An uplift of the ocean bottom caused by a submarine earthquake can trigger acoustic-gravity waves that travel at near the speed of sound in water and thus may act as early tsunami precursors. We study the spatiotemporal evolution of the pressure field induced by acoustic-gravity modes during submarine earthquakes, analytically. We show that these modes may all induce comparable temporal variations in pressure at different water depths in regions far from the epicenter, though the pressure field depends on the presence of a leading acoustic-gravity wave mode. Practically, this can assist in the implementation of an early tsunami detection system by identifying the pressure and frequency ranges of measurement equipment and appropriate install...