Today’s Agenda
• Gravity
- Background & basic intuition
- Data reduction
- Use of the instrument
- See textbook for detailed quantitative discussion
- Measure height of Millikan or S. Mudd
What and Why
Gravity measures the distribution of mass
• Variations in density
• Variations in interface locations
• Very non-­unique => use in conjunction with other constraints (geologic, seismic,…)
Targets include:
• Topographic compensation -­ the crustal root (long λ)
• Fault structures/offsets (best for dip-­slip faults or large offset strike-­slip)
• Aquifers
• Tunnels
• Buried bodies (tunnels, ore deposits,…)
• Basin sediment fill depth (together with magnetic field variations)
ρ2
ρ1
ρ1
ρ2
ρ1
ρ3
ρ2
Δρ is what matters!
http://gravmag.ou.edu/grav_interp/grav_interp.html
Data Availability
Global: Satellite tracking/altimetry (over water)/other
• ~2 km resolution over oceans
• ~100 km resolution from harmonic models
• None of these useful for our purposes
Regional: Carry the gravimeter around
• Land, boat, plane
• Arbitrary resolution
Basic intuition
Everything we need can be worked out from basic potential field theory (see Reynolds or Turcotte & Schubert):
∇ 2V = 0
where V is the gravitational potential
€
∂V
gz = −
∂z
where gz is the vertical component of the gravity field
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Units: 1 mGal = 10-­5 m/s2
1 g.u. = 0.1 mGal = 10-­6 m/s2
Attraction from harmonic distribution of mass I
Z
X
Δρhk cos( kx ) = Δσ k cos( kx )
Why?
• Intuition
• Decompose any mass distribution into individual harmonic (Fourier) contributions and sum linearly
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• Example: Topography, density variations on a 2D or 3D grid
Attraction from harmonic distribution of mass II
We won’t derive solutions today, but you can imagine integrating δg =
Z
Gδm
rˆ
2
r
X
Δρhk cos( kx ) = Δσ k cos( kx )
over our given harmonic mass distribution. The vertical component is:
€
€
−kz
gz = 2πGΔρhk cos( kx )e
= 2πGΔσ k cos( kx )e−kz
where k = 2π/λ, h<<λ
The e
€ -­kz term corresponds to the 1/r2 decay with distance
(the peaks and troughs tend to cancel when viewed from increasingly large distances) Attraction from harmonic distribution of mass III
gz = 2πGΔρhk cos( kx )e−kz = 2πGΔσ k cos( kx )e−kz
At long λ, k⇒0
gz = 2πGΔρh = 2πGΔσ
€
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•
•
No z-­dependence
Attraction from € an infinite sheet of mass
Will be used in making the Bouguer correction
For z≠0
• If z increases (or k increases ⇔ λ decreases): e-­kz ⇒0
• Upward continuation: e.g., ΔZ=λ, e-­2π=0.002 (faint signals)
• Downward continuation: e.g., ΔZ=-­λ, e2π=500 (amplification of short λ noise)
Measurements
Absolute
• Needs known reference station to tie to
Relative
• Small scales are fast (what we typically do)
Common corrections before getting to “interesting” signal
1. Instrument drift: return to base station regularly
2. Tides: return to base station regularly
3. Centrifugal acceleration: keep track of latitude (we wont)
4. Free Air
5. Bouguer
Free Air Correction
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•
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•
Bring all observations to a common horizontal reference surface (E.g., a datum or geoid)
Effect of distance not mass
Results in “Free Air Gravity”
Commonly used for airborne and marine data
Gm
Gm
g0 = 2 ; gh =
R
(R + h) 2
δgFA gh − g0 2g0
=
≈
= −0.3086 mGals/m
h
h
R
•
€
•
The L&R gravimeter has a sensitivity of ~3µGal
δgFA R
h=
≈ 1cm
2g0
Careful with surveying (know your target)
Bouguer Correction
Effect of mass (not distance) -­ Assume λ>>h, ignores dh/dx +… gB = 2πGΔρh
Δρ
h
h
€
What density to use? Standard is 2670 kg/m3
You can also use this to estimate density
• Nettleton’s Approach -­ Assume all variations are due to topography
• Clever differencing of gravity profiles is sometimes useful
-­ Over hill vs through roadcut (removes regional trend)
x
More sophisticated approaches
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Terrain corrections (graphical)
Parker’s Method (finite amplitude harmonic summation)
Common Shapes (Burger et al., Sec. 6.5):
- Vert/Horiz sheet/cylinder - quarter sheets
- spheres, polygons
- Talwani (2D and 2.5D prisms -­ Matlab)
ρ2
ρ1
ρ1
ρ2
ρ1
ρ3
ρ2
Δρ is what matters!
http://gravmag.ou.edu/grav_interp/grav_interp.html
Locoste & Romberg Model G Gravimeter
Love and Care
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•
•
$$,$$$ Needs power to keep isothermal -­ keep plugged in somewhere Highly sensitive spring and mass system (read Reynolds)
-­ Do not walk around without clamping the mass
Getting data
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Level gravimeter (you want all of gz)
Unclamp
Get a reading by adjusting the spring s.t. mass is in the “right place”
-­ can be tedious if you have changed location a lot
A manual raw data point is derived by reading off the “counts” and converting to mGals using a calibrated table (or Matlab)
Typically, take a manual reading at a base station
-­ use force-­feedback system to record relative values
-­ Careful with on/off of feedback during a survey
Reclamp (DON’T BE SHY ABOUT CHECKING YOUR COLLEAGUES)
Return to base station every ~30 minutes for tide/drift correction.
Homework
1. Read the chapter on gravity in Reynolds
2. Use gravity to estimate the height of each floor of Millikan or S. Mudd. Make sure to correct for any tidal/drift effects by interpolating differences in the base station measurement at the beginning and end of the survey. Since tides/drift are slowly varying over hours, you can empirically correct for them by remeasuring a base station at different times to estimate the correction as a function of time (by interpolation).
3. Given the coordinates of Caltech (34:09N, 118:10W, 250 m.a.s.l.) and the Landers EQ rupture, and assuming all other factors are constant, which area do you expect will have a higher vertical component of the gravity field (show your analysis)? (Hint: compare effects of latitude correction vs. FA correction).