Efficient computation of global high-resolution Jacobian matrices for analytical inversions of satellite observations of greenhouse gases: Application to the Integrated Methane Inversion (IMI) version 2.3

Zhang, Dandan; Jacob, Daniel J.; Lundgren, Elizabeth W.; Sulprizio, Melissa P.; Balasus, Nicholas; Estrada, Lucas A.; Reidy, Emily; Varon, Daniel J.

doi:10.5194/egusphere-2026-4399

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https://doi.org/10.5194/egusphere-2026-4399

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05 Oct 2026

| 05 Oct 2026

Status: this preprint is open for discussion and under review for Geoscientific Model Development (GMD).

Efficient computation of global high-resolution Jacobian matrices for analytical inversions of satellite observations of greenhouse gases: Application to the Integrated Methane Inversion (IMI) version 2.3

Dandan Zhang, Daniel J. Jacob, Elizabeth W. Lundgren, Melissa P. Sulprizio, Nicholas Balasus, Lucas A. Estrada, Emily Reidy, and Daniel J. Varon

Abstract. Dense satellite observations of atmospheric composition call for global high-resolution inversions to infer surface fluxes from the observations, using a chemical transport model (CTM) as forward model to relate the fluxes to the observed concentrations. The inversion can be done analytically when the relationship of surface fluxes to the observable atmospheric concentrations is linear. The main cost of analytical inversions is the explicit construction of the Jacobian matrix describing the CTM sensitivity of atmospheric concentrations to the gridded surface fluxes (state vector elements) and requiring perturbation simulations for all individual state vector elements. This cost has so far limited global analytical inversions to coarse resolution (~200 km) despite the availability of fine-resolution (~10 km) satellite observations. Here we show how the stretched-grid capability of the high-performance GEOS-Chem CTM (GCHP) operating on a cubed-sphere grid can considerably reduce the computational and storage costs of global Jacobian construction by achieving high resolution in the near-field of the state vector elements (target face) while gradually relaxing to coarse resolution at the antipode. Application to a stretch factor of 10 on the C36 cubed-sphere GCHP grid (C36S10) enables generation of a compact global Jacobian archive at 25-km resolution with 100 times lower computational and storage costs than a regular-grid C360 GCHP with global quasi-uniform 25-km resolution. We have implemented this capability in the Integrated Methane Inversion (IMI) version 2.3. Beyond its use for global inversions, the precomputed C36S10 Jacobian matrix archive can be used for regional inversions, which in the IMI rely on a nested rectilinear grid with boundary conditions at the edges. Conservative remapping of the stretched-grid Jacobian archive to the rectilinear grid reduces the cost of these regional inversions by an order of magnitude by skipping the Jacobian calculation. We demonstrate this capability with a 1-month regional inversion of TROPOMI methane satellite data over the South-Central US where the remapped C36S10 Jacobian reproduces closely the standard IMI results (posterior fluxes) at 0.25°×0.3125° resolution.

Received: 21 Jul 2026 – Discussion started: 05 Oct 2026

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Dandan Zhang, Daniel J. Jacob, Elizabeth W. Lundgren, Melissa P. Sulprizio, Nicholas Balasus, Lucas A. Estrada, Emily Reidy, and Daniel J. Varon

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RC1:
'Comment on egusphere-2026-4399', Anonymous Referee #1, 13 Oct 2026 reply
General Comment
The authors apply the stretched-grid capability of the GEOS-Chem chemical transport model to reduce precomputations and prior storage for analytical atmospheric inversion. While line 54 claims novelty, the core methodology relies on an established technique introduced over two decades ago (see, e.g., doi:10.5194/acp-5-3173-2005). The omission may not be isolated if we note that approximately two-thirds of the cited references originate from the authors' research group. Without a distinct methodological advance or a broader scientific takeaway message, it is difficult to identify the primary contribution of this work for the wider atmospheric science community.
Specific Comments
47: I thought TPCORE was usually non-linear.

53: one can also construct it row-by-row

87: how do the authors treat the uncertainty of the initial state?

88, 92: the incorporation of an empirical gamma factor appears conceptually inconsistent with a strict Bayesian formulation.

110: this statement is incorrect, since oceanic emissions extend beyond offshore oil and gas sources to include significant natural biogenic fluxes

171: what is the point of increasing the spatial resolution so much while keeping a monthly temporal resolution?

183: did the Jacobian computations cover the full year (i.e. 1 month for the first month, 2 months for the second one, etc.)?

217: maybe, outside of this group, some solutions have been designed over the years (see, e.g., https://doi.org/10.5194/gmd-6-583-2013)?

226-227: how useful is this anecdote, which is quite developed, to outside readers?

Reply
Citation: https://doi.org/10.5194/egusphere-2026-4399-RC1
- AC1: 'Reply on RC1', Dandan Zhang, 23 Oct 2026 reply
  
  Reviewer 1
  General Comment
  The authors apply the stretched-grid capability of the GEOS-Chem chemical transport model to reduce precomputations and prior storage for analytical atmospheric inversion. While line 54 claims novelty, the core methodology relies on an established technique introduced over two decades ago (see, e.g., doi:10.5194/acp-5-3173-2005). The omission may not be isolated if we note that approximately two-thirds of the cited references originate from the authors' research group. Without a distinct methodological advance or a broader scientific takeaway message, it is difficult to identify the primary contribution of this work for the wider atmospheric science community.
  Reply: Thank you for drawing our attention to the prior study by Peylin et al. (2005). We now cite it in the Introduction for the row-by-row construction of the Jacobian matrix (not applicable in our case because of the satellite observation density) and for the zoomed mesh. Beyond that, the Peylin et al. (2005) methods are very specific to sparse observations and limited-time inversions, whereas we are dealing with global inversion of high-density data.
  We have tried to be less Harvard-centric in our references in revision.
  We have added text as follows.
  “Construction can be done row by row using the CTM adjoint (Henze et al., 2007; Maksyutov et al., 2021; Zheng et al., 2024) when observations are sparse (Peylin et al., 2005; Seibert and Frank, 2004; Wu et al., 2011) but not for dense satellite observations (Berchet et al., 2021; Thompson et al., 2025).”
  “This variable-resolution approach is particularly advantageous for Jacobian matrix construction, allowing fine resolution in the near field and coarser resolution in the far field (Bergamaschi et al., 2009; Peylin et al., 2005; Zheng et al., 2021), while retaining a continuous global grid for mass conservation (Lauvaux et al., 2012; Schuh et al., 2021; Zheng et al., 2024).”
  Specific Comments
  47: I thought TPCORE was usually non-linear.
  Reply: You are right and this is now mentioned. As described in Estrada et al. (2025), we make our Jacobian perturbations sufficiently large that it doesn’t matter. We have modified the text as follows.
  “Some small nonlinearity can be introduced by the TPCORE advection algorithm in GEOS-Chem (Lin and Rood, 1996), behaving as noise in the fields. This noise is minimized by applying large perturbations (Estrada et al., 2025), and the remaining noise is part of the model transport error that gets incorporated into the observational error (Brasseur and Jacob, 2017).”
  53: one can also construct it row-by-row
  Reply: That is practical only with a small number of observational elements, as in Peylin et al. (2005). We have modified the text as follows:
  “Construction can be done row by row using the CTM adjoint (Henze et al., 2007; Maksyutov et al., 2021; Zheng et al., 2024) when observations are sparse (Peylin et al., 2005; Seibert and Frank, 2004; Wu et al., 2011) but not for dense satellite observations (Berchet et al., 2021; Thompson et al., 2025).”
  87: how do the authors treat the uncertainty of the initial state?
  Reply: Unlike in Peylin et al. (2005), initial conditions do not need to be included in the state vector because we have global observations to initialize our global inversion. We now describe the initialization of the forward model in Section 2:
  “Regional inversions in the IMI optimize boundary conditions as part of the state vector (Nesser et al., 2025). Prior estimates of these boundary conditions for inversion of TROPOMI data are from smoothed TROPOMI fields (Estrada et al., 2025) and these fields are also used as initial conditions in regional and global inversions. Initial conditions are not optimized as part of the state vector because they are by nature consistent with the observations themselves.”
  88, 92: the incorporation of an empirical gamma factor appears conceptually inconsistent with a strict Bayesian formulation.
  Reply: This is now acknowledged, though we do not use it in our regional example. We have revised the sentence to:
  “The regularization parameter γ is unity in the strict Bayesian formulation but may need to be adjusted to lower values to avoid overfitting to observations as diagnosed by departure of the posterior cost function from the expected chi-square distribution (Lu et al., 2021), which may arise from unaccounted observational error correlations.”
  We actually do not use it and now mention this in the caption of Figure 5:
  “…and regularization parameter γ = 1 (equation (1)).”
  110: this statement is incorrect, since oceanic emissions extend beyond offshore oil and gas sources to include significant natural biogenic fluxes
  Reply: Oceanic emissions of methane are considered negligible. This is now stated in the text:
  “The state vector includes all significantly emitting grid cells in the prior bottom-up estimate, which in the case of methane excludes oceans (except for offshore oil/gas) and barren land where emissions are assumed to be negligible…”
  171: what is the point of increasing the spatial resolution so much while keeping a monthly temporal resolution?
  Reply: The reviewer must be thinking of CO₂ rather than methane inversions, and surface rather than satellite observations. We have added to the text to draw distinction between common practices in methane and CO₂ inversions:
  “The state vector as chosen here is for annual emissions, with no optimization of the seasonality from the prior emission estimates. This choice prioritizes spatial over temporal resolution as is common in methane inversions, although other inversion approaches have optimized monthly methane emissions (Qu et al., 2021; Saito et al., 2026; Yu et al., 2023). Global CO₂ inversions generally optimize surface fluxes at monthly or finer temporal resolution but at coarser spatial resolution (Liu et al., 2021; Lloret et al., 2024). Our method could be adapted to optimize monthly fluxes with a similar state vector dimension by decreasing the stretch factor, with a variable stretch factor allowing higher spatial resolution over land regions where it is most desired.”
  183: did the Jacobian computations cover the full year (i.e. 1 month for the first month, 2 months for the second one, etc.)?
  Reply: The global Jacobian sensitivity is calculated as the sensitivity of observations to annual emissions. We clarified in Section 3:
  “Here we use the C36S10 stretched-grid GCHP, representing a C36 global grid with stretch factor S = 10, for Jacobian construction over a full-year (2024) inversion time window, in order to optimize global annual emissions at 25-km resolution.”
  “The state vector as chosen here is for annual emissions, with no optimization of the seasonality from the prior emission estimates. This choice prioritizes spatial over temporal resolution as is common in methane inversion, although other inversion approaches have optimized monthly methane emissions (Qu et al., 2021; Saito et al., 2026; Yu et al., 2023). Global CO₂ inversions generally optimize surface fluxes at monthly or finer temporal resolution but at coarser spatial resolution (Liu et al., 2021; Lloret et al., 2024). Our method could be adapted to optimize monthly CO₂ fluxes with a similar state vector dimension by decreasing the stretch factor, with a variable stretch factor allowing higher spatial resolution over land regions where it is most desired.”
  217: maybe, outside of this group, some solutions have been designed over the years (see, e.g., https://doi.org/10.5194/gmd-6-583-2013)?
  Reply: We thank the reviewer pointing out this prior study, which we now reference:
  “…due to the need to invert matrices (Bocquet et al., 2011; Yadav and Michalak, 2013). A reduced-rank or reduced-dimension approach is needed (Bousserez and Henze, 2018; Nesser et al., 2021; Yadav and Michalak, 2013).”
  226-227: how useful is this anecdote, which is quite developed, to outside readers?
  Reply: This is not an “anecdote” in that we are describing a new capability for the IMI, where regional monthly inversions are extensively used. But we now point out that the regridding demonstration would apply to any regional inversion not just in the IMI.
  In Introduction:
  “We further show that the compact global Jacobian matrix can be conservatively remapped for use in regional inversions anywhere in the world with the standard rectilinear configuration of the IMI or any other grid (Law et al., 2003; Monteil and Scholze, 2021), avoiding costly Jacobian construction for these regional inversions.”
  In Section 4:
  “Regional inversions may use a grid different than our C36S10 grid, so that remapping of our precomputed Jacobians to the destination regional grid is necessary. This is easily achieved because the Jacobian terms are additive on an area basis. For example, regional inversions in the IMI rely on the nested rectilinear grid of GCC model.”
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  Citation: https://doi.org/10.5194/egusphere-2026-4399-AC1

Dandan Zhang, Daniel J. Jacob, Elizabeth W. Lundgren, Melissa P. Sulprizio, Nicholas Balasus, Lucas A. Estrada, Emily Reidy, and Daniel J. Varon

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https://doi.org/10.5194/egusphere-2026-4399-supplement

Dandan Zhang, Daniel J. Jacob, Elizabeth W. Lundgren, Melissa P. Sulprizio, Nicholas Balasus, Lucas A. Estrada, Emily Reidy, and Daniel J. Varon

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Short summary

Dense satellite observations call for global high-resolution inversions to infer surface fluxes. The computational cost for explicit Jacobian construction has limited global analytical inversions to coarse resolution despite the availability of fine-resolution satellite observations. Here we show the stretched-grid high-performance GEOS-Chem model can considerably reduce the computational and storage costs while closely reproducing the posterior fluxes from the standard inversion results.


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