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GSC307
Introduction to Global Geophysics
Gravity and the Shape of the
Earth
Today
•  Lecture:
–  Gravity
–  Short extra “Lab” session at end of class
•  Lab 3 will be due before end of class today
•  Exam 1: during Lab time on Thursday!
–  Covers all lecture and lab material
–  Bring: ruler, protractor, calculator, graph paper
•  Lecture on Thursday: opportunity to ask
questions on Exam 1 material
Cal Poly Pomona
4/28/15
GSC307
Introduction to Global Geophysics
Cal Poly Pomona
GSC307
Introduction to Global Geophysics
International Reference Ellipsoid
A
•  = standardized reference
figure of the Earth
•  describes variation of gravity
with latitude: normal gravity
formula
5
5
5
The Geoid
•  Geoid = equipotential surface
of gravity that coincides with
mean sea level
•  NOT surface of constant
gravity!
EQUIPOTENTIAL
SURFACE
2
B
HO
gn = ge (1+ β1 sin λ + β 2 sin 2 λ )
AL
2
ON
RIZ
VE
RT
IC
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TA
L
€
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4/28/15
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GSC307
Introduction to Global Geophysics
GSC307
Introduction to Global Geophysics
Geoid
Height
Anomalies
•  Geoid reflects true mass distribution and differs from IRE
•  Bulge in geoid (potential is achieved further from the Earth’s
center) wherever mass excess under ellipsoid (positive gravity
anomaly, gravity is strengthened locally)
•  Wherever mass deficit under ellipsoid (negative gravity
anomaly): trough in the geoid (km’s in topography -> m’s in
geoid height)
4/28/15
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Correction of Gravity Measurements
•  Gravity measurements:
–  Objective: look for deviations from
reference value
–  Due to structures with anomalous
subsurface density
–  Problem: measurements are (usually)
not made on reference ellipsoid
–  Solution: “reduce”/correct
measurements to “bring” them on
ellipsoid
–  (Additional problem: non-unique)
4/28/15
GSC307
Introduction to Global Geophysics
Cal Poly Pomona
GSC307
Introduction to Global Geophysics
Gravity Corrections
Terrain correction:
Bouguer correction: Free-air correction:
•  Compensates for
reduction of g due to
terrain around
measurement site
•  Complex calculation
•  Compensate for
gravitational
attraction of plate of
constant thickness h
•  Compensates for
elevation of
measurement site
w.r.t. ellipsoid
A
Terrain
Correction
HILL
1
0
HILL
VALLEY
VALLEY
H1
H0
2
REFERENCEELLIPSOID
2
1
B
0
"/5'5%20LATE
2
H0
REFERENCEELLIPSOID
H1
"/5'5%20LATE
2
1
•  Gravity measurementCmay be0 perturbed by mountains and
H1
valleys
H0
•  Presence of hill-top or valley both
reduce
measured
gravity
2
2
REFERENCEELLIPSOID
•  To compensate for this, calculate and add a correction to
your measurement inD
both cases
0
2
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1
REFERENCEELLIPSOID
2
Cal Poly Pomona
2
A
HILL
HILL
1
0
VALLEY
VALLEY
H1
H0
GSC307
Introduction to Global Geophysics
REFERENCEELLIPSOID
2
GSC307
Introduction to Global Geophysics
2
1
B
Bouguer Correction
0
Bouguer
Correction
"/5'5%20LATE
2
H1
H0
"/5'5%20LATE
REFERENCEELLIPSOID
2
1
C
ΔgBP = 2πGρh
0
H1
H0
2
2
REFERENCEELLIPSOID
•  After leveling topography,
D there is still a fictive uniform layer
of rock between station and reference
ellipsoid.
1
0
REFERENCEELLIPSOID
2
•  Take layer to be flat disk, called2Bouguer
plate, compute
gravitational for known thickness and density: ΔgBP.
•  If station is above sea level this correction must be subtracted
from measured gravity, if below, we have to fill space above it
with rock, this requires increasing measured gravity.
4/28/15
Substituting numerical values gives: 0.0419x10-3 ρ mgal per
meter, with density in kg/m3, typically about 0.1 mgal per m.
Also can be used in marine surveys to correct for the oceans, by
replacing the sea water with rock.
€
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A
4/28/15
HILL
1
0
Cal Poly Pomona
HILL
VALLEY
VALLEY
H1
H0
2
REFERENCEELLIPSOID
2
1
B
0
"/5'5%20LATE
GSC307
Introduction to Global Geophysics
2
C
Free Air
Correction
H0
REFERENCEELLIPSOID
H1
"/5'5%20LATE
2
Free Air Correction
0
H1
H0
2
REFERENCEELLIPSOID
2
D
0
2
1
REFERENCEELLIPSOID
2
•  Finally, measured gravity must be corrected for elevation
above ellipsoid.
•  Gravitational attraction decreases with increased distance.
•  If station is located above ellipsoid, correction, ΔgFA, has to
be added to measurement.
•  If it is located below, correction is subtracted.
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Introduction to Global Geophysics
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Cal Poly Pomona
•  Free air correction assumes
no material between
measurement point and sea
level
ΔgFA = −
2h
g0
R
–  ~0.3086 mgals per meter of elevation
€
4/28/15
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GSC307
Introduction to Global Geophysics
GSC307
Introduction to Global Geophysics
Combined Elevation Correction
•  Free Air and Bouguer
corrections may be
combined in one
elevation correction
–  two corrections are
always of opposite sign
–  using typical density:
0.197 mgal per m,
add if station is above
ellipsoid, subtract if
below
4/28/15
BASICLAVAS
Gravity Anomalies
•  Discrepancy between measured, corrected gravity
and theoretical gravity (given by normal gravity
formula) is called gravity anomaly
METAMORPHIC
ROCKS
DOLOMITE
–  due to density of Earth’s interior not being
homogeneous
GRANITE
LIMESTONE
SHALE
SANDSTONE
n $ENSITY KGM
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GSC307
Introduction to Global Geophysics
4/28/15
Cal Poly Pomona
GSC307
Introduction to Global Geophysics
Isostasy
Isostasy
In 18th and 19th century Andes and Himalayas were found to not
deflect plumb lines as much as expected.
Explanation: isostasy: mass deficiency beneath the mountains
compensates for the mountain itself
4/28/15
•  Bouguer anomaly: measurement corrected for
free-air, Bouguer plate, terrain and latitude
•  Free air anomaly: measurement corrected for freeair, terrain and latitude
=> May look quite different across same structure
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Over geologic time, earth's mantle behaves like viscous fluid
When topography is created through crustal thickening, mantle
slowly flows out from beneath thickened region to compensate
Mantle thus behaves similar to water when ice block is placed on
its surface - water beneath ice flows outward and upward, ice
sinks downward until equilibrium is reached
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GSC307
Introduction to Global Geophysics
GSC307
Introduction to Global Geophysics
Airy Model of
Isostasy
Isostasy
http://www.geo.cornell.edu/
hawaii/220/PRI/isostasy.html
The water flows from zone of higher pressure to lower pressure
zones until pressure acting on water at a depth of compensation
is exactly equal everywhere
Different models for the Earth achieve compensation in different
ways
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GSC307
Introduction to Global Geophysics
• 
• 
• 
• 
Upper layer and substratum have constant densities
Isostatic compensation through deep roots
Mountains have roots, ocean has “anti-root”
Thickness of root varies in proportion to elevation of
topography
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GSC307
Introduction to Global Geophysics
Airy Model
Root of San Gabriel Mountains
Compensation Depth
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•  Compensation depth: depth
below which all pressures are
hydrostatic (above which mass
columns are equal)
•  Can be used to calculate root
thicknesses, for example
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GSC307
Introduction to Global Geophysics
GSC307
Introduction to Global Geophysics
Pratt Model of
Isostasy
Investigation of Isostasy
•  Depth at base of upper layer is
level
•  Isostatic compensation through
lateral variations in density
(differential expansion)
•  Mountains consist of and are
underlain by material of low
density
•  Oceans are underlain by high
density material
Both models represent local
isostatic compensation.
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Compensation Depth
Cal Poly Pomona
GSC307
Introduction to Global Geophysics
•  If a region is in isostatic equilibrium, there is no
excess or lack of mass above the compensation
depth
•  Free air anomaly includes the effect of all the
mass variations at a measuring point, whereas the
Bouguer anomaly has had the predicted effect of
the mass of the topography removed
–  Which one will tell you if isostatic equilibrium has been
achieved?
4/28/15
Cal Poly Pomona
GSC307
Introduction to Global Geophysics
Gravity Anomalies and Isostasy
Gravity Anomalies and Isostasy
•  If topography is compensated:
•  If topography is compensated:
–  mass excess above sea level is canceled by
mass deficiency below it
⇒ Free air anomaly is small
⇒ Bouguer anomaly large and negative
–  mass excess above sea level is
canceled by mass deficiency
below it
⇒ Free air anomaly is small
⇒ Bouguer anomaly large and
negative
•  If NOT compensated:
⇒ Free air anomaly is large and
positive
⇒ Bouguer anomaly zero
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In case topography is NOT
compensated:
⇒ Free air anomaly is large and positive
⇒ Bouguer anomaly zero
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GSC307
Introduction to Global Geophysics
GSC307
Introduction to Global Geophysics
Mars
Crossing the Mid Atlantic Ridge
•  Despite topographic rise
associated with ridge,
free-air gravity anomaly is
relatively flat and ~ zero
across whole structure.
•  By contrast, Bouguer
gravity anomaly has local
dip across the ridge axis.
•  Explanation?
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A!IRY
OCEAN
D
Free air gravity
SE A LE VE L
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MOUNTAIN
H
CRUST
MANTLE
#
#g
B0RATT
D
T
R
R
M
Cal Poly Pomona
H
C
R
•  Mars’ rough southern
hemisphere has flat Free Air
anomaly => near-isostatic
compensation
•  whereas smooth northern
plains display wider range
=> stronger surface layer
than in the south.
OCEAN
SE A LE VE L
H
MOUNTAIN
C
H
$
C
CRUST
GSC307
#
Introduction to Global Geophysics
MANTLE
M
C6ENING-EINESZ
MOUNTAIN
SE A LE VE L
#g
GSC307
Introduction to Global Geophysics
Hawaii
Vening Meinesz
Model
T
KM
CRUST
MANTLE
LOCAL
COMPENSATION
REGIONAL
COMPENSATION
•  Regional isostatic compensation
•  Upper layer behaves like an elastic plate and
distributes load of a surface feature over a wider
distance
•  Bending of plate depends on its elastic properties
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GSC307
Introduction to Global Geophysics
GSC307
Introduction to Global Geophysics
Isostatic Glacial Rebound
•  When load (like ice) is
added, it depresses
underlying crust
Glacial Rebound
Study of rate of rebound (~5-10
mm/yr) allows for estimates of
asthenospheric viscosity
–  May take thousands of
years
•  When ice melts, crust
slowly returns to its
original shape by
isostatic rebound
4/28/15
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GSC307
Introduction to Global Geophysics
4/28/15
Cal Poly Pomona
GSC307
Introduction to Global Geophysics
More Questions
Example Exam Questions
•  Type of Triple
Junction?
•  The study of the Earth’s shape is called ________
•  A planet cannot have a perfectly circular orbit
around a star according to Kepler’s second law:
TRUE / FALSE
•  What is the geoid?
•  Draw a cross-section through a subduction zone
4/28/15
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GSC307
Introduction to Global Geophysics
And Also…
•  Be prepared for velocity space diagrams!
–  Triple Junctions
•  Understand magnetic anomalies and how/why
they form and are related to the timescale
•  Understand measurements and corrections of
gravity
•  Be able to apply isostasy to calculate densities or
layer thicknesses
•  Bring ruler, protractor, calculator, color pencils
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