We have developed a nonparametric matched filter-bank spectrum estimator, referred to as the forward and backward (FB) amplitude and phase estimation (FBAPES) method, to perform dispersion analysis of borehole array sonic measurements. Different from conventional dispersion analysis methods, the amplitude and phase estimation method estimates the dispersive properties of wave modes by simulating the statistical character of interferences from noise and other modes taking place at the same frequency, designing an adaptive filter to suppress these interferences, and fitting the filter output to a spatial sinusoid with a designated slowness. Such an adaptive filter with low side-lobe response gives rise to a high-resolution fitness function, with clear and narrow ridges, which significantly facilitates identification of different modes. At each frequency, for a data set with a moderate noise level, slowness of all modes hidden in the recorded waveforms can be identified without prior knowledge of the exact number of wave modes. We apply an FB method to improve the robustness of the estimation. We used synthetic and field examples to verify the advantages of the method over the weighted spectral semblance and extended matrix pencil methods. Resolution and noise analyses confirmed the high stability and resolution performance of the FBAPES method.