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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
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Spring tideNeap tide
DiurnalSemi-diurnal
Spring tide
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Coriolis Force r
FCoriolis= 2mr
V r
Vertical component decreases near the pole
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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
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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!
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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
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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*
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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'
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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
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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
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-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)