Quantitative Evaluation of Spatial Interpolation Models Based on a Data-Independent Method - Advances in Data, Methods, Models and Their Applications in Geoscience - page 55

correlations between the known points, while the combined functions are the performance
of interpolations in mathematics, both of which constitute the commonality of interpolation
functions in mathematics and physics. Therefore, they can be unified as one general
interpolation model, just as follows:

Z  wZ  wZ   ... wZ   m WZ  m (1)

p 11 22 n n

where Z p is the estimated value of an interpolated point P(x , )

p y , Z denotes a sample
point with w i indicating its corresponding weight, m presents a constant, and n is the total
number of sample points.
In this united model shown as Formula 1, any interpolation function can be regarded as a
linear combination of sample points, with the difference of rules for weight allocation. In other
words, the determination of the weight vector W is essential and critical for interpolations. For
example, IDW determines its weight according to the distance between sample points directly,
while NNI employs Thiessen polygons and Kriging uses semivariable functions instead. As
for the moving curved surface fitting interpolation, though, the weight function is not obvious,
surface-fit functions are employed to allocate weights, implying the spatial relationships of
data points. The united interpolation models of the eight interpolation algorithms discussed in
this study have been separately listed in Tab. 1.
It has been proved in Tab. 1 that in spite of various interpolation algorithms and models,
they have the same intrinsic interpolation mechanism, and any common interpolation
method can be transformed into a united model. From the mathematical mechanism, any
spatial interpolation is actually a process of assigning weights to sample points, and

Interpolation
models

Parameter
Specification

Interpolation functions Weight vector(W) Constant (m)

d : the
distance
between P 0
and P i ;
k : a power
parameter

IDW, Inverse
Distance
Weighted

 k

0

d

n

i

 

w 

Z wZ

i

n

p ii



 k

d

i  1 i

i  1

n n



 

Z  wz m m [  ]

w

mw


(1 )

Kriging

p i i i

i  1 i 1

d : the
distance
between P 0
and P i ;
()
i

Rd : the
principal
curvature
function;
(,)

MC,
Minimum
Curvature

n



Z  R d ()  T x y (,)

w

Txy (,)

p i i

i  1

Txy : a
'trend'
function