Exponential scaling

Exponential scaling functions have some ideal mathematical properties. Take the Z-transform of a time function a_t:  

$$A(Z) \eq a_0\ +\ a_1 \,Z\ +\ a_2 \,Z^2\ + \ ...$$ (9)

The exponentially gained time function is defined by  

$$\uparrow A (Z) \eq a_0\ +\ a_1 \,e^{\alpha} \,Z\ +\ a_2 \,e^{ 2 \alpha } \,Z^2\ +\ ...$$ (10)

The symbol $\uparrow$ denotes exponential gain. Mathematically, $\uparrow$ means that Z is replaced by $e^{\alpha} Z$.Polynomial multiplication amounts to convolution of the coefficients:  

$$C(Z) \eq A(Z) \ B(Z)$$ (11)

By direct substitution,  

$$\uparrow C \eq ( \uparrow A ) \ ( \uparrow B )$$ (12)

This means that exponential gain can be done either before or after convolution. You may recall from Fourier transform theory that multiplication of a time function by a decaying exponential $\exp (- \alpha t)$ is the equivalent of replacing $- i \omega$ by $- i \omega + \alpha$ in the transform domain.

Specialize the downward-continuation operator $\exp (ik_z z)$ to some fixed z and some fixed k_x. The operator has become a function of $\omega$ that may be expressed in the time domain as a filter a_t. Hyperbola flanks move upward on migration. So the filter is anticausal. This is denoted by  

$$A(Z) \eq a_0\ \ +\ \ a_{-1} \ {1 \over Z }\ \ +\ \ a_{-2} \ {1 \over Z^2}\ \ +\ \ ...$$ (13)

The large negative powers of Z are associated with the hyperbola flanks. Exponentially boosting the coefficients of positive powers of Z is associated with diminishing negative powers--so $\uparrow A$ is A with a weakened tail--and tends to attenuate flanks rather than move them. Thus $\uparrow A$ may be described as viscous.

From a purely physical point of view cosmetic functions like gain control and dip filtering should be done after processing, say, $\uparrow (AB)$.But $\uparrow (AB)$ is equivalent to ($\uparrow A)$($\uparrow B)$,and the latter operation amounts to using a viscous operator on exponentially gained data. In practice, it is common to forget the viscosity and create A($\uparrow B)$.Perhaps this means that dipping events carry more information than flat ones.