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Unformatted text preview: What determine the angle of the refraction into the rock2 layer ? v1 v2 T T 1 Ѳ 1 Ѳ 1 Ѳ 2 Ѳ 2 Ѳ 2 Ѳ 1 A B T T 1 λ 1 λ 2 1 1 1 1 2 2 2 2 sin( ) sin( ) sin( ) sin( ) AB AB AB AB λ λ θ θ λ λ θ θ = = = = > > 1 2 1 2 1 2 1 2 1 1 2 2 sin( ) sin( ) sin( ) sin( ) * * v v v f v f θ θ θ θ λ λ λ λ = = = = > 1 1 1 2 2 2 sin( ) sin( ) v v θ λ θ λ = = The ratios of the velocities and wavelengths and sin(angles) are all equal! Otherwise, the wavefield would ‘tear’ apart. Derivation of Snell’s Law Tracing rays using Snell’s Law in multiple layer medium Tracing a raypath through multiple layers is simple. It is just the process of using Snell’s Law sequentially each successive interface. Note that the angle (i 1 ) at the top and the bottom (i 1 ’ ) of a layer is the same. If the lower layer’s velocity increases ( v 2 > v 1 ), the ray refracts AWAY from the interface normal. In the converse, the ray refracts TOWARDS the normal. If the deeper layers have a monotonic increase in velocity, the ray will continue to flatten out with depth. Eventually, the ray will reach its turning depth where it goes exactly horizontal and will start to 1 1 2 2 sin ( ) sin ( ) v v θ θ = Using Snell’s Law for a spherical geometry (not Cartesian) 1 1 2 2 1 2 sin( ) sin( ) r r v v θ θ = Note that the angle at the top and the bottom of a spherical shell are NOT the same! This is because the ‘layers’ are curved. But, if one is just calculating the angles on either side of an interface, then the two radius values (r 1 and r 2 ) are the same and the Cartesian form of Snell’s Law is operative (i.e., the radius scaling cancels out in the spherical Snell’s Law. Radial Earth velocity models, Ray paths, and traveltime curves s 1 s 2 (a) Representing the earth’s velocity structure as many shells of constant velocity. (b) Tracing raypaths through the constant velocity shells. Each ray has a different ‘ takeoff’ angle . (c) Traveltime curve. The changes is the slope (slowness units s/m) of the curve are directly related to the changes in velocity with depth. The farther the distance a ray travels, the deeper the ray dives before turning around to come back to the surface. 4 3 p s V V κ μ μ ρ ρ + = = The compressional (bulk) and shear (rigidity) modulus: P and S wave velocity ( ) restoring elastic force stress V massof parcel = ( 29 . : , , P Pisthe pressureapplied to dv v thesphere visvolume dvisvolumechange κ = : , * , . F F is applied shear force A d Ais area d shear angle μ θ θ = Average radial velocity structure of Earth The V p and V s velocity profile of the Earth. The crust, mantle lithosphere, upper mantle, transition zone, lower mantle, outer core, inner core are the primary divisions of the planet’s velocity profile. This was not know until the 1950’s....
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 Winter '12
 JOHN
 Velocity, Wave mechanics, ray refractssin

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