Download Chapter 7 Modelling the Spatial Distribution of Volcanoes: An

Chapter 7
Modelling the Spatial
Distribution of Volcanoes: An
Example From Armenia
Jennifer N. Weller, Andrew J. Martin, Charles B. Connor, L.J. Connor,
Arkadi Karakhanian
Geoscientists worldwide are faced with the task of assessing hazards associated with point-like features such as volcanoes and earthquake epicentres on
various temporal and spatial scales. Commonality among these phenomena
exists because the analysis of their spatial distribution and geologic setting
can be used to estimate hazards quantitatively. Often, these geologic hazard
assessments must evaluate the likelihood of very infrequent events that have
high consequence. For example, in the last two decades long-term probabilistic volcanic hazard assessment has been used in siting nuclear facilities
(McBirney and Godoy 2003). Such assessments have been conducted at
Yucca Mountain, Nevada, USA (Crowe et al. 1982; Connor et al. 2000),
the Muria Peninsula, Indonesia (McBirney et al. 2003), near Yerevan, Armenia (Karakhanian et al. 2003), and in Japan (Martin et al. 2004). A
central issue in all of these assessments is the likelihood of a new volcano
forming by eruptions in close proximity to the facility. At such facilities,
hazards with probabilities on the order of 10−6 to 10−8 per year are often
considered high (Connor et al. 1995, Martin et al. 2004) because overall
the risks associated with such facilities must be very low.
Similar assessments are used in Auckland, NZ, to estimate the probability of new volcanoes or volcanic vents forming in urban centres built in
regions of active volcanism (e.g., Magill and Blong 2005), and to forecast
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the location of new lava flow vents on large volcanoes, such as Mt. Etna
(Wadge et al. 1994). Accurately forecasting probable vent locations is a
crucial step in forecasting lava flow hazards. Even the most sophisticated
flow and transport models will fall short if a probabilistic study of potential
vent location is poorly implemented. This is because the location of potential vents exerts such strong control on eventual lava flow-paths down the
flanks of the volcano.
Hazards associated with new vent or volcano formation place a high
priority on statistical model development. Geological hazard assessments
for volcano and volcanic vent distributions should present robust estimates
of hazard rates, based on the spatial density and frequency of past events
formed by geological processes that are expected to persist and continue
during the future time period of interest. The challenge presented by these
hazards is that observed spatial distribution of volcanoes is only one realization of the potential distribution of volcanoes. The processes controlling
the spatial distribution of volcanoes, such as the generation and ascent of
magma, are unobserved. Hence it is impossible to forecast the future distribution of vents deterministically, and statistical models of vent distribution
must be developed.
This chapter, presents two kernel functions, the Gaussian and the adaptive Gaussian (Silverman 1986; Wand and Jones 1999), to analyze the spatial distribution of volcanoes. One goal of this analysis is to characterize
the spatial distribution of past events with a mathematical model. Another goal is to assess the possibility of future volcanism based on a known
distribution of events. A third objective involves discussing limitations of
the current statistical methods in use. Volcano location data from Armenia are used with the added goal of calculating the hazard associated with
future volcanic eruptions at the Armenian Nuclear Power Plant (ANPP)
(Karakhanian et al. 2003), using each of these methods.
Application of kernel estimation techniques provides a leading order
estimate of the conditional probability of volcanism occurring in an area
near the ANPP, given that volcanism occurs in the region. Unfortunately,
this model cannot be easily extended to a spatio-temporal model because
very little data have been collected about the timing of volcano formation
in Armenia, as in many volcanic arcs. This chapter considers how uncertainty in the bandwidth and uncertainty resulting from poorly constrained
rates of regional volcanism influences the probability that volcanism will
impact the ANPP site area. Potential ways of enhancing this assessment,
through improved data quality and statistical modelling, is discussed in the
concluding remarks.
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7.1
Nature of Volcano Distribution in Space
and Time
The spatial distribution of volcanoes is dictated by processes of magma
generation within the Earth’s mantle and magma ascent through the mantle,
ductile lower crust, and brittle upper crust. Each of these processes is
understood currently through inferences drawn from the geochemistry of
volcanic rocks, the presence and distribution of geophysical anomalies, such
as slow seismic wave propagation in areas enriched in magma (Zhao 2001),
and geologic features, such as igneous plutons and dykes, that indicate
styles of magma transport, and which have been exhumed by deformation
and erosion. These observations have helped geologists construct physical
models of magma generation and ascent in a variety of tectonic settings (e.g.,
Marsh 1979; Tamura et al. 2002; Condit and Connor 1996). Conversely,
studies of the spatial distributions of volcanoes have been used to elucidate
the geologic processes (Connor et al. 1992; Lutz and Gutmann 1995).
Magma generation is not uniform throughout the Earth’s mantle, but
rather, occurs under a limited set of circumstances. For example, subduction at some plate margins changes the chemical composition of the mantle,
particularly, by introducing volatiles, and alters the thermal structure of
the mantle, resulting in convection. These physical changes result in partial melting of the mantle to produce magma, which rises buoyantly toward
the surface. In other tectonic settings, the development of thermal plumes,
or decompression of the mantle through crustal extension, results in partial melting of the mantle, and generation of magmas. On a smaller scale,
gravitational instability in melt regions, density stratification of the crust,
differential stress in the crust, distribution of fractures, faults or rift zones,
and similar processes further influence magma ascent. Magma generation
and the ‘plumbing’ of the volcanic system are not directly observed in a
given area of active volcanism, but, at best, inferred from geophysical and
geochemical data, and physical models of magmatism (e.g., Marsh 1979;
Conder et al. 2002; Keskin 2003).
The frequency with which new volcanoes or volcanic vents form is controlled by the rates of these underlying processes. For example, the rate
of subduction of oceanic lithosphere at plate boundaries may play a role
in the rate of volcanism in an overlying volcanic arc (Conder et al. 2002).
In some extensional tectonic settings, such as the Basin and Range of the
western United States, the rate of crustal extension may exert a leading
order control of the rate of formation of cinder cones (e.g., Bacon 1982) and
other types of volcanic vents.
The observed spatial distribution of volcanoes is just a sample of an
underlying and unobserved distribution of the likelihood of volcano forma3
tion in time and space. This underlying distribution is a function of various
physical properties, such as variation in chemical composition of the mantle,
distribution of heat and heat flow, and nature of the crust. This underlying distribution is continuous, as these physical variations in the mantle
and crust are continuous. The fact that these geologic observations and
geological models are based on poorly resolved physical processes suggests
statistical models that can be used to forecast the process of volcano formation. In this sense, the spatial distribution of volcanoes can be thought
of as a point process. A point process is a theoretical construction for generating random events according to some probability model. The observed
spatial distribution of volcanoes in any particular region is a single realization of this point process. Because this realization is comprised of a limited
number of volcanoes, it may or may not provide a complete picture of the
underlying distribution of the likelihood of volcano formation in time and
space.
The physics of magma generation and ascent suggest that the distribution of volcanic events is a Poisson process. Each event, the formation of a
new volcano, is discrete in time and space. The probability of a new volcano
forming at a particular time and place is not in any way influenced by the
formation of other volcanoes in the past, but rather is only a function of
the underlying processes that influence magma generation and ascent. In
other words, a batch of magma that ascends to the surface and forms a
new volcano does not itself make it more or less likely that another batch
of magma will ascend in another location and form a new volcano. This assumption is valid because magmas, erupted at volcanoes, almost always are
small-volume fraction partial melts, their volume is very small compared to
the total volume of mantle which can potentially produce magma. In other
words, eruptions are so rare that the mantle is not depleted by previous
eruptions.
A Poisson model may not be appropriate for all cases. The formation
of parasitic cones, small vents on the flanks of a larger volcano, are not
independent of the formation of the central vent. Also, in extreme cases
of the largest volume silicic calderas, nonrefractory elements in the mantle
may actually be depleted to the extent that future patterns of volcanism
are influenced by the silicic eruption itself.
For a spatial Poisson process, the probability of a new event occurring
in a small area within a much larger region is related only to the spatial
intensity, (events expected per unit area). For spatio-temporal processes,
the probability of a new event occurring in a small area and within some time
interval is related to the spatio-temporal recurrence rate (events expected
per unit area per unit time). As mentioned previously, given the lack of
age determinations for Armenian volcanoes, the analysis here is limited to
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the spatial distribution of volcanoes, which involves estimation of spatial
intensity.
Physical models of magma genesis and observed volcano distributions
indicate that volcanoes cluster on a variety of spatial scales. Such clustering
is expected, for example, if magma generation is related to convective instability in the mantle (e.g., Tamura et al. 2002) and variable rates of tectonic
processes. Clustering suggests that nonhomogeneous Poisson models should
be used to forecast volcano distribution (Connor and Hill 1995). That is,
spatial intensity varies across the map region.
7.2
Spatial Distribution of Volcanoes
in Armenia
The spatial distribution of Quaternary volcanism in Armenia is known from
several decades of geologic mapping by scientists from the Armenia National Academy of Sciences and related organizations (see Karakhanian et
al. 2003). Volcanism in Armenia is related to the comparatively rapid (18
mm/year) convergence of the Arabian and Eurasian Plates, causing uplift of the Caucasus Mountains and partial melting of mantle enriched by
pre-collision subduction (Keskin 2003). In the Quaternary, 554 basaltic to
andesitic volcanoes developed in response to magma generation in this collisional zone (Figure 7.1). Most of these volcanoes are monogenetic. Monogenetic activity is characterized by the formation of a new volcano, such as
a cinder cone or lava dome, and a duration of volcanic activity, typically,
of less than 100 years (Connor and Conway 2000). Renewed volcanism in a
monogenetic volcanic field builds a new monogenetic volcano, rather than
reactivating an older volcano.
Armenia is tectonically active and subsidence and deposition of sediments occurs in basins which host much of the volcanism. This means that
some Quaternary volcanoes, especially small-volume cinder cones and lava
domes, may be completely buried and hence unrecorded in the database.
Hopefully, the estimate of spatial intensity and its variation across the map
region is not adversely impacted by burial of Quaternary-aged vents.
The ANPP is located in the north-western part of the Ararat Depression, a transtensional sedimentary basin formed in response to regional
right-lateral strike slip along major faults demarcating the boundary between the Arabian and Eurasian tectonic plates (Figure 7.1). Mt. Ararat
and Mt. Aragatz, two polygenetic volcanoes, erupting multiple times throughout history, formed on the margins of this basin. Monogenetic volcanoes,
widely dispersed within this basin, create an area known as the Shamiram
Volcanic Plateau. This area includes 38 monogenetic cinder cones located
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1.3 to 6 km north of the ANPP (Figure 7.2). In eastern Armenia, Quaternary volcanoes are concentrated in a NW-trending zone and volcano density
is particularly high on the Ghegam Ridge. Some of these volcanoes have
been dated as Holocene and one of their Late Pleistocene valley lava flows
terminates 25 km east of the ANPP plant site. The most recent volcanic
eruptions on the Ghegam Ridge have been dated between 4400 and 4500
± 70 year BP based on archaeological evidence and radiocarbon age determinations (Karakhanian et al. 2003). These are the most recent known
monogenetic eruptions in Armenia.
7.3
Application of Gaussian and Adaptive
Gaussian Kernels
A model of the spatial distribution of volcanoes in Armenia should capture
several specific features. First, as seen in most tectonic settings, volcanoes
cluster. Second, volcano density within clusters is much higher by orders
of magnitude, than elsewhere in the volcanic arc. Third, the volcanic clusters may be elongate, as in the Ghegam Ridge, or more circular, as in the
cluster of vents located just north of the ANPP. Finally, co-existing with
these prominent clusters, is an area of lower volcano density, the Shamiram
Volcanic Plateau area. The following sections, describe the key steps and
assumptions in our analysis: i) the development of the database of volcanic events, ii) the application of the Gaussian kernel function to model
volcano distribution, and iii) an improvement on the Gaussian kernel, the
adaptive Gaussian kernel function.The adaptive kernel estimator is a wellknown density estimator, but to our knowledge, new to geologic hazard
assessment.
7.3.1
Defining Events
The first and perhaps most important step in developing a probability model
for volcanic hazard assessment is to explicitly define events in the region of
interest (Connor and Hill 1995; Martin et al. 2004). This is important for
a number of reasons. For instance, an event could be the epicentre of an
earthquake (e.g., Woo 1996), or the location of a collapse sinkhole in a karst
terrain, or any other process that can be isolated to a precise location. In
the hazard assessment presented in this chapter, the definition of an event is
limited to the formation of a new volcano, and is estimated directly from the
known distribution of Quaternary volcanoes. This assessment specifically
does not look at the spatial or temporal distribution of volcanic eruptions
at existing volcanoes or at the volcanic hazards associated with eruptions,
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such as lava flows, lahars, or pyroclastic flows.
The volcano dataset used in this study includes both monogenetic volcanoes (one eruptive episode only) and polygenetic volcanoes (volcanoes
that have more than one eruptive episode). Volcano types include cinder
cones, domes, composite volcanoes, maars, and calderas (see Mader, this
volume). Modelling assesses the probability of any new volcanic vent forming within the area and therefore, treats individual vents, equally, regardless
of volcano type. Alternative methods for defining volcanic events can be
used in probabilistic volcanic hazard assessments. For example, polygenetic
volcanoes may be weighted more heavily depending on the number and
volume of past eruptions (Martin et al. 2004). Also, closely spaced and
similarly aged vents can be grouped together as a single event (Connor et
al. 2000), as the formation of volcanoes along these vent alignments may
not be independent events. Furthermore, only cones younger than a specific
age, for example those formed in the last 100,000 years, might be included
in the analysis as volcanic events or these could be weighted more heavily.
Because vents may be grouped into single volcanic events in varying ways
depending on their timing, distribution, and episodes of activity, data sets
can be defined in different ways that would change the estimates of spatial
intensity of volcanism. Partially because the additional data required by
alternative definitions does not yet exist for Armenia, a simple event classification scheme is assumed. The scheme delineates each mapped volcano
(Figure 7.1) as a single event.
7.3.2
Modelling Spatial Intensity of Volcanism
Once the event is explicitly defined, the next step is the mathematical
development of the probability model. As mentioned previously, volcano
clustering is a common feature in all volcanic settings. Application of the
Clark-Evans test confirms that volcano clusters occur in the Armenian data
set. The test shows that volcanic events in Armenia are clustered across
a variety of scales with > 99 percent confidence (Blyth and Ripley 1980;
Cressie 1991). Based on the results of this test, the underlying process is
assumed to be a nonhomogeneous Poisson process and the probability of an
event occurring within a small area, given that a new volcano forms in the
region (the event) is given by:
P [N ≥ 1] = 1 − e−λs A
(7.1)
where N is the number of events occurring within the small area of interest,
A (e.g., the area within which volcanism would impact the ANPP, should
it occur), and λs is the spatial intensity of volcanism within A. Note that
it is assumed in equation 7.1 that λs is constant over the area A, which is
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much smaller than the map region, X, Y , within which λs may vary. If this
assumption is not reasonable, then one must integrate λs over the region A.
Pioneering work by Diggle (1985) and Silverman (1986) has led to the
development of kernel estimation techniques that have been used previously
in studies to estimate spatial intensity of volcanism, including the Pinacate
Volcanic Field, Mexico (Lutz and Gutmann 1995), the Yucca Mountain
region, USA (Connor and Hill 1995), and the Tohoku region, Japan (Martin
et al. 2004). Kernel estimation techniques are widely used in seismology,
for forecasting epicentre locations (see Woo 1996). In this technique, spatial
variation in λs is a function of the distance to nearest-neighbour volcanoes
and a smoothing parameter, h, also known as the kernel bandwidth (Wand
and Jones 1999). The kernel function is a probability density function that
is symmetric about the origin and the event location (in this case volcano
location), and spreads probability away from the event (Diggle 1985). The
kernel model is useful because: i) probability maps can be made allowing
ease of comparison with other geologic information; ii) there is no need
to define zones of volcanic activity, as is required in homogenous Poisson
estimates; and iii) uncertainty in the distribution of individual events is
easy to assess.
Different kernel functions can be used to estimate the probability including the Cauchy kernel (Martin et al. 2004), the Epanechnikov kernel
(Lutz and Gutmann 1995), and the Gaussian kernel (Connor and Hill 1995).
It is widely agreed that the shape of the kernel function chosen in this type of
analysis generally has a trivial impact on probability calculations compared
to other factors, such as the bandwidth (Wand and Jones 1999). Here,
the Gaussian kernel function was chosen. This kernel has the properties
of smoothing intensity from a maximum centred on each volcano, radial
symmetry, and distribution tail that is nonzero for all distances from the
volcano. Moreover, the Gaussian distribution arises in problems of heat and
mass transfer, such as might be expected in volcanic systems, because these
are diffusion processes. Given that the data span an area, the bivariate
Gaussian kernel is used to estimate the intensity over a two-dimensional
surface:
N
1 X − 21 [ dhi ]2
λ̂s (x, y) =
e
(7.2)
2πNh2 i=1
where di is the distance from the point x, y where λ̂s is estimated, to the ith
volcano location, h is the kernel bandwidth, and N is the number of volcanoes (points) in the observed distribution. As N occurs in the denominator,
λ̂s is a probability density function, which will integrate to unity if the map
region X, Y is sufficiently large.
Probability estimates made using equations 7.1 and 7.2 depend on the
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value chosen for the bandwidth, h. Using a bivariate Gaussian kernel, events
will have a high estimated probability in proximity to existing volcanoes
if the value chosen for bandwidth is small, but low estimated probability
away from the volcano. On the other hand, a large value for the bandwidth will yield a more uniform estimate of probability distribution across
the region. Thus, selection of bandwidth is in many ways similar to selection of bin-width for histograms. In the Gaussian kernel, the bandwidth
is equivalent to the standard deviation of a symmetric, bivariate Gaussian
distribution. Therefore, the kernel function depends on the assumption that
the bandwidth is estimated in a geologically and/or statistically significant
way (Connor and Hill 1995).
One approach to bandwidth estimation is to compare the observed
nearest-neighbour distance between volcanoes with the expected distribution of nearest-neighbour distances. Based on the bivariate Gaussian kernel,
the cumulative distribution function (cdf) for fraction of volcanoes located
within distance, D, of their nearest neighbour is:
D
(7.3)
Λ̂s = erf √
2h
This model, with values h = 1 km, 1.5 km, and 2 km, is plotted together
with the observed distance to nearest neighbour volcanoes in Figure 7.3.
Although the observed distribution of distance to nearest-neighbour volcano
lies between these models, it is clear there are differences in the slopes of the
observed and modelled cdfs. At D > 0.9 km the tail of the observed cdf is
thicker than the modelled cdfs. That is, the distance to nearest neighbour
volcano in low density areas is greater than expected when compared to the
cdfs for nearest-neighbour distance from a bivariate Gaussian distribution.
This difference suggests that an adaptive kernel might be more appropriate to model volcano distribution. The adaptive Gaussian kernel allows
the bandwidth to change as a function of distance from volcanoes. Where
the distance, D, between the location of our spatial intensity estimate,
λ̂s (x, y), and mapped volcanoes is large, say D > 0.9 km, the bandwidth
is allowed to increase. Conversely, where λ̂s (x, y) is close to a volcano,
D ≤ 0.9 km, the bandwidth should decrease. A spatially varying bandwidth is adopted by calculating the geometric mean intensity from the spatial intensity, previously estimated using a fixed bandwidth, and within a
threshold distance, D, of mapped volcanoes:

 PM
log
λ̂
(x,
y)
s
i=1
 , D ≤ 0.9 km
log G = 
(7.4)
M
where M is the number of grid cells within the map region X, Y that are
9
closer to volcanoes than D. Then:
ha (x, y) = h
"
G
λ̂s (x, y)
# 21
(7.5)
N
X − 1 di 2
1
λ̂s (x, y) =
e 2 [ ha (x,y) ]
2πNha (x, y)2 i=1
(7.6)
Thus λ̂s (x, y) is found by iteration, beginning with a fixed bandwidth (Equation 7.4), and adjusting the bandwidth using a threshold density derived
from inspection of Figure 7.3.
Fixed bandwidth and adaptive kernels are compared by contouring
λ̂s (x, y) in Figure 7.4. Note that the fixed bandwidth kernel accurately
models the density of volcanic vents in areas of high volcano density, such as
in the Gegham ridge area. In areas of lower volcano density, the h = 1.5 km
fixed bandwidth model produces abrupt variations in the estimated spatial
intensity associated with the occurrence of individual volcanoes. In contrast,
the adaptive bandwidth model retains areas of high vent density, but also
models vent density in sparse regions with a smoother surface. The effect
of this smoothing is particularly clear in the area of the Shamiram Volcanic
Plateau, where vent density is comparatively low overall.
The estimate of λ̂s (x, y) at the site of the ANPP naturally varies as a
function of bandwidth (Figure 7.5). For both fixed and adaptive bandwidth
estimates, λ̂s (x, y) is maximum around h = 1.5 km, reflecting the close
proximity of the ANPP site to a cluster of monogenetic vents. Taking A =
50 km2 to account for the potential of lava flows forming at vents upslope
from the site that might inundate the area, and λ̂s (x, y) ≤ 4.5 × 10−4 km−2 ,
P [N ≥ 1 | event ] ≤ 0.02 (Equation 7.1).
7.4
Including the Rate of Volcanic Activity
Similarly, the probability of a volcanic event occurring anywhere in the
region within a given time interval is:
P [N ≥ 1 | event ] = 1 − e−λt ∆t
(7.7)
where ∆t is the time interval of interest and λt is the recurrence rate of
volcanism in the region. Where ages of volcanic events are well constrained,
and consistent with the geologic arguments outlined previously, λt is likely
nonhomogenous (Ho 1991; Condit and Connor 1996) and can vary by orders
of magnitude through time. Unfortunately, as is frequently the case worldwide, few age determinations have been made of volcanoes in Armenia. As
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a result, λt is naively based on temporal homogeneity as a starting point
and a simple estimate for the temporal recurrence rate,
λ̂t =
N −1
to − ty
(7.8)
where, N is the total number of volcanoes, to is the age of oldest event, and
ty is the age of the youngest event. Assuming that volcanoes used in this
study (N = 554) are all of Quaternary age, to is set equal to 1.8 million years
and ty is set to 0, or present, as the youngest eruption in the database is
thought to be approximately 4400 years. This gives λ̂t = 3×10−4 events per
year. In other words, approximately one event every 3300 years is expected,
acknowledging the fact that there is great uncertainty in this estimate, given
the lack of radiometric age determinations in the region.
7.4.1
Probability Estimates
Multiplying equations 7.1 and 7.7 gives the probability that the ANPP will
be affected by volcanism. If A = 50 km2 and ∆t = 1 year, using our range
of λ̂t and λ̂s (x, y) yields probabilities P [N ≥ 1] ≤ 6 × 10−6 per year, which,
whilst apparently low on human timescales, is high enough to be of concern
based on current International Atomic Energy Agency guidelines (McBirney
and Godoy 2003). Once again, this probability estimate is based on the
assumption that changes in the rate of volcanic activity do not accompany
shifts in the locus of activity. In an arc setting such as Armenia, and
given the persistence of volcanism in the arc throughout the Quaternary,
this makes sense for short time scales. Conversely, more confidence could be
placed in the model, and spatio-temporal models might become appropriate,
if a comprehensive radiometric dating program were completed.
7.5
Concluding Remarks
Von Mises (1957), in his treatise on probability and statistics, shows that it
is possible to develop an accurate model of recurrence, given enough events,
that is, provided the experiment can be performed many times. Geologic
hazard assessments typically have comparatively few events to work with.
In this case, the observed distribution of volcanoes, and models developed
to estimate hazards solely based on these data, are inherently uncertain.
The risk in using these models is that the distribution of past events may
poorly reflect the distribution of potential future events. Any additional
information that sheds light on the spatial distribution is therefore worth
analyzing. By assuming that the frequency of volcanic events is a physical
11
property of some magmatic system, the limiting value of that frequency is
basically unknown, and so probability estimates may span orders of magnitude.
Such issues might be further addressed through quantitative assessment of uncertainty. Assessment of model uncertainty in the application of
nonparametric models is not highly developed. Most literature (e.g., Wand
and Jones 1999) recommends a visual fit to the data to make an appropriate
choice of bandwidth, especially for multivariate cases. Clearly this approach
leaves something to be desired in hazard assessment, as large variation in
λ̂s (x, y) occurs with bandwidth. Uncertainty in the probability estimate for
the ANPP is partially characterized by modifying the kernel function with
a range of bandwidths ((Figure 7.5)) but this assessment does not fully address model uncertainty or affect uncertainty due to the limited number of
events. Bootstrap methods may offer additional insight into model uncertainty, but the best way to build model confidence may lie in incorporation
of additional geologic information via Bayesian methods.
The underlying geological process controlling the spatial patterns of
volcanism is assumed to approximate a Poisson process and persist into the
foreseeable future. The actual geological processes involved are multiple
and complex. Volcanism describes an arc across Armenia (Figure 7.1) that
is sub-parallel to the Alpine-Himalayan collision belt with uplift occurring
across Armenia as a result of the northward motion of the Arabian plate
with respect to Eurasia (Philip et al. 2001); The rate of convergence of
these two plates is 18 − 19 mm/year based on REVEL 2000 models (Dixon
and Mao 2002); Volcanism across the region is linked to subduction and
subsequent collision, and may result from slab steepening and break off,
which provides a viable mechanism for magma generation (Keskin 2003).
On a more local scale, volcanism is closely linked to N-S compression and EW extension (Philip et al. 2001) and the main geologic structures produced
in this tectonic setting are north-west trending right-lateral strike-slip faults
which in turn produce areas of transtension that create pull-apart basins
within which volcanism is localized.
Such pull-apart basins can be delineated by mapping, for example,
anomalies in the Earth’s gravity field, caused by density variations between
the sediments filling the basins and the surrounding crust (Tsuboi 1979).
A geologic model, such as one based on gravity anomalies, may allow probability of future events to be modified to account for our understanding of
the geologic setting of volcanism, as well as possible changes in the underlying processes over the future time period interest. Other pertinent geologic information could be used either alone or in combination with other
data to enhance probabilistic hazard assessments. These include seismic
tomographs (Martin et al. 2004), geochemical data (Condit and Connor
12
1996), fault and other structural data, and magnetic data. Using additional
datasets, for example in a Bayesian framework (see Martin et al. 2004) may
help us to fine tune the point process models to more accurately reflect the
underlying geological processes that control the spatial locations of future
volcanism. Formal mathematical coupling of geological models and statistical models based on historic frequencies is still in its infancy and an area
of ongoing research.
Further Reading
Silverman (1986), Cressie (1991), and Wand and Jones (1999) are required reading for those applying kernel density estimation to geologic hazard problems.
Additional information on specific applications in volcano hazard assessment, see
Martin et al. (2004) and Connor et al. (2000), and references therein.
Acknowledgements
The authors thank the Armenia National Academy of Sciences for field assistance
and data. Reviews by Stuart Coles, Sarah Kruse, Chris Tsokos, Rob Watts, and
an anonymous reviewer improved the manuscript. The work reported here was
supported by a NATO grant and the University of South Florida. Two of the
authors JNW and CBC received funding from the Environmental and Statistics
Programme funded jointly by NERC/EPSRC, UK, to attend the “Statistics in
Volcanology Workshop” held at the University of Bristol in March 2004.
13
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16
45˚
46˚
Armenia
N
44˚
41˚
Shamiram
Plateau
41˚
Ghegam
Ridge
▲
Azerbaijan
Mt Aragats
ANPP
Lake
Sevan
*
Yerevan
40˚
◊
40˚
▲Mt Ararat
Turkey
km
0
50
44˚
Iran
45˚
46˚
Figure 7.1: Quaternary volcanoes (circles) mapped in Armenia and Quaternary faults (dashed lines) result from convergence of the Arabian and
Eurasian tectonic plates at about 18-19 mm/year based on REVEL 2000
models. The ANPP is located west of the city of Yerevan and at the south
edge of a dense cluster of 38 monogenetic volcanoes, part of the Shamiram Volcanic Plateau. The Shamiram Plateau and the ANPP are located
within the fault-controlled Ararat Depression, between the Aragatz and
Ararat Quaternary stratovolcanoes. Another dense volcano cluster, containing the youngest known monogenetic volcanism in Armenia is on the
Ghegam Ridge, west of Lake Sevan. International borders are shown in
grey.
17
Figure 7.2: Photograph of the ANPP. A cluster of cinder cones located
north of the ANPP is seen in the background. Mt. Aragats is to the left of
the photo.
Fraction of Total Volcanoes
1.0
.9
.8
.7
.6
.5
.4
.3
.2
.1
0
1.5
1
2
0
1
2 3 4 5 6 7 8 9 10
Nearest Neighbor Distance (km)
Figure 7.3: Cumulative density distribution of distance to nearest-neighbour
volcano (heavy line) and expected near-neighbour distance calculated using
fixed bandwidth Gaussian kernels of h = 1 km, 1.5 km, and 2 km. Note the
thickening of the tail of the observed distribution relative to the models at
near-neighbour distances greater than approximately 0.9 km.
18
km
41˚00'
0
40˚30'
(a)
10
40˚30'
45˚00'
40˚00'
40˚30'
39˚30'
km
0
43˚30'
44˚00'
44˚30'
50
46˚00'
45˚30'
45˚00'
46˚30'
km
0
10
40˚00'
44˚00'
44˚30'
km
40˚30'
41˚00'
0
10
(b)
40˚30'
45˚00'
40˚30'
40˚00'
39˚30'
km
0
43˚30'
44˚00'
44˚30'
50
45˚30'
45˚00'
46˚00'
46˚30'
km
0
40˚00'
44˚00'
10
10-4
Likelihood per km2
10-3
44˚30'
Figure 7.4: Contour plots of volcano density modelled using fixed bandwidth
(a) and adaptive bandwidth (b) using contour intervals of 1 × 10−6 km−2 ,
1 × 10−5 km−2 , 1 × 10−4 km−2 , and shaded in high density regions. Insets
show contoured spatial intensity in the Ghegam Ridge (upper right) and
ANPP (lower left) areas. ANPP location indicated by solid square.
19
Likelihood per km2
5x10-4
4x10-4
3x10-4
2x10-4
1x10-4
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
Bandwidth (km)
Figure 7.5: Change in likelihood of new volcano formation at the ANPP
site for fixed bandwidth (dashed line) and adaptive kernel models (solid
line), estimated using equations 7.2 and 7.6, respectively, for a range of
bandwidths, h.
20