Application of a weighted likelihood method to hypocenter determination
© The Society of Geomagnetism and Earth, Planetary and Space Sciences (SGEPSS); The Seismological Society of Japan; The Volcanological Society of Japan; The Geodetic Society of Japan; The Japanese Society for Planetary Sciences; TERRAPUB. 2013
Received: 25 June 2012
Accepted: 28 July 2013
Published: 6 December 2013
The method of least squares is a standard approach to hypocenter determination in seismology. However, this method is not useful for data contaminated by systematic errors. To address this problem, we propose a weighted likelihood method (WLL) rather than a weighted least-squares method (WLSQ). Assuming a normally distributed random error and systematic errors, both methods give the same solution; however, variances of random errors estimated by WLSQ are much smaller than those estimated by WLL. Examining reasonable random errors, we simulate a case of systematic errors varying linearly with a given parameter, where the number of unknown parameters is reduced to one for simplification. We assume that a systematic error, two different arrays of stations, and three different weights are functions of distance. In the cases where biases affected by systematic errors are adequately reduced, the variances of random errors estimated by WLL become roughly equal to that assumed, but those estimated by WLSQ are much smaller than that assumed. This result implies that WLL is a better approach than WLSQ for data contaminated by systematic errors.
The method of least squares is a standard approach to the approximate solution of overdetermined systems. Use of this method can reduce the effect of random errors. The best solution in the least-squares method minimizes the sum of squared residuals—a residual being the difference between an observed value (arrival time of a seismic wave) and the fitted value provided by a model (origin time and location of an earthquake and a seismic velocity model). Hypocenter determination employs the nonlinear least squares method and is implemented by iterative refinement. If it is assumed that random error variances largely vary among observation stations, the weighted least-squares method (WLSQ) can be used to determine the hypocenter more reliably.
The residual is caused by both errors in the measurement of the arrival time and errors in the seismic velocity model for calculating theoretical arrival times. The former is a random error, but the latter is a systematic one. For a nationwide network in Japan, such as those operated by the Japan Meteorological Agency (JMA) and the National Research Institute for Earth Science and Disaster Prevention (NIED), a simple velocity model for hypocenter determination often involves systematic errors in the calculated travel time, since seismic velocities vary from region to region. JMA and NIED currently adopt WLSQ, in which the weight of each observation depends on its hypocentral distance. Their procedure, which may be a better approach than a simple least-squares method, obviously violates the condition that the least-squares method must be applied to data without systematic errors. Therefore, this procedure may give unreliable solutions, which has not yet been addressed.
In this study, we presume both normally distributed random errors and systematic errors and propose the use of the weighted likelihood method (WLL) (Hu and Zidek, 2002; Wang and Zidek, 2005) to address this problem. First, we demonstrate that WLSQ and WLL provide the same solution; however, the variance of random errors estimated by WLL exceeds that estimated by WLSQ. In order to clarify which method estimates more reasonable errors, both methods are applied to data contaminated with systematic errors in order to simulate hypocenter determination with an unsuitable velocity model. The solutions with both methods could be analytically obtained for the simulated data. This paper compares the variance of random errors estimated by the maximum likelihood method with the assumed ones. This comparison indicates that WLSQ underestimates the variance of the random errors and is unreliable as an approach to this problem.
2.1 Weighted least squares
2.2 Weighted likelihood method
When the same weights are applied in WLL and in WLSQ, the same solutions of c i are obtained. However, the maximum likelihood estimates of σ2 are different.
In applying WLL to hypocenter determination, the only difference from WLSQ is related to the formula for σ2, where Eq. (10) can be compared with Eq. (5). When NIED and JMA determine the hypocenter, a weight with epicentral distance may be introduced to reduce bias due to systematic errors in the calculated travel times. The structure of seismic wave velocities in Japan cannot be modeled with a simple layered structure, as employed by NIED and JMA. Therefore, using arrival times of more distant stations in the determination results in systematic errors, and bias of the determined hypocenter. In this section, we examine which of WLSQ or WLL is better for hypocenter determination affected by systematic errors from an inappropriate velocity structure.
We schematically simulate this problem assuming simple conditions. We assume that an earthquake occurs on the surface of a uniform medium and that stations are densely deployed in a line from the epicenter in the first case, and on a two-dimensional surface in the second case. A systematic error due to an inappropriate velocity structure is as large as a random error at a short distance, and becomes much larger at more distant points. It is assumed that the epicenter of the earthquake is rather well constrained but the origin time of the earthquake is not, since the dense stations are well distributed. Thus, the problem of hypocenter determination is reduced to the problem of determining the origin time of the event.
In the present study, we can estimate the expected values of Eqs. (17) and (18) with a probability density function for the epicentral distance of stations, S(r), and the assumption that the random error of ε i is normally distributed with a mean 0 and variance δ2. A weight function W(r) is applied and different limits of epicentral distance, u, are considered.
In order to derive an analytical solution of Eq. (17), we assume a two-step random value generation.
Step 1. Random generation of n stations within a distance of u.
Step 2. For each set of stations, generate a large number (l) of series of random errors, (ε ij : where j refers to the number of trials, j = 1, 2,…l).
The assumed systematic error, B(r), increases linearly with epicentral distance. WLSQ likely misapplies a weighting function that decays as the inverse of the second order of the epicentral distance, since the square of the systematic error increases in proportion to the second power of distance.
In Figs. 1(a), 1(b), 1(d) and 1(e), biases caused by systematic errors are not adequately reduced, where departs from 0. In these cases, exceeds the standard deviation of the assumed random error. In both station arrays, it is clear that the most quickly decaying weight function (Figs. 1(c) and 1(f)), gives the best solutions, where becomes at most 1 even in cases including stations up to 10 units away. This suggests that biases are adequately reduced. In these cases, becomes reasonable at around 1, mostly equal to the standard deviation of the assumed random error. In contrast, values are much smaller than that assumed. Therefore, WLSQ underestimates the standard deviation of random errors, suggesting an invalid application of WLSQ to the present issue.
Table 2 summarizes and for u = 10. Results obtained from the weight function decaying as the inverse of the seventh power are added as a more quickly decaying weight function, which is the case of NIED. In this case, and become slightly better than those of other cases, where becomes smaller and approaches 1. These examples suggest that if exceeds the standard deviation of the assumed random error, biases caused by systematic errors would not be adequately reduced.
4. Discussion and Summary
In general, as the number of stations increases (fixed distance limit), the results obtained by the simulation approach those of the analytical solution. Qualitative relations among and obtained from the Monte Carlo simulation do not largely differ from those of the analytical solutions, except in the case of a few tens of stations. Although not shown here, the agreement between the simulation and the analytical solution in other cases (the other station array and/or different weights) becomes better than that of Fig. 2. We have only discussed qualitative relations between the parameters, so an analytical solution is an appropriate approach to this issue.
However, JMA introduced a similar weighting depending on the hypocentral distance except for the order of power:
For S-waves (W s ): W s = W p /3.
Comparing weights in these methods with those of our simulation indicates that NIED, using a weight function decaying faster than the inverse fourth power, probably succeeds in reducing the bias by systematic errors (Table 2). But, JMA may fail to reduce bias with a weight function decaying inversely with the second power of distance. Both NIED and JMA must underestimate standard errors of hypocenters, which are calculated based on . Standard errors of hypocenters are often critical in considering reliable hypocenter distributions of clustering earthquakes and their tectonic implications.
For deeper earthquakes, a qualitative consideration could be given as follows. At short epicentral distances compared with the focal depth, systematic errors reach a level depending on the focal depth. On the other hand, at longer distances, systematic errors increase with distance mostly similar to those of shallow earthquakes. These result in smaller variances of residuals than those of shallower earthquakes, since a range of systematic errors becomes smaller than those of shallower earthquakes. This implies that a significant bias may remain, even if approaches the variance of random error. Such aspects should be discussed further as separate studies, since station arrays, the number of unknown parameters, and other factors must be rearranged to address these issues.
The present paper focuses on only the preliminary formulation of applying WLL to hypocenter determinations. In practical applications of WLL, an optimal weight function should be determined taking into consideration the results in the case of various kinds of earthquakes in different areas and depths (Hu and Zidek, 2002). WLL can be applied to hypocenter determination with a minor revision of current methods which are based on misapplications of WLSQ.
We propose the use of WLL rather than WLSQ for hypocenter determination. Both methods give the same solution; however, the variance estimated by WLSQ is much smaller than that estimated by WLL. Our simulation indicates that a weight function that decays faster with distance gives a better solution, which could be realized by WLL. In contrast, such flexible weight functions are not justified by WLSQ, since weights should be inversely proportional to the variances of errors. In the cases where biases affected by systematic errors are adequately reduced, the variances of random errors estimated by WLL become roughly equal to the given one, but those estimated by WLSQ are much smaller than the given one. Therefore, WLSQ should not be used to address systematic errors in hypocenter determination. We conclude that WLL is a better approach than WLSQ for data contaminated by systematic errors.
The author thanks two anonymous reviewers for their critical reading and comments on this manuscript.
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