|The Author has addressed all the comments provided by the Reviewers in the point-by-point reply to Reviewers comments; he clarified some aspects related the scope of this work and the methodology implemented to improve the general discrete multiplicative random cascade structure. However, some major and minor questions are still open and are not fully addressed in the revised manuscript.|
1. The major gap of this work is, based on my opinion, the lack of consideration of the non-stationarity property of MRC models. Which is the theoretical and practical relevance of a model that reproduces a "uncontrolled" non-stationary process? Changes to the model structure proposed in this work allow to better reproduce the autocorrelation function only on average (in time), since the autocorrelation function is not the same for each time step, i.e. it depends on the position in time at the fine resolution of cascade simulation; so, what happens at each time step? Ensemble simulation and autocorrelation estimation could help to clarify this issue. Hence, based on my opinion the Author should first fix the stationarity issue and then think about improving the stationary autocorrelation (independent from time to time) reproduction. It is also possible that the proposed model improvement already has an effect on the theoretical stationarity of the simulated process. If the Author does not want to tackle this complicated issue, he should at least mention and discuss it explicitly in the introduction of the revised manuscript, so that potential readers are aware of the important limitation of the proposed model.
2. A second major gap is the lacking of the derivation of the theoretical autocorrelation function given the modifications applied to the cascade, such as e.g. the different branching number at the first and subsequent levels etc. The dependence of the theoretical autocorrelation function on the model parameters (including the branching number) could help the reader understand the complexity of the model and the parameters that need to be calibrated. Further, it is not clear if the scaling (fractal or multifractal) property that is at the core of MRC model is conserved by the modifications introduced to the model. A theoretical comparison with the classical MRC model, that could be recalled at the beginning of the method section, would help for this.
3. The definition of the generator of the MRC is in the classical MRC models generally requires to introduce a probability model for the random variable W (the multiplicative weight). Can the Author explicitly define this probability model? If it is described by the probability values P in Eqs. 2 and 3 or 4 and 3 (it is an empirical distribution function), this means that each of the probability value (minus one thanks to the sum equal to 1) is a free parameter that needs to be calibrated from the observation. Further, parameters seems to be doubled since they are estimated based on two volume classes, separated by the 0.998 quantile of daily total amounts. Given this high threshold, the number of values used to calibrate the parameters of the lower class are expected to be much larger than those pertaining to the upper class, so that the uncertainty characterizing the two classes of parameter values are strongly different. Further, the threshold of separation of volume classes changes from the first to the subsequent disaggregation levels and then with the position. Are those considerations correct? Are the parameters summarized in table 2 all free parameters that need to be calibrated? In such a case, the model is strongly over-parametrized with respect to classical MRC models that are characterized by a few parameters. Indeed, the fact that the parameters are estimated from observations means that they are calibrated to reproduce the observed realization of the process. What happens if the model calibrated on the available dataset is applied and tested against a different dataset (another possible realization of the same stochastic process)? Since the model is over-parametrized it is strongly needed to test it for out-of-sample observation. This is also a very important point, that was not clear in the first version of the manuscript and that needs to be addressed before the manuscript can be considered for publication in HESS.
4. As for the branching numbers chosen in this model, I would avoid to introduce a “physical” explanation of the assumption of 8-hours wet/dry periods, since it is quite arbitrary the choice of this temporal duration. However, if the Author has a clear reference that justify this choice independently of the climatic location, this should be cited.
Some specific comments follow, where page and line numbers refer to the revised version of the manuscript.
Abstract. Since it is recognized that MRC can underestimate or overestimate the autocorrelation function, the Author should be more general in the abstract mentioning both the problems. Further, method C is mentioned in the abstract before its definition; please fix it.
Please specify if resampling is only an alternative method for improving autocorrelation estimation with respect to basic MRC models or if it is introduced here for a different scope.
P 2. L. 23-25. Since this is the common practice, the Author could be more specific on the transformation that alter the theoretical properties of the target resolution rainfall time-series. As it stands, the comment is still vague.
p. 3, l. 8-10. The point here is to what extent theoretical properties are reproduced numerically; it also depends on how numerical assessment is performed, if by ensemble or time-series simulations. Further, it is expected that higher the number of disaggregation levels the better is the accuracy of the model in reproducing the autocorrelation function (if the multifractal character holds true across all the scales).
p. 8, l. 9-11. Bounded cascades allow the multiplicative weights W to depend on the cascade level and converge to unity as the cascade proceeds, so that the simulated random process becomes smoother on smaller scales (that’s the meaning of “bounded”). In the literature, bounded random cascades have been applied to the stochastic fine graining of rainfall observations into high resolution data both in the canonical and micro-canonical form (e.g. Menabde et al. 1997). Hence, bounded cascade are not introduced to reproduce the multifractal behavior, since this is reproduced even using unbounded cascades. Please clarify which is the rationale and the main objective of bounded cascades according to previous literature.
p. 21, l. 1-2. Only the validation or also the calibration of the model used all these information/summary statistics?
p. 22, l. 7. Which optimization?
p. 30, l. 3. The term “diurnal cycle” is introduced here for the first time. What does the Author mean with this term and what does he refer to? This could be mentioned before, at the beginning of the methodology section.
p.30, l. 18. I suggest to use “quantities” or “statistical summary” instead of “parameters” to avoid confusion.
p.30, l. 23-24. Which is the scaling behavior mentioned here? This is not introduced and mathematically formulated in the manuscript. This is related also to the mathematical formulation of the autocorrelation function as requested in my previous major comment 2.
p. 33, l. 21-24. Since n=30 is a unexpected small number, I’m wondering if the convergence of the stochastic properties is tested on 30 realizations after averaging in time or not (i.e. for each time step or specific time quantities). Please clarify this issue.