Measurement method for isoseismal major and minor axes based on ellipse fitting algorithm

As the fundamental data for establishing the seismic intensity attenuation relationship, the reliability of isoseismal data directly affects the accuracy and reliability of the intensity attenuation relationship. The acquisition of traditional isoseismal parameters is based on direct measurement, which inevitably introduces a degree of subjectivity. A more objective approach to measuring the parameters of isoseismals has been proposed. However, it is inevitable that conceptual differences will exist due to algorithmic differences and the lack of links between the novel algorithm and commonly used direct measurement methods. In view of this, this paper introduces a novel approach to measuring the major and minor axes of isoseismals. The mode of isoseismal measurement and the concepts from the direct measurement method are incorporated into the ellipse fitting algorithm as mathematical constraints, thereby providing a systematic method for measuring the radii of isoseismal major and minor axes. In order to reduce the subjective influence of isoseismal major and minor-axis measurements, the radii of isoseismal major and minor axes are measured using the best-fitting ellipse of the isoseismal. This approach is based on the consistency and inheritance of the methods and concepts of the direct measurements. The resulting data provide the basic data for the establishment of intensity attenuation relationship.

The measurement modes of isoseismals are classified into two types according to whether the major-axis strike is fixed or not. In order to accommodate this distinction, the strike of major axis is also incorporated into the algorithm as a constraint. Additionally, the area of isoseismal is incorporated into the algorithm as a constraint. This leads to the proposal of four ellipse fitting algorithms in total: ① An unconstrained fitting algorithm, which implies that no additional constraints are imposed during the model fitting process. ② A fitting algorithm with major-axis strike constrained, which involves the addition of a priori major-axis strike information to the model fitting process. ③ A fitting algorithm with area and major-axis strike constrained, which incorporates both a priori major-axis strike and area into the model fitting process. ④ A fitting algorithm with area constrained, which entails the addition of a priori model area to the model fitting process.

It should be noted that the results of the algorithm will degrade from an ellipse to a circle when the isoseismal shape is circular （i.e. a special ellipse）. Therefore, it is necessary to construct circular isoseismal and simulate different levels of sampling noises so as to test the applicability of the algorithm. The results of the algorithm test demonstrate that all four algorithms yield satisfactory outcomes when the noise is minimal. However, the algorithm with area constraints exhibits a significantly superior performance compared with the algorithm without area constraints when the noise is elevated. Nevertheless, the applicability of the algorithms for degradation to a circle is deemed acceptable, given that extreme noise cases are unlikely to occur in practical scenarios.

The isoseismal data used by the ellipse fitting algorithm is subject to sampling bias, which may affect the output of algorithm. Therefore, it is necessary to analyze the impact of data sampling differences on the results of algorithm. The robustness of the algorithm was evaluated by randomly sampling points of isoseismal to simulate different sampling scenarios. The results indicate that the parameters show normally distributed, with the majority falling within one standard deviation of the mean value. Furthermore, the algorithm is robust. For the unclosed isoseismal with complex shapes, the calculation results are relatively discrete, and thus require separate verification. Furthermore, the measurements obtained by the ellipse fitting algorithm are consistent with previous results and can be used to establish the intensity attenuation relationship.