## 2 Materials and Methods
### Modeling Multivariate Spatial Dependence between SIF and XCO\({}_{2}\)
The locations of the basis functions and the extent of their spatial supports are displayed in Figure 4.  
After the estimated spatial trend is removed, the behavior of the small-scale spatial variation is modeled based on the vectors of spatial residuals \(\mathbf{R}_{i}=(R_{i1},\ldots,R_{in_{i}})^{T}\), with elements computed from the data as  
\[R_{il}=Z_{il}-\widehat{\mu}_{i}(\mathbf{s}_{il});\quad l=1,\ldots,n_{i},\ i=1,2. \tag{5}\]  
To mitigate bias and trend contamination often found in the residuals ([40], Section 2.2.5), we standardize each set of values using the empirical mean, \(\hat{\mu}_{R,i}\), and standard deviation, \(\hat{S}_{R,i}\), as follows:\[R_{il}^{*}=\frac{R_{il}-\hat{\mu}_{R,i}}{\hat{S}_{R,i}},\quad\text{for}\quad\hat{ \mu}_{R,i}=\frac{1}{n_{i}}\sum_{l=1}^{n_{i}}R_{il},\quad\text{and}\quad\hat{S}_{ R,i}=\sqrt{\frac{1}{n_{i}-1}\sum_{l=1}^{n_{i}}(R_{il}-\hat{\mu}_{R,i})^{2}}\;, \tag{6}\]  
\(l=1,\dots,n_{i}\), and \(i=1,2\). We then model the standardized spatial residuals as  
\[R_{il}^{*}=\eta_{i}^{*}(\mathbf{s}_{il})+\omega_{i}^{*}(\mathbf{s}_{il})+e_{il }^{*};\quad l=1,\dots,n_{i},\;i=1,2, \tag{7}\]  
where \(\eta_{i}^{*}(\cdot)\) is the smooth small-scale variation and \(\omega_{i}^{*}(\cdot)\) is the micro-scale variation with mean zero and variance \(\sigma_{\omega_{i},i}^{2}/\hat{S}_{R,i}^{2}\) on the standardized scale for the standardized residuals \(\{R_{il}^{*}\}_{il}\), and \(\epsilon_{il}^{*}=\epsilon_{il}/\hat{S}_{R,i}\) are spatially varying independent measurement errors on the standardized scale, obtained from the spatially varying independent measurement errors \(\{\epsilon_{il}\}_{il}\) as defined for Equation (1). The smooth small-scale-variation processes \(\eta_{1}^{*}(\cdot)\) (for SIF) and \(\eta_{2}^{*}(\cdot)\) (for XCO\({}_{2}\)) are the focus of our multivariate spatial modeling efforts. Between any two spatial locations \(\mathbf{s},\mathbf{u}\in D\), we model the underlying spatial dependence and cross-dependence in the standardized small-scale-variation processes with a bivariate covariance function \(C_{ij}^{*}(\mathbf{s},\mathbf{u})\equiv\text{cov}(\eta_{i}^{*}(\mathbf{s}),\, \eta_{j}^{*}(\mathbf{u}))\), for \(i,j\in\{1,2\}\). For \(C_{ij}^{*}(\cdot\,,\cdot)\), we selected the full bivariate Matem model [41; 42], popular for its flexible parameterization of spatial smoothness. For spatial lag \(h=||\mathbf{s}-\mathbf{u}||\), the model is symmetric and isotropic:  
\[ \begin{split} C_{ij}(h)=C_{ji}(h)&=\rho_{ij}\sigma _{il}\sigma_{jl}M(h;\,v_{ij},\,\ell_{ij}),\\ M(h;\,v_{ij},\,\ell_{ij})&=\frac{2^{1-v_{ij}}}{ \Gamma(v_{ij})}\left(\frac{h\sqrt{2v_{ij}}}{\ell_{ij}}\right)^{v_{ij}}K_{v_{ ij}}\left(\frac{h\sqrt{2v_{ij}}}{\ell_{ij}}\right),\end{split}  \tag{8}\]  
Figure 4: Solid lines outline the spatial supports of the bisquare basis functions for the spatial trend that we use for the large-scale variation of both SIF and XCO\({}_{2}\), while dashed lines outline the spatial supports of additional basis functions that we use for XCO\({}_{2}\). Note that the spatial supports are circular (the bisquare function is radially symmetric), but they appear as ovals due to the mapping projection. The red rectangle outlines the spatial domain from which OCO-2 data are taken. for \(i,j\in\{1,2\}\), where \(\sigma_{ii}>0\) is a scale parameter and \(\rho_{ij}\) is a (cross-) correlation coefficient, with \(\rho_{ij}=1\) when \(i=j\) and \(\rho_{ij}\in[-1,1]\) otherwise. The function \(M(\cdot\,;\,v_{ij},\ell_{ij})\) is the Matern correlation function [43; 44], with \(v_{ij}>0\) and \(\ell_{ij}>0\) denoting the smoothness and correlation length, respectively. Further, \(\Gamma(\cdot)\) is the gamma function and \(K_{v_{ij}}\) is a modified Bessel function of the second kind of order \(v_{ij}\) ([45], Section 10.2). As detailed in [41], the values of \(\{v_{ij}\}_{ij},\{\ell_{ij}\}_{ij},\) and \(\{\rho_{ij}\}_{ij}\) that result in a positive-semidefinite bivariate covariance function belong to a constrained parameter space.