Time series applied to volcanic data:
A review and application to
Fuego volcano.
Rüdiger Escobar Wolf
MTU, September 2010.
Outline:
1. What are time series and where do they come from?
2. In what way are time series useful and what can they tell
us?
3. How can we analyze time series and what methods are
out there?
4. The Fuego case: different domains and different time
scales.
5. The larger context of Fuego
What are time series and where do
they come from?
• A variable “varying” in time.
• The domain of the variable: physical
continuous magnitudes
Seismic… velocity.
Seismic… RSAM
GOES… Thermal
But also discrete or even categorical data
Where do time series come from?
• Measuring and recording the time-varying
“variable”.
• Discretization.
• Sampling resolution.
• Reliability & uncertainty.
• How well do we know the variable? I. e. the
size of an eruption?
Degradation with time and size.
• Older and smaller events are more difficult to
record.
• Dataset or “catalogue” completeness. How far
back do I think my dataset is reliable for a given
size of event?
• Implications for the statistical properties of the
dataset.
4
VEI
3
2
1
1500
1600
1700
1800
Year
1900
2000
Measuring and recording the time
variable
• Sampling resolution.
• Reliability & uncertainty.
• Aggregating data.
Dating…
• Some geologist are so boring that they end up
dating rocks…
• Dating prehistoric events: 40Ar/39Ar, 14C, K-Ar,
U series, etc…
• Point vs. interval data, and how to combine
them.
14C
14C
40Ar/39Ar
The time of historical events
• Chronicles and their interpretation
• Facts, myths and everything in between.
The year of [fifteen hundred] eighty one, on December twenty six, the volcano
started to throw fire more than usual, and it was so much what it threw, and with such
a fury, the next day of December twenty seven, through a mouth that it has in the
highest part, that from the abundant ash that came out, the air became black and
thick, such that people couldn’t see each other [in Antigua Guatemala?]…
…that ash reached many leagues from Guatemala, in the province of Xoconusco,
where the trees were found to be covered by it…
The next month of January, at the beginning of the year [fifteen hundred] eighty two,
on the fourteenth of that month, the same volcano started to throw so much fire, that
a great mishap was feared, because in the twenty four hours that the fury lasted, one
couldn’t see anything from the volcano but rivers of fire and very large rocks made
embers, which came out of the volcanoes mouth and came down with enormous fury
and impetus…
That fire caused much damage from the coast to the southeast, where it ruined a
pueblo de indios named San Pedro, two leagues from [Antigua] Guatemala, although
there were no deaths, because it happened during the day, and prevented by fear, all
the indios escaped with time, abandoning their homes…
Ciudad – Real , Antonio de, 1873, Relacion breve y verdadera de algunas cosas de las muchas que sucedieron al Padre Fray Alonso
Ponce en las provincias de la Nueva España, siendo comisario general de aquellas partes. Tomo I. Imprenta de la Viuda de Calero.
Madrid.
The problem of defining discrete
events for continuous variables
• Different approaches:
• Values over threshold
• Local peaks
In what way are time series useful and
what can they tell us?
• From very straightforward, i. e. simple
trends…
• To very complex, i. e. cyclic behavior, etc.
Seismic… RSAM
Thermal GOES
Cyclic behavior at Fuego 2002 – 2007?
2002
2003
2004
2005
2006
2007
Correlation between time series
• If they share some phenomenological
(causal?) relationship, chances are that they
may vary together (co-vary)
• Volcanologists (and other geo-scientists) try
this a lot!
Seismic and thermal
Stochastic nature and statistical
structure of some time series data
• Some element of randomness but not
completely unpredictable
• The question is “what can be predicted or
forecasted”?
• Parametric vs. Non-parametric
Probability distributions
and fitting of data
• What is the “distribution of the population
from which the sampled set comes from”?
• How to fit?
• MLE.
The recent (2002 – 2007)
Fuego dataset
Assumptions and elegant lies…
• Can we just assume certain things about the
distribution of the population? E. g. Normality,
independence, Poisson process?
• Is this assumption valid/justified? How do we
know?
• Stationary vs. Non-stationary time series.
A note on time series
and early warning…
• The “ideal” model of increasing risk,
acceptable risk threshold and
warning/response to that threshold.
• Some “real world details”.
Crisis time path (development) posibilities
1
Perceived risk (probability?)
A lethal
eruption
WILL
happen
soon.
A lethal
eruption
will NOT
happen
soon.
II
Tends to…
III
Peak and decrease
curve. This case reflects
and initial increase in
the perceived risk,
followed by a rapid
decrease, either due to
a decrease of the
observed activity, or
due to a process of
desensitizing.
Highly concave curve. This case is
unlikely to be maintained for a
prolonged period of time because of
accustomization and desensitizing. It
usually tends to become case III over
time.
IV
Linear increase curve.
This case represents a
steady increase in the
perceived risk, pointing
towards the actual
occurrence of the
eruption. This is arguably
the best case.
Highly convex
curve. This case
represents the
“sudden” or
“surprise”
scenario, in which
warning and
evacuation actions
can be severely
limited (even
impossible) due to
the short time
available to carry
them out. This is
arguably the worst
case.
I
0
Time
Progressively narrowing time window
for evacuation. As the window narrows The lethal
the options for action decrease and the
event
evacuation becomes more difficult. HAPPENS
Crisis time path (development) posibilities
1
Perceived risk (probability?)
A lethal
eruption
WILL
happen
soon.
In real life, the changes in perceived risk don’t happen as a continuously
varying function of time. They tend to happen as jumps or drops
(discontinuities) associated to the occurrence of key events and
findings (e. g. the initiation of the eruption or the issue of a warning).
A lethal
eruption
will NOT
happen
soon.
0
Time
Progressively narrowing time window
for evacuation. As the window narrows The lethal
the options for action decrease and the
event
evacuation becomes more difficult. HAPPENS
Crisis time path (development) posibilities
A lethal
eruption
WILL
happen
soon.
1
Perceived risk (probability?)
II
A lethal
eruption
will NOT
happen
soon.
III
IV
I
0
Time
Progressively narrowing time window
for evacuation. As the window narrows The lethal
the options for action decrease and the
event
evacuation becomes more difficult. HAPPENS
How can we analyze time series and
what methods are out there?
• Organizing dataset for analysis.
• Database for large, multidimensional
datasets… or just a simple table (the simplest
database) for small, low dimensionality
datasets.
Tools and platforms…
• From the very basic (and limited!): Excel…
• ...to the more complex (and powerful!)
Matlab, R, etc…
• Level of automatization and available (built in)
tools…
Depending on how big and good your dataset is you
can apply different tools.
• Borrowing tools from signal processing
community (electronic and communications
engineering, seismologists, etc):
• Time vs. frequency domain.
• Auto and cross-correlation.
• Fourier and Laplace transformations.
Focus on cyclic behavior
and correlation
Stochastic structure
and statistical methods
• Parametric: Choosing a distribution and
searching for a “best fit”.
• Hypothesizing on why they fit the distribution
or how the fit can be interpreted, for instance
in physical terms.
The recent (2002 – 2007)
Fuego dataset
Parameter
k = 1.55 > 0
Aging process?
The Fuego case:
Different domains and different
time scales.
Summarizing…
• Prehistoric data: very sparse and uncertain…
• goes back to 230 ka, but most relevant for the last 3.5
ka.
• A “fairly good” (detailed and time precise) historical
record.
• Some issues with interpretation for assessing the size,
explosivity, and other relevant characteristics of the
events.
A much more detailed record of recent
(since ~1960’s) activity.
Combined data sources:
• Pre-historic: dating and deposit assessment.
• Historic – older: Accounts from witnesses and
scientific reports.
• Historic – recent: INSIVUMEH bulletins,
OVFUEGO / INSIVUMEH records of lava flow
lengths and number of daily eruptions, GOES
and MODIS / MODVOLC thermal data, RSAM
from INSIVUMEH and J. Lyons, SE-CONRED
bulletins and personal notes.
Some caveats…
• Non-stationary: turning the volcano “on” and
“off”.
• Trends and non-homogeneous processes.
• How can we account for that in the time series
analysis?
Cyclic behavior at Fuego 2002 – 2007?
2002
2003
2004
2005
2006
2007
Clustering of eruptions since 1524?
4
VEI
3
2
1
1500
1600
1700
1800
Year
1900
2000
Thanks!
Questions?
Discussion?…