Comprehensive multivariate characterization of tropical cyclone and its district-level exposure characteristics over India
Univariate return periods of tropical cyclones
As illustrated in Fig. 6 in the “Methods” section, following a quasi-Lagrangian approach, we prepared a continuous and consistent input data required for the study, i.e., maximum wind speed (MWS), overland total precipitation (TP), and sea surface temperature (SST) for each timestep of 580 tropical cyclones (TCs) in the North Indian Ocean. A set of marginal cumulative distribution functions (CDFs) was constructed for each individual TC parameter to characterize their statistical distributions. The MWS was best described by a Generalized Extreme Value distribution (having shape parameter k = 0.2364, scale parameter σ = 12.4071, location parameter μ = 31.8696) and TP by a Gamma distribution (shape parameter k = 0.4969, scale parameter σ = 4890.2984). Fit assessments are detailed in Supplementary Table S3 and Supplementary Fig. S1.
Univariate return periods for MWS and TP calculated using Eq. 5 (refer to the “Methods” section) for 580 TCs in the North Indian Ocean revealed distinct variability between individual hazards (Fig. 1). Specifically, 294 TCs had a higher TP return period, while the remaining 286 TCs had a higher MWS return period, showing that neither TP nor MWS is always higher. Notably, 41 TCs in the past 70 years exhibited MWS return periods over tenfold those of TP, whereas 37 TCs exhibited the reverse. For example, Cyclone GAY (November 1989, Bay of Bengal) had extreme MWS (Category 5) but minimal over-land TP (return period = 0.16 years), whereas Cyclone 1997232N20088 (August 1997, Bay of Bengal) exhibited low MWS (return period = 0.18 years) but extreme TP (return period = 10 years) over land. These results underscore that univariate analyses may misrepresent joint hazard characteristics.
Fig. 1: Univariate return period for cyclonic hazards in the North Indian Ocean.
Univariate return period of a Maximum Wind Speed (MWS) and b over land Total precipitation (TP) observed by the 580 historical cyclone events during 1951–2020 in the Bay of Bengal (blue) and Arabian Sea (violet) basin. The distribution of return period of observed TCs during 1951–2020 based on c maximum wind speed and d total precipitation.
Joint return period of TCs
To capture the dependence structure among TC characteristics, copula functions were employed. Three Archimedean copula models (Clayton, Frank, Gumbel) were evaluated using the Kolmogorov–Smirnov (K–S) test at a 5% significance level, with the Clayton copula emerging as the best fit based on multiple goodness-of-fit criteria, including AIC-BIC and OLS scores (Supplementary Table S4). The estimated parameter for the selected Clayton copula (θ = 2.3474) indicates moderate lower-tail dependence, suggesting a strong association between low values of the marginal distributions of MWS and TP. To assess the robustness of this copula choice, a nonparametric bootstrap procedure with 1000 resamples was performed, resulting in a 95% confidence interval of [1.6253, 3.1171], confirming the stability and statistical significance of the parameter estimate. The corresponding probability density and cumulative distribution functions are presented in Supplementary Fig. S3.
Using Eqs. (6–7) of the “Methods” section, two-dimensional joint return periods, \({RP_\cup}\) (X1 ≥ x1 or X2 ≥ x2, i.e., either MWS or TP exceeding the threshold), and RP∩ (X1 ≥ x1 and X2 ≥ x2, i.e., both exceeding thresholds) were computed for 580 historical tropical cyclones (Fig. 2). All \({RP_\cup}\) return periods (Fig. 2a) were below 1 year, except for TC ‘1972251N15097’ (September 1972, Bay of Bengal, \({RP_\cup}\) = 1.23 years). In contrast, the RP∩ return period varied significantly, with 246 cyclones exceeding 1 year, 144 exceeding 2 years, 27 exceeding 10 years, 7 exceeding 25 years, and 2 exceeding 50 years. The maximum RP∩ was 53.3 years (Fig. 2b) for Category 5 TC ‘1990124N09088’ (1990), which exhibited moderate precipitation over land.
Fig. 2: Joint return period for cyclonic hazards in the North Indian Ocean.
a Joint return period \({RP_\cup}\) (X1 ≥ x1 or X2 ≥ x2) of extreme MWS and TP observed by the cyclone events, and b Joint return period RP∩ (X1 ≥ x1 and X2 ≥ x2) of MWS and TP observed by the cyclone events during 1951–2020 in the Bay of Bengal (blue) and Arabian Sea (violet) basin. The distribution of joint return period c \({RP_\cup}\) of observed TCs based on either MWS or TP, and d RP∩ of observed TCs based on both MWS and TP. Contours of joint return period e \({RP_\cup}\) and f RP∩ for historical TCs in NIO (1951–2020).
Further exploration of hazard characteristics revealed that high RP∩ was associated with cyclones where both MWS and TP were moderately high, whereas high \({RP_\cup}\) occurred when either hazard was severe (Fig. 2e, f). TCs with only one severe hazard did not exhibit high RP∩. These findings underscore the limitations of single-hazard analyses, which may misrepresent joint hazard risks, particularly for cyclones with one extremely severe hazard and another moderately serious hazard.
Supplementary analyses characterize average sea surface temperature (SST) during TC events in the NIO. The study examines mean SST along TC tracks within a 3° radius and explores its frequency analysis using a Gamma fit (Fig. S2). Joint return period contours were mapped to assess relationships between SST, maximum wind speed, and total precipitation, revealing that high SST does not necessarily align with extreme MWS or TP (Fig. S4). The results suggest that moderate wind speed and high rainfall are common during high SST conditions. A nonparametric correlation analysis further indicates weak relationships between SST and TC intensity metrics, with relatively stronger correlations observed for basin-averaged SST (SSTb) and SST on the day of cyclogenesis (SSTcb) (Fig. S5). These findings, supported by composite SST plots and correlation matrices, provide deeper insights into the interplay between SST and TC characteristics in NIO.
District-wise tropical cyclone exposure characteristics
Notably, tropical cyclone (TC) characteristics alone do not capture their local impacts over a location. To quantify the spatial variations in TC hazard propensity, we performed a district-wise analysis of cyclone exposure characteristics over a longer historical period (1901–2020) across India. Adopting a quasi-Eulerian framework (as graphically illustrated in Fig. 7a, “Methods” section), the frequency and duration of TC exposure were estimated for each district, providing a spatial assessment of TC influence.
Figure 3a illustrates the historical tracks of tropical cyclones over the North Indian Ocean (NIO) from 1901 to 2020, where the color of each track represents the peak maximum sustained wind speed achieved by the cyclone. The spatial distribution of cyclone frequency (Fig. 3b) reveals distinct regional patterns across districts in India. Regions including the state of Odisha, West Bengal, Jharkhand, Chhattisgarh, and Madhya Pradesh experienced more than two cyclones per year (i.e., over 240 cyclones in 120 years). Specifically, districts in Odisha such as Mayurbhanj, Baleshwar, Kendujhar, Bhadrak, Kendrapara, Jajapur, and Sundargarh, and districts in West Bengal such as Pashchim & Purba Medinipur and South & North 24 Parganas, as well as Pashchimi Singhbhum in Jharkhand, witnessed more than three cyclones annually. Other states, including Bihar, Uttar Pradesh, Rajasthan, Andhra Pradesh, and Maharashtra, experienced one to two cyclones annually. Districts in Andaman and Nicobar, Telangana, Tripura, Mizoram, Gujarat, Puducherry, and Tamil Nadu reported one cyclone on average every one to two years. Interestingly, inland states such as Jharkhand, Chhattisgarh, Madhya Pradesh, Bihar, Uttar Pradesh, Rajasthan, and Maharashtra exhibited annual frequencies ranging from one to three cyclones per year, indicating a broader regional influence of tropical cyclones beyond just coastal areas.
Fig. 3: Tropical cyclone exposure characteristics in the North Indian Ocean.
a Tropical cyclone tracks in the North Indian Ocean (NIO) from 1901 to 2020 considered for this study. The color of lines denotes the peak maximum sustained wind speed reached by the cyclone. b Number of cyclone exposures per year, and c Average duration of cyclone exposure per year for each district in India (1901–2020). d Shannon surprise to cyclone events for districts in India (1901–2020). Administrative boundaries are represented by black lines of varying thickness: country borders are the thickest, followed by state borders, with district boundaries shown using the thinnest lines.
Beyond cyclone frequency, the temporal extent of exposure provides crucial insights into the intensity and persistence of cyclone impacts at the district level. Figure 3c presents the annual average duration of cyclone exposure by districts in India during period 1901–2020, defined as the presence of cyclone centers within 100 km of a district (as graphically illustrated in Fig. 7a). Districts in Odisha, West Bengal, Jharkhand, Chhattisgarh, and Madhya Pradesh experienced prolonged exposure exceeding 48 h (2 days) annually under strong cyclone influence. Within these states, districts such as Mayurbhanj and Baleshwar in Odisha and Pashchim Medinipur, South and North 24 Parganas, and Purba Medinipur in West Bengal endured over 72 h (3 days) of annual cyclone exposure. Other districts in Odisha, including Kendujhar, Sundargarh, Kendrapara, Bhadrak, Jajapur, Puri, and Cuttack, as well as districts in Jharkhand (e.g., Pashchimi Singhbhum, Purbi Singhbhum, and Ranchi), and West Bengal (e.g., Bankura, Puruliya, Barddhaman, Haora, and Hugli), experienced cyclones over 48 h (2 days) annually. States such as Bihar, Uttar Pradesh, Chhattisgarh, Rajasthan, and Madhya Pradesh experienced over 24 h (1 day) of annual exposure, whereas Andhra Pradesh, Maharashtra, Andaman and Nicobar, Telangana, Lakshadweep, Gujarat, and Tamil Nadu experienced TC exposure over 12 h annually.
Shannon surprise to the cyclone event
To assess temporal variability in cyclone occurrence and its implications for risk perception and preparedness, we applied the Shannon surprise metric. This approach quantifies the rarity of cyclone events at a given location, identifying regions with high unpredictability in cyclone occurrence. Figure 3d presents the Shannon Surprise metric for cyclone events for all districts in India (1901–2020), which quantifies the rarity of cyclone occurrences across districts. States including Jammu and Kashmir, Himachal Pradesh, Punjab, Chandigarh, Uttarakhand, Sikkim, Arunachal Pradesh, and Nagaland encountered the highest surprise (Shannon Surprise > 6, with fewer than 10 cyclones in 120 years) from tropical cyclone events. Some of these locations, like Sikkim, Jammu and Kashmir, Himachal Pradesh, Chandigarh, and Arunachal Pradesh, tropical cyclone return periods were rarer than 30 years (surprise > 7.4).
Conversely, states such as Odisha, West Bengal, Jharkhand, Chhattisgarh, Madhya Pradesh, Uttar Pradesh, Andhra Pradesh, Bihar, Rajasthan, and Maharashtra experienced the least surprise (Shannon Surprise
Notably, intra-state variability in surprise values highlights differential cyclone exposure within the same state. For instance, northern Bihar and Uttar Pradesh exhibited high surprise, whereas southern districts of these states experienced more frequent cyclone occurrences. Similarly, the states of West Bengal, Madhya Pradesh, Haryana, Assam, Punjab, Odisha, Jharkhand, Chhattisgarh, Andhra Pradesh, Rajasthan, and Maharashtra displayed high spatial variation of surprise to tropical cyclones among their districts, suggesting non-uniform risk distribution within states. Table S5 in Supplementary Document lists state-wise the name of the district with minimum and maximum surprise and Shannon surprise value.
Surprise from non-stationarity
To assess temporal variability in cyclone occurrence and its implications for risk perception and preparedness, we applied statistical distance measures to quantify deviations from expected cyclone patterns at specific locations. This approach identified regions where traditional assessments may underestimate risk due to temporal shifts in cyclone behavior. Given the evolving climate conditions, assessing changes in cyclone event frequency over time is critical for understanding potential non-stationarity. Statistical distances were employed to measure surprise from non-stationarity in the distribution of cyclone event frequency in this study, that is, to quantify changes in cyclone event frequency between the historical (1901–1980) and recent (1981–2020) periods. Figure 4 presents the results of five statistical distance measures (“Methods” section, Eqs. 10–14): Kullback–Leibler (KL) Divergence, Hellinger Distance (HD), Bhattacharyya Distance (BD), Total Variation Distance (TVD), and Wasserstein Metric (WM or Earth Mover’s Distance). The results reveal significant geographic variability in the magnitude of changes across India. Out of 667 districts in India, 376 districts experienced at least one cyclone in a year for more than 30 years during the study period (1901–2020) (Fig. 4a). Other districts with fewer than 30 distinct years of cyclone events were excluded from statistical distance calculations.
Fig. 4: District-wise statistical distances.
a Number of years with at least one TC event in the period 1901–2020. Statistical Distances Measuring Surprise b Kullback–Leibler (KL) Divergence, c Hellinger Distance (HD), d Bhattacharyya Distance (BD), e Total Variation Distance (TVD), and f Wasserstein Metric (WM) from non-stationarity in cyclone event frequency distribution for cyclone event frequencies between the historical (1901–1980) and recent cyclone (1981–2020) event for all districts. Administrative boundaries are represented by black lines of varying thickness: country borders are the thickest, followed by state borders, with district boundaries shown using the thinnest lines.
The Kullback–Leibler (KL) Divergence (Fig. 4b) indicates high surprise (KL > 0.5) due to changes in cyclone frequency distribution in approximately 3% of districts, particularly in Chhattisgarh, Rajasthan, Andhra Pradesh, Tamil Nadu, Tripura, Jharkhand, and Odisha, suggesting notable shifts in cyclone frequency distribution. Thirty-seven districts showed moderate surprise (KL Divergence > 0.2), and about one-fifth of all districts considered exhibited low surprise (0.09 4c) highlights significant changes in cyclone frequency in 17 districts, with the highest surprise values (HD > 0.3) in Chhattisgarh, Tamil Nadu, Jharkhand, Odisha, Bihar, Uttar Pradesh, Tripura, Andhra Pradesh, and West Bengal. Eighty-one districts (about one-fifth of all districts considered) experienced moderate surprise (0.2 4d) further corroborates these findings, with 15 districts in Uttar Pradesh, Bihar, West Bengal, and Gujarat exhibiting substantial deviations (BD > 2) from historical cyclone frequency distributions. About half of all districts (183 districts) exhibited low surprise (1.5
Additional measures, such as Total Variation Distance (TVD, Fig. 4e) and the Wasserstein Metric (WM, Fig. 4f), provide further confirmation of non-stationarity, with districts in Tripura, Chhattisgarh, Madhya Pradesh, Tamil Nadu, Andhra Pradesh, Jharkhand, Odisha, Andaman & Nicobar Islands, Rajasthan, Uttar Pradesh, Bihar, and Puducherry experiencing significant alterations (TVD > 0.1 and WM > 0.01) in cyclone frequency. Fourteen districts exhibited moderate surprise (0.05
All five statistical distance measures consistently indicate a positive shift in cyclone frequency distributions from the historical (1901–1980) to recent (1981–2020) periods, indicating a directional shift away from historical stationarity. These changes are particularly pronounced in eastern, coastal, and some central districts of India. While many districts remain stable, a notable number exhibit moderate to high changes, highlighting emerging cyclone risk in previously less-affected areas. These findings emphasize the non-stationarity of cyclone hazards and the need for dynamic, region-specific risk assessments.
Fraction attributable risk (FAR)
To quantify changes in the probability of extreme cyclone exposure, as graphically illustrated in Fig. 7b, we assess the fraction attributable risk (FAR) of districts experiencing annual cyclone exposure exceeding 12 and 24 h (Fig. 5). Compared to the historical period (1901–1980), several districts exhibit substantial increases in exceedance probability in the recent period (1981–2020). On average, a district experiences between one cyclone every four years to up to four cyclones in a single year, depending on its geographic location (Fig. 3b). Typically, a single TC event contributes between 3 and 27 h of exposure, with an average of approximately 14 h.
Fig. 5: District-wise Fraction Attributable Risk (FAR) for tropical cyclone exposure across India.
Fraction Attributable Risk (FAR) for hazards from cyclone events with durations more than a 12 h and b 24 h for recent cyclone events (1981–2020) compared to the historical (1901–1980) event for all districts. c Distribution (histogram) of FAR values across districts in India for cyclone event with duration more than 12 h (red) and 24 h (blue). Administrative boundaries are represented by black lines of varying thickness: country borders are the thickest, followed by state borders, with district boundaries shown using the thinnest lines.
For annual cyclone exposure exceeding 12 h (Fig. 5a), districts in Tamil Nadu (Virudunagar, Perambalur, Madurai, Tiruppur), Puducherry (Karaikal), and Mizoram (Champhai) experienced the most significant increases, with FAR values exceeding 0.4, corresponding to a more than 66% increase in risk. Overall, 22 districts recorded an increase of more than 25% in cyclone risk (FAR > 0.2), while 18 districts observed a moderate to small increase in risk (FAR between 0 and 0.2). Conversely, a substantial decrease in risk was observed in several regions, particularly in Uttar Pradesh, Bihar, Rajasthan, West Bengal, Gujarat, and Madhya Pradesh, where FAR values were negative, indicating a reduction in cyclone exposure risk. 41 districts experienced up to a 33.33% reduction in risk (FAR between −0.5 and 0), 94 districts observed up to a 50% decrease (FAR between −1 and 0), and 72 districts had more than a 50% decrease in risk (i.e., FAR
For annual cyclone exposure exceeding 24 h (Fig. 5b), an even stronger spatial variation emerges. Districts in states such as Mizoram (Champhai and Aizawl), Tamil Nadu (Virudunagar, Madurai, Sivaganga, Pudukkottai, and Ramanathapuram), and Puducherry (Karaikal) experienced FAR values exceeding 0.5, indicating a more than 100% increase in exceedance probability. In contrast, 54 districts observed less than a 50% decrease in risk (FAR between −1 and 0), 233 districts exhibited a significant decline in risk (50%–90% decrease, FAR between −1 and −9) and 57 districts saw a decrease of more than 90% (FAR
