| 08 Sep 2026
The Lozano Closure: A TOA-Normalised Spectral Mapping Between Broadband Shortwave Radiation and PAR
Abstract. Accurate reconstruction of photosynthetically active radiation (PAR) from broadband shortwave (SW) measurements remains central to biosphere–atmosphere studies and ecosystem modelling. Most existing approaches rely on empirical regressions, machine-learning techniques, or site-specific calibration, often treating global and diffuse PAR separately. Here, we introduce the Lozano Closure, a physically constrained transmissivity framework that links broadband shortwave radiation and PAR through a top-of-atmosphere (TOA)-normalised spectral mapping.
The formulation expresses global PAR transmissivity as a power-law function of broadband transmissivity via a spectral exponent α, and diffuse PAR through an additional partition exponent β. Both exponents are derived from log–log transmissivity relations rather than fitted directly to PAR. Across multiple climatic regimes, the non-trivial solution yields pooled r² ≈ 0.98–0.99 for global PAR and 0.92–0.98 for diffuse PAR without site-specific calibration or auxiliary meteorological inputs.
The exponent α exhibits quasi-stationary behaviour consistent with its interpretation as a band-integrated optical-depth ratio, whereas β reflects the greater regime sensitivity of diffuse partitioning. Sensitivity analysis demonstrates that α depends structurally on the adopted extraterrestrial spectral normalisation, highlighting the importance of coherent TOA definitions for universal implementation.
The Lozano Closure provides a minimal, spectrally consistent alternative to empirical and machine-learning approaches, achieving competitive performance while maintaining cross-site transferability and physical interpretability. The framework is extensible to other spectral bands and offers a pathway toward spectrally coherent sub-band reconstruction within broadband radiation models.
The Lozano Closure: A TOA-Normalised Spectral Mapping Between Broadband Shortwave Radiation and PAR by Ismael López Lozano, Ekaterina Ezhova, Inamculada Foyo-Moreno, Inmaculada Alados, Markku Kulmala
General comments: The authors present an empirical method of correlating photosynthetically available radiation (PAR) with broadband downwelling solar irradiance, which is more frequently available, for both the global (direct and diffuse) and diffuse components. This approach is relatively straightforward and is based on the assumption of an exponential dependency between spectral irradiance and atmospheric optical depth. If I understand correctly, this is what the authors refer to as a 'physical' approach. Using a set of observation sites that measure both PAR and broadband irradiance, the authors demonstrate that a site-average correction factor for global and diffuse PAR is highly effective across all sites. This certainly deserves publication. However, I find the terminology somewhat problematic. The argument is sometimes rather lengthy, and some statements seem illogical and tautological. There are also far too many references, sometimes as many as three in every sentence. While I find the topic and findings interesting, I would urge you to substantially simplify and shorten the text. I find it irritating when people name a procedure after themselves ('Lozano Closure'). I do not see how the comparison of observed and parameterised PAR constitutes a closure. For a closure, you measure A, calculate B(A) and compare it with a measured B. Some terms are completely incomprehensible to me. For example, "structured and bounded mapping", "monotonic collapse", "secondary modulation", "dispersion of residuals", "heteroscedasticity", "diffuse fractions" and "secondary geometric modulation". I apologise if my review seems rather harsh. Despite my background being in cloud radiative transfer, I had real problems following some of the arguments. I hope that the criticisms below help to improve the readability.
Specific comments:
lines 41-42: "Within this context, the PAR spectral interval is particularly sensitive to changes in aerosol optical properties (Lozano et al., 2021).": The PAR interval is by definition a constant spectral interval. Do you mean PAR itself?
43: "and particularly of its diffuse component -": why the diffuse component? I would think that the global PAR is most relevant for photosynthesis.
67: "preserving transmissivity structure": sorry, this is not understandable
103-104: "geometrically normalised surface irradiance": What is this? Normalized by solar zenith angle?
110: " behave as low-dimensional functions": if it depends on transmission only, it is a one-dimensional function...
112-113: " structured and bounded mappings": please define what you mean by mapping and how it is structured and bounded!
118: " monotonic collapse": What is this?
118-119: "atmospheric attenuation behaves predominantly as a transmissivity-driven process.": attenuation and transmissivity is essentially the same thing.
271: " Where kt,B is the clearness index in band B": What is the difference between a clearness index and transmissivity?
287: " This provides the physical basis for introducing an effective mapping between SW and PAR transmittances." no, in my eyes this is the reason why a correlation between PAR and SW is difficult to obtain without additional information like water vapor path.
288: " closure assumption that their effective optical depths are proportional": Again, I don't understand why this is justified. Is the molecular absorption in SW similar to that in PAR, but stronger?
293-294: " The exponent ? therefore represents the ratio between effective broadband optical depths in the PAR and SW spectral ranges".: Well, yes, this is what Eq. (10) says, but it's only an assumption. It's circular reasoning.
300: "Please": Typo?
305: " This formulation preserves the physical bounds 0 < kt,SW < 1, ensures monotonicity, and maintains scale invariance in transmissivity space. It constitutes the minimal non-trivial solution consistent with the multiplicative structure of atmospheric attenuation.": Sorry, this sounds like a profound mathematical conclusion, but it's actually just a trivial consequence of Beer's law, i.e. the use of exponential functional behaviour.
310: "where kd,PAR is the diffuse fraction in the PAR spectral range.": doubling, same statement as in first part if the sentence
313: "Eq. (15)": But this is another assumption and does not follow from physics.
328: • "Secondary modulation of LC" What is a "secondary modulation"?
336-337: " These metrics quantify, respectively, systematic bias, overall dispersion of residuals, normalised error magnitude, and explained variance." Totally incomprehensive!
352: " and physically constrained (non-negative, TOA-limited)": I don't see the physics here. What do you mean by "TOA-limited"? That it cannot be larger than I_0?
358-360: " it indicates a systematic spectral mismatch between broadband and PAR attenuation arising from the trivial solution, where identical effective attenuation is assumed for both bands despite the wavelength- dependent effects of clouds, aerosols, and atmospheric gases.": this has been stated on line 354, already...
365-366: " that this pattern is robust across ecosystems and climates": what exactly do you mean by "robust", same linear bevaviour and same bias? Please specify!
403-404: " This behaviour is consistent with a spectral mapping effect: for a given SW transmissivity, the effective PAR transmissivity is not identical, and the mismatch projects as a multiplicative scaling error.": this is not an explanation but just says that a bias is a bias.
404: " heteroscedasticity": what is this? Please explain. You did explain basic statistical properties in table 2 but not this much more complex entity...
414-415: " This reveals that the relationship between diffuse fraction in broadband SW and in the PAR band is not identity-like; rather, it behaves as a systematic mapping between spectral regimes.": again, this sentence just states that a bias is a bias.
416: " The tendency toward smaller residuals at high diffuse fractions": this is not shown in Fig 5, where the residuals are shown vs transmissivity. Or what do you mean by "diffuse fraction"?
436: " exhibits a slightly lower r2 value (0.983 vs 0.984...": These numbers are statistically identical.
437: " the deviation between the fitted line and the 1:1 reference is reduced, and the slope approaches unity, indicating a reduction in systematic bias.": no, not with nearly identical r^2.
452-454: " indicating that the systematic bias observed in the trivial solution is largely removed. The linear alignment between measured and modelled diffuse PAR is preserved across the full range of conditions.": I find it interesting that a mean (pooled) diffuse PAR fitting works for all sites. This requires further explanation. Are cloud occurrence and water vapour profiles more or less the same for all sites?
460:. "6.4 Secondary geometric modulation": you mean "solar zenith angle dependent residulas"? I don't understand what the "Secondary" means in the subtitle?
How are equations (17) and (18) motivated?
488: "Figure 7a presents measured versus modelled global PAR using the geometrically modulated exponent ?(m). ": but which epsilon did you assume here?
516-517: " The spectral exponent ? derived in Section 6 can be interpreted as an effective ratio of band-integrated optical depths, linking SW transmissivity to PAR transmissivity.": no, not "interpreted". It is exactly defined this way!
518: " (Chandrasekhar, 2013": This book is from 1960.
519-520: "differences in extinction efficiency between the PAR range (400 - 700 nm) and the broader shortwave spectrum give rise to a systematic scaling relationship between their corresponding transmissivities.": Again, why should differences be used to explain a correlation? I don't understand the logic of this statement.
529-530: " Although scattering is more efficient at shorter wavelengths, its spectral dependence differs from that of gaseous absorption, leading to a distinct weighting of extinction across the solar spectrum.": I cannot follow this line of reasoning. What does scattering have to do with gaseous absorption?
530-531: As a result, SW transmissivity integrates absorption features not present within the PAR band,": You just wrote this in slightly different words in the sentence above.
574-579: " However, explicitly parameterising ? as a multivariate function would introduce additional predictors that reduce the universality and transferability of the closure. The guiding principle of the present formulation is to derive a spectrally consistent, TOA-normalised mapping that operates without site-specific calibration or auxiliary atmospheric inputs. Under this constraint, a stationary cross-site estimate of ? captures the dominant average spectral-partition behaviour while preserving structural simplicity.": but this is just an assumption or a hope and not justified by physics or observation.
580: " that this reduced formulation substantially removes the structural offset present in the trivial solution": ???
583-584: " conditionally when additional atmospheric state variables are available. Thus, ? should be interpreted as a band-effective diffuse partition descriptor under the universality constraint adopted in this work.": This simply states that if beta varies locally, additional predictors are needed to explain these variations. This is a trivial statement.
587: Please explain " kt-based".
592: "empirical relationships": to which parameter?
642: "produced a systematic shift in model behaviour when ? was held fixed": well, yes, because I_0 has changed. This is trivial.
645-646: " These results indicate that ? encodes the spectral mapping between broadband and PAR transmittances conditional on the adopted extraterrestrial spectral reference": this is just a very complicated way of stating that X*exp(alpha*tau) depends on X.
647: " This finding highlights the importance of coherent extraterrestrial normalisation for universal application of transmissivity-based models. Future.": To me, that's not a finding, but rather an assumption that applications are universal.