Adaptive Statistical Spatial Downscaling of Precipitation Supported by High-Resolution Atmospheric Simulation Data for Mountainous Areas of Nepal

1.   Introduction

With the progress of land surface process research, land simulations are being developed toward high resolutions. Generally, hydrological and ecological models require kilometer-scale resolutions (Ji and Yuan, 2018; Yuan et al., 2018); for some special surfaces (glaciers, lakes, etc.), hyper-resolution of 100 m or even 10 m is required. A main limitation of current hyper-resolution simulations is the lack of reliable scale-matching forcing data, especially in high-altitude mountainous areas (Immerzeel et al., 2014). Among commonly used atmospheric forcing data, precipitation is influenced by multiple processes and has high spatial and temporal variability. Meanwhile, precipitation greatly affects the water and energy cycles, and thus, its accuracy is most concerned in land hydrological modeling.

At present, most gridded precipitation datasets have a resolution of 10 km or coarser, and their errors are large in mountainous areas. Grid datasets based on station interpolation may have great uncertainties due to the sparse distribution of stations in complex terrain areas and their poor spatial representation. Satellite remote sensing, on the other hand, generally underestimates heavy precipitation and overestimates little precipitation, and cannot well reflect the characteristics of orographic precipitation as well as retrieve solid precipitation (e.g., Gao and Liu, 2013; Anjum et al., 2019; Wang et al., 2019). Intensive observations may be available in low altitudes of complex terrain regions, but satellite precipitation products calibrated against these station observations may still much underestimate precipitation amount in high altitudes (Jiang et al., 2022). Atmospheric reanalysis data can represent large-scale precipitation characteristics but cannot show fine-scale precipitation variability in complex terrain due to their coarse resolutions. Therefore, there is a need to develop dynamical or statistical downscaling methods for precipitation data so as to achieve hyper-resolution and high accuracy in mountainous areas.

Dynamical downscaling of reanalysis data by regional climate models shows that increasing the resolution of atmospheric models can better simulate small- and medium-scale processes and significantly improve the accuracy of precipitation simulations (Gao et al., 2015; Lin et al., 2018). Maussion et al. (2014) used Weather Research and Forecasting (WRF) model to downscale the NCEP final analysis (NCEP-FNL) in high mountain Asia (HAR), which reproduces the observed spatial and seasonal characteristics of precipitation. Compared with remotely sensed precipitation data, HAR can reflect the high amount of solid precipitation in winter and spring in Karakorum (Li et al., 2020). Many studies have found that convection-permitting modeling can better reflect precipitation amount, frequency, intensity, spatial variability (Collier and Immerzeel, 2015; Lin et al., 2018; Gao et al., 2020; Ou et al., 2020; Wang et al., 2020a; Li et al., 2021), and precipitation recycling ratio (Zhao et al., 2021). Numerous studies have demonstrated that high-resolution atmospheric simulations have achieved considerable accuracy, even surpassing observational networks in terms of monitoring snowfall in mountainous areas (Lundquist et al., 2019) and can be used to estimate the spatial variability of precipitation in complex terrain (Ouyang et al., 2021). In recent years, hyper-resolution land–atmosphere coupled simulation has also been tried (Collier et al., 2013; Bonekamp et al., 2019). However, atmospheric simulations at kilometer scale are very computationally intensive and difficult to run on a large domain for long periods, and may even lead to model crash at kilometer-resolution or finer in regions with large altitudinal gradients (Steppeler et al., 2002). Therefore, there is a need to develop computationally efficient methods to downscale data from a coarser resolution to a finer resolution of hundreds or even tens of meters.

Statistical downscaling has been widely studied and developed because of its computational efficiency in producing high-resolution data. For precipitation, the complexity varies among different downscaling methods, including empirical statistical and physical statistical models (Pandey et al., 2000; Liston and Elder, 2006). Empirical statistical models establish the relationship between forecast values and observed values through linear or nonlinear regression (Kilsby et al., 1998; Beckmann and Adri Buishand, 2002; Tareghian and Rasmussen, 2013). Among nonlinear regression methods, machine learning has become popular for downscaling in recent years because of its high efficiency and the ability to integrate multiple-source information (Mei et al., 2020; Wang et al., 2020b; Hong et al., 2021; Zhang et al., 2021), but it is not constrained by atmospheric dynamical processes. Jiang et al. (2021) trained a machine-learning model with reanalysis data as the input and a high-resolution simulated precipitation dataset as the target to obtain long-term high-resolution precipitation data, but the resolution cannot surpass the target data (~3 km in this study). Therefore, combining atmospheric dynamical processes and simultaneously reducing the dependence on observations may contribute to the development of a new statistical downscaling method.

The purpose of this study is to establish the statistical relationship between precipitation and topography for the statistical downscaling method of MicroMet (Liston and Elder, 2006). The idea is to use high-resolution atmospheric simulation data instead of station data to determine a key parameter in MicroMet so as to improve the accuracy of the downscaling results. The paper is organized as follows: Section 2 introduces the study area and the data used. Section 3 describes the downscaling method and the validation experiment in details. Section 4 presents the results of the validation experiments. Section 5 discusses on the scale dependence of the key parameter and the factors affecting downscaling accuracy. The results are summarized in Section 6.

2.   Study area and datasets

2.1   Study area

In this study, Nepal is taken as the research area to verify the effectiveness of the precipitation downscaling model. Nepal is located on the southern slope of the Central Himalayas, 80% for mountainous areas and 20% for plain areas. The topography is highly undulating, with altitudes from the lowest 60 m above sea level (ASL) to the highest 8848 m ASL within a horizontal range of 150–200 km from north to south (Fig. 1). During the summer months (June–September), the climate is under the influence of the South Asian monsoon and receives a large proportion of the annual precipitation. In winter, precipitation amount is small from December to February, and snowfall is dominant in the high-altitude mountainous areas. The rest of the year is hot and dry.

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Fig  1.  Topography and observing station distribution in Nepal. Blue and black dots indicate stations below and above 2500 m ASL, respectively.

2.2   Datasets

2.2.1   Observed data

In this study, observed precipitation data are used for the parameter estimation, model input, and validation of downscaling results (details in Section 3.2). Daily precipitation observed at stations were obtained from the Department of Hydrology and Meteorology (DHM), Government of Nepal. The station data are relatively rich compared to many mountainous regions. There are a total of 524 stations with above 90% below 2500 m ASL and less than 10% above 2500 m ASL in Fig. 1. Observations at some stations are not regularly conducted, and some data missing at remote stations due to maintenance difficulties. After quality control, only the stations with missing values no more than 5 days in a month were selected and their observations were processed into monthly mean precipitation rate. In this study, we use the data during 2013–2015 to support the establishment of the new downscaling method.

2.2.2   High-resolution atmospheric simulation data

In this paper, we use high-resolution simulation data but not station data to estimate the precipitation adjustment factor (PAF). The high-resolution simulated data used are obtained through dynamically downscaling the ECMWF reanalysis version 5 (ERA5) data (Hersbach et al., 2020) by WRF model with a grid spacing of approximately 3 km for the Tibetan Plateau (TP) and its surroundings (Zhou et al., 2021). The downscaling results outperform the ERA5 and HAR v2 in term of smaller biases and root-mean-square errors (RMSEs) as well as higher spatial pattern correlation coefficients (CCs) for precipitation. Furthermore, it more realistically reproduces observed diurnal variation characteristics of precipitation in the interior TP. Because of huge computational load, the high-resolution simulation data are now available only for 2013 and 2018. In this study, the simulation data in 2013 are used to estimate a key parameter in MicroMet.

2.2.3   Topography data

Topography data are important input to the MicroMet model. The grid altitudes for the 3-km WRF and 1-km target grids are produced by area-weighted interpolation of the USGS (US Geographical Survey)_30s (~900 m) altitude data.

3.   Statistical downscaling algorithm and validation experiment design

This section first introduces the precipitation downscaling algorithm in MicroMet model, and points out the potential disadvantages in the algorithm when applied to mountainous region. Then, an approach using high-resolution simulation data to determine a key parameter of MicroMet is proposed.

3.1   Precipitation downscaling algorithm in MicroMet model

MicroMet is a downscaling model used widely (Pan et al., 2014; Mernild et al., 2017; Zhang et al., 2020). It is an intermediate-complexity, quasi-physically based meteorological model used to produce high-resolution meteorological forcing data, including temperature, relative humidity, wind direction and speed, radiation, surface pressure, and precipitation (Liston and Elder, 2006). For precipitation downscaling, the model requires at least one available precipitation data within or near the area as input for each time step. The input data can be either station data or gridded data at a coarse resolution and the output results are spatial gridded data with a target resolution (usually a fine resolution). There are two steps for the downscaling.

First, the model interpolates the altitude and precipitation data at input stations, respectively, by the Barnes objective analysis scheme (Barnes, 1964; Koch et al., 1983) to form the gridded data at the target resolution. The gridded altitude (${z}_{0}$) and precipitation (${P}_{0}$) at the target resolution are a reference of the downscaling and will be used in the second. Note that the gridded altitude is different from actual altitude. The scheme uses a Gaussian distance weighting function to calculate the weight of an input point ($w$) as follows,

where $r$ is the distance between that input point to the interpolated grid, $f\left(\Delta n\right)$ defines a filter parameter that determines the smoothness of the interpolation result with its specific function below,

where $\Delta n$ is the average spacing of the input data (Koch et al., 1983).

Second, the gridded precipitation from the first step is corrected with altitude effect. The gridded altitude (${z}_{0}$) is different from the actual altitude of the grid ($z$), so the gridded precipitation is corrected with an adjustment factor to account for the altitude difference by the nonlinear function below,

where $P$ (mm day⁻¹) is the corrected precipitation rate, $z$ (km) is the real altitude in the target grid, and $X$ (km⁻¹) is the so-called precipitation adjustment factor (hereafter PAF). In general, the distribution of ${P}_{0}$ is smoother than that of $P$ because the altitude at each grid is not used in Step 1 but its effect is accounted in Step 2.

Figure 2 shows the relationship between the correction rate of interpolated precipitation [the bracketed part in Eq. (3)] and the altitude difference $z$ − ${z}_{0}$ for different PAFs. When PAF is positive, precipitation increases with increasing altitude; and vice versa. The larger the absolute value of PAF, the larger the variability of the correction rate with altitude difference. As a key parameter of the MicroMet, PAF varies in different seasons and geographical locations.

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Fig  2.  Schematic diagram of the relationship between precipitation correction rate and altitude difference for different PAFs (denoted by $X$; Liston and Elder, 2006).

The value of PAF is estimated based on observation data (Thornton et al., 1997). Given observed precipitation ${p}_{i}$ and ${p}_{j}$ at two stations with altitudes ${z}_{i}$ and ${z}_{j}$, the statistical relationship below can be derived according to Eq. (4),

Therefore, $X$ is obtained through optimization with the least square method by minimizing the cost function below,

where ${y}_{n}= \dfrac{{p}_{i}-{p}_{j}}{{p}_{i}+{p}_{j}}$, ${x}_{n}={z}_{i}-{z}_{j}$, $n$ is a certain pair of stations as any two stations can form a pair, and $m$ is the total number of the pairs of stations.

In the MicroMet model, the default values are derived through minimizing Eq. (5) based on a network of observations in the northwestern United States (Thornton et al., 1997). The default values may not be applicable to other regions. Even though the PAF can be user-defined in the model, it is still difficult to determine the PAF in regions with sparse stations. Since the PAF varies with geographical location, a single value of PAF also faces applicability problems when downscaling for a large domain.

3.2   PAF estimation method

The main difficulty with MicroMet is to determine the PAF, especially for mountainous areas. There are two challenges: one is the lack of station data for estimating the parameter in mountainous areas, and the other is yet unclear about the spatial scale of the parameter variation.

In this study, Nepal is selected as study area, where the terrain is complex and the stations are relatively dense. We propose to use high-resolution atmospheric simulation data to determine the PAF. Although high-resolution simulation results may have systematic biases compared with observations, they can roughly reflect the statistical characteristics of precipitation in complex terrain (Ouyang et al., 2021). Therefore, in this study, a high-resolution simulated precipitation in Zhou et al. (2021) is used to estimate the PAF. This parameter is estimated by dividing the area at a certain resolution to obtain the spatial distribution of the PAF at the corresponding resolution. The gridded parameter values are stored in an external input file of MicroMet. Meanwhile, the MicroMet model is modified so that it can call the corresponding PAF values according to individual areas.

To investigate the importance of spatial variation in the PAF, we compare the downscaling results for two scenarios: one assumes a single value for the PAF across the whole region, and the other assumes spatially varying PAFs, as detailed below. Both PAFs in the two scenarios are derived based on monthly mean precipitation rates in July–August–September (JAS) of 2013. The estimated PAF is applied to 2013–2015 in order to investigate the interannual transferability of PAF.

(1) A single value for the PAF across the whole region. Three single-value schemes are used, namely, the default value (De) of MicroMet model, the estimated one based on station data (St) according to Eq. (5), and the estimated one with the WRF-based high-resolution simulation data at all WRF grids (Wa) in Nepal. To calculate Wa, the precipitation and altitude in Eq. (5) are the monthly mean values at individual WRF grids. Note that the St value is not true since this estimation depends on the density of stations.

(2) Spatially varying PAFs. This estimation was only applied to WRF simulation data because limited station data cannot support the calculation of the varying PAFs. The PAF is estimated by using the precipitation data of WRF grids at 6 resolutions of 0.3°, 0.9°, 1.2°, 1.5°, 3.0°, and 4.5°, respectively; then, we evaluated which resolution is reasonable for the PAF estimation. Every resolution uses WRF simulation data at all grids within each square window to estimate PAF. The estimated value is only applied to the trisecting center of the window, and the spatial distribution of the PAFs for the whole region was obtained by sliding calculation window, as shown in Fig. 3.

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Fig  3.  Schematic diagram of sliding estimation for the PAF with WRF simulation data. The red dashed box is the calculation grid for a specific resolution, the blue dashed box indicates an eastward moving grid for the estimation, and the green dashed box indicates a southward grid for the estimation, and the estimated value of each grid is applied to the trisecting area (shaded) of the grid.

3.3   Downscaling validation method

MicroMet downscaling results depend on both PAF and input precipitation data. It is desirable to input as many station data as possible to ensure the downscaling accuracy. Meanwhile, a portion of station data have to be retained as independent validation data to verify the rationality of the interpolation. Therefore, cross-validation method is used. First, the observation stations were divided into input and validation stations in the ratio of 9 : 1 for each month (the numbers of input and validation stations are list in Table 1) through random sampling. Then, the observed precipitation at the input stations was downscaled to a 1-km grid by MicroMet using the PAF estimated by different schemes above, and the downscaling accuracy was evaluated by comparing the precipitation values between the observed one at the independent validation stations and the downscaled one at collocated grids. The RMSE and spatial CC between observed and interpolated precipitation are calculated for each random sampling. This process is repeated 10 times (hereafter 10 replicates). The averaged RMSE and CC of the 10 replicates are used to determine the optimal PAF estimation scheme.

Table  1.  Statistics on the numbers of input and validation stations in JAS 2013

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Station	July	August	September
Input	255	247	252
Validation	29	28	29

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4.   Results

This section compares the differences in downscaling results for different PAF scenarios. Based on the spatial variability of the PAF and early precipitation studies in Nepal, the optimal spatial scale for estimating PAF is determined and interpreted. We further analyzed the inter-annual transferability of PAF estimates.

4.1   Estimates of single PAF and relevant downscaling results

The values of the three single PAF schemes for JAS 2013 are shown in Table 2. The three schemes are introduced in Section 3.2. The Wa values are similar to the De values in MicroMet model for the three months, but they differ significantly from the St values.

Table 3 shows the RMSE and CC of the downscaled results in each month of JAS and their mean using each of the three single PAF schemes. The results using the PAF estimated with station data are better in terms of both RMSE and CC than the other two. This indicates that the station data are crucial for the reasonable estimation of a single PAF. The poor performance of the WRF-estimated single PAF may imply that the high-resolution simulation data cannot show the large-scale statistical characteristics of precipitation in this region.

Table  2.  PAF (km⁻¹) estimates through the three single-value schemes. “De” is derived from a network of observations in the northwestern United States (Liston and Elder, 2006), “St” is estimated from station data in Nepal, and “Wa” is derived from WRF-3km simulation data at all grids for Nepal

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Month	De	St	Wa
July	0.2	0.11	0.16
August	0.2	0.08	0.23
September	0.2	−0.01	0.28

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Table  3.  RMSE (mm day⁻¹) and CC for downscaling by using each of the three single-value PAF schemes in JAS 2013 and their mean (the bold numbers are optimal values)

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Period	Metric	De	St	Wa
July	RMSE	8.17	6.95	7.49
	CC	0.57	0.63	0.60
August	RMSE	7.45	6.15	8.16
	CC	0.51	0.58	0.48
September	RMSE	4.76	4.01	5.45
	CC	0.42	0.51	0.38
Mean	RMSE	6.79	5.70	7.03
	CC	0.50	0.57	0.49

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4.2   Estimates of spatially varying PAF and relevant downscaling results

The estimation method of spatially varying PAF has been introduced in Section 3.2. Figure 4 shows the spatial distribution of the estimated PAF at different resolutions for July 2013. The spatial distribution of the PAF varies significantly at different resolutions. At 0.3° resolution (Fig. 4a), the PAF in Nepal is predominantly positive. When the resolution gets coarse to 0.9°, negative values arise and dominate the area above 2500 m ASL (Fig. 4b), which obviously follows the topographic features. When the resolution is between 0.9° and 1.5° (Figs. 4c, d), the factor values above 2500 m ASL are almost negative, and their spatial features from 0.9° to 1.5° resolutions are roughly the same. As the scale continues to increase, the areas with negative values begin to expand to lower altitudes and cover over almost all Nepal (Figs. 4e, f). A number of early studies on the Himalayas show that precipitation increases/decreases below/above 2500 m ASL, respectively (Singh and Kumar, 1997; Bookhagen and Burbank, 2006; Ichiyanagi et al., 2007; Salerno et al., 2015; Yang et al., 2018; Chen et al., 2021; Lin et al., 2021), which supports the spatial variation of the PAF with 0.9°–1.5° resolutions. This implies that an interpolation with the spatially varying PAF with 0.9°–1.5° resolutions may produce reasonable downscaling results.

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Fig  4.  Spatial distributions of the estimated PAF at each spatial resolution in Nepal region in July 2013. The first number in the subtitle of each panel is the resolution used for the PAF calculation and the second number is the resolution that the value is used in Fig. 3. The PAF calculation is based on WRF-3km simulation data. The green line is the contour of 2500 m ASL.

Table 4 shows the mean RMSEs and spatial CCs of the precipitation downscaling results in JAS 2013 and their mean using PAFs estimated at different resolutions. Generally, the downscaling results with the spatially varying PAF estimates in the range 0.9°–3° perform better than downscaling at scales that are too small (0.3°) or too large (4.5°). This is further shown in Fig. 5.

Table  4.  RMSE (mm day⁻¹) and CC for precipitation downscaling by using the spatially varying PAF determined at different resolutions in JAS 2013 and their mean (the bold numbers are optimal values) using station data as input

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Period	Metric	WRF-0.3°	WRF-0.9°	WRF-1.2°	WRF-1.5°	WRF-3°	WRF-4.5°
July	RMSE	8.42	6.51	6.56	6.58	6.64	6.72
	CC	0.56	0.67	0.67	0.66	0.65	0.65
August	RMSE	8.64	5.76	5.86	5.81	5.94	6.07
	CC	0.44	0.62	0.59	0.59	0.57	0.56
September	RMSE	6.41	4.23	4.04	4.06	3.98	4.07
	CC	0.28	0.47	0.50	0.49	0.51	0.49
Mean	RMSE	7.82	5.50	5.49	5.48	5.52	5.62
	CC	0.43	0.59	0.59	0.58	0.58	0.57

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Fig  5.  Variations of (a, c) RMSE (mm day⁻¹) and (b, d) CC of precipitation downscaling results with resolution for the PAF estimation during (a, b) season (JAS; s1–s10 denote the 10 replicates) and in (c, d) each month of JAS 2013, but averaged over the 10 replicates. The evaluation is based on validation stations (see Table 1 for the number of validation stations).

Figures 5a and 5b show the variation of RMSEs and CCs of the downscaling results with the resolution of the PAF estimation for the JAS in 2013, respectively. The results of the 10 validation replicates, with each replicate having different input stations and validation stations determined randomly (see Section 3.3), are shown. It can be seen that the RMSE and CC among different replicates vary obviously, indicating impacts of spatial variability of precipitation at the stations on the evaluation. Nevertheless, RMSE for all samplings decreases and CC increases significantly as the resolution changes from 0.3° to 0.9°, RMSE increases and CC decreases slightly as the resolution changes from 3° to 4.5°, while RMSE and CC change little in the resolution range of 0.9°–3°.

Similarly, Figs. 5c and 5d show the variations of RMSE and CC varying with resolution, but the results are averaged over the 10 replicates in each month of JAS 2013, respectively. It can be seen that the variations of RMSE and CC with resolution have similar characteristics in different months. The optimal resolution is between 0.9° and 3°. For convenience, the results of the sliding estimate of 1.5° are suggested in practical applications of the model.

Comparing between the results using the single-value PAF (Table 3) and the spatially varying PAF (Table 4), we can see that the precipitation downscaling results with the spatially varying PAF estimates in the range 0.9°–3° perform comparable to or even better than that with the St single-value (i.e., PAF estimated from dense station data) and clearly better (less RMSE and higher CC for JAS) than that with the De and Wa single-value scenario (see Table 3). Therefore, the WRF high-resolution simulation data can be used to estimate the spatially varying PAF if an area of interest has scarce observations, and the spatially varying PAF scheme can better reflect the differences of local precipitation characteristics and thus more effectively improve the accuracy of the downscaling results.

Figure 6 shows the relative difference between the precipitation data downscaled by using the spatially varying PAF at 1.5° (1.5°_0.5°) and the St PAF. All station data are used as input in the downscaling. As shown in Fig. 6, there is no significant difference between the two downscaled results in the area below 2500 m ASL, while the results above 2500 m ASL are significantly different. This is because the fact that precipitation above 2500 m ASL decreases with altitude is shown in the WRF-3km data-based PAF but not in the station data-based PAF. Therefore, when a focused area is large and the stations are sparse, a single PAF has obvious disadvantage in reflecting the spatial variation of precipitation. The PAF determined at 0.9°–3° resolutions can better characterize its spatial feature.

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Fig  6.  Relative difference (%) between the precipitation downscaling results using the PAF estimated with WRF-3km simulation data (1.5°_0.5°) and that estimated from the station data in JAS 2013. The brown line is the contour of 2500 m ASL.

4.3   Interannual transferability of PAF

The results in Section 4.2 shows that the performance of spatially varying PAF estimated from high-resolution atmospheric simulation data greatly supports the downscaling. It is important to clarify the interannual transferability of the PAF, so the PAF estimated in 2013 was applied to the downscaling in 2014 and 2015. Figure 7 shows the comparison of the three-month mean RMSE and CC for the downscaling result among different PAF schemes. Because the results with the spatially varying PAF estimation in the range 0.9°–3° perform similarly, which is the same as 2013, Fig. 7 only shows the evaluation results for three spatially varying PAFs (WRF-0.3°, WRF-1.5°, and WRF-4.5°) and three single PAFs (De, St, and Wa).

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Fig  7.  Three-month mean (a, c) RMSE (mm day⁻¹) and (b, d) CC for precipitation downscaling in JAS (a, b) 2014 and (c, d) 2015 using the PAF estimated with station data or WRF-3km simulation data in 2013.

As shown in Fig. 7, the result with spatially varying PAF at 1.5° has a slightly higher RMSE and a slightly lower CC than that with St PAF, but it still performs better than the other schemes. To understand the interannual variability of PAF, Table 5 shows the comparison of the single value PAF estimated from station data during 2013–2015. It can be seen that the station-based PAF values change little in 3 yr. The spatially varying PAF representing the altitudinal dependence of precipitation within a sub-region (or grid) may have greater interannual variability than a single-value PAF representing that in a region. As a result, when PAF values determined for 1 yr are used for another year, the former may perform better in precipitation downscaling than the latter. Nevertheless, their error metrics have small differences (less than 0.2 in RMSE and 0.02 in CC), which justifies the reasonability with the different spatial PAFs.

Table  5.  Single-value PAF (km⁻¹) estimated with station data during 2013–2015

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Year	July	August	September
2013	0.11	0.08	−0.01
2014	0.11	0.05	0.10
2015	0.02	0.05	0.02

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5.   Discussion

5.1   Scale-dependence of the PAF

As shown in Fig. 4a, the 0.3°_0.1° scheme yields positive PAF in the area above 2500 m, which is significantly different from other schemes with coarser resolutions. This indicates that the PAF is scale-related. Thus, we speculate that the positive PAF from the 0.3°_0.1° scheme in the area above 2500 m may reflect the topographic variation of precipitation at small scales, while the 1.5°_0.5° scheme reflects the topographic variation of precipitation at larger scales.

In order to explore the scale effect on PAF, three areas with a negative value of PAF at 1.5°_0.5° scheme (Fig. 8a) and three areas with a positive value of PAF at 0.3°_0.1° scheme (Fig. 8b) were selected in western, central, and eastern Nepal, respectively. The observed precipitation at the stations within each area and the simulated precipitation at the collocated WRF-3km grids were used to estimate the PAFs separately.

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Fig  8.  Scale-dependence PAF over (a) three 1.5° × 1.5° areas and (b) three 0.3° × 0.3° areas. The background is the PAF distribution estimated under two resolutions with observation stations denoted by black dots.

Table 6 shows the PAFs estimated from the station data and from the simulated precipitation at the collocated grids at two scales, respectively. The values of the PAF estimated by the two types of data are relatively close to each other in the three 1.5° × 1.5° areas, but more different to each other in the three 0.3° × 0.3° areas. There are many aspects that may lead to the differences in the estimated PAFs, such as that the WRF-3km simulation is not sufficient to statistically reflect the distribution of precipitation at small scales, or that the small number of stations and the small altitude span between the stations within a small area may lead to a certain randomness in the calculated values. A further discussion on this issue requires very dense observations and is beyond the scope of this study.

Table  6.  Estimated PAFs (km⁻¹) at three 1.5° × 1.5° areas in Fig. 8a and three 0.3° × 0.3° areas in Fig. 8b. The estimation is based on station data and collocated WRF-3km grid data

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Resolution	Data	West	Middle	East
1.5° × 1.5°	Station	−0.10	−0.17	−0.05
	WRF grid	−0.11	−0.13	−0.22
0.3° × 0.3°	Station	0.18	−0.47	0.12
	WRF grid	0.34	0.26	1.12

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In a word, based on high-resolution atmospheric simulation results, the estimates of the PAF are reasonable for a large scale (~100 km or above) but need further investigations for a smaller scale (~30 km or below).

5.2   Factors affecting the downscaling accuracy

In addition to PAF, the number and position of input stations also affect the accuracy of downscaling. To explain this, we repeat the cross-validation experiment with the ratio of input and validation stations changing from 9 : 1 to 8 : 2. Figure 9 compares the JAS mean RMSE (Fig. 9a) and CC (Fig. 9b) for downscaling in 2013, with the two ratios. It can be seen that the result with spatially varying PAF at an optimal resolution still has smaller RMSE and higher CC than that with a single PAF at the ratio of 8 : 2. The optimal resolution for spatially varying PAF is between 0.9° and 3° and is the same as the ratio of 9 : 1, indicating that our conclusion is not sensitive to the ratio.

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Fig  9.  (a) RMSE (mm day⁻¹) and (b) CC for precipitation downscaling in JAS 2013 at two ratios (9 : 1 vs 8 : 2) of input stations to validation stations in the cross-validation.

Nevertheless, RMSE gets slightly larger at the ratio of 8 : 2 than that at the ratio of 9 : 1, because of fewer input stations in the downscaling. There are 255 and 227 input stations at the ratios of 8 : 2 and 9 : 1, respectively . Figure 10 shows the difference of downscaling results between two samplings at the ratios of 9 : 1 and 8 : 2, respectively, in July 2013. The common input stations at the two cases, as denoted by the dots are shown in Fig. 10a, and their exclusive input stations, as denoted by triangles for 9 : 1 and squares for 8 : 2, are shown in Fig. 10b. It shows that the areas with relatively small differences are close to common stations while the areas with relatively big differences are close to exclusive stations (mainly belonging to the case of 9 : 1), indicating that the downscaling accuracy highly depends on the number of input stations. This is not surprising, because PAF mainly explains the precipitation difference caused by the altitude difference, but it cannot overcome the downscaling error caused by the insufficient number of stations.

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Fig  10.  Differences between precipitation downscaling results of two samplings at the ratio of (a) 9 : 1 and (b) 8 : 2 in July 2013. The dots in (a) indicate common input stations of the two samplings. The triangles and squares in (b) indicate exclusive stations for 9 : 1 and 8 : 2.

6.   Summary

Understanding precipitation distribution in complex-terrain regions is an important topic in mountain hydrology, but it is extremely difficult to obtain reliable data. MicroMet is a widely-used semi-physically based statistical downscaling model. A key parameter in the model is the PAF of precipitation, which reflects the altitude dependence of precipitation. It is hard to estimate this parameter in complex-terrain regions due to sparse observations. In this study, we propose to estimate its value based on high-resolution dynamic downscaling results. In principle, the estimation through this way does not rely on station data and thus is not limited by regions and scales. Therefore, this is an adaptive method for precipitation interpolation or downscaling in complex-terrain regions. The effectiveness of this method is demonstrated in Nepal Himalaya, where the terrain is very complex.

We evaluated different schemes for estimating the PAF. The results demonstrate that high-resolution simulated precipitation data can be used as an alternative to observed data to estimate the PAF. The optimal resolution for the estimation in Nepal is about 1.5°, at which the estimated PAF value is approximately consistent with the characteristics of precipitation varying with altitude observed in early studies in this region. That is, precipitation in Nepal increases with altitude below 2500 m ASL and then decreases with altitude above 2500 m ASL. When the estimated PAF is used to downscale precipitation in Nepal, the downscaling results have higher accuracy than the results using the default PAF values and perform comparable to or better than those using the single PAF determined from the relatively dense station data.

We further discussed the scale effect of the PAF. It is revealed that the PAF estimated by station data and simulated data shows significant differences in small domains (below 30 km) but is similar in large domains (~100 km and above), indicating that the estimation of the PAF from high-resolution simulation data is applicable at least for such large domains. A further analysis shows that the number and distribution of input stations are crucial to the downscaling and PAF cannot fully overcome the downscaling error caused by the insufficient number of stations.

In summary, high-resolution atmospheric simulation data can be used to support statistical downscaling of observed precipitation data, which can achieve acceptable accuracy. In the future, this statistical downscaling method may be used for hyper-resolution statistical downscaling, which is crucial for hydrological studies in complex mountainous watersheds, especially for glacier mass balance modeling studies.