Linear events

A seismic event is called a linear event if it follows a linear trajectory. Usually such an event is mathematically described as:

w(t-px),

where w is the wavelet of the event, and p is the dip of the event. This description implies that the amplitudes of the wavelets are constant along the event. However field data show that wavelets along seismic events have slowly varying amplitude. Therefore, without loss of generality, we can describe a linear event by allowing its amplitude to vary exponentially as follows:

$$e^{\sigma x}w(t-px),$$

where $\sigma$ is limited to being a real number that determines the rate of amplitude variation. An event with a complicated amplitude function is represented as a superposition of multiple linear events with the same dip. Since these amplitude variations are usually smooth, we expect that the magnitude of $\sigma$ is small, and that the amplitude functions can be represented by a few exponential functions.