SEISMOLOGY, Lecture 1

8/22/2019 SEISMOLOGY, Lecture 1

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PEAT8002 - SEISMOLOGY

Lecture 1: Introduction and review of vector

operators

Nick Rawlinson

Research School of Earth SciencesAustralian National University
http://find/http://goback/


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Introduction
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What is Seismology?

Seismology is the study of elastic waves in the solid earth.Seismic wave theory, observation and inference are the

key components of modern seismology.

Seismology is motivated by our ability to record ground

motion caused by the passage of seismic waves.

(energy source)source

(seismogram)receiver

(elastic properties)medium
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A Propagating Seismic Wave

A T i l S i
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A Typical Seismogram

EP SPP PcS SS RL

E-W component of a seismic wavetrain recorded inShanghai, China from a magnitude 6.7 earthquake in the

New Britain region, PNG on 6th February 2000.

E th St t d P f S i l
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Earth Structure and Processes from Seismology

Seismology is the most powerful indirect method for

studying the Earths interior.

The existence of Earths crust, mantle, liquid outer core

and solid inner core were inferred from seismograms many

decades ago.

mantle

outer core

inner core

crust

1890: Liquid core

identified by Oldham

1909: Base of crust

identified by Mohorovicic1932: Inner core inferred

by Lehmann

E i R
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Economic Resources

Seismology can be used to reveal the location of economic

resources like hydrocarbon and mineral deposits.

T hi I i
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Tomographic Imaging

Seismic imaging can be performed at a variety of scales

(metres to 1,000s km) to reveal information aboutstructure, composition and dynamic processes.

6

6.5

6.5 6.5

7

7

7

7.5

8

8

0

50

100

150

DEPTH

[km]

70 W 69 W 68 W 67 W

W E

4.5 5.0 5 .5 6 .0 6.5 7 .0 7.5 8 .0 8 .5 9.0

Vp [km/s]

[From Graeber& Asch,1999]

22.75 S

Vp [km/s]

DEPTH

(km)

Tomographic Imaging
http://goback/http://find/http://find/http://goback/


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Tomographic Imaging

2.0

2.4

2.8

3.2

3.6

110

110

120

120

130

130

140

140

150

150

160

160

40 40

30 30

20 20

10 10

Rayleighw

avegroupvelocity(km/s)

0.2 Hz

Tomographic Imaging


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Tomographic Imaging

/ /

Perturbation [%]

Earthquake Processes
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Earthquake Processes

Seismograms are used in:

earthquake location

source mechanism studies

estimating deformation

measuring plate tectonic processes

hazard assessment

nuclear monitoring

Earthquake Location
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Earthquake Location

0

0

60

60

120

120

180

180

-120

-120

-60

-60

0

0

-60 -60

0 0

60 60

2002/05/28 to 2003/05/28

Focal Mechanisms
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Focal Mechanisms

Cascadia events

Seismic Hazard


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Seismic Hazard
http://find/


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Review of Vector Operators

In this course, we work with vector and tensor functions

that describe elastic deformation in solids. To do so, we

need to use the vector operators grad, div and curl.

Scalars Vectors and Tensors


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Scalars, Vectors and Tensors

At each point in a continuum, we can define a scalar

function that varies with position x and time t, and

describes some property of the medium e.g. temperature

T(x, t).

Similarly, we can define a vector function that varies with

(x, t) e.g. velocity v(x, t) = (vx, vy, vz).Each component of a vector can itself be a vector, in which

case we can define a tensor function. Examples of 3 3tensor functions include stress, strain and strain rate.

=

xx xy xzyx yy yzzx zy zz

The Gradient Operator
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The gradient operator is a vector containing three partialderivatives [/x,/y,/z]. When applied to a scalar, itproduces a vector, when applied to a vector, it produces a

tensor.

T = [T/x, T/y, T/z]

v =/x/y/z

vx vy vz =

vx/x vy/x vz/xvx/y vy/y vz/xvx/z vy/z vz/z

The gradient vector of a scalar quantity defines the

direction in which it increases fastest; the magnitude

equals the rate of change in that direction.



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x

Gradient of a scalar field

f(x,y)

f

The Divergence Operator
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The divergence operator

has the same form as

, but

has the opposite effect on the rank of the quantity on which

it operates. Applied to a vector it produces a scalar;

applied to a tensor it produces a vector.

v = /x /y /z vxvyvz

= vxx

+ vyy

+ vzz

The divergence of a vector field may be thought of as the

local rate of expansion of the vector field.

v = limV0

1

V

v ndS

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This formula describes an integral over surface Sof a

small element with volume V. n is the unit outward normal

on the surface.The physical significance of the divergence of a vector field

is the rate at which density" exits a given region of space.

For example, if u is the velocity of an incompressible fluid,

then

u = 0 - fluid particles cannot bunch up".

S

Divergence of a 2D vector field

V

n

(x,z)v

Curl of a Vector Field
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The other important vector operator is curl, . It can berepresented as a matrix operating on a vector field.

v = det

i j k

/x /y /z

vx vy vz =

vz/y vy/zvx/z

vz/x

vy/x vx/yThe curl may be thought of as the local curvature of the

vector field.

( v)i = limA0

niA

v dl



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The integral is taken around the perimeter of the small

area element A which is perpendicular to n, the unit

normal vector to v. Since there are three orthogonalorientations for the area element, the curl has threecomponents.

The physical significance of the curl of a vector field is the

amount of rotation" or angular momentum of the contents

of a given region of space.

A

Curl of a 3D vector field

n

dl

(x,y,z)v

Example 1
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p

Scalar function f =

(x 1)2 + (y 1)2 in the interval

0

x

2, 0

y

2.

0.2

0.4

0.4

0.6

0.6

0.6

0.8

0.8

0.8

0.8

1

1

0

1

2

y

0 1 2

x

Example 1
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p

f = x 1

(x 1)

2

+ (y 1)2,

y 1

(x 1)2

+ (y 1)2

0.2

0.4

0.4

0.6

0.6

0.6

0.8

0.8

0.8

0.8

1

1

0

1

2

y

0 1 2

x

Example 1
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p

f = 1(x 1)2 + (y 1)2= 2f = Laplacian

1.2

1.2

1.2

1.2

1.4

1.4

1.4

1.6

1.6

1.6

1.8

1.8

1.8

2

2

2.2

2.2

2.4

2.4

2.6

2.6

2.8

2.8

3

3

3.23

.2 3.43.4

3.63.8

44.24.4

4.64.8

5

5.2

5.45.65.8

66

.26.4

6.6

6.877.2

7.4

7.6

7.888.28.48.68.899.29.49.69.8

1010.210.410.610.81111.211.411.611.812

12.212.412.612.81313.213.413.613.81414.

214.414.614.81515.2

15.4

15.6

15

.816

16

.2

16

.4

16

.616.8

17

17.2

17.4

17.6

0

1

2

y

0 1 2

x

Example 1
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p

f ="0,0, x y1(x1)2+(y1)2! y x1(x1)2+(y1)2!#=0

For any scalar functions f,gand any vectors u, v

f = 0 v = 0(fg) = fg+ gf (f/g) = (gf fg)/g2

(fv) = f

v + v

f

(f

g) = f

2g+

f

g

(fv) = f v + f v (u v) = v u u v

Example 2
http://find/


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p

Scalar function f = (x 1) exp[(x 1)2 (y 1)2] in theinterval 0

x

2, 0

y

2.

0.4

0

.2

0

0.2

0.4

0

1

2

y

0 1 2

x

Example 2
http://find/


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p

f = exp[

(x

1)2

(y

1)2]1 2(x 1)

2,

2(x

1)(y

1)

0.4

0

.2

0

0.2

0.4

0

1

2

y

0 1 2

x

Example 2
http://find/


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f = 4(x

1) exp[

(x

1)2

(y

1)2](x

1)2 + (y

1)2

2

2

2

1

1

0

1

1

2

2

0

1

2

y

0 1 2

x

Example 3
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Vector function v = [ycos x, xcos y] in the interval0

x

2, 0

y

2.

0

1

2

y

0 1 2

x

Example 3


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v = ysin x xsin y

2

1

0

1

2

y

0 1 2

x

Example 3
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v = [0, 0, cos y

cos x]

0

1

2

z

0 1 2

x

0

1

2

z

0 1 2

y

y=1.0 x=1.0
http://find/

SEISMOLOGY, Lecture 1

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