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Pore
Pressure Definition: Interpolation Methods
If one of the three Water Pressure Grid options is selected, then
the user may also select the method of interpolation, which is used
to obtain the water pressure at any point in the soil from the grid
data. If the strength type for a soil is Discrete Function, then
interpolation is done to determine the strength of the soil at any
point within the soil region. In Slide, the following methods
are available: Thin Plate Spline, Chugh, Modified Chugh, Local Thin-Plate
Spline, TIN triangulation, Inverse Distance, and Linear by Elevation.
Thin-Plate Spline
The Thin-Plate Spline method utilizes the concept of an infinite
thin elastic plate under tension to determine a spline surface (a
smooth 3-dimensional surface which fits through all of the data
points). The spline surface is used to determine the sample value
at any location. See Franke (1985).
Chugh's Method
This is an interpolation method based on finding the nearest water
pressure grid point in each of the four quadrants with origin centered
at the point where the interpolation is required (e.g. at the midpoint
of the base of a slice). A plane is then fit through each combination
of three quadrant grid points and an interpolation is performed
for each plane. This results in four interpolations, which are then
averaged to obtain the final interpolated value at the desired point.
See Chugh (1981).
Modified Chugh
This is based on the Chugh method, with the additional requirement
that the interpolation point must be WITHIN the triangle formed
by any combination of three quadrant points. If the interpolation
point is NOT within a triangle, then this combination of quadrant
points is NOT used. This check insures that EXTRAPOLATED values
are not included in the average interpolated value. This avoids
numerical inaccuracies which sometimes occur with the Chugh method,
due to excessively large extrapolated values.
Local Thin-Plate Spline
The Local Thin-Plate Spline method is an extension of the Thin-Plate
Spline interpolation technique for use with a large number of data
points (>200). The only difference between the methods is that instead
of using all the data points for the interpolation, the Local version
takes a maximum of 10 closest points to the sample point and fits
a spline surface through them. If there are less than 10 data points,
then this method defaults to the non-local version. The local spline
surface is then used to determine the sample value.
TIN Triangulation
TIN (Triangulated Irregular Network) triangulation takes the data
points and triangulates them using the Delaunay triangulation method.
To calculate the value at a sample point, the program first determines
which triangle the point lies within. If the point lies outside
the convex hull of the data, then the secondary interpolation method
is used. The convex hull is a convex polygon with data points for
vertices that wrap around the perimeter of the data points. Once
the triangle that contains the sample point is found, the interpolated
value is calculated using linear interpolation. This is done by
calculating the plane equation that fits through the 3 data points
at the triangle vertices, then solving for the value using the coordinates
of the sample point and the plane equation.
Inverse Distance
The Inverse Distance Interpolation method weights every data point
according to its distance to the sample point. This scheme is also
known as the Shepard method (Shepard, 1968) and can be written in
the form:
where P is the location of the point to be interpolated, F(P) is
the interpolated value, Pi the location of the scattered data, Fi
are the scattered data values, and ||P-Pi||2 represents the distance
from P to Pi. The main deficiencies of this method are, 1) the local
extrema are typically located at the data points and this results
in poor shape properties, and 2) undue influence of points which
are far away. See Shepard (1968).
Linear by Elevation
The Linear by Elevation method only utilizes the elevation (y-coordinate)
of each data point. The method simply determines the closest data
point (elevation) above the sample point and the closest data point
(elevation) below the sample point, and linearly interpolates the
sample value based on these two data points. Interpolation is done
in the vertical direction only. This method is meant for horizontally
bedded soils where data varies by depth only. It is extremely useful
for cases where a complicated strength or pore pressure profile
exists in only the vertical direction. The x-coordinate of the data
points is not used in the interpolation process. However, it is
used to display the data on your model.
Secondary Interpolation Method
In methods such as the TIN Triangulation and the Chugh method, cases
exist where interpolation of a sample point can not be performed.
In the case of TIN triangulation, an interpolation value can not
be calculated if the sample point lies outside the convex hull of
the user-defined data points. In the Chugh method, if a data point
does not exist in all four quadrants surrounding the sample point,
then an interpolated value can not be calculated. In both these
cases, a secondary interpolation method is used. By default in Slide,
the Local Thin-Plate Spline method is used as the secondary interpolation
method. This insures that an interpolated value is always calculated
at a sample point.
Display of Interpolation Contours
The Slide Interpeter allows you to view contours of the
approximate results of the interpolation process, directly on the
model. The interpolation results should always be looked at to ensure
that the interpolation correctly simulates your field data. If it
does not, then use more data points or a different interpolation
technique. Since no interpolation method is guaranteed to work for
all datasets, different methods should be tried in order to determine
the best method for your data.
Contours of Discrete Strength function interpolation.
The strength was interpolated using 13000 data points generated
from a finite-element model. Notice the complex distribution of
strength.
References
Franke, Richard. (1985), Thin plate splines with tension, Computer
Aided
Geometric Design 2 87 -
95, North-Holland.
Chugh, A.K. (1981), Pore Water Pressure
in Natural Slope, International
Journal
for Numerical and Analytical Methods in Geomechanics, Vol. 5, 449
- 454,
John Wiley & Sons Ltd.
Shepard, D. (1968), A two dimensional interpolation function
for irregularly
spaced
data, Proc. 23rd Nat. Conf. ACM, 517?524.
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