Environmental Engineering Reference
In-Depth Information
h
þ c 1
i þ s 2
h
þ c þ ln
i
r 2
4
R ðÞ¼ 1
2p
r
2s
r
s
r
2p
ð 3 : 3 Þ
ln
K 0
where:
r is the distance between the predicted point and the sample data points; c is
defined as a constant which is equal to 0.577215; s 2
is the Weight parameter; K 0 is
the modified function.
Inverse Distance Weighted
One of the most commonly used interpolation techniques of scatter points is IDW.
The IDW method is based on the assumption that the interpolating surface should
be influenced mostly by the nearby points and less by the more distant points
( http://webhelp.esri.com ) . IDW deterministic interpolation technique applies the
idea that influence decreases with increasing the distance from particular points.
Burroughs and McDonnell [ 4 ], Philip and Watson [ 15 ], Watson and Philip [ 15 ]
realized that ''the value at un-sampled points is a distance-weighted average of
values at sampled locations within a defined neighborhood surrounding the un-
sampled point''. In this sense, Johnston [ 6 ] found that the points closer to the
prediction location will have more influence on the predicted value than points
located farther away [ 6 ].
To fill an elevation matrix, IDW technique uses weighted averaging techniques.
This means that the interpolated values of a cell are determined from the values of
nearby data points, taking into account the distance of the cell from those input
points [ 17 ]. The equation of IDW for interpolating the elevation is shown as:
b Zs ðÞ¼ X
N
k i : z ð s i Þ
ð 3 : 4 Þ
i ¼ 1
where:
b Zs ðÞ is the predicted value at the unsampled points S 0 ; N is the number of
measured sample points within the neighborhood defined for S 0 ; k i are the dis-
tance-dependent weights associated with each sample point; and z k i
is the
observed value at location S i . Weights are calculated using:
d p
i0
k i ¼
ð 3 : 5 Þ
P
N
d p
i0
i ¼ 1
X
N
k i ¼ 1
ð 3 : 6 Þ
i ¼ 1
where:
d i0 is the distance between the prediction values at unknown location S 0 and the
measured values at known location S i ; P is the power parameter which is applied
to define the rate of reduction of the weights as the distance increases [ 4 , 6 , 18 ].
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