1 Introduction to SeismologyGeology 5640/6640 Introduction to Seismology 20 Jan 2017 Last time: “Review” of Basic Principles; Vectors • Diffractions are predicted by Huygen’s Principle • But most modeling uses Ray Theory (i.e., the normal to the wavefront propagation direction) • Reflections are wave energy that “bounces off” a surface dividing media with different velocity properties, remaining in the original medium Refractions are energy transmitted through the boundary into the second medium • Conversions from P to S particle motion (or vice versa) almost always occur at these boundaries • All of these obey Snell’s Law… Read for Mon 23 Jan: S&W (§ ) © A.R. Lowry 2017
2 Seismo-Math & Notation (“Introduction” to Vectors)z x3 y x2 x x1 We’ll use these notations for Cartesian coordinate systems interchangeably… Vectors have direction and magnitude; here we may denote as boldface u or with an overbar or arrow . A hat-sign denotes a unit direction vector .
3 A vector v from point Q at coordinate (0, 0, 0) to point P at coordinate (a, b, c) may be denoted by any of z P (a,b,c) c Q (0,0,0) a y b x More generally, v between Q at (X1,Y1,Z1) and P at (X2,Y2,Z2) can be expressed as any of: Length of a vector is the square-root of the summed-squared elements, i.e.,
4 Addition and Subtraction of vectors is performedelement-wise. If: Then: (You could do this graphically too). A scalar multiplied by a vector gives: and
5 Properties of Vector Addition and Multiplication:Commutative: Associative: Distributive: Some Special Vectors: Null: Unit: has length 1: Note a unit vector in any direction can be defined as:
6 Example Unit Vectors: Thus we can write any vector in the form, e.g.: Part of this course will include an “introduction” to linear algebra… This sort of representation of vector multiplication is standard and will crop up a lot in this class!
7 Note that other notations are common…This is figure A.3-1 from the appendices of Stein & Wysession. , and are often used to denote three (arbitrarily- oriented) orthogonal directions, which may or may not equal , and . A normal is a unit vector that is perpendicular to a plane or surface. Recall that means !
8 Multiplying Vectors: A dot product represents the length of one vector multiplied by length of its projection onto another (i.e., a measure of how parallel they are). Hence, the dot product is a scalar value. We could also write this as: The latter, in which the repeated index implies the summation, is called Indicial Notation (or Einstein summation notation) & shows up a lot in seismology! Given and ,
9 The dot product is easily shown to have associative,commutative and distributive properties… It is also useful for calculating the angle between two vectors: We can also use it to calculate the projection of some vector b onto the direction of another vector a:
10 The Cross Product of two vectors is defined as:This produces a new vector that is orthogonal (perpendicular) to the plane spanned by a and b (with length = area of the parallelogram described by a and b). We can also express this as the determinant of the matrix: (Here, the straight lines surrounding the matrix denote the determinant). The determinant of a 3x3 matrix can be derived from 2x2 matrix determinants as shown at left:
11 The cross product c = ab obeys the right-hand rule:And the length of c is defined by ||c|| = ||ab|| = ||a|| ||b|| sin. Hence, the cross-product can be thought of as a measure of how orthogonal two vectors are. The length ||ab|| also equals the area of a parallelogram defined by a and b: ||c||
12 The cross-product is: Anti-commutative: ab = –ba Distributive: a(b + c) = ab + ac Associative: (ab)c = a(bc)