Nonlinear versus linear local earthquake location and uncertainty calculation using simulated data

A precise earthquake location and location error estimation is a crucial element in many seismological applications such as local earthquake tomography, seismicity and seismic hazard assessment. Location error estimates may also be crucial to establish whether the hypocenter trend of an earthquake sequence really marks the seismogenic structure or simply reflects ill-conditioning of the location process.So far many methods have been introduced to locate earthquakes. Earthquake location methods have undergone many changes by Geiger’s (1912) principles. One of the first programs based on Geiger’s principles is Hypo71 (Lee and Lahr, 1972), which has already been used in many studies. The basic theory of Geiger (1912) is using Taylor series expansion of the travel time function of source to station. In order to simplify the earthquake location problem, Geiger used only the first term of Taylor series expansion that led to a straight-line equation. Therefore, they are known as linearized relationships. Using the linearized relationships results in a decrease in the accuracy of earthquake location due to losing the higher terms of Taylor series; it may lead to failure in determining the location of earthquakes using a suboptimal network, e.g. where earthquakes are located outside the seismic network. Because of the nonlinearity of the earthquake location problem, all of the algorithms and methods based on theses linearized relationships solve the earthquake location equation iteratively. Thurber (1985) showed that when the depth of earthquake is smaller than the closest distance to station, determining the focal depth is not possible in the linearized methods. Furthermore, for using the higher terms of Taylor series, it is necessary to calculate higher degree derivatives, which are very complex, and sometimes impossible, using a three-dimensional velocity model. However, the non-linear earthquake location problem can also be solved directly by a range of probabilistic algorithms (Tarantola and Valet, 1982). Tarantola and Valet (1982) presented a method that determined the location of earthquakes with fully non-linear relationships without any need to calculate the partial derivatives. The basic theory of nonlinear probabilistic method to determine the location of the earthquakes was introduced by Tarantola and Valet (1982) and Tarantola (1987).Reporting a reliable uncertainty for the location of an earthquake is one of the most important parts of earthquake location, so that presenting the epicenter and depth for an earthquake without the uncertainty is completely meaningless. Moreover, knowing the uncertainty of a location is very important in many other studies such as seismicity and tomography. Thus all the methods and algorithms designed to earthquake location; present an uncertainty for the depth and epicenter of the location. Calculation of uncertainty in an entire earthquake location problem, such as Hypo71 (Lee and Lahr, 1972) based on Geiger’s principles and NonLinLoc (Lomax et al., 2000) is by calculation of a covariance matrix. The basic premise in these methods is that theuncertainties of the observed arrival times and their relationship with the predicted travel times are assumed to be Gaussian (bell-shaped). A bell-shaped error in the time of receipt will be achieved only if the error is observed at the time and is calculated from a random and independent model. However, apart from errors that result from picking the seismic phases (in arrival times); the biggest error in an earthquake location is given by the seismic network. Bondar et al. (2004) identified four main network criteria for epicenter accuracy: (1) the number of phases used in per location; (2) the distance to the closest station; (3) the azimuthal gap; and (4) the secondary azimuthal gap.Thus, many studies are done to find optimal conditions for the use of a network station, e.g. Chatelain et al. (1980), Kissling et al. (1988), Gomberg et al. (1990). Based on the relocation of explosions, Bondar et al. (2004) introduced four characteristics for an optimal seismic network to achieve a location within a 95% confidence level and under 5 km error in depth and epicenter: (1) there are 10 or more stations, all within 250 km, (2) an azimuthal gap of less than 110, (3) a secondary azimuthal gap of less than 160, and (4) at least one station within 30 km.Another source of related errors is to use an inappropriate velocity model of the seismic waves to predict the travel times from source to stations.In this work, to investigate the calculation of uncertainty in different location methods, we compared the performances of nonlinear and linear earthquake location methods with synthetic data by simulation of three clusters of earthquakes in Central Alborz region where the location problem was ill-conditioned. Comparisons were made between the non-linear probabilistic algorithm named NonLinLoc and linear location method known as Hypo71. We studied the performance of these algorithms under different suboptimal network conditions including primary and secondary (largest azimuthal gap by removing single station) azimuthal gaps, an inappropriate velocity model, phase-reading error and the distance to the nearest station using various synthetic tests in the same network-geometry conditions of the real earthquake sequences in Mosha, Firuzkuh and Qom regions.We found out that in the suboptimal network conditions, the location error estimates from Hypo71 were, in general, less accurate than NonLinLoc's and NonLinLoc solutions were more reliable. For earthquakes occurred inside a dense seismic network, we concluded that linearized methods produced lower quality location error estimates with no overall bias in the hypocentral coordinates compared to non-linear methods.