## 2 Materials and Methods
### Statistical Validation of the coSIF Data Product  
Cross-validation is a widely used method for evaluating the predictive performance of statistical models [49]. The method seeks to evaluate model predictions by withholding some subset of the available data when fitting the model and making predictions, using the withheld subset for validation, and repeating with other subsets. Often, one does leave-one-out cross-validation, where data are withheld one at a time, or \(K\)-fold cross-validation, where data are withheld in \"folds\". However, there are two important considerations in the spatial setting. First, fitting multiple spatial models to different subsets of data can quickly become computationally prohibitive if the number of subsets is large. Second, the spatial arrangement of the withheld validation data is important. For example, the locations of the withheld data may be chosen at random, or they may be chosen so that a region or block of data are missing. The latter reflects a common scenario in remote-sensing applications where a satellite's observations are hindered by blocks of clouds or retrieval malfunctions. In a spatial context, missing blocks of data also lead to a more rigorous validation test, since spatial prediction over large spans of unobserved regions is generally a harder problem than that over small unobserved regions [50]. Given the challenges of cross-validation in the spatial setting, we instead evaluate the performance of geostatistical methods (cokriging and kriging) for producing our data product through validation (rather than cross-validation) within blocks of withheld SIF data. Specifically, for a given block with spatial support \(A\), the SIF validation data \(\mathbf{Z}_{1A}=(Z_{11},\dots,Z_{1n_{A}})^{T}\) associated with spatial locations \(\{\mathbf{s}_{1},\dots,\mathbf{s}_{n_{A}}\}\subset A\) are _withheld_ from the available data, leaving \(\mathbf{Z}^{-A}\), of dimension \((n_{1}-n_{A})+n_{2}\), for use in the prediction framework. Because the withheld validation data, \(\mathbf{Z}_{1A}\), contain measurement error, we seek to predict \(\mathbf{Z}_{1A}\) rather than the noiseless latent vector \(\mathbf{Y}_{1A}\) in the validation setting [28]. It is easy to see that the predicted values, \(\widehat{\mathbf{Z}}_{1A}=\left(\widehat{Z}_{11},\dots,\widehat{Z}_{1n_{A}} \right)^{T}\), are identical to \(\widehat{\mathbf{Y}}_{1A}\) given by Equation (13). However, the RMSPEs of the elements of \(\widehat{\mathbf{Z}}_{1A}\) are different from those of \(\widehat{\mathbf{Y}}_{1A}\). Specifically,  
\[RMSPE\left(\widehat{Z}_{1a}\right)=\sqrt{RMSPE\left(\widehat{Y}_{1}(\mathbf{ s}_{a})\right)^{2}+\sigma_{e,1a}^{2}}\;;\quad a=1,\dots,n_{A}. \tag{15}\]  
To assess the advantage of leveraging geostatistical dependence between SIF and XCO\({}_{2}\), we use both a cokriging and a kriging version of this procedure to produce predictions of the withheld data. To assess the advantage of incorporating spatial dependence within SIF in the cokriging and kriging models, we compare the validation predictions from these geostatistical models to those from a trend-surface-only model (a special case of kriging) that does not leverage spatial dependence. Kriging and trend-surface predictions are based on SIF data alone, whereas cokriging predictions are based on SIF and XCO\({}_{2}\) data. Further details are given in Appendix A.2. We select a number of criteria to evaluate these predictions at validation locations. To characterize the marginal accuracy of the predictions, we consider average prediction error, or bias (BIAS), and root-average-squared prediction error (RASPE), given by  
\[BIAS \equiv\frac{1}{n_{A}}\sum_{a=1}^{n_{A}}\left(\widehat{Z}_{1a}-Z_ {1a}\right), \tag{16}\] \[RASPE \equiv\sqrt{\frac{1}{n_{A}}\sum_{a=1}^{n_{A}}\left(\widehat{Z}_ {1a}-Z_{1a}\right)^{2}}\;. \tag{17}\]  
A benefit of a geostatistical prediction framework is that, at each prediction location \(\mathbf{s}_{a}\in A\), we obtain a predictive distribution, say \(F_{1a}\), rather than a predicted value alone.