## 4 Discussion  
Information related to CO\({}_{2}\) concentration flows both to and from the land surface [4, 45]. The original AIRS and EC time series show that the global signal of CO\({}_{2}\) concentrations in the mid-troposphere exhibits less amplitude or variation on daily to annual scales than surface concentrations (Figure 1). This finding is supported by [PERSON] _et al._[39] and can be explained by the different processes governing the diurnal variability of CO\({}_{2}\) at the surface versus those operating between the ABL and the free troposphere (FT) to influence concentrations in the mid-troposphere [3, 20]. Using an information theory-based relative entropy approach combined with wavelet multi-resolution analysis, we have identified the dominant temporal scales of ABL-FT exchange of CO\({}_{2}\) and of AIRS utility for revealing source/sink dynamics in this region. While correlations of the wavelet multi-resolution analysis show agreement for all sites on the 512-day timescale, the addition of relative entropy gives insight into similarities on shorter timescales. Results from relative entropy indicate that AIRS could be used as a proxy for all land cover types on the 32-day (monthly) and longer timescales. For land cover types experiencing woody encroachment like KFS and K4B, AIRS may be representative of CO\({}_{2}\) concentrations at shorter timescales depending on how we interpret the influences of site characterization and errors associated with missing daily values. According to the wavelet multi-resolution analysis, the highest positive correlations between mid-tropospheric and surface CO\({}_{2}\) concentrations occur at 256 days for site KFS and at 512 days for sites KFS, KZU, and K4B (Figure 2). While correlations at the 512-day may be influenced by lower reliability at longer timescales in wavelet multi-resolution analysis [43, 44], it is not surprising that surface and mid-tropospheric CO\({}_{2}\) concentrations agree on longer timescales when ABL-FT exchangeshave become more integrated. The more significant correlations for KZU at the 512-day timescale are most likely related to greater soil respiration found in annually burned grasslands, especially in years of high precipitation, as compared with sites KFS and K4B experiencing woody encroachment [22]. [PERSON] _et al_. [46] have shown that 85\(\%\) of respiration in a tallgrass prairie plot, such as KZU, can be attributed to soil respiration. A higher interannual rate of soil respiration makes KZU less of a CO\({}_{2}\) sink, more closely reflecting the rising global trend seen in the AIRS measurements. Positive and negative correlations at 64 days in Figure 2c may illustrate the difference in seasonal dynamics between AIRS and the annually burned site of KZU versus sites KFS and KZU experiencing woody encroachment. Based on the findings of [PERSON] _et al_. [5, 17], the continental-scale gradient of CO\({}_{2}\), approximately 4 ppm between 30\({}^{\circ}\)N and 60\({}^{\circ}\)N, during the growing season may be related to this difference. A multi-month communication lag between ABL CO\({}_{2}\) and FT concentrations could manifest as a delay between the ABL carbon dynamics accompanying the onset of the growing season and the manifestation of this onset in the FT CO\({}_{2}\) values. This relationship between AIRS and sites KFS and K4B at the 64-day scale is also evident in the measure of relative entropy (Figure 3a & b), though the direction of information divergence is seen from KFS to AIRS (Figure 3a) and from AIRS to K4B (Figure 3b). Results of low relative entropy for sites with different land cover types indicate timescales when AIRS is most useful for inferring source/sink dynamics at the surface (Figure 3a, c). AIRS CO\({}_{2}\) concentrations diverge least from surface CO\({}_{2}\) concentrations at site K4B, a lowland area with mixed C\({}_{4}\) grasses and C\({}_{3}\) forbs experiencing woody encroachment.