Robust Estimations of Current Velocities with Four-Beam Broadband ADCPs

An extended explanation of the hypothesis and equations traditionally used to transform between four-beam ADCP radial beam velocities and current velocity components is presented. This explanation includes a dissertation about the meaning of the RD Instrument error velocity and a description of the standard beam-to-current components transformation as a least squares solution. Afterward, the variance-covariance matrix associated with the least squares solution is found. Then, a robust solution for transforming radial beam velocities into current components is derived under the formality of a weighted least squares approach. The associated variance-covariance matrix is also formulated and theoretically proves that the modulus of its elements will be generally lower than the corresponding modulus of the variance-covariance matrix associated with the standard least squares solution. Finally, a comparison between the results obtained using the standard least squares solution and the results of the weighted least squares method, using a high-resolution ADCP dataset, is presented. The results show that, in this case, the weighted least squares solution provides variance estimations that are 4% lower over the entire record period (8 days) and 7% lower during a shorter, more energetic period (12 h).