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Extreme weather events are important factors in designing infrastructures in Hong Kong (Peterson, 1981; Lam and Leung, 1994), given their substantial impact on human life. In order to rationalize and standardize the expressions for extreme weather events so that individuals, countries and regions can calculate the related indices in exactly the same way such that their analyses can fit together seamlessly, the Expert Team on Climate Change Detection, Monitoring and Indices (ETCCDMI) working under the joint World Meteorological Organization (WMO), Commission for Climatology (CCl)/World Climate Research Programme (WCRP) and Climate Variability and Predictability (CLIVAR) project (Peterson et al., 2001; Peterson, 2005) have been coordinating an international effort to develop, calculate and analyse a suite of indices. These indices are mainly based on daily temperature values or daily precipitation amounts. Some of them are based on fixed thresholds that are of relevance to particular applications.

This study examined the long-term trends in the frequency and intensity of extreme temperature and rainfall events in Hong Kong using a suite of 27 extreme indices adopted from the core indices developed by ETCCDMI with appropriate modifications to suit the sub-tropical climate of Hong Kong.

2. Meteorological observations and extreme indices

2.1. Meteorological observations

The Hong Kong Observatory Headquarters (HKOHq), located at the urban centre of Hong Kong, has been conducting temperature and rainfall observations continuously since 1885 except for a break during World War II (1940–1946). Leung et al. (2004) and Lee et al. (2006) studied the long-term trends of the annual mean temperature and annual total rainfall recorded at HKOHq. Their results indicated that both the annual mean temperature and total rainfall have increased over the past century under the effect of global warming and local urbanization. In this study, the daily maximum and minimum temperatures and hourly rainfall data recorded at HKOHq from 1885 to 2008 (except 1940 to 1946) were used to calculate and analyse the extreme indices for extreme weather events in Hong Kong.

Before conducting detailed analysis of the extreme indices, the metadata of the temperature and rainfall data collected at the HKOHq used for this study were examined for their homogeneity. According to the metadata for temperature and rainfall measurements from 1885 to 2008, there were slight changes (within 20 m) in the locations of the instruments used for temperature measurements in 1912 and 1933 as well as rainfall collection in 1933 (HKO, 1914; HKO, 1934; Lee et al., 2006). The standard normal homogeneity test (Alexandersson, 1986) was used to test the homogeneity of the time series of the annual mean temperature and annual total rainfall at the HKOHq from 1885 to 2008. Results showed that there was no break points in the years 1912 and 1933 at the 5% level, indicating that the impact of the minor changes in the measurement positions to the homogeneity of temperature and rainfall data collected could be considered negligible.

2.2. Extreme indices

A suite of 27 extreme indices to indicate the trend and significance of extreme temperature and rainfall events in Hong Kong were selected for this study. Of the 27 indices, 21 of them were adopted from the list of extreme indices proposed by the ETCCDMI (http://cccma.seos.uvic.ca/ETCCDI/indices.shtml) with appropriate modifications to suit the sub-tropical climate of Hong Kong.

2.2.1. Temperature-related extreme indices

CD12, SU33 and TR28, respectively represent the annual counts of cold days (daily minimum temperatures ≤ 12.0 °C), very hot days (daily maximum temperature ≥ 33.0 °C) and hot nights (daily minimum temperature ≥ 28.0 °C) in Hong Kong.

TX_{x}, TN_{x}, TX_{n} and TN_{n} are the annual highest maximum temperature, highest minimum temperature, lowest maximum temperature and lowest minimum temperature recorded every year, respectively.

TN10p is an index measuring the percentage of time with daily minimum temperatures lower than the 10th percentile of minimum temperatures calculated for each calendar day (with reference to the climatological normal) using a running 5-day window. This is a measure of the percentage of cool nights in a year. Similarly, TX10p is an index showing the percentage of cool days. TN90p and TX90p are the indices corresponding to the percentage of warm nights and warm days in a year, respectively.

The warm spell duration index (WSDI) is the annual count of days with at least six consecutive days with the daily maximum temperature above the 90th percentile of the maximum temperature calculated for each calendar day (with reference to the climatological normal) using a running 5-day window. The cold spell duration index (CSDI) is the annual count of days with at least six consecutive days with the daily minimum temperature below the 10th percentile of the minimum temperature calculated for each calendar day (with reference to the climatological normal) using a running 5-day window.

A list of the 13 temperature-related extreme indices, their definitions and modifications from those proposed by ETCCDMI, if any, are summarized in Table AI.

2.2.2. Rainfall-related extreme indices

As about 85% of the annual rainfall was recorded in the summer months from April to September in Hong Kong, in this study, some of the rainfall-related extreme indices are compiled using the rainfall data recorded from April to September only.

RX1hour, RX2hour, RX3hour, RX6hour, RX12hour, RX1day and RX5day are the annual maximum 1-h, 2-h, 3-h, 6-h, 12-h, 1-day and 5-day rainfall amounts recorded every year.

The simple daily precipitation intensity index [SDII(4–9)] is obtained by dividing the total rainfall from April to September by the number of wet days (daily rainfall ≥ 1 mm) from April to September in a year, while R30 is the annual count of days with daily rainfall ≥ 30 mm.

The consecutive dry days (CDD) and consecutive wet days (CWD) indices are the annual maximum length of dry and wet spells, respectively counting the maximum number of consecutive days with daily rainfall less than and more than 1 mm, respectively between April and September. PRCPTOT(4–9) is the annual total rainfall recorded in wet days (daily rainfall ≥ 1 mm) from April to September.

R95p and R99p are annual total precipitation due to events exceeding the daily 95th and 99th percentile of the climatological normal, respectively. They are measures of the contribution of extreme rainfall events to the total rainfall in a year.

A list of the 14 rainfall-related extreme indices, their definitions and modifications from those proposed by ETCCDMI, if any, are summarized in Table AII.

3. Analysis methodology

3.1. Linear regression analysis

This study adopted the linear regression method to determine the trend of the extreme indices. The t-test was applied to test the statistical significance of the trends at the 5% significance level (Karl et al., 1993; Easterling et al., 1997; Storch and Zwiers, 1999).

3.2. Time-dependent generalized extreme value distribution fitting

The long-term trends of the variation of the probability of occurrence of the extreme indices determined by their annual maximum or minimum values, namely TX_{x}, TN_{x}TX_{n}, TN_{n}, RX1hour, RX2hour, RX3hour, RX6hour, RX12hour, RX1day and RX5day were determined using the time-dependent generalized extreme value (GEV) distribution technique (Coles, 2001). In a GEV distribution, there are three model parameters such as the location parameter µ, the scale parameter σ and the shape parameter ξ. For the time-dependent GEV technique, the three parameters are regarded as time-dependent quantities varying linearly with time. The principle of the time-dependent GEV distribution technique is discussed in Appendix A1.

4. Results and discussion

4.1. Temperature-related extreme indices

Our analysis showed that all 13 temperature-related extreme indices had either increasing or decreasing trends in the period from 1885 to 2008 which are statistically significant at the 5% level. The four indices of the annual highest or lowest daily maximum and minimum temperatures (TX_{x}, TN_{x}, TX_{n} and TN_{n}) were found to increase by 0.09—0.14 °C per decade. Concurrently, the indices on very hot days (SU33), hot nights (TR28), warm days (TX90p) and warm nights (TN90p) increased by 0.9 days, 1.2 nights, 0.8 and 1.1% per decade respectively, whereas those for cold days (CD12), cool days (TX10p) and cool nights (TN10p) decreased by 1.2 days, 0.8 and 1.0% per decade, respectively. Furthermore, the WSDI was found to increase by 0.4 days per decade, whereas the CSDI decreased by 1.5 days per decade. Table I summarizes the analysis results of the long-term trends of the 13 temperature-related extreme indices.

Table I. Long-term trends of the 13 temperature-related extreme indices from 1885 to 2008

Indices

Trend (per decade)

Significance at the 5% level (yes/no)

CD12

− 1.2 days

Yes

SU33

+ 0.9 days

Yes

TR28

+ 1.2 nights

Yes

TX_{x}

+ 0.09 °C

Yes

TN_{x}

+ 0.11 °C

Yes

TX_{n}

+ 0.10 °C

Yes

TN_{n}

+ 0.14 °C

Yes

TN10_{p}

− 1.0% of days

Yes

TX10_{p}

− 0.8% of days

Yes

TN90_{p}

+ 1.1% of days

Yes

TX90_{p}

+ 0.8% of days

Yes

WSDI

+ 0.4 days

Yes

CSDI

− 1.5 days

Yes

Table II shows the results of the likelihood ratio tests for the significance of time-dependency of the three parameters µ, σ and ξ for the GEV distributions of TX_{x}, TN_{x}, TX_{n} and TN_{n} from 1885 to 2008. It showed that the GEV distributions of extreme values of these four extreme temperature indices generally had significant time-dependency.

Table II. Significance of time-dependency of µ, σ and ξ of the GEV distributions of TX_{x}, TN_{x}TX_{n} and TN_{n} from 1885 to 2008

Weather elements

Significance of time-dependency (at 5% level)

µ

σ

ξ

TX_{x}

Yes

No

No

TN_{x}

Yes

Yes

Yes

TX_{n}

Yes

No

No

TN_{n}

Yes

No

Yes

Figures 1 and 2 show the long-term trends of the 10-year and 100-year return values of TX_{x} and TN_{x} as well as TX_{n} and TN_{n} from 1885 to 2008, respectively. The rates of increase in the return values for fixed return periods for TX_{x}, TN_{x}, TX_{n} and TN_{n} ranged from about 0.1 to 0.3 °C per decade. All trends were statistically significant at the 5% level.

Figures 3 and 4 show the plots of return values against return periods in 1900 and 2000 for TX_{x} and TN_{x}, and TX_{n} and TN_{n}, respectively. The return period for TN_{n}≤4.0 °C lengthened from 6 years in 1900 to 163 years in 2000. However, the return periods for TN_{x} ≥ 30.0 °C and TX_{x} ≥ 35.0 °C shortened from > 100 and 32 years in 1900 to 51 and 4.5 years in 2000, respectively.

The observed long-term trends of the 13 temperature-related extreme indices in Hong Kong were consistent with the assessment made in the Fourth Assessment Report of the Intergovernmental Panel on Climate Change (IPCC, 2007) that less frequent cold days and more frequent hot days and hot nights have been associated with global warming (IPCC, 2007). Besides global warming, the observed long-term trends of these indices in Hong Kong could also be attributed to the local urbanization as the Hong Kong observatory was situated at the heart of Kowloon which had significant development in the past 50 years or so (Leung et al., 2004; Lee et al., 2006; Wu et al., 2008). During the pre-World War II period, the annual mean temperatures at HKOHq followed more or less the same trend as the global annual mean temperatures (Figure 5). In the post-war years, there were two periods with notable temperature rises at HKOHq. The first period was from late-1940s to mid-1960s and the second period began in the 1980s. A faster rate of warming at HKOHq compared to the global trend since the mid-1980s can be attributed to the effects of high density urban development. From 1947 to 2008, the annual mean temperatures at Macao which can be considered as a rural station (Fong et al., 2009), rose by an average of 0.10 °C per decade (but insignificant at the 5% level), slightly less than the global rate of 0.14 °C for the same period. In contrast, at the heart of urban Kowloon, HKOHq recorded an average rise of 0.16 °C per decade (significant at the 5% level), with an additional rise of 0.06 °C per decade as compared with Macao because of urbanization.

It can also be seen that the trends of indices related to minimum temperatures (CD12, TR28, TNx, TNn, TN10p, TN90p and CSDI) were in general larger than those related to maximum temperatures (SU33, TXx, TXn, TX10p, TX90p and WSDI). This is in line with the findings in other studies that urbanization has more prominent effect to night-time temperatures than daytime temperatures as the reduction in the sky view factor arising from the increase in the density as well as in the height of the high rise buildings reduces the release of heat back to the space during night-time (Kalande and Oke, 1980; Oke, 1982; Grimmond, 2007) and reduces the amount of solar radiation reaching the ground surface during daytime (Karl et al., 1993; Zhou et al., 2004).

4.4. Rainfall-related extreme indices

Results of our study showed that for indices reflecting extreme rainfall amounts in fixed time intervals (1 h to 5 days), the short term extreme rainfall amounts within 1 and 2 h had statistically significant increasing trends from 1885 to 2008. The annual maximum 1- and 2-h rainfall (RX1hour and RX2hour) increased by 1.7 and 2.1 mm per decade, respectively. However, the annual extreme rainfall amounts in three hours or longer showed insignificant trends at the 5% level.

SDII(4–9) and PRCPTOT(4–9) which respectively indicate the average rainfall amounts in wet days of a year and the total rainfall in wet days in summer months increased by 0.5 mm per day per decade and 27 mm per decade respectively, both statistically significant at the 5% level. The annual number of days with rainfall exceeding 30 mm as indicated by R30 also had a statistically significant (at 5% level) rising trend of 0.2 days per decade from 1885 to 2008. For the rainfall amount in heavy rain days, R95p and R99p increased by 22 and 11 mm per decade from 1885 to 2008. Both trends were statistically significant at the 5% level. The results indicated that the contribution of extreme rainfall events to the annual rainfall amount increased with time. Concurrently, CDD(4–9) also increased at a rate of about 0.3 days per decade from 1885 to 2008. The trend was significant at the 5% level. However, CWD(4–9) decreased by 0.1 day per decade but the trend was statistically insignificant at the 5% level. Table III summarizes the analysis results of the long-term trends of the 14 rainfall-related extreme indices.

Table III. Long-term trends of the 14 rainfall-related extreme indices from 1885 to 2008

Indices

Trend (per decade)

Significance at the 5% level (yes/no)

RX1hour

+ 1.7 mm

Yes

RX2hour

+ 2.1 mm

Yes

RX3hour

+ 1.7 mm

No

RX6hour

+ 1.1 mm

No

RX12hour

+ 1.5 mm

No

RX1day

+ 2.3

No

RX5day

+ 4.0

No

SDII

+ 0.5 mm/day

Yes

R30

+ 0.2 days

Yes

CDD(4–9)

+ 0.3 days

Yes

CWD(4–9)

− 0.1 days

No

R95_{p}

+ 22 mm

Yes

R99_{p}

+ 11 mm

Yes

PRCPTOT(4–9)

+ 27 mm

Yes

The results of the likelihood ratio tests for the significance of time-dependency of the three parameters µ, σ and ξ for the GEV distributions of RX1hour, RX2hour, RX3hour, RX6hour, RX12hour, RX1day and RX5day from 1885 to 2008 showed that only the GEV distributions of the extreme values of RX1hour, RX2hour and RX3hour had significant time-dependency (Table IV).

Table IV. Significance of time-dependency of µ, σ and ξ of the GEV distributions of RX1hour, RX2hour, RX3hour, RX6hour, RX12hour and RX1day from 1885 to 2008

Weather elements

Significance of time-dependency (at 5% level)

µ

σ

ξ

RX1hour

Yes

No

No

RX2hour

Yes

No

No

RX3hour

Yes

No

No

RX6hour

No

No

No

RX12hour

No

No

No

RX1day

No

No

No

RX5day

No

No

No

Figure 6 shows the long-term trends of the 10-year and 100-year return values of RX1hour, RX2hour and RX3hour from 1885 to 2008. The return values for fixed return periods for RX1hour, RX2hour and RX3hour were found to increase by 1.1, 1.8 and 2.0 mm per decade respectively, all significant at 5% level.

Figure 7 shows the plots of return values against return periods in 1900 and in 2000 for RX1hour, RX2hour and RX3hour. The return period for RX1hour of 100 mm or above was found to shorten from 37 years in 1900 to 18 years in 2000. For RX2hour of 150 mm or above and RX3hour of 200 mm or above, the return periods were found to shorten from 32 years and 41 years to 14 years and 21 years, respectively.

Similar to the 13 temperature-related extreme indices, the observed increase in the frequency of occurrence and intensity of short-term heavy rainfall events could be attributed to both global warming and local urbanization. Global warming could enhance the surface evaporation and increase the moisture holding capacity in the atmosphere, resulting in higher chance of the occurrence of heavy rainfall events (IPCC, 2007; Archer and Rahmstorf, 2010). Furthermore, a number of recent studies have suggested that the ‘urban heat island’ effect can induce more precipitation in the urban areas (Chow, 1986; Shepherd, 2002; Dixon and Mote, 2003). The attribution to local urbanization could be supported by the observed higher rising trend of the annual rainfall over urban areas than the other parts of the territory in the study on the regional characteristics of rainfall recorded by the raingauge network of the Hong Kong Observatory over different parts of Hong Kong from 1956 to 2005 (Mok et al., 2006).

5. Conclusions

The long-term trends of the occurrences of extreme temperature and rainfall events in Hong Kong were studied using a set of 27 extreme indices based on the meteorological data recorded at the HKOHq from 1885 to 2008.

In conjunction with the observed rising trend of the annual mean temperature, the annual lowest and highest minimum temperatures as well as the annual lowest and highest maximum temperatures in Hong Kong also had statistically significant rising trends, ranging from about 0.09 °C to 0.14 °C per decade. As a result, the annual number of very hot days (SU33) and hot nights (TR28) increased significantly, while the annual number of cold days (CD12) decreased significantly. Furthermore, the annual percentage of days with daily minimum temperature lower than the 10th percentile (TN10p) had a statistically significant decreasing trend, whereas the annual percentage of days with daily minimum temperature higher than the 90th percentile (TN90p) and the annual percentage of days with daily maximum temperature higher than the 90th percentile (TX90p) had statistically significant rising trends in the same period. In this respect, the warm spell duration as reflected by the annual count of days with at least six consecutive days with daily maximum temperature higher than the 90th percentile (WSDI) was found to increase by about 0.4 days per decade, whereas the cold spell duration as reflected by the annual count of days with at least six consecutive days with daily minimum temperature lower than the 10th percentile (CSDI) decreased by about 1.5 days per decade. Results of the time-dependent GEV distribution analysis showed that the return period for TN_{n}≤4.0 °C had lengthened from 6 years in 1900 to 163 years in 2000. However, the return periods for TN_{x} ≥ 30.0 °C and TX_{x} ≥ 35.0 °C shortened from > 100 years and 32 years in 1900, to 51 years and 4.5 years in 2000, respectively. These results indicated that days with extreme high temperature have become more frequent, whereas those with extreme low temperature have become more infrequent.

Regarding extreme rainfall events, there was an observed increase in the frequency of occurrence of heavy rain events in Hong Kong since 1885 and the contribution of heavy rain to the annual rainfall amount was also increasing with time. Moreover, there was an increasing trend in the maximum number of consecutive days with daily rainfall less than 1 mm during the summer months between April and September (CDD), indicating a lengthening of dry spell duration in the summer months. The time-dependent GEV distribution analysis revealed that the return periods for the 1, 2 and 3-hourly extreme rainfall had decreased significantly from 1885 to 2008. The return period for an hourly rainfall of 100 mm or more was found to have shortened from 37 years in 1900 to 18 years in 2000. These trends suggested that extreme short-term rainfall events in Hong Kong had become more frequent and the rainfall in Hong Kong tended to concentrate in more intense episodes with longer periods of little precipitation in between.

Appendix

Table AI.. Definition of the 13 temperature-related extreme indices used in this study

Extreme indices

Definition

Remarks/deviation from those developed by ETCCDMI (if any)

Unit

CD12 (cold days)

Annual number of days with minimum temperature ≤12.0 °C

Not proposed by ETCCDMI; days with minimum temperature ≤12.0 °C are defined as cold days in Hong Kong

Days

SU33 (very hot days)

Annual number of days with maximum temperature ≥ 33.0 °C

Threshold changed from 25 to 33 °C; days with maximum temperature ≥ 33.0 °C are defined as very hot days in Hong Kong

Days

TR28 (hot nights)

Annual number of days with minimum temperature ≥ 28.0 °C

Threshold changed from 20 to 28 °C; days with minimum temperature ≥ 28.0 °C are defined as hot nights in Hong Kong

Days

TX_{x}

Annual highest maximum temperature

Same as those developed by ETCCDMI

°C

TN_{x}

Annual highest minimum temperature

TX_{n}

Annual lowest maximum temperature

TN_{n}

Annual lowest minimum temperature

TN10p (cool nights)

Percentage of days when daily minimum temperature < 10th percentile:

Reference period changed from 1961–1990 to 1971–2000

% of days

Let TN_{ij} be the daily minimum temperature on day i in period j and let TN_{in}10 be the calendar day 10th percentile centred on a five-day window for the base period 1971–2000. (The percentage of time is determined where: TN_{ij} < TN_{in}10)

TX10p (cool days)

Percentage of days when daily maximum temperature < 10th percentile:

Reference period changed from 1961–1990 to 1971–2000

% of days

Let TX_{ij} be the daily maximum temperature on day i in period j and let TX_{in}10 be the calendar day 10th percentile centred on a 5-day window for the base period 1971–2000 (the percentage of time is determined where: TX_{ij} < TX_{in}10)

TN90p (warm nights)

Percentage of days when daily minimum temperature > 90th percentile:

Reference period changed from 1961–1990 to 1971–2000

% of days

Let TN_{ij} be the daily minimum temperature on day i in period j and let TN_{in}90 be the calendar day 90th percentile centred on a five-day window for the base period 1971–2000 (the percentage of time is determined where: TN_{ij} > TN_{in}90)

Table AII.. Definition of the 14 rainfall-related extreme indices used in this study

Extreme indices

Definition

Remarks/deviation from those developed by ETCCDMI (if any)

Unit

RX1hour

Annual maximum 1-h rainfall

Not developed by ETCCDMI but have significant social and economical impact in Hong Kong

Total rainfall between April and September divided by the number of wet days (daily rainfall ≥ 1 mm) between April and September

Changed from whole year to the period from April to September

mm

R30

Annual count of days with daily rainfall ≥ 30 mm

Same as that developed by ETCCDMI

mm

CDD(4–9) (Consecutive dry days)

Maximum length of dry spell between April and September, maximum number of consecutive days with daily rainfall < 1 mm: Let RR_{ij} be the daily rainfall amount on day i in period j. Count the largest number of consecutive days where: RR_{ij} < 1 mm

Changed from whole year to the period from April to September

Days

CWD(4–9) (Consecutive wet days)

Maximum length of wet spell between April and September, maximum number of consecutive days with daily rainfall ≥ 1 mm: Let RR_{ij} be the daily rainfall amount on day i in period j. Count the largest number of consecutive days where: RR_{ij} ≥ 1 mm

Changed from whole year to the period from April to September

Days

R95p

Annual rainfall amount because of extreme rainfall days (>95th percentile):

Reference period changed from 1961–1990 to 1971–2000

mm

Let R_{j} be the sum of daily rainfall amount for period j and let R_{wj} be the daily rainfall amount on wet day w (rainfall ≥ 1 mm) in period j and R_{wn}95 the 95th percentile of rainfall on wet days in the 1971–2000 base period. Then R95_{pj} is determined as the sum of R_{wj} at days with R_{wj} > R_{wn}95.

R99p

Annual rainfall amount because of extreme rainfall days (>99th percentile):

Reference period changed from 1961–1990 to 1971–2000

mm

Let R_{j} be the sum of daily rainfall amount for period j and let R_{wj} be the daily rainfall amount on wet day w (rainfall ≥ 1 mm) in period j and R_{wn}99 the 99th percentile of rainfall on wet days in the 1971–2000 base period. Then R99_{pj} is determined as the sum of R_{wj} at days with R_{wj} > R_{wn}99.

PRCPTOT(4–9)

Total rainfall in wet days (daily rainfall ≥ 1 mm) between April and September

Changed from whole year to the period from April to September

mm

TX90p (warm days)

Percentage of days when daily maximum temperature > 90th percentile:

Reference period changed from 1961–1990 to 1971–2000

% of days

Let TX_{ij} be the daily maximum temperature on day i in period j and let TX_{in}90 be the calendar day 90th percentile centred on a five-day window for the base period 1971–2000. (the percentage of time is determined where: TX_{ij} > TX_{in}90)

WSDI (warm spell duration index)

Annual count of days with at least six consecutive days when daily maximum temperature > 90th percentile:

Reference period changed from 1961–1990 to 1971–2000

Days

Let TX_{ij} be the daily maximum temperature on day i in period j and let TX_{in}90 be the calendar day 90th percentile centred on a 5-day window for the base period 1971–2000. Then the number of days per period is summed where, in intervals of at least six consecutive days: TX_{ij} > TX_{in}90

CSDI (cold spell duration index)

Annual count of days with at least six consecutive days when daily minimum temperature < 10th percentile:

Reference period changed from 1961–1990 to 1971–2000

Days

Let TN_{ij} be the daily minimum temperature on day i in period j and let TN_{in}10 be the calendar day 10th percentile centred on a 5-day window for the base period 1971–2000. Then the number of days per period is summed where, in intervals of at least six consecutive days: TN_{ij} < TN_{in}10

A1 Time-dependent generalized extreme value distribution technique

In extreme value theory, the maximum of a sequence of observations (such as annual maxima data), under very general conditions, is approximately distributed as GEV distribution which comprises three asymptotic classical extreme value models, namely Gumbel, Frechet and Weibull (Fisher and Tippett, 1928; Jenkinson, 1955; Gumbel, 1958). The corresponding cumulative distribution function and probability density function of the GEV are given in Equations (A1) and (A2), respectively.

(A1)

(A2)

where µ, σ and ξ are the location, scale and shape parameters, respectively. When ξ approaches 0, the distribution becomes the Gumbel type with the following cumulative distribution function:

(A3)

For ξ< 0 and ξ> 0, the distributions are known as Frechet and Weibull, respectively.

To obtain the parameter µ, σ and ξ, the GEV distribution is fitted to the annual extreme values. If the sample size is larger than 25, the method of maximum likelihood method (Cox and Hinkley, 1974; LeDuc and Stevens, 1977; Kharin and Zwiers, 2005) is adopted for the fitting in the GEV distribution in Equation (A1).

Suppose x_{1}, …, x_{n} are the annual maxima of the set of observations (e.g. annual maximum temperature), the likelihood L is the product of the densities of Equation (A2) for x_{1}, …, x_{n}. Mathematically,

(A4)

The estimates of µ, σ and ξ, say µ_{0}, σ_{0} and ξ_{0}, which maximize the likelihood L are called the maximum likelihood estimators.

As the extreme values of different meteorological elements may exhibit trends with respect to time, i.e. a time-dependent GEV distribution (Kharin and Zwiers, 2005; Feng et al., 2007), the GEV parameters µ, σ and ξ can be assumed as time-dependent quantities which vary linearly with time (t) as

(A5)

The standard likelihood ratio test was conducted to determine whether the trend estimated from the fitting of a time-dependent GEV distribution was significant against the null hypothesis of a time-independent GEV distribution with constant µ, σ and ξ. Assuming L_{0} and L_{1} are the log likelihood of the time independent (constant µ, σ and ξ) and the alternative time-dependent GEV distributions respectively, the null hypothesis is rejected at 5% significant level when

(A6)

where v is the difference in the number of estimated parameters (Coles, 2001) between the two test models. For the χ^{2} distribution with v = 1, the χ^{2}(v, 0.95) = 3.84. Details of the maximum likelihood method and the likelihood ratio test are documented in Kharin's study (Kharin and Zwiers, 2005).

After fitting the GEV distributions with the annual extreme values and determining the relevant maximum likelihood estimators, the N-year return values X_{N} can be estimated from the cumulative distribution function as: