Construction of a Time-Variant Integrated Drought Index Based on the GAMLSS Approach and Copula Function
, Libing Zhang 1,2, Yi Cui 1,2 and Fang Tong 1
Round 1
Reviewer 1 Report
I cannot understand the utility of the GAMLSS model. The article did not put emphasis on the probability distribution based on a multi-variable function.
In fact, the article uses the copulas operation in order to construct a joint probability distribution density function. Further based on the return period the drought categories can be determined.
The author must explain more analytically the selection of the copulas parameters and if the authors tested some basic copulas function for this reason. Any test of fitness regarding the joint probability function?
Author Response
- Reviewer #1:
- I cannot understand the utility of the GAMLSS model. The article did not put emphasis on the probability distribution based on a multi-variable function. In fact, the article uses the copulas operation in order to construct a joint probability distribution density function. Further based on the return period the drought categories can be determined. The author must explain more analytically the selection of the copulas parameters and if the authors tested some basic copulas function for this reason. Any test of fitness regarding the joint probability function?
Response: Thanks very much for the reviewer’s valuable comments, yes, the main context of the manuscript includes three parts, i.e., the derivation of time-variant PDFs of single drought indicators through GAMLSS method, the derivation of time-variant integrated drought index CTVDI through Copulas function and inverse operation, the application of time-variant integrated drought index CTVDI in drought recognition and extraction of drought characteristic indicators. Regarding to the application of Copulas function in the derivation of integrated drought indicator CTVDI and joint probability distribution of drought duration and drought severity, (1) we just utilized the widely-applied binary Archimedean Copula function for its advantages of rigorous logic structure, simple calculation formula and less estimated parameters, which primarily includes three types of Copula joint functions, i.e., GH Copula, Clayton Copula and Frank Copula, and this can be referred from line 248-253 and references [31-33] for details. (2) previous studies have verified that the GH Copula function has better fitting performances for describing the correlation of upper sample scatters, which can be referred from line 475-478 and reference [35] for more details. Therefore, the GH Copula function was selected to derive the joint probability distribution of drought duration and drought severity in this study. And (3), it has also been verified that the exponential distribution and Gamma distribution patterns can effectively describe the distribution features of drought duration and drought severity respectively in reference [35], and P-III distribution was also recommended to reveal the distribution features of drought duration and drought severity in reference [19]. Thus, we applied the exponential and P-III distribution patterns to derive the PDF of drought duration, and Gamma and P-III distribution patterns to derive the PDF of drought severity in this study, and this can be referred from line 465-472 in the revised manuscript.
Author Response File: 
Author Response.pdf
Reviewer 2 Report
Article: Construction of time-variant integrated drought index based on GAMLSS approach and Copulas function
Number: water-2233296
The manuscript was well prepared and organized, but I have some notes that should be taken into consideration, summarized as follows:
Corrected title: Construction of a time-variant integrated drought index based on the GAMLSS approach and Copulas function
As you mentioned, the time-variant integrated drought index can better reveal the characteristics of the drought evolution process compared to a single drought index, in which the determination of time-variant parameters as well as its combination scenarios are crucial for the derivation of time-variant PDFs of different single drought indicators but present difficulties for the application of the GAMLSS method.
How do you judge the reliability and applicability of the index so that it is better than individual indices? Other than statistical verification.
Despite the excessive implementation steps in terms of time and effort, the ease factor of an index can often be an advantage, so what makes the new index more important than the others mentioned? Do you have a reason for that? Response studies and risk assessment interpreted by dual indicators are of great importance in the case of meteorological indices. Can this index be more sensitive to the assessment of losses? Should be applied like this process and then discussed in depth.
You selected five types of two-parametric distribution functions. Why did you not use three-parametric distribution functions such as the log-logistic probability distribution?
What is the threshold value from which the characteristics of the drought were studied? Through the run theory, more details about the run theory should be clarified.
Section 2.1: Why you used spatial kriging interpolation calculation and not other methods
line 116–123, you should add citations to this sentence.
Check all abbreviations, especially in lines 56–70. Add the full name the first time.
There are some typos in the texts; the English and grammar check should be edited.
Author Response
- Reviewer #2:
The manuscript was well prepared and organized, but I have some notes that should be taken into consideration, summarized as follows:
- Corrected title: Construction of a time-variant integrated drought index based on the GAMLSS approach and Copulas function
Response: Thanks very much for the reviewer’s valuable comment, the title of the manuscript has been updated in the revised manuscript.
- As you mentioned, the time-variant integrated drought index can better reveal the characteristics of the drought evolution process compared to a single drought index, in which the determination of time-variant parameters as well as its combination scenarios are crucial for the derivation of time-variant PDFs of different single drought indicators but present difficulties for the application of the GAMLSS method. How do you judge the reliability and applicability of the index so that it is better than individual indices? Other than statistical verification.
Response: Thanks very much for the reviewer’s valuable comments, yes, the determination of time-variant parameters as well as its combination scenarios are crucial for the derivation of time-variant PDFs of different single drought indicators. Therefore, the reliability verification of the integrated drought index CTVDI derived through two time-variant PDFs of precipitation (P) and soil moisture (SM) are discussed from three aspects, as follows, (1) we applied the Generalized Akaike Information Criterion (AIC) to recognize the optimal fitting function of distribution parameters μt and σt as well as their combination scenario, which can be referred from line 236-239. (2) we utilized the linear and Kendall correlation coefficients to comparatively testify the distribution consistency between CTVDI and P as well as SM indicators, which can be referred from Tab. 4, line 426 and 437 in the revised manuscript. And (3), we also compared the historical drought recognition results through the time-variant integrated drought index CTVDI with historical statistics, which presented good consistence especially for the typical severe drought events in 1966, 1978 and 2001, which can be referred from Tab. 7 and line 513-526 for details in the revised manuscript.
Nevertheless, just as the review’s comment, the abovementioned analysis are all applicability discussion of the time-variant integrated drought index CTVDI from the perspective of statistics, and further exploration focusing on the sensitivity analysis of time-variant drought index CTVDI compared with other single drought indicator will be conducted in the future work, which have also been presented in the conclusion section, and can be referred from line 566-575.
- Despite the excessive implementation steps in terms of time and effort, the ease factor of an index can often be an advantage, so what makes the new index more important than the others mentioned? Do you have a reason for that? Response studies and risk assessment interpreted by dual indicators are of great importance in the case of meteorological indices. Can this index be more sensitive to the assessment of losses? Should be applied like this process and then discussed in depth.
Response: Thanks very much for the reviewer’s valuable suggestions, as the descriptions mentioned for the review’s last comment, this manuscript primarily focuses on the determination of time-variant parameters as well as its combination scenarios, the derivation of integrated time-variant drought index CTVDI and its reliability verification from the perspective of statistics. Besides, much work focusing on the sensitivity analysis of time-variant drought index compared with single drought indicator will be conducted in the future work, and this has also been provided in the conclusion section of the revised manuscript, and can be referred from line 566-575. In addition, I think it’s true that dual drought indicators will be more accurate and sensitive in the application field of meteorological drought hazard system analysis, but is not suitable in the field of drought disaster risk analysis. Because drought disaster risk analysis places more emphasis on the uncertainty quantification of socio-economic losses caused by drought events, and it is a challenge to derive the time-variant PDFs of quantitative indicators of drought disaster loss, which always has short sample series length and poor statistical characteristics.
- You selected five types of two-parametric distribution functions. Why did you not use three-parametric distribution functions such as the log-logistic probability distribution?
Response: Thanks very much for the reviewer’s valuable comment, considering the calculation simplicity and inspired by reference [26] and [28], five types of two-parametric distribution functions including NO, LOGNO, WEI, GA and GU were applied to derive the PDFs of different single drought indicators in this study, and more discussion focusing on the application of three-parametric distribution functions to derive the PDFs of dingle drought indicators will be conducted in the future work.
- What is the threshold value from which the characteristics of the drought were studied? Through the run theory, more details about the run theory should be clarified.
Response: Thanks very much for the reviewer’s valuable suggestions, more details related to run theory has been provided in the revised manuscript, and the threshold values of parameters R1, R2 and R3 of run theory are 0, -0.5 and -1 respectively to recognize historical drought process samples, which can be referred from line 292-302 in the revised manuscript.
- Section 2.1: Why you used spatial kriging interpolation calculation and not other methods
Response: Thanks very much for the reviewer’s valuable comment, the related citation has been provided in the revised manuscript, meanwhile, the spatial kriging interpolation approach is the most typical and widely applied method to access the spatial data series in hydrology research field, we selected the spatial kriging interpolation approach to access the historical precipitation and soil moisture data series of city scales through Arc GIS tool, and some other interpolation methods to access the spatial data sets and their differences will be discussed in the future work.
- line 116-123, you should add citations to this sentence.
Response: Thanks very much for the reviewer’s valuable suggestions, the citations of the related descriptions of line 121-125 have been added, more details can be referred from a-a in the revised manuscript.
- Check all abbreviations, especially in lines 56-70. Add the full name the first time.
Response: Thanks very much for the reviewer’s valuable comments, we have carefully checked all the abbreviations of the manuscript, and their full names all have been provided when been used for the first time in the revised manuscript.
- There are some typos in the texts; the English and grammar check should be edited.
Response: Thanks very much for the reviewer’s valuable comment, the revised manuscript has been carefully checked and modified by authors and other cooperators with research background in English-speaking countries, and many writing, grammar and language mistakes have been modified and improved, which have been colored in red in the revised manuscript.
Author Response File: Author Response.pdf
Round 2
Reviewer 2 Report
I don't have any additional comments. I think the article is suitable for publication in present form.
