Some tide gauge records, including tsunami fluctuations, were obtained from sites near and far from the source region, such as in Panama, Japan, and the USA. For example, tide gauge stations at Honolulu captured the tsunami with heights of several tens of centimeters (National Geophysical Data Center/World Data Service). However, the details of the tsunami arrival and fluctuations at some of these tide gauge stations were not clearly visible because chaotic waveforms with noise comparable to the observed wave height were mixed into the observed waveforms. Thus, we decided to use only the tide gauge data from Honolulu and Naos Island (Fig. 1), where the tsunami was clearly visible in the data, as transcribed by Honda et al. (1908) and Soloviev and Go (1975).

First, these tsunami waveforms were estimated from the brightness matrix of images obtained through the digitization and pixelization of the documents of Honda et al. (1908) and Soloviev and Go (1975). The time resolutions of the estimated data from Honolulu and Naos Island were expected to be 1.9 and 1.6 min, respectively, and height resolutions were expected to be 0.8 and 1.5 cm, respectively, in view of the relationship between the data scale and pixel resolution of the matrix. Time series datasets with an interval of 0.5 min were made based on the estimated data. Fourier analysis with a band pass filter of 10–120 min was applied to the datasets at both stations to estimate the water surface elevation change due to the tsunami. The waveforms estimated through the above process were then determined to be the observed waveforms of the tsunami.

The inverse analysis of this study estimated the constrained slip distribution in space using a smoothing factor, as described in Tanioka et al. (2004). The intensity of the smoothing factor for the analysis was determined with a trial-and-error step. Furthermore, the weight coefficients between the data at the Honolulu and Naos Island stations were set to 1.5 and 1 for the analysis because the observed tsunami height at Honolulu was significantly smaller than that at Naos Island.

The earthquake was first assumed to consist of 45 subfault zones (each with a length and width of 50 km) that covered the entire source area, as suggested by Kelleher (1972) (Fig. 1). Strike/dip angles and the depth of each subfault were assumed based on slab depths and their coordinates (Bird 2003; Hayes et al. 2012). A rake angle was assumed and fixed for all subfaults based on a subducting angle to the trench (Chlieh et al. 2014). The vertical displacement of the ocean floor due to each subfault was first estimated based on Okada (1985), and sea surface deformation was assumed to be equal to that displacement. Furthermore, the effect of horizontal displacement within the bathymetry on the sea surface deformation was considered based on Tanioka and Satake (1996) using computation bathymetry data with a 90 arc-s spatial resolution obtained by resampling the General Bathymetric Chart of the Oceans data with a 30 arc-s resolution (Smith and Sandwell 1997; Becker et al. 2009). The propagation of each sea surface deformation generated by each subfault was then simulated with a time step of 5 s to specify the tsunami waveforms at each tide gauge station. The occurrence time of the earthquake was assumed to be 15:30 (UTC) on January 31, 1906, based on Soloviev and Go (1975), and the rupture/rise times of the earthquake were not taken into consideration. For the Honolulu and Naos Island stations, the propagation computations were conducted based on the linear long wave model, which considers no dispersion, and the linear Boussinesq model, which considers high frequency dispersion. Both the high and the low frequency dispersion effects are expected to significantly influence the waveforms at Honolulu. Thus, all dispersion effects are introduced into the waveforms simulated by the linear long wave model based on the phase correction method developed by Watada et al. (2014). In contrast, only the high frequency dispersion effect is expected to influence the waveforms of Naos Island with respect to distance relations. Furthermore, although the phase collection method can yield all dispersion effects, some assumptions are required, such as propagation distance. Therefore, the linear Boussinesq model is more suitable and useful than the phase collection method for estimating the waveforms of Naos Island. The governing equations for the propagation computations are as follows (Goto 1991):

$$ \frac{\partial \eta }{\partial t} + \frac{1}{R\cos \lambda }\left[ {\frac{\partial }{\partial \lambda }M\cos \lambda + \frac{\partial N}{\partial \theta }} \right] = 0 $$

(1)

$$ \frac{\partial M}{\partial t} + \frac{gh}{R}\frac{\partial \eta }{\partial \lambda } = - fN + \frac{1}{R}\left[ {\frac{\partial }{\partial \lambda }\left\{ {\frac{{h^{3} }}{3}F_{1} + \frac{{h^{2} }}{2}F_{2} } \right\} - \frac{\partial h}{\partial \lambda }\left\{ {\frac{{h^{2} }}{2}F_{1} + hF_{2} } \right\}} \right] $$

(2)

$$ \frac{\partial N}{\partial t} + \frac{gh}{R\cos \lambda }\frac{\partial \eta }{\partial \theta } = fM + \frac{1}{R\cos \lambda }\left[ {\frac{\partial }{\partial \theta }\left\{ {\frac{{h^{3} }}{3}F_{1} + \frac{{h^{2} }}{2}F_{2} } \right\} - \frac{\partial h}{\partial \theta }\left\{ {\frac{{h^{2} }}{2}F_{1} + hF_{2} } \right\}} \right] $$

(3)

$$ F_{1} = \frac{1}{R\cos \lambda }\left\{ {\frac{{\partial^{2} }}{\partial t\partial \lambda }\left( {u\cos \lambda } \right) + \frac{{\partial^{2} v}}{\partial t\partial \theta }} \right\} $$

(4)

$$ F_{2} = \frac{1}{R\cos \lambda }\left\{ {\frac{\partial }{\partial t}\left( {u\cos \lambda \frac{\partial h}{\partial \lambda }} \right) + \frac{\partial }{\partial t}\left( {v\frac{\partial h}{\partial \theta }} \right)} \right\} $$

(5)

where *η* is the water surface elevation, *M* (= uh) and *N* (= vh) are the flux in the *λ* and *θ* directions, *u* and *v* are the velocity in the *λ* and *θ* directions, *h* is the still water depth, g is the acceleration due to gravity, *R* is the radius of the Earth, *f* is the Coriolis parameter, *λ* is the latitude, *θ* is the longitude, and t is the time. The linear long wave model is obtained using the above equations with an assumption that F1 and F2 are zero; the linear Boussinesq model is obtained when the above equations are fully solved. The linear long wave and the Boussinesq models were discretized based on the concepts of Goto et al. (1997) and Saito et al. (2014); however, the latitude dependence of grid size should be taken into consideration.

Here, the phase correction method requires a propagation distance and average water depth during tsunami propagation. The distance was assumed to be the arc distance along the great-circle path through the subfault location and the tide gauge station, as defined by Watada et al. (2014), who used a depth of 4 km as the average water depth during the propagation. However, arrival times of the waveforms simulated by the linear long wave model at the Honolulu station assumed a range of approximately 12–13 h. When the tsunami propagated following the linear long wave theory over distance and time, the equal average depth was comparable to 1.5–1.7 km with respect to the dispersion relation in the long wave condition. Therefore, the equal average depth for the phase correction was estimated and applied based on the relationship between the simulated arrival times of waveforms and the propagation distance using the dispersion relation in the long wave condition. Furthermore, modified dispersion relation curves corresponding to the equal average water depth were estimated by linear interpolation based on the dispersion relation curves with depths of 2, 4, and 6 km shown in Watada et al. (2014).