12 Quantification and Regionalization of Benthic Reflux
442
rain rate of non-reactive compounds. A reversal of
the effect, the reduction of the burial rate due to
decreasing input of non-reactive phases, may be
induced by physicochemical dissolution of
carbonate below the CCD (see Chapter 9).
Figure 12.12 demonstrates both effects schematically as simplified balances. The dissolution of
carbonate related to the degradation of organic
material is neglected in this case.
12.4 Conceptual Approaches and
Methods for Regional
Balancing
As pointed out in the preceding sections, the
establishment of basin-wide or global balances of
benthic material cycles always requires extrapolation procedures, because all available data
sets are limited in size and usually consist of inhomogeneously distributed point data. Various
conceptual approaches are available to approach
this goal. In this section, fundamental aspects to
the most frequently used methods will be briefly
presented.
12.4.1 Statistical Key Parameters
and Regression Analysis
The probably easiest method of regional balancing is based on the calculation of statistical key
parameters, such as the mean value or the median
value. In this context, the key parameters are
always related to geographically defined regions.
Therefore, the regions must be defined prior to
statistically evaluating the presumed representative data. The definition of the region is not the
result of the evaluation process. The knowledge
of the regional distribution in each region is
consequently an important pre-condition to the
application of the procedure. The sufficient fulfillment of these conditions permits balancing performance on a global level (e.g. Nelson et al. 1995).
The options and conclusions to be drawn from
regression analysis are essentially more diverse.
Data sets of two or more parameters are compared
with each other (cf. Figs. 12.11 or 12.18a/b). Such
mathematical descriptions of relations between
the single measured values and potential control
parameters enable the statistical extrapolation of
isolated measured values across the area (see
Sections 12.5.1 and 12.5.2). Furthermore, they can
be employed for the characterization and
demarcation of provinces, i.e. region-dependent
validity assessment for the correlations to be
investigated (cf. Section 12.3.1).
12.4.2 Variograms and Kriging
Fundamental publications related to this complex
geostatistical procedure originate from Krige
(1951) and Matheron (1963). In the broadest
sense, geostatistics deal with the spatial variability of location-dependent variables. The central
question is simply: are single and spatially
dispersed values capable of giving a representation of coherent structures, i.e. do the results of
local measurements show a spatial correlation? A
solution to this fundamental problem can be found
by means of variogram analysis. In contrast to
regression analysis, only the relationships of one
parameter are examined (e.g. values of benthic O 2
depletion or TOC content). Information on eventual control parameters, such as the biodegradable
amount of organic substance, as mentioned in the
example above, are not required. As long as the
single values are interrelated, their mathematical
description by the various kriging methods will
serve to provide the desired spatial interpolation.
Hence, it is evident that the proper application of
kriging methods requires a rigorous pre-analysis
by variogram analysis. Below, the essential theoretical basics are briefly outlined. A more advanced
background to this field of study can be obtained
by specific literature (e.g. Journel and Huijbregts
1978; Davis 1986; Akin and Siemes 1988; Schlüter
1996; Wackernagel 1996).
The construction of iso-linear maps implicates
that the individual values obtained from local
measurements are related to each other. These
interrelations are usually subject to directional
changes, i.e. closely neighbored values are more
similar than distant ones. Variogram analysis is
designed to identify such ‘trend structures’
showing decreasing similarity with increasing
distance. In particular, variograms include vector
functions designed to determine the one-half,
medium, and squared differences between measurements conducted at two discrete locations
(variances). The calculation of these variances will
consequently consider the distance between the
points. Thus, distance clusters need to be defined
to permit an allocation (Eq. 12.2) of each variogram
value (as a sum function - γ (h) ) to a number of
point pairs (n (h) ) defined by the chosen distance
442
rain rate of non-reactive compounds. A reversal of
the effect, the reduction of the burial rate due to
decreasing input of non-reactive phases, may be
induced by physicochemical dissolution of
carbonate below the CCD (see Chapter 9).
Figure 12.12 demonstrates both effects schematically as simplified balances. The dissolution of
carbonate related to the degradation of organic
material is neglected in this case.
12.4 Conceptual Approaches and
Methods for Regional
Balancing
As pointed out in the preceding sections, the
establishment of basin-wide or global balances of
benthic material cycles always requires extrapolation procedures, because all available data
sets are limited in size and usually consist of inhomogeneously distributed point data. Various
conceptual approaches are available to approach
this goal. In this section, fundamental aspects to
the most frequently used methods will be briefly
presented.
12.4.1 Statistical Key Parameters
and Regression Analysis
The probably easiest method of regional balancing is based on the calculation of statistical key
parameters, such as the mean value or the median
value. In this context, the key parameters are
always related to geographically defined regions.
Therefore, the regions must be defined prior to
statistically evaluating the presumed representative data. The definition of the region is not the
result of the evaluation process. The knowledge
of the regional distribution in each region is
consequently an important pre-condition to the
application of the procedure. The sufficient fulfillment of these conditions permits balancing performance on a global level (e.g. Nelson et al. 1995).
The options and conclusions to be drawn from
regression analysis are essentially more diverse.
Data sets of two or more parameters are compared
with each other (cf. Figs. 12.11 or 12.18a/b). Such
mathematical descriptions of relations between
the single measured values and potential control
parameters enable the statistical extrapolation of
isolated measured values across the area (see
Sections 12.5.1 and 12.5.2). Furthermore, they can
be employed for the characterization and
demarcation of provinces, i.e. region-dependent
validity assessment for the correlations to be
investigated (cf. Section 12.3.1).
12.4.2 Variograms and Kriging
Fundamental publications related to this complex
geostatistical procedure originate from Krige
(1951) and Matheron (1963). In the broadest
sense, geostatistics deal with the spatial variability of location-dependent variables. The central
question is simply: are single and spatially
dispersed values capable of giving a representation of coherent structures, i.e. do the results of
local measurements show a spatial correlation? A
solution to this fundamental problem can be found
by means of variogram analysis. In contrast to
regression analysis, only the relationships of one
parameter are examined (e.g. values of benthic O 2
depletion or TOC content). Information on eventual control parameters, such as the biodegradable
amount of organic substance, as mentioned in the
example above, are not required. As long as the
single values are interrelated, their mathematical
description by the various kriging methods will
serve to provide the desired spatial interpolation.
Hence, it is evident that the proper application of
kriging methods requires a rigorous pre-analysis
by variogram analysis. Below, the essential theoretical basics are briefly outlined. A more advanced
background to this field of study can be obtained
by specific literature (e.g. Journel and Huijbregts
1978; Davis 1986; Akin and Siemes 1988; Schlüter
1996; Wackernagel 1996).
The construction of iso-linear maps implicates
that the individual values obtained from local
measurements are related to each other. These
interrelations are usually subject to directional
changes, i.e. closely neighbored values are more
similar than distant ones. Variogram analysis is
designed to identify such ‘trend structures’
showing decreasing similarity with increasing
distance. In particular, variograms include vector
functions designed to determine the one-half,
medium, and squared differences between measurements conducted at two discrete locations
(variances). The calculation of these variances will
consequently consider the distance between the
points. Thus, distance clusters need to be defined
to permit an allocation (Eq. 12.2) of each variogram
value (as a sum function - γ (h) ) to a number of
point pairs (n (h) ) defined by the chosen distance