Upload
others
View
7
Download
0
Embed Size (px)
Citation preview
Probabilistic source inversion of static displacement data using neural networksPaul Käufl, Andrew Valentine & Jeannot TrampertDep. of Earth Sciences, Utrecht University, The Netherlands
Contact: [email protected]
Introduction
We present a neural network based method forpoint source inversion in a Bayesian framework.
We demonstrate the method by inverting co-seismic displacement measurements providedby real-time GPS networks. The static offset is
the remaining displacement after an earthquakehas occurred. We aim to investigate to whatextent this relatively novel observable can con-strain point source estimates.
Our method has potential for earthquake early
warning (EEW) systems, since it can rapidly pro-vide source parameter estimates together withuncertainty bounds. Once a trained network isavailable an inversion takes only a few millisec-onds on a desktop computer.
Concept
Figure 1: The solution to the inverse problem is the posteriorprobability distribution p(m|d = dobs) (Tarantola, 2005) —the conditional distribution p(m|d) evaluated at the observa-tion dobs. In the general case, this distribution is unknown.
Figure 2: However, we can find samples of the conditionaldistribution p(m|d) ! p(m)p(d|m) given a prior distribu-tion on the model parameters p(m), and a likelihood functionp(d|m), that is a data noise model and a forward modellingcode.
Figure 3: A mixture density network (MDN) forms a smooth,parametric approximation of the conditional probability den-sity based on a set of training samples. Once an observationis available, we can evaluate this approximation and retrievethe posterior distribution.
Training set & synthetic data
The training set is formed by 80.000 deviatoricpoint sources drawn from a uniform prior distri-bution. Synthetic static displacements are cal-culated in a layered, isotropic, elastic mediumusing a propagator matrix method developed byO’Toole and Woodhouse (2011).
An efficient point source parametrization
Uncertainties translate to distances in param-eter space. A meaningful parametrization isthus of paramount importance for probabilisticinversions. We work in the geometric momenttensor domain introduced by Tape and Tape(2012) with parameters as follows:
b isotropic component, 0 for deviatoricsources (fixed)
! non double-couple component, 0 for apure double couple
" strike# rakeh cosine of dipMW moment magnitude
Demonstration: The 2010 El Mayor-Cucapah Event
After training and testing the networks, wepresent observations for a MW 7.2, 2010 event
in Baja California, which yields 1-D posteriormarginal distributions.
!!/6 0 !/6"
!!/6
0
!/6
True
valu
e
0 ! 2!
# (strike)
0
!
2!
!!/2 0 !/2$ (rake)
!!/2
0
!/2
0 0.5 1.0
h (cos(dip))
0
0.5
1.0
6.5 7.25 8.0MW
6.5
7.25
8.0
True
valu
e
2 12 22
depth [km]
2
12
22
31.2 31.95 32.7
lat ["]
31.2
31.95
32.7
!117 !116 !115
lon ["]
!117
!116
!115
0.0
0.3
0.6
0.9
1.2
1.5
1.8
2.1
2.4
2.7
3.0
nats 0.0 0.8
"
0
300
600
900
1200
0 1
#
0
50
100
150
200
0 1
$
0
80
160
240
0.0 0.6
h
0
150
300
450
0.0 1.5MW
0
50
100
150
200
0.0 0.8
depth
0
300
600
900
1200
0 3
lat
0
40
80
120
160
0 3
lon
0
40
80
120
160
Figure 4: The network performance is assessed using another 4000 examples — the test set. Left: Predicted mode ofthe distribution vs the true value. Right: Histograms of the information gain — the distance between prior and posteriordistribution (colour coded in the left plot). A small information gain indicates that the posterior closely resembles the prior.Not all parameters can be predicted well.
!!/6 0 !/6"
0.0
0.5
1.0
1.5
2.0
GCMT
USGS CMT
USGS W phase
Melgar et al (2011)
O’Toole et al. (2013)
Wei et al. (2011)
0 ! 2!# (strike)
0.00
0.06
0.12
0.18
0.24
!!/2 0 !/2$ (rake)
0.0
0.4
0.8
1.2
1.6
0 0.5 1
h (cos[dip])
0.0
0.8
1.6
2.4
3.2
6.5 7.25 8.0MW
0
1
2
3
4
2 12 22
depth [km]
0.000
0.025
0.050
0.075
0.100
31.2 31.95 32.7lat ["]
0
2
4
6
8
!117 !116 !115
lon ["]
0.0
2.5
5.0
7.5
10.0
!!/6 0 !/6
"
0.0
0.5
1.0
1.5
2.0
0 5% 10% 20%
0 ! 2!
# (strike)
0.00
0.08
0.16
0.24
0.32
0.40
!!/2 0 !/2
$ (rake)
0.0
0.4
0.8
1.2
1.6
0 0.5 1
h (cos[dip])
0.0
0.6
1.2
1.8
2.4
6.5 7.25 8.0MW
0
1
2
3
4
2 12 22
depth [km]
0.000
0.025
0.050
0.075
0.100
31.2 31.95 32.7
lat ["]
0
2
4
6
8
!117 !116 !115
lon ["]
0.0
2.5
5.0
7.5
10.0
Figure 5: Inversion results for the 2010 Mw 7.2 El Mayor-Cucapah event. Vertical lines denote the position of otherpublished moment-tensor point source solutions (Melgaret al., 2012; O’Toole et al., 2012; Wei et al., 2011). Priordistributions are shown in green. Our uncertainty estimatescomprise most other solutions.
Figure 6: Influence of different amounts of perturbations ofthe 1-D crustal Earth model on the posterior marginals. Thesensitivity with respect to the Earth model seems minimal.Most parameters are unaffected up to unrealistically largevariations of 20%.
References
Melgar, D., Y. Bock, and B. W. Crowell (2012, February).Real-time centroid moment tensor determination for largeearthquakes from local and regional displacement records.Geophysical Journal International 188(2), 703–718.
O’Toole, T. B., A. P. Valentine, and J. H. Woodhouse (2012).Centroid-moment tensor inversions using high-rate GPSwaveforms. Geophysical Journal International 191(1),257–270.
O’Toole, T. B. and J. H. Woodhouse (2011). Numericallystable computation of complete synthetic seismograms in-cluding the static displacement in plane layered media.Geophysical Journal International 187 (3), 1516–1536.
Tape, W. and C. Tape (2012, July). A geometric set-ting for moment tensors. Geophysical Journal Interna-tional 190(1), 476–498.
Tarantola, A. (2005). Inverse Problem Theory. Number 4.SIAM.
Wei, S., E. Fielding, S. Leprince, A. Sladen, J.-P. Avouac,D. Helmberger, E. Hauksson, R. Chu, M. Simons, K. Hud-nut, T. Herring, and R. Briggs (2011, July). Superficial sim-plicity of the 2010 El Mayor–Cucapah earthquake of BajaCalifornia in Mexico. Nature Geoscience 4(9), 615–618.
Conclusions
A probabilistic neural network inversion yields re-alistic uncertainty estimates and is able to dealwith non-unique and multi-modal mappings.Computational demands are low, once a trainednetwork is available, making the method suitablefor EEW purposes.Static displacements provide robust information
on location and magnitude and show little sensi-tivity to the crustal model.The flexible treatment of input data makes itpossible to perform joint inversions of differentdata types, such as displacement waveformsand strong-motion accelerograms. A potentialthat still remains to be explored.