## 3 Results and Discussion
### Validation of coSIF and Comparison with Simpler Methods  
We evaluate the cokriging method for producing the coSIF data product by performing validation on data withheld in spatially contiguous blocks. As discussed in Section 2.3, the bivariate spatial covariance model used for cokriging is able to leverage cross-spatial dependence between SIF and XCO\({}_{2}\), beyond the spatial dependence within SIF alone. Hence, we perform the same validation procedure for the special case of kriging predictions of SIF (where no relationship between SIF and XCO\({}_{2}\) is assumed). Prior to obtaining kriging predictions, we re-fit the univariate spatial covariance model for SIF using the SIF-only semivariogram. We also compare these geostatistical predictions to the special case of trend-surface-only predictions (where it is assumed further that there is no spatial dependence in the SIF residuals); see Appendix A.2. As opposed to leave-one-out cross-validation, or \(K\)-fold cross-validation with subsets of validation data selected at random, validation with subsets of data missing in blocks is especially challenging in the geostatistical setting, since spatial predictions depend heavily on the quantity, proximity, and precision of nearby data. Here, we separately consider two 5-degree by 5-degree validation blocks outlined by the red squares in Figure 3. One of these blocks has extents [\(-95^{\circ}\)E, \(-90^{\circ}\)E, \(40^{\circ}\)N, \(45^{\circ}\)N] and covers a large section of Iowa in the center of the North American Corn Belt where SIF activity is especially prominent in July. The other block has extents [\(-104^{\circ}\)E, \(-99^{\circ}\)E, \(36.5^{\circ}\)N, \(41.5^{\circ}\)N] and consists predominately of cropland in the plains of Colorado and Kansas where SIF activity is more moderate in July. The spatial supports of these two validation blocks (Corn Belt and Cropland) are representative of large unobserved regions that realistically can occur in OCO-2 datasets (e.g., Figure 3). Recall from Section 2.5 that we produce a prediction, \(\widehat{Z}_{1a}\), and prediction standard error, \(RMSPE\left(\widehat{Z}_{1a}\right)\), at each validation location \(\mathbf{s}_{a}\), for \(a=1,\ldots,n_{A}\), in a given validation block with spatial support \(A\) where data are withheld from predictions. The validation standardized residuals are given by  
\[\left\{\frac{\widehat{Z}_{1a}-Z_{1a}}{RMSPE\left(\widehat{Z}_{1a}\right)}:\ a=1,\ldots,n_{A}\right\}. \tag{21}\]  
As a diagnostic check, it is important to explore whether these residuals are standard-Gaussian distributed, which is an indicator that the model is a good fit to the data in our setting. Quantile-quantile (Q-Q) plots of validation standardized residuals in July 2021 are shown in Appendix B (Figure 11) for the two validation blocks. For both blocks, the Q-Q plots that result from using the cokriging, kriging, and trend-surface-only methods are all quite similar and show that the residuals are approximately Gaussian, although each set of residuals exhibits slightly heavy-tailed behavior relative to the standard Gaussian distribution. This initial diagnostic result indicates that more sensitive validation metrics are needed to distinguish between the different prediction methods. We now evaluate the validation metrics outlined in Section 2.5 (BIAS, RASPE, INT, DSS, and MDSS). Table 1 presents these metrics for the validation predictions and prediction standard errors produced for July 2021 using the cokriging, kriging, and trend-surface-only methods in both validation blocks. With the exception of average prediction error (BIAS), which should be as close to zero as possible, all metrics are negatively oriented scores (lower is better). We find that BIAS is reasonably negligible for all methods in both validation blocks. The root-average-squared prediction error (RASPE) also exhibits similar results across the different prediction methods.