In EnKF-based CO₂ inversion systems, the information from observations is primarily transferred through correlations between observed concentrations and unobserved fluxes. At the same time, atmospheric transport, as well as the assumed uncertainties in both fluxes and observations, also play fundamental roles by shaping the uncertainty of modeled concentrations and the resulting observational influence. In the present study, we specifically focus on the spatial extent of observational constraints on fluxes. Thus, analyzing the correlation structures between CO₂ concentrations and fluxes is essential. For notational convenience, we occasionally use M to denote CO₂ concentration in the following, with the subscript TOT indicating total concentration, as distinct from individual tracers such as ANT, BIO, and OCE. To illustrate the typical spatial influence of fluxes, we compute the correlation between the prior flux state at the initial time and the resulting ensemble concentrations after one week of transport (at hour 168). This choice partially reflects the sequential update logic of an EnKF (Houtekamer and Mitchell, 2001), where observations constrain fluxes within a time window one by one. While this provides a representative snapshot, the correlation structures are qualitatively similar at other times (temporal evolution is analyzed in Sect. 3.2). This study uses results from 500-member simulations to characterize these correlation structures while results with smaller ensemble sizes are exemplified in Fig. S6.

In the 500-member simulations, the spatial correlation patterns are demonstrated, showing large values near the observation sites and minimal noise-like values (i.e., randomly mixed positive and negative values) in remote areas if the sites are influenced by a specific flux (Figs. 1 and S7). For example, in January, the LAN, HAT, and WLG sites exhibit strong correlations between surface MTOT and nearby FANT (Fig. 1b), FOCE (Fig. 1c), and FBIO (Fig. 1d), respectively. Similarly, in July, the SDZ, HAT, and WLG sites exhibit strong correlations (Figs. S7Co, 1i, and 1j). Many other sites also show significant correlations, suggesting the feasibility of using surface MTOT observations to invert fluxes. In general, the correlation between surface MTOT and nearby FANT is more potent in winter (January) than in summer (July), with notable correlations primarily observed in the eastern China, Korea, and Japan (Figs. S7A and C), where the FANT is large and concentrated. In contrast, the influence of FBIO is more pronounced in the western region, with strong correlations present in both winter and summer (Figs. S7E and G). At island sites, the influence of FOCE can be significant, especially in summer when the monsoon is directed landward. The specific spatial morphology of correlations at individual sites – such as their asymmetry and elongation direction – clearly reflects the modulating influence of local atmospheric transport processes.

Conversely, the overall spatial correlations pattern exhibits a relatively regular distribution, showing maximum values near the sites and a stable decline with distance. The locations of the strongest correlations are typically at or near the observation sites, mainly downwind of the monsoon directions. However, some sites exhibit “remote” correlations for surface MTOT (e.g., Figs. S7Ah and Bo). One notable characteristic of the spatial patterns is that the values are monotonic, which has an important implication. Observation at specific site can only influence a neighborhood's fluxes by increasing or decreasing them overall. In contrast, prior fluxes in this neighborhood may exceed the true values in some locations while falling short in others. Therefore, only a sufficient combination of observations can potentially recover the true fluxes from the prior estimates by both increasing and decreasing the fluxes across the neighborhood.

Regarding the measurement heights, the correlation patterns between MTOT at different heights, including XCO₂, and fluxes are generally consistent with those derived from surface MTOT (compare Figs. 1 and S8). This strong similarity implies that, in terms of diagnosing the information transfer pathway, the flux signals captured by higher-altitude MTOT or XCO₂ are dominated by similar source regions as the surface MTOT. Given the similarity of the correlation patterns across observation heights, the following analysis focuses primarily on surface concentrations. However, it should be emphasized that the difference in the observation innovations and transport errors can affect the relative influence of surface MTOT versus higher-altitude MTOT or XCO₂ in the inversion.

The spatial pattern of concentration-flux correlations is statistically similar to that of flux autocorrelation (compare Figs. 1d and S2a), which shows a clear decline with distance. Although the specific shapes of correlations vary across sites, the average decay scale of their spatial influence seems aligns closely with the autocorrelation length of the prior fluxes. For instance, the strong correlation areas for MTOT-FOCE are larger than those for MTOT-FANT or MTOT-FBIO, consistent with its longer autocorrelation length 1000 km of FOCE than the 600 km of the other fluxes (e.g., compare Fig. S7Cp and Dc). To further prove this, we introduce two additional simulation cases: ONLINE, which utilizes an online perturbed FBIO with land-cover-based autocorrelation patterns and has longer correlation length scales, and OFF100, which employs a shorter spatial correlation length of 100 km (see Table 1). Figure 2 illustrates the spatial correlations between surface MTOT and FBIO for these two cases, which can be compared to Figs. 1d–f and j–l for the OFF600 case.

As expected, the ONLINE case exhibits longer correlation scales in space than the OFF600 case, and the spatial patterns of correlations are predominantly influenced by land cover. Similarly, the OFF100 case shows a shorter correlation scale than the OFF600 case, consistent with the 100 km correlation length. These two additional cases clearly illustrate that the autocorrelation lengths of fluxes largely determine the basic spatial patterns of the concentration-flux correlations, despite that transport processes may translate the high-value areas and deform the shapes of the patterns to some extent. This property provides a more precise explanation of why the autocorrelation scales of the fluxes are critical in flux inversion, as CarbonTracker identifies them as one of the two primary “tuning knobs” (Jacobson et al., 2023). This is because these scales determine the possible spatiotemporal extent to which a concentration observation can contribute to inverting the fluxes.

In addition to the scale of correlation, the strength of the correlation between concentrations and fluxes is also influenced by the autocorrelation length of the fluxes. Notably, both the correlation scale and strength diminish simultaneously in the OFF100 case compared to the OFF600 case (e.g., compare Figs. 1j and 2j). Moreover, the correlation strength in the ONLINE case is greater than in the OFF600 case. This weaker correlation in cases with shorter correlation lengths for fluxes and a stronger correlation in cases with more considerable correlation lengths suggests a form of “resonance” of fluxes at different locations during transport, enhancing the maximum correlations observed. This “resonance” effect essentially stems from the integrative amplification of spatially and temporally coherent signals, which is also enabled by strong autocorrelations, by atmospheric transport. More technically, the effect reflects the constructive aggregation of contributions from spatially correlated flux perturbations to the modeled concentration response, such that concentration-flux covariance structures align more effectively with the transport-induced sensitivity pattern. When fluxes are strongly positively correlated over a large region (i.e., have a long autocorrelation lengths), the CO₂ signals released from these areas exhibit similar phases and variation trends during transport. Integrated by atmospheric flow and arriving at the observation site, their contributed concentration fluctuations add up coherently, thereby producing a stronger concentration-flux correlation. Conversely, if the spatial correlation scale of the fluxes is short, the emission signals from different upwind grid cells are independent of each other and randomly cancel out, resulting in a weak net signal at the site and consequently a lower correlation.

These two effects, directly or indirectly imposed by flux autocorrelations, make it more challenging to configure the space–time autocorrelation functions of fluxes (i.e., to specify the prior covariance matrix B). While data-driven methods that construct correlation functions directly from observations tend to yield shorter (i.e., <100 km) spatial correlation lengths (Kountouris et al., 2015, 2018) in Europe, our results indicate that a shorter correlation length not only reduces the correlation scale but also weakens the correlation strength, further amplifying the limitations of sparse observations.

Therefore, in the context of sparse surface observations, it is not recommended to strictly follow suggestions that advocate very short spatial autocorrelation lengths (e.g., 100 km). This issue is more clearly illustrated in Fig. 3, which shows the sum of localized (Gaspari and Cohn, 1999) concentration–flux correlations across all the 24 sites. When a spatial autocorrelation length of 600 km is used, the summed correlations cover most of the domain except South Asia and some weak-flux regions (Figs. 3a and c), indicating that fluxes over most regions can be constrained by observations from these 24 sites. In contrast, using a much shorter spatial autocorrelation length results in extensive blank areas, indicating that fluxes over most regions cannot be effectively constrained by these observations (Figs. 3b and d).

Consequently, for many applications it may be reasonable not to strictly follow observation-derived correlation lengths, but instead to adopt longer ones (e.g., Chandra et al., 2022; van der Laan-Luijkx et al., 2017).This study demonstrates that, for the current sparse surface observation network in East Asia, adopting a flux spatial autocorrelation length of approximately 600 km is appropriate. It is important to clarify that this length scale is not directly “measured” from the observations but rather serves as a parametric representation based on the physical understanding of flux spatial continuity in the region. The specification of the spatial correlation length L in the prior covariance matrix B serves two distinct purposes. The first is to represent the assumed spatial structure of flux errors. The second, related to the first and revealed by our results, is to regulate the propagation of observational information. Under a sparse observation network, the latter can become the dominant constraint. Mathematically, in the Kalman gain expression K=BHT(HBHT+R)-1, the term BHT governs how observational increments propagate to state variables. Even when atmospheric transport (H) physically links distant locations, prescribing a short correlation length (e.g., 100 km) forces the prior covariances to near zero for separations beyond approximately 300 km. This mathematical cutoff is independent of transport; the Kalman gain simply has no pathway to connect the observation to that distant flux, regardless of how well the transport model represents the physical connection. Here, it is important to distinguish between two aspects of concentration–flux correlations: the location of the correlation maximum (determined by atmospheric transport) and the spatial extent of the high-correlation region (determined by the prescribed autocorrelation length L). Even with a short L, transport can shift the correlation maximum far from the flux source. However, a short L causes the high-correlation region to be narrow, so that sparse observations can only constrain a small area around the location with strongest correlation. A longer L broadens the high-correlation region, enabling the same sparse network to constrain a much wider domain, as shown in Fig. 3. Thus, the recommended 600 km length is not a claim about the true spatial scale of flux errors, but a mathematical requirement to keep the information pathway open under sparse observations, allowing the inversion to utilize the longrange information that transport physically provides.

Denser XCO₂ observations may alleviate the limitations on autocorrelation length scales imposed by surface observation sparsity; however, the spatiotemporal sampling of XCO₂ is irregular, and its information “content” is generally lower than that of surface MTOT observations. It therefore remains unclear to what extent these limitations can be mitigated. Nevertheless, when only surface MTOT observations are used, the flux autocorrelation length should not be too short. In the future, as observational density increases, this optimal scale may change, which could enable more details spatial pattern of the inversion.

To provide a consistency check of the diagnostic results regarding the correlation length, we further performed a set of simplified inversion experiments within an observing system simulation experiments (OSSE) framework (see Sect. S2 in the Supplement). Although not designed as a comprehensive evaluation of inversion performance, these experiments show (Fig. S9) that when only sparse surface observations are used, a short spatial autocorrelation length (100 km) leads to minimal and highly localized posterior adjustments, whereas a longer length scale (600 km) enables broader propagation of observational constraints and improves agreement with the true flux. The inclusion of satellite observations increases overall constraint, but does not fully compensate for overly short correlation lengths. These results are consistent with the mechanism-diagnostic findings presented above.

In addition to the characteristic correlation length scale, the form of the prescribed correlation function (e.g., Gaussian, exponential, and hyperbolic) also influences how observational constraints propagate. In principle, correlation functions with longer tails can extend the theoretical reach of observational influence to more remote fluxes. However, our sensitivity tests (Fig. S10) show that this extension is limited in practical EnKF applications. Although slowly decaying functions increase weak long-range correlations, the resulting concentration–flux correlations at remote distances rapidly decrease to values that are difficult to distinguish from ensemble sampling noise. At the same time, the longer tail redistributes the correlation structure and tends to weaken nearby correlations, where observational signals are strongest and most robust. Therefore, the choice of correlation-function shape involves a practical trade-off between slightly increasing the theoretical adjustment range and preserving strong local constraints.

More broadly, similar to the choice of a longer correlation length, this trade-off reflects a balance between physical realism and inversion efficiency. Ideally, prior correlation functions (including their shape, correlation length, and related properties) should be informed by process understanding or model-data mismatch statistics, which may favor shorter or more heterogeneous structures. However, under sparse observational coverage, such physically realistic choices may fail to sufficiently propagate observational information to the target fluxes. Conversely, stronger and smoother correlation structures may improve the mathematical efficiency of the inversion by allowing sparse observations to constrain wider regions, albeit at the cost of reduced physical specificity. The results presented here should therefore be interpreted not as prescribing a universally optimal prior structure, but rather as illustrating the practical consequences of this balance under current observational limitations. Although the resulting balance may be region-specific, the methodology used to identify an appropriate configuration is straightforward and can be readily adapted to regions beyond our study domain.

The “resonance” effect, whereby coherent flux signals amplify correlations, implies a complementary phenomenon that can influence the spatial patterns of correlations: correlation “dilution” can occur when neighboring or coexisting flux types are uncorrelated. This is particularly relevant in atmospheric inversions, where multiple uncertain flux types jointly influence observed CO₂ concentrations. In FBIO inversions, FANT is usually fixed to reduce the degree of freedom. This is a practical necessity because observations are limited and sparse. However, as mentioned earlier, this approach raises theoretical concerns since multiple flux types influence CO₂ concentrations and may exhibit correlation “dilution”. Fixing one flux may alter the correlation between other fluxes and CO₂ concentrations, effectively removing the “dilution”.

To assess the impact of fixing FANT on the correlations between FBIO and CO₂ concentrations, as well as the implications of using the MANT tracer, we modeled three different CO₂ tracers corresponding to the three types of fluxes: MANT, MBIO, and MOCE. Figure 4 illustrates the correlations of tracers with their corresponding fluxes. Compared to the correlations obtained with all fluxes varying simultaneously (Fig. 1), fixing non-target fluxes leads to widespread and often substantial increases in correlation strength (e.g., compare Figs. 1a and 4a). These increases are most pronounced where MTOT correlations were originally weak – for instance, especially for FANT in July, FBIO in January, and FOCE in both January and July. Conversely, the increases in correlations for FANT in January and FBIO in July are less significant, as their correlations with for MTOT are already strong at many sites.

The significant enhancement of correlations upon fixing other fluxes confirms that correlation “dilution” is a real effect in multiflux systems. A key implication is that simply fixing FANT in a FBIO inversion may be methodologically problematic: the observational information that actually constrains FANT could be misattributed to FBIO, distorting its solution. Conversely, at sites where MTOT already shows strong correlations with a given flux (e.g., FANT in January or FBIO in July), the correlation increases due to fixing other fluxes is minimal. This indicates that where information is intrinsically strong, random errors or uncertainties in coexisting fluxes have negligible “diluting” impact.

In summary, fixing FANT in the inversion of FBIO can introduce disadvantages without offering significant benefits (except for numerical calculation). Nevertheless, these disadvantages will disappear if CO₂ tracer measurements are used instead of MTOT. In such cases, when MTOT are underestimated due to the underestimation of FANT, the MBIO tracer remains unaffected, making the strong correlation beneficial. Currently, there is no simple MBIO tracer, but the MANT tracer has been successfully applied in FANT inversions (Basu et al., 2016, 2020). However, some sites exhibit low correlations even for tracers (e.g., Fig. 4i), highlighting the limitations of using tracers. Conversely, as mentioned above, sites that show minimal correlation increases due to already strong correlations (e.g., Fig. 4b) may provide an opportunity to invert FANT effectively using MTOT observations. It should be noted, however, this opportunity is not guaranteed but only works for some circumstances.

The temporal dynamics of the information transfer pathway are diagnosed by analyzing the time series of the Maximum Regional Correlation (MRC) within a ∼1200×1200 km² domain surrounding each observation site. In Fig. 5, each line traces the evolution of the MRC between a flux impulse, of FANT, FOCE, and FBIO emitted at a specific hour, and the MTOT at the site over subsequent hours. A key observation is the extensive overlap of these lines in many panels, indicating that the concentration at a given time exhibits similar correlation strength with flux impulses from many different prior hours. Conversely, where lines are separated, they reflect distinct correlation levels for fluxes emitted at different times for same source. Unlike the monotonic decay observed in space, these temporal MRC series frequently exhibit sustained plateaus or complex fluctuations patterns, changing with seasons and height. The MRCs for different flux types often vary in a complementary manner, as one weakens, another strengthens, particularly in regions of moderate correlation (e.g., Fig. 5c, f, and i). This anti-phase behavior suggests that shifting transport pathways alternately enhance the influence of different source regions at the receptor site.

The MRC evolution for ocean fluxes (FOCE) shows distinct regimes across the observation network, reflecting the interplay of geography, atmospheric transport, and source-receptor connectivity. At predominantly remote oceanic or downwind coastal locations (e.g., Figs. 5o and S11 Ee), FOCE exhibits dominant MRC. This occurs because these sites are persistently influenced by marine air masses, where the ocean source is strong and contiguous. However, its absolute MRC magnitude is moderated by the simultaneous presence and competition from co-varying terrestrial fluxes. A clear example is the Nansha site (e.g., Fig. S11Ee), where the column airmass is extremely minimally diluted by terrestrial fluxes, allowing a clear and stable oceanic signal to dominate the concentration variability; In contrast, at deep inland or strongly continentally influenced sites, the MRC for FOCE is relatively low or even near zero. At first three stations in Fig. S11, where the marine air masses rarely reach the receptor under typical transport regimes, the MRC is zero. The majority stations, typically situated in coastal transition zones or eastern China (Fig. S11d, f, g, j, and k), show low, stable and dense MRC, displaying a characteristic of baseline ensemble. At these locations, the ocean-derived CO₂ contributes a persistent, near-constant background concentration.

The current features of site correlations related to ocean sources, which align intuitively with geography and climate, are largely attributable to and directly validate the reasonable prior's setting on perturbing monthly pCO₂ fields. Ocean surface pCO₂ and its fluxes vary slowly on synoptic scales, with dominant variations occurring on seasonal scales. Setting the prior temporal autocorrelation to 30 days essentially encodes this physical understanding into the inversion system. Consequently, the simulated information transfer pathways show that at sites influenced by the ocean, ocean signals manifest as a long-term, stable statistical entity. The reasonableness of site characteristics, such as which sites exhibit strong, moderate, or weak ocean influence, is precisely the expected outcome of this reasonable prior setting when processed through a realistic atmospheric transport model. If the ocean prior were set to a short decay time (e.g., 3 d), the system would assume that ocean flux information from a few days ago is nearly irrelevant, even at actual ocean sites. This would result in a loss of stable high correlations at ocean sites, replaced by rapidly decaying and highly fluctuating correlations and a disappearance of the “ocean background source” concept in the information transfer pathways, which would entirely contradict our fundamental physical understanding of ocean carbon exchange.

In contrast, the MRC for anthropogenic fluxes (FANT, grey lines) display highly heterogeneous and site-specific trajectories, including complex oscillations. This diversity should stem from the interaction between its prior specification and atmospheric dynamics, establishing a statistical tendency for information from older fluxes to weaken. In some extreme cases, the variation can show a decay trend rather than an oscillation (e.g., Figs. 5b and more prominently, Fig. S11Aw). However, at most of stations, this underlying decay of FANT is powerfully modulated by transport processes, such as shifting winds, boundary layer effect, creating complex fluctuations. A visible decay trend emerges only where transient transport variability is minimal (e.g., Fig. 5b), allowing the prior's statistical attenuation to be expressed clearly in the temporal trajectory of a single flux impulse's influence.

The biospheric fluxes (FBIO, cyan lines) occasionally exhibit a separation into two parallel strata with distinct correlation levels, within which lines follow coherent, wavelike paths (Fig. 5p, see also Fig. S11). This pattern is a visual encoding of the prior's structural definition: daytime (GEE) and nighttime (RES) fluxes are treated as independent processes. The information pathway faithfully preserves this prescribed discontinuity, leading to the clustered strata. The coherent oscillations within each stratum reflect the diurnal and synoptic-scale variability of these biological signals as they are transported, illustrating how a process-informed prior structure is carried through the physical system. These non-monotonic behavior highlights the critical modulating role of atmospheric transport dynamics.

The similarities in MRC between CO₂ concentration and different fluxes at various hours enable the inversions of fluxes in a period using a single observation, thereby reducing the need for continuous monitoring at a site. This aligns with the discussion by Chevallier et al. (2012), which suggests that the temporal density of observations is much less informative than spatial coverage. However, high temporal density can be beneficial due to the variations in correlations between CO₂ concentration at various hours and a particular flux. If an observation with weak correlations to fluxes is used by chance, the inverted flux will not be optimal. High-density temporal observations can enhance the reliability of the information because a strong correlation at neighboring hours can complement a weak correlation at one hour. This is particularly important when strong correlations are rare and may be missed by low-density observations (e.g., Fig. S11Fi). Furthermore, obtaining a reliable estimation of fluxes from prior fluxes requires a sufficient combination of observations, as discussed in Sect. 3.1.1. Thus, high-density temporal observations can improve the robustness of flux inversions by reducing the influence of inaccuracies in concentration modeling that arise from factors other than uncertain fluxes, ultimately leading to more accurate and reliable results.

It is observed in Sect. 3.2.1 that the MRC at an observation site for different flux types often vary in a complementary manner. This can be more clearly illustrated when we spread the “observation sites” on the whole domain: we calculate the MRC for an observation site in Fig. 5, and this calculation is then applied across all model grids to generate Fig. 6 for specific times.

Like the temporally complementary variation of different fluxes in Fig. 5, the three types of fluxes exhibit spatially complementary distributions in Fig. 6. In January, most areas in China, South Korea, and Japan show relatively strong correlations between surface CO₂ and FANT, with numerous small hotspots in India and Central Asia (Fig. 5a–c). Other land areas generally show strong correlations between surface CO₂ and FBIO (Fig. 5g–i). In the continental seas near China, South Korea, and Japan, the correlations between surface CO₂ and FANT are also strong. In contrast, the southern continental seas display stronger correlations between surface CO₂ and FBIO. In contrast, the open ocean, distant from the continent, shows strong correlations between surface CO₂ and FOCE (Fig. 5m–o). In July, the overall patterns are similar, but the influence of FANT becomes concentrated in smaller areas (Fig. 5d–f). Meanwhile, the influence of FBIO increases significantly in southern China, Korea, and Japan (Fig. 5d–f); however, the two-week-average does not increase very significantly due to the larger diurnal variations of correlations in July (compare Fig. 5a and c, FBIO). Over the ocean, the influence of FOCE becomes much stronger than in January, especially in the Indian Ocean and along the western margin of the Pacific Ocean (Fig. 5p–r).

The complementary distributions of MRCs for different types of fluxes implies the signal “dilution” discussed in Sect. 3.1.3 from another point of view. When some type of flux (e.g., FANT) is fixed, the MRC of that type of flux will be zero, and therefore the other type of flux will fill the absent MRC. Consequently, there will be signal misattributions.

There is another implication of Fig. 6. A large value means a strong correlation between surface CO₂ and nearby fluxes, suggesting that observations from these locations can be effectively used to invert fluxes. In other words, observation data collected from areas with strong correlations provide more valuable information about fluxes, making these locations ideal for establishing measurement sites. Conversely, sites located in areas with weak correlation are less informative for flux inversion. Since atmospheric transport plays a critical role in determining the locations of high correlations, and because atmospheric dynamics can vary rapidly and substantially (see also the January–July contrast in Fig. 6), the optimal measurement locations may change quickly; therefore, our findings should not be regarded as definitive. However, the complementary distributions of correlations for different fluxes highlight the potential for identifying new measurement sites based on the correlation analysis.