1 Introduction

The observed earthquake ground motions, in terms of amplitudes and frequency content, at recording sites, are influenced by the source, path and site characteristics. The effects of travel path on earthquake ground motion depend on the attenuation characteristics of the medium through which the seismic waves propagate; attenuation is an important parameter that governs the amplitudes and frequency content observed on the seismograms. The knowledge of the attenuation of seismic waves is essential for determining earthquake source parameters, predicting the strong ground motions due to earthquakes and estimating the seismic hazard of a region. This attenuation can be quantitatively defined by the inverse of the dimensionless quantity known as quality factor Q, which is a ratio of stored energy to dissipated energy during one cycle of the wave (e.g., Johnston and Toksöz 1981). The seismic wave attenuation is attributed to three factors: (1) anelastic absorption that mainly depends on temperature, fluid content and chemical compositions; (2) scattering of seismic waves, which is ascribed to generated small-scale velocity fluctuations; and (3) focusing of seismic waves due to propagation in 3-D structures. Because of these factors, amplitudes of seismic waves decay faster than predicted by geometrical spreading of wave fronts (Pandit et al. 2011). Therefore, attenuation provides important information about the structure of the earth (Calvet et al. 2013). Mak et al. (2004) observed that for high Q values (Q >600) the region is tectonically stable, while for low Q values (Q <200) the region is seismically active, and for Q values between (200 and 600), the region possesses moderate tectonic activity.

Due to the presence of various scale inhomogeneities in the earth, the observed ground motions in the vicinity of earthquakes often decay slowly leaving a coda following the direct, body and surface waves. These coda waves of local earthquakes were interpreted as backscattering waves from numerous randomly distributed heterogeneities in the earth crust (e.g., Aki 1969; Aki and Chouet 1975; Rautian 1976). The models were developed to use coda waves for the estimation of attenuation relations. The single scattering model considered the scattering as a weak process without loss of seismic energy by scattering, whereas the multiple scattering model the seismic energy transfer considered as a diffused process. Using coda waves, numerous studies have been conducted to determine the attenuation relation for different regions of the world (Ambeh and Fairhead 1989; Catherine 1990; Atkinson and Meeru 1992; Mandal and Rastogi 1998; Gupta et al. 1998; Mandal et al. 2001; Gupta and Kumar 2002; Paul et al. 2003; Kumar et al. 2005, 2007; Sharma et al. 2008; Sahin 2008; Raghukanth and Semala 2009; Dobrynina 2011; Gupta et al. 2012 and Calvet et al. 2013). Mukhopadhyay et al. (2006) computed the intrinsic and scattering attenuation characteristics using the coda waves of local earthquakes for the northwest Himalayas. Sato and Fehler (1998) estimated that the heterogeneities in the lithosphere were observed from coda-Q from local earthquakes on the scale of 0.1–10 km.

Himalaya, the highest mountain chain on the earth, is characterized by the concentration of interplate seismicity and high rate of upliftment as well as convergence (Molnar and Chen 1983; Nakata 1989; Demets et al. 1990). Himachal Lesser Himalaya, forming part of the northwest Himalaya, is bound to the west by the Kashmir Himalaya and to the east by the Garhwal Himalaya. Because of non-availability of digital data set, limited work has been carried out to estimate attenuation properties of the medium. A six-station local network of sensitive digital seismographs has been deployed in the Bilaspur region. The digital recordings from this network have been used to estimate the quality factor of the lithosphere of the Bilaspur region.

1.1 Geology, tectonics and seismicity of the Himachal Lesser Himalaya

The formation of the Himalaya has been attributed to the continent–continent convergence of the India and Eurasia lithospheric plates (e.g., Le Fort 1975; Seeber et al. 1981). From geological considerations, the Himalaya has been segregated into six parts: (1) the first part comprises basement rocks, (2) the second part represents inner crystalline nappe, (3) the third part is composed of rock types belonging to Simlas and Jaunsars, (4) the rock formations primarily belonging to Blainis, infra-Krols, Krols, Shalis and Tals constitute fourth part, (5) the lower tertiary containing rock formations like Subathus, Dagshais and Kasaulis comprises fifth part and (6) the sixth part represents sedimentary formations belonging to Siwalik group (e.g., Saxena 1971). The Himalaya is divided into four tectonic domains viz., the Sub-Himalaya (Siwalik), the Lesser Himalaya, the Great Himalaya and the Tethys Himalaya or Tibet Himalaya. The Indus-Tsangpo suture zone (ITSZ) constitutes the northern boundary of the India plate. Trans-Himadri fault defines the boundary between the Tethys Himalaya and the Great Himalaya. This fault was earlier called the Malari Thrust (Valdiya 1979; Valdiya et al. 1984) and later on renamed as Trans-Himadri fault (Valdiya 1987, 1989). The main central thrust (MCT) defines the boundary of the Great Himalaya and the Lesser Himalaya. The main boundary thrust (MBT) constitutes the boundary between the Lesser Himalaya and the Sub-Himalaya. The Himalaya frontal thrust (HFT) is located to the south of the Sub-Himalaya and separates it from the Indo-Gangetic Plains. The region between the MBT and the HFT is traversed by several subsidiary thrusts such as the Jawalamukhi Thrust (JT) and the Drang Thrust (DT). Within the above tectonic background, the study area is located in the Himachal Lesser Himalaya that forms the northwestern part of the Himalaya. Figure 1b depicts the segments of the MCT and the MBT, along with several other local tectonic features mapped in the study area. The tectonic features located in the close proximity of the Koldam site include the DT and the MBT. Geological mapping has indicated that at several places the MBT and the JT exhibited neotectonic activity (e.g., Srikanita and Bhargava 1998).

Fig. 1
figure 1

a Map of India showing the study region (rectangular box). b Map showing tectonic features around Bilaspur region of the Himachal Lesser Himalaya (After GSI 2000). Tectonic features (lines) are shown and include: MBT; MCT; DT; JT. Filled triangles depict the network stations

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The earthquake activity of the region is primarily ascribed to the convergence of the Indian and Eurasian tectonic plates. The study region falls in seismic zone V as per the seismic zoning map of India (IS: 1893-(Part l) 2002: General Provisions and Buildings). The region has witnessed many moderate- to large-sized earthquakes in the last more than 100 years. Prominent earthquakes that occurred in the Himachal Lesser Himalaya encompassing the Bilaspur region are listed in Table 1.

Table 1 Prominent earthquakes occurred in the Himachal Lesser Himalaya encompassing the Bilaspur region
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2 Methodology

This methodology adopted to estimate the quality factor (Q) of the Bilaspur region is based on well-known single backscattering model (Aki and Chouet 1975). This model is based on the premise that coda waves are backscattered body waves generated by randomly distributed heterogeneities in the earth’s crust and upper mantle. The size of scatterers is considered greater than the wavelength, and no velocity change or multiple scattering is allowed in the medium. Kopnichev (1977) and Gao et al. (1983) demonstrated that the coda waves observed at short lapse times (<100 s) are because of the single scattering, whereas those at long lapse times (>100 s) are due to multiple scattering. Based on the single backscattering model, the coda wave amplitude A(f, t) for a narrow bandwidth signal centered at frequency f and at lapse time t is given as (Aki and Chouet 1975):

$$ \;A(f,t) = S(f){t^{ - \alpha }}\exp \left( {\frac{ - \pi ft}{{{Q_{\text{c}}}}}} \right) $$
(1)

where S(f) represents the source function at frequency f, α is the geometrical spreading parameter, which is taken as ‘0.5’ and ‘1’ for surface waves and body waves, respectively, and Q c represents the quality factor of coda waves. Equation (1) can be written as:

$$ \ln \,\,(A(f,t),\;\;t) = \;\ln (S(f)) - \;\left( {\frac{\pi ft}{{{Q_{\text{c}}}}}} \right) $$
(2)

Relation (2) allows estimation of the Q c from the slope of the straight line, fitted between ln(A(f, t)) versus time t adopting least-squares method. According to Rautian and Khalturin (1978), the above relations are valid for lapse times greater than twice the S-wave travel time. The frequency-dependent relation of Q c is described by the power law: Q c(f) = Q 0 (f)n, where Q 0 is the value of Q c at 1 Hz, and n represents the degree of frequency dependence of Q c. The logarithm of this equation allows estimation of n and Q 0 using a simple linear regression.

2.1 Data set

A six-station local seismological network has been deployed around the Koldam located in the Himachal Lesser Himalaya (Fig. 1b. The local earthquake data collected through this network are interpreted to study the local seismicity and to map seismotectonic sources around the Koldam site. In this network, three short-period seismometers CMG-40TD1 (Guralp Systems Limited, UK) and two broadband seismometers CMG-3ESPC (Guralp Systems Limited, UK) are used as sensors to sense the three components of ground motion. The output from each sensor is coupled to a 24-bit portable data acquisition system (DL-24), and a global positioning system (GPS) is used to synchronize data samples to UTC or IST. The digital data are acquired at a rate of 100 samples per second (sps). In addition, one station is operated in an analog mode with short-period vertical-component seismometer L4C (Serceal, UK). Site characteristics and geographical coordinates of the recording stations are listed in Table 2.

Table 2 Site characteristics and geographical coordinates of the recording stations
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2.2 Data analysis

The digital data collected from all the stations have been converted to Seisan format with conversion programs available in SEISAN software (Havskov and Ottemoller 2003). Hypocenter parameters of local events have been estimated adopting HYPOCENTER program (Lienert et al. 1986; Lienert and Havskov 1995) and using the velocity model as listed in Table 3. The standard errors in the estimates of epicenter (ERH), focal depth (ERZ) and origin times (RMS residuals) are as follows: ERH ≤5.0 km; ERZ ≤5.0 km; and RMS ≤0.5 s. From May 2013 to March 2014, digital recordings of 41 local events from four stations (GOHR, KUNH, NERI and SKND) out of six stations are used in the study (Fig. 2). The digital data from Band and Nihri stations could not be used in this study because of low signal to noise ratio (SNR). All the local events occurred at epicentral distances of <100 km, and their magnitudes are in the range between 0.5 and 2.9.

Table 3 Velocity model used in the study
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Fig. 2
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Epicenters of events used in the study

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A MATLAB code has been developed based on the single backscattering model of Aki and Chouet (1975) for the estimation of coda-Q. The guidelines of CODAQ subroutine of SEISAN have been followed for the development of the code (Havskov and Ottemoller 2003). The waveforms with SNRs above 3 are selected for analysis. The correlation coefficients are also used as a second selection criterion because Q values with small correlation coefficients lead to a poorly constrained Q–f relation and consequently a less reliable estimate of Q 0 and n values. It is suggested that correlation coefficient should be >0.7 to obtain the reliable values of Q c. Further, the vertical components of coda waves have been used to estimate Q c because it has been shown that the coda analysis is independent of the components of the ground motion analyzed (Hoshiba 1993).

Origin times and coda arrival times have been estimated from arrival times of P- and S-waves. Adopting Butterworth filter, the seismograms have been band-pass filtered at seven frequency bands, viz., 1–2, 2–4, 4–8, 6–12, 8–16, 12–24 and 16–32 Hz (Table 4). For each band-pass filtered earthquake time histories, signal is selected from coda arrival to coda duration considered for analysis. The elimination of contamination caused by direct S-wave is essential for reliable Q c determination (Rautian and Khalturin 1978; Herraiz and Espinosa 1987). Therefore, direct S-waves from filtered earthquake time history are eliminated by taking the beginning of coda waves as twice the arrival time of S-wave from the origin time (Rautian and Khalturin 1978). The length of coda window is also important to get the stable solution. It is suggested that minimum window length should be 20 s (Havskov and Ottemoller 2005), and there is no maximum upper limit. In order to get stable and reliable solutions, three lapse time windows (20, 30 and 40 s) have been used. Coda waves selected in a specified time window are corrected for geometrical spreading. The corrected amplitudes are multiplied by ‘t α’ to account for geometrical spreading, and for local earthquakes, the value of α = 1. The envelopes of the coda waves are estimated from root mean square (RMS) coda amplitudes computed using a moving window of 2 s with overlap of 1 s. The natural log of RMS amplitudes is plotted with t, and a linear equation is fitted to the data. The slope of the line provides coda-Q estimates at a particular central frequency (f c) as depicted in Fig. 3.

Table 4 Various central frequencies with low-cut and high-cut frequency bands used for filtering
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Fig. 3
figure 3

Plot of event recorded at GOHR station on November 27, 2013. a Unfiltered data trace with coda window, b–e band-pass filtered displacement amplitudes of coda window at 1–2, 4–8, 8–16 and 16–32 Hz, respectively, and the RMS amplitude values multiplied with lapse time along with best square fits of selected coda window at central frequencies of 1.5, 6, 12 and 24 Hz, respectively. The Q c is determined from the slope of best square line. Abbreviations P, P-wave arrival time; S, S-wave arrival time

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3 Results and discussions

The frequency-dependent Q c relations for each recording site as well as average relations for the Bilaspur region have been estimated. The spatial variation of estimated Q c is studied, and its comparison with other seismically active regions of the India and world has been made. The 41 local events are grouped in three types according to the epicentral distance (R) range: near range (R < 30 km); medium range (30 ≤ R < 60 km); and distant range (R ≥ 60 km) (Table 5a–c).

Table 5 (a), (b), (c) shows the mean value with standard error at different frequencies and different lapse times (20, 30 and 40 s) for all stations which are distributed epicentral distance wise having (near range (R < 30 km), medium range (30 ≤ R < 60 km) and distant range (R ≥ 60 km)
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3.1 Frequency and lapse time dependence of Q c

The coda-Q estimated at central frequencies 1.5, 3, 6, 9, 12, 18 and 24 adopting three lapse time windows (LTWs) of 20-, 30- and 40-s durations is found to increase with increasing frequency and lapse time. We estimated frequency-dependent Q(f) relations for each of the three distance ranges as well as for the entire data set because such Q c relations are known to provide average attenuation characteristics of the medium properties of the localized regions (e.g., Mukhopadhyay et al. 2006). The mean values of Q c with standard errors for 20-, 30- and 40-s LTWs, and for different distance, ranges with respect to each central frequency are listed in Table 5a–c. From Table 5a, it is evident that Q c values are high at higher frequency ranges, which demonstrates the homogeneous character of the medium at deeper level. Similar patterns are observed from the Q c values listed in Table 5b, c. A power law Q c = Q 0 f n, where Q 0 is the value of Q c at 1 Hz and n is frequency dependence coefficient (Aki 1980), is fitted to the data, for different LTWs. For 20-s LTW, the power law provided the average values of Q 0 = 47 ± 26 and n = 1.32 ± 0.1 for (R < 30 km); Q 0 = 64 ± 16 and n = 1.21 ± 0.04 for (30 ≤ R < 60 km); and Q 0 = 55 ± 7.73 and n = 1.29 ± 0.03 for (R ≥ 60 km), respectively. Similarly, the values of Q 0 and n have been estimated for 30- and 40-s LTWs are tabulated in Table 6.

Table 6 Average value of quality factor Q 0 and n at different lapse time with distribution of epicentral distance
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The average estimates of Q c obtained by taking the mean of Q c values at different stations are listed in Table 7. Figure 4 depicts that at all the stations the variation of Q c with frequency has a linear trend for 20-, 30- and 40-s LTWs. Furthermore, the Q 0 values are found to increase with lapse time for all stations, and n values are found to decrease with the LTW (Table 7). The increase in Q c and decrease in n seem to signify the variation of attenuation character with depth as larger time window provides the effect of deeper part of the earth. Hence, it can be interpreted that when Q c increases and n decreases, then heterogeneity decreases with depth in the study area (Mukhopadhyay and Tyagi 2007). According to the seismic zoning map of India, the study region falls in seismic zone V which represents the high seismicity of the region, and this is also reflected in the high estimated value of ‘n.’ For the study region, Q c estimates follow the frequency-dependent relations: Q c = (70.3 ± 20.27) f (1.23±0.05) for 30-s LTW.

Table 7 Quality factor at each station and their average estimated values with different LTWs considered
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Fig. 4
figure 4

Plots of quality factors and central frequencies for all the five stations (a–d) and average with linear regression frequency-dependent relationship, Q c = Q 0 f n at different lapse time 20, 30 and 40 s

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3.2 Spatial variation of Q c

A large number of studies have demonstrated the dependence of Q c with lapse time (e.g., Roecker et al. 1982; Kvamme and Havskov 1989; Ibanez et al. 1990; Woodgold 1994; Akinci et al. 1994; Gupta et al. 1996; Mukhopadhyay and Tyagi 2007). Further, the lapse time has been related to the area of sampling by coda waves. According to Pulli (1984), for a single scattering model, the coda wave attenuation represents the average decay of amplitudes of backscattered waves on the surface of ellipsoid with earthquake of source and station as foci. The coda waves at a station consist of the combination of several scattered phases, which do not represent a single ray. Hence, Q c represents the average attenuation of the region comprises ellipsoidal volume with depth, h = h av + D2, where D2  = \( \sqrt {D{1^2} - {\varDelta^2}} \) is the small minor axis of ellipsoid for epicentral distance Δ and the average focal depth h av of the events. The large semi-axis D1 is the surface projection of ellipsoid with hypocenter and station as foci and can be defined as vt/2. The average lapse time is given by the relations: \( t = {t_{\text{st}}} + \frac{W}{2} \), where t st is beginning of the lapse time and W is the window length. The depths calculated for the ellipsoidal volume at different stations are given in Table 8. The increase in Q c with lapse time is attributed to factors like considering nonzero source receiver distance, anisotropic scattering and assumption of single scattering model instead of multiple scattering (Woodgold 1994). For the three LTWs considered in this study, the starting time of coda wave is taken twice the S-wave travel time, and only backscattered waves are recorded in this time window (Aki and Chouet 1975). Further, for all the analyzed local events, the LTW length is <100 s, and the multiple scattering effects are not important for local events for lapse time <100 s (Gao et al. 1983). Hence, in the studied region, the variation of Q c with lapse time is because of the variation of attenuation with depth and indicates that medium homogeneity increases with depth.

Table 8 The maximum depth of ellipsoid volume
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3.3 Comparison of Q c with other regions of the India and the world

The single scattering or multiple scattering models have been used to estimate Q c in different regions of the world (e.g., Aki and Chouet 1975; Sato 1977; Roecker et al. 1982; Pulli 1984; Wu 1985; Jin and Aki 1988; Havskov et al. 1989; Ibanez et al. 1990; Pujades et al. 1991; Canas et al. 1991; Akinci et al. 1994; Latchman et al. 1996; Zelt et al. 1999). In Fig. 5a, the Q c estimates for Bilaspur region of the Himachal Lesser Himalaya are compared with some of the regions of the world (Li et al. 2004; Havskov et al. 1989; Hellweg et al. 1995; Rovelli 1982; Mak et al. 2004; Mahood and Hamzehloo 2009; Rahimi and Hamzehloo 2008; Pujades et al. 1991; Barros et al. 2011). For comparison, frequency-dependent relations Q c = (70.3 ± 20.27) f (1.23±0.05) at 30-s LTW are considered, and it is found that average variation of Q c for Bilaspur region is very close to Amazon Craton (Brazil). This may be due to the some kind of geological similarity between Bilaspur region and Amazon Craton (Brazil). The rock types of the Himachal Lesser Himalaya primarily comprise metasedimentary, sedimentary, igneous and metamorphic rocks, whereas the Amazon Craton of Brazil comprises of granitic rocks and Phanerozoic terrains with sedimentary rocks of the Parecis basin.

Fig. 5
figure 5figure 5

a, b Comparison of Q c values for Bilaspur region of the Himachal Lesser Himalaya with the existing Q studies worldwide and in India

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The Himalaya is seismically and tectonically very active. Since 1995, several studies on the estimation of seismic wave attenuation were undertaken (e.g., Gupta et al. 1995, 1998, 2012; Gupta and Kumar 2002; Paul et al. 2003; Kumar et al. 2005; Ramakrishna et al. 2007). Figure 5b shows the comparison of Q c estimates of the Bilaspur region with the Q c estimates from various other Indian regions. The variation of Q c with frequency for the Bilaspur region is almost similar to that of Kachchh region. Table 9 shows the comparison of Q c at 1 Hz (Q 0) for the Bilaspur region with the other Indian regions. It is found that ‘Q 0’ has minimum value for the Bilaspur region. This means that the Bilaspur region is the most attenuating among the regions compared. The ‘n’ value for the Bilaspur region is also maximum than any other Indian region and some of the regions of the world (Table 9). This indicates that the Bilaspur region is highly heterogeneous in character and that the lowest ‘Q 0’ is observed for this Indian region. This may be due to the presence of crisscrossed fractures, intrusions and heterogeneities of varying scales attributed to the Himalaya orogeny.

Table 9 Comparison of attenuation quality factor ‘Q 0’ and ‘n’ with other regions of the world and India
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4 Conclusion

The frequency-dependent coda-Q relations for the Bilaspur region of the Himachal Lesser Himalaya have been estimated considering LTWs of 20-, 30- and 40-s durations. The Q c estimates are computed in the frequency range from 1.5 to 24 Hz, using a data set of 41 local events. The variation of Q c with distance is investigated by dividing the data set into near (R < 30 km), medium (30 ≤ R < 60 km) and distant ranges (≥60 km), respectively.

For 20-s LTW, the Q c estimates vary from 87 ± 4 (at 1.5 Hz) to 2716 ± 236 (at 24 Hz) for R < 30 km; 89 ± 10 (at 1.5 Hz) to 2706 ± 242 (at 24 Hz) for 30 ≤ R < 60 km; and 96 (at 1.5 Hz) to 3274 ± 260 (at 24 Hz) for R ≥ 60 km. Similarly, for 30-s LTW, the Q c estimates vary from 98 ± 6 (at 1.5 Hz) to 2971 ± 123 (at 24 Hz) for R < 30 km; 120 ± 10 (at 1.5 Hz) to 2990 ± 216 (at 24 Hz) for 30 ≤ R < 60 km; and 130 ± 12 (at 1.5 Hz) to 3082 ± 165 for R ≥ 60 km), while for 40-s LTW, Q c estimates vary from 118 ± 9 (at 1.5 Hz) to 3013 ± 132 (at 24 Hz) for R < 30 km; 116 ± 6 (at 1.5 Hz) to 2978 ± 122 (at 24 Hz) for 30 ≤ R < 60 km; and 166 (at 1.5 Hz) to 3356 ± 162 (at 24 Hz) for R ≥ 60 km. It is found that Q c values are high at higher frequencies and show the homogeneity at deeper zones.

For 30-s LTW, the frequency-dependent relations Q c = (70.3 ± 20.27) f (1.23±0.05) have been obtained for the Bilaspur region considering entire data set. Comparison of Q c estimates for the Bilaspur region with some of the seismically active region of the world has shown that the average variation of Q c for this region is very close to Amazon Craton (Brazil) due to similar lithologic setup. From the comparison of ‘Q 0’ and ‘n’ obtained for the Bilaspur region with other Indian regions, it is found that the Bilaspur region is most attenuating and highly heterogeneous in nature. Further, the variation of Q c with frequency for the Bilaspur region is almost similar to that of Kachchh region. The various attenuation relations developed for the Bilaspur region shall be useful for computing earthquake source parameters, simulating earthquake strong ground motions and for seismic hazard assessment of the region.