P1: Vendor/FNV P2: FLF
Mathematical Geology [mg] PL235-228125 September 29, 2000 11:14 Style file version June 30, 1999
56 Gon¸calves
but agree closely in the f (α) range related to each considered q value. This is a
consequence of the large number of lowvalues of arsenic in the distribution,which
is consistent with the local background for this element of approximately 90 ppm
in the area (Gon¸calves, 1996). Comparative histograms for the real data and model
were constructed and shown in Figure 6. All concentrations were normalized to
their maximum concentration so that the distributions could be compared. Model
concentrations were generated with φ = 2.5, K
1
= 1.3, and n = 5 (fifth-order
cells). Distributions differ mostly in the lower values (less than 0.1), where the
generated data showsspikesin some classes of concentrations. In the exampledata
set, a decrease in frequency is observed for each class of concentrations up to 0.1.
A major differenceappears in the first class of concentrations, which in the arsenic
data corresponds to the values close to the detection limit. Such a truncation is not
a model property, where a continuum of concentrations may be ideally achieved
as long as n →∞. However, setting a relative detection limit identical to the
real data set did not improve the model significantly (Fig. 6C). Values of relative
concentration greater than 0.1 are replicated fairly well, which justifies the closer
agreement between both model and experimental multifractal spectrums (Fig. 5).
Noticeable differences are observed in the frequencies of concentrations in the
intervals (0.2,0.3], and (0.4,0.5].
Another way to test model fitness to data would be the use of the q/D
q
plot (Fig. 2) but this proved to be less sensitive than the multifractal spectrum
in highlighting the differences (here referring to q < 0). Another problem arises
because of Equation (7), and that is the uncertainty in the estimates of τ (q) for
q > 0, especially close to 1 that make the D
q
estimation to diverge for q → 1, in
particular
ˆ
D
q
→−∞as q → 1−, and
ˆ
D
q
→+∞as q → 1+.
The proposed model may closely characterize at least some geochemical
distributions,providedthat the whole population is sufficientlyrepresented in both
extremesof the range of values. Therefore, in this case, the enrichment factor gives
themagnitudeoftheefficiencyoftheconcentrationprocessinrelationtotheglobal
average of arsenic present in the soils, which may attain a maximum concentration
of 1280 ppm. On the other hand, the K
1
constant may model the distribution of the
lower arsenic concentrations in the several small scattered veins that appear in the
sampled area, generally in the range of 200–500 ppm in the rocks. Examples of
these small structures are the lower value anomalies close to the sheared Caradoc
quartzites between sectors II and IV, and just to the north of sector IV as well
(Figs. 3 and 4). These last areas have correspondingly smaller concentrations in
the soils, down to 90 ppm, which represents the local background.
The existence of irregularly spaced data often implies the need for interpola-
tion of values onto a regular grid, which can be done via, e.g., kriging. Ordinary
lognormal kriging was used to estimate arsenic values,here followingthe method-
ology of Journel and Huijbregts (1978), interpolating a point every 100 m and
producing a more regular grid in order to test the differences in the multifractal