To construct an emulator for the forward model F(x), we employ Gaussian process (GP) regression to predict a label z*∈R at a new state x*∈Rm. A GP is defined by a kernel function (defined explicitly later) k(x,x′): X×X→R, where in the cases studied in this work X=Rm. We denote by Γ[X,X] the matrix of all kernel function evaluations over the training data X∈Rm×N of N points with the entries Γ[X,X]i,j=k(xi,xj), where xi and xj are the ith and jth training data points, respectively. Furthermore, Γ[x*,X] denotes the vector of kernel evaluations of state x* against all training points X. Using the training data together with vector of corresponding labels z∈RN, a GP prediction of label (or function value) at a new state x* is given by 1z*≡GP(x*)=Γ[x*,X]Γ[X,X]+σI-1z, where we have assumed without loss of generality that the training data are centered, and thus the GP has a zero mean. We add that the term Γ[X,X]+σI-1z does not depend on the new input state, so it can be precomputed. This makes the predictions take minimal computational time by avoiding inverting a potentially large matrix Γ[X,X]+σI-1.

In GP literature, the variance term σI is usually taken to be the measurement error or local-scale unexplained variability in the training labels z. However, since we are interested in reproducing the outputs of a computer code, the “measurements” are exact, and hence there is no measurement error. It was shown in that learning the parameters of GP models from noiseless data can lead to unstable predictive models and numerical singularities. For this reason, we treat σ as a regularization parameter, which captures the empirical mismatch between the model and the actual data, and optimize it together with other kernel parameters.

In addition to point predictions, GP prediction can be associated with prediction uncertainty (the posterior variance of the GP), given by 2σ*=k(x*,x*)-Γ[x*,X]Γ[X,X]+σI-1Γ[x*,X]T. The ability to include prediction uncertainties sets GP regression apart from many modern NN-based machine learning methods, which only provide a point estimate as a prediction. Large prediction variance can be an indication of departure from the support of a training data set, indicating that GP is likely to lose its prediction skill. Additionally, uncertainty from the predictions can be propagated forward and accounted for in further applications of GP-based emulators.

The GP formulas presented here rely on conditional Gaussian distributions and thus have a similar structure to that of optimal interpolation (OI; e.g., p. 157 of ) and related iterative optimal estimation (OE) algorithms (e.g., Eq. ), both of which are widely used methods in atmospheric remote sensing. OI and OE use Gaussian assumptions to derive a mean and covariance for the posterior distribution as a solution to an inverse problem, i.e., data assimilation or a retrieval. In Gaussian process regression, the target function (here, the forward model) is represented similarly by a Gaussian distribution that has a mean (prediction) and covariance (error estimate of the prediction).

Our interest will be in replicating the results of a gradient-based optimization problem. Hence, in addition to fast evaluations of F(x), we would also benefit from fast derivatives obtained from closed-form expressions. Combining Eqs. () and (), we get 3ddx*z*=ddx*Γx*,XΓX,X+σI-1z, which describes taking the derivative of Eq. () with respect to x*.

While other machine learning methods, such as artificial neural networks and multilayer perceptrons, can in principle be differentiated, computing the derivatives of a large architecture is computationally more demanding than evaluating Eq. (), which motivates our use of GP regression. Other similar approaches (e.g., radial basis function networks) can be shown to be universal approximators as well and could be used in place of GPs. As will be shown, our approach yields a fast and accurate predictor that is intuitive and relatively easy to implement, so comparison against other machine learning methods will not be pursued further in the scope of this work.

A crucial modeling choice in GP regression is specification of a kernel function. This task involves either expert knowledge of the domain structure or some iterative trial-and-error search. In our application, we have empirically observed that a kernel function consisting of the sum of Matérn and linear kernels yields excellent predictive performance. This is likely due to a locally near-linear behavior commonly assumed with the OCO-2 forward model being captured by the linear kernel, together with a largely flexible Matérn term that is known to capture a large variety of nonlinear effects. The Matérn kernel is a more expressive choice of kernel compared to the usual Gaussian/radial basis functions used by default in Gaussian process regression, which tend to be “too smooth” to capture more abrupt changes in the function that is being approximated. Furthermore, such a kernel can also be differentiated in closed form. The kernel function used throughout this work is given by 4k(x,x′)=α11+3l‖(x-x′)‖Wexp⁡-3l‖(x-x′)‖W+α2(Wx)T(Wx′), where ‖(x-x′)‖W=(x-x′)TW2(x-x′), W=diag(w) is a diagonal matrix, w∈Rm is a vector of weights, l∈R is a length scale parameter, and α1 and α2∈R+ are positive weights that are restricted to sum to 1.

To compute Jacobians, we need an expression for the derivative of the kernel function ddx*Γ[x*,X] in Eq. (). This can be computed in closed form from Eq. () using known matrix identities. The derivation of a closed-form expression can be found in Appendix .

Prediction quality of GP regression depends on identifying the hyperparameters θ that best fit the training data. In our case, following the form of our kernel function, we have θ=w,l,σ,α1,α2. Hyperparameters are commonly learned via optimization, using maximum likelihood estimation (MLE, ). This amounts to minimizing 5L(θ)=-12log⁡detΓθ-12zTΓ(θ)-1z, where Γ(θ)=Γ[X,X] evaluated at parameter values θ. Although this method is usually robust and performs well, GP applications with high-dimensional inputs and a large amount of training data are known to be challenging due to inverse matrix and log-determinant calculations. Numerous approaches have been suggested to tackle this problem (e.g., local approximations, ). Inspired by the kernel flow approach where kernel parameters are learned by minimizing a relative reproducing kernel Hilbert space (RKHS) norm, we propose a cross-validation RMSE-based method to be used in this work. Intuitively, RKHS norm is a way to measure the smoothness of a function approximation achieved with the kernel method. While smooth methods generally yield discretization-invariant predictors, we propose to directly minimize the prediction error instead. The intuition of this approach is to iteratively select small mini-batches of the training data set and individually leave points out one by one while using the rest of the mini-batch to predict the left-out values (via Eq. ). Learning the kernel parameters that minimize this prediction error leads to globally good predictions given new inputs. This approach leverages the known screening effect associated with Matérn kernels, where the effects of faraway points on prediction accuracy diminish, and only close-by points are necessary for prediction accuracy. The same intuition is the basis of numerous nearest neighbors GP methods (e.g., ). The upside of our approach is the ability to select small mini-batches on each training iteration, allowing for faster computations while avoiding expensive log-determinant calculations and inverting the large covariance matrices required in MLE. We will later show that our proposed method converges reliably and yields excellent predictions.

We start by selecting a mini-batch Xbatch and zbatch of size Nbatch by randomly sampling from training data. We define a leave-one-out (LOO) cross-validation loss function with respect to L2 error (also known as RMSE) by first considering taking out one data point from the training data and using the rest to predict it. This can be achieved by modifying the GP prediction formula from Eq. () and leaving out the ith data point. This is achieved via a rank-one downdate Γ̃(θ)-1-Γ̃(θ):,i-1Γ̃(θ):,i-TΓ̃(θ)i,i-1 to remove the effect of the ith data point from the inverse covariance matrix Γ̃(θ)-1. (See , and , for details.) The modified LOO prediction formula is then given by 6GP̃(θ,i)=Γ̃(θ):,iTΓ̃(θ)-1-Γ̃(θ):,i-1Γ̃(θ):,i-TΓ̃(θ)i,i-1zbatch, where Γ̃(θ)=ΓXbatch,Xbatch is the Nbatch×Nbatch covariance over the mini-batch evaluated at parameter values θ, Here, the notation Γ̃(θ):,i means all rows of the ith column. We then define the final loss function by using Eq. () to predict zi (the ith training label removed from the mini-batch) as 7ρ(θ)=∑i=k1kpGP̃(θ,i)-zi2+ϵ‖θ0-θ‖k, where i∈[k1…kp]⊂1…Nbatch is a subset of p≤Nbatch indices denoting elements of the mini-batch selected for prediction, which can be chosen as, for example, the entire mini-batch or the p nearest neighbors of the center point of the mini-batch. The regularization term, with error norm ‖⋅‖2, some penalty magnitude ϵ, and mean θ0, is included to ensure that kernel amplitude parameter values do not grow uncontrollably. This is done since we have observed empirically that letting non-identifiable parameters grow during optimization can lead to the optimizer getting “stuck”, whereas this problem is not observed when regularizing the loss function. One may, for example, set θ0 to be a vector of 1's.

We can now optimize the kernel parameters iteratively by repeatedly selecting mini-batches and updating θ along the gradient of ρ(θ), which is obtained by automatic differentiation using Julia's Zygote package . We note that closed-form kernel derivatives could be used here as well, but since automatic differentiation with mini-batch sizes we use uses negligible computational time, we will not pursue this idea further in this work. We note that as the mini-batch is selected at random, this method can be viewed as stochastic gradient descent. For this reason, we use the adaptive moment estimation (ADAM, ) optimizer to find the optimal value. Use of a momentum-based optimizer is further recommended in this application as we have observed that the cost function often has several local minima. The optimization procedure is summarized in Algorithm . The final parameter value can be selected to be the one corresponding to the smallest loss function value achieved during training.

As we aim to reproduce the performance of a function represented as computer code, we take advantage of the freedom to use a space-filling design for x in Rm for training data creation. We first span the unit cube [0,1]m with a Sobol' sequence of N points. In practice we employ Julia's Sobol.jl package for this step. Then, using information about the minimum and maximum physically feasible value of each input dimension, we scale the unit cube to span the whole state space. During research, we tested other methods like random sampling and Latin-hypercube-based methods, which turned out to leave “holes” in training data set, meaning non-constant predictive performance over the entire data set. Sobol' sequences, meanwhile, span the entire input space more evenly. Sobol' sequences are a space-filling design akin to Latin hypercubes, providing optimality (observed experimentally) in generation of training data. We further evaluate the computational model F(x) at each training point, obtaining states X∈RN×m and model outputs Y∈RN×n.

OCO-2 is a NASA-operated satellite mission dedicated to providing data products of global atmospheric carbon dioxide concentrations . The satellite is pointed towards Earth as it measures solar light reflected by Earth's surface and atmosphere, recorded as radiances. The OCO-2 instrument itself is composed of three spectrometers that measure light reflected from Earth's surface in the infrared part of the spectrum in three separate wavelength bands. These bands are centered around 0.765, 1.61, and 2.06 µm and are called the O₂ A-band (O2), the weak CO₂ band (WCO2), and the strong CO₂ band (SCO2), respectively. Each observation consists of 1016 radiances on separate wavelengths from each band (for more information, see, e.g., ). These measurements are then used to infer a state vector containing information on atmospheric properties like CO₂ concentration on 20 pressure levels, surface pressure, temperature, and aerosol optical depth (AOD). The state vector also includes surface properties like albedo and solar-induced chlorophyll fluorescence (SIF). The primary scalar quantity of interest is the column-averaged CO₂ concentration (XCO₂).

A key part to inferring XCO₂ from observed radiances is the construction of a computational atmospheric radiative transfer model which describes how solar radiation is propagated, reflected, and scattered by Earth's surface and atmosphere. Together with an instrument model, this computer code is known as the full-physics (FP) model, referred to in this work as 8y=F(x,b), where y is the output of the FP model (a wavelength-by-wavelength radiance), x is a state vector containing atmospheric and surface information, and b denotes model parameters held fixed during data processing. A thorough description of the FP model is given in the ATBD . To motivate our emulation approach, we will here describe parts of the forward model physics, which is not intended to be a full description of the included physics. Rather, we leverage this information to better design our emulator.

Part of the radiance comes from absorption of radiation by atmospheric molecules, given by 9I(λ)=f0(λ)cos⁡(τ0)⋅R(λ,θ,θ0,φ-φ0)exp⁡-g(λ), where λ is wavelength, the jth wavelength corresponds to the jth entry of radiance y, f0(λ) is the solar flux at the top of the atmosphere, R(λ,θ,θ0,φ-φ0) is the reflectance of the surface, g(λ) is an integral over radiation path length that sums over for all modeled absorbers, θ and φ are the observation zenith and azimuth angles, and θ0 and φ0 are the corresponding solar zenith and azimuth angles. Observation and solar angles have a significant effect on the observed and modeled radiances, which will be important later in this work.

After calculating the absorption with Eq. (), equations further describing atmospheric scattering are employed to solve for atmospheric radiative transfer (RT), which describes the total effect of atmosphere and surface on the scattered photons. The FP framework further includes an instrument model, which describes the effects of the observing system to the top-of-the-atmosphere radiances. These effects include instrument Doppler shift, spectral dispersion, and convolution with the instrument line shape (ILS) function, reducing the resolution from the finer RT grid to the coarser observational grid. On an abstracted level, this corresponds mathematically to 10IOBS(λ)=C1(λ)∫-∞+∞RT(λ′)ILS(λ,λ′)dλ′+C2(λ), where C1(λ) and C2(λ) denote the instrument effects other than convolution that can be expressed as multiplication and addition. Generally speaking, the instrument effects depend on different physical properties that can vary between detector arrays, while the RT portion of the forward model is constant within the instrument. This observation motivates us to focus on emulating the outputs of the RT, referred to as monochromatic radiances, after which instrument functions can be applied appropriately after the fact. Looking forward to operational integration of our emulator, this will reduce the complexity of the emulated system and arguably make our task easier.

The state vector elements comprising x for the FP model are summarized in Table . Notably, we have divided the table into two parts. The upper half lists the previously mentioned atmospheric and surface state vector elements that affect the RT part only, and the rest having to do with the instrument effects are in the lower half. This collection includes scaling factors for empirical orthogonal functions (EOFs) that capture unmodeled offsets in the observed radiances .

In addition to state vector elements, the FP model is parameterized by a set of parameters that are held fixed based on auxiliary information, such as laboratory measurements or meteorological data sets. These parameters include instrument calibration details, spectroscopy properties for absorbing gases, land elevation, and aerosol microphysical parameters. These aerosol parameters arise from the selection of two dominant aerosol types as a function of space and time. All aerosol types have different optical properties. This choice is determined a priori by global maps based on meteorological knowledge and measurements (see Fig. ). The possible dominant aerosol types are dust (DU), sulfate (SO), sea salt (SS), organic carbon (OC), and black carbon (BC). While constructing the emulator, we will consider data sets with a fixed pair of dominant aerosol species in order to decouple their physical effects from the rest of state vector. Separate emulators can then be constructed for each pair of aerosol species, and a selection of which one to use can be done by matching the measurement location with the appropriate types.

This work develops a proof-of-concept version of OCO-2 forward model emulator for a simulated case. For this reason and ease of access, we implement our simulations using the Reusable Framework for Atmospheric Composition (ReFRACtor, ). ReFRACtor is an extensible multi-instrument atmospheric composition retrieval framework that supports and facilitates combined use of radiance measurements from different instruments in the ultraviolet, visible, near-infrared, and thermal-infrared. It has been open-source since 2014 when it was first developed as the Level-2 processing code for OCO-2. Since 2017 the development team has worked to create a more general framework that supports more instruments and spectral regions. This framework has been developed to provide the broader Earth science community a freely licensed software package that uses robust software engineering practices with well-tested, community-accepted algorithms and techniques. ReFRACtor is geared not only towards the creation of end-to-end production science data systems, but also towards scientists who need a software package to help investigate specific Earth science atmospheric composition questions. Although ReFRACtor includes an implementation of a version of the OCO-2 production algorithm, the two have drifted since the initial intercomparison work was done. At that time it was validated against the B9.2.00 version of the software. For the most part mainly bug fixes have been kept in sync between the two versions. Additionally the core radiative transfer algorithms are the same, which justifies the use of ReFRACtor for constructing our emulator at this stage. Some minor additional algorithmic features made their way into the ReFRACtor version of OCO-2 from the production version. For the most part the major discrepancy will be due to changes in configuration values not implemented in ReFRACtor. These include values such as a priori and covariance versions, EOF data sets, ABSCO versions, and the solar model.

Inferring XCO₂ from measured radiances is an ill-posed inverse problem, which is referred to as performing a retrieval. The relationship between measurement and state is first modeled as 11y=F(x)+ε, where data y∈Rn are a radiance vector; unknown x∈Rm is the state vector, F; Rm→Rn is the OCO-2 FP model; and ε∈Rn is the measurement uncertainty. For completeness, we summarize the operational retrieval algorithm used in OCO-2 processing. The retrieval proceeds with solving the inverse problem by using Bayesian formulation, in which the additive error ε and the prior for x are assumed to be Gaussian such that 12ε∼N0,Sε,x∼Nxa,Sa. The measurement error covariance matrix Sε is assumed to be diagonal, with elements for each wavelength j given by 13σj2=k1yj+k2, where k1 and k2 are calibration parameters adjusted by the instrument calibration team. The a priori covariance is taken to be diagonal for non-CO₂ parameters, and the CO₂ profile is assumed to have a correlation structure shown in Fig. , which promotes continuous concentration profiles and limits the variability higher up in the atmosphere.

The retrieval is operationally carried out using iterative gradient-based methods to solve for the maximum a posteriori estimate, which is equivalent to minimizing the cost function: 14x^=argminxy-F(x)TSε-1y-F(x)+x-xaSa-1x-xa.

This optimization problem is solved using the Levenberg–Marquardt algorithm, in which at iteration i the state is updated according to 15(1+γ)Sa-1+KiTSε-1Kidxi+1=KiTSε-1y-F(xi)+Sa-1xa-xi, where γ is a damping parameter and Ki is the Jacobian of F(x) at iteration i. After each iteration, before updating the state, the effect of forward model nonlinearity is assessed by computing the quantity 16R=ci-ci+1ci-cFC, where ci is the value of the cost function () at iteration i, ci+1 is similarly at iteration i+1, and cFC is the cost function value assuming that F(xi+dxi+1)=F(xi)+Kidxi+1 (i.e., a linear update). Based on the value of R, one of the following is executed:

R ≤ 0.0001: γ is increased by a factor of 10. State is not updated.

As GPs tend to perform worse with increasing input dimension and because the standard GP formulation is developed for one-dimensional outputs, we will need to reduce the dimension of both the atmospheric state and the radiance. For the atmospheric state x, we leverage the fact that OCO-2 measurements are made at three separate wavelength bands, which leads to the state vector having band-specific elements, which can be ignored when dealing with other bands. This partition has been summarized in Table . Earlier work by considered cross-band correlations while emulating OCO-2 radiances, but the authors finally showed that the bands are distant enough from one another in wavelength space that they can be treated independently. With this insight, we proceed by constructing separate GPs for each band and using only the sensitive dimensions of x as inputs. We further notice that the 20-element CO₂ profile is continuous and can be expressed as loadings obtained using principal component analysis (PCA). The most straightforward way to do this is by truncated singular value decomposition (SVD) of the empirical covariance matrix of state vectors . To accomplish this, we use a simulation distribution derived by for one selected set of realistic geophysical conditions as a basis for our experiments and perform SVD on the covariance matrix of this distribution. Analysis of singular value decay suggests that the CO₂ profile can be represented with just four principal components, which we collect to a matrix Px as the four leading singular vectors. We then project the CO₂ profile to a principal component space and further standardize the states by using the mean and variance of the simulation distribution, leading to 19x̃=1σxdiag(PxT,Ic)x-μp,0c-μx, where μp is the CO₂ profile mean, μx and σx are the state mean and state standard deviation, diag(A,B) denotes a block diagonal matrix with blocks A and B, Ic is a c=m-16-dimensional identity matrix (as the profile is represented by 4 dimensions instead of original 20), and [μp,0c] is a stacked vector of CO₂ profile mean and a c-dimensional zero vector.

Next, we generate training data using a Sobol' sequence (see Sect. ). For this study, we can omit dispersion, EOF and SIF parts of the state vector (see Table ), and fix them to the prior mean. This follows from the discussion in Sect.  focusing on monochromatic radiances. Omitting dispersion simplifies computations as the wavelength grid would otherwise shift, making SVD for radiance dimension reduction hard. solved this problem by employing functional principal component analysis, while we can proceed with ordinary SVD. The empirical orthogonal functions (EOFs) are included in the operational retrieval to reduce fit residuals and therefore make convergence analysis easier. These have no direct impact on our study and can be safely omitted. Furthermore, the SIF parameters are fit on the O2 band only as part of the instrument effects, and we do not include them in the emulation for this reason. As is evident from Eq. (), the measurement geometry has a significant impact on the output of the FP model. For this reason we include three extra parameters, θ, θ0, and φ-φ0, to our training data vector. Sufficient and realistic limits to these parameters are obtained from the simulation distribution of by considering a 4σ interval around the mean values. In all, we now have m=4+21+3=28 for input space, coming from profile PCs, other included state vector elements, and geometry. We create a Sobol' sequence of 20 000 points for training and scale all dimensions of the hypercube to [-4,4], corresponding to 4 standard deviations on the normalized x̃ basis. We further obtain the training data set in original space by reversing the transformation ().

Training data Y (radiances) are obtained by evaluating the FP model on each x from the scaled Sobol' sequence. For this work, we choose a single realistic land nadir measurement to represent physical parameters not included in state vector x. We perturb sampling geometry to reflect relevant solar and instrument angles. For a real-world application this approach can be extended to include different scenes and other location-dependent parameters. To obtain the labels z, we similarly perform truncated SVD on the radiances Y separately on each wavelength band B∈[O2,WCO2,SCO2] and collect the leading nB singular vectors in matrices PB. The four leading singular vectors for each band are included in Fig.  to present the kinds of features the most significant principal components, or basis vectors, encode. While this decomposition could hold additional information on physical processes behind the radiative transfer model, we do not pursue such an analysis further in this work. With additional standardization of the variables, we obtain the following transformations for each wavelength band B: 20zB̃=1σzPBTyB-μB-μz, where μB is the radiance mean, and μz and σz are the principal component mean and standard deviation for band B.

The quality of this approximation is assessed by plotting the reconstruction PBPBT(yB-μB)+μB over a holdout data set not used in computing the SVD. We illustrate in the upper panel of Fig.  the distribution of relative reconstruction error from this data set. We have further applied the instrument function to each residual and further divided them by the measurement error standard deviation given by Eq. (). This metric is justified by the rationale that if the reconstruction error is less than or comparable to measurement error on the radiances, no significant amount of information is lost.

The final emulator g(x) can now be summarized in Fig. , where GPi(x̃B) is the GP prediction given by Eq. (), and the indices i, j, and k run through the number of principal components included in a given band. The effect of this choice will be examined further later in this work. The state is first normalized according to Eq. () for each band B, after which the GP-predicted principal component loadings are assembled back to radiances using relation in Eq. (). When evaluating the emulator, each index is independent and can then be computed in parallel.

Having obtained training data X̃ and z, we can now proceed in optimizing the kernel parameters as described in Sect. . We prescribe an individual GP per output parameter zi. We have N = 20 000 for training data size, and we set M = 100 for mini-batch size, set p = 5 for the number of prediction points per mini-batch, and run the ADAM optimizer for 5000 iterations with a small learning rate – in our case 0.02. We initialize all other parameters at 1, except for linear component weight at 0 and the nugget at 1×10-6 for x̃. As outlined in Sect. 3.3, we further reduce the dimension of the input space by selecting only the indices that a given wavelength band is sensitive to, given by Table .

For testing the performance of the algorithm, we draw a random sample Xtest from the same simulation distribution from as independent test data, which is then used to evaluate the FP model to create radiances Ytest. For test data, we fix dispersion, EOFs, and SIF at prior values as before. Example behavior of the loss function together with evolution of the kernel parameter values and true vs. predicted z values is shown in Fig. . We see that during training, the loss function values converge to a small value close to 0. The fluctuations of loss function values are due to the ADAM optimizer effectively being a stochastic gradient descent method: each mini-batch is sampled randomly, which causes consequent steps, possibly resulting in higher values, while the running average still always decreases. We can also see the evolution of different kernel parameters: the weight of the linear component, for example, was initially set very close to 0. During training, the algorithm correctly identifies the significance of the linear term, and the relative importance of this term overtakes the nugget (corresponding to random noise). Resulting predictions of principal component loadings can be seen to land very tightly on the one-to-one line, indicating good performance over the whole test data. The distribution of true vs. predicted z values for each component on each wavelength band is further illustrated in Fig. . We conclude that the predicted values correspond the true values most of the time. Some principal components show a larger spread on prediction errors (e.g., O2 PC 9 or WCO2 PC 9). These principal components can be redundant and do not contain meaningful information about the radiance, since the total predictive performance still remains very accurate.

Finally, we assemble the predicted z values back to radiances and compute the relative differences with the test data, shown in the upper panel of Fig. . On the lower panels, as before, we apply the instrument function to these residuals and divide by the measurement error standard deviation to underline that the desired performance would be to make less prediction error than measurement error.

After constructing the emulator obtaining radiances as outputs, we can further apply Eq. () to compute the Jacobians ddx̃z. We can then reverse the normalizing transformations on both x̃ and y and further apply the instrument functions to our Jacobians to get back to the operational observation units. The Jacobians obtained by evaluating both the FP model and the emulator on an example state vector together with the resulting profile averaging kernels are shown in Fig. . Accurate averaging kernels are important for downstream usage of the retrieved XCO₂, as it is used, for example, in flux inversion models to obtain the vertical sensitivity of XCO₂ to modeled atmospheric CO₂ profiles. We note that we have normalized the Jacobians and averaging kernels by maximum values of each row in the matrix for visual clarity. Although not perfectly similar, we conclude that these two outputs share significant similarity. The main difference in the averaging kernels mainly results from the choices of modeling concentration profiles by principal component loadings.

As noted in previous work by , an emulator provides substantial appeal in terms of computational efficiency. For the current work, the average computational times for model evaluation and Jacobians are summarized in Table  on a 2023 MacBook Pro. It is worth mentioning that ReFRACtor computes Jacobians via automatic differentiation, while our emulator does this analytically. Three cases are contrasted: the standard ReFRACtor FP evaluation, the emulator for monochromatic radiances plus ILS, and the emulator alone.

In recent years, the uncertainty quantification and statistics community has benefited enormously by utilizing the surrogate model by to explore the OCO-2 retrieval in numerous applications . We remark that for similar purposes, our emulator can be used as an even faster surrogate. As we see from Table , the majority of the computational cost for the emulator comes from the instrument effects, which are part of the ReFRACtor software. If one is not interested in including the effects of dispersion, SIF, and EOFs during the retrieval, we notice from Eq. () that instrument corrections to RT amount to multiplication, addition, and convolution, which is associative with respect to multiplication. We can then write the emulator as 21g(x)=ILSP^η(x)=ILSP^η(x), where g(x) is the overall emulator, ILS() is a function applying the instrument corrections from Eq. (), P^ is a projection matrix consisting of radiance basis functions that correspond to transforming predicted labels z back to radiances y following the last step in Fig. , and η(x) is the emulator predicting labels z from inputs x. Done this way, we can evaluate the instrument corrections on the basis vectors once, after which OE or MCMC can proceed an order of magnitude faster (according to Table ).

Previously in this work we did not prescribe a certain number of principal components to use in radiance dimension reduction. Figure  illustrates the retrieved XCO₂ root mean square error (RMSE) and mean absolute error (MAE) against the true known value, together with Fig. , illustrating radiance reconstruction and prediction RMSE and MAE similarly to Figs.  and , all as a function of the number of PCs used. We can collectively deduce that using more than 25 principal components per band does not yield any additional performance benefits. We remark that compared to the earlier work by , who argued for one to three principal components per band, our results show that many more components are needed for accurate retrievals. This highlights the importance of empirically checking the effect of dimensionality reduction and not relying on rules of thumb such as conserving 95 % of the variability.

To assess the effect of changing the dominant aerosol types on the performance of the retrievals, we repeat the training and retrieval procedure described in this section with two separate pairs of dominant aerosol types. Firstly, we consider dust (DU) and sea salt (SS) and, secondly, DU and sulfate (SO). These are among the most common aerosol combinations encountered in the OCO-2 operations. We repeat the retrievals for both cases with additional MD adjustment as before. Results of this experiment are summarized in Fig. . We conclude that the proposed method is robust in changing physical conditions, which indicates fitness for further operational integration.