Acoustic communication in the ocean with carrier frequency in the range 5--50 kHz has been performed in both deep and shallow water, and the communication channels can be sorted according to their time and frequency dispersion. Some of the underwater communication channels are significantly spread in both frequency and time; thus they are doubly spread channels. Scattering function estimates computed from real data of different doubly spread channels are presented, and physical scenarios that would produce the different channels are identified by simulations using ray trace and time variant FIR filters. The frequency dispersion depends in particular on relative speed between the transmitter and channel scatterers, or between transmitter and receiver for direct paths, and the eigenrays of some of the physical scenarios can have significantly different Doppler spreads. Current receivers are unable to establish reliable communication over such channels. A new receiver is proposed which can successfully demodulate signals which have propagated through some of the doubly spread channels with multiple Doppler spreads. The receiver makes use of filter taps spaced in both the delay and the Doppler dimensions. Its capabilities and limitations are demonstrated on both real and simulated data.