Geology Reference
In-Depth Information
i.e. how many paths over how many vertices
lead from any one vertex to another) (Roberts,
1976). This paper is concerned only with weighted
adjacency matrices, which is the transition fre-
quency/probability matrix of our Markov model.
Graph
Digraph
Processing steps to obtain neighbourhood
statistics
Weighted digraph
Signed digraph
Maps derived from the satellite images and the
outcrop photograph (gridded to a pixel-resolution
of 10
10 cm) were evaluated by counting the
Moore neighbourhood (all eight pixels adjoin-
ing any given pixel) of every pixel. Pixels were
colour coded according to the image classifi cation
and therefore each pixel class corresponded to a
facies. Only at the edges of the outcrop or satel-
lite image, or next to masked areas, could pix-
els neighbour 'nothing' (white or black on the
image), so all 'colour-to-white/black' adjacencies
were excluded from further analysis, since they
were only relevant to the specifi c outcrop/satel-
lite image situation, but cannot be generalized.
Throughout this paper, facies remain encoded by
pixel-classes or pixel-colours and the terms can be
interchanged.
From the raw counts of neighbourhood fre-
quency, the transition frequency matrix that
was then transferred into a transition probabil-
ity matrix was populated. In order to decide an
adequate gridding size for the outcrop photograph,
it was gridded to several sizes, the transition
probabilities were evaluated by neighbour count-
ing, the transition probability matrices were
made and their fi xed probability vectors com-
pared. Differences in fi xed probability vector
meant that facies had been lost due to too coarse
gridding.
Fig. 2.
Examples of graphs defi ned by the sets of
p
= 5
vertices {a,b,c,d,e} and
q
= 5 edges {ab,bc,cd,de,eb}. The
digraphs identify an ergodic set {b,c,d,e}. An ergodic set is
a combination of vertices from which, once entered, there
is no escape since no paths lead out of it. Therefore, once
the ergodic set is entered, the transient states outside this
set will never again be visited.
Graph theory
Graph theory, like fi nite Markov chains, is also
fi rmly founded in discrete mathematics and has,
since its invention by Euler in 1736 to settle the
famous Königsberg Bridge problem, been used
widely in the natural and engineering sciences
(Roberts, 1976). It is in wide use in population
(Caswell, 1982, 2001; Wardle, 1998) and land-
scape ecology (Cantwell & Forman, 1993; Urban &
Keitt, 2001; Urban, 2005). Graphs allow a con-
venient visualization of any type of problem that
has a spatial or temporal component that can be
visualized as states being connected by transitions.
Due to this property, graph theory is also closely
connected to matrix theory, and all graphs can be
conveniently expressed as matrices. Each facies is
expressed in a graph (Fig. 2) as a circle, called a
'vertex' or 'node'. The fact that facies neighbour
each other, or transit into each other, can be shown
on the graph by a line, or 'edge' that connects ver-
tices. These connections can be given a direction
in a digraph (short for directed graph), be assigned
weights (such as in the presented case here tran-
sitional probabilities) in a weighted digraph, or
be simply assigned positive or negative signs in
a signed digraph (Fig. 2). Each graph has an adja-
cency matrix (indicating the number of paths
between the vertices), a reachability matrix (indic-
ating the linkage between vertices, i.e. whether
any paths exist between vertices) and a distance
matrix (indicating the distances between vertices,
ANALYSES AND RESULTS
Distribution of facies in the recent Arabian Gulf
and comparison with Miocene Leitha Limestone
The classifi ed Ikonos imagery of the Jebel Ali
carbonate ramp draped over the digital elevation
model (Fig. 3) shows three shore-parallel zones. In
the fi rst 500 m from shore occurs a zone without
coral growth, instead dominated by sand, seagrass
and algae. A middle zone from 500 to 1000 m off-
shore consists mainly of sand and hardground. A
deep zone is characterized by coral growth of vari-
able density, interspersed by dense algae.
