The seismicity of Popocatépetl is dominated by long-period and very-long period signals associated with hydrothermal processes and magmatic degassing. We model the source mechanism of repetitive long-period signals in the 0.4–2 s band from a 15-station broadband network by stacking long-period events with similar waveforms to improve the signal-to-noise ratio. The data are well fitted by a point source located within the summit crater ∼250 m below the crater floor and ∼200 m from the inferred magma conduit. The inferred source includes a volumetric component that can be modeled as resonance of a horizontal steam-filled crack and a vertical single force component. The long-period events are thought to be related to the interaction between the magmatic system and a perched hydrothermal system. Repetitive injection of fluid into the horizontal fracture and subsequent sudden discharge when a critical pressure threshold is met provides a non-destructive source process.
 Popocatépetl Volcano is a 5,452-m-high andesitic stratovolcano located in the central region of the Mexican Volcanic Belt. Since its reawakening in December 1994, seismicity at Popocatépetl has been dominated by long-period (LP) and very-long period (VLP) signals accompanying emissions of ash and gas and the extrusion and destruction of lava domes [Arciniega-Ceballos et al., 1999; Arciniega et al., 2003; Chouet et al., 2005].
 LP seismicity at many active volcanoes exhibits stable source mechanisms associated with repetitive non-destructive processes [Kumagai et al., 2002; Nakano et al., 2003; Kumagai et al., 2005; Waite et al., 2008]. Previous analyses of LP seismicity at Popocatépetl led to the identification of three event families [Arciniega et al., 2008], among which the Type-I family was found to be prevalent. Modeling the source processes of the LP events observed at Popocatépetl is challenging due to low signal-to-noise ratios, commonly emergent arrivals, and the strong effects of topography and heterogeneous structure at periods shorter than ∼1 s. In order to derive a robust quantification of the source mechanism using waveform inversion a well-recorded signal is required. The repetitive character of this seismicity naturally lends itself to the application of waveform stacking to enhance the signal-to-noise ratio, providing a clear signal at all stations that is representative of a typical event.
 In this study we focus on quantifying the source mechanism of LP signals using waveform stacks representing the Type-I event. We present a brief description of the broadband network, the seismic data, and the stacking procedure used to enhance the LP signal-to-noise ratio. We then describe the application of a waveform inversion method to the stacked signals in the period band 0.4–2 s and conclude with a discussion of the implications of the LP source mechanism for volcanic processes at Popocatépetl.
 We use data recorded by a seismic network consisting of 15 broadband seismometers deployed on the upper flanks of Popocatépetl Volcano (Figure 1), at elevations ranging from 3300 to 4470 m, and distances from the volcano summit from 1.6 to 5.4 km [Arciniega et al., 2008]. The network was in operation for 7 months beginning on 25.11.99. During this time eruptive activity was dominated by periodic emissions of gas and ash and episodic passive effusion of lava. The occurrence of degassing bursts ranged from a few events to hundreds of events per day.
 Type-I events [Arciniega et al., 2008, Figure 4a] represent the 1-min-long degassing bursts observed during the first few months of the experiment. These events exhibit a compressional P-wave arrival at all stations and are distinguished by a repetitive harmonic wave train with dominant periods in the 1.4–1.9 s range that last for 5–10 s. An airwave arrival, particularly evident on the horizontal components, arrives at all of the stations within 5–15 s of the initial P-wave arrival. A very-long-period wavelet with a period of ∼30 s accompanies the larger LP events.
 We identify events during the time period 1 December 1999–5 February 2000 using cross-correlation. This period covers the operational time of station PPO2 (Figure 1) which we use as the reference station for identifying events. An event occurring at the end of December, 2000 was selected as the master event. Using a 6-s-long window that includes 1 s of noise prior to the P-wave arrival, we compute cross-correlation coefficients with the master event through the continuous seismic record of the vertical component of station PPO2. A total of 584 events with correlation coefficients >0.8 were identified.
 Waveforms for all matching events were then aligned, summed, and normalized by the respective number of events observed on each component to provide network-wide stacks. All of the stations have the same number of events contributing to each stack with the exception of station PPP3, which became operational after 14 January 2000. Temporal variability in event amplitudes after PPP3 was installed suggests a possible overestimate in the normalized stack amplitudes of a few percent may occur at station PPP3. This temporal variation is small, and not corrected for at station PPP3. The instrument response was then removed from each stack, providing the representative ground displacement. Choosing events with correlation coefficients greater than 0.9 decreases the number of available waveforms for the stacks by a factor of 10 with a subsequent increase in pre-event noise due to the lower number of events. As demonstrated below, the source-time functions and source locations derived from waveform inversions for various choices of correlation coefficient are similar. This indicates that the source remained stable for the considered two-month-long analysis, and that by including more events the signal-to-noise ratio in the stack is increased, thus improving the quantification of the source process.
Figure 2 shows the resulting stacked waveforms (black lines) with correlation coefficients >0.8 for eight stations closest to the summit crater. The waveforms are similar between individual components at different stations, and the upward polarity of first motion on the vertical components is suggestive of a volumetric source.
3. Waveform Inversion
 We perform full-waveform inversion of the LP signals assuming a point source embedded in a homogeneous elastic medium that accounts for the extreme topography of Popocatépetl. We follow the procedures outlined inWaite et al. . Minimization of the residual error between the data and synthetics calculated using finite differences [Ohminato and Chouet, 1997] provides the source centroid location and source mechanism.
 The 3-D velocity structure of Popocatépetl is unknown so our calculations are performed for a simple homogeneous velocity model discretized at 50 m with compressional wave velocityVp = 4 km s−1, shear wave velocity Vs = 2.3 km s−1, and density ρ = 2650 kg m−3. With these velocities, the corresponding wavelengths in the period band 0.4–2 s range from 1 to 8 km. The wavelength of shear waves at the shortest-period of 0.4 s is 1 km, and at these wavelengths the effects of km-scale structural heterogeneities may be important for stations located at distances of 3 km or more from the crater. Although not considered here, the choice of velocity structure will affect the amplitude of source mechanisms. An overestimation of the source amplitude occurs when using velocities faster than the actual velocity, and an underestimation of source amplitude occurs when using velocities slower than the actual velocity [Chouet et al., 2003]. Spurious single force components may also occur due to unknown velocity structure, particularly when stations greater than two to three wavelengths from the source are included in the inversion [Bean et al., 2008].
 A search for the location of the best fit source centroid is carried out over a volume centered on the summit crater, extending 1.4 km in the east-west and north-south directions, and spanning a depth range from the crater floor at 5.1 km elevation down to 2.7 km elevation. This portion of the domain encompasses the source region corresponding to the scatter in hypocentral distributions of LP seismicity [Arciniega et al., 2008] and includes the best fit centroid location of the source of VLP signals associated with Vulcanian explosions [Chouet et al., 2005].
 We first consider source mechanisms consisting of either three single force components, or six moment tensor components, or six moment tensor components and three single force components. The physical significance of the resulting source mechanism is based on the residual error, relevance of the free parameters in each model (evaluated using Akaike's Information Criterion (AIC) [Akaike, 1974]), and the plausibility of the resulting moment tensor.
 Based on residual error and AIC, the spatial search for the three types of source mechanism indicates the data is best fit using the mechanism composed of six moment tensor and three single force components (see Table 1). Including the more distant stations (PPJ4, PPP4, PPQ3, PPQ5, PPX3, and PPX4) in the waveform inversions results in a significant increase in residual error and does not provide an interpretable moment tensor. As indicated above, structural heterogeneity is not accounted for in our Green's functions and the effect of unknown structure increases with station distance. Therefore, we exclude these stations in the inversions. Restricting the waveform stacks to events with correlation coefficients >0.85 reduces the number of events to 295 with no significant change in the source location or error statistics (see Table 1). Restricting the waveform stacks to events with correlation coefficients >0.9 reduces the number of considered events to 66. The observed residual error (Table 1) increases due to the lower number of events and subsequent decrease in signal-to-noise ratio. The source location and source-time functions remain similar to those found for the other stacks.
Spatial locations X, Y, and Z are represented as nodes in the model space with node spacing equivalent to 50 m. The center of the crater floor is node 184,166,214 where Z(214) = 5100 m elevation.
Correlation coefficient used to obtain the event stack.
6 moments + 3 forces
6 moments + 3 forces
6 moments + 3 forces
1 crack + 3 forces
2 cracks + 3 forces
6 moments + 3 forces
 The source centroid is located 250 m below the southern portion of the crater floor with a relative spatial uncertainty of ±250 m (all nodes within 2% of the minimum residual error). Figure 2shows the waveform fits (green lines) for the best-fit model composed of moments and forces. The largest misfits are observed for the north component at PPJ3, and the east component at PPQ2 (marked in blue inFigure 2). PPJ3 is about 3.5 km distant from the source and unknown structural heterogeneity may contribute to this misfit. We are unable to explain the high-amplitude signal on the east component of PPQ2, and do not include this component in the waveform inversion.
 The moment tensor for the best fit model contains in-phase energy dominantly seen in the dipole components, indicating the mechanism represents a volumetric source; the volumetric source is accompanied by a single dominantly-vertical force. A tensile crack has an equivalent force system made of three vector dipoles with amplitudesλΔV, λΔV, and (λ + 2μ)ΔV, where λ and μ are the Lamé constants of the host rock, ΔV represents the volume change associated with the crack opening or closing, and where the dominant dipole is oriented normal to the crack plane [Aki and Richards, 1980; Chouet, 1996]. For a Poisson solid (λ = μ) the ratios of the principal axes for a horizontal crack are [1:1:3]. The point-by-point eigenvalue decomposition of the moment tensor shows the mechanism is stable with time and provides mean eigenvalues [1.3:1.5:3.0] scaled by 2.1 · 1011 Nm. The eigenvalue ratios suggest the source mechanism consists of a nearly horizontal crack, with a possible secondary component. The mean orientation of the maximum eigenvector (where the azimuth, ϕ, is measured counter-clockwise from east and the plunge,θ is measured from vertical) is ϕ = 377° and θ= 14°. The matrix addition of eigenvalue ratios representing a sub-horizontal crack with 80% of the source energy, and a sub-vertical crack with 20% of the source energy provides ratios [1.2:1.6:3.0] after normalization. These estimates are similar to the observed ratios and suggest the source mechanism is dominantly a horizontal crack.
 To test the possibility that the source mechanism may consist of a crack or combination of cracks, we conduct inversions using Green's functions representing one or a combination of two cracks plus three single forces. The crack moment tensors are calculated using equation (15) of Chouet  assuming λ = μ. Using a spatial grid search similar to that used for the previous models, and by varying the orientation of each crack model until a minimum residual error is reached at each node, we find the source orientation and location that best represent the single crack and dual crack systems (Table 1). Each of these models contain a horizontal component with an orientation similar to that observed from the inversion using six moments and three single forces (single crack: ϕ = 21°, θ ≃ 0° and dual crack: ϕ1 = 1°, θ1 = 14° and ϕ2 = 284°, θ2 = 46°). The spatial location of these models (see Table 1) is constrained to within ±250 m of the minimum error location for each mechanism. The single crack model and dual crack models are located below the eastern portion of the crater floor at depths of 150 and 450 m, respectively.
 A comparison between waveform inversion of the master event with the inversions for stacked waveforms is shown in Table 1. The inversion for a single event results in higher residual error and AIC, a very large range in possible source location, and a moment tensor that is not interpretable. This is due primarily to the low signal-to-noise seen in an individual event, and underscores the importance of waveform stacking.
 While the single crack model with three single forces has higher residual error and AIC values, it provides the simplest source-time function that is representative of all of the considered models.Figure 3ashows the source-time function of the single crack model expressed as volume change for the crack mechanism providing the minimum residual error, andFigure 3b shows the associated three single forces. Figure 3c shows the modeled crack mechanism. The waveform fits for the single crack model are shown as red traces in Figure 2.
 The shallow depth, horizontal structure, and eigenvalue ratios suggest the source is a steam-filled crack. An application of the Sompi method [Kumazawa et al., 1990] to the vertical force component of the single crack model provides a Q value of ∼5 for the decaying oscillation of the single force. Such attenuation may be attributed to steam, bubbly water, or bubbly magma [Kumagai and Chouet, 2000; Kumagai et al., 2005]. The shallow source depth precludes the presence of a bubbly magma in this andesitic system, suggesting the fluid in the crack consists of steam or steam mixed with bubbly water. The source is positioned above all of the stations, implying the upward polarity of first motions seen on the vertical components of the network (Figure 2) represent initial deflation (depressurization) of the source. The source-time function for the crack model (Figure 3a) shows an initial deflation followed by a short sequence of inflation and deflation cycles. The deflation can be interpreted as a rapid loss of pressure in the crack when a pressure threshold is reached, allowing fluids to quickly migrate out of the crack. Resonance is then sustained in the crack for 5–10 s. The vertical single force exhibits an initial upward motion followed by cycles synchronous with those seen in the crack. This behavior is similar to that seen for LP events at Mount St. Helens, where the oscillations were attributed to the overlying magma column responding to the crack deflation [Waite et al., 2008]. The volume change seen in the crack is 1.6 m3. This value reflects the representative average event obtained from the stacked waveforms, while the absolute range of event amplitudes contributing to the stack indicates volume changes ranging from 0.5–8.6 m3.
 Previous moment tensor analysis of VLP waveforms associated with Vulcanian degassing at Popocatépetl pointed to the expansion and compression of a sill ∼1500 m below the crater floor coupled with smaller components of expansion and compression of a dike, which was interpreted to represent the primary conduit from the sill to the summit crater [Chouet et al., 2005]. The LP source is within a few hundred meters of the surface projection of the dike, and distinctly separated from the VLP source (Figure S1, auxiliary material). The close proximity and orientation of the LP source to the magmatic dike is similar to shallow hydrofractures observed at the Inyo Domes, California where horizontal and 7-40-cm-thick pyroclastic-filled fractures extend up to a few hundred meters from the central magmatic conduit at depths of 300–500 m [Heiken et al., 1988]. LP sources observed at Kusatsu-Shirane, Japan [Kumagai et al., 2002], Kilauea, Hawaii [Kumagai et al., 2005], and Mount St. Helens [Waite et al., 2008] exhibit source mechanisms at shallow depths similar to our derived mechanism and have been interpreted as being hydrothermally-driven horizontal fractures. These geologic and geophysical observations support our interpretation that the LP events originate from a shallow hydrothermal crack that is repeatedly pressurized due to a degassing magma column impinging on a shallow hydrothermal system. The rapid release of fluids occurs in a repeating, non-destructive manner as indicated by the similarity of events over time.
 Obtaining a robust quantification of the source properties of similar repeating LP seismicity at Popocatépetl Volcano is achieved through the identification of events using cross-correlation and the representation of a typical event obtained through waveform stacking. The enhancement of the signal provides clear evidence, through full waveform inversion, that the source is a horizontal steam-filled fracture located ∼250 m below the crater floor and subjacent to the magma conduit. Including more events in the stacked signal (reducing the constraint on cross-correlation coefficients) provides a clearer view of the source process compared to using fewer events with higher correlation. The technique described in this work provides a guide to deriving LP source mechanisms when the events have low amplitudes and occur in a noisy environment.
 This work was partially supported by grants from CONACYT 101515 and UNAM-PAPIIT IN106111. We thank Matthew Haney and two anonymous reviewers for their comments.
 The Editor thanks two anonymous reviewers for their assistance in evaluating this paper.