Digital Signal Processing Reference
In-Depth Information
complex, the structures based on regular shapes cannot accurately describe the
irregular geometry and topology of pore space, which has become an obstacle for the
study of transport properties in porous media [3].
Information reconstruction for unknown regions is quite important and significant to
the study of porous media. When reconstructing the unknown information,
interpolation methods use some scattered points to estimate the unknown attributes of
unsampled nodes to build an accurate and complete mathematical model according to
some rules from math, physics and so on.
Although a number of interpolation methods were introduced, the accurate
reconstruction of information is still difficult to be realized. Interpolation methods are
mainly two types: “definite” methods and “indefinite” methods. The “definite” here
means that the forms, parameters and results of interpolation functions are mostly
definite. “Definite” methods include the inverse distance weighting method, the
triangular mesh method, the basis function method, etc. The “indefinite” means that the
forms of interpolation functions are indefinite and the selection of parameters in
interpolation functions depends on the principles of statistics [1, 2]. The main
“indefinite” interpolation methods are kriging and stochastic simulation in
geostatistics. Kriging and stochastic simulation, both based on variogram which only
describes the relations between two points in space and cannot reconstruct complex
patterns such as curvilinear shapes, are called two-point geostatistics [3, 4].
Therefore, indefinite algorithms can normally be classified into two categories [5,
6]. The first kind is that the interpolation algorithms which yield a unique interpolated
result. These interpolation algorithms are usually low-pass filters which tend to smooth
out local details of the spatial variability of the simulated variable. They provide a local
measure of uncertainty, e.g., a kriging variance.
Stochastic techniques which produce multiple possible realizations of the spatial
distribution are the second type. Typically, these stochastic algorithms are full-pass
filters which reproduce the full spectrum of the spatial variability. Fluctuations
between the realizations of multiple stochastic results provide a visual and quantitative
measure for the uncertainty about the underlying characteristics in the possible results.
Two-dimensional cross-sections of porous media are, in contrast to 3D images
generated by direct imaging, often readily obtained and available at a high resolution,
but they cannot include the three-dimensional information of porous media. Therefore,
a method using MPS (multiple-point geostatistics) and three-dimensional volume data,
which were obtained by synchrotron microtomography and were used as a 3D training
image, is proposed to reconstruct porous media. “Training image” originally is a
geological term which is used to describe the anisotropy in formation, the trend and
distribution of geological bodies, etc. A training image of porous media includes
different characteristics of pore spaces. After extracting these characteristics from a
training image and then combining them into a “characteristics database”, we can store
the characteristics in a data structure called “search tree”, which will be used again in
the simulation process to “copy” those characteristics into reconstructed results
according to the probability principle. Experimental results show that the method is
appropriate and practical.
