ROUGHENERS AND SMOOTHERS

I remind you of the ``standard trick'' used by Bill Symes, Bill Harlan, and probably many others. In geophysical fitting we generally have two goals, the first being data ``fitting'' and the second being the ``damping'' or ``regularization'' goal.

$$\begin{eqnarray} 0 \quad\approx\quad \bold r_d &=& \bold F \bold m - \bold d \\ 0 \quad\approx\quad \bold r_m &=& \lambda \bold R \bold m\end{eqnarray}$$ (1) (2)

where $\bold R$ is a roughening operator. Since the roughening operator $\bold R$ is somewhat arbitrarily chosen, presumably we can know an inverse to it, a smoothing operator $\bold S$so that $\bold R\bold S = \bold I$.In the simple one-dimensional case, the roughener is a first or second derivative operator and the smoother is a single or double integral. (I'm not so sure that the multidimensional case is as easy as we often like to pretend). We define a new variable $\bold x$ by $\bold m = \bold S \bold x$and have a set of regression goals for which iterative methods generally converge faster:

$$\begin{eqnarray} 0 \quad\approx\quad \bold r_d &=& \bold F \bold S \bold x - \bold d \\ 0 \quad\approx\quad \bold r_m &=& \lambda \bold x\end{eqnarray}$$ (3) (4)

After solving for $\bold x$ we easily find $\bold m = \bold S \bold x$.

A problem arises when we do not start from $\bold m=0$.Then before we can get started we need to find $\bold x_0$by solving $\bold m_0= \bold S \bold x_0$.