Spatiotemporal patterns of precipitation based on the Bayesian maximum entropy method in a typical catchment of the Heihe River watershed, northwest China

ABSTRACT Precipitation patterns are vital to water resource management and hydrological research, especially in the upper reaches of inland rivers in arid and semiarid areas. However, estimating spatiotemporal precipitation patterns at a basin scale is challenging due to limited observations. In this study, spatiotemporal patterns of precipitation amount, frequency, duration, and intensity at different time scales from 2014 to 2019 are estimated using the Bayesian maximum entropy method in the Tianlaochi catchment of the Heihe River watershed, northwest China. The study's results show that the annual average precipitation amount was 535.9 mm from 2014 to 2019, with precipitation amount between May and September accounting for 85.9% of the annual precipitation amount. For daily precipitation, the average frequency rate of light precipitation is highest at 59.55%, however, the average contribution rate of moderate precipitation is highest at 50.33%. The spatial distribution of precipitation is characterized by high-value areas concentrated in the central valley and low-value areas located at the catchment's outlet. The most important driving factors of precipitation patterns are elevation, relative humidity, and wind direction. These outcomes can be used to establish accurate hydrological models in the catchment and provide support for water resource management in the Heihe River watershed.


Introduction
Precipitation is one of the important components in the hydrological cycle (Tang et al. 2020). Precipitation is a starting gear to other hydrological processes, such as canopy interception, evapotranspiration, soil infiltration, and surface runoff; even more, it has great impacts on bio-geochemical cycles and climatic variations (Kaptué et al. 2015;Arulraj and Barros 2019;Roushangar and Alizadeh 2019). Therefore, it is the linkage between atmosphere and biosphere. Precipitation ensures plant growth, maintains ecosystem balance, and improves socio-economic development (Réchou et al. 2019). There is significant heterogeneity of precipitation in space and time. Thus, an accurate estimation of precipitation spatiotemporal patterns is vital to water resources management and ecoenvironment protection, especially in the arid and semiarid regions, where the water resources are extremely critical to regional development (Ning et al. 2015).
Accumulated precipitation amount reflects precipitation patterns, besides, the frequency, duration, and intensity are essential characteristics of precipitation to comprehensively understand the hydrological cycle (Li et al. 2016). For example, steady and moderate rainfall benefits the growth of plants to the maximum extent, while the smashing heavy rainfall causes disasters like flooding, water and soil erosion, landslide, to the disadvantage of the plants growth (Trenberth, Fasullo, and Shepherd 2015). With climate change, extreme precipitation events characterized as short duration and strong intensity are increasing at the global scale in recent years, which leads to flash floods (Westra et al. 2014). Precipitation characteristics have an integrated influence on the hydrological cycle, and frequency and intensity are especially important to surface hydrology (i.e. runoff, evaporation) (Dai, Giorgi, and Trenberth 1999). The variations of precipitation patterns in the upstream river are critical to the water recharging of the whole watershed and can influence the economic development and ecological health of the middle and lower reaches deeply (Peng et al. 2014).
There are several methods to measure precipitation, like rain gauge, radar, and meteorological satellite, corresponding to small, moderate, and large spatial scales, respectively (Casas, Rodríguez, and Redaño 2009;Ko, Lee, and Lee 2018;Kumar Singh et al. 2019). The global precipitation products with continuously temporal-spatial records based on gauge, satellite, radar, and their reanalysis have been widely used all over the world. At present, the highest spatial resolution of gaugebased, satellite, and reanalysis precipitation products is 0.5°× 0.5° (Sun et al. 2018), 0.04°× 0.04° ( Sorooshian et al. 2000), and 38 km × 38 km (Saha et al. 2010), respectively. These products support the researches of hydrological processes and climate change on large scales (Alexander et al. 2020). However, for small-scale studies, especially in mountain regions with complex underlying surface and fluctuant terrain, global precipitation products have high error and uncertainties (Grose et al. 2019). Therefore, in situ rain gauge measurements are necessary and reliable for hydrological cycle research in mountain regions.
The spatial distribution of precipitation is an important input for distributed hydrological model (Cristiano et al. 2019). There are several methods to realize the spatial estimation of precipitation. Traditional methods, such as inverse distance weighted interpolation and linear regression, are simple and quick to estimate spatial variations, but they do not account for spatial dependencies of observations (Yuan et al. 2019;Dhamodaran and Lakshmi 2020). The geostatistical methods, such as kriging and cokriging, have solved the above problem, but they do not consider the uncertainties of data (Li et al. 2016). The Bayesian maximum entropy (BME) method, known as the modern geostatistical method, not only considers the spatial dependencies of observations, but also estimates the underlying spatiotemporal uncertainties effectively (Christakos and Li 1998;Bayat et al. 2013). BME can obtain more reliable estimations than other traditional geostatistical methods by incorporating general and site-specific knowledge, such as physical laws, expert knowledge, and observation data (Christakos 1990). Therefore, BME has been widely applied in various science fields for spatiotemporal estimations, such as hydrology, meteorology, ecology, and sociology (He and Kolovos 2017). BME in precipitation estimation has been proved to be more accurate than other methods, especially when the precipitation observation stations distribute sparsely and discretely. For example, Bayat et al. (2013) found that the BME method performed better than ordinary kriging when used to investigate spatiotemporal variations of annual precipitation in the Namak Lake watershed of Iran. Shi et al. (2015) proved that the BME method generated more realistic precipitation maps representing the local precipitation disparities than ordinary kriging and cokriging did. Usually, the precipitation estimation based on BME is focused on the annual scale (Bayat et al. 2013;Bayat et al. 2014;Wang et al. 2017), only a few studies are involved in monthly and seasonal scales (Shi et al. 2015), there is no estimation on the daily scale yet.
However, spatial variation of precipitation on daily scale is important to reflect hydrological processes and soil erosion risks at the precipitation event scale Zubieta et al. 2017). Therefore, spatial estimations of precipitation on different time scales (annual, seasonal/monthly, and daily) are necessary to understand hydrological processes comprehensively.
The Heihe River is the second largest inland river in China, originating from the middle part of the Qilian Mountains. It supports the socio-economic development of the Hexi Corridor oasis in the middle reaches and sustains the stability of the riparian ecological systems in the lower reaches (Peng et al. 2014). However, there are two serious water-related issues for Heihe River Watershed: water resource shortage in the middle reaches and ecological deterioration in the lower reaches. Hence, water recharge in its upper reaches is nearly water resource in the entire watershed because of spare precipitation in the middle and lower reaches. So, the research of precipitation spatiotemporal patterns in the upper reaches of the Heihe River is critical to the ecological protection and economic development of the Hexi region.
Given the background above, this study aims to explore the precipitation spatiotemporal patterns (including precipitation amount, frequency, duration, and intensity) based on the BME method at annual, seasonal/monthly, and daily time scales in a typical catchment in the upstream Heihe River (Tianlaochi catchment). The specific objectives are: (1) to analyze the temporal patterns of precipitation in Tianlaochi catchment, (2) to estimate the spatial distribution of precipitation at different time scales based on BME method, (3) to explore the major driving factors of precipitation patterns under different time scales by Grey relational analysis (GRA). The novelty of this study is to explore the effective analysis methods of station-observed precipitation data, and to realize the continuously spatial estimation of precipitation in a catchment scale with limited observations.

Study area
Tianlaochi catchment (38°23 ′ 57 ′′ -38°26 ′ 56 ′′ N, 99°53 ′ 50 ′′ -99°57 ′ 10 ′′ E) is located at the upstream of Heihe River (Zang et al. 2021). The area of the catchment is 12.8 km 2 , and the elevation ranges from 2660 to 4419 m above sea level. It is the typical alpine climate characterized by short and warm summer and long and cold winter (Liu et al. 2019). The annual average temperature is 0.6°C, the highest and lowest temperature are 12.1°C (July) and −13.1°C (January), respectively. The annual average evaporation is 1080 mm and the annual average relative humidity is 59%. The annual average precipitation is 535.9 mm. Nearly 90% of precipitation happens from May to September in the form of rainfall and the rest 10% occurs from mid-September to mid-May in the form of snowfall (Peng et al. 2014). The vertical zonality of vegetation types is significant, and it differs from north-facing (shady) to south-facing (sunny) slopes, which is shown in Figure 1. On the sunny slope, the area with elevation <2700 m is steppe (i.e. temperate grassland dominated by plants in the grass family, the bean family, and the sedge family) (Schönbach et al. 2010), and the area with 2700-3250 m is Qilian juniper forest (one kind of forests dominated by Juniperus przewalskii). On the sunny and semi-sunny slopes, the area with 2900-3100 m is an interlaced subalpine meadow (dominated by Pedicularis resupinata, Polygonum viviparum, Anethum graveolens, Elymus nutans and Potentilla anserine) and Qilian juniper. On the shady slope, the area with 2660-3540 m is Qinghai spruce forest (one kind of forests Picea crassifolia), the area with 3250-3750 m is subalpine shrub (dominated by Dasiphora fruticosa and Caragana jubata), and the area >3800 m is covered by bare rock.

Data collection
There are 20 rain gauges and 6 meteorological stations in Tianlaochi catchment. HOBO RG3-M bucket-tipping rain gauges (Onset Computer Corporation, Bourne, Massachusetts, USA) with 20 cm inside diameter and 0.2 mm resolution, have been installed to observe precipitation. Every rain gauge is fenced by 2 m × 2 m barbed wire to avoid disturbance from animals. HOBO U30 meteorological stations (Onset Computer Corporation, Bourne, Massachusetts, USA) have been installed at different vegetation types to obtain meteorological parameters, such as precipitation amount, air temperature, wind speed, wind direction, solar radiation, and relative humidity. The geographic locations of each rain gauge and meteorological station are shown in Figure 1, and the basic information of them is listed in Table 1. Hereinafter, rain gauge and meteorological station are unified as stations. The time interval of stations is set as 30 min. Daily precipitation observations are the sum of records for the whole day. A rainy day is a day with a total precipitation depth ≥0.2 mm (Kang et al. 2018). The precipitation data used in the study is from 2014 to 2019 for continuous observation, the detailed record information of each station at a monthly scale is shown in Figure 2.
Precipitation data include the amount, frequency, duration, and intensity of precipitation in the study. Precipitation amount (mm) is the depth of water falling to the ground in specified time periods (e.g. year, month, day). Precipitation frequency is defined as the number of days that precipitation amount is ≥0.2 mm (d) in one month or one year. Precipitation duration is the number of precipitation hours in different time periods (h) (Chen et al. 2009). Precipitation intensity refers to the accumulated precipitation amount per hour (mm/h).

Quality control of precipitation data
The precipitation data are not consistent, and many missing records exist here due to instrument errors and exhausted power ( Figure 2). Therefore, all daily, monthly, and annual precipitation data are subject to quality control (QC). QC involves an extreme check and an internal consistency check (Shen and Xiong 2016). The extreme check is carried out through the statistical analysis method. First, the first quartile (Q1) and the third quartile (Q3) of precipitation data at the same time period   are calculated. Next, the interquartile range (IQR) is calculated based on Q1 and Q3, which is shown as follows: And then, the lower limit of outlier (LLO) and the upper limit of the outlier (ULO) are calculated as follows: For precipitation data at different time scales, the records less than LLO and greater than ULO are deleted in the study. The internal consistency check aims at recognizing the erroneous data caused by incorrect units, reading or coding. Data that are recorded at least 80% of all data present are used to further analysis in the study (Karl and Knight 1998;Zhai et al. 2005).
After QC, there are 602 daily precipitation records. Based on the precipitation grades divided by Gansu Meteorological Bureau, the daily precipitation grades are as follows: light precipitation < 5.0 mm, moderate precipitation (5.0-15.0 mm), heavy precipitation (15.0-30.0 mm), and intense precipitation (30.0-70.0 mm) (Jia 2012). For only two intense precipitation events recorded on 21 August 2017 and 14 June 2019, respectively, intense precipitation was not included in the analysis.

Methodology
In order to study the precipitation spatiotemporal patterns in Tianlaochi catchment, several steps were included as follows. Firstly, the record precipitation data was preprocessed by quality control to avoid the influence of outliers and missing data (see Section 2.3). Secondly, the BME method was used to estimate the spatial distributions of precipitation at different time scales, and the principle of it was exhibited in Section 3.1. Thirdly, the estimation accuracy of BME maps was assessed by crossvalidation (see Section 3.2). Finally, the driving factors of precipitation patterns were analyzed by GRA (see Section 3.3).

Bayesian maximum entropy interpolation
A natural variable that varies across space and time can be regarded as a spatiotemporal random field (S/TRF) (Christakos and Serre 2000). S/TRF is the important theoretical basis of BME (Christakos 1991). It includes three major components (Christakos 2002), that are where p refers to the space/time point of both observed and estimated precipitation observation stations, s represents the spatial position (s = (X, Y)), X and Y denoting longitude (°) and latitude (°), respectively, and t denotes time (i.e. year, month, or day). p map refers to the random fields based on observation points p m and estimation points p k .
(1) Space/time random variable (precipitation in the study) can be denoted as follows: where x map is the observed precipitation values.
(1) Possible realization of the S/TRF within the spatial mapping context can be represented as below: where x map is the estimated values of precipitation based on the known information. BME takes different types of knowledge bases and data into account and leads to more informative and accurate results than traditional spatial interpolation methods (Christakos 1990;Bayat et al. 2014;He, Christakos, and Jankowski 2019). The knowledge bases of BME are divided into general knowledge (K G ) and site-specific knowledge (K S ). K G is characterized by physical laws, primitive equations, scientific theories, phenomenological relationships, etc. In the study, K G denotes the mean trend and spatiotemporal covariance of precipitation at different time scales. K S refers to site-specific data, uncertain measurements, local charts, heuristics, etc. (Christakos 2002;Douaik, Van Meirvenne, and Toth 2005), including the hard and soft data. Hard data are the measurements of the natural variable obtained from real-time observation devices, lab experiments, monitoring stations, etc. (Christakos and Li 1998). A set of measured values X i at points S i (i = 1, 2, 3, . . . , n) can be denoted as the following vector Soft data are the description of experience, knowledge, intuition, etc. (Christakos and Li 1998). A set of soft data X i at pointsS i (i = 1, 2, 3, . . . , m) can be denoted as the following vector Soft data include interval data, probabilistic data, and functional data (Christakos and Li 1998). In total, four soft data types are included in BMEGUI software, that are soft uniform data, soft Gaussian data (probabilistic data), soft Triangular data, and soft Truncated Gaussian data (Table 2). In the study, hard data refer to precipitation data recorded by stations, soft data are probabilistic data obtained by the mean value and standard deviation of the true value (Bayat et al. 2014). For an exact point with missing precipitation data at different time scales (daily, monthly, and annual), soft data are used to supply the missing data at the corresponding time scales by the mean value and standard deviation of hard data from other points with true records. The BME process includes the prior, meta-prior, and posterior stages. In the prior stage, under the constraint conditions from K G , the probability density function (PDF) that maximizes the information entropy is obtained, namely, prior PDF f G (X map ). In the meta-prior stage, additional auxiliary information in appropriate forms is collected and organized to produce soft data, and hard and soft data are considered to obtain K S . In the posterior stage, prior PDF and K S both are applied to obtain posterior PDF f k (x k ) at unmeasured location p k (k = i), which is calculated as follows: where f S (x soft ) is the PDF of the soft data. The spatiotemporal estimation of precipitation based on the BME method was implemented by the BME graphical user interface (BMEGUI) 3.0 free software (http://www.unc.edu/depts/case/ BMEGUI/). And the covariance modeling parameters for precipitation amount, frequency, duration, and intensity at different time scales are listed in Tables S1-S4.

Cross-validation
Cross-validation is the estimation of the predictive power of models. The available data are split into a train set and a test set, the train set is used to train the model, and the test set is used to compute a prediction error (Varoquaux et al. 2017). In the study, cross-validation is used to estimate the accuracy of precipitation spatiotemporal estimation, which is suited for the accuracy validation of sparse points (Luo et al. 2020). All observed and estimated values of precipitation observation stations are normalized by the mean normalization methods (Zhang, Cheng, and Liu 2014). At first, the value of one precipitation observation station is omitted from the dataset randomly. Then, the BME estimated values for the station are compared with the observed values. In order to compare the estimated and observed values, five statistical indicators are used in the study, namely, percent average estimation error (PAEE), normalized mean square error (NMSE), mean absolute error (MAE), mean square error (MSE), and correlation coefficient (R 2 ) (Bayat et al. 2014). The formulas of these indicators are as follows: where X is the average of the observed values (i.e. the normalized precipitation amount, frequency, duration, and intensity in the study), N is the number of the stations with observed values, X( S i ) are the observed values in the station s = (X, Y), X( S i ) are the estimated values in station s = (X, Y), abs is the returns absolute value.
where s 2 is the variance of the observed value.
These five indicators are used to assess the accuracy of precipitation estimation results. The accuracy is higher when PAEE, NMSE, MAE, and MSE are closer to zero and R 2 is closer to 1.

Grey relational analysis
GRA is a systematic analysis method based on the Grey System theory (Rajesh and Ravi 2015). Grey relational grade (GRG) is used to represent the degree of influence between a compared data series and the reference data series (Huang and Xu 2016). The value of GRG is between 0 and 1. The relationship is stronger when the GRG value is higher.
In order to explore the influence factors of precipitation patterns in the Tianlaochi catchment, GRA was used in the study. The reference data series are the precipitation data at different time scales. The compared data series include the meteorological factors and topographical factors. The meteorological factors include temperature (°C), relative humidity (%), solar radiation (W/ m 2 ), wind direction (°), and wind speed (m/s), coming from the meteorological stations. The topographical factors include elevation (m), slope (°), and aspect (°), extracted by 2-m DEM data of Tianlaochi catchment using ArcGIS 10.2 software.
The original data series containing m factors with n samples are represented as X i = {X i (k)}, where i = 0, 1, 2, … , m-1, k = 1, 2, … , n. X 0 is the reference data series, and the others are the compared data series. There are three steps to calculate GRG. At first, the original data series is generalized by the mean normalization methods. Then, the relational coefficient j i (k) is computed using the formula as follow: where min i min k |X 0 (k) − X i (k)| represents the minimum distance between the reference data series and the compared data series, max i max k |X 0 (k) − X i (k)| represents the maximum distance between the reference data series and the compared data series. ρ is the distinguishing coefficient used to adjust the difference of the relational coefficient, which ranges from 0 to 1, usually, equals to 0.5 (Xiong, Liu, and Xiong 2009). Finally, the average value of relational coefficient GRG (g i ) is obtained as follow: where the higher GRG presents stronger relationship between precipitation and influence factors.

The spatiotemporal patterns of annual precipitation
The temporal variations of annual precipitation from 2014 to 2019 are shown in Figure 3 The interpolated spatial distributions of annual average precipitation amount, frequency, duration, and intensity are shown in Figure 4. The annual average precipitation amount varies from 413.0to 760.6 mm (Figure 4(a)). The low-value areas are concentrates in the outlet of the catchment and high elevation areas of sunny slopes. In contrast, the high-value areas are located on shady slopes with Qinghai spruce forest at lower elevations and subalpine shrub environments at higher elevations. The annual average precipitation frequency varies from 66.8 d to 94.6 d; the high-value areas are mainly located in valleys, whereas the low-value areas are distributed across high elevation areas of shady slopes (Figure 4(b)). The annual average precipitation duration varies from 205.2 to 356.8 h and increases with increasing elevation (Figure 4(c)). The annual average precipitation intensity varies from 1.7 to 1.9 mm/h, increasing from west to the east across the catchment (Figure 4(d)).

The spatiotemporal patterns of monthly precipitation
The temporal variations and statistical analysis of monthly precipitation are shown in Figures 5  and 6, respectively. In general, the change trends of monthly precipitation amount, frequency, duration, and intensity are similar, with high values of these parameters recorded from May to September and low values during the other months. The monthly precipitation amount from May to September accounts for 85.9% of the total annual precipitation amount. Across the studied period, the monthly variation ranges of June and August are more than 50 mm, whereas those of May, July, and September are about 25 mm. The precipitation amounts in January, February, March, November, and December are typically less than 10 mm. The monthly precipitation frequency and duration from May to September account for 75.5% and 82.8% of the total annual precipitation frequency and duration, respectively. In addition, the monthly average precipitation intensity during May to September is 2.0 mm/h, while that recorded during the other months is 1.5 mm/h. To summarize the spatial distribution of monthly precipitation, we show the spatial distributions of them into four seasons (i.e. DJF: December-January-February, MAM: March-April-May, JJA: June-July-August, SON: September-October-November) (Figure 7). The spatial distributions of monthly precipitation amount, frequency, duration, and intensity are shown in Figures S3-S6, respectively. In general, the high-value areas of precipitation in the four seasons are concentrated in the center of the catchment, whereas the low-value areas are mainly located at the outlet of the catchment. In terms of precipitation amount (Figure 7(a-d)), the values in the DJF, MAM, JJA, and SON periods are 4.2-7.0 mm, 89.6-141.0 mm, 308.4-385.1 mm, and 90.6-122.7 mm, respectively. The precipitation frequency values (Figure 7(e-h)), in JJA exceed 44.7 d, whereas the value in DJF is less than 2.9 d, and the values in MAM and SON are both around 22 d. Regarding precipitation duration (Figure 7(i-l)), the value in JJA is greater than 137.5 h, while the value in DJF is less than 5.9 h; the values in MAM range from 82.7 to 101.9 h and those in the SON range from 54.8 to 73.0 h. Finally, the values of precipitation intensity (Figure 7(m-p)) vary from 0.8 to 1.2, 1.6 to 1.9, 2.1 to 2.3, and 1.4 to 1.6 mm/h in DJF, MAM, JJA, and SON, respectively.

The spatiotemporal patterns of daily precipitation
The frequency rate (i.e. the ratio of light/moderate/heavy precipitation days to total precipitation days per year) and the contribution rate (i.e. the ratio of light/moderate/heavy precipitation amount to total precipitation amount per year) of daily precipitation for the light, moderate, and heavy grades are shown in Figure 8. The order of frequency rate from high to low, is light, moderate, and heavy precipitation; the frequency rate of light precipitation is more than 50%, while its contribution rate is only about 20%. The contribution rate of moderate precipitation is the highest, accounting for about 50% of the total. When the contribution rate of light precipitation is more than that of heavy precipitation, the annual precipitation is relatively lower, such as in 2016, 2017, and 2018. In contrast, when the contribution rate of light precipitation is less than that of heavy precipitation, the annual precipitation is relatively higher, such as that recorded in 2014, 2015, and 2019.
The average frequency rate and contribution rate of light, moderate, and heavy precipitation for each month are shown in Figure 9. In January, February, March, November, and December, only light precipitation is recorded. In April and October, there is light and moderate precipitation, whereas heavy precipitation only occurs from May to September. The contribution rate sums of moderate and heavy precipitation exceed the frequency rate sums of these classes from April to September. The maximum frequency and contribution rates of heavy precipitation occur in July, corresponding to the largest precipitation amount.
The spatial distributions of daily precipitation amount, duration, and intensity for the three precipitation grades are shown in Figure 10. In terms of the precipitation amount and duration, the high-value areas are concentrated in the center and southwest part of Tianlaochi catchment, while the low-value areas are located at the outlet and in the eastern highlands of the catchment. Regarding precipitation intensity, the high-value areas are located in the eastern highlands and the southern part of the outlet, whereas the low-value areas are distributed at the center and the outlet.

Validation of spatial distribution maps
The accuracy validation of the spatial estimations for precipitation amount, frequency, duration, and intensity at different time scales are shown in Table 3. In general, the accuracies are high. The values of PAEE, NMSE, MAE, and MSE are low, with mean R 2 values of 0.69, 0.76, 0.83, and 0.79 for precipitation amount, frequency, duration, and intensity, respectively. In detail, at an annual scale, the accuracy levels are higher for precipitation amount, frequency, duration, and intensity. At a seasonal scale, the accuracies in DJF are relatively low, representing the higher values of PAEE, NMSE, MAE, and MSE, and lower values of R 2 . The reason for this trend is the comparatively low precipitation recorded in DJF, with the scarcity of records limiting the accuracy of the BME estimates, while, accuracies are high in the other seasons. On a daily scale, the accuracy of the heavy precipitation data are relatively good, which may result from the stronger sensitivity of precipitation sensors to heavy precipitation. In summary, the accuracy of the study's BME estimation maps is overall reliable in the catchment.

The driving factors analysis
To identify the main factors driving spatiotemporal precipitation patterns, GRA is used in this study, the results of which are shown in Tables 4 and 5. A higher gray relational coefficient means a stronger driving force of a given factor on precipitation patterns. For topographic factors, elevation is always the most important driving factor at different time scales, due to substantial elevation changes in the studied catchment. In contrast, the most important meteorological factor varies for different time scales. The relative humidity is the most important factor driving changes

Solving precipitation data with uncertainties
Accurately estimating the spatiotemporal patterns of precipitation in mountainous regions is a crucial and challenging task in hydrological applications due to the paucity and uneven spatial distribution of stations (Bayat et al. 2013). To address this problem, the rain gauges and meteorological stations in the study area were distributed as evenly as possible. In total, there are 26 stations, covering all vegetation types, different elevation grades, as well as different slopes and aspects within the catchment. Nonetheless, the spatial distribution of stations is sparse. In addition, the time series of records are intermittent due to human and instrumental error (e.g. lack of power, overwriting of records). Therefore, uncertainties in precipitation data are unavoidable. In this study, two methods were used to address data uncertainties. One is QC, which can be used to remove suspect data based on statistical analysis, an approach that has been proven to be necessary and effective by other studies (Li et al. 2016;Shen and Xiong 2016). We also apply spatial estimation based on the BME method, which is especially applicable to the interpolation of results from sparse observation stations with missing data (Lee, Yeatts, and Serre 2009;de Nazelle, Arunachalam, and Serre 2010). This approach can estimate the missing data through different types of soft data and has been widely applied to estimate the spatial distribution of precipitation with excellent results (Bayat et al. 2014;Wang et al. 2016;Zhang et al. 2016).

The significance of spatiotemporal precipitation patterns
Constraining precipitation patterns is crucial for identifying hydrological extremes (i.e. floods and droughts) (Muñoz-Jiménez et al. 2018;Wolski et al. 2020), developing distributed hydrological models (Nezlin and Stein 2005;Gu et al. 2020), managing agriculture, and protecting eco-environment . In arid and semiarid regions, where water resources are scarce and highly sensitive to climate change, precipitation represents a primary factor influencing hydrological cycle processes and economic development (Ruiz-Sinoga et al. 2012;Lewandrowski et al. 2017). Inland rivers support the development and survival of society in arid and semiarid regions. As the second largest inland river basin in northwest China, the Heihe River watershed is important to the development of the Hexi corridor oasis. However, increasing population and cropland area in the river's middle reaches have increased the consumption of water, thus decreasing the input of water to the lower reaches which, in turn, has led to numerous eco-environmental problems (Zeng et al. 2012;Nian et al. 2013). The annual precipitation in the upstream, middle stream, and downstream regions of the Heihe River Basin are more than 350, 100-250, and less than 50 mm, respectively (Hu and Jia 2015). Accordingly, precipitation recharge in the upstream region is nearly represents the water resource of the entire watershed (Lan et al. 2002). Against a backdrop of climate change and socio-economic development, the allocation of water resources between the middle and lower reaches depends on surface runoff in the upper reaches. Accurate prediction of surface runoff through hydrological models is essential; thus, as an important input, the spatiotemporal patterns of precipitation in upstream areas play a vital role in estimating surface runoff. Therefore, estimating spatiotemporal patterns of precipitation in Tianlaochi catchment from sparse data illustrates an effective method for understanding water resources in the upper reaches of the Heihe River watershed.

The factors driving precipitation patterns
Precipitation patterns are driven by numerous factors, including topographic elements, meteorological conditions, atmospheric circulation, convective activity, human activities, and underlying surface changes (Roe 2005;Muñoz-Jiménez et al. 2018;Gu et al. 2020). Based on the availability of data and the nature of the study area, the driving forces of various topographic factors and meteorological conditions on precipitation patterns were analyzed by GRA (Table 4 and Table 5). For topographic factors, elevation was identified as the most important factor. The elevation range reaches Relative humidity Elevation 1759 m in the study area, leading to typical orographic precipitation ( Figure 11). The annual precipitation amount by area initially increases with the increasing elevation, then decreases with further elevation increase, and finally stabilizes (Figure 11(b)). The maximum precipitation amount occurs in areas at elevations from 3100 to 3400 m. Grose et al. (2019) also found that topography plays a key role in projected rainfall change in mid-latitude mountainous regions. In terms of meteorological factors, the most important factor varies over time. Broadly, relative humidity, wind direction, temperature, and solar radiation represent important meteorological factors controlling precipitation patterns at a basin scale. Armon, Morin, and Enzel (2019) also found that meteorological conditions exert a significant influence on the occurrence of regional and local rainstorms.

Limitations and implications
This study has several important implications for water resource management, in addition to some limitations. First, soft data used to conduct BME estimation were only derived from precipitation observation data in the study. Other soft data could also be applied based on the relationships between precipitation and other parameters, such as elevation, temperature, and water vapor pressure, which may yield further insights . Second, only meteorological and topographic factors were considered in the analysis of drivers of precipitation patterns, however, these necessarily reflect only some environmental effects on precipitation (Fang et al. 2013;Long, Zhang, and Ma 2016). Other factors, such as general atmospheric circulation, El Niño-Southern Oscillation, convective activity, human activities, and underlying surface changes, can also influence precipitation patterns and should be considered in future studies (Yu et al. 2014;Roushangar, Alizadeh, and Adamowski 2018). Finally, there are both liquid and solid precipitation phases in the study area, i.e. rainfall and snowfall, which account for 90% and 10% of the total precipitation amount, respectively (Peng et al. 2014;Tang et al. 2020). However, this study regards the precipitation as an entirety without identifying its phase, which may lead to some uncertainties.

Conclusion
This study analyzed the spatiotemporal patterns of precipitation characteristics (amount, frequency, duration, and intensity) of Tianlaochi catchment in the upper reaches of the Heihe River watershed from 2014 to 2019. The BME method was used to estimate spatial patterns of precipitation characteristics, with the temporal patterns of precipitation analyzed based on annual, seasonal (i.e. DJF, MAM, JJA, and SON), and daily (i.e. light, moderate, and heavy precipitation within a day) time scales. Additionally, the driving factors of precipitation patterns were explored by the GRA method. The main conclusions of this study are as follows. (1) The BME method is effective at estimating spatial patterns of precipitation in mountainous catchments with sparse stations and missing data on different time scales, especially at an annual scale.
(2) Moderate precipitation at higher frequency rates has the largest contribution rate in the studied area.
(3) Elevation is the most important driving factors of precipitation patterns in the small studied catchment. Overall, this study provides an effective method for estimating precipitation patterns in mountainous regions with limited observations and can support water resource management in arid and semiarid regions.