Nonlinear grid search

Linear traveltime inversion is very efficient in locating seismic sources. However, the traveltime function is inherently a nonlinear function especially for heterogeneous me- dia. Thus, the linearized location method may fail to find the correct source location or to estimate the location uncertainties when the inverse problem is ill-posed. Since it is so convenient and efficient to calculate the traveltimes of different seismic phases, a simple strategy to locate seismic source is to perform an exhaustive grid search over all possible source locations and origin times. In this way, one can obtain a probabilistic location result with complete uncertainty estimates (Lomax et al., 2000). The first step is to calculate a traveltime table for every grid point in the velocity model. The hypocenter location and origin time of a source can be determined by finding a point and origin time which has the best agreement with the observed arrival times. The best agreement is often defined as the minimum overall arrival time residual for all stations,

i.e. xs, ys, zs, t0= minPN_i=1[ti− Ti(x, y, z, t)]2/N (xs, ys, zs, t0 are source locations and

origin time, ti is the observed arrival time for station i, Ti(x, y, z, t) is the calculated

arrival time for a point in the model and N is the total number of stations). The mean squared residual is often used since it leads to simple forms in the minimization problems and also works well when the residuals are induced by Gaussian noise. How- ever when residuals contain individual large outliers or are not of similar size, other residual formats such as root mean squared residual, absolute residual or weighted residual should be used to reduce the influence of large outliers and the non-Gaussian distribution in the input data.

The traveltime residuals can also be used to estimate the location uncertainties. A complete distribution of the residual space can be determined through a full grid search, which can be further used for uncertainty estimation. Figure 1.4 shows hypocenter lo- cation results and the corresponding uncertainties of two seismic events. When seismic location is well-constrained (Figure 1.4 left panel), location uncertainties obtain by grid

(a)

(b)

Figure 1.4: Location uncertainties shown by density scatterplots of grid search. Map view in the x-y direction and vertical cross sections in the x-z and y-z directions are shown. Stars and intersection of dashed lines indicate maximum likelihood hypocenter location. The Gaussian estimates of the hypocenter location and uncertainties are shown by the black dots and the ellipsoids, and projection of the 68% confidence ellipsoid is shown in the figures. Left panel (a) shows the results of a relatively well-constrained earthquake location. Right panel (b) shows the results of a a poorly-constrained earthquake location. Intersection of solid lines indicates hypocenter location in the catalogue. Black triangles show station locations. Figure modified from Husen et al. (2003).

search (scatter points) correspond well with the location uncertainties obtained by a Gaussian assumption (ellipsoid). However, when seismic location is poorly-constrained because of the poor coverage of the monitoring geometry (Figure 1.4 right panel), using a Gaussian assumption cannot get a good estimation of location uncertainties and the location results can also be biased in this situation. Grid search methods can obtain an accurate hypocenter location and a complete distribution of location uncertainties.

The exhaustive grid search method is feasible for local seismic location or small models. However, large models or large number of seismic events will make this method inefficient. In order to make the grid search method more efficient, global optimization algorithms such as the genetic algorithm (Kennett & Sambridge, 1992, Sambridge & Gallagher, 1993, Billings et al., 1994), the simulated annealing (Billings, 1994), the

differential evolution (Ruˇzek & Kvasniˇcka, 2001) and the neighbourhood algorithm

(Sambridge & Kennett, 2001) have been used for a stochastic search in the direct grid search method. The direct utilization of the global optimization algorithms in the grid search method can generate stochastic sampling of the parameter space, but cannot obtain a global or well distributed sampling of the parameter space, which cannot produce complete, probabilistic location results and uncertainty estimations (Lomax et al., 2000).

For conventional grid search methods, the misfits between observed and theoreti- cally calculated data are used to evaluate location results at each grid node, and the

§1.3 Developments in local seismic location techniques 17

grid node which has the lowest misfit is selected as the best hypocenter location. In contrast to conventional grid search methods, Tarantola & Valette (1982) proposed a probabilistic inversion approach based on Bayes theorem (Stone, 2013) to locate seismic events within the framework of the grid search method. In the probabilistic inversion approach, the observed arrival time measurements together with a-priori model pa- rameter information are utilized to compute a complete probability density function of the model parameters (hypocenter coordinates) (Moser et al., 1992, Wittlinger et al., 1993). Both an exhaustive grid search or Monte-Carlo search can obtain a global or well-distributed sampling of the model space, and can thus obtain complete, proba- bilistic locations. However, the computational effort is huge when the model space is too large. In order to increase the computational efficiency and also to obtain a well- distributed sampling of the model space at the same time, Lomax et al. (2000) apply the Metropolis-Gibbs sampling technique and simulated annealing in the grid search method. As an importance sampling technique, the Metropolis-Gibbs algorithm adapts the sampling by incorporating information obtained from all previous data samples so that the sampling density will follow the distribution of the target function (Lomax et al., 2000). The obtained probabilistic hypocenter solution is able to depict the loca- tion uncertainty due to Gaussian-distributed picking, the traveltime calculation errors, the network-event geometry and the incompatibility of the phase identifications.