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N. Cressie 439regional and seasonal nature of CO 2 becomes obvious by looking at thesemaps. Uncertainty is quantified by the predictive standard deviations, andtheir heterogeneity (due to different atmospheric conditions and different samplingrates in different regions) is also apparent from the animated maps.It is worth pointing out that the “smoothed” data, ˜Z ≡{̂XCO2(x i ,y i ; t i ):i =1,...,n}, are different from the original radiances, Z ≡{Z(x i ,y i ; t i ):i =1,...,n}. Thus,[Y | ˜Z,θ] isdifferentfrom[Y |Z, θ]. Basing scientific inferenceon the latter, which contains all the data, is to be preferred, but practicalconsiderations and tradition mean that the information-reduced, ˜Z = f(Z),is used for problems such as flux estimation.Since there is strong interest from the carbon-cycle-science community inregional surface fluxes, horizontal averaging should be a more interpretablesummary of Y than vertical averaging. Let g 1 {Y(x, y; t)} denote the surfaceCO 2 concentration with units of mass/area. For example, this could be obtainedby extrapolating the near-surface CO 2 information in Y 0 (x, y; t). Thendefine∫/ ∫Y (x, y; t) ≡ g 1 {Y (u, v; t)} dudv dudvR(x,y)R(x,y)andg H (Y ) ≡{Y (x, y; t) :(x, y) ∈ D g ,t∈ D t }, (38.17)where the subscript H signifies horizontal averaging, and where R(x, y) isa pre-specified spatial process of areal regions on the geoid that defines thehorizontal averaging. (It should be noted that R could also be made a functionof t, and indeed it probably should change with season.) For a pre-specifiedtime increment τ, define∆(x, y; t) ≡Y (x, y; t + τ) − Y (x, y; t)τwith units of mass/(area × time). Then the flux field isg F (Y ) ≡{∆(x, y; t) :(x, y) ∈ D g ,t∈ D t }. (38.18)At this juncture, it is critical that the vector of estimated CO 2 in the column,namely, Y ∗ 0(x i ,y i ; t i ), replaces ̂XCO2(x i ,y i ; t i )todefinethesmootheddata, ˜Z. Then the data model [˜Z|Y,θ] changes, but critically the spatiotemporalstatistical model for [Y |θ] isthesameasthatusedforverticalaveraging.Recall the equivariance property that if Y is sampled from the predictivedistribution (38.6) or (38.8), the corresponding samples from g H (Y ) andg F (Y )yieldtheircorrespondingpredictivedistributions.TheHMparadigmallows other data sources (e.g., in situ TCCON measurements, data fromother remote sensing instruments) to be incorporated into ˜Z seamlessly; see,e.g., Nguyen et al. (2012).,