## Abstract

This study presents a method to linearly evaluate the rainfall frequency–intensity distribution, which is an important component of climatological rainfall characteristics. To grasp and represent the key information of the rainfall frequency distribution by intensity, a two-parameter double exponential function is formulated and fitted to the hourly rainfall observation at each station. The values of the two parameters are estimated by transforming the distribution to a linear pattern. The two parameters determine the location and shape of the fitted distribution curve, and they have different modulating effects in different intensity categories, one governing the low-intensity section and the other dominating the intense rainfall. Through analysis of the estimated parameters, essential features of rainfall distribution can be obtained and assessed. The proposed method is applied to analyze the climatology and long-term variation of the late-summer rainfall in China. It is found that topography and monsoon circulation are two major factors controlling the rainfall frequency–intensity distribution. At stations with high surface altitudes and complex orography, the frequency of light rain is extremely high and the number of intense rainfall events is relatively small. In the plain areas of eastern China, especially those influenced by the main monsoon rain belt, heavy rainfall is more frequent. By tracking the displacement of the parameter pairs, the decadal changes in rainfall frequency–intensity distribution can be clearly visualized and evaluated on a plane constructed by the two parameters. The southern flooding and northern drought pattern can be attributed to the changes in light and moderate rainfall, while the intense rainfall exhibits opposite trends.

## 1. Introduction

Rainfall is one of the most important factors that determine the climate of a region, and advancing the knowledge of observed rainfall characteristics is always a key concern for meteorologists and climatologists. Amount, frequency, and intensity are three primary and widely used parameters to describe the rainfall features. Taking the late-summer [July and August (JA)] rainfall in contiguous China for example, we can get an overall picture of rainfall characteristics based on the three variables. As shown in Fig. 1a, there are two major belts with large rainfall amount, one extending from northeastern China to southwestern China and the other in southern China. The largest rainfall frequency center is located in southwestern China and fewer rainfall hours are recorded in northern and eastern China (Fig. 1b). The hourly rainfall intensity decreases from southeast to northwest China (Fig. 1c). However, an important rainfall climatological feature cannot be identified by these plots directly. Stations SuoLun (SL) and LaoHeKou (LA), marked by black circles in Fig. 1a, have similar values of the three rainfall variables. The differences (LA minus SL) in climatological JA rainfall amount, frequency, and intensity between the two stations are only 17.00 mm JA^{−1}, −1.94 h JA^{−1}, and 0.15 mm h^{−1}. Nevertheless, the rainfall frequency distributions for various intensities at the two stations are distinctively different. At station SL (blue lines in Fig. 1d), more rainfall hours are located in the low-intensity section. For rainfall with hourly intensity less than 10 mm h^{−1}, there are 437 more hours at station SL than LA. The red dashed line in Fig. 1d presents a higher tail, indicating many more rainfall hours in the high-intensity section at station LA. For rainfall stronger than 15 mm h^{−1}, the frequency at station LA (130 h) is more than twice that at station SL (61 h). The rainfall frequency–intensity distribution, as presented by the location and shape of the lines in Fig. 1d, is essential for characterizing the local climate. The combination of frequency and intensity is as vital as the other parameters (Trenberth et al. 2003). Light rainfall and heavy rainfall correspond to different mechanisms and exert distinct influences on natural ecosystems, runoff, agriculture, and human society. Grasping the proportion of rainfall with various intensities in addition to the rainfall amount, frequency, and intensity can provide useful information for understanding local climatic backgrounds, validating numerical models, and evaluating potential environmental risks.

As shown in Fig. 1d, the accumulated number of rainfall hours is large in the weak section and tends to zero in the intense section. The extreme large difference in the head and tail of the distribution adds considerable difficulty to the evaluation of rainfall intensity structure. Yu and Li (2012) found that the widely used exponential function cannot accurately describe the major features of the frequency–intensity distribution because of the high frequency of weak rainfall and the dramatic decrease as intensity increases. In this paper, a new method to objectively and linearly evaluate the rainfall frequency–intensity distribution is presented and applied to show features of rainfall in China. The other parts of the paper are organized as follows: section 2 gives a brief description of the rainfall dataset used in this study. The method is proposed in section 3 and two applications are shown in section 4. Section 5 summarizes the major points of this paper.

## 2. Data description

This study is based on the dataset of hourly rain gauge records for the period from 1961 to 2010 at 465 meteorological stations throughout contiguous China. This dataset was obtained from the national climatic reference network and national weather surface network of China and was collected and quality controlled by the National Meteorological Information Center of the China Meteorological Administration. To ensure the accuracy of the hourly rainfall data, further quality control has been applied. By accumulating the hourly data, the daily rainfall amount *R*_{h} was calculated. The hourly data–based *R*_{h} was then compared with the record of daily rainfall amount *R*_{d}, which was measured by the nonrecording gauges at the same station. If |*R*_{h} − *R*_{d}| > *R*_{d}/10, the records of hourly data in this day were regarded as missing values. Only data for July and August were used in this study. If more than 10% of hourly records in one JA period were missing, the year was marked as a missing value.

## 3. Methodology

Station HanZhong (H), marked by a black circle in Fig. 1b, is used as an example to illustrate the method to analyze the rainfall frequency–intensity distribution. All the JA rainfall hours at station H during 1961–2010 are binned by hourly intensity with intervals of 1 mm h^{−1}. The accumulated rainfall hours in each category are then normalized to 100 JA periods. The distribution of rainfall frequency by hourly intensity, which is shown in Fig. 2a by red dots, exhibits high values in the weak rainfall section and falls to the bottom quickly as the intensity increases. The frequency of rainfall with intensity of 1–2 mm h^{−1} is 2594 h in the 100-JA period and the frequency in the 2–3 mm h^{−1} category drops to only 1398 h. The rainfall weaker than 5 mm h^{−1} accounts for 91.1% of the total frequency. The frequency in the high-intensity section is quite small, with only 376 h with intensity higher than 10 mm h^{−1}. Based on the distribution features of the red dots, the following two-parameter double exponential function [Eq. (1)] is used to fit the frequency–intensity curve

Here *I* represents hourly rainfall intensity and Fr(*I*) is for frequency. The *α* and *β* are two parameters that can be obtained with the observed frequency–intensity distribution. Taking the natural logarithm twice at both sides of Eq. (1), we can get

The logarithm and double logarithm of rainfall frequency at various intensity categories are shown by black asterisks and blue triangles in Fig. 2a. The blue triangles fall exactly along a straight line before the intensity exceeding 15 mm h^{−1}. Since most (99.1%) of the rainfall hours are weaker than 15 mm h^{−1}, the linearity of the blue triangles can be considered as an important feature of the frequency–intensity distribution. The essential information about the major part of the rainfall structure can be presented by the corresponding line, which enables us to analyze and evaluate the frequency–intensity distribution linearly. By the least squares fitting of Eq. (2), *α* and *β* can be determined. In this example, *α* = 2.13 and *β* = 14.75. Then, the corresponding fit lines for different markers (the blue straight line and black and red curved lines in Fig. 2a) are drawn with fixed *α* and *β*. Both the blue and black lines fit the scatter points fairly well, especially where the intensity is not too high. The red line calculated from exp[exp(*α* − *I*/*β*)] − 1 also closely matches the distribution of red dots and represents the notable features such as the high values and sharp decrease in the weak section and the low and flat tail in the high-intensity section.

To further test whether the double exponential function involving only two parameters can accurately describe the rainfall frequency–intensity distribution, the quantiles of the observation at station H are plotted against the corresponding quantiles deduced from Eq. (1). As shown in Fig. 2b, most circles positioned by the quantiles of the two datasets range along the *y* = *x* line, indicating that the two distributions are numerically similar and Eq. (1) can represent the location and shape of the observed rainfall distribution fairly well. It is to be noted that there are some circles in the upper-right corner deviating from *y* = *x*. Both the deviated circles, which are beyond the 99.9th percentile, and the blue triangles apart from the straight line (Fig. 2a) can be attributed to extreme rainfall events. Since the extreme events occur rarely (only 3 h with intensity higher than 50 mm h^{−1} are recorded in the 50-yr period) and have an insignificant contribution to the climatological frequency–intensity distribution, their influences are not considered in this paper.

The two parameters *α* and *β* determine the slope and the intercept of the blue line in Fig. 2a and carry the key information of the frequency–intensity distribution. Larger *α* denotes a higher position of the *y* = *α* − *x*/*β* lines and larger *β* corresponds to more flat lines. Nevertheless, the relationship between the parameter values and the pattern of the modeled frequency–intensity lines demands more discussion. Figure 3 shows the graphs of the double exponential function, *y* = exp[exp(*α* − *x*/*β*)] − 1, and illustrates the distinct effects of *α* and *β*. Given a fixed *β* (=50), the left end of the function line rises as *α* increases gradually from 1.6 to 2.4 (Fig. 3a). The variation in *α* leads to large differences in *y* when the magnitude of *x* is small. As *x* increases, the effect of *α* decreases quickly and tends to zero when *x* tends to infinity. Figure 3b presents the lines with *β* varying from 20 to 100 and *α* set to 1.985 (to make sure that Fig. 3b has the same *y* axis as Fig. 3a). The five lines are arranged from bottom to top as *β* increases. The change in *y* resulting from the variation of *β* has large values in a certain range of *x*. It diminishes to zero at *x* = 0, and all the curves converge at *y* = exp[exp(*α*)] − 1. Because of the small values in the tail of the double exponential function, the distances between the five lines reduce gradually as *x* increases in the large value section. Taking the value of *y* into consideration, the dashed line in Fig. 3b shows the distance between the lower two lines divided by the value of the lowest line, (*y*_{2} − *y*_{1})/*y*_{1}. Here *y*_{1} (*y*_{2}) is the first (second) line from the bottom. The effect of *β* on the relative value of *y* increases with *x*. The similar dashed line in Fig. 3a exhibits a contrary trend, indicating that the *α*’s influence reduces with *x*. The two parameters *α* and *β* modulate different parts of the line of the double exponential function. In the frequency–intensity distribution, *α* is more closely related to the frequency of weak rainfall, while *β* can be used to assess the frequency of intense rainfall. In the following section, the two parameters are applied to evaluate the frequency–intensity distribution of rainfall in China.

## 4. Application to late-summer rainfall in China

### a. Climatic rainfall features

The 1961–2010 accumulated JA rainfall frequency–intensity distributions at all the 465 stations are calculated and then normalized to 100 JA periods. By fitting the double exponential function [Eq. (1)] to the frequency structure, the parameter pairs are determined. Through representing the parameter pair at each station as a position on the *α*–*β* plane (Fig. 4a), we can easily visualize the overall features and categorize stations into different groups. The parameter *α* ranges from 1.88 to 2.38, with the median being 2.03. For *β*, the minimum is 3.55, the maximum is 39.38, and the median is 21.09. As shown in Fig. 4a, there are three regions with white backgrounds: a rectangle in the center, a horizontal bar, and a vertical bar. The rectangle is bordered by the first and the third quartiles of *α* (1.99–2.08) and *β* (16.29–24.80). The horizontal (vertical) bar is positioned by the 40th and the 60th percentiles of *α* (*β*). Using the two bars as boundaries and excluding the rectangle, four regions can be partitioned from each other and are colored red, sienna, blue, and green, respectively.

Since the distribution of black circles resembles a belt extending from upper left to lower right, most stations are located in the sienna and green regions. There are 119 stations in the sienna zone, which are characterized by large *α* and small *β*, that is, more weak rainfall hours and less intense rainfall. The spatial distribution of these stations is shown by sienna squares in Fig. 4b. Most of the 119 stations are located in mountain areas with high altitudes, such as the Tibetan Plateau, the highland in north China, and the Changbai Mountain Range in the northeastern China. The orographic forcing favors the high frequency of rainfall occurrence, and the difficulty in water vapor transportation in high terrain regions limits the intensity of precipitation. Therefore, light rainfall is dominant at the sienna stations, and the frequency–intensity curves are featured by a much steeper decline and a lower tail. Comparing the positions of sienna squares with the climatological rainfall frequency (Fig. 1b), it can be found that the extreme high rainfall frequency in the southern Tibetan Plateau and its southeastern extension area is contributed by the large number of weak rainfall events.

The stations grouped in the green zone have small *α* and large *β* values, indicating that the proportion of light rainfall is relatively low and the occurrence of intense precipitation tends to be frequent. Most of these stations, denoted as green circles in Fig. 4b, are located in central eastern China. There is a definite dividing line between the sienna squares and green circles, and the two groups of stations can be separated geophysically. Rainfall in central eastern China is largely affected by the East Asian summer monsoon, which exhibits significant seasonal advance and retreat in a stepwise way (Tao and Chen 1987; Ding 1992; T. Chen et al. 2004; Ding and Chan 2005). The monsoon rain belt stagnates over north China during the period of mid-July to early or mid-August. Consistent with the location of the monsoon rain belt, 74 of the 112 stations in the green zone are concentrated in the plain area north of 30°N. Favorable large-scale circulation and plentiful water vapor transportation lead to long-duration (lasting more than 6 h) rainfall events in north China (Yuan et al. 2010), and also ensure the relatively high proportion of intense rainfall.

The red dots in Fig. 4b are stations in the red zone, with large values of both *α* and *β*. The 20 stations are either on the windward slope of mountains or in the coastal areas, and all of them can be identified as relatively large centers of rainfall amount, frequency, and intensity (Figs. 1a–c). As shown in Fig. 1c, high rainfall intensity centers are located in both north China and the coastal area in south China. Although the two regions have similar climatological intensities, the frequency–intensity distributions are different. The southern coastal region is subjected to various kinds of precipitation events, such as local showers in late afternoon related to high surface air temperature and large amount of water vapor, and extremely heavy rainfall from tropical cyclones (Wu et al. 2005). Therefore the frequencies of both light and intense rain are high in this region, and the mean rainfall intensity is large. Only nine stations are categorized in blue zone, which features a low frequency of both weak and intense rainfall. The blue triangles are located among the sienna squares. Except for one red dot in northeastern China, all the red-zone stations are situated south of 33°N, and all the blue triangles are north of 33°N. The separation of the blue zone and the red zone is one advantage of the double exponential function. The major difference between these two zones is the frequency of hourly rainfall, both in low- and high-intensity sections. Other wet-hour-based probability density function distributions might have difficulty distinguishing the differences caused by the proportion of no-rain hours.

Through analyzing the values of *α* and *β*, some features of the frequency–intensity distribution are revealed. The new features are closely related to topography and climatological circulation, which are key factors influencing the characteristics of rainfall events.

### b. Decadal changes in rainfall features

Besides presenting climatic features, the two-parameter double exponential function method can also be used to diagnose decadal changes in rainfall characteristics by tracking the parameter values. As documented previously in numerous studies, the late-summer precipitation in central eastern China has undergone significant changes in the last half of the twentieth century, which are characterized by the increased rainfall over the middle-lower reaches of the Yangtze River valley and decreased rainfall over north China (Hu 1997; Xu 2001; Hu et al. 2003; Yu et al. 2004; L. Chen et al. 2004; Ding et al. 2008). To evaluate the frequency–intensity distribution changes associated with the southern flooding and northern drought (SFND), two stations are chosen from the Yangtze River valley (S; NanXian) and north China (N; WeiXian), and marked by black circles in Fig. 1c. Following Yu et al. (2010), the decadal changes in hourly rainfall frequency at the two stations are analyzed by calculating the difference between 1986–2005 and 1966–85. The rainfall frequency at station N has decreased 24.50 h in each JA, and the frequency at station S has increased 31.50 h. The changes at both stations are statistically significant at the 0.05 level, according to the Student’s *t* test.

The parameters of Eq. (1) are calculated from the observed rainfall in a 20-yr sliding window, which starts from 1966 and ends in 2010 with a step size of 5 years. On the *α*–*β* plane shown in Fig. 5, the parameter pairs for different time periods are presented by markers with different colors. The squares are for station N and the dots for station S. The first four phases of station N are arranged in a straight line, and large displacements are found between the four squares. As the phases drop from upper left to lower right on the *α*–*β* plane, the *α* decreases from 2.14 to 1.97 and the *β* increases from 11.48 to 18.72. On the contrary, round dots signifying station S climb from lower right to upper left. Except in the fourth phase (1981–2000), the *α* presents an increasing trend and the *β* decreases across phases. Even though the frequencies at both stations show significant decadal changes, the two parameters change in opposite directions, indicating there is no overall uplift or subsidence for the frequency–intensity line. With the decreasing (increasing) *α*, the frequency of weak rainfall has decreased (increased) at station N (S), while the intense rainfall frequency experiences a reversed trend. The decrease (increase) of the total rainfall amount at station N (S) comes from the weak rainfall. This is consistent with the results of Yu et al. (2010) that the SFND pattern over the central eastern China is mostly attributed to changes in rainfall with relatively low intensity. Another point to be noted is that the last two squares and the sixth dot reverse the direction. In the last two decades, the frequency of intense rainfall has decreased (increased) in station N (S). There might also be a turning point for the mechanisms governing the decadal rainfall changes.

By comparing the phases of the two stations, it is found that the differences between stations N and S have changed in the last several decades. In the first phase (1966–85), the dark-blue square (dot) is located at the upper-left (lower right) corner of the *α*–*β* plane, indicating the relatively large proportion of weak (intense) rainfall. The distribution curve of station N has a higher head (weak section) and a lower tail (intense section) than that of station S. In the following four phases, the “distance” between the square and the dot reduces step by step. In the last three phases, the squares are lower and to the left of the dots; that is, both the *α* and the *β* of station N are smaller than those of station S. Correspondingly, the relative positions of the two distribution curves have shifted, and the station N’s curve is completely below the curve of station S.

Also to be noted, Fig. 5 provides a concise statistical summary of how the precipitation structure evolves. The advantage of this *α*–*β*-plane-based diagram will be more evident in the comparison of multimodel results or a series of sensitive numerical experiments, since it allows much information to be compacted and presented in one plot.

## 5. Summary

A two-parameter double exponential function is used to represent the rainfall frequency–intensity distribution, and the method is simple and easy to carry out. The essential information of the distribution pattern is transformed into a straight line and can be linearly expressed by two parameters. The ability of the double exponential function and parameter estimates to adequately fit the empirical distribution of observed rainfall allows for the interpretation of the parameter pairs as a compact summary of the frequency–intensity distribution. The values of the two parameters control the specific pattern of the distribution curve, and each parameter dominates different ranges of intensities. This valuable characteristic makes the *α*–*β* plane physically meaningful and the climatological status (shifts) of the rainfall frequency–intensity structure can be explicitly presented by the positions (tracks) of points on the *α*–*β* plane. The application of the proposed method reveals some new features of rainfall in China. First, the stations in highly elevated land areas tend to have a large amount of light rainfall events and a small proportion of heavy rainfall. Meanwhile, in the eastern plain regions, especially those under the control of the major monsoon rain belt, the frequency of intense (weak) rainfall is relatively high (low). Second, the SFND in recent decades is closely associated with the shifts in the frequency–intensity distribution. Both the flooding trends in the mid-lower reaches of the Yangtze River valley and the decreased rainfall over north China are contributed by changes in light and moderate rainfall.

## Acknowledgments

This research was supported by the Major National Basic Research Program of China (973 Program) on Global Change under Grant 2010CB951902, the Basic Scientific Research and Operation Foundation of the CAMS (2013Z004), the Jiangsu Collaborative Innovation Center for Climate Change, and the National Natural Science Foundation of China under Grant 41221064 and 41322034.

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