A Proof-of-Concept Algorithm for the Retrieval of Total Column Amount of Trace Gases in a Multi-Dimensional Atmosphere
Abstract
:1. Introduction
2. Retrieval Algorithm
- an inverse model,
- a solution method for solving the inverse problem, and
- a linearized multi-dimensional radiative transfer model for computing the forward model and the Jacobian matrix at each iteration step.
2.1. Inverse Models
- the Differential Radiance Model with Internal closure (DRMI), in which the measured and simulated differential spectral signals are fitted, and
- the Differential Radiance Model with External closure (DRME), in which the measured differential spectral signal and a simulated spectral signal with its smooth component extracted are fitted.
- In DRME, we assume the linearized model [19]
- We adopt an approximate inverse model in which we compute the spectral signal by an one-dimensional radiative transfer model, and (similar to the Ring and polarization correction spectra) introduce a correction spectrum that accounts on three-dimensional effects. To define the correction spectrum, we take into account that, for example, in DRME, we can write
2.2. Solution Method
- In contrast to the method of Tikhonov regularization [22], the regularization parameter is not constant during the iterative process. Instead, the regularization parameters are the terms of a decreasing (geometric) sequence, i.e., with . In this way, the amount of regularization is gradually decreased during the iterative process.
- For iterative regularization methods, the number of iteration steps k plays the role of the regularization parameter, and the iterative process is stopped after an appropriate number of steps in order to avoid an uncontrolled expansion of the errors in the data. The stopping rule used in this study is the discrepancy principle [23], according to which, the iterative process is terminated after steps such that
- The regularization matrix is chosen as a diagonal matrix, that is, the penalty term is taken as
2.3. Linearized Radiative Transfer Model
- The linearized forward approach relies on an analytical computation of the derivatives. The method is accurate and has the advantage that no assumptions rather than those of the forward model have to be made. However, the method is time consuming and memory demanding when the number of parameters to be retrieved is large. The reason is that not only the source function has to be stored as a spherical harmonic series at each grid point, but also its derivatives with respect to the atmospheric parameters of interest.
- The linearized forward-adjoint approach relies on the application of the adjoint radiative transfer theory. The method requires less storage for derivatives calculation, is much faster, but relatively less accurate. The main reason for this lower accuracy is that different interpolation schemes are used for radiance and derivative calculations.
- Correlated k-distribution method. Consider a discretization of the spectral interval into a set of equally spaced wavelengths with the discretization step , and assume that the transmission within a spectral interval depends only on the distribution of the gas absorption coefficient within the spectral interval [28]. Letting be the cumulative density function of in the spectral interval , the inverse distribution function, and a set of quadrature points and weights in the interval , the spectral signal (1) and its partial derivative with respect to the total column are computed as
- Principal component analysis. At wavelength , the integrated signal is related to the integrated signal calculated by a simplified (approximate) radiative transfer model through the relationThe correction factor in Equation (32) is actually the quantity that is calculated by means of the principal component analysis. To summarize this approach, we assume that for each wavelength , the spectral variability of the optical parameters can be described by a vector , defined byTo compute the derivative of the integrated signal with respect to the total column , we may use the principal component analysis to calculate the derivative correction factor , defined by [29]Alternatively, taking the derivative of Equation (33) with respect to , i.e.,
3. Numerical Simulations
- The domain of analysis is a rectangular prism of lengths km and km. The discretization steps along the horizontal directions are km. Along the vertical direction, the atmosphere between 0 and 50 km is discretized with a step of 0.5 km between 0 and 3 km, 0.1 km between 3 and 4 km, 0.5 km between 4 and 10 km, 1.0 km between 10 and 14 km, 2 km between 14 and 30 km, and 5 km between 30 and 50 km.
- A homogeneous cloud is placed between 3 and 4 km. The cloud extinction field is given by , where and is the indicator function (note that takes the values 1 and 0 inside and outside the cloud, respectively). The cloud phase function is a Henyey–Greenstein phase function [30] with the asymmetry parameter , and the cloud single-scattering albedo is 0.99. Eight cloudy scenes are generated by a two-dimensional broken cloud model [14] with a cloud fraction of about The extinction field is smoothed at the boundary of a cloudy region in order to avoid abrupt changes in the horizontal plane. The indicator functions corresponding to the eight cloudy scenes are illustrated in Figure 1a–h. For two-dimensional geometries, slices at km are selected from the cloud fields 1, 2, 3, 5, 6, and 8. The corresponding indicator functions are shown in Figure 2. Note that the slices corresponding to the cloudy scenes 4 and 7 are similar to the slice corresponding to the cloudy scene 2, and are therefore omitted.
- The number of discrete zenith and azimuth angles are and , respectively, the solar and instrument zenith angles are and , respectively, and the relative azimuth angle is .
- A Lambertian reflecting surface with the surface albedo is considered.
- The footprint of the detector is a square of length centered at , and , and a wavelength-dependent slit function corresponding to the TROPOMI instrument is assumed.
- In addition to the scattering and absorption by the cloud, molecular Rayleigh scattering and the absorption by , ozone (), oxygen dimer (), and water vapor () are considered. The measurement spectral grid roughly resembles the TROPOMI’s spectral resolution and consists of 119 spectral points between 425 nm and 450 nm.
- For a clean scenario, we use the a priori partial column profile of [17] illustrated in Figure 3 to generate the true (exact) partial column profile. In this regard, denoting the a priori partial columns of gas g by , we choose the true partial columns as , with and . The true total column of gas g is then computed as ; thus, .
- For , we generate the simulated spectral signal by means of SHDOM.
- For cubic smoothing polynomials, i.e., , we determine the coefficients of the polynomial as the solutions of the least-squares problem (13).
- We compute the noisy spectral signal as , where the measurement errors are assumed to be independent Gaussian random variables with zero mean and standard deviation , where SNR is the signal-to-noise ratio. It should be pointed out that in view of the approximation
- We include the Ring correction spectrum illustrated in Figure 4 in the retrieval, and choose the a priori and true Ring amplitudes as and , respectively. Note that the inelastic scattering is described by a first-order Rayleigh scattering model, i.e., by applying a first-order iteration scheme to the one-dimensional radiative transfer equation for inelastic scattering [31].
- For DRMI, we compute the measured differential spectral signal as
- an adaptive grid with a splitting accuracy of ,
- the principal component analysis with , and the derivative correction factor as in Equation (39),
- periodic boundary conditions,
- The regularization parameters are the terms of a geometric sequence with ratio and initial value . Thus, at the first iteration step, the regularization parameter is .
- The weighting factors specifying the contribution of each component of the state vector into the regularization matrix are , , , and for all . By this choice, the total columns of the auxiliary gases are stronger constrained to the a priori than the total column of and the Ring correction spectrum.
- The control parameter in the discrepancy principle Equation (24) is .
3.1. Test Example 1
- the inverse problem is severely ill-posed (for the DRMI model, the condition number of the Jacobian matrix at the initial guess is ) and
- there is a strong correlation between the and Ring effect signatures (when the Ring correction spectrum is not included in the retrieval, the condition number is ; thus, decreases by three order of magnitude).
- At the initial guess, the residual corresponding to DRMI is much smaller than that corresponding to DRME. This occurs because the discrepancies between the differential spectra are usually small.
- In DRMI, the residual decreases very fast at the first iteration step and then more steadily, while in DRME, the residual gradually decreases.
3.2. Test Example 2
4. Discussion
- The differential radiance models with internal and external closures yield accurate results with reasonable computation times (of about 35–40 min).
- An inverse model based on an approximate computation of the partial derivative leads to a reduction of the computation time by about 25%, but to large relative errors.
- Provided that the regularization parameter is optimally chosen, reasonable accurate results with a computation time of about 6 min can be obtained when the iterative process, corresponding to the differential radiance model with external closure, is stopped after one iteration step. The fact that the optimal value of the regularization parameter depends on the cloudy scene, makes it more difficult to apply this one-step retrieval algorithm, or equivalently, DOAS-type models.
- Although accurate, the retrievals based on the differential radiance model with external closures are inefficient; the computation time is of about 14 h and 15 min for a full-step retrieval algorithm, and 2 h and 30 min for an one-step retrieval algorithm. The one-step retrieval algorithm is less accurate than in the case of two-dimensional geometries, but the results are still acceptable.
- The application of a fast one-dimensional radiative transfer model to retrieve the column amount for a three-dimensional cloudy scene leads to relative errors up to 15%. These errors can be reduced when a differential correction spectrum due to three-dimensional effects is included in the retrieval.
- to construct a database for the spectral signal and its derivatives , and to use these spectra in an one-step, three-dimensional retrieval algorithm, or
- to construct a database for the differential correction spectra accounting on three-dimensional effects , and to use these spectra in a full-step, one-dimensional retrieval algorithm.
Author Contributions
Funding
Conflicts of Interest
Appendix A
- With delivered by the reference sector method, and and determined at Step 1, solve the nonlinear equation
Appendix B
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Cloudy Scene | DRMI | DRMI Approx. Derivatives | ||
---|---|---|---|---|
Rel. Errors | CPU | Rel. Errors | CPU | |
1 | 39:51 | 32:24 | ||
2 | 40:12 | 32:48 | ||
3 | 39:14 | 31:06 | ||
5 | 38:54 | 30:44 | ||
6 | 39:32 | 31:16 | ||
8 | 40:06 | 32:33 |
Cloudy Scene | DRME | DRME (One Iteration) | ||
---|---|---|---|---|
Rel. Errors | CPU | Rel. Errors | CPU | |
1 | 35:10 | 5:33 | ||
2 | 36:47 | 5:54 | ||
3 | 34:57 | 5:12 | ||
5 | 34:23 | 5:01 | ||
6 | 35:06 | 5:28 | ||
8 | 36:23 | 5:48 |
Cloudy Scene | DRME | DRME (One Iteration) | ||
---|---|---|---|---|
Rel. Errors | CPU | Rel. Errors | CPU | |
1 | 14:15:31 | 2:34:03 | ||
2 | 14:21:43 | 2:36:42 | ||
3 | 14:15:57 | 2:32:26 | ||
5 | 14:15:23 | 2:32:11 | ||
6 | 14:14:36 | 2:33:38 | ||
8 | 14:20:13 | 2:34:16 |
Cloudy Scene | DRMI | DRMI (Correction. Spectrum) | ||
---|---|---|---|---|
Rel. Errors | CPU | Rel. Errors | CPU | |
1 | 2:21 | 2:42 | ||
2 | 2:32 | 2:58 | ||
3 | 2:14 | 2:36 | ||
4 | 2:04 | 2:24 | ||
5 | 2:27 | 2:47 | ||
6 | 2:25 | 2:45 | ||
7 | 2:25 | 2:46 | ||
8 | 2:30 | 2:50 |
Cloudy Scene | DRME | DRME (Correction. Spectrum) | ||
---|---|---|---|---|
Rel. Errors | CPU | Rel. Errors | CPU | |
1 | 2:10 | 2:16 | ||
2 | 2:12 | 2:18 | ||
3 | 2:05 | 2:11 | ||
4 | 1:52 | 2:01 | ||
5 | 2:16 | 2:22 | ||
6 | 2:14 | 2:18 | ||
7 | 2:13 | 2:19 | ||
8 | 2:21 | 2:17 |
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Doicu, A.; Efremenko, D.S.; Trautmann, T. A Proof-of-Concept Algorithm for the Retrieval of Total Column Amount of Trace Gases in a Multi-Dimensional Atmosphere. Remote Sens. 2021, 13, 270. https://doi.org/10.3390/rs13020270
Doicu A, Efremenko DS, Trautmann T. A Proof-of-Concept Algorithm for the Retrieval of Total Column Amount of Trace Gases in a Multi-Dimensional Atmosphere. Remote Sensing. 2021; 13(2):270. https://doi.org/10.3390/rs13020270
Chicago/Turabian StyleDoicu, Adrian, Dmitry S. Efremenko, and Thomas Trautmann. 2021. "A Proof-of-Concept Algorithm for the Retrieval of Total Column Amount of Trace Gases in a Multi-Dimensional Atmosphere" Remote Sensing 13, no. 2: 270. https://doi.org/10.3390/rs13020270