In order to obtain the complete posterior earthquake location pdf and estimate reliable location uncertainties, in the present work was used the software NonLinLoc (henceforward NLL; Lomax et al., 2000). NLL performs a search of the solution space in a 3-D model and estimates the posterior location pdf for the spatial (x, y, z) coordinates of the hypocentre and the maximum likelihood origin time. It benefits from a probabilistic nonlinear approach to the earthquake location problem, thus producing more consistent estimation of uncertainties.
Following there is a brief description of how NLL works and its 3 options of search algorithms:
Firstly, given an input velocity model, the program creates a 3-D model grid considering previously defined parameters like grid spacing, number of nodes and origin of the grid. Then, NLL generates travel-time and take-off angle grids for each
Chapter 2 – Theory and Methods
phase type (i.e. P, S) at each one of the stations considered in the seismic network. These travel-time grids will then be used when searching for an optimum solution. Lastly, the code locates the event(s) defined by phase picks, by using a preferred direct- search algorithm over the 3-D volume. The chosen sampling algorithm in NLL can be (i) nested grid-search, (ii) Metropolis-Gibbs or (iii) Oct-tree.
(i) Nested grid-search algorithm: the nested grid-search algorithm performs a systematic search over the 3-D volume obtaining a misfit function, an optimal hypocentre and the posterior location pdf. The algorithm searches successively finer from an initial grid into subsequent ‘nested’ grids located within the previously searched grid until reaching an optimum solution. Unfortunately this method requires great computer power and is highly time-consuming. Moreover, it requires a careful selection of the grid size and node spacing: if the final sampled grid is too large in relation with the size of the pdf, then the resolution would be too low; instead, if the final grid is too small, it would not contain the totality of the space used by the pdf, thus truncating it.
It is convenient now to introduce the concept of importance sampling. An importance samplingalgorithm is a sampling algorithm that increases its efficiency in targeting a function by choosing a sampling density which follows the function as closely as possible. Although in our case the target function (earthquake location pdf) is unknown, the efficiency of the search can still be improved by adjusting (or adapting or evolving) the sampling by incorporating information gained from previous samples, so that the sampling density tends towards the target function, reducing the required computer power and processing time (Lomax et al., 2000). The Metropolis-Gibbs and Oct-tree sampling algorithms are examples of importance sampling.
(ii) Metropolis-Gibbs sampling algorithm: The Metropolis-Gibbs sampling performs a guided random search over the 3-D volume obtaining a set of samples that follow the location pdf. These samples contain an estimate of the optimal hypocentre and an image of the posterior location pdf. Therefore, the search is directed over the spatial solution towards regions of high likelihood for the location pdf. This algorithm is much faster (around 100 times faster) than the grid-search, although its downside is that being a
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stochastic search it may provide inconsistent recovery of a very irregular pdf with multiple maxima, missing important features of the pdf’s space (Lomax et al., 2000). Further, it also requires careful selection of sampling parameters.
(iii) Oct-tree sampling algorithm: the Oct-tree importance sampling method (Lomax and Curtis, 2001) provides a complete estimation of the earthquake location pdf over a 3-D volume in a more efficient way than the previous two algorithms. It takes its name from being a hierarchical tree data structure in which each node has exactly eight child- cells or children nodes.
The Oct-tree algorithm performs a recursive subdivision and sampling of rectangular cells over the 3-D volume converging to a cascade of oct-tree structures that contain pdf values. The higher the pdf values in a sampled region (low misfit) the larger the number of smaller cells around that region. In this way, the method provides an importance sampling of the true pdf, representing it in a consistent manner.
The relative probability (Pi) that an earthquake location is in a given cell i is approximately given by
Pi = vi× pdf(xi) (2.8; Lomax and Curtis, 2001) where vi is the volume of the cell i and xi is the centre of the cell i. The algorithm starts sampling the full search space on an initial crude regular grid obtaining the misfit value at the centre of each grid cell and calculating its probability Pi. Then, the cell with the highest probability is divided into eight new children cells and the process is repeated recursively.
The advantages of the Oct-tree sampling method in comparison with the previous two algorithms are its simplicity, speed, stability and completeness. Because of this, the Oct- tree sampling method was chosen to perform direct-search earthquake location on the seismic sequence analysed in Chapter 4 and in Chapter 3, providing in this way complete and accurate hypocentral solutions and reliable estimations of uncertainties.
Chapter 2 – Theory and Methods