Abstract
This chapter provides specific examples for the first and second derivatives of an objective functional with respect to selected structural and source parameters. We discuss the general characteristics of Fréchet kernels in the context of single scattering from within a first-order influence zone. With the help of the adjoint method we then derive explicit formulas for Fréchet kernels in isotropic, radially anisotropic and visco-elastic media. First derivatives with respect to source parameters are shown to be intimately related to the time-reverse imaging of seismic rupture processes. A strong focus is on the interpretation of the Hessian especially in terms of resolution and trade-off kernels. We demonstrate the relation between Hessian kernels and several forms of second-order scattering. We give a recipe for the computation of second derivatives, followed by a collection of Hessian kernels for a variety of structural parameters.
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© 2011 Springer-Verlag Berlin Heidelberg
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Fichtner, A. (2011). First and Second Derivatives with Respect to Structural and Source Parameters. In: Full Seismic Waveform Modelling and Inversion. Advances in Geophysical and Environmental Mechanics and Mathematics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-15807-0_9
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DOI: https://doi.org/10.1007/978-3-642-15807-0_9
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