Geochemistry, Geophysics, Geosystems
Supporting Information for
Constraints on shear velocity in the cratonic upper mantle from Rayleigh wave phase
velocity
Aaron C. Hirsch
Department of Earth and Environment, Boston University, 675 Commonwealth Avenue, Boston,
MA 02215, USA. Now at Graduate School of Oceanography, University of Rhode Island, 215 S.
Ferry Road, Narragansett, RI 02882, USA (aaron_hirsch@my.uri.edu)
Colleen A. Dalton
Department of Earth, Environmental and Planetary Sciences, Brown University, 324 Brook
Street, Providence RI 02912, USA (colleen_dalton@brown.edu)
Jeroen Ritsema
Department of Geological Sciences, University of Michigan, 1100 North University Avenue, Ann
Arbor, Michigan 48109, USA (jritsema@umich.edu)
Contents of this file
Figures S1 to S6
Description of using misfit-based criterion to identify acceptable models
Figures S7 to S11
Introduction
Included are additional figures that expand graphically on the results from the main body of text.
This includes the geotherm model variables that represent an acceptable fit, geotherm generated
forward model results of specific surface heat flow values, shear velocity minimums from the
geotherm-generated results, and model results of assumption testing. A section, with five
additional figures, that describes results obtained using a misfit-based selection criterion rather
than an error-bar-based selection criterion is also included here.
1
Figure S1. (a) GTR1 regions [Jordan, 1981]. Region “S” (Precambrian cratons) is shown in red,
the “P” and “Q” continental regions have been grouped together and are colored yellow, and the
three oceanic regions have been grouped together and are colored blue. White lines connecting
white triangles show the ten locations studied by Lebedev et al. [2009], and white polygons show
the approximate areas of focus by Pedersen et al. [2009]. (b) For comparison, the regionalization
scheme of CRUST1.0 [Laske et al., 2013], with the three Archean and one early/mid Proterozoic
region colored red, the remaining continental regions colored yellow, and the oceanic regions
colored blue. (c) Hit count map, in 5ox5o cells, calculated using minor-arc great-circle paths from
100-s Rayleigh wave data set of Ritsema et al. [2011].
2
Figure S2. (a) Values of mantle potential temperature and surface heat flow used for the
calculation of cratonic geotherms, which yield 1-D Earth models that provide an acceptable fit to
the observations. Color scale shows the number of acceptable models for each combination of
parameters. (b) As in (a) but showing values of bulk crustal heat production.
3
Figure S3. (a) Rayleigh wave phase velocity for 3750 a priori geotherm-generated models with
surface heat flow = 25 mW/m2. (b) As in (a) but for 3750 models with surface heat flow = 35
mW/m2.
4
Figure S4. Red curves show the normalized histogram of the depth of the upper-mantle
minimum in shear velocity for all 26,250 geotherm-generated Earth models tested. Blue curve
shows the normalized histogram for the 165 a posteriori models.
5
Figure S5. A posteriori distribution of the randomly generated models with perturbations
confined to the depth range 37-400 km with magnitude < 30%. (a) For the 10,000 a priori
models from which this a posteriori distribution is selected, VPV and VPH are set equal to 1.85
times VSV and VSH, respectively. (b) For the 10,000 a priori models from which this distribution
is selected, the VP/VS ratio is set equal to its value in reference model STW105. The VS values
corresponding to the maximum of the distribution are shown for scenario (a) in white and for
scenario (b) in yellow. (c) For the 10,000 a priori models from which this distribution is selected,
the perturbations to STW105 are generated using a degree-8 polynomial instead of a degree-4
polynomial. Cyan curve shows maximum of distribution.
6
Figure S6. (a) Voigt-averaged shear-velocity profiles for the 1275 geotherm-generated models
that provide a satisfactory fit to the observations. Unlike the profiles shown in Fig. 6a, for which
the ratio VP/VS = 1.85, the Earth models from which these profiles were selected have VP/VS =
1.73. Thick green curve shows the average of these profiles and thick blue curve shows the
average of acceptable geotherm-generated models with VP/VS=1.85. (b) As in Fig. 6a. Thick
blue and green curves as described in (a). (c) Phase-velocity curves corresponding to the shearvelocity profiles in (a).
7
Using a misfit-based criterion to identify acceptable models
Our preferred criterion for identifying successful models is to require that the predicted Rayleigh
wave phase velocity at all periods falls within the error bars determined from the observations. In
this section we describe the effect on our results if we use a different selection criterion. In this
case, we identify successful models on the basis of chi-squared misfit:
𝑁𝑃
𝜒2
1
1
2
=
∑ 2 [𝑐𝑖𝑜𝑏𝑠 − 𝑐𝑖𝑝𝑟𝑒𝑑 ]
𝑁𝑃 𝑁𝑃
𝜎𝑖
𝑖=1
where NP is the number of periods (here, 16), σi is the observational uncertainty for the ith period
(here, the calculated standard deviation), and ci is phase velocity, with superscripts obs and pred
to indicate observed and predicted values, respectively. The center of the observed range is used
for the ciobs values. Any model that yields values of χ2/NP ≤ 1 is considered successful.
This misfit-based selection criterion identifies all of the models identified by our error-bar-based
selection criterion plus additional models not identified by our error-bar-based selection criterion.
Table S1 lists the number of additional successful models identified by the misfit-based criterion
for each of the four depth-range scenarios, with maximum deviations of 20% and 30% treated
separately, each of which consists of 10,000 trial models.
Table S1.
Depth-range scenario
37-200 km
20% deviation
922
30% deviation
486
37-300 km
516
277
37-400 km
273
159
37-500 km
168
94
Figs. S7-S10 summarize the characteristics of the Earth models are identified as successful using
the misfit-based criterion but not the error-bar-based criterion. When the perturbations are
confined to 37-200 km, the misfit-based criterion allows many additional models that fall below
or above the observed range at periods ≤ 50 s (Fig. S7a,b). A number of models also fall below
the observed range at intermediate periods (60-175 s). In terms of isotropic shear velocity, the
misfit-based criterion permits values smaller and larger by <0.2 km/s at all depths (but especially
at depths < 75 km) than the range defined when the error-bar-based criterion is used (Fig. S7c).
Similar conclusions are reached when the perturbations are confined to 37-300 km; this scenario
also permits additional models that exceed the observed range at periods ≥ 175 s (Fig. S8). As a
result of these models being included, the misfit-based criterion permits higher isotropic shear
velocity by <0.3 km/s at depths > 150 km than the range defined when the error-bar-based
criterion is used. When the perturbations are confined to 37-400 km and 37-500 km a slightly
different trend emerges. Models exceed the observed range at the shortest and longest periods,
and models fall below the observed range at mostly intermediate periods (Figs. S9 and S10). This
results in some shear-velocity values lower by ~0.1 km/s at depths 100-250 km and higher by
<0.2 km/s at depths > 250 km than the range defined when the error-bar-based criterion is used.
We apply a k-means clustering scheme to the suite of successful models generated using the
misfit-based criterion and compare the profiles identified from this process with those identified
when using the error-bar-based criterion (Fig. S11). The cluster profiles identified when using
8
the misfit-based criterion do not differ significantly from those identified when using the errorbar-based criterion for the 37-200 km and 37-300 km depth-range scenarios. The profiles for 37300 km achieve slightly higher shear velocities at depths > 150 km when the misfit-based
criterion is used, in agreement with the distribution of individual shear-velocity profiles (Fig.
S8c).
The cluster profiles for the 37-400 km depth-range scenario are shown in Fig. S11c. With the
misfit-based criterion, three of the four cluster profiles contain a low-velocity zone (at depths of
~150 km, 200 km, and 300 km). The profile containing the shallowest low-velocity zone (150
km) does not have an obvious counterpart in the error-bar-based cluster profiles and presumably
appears as a result of the lower shear-velocity values permitted by the misfit-based criterion in the
depth range 100-250 km. A similar conclusion is reached for the 37-500 km scenario.
To summarize this section, identifying acceptable models on the basis of the summed misfit over
all periods instead of the requirement that predicted phase velocity falls within the error bars of
the average cratonic value permits a larger number of successful models. Predicted phase
velocity for these models tends to fall below the observed range at intermediate periods and fall
above the observed range at the shortest and longest periods. As a result, some successful Earth
models contain lower shear velocity at depths 100-250 km and/or higher shear velocity at depths
< 80 km or depths > 250 km than the error-bar-based models. However, most of the successful
Earth models identified with the misfit-based criterion and not the error-bar-based criterion depart
from the range defined by the error-bar-based models for only a small portion of the total depth
range. Using the misfit-based criterion does not identify, for example, an entirely new class of
Earth models that the error-bar-based criterion rejects. Instead, the misfit-based criterion results
in only minor adjustments to the characteristics of the suite of successful Earth models.
Conclusions drawn about the thermal structure of the cratonic upper mantle should therefore not
be sensitive to the choice of selection criterion.
Figure S7. Characteristics of Rayleigh wave dispersion curves for Earth models that are
identified as successful using the misfit-based criterion and not using the error-bar-based
criterion. For the 37-200 km depth-range scenario, (a) Predicted Rayleigh wave phase velocity
plotted relative to the center of the observed range. Thick black curves show the full observed
range. Curves that fall below the observed range at at least one period are colored green; curves
that never fall below and at some period exceed the observed range are colored blue. (b)
Percentage of models in (a) with phase-velocity values that fall above or below the observed
range at each period. (c) Voigt-averaged isotropic shear velocity. Models are colored according
to the scheme in (a). Thick black curves define the range of successful shear-velocity values
identified using the error-bar-based criterion (i.e., Fig. 7a).
9
Figure S8. As in Fig. S7 but for the 37-300 km depth-range scenario.
Figure S9. As in Fig. S7 but for the 37-400 km depth-range scenario.
Figure S10. As in Fig. S7 but for the 37-500 km depth-range scenario.
10
Figure S11. As in Fig. 10, result of applying k-means cluster analysis to the acceptable randomly
generated models. (a) 37-200 km; (b) 37-300 km; (c) 37-400 km; (d) 37-500 km. Each profile is
an average of the individual profiles that belong to each cluster. Solid curves: cluster profiles
determined from successful models identified with misfit-based approach. Dashed curves: cluster
profiles determined from successful models identified with error-bar-based approach.
11