Gravity Data Acquisition

Establishing base stations

tie to IGSN71 or other absolute reading if possible; below are some OK stations; click to see values:

establish local base
- close to or within survey area
- stable, permanent, easy to find
- good control on x, y, z (benchmark, road intersection, airport)
can establish new base by looping from old base

Determining Elevations

surveying (costs >= gravity survey!)
topo sheet
inertial guidance system
altimeter
GPS, DGPS: GPS Primer
RTK dual-frequency DGPS (2 units with real-time communication/correction between them) gives cm-level accuracy in seconds; cost (Feb. 2001) is about $45K
DEMs 7.5-Minute DEM: 30x30-meter data spacing
- Sample DEM: Contours, Color Shaded, Shaded Relief, Map
More on DEMs (10 m vs. 30 m)

Determining Horizontal Position

same as elevation
GPS: ~$100 GPS gives x, y to about 10 m, elev. to about 100-500 m

Adjusting Observed Gravity

Tidal correction

secular variations in g are (generally) undesirable
effect of Sun about 50% of Moon
each has 12-hour period (front and back bulge),
but tidal inequality...
2 effects:
- pull of bodies on meter
- distortion of Earth (solid earth tide); adds about 12% to this effect
total magnitude about 0.2, 0.3 mgal
to correct:
- fixed recording gravimeter
  - located in or near survey area
  - subtract variations from survey data
  - probably most accurate correction, but $$
- tide tables (gravity)
  - read tidal correction for given time and location
  - many not apply well near water
  - no longer published
- calculate tidal effect
  - computer program yields correction for time and location
  - incorporated into Scintrex meter
- include in drift correction
data from Sandia's Lacoste and Romberg Model G meter, sitting in Gilbert's lab, connected to chart recorder (light line); dark line from my tide computer program

http://www.livescience.com/environment/070228_beijing_anomoly.html

Drift Correction

secular variations in g are undesirable
instrument drift (DT, DP, creep), tides
assumptions
- changes are smooth and slow
- changes are independent of location
drift estimated by reoccupation of (base) station
must reoccupy every 2 hours or so:

using same base for entire survey not practical (or necessary):

Station

Time

Dial Divisions

Drift Rate

Elapsed Time

Correction

Corrected Reading

Base

11:20

762.71

GN1

11:42

774.16

GN2

12:14

759.72

GN3

12:37

768.95

GN4

12:59

771.02

Base

13:10

761.18

shift correction; the next day...

Station

Time

Dial Divisions

Shift

Drift Rate

Elapsed Time

Correction

Corrected Reading

Base

10:20

763.68

GN5

10:42

775.16

GN6

11:14

765.42

GN7

11:37

765.35

GN8

11:59

770.32

Base

12:10

760.28

Solution to above; don't look at this until you've tried it yourself!
3-point drift correction
- assume Dg = at² + bt + c
- evaluate a, b and c using Dg known at one station at t₁, t₂ and t₃
- _{How Excel stores dates and
  times}
- Excel spreadsheet example

Latitude correction

Geodetic Reference System Formulae refer to theoretical estimates of the Earth's shape
From these GRS formulae we obtains International Gravity Formulae (IGF)
Several different formulae have been adopted over the years
In these equations, is geographic latitude and is commonly referred to as theoretical gravity or normal gravity
- First internationally accepted IGF was 1930:
- This was found to be in error by about 13 mgals; with advent of satellite technology, much improved values were obtained.
- The Geodetic Reference System1967 provided the 1967 IGF:
- Most recently IAG developed Geodetic Reference System 1980, leading to World Geodetic System 1984 (WGS84); in closed form it is:
The IGF value is subtracted from observed (absolute) gravity data. This corrects for the variation of gravity with latitude

Latitude correction: short form

note that over small range, curve is nearly straight-line slope

lat_short.gif (1538 bytes)

where f is a typical latitude for the (small) field area
miles north, in above formula
subtract this amount for each mile north

Free-air correction

"flag-pole" correction
accounts for decrease in g(r) due to change (increase) in elevation (r)
does not take in into account mass which may be elevating you
could pick datum (e.g., sea level), compute "exact" difference, but over small changes in elevation, h, change is nearly linear
note that, once IGF is removed, we take elevations relative to sea level, not center of Earth!

Taylor Series

this is the free-air effect: g decreases (hence sign) with elevation
alternative "derivation":
how big an effect? take g = 9.83 m/s²; r = 6371 km,
correction: add 0.3086 times elevation in meters

Bouguer correction

"bulldozer" correction

accounts for mass (rock) responsible for elevation change (between observation point and sea level, usually)

depends on density of material holding you up (Bouguer density)

gravity due to infinite slab, thickness h (m), density r (g/cm³):

slab holding you up increases g; must subtract this effect out

requires knowledge of h and r

Burger Table 6-2

Selecting Reduction Density

while sea level may be datum, variation in elevation occurs in near subsurface
methods for selecting Bouguer density:

1. Standard Bouguer density = 2670 kg/m³

nominally average crustal density
ensures continuity between surveys

2. Direct measurement

collect samples, core, drill samples, hand samples
inaccessibility; may not be representative

3. Geologic map to get rock type; get values from tables, graphs, etc.

Rock Type	Density
ice	880 - 920
sea water	1010 - 1050
shale	1950 - 2700
limestone, dolomite	2500 - 2850
sandstone	2100 - 2600
soil & alluvium	1650 - 2200
rock salt	1850 - 2150
felsic igneous rocks	2550 - 2750
mafic igneous rocks	2700 - 3000
ultramafic rocks	3000 - 3300

4. Density profile (Nettleton method)

collect closely-spaced g readings over topographic feature
make latitude, free-air correction
make Bouguer correction, with various values of r
find r which gives least correlation with topography

5. Logs

gamma-gamma density logs
- g-ray source (usually Cobalt), geiger counter
- when g-rays (photons) collide w/ electrons in rock, Compton scattering causes them to be reflected
- intensity of reflected beam depends on density of electrons, which depends on density

neutron density logs

seismic velocity logs (due to rough correlation of density w/ velocity)

borehole gravity meter (see www.edcon.com)

measures Dg to 0.01 mgal
must overcome small hole size, hostile environment, self-leveling
depth of investigation ~ point separation
best method for getting true formation density

6. Linear regression (least squares) method

assumes no correlation between topography and subsurface density (i.e., anomalies are randomly distributed with respect to elevation)
therefore correlation between topography and g will be due to Bouguer slab
plot Dg_fa vs. elevation, h
fit line through points
slope will approximate 2pGr; solve for r (Bouguer density)

Least Squares Fit

data pairs: x₁, y₁; x₂, y₂;...x_n, y_n, where n is the total number of data points
straight line: y = mx + b, where m is the slope, b the intercept
minimize the sum of the squares of the residuals (SSR):
setting derivatives = 0 gives the normal equations:
these two equations are used to solve for unknowns, m and b. For homework, show that:
Problems with linear regression method for getting density:
if density varies with elevation, will get curve

Bouguer correction at sea, underground

surface survey
- first term replaces water with crust
- second term Bouguer corrects to sea level
underwater survey (for accurate value at sea)
underground survey

Other corrections to gravity at sea

not a stable platform as in land gravity
accelerations can be of order , >> accuracy desired
corrections also apply to airborne gravity

Acceleration correction

ship can't accelerate up or down very long, so average over t eliminates a_z (not so for aircraft)
must know a_x, a_y (accelerometers, gyroscope to keep accelerometers level)
natural period of land g meter typically ~10s, but waves also ~10s, so ship gravimeter must have much longer period

Eotvos correction (Nettleton, p. 116 - 118)

important for shipborne and airborne gravity
component of velocity in E direction increases apparent angular velocity (Coriolis Force)
biggest single source of error in shipborne gravity comes from error in V, a (although GPS, particularly DGPS, probably helps significantly)
centrifugal acceleration
component in vertical direction
effect in change in angular velocity, w (Dw will be small compared to w on ship or even plane)
if velocity is V, E component is
angular velocity due to this motion is
so, da_V is given by
there is also a simple acceleration in vertical direction due to total velocity, V
Total Eotvos Correction
For V in knots, correction in mgals
Example: 10 knots E at equator => 75 mgals!

Terrain correction

Bouguer correction assumes infinite slab; terrain correction corrects for this erroneous assumption!

Bouguer correction works for gently sloping surfaces, like topography on basement

error < 3% for slope < 1/5 (see Adams and Hinze, vol. 3, SEG Geotech. & Environ. Gphy., p.99)

always positive

requires detailed info on elevation

size of correction depends on relief

hammer_zones.gif (30798 bytes)

Burger Table 6-3

Hammer Terrain Correction Chart

Terrain Correction with DEMs

advent of DEMs has made medium to far-field correction much easier
short-field correction may be done using "newly available, reflectorless laser rangefinders. Such rangefinders permit a detailed digital terrain data to be acquired in the vicinity of a gravity station within only 2-3 minutes, permitting the gravity meter operator to acquire the terrain data needed ... at the same time as the gravity measurements are made."
Geodesy Group at Curtin University - Terrain Effect
Geophysical Software
Sample DEM: Contours, Color Shaded, Shaded Relief, Map

A nautical mile is accepted for use with the International System of Units (SI), but it is not an SI unit. It used for maritime, aviation purposes and is commonly used in international law and treaties, especially regarding the limits of territorial waters.

The international standard definition is: 1 nautical mile = 1852 meters exactly.

The nautical mile was historically defined as a minute of arc along a meridian of the Earth. It can therefore be used for approximate measures on a meridian as change of latitude on a nautical chart.

Nations had different definitions of the nautical mile. International agreement was achieved in 1929, when the International Extraordinary Hydrographic Conference held in Monaco adopted a definition of one (1) international nautical mile as being equal to 1,852 metres exactly. This value is very close (within 0.01 percent) to the average length of one minute of latitude (1852.235 m).

Free-air, Bouguer, Isostatic Anomaly [KNOW THESE DEFINITIONS]

for these definitions, elevations are elevations above mean sea level (AMSL)
latitude and free-air correction virtually always made, giving the Free-Air Anomaly (FAA)
for environmental/exploration work, some Bouguer correction is made
for gravity in mgals and elevation in meters, then
The Standard Bouguer Anomaly uses Bouguer density of 2.67 g/cm³
note that, once r is chosen, FAC and BC can be combined into one correction: the elevation correction
the Complete Bouguer Anomaly also includes the terrain correction
sometimes make a correction for isostasy: Isostatic Anomaly

Estimating Survey Error

important even before survey contemplated!

dependent errors add algebraically

independent errors add vectorially

sources of error:
- meter dial
- meter consistency
- drift
- latitude
- elevation (Free-air and Bouguer dependent; partially cancel)
- terrain
- others?

Station	Time	Dial Divisions	Drift Rate	Elapsed Time	Correction	Corrected Reading
Base	11:20	762.71
GN1	11:42	774.16
GN2	12:14	759.72
GN3	12:37	768.95
GN4	12:59	771.02
Base	13:10	761.18