the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
| 24 Mar 2025
CLEAR: a new discrete multiplicative random cascade model for disaggregating path-integrated rainfall estimates from commercial microwave links
Abstract. A novel disaggregation algorithm for commercial microwave links (CMLs), named CLEAR (CML Segments with Equal Amounts of Rain), is proposed. CLEAR utilizes a multiplicative random cascade generator to control the splitting of link segments, with the generator's standard deviation dependent on the rain rate and segment length. Spatial consistency during the splitting process is maintained using rain rate information from neighboring CMLs. CLEAR is evaluated on a network of 77 CMLs in Prague. The performance is assessed first using simulated rainfall fields and second through a case study with real attenuation data from the network to demonstrate its applicability in real-world scenarios. Results from the virtual rainfall fields indicate good overall performance, including the generation of realistic spatial patterns. CLEAR effectively estimates maximal and minimal rain rates along CML paths and outperforms a commonly used benchmark algorithm. The stochastic nature of CLEAR allows it to represent uncertainty as an ensemble of rain rate distributions along CML paths. However, the generated ensembles significantly underestimate overall variability along the paths. Additionally, the case study on real data highlights challenges associated with uncertainties in CML quantitative precipitation estimates, which are common across all methods. In conclusion, CLEAR contributes to generating more representative rainfall distributions along CMLs, which is critical for spatial reconstruction of rainfall fields from path-integrated CML data. It also has the potential to reduce errors in CML quantitative precipitation estimates caused by assuming uniform rain rates along CML paths.
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The requested preprint has a corresponding peer-reviewed final revised paper. You are encouraged to refer to the final revised version.
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Preprint
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The requested preprint has a corresponding peer-reviewed final revised paper. You are encouraged to refer to the final revised version.
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Journal article(s) based on this preprint
Interactive discussion
Status: closed
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RC1: 'Comment on egusphere-2025-487', Anonymous Referee #1, 27 Apr 2025
This paper suggests a novel approach to disaggregate path-average rain rate aver CML. The topic is relevant and interesting for the community. The paper is very “clear” ;-) to read. Indeed the methodology is presented in a straightforward way, results displayed and discussed in convincing way, and current limits are also properly addressed. In general, I think that the paper only require minor modifications before it is suitable for publication.
Minor specific comments:
- l. 97: “L1” should be in italic.
- l. 110-114: it is indeed one approach, but there are others. Did authors carried out some tests to opt for this approach ? This should be clarified.
- Section 2.6: maybe say few words on the uncertainty associated to radar QPE. Is it really a reliable reference ? (few things are said l. 345 on the topic)
- l. 176-177: could authors clarify why does the parameters of the cascade generator depend on link orientation and length ?
- l. 180: the whole point is in “realistically”… How sure are you that your rainfall simulations are “realistic” ? I also agree that the other approach also has intrinsic limitations as pointed out by the authors. Maybe carrying out the calibration with both method and highlighting the differences would be more convincing.
- Fig. 3 (right): It would be interesting to find a way to always display the differences according to rain rate.
- Section 2.8: please justify better the choice of GMZ as benchmark. It seems as a simplistic approach with regards to what is suggested in this paper, hence not surprisingly it performs worse.
- L. 245 – 248: I understand the point, but it would be interesting to show quantitatively the increase in performance in areas with denser CML concentration.
- Section 5.1: wouldn’t it be possible to use empirical curves for SD as a function of rain rate instead of a modelled one
Citation: https://doi.org/10.5194/egusphere-2025-487-RC1 - AC1: 'Reply on RC1', Martin Fencl, 10 Jul 2025
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RC2: 'Comment on egusphere-2025-487', Anonymous Referee #2, 24 May 2025
The authors present a new method to distribute mean rainfall rates from a commercial microwave link (CML) network along the CML line of sight. The underlying random cascade model considers spatial coherence within the CML network and provides uncertainty estimates. Required model parameters that determine the uniformity of the rainfall distribution along each transect are fitted to three virtually generated rainfall fields and the Prague CML network. Finally, the method is compared to the method by Goldshtein et al. (2009) for the same three virtually generated rainfall fields and radar observations of a single two-hourly rainfall event in Prague on 21 September 2014. The results show improved reconstruction capabilities by the proposed method.
While the newly developed method might be beneficial compared to existing methods, the performance evaluation presented in this work is not convincing. Below are several general and specific comments that the authors must address in a revised manuscript.
General comments
1. Sample size
This work is based on a tiny sample size of just four rainfall events: 3.5 hours of virtually generated rainfall fields and 2 hours of radar observations in Prague. This sample size does not provide a statistical basis for model evaluation and intercomparison with the Goldshtein method. I suggest using at least a full annual cycle of weather radar and CML data to capture a larger number of heavy precipitation events. This will allow for a more differentiated comparison with the Goldshtein method under different seasons and precipitation regimes. Also, it provides insights into whether the proposed method is even valid for an entire season or if the underlying parameters of the SD model need to be adjusted for convective/stratiform types.2. Virtual rainfall fields
This work mainly relies on virtual rainfall fields generated following the method by Schleiss et al. (2012). Most readers probably won't be familiar with this method. Therefore, I suggest adding a short description of how the method works and which parameters are required to generate rainfall fields. It should be discussed if any assumptions used to generate the rainfall fields are similar to assumptions of the proposed CML disaggregation method. Also, the spatiotemporal structure is described as "realistic," but no evaluation against any observations, e.g., radar, is performed. Especially, spatial features between 100 to 1000 m might lack any reference as those scales are typically below the horizontal resolution of weather radars.3. Model evaluation
There is a lack of independent rainfall fields for the intercomparison of the proposed method with the Goldshtein method. This work fits the parameters of the SD model to the virtual rainfall fields and then uses the same rainfall fields for the model intercomparison. Therefore, it is no surprise that the proposed method outperforms the Goldshtein method, e.g., the RMSE of capturing the maximum rainfall is about 70% lower (Fig. 5). The results for the independent two-hourly radar field show similar results as the Goldshtein method and path-averaged rainfall. I suggest using independent radar observations for evaluation over a long time period (not just two hours on 21 September 2014). Moreover, I was wondering if high-resolution atmospheric models (large-eddy simulations) could provide a source of rainfall fields that could be used instead of the statistically generated rainfall fields.4. Structure
I strongly encourage the authors to revise the structure of this paper from chapters 2-5:
- Split "Material and methods" into separate "Data" and "Methods" sections. This will avoid the current jumps between methodology and data sections.
- Move the methodology details from the appendix to the respective location in the method section. I do not see a reason for describing these steps in the appendix.
- Merge the two result sections ("results" and "case study") or rename the "results" section to avoid confusion on where to find all results.
- Avoid presenting new results in the discussion section.
Specific comments
line 46: Word missing: "distribution of"
line 53: Add a new line before "This paper addresses"
line 58-59: Provide references.
line 60: This sentence is incomplete. Add information on the reference CLEAR gets compared to.
line 77: This should be mentioned in the code and data availability section and not here.
Equation 4: The equation is not correct. Dividing by R0 and knowing that R1 = R2 (line 96) leads to W1 = W2.
line 104: Repeats line 97
line 459: How does the choice of 4km x 4km initial grid size affect the final prediction? Is there a way to compute this initial grid size for a given CML network?
line 465: "contain"
line 467: I do not understand this paragraph and what is meant by "24 regularly spaced positions". Please rewrite this and clarify.
line 118: How does the choice of the threshold affect the predictions, and which value is used here?
line 124: References to specific scripts are not very useful for the reader. If those are relevant to understanding this paper, I suggest pseudocode.
line 127: Provide a more meaningful section header instead of "Approach".
line 128: This CML data has to be introduced and described first.
line 130: See comment on line 124.
line 131: GMZ not introduced.
line 151: Move this to the "code and data availability" section.
line 157: What is meant by "the case studies"? I don't think any case studies (time, place) were introduced yet. Also, which time period does the radar data cover? Mention this in the text with basic statistics on the rainfall amount, maximum rain rates, and number of rain events.
line 180: How was it ensured that the simulated rainfall fields correspond to the "local climatology"? And for which place were they tailored to?
line 188: There is no "=" sign in equation 3 and thus no left hand side, please check.
line 191: See comment on line 124.
Figure 2: Typo in the caption: "sum"
Figure 3: What causes the "wave" pattern between 0-10 mm/h? Could radar data provide a way to verify this empirical relationship, at least for the longer CML links?
line 201: Are these the same rainfall fields that were used to fit the SD model?
line 232: Remove "(and in general)". While extreme events are rare in reality, it should be ensured that their number is sufficient for the evaluation.
line 246: How do I see if the CML link in that figure is isolated? Ideally, indicate those links in the map.
Figure 5: Is this strong improvement in capturing the rain rate maxima expected? While the reference algorithm achieves an RMSE of about 6, the method presented here achieves an RMSE of 2.7, i.e., more than 50% reduction in RMSE. This is not discussed sufficiently in the text and I am afraid that this is due to the fact that SD parameters were fitted to the same rainfall fields.
Figure 8: Highlight the point density as color shading to improve the readability of this plot.
line 311: Please define measurement and model uncertainty to avoid confusion. I would not use them interchangeably as the measurement uncertainty is typically associated with the input (path-averaged rain rate) and the model uncertainty with the model parameter uncertainty (SD model, coherence rule, ...).
line 323: Move this section to data and methods. There is no reason for the separate sections "2.4 CML data" and "4.1 CML data and rainfall retrieval". Also, mention the exact time of the rainfall event in UTC.
line 340: Provide the actual time instead of "time step 21".
line 343: Rewrite the sentence and describe what "locations are better" means.
line 344: It is true that, in this case, CLEAR seems to perform better than GMZ. However, even the path-averaged product provides better minima and maxima in both RMSE and R^2 compared to GMZ. And the R^2 of the minima and maxima from the path-averaged product is equal to CLEAR. While the sample size of just two hours of radar data is tiny, one would still expect to see similar trends as in Fig. 5 for the simulated rainfall fields.
Figure 11: The transparency of points in column a is higher than in the other columns. All three data sets should be presented the same way for a fair comparison.
Figure 12: What is the difference between this figure and Figure 3, except that the empirical SD is shown per event? I would suggest combining both figures to show that, e.g., event 2 causes the "wave" pattern from 0-10 mm/h.
line 391: I assume this analysis is not shown here? Using a year of radar data is a much more promising approach than the approach in this paper. I am surprised that overfitting occurred for an entire year of data; please specify.
line 394: Explain the "non-stationarity issues".
line 405: The discussion chapter should not contain "two additional analysis". Instead, it should interpret, explain, and contextualize the results presented before.Citation: https://doi.org/10.5194/egusphere-2025-487-RC2 - AC2: 'Reply on RC2', Martin Fencl, 10 Jul 2025
Interactive discussion
Status: closed
-
RC1: 'Comment on egusphere-2025-487', Anonymous Referee #1, 27 Apr 2025
This paper suggests a novel approach to disaggregate path-average rain rate aver CML. The topic is relevant and interesting for the community. The paper is very “clear” ;-) to read. Indeed the methodology is presented in a straightforward way, results displayed and discussed in convincing way, and current limits are also properly addressed. In general, I think that the paper only require minor modifications before it is suitable for publication.
Minor specific comments:
- l. 97: “L1” should be in italic.
- l. 110-114: it is indeed one approach, but there are others. Did authors carried out some tests to opt for this approach ? This should be clarified.
- Section 2.6: maybe say few words on the uncertainty associated to radar QPE. Is it really a reliable reference ? (few things are said l. 345 on the topic)
- l. 176-177: could authors clarify why does the parameters of the cascade generator depend on link orientation and length ?
- l. 180: the whole point is in “realistically”… How sure are you that your rainfall simulations are “realistic” ? I also agree that the other approach also has intrinsic limitations as pointed out by the authors. Maybe carrying out the calibration with both method and highlighting the differences would be more convincing.
- Fig. 3 (right): It would be interesting to find a way to always display the differences according to rain rate.
- Section 2.8: please justify better the choice of GMZ as benchmark. It seems as a simplistic approach with regards to what is suggested in this paper, hence not surprisingly it performs worse.
- L. 245 – 248: I understand the point, but it would be interesting to show quantitatively the increase in performance in areas with denser CML concentration.
- Section 5.1: wouldn’t it be possible to use empirical curves for SD as a function of rain rate instead of a modelled one
Citation: https://doi.org/10.5194/egusphere-2025-487-RC1 - AC1: 'Reply on RC1', Martin Fencl, 10 Jul 2025
-
RC2: 'Comment on egusphere-2025-487', Anonymous Referee #2, 24 May 2025
The authors present a new method to distribute mean rainfall rates from a commercial microwave link (CML) network along the CML line of sight. The underlying random cascade model considers spatial coherence within the CML network and provides uncertainty estimates. Required model parameters that determine the uniformity of the rainfall distribution along each transect are fitted to three virtually generated rainfall fields and the Prague CML network. Finally, the method is compared to the method by Goldshtein et al. (2009) for the same three virtually generated rainfall fields and radar observations of a single two-hourly rainfall event in Prague on 21 September 2014. The results show improved reconstruction capabilities by the proposed method.
While the newly developed method might be beneficial compared to existing methods, the performance evaluation presented in this work is not convincing. Below are several general and specific comments that the authors must address in a revised manuscript.
General comments
1. Sample size
This work is based on a tiny sample size of just four rainfall events: 3.5 hours of virtually generated rainfall fields and 2 hours of radar observations in Prague. This sample size does not provide a statistical basis for model evaluation and intercomparison with the Goldshtein method. I suggest using at least a full annual cycle of weather radar and CML data to capture a larger number of heavy precipitation events. This will allow for a more differentiated comparison with the Goldshtein method under different seasons and precipitation regimes. Also, it provides insights into whether the proposed method is even valid for an entire season or if the underlying parameters of the SD model need to be adjusted for convective/stratiform types.2. Virtual rainfall fields
This work mainly relies on virtual rainfall fields generated following the method by Schleiss et al. (2012). Most readers probably won't be familiar with this method. Therefore, I suggest adding a short description of how the method works and which parameters are required to generate rainfall fields. It should be discussed if any assumptions used to generate the rainfall fields are similar to assumptions of the proposed CML disaggregation method. Also, the spatiotemporal structure is described as "realistic," but no evaluation against any observations, e.g., radar, is performed. Especially, spatial features between 100 to 1000 m might lack any reference as those scales are typically below the horizontal resolution of weather radars.3. Model evaluation
There is a lack of independent rainfall fields for the intercomparison of the proposed method with the Goldshtein method. This work fits the parameters of the SD model to the virtual rainfall fields and then uses the same rainfall fields for the model intercomparison. Therefore, it is no surprise that the proposed method outperforms the Goldshtein method, e.g., the RMSE of capturing the maximum rainfall is about 70% lower (Fig. 5). The results for the independent two-hourly radar field show similar results as the Goldshtein method and path-averaged rainfall. I suggest using independent radar observations for evaluation over a long time period (not just two hours on 21 September 2014). Moreover, I was wondering if high-resolution atmospheric models (large-eddy simulations) could provide a source of rainfall fields that could be used instead of the statistically generated rainfall fields.4. Structure
I strongly encourage the authors to revise the structure of this paper from chapters 2-5:
- Split "Material and methods" into separate "Data" and "Methods" sections. This will avoid the current jumps between methodology and data sections.
- Move the methodology details from the appendix to the respective location in the method section. I do not see a reason for describing these steps in the appendix.
- Merge the two result sections ("results" and "case study") or rename the "results" section to avoid confusion on where to find all results.
- Avoid presenting new results in the discussion section.
Specific comments
line 46: Word missing: "distribution of"
line 53: Add a new line before "This paper addresses"
line 58-59: Provide references.
line 60: This sentence is incomplete. Add information on the reference CLEAR gets compared to.
line 77: This should be mentioned in the code and data availability section and not here.
Equation 4: The equation is not correct. Dividing by R0 and knowing that R1 = R2 (line 96) leads to W1 = W2.
line 104: Repeats line 97
line 459: How does the choice of 4km x 4km initial grid size affect the final prediction? Is there a way to compute this initial grid size for a given CML network?
line 465: "contain"
line 467: I do not understand this paragraph and what is meant by "24 regularly spaced positions". Please rewrite this and clarify.
line 118: How does the choice of the threshold affect the predictions, and which value is used here?
line 124: References to specific scripts are not very useful for the reader. If those are relevant to understanding this paper, I suggest pseudocode.
line 127: Provide a more meaningful section header instead of "Approach".
line 128: This CML data has to be introduced and described first.
line 130: See comment on line 124.
line 131: GMZ not introduced.
line 151: Move this to the "code and data availability" section.
line 157: What is meant by "the case studies"? I don't think any case studies (time, place) were introduced yet. Also, which time period does the radar data cover? Mention this in the text with basic statistics on the rainfall amount, maximum rain rates, and number of rain events.
line 180: How was it ensured that the simulated rainfall fields correspond to the "local climatology"? And for which place were they tailored to?
line 188: There is no "=" sign in equation 3 and thus no left hand side, please check.
line 191: See comment on line 124.
Figure 2: Typo in the caption: "sum"
Figure 3: What causes the "wave" pattern between 0-10 mm/h? Could radar data provide a way to verify this empirical relationship, at least for the longer CML links?
line 201: Are these the same rainfall fields that were used to fit the SD model?
line 232: Remove "(and in general)". While extreme events are rare in reality, it should be ensured that their number is sufficient for the evaluation.
line 246: How do I see if the CML link in that figure is isolated? Ideally, indicate those links in the map.
Figure 5: Is this strong improvement in capturing the rain rate maxima expected? While the reference algorithm achieves an RMSE of about 6, the method presented here achieves an RMSE of 2.7, i.e., more than 50% reduction in RMSE. This is not discussed sufficiently in the text and I am afraid that this is due to the fact that SD parameters were fitted to the same rainfall fields.
Figure 8: Highlight the point density as color shading to improve the readability of this plot.
line 311: Please define measurement and model uncertainty to avoid confusion. I would not use them interchangeably as the measurement uncertainty is typically associated with the input (path-averaged rain rate) and the model uncertainty with the model parameter uncertainty (SD model, coherence rule, ...).
line 323: Move this section to data and methods. There is no reason for the separate sections "2.4 CML data" and "4.1 CML data and rainfall retrieval". Also, mention the exact time of the rainfall event in UTC.
line 340: Provide the actual time instead of "time step 21".
line 343: Rewrite the sentence and describe what "locations are better" means.
line 344: It is true that, in this case, CLEAR seems to perform better than GMZ. However, even the path-averaged product provides better minima and maxima in both RMSE and R^2 compared to GMZ. And the R^2 of the minima and maxima from the path-averaged product is equal to CLEAR. While the sample size of just two hours of radar data is tiny, one would still expect to see similar trends as in Fig. 5 for the simulated rainfall fields.
Figure 11: The transparency of points in column a is higher than in the other columns. All three data sets should be presented the same way for a fair comparison.
Figure 12: What is the difference between this figure and Figure 3, except that the empirical SD is shown per event? I would suggest combining both figures to show that, e.g., event 2 causes the "wave" pattern from 0-10 mm/h.
line 391: I assume this analysis is not shown here? Using a year of radar data is a much more promising approach than the approach in this paper. I am surprised that overfitting occurred for an entire year of data; please specify.
line 394: Explain the "non-stationarity issues".
line 405: The discussion chapter should not contain "two additional analysis". Instead, it should interpret, explain, and contextualize the results presented before.Citation: https://doi.org/10.5194/egusphere-2025-487-RC2 - AC2: 'Reply on RC2', Martin Fencl, 10 Jul 2025
Peer review completion
Journal article(s) based on this preprint
Data sets
Data and codes underlying the publication: "CLEAR: a new discrete multiplicative random cascade model for disaggregating path-integrated rainfall estimates from commercial microwave links Martin Fencl and March Schleiss https://doi.org/10.4121/5c4ad375-4e88-402b-ac46-d27bb47250c3.v1
Model code and software
Data and codes underlying the publication: "CLEAR: a new discrete multiplicative random cascade model for disaggregating path-integrated rainfall estimates from commercial microwave links Martin Fencl and March Schleiss https://doi.org/10.4121/5c4ad375-4e88-402b-ac46-d27bb47250c3.v1
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Marc Schleiss
The requested preprint has a corresponding peer-reviewed final revised paper. You are encouraged to refer to the final revised version.
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