# Contiguous rupture areas of two Nankai Trough earthquakes revealed by high-resolution tsunami waveform inversion

## Abstract

[1] We have developed a new method for inverting tsunami waveforms that reveals considerable detail in megathrust slip during subduction zone earthquakes. Previous methods have ensured compliance with the shallow-water theory used to compute tsunami waveforms by using large subfaults that generate only long-wavelength seafloor deformation. We show that a better approach is to use small subfaults coupled with a smoothing criterion that ensures compliance with the shallow-water approximation. In an application of the method to historical earthquakes in the Nankai Trough, southwestern Japan, we find that the areas with slip > 1 m for the earthquakes of 1944 and 1946, which ruptured adjacent segments of the subduction zone, are separated by a sharp, non-overlapping boundary. This establishes that interseismic accumulation of strain energy extends very close to the boundary between rupture zones, and strongly suggests that this boundary is associated with a physical barrier to rupture.

## 1. Introduction

[2] A tsunami is an oceanic gravity wave generated by a vertical displacement of the sea floor, most often caused by a submarine earthquake or landslide. Most large tsunami are caused by large, shallow earthquakes along submarine subduction zones. The dimensions of faulting caused by magnitude-8 class earthquakes is typically of 100 km or more, so the wavelength of tsunami generated by such earthquakes is generally much greater than the water depth, and shallow-water (long-wave) theory is appropriate for calculating tsunami propagation.

*Satake*[1989] proposed a formal inversion method for tsunami waveform data which has been applied to many earthquakes in the Alaska-Aleutians [e.g.,

*Johnson*, 1998], northern Japan [e.g.,

*Satake*, 1989;

*Hirata et al.*, 2003;

*Tanioka et al.*, 2004] and the Nankai trough [

*Satake*, 1993;

*Tanioka and Satake*, 2001;

*Baba et al.*, 2002]. In the method, the fault plane is first divided into several subfaults and deformation on the ocean bottom computed for each subfault with a unit amount of slip. Using this as an initial condition, tsunami waveforms are numerically computed based on the shallow-water theory. The observed waveforms are expressed as a superposition of the computed waveforms as follows,

**d**is a vector containing the observed tsunami waveforms,

**A**is a matrix whose columns consist of computed waveforms for each subfault, and

**x**is a vector whose components are the amount of slip on each subfault. The slip on each subfault can be estimated by a least-squares inversion of the above equation [see

*Satake*, 1989].

[4] The shallow-water theory limits the type of slip distribution which can be used in the tsunami calculation to those which generate no short-wavelength changes in seafloor topography. Previous studies attempted to accommodate this constraint by using large subfaults (e.g., 45 × 45 km subfaults used by *Tanioka and Satake* [2001] and *Baba et al.* [2002]), since large subfaults will generate long-wavelength seafloor disturbances. This approach has at least two disadvantages: (1) short-wavelength tsunami are still generated where adjacent shallow subfaults have differing amounts of slip; (2) resolution of details in the slip pattern is not optimum – e.g., peaks and troughs in the slip pattern are spread out over the large subfaults, even when the data and theory might allow for finer resolution.

[5] We attempt to address these problems by using very small subfaults which allow fine detail to be resolved in the slip distribution, but we include in the inversion a smoothing criteria which ensures that the combinations of small subfaults always constitute a smooth slip distribution. Although the tsunami calculation for each subfault may not be consistent with shallow-water theory, linearity of the shallow-water wave equations allows these (potentially inaccurate) solutions to be surperposed to produce a solution that is consistent with shallow-water theory.

## 2. Tsunami Waveforms Inversion With Finer Subfaults

[6] An overly large number of model parameters (number of subfaults) will give rise to instability in the inversion. The solutions obtained by using many small subfaults without a smoothness constraint exhibit unrealistically large and rapid fluctuations in slip, and spectra of the vertical seafloor deformation have peaks at wavelengths shorter than 80 km (dashed curves in Figure 1). The conventional limit of the tsunami wavelength on the assumption of the shallow-water theory is about 80 km at the Nankai trough since the deepest seafloor depth is about 4000 m. Thus, the vertical deformation of the seafloor calculated from slip distributions obtained by inversion without a smoothing constraint is not consistent with shallow-water theory.

*x*

_{ij}is the slip on the

*ij*th subfault,

*i*is the subfault index along strike and

*j*is the index down dip. This is written in vector form as

**S**is a matrix which represents the smoothness constrain. This equation is introduced into the observation equation (1), and we get

## 3. Application to the Nankai Earthquakes

[8] We apply the new inversion method described above to tsunami records of the 1944 Tonankai and 1946 Nankai earthquakes. These events were M8-class interplate earthquakes which ruptured adjacent segments of the Nankai trough. Their slip distributions have already been estimated from tsunami waveforms inversion using large subfaults (45 × 45 km) without a smoothness criterion [*Tanioka and Satake*, 2001; *Baba et al.*, 2002]. Here, we divide a plate boundary that was estimated by seismic survey results [*Baba et al.*, 2002] into smaller subfaults distributed over the slip areas of the respective events. The dimension of the subfault is set to be 10 × 10 km because of the data sampling interval of 1 minute in this study. That is, tsunami propagates about 10 km per 1 minute at the water depths of 2000 m, where the major slips of the earthquakes occurred under the seafloor. The depth and dip of each subfault is defined based on the plate model. Other fault parameters (strike and rake), as wells as the Green's function calculations and observed data are the same as those of *Baba et al.* [2002] for the 1944 Tonankai earthquake and *Tanioka and Satake* [2001] for the 1946 Nankai earthquake, respectively. The non-negative least squares method [*Lawson and Hanson*, 1974] is used to solve equation (4) for the set of positive subfault slips, which is also identical to those of the previous studies. Thus, the only difference between the present study and the previous ones is the use of smaller subfaults and a smoothness criterion. We repeatedly invert the tsunami waveforms using equation (4), by varying the weight between the smoothness constrain and the observed data, and the calculated wavelength of the seafloor vertical deformation is checked until the smallest α leading to a solution consistent with shallow-water theory is found (Figure 1). The smallest weights resulting in vertical seafloor deformation consistent with the shallow-water theory are 2.7 and 4.0 for the 1944 Tonankai and 1946 Nankai earthquakes, respectively. The smallest value of α is generally dependent on the length of data vector (d), and it tends to become larger when the length of data vector is large.

[9] The obtained slip distributions of the 1944 Tonankai and 1946 Nankai earthquakes are shown in Figure 2. The basic slip patterns are not changed in comparison with previous estimations, since the observed tsunami waveforms and time windows for the inversions are exactly the same as those of the previous studies. However, the slip distributions become smoother, and the boundary between the rupture and non-rupture areas are more clearly imaged. An important improvement of the new method is the imaging of the rupture boundary. In the previous studies, the location of the slip boundary has been dependent largely on the location of the large subfaults that are set arbitrarily in order to cover the slip area. In the new method the slip boundary is determined by the fit to the data.

[10] The agreement between the observed and computed tsunami waveforms is much better than that of the previous study (Figure 3). For the 1944 Tonankai event, the RMS residual is 10.5 cm, and the variance reduction is 76 % which is larger than the 68 % of the previous study. For the 1946 Nankai earthquake, the RMS residual is 11.7 cm and the variance reduction is 85 %.

## 4. Discussion

[11] As shown in Figure 2, regions of significant slip (over 1 m) for both events do not overlap, and yet are very close to each other. To our knowledge this is the first time rupture areas of adjacent subduction zone earthquakes have been imaged at such high resolution. The rupture of the 1944 event stopped off Cape Shiono, and then virtually all of the area to the west which failed to rupture during this event ruptured 2 years later during the 1946 event. Cape Shiono has long been recognized as the demarcation point for Nankai Trough megathrust earthquakes, which typically rupture the subduction zone either to its West or to its East [see, e.g., *Ishibashi and Satake*, 1998]. As is the case for similar “rupture barriers” in the Aleutian Arc [*Ryan and Scholl*, 1993], the Sunda Arc [*Newcomb and McCann*, 1988], the nature of the rupture barrier in the Nankai Trough is poorly understood. It may be due to change in dip of the plate interface, as suggested by *Cummins et al.* [2002], but may also be due to a region of weakened material in the upper plate that is incapable of accumulating interseismic strain energy. It may even be that the accumulation and release of strain energy along the plate boundary is random, and the rupture barrier is only an artifact due to chance concentration of strain energy east and west of Cape Shiono.

[12] Our result clearly establishes that the interseismic accumulation of strain energy occurs very close to the rupture barrier off Cape Shiono. We feel this strongly supports the existence of an actual rupture barrier, and that any weak or stably sliding zone off Cape Shiono must be only tens of km in width at most. The more likely scenario would seem to be that of *Cummins et al.* [2002], who suggest that a change in plate dip gives rise to a rupture barrier, although some other change in the geometry or character of the plate interface off Cape Shiono may act as a barrier to rupture.

[13] Other recent studies have used the positive correlation between the rupture patterns of Nankai Trough earthquake rupture and crustal structure to suggest that a relationship exists between the two, and in general our results support these suggestions. *Park et al.*'s [2002a] analysis of seismic reflection profiles and outer ridge topography suggested that a steep landward-dipping splay fault off the Kii Peninsula may have activated during 1944 Tonakai earthquake. The considerable slip (>1 m) near the trench of the 1944 slip model that appears off Kii Peninsula is in better agreement with the spatial extent of the splay fault than that of the previous studies (Figure 2).

[14] *Kodaira et al.* [2000] imaged a seamount subducting beneath the accretionary prism off Cape Muroto. The Deep Strong Reflector was identified by *Park et al.* [2002b] in multichannel seismic reflection surveys. Both of these features coincide with areas of low slip in previous models of earthquake rupture, but the low resolution of these models made it difficult to determine whether the positive correlation was due to the fortuitous location of these features near subfault boundaries. The high-resolution slip distributions obtained in the present study conclusively demonstrate that these features truly coincide with centers of low slip in the rupture model (Figure 2), and are not artifacts due to model under-parameterization.

[15] In some other subdcution zones, several authors [e.g., *Schwartz*, 1999] have shown that significant slip was centered in areas that experienced low slip in the previous event, suggesting that the slip pattern varies markedly for earthquakes which sequentially rupture a given subduction zone segment. However, at the Nankai trough, the features of our slip models described here support the alternate hypothesis that slip distributions of subduction zone earthquakes should be similar to the previous earthquake since crustal structure has an important effect on either megathrust rupture propagation or the interseismic accumulation of elastic strain, or both.

## Acknowledgments

[16] We thank Yuichiro Tanioka for comments and providing us with his tsunami analysis tools and waveforms for the 1944 Tonankai and 1946 Nankai earthquakes. Reviews by anonymous reviewers are greatly appreciated. We used Generic Mapping Tools [*Wessel and Smith*, 1998] to calculate the spectrums and the figures in the paper.