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

By Guust Nolet

The 1st textbook to supply an intensive creation to seismic tomography for complicated scholars and learn practitioners.

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C c This is only the change in the travel time dt along the small segment. The change in total travel time T between two points A and B is obtained by integrating dt 30 Ray theory for seismic waves R S r(s) + δ r (s) r (s) Fig. 8. Fermat’s principle: perturbing the true minimum time ray from the actual trajectory r(s) (solid line) to a simplified path r(s) + δr(s) (broken line) influences the time integral only to second order in δr. If the perturbed ray is computed for a spherically symmetric Earth model, this greatly improves the efficiency of the calculations.

For a wave in a two-dimensional membrane, how does the amplitude vary with r? 5 The ray approximation What if the medium is not homogeneous? 11) numerically. ). 12). 22) where f (ω) now denotes the spectrum of the source. 62). 21). At location r = (x1 , x2 , x3 ) let us denote the delay by τ (r): P (r, t) = A(r)δ[t − τ (r)] . 18) does not hold. However, if the change in ρ is gradual, its derivatives will be small and we expect a solution that still behaves like a wave. Its velocity may be faster or slower depending on the location, but in the absence of sharp changes we expect an absence of reflections, such that we retain a wavefront, albeit not the spherical front given by τ = r/c that we encountered in the homogeneous case.

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) . 27) numerically. This is comparatively easy in the case of layered or spherically symmetric media. On the other hand, if the seismic velocity is also a function of one or two horizontal coordinates, it may be very difficult.

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