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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