The source mechanism of long-period (LP) events is examined using synthetic waveforms generated by the acoustic resonance of a fluid-filled crack. We perform a series of numerical tests in which the oscillatory signatures of synthetic LP waveforms are used to determine the source time functions of the six moment tensor components from waveform inversions assuming a point source. The results indicate that the moment tensor representation is valid for the odd modes of crack resonance with wavelengths 2L/n, 2W/n, n = 3, 5, 7, …, where L and W are the crack length and width, respectively. For the even modes with wavelengths 2L/n, 2W/n, n = 2, 4, 6, …, a generalized source representation using higher-order tensors is required, although the efficiency of seismic waves radiated by the even modes is expected to be small. We apply the moment tensor inversion to the oscillatory signatures of an LP event observed at Kusatsu-Shirane Volcano, central Japan. Our results point to the resonance of a subhorizontal crack located a few hundred meters beneath the summit crater lakes. The present approach may be useful to quantify the source location, geometry, and force system of LP events, and opens the way for moment tensor inversions of tremor.
 Long-period (LP) seismicity, including LP events and tremor, is widely observed at active volcanoes and related areas in association with heightened volcanic activity. The LP event is characterized by decaying harmonic oscillations except for a brief time at the event onset, while tremor is marked by sustained harmonic oscillations. The characteristic frequencies of these signals share a similar typical range between 0.5 and 5 Hz [Chouet, 1996], which may represent resonances of a fluid-filled resonator in a magmatic or hydrothermal system. Differences between the LP event and tremor may be attributed to differences in their temporal excitation, which is time-localized in the LP event and sustained for tremor. LP seismicity may be viewed as the dynamic response of a fluid-filled resonator to pressure disturbances associated with mass transport and/or magmatic heat [Chouet, 1996]. The quantitative interpretation of these signals is thus critically important to our understanding of physical processes beneath volcanoes, which is crucial to the assessment and mitigation of volcanic hazards, and to eruption prediction.
 Although the above studies are primarily focused on the resonance frequencies of LP events, the LP waveforms themselves can also be used to quantify the source mechanisms of LP events and tremor. Nakano et al.  proposed a method to estimate the effective excitation function, that is the apparent excitation observed at an individual receiver, by applying an autoregressive (AR) filter to the LP waveform. M. Nakano et al. (Source mechanism of long-period events at Kusatsu-Shirane Volcano, Japan, inferred from waveform inversion of the effective excitation functions, submitted to Journal of Volcanology and Geothermal Research, 2002, hereinafter referred to as Nakano et al., submitted manuscript, 2002), later performed waveform inversions of these excitation functions to estimate the source mechanism of LP events observed at Kusatsu-Shirane Volcano, central Japan. The present paper complements these two studies and uses the oscillatory signatures of the LP waveforms, which were removed by AR filtering in the former studies, to estimate the source properties of LP events.
 It is relatively easy to perform waveform inversions of the oscillatory signatures of LP events. Such a study was performed by Aoyama and Takeo  for an LP event observed at Asama Volcano, Japan. Aoyama and Takeo  assumed a point source and determined the six moment tensor components of the source from a waveform inversion of a few cycles of oscillations in the LP waveforms. The interpretation of results from such waveform inversions may not always be straightforward, however, because the oscillatory signatures usually consist of a superposition of different resonance modes which can display complex spatial patterns of seismic wave excitation at the wall of a resonator of finite size [Chouet, 1992, Figure 2]. Detailed examination of the point source representation of such resonance pattern is required for a correct interpretation of results from waveform inversions. In this paper, we perform waveform inversions of synthetic LP signatures radiated by a fluid-filled crack model [Chouet, 1986, 1988, 1992] to test the point source representation of LP events. We then apply our inversion method to the oscillatory signatures of an LP event observed at Kusatsu-Shirane Volcano to demonstrate the usefulness of this approach.
2. Numerical Tests
 We use the crack model of Chouet [1986, 1988, 1992] to synthesize LP waveforms. The parameters of the crack are α/a, b/μ, W/L, and the crack stiffness C = (b/μ)(L/d), in which α is the compressional wave velocity of the rock matrix, a is the sound speed of the fluid, b is the bulk modulus of the fluid, μ is the rigidity of the solid, and W, L, and d are the crack width, length, and aperture, respectively. The ratios α/a and b/μ are related to ρf (density of the fluid) and ρs (density of the solid) through the relation
for Lame's coefficients λ = μ. We use a rectangular crack with a fixed width (W) to length (L) ratio of 0.5, and fixed length to aperture ratio L/d = 104. We further assume the ratios α/a = 7 and ρf/ρs = 0.018, or equivalently, b/μ =1.1 × 10−3 and C = 11, appropriate for a crack containing a misty gas (water droplet-H2O gas mixture) with gas-weight fraction of 50% [Kumagai and Chouet, 2000].
Figure 1a shows the locations of receivers used in our numerical tests of LP waveform inversion. We consider 14 three-component receivers distributed around the crack. We use the discrete wave number method [Bouchon, 1979; Chouet, 1982] to synthesize the three components of the ground velocity response to the excitation of a vertical crack embedded in a homogeneous half-space with compressional wave velocity α = 2700 m/s, shear wave velocity , and density ρs = 2500 kg/m3. The crack is set in the plane y = 0 and extends between the depths of 200 and 480 m, and from −70 to +70 m in the x-direction (Figure 1b). We use a crack excitation in the form of a step ΔP in pressure applied over a small area ΔS of the crack wall located at the center of the crack (see inset in Figure 2a). Figures 2a and 2b illustrate the vertical velocity waveform and corresponding amplitude spectrum obtained at S7 for the crack resonance triggered by a pressure step ΔP = 105 Pa. The dominant spectral peak near 2 Hz in Figure 2b represents the transverse mode with wavelength 2W/3.
 The synthetics calculated at the 14 receivers are inverted to reconstruct the source time functions of the six moment tensor and/or three single force components following the method developed by Ohminato et al. , in which we apply a slight procedural modification as given by Nakano et al. (submitted manuscript, 2002). In this method, the source time functions of the moment tensor and/or single force components are composed of a superposition of successive subevents generated with an elementary source time function at a fixed point source. A grid search is conducted around the initial point source location to find the best fit source location. This method was successfully applied to analyses of the source mechanisms of very-long-period (VLP) signals with periods longer than a few seconds observed at various volcanoes [Ohminato et al., 1998; Nishimura et al., 2000; Kumagai et al., 2001]. In the present study, we use a one-cycle cosine function as the elementary source time function S(t),
where tp represents the characteristic period. Compared to other possible elementary functions such as a smooth step or a Gaussian, a one-cycle cosine was found to be the most appropriate elementary function for inversions of the oscillatory signatures in LP waveforms. We determine the value of tp that best fits the waveform data in each individual inversion. A simple homogeneous half-space is assumed in the calculations of Green's functions.
 In our first test, we low-pass filter the three-component seismograms at 6 Hz and perform the waveform inversion for six moment tensor components assuming a point source whose location coincides with the center of the area ΔS over which the pressure transient is applied. Figure 3 shows the source time functions obtained by inversion. A cosine function with characteristic period tp = 0.5 s was found to be appropriate for this inversion. Waveform fits obtained for the vertical components of displacement are shown in Figure 4. The waveform data are nicely explained by the moment tensor solution in Figure 3, which is characterized by dominant volumetric change components (Mxx, Myy, and Mzz) and by oscillatory signatures superimposed on step increases in these components. The slight decreasing trends found in the volumetric change components after the initial step offsets are numerical artifacts caused by the coarse spatial discretization used in our finite difference calculations of the crack model. These drifts can be eliminated by the selection of a finer grid; however, such calculations require extensive computer time and were not attempted as they are not critical to the point being made here. The step amplitude of the Myy component is roughly three times larger than those of the Mxx and Mzz components, consistent with an expansion of the vertical crack in the y-direction. This result is in clear agreement with the step increase in pressure used for the crack excitation (Figure 1), pointing to a successful recovery of the source excitation in this inversion. It should be noted that the amplitudes of oscillations in the Mxx, Myy, and Mzz components show similar ratios to the step amplitude ratios.
 In our second test, we band-pass filter the seismograms between 1 and 3.5 Hz to remove the static components of the signals associated with the step in pressure and to emphasize the oscillatory features of the signals. We then perform an inversion of these waveforms assuming six moment tensor and three single force components to investigate the effect of a single force in the representation of the LP source (Figure 5). We also perform an inversion assuming six moment tensor components only (Figure 6). An elementary cosine function with characteristic period tp = 0.25 s was found to provide the best fits to the waveform data in these inversions, in which the best fit source locations coincide with the center of the crack excitation. As shown in Figures 5 and 6a, the differences between the two moment tensor solutions are very slight, indicating that a single force is not required in our description of the source mechanism. This is quite reasonable since mass advection, which is at the origin of the single force [Takei and Kumazawa, 1994], is not present in our model. The moment tensor solution in Figure 6a is dominated by oscillatory volumetric components in which the Myy component dominates over Mxx and Mzz. In Figure 6b, we plot the eigenvectors whose lengths are proportional to their respective eigenvalues calculated for the moment tensor solution shown in Figure 6a during the interval 3–6 s, in which we make no distinction between expansion and contraction for simplicity. The amplitude ratios are 1:3:1 for the x(N), y(E), and z(down) directions, respectively, indicating expansion and contraction of the vertical crack in the y-direction, a feature expected for this source.
 Our third test assesses the effect of source mislocation. We perform an inversion for a point source located 140 m in the positive y-direction from the best fit source location. We again use a one-cycle cosine function with tp = 0.25 s as the elementary source time function. The results of this inversion are shown in Figure 7. The amplitude ratios of the three principal axes of the moment tensor remain 1:3:1 (Figure 7b), but the directions of the principal axes are slightly rotated (compare Figure 7b with Figure 6b). Similar inversions performed for other source mislocations show that the directions of the principal axes change systematically with the horizontal mislocation of the assumed point source. The amplitude ratios and directions of the principal axes are not seriously affected by vertical mislocations of the source, but their amplitudes increase as the depth of the mislocated point source increases.
 In our fourth test, we decrease the number of receivers to evaluate the minimum number of receivers required to guarantee an adequate degree of resolution of the source mechanism in the waveform inversion of the LP event. We use the seismograms from the seven receivers S1-S7 (Figure 1a), whose spatial distribution mimics that used in the seismic network at Kusatsu-Shirane Volcano [Ida et al., 1989]. We use the three-component seismograms from S1, S4, S5, and S7, and vertical-component seismograms from S2, S3, and S6, as in the Kusatsu-Shirane network. We also perform a waveform inversion using only synthetic three-component seismograms from the four receivers S1, S4, S5, and S7. An elementary cosine function with tp = 0.25 s is used for both inversions. The inversion results based on synthetics from the seven receivers, and inversion results based on synthetics from the four receivers only, are shown in Figures 8a and 8b, respectively. As shown in Figures 6, 8a, and 8b, the scatter in the eigenvectors increases as the number of stations decreases. This increasing scatter is a consequence of the lack of resolution due to an insufficient spatial sampling of the wave field. There are significant differences in the moment tensor solution in Figure 8b compared to those in Figures 6 and 8a, suggesting that more than four stations featuring three-component sensors may be required for an accurate moment tensor inversion.
 Our fifth test investigates the dependence of the inversion results on the resonance mode. To examine other resonance modes in the waveform inversion, we keep the same crack geometry as in our earlier tests and calculate the crack excitation for a pressure step applied over a narrow strip located at a distance L/4 from the top of the crack and extending across the entire width of the crack (see inset in Figure 2c). The synthetic vertical velocity seismogram and corresponding amplitude spectrum observed at S7 are shown in Figures 2c and 2d, respectively. As illustrated in Figure 2d, the longitudinal modes with wavelengths L and 2L/5 are dominant in this crack excitation.
 The synthetic seismograms from the 14 receivers are first band-pass filtered between 1 and 3.5 Hz to emphasize the 2L/5 mode. We perform the waveform inversion assuming six moment tensor components, in which we use a one-cycle cosine function with tp = 0.25 s as the elementary source time function. The results (Figure 9a) indicate that the volumetric components are dominant and that the Myy component dominates over the Mxx and Myy components. As demonstrated in the eigenvectors in Figure 9b, the amplitude ratios are consistent with expansions and contractions of the vertical crack in the y-direction. Next, we use the seismograms band-pass filtered between 0.1 and 1 Hz to emphasize the L mode. We perform the waveform inversion assuming six moment tensor components, in which we use a one-cycle cosine function with tp = 2.0 s as the elementary source time function. The results (Figure 9c) again point to a dominance of volumetric components. As illustrated in Figure 9d, however, the amplitude ratios of the three principal axes of the moment tensor are not consistent with the vertical crack. Waveform fits obtained for the vertical components of the displacement seismograms for the L mode are shown in Figure 10, where discrepancies in the phase are apparent in most of the traces. The inconsistency is related to the mode of oscillation considered and its implications are discussed in more detail later in the paper.
 Although an inconsistency is apparent in the results obtained for the L mode, our numerical tests suggest that a point source moment tensor inversion of the oscillatory signatures of LP waveforms may provide useful information on the geometry and force system of the LP event, provided the inversion is based on more than four three-component receivers surrounding the source. Our inversion results are obtained for synthetic waveforms calculated for fixed values of the crack parameters. As the complex frequencies of oscillating signatures in synthetic waveforms are sensitive to the values of the crack parameters [Kumagai and Chouet, 2000], the values selected for the crack parameters affect the choice of the appropriate tp used for waveform inversions. However, this does not affect our main conclusions based on the numerical tests discussed above. In the following, we apply this inversion scheme to an LP event observed at Kusatsu-Shirane Volcano, Japan.
3. Application to an LP Event at Kusatsu-Shirane
 Kusatsu-Shirane is a composite andesitic volcano located in central Japan (Figure 11). LP events characterized by nearly monochromatic oscillation signatures have been frequently observed at this volcano [Hamada et al., 1976; Fujita et al., 1995; Nakano et al., 1998]. Kumagai et al.  performed a detailed study of the temporal variations in the complex frequencies of LP events observed at this volcano during the period August 1992 through January 1993, during which significant variations in both frequency and Q factor were observed. They used the acoustic properties predicted by the crack model in their interpretation of the temporal variations, and showed that such variations can be consistently explained by the dynamic response of a crack containing hydrothermal fluids to a magmatic heat pulse. These results point to the existence of a hydrothermal crack at the source of LP events beneath Kusatsu-Shirane Volcano.
Figure 11 shows the seismic network at Kusatsu-Shirane, which features three-component seismometers at stations JIE, YNE, YGW, and AIM, and vertical-component seismometers at stations JIW, MZW, and YNW. The seismometers have a natural frequency of 1 Hz and are critically damped. All the records are sampled at 120 samples/s/channel [Ida et al., 1989]. A vertical velocity seismogram and corresponding amplitude spectrum observed at JIE for an LP event that occurred on 22 October 1992, are shown in Figures 12a and 12b, respectively. This LP event shows a monochromatic oscillatory signature with a dominant spectral peak near 1.5 Hz. Based on the interpretation of Kumagai et al. , this event occurred in a hydrothermal crack filled with a misty gas during a period of gradual drying up of the gas in response to magmatic heat.
 After deconvolution for instrument responses the seismograms are band-pass filtered between 0.8 and 5 Hz. We perform the waveform inversion of the band-passed data assuming six moment tensor components, in which we use a one-cycle cosine function with tp = 0.25 s as the elementary source time function. Green's functions are calculated for a homogeneous half-space with compressional wave velocity α = 2500 m/s [Ida et al., 1989], shear wave velocity , and density ρs = 2100 kg/m3. As waveform data are not available for the NS component at YGW and vertical component at MZW, our inversion relies on 13 seismograms only. We conduct a grid search to find the best fit point source position, which is located at a depth of about 200 m between the Yugama and Mizugama craters (see Figure 11).
 The source time functions of the six moment tensor components are shown in Figure 13a, and the corresponding waveform matches are shown in Figure 14. The waveform fits are fairly good except for the EW component at AIM. The moment tensor solution in Figure 13a is dominated by volumetric components with oscillatory signatures, in which the Mzz component dominates over Mxx and Myy components. The eigenvectors calculated for the moment tensor solution in Figure 13a during the interval 4–8 s, are plotted in Figure 13b. The lengths of the vectors are proportional to their respective eigenvalues. Although the results display significant scatter, the amplitude ratios of the three principal axes are roughly represented by 1:1:3, with a dominant axis slightly inclined from the vertical direction. Our grid search shows that the amplitude ratios of the three principal axes remain stable for different source locations, but suggests a slight dependence of the orientations of the principal axes on the horizontal source location, as previously demonstrated in our tests of the effects of source mislocations on inversions of synthetic waveforms (see Figure 7). Our results are therefore consistent with the resonance of a subhorizontal crack at the source of the LP event.
 Synthetic waveform inversions performed for the 2W/3 and 2L/5 modes both yield amplitude ratios for the three principal axes of the moment tensor that consistently point to a crack. In contrast, inversions conducted for the L mode were found to be inconsistent with the expected source mechanism for a crack. The cause of this inconsistency may be rooted in the parity of the mode considered for inversion. The even modes of the crack oscillation have wavelengths 2L/n, 2W/n, n = 2, 4, 6, …, while the odd modes have wavelengths 2L/n, 2W/n, n = 3, 5, 7, … [Chouet, 1992]. Associated with each odd mode is a net volume change in the crack during each half cycle of resonance, which is detected as either a crack expansion or crack contraction. No such net volume change exists in the even modes (see Figure 15). In the L mode for example, the resonance may be viewed as a combination of expanding and contracting cracks, which can be represented by a couple of the moment tensor, i.e., a third-order tensor. Higher-order tensors are also required to represent even modes with shorter wavelengths. This property suggests that a moment tensor inversion based on a point source may not be appropriate for even modes.
 A more general representation of the crack resonance modes is given by Takei and Kumazawa . In their representation, the phenomenological source of the seismic waves is given by the traction on the boundary between the source and surrounding medium. The traction is expressed in terms of generalized spherical harmonics components, each of which is identified with the harmonic order ℓ and modal type k (one toroidal and two spheroidal modes denoted by T, S1, and S2, respectively). In this representation, not only a single force (S1 with ℓ = 1) and moment tensor (a combination of S2 with ℓ = 0 and S1 with ℓ = 1) but also higher-order tensors are systematically described. Based on this representation, the L mode may be described by a combination of S2 with ℓ = 1 (pressure dipole) and S1 with ℓ = 3. Accordingly, even modes with shorter wavelengths can be described by S1 and S2 with higher harmonic orders, which have smaller excitation efficiencies of seismic waves according to Takei and Kumazawa . A generalized source representation is thus clearly required for the even modes. The Akaike's Information Criterion [Akaike, 1974] can be used to evaluate the harmonic orders and modal types in a waveform inversion. However, such inversion is not easy to perform as the number of unknown parameters for this source rapidly increases with the decreasing wavelengths of the even modes, therefore requiring the use of a very dense seismic network. In light of the smaller excitation efficiency of seismic waves in the even modes [Takei and Kumazawa, 1995], a waveform inversion of the oscillatory signatures of LP events based on six moment tensor components is practically justified.
 We successfully applied a waveform inversion to determine the moment tensor of an LP event observed at Kusatsu-Shirane. We determined the best fit source location based on a simple homogeneous half-space. Our results point to a subhorizontal crack located at a depth near 200 m beneath the summit crater lakes. As indicated in Figure 13b, however, the directions of the eigenvectors and amplitude ratios of the principal axes of the moment tensor both display considerable scatter. Our numerical tests suggest that this scatter may be attributed in large part to a lack of resolution due to an inadequate spatial sampling of the wave field. Our lack of knowledge of the three-dimensional velocity structure beneath Kusatsu-Shirane, and neglect of topography in the calculations of Green's functions probably also contribute to the scatter. A horizontal mislocation of the source may further affect the directions of the dominant axes of the moment tensor. A consideration of the three-dimensional velocity structure and topography of Kusatsu-Shirane in our calculations of Green's functions might affect both the location of the best fit source and direction of the dominant dipole components of the source. Nevertheless, our present results clearly point to a crack geometry for the LP source, and further suggest that the dominant mode of resonance observed is most likely an odd mode. This finding may be regarded as the first direct and solid evidence supporting the idea that LP events originate in the resonance of a crack, thus justifying the assumptions made by Kumagai et al.  in their interpretation of temporal variations in the complex frequencies of LP events observed at Kusatsu-Shirane.
 Our inversion of the oscillatory signatures in the tail of LP waveforms relies on the assumption of a source mechanism consisting of six moment tensor components only. This assumption is justified in light of the physical consideration that mass advection is not associated with the resonance, as supported by our numerical tests. However, this does not preclude the existence of a single force active at the event onset. Fluid movements and associated pressure disturbances may provide excitation mechanisms for crack resonance, which may generate a single force at the event onset. To address this issue, a useful approach is that developed by Nakano et al. . In this approach, the oscillatory signatures of LP events are removed by application of an AR filter to estimate the apparent excitation functions of the individual LP waveforms at the event onset. Waveform inversions of these excitation functions (Nakano et al., submitted manuscript, 2002) may be more useful for an assessment of the excitation mechanisms of LP events including both moment tensor and single force. However, one disadvantage of the method of Nakano et al. is that it requires the additional task of filtering based on an AR model prior to performing the inversion. To properly remove the oscillatory signatures of LP waveforms, the characteristic properties of the resonator system must be known. Nakano et al.  assumed that the excitation of LP events is time-localized and used the decaying oscillations in the tail of the LP waveform to determine the characteristic properties of the resonator system. As demonstrated in the present study, quantitative information about to the source location, geometry, and force system can be obtained from waveform inversions of the oscillatory signatures of LP waveforms in a relatively convenient way. Although these two approaches may be regarded as complementary, a clear advantage of the present method is that it does not require any assumption about the excitation function, which enables the application of this inversion scheme not only to LP events but also to tremor. Tremor is more generally observed at active volcanoes compared to LP events. This opens the way to a more quantitative approach in analyses of LP seismicity, which should help with the task of interpreting magmatic and hydrothermal activities beneath volcanoes.
 Synthetic LP waveforms generated with the fluid-filled crack model were used to examine the source location and source mechanism of LP events. Numerical tests based on waveform inversions of the oscillatory signatures in synthetic LP waveforms indicate that moment tensor inversions assuming a point source are valid for the odd modes. A generalized source representation using higher-order tensors is required for the even modes. In light of the small excitation efficiency of seismic waves in the even modes, the moment tensor inversion may be generally applicable to LP events. Our numerical tests also suggest that more than 4, and ideally 10 to 15, three-component stations surrounding an LP source are required for an accurate description of the moment tensor. We applied the waveform inversion to an LP event observed at Kusatsu-Shirane Volcano, central Japan. Our inversion results point to the resonance of a subhorizontal crack located at a depth near 200 m below the summit crater lakes. The present paper demonstrates the usefulness of waveform inversions of the oscillatory signatures of LP events for a quantification of their source location, geometry, and force system, and opens the way for moment tensor inversions of tremor.
 We thank Yasuko Takei for valuable discussions on the generalized source representation. We are also grateful to Yoshiaki Ida, Jun Oikawa, and members of Earthquake Research Institute, University of Tokyo, for sharing the waveform data from Kusatsu-Shirane Volcano. Comments from two anonymous reviewers and an anonymous associate editor helped improve the manuscript.