On the solution of the multiple collocation problem

A new solution method is given for the general multiple collocation problem formulated in terms of the covariance equations. By a logarithmic transformation, the covariance equations reduce to ordinary linear equations that can be handled using standard methods. Solution by matrix inversion has the advantage that the analytical solutions can be reconstructed. The method can be applied to each determined or overdetermined subset of the covariance equations. It is demonstrated on quintuple collocations of ocean surface vectors winds obtained from buoys, three scatterometers and model forecasts, with representativeness errors estimated from differences in spatial variances. The results are in good agreement with those from quadruple collocation analyses reported elsewhere. The average of the solutions from all determined subsets of the covariance equations equals the least-squares solution of all equations. The standard deviation of all solutions from determined subsets agrees with the accuracy found in earlier triple and quadruple collocation studies, but the difference between minimum and maximum value is much larger. It is shown that this is caused by increased statistical noise in more complex solutions. The averages of the error covariances are close to zero, with a few exceptions that may point at small deficiencies in the underlying error model. Precise accuracy estimates are needed to decide to what extent statistical noise explains the spreading in the results and what is the role of deficiencies in the underlying error model.