An MCMC Bayesian full moment tensor inversion constrained 2 by first-motion polarities and double couple percent 3 4

by first-motion polarities and double couple percent 3 4 Mehrdad Pakzad1, Mahnaz Khalili2, Shaghayegh Vahidravesh2 5 1Institute of Geophysics, University of Tehran, Tehran, 1435944411, Iran 6 2Graduate Student of Geophysics, Institute of Geophysics, University of Tehran, Tehran, 1435944411, Iran 7 Correspondence to: Mehrdad Pakzad (pakzad@ut.ac.ir) 8 9 Abstract. Monte Carlo Markov chain (MCMC) samplings can obtain a set of samples by directed random walk, 10 mapping the posterior probability density of the model parameters in Bayesian framework. We perform earthquake 11 waveform inversion to retrieve focal angles or the elements of moment tensor and source location using a Bayesian 12 MCMC method with the constraints of first-motion polarities and double couple percentage using full Green functions 13 and data covariance matrix. The algorithm tests the compatibility with polarities and also checks the double couple 14 percentage of every site before the time-consuming synthetic seismogram computation for every sample of moment 15 tensor of every trial source position. Other than large earthquakes, the method is especially suitable for weak events 16 (M < 4) that their focal mechanisms cannot be well-constrained by polarities or seismograms alone, unless a dense 17 local network is available; something that is generally occasional. Twoand one-station solutions show more 18 agreement with all-station solution if polarity and DC% constraints are employed. In order to examine the validity of 19 the method, two events with the independent focal mechanism solutions are utilized. Furthermore, we also calculate 20 data covariance matrix from pre-event noise and Green function uncertainty to obtain the errors of focal mechanisms. 21 22

and Wéber 2018. Comparing to waveform data, the information 53 content of first-motion polarities of body waves is low, that is why a dense coverage of focal sphere is required for a 54 reliable result. On the other hand, for high frequency weak events, available velocity distributions are usually not 55 detailed enough to model their waveforms and retrieve the focal mechanisms, that is, waveforms can be modelled 56 convincingly just for relatively close stations to receive a quite dependable focal mechanisms solution for near station 57 earthquakes. However, seismic networks are not usually dense enough to make sufficient data available for inversion.

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Therefore, combining polarity data with near-station records can be helpful. In the case of a small event occurrence 59 and with low number of stations, the objective cannot be more than to retrieve its DC focal mechanism with the 60 uncertainty. Earthquakes source inversion is relevant to the location determination and also velocity models.

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As to the constraints, various methods adopted them for retrieving focal mechanisms of weak events in sparse The PPD is computed using the Bayesian rule to the parameters , that can be strike, dip and rake or elements of MT 77 and , the location; given polarities, and waveforms,

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The inverse problem is linear in and nonlinear in , which results in complex structure of the joint posterior 94 distribution of the model parameters. In their waveform-based Bayesian full moment tensor inversion, Gu et al. (2018) 95 designed an MCMC approach to incorporate variation in into the problem. They first obtain the marginal posterior 96 probability distribution ( * | ) for any given and use it to calculate the Metropolis acceptance ratio.

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The computational cost of running the code on a 2.60 GHz Dual-Core CPU, 4G memory PC, for 1000 iterations in We perform several synthetic tests to confirm the validity of the method. The configuration of the stations in the 143 synthetic tests is identical to the recording stations of Sargans earthquakes used also as an application (Fig. 1).

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The results with different Signal-to-noise ratio (SNR) are shown in Table 1 Table 1 we utilize all stations in the inversion while in Tables 155 2 and 3 we just used two nearby stations, LIENZ and SGT04. Table 2 and 3 differ in using the constraints of polarity 156 and DC% > 70 in Table 2. The results presented in Table 1 and Table 2 are more close to each other; although in the 157 latter, we just used two stations. On the other hand, the results in Table 3 shows that the solutions deteriorate more, in 158 terms of Kagan angles and deviatoric part due to the lack of polarity and DC% constraints. For example, for SNR 159 equal to 0.5, Kagan angle is 8⁰, in case of applying the constraints, while it increases to 30⁰ otherwise. For the case of 160 using all stations, we calculate the location as model parameter (Table 1) while for two-station cases we fix the location 161 to the one obtained for all-station computation (Table 2 and 3).

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The outer chain consists of drawing samples by the adaptive Metropolis method and calculating the marginal posterior 163 probability for any given location and performing the acceptance test which is a Metropolis test. The iteration is 164 repeated for 1000 times, however after few hundred steps, the optimum location is found. In the synthetic test without  195 Figure 4 shows the random walk in the focal angles' solution space utilizing all stations with no usage of noise (first 196 row in Table 1). The strikes, dips and rakes are calculated from the actual random walk in MT space.

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We present the results of applying the method on two small (Mw 3.6 and 3.8) events with available independent focal 207 mechanism solutions. The first earthquake, which was also used in the synthetic tests above, is a Switzerland event

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Firstly, we test the method using all stations and all polarities. We filtered the records in frequency range 0.02 to 0.15

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Hz by Butterworth filter and inverted in the displacement domain. The results are presented in Fig. 5 to 10 and Table The selected for Sargans earthquake is 35 (Fig. 6). Actually, there is a range of values that gives identical location

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The DC solution of Sargans earthquake is a strike-slip mechanism. It is obtained for full , that is, considering both 233 data and theoretical uncertainties and in the displacement domain (Fig. 9). For this event data uncertainty is dominant Here we apply the method on the second event happened around the town of Malard near Tehran, Iran, with MW 3.8,

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on December 26, 2017 at 21:24:34 UTC (Fig 11). The reference solution of this event is our solution, that is, the

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In order to apply the method on this earthquake, we utilize 21 first-motion polarities from broadband and short-period