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For layered models with a low-velocity half-space, Rayleigh waves can exhibit inversely dispersive patterns and thus are advantageous for providing useful information on the seismic properties of such structures. However, the calculation of dispersion curves usually encounters some difficulties such as root skipping or numerical instability due to the presence of leaky modes. The full-wavefield method and spectral element method can offer better solutions but at the cost of a significant computational effort. For practical applications, most researchers treat the zeros of the real part of the secular function as an approximation to the leaky modes, which may lead to relatively large errors that cannot be ignored. Instead, in this paper, we propose a simple and efficient method to approximate the leaky mode based on the local minima of the absolute value of the secular function. Compared with the dispersion curve generated from the real part of the secular function, the proposed method can provide a more accurate approximation. We further apply a new efficient Monte Carlo search technique to two synthetic datasets and a permafrost example. Results show that our forward and inverse methods are effective, efficient, and robust. Moreover, our numerical experiments indicate that the modified Thomson-Haskell method is more suitable for calculating leaky modes compared with the fast delta method and the generalized reflection/transmission method, and that the inversion results are less sensitive to the effect of the limited spread length on the measured dispersion curves. The present study emphasizes the high efficiency of the proposed method in imaging permafrost structures and foundation features of buildings or roads. • A fast solver for calculating the apparent modes due to a low-velocity half-space. • The modified Thomson-Haskell method is more suitable for calculating the leaky modes. • Application for imaging permafrost structures. [ABSTRACT FROM AUTHOR]