The present study investigates the tsunami generation process by using 3-D numerical simulations and the linear potential theory. First, we evaluate the relation between sea-bottom elevation and sea-surface elevation as function of source size L, sea depth H and source duration T, based on 3-D numerical simulations. The surface elevation decreases with increasing sea depth and source duration. The difference between the sea-bottom and the sea-surface elevation appears when the source size is smaller than approximately 10 times the sea depth for a short source duration. The linear potential theory can precisely predict the numerical simulation results. Based on the theory, we can consider the tsunami generation as two spatial lowpass filter processes, in which the cut-off wavenumbers are given by the sea depth and the source duration. The criteria for small source size and short source duration are given as L < 13H and T < L/(8c), respectively, where c is the phase velocity of the tsunami. We then simulate the tsunami generation of the 1896 Sanriku tsunami earthquake, Japan. The simulated sea-surface elevation is significantly different from the sea-bottom elevation, which suggests the need for correction of the sea depth and source duration for the precise evaluation of the initial water-height distribution. To include these effects in 2-D simulations, we can use the impulse response function and add the fractional sea-surface uplift within the time step to the sea surface, for each time step.