1. GEO 5/6690 Geodynamics 24 Oct 2014 © A.R. Lowry 2014 Read for Wed 5 Nov: T&S 105-130 Last Time: Flexural Isostasy Generally, loading will occur both by surface mass flux processes and internal mass flux. If D and radial density structure are known, can solve for surface loads and internal mass loads (useful for understanding the processes that create topo and gravity variations!) (unknown loads) x (flexural & gravity response) = (observations) However to get D (or equivalently T e ) we need more information. Forsyth’s method: assume loads are uncorrelated & find T e that predicts coherence  2 of gravity & topography most closely matching observed coherence Example: Loading analysis in western North America

2. Next Journal Article(s) Reading: For Monday Nov 3: Audet & Bürgmann (2011) Dominant role of tectonic inheritance in supercontinent cycles. Nature Geosci. 4 184-187. (Tyler will lead) Important to think about: What are the possible reasons for a directional dependence of T e ? Also, read the abstract and conclusions (and look at the figures) of: Kirby & Swain (2014) On the robustness of spectral methods that measure anisotropy in the effective elastic thickness. Geophys. J. Int. 199(1) 391-401.

3. Possibility we considered at the time:

4. Becker et al. (2014) Earth Planet. Sci. Lett. did similar analysis with seismic studies using EarthScope data Lowry & Pérez-Gussinyé (2011) Nature; Levander & Miller (2012); Schmandt & Humphreys (2010) Match between residual and dynamic topography from present-day mantle flow: Correlation ~0.6

5. Example: Tharsis Rise, Mars: Martian topography is dominated by (1) a north-south hemispheric “crustal dichotomy” and (2) the Tharsis rise, average elevation 5000 m covering 20% of the planet The geoid is the shape of the gravity field. The 2000 m geoid anomaly over Tharsis is the largest in the solar system!

6. The Tharsis Rise Loading Controversy: Surface topography constructed by volcanism? Thermal/chemical buoyancy of a single mantle plume? [e.g., Willemann & Turcotte, 1982; Solomon & Head, 1982] [e.g., Sleep & Phillips, 1979; Harder & Christensen, 1996; Harder, 2000] Probably some combination of both!

7. Ratios of Geoid/Topography in the Spherical Harmonic Domain: Internal loading is a significant fraction of total!

8. Method: In the spherical harmonic domain, solve for: surface load height h S ilm lithospheric flexure w ilm internal load mass  ilm Using equations for observed topography h ilm and geoid N ilm including: the definition of surface load finite amplitude geoid calculation flexure of a thin elastic shell over a self-gravitating, viscous sphere Solve for each harmonic coefficient i,l,m Iterate: Set w n ilm = 0 on the first iteration and update on subsequent iterations Relations depend on: crustal density  0 mantle density  1 crustal thickness T c load radius R I lithosphere thickness T e asth. viscosity 

9. Important Point to Note: Using these three equations in three unknowns, we can devise an isostatic model that EXACTLY fits both the gravity and the topography data using ANY choice of reference density structure, lithospheric thickness and internal load depth. Hence, we must either (1)Explore the range of possible solutions by using all of the possible range of density and other parameters, or (2) Use some additional measurement or constraint to estimate those parameters independently.

10. Effects of lithosphere thickness and internal load depth on the estimate of internal load contribution

11. Effects of crustal density and crustal thickness on the estimate of internal load contribution

12. Constraining Parameter Space: Correlation of the load estimates is very sensitive to parameter-induced error, because errors in one load must be balanced by error in the other Hence we assume surface and internal load fields are uncorrelated, and search for model params that minimize the correlation of the two!

13. Using the model parameterization that minimizes load coherence, the “best” estimate of surface loading has an average thickness of 17 km within the Tharsis rise, and average flexure is 12 km. The averaged internal load is buoyant but small.