1
Applied Geophysics – Corrections and analysis
Corrections and analysis
Reading: Today: p22-39Next Lecture: p39-64
Gravity:
Applied Geophysics – Corrections and analysis
Gravity corrections
Measure gravity variations at stations around loop
Correct for drift
Observations still subject to extraneous effects unrelated to subsurface geology
Must make corrections…
1. Latitude correction
2. Free-air correction
3. Bouguer correction
4. Terrain correction
2
Applied Geophysics – Corrections and analysis
Latitude correction
)2sin0000059.0sin0053024.01(780318.9 22 φφφ −+=g
…correct for the spheroid
Geodetic Reference System (GRS-1967) formula
φφ 2sin12.8=C
m/s2
For smaller scale studies we can simplify:
g.u./km-N-S
Correction is greatest at mid-latitudes. For 0.1 g.u. accuracy need relative N-S distance to 12 m
Add or subtract?
Applied Geophysics – Corrections and analysis
Free-air correction
hCF 086.3=
2E
E
RGM
g =Accounts for the 1/r2 decrease in gravity with distance from the center of the Earth, recall:
We calculate:
For 0.1 g.u. accuracy stations elevation is needed to 4 cm.
Add or subtract?
where h is the elevation in meters
3
Applied Geophysics – Corrections and analysis
Bouguer correction
ρhCB ∆= 000419.0
∆h
Accounts for rock thickness between current and base station elevation
Treat the rock as an infinite horizontal slab:
For 0.1 g.u. accuracy station elevation is needed to 9 cm
Add or subtract?
where ∆h is in m and ρ is in km/m3
Applied Geophysics – Corrections and analysis
Terrain correctionWhen Bouguer correction is inadequate, also use terrain correction
hill
valley
Add or subtract?
Approaches to correctionRectangular gridHammer segments … both use elevation differences
between station and surround
4
Applied Geophysics – Corrections and analysis
Hammer correctionTerrain correction
Applied Geophysics – Corrections and analysis
Free-air anomaly
A gravity “anomaly” suggests the difference between a theoretical and observed value.
Tie survey to base station where absolute gravity is known
Then the free-air anomaly is:φgCgg FobsF −+=∆
Have not corrected for topography
• Onshore: anomaly map similar to topography
• Free-air anomaly mainly used for offshore
5
Applied Geophysics – Corrections and analysis
Bouguer anomaly
TBFobsB CCCCgg +−++∆=∆ φ
Apply all the corrections:
Smaller scale engineering/environmental surveys:
• Not tied to absolute gravity
• Use corrections with accuracy necessary
…determined by the size of the target signal
…watch the signs!
Applied Geophysics – Corrections and analysis
Field determination of density
Why do we need rock density?
Nettleton method: density profile
Other methods:
• The same problem can be formulated mathematically and solved by least squares
• Borehole logging
6
Applied Geophysics – Corrections and analysis
Analysis and interpretation
Once we have made our gravity observations, corrected for surface effects,
we attempt to deduce sub-surface structure
Considerations:
• Anomaly profile (2D structure) or map (3D structure)?
If anomaly length > twice the width a 2D interpretation is OK
• Ambiguity
There are an infinite number of structures that could generate the observations
• Forward calculation
“Guess” at structure, calculate the anomaly and compare. Simple formula for depth and size of geometric shapes.
• Inverse modeling
Directly invert for structure, choose constraints on the geometry …don’t forget that ambiguity.
Applied Geophysics – Corrections and analysis
Buried sphere
( )[ ] 23222
3
11
34
zxzGRgz
+
∆=∆
ρπ
Analytic expressions for simple geometric shapes
e.g. a buried sphere
21302.1 xz =Depth rule
Note: it is only the density contrast that is important
7
Applied Geophysics – Corrections and analysis
Gravity anomaly map
Applied Geophysics – Corrections and analysis
Simple shape anomalies
8
Applied Geophysics – Corrections and analysis
2D vertical column
∆=∆
1
2ln2 rrbGgz ρ
Gravity anomaly due to a two dimensional vertical column:
Applied Geophysics – Corrections and analysis
2D vertical columns
∑
∆=∆
i i
iiiz r
rbGg 1
2
ln2 ρ
Gravity anomaly due to a series of vertical columns:
∆=∆
1
2ln2 rrbGgz ρ
Gravity anomaly due to a two dimensional vertical column:
9
Applied Geophysics – Corrections and analysis
Spreadsheet: Grav2Dcolumn
Gravity Anomalies: 2D forward calculation for rectangular parallelepipeds with greater vertical extent than horizontalsee Dobrin and Savit eq 12-34
Define density structure
Adjust bold numbers…coulum center (km)
density contrast (g/cm3)
top (km)
bottom (km)
error check
0 0.5 0 0 OK1 0.5 0 0 OK2 0.5 0 0 OK3 0.5 0 0 OK4 0.5 2 8 OK5 0.5 0 0 OK6 0.5 0 0 OK7 0.5 0 0 OK8 0.5 0 0 OK9 0.5 0 0 OK10 0.5 0 0 OK11 0.5 0 0 OK12 0.5 0 0 OK13 0.5 0 0 OK14 0.5 0 0 OK15 0.5 0 0 OK16 0.5 0 0 OK17 1 0 0 OK18 0.5 0 0 OK19 0.5 0 0 OK20 0.5 0 0 OK
Calculated gravity anomaly
0.00
0.50
1.00
1.50
2.00
2.50
3.00
3.50
4.00
4.50
0 2 4 6 8 10 12 14 16 18 20distance (km)
dgz
(mG
al)
0
1
2
3
4
5
6
7
8
9
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20de
pth
(km
)
Applied Geophysics – Corrections and analysis
Spreadsheet: Grav2Dcolumn
Gravity Anomalies: 2D forward calculation for rectangular parallelepipeds with greater vertical extent than horizontalsee Dobrin and Savit eq 12-34
Define density structure
Adjust bold numbers…coulum center (km)
density contrast (g/cm3)
top (km)
bottom (km)
error check
0 0.5 0 0 OK1 0.5 0 0 OK2 0.5 7 9 OK3 0.5 6 10 OK4 0.5 5.5 9.5 OK5 0.5 5 9 OK6 0.5 4.7 8 OK7 0.5 4.5 7 OK8 0.5 4.4 6 OK9 0.5 4.3 5.5 OK10 0.5 0 0 OK11 0.5 0 0 OK12 0.5 0 0 OK13 0.5 0 0 OK14 0.5 0 0 OK15 0.5 0 0 OK16 0.5 0 0 OK17 1 1 2 OK18 0.5 0 0 OK19 0.5 0 0 OK20 0.5 0 0 OK
Calculated gravity anomaly
0.00
1.00
2.00
3.00
4.00
5.00
6.00
7.00
8.00
9.00
10.00
0 2 4 6 8 10 12 14 16 18 20distance (km)
dgz
(mG
al)
0
2
4
6
8
10
12
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
dept
h (k
m)
10
Applied Geophysics – Corrections and analysis
Ambiguity - I
Applied Geophysics – Corrections and analysis
Ambiguity - II
( )[ ] 23222
3
11
34
zxzGRgz
+
∆=∆
ρπ
11
Applied Geophysics – Corrections and analysis
Spreadsheet: Grav2Dcolumn
Gravity Anomalies: 2D forward calculation for rectangular parallelepipeds with greater vertical extent than horizontalsee Dobrin and Savit eq 12-34
Define density structure
Profile 1 Profile2Adjust bold numbers… Adjust bold numbers…
coulum center (km)
density contrast (g/cm3)
top (km)
bottom (km)
error check
density contrast (g/cm3) top (km)
bottom (km)
error check
0 0.3 0 0 OK 0.9 0 0 OK1 0.3 4 4.4 OK 0.9 0 0 OK2 0.3 4 4.4 OK 0.9 0 0 OK3 0.3 4 4.3 OK 0.9 0 0 OK4 0.3 4 4.3 OK 0.9 0 0 OK5 0.3 4 4.4 OK 0.9 0 0 OK6 0.3 4 4.4 OK 0.9 0 0 OK7 0.3 4 4.5 OK 0.9 0 0 OK8 0.3 4 4.6 OK 0.9 0 0 OK9 0.3 4 4.7 OK 0.9 0 0 OK10 0.3 4 4.8 OK 0.9 8 12 OK11 0.3 4 4.7 OK 0.9 0 0 OK12 0.3 4 4.6 OK 0.9 0 0 OK13 0.3 4 4.5 OK 0.9 0 0 OK14 0.3 4 4.4 OK 0.9 0 0 OK15 0.3 4 4.4 OK 0.9 0 0 OK16 0.3 4 4.3 OK 0.9 0 0 OK17 0.3 4 4.3 OK 0.9 0 0 OK18 0.3 4 4.4 OK 0.9 0 0 OK19 0.3 4 4.4 OK 0.9 0 0 OK20 0.3 0 0 OK 0.9 0 0 OK
Gravity anomaly
0.00
0.50
1.00
1.50
2.00
2.50
0 2 4 6 8 10 12 14 16 18 20distance (km)
dgz
(mG
al)
Prof ile 1Prof ile 2
Profile 1
0
2
4
6
8
10
12
14
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
dept
h (k
m)
Profile 2
0
2
4
6
8
10
12
14
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20