We thank the anonymous referee for the comments for improvement. In the following, we present the replies to each comment.

****** Comment 1

we have used quantiles instead of the classical nomenclature `non-exceedance probability' in the description of the algorithm to compute the cdf of the rainfall field. Surely we should have made it clear before doing so. 

****** Comment 2

The referee points out that this work involves only spatial simulation, and the temporal part is not discussed after the introduction, which is true. In the following, we present our understanding concerning the specific issue `How to obtain QPE (Quantitative Precipitation estimates) at a high spatio-temporal resolution, which should have been discussed briefly in the manuscript. 

First, it should be clarified that what we are doing in this work is not now-casting, but rather hind-casting. Given the observed radar estimates and station data (some weather condition that has already existed), we try to estimate the true rainfall field. Since we are not forecasting, we do not simulate time series of precipitation. With the high spatial resolution of radar estimates, e.g., (1 km X 1 km), or even (500 m X 500 m), the acquisition of QPE with high spatial resolutions is possible. Then how to obtain QPE with relatively high time resolutions? Though this work discusses only spatial simulation (or QPE in hind-cast mode), the hind-cast can be made at fine time resolution (every 10 min, or even 5 min). Given the above, one could say the QPE is obtained at high spatio-temporal resolutions.

It should be noted that the term `hind-cast' is relative to the estimation of precipitation. The resultant QPE is still useful for early flood detection. Nonetheless, it should be admitted that QPF (Quantitative Precipitation Forecasts) is more useful for pluvial floods forecasting, where approaches to model the temporal evolution of precipitation fields in the future are of interest (i.e., simulation of time series of precipitation as mentioned by the referee). However, most of the nowcast models are only radar-based. Due to the indirect measurement of the intensities, radar data are prone to various sources of errors which usually cause an underestimation of the rainfall intensities (Berne & Krajewski, 2013), and underestimated intensities fed to the nowcast will not be very useful for the urban flood application. In the recent work by Shehu & Haberlandt (2020), the authors indicate that the predictability of the nowcast model is expected to be extended by improving the rainfall field fed into the model, which might become a research hotspot in the future.

****** Comment 3

First, we would love to answer the question `What is the value of radar data as additional information to build the cdf?' The additional information of radar data in the construction of the cdf is that it provides a hint on how representative are the samples (station data) for the entire precipitation field. Whether the samples have missed the extremes of the precipitation field? Whether the dry area has been missed (could happen during a stratiform event that is large)? Then a problem arises: what makes the samples representative? A representative set of samples means the gauge respective (radar) quantiles, or the gauge respective non-exceedance probability (in the uniformed radar data), cover the entire range [0,1]. Whether the sample size is large or small, radar data can always provide this sort of information. Yet admittedly, with large sample size, it is more likely to have representative samples.

In practice, for mesoscale hydrological studies often only small sample sizes of irregular distributed recording rainfall stations are available (e.g. about 10 stations). To obtain an accurate cdf using the proposed method, enough station data should be available. Yet there are possibilities to improve the applicability of the method by increasing the sample size in space and time.

    (1) Increase the sample size in space. In the work by Yan & Bárdossy (2019), a method to decrease the wind-induced discrepancy of radar and gauge data is introduced. For small domains where only a few rain gauges (say 10) are available, one can assume the uniform movement of the rain parcels. Under this consumption, one can displace the radar grid using a vector that increases the radar-gauge agreement, and 10 new pairs can be found in the displaced radar grid. If N vectors have been used to displace the radar grid, then one can obtain (10 * N) new pairs, as normally more than one such vector can be found. One can pool these pairs and compute the cdf. It should be noted that using the above method, one only enriches the original 10 pairs (station data, quantile/non-exceedance probability) in the second axis, namely the y-axis in the cdf-plot.

    (2) Increase the sample size in time. One can also pool the radar-gauge pairs from a fixed time window by assuming stable distribution in the relevant time.

****** Comment 4

It is correct that another simulation method could have been used after the estimation of the cdf, for example, phase annealing (Yan et al., 2020). There are a bunch of unconditional simulation methods. Yet to our knowledge, methods for conditional simulation are rare. We have used random mixing in this work due to the following reasons (which could be discussed briefly in the manuscript):

    (1) The relatively high efficiency. It took around 5.87 sec to generate a realization for Scenario: 6 x 6 rain gauges, which is comparable to the time consumption of Kriging with external drift (3.08 sec). However, with a simulation method, one naturally wants to generate a bunch of realizations to see certain statistical properties. In that case, the total time consumption is much longer than an interpolation method. Note that the above-mentioned time consumptions are based on the performance of a normal laptop.

    (2) The code availability. In the recent work by Hörning & Haese (2021), the authors present a Python package for conditional simulation of spatial random fields using Random mixing. 

****** Literature

Berne, A., Krajewski, W.F., 2013. Radar for hydrology: unfulfilled promise or unrecognized potential? Adv. Water Resour. 51, 357–366. 

Hörning, S.,  Haese, B., 2021. RMWSPy (v 1.1): A Python code for spatial simulation and inversion for environmental applications. Environmental Modelling and Software, 138(12), 104970.

Shehu, B., Haberlandt, U., 2020. Relevance of merging radar and rainfall gauge data for rainfall nowcasting in urban hydrology. Journal of Hydrology, 125931.

Yan, J., Bárdossy, A.. 2019. Short time precipitation estimation using weather radar and surface observations: with rainfall displacement information integrated in a stochastic manner. Journal of Hydrology.

Yan, J., Bárdossy, A., Hörning, S., Tao, T., 2020. Conditional simulation of surface rainfall fields using modified phase annealing. Hydrol. Earth Syst. Sci., 24, 2287–2301.

First, we express our gratitude to Prof. Uijlenhoet for his careful evaluation and valuable suggestions for the improvement of this manuscript. Looking through the comment, we decided to separate the corresponding Author Comment (AC) into two parts: in the first part we present our reply concerning the general remarks from the referee; in the second part, the reply concerning the specific remarks is present. As for the editorial remarks, we thank the referee again and consider making improvement according to each of the editorial remark.

In the general remarks, Prof. Uijlenhoet indicates that there are two limitations, which should be stressed more clearly in the paper. In the following, we present our thinking towards the two issues.

(1) “it is a pure simulation study ...” There are no radar and rain gauge data employed in this study. We have used synthetic rainfall fields for the verification, and the radar and rain gauge data used for application of this approach are derived from the synthetic rainfall fields.

The motivation of doing so is that one can hardly verify the accuracy of a method on the whole without the knowledge of the true rainfall field. There are studies that employ leave-n-out cross validation for the verification, where the QPE is obtained without the knowledge of the n gauge observations, and the estimates at the relevant gauges are compared with the concealed observed values. In this specific setup, one only verifies the accuracy at several points, and the verification of the QPE in terms of the statistics of the entire field (e.g., mean, regional extreme, etc.) is impossible due to the lack of knowledge of the true rainfall field. However, compared to the accuracy at limited points, the accuracy of the QPE in terms of the overall statistics is more important for many applications. Due to the fact that the true rainfall fields are impossible to acquire at the moment, we have used the synthetic ones.

Certainly, synthetic data can only partially represent the reality. As indicated by the referee (Page 2, the 5th bullet in the file “hess-2021-56-RC2-supplement.pdf”), we only introduce random error in the radar estimates, and the systematic errors are not considered. This remind from the referee is very important. We think the practice of “verification using synthetic data” is appropriate on condition that the synthetic data can represent the reality to some extent (To what extent? At this moment we cannot be more concrete). The specific comment from the referee reminds us that we should improve the synthetic data in terms of the representativeness for the reality, or at least try to consider the range-dependent error, which could become an interesting topic in the future study. Besides, we take the referee’s suggestion that we should reflect the fact of using the synthetic data more clearly in the abstract and introduction of this paper.

(2) “the study only considers the estimation of spatial rainfall fields, and completely neglecting the temporal aspect of QPE, which is important for hydrological applications.” We have noticed that this issue has been raised again and consider to modify the title of this manuscript to reflect the focus of this study in a more straightforward way, for example, “Simulation of spatial rainfall fields …”, and give a clearer description in the abstract and/or introduction.

This paper only discusses the estimation of spatial rainfall fields. Yet due to the high temporal resolutions of both radar and rain gauge data, the spatial rainfall fields can be estimated at high temporal resolutions, which ends up with QPE at high temporal resolutions. It should be stressed out that such QPEs are obtained in a hind-cast mode, namely, we try to approach the true rainfall field given the observed radar and rain gauge data (some weather condition that has already existed). Unlike the acquisition of QPF (Quantitative Precipitation Forecasts), where modeling of temporal evolution of the precipitation field is of interest, the proposed approach does not involve any temporal simulation.  Yet we think that the approach has great potential to improve the quality of QPF. The predictability loss of a nowcast model is caused mainly by (a) the inability of radar to capture the true rainfall field and (b) because the Lagrangian Persistence is unable to model the temporal evolution of the rainfall field (Shehu & Haberlandt, 2020). From this point of view, the approach could be used to improve the rainfall field fed into the nowcast model.

Furthermore, the skill of radar-based nowcasting has been experiencing an evolution from the deterministic to probabilistic frame work to estimate the predictive uncertainty (e.g., Pierce et al., 2012; Shehu 2020). A common approach is based on stochastic simulation in which correlated noise fields are used to perturb a deterministic nowcast (e.g., Seed, 2003; Germann and Zawadzki, 2004; Bowler et al., 2006). What if we feed the nowcast model with multiple realizations of the current rainfall field, for example, the product of the proposed approach? If that is the case, maybe one no longer needs the perturbation for the deterministic nowcast. Besides, if the nowcast community can embrace a change from the deterministic to probabilistic, why not the hindcast community?

Literature

Bowler, N. E., Pierce, C. E., and Seed, A.W.: STEPS: A probabilistic precipitation forecasting scheme which merges an extrapolation nowcast with downscaled NWP, Q. J. Roy. Meteorol. Soc., 132, 2127–2155, 2006.

Germann, U. and Zawadzki, I.: Scale Dependence of the Predictability of Precipitation from Continental Radar Images, Part II: Probability Forecasts, J. Appl. Meteorol., 43, 74–89, 2004.

Seed, A. W.: A Dynamic and Spatial Scaling Approach to Advection Forecasting, J. Appl. Meteor., 42, 381–388, 2003.

Shehu, B. and Haberlandt, U.: Relevance of merging radar and rainfall gauge data for rainfall nowcasting in urban hydrology. Journal of Hydrology, 125931, 2020.

Pierce, C., Seed, A., Ballard, S., Simonin, D., and Li, Z.: Nowcasting, in: Doppler Radar Observations – Weather Radar, Wind Profiler, Ionospheric Radar, and Other Advanced Applications.

As Prof. Uijlenhoet did not number the specific remarks and listed them in the order that they appear in the manuscript, for consistency, we use line/figure/table number to refer to the corresponding remark hereafter.

Considering the relatively large amount of the specific remarks, we have classified them into two categories: (a) the deficiencies raised where the corresponding replies are not that necessary, but the relevant improvements should be made, such as the inappropriate reference (Remark L. 45), lack of reference (Remark L. 192), ambiguous statement (Remark L. 295-296), suspicion of using undefined abbreviations (Remark Fig. 9, Table 1); (b) the kind of deficiencies/ambiguities that need to be specified or discussed. We have summarized the second kind of remarks into the following 6 topics:

1) The role of random mixing (Remark L. 135-204) and the role of the marginal distribution function in the approach to obtain spatial rainfall fields, respectively, as well as the issue raised concerning the intermittency in the computation of the marginal distribution function (Remark L. 91-97).

Random mixing (RM) is an excellent tool that performs conditional simulation in Gaussian space, but it is not irreplaceable. Another simulation method could be used instead. RM is employed in this work due to (a) the relatively high efficiency (see Comment 4 in RC1 for detailed information on the computational time), which makes mass production of realizations possible, and (b) code availability. A Python package for conditional simulation of spatial random fields using RM is available (Hörning & Haese, 2021), where the authors give practical demonstrations on the usage of the method.

The marginal distribution function, on the other hand, is the foundation of the approach. The statistics of intermittent precipitation are non-Gaussian, and such properties restrict the usage of well-established stochastic models that assume Gaussianity (including RM) (Pulkkinen et al., 2019). Specifically, the marginal distribution function serves two functions: (a) data transformation: the simulated Gaussian fields are transformed to precipitation fields utilizing the normal-quantile transformation (Bogner et al., 2012), where the marginal distribution function is required; (b) definition of the constraints in Gaussian space.

The referee has raised an issue concerning the intermittency in the computation of the marginal distribution function (Remark L. 91-97). We fully agree that radar and rain gauges have different space-time sampling properties, and one should not expect the same probability of finding zeros by both sensors. Yet one should choose from the two sources of information to evaluate the intermittency. We have used radar data, as the intermittency computed from a limited number of rain gauge observations is less reliable. The issue raised by the referee is a redundancy error. The starting point (0, the ratio of the dry area /intermittency), where the marginal distribution function starts to be positive, has already been set. Practically, we have not used the sampled zeros (at the gauge locations) by both sensors in the computation of the marginal distribution function. The enforcement (L. 91-97 ) is not necessary in the first place. Nevertheless, the redundant step does not influence the estimated marginal distribution function.

2) The interchangeable use of “correlation function” and “variogram” (Remarks L. 118-124 & L. 212-213).

Thank you for the reminder. Otherwise, we might have ignored the lack of clarity before the interchangeable use. We have made simple statistics: the word “variogram” has appeared five times and is relatively concentrated (L. 120-123 and the caption in Fig. 2), while the “correlation function” showed up 14 times. Hence it might be easier if we stick with “variogram”. Yet on second thought, we found the complete cutoff of “variogram” is not easy. The simulation is developed under the assumption that the correlation function of the process is stationary, and the correlation between the process at any two locations is only a function of the vector connecting the two locations (i.e., second-order stationarity). Rather than estimating the covariance function, it is common in geostatistics to work with the variogram, i.e., the variance of the difference as a function of the vector. It has been shown that the estimation of the variogram is more stable than the estimation of the correlation function directly (Calder & Cressie, 2009). Namely, one applies the simulation using the correlation function as the measure of spatial dependence, yet the spatial dependence of the simulated product is normally examined on its variogram. Thus, we consider preserving the part where the “variogram” present, but providing a clearer description to explain why “variogram” instead of “correlation function”.

The referee has expressed concern about the distinction between the empirical variogram (evaluated from the truncated Gaussian field) and the true variogram. The relevant discussion is missing in the manuscript. It is not a problem for the approach. RM is a geostatistical simulation method, and the fundamental element is the spatially correlated random field. Similarly, as the case in Kriging, the choice of variogram has a limited effect on the results (Verworn & Haberlandt, 2011). Further, the variogram computed from the truncated Gaussian field (transformed radar data) gives an excellent hint to approach the true variogram.

3) The target of the proposed approach (Remark L. 217-218).

The proposed approach is aimed at estimating spatial rainfall fields of short accumulation time: 15 min, 10 min, or even 5 min (if the good quality of rain gauge data can be maintained at such fine time resolution). In the last case, there is no aggregation of radar data; thus, there should be no dispute about employing the log-normal distribution as the model of the CDF of the rainfall field. Further, we think that slight aggregation of radar data (say 2 or 3 timesteps) should not change the type of distribution function remarkably. Under the above assumption, we think it is not inappropriate to use the log-normal distribution function as the model of the CDF of the rainfall field. Admittedly, the statement (L. 217-218 in the manuscript) is ambiguous, and a specification should be provided.

4) Concern raised due to the employment of synthetic data (Remarks L. 232-233 & L. 238-240 & L. 265).

This study is a pure simulation study, where one has full control over the stochastic process due to the employment of the synthetic data (as the true rainfall field). Undoubtedly, synthetic data can only partially represent reality, which is one limitation of the approach. Specifically, this study only considered the random error in radar estimates and did not consider systematic effects (e.g., range dependent error) in radar estimates (Remark L. 232-233). The basic assumption of the proposed approach is that the field pattern indicated by radar is similar to the pattern of the precipitation field on the ground. If the systematic effects are prevailing in the radar estimates such that the assumption is not valid (Remark L. 238-240), then the proposed approach is no longer applicable. The above limitation should be discussed in the manuscript, see Topic (6).

Similarly, with a pure simulation study, one has the opportunity to decide the layout of the rain gauge network as well as the corresponding density, which brings both pros and cons.  The advantage is that one has full control over the stochastic process, which makes the sensitivity study possible. While the drawback is that one can hardly comprehensively model the rain gauge network encountered in practice, which is usually irregularly distributed with various densities. Now that we have full control, and also because the real-world gauge network is too complicated to model, we made things as simple as possible: square domain and uniformly distributed rain gauges (concerning Remark L. 265). Yet admittedly, a brief discussion is necessary where relevant.

As the temporal aspect of QPE has been discussed in the reply concerning the general remarks, we just skip Remark Fig. 5.

5) The distinction between the proposed method and the Kriging methods involved in the study (Remarks L. 252-253 & L. 306-307) and how the results from the proposed method can be utilized (Remark L. 314).

First, the choice of the two parameters (Remark L. 252-253). The choice of the two parameters is indeed arbitrary because it makes no difference to the results for the proposed approach. In the algorithm, the radar estimates are converted to quantiles, and the corresponding quantile map (or field pattern in the nomenclature of the manuscript) is incorporated. The monotonic transformation (the power function) does not change the quantile map; thus the results of the approach are not affected. Yet the choice of the two parameters does matter for the Kriging methods involved in the study, e.g., an underestimation of the radar estimates leads to an underestimation in the kriging results. As it is not essential for the proposed approach, we do not care too much about the choice of the two parameters. Further, as suggested by Berne & Krajewski (2013) and Curry (2012), radar data are prone to underestimate the precipitation, and underestimated precipitation will not be very useful for many hydrological applications; thus, we have modeled a case when radar underestimates the rainfall.

Considering the distinction of the results from the proposed approach and the Kriging methods (Remark L. 306-307), with the Kriging methods, one obtains an estimate (one rainfall field) that tends to underestimate the peak and overestimate the small, i.e., more middle-ranged rainfall values are present in the estimate. In the algorithm of Kriging, the marginal distribution function is not required. The Kriging results are obtained via geostatistical interpolation where the stationarity of the regional variable is assumed. However, if one evaluates the empirical distribution function from the Kriging-estimates, the departure of the empirical distribution from the true distribution function can be observed.

By comparison, with the proposed approach, one can obtain a bunch of realizations (estimates). The individual estimate gives relatively accurate statistics (mean, variance, covariance). Yet practically, one can hardly obtain a single realization with accurate statistics, and simultaneously with the precise positions of the rainfall peaks. Namely, one gets the peaks accurately located but erroneous statistics in the mean realization, while accurate statistics but inaccurate positions of the peaks in the individual realizations (concerning Remark L. 314). Nevertheless, the realizations are still useful. In this fully controlled setup, we know exactly how the true rainfall field looks like. Yet when the true field is unknown, one possibility is that we simulate a bunch of realizations that are statistically identical to the true field and feed each realization to applications, e.g., hydrological models, nowcast models. The corresponding result is a range of possible outcomes, which can represent the estimation uncertainty.

6) The lack of a discussion section (L. 405).

It is necessary to have a discussion section, where the assumptions and the associated limitations, the potential suitability of the approach, and the possible improvement in the future can be discussed. We sincerely thank the referee for the valuable suggestion.

Literature

Berne, A. and Krajewski, W.F.: Radar for hydrology: unfulfilled promise or unrecognized potential? Adv. Water Resour. 51, 357–366, 2013.

Bogner, K., F Pappenberger, and Cloke, H. L.: Technical note: the normal quantile transformation and its application in a flood forecasting system. Hydrology and Earth System Sciences,16,4(2012-04-02), 16, 1085-1094, 2012.

Calder C.A. and Cressie N.: Kriging and Variogram Models, in: International Encyclopedia of Human Geography, edited by: Rob, K. and Nigel, T., 49-55 pp., Elsevier, 2009. https://www.sciencedirect.com/science/article/pii/B9780080449104004612

Curry, G.R.: Radar essentials: A concise handbook for radar design and performance analysis. Radar Essentials: A Concise Handbook for Radar Design and Performance Analysis. SciTech Publishing, 2012. DOI:10.1049/SBRA029E.

Hörning, S. and Haese, B.: RMWSPy (v 1.1): A Python code for spatial simulation and inversion for environmental applications. Environmental Modelling and Software, 138(12), 104970, 2021.

Pulkkinen, S., Nerini, D., Hortal, A., Velasco-Forero, C., and Foresti, L.: Pysteps: an open-source python library for probabilistic precipitation nowcasting (v1.0). Geoscientific Model Development, 12(10), 4185–4219, 2019.

Verworn, A. and Haberlandt, U.: Spatial interpolation of hourly rainfall-effect of additional information, variogram inference and storm properties. Hydrol. Earth Syst. Sci. 15 (2), 569–584, 2011.