Triple collocation: Beyond three estimates and separation of structural/non-structural errors

•Unique least squares solution to collocation problem for any number of estimates•Two-level collocation procedure to separate structural/non-structural errors•Allow part of the errors to be correlated•Formulated as a Pythagorean constraint problem in Hilbert space•Error assessment experiments on nine soil moisture products

This study extends the popular triple collocation method for error assessment from three source estimates to an arbitrary number of source estimates, i.e., to solve the “multiple” collocation problem. Past efforts on multiple collocation fall short of delivering a well-defined, well-interpreted and unique solution thus may result in inconsistent conclusions. Here the error assessment problem is solved through Pythagorean constraints in Hilbert space, which is slightly different from the original inner product solution but allows us to define a unique solution to the multiple collocation. The Pythagorean solution is fully equivalent to the original inner product solution for the triple collocation case. The multiple collocation turns out to be an over-constrained problem and a least squares solution is presented. As the most critical assumption of uncorrelated errors will surely fail in most multiple collocation problems, we propose to divide the source estimates into structural groups based on their production process and treat the structural and non-structural errors separately. Such error separation allows the source estimates to have their structural errors fully correlated within the same structural group, which is much more realistic than the original assumption. A new error assessment procedure is developed which performs the collocation twice, each for one type of errors (structural and non-structural), and then sums up the two types of errors. The new procedure is also fully backward compatible with the original triple collocation technique. Error assessment experiments are carried out for surface soil moisture data from multiple remote sensing models, land surface models, and in situ measurements.