Moment Magnitude Homogenization Relations in the South Korean Region from 1900 to 2020

Abstract

Utilizing several international and local earthquake catalogs, a set of regression models converting magnitude types typically found in the South Korean region to moment magnitude M_W is presented. These linear regressions were performed using ordinary and total least squares techniques. A standard deviation applied to the residuals was used as a metric for ranking the M_W regressions. This resulted in M_W regressions from M_V,JMA, M_D,JMA, M_S,MOS, and M_S,BJI having the lowest standard deviations, while M_L,IDC showed the highest standard deviation. Other magnitude types ranked differently depending on regression type. An examination of the residuals showed a linear model was appropriate for most magnitude types, with the exception of M_L,KMA and M_L,IDC. Residuals suggested a bilinear model worked well for M_L,KMA, while M_L,IDC showed a monotonic trend.

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, 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, M_S, short-period body wave magnitude, m_b, the local magnitude, M_L, also commonly known as the magnitude on the Richter scale [13,14,15,16]. Notably, 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 M_w from M_s, m_b, 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 Pujols [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_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_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.

2. Materials and Methods

Several earthquake catalogs will be required to develop appropriate magnitude homogenization relationships for the South Korean region. Specific catalogs used in this study include:

• International Seismological Centre (ISC), Reviewed ISC Bulletin. This is a global catalog of seismic events recorded from all over the world starting from 1900. The ISC bulletin is updated periodically [25,26]. Many international seismological agencies submit data to the ISC. These agencies include the United States Geologic Survey National Earthquake Information Center (NEIC) as well as the Commission for the Comprehensive Nuclear-Test-Ban Treaty Organization International Data Center (IDC). The NEIC was formerly known as the National Earthquake Information Service (NEIS), while the IDC was formerly known as the Experimental International Data Center (EIDC).
• Global Centroid Moment Tensor catalog (GCMT). This is an earthquake moment tensor catalog for global earthquakes [27]. Data starts from 1976.
• ISC-GEM Global Instrumental Earthquake Catalog. This catalog focuses primarily on large earthquakes in the ISC Bulletin, where the data is more reliable, to improve, extract, and constrain certain earthquake parameters [19,28,29,30]. When available, ISC-GEM uses M_W stated by GCMT. Data starts from 1904.
• Korea Meteorological Administration (KMA). This is a local catalog of seismic events local to the Korean peninsula [31]. Most of the data is recorded as M_L due to historical reasons [21]. Data starts from 1978. This online database is in Korean.

A search is made within these catalogs to compile a set of earthquakes relevant to the South Korean region. Events from the beginning of 1900 until the end of 2020 were considered. The search region was restricted to within 200 km of the mainland border and the islands of South Korea within 31° to 41° N and 122° to 134° E. Depth was limited to 40 km as any earthquake deeper could be below the crust and from the mantle. Earthquakes originating in the mantle have different wave propagation behavior and mechanisms than those within the crust, concerns typically not accounted for in seismic hazard analyses. Additionally, when the option was available, a minimum magnitude of 2.5 was selected as it is assumed earthquakes with magnitudes at 2.5 or less would not influence magnitude homogenization model development. It is rare to see an earthquake catalog with listed events at M_W below three. Moreover, there will be no distinction between a main shock and foreshocks or aftershocks.

Using the compiled earthquake catalog, earthquakes with a listed M_W in addition to other magnitude types will be plotted. Data of the same magnitude type from related seismological agencies will be grouped into one representative agency. For example, M_S from NEIC will include M_S data from NEIC as well as NEIS, and m_b data from IDC will include m_b data from both IDC and EIDC.

Continuing with the previous examples, this suggests a few agencies will have listed events with an M_W, such as ISC-GEM and GCMT. Data from GCMT is used as the base reference [19,20] and therefore this study will treat M_W,GCMT as the base M_W. Additionally, although not an international seismological agency, NIED is considered to have accurate constraints of the moment tensors for earthquakes in Japan and the surrounding regions [19,20]. Therefore, earthquakes listed with a M_W from NIED will be included in a base list. This base list will contain M_W data as the dependent variable in regression, with M_W,GCMT having first priority, M_W,ISC-GEM having second priority, and M_W,NIED having third priority. Earthquake data where there are five or fewer earthquakes from a single representative agency of one non-M_W magnitude type will not be considered in the regression.

Two regression approaches are considered; ordinary least squares (OLS) and total least squares (TLS) regression. Whereas OLS regression minimizes the residuals of the dependent variable, TLS regression attempts to minimize a distance metric that is a function of residuals between the dependent and independent variables. When this function is the Euclidean distance, the method is called orthogonal regression. TLS herein refers to orthogonal regression. An effective TLS technique, Deming regression, is not considered here as the bivariate method requires errors with known variances. Although ISC provides a magnitude error estimate in their Bulletin, other seismological agencies do not. The reader should note that when the ratio of the known variances is one, Deming regression is the same as orthogonal regression. The resultant regressions for M_W will be termed M_W,proxy herein. Regressions will not be performed on agencies that list four or fewer earthquakes for a specific magnitude type.

To help prioritize magnitude relations, the standard deviation will be used as a metric. For OLS, σ_OLS is defined as the standard deviation of the residuals. For TLS, σ_TLS is defined as the standard deviation of the Euclidean distances between data points and the regressed model. These standard deviations will serve as a metric to help prioritize and rank the magnitude homogenization relationships.

3. Results

The resultant earthquake catalog has 1544 earthquakes with contributions from multiple agencies.

Table 1. Seismological agencies that supplied relevant earthquake event data for this study.

Code	Agency	Country
BJI	China Earthquake Networks Center	China
CGS	Coast and Geodetic Survey of the United States	U.S.A.
EIDC	Experimental International Data Center	U.S.A.
GCMT	The Global Centroid Moment Tensor Project	U.S.A.
IDC	International Data Centre	Austria
ISC	International Seismological Centre	U.K.
JMA	Japan Meteorological Agency	Japan
KEA	Korea Earthquake Administration	North Korea
KMA	Korea Meteorological Agency	South Korea
MOS	Geophysical Survey of Russian Academy of Sciences	Russia
NIED	National Research Institute for Earth Science and Disaster Prevention	Japan
NEIC	National Earthquake Information Center	U.S.A.
NEIS	National Earthquake Information Service	U.S.A.
PEK	Peking	China

describes the seismological agencies and their country of operation. As previously shown, these agencies are typically described by a three or four letter code. Note that BJI and PEK are treated as the same agency, while NEIC, NEIS, and CGS are also treated as the same agency.

Table 2. Number of events with a listed M_w categorized by agency.

Agencies	Number of Events with a Mw	Number of Events with a Mw,GCMT
GCMT	15	15
ISC-GEM	14	5
NIED	135	9
NEIC	8	7
JMA	7	5

describes the base list and shows the number of events with an assigned M_W per seismological agency. There are 15 and 14 events in GCMT and ISC-GEM, respectively, with an overlap of five events. These five events shared the same magnitudes. NIED listed 135 events with a reported M_W, however only nine events shared a M_W,GCMT listing. As a result, there are 149 events with a listed M_W by ISC-GEM, GCMT, or NIED. Surprisingly, there are only eight and seven events with a reported M_W by NEIC and JMA, respectively. Table 2 also lists the number of earthquakes listed by ISC-GEM, GCMT, NIED, NEIC, and JMA that share an event listed in GCMT. These data suggest most of the ISC-GEM and NIED data are not listed within the GCMT database.

A plot of M_W versus M_W,NEIC and M_W,JMA is presented in Figure 1 to help ascertain how correlated the M_W data set are. Data compiled from a previous study in Africa is also shown in the background of Figure 1 to show that the dataset are within previous work [20]. Overall, M_W,NEIC and M_W,JMA show a general one-to-one relation, but there is some slight scatter. Performing linear and total squares regression on the data set results in:

M_w,proxy = 1.119 M_W,NEIC − 0.636; σ_OLS = 0.166; 4.6 ≤ M_W,NEIC ≤ 6.4

(1a)

M_w,proxy = 1.158 M_W,NEIC − 0.842; σ_TLS = 0.110; 4.6 ≤ M_W,NEIC ≤ 6.4

(1b)

M_w,proxy = 1.000 M_W,JMA − 0.000; σ_OLS = 0.045; 4.1 ≤ M_W,JMA ≤ 5.5

(2a)

M_w,proxy = 1.007 M_W,JMA − 0.036; σ_TLS = 0.045; 4.1 ≤ M_W,JMA ≤ 5.5

(2b)

With coefficients of determination R² = 0.941 and 0.986 for Equations (1) and (2), respectively. The regression coefficients are similar with the σ_OLS being greater than σ_TLS for both relationships. These relationships also suggest M_W,JMA is a better indicator of M_W than M_W,NEIC.

To help increase the base list of M_W, some statistical hypothesis testing is conducted to determine if Equations (1a) and (2a) are similar to a one-to-one line (slope = 1, intercept = 0) representing what M_W estimates should be. Hypotheses testing performed on Equation (1a) found no significant difference against a slope of 1, t_crit(6,0.05) = 2.45, p = 0.34 and no significant difference against an intercept of 0, t_crit(6,0.05) = 2.45, p = 0.33. Hypotheses testing performed on Equation (2a) found no significant difference against a slope of 1, t_crit(4,0.05) = 2.78, p = 0.82 and no significant difference against an intercept of 1, t_crit(4,0.05) = 2.78, p = 0.82. This implies the relationships described by Equations (1a) and (2a) are not different from a one-to-one line. Hypotheses testing is not conducted on the TLS results as the residuals are not well-defined within the framework. As such, it is assumed the results from statistical hypothesis testing encompass the results from TLS. Due to this, the M_W,NEIC and M_W,JMA data are considered as M_W and placed in the base list, with M_W,JMA having priority over M_W,NEIC. In actuality, only one additional event was added to the base list, for a total of 149 earthquakes with a listed M_W. Given a baseline M_W,

Table 3. Number of earthquakes with a listed M_w as well as another magnitude type categorized by agency.

Agencies	Number of Equivalent Mw–M Pairs Used in Regression
ISC	79
NEIC	45
IDC	139
JMA	132
BJI	80
MOS	43
KEA	14
KMA	46

presents the number of events that contain M_W and any magnitude (i.e., M_S, m_b, M_L etc.) according to agency. The table shows most of the data for analyses are from IDC and JMA with very limited data sets from KEA, a North Korean agency.

A plot of M_w and M_S pairs from multiple agencies is shown in Figure 2. M_S data from ISC, NEIC, BJI, MOS, and IDC is shown in Figure 2a–e, respectively. The figures show relatively little scatter and the data appear linear with R² = 0.935, 0.725, 0.940, 0.899, and 0.908 for ISC, NEIC, BJI, MOS, and IDC, respectively. Furthermore, the M_S data does not show signs of saturation, a phenomenon where magnitude does not change as M_W continues to increase. Mathematically, M_S saturation tends to reveal itself with earthquakes approaching M_S~8. On the lower end of the magnitude range, the figure shows NEIC, BJI, and MOS to have no data pairs M_S < 4, while ISC and IDC have data pairs for M_S < 4. For comparison, the ISC data generally plots closer to the right-side boundary of a global earthquake catalog data set [19].

OLS regressions on the M_S data result in the following relationships:

M_w,proxy = 0.723 M_S,ISC + 1.661; σ_OLS = 0.184; 3.0 ≤ M_S,ISC ≤ 7.0

(3a)

M_w,proxy = 0.550 M_S,NEC + 2.772; σ_OLS = 0.237; 4.4 ≤ M_S,NEIC ≤ 6.7

(3b)

M_w,proxy = 0.815 M_S,BJI + 0.859; σ_OLS = 0.158; 3.9 ≤ M_S,BJI ≤ 6.5

(3c)

M_w,proxy = 0.681 M_S,MOS + 1.820; σ_OLS = 0.149; 4.0 ≤ M_S,MOS ≤ 6.9

(3d)

M_w,proxy = 0.777 M_S,IDC − 1.565; σ_OLS = 0.198; 2.6 ≤ M_S,IDC ≤ 6.5

(3e)

Additionally, TLS regressions on the M_S data result in the following relationships:

M_w,proxy = 0.741 M_S,ISC + 1.579; σ_TLS = 0.149; 3.0 ≤ M_S,ISC ≤ 7.0

(4a)

M_w,proxy = 0.601 M_S,NEIC + 2.513; σ_TLS = 0.206; 4.4 ≤ M_S,NEIC ≤ 6.7

(4b)

M_w,proxy = 0.836 M_S,BJI + 0.756; σ_TLS = 0.122; 3.9 ≤ M_S,BJI ≤ 6.5

(4c)

M_w,proxy = 0.706 M_S,MOS + 1.698; σ_TLS = 0.122; 4.0 ≤ M_S,MOS ≤ 6.9

(4d)

M_w,proxy = 0.807 M_S,IDC − 1.453; σ_TLS = 0.155; 2.6 ≤ M_S,IDC ≤ 6.5

(4e)

Figure 2 shows that both OLS and TLS regressions produce similar relationships for the magnitude ranges presented.

Similar to Figure 2, a plot of M_w and m_b pairs from multiple agencies is shown in Figure 3. m_b data from ISC, NEIC, BJI, MOS, and IDC is shown in Figure 3a–e, respectively. The data show some scatter but appear linear with R² = 0.891, 0.806, 0.814, 0.821, and 0.804 for ISC, NEIC, BJI, MOS, and IDC, respectively. The data also hint at potential magnitude saturation at m_b ~ 6, which is what is expected. Similar to the M_S data, the figure shows NEIC, BJI, and MOS to have no data pairs m_b < 4, while ISC and IDC have data pairs for m_b < 4. Additionally, the ISC data generally plot closer to the right-side boundary of a global earthquake catalog data set [19], similar to that shown in Figure 2.

OLS regressions on the m_b data result in the following relationships:

M_w,proxy = 0.898 m_b,ISC + 0.649; σ_OLS = 0.198; 3.3 ≤ m_b,ISC ≤ 5.9

(5a)

M_w,proxy = 1.064 m_b,NEIC − 0.233; σ_OLS = 0.263; 3.7 ≤ m_b,ISC ≤ 5.9

(5b)

M_w,proxy = 1.360 m_b,BJI − 1.555; σ_OLS = 0.260; 4.0 ≤ m_b,BJI ≤ 5.8

(5c)

M_w,proxy = 1.057 m_b,MOS − 0.312; σ_OLS = 0.224; 4.1 ≤ m_b,MOS ≤ 6.2

(5d)

M_w,proxy = 1.090 m_b,IDC + 0.113; σ_OLS = 0.281; 3.1 ≤ m_b,IDC ≤ 5.1

(5e)

Additionally, TLS regressions on the m_b data result in the following relationships:

M_w,proxy = 0.948 m_b,ISC + 0.430; σ_TLS = 0.157; 3.3 ≤ m_b,ISC ≤ 5.9

(6a)

M_w,proxy = 1.201 m_b,NEIC − 0.874; σ_TLS = 0.174; 3.7 ≤ m_b,ISC ≤ 5.9

(6b)

M_w,proxy = 1.572 m_b,BJI − 2.532; σ_TLS = 0.147; 4.0 ≤ m_b,BJI ≤ 5.8

(6c)

M_w,proxy = 1.185 m_b,MOS − 0.946; σ_TLS = 0.150; 4.1 ≤ m_b,MOS ≤ 6.2

(6d)

M_w,proxy = 1.242 m_b,IDC − 0.470; σ_TLS = 0.183; 3.1 ≤ m_b,IDC ≤ 5.1

(6e)

Figure 3 shows TLS regressions on NEIC, BJI, MOS, and IDC data to be steeper relative to OLS regressions most likely due to the 2005 March 20 M_W 6.6 Fukuoka earthquake.

Figure 4 plots M_W and M_L pairs from multiple agencies. Data from KMA, JMA, BJI, KEA, and IDC are shown in Figure 4a–e, respectively. In the ISC Bulletin, magnitudes listed as M should be treated equivalent to M_L [16]. Since the local magnitude in the JMA seismological bulletin is listed as M_JMA, the local magnitude used by JMA will be termed M_JMA herein for consistency. Data from JMA, BJI, and KEA appear linear with decent scatter having R² = 0.895, 0.838, and 0.930 for JMA, BJI, and KEA, respectively. The KMA data appear relatively scattered with R² = 0.780 while the IDC data appear the most scattered with R² = 0.408.

The figures show there is a lack of M_L data pairs for BJI and KEA at M_L < 4, while KMA, JMA, and IDC are showing a good amount of data for M_L < 4. Interestingly, Figure 4a appears to show a type of magnitude saturation where, as M_L decreases, M_W stays within a small range. Considering this, a bilinear relationship is proposed where an OLS and TLS regression is applied on magnitude pair data above M_L = 3.6, but kept constant for M_L < 3.6. In this approach, R² for data where M_L < 3.6 is 0.810, a moderate increase from 0.780, but still suggesting a linear relationship.

OLS regressions on the M_L data result in the following relationships:

M_w,proxy = 0.968 M_L,KMA + 0.199; σ_OLS = 0.257; 2.5 ≤ M_L,KMA ≤ 3.6
M_w,proxy = 3.6; 3.6 < M_L,KMA < 5.8

(7a)

M_w,proxy = 0.866 M_JMA + 0.339; σ_OLS = 0.185; 3.3 ≤ M_JMA ≤ 7.0

(7b)

M_w,proxy = 1.026 M_L,BJI − 0.584; σ_OLS = 0.210; 4.4 ≤ M_L,BJI ≤ 6.0

(7c)

M_w,proxy = 0.888 M_L,KEA + 0.184; σ_OLS = 0.188; 3.8 ≤ M_L,KEA ≤ 6.0

(7d)

M_w,proxy = 0.806 M_L,IDC + 1.514; σ_OLS = 0.501; 2.5 ≤ M_L,IDC ≤ 4.6

(7e)

Additionally, TLS regressions on the M_L data result in the following relationships:

M_w,proxy = 1.085 M_L,KMA − 0.314; σ_TLS = 0.190; 2.5 ≤ M_L,KMA ≤ 3.6
M_w,proxy = 3.6; 3.6 < M_L,KMA < 5.8

(8a)

M_w,proxy = 0.911 M_JMA + 0.159; σ_TLS = 0.138; 3.3 ≤ M_JMA ≤ 7.0

(8b)

M_w,proxy = 1.133 M_L,BJI − 1.139; σ_TLS = 0.143; 4.4 ≤ M_L,BJI ≤ 6.0

(8c)

M_w,proxy = 0.919 M_L,KEA + 0.041; σ_TLS = 0.139; 3.8 ≤ M_L,KEA ≤ 6.0

(8d)

M_w,proxy = 1.434 M_L,IDC − 0.662; σ_TLS = 0.341; 2.5 ≤ M_L,IDC ≤ 4.6

(8e)

In addition to M_W and M_JMA, JMA also uses a displacement magnitude, M_D,JMA, and a velocity magnitude, M_V,JMA [32,33,34]. Earthquakes with these JMA displacement and velocity-based magnitudes and a corresponding M_W are plotted in Figure 5. The data for M_D,JMA is not as plentiful, ranging from M_D,JMA~4.0 to 6.0. On the other end, M_V,JMA appears to be associated with events at the lower magnitude range, with M_V,JMA~3.5 to 5.0. Both magnitudes show linear behavior and little scatter.

OLS regressions on these magnitude types result in the following relationships:

M_w,proxy = 0.684 M_V,JMA + 0.916; σ_OLS = 0.075; 3.5 ≤ M_V,JMA ≤ 5.0

(9a)

M_w,proxy = 0.916 M_D,JMA + 0.312; σ_OLS = 0.109; 4.2 ≤ M_D,JMA ≤ 5.8

(9b)

Additionally, TLS regressions on these magnitude types result in the following relationships:

M_w,proxy = 0.934 M_D,JMA + 0.226; σ_TLS = 0.080; 3.5 ≤ M_V,JMA ≤ 5.0

(10a)

M_w,proxy = 0.694 M_V,JMA + 0.879; σ_TLS = 0.062; 4.2 ≤ M_D,JMA ≤ 5.8

(10b)

As stated previously, the σ’s will be used to prioritize magnitude relations. Note that σ_TLS treats the residual as the shortest distance from data points to the regressed line.

Table 4. Comparison of the different rankings of magnitude type regressions based on σ_OLS and σ_TLS.

Priority	Magnitude Type	σOLS	Magnitude Type	σTLS
1	MV,JMA	0.075	MV,JMA	0.062
2	MD,JMA	0.109	MD,JMA	0.080
3	MS,MOS	0.149	MS,MOS	0.122
4	MS,BJI	0.158	MS,BJI	0.122
5	MS,ISC	0.184	MJMA	0.138
6	MJMA	0.185	ML,KEA	0.139
7	ML,KEA	0.188	ML,BJI	0.143
8	MS,IDC	0.198	mb,BJI	0.147
9	mb,ISC	0.198	MS,ISC	0.149
10	ML,BJI	0.210	mb,MOS	0.150
11	mb,MOS	0.224	MS,IDC	0.155
12	MS,NEIC	0.237	mb,ISC	0.157
13	ML,KMA 1	0.257	mb,NEIC	0.174
14	mb,BJI	0.260	mb,IDC	0.183
15	mb,NEIC	0.263	ML,KMA1	0.190
16	mb,IDC	0.281	MS,NEIC	0.206
17	ML,IDC	0.501	ML,IDC	0.341

¹ Regression is bilinear.

compares the ranking of each relationship in ascending order for OLS and TLS. The table shows that based on σ, M_V,JMA, M_D,JMA, M_S,MOS, and M_S,BJI have the highest, second, third, and fourth highest priorities, respectively. However, after these first four magnitude types, there appears to be a difference in rankings depending on the type of regression used. For example, the M_S,ISC to M_W relationship has the fifth lowest σ_OLS, but is ranked ninth when considering σ_TLS. The largest discrepancy appears to be with m_b,BJI, where it is ranked fourteenth under OLS regression but ranked eighth under TLS regression. Interestingly, the magnitude type by the most local agency, M_L,KMA, ranked thirteenth under OLS and fifteenth under TLS conditions. The table also suggests m_b data are more scattered relative to M_S data as both OLS and TLS regressions show higher σ’s.

4. Discussion

In terms of performance, residuals from M_S regressions are plotted in Figure 6. These plots show the OLS and TLS residual biases for each regression are essentially zero. Additionally, TLS residuals also show less variability relative to OLS residuals. The regression residuals for M_S,IDC appear more scattered especially at M_S,IDC < 4, with three events beyond 2σ for both OLS and TLS regressions. Even so, these plots suggest a linear model can be appropriate for M_S regressions.

Residuals from m_b regressions are plotted in Figure 7. These plots show the OLS and TLS residual biases for each regression are close to zero, however there appears to be considerable scatter within the residuals for each regression, with each regression having about three events beyond 2σ for OLS regressions. Interestingly, the TLS regression for m_b,NEIC in Figure 7b had no events outside of 2σ. Although the figure shows decent scatter across magnitudes, there does not appear to be any trend, suggesting a linear model can be appropriate for m_b regression.

Residuals from M_L regressions are plotted in Figure 8. Residual bias from OLS and TLS regressions are zero with TLS residuals showing less variability relative to OLS residuals. The proposed regression for M_L,KMA was bilinear to account for the stagnation at lower magnitudes and the residuals appear to agree in Figure 8a. It is not known if this saturation effect is a characteristic of lower magnitudes or an artifact of M_L calibrated for Japan [21,35]. Interestingly, there are several events between 4.5 < M_L,KMA < 5 where the bilinear model appears to underestimate magnitudes. Residuals for M_JMA also appear to show events at M_JMA~3.7 and 5.2 to be underestimated as shown in Figure 8b, with five events beyond 2σ for both OLS and TLS regressions, the highest amongst all models. However, this may be due in part to the relatively higher number of earthquakes assigned this magnitude type. Figure 8e shows the M_L,IDC residuals that appear the most scattered across the magnitude range, resulting in relatively large σ. Even with a large σ, there are four and three events beyond 2σ for OLS and TLS regressions, respectively. Moreover, the decreasing behavior with M_L,IDC suggest a linear model for regression may not have been appropriate. As such, the proposed regression relationship herein is not recommended for homogenization purposes but is presented for completeness.

Interestingly, the residuals for M_V,JMA and M_D,JMA appear the most linear of all the regressions presented herein as shown in Figure 9. This may be perhaps due to the limited number of earthquake events. However, the reason why there are so few events is that JMA started using M_D,JMA and M_V,JMA after discontinuing the use of M_JMA in 2016. Nonetheless, all the data points fall within 2σ and suggest a linear model to be appropriate to the data.

Overall, these findings reinforce the notion that M_W estimates from magnitude type regressions are dependent on regression type. A linear trend model appears to serve the data well with perhaps the exceptions of M_L,KMA and M_L,IDC. For magnitude types that exhibited more linear behavior, either OLS or TLS regression would yield similar results. Differences between OLS and TLS relationships appear when the data show events that seem to be significant outliers, relative to a linear trend.

An additional comparison is made between the proposed homogenization regressions herein against regressions from previous studies in Figure 10. As previously mentioned, a study based on global ISC data provided regressions for M_S,ISC and m_b,ISC [19]. Another study focused on the continent of Africa also offered homogenization relationships M_S,ISC, m_b,ISC, M_S,NEIC, and m_b,NEIC [20]. Both studies derived their relationships using orthogonal regression, which is comparable to the TLS approach used here. Figure 10a,b show the residuals to both global and African relationships for M_S,ISC and m_b,ISC, respectively. The global relationship appears to overestimate M_S,ISC for the compiled data and shows a linearly increasing trend for the m_b,ISC residuals, suggesting an inappropriate model. Figure 10c,d show the residuals from the African relationships for M_S,NEIC and m_b,NEIC, respectively. The residuals from the African model plot slightly above and away from the TLS M_S,NEIC residuals, while the residuals from the African model plot mostly lower and away from the TLS m_b,NEIC residuals.

A South Korean study presenting a local magnitude calibration also showed magnitude type relationships between M_W, M_L, and M_L,KMA [21]. They separated between horizontal and vertical M_L derivations, but each one showed similar results. The residuals from the horizontal model are presented in Figure 10e. These residuals show significantly more scatter relative to the TLS residuals shown herein and seem to suggest bias, especially at M_L < 3.6. As mentioned previously, KMA used an M_L scale calibrated for Japan, which may be a source for much of observed scatter. Additionally, the M_W relationships provided in the South Korean study were compared against their proposed M_L scale.

5. Conclusions

Several relationships estimating M_W from various magnitude types typically encountered in the South Korean region were presented. These relationships are based on ordinary and total least square regression techniques. Of the relationships, the ones utilizing M_V,JMA, M_D,JMA, M_S,MOS, and M_S,BJI performed the best in terms of a standard deviation metric, while the relationship utilizing M_L,IDC had the highest standard deviation. An inspection of the data showed most of the regressions with higher standard deviations tended to be of the m_b type. The M_L,KMA regression dataset showed little change with M_W as M_L,KMA decreased, which led to a bilinear regression model. Still, the resultant standard deviation was relatively high.

An examination of the residuals showed the linear regression model, whether ordinary or total least squares, was appropriate for many magnitude types. For M_L,KMA, a bilinear model constrained the residuals, while the decreasing pattern of the residuals for M_L,IDC suggested a linear model was perhaps not a good fit to the regression dataset. As a result, it was not recommended for use in homogenization activities. Moreover, a comparison of the residuals were made against homogenization relationships derived from global, African, and South Korean data. Residuals from these models showed more scatter, bias, as well as obvious trends, suggesting the proposed homogenization relationships herein are more appropriate for the South Korean region.