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Last Time: Ground Penetrating Radar• Radar reflections image variations in Dielectric constant r (= relative permittivity) 3-40 for most Earth materials; higher when H2O &/or clay present• Radar attenuation similar to seismic: where:
higher for clay, silt, briny pore fluids• Velocity is not estimated (rather depth is approx. from ~V) (but can be estimated from diffraction moveout!!!)• Standard “processing” includes static corrections for elevation, filtering, AGC
Geology 5660/6660Applied Geophysics
03 Mar 2014
© A.R. Lowry 2014For Fri 7 Mar: Burger 349-378 (§6.1-6.4)
€
I =I 0e−αr
€
α =ω μ2
σ 2
ε 2ω2+1 −1
⎛
⎝ ⎜ ⎜
⎞
⎠ ⎟ ⎟≈
σ
2
μ
ε
€
V =c
ε rμ
€
R ≈ε1 − ε2
ε1 + ε2
≈V2 −V1
V2 +V1
Introduction to Gravity
Gravity, Magnetic, & DC Electrical methods are all examples of the Laplace equation of the form:
2u = f (sources),
where u is a potential, is the gradient operator
Notation: Here, the arrow denotes a vector quantity; the carat denotes a unit direction vector.Hence, the gradient operator is just a vector form of slope…
Because Laplace’ eqn always incorporates a potential u, we call these “Potential Field Methods”.
→
^
€
r∇
€
r∇≡ ˆ x
∂
∂x+ ˆ y
∂
∂y+ ˆ z
∂
∂z
Gravity
We define the gravitational field as
And by Laplace’ equation,
(1)
given a single body of total mass M; hereG is universal gravitational constant = 6.672x10-11
Integrating equation (1), we have
(2)
Nm2
kg2
€
rg
€
r∇u =
r g
€
∇2u =r
∇ •r g = −4πGM
€
rg = −4πG ρdV
Vol
∫∫∫
IF the body with mass M is spherical with constant density, equation (2) has a solution given by:
Here r is distance from the center of mass; is the direction vector pointing toward the center.
Newton’s Law of gravitation:
So expresses the acceleration of m due to M! has units of acceleration Gal in cgs (= 0.01 m/s2)
On the Earth’s surface,
€
rg = ˆ r
GM
r 2
€
ˆ r
€
rF = ˆ r
GMm
r 2= m
r g
€
rg
€
rg
€
rg = ˆ r
GME
RE2
= ˆ r 6.67 ×10−11
( ) 5.97 ×1024( )
6.38 ×106( )
2= 9.8 m/s2
HOWEVER, is not radially symmetric in the Earth… so is not constant!
Gravity methods look for anomalies, or perturbations, from a reference value of at the Earth’s surface:
01
gref
gobs
€
rg
€
rg
Global Free-Air Gravity Field from GRACE
Example:
Image from UT-CSR/NASA
Gravity Measurements:
I. Absolute Gravity: Measure the total field time of a falling body
prism
vacuum
laser
~2m
• Must measure time to ~10-11 s; distance to ~10-9 m for 1 μgal accuracy!
• Nevertheless this is the most accurate ground-based technique (to ~3 μgal)
• Disadvantages: unwieldy; requires a long occupation time to measure
Gravity Measurements:II. Relative Gravity: Measures difference in at two locations.
• Pendulum: difference in period T:
Errors in timing of period T ~0.1 mgal
• Mass on a spring: Mg = kl or g = kl/M
Worden and Lacoste-Romberg are of this type
(“zero-length” spring of L-R yields errors around 6 μgal)
l
mass M
length l
spring constantk
€
rg
€
T = 2πk
g≈ 2π
l
g
Gravity Measurements:III. Satellite Gravity: Measure (from space) the height of an equipotential surface (called the geoid, N)
relative to a reference ellipsoid.
QuickTime™ and aTIFF (Uncompressed) decompressor
are needed to see this picture.
Gravity Measurements:
• Ocean Altimetry: Measure height of the ocean surface using radar or laser (e.g., JASON)
• Satellite Ranging: Satellite orbits follow the geoid
Measure orbits by ranging from the ground to the satellite or ranging between two satellites (e.g., GRACE)
N
III. Satellite Gravity: Measure (from space) the height of an equipotential surface (called the geoid, N) relative to a reference ellipsoid
Global Free-Air Gravity Field from GRACE
Example:
Image from UT-CSR/NASA
GRACE designed for time-variable gravity, mostly hydrosphere