Empirical Greens Functions: estimating uncertainties of earthquake source spectra

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Empirical Greens Functions estimating
uncertainties of earthquake source spectra

• Germán Prieto
• IGPP, U.C. San Diego

May 25, 2005 IGPP at UCLA Los Angeles
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Earthquake Scaling
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The Question
Are the spectra of larger and smaller earthquakes
similar? How can we accurately describe the
source spectra for different sized events? How
can we describe the uncertainties of source
spectrum estimation when using state-of-the-art
techniques such as EGFs?
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EQ Physics Outline

• Introduction
• The Empirical Greens Functions (EGF)
• Weighted and Combined EGF
• An Example
• Some conclusions

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Energy Partitioning
Stress
ER Radiated Energy EG Fracture Energy EF Frictiona
l Heat Dc Critical Distance s0 s1 Initial final
stress Ds Stress Drop sf Frictional strength
Slip
Partition of energy for the earthquake process is
still unanswered. Do large and small earthquakes
follow this simple model?
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Stress and Radiated Energy
Simple models follow self-similar
process. Differences in mechanics of EQ may be
explained by normal stress reduction (Brune,
1993), acoustic fluidization (Melosh, 1979),
shear melting (Kanamori and Heaton, 2000), etc.
Is the stress drop close to the absolute stress
level (weak faults) or just a small portion
(strong faults)?
(from Mori et. al, 2003)
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Spectral Analysis 101
Time Series
Spectrum
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Spectral Shapes

• Source spectrum
• constant Q (Abercrombie, 1995), variable Q
  (Boatwright, 2002)
• Multiple Empirical Greens function (Ide, et al.
  2001)
• Coda Wave envelopes (Mayeda and Walter, 1996)
• Spectral stacking (Prieto et al. 2004)
• In this study
• focus on estimating the relative spectral shape.
• Use empirical greens function idea

Estimate the source spectrum, rather than
parameters derived from the spectrum
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EQ Physics Outline

• Introduction
• The Empirical Greens Functions (EGF)
• Weighted and Combined EGF
• An Example
• Some conclusions

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Empirical Greens Functions
Data recorded at a seismic station d(t) is a
function of the source time function s(t)
convolved () with the path and attenuation
effects a(t), also known as the transfer function.
Convolution is multiplication in the frequency
domain
Implicit here is that the aftershock have the
same focal mechanism as the main event. There is
no guarantee.
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Empirical Greens Functions
Large event
y(t) s(t) a(t)
Small event
x(t) d(t) a(t)
A d function in time, is a constant in frequency
domain, it doesnt change the relative shape of
the spectrum.
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Empirical Greens Functions
We take the small event x(t) to represent the
transfer function of the path between source and
receiver. The signal of the larger event is then
In utopian conditions, this could be solved by
dividing the Fourier transforms (the spectra of a
delta function d(t) is a constant)
In practice this does not work
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Empirical Greens Functions
Reason 1 The X(f) will have numerous zeroes, that
do not correspond to zeroes of Y(f), due to noise
in signals.
Reason 2 Rupture mechanics of the smaller event
may be more complicated than initially thought.
That is, the approximation to a delta function is
not admissible.
Reason 3 Anything you can think of. Impulse
response changes (groundwater change),
contamination between phases, etc.
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Empirical Greens Functions
Reason 1 Zeroes in X(f) Use Multitaper Spectrum
Estimates Use more than one EGF.
Reason 2 d(t) approximation fails Use only a
frequency range where the approximation is
accurate. Use different events for different
frequency bands.
Reason 3 Other reasons Try covering the focal
sphere as much as possible, cover a decent time
range.
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d(t) approximation
The Haskell Source model (Self-similar events)
Source Spectra
Far-field Displacement Pulse
Preferred EGF

• A d function in time, is a constant in the
  frequency domain.
• Even for small events the approximation breaks
  down at high frequencies.
• Creates consistently biased results (slower
  fall-off if used as EGF).

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The largest EGF criteria

• A flat spectrum is effectively a delta function
  in the time domain.
• As one gets closer to the fc the approximation
  breaks down.
• Use simple model to define fmax of deconvolution.

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Deviations from a d function

• Small deviations of scaling, smaller stress drop
  (red line) or misjudged seismic moment (blue
  line) change corner frequency.
• Using 1/5fc as your fmax, accounts for this
  small variations, resulting in an accurate EGF.
• It can also account for different source models.
  This factor, the misfit is used as a measure of
  degrees of freedom.

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SNR for different frequencies

• At low frequencies, earths noise (microseism
  noise) obscures the signal for small events.
• One would effectively deconvolve noise, biasing
  results.
• No reason to toss out very small events, M2 EQs
  can be used at low frequencies, and M0-1 up to
  10 - 50 Hz.

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Choosing the EGF for frequency f

• Use weighted and combined EGF to construct the
  source spectrum.
• At high frequencies, use the EGF to about 1/5fc.
• At low frequencies, use SNRgt5, thus using only
  the larger EGFs.
• The shape of the spectrum is preserved at all
  frequencies.

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EQ Physics Outline

• Introduction
• The Empirical Greens Functions (EGF)
• Weighted and Combined EGF
• An Example
• Some conclusions

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Mean-square error criteria
The mean-square error is a measure of how close
to q is our guess. Simplifying, the mse can
be divided into two parts The variance and the
bias terms. The EGF approach is always biased,
depending on closeness to the fc of the EGF
used. The variance term is easily obtained from
the Multitaper spectrum estimation codes, and the
bias is estimated by assuming a given model for
the M0/fc scaling.
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Tracking the uncertainties
In order to obtain correct estimates of
uncertainties, we use the theory of propagation
of errors. For example, the basic idea of
division of two independent random variables,
gives Standard deviation estimates sX, sY, sS
of the random variables X, Y, S. Other rules have
to be followed for summing, scaling, etc.
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Bias
Ideal Case
Our Case
The bias (power spectral units)
The bias term reflects the effect of the corner
frequency fc of the EGF used. The closer you are
to fc the worse your d(t) approximation is.
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Weighted and combined EGF
Lets assume we can write the source spectrum
S(f) as a linear combination of the different egf
corrected spectra, the mse is Where we
introduced the Lagrangian multiplier l to add a
second constraint. Differentiating and setting to
zero we end with a simple linear problem
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EQ Physics Outline

• Introduction
• The Empirical Greens Functions (EGF)
• Weighted and Combined EGF
• An Example
• Some conclusions

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Comparison to Regional methods
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Seven ML 4 - 6
Northridge
Joshua Tree/Landers
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Corner frequency
fc-3 line
Anza results
New analysis of larger earthquakes
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EGF spectra of larger events
Note shallow falloff here, real or EGF problem?
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Study Area

• 10 stations (8 effec. used)
• Vel. and ground motion sensors.
• 100 samples/s
• Max S-R distance 50 km.
• 169 EGF ML 0 - 2.9
• 3 months after mainshock
• Mainshock ML 5.1 (Red) Oct. 31 2001
• Max. interevent distance 2km.
• Max. 5km. for M gt 2

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Spectrum estimation
M5.1

• Spectral Window 12s.
• Includes P and S waves
• SNR from a similar pre-event waveform

M2.9
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Weighted EGF

• A weighted and combined EGF leads to reliable
  estimates.
• Weighting based on SNR and expected 1/5fc.

EGF corrected spectra
Venkataraman et. al. (2002)
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The source spectrum
Model 1 fc 1.96 Hz n 2.25
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Source scaling
Fall-off is better constraint. Better EGF used.
ML 5.1
Mw 4.7
Small Anza Eqs.
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Source scaling
Anza M5.1 5 - 95
Small Anza Eqs.
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Some conclusions

• The use of EGF is always biased to lead to
  shallow fall-off of the source spectrum.
• A weighted and combined or sectioned EGF may
  accomplish better results, taking into account
  both the low frequency SNR and the corner
  frequency of the EGF used.
• By the use of combined EGFs it is possible to
  estimate uncertainties of the source spectrum.
• Similarity of the source spectra may not hold for
  the larger earthquakes, of course more analysis
  of different events is needed.
• Better analysis of spectra, and specifically
  uncertainties are needed to better constraint the
  scaling of earthquakes.

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