Gravity: Corrections and...

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Applied Geophysics – Corrections and analysis Corrections and analysis Reading: Today: p22-39 Next 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

Transcript of Gravity: Corrections and...

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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

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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

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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

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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

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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

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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

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Applied Geophysics – Corrections and analysis

Gravity anomaly map

Applied Geophysics – Corrections and analysis

Simple shape anomalies

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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:

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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)

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Applied Geophysics – Corrections and analysis

Ambiguity - I

Applied Geophysics – Corrections and analysis

Ambiguity - II

( )[ ] 23222

3

11

34

zxzGRgz

+

∆=∆

ρπ

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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