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434 Environmental informaticsthe data Z paint an imperfect and incomplete picture of Y .Often,thefirsttool that comes to a scientist’s hand is a “data smoother,” which here I shallcall f. Suppose one definesỸ ≡ f(Z); (38.11)notice that f “de-noises” (i.e., filters out highly variable components) and“fills in” where there are missing data. The scientist might be tempted tothink of Ỹ as data coming directly from the process model, [Y |θ], and useclassical statistical likelihoods based on [Y = Ỹ |θ] tofitθ and hence themodel [Y |θ]. But this paradigm is fundamentally incorrect; science shouldincorporate uncertainty using a different paradigm. Instead of (38.11), supposeIwrite˜Z ≡ f(Z). (38.12)While the difference between (38.11) and (38.12) seems simply notational,conceptually it is huge.The smoothed data ˜Z should be modelled according to [ ˜Z|Y,θ], and theprocess Y can be incorporated into an HM through [Y |θ]. Scientific inferencethen proceeds from [Y | ˜Z] in a BHM according to (38.6) or from [Y | ˜Z, ˆθ] in anEHM according to (38.10). The definition given by (38.12) concentrates ourattention on the role of data, processes, and parameters in an HM paradigmand, as a consequence, it puts uncertainty quantification on firm inferentialfoundations (Cressie and Wikle, 2011, Chapter 2).Classical frequentist inference could also be implemented through amarginal model (i.e., the likelihood),∫[ ˜Z|θ] = [ ˜Z|Y,θ] × [Y |θ]dY,although this fact is often forgotten when likelihoods are formulated. As aconsequence, these marginal models can be poorly formulated or unnecessarilycomplicated when they do not recognize the role of Y in the probabilitymodelling.38.5 EI for spatio-temporal dataThis section of the chapter gives two examples from the environmental sciencesto demonstrate the power of the statistical-modelling approach to uncertaintyquantification in EI.38.5.1 Satellite remote sensingSatellite remote sensing instruments are remarkable in terms of their opticalprecision and their ability to deliver measurements under extreme conditions.