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