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- 1. The earthquake location problem István Bondár Research Centre for Astronomy and Earth Sciences, Institute for Geological and Geochemical Research, Hungary Data Analysis Tools and Methods in Seismology Scientific Institute, Mohammed V University, Rabat, Morocco, 14 – 16 November 2022
- 2. Event location Why locate events? • We want to know where an earthquake stroke so that we can send help to the right place • Earthquakes reveal tectonic and plate boundaries Accurate event locations are important for • Seismic hazard and risk analyses • Seismic source zones! • Seismotectonic studies (tomography) • Monitoring research nukes, induced seismicity 2
- 3. The location problem Find the location, depth and origin time (hypocentre) that minimizes the difference between observations and predictions of phase arrival times as measured at a network of seismographic stations. The predicted travel time for a phase arrival is a function of the station and source coordinates as well as the velocity model Predicted travel times may be adjusted by various corrections, to account for the ellipticity of the Earth and topographical effects, as well as path corrections to account for three-dimensional velocity heterogeneities Travel time is a non-linear function of event location Event location is a non-linear inversion problem 3 di = ti obs -ti pred = ti obs -(t +ti model +ti corr )
- 4. Method of circles 4 ts -tp = D vs - D vp D = vpvs (ts -tp ) vp -vs = vp (ts -tp ) 3 -1 • The difference between S and P arrival time provides an estimate of distance from the station to the epicentre (assuming a P/S ratio of √3) • For typical values of P velocities, 8 to 10 times the S - P arrival time difference gives a reasonable estimate of the epicentral distance • The intersection of the circles drawn around the stations with the corresponding radius of the epicentral distance defines the most likely position of the epicentre • No constraint on source depth, S picking errors could be large
- 5. Geiger’s method Geiger (1910) • If the initial source coordinates are sufficiently close to the true hypocentre, the residuals can be expanded in a Taylor series with the higher order terms considered to be negligible • Yields a linear system of N equations (N=number of arrival-time observations) with M ≤ 4 model parameters Nearly all linearized inversion methods are based on Geiger’s method Very sensitive to the initial hypocentre guess 5 di = ¶ti ¶x Dx + ¶ti ¶y Dy+ ¶ti ¶z Dz +Dt di = ti obs -ti pred = ti obs -(t +ti model +ti corr )
- 6. Linearized Inversion Algorithms Assuming independent, normally distributed data, the likelihood function is maximized • Equivalent to solving the equation: • where G is the design matrix containing the partial derivatives of N data by M model parameters, m is the (Mx1) model adjustment vector, [Δx, Δy, Δz, Δτ]T, d is the (Nx1) vector of time residuals and Cd is the data covariance matrix describing the uncertainties in the data. • Solve with iterative least squares: 6 £(m) = exp - 1 2 d -Gm ( ) T Cd -1 d -Gm ( ) ì í î ü ý þ mk+1 = mk + mest mest = GT G ( ) -1 GT d Gwm = WGm = Wd = dw; W = Cd -1/2
- 7. • Let the SVD of the general inverse of Gw be • where 𝚲w is the (NxN) matrix of singular values (eigenvalues), Uw is an (MxN) orthonormal matrix and Vw is an (NxN) orthornormal matrix whose columns are the corresponding eigenvectors of 𝚲w. • The model covariance matrix defines a four-dimensional error ellipsoid, whose projections provide the two-dimensional epicenter error ellipse and the one-dimensional estimates of depth and origin time uncertainties • Formal uncertainties are scaled to the pth percentile confidence level • Formal uncertainties measure precision, not accuracy • Location bias due to systematic measurement and model errors cannot be accounted for by the error ellipsoid 7 Formal uncertainties Gw -1 = VwLw -1 Uw T Cm = Gw -1 CdGw -1T = VwLw -2 Vw T m-mh ( ) T Cm -1 m-mh ( )=kp 2
- 8. 8 Travel time tables • Global radially symmetric 1D velocity models are routinely used • Local velocity models used by local networks • Travel-time corrections • WGS84 ellipsoid • Station elevation above the ellipsoid • Water depth for pwP • Path corrections to account for 3D velocity heterogeneities • 3D models are approaching the resolution to be useful in event location • Raytracing is slow and expensive
- 9. 9 Error budget Measurement errors • Errors in picking the onset times of phases • Phase identification errors • Waveform correlation technique may reduce picking errors to subsample level Model errors • Travel time prediction errors due to unmodeled velocity structures • Systematic errors introduce location bias and can only be reduced by introducing better velocity models • May cause correlated travel time prediction errors Ignoring the higher order terms in the Taylor expansion • Typically negligible compared to measurement and model errors The error budget is described by the data covariance matrix • The data covariance matrix is diagonal if the observations are independent • Some early algorithms even assume that all picking errors are the same: C-1=1/𝜎
- 10. 10 Picking errors Often modelled as independent, Gaussian processes • the distribution of travel-time residuals is skewed and suffers from heavy tails • onset times of seismic waves traveling along the same ray paths are systematically picked late with decreasing signal-to-noise ratio. • later phases are typically picked with larger errors lognormal Weibull
- 11. 11 Picking phases • We need to pick the arrival time of phases to be able to locate an event • At a bare minimum we need to pick the first-arriving P phases • Only pick phases that are clearly visible • Pg is often energetic, impulsive phase • Pn is often emergent and for small events it is easy to miss • Make an effort to pick later phases, they carry a lot of information • If you work with an array, use fk • gives you an estimate for slowness and back-azimuth • With three component stations make use of all components, not just the vertical channel
- 12. 12 Picking phases, continued • Use the most appropriate time window and frequency band to pick a phase • time-bandwidth product is the duration of the signal and its spectral width • uncertainty principle: the time-bandwidth product is constant • short duration signals have higher frequency content than long signals • In the automatic processing of waveforms use a detector that incorporates the Akaike information criteria (AIC) to determine the onset time of a phase • When manually picking phases try to compensate for the lateness of pick (the true phase onset is increasingly obscured by noise with decreasing SNR) • Try to pick later phases, they help getting better depth and location • Beware of predicted phase arrival times! • They only give you hints when a particular phase is supposed to arrive according to your velocity model and the trial hypocenter. • The Earth is 3D and the 1D travel-time predictions are often wrong
- 13. 13 Phase identification errors Waveforms of nuclear explosions carried out at the Nevada test site recorded at Elko, Nevada about 400 km distance. The explosions were detonated within 15 km of one another. The waveforms are bandpass filtered between 1 and 3 Hz and shown with increasing event magnitude. While for the two larger explosions the first- arriving Pn can be easily picked, for the two smaller explosions the first-arriving Pn is completely masked by noise and the more energetic, later arrival Pg could be erroneously picked as the first arrival.
- 14. 14 The importance of later phases • Later phases provide better control in event location • They carry information on depth and the Earth’s inner structure • Pick P-type phases on the vertical component and S-type phases on the rotated radial and transverse components • Only pick phases that you are confident that they are really there • Never, ever pick predicted phases! Mariana trench event recorded at HRV at 110°
- 15. 15 Teleseismic Events • Why bother with teleseismic events if I’m a local network operator? • Receiver function analysis uses teleseismic events to determine the local velocity structure beneath your station • SKS splitting uses teleseismic events to determine the anisotropic structure of the lithosphere beneath your network • Contributing event locations and picks to ISC, NEIC, EMSC helps them to accurately locate events globally, and regions lacking local networks, such as oceans • Your picks will be used in global and regional tomography studies • Eventually you’ll have better velocity models to work with
- 16. 16 Nonlinear methods Geiger’s method can easily stuck in a local minimum Nonlinear inversion methods explore the search space For each trial hypocentre the misfit is calculated (forward problem) and after a number of iterations the best solution is retained Requires no derivatives Does not guarantee to find the absolute minimum Slow (calculating the forward solution a zillion times takes time) Exhaustive grid search is not viable Search strategies to sample the a posteriori location probability density function Natural neighbour Markov Chain Monte Carlo Metropolis-Gibbs sampling Genetic algorithm Simulated annealing Difficulties to map the uncertainty contours; still slow for routine operations
- 17. 17 Multiple event location methods Simultaneous analysis of arrival times that are associated with an entire event cluster is more robust and provide more information to constrain the unknowns Hypocentres, travel time corrections, phase names, picking error Always an underdetermined problem, as there are always more unknowns than equations Can only provide relative locations (event pattern) But not absolute locations (the entire cluster can be shifted) Cluster connectivity is important but difficult to measure Double difference (hypoDD) Exploits waveform correlation to get extreme precision differential arrival times Obtains highly precise relative event locations in a local network Double difference: the residual between observed and calculated differential travel times between two events Bayesian nonlinear Markov Chain Monte Carlo (bayesloc) Can deal with arbitrarily large data sets
- 18. 18 Ground truth events • Ground Truth (GTx) • An event hypocentre with an accuracy known at a high (95%) confidence level; x stands for the GT accuracy • GT events are needed to develop, test and validate • Travel-time predictions from 3D velocity models • New association and location algorithms • Real GT0 events are scarce • Systematic identification and collection of GT events began at the PIDC and the DoE labs in the 1990s • Since 2008 the ISC hosts and maintains the GTDB • Coordinated by the IASPEI Working Group on Reference Events • iLoc applies the Bondár and McLaughlin (2009) ground truth selection criteria for every event it locates
- 19. GT5 selection criteria Bondár et al. (2004): Bondár and McLaughlin (2009): - nsta ≥ 10 within 2.5° - stations within 150 km - nsta ≥ 1 within 30 km - nsta ≥ 1 within 10 km - gap ≤ 110° - ΔU ≤ 0.35 - sgap ≤ 160° - sgap ≤ 160° - maxdist ≥ 10° - maxdist ≥ 3-10° The GT5 selection criteria • Developed to identify GT5 candidate earthquakes in bulletins • Introduced the notions of secondary azimuthal gap and ∆U to account for large azimuthal gaps and correlated ray paths • Focus on local networks to avoid cross-over distances • Require a nearby station to provide depth resolution • The more stringent 2009 criteria are currently used • Serve as guidelines – they do not substitute the need for more involved studies
- 20. Network quality metrics Correlated stations Large gap unif esaz b N i unif U N b unif esaz U i i i ; 360 ; 1 0 , 360 ) ( 4
- 21. IASPEI Reference Event List • Semi-annual selection of GT events at the ISC • Currently 11,567 events, GT5 earthquakes and GT0-GT5 explosions
- 22. ISC website