Last Time: Ground Penetrating Radar Radar reflections image variations in Dielectric constant r ( = relative permittivity ) 3-40 for most Earth materials;

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