Seismic attenuation in the Middle America Region and the frequency dependence of intrinsic Q


  • L. A. Dominguez,

    Corresponding author
    1. Department of Earth and Space Sciences, University of Los Angeles, Los Angeles, California, USA
    • Corresponding author: L. A. Dominguez, Department of Earth and Space Sciences, University of California, Los Angeles, 3806 Geology Bldg., Los Angeles, CA 90095, USA. (

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  • P. M. Davis

    1. Department of Earth and Space Sciences, University of Los Angeles, Los Angeles, California, USA
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[1] We evaluate the mechanisms of attenuation for the Middle America (MA) region in the range of 1.0≤f≤15 Hz. Our analysis focuses on reconciling laboratory experiments, which suggest that in this frequency range, intrinsic attenuation has a weak to zero frequency dependence, with results obtained using the Multiple Lapse Time Widow (MLTW) method. The MLTW method is designed to separate the contributions of scattering by heterogeneities and intrinsic attenuation from total attenuation assuming spherical geometrical spreading and isotropic scattering. Application of the MLTW method suggests that total attenuation in the MA region is concentrated towards the upper crust and is relatively constant throughout the region. It also shows a strong frequency dependence of the energy loss component of the attenuation (apparent intrinsic attenuation). We test the effect that forward and back scattering operators, distribution of the sources, and absorption of energy into the mantle has on the separation of the components of attenuation. We conclude that the inclusion of the frequency-dependent leakage of energy towards the mantle provides a mechanism to satisfy both laboratory experiments and estimates based on the MLTW method. For the Middle America region, we find that the intrinsic attenuation math formula.

1 Introduction

[2] Numerous authors have investigated the mechanisms of attenuation in southern and central Mexico. Table 1 summarizes previous studies for this geographical area. Canas and Egozcue [1988] estimated the seismic attenuation of Lg waves in southern Mexico using the coda Q method for a low frequency band around 1 Hz. Castro and Anderson [1990] calculated the attenuation of S- waves along the subduction zone, and considered changes in the geometrical spreading as a function of distance. Ordaz and Singh [1992] investigated the seismic amplification factors in the hill zone of Mexico City, and estimated the quality factor of QS=273 f0.66 for the forearc region. Singh and Iglesias [2007] reported remarkably lower values for the central part of the volcanic arc, Q(f)=98 f0.72. Castro and Mungia [1993] calculated the attenuation for P- and S- waves for the Oaxaca region. In a later paper, Castro and Mungia [1994] compared results of attenuation between the Oaxaca and Guerrero regions. Comparison of these results show that in the frequency range of 1–30 Hz, attenuation in Oaxaca is between 3.7 to 5.0 times larger than the attenuation in Guerrero. Likewise, Yamamoto and Quintanar [1997] and Ottemöller and N. M. Shapiro [2002] observed lateral variations on the Q of Lg waves along southern Mexico. These studies concluded that propagation of Lg becomes inefficient in the Gulf of Tehuantepec and along the costal plains of the Gulf of Mexico. Contrary to Castro and Mungia [1994], they found that attenuation is higher in the Oaxaca region compared with the Guerrero region. García and Singh [2009] found changes in the attenuation between inland paths and paths along the Pacific coast for interplate events. This study suggests that the structure of the geometry and structure of subduction zone cause changes in the geometrical spreading. Cruz-Jiménez and Chávez-Gracía [2009] used 2D numerical simulations to analyze the influence of the Trans-Mexican Volcanic Belt (TMVB) and the geometry of the subduction zone for trajectories parallel and perpendicular to the trench for the Colima and Guerrero regions. Their results suggest that differences in the geometry of the subducting slab do not significantly influence the attenuation. They propose that attenuation is more effectively affected by the location of the TMVB. Margerin and Campillo [1999] applied a simple layered model to the set of stations along the coast of Mexico and concluded that a frequency-independent quality factor of QInt=1000 can adequately fit coda attenuation QCoda estimates in the frequency range of 1≤f≤15 Hz. They suggested that leakage of energy into the mantle at low frequencies ∼1 Hz heavily affects the coda attenuation, while high frequencies are mostly determined by the intrinsic attenuation. In brief, these studies suggest that attenuation of Lg, S and P waves are frequency dependent if interpreted as a combination of the inelastic and dissipative properties of the medium.

Table 1. Previous Studies
Oaxaca0.7< f<1.7 HzQLg=208 f0.4Canas and Egozcue [1988]
Chiapas0.7< f<1.7 HzQLg=211 f0.4Canas and Egozcue [1988]
Guerrero0.1< f<40 HzQS=278 f0.92Castro and Anderson [1990]
Guerrero0.2< f<10 HzQS=273 f0.66Ordaz and Singh [1992]
Oaxaca1.0< f<25 HzQP=22 f0.97Castro and Mungia [1993]
Oaxaca1.0< f<25 HzQS=56 f1.01Castro and Mungia [1993]
Guerrero1.0< f<15 HzQInt=1000Margerin and Campillo [1999]
TMVB1.6< f<8 HzQLg=226 f0.486Ottemöller and N. M. Shapiro [2002]
Guerrero1.6< f<8 HzQLg=193 f0.872Ottemöller and N. M. Shapiro [2002]
Oaxaca1.6< f<8 HzQLg=228 f0.895Ottemöller and N. M. Shapiro [2002]
Central TMVB0.25< f<8 HzQLg=98 f0.72Singh and Iglesias [2007]

1.1 Attenuation Components and Frequency Dependance

[3] Correct evaluation of the different components of attenuation is important for studies of propagation of seismic waves in the upper crust at high frequencies (f>1 Hz). Typically, three major factors are associated with the decay of seismic waves: (1) geometrical spreading which describes the shape of the expanding wavefront, (2) intrinsic or inelastic attenuation which accounts for conversion of energy into heat and (3) scattering attenuation due to redistribution of energy in the medium. Nearly every study that attempts to separate these contributions finds that both scattering and intrinsic attenuation vary with increasing frequency as fγ where γ∼1 (e.g., Table 1). However, laboratory experiments suggest that intrinsic Qi has a weak to zero frequency dependence, i.e., (γ∼0). In a classic paper, Knopoff [1964] first reviewed the frequency dependence of math formula in homogenous materials concluding that conversion of energy into heat in solid is nearly independent of frequency while in liquids is proportional to frequency, f. Faul and Fitz Gerald [2004] observed that cancellation between negative and positive dependencies of dissipation leads to a nearly frequency-independent behavior in partially molten regions of the upper mantle. Experimental work on essentially melt-free polycrystalline olivine shows a generally systematic mild dependence on period across the range 1–1000 sec increasing with increasing period approximately as the 1/4– 1/3 power of period [Jackson and Faul, 2010].

[4] We use data from broadband seismic networks in Mexico to test whether the differences can be resolved by modifying the standard scattering models: (1) by depth dependent distributions of scatterers; (2) forward or back scattering; and (3) leakage of energy into the mantle. We find that (3), adding the leakage term reconciles estimates based on seismic data with laboratory experiments and radiative transfer theory [Hoshiba et al., 1991], resulting in an average frequency-independent intrinsic attenuation for the Middle America region and frequency-dependent scattering and leakage attenuations.

2 Data

[5] We carried out the evaluation and separation of the components of attenuation along two profiles: (1) perpendicular to the trench for 25 stations from the Middle America Subduction Experiment (MASE) array, and (2) along the trench for permanent stations. The local seismological service (SSN) reported a total of 1354 local events during the MASE recording period 2005–2007. The magnitudes of that reported in the local catalog range between 2.9≤Mw≤5.4. Figure 1 shows the location of the stations analyzed and the local seismicity color-coded as a function of depth. Note the irregular distribution of the seismicity along the trench. In the southeast, along the Isthmus of Tehuantepec, seismicity extends from the trench up to ∼300 km inland. This area corresponds to the steeply dipping section of the slab where the seismicity progressively deepens from the trench up to 250 km [Melgar and Pérez-Campos, 2010; Kim et al., 2011; Chen and Clayton, 2012]. Further West, the seismicity is concentrated in a narrow band of ∼100 km between the trench and inland near the coast, where most of the seismicity occurs in the upper ∼60 km. Beyond this area, between the coast and the TMVB, seismic activity scatters which coincides with the flat part of the subducting slab [Pérez-Campos et al., 2008; Kim et al., 2012]. In this section, slow slip events have been widely documented [Lowry et al., 2001; Kostoglodov et al., 2003; Yoshioka et al., 2004; Franco et al., 2005; Vergnolle et al., 2010; Radiguet et al., 2011; Walpersdorf et al., 2011; Husker et al., 2012] suggesting a weakly coupled interface. Blue triangles indicate stations from the MASE array, and green triangles denote permanent stations used in this study. Only seismic records with a signal to noise ratio higher than 3 and hypocentral distance between 40≤dhypo≤140 km are used. The amplitude of the signal in this case corresponds to the integrated energy density in a window between 30 and 45 sec from the arrival of the S- wave.

Figure 1.

Map of seismic stations and seismicity in Mexico. Blue triangles show the location of the MASE array; green triangles indicate the location of the permanent local stations used in this study. Circles indicate the seismicity between 2005 and 2007 color-coded by depth. TMVB—Trans-Mexican Volcanic Belt.

3 Separation of the Components of Attenuation Based on the MLTW Method

[6] The Multiple Lapse Time Window Method (MLTW) initially formulated by Hoshiba and Sato [1991], uses temporal and spatial variation of seismic coda to separate the contribution of intrinsic and scattering attenuation. It is based on the radiative transfer theory approach introduced by Wu and Aki [1985] and Zeng [1993]. The MLTW method has been applied to numerous regions around the world such as central California, Long Valley, and Hawaii [Mayeda et al., 1992]; Kanto-Tokai region Japan [Fehler et al., 1992]; Northern Chile [Hoshiba et al., 2001]; West Greece [Tselentis, 1998]; South Korea [Chung et al., 2010]; Southern California [Jin et al., 1994]; Cajon Pass, California [Adams and Abercrombie, 1998]; Turkey [Akinci and Eyidoǧan, 2000]; Italy [Bianco et al., 2005; Del Pezzo et al., 2011]; central Andes [Badi et al., 2009]; Spain [Akinici et al., 1995]; Canada [Feustel et al., 1996], among other studies.

[7] We apply the MLTW method to seismograms from the stations in Figure 1. Separation of the components of attenuation is achieved by solving the following energy conservation equation [Hoshiba, 1991],

display math(1)

where Ei=1,2, the normalized energy density in three time intervals of length T0=15 sec beginning with the arrival of the S- waves. math formula, math formula are the energy loss and scattering, attenuation, respectively. g is the scattering coefficient math formula, δ is the Kronecker's delta, r is the distance between the source and the receiver, v is the average S- wave velocity, and ω the angular frequency. ES(r,t) are the synthetic energy density envelopes obtained using Monte Carlo scattering simulations, and W0 is the total energy used in the simulations. The simulations do not include loss mechanisms, and so are modified to fit the data by adjusting QLossin the exponential term and by scaling the scattering coefficient, g. The product 4πr2 on the left hand side accounts for the energy decay due to geometrical spreading for the simulated energy curves. The first term on the right-hand side is the direct energy that travels to the receiver at the shear wave velocity. It is less than the total energy due to the combined effect of scattering and energy loss. The second term is the scattered energy that arrives after the direct energy as part of the coda after correcting for energy loss and geometrical spreading. Useful combinations of these parameters include the seismic albedo, math formula, i.e., the fraction of total attenuation attributable to scattering, total attenuation, math formula and extinction length Le=vQTot/ω, which is the distance over which the initial energy decays by a factor of exp(−1).

[8] In the original description of the method [Hoshiba, 1991], the loss term is attributed solely to conversion of energy into heat or intrinsic attenuation math formula. In the following sections, we will examine the contribution of other mechanisms that remove energy from the system, such as leakage of energy into the mantle. In the meantime, we test the hypothesis that math formula.

[9] Figure 2 illustrates the energy density windows analysis. The frequency dependence of Q is obtained by application of the MLTW method for data separated into different frequency ranges. First, we apply a bandpass filter at different frequencies. We used a 4−pole Butterworth filter with central frequencies at fc=0.75,1.0,1.5,2.0,3.0,4.5,6.0,9.0,12.0,15.0 Hz. Corner frequencies are estimated as function of the central frequency as fhigh,low=(1±1/3) fc. The envelope of the seismogram is obtained by sampling the maximum absolute value in n windows of width Tw=1/flow. Then, we compute the normalized energy density for three 15 sec consecutive time windows after the theoretical S- wave arrival. Each envelope is normalized by dividing by the coda mean amplitude evaluated in a 5 sec window at fixed lapse time, in this case 45 sec. Coda normalization is applied to remove the site and source effects, which makes possible to compare events with different magnitude, and eliminate the local amplification factor due to the near surface conditions [Aki, 1969; Phillips and Aki, 1986].

Figure 2.

Example of the normalized energy analysis. (Top) The raw data for an arbitrary station, the vertical line indicates the S-wave arrival. (Middle) Filtered data between 1 and 2 Hz. Each seismogram is filtered using a four-pole Butterworth with bandwidth 2fcentral/3. (Bottom) Root mean square amplitude of the filtered signal. Shaded areas indicate the intervals of integration for each of the consecutive time windows.

[10] Once we have a significant number of empirical energy densities over a regional distance range (d<120 km), we estimate the best fitting parameters by grid search in the space (B0,Le−1). Figure 3a illustrates an example for station CARR in the frequency band of 4–8Hz. Each marker in the left panel corresponds to the normalized energy for a given event, for each of the three time windows indicated by the shape of the marker. In each case, we divide the available data within 30 bins. Solid lines show the average energy in a sliding window of ∼ 6−8km with 50% overlap. Dotted lines show the theoretical curves obtained by fitting the data to the numerical model based on equation (1). Figure 3b shows the grid search for this particular case, color contours shows the normalized χ2 misfit. The ellipsoid indicates the 95%confidence interval around the best fitting solution (star). Solid labeled contours show the solution space that yields to constant energy loss estimates (QLoss = QInt = constant). The separation of the components of attenuation is calculated using the following equations,

display math(2)


display math(3)

As laboratory experiments suggest, we seek to examine the frequency dependencies of both the scattering model and the geometrical spreading to see what modifications of the terms in equation (1) might be necessary to keep (QLoss=constant) assuming that this term is roughly equivalent to the intrinsic attenuation (QLossQInt) as commonly stated in the MLTW method.

Figure 3.

(a) The fit of equation (1) for the three time intervals for station CARR (MASE Array) in the interval 4–8Hz: 0–15sec squares, 15–30sec diamonds, and 30–45sec triangles. Solid lines show the average energy in a moving window of ∼6–8km; dotted lines correspond to the best fitting model obtained by grid search. (b) Parameter space: The star indicates the pair (B0,Le−1) that best fits the data; the ellipsoid shows the 95% confidence interval; and the labeled lines indicate the iso-contours of constant energy loss QLoss.

3.1 Forward or Back Scattering

[11] The original formulation of the MLTW method [Hoshiba, 1991] assumes that multiple isotropic scattering in the Rayleigh regime [Wu, 1989] is the dominant mechanism in the generation of coda envelopes. Over the past several decades, diverse studies have examined different non-isotropic scattering modes to model the coda wave decay. Aki [1969] described coda as the result of single backscattering of surface waves. Following work that showed that isotropic body waves dominate coda waves [Aki and Chouet, 1975]. Sato [1982] examined the assumption of isotropic scattering and suggested that early parts of the coda are dominated by forward scattering. Numerical simulations such as, Sato [1994, 1995]; Gusev and Abubakirov [1996]; Hoshiba [1995]; Margerin and Campillo [2000]; Sato [2008], included the effect of non-isotropic scattering on the computation of synthetic energy envelopes. These simulations find that broadening of the energy envelopes can be explained by forward scattered body waves.

[12] To examine non-isotropic scattering, we modified the MLTW method as suggested in Hoshiba [1995] to test how the frequency dependence of QLossdepends on the nature of the scattering propagation mode. The angular dependence factor is added to the simulations by assuming a medium with a Gaussian autocorrelation function. Hence, the product of the probability density function (equation (1) in Hoshiba and Sato [1991]) by the expression,

display math(4)

provides a non-isotropic radiation pattern controlled by the parameter μ. Where θis the angle between the incident field and the scattered wave and μis a constant that determines the preferential direction of propagation. If μ>0 forward scattering is stronger while negative values indicate the opposite situation, μ=0 corresponds to the isotropic case. Synthetic envelopes were obtained as described in Hoshiba and Sato[1991]. Figure 4shows the results of estimating energy loss and scattering attenuation using a non-isotropic model of scattering. Selection of the either forward or backward scattering has little influence in the estimates of the loss attenuation. In the frequency range of 2.0≤f≤15 Hz, the loss attenuation can be written as math formula for all cases. Increments of the parameter μis proportional to a positive change in the estimates of the scattering attenuation and therefore to the total attenuation. Figure 4d indicates how the solution space (B0,Le−1) is affected by the selection of the mode of propagation. A positive trend in both the extinction length and seismic albedo follows the increment in μ. The shape of the energy loss attenuation curves math formula shows a Gaussian form with frequency, similar Gaussian variation in attenuation has been reported by [Padhy, 2005] and is thought to arise from a Gaussian distribution of scatterers in space. In summary, the non-isotropic scattering mode does not affect the frequency dependence of any of the attenuation components. Neither forward nor backward scattering can solely resolve our hypothesis of frequency-independent intrinsic attenuation. In the next sections, we evaluate the consequences of varying both the depth range of the input data and the different components of the energy balance equation (1).

Figure 4.

Separation of intrinsic and scattering attenuation using different models of scattering for station CAIG. (a) Scattering attenuation math formula; (b) energy loss attenuation math formula; (c) total attenuation math formula, (d) solution space (B0,Le−1).

3.2 Depth Dependence

[13] An immediate improvement to the modeling of coda waves in a realistic earth model is the addition of a layered structure. Several authors have investigated the effect of assuming a layer overlying a half space in the frame of radiative transfer theory. Abubakirov and Gusev [1990] demonstrated that the mean free path increases with depth. Margerin and Campillo [1998] showed that the layered structure of the earth contributes to the trapping of energy in the crust, which trades off with leaking of energy into the mantle. Hoshiba [1997] obtained synthetic energy envelopes by calculating the transmission and reflection coefficients of energy particles in a random walk. Yoshimoto [2000] and Del Pezzo and Bianco [2010] found that the assumption of a half space leads to an overestimation of the scattering coefficient g. These studies highlight the role of the mantle in the transport of energy in the crust and the bias on the estimates of attenuation.

[14] We follow the direct simulation Monte Carlo (DSMC) method [Yoshimoto, 2000] to compute synthetic energy envelopes in a two layer model. Figure 5 shows the velocity profile and scattering coefficient as a function of depth used in the modeling. The model represents a 40km thick crust [Pérez-Campos et al.2008] with a velocity gradient overlying a transparent mantle with constant velocity. We obtained synthetic envelopes for eight characteristic models with scattering coefficients in the range of 0.01≤ g ≤0.5 km−1for the crust and g≡0 for the mantle. When a package of energy propagating in the crust reaches the crust-mantle interface at an angle lower than the critical angle, it is removed from the simulation and then lost. The effect of the free surface is also included within the simulations.

Figure 5.

Schematic model of the sources of attenuation of the MLTW-L method (section 4) and velocity and scattering coefficient profile used in section 3.3. This model assumes that the mean free path (g=0) in the mantle yields to infinity and negligible scattering attenuation.

[15] Figure 6 compares the results obtained for a half space and the two layered model for stations along the MASE array. We subdivided the incoming data into four different subsets corresponding to different depth ranges to examine the depth dependence of the sources. The marker shape indicates the type of the attenuation: Circles correspond to the total attenuation, triangles to the energy loss attenuation, and squares indicate scattering attenuation. We used the expression Q(f)=Q0 fγ to fit the Q estimates in the range of 3≤ f≤20 Hz, where Q0is the extrapolated attenuation at 1Hz. Different colors indicate the selection of the events as a function of depth. Black markers were obtained by analyzing sources in the upper crust 0≤d≤20 km; red markers indicate crustal events with depths of 0≤d≤40 km; blue markers correspond to a depth range of 0≤d≤100 km, and green markers constraint the focal depth to a range of 20≤d≤100 km. Solid markers indicate that the model used to compute the synthetic envelopes corresponds to an infinite medium with an even distribution of scatters. Open markers show the results using the two-layer model.

Figure 6.

Summary of results using the MLTW for different depth ranges for the model Q(f)=Q0fγ for frequencies 3≤f≤15 Hz. Solid markers indicate the results obtained using an infinite medium while open markers correspond to a two-layer model. Depth ranges are color-coded: Black 0≤depth≤20 km; red 0≤depth≤40 km; blue 0≤depth≤100 km; and green 20≤depth≤100 km. Type of attenuation is indicated by the shape of the markers: Total attenuation QTot(Q0,γ) circles, energy loss QLoss(Q0,γ) triangles, and scattering attenuation QSct(Q0,γ) squares.

[16] Using a layered medium for the inversion leads to higher values of quality factor Q0 by roughly a factor of 2 in all components. Furthermore, the event classification reveals mild variations of the results regardless of the focal depth of the sources. In all cases, the parameter γ shows little fluctuations around unity, which indicates strong frequency dependence. This is a commonly observed result for the total and scattering attenuation as well as intrinsic attenuation. Our results are in agreement with borehole observations using the same method [Adams and Abercrombie, 1998].

3.3 Leakage

[17] The assumption that the mantle acts as an absorbing medium with infinite mean free path (g=0), produces an energy-loss type attenuation effect that removes energy from the simulations (see Figure 6). In the previous sections, we found that application of the MLTW yields energy-loss attenuation with high frequency dependence. If we assume that the energy loss is purely due to the conversion of energy into heat, these results contradict laboratory experiments [Knopoff, 1964; Faul et al., 2004; Jackson and Faul, 2010] and field observations [Hasegawa, 1985; Margerin et al., 1999; Davis and Clayton, 2007] which suggest a weakly frequency-dependent intrinsic quality factor of the order QInt∼1000. Of all the alternatives analyzed in the previous sections using the MLTW method, we discovered that assuming a spherical geometrical spreading with a frequency dependent QLoss(f) and QSct(f) provided the best fit to the data. To reconcile the experimental constraints, i.e., keeping intrinsic attenuation constant, we separate the exponential decay in equation (1) into energy losses caused by intrinsic attenuation and loss from leakage towards the mantle so that

display math(5)

Leakage is the result of scattering in the vertical direction into the mantle where the energy is absorbed rather than back-scattered. This will include both downwards-scattered energy and upwards-scattered energy that reflects from the free surface.

[18] Energy loss associated with leakage is then expected to have the same exponential dependence on frequency as the scattering operator. Margerin and Campillo [1999] found that leakage follows an exponential decay and demonstrated that a frequency dependent Qi is not required when considering mantle absorption. The modification to equation (1) leaves the above fits to the data exactly the same, but the interpretation is different. The modified energy conservation law for the MLTW is then rewritten as,

display math(6)

This model (hereafter referred as MLTW-L) implies that leakage of energy in the original implementation may have been misinterpreted as frequency-dependent intrinsic attenuation. We reinterpret our results by keeping the geometrical spreading constant and assuming constant intrinsic attenuation, which must have a value less than, or equal to, the high frequency (asymptotic) value of total attenuation. Figure 7 shows the analysis for station CAIG. Figure 7a shows the results obtained by direct application of the MLTW, while the Figure 7b shows the reinterpreted results, MLTW-L, assuming that the intrinsic quality factor is QInt=2000. The curves show strong leakage of energy at frequencies between 1≤f≤4.5 Hz, which fades at higher frequencies. In the original formulation, the loss term was attributed solely to intrinsic attenuation math formula. However, we conclude that leakage of energy into the mantle must be included.

Figure 7.

Reinterpretation of the results for station CAIG. (a) Results for the direct application of the MLTW method. (b) Results assuming that the intrinsic attenuation is independent of frequency and leakage of energy towards the mantle according to equation (6).

4 Application of the MLTW-L Model to MA Data

[19] In this section we apply the modified form of the MLTW method (MLTW-L) to the data from the MASE array, perpendicular to the trench, and stations of the Mexico Seismic Network, parallel to the trench. This involves interpreting what was formerly designated as intrinsic attenuation in traditional application of the MWLT method to be the combined effect of the energy leakage into the mantle and intrinsic attenuation. Figure 8 shows the frequency dependence of the seismic albedo B0 and the inverse extinction length Le−1for 25 stations from the MASE array, i.e., perpendicular to the trench. We reinterpret seismic albedo as the fraction of total attenuation attributable to scattering, where total attenuation includes scattering, leakage and intrinsic attenuation, i.e., math formula. As suggested by laboratory results, we will assume that the intrinsic attenuation is constant math formula. Seismic albedos among all stations between stations ACAP and PETA show similar values at the different frequencies. Low frequencies show higher seismic albedo, suggesting that attenuation is mostly dominated by the effect of heterogeneities in the crust, while attenuation at higher frequencies is dominated by loss terms such as mantle leakage, conversion of energy into heat and anelastic deformation of the rocks. Between stations UICA and MAXE, estimation at low frequencies (0.75 and 1.0 Hz) becomes unstable. This is predominantly caused by the low signal to noise ratio due to a change in event size and the near surface lithologies. On the other hand, the extinction length shows low frequency dependence regardless of the location of the station.

Figure 8.

(a) Seismic albedo B0 as a function of frequency for the MASE array. Each line denotes a different frequency. (b) Extinction length Le−1 as a function of frequency. Stations are sorted by latitude, ACAP is the closest to the coast.

[20] Figure 9 shows the leakage math formulaand scattering attenuation math formulafor the same set of stations. We observe that the leakage attenuation at low frequencies (0.75≤fcentral≤3.0 Hz) remains relatively constant. Conversely, scattering attenuation at higher frequencies (f>3 Hz) decreases as frequency increases. In both cases, attenuation varies smoothly between stations. In general, the MLTW-L model finds that most parts of the spectrum are dominated by energy loss due to leaking of energy into the mantle and scattering in the crust. The frequency dependence significantly changes between low (f<3 Hz) and high frequencies (f>3 Hz).

Figure 9.

(a) Leakage attenuation math formula as a function of frequency for the MASE stations. Each line corresponds to a different frequency band. (b) Scattering attenuation math formula as a function of frequency. Stations are sorted by latitude. ACAP is the southern most station.

[21] Average results for all stations perpendicular to the trench are shown in Figure 10. The frequency dependence of the variables B0and Le−1disappears for frequencies larger than 3Hz as shown in Figures 10a–10c. Figure 10d shows the average attenuation curves for the independent estimates of leakage, scattering and total attenuation.

Figure 10.

Average results for the MASE array. (a) Seismic albedos B0 as a function of frequency; (b) extinction length Le−1 and (c)scattering coefficient g. (d) Shows the average attenuation math formula, math formula and math formula for all stations.

[22] Scattering attenuation along the coast was estimated at stations of the Mexico Seismic network (MMIG ZIIG CAIG PNIG, Figure 1). In general, the average attenuation values are similar within the one sigma error bars to values perpendicular to the trench. Some of the estimates of scattering attenuation are lower. Although the difference between these profiles is statistically weak, it is consistent for all stations along the coast as shown in Figure 11. The solid line corresponds to the average results for the MASE array (Figure 10). For leakage and scattering attenuation, both profiles perpendicular and parallel to the trench fit the data remarkably well with the exception of the ZIIG station at 0.75 Hz. This is likely caused by the reduced number of events recorded in the station and the usually low signal to noise ratio at low frequencies.

Figure 11.

Comparison between the attenuation along the MASE array and stations along the coast. (a) Leakage attenuation math formula, (b) scattering attenuation math formula, (c) total attenuation math formula.

5 Discussion

[23] We evaluated the sources and frequency dependence of the attenuation in the MA region using the MLTW method. In the first instance, we redefine the intrinsic attenuation term from the original formulation of the MLTW as the combination of all possible energy-loss mechanisms. This assumption led to the commonly observed result that the energy loss usually associated to the intrinsic attenuation is frequency dependent. Therefore, we analyze a set of alternatives to resolve this apparent contradiction between the laboratory experiments, which suggests that the intrinsic attenuation is frequency independent and results from application of the MLTW method. A priori assumption of the MLTW method indicated that the scattering mechanism is dominantly isotropic. Therefore, we tested different scattering operators such as forward and backward scattering. We assume a medium with a Gaussian autocorrelation function to examine the effect of non-isotropic scattering. Our analysis concludes that the selection of the preferential mode of propagation has little influence in the final estimates of energy loss attenuation and does not remove the frequency dependence. Furthermore, we compared attenuation measurements based on preselection of the events over different depth ranges, and a two-layer crustal model with a velocity gradient [Yoshimoto, 2000]. Our estimates show similar results regardless of separation of the sources by depth. This suggests that most of the attenuation by scattering is mainly generated in the upper part of the crust, which is in agreement with borehole studies in southern California and Japan [Adams and Abercrombie, 1998; Kinoshita, 2008]. Based on the standard MLWT model, we obtained energy loss and scattering attenuation curves for all MA stations with strong frequency dependence. Our estimates of the parameter γ in Q=Q0fγ, are close to unity. The selection of the synthetic scattering model used in the calculation of the energy curves produces small variations in the estimates of the parameters Q0 and γ. The two-layer model results in higher values ofTotQ0 compared with the infinite medium. The opposite case occurs for the estimates ofLossQ0 andSctQ0 where the infinite model gives rise to slightly higher values. These differences are not statistically significant for the energy loss and total attenuation parameters, but have a stronger influence on the estimates of seismic albedo and consequently scattering attenuation.

[24] We proposed that adding an exponential decay to the scattering component of the energy in the MLTW model accounts for the energy leakage by scattering into the mantle. This extra term allows us to keep the intrinsic attenuation frequency independent and takes into account the layered structure of the scattering properties of the crust and mantle.

[25] Application of the MLTW-L model to determine the scattering properties of the Middle America region at high frequencies (>1 Hz) shows similar results in both directions of propagation: perpendicular and parallel to the trench. For the inland array, the estimates of (B0, Le−1) of the MLTW-L between stations ACAP and PETA are comparable at all frequencies. At low frequencies (≤3 Hz), the northernmost stations (UICA to MAXE) exhibit higher values in the estimates of seismic albedo and lower values for the extinction length. These differences can be explained as arising from two possible mechanisms: (1) changes in the near surface lithology, and (2) bias due to low signal to noise ratios. We favor (2) as the explanation, because at the northern stations, many of the events have smaller magnitudes compared with the events in the vicinity of the trench and so signal to noise is less. In general, results are similar to previous estimates in the region mentioned in section 1 (Canas and Egozcue [1988], Castro and Anderson [1990], Ordaz and Singh [1992], Castro and Mungia [1993], Castro and Mungia [1994], Ottemöller and N. M. Shapiro [2002]), but the interpretation is different.

[26] Differences in attenuation are more apparent in the estimation of math formulabetween the profiles perpendicular and parallel to the coast. For the stations parallel to the coast, scattering attenuation is slightly smaller compared with those along the perpendicular profile. This might be explained by the wedge-type shape of the crust near to the trench that causes changes in either the mode of propagation or the distribution of heterogeneities.

6 Conclusion

[27] As has been found in most regions of the world, the MLTW method applied to seismic data from Mexico gives rise to frequency-dependent scattering and intrinsic attenuation. Intrinsic attenuation varies with a frequency exponent of γ≈−1. Values perpendicular and parallel to the trench are similar with scattering attenuation slightly lower parallel versus perpendicular. Given that experimental data find that intrinsic attenuation of crustal and mantle rocks is approximately frequency independent, we examine whether the apparent frequency dependence found from application of the MLWT method arises from the difficulty of separating the two phenomena. We tested for layered scatterers, forward and backward scattering and crustal leakage. Of these, we favor crustal leakage as the simplest modification to the MLTW model that both fits the data best, and satisfies the physical constraint of frequency independent QInt. This reinterpretation shows that leakage is an important contributor to the total attenuation for frequencies between 1≤f≤15 Hz, and that intrinsic attenuation is nearly constant, math formula. Additionally, classification of the events into different depth ranges produces small variations in the results, which suggests that near surface attenuation dominates the average value of Q over the crustal volume. We conclude that most of the attenuation and scattering is generated in the near to the surface layers and that loss by scattering into the deeper layers is frequency dependent but intrinsic Q is not.


[28] The authors would like to thank Prof. Edoardo Del Pezzo and an anonymous reviewer for their valuable comments, which helped improve the manuscript and the interpretation of the results. This research was supported by the Southern California Earthquake Center. SCEC is funded by NSF Cooperative Agreement EAR-0529922 and USGS Cooperative Agreement 07HQAG0008. The SCEC contribution number for this paper is 1644.