Generate the velocity PDD

1. Some advantages of the maximum-likelihood earthquake location Tatiana Toteva and Tim Long School of Earth and Atmospheric Sciences, Georgia Institute of Technology The maximum likelihood estimate of earthquake location is a robust alternative to earthquake location methods that use the conventional iterative least-squares technique. In maximum-likelihood earthquake location, the joint probability density distribution of the data and model define the probability that any proposed latitude, longitude, depth and origin time is a valid solution. We use a simple grid search method to find the solution with the maximum probability. If the arrival times exhibit a normal distribution of errors about a mean value and if they are weighted in inverse proportion to their uncertainty, then the conventional and maximum likelihood solutions are identical. The data errors though are not restricted to Gaussian and the velocity structure can be represented by an arbitrary probability distribution. The uncertainty in the velocity and in the arrival times can be incorporated as a priori information and it leads to a more stable solution. In this study we define for each phase a probability density distribution that includes all the possible arrival times and their observed uncertainty. Tests on synthetic and real data show that multiple arrival picks does not affect the correct determination of the maximum likelihood point. The uncertainty in the solution is proportional to the uncertainty in the velocity and the arrival time data. • Why is the earthquake location problem important? • The problem of accurate earthquake location is of critical importance in seismology. It forms a base for different seismological research • seismotectonic studies • seismic hazard evaluation • nuclear test ban treaty • hypocenter migration monitoring etc. What are the uncertainties? The uncertainty in the solution is controlled by a few possible sources of location errors. The incorrect synchronization of the absolute time at the seismometer will lead to a systematic error in the arrival time. The noise in the data causes observational (random) errors. These are usually errors associated with incorrect phase identification and/or arrival time measurement. Since the velocity structure is not exactly known, there could be also model errors. They are a consequence of the assumptions for a layered velocity model, wave propagation along a straight line etc. Insufficient and/or irregular station distribution also contributes to the uncertainty in the location solution. Algorithm for maximum likelihood location The traditional approach for finding the earthquake location is based on the Geiger’s method for linearizing the problem. The input data are the arrival times, measured from the seismic records at different seismic stations. The velocity model is not known exactly and it is usually assumed to be layered. The model parameters (latitude, longitude, depth of the earthquake) are defined using the Generalized Inverse technique, finding the Least Squares Error. Here we propose a maximum likelihood method that finds the most likely solution instead of the least error. We implement a simple grid search over latitude, longitude, velocity, depth and origin time. Instead of using exact arrival times, we use Probability Density Distribution for the arrival time as an input. The velocity model is represented by Probability Density Distribution for the velocity. R E S U L T S The uncertainty in the velocity model does not affect the maximum likelihood point, but it changes the uncertainty in the solution. Bigger velocity uncertainty causes bigger uncertainty in the solution. INPUT The input is the arrival time of each phase. Sometimes the arrival time of the seismic wave is not clearly readable. In this case, the algorithm allows the interpreter to use two or more possible arrival times. Then the PDD for the arrival time will have irregular shape. If the arrival time is clear, then its PDD can have Gaussian shape. The arrival time uncertainty also does not affect the finding of the correct solution, but it was found to be proportional to the uncertainty in the solution. It also was found to have much bigger effect on the uncertainty of the solution, then the velocity uncertainty does. If multiple arrival times are used, in case of unclear first arrival, the algorithm finds the correct location, but the uncertainty in the solution is affected. One of the disadvantages of the traditional Least Squares method for earthquake location is that insufficient data may lead to erroneous solution. Usually at least five phases of three stations are necessary. Here we demonstrate how the maximum likelihood algorithm performs if used only three arrival times from three stations. The correct location is found, but the uncertainty in the solution increases significantly. Simple grid search is implemented over latitude, longitude, depth of the earthquake, as well as velocity and origin time. Initial guess is made for the earthquake parameters and the search is implemented with a step big enough to not miss the maximum likelihood point. Then the search is repeated over a smaller region with smaller step and so on until the maximum probability PDD (φ, λ, h,T0, v) for the location is found. C O N C L U S I O N S We describe an algorithm that is based on the maximum likelihood method for locating earthquakes. Additional information for the velocity and arrival time uncertainty is incorporated and this leads to a more precise and stable location. The traditional Least Squares (LS) approach has a few limitations. One is related to “outliers” in the data. An outlier usually is a wrong arrival time. With the maximum likelihood algorithm, more then one arrival time can be used if the data are too noisy. Another limitation is the velocity model. Instead of using a layered model here we use a continuous velocity Probability Density Distribution, so the most likely velocity is used to find the location. With the maximum likelihood method the correct location is found even if there is not enough data, which is another limitation of the traditional method. The algorithm that we propose here is not as fast as the traditional LS method. This makes it inapplicable to near-real time locations. There are two aspects of this limitation – developing an automatic technique for the iterations and using a much faster processor. Generate the velocity PDD The shape of the velocity PDD depends on our knowledge about the velocity structure for the region of interest. If the velocity structure is well known and relatively homogeneous, then the velocity PDD can have Gaussian shape. If the region is highly heterogeneous and/or with unknown velocity structure, then the PDD function will have irregular shape. It can be generated from published data in the literature. In our case we used the velocity model of Kaufmann.