nonlinloc non-linear, probabilistic, global-search earthquake location in 3d media sudipta sarkar...
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NonLinLocNonLinLocNon-Linear, Probabilistic, Non-Linear, Probabilistic, Global-Search Earthquake Global-Search Earthquake
Location in 3D MediaLocation in 3D MediaSudipta SarkarSudipta Sarkar
MIT Workshop for Advanced MIT Workshop for Advanced Methods on Earthquake Methods on Earthquake
LocationLocationJune 26, 2007.June 26, 2007.
ReferenceReference Research Software - maintained and
developed by: Dr. Anthony Lomax
Software is distributed under the GNU Public License
Continuously evolving with feedback/help from users at various research orgs, including MIT.
http://alomax.free.fr/nlloc/index.html
The NonLinLoc CodesThe NonLinLoc Codes Vel2GridVel2Grid: : Given a velocity model description, Given a velocity model description,
creates a model grid. Outputs a 3D Grid.creates a model grid. Outputs a 3D Grid. Grid2TimeGrid2Time: : Given a 3D Model Grid, calculates Given a 3D Model Grid, calculates
travel times from a point within a 3D Grid to all travel times from a point within a 3D Grid to all other points within the grid. Outputs a set of 3D other points within the grid. Outputs a set of 3D Grids.Grids.
NLLocNLLoc: : Determines the location for one or more Determines the location for one or more events within a 3D Grid using a systematic grid-events within a 3D Grid using a systematic grid-search, a stochastic, Metropolis-Gibbs search, or a search, a stochastic, Metropolis-Gibbs search, or a hybrid "Oct-Tree" method. Outputs misfit or hybrid "Oct-Tree" method. Outputs misfit or probability density function (PDF) on a 3D Grid, probability density function (PDF) on a 3D Grid, description of best hypocenter, and other results.description of best hypocenter, and other results.
A Typical Location Workflow A Typical Location Workflow with NonLinLocwith NonLinLoc
1.1. Obtaining seismic phase picks in a supported format. Obtaining seismic phase picks in a supported format. Example Phase File formats: NLL, Hypo71, HypoEllipse, Example Phase File formats: NLL, Hypo71, HypoEllipse, HypoDD, SimulPS, NEIC, SEISAN, ETH, Nordic, NCSN, HypoDD, SimulPS, NEIC, SEISAN, ETH, Nordic, NCSN, ISC, CSEM, etc.ISC, CSEM, etc.
2.2. Determining a 3D search region and velocity model for Determining a 3D search region and velocity model for this region.this region.
3.3. Using Using Vel2GridVel2Grid or other software to produce a velocity or or other software to produce a velocity or slowness model 3D grid file for the search region for each slowness model 3D grid file for the search region for each phase type (i.e. P or S).phase type (i.e. P or S).
4.4. Using Using Grid2TimeGrid2Time to produce travel-time 3D grid files for to produce travel-time 3D grid files for each phase type at each station.each phase type at each station.
5.5. Using Using NLLocNLLoc to locate each event defined by the phase to locate each event defined by the phase picks.picks.
6.6. Processing and plotting the location results.Processing and plotting the location results.
Phase PicksPhase Picks
Pick FilePick File
Vel2GridVel2GridThe Vel2Grid program converts analytic or other velocity model specifications into a 3D grid file containing velocity or slowness values.
Velocity (km/s)
Dep
th (
km)
Grid2TimeGrid2Time Given a velocity model grid, Given a velocity model grid, Grid2TimeGrid2Time calculates calculates
the travel-time from a source point in a 3D grid to the travel-time from a source point in a 3D grid to all other points in the grid.all other points in the grid.
The travel-time calculation is done using the The travel-time calculation is done using the Eikonal finite-difference scheme ofEikonal finite-difference scheme of Podvin and Podvin and LecomteLecomte.. [[RefRef. Podvin, P. and Lecomte, I., 1991, Finite . Podvin, P. and Lecomte, I., 1991, Finite difference computation of traveltimes in very contrasted velocity difference computation of traveltimes in very contrasted velocity models: a massively parallel approach and its associated tools., models: a massively parallel approach and its associated tools.,
Geophys. J. Int.Geophys. J. Int., 105, 271-284.], 105, 271-284.] This method relies on a systematic application of Huygen's principle in the finite This method relies on a systematic application of Huygen's principle in the finite
difference approximation. Such approximation explicitly takes into account the difference approximation. Such approximation explicitly takes into account the existence of different propagation modes (transmitted and diffracted body waves, head existence of different propagation modes (transmitted and diffracted body waves, head waves). Local discontinuities of the time gradient in the first arrival time field (e.g. waves). Local discontinuities of the time gradient in the first arrival time field (e.g. caustics) are built as intersections of locally independent wavefronts. As a caustics) are built as intersections of locally independent wavefronts. As a consequence, the proposed method provides accurate first travel times in the consequence, the proposed method provides accurate first travel times in the presence of extremely severe, arbitrarily shaped velocity contrasts.presence of extremely severe, arbitrarily shaped velocity contrasts.
NLLocNLLoc The earthquake location algorithm implemented in The earthquake location algorithm implemented in
the program NLLoc follows the the program NLLoc follows the probabilistic probabilistic formulationformulation of inversion presented in of inversion presented in Tarantola and Tarantola and Valette (1982)Valette (1982) and and Tarantola (1987)Tarantola (1987). . [[RefRef. Tarantola, A. and Valette, B., 1982, Inverse problems = quest for . Tarantola, A. and Valette, B., 1982, Inverse problems = quest for information., information., J. Geophys.J. Geophys., 50, 159-170. , 50, 159-170. Tarantola, A., 1987, Tarantola, A., 1987, Inverse problem theory: Methods for data fitting and Inverse problem theory: Methods for data fitting and model parameter estimationmodel parameter estimation, Elsevier, Amsterdam, 613p.], Elsevier, Amsterdam, 613p.]
Information on Data and Model, Prior PDF, Forward Information on Data and Model, Prior PDF, Forward relationshiprelationship
Bayesian Framework Posterior Density Posterior Density Function (PDF)Function (PDF)
• The maximum likelihood (or minimum misfit) point of the complete, non-linear location PDF is selected as an "optimal" hypocenter.
ETH Example: Irregular, extensive PDF
Figure Courtesy: Anthony Lomax
ETH Example: Elongated PDF
Figure Courtesy: Anthony Lomax
?
Difficulties: Depth-origin time trade-off
P, SP, SP+S
Δ < depth
accurate velocity model (3D?)
ETH Example: well located event
location PDF
epicentre
Figure Courtesy: Anthony Lomax
Likelihood/Misfit FunctionsLikelihood/Misfit Functions The LS-L2 Likelihood FunctionThe LS-L2 Likelihood Function
Most earthquake location algorithms are based on an L1 or L2 Most earthquake location algorithms are based on an L1 or L2 norm of the misfit between observed and calculated travel times norm of the misfit between observed and calculated travel times for each observation, given a nominal error for each observations. for each observation, given a nominal error for each observations.
An outlier observation has a residual greater than its An outlier observation has a residual greater than its nominal error. nominal error.
The Equal Differential Time (EDT) Likelihood FunctionThe Equal Differential Time (EDT) Likelihood Function An alternative to the LS-L2 likelihood function that is very robust An alternative to the LS-L2 likelihood function that is very robust
in the presence of outliers.in the presence of outliers. With both the EDT and LS-L2 likelihood functions, the errors With both the EDT and LS-L2 likelihood functions, the errors
in the observations (seismic wave arrival times) and in the in the observations (seismic wave arrival times) and in the forward problem (travel-time calculation) are assumed to be forward problem (travel-time calculation) are assumed to be Gaussian. This assumption allows the direct, analytic Gaussian. This assumption allows the direct, analytic calculation of a maximum likelihood origin time for the LS-calculation of a maximum likelihood origin time for the LS-L2 likelihood function, while the EDT determination is L2 likelihood function, while the EDT determination is inherently independent of any origin time estimate. Thus the inherently independent of any origin time estimate. Thus the 4D problem of hypocenter location reduces to a 3D search 4D problem of hypocenter location reduces to a 3D search over latitude, longitude and depth. over latitude, longitude and depth.
)(xx misfitfpdf Probability Density Function
Probabilistic, global-search event location
Probabilistic, global-search event location
iobs
ii TcalcTobs
etpdf2
2
2
1
0,
xx
xRMS/L2
RMS/L2 vs EDT Probability Density Function
RMS/L2 vs EDT Probability Density Function
“satisfy all the observations”
2
,
Nbaba
ba
TTcalcTTcalcTobsTobs
obsobs
epdfxxxx
x
EDT (Equal Differential Time)
“satisfy the most pairs of observations”
• independent of origin time
RMS/L2 vs EDT with outlier data RMS/L2 vs EDT with outlier data
EDT
RMS/L2
RMS/L2
perfect data (6 obs) 1 outlier data (err=10)
all residuals ~
all residuals ~ 0
all residuals ~ 0
EDT
residual outlier ~ 10
other residuals ~ 0
Global-Search methods: Grid searchGlobal-Search methods: Grid search
PDF image multiple minima efficiency
PDF image multiple minima efficiency
Global-Search methods: Directed walk
Global-Search methods: Directed walk
simulated annealingsimulated annealingmetropolis methodsmetropolis methodssimplex…simplex…
PDF image multiple minima efficiency
PDF image multiple minima efficiency
Search methods: Importance sampling
Search methods: Importance sampling
[Genetic algorithm][Genetic algorithm]Neighbour methodsNeighbour methodsOct-treeOct-tree
PDF image multiple minima efficiency
PDF image multiple minima efficiency
Oct-Tree importance sampling:
Fast, probabilistic earthquake location in 3D modelsAnthony Lomax
Andrew Curtis
The Oct-Tree method
xx misfitke 2/1
Probability solution is in a cell with
volume Vi :
xii VP
Solution probability density function
(PDF):
Pmax
Pj
Pk
Pmin
Ordered list of probability values for
all previously sampled cells:
Grid of sampled cells(Oct-Tree in 3D):
The Oct-Tree method
Sub-division of highest probability cell:
1 sample 8 new samples
cell volume cell volume / 8
Oct-Tree sampling procedure
a) true PDF b) initial sampling
c) subdivision
d) subdivision e) subdivision f) many subdivisions
Oct-tree search: Discussion
• Much faster than grid-search (factor 1/1000)
• More global and complete than Metropolis
• Very few parameters (initial grid size, number samples)
resolve PDF image find multiple minima efficiency
resolve PDF image find multiple minima efficiency
Summary: NonLinLoc event locationSummary: NonLinLoc event location
3D & complex models non-linear more complete than linearized loc
3D & complex models non-linear more complete than linearized loc
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