A Bayesian approach to modelling the relationship between local and
moment magnitudes (ML and MW) for UK earthquakes
Sarah
1
Touati ,
Richard
Problem: Unifying UK earthquake
magnitudes Local magnitude (M ) is
L
3.5
2.5
3.0
Mw
4.0
4.5
derived from the maximum
amplitude of seismic waves
generated from the
earthquake, as measured
on a seismogram. This
saturates at around
magnitude 6: further
increases in size no longer
produce the same
increments in amplitude.
2.5
3.0
3.5
4.0
ML
4.5
5.0
Moment magnitude (MW),
by contrast, is a proxy
measure for the energy
released by an earthquake,
which is determined
through spectral analysis,
and requires more modern,
broadband instruments. It
does not saturate as ML
does.
ML and MW have different physical meanings, but theory suggests ML should be equal to
MW at magnitudes lower than the saturation threshold (the MW scale was devised with
the intention of numerical equivalence with another previous magnitude scale, MS).
However, real data sets frequently differ from this theory, as can be seen in the plot
above, which shows the 111 UK earthquakes for which both ML and MW are available.
Ultimately we wish to estimate MW for all events in the catalogue; we therefore need to
understand and model the relationship between these two quantities, in the presence of
measurement errors in both.
Results: UK data
OpenBUGS reads in the model (an
encoding of the nodes and
connections above, as well as the
priors specified), the data, and the
initial values, and simulates the
joint posterior over four MCMC
chains with 11,000 iterations
(discarding the first 1,000).
To produce a predictive plot of the
true MW for any given ML (within a
certain range), we also create a
grid of hypothetical measured ML,
for which the algorithm also
produces posterior MW estimates
at the same time as doing the
inference on the data. The results
of these are shown on the right,
for the three different models
tried, along with regression fits.
These may be viewed as a ‘look-up
table’, giving posterior quantiles of
MW for a given measured ML.
Deviance Information Criterion
(DIC) is one way of comparing the
goodness of fit of different
models; in our case, DIC favours
the single straight line model,
indicating that there is little
evidence of ML saturation in the
magnitude range of our data. The
quadratic and double straight line
models show a large “fanning out”
of the MW quantiles towards large
magnitudes, displaying the large
uncertainty in the parameters
governing curvature – which
appears to be mostly upward, as
expected from theory.
We note also that for the single
straight line model, the Bayesian
posterior mean slope and
intercept are closer to the
theoretical values of 1 and 0,
respectively, than the regression
fits. This is apparent from the
slopes of the lines in the top plot,
and is not caused by the priors,
which, although centred on the
theoretical values, had large
standard deviations of 0.5. Thus,
some (although perhaps not all) of
the discrepancy between MW and
ML seems to be accounted for in a
Bayesian analysis taking into
account the errors and all prior
information.
Straight line (2 parameters)
2
Chandler ,
1University
Ian
1
Main ,
Roger
3
Musson
of Edinburgh, School of GeoSciences
2University
College London, Department of
Statistical Science
3British
Geological Survey, Edinburgh
Method: Bayesian inference
ML are scattered around the
model values due to variations
in properties of the recording
network and Earth’s crust
ML measurement
error – specified
generally for all
events
MW measurement
error – specified for
each event in the
catalogue, and
allowed to be
uncertain through
stochastic parameter
ν
Model
parameters,
along with MW,
determine ML
‘True’
values
underlie the
measured
quantities
Modelling the system
Priors
Bayesian inference, through an MCMC
inference engine such as OpenBUGS, is
particularly well-suited to this kind of problem.
This is done by encoding a conceptual model
of the system being modelled, shown above as
a directed graph. Arrows indicate the flow of
causality. Since MW is the more fundamental
quantity, we take it as the starting point, with
ML dependent on it through the model β (with
some stochastic variation around this). We
distinguish between the measured magnitudes
(written with tildes, on the right) and the true
underlying values, to which the former are
related by the addition of a measurement
error.
Priors are set on all unobserved stochastic nodes as follows:
•
β: Gaussian prior with standard deviation of 0.5
•
MW: Gutenberg-Richter (exponential) prior. The ‘b’
parameter has its own hyperprior that covers a
plausible range of values.
•
Measurement errors: these are not taken as fixed, but
are also stochastic variables. Expressed in terms of a
‘precision’ (1/variance), all errors have gamma priors,
set to allow errors to vary around the level quoted in
the catalogue.
Initial values
Initial values must be set for unobserved variables. Four
different chains are run, using different initial values that are
plausible but dispersed from each other.
Results: Simulation
Quadratic (3 parameters)
Double straight line (4 parameters)
To test the performance of our method and compare it with regression methods (least squares and orthogonal
regression), we simulate hypothetical sets of 100 events, with equal ML and MW, and with three different levels of
Gaussian measurement error added: standard deviation 0.1 (low), 0.3 (medium), and 0.5 (high). We simulate
10,000
realisations
for each
situation.
Then we use
each type of
inference to
estimate the
slope and
intercept. The
results are
shown in the
plot (left). We
note that the
Bayesian
inference can
be highly
accurate and
with low
spread, even
when
measurement
errors are
large, if it is
possible to be
confident
(and right!)
about the
parameters
beforehand.
Even when
priors are less
confident,
Bayesian
analysis
performs
better than
least squares.