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Unlike the classical statistical approach, geostatistics takes into account the
spatial dependence of variables. Namely, geostatistical methods of interpolation
start from the assumption that by knowing the value of a property at known points,
it is possible to establish its value at unknown points as well. Assuming that the
samples are representative and consistent, the values of the corresponding variable
at a new salt location can be obtained using the appropriate interpolation method.
Interpolation methods include a set of procedures for creating assumed values of
interest. The methods that can be applied to create a 3D model are:
• Finite element methods
• Interpolation methods using moving surfaces (interpolation with local
polynomials, inverse distance method, oblique plane method)
• Methods with variational approach (minimum curvature method, spline method)
• Geostatistical methods (collocation method, linear prediction by least squares
method, kriging method)
• Methods of forming TIN models based on input data and interpolation of heights
for grid points from TIN data structure
• Methods of data pre-processing using TIN, with the aim of forming an additional
set of data (structural terrain lines and points in parts with rare starting data) and
their use for interpolation of 3D models using some of the previous procedures
Choosing the appropriate interpolation method is not easy and requires a good
knowledge of the characteristics of the terrain being modelled, the characteristics of
the input data as well as the characteristics of the available interpolation methods.
For interpolation of heights in points of 3D models, all methods can be used which,
on the basis of heights given in arbitrarily distributed points, the height for any set
point that falls within the area are covered by the input data. Some of these methods
have been specially developed for interpolation of heights at grid points.
6.6.2.1 Basic Principles of Variogram Calculation
The introduction showed us how geostatisticians observe the spatial dependence of
sampled or analytical values. The variogram is the basic means of evaluation, quantification and spatial dependence. In practice, the variogram represents the root
mean square of the difference of two values calculated as a function of the distance
of these values. It is the basis of all geostatistical calculations (Wellmer 1998).
The set of all data pairs at the same distance is called the class, and by merging
the values for each class, the curve of the experimental variogram is obtained. There
are four parameters that can be read from the variogram, and they are (1) deviation
(“Nugget”), (2) threshold (“Sill”), (3) range (“Range”) and (4) distance (“Distance”).
(1) Deviation (“Nugget”) is a positive value on the y-axis, where the variogram
curve intersects that axis. (2) The threshold (“Sill”) is the value corresponding to the
variance. Once the curve reaches the threshold, it stops growing properly and generally begins to oscillate around the threshold. (3) Range is the distance along the
x-axis from zero to the point where the variogram curve intersects the threshold.
6 Evidence for Atmospheric Depositions Using Attic Dust, Spatial Mapping…
Unlike the classical statistical approach, geostatistics takes into account the
spatial dependence of variables. Namely, geostatistical methods of interpolation
start from the assumption that by knowing the value of a property at known points,
it is possible to establish its value at unknown points as well. Assuming that the
samples are representative and consistent, the values of the corresponding variable
at a new salt location can be obtained using the appropriate interpolation method.
Interpolation methods include a set of procedures for creating assumed values of
interest. The methods that can be applied to create a 3D model are:
• Finite element methods
• Interpolation methods using moving surfaces (interpolation with local
polynomials, inverse distance method, oblique plane method)
• Methods with variational approach (minimum curvature method, spline method)
• Geostatistical methods (collocation method, linear prediction by least squares
method, kriging method)
• Methods of forming TIN models based on input data and interpolation of heights
for grid points from TIN data structure
• Methods of data pre-processing using TIN, with the aim of forming an additional
set of data (structural terrain lines and points in parts with rare starting data) and
their use for interpolation of 3D models using some of the previous procedures
Choosing the appropriate interpolation method is not easy and requires a good
knowledge of the characteristics of the terrain being modelled, the characteristics of
the input data as well as the characteristics of the available interpolation methods.
For interpolation of heights in points of 3D models, all methods can be used which,
on the basis of heights given in arbitrarily distributed points, the height for any set
point that falls within the area are covered by the input data. Some of these methods
have been specially developed for interpolation of heights at grid points.
6.6.2.1 Basic Principles of Variogram Calculation
The introduction showed us how geostatisticians observe the spatial dependence of
sampled or analytical values. The variogram is the basic means of evaluation, quantification and spatial dependence. In practice, the variogram represents the root
mean square of the difference of two values calculated as a function of the distance
of these values. It is the basis of all geostatistical calculations (Wellmer 1998).
The set of all data pairs at the same distance is called the class, and by merging
the values for each class, the curve of the experimental variogram is obtained. There
are four parameters that can be read from the variogram, and they are (1) deviation
(“Nugget”), (2) threshold (“Sill”), (3) range (“Range”) and (4) distance (“Distance”).
(1) Deviation (“Nugget”) is a positive value on the y-axis, where the variogram
curve intersects that axis. (2) The threshold (“Sill”) is the value corresponding to the
variance. Once the curve reaches the threshold, it stops growing properly and generally begins to oscillate around the threshold. (3) Range is the distance along the
x-axis from zero to the point where the variogram curve intersects the threshold.
6 Evidence for Atmospheric Depositions Using Attic Dust, Spatial Mapping…