Geophysics/Tectonics

GLY 325

EarthPhysicalProperties

Mesosphere(hot,plastic,strong)

Inner Core(Solid)

Lithosphere(cool,brittle,rigid)

6400 km

5000 km

0 km

100 km

Outer Core(Liquid)

Asthenosphere(hot,plastic,weak)

2900 km

300 km

Geophysical Surveys

Active or Passive--

Passive geophysical surveys incorporate measurements of naturally occurring fields or properties of the earth (i.e., earthquake seismology, gravity, magnetics, radiometric decay products, Self Potential (SP), Magnetotelluric (MT), and heat flow).

Active geophysical surveys impose a signal on the earth, and then the earth’s response to this signal is measured (i.e., seismic reflection/refraction, DC resistivity, Induced Polarization (IP)and Electromagnetic (EM)).

Geophysical Technique

Measured Earth Property

Earth Property Effecting Signal

Seismic

Potential

Field

Heat Flow

Natural Source:

Earthquake

Ground Motion (Displacement,

Velocity or Acceleration)

Seismic Velocity (V ) and Attenuation (Q )

Seismic Velocity (V )

Acoustic Impedance (Seismic Velocity, V,

and Density, )

Density ()

Magnetic Susceptibility () and Remanent

Magnetization (Jrem)

Thermal Conductivity (k ) and

Heat Flow (q )

Gravitational Acceleration (g )

Strength and Direction of Magnetic Field (F )

Geothermal Gradient (T/z )

Refraction

Reflection

Gravity

Magnetics

Con

trol

led

Sou

rce

Elastic Waves

Elastic Behavior -- the ability of a material to immediately return to its original size, shape, or position after being squeezed, stretched, or otherwise deformed. The material follows Hooke’s Law ( = C).

Plastic (Ductile) Behavior -- a permanent change in the shape, size, etc., of a solid that does not involve failure by rupture.

Stress -- force per unit area ()

Strain -- change in shape or size ()

Elastic Waves

A material’s behavior can be plotted on a stress/strain diagram:

Hooke’s Law ( )

Elastic Behavior

Ductile Behavior

Increasing Strain

Incr

easi

ng S

tres

s

Elastic Limit

Elastic Waves

However, an important factor we haven’t talked about is strain rate ( ). A material will have a different stress/strain diagram with varying .

Strain

Str

ess

Slow rate of shape change

(low strain rate)

Fast rate of shape change

(high strain rate)

Strain

Str

ess

Elastic Waves

The deformation of the lithosphere (folding) is a slow strain-rate process (ductile), while the propagation of seismic waves is a fast strain-rate process (elastic).

Strain

Str

ess

Fast rate of shape change

(High strain rate)

Slow rate of shape change

(Low strain rate)

Strain

Str

ess

Elastic Waves

So in general, all seismic waves are elastic waves and propagate through material through elastic deformation.

We can design computer models of earth materials to behave elastically, and demonstrate not only how seismic waves propagate, but also the form in which they propagate.

The following models plot in color the displacement of material as seismic energy passes.

Elastic Waves, as waves in general, can be described spatially...

…or temporally.

Elastic Waves

The controlling factors in the propagation of seismic wave are the physical properties of the material through which the seismic energy is travelling.

The specific properties are called the elastic constants:

Bulk Modulus (k) -- describes the ability to resist being compressed.

Shear Modulus (µ) -- describes the ability to resist shearing.

Elastic Waves

It turns out that k and µ can be difficult to measure, so other elastic constants relating the two were derived:

Young’s Modulus (E) -- describes longitudinal strain in a body subjected to longitudinal stress.

Poisson’s Ratio () -- describes transverse strain divided by longitudinal strain in a body subjected to longitudinal stress.

Elastic Waves

Lame’s Constant () -- interrelates all four elastic constants and is very useful in mathematical computations, though it doesn’t have a good intuitive meaning.

It’s important for you to know the terms and what they represent (when appropriate) because we will be using them in labs.€

=k −2μ

3=

νE

(1+ ν )(1− 2ν )

The Wave Equation

We’ll look at the scalar wave equation to mathematically express how elastic strain (dilatation, ) propagates through a material:

2 = ( + 2) 2 t2

where

xxyyzz

and 2 is the Laplacian of , or

xyz

Elastic Waves

When solving the wave equation (which describes how energy propagates through an elastic material), there are two solutions that solve the equation, Vp and Vs . These solutions relate to our elastic constants by the following equations:

Elastic Waves

It turns out that Vp and Vs are probably familiar to you from your introductory earthquake knowledge, since they are the velocities of P-waves and S-waves, respectively.

So, now you know why there are P- and S-waves--because they are two solutions that both solve the wave equation for elastic media.

P S

The Wave Equation

The wave equation can be rewritten as

2 = 2 t2

where = ( + 2)/, or alternatively as

2 = 2 t2

where = /

And you’ll recognize the physical realization of these equations as = P-wave and = S-wave velocity.

The Wave Equation

Since the elastic constants are always positive, is always greater than , and

/ = [/(+2)]1/2 = [(0.5-)/(1-)]1/2

So, as Poisson’s ratio, , decreases from 0.5 to 0, / increases from 0 to it’s maximum value 1/√2; thus, S-wave velocity must range from 0 to 70% of the P-wave velocity of any material.

The Wave Equation

These first types of solutions–P-waves and S-waves–are called body waves. Body waves propagate directly through material (i.e. its “body”).

I. Body Waves a. P-Waves

1. Primary wave (fastest; arrive first)2. Typically smallest in amplitude 3. Vibrates parallel to the direction of

wave propagation. b. S-Waves

1. Secondary waves (moderate speed; arrives second)2. Typically moderate amplitude2. Vibrates perpendicular to the direction

of wave propagation.

The Wave Equation

The other types of solutions are called surface waves. Surface waves travel only under specific conditions at an interface, and their amplitude exponentially decreases away from the interface.

II. Surface waves (slowest)1. Arrives last2. Typically largest amplitude2. Vibrates in vertical, reverse elliptical motion (Rayleigh) or shear elliptical

motion (Love)

The Wave Equation

The three types of surface waves are:

1) Rayleigh Waves–form at a free-surface boundary. Air closely approximates a vacuum (when compared to a solid), and thus satisfies the free-surface boundary condition. Rayleigh waves are also called “ground roll.”

2) Love Waves–form in a thin layer when the layer is bound below by a seminfinite solid layer and above by a free surface.

3) Stonely Waves–form at the boundary between a solid layer and a liquid layer or between two solid layers under specific conditions.

The Wave Equation

For a “typical” homogeneous earth material, in which Poisson’s ratio = 0.25 (also called a Poisson solid), the following relationship should be remembered between P-wave, S-wave, and Rayleigh wave velocities:

VP : VS : VR = 1 : 0.57 : 0.52

In otherwords, VS is about 60% of VP, and VR is about 90% of VS.

But remember, this only is a guide...

The Wave Equation Modeled

The wave equation explains how displacements elastically propagate through material. In models, colors represent the displacement of discrete elements (below: yellow–positive, purple–negative) away from their equilibrium position.