PEAT8002 - SEISMOLOGY
Lecture 1: Introduction and review of vector
operators
Nick Rawlinson
Research School of Earth Sciences
Australian National University
Introduction
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.
(seismogram)
receiver
(energy source)
source
medium
(elastic properties)
A Propagating Seismic Wave
A Typical Seismogram
P
PP
PcS S
SS
L
R
E
E-W component of a seismic wavetrain recorded in
Shanghai, China from a magnitude 6.7 earthquake in the
New Britain region, PNG on 6th February 2000.
Earth Structure and Processes from Seismology
Seismology is the most powerful indirect method for
studying the Earth’s interior.
The existence of Earth’s crust, mantle, liquid outer core
and solid inner core were inferred from seismograms many
decades ago.
crust
mantle
outer core
inner core
1890: Liquid core
identified by Oldham
1909: Base of crust
identified by Mohorovičić
1932: Inner core inferred
by Lehmann
Economic Resources
Seismology can be used to reveal the location of economic
resources 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 about
structure, composition and dynamic processes.
W
E
0
6
6.5
6.5
7
6.5
DEPTH [km]
DEPTH
(km)
50
7
7
7.5
8
8
100
22.75 S
150
4.5
5.0
5.5
6.0
6.5
7.0
7.5
8.0
8.5
9.0
Vp
Vp [km/s]
[km/s]
70 W
69 W
68 W
[From Graeber & Asch, 1999]
67 W
Tomographic Imaging
130˚
120˚
110˚
−10˚
140˚
150˚
160˚
−10˚
3.2
2.8
−30˚
−30˚
2.4
2.0
−40˚
−40˚
0.2 Hz
110˚
120˚
130˚
140˚
150˚
160˚
Rayleigh wave group velocity (km/s)
3.6
−20˚
−20˚
Tomographic Imaging
δφ/φ
δβ/β
Perturbation [%]
-1.5
0
1.5
Earthquake Processes
Seismograms are used in:
earthquake location
source mechanism studies
estimating deformation
measuring plate tectonic processes
hazard assessment
nuclear monitoring
Earthquake Location
0˚
60˚
120˚
180˚
-120˚
-60˚
0˚
60˚
60˚
0˚
0˚
-60˚
-60˚
0˚
60˚
120˚
180˚
2002/05/28 to 2003/05/28
-120˚
-60˚
0˚
Focal Mechanisms
Cascadia events
Seismic Hazard
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
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 × 3
tensor functions include stress, strain and strain rate.


σxx σxy σxz
σ = σyx σyy σyz 
σzx σzy σzz
The Gradient Operator
The gradient operator ∇ is a vector containing three partial
derivatives [∂/∂x, ∂/∂y , ∂/∂z]. When applied to a scalar, it
produces a vector, when applied to a vector, it produces a
tensor.
∇T = [∂T /∂x, ∂T /∂y , ∂T /∂z]




∂/∂x ∂v
/∂x
∂v
/∂x
∂v
/∂x
x
y
z
∇v = ∂/∂y  vx vy vz = ∂vx /∂y ∂vy /∂y ∂vz /∂x 
∂/∂z
∂vx /∂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.
The Gradient Operator
Gradient of a scalar field
y
∆
f(x,y)
x
f
The Divergence Operator
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.
 
vx
∂vy
∂vx
∂vz
+
+
∇ · v = ∂/∂x ∂/∂y ∂/∂z vy  =
∂x
∂y
∂z
vz
The divergence of a vector field may be thought of as the
local rate of expansion of the vector field.
{
1
∇ · v = lim
v · n̂dS
V →0 V
The Divergence Operator
This formula describes an integral over surface S of a
small element with volume V. n̂ 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".
n
S
V
v (x,z)
Divergence of a 2−D vector field
Curl of a Vector Field
The other important vector operator is curl, ∇×. It can be
represented as a matrix operating on a vector field.


i
j
k ∂vz /∂y − ∂vy /∂z
∇ × v = det ∂/∂x ∂/∂y ∂/∂z = ∂vx /∂z − ∂vz /∂x 
vx
vy
vz ∂vy /∂x − ∂vx /∂y
The curl may be thought of as the local curvature of the
vector field.
I
n̂i
(∇ × v)i = lim
v · dl
A→0 A
Curl of a Vector Field
The integral is taken around the perimeter of the small
area element A which is perpendicular to n̂, the unit
normal vector to ∇ × v. Since there are three orthogonal
orientations for the area element, the curl has three
components.
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.
n
dl
A
v (x,y,z)
Curl of a 3−D vector field
Example 1
p
Scalar function f = (x − 1)2 + (y − 1)2 in the interval
0 ≤ x ≤ 2, 0 ≤ y ≤ 2.
2
1
1
0.8
8
0.
0.6
1
1
0.2
0.4
1
0.8
y
0.6
0.4
0.6
0.
8
1
0
0
1
x
2
Example 1
"
x −1
y −1
∇f = p
,p
(x − 1)2 + (y − 1)2
(x − 1)2 + (y − 1)2
2
1
1
0.8
8
0.
0.6
1
1
0.2
0.4
1
0.8
y
0.6
0.4
0.6
0.
8
1
0
0
1
x
2
#
Example 1
∇ · ∇f = p
1
(x −
1)2
+ (y − 1)2
2
= ∇2 f = Laplacian
1
1.2
1.4
1.
2
1
1.8.6
2
2..2 2
2. 4
8
3
3.
2
1.6
1
2.8
1
6676...42
.825
65.4.8
54
115
16.68.
246
17 .24
1114.8
5..26
111455.6
14
.6
.234...4
1112113214
286
84642.8 .4.865
110.4
.9.4
110.6
11
0
10.2
9.8
9.6
2
9.
8.6
889.8
7.27 7
.428 8
111767 121.2.
1
2.
1.2
4
1.4
4
2.2
2
1.8
1.
1
2.6
y
3.4
6
2. 3.2.4.364.8
333 44.42.6.44..82
5
1
8
1.
1.6
1.2
0
0
1
x
2
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
∇(fg) = f ∇g + g∇f
∇ · (f v) = f ∇ · v + v · ∇f
∇·∇×v=0
∇(f /g) = (g∇f − f ∇g)/g 2
∇ · (f ∇g) = f ∇2 g + ∇f · ∇g
∇ × (f v) = ∇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 the
interval 0 ≤ x ≤ 2, 0 ≤ y ≤ 2.
0.2
−0.2
1
0.4
.4
−0
y
0
2
0
0
1
x
2
Example 2
n
o
∇f = exp[−(x−1)2 −(y −1)2 ] 1 − 2(x − 1)2 , −2(x − 1)(y − 1)
0.2
−0.2
1
0.4
.4
−0
y
0
2
0
0
1
x
2
Example 2
n
o
∇·∇f = 4(x−1) exp[−(x−1)2 −(y −1)2 ] (x − 1)2 + (y − 1)2 − 2
2
0
−1
−2
2
2
−2
y
1
1
−1
1
0
0
1
x
2
Example 3
Vector function v = [y cos x, x cos y ] in the interval
0 ≤ x ≤ 2, 0 ≤ y ≤ 2.
y
2
1
0
0
1
x
2
Example 3
∇ · v = −y sin x − x sin y
2
0
y
−2
1
−1
0
0
0
1
x
2
Example 3
∇ × v = [0, 0, cos y − cos x]
y=1.0
x=1.0
2
z
z
2
1
1
0
0
0
1
x
2
0
1
y
2