1. 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

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

3. 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.

4. 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

5. 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.

6. 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

7. 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)

8. 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

9. 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

10. 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

11. 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…

12. 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%.

13. 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!)

14. 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