Spatial distribution of extreme precipitation in the Tibetan Plateau and effects of external forcing factors based on Generalized Pareto distribution

The daily precipitation data of the years 1955–2017 from May to September were retrieved; then a Generalized Pareto Distribution (GPD) and maximum likelihood methods were adopted to understand trends and calculate the reappearance period of heavy precipitation in the Tibetan Plateau (TP). The daily precipitation values at 22 stations in the TP were found to conform to the model, and theoretical and measured frequencies were consistent. According to the spatial distribution of the maximum precipitation value, the extreme values of Shigatse and Lhasa showed large fluctuations, and the probability of record-breaking precipitation events was low. In the western part of Nagqu, the probability of extreme precipitation was relatively low, and that of record-breaking precipitation was relatively high. The peak values of extreme precipitation in the flood season in the TP generally exhibited a decreasing trend from southeast to northwest, and the extreme value of the flood season that reappeared in the southeast region was approximatelytwice that of the northwest region. The maximum rainfall in most areas will exceed 20 mm in the next 5–10 years, and the maximum rainfall in Shigatse will reach 52.7 mm. After 15 years of recurrence in various regions, the peak rainfall in the flood season has become low. Most of the regions in the model have different responses to ENSO and Indian Ocean monsoon indices with external forcing factors.


INTRODUCTION
Regional floods, droughts, high temperatures, snow and ice storms and other extreme events have occurred frequently due to global climate warming, especially since the 1980s (IPCC ; Liu et al. ). Extreme precipitation is one of the most concerning and influential natural threats, and it is the focus of climate prediction research (Ding et al. ). To study extreme precipitation and to improve prediction accuracy are crucial for establishing a modern public safety system and disaster prevention and mitigation strategy.
Many scholars have used extreme value theory to study extreme climate. Leadbetter et al. ()  Zhao & Zhai () improved the GPD model and got a better extremum distribution parameter, and the result is better than that of the linear moment method.
Linear correlation between the atmospheric circulation and precipitation was proposed in many studies. The circulation anomaly not only increase land surface temperature, but also changes global precipitation (Wallace et al. ). Some studies have indicated that when strong El Nino-Southern Oscillation (ENSO) events occur, through remote correlation, persistent droughts and floods will occur in some parts of the world, and extremely heavy precipitation events will occur in The Tibetan Plateau (TP) is located in a geographical location that is the main and sensitive to global climate change. The occurrence of extreme events is an important response to regional climate change. Few studies have forecasted precipitation in the flood season by using the GPD method; therefore, the distributed parameter model of GDP, mainly to exceed the threshold data as sample data modeling, starting from extreme climate distribution, revealed the development rule of extreme precipitation to explore its probability. GPD is expected to provide an improved analysis of extreme events and provide a scientific basis for disaster prevention and mitigation in this article.

Study area
The TP, has average altitude of 4,000 m, and a semi-arid monsoon climate in the plateau temperate zone. Due to the high altitude, the temperature of the TP is much lower than in other the regions of the same latitude. The average temperatures during spring, summer, autumn, winter are À11.2 C to À2.5 C, 7.1 C to 18.5 C, À2.4 C to 12.9 C, and À13.4 C to 5.0 C, respectively. The annual precipitation is 66.3-894.5 mm. The annual precipitation in most regions of Nyingchi is >600 mm, while that in Ali is <200 mm. With the alternating influence of the west wind in winter and the southwest monsoon in summer, the dry season and rainy season are clear. The rainy season is from May to September, and it generally accounts for approximately 90% of the annual precipitation. Furthermore, the precipitation is extremely uneven in different regions, gradually decreasing from Nyingchi to Ali.
The topography of the TP is relatively complex, and it is mainly composed of high mountains and valleys. It has various vegetation types. The southeast of the TP is at a low altitude and mainly includes broadleaf forest, coniferous forest, and shrubs. The north and northwest regions have meadows, steppe, and desert.

Data
The study data was taken from the Information Network Center of Tibet Meteorological Bureau. All data were tested for temporal and spatial consistency.
The study data consisted of daily precipitation data of 22 stations in the TP over the past 50 years  from May to September. Observation data of some weather stations were from in 1955, 1961, and 1978. The circulation index and sea surface temperature (SST) index were obtained from NOAA.

Density and distribution function
The GPD can directly use the original data over the years to set the threshold value artificially. After setting the threshold value, the maximum or minimum value exceeding the threshold value every year was extracted based on the criterion known as 'peaks over threshold' (Beirlant et al. ).
The distribution function of the GPD is obtained as follows: The density function of the GPD is obtained as follows: The GPD includes three main parameters, namely location ξ, which is similar to the mathematical expectation of general distribution, scale parameter a, and shape parameter k. When k < 0, ξ < x < ∞; when k > 0, GPD is an exponential distribution; and k ¼ 1, GPD is uniformly distributed on the interval [ξ, ξ þ a].

Maximum likelihood parameter estimation
The maximum likelihood is calculated as follows: The logarithmic likelihood is calculated as follows: This is an increasing function of ζ, and the estimated value of ζ is the minimum value X 1 of the observation series. According to the Newton iteration method, other parameters can be obtained by combining the following three equations:

Recurrence period
Precipitation fraction at time T in a certain recurrence period is as follows: x ¼ξ þâ k (1 À T Àk ) (9)

Data processing and statistics
Matlab was used for statistical analysis of the data. Data that did not follow a normal distribution were transformed. The correlation coefficient was tested by using the t test, and p < 0.05 indicated a significant result.

Selection of threshold value
Threshold is the premise of parameter estimation. If the threshold is high, the estimated parameter variance will be large; if it is low, the function condition cannot be satisfied.
The estimator will become biased, leading to function divergence. We adopted the method of dynamic threshold estimation; that is, within a certain threshold range, the shape and scale parameters of the equation larger than the threshold should remain unchanged.
The observation records of most stations in the TP began in the mid and late 1950s. To meet the equation requirements, we selected 22 stations with complete observation records and data of 30-50 years. First, by arranging the observed data in a descending order, the threshold value between 90th and 99th percentiles can be calculated.
Then, according to the Equations (6)-(8), the corresponding shape and scale parameters can be calculated.
The result of the Lhasa station (Table 2) showed that the shape parameter with the observed value between the 93rd and 98rd percentiles was stable at approximately 6.5. However, for the 98 th percentile (Kolmogorov-Smirnov, K-S), the value is the highest at approximately 0.087. Therefore, we choose the observed value of the 98th percentile as the threshold value.

Model fitting effect test
To test whether the model conformed to the known theoretical distribution function, the Kolmogorov method was used in this study. Most sites with p > 0.05 confirmed the null hypothesis (Figure 2(a)). The daily strong precipitation during the flood season of the TP basically conforms to the model. Although differences were observed between stations, the GPD fitting curve showed that the theoretical frequency was consistent with the measured frequency.
We selected Lhasa, Sênggê Zangbo (Ali), and Kongga stations as representative stations, with their records starting from 1955, 1961, and 1978, respectively (Figure 2 The results showed that the data length does not affect the statistical inference of precipitation extremum, and the longer the data period, the better the fitting results. We calculated the errors corresponding to different cumulative frequencies (Table 3). When the cumulative frequency was low, almost no error occured. With an increase in the cumulative frequency, the error gradually increase. In particular, when the cumulative frequency of Kongga station was 0.9, the error reached 1.1, indicating that the length of the research period does not affect the simulation effect, and the random error of the equation itself might be one of the reasons for the large error.   As the second important parameter of the model, the shape parameter represents the different tail distribution characteristics of the model. Table 4 shows that the positive area of shape parameters mainly consists of Bangor, Amdo, and other areas, where the possibility of record-breaking precipitation events is higher than in other areas. The negative values of shape parameters were mainly in Shigatse, Lhasa, and other areas along the river, followed by stations in Qamdo, where the probability of precipitation breaking records is relatively low. According to the distribution of corresponding scale parameters, the extreme value variation rate of precipitation along the river and in Nyingchi and other areas is high, and the dispersion degree is larger than the mean value; thus, the probability of breaking the record is low. This is consistent with the synoptic principle that the larger the wavelength, the smaller the frequency.
The number of precipitation days in the west of Nagqu is low, and the precipitation variation rate was low. Once precipitation occurs, the extreme state is likely to be a recordbreaking behavior. Thus, the more frequent the summer precipitation, the lower the possibility of the extreme value to break the record. In arid areas, the climatic probability of  Maizhokunggar À0.14 6.38 Zidane À0.14 5.84 precipitation is usually low, so the probability of breaking records is high.

Estimation of recurrence period
The TP flood season precipitation extreme value return level generally presents the southeast northwest high low, from southeast to northwest gradually decreasing, and southeast to reproduce the extreme flood season is about two times of in northwest China, but also reflects Nyingchi the most serious secondary disasters caused by heavy rainfall, the agreement with the observation (Table 5 to the scale and shape parameters, respectively.

GPD1
: GPD2: where t represents time, a 0 and k 0 represent the linear growth trends of the scale and shape parameters, respectively, and a 1 t and k 1 t represent the covariables of linear increment. To ensure that the data are positive, scale and shape parameters were transformed into their logarithmic forms. In GPD1, the shape parameter was fixed, and the scale parameter was added with an external forcing factor.
In GPD2, the scale parameter was fixed, and the shape parameter was added with an external forcing factor.
To assess whether these models improved on the original model, a test judgment must be made. The method and pro-  D obeys the chi-square distribution. If at the significance level of 5%, D > C 1-a , and C 1-a is the distribution quantile of ξ 2 k , then the null hypothesis, namely M 1 , is better able to describe the data.
The results showed that the scale and shape parameters changed when the two external forcing factors were added (Table 6). When ENSO was added, the change amplitude of the shape parameter was higher than that of the scale parameter. The range of the scale parameter was À0.2 to 0.33.
The scale parameters of Burang, Gyangze, Lhari, Lhunze, Maizhokunggar, Markam, Shigatse, and Zidane exhibited an increasing trend. However, the scale parameter did not change in Sênggê Zangbo. Thus, ENSO has no effect on the scale parameter in the Sênggê Zangbo area. After ENSO was added to the shape parameter, the value increased. This indicated that the probability of breaking the extreme precipitation value in most areas also increased.
After adding the ID influence factor, the scale parameter exhibited a negative value in most areas, and the range was À0.16 to 0.087. The shape parameters in most areas exhibited limited numerical change after the ID factor was added.
In general, the southeastern region of the TP is strongly affected by ENSO, whereas the southern marginal region is strongly affected by ID. Along the Yarlung Zangbo River to Qamdo, both indexes have a substantial influence, with a gradual decline from southeast to northwest.
The effect of these two forcing factors can be ignored in Ali due to its geographical location. The summer precipitation is only slightly affected by SST, which may be caused by the southward movement of cold air from the north or the small amount of water vapor carried by the westerly belt. Moreover, Nyingchi is located in the eastern section along the Yangtze River, and the external forcing effect of ENSO is smaller than that of the ID factor. Thus, ENSO is not the most important external factor affecting the extreme precipitation value, mainly because the Indian Ocean SST anomaly affects the Indian Ocean monsoon and then affects the distribution of water vapor transport and water vapor flux divergence over the south of the TP through circulation adjustment, finally changing the precipitation pattern. Considering the ENSO and ID factors, their occurrence mechanism and geographical location determine their regional effects on summer extreme precipitation. Different physical mechanisms lead to different climate effects. In the analysis of summer extreme precipitation, the correlation between multiple circulation factors and sea-air coupling is also considered.

CONCLUSION
The probability distribution model of GPD was introduced to simulate the regularity of heavy precipitation during the flood season in the TP. The results showed that the daily heavy precipitation at 22 stations in the TP conformed to the model, and the theoretical frequency was consistent with the measured frequency.
The high-value regions for the scale parameter were located in Lhasa, which indicates that the extreme value fluctuated greatly. Conversely, the small-value region was located in the west of Nagqu, indicating that the extreme value fluctuated only slightly. The positive region for the shape parameter was mainly located in Nagqu, suggesting that the probability of a record-breaking precipitation event is relatively high. The negative area was mainly in Shigatse, Lhasa, and other areas along the river line, followed by stations in Qamdo, where the probability of extreme precipitation is relatively low.
In the TP, the recurrence of extreme precipitation during the flood season exhibited a gradually diminishing trend from southeast to northwest, from the level of 5 years and 10 years reappear flood extreme value, in addition to Sênggê Zangbo; other regions with extreme rainfall receive >20 mm rainfall. In the TP, the extreme precipitation threshold is 52.7 mm; Lhasa and Nyingchi also have a high extreme value distributions. For a 15-year recurrence, the increase in rainfall extremity in flood season was slow.
Most areas are affected by the external forcing factors of ENSO and ID, which also indicates that these two indexes have a strong influence on extreme precipitation in the TP during summer. In Nyingchi, the forcing effect of ID was greater than that of ENSO.