1. MIT Workshop for Advanced Methods on Earthquake Location
NonLinLoc Non-Linear, Probabilistic, Global-Search Earthquake Location in 3D Media
Sudipta Sarkar
MIT Workshop for Advanced Methods on Earthquake Location
June 26, 2007.

2. Reference
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.

3. The NonLinLoc Codes
Vel2Grid: Given a velocity model description, creates a model grid. Outputs a 3D Grid.
Grid2Time: Given a 3D Model Grid, calculates travel times from a point within a 3D Grid to all other points within the grid. Outputs a set of 3D Grids.
NLLoc: Determines the location for one or more events within a 3D Grid using a systematic grid-search, a stochastic, Metropolis-Gibbs search, or a hybrid "Oct-Tree" method. Outputs misfit or probability density function (PDF) on a 3D Grid, description of best hypocenter, and other results.

4. A Typical Location Workflow with NonLinLoc
Obtaining seismic phase picks in a supported format. Example Phase File formats: NLL, Hypo71, HypoEllipse, HypoDD, SimulPS, NEIC, SEISAN, ETH, Nordic, NCSN, ISC, CSEM, etc.
Determining a 3D search region and velocity model for this region.
Using Vel2Grid or other software to produce a velocity or slowness model 3D grid file for the search region for each phase type (i.e. P or S).
Using Grid2Time to produce travel-time 3D grid files for each phase type at each station.
Using NLLoc to locate each event defined by the phase picks.
Processing and plotting the location results.

5. Phase Picks

6. Pick File

7. Vel2Grid
The Vel2Grid program converts analytic or other velocity model specifications into a 3D grid file containing velocity or slowness values.
Velocity (km/s)
Depth (km)

8. Grid2Time
Given a velocity model grid, Grid2Time calculates the travel-time from a source point in a 3D grid to all other points in the grid.
The travel-time calculation is done using the Eikonal finite-difference scheme of Podvin and Lecomte. [Ref. Podvin, P. and Lecomte, I., 1991, Finite difference computation of traveltimes in very contrasted velocity models: a massively parallel approach and its associated tools., Geophys. J. Int., 105, ]
This method relies on a systematic application of Huygen's principle in the finite difference approximation. Such approximation explicitly takes into account the 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. caustics) are built as intersections of locally independent wavefronts. As a consequence, the proposed method provides accurate first travel times in the presence of extremely severe, arbitrarily shaped velocity contrasts.

9. NLLoc
The earthquake location algorithm implemented in the program NLLoc follows the probabilistic formulation of inversion presented in Tarantola and Valette (1982) and Tarantola (1987) [Ref. Tarantola, A. and Valette, B., 1982, Inverse problems = quest for information., J. Geophys., 50, Tarantola, A., 1987, Inverse problem theory: Methods for data fitting and model parameter estimation, Elsevier, Amsterdam, 613p.]
Information on Data and Model, Prior PDF, Forward relationship
Bayesian Framework
Posterior Density Function (PDF)
The maximum likelihood (or minimum misfit) point of the complete, non-linear location PDF is selected as an "optimal" hypocenter.

11. ETH Example: Irregular, extensive PDF
Figure Courtesy: Anthony Lomax

12. ETH Example: Elongated PDF
Figure Courtesy: Anthony Lomax

13. ? Difficulties: Depth-origin time trade-off Δ < depth P+S P, S P, S
accurate velocity model (3D?)

14. ETH Example: well located event
epicentre
location PDF
Figure Courtesy: Anthony Lomax

15. Likelihood/Misfit Functions
The LS-L2 Likelihood Function
Most earthquake location algorithms are based on an L1 or L2 norm of the misfit between observed and calculated travel times for each observation, given a nominal error for each observations.
An outlier observation has a residual greater than its nominal error.
The Equal Differential Time (EDT) Likelihood Function
An alternative to the LS-L2 likelihood function that is very robust in the presence of outliers.
With both the EDT and LS-L2 likelihood functions, the errors in the observations (seismic wave arrival times) and in the forward problem (travel-time calculation) are assumed to be Gaussian. This assumption allows the direct, analytic calculation of a maximum likelihood origin time for the LS-L2 likelihood function, while the EDT determination is inherently independent of any origin time estimate. Thus the 4D problem of hypocenter location reduces to a 3D search over latitude, longitude and depth.

16. Probabilistic, global-search event location
Probability Density Function
The goal of probabilistic, global-search event location is to efficiently determine a complete image of the location Probability Density Function (PDF). The PDF is formed by some function f of the misfit between observed and predicted data, and includes information about the uncertainties in the observed data and in the forward calculation used to generate predicted data.
The PDF may be visualized as contours of confidence (left), as a scatter density plot (right), or by other means. The maximum likelihood hypocenter (star) may be of interest for certain analyses.

17. RMS/L2 vs EDT Probability Density Function
“satisfy all the observations”
EDT (Equal Differential Time)
With an L2 norm and least-squares misfit function, the PDF represents the probability that all the observations are satisfied. In contrast, the EDT PDF represents the probability that the largest number of observations are satisfied. Thus the EDT function is more robust when some of the observations are outliers and cannot be satisfied. The EDT formulation is also independent of origin time since it uses only time differences.
“satisfy the most pairs of observations”
independent of origin time

18. RMS/L2 vs EDT with outlier data
perfect data (6 obs)
1 outlier data (err=10)
RMS/L2
RMS/L2
all residuals ~ 0
all residuals ~ 
EDT
EDT
With perfect data (left panels), both RMS/L2 and EDT find the same maximum likelihood location. With 1 outlier datum (right panels), the RMS/L2 maximum likelihood location is biased to the northwest, the high probability region of the PDF does not overlap the true location, and the residuals for all data are similar and large. Thus the solution is biased but the solution statistics give no indication of this nor do they give and understanding of the cause of the bias. In contrast, the EDT maximum likelihood location is the same as with perfect data, and the correct, very large residual for the outlier data is determined.
residual outlier ~ 10
other residuals ~ 0
all residuals ~ 0

19. Global-Search methods: Grid search
 PDF image
 multiple minima
 efficiency

20. Global-Search methods: Directed walk
 PDF image
 multiple minima  efficiency
simulated annealing
metropolis methods
simplex…

21. Search methods: Importance sampling
 PDF image
 multiple minima  efficiency
[Genetic algorithm]
Neighbour methods
Oct-tree

22. Oct-Tree importance sampling:
Fast, probabilistic earthquake location in 3D models
Anthony Lomax
Andrew Curtis

23. The Oct-Tree method Solution probability density function (PDF):
Grid of sampled cells
(Oct-Tree in 3D):
Probability solution is in a cell with volume Vi :
Pmax Pj Pk
Pmin
Ordered list of probability values for all previously sampled cells:

24. The Oct-Tree method Sub-division of highest probability cell: 1 sample
8 new samples
cell volume
cell volume / 8

25. Oct-Tree sampling procedure
a) true PDF
b) initial sampling
c) subdivision
d) subdivision
e) subdivision
f) many subdivisions

26. 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)

27. Summary: NonLinLoc event location
 resolve PDF image
 find multiple minima
 efficiency
 3D & complex models
 non-linear
 more complete than linearized loc