Download A Breviary of Seismic Tomography: Imaging the Interior of by Guust Nolet PDF

By Guust Nolet

The 1st textbook to supply an in depth advent to seismic tomography for complex scholars and examine practitioners.

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Extra info for A Breviary of Seismic Tomography: Imaging the Interior of the Earth and Sun

Example text

The only method that is guaranteed to yield the fastest raypath in every circumstance is Dijkstra’s method, first used in seismic tomography by Nakanishi and Yamaguchi [224] but better known in the efficient variant developed by Moser [221] as the ‘shortest path method’. Unfortunately, the stability of the shortest path method is offset by two important disadvantages: its travel time is only approximate and in case more than one ray arrives in the receiver, only the fastest ray can be found. g.

61) implies that a delta function delayed by τ seconds acquires a positive phase ωτ in its spectrum: ∞ −∞ δ(t − τ )eiωt dt = eiωτ . 64) As a consequence, delaying an arbitrary signal by τ seconds will increase the phase in its spectrum by ωτ : ∞ −∞ f (t − τ )eiωt dt = F (ω)eiωτ . 65) Heaviside’s step function h(t) is defined as a function that is 0 for t < 0 and 1 for t > 0 and has a Fourier transform ∞ h(ω) = eiωt dt = 0 eiωt iω ∞ 0 = eiω∞ − 1 . iω With a little bit of damping, modelled by a small positive imaginary component of ω, eiω∞ → 0 and h(ω) = − i 1 = .

3. Ray and wavefront geometry pend on the frequency ω, so that diffracted waves can not be handled adequately (diffracted waves are generally dispersive). 23) generally does a good job of representing the major characteristics of the pulse. In Chapter 5 we shall see that the effects of attenuation – which are frequency dependent – can easily be incorporated into ray theory. 22), divide by ω2 , set f = 0 outside the source region and retain only terms up to order 1/ω we find: −Aeiωτ = κ i ∂ ∂xi iA ∂τ 1 iA ∂ 2 τ 2i ∂A ∂τ 1 + + − 2 ω ∂xi ρ ω ∂xi ρω ∂xi ∂xi ρ 1 ρ ∂τ ∂xi 2 A eiωτ.

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