By Guust Nolet
The 1st textbook to supply an in depth creation to seismic tomography for complicated scholars and learn practitioners.
Read Online or Download A Breviary of Seismic Tomography: Imaging the Interior of the Earth and Sun PDF
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Additional info for A Breviary of Seismic Tomography: Imaging the Interior of the Earth and Sun
67) repeatedly in this book. An important equality is given by Parseval’s theorem. Let u(t) and v(t) be two time series, which we here assume to be real. The product of the two is proportional to amplitude squared, or energy: E= = ∞ u(t)v(t)dt −∞ ∞ u(t) −∞ 1 = 2π 1 = 2π 1 2π ∞ −∞ ∞ ∞ v(ω)e−iωt dωdt −∞ ∞ u(t)e−iωt dt v(ω)dω −∞ u∗ (ω)v(ω)dω . 68) −∞ A special case is when u(t) = v(t): ∞ −∞ u(t)2 dt = 1 2π ∞ u(ω)u∗ (ω)dω . 2). With that one easily establishes a simplified version of Parseval for real signals u(t) and v(t), involving only the positive spectrum: ∞ −∞ u(t)v(t)dt = 1 Re π ∞ u∗ (ω)v(ω)dω .
1). Note that φ is not necessarily a monotonously decreasing function of i, and that the iteration may fail if the interval contains a point where dφ/di = 0. In tomographic inversions with 106 or more data it is almost inevitable that some rays fail to converge. Rejecting them usually has no significant effect on the resulting image, while trying to salvage them may consume an inordinate amount of time. 2 Ray bending Tracing rays between fixed sources and receivers in three-dimensional media is harrowingly difficult.
24 Ray theory for seismic waves φ = π/2 i ds rdφ dr r φ φ=0 Fig. 5. The polar coordinate system for a ray in the equatorial plane. since ∇(1/c) is oriented in the direction of rˆ . Thus rˆ × p is a constant vector, the length of which we denote by p, as in the Cartesian case, but Snel’s law now takes a different form: r sin i p= (spherical). 4b we see that for the spherical slowness: dT . 32) with the pathlength s as an independent variable, we still lack an equation for the angle of incidence i as a function of s.