# Shear-Wave Velocity Model from Site Amplification Using Microtremors on Jeju Island
## 3 Methods  
The inversion of HVSR curves for Vs profiles presents a costly endeavor, primarily due to the extensive and labor-intensive search required in the parameter space. This complexity is further compounded by the prolonged convergence times or the potential failure to converge, a consequence of the highly ill-posed nature of the nonlinear inversion problem, such as determining the Vs structure [35; 36]. To overcome this challenge and improve the reliability of the results, a key strategy adopted in this study involved the independent use of two inversion algorithms for the same observed HVSR curve. Thus, an HVSR inversion scheme was executed using Monte Carlo (MC) simulations and simulated annealing (SA) algorithms. These algorithms are renowned for their effectiveness in computing nonlinear inversion processes [37; 38; 39; 40; 41]. An additional strategy involved permitting a broad spectrum of variances in the observed HVSR curves during the inversion process. To address the ill-posed inversion condition, the inversion scheme was applied across a range of HVSR curve variances (0.25, 0.5, 0.75, 1.0, 1.25, 1.5, 1.75, 2.0, 2.25, and 2.5). This approach facilitated independent observation of the HVSR for each algorithm, followed by a comprehensive evaluation. As a third strategy, to refine the initial parameter space exploration and intensify the search for viable solutions, initial boundary limits for the parameters were established based on previous studies conducted in the study area. Incorporating insights from all prior studies, initial values for Vs, layer thickness, and density were assigned for the inversion process. The initial ranges of the inversion parameters for the sites under investigation are enumerated in Table 3 [26; 27; 28; 29; 30; 31; 32]. Figure 1: Locations of Pohang earthquake series, Kumamoto earthquake series (red circles), and nine seismic sites (green triangles). To address the complexities inherent in a highly ill-posed nonlinear inversion process, two critical steps were implemented to establish the initial shear-wave velocity profile for the inversion. The first step involved determining the initial values of the free parameters for an efficient nonlinear inversion, constrained within the predetermined search limits of the layer parameters, using a random search method. Based on these initial values, the second step entailed searching the parameter spaces for global minimum misfit. Previous sensitivity analyses have highlighted that the HVSR curve is predominantly influenced by the S-wave velocity, Vs, and the layer thickness [19; 42]. The shear velocity profile in the Jeju region was examined by evaluating the misfit between the observed and synthetic HVSR curves, including an assessment of their shape characteristics such as the peak amplitude, number of peaks, and specific frequency. The misfit value is quantified as follows:  
\[\Delta_{HV}=\sum_{i=1}^{n}\frac{\big{(}HV_{obs}(f_{i})-HV_{th}(f_{i})\big{)}^ {2}}{(HV_{obs}(f_{i}))^{2}}, \tag{1}\]  
where \(\Delta_{HV}\) denotes the misfit value, \(HV_{obs}(f_{i})\) indicates the HVSR of the observed environmental noise, and \(HV_{th}(f_{i})\) represents the synthetic noise. The synthetic HVSR curves were generated from the simulated values utilizing the principles of diffuse-field theory. The foundational work of [PERSON] et al. [37], which models microtremor HVSR for surface receivers, was adapted to the synthetic HVSR in this study. Under the basic assumption, the medium is conceptualized as a series of horizontal uniform layers overlying a homogeneous infinite half-space, the latter representing a rigid basement. [PERSON] et al. [43] demonstrated that configuring the shear velocity profile with approximately eight layers yields reliable results for specific types of velocity structures, notably those with a cover layer thickness of several hundred meters, although this is contingent upon the varied conditions of the site. Inspired by previous research [26; 27; 28; 29; 30; 31; 32] and the insights of [PERSON] et al.