the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
| 20 Nov 2024
Data-Informed Inversion Model (DIIM): a framework to retrieve marine optical constituents in the BOUSSOLE site using a three-stream irradiance model
Abstract. Within the New Copernicus Capability for Trophic Ocean Networks (NECCTON) project, we aim to improve the current data assimilation system by developing a method for accurately estimating marine optical constituents from satellite-derived Remote Sensing Reflectance. We developed and compared two frameworks by implicitly inverting a semi-analytical expression derived from the classical Radiative Transfer Equation. First, we used a Bayesian estimation, which provided retrievals of the optical constituents along with their uncertainties. Moreover, using historical in-situ measurements together with a Markov Chain Monte Carlo (MCMC) algorithm to adjust the model parameters, we were able to reduce the root mean square Error (RMSE) between the retrieved data and in-situ observations. Second, we employed the Stochastic Gradient Variational Bayes (SGVB) framework to efficiently approximate the Maximum Posterior (MAP) estimates of the optical constituents while simultaneously finding the Maximum Likelihood Estimate (MLE) of the model parameters. This approach resulted in faster computations of the optical constituents compared to Bayesian estimations, with equivalent RMSE values between the retrieved data and in-situ observations. We showed that both, the MCMC and SGVB based algorithms, were able to find sets of optimal parameters, which, due to correlations between them, are not unique. We conclude that both methods are consistent with the Radiative Transfer Equation. The first method provides reliable uncertainty estimations, while the second offers a faster alternative to standard inversion techniques, making it suitable for inversion and model optimization problems where MCMC algorithms are intractable.
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CEC1: 'Comment on gmd-2024-174 - No compliance with the policy of the journal', Juan Antonio Añel, 26 Dec 2024
Dear authors,
Unfortunately, after checking your manuscript, it has come to our attention that it does not comply with our "Code and Data Policy".
https://www.geoscientific-model-development.net/policies/code_and_data_policy.htmlin your Code and Data Availability section you provide a handful of links for sites containing information related to your manuscript. However, such sites are not acceptable for scientific publication. They include a GitHub site and Google drives. You have included the DOI for a Zenodo repository containing part of the information, after an initial request from the Topical Editor; however, the Zenodo repository that you provide does not contain the necessary information to replicate the work that you present in your manuscript. For example, when trying to run the Python notebook on model availability, it tries to connect to the external GitHub site (moreover under certain circumstances/configurations it fails to do it).
I have to be clear here. All the models, code and data necessary to replicate your work must be contained in the Zenodo repository and must be standalone, that is, work without need for additional download of software or data from third party sites (excluding common libraries such as pandas, numpy, etc. that should be listed with their versions). Also, the Code and Data Availability section in your manuscript contains several references to sites that do not comply with our policy. This section is not there to advertise the "last version" of a software or the work of the authors, but to provide the specific information that assures the compliance with the principle of scientific reproducibility. Including such unuseful information only introduces unnecessary noise and complexity in the assessment of the compliance of your submitted manuscript. Therefore, please, remove from this section all the information about sites that do no comply with the policy for permanent archival of code and data.
Therefore, the current situation with your manuscript is irregular. Please, publish your code in one of the appropriate repositories and reply to this comment with the relevant information (link and a permanent identifier for it (e.g. DOI)) as soon as possible, as we can not accept manuscripts in Discussions that do not comply with our policy. Also, please include the relevant primary input/output data. Also, you must include a modified 'Code and Data Availability' section in a potentially reviewed manuscript, containing the DOI of the new repositories.
I note that if you do not fix this problem, we will have to reject your manuscript for publication in our journal.
Juan A. Añel
Geosci. Model Dev. Executive EditorCitation: https://doi.org/10.5194/gmd-2024-174-CEC1 -
AC1: 'Reply on CEC1', Carlos Enmanuel Soto Lopez, 08 Jan 2025
Dear Prof. Juan A. Añel
Geosci. Model Dev. Executive Editor
After careful reviewing of the code and data availability, we have uploaded in Zenodo a standalone version of the code used to reproduce the results from the article, as well as the data needed to reproduce it. In this new upload, we make no mention of other sites, except the standard python3 libraries:
- networkx>=2.8.8
- torch
- torchvision
- torchaudio
- ConfigSpace>=1.1.1
- constrained-linear-regression>=0.0.4
- matplotlib>=3.7
- matplotlib-inline>=0.1.7
- mdurl>=0.1.2
- multiprocess>=0.70.16
- numpy>=1.24
- pandas>=2
- ray>=2.1
- scipy>=1.1
- seaborn>=0.13.2
- sympy>=1.12
- pvlib>=0.11.0
- tqdm>=4.66.5
- pyarrow
- tensorboardX
We have also prepared a README with instructions on how to install the required external packages with a Makefile. We also included a Python script (reproduce.py). By launching reproduce.py, the user is guided to reproduce the calculations and figures in the manuscript, with the option to run all calculations (-a), only the Bayesian computations (-b), only the sensitivity analysis and MCMC analysis (-m), or only the calculations related to training the neural network (-n).
I'll send a version of the manuscript with a different 'Code and Data Availability' section, as well as the DOI identifier updated in the Citations section via email.
DOI from the new code and data availability: 10.5281/zenodo.14609747.
Link to the code and data availability: https://doi.org/10.5281/zenodo.14609747
Best regards,
Carlos Enmanuel Soto Lopez
Citation: https://doi.org/10.5194/gmd-2024-174-AC1
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AC1: 'Reply on CEC1', Carlos Enmanuel Soto Lopez, 08 Jan 2025
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RC1: 'Comment on gmd-2024-174', Anonymous Referee #1, 15 Apr 2025
This paper presents a new inversion setup for recovering marine optical constituents (chl_a, non-algal particles and CDOM) and optimal parameters for a three-stream marine radiative transfer model using satellite reflectance measurements and in situ data from the BOUSSOLE buoy located in the oligotrophic sector of the Ligurian Sea (western Mediterranean). Two inversion frameworks, one based on a Bayesian estimation approach including an MCMC algorithm and the other based on a neural network inversion method, are developed and tested using time series of observations collected during 2005-2012. The results of the demonstration illustrate how the methods work. The advantages and limitations of both inversion methods are briefly assessed and discussed.
The inversion of sea surface remote sensing reflectance data is a difficult and important subject that has been investigated for years, and the work presented here is one more step towards the direct assimilation of satellite reflectances into marine biogeochemistry models.
The article provides details about the two proposed algorithms, which are original and potentially useful to other users, but which may question about the robustness of the methods given the many ad hoc algorithmic adjustments needed to ensure convergence. This may also question reproductability of the inversions in different conditions (different sites, data sets etc.). Sections 4 and 5 in particular are difficult to understand and require significant clarifications before they can be published. The benefits of the method should be better discussed and argued in relation to the methods currently used, and in particular with respect to the Copernicus marine products available today. The consistency of the parameter estimation results is also questioned (see section 5 comments).
A major revision is necessary before considering publication in GMD. Below are mentioned a list of major recommendations for the various sections of the paper, as well as more specific but important comments and questions to be addressed to improve understanding of the paper. On top of this revision, some careful check of the language will be needed to avoid approximate wording as found in the original manuscript.
Major comments
- Section 1. The authors refer to inversions of semi-analytical expressions of the RTE (l33-l34). As the term is mentioned in the abstract but not used afterwards, the reader could have doubts as to whether this is actually the case in the paper. If the complete RTE model (i.e. discretized on the vertical) is not used afterwards, this should be explained in the introduction. Similarly, it should also be stated from the begining if the aim of the paper is to estimate IOPs at the sea surface only (assuming vertical homogeneity of marine optical components), and not throughout the water column.
- Section 2. I suggest to move this section after section 3 and explain more clearly how the different data are related to the RTE model variables, surface boundary conditions etc. In addition I would suggest introducing a schematic diagram of the water column including the different fluxes, irradiances and variables of interest, data collected etc. to improve readability.
- Section 3. It looks that the marine optical constituants are assumed to be constant on the vertical. It should be clarified if vertical homogeneity is assumed as this is far from a trivial assumption. It could significantly limit the applicability of the framework to locations where such an assumption is not verified, e.g. when shalow sub-surface chlorophyll maximum concentration occur. A discussion should be added explaining how the scheme could be extended for inhomogeneous water columns. As for vertical homogeneity of optical constituents, it seems that temporal persistence is assumed over the daily cycle. Please confirm and provide some justification.
- Section 4, and section 4.4.2 in particular, is very difficult to understand and needs to be clarified. More generally, the sequence of algorithmic operations in the Bayesian inversion framework and associated partitioning of the training data set need to be better explained, so that the reader understands the order in which the latent variables and the model parameters to be optimized are calculated. This clarification is necessary to fully understand the applicability of the method and the results in section 5. The algorithmic complexity of the sequence also needs to quantified and explained more transparently. The list of optimized paramaters is confusing in places. Please clarify the actual list of optimized parameters referring to the list displayed in Table 1 and 2.
- As presented, the overall convergence of the method does not seem very robust and is subject to numerous adjustments depending on the dataset and DYFAMED site considered. Section 4.4.2: The sentence “we choose to perturb our parameters in such a way that we end up with a fourteen parameter space” is not understandable. What does the hyper-parameter alpha q (line 283) represent? The parameter perturbation method needs to be better explained (also related to Table 5). The last sentence of section 4.4 “... and sampled only uncorrelated values” is rather cryptic.
- There are no objective elements in the paper to guide the choice between the MCMC vs. SGVB methods. The GMD framework calls for guidelines to direct the user towards the most suitable method (Bayesian approach or neural network) instead of experimenting (as seems to be proposed) until a satisfactory converged solution is found.
- Section 5. The presentation of the results (section 5) is confusing. It is not clear what is the purpose of the sensitivity study introduced at the beginning of the section, since its main conclusion (the use of a single set of parameters over one year is sub-optimal) is not used thereafter anyway. This part should be removed, or better justified. What means the new bphy,Int parameter introduced here (Figure 3 and Table 5) ? A new structuring of this section into three parts (results of the MCMC method, results of the SGVB method, comparison of the 2 methods) should be considered, including a more in-depth evaluation and interpretation of the results. For example, the reason why the IOP values found with SGVB differ very slightly from the original values compared with MCMC requires substantiated explanations.
- Figure D1 does not include a comparison with the chl_a concentration estimated using conventional ocean colour (OC) algorithms available in the CMEMS catalogue. It is strongly recommended to add these figures in the plot and provide comments about their consistencies, as one of the potentially key advantages of the proposed method is to be used in operational setups.
- Section 6. The question of the reproductibility of the inversions in other sites needs to be addressed explicitly in a more convincing manner. The discussion refers to some generic aspects (e.g. use of neural networks in earth sciences) but does not address how useful the proposed tools will be for other sites.
- One key missing item is a discussion of the interest of the proposed approach with restect to existing ones (e.g. in the Copernicus framework). How accurate/comparable are the results of the inversion (e.g. D1a, others ?) compared to standard Copernicus products ? Please provide some quantitative comparisons to assess this aspect.
Specific comments and questions
- Abstract, « we conclude that both methods are consistent with the Radiative Transfer Equation » : what is really meant by this sentence ?
- MCMC = Markov Chain Monte Carlo (abstract, ) or Markov State Monte Carlo (section 4) : please unify the nomenclature or explain nuances
- Line 43-l44 and elsewhere : replace « density » of optical constituents by « concentration » ?
- Line 55 : It allows for …
- Line 85 « After filtering … » : please revise and clarify this sentence
- Line 90 : below instead of above ?
- Line 93-94 : what is the « heigh vertical variability » ? Please revise the sentence and better explain the rationale behind the choice of measurements at a depth of 9m
- Equation (1) : d/dh missing in second and third equations
- Section 3.1 : an equation is missing to describe how PAR is related to the direct, scattered and upward irradiances.
- Section 4.2. It is not clear if the process to adjust the alpha hyperparameter defining the model error is dependent on the accuracy of the surface boundary conditions (OASIM). Please provide comments and clarification.
- Line 180 : x(lambda) is a 5-component vector (including Eu) while it is a 4 component vector in eq(11). Please explain.
- Line 217: I don’t understand why in situ observations are available for 3 wavelengths only. This is not consistent with what is said in section 2.3. Please update section 2.3 accordingly.
- Line 238: his -> the
- Line 256: please explain the “standard error propagation scheme” used to compute the uncertainties.
- Figure 1: the Estimate of the optimal parameters (“Lambda tilde”) is not shown in the diagram.
- Line 346: inpute -> input.
- Table 4: why only 9 parameters are shown here ?
- End of Appendix A: “For completeness, … has to be exchanged …”
- Line 518: “… these two errors”.
- Figure D1. Why 2012 time series ? do you mean 2005-2013 time series ?
Citation: https://doi.org/10.5194/gmd-2024-174-RC1 -
AC2: 'Reply on RC1', Carlos Enmanuel Soto Lopez, 29 May 2025
Dear Reviewer,
Please find attached a point-by-point response to your comments, together with a description of
the changes made to the original manuscript (DIIM_REPLY_REW.pdf).We sent to the editors a revised version of the last update to the manuscript entitled “Data-Informed Inversion Model
(DIIM): a framework to retrieve marine optical constituents in the BOUSSOLE site using a three-stream
irradiance model ”, with the most important changes highlighted in blue, hopping for it to be available soon.Finally, we also write here the point by point reply (same as reply found in the pdf uploaded).
Yours sincerely,
Carlos Enmanuel Soto LopezPoint by point reply:
- Reviewer comment: Section 1. The authors refer to inversions of semi-analytical expressions
of the RTE (l33-l34). As the term is mentioned in the abstract but not used afterwards, the reader
could have doubts as to whether this is actually the case in the paper. If the complete RTE model
(i.e. discretized on the vertical) is not used afterwards, this should be explained in the introduction.
Similarly, it should also be stated from the beginning if the aim of the paper is to estimate IOPs at the
sea surface only (assuming vertical homogeneity of marine optical components), and not throughout
the water column.
Author reply: We have added information in the Introduction about the type of model used. The
model is based on an expression derived from the Radiative Transfer Equation (RTE), with some
terms—such as the chlorophyll-to-carbon ratio (see manuscript, Eq. 3)—estimated empirically. This
makes it a semi-analytical expression. We have also clarified that the inversion is intended solely to
approximate the inherent optical properties (IOPs) at the sea surface. - Reviewer comment: Section 2. I suggest moving this section after section 3 and explain more
clearly how the different data are related to the RTE model variables, surface boundary conditions,
etc. In addition, I would suggest introducing a schematic diagram of the water column, including the
different fluxes, irradiances, and variables of interest, data collected, etc, to improve readability.
Author reply: As suggested, we have moved this section to follow the model description and
expanded on the relationship between the OASIM data and the boundary conditions. We also
provided more detail on the origin of the data, noting that it was previously used to assess the validity
of the OASIM model in the Mediterranean Sea [4]. Finally, we also included a diagram describing the
different fluxes involved in the RTE. - Reviewer comment: Section 3. It looks that the marine optical constituents are assumed to be
constant on the vertical. It should be clarified if vertical homogeneity is assumed, as this is far from a
trivial assumption. It could significantly limit the applicability of the framework to locations where
such an assumption is not verified, e.g. when shallow sub-surface chlorophyll maximum concentration
occurs. A discussion should be added explaining how the scheme could be extended for inhomogeneous
water columns. As for vertical homogeneity of optical constituents, it seems that temporal persistence
is assumed over the daily cycle. Please confirm and provide some justification.
Author reply: The equations assume a homogeneous, infinitely deep layer. By making this
assumption, the system of equations becomes linear and can be solved analytically. The assumption
is justified for deep case 1 waters (dominated by phytoplankton), like the one studied in the present
work. During winter, the chlorophyll concentration in the first layer is approximately constant due to
mixing (see [5], Fig. 1), while most of the downward irradiance comes from the first 10 to 20 meters
(see [6], Fig. 1 and Fig. 2). During summer, there is no mixing, but still there is a region, around 20
to 50 meters, with constant chlorophyll concentrations, making the assumption justified. For coastal
areas, we are considering extending the model to include the process of light reflection at the sea floor,
which will be the matter for future work. Concerning the temporal persistence, we are computing
daily averages of the different concentrations (averaging over hours with light), making it coherent
with satellite data retrievals. We added all this information to the modified manuscript. - Reviewer comment: Section 4. Section 4.4.2 in particular, is very difficult to understand and
needs to be clarified. More generally, the sequence of algorithmic operations in the Bayesian inversion
framework and associated partitioning of the training data set need to be better explained, so that the
2/8 reader understands the order in which the latent variables and the model parameters to be optimised
are calculated. This clarification is necessary to fully understand the applicability of the method and
the results in section 5. The algorithmic complexity of the sequence also needs to be quantified and
explained more transparently. The list of optimized parameters is confusing in places. Please clarify
the actual list of optimised parameters, referring to the list displayed in Table 1 and 2.
Author reply: We substantially re-wrote and re-organized section 4 to make the explanation of how
the algorithm works clearer, including tables of step by step explanations for the Bayesian inversion
and Monte Carlo algorithms (Alg. 1, Alg. 2, and Alg. 3). Concerning the algorithm complexity, the
aim is to understand how the computational time increases as a function of the dimensionality of the
problem. The general consensus is that under certain conditions, the computational time increases
polynomially, but in general, the increase could be exponential [1]. Indeed, the complexity estimation
for this kind of algorithms is an active area of research, and could be in itself the subject of another
work.
Reviewer comment: As presented, the overall convergence of the method does not seem very robust
and is subject to numerous adjustments depending on the dataset and DYFAMED site considered.
Section 4.4.2: The sentence “we choose to perturb our parameters in such a way that we end up with
a fourteen-parameter space” is not understandable. What does the hyperparameter alpha q (line 283)
represent? The parameter perturbation method needs to be better explained (also related to Table 5).
The last sentence of section 4.4 “... and sampled only uncorrelated values,” is rather cryptic.
Author reply: Motivated by the reviewer’s comment on the non-robustness of the method, weanalyzed the two proposed algorithms separately, looking for the underlying reasons explaining their
different estimates.
The mean value of the retrieved optical constituents for both methods where consistent with each
other, in the sense that the Variational Bayes method gives values within the uncertainty of the
Bayesian approach. The robustness of the method is confirmed by the statistics on the test data, which
were not used for hyperparameter tuning or neural network training, as both approaches produce
values close to the observations. Additional evidence of robustness comes from the comparison with a
state-of-the-art algorithm for chlorophyll estimation from satellite observations, presented at the end
of the Results section.
On the other hand, the forward model optimization presented differences between the MCMC mean
values and those obtained with the SGVB estimator. Upon analyzing the results, we identified
what we believe to be the main reasons behind these discrepancies. On one hand, the loss functions
used in the two methods are different due to the regularization term used for the neural network
training. On the other hand, the algorithm used to train the neural network relies on standardized
data and minibatch minimization, two common practices in neural network training that help improve
generalization and prevent overfitting by approximating the full dataset gradient with that of smaller
subsets. For this resubmission, we therefore also applied the standardization to the data used in the
MCMC algorithm to ensure a fairer comparison. This change made the final parameters estimated
by both methods closer to each other with respect to the previous experiments, in a way that 22 of
the 24 perturbed parameters were within the uncertainty of the MCMC estimates; Nonetheless, the
parameters obtained through simultaneous training of the neural network using the SGVB estimator
demonstrated better generalization properties.
In the author’s view, the fact that both methods give slightly different results in finding the optimal
parameters is not an indicator of the incompatibility or non-robustness of the methods. Indeed,
MCMC estimates the probability density of the parameters given the training data, i.e. the posterior.
This distribution reflects the region of parameter space where the optimal parameters are likely to
lie, while its mean provides a point estimate of those parameters. Instead, the SGVB estimator
exploits minibatch minimization and the training of a Neural Network to estimate the MLE of the
parameters, a different estimator with slightly different assumptions (we never linearized the Forward
function, for example). Both algorithms succeed in finding a satisfactory estimator of the optimal
parameters, minimizing the RMSE on the test data. While MCMC methods are more commonly
used, our goal was to demonstrate the validity and efficiency of the SGVB estimator in marine
inversion problems. - Reviewer comment: There are no objective elements in the paper to guide the choice between the
MCMC vs. SGVB methods. The GMD framework calls for guidelines to direct the user towards the
most suitable method (Bayesian approach or neural network) instead of experimenting (as seems to be
proposed) until a satisfactory converged solution is found.
Author reply: We added a few comments in the Conclusion section detailing about the advantages
and disadvantages of both methods. In summary, we recommend the MCMC and traditional Bayesian
approaches for their theoretical simplicity and interpretability, and their ability to estimate the
uncertainty of the results. The Variational Bayes approach is more effective for intractable problems,
where the posterior is costly to compute, returning good estimates of the MLE. At the moment,
our findings are that the method doesn’t return reliable uncertainty quantification, but still offers a
good alternative for the latent variable estimation (estimation of the optical constituents), as well as
for the model optimization. In the paper, we compared the results of both methods, showing their
equivalence, since the dimensionality of the problem allows it, but in most of the state-of-the-art
forward models, the latent space, as well as the parameter space, is usually much larger, and the
standard methods are not any more computationally feasible. - Reviewer comment: Section 5. The presentation of the results (section 5) is confusing. It is
not clear what the purpose of the sensitivity study introduced at the beginning of the section is, since
its main conclusion (the use of a single set of parameters over one year is sub-optimal) is not used
thereafter anyway. This part should be removed, or better justified. What does the new bphy,Int
parameter introduced here (Figure 3 and Table 5)? A new structuring of this section into three parts
(results of the MCMC method, results of the SGVB method, comparison of the 2 methods) should be
considered, including a more in-depth evaluation and interpretation of the results. For example, the
reason why the IOP values found with SGVB differ very slightly from the original values compared
with MCMC requires substantiated explanations.
Author reply: We re-structured the results of the section into four parts: the first one focuses on
the Bayesian retrieval of the optically active constituents on the surface of the sea and the uncertainty
estimation; the second on the parameter optimization; the third on the comparison between the
Bayesian outputs and the Variational Bayes approach; the last one compares our results with a state
of the art algorithm for satellite sea surface chlorophyll a estimation. We expect this rewriting to
make the content of the section more clear.
Regarding the sensitivity study, we provided additional motivation by framing it as a measure of the
seasonal variability of the parameters. Moreover, it allowed us to analyze the differences between
the two optimization methods. In particular, it revealed that among the two parameters showing
the greatest discrepancy between methods, only one played a significant role in optimizing the
particulate backscattering coefficient. Since these measurements were the noisiest ones, we speculate
that overfitting could have misled the MCMC algorithm. - Reviewer comment: Figure D1 does not include a comparison with the chlorophyll a concentration
estimated using conventional ocean colour (OC) algorithms available in the CMEMS catalogue. It is
strongly recommended to add these figures in the plot and provide comments about their consistency,
as one of the potentially key advantages of the proposed method is to be used in operational setups.
Author reply: To compare our inversion results with a conventional ocean color algorithm for
estimating chlorophyll-a concentration, we added a dedicated subsection in the Results. In this
analysis, we performed a comparison over an extended region near the BOUSSOLE buoy, using the
MedOC4.2020 algorithm [2]. The reference dataset consists of measurements obtained via High-
Performance Liquid Chromatography (HPLC) [3]. The comparison showed the consistency between
both methods (See manuscript, Fig. 10). - Reviewer comment: Section 6. The question of the reproducibility of the inversions in other sites
needs to be addressed explicitly in a more convincing manner. The discussion refers to some generic
aspects (e.g. use of neural networks in earth sciences) but does not address how useful the proposed
tools will be for other sites.
Author reply: We plan to asses the validity of the inversion on the rest of the Mediterranean in a
future work. At the moment, we presented a validation only on a region close to the BOUSSOLE
buoy of 4 × 4 degrees, in the North West Mediterranean Sea, where conditions are similar to the
assumptions made for this work.
Reviewer comment: Specific comments and questions: Abstract, ≪ we conclude that both
methods are consistent with the Radiative Transfer Equation ≫ : what is really meant by this sentence
?
Author reply: We changed this sentence; what we meant was that the SGVB estimator was
estimating an inversion of the RTE more than just adjusting a NN to data. We agreed that our results
were not primarily focused on that aspect and therefore replaced it with the comparison between our
inversion method and the state-of-the-art algorithm for chlorophyll-a concentration, as we believe this
comparison is more relevant and deserves to be highlighted in the abstract.
Reviewer comment: MCMC = Markov Chain Monte Carlo (abstract, ) or Markov State Monte
Carlo (section 4) : please unify the nomenclature or explain nuances
Author reply: The correct name is Markov Chain Monte Carlo. - Reviewer comment: Line 43-l44 and elsewhere: replace ≪ density ≫ of optical constituents by
≪ concentration ≫ ?
Author reply: Thank you for pointing that out. - Reviewer comment: Line 55 : It allows for . . .
Author reply: Thank you for bringing that to my attention. - Reviewer comment: Line 85 ≪ After filtering . . . ≫ : please revise and clarify this sentence
Author reply: First, we removed any data coming from the buoy reporting an absolute tilt higher
or lower than 10 degrees. We also removed the data recorded at a depth more than 2 m below the
nominal values (4 m and 9 m, depending on the instrument of measurement). Also, the downward
light attenuation coefficient data were filtered with an analog high-pass filter, using the package SciPy
from the programming language Python, filtering the noise with a frequency less than 4 hours. Finally,
we proceeded to average the daily values. - Reviewer comment: Line 90 : below instead of above ?
Author reply: Thank you, we meant below. - Reviewer comment: Line 93-94 : what is the ≪ heigh vertical variability ≫ ? Please revise the
sentence and better explain the rationale behind the choice of measurements at a depth of 9m
Author reply: We made it clearer now. The measurements were taken at 9 meters deep. Because
there is low variability, the chlorophyll measurements were considered as sea surface measurements.
However, the downward light attenuation coefficient has a high variability, so we can not consider it
as a measure at the sea surface. - Reviewer comment: Equation (1) : d/dh missing in second and third equations
Author reply: I really appreciate all these comments—thank you. Yes, I’ve made the corrections. - Reviewer comment: Section 3.1 : an equation is missing to describe how PAR is related to the
direct, scattered and upward irradiances.
Author reply: Thank you for pointing this out. You are right an equation describing how PAR
relates to the direct, scattered, and upward irradiances was missing. We have now included it in
Section 3.1. - Reviewer comment: Section 4.2. It is not clear if the process to adjust the alpha hyperparameter
defining the model error is dependent on the accuracy of the surface boundary conditions (OASIM).
Please provide comments and clarification.
Author reply: Alpha is a hyperparameter. In linear regression problems, it typically represents the
inverse of a regularization term and is often set to a small value so as to not affect the maximum
likelihood estimate (MLE). In this resubmission, we discussed in the Results section about the
prior. Since it is an informative prior, without it, the estimated uncertainty increases by one order
of magnitude (see manuscript, Results section). Therefore, the choice of alpha must balance two
objectives: First, it must ensure robustness (i.e., small variations in alpha do not significantly affect
the results) and second, it has to yield realistic uncertainty estimates, meaning that the reported
uncertainty should, on average, match the observed discrepancies between model predictions and
actual data. We have revised the Appendix to clarify this point. - Reviewer comment: Line 180 : x(lambda) is a 5-component vector
(including Eu) while it is a 4 component vector in eq(11). Please explain.
Author reply: The correct number is 4 components. - Reviewer comment: Line 217: I don’t understand why in situ observations are available for 3
wavelengths only. This is not consistent with what is said in section 2.3. Please update section 2.3
accordingly.
Author reply: My apologies if it was not clear. I added a small clarification at the end of the data
acquisition section. Taking into account the assumptions and data availability, the in-situ observations
considered are sea surface chlorophyll, 9 meters deep downward light attenuation coefficient in 5
wavelengths, (412.5,442.5,490,510,555) nm, and sea surface particulate backward scattering coefficient
at 3 wavelengths (442,490,510) nm. - Reviewer comment: Line 238: his -¿ the
Author reply: This section was re-worked. - Reviewer comment: Line 256: Please explain the “standard error propagation scheme” used to
compute the uncertainties.
Author reply: I expanded on the equations. I was referring to the error propagation equations:
∆F (⃗x) 2 = ∇x F (⃗x )Σx∇xF (⃗x )T , where ∆F (⃗x ) is the error of a function F (⃗x ), ∇xF (⃗x ) is the Jacobian,
and Σx is the covariance matrix of x. In our case, Σx = Σzd∗ . These equations assume that each
component of x is not correlated with the others, and, in this respect, is only an approximation for
nonlinear functions. - Reviewer comment: Figure 1: the Estimate of the optimal parameters (“Lambda tilde”) is not
shown in the diagram.
Author reply: They are the parameters from p\hat{Λ}(y, H|x, z) which is the likelihood given the estimated
latent variable z and the boundary conditions x . In other words it is the probability of observing the
Remote Sensing Reflectance, and observations, as a function of the estimated parameters \hat{Λ} , given the
boundary conditions and the state of the estimated state of the ocean. - Reviewer comment: Table 4: why only 9 parameters are shown here ?
Author reply: We perturbed 15 lambda dependent parameters, and 9 non lambda dependent. The
table shows the 9 that were not dependent on lambda, and Fig. 10 (in the new manuscript) has
the final values for the lambda dependent ones. On the other hand, on other tables, we report the
perturbation factors δi. We rewrote the section where we explain how the parameters were perturbed.
More specifically:
The values of the λ dependent vector of dimension five representing the phytoplacton-specific absorption
coefficient aphy were perturbed as: a∗phy = δaphy aphy0 with δaphy a learnable scalar, and aphy0 the
literature values. We chose it like this to maintain the shape of the function aphy(λ) unperturbed.
For the carbon-specific scattering and backscattering coefficients bphy(λ) and bb,phy(λ), we first linearly
interpolated them with the literature values, and perturbed the tangent and the intercept of the linear
interpolations, bphy(λ)∗ = δbphy,int b0phy,int + δbphy,T b0phy,Tλ.
The parameters dCDOM, br,NAP, SCDOM, Θminchla, Θ0chla, β, σ, Qa and Qb perturbations consisted in per
parameter scalar multiplications. All the other parameters were left unperturbed.
In this way, we perturbed 24 parameters, 9 of them by multiplying them for a scalar δi, i equal to
each of the perturbed parameters, the five components of aphy by multiplying them by the same scalar
δaphy , and finally, bphy(λ) and bb,phy(λ) by linearly interpolating them, and perturbing the tangent
and the intercept of each of them, making a total of 14 perturbation factors. - Reviewer comment: End of Appendix A: “For completeness, . . . has to be exchanged . . . ”
Author reply: We agree, thank you for pointing it out. - Reviewer comment: Line 518: “. . . these two errors”.
Author reply: We made some adjustments to Appendix B. - Reviewer comment: Figure D1. Why 2012 time series? Do you mean 2005-2013 time series ?
Author reply: Thank you for pointing it out, we fixed it as 2005-2013 time series.
References
[1] Alexandre Belloni and Victor Chernozhukov. “On the computational complexity of MCMC-based
estimators in large samples”. In: (2009).
[2] S Colella et al. EU Copernicus Marine Service Quality Information Document for the Ocean Colour
Mediterranean and Black Sea Observation Product, OCEANCOLOURM EDB GCL3N RT009143, Issue4.1, M erc
doi: https://doi.org/10.48670/moi-00299, (Accessed on 05-23-2025).
[3] V Di Biagio, S Campanella, and G Cossarini. In situ dataset for initialization and validation of the
Copernicus Med-MFC biogeochemical model system (MedBGCins). doi: https://doi.org/10.5281/zenodo.1548996
[4] Paolo Lazzari et al. “Assessment of the spectral downward irradiance at the surface of the
Mediterranean Sea using the OASIM ocean-atmosphere radiative model”. In: Ocean Science
Discussions 2020 (2020), pp. 1–39.
[5] A. Mignot et al. “From the shape of the vertical profile of in vivo fluorescence to Chlorophyll-a
concentration”. In: Biogeosciences 8.8 (2011), pp. 2391–2406. doi: 10.5194/bg-8-2391-2011.
url: https://bg.copernicus.org/articles/8/2391/2011/.
[6] JJ Simpson and TD Dickey. “The relationship between downward irradiance and upper ocean
structure”. In: Journal of Physical Oceanography 11.3 (1981), pp. 309–323.
- Reviewer comment: Section 1. The authors refer to inversions of semi-analytical expressions
-
RC2: 'Comment on gmd-2024-174', Anonymous Referee #2, 03 Jun 2025
The manuscript is well written, the proposed methods are well explained and in general everything is clear. Although the methods employed are not new, using them is well founded for this particular application. In my opinion the manuscript is worth of be published addressing the following typos and comments.
One note before my comments. I read the comments and replies of the previous referee after I reviser the paper. I mostly agree with his/her comments, I see they were addressed, and it seems there is already a revised version of the paper, but I do not have access to it (or I don't know how to view it). In any case, since my comments are minor and do not coincide with the other referee, there you have them, but take into account that I read the preprint version named gmd-2024-174.pdf, no the new one.
- Line 89, "data filtered with an analog high pass filter, using the package SciPy". It cannot be an analog filter. Maybe the prototype is analog, but it must be a digital filter. Please explain/correct this.
- Line 130, respectably => respectively
- Line 249, in eq. 19 K is not defined.
- Line 313, DK divergence => DKL divergence
- Line 347, the DK divergence => the DKL divergence
- Line 357, Cholezky => Cholesky
- Line 371, qϕ(z,y,x) => qϕ(z|y,x)
Figures:
- Fig. 1, What is exactly "model of the measurements". Maybe you can add R_{rs}^{MODEL} and H(ˆZ,X;Λ).
- Fig. 2 caption: qϕ(z|z,y) should be qϕ(z|x,y), and "dose that learn the Cholezky" => those? that learn the Cholesky.
- Fig 5. (Related to it) Please give more explanation on: Why MCMC sub or over estimates? Why the SGVB does not have a confidence interval if the method provides it? And why do you think is the cause the SGVB uncertainty fails?
Citation: https://doi.org/10.5194/gmd-2024-174-RC2
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