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signifi cance of differences between the matrices
was evaluated. The signifi cant differences
(
p
< 0.001) confi rmed the presence of the simple
Markov property. The expected TPM was raised
to higher powers (up to the fi fth) and again the
observed against the expected TPM was tested for
higher-order Markov property with Chi-square
tests that were signifi cant (
p
< 0.001). Since pixel-
size would exert strong control on any higher
order Markov property (the smaller the pixel
and the bigger the patches, the higher the order),
this was not investigated further. To account for
pixel-size effects, the matrix was also treated as
embedded Markov chain not allowing self trans-
itions and the Markov property was found again
(Chi-square test,
p
< 0.05).
Clearly, the quantitative expression of the
spatial facies neighbourhood pattern in the
Arabian Gulf was a regular Markov chain (
d
=
1
,
higher powers of the matrix without zero entries).
From this, a fi xed probability vector was found
(Kemeny & Snell, 1960; Horn, 1975; Roberts,
1976) because if
n
is a row vector and
P
is the
s
roughly matched by the frequency of facies seen
in outcrop, provided equivalent modern and
fossil systems that are time-transgressive were
found (i.e. comparable environments - it would
be nonsensical to compare environments that do
not have similar spatial or ecological dynamics).
So, if the conjecture that the FPVs of ancient and
modern vertically and horizontally evaluated
sedimentologically similar systems can be simi-
lar was true, similar transitions and total facies
frequencies should be found both in the Fenk
Quarry (Fig. 6) and the Arabian Gulf satellite
image.
Several caveats have to be applied to this
analysis. First, not all facies co-occurred in the
recent and the fossil landscapes, although many
did: dense corals (fossil equivalent - coral frame-
stone) and sparse corals (fossil equivalent -
coral/oyster fl oatstone, coral rudstone), as well
as the sandy facies (Arabian Gulf - sand, algae,
seagrass; fossil equivalent - corallinacean cal-
carenite). No equivalents existed in the modern
Arabian Gulf for the
Isognomon
oyster banks,
which acted as hardgrounds for coral settlement
(the only equivalent would be
Pinctada radiata
oyster beds within the Arabian Gulf algae facies,
however, recent oysters are much smaller than
those of the Miocene) and the rhodolith facies
(rhodoliths were uncommon in the recent study
area). It was possible to directly compare the
percentage contribution of the following facies:
dense coral (Miocene - framestone; recent - dense
live and dense dead coral), sparse coral (Miocene -
coral rud- and fl oatstones; recent - sparse corals),
sandy facies (Miocene - corallinacean calcarenite;
recent - hardgrounds, sand). Other facies correl-
ated approximately, i.e. the Miocene
Isognomon
oyster beds were included into the hardgrounds
because their dense shell-layers acted as hard-
ground and substrate for coral framework initia-
tion. The Miocene marls could be compared (with
some liberty) to the recent seagrass beds, which
usually occurred on mud or muddy sand.
Also sampling intensity of the outcrop and
grid-size of the two-dimensional outcrop map is of
equal importance as pixel-size in the landscape.
We sampled the image at 10 cm intervals (image
gridded to 10 cm pixels), since no signifi cant dif-
ferences (ANOVA,
F
= 2.43, df = 5.42,
p
> 0.05)
existed between the FPVs obtained from sampling
at 50, 20, 10, 5 and 2 cm. Since we counted the
transitions in the Moore neighbourhood of each
pixel, increased within-facies (self-) transitions
were balanced by equally increased among-facies
s
transition matrix,
n
(
t
+ 1) =
n
(
t
)·
P
, and after
m generations
n
(
t
+
m
) =
n
(
t
)·
P
m
. As
m
increases,
n
converges to a stable distribution, the fi xed
probability vector
n
*, which is the solution of
s
linear equations
n
* =
n
*·
P
.
The fi xed probability vector (FPV) expresses
the likelihood with which every state is encoun-
tered independent of the state the Markov chain
is started in (Roberts, 1976). If the fi xed probabil-
ity vector expresses the likelihood of successional
stages occurring in space, it could also express
their likelihood of occurring through time. If a
point is twice as likely to fall within stage A (or
facies A), because A is twice the size of stage
(facies) B (to be precise: stage A's spatial expres-
sion in facies A is twice the size of stage B's spa-
tial expression in facies B), then through time,
everything else being equal, a point will also prob-
abilistically be encountered twice more often in
stage (facies) A than in stage (facies) B. The Law of
Large Numbers for regular Markov chains defi nes
the FPV as representing the fraction of time that
the process can be expected to be in state
s
j
for
a large number of steps (Kemeny & Snell, 1960).
From this it is proposed that the spatial transi-
tions in any landscape could be useful as a proxy
for the temporal transitions and vice versa.
If this were indeed so, the frequency of the
facies (which is equivalent to the number of
transitions into and within a facies) visible in the
horizontal modern landscape should be at least
