1 Introduction

Attenuation of high-frequency (> 1 Hz) seismic waves is one of the major components in evaluating the seismo-tectonic feature and seismic hazard assessment of the region. The attenuation of seismic waves is expressed using the dimensionless quality factor Q, defined as the fractional loss of energy per unit cycle of the wave. One method for estimating coda-Q (Q c) is from the decay rate of coda waves, waves scattered from randomly distributed heterogeneities of the medium (Mamdouh et al. 2013; Mukhopadhyay and Tyagi 2007; Kumar et al. 2005). The parameter Q c represents the total attenuation properties of the earth’s crust, that includes both intrinsic or anelastic absorption and scattering attenuation due to random heterogeneities of the medium (Lee 1999).

Numerous studies have been done worldwide to understand the attenuation characteristics of the medium using coda waves (Aki and Chouet 1975; Badawy and Morsy 2012; Boulanouar et al. 2013; Biescas et al. 2007; Giampiccolo et al. 2002; Gupta et al. 1998; Parvez et al. 2008; Sato et al. 2012). The results from these studies show that the values of Q c vary with both frequency (Q c = Q 0 f n, Q 0 is the value of Q c at 1 Hz (Singh and Hermann 1983) and n is frequency-dependent exponent) and lapse time. The lapse-time dependence of Q c is attributed to the effect of both multiple scattering and depth (Masahiro 1992; Barros et al. 2011). The values of Q c obtained with a single scattering model show that Q 0 is less (< 200) and n is high (≥ 1.0) for tectonically active regions, while Q 0 is more (> 200) and n is low (< 1) for stable areas (Sato et al. 2012).

Very little work has been done on seismic wave attenuation in northern Morocco, a region characterized by active subduction with complex tectonics (Boulanouar et al. 2013, 2016). Boulanouar et al. (2013) showed the spatial variation of coda wave attenuation using aftershocks of the Al Hoceima earthquake (Ml = 6.4) of 24 February, 2004.

Here, we study the attenuation proprieties of the medium using coda waves beneath northern Morocco (Fig. 1), using the single back-scattering model of Aki and Chouet (1975) by analyzing a large number of three-component broad-band seismograms of local earthquakes during 2008. We also show dependence of Q c on frequency and lapse-time, examine the Q c estimates of three-components of motion, compare our results with those of other studies obtained with similar method, and finally correlate the results with the seismo-tectonic features of the region.

Fig. 1
figure 1

Map of the northern Morocco showing the ray traces used in this study (blue ones belongs to the western Mediterranean seismic network and the red ones to the IberArray network)

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2 Data

The seismograms used in this study are from 66 local earthquakes (Table 1) recorded by 26 three-component broad-band stations (Table 2) equipped with Trillium 40 sensors (100 Hz). Figure 1 shows the map of the study area with the location of events and stations used in this study. The seismic stations are maintained by the Scientific Institute of Rabat. The magnitudes of the analyzed events range from 0.7 to 3.4. The depth and epicentral distance of the events vary from 0 to 120.8 km, and from 12.4 to 289.0 km, respectively. Data were recorded digitally at 100 samples per second. Only good quality seismograms with signal-to-noise ratio greater than 3.0 were processed. The events used in this work (Fig. 1) are previously located using the (Instituto Geografico National) (IGN) reference velocity model and selected from earthquake location database used in the local earthquake tomography of the Betic-Rif system, western Mediterranean (El Moudnib et al. 2015).

Table 1 List of earthquakes used for estimation of coda, Q
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Table 2 List of seismic stations that belong to the Morocco network
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3 Method of Analysis

The coda decay rate of local earthquakes is found to be independent of source-receiver distance (Aki 1969). Assuming the single back-scattering model for coda generation for coincident source and receiver, the amplitude of band-pass filtered coda envelope as a function of lapse time from origin time “t” can be expressed as (Aki and Chouet 1975):

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

where S(f) is the coda source factor at frequency f, Q c is the quality factor for coda wave, t −1 accounts for the geometrical spreading for body waves. By taking the natural logarithms of Eq. (1) and rearranging terms, we obtain the following equation:

$$\ln \left[ {A(f,t) \times t} \right] = C - b \times t ,$$
(2)

where b and C equal to – πf/Q c and ln[S(f)], respectively. The Q c values are calculated from the linear regression of \(\ln \left[ {A(f,t) \times t} \right]\) versus t for each frequency band centered at 0.75, 1.5, 3.0, 6.0 and 12.0 Hz (Table 3). Coda envelope can be obtained by taking the root mean squares (rms) of the signal in a sliding window. The linear regression was performed for four lapse time windows of 30–60 s starting from 2ts (ts is the S-wave arrival time) (Rautian and Khalturin 1978) to examine the lapse time dependence of Q c. Finally, the power-law form of Q c = Q 0 f n is fitted for the both the vertical and horizontal components for each station, where Q 0 is the Q c-value at 1-Hz and n is the frequency relation parameter. Figure 2 shows an example of data processing used for determining Q c at different frequency bands, using a vertical component seismogram from a representative event recorded at station CHAF.

Table 3 Parameters of a band-pass filter showing lower cut-off, higher cut-off and central frequencies
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Fig. 2
figure 2

An example of data analysis for coda-Q estimation using the CODAQ module of Havskov and Ottemoller (2003). The seismogram from the event on 08th June, 2008 recorded at the station CHAF is band-pass filtered at five frequency bands centered at 0.75, 1.5, 3.0, 6.0 and 12.0 Hz. The coda wave portion of the 60 s duration is indicated by arrows (↑). The slope of the coda envelope fitted (solid line based on Eq. 2) is used to estimate Q c. H depth (km), M coda magnitude, TP P onset time, TC start of coda window (in sec) measured from origin time, WIN window length, ST start of coda window in terms of S travel time, tc = 2 × ts, F central frequency in Hz, CO correlation coefficient, and S/N signal-to-noise ratio

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4 Tectonic Setting

Northern Morocco belongs to the Iberia-Africa collision zone, which is one of the complex deformation zones in the western Mediterranean (Khattach et al. 2013; Timoulali et al. 2014). The section beneath this collision zone is characterized by a diffuse and moderate seismicity, in comparison to the neighboring zones with large earthquakes in the east and the west (Cherkaoui and Hassani 2012; Inmaculada et al. 2003). The area under study experienced two moderate earthquakes in last decades (Boulanouar et al. 2016; Medina and Alami 2006) (Al Hoceima 2004, M ~ 6.4, Al Hoceima 1994, M ~ 5.8). The area is characterized by shallow (< 30 km) seismic activity that is concentrated in the Al Hoceima region and in the region immediate south to the Nador city. The annual frequency of the felt earthquakes (intensity > V) is 2.4 (Cherkaoui et al. 1990). The intermediate depth earthquakes occur in the region ranging from north–south to west of the Alboran Sea.

5 Results and Discussion

In this study, we estimate attenuation properties of the medium using coda waves in 1.5–12 Hz in northern Morocco. For each component of motion, the average values of frequency-dependent Q c for the whole study area are obtained by averaging the Q c values over all the stations for each frequency (Table 4). The estimates are \(Q_{\text{c}} = (143.75 \pm 1.09)f^{(0.864 \pm 0.006)}\), \(Q_{\text{c}} = (140.12 \pm 1.08)f^{(0.85 \pm 0.005)}\) and \(Q_{\text{c}} = (140.42 \pm 1.81)f^{(0.902 \pm 0.004)}\) for vertical (Z), north–south (N–S), and east–west (E–W) component of motion, respectively. Table 5 lists the frequency-dependent power-law forms of Q c for the three-components of motion for each lapse time (Fig. 3).

Table 4 Average estimates of Qc along with their error bars (1 standard deviation) for different frequency bands and lapse times for the three components of motion
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Table 5 The power-law form of Q c (f) = Q 0(f)n obtained for different lapse time windows for the three components of motion
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Fig. 3
figure 3

Typical example of three-component velocity seismograms recorded at station CHAF of the network from the event on 21 January 2008, 02:24:03:300 h. BHZ, vertical component; BHN, north–south component; BHE, east–west component. The line marked ‘O’ represents the origin time of the event

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5.1 Lapse Time Dependence of Q c

To test the lapse-time dependence of Q c in northern Morocco, we estimate the Q c values at four different lapse time windows of length 30, 40, 50 s and 60 s starting from 2ts. Figures 4, 5, 6 show the frequency-dependent Q c relations for each lapse-time for the Z, N–S and E–W component of motion, respectively. It is shown from Figs. 4, 5, 6 that as frequency increases, the values of Q c increase with lapse time similar to findings for several other regions (Ibanez et al. 1990; Del Pezzo and Patané 1992; Akinci et al. 1994; Gupta et al. 1998; Giampiccolo et al. 2002). Figure 7 shows the lapse time dependence of Q −10 and the exponent n, where both decrease with increase in lapse time. As an increase in large time samples the deeper part of the Earth, it is expected that both Q −10 and n decrease with increasing depth, because heterogeneity of the medium generally decreases with depth. The higher Q 0 (Q c at 1 Hz) is expected for larger lapse times as coda is mainly dominated by the energy that has propagated in the deeper portions of the Earth with lower-attenuation zones. Thus, the observed lapse-time dependence of Q c can be interpreted as due to increase of Q c with depth. Depth-dependent variation in attenuation indicates the variation in level of heterogeneity with depth. The longer the analysis window, the larger is the sample of ellipsoidal volume of the lithosphere. Thus, the variation of Q c with lapse time can be explained by comparing different sampling volumes of the earth medium. Wennerberg (1993) showed that such a trend in Q c can be explained by variation in intrinsic or scattering attenuation in the expanding volume of earth medium sampled by coda waves, assuming multiple scattering of S-waves. Our method is based on the single back-scattering of seismic S-waves. For evaluating the relative contributions of intrinsic and scattering attenuations to the total attenuation, we need a multiple scattering model of coda wave generation, however. We plan to do such analysis with a widely used multiple lapse time window analysis (MLTWA) method, as a scope of future study (Masahiro 1992; Tuvè et al. 2006; Woodgold 1994).

Fig. 4
figure 4

Lapse time dependence of Q c(f) for the vertical (Z) component of motion, as shown in the uppermost four plots. They are again shown in one plot for comparison (bottom figure)

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Fig. 5
figure 5

Same as Fig. 4, but for the N–S component of the motion

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Fig. 6
figure 6

Same as Fig. 4, but for the E–W component of the motion

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Fig. 7
figure 7

Comparison of average values of 1/Q 0 and n with lapse time for vertical component of motion

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We cannot attribute the lapse time dependence of Q c entirely to the effect of depth, however. The lapse-time dependence of Q c is also expected in cases where (1) the distribution of scatterers is heterogeneous, with more scatterers being present at shallower depths, (2) the intrinsic Q of the medium is not constant, or (3) multiple scattering occurs in the late coda. For local events with a short lapse time window (less than 100 s), the effect of multiple scattering is not significant (Kopnichev 1977; Gao et al. 1983). For regional events with long lapse times (greater than mean free time), however, a contribution of multiple scattering to the lapse-time dependence of Q c cannot be neglected (Padhy and Subhadra 2010). For example, Del Pezzo et al. (1991) have shown for the Etna volcano data that the single back-scattering model cannot be used to explain the lapse-time dependence of coda waves. The length of lapse-time window for multiple scattering to become significant depends on the scale length of the heterogeneities of the medium. In this study, however, the lengths of lapse-time window are less than 100 s, so the use of single scattering model is acceptable for the study area.

The average Q c relations for the study area for the vertical component of motion are \(Q_{\text{c}} (f) = (97.6 \pm 1.05)f^{(0.96 \pm 0.03)}\), \(Q_{\text{c}} (f) = (139.5 \pm 1.08)f^{(0.86 \pm 0.05)}\), \(Q_{\text{c}} (f) = (156.1 \pm 1.09)f^{(0.86 \pm 0.06)}\) and \(Q_{\text{c}} (f) = (180.3 \pm 1.13)f^{(0.81 \pm 0.08)}\) for lapse-time windows of 30, 40, 50 and 60 s, respectively. The frequency dependence of Q c is related to scale length of heterogeneities of the medium. The strong frequency dependence could be related to a high degree of heterogeneity of the medium (Mayeda et al. 1992).

The Q c values are considered to represent an average over the ellipsoid volume sampled by coda waves for a given lapse-time (Pulli 1984). For a given average lapse-time t, the surface projection of such ellipsoid with foci at the source and receiver is defined by

$$\frac{{x^{2} }}{{(vt/2)^{2} }} + \frac{{y^{2} }}{{(vt/2)^{2} - (r/2)^{2} }} = 1 ,$$
(3)

with the semi-major axes a = vt/2 and semi-minor axes \(b = \sqrt {(vt/2)^{2} - (r/2)^{2} }\), v is S-wave velocity, r is the source-receiver distance, and x and y are the surface coordinates (Pulli 1984). The average lapse time t is taken as \(t = t_{\text{start}} + \frac{W}{2}\), where t start is the starting time of the coda window and W is the coda window length. The maximum depth of sampling that is related to semi-major axes is about 105 km assuming an average S-wave speed of 3.5 km/s in the medium. Our analysis assumes a uniform velocity model of the whole medium beneath the region, although in reality velocity varies with depth. Thus, we think that our Q c estimates represent the average attenuation properties of the whole crust and part of the upper mantle beneath northern Morocco.

For an average S-wave velocity of 3.5 km/s, the length of the larger semi-major axis is approximately 53 and 105 km for 30 and 60 s lapse time, respectively, which clearly shows how multiple scattering, which is expected to occur at large lapse-times, reduces the volume sampled by coda waves, although the effect of multiple scattering seems to be minimum in this case. Our findings are similar to those of Ibanez et al. (1990) who compared the volume sampled by the scattered waves for a fixed lapse time (70 s) for seismic sources located at different depths. They observed that for events at greater depths, the larger semi-major axis is about 150 km long, decreasing to about 110 for shallow earthquakes in Granada, southern Spain. Thus, the effects of multiple scattering shorten the sampling volume of the medium.

5.2 Variation of Q c for the Three Components of Motion

The average values of Q c for the three components of motion are plotted in Fig. 8. The Q c(f) values for the three components of motion are similar at all frequencies analyzed, which is expected because of averaging over a larger volume of the medium sampled by coda waves (Fig. 8 and Table 5). It might also be possible that different radiation patterns may give rise to differential energy decay pattern in three components of motion, but their effect is negligible for coda waves at high frequencies. Sato (1984) has shown that in case of single scattering, the directional partition of energy in S-coda depends on the focal mechanism and hence on radiation pattern for a point source dislocation. These differences in source radiation pattern are averaged out in later coda arrivals, which make the Q c values insensitive to the component of ground motion. A similar estimate of Q c, that is independent of the component of ground motion, is found for volcanic micro-earthquakes at CampiFlegrei (Del Pezzo et al. 1985).

Fig. 8
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Plots summarizing the lapse time dependence of Q c(f) for all the three-components of motion

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5.3 Comparison with Other Results

In general, the estimates of Q c can be affected by differences in parameters such as coda window length and coda onset time, distance and depth of events used, frequency range, use of body or surface waves, and the type of model used to determine Q c (Table 6).

Table 6 Worldwide Q studies done in active and stable areas
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Here, we compared Q c values for different areas obtained from coda waves for similar coda onset time and lapse-time that sample approximately similar volumes of the medium sampled (Fig. 9). To minimize the influence of the use of different model parameters on Q c measurements, we compared our results with others using the similar method. The comparison shows that the study area is seismically (1) more active than New England, central and northeastern USA (Phillips and Aki 1986), (2) less active than east central Iran, Yunnan in China, Washington State, Parkfield in USA, Zagros, and southern Iberia, and (3) similar to Garhwal Himalaya in India and south central Alaska. Such a regional variation in Q c can be interpreted in terms of (1) vertical and lateral variation in structural heterogeneities among the regions (Aki 1969; Steck et al. 1989; Fehler et al. 1992), (2) velocity contrast in a layered structure (Hoshiba 1997; Margerin et al. 1998), and (3) near-surface effects, focusing and defocusing of seismic waves (Frankel and Wennerberg 1987).

Fig. 9
figure 9

Comparison of the coda-Q c values of the Northern Morocco region with those of other regions of the world (Ma’hood and Hamzehloo 2009; Singh and Hermann 1983)

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Regions with active tectonics are generally associated with low Q 0 (less than 200) values, and vice versa such as the Yunnan Province in China (Q 0 = 49 and n = 0.95, Li et al. 2004), Zagros continental collision zone in southwestern Iran (Q 0=99 and n = 0.84, Rahimi and Hamzehloo 2008), Granada zone (southern Spain) (Q 0 = 155 and n = 0.89, Ibanez et al. 1990) and the state of Washington (Q 0 = 63 and n = 0.97, Havskov et al. 1989). On the contrary, stable regions are characterized by relatively high values of Q 0 (more than 200), such as New England (Q 0 = 460, n = 0.45; Pulli 1984), N. Iberia (Q 0 = 600, n = 0.45; Pujades 1991). The value of the exponent n increases with the intensity of tectonic activity and is more than 1.0 for tectonically active regions.

6 Conclusions

In this work, the attenuation proprieties of high-frequency (> 1 Hz) coda waves are studied for northern Morocco. The average frequency-dependent Q c values for three components of ground motion are found to be similar, suggesting the average property of the medium sampled by coda waves over a large sampling volume. The estimates of Q c and their frequency dependences are found to be similar to those for active regions of the world like Garhwal Himalaya and south central Alaska. Lapse-time dependence of Q c could be related to the depth-dependent variation in attenuation and hence in level of heterogeneity of the medium. Strong frequency dependence suggests high degree of heterogeneity of the medium associated with complex subduction structure of the medium. On comparing the estimated Q c values with other regions indicates that the northern Morocco is seismically active characterised by low values of Q c and hence by relatively high attenuation. These results will be useful in characterizing the path effect that is necessary for ground motion modelling in the region.