The manuscript is generally well written and structured and the equations appear to be correct (I checked quite a few). However, it lies in the nature of such a relatively mathematical manuscript addressing an optimisation problem that the soundness of the approach remains difficult to judge, unless sufficient and convincing validation results are provided. In my opinion, the presented validation results are insufficient and, therefore, I recommend a major revision.

The manuscript solely relies on SURFRAD sites covered by vegetation (mainly grass). However, the emissivity variation of green and dry grass is relatively limited. The manuscript needs to include bare-soil cases to demonstrate that the developed method can reliably retrieve broadband emissivity over its full, naturally occurring  range. This should be demonstrated with a combination of retrievals from simulated data as well as suitable in-situ data, e.g., from SURFRAD station 'Desert Rock' (DRA) and bare-soil BRSN stations.

Specific points:

line 19: 'broadband' should be moved from infront 'LST' to 'emissivity products'

line 86: '... provides a physical meaningful lower bound ...' (apparent temperature is always <= LST)

line 210: eq. (14) limits the strength of the signal from which information on LST and emissivity has to be retrieved. Therefore, convincing validation results  are of utmost importance.

Many occurrences: you probably want to write 'radiance' instead of 'irridiance' (please check everywhere)

line 629: typo 'SURFRAD'

table 1 and figure 6: 'Case index' refers to the Filenames in table 1. However, in table 1 no index is provided. Please clarify.

figure 4 and 5 and elsewhere in text: do you really want to write 'dt' or simply 't'?

figure 6: if the case index in this figure refers to the file numbers in table 1, I recommend labelling the figure with the corresponding SURFRAD station abbreviations. Then it also becomes clear, that the emissivity over the same site varies considerably over only a few days: over grass, this is unrealistic.

 Table 2: also add 'case index'

Table 4: the columns with the absolute bias / error values can be deleted.

Summary

The paper describes a joint retrieval of LST and broadband LSE, performed only when the inversion is sufficiently well conditioned, with explicit uncertainty quantification. The LST inversion is accepted only when the measurement geometry and irradiance observations yield a sufficiently well-conditioned solution. A linearized emissivity provides a physically motivated first-guess for the Newton-Raphson method, strictly as a diagnostic and initialization aid, not as the final retrieval. The uncertainty in ε0 increases sharply under weak irradiance contrast (poorly conditioned). This divergence, and not the magnitude of ε0, serves as a diagnostic indicator.

Overall Assessment

The manuscript is well written and presents an interesting joint emissivity-surface temperature retrieval concept, supported by a sophisticated diagnostic framework and a well-structured approach to uncertainty propagation and decomposition. However, several interpretations in the Results section appear broader than what the retained dataset directly supports regarding robustness, neutrality of temporal pairing, and stability across conditioning regimes. The authors should report the effective sample sizes at each workflow stage: candidate pairs, rejected pairs, retained pairs, and successful Newton-Raphson retrievals, and qualify the conclusions accordingly.

While the manuscript has limitations in validation strength, the uncertainty propagation and decomposition framework is interesting and goes beyond standard bulk-error metrics. I therefore recommend Major Revision to strengthen the validation and better align the strength of the conclusions with the supporting evidence.

Title

“Conditioning-controlled retrieval” sounds non-standard, and it makes me wonder: “numerical conditioning” or “conditioning on auxiliary parameters”? Perhaps rephrase “Conditioning-aware …” or “Stability-aware …”

e. g., Stability-Aware Retrieval of Land Surface Temperature and Broadband Surface Emissivity from Paired Longwave Irradiance Measurements.

Abstract/Clarification

Essentially, the challenge begins with an underdetermination problem: N obs and N+1 unknowns. The authors introduce additional constraints, including quasi-steady LST, temporal pairing, Newton-Raphson convergence rate (Pk), and Jacobian (J) strength, to increase the effective information content. Even after the problem is regularized, it is not solvable in all cases because J may become rank-deficient or poorly conditioned, and the condition number (k) may become large, leading to instability.

The paper attributes the retrieval challenge mainly to low irradiance contrast. However, as stated above, the inversion is intrinsically ill-posed (N+1 unknowns). Low irradiance contrast further degrades the conditioning of an already weakly identifiable inverse problem by reducing the sensitivity of the measurements to the unknowns. It would help distinguish the structural underdetermination of the inverse problem from the subsequent deterioration in numerical conditioning under low-contrast conditions (ill-posedness and poor conditioning). The authors mention this later in 38-40.

Introduction

35: If I recall correctly (I may be wrong), F. Nerry, F. Becker, and later others do explicitly propagate emissivity uncertainty, at least using first-order analytical error propagation.

Rephrase. E.g.: While previous works have examined the sensitivity of LST to LSE uncertainty, explicit uncertainty propagation within a joint inversion framework has received comparatively less attention.

2. Physical and conceptual background

91-98: Nicely explained and is at the heart of the approach.

3. Emissivity and surface temperature retrieval

Streamline at the authors’ discretion:

Much of Section 3 is scientifically useful, but some of the linear algebra and inverse-problem material is presented at a level of detail that may be excessive for the target audience. Since concepts such as Jacobian-based sensitivity, covariance propagation, conditioning, and convergence diagnostics are standard in numerical linear algebra and inverse-problem theory, this section could be shortened, reorganized, or moved to an appendix, if permitted by the journal, to improve readability.

Fig. 1 nicely summarizes the workflow.

1. Select an initial t1, t2 pair within a constrained time window
2. Apply smoothing to the upwelling and downwelling irradiances
3. Compute blackbody-equivalent temperatures Tbb at t1 and t2, then compute Tbb_avg
4. Decision: accept pair only if quasi-steady condition is met; otherwise reject or adjust
5. Compute first-guess / linearized emissivity ε0
6. Run Newton-Raphson (NR) retrieval with Iter_max=8
7. Apply NR diagnostics: residual, Jacobian strength, observed convergence order.
8. Compute the emissivity-induced temperature correction ΔT
9. Compute corrected surface temperature: Ts = Tbb_avg - ΔT
10. Propagate uncertainty and decompose contributions
11. Validate against in situ/reference temperature

165: Either write Newton-Raphson method (traditional/clear) or Newton method (modern optimization terminology). Avoid switching between them unless there is a reason.

3.2.2. Determination of the initial (linearized) emissivity estimate

215: optimizes numerical conditioning -> improves numerical conditioning.

Per description, the filter does not directly optimize the conditioning (i.e., manipulate the Jacobian). Rather, it screens pairs likely to yield poorly conditioned retrievals.

219: Consider rephrasing

 “well-posed data pairs” -> “data-pairs that render the inverse problem well posed”

3.3. Newton diagnostics for emissivity inversion

223: “local curvature of the inversion” -> “local sensitivity”

265: Please verify. “the local curvature of the objective function”. I think it should be “the local rate of convergence of the iteration”.

The diagnostic Pk is estimated from successive Newton-Raphson updates and does not directly quantify curvature, which would require second derivatives. This does not affect the workflow, but the current wording may be misleading

3.3.4. Number of Newton iterations

Eqn 20: replace curvature with sensitivity/conditioning (as in 3.3)

Also consider: A Jacobian can be rank-deficient, weak, or nearly singular, but “flat” is not standard terminology for a Jacobian.

3.6. Number of Newton iterations

573: “high-frequency:” unclear. Did you mean ‘selected pairs within a short interval’ or ‘temporally close pairs selected from high-frequency irradiance measurements’?

4. Validation Framework

Concerns/Clarifications

• The reference surface temperatures are useful, but not fully independent because they are computed using SURFRAD upwelling longwave irradiance, which is also used in the proposed retrieval. Is this better described as benchmarking rather than independent validation? Although this is discussed later around lines 876-882, the phrase “independent in-situ measurements” should be clarified when first introduced. If the reference remains derived from SURFRAD irradiance combined with MODIS-based broadband emissivity, it is not fully independent.
• The authors state: “A total of thirty-nine data files acquired across different years, seasons, and atmospheric states were analyzed.” What is the effective validation sample size after temporal pairing, quasi-steady filtering, and Newton diagnostic screening? This is unclear from the current write-up.
• Section 4.2 employs MODIS-derived emissivities to construct the reference surface temperatures used for validation. Although these emissivities are not used within the retrieval itself, the resulting validation benchmark is subject to many of the same limitations discussed in Section 2. Please clarify the assumptions in the narrowband-to-broadband emissivity conversion and quantify, or at least discuss, the uncertainty it may introduce.

5. Discussions

636: The manuscript refers to validation against “independent in-situ measurements.” Please clarify what constitutes the independent ref. (same as the comment in Section 4.).

645-648: Please see previous comment on effective validation sample size. The logic in Fig. 2 is sound, but the interpretation depends on the effective sample size. It shows that for the analyzed retrievals, residuals remain small over the observed range of J values. I am unsure about the broader conclusion.

Fig 2/Table 1: These results show near-linear convergence for all retained cases, which is encouraging. However, the diagnostic does not appear to distinguish well-conditioned from weakly conditioned retrievals. Were all retained cases effectively well-conditioned, or were weakly conditioned cases screened out before this analysis?

Fig 5: The data do not show an obvious dependence of emissivity on temporal separation. However, the distribution is heavily concentrated at very short separations (<2 min), with relatively few observations at longer intervals. It is therefore difficult to conclude that adaptive temporal pairing is a “neutral conditioning mechanism” solely from this plot. The conclusion should be tempered or supported with a larger, more evenly distributed sample.

Fig 6: Please clarify how many candidate and retained retrievals were analyzed and whether the reported stability holds outside the screened cases.

In this section, many conclusions rely on the retained retrieval sample. It would be helpful if the manuscript reported candidate pairs, rejected pairs, retained pairs, and successful Newton-Raphson retrievals for each site/file before making broad claims about robustness.

735: local curvature -> local sensitivity? (unless using 2nd derivative)

5.3.1 Contrast and SNR

Interesting and physically plausible. Please clarify whether these interpretations are supported by the present dataset or represent expected behavior of the retrieval framework more generally. Overall, this is a useful discussion.