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Let be an abstract vector containing as components
the water depth over a 2-D spatial mesh.
Let be an abstract vector whose successive components
are depths along the vessel tracks.
One way to grid irregular data is to minimize the length
of the residual vector :
| |
(20) |

where is a 2-D linear interpolation (or binning) operator
and is the data residual.
Where tracks cross or where multiple data values end up in the same bin,
the fitting goal (20) takes an average.
Figure
is a display of simple binning of the raw data.
(Some data points are outside the lake.
These must represent navigation errors.)
Some model-space bins will be empty.
For them we need an additional ``model styling'' goal,
i.e. regularization.
For simplicity we might minimize the gradient.

| |
(21) |

where and is the model space
residual.
Choosing a large scaling factor will tend to smooth
our entire image, not just the areas of empty bins.
We would like to be any number small enough
that its main effect is to smooth areas of empty bins.
When we get into this further, though, we'll see that
because of noise
some smoothing across the nonempty bins is desireable too.

** Next:** Preconditioning for accelerated convergence
** Up:** ELIMINATING NOISE AND SHIP
** Previous:** ELIMINATING NOISE AND SHIP
Stanford Exploration Project

4/27/2004