# Effects of water domains on seismic wavefields: A simulation case study at Taal volcano, Philippines

- Yuta Maeda
^{1}Email author and - Hiroyuki Kumagai
^{1}

**65**:650020085

https://doi.org/10.5047/eps.2012.07.004

© The Society of Geomagnetism and Earth, Planetary and Space Sciences (SGEPSS); The Seismological Society of Japan; The Volcanological Society of Japan; The Geodetic Society of Japan; The Japanese Society for Planetary Sciences; TERRAPUB 2013

**Received: **9 September 2011

**Accepted: **10 July 2012

**Published: **6 March 2013

## Abstract

We examined the effects of water domains on seismic wavefields at the Taal volcano, Philippines, where there is a summit crater lake on an active volcanic island surrounded by a caldera lake. We conducted forward waveform simulation and synthetic waveform inversion using the 3D finite-difference method for various models with and without the water lakes. Our study demonstrated the existence of both near-and far-field effects of the water domains. The near-field effect is related to source excitation affected by a water domain, and the far-field effect is caused by dispersion of Rayleigh waves propagating through a water domain. Our study also showed that the replacement of water by a vacuum rather than by a solid medium provides a better approximation of a water domain. A water domain has considerable effect on seismic waveforms and must be properly treated when the waveform period is shorter than 2 s, even if recording stations lie in the near field. If the waveform period is longer than 5 s, a water domain has a minor effect on seismic waveforms and may be replaced by a vacuum.

## Key words

## 1. Introduction

Long-period (LP: 0.2–2 s) and very-long-period (VLP: 2–100 s) seismic signals are commonly observed at volcanoes worldwide. They are thought to be generated by volumetric changes and movements of volcanic fluids, and are thus important for understanding pre-eruptive fluid processes in volcanoes. To quantify LP and VLP sources, waveform inversion techniques developed by Ohminato *et al.* (1998), Nakano and Kumagai (2005), and Auger *et al.* (2006) have been commonly used. In waveform inversion analyses, accurate estimations of Green’s functions are critically important.

Waveform inversion studies have been conducted at volcanoes with crater lakes and volcanic islands surrounded by sea water, such as Kilauea (Ohminato *et al.*, 1998), Aso (Legrand *et al.*, 2000), Miyake (Kumagai *et al.*, 2001; Kobayashi *et al.*, 2009), Kusatsu-Shirane (Nakano *et al.*, 2003), Usu (Yamamoto *et al.*, 2002), Stromboli (Chouet *et al.*, 2003), Hachijo (Kumagai *et al.*, 2003), and Satsuma-Iwojima (Ohminato, 2006). In these studies, water domains were usually ignored and represented either by solid or vacuum domains in calculations of Green’s functions, but without stated justification except for the study of Chouet *et al.* (2003), who concluded that the effect of sea water was negligible within a VLP band at Stromboli.

The influence of water domains on synthetic seismograms has been more extensively studied in strong ground motion seismology. Using a 2D boundary element method, Hatayama (2004) synthesized Rayleigh waves at an epicentral distance of 60 km from a seismic source at a depth of 5 km with, and without, including a sea layer. He concluded that the sea has an important effect on the Rayleigh wave if the source duration is shorter than 3 s and the water depth is greater than 800 m. He also concluded that the replacement of sea water by a vacuum provides a better approximation for the sea layer than does the substitution of a solid medium for sea water. Petukhin *et al.* (2010) used a 3D finite-difference method to calculate synthetic seismograms of large interplate earthquakes by using a 3D velocity and density structure with, and without, an oceanic layer. They concluded that the oceanic layer has a large influence on the Rayleigh waves for a shallow (∼5-km depth) earthquake, whereas it has negligible influence on synthetic seismograms for a deep (> 10-km depth) source.

Although these strong-motion studies considered earthquake sources at depths greater than 5 km, LP and VLP events at volcanoes often occur at depths of 1 km or shallower. Seismic stations at volcanoes are located at source distances from a few hundred meters to several kilometers. Volcanoes exhibit steep and complex topographies. The effects of water domains on seismograms at volcanoes in such complex situations may depend on station configurations and the periods, or wavelengths, of LP and VLP signals, and should be evaluated by numerical simulation approaches.

*et al.*, 2009). Most of the Taal eruptions have taken place after quiescent periods of less than 30 years, but more than 30 years have passed since the last eruption in 1977. This implies a high risk of near-future eruptions at Taal. To improve the monitoring and understanding of the volcano, a joint Japan-Philippine team installed five broadband seismometers at Taal in November 2010, as well as infrasonic and geomagnetic sensors and GPS receivers.

Taal holds a caldera filled with water (Taal Lake: TL), which has EW and NS dimensions of 15 and 25 km, respectively, and a maximum depth of 200 m (Fig. 1(b)). The currently-active crater is the Volcano Island (elevation 311 m) surrounded by TL. The active crater forms the Main Crater Lake (MCL) with a diameter of 1.2 km and a maximum water depth of 80 m (Fig. 1(c)). Taal thus provides a good test field to evaluate the effects of water domains on seismic waveforms. Such evaluations are important for the correct calculations of Green’s functions, which are used in broadband seismic monitoring at Taal based on the waveform inversion approach.

In this study, we evaluated the effects of the two water domains on seismic wavefields at Taal volcano through forward waveform simulation and synthetic waveform inversion. We calculated synthetic seismograms using the finite-difference method, assuming the following three models: the water domains treated as water (water model), the water replaced by a vacuum (vacuum model), and a solid medium (solid model). In Section 2, we describe the finite-difference method used to calculate synthetic seismograms. In Section 3, we compare forward synthetic seismograms calculated for the three models at various epicentral distances to evaluate the effects of the water domains on the seismic wavefields. Then, in Section 4, using synthetic observed seismograms and Green’s functions at real seismic station locations, we examine the effects of the water domains on waveform inversion. We discuss our results in Section 5, and present our conclusions in Section 6.

## 2. Calculation of Synthetic Seismograms

We used a second-order finite-difference method (FDM) to calculate synthetic seismograms. In the FDM code, we used algorithms proposed by Maeda *et al.* (2011), and Okamoto and Takenaka (2005). The algorithm of Maeda *et al.* (2011) uses the staggered grid cell proposed by Ohminato and Chouet (1997) to calculate the synthetic seismograms for models with arbitrary 3D topography and structure. An efficient absorbing boundary, known as a perfectly matched layer (PML; Chew and Liu, 1996; Festa and Nielsen, 2003), is used to efficiently calculate long synthetic seismograms. The algorithm of Okamoto and Takenaka (2005) allows the inclusion of a water domain, which is realized by setting Lame’s coefficient *µ* to zero for the cells representing the water domain. The solid-water boundary is placed at the interfaces of the grid cells, where *µ* is set to zero, so that the tangential stress is zero and the normal velocity is continuous at the boundary. An average of the solid and water densities is used as the density at the solid-water boundary. Okamoto and Takenaka (2005) theoretically showed that the normal stress component is continuous through the boundary when the averaged density is used in the second-order finite-difference equations. Okamoto and Takenaka (2005), and Nakamura *et al.* (2011), used numerical tests to justify their use of these boundary conditions. We used Cartesian coordinates *x*, *y*, and *z* corresponding to E, N, and upward, respectively, throughout this study.

*P*-wave velocity

*V*

_{ p }= 1450 m/s, an

*S*-wave velocity

*V*

_{ s }= 0 m/s, and a density

*ρ*= 1020 kg/m

^{3}. For the solid half space, we assumed

*V*

_{ p }= 3500 m/s, , and

*ρ*= 2400 kg/m

^{3}. We used the following Ricker wavelet as the input source time function, where

*t*is time, and

*T*

_{ p }and

*T*

_{ s }are constants related to the duration and peak time of the wavelet, respectively. We used an isotropic source with

*T*

_{ p }= 1s and

*T*

_{ s }= 2 s at a depth of 220 m below the water surface. Figure 2 compares the vertical velocities on the water surface calculated by the FDM code (solid lines), using a uniform 40 m grid and a time step of 0.004 s, with those calculated by the discrete wavenumber method (DWM) code of Takeo (1985) (dotted lines). In the DWM calculation, we used a small value of an S-wave velocity for the water layer (

*V*

_{ s }= 0.01 m/s) because the DWM code could not treat

*V*

_{ s }= 0 m/s. We used water-layer thicknesses of 40 m (Figs. 2(a) and 2(b)) and 160 m (Figs. 2(c) and 2(d)), which are the representative depths of MCL and TL, respectively. Note that the 40-m-thick water-layer is expressed by only one grid cell in the FDM calculation. In Fig. 2, we also indicate misfits among the waveforms, defined as where and denote vertical velocity waveforms calculated by the FDM and DWM codes, respectively,

*N*denotes the number of data samples in each trace, and Δ

*t*denotes a time step (0.004 s). We obtained the misfit value of 0.03 for an epicentral distance of 1 km and a water-layer thickness of 40 m (Fig. 2(a)), and 0.36 for an epicentral distance of 10 km and a water-layer thickness of 160 m (Fig. 2(d)). These misfit values will be used to evaluate our synthetic results later.

## 3. Forward Synthetic Tests

In our forward waveform simulation tests, we assumed an isotropic Ricker wavelet (Eqs. (1) and (2)) source beneath the MCL (Fig. 1(c)). We calculated synthetic seismograms at real and virtual station locations (Figs. 1(b) and 1(c)). We used a 10-m mesh digital elevation model (DEM) containing bathymetries of TL and MCL. Delmelle *et al.* (1998) estimated the water surfaces of TL and MCL to be 2 and 4 m above sea level, respectively; we therefore set them at sea level. We used the computational domain down to a depth of 4 km below sea level, surrounded by 800-m-thick PML domains. We used the same grid size, time step, *P-*and *S*-wave velocities, and densities of the water and solid medium as those in the previous section.

*y*-component velocity

*V*

_{ y }(upper panels), indicating that the wave amplitudes are highly weakened near the water surfaces. In TL, there is a ring-like pattern corresponding to wave packets, indicating that the wave energies are partly transmitted to the water domains. When we look at the snapshots of the vertical velocity

*V*

_{ z }(lower panels), the wave amplitudes in the lake regions are not weakened. This difference among the components may be attributed to the boundary conditions. For the normal component, the velocity is continuous on the solid-water interfaces and is not zero on the water surfaces. On the other hand, the tangential velocity components are not continuous on the solid-water interfaces and become zero on the water surfaces. The latter condition arises from the equation of motion with the pressure

*p*= 0 on the water surfaces. We note that the snapshots of

*V*

_{ y }in Fig. 3, in the lake region, are not zero because they are at half grids below the water surfaces in the staggered grid scheme.

*T*

_{ p }= 2 s and a source depth of 220 m. Figure 4 compares the waveforms obtained using these two domains. We can see no distinct difference at the main portions of the waveforms and only a minor difference in the later portions. We quantify the waveform differences obtained using the two domains through misfits, defined as where and denote synthetic velocity components in the

*i*-direction at a station calculated with the water model using the small and entire domains, respectively. We calculated the misfit value at each station shown in Fig. 1(b). The largest misfit was 0.13 at the northernmost virtual station (VSN) and the second largest was 0.082. Both of these were small enough to have a negligible effect on the following conclusions, which justify the use of the small domain in the following calculations.

*Tp*= 1 s and a source depth of 220 m. At VTTK (Fig. 5(b)), clear phase shifts among the three models are evident in the Rayleigh wave portion, whereas the waveform differences among the three models are small in the

*P*-wave portion. At VTDK (Fig. 5(a)), differences in amplitude among the three models appear for the entire duration of the main wave packet, but the phase shifts are not observed.

*V*

_{ p }= 2000 m/s, layer 2 ranging from 500 to 2000 m below sea level with

*V*

_{ p }= 3100 m/s, and a half space below layer 2 with

*V*

_{ p }= 4300 m/s. We used and

*ρ*= 1700 + 0.2

*V*

_{ p }(Gardner

*et al.*, 1974). We used the same source as used in the previous test. Figure 6(b) compares synthetic waveforms at VTTK calculated for the three models with the layered solid structure. A large amplification, and a phase delay of the Rayleigh wave, is prominent in the water model. This result indicates that the water domains have a large effect on the Rayleigh wave portion at far-field stations regardless of a homogeneous, or a layered, solid structure. At VTDK (Fig. 6(a)), we can see amplitude differences among the three models in the entire duration of the main wave packet but the phase shifts are small. These characteristics at near-field stations are consistent with the simulation results obtained using the homogeneous solid structure (Fig. 5(a)). Given these conclusions, we hereafter show the results obtained using the homogeneous solid structure for simplicity.

*i*-direction calculated with the vacuum and solid models, respectively. We calculated the misfits between the solid and water models (

*δ*

_{sw}) and between the vacuum and water models (

*δ*

_{vw}) for ten combinations of source duration and depth, which we plotted as functions of epicentral distance (Fig. 7).

For all ten cases, the misfits between the vacuum and water models (*δ*_{vw}) are generally smaller than those between the solid and water models (*δ*_{sw}), indicating that the vacuum model provides a better approximation of the water domain than the solid model. The misfits *δ*_{sw} and *δ*_{vw} for the shallower source (Figs. 7(a)–7(e)) are larger than those for the deeper source (Figs. 7(f)–7(j)). The maximum values of *δ*_{sw} for *T*_{
p
} = 1 and 2 s and source depth of 220 m are 1.41 and 0.68, respectively (Figs. 7(a) and 7(b)), indicating that the water domains markedly affect the seismic waveforms when the period is shorter than 2 s. The misfit *δ*_{sw} gradually decreases with increasing source duration *T*_{
p
} (Figs. 7(c) and 7(d)) and becomes smaller than 0.23 for *T*_{
p
} = 5 s (Fig. 7(e)). Similar characteristics are also visible for misfit *δ*_{vw} (Figs. 7(a)–7(e)) and for a source depth of 700 m (Figs. 7(f)–7(j)). We also conducted calculations for *T*_{
p
} = 10 s, which resulted in similar misfit values to those for *T*_{
p
} = 5 s. These results consistently indicate that the effects of the water domains are relatively small when the period is longer than 5 s.

For *T*_{
p
} = 1 and 2 s for stations not on the island (epicentral distance >5 km), the waveform misfits *δ*_{sw} and *δ*_{vw} increase as the epicentral distance increases (Figs. 7(a), 7(b), 7(f), and 7(g)). These increasing trends suggest that the waveform misfits among the three models represent the effect of TL. Figure 7 also indicates that the misfits at VTTK are larger than those at the virtual station VSN located at a similar epicentral distance (10 km). The TL portion on the ray path between the source and VTTK was longer than that between the source and VSN (Fig. 1), which caused the larger misfit values at VTTK. On the other hand, for *T*_{
p
} = 1 s and a source depth of 220 m (Fig. 7(a)), the misfits at the closest station are relatively large and decrease as the epicentral distance increases within the island (epicentral distance ≤ 4 km), which we consider to be the effect of MCL. These results indicate that the water domains have both near- and far-field effects.

In Section 2, we evaluated the waveform misfits *δ*_{FD} (Eq. (3)) between the FDM and DWM for water-layer thicknesses of 40 and 160 m (Fig. 2), which correspond to typical depths of MCL and TL, respectively. In the evaluation, we used a source with *T*_{
p
} = 1 s at 220 m depth, which were the same as those used in Fig. 7(a). Hence, the misfits *δ*_{FD} may be used as rough estimations of the FDM calculation errors in Fig. 7(a). At an epicentral distance of 1 km, *δ*_{FD} for a water-layer thickness of 40 m (0.03, Fig. 2(a)) may be appropriate for the rough error estimation because the waveforms are mainly affected by MCL. This error (0.03) is small compared to the waveform misfits *δ*_{sw} and *δ*_{vw} at the same epicentral distance (0.15–0.38, Fig. 7(a)). At an epicentral distance of 10 km, the waveforms are mainly affected by TL so that *δ*_{FD} for a water-layer thickness of 160 m (0.36, Fig. 2(d)) may be appropriate for the error estimation. This relatively large error (0.36) is mainly attributed to the Rayleigh wave portion of the waveform propagating through the water domain (Fig. 2(d)), which may be caused by our use of only a few vertical grid cells in the water domain. However, the misfits *δ*_{sw} and *δ*_{vw} at the same epicentral distance (0.81–1.42, Fig. 7(a)) are 2–4 times larger than this error (0.36). Therefore, our interpretation of the near- and far-field effects may not be seriously affected by the FDM calculation errors.

*δ*

_{ww}for

*T*

_{ p }= 1 and 2 s, respectively. The maximum value of

*δ*

_{ww}is 0.49 for

*T*

_{ p }= 1 s, indicating that the synthetic seismograms are affected by the treatment of the solid-water boundary.

## 4. Waveform Inversion of Synthetic Data

### 4.1 Inversion procedure

We conducted waveform inversion using synthetic waveforms to examine how the water domains affect inversion solutions. Synthetic observed seismograms at the real station locations were calculated using the water model, and the three models (water, vacuum, and solid models) were used for calculations of Green’s functions, in which we used the homogeneous solid structure. We conducted a least-squares inversion for six moment tensor time functions based on the method of Auger *et al.* (2006), in which waveform inversion is performed in the frequency domain to reduce the computational time and computer memory requirements.

We synthesized observed seismograms assuming a point vertical tensile crack source with an N-S opening direction. We set the crack source beneath MCL at 220 m below sea level, under the red solid circle in Fig. 1. We used a Ricker wavelet characterized by *T*_{
p
} = 2 s and *T*_{
s
} = 4 s as the source time function. Source depths of 100–300 m have been commonly estimated for LP and VLP events at various volcanoes (e.g., Hidayat *et al.*, 2002; Chouet *et al.*, 2003; Nakano *et al.*, 2003; Maeda and Takeo, 2011). Because the Eurasian plate subducts eastward (Galgana *et al.*, 2007; Ku *et al.*, 2009), a tensile crack with an N-S opening direction is a plausible source mechanism at Taal volcano.

List of station configurations for synthetic inversions conducted in this study. Stations used in inversions are indicated by asterisks (*).

Station network | ||||||
---|---|---|---|---|---|---|

VTDK | VTCT | VTBM | VTNP | VTBC | VTTK | |

A | * | * | * | * | * | |

B | * | * | * | * | ||

C | * | * | * | * |

### 4.2 Inversion results

*M*

_{1},

*M*

_{2}, and

*M*

_{3}indicate the largest, intermediate, and smallest eigenvalues for the principal axes, respectively, and

*α*

_{1}indicates the angle between

*M*

_{1}and the

*y*-axis. If the inversion solution is perfect,

*M*

_{2}

*/M*

_{1}=

*M*

_{3}

*/M*

_{1}= 1/3 and

*α*

_{1}= 0 in Fig. 9, and the tensile directions shown in Fig. 10 are oriented in the N-S direction. Since we defined the eigenvalues to satisfy |

*M*

_{1}(

*t*)|

*> |M*

_{2}(

*t*)|

*> |M*

_{3}(

*t*)| at each time

*t*, jumps of the eigenvalues occurred at times when the order of the eigenvalues changed (Fig. 9), although the moment tensor solutions were continuous with time (Fig. 8). When the water model was used to calculate Green’s functions, the inversion solutions showed almost perfect recoveries, as expected (Figs. 8(a), 9(a), and 10(a)). The inversion solutions obtained using the vacuum and solid models show deviations from the input source. In particular, the deviations of

*M*

_{ zz }(Figs. 8(b) and 8(c)) and

*M*2 (Figs. 9(b) and 9(c)) are large for both of these models. In the inversion result obtained using the solid model, the crack deviates from the vertical (gray arrows in Fig. 10(c)). The deviations of the solutions obtained using the solid model are clearly larger than those obtained using the vacuum model.

*i*= 1, 2, 3, and the superscripts ‘inv’ and ‘input’ indicate eigenvalues for the inversion results and the input source, respectively. Here,

*k*

_{1}Δ

*t*and

*k*

_{2}Δ

*t*define the time window shown in Figs. 8 and 9 (i.e.

*k*

_{1}Δ

*t*= 2 s and

*k*

_{2}Δ

*t*= 6 s). We compared

*E*

_{ i }obtained from the three models with various station configurations (Fig. 11). For the water model, the errors are smaller than 0.04 for all parameters and station configurations. The errors for the solid model are generally larger than those for the vacuum model. For example, the errors in

*M*

_{2}(

*E*

_{2}) using station configuration A are 0.30 and 0.77 for the vacuum and solid models, respectively.

## 5. Discussion

Our forward waveform simulation revealed that water domains have a considerable effect on seismic waveforms when the period is shorter than 2 s (Fig. 7), whereas their effects on waveforms are small when the source duration is longer than 5 s. Our simulation also showed that the water model is better approximated by a vacuum model than a solid model.

The seismograms at distant stations beyond the Volcano Island are dominantly affected by the outer lake (TL). Synthetic seismograms at station VTTK (Fig. 5(b)) display relatively large waveform differences among the three models for the Rayleigh wave portions of the waveforms. At station VTTK, amplitude differences among the models are minor, but a phase delay of the Rayleigh wave is prominent in the water model (Fig. 5(b)). This phase delay may be caused by dispersion of the Rayleigh wave propagating through the water domain in TL. Although Fig. 5(b) was obtained using the homogeneous structure for the solid domain, our simulation using the layered structure for the solid domain (Fig. 6) also showed that the water domains affect the Rayleigh wave portions of the waveforms at far-field stations.

Hatayama (2004) performed 2D waveform simulations to examine the effect of an oceanic layer on waveforms with periods of 0.18–6 s observed at an epicentral distance of 60 km using both homogeneous and layered solid structures. Petukhin *et al.* (2010) performed 3D waveform simulations for periods of a few seconds, and epicentral distances of ∼20 km, to examine the effect of an oceanic layer using a 3D layered solid structure of the Kinki region, Japan. Both studies concluded that (1) the oceanic layer affects the Rayleigh wave portion of the waveforms at far-field stations, and (2) the effect of the oceanic layer becomes minor with increasing source depth. Hatayama (2004) also concluded that the water layer is better approximated by vacuum than solid. These conclusions are consistent with our results for the far-field waveforms (Figs. 5(b), 6(b), and 7).

On the other hand, we showed that seismograms at stations close to the source are affected by MCL (Fig. 7). At these stations, body and surface waves arrive almost simultaneously because the hypocentral distances are small. The waveform differences among the three models at station VTDK (Fig. 5(a)) are visible for the entire duration of the main wave packet. At station VTDK, the seismic amplitudes are smallest for the solid model; those of the water and vacuum models are almost the same. The waveform misfits at stations on the island for *T*_{
p
} = 1 s and source depth 220 m tend to be larger at nearer stations (Fig. 7(a)). On the other hand, when the source is deeper (700 m), this tendency is not observed (Fig. 7(f)). Thus, we consider that the waveform misfits at stations on the island reflect the near-field effect of MCL. Our simulations thus demonstrate that water domains have both near- and far-field effects.

Although the waveforms calculated by the FDM code may have some errors (Fig. 2), the waveform misfits among the three models (Fig. 7) are larger than the estimated errors at near- and far-field stations (Section 3) so that our interpretation of the near-and far-field effects may not be seriously affected by FDM calculation errors.

Our inversion tests support our interpretation of the simulations. The inversion solutions obtained using the solid model were overestimated (Fig. 8(c)), which can be explained by the smallest near-field seismic amplitudes in the solid model. On the other hand, the inversion test using the vacuum model provided solutions similar to the input source (Fig. 8(b)). For the solid model, the use of data from station VTDK worsened the errors in the inversion results (Fig. 11), which is explained by the near-field effect. When we eliminated the structural effect of MCL, however, the errors in the inversion results were generally smaller when using data from stations closer to the source (Fig. 12). This is explained by the far-field effect at distant stations, where the waveform differences increase due to the dispersion of the Rayleigh wave.

Chouet *et al.* (2003) compared synthetic seismograms at the Stromboli volcano, Italy, with and without a sea layer, and concluded that the effect of the sea layer was minor. Their results are consistent with ours, because (1) they used water and vacuum models, not a solid model, for their comparison, (2) the dominant periods of the waveforms in their study were around 5 s, and (3) all of the stations they used were on an island surrounded by sea.

## 6. Conclusions

We examined the effects of water domains on seismic wavefields at Taal volcano, Philippines, a volcano with both crater and caldera lakes. We conducted forward waveform simulation and synthetic waveform inversion using the 3D finite-difference method with and without water lakes. Our results suggest that the water domains have both near- and far-field effects on seismic wavefields. The near-field effect is related to source excitation affected by a water domain, whereas the far-field effect is caused by the dispersion of Rayleigh waves propagating through a water domain. Our study also showed that the replacement of water by a vacuum, rather than a solid medium, provides a better approximation of a water domain. A water domain has a marked effect on seismic waveforms and the results of waveform inversion when the waveform period is shorter than 2 s. If this is the case, a water domain must be properly accounted for, even if the seismic stations lie in the near field. We also showed that the effect of a water domain is small for waveform periods longer than 5 s.

## Declarations

### Acknowledgments

We thank Taro Okamoto for providing numerical data to check the FDM calculations. The DEM of Taal volcano that we used was provided by the Philippine Institute of Volcanology and Seismology (PHIVOLCS). This study was financially supported by the Japan Science and Technology Agency (JST) and the Japan International Cooperation Agency (JICA). Comments from two anonymous reviewers helped to improve the manuscript.

## Authors’ Affiliations

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