We investigate the feasibility of using truncated perturbation expansions higher than first order to compute the effect of structural variations relative to a laterally homogeneous reference on broadband (e.g., 20 to 250 sec period) Rayleigh-wave velocities and eigenfunctions. Feasibility is a function of speed, accuracy, and ease-of-use. We discuss the physical meaning of relevant terms in the expansion and posit and test an expansion, referred to as quasi-third-order theory, that consists of all boundary and volume self-terms through third-order and at second-order boundary-volume cross-terms between all boundaries and adjoining volumes that the boundary intersects. We set accuracy criteria at 0.5% for group and phase velocities and several percent for vertical eigenfunctions. For the magnitude of crustal and upper mantle heterogeneities found across Eurasia, first-order perturbation theory meets these criteria for group and phase velocities only above about 80 sec period but meets the eigenfunction criterion down to about 30 sec period. The use of quasi-third-order theory for phase and group velocities and the first-order theory for eigenfunctions is fast (about two orders of magnitude faster than the flat-earth and spherical-earth eigenfunctions codes used for comparison), relatively easy to use, and should meet the accuracy criteria required in most inversions down to about 30 sec period. If accuracy standards are more stringent than those set here, if there are structural variations larger than those considered here, or if the application requires inversion below about 30 sec period, then it would be advisable to regionalize the area of study and to introduce more than one reference model.

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