Multifractals and Local Singularity Analysis - Geomathematics: Theoretical Foundations, Applications and Future Developments - page 440

Cheng ( 1999 , 2005 , 2008 ) proposed a model for incorporating local spatial
association and singularities for interpolation of exploratory data. Recent 2-D
applications using stream sediment geochemical data, including Cheng and
Agterberg ( 2009 ), Zuo et al. ( 2009 ) and Xiao et al. ( 2012 ), results in relatively
small anomalies consisting of minimal singularities on maps. Some of these
anomalies are spatially correlated with known mineral deposits and others
present new targets for further mineral exploration. On conventional contouring
maps such anomalies are smoothed out. Although local singularity mapping uses
the same data as those on which element concentration contour maps are based,
the new local singularity patterns are strikingly different in appearance. The
reason is that only data from points within very small neighborhoods are used for
estimating the singularities and data are weighted in a manner that differs from
weighting in most contouring methods. Understanding the behaviour of the
semivariogram or spatial autocorrelation function over very short distances is
important for the study of localized maxima and minima in the variability
patterns of element concentration values. Local singularity analysis, which has
become important for helping to predict occurrences of hidden ore deposits, is
based on the equation x

E αE where x , as before, represents element con-
centration value, c is a constant,

¼ c ·

is a normalized distance
measure such as block cell edge, and E is the Euclidian dimension. The singu-
larity is estimated from observed element concentration within small neighbor-
hoods in which the spatial autocorrelation is largely determined by functional
behavior near the origin that is difficult to establish by earlier geostatistical
techniques.
Iterative algorithms proposed by Chen et al. ( 2007 ) and Agterberg ( 2012a , b ) for
local singularity mapping when E

α is the singularity,

E

1 will be discussed in Sect. 11.6 . Local
singularities obtained by application of these algorithms provide new information
on the nature of random nugget effects versus local continuity due to clustering of
ore crystals as occurs in many types of orebodies and rock units. Other applications
of local singularity analysis include Xie et al. ( 2007 ).
In Cheng's ( 1999 , 2005 ) original approach, geochemical or other data col-
lected at sampling points within a study area are subjected to two treatments. The
first of these is to construct a contour map by any of the methods such as kriging
or inverse distance weighting techniques generally used for this purpose.
Secondly, the same data are subjected to local singularity mapping. The local
singularity

¼

then is used to enhance the contour map by multiplication of the
contour value by the factor

α

E α 2 where

1 represents a length measure. In
Cheng's ( 2005 ) application to predictive mapping, the factor

E <

E α 2 is greater than
2 in places where there had been local element enrichment or by a factor less than
2 where there had been local depletion. Local singularity mapping can be useful
for the detection of geochemical anomalies characterized by local enrichment
even if contour maps for representing average variability are not constructed
( cf . Cheng and Agterberg 2009 ; Zuo et al. 2009 ).