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The authors consider underwater acoustic (UWA) channel estimation based on sparse recovery using the recently developed homotopy algorithm. The UWA communication system under consideration employs orthogonal frequency-division multiplexing (OFDM) and receiver preprocessing to compensate for the Doppler effect before channel estimation. The authors first extend the original homotopy algorithm which is for real-valued signals to the complex field. The authors then propose two enhancements to the sparse recovery-based UWA channel estimator by exploiting the UWA channel temporal correlations, including the use of a first-order Gauss-Markov model and the recursive least-squares algorithm for channel tracking. Moreover the authors propose a scheme to optimise the pilot placement over the OFDM subcarriers based on the discrete stochastic approximation. Simulation results show that the homotopy algorithm offers faster and more accurate UWA channel estimation performance than other sparse recovery methods, and the proposed enhancements and pilot placement optimisation offer further performance improvement.