Recently, we proposed the so-called optimal nearly analytic discrete method (onadm) for computing synthetic seismograms in acoustic and elastic wave problems (Yang et al 2004). In this article, we explore the theoretical properties of the onadm including the stability criteria of the onadm for solving 1D and 2D scalar wave equations, numerical dispersion, theoretical error, and computational efficiency when using the onadm to model the acoustic wave fields. For comparison in the 1D case, we also discuss numerical dispersions and stability criteria of the so- called Lax–Wendroff schemes with accuracy of O(Δt4, Δx8) and O(Δt4, Δx10) and the pseudospectral method (psm). We then apply the onadm to the heterogeneous case in synthetic seismograms. Promising numerical results illustrate that the onadm provides a useful tool for large-scale heterogeneous practical problems because it can effectively suppress numerical dispersions caused by discretizing the wave equations when too-coarse grids are used. Numerical modeling also indicates that simultaneously using both the wave displacement and its gradients to approximate the high-order derivatives is important for decreasing the numerical dispersion and source-generated noise caused by the discretization of wave equations because wave- displacement gradients include important seismic information.