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Inverse
Distance Weighted Interpolation
One of
the most commonly used techniques for
interpolation of scatter points is inverse
distance
weighted (IDW) interpolation.
Inverse distance weighted methods are based on the
assumption that the
interpolating
surface should be influenced most by the nearby
points and less by the more distant points.
The interpolating surface is a weighted
average of the scatter points and the weight
assigned to each
scatter point
diminishes as the distance from the interpolation
point to the scatter point increases.
Several options are available for
inverse distance weighted interpolation. The
options are selected using
the Inverse
Distance Weighted Interpolation Options dialog.
This dialog is accessed through the Options
button next to the Inverse distance
weighted item in the 2D Interpolation Options
dialog. SMS uses
Shepard's Method for
IDW:
Shepard's Method
The
simplest form of inverse distance weighted
interpolation is sometimes called
(Shepard 1968). The equation used is as
follows:
where
n is the number of scatter points in the set, fi
are the prescribed function values at the scatter
points (e.g. the data set values), and
wi are the weight functions assigned to each
scatter point. The
classical form of
the weight function is:
where p is an arbitrary positive real
number called the power parameter (typically, p=2)
and hi is the
distance from the scatter
point to the interpolation point or
where (x,y) are the coordinates of the
interpolation point and (xi,yi) are the
coordinates of each scatter
point. The
weight function varies from a value of unity at
the scatter point to a value approaching zero as
the distance from the scatter point
increases. The weight functions are normalized so
that the weights
sum to unity.
The effect of the weight function is
that the surface interpolates each scatter point
and is influenced most
strongly between
scatter points by the points closest to the point
being interpolated.
Although the weight
function shown above is the classical form of the
weight function in inverse distance
weighted interpolation, the following
equation is used in SMS:
where hi is the distance from the
interpolation point to scatter point i, R is the
distance from the
interpolation point
to the most distant scatter point, and n is the
total number of scatter points. This
equation has been found to give
superior results to the classical equation (Franke
& Nielson, 1980).
The weight function
is a function of Euclidean distance and is
radially symmetric about each scatter point.
As a result, the interpolating surface
is somewhat symmetric about each point and tends
toward the
mean value of the scatter
points between the scatter points. Shepard's
method has been used
extensively
because of its simplicity.
Computation
of Nodal Function Coefficients
In the
IDW Interpolation Options dialog, an option is
available for using a subset of the scatter points
(as
opposed to all of the available
scatter points) in the computation of the nodal
function coefficients and in
the
computation of the interpolation weights. Using a
subset of the scatter points drops distant points
from consideration since they are
unlikely to have a large influence on the nodal
function or on the
interpolation
weights. In addition, using a subset can speed up
the computations since less points are
involved.
If the Use subset
of points option is chosen, the Subsets button can
be used to bring up the Subset
Definition dialog. Two options are
available for defining which points are included
in the subset. In one
case, only the
nearest N points are used. In the other case, only
the nearest N points in each quadrant
are used as shown below. This approach
may give better results if the scatter points tend
to be clustered.
The Four
Quadrants Surrounding an Interpolation
Point.
If a subset of the
scatter point set is being used for interpolation,
a scheme must be used to find the
nearest N points. Two methods for
finding a subset are provided in the Subset
Definition dialog: the
global method
and the local method.
Global Method
With the global method, each of the
scatter points in the set are searched for each
interpolation point to
determine which
N points are nearest the interpolation point. This
technique is fast for small scatter point
sets but may be slow for large sets.
Local Method
With the local
methods, the scatter points are triangulated to
form a temporary TIN before the
interpolation process begins. To
compute the nearest N points, the triangle
containing the interpolation
point is
found and the triangle topology is then used to
sweep out from the interpolation point in a
systematic fashion until the N nearest
points are found. The local scheme is typically
much faster than
the global scheme for
large scatter point sets.
Computation
of Interpolation Weights
When computing
the interpolation weights, three options are
available for determining which points are
included in the subset of points used
to compute the weights and perform the
interpolation: subset, all
points, and
enclosing triangle.
Subset of Points
If the Use subset of points option is
chosen, the Subset Definition dialog can be used
to define a local
subset of points.
All Points
If the Use all
points option is chosen, a weight is computed for
each point and all points are used in the
interpolation.