Temporal Scaling Characteristics of Sub‐Daily Precipitation in Qinghai‐Tibet Plateau

The Qinghai‐Tibet Plateau (QTP) is highly susceptible to destructive rainstorm hazards and related natural disasters. However, the lack of sub‐daily precipitation observations in this region has hindered our understanding of rainstorm‐related hazards and their societal impacts. To address this data gap, a new approach is devised to estimate sub‐daily precipitation in QTP using daily precipitation data and geographical information. The approach involves establishing a statistical relationship between daily and sub‐daily precipitation based on data from 102 observation sites. This process results in a set of functions with six associated parameters. These parameters are then modeled using local geographical and climatic information through a machine learning algorithm called support vector regression. The results indicated that the temporal scaling characteristics of sub‐daily precipitation can be accurately described using a logarithmic function. The uncertainty of the estimates is quantified using the coefficient of variance and coefficient of skewness, which are estimated using a logarithmic and linear curve, respectively. Additionally, the six parameters are found to be closely linked to geographical conditions, enabling the creation of a 1‐km parameters data set. This data set can be utilized to quantitatively describe the probabilistic distribution and extract key information about maximum precipitation duration (from 1 to 12 hr). Overall, the findings suggest that the generated parameters data set holds significant potential for various applications, including risk analysis, forecasting, and early warning for rainstorm‐related natural disasters in QTP. The innovative method developed in this study proves to be an effective approach for estimating sub‐daily precipitation and assessing its uncertainty in ungauged regions.


Introduction
High mountain regions (HMR) are highly vulnerable to destructive rainstorm hazard and related natural disasters, including floods, landslides, erosion, mud and debris slides (Hock et al., 2022;Yao et al., 2013).These hazards pose a significant threat to human lives, properties, the environment, and ecosystems in both HMR and downstream regions (Terzi et al., 2019;Viviroli et al., 2007).Moreover, due to the high elevations, the temperature in HMR usually has a higher increasing rate compared to plain regions (Pepin et al., 2015(Pepin et al., , 2022)), thus HMR are more sensitive to the impacts of climate change, due to which the risks of rainstorm-related natural disasters keep increasing and their potential influences are aggravating (Immerzeel et al., 2019).However, effective control and prevention of rainstorm-related natural disasters in HMR is a big challenging issue.On the one hand, it is due to the unfavorable characteristics of these disasters, such as sudden occurrence along with high-intensive rainfall, rapid formation and evolution of disasters, and strong destructive outcomes, leading to high difficulty in simulation and forecasting of natural disasters in HMR.On the other hand, due to the high elevation and complex topography and harsh geographical conditions, field hydro-climatic observations, especially short-duration rainfall observations, are barely available in HMR, hindering the studies of natural disasters.Hence, how to accurately evaluate the risks of rainstorm-related natural disasters in HMR, as well as their reliable forecasting and early warning, is a high requirement but also an extraordinarily challenging task.
Qinghai-Tibet Plateau (QTP) is a typical HMR with damaging natural disasters occurring frequently.In the region, short-duration heavy precipitation at sub-daily scales is an important trigger not only for flash flooding disasters but also for landslide and debris flows (Zhao et al., 2022).Over the last two decades, more than 2,000 rainstorm-induced disasters occurred across the region, and they affected 7.2 million people and caused economic losses of 13.9 billion RMB (Sun et al., 2021).However, QTP is also a typical ungauged region, and short-duration precipitation data with long observation periods are limited, leaving a large data gap for supporting rainstormrelated natural disaster studies in the region.
Satellite remote sensing data and reanalysis data are important alternative sources for obtaining precipitation data in ungauged regions.The two data sources provide good spatial coverage, however, their quality relies heavily on the inversion techniques and requires bias correction with the help of rain gauges (Lu & Yong, 2018).Several daily precipitation data sets are available for QTP, and previous studies made deep comparisons and improvements in their quality (Y.Ma et al., 2018;Tong et al., 2014).These daily data, however, are too coarse to meet the high time-resolution request at sub-daily scales for the rainstorm-related natural disaster studies in the region (Jiang et al., 2023;Wu & Gao, 2013).Although there have been several satellite products for sub-daily precipitation in QTP, their quality is much lower than that of daily precipitation products (Z.Ma et al., 2022;Sun et al., 2018;G. Tang et al., 2016), due to the lack of field sub-daily precipitation observations for calibration, and thus they cannot reliably support the rainstorm-related natural disaster studies (Casanueva et al., 2020).
Temporal downscaling is another way to generate sub-daily precipitation data from daily precipitation, and dynamical downscaling and statistical downscaling are two approaches used widely (J.Tang et al., 2016;Yan et al., 2021;Yin & Chen, 2020).Dynamical downscaling is computationally expensive because of a large amount of computation and is greatly influenced by the boundary conditions provided by global climate models or regional climate models (Maraun et al., 2010;Ou et al., 2020).Comparatively, statistical downscaling is based on the establishment of statistical relationships between local observations and large-scale climatic (sometimes also geographical) variables, and is computationally effective (Benestad et al., 2008;Chen et al., 2005;Hanssen-Bauer et al., 2005).Thus, this approach is an effective alternative for generating high-time-resolution precipitation data in ungauged regions (Wilby et al., 1998).
In the framework of statistical downscaling, many machine learning and deep learning methods were used to explore the connections between low-and high-time resolution scales, such as the artificial neural network method (Vu et al., 2015), fractal-multifractal method (Maskey et al., 2019), multiplicative cascade method (Müller & Haberlandt, 2018), extreme learning machine method (MoradiKhaneghahi et al., 2019), and stochastic disaggregation method (Socolofsky et al., 2001;Takhellambam et al., 2022).The above temporal downscaling methods usually pursue a statistically perfect fit; however, their effectiveness depends not only on the data availability but also on physical mechanisms of precipitation.Some studies used statistical methods for temporal downscaling based on probabilistic distribution theory, such as exponential distribution (Benestad et al., 2012(Benestad et al., , 2021))and generalized extreme value distribution (Lima et al., 2018).The relative consistency in precipitation frequencies at different timescales is employed to make a temporal downscaling, but it is limited to the context at a specific frequency.Besides, other studies also used the linear scaling methods for temporal downscaling from daily to hourly scales (Cattoën et al., 2020;Lee & Jeong, 2014;Lee & Park, 2017), but ignoring the underlying nonlinear relationship.
The quality of any empirical temporal downscaling strongly depends on the number of field observations and the statistical methods used.Compared to the complex and indescribable relationship between daily and sub-daily precipitation generated by machine learning and deep learning methods in previous studies (Müller & Haberlandt, 2018;Socolofsky et al., 2001;Takhellambam et al., 2022), it could be a more effective and feasible strategy to establish a describable equation of the relationship of precipitation between daily and sub-daily scales, to quantify the temporal scaling characteristics of sub-daily precipitation.The motivation for this strategy is the observation and expectation that the temporal scaling characteristics of sub-daily precipitation has simple and smooth shapes with regional similarities (Parding et al., 2023).Based on the describable equation established, it would be favorable to understand the spatial pattern of sub-daily precipitation and its influencing factors in a region.
The main objective of this study is to generate a high-resolution gridded parameters data set to quantify the temporal scaling characteristics of sub-daily precipitation and the associated uncertainty in the Qinghai-Tibet Plateau, as a basis of estimating sub-daily precipitation in the region.To do this, the hourly precipitation data measured at 102 weather stations in QTP are used, and a logarithmic function, with two key parameters, is selected to describe the temporal scaling characteristics of sub-daily precipitation at these stations.Furthermore, the influences of the geographical and climatic conditions on the two key parameters at these stations are further explored, based on which a 1-km gridded data set of the two key parameters is generated in QTP.To the best of our knowledge, exploring the temporal scaling characteristics of sub-daily precipitation in QTP is a novel approach.The parameters data set provides an effective and practical approach to estimating sub-daily precipitation and the associated uncertainty for areas with available daily precipitation data, supporting the rainstormrelated natural disaster studies in QTP.

Study Area
The Qinghai-Tibet Plateau, well known as "the Third Pole," is located in the south-central part of the Eurasian continent, between 25°59′N-40°1′N and 67°40′E-104°40′E and covering an area of over 3 million km 2 (Y.Zhang et al., 2021).It is the highest (average altitude >4,000 m asl) and largest plateau in the world (G.Zhang et al., 2013).The QTP is characterized by plains, valleys, and mountains, and the surface topography changes from very gentle to highly steeped conditions (0-78°) (Sajadi et al., 2022).Under the influences of regional monsoon systems, precipitation from May to September accounts for more than 70% of annual precipitation in most parts of QTP (Kukulies et al., 2020;Zhu et al., 2020).
The east and south parts of QTP, with extremely valley topography, are the origins of several large Asian rivers including the Yellow River, Yangtze River, Mekong River, Salween River Basin, Brahmaputra River and others (Liu et al., 2020).The climatic conditions in these river valleys are controlled by the regional monsoon systems, and thus rainstorms frequently occur in summer, generating serious flash floods, landslides, mudslides and other natural disasters, as one of the hotspots for natural disaster studies on Earth.Unfortunately, the population and economy are also concentrated in these river valleys and thus are prone to destructive rainstorm-related natural disasters (Cui & Jia, 2015;W. Ma et al., 2021).Under the influences of climate change, the occurrence frequency and intensity of natural hazards keep increasing, which poses serious threats to people's lives and properties in QTP.
In this study, we mainly concern the east and south parts of QTP, and it includes eight basins: the Hexi Corridor Basin, the Qaidam Basin and the origin of the Yellow River Basin in the northeast, the origin of the Jinsha River Basin, the Mekong River Basin and the Salween River Basin in the southeast, the Yarlung Tsangpo River Basin (as the upper reach of the Brahmaputra River Basin), and the origin of the Ganges River Basin in the south (see Figure 1).

Hourly Precipitation Data
The hourly precipitation data measured at 102 weather stations (see Figure 1) in QTP are used for this study.They were collected from the China Meteorological Administration (CMA).All the data have been quality checked to ensure their reliability for scientific studies.The weather stations are mainly located in the eight basins concerned in this study, where rainstorm-related natural disasters occur frequently (Wang et al., 2021;Yang et al., 2022).However, there are limited stations in the other regions of QTP, where historical occurrences of rainstorm-related natural disasters were much less frequent than in the river valley regions.These hourly precipitation data have different measured periods, with different starting years from 1951 to 2012, but with the same ending year of 2018.For each station, all the measured data are used to investigate the relationship between daily and sub-daily precipitation.Considering that rainstorm hazard and related natural disasters mainly occur in summer in QTP, the hourly precipitation data from May to September in each year are used for this study.
Based on the Universal Time Coordinated (UTC) time, the daily precipitation (denoted as P 24 ) is calculated as the sum of hourly precipitation from 0:00 UTC to 24:00 UTC.Only wet days with cumulative precipitation exceeding 1.0 mm are counted here.For certain station, the maximum magnitude of t-hour consecutive precipitation (denoted as P t , with t = 1, 2, …, 12 hr, respectively) in a wet day is calculated to establish its P t /P 24 ∼ t curve; after that, the average curve of these P t /P 24 ∼ t curves among all wet days is calculated, to quantify the temporal scaling characteristics of sub-daily precipitation at the station.
Moreover, the uncertainty of the average P t /P 24 ∼ t curve is quantified by using the indicators of coefficient of variance (C v ), and coefficient of skewness (C s ), which can also be estimated from all the P t /P 24 ∼ t curves at a station.The different P t /P 24 ∼ t curves at the 102 weather stations are further compared to investigate the spatial difference.
The ratios of sub-daily precipitation (t = 1, 3, 6, 12 hr) to daily precipitation at these stations are used as examples to present the regional similarities.Figure 2 clearly indicates that the ratio of sub-daily precipitation to daily precipitation is larger in the northern part than in the southern part of the study area.For example, P 1 accounts for less than 46% (Figure 2a) of P 24 in the southeastern part of QTP, but more than 46% in other regions.The spatial distribution of P 3 /P 24 ratio (Figure 2b) is similar to that of P 1 /P 24 .Besides, the P 6 /P 24 (Figure 2c) and P 12 /P 24 (Figure 2d) ratios are larger in the northern part than in the southern part of the study area, implying that heavy precipitation generally has a longer time duration in the southeastern QTP with wet climatic conditions.Overall, the duration of precipitation events is usually no longer than 12 hr in the study area, as the P 12 /P 24 ratios are larger than 86% at all stations.Note that it is the reason of setting the longest t = 12 hr in this study.
The regional similarities of P t /P 24 shown in Figure 2 imply the potential influence of geographical and climatic conditions on the sub-daily precipitation ratios.This motivates investigating the temporal scaling characteristics of sub-daily precipitation, as a new and feasible strategy developed in this study to estimate sub-daily precipitation in QTP.

Geographical and Climatic Data
The digital elevation model (DEM) data, with a 1-km resolution, was used to extract the values of the geographical variables including longitude, latitude, and altitude in the study area.The DEM data were obtained from the United States Geological Survey.Besides, the daily data of precipitation, surface temperature and relative air humidity were obtained from CMA, and their mean values were calculated.The three geographical variables and three climatic variables are used to further investigate their influences on the temporal scaling characteristics of sub-daily precipitation in QTP.

Fitting of P t /P 24 ~t Curve
In this study, the linear, logarithmic, exponential, power and inverse functions are used and compared, to choose the most suitable function for describing the temporal scaling characteristics of sub-daily precipitation in QTP.These functions are shown in Table 1.
The indicator of coefficient of determination (R 2 ) is used to evaluate the goodness of fit of these functions: where f(t) is the average P t /P 24 ∼ t curve calculated from the observed data at a station, and f (t) is the mean of f(t); f(t)′ is the corresponding P t /P 24 ∼ t curve estimated by using certain fitting function in Table 1.R 2 with a value closer to 1 indicates a better fitting result by certain function.The p-value of 0.01 from the F-statistic test is further used to judge the statistical significance of the  fitting result at the 1% significance level.The fitting function with the largest R 2 value is chosen as the best one to describe the temporal scaling characteristics of sub-daily precipitation.
Further, C v t and C s t at each t-duration (with t = 1, 2, …, 12 hr) are calculated as: where f i (t) is the P t /P 24 ∼ t curve calculated from the observed data in ith day, with the total number n of wet days, and f(t) is its average P t /P 24 ∼ t curve.Note that these weather stations have different number n of wet days.Based on it, the C v t ∼ t curve and C s t ∼ t curve at each station can be obtained to quantify the uncertainty of the average P t /P 24 ∼ t curve.

Explanation of P t /P 24 ~t Curve
Considering the potential influences of geographical and climatic conditions on the temporal scaling characteristics of sub-daily precipitation, three geographical variables (longitude, latitude and altitude) and three climatic variables (average daily precipitation, surface temperature and relative air humidity) are used to explain the two key parameters in the chosen function.Here, the multiple linear regression (MLR) model, support vector regression (SVR) model and random forest regression (RFR) model, which are widely used for exploring the connections among variables (Chen et al., 2017;Ganguli & Reddy, 2014;Sachindra et al., 2018;Sajadi et al., 2022;Yan et al., 2022), are applied to establish the relationship between the six explanatory variables and the two key parameters.It is noted that the three models have been widely applied for spatial interpolation of monthly and daily precipitation in QTP (He et al., 2023;Jing et al., 2016;Z. Ma et al., 2018;Shi & Song, 2015), while their suitability and applicability for handling spatial interpolation of hourly precipitation in the region will be investigated in this study.The details of the three models can be found in Feidas et al. (2014) and Rodriguez-Galiano et al. (2015).Moreover, the normalized Taylor diagram, with three statistics of normalized standard deviation (σ nor ), normalized centered root-mean-square error (CRMSE nor ), and Pearson correlation coefficient (r), is used to evaluate the performance of the three models (Kärnä & Baptista, 2016;Taylor, 2001).
In the modeling practice, the data at 68 weather stations (accounting for 67% of the total stations) are used for calibration, and the data at the residual 34 weather stations (accounting for 33% of the total stations) are used for validation.Based on the results in the normalized Taylor diagram, the most suitable model can be chosen and established.After that, the data of the geographical and climatic variables in each 1-km grid are used as inputs to run the established model, to obtain the two key parameters in all grids.Based on it, the 1-km gridded parameters data set covering the whole QTP can be generated.

Temporal Scaling Characteristics of Sub-Daily Precipitation
Taking a station (with the star symbol shown in Figure 1) as an example, its P t /P 24 ∼ t curves in all wet days are calculated and described by using boxplots, with its average P t /P 24 ∼ t curve shown as blue triangle points in Figure 3a.It presents that the average P t /P 24 value keeps increasing with t.Moreover, the dispersible characteristics (i.e., box length in Figure 3a and quantified by C v t in Figure 3b) of P t /P 24 with smaller t-duration is bigger, implying larger uncertainty.Along with the increase of t-duration, the dispersible characteristics of P t /P 24 become smaller, corresponding to decreasing C v t value, but P t /P 24 indicates more obviously negative skew characteristics (quantified by C s t in Figure 3b).
The average P t /P 24 ∼ t curve is then fitted by using the five functions in Table 1. Figure 3a shows obvious difference in their fitting performances.The fitting curves by using the linear function and exponential function Earth's Future 10.1029/2024EF004417 tend to be linear, which are far from the shape of the average P t /P 24 ∼ t curve.
The other three functions provide similar fitting results, and the result from the logarithmic function is visually the best.
The performances of fitting results by using the five functions at the 102 stations are evaluated, and the results are shown in Figure 4a.It indicates that the logarithmic function provides the best fitting results, with an average R 2 value of 0.98 and a value range of 0.86-0.99.More importantly, the uncertainty of the fitting result by using the logarithmic function is the smallest (with the shortest box length in Figure 4a).Comparatively, the fitting results of the inverse function and power function have the average R 2 value of 0.95 and 0.94, respectively, which are a little worse than that of the logarithmic function.However, the linear function and exponential function performed badly and cannot catch the shape of the P t /P 24 ∼ t curve.It proves the nonlinear relationship between daily and sub-daily precipitation in QTP, just as shown in Figure 3a.While the approach of exploring the linear scaling relationship between daily and hourly precipitation, as done in previous studies (Cattoën et al., 2020;Lee & Jeong, 2014;Lee & Park, 2017), is not suitable for the study area.
Based on the above results, the logarithmic function of P t / P 24 = a mean × ln t + b mean was chosen to describe the temporal scaling characteristics of sub-daily precipitation in QTP.In the function, parameter b mean reflects the ratio between 1-hr and 24-hr precipitation, that is, b mean = P 1 /P 24 ; a larger a mean means a higher P t /P 24 ratio at certain t-duration.Among the 102 stations, parameter a mean has a value range of 0.13-0.21and parameter b mean has a value range of 0.37-0.66,as shown in Figures 5a and  5b, respectively.Regarding the spatial distribution, parameter a mean (b mean ) has a southeast-northwest decreasing (increasing) pattern.The distributions of the two parameters reflect the spatial pattern of temporal scaling characteristics of sub-daily precipitation.Based on the physical meanings of the two parameters, it is known that heavy precipitation events with short durations usually occurred in western and central QTP with dry climatic conditions; comparatively, longer-duration precipitation more likely occurred in the southeastern QTP with wet climatic conditions.This spatial pattern is consistent with that in Figure 2, supporting the reliability of the chosen logarithmic function for describing the temporal scaling characteristics of sub-daily precipitation.
In addition, the C v t ∼ t curves and C s t ∼ t curves at the 102 weather stations are also fitted by using the five functions.Results show that the logarithmic function (C v t = a cv × ln t + b cv ) gives the best fitting results for the 4b), and the linear function (C v t = a cs × t + b cs ) provides the best fitting results for the C s t ∼ t curves (Figure 4c).The parameters a cv , b cv , a cs , and b cs at the 102 stations have the value ranges of 0.15-0.10,0.33-0.53,0.41-0.14,0.21-0.92,respectively, with their spatial distributions shown in Figures 5c-5f.Larger absolute values of parameters a cv and b cv in south part of QTP imply bigger dispersible characteristics of sub-daily precipitation compared to that in the northern part.Besides, larger absolute values of parameter a cs (i.e., the slope of the chosen linear function) in the north reflect more obviously skew characteristics of sub-daily precipitation compared to that in the south part.

Relationship Between P t /P 24 ~t Curve and Explanatory Variables
The three geographical variables and three climatic variables are used to explain the key parameters of a mean and b mean in the chosen logarithmic function and their spatial distribution in Figure 5, to investigate the geographical and climatic influences on the spatial pattern of the temporal scaling characteristics of sub-daily precipitation in QTP.These explanatory variables can basically reflect both the moisture availability and redistribution of water vapor, which to a large extent affect precipitation amount and its spatial distribution (Jones et al., 2010;Lepore et al., 2016;Li et al., 2021).Figure S1 in Supporting Information S1 shows the different correlation degrees and directions of these explanatory variables with a mean and b mean , reflecting their different effects.Three models (MLR, SVR, RFR) are used to establish the relationship between the explanatory variables and each parameter.To explore the different effects of the geographical and climatic conditions on the two parameters, three types of explanatory variables are set here: three geographical variables (denoted as "geo"), three climatic variables (denoted as "clim"), and all the six variables (denoted as "geoclim").The modeling results are evaluated by the normalized Taylor diagram and are shown in Figure 6.In this diagram, the evaluation metric of modeling result being closer to the original value (the black point) indicates a better accuracy (Xiao et al., 2020).It shows that no matter which types of explanatory variables are used as inputs, the MLR model performed badly, indicating that the relationships between the two parameters and the six explanatory variables cannot be explained well by the linear model, implying their nonlinear relationships.Comparatively, the SVR model performed well.For parameter a mean , the value of r and CRMSE nor are 0.82 and 0.58 in the SVR(geo) model.For parameter b mean , the value of r and CRMSE nor are 0.86 and 0.51 in the SVR(geoclim) model.Besides, the RFR model performed worse than SVR, which may be due to the small data sample used here, as the former usually requires a large sample size to ensure accurate modeling results (Millard & Richardson, 2015).Comparatively, the SVR model has a high generalization capability, particularly with respect to limited samples (Shao & Lunetta, 2012), which may explain its better performance compared to RFR.Furthermore, Figure 6 indicates the different effects of the geographical and climatic variables.For parameter a mean , the SVR(geo) model performs better than SVR(clim) model and SVR(geoclim) model (Figure 6a), suggesting that climate variables cannot give more useful explanations for the spatial pattern of a mean .While for parameter b mean , the SVR(geoclim) model outperforms SVR(geo) model and SVR(clim) model (Figure 6b).Although the three climatic variables contribute a part to the modeling results of parameter b mean , it should be noticed that their values may have uncertainty due to the short data periods used in some weather stations.Although some other satellite-or reanalysis-based data sources can be used to estimate the values of the three climatic variables, their accuracy may have big biases in QTP (Hu et al., 2019;Simmons et al., 2010;J. Zhang et al., 2021), and their spatial resolution is much coarser than 1-km resolution concerned in this study.Comparatively, the values of the three geographical variables are stable, and the geographical conditions can to some extent reflect the climatic conditions.The SVR(geo) model with the three geographic variables gives the r of 0.77, σ nor of 0.71 and CRMSE nor of 0.64 for parameter b mean , whose accuracy is thought acceptable in such an ungauged region of QTP.
By considering the different performances of the three models, as well as the different influences of these explanatory variables and their data availability and reliability, the SVR model is selected, and the geographic variables of longitude, latitude, and altitude at 1-km grid are used as inputs, based on which the 1-km gridded parameter a mean and b mean in the whole QTP are generated, as shown in Figure 7.The value range of a mean and b mean in the whole QTP is from 0.14 to 0.20 and from 0.38 to 0.65, respectively.For parameter a mean , its high values appear in the eastern, southeastern, and southern boundary areas of QTP (Figure 7a), where natural disasters induced by rainstorm events occurred frequently (Wang et al., 2021).While the value of a mean is relatively low in central QTP, including the Qaidam Basin, the headwater regions of the Jinsha River Basin, the Mekong River Basin, the Salween River Basin, and the mid part of the Yarlung Tsangpo River Basin.For parameter b mean , its high values appear in the Qaidam Basin and small values in southeast QTP (Figure 7b).As a result, the two parameters of a mean and b mean can determine the logarithmic P t /P 24 ∼ t curve at each grid, as a basis of estimating t-hour precipitation, especially for those areas with only daily precipitation records available.

Uncertainty Quantification
For quantifying the uncertainty of the logarithmic P t /P 24 ∼ t curve, we estimated the logarithmic C v t ∼ t curve and the linear C s t ∼ t curve (as shown in Figure 3b) by using the MLR, SVR, and RFR model, and the three types of explanatory variables are used again.The correlations of these parameters with the six explanatory variables can be found in Figures S2 and S3 of the Supporting Information S1.The results in Figure S4 of the Supporting Information S1 present the different performances of the three models, among which the SVR model performed well, being similar to those in Figure 6.
By considering the data availability and reliability again, the SVR model with the inputs of three geographical variables is chosen to estimate the key parameters in the logarithmic C v t ∼ t curve and in the linear C s t ∼ t curve, based on which the 1-km gridded parameters of a cv , b cv , a cs , and b cs in the whole QTP are generated, and their spatial distribution are shown in Figure 8.For the two parameters a cv and b cv in logarithmic C v t ∼ t curve, their value ranges are 0.16-0.10 and 0.35-0.52,respectively.High values of a cv appear in the Qaidam Basin and central QTP, and low values of a cv appear in the eastern, southeastern, and southern boundary areas of QTP (Figure 8a).The spatial distribution of b cv is opposite of that of a cv , with low values appearing in the Qaidam Basin and high values in eastern, southeastern and southern QTP.For the two parameters a cs and b cs in linear C s t ∼ t curve, their values ranges are 0.41-0.13and 0.17-0.88,respectively.High values of a cs appear in the By using the estimated four parameters in Figure 8, the probabilistic distribution of the P t /P 24 value at any tdurations in each grid can be mathematically described, and it can be a reliable basis of quantitative uncertainty evaluation of the sub-daily precipitation (i.e., P t ) estimated from Figure 7, especially for short time-duration with high uncertainty (as the example shown in Figure 3).Therefore, the 1-km gridded data set of the six key parameters can catch the essential information of the complex extreme precipitation conditions over QTP.

Main Contributions of This Study
QTP is a typical ungauged region, and the precipitation data at sub-daily scales for the studies of natural disasters are limited.For filling the data gap of sub-daily precipitation in QTP, this study developed a new strategy to explore the temporal scaling characteristics of sub-daily precipitation in the region, by considering the influences of geographical and climatic conditions.Based on it, a 1-km gridded parameters data set for estimating sub-daily precipitation in the whole QTP was generated.The results indicated that a logarithmic function with two parameters of a mean and b mean can well describe the temporal scaling characteristics of sub-daily precipitation (denoted as P t /P 24 ∼ t curve) at the 102 weather stations in QTP.Besides, the logarithmic C v t ∼ t curve with the parameters of a cv and b cv , and linear C s t ∼ t curve with the parameters of a cs and b cs , can quantitatively evaluate the uncertainty of the P t /P 24 ∼ t curve.Moreover, each of the above six parameters is closely related to the geographical conditions, and their relationships can be well established by the SVR model.By using the model, three geographic variables of longitude, latitude, and altitude at 1-km grid are used as inputs, based on which the 1-km gridded data set for the six parameters in the whole QTP is generated.The parameters data set reflects the essential information of the complex extreme precipitation conditions over QTP, and thus it can provide a feasible approach for estimating sub-daily precipitation, supporting the studies of rainstorm-related natural disasters in the region.Some satellite data and reanalysis data of daily precipitation covering the whole QTP are available.Based on them, the maximum t-duration precipitation (t = 1, 2, …, 12 hr) in local areas can be estimated by using the logarithmic P t /P 24 ∼ t curve established.Particularly, the probabilistic distribution of the maximum t-duration precipitation can also be quantitatively described by using the logarithmic C v t ∼ t curve and linear C s t ∼ t curve established.Following the probabilistic distribution, it would be favorable and flexible for extracting the key tduration precipitation information at any concerned significance levels (or confidence intervals) by researchers and engineers, as a necessary basis for risk analysis and also for forecasting and warning of rainstorm-related natural disasters in QTP.
It should be emphasized that this study revealed the simple and smooth temporal scaling characteristics of subdaily precipitation (see Figure 3a), as well as their regional similarities (see Figures 2,5,7,and 8) due to the effects of geographical and climatic conditions in QTP.Thus, the strategy developed in this study to explore the describable temporal scaling characteristics of sub-daily precipitation is thought as an innovative approach for temporal downscaling, as it is simpler, easier to understand, and more effective compared to those conventional methods which tried to establish the complex and indescribable relationship between daily and sub-daily precipitation (Müller & Haberlandt, 2018;Socolofsky et al., 2001).This innovative strategy is recommended for estimating sub-daily precipitation and evaluating its uncertainty in those regions with the lack of short-duration precipitation data.

The Uncertainty Associated With the Parameters Data Set
The reliability of the parameters data set generated in this study could be influenced by two sources of uncertainty: the quantity of field hourly precipitation observations and the selection of explanatory variables.In our study, we incorporated all the available hourly precipitation data from the 102 weather stations, ensuring that there was no overlap in their data periods.This approach was implemented to capture the maximum amount of relevant statistical characteristics of hourly precipitation and to utilize the available information to the fullest extent.The hourly precipitation data records are longer than 10 years at 98 stations, and longer than 30 years at 38 stations.The different data lengths may influence the stability of the P t /P 24 ∼ t curves.To clarify it, we used the same example station as that in Figure 3, and calculated the P t /P 24 ∼ t curves corresponding to six data periods.Unexpectedly, Figure 9 shows that the P t /P 24 ∼ t curves are consistent despite different data records being used.It may be due to the intrinsic stability and consistency of the temporal scaling characteristics of sub-daily precipitation, which is controlled by its specific geographic and climatic conditions.Thus, it is thought that the data lengths have a weak influence on the stability of the P t /P 24 ∼ t curves.This stability on the one hand proves the advantage of the innovative strategy developed in this study, on the other hand, it reflects the reliability of the parameters data set generated in Figures 7  and 8.
The number of weather stations may also impact the reliability of the parameters data set generated.In this study, 100 stations are located in the river valley region of the eight basins, based on which the parameters data set generated in the region could be reliable.However, the central and western parts of QTP only include two weather stations (see Figure 1), where rainstorm-related natural disasters historically occurred much less frequently than in the river valley regions (Wang et al., 2021).To present the big picture, we used the SVR model calibrated in the river valley region to estimate the six parameters at each 1-km grid in the whole QTP.In the future, more field precipitation observations with long periods are still required to further verify the of the parameters data generated, especially for the central and western parts of QTP.
Another issue is about the types of precipitation.Considering that rainstormrelated natural disasters in QTP mainly occur in summer, in this study we mainly analyzed the hourly rainfall data from May to September, without considering sleet and snow phenomena in those high areas above the snow line (You et al., 2020;Yuan et al., 2022).However, it should be noticed that the population and economy are mainly distributed below and far from the snow line.Thus, the issue would not influence the practical application of the parameters data set generated for the studies of rainstorm-related natural disasters in QTP.
Moreover, in this study, we mainly used the geographical variables of longitude, latitude and altitude to explain the six key parameters.As explained in Section 4.2, although the three climatic variables contribute a part to the modeling results of some parameters, their effects are much weaker than those of the three geographical variables.More importantly, high spatial resolution (1-km) data of the three climatic variables are unavailable to support the generation of 1-km gridded parameters data set.Alternatively, the modeling accuracy (see Figure 6) based on the three geographical variables is thought to be acceptable, based on which the parameters data set generated in Figures 7 and 8 are thought to be reliable.In the future, more explanatory variables can be used to improve the quantification results of temporal scaling characteristics of sub-daily precipitation in the study area.
Besides, due to the limitation of available weather stations, some related issues such as the effects of altitudinal precipitation gradient (Jiang et al., 2022) and diurnal characteristics of sub-daily precipitation (Li, 2018;Zipser & Xu, 2011), cannot be fully taken into account in this study.Future studies considering these could further improve the quality of the parameters data set generated.Finally, it is pointed out that the durations of historical extreme rainfall and flooding disasters in QTP are from minutes to hours, and thus the precipitation information at subhourly scales is another open issue that should be explored in the future.

Conclusions
The Qinghai-Tibet Plateau (QTP) is known for its high vulnerability to destructive rainstorm hazards and related natural disasters.However, the study of these natural disasters in the region has been hindered by limited observed short-duration precipitation data.To address this data gap, we conducted an investigation into the temporal scaling characteristics of sub-daily precipitation in the QTP, and we further examined the influence of geographical and climatic conditions on this temporal scaling characteristics of sub-daily precipitation.As a result of our research, we have generated a comprehensive 1-km gridded parameter data set that can be used to estimate sub-daily precipitation throughout the entire QTP.The data set includes six key parameters, among which the first two parameters of a mean and b mean determine the temporal scaling characteristics of sub-daily precipitation described by a logarithmic P t /P 24 ∼ t curve, and the last four parameters (a cv , b cv , a cs , b cs ) quantify the uncertainty of the temporal scaling characteristics of sub-daily precipitation.Overall, the parameters data set can quantitatively describe the probabilistic distribution of the maximum t-duration precipitation.Based on the probabilistic distribution, it can easily and flexibly extract the key t-duration precipitation information at any desired significance level, providing a fundamental basis for the risk analysis and forecasting/early warning of rainstormrelated natural disasters in QTP.Finally, we would like to emphasize the significance of developing a quantifiable equation to describe the temporal scaling characteristics of sub-daily precipitation as an innovative approach to temporal downscaling.This methodology not only serves as a new and effective criterion for assessing the quality of various data sources but also enables the evaluation of bias correlation.To further validate the application and effectiveness of the innovative strategy devised in this study, it is recommended to conduct additional case studies in different with diverse geographical and climatic conditions.

Figure 1 .
Figure 1.Locations of the Qinghai-Tibet Plateau (QTP) and the 102 weather stations used in this study.The station with the star symbol is used as an example for the analysis in Section 4.1.I: the Hexi Corridor Basin; II: the Qaidam Basin; III: the origin of the Yellow River; IV: the origin of the Jinsha River Basin; V: the origin of the Mekong River Basin; VI: the origin of the Salween River Basin; VII: the Yarlung Tsangpo River Basin; VII: the origin of the Ganges River Basin.

Figure 3 .
Figure 3. (a) Boxplots of the P t /P 24 ∼ t curves at an example station and their average value (blue triangle points), and the fitting curves of five functions; and (b) coefficient of variance (C v t ), and coefficient of skewness (C s t ) of the P t /P 24 (t = 1, 2, …, 12 hr) curves.The location of the example station (with the star symbol) is shown in Figure 1.

Figure 4 .
Figure 4. Boxplots of R 2 values of the five functions for fitting the P t /P 24 ∼ t curves (a), C v t ∼ t curves (b) and C s t ∼ t curves (c) at the 102 weather stations considered in this study.The blue solid line and red diamond point refer to the median and mean value of R 2 , respectively.

Figure 5 .
Figure 5. Spatial distributions of parameters a mean (a), b mean (b), a cv (c) b cv (d), a cs (e), and b cs (f) at the 102 weather stations in Qinghai-Tibet Plateau (QTP).Regarding these parameters, the logarithmic function of P t /P 24 = a mean × ln t + b mean is chosen to describe the P t /P 24 ∼ t curve; the logarithmic function of C t v = a cv × ln t + b cv is chosen to describe the C v t ∼ t curve; and the linear function of C t s = a cs × t + b cs is chosen to describe the C s t ∼ t curve.

Figure 6 .
Figure 6.Normalized Taylor diagrams showing the performances of the MLR, SVR, RFR models for estimating parameter a mean (a) and b mean (b).Normalized standard deviation (σ nor ) is on the radial axis; correlation coefficient (r) is on the angular axis; and green dashed lines indicate CRMSE nor .The suffixes "geo," "clim", and "geoclim" refer to models with the input data of geographical variables, climatic variables, and a combination of them, respectively.The parameters a mean and b mean are the same as that in Figure 5.

Figure 7 .
Figure 7. 1-km gridded data set of the parameters of a mean (a) and b mean (b) over QTP.The two parameters determine the logarithmic P t /P 24 ∼ t curve at each grid, and they were generated by using the SVR model, with the geographic variables of longitude, latitude, and altitude as inputs.The eight basins are the same as those in Figure1.

Figure 8 .
Figure 8. Spatial distribution of the parameters a cv (a), b cv (b), a cs (c), and b cs (d) in 1-km grids in QTP.The four parameters are the same as those in Figure 5.They were generated by using the SVR model, with the geographic variables of longitude, latitude, and altitude as inputs.The eight basins are the same as those in Figure 1.

Figure 9 .
Figure9.The average P t /P 24 ∼ t curves calculated by using six data periods at the example station, with its location (star symbol) shown in Figure1.

Table 1
Types of Functions Used in This Study for Fitting the P t /P 24 ∼ t Curve a f(t) = a * ln t + b Exponential function f(t) = a * e btPower functionf(t) = a * t b Inverse function f(t) = a * (1/t) + ba Here f(t) is P t /P 24 , and t is hour order, with t = 1, 2, …, 12 hr concerned in this study; a and b are two key parameters in the corresponding function.