Probable maximum tropical cyclone parameters for east and west coast of India

In this discussion, the authors will demonstrate that the large discrepancy of results in Kumar et al. (Nat Hazards 1–19, 2022) using different fitting methods is a cause of inappropriate implementation of some methods. The authors will also show the significant overestimate of the higher return period wind speeds using the basin-wide maximum sustained wind speeds compared to the results using simulations.


Introduction
The paper by Kumar et al. (2022, https:// doi. org/ 10. 1007/ s11069-022-05773-2), referred to as the "paper" thereafter, discusses the probable maximum tropical cyclone (PMTC) parameters for India's east and west coasts using historical cyclone information. The recommended PMTC parameters are questionable due to ignorance of the location of the maximum sustained wind speed, inappropriate fitting methods, and inconsistency between using future projected Sea Surface Temperature and historical records.
Tropical cyclone wind hazards have been assessed for the North Atlantic regions (e.g., Vickery et al. 2000Vickery et al. , 2009Li and Hong 2014), western North Pacific regions (e.g., Hong 2015, 2016), east Indian coastal line , east coastal areas of Arabian Peninsula , and Australia (Harper and Mason 2016). These works considered information from the best track dataset in the physics-based simulation, but none used the maximum sustained wind speed as the input. Using the technique introduced by Li and Suresh Kumar (2021), this discussion investigates the impact of using the maximum sustained wind speeds on a site-specific wind hazard assessment. Moreover, this discussion shows the performance of the Lieblein BLUE method with considering the Block Maximum values and the ordinary least square method with considering the independent storm method.

Data and probability distribution
The cyclone information obtained from the International Best Track Archive for Climate Stewardship (IBTrACS) (Knapp et al. 2018) for the North Indian Ocean since 1952 was used in the following analysis. Several locations along the eastern coast of India were selected (Fig. 1a), with all historical cyclones within a radius of 250 km for each site (Fig. 1b).
The first practice was to appreciate the proper probability distribution of the maximum sustained wind speeds and the central pressure deficit for cyclones in the Bay of Bengal (Fig. 1c, including historical cyclones within a radius of 1000 km centered at the Bay of Bengal). Some landfalling storms are included but would not affect the conclusions as their intensities decay significantly.
Several methods can be used for extreme value analyses, such as block maximum and independent storm approaches. Gumbel distribution and Generalized Extreme Value (GEV) distribution can be considered with the block maximum method ) and independent storm approach (Cook and Harris 2004 Past studies (e.g., Cook and Harris 2004) have shown that either wind speeds (w = 1) or wind pressures (w = 2) can follow a Gumbel distribution, while the variation of the w depends on the best fit to the data.
For determining the Gumbel distribution parameters, past studies Li 2013, 2014); , have shown that the Generalized Least Square (GLS) method, also known as Lieblein BLUE, is the most appropriate method. Given the BLUE coefficients, the Gumbel distribution parameters can be determined by, where the c u,i and c a,i are BLUE coefficients, and x i:n is the i-th ascendingly ordered extreme value in total samples of n.
The work by Lieblein (1974) only provides BLUE coefficients up to 16 samples, with an extension by Cook (1985) to 24 samples and by  to 100 samples.  provided improved plotting positions derived from the exact BLUE coefficients up to a few hundred, which can be used with the ordinary least square method and have comparable performance to BLUE. The plotting position can be expressed by, The ordered extreme variables are defined by, When the sample size becomes small, the sample size effect needs to be considered Li 2018).
The ordinary least square (OLS) method was considered to fit Eq. (2) with the plotting positions determined by (Harris 2009 where y (1) associates with the largest value, N is the number of years of the total records, and i denotes the i-th descendingly ordered extreme variables.
3 Fitting distribution for the peak storm central pressure deficit and the peak maximum sustained wind speeds The peak values of the maximum sustained wind speeds and central pressure deficit associated with each cyclone were extracted and presented as a circle in Fig. 2. Similarly, the annual maximum values for the sites were extracted and shown as squares in the same plots. As can be observed, the extracted values mostly overlap in the plots. The independent storm method has more samples at the lower tail of the distribution. The fitted Gumbel distribution, using BLUE, is presented as a dashed line in the plots. The fitted penultimate extreme value distribution, using OLS, is shown as a solid line. For the central pressure deficit, the best-fitted w is 0.993. Therefore, w = 1 is finally used. For the wind speeds, the best-fitted w = 1.18 is adopted in the analysis below. The results show that the difference between the two fitted distributions is very small. The estimated 10,000-year return period central pressure deficit is 258 mbar and 247 mbar for using Eq. (1) and Eq. (2), respectively.
The difference between these two values is only about 4%. The estimated 10,000-year return period wind speed is 554 km/h and 534 km/h using Eqs. (1) and (2), respectively. Again, the difference is only about 4%. These differences appear very small for such a high return period. These values are generally comparable to the best values presented in the paper for the Bay of Bengal, which are 273 mbar and 559 km/h. However, the paper showed that the difference between various methods, especially the "graphical method," which is the least square method, and the "numerical method," which applies limited BLUE coefficients, can be substantial.

Higher return period cyclone wind speeds for coastal cities along the eastern coastal line of India
As mentioned in the paper, the extremely large values of the 10,000-year return period wind speeds could be implausible. The following discussion extended the analysis by Li and Suresh Kumar (2021) to a higher return period. Details of the cyclone risk assessment method can be referred to their work. Three exercises were carried out. The first analysis used the technique presented by Li and Suresh Kumar (2021) to predict the extreme cyclone-induced wind speeds at the sites presented in Fig. 1a and considered Eq.
(2) with varying w. The second analysis used the maximum sustained wind speeds within the 250 km radium centered at each site and considered Eq.
(2) with w = 1. The third analysis is similar to the second but uses w = 2. A few examples of the simulated data (for analysis 1) and maximum sustained wind speeds (for analyses 2 and 3) are presented in Fig. 3 with corresponding distribution fittings. Li and Suresh Kumar (2021) found that a w = 1 is adequate for most sites to fit the maximum sustained wind speeds. However, it can lead to an implausible large return period value at a very high return period. Therefore, fittings with w = 2 are only included for comparison purposes. Figure 4 presents the results obtained using the maximum sustained wind speeds with w = 1 shown as a solid red line with the circle, and the results with w = 2 are presented as dash lines. Results using the simulation method , denoted as Prediction, are shown as a solid black line with uncertainty bounds (+ -/10%) as black dot lines. There exhibit significantly large differences among the results from these three analyses. Using w = 1 for fitting the maximum sustained wind speeds can lead to extremely large values, although it is the value close to the best-fitted parameter. For the 1000-year return period, the results for w = 2 and using the maximum sustained wind speed have an average value of 273 km/h and a maximum value of 297 km/h. These values become 318 km/h and 346 km/h for the 10,000-year return period.
These results are close to the recommended value for the Bay of Bengal in the paper. However, Fig. 4 shows that these results are substantially higher than the results using the simulation technique. Considering the upper uncertainty bound, the maximum value for the simulation results is about 216 km/h and 286 km/h for the 1000-year and 10,000-year return periods, respectively. The average values become 170 km/h and 206 km/h for the 1000-year and 10,000-year returns, respectively. There is a 20% to 25% difference between the maximum value estimated using the simulation site-specific technique and the maximum value using the maximum sustained wind speed. The differences become about 40% if the average value of the 10,000-year return period wind speeds for all sites are compared. These differences appear substantially large for engineering practice.

Remarks and conclusions
Wind hazard assessment is critical for structure and infrastructure near the coast. However, considering possible climate change adaptation could require a potential increase of design conditions in future climate (Li et al. 2022;Li 2023) and the urgency to reduce the carbon footprint of grey structures, e.g., concrete and steel structures, appropriate design value that meets the target safety requirement shall be deliberately determined. The storm surge close to the coastline is mostly associated with the cyclones near the coastline but less correlated to the storms far from the coastline. Therefore, it is still important to use the storm information close to the coastline and better to apply simulation techniques (e.g., Vickery et al. 2008;Li and Hong 2014, 2015Li and Suresh Kumar 2021) to model the nearshore tropical cyclone wind forces used by the storm surge modeling. Using maximum sustained wind speeds for a specific site wind hazard assessment can lead to substantial conservatism. Funding The authors declare that they received no external funding for this research.

Conflict of interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported.

References
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