Unwrapping surface-skimming Doppler sonar data, and separating waves from currents

In the spring of 2017, our group deployed a surface-skimming Phased-Array Doppler Sonar (PADS) to provide estimates of surface waves and currents. Here I discuss a methodology developed to “unwrap” the aliased Doppler sonar signals, and at the same time separate the wave motions (which are mostly responsible for the aliasing in the first place) from the residual “underlying currents” (largely associated with Langmuir circulation, but also internal waves, inertial currents, fronts, and whatever else is there). The approach involves starting with the lowest frequency waves, and iteratively increasing the range of frequencies included, while “unwinding” the raw complex coherences by the accumulated estimate of wave velocities at each iteration. On each pass, even the lowest frequency estimates are improved upon, until eventually a robust “end point” is reached, at which point the wave orbital velocities (stored separately)have been effectively “subtracted” (or “back-rotated out”) from the complex coherences, providing `residual' Doppler shifts. While these residuals do include Doppler noise, they also include signals of key interest (as listed above), which can be retrieved with far less averaging than would be required with the waves included.