1. 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 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

2. 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

3. Looking closer
though there
are some
interesting
patterns

5. 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…

6. 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.

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

8. 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)

9. 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….

10. 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…

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

12. 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…

13. 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…

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

15. 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

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

17. • 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

18. Final SNARF 1.0
GIA field…

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

20. Solution Statistics: Prefit: WRMS (horizontal): 1.22 mm/yr
WRMS (vertical): mm/yr
WRMS (all): mm/yr
Postfit:
WRMS (horizontal): mm/yr
WRMS (vertical): mm/yr
WRMS (all): mm/yr

21. 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!)

22. Post-Glacial Rebound (PGR) (aka Glacial Isostatic Adjustment, GIA)
Reading: Wahr course notes, Chapter 7 ( )
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.

23. 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.

24. 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