The geostatistician takes another approach, called krigging. Instead of 
solving a global optimization problem the geostatistician
solves a series of small inversion problems.
They find all of the points $m_1 .... m_n$ around the point they
are trying to estimate $m_0$. The vector distance between all data points 
$\bf d_{ij}$ 
and each data point and the estimation point $\bf d_{i0}$ are then computed.
Using the predefined covariance function estimate $C$, a covariance
value is then extracted 
between all known point pairs $C_{ij}$ and
between known points and
estimation point $C_{i0}$ at the given distances $\bf d_{ij}$  and
$\bf d_{i0}$   (Figure~\ref{fig:covar-def}).
They then solve for the weights   ($w_1 ... w_n$) by solving the set
of equations implied by
\begin{equation}
\left[
\begin{array}{cccc} 
C_{11}  &...& C_{1n} & 1 \\
.  &...& . & . \\
.  &...& . & . \\
.  &...& . & . \\
C_{n1}  &...& C_{nn} &  1 \\
1 &...& 1 & 0 
\end{array}
\right]
\left[
\begin{array}{c}
w_1 \\
. \\
. \\
. \\
w_n \\
\mu 
\end{array}
\right]
=
\left[
\begin{array}{c}
C_{10} \\
. \\
. \\
. \\
C_{n0} \\
1 
\end{array} \label{eq:krig}
\right] .
\end{equation}
Estimating $m_0$ is then simply,
\beq
m_0= \sum_{i=1}^{n} w_i m_i.
\eeq
To guarantee that the matrix in equation (\ref{eq:krig})  
is invertible geostatisticians approximate
the covariance function
 through a linear combination of a limited set of 
functions.
Each function is described by:
\begin{description} 
\item [{\it range}], the distance at which the covariance function is
essentially gone to 0,
\item [{\it anisotropy}], the amount the covariance function depends
on radial angle, and
\item [{\it orientation}], the major orientation axis of the covariance function.
\end{description}
These parameters are used to describe either a spherical, exponential,
Gaussian, or power model
that are guaranteed to produce a positive definite covariance matrix.
\plot{covar-def}{width=6.0in,height=3.0in}{Definition of the terms
in equation (\ref{eq:krig}). A vector is drawn between two points.  The covariance 
at the angle and distance describing the vector is then selected.}


The left panel of Figure~\ref{fig:geostat-mis} shows the geostatistical
approximation of the model covariance for our missing data problem.
This approximation does a good job characterizing the primary dip
of the covariance function, but
does not  accurately describe the {\it range} of the covariance function.
The right panel of Figure~\ref{fig:geostat-mis} shows the result of applying 
krigging
to the same missing data problem.  The result is fairly comparable to the
geophysics result. Instead of adding a second dip at every location we have
only forced the primary dip. 
Because this approximation adds insufficient {\it range} for the
covariance function our answer returns to the local average
between the two wells with the greatest separation.
Overall the result is as unsatisfactory as the geophysics approach to the
problem.
\plot{geostat-mis}{width=6.0in,height=3.0in}
{Left panel shows the approximated covariance matrix, right
panel is the result of solving equation (\ref{eq:krig}) at every
unknown model point.}
