We employ high-resolution large-eddy simulations to generate realistic plume fields under controlled conditions where the true emission rates and atmospheric chemistry are known. The simulation domain spans 16.4 km × 8.2 km horizontally and extends 4.1 km vertically, covering approximately half of TANGO's along-track viewing domain while ensuring the entire plume remains within the field of view. The computational grid comprises 256×128×64 cells, yielding 64 m horizontal and vertical sampling. This is approximately five times finer than TANGO's native 300 m resolution, ensuring proper representation of the turbulent cascade and inertial sub-range while providing realistic small-scale variability that will be smoothed by the instrument spatial response function.

Our base case is a synthetic emission source (x0,y0,z0)=(1024,4096,200)m that represents a mid-sized coal-fired power plant. Emissions are distributed spatially using Gaussian profiles with 50 m standard deviation to represent dispersion from a stack. The source emits CO₂, NO, NO₂, carbon monoxide (CO), and propene (C₃H₆) with rates representative of typical coal combustion: CO2=9.0 (12.5 Mt yr⁻¹), NO =0.0203, NO₂ =0.00107, CO =0.001135, and C₃H6=0.000119 kmol s⁻¹ . A constant west-to-east wind of 5 m s⁻¹ is imposed uniformly throughout the domain, and the background atmosphere is initialized with 30 ppb O₃, which drives photochemical conversion of NO to NO₂ through the reaction NO + O₃ → NO₂ + O₂, while background NO and NO₂ concentrations are initialized to 0.05 and 0.1 ppb, respectively. After allowing sufficient time for turbulence to develop and reach a quasi-steady state, we extract 10 independent plume realizations at 1 min intervals. This temporal sampling strategy serves two purposes: providing multiple plume structures for uncertainty analysis, and enabling assessment of how temporal offsets between CO₂ and NO₂ observations affect reconstruction performance when the satellites do not measure simultaneously.

To investigate performance across different emission strengths, we generated two additional simulation sets by uniformly scaling all emission rates of our base case (CO₂, NO, NO₂, CO, and C₃H₆) by constant factors while keeping all other atmospheric and meteorological conditions identical. This approach maintains realistic emission ratios and plume chemistry while varying signal strength. This results in the four plume cases in this study corresponding to CO₂ emission rates of 1.44 (2.0 Mt yr⁻¹), 1.8 (2.5 Mt yr⁻¹), 3.6 (5.0 Mt yr⁻¹), and 9 kmol s⁻¹ (12.5 Mt yr⁻¹), representing small, medium, and large industrial facilities within TANGO's target range. By scaling the emission of all species proportionally rather than adjusting CO₂ alone, the NO_x / CO₂ emission ratios remain fixed across cases. Note, however, that the chemical regime is not strictly invariant: higher NO_x emissions lead to stronger O₃ titration, which non-linearly affects the NO → NO₂ conversion timescale. The CO₂ scaling is purely linear and affects results primarily through the signal-to-noise ratio.

Figure  shows a representative snapshot from the LES for the 12.5 Mt yr⁻¹ emission case. Panel (a) displays CO₂ column density, peaking at the source location with characteristic turbulent plume structure. Panel (c) shows NO column density, also maximizing at the source. Panel (b) shows NO₂ column density, with maximum enhancement several kilometers downwind due to chemical conversion from NO to NO₂ during advection. Panel (d) indicates O₃ depletion along the plume from NO titration reactions. The spatial offset between CO₂ and NO₂ enhancement maxima demonstrates the coupled turbulent-chemical processes that TANGO will observe.

To emulate TANGO measurements, MicroHH fields are processed through an observation operator that includes (1) vertical integration to obtain column densities, (2) convolution with a 2D Gaussian point spread function (300 m FWHM) to simulate instrument spatial response, and (3) data sampling on 300m×300m pixels. CO₂ and NO₂ can be sampled on spatially offset grids or from different temporal snapshots, simulating asynchronous observations expected from the two-satellite configuration. Finally, Gaussian measurement noise is added based on TANGO mission requirements: 0.576 mol m⁻² for CO₂ (approximately 1 % of the 420 ppm atmospheric background column) and 3.75×10-5 mol m⁻² for NO₂. Considering the maximum plume enhancements with respect to the atmospheric background, the NO₂ measurements exhibit approximately four times higher signal-to-noise ratio compared to CO₂. Figure a, b displays an example plume for the 12.5 Mt yr⁻¹ case after down-sampling to 300 m resolution and adding TANGO-representative measurement noise. These synthetic observations serve as the basis for Monte Carlo experiments as well as linear error propagation analyses to evaluate emission estimation performance and quantify uncertainties under various operational scenarios.

To account for atmospheric conversion of NO to NO₂ and dilution downwind of an industrial source, the CO₂ / NO₂ ratio is modeled by an exponential function : 1F(s;x)=m1exp⁡-sτd+m0, here, s denotes the downwind Euclidean distance from the emission source, τd is the characteristic spatial decay length scale, m1 is the initial amplitude of the ratio, and m0 represents the asymptotic background value. The exponential parameterization represents the dominant physical processes: elevated CO₂ / NO₂ ratios in the vicinity of the source (arising from direct CO₂ emissions and the initially limited formation of NO₂), followed by an exponential decrease as NO is oxidized to NO₂ and both species are subject to turbulent dilution. The three parameters τd, m0, and m1 that are summarized in a state vector 2x=(τd,m0,m1)T, provide quantitative, observation-based characterization of plume chemical evolution. The spatial decay parameter τd yields an effective chemical timescale when divided by the wind speed, the apparent source ratio m1+m0 characterizes the CO₂ / NO₂ ratio at the emission location, and m0 represents the background CO₂ / NO₂ ratio downwind. Physical interpretation of these parameters is discussed in Sect. .

The functional form of Eq. () is physically motivated as follows. Freshly emitted NO_x is dominated by NO, so the NO₂ concentration is initially suppressed relative to its photostationary-state value. As NO is oxidized to NO₂, the CO₂ / NO₂ ratio decreases from an initially elevated value toward a near-field asymptote m0. The exponential term in Eq. (1) describes this approach to photostationary-state partitioning, while CO₂ acts as a conserved tracer on these scales. We emphasize that this is an empirical near-field approximation whose validity rests on the assumption that NO_x loss through secondary chemistry – primarily NO₂ + OH → HNO₃ – remains negligible within the analysis range. Any residual NO_x loss is implicitly absorbed into the fitted value of m0, which should therefore be interpreted as an observational near-field diagnostic rather than a true physical ratio. This assumption is supported by the MicroHH simulation used in this study, which explicitly accounts for secondary chemistry losses. As shown in Fig. 2c, the modeled CO₂ / NO₂ ratio remains constant across the far field within the simulation domain, confirming that NO_x loss through secondary chemistry is negligible on the scales considered here. This is consistent with Fig. 8 of and validates the assumption within the mission requirements. We note that applied the same functional form to the NO₂ / NO_x ratio, where the ratio rises from near zero at the source to a photostationary-state plateau. The advantage of the CO₂ / NO₂ ratio is that the relevant chemistry parameters can be determined by a fit, whereas they must be fitted or calibrated from additional information when working with the NO₂ / NO_x ratio.

To fit the model in Eq. () to observations, we first define the unitless ratio 3yi=[CO2]i[NO2]i, as observable where [CO2]i and [NO2]i are the total column densities of the corresponding trace gases at spatial sampling point i. Assuming uncorrelated Gaussian measurement errors in CO₂ and NO₂ with standard deviations σi,CO2 and σi,NO2, the precision in yi is obtained via standard error propagation: 4σyi2yi2=σi,CO22[CO2]i2+σi,NO22[NO2]i2. For the measurement vector y=(y1,⋯,yN), the corresponding error covariance matrix is given by 5Sy=diag(σy12,…,σyN2). In the following, we assume that the measured ratio y can be represented by our model F(si;x), evaluated at the distance si between the sampling point i of the observation and the source location. This implicitly neglects a potential model bias arising from the fact that F describes the ratio of the underlying physical CO2 and NO2 fields, whereas the measurement y is a ratio of observed quantities whose numerator and denominator each correspond to a convolution of the true physical fields with the instrument's spatial response function. The higher the spatial resolution of the sensor, the smaller this bias is expected to be. In the following, we assume that for the TANGO spatial resolution of 300 m this error can be neglected; however, we will revisit this assumption when discussing systematic biases in the retrieved components of the state vector. Under this assumption, the state vector in Eq. () is estimated by minimizing the weighted least-squares cost function : 6xret=argminx(y-F(x))TSy-1(y-F(x)), where F(x) is the forward model vector with elements F(si;x) for each pixel. The minimization is performed using a Gauss–Newton iterative scheme, where a linearized least squares problem is solved per iteration step. After convergence, the solution is 7xret=Gy with the gain matrix 8G=(JTSy-1J)-1JTSy-1 and the parameter covariance matrix 9Sx=(JTSy-1J)-1. The forward model Jacobian J with 10Ji,k=∂F(si;x)∂xk is calculated at the expansion point of the last iteration step.

To ensure robust fitting, we filter out pixels in the measurement vector y with insufficient signal-to-noise: specifically, pixels where 11[NO2]iσi,NO2<2or[CO2]iσi,CO2<2 or where the unit-less ratio uncertainty σyi exceeds a predefined threshold. In this study we choose a threshold of 50 % of the ratio value. Only pixels passing these criteria are included in the measurement vector. This signal-to-noise selection criterion is functionally equivalent to a plume mask applied jointly to the CO₂ and NO₂ fields, as is also evident from Fig. c, d. Note that the same NO₂-derived mask used in the masking approach can optionally be applied to the reconstruction approach here as well. Figure d shows the modeled CO₂ / NO₂ ratio using Eq. () for the 12.5 Mt yr⁻¹ base case, illustrating how the model captures the essential chemistry and transport processes represented in the MicroHH simulation shown in Fig. c. The model is applied directly to the CO₂ / NO₂ ratio, so finite spatial resolution mainly affects the source ratio m1+m0, while the decay scale τd and m0 characterization is discussed in Sect. .

After the model fit, a CO₂ field can be derived from the NO₂ measurements by 12[CO2]irecon=F(si;xret)⋅[NO2]i.

The error covariance matrix Srecon of the reconstructed [CO2]recon field is given by 13Srecon,ij=[NO2]i[NO2]jJiSxJjT+δijF2(si;xret)σi,NO22,i,j=1,…,N, where Ji denotes the ith row of the Jacobian matrix J, Sx is the covariance matrix of the fitted model parameters, σNO2 is the NO₂ measurement noise standard deviation, and δij is the Kronecker delta. The second term on the right-hand side of the equation describes uncorrelated CO₂ errors due to a scaling of corresponding NO₂ errors, whereas the first term describes correlated CO₂ errors due to the errors on the model parameter x.

Figure e shows the reconstructed CO₂ field for the 12.5 Mt yr⁻¹ case, which exhibits reduced noise compared to the original CO₂ measurement. The difference between the reconstructed and original CO₂ fields (Fig. f) primarily reflects the filtered measurement noise in the first term in Eq. (). The implications of this noise reduction for integrated emission uncertainty are evaluated in Sect. . By design, the ratio model can represent only those spatial and temporal structures that are describable by the three fitted parameters. Consequently, systematic errors that are present in the CO₂ column density product but absent in the corresponding NO₂ field are substantially mitigated by the fitting procedure. For example, retrieval artifacts – such as striping or surface-related structures – that affect CO₂ and NO₂ differently are partially suppressed.

To assess the effective information content of the reconstruction, we compute the sensitivity of CO2recon to the true CO₂ signal using a pixel-wise sensitivity matrix, analogous to an averaging kernel : 14Aij=∂[CO2]irecon∂[CO2]jtrue=[NO2]i[NO2]j⋅J⋅Gij. Matrix A quantifies how strongly [CO2]recon responds to the true [CO2]true signal: 15[CO2]recon=A⋅[CO2]true. The complementary term 1-Aii defines the fraction of the signal not captured by the model, representing the null-space error of the reconstruction approach . This sensitivity analysis quantifies the extent to which the reconstructed CO₂ field faithfully represents the true atmospheric state versus being influenced by the NO₂ observations and model constraints.

Figure  illustrates the application of the averaging kernel A for the 12.5 Mt yr⁻¹ base case. To stress-test the method, we define the true CO₂ field as a plume from 9 min before the time step used to derive A (Fig. a); this represents an extreme case, as the temporal separation between CO₂ and NO₂ observations from TANGO will be less than 1 min in practice. When A is applied to this earlier plume according to Eq. (), the reconstructed field (Fig. b) exhibits turbulence patterns from the later NO₂ plume used to construct A, demonstrating that reconstruction always responds with the turbulence features of the NO₂ field rather than the true CO₂ field. The difference field (Fig. c) reveals this temporal mismatch in turbulent structures. This represents a fundamental limitation of the method: reconstruction performance depends critically on temporal alignment between CO₂ and NO₂ observations, degrading when CO₂ and NO₂ plume structures begin to de-correlate. Although the 9 min separation used here is far larger than what TANGO will experience, it serves to clearly expose this sensitivity. Quantitative evaluation of temporal separation effects, including sub-minute separations representative of TANGO, is presented in Sect. .

As a simpler alternative to full reconstruction, NO₂ observations can be used to define a plume mask that identifies the spatial extent of the emission plume. A threshold is applied to the NO₂ field to identify pixels within the plume, and the CO₂ emission is then estimated by integrating only those CO₂ pixels that fall within this mask. This approach avoids the complexity of ratio fitting while still leveraging NO₂'s superior signal-to-noise ratio for plume detection.

The masking threshold is chosen to balance two competing effects: setting the threshold too low includes background noise and increases random error, while setting it too high truncates the true plume extent and introduces systematic bias. An optimal threshold can be determined empirically by minimizing the total error (combining bias and random uncertainty). Performance evaluation and threshold optimization are presented in Sect. .

This NO₂-based masking approach is conceptually related to the neighboring-pixel background estimation methods of and , both of which use spatial information from adjacent pixels to define the plume boundary and background level. The use of NO₂ for plume detection has been explored in previous multi-species emission studies . Here, our threshold is applied to the co-emitted NO₂ field rather than to the target species CO₂ itself, exploiting NO₂'s superior signal-to-noise ratio for spatially coherent plume delineation independent of CO₂ noise.

To estimate the total emission rate E, we employ the IME method. Fundamentally, the emission rate equals the total plume mass divided by the plume residence time: E=M/ttrans, where ttrans=L/U is the plume transit time, L is the plume length along the wind direction, and U is the effective mass-weighted wind speed. Discretizing to pixels yields : 16E=Δattrans∑i∈I[CO2]i, where I denotes pixels within the plume boundary and Δa is the pixel area (assumed constant). We note that this distance-based formulation differs from the implementation, which uses L=A and large-eddy simulation calibration of the effective wind speed .

For the reconstruction approach, [CO2]i is replaced by the reconstructed field [CO2]irecon from Eq. (); pixel selection is performed through the signal-to-noise filter in Eq. (11), which is functionally equivalent to a plume mask. For the masking approach, the original CO₂ observations are used, with the summation restricted to pixels identified by the NO₂-derived mask.

The precision σE of the flux estimate E is derived from the error covariance of the CO₂ field by 17σE2=KCO2TSCO2KCO2, where Ki,CO2=∂E/∂[CO2]i=Δa/ttrans is the derivative linking each pixel's CO₂ value to the total emission, and SCO2 is the covariance of the CO₂ field. For reconstruction, SCO2=Srecon from the previous subsection, which includes spatial correlations introduced by the ratio fitting. For masking, SCO2 represents the diagonal measurement error covariance of the original CO₂ observations.

Critical to proper uncertainty quantification is rigorous error propagation accounting for the full error covariance matrix, including off-diagonal spatial correlations. For reconstruction-based methods, these correlations arise from the ratio fitting process and can dominate the integrated emission uncertainty even when pixel-level noise appears reduced. The impact of these correlations on emission uncertainty is quantified in Sect. .

Table  presents comprehensive emission estimation results for both NO₂-based masking and CO₂ / NO₂ reconstruction approaches across all four emission strengths under ideal conditions – perfect spatial and temporal co-location. Performance depends strongly on source strength, revealing a clear hierarchy in method applicability.

At the lowest emission strength of 2.0 Mt yr⁻¹, both approaches show substantial limitations. The NO₂ masking approach retrieves 1.22 kmol s⁻¹ with a bias of -15.3 % and precision of 18.5 % (Monte Carlo) or 11.8 % (analytical). The reconstruction approach performs worse, retrieving 1.06 kmol s⁻¹ with a bias of -26.6 % and precision of 19.0 % (Monte Carlo) or 12.8 % (analytical). These substantial biases reflect persistent ratio fitting errors when the CO₂ signal is weak relative to measurement noise. The insufficient CO₂ signal-to-noise ratio prevents reliable characterization of the ratio evolution, leading to systematic underestimation of the true emission in a non-linear least squares fit. Chemistry parameters at this emission level exhibit uncertainties exceeding 100 % for m1 and τd, rendering them uninterpretable. This emission strength falls below TANGO's operational detection threshold. At 2.5 Mt yr⁻¹, representing TANGO's nominal detection limit, the NO₂ masking approach shows marked improvement: 1.65 kmol s⁻¹ retrieved emission with -8.1 % bias and 14.0 % precision (Monte Carlo) or 10.1 % (analytical). In contrast, the reconstruction approach still exhibits substantial bias: 1.41 kmol s⁻¹ with -21.4 % bias and 14.6 % precision (Monte Carlo) or 11.1 % (analytical). While both methods achieve similar precision (14 %–15 %), the masking approach demonstrates superior bias characteristics, making it operationally preferable for weak sources near the detection threshold. Chemistry parameters remain poorly constrained, with m1 uncertainty at 77.8 % and τd uncertainty at 78.3 %, precluding meaningful atmospheric interpretation.

At an emission rate of 5.0 Mt yr⁻¹, both retrieval strategies achieve acceptable performance. The masking method yields an inferred flux of 3.67 kmol s⁻¹ with a bias of +1.9 % and a precision of 7.6 % (Monte Carlo estimate) or 6.8 % (analytical estimate). The reconstruction method yields 3.44 kmol s⁻¹ with a bias of -4.3 % and a precision of 7.3 % (Monte Carlo) or 6.9 % (analytical). For both approaches, the precision is on the order of 7 %, and the biases are reduced to within ± 5 %. This emission level thus marks a critical threshold at which the CO₂ signal becomes sufficiently strong to enable robust characterization of concentration ratios. At this emission magnitude, the inferred chemistry parameters become physically interpretable: m1=3799.7 with a relative precision of 25.7 %, m0=729.9 with 14.3 % precision, and τd=2011.2 m with 32.3 % precision. Although the associated uncertainties remain considerable, they are sufficiently constrained to allow qualitative evaluation of the plume chemistry evolution. At the highest emission strength of our study of 12.5 Mt yr⁻¹, both methods achieve robust performance. The masking approach retrieves 9.22 kmol s⁻¹ with +2.4 % bias and 3.4 % precision (Monte Carlo) or 3.1 % (analytical). The reconstruction approach retrieves 9.40 kmol s⁻¹ with +4.5 % bias and 3.1 % precision (Monte Carlo) or 2.8 % (analytical). Both approaches achieve approximately 3 % emission precision with small biases below 5 %, demonstrating the fundamental method capabilities under optimal signal conditions. Chemistry parameters are well-constrained: m1=5349.5 with 7.4 % uncertainty, m0=904.3 with 6.2 % uncertainty, and τd=2302.9 m with 9.7 % uncertainty, enabling quantitative characterization of atmospheric chemistry processes. Across all emission strengths, Monte Carlo and analytical uncertainties agree within 1–2 percentage points, validating the linear error propagation framework described in Sect. . The final column of Table  shows that fixing chemistry parameters to true values (known from noise-free simulations) reduces emission uncertainty by approximately a factor of two for the 12.5 Mt yr⁻¹ case (1.6 % versus 3.1 %), demonstrating best-case performance with perfect prior knowledge – unavailable operationally.

Direct comparison between NO₂-based masking and CO₂ / NO₂ reconstruction reveals a counter-intuitive central finding: despite producing visually cleaner CO₂ fields with suppressed pixel-scale noise (Fig. ), reconstruction does not improve emission precision relative to simple masking. At low emissions (2.0 and 2.5 Mt yr⁻¹), masking shows superior bias characteristics (-15.3 % and -8.1 %) compared to reconstruction (-26.6 % and -21.4 %). At intermediate and high emissions (5.0 and 12.5 Mt yr⁻¹), both achieve comparable accuracy with biases within ± 2 %–5 %. Precision remains essentially equivalent across all emission strengths: approximately 19 %, 14 %, 7 %, and 3 % for the four cases, respectively. For emission estimates, this precision equivalence despite markedly different noise appearance of the plume requires careful explanation. The reconstruction process fits a three-parameter exponential function to the observed CO₂ / NO₂ ratio, which cannot capture high-frequency pixel-scale fluctuations. The uncorrelated CO₂ measurement noise enters the ratio and maps onto the fitted parameters (m0, m1, τd). When the fitted ratio is multiplied by NO₂ to reconstruct CO₂ (Eq. ), pixel-scale noise variability is suppressed (first term in Eq. ), but spatial correlations of noise spanning the entire plume extent are introduced. The uncorrelated errors due to direct mapping of the NO₂ error (second term in Eq. ) is not a dominant error term.

For spatial integration in the IME method, the error correlations dominate total uncertainty because the individual errors on the CO₂ columns coherently add rather than cancel. In contrast, uncorrelated pixel noise in the original CO₂ field (used by the masking approach) averages out during integration, following σtotal∝N scaling. This behavior applies generically to any method using NO₂ to smooth, regularize, or reconstruct CO₂ fields. A critical implication of this is that such methods must provide complete error covariance matrices to demonstrate genuine error reduction. Computing emission precision neglecting spatial correlations introduced by the fitting or smoothing process, can severely underestimate the precision of the emission estimates. In our reconstruction approach, accounting for the full covariance matrix (off-diagonal terms in Srecon) increases the integrated emission uncertainty by more than a factor of 60 compared to cases ignoring error correlation.

Coming back to the plume masking approach, Fig.  quantifies the trade-off between emission bias and precision for different masking thresholds for the 12.5 Mt yr⁻¹ case. Two masking strategies are compared: applying a threshold to CO₂ observations (blue) versus NO₂ observations (red), with thresholds defined as multiples of the respective noise standard deviations (σCO2=0.576, σNO2=3.75×10-5 mol m⁻²). Without masking (threshold <0), plume mass is unbiased but exhibits the largest random error as the entire background noise is integrated (Fig. b). Thresholding CO₂ reduces precision but rapidly increases bias: low thresholds preferentially exclude negative noise creating positive bias (Fig. a), while high thresholds truncate true signal causing negative bias. Figure c shows total error defined by etot=b2+σ2 with precision σ and bias b. It demonstrates that no single CO₂ threshold achieves robust performance across the full bias-precision trade-off.

Figure  illustrates spatial effects of a NO₂ masking strategies for the CO₂ plume mask. The true plume mask (white contour, 0.008 mol m⁻² threshold from noise-free CO₂) serves as reference. Panel (a) with CO2>0 shows isolated noise features inflating integrated mass. Panel (b) with CO2>1.5σCO2 suppresses noise but truncates plume edges. Panel (c) demonstrates NO₂-derived masking: a threshold of 3σNO2 yields <1% bias and approximately 2 % random error, closely following the true plume extent while avoiding both noise contamination and premature truncation. This NO₂-based approach represents approximately a factor-of-two improvement in precision compared to analyzing CO₂ without NO₂ guidance. Thus for TANGO emission quantification, we recommend NO₂-based masking: it is computationally simpler than reconstruction, exhibits superior bias characteristics for weak sources below 5 Mt yr⁻¹, and achieves equivalent precision across all emission strengths. The main value of the reconstruction approach lies in enabling chemistry parameter retrieval, discussed in the following subsection.

As outlined in Sect. , the fit of the model in Eq. () to CO₂ and NO₂ observations simultaneously retrieves three chemistry parameters: the spatial decay scale τd, the amplitude m1, and the background ratio m0, with the apparent source ratio given by m1+m0 (Table ). These parameters describe plume evolution processes, complementing emission estimates. The quality of the retrieved parameters depends critically on emission strength, following the same hierarchy observed for emission quantification. At 2.0 and 2.5 Mt yr⁻¹, chemistry parameters are uninterpretable due to errors exceeding 70 %–100 %. Weak CO₂ signals provide insufficient information to constrain the ratio evolution. At 5.0 Mt yr⁻¹, parameters become interpretable: m1=3799.7 (25.7 % precision), m0=729.9 (14.3 % precision), and τd=2011.2 m (32.3 % precision). While uncertainties remain substantial, they permit qualitative assessment of plume chemistry. At 12.5 Mt yr⁻¹, robust retrieval is achieved: m1=5349.5 (7.4 % precision), m0=904.3 (6.2 % precision), and τd=2302.9 m (9.7 % precision), enabling quantitative characterization. Overall, the interpretation of these parameters even for moderate precisions requires careful consideration of observational constraints and potential biases.

The spatial decay parameter τd exhibits the clearest interpretation among the three fitted parameters. Retrieved values of 2100–2300 m for the medium and large emission cases substantially exceed TANGO's 300 m point spread function, indicating that τd represents a genuine atmospheric process which is little affected by the forward model error that we discussed in Sect. . The decay scale characterizes the spatial evolution of the CO₂ / NO₂ ratio along the plume: higher values indicate slower ratio changes, while lower values indicate rapid decay. When combined with information on the wind speed U in the plume direction, τd converts to an effective chemical timescale τs=τd/U that quantifies the effective rate of NO → NO₂ conversion relative to the conservative tracer CO₂.

In the simulation, the background O₃ concentration is 30 ppb and the rate coefficient for NO + O₃ → NO₂ + O₂ is k=1.9×10-14 cm³ molec.⁻¹ s⁻¹, giving a theoretical clean-air NO lifetime of τchem=(k[O3])-1≈70 s. The competing photolysis reaction NO2+hν→NO+O partly counteracts this conversion, with a daytime timescale of ∼300s, so the net NO→NO2 conversion in clean air is slower than the 70s lower limit. In practice, however, the fresh plume rapidly titrates local O₃, reducing its concentration and substantially lengthening the effective conversion timescale. The degree of this ozone depletion is controlled by turbulent mixing: faster dilution replenishes background O₃ more quickly, shortening τs, while slower dilution sustains O₃ depletion and extends it. The retrieved τs therefore encodes both chemistry and plume dynamics. For the simulation, the imposed boundary wind is 5 m s⁻¹, but surface friction reduces the effective mass-weighted wind speed within the plume to approximately 4 m s⁻¹, yielding τs=τd/U≈2300m/4ms-1≈575 s.

The apparent source ratio m1+m0 nominally represents the CO₂ / NO₂ ratio extrapolated back to the emission source location. This does not correspond to the true CO₂ / NO_x emission ratio, since in the MicroHH setup 95 % of the emitted NO_x is assumed to be released as NO rather than NO₂. For the 12.5 Mt yr⁻¹ case, the retrieved value is m1+m0=5349.5+904.3=6253.8. In principle, this could reveal information about combustion characteristics, fuel composition, or emission control technologies. However, this parameter is biased by more than 25 % at TANGO's 300 m spatial resolution, preventing direct interpretation as the true emission ratio without careful correction. This bias arises from the limited spatial resolution of satellite instruments, which cannot fully resolve the narrow plumes near the source. Where the ratio model describes the ratio of “real” fields, the observed ratio is affected by the instrument spatial response that blurs the observed CO₂ and NO₂ field independently. This affects the accuracy of the retrieved quantity m1+m0 which is mainly determined at the source location of the plume. Consequently, m1+m0 cannot be directly interpreted as the true CO₂ / NO₂ emission ratio without accounting for these resolution-dependent biases.

Finally, we discuss the background ratio m0, which represents the empirical CO₂ / NO₂ ratio in the far downwind portion of the plume within the near-field validity range of the model – i.e., where the NO → NO₂ conversion is essentially complete but NO_x loss through secondary chemistry remains small. Retrieved m0 values range from approximately 730 to 900 for the larger emission cases (Table ). Given its definition, one might wish to use m0 to infer the true background CO₂ / NO_x emission ratio. However, any conversion of m0 to this underlying emission ratio depends on atmospheric parameters that are not directly constrained by the TANGO observations, such as the degree of O₃ depletion within the plume. Despite this limitation in interpretability, m0 remains valuable as an empirical diagnostic: it represents the CO₂ / NO₂ ratio far downwind where the plume has diluted sufficiently for background air to dominate, restoring the ambient CO₂ / NO₂ ratio. Systematic variations in m0 across different locations or atmospheric regimes can therefore provide insight into differences in background composition and plume-chemistry evolution, even if its absolute value cannot be unambiguously related to intrinsic emission characteristics. Within the plume lengths analysed here (<15 km), NO_x oxidation via NO₂ + OH (lifetime ∼2 h) is a secondary effect; however, its influence is implicitly absorbed into the fitted parameters, in particular m0, which encodes the far-field CO₂ / NO₂ ratio where such processing has already modified the NO₂ / NO_x partitioning.

Up to this point, the CO₂–NO₂ synergy has been analyzed under the idealized assumption of perfectly aligned observations of both species. For TANGO, however, due to the configuration comprising the two CubeSats TANGO-Carbon and TANGO-Nitro, the CO₂ and NO₂ measurements are neither spatially nor temporally fully co-located. To assess the robustness of our methodology under realistic TANGO operational conditions, we again consider the 12.5 Mt yr⁻¹ base case and conduct an extensive Monte Carlo experiment: 10 independent MicroHH plume snapshots are each combined with 500 distinct noise realizations, resulting in 5000 total realizations per configuration. Both the masking and reconstruction strategies are then evaluated for their sensitivity to spatial and temporal misalignment between the CO₂ and NO₂ observations, simulated by sampling CO₂ and NO₂ on shifted grids and by selecting plume snapshots with varying time separations.

We demonstrate the method on real satellite data using EnMAP observations over three industrial sources from : Matla (5 October 2023), and two power plants close to Riyadh (PP10, 11 July 2023 and PP9, 15 July 2023). EnMAP provides hyperspectral measurements at 30 m spatial resolution, enabling simultaneous CO₂ and NO₂ retrievals from the same overpass. Both retrievals exhibit artifacts such as along-track stripes and surface-related patterns from radiometric calibration uncertainties and spectral interference, more pronounced in CO₂ than NO₂. Site characteristics, observation dates, and wind conditions are summarized in Table . Wind speeds vary from 2.79 (Matla) to 8.76 m s⁻¹ (PP9), affecting plume dispersion and NO₂ plume evolution in downwind direction. Because did not provide retrieval error estimates, we estimated measurement noise from the standard deviation of non-plume background pixels in each scene: 0.65 mol m⁻² for CO₂ and 2.92×10-4 mol m⁻² for NO₂ at Matla, with similar values for the other sites. However, applied destriping and smoothing prior to emission quantification. Consequently, these noise estimates are likely too low, and the assumption of uncorrelated pixel-to-pixel errors is likely invalid. The true error structure is dominated by spatially correlated residuals from preprocessing, which cannot be reliably characterized from background variability alone.

Figure  shows application of both masking and reconstruction approaches to the three sites. Row (a–c) displays original EnMAP CO₂ retrievals with clear plume structures but substantial noise and surface artifacts. Row (d–f) shows reconstructed CO₂ fields after CO₂ / NO₂ reconstruction, exhibiting smoother plumes with reduced interference. Row (g–i) shows NO₂ retrievals with lower noise but some remaining artifacts. Row (j–l) displays differences between original and reconstructed CO₂ fields, highlighting noise and bias reduction achieved through the ratio-based approach. As demonstrated earlier, however, this visual noise reduction does not translate into a reduced error on the retrieved emission flux compared to using the original CO₂ field directly. Table  compares emission estimates and retrieved reconstruction parameters across the three sites. We note that used the cross-sectional flux method and reported maximum values within 10 km from the source, methodologically different from our IME method. We averaged their cross-sectional flux estimates over the 10 km downwind range to obtain representative values more comparable to spatially integrated IME results.

For Matla and PP10, emissions from CO₂ measurements with NO₂ masking (26.9 and 47.4 Mt yr⁻¹) and reconstructed fields (25.3 and 45.2 Mt yr⁻¹) agree well with averaged published values (28.1 and 46.0 Mt yr⁻¹). For PP9, CO₂ with NO₂ masking yields 55.6 Mt yr⁻¹ and the reconstructed field gives 49.6 Mt yr⁻¹, both consistent with the averaged published value of 52.6 Mt yr⁻¹. These results demonstrate that both masking and ratio-based reconstruction are applicable to real EnMAP measurements, successfully reproducing previously published emission estimates. Retrieved reconstruction parameters provide additional insight into plume chemistry and atmospheric conditions. The apparent CO₂ / NO₂ source ratio (m1+m0) is highest for Matla (3391), while the two Riyadh power plants exhibit similar values (PP10: 2410, PP9: 2834). These differences may reflect variations in combustion efficiency, fuel composition, or flue gas treatment, though they must be interpreted cautiously due to the resolution-dependent biases discussed in Sect.  (bias >25% expected at finite spatial resolution).

The spatial decay parameter τd shows substantial variation: Matla exhibits 1744 m, PP10 shows 3188 m, and PP9 displays 1451 m. When converted to temporal scales using observed wind speeds (Table ), PP9 exhibits the fastest effective chemical timescale (τs=τd/U=166 s), whereas for Matla and PP10, τs values are similar with τs=626 and 744 s, respectively. The faster evolution of the CO₂ / NO₂ ratio, indicates different chemical boundaries for the PP9 plume compared to the other two plumes. The background ratio m0 also varies across sites. Again, Matla and PP10 show similar results with m0=1007 and 941, respectively, whereas PP9 deviates from this with m0=1274. This supports our conclusion based on τs, although we are aware that a more detailed chemical analysis is required to interpret the results. The large spread in τs across sites (166–744 s) likely reflects differences in boundary-layer turbulence intensity: stronger turbulence entrains background O₃ into the plume more rapidly, shortening the effective NO → NO₂ timescale. The notably short τs at PP9 (166 s, close to the clean-air theoretical minimum of ∼ 70 s) is consistent with its high wind speed and vigorous mixing, whereas the longer timescales at Matla and PP10 suggest more sustained O₃ depletion under calmer conditions. Boundary-layer turbulence diagnostics from atmospheric reanalysis products (e.g., CAMS) could be used to test this interpretation quantitatively, but this is beyond the scope of the present study. These results from EnMAP suggest that all the chemistry parameters retrieved from the data remain interpretable even in real EnMAP observations when combined with ancillary wind information, despite complexities from finite spatial resolution and retrieval artifacts.

Quantitative error estimation in EnMAP data is severely limited by preprocessing applied to the products provided by . Input noise estimates from background pixels likely underestimate true uncertainties due to smoothing and destriping that introduce spatial correlations. Consequently, our analytically propagated errors (0.04–0.06 kmol s⁻¹, or approximately 0.1 % relative uncertainty) appear unrealistically small compared to published uncertainties from (6.38–14.64 kmol s⁻¹, or approximately 20 %–30 % relative uncertainty). Our tabulated uncertainties should be regarded as lower bounds reflecting only propagated measurement noise under assumptions of perfect functional form and error-free ancillary data, not accounting for systematic errors, model limitations, or preprocessing-induced correlations.

We note that our uncertainty analysis focuses on measurement noise and co-location errors, and does not include wind speed uncertainty. Wind speed enters the IME estimate linearly through the transit time ttrans=L/U, so errors in the effective wind speed translate directly to proportional errors in the flux: a 30 % error in wind speed yields a 30 % error in the inferred emission rate. Wind speed uncertainty is often the dominant error source in operational satellite-based emission quantification e.g.,.

A clear limitation is that reconstruction assumes NO₂ measurements are free of artifacts. Any spurious features in NO₂ – such as striping or calibration errors – are directly imprinted onto reconstructed CO₂ fields through the ratio multiplication (Eq. ), making reconstruction accuracy highly sensitive to NO₂ data quality. This is evident in Fig. , particularly panel (i), where remaining NO₂ artifacts propagate into the reconstructed CO₂ field. Nonetheless, successful application to EnMAP observations demonstrates practical feasibility of the CO₂ / NO₂ synergy approach on real satellite data. Emission estimates agree with independent published results, and chemistry parameters exhibit physically reasonable values with systematic site-to-site variations. For TANGO, with optimized instrument design, consistent measurement characteristics, lower noise levels, and availability of full retrieval error covariances, we expect more robust performance and reliable uncertainty quantification when complete error propagation is applied. The method is ready for deployment in TANGO and other multi-species satellite missions including GOSAT-GW and CO₂M.