Geology Reference
In-Depth Information
(a)
Holocene
Arabian Gulf
(b)
(c)
Holocene
Arabian Gulf
Fig. 9.
Model of the Arabian Gulf
biostrome expressed as a regular Markov
chain allowing transitions into all direc-
tions, thus consisting of a single ergodic
set. The sandy facies (set B
1
) are con-
nected in both directions via subma-
trices B
12
and B
21
to the hardground
facies (set B
2
). (a) Weighted digraph,
(b) structure of the matrix and (c) the
populated matrix. Note that the sequence
of vertices changed from Fig. 8 due to
rearrangement on the digraph.
H
S
S
A
which the chain can enter, but no longer leave.
It was assumed that all sand would eventually
become hardground and enter the coral loop (the
hardground-coral loop being the ergodic set), a
situation expected in a reef-building scenario.
While an FPV can be calculated for an ergodic
Markov chain, the
n
i
corresponding to the
transient states (facies) consist of zeros. Transient
states are those to which the chain cannot return
once it enters the ergodic set because no trans-
ition into their direction exists. They are absorbed
into the ergodic set without option to leave, thus
eventually decreasing their frequency to zero.
This suggests two things: (1) a different outcome
from the regular model since the entire loop of
sand-dependent facies will reduce to zero; (2) an
earlier stated restriction for the derivation of
the temporal model is violated, namely that the
FPV of the temporal model should be similar to
that of the spatial model. This constraint was
relaxed to allow exploration of the ergodic Markov
chain and required only all non-zero
n
i
of the FPV
to not signifi cantly differ from their counterparts
in the spatial FPV.
The original temporal TPM was rewritten in its
canonical form and the ergodic set reduced to a
single absorbing state (Fig. 11d and e) allowing
treatment as an absorbing Markov chain. The lat-
ter is an ergodic chain in which the ergodic set is
reduced to a single state. Once this state is entered,
no exit is possible - the chain is 'absorbed' into
it. From the canonical matrix 'landscape drift'
(in analogy to the 'ecological drift' of Hubbell
(2001)) was calculated, which is the number of
times the system resides in any transient state
(facies)
s
ij
before absorption:
Q
)
1
N
= (
I
where
I
is the absorbing matrix (an identity matrix
of the appropriate dimension corresponding to
Q
) and
Q
is the matrix of transient states.
is a
column vector of ones and the fi xation times (the
time until absorption - where time is measured in
matrix steps) were calculated as
ξ
(
N
) =
N
ξ
, with a variance of
Var(
) = (2
N
I
)
sq
