- Introduction
- Isostatic response functions
- Results for western North America
- Acknowledgements
- References

The flexural rigidity of continental lithosphere can be estimated from isostatic transfer functions (admittance, coherence) relating topography and gravity anomalies using the equation for the flexure of a thin elastic plate. There is growing evidence that such transfer functions are anisotropic, and inferred weak directions correlate with principal directions of crustal stress, indicating either more complicated models of loading, and/or anisotropic rigidities. Here we derive isostatic response functions using the equations for the flexure of a thin elastic plate that incorporate the effects of in-plane loading. We then calculate local 1-D and 2-D wavelet admittance and coherence functions in western North America, and invert for either rigidity anisotropy or in-plane force.

Flexural isostasy describes the condition that loads must be
supported at some depth within the lithosphere via elastic plate
flexure. The flexural rigidity
, where is Young's modulus
and is the elastic thickness, is a rheological property that governs
the resistance of the plate to bending. A popular method of estimating
the flexural rigidity is based on calculating transfer functions
relating gravity and topography and inverting
using isostatic response functions obtained from plate flexure equations
(*Forsyth*, 1985).
In the case where an isotropic plate is loaded both horizontally
by in-plane force and surface load , the flexure equation is written

where and are rigidities in two perpendicular directions, and is the torsional rigidity approximated by . The vertical load at the surface is given by

where is gravitational acceleration, is the topography, and are crustal and mantle density, respectively, and we will use the shorthand . In-plane forces are given by

where , represent axial loads (compression is negative) in the and directions, respectively, and denotes shear loading. Solving these equations in the Fourier domain yields linear isostatic response functions relating Moho deflection to surface topography that take the form

where , and . The functions and correspond to

For subsurface loading the isostatic function is

Isostatic response functions are combined to form theoretical admittance () and coherence () functions between Bouguer gravity and topography

(29.2) | |||

(29.3) |

where is the ratio between surface and subsurface loads, is the gravitational constant, and is the depth of compensation, taken at the Moho.

We calculated the wavelet admittance and coherence in western North America
following the method of *Audet and Mareschal* (2007)
and inverted the corresponding isostatic quantities to yield
estimates of for the isotropic case, and
and
(i.e. the direction of ) for the orthotropic case. We give results in
terms of elastic thickness using the relation
,
where can be either , or
(Figure 2.52). Low values (30 km) are found across
most of western North America, increasing toward the continental interior.
In the northeastern craton, values can reach 100 km.
Maps of and follow the same general
patterns as the isotropic , whereas
is oriented dominantly SW-NE, except in the highly deforming regions
of western United States, where it is highly variable in both magnitude and
direction.

We further compared the weak direction with orientations of
maximum horizontal compressive stress
from the World Stress Map project `http://dc-app3-14.gfz-potsdam.de/`.
We re-sampled stress indicators onto the grid and calculated
the angular difference between both directions (Figure 2.53a,b).
There is good agreement in the Canadian Cordillera and near the
coast of California, both regions where stress regime is compressional.
An exception is the arc and forearc in the Pacific Northwest, where compressive
stress directions are parallel to the coast whereas weak directions
are perpendicular, perhaps reflecting more complex loading near the
subduction zone. Angle difference is large () in extensional regimes,
such as Basin and Range and western Colorado Plateau.

These correlations allow us to use isostatic functions for the isotropic plate
with axial loading to fit the 2-D coherence and admittance in order to
estimate load magnitudes where angular difference is within 30 .
We use isotropic and obtained previously, and estimate
total axial load (Figure 2.53c). Preliminary
results indicate loads on the order of 100-600 MPa, which are up to
three times larger than estimates from dynamical models of deformation
using a uniformly thick (100 km) elastic plate (*Humphreys and Coblentz*, 2007).
Such large discrepancy also suggests that, in addition
to lithospheric stress, significant rigidity anisotropy must be involved
in producing anisotropy in the observed transfer functions. Lastly we note
that shear loads were not modeled at this point, which will be the focus of future efforts.

This work was funded by the Miller Institute for Basic Research in Science (UC Berkeley).

Audet, P., and J.C. Mareschal, Wavelet analysis of the coherence between Bouguer
gravity and topography: Applications to the elastic thickness anisotropy in the
Canadian Shield, *Geophys. J. Int., 168*, 287-298, 2007.

Forsyth, D.W., Subsurface loading and estimates of the flexural rigidity
of continental lithosphere, *J. Geophys. Res., 90*, 12, 623-12, 632, 1985.

Humphreys, E.D., and D. Coblentz, North American dynamics and western U.S.
tectonics, *Rev. Geophys., 45*, RG3001, 2007.

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