be detected within vent clusters, therefore, it is worthwhile to isolate vent clusters for analysis, rather than simply averaging activity across the whole field (Conway et al., 1998). The impetus for cluster formation is unknown, however, they could be related to structural weaknesses in the crust. The use of robust cluster identification methods may help explain vent emplacement processes within a volcanic field (Connor et al., 1992). Cluster analysis is preferred over vent density contour analyses, as vent density contour analyses require grid spacing inputs, which are subjective and may impact the results dramatically (Connor et al., 1992). Cluster analyses should be undertaken before looking for alignments within a field. Connor et al. (1992) explain that this is due to the inability of alignment analyses to adequately account for heterogeneity in vent distributions, and that this weakness may negatively impact the results. Once clusters are identified, they may give meaning to other seemingly random distributions within a volcanic field (e.g., geochemical data), may be linked to faults and other structural and geophysical features, and can be incorporated into alignment analyses to minimize the effect of clustering on vent alignment determinations (Connor, 1990).

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These methods require a minimum of assumptions, but require many iterations to identify clusters that do not change with search radii. Connor et al. (1992) used a uniform kernel density fusion cluster analysis on the Springerville Volcanic Field in Arizona. In this cluster analysis method, vents that lie within the search radius of each vent are added to a matrix, which is then used to link vents to one another with an algorithm. A density fusion calculation is then used to identify clusters. This method is quite robust; even overlapping clusters may be identified, given the correct search criteria. However, some vents may be assigned to the wrong cluster, especially if a nearby cluster is larger. This is particularly problematic if clusters are not well defined or vent density is low. Search radii must be input. The Hopkins F, Clark-Evans, and K function tests distinguish between clustered and random vent distribution. Researchers may combine these tests with cluster analyses to describe the spatial distribution of volcanic fields. After applying the Hopkins F test, the Clark-Evans test, and the K function test to determine if clusters were present in the Yucca Mountain Field in Nevada, a weighted-centroid cluster analysis was performed by Connor and Hill (1995) to identify these clusters. They then plotted vents versus distances to other vents to illustrate which vents were clustered and to examine how the distances within and between clusters were distributed (Figure 4). These distances were instrumental in choosing an appropriate smoothing constant (h) in the spatio-temporal kernel estimation method (see “Spatio-Temporal Models,” below).

Figure 4. Connor and Hill (1995) used combination of simple distribution algorithms and cluster analysis to identify vent clusters and distances in the Yucca Mountain Volcanic Field, Nevada, USA. This information may be used to determine suitable inputs to spatio-temporal models. Connor (1990) suggests that vent alignment analyses (such as two-point azimuth, Hough transform, and two-dimensional Fourier transform methods) used in conjunction with cluster analysis produces robust results. He modified the uniform kernel density fusion cluster analysis to look for clusters of cinder cones. Affecting the results of this method are the search radius, spatial density of cinder cones, and any space between cinder cones. If clustering is a factor in

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the field, the cinder cone clusters will not be altered drastically by greatly varying the search radius. There are, however, difficulties when applying statistical probabilities to the results of this method.

VENT ALIGNMENTS Vent lineaments are a common feature in volcanic areas. Alignments may indicate crustal weaknesses such as faults, major stress orientations, and dike propagation processes. Correlating crustal structures with alignments in volcanic fields may aid in probabilistic hazard estimations. Several statistical methods are used to determine the presence, location, and orientation of alignments within volcanic fields and vent clusters (Hammer, 2009). Hammer (2009) reviewed point alignment methods used in other studies. He divided point alignment statistical analyses into three groups: azimuthal methods (e.g., 2-D azimuth and the modified Lutz and Gutmann (1995) azimuth method), spatial transformation (e.g., Hough Transform, 2D Fourier), and strip methods (created by Zhang and Lutz, 1989; adapted by Arcasoya et al., 2004).

Azimuth Methods Two-point Azimuth or ‘Azimuthal’ Method This method tests the likelihood that any alignments in a group of points are random versus a product of a regional stress field (Connor et al., 1992; Wadge and Cross, 1988; Lutz, 1986). Lineaments detected above the threshold using this method should be examined further with field or statistical work. This method provides several improvements over other statistical methods. Patterns within the field, its shape, and every possible pairing of points are assessed, and structural features with directional orientations do not have to be known or incorporated into the analysis, as the points alone communicate any structural influences or patterns (Lutz, 1986; Connor et al., 1992). A downfall of this method is the exaggeration of the number of azimuths detected in an elongated field of points along the axis of the elongation. In instances where the shape of the field of points is not a perfect circle, the effects of the elongation would have to be corrected using Monte Carlo simulations to compare random distributions of points in a field with a similar shape to the test field (described in Lutz, 1986 and Wadge and Cross, 1988). This method is best used in conjunction with another method (most commonly Hough Transform, see below) to mitigate the effects of point distribution heterogeneities found within volcanic fields (Connor, 1990). This method also does not locate vent lineaments, only the major orientations of anisotropy within the field (Connor et al., 1992), however, in 1995, Lutz and Gutmann presented a modified version of the azimuthal method that distinguishes alignments on several scales and also identifies the locations and orientations of the alignments. Lutz (1986), the creator of this method, recommends an upper limit of several hundred points due to the computational effort involved. The AVF has approximately 50 points (vents), so this method should be applicable in that sense. To use this method, we must know the vent locations accurately to construct azimuthal histograms. A representative confidence interval (usually 95 or 99%) and a directional bin interval (usually 10%) must be chosen.

Spatial Transformation Methods

Spatial transformation methods, as defined by Hammer (2009), simply alter the way that point patterns are viewed, aiding in the identification of alignments. Alignments detected in this manner have no statistical significance. These methods are best used in conjunction with azimuthal methods to identify not only the presence and orientation of alignments, but their locations as well.

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Two-dimensional Fourier Transform Two-dimensional Fourier Transform analysis is similar in purpose to the two-point azimuth analysis. In the two-dimensional Fourier Transform, spatial trends are represented by large amplitude spectra. The distribution of the spectra is indicative of significant trends in the data set (Connor, 1990). Clusters of vents with only a few cones cannot be analysed with this method, and therefore may not be applicable to the AVF, with its small number of vents (Connor, 1990). Hough Transform The Hough Transform Method is an algorithm used to detect lines from a field of points. The location of points within a plane can be described by a radius from the origin, and an angle relative to x in that same grid space. Line segments can be described using x, y, r, and theta as variables with a parametric equation. Each (x, y) coordinate in grid space corresponds to a sinusoidal curve in the Hough transform parametric space. Points that can be connected by a line (thereby yielding azimuthal directions) intersect at an (r, theta). That (r, theta) point is translated into an orientation (Connor, 1990). The Hough Transform is more computationally efficient than the two-point azimuth method, and can detect small, local-scale lineaments (Connor, 1990). However, it is affected by the number of vents and shape of the vent field (Connor et al., 1992). The more vents and the more elongate the shape of the field, the more false alignments will be detected (Ho, 1991). Used in conjunction with the two-point azimuth method, these disadvantages can be overcome. Lutz and Gutmann (1995) used the two-point azimuth method with kernel density estimation to identify the location and alignment of vents on several scales in the Pinacate volcanic field in Mexico. Connor et al. (1992) used the two-point azimuth method to detect anisotropies in vent clusters, and then the Hough Transform method to ascertain the location of those lineaments within each cluster in the Springerville Volcanic Field in Arizona. Connor (1990) used the two-point azimuth, 2D Fourier, and the Hough Transform to identify lineaments within clusters in the TransMexican volcanic belt. They also were able to link geochemically unique lavas to these lineaments. Wadge and Cross (1988) used the Hough Transform and two-point azimuth method to find alignments in the Michoacan-Guanajuato volcanic field in Mexico. Connor et al. (1992) examined the vent distribution in the Springerville Volcanic Field in Arizona. They used the two-point azimuth and Hough Transform methods to look at alignment after first identifying clusters within the field.

Strip Methods Zhang and Lutz (1989) created the strip method while examining the relationship of structural features to igneous complexes and kimberlite emplacement in Africa. They compared the densities of points within parallel strips to the densities expected from a random (Monte Carlo) distribution. Strips were rotated, and azimuths of significant trends detected were assembled in a histogram. This method was adapted by Arcasoya et al. (2004) by using GIS to detect lineaments in the Cappadocian Volcanic Province in Turkey (Figure 5).

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Figure 5. The strip method used in conjunction with GIS to detect vent alignments by Arcasoya et al. (2004) in the Cappadocian Volcanic Province in Turkey. Small dots represent vent locations; solid lines are alignments within each strip that connects farthest points. The strip method of identifying vent alignments has many variations depending on the characteristics of the volcanic field, the desired robustness of the results, and the amount of information about the field that is readily available. To overcome the weakness of assumed homogeneity found in all of the above methods, Amorese et al. (1999) advanced the strip analysis method by developing the ‘binomial blade’ method while studying earthquake distribution in France. They divided the strips into subregions and used an exact binomial test to determine the significance of the results. This method does not assume spatial homogeneity, determines the orientations and locations of alignments, does not require grid and grid spacing determinations, and also assigns significance to the results (Amorese et al., 1999; Hammer, 2009). However, Hammer (2009) identified several downfalls to the Amorese et al. method: it requires circle radius and blade width inputs, and can identify alignments that do not exist due to over-testing and problems with the significance results. Hammer (2009) further modified strip methods by testing the Amorese et al. (1999) binomial blade method using three case studies and identifying ways to overcome the above weaknesses. He concludes that the original ‘binomial blade method’ is best used for mapping lineaments and for post-processing, while the ‘strip width parameter’ is good for exploratory data analysis. For more robust analyses, he maintains that the ‘multiple blade’ and ‘multinomial sector’ methods are best used. For studies that want to reduce the number of variables, the ‘continuous sector’ method only requires one input parameter (Hammer, 2009).

SHIFTS IN LOCUS OF ACTIVITY Shifts in vent centres are common in basaltic fields and they sometimes migrate in a consistent direction. For example, in the SP cluster of the San Francisco Volcanic Field, the locus of activity has moved from west to east at a rate of 2.9 cm/yr, which correlates well with the

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movement of the North American plate (Conway et al., 1998). Reliable eruption ages are keys to calculating this characteristic of monogenetic fields. Even with good age control, however, these shifts may or may not be easily explainable.

SPATIO-TEMPORAL MODELS Many of the methods mentioned above are used to examine trends in volcanic fields before using these components to create an integrated model that can encapsulate all of this information. These integrated models can be used to predict future eruption timing and location and to create illustrations of recurrence rates per unit area, or probability estimate maps. Probability estimate maps are perhaps the most useful tool for predicting future activity and designing evacuation and risk mitigation planning in the absence of earthquakes, increased gas emissions, or other good indicators of future eruptions (Connor and Hill, 1995). Generally, two methods are available to estimate the probability of future events: the spatio-temporal nearest-neighbour model and the kernel estimation technique. The spatio-temporal nearest-neighbour model is used to estimate local recurrence rates as a function of distance to and ages of volcanic events and the kernel estimation technique estimates the local intensity of volcanic events (Martin et al., 2004).

Spatio-temporal Nearest Neighbour Recurrence Rate Model This model is useful because it takes into account the waxing and waning of the volcanic activity at a field, and how the location of the vents, petrology, and magnitude of events changed through time (Condit and Connor, 1996). It is also useful because it weights the most recent events more heavily than earlier events (Martin et al, 2004). However, this model requires accurate ages for each event, which may not be available (such as in the AVF). This model is often used in conjunction with kernel estimation techniques (Martin et al., 2003; Condit and Connor, 1996). From the spatio-temporal recurrence rate model, probability maps may be constructed (Condit and Connor, 1996).

Kernel Technique As discussed earlier in this report, the kernel technique was first used to estimate vent density (Lutz and Gutmann, 1995; Connor and Hill, 1995). The kernel function technique defines the intensity of volcanic events at a point in the field using a smoothing constant, h, and a distribution function to describe the distance to nearby volcanoes. The Epanechnikov, Cauchy, and Gaussian kernel functions are the most used functions, the choice of which depends on the characteristics of the volcanic field in question (Martin et al., 2003). The Gaussian function should be used with most small volcanic fields like the AVF (Martin et al., 2004). The Gaussian kernel function bases its predictions on the idea that subsequent eruptions will occur close to existing volcanoes. Although the function used is important, the smoothing constant, h, exerts the most control over the results of this technique and if not chosen carefully, can affect the results dramatically (Martin et al., 2003; Martin et al., 2004). In the kernel estimate technique, a small h restricts the probability of future eruptions occurring to areas close to the existing volcanoes, while a larger h spreads that probability across the entire field (Connor and Hill, 1995). Spatial cluster analyses may help define h. The shape of the field is also an assumption in the kernel model, but was found through experimentation to be less important than h by many researchers (e.g., Lutz and Gutmann, 1995; Connor and Hill, 1995). This method is the least sensitive to shifts in the locus of volcanism, as compared to the spatio-temporal nearest neighbour and nearest neighbour kernel models (Martin et al., 2004).

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The advantage to this method is that exact ages of all the events do not have to be known; instead it just requires the locations of the events. However, this method does not weight the newer events differently than older events. This model requires a choice of density function that best represents the distance to nearby volcanoes, an adequate smoothing coefficient, and grid spacing input. The Gaussian function most accurately represents the distribution of volcanoes in monogenetic basaltic fields, especially small fields like the AVF, although Martin et al. (2004) also used the Cauchy distribution. Condit and Connor (1996) used results from temporal and vent density analyses to develop a spatio-temporal near neighbour (recurrence rate per unit area) model to explain the trends found in the Springerville Volcanic Field. They used a Poisson distribution to estimate probability, and then compared the expected and average recurrence rates and calculated the probability that the model was correct using a null hypothesis. Connor and Hill (1995) modelled the distribution and timing of areal basaltic volcanism near Yucca Mountain in Nevada using three non-homogeneous Poisson methods to analyse vent distributions and to estimate the recurrence rates: the spatio-temporal nearest neighbour, kernel, and nearest-neighbor kernel. The nearest neighbour kernel estimate method replaces h with a variable representing each distance to a nearby volcano. All three methods use the location of past events to predict future events, but use either the distance to, time since, or the behaviour of nearby volcanoes to predict those recurrence rates. They used two data sets, and defined an ‘event’ two different ways in order to cover all the possible behaviours. Using these three methods, Connor and Hill (1995) found that the Yucca Mountain region cinder cones were definitely clustered, the spatial variation of eruption probabilities was highly variable across small distances, and that the models estimated eruption rates between approximately 1 x 10-4 to 5 x 10-4 years over the next 10,000 years. Martin et al. (2003) analysed the Higashi-Izu and the Kannabe-Oginosen volcanic fields in Japan using a spatial model, a spatio-temporal model, and a modified spatio-temporal model. The kernel technique was used to estimate vent density, and the Epanechnikov kernel function was then used to estimate the spatial recurrence rate. Probability estimates were then made for each point by multiplying the local spatial recurrence rate by the temporal recurrence rate. This method may be tested by removing the last few events from the calculations and predicting their locations. The problem with this method is that the temporal recurrence rate weights recent events equally with older events. To address this shortcoming of the temporal recurrence rate, they also used the spatio-temporal model, which calculates local temporal recurrence rates as well as spatial using the nearest neighbour method. They also modified the spatio-temporal model to include lengths and orientations of shallow structural features such as dikes, lineaments, and faults. The modification does not alter the probabilities of a future eruption, but does impact the distribution of those probabilities across the volcanic fields (Martin et al., 2003). Probabilities of future eruption in the next 10,000 years range from 3 x 10-1 to 5 x 10-1, which is roughly similar to Springerville Volcanic Field and the Yucca Mountain Region probability estimates (Martin, et al., 2003; Condit and Connor, 1996; Connor and Hill, 1995). Conway et al. (1998) used the average recurrence rate and a simple Poisson model to determine that the SP cluster in the San Francisco Volcanic Field has a 95% chance of another eruption in the next 22 to 26 ky. A Gaussian kernel technique was then used to model the spatial distribution of vents that formed in the last 300 ky. They used the average vent density of the field to calculate the smoothing constant, h. They found that the next eruption would occur in a certain 250 km2 area with a 95% confidence interval.

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Alternative Methods MARKOV CHAINS Bebbington (2007) notes that Markov chains may be used to predict spatial and temporal elements of future eruptions. Eruption size can be taken into account with this method. In a temporal Markov chain, it is assumed that previous ‘states’ of the volcano (eruption vs repose time) will have no effect on the current state. The volcano may remain in its current state, or transition to the next state (assumed to be random), however, the Markov chain can only predict eruptions based on a probability estimate describing its current state. When available, a spatial component may be added to the Markov chain method to provide a more complete picture. FRACTALS Chaos and fractals are a type of time and size predictable model and can be used to describe volcano behaviour in comparison with other volcanoes, group volcanoes by type, and determine the modelling technique to be applied to each group. Fractals may identify clusters of eruptions and other temporal patterns. Markov and fractal methods assume that the pattern of eruptions is consistent through time. This does not adequately reflect the behaviour of most volcanoes in the long term.

USING GIS TO ASSESS POINT DISTRIBUTION Bishop (2007) used custom-developed Arcview Extensions to assess the spatial patterns found in the Mount Gambier volcanic sub-province in Australia. Through these techniques, Bishop detected a migration of activity, alignments, and patterns that changed on a variety of scales. Moreover, Bishop (2009) classified various landforms found within a monogenetic volcanic field according to complexity, and suggested that GIS may be useful to analyse the spatial distribution of these classified landforms to gain insight into any tectonic or other processes at work, and to examine spatial diversity of a volcanic field. Arcasoya et al. (2004) also used GIS to detect lineaments in the Cappadocian Volcanic Province in Turkey.

BAYESIAN INFERENCE FOR LONG-TERM ERUPTION FORECASTING Bayesian inference is one way to integrate additional datasets into spatio-temporal models, which usually only take into account the timing and spatial distribution of volcanic events. Geochemistry, gravity anomaly maps, and other data sets may improve long-term probabilistic assessment models and refine estimates of the timing and location of future events. These additional data sets can be represented in a probability density function in Bayesian inference (Martin, et al, 2004). Additional data sets may further constrain sites of future activity (Condit and Connor, 1996; Connor et al., 2000). A study by Martin et al. (2004) included geophysical data in a probabilistic assessment of future eruptions at Tohoku volcanic arc in Japan using Bayesian inference. The first step of probability analysis using Bayesian inference involves constructing a spatio-temporal model. Martin et al. (2004) used the Clark-Evans near-neighbour test to compare whether the vents in the field were truly random or showed some sort of pattern. The test showed that the vents in the field were clustered (Figure 6).

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Figure 6. Martin et al. (2004) incorporated geophysical data into a spatio-temporal model using Bayesian inference to construct a probability density map for the Tohoku volcanic arc in Japan. Any spatial data set may be included in a spatio-temporal model using Bayesian inference. Martin et al. (2004) also approached the dilemma of defining a volcanic ‘event’ by using two definitions so that the results could be contrasted and compared and used as constraints on the true probability of future events. In the first, each centre was weighted equally, and in the second, each centre was weighted by eruptive volume in order to incorporate polygenetic behaviour into the calculation. This results in a conservative probability estimate for the unweighted scenario, and higher probabilities near the weighted centres in the second case. Once Martin et al. (2004) found a suitable method of representing the spatial and temporal heterogeneity of the Tohoku Volcanic Field and had calculated the probability of future events at each location using a Poisson distribution calculation, they then used this as a template on which to incorporate additional data sets (such as geophysical information) for the area in question. This requires one to develop a likelihood function describing the data using expert knowledge and assigning meaning to variations in the dataset, paying particular attention to how that data point may relate to future events (in their example, low P velocity perturbations may correlate to an area of partial melting and therefore be a source of future volcanism, increasing the probability of an event near that location). Martin et al. (2004) demonstrate that any spatial data set may be converted into a likelihood function. A 2D surface is interpolated from the data and the likelihood parameters are generated and applied across that surface at the same grid spacing as used in the kernel technique employed earlier in the analysis. It is assumed that the geophysical and chemical characteristics of more recent past events serve as a guide for assigning meaning to the particular environment that may produce a future event. Again, in their example, if 80% of the past events occurred in an area where the geothermal gradient is greater than 100 K/km, the user can describe this ratio mathematically so that this may be used to calculate the probability

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of future events occurring in the areas with similar gradients. In this way, a likelihood function may be developed. Using Bayes’ theorem, the previously calculated probability density function (from spatial and temporal information alone) is combined with the likelihood function calculated from the geophysical data. The probability of a new event can then be calculated for each grid point using the same Poisson distribution calculation described earlier. The major advantage of Bayesian inference as a predictive tool is that the model becomes increasingly refined as more information about the volcanic field becomes available. Martin et al. (2004) used GMT (Generic Mapping Tools) to create probability maps, but GIS or another mapping or visualization programme could also possibly be used. Marzocchi et al. (2010) modified the Bayesian method by adding an event tree model for volcanic hazards (BET_VH). In this method, an ‘event tree’ of every possible outcome relating to the volcano in question is created; user input and monitoring data is then used to estimate the probability of each volcanic event (Figure 7). A real-life example of tephra fallout from Campi Flegrei was used to show that uncertainties in data can drastically affect the long-term volcanic hazard assessment (Selva et al., 2010).

Figure 7. Example of an Bayesian Event Tree for volcanic hazards (Marzocchi et al., 2010).

STUDIES ON THE AUCKLAND VOLCANIC FIELD The Auckland Volcanic Field is approximately 360 km2 in area and contains 49 to 52 known vents. The field has been active for roughly 250,000 years, and there is evidence for near contemporaneous activity at multiple vents at various points in the field’s history. Additionally, there is evidence that the most recent eruptions were polygenetic, behaviour that has been recognized at other (predominantly) monogenetic basaltic fields (Connor et al., 2000; Allen and Smith, 1994; Magill et al., 2005; Needham, 2009; Molloy et al., 2009). The graywacke basement to the east of Rangitoto is assumed to be the eastern boundary for the field (Magill et al., 2005; Allen and Smith, 1994), however, there has been no consensus on any spatial or temporal trends within the field in investigations done thus far.

Eruption Magnitude and Frequency Molloy et al. (2009) studied eruption recurrence rates in the AVF based on tephra layers obtained from three maar cores in Auckland. They were able to correlate tephra layers to well known, well-dated Central North Island eruptions and thereby, place time and magnitude constraints on the basaltic tephra layers from lesser known AVF eruptions. They found an average frequency of one eruption every 3.5 ky and an average recurrence rate of 2.9 x 10-4 events per year in the past 80 ky in the AVF, which is comparable to other monogenetic basaltic fields (Table 1) (Connor et al., 2000; Molloy et al., 2009). Molloy et al. (2009) also used a

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survival analysis approach using data collected from sediment cores in the Auckland maars to assess the threat of a certain thickness of tephra fall from the Central North Island volcanoes. They emphasized the greater risk from tephra produced at distant volcanoes than that of the risk of volcanic eruption from the AVF. Using mostly average recurrence rate with the dates available at the time, many of which have since been reassessed or deemed unusable (Lindsay and Leonard, 2009), Allen and Smith (1994) report that several researchers have calculated probabilities of an eruption as 2 to 8 percent per century. Allen and Smith also mention that these calculations cannot be realistically used due to lack of detail. They report that evidence for two eruptions from one magma batch was found for some events, thus ruling out the use of simple frequency models to explain future behaviour of the AVF. Several researchers (Cassidy et al.,1999; Cassidy, 2006; Cassata et al., 2008) note that there is strong evidence for eruptive activity at several vents within a close span of time, or perhaps even simultaneously. Molloy et al. (2009) found a ‘flare up’ of activity from 35 to 24 ky; 16 of the 24 tephra layers from the AVF that have been deposited over the last 80 ky were associated with this flare up. However, this vigorous activity does not necessarily correlate with increased magmatic output. Cassidy et al. (1999) also suggest that volcanic activity at each vent ceased within a few years—no more than 100 years--based on magnetic anomaly data. Molloy et al. (2009) calculated repose times from sedimentation rates in the maars; repose intervals varied from less than 0.5 ky to as much as 20 ky. Clearly, any model assessing the AVF should take this waxing and waning of activity and the temporal and magnitude variations into account. Bebbington and Cronin (submitted) emphasized these characteristics in their study of the AVF and incorporated it into their spatio-temporal event-order model. They used tephra thicknesses found in cores around the field, the eruptive volume, as well as all known stratigraphic and measured ages, to estimate probable eruption order and therefore, ages for each volcano. They found that while the location of successive events does not occur in clusters, the timing of eruptions does tend to occur in ‘flare-ups’ of activity. They use a triggering model, or a point process intensity model, to represent the temporal pattern, with previous eruptions increasing the probability of a future eruption in the near future. Ongoing research by Phil Shane and Aleksandra Zalwana-Geer involves examining the crypto- and macro-layers of tephra in sediment core obtained from Auckland maars. These data will be valuable as they will likely provide the basis for recurrence rates used in any probabilistic hazard method used to assess possible future eruptions in the AVF. The tephra cores could also fine-tune the age database compiled by Lindsay and Leonard (2009).

Spatial Heterogeneity In the literature, no consistent findings regarding spatial vent patterns in the AVF are reported. Cassidy et al. (1999) reports that geophysics support a NNW-SSE structural trend, while Allen and Smith (1994) write that the location distribution of the eruption centres is random and that no geophysical control is evident from the many studies done. Allen and Smith (1994) also find that the maximum distance between any two centres is not more than 12 km from the previous eruptions, therefore, the next eruption will most likely be within 12 km of Rangitoto. Using Allen and Smith (1994), Magill et al. (2005) found evidence for spatial and temporal clustering using spatial point pattern analysis, including the Ripley’s K-function test, including major alignments trending SW-NE and minor lineaments trending NW-SE within the clusters. Bebbington and Cronin (submitted) did not find the clusters or evidence for the clustering distance reported in Magill et al. (2005); they cite that the clustering found in Magill et al. was more of an artefact of the Allen and Smith (1994) ordering scheme than an actual tendency of

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eruption centres to cluster. Using their revised event-order model, Bebbington and Cronin (submitted) analysed the distances and azimuths between chronological eruption pairs, finding that there was a preference against the most recent azimuthal direction in future eruptions. They also observed that the distances between pairs of eruptions were random (i.e., previous distances gave no indication of future distances), that there is no steady-state pattern of eruption location in the AVF, and that the timing of activity in the field may pulse irregularly. According to the Bebbington and Cronin event-order model, the location of future eruptions seems to occur independently of previous eruptions. Therefore, they conclude, the temporal and spatial components of the model may be assessed separately. They represented the spatial heterogeneity of the AVF with a density kernel model. Their results indicate that there is a ‘strong’ NE-SW alignment that parallels the Taupo Volcanic Zone to the south and the angle of the subduction zone to the east. These matching alignments may indicate that the subduction zone has some influence on AVF volcanism in at depth. von Veh and Németh (2009) used the nearest-neighbour azimuth and Hough Transform methods to assess previous vent alignments and to better predict the next vent location along those alignments. In the azimuth calculations, it was found that the clustering of vents in the centre of the volcanic field causes the mean distance between vents to be shorter than expected. They found major ENE-NNE and minor NNW-NW trends. Ongoing work by Julie Rowland and Nicolas LeCorvec involves examining the Auckland Volcanic Field for any likely structural controls on the location of previous and future vents. These data will be used in conjunction with the petrological study of the chemistry of the AVF eruptive products by Lucy McGee and Ian Smith to ascertain likely magma pathways, depths of origin and storage, progression of chemistry through time, and any tell-tale isotopic signatures. The chemistry may indicate to the structural geologists where to look for any changes in the crust that may control the vent locations. For a comprehensive review of the structural history of the Auckland Volcanic Field, see Irwin (2009).

Analogue Fields Allen and Smith (1994) and Magill et al. (2005) postulate that the South Auckland Volcanic Field (SAVF) should be a very close analogue to the AVF due to their similar lava chemistry and eruption styles, and that trends seen in that volcanic field can be expected in the AVF. If this holds true, the AVF is in its infancy, as the SAVF was active for approximately 1 million years, and produced 70 centres.

CONSIDERATIONS The DEVORA programme intends to answer the following questions regarding the probabilistic hazard from the AVF:

• What is the distribution in time of past eruptions affecting Auckland? • What is the likelihood and size of future eruptions affecting Auckland? • What are likely styles, sizes and hazards of future eruptions? • Where are we in the lifespan of the Auckland Volcanic Field? • How do we usefully calculate probabilistic volcanic hazard for Auckland? • What is the probabilistic volcanic hazard? • How intensive should the monitoring be to provide adequate warning of an AVF

eruption? To answer the questions pertaining to probabilistic volcanic hazard estimations for Auckland, any successful model will have to take into account the following considerations and questions:

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Cryptotephra & Geochronology

• What constitutes an ‘event?’ One event per vent and one vent per event? What then of Rangitoto, which may have had, by recent accounts, two separate eruptions of two separate magma batches? Or the evidence that suggests two or more vents were active at the same period in history?

• How is the AVF waxing/waning of activity (the ‘flare up’) incorporated?

• Are there clusters of vents?

• How do we predict the style or size of eruption or the likelihood of multiple vents?

• What is the frequency of ash fall from distant locations (this will inform distal hazard and

risk models)?

Petrology

• How do we predict the style or size of eruption or the likelihood of multiple vents?

• Is there a pattern of magma output (i.e., is the field time- or volume-predictable)? Is the magma output rate variable or constant?

• Does activity in each detected cluster wax and wane, as well as within the field as a

whole?

• Is the geochemical makeup of each vent significant in light of these alignments and/or clusters? Are there patterns?

• Do the centres formed during the 30 ka flare-up have similar geochemical

characteristics?

Structure of the Crust

• Has there been a shift in locus in the field? Can it be explained by plate movement or tectonic patterns?

• Is vent distribution random, clustered, or ordered?

• Are there localized vent lineaments as well as regional scale alignments? Are they

similar in direction? Are vent locations being controlled by shallow, small faults/weaknesses or deep-seated, large structures?

• Are any alignments consistent with geophysical indicators or trends?

• What is the shape of field; is one axis longer than the other?

• What is the relationship between the SAVF and the AVF?

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Analogue/Miscellaneous

• Which analogue fields most resemble the AVF in tectonic/structural setting, shape, behaviour, number of events, etc.? Can we use these fields as a template for future activity in the AVF?

• Is the South AVF a true analogue? If so, what studies have been done on the SAVF?

• Small sample number of vents (49-52 vs hundreds to thousands)—can it be described

categorically as a ‘small, long-lived(?) monogenetic basaltic field?’

• Stage of field/age—where are we in the field’s lifetime?

RECOMMENDATIONS Assessing the volcanic field for simple temporal patterns, magma output, geochemical trends, and other notable behaviours is the first step of developing a robust spatio-temporal model. In addition, Connor (1990) and Connor et al. (1992) emphasized the importance of identifying clusters prior to conducting alignment analyses, both of which should also be inputs to the integrated model. For alignment analyses, the Hough Transform and 2-point azimuth methods are best used in conjunction with one another to overcome their individual weaknesses (large computational requirements versus the field shape and number of vents, respectively). Wadge and Cross (1988) used the two-point azimuth method for regional scale pattern detection because it took into consideration the field shape; they used the Hough Transform method to examine local-scale directional patterns because it was more computationally efficient. Connor (1990) also used both methods, in addition to the 2D Fourier and vent clustering, in his analysis of the TransMexican Volcanic Belt. Hammer (2009) suggests using an updated ‘strip method’ using one of three modifications to analyse alignments in point patterns, depending on the goal of the analysis. The following outlines possible steps that could be used to assess the AVF during the development of an appropriate spatio-temporal model. This is modeled after methods described in Martin et al. (2004), Condit and Connor (1996), and suggestions made in Hammer (2009). However, this model only predicts probabilities of future event locations, not timing. We may want to use a spatio-temporal model if we can refine ages significantly. The outcome of each would be determined from the answers to the questions and considerations mentioned earlier.

1) Magma output—are there any simple trends such as volume/area or timing (repose) patterns? These could be determined by linear regression models. It should be possible to use the ArcGIS and fieldwork to determine distances and areas/volumes of each eruption to analyse the AVF statistically.

2) Use maximum/minimum recurrence rates and repose intervals through time to constrain recurrence rate.

3) Clark-Evans test or other nearest neighbour test to determine if points are random or clustered.

4) Identify vent clusters: o using a vent density technique, such as kernel technique, plotting fraction of

volcanic vents versus distance to nearest neighbour to determine optimum h values (Gaussian distribution most accurately describes small volcanic fields)

o Use 2-point azimuth analysis to assess significance of aligned vent orientations

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o Hough transform method to identify the location of vent alignments, OR, o Use a strip method with modification to analyse for alignments, to eliminate the

assumption of spatial homogeneity 5) Construct probability data sets for Bayesian inference or BET_VH: geochemistry;

timing/ashfall; structural data; gravity anomalies; and other geophysical indicators. Threats from ash fall from distant volcanoes on the North Island may be incorporated into this model.

6) It may be possible to ‘test’ this model by removing recent events (Rangitoto) and trying to predict their timing and/or location.

These suggestions and the content of this report is intended for use in assessing the AVF for long-term hazard modelling; short-term modelling aspects (as in Exercise Ruaumoko) will need to be considered in a separate analysis.

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