Transcript Bayesian maximum entropy solution of the stochastic

Determination of the optimum
spatiotemporal sampling
density to map soil pollution in
an intensive mining area
by
K. Modis, National Technical University of Athens
and
K. Vatalis, Technological Educational Institution of W Macedonia
CEMEPE 2011 – 22-06-2011, Skiathos, GREECE
Modis et al. "Determination of the optimum spatiotemporal sampling density to map soil pollution"
Contents





Scope
Theory
Data
Results
Conclusions
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Modis et al. "Determination of the optimum spatiotemporal sampling density to map soil pollution"
Scope
3
Modis et al. "Determination of the optimum spatiotemporal sampling density to map soil pollution"
Objective

The objective of this work
is
to
apply
recently
established
theoretical
results
(Modis
and
Papaodysseus, 2006) in
order to estimate the
optimum sampling grid to
map soil pollution in the
lignite mining and industrial
area of Ptolemais.
Scope
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Modis et al. "Determination of the optimum spatiotemporal sampling density to map soil pollution"
Introduction

The theoretical results are
based on the sampling
theorem of information
which states that a band
limited random waveform
can be reconstructed by its
samples if the sampling rate
is greater than a critical
value depending on the
waveform characteristics
Scope
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Modis et al. "Determination of the optimum spatiotemporal sampling density to map soil pollution"
Introduction


In simple words, the methodology followed is an
extension of information theory applied to the
geosciences practice.
But, does it looks reasonable that there might be a
limit in sampling density, and more samples will
add little to the quality of the approximation?
Scope
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Modis et al. "Determination of the optimum spatiotemporal sampling density to map soil pollution"
Introduction

The answer in fact is yes, under the condition that
the frequency content of the study variable is
limited. Band limitedness ensures slow variation.
And it is easier to sample a slow varying
phenomenon than a rapidly varying one.

There are two important facts about our approach …
Scope
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Modis et al. "Determination of the optimum spatiotemporal sampling density to map soil pollution"
Fact 1

Our approach is the only
alternative up to date for
the estimation of a
theoretical limit to the
sampling density, above
which
there
is
no
significant improvement to
the accuracy of mapping.
Scope
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Modis et al. "Determination of the optimum spatiotemporal sampling density to map soil pollution"
Fact 2

It can be shown that, in
the case of an existing
sampling grid, if the
sampling density is
close to the ideal one,
most
interpolation
algorithms converge to
reality.
Scope
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Modis et al. "Determination of the optimum spatiotemporal sampling density to map soil pollution"
Theory
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Modis et al. "Determination of the optimum spatiotemporal sampling density to map soil pollution"
Bandlimited waveforms


According to the aforementioned sampling
theorem, only bandlimited waveforms can be
reconstructed by their samples.
Modis and Papaodysseus (2006) have shown that
earth phenomena represented by a covariance
model with a sill (e.g. Spherical scheme) are
approximately bandlimited.
Theory
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Modis et al. "Determination of the optimum spatiotemporal sampling density to map soil pollution"
Linear covariance
1
R(h)
1
γ(h)
For
example,
the
fourier transform of a
linear
variogram/
covariance model with
a sill,
-a
0
a
h
-a
0
a
h
(b)
(a)
a
is approximately
bandlimited
S(ω)

0
Theory
2π/a
ω
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Modis et al. "Determination of the optimum spatiotemporal sampling density to map soil pollution"
Spherical covariance

Or,
the
fourier
transform
of
the
spherical
variogram/
covariance model with
a sill,
-1,5
-0,5
0,5
1,5
is also approximately
bandlimited
Theory
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Modis et al. "Determination of the optimum spatiotemporal sampling density to map soil pollution"
The critical sampling density


In that case, it is also shown that the critical
sampling interval equals half the range of
influence of the underlying covariance model.
According to the above, the estimation of the
critical sampling density of a spatial phenomenon
is done in two steps:
Theory
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Modis et al. "Determination of the optimum spatiotemporal sampling density to map soil pollution"
Step 1

At
the
structural
analysis stage, the
variogram/ covariance
model is estimated. If it
is a model without a
sill, the process stops
here.
Theory
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Modis et al. "Determination of the optimum spatiotemporal sampling density to map soil pollution"
Step 2

At the estimation stage, the
critical sampling interval is
estimated by halving the
range of influence of the
underlying
variogram
model.
Theory
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Modis et al. "Determination of the optimum spatiotemporal sampling density to map soil pollution"
Data
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Modis et al. "Determination of the optimum spatiotemporal sampling density to map soil pollution"
The Ptolemais lignite area

The
wider
lignite
opencast mining and
industrial
area
of
Ptolemais, is located in
western Macedonia, in
the northern part of
Greece.
Data
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Modis et al. "Determination of the optimum spatiotemporal sampling density to map soil pollution"
Lignite and Pollution


Some of elements contained in lignite (e.g. As,
Cd, Cr, Hg, Ni, Pb, Se, Bi and U) can be
potentially harmful to the environment.
Although these elements are present in small
concentrations in the coal basin (generally a few
ppm), the vast amount of coal that is burned
annually mobilizes tons of these pollutants.
Data
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Modis et al. "Determination of the optimum spatiotemporal sampling density to map soil pollution"
Sampling


During the period from
early 2003 to 2006, 101
soil samples from a set
of 48 locations in the
wider area were taken at
various time periods.
A number of yearly
samples are available for
each location, varying
from 1 to 4.
Data
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Modis et al. "Determination of the optimum spatiotemporal sampling density to map soil pollution"
Sampling

The samples were collected
from a depth of 10-30 cm
and analyzed for a variety
of characteristics, including
the presence of As, Cr, Ni,
Bi and other heavy metals.
Data
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Modis et al. "Determination of the optimum spatiotemporal sampling density to map soil pollution"
Allowable Concentrations

A first processing of the data reveals that
51% of the samples exceed the TAC for As,
14% for Cr, 19% for Ni and 61% for Bi.
mean
stdev
min
median
max
TAC
pollutant
Data
As
12.30
10.10
2.40
7.10
42.20
7.00
Cr
17.50
19.90
0.60
6.50
60.50
55.00
Ni
10.10
10.50
0.40
6.50
57.50
20.00
Bi
12.70
16.10
0.00
0.30
43,50
0,20
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Modis et al. "Determination of the optimum spatiotemporal sampling density to map soil pollution"
Results
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Modis et al. "Determination of the optimum spatiotemporal sampling density to map soil pollution"
Step 1. Structural analysis

Using the data derived
from the sampling
campaign concerning
As, Cr, Ni and Bi
concentrations,
the
covariance functions of
these variables were
calculated.
Results
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Modis et al. "Determination of the optimum spatiotemporal sampling density to map soil pollution"
Step 2. Sampling rate estimation


Covariance graphs show a spatial range of
influence from 3000 to 5000 m in all elements,
while the temporal range varies from 2 to 4
years.
Using our formula to estimate the optimal
sampling grid, we get a value of 1500 to 2500
m for the spatial part and 1 to 2 years for
temporal, according to model.
Results
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Modis et al. "Determination of the optimum spatiotemporal sampling density to map soil pollution"
Results


From the above analysis, it
is apparent that the area,
while
locally
over
sampled, is generally
under-sampled.
The optimum sampling
grid might have been
obtained with a more or
less similar cost.
Results
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Modis et al. "Determination of the optimum spatiotemporal sampling density to map soil pollution"
Results

From the temporal point of view, it seems that the
area is sufficiently sampled with the annual
samples.
Results
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Modis et al. "Determination of the optimum spatiotemporal sampling density to map soil pollution"
Conclusions
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Modis et al. "Determination of the optimum spatiotemporal sampling density to map soil pollution"
Conclusions


The design of the sampling program can be
formulated as an optimisation problem.
The selection of the appropriate sampling
network and grid size can contribute to save
money and time as well as to maximize the
obtained information.
Conclusions
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Modis et al. "Determination of the optimum spatiotemporal sampling density to map soil pollution"
Thank you!
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