Plate Motions - Quantitative Plate Tectonics - page 61

frame, to the position that this block had at time T
in a paleotectonic reference frame . An important
example of this class of frames is represented by
the paleomagnetic frames mentioned in Sect. 2.3 ,
but one could wish to refer the reconstructions
to a set of hot spots (e.g., Müller et al. 1993 )
or even use a present day continent, for example
N. America or Eurasia, as a reference frame
for plate reconstructions. The existence of total
reconstruction matrices is again a consequence of
Euler's theorem. The corresponding Euler pole is
called a total reconstruction pole .Atstep#1of
Algorithm 2.1, the total reconstruction matrix is
initialized by the identity matrix I , and the current
node, c , is set to be the starting node. At step #2,
avariable p is assigned the parent of the current
node in the tree structure. At the next step, if the
current node c coincides with the root of the tree,
so that p D 0, then the iteration stops and the final
reconstruction matrix is updated by adding the
transformation of the root node with respect to the
paleotectonic reference frame, R c . At step #4, the
current rotation matrix is updated by adding the
relative rotation of the current node with respect
to its parent. Then, at the next step, we move
upwards to the next higher level by assigning
the current node its parent and the sequence is
restarted. On exit, this algorithm furnishes the
total reconstruction matrix of n at time T in the
variable R n ( T ).
As an example, the application of this algo-
rithm to the circuit of Fig. 2.28 would give the
following total reconstruction matrices:
8
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where the rotation S ij is calculated using the
reduced angle (Eq. 2.41 ). Now we can address
the problem of complementing the kinematic
representation of a set of tectonic plates through
velocity and acceleration fields. We know that the
linear velocity v ( r ) at the location represented by
a position vector r can be calculated easily start-
ing from an Euler vector ¨ (Eq. 2.17 ). Therefore,
the problem of representing velocity fields can
be reduced to the problem of determining the
instantaneous axis of relative rotation between
two plates sharing a boundary at time T , indepen-
dently from the eventuality that these are conju-
gate plates or not. Furthermore, it is occasionally
necessary to determine absolute velocity fields
in the selected paleotectonic reference frame.
Clearly, in the case of relative velocity fields
between conjugate plates the calculation should
be simplified by the fact that the relative motions
are rotations about fixed axes at constant angular
velocities. However, even in this eventuality it is
necessary to take into account that the rotation
axis of a stage pole is fixed with respect to a
plate that is considered at rest in the present
day geographic frame. Therefore, the axis must
be rotated according to the total reconstruction
matrix of this plate at time T before it can be used
for calculating velocity vectors. Let n ij (0) be the
unit vector of the rotation axis associated with a
stage rotation S ij ( T k 1 , T k ). If T k 1 T T k ,and
R j ( T ) is the total reconstruction matrix of the
reference plate at time T , then the orientation of
this axis at time T will be given by:

R A .T/ D R C .T/R AC .T/
R B .T/ D R C .T/R BC .T/
R D .T/ D R C .T/R DC .T/
R E .T/ D R C .T/R DC .T/R ED .T/

n ij .T/ D R j .T/n ij .0/ (2.48)

: At this point, to form a complete Euler vector

we still need to assign an angular velocity ¨ at
time T . This task can be easily accomplished,
because during a stage the relative angular veloc-
ity between two plates is assumed to be approx-
imately constant, thereby we can always deter-
mine this quantity starting from the stage angle
k and the temporal boundaries T k 1 and T k .It
results:

To calculate the set of finite reconstruction
matrices R ij ( T ) associated with a plate circuit at a
given intermediate time T , algorithm 2.2 uses the
components of these transformation matrices at
stage boundaries. If T k 1 T T k , then the cor-
responding finite reconstruction of plate i relative
to plate j is given by:

k
T k T k1

¨ D (2.49)

R ij .T/ D S ij .T k1 ;T/R ij .T k1 / (2.47)