outline construction of gravity and magnetic models principle of superposition (mentioned on week...
TRANSCRIPT
Outline
Construction of gravity and magnetic models Principle of superposition (mentioned on week 1) Anomalies
Reference models Geoid Figure of the Earth Reference ellipsoids
Gravity corrections and anomalies Calibration, Drift, Latitude, Free air, Bouguer, Terrain,
Aeromagnetic data reduction, leveling, and processing
Gravity anomalies The isolation of anomalies (related to unknown local
structure) is achieved through a series of corrections to the observed gravity for the predictable regional effects
According to Blakely (page 137), it is best to view the corrections as superposition of contributions of various factors to the observed gravity (next slide)
Gravity anomaliesObserved gravity = attraction of the reference
ellipsoid (figure of the Earth)
+ effect of the atmosphere (for some ellipsoids)+ effect of the elevation above sea level (free
air)+ effect if the “average” mass above sea level
(Bouguer and terrain)+ time-dependent variations (drift and tidal)+ effect of moving platform (Eötvös)+ effect of masses that would support
topographic loads (isostatic)
+ effect of crust and upper mantle density (“geology”)
If we model and subtract these terms from the data…
…then the remainder is the “anomaly” (for example, “free air” or “Bouguer” gravity)
Geoid and Reference Ellipsoid Geoid is the actual equipotential surface at
(regional) mean sea level Reference ellipsoid is the equipotential
surface in a uniform Earth Much more precisely known from GPS and satellite
gravity data Recent recommendations are to reference all
corrections to the reference ellipsoids and not to the geoid
Hydrostatic rotating Earth The surface of static fluid has a constant potential :
22
polar2 2 23 3
1 1 1, sin
2 2 2
rrU r GM r const GM
R R
22 3
2
polar
1 sin 1r R
r GM
2 3
polar 2polar2 3
2
1 sin2
1 sin
r Rr r
GMRGM
Therefore:
Gravity potentialCentrifugal potential
Conventionally, the equatorial radius is used for referencing:
21 coser r f
Gravity flattening Because of increased radius and rotation, gravity
is reduced at the equator:
21 coseg g
b is called the “gravity flattening”:
p e
e
g g
g
Reference Ellipsoids International Gravity Formula
Established in 1930; IGF30 Updated: IGF67
World Geodetic System (last revision 1984; WGS84) Established by U.S. Dept of Defense Used by GPS So the gravity field is measured above the atmosphere The difference from IGF30 can be ~100 m
A number of other older ellipsoids used in cartography
Also note the International Geomagnetic Reference Field: IGRF-11
Gravity flattening and the shape of the Earth Exercise: from the expressions for the Earth’s
figure and gravity flattening, show that the radius can be estimated from measured gravity as:
251 cos
2ee
gr r m
g
Multi-year drift of our gravity meter During field schools, the G267 gravimeter usually drifts by 0.1-0.2 mGal/day
Bullard B correction Necessary at high elevations (airborne gravity) Added to Bouguer slab gravity (subtracted from Bouguer-
corrected gravity) to account for the sphericity of the Earth
3 7 2 14 3 mGal (with i1.464 10 3.533 10 4.5 10 n meters) B h h hB h
Elevation above reference ellipsoid, h (m)
Bulla
rd B
corr
ect
ion (
mG
al)
Instrument Drift correction During the measurement, the instrument is used at
sites with different gravity gs and also experiences a time-dependent drift d(tobs)
Therefore, the value measured at time tobs at station s is:
obs obss su t g d t For d(tobs), we would usually use some simple
dependence, for example: a polynomial function
0
nk k
kk
d t a t t
where d0 is selected to ensure zero mean: <d(t)> = 0, that is:
00
nk
kk
d t a t d
(*)
Instrument Drift correction (cont.) Equation (*) is a system of linear equations with
respect to all gs and ak:Lm u
where m is a vector of all unknowns:
1
2
0
1
...
...
g
g
a
a
m
Instrument Drift correction (cont.)
… u is a vector of all observed values:
1 1
1 2
1
...
...
n m
n m
u t
u t
u t
u t
u
Instrument Drift correction (cont.)
… and matrix L looks like this:
21 1
22 2
23 3
24 4
25 5
0 1 ...
0 1 ...
1 0 ...
1 0 ...
1 0 ...
... ... ... ... ...
t t t t
t t t t
t t t t
t t t t
t t t t
L