Post on 14-Dec-2015
Problem 1
The corrections can be larger than the anomaly
Stat. TimeT
Dist. (m)
Elev. (m)
Reading(dial units)
Base readingat time T
Drift corr’d anom. (gu)
LC(gu)
FAC(gu)
BC(gu)
Free airanom(gu)
Boug.anom.(gu)
BS 0805 0 0 2934.2 0 0 0
1 0835 20 10.37 2931.3 2934.49 -12.10 -0.16 32.00 -11.73 19.74 8.01
2 0844 40 12.62 2930.6 2934.57 -15.05 -0.32 38.95 -14.28 23.58 9.3
3 0855 60 15.32 2930.4 2934.68 -16.23 -0.48 47.28 -17.34 30.57 13.23
4 0903 80 19.40 2927.2 2934.76 -28.67 -0.63 59.87 -21.95 30.57 8.62
BS 0918 0 0 2934.9 0
Note that the most significant part of the free-air anomaly appear to due to the attraction of the extra material beneath the survey stations, and that when the Bouguer correction is applied the remaining anomaly is quite small and positive – i.e. the rocks below these stations are slightly denser than those below the base station.
Problem 2
The width of the anomaly is related to the depth of burial of the target body
Set K = 2GΔρ2πb2 and determine Δg at x = 0 and x = x1/2
at x = 0 Δgmax = Kz/(0 + z2) and at x = x1/2 Δgmax/2 = Kz/(x1/2
2 + z2)
equating these two we get:
1/2 x Kz/(0 + z2) = Kz/(x1/22 + z2)
Hence 2z2 = x1/22 + z2 and zmax = x1/2
Problem 3
x (m)g (gu) x (m) g (gu)
-4 -0.008 -4 -0.015
-3 -0.014 -3 -0.025
-2 -0.03 -2 -0.051
-1 -0.102 -1 -0.1275
0 -0.51 0 -0.255
1 -0.102 1 -0.1275
2 -0.03 2 -0.051
3 -0.014 3 -0.025
4 -0.008 4 -0.015
Why is it Zmax?If the body were deeper, the width of the anomaly would increase. But you could obtain the same gravity anomaly with other bodies placed at shallower levels in the section
Lecture 2 Gravity 2
Formulae for a horizontal slab
Burial depth on width of anomaly
Modelling gravity data. Non-uniqueness
Calculation of the total mass excess/deficit
Filtering/processing of gravity data
Problems 4-5
Infinite horizontal slab
2g Gt
Example, Δρ = 400 kgm-3 t = 50m
δg = 2 x 3.14 x 400 x 6.67 x 10-11 x 50 = 8.38 x 10-6 ms-2 = 8.38 gu
Independent of depth to slab
12 tan2
xg G t
z
2g Gt
δg = 0
Change occurs more rapidly when slab is closer to the surface
Sets of theoretical curves are very useful
= 8.38 gu
Semi-infinite thick horizontal slab
12 2 1 1
2
2 1r
g G z z x nr
Start with a simple model – gradually make it more sophisticated
Modelling and non-uniquenessAn infinite number of subsurface density models will fit the gravity data equally as well
Can we constrain Δρ?
Can drill holes constrain parts of the model?
Do we have geological knowledge that limits therange of possible density models?
Do we have other geophysical data that can distinguish between density models?
Forward modelling (by nature is often subjective)Guess model, see if it fits the observed data, change model until it does fit the data
Forward modelling example
matics of crater formation
Younger, thinner Proterozoic crust is adjacent to with the thickened Archean craton, NE India
Modelling and non-uniqueness
Four models of Chicxulub (a,b,c from gravity, magnetic and borehole data)
Forward modelling cont.
Debate on shape of stratigraphic uplift
Shape tells us about thekinematics of crater formation
Inverse modelling – strive to make objectiveSet up an automated procedure to test a wide range of models and chose a range that fit the data reasonably well.
e.g. allow density to vary between 0.05 and 0.2 in steps of 0.01 g/cm3
and allow z (burial depth) to vary between 2 and 5 km in steps of 0.1 km
Mass excess/deficitGauss’s theorum: The integral of the divergence of a vector field over a region of space is equal to the outward normal component of the field over the surface enclosing the region.
gdxG
Me21
1
2Me Area under curve
G
The mass excess calculation is unique
Meaning: For any particular body, the total gravity anomaly is the samei.e the area (or volume) under the curve is the same
In 2D the mass excess per unit length is:
In 3D this would be:
gdxdyG
Me21 curveundervolume
GMe
21
Me = 1/2πG x ∑ Δg x Δa
where Δa is the area
∑ Δg = Δg1+ Δg2 + Δg3 ... etc
Regional versus residual gravity anomalies
Regional – long wavelength features due to deep seated structuresResidual – short wavelength features due to shallow structures
z = 0
z = -s
z = s
z = 2s
We measure g on surface z = 0 at stations that are a distance x apart
We can calculate the value of g on any surface we like z = s, -s etc
gm = k1g0 + K2(g1+g-1) + K3(g2 + g-2)… etc
ki depends on s and x
g
gm
g0 g1 g2 g3g-1g-2
x