Regression relationships for conversion of body wave and surface wave magnitudes toward Das magnitude scale, Mwg

A reliable and standardized estimation of earthquake size is a fundamental requirement for all tectonophysical and engineering applications. Several investigations raised questions about the determinations of smaller and intermediate earthquakes using Mw scale. Recent investigations (Das et al. in Bull Seismol Soc Am 108(4):1995–2007, 2018b) show that the moment magnitude scale Mw is not applicable for lower and intermediate ranges throughout the world and does not efficiently represent the seismic source potential due to its dependence on surface wave magnitudes; therefore, an observed seismic moment (M0)-based magnitude scale, Mwg, which smoothly connects seismic source processes and highly correlates with seismic-radiated energy (Es) compared to the Mw scale is suggested. With the goal of constructing a homogeneous data set of Mwg to be used for earthquake-related studies, relationships for body wave (mb) and surface wave magnitudes (Ms) toward Mwg have been developed using regression methodologies such as generalized orthogonal regression (GOR) (GOR1: GOR relation is expressed in terms of the observed independent variable; and GOR2: GOR relation is used inappropriately in terms of theoretical true point of GOR line) and standard least-square regression (SLR). In order to establish regression relationships, global data have been considered during 1976–2014 for mb magnitudes of 524,790 events from the International Seismological Centre (ISC) and 326,201 events from the National Earthquake Information Center (NEIC), Ms magnitudes of 111,443 events from ISC along with 41,810 Mwg events data from the Global Centroid Moment Tensor (GCMT). Scaling relationships have been obtained between mb and Mwg for magnitude range 4.5 ≤ mb ≤ 6.2 for ISC and NEIC events using GOR1, GOR2 and SLR methodologies. Furthermore, scaling relationships between Ms and Mwg have been obtained for magnitude ranges 3.0 ≤ Ms ≤ 6.1 and 6.2 ≤ Ms ≤ 8.4 using GOR1, GOR2 and SLR procedures. Our analysis found that GOR1 provides improved estimates of dependent variable compared to GOR2 and SLR on the basis of statistical parameters (mainly uncertainty on slope and intercept, RMSE and Rxy) as reported in Das et al. (2018b). The derived global scaling relationships would be helpful for various seismological applications such as seismicity, seismic hazard and Risk assessment studies.


Introduction
The earthquake magnitude scale is one of the most fundamental earthquake source parameters used for measuring the strength of an earthquake. In 1935, the first earthquake magnitude scale (local magnitude: M L ) was introduced by Richter (1935) for earthquakes in Southern California. After 10 years, Gutenberg (1945a) extended the local magnitude scale to measure earthquakes at long distances and defined the earthquake magnitude scale as "M s " called the surface-wave magnitude, considering the surface wave (period between 17 and 23 s) of a seismic signal.
The concept of body wave magnitude was first proposed by Gutenberg (1945a, b) and later redefined by Gutenberg and Richter (1956). Long period body wave magnitude (mB) determination is based on the ratio of maximum amplitude to period of P or S waves with periods up to about 10 s recorded by intermediate-to long-period instruments (Das et al. 2011). The body wave magnitude (m b ) is determined using P waves around 1 s, and time periods considered for mB are several to 10 s. (Das et al. 2011). The surface wave magnitude scale does not work for higher earthquake size because of saturation. The saturation of surface wave magnitude occurs when fault rupture dimension of an earthquake exceeds the wavelength of the earthquake wave used for magnitude estimation. Kanamori (1977) and Hanks and Kanamori (1979) developed another scale called M w scale that is often considered as non-saturating magnitude scale. M w scale is mainly developed on the following issues: (1) M w scale is only adequate for larger earthquake (Kanamori 1977); (2) The validation step of the development of M w scale is solely performed on Southern Californian Seismicity (Tables 1 and Table 2 of Hanks and Kanamori 1979); thus, applicability of M w scale ≥ 3.0 is only valid for Southern California not for worldwide. (3) M w scale is based on the relationship between Energy and Surface wave magnitude, and thus, it is not closely related with source; (4) Direct observed seismic moment record is not considered in the development of M w scale only based on substitution; (5) M w scale used constant stress drop which is applicable for upper crust; therefore, applicability of M w scale is also limited to the upper crust.
Despite the popularity of M w , it provides limited information about the earthquake source, especially regarding its high-frequency content (e.g., Beresnev 2009) which is more relevant for the evaluation of an earthquake's shaking potential. It has also been observed by many authors that using only one scale, i.e., M w scale does not serve the purpose for measuring the actual size of earthquakes due to its inherent limitations (e.g., Kanamori 1977;Choy and Boatwright 1995;Kanamori and Brodsky 2004;Bormann et al. 2009;Wason et al. 2012;Das 2013;Das et al. 2013, 2014b, 2018b, Lin et al. 2020. Recently, an advanced unsaturated earthquake magnitude scale, i.e., Das magnitude scale M wg , has been reported (Das et al. 2019) which circumvents the limitations of M w scale. The M wg scale is mainly based on observed seismic moment record using worldwide data. The M wg is based on low-and high-frequency spectra of seismic signal. The M wg scale is well connected with radiated energy and observed magnitude scales (e.g., m b , M s , M e ). M wg scale is directly proportional to the logarithm of the observed seismic moment, and thus, it is related to seismic source process, and it is not saturated for large magnitude earthquakes. Therefore, M wg scale depicts a uniform behavior for wider magnitude ranges. Thus, it is preferred to compile earthquake catalogs with all magnitudes expressed in this unified scale M wg for the purpose of seismic hazard assessment and other important seismological studies associated with seismicity.
Toward preparation of a homogeneous earthquake catalog, it is generally required to express different magnitudes as one magnitude (M wg ) using regression relationships. We provide regression relationships of entire world for body and surface wave magnitudes toward seismic moment scale M wg using SLR (Standard Least-square Regression), GOR2 (General orthogonal Regression: suggested in Fuller 1987 andCarroll andRuppert 1996) and GOR1(General Orthogonal Regression suggested in Das et al. 2018b) as suggested in recent literature (Das et al. 2011(Das et al. , 2016Das et al. 2014aDas et al. , b, 2018aFuller 1987;Nath et al. 2017;Ristau 2009;Wason et al. 2012).

Seismic moment magnitude (M wg ) and moment magnitude (M w ) scales
To understand the magnitude scales based on M o detailed background of M wg and M w scales is given below. Kanamori (1977) defined a magnitude scale (Log W 0 = 1.5 M w + 11.8, where W 0 is the minimum strain energy) for great earthquakes using Gutenberg Richter Eq. (1). Kanamori (1977) used W 0 in place of E s (dyn.cm) and consider a constant term (W 0 /M o = 5 × 10 -5 ) in Eq. (1) and estimated M s and denoted as M w (dyn.cm). It is important to note that the energy Eq. (1) is derived by substituting m = 2.5 + 0.63 M in the energy equation Log E = 5.8 + 2.4 m (Richter 1958), where m is the Gutenberg unified magnitude and M is a least squares approximation to the magnitude determined from surface wave magnitudes. After replacing the ratio of seismic Energy (E) and Seismic Moment (M o ), i.e., E/M o = 5 × 10 -5 , into the Gutenberg-Richter energy magnitude Eq. (1), Hanks and Kanamori (1979) provided Eq. (2):

M w scale
Note that Eq. (2) was already derived by Kanamori (1977) and termed it as M w . Eq. (2) was based on large earthquakes; hence, in order to validate Eq. (2) for intermediate and smaller earthquakes, Hanks and Kanamori (1979) compared this Eq. (2) with Eq. (1) of Percaru and Berckhemer (1978) for the magnitude 5.0 ≤ M s ≤ 7.5 (Hanks and Kanamori 1979). Note that Eq. (1) of Percaru and Berckhemer (1978) for the magnitude range 5.0 ≤ M s ≤ 7.5 is not reliable due to the inconsistency of defined magnitude range (moderate to large earthquakes defined as M s ≤ 7.0 and M s = 7-7.5) and scarce data in lower magnitude range (≤ 7.0) which rarely represents the global seismicity (e.g., see Figs. 1A, B, 4 and Table 2 of Percaru and Berckhemer 1978).
In order to validate Eq. (2) for the lower magnitude range, Eq.
(2) is compared with the Sothern California M o and M L relationship for the magnitude range 3.0 ≤ M L ≤ 7.0. It is natural that the relationship between M 0 and M L will vary due to different seismotectonic and geological setting (e.g., Hutton and Boore 1987;Choy and Boatwright 1995;Ristau et al. 2003;Keir et al. 2006). As stated above and in earlier studies (e.g., Das et al. 2019), validation of M w scale was performed on Californian seismicity; therefore, the M w scale (1) Log E s = 1.5M s + 11.8.
(2) Log M o = 1.5M s + 16.1 1 3 (2/3 log M o −10.7) provided by Hanks and Kanamori (1979) is only applicable for certain region. Hanks and Kanamori (1979) also referred to their formulation that M w is uniformly valid for M w ≥ 7.5 as pointed out by Kanamori (1977). Therefore, its use worldwide is not appropriate for magnitude ranges < 7.5. This can also be easily understood from the strong deviations of M w with m b and M s scales (Figs. 1, 2 and 3 of Das et al. 2019). It is important to note that the comparison between M w scale and observed other magnitude scales (M L and M s ) was the main criteria for validation of M w scale in Hanks and Kanamori (1979).  , (x t , y t ), t = 1, 2, 3) and estimated points (i.e., (X t , Y t t ), t = 1, 2, 3) on the fitted regression line (solid black line) for a set of three observed points (X 1 , Y 1 ), (X 2 , Y 2 ), and (X 3 , Y 3 ); a standard linear-square regression (SLR) line; b general orthogonal regression (GOR) line varies generally from few bars to 125. The variability of Δσ is significant; therefore, stress drop cannot be assumed to be constant (See Table 2, Percaru and Berckhemer 1978). Hence, depending on the value of constancy, M w value for a given earthquake will change significantly. Furthermore, the value of constancy (E s /M o ) in the derivation of M w scale is only applicable for shallow earthquakes. Thus, M w scale is only applicable for shallow earthquakes mainly for two reasons: (1) M w scales are primarily derived from surface wave scale because the fundamental equation used for obtaining the M w scale was a relationship between Energy and Surface wave magnitude (Log E S = 1.5M s + 11.8), and (2) Used constant value (E s /M o = 5*10 -5 = Δ 2 ) in the development of M w scale is only applicable for shallow earthquakes. The unsaturated M w is globally valid for large earthquake as M w scale was based on equations Log E S = 1.5M s + 11.8 and Log M 0 = 1.5M s + 16.1 (Details are given in Richter 1958;Kanamori 1977;Purcaru and Berckhemer 1978).

M wg scale (Das magnitude scale)
In  magnitudes with 1316 events have been collected from ISC, NEIC, respectively. Furthermore, 397 seismic-radiated energy data have been collected from Choy and Boatwright (1995). Das et al. (2019) derived simple least-squares fitting relationship between M 0 and m b using 25,708 global events, which is as follows: As discussed above, M w scale was developed from the equation Log E S = 1.5M s + 11.8. In this equation, M s is saturated around 8.6; thus, applicability of M w is valid up to 8.6. But, if E s is known independently and put in this equation, then M s would not saturate (Hanks and Kanamori 1979). In Eq. (3), M 0 knows independently; therefore, substituting M o in left-hand side of Eq. (3) will produce M wg and it will not saturate, as given in Eq. (4).
Both M w and M wg are in terms of M o (dyn cm), so they are physics based and will not saturate. One can easily estimate M o from M w scale by using Eq. (5) Tohoku-Oki earthquake (March 11th, 2011) has an M w value of 9.1 and caused serious damage. Using Eq. (5), Log M o of Tohoku-Oki earthquake will be 29.72509452, and then, using Eq. (4) one can estimate M wg as 9.2.
The M wg scale is highly correlated with radiated energy E s (Fig. 1) and observed magnitudes (e.g., m b , M s , M e ) as reported in Das et al. (2019).

Methodology
General orthogonal regression (GOR) yields a linear relationship between dependent (y t ) and independent (x t ) variables based on observed data (X t , Y t ) having errors in both the variables (Madansky 1959;Kendall and Stuart 1979;Das et al. 2018b). In the conventional GOR (GOR2) procedure, estimate of y t is obtained by substituting X t (instead of x t ) in the GOR relation ( y t = 0 + 1 x t ). The conventional procedure produces a bias in the estimate as demonstrated in recent publications (e.g., Wason et al. 2012;Das et al. 2014aDas et al. , b, 2018a. Hence, this problem was corrected by adding one additional step in the estimation procedure as explained in Wason et al.(2012) and Das et al. (2018b) and thus, the application of GOR2 must include the suggested correction to overcome the limitations of GOR2 (see Wason et al. 2012;Das et al. 2014aDas et al. ,b, 2018a. Carroll and Ruppert (1996) had also reported about the misuse of GOR2 and cautioned for the over estimation of regression slope. The difference between Carroll and Ruppert (1996) and Das et al. (2018b) is that Das et al. (2018b) used error variance value given by Fuller (1987), but Carroll and Ruppert (1996) modified the error variance value (η) to adjust the overestimation of slope. Das et al. (2018b) adjusted the overestimation of slope through an intermediate step in which GOR is corrected. The corrected GOR as described in Das et al. (2018b) is henceforth denoted as GOR1. In order to better understand the limitations of GOR, a graphical representation of GOR has been discussed below. The methodology GOR2 is called as OR (Orthogonal Regression) when η is considered to be 1.

A graphical representation of GOR
The graphical representation of GOR is provided below for easy understanding the limitations involved in the GOR method. It is observed that GOR inherent problem is not well addressed in the existing literature before the study of Das et al. (2012); therefore, to provide a clear view on GOR, two different cases are discussed below.

Case I
Let us consider a SLR line obtained from data pairs (X 1 , Y 1 ), (X 2 , Y 2 ) and (X 3 , Y 3 ), and the corresponding theoretical true points on the SLR line are (x 1 = X 1 , y 1 ), (x 2 = X 2 , y 2 ) and (x 3 = X 3 , y 3 ), respectively. Note that these theoretical true points on the line are used to derive the best fitting SLR line by minimizing the vertical residuals. On substituting the independent observed variables X 1 , X 2 , X 3 in the obtained SLR line, one can achieve the theoretical true points ((x 1 = X 1 , y 1 ), (x 2 = X 2 , y 2 ) and (x 3 = X 3 , y 3 )) that were used in the derivation of the best fitting SLR line. In Fig. 2a, ED 1 , ED 2 and ED 3 are the Euclidean distances between (X 1 , Y 1 ) and (x 1 = X 1 , y 1 ), (X 2 , Y 2 ) and (x 2 = X 2 , y 2 ), and (X 3 , Y 3 ) and (x 3 = X 3 , y 3 ), respectively. These Euclidean distances are used during the development of the SLR line. Let AD 1, AD 2, AD 3 be the achievable distances after substitution of X 1 , X 2 , X 3 in the obtained SLR relation. Note that in SLR, distances used during minimization for building the line can also be achieved in the estimations, i.e., ED1 = AD 1, ED 2 = AD 2, ED 3 = AD 3 (Fig. 2a).

Case II
Consider a GOR line obtained using observed data pairs (X 1 , Y 1 ), (X 2 , Y 2 ), and (X 3 , Y 3 ), with errors in both the variables. The theoretical true points (true Points) of these data pairs on the GOR line, i.e., (x 1 , y 1 ), (x 2 , y 2 ) and (x 3 , y 3 ) are given by minimizing the Euclidean distance (statistical Euclidean distance or weighted orthogonal distance). By substituting observed values X 1 , X 2 , X 3 in the obtained GOR line, the corresponding theoretical true points cannot be achieved unlike in the Case I. Instead of obtaining the theoretical true points, totally different points on the GOR line are achieved (see Fig. 2b). This issue can also be understood using Euclidean distance concept.
Hence, GOR2 introduced bias in the estimation as it is not possible to get the corresponding true point on the direct substitution of any observed value of the independent variable in the GOR line. Therefore, the Squared Euclidean distance (Fuller 1987, F1.3.14) is not applicable in case of GOR.

Conversion relationships among different magnitude scales
The GOR relationship requires error variance ratio values (η) for its derivation. To find the uncertainties associated with m b , M s and M wg magnitude determinations, standard deviation of the differences between the magnitude determinations by two different agencies for a same type of earthquake magnitude have been estimated. In the case of m b or M s , the standard deviations have been estimated from the differences between observed ISC and NEIC data, and for M wg , differences have been obtained between observed GCMT and NEIC data. The comparative standard deviations associated with different magnitude types are estimated to be 0.09, 0.11, 0.12 and 0.2 for M wg , M w , M s and m b , respectively. These values are consistent with other reported values suggested in earlier studies (e.g., Kagan 2003;Das et al. 2011). The knowledge of error variance ratio (η) is very critical for performing GOR relations. The use of equation error in estimating η has not addressed in GOR equations performed in earlier seismological studies, except in Das et al. (2018b). Carroll and Ruppert (1996) suggested to use = e + q, u (where q, denotes equation error, see Table 1 of Carroll and Ruppert 1996) instead of using ( = e u ) because equation error q, is not considered. As equation error calculation is not straight forward, therefore, we use Fuller (1987) Fig. 3.
Out of all, 245,899 events have m b values both from ISC and NEIC. The GOR1, GOR2 and SLR relationships with η = 1 between m b, ISC and m b, NEIC are as given below (Fig. 3).
The developed corresponding relationships between M s, ISC and M s, NEIC are as follows (Fig. 3) The GOR1, GOR2 and SLR relationships with η = 1 between M wg, GCMT and M wg, NEIC . are as given below (Fig. 3). Furthermore, GOR1, GOR2 and SLR relationships have been derived between m b , NEIC and M wg, GCMT for the magnitude range 4.5 ≤ m b,NEIC ≤ 6.2 for different η values. The variations of slope and intercept coefficients with respect to η values are shown in Fig. 4, and the GOR1, GOR2 and SLR relationships between body wave magnitude of NEIC and seismic moment magnitudes for η = 0.2 are given as follows: The plots of the all regression relationships  are shown in Fig. 5. In conversion from m b,NEIC to M wg , GOR1 provides estimates closer to SLR, but have lower uncertainties in slope and intercept along with standard deviations and correlation coefficients compared to GOR2 and SLR.

Surface wave magnitudes to seismic moment magnitudes
It has been found that the 21,474 global M s events used in this study for magnitude range 3.1 ≤ M s ≤ 8.4 depict a bilinear trend the same was also suggested by Wason et al. (2012). Hence, the magnitude range has been subdivided into two parts, that is, 3.1 ≤ M s ≤ 6.1 and 6.2 ≤ M s ≤ 8.4, assuming that the distribution is linear in these respective magnitude ranges (Fig. 6a).
In order to convert M s to M wg , 19, 826 events have been considered in the magnitude range 3.1 ≤ M s ≤ 6.1 and 1639 events in the magnitude range 6.2 ≤ M s ≤ 8.4. GOR1, GOR2 and SLR relationships between M s and M wg with η = 0.56 for the magnitude range 3.1 ≤ M s ≤ 6.1 are given below. The developed corresponding relations for 6.2 ≤ M s ≤ 8.4 are as follows: The plots of regression relationships from Eqs. 21-26 are shown in Fig. 6. It is observed from Fig. 6 that GOR1 lies in the middle between GOR2 and SLR. GOR2 shows overestimation values in almost all the ranges of Fig. 6c, and SLR shows underestimation values in the majority portion of both the magnitude ranges of M s . The variations of slope and intercept values with respect to η are shown in Fig. 7. It has been observed from Fig. 7 that for η > 3.5, both the GOR2 and GOR1 show nearly equivalent results.

Discussions and conclusions
The M wg is an unsaturated magnitude scale, based on both low-and high-frequency spectra of a seismic signal using observed M o , connects smoothly the seismic source potential and seismic-radiated energy and is applicable for wider magnitude ranges (> 3.5) worldwide. Existing low-frequency spectra based M w are only applicable for ≥ 7.5 worldwide (Kanamori 1977, Table 1 and Table 2 of Hanks and Kanamori 1979;Das et al. 2019). The use of M w scale for ≥ 3.0 is only applicable for Southern California; however, each part of entire globe has different tectonic environment and geological setting; therefore, use of M w scale for ≥ 3.0 in entire globe will have adverse effects on Seismicity, Earthquake Hazard Assessment, Early Warning System, and other related seismological studies.
Therefore, a uniform earthquake catalog in terms of M wg applicable for lower, intermediate and higher magnitude ranges and globally valid, is critically important for any seismological or geophysical studies. In view of this, scaling relationships between magnitudes (m b /M s ) and M wg have been derived considering the entire world dataset. The m b magnitude data for 5,24,790 events from the ISC and 3,26,106 events from the NEIC, the M s magnitude data for 1,11,443 events from the ISC and 16,048 events from the NEIC, along with M wg values for 41,810 events from the GCMT during the period 01 January 1976-31 December 2014 have been considered.
In order to compare estimation techniques of different magnitudes from different agencies (e.g., ISC, NEIC and GCMT), GOR1, GOR2 and SLR relationships have been obtained. Maximum absolute difference between ISC and NEIC body wave magnitude estimations is found to be 0.12, 0.58, and 0.234 corresponding to GOR1 (Eq. 6), GOR2 (Eq. 7) and SLR (Eq. 8), respectively. It is observed that absolute average difference between observed m b,ISC and m b,NEIC differs by 0.16 m.u where as their average difference is 0.09 m.u. A similar bias between these has also been reported by Das et al. (2011) andUtsu (2002). Present analysis indicates that m b values obtained by ISC and NEIC are not equivalent as reported in earlier studies (e.g., Das et al. 2011). As surface wave magnitude estimations by ISC and NEIC use the same technique, so it is expected that the magnitude determinations from these databases should be more or less equivalent (Utsu 2002;Das and Wason 2010;Das et al. 2011). The equivalence between the two M s estimates (ISC &NEIC) has been verified for the magnitude range 2.8 ≤ M s,NEIC ≤ 8.8 through GOR1, GOR2 and SLR relations. Almost all these methods show M s estimates by ISC and NEIC are equivalent and could be treated them as unified dataset. Maximum absolute difference between ISC and NEIC for Surface wave magnitude estimations is found to be 0.31, 0.01, and 0.07 corresponding to GOR1 (Eq. 9), GOR2 (Eq. 10) and SLR (Eq. 11), respectively. It is found that absolute average difference between observed M S,ISC and M S,NEIC differs by 0.098 m.u where as their average difference is −0.003 m.u.
For conversion of m b,ISC to M wg , GOR1, GOR2 and SLR relationships have been derived using ISC and GCMT data for 36,767 events of magnitude range 4.5 ≤ m b,ISC ≤ 6.2 with η = 0.2. Furthermore, using three regression methodologies, slope and intercept parameters have also been obtained in the magnitude range 4.5 ≤ m b,ISC ≤ 6.2 for η ≤ 5.0 (Fig. 4). However, previous studies for m b,ISC to M w conversion were based on two methods: ISR (Inverted Standard Regression) and SLR relations between m b,ISC and M w (Das et al. 2011;Scordilis, 2006). Regression coefficients computed for m b,ISC to M wg are having lesser uncertainties in GOR1 compared to GOR2 and SLR. Correlation coefficient (Rxy) and standard deviation (RMSE) values for conversion of m b,ISC to M wg are found to be improved in case of GOR1 compared to .
Similarly, GOR1, GOR2 and SLR relationships between m b,NEIC and M wg in the magnitude range 4.5 ≤ m b,NEIC ≤ 6.2 were obtained using data for 20,863 events with η = 0.2 (Fig. 3). Equations 15-20 indicate that connection between m b and M wg values is comparatively better than the connection between m b and M w values, and the same is already proven in Das et al. (2019).
Maximum differences between observed m b,NEIC and estimated M wg values are found to be 0.1, 0.6 and 0.2 m.u corresponding to GOR1, GOR2 and SLR methods. The magnitude interval range (4.5 ≤ m b,ISC /m b,NEIC ≤ 6.2) adopted in this study is primarily based on the completeness of the dataset so that more reliable relationship could be developed Das et al. 2018b). However, one can extend the relationship < 4.5 for estimating indicative results.
For M s to M wg conversion, this study depicts a bilinear trend (Fig. 6a) as was also suggested by Wason et al. (2012) for M s to M w conversion. Therefore, the magnitude interval has been splitted into two parts: 3.1 ≤ M s ≤ 6.1 and 6.2 ≤ M s ≤ 8.4, considering the distribution is linear in these respective magnitude ranges. For conversion of M s magnitudes to M wg , GOR1, GOR2 and SLR relationships have been derived for magnitude ranges 3.1 ≤ M s ≤ 6.1 (using 19,826 events) and 6.2 ≤ M s ≤ 8.4 (using 1639 events). The observed M s values for magnitude ranges 3.1 ≤ M s ≤ 6.1 are found to be lesser than the estimated M wg by all three methods up to magnitude ≤ 5.5. The maximum observed difference between observed M s and estimated M wg by GOR1, GOR2 and SLR is found to be 0.7, 0.6 and 0.8, respectively, mainly in the lowest magnitude bin. However, these values are found to be 0.2, 0.18, and 0.23 for highest magnitude range, i.e., at magnitude 6.1. In the range of 6.2 ≤ M s ≤ 8.4, the observed M s magnitudes are found to be lesser than the estimated M wg by all three methods up to magnitude ≤ 7.4. The maximum observed differences between observed M s and estimated M wg by GOR1, GOR2 and SLR procedures are found to be 0.19, 0.25 and 0.18, respectively, mainly in the lowest magnitude range. However, these values are found to be 0.03, 0.23, and 0.13 for the highest magnitude range, i.e., at magnitude 8.4. In order to know the regression parameters (e.g., slope and intercept) for different η values of two different types of regression relationships, a graph is presented in Fig. 7. A drastic variation in slope is observed up to η = 0.8 in case of GOR2 as compared to GOR1. However, the slope remains same for SLR method because there is no used of η for the SLR relation.
In statistical science, superiority of the regression models is generally based on the degree of uncertainty in the regression coefficients (i.e., slope, intercepts) and values of standard deviation (RMSE) and correlation coefficients (Rxy). For developing relationships between m b or M s and M wg , GOR1 yields, in general, lesser errors in slope and intercept compared to GOR2 and SLR and provides better correlation coefficient and standard error values as compared to the other two procedures. It is important to note that GOR estimation requires η values and its (η) determination is not always appropriate, so there is a high possibility that the η value contains uncertainty. Since the GOR2 line has a higher steep than GOR1 (Figs. 4, 7), therefore, GOR2 is more sensitive with respect to η values . Furthermore, it is also observed that, in general, the estimates given by the GOR1 procedure lie in between the estimates of GOR2 and SLR (Figs. 5, 6) for m b and M wg , M s and M wg conversions.
The regression relationships developed in this study in terms of M wg based on global data are beneficial for preparing unified earthquake catalogs for any earthquake prone regime in the absence of local/regional regression relationships, as the earthquake catalog for most seismic prone areas are not homogenous in magnitude types. The GOR1 relationships developed in this study with smaller uncertainties compared to GOR2 and SLR are preferred for conversions as they transmit lower level of uncertainty in the seismic hazard assessment and seismicity studies.