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different interpolation models have different patterns of weight allocation. While
concerning the meaning of geography, the essence of interpolations lies in the spatial
correlations between unmeasured points and sample points, reflected during the course of
weight allocation. Both sides of mathematics and geography mentioned here can not only
give a hypostatic explanation for the spatial interpolation physical mechanism, but can
also provide certain guidance for further analysis and evaluation of spatial interpolation
models.
3. Methods and procedures
In order to achieve the objectives proposed above, a data-independent experiment has been
carried out, which allowed us to quantitatively analyze and evaluate different spatial
interpolation models. Fig. 1 shows the flowchart of the whole process employed for our
experiment. More specific procedures are illustrated as follows: (1) Constructing a
mathematical surface with a known-formula; (2) Discretizing the mathematical surface and
then randomly sample N points from those discrete ones; (3) Adding errors with varying
levels to the randomly sampling points, so that we can get discrete points with the same
distribution but varying error-levels; (4) Making interpolated operations separately on the
sampling points without errors and the ones with varying error-levels, using the eight
interpolation models mentioned above; (5) Analyzing and evaluating the results acquired
from different interpolation algorithms according to different evaluation indices.
It is noted that all of the eight interpolation algorithms applied in this study are fulfilled by
Surfer 8.0, a powerful contouring, gridding and 3D surface mapping package. Another
aspect should be indicated is about the parameter-setting during interpolation. The
parameters here mainly consist of three kinds: (1) a search neighhood including its search
radius and the number of sampling points, which should be set for the local interpolation
methods such as LP, IDW, MSM and TPS; (2) the maximum residual and the maximum
number of cycles when gridding with MC method; (3) variogram models like linear,
gaussian and logarithmic models for Kriging interpolator. Except variogram models used in
Kriging interpolation, the parameters in the others interpolation methods are control
parameters and can be set as default of Surfer 8.0, for they have no effect on weight
allocation. While for Kriging, the choice of variogram models has a close connection with
weight allocation and may affect the results of interpolation. Through repeated tests and
validations, the linear model is selected in this study.
3.1 Design of mathematical surfaces
In this study, we took the similar approach as reported by Zhou and Liu (2002, 2003 &
2004) by employing pre-defined standard surfaces for testing and comparing selected
algorithms. As a result, the 'true' attribute value of any point on the standard surfaces
which are pre-defined by known mathematical formulas can be acquired without errors.
Our focus is on the difference between the values calculated by interpolation methods and
the 'true' valuesto compare these interpolation algorithms objectively. According to the
complexity of the surfaces, three surfaces have been selected for test, namely a simple
surface, a more complex surface and a Gauss synthetic surface, which are defined by the
equations below:
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