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Using Quantile Regression to Detect Relationships between Large-scale Predictors and Local Precipitation over Northern China


doi: 10.1007/s00376-014-4058-7

  • Quantile regression (QR) is proposed to examine the relationships between large-scale atmospheric variables and all parts of the distribution of daily precipitation amount at Beijing Station from 1960 to 2008. QR is also applied to evaluate the relationship between large-scale predictors and extreme precipitation (90th quantile) at 238 stations in northern China. Finally, QR is used to fit observed daily precipitation amounts for wet days at four sample stations. Results show that meridional wind and specific humidity at both 850 hPa and 500 hPa (V850, SH850, V500, and SH500) strongly affect all parts of the Beijing precipitation distribution during the wet season (April-September). Meridional wind, zonal wind, and specific humidity at only 850 hPa (V850, U850, SH850) are significantly related to the precipitation distribution in the dry season (October-March). Impacts of these large-scale predictors on the daily precipitation amount with higher quantile become stronger, whereas their impact on light precipitation is negligible. In addition, SH850 has a strong relationship with wet-season extreme precipitation across the entire region, whereas the impacts of V850, V500, and SH500 are mainly in semi-arid and semi-humid areas. For the dry season, both SH850 and V850 are the major predictors of extreme precipitation in the entire region. Moreover, QR can satisfactorily simulate the daily precipitation amount at each station and for each season, if an optimum distribution family is selected. Therefore, QR is valuable for detecting the relationship between the large-scale predictors and the daily precipitation amount.
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Manuscript received: 31 March 2014
Manuscript revised: 17 July 2014
通讯作者: 陈斌, bchen63@163.com
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Using Quantile Regression to Detect Relationships between Large-scale Predictors and Local Precipitation over Northern China

  • 1. Key Laboratory of Regional Climate-Environment Research for Temperate East Asia, Institute of Atmospheric Physics, Chinese Academy of Sciences, Beijing 100029 xzh@tea.ac.cn

Abstract: Quantile regression (QR) is proposed to examine the relationships between large-scale atmospheric variables and all parts of the distribution of daily precipitation amount at Beijing Station from 1960 to 2008. QR is also applied to evaluate the relationship between large-scale predictors and extreme precipitation (90th quantile) at 238 stations in northern China. Finally, QR is used to fit observed daily precipitation amounts for wet days at four sample stations. Results show that meridional wind and specific humidity at both 850 hPa and 500 hPa (V850, SH850, V500, and SH500) strongly affect all parts of the Beijing precipitation distribution during the wet season (April-September). Meridional wind, zonal wind, and specific humidity at only 850 hPa (V850, U850, SH850) are significantly related to the precipitation distribution in the dry season (October-March). Impacts of these large-scale predictors on the daily precipitation amount with higher quantile become stronger, whereas their impact on light precipitation is negligible. In addition, SH850 has a strong relationship with wet-season extreme precipitation across the entire region, whereas the impacts of V850, V500, and SH500 are mainly in semi-arid and semi-humid areas. For the dry season, both SH850 and V850 are the major predictors of extreme precipitation in the entire region. Moreover, QR can satisfactorily simulate the daily precipitation amount at each station and for each season, if an optimum distribution family is selected. Therefore, QR is valuable for detecting the relationship between the large-scale predictors and the daily precipitation amount.

1. Introduction
  • Global warming caused by an accelerated increase of greenhouse gas concentrations has an important influence on the past and present climate. This may further affect hydrology and the management of water resources (Yang et al., 2010). In particular, arid and semi-arid areas of China are very sensitive to large-scale climate changes and are suffering from water scarcity. These areas are more prone to drought, because their precipitation amount critically depends on a few precipitation events (An et al., 2006). Drought is a recurring extreme climate event characterized by below-average precipitation over a period of months to years; and precipitation is considered a fundamental factor for drought (Dai, 2011). Therefore, proper assessment of future precipitation scenarios is an important issue for climate change impact studies in arid and semi-arid areas of the country.

    Global climate models (GCMs) are widely used for sim-ulating present and future climate. However, they are largely inappropriate for many climate impact assessment studies at the regional or local scale because of their coarse spatial resolutions (Wilby et al., 2004). Downscaling techniques attempt to bridge this scale gap by translating GCM simulations into regional- and local-scale variables or predictands (Hanssen-Bauer et al., 2005; Benestad et al., 2008; Fan et al., 2013). Downscaling is based on the idea that regional/local climate is conditioned by the large-scale climatic state and regional/local physiographic features such as topography, land-sea distribution, and land use (Wilby et al., 2004). There are two commonly used downscaling techniques——dynamical and statistical. Dynamical downscaling is a regional climate model (RCM) nested within a GCM to represent atmospheric physics with a higher resolution. The basic concept of statistical downscaling is to identify statistical relationships between observed large-scale and regional/local climate. A major advantage of statistical downscaling techniques over RCMs is that they are computationally inexpensive and once the statistical functional relationship is established by historical observations, future regional/local climate scenarios can be easily projected by applying it to various GCM simulations (Fan et al., 2005; Wilks, 2010). When performing statistical downscaling, there are two main challenges: how to determine the functional relationships between large-scale and local-scale climate, and how to select the predictor variables (Tareghian and Rasmussen, 2012).

    Figure 1.  Locations of stations in the study and annual mean precipitation amount (Pmean, units: mm) during 1961-90. Black lines represent contours of 200 mm and 500 mm.

    The most common form of quantitative relationship between a predictand and large-scale predictor variables is one in which the predictand is a function of predictors. However, other types of relationships have also been used, for instance between predictors and parameters of the statistical distribution of the predictand or frequencies of extremes of the predictand (Wilby et al., 2004). However, there are very few studies that explore the relationship between predictors and all parts of the conditional distribution of the predictand. (Koenker and Bassett, 1978) developed a regression-type model, called quantile regression (QR), which can be applied to estimate the functional relationship between predictor variables and any quantile in the distribution of the response variable. QR has been widely used in a number of social studies (e.g., McGuinness and Bennett, 2007). However, this technique has only been recently introduced in a few climate change studies (Barbosa, 2008; Tareghian and Rasmussen, 2012, 2013; Lee et al., 2013). For example, (Tareghian and Rasmussen, 2013) used QR to investigate the relationship between predictors and any quantile of the distribution of daily precipitation and used the estimated quantiles to determine the conditional distribution of daily precipitation for a given wet day. Their study found that compared with traditional linear regression, QR produces different rates of changes in different quantiles of the response variable, providing a more complete view of relationships between variables that are missed by other regression methods. QR is therefore suitable for the study of changes in frequency of extremes over time. Another advantage is that there is no need to make any assumptions about the distribution a predictand should follow, in contrast to a generalized linear model and weather generator, which should assume the distribution family of a variable (Wilks, 2010; Yang et al., 2010). In addition, QR has considerable flexibility in selecting large-scale predictors because different subsets of predictors can be selected for various parts of the conditional distribution. QR has received considerable attention in the statistical literature, but less so in climate change studies, especially in China. The present study uses QR to explore the statistical connection of all parts of a precipitation distribution with large-scale atmospheric variables, toward the development of a statistical downscaling model based on QR in arid and semi-arid regions of China for the future.

2. Study area and data
  • The study area focuses on the semi-arid and arid regions of China. Given that these regions are mainly north of 35°N (Ma et al., 2005), 238 meteorological observation stations at latitudes higher than 35°N, together with complete daily precipitation time series for 1961-2008 were selected for analysis (shown in Fig. 1). Daily precipitation data were collected from the China Meteorological Data Sharing Service System (http://data.cma.gov.cn). Stations with annual mean precipitation (Pmean) less than 200 mm (red squares) are west of contour Pmean =200 mm (P200), which is typically defined as the arid region. Stations with Pmean of 200-500 mm (blue dots), between P200 and contour Pmean =500 mm (P500) were regarded as the semi-arid region, according to (Chen et al., 2010). Generally, a sufficient number of samples are required to build a statistical downscaling model of precipitation (Wilby et al., 2004). Given the small number of precipitation events in the target region, we did not use the standard four seasons but divided the year into two seasons to increase the sample size. In China, precipitation generally concentrates during April to September (Zhai et al., 2005). Therefore, we used those six months as the wet season and October to March as the dry season. The threshold of 0 mm d-1 was used to classify a day as dry or wet.

    Observed daily large-scale predictors are commonly replaced by reanalysis data, owing to their insufficient availability and coverage. Reanalysis data used herein were extracted from the National Center for Atmospheric Research (NCAR)/National Centers for Environmental Prediction (NCEP) reanalysis dataset, with resolution 2.5°× 2.5° for the period 1961-2008 (Kalnay et al., 1996).

3. Methodology
  • QR was proposed as an extension to traditional linear regression (Koenker and Bassett, 1978). Traditional linear regression seeks a model for the conditional expectation of a response variable (Y), whereas QR is concerned with identification of models for quantiles of interest in the conditional distribution of the response variable. The traditional linear regression function has the form Y=βX+α, with β representing the regression coefficient and α the intercept, where X is a vector of explanatory variables. Parameters α and β are generally assessed by ordinary least squares estimation of the mean of Y conditional on X(E[Y|X]), and are obtained by minimizing the sum of squared residuals.

    QR can be easily interpreted by replacing E[Y|X] with any given quantile of the response variable Q[τ|X]. For each quantile, τ, the linear QR can be described as Y=f'(αττ,X), with βτ denoting the QR coefficient and ατ the intercept for each τ. Both parameters vary, depending on the τth quantile being considered. In contrast to traditional linear regression, an estimate of the two parameters can be obtained from the conditional quantile function by minimizing the sum of the absolute errors:

    \begin{equation} \widehat{\beta_\tau}={\rm arg\;min}\sum_i\rho_\tau[y_i-f'(\alpha_\tau,\beta_\tau,x_i)] , \end{equation}

    where (xi,yi) is a given set of observations of X and Y; ρτ( ) is the tilted absolute value function defined as follows (Barbosa, 2008):

    \begin{equation} \rho_\tau(u)=\left\{ \begin{array}{l@{\quad}l} u(\tau-1) & u<0\\ u\tau & u\geqslant 0 \end{array} \right. . \end{equation}

    Detailed information on QR can be found in the literature (Barbosa, 2008; Tareghian and Rasmussen, 2012; Lee et al., 2013). The "quantreg" package in the R language was run for QR analysis.

  • To assess uncertainty in the coefficient estimates of QR, a bootstrap method was used. This method permits acquisition of a distribution of regression coefficients instead of a single estimate. There has been widespread application of the method to compute standard errors and estimate confidence intervals in the implementation of QR.

    The bootstrap method is a Monte-Carlo method for estimating the sample distribution of a parameter estimator that is calculated from some population of size n. (Efron, 1982) first introduced the method for non-linear median regression. The method was further developed and applied in QR settings (De Angelis et al., 1993), and is based on resampling from the empirical distribution of the residual vector:

    \begin{equation} \widehat{u_\imath}=y_i-f'(\alpha_\tau,\beta_\tau,x_i),\quad i=1,\ldots,n . \end{equation}

    Drawing bootstrap samples u1*,…,un* with replacement from the estimated empirical distribution and using Eq. (3), we obtain

    \begin{equation} y_i^*=f'(\alpha_\tau,\beta_\tau,x_i)+u_i^* . \end{equation}

    Then, a bootstrapped regression coefficient is calculated using

    \begin{eqnarray} \beta_\tau^*&=&{\rm arg\;min}\;\sum_{i|y_i^*<f'(\alpha_\tau,\beta_\tau,x_i)}(1-\tau)[y_i^*-f'(\alpha_\tau,\beta_\tau,x_i)]+\nonumber\\ &&\sum_{i|y_i^*>f'(\alpha_\tau,\beta_\tau,x_i)}\tau[y_i^*-f'(\alpha_\tau,\beta_\tau,x_i)] . \end{eqnarray}

    This process is repeated m times, and \(\hat\beta_\tau,1^*,\ldots,\hat\beta_\tau,m^*\) are obtained, so the asymptotic variance of \(\hat\beta_\tau\) can be estimated (Koenker, 2005).

    However, this residual bootstrap method is only appropriate under the independent and identically distributed error assumption, which is rarely satisfied (Koenker, 1994). The (x,y)-pair bootstrap is an alternative method. In contrast to the residual bootstrap, the (x,y) pairs are resampled n times with replacements from the joint empirical distribution of the sample producing a new sample of size n, (xi*,yi*). According to Eq. (5), a regression coefficient is estimated for each bootstrap sample of (xi*,yi*). Similar to the residual bootstrap, the above process is repeated m times, and the asymptotic variance of \(\hat\beta_\tau\) can be estimated. An advantage over the residual bootstrap is that the (x,y)-pair bootstrap can be applied for independent but not identically distributed error terms. Therefore, confidence intervals and p-values for the coefficients were estimated using the (x,y)-pair bootstrap QR in the present work.

  • The choice of predictor variables is one of the most influential steps in the development of statistical downscaling models. Circulation-related predictors (i.e., mean sea level pressure, geopotential height, and wind component) are usually chosen as candidate predictors, because their long series are available from various reanalysis products and they can be modeled by GCMs with high skill (Liu et al., 2011). In addition to circulation-based predictors, (Hellstr\"om et al., 2001) suggested that inclusion of humidity-related predictors is important for capturing climate change signals. They demonstrated that a change of specific humidity appears to be the major contributor to long-term rainfall change in the future, relative to the change of relative humidity. Temperature is often used to downscale precipitation (Maraun et al., 2010). Many previous studies (Huth, 2002; Fan et al., 2013) have favored upper-air predictors, as these are usually better simulated by GCMs than near-surface variables. Therefore, a total of 11 variables at two pressure levels (500 hPa and 850 hPa) were selected as candidate predictors (Table 1).

    The choice of predictor domain should also be given high priority in statistical downscaling methods, because it may affect downscaled results (Benestad, 2001). (Brinkmann, 2002) found that single-grid predictors are not necessarily related to an underlying location; therefore, a multi-grid predictor domain should be selected for each station during statistical downscaling. (Liu et al., 2011) believed that potential grids should include the grid nearest the target station and those surrounding it. In our study, predictors were calculated using a weighted average of grid cells near the target station. We used 360 km as a radius to define grid cells near the target station. The weight for grid k can be obtained using

    \begin{equation} w[(x_s,y_s),(x_k,y_k)]=\min\{360-d[(x_s,y_s),(x_k,y_k)],0\} , \end{equation}

    where (xs,ys) are the coordinates of the target station, and (xk,yk) are the center coordinates for grid k, and d[(xs,ys), (xk,yk)] is the distance between them (in km). A detailed description is given by (Beuchat et al., 2012). Weighted averages of predictors were standardized for future analysis by subtracting the mean and dividing by the standard deviation over the entire study period.

  • We used QR to build a connection between the large-scale predictors and daily precipitation amount. Regression lines at certain quantiles were fitted by QR to estimate the impact of a single predictor on the quantile of precipitation amount. The 11 standardized predictors were also taken as inputs to the QR models for evaluating their relative contributions to precipitation amount at various quantiles. Confidence intervals and p-values for the QR coefficients were estimated using the (x,y)-pair bootstrap. If the p-value of the regression coefficient of a predictor at a given quantile was less than 0.05, the impact of the predictor on the precipitation amount at the quantile was deemed as significant at the 5% level.

    Simulation of the precipitation amount is a key step in downscaling daily precipitation if a day is wet. Therefore, we concentrate on describing how to model the precipitation amount on a given wet day using QR. Fundamentally, an appropriate probability distribution of the daily precipitation amount for any wet day should be searched for, conditioned by the set of predictor variables. Then, a random value is drawn from this conditional distribution to generate the precipitation amount for the day (Wilks, 2010; Yang et al., 2010). To obtain a detailed representation of a precipitation distribution for a wet day, the QR method was used for a range of quantiles, corresponding to non-exceedance probabilities 0.01 to 0.99, in steps of 0.01. A regression model was constructed using significant predictor variables as inputs and then a quantile estimate was produced for each probability level (0.01,0.02,…,0.99) using the predictors for a given wet day. The quantile estimates should be fitted by a distribution family and then the precipitation amount on the wet day is generated by drawing a random value from the fitted precipitation distribution. As an example, Fig. 2 shows quantile estimates of Beijing precipitation amounts (red dots) for a particular wet day and a fitted curve of these estimates using a gamma distribution (black line). These estimated quantiles can be referred to as a result of sample variability and an approximation of a cumulative distribution function (CDF) of daily precipitation. The above process was repeated for all wet days in the study period, resulting in estimated precipitation amounts for those days. Simulated and observed precipitation amounts were compared to measure the ability of fitting the precipitation amount using QR.

    Four families of distributions were used (gamma,\,Weibull, exponential, and log-normal) and their results were compared. To estimate the distribution parameters, two methods are conventionally used: maximum likelihood and method of moments (Xue and Chen, 2007). Here, maximum likelihood was used to estimate the distribution parameters. In a strict sense, a goodness-of-fit test should be conducted to find which distribution family best fits the quantiles on any given wet day (Tareghian and Rasmussen, 2013). However, we simplified the procedure by using the same distribution type for all wet days. The four distribution families were thus applied consistently, to determine which family provided the overall best results for any station and season.

  • The quantile-quantile (Q-Q) plot is a graphical method that is commonly used to compare two probability distribu-tions by plotting their quantile estimates against each other. If the two distributions are alike, points in the Q-Q plot will approximately lie on the y=x reference line (Xue and Chen, 2007). The Q-Q plot was used to assess the performance of the QR models in fitting precipitation amounts.

    Figure 2.  Predictions for precipitation quantiles from 0.01 to 0.99 in steps of 0.01, by QR (red dots) and curve fitted by gamma distribution (black line), for Beijing on a given wet day.

    Figure 3.  Relationships between Beijing daily precipitation and V850 (a) and SH850 (b) for the wet season. Solid lines denote regression for selected quantiles; dashed line represents regression based on least square estimation.

    Figure 4.  The same as Fig. 3, except for the dry season.

    Figure 5.  Quantile regression (black dashed lines) and traditional linear regression (TLR) (solid red lines) coefficients of Beijing daily precipitation for the wet season, and their 95% confidence bounds [gray shading (QR) and dashed red lines (TLR)] for 11 predictors. The horizontal axis indicates quantile.

    Figure 6.  The same as Fig. 5, except for the dry season.

    Figure 7.  Spatial distribution of stations for which 90th quantile regression coefficients (Coef) were statistically significant at a 5% level for each predictor for the wet season. Purple lines represent contours of annual mean precipitation of 200 mm and 500 mm (see Fig. 1).

    To quantitatively measure goodness-of-fit, an average distance (AD) from the scattered points to the y=x reference line in the Q-Q plot was exploited according to the following equation (Li et al., 2012):

    \begin{equation} {\rm AD}=\dfrac{1}{N}\sum_{j=1}^N|x_{{\rm OBS},j}-x_{{\rm DS},j}|\sin\left(\dfrac{\pi}{4}\right) , \end{equation}

    where x OBS,j and x DS,j are observed and simulated quantiles for point j.

4. Results
  • In this section, two large-scale predictors, V850 and SH850, were used to illustrate how QR was applied to investigate the relationship between a predictor and the Beijing daily precipitation amount. First, scatter plots were drawn to show how the amount was affected by V850 and SH850, and then certain QR lines were fitted to examine the influence of the predictors on selected quantiles.

    In the wet season, the increase of daily precipitation amount at Beijing was seen with V850 increase in the 75th, 90th, 95th, and 99th quantiles and the mean state fit by linear regression based on least square estimation. The regression slope in the 99th quantile was much steeper than those in the mean state and 90th and 95th quantiles. However, precipitation in the 10th and 50th quantiles was obviously not influenced by V850 (Fig. 3a). Similarly, SH850 had a positive contribution to wet-season precipitation in the 50th, 75th, 90th, 95th, and 99th quantiles and the mean state, whereas there was almost no effect on the 10th quantile. SH850 had a more significant influence on precipitation in the upper-tail quantiles than in the other quantiles (Fig. 3b).

    In contrast to the wet season, V850 showed almost no contribution to precipitation in the various quantiles at Beijing for the dry season. Figure 4a shows that regression lines fit by QR are almost parallel, suggesting that the precipitation variability in these quantiles did not vary dramatically with V850. Similar to the wet season, SH850 had a positive role in producing dry-season precipitation in the 50th, 75th, 90th, 95th, and 99th quantiles and the mean variability. Precipitation in extreme quantiles was especially influenced by SH850 (Fig. 4b).

    In summary, there were different impacts of a single predictor variable on the precipitation amounts in various quantiles. This result highlights a superiority of QR. That is, the influence of a large-scale predictor variable on precipitation amount in different quantiles, especially extreme ones, can be observed intuitively by fitted regression lines. Consequently, it is easy to understand the physical process involved.

  • The 11 predictor variables were taken as inputs to the QR model for exploring the relationship between the Beijing precipitation amount and the large-scale predictors. Regression coefficients (black dashed lines) and their confidence intervals (gray shading) for quantiles 0.1 to 0.9 (in steps of 0.1) were estimated using the bootstrap method in QR and are displayed in Figs. 5 and Figs.6.

    For the wet season, confidence envelopes of the regression coefficients of predictors V850, SH850, V500, and SH500 almost do not cross the zero line, suggesting that the impact of these four predictors on the entire distribution of precipitation is statistically significant. Moreover, a more pronounced effect is observed with quantile increase. This result implies that meridional water vapor transport into Beijing is important to its wet-season precipitation, and that heavy precipitation is associated with an enhanced water vapor supply. In contrast, U850 and T850 show no significant influence on wet-season extreme precipitation at Beijing, in spite of an obvious impact in the middle quantiles. H850 also makes a significant contribution to the extreme precipitation (Fig. 5).

    For the dry season, U850, V850, and SH850 are the three major influences on the entire distribution of precipitation at Beijing. This means that both zonal and meridional water vapor transports influence dry-season precipitation. The higher the quantile, the stronger its impact, but the greater the uncertainty. In addition, T850 had a significant effect on Beijing dry-season extreme precipitation (Fig. 6).

    Through comparison of the two aforementioned seasons, we found that meridional wind and specific humidity at both 850 hPa and 500 hPa levels consistently affected the wet-season precipitation, whereas only those at 850 hPa were related to the dry-season precipitation. Further, more predictor variables could be retained to explain the wet-season precipitation than the dry-season precipitation. This is probably because the wet-season precipitation is predominantly convective and more complex, whereas the dry-season precipitation is mostly caused by synoptic-scale systems (Tareghian and Rasmussen, 2013). The QR coefficients of the large-scale predictors were near zero with the quantile close to zero, which suggests that impacts of the large-scale predictors on light precipitation in the two seasons are minor.

    To summarize, some predictor variables were important across a range of quantiles, and some were important only for certain parts of the conditional distribution. Still, others were not important for all parts of that distribution. Thus, it is reasonable to select a set of different predictors to develop a different QR model when simulating precipitation amount. Compared with conventional statistical downscaling of precipitation, such as a generalized linear model or a linear regression model with a fixed set of predictors for the total precipitation distribution (Hanssen-Bauer et al., 2005), models based on QR can make more effective use of available information.

  • Figure 8.  The same as Fig. 7, except for the dry season.

    QR was used to evaluate relationships between extreme precipitation (90th quantile) and the 11 candidate predictors for the 238 stations in the study region. If the p-value of the regression coefficient of a given predictor at a station evaluated via the bootstrap method was less than 0.05, this predictor could be considered to have a significant effect on extreme precipitation at that station (at a 5% significance level). Such a station is labeled in black (for negative coefficient) or red (for positive coefficient) dots in Fig. 7. This figure indicates that overall, SH850 and V850 were the most important predictors of wet-season extreme precipitation. SH850 had an outstanding effect on wet-season extreme precipitation over the entire study region, except at a few stations in southern Xinjiang. SH500, V500, and V850 had an obvious effect on the extreme precipitation mainly at stations east of line P200. This finding implies that the northward transport of water vapor has the strongest relationship with extreme precipitation at these stations. (Zhou and Yu, 2005) asserted that anomalous precipitation is directly related to moisture supply and that a large amount of water vapor is advected from adjacent oceans into the East Asian monsoon region by large-scale monsoon circulations. (Simmonds et al., 1999) concluded that water vapor transport by the Indian monsoon and southeastern Asian monsoon effect northeastern China; however, transport by midlatitude westerlies has a significant impact only in northern China, but this is very weak. This was also observed in our study. The effect of T850 on extreme precipitation was negative at stations mainly east of line P200. Effects of T500 on extreme precipitation were opposite between stations west of that line (positive) and east of it (negative). Compared with the predictor variables at 850 hPa, those at 500 hPa had a much weaker impact on extreme precipitation in the study region. Similarly, SH850 and V850 were the major contributors to dry-season extreme precipitation there (Fig. 8). This also suggests that the meridional transport of water vapor has a major influence on dry-season extreme precipitation.

    Figure 9.  Q-Q plots of observed versus simulated wet-season precipitation (units: mm) using four distribution families at four selected stations: (a) \"Ur\"umqi, (b) Hetian, (c) Beijing, and (d) Harbin.

    In general, different significant predictors affect the extreme precipitation at different stations; and V850 and SH850 are the two major predictors of the extreme precipitation in the two seasons. T850 had an important and negative effect on extreme precipitations at stations mainly in the semi-arid and semi-humid areas, but no significant effect was observed on the extreme precipitation in the arid region. This agrees with the conclusion of (Zhai and Pan, 2003). They studied spatial and temporal characteristics of precipitation extremes over northern China during the second half of the 20th century, and concluded that in the context of global warming, the number of heavy rain days had a clearly decreasing trend on the Northeast China Plain and in North China.

  • Four of the 238 stations were selected to illustrate how to generate daily precipitation on wet days using the procedure in section 3.4. Figure 9 shows simulated results during the entire wet season at the four sample stations, in the form of Q-Q plots. Despite an acceptable fit for lower values of precipitation, the four distribution families reveal differences in fitting greater values. For \"Ur\"umqi Station, the gamma distribution model for precipitation provided a better fit. For Hetian and Beijing, the gamma and Weibull distribution models had a more reasonable performance. For Harbin, the Weibull distribution model performed more satisfactorily.

    For the dry-season precipitation, the Weibull distribution model exhibited a performance superior to those of the other distributions at \"Ur\"umqi and Harbin, whereas those of the gamma distribution indicated more realistic performance at Hetian and Beijing (Fig. 10).

    Figure 10.  The same as Fig. 9, except for the dry season.

    Table 2 shows that for the four sample stations, AD values corresponding to the four distribution types had differences. The smaller the AD value of the distribution family, the better the QR model of that family simulated the precipitation amount. Model skill depended on the station and distribution family used. For the wet season, the Weibull distribution model performed better than the other distribution families at Harbin, whereas the gamma distribution model performed better at the other stations (Fig. 9). For the dry season, the Weibull distribution model gave better results than the others at \"Ur\"umqi, whereas the gamma distribution model performed best at the other three stations. Compared with the wet season, the AD values of the dry season were much smaller. This may be partly because the dry-season precipitation tends to be less extreme and therefore easier to model.

5. Conclusion
  • In this paper, QR was introduced to examine the predictor-precipitation relationship over northern China from a new perspective. This method provides a more complete analysis of the predictor-precipitation relationship, because it focuses on the full spectrum of the daily precipitation distribution rather than the mean state. The main conclusions are summarized as follows.

    The superiority of QR is such that each predictor has a different influence on various parts of the precipitation distribution for the wet and dry seasons. For example, SH850, SH500, V850, and V500 had major impacts on all parts of the distribution of Beijing wet-season precipitation, whereas U850, V850, and SH850 were major contributors to the dry-season precipitation. Effects of these predictors on precipitation amount strengthened with increasing quantile, but were minor for light precipitation.

    Relationships between the large-scale predictors and extreme precipitation over northern China were also examined using the QR method. We observed that SH850 had a remarkable influence on wet-season extreme precipitation region-wide, and the influences of V850, V500, and SH500 were significant mainly in semi-arid and semi-humid areas. SH850 and V850 were the two important predictors for dry-season extreme precipitation region-wide. T850 had an important and negative impact on extreme precipitation in semi-arid and semi-humid areas, but no significant effect in the arid region. This can be interpreted as an impact of global warming.

    QR was used to simulate daily precipitation amounts for wet days at selected stations. Performance of the QR model in simulating daily precipitation amounts was evaluated and compared using Q-Q plots and the AD values. The gamma and Weibull distributions gave more realistic results relative to others for both seasons. These results suggest that QR can simulate the precipitation amount if an optimum distribution family is chosen. Therefore, QR is very useful for identifying the association of all parts of the conditional distribution of daily precipitation with the large-scale predictors, and should be considered in the study of climate change.

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