SUMMARY Compared with surface waves, guided waves are rarely applied in near-surface investigation. The main reason may lie in the complexity of their dispersion curves. Besides, the study and understanding of guided wave dispersion characteristics are now also inadequate and not deep enough. In this paper, we derived the complete theoretical dispersion curves of P–SV-wave and pure P-wave systems in layered media based on the transmission matrix method and obtained the relative displacement amplitude coefficients at the free surface as a function of frequency and phase velocity for both surface and guided waves. By assigning the value of relative displacement amplitude coefficient to the corresponding point (f,v) on dispersion curve, we got a multi-information diagram called relative amplitude dispersion map (RADM). As a unified description of surface and guided waves, RADM not only shows the velocity–frequency relationship but also represents the polarized energy ratio at the free surface by display colours. The accuracy of RADM was proved by synthetic seismic records, in which RADMs fit well with the corresponding dispersion energy of surface and guided waves. In addition, we designed six models with different Poisson's ratio (PR) and different layer numbers for comparison. It shows that the dispersive vertical-to-horizontal amplitude ratio of guided waves is complex and discontinuous in RADM, which brings great difficulty for mode identification and even affects the subsequent inversion. Tests also show that for high PR layers, the trends of guided P–SV-wave dispersion curves are basically consistent with those of pure P wave. With the decrease of PR, dispersion curves of guided P–SV wave gradually deviate from those of pure P wave. However, RADMs can be greatly consistent with the dispersion energy in either case. This is of great significance for the inversion of near-surface P and S velocities by using dispersion relationships of multimode surface and guided waves.

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Frequency-domain finite-difference (FDFD) modeling plays an important role in exploration seismology. However, a major disadvantage of FDFD modeling is the computational cost, especially for large-scale models. By compactly distributing nonzero strips, the elongated stencil helps to generate a narrow-bandwidth impedance matrix, improving computational efficiency without sacrificing numerical accuracy. To further improve the accuracy and efficiency of modeling, we have developed an optimal FDFD method with an elongated stencil for 2D acoustic-wave modeling. The Laplacian term is approximated using the directional-derivative method and the average-derivative method. The dispersion analysis indicates that this elongated-stencil-based method (ESM) achieves higher accuracy than other finite-difference methods with the elongated stencil, and it is more suitable for large grid-spacing ratios. To keep the phase-velocity error within 1%, 15-point and 21-point schemes in the ESM only require approximately 2.28 and 2.19 grid points per wavelength, respectively, when the grid-spacing ratio, namely, the ratio of directional sampling intervals, is not less than 1.5. Moreover, we also adopt a variable-stencil-length scheme, in which the stencil length varies with the velocity, to further reduce the computational cost in frequency-domain modeling. Several numerical examples are presented to demonstrate the effectiveness of our ESM.

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