Geology 5660/6660Applied Geophysics

17 Jan 2014

© A.R. Lowry 2014Read for Wed 22 Jan: Burger 21-60 (Ch 2.2–2.6)

Last time: The Wave Equation; The Seismometer• The elastic wave equation: Assumes an isotropic solid Assumes elastic constitutive law = c Stress/strain relations assume infinitesimal strain Rheology is linear elastic: (Hooke’s Law) The wave equation:

• Velocities are more sensitive to & than to ; are sensitive to porosity, rock composition, cementation, pressure, temperature, fluid saturation

€

Vp = α =λ + 2μ

ρ

€

Vs = β =μ

ρ

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∂2u

∂t2=α 2 ∂θ

∂x+ β 2∇ 2u

Seismic ground motions are recorded by a seismometer or geophone.Basically these consist of:• A frame, hopefully well-coupled to the Earth,• Connected by a spring or lever arm to an• Inertial mass.• Motion of the mass is damped, e.g., by a dashpot.• Electronics convert mass movement to a recorded signal (e.g., voltage if mass is a magnet moving through a wire coil or vice-versa).

Instrumentation

Mframe

spring

mass

dashpot

isometric view

cross-sectional view

Geophone:• Commonly-used by industry, less often for academic, seismic reflection studies• Often vertical component only• Often low dynamic range

Undamped responseof mechanical system

Response afterelectronic damping

10 Hz “natural frequency”

10 10020 500200

A Seismometer differs mostly in cost/componentry… 3-c, > dynamic range

Recall that an idealized mass on a spring is a harmonic oscillator: Position x of themass follows the form x = A cos (t + )where A is amplitude, t is time, is the natural frequency of the spring, and is a phase constant (tells us where the mass was atreference time t = 0). x

AT = 2/

In the frequency domain this is a delta-function:

Understanding the Frequency Domain:

Signal recorded by a seismometer is a convolution of the wave source, the Earth response, and the seismometerresponse.

WaveSource

EarthResponse

SeismometerResponse

where denotes convolution:

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( f ⊗g) = f (t − x)g(x)dx−∞

∞

∫Example:

=

So, want seismometer response tolook as much as possible like asingle delta-function in time: t = 0

Seismometer response is given by:

€

d 2i

dt2+2hω0

di

dt+ω0

2i =K

R

d 3x

dt3

where i is current, 0 is “natural frequency” of the spring-mass system oscillation, K is electromagnetic resistance to movement of the coil, R is electrical resistance to current flow in the coil, & x is movement of the coil relative to the mass.

The damping factor h is given by: where is the mechanical damping factor.

Hence we choose K and R to give a time response thatlooks as much as possible like a delta-function (= a flatfrequency response):

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2hω0 =τ

M+

K 2

MR

This corresponds to 0/h = 1: Critical damping

Critically Damped

Overdamped

Underdamped

Seismometers typically are designed to be slightlyoverdamped (0/h = 0.7).

Seismometer Damping

Source Function

An explosion at a depth of 1 km & t = 0is recorded by a seismometer at the surface with the damping responseshown. What will the seismogram looklike?

V = 5 km/s

(Note: for really big signals, can get more robust operationand lower frequencies from other types of instruments… E.g. GPS!)

4 April 2010 M7.2 Baja California earthquake

(Note: for really big signals, can get lower frequencies fromother types of instruments… E.g. GPS!)

10 August 2009 M7.6 earthquake north of Andaman

GPS Displacement

Seismometer Displacement

Huygen’s Principle:

Every point on a wavefront can be treated as a point source for the next generation of wavelets. The wavefront at a timet later is a surface tangent to the furthestpoint on each of these wavelets.

We’ve seen this before…This is useful because the extremal pointshave the greatest constructive interference

Fermat’s Principle (or the principle of least time):

The propagation path (or raypath) between any two pointsis that for which the travel-time is the least of all possiblepaths.

Recall that a ray is normal to a wavefront at a given time:

A key principle because most of our applications will involvea localized source and observation at a point (seismometer).

•

•

V = fast V = slow

least time in slow

least time in fast

Fermat’s principle leads to Snell’s Law:

Travel-time is minimized whenwhen the ratio of sines of theangle of incidence (anglefrom the normal) to a velocity boundary is equal to the ratio of the velocities, i.e.,

straight line

least time

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sinθ1

sinθ2

=V1

V2