1. Gravity Data Reduction
In decreasing order, gravity is affected by
Latitude (due to shape and rotation
Elevation
Local topography
Lunar and solar tidal forces
Local density variations
The first four must be corrected for before we can “see” the local
density variations in the gravity data
All but the first of these effects must be corrected for before we can \see" the local density variations in the
gravity data.

2. Gravity Data Reduction
In the sections below we will go through each of these correction factors in more detail, giving derivations and formulas. You should refer back to equation (2.16) throughout the following:
Notes: i) All corrections have a positive sign by convention
ii) Assumes drift and Eötvös correction are already carried out

3. Gravity surveys are always relative to some reference value:
For absolute gravity measurements, you will usually give your data in terms of the difference from the reference spheroid (i.e. gt=gr)
For local surveys we simply establish a “baseline” gravity value to use as the “theoretical” background value. Usually this baseline value is far away from the target.
When carrying out a latitude correction there are two separate considerations:
1. Are the data being tied into absolute gravity measurements (available worldwide)? If this is the case then it is crucial that your survey be meaningful in absolute terms. Thus you will want at least one reference point in your survey to be an absolute gravity recording, and for this point you will need to use the correct value of gt (the reference spheroid) in equation (2.16). From then on all your measurements can be relative measurements with respect to this base value. The equation for the reference spheroid was given earlier as
gr = 978; 031:8 (1+ 0: sin2
2. For local changes in latitude equation (2.7) is somewhat inaccurate, since the latitude changes will be very small indeed. Instead we work out from this what the change in gt is for small changes in latitude by using a rst order Taylor expansion formula:
(2.18)
It is, however, more convenient to measure distance, rather than latitude changes: Since an arc-length in a North-South direction ds = Red, therefore if we move a distance s in a North-South direction the gravity will change by an amount
= 0:811 sin 2 mgal/km (2.19)
(per km of movement toward the equator).
Note that this quantity is zero at the equator and poles. It works out to 0.01 mgal every 13 m at a latitude of 45.

4. Latitude corrections:
Because of the earth’s shape, and its rotation, changes in latitude alone create significant changes in gravity. This must be estimated and corrected.
In order to estimate the change with latitude, we use
and , thus
When carrying out a latitude correction there are two separate considerations:
1. Are the data being tied into absolute gravity measurements (available worldwide)? If this is the case then it is crucial that your survey be meaningful in absolute terms. Thus you will want at least one reference point in your survey to be an absolute gravity recording, and for this point you will need to use the correct value of gt (the reference spheroid) in equation (2.16). From then on all your measurements can be relative measurements with respect to this base value. The equation for the reference spheroid was given earlier as
gr = 978; 031:8 (1+ 0: sin2
2. For local changes in latitude equation (2.7) is somewhat inaccurate, since the latitude changes will be very small indeed. Instead we work out from this what the change in gt is for small changes in latitude by using a rst order Taylor expansion formula:
(2.18)
It is, however, more convenient to measure distance, rather than latitude changes: Since an arc-length in a North-South direction ds = Red, therefore if we move a distance s in a North-South direction the gravity will change by an amount
= 0:811 sin 2 mgal/km (2.19)
(per km of movement toward the equator).
Note that this quantity is zero at the equator and poles. It works out to 0.01 mgal every 13 m at a latitude of 45.

5. “Free air” correction:
and
Therefore
is the correction per metre of elevation change.
Free air correction
Once we have corrected the readings for motion (the Eotvos correction), for drift, and for latitude, our readings are now responding to local eects. However, not all of these local effects are related to geology. The first local effect we want to remove is the effect of any elevation changes in the instrument during the survey.
We know from Newton's law of gravitational attraction that as we move away from the centre of the earth the gravitational attraction falls o as the square of the distance. If we climb a hillside while gravity surveying, we are moving away from the centre of the earth: we can work out the exact amount of the gravity effect using a similar rst order method similar to that used above:
and the constant term,
2gRe= 0:3086 mgal/m
(i.e., per metre change of elevation). Note that if r is positive (increases in elevation), g is negative (the meter reads less gravity), hence the correction
gFA = +0:3086 mgal/m: (2.20) Note that for an eld accuracy of 0.01 mgal, the elevation must be known to within 3 cm. Note also that we ignore any excess mass (i.e. due to the hillside we are climbing) at this stage - for this reason the correction is known as the \free air" correction. Free air \anomaly": If no further corrections are carried out, then the result is known as the \free air anomaly". Sometimes the free air anomaly is presented, without further corrections.

6. “Bouguer” correction:
Free air correction does not account for excess mass of hillsides, or mass deficiency of valleys
Bouguer correction (approximately) accounts for change in mass, by assuming we replace terrain by an infinite slab
Thus the Bouguer correction is an over-correction
Thanks to the inverse square law, which in the limit approaches zero rapidly, the correction is reasonable
The small errors introduced are corrected in the next stage (the Terrain correction”)
Bouguer correction
In the previous correction factor no account was taken of the excess mass of the hillsides we climb, or the mass deciency due to valleys we descend into. The purpose of the Bouguer correction is to account for this change in mass close to the gravimeter. Taking into account complex topography seems like a dicult task, but the Bouguer correction is based on a simple approximation: In the Bouguer correction we assume that we may replace the terrain underneath (or above) the gravity meter by an innite slab of country rock (or air) | quite a drastic assumption at rst sight! As a result, the Bouguer correction will inevitably be an over-correction (there will always be less of mass anomaly than that contained in an innite slab The error in the Bouguer correction is smaller than might initially be thought, thanks to the inverse
square law| the eect of the approximation is less and less important as we calculate the contributions for far distances, since
approaches zero rapidly as the distance, r increases.

7. “Bouguer” correction:
Formula for the gravitational attraction of an infinite (cylindrical) slab is
where ρ is the density (in gm/cm3) of the excess mass (or mass deficiency). Often the approximation ρ ~ 2.67 gm/cm3 is used, in which case:
The formula for the gravitational attraction of an innite slab is derived by integrating over a cylindrical volume and letting the limits go to innity (your G249 notes contain the derivation, which is tedious but straightforward). The result is
gB = 2G = 0:0419 mgals/m (2.21)
where is the density (in gm/cm3) of the excess mass of the innite slab (i.e., of the hillside), or the density of the missing mass (in the case of a valley). For accurate corrections this quantity must be estimated externally | often an approximate value is used of 2:67gm/cm3 (the average density of crustal rocks). Note, if we take the approximate value of 2:67 and combine the free air correction, equation (2.20) with the Bouguer correction (2.16), we obtain a formula for the combined elevation correction of gelevation = (0:3086
Bouguer anomaly The gravity field we obtain at this stage is referred to as the \Bouguer anomaly"; again this is sometimes presented without further correction.

8. “Terrain” correction:
An overlay (a “graticule”) is used to compute the contributions in various sectors of a topographic map
For modern, large scale work this is automated digitally together with the digital topo maps
Terrain correction
Once the latitude, free-air and Bouguer corrections have been carried out, this result is often suciently accurate for interpretation purposes (see Figure 2.9, below). However, in very rugged terrain the gravitational attraction of nearby hillsides (or the missing gravity due to nearby valleys) will aect the gravity enough to disturb the interpretation. The Terrain correction is made in order to correct for the small errors left over after the Bouguer correction (due to the innite slab approximation).

9. “Terrain” correction:
To make a topographic correction requires access to a detailed topographic map. The basic principle is that a \graticule" (an overlay, see Figure 2.10) is used to compute the contributions in various sectors of the map by averaging the elevation within each sector and summing the results. For small scale surveys this can be done manually, for large scale digital work the terrain correction can be automated using modern GIS software.

10. the result after the free air correction
Free air anomaly
gFA
the result after the free air correction
sometimes useful in interpretation (see examples)
To make a topographic correction requires access to a detailed topographic map. The basic principle is that a \graticule" (an overlay, see Figure 2.10) is used to compute the contributions in various sectors of the map by averaging the elevation within each sector and summing the results. For small scale surveys this can be done manually, for large scale digital work the terrain correction can be automated using modern GIS software.

11. “Isostatic” correction:
Very large scale features (i.e., mountain ranges) will have corresponding isostatic, low density, roots in the base of the crust
Resulting mass deficiency causes a gravity low on the Bouguer map
Bouguer slab effect of the crustal root can also be calculated, and removed
Generally only used on a large scale
Terrain correction
Once the latitude, free-air and Bouguer corrections have been carried out, this result is often suciently accurate for interpretation purposes (see Figure 2.9, below). However, in very rugged terrain the gravitational attraction of nearby hillsides (or the missing gravity due to nearby valleys) will aect the gravity enough to disturb the interpretation. The Terrain correction is made in order to correct for the small errors left over after the Bouguer correction (due to the innite slab approximation).

12. Gravity interpretation and gravity modelling

13. Gravity interpretation and gravity modelling
The rst step in any initial examination of a gravity prole, or a gravity map is to attempt to separate regional trends from local anomalies. Generally, in mineral prospecting and in engineering gravity surveys it is the local anomalies that are the primary objectives. The principle is illustrated in Figure The reason that this works is that large scale (i.e., long wavelength) anomalies generally arise from deep crustal or upper mantle sources; shallower anomalies have a much more rapidly varying, short wavelength signature. Hence the separation illustrated in Figure 2.12 has the eect of isolating eects due to relatively shallow density anomalies.
First step: separation of regional trends / local anomalies
large scale (long wavelength) features generally arise from deep crustal or upper mantle sources
shallower density variations have more rapidly varying, short wavelength signatures

14. Gravity modelling Further analysis requires some basic relations
simple models, such as the spherical density anomaly are very helpful
To make any further, quantitative interpretation of the anomaly requires an understanding of the basic relations between density anomalies and the resulting gravity anomalies. Such an understanding is gained by analyzing very simple models, such as the model of a spherical anomaly shown below in Figure 2.13.

15. Gravity modelling Point mass (or sphere): Vertical component:
where G is the gravitational constant and r is the distance to the centre of the sphere. Note that in this case the mass Me is the excess mass (or, if negative, the mass deciency) and the gravitational acceleration g is the gravity anomaly. From the Figure above we have r2 = x2 + y2 + z2 and cos = z=r . We are interested in the vertical component of the acceleration, gz = g cos : Thus

16. Gravity modelling Vertical component: Mass excess: and Thus:
The mass excess, Me can be calculated from the size of the sphere and the density contrast with the surrounding material, :
Me =43a3: (2.24)
If we use this in equation (2.23), and substitute the numerical value of G = 6:67 (where rho is measured in gm/cm3 and x; y and z are measured in metres).
Thus:

17. Gravity modelling At x=y=0:
On this Figure two gravity readings have been identied: gz(max), the maximum reading (directly over the top of the body), and readings at exactly half of this maximum reading. The x coordinate at which the reading falls to the half-maximum, in either direction, is known as the \half-width" of the anomaly, x1=2. By inserting x = y = 0 in equation (2.25), we find
Plot the function against the location of the gravimeter, x

18. Gravity modelling: depth estimation
The “half-width”:
On this Figure two gravity readings have been identied: gz(max), the maximum reading (directly over the top of the body), and readings at exactly half of this maximum reading. The x coordinate at which the reading falls to the half-maximum, in either direction, is known as the \half-width" of the anomaly, x1=2. By inserting x = y = 0 in equation (2.25), we find

19. Gravity modelling: depth estimation
This:
must fall by a factor of 2,
or:
On this Figure two gravity readings have been identied: gz(max), the maximum reading (directly over the top of the body), and readings at exactly half of this maximum reading. The x coordinate at which the reading falls to the half-maximum, in either direction, is known as the \half-width" of the anomaly, x1=2. By inserting x = y = 0 in equation (2.25), we find

20. Gravity modelling: depth estimation
Solving for the half-width:
The depth of the centre of the body is about 1.3 times the half-width of the anomaly
this does not depend on the density contrast

21. Non-uniqueness Each body yields the same gravity profile
the spherical model is the deepest
Thus we should write:
Relationship (2.28) was derived for a perfect sphere model. It is important to note that such a model gives the narrowest possible anomaly (remember, the formula is identical to that of a point mass). If the body were broader in shape, then a similar half-width could be obtained from a shallower body (see Figure 2.14). Thus we should write

22. Alternative depth estimation
deeper anomalies generate wider, gentler shapes
signature from shallow anomalies varies more rapidly
suggests the derivative dg / dx may be useful
Differentiating the equation and finding the maximum, we obtain
for “3-D” shapes
for “2-D” shapes
An alternative depth estimate may be generated by looking at the slope of the gravity prole. Clearly deeper anomalies generate wider, gentler anomaly shapes (hence, the signature from the deep crust and upper mantle is very long wavelength, while the signature from shallower anomalies varies much more rapidly. If we dierentiate equation (2.25) and extract the maximum slope (the maximum value of dg=dx) we obtain another limiting depth estimate
: (2.30)
Equation (2.30) is suitable for \3-D" anomalies (i.e., those with roughly circular gravity contours). In cases in which the contours are elongated (i.e., there is an apparent strike direction), there is an equivalent \2-D“ formula, derived from cylindrical anomaly shapes:

23. Excess mass estimation
Re-arrange Newton’s law:
Substitute the depth estimate (from half-width):
Note: this is excess mass. To compute actual mass use:
(ρ1 is the bulk density of the target, ρ2 is the bulk density of the host rock).
We can also estimate, very simply, the total excess mass contained in the anomaly: if we re-arrange Newton's law (2.22), and substitute x = y = 0 we have
: (2.32)
Further substitute our expression (2.28) for the depth into the result, we obtain2
(2.33)
(the half-width is measured in meters, and gzmax is measured in mgal). This estimate of the excess mass is quite robust (i.e. the anomaly may look quite dierent from the gure, but the half-width and the maximum reading are still useful in the equation). However it is important to understand th at the gravity survey says next to nothing about the concentration of this excess mass: the mass may be concentrated in a very small region, or spread out over a very large sphere { either way the anomaly remains the same. It should be noted that it is always the excess mass we calculate with formulas such as equation (2.33).
To compute the actual mass from the excess mass we would use (2.34) where 1 is the bulk density of the anomalous body, and 2 is the density of the host rock. Clearly the accuracy of the density estimates is the limiting factor on calculations of total mass.

24. Density contrast estimation
density contrast, Δρ cannot be determined uniquely from the data
we can put a limit on the smallest contrast that could explain the data
the largest possible sphere for a given depth extends to the surface:
We can also estimate, very simply, the total excess mass contained in the anomaly: if we re-arrange Newton's law (2.22), and substitute x = y = 0 we have
: (2.32)
Further substitute our expression (2.28) for the depth into the result, we obtain2
(2.33)
(the half-width is measured in meters, and gzmax is measured in mgal). This estimate of the excess mass is quite robust (i.e. the anomaly may look quite dierent from the gure, but the half-width and the maximum reading are still useful in the equation). However it is important to understand th at the gravity survey says next to nothing about the concentration of this excess mass: the mass may be concentrated in a very small region, or spread out over a very large sphere { either way the anomaly remains the same. It should be noted that it is always the excess mass we calculate with formulas such as equation (2.33).
To compute the actual mass from the excess mass we would use (2.34) where 1 is the bulk density of the anomalous body, and 2 is the density of the host rock. Clearly the accuracy of the density estimates is the limiting factor on calculations of total mass.
here the same mass is distributed over the largest possible volume
this represents the smallest possible density contrast

25. Next lecture: Complex models
modelling may be carried out analytically (for standard shapes)
more commonly today modelling is carried out numerically
relationship between gravity profile, depth, shape, density contrast is always non-unique
concept of a “limiting depth” is usually helpful: narrow, short wavelength features cannot come from deep models
excess mass can always be uniquely determined
for gridded data, we can use
The spherical and cylindrical models used in the previous section may seem rather limited in application, however we may draw several general conclusions from these models: Modelling for more complex anomaly shapes can be carried out analytically (for standard geometrical
shapes), or, more commonly by numerical computing methods. Telford et al. contains equations for gravity prole shapes for a wide range of density anomaly shapes (cylinders, thin sheets, faulted beds, dikes, etc).
The relationship between the gravity prole and the depth and shape of the density anomaly is always non-unique | alternative models can always be proposed that will t the data, but the concept of a limiting depth is helpful. Narrow, short wavelength anomalies cannot come from deep structures.
Excess mass (related to total tonnage of potential ore for example) can always be uniquely determined, provided the entire Bouguer anomaly has been captured on the gravity map. In cases of complex contour shapes the gravity survey may be divided into grid squares. If the area of each grid square
is ai and the average gravity anomaly within each grid square is gi, then the total excess mass is given by
(2.35)
(this result is derived from Gauss' Theorem, see Telford et al, P9).
Analytical results using the same methods we used above for the sphere are available for a wide range of anomaly shapes in Telford et al. (horizontal rod, vertical cylinder, thin dipping sheet, horizontal sheets). While the mathematical developments are tedious and perhaps no longer required, it is worth looking at the results of these calculations to get a feel for the controlling factors on gravity proles. More complex shapes can be modelled using computer programs which usually operate by numerically dividing up the body into rectangular cells (or, in 3-D, into volume elements) and summing the point mass
contributions from all of the cells.