But seismic amplitudes are proportional to the square root of the energy. So the basic geometry of energy spreading predicts only a single power of time for the spherical divergence correction. An additional power of t arises from a simple absorption calculation. Absorption requires a model. The model I'll propose is too simple to explain everything about seismic absorption, but it nicely predicts the extra power of t that experience shows we need.
Many people go astray when they model real seismic data by such a monochromatic wave.
A better model is that the seismic source is broad band, for example an impulse function. Because of absorption, high frequencies decay rapidly, eventually leaving only low frequencies, hence a lower signal strength. The seismic energy available for the creation of an impulsive time function is just proportional to the area under the damped exponential function of frequency. As for the phase, all frequencies will be in phase because the source is assumed impulsive and the velocity is assumed constant. Integrating the exponential from zero to infinite frequency provides us with an inverse power of t thus completing the justification of a t 2 divergence correction.