Optimal sampling distribution and weighting for ensemble averages

In many current-measuring applications, limited energy constrains the number of velocity measurements that can be taken, and limited data storage capacity further constrains the number of measurements that can be recorded. Consequently, samples are combined into ensemble averages before recording, and the number of samples available per ensemble average is limited. Unlike traditional inherently-integrating current meters, which ideally record a uniformly weighted average of the velocity over the interval between recorded measurements, a sampling instrument like an acoustic Doppler current profiler (ADCP) is free to arbitrarily weight the contribution of the velocity over time to each recorded measurement. This is possible through the choice of sample timing and the use of weighting factors (windowing) in the ensemble average, subject to the constraint on the ratio of the average sample rate to the average recording rate. Examples of choices of sampling distributions include uniform intervals, pseudorandom intervals, and short bursts of uniformly spaced samples with uniform or pseudorandom intervals between bursts. This paper attempts to put the choice of sampling distribution and weights on a rational basis. An “optimal” choice should minimize either expected or worst-case spectral aliasing into the frequency range of interest to the data user, given limited a priori knowledge of the typical shape of the velocity spectrum and the statistics of the parameters that describe it. An optimal choice should also be robust to data dropout at a given probability, assuming independence of dropout among samples. An example is given where waves must be averaged out to measure tidal currents as accurately as possible