the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
| 03 Aug 2026
1D-VAR system for a ground-based microwave radiometer: application of an inter-channel observation-error covariance matrix
Abstract. This study presents the specification of an inter-channel observation-error covariance matrix (R) and evaluates its effects on One-Dimensional Variational (1D-VAR) retrievals from a ground-based microwave radiometer (MWR) using RTTOV-gb as the observation operator. In the 1D-VAR, the accurate characterization of the background error covariance matrix and R is crucial, as they determine the relative weighting of background and observational uncertainties and thus directly control retrieval performance. However, realistic specification of R remains challenging due to the complexity of error sources, including instrument noise, forward-model error, and representativeness error. Consequently, many previous studies have adopted a simplified diagonal approximation. In this study, a correlated R is specified and compared with its diagonal configuration to quantify the effects of inter-channel error correlations on 1D-VAR performance. Radiosonde validation shows root mean square error (RMSE) reductions of 1.02 % for temperature and 4.52 % for humidity relative to the diagonal configuration below 1000 m. Using the correlated configuration, the 1D-VAR retrieval outperforms the Numerical Weather Prediction (NWP) model used as the background in the lowest 1000 m, achieving RMSE reductions of up to 23 % for temperature and 12 % for humidity, indicating an improved representation of near-surface variability during the intensive observation periods. The correlated configuration improves convergence efficiency, with 5.7 % of successful retrievals converging in a single iteration (compared to 0.1 % for the diagonal configuration), reducing the total processing time by approximately 2.6 %. In addition to enhanced computational efficiency, the correlated R maintains comparable retrieval accuracy in the lower troposphere. Overall, explicitly accounting for inter-channel observation-error correlations enhances convergence efficiency and provides modest improvements in retrieval accuracy below 1000 m, highlighting the importance of realistic observation-error covariance specification for lower-tropospheric thermodynamic profiling.
Competing interests: At least one of the (co-)authors is a member of the editorial board of Atmospheric Measurement Techniques.
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Status: final response (author comments only)
- RC1: 'Comment on egusphere-2026-3545', Anonymous Referee #1, 01 Sep 2026
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RC2: 'Comment on egusphere-2026-3545', Anonymous Referee #2, 30 Oct 2026
Reviewer Comments
This manuscript investigates the impact of accounting for inter-channel observation-error correlations in a one-dimensional variational (1D-VAR) retrieval system for temperature and humidity profiles using ground-based microwave radiometer (MWR) observations. Comparing retrievals obtained with a diagonal observation-error covariance matrix (DIAG) and a full covariance matrix (FULL) is of practical interest for the effective use of MWR observations.
However, important questions remain regarding the estimation and physical interpretation of the observation-error covariance matrix, the rationale for introducing FULL.wt by empirically weakening the correlations, and the statistical robustness of the reported improvements. Further clarification is also needed before a reduction in the number of iterations can be interpreted as improved numerical convergence. The major issues below should be addressed, and the conclusions should be brought into closer alignment with the evidence presented.
Major comments
- Section 3.4, Eqs. 7–8: The observation-error covariance matrix is constructed as R = E + M + F, where E, M, and F represent instrument noise, representativeness error, and radiative transfer model error, respectively. However, it is unclear whether these estimates contain overlapping error contributions. Brightness-temperature variability during clear-sky periods may include genuine atmospheric variability. Similarly, differences between observations separated by 391 s contain both atmospheric variability and instrument noise at the two observation times. If the instrument errors are temporally uncorrelated and have the same covariance E, their contribution to the covariance of the differences is 2E. Please clarify whether this contribution was removed when estimating M in Eq. 8, justify the use of variability over 391 s as an estimate of model representativeness error, and explain why cross-covariances among the underlying instrument, representativeness, and radiative transfer model errors can be neglected.
- Section 4.3, Fig. 7: Because the original FULL configuration produces larger temperature RMSEs than DIAG between 400 and 1000 m, the authors introduce FULL.wt by multiplying the off-diagonal elements of the V-band covariance submatrix by 0.05. However, the procedure used to select this factor and its physical justification are unclear. Please identify the cases used to select the factor, present a sensitivity analysis, and compare DIAG, the original FULL, and FULL.wt using independent cases that were not used for tuning. If independent validation is not feasible, the adjustment should be explicitly characterized as exploratory and the claims of improved performance should be qualified accordingly. Please also distinguish the original FULL from FULL.wt consistently throughout the manuscript, including the conclusions.
- Sections 4.2–4.3, Figs. 6–7: The reported reductions in RMSE for FULL relative to DIAG below 1000 m are modest (1.02% for temperature and 4.52% for humidity) and are based on 41 radiosonde validation cases. It is unclear whether these differences exceed sampling variability. Please quantify the uncertainty in the RMSE differences, for example by using a block bootstrap that accounts for temporal dependence among cases. If the uncertainty estimates do not provide clear evidence of improvement, the conclusions should acknowledge this limitation rather than presenting the differences as established improvements.
- Section 3.1, Eqs. 4–6; Section 4.3: The difference in iteration counts between FULL and DIAG is interesting. However, its interpretation depends on the stopping criteria used in the minimization. Please specify the numerical threshold corresponding to the symbol “≪” in Eq. 4 and clarify which criteria determine iterative convergence and which are applied only for post-retrieval quality control. If the stopping criteria depend on R, please assess whether comparable levels of optimization convergence are achieved using a common diagnostic, such as the gradient norm or the magnitude of the state-vector increment. For the reported approximately 2.6% reduction in elapsed runtime, please also describe the benchmarking conditions, number of runs, and variability in timing.
- Section 3.4, Table 1, Fig. 4: The extent to which the estimated inter-channel correlations represent actual observation errors has not been sufficiently assessed. Please discuss uncertainty in the estimated correlation structure and, in particular, why the V-band correlations require such substantial attenuation in FULL.wt. Although O–B covariance cannot be directly equated with R because it also contains background-error contributions, supplementary consistency diagnostics could be informative. Please also confirm that the covariance matrices used for FULL and FULL.wt are positive definite, report their condition numbers (or those of the corresponding correlation matrices), and describe any regularization applied.
- Section 3.5: The 1D-VAR solution depends on the relative weighting specified by B and R. Please clarify how B was estimated, the period used to derive its statistics, its vertical correlation structure, whether temperature–humidity cross-correlations are included, and how it was interpolated to 101 levels. At a minimum, the implications of the B specification for the interpretation of the results should be discussed.
- Sections 2.3.2 and 4.1; Figs. 2 and 5: A reduction in the O–R RMSE relative to the O–B RMSE is to some extent expected in a 1D-VAR system that optimizes the fit to the same observations. It should therefore not, by itself, be presented as independent evidence of retrieval accuracy or robustness. Conclusions regarding retrieval accuracy should be based primarily on the radiosonde validation. In addition, please specify the period used to derive the bias-correction coefficients from background-simulated brightness temperatures, the period over which the correction was applied, and the sample size, and clarify whether R was estimated consistently with the bias-corrected observations.
- Section 5: Several statements in the conclusions appear stronger than warranted by the results. In particular, the reported reductions in iteration counts and runtime do not, by themselves, establish improved numerical stability. The claim that FULL misinterprets useful channel signals as noise also requires supporting evidence, as the RMSE differences alone do not demonstrate this mechanism. Furthermore, the statement that the retrieved profiles better represent boundary-layer transition processes than the background should be supported by an independent evaluation or expressed more cautiously. Please revise the conclusions to distinguish directly demonstrated findings from possible interpretations, taking into account the limitations discussed in Comments 2–4.
Minor comments
- Sections 2.2 and 4.2: The numbers of radiosonde launches reported for the two IOPs (56 and 82) sum to 138, whereas Section 4.2 reports 137. Please reconcile these numbers and clarify whether any case was excluded.
- Section 3.1, line 162: The 0.4 K threshold for accepting the background profile without iteration is described as empirically determined from the O–B RMSE distribution. Please report the fraction of cases for which the background was accepted directly and discuss whether this procedure affects the comparison between FULL and DIAG, since these cases are insensitive to the choice of R.
- Section 3.1, Eq. 6, lines 176–178: The χ² threshold of 100 is adopted following Hewison (2007), and the authors state that it provides a reasonable balance between convergence and retrieval performance. Please briefly explain how this balance was assessed in the present study. Because the χ² statistic depends on R, please also clarify whether FULL and DIAG were evaluated on the same set of retrieval cases.
- Table 1: The values are expressed in K, whereas the caption refers to diagonal covariance elements, which would have units of K². Please clarify whether the table reports standard deviations and revise the caption accordingly.
- Figure 4: Please provide the correspondence between channel numbers and frequencies to facilitate interpretation.
- Figure 5: Panel b is labeled 25.44 GHz in the figure but 27.84 GHz in the text and caption. Please correct this inconsistency.
- Line 321: Please explain the selection criteria leading to the sample of 24,312 retrievals.
- Lines 393–396: The discontinuities in O–B following LN2 calibration are attributed to condensation and nonuniform LN2 coverage of the absorber. However, Section 2.3.2 appears to describe these as possible explanations based on earlier studies. If these mechanisms were directly observed in the present study, please provide supporting evidence. Otherwise, revise the wording to present them as possible causes rather than established mechanisms.
- Lines 396–397: The statement “biased data were removed, and outliers better satisfied the 1D-VAR assumptions” is unclear. According to Section 2.3.2, systematic biases were corrected, while outliers exceeding three standard deviations were excluded. These are distinct procedures, and the remaining observations are not necessarily consistent with the Gaussian-error assumption. Please revise the statement to accurately describe the processing and avoid implying that the assumptions of 1D-VAR have been verified.
Citation: https://doi.org/10.5194/egusphere-2026-3545-RC2
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Overview:
This paper is a nice continuation of previous studies on 1D-VAR retrievals using HATPRO MWR observations, particularly in its examination of an observation error covariance matrix that accounts for inter-channel error correlations. The authors carefully applied data screening to the MWR measurements collected in Seoul, Korea, and developed a 1D-VAR retrieval procedure that includes the 1D-VAR framework, an observation error covariance matrix incorporating radiometric noise from the instrument, radiative transfer model error, and representativeness error, as well as a background error covariance matrix for the retrieval of temperature and water vapor profiles. They used LDAPS data as the background and radiosonde observations to validate the retrieval results.
The main objective of this study is to demonstrate the need to account for correlations between MWR channels when constructing the observation error covariance matrix. Given the important role of the observation error covariance matrix in the retrieval procedure, how to construct an observation error covariance matrix that properly accounts for off-diagonal terms is one of the key aspects of this study.
The authors’ efforts are appreciable, and the subject is suitable for publication in Atmospheric Measurement Techniques. However, more details should be adjusted and included in the manuscript. I recommend publication after major revisions.
Comments on the manuscript: