Crustal Stress and Mechanical Anisotropy of the Lithosphere in Western North America

Pascal Audet


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.

Figure 2.52: Elastic thickness results for western North America. a) Isotropic $T_e$; b) Direction of $T_{min}$ ($\beta$); c) $T_{min}$; d) $T_{max}$. Shading in b) is given by $(T_{max}-T_{min})/T_{max}$. Grey lines indicate major tectonic boundaries.
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Figure 2.53: Comparison between $\beta$ and orientation of maximum horizontal compressive stress. a) shows crustal stress indicators (red bars) and co-located weak directions (black bars); b) is the angular difference between both indicators; and c) is the estimated in-plane load necessary to fit the observed 2-D admittance and coherence.
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Isostatic response functions

Flexural isostasy describes the condition that loads must be supported at some depth within the lithosphere via elastic plate flexure. The flexural rigidity $D\propto ET_e^3$, where $E$ is Young's modulus and $T_e$ 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 $n$ and surface load $q$, the flexure equation is written

D\nabla^4w({\bf r})=q({\bf r})+n({\bf r}),
\end{displaymath} (29.1)

where $w$ is plate deflection at the Moho, and ${\bf r}=(x,y)$. For an orthotropic plate, equation (2.13) becomes

\left[D_x\frac{\partial^4}{\partial x^4}+2B\frac{\partial^4}...
...rtial^4}{\partial y^4}\right]w({\bf r})=q({\bf r})+n({\bf r}),

where $D_x$ and $D_y$ are rigidities in two perpendicular directions, and $B$ is the torsional rigidity approximated by $B\approx \sqrt{D_xD_y}$. The vertical load at the surface is given by

q({\bf r})=\rho_cgh({\bf r})-(\rho_m-\rho_c)gw({\bf r}),

where $g$ is gravitational acceleration, $h$ is the topography, $\rho_c$ and $\rho_m$ are crustal and mantle density, respectively, and we will use the shorthand $\Delta\rho = \rho_m-\rho_c$. In-plane forces are given by

n({\bf r})=\left[N_x\frac{\partial^2}{\partial x^2}+N_y\frac...
..._{xy}\frac{\partial^2}{\partial x\partial y}\right]w({\bf r}),

where $N_x$, $N_y$ represent axial loads (compression is negative) in the $x$ and $y$ directions, respectively, and $N_{xy}$ 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

\Theta({\bf k})=\frac{\rho_c}{\Delta\rho}\left[1+\frac{\psi(...
...frac{\zeta(N_x,N_y,N_{xy},{\bf k})}{\Delta\rho g}\right]^{-1},

where ${\bf k}=(k_x,k_y)$, and $\vert{\bf k}\vert=k$. The functions $\psi$ and $\zeta$ correspond to

\psi = \left\{ \begin{array}{ll}
Dk^4 & \mbox{isotropic pla...
...k_y^2\right)^2 & \mbox{orthotropic plate}\,.

\zeta = \left\{ \begin{array}{ll}
0 & \mbox{no in-plane for...
...N_{xy}k_xk_y & \mbox{with in-plane force}\,.

For subsurface loading the isostatic function is

\Phi({\bf k})=\frac{\rho_c}{\Delta\rho}\left[1+\frac{\psi(D_...
...ho_c g}+\frac{\zeta(N_x,N_y,N_{xy},{\bf k})}{\rho_c g}\right].

Isostatic response functions are combined to form theoretical admittance ($Q$) and coherence ($\gamma^2$) functions between Bouguer gravity and topography
$\displaystyle Q({\bf k})=2\pi \Delta \rho G e^{(-\vert{\bf k}\vert z_c)}\frac{\left(\Theta+\Phi f^2\Theta^2\right)}{\left(1+f^2 \Theta^2\right)},$     (29.2)
$\displaystyle \gamma^2({\bf k})=\frac{(1+\Phi\Theta f^2)^2}{(1+f^2\Theta^2)(1+f^2\Phi^2)},$     (29.3)

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

Results for western North America

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 $D$ for the isotropic case, and $D_{min},D_{max}$ and $\beta$ (i.e. the direction of $D_{min}$) for the orthotropic case. We give results in terms of elastic thickness using the relation $D=\frac{ET^3}{12(1-\nu^2)}$, where $T$ can be either $T_e$, or $T_{min},T_{max}$ (Figure 2.52). Low values ($T_e<$30 km) are found across most of western North America, increasing toward the continental interior. In the northeastern craton, $T_e$ values can reach 100 km. Maps of $T_{min}$ and $T_{max}$ follow the same general patterns as the isotropic $T_e$, whereas $\beta$ 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 We re-sampled stress indicators onto the $T_e$ 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 ($>45$$^{\circ}$) 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$^{\circ}$ . We use isotropic $T_e$ and $\beta$ 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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