Revisiting Seismic Energy of Shallow Tremors: Amplifications Due To Site and Propagation Path Effects Near the Nankai Trough

We investigated the effects of the propagation path and site amplification of shallow tremors along the Nankai Trough. Using far‐field S‐wave propagation from intraslab earthquake data, the amplification factors at the DONET stations were 5–40 times against an inland outcrop rock site. Thick (∼5 km) sedimentary layers with VS of 0.6–2 km/s beneath DONET stations have been confirmed by seismological studies. To investigate the effects of thick sedimentary layers, we synthesized seismograms of shallow tremors and intraslab earthquakes at seafloor stations. The ratios of the maximum amplitudes from the synthetic intraslab seismograms between models with and without thick sedimentary layers were 1–2. This means that thin lower‐velocity (<0.6 km/s) sediments just below the stations primarily control the estimated large amplifications. Conversely, at near‐source (≤20 km) distances, 1‐order amplifications of seismic energies for a shallow tremor source can occur due to thick sedimentary layers. Multiple S‐wave reflections between the seafloor and plate interface are contaminated in tremor envelopes; consequently, seismic energy and duration are overestimated. If a shallow tremor occurs within underthrust sediments, the overestimation becomes stronger because of the invalid rigidity assumptions around the source region. After 1‐order corrections of seismic energies of shallow tremors along the Nankai Trough, the scaled energies of seismic slow earthquakes were 10−10–10−9 irrespective of the region and source depth. Hence, the physical mechanisms governing seismic slow earthquakes can be the same, irrespective of the region and source depth.

The source parameters of seismic slow earthquakes have been extensively studied worldwide to discuss their physical characteristics.The seismic moments of seismic slow earthquakes can be obtained from an analysis of the VLFE frequency bands (Ide & Yabe, 2014;Ito et al., 2009;Maury et al., 2016Maury et al., , 2018;;Sugioka et al., 2012;Takemura, Baba, Yabe, Emoto, et al., 2022;Takemura, Obara, et al., 2022;Takeo et al., 2010).Their estimations are stable for both shallow and deep VLFEs.However, because seismic wave scattering due to small-scale (<several kilometers) heterogeneities becomes dominant at frequencies above 1 Hz (e.g., Sato et al., 2012), the source parameters of tremors cannot be deterministically estimated via waveform-based methods.Thus, the seismic energies in the tremor band have been estimated from smoothed velocity envelopes assuming far-field body waves in an infinite homogeneous medium (e.g., Annoura et al., 2016;Maury et al., 2018;Wech, 2021;Yabe & Ide, 2014).The scaled energy, the ratio of the seismic energy to the seismic moment, characterizes the dynamics of earthquake faulting (Kanamori & Rivera, 2006).Owing to the observational gap of an intermediate (0.1-1 Hz) frequency band due to microseism noise (Fig. 8 of Ide, 2014), sufficient seismic energy cannot be evaluated within this frequency band.Thus, the scaled energy of seismic slow earthquakes can be calculated as the ratio of the seismic energy of a tremor/LFE divided by the seismic moment of the accompanying VLFE.
Slow earthquakes have been detected in several regions of Japan.Deep slow earthquakes occur at depths of 30-40 km, near the interface of the subducted Philippine Sea Plate.These signals were observed in the inland seismic networks (Aoi et al., 2020).The observed seismic moment rates of the deep VLFEs and the energy rates of the deep tremors are 10 11 -10 12 Nm/s and 10 1 -10 3 J/s, respectively.The scaled energy of deep slow earthquakes ranges from 10 10 to 10 9 (Ide et al., 2008;Ide & Maury, 2018;Ide & Yabe, 2014), significantly less than that of ordinary earthquakes (approximately 3 × 10 5 ; e.g., Ide & Beroza, 2001).Such a 4-order difference in the scaled energy between ordinary and slow earthquakes also suggests different governing mechanisms for both slip phenomena.
Permanent networks of ocean bottom seismometers (OBSs) have been in development since 2010 (see Aoi et al., 2020).These networks enable us to investigate the source properties of shallow VLFEs and tremors near the Nankai Trough and Japan Trench.The depths of shallow seismic slow earthquakes are ≤10 and 10-20 km, respectively.High-frequency seismograms at OBSs contain large site amplifications due to the low-velocity sediments beneath the OBSs. Figure 1a shows sample waveforms at the offshore DONET (M.KMD13) and inland F-net (N.KISF) stations during an intraslab earthquake.After correcting for the geometrical spreading of the body waves, the maximum amplitude at M.KMD13 was still approximately eight times larger than that at N. KISF.This was due to site amplification at M.KMD13.
Site amplification factors for an inland rock site have been estimated to accurately estimate the physical properties of offshore earthquake phenomena using OBSs.Because the signals of shallow tremors are too weak at inland rock sites (see Figure 1 of Takemura, Hamada, et al., 2023), site amplifications are typically estimated based on near-vertical incident body waves from intraslab earthquakes (blue arrows in Figure 1b).Although vertical Swave amplitudes at OBSs tend to be weakly amplified, amplification factors of 5-30 against an inland rock site in horizontal components have been observed in previous studies (Kubo et al., 2018(Kubo et al., , 2020;;Takemura, Emoto et al., 2023;Yabe et al., 2019).These site amplification factors include the effects of thick sedimentary layers with V S of 0.6-2 km/s and thin sediments of V S < 0.6 km/s just below OBSs (see Figure 1b).Thick sedimentary layers beneath the DONET stations have been confirmed in seismological studies (e.g., Akuhara et al., 2020;Kamei et al., 2012;Tonegawa et al., 2017).The resolutions of sedimentary layers estimated based on seismological methods are several hundred meters.Thus, thicknesses of unmodeled thin lower-velocity (V S < 0.6 km/s) sediments may be less than several hundred meters.The propagation paths between intraslab earthquakes and shallow tremors were significantly different (Figure 1b), but the obtained site amplifications were used in site corrections for shallow tremor waveforms.After site corrections, the seismic energies of the shallow tremors were obtained in the same manner as those of the deep tremors (Nakano et al., 2019;Yabe et al., 2019Yabe et al., , 2021)).The seismic energy rates of the shallow tremors range from 10 3 to 10 6 J/s.The scaled energies of shallow tremors exhibited regional differences: 10 9 -10 8 off Cape Muroto and southeast off the Kii Peninsula and 10 10 -10 9 off the Kii Channel and along the Japan Trench.There is a depth difference in the scaled energies (0-1 order difference) beneath and off the Kii Peninsula.This depth difference in scaled energy could be considered a result of differences in temperature and pressure at shallow (<150°C, <0.2 GPa) and deep (>300°C, 1 GPa) depths (Yabe et al., 2019).
Characteristics of high-frequency seismic waves around shallow plate boundaries are complicated.The bottomright panel of Figure 1a shows a sample waveform of a shallow tremor.This long-duration and spindle-shaped envelope is caused not only by complicated long-duration moment rate functions but also by envelope broadening due to the trapping of seismic waves within the thick sedimentary layer (Takemura, Emoto, et al., 2023;Takemura, Yabe, et al., 2020).This envelope broadening can be significant if seismic sources are located just below thick sedimentary layers.Typical ordinary earthquakes tend to occur at deeper depths, and thus, envelope broadening due to thick sedimentary layers becomes weak.Using a local 3D sedimentary model in this region, Takemura, Yabe et al. (2020) demonstrated that the model simulation could reproduce the observed envelope shape of a shallow tremor.Observations of shallow slow earthquakes are still limited in the world (summarized in Takemura, Hamada, et al., 2023), but similar phenomena can be expected in near-source OBS observations of shallow tremors within Nankai, Mexico, Costa Rica, and Hikurangi subduction zones.However, the latter effects have yet to be incorporated into conventional methods of seismic energy estimation.
In this study, to better understand seismic slow earthquakes at shallow depths, we investigated the effects of thick sedimentary layers on high-frequency seismic waves at OBSs using both observed DONET and synthetic seismograms.The Results section first provides site amplifications at DONET stations estimated via the conventional method using observed intraslab earthquakes.These site amplifications could include both effects of thick lowvelocity and thin lower-velocity sediments.We synthesized high-frequency seismograms at the OBSs from an intraslab earthquake and a shallow tremor using a wavenumber integration program code and local onedimensional (1D) velocity models.The shallow tremors along the Nankai Trough occur around the basement of thick sedimentary layers (approximately 8 km), while intraslab earthquakes tend to be located at depths of 20-40 km.We investigated the depth-dependent propagation path effects of thick sedimentary layers using synthetic seismograms with and without thick sedimentary layers.Synthetic seismograms clearly demonstrate amplification differences due to path effects due to source depth.A comparison between the estimated site amplifications from observations and the effects of thick sedimentary layers from synthetics provided the major cause of the site amplifications at OBSs.We then evaluated the amplification of seismic energies for shallow tremors caused by thick sedimentary layers.
In the Discussions section, we discuss seismic energy amplifications due to an invalid assumption of heterogeneities around the seismic source.Determining the source depths of shallow slow earthquakes remains challenging, but the rigidity in seismic energy estimation has been typically assumed to be 33 GPa as the crustal property.Seismic and scaled energies of seismic slow earthquakes were estimated in various regions of Japan.Based on the resultant seismic energy amplifications of shallow tremors, we revisited the scaled energies of seismic slow earthquakes in Japan.

Data and Methods
We used continuous velocity seismograms recorded at the DONET (National Research Institute for Earth Science and Disaster Resilience, 2019) and F-net (National Research Institute for Earth Science and Disaster Resilience, 2019) stations.Almost all the F-net broadband seismometers were deployed at outcrop rock sites; thus, Fnet data can be used as a reference for site correction.Takemoto et al. (2012) demonstrated that the values of Swave amplification factors are similar at all F-net stations for various frequency ranges.Each DONET node contains four to five seismic stations.Detailed information on both networks is available in Aoi et al. (2020).We did not use the M.KMA and M.KME nodes because of their distances from the shallow tremor sources.We also did not use unburied DONET stations (KMC11 and KMC12).To estimate the site amplifications at the DONET stations, we used data from 140 intraslab earthquakes from April 2016 to December 2022.The origin time, hypocenter locations, and magnitudes were obtained from a unified hypocenter catalog provided by the Japan Meteorological Agency (JMA).The JMA magnitudes ranged from 3.0 to 5.1.We measured the maximum S-wave amplitudes at the F-net and DONET stations.We estimated the site amplification factors of the DONET stations based on the method by Yabe et al. (2019).Assuming far-field body wave propagation in a homogeneous media, the S-wave amplitude at the j-th station from the i-th intraslab earthquake can be expressed as follows: where S i is a source term, R ij is hypocentral distance, α is the attenuation factor of πf/QV S , and G j is a site amplification factor at the j-th station.We set the site amplification factor of the N.KMTF to 1.This equation can then be solved using the least-square method.
We synthesized seismograms assuming a 1D velocity structure model to investigate the propagation path effects near the Nankai Trough.The 1D P-wave model around the DONET stations by Nakano et al. (2013) was used.This P-wave velocity model is optimal for hypocenter location analysis in this region.The S-wave velocity, density, and anelastic attenuation were obtained by assuming the empirical laws proposed by Brocher (2005Brocher ( , 2008)).We named this model "DONET1D" (Figure 2a).In DONET1D, the interface of the Philippine Sea Plate is located at a depth of 8.07 km.DONET1D agreed with the 1D S-wave velocity models beneath M.KMB06 and M. KMD13 by Tonegawa et al. (2017) (blue dashed lines in Figure 2a).Thick (∼5 km) sedimentary layers with V S of 0.6-2.3km/s exist beneath M.KMB and M.KMD.Using the local 3D model, Takemura, Yabe et al. ( 2020) and Takemura, Emoto, et al. (2023) demonstrated that the major cause of spindle-shale high-frequency seismogram envelopes at OBSs southeast off the Kii Peninsula is thick sedimentary layers, rather than other heterogeneities (such as bathymetry, the subducted Philippine Sea Plate, seawater, and small-scale velocity heterogeneities).In addition, shallow tremors and VLFE epicenters are located around the M.KMB and M.KMD nodes (Nakano et al., 2018;Takemura, Obara, et al., 2022;Tamaribuchi et al., 2022).Thus, modeling using DONET1D can provide the average characteristics of high-frequency seismic wave propagation within shallow tremor regions along the Nankai Trough.
To investigate the effects of thick sedimentary layers, we prepared another 1D model, DONET1D' (Figure 2b) in which the physical parameters of the sedimentary layers were replaced with those of the oceanic crust.Green's functions using both 1D models can be evaluated by employing the wavenumber integral calculations using the open-source code "Computer programs in Seismology" (CPS; Herrmann, 2013).The seismic sources were assumed to be a low-angle thrust mechanism (strike/dip/rake = 270°/10°/90°) at a depth of 8.07 km and a normal fault mechanism (strike/dip/rake = 300°/45°/ 120°) at a depth of 40 km.These are the typical mechanisms of shallow tremors and intraslab earthquakes in this region.The seismic moment Mo was fixed at 3.98 × 10 13 Nm (moment magnitude Mw 3.0).
Simulations within the same models using the open-source finite-difference method code OpenSWPC (Maeda et al., 2017) were also conducted to evaluate seismic wavefields in 3D volumes.The 3D simulation model covered 105 × 30 × 75 km 3 and was discretized using a uniform grid of 0.015 km.The 64-s seismic wave propagation was calculated using 80,000 time steps.In the OpenSWPC simulations, we assumed a single-cycle Küpper wavelet with a duration of 0.25 s rather than an impulse source time function (STF) to reduce numerical instability.Short-duration STFs were assumed in both CPS and OpenSWPC synthetics.The simplest tests with a short-duration test can represent seismic wave propagation from a shallow LFE source.Although short-duration seismic slow earthquakes have recently been reported (Toh et al., 2023), this assumption may be invalid for realistic tremor synthetics.Therefore, we examined the effects of complicated STFs using the Brownian slow earthquake (BSE) model (Ide, 2008;Ide & Maury, 2018).
Using theoretical S-wave traveltimes (T S ) in 1D models, we measured the maximum S-wave amplitudes for each filtered velocity seismogram from times starting at T S -1 to reduce the effects of the zero-phase Butterworth filter.The seismic energies were calculated using smoothed velocity envelopes as a typical tremor analysis.We could not identify P and S phases from the spindle-shape tremor waveforms (Figure 1a).The vector sum of the threecomponent envelopes of filtered waveforms at frequencies of 2-8 Hz was calculated.A 5-s moving average was applied to obtain smooth envelopes.Owing to the lack of clear P-and S-wave onsets, smoothed envelopes are typically used as S-waves for the location and energy analyses of the tremors.The half-value width, τ (t 2 t 1 ), of the smoothed envelope was measured as the source duration.The normalized seismic energy E ij /C ij at the j-th station was calculated using the following equation: where R ij is the hypocentral distance from the i-th source to the j-th receiver.The t 2 and t 1 are times of half-value width starting and ending, respectively.The constant, C, is expressed as follows: where V S is the S-wave velocity, ρ is density f C is the central frequency, and Q is a quality factor.In previous studies, V S and ρ were typically fixed as 3.5 km/s and 2.7 g/cm 3 , respectively.These values are based on the assumption of far-field body wave propagation in an infinite homogeneous medium with a rigidity of 33 GPa.The effects of the source radiation pattern were also typically neglected because of high-frequency seismic wave propagation at regional distances (Takemura, Yabe, et al., 2020;Takemura et al., 2009Takemura et al., , 2016;;Trugman et al., 2021).We assumed isotropic radiation in energy estimation similar to the conventional method for tremor analysis, but synthetics could include the effects of the four-lobe source radiation pattern.We evaluated E ij /C ij for DONET1D and DONET1D' because C ij became common at stations with the same distances.The estimated Q at 2-8 Hz was approximately 800 (results shown in the next section) and was not dominant in the energy estimation.

Site Amplifications at DONET OBSs Based on the Conventional Method
Figure 3 shows the estimated site amplification factors at the F-net and DONET stations.Both vertical and horizontal site amplifications at N.KMTF (bold diamonds) were fixed as 1.We estimated the site amplifications of the 1-2, 2-4, 4-8, and 2-8 Hz (tremor band).Our site amplification factors agree well with those in previous studies (Kubo et al., 2018;Yabe et al., 2019).The amplification factors of the horizontal component range from 5 to 40, while those of the vertical component range from 0.5 to 3, except for the stations near the Nankai Trough.Differences between the horizontal and vertical components were also reported for the S-wave coda H/V ratio (Takemura, Emoto, et al., 2023).The estimated Q values at 1-2, 2-4, 4-8, and 2-8 Hz were 254, 481, 933, and 795, respectively.The estimated site amplifications and Q values were obtained from the Zenodo repository (see "Open Research").

Characteristics of Synthetic Seismograms With/Without Thick Sedimentary Layers
Figure 4 shows the synthetic velocity seismograms of DONET1D and DONET1D'.A bandpass filter with frequencies of 2-8 Hz was used.The seismic waves from a shallow tremor source were effectively trapped within thick sedimentary layers (Takemura, Yabe, et al., 2020); consequently, the onset of the P-and S-waves became unclear, and strong envelope broadening occurred (Figure 4a).The group velocity of traces at distances ≥20 km fluctuated around 2.3 km/s, which was typically within the velocity range for shallow tremors in this region ( Akuhara et al., 2023).Reverberations within the sedimentary and seawater layers cause unclear onsets of individual phases (Movie S1).In the model without sedimentary layers (Figure 4b and Movie S2), clear P-and Swave onsets were observed.sP converted and multiple reflected waves from the sea surface were observed.For a slab earthquake source (Figures 4c and 4d and Movie S3-S4), although the traveltimes of the P-and S-waves were delayed because of the thick sedimentary layer, P and S wavetrains were clearly identified in both models.

Amplifications of Maximum S-Wave Amplitudes and Seismic Energies
Examples of the filtered seismograms for a shallow tremor source are shown in the top panels of Figure 5. Unclear P-and S-wave onsets and envelope broadening are observed in DONET1D (blue lines).Envelope broadening in the smoothed envelopes of DONET1D (blue bold lines) is caused by the contamination of reflected S-waves between the seafloor and the basement of the sedimentary layers (plate interface).In DONET1D' (red lines), Pand S-wave signals were clear and impulsive.The reflection phases from the sea surface were repeatedly confirmed after S arrival.The smoothed envelopes in DONET1D' (red bold lines) contain not only the S-wave content but also those of P-and reflected waves from the sea surface.At an epicentral distance of 20 km, the maximum amplitude of the smoothed envelope in DONET1D was several times larger than that of DONET1D'.This amplitude difference decreased at 40 km; however, the envelope duration in DONET1D remained longer.
Figure 6 shows the maximum S-wave amplitudes at 1-2, 2-4, and 4-8 Hz for the shallow tremor and intraslab earthquake.The blue and red symbols represent the results from DONET1D and DONET1D', respectively.For an intraslab earthquake, the differences in the horizontal maximum S-wave amplitudes between DONET1D and DONET1D' were practically constant irrespective of the distance.The differences in an intraslab source between DONET1D and DONET1D' decreased with increasing frequency.The ratios of the horizontal maximum S-wave amplitudes between DONET1D and DONET1D' (amplification due to thick sedimentary layers) were approximately 1-2 at 1-2 and 2-4 Hz, implying that the observed large horizontal amplifications of maximum S-wave amplitudes against N.KMTF (Figure 3) were mostly caused by thin lower-velocity (V S < 0.6 km/s) sediments just below the DONET stations (brown areas in Figure 1b).These distance-independent differences can easily be corrected using the conventional method.However, complicated differences in the maximum S-wave amplitudes between the models appeared for a shallow tremor source (Figure 6b).At 1-2 and 2-4 Hz, the horizontal S-wave amplitudes were 2-13 times amplified at distances of 5-20 km (near-source OBSs).The differences in horizontal  S-wave amplitudes also decreased with increasing frequency and distance.The effects of thick sedimentary layers on the maximum S-wave amplitudes for shallow tremors and intraslab earthquakes differed completely.
Figure 7 shows the normalized seismic energy E/C at each distance.Although an impulse STF was assumed in the CPS synthetics, a 10-s half-value width (dashed lines in the right panels in Figure 7) was expected because of the 5-s moving average smoothing.As C is common at stations at the same distance, the ratios of the seismic energies of DONET1D and DONET1D' (amplification factor for seismic energy) (Figures 7c and 7d) reflect the amplification factors of the seismic energies due to the thick sedimentary layer.The differences for an intraslab earthquake (Figures 7a and 7c) were also nearly constant (2-3 times), irrespective of the distance.Distancedependent features of seismic energy amplification were observed for the shallow tremor source (Figures 7b  and 7d).At distances of ≤5 km (the region just above a source), we observed an amplification factor of approximately 4. This is slightly larger than an intraslab source (3.3) but can be considered a vertical incident amplification factor.Large (>5) seismic energy amplifications were observed at distances of 5-20 km.Reflected S-waves from the sediment/oceanic crust boundary (plate boundary) appeared repeatedly (Movie S1) in DONET1D, although such phases were not observed in DONET1D' (Movie S2).The smoothed velocity envelopes in DONET1D contained such reflections; consequently, large energy amplifications occurred at distances of 5-20 km.
According to previous studies of shallow slow earthquakes (Nakano et al., 2018;Sugioka et al., 2012;Takemura et al., 2019), we considered that shallow tremors could be molded by a low-angle thrust faulting mechanism around the plate boundary.In this case, S-wave energies can be weakened at distances of 5-20 km due to the four- lobe S-wave source radiation pattern (red symbols in Figures 6b and 7b).Four-lobe S-wave amplitude pattern is gradually distorted as increasing frequency and distance because of seismic wave scattering due to small-scale heterogeneities (e.g., Imperatori & Mai, 2013;Morioka et al., 2017;Takemura et al., 2009Takemura et al., , 2016)).Takemura, Yabe, et al. (2020) demonstrated that in the cases of seismic sources just below or within a thick sedimentary layer, trapping seismic wave energies within thick sedimentary layers also distort the four-lobe S-wave amplitude pattern.Thus, significant amplifications of seismic energies for shallow tremors appear at distances of 5-20 km.However, we note that this feature was not changed in the model with exponential-type small-scale random fluctuations (Figure S1 in Supporting Information S1), which distort the four-lobe S-wave radiation patterns.This means that the effects of thick sedimentary layers can be dominant.At distances >20 km, S-waves propagated horizontally, and the amplification of seismic energy weakened with increasing distance.Seismic energy amplifications of a shallow tremor at distances less than 5 km are similar to those for an intraslab earthquake.
Stronger attenuation than the empirical law has been estimated at shallower depths (e.g., Abercrombie, 1997;Eberhart-Phillips et al., 2014;Yoshida et al., 2023).Thus, we additionally synthesized seismograms using DONET1Dq, where the Q S value within the shallowest layer was replaced from 39.8 to 20 (Figure S2 in Supporting Information S1).Amplifications of seismic energies become weaker from the original DONET1D, but propagation-path amplification tends to be dominant in shallow tremor seismograms at distances of 5-20 km.
To evaluate the effects of the STFs, we synthesized them based on the BSE model (Ide, 2008;Ide & Maury, 2018).We prepared 200 BSE model STFs with a characteristic time α of 0.01 s 1 , which were normalized as each seismic moment of 1.These STFs were convolved using Green's functions in DONET1D and DONET1D'.The resultant ratios of the seismic energies of DONET1D and DONET1D' are illustrated in Figure 8a. Figure 8b shows two examples of normalized BSE model STFs.The duration of the prepared STFs ranged from 1 to 54 s (Figure 8c).Large amplifications of seismic energies at distances of 5-20 km were commonly observed (Figure 8a).The strength of the envelope broadening appears to depend on the source duration (solid, dashed, and dotted lines in Figure 8d).Parameters τ and τ 0 are the half-value widths of the synthetic envelopes from DONET1D and DONET1D', respectively.The BSE model STFs with shorter durations exhibited strong envelope broadening (large (τ τ 0 )/τ 0 ), as shown by the results of an impulse STF (bold blue line in Figure 8d).With increasing source duration, the effects of envelope broadening caused by the thick sedimentary layer tended to be relatively weak (blue dashed and dotted lines).For the longest duration STF case (blue dotted line), nearly similar half-value widths were found in both models.It indicates that if source durations are sufficiently longer than the envelope widths of Green's functions, overestimations of source durations are negligible.

Discussion
The characteristics of seismic energy amplification caused by thick sedimentary layers differ between intraslab earthquakes and shallow tremors.The conventional methods cannot correct large amplifications at distances of 5-20 km.Owing to the signal-to-noise ratio of shallow tremors at OBSs, near-source (≤20 km) OBSs are typically analyzed.Based on seismic energy amplifications due to thick sedimentary layers (Figure 7d), we should correct additional 0.5-1 order amplifications in seismic energies of shallow tremors in previous studies along the Nankai Trough.This amplification correction is valid when shallow tremors occur at the plate interface.

Effects of Invalid Rigidity Assumption on Seismic Energy Estimation
Slow earthquake phenomena are considered slip phenomena at the plate boundary.Although the precise determination of the source depths of shallow slow earthquakes remains challenging, shallow VLFEs tend to be located within underthrust sediments around the décollement (Akuhara et al., 2020;Sugioka et al., 2012;Yamamoto et al., 2022).Underthrust sediments are considered to have low seismic velocity (1-2 km/s).Is the assumption of 33 GPa rigidity (V S = 3.5 km/s and ρ = 2.7 g/cm 3 ) in the conventional seismic energy estimation valid?To answer this question, we synthesized Green's functions at depths of 6.0 and 9.0 km.The former and latter sources are located within the underthrust sediment (V S = 2.3 km/s) and oceanic crust layer 2 (V S = 3.3 km/ s).As in previous synthetics, we fixed a focal mechanism and a seismic moment of 3.98 × 10 13 Nm.
In all synthetics, relatively large amplifications were observed at distances of 5-20 km (Figure 9).Based on these results, amplifications caused by path effects of the thick sedimentary layers were dominant at distances of 5-20 km because the reflected S-waves from the plate interface had sufficient amplitudes.In such cases, the energies of the reflected S-waves are contaminated within a half-value width time window of smooth envelopes; consequently, the seismic energies of shallow tremors tend to be overestimated.
Although the effects of multiple S-wave reflections commonly appear at distances of 5-20 km, seismic energy amplification increases with decreasing source depth.This is because of the differences in rigidity between DONET1D and DONET1D'.The rigidity of DONET1D' was constant (28 GPa) at depths shallower than 11 km, but the rigidities of the source regions at depths of 6 and 9 km in DONET1D were 12 and 28 GPa, respectively.Although a double-couple source could not be strictly assumed at the plate boundary, the intermediate features between the 6-and 9-km sources appeared in an 8.07 km source.These rigidity differences could be another cause of seismic energy amplification from the method assuming far-field S-wave propagation in an infinite medium with a rigidity of 33 GPa.The precise spatial distribution of rigidity is also important for seismic moment estimation (Figures 4, 5, and 7 in Takemura et al., 2021).Although the seismic moment is proportional to the observed amplitudes, the seismic energy is calculated by temporal integration of the square velocity amplitudes.Thus, the effects of incorrect rigidity assumptions are more severe in the seismic energy estimation.

Propagation Path Amplifications Along the Japan Trench
The seismic and scaled energies of slow earthquakes were also evaluated along the Japan Trench, offshore regions of northeastern Japan, and Hokkaido (Yabe et al., 2021).These were also calculated using the conventional method of the OBS network (S-net).We also synthesized velocity seismograms using a 1D velocity model around the Japan Trench to validate their estimations.The 1D model was constructed from a 1D depth profile at 143.6°E and 40.0°N from the local 3D model of Koketsu et al. (2012).The region at 143.6°E and 40.0°N is approximately the centroid of tremor activity.We refer to this model as NEJP1D.We also constructed NEJP1D' in which the physical parameters of the sedimentary layers in NEJP1D were replaced with those of the crust.The source of the tremor was located at a depth of 12.85 km, the plate interface.The ratios between NEJP1D and NEJP1D' (Figure 10a) were stable (3.5-7) compared with those along the Nankai Trough (Figure 7d).We also plotted energy amplifications for an intraslab earthquake (diamonds in the middle panel of Figure 10a).Because of differences in focal mechanisms and incident angles to the basement, amplifications for an intraslab earthquake were more stable and larger.However, distant-independent features commonly appeared in both cases.The differences in seismic energy amplifications between tremors and intraslab earthquakes were weaker compared with those along the Nankai Trough (Figure 7a).Thus, the conventional method practically works in the Tohoku region.This is because the tremors occurred sufficiently deeper than the basement of the sedimentary layer (Figure 10b).The differences in the propagation paths between tremors along the Nankai Trough and Japan Trench are illustrated in Figures 1b and 10b.
Near-source (≤20 km) OBS data are often selected in seismic energy estimations of shallow tremors due to the signal-to-noise ratio.Based on the above synthetic studies and the selected use of near-source OBSs, we conclude that approximately 0.5-1.3order overestimations can occur in the seismic energy estimation of shallow tremors along the Nankai Trough (Figures 7 and 9).Similar overestimations from near-source OBSs are expected for shallow slow earthquakes in the regions of Hikurangi, Mexico, and Costa Rica, where shallow slow earthquakes are located near the trough or shallower (≤10 km) depths (Baba et al., 2021;Plata-Martinez et al., 2021;Todd et al., 2018;Walter et al., 2013).Although slow earthquakes are generally phenomena of faulting on the plate boundary, amplifications are typically more severe if shallow tremors occur within sedimentary layers.

Scaled Energy of Seismic Slow Earthquakes at Deep and Shallow Depths Around Japan
Based on the above results, we revisited the scaled energy of slow earthquakes.Figure 11 shows the relationships between the seismic energy and moment rates for slow earthquakes in various regions and depths.Deep slow earthquakes in the Nankai, Mexico, and Cascadia subduction zones were obtained from previous studies (Ide, 2016;Ide & Maury, 2018;Ide & Yabe, 2014).The relationship between the moment and seismic energy rates of slow earthquakes along the Japan Trench was from Yabe et al. (2021).Based on the effects of the thick sedimentary layer near the Nankai Trough, we performed a 1-order correction for the seismic energy rates of shallow tremor and a 0.3-order correction for the seismic moment rate of shallow VLFEs along the Nankai Trough from the results in Yabe et al. (2019Yabe et al. ( , 2021)).The 0.3 order corrections of the seismic moment rates of the shallow VLFEs were determined by the rigidity difference between the oceanic crust (28 GPa) and underthrust sediments (12 GPa).The rigidity of the oceanic crust is almost twice that of underthrust sediments, and a two-fold amplification of the VLFE signals is expected.Temperature and lithostatic pressure at deep depths are 150-500°C and 0.7-1.7 GPa, which are significantly larger than those at shallower depths (<150°C and <0.2 GPa) (Behr & Bürgmann, 2021;Saffer & Wallace, 2015;Syracuse et al., 2010).Even for these large differences in tectonic environments, we concluded that the scaled energies of seismic slow earthquakes range from 10 10 to 10 9 , irrespective of region and depth (filled symbols in Figure 11).
An Mo ∝ T upper-bound scaling law for deep slow earthquakes in various subduction zones was suggested (Ide & Beroza, 2023).However, the relationships of detectable shallow VLFEs between seismic moments and durations along Nankai Trough (Sugioka et al., 2012;Takemura et al., 2019;Takemura, Obara, et al., 2022) are slightly different with an Mo ∝ T upper-bound scaling law for deep slow earthquakes.The detectability of VLFEs along the Nankai Trough was evaluated in Takemura et al. (2024).Shallow VLFEs lie between scaling laws of ordinary (Mo ∝ T 3 ) and deep slow (Mo ∝ T ) earthquakes.Duration ranges were not different at different depths, but the seismic moments of shallow VLFEs were 1-2 orders larger than those at deeper depths (Ide et al., 2008).Other differences between shallow and deep slow earthquakes (durations and recurrent intervals of slow earthquake episodes, migration speeds, etc.) were summarized in a recent review paper (Takemura, Hamada, et al., 2023).Despite the different scaling laws between deep and shallow slow earthquakes, our study suggests that the scaled energies of seismic slow earthquakes are common (10 10 -10 9 ), irrespective of depth and region.What factors cause the different distributions of seismic moments and durations at shallow and deep depths?Source analysis of seismic slow earthquakes under valid assumptions should be addressed in future studies to answer this question.Integrating seismological, geodetic, geological, and experimental studies is indispensable for investigating the source physics and tectonic environments of slow earthquakes.

Conclusions
Recent studies on high-frequency seismic wave propagation have revealed that the effective trapping of seismic waves within thick sedimentary layers affects the waveforms observed at OBSs, even for near-source distances.Large envelope broadening and amplification are expected in high-frequency seismograms of OBSs.Thus, in this study, we investigated the effects of the propagation path and site amplification on seismic energy estimations, especially for shallow tremors along the Nankai Trough.
Assuming near-vertical incidents to OBSs and far-field S-wave propagation, we estimated frequency-dependent site amplifications at DONET stations; the amplification factors of DONET stations in the horizontal component ranged from 5 to 40.The synthetics for an intraslab earthquake assuming a local 1D velocity model with V S ≥ 0.6 km/s are only 1-2 times the amplifications from a 1D model without sedimentary layers.This indicates that large amplifications at the DONET stations were primarily controlled by thin lower-velocity (<0.6 km/s) sediments just below the DONET stations, and for a shallow tremor source, 5-10 times the amplifications of seismic energy due to thick sedimentary layers appeared at near-source (≤20 km) distances.This amplification was caused by multiple reflected S-waves from the plate interface.Because the S-phase cannot be identified from typical tremor waveforms, smoothed velocity envelopes have been widely used in seismic energy analysis.In this case, multiple reflected S-waves were contaminated.If shallow tremors occur within underthrust sediments, the assumption of far-field S-wave propagation in an infinite medium with a rigidity of 33 GPa is invalid.The incorporation of precise rigidity around the source region is required.
Overestimations owing to thick sedimentary layers often occurred in the seismic energy estimations of shallow tremors at sources shallower than 10 km.Similar overestimations using near-source (≤20 km) OBSs potentially occur in regions of Hikurangi, Costa Rica, and Mexico.Based on propagation path amplification at near-source OBSs and the invalid rigidity assumption, approximately 0.5-1.3order overestimations can occur in the seismic energy estimation of shallow tremors along the Nankai Trough based on the conventional method.After correcting for overestimations in previous studies, the scaled energies of shallow seismic slow earthquakes along the Nankai Trough and Japan Trench and deep seismic slow earthquakes in various regions range from 10 10 to 10 9 .This means that the physical mechanisms governing seismic slow earthquakes can be the same, irrespective of region and source depth.

Data Availability Statement
We used DONET (National Research Institute for Earth Science and Disaster Resilience, 2019a) and F-net (National Research Institute for Earth Science and Disaster Resilience, 2019b) data.The Python package, HinetPy (Tian, 2020), was used to download the data.CPS (Herrmann, 2013) and OpenSWPC (Maeda et al., 2017) were used for waveform synthesis.Seismic analysis codes (Goldstein & Snoke, 2005), obspy (Beyreuther et al., 2010), scipy (Virtanen et al., 2020), numpy (Harris et al., 2020), and Generic Mapping Tools (Wessel et al., 2013) were used for waveform analysis and image creation.The catalog of ordinary earthquakes used to estimate site amplification was obtained from the JMA (https://www.data.jma.go.jp/eqev/data/bulletin/ index.html).The catalogs of slow earthquakes along the Nankai Trough were referred from the "Slow earthquake database" (Kano et al., 2018).Estimated site amplification factors at DONET stations and Movies S1-S4 are available at a Zenodo repository (Takemura, 2023).

Figure 1 .
Figure 1.(a) Examples of high-frequency seismograms at offshore (M.KMD13) and inland (N.KISF) stations.The open diamonds represent the locations of other DONET stations.The upper and middle right panels are high-frequency (>1 Hz) EW-component seismograms during an intraslab earthquake, which occurred at a depth of 39 km at 3:19 on 22 May 2020 (JST).The right bottom panel shows a high-frequency (>1 Hz) EW-component seismogram during a shallow tremor, which occurred at 10:19 on 12 December 2020 (JST).The blue star and red circle represent epicenters of an intraslab earthquake and a shallow tremor, respectively.(b) Schematic illustration of propagation paths from seismic sources to OBSs southeast of the Kii Peninsula region.

Figure 2 .
Figure 2. Assumed 1D velocity structure models.(a) DONET1D model constructed from the 1D P-wave model of Nakano et al. (2013) and empirical laws of velocity structures(Brocher, 2005(Brocher, , 2008)).(b) DONET1D', where physical parameters within sedimentary layers are replaced with those within the oceanic crust.The red and blue colors represent P-and S-waves, respectively.The dashed lines are density as a function of depth.The blue dotted lines in (a) are S-wave velocity models beneath M.KMB06 and M.KMD13 (locations shown in the map of Figure1a) byTonegawa et al. (2017).

Figure 3 .
Figure 3. Spatial variations of site amplification factors at each frequency band.The upper and bottom panels are site amplification factors for vertical and horizontal components, respectively.The gray circles in the upper right panel are epicenters of slab earthquakes used in estimating site amplification factors.The diamond enclosed by the bold line is the reference site N.KMTF (the site amplification factor of N.KMTF was fixed as 1).The location of N.KMTF is represented by the bold diamonds.

Figure 4 .
Figure 4. Radial component velocity seismograms synthesized using CPS.The seismic sources in (a, b) and (c, d) are a shallow tremor and intraslab earthquake, respectively.The source time functions of each case are an impulse.(a, c) DONET1D and (b, d) DONET1D'.We applied a bandpass filter of 2-8 Hz, and maximum amplitudes at each trace were normalized.

Figure 5 .
Figure 5. Examples of Green's functions at distances of 20 and 40 km for a shallow tremor source.The radial component velocity traces were filtered with a passed frequency of 2-8 Hz.The bold lines are smoothed velocity envelope traces.The black dashed lines represent theoretical S-wave travel times in each model.

Figure 6 .
Figure 6.Maximum S-wave amplitudes at each frequency band from (a) intraslab earthquake and (b) shallow tremor sources.The blue and red symbols are the maximum S-wave amplitudes in DONET1D and DONET1D', respectively.The filled and open symbols are those in horizontal and vertical components, respectively.

Figure 7 .
Figure 7. Normalized seismic energies of (a) intraslab earthquake and (b) shallow tremors.Normalization factor C includes physical parameters and anelastic attenuation.The blue and red symbols represent the results of DONET1D and DONET1D', respectively.Ratio of seismic energies between DONET1D and DONET1D' for (c) intraslab earthquake and (d) shallow tremor sources.

Figure 8 .
Figure 8. Ratio of seismic energies of shallow tremor using the Brownian slow earthquake (BSE) model with a characteristic time of α = 0.01 s 1 .(a) The ratio of seismic energies between DONET1D and DONET1D' for the BSE model and impulse STF (Green's function), (b) examples of BSE model STF, and (c) durations of used BSE model STFs.The time-series of moment rates for two examples of BSE model STFs are illustrated in (b).The seismic moments of both examples are normalized as 1.The light blue lines in (a) represent ratios of seismic energies between DONET1D and DONET1D' for individual BSE model STFs.The blue bold line is the same as in panel (b).(d) Estimated half-value width ratio between DONET1D (τ) and DONET1D' (τ 0 ).The blue bold, dashed, and dotted lines in (d) are ratios of impulse STF, BSE model STF with a duration of 17 s, and the longest BSE model STF (54 s), respectively.

Figure 9 .
Figure 9. Seismic energies of shallow tremors at depths of 6 (within the sedimentary layer), 8.07 (plate boundary), and 9 km (within the second layer of the oceanic crust).

Figure 10 .
Figure 10.Seismic energies of synthetic tremor envelopes along the Japan Trench.(a) Seismic energies (E/C), ratio of seismic energies, and half-value width (duration) of NEJP1D and NEJP1D'.In the middle panel of (a), ratio of seismic energies of an intraslab earthquake (40 km depth) between NEJP1D and NEJP1D' are also plotted by diamonds.(b) NEJP1D model.The red and blue colors represent P-and S-waves, respectively.The dashed lines represent density as a function of depth.The right panel in (b) is the map around the Japan Trench.Tremor epicenters are referred from Nishikawa et al. (2019).The bottom panel in (b) is a schematic figure of seismic wave propagation from the tremor off Tohoku.

Figure 11 .
Figure 11.Revisited relationships between the seismic energy rates of tremors and seismic moments of accompanying VLFEs.The gray diamonds indicate deep seismic slow earthquakes in Mexico, Cascadia, and Nankai subduction zones (Ide, 2016; Ide & Maury, 2018; Ide & Yabe, 2014).The orange triangles indicate the seismic moment and energy rates of seismic slow earthquakes off Tohoku from Yabe et al. (2021).The light blue open circles indicate the original results of shallow slow earthquakes along the Nankai Trough(Yabe et al., 2019(Yabe et al., , 2021)).The blue-filled circles indicate corrected relationships between seismic moment and energy rates for shallow slow earthquakes along the Nankai Trough.