Geoscience Reference
In-Depth Information
this surface appear smooth the values of the grid nodes are
adjusted so that the second derivative of the gridded sur-
face varies smoothly, i.e. it has minimum curvature. Some
types of geophysical data, e.g. gravity and magnetic
fields,
are smoothly varying, but there is no direct physical basis
for using smoothness as a basis for interpolation.
Data - irregular spacing
2
1
2.7.2.1
Statistical interpolation
The statistical approach involves calculating some form of
average of the measurements in the window. The median
value has the advantage of being immune to the effects of
outliners (extreme values) in the data series, presumed to
be noise. Arithmetic or geometric means can incorporate a
system of weights, one for each data point scanned and
which, for example, vary inversely proportionally to the
distance to the data points. Therefore points closer to the
grid node exert a greater in
3
2
3
1
Component splines
2
3
1
uence on the interpolated
value than those further away. An extreme form of
weighted averaging is to assign a value to the node that is
equal to that of the closest data point. This is known as
nearest neighbour gridding and it can be effective if the
data are already very nearly regularly spaced. If this is not
the case the resultant dataset can have an unacceptably
'
Final curve - regular spacing
Figure 2.15
1D splining. Cubic splines are
fitted to each pair of
the irregularly spaced data points so that the gradients of connecting
splines are the same at the joining point (
Δ
1
etc). The new values,
interpolated at regularly spaced intervals, are obtained from
the splines.
appearance. All of the gridding algorithms
described above are suitable for both randomly distributed
and line-based data, and minimum curvature adjustment
can be applied.
One of the more sophisticated statistical interpolation
methods is kriging, which is widely used in mineral-
weighted moving averages, where low-valued data points
are increased and high values are decreased using
smoothing factors or kriging coefficients (weights) depend-
ent on both the lateral dispersion of the data points and
their values. The method is not commonly used in geo-
physics, although it can be very useful for small datasets
having an uneven distribution of data points of
blocky
'
position with weights and distorted so that it passes
smoothly through the points to be connected or interpol-
ated. The
flexible strip naturally assumes a form having
minimum curvature, since elasticity tries to restore its
original straightness but is prevented from doing so by
the constraining weights. The curve formed is a cubic
polynomial and is known as a cubic spline.
Splining in the numerical sense is a line-fitting method
that produces a smooth curve (De Boer,
2001
). Many types
of polynomial functions can be used as splines, but cubic
polynomials have less possibility of producing spurious
oscillations between the data points, a characteristic of
some other functions. The cubic spline consists of a series
of cubic functions each
fitted to pairs of neighbouring data
points. They join smoothly at their common points, where
the functions have the same gradients and curvature (i.e.
the same
first and second derivatives). New data values
are calculated in the data intervals using the respective
function. 1D interpolation using cubic splines is illustrated
in
Fig. 2.15
.
The motivation for splining is a
large
dynamic range.
2.7.2.2
Function-based interpolation
An advantage of function-based interpolation methods is
that particular behaviour of the measured parameter can
be incorporated into the interpolation process, most com-
monly smoothness. By far the most common function-
based interpolation methods use splines. A spline, in its
original sense, is a thin strip of
flexible material used pre-
computer drafting to draw smooth curves. A physical
model applicable to 1D data is the drafting spline held in
'
'
smooth
curve. The smoothness of splines may actually be a disad-
vantage, since if a parameter varies abruptly the require-
ment for smoothness may result in spurious features
infiltrating the interpolated data. Normally there is some
pleasingly
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