Echos get weaker with time.
To be able to see the data at late times,
we generally increase data amplification with time.
I have rarely been disappointed by my choice
of the function *t ^{2}* for the scaling factor.
The

The first of the two powers of *t* arises because we are
transforming three dimensions to one.
The seismic waves are spreading out in three dimensions,
and the surface area on the expanding spherical wave increases
in proportion to the radius squared.
Thus the area on which the energy is distributed is increasing
in proportion to time squared.
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.
For the model we assume:

1 | One dimensional propagation |

2 | Constant velocity |

3 | Constant absorption Q^{-1} |

4 | Reflection coefficients random in depth |

5 | No multiple reflections |

6 | White source |

These assumptions immediately tell us that a monochromatic wave
would decrease exponentially with depth,
say, as where *t* is travel-time depth
and is a decay constant which is inversely
proportional to the wave quality factor *Q*.
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.
At propagation time *t* the original white (constant) spectrum is replaced
by the forementioned function which is
a damped exponential function of frequency.
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.

It is curious that the shape
of the expected seismogram envelope *t ^{-2}* does
not depend on the dissipation constant .But changing the spectrum of the seismic source does change
the shape of the envelope.
It is left for an exercise to show that a seismic source
with spectrum implies a divergence
correction .

The seismic velocity increases with depth, so sometimes people who know the velocity may improve the divergence correction by making it a function of velocity (and hence offset) as well as time.

In reality it may be fortuitous that *t ^{2}* fits data so well.
Actually,

10/31/1997