Given an n-dimensional state vector of gridded fluxes x and an m-dimensional vector of observations y, both with normally distributed errors, the optimal flux estimate x^ is obtained by minimizing a Bayesian cost function 1J(x)=(x-xA)TSA-1(x-xA)+(y-F(x))TSO-1(y-F(x)), where xA and SA are the prior flux estimate and error covariance matrix, respectively, SO is the observing system error covariance matrix representing uncertainties in the observations and the transport model, and F is the transport model (Brasseur and Jacob, 2017). We assume as is standard that the transport model initial conditions are given by a sufficiently long spin-up simulation driven by the BCs so that the initial conditions are consistent with the BCs. If the transport model is linear, then F(x)=Kx+c where K=∂F/∂x is the Jacobian matrix. The vector c represents the model background defined by the transport of the BCs to the same spatiotemporal locations as the observations. If the BC concentrations are optimized as part of the inversion, information about the background is instead contained in the state vector and in the columns of the Jacobian matrix so that c=0.

We can write the analytical solution for the cost function minimum that yields the optimal (posterior) flux estimate x^ as 2x^=xA+G(y-(KxA+c)) where 3G=∂x^∂y=KTSO-1K+SA-1-1KTSO-1 is the gain matrix that represents the sensitivity of the posterior fluxes to the observations. We can separate the model-observation difference into the contributions from the errors in the prior fluxes relative to the true fluxes (xT) and the errors in the observing system (ε=y-(KxT+c)) so that 4x^=xA+A(xT-xA)+Gε where A=∂x^/∂x=GK is the averaging kernel matrix, a measure of inversion information content that gives the sensitivity of the posterior fluxes to the true fluxes. The trace of the averaging kernel matrix gives the degrees of freedom for signal (DOFS), the number of independent pieces of information quantified by the inversion (Rodgers, 2000).

The posterior error induced by a BC error (εC) is derived by comparing the posterior fluxes produced with the true BC (cT) to an inversion with the BC error (c=cT+εC): 5Δx^=-GεC. We define Eq. (5) as the diagnostic that estimates the effect of BC bias on the posterior fluxes given an explicitly constructed Jacobian matrix and assumed BC error statistics, which can be estimated from observed variability of background concentrations. BC-induced posterior errors are controlled by the gain matrix, which is a function of the observing system errors, prior flux errors, and transport as represented by the Jacobian matrix.

To better understand how observing system errors, prior flux errors, and transport determine the influence of BC biases on posterior flux estimates (Eq. 5), we consider a one-dimensional horizontal transport model with constant wind speed U. Figure 1a depicts this model for n grid cells of length L. The Jacobian matrix for this transport model is derived through steady-state mass balance (Sect. S1 in the Supplement) to be lower diagonal with constant values given by 6τ=ατ′ where α converts between the flux units and the observation units and τ′ is the grid cell residence time calculated as τ′=L/U. The trace gas sources in the model are the BC c and fluxes x=[x1,x2,…,xn]T. We assume that the m observations are uniformly distributed in space and time with constant uncorrelated error variance σO2. We solve the inversion (Eq. 2) using the averaged observations in each grid cell so that the observing system error covariance matrix SO is diagonal with constant error variances σO2/mg where mg=m/n is the number of observations in each grid cell. We assume constant prior fluxes xA and diagonal prior error covariance matrix SA with constant error variances σA2. We define the dimensionless, domain-average information ratio R of the prior error variances in concentration units to the observing system error variances 7R=mgτσAσO2. The information ratio increases with decreasing observing system error standard deviation, increasing prior error standard deviation, and increasing residence time so that the observations are more sensitive to the fluxes relative to the BC. Large values of the information ratio (R≫1) represent the case where the prior errors are larger than the observing system errors so that the posterior fluxes are strongly constrained by the observations (observation-rich). Small values (R≪1) correspond to inversions that are limited by the number or uncertainty of the observations (observation-limited), including the common case of spatially heterogeneous observations. Consider an illustrative inversion of a methane-like trace gas with τ=1.4h (corresponding to U=5ms-1 and L=25 km), σA=12.5ppbh-1 (corresponding to 50 % uncertainty on relatively large prior emissions of 25 ppb h-1), and σO=10ppb. In this case, an average of mg=200 observations per grid cell are needed to achieve R=1. The information ratio for an inversion can be increased by increasing the inversion duration to include more observations or by coarsening the resolution of the gridded fluxes optimized by the inversion.

The effect of a constant BC bias εC on the posterior fluxes is calculated as a function of the information ratio with the diagnostic (Eq. 5). We derive (Sect. S2 in the Supplement) the BC-induced error in the limiting cases of inversions that are observation-limited (R≪1) or observation-rich (R≫1): 8Δx^j=-τ-1εC(n-j+1)R,R≪1-τ-1εCR-j+1,R≫1,j={1,…,n}.

Figure 2 shows the BC-induced error in these limiting cases (panel a) and for intermediate values of the information ratio calculated using Eq. (5) (panel b) for an illustrative inversion with n=10, εC=10 ppb, and τ=1.4 h corresponding to U=5ms-1 and L=25km. In all cases, BC-induced errors decrease across the domain with the rate of decay bounded by the limiting cases in Eq. (8). In the observation-limited case, the BC-induced errors decrease linearly with the distance from the upwind boundary. The total BC-induced error is relatively small because of the strong constraint from the prior estimate. As the information ratio increases from R≪1, BC-induced errors increase because of the decreased prior constraint. The errors increase the most in the most upwind grid cells. As the information ratio approaches one, the total error converges to the total flux needed to explain the error in the BC (τ-1εC). Further increases in the observational constraint decrease the length scale over which the BC bias influences the posterior solution but not the total BC-induced error. In the observation-rich case, the BC-induced errors decay geometrically as the distance from the upwind boundary increases so that the BC-induced errors are limited to the most upwind grid cell.

We also consider the effect of inverse methods to decrease BC biases within the constant parameter, one-dimensional inverse model. We compare the effect of optimizing the BC concentrations (boundary method) or fluxes outside of the domain of interest (buffer method). The boundary method optimizes one or more terms corresponding to BC concentrations along with the domain fluxes (Lauvaux et al., 2012; Wecht et al., 2014; Hancock et al., 2025). The buffer method expands the inversion domain to optimize fluxes in outlying grid cells, which are allowed to vary unphysically to absorb BC biases and are excluded from the final analysis (Shen et al., 2021; Varon et al., 2022). Both approaches increase the state vector dimension, thereby altering the Jacobian matrix and prior error covariance matrix. The buffer method also applies large prior error standard deviations to the buffer grid cells, often by aggregating the buffer grid cells into large clusters.

We derive the conditions under which the boundary and buffer methods are equivalent for an inversion of a single grid cell. For the boundary method, we use a BC prior error standard deviation equal to the constant BC bias εC and update the Jacobian matrix to include the sensitivity of the grid cell average observation to the BC. For the buffer method, we add a buffer grid cell and scale its prior error standard deviation by a factor of p. We assume observations are equally distributed over both grid cells. We solve both inversions. The posterior fluxes are equivalent when 9p=εCτσAR+2. Assuming steady state and knowledge of the domain wind speed, the buffer scale factor p can be chosen so that buffer and boundary methods perform equivalently. The scale factor goes to infinity for very small values of the information ratio (observation limited) and to zero for very large values (observation rich). The dependence of the scale factor on residence time implies that the buffer method is more vulnerable to varying wind speeds than the boundary method.

Equation (5) can estimate the effect of BC-induced errors on the posterior solution once model transport is characterized by the construction of the Jacobian matrix. We estimate BC-induced errors to inform the choice of inversion domain before building the Jacobian matrix by applying Eq. (5) to a simple steady-state two-box model that generalizes a gridded flux inversion. Figure 1b and c depict the model. The domain has BC c and constant wind speed U. All grid cells have representative grid cell length L given the wind direction. We consider a grid cell (with index j) that is j grid cells from the domain boundary and define the upwind grid cell (with index j0) as the aggregate of the upwind j grid cells. We conduct an inversion of the average observations over these two grid cells y=[yj0yj]T regularized by prior fluxes xA=[xA,j0xA,j]T. We define the prior error covariance matrix as 10SA=σA,j0200σA,j2 and use an equivalent structure for the observing system error covariance matrix. We assume the observing system error standard deviations account for the reduction in error resulting from averaging the observations. The Jacobian matrix is derived by steady-state mass balance (Sect. S1) to be a function of τ (Eq. 6) given by 11K=τj0j1. We solve for the effect of a constant BC bias εC=[εCεC]T on the posterior flux in the grid cell of interest Δx^j by applying the diagnostic (Eq. 5) to compute Δx^=[Δx^j-1Δx^j]T. This defines the preview 12Δx^j=-(τ-1εC)Rj1+Rj+βRj0+RjRj0 where Rj0 and Rj refer to the information ratio (Eq. 7) in the upwind and selected grid cell, respectively, with τj0=jτ, and 13β=mg,jmg,j0+1. Here, mg,j0 and mg,j refer to the number of observations in the upwind and selected grid cell, respectively, so that β represents the relative observation density across the grid cells. For grid cells abutting the boundary, all upwind values are set to 0 so that the BC-induced error approaches τ-1εC when the grid cell information ratio is large and 0 when the grid cell information ratio is small. As distance from the boundary increases, the BC-induced errors on average decrease due to the accumulation of upwind fluxes and observations, which increases the upwind information ratio Rj0. Variability in the prior fluxes and observation density as quantified by the information ratio Rj is super-imposed on this decay.

By analogy to our one-dimensional model, we consider the observation-limited and observation-rich limiting cases assuming constant prior fluxes, prior flux standard deviations, observation density, and observing system errors. In this case, Rj0=j4Rj due to the dependence on the number of grid cells j of the upstream observation count (mg,j0=jmg), grid cell residence time (τj0=jτ), and prior flux uncertainty (assuming the uncertainties are uncorrelated, σA,j0=σA/j). The resulting BC-induced errors 14Δx^j=-(τ-1εC)R,R≪1-(τ-1εC)j-4R-1,R≫1 match the general form of the one-dimensional approximation (Eq. 8), but with the geometric decay in the observation-controlled limiting case approximated as quartic decay.

We quantify the sensitivity of posterior fluxes to BC biases by perturbing the true BC in our one-dimensional model, solving the inversion, and comparing the posterior fluxes to those generated by an inversion solved with the true BC (Nesser, 2025). The inversions apply no other sources of bias (including transport errors) to isolate the effect of BC biases. The domain-average information ratio R=0.2 for these demonstration inversions reflects the low DOFS achieved (5 for inversions with constant wind speeds and 7 for varying wind speeds) despite the uniformity and large number of observations (m=1000), consistent with real inversions (Varon et al., 2023).

Figure 3a shows the relative difference in the posterior fluxes, normalized by the prior fluxes, induced by a constant 10 ppb BC perturbation for an inversion with constant wind speeds. As expected from our theoretical analysis, the resulting error on average decreases as the distance from the upwind boundary increases. Exceptions to the decreasing trend result from normalizing by the prior fluxes. The preview (Eq. 12) accurately predicts the error as expected from the specification of the wind speed. The diagnostic (Eq. 5; not shown) perfectly quantifies the no correction method error as expected given the specification of the BC bias.

Figure 3a also shows the effect of the buffer and boundary methods, which are described in Table 1. The buffer method optimizes a buffer cluster of three native resolution grid cells. Increasing the buffer grid cell size decreases the magnitude of BC-induced errors but increases the computational cost. The prior uncertainties of the buffer element are increased by a factor of 10p where p is the scale factor for which the boundary and buffer methods are theoretically equivalent (Eq. 9). Increasing p improves performance, but the error reduction quickly asymptotes so that a factor of 10 is sufficient to maximize error reduction in all tested cases. The boundary method optimizes an average correction to the BC.

Both the boundary and buffer methods reduce the error induced by the constant BC perturbation. The buffer method achieves larger uncertainty reductions than the boundary method as expected due to the choice of scale factor on the prior uncertainty. The uncorrected errors for the boundary method result from the decreased information available to correct the upwind fluxes, while the residual errors for the buffer method result from uncorrected BC biases. In all cases, the residual error magnitude, as measured by the root mean square error (RMSE) of the posterior fluxes compared to the inversion solved with the true BCs, is constant as a function of BC perturbation. This implies that the correction methods may increase posterior errors for grid cells close to the boundary in inversions with small BC biases. Excluding these grid cells from final analysis will minimize the effect of BC biases on the inversion.

The consistent performance of the metrics and of the correction methods results in part from the use of constant wind speeds, which is aligned with the assumptions used to derive the preview and the equivalence between the correction approaches. Figure 3b shows the effect of varying wind speeds. The BC-induced bias consists of decaying upwind biases and structured downwind biases. The preview (Eq. 12) accurately captures the average error decay over the domain, though it is unable to predict the structure of the downwind biases. The diagnostic (Eq. 5; not shown) perfectly predicts both the upwind and downwind biases due to its representation of transport and perfect knowledge of the BC bias. Both correction methods decrease the upwind biases, but only the boundary method decreases the downwind biases. Moreover, the RMSE of the posterior fluxes compared to the inversion solved with the true BCs is constant as a function of perturbation magnitude for the boundary method while the RMSE increases with perturbation magnitude for the buffer method. The buffer method is less able to decrease the variability of the BC-induced biases due to its dependence on wind speed (Eq. 9).

Varying BC biases are unresolvable within inversions without prior knowledge of their structure. We consider variable BC biases to identify the types of biases that are most important to avoid, to understand the impact of correction methods, and to demonstrate the performance of the preview and diagnostic when the structure of the BC bias is not known. Because varying biases can be represented as the sum of periodic functions, we represent them with periodic BCs with varying y intercept, amplitude, and period number (Sect. S3 in the Supplement). The correction methods are implemented as in the constant perturbation case (Table 1 and Sect. 3.1) but with three BC elements optimized instead of one in the boundary method. This marginally increases the computational cost of the inversion but improves performance.

As in the inversion with a constant BC error and varying wind speeds, the periodic BC-induced bias consists of decaying upwind biases and structured downwind biases (Fig. S1 in the Supplement). The upwind biases are driven by the mean bias of the periodic perturbation over the inversion period. The spatial structure of the downwind biases is driven by the frequency of the BC error and the magnitude by the amplitude. This suggests that the correction methods will improve the upwind errors but not the downwind errors and that decreasing the amplitude of BC errors may reduce biases that are otherwise unresolvable.

Figure 4 shows the relative difference in the posterior fluxes, normalized by the prior fluxes, induced by a demonstration periodic perturbation with mean bias of 4 ppb over the inversion period (panel a and Sect. S3) for inversions with constant and varying wind speeds (panels b and c, respectively). The upwind biases are controlled by the mean bias with the error on average decreasing with the distance from the upwind boundary. The preview and diagnostic (Eqs. 13 and 5, respectively), which are calculated using the mean bias, capture these biases. The diagnostic also captures the downwind biases attributable to transport. Similarly, both correction methods decrease the upwind bias. The boundary method achieves modest error reductions because the BC-induced error is of similar magnitude to the error generated by the loss of information content due to the BC optimization. Downwind biases are only decreased by the boundary method when three or more BC elements are optimized.

BC-induced posterior biases are a function of the relative constraint from the observations compared to the prior fluxes as measured by the information ratio (R, Eq. 7) and the grid cell residence time (τ, Eq. 6). To determine how changes in these quantities influence BC-induced biases, we define the influence length scale for our one-dimensional model as the number of grid cells required for the BC-induced error normalized by the prior fluxes to decrease below 0.25 (Δx^/xA<0.25). The influence length scale captures the strong dependence of the BC-induced errors on the number of grid cells from the domain boundary and reflects the additional computational cost associated with optimizing these grid cells. We vary the mean prior flux (from 20–50 ppbh-1) and associated prior error standard deviations, observation density (from 50–200 observations per grid cell), and mean wind speed (from 3–10 ms-1) and calculate the domain average information ratio and residence time for each inversion. Uncertainty is quantified as the standard deviation of the influence length scale across 100 inversions solved with random prior fluxes. All inversions use varying wind speeds and a 10 ppb BC perturbation, though the results are robust for constant wind speeds.

Figure 5a shows the influence length scale as a function of the information ratio for an inversion with no correction method. As predicted from the one-dimensional model solved with constant parameters (Eq. 8 and Fig. 2), the influence length scale decreases with the information ratio. The largest rate of change occurs in the observation-limited regime as the observational constraint shifts the BC-induced errors into the upwind grid cells. In the observation-rich case, the influence length scale is relatively constant as a function of the information ratio. Figure 5b shows the inverse dependence of the influence length scale on grid cell residence time as predicted by Eq. (8). As residence time increases, the influence length scale decreases because the relative contribution of the fluxes compared to the BC increases in the observed concentrations, improving the observational constraint.

Finally, we consider the performance of the metrics and correction methods. The preview (Eq. 12) on average captures the dependence of the BC-induced error (no correction method) on the information ratio and grid cell residence time. In almost all cases, it marginally overestimates the influence length scale so that it represents an upper bound on error. As expected from the prescription of the BC perturbation, the diagnostic (not shown) perfectly predicts the influence length scale associated with no correction method. Both correction methods display functionally no dependence on the information ratio and grid cell residence time. The boundary and buffer methods comparably reduce the influence length scale in all cases.