Probabilistic source inversion of static displacement data using neural networksPaul Käufl, Andrew Valentine & Jeannot TrampertDep. of Earth Sciences, Utrecht University, The Netherlands

Contact: p.j.kaufl@uu.nl

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.

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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.

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Melgar et al (2011)

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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.