# Spatial Retrievals of Atmospheric Carbon Dioxide from Satellite Observations

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## Abstract

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_{2}) with application to the Orbiting Carbon Observatory-2/3 (OCO-2/3). Formally, the retrieval state vector is extended to include atmospheric and surface conditions at many footprints in a small region, and a prior distribution that assumes spatial correlation across these locations is assumed. This spatial prior allows the length-scale, or range, of spatial correlation to vary between different elements of the state vector. Various single- and multi-footprint retrievals are compared in a simulation study. A spatial prior that also includes relatively large prior variances for CO

_{2}results in posterior inferences that most accurately represent the true state and that reduce the correlation in retrieval error across locations.

## 1. Introduction

_{2}) concentration that is improving quantitative inferences for the carbon cycle [3]. The recently launched OCO-3 satellite facilitates small-area investigations over areas such as megacities [4]. These high-resolution satellite products can ultimately be used in flux inversion systems to infer carbon sources and sinks, potentially at regional scales [5,6]. The end-to-end processing pipeline from satellite spectra (Level 1) to inferred carbon fluxes (Level 4) includes multiple stages of inference that require robust uncertainty quantification [7].

_{2}concentrations at different altitudes, along with other atmospheric and surface constituents within the instrument field of view, or footprint, which is $1.3\times 2.25$ km in nadir mode. The OCO-2 retrieval algorithm infers the state vector from the observed spectrum by solving an inverse problem involving a physical forward model, which relates the state vector to the spectrum. The forward model is combined with a prior assumption on the state in a Bayesian framework known as optimal estimation (OE) in the remote-sensing literature [8]. Operational algorithms for OCO-2/3 perform this retrieval one location (footprint) at a time [9,10]. Since this retrieval approach produces Level 2 data products, we will refer to this operational retrieval as the L2 retrieval.

_{2}, along with surface pressure and aerosol profiles. The observed satellite spectra alone do not provide sufficient information to infer the full unknown state, making the retrieval an ill-posed inverse problem. The OE methodology uses prior information to provide regularization. The primary quantity of interest (QOI) is XCO

_{2}, the column-averaged dry-air mole fraction of CO

_{2}. A ground-based network known as the Total Carbon Column Observing Network (TCCON) [11] provides validation for the OCO-2/3 retrievals. Further, since atmospheric CO

_{2}varies smoothly (horizontally) in space over long ranges in areas with minimal sources/sinks, such as the oceans of the Southern Hemisphere, analysis of retrievals in small spatial areas provides insight into the potential spatial correlation of retrieval errors [12,13,14]. The inherent spatial dependence in the true geophysical process of interest and the presence of spatially correlated retrieval errors motivate interest in a retrieval methodology involving multiple locations simultaneously. Here, we propose the development and testing of a spatial retrieval algorithm that carries out a joint retrieval for all pixels in a given region by exploiting the fact that CO

_{2}values at two neighboring pixels should be very similar to each other; that is, the CO

_{2}field exhibits strong spatial dependence, particularly in the absence of strong sources and sinks.

_{2}values at each pixel should be closer to the truth. (2) We expect the derived uncertainties to be more reflective of the actual discrepancies between the retrieved and true values. (3) Spatial retrievals allow a characterization of the spatial dependence in the retrieval errors. (4) We expect that this spatial dependence in the errors is reduced, which is very important for follow-up analyses, such as flux inversion. (5) As spatial retrievals allow regularization of the retrieval problem in the “horizontal” direction, they might allow a relaxation of the regularization along the dimension of the state vector, hence making the retrieval less dependent on a priori assumptions related to the state.

_{2}concentration near the surface in the presence of sources and sinks may have a different correlation range than concentration higher in the atmosphere where it is well mixed. Further, other state vector components, such as surface albedo, are likely to vary on spatial scales different from those of atmospheric constituents.

_{2}concentrations at different pressure levels. Modeling the cross-covariance in the spatial structure is a major contribution of our article in this regard. In statistical terms, the constraint-based spatial retrieval formulation [15] corresponds to a Gaussian Markov random field (GMRF), specifically the conditional autoregressive (CAR) model [23]. A GMRF prior is also considered in hierarchical Bayesian retrievals of aerosol optical depth (AOD) from spectra observed by the Multi-angle Imaging Spectro Radiometer (MISR) [24,25]. The intrinsic GMRF implied through these spatial priors produces a somewhat inflexible spatial correlation structure, and the models we investigate offer additional flexibility.

_{2}using a reduced-order linear model. Section 4 provides concluding remarks and prospects for future work.

## 2. Spatial Retrieval Methodology

_{2}concentration from the satellite spectra in the operational retrieval configuration and provide an extension to a multi-footprint spatial retrieval.

#### 2.1. Model and Notation

_{2}concentration, other variable and unknown atmospheric constituents are included in the state vector for OCO-2. The additional elements include atmospheric aerosols, surface albedo, and surface air pressure. The numerical investigation in Section 3 examines an implementation with $p=39$ [20]. The collection of state vectors in the small area is assembled into an $np$-dimensional vector,

_{2}A-band at 0.76 μm, the weak CO

_{2}band at 1.61 μm, and the strong CO

_{2}band at 2.06 μm. Each band corresponds to a different spectrometer on OCO-2 and has up to 1016 channels. The vector $\mathbf{y}\left(\mathcal{S}\right)$ represents the set of all spectra at locations in $\mathcal{S}$. The physical relationships between the state and the spectra are contained in a forward model $\mathbf{F}(\xb7)$. In general, the forward model is nonlinear.

#### 2.2. The Spatial Objective Function

_{2}at different pressure levels and the other state variables). This prior covariance matrix controls the regularization along the dimension of the state vector, and can be viewed as a “vertical” regularization for the CO

_{2}profile. Since the forward model is nonlinear, the objective function is optimized numerically and separately for each i using an algorithm, such as gradient descent, Gauss–Newton, or Levenberg–Marquardt [8].

_{2}and other variables in the state vector. This allows us to borrow strength over space. These off-diagonal blocks can be viewed as a “horizontal” regularization. This spatial retrieval strategy has been implemented in multi-footprint retrievals for aerosols in particular [15,16]. These approaches have typically achieved this regularization with spatial smoothness constraints rather than with a spatial statistical model. Even so, the spatial smoothness constraints can be viewed as specific structures for the prior covariance matrix $\Sigma $. More generally, parameterizing the cross-correlations offers additional flexibility in the retrieval, but can be challenging. A strategy for the OCO-2 small areas is discussed in Appendix A.1.

#### 2.3. State Vector

_{2}, surface air pressure, atmospheric aerosols of varying types, and wavelength-dependent surface albedo in the three spectral bands. Further details on this configuration can be found in [20]. This slightly simplified state vector omits some elements in the full physics state vector [10].

_{2}varies vertically. Near the surface, atmospheric CO

_{2}can be highly variable in the presence of surface sources or sinks. At higher altitudes, away from direct sources and sinks, CO

_{2}can have larger correlation length scales [28,29]. Aerosols are similarly sensitive to atmospheric transport, but have generally shorter residence times. Surface pressure and clouds have spatial scales connected to weather systems. Surface albedo over land can have short correlation length due to heterogeneity in surface types. A multi-footprint retrieval methodology should have the capability for spatial dependence that varies with the state vector element.

_{2}, which is the column-averaged dry-air mole fraction of CO

_{2}. This quantity is a weighted average of the CO

_{2}vertical profile portion of the state vector,

_{2}, given a Gaussian distribution for $\mathbf{x}\left({\mathbf{s}}_{i}\right)$, is given by:

#### 2.4. A Tractable Linear Model

- If ${\mathbf{F}}_{\mathcal{S}}$ is block-diagonal, then so is ${\mathbf{F}}_{\mathcal{S}}^{\prime}{\mathbf{V}}^{-1}{\mathbf{F}}_{\mathcal{S}}$.
- If $\Sigma $ is block-diagonal, then so is ${\Sigma}^{-1}$.
- If both are block-diagonal, then so are ${\Sigma}_{x|y}^{-1}$ and ${\Sigma}_{x|y}$. This would imply that dependence across footprints is being ignored. Further, the posterior covariances for individual footprints will typically be incorrect.

#### 2.5. Considerations for Degeneracy

## 3. Numerical Study

#### 3.1. Simulation and Retrieval Configuration

_{2}and aerosols.

_{2}, as well as for other key state vector elements, such as surface pressure.

- Operational, ${\Sigma}_{a}={\mathbf{I}}_{64}\otimes {\Sigma}_{a,0}$, where ${\mathbf{I}}_{64}$ is an identity matrix with a dimension matching the number of spatial locations. The OCO-2 operational prior covariance for a single footprint, ${\Sigma}_{a,0}$, is used at all locations, assuming no spatial correlation. This is essentially a single-footprint retrieval. In this case, the prior standard deviations for the CO
_{2}profile are substantially larger than those in ${\Sigma}_{T}$ (see Figure 3 of [30]). - Spatial, ${\Sigma}_{a}={\mathbf{S}}_{a}{\mathbf{C}}_{a}{\mathbf{S}}_{a}$. The within-footprint operational correlation structure is extended between footprints by averaging parameters (see (A1) in Appendix A.1), yielding a multivariate spatial correlation matrix ${\mathbf{C}}_{a}$. This is combined with the standard deviations used in the operational retrieval, represented in the diagonal matrix ${\mathbf{S}}_{a}$.
- True, ${\Sigma}_{a}={\Sigma}_{T}$. The prior covariance is set to the true data-generating spatial covariance.

#### 3.2. Results

_{2}are illustrated in Figure 5 and Figure 6. The spatial retrievals result in smaller retrieval-error standard deviations, which is illustrated through XCO

_{2}credible intervals for a portion of the October 2015 Lamont template in Figure 5. Science investigations that use OCO-2 data involve combining retrievals in various ways [3], and an understanding of the correlation of retrieval errors is often critical. Figure 6 displays a series of correlation matrices of the XCO

_{2}retrieval error for the three choices of prior covariance. The operational single-footprint approach yields errors that are strongly spatially correlated, while the spatial prior reduces spatial correlations in the error for the October 2015 Lamont case. Figure 7 summarizes the mean absolute error (MAE) for XCO

_{2}by location for the Lamont template. While the errors are relatively uniform across the small area, they are smallest in magnitude near the center, where the spatial retrieval provides the most information from surrounding locations.

_{2}across multiple simulations in Table 3 by computing the mean squared error (MSE) and two forms of a mean log score. The log score evaluates the likelihood of the generated truth given the posterior distribution implied by the retrieval (e.g., [31]). These scores are examples of proper scoring rules or metrics that characterize predictive distributions. In practical terms, proper scoring rules are optimized when predictive distributions are as narrow as possible while still capturing the true state for a sufficient percentage of the time. The marginal log score uses the variance of the XCO

_{2}estimate for each location, ignoring the off-diagonal covariance terms relating XCO

_{2}measurements at different locations. For example, with 100 generated samples ${\mathbf{x}}_{\mathcal{S}}$, the joint $\mathcal{L}$ and marginal log scores $m\mathcal{L}$ are

_{2}posterior covariance ${\Sigma}_{\begin{array}{c}\hfill {\mathrm{XCO}}_{2}\end{array}|y}^{\left(k\right)}$. Since the operational prior assumes independence across footprints, the joint and marginal scores will be the same for this prior choice. For all three templates, the spatial prior yields the most desirable outcomes for both the marginal and joint scores. The scores are consistent with Figure 4, showing that using the true covariance as the prior is far too optimistic due to the prior mean misspecification.

_{2}is poorly estimated. Performance for the full state is consistent with the XCO

_{2}estimates. The important takeaway from this simulation is that the improvement in posterior accuracy and precision due to a spatial model is closely related to the covariance parameters. As the parameters strengthen cross-correlation, the benefit of a spatial model becomes more obvious.

## 4. Discussion

_{2}from remote-sensing observations over small spatial areas. The statistical properties of the retrieval approach were investigated with simulation experiments with a realistic simplified physical forward model. A prior that combines spatial correlation across footprints with a large prior variance for individual levels of the CO

_{2}vertical profile often provides improved precision over single-footprint retrievals (Figure 5). In addition, the multi-footprint approach can reduce the spatial correlation of retrieval errors, which enhances utility for downstream scientific use of the retrieval results [3].

_{2}and other trace gases. Our simulations suggest that the actual spatial dependence as well as the assumed spatial dependence in the prior retrieval distribution have an impact on the retrieval precision and magnitude of spatial correlation in retrieval errors. Widespread implementation would need additional investigation of these interactions, as well as of the role of the within-footprint correlation structure. The multi-footprint retrieval may have added value in situations that are challenging for the current single-footprint retrieval, including low signal-to-noise situations, such as dark surfaces and high-latitude observations [32].

_{2}[14]. The additional parameters necessary for spatial retrievals could be estimated offline using atmospheric transport models [34] if the spatial resolution is suitable.

_{2}. The use of the larger-variance spatial prior in Section 3.1 provides an initial investigation of this idea, but further adjustments to the within-footprint correlation structure could prove useful. This investigation would be valuable for the CO

_{2}retrieval problem as well as for the retrieval of vertical profiles of other trace gases, such as O

_{3}and CH

_{4}.

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

AirMSPI | Airborne Multiangle SpectroPolarimetric Imager |

AOD | Aerosol Optical Depth |

CAR | Conditional Autoregressive |

GMRF | Gaussian Markov Random Field |

GOSAT | Greenhouse Gas Observing Satellite |

GP | Gaussian Process |

MAE | Mean Absolute Error |

MAIA | Multi-Angle Imager for Aerosols |

MISR | Multi-angle Imaging SpectroRadiometer |

MSE | Mean Squared Error |

OCO-2/3 | Orbiting Carbon Observatory-2/3 |

OE | Optimal Estimation |

PARASOL | Polarization and Anisotropy of Reflectances for Atmospheric Science coupled with |

Observations from a Lidar | |

PC | Principal Component |

PMA | Pointing Mirror Assembly |

POLDER | Polarization and Directionality of the Earth’s Reflectances |

QOI | Quantity of Interest |

REML | Restricted Maximum Likelihood |

SAM | Snapshot Area Mode |

TCCON | Total Carbon Column Observing Network |

## Appendix A

#### Appendix A.1. Spatial Statistical Model Estimation

_{2}vertical profiles) as well as spatial correlation across footprints in a small area. This multivariate spatial statistical model is constructed to represent the small areas within the single OCO-2 orbits studied in Section 3. Since the areas considered are small in extent and to maintain focus on the statistical complexity of the multivariate nature of the problem, the spatial covariance of the state is taken to be isotropic. Therefore, the model is $\mathbf{x}(\xb7)\sim GP\left(\mathbf{\mu}\right(\xb7),\mathbf{C})$, where $GP$ is a Gaussian process with mean function $\mathbf{\mu}$ and cross-covariance function $\mathbf{C}$. For a given small area, the main challenge is to estimate the cross-covariance function $\mathbf{C}$.

_{2}vertical profile, surface pressure, albedo, and aerosols. Where necessary, the estimation was constrained to the nearest positive definite matrix [38]. These computations were carried out using the Matrix package in the R statistical computing environment [39].

- Run a single-footprint simulation experiment of the full retrieval system for the location of interest.
- Estimate the retrieval error covariance $\Omega $ from the simulation results.
- Assemble OCO-2 retrievals for orbits in the month of interest within 300 km of the TCCON site.
- Estimate the within-footprint covariance $\mathbf{G}$ from the OCO-2 retrievals.
- Estimate the spatial correlation parameters ${\lambda}_{k}$ and ${\nu}_{k}$ from the OCO-2 retrievals, one state vector element at a time.

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**Figure 1.**

**Left**: Portion of an Orbiting Carbon Observatory-2 (OCO-2) orbit representing a small area for which a spatial retrieval is investigated. The area in green represents the small area.

**Right**: Zoomed-in view of the locations of the centers of $n=64$ individual footprints for the small area.

**Figure 2.**Estimated Matérn spatial correlation range parameters in kilometers for all state vector elements in three Total Carbon Column Observing Network (TCCON) templates. Estimates have been truncated above at 100 km.

**Figure 3.**True mean vectors ${\mathbf{\mu}}_{T}$ and operational retrieval prior mean vectors ${\mathbf{\mu}}_{a}$ for the CO

_{2}vertical profile at the three TCCON templates.

**Figure 4.**Log mean squared error (MSE) for the retrieval error of the state vector components, averaged across 100 samples and 64 locations per sample. State vector elements are grouped into the CO

_{2}vertical profile (

**top**) and all other elements (

**bottom**). All results are for the October 2015 Lamont template.

**Figure 5.**Example of pointwise 95% posterior credible intervals for XCO

_{2}(in ppm) at a subset of the 64 locations for the October 2015 Lamont template. For spatial retrievals, the posterior intervals were narrower and centered closer to the true XCO

_{2}values.

**Figure 6.**XCO

_{2}retrieval error correlations across locations for 100 samples for the October 2015 Lamont template. Typical operational priors lead to predictions with errors that have strong spatial correlation.

**Figure 7.**Mean absolute error (MAE) for XCO

_{2}(in ppm) across locations for 100 samples for the October 2015 Lamont template for retrievals using the spatial prior.

**Table 1.**Elements of the state vector for the reduced-order model of [20].

Collection | Number of Elements |
---|---|

CO_{2} Vertical Profile | 20 |

Surface Pressure | 1 |

Surface Albedo | 6 = 2 per band × 3 bands |

Aerosols | 12 = 3 per type × 4 types |

**Table 2.**True mean vectors ${\mathbf{\mu}}_{T}$ and operational retrieval prior mean vectors ${\mathbf{\mu}}_{a}$ for selected state vector elements at the three TCCON templates. The CO

_{2}profile means are displayed in Figure 3. Aerosols are represented with two location-specific types plus cloud ice and water.

Lamont | Wollongong | Wollongong | ||||
---|---|---|---|---|---|---|

Oct 2015 | Dec 2016 | Jun 2017 | ||||

State Vector Element | ${\mathbf{\mu}}_{\mathbf{T}}$ | ${\mathbf{\mu}}_{\mathbf{a}}$ | ${\mathbf{\mu}}_{\mathbf{T}}$ | ${\mathbf{\mu}}_{\mathbf{a}}$ | ${\mathbf{\mu}}_{\mathbf{T}}$ | ${\mathbf{\mu}}_{\mathbf{a}}$ |

XCO_{2} [ppm] | 396.34 | 395.72 | 398.84 | 400.76 | 399.98 | 402.02 |

Surface Pressure [hPa] | 986.36 | 983.60 | 949.81 | 945.89 | 953.18 | 952.27 |

Strong CO_{2} Mean Albedo | 0.194 | 0.118 | 0.147 | 0.183 | 0.133 | 0.097 |

Strong CO_{2} Albedo Slope | $8.05\times {10}^{-5}$ | 0 | $1.06\times {10}^{-4}$ | 0 | $1.80\times {10}^{-5}$ | 0 |

Weak CO_{2} Mean Albedo | 0.204 | 0.193 | 0.213 | 0.223 | 0.212 | 0.231 |

Weak CO_{2} Albedo Slope | $-2.49\times {10}^{-5}$ | 0 | $-2.52\times {10}^{-5}$ | 0 | $-2.25\times {10}^{-5}$ | 0 |

O_{2} A-Band Mean Albedo | 0.300 | 0.258 | 0.261 | 0.338 | 0.252 | 0.232 |

O_{2} A-Band Albedo Slope | $-1.43\times {10}^{-4}$ | 0 | $-1.15\times {10}^{-4}$ | 0 | $-1.63\times {10}^{-4}$ | 0 |

Aerosol Type 1 | Sulfate | Sulfate | Sulfate | |||

Log Optical Depth | −3.72 | −3.72 | −4.26 | −4.09 | −4.80 | −4.89 |

Profile Height | 0.83 | 0.90 | 0.79 | 0.90 | 0.93 | 0.90 |

Log Profile Thickness | −2.65 | −3.00 | −2.32 | −3.00 | −3.49 | −3.00 |

Aerosol Type 2 | Dust | Sea Salt | Sea Salt | |||

Log Optical Depth | −6.13 | −4.72 | −5.27 | −4.11 | −5.36 | −4.95 |

Profile Height | 0.72 | 0.90 | 0.82 | 0.90 | 0.91 | 0.90 |

Log Profile Thickness | −2.50 | −3.00 | −3.19 | −3.00 | −3.76 | −3.00 |

Cloud Ice | ||||||

Log Optical Depth | −5.26 | −4.38 | −5.16 | −4.38 | −5.90 | −4.38 |

Profile Height | 0.17 | 0.15 | 0.23 | 0.16 | 0.01 | 0.20 |

Log Profile Thickness | −3.22 | −3.22 | −3.22 | −3.22 | −3.22 | −3.22 |

Cloud Water | ||||||

Log Optical Depth | −5.13 | −4.38 | −4.89 | −4.38 | −5.10 | −4.38 |

Profile Height | 0.86 | 0.75 | 0.86 | 0.75 | 1.08 | 0.75 |

Log Profile Thickness | −2.30 | −2.30 | −2.30 | −2.30 | −2.30 | −2.30 |

**Table 3.**Performance of multiple retrieval approaches from three simulation experiments. The joint and marginal log scores are defined in Equations (3) and (4) and summarize the retrieval of XCO

_{2}. The mean squared error (MSE) is shown for XCO

_{2}and the full state vector, as well as the mean absolute error (MAE) for the full state vector. Results in bold indicate the optimal performance for each metric.

XCO_{2} | Full State | |||||
---|---|---|---|---|---|---|

Site | Method | Marginal Log Score | Joint Log Score | MSE | MAE | MSE |

Lamont | True | −1318 | −Inf | 0.46 | 0.73 | 0.75 |

Oct 2015 | Operational | −90 | −90 | 0.63 | 0.51 | 0.62 |

Spatial | −22 | 25 | 0.03 | 0.24 | 0.27 | |

Wollongong | True | −44863 | −Inf | 16.07 | 4.72 | 4.74 |

Dec 2016 | Operational | −87 | −87 | 0.88 | 0.76 | 0.78 |

Spatial | −23 | 12 | 0.11 | 0.43 | 0.46 | |

Wollongong | True | −11818 | −Inf | 6.25 | 3.02 | 3.12 |

Jun 2017 | Operational | −61 | −61 | 0.20 | 0.53 | 0.56 |

Spatial | −12 | 0.2 | 0.04 | 0.44 | 0.48 |

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**MDPI and ACS Style**

Hobbs, J.; Katzfuss, M.; Zilber, D.; Brynjarsdóttir, J.; Mondal, A.; Berrocal, V.
Spatial Retrievals of Atmospheric Carbon Dioxide from Satellite Observations. *Remote Sens.* **2021**, *13*, 571.
https://doi.org/10.3390/rs13040571

**AMA Style**

Hobbs J, Katzfuss M, Zilber D, Brynjarsdóttir J, Mondal A, Berrocal V.
Spatial Retrievals of Atmospheric Carbon Dioxide from Satellite Observations. *Remote Sensing*. 2021; 13(4):571.
https://doi.org/10.3390/rs13040571

**Chicago/Turabian Style**

Hobbs, Jonathan, Matthias Katzfuss, Daniel Zilber, Jenný Brynjarsdóttir, Anirban Mondal, and Veronica Berrocal.
2021. "Spatial Retrievals of Atmospheric Carbon Dioxide from Satellite Observations" *Remote Sensing* 13, no. 4: 571.
https://doi.org/10.3390/rs13040571