Geodesy & Crustal Deformation Geology 6690/7690 Geodesy & Crustal Deformation 1 Dec 2017 Last time: Earth elastic response to loading • Mass changes near the Earth’s surface can exceed 1 m of water equivalent thickness on timescales of order months to years. Largest variations are in continental hydrology, followed by ocean mass, atmospheric pressure, & other. • Deformation is described in the spherical harmonic domain as a linear function of degree l by the load Love numbers hl, kl, ll. The geoid uses the load Love number for gravitational potential, kl, as to account for ~30% perturbation of gravity associated with Earth deformation response to mass loads! © A.R. Lowry 2017

Reading for Monday (4 Dec): Lau et al. (2017) Tidal tomography constrains Earth’s deep-mantle buoyancy. Nature 551 321–336. (Last one so I’ll do!)

Sphericity of the Earth matters for this deformation so we generally express the response as that of a radially- symmetric spherical Earth, in which case we can describe the Earth response using l-dependent coefficients (called Love numbers) in the spherical harmonic domain. Three coefficients are useful, describing the response of gravitational potential (kl), surface vertical displacement (hl), and surface horizontal displacement (ll) to a unit mass load. The total geoid for example can be described as for mass change  occurring precisely at the Earth’s surface.

These of course depend on Earth elastic properties. Below are vertical displacement Love numbers for continental PREM, PREM substituting Crust2.0 properties from North Dakota, and an attempt to create a very compliant end member by substituting 10 km of unconsolidated sediment into the Crust2.0 model case. PREM PREM + Crust2.0 in North Dakota PREM + 10 km sediment

Spherical harmonic amplitudes for given amplitudes of the surface mass field, Mlm, can be calculated as: Vertical displacement: Horizontal displacement: Gravitational potential: These can be converted to displacements as a function of location  via an inverse spherical harmonic transform… Note here that for horizontal displacements the mass field is vector-transformed to separate fields representing the variations in -only and -only directions to get N, E.

The solution for a disk of radius  is relatively simple, given by (e.g. for vertical): where Legendre coefficients l are: Here tested a discrete spherical harmonic solution (blue) against the disk solution (red) From Chamoli et al., JGR 2014

Example: Devil’s Lake basin, North Dakota: In 2004 this closed basin was nearly full to bursting (15 m change in lake level) but not quite… State of ND has since constructed an outlet. (wells)

railroad bed suggested ~30 cm of subsidence, broadly distributed! Example: Devil’s Lake basin, North Dakota: Will Gosnold (UND) comparison of ca. 2003 GPS heights to ca. 1948 leveling along a railroad bed suggested ~30 cm of subsidence, broadly distributed! Too much to explain with elastic response to water depth in the lake alone…. PREM PREM + Crust2.0 PREM + 10 km sediment

Note: Although the magnitude of Love numbers increases with decreasing scale, the increase is small relative to the 2l + 1 factor in the denominator of the relation to amplitude of the height change… So larger-scale loads have larger height response.  ~ 40,000/l km

Red circles are observed D-height; cyan are corrected for a model of GIA Viscoelastic response would be negligible (<0.1 mm). Post-glacial rebound in the region makes some difference but not nearly enough (only up to 5 mm…) However can match it approximately if the water thickness is about twice the lake depth, and spread over a much broader area. PREM+CRUST2.0 PREM+10 km seds

Common for such large signals to be interpreted as indicating Bevis et al., Geophys. Res. Lett., 2005 Common for such large signals to be interpreted as indicating anomalous low elastic properties (e.g., 7 cm (!) annual height variation at Manaus GPS site in the Amazon basin) but it’s much more likely to represent “hidden” mass, plus perhaps poroelastic deformation…

Important that spherical harmonic domain modeling assumes radially-varying Earth properties only (i.e., no lateral heterogeneity!) But for local/regional problems one can use the local/regional elastic and density properties. Here, modeled variations from PREM using Crust2.0 in radially symmetric (top) and fully 3D (middle) modeling… Differences from PREM are typically ±20%, but differences between locally-1D and fully 3D are generally <10%.

Solid Earth Tidal Deformation (cf John Wahr’s class notes pp 251–271) Earth tides are deformation of the solid Earth in direct response to the change in gravitational potential of (primarily) the sun & moon resulting from rotation of the Earth and variations in orbits. (This is distinct from the solid-Earth deformation in response to ocean tidal mass). • Can be several tens of cm peak-to-peak • Not observed by most ground-based instruments (with important exceptions of strain-gauges & tiltmeters!) • Is observed by space-based positioning systems (e.g. GPS/GNSS!)

Deformation results from differences in the gravitational attraction at different spots within the Earth… Complexity added by Earth rotation coupled with change in orientation of Earth rotational axis relative to lunar orbit (±23.5° from equator over a lunar cycle). This leads to dominant periods of 12, 24 and ∞ hours (modulated by other periods of 1/10 days up to 1/18.6 years!) Total gravitational attraction Difference from mean gravitational attraction