Abstract
This study aims to develop a stochastic method (SM_GSTR) for generating short-time (i.e., hourly) rainstorm events at all grids (named gridded rainstorm events) in a region. The proposed SM_GSTR model is developed by the non-normal correlated multivariate Monte Carlo simulation (MMCS) method with the statistical properties and spatiotemporal correlation structures of the four event-based gridded rainfall characteristics. The radar-based rainfall data on 20 typhoon events at 336 grids in a basin located in north Taiwan, Nankan River watershed, are used in the model development and demonstration. The results from the model demonstration indicate that the proposed SM_GSTR model can reproduce a great number of gridded rainfall characteristics, of which, the statistical properties in time and space have a good fit to those from the observations in association with the acceptable deviation; thus, it can reasonably emulate the behavior of the rain field in both time and space. It is expected that the resulting massive rainfall-induced disasters (e.g., inundation and landslide) from the physical-based numerical model with the simulated gridded rainstorms by the proposed SM_GSTR model can be applied to establish an alternative artificial intelligence (AI) model for effectively forecasting the hydrologic variables (e.g., runoff and water-level).
Highlights
A stochastic model for generating a rain field is develop under consideration of correlation of rainfalls in time and space.
The rainstorm event consists of five gridded rainfall characteristics.
The resulting rain field is composed of simulated rainstorm at all grids.
A correlated multivariate Monte Carlo method is applied.
The statistical properties of simulated rainstorms at all grids can be persisted.
INTRODUCTION
Recently, due to climate change and the occurrence of extreme rainstorm events with high resolution in time and space, investigations related to rainfall-induced disasters have played an essential role in relevant hydrologic applications, such as flood forecasts (e.g., Chang et al. 2014; Bhola et al. 2018; Zanchetta & Coulibaly 2020), the determination of rainfall thresholds (e.g., Wu et al. 2015; Chang et al. 2018; Ran et al. 2018) and the regulation of the flood risk map (e.g., Amarnath 2014; Doroszkiewicz et al. 2019; Aryal et al. 2020). For example, the results from numerical simulation show that the inundation depths at the locations and corresponding flooding areas are important and advantageous in relation to the hydrological/hydraulic analyses regarding the quantification and evaluation of the flood-induced damage (Smith 1997). However, it remains generally difficult to measure the real-time practical inundation-depth hydrograph and even the induced flooding area due to the limitation of measurement equipment or hindrance in data acquisition, processing, and analysis (e.g., Amarnath 2014; Park et al. 2018); this situation possibly leads to a lack of reliable references regarding the fine spatiotemporal resolution required for the parameter calibration of the hydraulic numerical models (Shen et al. 2019). Despite the fact that the possible maximum flooding area caused by a rainstorm event can be estimated in accordance with the observed maximum inundation-depth recorded at particular locations used to calibrate the parameters of a hydraulic numerical simulation model, the resulting temporal and spatial hydrological features (e.g., inundation depth) include uncertainties with high likelihood attributed to insufficient long-term observations (Hardesty et al. 2018). In order to successfully carry out high-resolution inundation simulation in time and space, the rainfall feature with high spatiotemporal resolution (i.e., gridded precipitation or high-density gauged rainfall) should be taken into account (e.g., Lovat et al. 2019; Try et al. 2020; Zanchetta & Coulibaly 2020). Although improving the spatial density of the rain-gauge networks can effectively capture the rainfall features representing the components of the hydrological cycle, the high cost of equipment and associated maintenance require consideration (Terink et al. 2018). Therefore, rain-field simulation modeling is required in implementing flood and inundation simulations through the hydrologic/hydraulic model (e.g., Wheater et al. 2005; Paschalis et al. 2013; Jiang & Tatano 2015; Wu et al. 2015; Chang et al. 2018; Zanchetta & Coulibaly 2020).
Numerous methods for reproducing rain fields are developed on the basis of physically based models and stochastic modeling, respectively; the commonly used physically based models are the general circulation models (GCMs) which predominantly simulate and predict climatic features (e.g., precipitation and temperature) on a global scale (e.g., Reed 1986; Moron et al. 2008; Khan et al. 2018; Ahmed et al. 2019; Liu et al. 2020). Nevertheless, GCMs commonly generate gridded precipitation based on the project-climate conditions with large uncertainties caused by the concept assumption, model structure, and calibrated parameters (e.g., Yip et al. 2011; Khan et al. 2018).
Regarding the stochastic approaches used in rainfall simulation, the statistical moments of the point-based precipitation (i.e., gridded rainfall characteristics) are necessary to reproduce rainfall processes (e.g., Wheater et al. 2005; Yang et al. 2005; Li et al. 2012; Leblois & Crutin 2013; Paschalis et al. 2013; Jiang & Tatano 2015; Gao et al. 2018; Park et al. 2018; Qiang et al. 2020), For example, Wheater et al. (2005) utilized the generalized linear models (GLMs) to generate the daily rainfall using the corresponding spatial structure extracted from observations; they are then combined with the estimated hourly based sequence by a single-site Poisson process under the assumption of temporal stationarity, enabling the simulated hourly rainfall series at different locations in a region to be obtained. Leblois & Crutin (2013) presented an approach for the simulation of the rainfall-field sequences within the advection field through a classical Gaussian regional simulation technique and the turning-band method. Paschalis et al. (2013) developed a stochastic model for simulating high-resolution spatiotemporal rainfall (named STREAP) by means of the multivariate stochastic simulation method with the spatial and temporal statistical structure of observed radar-derived rainfall data under the assumption of isotropicity in space (i.e., the identical rainfall accumulation for the entire spatial domain) and the isotropic-simulated rainfall fields. Yang et al. (2005) adopted the observations of gauged daily rainfall to identify the best-fit GLM and to quantify the statistical properties, including the spatial correlation structure regarding daily rainfall, the joint distribution for the wet-day pattern of rainfall occurrence and the probability distribution of daily rainfall amount at wet sites used in the rain-field simulation. Gao et al. (2018) presented a daily rainfall simulation model by producing the rainfall events comprising simulated rainfall characteristics in terms of rainfall duration, storm depth, and dimensionless storm pattern defined by Wu et al. (2006) at gauged sites. In Gao et al.’s method, the generation of daily rainfall is implemented through the non-normal correlated multivariate Monte Carlo simulation (MMCS) approach with their copula dependence functions, developed based on the Nataf transformation in association with a linear correlation matrix of random variables (Lebrun & Dutrfoy 2009). By hypothesizing isotoropicity in rainfall characteristics within a minor region, the resulting rainfall events exhibit similar behavior in time to those at the gauges. Qiang et al. (2020) proposed a rainstorm generator for reproducing the rainfall process at various locations by calculating the areal rain field using the interpolation approach with gauged simulated rainfall under consideration of the stochastic dependence matrix in space in terms of a copula function (Jiang & Tatano 2015; Gao et al. 2018); furthermore, Qiang's rainstorm generator can be extended to simulate rain fields in response to climate change by adjusting the parameters of the marginal probability distributions. In addition to stochastic modeling with the Monte Carlo simulation, Park et al. (2018) proposed a hybrid stochastic rainfall model to generate rainfall characteristics. In Park et al.’s model, the monthly rainfalls are first simulated using the seasonal autoregressive integrated moving average approach, and the monthly rainfalls are disaggregated into the hourly based rainfalls through the modified Bartlett-Lewis rectangular pulse (MBLRP) model. Apart from the simulation models with prior statistical properties of rainfall features, Oriani et al. (2014) developed a nonparametric approach to generate the rainfall sequence through direct sampling technique without taking into account the aforementioned prior statistics. Origni's method can provide the time series of daily rainfall and the associated temporal patterns by means of the autoregressive and Markov chain-based methods on the basis of the training image obtained from the previous rainfall record; however, it has inherent limitations in representing extremes (Li et al. 2012).
Of the numerous rainfall-simulation models developed, most focus on simulating the correlated rainfall amount/depth for a specific rainfall duration at various locations in the region of interest under an assumption of the isotropic storm pattern in terms of the identical rainfall distribution in time (e.g., Paschalis et al. 2013; Jiang & Tatano 2015; Yang et al. 2005; Gao et al. 2018; Qiang et al. 2020). That is to say, the past investigations mainly emphasized the assessment regarding the effects of temporal and spatial changes in rainfall on the relevant hydrologic and hydraulic analysis. However, in reality, in addition to the spatial or temporal uncertainties in the rainfall, the spatiotemporal resolution and variation in the rain field possibly influences the urban/watershed hydrologic responses required in flood-hydrological modeling and the relevant induced disaster prevention and mitigation (e.g., Reed et al. 2007; Leblois & Crutin 2013; Jiang & Tatano 2015; Park et al. 2018). Therefore, this study aims to develop a stochastic model for generating gridded short-term (i.e., hourly) rainstorm events (called SM_GSTR model) by taking into account the spatiotemporal dependence in the rain field. It is expected that hydraulic dynamic modeling with the resulting rainstorms at all grids (i.e., gridded rainstorms) from the SM_GSTR model can provide more detailed information on rainfall-induced forecasts with fully high resolution in time and space for the decision-making and infrastructure-planning of disaster early warning and proofing systems.
METHODOLOGY
Model concept
This study intends to develop a stochastic model for generating a large number of gridded rainstorm events, named the SM_GSTR model. In detail, the proposed SM_GSTR model is established based on Wu's method (Wu et al. 2006) involving a stochastic model for generating event-based rainstorm events comprising the at-site rainfall characteristics, including rainfall duration, storm depth, and storm pattern without considering their spatial correlations among rainfall characteristics. Also, the spatial heterogeneity regarding rainfall significantly impacts the shape of the hydrological response, especially for the combination of catchment features and rainfall characteristics (Volpi et al. 2012), By doing so, this study modifies Wu's model by taking into account both spatial and temporal correlations regarding rainfall characteristics extracted from the gridded rainfall data (called gridded rainfall characteristics). It is noticed that the main difference of the proposed SM_GSTR model from Wu's method is the correlation structure in both time and space considered in the generation of rainstorm events at all grids within a region.
In developing the proposed SM_GSTR model in accordance with the model framework, the observations of the rainfall characteristics at all grids within the study area, defined as gridded rainfall characteristics, are extracted from historical hourly rainfall data. Uncertainty analysis is then carried out to quantify the corresponding statistical properties regarding time and space, including the first four statistical moments and correlations; in so doing, the resulting statistical properties of gridded rainfall characteristics are used to reproduce the short-term rainstorm events at all grids in a region, composed of the simulated gridded rainfall characteristics by the non-normal correlated MMCS method (Wu et al. 2006). The aforementioned relevant methods and concepts are addressed below.
Brief introduction to stochastic generation of at-site rainstorms
At-site rainfall characteristics
∈[0, 1] represents the dimensionless time; d is the storm duration; 
∈[0, 1] stands for the cumulative dimensionless rainfall; 
is the incremental dimensionless rainfall amount; Dt is the cumulative rainfall depth at time t; and Dd denotes the total rainfall depth.
Simulation of at-site rainfall duration and storm depth
and 
, the standard deviations 
and 
, and the correlation coefficient 
; Zi and Zj are corresponding bivariate standard normal variables to the variable Xi and Xj with the correlation coefficient 
and the joint standard normal density function ϕij(·). In referring to Equation (2), the MMCS method can generate multivariate with high correlation in time or space on the basis of the selected marginal distributions and correlation coefficients (Chang et al. 1997).
in the original space to 
in the normal space by:
, one can obtain the corresponding random variates in the original space by the following inverse transformation:
is the marginal cumulative distribution function (CDF) of random variable Xi in the original space, and Ф(·) is the standard normal CDF. The framework of generating the correlated multivariate by means of the MMCS method can be referred to in Figure 2.
Flowchart of generating the correlated multivariate by the non-normal correlated multivariate Monte Carlo simulation (MMCS) method (Tung & Yen 2005).
Flowchart of generating the correlated multivariate by the non-normal correlated multivariate Monte Carlo simulation (MMCS) method (Tung & Yen 2005).
Simulation of storm pattern
In Wu's method, the rainfall duration and storm depth regarded as the spatial rainfall characteristics can be directly generated by the MMCS method. However, the storm pattern is defined as the time distribution of rainfall, comprising dimensionless rainfall in which the time-axis is divided into 12 increments (Wu et al. 2006). Accordingly, two special constraints of dimensionless rainstorm patterns must be observed: (1) the summation of dimensionless rainfall must be equal to 1: P1/12 + P2/12 + …+ P12/12 = 1 as 0 ≤ F1/12 ≤ F2/12 ≤ …≤ F12/12 = 1; and (2) the dimensionless rainfall must be between 0 and 1: 0 ≤ Pτ ≤ 1 for τ =1/12, 2/12, …, 12/12.
should be correlated non-normal variables, they would be treated as the unconstrained variates 
without the above special constraints through the log-ratio transformation (called the log-ratio transformation) as follows:
is the specified dimensionless time with 
being the associated non-negative dimensionless rainfall. According to Equation (6), it is observed that the transformed variable 
ranges from 
to 
and that it can avoid the numerical problem.
of the dimensionless rainfall, excluding the one at 
they are then transformed into the dimensionless rainfalls through the following equations:
, Wu et al. (2006) utilized the Johnson distribution system, which is more flexible in fitting the rainstorm pattern than are other distributions (Fang & Tung 1996), to standardize the log-ratio variates instead of the standard normalization equation, i.e., Equation (6), used in the non-normal correlated MMCS with Equation (2). It is well-known that the Johnson distribution is a system of frequency curve consisting of four parameters (Johnson 1949):
and 
are the location and scale parameters, respectively. In general, the Johnson distribution system is grouped into three types:
Of the three types of Johnson distribution, there is very good agreement between the measured and estimated rain variables using the Johnson distribution system, especially for the bounded system (
) (D'Adderio et al. 2016). Hill et al. (1976) presented an algorithm to identify the best-fit type of the Johnson distribution system and calibrate the associated parameters listed in Equation (10) by means of matching the first four product-moments of the random variables.
Consequently, the framework of generating at-site rainfall characteristics through Wu's method includes five steps: (1) quantification of statistical properties of at-site rainfall characteristics; (2) generation of rainfall duration and storm depth; (3) simulation of storm patterns composed of generated dimensionless rainfalls; and (4) reproduction of at-site rainstorms by combining the simulated at-site rainfall characteristics as shown in Figure 3.
Development of the proposed SM_GSTR model
Definition of gridded rainfall characteristics
and 
stand for the dimensionless rainfall at the dimensionless time kth and resulting areal average for the jth grid; 
is the cumulative dimensionless rainfall at the dimensionless time 
for the jth grid;
is the areal average of the dimensionless rainfall at the dimensionless 
; and 
serves as the difference in the cumulative dimensionless rainfall at the dimensionless time for the jth grid from the corresponding areal average. Note that the areal average 
is calculated through the following equation:
serves as the number of grids in the steady area; 
stands for the cumulative dimensionless rainfall for the dimensionless time 
at the jth grid. Since the 
and 
can be treated as the temporal and spatiotemporal variates, respectively, the deviation 
is accordingly regarded as the spatiotemporal variable.In summary, this study defines gridded short-term rainstorms in terms of gridded rainfall characteristics, including the rainfall duration of events, the rainfall depths at all grids (i.e., gridded rainfall depths), and the areal average storm pattern consisting of cumulative dimensionless rainfalls and the associated deviation at different dimensionless times. Figure 4 presents the process of extracting the aforementioned gridded rainfall characteristics from the observed gridded hyetographs regarding historical rainstorms.
Graphical process of extracting the gridded rainfall characteristics from observed hyetographs of rainstorm events.
Graphical process of extracting the gridded rainfall characteristics from observed hyetographs of rainstorm events.
Stochastic generation of gridded rainfall characteristics
With respect to Wu's method, the non-normal correlated MMCS method (Wu et al. 2006) is utilized to generate the highly temporally correlated rainfall characteristics. By doing so, the proposed SM_GSTR model also applies the MMCS method to simulate a number of the gridded rainfall characteristics in response to the uncertainties in time and space. In detail, the process of generating the gridded rainfall characteristics by means of the proposed SM_GSTR model can be grouped into four parts: the simulation of the rainfall duration and rainfall depths at all grids (i.e., gridded rainfall depths), the simulation of the areal average storm patterns, and the generation of the deviation of gridded storm pattern. The detailed concept of the aforementioned generation process is introduced in the following:
- 1)
Rainfall duration and gridded rainfall depths: Regarding the proposed SM_GSTR model, during a simulated rainstorm event, the corresponding rainfall duration and a number of the rainfall depths at the grids of interest (i.e., gridded rainfall depths) are correlated variables, which differ from Wu's method for simulating at-site events. Therefore, in the case of N grids used in the field, the MMCS model is adopted herein to simultaneously simulate (N + 1) variables, including rainfall duration and storm depths of the N grids.
- 2)Simulation of the areal average storm pattern: Within the SM_GSTR model, similar to Wu's method, the dimensionless rainfalls
are simulated by the MMCS method with Johnson distribution, and the corresponding areal average values 
are obtained through the following equation:(14)
at all grids could be also reproduced through the MMCS method with the conventional marginal probability functions. Eventually, by combining the simulations of the areal average of the cumulative dimensionless rainfalls with the corresponding simulated deviation at all grids 
(i.e., gridded deviation of gridded storm pattern) for various dimensionless times, the generated grid-based storm pattern comprising the cumulative dimensionless rainfalls can be obtained through the following equation:
stands for the generated cumulative dimensionless rainfall at the dimensionless time τ for the jth grid; 
denotes the simulated areal average of cumulative dimensionless rainfall at the dimensionless time τ and 
represents the generated deviation of the cumulative dimensionless rainfall at the dimensionless time τ for the jth grid. However, in the case of the unreasonable simulated cumulative dimensionless rainfall of storm pattern obtained through Equation (15), i.e., 
, this study modifies the 
through the optimization method with the flooding equation to find out an adjusting factor of the gridded mean deviation:
and 
stand for the modified simulation of cumulative dimensionless and the corresponding adjusting factor, respectively. In Equation (16), the adjusting factor of the gridded deviation (
) can be determined by the optimization method, such as the genetic algorithm (GA), subject to a condition as:
Eventually, using a great number of simulations of the five gridded rainfall characteristics, the resulting hyetographs of simulated rainstorm events at all grids in a region of interest can be obtained through the graphical process as shown in Figure 5.
Graphical illustration of combining the generated gridded rainfall characteristics as the gridded rainstorm events.
Graphical illustration of combining the generated gridded rainfall characteristics as the gridded rainstorm events.
Framework of modeling stochastic generation of gridded rainstorms
To sum up the concepts introduced above, the proposed SM_GSTR model includes (1) the characterization of gridded short-term rainstorm events; (2) the quantification of uncertainties of gridded rainstorms in time and space; and (3) the stochastic generation of spatiotemporally correlated rainfall characteristics. Note that in generating the gridded rainfall characteristics, especially for the rainfall durations and gridded rainfall depths, via the proposed SM_GSTR model, the conditions of non-negative gridded rainfall characteristics should be given in order to avoid the unreasonable simulations, i.e., the negative rainfall durations, rainfall depths, and dimensionless rainfalls of storm patterns. The detailed framework of developing the proposed SM_GSTR model can be addressed (see Figure 6) as follows:
Step [1]: Collect the grid-based precipitation of numerous rainstorm events in the study area and extract their gridded rainfall characteristics, i.e., rainfall duration, gridded rainfall depth, areal average of cumulative dimensionless rainfall and the associated deviation at various dimensionless times.
Step [2]: Implement uncertainty analysis for the gridded rainfall characteristics to quantify the uncertainties in terms of the statistical moments and correlation coefficients in time and space.
Step [3]: Generate the event-based rainfall duration and the gridded rainfall depths using the non-normal correlated MMCS method based on their statistical properties and correlation coefficients in time and space.
Step [4]: Generate the areal average of the dimensionless rainfalls based on the statistical moments and their temporal correlation coefficients; then, cumulate the simulated areal averages as the areal average cumulative dimensionless rainfalls.
Step [5]: Simulate the deviation of cumulative dimensionless rainfall at all grids (i.e., gridded deviation) for each dimensionless time based on the corresponding statistical moments and correlation coefficients.
Step [6]: Combine the simulated average of the cumulative dimensionless rainfall at each dimensionless time at Step [4] with the corresponding generated grid-based deviation obtained at Step [5] as the cumulative dimensionless rainfall at each dimensionless time at all grids in the study area.
Step [7]: Combine the resulting simulations of gridded rainfall characteristics from the above steps as the gridded short-term rainstorm events. The graphical process of establishing the proposed SM_GSTR model can be seen in Figure 6.
Framework of developing the proposed SM_GSTR model.
STUDY AREA AND DATA
Introduction to study area
The study area, the Nankan River, in which the corresponding length and the drainage area are 31 km and 224 km2, respectively, is in Taoyuan County of northern Taiwan (see Figure 7); also, on average, the slope approximates 0.0077. In addition, it flows through the Guishan, Taoyuan, and Luzhou Districts in Taoyuan City. Within the Nankan River watershed, there are six riverside parks and three branches, Dongmon Creek, KengZi Creek, and Qiedong Creek. Of these branches, the flooding risk at the Dongmon Creek, previously inundated, has been noticeably reduced by increasing the number of the detention ponds and the cross-section area in the river channel. Additionally, it is well-known that Taiwan Central Weather Bureau (CWB) can provide the quantitative precipitation estimation (QPE) with the spatial resolution of 1.5 km × 1.5 km; hence, there are 336 grids (called QPE grids) located in Nankan River watershed as shown in Figure 7 and the corresponding gridded hourly rainfall data can be utilized in the development and demonstration of the proposed SM_GSTR model.
Locations and DEM as well as QPE grids (blue circle) within the study area of Nankan River watershed (yellow region).
Locations and DEM as well as QPE grids (blue circle) within the study area of Nankan River watershed (yellow region).
Spatiotemporal analysis of the study data
Since the purpose of this study is to develop a stochastic model for simulating a large number of rainstorms with high resolution in time and space, the hourly rainfall datal of 20 rainstorm events at 336 grids within the study area (Nankan River watershed) (see Table 1) are selected as the study data.
Summary of 20 historical rainstorm events in the Nankan River for model development and application
| Event | Beginning time | Ending time | Duration (hours) | Areal average rainfall depth (mm) |
|---|---|---|---|---|
| 1 | 2005/7/17 10:00 | 2005/7/18 18:00 | 33 | 139.8 |
| 2 | 2005/8/4 09:00 | 2005/8/6 00:00 | 40 | 230.6 |
| 3 | 2005/8/31 07:00 | 2005/9/1 08:00 | 26 | 125.5 |
| 4 | 2005/10/1 23:00 | 2005/10/2 18:00 | 20 | 54.5 |
| 5 | 2008/7/26 22:00 | 2008/7/28 22:00 | 49 | 105.8 |
| 6 | 2008/9/12 02:00 | 2008/9/15 00:00 | 71 | 403.2 |
| 7 | 2010/10/21 00:00 | 2010/10/22 13:00 | 38 | 120.8 |
| 8 | 2012/6/11 21:00 | 2012/6/12 23:00 | 27 | 366.7 |
| 9 | 2012/6/14 11:00 | 2012/6/15 06:00 | 20 | 18.5 |
| 10 | 2012/8/26 09:00 | 2012/8/27 08:00 | 24 | 29.8 |
| 11 | 2013/5/11 01:00 | 2013/5/13 01:00 | 49 | 184.8 |
| 12 | 2013/7/12 16:00 | 2013/7/13 14:00 | 23 | 75.4 |
| 13 | 2013/10/4 14:00 | 2013/10/6 23:00 | 58 | 56.3 |
| 14 | 2014/5/20 20:00 | 2014/5/22 00:00 | 29 | 195.9 |
| 15 | 2014/7/22 21:00 | 2014/7/24 03:00 | 31 | 43.4 |
| 16 | 2014/9/21 16:00 | 2014/9/22 12:00 | 21 | 83.5 |
| 17 | 2015/8/7 08:00 | 2015/8/8 13:00 | 30 | 184.9 |
| 18 | 2015/9/27 14:00 | 2015/9/29 05:00 | 40 | 138.3 |
| 19 | 2016/9/26 10:00 | 2016/9/28 04:00 | 43 | 106.3 |
| 20 | 2017/6/2 10:00 | 2017/6/4 04:00 | 39 | 271.2 |
| Event | Beginning time | Ending time | Duration (hours) | Areal average rainfall depth (mm) |
|---|---|---|---|---|
| 1 | 2005/7/17 10:00 | 2005/7/18 18:00 | 33 | 139.8 |
| 2 | 2005/8/4 09:00 | 2005/8/6 00:00 | 40 | 230.6 |
| 3 | 2005/8/31 07:00 | 2005/9/1 08:00 | 26 | 125.5 |
| 4 | 2005/10/1 23:00 | 2005/10/2 18:00 | 20 | 54.5 |
| 5 | 2008/7/26 22:00 | 2008/7/28 22:00 | 49 | 105.8 |
| 6 | 2008/9/12 02:00 | 2008/9/15 00:00 | 71 | 403.2 |
| 7 | 2010/10/21 00:00 | 2010/10/22 13:00 | 38 | 120.8 |
| 8 | 2012/6/11 21:00 | 2012/6/12 23:00 | 27 | 366.7 |
| 9 | 2012/6/14 11:00 | 2012/6/15 06:00 | 20 | 18.5 |
| 10 | 2012/8/26 09:00 | 2012/8/27 08:00 | 24 | 29.8 |
| 11 | 2013/5/11 01:00 | 2013/5/13 01:00 | 49 | 184.8 |
| 12 | 2013/7/12 16:00 | 2013/7/13 14:00 | 23 | 75.4 |
| 13 | 2013/10/4 14:00 | 2013/10/6 23:00 | 58 | 56.3 |
| 14 | 2014/5/20 20:00 | 2014/5/22 00:00 | 29 | 195.9 |
| 15 | 2014/7/22 21:00 | 2014/7/24 03:00 | 31 | 43.4 |
| 16 | 2014/9/21 16:00 | 2014/9/22 12:00 | 21 | 83.5 |
| 17 | 2015/8/7 08:00 | 2015/8/8 13:00 | 30 | 184.9 |
| 18 | 2015/9/27 14:00 | 2015/9/29 05:00 | 40 | 138.3 |
| 19 | 2016/9/26 10:00 | 2016/9/28 04:00 | 43 | 106.3 |
| 20 | 2017/6/2 10:00 | 2017/6/4 04:00 | 39 | 271.2 |
Event-based rainfall duration and storm depth
Figure 8 represents the hyetographs of the 20 selected radar-based rainstorm events (2005–2017) provided by Taiwan Central Weather Bureau. In view of Table 1 and Figure 7, the rainfall duration ranges from 20 hours to 70 hours; and, on average, the gridded rainfall depths are located between 40 and 410 mm, implying that the rainstorm events with various rainfall intensities can be used to evaluate the effect of rainstorm scale on the hydrologic response. Also, Figure 9 shows the gridded rainfall depths of 20 rainstorms, indicating that the spatial distribution of the gridded rainfall depths exhibits an obvious change with the rainstorm event considered. For example, in regard to most of the rainstorm events (e.g., EV3, EV13, and EV18), their rainfall depths at all grids gradually approach two centers; namely, the maximum rainfall depths of most rainstorm events can be found approximately near two grids; it can also be found that the rainfall depth for rainstorms (e.g., EV10, EV13, and EV15) at all grids are distributed normally in the study area. The above results indicate significant spatial uncertainty and variation in the gridded rainfall depth in the study area.
Gridded rainfall series of 20 observed rainstorm events used in the model development and application.
Gridded rainfall series of 20 observed rainstorm events used in the model development and application.
Spatial distribution of gridded rainfall depth of 20 observed rainstorm events.
Gridded storm patterns and associated deviation
In this study, the gridded storm pattern consisting of the cumulative dimensionless rainfalls at various dimensionless times is grouped into two components: the areal average storm pattern and associated deviation at all grids (i.e., gridded deviation) defined as the gridded rainfall characteristics. According to the framework of developing the proposed SM_GSTR model, the uncertainty quantification should be carried out to quantify the corresponding statistical properties in advance (see Figures 10 and 11). From Figure 10, it can be seen that there are obviously different varieties of the areal average storm patterns in the study area; hence, the temporal variations in the storm pattern should be considered. Also, Figure 11 presents the spatial varying trend of gridded deviation regarding the storm pattern. It indicates that the gridded deviation exhibits an obvious varying trend in space; namely, its spatial distribution changes with the dimensionless times and rainstorms. Accordingly, the gridded deviation of storm pattern possibly involves uncertainty in time and space.
The areal average of cumulative dimensionless rainfalls (storm pattern) derived from 20 observed rainstorm events.
The areal average of cumulative dimensionless rainfalls (storm pattern) derived from 20 observed rainstorm events.
Gridded deviation of cumulated dimensionless rainfalls of 20 observed rainstorm events at the dimensionless time 
.
Gridded deviation of cumulated dimensionless rainfalls of 20 observed rainstorm events at the dimensionless time .
To sum up the above results, the gridded rainfall characteristics could be regarded as highly spatially and temporally correlated variables in association with the uncertainties of different degrees. By doing so, as simulating the rain field with high resolution in time and space, it is essential to take into account the uncertainties of gridded rainfall characteristics and their dependence structure in time and space.
RESULTS AND DISCUSSION
To consider the spatial and temporal variations and correlations in the rain field, a stochastic modeling for generating a great number of gridded rainstorms is established based on the model development framework (see Figure 6). In detail, the uncertainties in the gridded rainfall characteristics extracted from the radar-based rainfall data (i.e., QPE) in the section ‘Introduction to study area’ can be quantified to obtain the spatial and temporal statistical properties; massive gridded rainfall characteristics can then be reproduced through the MMCS method. Eventually, a great number of gridded rainstorms involved in the rain field can be obtained by integrating the gridded rainfall characteristics. The relevant results are evaluated and discussed below.
Uncertainty analysis for grid-based rainfall characteristics
Event-based rainfall and gridded rainfall depth
Figure 12 shows the statistical properties of the rainfall duration and gridded rainfall depth; on average, the durations of gridded rainstorm events frequently occurring in the Nankan River watershed is about 35 hours; whereas the lower and upper bounds of 95% confidence interval are 20 hours and 53 hours, respectively, indicating that 1-day and 2-day events frequently take place in the study area, the Nankan River watershed. In the case of the statistical properties of gridded rainfall depth, its average and standard deviation ranges between 80 and 220 mm as well as 82 and 177 mm. Although its lower bound of 95% confidence interval approximates 20 mm, its upper bound has a wide range from 284 to 336 mm, respectively. This implies that even for the same rainstorm event, the variation in rainfall depth at all grids in the study area obviously varies with the locations of the grids.
Statistical properties of gridded rainfall characteristics (rainfall duration and gridded rainfall depth).
Statistical properties of gridded rainfall characteristics (rainfall duration and gridded rainfall depth).
Areal average of storm pattern and gridded deviation
For the statistical properties of cumulative dimensionless rainfall of the areal average storm pattern as shown in Figure 13 and Table 2, a wider 95% confidence interval can represent that various varieties of storm patterns (e.g., central, advanced, and delayed-shaped patterns) take place in the study area. Also, Table 2 lists the correlation coefficients among the dimensionless rainfall (about −0.3 to 0.97); it shows that the areal averaged cumulative dimensionless rainfalls are strongly correlated with different dimensionless times. In addition, the corresponding statistics of its deviation at each grid should be considered, as shown in Figure 13. Moreover, Figure 3 illustrates the mean and standard deviation of the gridded deviation of storm patterns for the dimensionless time selected (
, 
, 
, and 
). It can be known that the aforementioned statistical moments significantly change with the dimensionless time in which the large moments can be found at the central dimensionless rainfall (
5); whereas, the cumulative dimensionless rainfall at the beginning (
3) and ending time steps (
9) are obviously smaller than ones at the remaining time steps. Also, since the correlation coefficients among the gridded deviations of cumulative dimensionless rainfalls range approximately from −0.87 to 0.95, the gridded deviation should be treated as the spatially correlated variables. By doing so, in the proposed SM_GSTR model, the areal average and gridded deviation of storm patterns must be generated under consideration of the correlation coefficients in time and space given in advance.
Correlation coefficients among dimensionless rainfall Pτ at various dimensionless time τ1–τ12
| Dimensionless time τ | Pτ1 | Pτ2 | Pτ3 | Pτ4 | Pτ5 | Pτ6 | Pτ7 | Pτ8 | Pτ9 | Pτ10 | Pτ11 | Pτ12 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Pτ1 | 1 | |||||||||||
| Pτ2 | 0.016 | 1 | ||||||||||
| Pτ3 | −0.2 | 0.87 | 1 | |||||||||
| Pτ4 | −0.243 | 0.792 | 0.964 | 1 | ||||||||
| Pτ5 | −0.244 | 0.749 | 0.93 | 0.966 | 1 | |||||||
| Pτ6 | −0.231 | 0.736 | 0.907 | 0.938 | 0.964 | 1 | ||||||
| Pτ7 | −0.187 | 0.646 | 0.804 | 0.831 | 0.847 | 0.88 | 1 | |||||
| Pτ8 | −0.205 | 0.481 | 0.642 | 0.687 | 0.702 | 0.722 | 0.807 | 1 | ||||
| Pτ9 | −0.142 | 0.322 | 0.417 | 0.451 | 0.459 | 0.474 | 0.553 | 0.702 | 1 | |||
| Pτ10 | −0.209 | 0.186 | 0.282 | 0.288 | 0.273 | 0.28 | 0.275 | 0.269 | 0.458 | 1 | ||
| Pτ11 | −0.176 | −0.037 | 0.046 | 0.028 | 0.003 | −0.029 | −0.103 | −0.211 | −0.102 | 0.076 | 1 | |
| Pτ12 | 0.02 | 0.017 | 0.015 | 0.013 | 0.021 | 0.026 | 0.035 | 0.008 | −0.015 | 0.007 | −0.006 | 1 |
| Dimensionless time τ | Pτ1 | Pτ2 | Pτ3 | Pτ4 | Pτ5 | Pτ6 | Pτ7 | Pτ8 | Pτ9 | Pτ10 | Pτ11 | Pτ12 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Pτ1 | 1 | |||||||||||
| Pτ2 | 0.016 | 1 | ||||||||||
| Pτ3 | −0.2 | 0.87 | 1 | |||||||||
| Pτ4 | −0.243 | 0.792 | 0.964 | 1 | ||||||||
| Pτ5 | −0.244 | 0.749 | 0.93 | 0.966 | 1 | |||||||
| Pτ6 | −0.231 | 0.736 | 0.907 | 0.938 | 0.964 | 1 | ||||||
| Pτ7 | −0.187 | 0.646 | 0.804 | 0.831 | 0.847 | 0.88 | 1 | |||||
| Pτ8 | −0.205 | 0.481 | 0.642 | 0.687 | 0.702 | 0.722 | 0.807 | 1 | ||||
| Pτ9 | −0.142 | 0.322 | 0.417 | 0.451 | 0.459 | 0.474 | 0.553 | 0.702 | 1 | |||
| Pτ10 | −0.209 | 0.186 | 0.282 | 0.288 | 0.273 | 0.28 | 0.275 | 0.269 | 0.458 | 1 | ||
| Pτ11 | −0.176 | −0.037 | 0.046 | 0.028 | 0.003 | −0.029 | −0.103 | −0.211 | −0.102 | 0.076 | 1 | |
| Pτ12 | 0.02 | 0.017 | 0.015 | 0.013 | 0.021 | 0.026 | 0.035 | 0.008 | −0.015 | 0.007 | −0.006 | 1 |
Statistical properties of gridded rainfall characteristics (areal average and gridded deviation of storm pattern).
Statistical properties of gridded rainfall characteristics (areal average and gridded deviation of storm pattern).
Evaluation of simulations of gridded rainfall characteristics
Using the proposed SM_GSTR method with the spatial and temporal statistical properties of gridded rainfall characteristics, the resulting 1,000 simulations can be obtained as shown in Figures 14 and 15. To evaluate the applicability of the proposed SM_GSTR method for simulating the gridded rainstorms, the statistical features calculated from 1,000 simulation at all grids within the study area are compared with the results from the observations (see Figures 16–21), including the mean, standard deviation, and 95% confidence interval. In theory, since the gridded rainfall characteristics in time and space are generated by using the non-normal correlated MMCS based on the Nataf-based algorithm, in which the correlation structure among simulations has been proven to be consistent with those from observations by Chang et al. (1997), the consistency of the correlation structures in time are evident regarding the gridded rainfall characteristics.
1,000 simulations of rainfall duration, gridded rainfall depths, and areal average storm pattern.
1,000 simulations of rainfall duration, gridded rainfall depths, and areal average storm pattern.
1,000 simulations of gridded deviation of storm pattern.
Comparison of statistical properties of observed and simulated rainfall durations.
Comparison of statistical properties of observed and simulated rainfall durations.
Comparison between statistical properties of observed and simulated gridded rainfall depth.
Comparison between statistical properties of observed and simulated gridded rainfall depth.
Correlation coefficients between the statistical properties of observed and simulated gridded rainfall depths.
Correlation coefficients between the statistical properties of observed and simulated gridded rainfall depths.
Comparison between statistical properties of the observed and simulated areal averages of cumulative dimensionless rainfalls.
Comparison between statistical properties of the observed and simulated areal averages of cumulative dimensionless rainfalls.
Correlation coefficients between the statistical properties of observed and simulated areal averages of cumulative dimensionless rainfalls.
Correlation coefficients between the statistical properties of observed and simulated areal averages of cumulative dimensionless rainfalls.
Comparison of statistical properties of observed and simulated deviation of gridded storm patterns.
Comparison of statistical properties of observed and simulated deviation of gridded storm patterns.
In this study, since the gridded rainfall events are reproduced by combining the simulated gridded rainfall characteristics, this study focuses on evaluating the performance in comparison to the correlation with the observed data by calculating the correlation coefficients among the statistical properties, including the mean, standard deviation, and 95% confidence interval (see Figure 22). In detail, the spatial correlation coefficients (i.e., the gridded rainfall depths and the gridded deviation of the cumulative dimensionless rainfall) and temporal correlation coefficients (i.e., the areal average of cumulative dimensionless rainfalls) are adopted to assess the consistency of correlations regarding the gridded rainfall characteristics. The relevant results and discussion are addressed below.
Correlation coefficients between the statistical properties of observed and gridded deviations of cumulative dimensionless rainfalls.
Correlation coefficients between the statistical properties of observed and gridded deviations of cumulative dimensionless rainfalls.
Rainfall duration and gridded rainfall depths
By observing Figure 16, the rainfall durations of 1,000 simulated gridded rainstorm events change from 3 hours to 81 hours. As compared with the statistics from observations, the average of the simulated duration (36 hours) approximates that from the observation (37 hours); and the standard deviation of the simulated duration (12 hours) is nearly equal to that from the observations (13 hours). It can be seen that the lower bound of the 95% confidence interval of the simulated duration (12 hours) is smaller than that from observation (20 hours); accordingly, a larger upper bound of 65 hours, than that from observation (53 hours), should be obtained. This difference implies that there noticeably exists variation in event-based rainfall duration; thus, the rainstorm events of different time scales are applied in the development of the proposed SM_GSTR model in response to the short-term effect on the simulation of gridded rainstorms.
In referring to Figure 17, it can be seen that the mean values of simulated gridded rainfall depth (on average 141.3 mm) is close to that from observation (about 164.2 mm); whereas the standard deviation of simulated gridded rainfall depth (41.2 mm–160.8 mm) has an acceptable relative difference (−0.47) to observed values (78.8 mm–176.8 mm). The lower bound of the simulated gridded rainfall depth (on average 74.1 mm) significantly departs from the observed values (15.7 mm), exhibiting good agreement between the simulations and observations in space; on the contrary, the upper bound (234.2 mm–781.4 mm) is slightly greater than the observations (267.7 mm–686.6 mm). Also, similar to the above statistical features, the change in the upper bound of simulated gridded rainfall depth approximately matches those from the observation.
As a result of the gridded rainfall depths being the spatial variables, the corresponding correlation coefficients result from the statistical properties of the gridded rainfall depth quantified using 1,000 simulations of rainfall depths at each grid (see Figure 18). From Figure 18, the correlation coefficient of the mean value and the upper bound of 95% confidence interval exceeds 0.8; this means that the averages of the simulated gridded rainfall depths are strongly related to the observations in space. Also, in spite of the larger relative errors regarding the standard deviation and lower bound of 95% confidence interval calculated from the simulations, about −1.47 and 5.04, their spatial correlation coefficients approximate 0.5.
To sum up the above results, the average rainfall depths at various grids have good agreement with those derived from observations. However, their standard deviations are mostly less than those from the observations which leads to a narrow 95% confidence interval consisting of the higher lower bound and smaller upper bounds, respectively, probably in order to persist in the correlation structures regarding the rainfall depths in space. In detail, as for the upper bound of the 95% confidence interval, the maximum (about 770 mm) can be found at the 336th grid and it slightly exceeds the observation (approximately 680 mm), implying that the proposed SM_STGR model is advantageous in the generation of heavy rainstorms probably applied in the investigations related to the flooding and inundation simulation. Consequently, the proposed SM_GSTR model can reproduce the simulations of event-based gridded rainfall depths with a high spatial correlation in comparison to those from the observations.
Gridded storm patterns
Similar to the gridded rainfall depths, the performance evaluation of simulating gridded storm patterns is accomplished by comparing the statistical properties and corresponding dependence in time and space between observation and simulations. Since the simulation of the gridded storm pattern is combined with the generated areal average of cumulative dimensionless rainfalls and associated deviation at dimensionless times, the performance assessment of simulating the gridded rainstorm can be achieved in comparison to the statistical properties of the areal average and associated gridded deviation, respectively. The relevant results are addressed and assessed below.
Areal average
Figure 19 shows the comparison of the statistical properties of the areal average of the cumulative dimensionless rainfalls from the simulations with those from the observation; the statistical properties of the simulations excellently match those from the cumulative dimensionless rainfall with the low relative errors being −0.095 (mean), −0.263 (standard deviation), 0.76 (upper bound), and −0.078 (lower bound). These results reveal that the proposed SM_GSTR model can reproduce the areal average of storm pattern with high reliability by permitting well the temporal statistical properties of the storm pattern.
Since the areal average of dimensionless rainfalls can be treated as the temporal variate, the correlation coefficients of their statistical properties can be used to assess the dependence between the observed and simulated areal averaged dimensionless rainfalls. Figure 20 presents the comparison in the correlation coefficients of the statistical properties, indicating that the above correlation coefficients mostly reach 0.95. This result reveals that the varying trend of the simulated areal average of dimensionless rainfalls in time exhibits a good match with the observations. This matching indicates that the proposed SM_GSTR model can generate a large number of areal averaged storm patterns of which the temporal dependence is consistent with the observations.
Gridded deviation
The assessment of generating the gridded deviations of the cumulative dimensionless rainfalls at the dimensionless times of 
are made as shown in Figure 21. It can be known that the mean and standard deviation of the simulated grid-based bias at the four dimensionless times have an acceptable agreement with those from observations in which the relative errors are on average −0.011 and −0.045, respectively. Moreover, the fitness of the upper bound of 95% confidence interval to the results from observations is better than the lower bound due to the lower bound of 95% confidence interval simulated gridded bias being mostly overestimated. However, it can be seen that the spatial change in the bounds of 95% confidence interval has good agreement with the observation as a result of the high correlation coefficient being 0.6.
In addition to the areal average of dimensionless rainfalls, this study quantifies the correlations among the gridded deviations of the cumulative dimensionless rainfall at the four dimensionless times as shown in Figure 22. Figure 22 shows that the mean value of the simulated gridded deviations at the four dimensionless times is evidently and strongly related to that from observation with the correlation coefficient being 0.9. As a result, even for the high statistical moments (the standard deviation and 95% confidence interval) (see Figure 22), there is good fitness to the results from the observed values of the gridded deviation, because the correlation coefficients between the simulated and observed deviation exceed 0.6.
Combination of areal average and gridded deviation
Furthermore, according to a graphical comparison of storm patterns at the specific grids for various simulated grid-based rainstorm events (see Figure 23), there are similar temporal changes in storm patterns at the six grids randomly selected in the Nankan River watershed. In detail, the six grids concerned exhibit similar types of storm patterns, with different gridded deviations. Especially for the neighboring grids (e.g., Grid185 and Grid188), the temporal varying trend of storm pattern at Grid185 is comparable to that at Grid188, and their corresponding difference is, on average, merely about 0.0016.
Graphical comparison of gridded storm patterns at ten grids concerning the 1st, 100th, and 1,000th simulated events.
Graphical comparison of gridded storm patterns at ten grids concerning the 1st, 100th, and 1,000th simulated events.
To sum up the above results, the proposed SM_GSTR method can take into account the statistical properties of gridded rainstorms, especially for the spatiotemporal correlation, to generate the time series of rainfall at all grids within a study area. Also, the resulting simulations of the gridded rainfall characteristics by the proposed SM_GSTR method are markedly consistent with the results from the observations, except for the standard deviation of the gridded rainfall depth and the upper bound of the 95% confidence interval. As a result, the proposed SM_GSTR method can well capture the behavior in time and space of gridded rainstorms.
Spatial comparison of simulations of gridded rainstorms
Eventually, the generated grid-based rainfall characteristics can be combined as the simulated hyetographs at all grids. Figure 24 illustrates the 1st, 100th, and 1,000th simulated rainstorms at 336 grids in the Nankan River watershed, comprising the generated gridded rainfall characteristics. It can be found that the above four gridded rainstorm events are significantly associated with different changes in space and time. Also, in the case of spatial distribution of hourly gridded rainfall for the 13-hour simulated event as shown in Figure 25, the rainfall reaches the maximum around the 11th hour and the grids with larger rainfall are mostly located near the lower right corner of the study area. It summarizes that the spatial change in the gridded rainstorm simulated by the proposed SM_GBTR model varies with time and resembles the spatial distribution of real rainstorms.
Hyetographs of the 1st, 100th, and 1,000th simulated gridded rainstorm events.
Spatial distribution of hourly rainfalls of the 100th simulated rainstorm event (duration = 13 hours).
Spatial distribution of hourly rainfalls of the 100th simulated rainstorm event (duration = 13 hours).
CONCLUSION
This study aims to present a stochastic method (SM_GSTR) for generating a large number of gridded short-time (hourly) rainstorm events within a watershed/urban area by means of the correlated MMCS approach (Wu et al. 2006). In the proposed SM_GSTR model, the rainstorms at all grids of interest consist of the four spatially and temporally correlated rainfall characteristics, the rainfall duration, gridded rainfall depths, and gridded storm pattern composed of the areal average storm patterns and the associated gridded deviation of storm pattern.
A basin located in north Taiwan, the Nankan River watershed, is selected as the study area and the associated grid-based precipitation data provided by Center Weather Bureau in Taiwan of 20 typhoon events are used to demonstrate its feasibility and applicability, in which the statistical properties of gridded rainfall characteristics are compared to those from the observations. The results from the model feasibility indicate that for the comparison of the statistical properties of gridded rainfall characteristics, the mean values of gridded rainfall depths and the mean and standard deviation of the areal average storm pattern have excellent agreement with those from their observation. Also, the correlation coefficients of the mean value and upper bound of 95% confidence interval of grid-based rainfall characteristics (about 0.8) are quite consistent with those from observations, whereas the correlation coefficient of the standard deviation and lower bound of 95% confidence interval (approximately 0.65) merely differ from observations attributed to an acceptable bias. It is expected that the proposed SM_GSTR can reproduce a large number of the gridded rainfall characteristics, of which the corresponding statistical features approximate those from observations with an acceptable and reasonable bias.
The proposed SM_GSTR model is established by modifying Wu's method under consideration of the temporal and spatial dependence structure of rainstorms with various temporal and spatial scales. However, the effect of short-range scale regarding the fractal behavior of rainfall should impact the variation in the precipitation and induced runoff (e.g., Kantelhardt et al. 2006). Therefore, the effect of multifractal temporal scale on the simulation of storm pattern would be evaluated in future work. Also, due to the overestimated lower bound of the 95% confidence interval regarding the gridded rainfall depths, the proposed SM_GSTR model might have difficulty generating the gridded rainfall depths of light rainstorms. Therefore, further work should be done by improving the SM_GSTR model under consideration of persistence of the standard deviations regarding the gridded rainfall depths.
In addition to improvement regarding the model development, it is well known that climate change impacts the evaluation in the hydrological analysis so strongly as to require more reliable estimates of future precipitation (Olsson et al. 2013). Nevertheless, the proposed SM_GSTR method reproduces a large number of high-resolution rain fields comprising the gridded rainstorm events based on the statistical properties of observed QPE data without considering the effect of the climatic conditions (e.g., temperature, moisture, and CO2) in response to climate change (Doroszkiewicz et al. 2019). Therefore, by integrating with a physically based climatic simulation model, such as the GCMs, the proposed SM_GSTR model would be capable of producing the near-real simulations of the gridded rainstorm events under the consideration of climate change (e.g., Qiang et al. 2020). Furthermore, the proposed SM_GSTR method would be combined with the relevant hydrologic/hydraulic numerical simulation models to reproduce massive rainfall-induced simulation scenarios with high resolution in time and space (e.g., flood and inundation); the resulting gridded rainfall-induced simulations could be applied in training an AI model for flood forecasts.
DATA AVAILABILITY STATEMENT
All relevant data are included in the paper or its Supplementary Information.
