This is called the epicentral angle therefore in a

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denoted by the angle subtended from the center of the earth. This is called the epicentral angle (∆). Therefore in (a), all the chords (A 1 – B 1, A 2 - B 2, etc ) have the SAME epicentral angle. And, if the travel-times along each of these chords (seismic raypaths) is the same, then this means that the earth velocity structure only varies with radius (spherically symmetric) and does not vary with azimuth. (b) same as (a) except for larger epicentral angle.
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Spherical symmetry: velocity only changes with radius! The properties (e.g., seismic velocity) along the radial lines that start at the center of the sphere are all are the same! This is spherical symmetry: also called radial spherical symmetry. V(r) V(r) V(r) V(r) V(r) V(r) V(r) V(r)
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Distance versus time travel-times for uniform and Velocity(r) earth Around 1920, enough travel- time measurements from earthquakes at different epicentral angles (distances) were measured. This is the observed time curve below. This showed that at greater distances (epicentral angles) the travel-times came in earlier than for a uniform velocity earth. Ah hah! This means that the earth’s velocity increases with depth and is NOT uniform!
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Plotting millions of P and S-wave travel-times reveals seismic phases This is how the outer liquid iron and solid iron inner core were discovered.
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Incident P-wave: P-Reflection, P-transmitted, S-reflection, S- transmitted When a P-wave hits a velocity boundary, it makes three P-waves: a refracted, a transmitted , and a reflected wave. But, it also forms a reflected and transmitted S-wave too! Snell’s law still works, except when the outgoing ray is a S-wave , the S-wave velocity must be used in Snell’s Law.
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Refraction of a (P or S) ray due to velocity change: Snell’s Law When a ray traverses a velocity contrast (change), the ray MUST refract. Why? Because otherwise the wavefront would ‘ tear ’ apart which the physics does not permit. Also, in most all cases a reflected wave is made. All waves refract: e.g., seismic, light, EM, water waves. The refraction law: Snell’s Law. Note that all angles between the ray and interface are reckoned with respect to the surface normal. 1 2 1 2 sin( ) sin( ) v v θ θ = For a wave reflected from a flat interface, the angle of incidence EQUALS the angle or reflection. incidence reflection θ θ = reflected incident transmitte d
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Snell’s Law derivation An explosion makes a wavefield at (S). The waves travel outwards as represented by the wavefront at different times. Where the wave hits the velocity interface, the waves refracts (and reflects). What determine the angle of the refraction into the rock-2 layer ?
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v1 v2 T 0 T 1 Ѳ 1 Ѳ 1 Ѳ 2 Ѳ 2 Ѳ 2 Ѳ 1 A B T 0 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!
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