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GEOMATICS, NATURAL HAZARDS AND RISK 2021, VOL. 12, NO. 1, 167–180 https://doi.org/10.1080/19475705.2020.1863269 A unified moment magnitude earthquake catalog for Northeast India a,b a Ranjit Das and Claudio Meneses Department of Computing & Systems Engineering, Universidad Cat olica del Norte, Antofagasta, Chile; National Research Center for Integrated Natural Disaster Management, Regi on Metropolitana, Chile ABSTRACT ARTICLE HISTORY Received 18 June 2019 Earthquake-related studies on seismicity and seismic hazard Accepted 8 December 2020 assessment need a homogenous earthquake catalog for the region studied. A homogenous earthquake catalog for Northeast KEYWORDS India region was compiled using derived regional and global Homogenous earthquake empirical relationships between different magnitudes and catalog; seismic moment; moment magnitude based on an improved error-corrected meth- Northeast India; generalized odology suggested in the recent literature. To convert smaller orthogonal regres- magnitude earthquakes, global empirical equations were derived sion; regression and used. A procedure is suggested to change different magni- tudes into moment magnitude. A homogenous earthquake cata- log of 9845 events was compiled for the time period 1897–2012. Entire magnitude range (EMR) was found to be the most reason- able method for estimating magnitude of completeness. Derived local and global empirical equations are useful for every seismic hazard or seismicity study. A complete and consistent homogen- ized earthquake catalog prepared in this study could provide good data for studying earthquake distribution in Northeast India. By carefully converting these original magnitudes into homogen- ized M magnitudes, an obstacle is removed for the consistent assessment of seismic hazards in Northeast India. Introduction Northeast India is one of the most seismically active regions on a world basis, lying within the geographical coordinate 20 –30 lat. and 87 –98 long. Two great earth- quakes, namely, Shillong earthquake in 12 June 1897 and Assam earthquake in 15 August 1950 occurred in this region a short time ago. Frequent moderate to inter- mediate magnitude earthquakes occur in the region. To properly understand an earthquake phenomenon in a region, a complete and consistent earthquake catalog is essential. In general, earthquake catalogs for regional CONTACT Ranjit Das ranjit244614@gmail.com Supplemental data for this article is available online at https://doi.org/10.1080/19475705.2020.1863269. 2021 The Author(s). Published by Informa UK Limited, trading as Taylor & Francis Group. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/ licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 168 R. DAS AND C. MENESES Table 1. Observational errors for body, surface and moment magnitudes. Errors Different magnitudes (standard deviations) Comments Body wave magnitudes (m ) 0.2 Error is estimated using the same events reported by ISC and NEIC Surface wave magnitudes 0.12 Error is estimated using the same events (M ) reported by ISC and NEIC Moment magnitude 0.09 Error is estimated using the same events (M ) reported by GCMT and NEIC seismicity are heterogeneous in magnitude types, whereas a homogeneous and com- plete earthquake catalog is a basic requirement for studying earthquake distribution in a region, as a function of space, time and magnitude. Because of the inherent com- plex nature of earthquake phenomena and variations in instrumental characteristics, network coverage and observational practice, different definitions and methodologies have been devised for determining the earthquake size that leads to different magni- tude scales. Due to use of different magnitude scales, an earthquake database becomes inhomogeneous in terms of earthquake size. So, in order to build a homogeneous earthquake catalog for a seismic region, the regression relationships used for changing different magnitude types into a preferred magnitude scale (i.e. to moment magnitude M as it does not saturate at higher magnitude levels) are critically important since bias introduced during the conversion propagates errors in the frequency magnitude distribution parameters and consequently in the seismic hazard estimates. Most of the regression relationships used for magnitude conversion is based on the assumption that one of the magnitudes (independent variable) is error-free. When both the mag- nitude types contain measurement errors and the use of the standard least-squares regression procedure leads to systemic errors as high as 0.3–0.4 (Castellaro et al. 2006) in magnitude conversion, it is inadequate and, even more important, it may result in catalog incompleteness. General orthogonal regression (GOR) analysis is more appropriate to estimate regression relationships between different magnitude types (Thingbaijam et al. 2008; Ristau 2009; Das et al. 2012a, 2012b, 2013, 2014a, 2014b, 2018a, 2018b). However, it is well addressed in different studies on GOR pro- cedure usability to obtain an unbiased estimate of the dependent variable (Das et al. 2012a, 2012b, 2013, 2014a, 2014b, 2018a, 2018b; Wason et al. 2012). In regards to the homogenization of an earthquake catalog for Northeast India using regression relationships, several authors have worked on empirical relationships between different magnitudes and moment magnitude scale (Thingbaijam et al. 2008, Yadav et al. 2009; Das et al. 2012a, 2012b; Anbazhagan and Balakumar 2019;Nath et al. 2017; Pandey et al. 2017). Thingbaijam et al. (2008) derived GOR for the conversion of body and surface wave magnitudes into moment magnitude, finding significant dispersion in the conversion of m into M . In a study conducted for this region, Yadav et al. b,ISC w (2009) used the standard regression technique for the conversion of m and M into M . b s w Das et al. (2012a, 2012b) derived regression relationships for m to M and M to M b w s w using standard regression, GOR and inverted standard regression, considering the error variance value (g) as 0.36 and 1, respectively. Considering the global earthquake database, these values are further modified as g ¼ 0.2 and 0.56 in a separate study by Das et al. (2014a, 2014b). Errors of different magnitude scales are shown in Table 1. GEOMATICS, NATURAL HAZARDS AND RISK 169 Table 2. Regression relations developed for Northeast India region and entire Globe. Magnitude Slope Intercept Slope Intercept Slope Intercept Regression relation Type range (GOR1) (GOR) (GOR2) (GOR2) SLR SLR m , to M g ¼ 0.2 Regional 4.8 m , 6.1 1.084 0.3106 1.448 2.223 1.0497 0.1282 b ISC w b ISC m , to M g ¼ 0.2 4.8 m , 6.1 1.104 0.495 1.513 2.653 1.0686 0.3061 b NEIC w b NEIC M to M g ¼ 0.56 4.1 M 6.1 0.615 2.32 0.645 2.168 0.5788 2.5136 s,ISC w s,ISC M to M g ¼ 0.56 4.2 M 6.1 0.699 1.878 0.732 1.713 0.6621 2.0634 s,ISC w s,NEIC m , to M g ¼ 0.2 Global 2.9 m , 6.1 0.828 1.167 1.457 2.11 0.7702 1.4698 b ISC w b ISC m , to M g ¼ 0.2 3.8 m , 6.1 0.895 0.729 1.604 3.001 0.8404 1.0194 b NEIC w b NEIC M to M g ¼ 0.56 3.0 M 6.1 0.637 2.241 0.669 2.082 0.5945 2.455 s,ISC w, s,ISC M to M g ¼ 0.56 3.6 M 6.1 0.675 2.037 0.718 1.821 0.6258 2.289 s,ISC w s,NEIC M to M g ¼ 0.56 6.2 M 8.4 0.973 0.196 1.08 0.501 0.9186 0.5588 s W s Intensity to M g ¼15–12 0.389 4.029 0.394 3.997 0.3615 4.244 Local magnitude g ¼ 1 5.0 M 6.6 1.193 0.943 1.318 1.617 1.120 0.555 Duration magnitude g ¼ 1 4.2 M 6.8 0.742 1.565 0.864 0.949 0.578 2.391 In this study, an improved GOR methodology was used, as suggested by Das et al. (2018b), for deriving regression relationships to change body and surface wave magni- tudes into moment magnitudes on a regional and global basis. A dataset of 9845 earth- quake events was used in terms of M in the magnitude range 1.6–8.7 belonging to the region studied (lat. 20 –30 and long. 87 –98 ) for the period 1897–2012. GOR regres- sion relationships using regional and global datasets for the conversion of m and M b s into M were derived. In the compiled earthquake catalog for Northeast India, most events occur in a lower magnitude range (m and M ), moment magnitude values being b s available only for a few events. In view of the scarce moment magnitude data in the lower magnitude side of Northeast India, global seismicity-based GOR relationships were used for the conversion of m and M into M . The homogenous catalog will remain b s w open for complementing it in the near future with advanced seismic moment scale M wg (Das Magnitude scale, Das et al. 2019) or energy magnitude (M ), and/or other related earthquake size measuring scales which allow a better characterization of the rupture mode and released seismic energy. The homogenous catalog will not only serve as a sound database for seismic risk assessment interests, but also for many other purposes. Following the earthquake catalog unification, declustering was also conducted, its com- pleteness being assessed in the next section. GOR methodology GOR methodology based on the minimization of Euclidean distance between the given observed points and the corresponding points on the GOR line (Madansky 1959; Kendall and Stuart 1979; Fuller 1987; Carrol and Ruppert 1996; Das et al. 2012a, 2012b, 2014a, 2014b, 2018a, 2018b; Karimiparidari et al. 2013; Wason et al. 2012; Goitom et al. 2017) was used. A critical detail of the GOR procedure is explained in Appendix. Regional regression relationships For the conversion of m , and m into M in the magnitude range b ISC b,NEIC w,GCMT 4.8 m m 6.1, the regression relationships were derived using newly b,ISC/ b,NEIC developed GOR with g ¼ 0.2 (Das et al. 2011), based on a dataset of 116 and 106 170 R. DAS AND C. MENESES Figure 1. Regional relations: (a) m , jM (b) m jM (c) M jM ,(d) M jM , Global rela- b ISC w, b,NEIC w, s,ISC w s,NEIC s,ISC tions: (e) m , jM (f) m jM (g) M jM ,(h) M jM , (i) M jM in the magnitude range b ISC w, b,NEIC w, s,ISC w s,NEIC s,ISC s w 6.2 M 8.4, (j) intensityjM , (k) local magnitude M jM and (l) duration magnitude M jM . s w L w D w GEOMATICS, NATURAL HAZARDS AND RISK 171 Figure 1. Continued. events for the period 1976–2007, respectively. The regression parameters obtained for these conversion relationships are shown in Table 2 and the plots are shown in Figure 1(a and b), respectively. 172 R. DAS AND C. MENESES Figure 2. Magnitude dependent space and time windows used for removal of aftershocks (dependents events). The asterisks below the window lines are considered to be dependent events. In addition, the regression relationships between M and M using 93 s,ISC w,GCMT events in the range 4.1 M 6.1, and between M and M using 57 s,ISC s,NEIC w,GCMT events in the range 4.2 M 6.1, were derived following the newly developed s,NEIC GOR procedure. The regression parameters for the corresponding regression relation- ships are shown in Table 2 and the plots are shown in Figure 1(c and d). Global regression relationships For the conversion of m , to M into the magnitude range 2.9 m , 6.1, b ISC w,GCMT b ISC and m to M in the magnitude range 3.8 m , 6.1, we derived the b,NEIC w,GCMT b NEIC GOR relationships with g ¼ 0.2 based on a dataset of 22,803 and 22,340 events, respectively, for the period 1976–2006. The regression parameters obtained for these conversion relationships are shown in Table 2 and the plots are shown in Figure 1(e and f). For the GOR relationship between M and M , 15,728 events were used s,ISC w,GCMT in the ranges 3.0 M 6.1. Similarly, 7579 events were used in the magnitude s,ISC range 3.6 M 6.1 for the GOR relationship between M and M s,NEIC s,NEIC s,GCMT. Furthermore, for the conversion of higher surface wave magnitudes, 2026 events were used, combining ISC and NEIC data into the magnitude range 6.2 M 8.4. The s GEOMATICS, NATURAL HAZARDS AND RISK 173 Table 3. Aftershock identification windows (Uhrhammer 1986) used in the present study. Magnitude 2.5 3 3.5 4 4.5 5 5.5 6 6.5 7 7.5 8 8.5 9 Distance (km) 2.7 4.0 6.0 9.0 13.4 20.0 29.9 44.7 66.8 99.9 149.3 223.2 333.6 498.7 Time (days) 1.2 22.6 26.1 30.1 34.7 40.0 46.1 53.2 61.3 70.7 81.6 94.1 108.5 125.1 regression parameters for the corresponding regression relationships are shown in Table 2 and the plots are shown in Figure 1(g–i), respectively. A magnitude–intensity relationship based on data from 29 earthquakes in India and nearby regions from 1897 to 2016, with independent MMI (I ) and moment magnitude (M ) identified from different sources, was developed as follows: M ¼ 0:389 I þ 4:029, w 0 where I is the maximum epicentral intensity (Figure 1(j)). Historical earthquakes with only intensity values were changed into M using the above empirical intensity relationship. Local magnitudes The relationship between local magnitude and moment magnitude, based on 100 earthquakes in Northeast India was derived for the time period 1976–2005. The derived GOR1 relationship with g ¼ 1 is given as follows: M ¼ 1:1926M 0:943: w L Duration magnitude The relationship between duration magnitude and moment magnitude, based on 376 global data earthquakes collected from ISC database, was derived as follows: M ¼ 0.742M þ 1.565451. w D Scheme for magnitude conversion into moment magnitude M The following scheme was followed for the conversion of different magnitude types into the unified moment magnitude M . (1) The unassigned magnitudes in the catalog by Gupta et al. (1986) are treated as M (Thingbaijam et al. 2008; Das et al. 2013). (2) When m and M magnitude types are reported for an event, magnitude type b s m 6.0 and above M are chosen. b s (3) In case of small m magnitude events, the appropriate global relationship is used in the absence of a regional regression relationship. (4) If only intensity data are available for an event, then M is obtained by using the relationship between intensity and M developed in this study. w 174 R. DAS AND C. MENESES Table 4. Magnitude of completeness by different methods. Methods M Maximum curvature 4.1 Fixed M ¼ minimum magnitude (min) 2.9 M 90 (90% probability) 4.5 Best combination (Mc95-Mc90- maximum curvature) 4.5 EMR method 4.2 M due b using Shi and bolt uncertainty 5 M due b using bootstrap uncertainty 4.1 M due b using Cao-Criteria 3.2 (5) Scordilis (2006) and Das et al. (2011) reported the equivalent of M and w,GCMT M . Therefore, in the absence of primary M , M values in the w,NEIC w,GCMT w,NEIC catalog are considered as almost identical proxies. Theunified catalog compiledherecan be obtained from theauthoron request. A sam- ple dataset of a homogenized catalog with 500 events is provided in the electronic version. Earthquake catalog declustering An earthquake sequence generally consists of foreshocks – main shocks – and after- shocks. The foreshocks and aftershocks, being dependent events, should be eliminated from the catalog to estimate seismic hazards. Several methods have been proposed for declustering a catalog (e.g. Gardner and Knopoff 1974; Reasenberg 1985; Uhrhammer 1986). Declustering was conducted by using Uhrhammer (1986), following a moving space and time window approach (Figure 2). The time and distance window is shown in Table 3. After declustering 3454 events removed from the catalog for the period 1897–2012, they were homogenized with GOR1 procedure. Determination of magnitude of completeness To find the completeness of the homogenized catalog after declustering, eight differ- ent methods were employed using Zmap software (Table 4). The methods most fre- quently used were the entire magnitude range method (EMR) (Ogata and Katsura 1993; modified by Woessner and Wiemer 2005), maximum curvature (MAXC) method (Wiemer and Wyss 2000), the goodness-of-fit test (GFT) (Wiemer and Wyss 2000), and M determination by b-value instability (Cao and Gao 2002). EMR method shows a reasonable value for the complete dataset, therefore, using EMR method for magnitude of completeness is better. Summary and conclusions This study aims to obtain a homogenized and complete earthquake catalog by devel- oping GOR conversion relationships, following an improved error-corrected method- ology from Das et al. (2018b). In this regard, regional regression relationships were derived for m to M from ISC and NEIC databases in the magnitude range b w 4.8 m m 6.1, using 116 and 106 events, respectively. Similarly, for the b,ISC/ b,NEIC conversion of M into M , regression relationships were derived for the magnitude s w GEOMATICS, NATURAL HAZARDS AND RISK 175 Figure 3. Annual recording earthquakes with magnitude 3.0 in the study region from three major global agencies: (a) GCMT, (b) ISC and (c) NEIC/USGS databases. ranges 4.1 M 6.1 and 4.2 M 6.1, using the dataset of 93 and 53 events s,ISC s,NEIC for ISC and NEIC, respectively. As regional relationships do not cover the smallest mag- nitude range and there is a big number of smaller magnitudes, global relationships were derived for the conversion of body and surface wave magnitudes into moment magni- tudes. In this regard, regression relationships were derived for the conversion of m b,ISC into M in the magnitude range 2.9 m , 6.1, and m to M in the w,GCMT b ISC b,NEIC w,GCMT magnitude range 3.8 m , 6.1. GOR relationships were derived with g ¼ 0.2 based b NEIC on a dataset of 22,803 and 22,340 events, respectively. For the regression relationship between M and M , data from 15,728 events s,ISC w,GCMT were used in therange3.0 M 6.1. Similarly, for the relationship between M s,ISC s,NEIC and M , 7579 events wereusedin the magnituderange3.6 M 6.1 In add- s,GCMT s,NEIC . ition, for the conversion of higher surface wave magnitudes, 2026 events, combining ISC and NEIC data in the magnitude range 6.2 M 8.4, were used. A complete and consistent homogenized earthquake catalog prepared using these relationships could provide good data for studying earthquake distribution in Northeast India. By carefully changing these original magnitudes into homogenized M magnitudes, an obstacle was removed for the consistent assessment of seismic hazards in Northeast India. Funding This research was supported by CORFO ING203016EN12-Grant, FONDECYT Grant 11200618 and CONICYT/FONDAP grant 15100017. Data availability statement Body and surface wave magnitudes of earthquake events for the entire world from ISC (International Seismological Center) U.K. (http://www.isc.ac.uk/search/Bulletin) (last accessed August 2012) and moment magnitudes from GCMT (Global Centroid Moment Tensor data- base, http://www.globalcmt.org/CMTsearch.html) (last accessed October 2012). For the region studied, data from 9845 earthquake events for the period 1897–2012 were compiled from various databases (e.g. ISC, NEIC, GCMT, IMD, NEIST). For the historical seismicity from 1897 to 1962, data were taken from the catalog by Gupta et al. (1986). For the 176 R. DAS AND C. MENESES period 1964–2010, data were compiled from global ISC and NEIC databases. Data for 1963 were taken from International Seismological Summary (ISS). In addition, GCMT and NEIC moment magnitude data were considered for the periods 1978–2012 and 1964–2012, respect- ively. The number of annual earthquakes recorded by ISC, GCMT and NEIC for Northeast India are shown in Figure 3. Some events for the period 1999–2006 from India Meteorological Department (IMD) seismological bulletins, New Delhi, and the catalog by Bapat et al. (1983) were also considered. In addition, data from global events were considered to develop regres- sion relationships to obtain the conversion of lower magnitude ranges not covered by regional conversion relationships. A sample of 500 homogenized events is shown in Electronic Supplement (Table 6). Disclosure statement No potential conflict of interest was reported by the authors. Acknowledgement This research was supported by CORFO ING203016EN12-Grant, FONDECYT Grant 11200618 and CONICYT/FONDAP grant 15100017. We also thank the Editors and the Reviewers for constructive comments. References Anbazhagan A, Balakumar A. 2019. Seismic magnitude conversion and its effect on seismic hazard analysis. J Seismolog. 23(4): 623–647. Bapat A, Kulkarni RC, Guha SK. 1983. Catalogue of Earthquakes in India and Neighborhood from Historical Period upto 1979 Roorkee, India: Indian Society of Earthquake Technology. Cao AM, Gao SS. 2002. Temporal variation of seismic b-values beneath northeastern Japan island arc. Geophys Res Lett. 29(9):48–1–48-3. Carrol RJ, Ruppert D. 1996. The use and misuse of orthogonal regression in linear errors-in- variables models. Am Stat. 95(1):58–74. Castellaro S, Bormann P. 2007. Performance of different regression procedures on the magni- tude conversion problem. Bull Seism Soc Am. 97(4):1167–1175. Castellaro S, Mulargia F, Kagan YY. 2006. Regression problems for magnitudes. Geophys J Int. 165(3):913–930. Das R, Wason HR, Sharma ML. 2011. Global regression relations for conversion of surface wave and body wave magnitudes to moment magnitude. Nat Hazards. 59(2):801–810. Das R, Wason HR, Sharma ML. 2012a. Magnitude conversion to unified moment magnitude using orthogonal regression relation. J Asian Earth Sci. 50:44–51. Das R, Wason HR, Sharma ML. 2012b. Homogenization of earthquake catalog for northeast India and adjoining region. Pure Appl Geophys. 169(4):725–731. Das R, Wason HR, Sharma ML. 2013. General orthogonal regression relations between body wave and moment magnitudes. Seismol. Res. Lett. 84: 219–224.. Das R, Wason HR, Sharma ML. 2014a. Reply to ‘Comment on “Magnitude conversion prob- lem using general orthogonal regression” by HR Wason, Ranjit Das and ML Sharma’ by Paolo Gasperini and Barbara Lolli. Geophys J Int. 196(1):628–631. Das R, Wason HR, Sharma ML. 2014b. Unbiased estimation of moment magnitude from body-and surface-wave magnitudes. Bull Seismol Soc Am. 104(4):1802–1811. Das R, Sharma ML, Wason HR, Gonzalez G, Chodhury D. 2019. A seismic moment magni- tude scale. Bull Seismol Soc Am. 109(4):1542–1555. Das R, Wason HR, Sharma ML. 2018a. Reply to “Comments on ‘Unbiased estimation of moment magnitude from body-and surface-wave magnitudes’ by Ranjit Das, H. R.Wason GEOMATICS, NATURAL HAZARDS AND RISK 177 and M. L.Sharma and ‘Comparative analysis of regression methods used for seismic magni- tude conversions’ by P. Gasperini, B. Lolli, and S. Castellaro” by Pujol. Bull Seismol Soc Am. 108(1):540–547. Das R, Wason HR, Gonzalez G, Sharma ML, Choudhury D, Lindholm C, Roy N, Salazar P. 2018b. Earthquake magnitude conversion problem. Bull Seismol Soc Am. 108(4):1995–2007. Fuller WA. 1987. Measurement error models. New York: Wiley. Gardner JK, Knopoff L. 1974. Is the sequence of earthquakes in southern California, with aftershocks removed, Poissonian? Bull Seismol Soc Am. 64(5):1363–1367. Goitom B, Werner MJ, Goda K, Kendall JM, Hammond JOS, Ogubazghi G, Oppenheimer C, Helmstetter A, Keir D, Kemp FI. 2017. Probabilistic seismic-hazard assessment for Eritrea. Bull Seismol Soc Am. 107(3). Gupta HK, Rajendran K, Singh HN. 1986. Seismicity of Northeast India region: PART I: the database. J Geol Soc India. 28:345–365. Karimiparidari S, Zare M, Memarian H, Kijko A. 2013. Iranian earthquakes, a uniform cata- logue with moment magnitudes. J Seismol. 17(3):897–911. Kendall MG, Stuart A. 1979. The advanced theory of statistics. Vol. 2, 4th ed. London: Griffin. Madansky A. 1959. The fitting of straight lines when both variables are subject to error. Am Statist Assoc J. 54(285):173–205. Nath SK, Mandal S, Adhikari MD, Maiti SK. 2017. A unified earthquake catalogue for South Asia covering the period 1900–2014. Nat Hazards. 85(3):1787–1810. Ogata Y, Katsura K. 1993. Analysis of temporal and spatial heterogeneity of magnitude fre- quency distribution inferred from earthquake catalogs. Geophys J Int. 113(3):727–738. Pandey AK, Chingtham P, Roy PNS. 2017. Homogeneous earthquake catalogue for Northeast region of India using robust statistical approaches. Geomatics Nat Hazards Risk. 8(2):1477–1491. Reasenberg P. 1985. Second-order moment of central california seismicity, 1969–1982. J Geophys Res. 90(B7):5479–5495. Ristau J. 2009. Comparison of magnitude estimates for New Zealand Earthquakes: moment magnitude, local magnitude, and teleseismic body-wave magnitude. Bull Seism Soc Am. 99(3):1841–1852. Scordilis EM. 2006. Empirical global relations converting MS and mb to moment magnitudes. J Seismol. 10(2):225–236. Thingbaijam KKS, Nath SK, Yadav A, Raj A, Walling MY, Mohanty WK. 2008. Recent seismi- city in Northeast India and its adjoining region. J Seismol. 12(1):107–123. Uhrhammer RA. 1986. Characteristics of northern and central California seismicity. Earthquake Notes. 57(1):21. (abstract). Wason HR, Das R, Sharma ML. 2012. Magnitude conversion problem using general orthog- onal regression. Geophys J Int. 190(2):1091–1096. Wiemer S, Wyss M. 2000. Minimum magnitude of complete reporting in earthquake catalogs: exam- ples from Alaska, the western United States, and Japan. Bull Seismol Soc Am. 90(4):859–869. Williamson J. 1968. Least-square fitting of a straight line. Can J Phys. 46(16):1845–1847. Woessner J, Wiemer S. 2005. Assessing the quality of earthquake catalogues: estimating the magnitude of completeness and its uncertainty. Bull Seismol Soc Am. 95(2):684–698. Yadav RBS, Bormann P, Rastogi BK, Das MC, Chopra S. 2009. A homogeneous and complete earthquake catalog for Northeast India and the adjoining region. Seismol Res Lett. 80(4): 609–627. Appendix Let us consider x and y as the true values and X and Y as the observed values of the inde- t t t t pendent and dependent variables, respectively. We further assume that X and Y have meas- t t urement errors of d and e as measurement errors, respectively. Therefore, we can write, X ¼ x þ d, (A1) t t 178 R. DAS AND C. MENESES Figure 4. Plots showing the theoretical true points (x , y ) corresponding given observed value (X , t t t Y ) using (a) SLR; (b) GOR. Plots also show the deviations (in case of SLR, e.g., y -y , and in case of t 1 1 GOR, e:g:, y -Y ) between theoretical true values y and estimated dependent variables on direct 1 t substitution of X in Equation (3). Plot also explain the Euclidean distance used during the deriv- ation of GOR does not maintain in the estimation (Das et al. 2018a). Y ¼ y þ e, (A2) t t and the regression model is Y ¼ a þ bx þ e, (A3) t t where b and a are the slope and intercept of linear relationship, respectively. 2 2 If, s , s and s are the sample covariances of Y , X and between Y , and X , then X Y t t t t Y X t t t t 2 2 2 s ¼ b r ^ þ gr ^ , (A4) Y x t t d 2 2 2 s ¼ r ^ þ r ^ , (A6) X x t t d s ¼ br ^ , (A5) X Y x t t t where the error variance ratio g ¼ (A7) From the above simultaneous Equations (A4)–(A6), we obtain qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2 2 2 2 2 s gs þ ðs gs Þ þ 4gs Y X Y X X Y t t t t t t b ¼ : (A8) 2s X Y t t XtYt GEOMATICS, NATURAL HAZARDS AND RISK 179 Table 5. Comparison of statistical Euclidean distances given by Fuller (1987), Williamson (1968) and Castellaro et al. (2006) using Fuller example 1.2.3. Y X S S S t t W F C 86 70 2.33804 2.33804 133.2682416 115 97 0.763141 0.763141 43.49904212 90 53 2.02E 06 2.02E 06 0.000115239 86 64 1.396907 1.396907 79.623681 110 95 0.093455 0.093455 5.326917235 91 64 0.248316 0.248316 14.15401839 99 50 1.975349 1.975349 112.5948638 The estimator for a can be obtained from the relation ^ a ¼ Y bX , (A9) t t where X and Y are the average observed values. t t GOR estimations are inappropriately used in most of the seismic literature, therefore, it is important to discuss the issues to provide a complete view on this subject. The discussions will be conducted using two cases (Das et al. 2018a, 2018b): Case 1. A Standard Linear Regression (SLR) line is derived considering data pairs (X , Y ), 1 1 (X , Y ) and (X , Y ), and the corresponding true points, that is, (x ¼ X , Y ), (x ¼ X , Y ) 2 2 3 3 1 1 1 2 2 2 and (x ¼ X , Y ), respectively, on the SLR fitting line, using vertical distance minimization. 2 3 3 These true points on the line are used for estimating the best fitting SLR line, minimizing ver- tical residuals. After substituting the independent variables X , X , X in the SLR line obtained, 1 2 3 the exact true points can be obtained (see Figure 4(a)). Thus, statistical distance, which is used in the derivation of SLR line, is also used in the estimation of the dependent variable for a given independent variable. Case 2. A GOR fitting line is derived considering the data pairs (X , Y ), (X , Y ) and (X , 1 1 2 2 3 Y ) with errors in both variables. The true points of these data pairs ((X , Y ), (X , Y ) and 3 1 1 2 2 (X , Y )) on the GOR line, that is, (x , y ), (x , y ) and (x , y ) are given by minimizing 3 3 1 1 2 2 3 3 Euclidean distance. In substituting X , X , X values in the GOR line obtained, the correspond- 1 2 3 ing true points cannot be obtained, unlike Case 1. Therefore, the conventional GOR procedure does not follow the statistical Euclidean distance criteria. In substituting, instead of obtaining the true points, totally different points on the GOR line are achieved (see Figure 4(b)). In view of the above discussion of Cases 1 and 2, the author recommended estimating firstly the true value of x through SLR relationship between true abscissa (x ) on GOR line i t and X ., and then obtain Y estimate by substituting an estimated x in the GOR relationship t t t or directly derive SLR relationship between X and true ordinate (y ). t t It is observed that Castellaro et al. (2006) and Castellaro and Bormann (2007) inappropri- ately used the Euclidean distance and derived the GOR line. Thus, their conclusions are invalid for GOR conventional type. Therefore, it is incorrect to use these studies for future references. To point out inaccuracy, Fuller (1987) Euclidean distance is denoted by S , Williamson (1968), Euclidean distance is denoted by S and Castellaro et al. (2006), and distance is denoted by S w C n 2 ðÞ Y b b X t t 0 1 S ¼ , (A10) r 2b r þ b r 1 1 e t¼1 d de 0 1 ! ! n 2 2 X x b x þ b Y ðÞ ðÞ t t t t 1 0 @ A S ¼ þ , (A11) r r d e t¼1 180 R. DAS AND C. MENESES ! ! n 2 ðÞ Y b x b t i 2 1 0 S ¼ ðÞ X x þ : (A12) i i i¼1 Table 5 shows that S and S are same, but S is quite different because Castellaro et al. F w C (2006) and Castellaro and Bormann (2007) used incorrect equations (details are given in Das et al. 2018a).
"Geomatics, Natural Hazards and Risk" – Taylor & Francis
Published: Jan 1, 2021
Keywords: Homogenous earthquake catalog; seismic moment; Northeast India; generalized orthogonal regression; regression
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