Revisiting measurement and representation errors in geophysical sciences: From classical collocation‐based measurement error estimation to “sampling‐aware field‐informed retrieval”, a method that explicitly accounts for representation errors

Abstract The quantities of interest in geosciences (e.g., atmospheric wind, rock porosity) typically exhibit structure across a wide range of spatial and temporal scales. As a result, it is insufficient to define measurement errors as some deviations from a reference truth. The main issue is that different instruments and models filter reality in distinct ways. Without a precise mathematical description of this filtering, the exact nature of measurement, model, and representation errors remains unclear. To address this, a measurement model that specifies which geophysical field is being sampled and where is proposed. Based on this model, formal definitions for measurement, model, and representation errors are given. The distinction between these types of errors is also discussed. A novel aspect of this study is the demonstration of conditions under which measurement and representation errors will be correlated with one another. Using this new framework, two common measurement error estimation methods, namely the triple‐cornered hat and triple collocation, are shown to be special cases of a more general approach, called sampling‐aware field‐informed retrieval, that explicitly accounts for representation error. Idealized experiments are conducted to illustrate how this general approach performs in various measurement error estimation scenarios involving representation errors. In the more realistic scenarios where observations originate from differently sized sampling windows and are not perfectly collocated with one another, it is shown that estimating measurement errors is possible only if representation errors are accounted for. It is also demonstrated that displacement errors must be taken into account to avoid correlations between measurement and representation errors. More generally, this study advocates that the formulation of a measurement model is necessary for the measurement and representation errors to be clearly defined.