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