This document discusses methods for locating earthquakes based on analyzing arrival times of seismic phases at stations. Accurate event location is important for various applications but also challenging due to uncertainties from measurement errors, velocity model inaccuracies, and the non-linear nature of the problem. The document reviews classical methods like Geiger's linearized inversion and also discusses more advanced non-linear techniques and using multiple events to better constrain locations. It emphasizes picking higher quality phase arrival times and using reference ground truth events to validate new location methods and velocity models.

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