The behavior of fluid systems is often characterized by their response to initial conditions (stability), external forcings (receptivity) or internal changes (sensitivity). The analysis of these quantities involves direct and adjoint global fields. An efficient and minimally-invasive matrix extraction method will be introduced that allows access to linearized and adjoint information directly from nonlinear simulations. This technique will then be used to investigate the mechanisms underlying the generation of tonal noise around airfoils; it allows the identification and analysis of each component (suction-side and pressure-side boundary layers, separation bubbles, wake) as to its contribution to the emitted acoustic spectrum. Furthermore, the adjoint fields locate the origin of dominant instabilities and point towards means of passive noise control.