Page 6 - i1052-5173-28-6
P. 6

log Exceedence       A            Intercept = 7.7                                                             composite; another element is then
                                                                                                              selected and randomly divided. Because
                1.0                                 B                                                         some minimum area should serve to sepa-
                                                                                                              rate lithospheric “plates” and smaller
                0.6 Slope = -0.21; R2 = 0.97             Sampled Model                                        structural elements, we assume a mini-
                0.2 Slope = -2.31; R2 = 0.95                                                                  mum area of 4,000 km2, about half the size
                                                                                                              of the smallest (Manus, 8,117 km2) plate in
                      5.5 6.0 6.5 7.0 7.5 8.0            5.5 6.0 6.5 7.0 7.5 8.0                              the Bird (2003) database. Moreover, owing
                                                                                                              to constraints imposed by length scales of
                                              log Area (km2)                                                  mantle convection (e.g., Lenardic et al.,
                                                                                                              2006), we assume a maximum plate area
                1000    C   Data “smaller plate” slope                                                        of 200 × 106 km2, about twice the area of
                 800        Data “larger plate” slope                                                         the largest (Pacific, 104 × 106 km2) plate.
Frequency        600        Model all regression slopes                                                       Given these two constraints, repeated
                                                                                                              annealing and division of members of the
                400                                                                                           population rapidly results in model size
                                                                                                              frequencies that are both stable with
                200                                                                                           respect to numbers of iterations and indis-
                                                                                                              tinguishable from the observed frequency
                      0.60  0.66  0.72 0.78 0.84                 0.90   0.96                                  distribution of modern plate areas (Fig.
                                                                                                              1A). The range of permissible area fre-
                       D                     Log-log Slope R2                                                 quencies afforded by this simple model of
                                                                                                              repeated random annealing and fragmenta-
Frequency       600                                      Data di erence in slopes                             tion completely overlaps the observed sizes
                500                                      Model di erences in slopes                           of Bird’s (2003) 52 plates.
                400
                300                                                                                           VERACITY OF PLATE
                200                                                                                           SUBPOPULATIONS
                100
                                                                                                                A single “broken sheet” hypothesis for
                      0 0.4 0.8 1.2 1.6       2 2.4 2.8 3.2 3.6         4 4.4                                 the generation of a continuum of plate
                                                                                                              sizes must also account for the widely held
                                  Differences in log-log Slopes                                               perception that plate areas somehow com-
                                                                                                              prise two or more subpopulations, each
Figure 2. Correlations and slopes of apparent linear trends in measured and model plate area fre-             scaled to some distinct tectonic processes.
quencies. (A) Plate areas from Gurnis et al. (2012) exhibiting an apparent inflection in slope at log         We suggest that what appear to be “popu-
area ~7.7 (~50 × 106 km2); slope difference is 2.11. (B) Two model area frequency distributions, each         lation-specific” segments in log-size ver-
comprising 20 randomly delimited plate areas with a total of 510 × 106 km2; red and blue lines repre-         sus log-exceedance plots (Fig. 1) are no
sent the two best-fit log-linear regressions that account for the largest amount of plate size vari-          more than coincidental trends in a sparsely
ance. (C) Frequency distribution of R2 values of 1,000 models of 20 randomly delimited plate areas            sampled continuum of plate areas. Two
(light yellow bars) compared to R2 values of smaller (red bar, red line in [A]) and larger (blue bar, blue    issues are relevant to the veracity of divid-
line in [A]) “populations” in the Gurnis et al. (2012) data. (D) Frequency distribution of apparent dif-      ing and interpreting curvilinear log-log
ferences in slopes of “smaller” (e.g., red lines in [B]) and “larger” (e.g., blue lines in [B]) plate areas   data arrays on the basis of apparent
among 1,000 models of 20 randomly delimited plate areas (tan bars) compared to that defined by                straight line segmentation. First, any
smaller (red line in [A]) and larger (blue line in [A]) plate “populations” in the Gurnis et al. (2012) data  model that includes a greater number of
(brown bar). Note that area-exceedance correlations of “small” and “large” plate areas in the                 subdivisions and a greater number of
observed data as well as differences in these slopes all fall well within the range of values expected        parameters (each line segment being
for the sparse sampling of a continuous broken sheet distribution of plate areas.                             described by some slope and intercept) will
                                                                                                              certainly result in better agreement with
frequency distributions and those observed    boundaries. Modern plate size frequencies                       data than one with fewer parameters (only
among measured plate areas; and (2) yield     are a snapshot of the time-integrated geo-                      the number of plates and total area com-
results that are in agreement with the        logic histories of the growth and decline                       prise the broken sheet representation).
apparent grouping of plate areas into the     in the numbers and sizes of all constitu-                       However, benefits from increases in good-
several subpopulations based on apparent      ents of the global plate population (e.g.,                      ness-of-fit are balanced by costs in model
linear trends in log-log plot of area versus  Morra et al., 2013).                                            complexity (e.g., Akaike, 1974), and
exceedance (e.g., Fig. 1). With respect to                                                                    greater numbers of model parameters run
differences between theoretical and             A straightforward model of such pro-                          counter to the heuristic perception that
observed plate areas, it seems apparent       cesses might simply presume that the                            simpler is better. Furthermore, any array
that increases in the size of any particular  observed lithospheric plate area frequency                      representing some sparsely sampled curvi-
plate might occur fairly continuously         distribution is a natural consequence of                        linear distribution will unavoidably exhibit
through marginal accretion during sea-        both the random division and random
floor spreading or more abruptly during       annealing of members of some initial pop-
the development of tectonic sutures at        ulation of plate areas. We effect such a
convergent margins, and that decreases        simulation with a population of n = 52
might occur continually during subduc-        plates (e.g., Bird, 2003), each with an ini-
tion, or relatively episodically during the   tial area of 9.8 × 106 km2 (A = 510 × 106
development of rifted or transform            km2). From this group, one pair is selected
                                              at random and annealed into a single

6 GSA Today | June 2018
   1   2   3   4   5   6   7   8   9   10   11