By Guust Nolet
The 1st textbook to supply an intensive advent to seismic tomography for complicated scholars and learn practitioners.
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Additional info for A Breviary of Seismic Tomography: Imaging the Interior of the Earth and Sun
The convolution c(t) between two signals u(t) and v(t) arises in filter theory and in physical systems with a memory, where the past value of the signal (or function) v(t) is multiplied with the current value of the signal u(t). Convolution is often denoted by an asterisk (∗) and is defined as: c(t) = u(t) ∗ v(t) ≡ ∞ −∞ u(τ )v(t − τ )dτ . 75) showing that convolution in the time domain is equivalent to multiplication in the frequency domain. e. multiplication in the time domain transforms to convolution in the frequency domain: u(t)v(t)eiωt dt = u(ω) ∗ v(ω) .
This allows us to write a wavefield P2 in terms of its value at that surface – away from any of its † A Green’s function at location r and time t from a source at r at time τ is written as G(r, t; r , τ ). This notation is somewhat cumbersome and not always needed. If the source is at the origin at time 0 , we shall often simplify notation and write G(r, t) or G(r, ω). For arbitrary source location r we write G(r, t; r ), or G(r, ω; r ). If delayed by τ seconds, the Green’s function becomes G(r, t; τ ) = G(r, t − τ ).
On the other hand, when we deal with normal modes we often define θ = 0 as the source location, so that θ is the epicentral distance. 42 Ray tracing station φ2 φ1 i1 Fig. 1. The shooting method: the first try uses angle i1 but overshoots the seismograph in the station, whereas i2 gives a ray that arrives at a distance that is too short compared to the epicentral distance φ of the station. 2). we can work out the dot product rˆs · rˆr with r = 1 and obtain: cos = cos ϑr cos ϑs + sin ϑr sin ϑs cos(ϕr − ϕs ) .
A Breviary of Seismic Tomography: Imaging the Interior of the Earth and Sun by Guust Nolet