Free Air and Bouguer Anomaly

Use the Gravity > Free Air, Bouguer Anomaly menu option (geogxnet.dll(Geosoft.GX.Gravity.FreeAirAndBouguerAnomaly;Run)*) to carry out the following components of a gravity reduction:

(i) Free Air anomaly calculation

(ii) Bouguer anomaly calculation

(iii) Complete Bouguer anomaly (Bouguer anomaly + terrain correction)

Free Air and Bouguer Anomaly dialog options

Application Notes

This GX assumes:

• A gravity database is currently loaded (see Import Gravity Survey).

• The tide and drift corrections have been applied to produce absolute gravity (see Gravity Corrections).

• That Latitude, Longitude ground locations are defined.

• The Latitude correction (the theoretical gravity at the station location) has been applied (see Latitude Correction).

• Free air anomaly results will be stored in the "FreeAir" channel in the database.

• The survey elevation data should be contained in an "Elevation" channel in the database.

1. Free Air Anomaly

The free air correction is calculated and added to the latitude corrected gravity. The expressions for the supported corrections are:

Where:

Hs: the station elevation (relative to the datum, positive up)in metres

L:  the latitude of the station

The last formula accounts for the non-linearity of the free air anomaly as a function of both latitude and height above the geoid.

If you would like to provide alternate equation(s), define them in the file "Gravity_Free_Air.lst" located in the folder %USERPROFILE%\Documents\Geosoft\Desktop Applications \etc.

Each expression should be defined on one line and should consist of a unique name followed by the complete expression using the following syntax:

Where:

Input_Gravity: the variable containing the input gravity channel; designated by the variable $sGravity.

In the expression, you can also use the variables:

$rEm:        the conversion factor to convert the units of elevation to metres

$sElevation: the elevation channel

$sLatitude:  the latitude channel

For example, the syntax of the standard free air correction equation can be written as:

2. Bouguer Anomaly

The Bouguer anomaly corrects the free air anomaly for the mass of rock that exists between the station elevation and the spheroid. It includes a correction for water depth and ice thickness if the "Water" and "Ice" channels are non-zero:

Bouguer Anomaly Correction Formula

For ground survey (including on lake surface) or shipborne (marine) survey:

For airborne survey:

Where:

Gb: Bouguer anomaly in milligals

Gf: Free air anomaly

D:  Bouguer density of the earth in g/cm³

Dw: Bouguer density of water g/cm³

Di: Bouguer density of ice in g/cm³

Hs: Station elevation in metres

Hw: Water depth in metres (including ice)

Hi: Ice thickness in metres

Gc: Curvature (Bullard B) correction

Hg: Ground (DEM) elevation at survey station location in metres

Water Channel

The "Water" channel contains the water depth at the gravity survey station location. These depths should be positive or zero.

Curvature (Bullard B) Correction Gc

The purpose of the curvature correction as a step in producing the Bouguer anomaly is to convert the geometry for the Bouguer correction from an infinite slab to a spherical cap whose thickness is the elevation of the station and whose radius (arc length) from the station is 166.735 km. Curvature correction depends on the station elevation (e.g. 1.1 mGal at 1000 m, 1.5 mGal at 2000 m, 1.2 mGal at 3000 m, 0.2 mGal at 4000 m, -1.5 at 5000 m and –3.9 at 6000 m), and latitude (ranging in dozens µGal). LaFehr’s formula [6], is used here for the curvature correction.

Figure 1: Schematic illustration (not to scale) of the spherical cap in Bullard B correction. The spherical cap surface radius of 166.7 Km, which is the outer radius or the Hayford-Boweie Zone O, is based on minimizing the difference between the effect of the cap and that of an infinite horizontal slab [LaFehr, 1991].

LaFehr expands the Bullard B correction calculation as:

Where:

BB: Bullard B correction

γ:  Gravitational constant

ρ:  Density

h:  Elevation (from sea level)

R:  Earth radius to the station:
     
   Where: 

								
       Ro: Earth’s sea level radius at the survey location
							

μ:  Dimensionless coefficient:
    
							

λ: Dimensionless coefficient: 				
								
    
 
								
   Where:
								
        
								
        
								
        
								
        
								
        
								
        
								
        
							
       α:  The half angle subtended at the earth’s center by the section of the earth’s surface at sea level for which the outer distance from the station is normally taken to be 166.7 km.

For context, below is what each of the Bullard corrections refer to:

• Bullard A: approximates the local topography (or bathymetry) by a slab of infinite lateral extent, constant density, and thickness equal to the elevation of the point with respect to the mean sea level.
• Bullard B: curvature correction, which replaces the Bouguer slab by a spherical cap of the same thickness to a distance of 166.735 km.
• Bullard C: terrain correction, which consists of the effect of the surrounding topography above and below the elevation of the calculation point [Nowell, 1999].

3. Complete Bouguer Anomaly

The Complete Bouguer anomaly corrects the Bouguer anomaly for irregularities of the earth due to terrain in the vicinity of the observation point. The terrain correction GXs can be used to calculate the terrain correction.

The complete Bouguer anomaly equation is:

Where:

Linear Correlation Coefficient

If more than one Bouguer anomaly channel is requested, a Linear Correlation Coefficient is calculated for each density and reported in a popup window. Normally, to select the density that yields the least correlation between the shape of the topography and the calculated Complete Bouguer anomaly, you would visually focus on sections where the topography channel varies a lot (hills & valleys) then check if the gravity follows a similar trend and disregard the correlation in the remainder of the data. With this in mind, to use the Linear Correlation Coefficient as a decision aid, calculate it for such subsets of data. Select carefully a subset where you know that the topography is evident in the gravity data and apply the Bouguer correction for multiple densities. Once you find the best density (least correlation, coefficient closest to zero) use it to calculate the Bouguer anomaly for the entire dataset.

The Linear Correlation Coefficient equation is:

Where:

: Complete Bouguer anomaly

: Elevation

The correlation coefficient values reside between -1 and 1. A correlation coefficient closer to 1 indicates a close correlation between the x (Complete Bouguer anomaly) & y (elevation) profiles. A coefficient closer to -1 indicates an inverse correlation. As the coefficient approaches 0, the correlation decreases. The correlation that is the closest to 0 will be indicated with a star symbol (*) – as the least correlated one.

The correlation coefficients for the specified densities are reported at the end of the process. Furthermore, these coefficients are saved in the project working folder under the unique file name: BouguerNettletonCrossCorrelationCoefficientsyyymmdd.log

Nettleton’s Method

Nettleton’s method is based on the observation that the use of an appropriate reduction density will remove a correlation of observed gravity with topography. In this method, a gravity profile over a study area, in which there are significant sharp variations in elevation, is reduced by using a range of Bouguer reduction densities. The reduction density with a minimum correlation with elevation would give anomalies most representative of the underlying geologic structure.

The following assumptions must hold for Nettleton's method to apply:

• Nettleton's estimate fails if the topography is dominated by deeper geological structures.
• A relatively homogeneous density distribution in the area of interest.
• Sufficiently large (> 100 m) elevation differences and a uniform distribution of gravity measurements with respect to horizontal positions 'and elevations.
• A sufficient model of the regional field to be separated from the data.

Gravitational Constant

The gravitational constant (G) is a physical constant that is difficult to measure with high accuracy and which has varied broadly over time. G is re-evaluated every four years approximately. The table below lists published figures along with the uncertainty and source.

Published G values

References