Recently developed chaotic charged system search was combined to feasible-based method to solve constraint engineering optimization problems. Using chaotic maps into the CSS increases the global search mobility for a better global optimization. In the present method, an improved feasible-based method is utilized to handle the constraints. Some constraint design examples are tested using the new chaotic-based methods, and the results are compared to recognize the most efficient and powerful algorithm.

Summary The linear Gauss-Markov model for waveform-based moment tensor inversion often relies on the overdetermined least-squares method. It needs a proper stochastic model of the observables for accurate and precise estimates of the unknown parameters. Furthermore, estimating the level and distribution of random errors in the observed waveforms is challenging due to assessing the minimum-variance unbiased estimator (MVUE). Hence, according to the considerable effects of random data errors in assessing the uncertainty of the moment tensor components, this paper aims to describe an MVUE of the data covariance matrix and its application on uncertainty quantification of the moment tensor. The used mathematical prescription allows us to use the covariance matrix for the three-component noise records at every station and all possible cross-correlations among the recorded noise wave field. To illustrate the proposed method’s performance, we conducted tests with synthetic data using configuration of the 2018 Mw 6.8 Zakynthos (Ionian Sea, Greece) earthquake. Both uncorrelated and correlated random noise traces were added to the synthetic waveform data in amounts between 5 and 20 percent of the maximum amplitude. In order to test the efficiency of the method, we considered three different structures of covariance matrix: (i) diagonal matrix (contains a variance of individual measurements at seismic stations), (ii) block-diagonal matrix (considering cross-covariance among three components at each station), (iii) full covariance matrix. Test results are presented by comparison of the moment tensor inversion outcomes with known noise levels of generated synthetic data and with synthetic focal mechanisms, the ability of the estimated full covariance matrix in illustrating the minimum variance of parameters (namely, minimum posterior uncertainties), unbiased of the parameters, and valued of the cross-correlations between the components of each station and also among stations. Finally, we applied the method to the real waveforms of the Zakynthos earthquake having inferred focal mechanism of strike/dip/rake angles 13/40/171(deg) with 33 per cent DC and -61 per cent CLVD. The focal mechanism solution has strike/dip/rake angles 19/34/177 (deg) with 69 per cent DC and -23 per cent CLVD when using our estimated full covariance matrix.