Seismology and Seismic Imaging

Transcription

1 Seismology and Seismic Imaging 4. Ray theory N. Rawlinson Research School of Earth Sciences, ANU Seismology lecture course p.1/23

2 The ray approximation Here, we consider the problem of how body waves (P and S) propagate through a medium in which the elastic parameters vary with spatial location. The elastic wave equation in a medium with spatially variable properties is ρü = λ( u) + µ [ u + ( u) T ] + (λ + 2µ) ( u) µ u The two terms containing λ and µ mean that P and S motions do not decouple in heterogeneous media. Seismology lecture course p.2/23

3 However, if the scale length of variations in λ and µ are large compared to the seismic wavelength, then P and S can be treated separately and the elastic wave equation is simplified. Even so, solving the elastic wave equation requires exhaustive computational effort. Ray theory is an alternative approach in which a point on the wavefront is tracked rather than the complete wavefield. Ray theory is extensively used due to its simplicity, speed and applicability to a wide range of problems. Seismology lecture course p.3/23

4 Ray theory is strictly valid for media whose length scale variation of λ and µ is much larger than the seismic wavelength (the high frequency assumption). At low frequencies, diffraction can be significant, and ray theory is not generally valid. Seismology lecture course p.4/23

5 The eikonal equation Consider the propagation of compressional waves in heterogeneous media. From before, we have: 2 Φ 1 α 2 2 Φ t 2 = 0 where Φ represents the scalar potential of a P-wave. Now assume a harmonic solution of the form: Φ = A(x)exp[ iω(t(x) + t)] where T(x) is a phase function which describes the arbitrary distribution in space of a surface of constant phase. Seismology lecture course p.5/23

6 Substitution of the above expression into the wave equation yields: T 2 1 α 2 = 2 A Aω 2 A similar expression can be obtained for S-waves (β instead of α). If we now make the high frequency approximation i.e. that ω is large enough that 1/ω 2 0, then T 2 = U 2 which is known as the eikonal equation. U = slowness = 1/velocity. Seismology lecture course p.6/23

7 T(x) = constant defines surfaces called wavefronts. T(x) defines raypaths. The function T(x) has units of time and simply represents the time required by the wavefront to reach x from some reference location x 0. Seismology lecture course p.7/23

8 The ray equation If we denote s as the arc length parameter along a ray and r as the position vector of the ray, then dr ds = T U dr s 1 s 0 r 0 r 1 0 Seismology lecture course p.8/23

9 Now let s examine how the surface of constant traveltime T varies along the ray: dt ds = T dr ds = T T U = U The above two equations may be combined to give: [ d U dr ] = U ds ds which is referred to as the ray equation and may be used to integrate the ray trajectory through an arbitrarily varying medium in 3-D space. Seismology lecture course p.9/23

10 Fermat s principle Fermat s principle states that the ray path between two points P and Q is a path of stationary time t PQ = t Q P Uds = extremum true path true path L Seismology lecture course p.10/23

11 To prove Fermat s principle, we need to show that when a ray path is perturbed, the effect on traveltime is second order δt PQ = δ Q P Uds = Q P δuds + Uδ(ds) r+dr + δr+d( δr) r+ δr δr dr+d( δr) perturbed ray path segment dr reference ray path segment δr +d( δr) r+dr r Seismology lecture course p.11/23

12 We can show that, ignoring higher order terms, this integral reduces to: δt PQ = Q P [ U d ds ( U dr )] ds δrds and the integrand equals zero by the ray equation. Hence, the first-order perturbation of traveltime due to a perturbation in the ray path is zero, and we have proven Fermat s principle. Seismology lecture course p.12/23

13 Snell s law We can use Fermat s principle to derive Snell s law, which describes the refraction of a ray path at an interface between media of different wavespeeds. B( x 2, z 2 ) v 2 v 1 i 2 t 2 O( X, 0) z=0 t 1 i 1 x 1 A(, z 1 ) x=x Seismology lecture course p.13/23

14 The total traveltime T between A and B is given by T = z (X x 1 ) 2 v 1 + z (X x 2 ) 2 v 2 Fermat s principle says that dt/dx = 0, so X x 1 v 1 z (X x 1 ) 2 + X x 2 v 2 z (X x 2 ) 2 = 0 This gives us Snell s law: sin i 1 v 1 = sin i 2 v 2 Seismology lecture course p.14/23

15 Rays in spherically symmetric media On a global scale, the structure of the Earth is, to a good approximation, spherically symmetric. In this case, slowness is a function of radius only: U(r). Let us start by looking at how the quantity ( r U dr ) ds varies along the raypath Seismology lecture course p.15/23

16 Ray parameter In other words, how does it vary with the arc length parameter s? [ ( d r U dr )] = dr ( ds ds ds U dr ) + r d ( U dr ) ds ds ds = 0 Thus, r ( U dr ) ds = constant so this quantity is preserved along the ray, and its direction is normal to the plane that contains dr/ds and r. Seismology lecture course p.16/23

17 The raypath therefore lies in a plane that contains the origin of the coordinate system. The magnitude of the preserved quantity p is referred to as the ray parameter: r U dr ds = ru sin i p is constant along the path for a particular ray. p = ru sin i is a general form of Snell s law for spherically symmetric media. The radius at which the ray bottoms out is given by r = p/u since this occurs when i = 90. Seismology lecture course p.17/23

18 Traveltime determination The eikonal equation in spherical coordinates with U(r) is ( ) 2 T + 1 ( ) 2 T = [U(r)] 2 r r 2 θ To solve the eikonal equation, assume a separation of variables, i.e. T(r,θ) = f(θ) + g(r) Substitution into the eikonal equation yields: T(r,θ) = kθ ± r r S r2 U 2 k 2 dr r where k 2 is a separation constant. Seismology lecture course p.18/23

19 Since T is traveltime, it must be positive and increase with distance along the ray. Take +ve for ascending ray since dr > 0. Take ve for descending ray since dr < 0. Therefore, the integral needs to be split up if the ray turns. S R r S θ r R Seismology lecture course p.19/23

20 It can be shown that k = p, the ray parameter, so T(r,θ) = pθ ± r r S r2 U 2 p 2 dr r If we differentiate the above expression with respect to θ, we get: ( ) T = p θ which tells us that the ray parameter can be determined from data if the source separation is known. Seismology lecture course p.20/23

21 The ray parameter can be measured from the gradient of a traveltime curve. T θ θ Seismology lecture course p.21/23

22 If we now make use of Fermat s principle, which can be stated here as T/ p = 0, we can differentiate our integral expression with respect to p to give T = ± r ru 2 r S r2 U 2 p 2dr If the ray begins and ends at the surface, then the ascending and descending contributions are equal. S R T 1 T 2 r e r 0 r e Seismology lecture course p.22/23

23 The total traveltime is thus given by (c =wavespeed): T = 2 re ru 2 re r 0 r2 U 2 p 2dr = 2 r 0 r/c r2 c 2 p 2dr If is the total angular distance covered by the ray, then: re pc/r = 2 r2 c 2 p 2dr since from before: r 0 θ = ±p r r S dr r r 2 U 2 p 2 Seismology lecture course p.23/23