1. Outline Construction of gravity and magnetic models Reference 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

2. 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)

3. Gravity anomalies
Observed 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)

4. 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

5. Hydrostatic rotating Earth
The surface of static fluid is at constant potential :
Gravity
potential
Centrifugal
potential
Therefore:
where:
Conventionally, the equatorial radius is used for referencing:

6. Gravity flattening
Because of rotation, gravity decreases with colatitude q:
where
so that the gravity at the pole equals
Parameter b is called “gravity flattening”:

7. 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

8. Gravity flattening and the shape of the Earth
Exercise: from the expressions for the Earth’s figure and gravity flattening, show that the radius at colatitude q can be estimated from measured gravity as:

9. Multi-year drift of our gravity meter
During field schools, the G267 gravimeter usually drifts by mGal/day

10. 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
Bullard B correction (mGal)
Elevation above reference ellipsoid, h (m)

11. 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:
(*)
For d(t), we would usually use some simple dependence; for example, a polynomial function:
where d0 is selected to ensure zero mean: <d(t)> = 0, that is:

12. Instrument Drift correction (cont.)
Equation (*) is a system of linear equations with respect to all gs and ak:
where m is a vector of all unknowns:

13. Instrument Drift correction (cont.)
… u is a vector of all observed values:

14. Instrument Drift correction (cont.)
… and matrix L looks like this:
First columns correspond to gravity stations
Last columns correspond to n drift correction terms
Rows correspond to recording times

15. Instrument Drift correction (finish)
Then, the Least Squares solution of this matrix equation is achieved simply by:
In Matlab, this can be written as:
Vector m contains all drift terms and all drift-corrected gravity values at all stations considered