1. Geodesy & Crustal Deformation
Geology 6690/7690
Geodesy & Crustal Deformation
6 Dec 2017
Last time: Earth elastic response to loading
• Deformation is described in the spherical harmonic
domain as a linear function of degree l by the load
Love numbers hl, kl, ll:
Vertical:
Horizontal:
Gravitational potential:
• Surface displacements response to a point load are down
and toward the load; vertical is sensitive to integral whereas
horizontal is sensitive to local gradient of mass
• Variability in elastic properties contributes to ~±20%
perturbations in surface vertical response
© A.R. Lowry 2017

2. 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!)

3. 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

4. Interesting aside… ~5° of lunar orbit inclination
and last ~1%
of Earth
mass from
early
bombard-
ment by
planetesimals …
Cuk et al,
Nature
2016!

5. Isostasy in a viscoelastic fluid (“Rebound”)
Given some initial sinusoidal deflection of the Earth’s
surface:
where k is wavenumber = 2/
what is the evolution of w through time?
Elevated beaches or
“strandlines” in Hudson’s
Bay region of Canada…

6. We impose:
(1) Conservation of fluid: (2D) v
x + x x z
Where:
u and v are flow
in x, z
directions...
Note this implies
the fluid is
incompressible u
z + z

7. We impose:
(2) Elemental force balance:
includes pressure forces,
viscous forces, and
body force (= gravity).
For pressure forces, the
imbalance is given
by the pressure
gradients: (2D)
p(z)x
x + x x z
p(x+dx)z
p(x)z
z + z
p(z+z)x

8. We impose:
(2) Cont’d: viscous forces:
The net viscous forces are
given by
(to first
order).
For a
Newtonian
fluid with
viscosity ,
zz(z)x
zx(z)x
x + x x z
xz(x)z
xx(x+x)z
xx(x)z
xz(x+x)z
z + z
zx(z+z)x
zz(z+z)x

9. The body force is simply xz in the z direction. Typically
we simplify pressure by subtracting a hydrostatic:
P = p – gz
Then (after some algebra; see T&S) we have the force
balance equations:

10. We can define a stream function (= potential of the
flow field) such that (for an incompressible fluid):
is flow velocity in the x-direction;
is flow velocity in the z-direction (vertical).
Solutions  must satisfy:
Substituting for u and v, we find the biharmonic equation has
(eigenfunction) solutions of the form:
and

11. Solving (with boundary conditions) we find that:
where the decay constant
Note: This solution is specific to the case of a viscoelastic half-
space! To zero-th order, would be better approximated by an
elastic layer over a viscoelastic halfspace, in which case one
would “filter” the viscoelastic response (multiply in the
spectral domain) by the amplitude response for the elastic
layer.

12. 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

13. Looking closer
though there
are some
interesting
patterns