57. Bird, P., Z. Liu, & W. K. Rucker (2008) Stresses that drive the
plates from below: Definitions, computational path, model optimization, and
error analysis, *J. Geophys. Res., 113*(B11), B11406,
doi:10.1029/2007JB005460, plus digital appendices.

We divide the torques on each
surface plate into 3 parts: lithostatic-pressure, side-strength, and
basal-strength. We compute each part for each of 52 plates using a thin-shell
finite-element model of the lithosphere with: topography, variable heat-flow, variable
crust and lithosphere thicknesses from seismic data, transient geotherms,
nonlinear rheology, and weak faults. We present an iterative method of
adjusting boundary conditions that results in correct plate velocities without
the need for models of deep mantle flow. Uncertainty remains because
side-strength torques, and therefore inferred basal-strength torques, depend on
the effective friction of faults. Therefore, we compute a two-parameter suite
of models with differing trench resistance and differing fault friction, and
evaluate their misfits relative to: seafloor spreading rates, geodetic
velocities, intraplate stress directions, and azimuths of seismic anisotropy.
The minimum misfit occurs at effective fault friction of 0.1 and trench resistance
2´10^{12} N/m. In this
preferred model, computed values of mean basal-strength traction systematically
increase for smaller plates. We analyze error sources and find that the
largest source is unmodeled variation in effective friction of plate-boundary
faults. Discounting highly uncertain results, we find mean basal shear
tractions of no more than 1 MPa for the 6 largest slabless plates: AF 0.2 MPa;
AN 0.1 MPa; NA 0.6 MPa; EU 1.0 MPa; SA 1.0 MPa; SO 0.9 MPa. The directions of
basal shear traction on these plates are generally forward, meaning subparallel
to absolute velocity. Basal-strength torques on plates with subducting slabs
represent the sum of net slab-pull and distributed basal shear traction; if
these torques are attributed to net slab-pull alone, net slab-pull is generally
toward the trench and of order 5´10^{12}
N/m. Thus, present plate motions on Earth appear to be driven primarily by
deep mantle convection, rather than by topography and associated lithostatic
pressures.

Figure 1. Mercator projection of most of the Earth5 thin-shell spherical finite-element grid used in this study. Green lines are continuum element boundaries; largest elements have 240-km sides. Red lines with dip marks are fault elements, along plate boundaries and within orogens: no tick = 90° dip; straight tick = 55°; box = 45°; open triangle = 20°; filled triangle = 14° (subduction zone).

Figure 2. Close-up view of the Aegean-Persian-Tibetan portion of the finite element grid of Figure 1, superimposed on a topographic base map with shaded relief. Fault symbols as in Figure 1.

Figure 3. Composite
heat-flow model obtained by merging age-dependent model heat-flow from *Stein
& Stein* [1992] (where seafloor age is known from *Mueller et al.*
[1997]) with interpolated heat-flow kriged from data of *Pollack et al.*
[1991, 1993] and/or published maps in some continental areas. Color scale is
selected to show detail in continental areas of relatively low heat-flow. All
spreading ridges were assigned uniform 0.3 W/m^{2} conductive heat flow
along their centers.

Figure 4. Travel-time
anomalies for vertically-incident S waves traveling though the upper mantle
above 400 km depth, from the S20RTS model of *Ritsema
& Van Heijst* [2000].
Corrected for variations in crustal structure, to give the upper-mantle
anomaly.

Figure 5. Model of total lithosphere thickness. A composite of continental thicknesses scaled from vertical-S-wave upper-mantle travel-time-anomalies (in Figure 4), and an age-dependent model in the ocean basins. Some extreme values were further adjusted to prevent model geotherms from having maxima higher than the assumed asthenosphere temperature.

Figure 6. Spreading
rates of mid-ocean ridges, based on identification of linear magnetic anomaly
bands. Most (277) data from Table 3 of *DeMets et al.* [1990], corrected
downward by 4.4% for the timescale adjustment of *DeMets et al.* [1994].
Back-arc spreading and microplate tectonics are represented by 35 additional
rates whose citations were given by *Bird* [2003].

Figure 7. Velocities
of geodetic benchmarks (GPS technique) from the dataset of *Kreemer et al.*
[2003], who excluded known coseismic and post-seismic anomalies. We further
filtered the dataset to remove benchmarks less than 100 km from subduction zone
trenches or less than 25 km from other plate boundaries, because within those
bands our approximate correction for effects of temporary fault locking and
elastic strain accumulation might be inadequate.

Figure 8.
Most-compressive horizontal principal stress directions (_{}) with
their respective 90%-confidence sectors. Obtained by interpolation of
non-seismic World Stress Map data, plus intraplate Harvard CMT moment tensor
orientations, to a regular global grid by the clustered-data technique of *Bird
& Li* [1996]. Results shown only where 90%-confidence sectors were ±45°
or narrower. Main map is Mercator projection; polar insets are orthographic.

Figure 9. Azimuths of
fast polarization of SKS waves (_{}) arriving at seismic
stations, from the global compilation of *Fouch & Rondenay* [2006].
Size and color of symbol are determined by the splitting time in s. Blue bands
extend 700 km inland from subduction zones, indicating that we did not use data
from these regions. Four additional rectangular regions around the East Africa
rift, Hawaii, Yellowstone, and Iceland hotspots were also edited out.

Figure 10. Surface
velocity field from the preferred finite-element model Earth5-049. Most plates
have correct velocity (according to PB2002 model of *Bird* [2003]) because
they were forced by boundary conditions, as explained in text. Three regions
of unphysical behavior may be seen around Tibet, in western North America, and
in the central Andes.

Figure 11. Fault heave rates from the preferred finite-element model Earth5-049. Equivalent to the velocity discontinuities in Figure 10. See caption of Figure 10 for comments on realism.

Figure 12. Intraplate long-term (anelastic) strain-rate from the preferred finite-element model Earth5-049. Excludes all straining due to slip rates of fault elements. Equivalent to those velocity gradients in Figure 10 which are not due to Eulerian plate rotation on a sphere. See caption of Figure 10 for comments on realism.

Figure 13. Vertical
integrals of stress anomaly (small tensor symbols), and vertical integrals of
maximum shear stress (colors) in the preferred finite-element model
Earth5-049. Color scale is designed to emphasize variations within the low-intensity
parts of the stress anomaly field. It is notable that the strongest predicted
stress fields do not cross plate boundary faults, but occur within oceanic
lithosphere where distinct plates are converging (IN-AU, SA-SC) or diverging
(AF-SO, NA-SA) in the PB2002 model of *Bird* [2003]. The realism of these
very strong intraoceanic stress fields is uncertain.

Figure 14.
Most-compressive horizontal principal stress directions (_{}) from
the preferred finite-element model Earth5-049. Bars are color-coded according
to predicted stress regime: red = normal-faulting; green = strike-slip; blue =
thrust-faulting. Compare to Figure 8.

Figure 15. Triplets of ficticious point forces which are equivalent to balanced lithostatic-pressure, side-strength, and basal-strength torques on each plate, in the preferred finite-element model Earth5-049. Each triplet of point forces is connected by a black leader line to the center of the relevant plate. For leader lines which extend toward the poles, see Fig. 16.

Figure 16. Polar orthographic projections of triplets of ficticious point forces, as explained in Figure 15. Each triplet of point forces is connected by a black leader line to the center of the relevant plate. For leader lines which extend toward the equator, see Figure 15.

Supplemental Figure. (This figure was not included in the published paper; I drew it later for a class.)