PEAT8002 - SEISMOLOGYLecture 1: Introduction and review of vector

operators

Nick Rawlinson

Research School of Earth SciencesAustralian National University

Introduction

What is Seismology?

Seismology is the study of elastic waves in the solid earth.Seismic wave theory, observation and inference are thekey components of modern seismology.Seismology is motivated by our ability to record groundmotion caused by the passage of seismic waves.

(energy source)source

(seismogram)receiver

(elastic properties)medium

A Propagating Seismic Wave

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 theNew Britain region, PNG on 6th February 2000.

Earth Structure and Processes from Seismology

Seismology is the most powerful indirect method forstudying the Earth’s interior.The existence of Earth’s crust, mantle, liquid outer coreand solid inner core were inferred from seismograms manydecades ago.

mantle

outer core

inner core

crust

1890: Liquid coreidentified by Oldham1909: Base of crustidentified by Mohorovicic1932: Inner core inferredby Lehmann

Economic Resources

Seismology can be used to reveal the location of economicresources like hydrocarbon and mineral deposits.

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

88

0

50

100

150

DE

PT

H [k

m]

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]

DE

PT

H (

km)

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˚

Rayleigh w

ave group velocity (km/s)

0.2 Hz

Tomographic Imaging

δφ/φ δβ/β

Perturbation [%]

-1.5 0 1.5

Earthquake Processes

Seismograms are used in:

earthquake locationsource mechanism studiesestimating deformationmeasuring plate tectonic processeshazard assessmentnuclear monitoring

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

Cascadia events

Seismic Hazard

Review of Vector Operators

In this course, we work with vector and tensor functionsthat describe elastic deformation in solids. To do so, weneed to use the vector operators grad, div and curl.

Scalars, Vectors and Tensors

At each point in a continuum, we can define a scalarfunction that varies with position x and time t , anddescribes some property of the medium e.g. temperatureT (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 whichcase we can define a tensor function. Examples of 3× 3tensor functions include stress, strain and strain rate.

σ =

σxx σxy σxzσyx σyy σyzσzx σzy σzz

The Gradient Operator

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 atensor.

∇T = [∂T/∂x , ∂T/∂y , ∂T/∂z]

∇v =

∂/∂x∂/∂y∂/∂z

[vx vy vz

]=

∂vx/∂x ∂vy/∂x ∂vz/∂x∂vx/∂y ∂vy/∂y ∂vz/∂x∂vx/∂z ∂vy/∂z ∂vz/∂z

The gradient vector of a scalar quantity defines thedirection in which it increases fastest; the magnitudeequals the rate of change in that direction.

The Gradient Operator

x

y

∆

Gradient of a scalar field

f(x,y) f

The Divergence Operator

The divergence operator ∇· has the same form as ∇, buthas the opposite effect on the rank of the quantity on whichit operates. Applied to a vector it produces a scalar;applied to a tensor it produces a vector.

∇ · v =[∂/∂x ∂/∂y ∂/∂z

] vxvyvz

=∂vx

∂x+

∂vy

∂y+

∂vz

∂z

The divergence of a vector field may be thought of as thelocal rate of expansion of the vector field.

∇ · v = limV→0

[1V

{v · ndS

]

The Divergence Operator

This formula describes an integral over surface S of asmall element with volume V. n is the unit outward normalon the surface.The physical significance of the divergence of a vector fieldis 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 2−D vector field

V

n

(x,z)v

Curl of a Vector Field

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 ∂/∂zvx vy vz

∣∣∣∣∣∣ =

∂vz/∂y − ∂vy/∂z∂vx/∂z − ∂vz/∂x∂vy/∂x − ∂vx/∂y

The curl may be thought of as the local curvature of thevector field.

(∇× v)i = limA→0

[ni

A

∮v · dl

]

Curl of a Vector Field

The integral is taken around the perimeter of the smallarea element A which is perpendicular to n, the unitnormal 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 theamount of “rotation" or angular momentum of the contentsof a given region of space.

A

Curl of a 3−D vector field

n

dl

(x,y,z)v

Example 1

Scalar function f =√

(x − 1)2 + (y − 1)2 in the interval0 ≤ 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

1

1

1

0

1

2

y

0 1 2

x

Example 1

∇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

1

1

1

0

1

2

y

0 1 2

x

Example 1

∇ · ∇f =1√

(x − 1)2 + (y − 1)2= ∇2f = Laplacian

1 1

1

1

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 3.4

3.4

3.63.844.24.44.64.8

5

5.2

5.45.65.8

66.

26.4

6.6

6.8

77.2

7.4

7.67.888.28.48.68.899.29.49.69.81010.210.410.610.81111.211.411.6

11.8

12

12.2

12.412.612.81313.213.413.613.81414.214.414.614.81515.215

.4

15.6

15.81616.2

16.4

16.616.8

1717.217.4

17.6

0

1

2

y

0 1 2

x

Example 1

∇×∇f ="

0,0, ∂∂x

y−1√

(x−1)2+(y−1)2

!− ∂

∂y

x−1√

(x−1)2+(y−1)2

!#=0

For any scalar functions f , g and any vectors u, v

∇×∇f = 0 ∇ · ∇ × v = 0

∇(fg) = f∇g + g∇f ∇(f/g) = (g∇f − f∇g)/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

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

∇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

∇·∇f = 4(x−1) exp[−(x−1)2−(y−1)2]{

(x − 1)2 + (y − 1)2 − 2}

−2

−2

−1

−1

01

1

2

2

0

1

2

y

0 1 2

x

Example 3

Vector function v = [y cos x , x cos y ] in the interval0 ≤ x ≤ 2, 0 ≤ y ≤ 2.

0

1

2

y

0 1 2

x

Example 3

∇ · v = −y sin x − x sin y

−2

−1

0

0

0

1

2

y

0 1 2

x

Example 3

∇× 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