The algebraic force-field expansion recently proposed [T. Sako, K. Yamanouchi, and F. Iachello, Chem. Phys. Lett. 299, 35 (1999)] is applied to fit the experimental vibrational term values of H2O and SO2 in the electronic ground (X) over tilde (1)A(1) state. The comparison of results of least-squares fits by the algebraic force-field expansion with those by the conventional force-field expansion shows that the convergence of the algebraic model is much faster than that of the conventional model and this rapid convergence becomes more significant when the Hamiltonian is expressed in local coordinates rather than in normal coordinates. It is also demonstrated that coordinate-space vibrational wave functions can be constructed directly through the experimental-level energy fit by the algebraic Hamiltonian expansion. From the nodal patterns of the vibrational wave functions constructed with the optimized Hamiltonian parameters of SO2, the bifurcation of the wave functions characteristic of the local-mode doublet states are identified in a vibrationally highly excited energy region. It is shown that the local-mode structure of the normal-mode limit molecule SO2 has the same origin as that for the local-mode limit molecule H2O.