Final Report to the Earthquake Commission on Project No. UNI/536: ‘Bayesian methods of earthquake focal mechanism estimation and their application to New Zealand seismicity data ’ David Walsh, Richard Arnold, John Townend. June 17, 2008 1 Layman’s abstract We investigate a new probabilistic method of estimating earthquake focal mech- anisms — which describe how a fault is aligned and the direction it slips dur- ing an earthquake — taking into account observational uncertainties. Robust methods of estimating focal mechanisms are required for assessing the tectonic characteristics of a region and as inputs to the problem of estimating tectonic stress. We make use of Bayes’ rule, a probabilistic theorem that relates data to hypotheses, to formulate a posterior probability distribution of the focal mech- anism parameters, which we can use to explore the probability of any focal mechanism given the observed data. We then attempt to summarise succinctly this probability distribution by the use of certain known probability distribu- tions for directional data. The advantages of our approach are that it (1) models the data generation process and incorporates observational errors, particularly those arising from imperfectly known earthquake locations; (2) allows explo- ration of all focal mechanism possibilities; (3) leads to natural estimates of focal mechanism parameters; (4) allows the inclusion of any prior information about the focal mechanism parameters; and (5) that the resulting posterior PDF can be well approximated by generalised statistical distributions. We demonstrate our methods using earthquake data from New Zealand. We first consider the case in which the seismic velocity of the region of interest (described by a veloc- ity model) is presumed to be precisely known, with application to seismic data from the Raukumara Peninsula, New Zealand. We then consider the case in which the velocity model is imperfectly known, with application to data from the Kawerau region, New Zealand. We find that our estimated focal mecha- nism solutions are for the most part consistent with all available polarity data, and correspond closely to solutions obtained using established methods. Addi- tionally, the distribution of focal mechanism parameters can be accurately and succinctly summarised by the parameters of the probability distributions we have examined. 2 Technical abstract We develop a new probabilistic (Bayesian) method for estimating the distribu- tion of focal mechanism parameters based on seismic wave polarity data. We investigate the use of generalised Matrix Fisher distributions for parameterising focal mechanism uncertainties. The advantages of our approach are that it (1) models the data generation process and incorporates observational errors, par- ticularly those arising from imperfectly known earthquake locations; (2) allows exploration of the entire parameter space; (3) leads to natural point estimates of focal mechanism parameters; (4) allows the inclusion of a priori informa- tion about the focal mechanism parameters; and (5) that the resulting posterior PDF can be well approximated by generalised Matrix Fisher distributions. We present here the results of our method in two situations. We first consider the case in which the seismic velocity of the region of interest (described by a ve- locity model) is presumed to be precisely known, with application to seismic data from the Raukumara Peninsula, New Zealand. We then consider the case in which the velocity model is imperfectly known, with application to data from the Kawerau region, New Zealand. We find that our estimated focal mechanism solutions are for the most part consistent with all available polarity data, and correspond closely to solutions obtained using established methods. Further, the generalised Matrix Fisher distributions we examine provide a good fit to our Bayesian posterior PDF of the focal mechanism parameters. Finally, we demonstrate how informative prior distributions on focal mechanism parame- ters can be incorporated into our model. 3 Publications/outputs relating to this project 1. Walsh, D., Arnold, R., and Townend, J., 2008. A Bayesian approach to determining and parameterising earthquake focal mechanisms, Geophys- ical Journal International, submitted. 2. Walsh, D., 2008. Directional statistics, Bayesian methods of earthquake focal mechanism estimation, and their application to New Zealand seis- micity data. Unpublished MSc thesis, Victoria University of Wellington. 3. Walsh, D., Arnold, R., and Townend, J., 2007. Bayesian Methods of fo- cal mechanism estimation, in Geological Society of New Zealand / New Zealand Geophysical Society Joint Conference, November 2007, Tau- ranga, Abstract. 4. Walsh, D., 2007. Directional statistics and Bayesian methods applied to earthquake focal mechanism estimation, in The New Zealand Mathe- matics and Statistics Post-Graduate’s Conference Programme, Novem- ber 2007, Queenstown, Abstract. Directional Statistics, Bayesian Methods of Earthquake Focal Mechanism Estimation, and their Application to New Zealand Seismicity Data A project submitted in fulfillment of the requirements for the Degree of Master of Science in Statistics by David Walsh School of Mathematics, Statistics and Computing Sciences Victoria University of Wellington P.O. Box 600 Wellington New Zealand Victoria University of Wellington 2008 Abstract A focal mechanism is a geometrical representation of fault slip during an earthquake. Reli- able earthquake focal mechanism solutions are used to assess the tectonic characteristics of a region, and are required as inputs to the problem of estimating tectonic stress. We develop a new probabilistic (Bayesian) method for estimating the distribution of focal mechanism pa- rameters based on seismic wave polarity data. Our approach has the advantage of enabling us to incorporate observational errors, particularly those arising from imperfectly known earth- quake locations, allowing exploration of the entire parameter space, and leads to natural point estimates of focal mechanism parameters. We investigate the use of generalised Matrix Fisher distributions for parameterising focal mechanism uncertainties by minimising the Kullback- Leibler divergence. We present here the results of our method in two situations. We first consider the case in which the seismic velocity of the region of interest (described by a velocity model) is presumed to be precisely known, with application to seismic data from the Raukumara Peninsula, New Zealand. We then consider the case in which the velocity model is imperfectly known, with application to data from the Kawerau region, New Zealand. We find that our estimated focal mechanism solutions for the most part are consistent with all available polarity data, and correspond closely to solutions obtained using established meth- ods. Further, the generalised Matrix Fisher distributions we examine provide a good fit to our Bayesian posterior PDF of the focal mechanism parameters, enabling the posterior PDF to be succinctly summarised by reporting the estimated parameters of the fitted distribution. ii Acknowledgments Firstly, a huge thank you to my supervisors, Richard Arnold and John Townend, for all their help over the course of this project. I was, it seemed to me, constantly full of questions, and they never failed to use their knowledge and expertise to guide me in the right direction. Cheers guys. Thanks to those who provided the data used in this project; Martin Reyners, Daniel Clarke, and Kevin Fenaughty through GeoNet. Thank you also to the Earthquake Commission, who funded this project (grant UNI/536). Thanks to all my fellow graduate students who I’ve had the pleasure of studying with over the last two years. Thank you to the staff in the Statistics and Operations Research group, particularly Megan Clark, who has always been supportive, and helped me pay the rent by employing me as a Teaching Assistant during my course of study. Thanks also to Ray Brown- rigg, our Statistical Computing Engineer, who was a great help in what was a computationally intensive project. Lastly, thanks to my friends and family for all their support throughout. iii Contents Abstract........................................ ii Acknowledgments................................... iii 1 Introduction 1 1.1 MotivationandObjectives . .. 1 1.2 FocalMechanisms ............................... 2 1.2.1 P-waveFirstMotions ......................... 3 1.2.2 S-waveInformation .......................... 6 1.2.3 FocalMechanismAmbiguities . 7 1.2.4 SolutionQualityandSourcesofError . ... 8 1.3 PreviousMethods ............................... 9 1.3.1 OptimisationMethods . 10 1.3.2 ProbabilisticMethods . 12 1.4 ContributionofthisThesis . ... 15 1.5 Outline ..................................... 15 2 Directional Statistics 16 2.1 CircularData .................................. 16 2.1.1 vonMisesDistribution . 16 2.2 SphericalData ................................. 17 2.2.1 vonMises-FisherDistribution . .. 18 2.2.2 BinghamDistribution . 19 2.3 OrientationData................................. 20 2.3.1 StiefelManifolds............................ 21 2.3.2 MatrixFisherdistribution . 22 2.3.3 8-ModeMatrixFisherdistribution . ... 35 2.3.4 GoodnessofFitTesting . 39 3 Bayesian Methods 41 3.1 Bayes’Theorem ................................ 41 3.2 Examples .................................... 42 3.2.1 LighthouseProblem . .. .. .. .. .. .. .. 42 3.2.2 EarthquakeHypocentreLocation . 44 3.3 ApplicationtoFocalMechanisms . ... 48 3.3.1 VelocityModelKnown