1 Introduction

The earthquake magnitude is regarded as the most directly measurable and simple parameter to specify quantitatively the size of an earthquake. The Richter local magnitude M L scale (Richter 1935) for an earthquake is still widely used in different parts of the world. Following Richter, multitude of magnitude scales, each defined in terms of amplitudes recorded on a particular type of seismograph (i.e., over a particular limited spectral band) have been introduced.

From the study of source mechanism by an elastic dislocation theory, Aki (1966, 1967) stated that the amplitude of a very long-period wave is proportional to the seismic moment, M 0, of an earthquake. Aki (1966) first measured the value of M 0 of the 1964 Niigata, Japan, earthquake. Ben-Menahem et al. (1969) also suggested that the far-field static-strain field is proportional to M 0. From then on, seismic moment was considered as a new parameter to specify the size of an earthquake. Based on M 0, moment magnitude M W has been defined by Hanks and Kanamori (1979). The moment magnitude scale (M W), as defined by Kanamori (1977), has an advantage of not getting saturated for larger earthquakes, unlike the Richter amplitude-based scales (e.g., Hanks and Kanamori 1979; Howell 1981; Ottemoller and Havskov 2003). This enables the wide acceptance of M W as a stable scale, for larger as well as small to moderate magnitude earthquakes (Lay and Wallace 1995).

The magnitude by definition quantifies the energy radiated over a particular fixed frequency band, M 0 is estimated seismically from the amplitude of far-field “long-period” seismic radiation (Aki 1966). Because the frequency distribution of radiated seismic energy changes with earthquake size (e.g., Aki 1967), magnitude scales suffer severe intrinsic limitations, such as saturation (Kanamori 1977; Hanks and Kanamori 1979; Hutton and Boore 1987) and discrepancies between the scales (Gutenberg and Richter 1956). Since the moment can be estimated from the recording of all suitable seismographs, the only limitation for M 0 is the difficulty in locating and properly processing on scale recording from seismograph with adequate long-period response.

Estimating the duration magnitude (M D) from signal duration is an empirical relation between the magnitudes of the earthquake and the duration of its recorded signal. This correlation has been consistently observed in different parts of the world. Lee et al. (1972) established an empirical formula for estimating the magnitude of local earthquakes using signal durations. A partially satisfactory theoretical basis for this correlation has been sought in terms of the properties of coda waves (Suteau and Whitcomb 1979; Lee and Stewart 1981). Castello et al. (2007), Sitaram and Bora (2007), and Al-Arifi and Al-Humidan (2012) derived an empirical relationship between the total signal duration of local earthquakes and magnitude for the area in Italy, NW Saudi Arabia, and northeast India, respectively.

Conversion of magnitude scales for small earthquakes plays an important role for seismic hazard calculations. Several studies have been made to examine region-specific relation between M W and M L (Ristau et al. 2003). Although the size of the events occurring in northeast India region is designated by M L in The India Meteorological Department (IMD) catalog, its consistency with moment magnitude (M W) is not examined or reported. Sitaram and Bora (2007) reported an empirical relation between M L and M D for northeast India and they observed that M L starts to saturate above magnitude 6.5. Recently Baruah et al. (2012) examined relationship between M W and M L for the northeastern region of India and they observed with a magnitude discrepancy 0.04 units.

In this paper, we aim to improve the scaling between M W and M L , M W and M D, M 0 and M L, and M 0 and M D for 162 earthquakes in Shillong-Mikir plateau and its adjoining region of northeast India by extending the M W estimates to lower magnitudes using the spectral analysis of P-waves from vertical component seismograms. We attempt to obtain the frequency-independent, long-period spectral level Ω 0 below the corner frequency of a displacement spectrum (Brune 1970, 1971). The long-period spectral amplitude of the source spectrum can be related to M 0 (Brune 1970), from which M W can be computed (Hanks and Kanamori 1979).

2 Tectonic setting

The northeastern region (NER) (see Fig. 1) of India is one of the most complex tectonic domain in the world which is manifested by the ongoing India-Asia collision to the north and Indo-Burmese subduction to the east (Bilham and England 2001; Kayal et al. 2012). During the last ~118 years since 1897, the region has experienced 20 large (M ≥ 7.0) and two great earthquakes (M ≥ 8.5); one on 12 June, 1897 (Oldham 1899) and the other on 15 August, 1950 (Tandon 1954). It may be mentioned that the 1897 great earthquake is the first instrumentally recorded event in India.

Fig. 1
figure 1

Map showing the major tectonic features of the study region (modified from Kayal et al. 2012). The great earthquake of 12 June, 1897 and 15 August, 1950 is shown by a larger red star. The digital broadband seismic stations are shown by the red triangles. The major tectonic features in the region are indicated: Main Central Thrust, Main Boundary Thrust, Kopili Fault, Dauki Fault, Dh F Dudhnoi fault, DT Dapsi thrust, OF Oldham fault, CF Chedrang fault, BS Barapani shear zone. Inset Map of India indicating the study region

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The NER, India comprises distinct geological units, like the Himalayan frontal arc to the north, the highly folded Indo-Myanmar mountain ranges or Burmese arc to the east, the Brahmaputra river alluvium in the Assam valley and the Shillong-Mikir plateau is sandwiched between these two arcs, and thick sediments of the Bengal basin to the south. Seismotectonics of the region has been the subject of several studies (e.g., Tapponnier et al. 1982; Kayal and De 1991; Kayal 2001, 2008; Nandy 2001). Bilham and England (2001), on the basis of geodetic and GPS data, interpreted the Shillong plateau as a pop-up structure bounded by two reverse faults and argued that the 1897 great earthquake was produced by a south dipping hidden fault at the northern boundary of the Shillong plateau, which they named Oldham fault, that extends from a depth of about 9 km down to 45 km. They further suggested that the Shillong plateau earthquakes are caused by the deformation of this pop-up structure between the Dauki fault and the Oldham fault (Fig. 1). The northwest-southeast trending Kopili fault separates the Shillong plateau from its fragment part, Mikir Hills. The Assam valley is an ENE-WSW trending narrow valley, which lies between Shillong plateau and the eastern Himalayan tectonic domains. Recent seismicity and tectonics of the region have been reviewed by Kayal et al. (2008, 2012), Baruah et al. (2011), and Bora and Baruah (2012), Bora et al. (2013); Bora et al. (2014). Prominent geological units of the northeastern region of India, however, remain geophysically less studied due to inaccessibility of the terrain.

3 Database

In this study, we use 162 best located earthquakes in the period from 2001 to 2010 by a network of broadband seismic stations in the Shillong-Mikir Plateau and its adjoining region (Fig. 2). These stations are operated by different agencies/Universities viz. National Geophysical Research Institute (NGRI)—Hyderabad, North East Institute of Science and Technology (NEIST)—Jorhat, Indian Institute of Geomagnetism (IIG)—Mumbai, and Gauhati University (GU). The selected events have higher signal-to-noise ratio (SNR) and distinct direct P and S phases. All broadband seismic stations are operated both in continuous mode and trigger mode, and the seismograms are recorded at 100 samples per second. To avoid aliasing effect, low-pass filters with corner frequencies of 35 Hz were applied. The recorded seismograms have been corrected using instrumental response based on the electrodynamic constant, critical damping, natural frequency of seismometers, and bit weight of unit gain of each recording unit for all stations. Because all the stations are located on hard rock, the site effects are neglected. The station parameters are given in Table 1. The hypocentral parameters are located by the HYPOCENTER location program of Lienert et al. (1986) based on crustal velocity model of Bhattacharya et al. (2008) and Baruah et al. (2011). Uncertainties involved in the estimates of epicenters show that about 85 % of the events are located with an error of <2 km in depth and epicenter, and the error in origin times is of the order of 0.3 s. Duration magnitude (M D) of these events are estimated in the range 3.0–4.96 (Lee et al. 1972; Sitaram and Bora 2007). The focal depth of the events are in the range from 5 to 44 km.

Fig. 2
figure 2

Hypocentral distributions of 162 earthquakes used in this study. To the right, different magnitude ranges of the epicenters are defined. M DA is the average duration magnitude

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Table 1 Station parameters along with abbreviations
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4 Theoretical consideration

Most earthquake source theories predict a far-field displacement spectrum that is constant at low frequencies and inversely proportional to some power of frequency at high frequencies (Haskel 1964; Savage 1966; Aki 1967; Brune 1970; Molnar et al. 1973). From a body wave spectra two quantities are obtained, the long-period spectral level, Ω 0, and the corner frequency, f c.

In agreement with the widely used theoretical models of seismic sources, the far-field displacement spectrum Ω(f) can be described by a one-corner frequency model

$$\varOmega (f) = \frac{{\varOmega _{0} }}{{\left[ {1 + ({\raise0.7ex\hbox{$f$} \!\mathord{\left/ {\vphantom {f {f_{\text c} }}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{${f_{\text c} }$}})^{{\gamma n}} } \right]^{{{\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 \gamma }}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{$\gamma $}}}} }},$$
(1)

where Ω 0 is the low frequency spectral level, f c is the corner frequency, n is the high frequency spectral fall-off, and γ is a constant. If γ = 1, Eq. (1) is the spectral shape proposed by Brune (1970).

The seismic moment (M 0) is estimated from the low frequency level (Ω 0) through the relation (Keilis-Borok 1959)

$$M_{0} = 4\pi \rho {v}_{\text p}^{3} R\frac{{\varOmega_{0} }}{{R_{\theta \phi } }},$$
(2)

where ρ = 2700 kg/m3 is the average crustal density of the study area, “v P” is the average P-wave velocity as a function of depth (Bhattacharya et al. 2008; Baruah et al. 2011; Bora et al. 2014), R is the source-to-receiver distance and \(R_{\theta \phi }\) is accounted for the radiation pattern of P-waves. The root mean square averages of radiation pattern coefficient \(R_{\theta \phi }\) = 0.52 is used for all the events in accordance with Boore and Boatwright (1984). Since the data utilized correspond to small earthquakes, the solution for the fault plane may not be unique, either by the first impulses technique or by calculating the moment tensor. As a result the use of average value is adequate since the stations have good azimuthal coverage. An average seismic moment <M 0> was determined from the average of the logarithmic values obtained at different stations, as proposed by Archuleta et al. (1982), following the equation

$$<M_{0}>\,= {\text{anti}}\log \left[ {\frac{1}{N}\sum\limits_{i = 1}^{N} {\log M_{{0_{i} }} } } \right],$$
(3)

where N is the number of stations used, and M 0 is the seismic moment determined from Eq. (2) for the ith record.

The obtained seismic moments (M 0) were used to estimate the moment magnitude (M W) using Hanks and Kanamori (1979) relationship:

$$M_{\text{W}} = \left( {\frac{{\log M_{0} }}{1.5}} \right) - 6.03,$$
(4a)

where M 0 is in (N·m). Hanks and Kanamori (1979) showed that moment magnitude M W is equivalent to the local magnitude M L. M W is not generally considered as the primary magnitude in earthquake catalogs and knowing how other magnitudes relate to M W in a particular region is a topic of great interest because of seismic hazard estimation.

In order to obtain empirical relations for the determination of duration magnitude (M D) for a given station is usually given in the form, according to Lee and Stewart (1981),

$$M_{\text{D}} = C_{0} + C_{ 1} { \ln }\,D_{\text{s}} + C_{ 2} D + C_{ 3} h,$$
(4b)

where D s is signal duration in s, D is epicentral distance in km, h is focal depth in km and C 0, C 1, C 2, and C 3 are empirical constants. These constants usually are determined by correlating signal duration with Richter magnitude for a set of selected earthquakes taking epicentral distance and focal depth into account. Duration magnitude is computed for each station and the average of the station magnitudes is taken to be the network duration magnitude (M D A) which is basically used in this study.

An empirical relation between M L and M D is reported by Sitaram and Bora (2007) for NER India, which is given by

$$M_{\text{D}} = \, \left( { 1.0 5\pm 0.0 1} \right)M_{\text{L}} + \, \left( { - 0. 1 7\pm 0.0 5} \right).$$
(5)

Further the M L is computed using the Eq. (5). Although the size of the events occurring in northeast India region is designated by M L in the Shillong catalog of the India Meteorological Division (IMD), its consistency with moment magnitude (M W) is not examined or reported.

5 Data analysis

The first step of the data processing was the selection of time windows for which P-wave spectra has to be computed. The selected spectra were filtered between 1 and 20 Hz using a four-pole band-pass Butterworth filter after correcting them for their individual baseline effect. The Fast Fourier Transform (FFT) of the P-wave signal was calculated using the vertical component of the ground motion. The P-wave spectra at each window was computed on a 1.28 s long (128 samples) starting from the P-onset, in order to avoid contamination from the S-phase. The time windows were tapered with a 10 % cosine taper at the beginning and end of the series to minimize the spectral energy leakage. Moreover, to avoid near-field effects and to fulfill the point-source approximation, only the stations having epicentral distances greater than 10 km were taken into account (Bora et al. 2013).

The spectral parameters are estimated by fitting every P-wave displacement spectrum with the theoretical general model for displacement spectra as expressed by Eq. (1). To avoid potential biases due to inspection by eye, an iterative non-linear best fitting search algorithm is used to minimize the difference between the theoretical and observed displacement spectra. The waveform analysis tool in seismology seismographer by Abdelwahed (2011) is used to estimate the spectral parameters. In this method of spectral fitting the marquard linearized least-square method is used to solve the non-linear problems (Press 1989). The method is able to fit the spectra efficiently without a trade off between the corner frequency and flat part. To secure the stability of our estimations, we performed different tests on a sample of spectra by varying the starting values of Ω 0, f c, and n. It has been assumed that the amplitude source spectrum has the shape of Brune’s model (1970). The best fits to the observed displacement spectra to the theoretical one at a number of stations are plotted in Figs. (3, 4, 5).

Fig. 3
figure 3

a Example of vertical component instrument corrected seismogram recorded at JPA station. b P-onset (128 samples). c Blue line indicates the observed spectrum and the red-dashed line is its best fitting.

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Fig. 4
figure 4

a Example of vertical component instrument corrected seismogram recorded at SHL station. b P-onset (128 samples). c Blue line indicates the observed spectrum and the red-dashed line is its best fitting

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Fig. 5
figure 5

a Example of vertical component instrument corrected seismogram recorded at TZR station. b P-onset (128 samples). c Blue line indicates the observed spectrum and the red-dashed line is its best fitting

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6 Scaling relationships

In this study, the seismic moment (M 0) of 162 events are estimated for small to moderate earthquakes (3.0 ≤ M D ≤ 5.0) originated in the Shillong-Mikir plateau and its adjoining region. The seismic moment ranges from 1.90 × 1013 to 2.00 × 1016 N m. The estimated seismic moment from this study was used to estimate the moment magnitude (M W) using Eq. (4a). These results are summarized in Table 2. Hanks and Kanamori (1979) showed that the moment magnitude M W is equivalent to the local magnitude M L. In this study, the M L is estimated using Eq. (5) and also the M L value obtained from Shillong catalog of the IMD. We perform linear regressions for the scalar seismic moment M 0 versus M L and M D, respectively. The seismic moment M 0 versus M L and M D plots are shown in Fig. 6; the fits are well constrained by the regression line:

$$\begin{aligned} & {\text{log }}\left( {M_{0} } \right) = \left( 1.46 \pm 0.07 \right)\,M_{{\text{L}}} + \left( 9.32 \pm 0.22 \right) \hfill \\ &r^{2} = 0.76\,{\text{and}}\,\sigma = 0.25, \hfill \\ \end{aligned}$$
(6)
$$\begin{aligned} &{\text{log }}\left( {M_{0} } \right) = \left( 1.39 \pm 0.06 \right)\,M_{{\text{D}}} + \left( 9.54 \pm 0.21 \right) \hfill \\ &r^{2} = 0.76\,{\text{and}}\,\sigma = 0.25, \hfill \\ \end{aligned}$$
(7)

where r and σ are the correlation coefficient and standard error, respectively.

Table 2 Hypocentral parameters and Seismic moment (M 0), Moment magnitude (M W), Local Richter magnitude (M L) of the earthquakes analyzed in this study
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Fig. 6
figure 6

Plot of average seismic moment (M 0) versus local magnitude (M L) and duration magnitude (M D). The line was obtained by least squares fitting the observations. The errors associated with the slope and the intercepts are the standard errors

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The dependency between M W versus M L and M D are presented in Fig. 7 and the corresponding regression lines are given in Eqs. (8, 9)

$$\begin{aligned} &M_{{\text{W}}} = \left( 0.98 \pm 0.04 \right)\,M_{{\text{L}}} + \left( 0.19 \pm 0.15 \right) \hfill \\ &r^{2} = 0.76\,{\text{and}}\,\sigma = 0.17, \hfill \\ \end{aligned}$$
(8)
$$\begin{aligned} &M_{{\text{W}}} = \left( 0.93 \pm 0.04 \right)\,M_{{\text{D}}} + \left( 0.35 \pm 0.14 \right) \hfill \\ &r^{2} = 0.76\,{\text{and}}\,\sigma = 0.17, \hfill \\ \end{aligned}$$
(9)

where r and σ are the correlation coefficient and standard error, respectively.

Fig. 7
figure 7

Scaling relationship between M W versus M L and M D for selected earthquakes and associated regression line

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7 Discussion and conclusions

This work represents the calculation of seismic moment of 162 local earthquakes in Shillong-Mikir Hills plateau and its adjoining region of northeastern India using the spectral analysis of P-waves. The estimated seismic moments (M 0) range from 1.90 × 1013 to 2.00 × 1016 N m.

The relationship between seismic moment M 0 versus M L and M D are examined for Shillong-Mikir Hills plateau and its adjoining region. The scaling (Eqs. 6 and 7) is similar to determinations obtained in different regions for logarithm of the seismic moment as a function of local magnitude, M L and duration magnitude, M D (e.g., Archuleta et al. 1982; Bakun and Lindh 1977; Bindi et al. 2001; Bora et al. 2013; De Luca et al. 2000; Dutta et al. 2003; Fletcher et al. 1984; Radulain et al. 2014; Castello et al. 2007). Thus, the slope value of 1.46 and 1.39 falls within the values previously obtained by other authors: ~1.1 (e.g., Bindi et al. 2001; De Luca et al. 2000; Fletcher et al. 1984) and ~2.5 (e.g., Bakun and Lindh 1977; Bora et al. 2013; Dutta et al. 2003; Radulain et al. 2014; Castello et al. 2007).

Also the relationship between M W versus M L and M D has been examined for the studied region of NE India. The regression lines (Eq. 8 and 9) obtained from this study are in agreement with the determinations obtained by other authors (e.g., Bakun and Lindh 1977; Archuleta et al. 1982; Fletcher et al. 1984; Wang et al. 1989; Trifu and Radulian 1991; De Luca et al. 2000; Bindi et al. 2001; Dutta et al. 2003; Radulain et al. 2014; Castello et al. 2007; Baruah et al. 2012) for the M L scale in a range of magnitude similar to that investigated here. Using Eqs. (8) and (9), we computed M L and MD for all the events. The M L values can be considered consistent with M W, within 0.1 magnitude units, in 90 % of the cases. The M D values can be considered consistent with M W, within 0.2 magnitude units, in 90 % of the cases. This authenticates our empirical approach. Higher discrepancies in some cases may be ascribed to the fact that superposition of relative amplitudes of earthquake phases hinders estimating magnitude values which was also observed by Giampiccolo et al. 2007. Thus, the estimated M W values in the earthquake catalog need to be examined with respect to M L and M D for an effective seismic hazards analysis and tectonic studies in the Shillong and Mikir plateau region of northeast India.

Accurate estimation of M W for small magnitude earthquakes is important for the prediction of ground motion at low magnitudes. For example, ground motion prediction equations for peak ground acceleration and velocity are usually expressed as functions of M W (e.g., Atkinson and Boore 2006; Akkar and Bommer 2007; Sokolov et al. 2008). For this reason, it is usual to find empirical formulae relating M W to the magnitude scales used locally in seismological practice. Empirical relations between M W and M L, M W and M D, and M 0 and M L are developed for local earthquakes that may be useful for seismic hazard assessment.

7.1 Data and resources

The seismograms which are used for this study were recorded by various stations of different organizations like North East Institute of Science and Technology (CSIR-NEIST), Jorhat, Assam, India; National Geophysical Research Institute (CSIR-NGRI), Hyderabad, India; Indian Institute of Geomagnetism (IIG), Mumbai, India; and Gauhati University, Guwahati, Assam, India. The catalog data from India Meteorological Department (IMD)—Shillong observatory were also used for the study.