Location of Earthquakes in Threedimensional Media Using a Divide and Conquer Method

1
Location of Earthquakes in Three-dimensional
Media Using a Divide and Conquer Method

• Gary L. Pavlis
• Peng Wang
• Indiana University
• Frank L. Vernon
• University of California, San Diego

2
Scientific Problem

• Earthquake location a first-order data processing
  problem in seismology
• Precision location and error appraisal critical
  to numerous science questions in earthquake
  physics and tectonics
• Computer methods for location standard practice
  since late 1960s BUT
• Universal use of digital recording changed data
  quality dramatically
• Modern computing capability makes new algorithms
  feasible
• Precision is mainly limited by inadequate
  knowledge of earth structure

3
Fundamentals

• Data
• Arrival times of seismic phases
• Array slowness vectors
• Estimate source coordinates, xi , and origin
  time, t

4
Earthquake Location is a Nonlinear Optimization
Problem

• Issues
• What do you seek to optimize?
• To linearize or not to linearize?
• Model errors versus measurement errors?
• Nonlinearity scales with wavefront curvature
• Not an issue at teleseismic distances
• Not an issue at regional distances unless errors
  are large
• Can be important for local networks

5
Single Event Location

• Uses iterative sequence h0dh1dh2 dhk
• At convergence dh1-gt0
• Iteration seeks some norm of residual vector, r,
  r (usually this is a robust variant of L2)
• Each event is located in isolation

6
Single Event Location Errors (Pavlis, 1986)

• dhA-g emodel n e

where

• dherror in hypocenter estimate
• A-g generalized inverse
• emodel t - tmodel
• n nonlinear error
• e measurement error

7
Multiple Event Location Methods

• Joint Hypocenter Determination (JHD) Douglas,
  1966
• Hypocentroidal Decompositon Jordan and Sverdrup,
  1981
• PMEL Pavlis and Booker, 1993 and SELM Pavlis
  and Hokanson, 1985
• Double differences Gott et al, 1994 Waldhauser
  and Ellsworth, 2000
• Some subtle variants of the above

8
Basic equations
Data
Hypocentral Parameters
Velocity Model Parameters
9
What is PMEL

• Uses only path anomaly terms (Use B ignore C)
• Local earthquake tomography does opposite (Use
  C ignore B)
• Solves matrix progressively
• Iteratively solves for locations
• Inverts for path anomaly terms
• Iterates till convergence

10
When is path anomaly approximation valid?
No
Yes
Paths very different
Paths are almost equal
Earthquake
Station
11
PMELGRID key concepts

• Builds on PMEL (Pavlis and Booker, 1982)
• Uses grid of control points
• Clustering algorithm associates events with grid
  of control points
• Repeated PMEL calculations on 104 to 106 control
  points
• Later is fundamental reason this is a high
  performance computing problem
• Builds travel time grids empirically
• Projectors
• Uses 3D model only to resolve bias problems, tt
  grid is derived from all data
• Automatically reverts to 3D model travel time in
  regions with no events
• Robust due to spatial localization strategy
• Designed as an automated tool for large scale
  relocation of any catalog at any scale (Global,
  regional, local)

12
Control grid concept in 3D

• Well have to shift to iView3D for this

13
Spatial Association
14
PMELGRID key concepts

• Builds on PMEL (Pavlis and Booker, 1982)
• Uses grid of control points
• Clustering algorithm associates events with grid
  of control points
• Repeated PMEL calculations on 104 to 106 control
  points
• Builds travel time grids empirically
• Projectors
• Uses 3D model only to resolve bias problems, tt
  grid is derived from all data
• Automatically reverts to 3D model travel time in
  regions with no events
• Robust due to spatial localization strategy
• Designed as an automated tool for large scale
  relocation of any catalog at any scale (Global,
  regional, local)

15
Computed station correction surfaces are
spatially coherent empirically computed travel
time grid
Anza control point grid (z5 km slice)
Anza hypocentroid grid (z5 km slice)
16
PMELGRID key concepts

• Builds on PMEL (Pavlis and Booker, 1982)
• Uses grid of control points
• Clustering algorithm associates events with grid
  of control points
• Repeated PMEL calculations on 104 to 106 control
  points
• Builds travel time grids empirically
• Projectors
• Uses 3D model only to resolve bias problems, tt
  grid is derived from all data
• Automatically reverts to 3D model travel time in
  regions with no events
• Robust due to spatial localization strategy
• Designed as an automated tool for large scale
  relocation of any catalog at any scale (Global,
  regional, local)

17
The absolute and relative error problem
Bias Absolute error
Relative error
Truth
Location Estimates
18
Absolute and relative errors

• Absolute errors (Pavlis, 1986)
• Are caused completely by velocity model errors
• Are smoothly varying in space
• Compensated in PMELGRID using projectors s
  PNs3d PRsdata
• Relative errors
• Measurement errors introduce random scatter that
  can be removed only by averaging (ubiquitous
  theme in science)
• Model errors introduce relative errors in more
  subtle ways (Pavlis, 1990) that PMELGRID handles
  correctly
• Consellation variations
• An error that scales with offset from the control
  point
• Result
• PMELGRID solutions minimize relative error terms
  correctly under the linear approximation
• Projectors separate knowable from unknowable
• Data constrain model strongly in regions with
  many events
• In areas with no coverage, automatically reverts
  to model-based travel times

19
Absolute/relative error simulation result
20
Applications to real data

• Have applied this to
• Anza (southern California)
• Southeast Alaska
• Krghystan (central Asia)
• Example run time Anza gt 30 days serial
• Lets look at location results for Anza in 3D
  with iView3d

21
An important advantage of PMELGRID is robustness

• Spatial localization allows more flexibility in
  outlier detection than global methods (e.g. local
  earthquake tomography)
• Robust methods used in PMELGRID
• Conventional M-estimators in event location
  algorithm (residual weighting)
• F-test compares rms of each event versus ensemble
  rms (corrected for trial removal) to discard
  entire events with high rms (multiple outliers)

22
Improvement in data fit
Bad news SE Alaska no improvement
Good news Anza -- huge improvement
Why large, sparse, network with variable data
quality dominated by emergent regional event
signals
Why small, quality network dominated by
impulsive signals
23
Parallel Processing
Pmelgrid (MPI distributed memory)
Dbpmel (SMP processors)
dbwriter1
Database View
dbreader
Memory Pig
dbwriter2
dispatcher
...
P m
P 2
P 3
P 1
W3
Wm
...
W1
W2
Output Function
Bottleneck
Workers
Source at http//www.indiana.edu/aug/contrib.html
24
PMELGRID Summary

• Uses grid of control points
• Clustering algorithm associates events with grid
  of control points
• Repeated PMEL calculations on 104 to 106 control
  points
• Builds travel time grids empirically
• Projectors
• Uses 3D model only to resolve bias problems, tt
  grid is derived from all data
• Automatically reverts to 3D model travel time in
  regions with no events
• Robust estimators can exploit localization
• Compute intensive but embarrassingly parallel
• Open source at http//www.indiana.edu/aug/contrib
  .html with CVS system to get most up to date bug
  fixes