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would be necessarily non-random and while it
would be a good tool to model landscapes from
cores and vice versa, it would not be an adequate
forward modelling tool for questions of order
and noise in sequences (Wilkinson
et al.
, 1997,
2003; Lehrmann & Goldhammer, 1999; Rankey,
2004; Burgess, 2006).
When using the derived temporal models to
predict changes in facies, as for the Landsat scene
in rising/falling sea-level scenarios presented
here, the fi xed probability vector gives informa-
tion about the contribution of facies in the new
environment but not about the time needed to
achieve this. It is possible to evaluate how many
matrix time-steps are needed (i.e. to what power
the TPM needs to be raised) until a distribution
of facies close to that of the FPV is obtained.
Using an absorbing model, it has been shown
that it is possible to calculate time until absorp-
tion of the transient facies (Fig. 11). However, the
timescale is always one of matrix multiplication
steps and careful calibration of a matrix step's
meaning in years is required. Such information
could be obtained in outcrop by dating the length
of time required for a facies transition predicted
from the spatially derived model. Here, no cores
through the living Arabian Gulf environment
were available, nor were reliable dating from the
Miocene outcrop and therefore time information
was lacking. Clearly more work on this aspect is
required.
The depth of sea-level fall and height of sea-
level rise need to be encoded in the transition
probabilities. Here, assumptions of sea-level rise
and fall were small, in the range of 1.5-2 m. The
biological, and to some degree the sedimentary
environments encountered on the banks studied are
relatively sensitive depth indicators and occurred
within relatively specifi c depth ranges (Figs 3
and 12). Therefore, they could expand and restrict
at the cost of each other and no new facies had
to be introduced. Models such as presented
here are limited to situations without dramatic
changes in facies arrangement and composi-
tion and require at least a certain facies-depth
relationship, which is not always unequivocal
(Rankey, 2004).
The application of Markov chains to sediment-
ary systems or to ecological succession is not new
(Krumbein, 1967; Pielou, 1969; Schwarzacher,
1969; Horn, 1975; Usher, 1979), and its concep-
tual link to Walther's Law for the forward mod-
elling of sedimentary landscapes has also been
noted before (Doveton, 1994; Parks
et al.
, 2000;
Elfeki & Dekking, 2005). What is new in this
analysis is the addition of graphs for the easy
visualization of functional pathways within the
system and derivation of a spatial model from
the temporal - or vice versa. This allows ecology
and sedimentology to be linked, since heed can
be paid to ecological dynamics that may need to
be introduced into the derived models of facies
dynamics via modifi ed transition probabilities.
This is an important new step to the existing tech-
niques used for forward modelling with Markov
chains - the visualization of spatial relationships
and the derived temporal functioning in graphs.
Rather than adapting the transition matrix via
weightings by scaling factors (Schwarzacher,
1969; Parks
et al.
, 2000) the graph-evaluation step
allows a quite realistic derivation of a temporal
matrix (or spatial matrix, depending on starting
point) from the spatial matrix. The fi xed point
vector is the link between the spatial and tem-
poral models and any modifi cation must result in
statistically similar FPVs. This essentially makes
the fi xed point vector a quantitative expression
of Walther's Law and the unifying link between
(matrix - and thus its encoded facies) space
and time.
ACKNOWLEDGEMENTS
We thank R.N. Ginsburg for being an inspiration
and an enthusiastic as well as patient teacher. He
planted more ideas in our minds than we our-
selves are aware of. He taught us how to think,
ask questions, and keep an open, yet critical,
mind. His contribution to our science is unri-
valled. And by making us ask the 'so what?'
question after every major thought or posit, he
made us go and think further and better than
we would have otherwise. Work on this paper
was sponsored among others by the Austrian
Science Foundation (FWF), Austrian Geological
Survey, the Dolphin Energy/WWF Coral Reef
Project, the Ocean Drilling Project and NOAA
grant NA16OA1443 to NCRI at NSU. E. Rankey
contributed many useful thoughts, discussions
and carefully edited the manuscript. W. E. Piller
helped with measured sections and evaluating
the Fenk outcrop. We thank M. Chandler, G. Rae,
R. al Mubarak, Th. Abdessalaam, A. al Cibahy
and F. Launay for unwavering support during
work in the Gulf. This is NCRI publication num-
ber 83 and a contribution of the NCRI Monitoring
Network.
