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\markright{Model Reparameterization --- W.S. Harlan}
\title{Regularization by Model Reparameterization}
\author{William S. Harlan}
\date{August, 1995}
\maketitle
Geophysical inversion frequently makes use of regularization,
such as the ``Tikhonov regularization''
used by Kenneth Bube and Bob Langan \cite{SEG-1994-0980}
for their ``continuation approach.''
I'd like to suggest an adjustment of the objective
function to allow faster convergence of
regularization and the continuation approach.
A damping term that discourages complexity can be
replaced equivalently by a change of variables to
model simplicity directly.
For an optimized inversion, an objective function typically
includes a norm of the difference
between a data vector $\svector{d}$ and a non-linear transform
$\svector{f}( \svector{m})$ of a model vector $\svector{m}$.
The global minimum of this norm is often flat, with
little sensitivity to large variations in the model.
For regularization (more than simple damping), a
linear operator $\stensor{D}$ is chosen
to remove simplicity and preserve complexity when applied
to the model vector as $\stensor{D} \cdot \svector{m}$.
Most examples use a roughening operator, such as a derivative, to suppress
long wavelengths and amplify short wavelengths.
A regularized objective function adds a norm of
this roughened model to the norm fitting the data:
\begin{equation}
\label{eq:orig}
\min_{\svector{\scriptstyle m}} ~~
J_1 (\svector{m} )=
\| \svector{d} - \svector{f}(\svector{m}) \|^2
+ c \| \stensor{D} \cdot \svector{m} \|^2 .
\end{equation}
This particular $l^2$ objective function is easily motivated
as a maximum {\sl a posteriori} estimate of the model given
the data. Additive noise is assumed to be Gaussian
and uncorrelated with zero mean. The model is assumed to be Gaussian
and zero mean, with an inverse covariance matrix equal to
\begin{equation}
\label{eq:cova}
\stensor{C}_m^{-1} \equiv E(\svector{m} \svector{m}^* )^{-1}
= \stensor{D}^* \cdot \stensor{D} .
\end{equation}
Asterisks indicate adjoints.
The assumption that model samples are correlated is equivalent
to the encouragement of simplicity. A constant $c$
adjusts the ratio of variances assumed for noise and the model.
Bube and Langan's continuation approach begins with a large constant
$c$, minimizes the objective function (\ref{eq:orig}) for a first model,
then reduces $c$ repeatedly for a tradeoff between simplicity
and accuracy in fitting the recorded data. They find
the simplest model possible to explain the data adequately,
without preventing the model from using complexity
to fit genuinely significant features of the data.
Informative details are added to the model when
justified by the data, without unnecessary distracting
details that are poorly determined from the data.
Each minimization of the objective function (\ref{eq:orig})
for a fixed constant $c$ typically uses a descent method such
as Gauss-Newton with conjugate gradients. The properties of the gradient
are important to the rate of convergence:
\begin{eqnarray}
\label{eq:gfir}
\svector{\nabla}_{\svector{\scriptstyle m}} J_1 (\svector{m}_0 ) &=&
\svector{\nabla} \svector{f} ( \svector{m}_0 )^*
\cdot [ \svector{d} - \svector{f}( \svector{m}_0 ) ]
\\
\label{eq:gsec}
&+& c \stensor{D}^* \cdot \stensor{D} \cdot \svector{m}_0 .
\end{eqnarray}
The model is perturbed with scaled sums of successive
gradients, evaluated for different reference versions $\svector{m}_0$
of the model.
The first term (\ref{eq:gfir}) is able to introduce fairly
arbitrary complexity into the model immediately and at any time, even if
such complexity will be suppressed at the global
minimum of objective function (\ref{eq:orig}). The second
term (\ref{eq:gsec}) must wait until the reference
model $\svector{m}_0$ has been revised in later iterations
to suppress this unnecessary complexity. Meanwhile, the
first term (\ref{eq:gfir}) of later iterations can continue
to introduce other unnecessary complexity into the model.
The second term removes complexity in the reference model,
not in the current perturbation.
Convergence is slow. Slow convergence is
a natural consequence of applying perturbations which
do not have any of the correlations assumed for the model
samples. Instead,
let us introduce the appropriate correlation into all gradient
perturbations.
Assume a new operator $\stensor{S}$ as a partial right inverse
of $\stensor{D}$, so that the two operators approximate
an identity: $\stensor{D} \cdot \stensor{S} \approx \stensor{I}$.
This operator should be designed to preserve simplicity
and suppress complexity, although without destroying complexity
entirely. If $\stensor{D}$ is a roughening operator like
differentiation, then $\stensor{S}$ should be a smoothing
operator like leaky integration.
More directly, define the simplification operator
as a factored form of the assumed covariance.
(Indeed, such a factorization always exists because
the covariance is positive semidefinite.)
\begin{equation}
\label{eq:covb}
\stensor{C}_m \equiv E(\svector{m} \svector{m}^* )
= \stensor{S}^* \cdot \stensor{S} .
\end{equation}
Minimization of the original objective function
(\ref{eq:orig}) is entirely equivalent to minimizing
the objective function with a new variable $\svector{m}'$,
where $\svector{m} = \stensor{S} \cdot \svector{m}'$:
\begin{equation}
\label{eq:revi}
\min_{\svector{\scriptstyle m}'} ~~
J_1 ( \stensor{S} \cdot \svector{m}' )=
J_2 ( \svector{m}' )=
\| \svector{d} - \svector{f}(\stensor{S} \cdot \svector{m}') \|^2
+ c \| \svector{m}' \|^2 .
\end{equation}
The second term reduces to a simple damping norm,
demonstrating that the new model $\svector{m}'$ now
has uncorrelated samples.
Although we optimize this new model $\svector{m}'$,
we keep and use the original
model $\svector{m} = \stensor{S} \cdot \svector{m}'$.
Continuation can adjust the constant $c$ as before,
with identical results (assuming complete minimization
of the objective functions [\ref{eq:orig}] and [\ref{eq:revi}]).
The revised gradient contains the desired correlation:
\begin{eqnarray}
\label{eq:nfir}
\svector{\nabla}_{\svector{\scriptstyle m}'} J_2 (\svector{m}'_0 ) &=&
\stensor{S}^* \cdot
\svector{\nabla} \svector{f}
( \stensor{S} \cdot \svector{m}'_0 )^*
\cdot [ \svector{d} - \svector{f}( \stensor{S} \cdot \svector{m}'_0 ) ]
\\
\label{eq:nsec}
&+& c \svector{m}'_0
\end{eqnarray}
The last operation appearing in the first term of this
gradient (\ref{eq:nfir}) is the adjoint $\stensor{S}^*$
of the operator $\stensor{S}$, both of which are
simplification operators.
(Many such operators are self-adjoint.)
Unlike the first term (\ref{eq:gfir}) of the original gradient,
the revised term (\ref{eq:nfir}) suppresses complexity
from each new perturbation direction.
The original term (\ref{eq:gfir})
contained arbitrary correlations. If (\ref{eq:gfir})
were entirely uncorrelated, then the revised
term (\ref{eq:nfir}) would have exactly the desired correlations
assumed by the covariance (\ref{eq:covb}).
The two objective functions produce different results when
optimization is incomplete.
A descent optimization of the original objective
function (\ref{eq:orig}) will begin with complex
perturbations of the model and slowly converge
toward an increasingly simple model at the global minimum.
A descent optimization of the revised objective
function (\ref{eq:revi}) will begin with simple
perturbations of the model and slowly converge
toward an increasingly complex model at the global minimum.
The latter strategy is more consistent with the
overall goal of the continuation approach.
A more economic implementation can use fewer iterations.
Insufficient iterations result in an insufficiently complex
model, not in an insufficiently simplified model.
I also prefer to adjust more than a single scale factor
$c$. Instead, assume a suite of simplification operators
$\stensor{S_i}$ which allow increasing complexity as the
index $i$ increases. (Furthermore $\forall \svector{m}_i'$ and
$j>i, \strut \exists ~ \svector{m}_j' \ni \stensor{S_j} \cdot \svector{m}_j'
= \stensor{S_i} \cdot \svector{m}_i'$.)
We then can optimize a suite of possible models,
$\{ \svector{m}_i = \stensor{S_i} \cdot \svector{m}'_i \}$
of increasing complexity as $i$ increases.
Use each optimized model $\svector{m}_i$ to initialize \strut the
next $\svector{m}_{i+1}$. As multigrid
methods have shown, we can thus improve our overall
convergence by optimizing the most reliable (smoothest)
global features in the model before attempting finer
detail.
Finally, I think it easier to choose a simplification
operator $\stensor{S}$ which describes the desirable
features of the model, rather than an operator $\stensor{D}$
which keeps only features thought to be undesirable.
I see some value in constructing both, however, to
check the consistency of assumptions.
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