Chapter 14
Magma memory recorded by
statistics of volcanic explosions
at the Soufrière Hills volcano,
Montserrat
O. Jaquet, R. S. J. Sparks, R. Carniel
Volcanic eruptions are commonly characterised by time series of events,
such as earthquakes and explosions, which can be analysed by statistical
techniques to interpret physical mechanisms of eruption and to be applied
to forecasting. We apply geostatistical methods (Chilès and Delfiner 1999)
to a time series of Vulcanian explosions that occurred at the Soufrière Hills
volcano, Montserrat (Figure 14.1) in the period 22 September to 21 October
1997. These techniques can be used to detect correlations in occurrences of
volcanic processes. Such correlations indicate that the processes are capable
of remembering their past activities and can be used to detect memory
effects in dynamic systems.
The sequence of 75 Vulcanian explosions at Soufrière Hills followed
a collapse of the andesite lava dome on 21 September 1997 (Druitt et al.
2002). The mean repose interval was 9.6 hours with a minimum of 2.8 hours
and a maximum of 33.7 hours (Connor et al. 2003). The explosions were
short-lived (tens of seconds), impulsive and energetic (Druitt et al. 2002)
with column heights between 5 and 15 km above sea level and individual
ejecta volumes up to 6.6 × 105 m3 . The explosion time series is complete and
precisely timed by seismic signals, allowing us to apply a stochastic approach
using the variogram (Chilès and Delfiner 1999; Jaquet and Carniel 2001)
1
to characterise the statistical behaviour. The data consist of the date and
time of an explosion (Tm ), the time interval between consecutive explosions
I(ti ), and the Real-time Seismic Amplitude Measurement (RSAM) (Endo
and Murray 1991) averaged over 10 minute intervals S(tj ). The data are
tabulated in Druitt et al. (2002) and are displayed in Figure 14.2. The
explosions are well-defined by pronounced spikes in RSAM values.
14.1
Stochastic Analysis
14.1.1
Methodology
Occurrences of volcanic activity present complex and irregular patterns.
However, such activity usually exhibits some structure in time linked to the
mechanisms of the volcanic system. Consequently, the behaviour of time
series monitored at active volcanoes is not expected to be completely random but should present some correlation or persistence in time. In general,
these time series are characterised by short data sets of several irregularly
sampled geophysical parameters with potential gaps due to data losses or
incompleteness of the record. However, most classical statistical techniques,
such as Fourier methods, require fairly long time series without recording
gaps. In addition, the collected data often indicate non-oscillatory or even
non-stationary behaviours of volcanic activity. For the analysis of these
particular time series, the probabilistic formalism of geostatistics (Matheron 1962) is helpful, because it offers the simple statistical and analytical
tool of the variogram for the detection and quantification of the behaviours
observed when monitoring volcanic activity. The variogram was introduced
initially for the study of turbulent flow (Kolmogorov 1941) and has mainly
been applied to spatial problems (Matheron 1962; Chilès and Delfiner 1999;
Davis 2003). The capabilities of variogram analysis for time series have
been shown by Jaquet and Carniel (2001, 2003, 2005) in volcanology and
by Rouhany and Wackernagel (1990) in hydrogeology.
Volcanic activity is described with specific models which reveal characteristics of the underlying volcanic processes. Models of stochastic processes,
i.e., 1D random functions in geostatistics, are used to describe volcanic time
series. This modelling choice implies that a time series describing volcanic
activity is interpreted as a realisation of a stochastic process V (t). Mathematically, a stochastic process can be considered as an indexed set of random variables. One draw from this set of random variables constitutes a
realisation of a stochastic process. If structure exists in the volcanic time
series, the random variables of the stochastic process are no longer considered independent. Since, in practice a unique time series of a given variable
is generally available, some simplifications are required for the inference of
2
the model. The idea is to assume some form of statistical homogeneity
with respect to the observed behaviour in time. Such an assumption leads
to the classical hypothesis of second-order stationarity where the first two
moments of the stochastic process:
E [V (t)] = m
E [(V (t) − m)(V (t + τ ) − m)] = C(τ )
(14.1)
are invariant under translations; i.e., the mean m is constant and the autocovariance C(τ ) depends only on the time interval (or lag) τ . E[ ] denotes
the mathematical expectation.
A weaker stationary hypothesis is the intrinsic hypothesis, which considers the following assumptions with respect to the increment, V (t + τ ) −
V (t), of the stochastic process (Chilès and Delfiner 1999):
E [V (t + τ ) − V (t)] = m(τ ) = 0
V ar [(V (t + τ ) − V (t))] = 2γ(τ )
(14.2)
where the mean of the increments (or drift) m(τ ) is assumed to be equal
to zero and the variance of the increments 2γ(τ ) depends only on the time
interval. Under this hypothesis an intrinsic stochastic process is defined as
a stochastic process with stationary increments and the variogram γ(τ ) is
defined as follows:
1 γ(τ ) = E (V (t + τ ) − V (t))2
2
(14.3)
For an intrinsic stochastic process, the existence of mean and variance of
increments (eq. 14.2) does not imply the existence of a constant mean or a
constant variance (eq. 14.1). As a result, under the intrinsic hypothesis, the
variogram may grow indefinitely rather than being bounded. This allows a
larger class of temporal behaviours to be described. But, in order to comply
with the intrinsic hypothesis where the drift is assumed zero (eq. 14.2), the
variogram must grow slower than a parabola (Wackernagel 2003):
lim
γ(τ )
= 0,
τ2
τ −→ ∞
(14.4)
In the stationary case (cf. eq. 14.1), the following relation holds:
γ(τ ) = C(0) − C(τ )
(14.5)
That is, in the stationary case the variogram is bounded by a limiting value
equal to the variance of the stochastic process (Figure 14.3). The corresponding time interval is named the time scale and beyond it the correlation
3
no longer persists. This time scale corresponds to the amount of past activity in memory and expresses the persistence of the behaviour for the time
series. The variogram of a time series delivers estimates of the degree of
dissimilarity at various scales in time. In particular, at short time intervals
(near the origin), the rate of the variogram increase provides information
about the continuity and the regularity of the time series (Figure 14.3).
In order to describe the variability of a time series, v(t), with the help
∗
of the variogram, the following measure of dissimilarity γαβ
is calculated
between pairs of data values v(tα ) and v(tβ ):
∗
γαβ
=
(v (tα ) − v (tβ ))2
2
(14.6)
This measure of dissimilarity is made dependant on the time interval:
γ ∗ (τ ) =
1
(v (tα + τ ) − v (tα ))2
2
(14.7)
Finally, by averaging all the data pairs with the same time interval, the
experimental variogram γ ∗ (τ ) of the time series is estimated using the following expression:
γ ∗ (τ ) =
nτ
1 X
(v (tα + τ ) − v (tα ))2
2nτ α=1
(14.8)
Where nτ is the number of data pairs separated by a given time interval. In
the case of irregular sampling intervals or gaps in the time series, the time
interval is defined according to classes. This means that for the calculation
of an experimental variogram value at a given (exact) time interval, the pairs
of data selected are separated by a time interval with a certain tolerance.
Generally, the tolerance is set equal to τ /2 and then the corresponding class
for the time interval is: τ ± τ /2.
Choosing authorised parametric models (i.e., conditionally negative
definite functions; Wackernagel, 2003) for the variogram permits the application of estimation and simulation methods (Chilès and Delfiner 1999)
for forecasting volcanic activity (Jaquet and Carniel, this volume). In addition, these parametric models constitute a catalogue of functions describing
the various behaviours of time series and they provide parameter estimates
quantifying the variability in terms of scale and intensity.
For the variogram, the time dependence between the observations renders it difficult to conceive statistical tests. Therefore, when a variogram
model is inferred, the hypothesis (e.g., stationarity versus intrinsic) is selected according to the observed behaviour of the experimental variogram,
combined with other available information (geological, etc.). This choice is
4
a decision made by the modeller and as long as it is not refuted by additional data or instrumental problems, the chosen hypothesis is maintained.
In conclusion, the variogram presents considerable advantages of simplicity
in its application. The variogram does not suffer from the bias (link to
the estimation of the mean from the data) inherent to the autocovariance
(Cressie and Grondona 1992; Haslett 1997) and, in addition, the variogram
allows for the description of a wider class of temporal behaviours.
14.1.2
Results
A Gaussian transformation was applied to data RSAM, S(tj ), which generates a time series, S(tj )Gauss , with a Gaussian distribution centred (mean
= 0) and reduced (variance = 1). This non-linear transformation reduces
the influence of extreme values and improves the detection and the parameterisation of time correlation. For the RSAM time series, nested persistent
behaviours were identified on the variogram (Figure 14.4(a)). That is, patterns in activity persist on at least two time scales, indicated by the line
segments fit to the variogram. This variogram behaviour was fitted by
weighted least squares method where more weight was given to the variogram behaviour at short time intervals. The chosen model corresponds
to the sum of two spherical models (Chilès and Delfiner 1999) in order to
match the linear behaviour observed with two different slopes close to the
origin:

h
i
h
i
3 τ
3 τ
1 τ3
1 τ3

b
+
b
+
b
,
τ ≤ a1
−
−

3
3
0
1
2
2 a1

i 2 a2 2 a2
h 2 a1
3
γM (τ ) =
, a1 < τ ≤ a2 (14.9)
b0 + b1 + b2 32 aτ2 − 21 τa3

2


b0 + b1 + b2
,
τ > a2
where γM (τ ) is the variogram model, the parameter b0 represents the intensity (i.e., a part of the variance which is equal to b0 + b1 + b2 ) of the
random component of the time series. This component is mainly related
to variability occurring below the sampling scale and to measurement errors. The parameters b1 and b2 correspond to the intensity of the stochastic
components for the time series. Finally, the parameters a1 and a2 are the
time scales for which the following values were estimated: a short memory
time scale of 4 hours and a long memory time scale of 87 hours. Beyond
the time scale a2 , the behaviour of the time series becomes uncorrelated in
time. RSAM values integrate all the seismic activity at the volcano as well
as secondary effects like rockfalls. The main contribution to the variability
of RSAM data is provided by the explosions. The explosion signature lasts
at an hourly scale (Figure 14.2) and corresponds to the short memory time
scale. Since RSAM values are a proxy of the energy of the system, the 87
5
hours time scale represents the period for which the energy of the system
tends to persist at similar levels.
The variogram for the time series I(ti ) was calculated, remembering that I(ti ) is the time interval between consecutive explosions (Figure 14.4(b)). It displays some statistical fluctuations at large time intervals;
but its linear behaviour close to the origin is well defined by a sufficient number of data pairs. An elementary spherical model was fit to the variogram
in and identified a time scale of approximately 60 hours. This time scale
represents the persistence in the timing of explosions; i.e., explosions occurring during this time scale are likely to occur with similar repose periods.
Both time series show evidence of persistent behaviour for the timing of
explosions, but on slightly different time scales.
14.2
Model Development
We consider the time scale of the repose periods as a memory indicator (of
the volcanic system) lasting over several explosions. Our analysis suggests
a schematic model for the Vulcanian explosions (Figure 14.5). A volcanic
conduit connects the magma chamber, estimated at approximately 5 km
depth (Barclay et al. 1998; Melnik and Sparks 2002), to a surface explosion
crater. The conduit narrows with depth and is divided up into several zones,
each of which represents a magma batch that will eventually erupt in an
explosion. Eight magma batches are shown as this is close to a suitable
number to give memory time scales of 60 to 87 hours for explosions spaced
on average at 9.6 hour intervals. When an explosion occurs all the deeper
magma batches are decompressed so that their evolution is influenced by
the explosion magnitude. We suggest that this is the physical explanation of
the system memory. When magma batch 1 (Figure 14.5) explodes all deeper
magma batches ascend the conduit during the repose interval, with batch
2 filling the uppermost parts of the conduit and a new batch of magma
entering at the base of the conduit from the chamber. We show below,
with illustrative calculations, that this new batch is not much affected by
the decompression due to explosions because it was part of a voluminous
magma chamber where the net decompression would be very small.
The volumetric magma discharge rate during the period of the time
series has been estimated to be 8.5 m3 /s (DRE = Dense Rock Equivalent)
(Druitt et al. 2002; Sparks et al. 1998). For a mean repose period of 9.6
hours the average volume of each batch is estimated at 3 × 105 m3 (DRE)
(Druitt et al. 2002; Formenti et al. 2003). Given the wide range of repose
periods and explosion magnitudes, there must have been a considerable
range of batch volumes around this average. Direct estimates of the vent
diameter and spine diameters during dome extrusion (Melnik and Sparks
6
2002; Watts et al. 2002) give a cross-sectional area of 700 m2 for the uppermost conduit. Taking the density of vesicle-free magma as 2650 kg/m3
then the decompression associated with explosive eruption of 3 × 105 m3 is
estimated as at least 11 MPa, which evacuates magma (DRE) from the uppermost 430 m of the conduit. The depth of evacuation must be greater,
because the magma is vesicular, but the gas has negligible mass, so this
vesicularity does not influence the decompression. The decompression, however, may be significantly larger than this value for two reasons. First, if
the conduit narrows with depth then the calculated magma depth and the
change in magmastatic pressure must both increase. We show below by
inference from the memory time scales that the conduit narrows with depth
by a factor that could be as much as two. Second, large overpressures are
thought to develop in the uppermost parts of the conduits during viscous
magma ascent (Melnik and Sparks 1999, 2005) and to account for explosions. Taking into account all these effects the typical drawdown for an
explosion will be in the range 1 to 2 km, in agreement with the analysis of
Formenti et al. (2003). Overpressures in the range 5 to 20 MPa may arise
based on modelling of conduit flow (Melnik and Sparks 1999, 2005) and
experimental studies of threshold pressures for explosions (Adilbirov and
Dingwell 1996; Spieler et al. 2004). The decompressions associated with
each explosion are thus estimated in the range 15 to 30 MPa. Alternatively
assuming a pressure of 130 MPa (Devine et al. 1998) for the pressure at
the bottom of the 5 km long conduit and 8 batches, gives a mean pressure
drop of order 16.2 MPa for each explosion. Magma rising through the conduit is thus affected by several decompressions by the time it arrives in the
uppermost conduit and explodes.
We now consider the effects of explosions on the magma chamber. Following an explosion the magmastatic weight of magma in the conduit is
reduced and hence the pressure difference driving flow (i.e., the difference
between the magma chamber pressure and the magmastatic pressure of the
magma column) is increased. Magma flows from the chamber into the conduit, attempting to re-establish the magmastatic weight before the next
explosion. The pressure change in the chamber depends on its volume and
the inflow and outflow of magma. Thus far (August 2005) Soufrière Hills
volcano has erupted 0.5 km3 of andesite magma, providing a lower bound
on chamber volume. For a typical magma batch of 3 × 105 m3 the fractional
volume change in a 0.5 km3 chamber is 6 × 10−4 . If the walls are rigid then
total chamber volume is fixed and the pressure will decrease due to magma
expansion. Soufrière Hills volcano magma is water-saturated and is likely to
contain excess volatiles as bubbles: based on evidence from melt inclusion
studies (Devine et al. 1998), phase equilibria experiments (Barclay et al.
1998; Rutherford and Devine 2003), and observations of excess SO2 emis7
sions (Edmonds et al. 2001). For water-saturated magma pressure changes
caused by outflux of magma into the conduit can be accommodated by gas
exsolution and expansion of existing gas bubbles. The volume of water
exsolved, Vge , can be related to the pressure change using solubility laws:
Vge = (xVc ρm ) /ρg
i
h
1/2
1/2
x = s Pi − Pf
(14.10)
(14.11)
where x is the gas weight fraction, s is the solubility coefficient (s =
4.1 × 10−6 Pa−1/2 ), Vc is the melt volume in the chamber, ρm is the melt
density (equals 2300 kg/m3 for rhyolite), ρg is the gas density, Pi is the
initial pressure and Pf is the final pressure. The volume change, Vb , for
bubbles occupying an initial volume Vbi can be estimated:
Vb = Vbi {(Pi /Pf ) − 1}
(14.12)
A decrease in pressure will result in an expansion of the melt phase by a
volume Vm according to:
Vm = Vc [1 − exp {(Pf − Pi ) /b}]
(14.13)
where b is the melt compressibility (∼ 15 GPa).
Thus the pressure change can be calculated by equating the three volumes (Vge +Vb +Vm ) with the total volume of magma evacuated (3 × 105 m3 ).
An upper bound on the pressure change can be calculated by assuming: a
volume of melt of 0.175 km3 in the chamber of volume 0.5 km3 , noting that
Soufrière Hills volcano andesite is crystal-rich (65% by volume); temperature of 855◦ C and initial pressure of 130 MPa, constrained by petrological
data (Barclay et al. 1998; Murphy et al. 2000; Devine et al. 2003), yielding
a gas density ρg = 170 kg/m3 ; that crystals are incompressible and wallrocks rigid; and that there are no pre-existing bubbles (Vbi = 0) and watersaturated melt. Using these parameters the pressure change is calculated at
0.68 MPa with gas exsolution contributing 97.4% of the change. If the melt
volume is increased to 1 km3 and there were 5% pre-existing bubbles the
pressure change is calculated at approximately 0.1 MPa with the relative
volume contributions being: Vge = 84.3%, Vb = 13.3%, Vm = 2.4%. The
pressure change can be expected to be even smaller than these estimates
since the chamber walls are not rigid and the chamber is an open system
(Murphy et al. 2000). Thus the effects of surface explosions on magma
chamber pressure are only a small fraction of the pressure changes in the
conduit; thus the chamber itself does not have a memory of the explosions.
The memory time-scale enables magma transit time through the conduit, average ascent velocities and conduit cross-section to be estimated. For
8
a conduit length of 5 km and transit times of 60 and 87 hours, magma speeds
of 0.023 m/s and 0.016 m/s are calculated, results in reasonable agreement
with the > 0.02 m/s estimate of Rutherford and Devine (2003), based on
the thicknesses of hornblende reaction rims. For the estimated flow rate of
8.5 m3 /s, cross-sectional areas of 370 m2 and 530 m2 are calculated, which
is much less than the observed surface cross-sectional area of 700 m2. We
conclude that the conduit is flared, becoming narrower at depth. Couch et
al. (2003) also inferred a narrower conduit at depth to explain the absence
of extensive low-pressure microlite crystallization in pumice samples.
The origin of the memory effect remains unexplained just by a sequence
of decompressions on rising magma. If each magma batch attains equilibrium after a decompression then no memory effect should be expected.
However, there are strong kinetic effects related to degassing, crystallization
and consequent rheological changes (Sparks 1997). Modelling studies (Melnik and Sparks 1999, 2005) and experimental investigations (Couch et al.
2003) indicate that gas exsolution, bubble growth and microlite crystallization are very sensitive to pressure history. In ascending from the chamber
andesite magma changes its viscosity by several orders of magnitude, with
large variations in viscosity at any fixed pressure as a consequence of very
small changes in gas exsolution and crystal content. Disequilibrium prevails
with, for example, kinetic time scales for microlite crystallization being several hours to days in the Soufrière Hills andesite (Couch et al. 2003). The
build up of internal pressure to the threshold conditions for an explosion
depends on the highly non-linear processes of gas exsolution, gas escape,
time evolution of rheological properties and crystallization. The threshold
for an explosion is sensitive to the porosity in the magma (Adilbirov and
Dingwell 1996). The build-up in pressure is also influenced by the escape of
gas, which is in turn controlled by the development of permeability in the
vesiculating magma (Takeuchi et al. 2005) with very non-linear relationships between porosity and permeability. Unfortunately a complete model
of these complex, interacting processes has not yet been developed so the
arguments for a kinetic cause of the memory can only be made qualitatively.
However the analysis of the same time series data by Connor et al. (2003)
shows that the distribution of repose periods fits a log logistic model, which
can be linked to a system with competing effects on the variation of overpressure between explosions. For any particular repose period and explosion
the magma batch will have been through a complex pressure and rheological
history that ultimately leads to exceeding the threshold for explosion. This
batch has shared part of this history with deeper batches and this common
experience results in a system memory.
9
14.3
Concluding Remarks
Statistical analysis of a time series of repose intervals and seismic amplitude for 75 Vulcanian explosions at the Soufrière Hills volcano, Montserrat
indicates that the system has memory. The time-scale is approximately 60
hours based on repose interval data and 87 hours based on seismic data.
The mean repose interval is 9.6 hours, so memory of the effects of an explosion is retained for about eight subsequent explosions. These results are
consistent with decompression of ascending magma. Each explosion empties the upper conduit, decompresses deeper magma by several MPa, and is
followed by magma ascent to refill the upper conduit. Magma will be decompressed by several explosions during episodic ascent. In contrast magma
in the voluminous chamber is little affected by explosions. The system has
memory of the pressure variation history of ascending magma, caused by
kinetic effects related to degassing, crystallization and rheological stiffening. Memory is controlled by magma transit time through the conduit, the
magnitudes of explosions and the magma volume occupying the conduit.
For the Soufrière Hills with a conduit length of 5 km and average magma
discharge rate of 8.5 m3/s, mean ascent speeds of 0.016 m/s and 0.023 m/s
and conduit cross-sections of 530 m2 and 370 m2 are calculated for memory
time scales of 60 and 87 hours. The conduit must narrow with depth since
the observed surface cross-section is 700 m2 .
Stochastic methods applied to volcanic time series can allow the recognition and interpretation of memory effects in terms of system dynamics.
Here we show that the memory of a series of explosions reflects the repeated decompression of magma ascending along a conduit. Such methods
can constrain physical attributes of a system, such as magma ascent speed,
and contribute to forecasting by establishing how sequential volcanic events
are correlated.
Further reading
For an in depth introduction and presentation of multivariate geostatistical models and methods, the reader may refer to the book by Wackernagel (2003). And
for an extensive and detailed mathematical presentation of geostatistics covering
the last four decades of theoretical developments, the book by Chilès and Delfiner
(1999) offers the complete story in one volume. For the ongoing odyssey of the
Soufrière Hills volcano, the reader should consult the impressive book by Druitt
and Kokelaar (2002) that includes thirty publications on the phenomenology and
the monitoring of the eruption.
Acknowledgements
The authors acknowledge support of the EU within the MULTIMO Project
10
(Energy, Environment and Sustainable Development Program, EU Contract n.
EVG1-CT-2000-00021). RSJS acknowledges a Royal Society Wolfson Merit Award.
We acknowledge Brian Baptie who kindly supplied the RSAM data and MVO
for permission to use the seismic data. We also acknowledge the reviews of
Chuck Connor and anonymous referees. Finally, part of this work was presented to the 2004 workshop entitled: “Statistics in Volcanology” which was part
of the Environmental Mathematics and Statistics Programme funded jointly by
NERC/EPSRC, UK.
11
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15
Atlantic
Ocean
N
Montserrat
Lesser
Antilles
Silver Hills
150 km
St. John's
Salem
Centre Hills
W.H. Bramble
Airport
Harris
Cork Hill
Exclusion Zone Long
Ground
St. George's
English's
Hill
Crater
Tar River
Soufrière Hills
Plymouth
St. Patrick's
South Soufrière
Hills
4 km
RSAM [counts]
Figure 14.1: Map of the Montserrat island, UK dependency which belongs
to the Lesser Antilles in the Caribbean Sea. The active volcano of the
Soufrière Hills is located in the southern part of the island, where current
activity has forced authorities to create an exclusion zone for the population.
400
200
0
0
100
22 Sep
26 Sep
200
30 Sep
300
400
500
600
4 Oct
8 Oct
12 Oct
17 Oct
700
21 Oct
Time [hours]
Figure 14.2: Sequence of 75 Vulcanian explosions (*) at Soufrière Hills
volcano and their seismic signature using RSAM measurements.
16
(a)
parabolic
regular and differentiable
(b)
continuous and non-differentiable
linear
(c)
high irregularity
discontinuous
(d)
nugget effect
absence of correlation
2
s
(e)
bounded variogram
stationary
a
(f)
unbounded variogram
intrinsic
Figure 14.3: Variogram behaviour at short time intervals in relation to the
regularity and the continuity of the time series (a-d). Variogram behaviour
at long time intervals for stationary and intrinsic cases (e-f) where s2 is the
variance and a is the time scale.
17
Variogram of I(ti) [hour]²
Variogram of S(tj)Gauss
(b)
(a)
1.25
1.00
1.00
0.75
0.75
0.50
0.50
0.25
0.25
0
0
50
100
76
30
93
83
76
150
200
93
89
20
73
68
64
67
61 56
49
61
64
57 60
57
93
10
106
10
50
0
0
77 71
250
Time interval [hour]
0
0
50
100
150
200
250
Time interval [hour]
Figure 14.4: Variograms and fitted models for the time series S(tj )Gauss (a)
and I(ti ) (b) during the sequence of explosions. The inset (a) displays the
variogram behaviour at a short time interval for the time series S(tj )Gauss .
The dashed horizontal line corresponds to the variance of the data and the
numbers on the graph (b) represents the number of data pairs (cf. eq. 14.8).
18
(a)
(c)
(b)
Figure 14.5: Schematic of a model to explain the memory effects during
magma ascent related to repetitive explosions. In (a), 8 magma batches
are depicted within the conduit, each of which corresponds to the average
volume of magma involved in an explosion. In (b), batch 1 explodes, decompressing all the deeper batches. In (c), magma rises and batch 2 reaches
the near surface prior to the next explosion and a new magma batch enters
the base of the conduit. Thus, on average, the persistence of memory in this
system is expected to include the eruption of eight separate magma batches
on average, or approximately 70-80 hr.
19