NonLinLoc Non-Linear, Probabilistic, Global-Search Earthquake Location in 3D Media

1. NonLinLocNon-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. • http://alomax.free.fr/nlloc/index.html

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 ofPodvin 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, 271-284.] • 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, 159-170. 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. Δ < depth P+S Difficulties: Depth-origin time trade-off P, S P, S ? accurate velocity model (3D?)

14. epicentre ETH Example: well located event 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

17. RMS/L2 vs EDT Probability Density Function RMS/L2 “satisfy all the observations” EDT (Equal Differential Time) “satisfy the most pairs of observations” • independent of origin time

18. EDT 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 residual outlier ~ 10 other residuals ~ 0 all residuals ~ 0

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

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

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

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

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

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

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

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