## 2 Methods
### Relative Entropy and Wavelet Multi-resolution Analysis  
Following [PERSON] [29], information theory-based relative entropy combined with a wavelet multi-resolution analysis was used to examine EC and AIRS time series. Surface and mid-tropospheric signals of CO\({}_{2}\) concentration were quantified as a function of temporal scale and compared at each level of decomposition, corresponding to scales of 2, 4, 8, 16, 32, 64, 128, 256, 512, and 1024 days. Wavelet multi-resolution analysis achieves a high level of localization in both the time and frequency domains through the dilation (\(m\)) and translation (\(n\)) of a \"mother\" wavelet (\(\psi\)), in this case a Daubechies least-asymmetric eight wavelet. The wavelet transform is found by  
\[W(m,n)=\lambda_{0}^{-m/2}\int_{-\infty}^{\infty}f(t)\psi(\lambda_{0}^{-m}t- nt_{0})dt \tag{1}\]  
where \(\lambda_{0}\) is the initial scale of decomposition, and the wavelet is defined by  
\[\psi_{m,n}(t)=\frac{1}{\sqrt{\lambda_{0}^{m}}}\psi\left(\frac{t-nt_{0}\lambda _{0}^{m}}{\lambda_{0}^{m}}\right) \tag{2}\]  
Wavelet multi-resolution analysis exhibits a \"zoom-in\" capability that allows for the identification of brief, high-frequency events and low-frequency variability in a time series [41]. Windows of localization are narrow when examining high-frequency signals and wide when looking at low-frequency signals. At each level of decomposition, the relative contributions of one time series to another can be obtained by reconstructing the original signal following Brunsell [29]\[f(t)=\sum_{n}\sum_{m}D_{m,n}\psi_{m,n}(t) \tag{3}\]  
where \(D_{m,n}\) represents the wavelet coefficients. The original time series \((X)\) at scale \(m\) can be reconstructed from the inverse wavelet transform of \(f(t)\) at scale \(m\). The complete time series can then be reconstructed based on these scale-wise reconstructions as  
\[X(t)\approx\overline{X_{m}(t)}+\sum_{m\geq m_{0}}X^{\prime}_{m}(t) \tag{4}\]  
using residual fluctuations \((X^{\prime})\) at each point \(t\). Wavelet subsignals or band-pass filtered renderings of EC tower and AIRS time series at each scale of decomposition were examined for temporal variations. At each temporal scale, correlation coefficients were calculated between original, undecomposed daily time series and wavelet decomposed versions of all three EC tower and corresponding AIRS time series. Correlations between wavelet decomposed versions of EC time series and wavelet decomposed versions of AIRS time series were also analyzed. A correlation of 0.2 and above, explaining at least 20\(\%\) of the variance, is considered to be indicative of a physical relationship. Information flow, gain, and transfer between land-atmosphere interactions can be quantified with information theory-based relative entropy [29, 31]. Indicating the distance between the probability density functions of variables \(x\) and \(y\), relative entropy is defined as  
\[R(x,y)=\sum_{i}p_{i}log\left(\frac{p_{i}}{q_{i}}\right) \tag{5}\]  
\(R\) is the additional amount of information needed to represent the probability density function (pdf) of \(p\) associated with \(x\) given the probability of \(y\) in bin \(i\) determined from the pdf of \(q\). Relative entropy can improve our understanding of surface-atmosphere interactions by identifying temporal scales where there is greatest similarity or exchange of information between time series of mid-tropospheric and surface CO\({}_{2}\). The relative entropy between original and wavelet decomposed versions of AIRS and EC tower observations was obtained, and the similarity or divergence between the pdfs of CO\({}_{2}\) concentration in the mid-troposphere versus at the surface was quantified as a function of temporal scale. Small values of relative entropy at a particular timescale signify that the pdf of the original signal is more closely approximated by the pdf of the decomposed version of the other signal at that timescale.