Geology Reference
In-Depth Information
vectors could not be tested for signifi cance due
to different size.
In the case of the Arabian Gulf biostrome, the spa-
tial model was expressed by a regular Markov chain
with eight vertices and
d
= 1 (Fig. 5b). Each vertex
could be reached by every neighbour (or itself)
in one, or any multiple of one, steps. Translated
into time, this would assume total reversibility
of any process. However, this is known not to be
the case in many ecological and sedimentological
processes. The FPV only informs about the pro-
portional share of each state's
n
i
occurrence. To
transform the matrix of spatial adjacencies into an
ecologically and sedimentologically meaningful
model of system functioning in time, we had to
take some liberties.
The digraph of spatial transitions (Fig. 5b)
showed the two previously demonstrated func-
tional groups in the Arabian Gulf (Fig. 3) and the
Fenk outcrop: unconsolidated facies (sand and
everything that grows on sand, in our case algae
and seagrass) and consolidated facies (hardground
and everything that grew on it, such as algae, the
coral biostrome and its degradation products such
as dead corals and corals covered by algae). Within
the unconsolidated facies, sand was considered as
a central node from which seagrass and algae, that
also transited into each other, could be reached.
This was based on observations of seagrass inter-
weaved with dense algae fl uctuating in time and
space. The treatment of sand and its transition
into hardground was key in defi ning the nature
of the model. It is known (Shinn, 1969; Uchupi
et al
., 1996) that sand in the southern Arabian
Gulf rapidly cements into hardgrounds due to the
high alkalinity of the seawater. Once consolid-
ated, the sand turns to rock and will not return to
sand in any appreciable quantity, with the excep-
tion of parts that are ground-up to sand in higher-
energy environments. Thus, in an imaginary core
through this landscape, consolidated sand will
forever remain consolidated sand, since any cover
by mobile sand sheets would not be detectable in
the geological record. But in the active landscape,
hardground can very well be covered by sand and
in the spatial digraph (Fig. 5b) there is a strong
(i.e. bilateral) connection. This suggested two
treatments: fi rst considering the transition sand-
hardground reversible with the argument that
when a patch of hardground is covered by sand it
is functionally sand and can revert to hardground
when the sand is removed, in which case the
Markov chain remains regular. This could be con-
sidered a more biological line of argument since
sand cover will make habitat unsuitable for fauna
and fl ora that require hardground. Second the
transition sand-hardground could be considered
irreversible (the 'geological' line of argument,
because even if covered by a layer of sand, the
hardground itself remains hard), which makes the
Markov chain ergodic and the transient unconsol-
idated sediments eventually become absorbed into
the hardground-coral set. This allows exploration
of two Markov models: one regular, one ergodic.
When constructing the TPM of temporal
dynamics in the regular model, we reordered the
sequence of vertices to make the system easier
to understand (Fig. 9). The resulting matrix con-
sisted of four submatrices: the 'unconsolidated'
facies (B2) in the lower right, the 'consolidated
facies' in the upper left (B1), transitions from B2
into B1 in the upper right (B12), and transitions
between the upper left and lower right matrices
(Fig. 9b and c). Entries into the matrix needed
to be similar to those in the spatial transition
matrix with some leeway to change entries since
not all spatial transitions existed in the ancient
example. The loops (i.e. the self-transition prob-
abilities) were kept constant whenever possible
because the loops had the highest values in the
temporal matrix. Loops are powerful determi-
nants of facies extent and stability which were
not appropriate to change. The between-facies
transitions were adjusted according to what tran-
sitions were considered more, or less, likely.
The FPV of the regular Markov chain that
represented the temporal model of Arabian Gulf
facies transitions was calculated and compared
with the values of the pixel-count. No statisti-
cally signifi cant differences were found (
t
-test,
p
< 0.01; after positive test for normality;
Fig. 10). Thus, the FPV of the temporal model,
which consisted of spatial adjacency modifi ed
into assumptions of temporal transitions, corres-
ponded to the actual sizes of the facies. This may
appear circular logic, but to arrive at the temporal
model transition, probabilities were changed and
some even set to zero. This opened the possi-
bility that FPVs could differ. However, Fig. 10
demonstrates that the regular Markov chain model
of temporal functioning resulted in landscape met-
rics almost identical to those obtained from pixel
counting.
What if uncertainty existed about the
reversibility of the sand-into-hardground trans-
ition? To test this, the regular model was modifi ed
into an ergodic Markov chain (Fig. 2b). In an
ergodic chain, a set (the ergodic set) exists into
