Earthquake location process has an important role in any seismological applications including seismic tomography etc. The relationship between the travel-time of seismic phases and earthquake hypocenter (latitude, longitude, depth) and origin-time is non-linear and many different linearized methods have been implemented in recent years to linearize the relationship. The principal underlying linearized methods is given by Geirger (1912) based on the Taylor series. However, attempts have been made to incorporate higher terms of the Taylor series in earthquake location process (e.g. Thurber, 1985). Using more higher-terms of the Taylor series provides more constrained solution at the expense of computation costs. Full-nonlinear earthquake location algorithm was also developed and discussed earlier e.g. by Tarantola and Vallette (1982) and Tranatola (1987). In this algorithm, the location of earthquake is defined using Probability Density Functions (PDF) of all possible points around the hypocenter. In this study we use a nonlinear probabilistic method based on global search methods. In this method, the calculations of the partial derivatives are not required, and there is a higher probability to converge to the global minima due to the nonlinearity of the problem. (E.g. Lomax et al, 2008). The optimal solution in this method can be found using different algorithms such as Metropolis -Gibs (Metropolis et al, 1953), grid search (e.g. Sambridge and Mosegaard, 2002) and Octtree (oct-tree) Importance sampling algorithm (Lomax and Curtis, 2001). The oct-tree importance sampling algorithm is very faster, more complete and simple in comparison to the other methods e.g. grid search and metropolis-Gibs algorithms. The application using oct-tree algorithm is provided to produce accurate, efficient and complete mapping of Probability Density Functions (PDFs) (e.g. Lomax and Curtis, 2001). In this study we use the oct-tree importance sampling to find the optimized solution.
In this study, we apply non-linear earthquake location method developed by Lomax et al. (2000) for local earthquakes during 2006-2010, for magnitude Mn?4 occurred in the Central Alborz region. Non-linear location method is based on a posterior Probability Density Functions (PDF) determined for the model parameters. This function represents a complete probabilistic solution for the earthquake location problem including information on the uncertainties due to phase-picking uncertainties, calculated travel-time and the network geometry. We perform different synthetic tests to evaluate the performances of non-linear method, where the location problem is ill-conditioned due to station geometry and phase picking error. In this regard we test the effect of azimuthal gap and distance to the nearest station using various synthetic tests conducted in this research.
In the synthetic tests conducted, we consider 4 events in different situations located outside of an assumed seismic network. The seismic network includes 8 stations with station-spacing of order of 15 km. The azimuthal gap of the events varies in the range of 225-317 degree. In these tests we also add noise with Gaussian distribution in arrival-times, in order to investigate the performance of the nonlinear location method in the presence of large azimuthal gap and noise in the data simultaneously. In other synthetic-tests to consider the performance of nonlinear location method due to distance of nearest station to earthquake, we perform tests using different situations in presence of noise in the data with different levels in arrival times. In first case we consider an event in a dense network with station spacing of order of 15 km, and in second case we expand the network to 150 km station spacing.
In this study to relocate earthquakes occurred in the Central Alborz region during 2006-2010, for magnitude Mn?4, we have used the data set of Iranian Seismological Center (IRSC), including the arrival times of P and S phases. In this regard we used three sub networks, Tehran, Sari and Semnan belonging to the IRSC network. Also in order to enhance the station coverage especially in the southern part of the Alborz region, we added data-set of two other stations from Isfahan sub network, namely KLH and ZEF stations.
Finally, we show the robustness of the non-linear location algorithm in the presence of outliers by analyzing the shape, size and position of the 68% confidence ellipsoid that can be calculated from the PDF to track the changes in the distribution of the PDF with changing station geometry.
We find that the non-linear method is robust in the presence of high azimuthal gap e.g. 300 degree and high Gaussian errors up to 1.0 sec, and is able to locate earthquakes with error less than 5 km. We relocate 16 earthquake occurred in the Central Alborz region with Mn ? 4.0 between 2006 -2010. Despite of high azimuthal gap and high station spacing in the dataset used in this study, 10 earthquakes located with horizontal error less than 3 km. In order to verify the quality of results, we compare the non-linear location results with those reported by IGUT and IIEES. The comparison shows that the nonlinear relocation solutions are, in most of cases, closer to IGUT solutions.