#
The Significance of Fast Radiative Transfer for Hyperspectral SWIR XCO_{2} Retrievals

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## Abstract

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## 1. Introduction

## 2. Materials and Methods

#### 2.1. The UoL Algorithm

#### 2.2. Fast Radiative Transfer

#### 2.2.1. Low-Streams Interpolation (LSI)

#### 2.2.2. Principal Component-Based Method (PCA)

#### 2.2.3. Linear-k

#### 2.3. XCO${}_{2}$ Retrievals from OCO-2

#### 2.4. Filtering and Bias Correction

## 3. Results

#### 3.1. Bulk Statistics

#### 3.2. Filter Thresholds and Bias Correction

#### 3.3. Final Comparison Using Bias-Corrected XCO${}_{2}$

#### 3.4. Computational Effort

## 4. Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

ACOS | Atmospheric Concentrations from Space |

AK | Averaging kernel |

CAMS | Copernicus Atmosphere Monitoring System |

CO${}_{2}$ | Carbon dioxide |

CT2019 | CarbonTracker, version 2019 |

EOF | Empirical orthogonal function |

GeoCarb | Geostationary Carbon Observatory |

GOSAT | Greenhouse Gases Observing Satellite |

LMDZ | Laboratoire de Météorologie Dynamique |

LSI | Low-streams interpolation |

MS | Multiple scattering |

NASA | National Aeronautics and Space Administration |

NIES | National Institute for Environmental Studies |

OCO-2 | Orbiting Carbon Observatory-2 |

PCA | Principal component analysis |

RT | Radiative transfer |

TOA | Top-of-atmosphere |

UoL | University of Leicester |

XCO${}_{2}$ | Column-averaged dry-air mole fraction of carbon dioxide |

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**Figure 1.**Global map of used OCO-2 scenes as well as their observation modes (LN = land nadir, LG = land glint, OG = ocean glint).

**Figure 2.**Footprint biases, calculated using the difference of the retrieved XCO${}_{2}$ between a given footprint and the frame median, where we only use frames where the retrievals for all eight footprint converged. The displayed values represent the medians of all differences for the eight footprints respectively. Along with the four retrieval sets, we also show the footprint biases from the ACOS OCO-2 V8 data product (land scenes).

**Figure 4.**Overview of iteration numbers and fit residuals for the O${}_{2}$ A-band. The ${\chi}^{2}$ statistic for the other two bands is very similar across the four retrieval sets. For ocean glint scenes, the four sets show slightly shifted ${\chi}^{2}$ populations, with low-streams interpolation (LSI) having the lowest distribution mode.

**Figure 5.**This figure depicts the distribution of prior optical depth for the CAMS-derived tropospheric aerosols, for three subsets of scenes. The three subsets were separated by the relative differences of the fit ${\chi}^{2}$ for the O${}_{2}$ A-band between linear-k and the PCA (5) runs. From the distributions, it is apparent that the linear-k fit ${\chi}^{2}$ were more similar to the PCA (5) retrieval set when the prior aerosol optical depths were small. The three subsets were roughly equally large and contained between 23,000 and 26,000 scenes each.

**Figure 6.**Map showing the range of retrieved (unfiltered) XCO${}_{2}$ for the four fast radiative transfer (RT) runs, gridded to ${3}^{\circ}\times {3}^{\circ}$ bins.

**Figure 7.**Example for parametric bias correction for land glint observations and the $\delta {\nabla}_{{\mathrm{CO}}_{2}}$ parameter. The orange, dashed line is obtained via a least-squares fit (see Table 3).

**Figure 8.**Geographical distribution of pair-wise differences between retrieval sets (gridded into ${3}^{\circ}\times {3}^{\circ}$ bins), as well as the corresponding histograms to the right of each map. (LN = land nadir, LG = land glint, OG = ocean glint).

**Table 1.**Statistics of the retrieved XCO${}_{2}$ differences ($\mathsf{\Delta}{\mathrm{XCO}}_{2}$), linear-k minus PCA (5), for different bins of prior tropospheric aerosol optical depth ${\tau}_{\mathrm{aero}}$. Footprint biases have already been removed according to Figure 2. $\sigma $ is the standard deviation of the differences, and $\tilde{\sigma}$ is a robust scatter of the differences, calculated as $\tilde{\sigma}=({P}_{95}-{P}_{5})/3.289$, where ${P}_{i}$ is the i-th percentile of the differences. $\tilde{\sigma}\approx \sigma $ (standard deviation) for a normally distributed set of numbers.

$\mathbf{\Delta}$XCO${}_{2}$ [ppm] | ||||
---|---|---|---|---|

$\mathbf{\Delta}$ | $\tilde{\mathit{\sigma}}$ | IQR | N | |

${\tau}_{\mathrm{aero}}\le 0.05$ | +0.03 | 0.28 | 0.45 | 23174 |

$0.05<{\tau}_{\mathrm{aero}}<0.15$ | −0.06 | 0.93 | 1.01 | 25639 |

${\tau}_{\mathrm{aero}}\ge 0.15$ | −0.33 | 1.45 | 1.62 | 26173 |

**Table 2.**Filter thresholds for the different fast RT methods and OCO-2 observation modes. The filters concerning the fit ${\chi}^{2}$ statistic and the number of both non-divergent and divergent iterations are applied to all three observation modes in addition to the ones listed below. Retrieval sets LSI, PCA (1) and PCA (5) use the same filters, whereas the linear-k filters are listed in the rightmost column. Values are in ppm.

All Observation Modes | ||||

LSI | PCA (1) | PCA (5) | Linear-k | |

${\mathrm{N}}_{\mathrm{iterations}}$ | ≤4 | ≤9 | ||

${\mathrm{N}}_{\mathrm{divergent}}$ | =0 | =0 | ||

${\chi}^{2}$ Band 1 | <$3.5$ | <$4.5$ | ||

${\chi}^{2}$ Band 2 | <$3.5$ | <$3.5$ | ||

${\chi}^{2}$ Band 3 | <$3.0$ | <$3.0$ | ||

Land Nadir Additional | ||||

LSI | PCA (1) | PCA (5) | Linear-k | |

$\delta {\nabla}_{{\mathrm{CO}}_{2}}$ [ppm] | ≥0, ≤13 | ≥$-1$, ≤13 | ||

Throughput | 17,291 (41.7%) | 17,541 (42.5%) | 17,417 (41.8%) | 16,587 (39.5%) |

Land Glint Additional | ||||

LSI | PCA (1) | PCA (5) | Linear-k | |

$\mathsf{\Delta}{p}_{\mathrm{surf}}$ [hPa] | ≥$-1.5,$ ≤$2.5$ | ≥$-1.5$, ≤$2.5$ | ||

$\delta {\nabla}_{{\mathrm{CO}}_{2}}$ [ppm] | ≥0, ≤20 | ≥$-1$, ≤20 | ||

Throughput | 17,247 (37.5%) | 17,641 (38.4%) | 17,419 (37.6%) | 18,119 (38.8%) |

Ocean Glint Additional | ||||

LSI | PCA (1) | PCA (5) | Linear-k | |

$\delta {\nabla}_{{\mathrm{CO}}_{2}}$ [ppm] | ≥$-10$, ≤4 | ≥$-10$, ≤4 | ||

Throughput | 21,867 (62.6%) | 21,345 (62.3%) | 21,638 (62.3%) | 22,615 (65.0%) |

**Table 3.**Bias correction coefficients and intercepts, applied sequentially for each fast RT method and observation mode. $\delta {\nabla}_{{\mathrm{CO}}_{2}}$ is in units of ppm, and $\mathsf{\Delta}{p}_{\mathrm{surf}}$ is in units of hPa. ${\rho}_{1}$ and ${\rho}_{2}$ are the retrieved surface albedo for band 1 and band 2, respectively.

Land Nadir | ||||

LSI | PCA (1) | PCA (5) | Linear-k | |

$0\le \delta {\nabla}_{{\mathrm{CO}}_{2}}\le 13$ | 0.126 | 0.132 | 0.136 | 0.117 |

Intercept | −0.70 | −0.67 | −0.66 | −0.57 |

Land Glint | ||||

LSI | PCA (1) | PCA (5) | Linear-k | |

$-1\le \delta {\nabla}_{{\mathrm{CO}}_{2}}\le 13$ | 0.111 | 0.121 | 0.127 | 0.095 |

$-1.5\le \mathsf{\Delta}{p}_{\mathrm{surf}}\le 2.5$ | −0.398 | −0.364 | −0.358 | −0.218 |

Intercept | −0.31 | −0.35 | −0.36 | −0.26 |

Ocean Glint | ||||

LSI | PCA (1) | PCA (5) | Linear-k | |

$-10\le \delta {\nabla}_{{\mathrm{CO}}_{2}}\le 4$ | 0.674 | 0.675 | 0.675 | 0.663 |

$0.90\le {\rho}_{1}/{\rho}_{2}<1.055$ | 3.236 | 3.280 | 3.236 | 4.009 |

Intercept | −3.42 | −3.43 | −3.37 | −4.16 |

**Table 4.**Statistics of retrieved, bias-corrected XCO${}_{2}$ between retrieval sets (scene matched). $\mathsf{\Delta}$ is the mean, $\tilde{\sigma}$ is a robust scatter defined in Table 1, and IQR is the inter-quartile range ${P}_{75}-{P}_{25}$ (${P}_{i}$ is the i’th percentile). Values are in ppm.

Land Nadir | Land Glint | Ocean Glint | |||||||
---|---|---|---|---|---|---|---|---|---|

$\mathbf{\Delta}$ | $\tilde{\mathit{\sigma}}$ | IQR | $\mathbf{\Delta}$ | $\tilde{\mathit{\sigma}}$ | IQR | $\mathbf{\Delta}$ | $\tilde{\mathit{\sigma}}$ | IQR | |

Before parametric bias correction | |||||||||

linear-$\mathit{k}$ - PCA (5) | −0.04 | 0.66 | 0.35 | −0.10 | 0.50 | 0.40 | −0.09 | 0.14 | 0.11 |

LSI - PCA (5) | −0.07 | 0.58 | 0.26 | −0.05 | 0.39 | 0.17 | −0.02 | 0.05 | 0.04 |

PCA (1) - PCA (5) | −0.03 | 0.47 | 0.17 | −0.01 | 0.25 | 0.09 | 0.00 | 0.02 | 0.02 |

After parametric bias correction | |||||||||

linear-$\mathit{k}$ - PCA (5) | −0.04 | 0.65 | 0.39 | −0.11 | 0.47 | 0.36 | 0.02 | 0.11 | 0.08 |

LSI - PCA (5) | −0.02 | 0.58 | 0.28 | −0.00 | 0.39 | 0.18 | 0.02 | 0.04 | 0.04 |

PCA (1) - PCA (5) | −0.02 | 0.47 | 0.18 | 0.01 | 0.25 | 0.10 | 0.00 | 0.02 | 0.02 |

**Table 5.**Statistics of the differences between bias-corrected retrievals and model-median truth data, where $\mathsf{\Delta}$ is the mean, $\tilde{\sigma}$ is a robust scatter defined in Table 1, and IQR is the inter-quartile range. Values are in units of ppm.

Land Nadir | Land Glint | Ocean Glint | |||||||
---|---|---|---|---|---|---|---|---|---|

$\mathbf{\Delta}$ | $\tilde{\mathit{\sigma}}$ | IQR | $\mathbf{\Delta}$ | $\tilde{\mathit{\sigma}}$ | IQR | $\mathbf{\Delta}$ | $\tilde{\mathit{\sigma}}$ | IQR | |

Before parametric bias correction | |||||||||

LSI | −0.12 | 1.85 | 1.95 | −0.02 | 1.79 | 2.03 | −0.31 | 2.18 | 2.38 |

PCA (1) | −0.07 | 1.87 | 1.90 | 0.04 | 1.78 | 1.98 | −0.27 | 2.17 | 2.36 |

PCA (5) | −0.03 | 1.88 | 1.90 | 0.05 | 1.78 | 1.99 | −0.27 | 2.17 | 2.35 |

linear-$\mathit{k}$ | −0.02 | 1.71 | 1.85 | −0.09 | 1.66 | 1.86 | −0.27 | 2.12 | 2.27 |

After parametric bias correction | |||||||||

LSI | −0.01 | 1.71 | 1.91 | 0.03 | 1.69 | 1.96 | 0.02 | 0.97 | 1.09 |

PCA (1) | −0.02 | 1.69 | 1.87 | 0.06 | 1.65 | 1.93 | 0.01 | 0.97 | 1.09 |

PCA (5) | 0.01 | 1.68 | 1.87 | 0.04 | 1.65 | 1.94 | 0.01 | 0.97 | 1.09 |

linear-$\mathit{k}$ | 0.01 | 1.66 | 1.83 | −0.04 | 1.59 | 1.82 | 0.04 | 0.95 | 1.07 |

**Table 6.**Comparison of computational effort for the various fast RT methods, where only scenes where considered which were successfully executed for all four sets ($N=112,073$). Times are in seconds. The columns show the mean execution time per iteration ($\mathsf{\Delta}$) and its scatter ($\tilde{\sigma}$, see Table 4), whereas the total execution time sums up all performed iterations.

Land Nadir | Land Glint | Ocean Glint | |||||||
---|---|---|---|---|---|---|---|---|---|

$\mathbf{\Delta}$ | $\tilde{\mathit{\sigma}}$ | Total | $\mathbf{\Delta}$ | $\tilde{\mathit{\sigma}}$ | Total | $\mathbf{\Delta}$ | $\tilde{\mathit{\sigma}}$ | Total | |

LSI | 127 | 31 | 17.94 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{6}$ | 136 | 31 | 21.66 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{6}$ | 112 | 15 | 14.16 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{6}$ |

PCA (1) | 133 | 30 | 18.64 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{6}$ | 156 | 32 | 24.44 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{6}$ | 148 | 31 | 18.55 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{6}$ |

PCA (5) | 169 | 33 | 23.93 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{6}$ | 246 | 42 | 39.10 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{6}$ | 245 | 39 | 30.92 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{6}$ |

linear-k | 99 | 31 | 15.31 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{6}$ | 109 | 29 | 18.76 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{6}$ | 93 | 18 | 12.74 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{6}$ |

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**MDPI and ACS Style**

Somkuti, P.; Bösch, H.; Parker, R.J. The Significance of Fast Radiative Transfer for Hyperspectral SWIR XCO_{2} Retrievals. *Atmosphere* **2020**, *11*, 1219.
https://doi.org/10.3390/atmos11111219

**AMA Style**

Somkuti P, Bösch H, Parker RJ. The Significance of Fast Radiative Transfer for Hyperspectral SWIR XCO_{2} Retrievals. *Atmosphere*. 2020; 11(11):1219.
https://doi.org/10.3390/atmos11111219

**Chicago/Turabian Style**

Somkuti, Peter, Hartmut Bösch, and Robert J. Parker. 2020. "The Significance of Fast Radiative Transfer for Hyperspectral SWIR XCO_{2} Retrievals" *Atmosphere* 11, no. 11: 1219.
https://doi.org/10.3390/atmos11111219