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where the facies are compatible. No signifi cant
differences existed between recent and Miocene
distributions (
t
-test,
p
< 0.001, after it tested posit-
ively for normality).
(a)
Holocene Arabian Gulf
Algae on
coral
Dense dead
coral
Seagrass
The space-time bridge
Based on Fig. 7, the patterns of spatial and tem-
poral facies successions (encapsulated in relative
facies frequencies as expressed by the FPV) were,
at least in the chosen example, very similar. To
create a model of how facies transitions work in
time and to evaluate resulting facies frequencies
if transition probabilities were changed, either
the vertical or the horizontal transition matrix
could be chosen for the construction of the model.
Because of better control and more informa-
tion about the living spatial structure, the recent
landscape was chosen to start with. A temporal
transition digraph was constructed from the
spatial adjacency digraph by modifying in- and
outdegrees whenever certain transitions were only
possible in space but not in time (or vice versa,
if one wanted to use the temporal transitions to
model spatial transitions). For example, sand
may directly neighbour dense coral horizontally
and vertically (Fig. 5b), but a successional trans-
ition from sand directly into dense coral is not
possible in a single step. Rather, sand has to fi rst
pass through hardground and sparse coral stages
(Fig. 8a and b).
Constraints for the temporal Markov model
were: (1) use of the same vertices (facies) in the
temporal digraph (matrix) as in the spatial digraph
(matrix) and (2) maintenance of the same num-
ber of vertices (= facies). Accuracy of the derived
model was assessed by (3) the FPV, which defi nes
the likelihood of facies encounter irrespective of
starting state. Therefore, differences between the
FPV from the temporal digraph (matrix) and the
actually observed size distribution of stages or
facies obtained by pixel-counting the classifi ed
satellite image should not be signifi cant. The FPV
of the temporal TPM should correspond with the
FPV of the horizontal (living, landscape) and ver-
tical (ancient, outcrop) TPMs.
If either the temporal or spatial model
was assumed to be more complex than the
other, then conditions (1) and (2) could be
relaxed to let the same vertices (facies) occur
but to allow additional vertices (facies) in the
matrix and FPV. The corresponding FPVs had
to conform to (3), i.e. not differ in correspond-
ing facies. In such a case, differences between
Dense live
coral
Sparse
coral
Algal lawns
(b)
Holocene Arabian Gulf
Algae on
coral
Dense dead
coral
Seagrass
Dense live
coral
Sparse
coral
Algal lawns
(c)
Sparse
oyster/coral
Sparse
coral
Marl
Sand
Oyster
Dense
coral
Rhodolith
Sparse coral
on sand
Fig. 8.
Construction of the temporal digraph from
the spatial digraph, obtained from the spatial TPM of
Arabian Gulf and Leitha Limestone facies (Fig. 5b). Edges
between vertices were deleted if a facies transition could
not occur (i.e. it calcarenite could not turn into sand
again). Transition probabilities are adjusted to maintain
row summability to unity. (a) A regular Markov chain
of the recent Arabian Gulf situation, and assumes that
sand can turn into hardground, which in turn can tran-
sit into sand again. (b) An ergodic Markov chain with a
transient set in the sandy facies (8,5,4) and an ergodic set
in the hardground facies (7,3,1,2,6). This model assumes
that once sand cements into hardground, it will never
turn into sand again. (c) The regular Markov chain rep-
resentation of facies dynamics in the Miocene Leitha
Limestone.
