Long waves are often assumed to model tsunamis, but the wavelength of the initial water height distribution produced by a large submarine earthquake, particularly in the direction perpendicular to the fault strike, is sometimes not much greater than the water depth. The resulting tsunami may have a dispersive character that cannot be simulated based on a conventional long-wave approximation. The 2004 earthquake off Kii Peninsula (M 7.4) on the southern coast of Japan indeed produced a dispersive tsunami that was recorded at two stations located off Shikoku. For the foreshock (M 7.1), on the contrary, a dominant dispersive tsunami was not recognized at these stations. Because dispersive waves show strong directional dependence with respect to the fault strike, the above difference indicates that the strikes of the main shock and the foreshock were different. We conducted a tsunami waveform inversion analysis based on the dispersive tsunami equations to estimate the initial water height distribution of the main shock. The estimated initial water height distribution overlapped with the aftershock region, suggesting that the fault strike was perpendicular to the trough axis, and the total displaced water volume was 1.7–2.0 × 10⁹ m³. When we used the conventional long-wave approximation, the estimated initial water height distribution extended considerably from the aftershock area, because artificial sources were needed outside the aftershock area to reproduce the observed dispersive waves.