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We present a method for extracting gravitational waves from numericalspacetimes which generalizes and refines one of the standard methods based onthe Regge--Wheeler--Zerilli perturbation formalism. [abridged] We then presentfully nonlinear three-dimensional numerical evolutions of a distortedSchwarzschild black hole in Kerr--Schild coordinates with an odd parityperturbation and analyze the improvement we gain from our generalized waveextraction, comparing our new method to the standard one. [abridged]  We find that, even with observers as far out as $R=80 M$--which is largerthan what is commonly used in state-of-the-art simulations--the assumption inthe standard method that the background is close to having Schwarzschild-likecoordinates increases the error in the extracted waveforms considerably. Evenfor our coarsest resolutions, our new method decreases the error by between oneand two orders of magnitudes. Furthermore, we explicitly see that the errors inthe extracted waveforms obtained by the standard method do not converge to zerowith increasing resolution. [abridged]  In a general scenario, for example a collision of compact objects, there isno precise definition of gravitational radiation at a finite distance, andgravitational wave extraction methods at such distances are thus inherentlyapproximate. The results of this paper bring up the possibility that differentchoices in the wave extraction procedure at a fixed and finite distance mayresult in relative differences in the waveforms which are actually larger thanthe numerical errors in the solution.Comment: 19 page

How far away is far enough for extracting numerical waveforms, and how much do they depend on the extraction method?