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438 Environmental informaticsThe mode should be considered to be just one possible summary of thepredictive distribution; its corresponding loss function is{L(Y,Ŷ )= 0 if Y = Ŷ,1 if Y ≠ Ŷ ;see, e.g., Berger (1985). I shall refer to this as the 0–1 loss function. That is,should even one element of the approximately 60-dimensional estimated statevector miss its target, a fixed loss is declared, no matter how close it is to themissed target. And the same fixed loss is declared when all or some of theelements miss their targets, by a little or a lot. From this decision-theoreticpoint of view, the predictive mode looks to be an estimate that in this contextis difficult to justify.The next phase of the analysis considers the dry air mole fraction (inppm) of CO 2 averaged through the column from Earth’s surface to the satellite,which recall is called XCO2. Let Y ∗ 0(x, y; t) denotethepredictivemodeobtained from (38.8), which is the optimal estimate given by (38.9) with the0-1 loss function. Then XCO2(x, y; t) isestimatedbŷXCO2(x, y; t) ≡ Y ∗ 0(x, y; t) ⊤ w, (38.15)where the weights w are given in OCO-2 ATB Document (2010). From thispoint of view, ̂XCO2(x, y; t) is the result of applying a smoother f to theraw radiances Z(x, y; t). The set of “retrieval data” over the time period D tare {̂XCO2(x i ,y i ; t i ) : i = 1,...,n} given by (38.15), which we saw from(38.12) can be written as ˜Z; and Y is the multivariate spatio-temporal field{Y(x, y; t) :(x, y) ∈ D g ,t ∈ D t },whererecallthatD g is the geoid and theperiod of interest D t might be a month, say.The true column-averaged CO 2 field over the globe is a function of Y ,viz.g V (Y ) ≡{XCO2(x, y; t) :(x, y) ∈ D g ,t∈ D t } , (38.16)where the subscript V signifies vertical averaging of Y through the column ofatmosphere from the satellite’s footprint on the Earth’s surface to the satellite.Then applying the principles set out in the previous sections, we need toconstruct spatio-temporal probability models for [ ˜Z|Y,θ] and [Y |θ], and eitheraprior[θ] or an estimate ˆθ of θ. This will yield the predictive distribution of Yand hence that of g V (Y ). Katzfuss and Cressie (2011, 2012) have implementedboth the EHM where θ is estimated and the BHM where θ has a prior distribution,to obtain respectively, the empirical predictive distribution and thepredictive distribution of g V (Y )basedon ˜Z.Thenecessarycomputationalefficiencyis achieved by dimension reduction using the Spatio-Temporal RandomEffects (STRE) model; see, e.g., Katzfuss and Cressie (2011). Animated globalmaps of the predictive mean of g V (Y ) using both approaches, based on AIRSCO 2 column averages, are shown in the SSES Web-Project, “Global Mappingof CO 2 ”(seeFigure2atwww.stat.osu.edu/∼sses/collab co2.html). The