peat8002 - seismology lecture 1: introduction and …rses.anu.edu.au/~nick/teachdoc/lecture1.pdf ·...
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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