Elevation Influence on Rainfall and a Parameterization Algorithm in the Beijing Area

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  • Corresponding author: Linye SONG, lysong@ium.cn
  • Funds:

    Supported by the National Natural Science Foundation of China (41605031), National Key Research and Development Program of China (2018YFF0300102 and 2018YFC1507504), and Beijing municipal science and technology plan (Z151100002115012).

  • doi: 10.1007/s13351-019-9072-3

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  • Using a high-density automatic weather stations (AWS) dataset of hourly rainfall observations, the present study investigates the relationship between rainfall and elevation in the Beijing area, and further proposes a rainfall amount dependent parameterized algorithm considering the elevation effect on rainfall on hourly timescale. The parameterization equation is defined as a segmented nonlinear model, which calculates the mountain rainfall as a function of valley rainfall amount. Results show that there exists an evident enhancement of rainfall amount by elevation effect in the Beijing area. In particular, larger rainfall amount is generally found in higher mountains, especially for slight rain and moderate rain. Furthermore, six representative station pairs located in valleys and on mountains respectively are selected to estimate the values of optimal parameters in the parameterization equation. The parameterization algorithm of elevation dependence can produce a reduction in the root-mean-square error and obtain a much closer mountain rainfall total to the observations compared with those using no elevation dependence. Furthermore, the spatial distribution of rainfall is more realistic and accurate in mountainous terrain when elevation dependence is considered. This study helps to understand the variability of rainfall with complex terrain in the Beijing area, and gives a possible way to parameterize rainfall–elevation relationship on hourly timescale.
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  • Fig. 1.  Topography of the Beijing area (Zh, shading; m) obtained by linear interpolation from the global 30'' elevation dataset provided by the United States Geological Survey. The locations of 226 stations in the Beijing area are shown with “+”. Numbers 1–6 indicate the locations of the six mountain–valley station pairs selected in the analysis (red dot: mountain station; blue dot: valley station).

    Fig. 7.  Scatter plot of hourly rainfall amounts at the valley and mountain for station pair 5. The red solid line shows the parameterized relationship using Eq. (1). The dashed black line is the diagonal line.

    Fig. 2.  Distributions of rainfall amount (mm yr−1) in (a) flooding season (May–September) and (b) non-flooding season (October–April), and percentage of rainfall amount in annual total (%) in (c) flooding season and (d) non-flooding season during 2009–16.

    Fig. 3.  The average rainfall amount for classes 1–9 defined in Table 2, using (a) all data samples, (b) light rain (< 2.5 mm h−1), (c) moderate rain (2.5–8.0 mm h−1), (d) heavy rain (8.0–16.0 mm h−1), and (e) torrential rain (> 16.0 mm h−1).

    Fig. 4.  Scatter plots of rainfall amount and elevation relationships for 226 stations. (a) All data samples, (b) light rain (less than 2.5 mm h−1), (c) moderate rain (within 2.5–8.0 mm h−1), (d) heavy rain (within 8.0–16.0 mm h−1), and (e) torrential rain (larger than 16.0 mm h−1). The linear regression lines (dashed lines) and correlation coefficients (CCs) between rainfall amount and elevation are also given, which are separately analyzed for the whole Beijing area (black), plain area (red), and mountain area (blue). * and ** indicate the CC significant at the 90% and 95% confidence levels, respectively.

    Fig. 5.  (a) Histogram of hourly rainfall data samples, separated by four-level rain: light (0–2.5 mm h−1), moderate (2.5–8.0 mm h−1), heavy (8.0–16.0 mm h−1), and torrential (larger than 16.0 mm h−1). Distributions of annual rainfall amount (mm yr−1) for light and moderate rain are given in (b) and (c), respectively.

    Fig. 6.  Box plots of hourly precipitation for each station pair located at mountain and valley. (a)–(f) represent the 1–6 station pairs listed in Table 1. For each box plot, the top (bottom) of the box indicates the 75% (25%) quartile, and the middle of the box gives the median value. The enhancement factors (EFs; mm h−1 km−1) are shown for each station pair.

    Fig. 8.  (a) Relationship between ratio A and parameter b for the six different locations, respectively. (b) The averaged analytical fitted line. Parameter a is set as 1.8 here.

    Fig. 9.  (a) RMSE and (b) totals of the mountain station rainfall estimated by no elevation dependence (red) and with elevation dependence (blue) using the parameterization algorithm in Eq. (1). The observed rain total (green) is also shown in (b).

    Fig. 10.  Relationship between original topography (x axis; m) and modified topography (y axis; m) calculated from Eq. (8) with Zmax = 1000 m and ∆Z = 500 m.

    Fig. 11.  Rainfall case of an hourly AWS interpolated field at 1700 LST 1 May 2015 in the Beijing area for (a) pure station interpolation and (b) station interpolation with the elevation effect. The topography is indicated by contour lines with a 200-m interval.

    Fig. 12.  Example of precipitation analysis at 0850 UTC 23 May 2016 based on the combination of real-time station data and radar data in the RMAPS-IN system. (a) Pure station interpolation using 0850 UTC AWS data, (b) radar derived precipitation using 0847 UTC radar quantitative precipitation estimation (QPE) data, (c) station interpolation with the elevation effect at 0850 UTC, and (d) product of final precipitation analysis.

    Table 1.  Information of the six representative station pairs shown in Fig. 1, including station identification, station name, elevation (z), vertical elevation difference (Δz), horizontal distance difference (Δx), the location, and relative direction of the valley station to the corresponding mountain station

    No.Station IDStation namez (m)Δz (m)Δx (m)Location
    1A1352Miaofengshan10727465542Central–western Beijing
    Northwest direction
    A1410Pusalu326
    2A1658Yunmengshan6342935812Northern Beijing
    Northwest direction
    A1604Liulimian341
    3A1511Bolitai6803495886Northeastern Beijing
    Northwest direction
    A1507Zhenluoying331
    4A1469Songshan8403658198Northwestern Beijing
    Southeast direction
    A1458Yeyahu475
    5A1360Baihuashan9954198956Southwestern Beijing
    Northeast direction
    A1353Qingshui576
    6A1417Hezijian8726577324Central–western Beijing
    Southeast direction
    A1404Gujiang215
    Download: Download as CSV

    Table 2.  The definition of different elevation classes

    ClassElevation (m)Station numberAverage elevation (m)
    10 ≤ Zst < 10013145.6
    2100 ≤ Zst < 20025136.2
    3200 ≤ Zst < 30013250.5
    4300 ≤ Zst < 40014343.4
    5400 ≤ Zst < 50012458.7
    6500 ≤ Zst < 60012553.3
    7600 ≤ Zst < 70011643.3
    8700 ≤ Zst < 10004894.8
    9Zst ≥ 100041282.3
    Download: Download as CSV

    Table 3.  Characteristics of rainfall between each station pair located at mountain and valley, including rainfall total, ratio of the mountain to valley rainfall totals (A), and average rainfall intensity difference during the flooding seasons in the period 2009–16. * and ** indicate that the average difference is significant at the 85% and 90% confidence levels based on the two-tailed Student’s t test, respectively

    No.Station pairRainfall total
    (mm)
    Ratio AAvg (RmRv)
    [mm (12 h)−1]
    1A13523141.61.110.14*
    A14102842.9
    2A16583683.01.130.19**
    A16043267.5
    3A15113997.81.090.17*
    A15073668.4
    4A14692738.41.180.22**
    A14582314.7
    5A13603172.61.250.33**
    A13532531.4
    6A14173166.61.100.15*
    A14042867.7
    Download: Download as CSV

    Table 4.  Estimated optimum parameter values for the selected six station pairs

    No.Station paira b (mm−1)Rc (mm h−1)RMSE (mm h−1)
    1A1352–A14101.80.700.570.900
    2A1658–A16041.80.490.820.760
    3A1511–A15071.70.430.811.085
    4A1469–A14582.10.810.680.657
    5A1360–A13532.40.900.780.817
    6A1417–A14041.60.370.810.897
    Download: Download as CSV

    Table 5.  Estimated optimum parameter values using the location-independent value of a = 1.8

    No.Station pairab (mm−1) Rc (mm h−1)RMSE (mm h−1)
    1A1352–A14101.80.700.570.900
    2A1658–A16041.80.490.820.760
    3A1511–A15071.80.590.681.086
    4A1469–A14581.80.381.050.659
    5A1360–A13531.80.231.740.819
    6A1417–A14041.80.720.560.898
    Download: Download as CSV
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Elevation Influence on Rainfall and a Parameterization Algorithm in the Beijing Area

    Corresponding author: Linye SONG, lysong@ium.cn
  • 1. Institute of Urban Meteorology, China Meteorological Administration, Beijing 100089, China
  • 2. Central Institute for Meteorology and Geodynamics, Vienna A-1190, Austria
Funds: Supported by the National Natural Science Foundation of China (41605031), National Key Research and Development Program of China (2018YFF0300102 and 2018YFC1507504), and Beijing municipal science and technology plan (Z151100002115012).

Abstract: Using a high-density automatic weather stations (AWS) dataset of hourly rainfall observations, the present study investigates the relationship between rainfall and elevation in the Beijing area, and further proposes a rainfall amount dependent parameterized algorithm considering the elevation effect on rainfall on hourly timescale. The parameterization equation is defined as a segmented nonlinear model, which calculates the mountain rainfall as a function of valley rainfall amount. Results show that there exists an evident enhancement of rainfall amount by elevation effect in the Beijing area. In particular, larger rainfall amount is generally found in higher mountains, especially for slight rain and moderate rain. Furthermore, six representative station pairs located in valleys and on mountains respectively are selected to estimate the values of optimal parameters in the parameterization equation. The parameterization algorithm of elevation dependence can produce a reduction in the root-mean-square error and obtain a much closer mountain rainfall total to the observations compared with those using no elevation dependence. Furthermore, the spatial distribution of rainfall is more realistic and accurate in mountainous terrain when elevation dependence is considered. This study helps to understand the variability of rainfall with complex terrain in the Beijing area, and gives a possible way to parameterize rainfall–elevation relationship on hourly timescale.

1.   Introduction
  • Rainfall events can exert significant impacts on the economy, industry, agriculture, and the daily lives of tens of millions of people. Therefore, research into precipitation characteristics and forecasting is of great importance for risk management, disaster prevention and mitigation (e.g., Hao and Ding, 2012; Duan et al., 2013; Zhang et al., 2013; Wang Y. et al., 2014; Song and Duan, 2015; Zhao et al., 2019a, b). In particular, heavy rainfall on the mountainous terrain may lead to serious secondary disasters, such as mud rock flow and landslides. As the political and cultural center of China with a very large population, Beijing often encounters rainfall disasters in its flooding season, which cause great casualties and losses. For example, a local short-duration heavy rainfall event occurred under favorable terrain conditions on 1 August 2002 in Miyun and caused a serious mud rock flow and flood fatalities (Wang et al., 2003). An internationally noted extreme event generated by a mesoscale convective system occurred from 1000 LST 21 July to 0400 LST 22 July 2012 with a record 460 mm of rain falling in Fangshan (Zhang et al., 2013). Wang N. et al. (2014) carried out a numerical simulation of the above “7. 21” extreme rainfall event using the Weather Research and Forecasting (WRF) model. They found that the local terrain on the southwest side of Beijing had a significant impact on the location of precipitation center. This extreme torrential rainfall event killed 79 people due to drowning, lightning strike, electrocutions, and landslides (Zhang et al., 2013), and caused direct economic losses of ¥61 billion (Zhou, 2012).

    Many previous studies indicated the importance of topography effect on the spatial distribution of rainfall in mountainous areas (e.g., Smith, 1979, 1989; Barry, 1992; Fu, 1992; Basist et al., 1994; Singh and Kumar, 1997; Weisse and Bois, 2001; Sun, 2005; Liao et al., 2007; Chen and Zhang, 2010). However, previous studies focused mainly on the long-term precipitation, such as monthly, annual, interannual, and daily timescales. The elevation effect was suggested to be most pronounced over long time periods. At shorter time periods, the relationship between rainfall and elevation was much more complex (Haiden and Pistotnik, 2009). Elevation influence on rainfall is crucial for realistic estimation of hourly rainfall distribution, especially in mountainous area with relative sparser gauge observations. However, this issue has not been fully considered.

    Located in the transition zone between the mountainous area and the North China Plain, Beijing is surrounded by the Taihang Mountains to its west and the Yanshan Mountains to its north-northeast. The spatial distribution and diurnal variation characteristics of summer rainfall in Beijing area have been widely investigated (e.g., Sun, 2005; Miao et al., 2011; Yang et al., 2013, 2017; Jiang et al., 2014; Liu et al., 2014; Zheng et al., 2014). However, much fewer studies have investigated the influence of elevation on rainfall in the Beijing area, especially on the hourly timescale. Previous studies only used hourly rainfall dataset from less than 30 stations, or simulated only one rainfall case using numerical model. For example, Yin et al. (2011) investigated the possible effect of topography on the characteristics of diurnal rainfall variations in the Beijing area by using 26 stations (6 stations belonged to neighboring Hebei Province), and showed that stations in the plain area exhibited typical night rain peaks while those in the mountainous areas exhibited clear afternoon peaks. Hourly rainfall data with spatially sufficient observations and enough record length are important for investigating the elevation influence on local precipitation distribution, and to form a parameterization algorithm of elevation dependence. In this study, we apply a high-density automatic weather station (AWS) dataset with 226 stations over Beijing during the period 2009–16, to investigate the elevation influence on rainfall in the Beijing area. The purpose of this paper is to answer the following questions: (1) What is the possible relation between the elevation and rainfall over the area of Beijing using hourly rainfall dataset during flooding season? (2) Based on the elevation influence, how to develop a parameterization algorithm suitable for the Beijing area?

    The paper is organized as follows. Section 2 introduces the dataset used in the study. Section 3 describes an elevation-considered parameterization algorithm proposed in this study. Section 4 presents the results of the elevation–rainfall relationship in the Beijing area, and the results of the estimated optimum orographic parameters. Applications of the parameterization algorithm in interpolating precipitation field and obtaining high-resolution precipitation analysis in a real-time nowcasting system are also given. Section 5 gives conclusions and discussion.

2.   Dataset
  • The data used in this study are from the hourly rain gauge observations in Beijing during the period from 1 January 2009 to 31 December 2016. The data are obtained from the Meteorological Information Center (MIC) of the Beijing Meteorological Service (BMS), and have been quality controlled by MIC. Hourly rainfall data are collected from 226 stations with information of accurate locations (Fig. 1). The spatial site distribution is denser in the central urban area than in the surrounding regions, and it is relatively sparse in the southwestern and north-northeastern mountainous regions.

    Figure 1.  Topography of the Beijing area (Zh, shading; m) obtained by linear interpolation from the global 30'' elevation dataset provided by the United States Geological Survey. The locations of 226 stations in the Beijing area are shown with “+”. Numbers 1–6 indicate the locations of the six mountain–valley station pairs selected in the analysis (red dot: mountain station; blue dot: valley station).

    The average distances between rain-gauge stations of real-time AWS network in the Beijing area are 3–5 km in the urban region, and 8–10 km in the rural region, respectively. However, the average station distance is approximately 15–20 km in the mountainous areas where there are fewer stations (Liu et al., 2014). The parameterization algorithm is performed by using station pairs located in valleys and on mountains, respectively. In this study, we chose a horizontal spatial scale of 5–10 km (meso-γ scale) to select such station pairs. There are two reasons for focusing on the local meso-γ-scale characteristics of rainfall from a valley floor to a mountain ridge. First, this has been proposed as the optimal scale for application of the elevation–precipitation relationship (Daly et al., 1994; Sharples et al., 2005; Haiden and Pistotnik, 2009). Second, distribution of the real-time AWS rain gauge stations in the Beijing area is dense enough to capture the characteristics of rainfall variations on the meso-β scale (approximately 20–100 km).

    Table 1 lists the information of six selected representative station pairs located in valleys and on mountains, respectively. Besides station identification and station name, the topographic elevation, difference in vertical elevation, difference in horizontal distance, and relative location of the valley station to the mountain station are also given in detail. Note that it is very important to choose suitable station pairs to study the relationship between rainfall intensity and elevation. In this study, the above six station pairs (Table 1) are chosen for two reasons: firstly, they are located in the mountainous area and can represent different parts of Beijing; secondly, they meet the requirement of spatial-scale definition (meso-γ scale) for parameterization algorithm application. In particular, the difference of horizontal distance between the valley station and mountain station is limited to approximately 7 km (5–9 km) and the difference of vertical elevation is limited to about 500 m (300–800 m) when selecting the station pairs. Stations of pair 5 are located in the mountainous southwestern part of Beijing, with the valley station to the relative northeast direction of the mountain station (Table 1 and Fig. 1). This station pair is in the interior high terrain area. Stations of pairs 1 and 6 are located in central–western parts of Beijing near the transition zone between the plains and the mountains, with different valley station directions of northwest and southeast to the corresponding mountain stations, respectively (Table 1 and Fig. 1). Stations of pairs 2 and 3 are located in the north and northeast parts of Beijing, with the valley stations to the relative northwest direction of the mountain stations. These four station pairs are in the area from the foreland towards the upslope areas. Stations of pair 4 are located in the northwest parts of Beijing, with the valley station to the relative southeast direction of the mountain station (Table 1 and Fig. 1). Another reason for selecting these station pairs is their relatively complete record. For each station pair, the rainfall amount data are used only when both observations at valley and on mountain are available.

    No.Station IDStation namez (m)Δz (m)Δx (m)Location
    1A1352Miaofengshan10727465542Central–western Beijing
    Northwest direction
    A1410Pusalu326
    2A1658Yunmengshan6342935812Northern Beijing
    Northwest direction
    A1604Liulimian341
    3A1511Bolitai6803495886Northeastern Beijing
    Northwest direction
    A1507Zhenluoying331
    4A1469Songshan8403658198Northwestern Beijing
    Southeast direction
    A1458Yeyahu475
    5A1360Baihuashan9954198956Southwestern Beijing
    Northeast direction
    A1353Qingshui576
    6A1417Hezijian8726577324Central–western Beijing
    Southeast direction
    A1404Gujiang215

    Table 1.  Information of the six representative station pairs shown in Fig. 1, including station identification, station name, elevation (z), vertical elevation difference (Δz), horizontal distance difference (Δx), the location, and relative direction of the valley station to the corresponding mountain station

3.   Parameterization algorithm
  • The parameterization algorithm proposed in this study is rainfall amount dependent. Previous study indicated that if the gradient between rainfall and elevation was parameterized as a function of rain rate, the error caused by using the annual rainfall gradient for each of the 12-h sub-periods can be significantly reduced (Haiden and Pistotnik, 2009). It has been successfully applied in a nowcasting system of the Integrated Nowcasting through Comprehensive Analysis (INCA; Haiden et al., 2009; Wang et al., 2017). In addition, previous studies have demonstrated a physical orographic rainfall process by the seeder–feeder mechanism (e.g., Bergeron, 1965; Smith, 1979, 1989; Robichaud and Austin, 1988; Cotton and Anthes, 1989; Purdy et al., 2005). Specifically, when the seed rain rate is low, it is the conversion that limits the rainfall increment caused by elevation effect. In this case, increasing the seed intensity will result in an approximately parabolic increment of rainfall as only a small part of the condensate is washed away. In contrast, when the seed rain rate is strong, it is the condensation that limits the orographic rainfall increment. In this case, increasing the seed intensity will result in a basically linearly additive rainfall increment as much of the condensate is washed away. Following these ideas, a parameterization algorithm considering the elevation influence on rainfall is designed to be rainfall amount dependent. The equation for the parameterization algorithm is defined as:

    $$ {R_{\rm m}} = \left\{ \begin{array}{l} {R_{\rm v}}\left({a - b{R_{\rm v}}} \right),\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;{R_{\rm v}} \leqslant {R_{\rm c}}\\ {R_{\rm v}} + \left({a - 1 - b{R_{\rm c}}} \right){R_{\rm c}},\;\;\;{R_{\rm v}} > {R_{\rm c}} \end{array} \right.. $$ (1)

    Here, Rm and Rv represent the hourly rainfall on mountain and at valley, respectively. Rc denotes the critical valley rainfall, which is a threshold in the segmented Eq. (1). When Rv is smaller (larger) than Rc, the relationship between valley rainfall amount and mountain rainfall amount is parabolic (linear). This parameterization equation provides the mountain rainfall as a segmented function of the valley rainfall amount.

    At Rv = Rc, the link between Rv and Rm transforms from a parabolic relation to a linear function (Fig. 7). To make sure that Eq. (1) is continuous at the threshold point Rc, the value of Rc should satisfy the criteria of ${\rm{d}}{R_{\rm m}}/{\rm{d}}{R_{\rm v}} = 1$. As such, it can be obtained that the threshold value Rc is related to the orographic parameters a and b, according to:

    Figure 7.  Scatter plot of hourly rainfall amounts at the valley and mountain for station pair 5. The red solid line shows the parameterized relationship using Eq. (1). The dashed black line is the diagonal line.

    $$ {R_{\rm c}} = \left({a - 1} \right)/\left({2b} \right). $$ (2)

    Therefore, if b is replaced by Rc and a, Eq. (1) can be transformed into:

    $$ {R_{\rm m}} = \left\{ \begin{array}{l} {R_{\rm v}} \{ {1 + \left({a - 1} \right)\left[{1 - {R_{\rm v}}/\left({2{R_{\rm c}}} \right)} \right]} \},\;\;\;{R_{\rm v}} \leqslant {R_{\rm c}}\\ {R_{\rm v}} + \left({a - 1} \right){R_{\rm c}}/2,\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;{R_{\rm v}} > {R_{\rm c}} \end{array} \right.. $$ (3)

    Here, in the case of very small values of Rv, the minor term can be ignored, and the simplified relationship between Rm and Rv can be expressed as:

    $$ {R_{\rm m}} = a{R_{\rm v}}. $$ (4)

    Therefore, physical meaning of parameter a represents the ratio between Rm and Rv when Rv is very small. Physical meaning of parameter b measures the decreasing strength of the ratio between Rm and Rv with increasing valley rainfall amount.

    Parameters a and b can be estimated by computing the root-mean-square error (RMSE) and the totals of mountain rainfall as forecasted by the valley rainfall using parameterization Eqs. (1) or (3). For a given parameter a and a known A, parameter b can be estimated by minimizing the RMSE of the mountain rainfall amount as forecasted by the valley rainfall amount. Here, A is defined as ratio of the total long-term mountain rainfall to the total valley rainfall:

    $$ A = {\left({{R_{\rm m}}} \right)_{{\rm{long}} {\text{-}} {\rm{term}}}}/{\left({{R_{\rm v}}} \right)_{{\rm{long}}{\text{-}}{\rm{term}}}}. $$ (5)

    To progress the relationship from station pairs into a gridded precipitation field, station topography Zst(i, j) is first developed by inverse distance weighted interpolation using AWS rain gauge station data. Here, (i, j) represents the net grid, with i for longitude and j for latitude directions. Then, using the link between the valley and mountain rainfall in Eq. (1), relative rainfall gradient field can be calculated as:

    $$ {G_{{\rm{elev}}}}\left({i,j} \right) \equiv \frac{1}{R}\frac{{\Delta R}}{{\Delta Z}} \approx \frac{1}{{{R_{\rm v}}\left({i,j} \right)}}\frac{{{R_{\rm m}}\left({i,j} \right) - {R_{\rm v}}\left({i,j} \right)}}{{\Delta Z\left({i,j} \right)}}. $$ (6)

    The orographic rainfall increment caused by the elevation effect can therefore be obtained by:

    $$ \Delta {R_{{\rm{elev}}}}\left({i,j} \right) = {G_{{\rm{elev}}}}\left({i,j} \right)\left[{{Z_h}\left({i,j} \right) - {Z_{{\rm{st}}}}\left({i,j} \right)} \right]{R_{\rm v}}. $$ (7)

    Here, Zh(i, j) is the 1-km resolution gridded topography obtained by linear interpolation from the global 30'' elevation dataset provided by the United States Geological Survey.

    The topography Zh(i, j) should be modified in the application of the parameterization algorithm. The altered topographic height is calculated as:

    $$ {\hat Z_h} \!=\! \left\{ {\begin{array}{*{20}{c}} {{Z_{\max }} - \Delta Z\exp \left({ - \dfrac{{{Z_h} - {Z_{\max }} + \Delta Z}}{{\Delta Z}}} \right),\;\;{Z_h} > {Z_{\max }} - \Delta Z}\\ \!\!{{Z_h},\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;{Z_h} \leqslant {Z_{\max }} - \Delta Z} \end{array}} \right.. $$ (8)
4.   Results
  • Using the hourly rain gauge observation dataset, the spatial distribution of the annual accumulated rainfall amount in the Beijing area is calculated. Here, the flooding season in the Beijing area is defined from May to September, which is also called warm season and receives majority of the annual total rainfall (Yin et al., 2011). The non-flooding season is referred to as the remaining months from October to April. It is found that during flooding season, maximum rainfall occurs in the northeastern and southwestern parts of Beijing (Fig. 2a). Precipitation decreases from the urban area to the northwestern part of Beijing. This distribution was suggested to be influenced by many factors, such as topography, prevailing wind direction, land use change, and urbanization (Sun and Yang, 2008; Chen and Zhang, 2010; Miao et al., 2011; Yang et al., 2013; Zheng et al., 2014). In particular, the high value of northeastern mountainous region may be related to the topography (Yang et al., 2017). However, during the non-flooding season, Beijing received much less rainfall (Fig. 2b). Because flooding season occupies more than 80% of the annual total rainfall in the Beijing area (Figs. 2c, d), we mainly focus on the flooding season in this study.

    Figure 2.  Distributions of rainfall amount (mm yr−1) in (a) flooding season (May–September) and (b) non-flooding season (October–April), and percentage of rainfall amount in annual total (%) in (c) flooding season and (d) non-flooding season during 2009–16.

    To investigate the possible role of elevation influence in the flooding season, the relationship between rainfall and terrain elevation is examined. The 226 stations in the Beijing area are divided into 9 classes according to different elevation ranges (Table 2). The average rainfall amount is analyzed for the 9 classes. Despite the high accumulated annual rainfall in the plain area from classes 1 to 4, it is found that the average rainfall increases with elevation in the mountainous area from classes 5 to 9 (Fig. 3a). The correlation coefficient (CC) between rainfall and elevation for all samples is −0.30, which is significant at the 95% confidence level according to the Student’s t test (Fig. 4a). It indicates a negative relationship of rainfall amount with the elevation. However, the CC in the high-lying mountainous area is 0.31, significant at the 95% confidence level (Fig. 4a). Hence, there exists a significant positive relationship between rainfall amount and elevation in mountainous area. Note that the large average rainfall amount in the low-lying plain area and the significant negative correlation in the whole area may be most likely caused by the urban heat island effect and the associated circulation in this region (Sun and Yang, 2008; Miao et al., 2011; Yang et al., 2013).

    ClassElevation (m)Station numberAverage elevation (m)
    10 ≤ Zst < 10013145.6
    2100 ≤ Zst < 20025136.2
    3200 ≤ Zst < 30013250.5
    4300 ≤ Zst < 40014343.4
    5400 ≤ Zst < 50012458.7
    6500 ≤ Zst < 60012553.3
    7600 ≤ Zst < 70011643.3
    8700 ≤ Zst < 10004894.8
    9Zst ≥ 100041282.3

    Table 2.  The definition of different elevation classes

    Figure 3.  The average rainfall amount for classes 1–9 defined in Table 2, using (a) all data samples, (b) light rain (< 2.5 mm h−1), (c) moderate rain (2.5–8.0 mm h−1), (d) heavy rain (8.0–16.0 mm h−1), and (e) torrential rain (> 16.0 mm h−1).

    Figure 4.  Scatter plots of rainfall amount and elevation relationships for 226 stations. (a) All data samples, (b) light rain (less than 2.5 mm h−1), (c) moderate rain (within 2.5–8.0 mm h−1), (d) heavy rain (within 8.0–16.0 mm h−1), and (e) torrential rain (larger than 16.0 mm h−1). The linear regression lines (dashed lines) and correlation coefficients (CCs) between rainfall amount and elevation are also given, which are separately analyzed for the whole Beijing area (black), plain area (red), and mountain area (blue). * and ** indicate the CC significant at the 90% and 95% confidence levels, respectively.

    As the levels of hourly rainfall intensities can be usually defined as light rain (0–2.5 mm h−1), moderate rain (2.5–8.0 mm h−1), heavy rain (8.0–16.0 mm h−1), and torrential rain (larger than 16.0 mm h−1), the frequency of AWS hourly rainfall samples used in this study is thus displayed in Fig. 5a according to the four levels. It shows that the vast majority of samples are within 0.1–2.5 mm h−1 threshold, with more than 320,000 samples (Fig. 5a). It accounts for 81.64% of all hourly rainfall data samples. Moderate rain samples within 2.5–8.0 mm h−1 threshold are the secondary largest, which occupy 12.96% of total data samples. However, the samples of heavy rainfall (8.0–16.0 mm h−1) and torrential rainfall (larger than 16.0 mm h−1) are much less as the hourly rainfall data display quite skewed distribution. They only account for 3.46% and 1.94% of all hourly rainfall data samples, respectively. The relation between rainfall amount and elevation is thus further analyzed for threshold 0–2.5, 2.5–8.0, 8.0–16.0, and larger than 16.0 mm h−1, respectively. Results show that there exists an obvious positive relationship between rainfall amount and elevation for light rain within 0–2.5 mm h−1 threshold. The rainfall amount is increasing as the elevation increases from class 1 to 9 (Fig. 3b). This characteristic also exists for moderate rain within 2.5–8.0 mm h−1 threshold, although the increasing trend is weaker than that for slight rain (Fig. 3c). Furthermore, the CCs between rainfall amount and elevation are 0.71 and 0.38 for slight rain and moderate rain, respectively, both significant at the 95% confidence level (Figs. 4b, c). The statistical relations are both positively significant in low-lying plain areas and high-lying mountainous areas, with the CCs of 0.45 and 0.78 for slight rain, 0.17 and 0.58 for moderate rain, respectively. The spatial distribution of rainfall amount displays an increasing characteristic from low elevation areas northwest to high elevation areas (Figs. 5b, c), further verifying the significant positive rainfall–elevation relationship for slight and moderate rainfall. However, for heavy (torrential) rainfall threshold of 8.0–16.0 mm h−1 (larger than 16.0 mm h−1), slight (obvious) decreasing is found as the elevation increases from class 1 to 9 (Figs. 3d, e). In the high-lying mountainous area, no significant relationship is found for threshold of 8.0–16.0 mm h−1 or larger than 16.0 mm h−1, with the insignificant CCs of only 0.04 and −0.01, respectively (Figs. 4d, e). Note that the CC is about −0.54 for rainfall larger than 16.0 mm h−1, which is statistically significant. This significant negative relationship may be mainly due to the fact that Beijing usually receives extreme rainfall events in its urban area, resulting in more torrential rain in low-lying plain area and negative correlation in the whole Beijing area (Miao et al., 2011; Yang et al., 2017).

    Figure 5.  (a) Histogram of hourly rainfall data samples, separated by four-level rain: light (0–2.5 mm h−1), moderate (2.5–8.0 mm h−1), heavy (8.0–16.0 mm h−1), and torrential (larger than 16.0 mm h−1). Distributions of annual rainfall amount (mm yr−1) for light and moderate rain are given in (b) and (c), respectively.

    The enhancement of rainfall with elevation effect is also analyzed in the six selected station pairs (Table 1, Fig. 1) located in valleys and on mountains, respectively. On one hand, the rainfall totals accumulated from hourly rainfall dataset are, in general, higher on the mountain station than at the corresponding valley station (Table 3, ratio A). The average hourly rainfall intensity difference between the mountain and valley stations is approximately 0.1–0.4 mm (12 h)−1. The difference values are small, which may be close to the precipitation measurement error scale of observation stations. Therefore, the rainfall differences between each mountain and valley stations are evaluated based on the two-tailed Student’s t test. In particular, the average differences are 0.14, 0.17, and 0.15 mm (12 h)−1 for station pairs 1, 3, and 6, significant at the 85% confidence level. In addition, the average differences are 0.19, 0.22, and 0.33 mm (12 h)−1 for station pairs 2, 4, and 5, significant above the 90% confidence level (Table 3). This suggests that rainfall intensity difference between the mountain and valley stations is not likely attributed to the measurement error. Furthermore, Fig. 6 shows the box plot of hourly precipitation for each station pair. A higher median value and 75% and 25% quartiles of hourly rainfall are observed on mountain station A1352 compared with those at valley station A1410 (Fig. 6a). This is also true for the other station pairs (Figs. 6bf). The enhancement factor (EF) is calculated, which is defined as the difference of median rainfall intensity between mountain and valley stations with the change of terrain elevation. The EFs are all positive, being 0.40, 0.68, 0.85, 0.27, 0.71, and 0.30 mm h−1 km−1 for the six mountain–valley station pairs, respectively. These results indicate an enhancement of hourly rainfall amount with increasing elevation.

    No.Station pairRainfall total
    (mm)
    Ratio AAvg (RmRv)
    [mm (12 h)−1]
    1A13523141.61.110.14*
    A14102842.9
    2A16583683.01.130.19**
    A16043267.5
    3A15113997.81.090.17*
    A15073668.4
    4A14692738.41.180.22**
    A14582314.7
    5A13603172.61.250.33**
    A13532531.4
    6A14173166.61.100.15*
    A14042867.7

    Table 3.  Characteristics of rainfall between each station pair located at mountain and valley, including rainfall total, ratio of the mountain to valley rainfall totals (A), and average rainfall intensity difference during the flooding seasons in the period 2009–16. * and ** indicate that the average difference is significant at the 85% and 90% confidence levels based on the two-tailed Student’s t test, respectively

    Figure 6.  Box plots of hourly precipitation for each station pair located at mountain and valley. (a)–(f) represent the 1–6 station pairs listed in Table 1. For each box plot, the top (bottom) of the box indicates the 75% (25%) quartile, and the middle of the box gives the median value. The enhancement factors (EFs; mm h−1 km−1) are shown for each station pair.

    The above analysis suggests an elevation enhancement of rainfall amount in the Beijing area. However, the rainfall–elevation relationship at hourly time scale is still difficult to parameterize because of its complex variability. Figure 7 shows an example of the scatter distribution of the rainfall amounts in the valley and on the mountain for stations of pair 5. At hourly time scale, the distribution of hourly rainfall amount is largely scattered. Using the segmented equation of parameterization algorithm described in Section 3, the following section will present the results of the optimum estimation of elevation parameters suitable for hourly rainfall in the Beijing area.

  • Elevation parameters a and b varied independently during the parameterization for all six station pairs in Table 1. Using the parameterization algorithm proposed in Section 3, the estimated optimum values of a and b are shown in Table 4. The ratio between the mountain and valley rainfall at the limit of a small amount of valley rainfall (i.e., a) is approximately 1.6–2.4 for the six station pairs. The optimal estimation of the parameter b is ranging among 0.37–0.90 (Table 4) for Beijing area. Furthermore, it should be noted that parameter a can be set to a location-independent value of 1.8 without significantly increasing the RMSE of the mountain rainfall (Table 5). The optimal values of parameter b also change correspondingly (Table 5).

    No.Station paira b (mm−1)Rc (mm h−1)RMSE (mm h−1)
    1A1352–A14101.80.700.570.900
    2A1658–A16041.80.490.820.760
    3A1511–A15071.70.430.811.085
    4A1469–A14582.10.810.680.657
    5A1360–A13532.40.900.780.817
    6A1417–A14041.60.370.810.897

    Table 4.  Estimated optimum parameter values for the selected six station pairs

    No.Station pairab (mm−1) Rc (mm h−1)RMSE (mm h−1)
    1A1352–A14101.80.700.570.900
    2A1658–A16041.80.490.820.760
    3A1511–A15071.80.590.681.086
    4A1469–A14581.80.381.050.659
    5A1360–A13531.80.231.740.819
    6A1417–A14041.80.720.560.898

    Table 5.  Estimated optimum parameter values using the location-independent value of a = 1.8

    Then, we apply Eq. (1) with a = 1.8 to the dataset in flooding season from 2009 to 2016 of all six station pairs separately, and then vary parameter b to calculate different estimated ratios, A. The results show that the derived relationship between A and b is generally similar for all the six station pairs (Fig. 8a). This indicates that the parameterization method is viable and reasonable. The similarity among the selected station pairs may be attributed to roughly similar climatic conditions at different locations within the Beijing area. The fitted relationship between A and b averaged for all six station pairs can be analytically expressed as:

    Figure 8.  (a) Relationship between ratio A and parameter b for the six different locations, respectively. (b) The averaged analytical fitted line. Parameter a is set as 1.8 here.

    $$ b\left(A \right) = \frac{1}{{{c_1}\left({A - 1} \right)}} - \frac{1}{{{c_2}}}. $$ (9)

    Here, c1 = 11.47 mm and c2 = 6.55 mm are obtained (Fig. 8b).

    For comparison, Fig. 9 shows the mountain rainfall RMSE values and mountain rainfall totals when forecasted by valley rainfall for using no elevation correction and applying the elevation dependence parameterization algorithm. It can be found that the parameterization of the elevation dependence produces a reduction of RMSE compared with no elevation dependence for all the six station pairs, although the reduction is slight (Fig. 9a). In addition, compared with no elevation dependence, the parameterization of the elevation dependence produces rainfall totals much closer to the observations (Fig. 9b). Note that the rainfall total is accumulated by hourly rainfall data. Hence, the results indicate a definite more accurate total rainfall distribution and a more accurate hourly rainfall distribution using the elevation dependence algorithm than those using no elevation dependence.

    Figure 9.  (a) RMSE and (b) totals of the mountain station rainfall estimated by no elevation dependence (red) and with elevation dependence (blue) using the parameterization algorithm in Eq. (1). The observed rain total (green) is also shown in (b).

  • The elevation algorithm with optimum parameters estimated by using rain gauge data in the Beijing area is applied to obtain a gridded precipitation field with the elevation effect interpolated from the stations. For the Beijing area, Zmax = 1000 m and ∆Z = 500 m are set empirically. The results of the modified topography calculated from Eq. (8) are given in Fig. 10. When the topography is less than 500 m, the original height is retained; however, it decreases exponentially when the topography is greater than 500 m (Fig. 10).

    Figure 10.  Relationship between original topography (x axis; m) and modified topography (y axis; m) calculated from Eq. (8) with Zmax = 1000 m and ∆Z = 500 m.

    The parameterization algorithm is a useful method for precipitation mapping in mountainous areas. A comparison of the precipitation interpolation fields calculated from the hourly rain gauge observation dataset for Beijing with and without the elevation effect demonstrates that the two precipitation fields are mostly the same on the plain, but the precipitation field with the elevation effect is more realistically distributed in the mountainous terrain, shown in Fig. 11 as a precipitation case at 1700 LST 1 May 2015. The rainfall increment by the elevation effect is clearly apparent, with a greater rainfall amount in the higher mountains (Figs. 11a, b). Furthermore, the distribution of rainfall is more realistic using the elevation parameterization algorithm, tracking along the mountain ranges (Fig. 11b). In the low-lying plain region, the precipitation distribution is generally the same as there is little influence of elevation in plain areas (Fig. 11). It should be noted that the rainfall amount dependent parameterization algorithm is increasingly important for the interpolated precipitation field in the mountainous terrain using sparse station rain gauge data (figure omitted).

    Figure 11.  Rainfall case of an hourly AWS interpolated field at 1700 LST 1 May 2015 in the Beijing area for (a) pure station interpolation and (b) station interpolation with the elevation effect. The topography is indicated by contour lines with a 200-m interval.

    Despite the application of precipitation mapping in mountainous areas, the above parameterization algorithm of elevation dependence can also be applied operationally to the Rapid-refresh Multi-scale Analysis and Prediction System-Integration (RMAPS-IN), which is a nowcasting system developed by Institute of Urban Meteorology. The RMAPS-IN system generates real-time 10-min updated hourly precipitation analysis based on observed rain-gauge and radar quantitative precipitation estimation. The parameterization of the elevation effect is applied to the station rainfall. Figure 12 shows a precipitation case of the RMAPS-IN precipitation fields at 0850 UTC 23 May 2016. Compared with pure station interpolation, the station interpolation with the elevation effect does not change the overall picture of the precipitation fields on the larger spatial scale, indicating the reasonability of the parameterization algorithm. However, on smaller spatial scales, the precipitation variation pattern due to individual valleys and ridges is found to be more precise and realistic with the elevation effect on the mountainous terrain (Figs. 12a, c). The distribution of precipitation analysis becomes finer using the rainfall amount dependent parameterized algorithm than the pure station interpolation. This is because the parameterization provides a better estimate of the orographic rainfall increment. In the non-mountainous area, the precipitation pattern is generally the same with and without the elevation effect as the elevation effect is very weak in the plain region (Figs. 12a, c). The radar derived rainfall data are further used in the RMAPS-IN system. The radar data are suitable to describe the space and time of rainfall, but the transformation of the measured radar reflectivity into the rain rate is not accurate enough as it may be influenced by various types of errors (Kitchen and Blackall, 1992; Hagen and Yuter, 2003; Meischner, 2004). The radar derived precipitation field with the 6-min interval at 0847 UTC 23 May 2016 showed a more accurate rainfall amount and distribution in the central–eastern plain area, while it failed to capture the rainfall features in the mountain areas, which may be mainly caused by mountain sheltering (Fig. 12b). In the real-time application of RMAPS-IN nowcasting system, the radar precipitation field can capture the elevation effect on rainfall to some extent. Hence, for the final precipitation analysis, the precipitation increments due to the radar field and the elevation effect in the station field are combined. To avoid double counting, the precipitation increments due to radar and station elevation dependence are added only if they had different signs. If they had the same sign, only the larger increment value is used (Haiden et al., 2009). Finally, the parameterization is successfully applied to form a more accurate and realistic spatial distribution of the final precipitation analysis (Fig. 12d).

    Figure 12.  Example of precipitation analysis at 0850 UTC 23 May 2016 based on the combination of real-time station data and radar data in the RMAPS-IN system. (a) Pure station interpolation using 0850 UTC AWS data, (b) radar derived precipitation using 0847 UTC radar quantitative precipitation estimation (QPE) data, (c) station interpolation with the elevation effect at 0850 UTC, and (d) product of final precipitation analysis.

    It should be noted that the rainfall amount dependent parameterized algorithm may overestimate rainfall intensity according to its rainfall enhancement property based on the seeder-feeder mechanism. Hence, the false alarm rate (FAR) is further calculated. By evaluating eight precipitation events in Beijing during summer seasons (22 June, 5 July, 20 July, 22 August 2017, and 13 June, 11 July, 21 July, and 5 August 2018), the false alarm rate is about 0.027, 0.034, 0.030, and 0.003 on average for thresholds of 1, 5, 10, and 20 mm h−1, respectively. It indicates that the empty report rate is low.

5.   Conclusions and discussion
  • This study investigates the relation between rainfall and elevation in the Beijing area using a high-density AWS dataset of hourly rainfall observations, and further proposes a rainfall amount dependent parameterization algorithm of the elevation effect on rainfall on hourly timescale. The results show that there exists an enhancement of rainfall amount by the elevation effect. In particular, larger rainfall amount is found in higher mountains. For light and moderate rains, the CCs between rainfall amount and elevation are significant at the 95% confidence level, indicating an increase of rainfall amount when the elevation increases in Beijing. The light rain within the 0–2.5 mm h−1 threshold is found to be most closely related with the elevation. The statistical relations are both positively significant in plain areas and mountainous areas, with the CCs of 0.45 and 0.78, respectively. Box analysis of six representative station pairs located at valleys and on mountains further suggests an increment of hourly rainfall amount in higher mountains.

    The six representative station pairs are further used to estimate the optimum values of the elevation parameters in the rainfall amount dependent parameterization algorithm. The parameterization equation is defined as a segmented nonlinear model, which calculates the mountain rainfall as a function of valley rainfall amount. The conceptual mode is based on the physics of the seeder–feeder mechanism, defining a parabolic increment for small rainfall amounts and a linear increment for larger rainfall amounts. Compared with the results using no elevation correction, improvements including a smaller RMSE and much closer rainfall totals are derived by using the parameterization algorithm of the elevation effect. Therefore, this is a useful method for distributing a given long-term rainfall increment to individual hourly rainfall intervals.

    The application of the rainfall amount dependent parameterization algorithm of the elevation effect on rainfall in Beijing suggests that this approach can better estimate the spatial distribution of station interpolated precipitation. It can be applied as a useful method for precipitation mapping in mountainous areas, which can capture the precipitation variation patterns due to individual valleys and ridges. On the other hand, it can also be applied in the real-time local nowcasting system. The results show that it can improve the final precipitation analysis of RMAPS-IN system, especially for mountainous terrain. However, some issues still need to be investigated. For example, the detailed physical explanation for the significant CCs between the rainfall amount and the elevation still needs to be explored in the future. In addition, other environmental factors, such as wind direction, heat-island circulation, and mountain slope angle, are not considered in this study. Despite the superiority of the parameterization algorithm compared with no elevation considered as demonstrated in this study, the parameterization has its limitations. The algorithm is not able to reflect the hourly rainfall event when the mountain station receives rainfall but no rainfall is observed at the valley station. Hence, the problem caused by mountain-only rainfall event still needs to be investigated in the future.

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