## 1 Introduction  
After a period of relatively low seismicity, South Korea suddenly experienced two moderate earthquakes, the 16 September 2016 Gyeongju Earthquake and the 15 November 2017 Pohang Earthquake [1, 2]. These events ignited multiple discussions on the vulnerability of South Korean infrastructure to seismic events, especially considering that both earthquakes occurred near nuclear power plants. The state-of-practice in evaluating how earthquakes can impact engineered infrastructure is seismic hazard analysis [3, 4, 5, 6, 7]. Seismic hazard analyses on major projects are typically conducted using a probabilistic approach, whereby stochastic models are used as components in solving the hazard integral. These models, such as modern ground motion prediction equations and recurrence relationships, are generally calibrated to moment magnitude, \(\mathrm{M_{W}}\)[8, 9, 10, 11, 12]. However, earthquake catalogs used to develop tools for use in probabilistic seismic hazard analysis tend to have earthquakes listed with differing magnitude types, such as surface wave magnitude, \(\mathrm{M_{S}}\), short-period body wave magnitude, \(\mathrm{m_{b}}\), the local magnitude, \(\mathrm{M_{L}}\), also commonly known as the magnitude on the Richter scale [13, 14, 15, 16]. Notably, \(\mathrm{M_{L}}\) requires a local calibration process that takes into account site and path effects that make it unique to a region [17]. Accordingly, using a variety of earthquake magnitude types in earthquake research or practical applications is cumbersome and computationally inconsistent as some agencies may use different standards for the same magnitude [18].  
To overcome this issue, seismologists and earthquake engineers conduct magnitude homogenization. Magnitude homogenization entails developing relationships between the various magnitude types to a single magnitude type. These relationships typically relate \(\mathrm{M_{w}}\) from \(\mathrm{M_{s}}\), \(\mathrm{m_{b}}\), \(\mathrm{M_{L}}\), or any other additional magnitude types relevant to the region of interest [19, 20, 21]. Magnitude homogenization is not a trivial matter as there are considerations with regression techniques and relationship priority, which might or might not influence calculations [20, 22, 23]. The reader is encouraged to refer to the work of [PERSON] [24] as it provides a good review on regression issues stemming from magnitude homogenization. The work essentially suggests Deming regression may not always be the best method in predicting earthquake magnitudes. Moreover, few low magnitude earthquake events list an M\({}_{\text{W}}\), making it difficult for certain types of application.  
Given this backdrop, this study proposes a variety of relationships to homogenize earthquake magnitudes estimated within the South Korean region to M\({}_{\text{W}}\). This involves developing an earthquake catalog, which should include events within and immediately outside of South Korea, approximately 200 km away, to allow the results to be applicable and consistent with seismic hazard studies. Regression techniques, such as ordinary least squares and total squares regression, will be applied to this expanded catalog to help understand how these techniques influence ranking and priority amongst the various magnitude types.