Geodesy & Crustal Deformation Geology 6690/7690 Geodesy & Crustal Deformation 8 Dec 2017 Last time: Earth tides; viscoelastic rebound • Earth tides are solid Earth deformation in direct response to time-space differences in gravitational attraction of the sun & moon (so, expressed in terms of the kl Love number) • Peak-to-peak deformation of order tens of cm, so these strains dominate strainmeter/tiltmeter data; must be corrected in GPS positions as well (& constrain deep mantle density!) • Evidence that the 12-24 period Earth tides influence slow fault slip (in borehole strainmeter data), shallow tremor frequency... • Force equilibrium balance for viscoelastic flow in a half-space  pressure forces plus viscous forces plus body force (gravity)… Plus conservation of fluid ( incompressible): © A.R. Lowry 2017

Calais et al. examined GPS velocities in the “stable” NoAm plate interior with main result: Patterns are broadly consistent with GIA. Calais et al., J. Geophys. Res. 111 2006

Looking closer though there are some interesting patterns

This from Wahr & Zhong on the GRACE website… The Gulf Coast has slightly less subsidence than nearby regions (reference frame issue?) but not that much…

On the likely depocenter of Pleistocene ice: Some past studies identify it with a free-air gravity anomaly throughout eastern Canada… For practical purposes, probably further west than where they put it.

Must be careful with gravity, as there’s a lot going on…

Secular (i.e., linear) change in the GRACE geoid is less ambiguous about where uplift is occurring than earlier data such as shorelines, tide gauges and lake levels… (Detlef & Ivins, J. Geodynamics, 2008)

GRACE geoid rate (Detlef & Ivins, J. Geodynamics, 2008) GPS vertical velocities show some general correspondence to the GRACE measurements… Difficult to see given differences in projections here, but there is a sampling issue with GPS. Sella et al. (GRL, 2004) used both campaign and continuous GPS data; get better northern spatial sampling than Calais et al….

Recall also that the strain response depends on assumptions about rheology of the lithosphere! Expect to see significant focusing of strain in zones of rheological weakness, even if those zones are far from the edges of the ice sheet. Note plate-like behavior and strain focusing along San Andreas for this model…

Rheological variations of course are much more complicated… Te proxy for lithospheric strength from Marta’s mapping

Note based on this might also expect to see a change in plate motions… None apparent in rotation pole (but wouldn’t necessarily expect it to!) Rather expect in angular velocity: Calais et al. get 0.202°/Myr, DeMets et al. got 0.216…

Recall importance of sampling and motion bias in defining a reference frame!!! Recognition of this led to an attempt to define a Stable North America Reference Frame (SNARF) that was rebound-free…

Ice-1 Lithospheric Thickness = 120 km UM = 0.8  1021 Pa s LM = 10  1021 Pa s

Ice-1 Ice-5g GRACE SNARF Approach • Rather than adopt an arbitrary (& undoubtedly incorrect) ice & Earth model pair that would introduce systematic errors, SNARF used statistical properties of ALL ice & Earth models available (at that time)… • GPS velocities were assimilated into an a priori GIA model based on a suite of predictions to yield an observation- driven model Ice-1 Ice-5g GRACE

First test: Assimilate a single vertical velocity into the model (without incorporating physics of viscoelastic response; just the spatial statistics of 1D rebound models!)

• Second test: Assimilate a subset of all NoAm vertical site velocities • The assimilated field (still with “no physics”) shares many of the features of a realistic GIA field

Final SNARF 1.0 GIA field…

Despite sampling- related differences in appearance of the GPS and GRACE expressions of GIA, GRACE and SNARF vertical look pretty similar…

Solution Statistics: Prefit: WRMS (horizontal): 1.22 mm/yr WRMS (vertical): 3.81 mm/yr WRMS (all): 1.74 mm/yr Postfit: WRMS (horizontal): 0.71 mm/yr WRMS (vertical): 1.30 mm/yr WRMS (all): 0.80 mm/yr

But, remember there is a significant difference in GIA response of a 1D versus a 3D Earth! This model assumed only radial changes in material properties so lends no insight into potential plate-like expression of transient motions associated with GIA. Eventually SNARF was abandoned due to its complexity; PBO products use NAM08 RF after Altamimi et al. (2012) (leaving plate motion & GIA conflated for us to figure out!)

Post-Glacial Rebound (PGR) (aka Glacial Isostatic Adjustment, GIA) Reading: Wahr course notes, Chapter 7 (235-249) Recall that for a spherical elastic Earth, we can describe the vertical, horizontal and gravitational response in the spherical harmonic domain via the load Love numbers hl, ll and kl respectively: Vertical displacement: Horizontal displacement: Gravitational potential: & the amplitudes can be inverse-transformed to spatial domain.

The Love numbers can be calculated for any arbitrary (radially symmetric) Earth model given known density and elastic parameters as a function of radius in the Earth (e.g., we commonly use PREM). If the Earth is (Maxwell) viscoelastic rather than truly elastic, the only change needed is to recognize that elastic parameters  & l are time-dependent. In the time-frequency domain, Here  is the viscosity.

Thus we can calculate a “brute force” viscoelastic response relatively easily by adding viscosity with radial depth to the elastic & density model, calculating multiple Love number curves representing a range of frequencies for a given viscosity profile, and transforming the load history to the time-spectral domain and back again in order to get time- dependent Earth surface response! 30 ka 20 ka 10 ka 0 ka