Formal path to the low-cut filter

This book defines many geophysical estimation problems. Many of them amount to statement of two goals. The first goal is a data fitting goal, the goal that the model should imply some observed data. The second goal is that the model be not too big or too wiggly. We will state these goals as two residuals, each of which is ideally zero. A very simple data fitting goal would be that the model m equals the data d, thus the difference should vanish, say $0\approx m- d$.A more interesting goal is that the model should match the data especially at high frequencies but not necessarily at low frequencies.

$$0 \quad\approx\quad -i\omega(m - d)$$ (6)

A danger of this goal is that the model could have a zero-frequency component of infinite magnitude as well as large amplitudes for low frequencies. To suppress this, we need the second goal, a model residual which is to be minimized. We need a small number $\epsilon$.The model goal is

$$0 \quad\approx\quad \epsilon \ m$$ (7)

To see the consequence of these two goals, we add the squares of the residuals

$$Q(m) \eq \omega^2 (m-d)^2 + \epsilon^2 m^2$$ (8)

and then we minimize Q(m) by setting its derivative to zero

$$0\eq {dQ\over dm} \eq 2 \omega^2 (m-d) + 2\epsilon^2 m$$ (9)

or  

$$m \eq {\omega^2 \over \omega^2+ \epsilon^2}\ d$$ (10)

which is a low-cut filter with a cutoff frequency of $\omega_0=\epsilon$.

Of some curiosity and significance is the numerical choice of $\epsilon$.The general theory says we need an epsilon, but does not say how much. For now let us simply rename $\epsilon=\omega_0$and think of it as a ``cut off frequency''.