Gravity Reduction

8/13/2019 Gravity Reduction

1/12

1

Reduction of Gravity Data

1. Tidal Correction Magnitude ~0.3 mgal, variation depends on latitude and time (i.e. position of moon &

sun w.r.t. latitude & longitude)

Moon's attraction ~ 2 times that of the sun's attraction Diurnal (24 hour period) & Semi-diurnal component are dominant Usually included in strument drift correction.2. Drift Correction ~0.1 mgal/week due to creep in the springs and is usually uni-directional

To correct for Drift & Tidal effects, stations need to be re-occupied and read at leastonce every 3-4 hours (best between 1-2 hours).

If meter movement is not clamped between readings or gravimeter is subjected tosudden motion, erratic changes in reading may occur. If the instrument is bumped, it

is wise to re-read a known station immediately.

Plot the readings at the base-station as a function of time, use straight lines to connectneighboring points and to interpolate for readings at any other time gd(t).

For gravity measurements at other stations, the Drift & Tide corrected reading is theobserved value at time t minus gd(t) , the interpolated reading at time t.

3. Etvos Correction for moving platforms A gravimeter in motion experience the Coriolis acceleration -

2

r

r

V (due to earth

rotation). The vertical component of Coriolis acceleration gives Etvos acceleration.

The correction term depends on the velocity of the ship V (in km/hr), its latitude and its heading :: gE = 4.040Vcos sin + 0.001211V

2

Error in Etvos Correction due to errors in V and is :gE( ) = 0.0705Vcos cos( )d + 4.040cos sin + 0.002422V( )dV

Sensitivity to Velocity Error greatest for E-W coarse, Sensitivity to direction Error greatest for N-S coarse


2/12

Spring tideNeap tide

DiurnalSemi-diurnal

Spring tide


3/12


4/12

Coriolis Force r

FCoriolis= 2mr

V r

Vertical component decreases near the pole


5/12

200121100404 V.sincosV.gE +=

latitude

heading

North

East

Equator N Pole

Correction is

largest for

E-W direction

and at the

equator

gE( )

= 0.0705V coscos

( )d+ 4.040cossin + 0.002422V

( )dV

latitude

heading

North

East

Equator N Pole

Error largest

for heading

~45o and at

equator


6/12

2

4. Latitude Effects:Since the earth is rotating, its shape is spheroidal (to 1storder approximation) due to

the centrifugal acceleration (2d where d is the perpendicular distance from the

rotation axis and is the angular speed of Earth rotation).

T he spheroid has radius r varying with latitude according to:

r( ) a(1 fsin2 )

where a is the radius of Earth at the equator, c is the radius of Earth at the pole and

the flattening of Earth f =a c

a. Since a-c~21 km , a~6378 km , therefore f~1/299.

Thegravity on the spheroidcan be derived from the gravitational potential U:

U GM

r+

GMa 2

r

3 J23

2sin

2 1

2

1

22r2 cos2

The 2nd term is due to the spheroidal shape of Earth where J2 is a constant

determined by the distribution of mass and the term in brackets is the 2 nd degree

harmonic giving the spheroidal shape. The 3rdterm is the Centrifugal potential.

Since

g = U = U

r

2

+ U

r

2

U

r=

GM

r2

3GMa2

r4

J23

2sin

2 1

2

2rcos 2

Its value on the spheroid r() is thenormal gravity given by:

( ) =geq 1 +psin2 ( ) + qsin2 2( ){ }

where geq=978.0327 gal ,p=0.0053024 and q=-0.0000059. (WGS1984)

Question:How much does the latitude effect of g vary between 2 stations 100 km

apart in the north-south direction?

To do this, first determine the horizontal gradient in the N-S direction and multiply by

the separation. As shown in the next page, the "northward gradient" is latititude

dependent. Around Calgary, the gradient is about 0.8 mgal/km. Therefore, 100 km

north of Calgary, the latitude effect gives an increase of 80 mgal.

Question:If we want the latitude effect to be less than 1/100 mgal, how accurate do

we have to determine the location of the stations? Answer: about 10 meters!


7/12

g(mgal)

latitude (degrees)

sea level1 km5 km

10 km

latitude (degrees)

northgradientofg(mgal/km)

g = 978.04 1+0.00529 sin2 5.9x10 6 sin2 2 0.3086H

gisin gal, is latitude &His height in km

horizontal gradient of g northwards

ggrad =

gs =

g

s

for gradient in mgal/ km

s= 180

R

= 8.9932

g

grad= 8795 5.8 10 5 cos sin 1.31 10 7 cos 2 sin 2

1. Variation of gravity with latitude and height

Variation of horizontal gradient of g with latitude

g (mgal) above a slab of mass of density (gm/cc) and height h (m)

0.3086h + 0.042 h = 0.191h if = 2.8gm /cc


8/12

3

Direction of Gravity Vector. The gravitational acceleration is directed toward the center

of Earth g*,but the centrifugal force is directed away from the axis of rotation, thus the

"effective gravity" vector g is NOT directed toward the center of Earth.

Geoid. The shape of the Earth is not exactly a spheroid given by r( ) a(1 fsin2 )

(see diagram below). The undisturbed surface of the ocean (no wind, no current, no tides

etc. and averaged over time) is an equipotential surface (everywhere perpendicular to the

direction of the plumb line) called the geoid. Mean sea level (with tides removed) is a

good approximation to the geoid, but differs by meters (due to winds & ocean currents).

Definition of geoid on land is complicated by the upward attraction of mass above.

Gravity measured at point P' above the geoid is projected along the vertical to point P on

the geoid. Gravity observations, reduced to the geoid (but NOT to the surface of the

spheroid) are calledgravity anomaly gP =gP Q = gP + ( P Q)

If gravity observations are reduced to the spheroid (P projected to Q on the spheroid

along the normal), they can be compared with the normal gravity giving gravity

perturbation

gP =gP P =gP Q Nr

where geoid height N is the distance

between P and Q.

reference spheroid

geoid

equipotential surface

that pass thru P'

P'

P

QN

R2R

gg*


9/12

4

5. Elevation Effect (Free Air Correction):As one moves away from the center of the Earth, gravity decreases. The rate of decrease

can be deduced by assuming a spherical earth: g =GM

r2

.Thereforeg

r=

2

r

GM

r2 =

2

rg

which gives a value of -0.3086 mgal/m If we take into the account of the spheroidal

shape and the height, theng

r= 0.30855 0.00022cos(2 ) + 0.73 107 h where h is in

meters. For most of our purpose, we just use -0.3086 mgal/m. If we measure the gravity

at some height above datum, then we subtract this gravity decrease to remove the

elevation effect. Thus the Free Air Gravity Anomaly (with latitude and elevation

effects removed) can be obtained from the observed value by :

gFA =gobs ( ) + 0.3086h .

For an accuracy of 0.01 mgal, elevation must be known to within 3 cm!

6. Bouger Correction for material between station level and datum.

gB =gB' 2go

REh + 2 G h = gB' (0.3086 0.042 )h

where the 2nd term on the right is due to Free Air effect and the 3 rd term is due to

downward attraction of a slab of thickness h. Since gB =gobs ( ) is the measured

value with latitude effects removed, the above equation can be rearranged to give the

reduction formula: gB' =gB + (0.3086 0.042 )h Therefore, Bouger Anomaly (with

latitude, elevation and topographic mass effects removed) is given by:

gBouger = gFA 0.042 h

The values of g for the above formulas are in mgal if h is in meters and in gm/cc. For

example, with =2.8 gm/cc, -0.3086h+0.042 h = -0.191h.

Datum plane

h

A

B

D

E

B'


10/12

5

The standard way of defining Bouger Anomaly at sea is to replace the water at sea

(density = 1 gm/cc) by rock material of density = 2.67 gm/cc. Thus, we are in effect

adding 0.042( w = 0.070w, where w is the water depth in meters. Due to

isostasy, the Free Air anomaly is small in most oceanic regions, but the Bouger correction

above is strongly positive, thus the Bouger correction term completely masks the effects

of subsurface density variations. Furthermore, the density of the seafloor is usually

greater than the assumed density of 2.67 gm/cc. Thus, the Bouger anomaly at sea is less

useful than on land. Today, most geophysicists use Free Air anomaly at sea but use

Bouger anomaly on land.

The density of the local topographyis required for the calculation of Bouger correction.

For flat or hilly areas, the uncertainty in density do not pose a big problem. But for

mountainous regions, the problem may be severe. Nettleton proposed to find the optimal

density that gives the minimum correlation between local topography and the Bouger

anomaly. For example, the hill on the left of the diagram below is not underlain by any

lateral density variation, Bouger anomaly reduced to zero when the correct density is

used. However, for the igneous intrusion on the right, Nettleton's method do not work

because a much larger density is required to minimize the Bouger anomaly.

Note that the assumptionsof Bouger Correction are: 1) slab is of uniform density; 2) of

infinite horizontal extent. The last assumption is valid only if distance from the edge >>

h. So at stations D & E (diagram in last page), Terrain Correctionis needed.

-20

-10

0

10

20

30

BougerAnomaly(mgal) Nettleton's Method

Free Air

=2

=2.7

=4.5

=0

-600

-400

-200

0

200

400

Topography(m)

Geology

=2.7=3.0

=2.7


11/12

6

7. Terrain CorrectionConsider the reduction at station D, The infinite horizontal slab of thickness h is

deficient in 2 aspects:

1) it neglects the upward attraction of the material above D (shaded part to the right)

2) it assumed that the valley to the left is filled with rocks, thus it overestimates the

downward attraction. To correct for this, an upward attraction must be applied.

Thus, both correction terms to 1 & 2 are of the same sign!

Today, terrain corrections are performed by computers. The procedure is as follows:

1) start with a good topographic map (contour interval < 10 m) around the station andextending beyond the survey area; station locations; average density of rocks.

2) with the station as the center, concentric compartments are placed on the topo map.3) For each compartment, estimate the average elevation with respect to the elevation

of the station. The terrain gravity due to this compartment with bounding radius r1&

r2 and angle is : gi = G r12 + 2 r2

2 + 2 + r2 r1[ ] . Note that

appears as a square, this is because, as explained above, the correction for extra mass

above or mass deficit below the station have the same sign! Also the correction is

small if r1 + r2( ) 2 > 20 .

4) Sum up the contributions from all compartments to give the total Terrain Correction.No provision for relief within 2 meters from the station as any 1 m relief can have

large (>0.04 mgal) corrections.

In areas of steep and erratic slopes, terrain corrections are usually not very accurate. So it

is better to have the gravity stations located away from sharp relief (whenever possible).

Datum plane

h

D

upward attractionoverestimate

of downward

attraction


12/12

-5

0

5

10

15

20

mgal

-10 -8 -6 -4 -2 0 2 4 6 8 10

Topography Effect

Latitude Effect (0.8 mgal/ km near Calgary)

Buried Sphere at 2km depth

-5

0

5

10

15

mgal

-10 -8 -6 -4 -2 0 2 4 6 8 10

X (km)

Reduction of Gravity Data

Topo & Lat Effects removed

Latitude Effect removed

TOTAL (Observed gravity)

Illustrate that Latitude & Topo Effects can mask

gravity anomaly of target (sphere)

Documents

Gravity Reduction