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2 results

  • 5 - Waves in Heterogeneous Media

  • Edward S. Krebes, University of Calgary
  • Book:
    Seismic Wave Theory
    Published online:
    15 March 2019
    Print publication:
    28 March 2019, pp 187-229

  • Summary

    The subsurface of the Earth consists typically of media that are not homogeneous, but rather heterogeneous (or inhomogeneous), such as sequences of homogeneous layers where each layer has a different density and wave velocity, or zones where the wave velocity and medium parameters vary smoothly with position. This chapter looks at the theory of how seismic waves propagate in such heterogeneous media. In addition, coverage includes applications of the theory to well surveys and logs (tools used in exploration seismology), the Wiechert–Herglotz method for computing how the seismic wave velocity varies with position from measurements of seismic wave travel times, the eikonal equation (an equation for computing wave travel times), zero-offset ray tracing (a relatively simple technique for roughly estimating the nature of the seismic data that might be recorded in regions of complex subsurface structure), diffracted seismic waves, acoustic waves in heterogeneous materials, and ray equations (for computing the paths of seismic waves in heterogeneous media).

  • Correctors and field fluctuations for thepϵ(x)-Laplacian withrough exponents : The sublinear growth case

  • Silvia Jimenez
  • Journal:
    ESAIM: Mathematical Modelling and Numerical Analysis / Volume 47 / Issue 2 / March 2013
    Published online by Cambridge University Press:
    11 January 2013, pp. 349-375
    Print publication:
    March 2013

  • A corrector theory for the strong approximation of gradient fields inside periodiccomposites made from two materials with different power law behavior is provided. Eachmaterial component has a distinctly different exponent appearing in the constitutive lawrelating gradient to flux. The correctors are used to develop bounds on the localsingularity strength for gradient fields inside micro-structured media. The bounds aremulti-scale in nature and can be used to measure the amplification of applied macroscopicfields by the microstructure. The results in this paper are developed for materials havingpower law exponents strictly between  −1 and zero.