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.

Triple collocation is an established technique for retrieving linear calibration coefficients and observation error variances of a physical quantity observed simultaneously by three different observation systems. The formalism is extended to an arbitrary number of systems, and representativeness errors and associated cross-covariances are included in a natural way. It is applied to quadruple collocations of ocean surface vector winds from two scatterometers (ASCAT-A, ASCAT-B, or ScatSat), buoy measurements, and NWP model forecasts. There are fifteen possible sets of quadruple collocation equations, twelve of which are solvable for the essential variables (calibration coefficients, observation error variances, and common variance) as well as two additional error covariances, each set leading to a different solution. A remarkable property of the quadruple collocation equations is proven: when the two additional error covariances from a particular solution are used to correct the corresponding observed covariances, all sets yield the same solution. Therefore the quadruple collocation equations by themselves give no information on the representativeness errors involved; these have to be estimated using other methods. The spreading in the solutions is a measure of the accuracy of the underlying error model. Variation of the scale at which the spatial variances are evaluated yields an optimal scale of 100 to 200 km. For the datasets used in this study the error in the scatterometer error variances is 0.03 to 0.05 ms-1, more than expected on statistical grounds. A more precise determination requires an error model better describing the data.