1. Introduction
As of late 2020, several space-based missions are in operation with the goal to quantify the concentration of carbon dioxide (CO
) in Earth’s atmosphere. These satellites are placed in sun-synchronous low-Earth orbits and carry spectrometers with high spectral resolution to measure reflected and backscattered spectra in the shortwave infrared (SWIR) wavelength region. The currently operating instruments are GOSAT [
1], launched in early 2009, OCO-2, launched in 2014, TanSat, launched in 2016, GOSAT-2, launched in 2018 and most recently OCO-3 [
2] which was launched in May 2019 and installed on the International Space Station. These five instruments share the same overall concept, which is based on the measurement of two CO
absorption bands in the SWIR near 1.6
m and 2.06
m along with the oxygen A-band at 0.756
m which can be used to constrain the reference oxygen column. While there are several other space-based instruments which obtain CO
absorption spectra in the thermal infrared region (TIR), such as Infrared Atmospheric Sounding Interferometer (IASI), we focus only on measurements from the SWIR and relevant instruments. SWIR measurements have sensitivity to the total CO
column down to the surface, whereas TIR measurement sensitivity is peaked at the middle and upper troposphere [
3,
4]. Both GOSAT and GOSAT-2 feature Fourier transform spectrometers (TANSO-FTS and TANSO-FTS-2) that point to specific locations for each single measurement. In contrast, the TanSat and OCO-2/3 instruments are grating-type spectrometers which sample the atmosphere in a push-broom manner with a small swath width of the order of 10 km. The two different instrument types have a direct impact on sampling density and total number of measurements collected in any given time range. For GOSAT [
5] and GOSAT-2 [
6,
7], the integration time of the two FTS instruments is about the same with 4 s for each measurement scene. OCO-2/3 and TanSat, on the other hand, collect 24 and 27 measurements per second respectively. While these satellites orbit Earth around 15 times per day, the instruments collect between ∼
(GOSAT) and ∼
(OCO-2) measurements on the sunlit side of the Earth. Generally, between a third and half of those measurements remain after screening for clouds. Routine data operations for OCO-2 involve running the retrieval algorithm on the order of 60,000 of the clearest selected scenes for every day. Future CO
-focused missions push that number even further up. The geostationary Carbon Cycle Observatory [
8], GeoCarb, is expected to measure well over a million cloud-free scenes over the Americas every day. The proposed CO2M mission [
9] will feature a constellation of at least three satellites and a larger measurement count than GeoCarb. The computational resources to process these measurements for operational services have thus required a significant increase and will continue to do so.
The retrieval algorithms are designed to extract the column-averaged dry-air volume mixing ratio of carbon dioxide (XCO
) using mostly a three-band approach. The two CO
absorption bands contain information on the column of carbon dioxide, whereas the O
A-band provides a reference for the total amount of dry air within the column of a particular measurement. Having two CO
bands separated by ∼0.4
m in wavelength in conjunction with the O
A-band also allows to partially account for scattering by aerosols. While there is more than one degree of freedom contained in the retrieved CO
profiles that can be exploited for explicit profile information, as was done by e.g., Kulawik et al. [
10] and Noël et al. [
11], most publicized data products based on SWIR-measurements collapse the profile into a column average. Other instruments allow for different approaches that retain more of the vertical information. The Atmospheric Chemistry Experiment (ACE), for example, achieves this through solar occultation measurements [
12]. Other instruments, such as IASI, have a different vertical sensitivity due to the spectral location of measured absorption bands in the TIR. Crevoisier et al. [
13], for example, retrieved the upper tropospheric part of the CO
column between 11 km and 15 km using absorption bands at wavelengths 4.3
m and 15
m.
XCO
retrievals are exceptionally challenging due to the imposed limits on systematic biases. Since one of the main applications of space-based XCO
is the estimation of surface carbon fluxes [
14], systematic biases in the XCO
values can result in biases in the inferred surface fluxes. Chevallier et al. [
15] have investigated the sensitivity of their flux inversion system and found that systematic biases on the order of ∼0.3 ppm can significantly alter the inferred surface fluxes. In order to both retain a significant number of measurement scenes as well as improve on the systematic biases, multiple scattering has to be included in the radiative transfer calculations of the retrieval forward model [
16,
17,
18,
19,
20,
21]. While e.g., Nelson et al. [
22] have explored the possibility of applying a clear-sky only retrieval, in which atmospheric scattering is not considered, they found that scene selection and post-retrieval filtering must be performed much more aggressively to keep retrieval errors low. As such, the preferred option at the time being is to include atmospheric scattering in contemporary XCO
retrieval algorithms, such as RemoTeC [
23], the University of Leicester (UoL) algorithm [
24,
25], the NIES algorithm [
26], WMF-DOAS [
27,
28], FOCAL [
29,
30], and the NASA Atmospheric CO
Observations from Space (ACOS) algorithm [
31].
Radiative transfer (RT) calculations are a significant computational burden on most retrieval algorithms when they include atmospheric scattering. Additionally, retrieval algorithms do not just require the calculated top-of-the-atmosphere (TOA) radiances, but also first-order derivatives of said TOA radiances with respect to various atmospheric or surface parameters, also called weighting functions. Having to compute these derivatives adds additional computational cost. Performing multiple scattering (MS) TOA radiance calculations in a purely monochromatic fashion would cause the total duration for a single retrieval to be several hours, which would make routine data operations prohibitively expensive and slow. The aforementioned retrieval algorithms thus employ so-called fast RT acceleration methods to obtain speed-ups of around two orders of magnitude to reduce the computation time for single retrievals down to merely a few minutes. In this work we focus on three specific acceleration techniques that are routinely employed by the following algorithms: RemoTeC uses linear-
k [
32], UoL uses a principal component-based method [
33], and ACOS uses low-streams interpolation [
34].
For the first time, we present a direct comparison of these three acceleration techniques in a like-for-like manner using XCO
retrievals from OCO-2 measurements. All three fast RT methods were implemented into the UoL retrieval algorithm in a unified manner, such that the only difference is the fast RT method itself, with all other parameters and retrieval inputs being the same. Previously, Connor et al. [
35] conducted a study to investigate various forward model errors in OCO-2 XCO
retrievals, but did not include the impact of radiative transfer itself. Our work is therefore the first to attempt to quantify forward model errors in retrieved XCO
due to radiative transfer acceleration techniques using the same retrieval algorithm.
4. Discussion
Figure 8 and
Table 4 contain the main result of our study. We show that after representative post-retrieval filtering and parametric bias correction, the overall XCO
bias between the various fast RT methods is generally below 0.05 ppm with an overall scatter of
0.70 ppm. The corresponding inter-quartile ranges (IQRs) do not exceed 0.40 ppm, and the differences between IQRs and
are larger for land nadir and land glint modes. This mostly reflects the long-tailed aspect of the distributions seen in
Figure 8 for land scenes. The parametric bias correction procedure has a notable effect when assessing the retrieved XCO
against the truth proxy which is used for the bias correction itself. Its impact is largest on ocean glint scenes where it reduces the scatter by over 50%. Comparing this to the pair-wise differences between retrieval sets, we observe that the impact is less across all observation modes and all compared pairs. We thus conclude that the bias correction procedure is roughly equally effective for all four retrieval sets, and, on average, adjusts the raw XCO
values in a similar way. This is also underlined by the bias correction coefficients in
Table 3 which show little difference between the four sets.
Putting the numbers of
Table 4 into context, we can refer to overall biases and scatter of some XCO
product against a global truth proxy. O’Dell et al. [
31] state biases and scatter for overpasses over ground-based stations, which range between
ppm and
ppm (ocean glint) to
ppm and
ppm (land nadir). The same statistics against a set of global models exhibit similar values for bias and scatter (not shown in O’Dell et al. [
31]). Given those numbers, we can see that the differences (
) of bias-corrected XCO
from retrievals using different fast RT methods are generally smaller than overall biases to the truth proxies, by almost an order of magnitude. The way in which the overall scatter between fast RT sets would enter the XCO
scatter of a retrieval–model truth comparison is not straightforward, mostly because a small, manual modification of filter thresholds can change those statistics. Focusing on mean biases alone, we can therefore assume that the choice of fast RT method would not cause a significant change in how a final XCO
product compares to truth proxies on a global level. This is implicitly already shown in Buchwitz et al. [
40], where several XCO
and XCH
retrieval sets were evaluated in a systematic way. However, the following caveats of our study have to be considered. We only observed a small set of OCO-2 scenes, which after coarse filtering leave the Sahara desert and South America regions mostly empty (see
Figure 8). Those regions are known to be particularly difficult for XCO
retrievals, as a result of either high aerosol loading due to mineral dust, or the presence of many small clouds that create biases via 3-D scattering effects [
62]. As a result of our limited scene selection we are also biased towards clearer scenes in terms of aerosols, however it does not strictly follow that we can consider our obtained differences between fast RT methods as a lower bound.
The four retrieval sets across three different fast RT methods group somewhat distinctly into LSI and PCA on one side, and linear-
k on the other side. As we have discussed in detail in
Section 2.2, this grouping is expected due to the way these fast RT methods are designed on a fundamental level. The biggest discrepancy is related to Jacobians which have strong multiple-scattering contributions, such as any aerosol Jacobians. As the forward model is non-linear, even slight modifications to Jacobians can lead to different results for both the final XCO
and the remaining portion of the state vector. We therefore employed a filtering and bias correction method for each individual retrieval set to systematically correct the raw XCO
values in a manner that resembles the procedure done in most publicly available XCO
data sets.
Table 4 shows that the bias correction procedure reduces the inter-set spread, which was low to begin with (
ppm). Looking at the values of the parametric bias correction coefficients in
Table 3, we observe that values across a certain parameter range for a certain observation mode are comparable between the fast RT methods. The linear-
k set behaves very similarly for land nadir observations and the
parameter. For land glint observations, we see more discrepancies for both the
and the
parameters. Finally, for ocean glint observations, the discrepancies pertain to the albedo ratio (
) parameter only. These observations are also consistent with the notion that linear-
k has a different sensitivity to aerosols—the retrieved CO
profile is linked to the retrieved aerosol profiles through interference errors [
35].
When we compare the bias-corrected XCO
values against the individually sampled model medians (see
Table 5), we observe that the linear-
k set exhibits the most favourable statistics across all observation modes, despite comparable and sometimes larger numbers of quality-filtered scenes (see
Table 2). When looking at the robust standard deviations as a measure of scatter, the linear-
k set of retrievals shows between 0.02 ppm (land nadir) and 0.05 ppm (land glint) lower values. While the differences of scatter values themselves are statistically significant at
20,000, we do not consider this as evidence of superior performance of linear-
k over the other methods. A more thorough and automated process involving scene filtering and bias correction would be required to make an unbiased assessment when the differences are less than a tenth of a ppm. The results, however, suggest that the linear-
k retrievals respond slightly more favorably to the filtering and bias correction process in our particular example. Despite that radiance residuals for PCA (1) can be twice as large as those of PCA (5) [
33], those differences do not impact the overall comparison against the truth proxy. This particular observation suggests that one principal component is sufficient for OCO-2 XCO
retrievals using the PCA-based method.
Bulk statistics, as stated in
Table 4 and
Table 5, of course, do not capture the entire characteristics of a pair-wise comparison of two XCO
retrieval sets.
Figure 8 show several regions in which significant differences occur. The overall most similar two sets of retrievals are the two runs using the PCA-based method. Only 124 out of 3049 grid cells (at
aggregation), or roughly 4.1%, exhibit a difference larger than 0.3 ppm. Interestingly, bias correction does not change that number much (123 out of 3049 without parametric bias correction). This itself is somewhat remarkable as the radiance residuals are significantly dependent on the number of principal components used in the radiance reconstruction process (see Somkuti et al. [
33]). When comparing the LSI set against the PCA (5) set, we observe more regional differences and the number of grid cells with differences larger than 0.3 ppm is 5.7% (171 out of 3020) after bias correction. Finally, comparing linear-
k against PCA (5) shows 14.7% of
grid cells (426 out of 2905) being more than 0.3 ppm apart (21.4% before bias correction). The different behavior in the comparisons linear-
k↔ PCA (5) and PCA (1) ↔ PCA (5) suggest that the radiance reconstruction of the various fast RT methods have a smaller impact than the reconstruction of the Jacobians.
We want to stress the shortcomings of our study to also highlight where our results are applicable, compared to where they are not. The small subset processed in our study (see
Section 2.3) does not allow for any small-scale analysis. This means our results cannot make any statements about discrepancies for e.g., city-scale XCO
collections seen in OCO-3 [
2]. Further, since we only use OCO-2 scenes which were already quality filtered, the selected scenes are overall biased towards lower aerosol loadings and are generally more well-behaved. This is evident in the number of iterations: more than 80% of all retrievals converge in four iterations or less. As already mentioned before, not all scenes necessarily fall into the “more clear” category, since we use the UoL algorithm setup, which, compared to the ACOS algorithm, sources different prior values for many state vector variables, most importantly aerosols. A look at
Figure 8 reveals that regional differences can surpass ∼0.3 ppm, even for the comparison between PCA (1) and PCA (5), which would likely cause differences in applications such as surface carbon flux inversions [
15]. Including a more extensive set, i.e., a longer time series, of OCO-2 scenes might change the pattern of regional differences as well as fill in areas which have no scenes at all in
Figure 8. Moreover, the choice of fast RT method could also result in a changed seasonal cycle due to the impact of seasonally varying aerosol distributions. We believe the impact of the reconstruction accuracy of aerosol Jacobians to be significant, however that conclusion could hold only for the UoL algorithm. Other algorithms, such as RemoTeC, characterize and retrieve aerosols using a different approach (see e.g., Wu et al. [
44]). Further, the aerosol sensitivity of OCO-2 is comparable to that of GOSAT and GOSAT-2, however not for instruments like CO2M as its spectral resolution is lower. In upcoming work, one could aim to disentangle the impact of weighting functions from the radiance reconstruction using a hybrid fast RT scheme: radiances could be reconstructed using one fast RT method, and Jacobians would be calculated using another method.
5. Conclusions
In this study, we implemented three contemporary fast radiative transfer (fast RT) methods into the UoL XCO
retrieval algorithm. This allowed us to explore the impact of the choice of fast RT method with all other components of the retrieval algorithm being the same. We ran four sets of retrievals (with two different settings for the PCA-based method) and performed post-retrieval filtering as well as parametric bias correction to ensure that each set of retrievals is individually treated as they would in any XCO
data set processing. The bulk differences in XCO
between the four sets show just a small overall bias
ppm, and the difference scatter being between
ppm (
being a robust standard deviation) for land nadir, and
ppm for ocean glint. Regional differences, however, can exceed 0.3 ppm (after bias correction) for 4% of the data points and can thus be considered significant biases. We find that land nadir observation modes exhibit the largest differences, and ocean glint scenes the lowest. In the comparison against CO
models, which were also utilized as truth proxies for the parametric bias correction, linear-
k shows the best performance, although the differences are too small to make a more general recommendation in the context of OCO-2 retrievals. Further studies, especially on small- and city-scale aggregates, need to be undertaken to understand the full impact of fast RT methods on XCO
retrievals in the shortwave-infrared wavelength region. For example, the estimation of point-source emission rates, which relies on the determination of XCO
(or XCH
) enhancements [
9,
63,
64], can be affected by biases amongst other sources of uncertainty, such as wind direction and wind speed. Given the similarity of results between the PCA (1), PCA(5) and LSI sets of retrievals, we believe the reconstruction accuracy of atmospheric weighting functions related to aerosols to be a major driver of the observed discrepancies between linear-
k and the other sets. Finally, the data quality filtering and bias correction approaches need to be formulated using a more autonomous and reliable technique to reduce the impact of manually chosen parameters and thresholds.