Third-Order Polynomial Normal Transform Applied to Multivariate Hydrologic Extremes

Round 1

Reviewer 1 Report

Notes and comments presented in the attached file Review424794.pdf

Comments for author File: Comments.pdf

Author Response

My replies to reviewer's comments are prepared in the attached file.

Author Response File: Author Response.docx

Reviewer 2 Report

The aim of this work is to present a procedure to estimate the coefficients of a multivariate TPNT by explicitly incorporating sampling errors associated with the sample moments and correlation coefficients. The method is applied to a data set of extreme rainfall used to construct IDF curves. The manuscript is within the interest field of the WATER Journal; however it presents many issues that should be solved before its final publication. The following points should be addressed:  

General comments:

A general comment is regarding the structure of the manuscript. In order to improve the structure I would suggest to include the subsections 1.1, 1.2, 2 (all subsections) and 3 (all subsections) under a Chapter called “METHODS”.

Regarding the data set used for the practical application it is very good how the authors have made it very transparent by including the whole data set along with some of the different in-between results, which makes it possible for readers to redo the exercise themselves if desired.

One important issue is regarding the available data set with consists of 27 years of maximum rainfall for different durations. With this data set the authors should only show results up to 100 years, but not extrapolate this short data set to higher return periods like 200 and 500 years.

Specific comments:

1.     Abstract – General comment: Some information is missing regarding the data used for testing the proposed methodology and evaluating the results. What durations and return periods were evaluated? What data was available? How is the method performing for different durations and return periods? How is the method´s general performance evaluated?

2.     Line 34: Replace “in comparing” by “in comparison”.

3.     Lines 35-37. The existing references related with multivariate distribution functions is much broader. Here only 4 references are mentioned. Please include more references and discuss about the advantages and limitations of existing models.

4.     Lines 38-42. To facilitate the reading of this section this paragraph should be placed after the paragraph describing the copula models.

5.     Lines 39-40. Please include a reference/s regarding the “approximated approaches”.

6.     Lines 41-42. Please include a reference/s for the statement “approximated approaches only retain partial information”.

7.     Lines 43-53. The Copula functions and introduced here along with some applications. From this paragraph it is not clear why these functions are not used in this work. What are the limitations/drawbacks of these functions that the authors decided not to use them here? Please include a comment.

8.     Line 54: Write “Scheme” in singular form.

9.     Line 55. Include references for NORTA and Nataf methods.

10.  Line 56. Replace “transform” by “transformation”.

11.  Line 56. Write “an individual variable”.

12.  Line 58. Include “the” next to “standard normal …”.

13.  Line 59. Replace “relation” by “a relationship”.

14.  Line 60. Write “an equivalent”.

15.  Lines 60-62. The notation of variables X_j and X_j│is not so intuitive. Why didn’t the authors use X_j and X_k ?

16.  Line 63-65. Here the authors explain that an orthogonal transformation of the variables in original space is necessary in order to uncorrelated them. If the transformed variables are uncorrelated, why is it necessary a multivariate analysis such as the Gaussian copula? Please clarify.

17.  Line 80. Write “variable” in its singular form.

18.  Line 82. Write “of the individual random …”.

19.  Line 85. References [26,30,31,32,33], be more specific regarding the application of multivariate TPNT.

20.  Line 88. In line 78 the authors mention that “To relax the information requirement on specifying the distribution functions TPNT can be used”. However here they claim that “most of the applications are done under the assumption of known marginal distributions”. Please clarify this issue as it can be confusing for the readers.

21.  Line 90: Rephrase “the amount of data is generally insufficient large to…” by something like “the amount of available data is generally limited to…”.

22.  Line 94-97: This sentence is too long and confusing and therefore could be splited and rephrased.

23.  Introduction - General comment: In the conclusions the following is stated: “a general framework is presented to optimally determine multivariate TPNT coefficients having the constraints that (1) preserves the statistical L-moments and correlations with explicit consideration of their associated sampling errors; and (2) complies one-to-one monotonicity increasing relation between quantiles of the original and normal transformed variables”. Some of this information was neglected in the general aims presented in the introduction and should be included.

24.  Line 105: The authors state “first four production of … variable” is this referring to the first foru moments? Please clarify.

25.  Line 107: Equation 3. Should the skew and excess coefficients refer to the original variable (X) or standardized one (X´)?

26.  Line 112: Replace “an” by “can”.

27.  Line 129: Use “transformation process” or “process of transformation”.

28.  Line 143: Replace “into” by “within the” or “during the”.

29.  Line 153: Replace “that by “the”.

30.  Lines 142-159: The title of this section refers to the multivariate case, however the whole explanation refers to the bivariate one. The authors could either extend the section and explain how the equations can be extended to a higher dimensional case, or change the name of this section.  

31.  Line 168: Equations 4 and 10 are very similar. Is it necessary to include both? If yes then the equation 10 should be relocated to section 1.2 in which the multivariate framework is introduced.

32.  Line 169: Change the font of the text.

33.  Line 177. Replace “lieu”.

34.  Line 179: Eliminate “in”.

35.  Line 180: Replace “can preserve” by “are able to preserve”.

36.  Line 181: Add a sentence presenting the following points. Otherwise eliminate the “:”.

37.  Line 182: Add “the” before individual.

38.  Lines 185, 205 and 207: There are “0” in the third lines. Should these values be there?

39.  Line 190: Add an “a” before variable.

40.  Lines 194-197. The set of equations is numbered with a, b, c, d. The authors should do the same with the other set of equations. (e.g. Equation 3, 4, 5, etc.).

41.  Lines 205 and 207: To clarify include a } at the end of the four equations and before the ≤.

42.  Line 291 – Figure 1: For a better connection between the text and the figure it would be helpful to include the numbers (1) to (4) into the flow chart. Furthermore as the procedure is recursive it would help the readers to include the recursive step in the flow chart.

43.  Line 344: Rephrase “for each duration independently”.

44.  Lines 345-350. The following reference could as well be revised by the authors and possibly included in this section with some comment. Reference: B. Gräler, S. Fischer und A. Schumann: Joint modeling of annual maximum precipitation across different duration levels, Discussion Paper SFB 823, 2016.

45.  Line 354: Rephrase ” in which 1992 data were not used because the gauge malfunctioned much of the time” e.g. “the year 1992 was excluded from the analysis due to long periods of registers with technical issues“.

46.  Line 355: Exclude the word “storm”.

47.  Line 365 – Table 1: The authors present the values of the original variable (X). However they could make use of this table to include information of the Standard Normal variables (Z).

48.  Line 366 – Table 2: Be consistent with the number of decimals shown here. The mean and first L-moment should be the same value.

49.  Line 368: This paragraph should be connected to the previous part of this section in a smoother way. It jumps to the parameter estimation but a connection with the previous presented results would improve the following of the message for the readers.

50.  Line 369: The equation 27 should refer to the original one which was equation 5.

51.  Line 385: Indicates LM/Mono, however the Table 4 in reference states LM + Mono. Be consistent.

52.  Line 379: Is Part(a) referring to Table 4? Please clarify.

53.  Line 384: Replace “rainfalls of which the two monotonicity” by “rainfalls for which the two monotonicity”.

54.  Line 391: Figures 2 and 3 have the exact same caption. Please be more specific in the caption of these different figures.

55.  Line 397 – Table 4: Is it possible to include an additional indicator regarding the crossing of different curves (Equation 28)? This would facilitate the reader to understand what is the advantage of the “LM/Mono/Corr/NC” model.

56.  Lines 399-409. This explanation should be relocated somewhere in the Methods Section, as part of the proposed Methodology and not in the Results section.

57.  Line 408: Revise “additional N  1 no-crossover constraints”.

58.  Line 419: Please revise the statement “intensity quantile” as it should be the value of intensity but not quantile, as this statement refers to one particular return period.

59.  Line 421: Replace “in a much higher return period range” by “for high return periods”.

60.  Line 424: Refer to Figure 4 (not 3).

61.  Line 448: The authors claim ”undulation diminishes somewhat”. Please use another term to describe this attribute. Furthermore the undulation is clearly present in all 4 presented models. This should be clearly stated somewhere in the manuscript (either in the results or conclusions) as some limitation of the presented method.

62.  Line 448: To show the improvement in terms of decrease of undulation it would be helpful to include a figure for one return period (for example 50 years) comparing the 4 models. Then the authors can discuss based on that figure the advantages of including additional constrains to the model.

63.  Line 451: The authors should avoid discussing about a return period of 5000 years with a data set of only 27 years … they should refer to up to 100 years. Please correct this and as well change Figures 2, 3 and 4 to refer to return periods according to the available data.

64.  Line 456: Change “too lower” by “too low” and please explain what is exactly meant by too low.

65.  Line 461: Change to “allows”.

66.  Line 464: Replace “random variables with mixture of non-normal distributions” with “random variables following non-normal distributions”.

67.  Line 465: replace “great majority of the” by “most of the”.

68.  Line 466: The sentence is not clear “… are to preserve …” should be replaced by another verb accordingly to the sentence.

69.  Line 468: Replace “(TPNT) method by which preservation of marginal L-moments and correlations among variables are made.” by “(TPNT) method which relies on the preservation of marginal L-moments and correlations among variables”.

70.  Line 474: Replace “of relevant” by “of relevance”.

71.  Line 476-477: “no-crossover constraints … are included” is not clear. Please briefly describe what is meant by no-crossover constrains.

72.  Line 478: Include “the” to monotonicity condition.

73.  Conclusions - General comment: It is not clear how the proposed method was evaluated. Please include a brief description of the procedure to conclude that the proposed method results in “suitable” coefficients for the multivariate TPNT.

Author Response

My replies to reviewer's comments are contained in the attached file.

Author Response File: Author Response.docx