MIT OpenCourseWare http://ow.mit.edu 1.510 Introdution to Seismology Spring 008 For information about iting these materials or our Terms of Use, visit: http://ow.mit.edu/terms.
1.510 Leture Notes 3.3.007 The Wave Equation φ = φ, for a P wave Often written φ φ = 0 or L ( φ) = 0 where L is an operator. i( k x ω Using d Alembert s Solution: φ( = A( e, where the wave number k indiates the diretion of the wave Ray Parameter and Slowness A useful way to haraterize a wave s ray path is via its slowness, the reiproal of the apparent veloity. Figure 1. The arrow is used for a ray and the dashed line is used for a wavefront. The wavenumber k indiates the diretion of the ray. The angle i is both the takeoff angle and the angle of inidene. We define the wave speed, = ds/dt, with horizontal wave speed, x = dx/dt, and vertial wave speed, z = dz/dt. Using Figure 1 we an relate the angle if inidene with the horizontal and vertial wave speed. ds dt sin( i) = = = p dx dx x ds dt os( i) = = = η dz dz z Here p is horizontal slowness, also known as the ray parameter, and η is vertial slowness. 1 sin( i) 1 os( i) p = η = x z The slowness vetor, s = ( p, η), is omposed of the horizontal and vertial slowness. Some properties of the slowness vetor: 1 1 s = p + η = p +η = 1
1.510 Leture Notes 3.3.007 However, the addition of squares of horizontal wave speed and vertial wave speed does not equal to squares of wave speed, +. In addition, we will examine ritial phenomenon in refletion and refration x z with the relation η = 1 p. In terms of wave number, eah omponent of wave number an be represented by horizontal and vertial slowness. ω ω kx = = ωp k = z = ωη x z Thus, wave number speed is related to the slowness vetor. k = ( k, k ) = ( ω p, ωη) = ω( p, η) = ωs x z Geometri Ray Theory Remember from plane wave superposition: φ( π π A(...) e i( k x ωt ) dk dk dω x y We will use a high frequeny approximation, the limit as ω, whih leads to geometri ray theory. We an gain insight into the behavior of the seismi waves by onsidering the ray paths assoiated with them. This approah, studying wave propagation using ray path, is alled geometri ray theory. Although it does not fully desribe important aspets of wave propagation, it is widely used beause it often greatly simplifies the analysis and gives a good approximation. Eikonal Equation eikon = image (Greek) Consider the following solution to the wave equation, φ = φ : i( k x ω φ( = A( e We hoose to work at a travel time, T (. iωt ( φ( = A( e Working to insert this expression bak into the wave equation: iωt iωt φ = Ae iωa Te A wave front onnets points of equal phase: equal phase ~ equal travel time, T, from the origin. iωt ( A ω A T i(ω A T + ωa T ) e φ = ) φ = ω i T Ae ω A ω A T iω Aω ( A T + A T ) = = Ak Real Imaginary ω ω note: k =, kβ = and for the general ase β For information on propagation, onsider just the real part. ω k =
1.510 Leture Notes 3.3.007 Aω A ω A T = 1 A T = Aω Apply the high frequeny approximation and take the limit as ω. For suffiiently large ω the right-hand side goes to 0. 1 = T for a P wave For the general ase: 1 1 T = T ( = ( 1 T ( = k = s = ( p, η) = slowness vetor ( What does it mean? Gradient of a wavefront at a position x (here defined as the travel time, surfae of equal phase) is equal to the loal slowness. The diretion of maximum hange of the wavefront defines the diretion of the wave propagation. What are the impliations? Rays are perpendiular to wavefronts. The slowness gives the gradient of the travel time, and the gradient of the travel time speifies the diretion of the ray. Eah time ( hanges, the gradient of T has to hange, and the diretion of propagation hanges at the same time. If one knows (, there is a way to reonstrut the diretion of the ray: eikonal ray traing. Warning: ω needs to be suffiiently large, but it does not need to be infinite for the eikonal equation to be a valid simplifiation of the wave equation. There are no fixed rules but some onditions of validity exist: Change in wave speed along the ray has to be small i.e. the distane over whih C( hanges has to be large ompared to the wavelength. Curvature, grad(t), must be small ompared to the wavelength. Small urvature Large urvature Extreme ase: refletion or infinite urvature. This an be studied as long as you onsider infinitely high frequenies infinitely narrow rays. Fermat s Priniple Consider the kineti and potential energy along an arbitrary path between two points, A and B. Stationary points of the integrated differene between KE and PE over all possible paths speify paths of least ation. There are an infinite number of paths from A to B, but there is only B one orret path: the one with the shortest travel time. Both the shortest and the longest paths are stationary points. 3
1.510 Leture Notes 3.3.007 Consider the mathematial formulation of the problem. Hene,. Fermat s Priniple implies Snell s Law. The travel time urve, plotted as a funtion of offset, is typially a hyperboli funtion. Near stationary points, the urve is usually fairly flat, whih implies that, near optimum or stationary points, the travel time is loally insensitive to slight variations in offset. Consequently, lose to a stationary point, small deviations in the ray path an be treated as seond order effets. For example The presene of a fast body embedded in a homogeneous medium auses the referene (optimal) ray path to deflet from its original path between A and B. But, as a onsequene of Fermat s Priniple, the hange in ray path produes a seond order effet on the arrival time. The effet on travel time of hanges in wave speed (elasti parameters) is first order. Impliation: We an generate referene ray paths assuming a homogeneous medium and use this referene model to simplify subsequent analysis for heterogeneous media. Important for travel time tomography. 4
1.510 Leture Notes 3.3.007 Snell s Law in a Spherial Medium Ray Equation 5
1.510 Leture Notes 3.3.007 Radius of Curvature 6