Corresponding author: T. Takahashi, Institute for Research on Earth Evolution, Japan Agency for Marine-Earth Science and Technology, 3173-25 Showa-machi, Kanazawa-ku, Yokohama 236-0001, Japan. E-mail: (email@example.com)
 Seismic waves at high frequencies (>1 Hz) show collapsed and broadened wave trains caused by multiple scattering in the lithosphere. This study analyzed the envelopes of direct S waves in southwestern Japan and on the western side of the Nankai trough and estimated the spatial distribution of random inhomogeneities by assuming a von Kármán type power spectral density function (PSDF). Strongly inhomogeneous media have been mostly imaged at shallow depth (0–20 km depth) in the onshore area of southwestern Japan, and their PSDF is represented as P(m) ≈ 0.05m−3.7 km3, with m being the spatial wave number, whereas most of the other area shows weak inhomogeneities of which PSDF is P(m) ≈ 0.005m−4.5 km3. At Hyuga-nada in Nankai trough, there is an anomaly of inhomogeneity of which PSDF is estimated as P(m) ≈ 0.01m−4.5 km3. This PSDF has the similar spectral gradient with the weakly inhomogeneous media, but has larger power spectral density than other offshore areas. This anomalous region is broadly located in the subducted Kyushu Palau ridge, which was identified by using velocity structures and bathymetry, and it shows no clear correlation with the fault zones of large earthquakes in past decades. These spatial correlations suggest that possible origins of inhomogeneities at Hyuga-nada are ancient volcanic activity in the oceanic plate or deformed structures due to the subduction of the Kyushu Palau ridge.
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 The Nankai trough off southwestern Japan is a convergent margin where the Philippine Sea plate is descending beneath the Eurasian plate (Figure 1). Major interplate earthquakes in this region are called Tokai, Tonankai, or Nankai earthquakes according to the fault segment on which they occur. Studies on earthquake occurrence history have shown that these earthquakes have recurrence intervals of 100–150 years [e.g., Ando, 1975], and their rupture propagation shows various patterns, for example, the individual rupture of one segment or nearly simultaneous or successive ruptures of contiguous segments.
 Seismic velocity structures in the Nankai trough have been intensively studied by active and passive source surveys to elucidate relations among seismic structures, fault segment distribution, and rupture propagation. For example, a wide-angle refraction survey revealed a subducted seamount on the eastern side of the rupture area of the 1946 Nankai earthquake [Kodaira et al., 2002]. Kodaira et al.  proposed that this subducted seamount acted as a barrier to rupture propagation during the earthquake. A reflection survey detected a splay fault branching from the plate boundary along the Tonankai segment [Park et al., 2002], and coseismic slip along this splay fault has recently been confirmed by core sample analysis [Sakaguchi et al., 2011].
 Studies of seismic wave scattering at high frequency (>1 Hz) have developed various methods to estimate the 2-D or 3-D distribution of random inhomogeneities (or of scattering coefficients) in the lithosphere. These studies have shown clear spatial correlations among random inhomogeneities, fault zones of large earthquakes, seismicity, and tectonic settings. For example, Asano and Hasegawa  estimated the 3-D distribution of scattering coefficients in the crust by an inversion analysis of coda waves (>1 Hz). Their method is applicable when inelastic attenuation can be assumed to be spatially uniform. They detected a strong scattering region and discussed that the strong scattering was related to a fractured structure due to the 2000 western Tottori earthquake (Mw 7.3) in southwestern Japan. P wave envelope analysis of teleseismic waves has been used to identify regional variations in scattering strength, which tend to be correlated with tectonic conditions [e.g., Nishimura et al., 2002; Kubanza et al., 2007].
Takahashi et al. [2009, 2011] proposed that an inversion analysis of peak delay times of direct waves (S waves) at regional distances could be used to estimate the 3-D distribution of random velocity inhomogeneities. The peak delay time tp is defined as the time lag from the S wave onset to the maximal amplitude arrival of root mean square (RMS) waveform envelopes. This quantity reflects the mean delay of seismic wave energy due to multiple forward scattering, and it is quite insensitive to intrinsic absorption due to medium inelasticity [e.g., Saito et al., 2002]. The inversion analysis of tp can be used to estimate distributions of the power spectral density function (PSDF) of random velocity inhomogeneities without a strong assumption of medium inelasticity. In northeastern Japan and the northern Izu-Bonin arc, strong inhomogeneities have been imaged near high-microseismicity regions and beneath Quaternary volcanoes [Takahashi et al., 2009, 2011]. The PSDFs of random inhomogeneities show different spectral gradients between volcanic areas and high-seismicity regions. In other words, the statistical properties of the strong inhomogeneities beneath Quaternary volcanoes, which probably reflect inclusions of volcanic rock or magma, are different from those of fractured structures originating from seismic activity.
 In this study, we applied the peak delay time analysis to southwestern Japan and the Hyuga-nada region of the Nankai trough by using seismograms recorded by onshore and offshore stations. Hyuga-nada is the sea area at the western margin of the Nankai trough fault segments (Figure 1). Large earthquakes in the Hyuga-nada area occurred in 1968 (M 7.5), October 1996 (Ms 6.7), and December 1996 (Ms 6.7) [Yagi et al., 1998, 2001], and the study area also contains many Quaternary volcanoes and source areas of non-volcanic tremor [e.g., Obara, 2002]. This area is appropriate for studying random inhomogeneity and its relation to tectonic conditions, fault segments, and crustal structures. Medium beneath the Quaternary volcanoes possibly has strong intrinsic attenuation [e.g., Takahashi, 2012]; therefore, tp analysis is more adequate than coda wave studies to achieve our aims. We examined medium characteristics by comparing the distribution of random inhomogeneities in the study area with their distribution beneath northeastern Japan and the northern Izu-Bonin arc.
2 Peak Delay Times in Von Kármán Type Random Media
 The S wave velocity in a medium is assumed as V(x) = V0(1 + ξ(x)), where V0 and ξ(x) are, respectively, the background S wave velocity and random fluctuation. We consider an ensemble of random fluctuation. The ensemble average of the fluctuation 〈ξ(x)〉 is set to 0.0. It is assumed that the PSDF of ξ(x) is represented by a 3-D von Kármán-type PSDF,
where Γ represents the gamma function and m is the spatial wave number. Parameters a, ε, and κ are, respectively, the characteristic length of inhomogeneity, the RMS of fractional fluctuation, and a parameter related to the spectral roll-off at large wave numbers (m > > 1/a). Small κ corresponds to weak spectral gradient, and means strong inhomogeneities at larger wave number [e.g., Sato et al., 2012, Figure 2.6].
 We consider the case that an impulsive wavelet is isotropically radiated from a point source in a 3-D von Kármán-type random medium. When the fluctuation is small (|ξ| ≪ 1) and a is longer than the wavelength of the incident wave, small-angle scattering in the forward direction is dominant. Because multiple forward scattering accumulates with increasing travel distance, the impulsive wavelet is gradually collapsed and broadened. This broadening process of the direct wave can be stably measured by using wave envelopes. Quantitative relations between this envelope broadening of seismic waves and random inhomogeneities are studied by Markov approximation of the parabolic wave equation [e.g., Sato, 1989].
Saito et al.  examined seismic wave envelopes of spherically outgoing scalar waves in von Kármán-type random media by assuming a spatially uniform random medium, in which the whole space is represented by a set of values of a, ε, and κ. According to Saito et al. , the peak delay time tp of the RMS envelope can be written as a function of the travel distance r and the central frequency f:
 Parameters bp(κ) and p(κ) increase monotonically from 0.10 to 0.38 and from 1.19 to 1.99, respectively, as κ increases from 0.1 to 1.0. C(κ) ranges from 0.56 to 2.31 and reaches its maximum value at κ = 0.6. Details of these parameters are summarized by Saito et al.  and Takahashi et al. .
 It can be seen that κ and ε2/(p(κ) − 1)a− 1 are constitutive factors for the peak delay time. The Markov approximation describes wave scattering and diffraction due to inhomogeneities at large wave numbers (m ≫ 1/a). The smaller wave number component of velocity inhomogeneity contributes to the fluctuations of onset time. If we take ensemble average of seismic waves of which onset times are not aligned, the peak delay time and envelope shape are distorted [Sato, 1982]. Therefore, this travel time fluctuation effect was explicitly removed in the Markov approximation [e.g., Saito et al., 2002]. Because of this process, the combined term of ε and a appears in our method. Note that the combined term ε2/(p(κ) − 1)a− 1 is related to amplitudes of PSDF at large wave numbers (m ≫ 1/a). For example, if κ is spatially uniform, large ε2/(p(κ) − 1)a− 1 region has large power spectral density at m ≫ 1/a.
 To take into account the 3-D spatial variation of random inhomogeneities, we partitioned the study area into many blocks. Random inhomogeneities along a seismic raypath (i.e., an unperturbed raypath) can be represented by the piecewise constant variation of P(m) from source to receiver. The peak delay time along such a raypath can be evaluated by using a recursive formula proposed by Takahashi et al. . This method uses equations (2)–(4) recursively from source to receiver. The basic idea of the recursive application is the successive replacement of nonuniform random media with a uniform medium. At every boundary between different random inhomogeneities, this method replaces the medium from the source to the boundary with a uniform random medium with the same PSDF as that across the boundary, and it introduces an equivalent travel distance to ensure that after the replacement tp remains the same at the boundary. Recursive application of this replacement procedure allows us to evaluate tp at a station by using κ and ε2/(p(κ) − 1)a− 1 values along the raypath. Takahashi et al. [2008, 2011] provide a detailed summary of this method.
 By using this recursive formula and numerical simulation, Takahashi et al.  showed that the frequency dependence of tp is controlled only by κ. If κ is 1.0 from the source to the receiver, then tp is the same among all frequency bands. On the other hand, if a region of smaller κ exists along the raypath, then tp becomes larger at higher frequencies. Parameter ε2/(p(κ) − 1)a− 1 does not affect the frequency dependence of tp. If we observe large tp at all frequency bands without clear frequency dependence, we can expect large ε2/(p(κ) − 1)a− 1 and large κ regions on their raypath. Accordingly, we need to analyze tp at multiple frequency bands to estimate both κ and ε2/(p(κ) − 1)a− 1. The results of synthetic inversion tests of tp have further suggested that an explicit constraint on the frequency dependence is necessary for stable estimation of both parameters [Takahashi et al., 2009]. Details of the inversion analysis are described in section 5.
 In this study, we analyzed velocity seismograms recorded by ocean-bottom seismographs (OBSs) and onshore seismic stations in southwestern Japan. Figure 2 shows the locations of the seismic stations (184 OBSs and 209 onshore stations), earthquake epicenters, and raypaths used in the analysis. OBSs were deployed in two different periods by the Japan Agency for Marine-Earth Science and Technology. From December 2008 to January 2009, a seismic survey in the Hyuga-nada region was conducted with 160 OBSs deployed along four lines (HY01–04) designed for seismic refraction surveys [Nakanishi et al., 2010]. Along each survey line, the OBSs were deployed at 5 km intervals. After 2 months of observation, 157 OBSs were retrieved successfully. All OBSs in this survey were equipped with a three-component, 4.5 Hz, short-period seismometer, and continuously recorded seismograms at a sampling rate of 200 Hz. Another OBS observation study was conducted in 2004 off Shikoku [Obana et al., 2006] with 30 short-period OBSs, each with a three-component sensor, which continuously recorded seismograms for 3 months at a sampling rate of 100 Hz. These OBSs were deployed approximately 20 km apart on average. After the observation period, 27 OBSs were recovered. Onshore stations belong to the Hi-net and F-net seismic networks, which are maintained by the National Research Institute for Earth Science and Disaster Prevention of Japan [Okada et al., 2004; Obara et al., 2005]. Hi-net stations have three-component 1 Hz seismometers, and F-net stations are equipped with broadband seismometers.
 We used data from earthquakes that occurred in and around the subducting Philippine Sea plate. We selected small and moderate-sized earthquakes with magnitudes of less than 5.0 to ensure sufficiently short rupture duration, because our method assumes an impulsive wave radiation at source. The focal depth range of these events was 35–200 km. We excluded shallow earthquakes because our forward modeling cannot take into account guided waves in the crust. Yamamoto et al.  determined the hypocenters of the earthquakes observed by the 157 OBSs in the Hyuga-nada region. They analyzed travel times of both onshore seismic stations and OBSs by taking account of the 3-D velocity structures at shallow depth beneath Hyuga-nada [Nakanishi et al., 2010] and using a 1-D velocity model for the deeper parts. Hypocenters of other earthquakes are from the hypocenter catalogue of the Japan Meteorological Agency (JMA). We evaluated the seismic raypaths by using the 3-D velocity structure of Matsubara et al. . Even though our forward modeling is based on a constant background velocity, we used the 3-D velocity structure in order to assign random inhomogeneities to appropriate locations in the crust and uppermost mantle.
 Hypocentral distance range is 100–300 km to secure the tp as an adequate measurement of accumulated scattering effect. The lower limit satisfies the condition that travel distance should be much longer than both the wavelength of incident waves and the characteristic scale of medium inhomogeneities. The upper limit was chosen so that the travel distance dependence of envelope broadening would have different characteristics at larger travel distances [Sato, 1989]. One of the possible origins of such different characteristic of tp is intrinsic attenuation. If the accumulated attenuation effect becomes significant, we cannot interpret tp as an accumulated scattering effect.
 We measured peak delay times of S wave from the RMS envelopes of horizontal components in the 4–8, 8–16, and 16–32 Hz frequency bands. Recent studies on the Markov approximation of vector waves in Gaussian random media [e.g., Sato, 2007] showed that radial component possibly has larger peak delay than transverse components for S wave from a point source. Even though the vector wave envelopes in von Kármán-type random media has not been fully clarified yet, we did not use the vertical component to avoid this effect. We calculated the moving average in a window with 10 times the width of the center period. We measured tp in seconds in a 30 s time window starting from the S wave onset. We sometimes shortened this time window manually to avoid noise or air-gun signals of active seismic surveys. At higher frequencies, we sometimes observed no clear S wave arrival because of strong scattering or attenuation. We excluded those data by visual inspection. Thus, the following analysis uses 35,173 data at 4–8 Hz, 33,958 at 8–16 Hz, and 22,127 at 16–32 Hz.
4 Characteristics of Observed tp
 Figure 3 shows an example of RMS envelopes at 4–8 Hz and 8–16 Hz observed by OBSs in the Hyuga-nada area. The focal depth of the earthquake (M 2.3) was 157 km according to the JMA earthquake catalogue. We can see a clear variation of the peak delay times along survey line HY01. Stations at the northwestern end of HY01 tend to show a clear S wave onset with relatively small tp (~1–2 s), whereas OBSs at the southeastern end show broadened wave trains and large tp (~5–10 s). Since the hypocentral distances are almost the same for all stations, this variation is possibly due to spatial variations of random inhomogeneities. The peak delay times at 8–16 Hz are similar to those at 4–8 Hz at all stations along the HY01 survey line. The weak frequency dependence of the envelopes suggests that κ in this area is close to 0.8–1.0 and that large ε2/(p(κ) − 1)a− 1 regions may exist on the raypaths of stations at the southeastern end of HY01. Figure 3 shows relatively clear P waves at both frequency bands. However, P waves of other earthquakes frequently show unclear peak arrivals because of bad S/N ratio or relatively large amplitudes of wave train between P and S waves. Consequently, we analyzed only S wave for a stable estimation of spatial distribution of random inhomogeneities.
 The frequency dependence of peak delay times, which reflects the distribution of κ, is measured by fitting a regression line log(tp[f Hz]/tp[fref Hz]) = Afreq + Bfreq log f to the data for each station [e.g., Obara and Sato, 1995]. The reference frequency fref is set to the central frequency band, 8–16 Hz. Normalization by tp at fref suppresses the travel distance effect of tp. Large Bfreq values suggest the existence of small-κ regions on some raypaths observed at a station. This study evaluated Bfreq at every seismic station by dividing the whole data set into three subsets according to focal depth. The focal depth dependence of Bfreq reflects the depth dependence of the κ distribution, and possibly improves the vertical resolution in the following inversion analysis.
 We examined the distributions of Bfreq (Figure 4a) and its standard deviation (Figure 4b) in three focal depth ranges: 35–50, 50–80, and 80–200 km. High Bfreq values were observed in volcanic areas of Kyushu and on its northwestern side. This correspondence with the volcano distribution is broadly comparable to that seen in northeastern Japan [Takahashi et al., 2009]. Bfreq was also large in the Chugoku area, even though there are few Quaternary volcanoes. At most OBSs, Bfreq tended to be ~0.0 except at stations with large , which suggests that most of the area from offshore Hyuga-nada to off-Shikoku may have large κ.
 The minimal value distribution of the peak delay times [Takahashi et al., 2007] gives some insight into the spatial variation of random inhomogeneities without requiring any assumptions about PSDF models of random inhomogeneities. This method is based on the simple idea that small tp means the absence of any strong inhomogeneities along the raypath. Following previous studies [e.g., Takahashi et al., 2007, 2011], we measured the quantity Δ log tp[f], which is defined as the deviation from the regression line: Δ log tp[f] = log tpobs[f] − (Adis[f] + Bdis[f]log r). This measured quantity reflects the accumulated scattering effect relative to the averaged value in the study area with consideration of the travel distance dependence of tp. Regression lines were estimated as log tp = 0.39 log r − 0.65 at 4–8Hz, log tp = 0.48 log r − 0.86 at 8–16Hz, and log tp = 0.45 log r − 0.73 at 16–32Hz. Bdis are much smaller than northeastern Japan (1.54 ~ 1.60) and northern Izu-Bonin arc (0.90 ~ 1.11) regardless of frequency bands. It implies that most of the study area has weak inhomogeneities and/or large κ. We partitioned the study area into blocks, each with a horizontal spatial extent of 0.10° × 0.10° and a depth of 20 km. In each block, we measured the minimal value of Δ log tp[f] among all raypaths that propagated in the block.
 We examined the minimal value distribution of Δ log tp[f] in the 4–8 Hz (Figure 5a) and 8–16 Hz (Figure 5b). In the figure, darker colors indicate larger Δ log tp[f], and to emphasize the lateral variation of Δ log tp[f], black is used to indicate Δ log tp[f] = 0.0. Even though data scatter can severely affect this mapping, we can see some remarkable characteristics in both frequency bands. At 4–8 Hz, large minimal values of Δ log tp[f] are imaged beneath Quaternary volcanoes and in the Hyuga-nada region. Regions with strong scattering beneath the Quaternary volcanoes in Kyushu tend to be clearer in the shallower depth range. Beneath Hyuga-nada, a region of strong scattering is imaged in the southwestern part of the OBS survey area covered by survey lines HY01–04. Another large tp region can be found beneath the northwestern part of the Shikoku region at 20–40 km depth. The minimal value distributions at 8–16 Hz are broadly similar to those at 4–8 Hz.
5 Inversion Procedure
 Inversion analysis of the peak delay time minimizes the residuals of tp in the three frequency bands while simultaneously placing a constraint on Bfreq. In outline, the inversion procedure is almost the same as that described by Takahashi et al. [2009, 2011] except for the parameter sampling algorithm. Here we summarize the objective functions and the parameter sampling algorithm.
 The inversion analysis has two steps. The aim of the first step is to obtain a rough estimation of the κ and ε2/(p(κ) − 1)a− 1 distributions with an explicit constraint on Bfreq. The objective function of this first step is defined as
 The first term is the residual of tp at reference frequency band fref. Variance of tp is assumed to be 1.0 s2 to make the objective function non-dimensional. The second term is the constraint on Bfreq, defined as
where j and D indicate the station and the focal depth range, respectively. Ns(D) is the number of Bfreq values observed at the Dth focal depth range. Lκ and Lε in the third term of equation (5) represent the Laplacian smoothing constraints for unknown parameters κ and ε2/(p(κ) − 1)a− 1, respectively. Coefficients , wκ, and wε are weighting factors for , Lκ, and Lε, respectively.
 The second step refines the distributions of κ and ε2/(p(κ) − 1)a− 1 by minimizing the residuals of tp in all frequency bands while keeping the residual value of Bfreq small. The objective function of this step is
 The first term is the sum of the squared residuals of tp in all frequency bands. This study introduces normalization of this residual by the number of raypaths because the numbers of available data differed significantly among the frequency bands. The second term is the smoothing constraint and is the same as the second term of the first step. The third term represents a constraint on the first-step results; w1stStep is the weighting factor of this term.
 Previously, Takahashi et al. [2009, 2011] used a genetic algorithm (GA) [Holland, 1975] to find the optimal solution. Some trial analyses with a GA in this study showed relatively stable convergence, but the reproducibility of the result was not sufficient because of the existence of local minima. Therefore, in this study, we used the exchange Monte Carlo (EMC) method [e.g., Hukushima and Nemoto, 1996], because this approach is applicable to the solution of inverse problems that have many local minima.
 The EMC is an algorithm that overcomes a shortcoming of the simulated annealing (SA) [Kirkpatrick et al. 1983]. SA assumes the posterior probability k exp(−s/T), where k is a normalization constant, s is an objective function, and T is temperature. Parameter sampling is conducted by the Metropolis and Hastings algorithm [Metropolis et al., 1953; Hastings, 1970] from an arbitrary point in the model space where the temperature is sufficiently high. The higher temperature allows global sampling over the whole model parameter space. As the parameter sampling proceeds, the temperature is gradually decreased for dense sampling near the global minimum. With this method, however, local minima may cause the parameter sampling to be halted before the global minimum is reached even if the temperature decrease is very gradual.
 The EMC algorithm conducts random walks at different temperatures in parallel without any changes in temperature. It thus ensures parameter sampling over the whole parameter space at high temperatures in parallel with dense sampling near the global minimum at low temperatures. If a sampling point at high temperature has a larger posterior probability than one at lower temperature, these sampling points will be exchanged following the same rule as with the Metropolis and Hasting algorithm. In other words, this method samples the parameter space under the posterior probability density , where k′ is a normalization constant and n is the number of temperatures. This approach has been successfully used for the inversion of maximal amplitudes to estimate the attenuation structure of S waves in northeastern Japan [Takahashi, 2012]. After some trial analyses, we used 32 temperatures as follows:
 These values were chosen to ensure both a sufficient number of exchanges between different temperatures and wide-area sampling in the parameter space at higher temperatures.
 We set the parameter sampling ranges to 0.1 ≤ κ ≤ 0.9 and − 6.5 ≤ log(ε2/(p(κ) − 1)a− 1) ≤ − 0.5. Note that the upper limit of κ is not 1.0 but 0.9. This is because p(κ), which controls the frequency dependence of tp, takes the same values for κ = 0.9 and κ = 1.0: p(0.9) = p(1.0) = 1.99. We generated the initial models for the inversion analysis by the following procedure. First, we performed a grid search for κ and ε2/(p(κ) − 1)a− 1 with the objective function of the first step (equation (5)) by assuming a spatially uniform structure. This grid search was conducted at 64 points evenly distributed in the κ − log(ε2/(p(κ) − 1)a− 1) space to find the best fit uniform structure. Then, we added random fluctuations to the best fit uniform model to make 32 different initial models. These models were used as the starting points of parameter sampling at different temperatures of the EMC algorithm. The block size in the inversion analysis was set to 0.25° × 0.25° horizontally and 20 km vertically. This block size is sufficiently large to define PSDF, in other words, lengths of horizontal and vertical directions are larger than characteristic scale a. The Markov approximation assumes that a is larger than the wavelength of incident waves. Consequently, block size should be much larger than wavelength of S wave at 4 Hz. The number of iterations of the EMC algorithm was 160,000 for the first step and 30,000 for the second step. The inversion result is the expectations of the model parameters evaluated at the end of the parameter sampling near the global minimum. We used 5000 iterations to evaluate these expectations.
 The initial grid search was conducted by using different values from 0.1 to 4.0. The initial uniform structure was estimated as κ = 0.65 and ε2/(p(κ) − 1)a− 1 = 10.0−3.6875 km−1 regardless of . The RMS residual of tp for the initial uniform structure was 2.15 s.
 Weighting factors wκ, wε, and were selected by performing trial analyses using various combinations of factors. We then plotted Lκ, Lε, and against their respective weighting factors (Figures 6a–6c). Here black and gray symbols are used to clarify mutual influences among constraints. For example, the black dots in Figure 6a mean that both Lε and have sufficiently converged to small values and gray dots mean that either Lε or is not fully converged. We then chose the weighting factors, wκ = 15.0, wε = 7.5, and = 1.0, as the smallest values that could achieve sufficient convergence of all constraints at the end of the first step.
 Weight w1stStep is selected on the basis of the residual changes of tp and Bfreq during the second step of the analysis. Figure 6d shows the changes in (δF) and the RMS residual of tp (δtp) during the second step for wκ = 15.0, wε = 7.5, and = 1.0, where positive values mean increases in the residuals. The weighting factor w1stStep should be chosen to ensure sufficient convergence of tp and the smoothness of unknown parameters while keeping small; that is, |δtp| should be as large as possible while keeping δF small. By considering the changes in δF and δtp in Figure 6d and the reproducibility of inversion result, we set w1stStep = 50.0. We then used these four weighting factors to obtain the final inversion results. The RMS residual of tp after the first step was 1.96 s, and it became 1.91 s at the end of the inversion analysis.
 We plot the spatial distributions of κ (Figure 7a) and ε2/(p(κ) − 1)a− 1 (Figure 7b), and the distributions of P(m = 15 km−1) evaluated from κ and ε2/(p(κ) − 1)a− 1 by assuming a = 5 km (Figure 7c), at 0–20, 20–40, and 40–60 km depth. The wave number m = 15 km−1 approximately corresponds to the central wave number of the analyzed frequency bands. P(m = 15 km−1) can thus be expected to represent faithfully a scattering strength for frequencies higher than 10Hz, since the diffraction and multiple forward scattering of seismic waves are mainly caused by the inhomogeneities of which spatial scale is longer than the wavelength of the incident wave. Note that the value assumed for a does not significantly affect the estimation of P(m) at m > > 1/a [e.g., Takahashi et al., 2008].
 Strongly inhomogeneous media with large P(m = 15 km−1) (>10−7 km3) and small κ (=0.3–0.5) were found in the Kyushu and Chugoku regions. In these regions, the PSDF was estimated as P(m) ≈ 0.05 m−3.7 km3. In the Kyushu region, strong inhomogeneities extend down to 40–60 km depth except in the Beppu-Shimabara rift zone. Strong inhomogeneities in the Chugoku region are widely distributed at 0–20 km depth and show low correlation with the Quaternary volcanoes distribution. This strongly inhomogeneous region becomes small with depth increasing. At 40–60 km depth, strong inhomogeneities are located only near the Quaternary volcanoes.
 Most of the other areas show weak inhomogeneity with small P(m = 15 km−1) and large κ (>0.5). The PSDF is approximately P(m) ≈ 0.005 m−4.5 km3. In Nankai trough, there is a slightly large P(m = 15 km−1) area in the southwestern part of Hyuga-nada at 0–20 km depth. PSDF in this area is estimated as P(m) ≈ 0.01 m−4.5 km3. This PSDF has the similar spectral gradient with the weakly inhomogeneous areas, but has larger power spectral density due to large ε2/(p(κ) − 1)a− 1 (~ 6.0 × 10− 4 km−1).
 Relatively large ε2/(p(κ) − 1)a− 1 values are also distributed at 20–40 and 40–60 km depth in western Shikoku and along the Pacific coastline of Kyushu. At 40–60 km depth, slightly large P(m = 15 km−1) regions were imaged in the western Shikoku region. This region is near the source areas of non-volcanic tremors [e.g., Obara, 2002, 2010] and slow-slip events [e.g., Hirose and Obara, 2010]. Our study area does not fully cover the tremor area along the subducted Philippine Sea plate, but this result suggests that determination of the distribution of random inhomogeneities may be important to understand relations between medium properties and the occurrence of non-volcanic tremor and slow slip events.
 Figure 8 summarizes the PSDF of this study and those in northeastern Japan and the northern Izu-Bonin arc [Takahashi et al., 2009, 2011]. The strong random inhomogeneity with large P(m = 15 km−1) (>10−7 km3) and small κ (=0.3 ~ 0.5) in the Kyushu and Chugoku regions (SW(1) in Figure 8) shows similar PSDF with the random inhomogeneities beneath Quaternary volcanoes in northeastern Japan and the northern Izu-Bonin arc. PSDF in slightly large P(m = 15 km−1) region in Hyuga-nada (SW(2)) is close to that in the fore-arc side (20–60 km depth) of the northeastern Japan, but κ is larger than northeastern Japan. Weak inhomogeneities in the other area of this study (SW(3)) are close to those in the fore-arc side in the northern Izu-Bonin arc (30–70 km depth).
7.1 Random Inhomogeneities in Onshore Area
 Strong inhomogeneities with small κ at 0–20 km depth were found in the Chugoku and Kyushu regions. In contrast, in the Shikoku region, inhomogeneities were notably weak and κ was large at the same depth range. Takahashi et al. [2009, 2011] showed that northeastern Japan and the northern Izu-Bonin arc are characterized by strong random inhomogeneities in the shallowest depth range (0–20 km depth in northeastern Japan and 0–30 km depth in the northern Izu-Bonin arc). However, these strong inhomogeneities at shallow depth have not yet been fully discussed, because their spatial correlations with other parameters are not clear. In this study, we were able to find a clear difference between the Chugoku and Shikoku regions and a good correlation between strong inhomogeneity at shallow depth and the distribution of volcanic rocks on the geological map of Japan [Geological Survey of Japan, AIST, 2011]. The Chugoku region is composed mainly of various volcanic rocks, whereas the Shikoku region is composed mainly of an accretionary complex, except for the northern edges of the island. Although the origin of the strong random inhomogeneities is still a matter for debate, we suppose that possible origins of the S wave inhomogeneities include deformation of layered media, random distributions of cracks and intrusions of various different rock types, and the presences of geofluids such as aqueous fluids or magma. Our result suggests that the presence of different volcanic rock types is one explanation for the presence of random inhomogeneities with small κ in the crust. The distribution of the minimal value of Δlogtp [f] (Figure 5) shows some fluctuation at 0–20 km depth, unlike P(m = 15 km−1) of random inhomogeneities. This difference suggests that strong inhomogeneities in the Chugoku region tend to be located in thin layers or in small regions in the 0–20 km depth range.
Carcole and Sato  estimated the lateral variation of scattering Q−1 [e.g., Sato et al., 2012, p. 66] in Japan by analyzing coda waves. Even though their method does not resolve the vertical variation of scattering Q−1, we found some clear correlations in this study. Their scattering Q−1 shows strong scattering in the Chugoku region and beneath the Quaternary volcanoes in the Kyushu region. These regions mostly correspond to the large P(m = 15 km−1) area at 0–20 km depth in this study. According to Carcole and Sato , the scattering Q−1 in the Shikoku region is the weakest in Japan. P(m = 15 km−1) is smaller in the Shikoku region than in all regions of northeastern Japan [Takahashi et al., 2009]. Note that coda waves are mainly affected by backward scattering rather than by forward scattering. That means that coda wave analyses have higher sensitivity to small-scale inhomogeneities. Therefore, the higher correlation of their scattering Q−1 with P(m = 15 km−1) than with ε2/(p(κ) − 1)a− 1 appears to be reasonable. We conclude that the results obtained by direct wave analysis and by coda wave analysis in this region are broadly consistent.
Sawazaki et al.  also estimated random inhomogeneities in the Chugoku and Shikoku regions, using an approach based on the Markov approximation. They assumed spatially uniform random inhomogeneities and a = 5 km, and estimated κ = 0.9 and ε = 0.08 by analyzing two earthquakes, both with a focal depth of about 40 km. Their estimation can be written as ε2/(p(κ) − 1)a− 1 ≈ 1.2 × 10− 3 km−1 and P(m = 15 km−1) ≈ 4.0 × 10− 8 km3. This P(m = 15 km−1) estimation is almost the same as our estimation at 20–60 km depth beneath the Chugoku region. However, their values of κ and ε2/(p(κ) − 1)a− 1 are different from our result which shows κ = 0.7–0.9 and ε2/(p(κ) − 1)a− 1 ≈ 3.0 × 10− 4 km−1. They found a strong trade-off between ε and κ, and their estimation ε2/(p(κ) − 1)a− 1 ≈ 1.2 × 10− 3 km−1 is larger than any regions of this study area. To resolve the trade-off between κ and ε2/(p(κ) − 1)a− 1, in this study we introduced an explicit constraint on the frequency dependence of tp in the first step [Takahashi et al., 2009]. Therefore, this study and that of Sawazaki et al.  agree in terms of the scattering strength at higher frequencies, but our approach has the advantage that it resolves the trade-off between two parameters of a von Kármán-type PSDF of random inhomogeneities.
 Strong random inhomogeneities near the Beppu-Shimabara rift zone are localized at shallow depth (0–40 km depth), whereas those in the 40–60 km depth range are mostly comparable in strength to the inhomogeneities in the surrounding regions. In contrast, small κ regions beneath the Quaternary volcanoes in northeastern Japan and in the northern Izu-Bonin arc occur to depths of 40–60 km or 50–70 km, respectively. According to the travel time tomography beneath the Kyushu area [Matsubara et al., 2008; Xia et al., 2008], the low-velocity region beneath the Beppu-Shimabara rift zone is distributed from the lower crust to a depth of about 100 km. Therefore, the low-velocity anomaly in this region is not highly correlated with the strong random inhomogeneities. In the northern Izu-Bonin arc, the correlation between random inhomogeneities and velocity structure is also low beneath Quaternary volcanoes. Two low Vs anomalies have been imaged beneath the volcanic front [Obana et al., 2010], whereas three small κ regions are distributed within and between these low Vs regions [Takahashi et al., 2011]. These observations suggest that random inhomogeneities and the velocity structure reflect different properties of the medium. If we suppose that the low-velocity anomaly beneath the rift zone is partly related to magma intrusion, our result implies that the magma inclusions in the deeper part are too small to be detected by seismic waves at 4–32 Hz, or that velocity fluctuations at spatial scales of a few hundreds of meters to a few kilometers are not strong enough to excite multiple forward scattering. In the future, we need to clarify the attenuation structure after removing scattering attenuation [Takahashi, 2012] to elucidate the properties of the medium in terms of the distribution of magma and other fluids beneath volcanoes.
7.2 Random Inhomogeneities in Nankai Trough
 The Hyuga-nada region shows a clear lateral change of ε2/(p(κ) − 1)a− 1 and P(m = 15 km−1) between its northeastern and southwestern parts. The velocity structure shows similar lateral variations. Detailed Vp structures along survey lines HY01–04 have been estimated by seismic refraction survey [Nakanishi et al., 2010]. Along HY03, the crust is thick at the southwestern end of HY03, but it becomes thin between HY01 and HY02. This relatively thick crust has been interpreted as the subducted Kyushu Palau ridge by considering the bathymetry and reflector distribution along the survey line. Yamamoto et al.  estimated the spatial extent of the subducted Kyushu Palau ridge from the 3-D velocity structure. The slightly large P(m = 15 km−1) region with large ε2/(p(κ) − 1)a− 1 is almost located in the subducted Kyushu Palau ridge of their study. On the fore-arc side of the volcanic front in the northern Izu-Bonin arc at 0–30 km depth [Takahashi et al., 2011], there is a small area of which random inhomogeneities (κ ~ 0.9 and ε2/(p(κ) − 1)a− 1 ~ 3.0 × 10− 3 km−1) is qualitatively similar to the slightly large P(m = 15 km−1) region. Even though the spatial resolution of the data in the northern Izu-Bonin arc was lower than that in this study, Takahashi et al.  pointed out the existence of a frontal arc high, a remnant of an ancient arc, in the large ε2/(p(κ) − 1)a− 1 and large κ region of the fore arc. Because the Kyushu Palau ridge was also produced by ancient volcanic activity on the oceanic plate [e.g., Okino et al., 1994], this study and that by Takahashi et al.  seem to agree with regard to the relation between random inhomogeneities and ancient volcanic activity. From another viewpoint, the large ε2/(p(κ) − 1)a− 1 and large κ may have been generated by crustal deformation due to the subduction of Kyushu Palau ridge. Crustal deformation due to compression may create a small a structure. Thus, in this case, ε2/(p(κ) − 1)a− 1 might be large because a is small. Note that the former idea, the association with ancient volcanic activity, cannot account for the difference between the subducted Kyushu Palau ridge and the small κ area in the Chugoku region. In the future, we will need to clarify differences in medium properties by examining various parameters, including seismic velocity, random inhomogeneities, attenuation, and geological characteristics, to elucidate the detailed generation processes of random inhomogeneities.
Yagi et al. [1998, 2001] estimated the postseismic and coseismic slip zones of large earthquakes in the Hyuga-nada region by seismic waveform and GPS data inversions. Furumura et al.  evaluated the source rupture area by forward modeling of the height of the tsunami generated by the 1707 Hoei earthquake. Their fault models do not overlap with the subducted Kyushu Palau ridge, as Nakanishi et al.  and Yamamoto et al.  have already pointed out. Our study has clarified that the medium in their fault zones does not show clear anomalies with regard to inhomogeneities. The high-seismicity region in northeastern Japan shows significantly strong random inhomogeneities, with the PSDF characterized by large ε2/(p(κ) − 1)a− 1 and large P(m = 15 km−1) [Takahashi et al., 2009]. Even though this high-seismicity region is not perfectly correlated with the observed microseismicity, we suppose that tp at 4–32 Hz has some sensitivity to cracks and fractured structures due to current active seismicity. Thus, the weak inhomogeneities in the fault zones in the Hyuga-nada region imply that the fractured structures generated by the aforementioned large earthquakes and their aftershocks have already healed, and they therefore do not act as S wave scatterers in this frequency band. Coda Q−1 sometimes shows temporal changes associated with huge earthquakes, and is recovered to their original state before earthquakes [e.g., Sugaya et al., 2009]. We propose that studies on the spatial distribution of random inhomogeneities and inelasticity may be important for monitoring the medium recovery state after earthquakes.
 By an inversion analysis of peak delay times of high-frequency S waves, we revealed the 3-D distribution of the PSDF of random velocity inhomogeneities in southwestern Japan and on the western side of the Nankai trough. Strong inhomogeneities with small κ (0.3–0.5) are widely distributed in the Kyushu and Chugoku regions at 0–20 km depth. We estimated the PSDF of these regions as P(m) ≈ 0.05 m−3.7 km3. This strong inhomogeneity shows a good correlation with the distribution of Quaternary volcanoes and volcanic rocks and suggests that the random distribution of volcanic rocks of various types is one of the origins of small κ and strong random inhomogeneities in the crust. Beneath the Beppu-Shimabara rift zone, strong inhomogeneity with small κ appears only at 0–40 km depth, in contrast to the volcanic areas in northeastern Japan and the northern Izu-Bonin arc. This difference may reflect differences in the magma generation process or the geofluid distribution between the rift zone and the arc regions. Weakly inhomogeneous media in other areas are characterized by large κ (>0.5). In these areas the PSDF is P(m) ≈ 0.005 m−4.5 km3. In the Hyuga-nada region, where the media are weakly inhomogeneous, there is a clear variation of ε2/(p(κ) − 1)a− 1 at 0–20 km depth. Large ε2/(p(κ) − 1)a− 1 (~6.0 × 10− 4 km−1) region has slightly large P(m = 15 km−1) and broadly corresponds to the subducted Kyushu Palau ridge. This slightly large P(m = 15 km−1) region may have been generated by ancient volcanic activity in the oceanic crust or by crustal deformation due to subduction of Kyushu Palau ridge.
 We appreciate two anonymous reviewers for their constructive comments. The OBS observation at Hyuga-nada was conducted by the Japan Agency for Marine-Earth Science and Technology as a part of “Research concerning Interaction between the Tokai, Tonankai and Nankai Earthquakes” funded by the Ministry of Education, Culture, Sports, Science and Technology, Japan. The authors are grateful to Makoto Matsubara for providing their 3-D velocity model [Matsubara et al., 2008]. We thank everyone at the National Research Institute for Earth Science and Disaster Prevention involved in the development and ongoing maintenance of Hi-net and F-net. Generic Mapping Tools software [Wessel and Smith, 1998] was used to produce the figures presented here, and Seismic Analysis Code software [Goldstein et al., 2003] was used for signal processing.