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For a given {\it misfit function}, a specified optimality measure of a model,its gradient describes the manner in which one may alter properties of thesystem to march towards a stationary point. The adjoint method, arising frompartial-differential-equation-constrained optimization, describes a means ofextracting derivatives of a misfit function with respect to model parametersthrough finite computation. It relies on the accurate calculation of wavefieldsthat are driven by two types of sources, namely the average wave-excitationspectrum, resulting in the {\it forward wavefield}, and differences betweenpredictions and observations, resulting in an {\it adjoint wavefield}. Allsensitivity kernels relevant to a given measurement emerge directly from theevaluation of an interaction integral involving these wavefields. The techniquefacilitates computation of sensitivity kernels (Fr\'{e}chet derivatives)relative to three-dimensional heterogeneous background models, thereby pavingthe way for non-linear iterative inversions. An algorithm to perform suchinversions using as many observations as desired is discussed.

The Adjoint Method Applied to Time-Distance Helioseismology