Evaluation of the Applicability of Three Methods for Climatic Spatial Interpolation in the Hengduan Mountains Region

Xiaoming Xu aKey Lab of Land Surface Pattern and Simulation, Institute of Geographic Sciences and Natural Resources Research, Chinese Academy of Sciences, Beijing, China
bUniversity of Chinese Academy of Sciences, Beijing, China

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Xueqin Zhang aKey Lab of Land Surface Pattern and Simulation, Institute of Geographic Sciences and Natural Resources Research, Chinese Academy of Sciences, Beijing, China

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Xiang Li aKey Lab of Land Surface Pattern and Simulation, Institute of Geographic Sciences and Natural Resources Research, Chinese Academy of Sciences, Beijing, China
bUniversity of Chinese Academy of Sciences, Beijing, China

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Abstract

An ideal spatial interpolation approach is indispensable for obtaining high-quality gridded climatic data in mountainous regions with scarce observations, particularly for the Hengduan Mountains Region (HMR) with dense longitudinal ranges and gorges. However, there is much controversy about the applicability of thin plate smooth spline (TPSS), cokriging, and inverse distance weighting (IDW) in mountainous regions. Here, we use the daily observations of temperature and precipitation at 125 stations in HMR and its surroundings from 1961 to 2018 and adopt three interpolation methods to map the annual average temperature and precipitation at a resolution of 500 m in HMR. Then, we assess the applicability of three interpolation methods in HMR from the perspectives of interpolation accuracy and effects. The evaluation implies a satisfactory interpolation accuracy of TPSS with the highest correlation and lowest error, whether for temperature (R2 = 0.92, RMSE = 1.2°C) or precipitation (R2 = 0.54, RMSE = 165.9 mm). In addition, the TPSS could better display the temperature (precipitation) gradient along elevation and depict dry valleys’ high-temperature and low-precipitation characteristics. Moreover, the satisfactory interpolation performance of TPSS mainly benefits from the screening of optimal TPSS model that varied primarily with the regional topography feature and meteorological observation density. The uncertainty of gridded climate datasets has become an urgent problem to solve in the complex terrain. This research illustrates the satisfactory applicability of TPSS for climatic spatial interpolation in HMR, providing theoretical support for high-precision interpolation in complex terrain, hopefully improving the regional weather forecasts and disaster warnings.

© 2022 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

This article is licensed under a Creative Commons Attribution 4.0 license (http://creativecommons.org/licenses/by/4.0/).

Corresponding author: Xueqin Zhang, zhangxq@igsnrr.ac.cn

Abstract

An ideal spatial interpolation approach is indispensable for obtaining high-quality gridded climatic data in mountainous regions with scarce observations, particularly for the Hengduan Mountains Region (HMR) with dense longitudinal ranges and gorges. However, there is much controversy about the applicability of thin plate smooth spline (TPSS), cokriging, and inverse distance weighting (IDW) in mountainous regions. Here, we use the daily observations of temperature and precipitation at 125 stations in HMR and its surroundings from 1961 to 2018 and adopt three interpolation methods to map the annual average temperature and precipitation at a resolution of 500 m in HMR. Then, we assess the applicability of three interpolation methods in HMR from the perspectives of interpolation accuracy and effects. The evaluation implies a satisfactory interpolation accuracy of TPSS with the highest correlation and lowest error, whether for temperature (R2 = 0.92, RMSE = 1.2°C) or precipitation (R2 = 0.54, RMSE = 165.9 mm). In addition, the TPSS could better display the temperature (precipitation) gradient along elevation and depict dry valleys’ high-temperature and low-precipitation characteristics. Moreover, the satisfactory interpolation performance of TPSS mainly benefits from the screening of optimal TPSS model that varied primarily with the regional topography feature and meteorological observation density. The uncertainty of gridded climate datasets has become an urgent problem to solve in the complex terrain. This research illustrates the satisfactory applicability of TPSS for climatic spatial interpolation in HMR, providing theoretical support for high-precision interpolation in complex terrain, hopefully improving the regional weather forecasts and disaster warnings.

© 2022 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

This article is licensed under a Creative Commons Attribution 4.0 license (http://creativecommons.org/licenses/by/4.0/).

Corresponding author: Xueqin Zhang, zhangxq@igsnrr.ac.cn

1. Introduction

Accurate gridded climate datasets are vital for understanding climate change and its impact on ecosystem services, hydrological processes, and natural disasters in mountainous regions with complex terrain and sparse ground observations. Temperature and precipitation are crucial drivers of climatic (Boucher et al. 2020; Wrzesien et al. 2018) and ecohydrological modeling (Beck et al. 2017; Dong et al. 2015; Du et al. 2012). Especially in mountainous regions with fragile environments and frequent geological disasters, accurate temperature and precipitation data are of vital significance for comprehending the mechanisms and processes of environmental evolution and then preventing and mitigating natural calamities (Khan et al. 2021; Kirschbaum et al. 2020; Pavlova et al. 2014; Stoffel et al. 2014). The primary sources of continuous spatial temperature and precipitation data are satellite products [e.g., Global Precipitation Climatology Project (Adler et al. 2017) and land surface temperature data from MODIS (Long et al. 2020)], interpolated data from ground observations [e.g., gridded surface air temperature homogenized dataset for China basing on thin plate spline interpolation (Xu et al. 2020)], and reanalysis data combing multisource observations and atmospheric model [e.g., fifth-generation reanalysis product of the European Centre for Medium-Range Weather Forecasts (Jiang et al. 2021)]. Although the satellite products have the advantage of directly obtaining spatial continuous climate data through the statistical relationship between remote sensing data and ground observations, their quality is significantly affected by topographical (Lin et al. 2016; Lu et al. 2019; Maggioni et al. 2016), weather (Rahman et al. 2018), and climate (Bai et al. 2018) conditions. Moreover, compared with ground observations, the ability of satellite-derived data is weaker in expressing extreme events that are crucial triggers of mountain disasters (Nastos et al. 2013; Nepal et al. 2021). For example, Integrated Multi-satellitE Retrievals for the Global Precipitation Measurement (GPM) mission (IMERG) and Tropical Rainfall Measuring Mission (TRMM) Multisatellite Precipitation Analysis 3B42 (TMPA), version 7, both overestimate the maximum daily rainfall in the warm season over Qinghai–Tibet Plateau (Ma et al. 2016). In contrast, the reanalysis data provide the best estimation of the atmospheric features based on the numerical weather prediction models rather than the actual atmospheric situation (Parker 2016). Besides, reanalysis data face enormous challenges in monitoring near-surface temperature (Scherrer 2020) and precipitation (Sun et al. 2018) in alpine regions. Thus, ground observing is still the most reliable method for mountain climate monitoring. However, the scarce observation stations pose enormous challenges for climatic spatial interpolation in mountainous regions with highly complex terrain, particularly for the Hengduan Mountains Region (HMR), characterized by dense longitudinal towering mountain ranges and deep gorges. Therefore, adopting a suitable spatial interpolation approach to gridding the relatively short and scarce surface observations is imperative.

Spatial interpolation refers to using known points’ values to estimate the unknown points, which can be classified into deterministic and geostatistical methods (Chang 2018). The deterministic interpolation methods estimate the values of unknown points directly based on the known surrounding points, including trend surface analysis (Bailey and Gatrell 1995; Davis 2002), Thiessen polygons (Thiessen 1911), inverse distance weighting (IDW; Shepard 1968), and kernel density estimation (Davis et al. 2011; Parzen 1962). Based on a stochastic model, geostatistical interpolation provides not only the optimal estimate (i.e., the deterministic part) but also the associated errors (i.e., the stochastic part, the variance of estimates) (Amini et al. 2019; Frazier et al. 2016; Wameling 2003), whereas deterministic interpolation only provides the former part. Geostatistical interpolation methods mainly refer to kriging and its derivatives (Burrough et al. 2015; Lucas et al. 2022), containing ordinary kriging (Cressie 1988), simple kriging (Webster and McBratney 1987), universal kriging (Armstrong 1984; Zimmerman et al. 1999). IDW is one of the most popular spatial interpolation methods (Li and Heap 2011). Although the interpolation accuracy is affected by the observation density and the topographic features, IDW is commonly employed for climatic spatial interpolation in regions with scarce observation due to its convenience and effectiveness (Yang et al. 2020; Longman et al. 2019). The theoretical basis of spatial interpolation methods is Tobler’s First Law of Geography: “everything is related to everything else, but near things are more related than distant things” (Tobler 1970). Based on spatial autocorrelation, some climate factors’ values are also controlled by other geographical factors, such as elevation (Thakuri et al. 2019; Kattel and Yao 2018; Navarro-Serrano et al. 2020), the distance to the coastline (DTC) (Buttafuoco and Lucà 2020; Chao et al. 2018; Makarieva et al. 2009), and slope and aspect (Kidron 2005). Hence, methods taking environmental factors as auxiliary variables are developed, such as cokriging and TPSS (Keller and Borkowski 2019). As the multivariate variant of ordinary kriging, cokriging makes predictions by adopting second or more variables (e.g., elevation) highly correlated with dependent variables in case of poor sampling and weak spatial autocorrelation of dependent variables (Krajewski 1987). As for the thin plate smooth spline (TPSS), one commonly used interpolation method adds multiple linear submodels based on the thin plate spline function. Consequently, based on TPSS, Hutchinson (1995) developed the Australian National University Spline (ANUSPLIN) software package for climatic spatial interpolation. ANUSPLIN is especially suitable for spatial interpolating climate elements in a long time series because it could generate multiple surfaces of climate factors simultaneously (Herrera et al. 2019; Serrano-Notivoli and Tejedor 2021).

Nevertheless, accurate climatic spatial interpolation remains an enormous challenge for regions with highly intricate terrain, diversified climate systems, and sparse ground observations (Camera et al. 2014; Saghafian and Bondarabadi 2008; Shan et al. 2020; Suparta and Rahman 2016). In particular, the climatic spatial interpolation in HMR has been a tremendous enduring challenge due to the complex topography and steep climatic gradients. Located in the transition zone between the Qinghai–Tibet Plateau (QTP) and the Sichuan basin, the HMR is a steep zone for climate, vegetation, and geomorphic features. In addition, the sparse meteorological observation stations mainly distributed in valleys could not fully represent the complexity and diversity of the climate in HMR. Furthermore, the HMR is rich in hydropower resources and has many major hydropower stations (e.g., Baihetan, Xiluodu, and Wudongde projects), attributed to the vast altitude difference and the dense staggered huge longitudinal ranges–valleys. In addition, other infrastructures, including the Sichuan–Tibet Railway under construction, are confronted with the frequent occurrence of diversified mountain hazards. Consequently, both scientific research and practical application demand high-resolution climate data.

Present research in HMR has adopted different climatic interpolation methods, such as IDW (Yu et al. 2019), kriging (Xu et al. 2022), spline (Zhu et al. 2012), and TPSS (Yin et al. 2020). However, the applicability of these spatial interpolation methods in HMR remains unclear. Hence, we assess the interpolation performance for temperature and precipitation of three climatic spatial interpolation methods (i.e., TPSS, cokriging, and IDW) in HMR with extremely complex terrain from the two perspectives of interpolation accuracy and effects, where the interpolation effects contain the capability of depicting temperature (precipitation) gradient with elevation and typical landscape (i.e., dry valleys).

2. Data and methods

a. Study area

Here we focus on the HMR located between 24.7° and 34.1°N and 96.6° and 104.7°E, covering an area of 4.4 × 105 km2 (Fig. 1). The vast majority of the HMR lies within Sichuan Province, Yunnan Province, and Tibet Autonomous Region, China, with areas accounting for 56.3%, 23.1%, and 18.3%, respectively. The rest 2.4% of the regions are scattered in Qinghai and Gansu provinces, China, and Myanmar, with a total area of 1.05 × 104 km2. More than 10 ethnic groups gather in HMR, including Tibet, Yi, Qiang, Bai, Tujia, Miao, Hui, Naxi, and Han nationalities.

Fig. 1.
Fig. 1.

Location and topography of HMR, the spatial distribution of meteorological observation stations in HMR (pink dots) and its surroundings (cyan dots), and vertical profile of elevation from northwest to southeast (from the starting point P to the ending point P′). The apogee of the HMR is sited in the Gongga Mountains. The boundary of QTP divides the HMR into two sections: IHQ (intersection of HMR and QTP, northwest of the thick red line) and AOQ (area in HMR but out of QTP, southeast of the thick red line).

Citation: Journal of Hydrometeorology 24, 1; 10.1175/JHM-D-22-0039.1

The HMR includes two geomorphic units: the intersection of HMR and QTP (denoted IHQ) and the area in HMR but out of QTP (denoted AOQ), distributed to the northwest and the southeast of the thick red line in Fig. 1, respectively. The HMR has a highly complex terrain with an average altitude of 3434 m above mean sea level (MSL) and a maximum elevation difference of 7206 m. The maximum elevation of 7556 m MSL in HMR appears in the Gongga Mountains. Compared with less than 10% in AOQ, areas with an elevation over 3000 m MSL account for 83.3% in IHQ. The vast elevation drop results in the belt around the QTP boundary being a steep zone of climate, vegetation type, and geological conditions. Towering mountains and deep valleys crisscross longitudinally from west to east in HMR, mainly including the Boshula Mountain Range–Gaoligong Mountains, Nu River, Taniantaweng Mountains–Nu Mountains, Lancang River, Mangkang Mountains–Yun Range, Jinsha River, Queer Mountains–Shaluli Mountains, Yalong River, Daxue Mountains–Xiaoxiang Range–Lunan Mountains, Dadu River, Qionglai Mountains–Jiajin Mountains–Daxiang Range, Min River, and Min Mountains.

The unique topographic features of HMR lead to a complex and diverse spatial distribution of temperature and precipitation. Deep longitudinal valleys are crucial channels for the humid airflow from south to north, while the north–south tall mountains are enormous barriers to the water vapor input from the east and west sides. Apart from the combined influence of the East Asian monsoon and the South Asian monsoon, the HMR is frequently affected by multiple mesoscale systems, including southwest vortex, Tibetan Plateau vortex, shear line, and mesoscale convective systems (MCS). These mesoscale systems’ generation, development, and movement are generally accompanied by rainfall, even precipitation extremes that are important trigger factors of mountain disasters.

The complex and diverse climate leads to various vegetation types in HMR. Sorted by coverage area, the main vegetation types in HMR include scrub (35.7%), coniferous forest (25.8%), meadow (15.7%), broadleaf forest (7.8%), cultural vegetation (4.9%), grass–forb community (4.8%), alpine vegetation (4.3%), and others (1.0%).

b. Data

1) Daily temperature and precipitation observations

We utilize a daily dataset of surface climate observation in China provided by the China Meteorological Administration (CMA; http://data.cma.cn/site/index.html). This climate dataset has undergone strict quality control, with the potential outliers removed. We extract the daily temperature and precipitation records of 73 meteorological stations in HMR and 52 in its surroundings, covering the period of 1961–2018. The 125 meteorological stations have relatively complete climate observed records, with the proportion of missing temperature and precipitation records only 3.2% and 4.2%. The missing values of the daily temperature (precipitation) were replaced by the multiyear mean values of the same daily sequence. Based on the filled climate records, we aggregated the annual temperature and precipitation by averaging the daily temperature and summing the daily precipitation, followed by calculating the multiyear mean temperature and precipitation values.

The horizontal and vertical distribution of meteorological stations is uneven. The density of observation stations is 3.2 per 104 km2 in AOQ and 1.1 per 104 km2 in IHQ (Fig. 1). Only two meteorological stations are over 4000 m MSL, named Luolong (56223 in WMO Integrated Global Observing System) and Shiqu (56038).

2) Altitude and coastline data

Except for latitude and longitude, the two indispensable independent variables of geographic information required in the spatial interpolation, we supplement two other independent variables, that is, altitude and DTC, that significantly impact the spatial climate distribution in HMR. First, we extract the altitude data from the Global Multiresolution Terrain Elevation Data (GMTED2010; https://topotools.cr.usgs.gov/gmted_viewer/viewer.htm), developed jointly by the U.S. Geological Survey (USGS) and the National Geospatial-Intelligence Agency (NGA). GMTED2010 is a widely used elevation dataset for global- and regional-scale applications because of its advantages in consistency and vertical accuracy over the previous generation of GTOPO30 (Sadeghi et al. 2021; Amatulli et al. 2018). The dataset has three spatial resolutions of 30, 15, and 7.5 arc s (about 1000, 500, and 250 m), respectively.

We obtain the DTC by calculating the Euclidean distance (Danielsson 1980) from the coastline in ArcGIS 10.7 (https://desktop.arcgis.com/en/arcmap/10.7/tools/spatial-analyst-toolbox/understanding-euclidean-distance-analysis.htm). The coastline data come from 1:1 000 000 National Basic Geographic Database (https://www.webmap.cn/commres.do?method=result100W) provided by the National Catalogue Service for Geographic Information. The overall current status of the coastline data is 2015, using the 2000 national geocoordinate system and the 1985 national elevation benchmark.

We resample the spatial resolution of the altitude and DTC to 500 m for unification. Correspondingly, the spatial resolutions of the interpolated temperature and precipitation surfaces are 500 m.

c. Methodology

1) Interpolation methods

For the spatial interpolation of temperature and precipitation in HMR, we perform TPSS with ANUSPLIN 4.3 (https://fennerschool.anu.edu.au/research/products/anusplin) and perform cokriging and IDW with ArcGIS 10.7 (https://desktop.arcgis.com/zh-cn/arcmap/10.7/get-started/setup/arcgis-desktop-quick-start-guide.htm).

(i) TPSS based on ANUSPLIN 4.3
TPSS can visualize the complex relationships between continuous predictors and response variables. ANUSPLIN is a suite of Fortran programs to calculate and optimize TPSS, suitable for dealing with datasets with unlimited climate stations (Hutchinson 1998). A general TPSS model is given by
Zi=f(xi)+j=1ρβiΨi(xi)+εi(i=1,2,,n;j=1,2,,ρ),
where Zi is the estimated value at the position of xi; f is an unknown smooth spline function to be estimated; the Ψi are a set of ρ known functions; the βi are a set of unknown parameters to be estimated.
(ii) Cokriging
Cokriging is a multivariable valuation method based on kriging. It adds auxiliary information highly correlated with dependent variables as covariates into kriging. Cokriging considers the spatial autocorrelation of dependent variables and the correlation between covariates and dependent variables. The theoretical model of cokriging is
Z(S0)=i=0NaiZ(Si)+j=0MbjY(Sj),
where Z(S0) is the interpolation result at the unknown point S0; Si and Sj are the positions of observation points of independent variables and covariates around S0; Z(Si) and Y(Sj) are the measured values of the independent variable Si and covariate Sj, respectively; ai and bj are the unknown weight of independent variable and covariate; N and M are the numbers of independent variables and covariate measurements around point S0.
(iii) IDW
IDW estimates values of unknown points by averaging nearby known points with distance-based weights (Shepard 1968). Its interpolation function is as follows:
Z(x0)=i=1ndiuZ(xi)i=1ndiu,
where Z(x0) is the estimation at the interpolated point x0; Z(xi) is the observation at a data point xi; n is the total number of the nearby known points; di is the distance between the measured point xi and the interpolated point x0; u is the power parameter deciding how fast the weight decreases with the distance.

2) Screening of optimal TPSS spatial interpolation model based on ANUSPLIN 4.3

This study uses four geographic factors of longitude, latitude, elevation, and DTC as independent variables to predict the spatial distribution of temperature and precipitation in HMR. The independent spline variables are fundamental and essential, while the covariates are optional. So, fixing the longitude and latitude as independent spline variables and utilizing elevation and DTC as independent spline variables or covariates, we design nine climatic spatial interpolation schemes in line with different combinations (Table 1).

Table 1

Scheme design of spatial interpolation based on ANUSPLIN 4.3.

Table 1

To screen the optimal TPSS climatic spatial interpolation model, we first debug the optimal attribute values of interpolation parameters (i.e., spline order, dependent variable conversion mode, and smoothing directive) based on the error values in the log files of ANUSPLIN 4.3. The ANUSPLIN provides two smoothing directives of generalization of the maximum likelihood (GML) and generalized cross validation (GCV). The judgments for the optimal values of interpolation parameters include (i) the fitting degree of freedom (signal) is less than half of the number of observation points, and (ii) the GCV is the smallest. Second, we calculate and compare the nine schemes’ error indicators [see section 2c(3) for details] to determine the optimal TPSS spatial interpolation model in HMR.

3) Cross validation

We utilize cross validation to objectively compare the interpolation accuracy of the three spatial interpolation methods. In a cross-validation procedure, partial data points are extracted as interest points, with the rest points estimating the value of these interest points (Picard and Cook 1984). Meanwhile, this procedure will be repeated several times to ensure the randomness of sampling.

Our study randomly samples 20 interest points from the 73 data points located in HMR. The sampling repeats 3 times, obtaining 60 interest points finally. In detail, we first assign each data point in HMR with a random number in the range of 0–1, which is generated by the “random” (RAND) function in Microsoft Excel. Second, we sort these 73 random numbers from the smallest to the largest. Then we sample the data points corresponding to the first 20 random numbers as interest points for the first experiment. Therefore, the data points corresponding to the 21–40 and 41–60 random numbers are sampled as interest points for the second and third experiments. This process ensures the randomness and nonrepetitiveness of sampling. For each experiment, we adopt three spatial interpolation methods (i.e., TPSS, cokriging, and IDW) to estimate the values of 20 interest points, using the remaining 105 data points (53 points in HMR and 52 points in its surroundings).

We evaluate the interpolation performance of three interpolation methods using five statistical metrics: 1) the Pearson correlation coefficient (CC), 2) the determination coefficient (R2), 3) the mean bias (MB), 4) the mean absolute error (MAE), and 5) the root-mean-square error (RMSE). The CC measures the linear correlation between the observations and predictions. The R2 statistically calculates the approximation of predictions to the observations. The closer the values of CC and R2 are to 1, the higher accuracy of the interpolation models. The MB represents the systematic bias of three interpolated methods, and the MAE and RMSE are used to assess the average magnitude of errors. The formulas of the five statistical metrics are as follows:
CC=i=1n(xix¯)(xix¯)i=1n(xix¯)2i=1n(xix¯)2,
R2=1i=1n(xixi)2i=1n(xix¯)2,
MB=1ni=1n(xixi),
MAE=1ni=1n|xixi|, and
RMSE=1ni=1n(xixi)2,
where xi represents the observation value, x¯ is the mean value of observations, xi means the estimated value, x¯ is the mean value of estimations, and n is the number of calculated points.

4) Comparison of interpolation effects

We analyze the interpolation effects of TPSS, cokriging, and IDW by comparing their depiction of temperature (precipitation) gradient with elevation and illustration of typical landscape (i.e., dry valleys). First, we map the annual average temperature (precipitation) interpolated by TPSS, cokriging, and IDW in HMR and enlarge them in the Shaluli Mountains to investigate the climate gradient from the floor to the top of the mountain. Then, using the Raster Calculator tool in ArcGIS 10.7, we calculate the estimated temperature (precipitation) differences between cokriging and IDW with TPSS in HMR, exploring the reasons behind their different interpolation performance.

According to the Comprehensive Scientific Expedition to the Qinghai–Tibet Plateau conducted by the Chinese Academy of Sciences during 1981–84, the boundaries of dry valleys in HMR were demarcated. It is well known that the dry valleys are characterized by higher temperatures and less precipitation than the surroundings. Therefore, we calculate the regional average value, maximum value, minimum value, and standard deviation (SDEV) of temperature (precipitation) in dry valleys to compare the applicability of TPSS, cokriging, and IDW in dry valleys. The formula of SDEV is as follows:
SDEV=1n1i=1n(xix¯)2.

3. Results

a. Optimal TPSS climatic spatial interpolation model

Based on the optimal attribute values of interpolation parameters, we obtained the optimal TPSS model by comparing five statistical metrics of the nine interpolation schemes in HMR (Table 2). Among the nine temperature interpolation schemes in HMR, the scheme I (shown in Table 1) had the best performance (CC = 0.99 and p < 0.01, R2 = 0.99, bias = 0°C, MAE = 0.4°C, and RMSE = 0.5°C). As for the precipitation interpolation, scheme B (see Table 1 for details) performed best among the nine interpolation schemes (CC = 0.94 and p < 0.01, R2 = 0.89, bias = −2.3 mm, MAE = 69.8 mm, and RMSE = 102.5 mm). In particular, the optimal TPSS spatial interpolation model for annual average temperature in HMR was a cubic spline function using longitude, latitude, elevation, and DTC as independent spline variables, with GML as the smoothing directive. On the other hand, the optimal interpolation model for precipitation was a cubic spline function with longitude and latitude as independent spline variables, elevation as the independent covariate, GCV as the smoothing directive, and square root as the dependent variable transformation.

Table 2

Five statistical metrics (CC, R2, MB, MAE, and RMSE) of nine TPSS spatial interpolation schemes for annual average temperature and precipitation in HMR. The symbol “**” means the correlation between the observations and predictions is statistically significant at the 1% significance level (p < 0.01). The optimal interpolated schemes with the maximum values of CC and R2 and the minimum values of MB, MAE, and RMSE are highlighted in bold.

Table 2

b. Comparison of interpolation performance

To objectively evaluate the applicability of TPSS, cokriging, and IDW for climatic spatial interpolation in HMR, we compared their performance in the spatial interpolation of annual average temperature and precipitation from two aspects: 1) quantitative evaluation of interpolation accuracy through five statistical metrics (CC, R2, MB, MAE, and RMSE); and 2) comparison of interpolation effects involving the depiction of the climatic gradient along elevation and typical natural landscape of dry valleys.

1) Interpolation accuracy

Regarding interpolation accuracy, TPSS performed better than cokriging and IDW for the annual average temperature and precipitation in HMR. We plotted scatterplots based on the observed and predicted temperature (precipitation), and then calculated CC, R2, MB, MAE, and RMSE to evaluate the interpolation accuracy of TPSS, cokriging, and IDW (Fig. 2). TPSS produced estimations of temperature with the highest correlation (CC = 0.96 and R2 = 0.73) and the lowest error (MAE = 0.8°C and RMSE = 1.2°C) among the three interpolation methods. However, TPSS slightly underestimated the annual average temperature with the MB of −0.2°C. Cokriging and IDW overestimated the annual average temperature with the MB of 0.3 and 0.5°C. As for the interpolation of annual average precipitation, TPSS (CC = 0.73, R2 = 0.54, MB = 0.8 mm, MAE = 123.6 mm, and RMSE = 165.9 mm) also outperformed cokriging and IDW.

Fig. 2.
Fig. 2.

Evaluation of the interpolation accuracy of TPSS (red points), cokriging (blue points), and IDW (green points) for annual average (a)–(c) temperature (marked by dots) and (d)–(f) precipitation (marked by triangles) in HMR. The scatterplots plot the predicted temperature or precipitation (vertical axis) against their observations (horizontal axis), where the black diagonal lines (1:1 line) represents a perfect fit for the observations. Five statistical metrics are calculated, involving CC, R2, MB, MAE, and RMSE.

Citation: Journal of Hydrometeorology 24, 1; 10.1175/JHM-D-22-0039.1

To evaluate the spatial interpolation capability of TPSS, cokriging, and IDW, we calculated the MB, MAE, and RMSE of the three interpolation methods in two subregions (i.e., the IHQ and AOQ, see more details in Fig. 1) (Table 3). Then we mapped the absolute errors of annual average temperature and precipitation during 1961–2018 at 60 points in HMR (Fig. 3). For the annual average temperature interpolation, TPSS, cokriging, and IDW all had less MAE and RMSE in flat terrain (i.e., AOQ) than in complex topography (i.e., IHQ). Besides, TPSS had the highest temperature interpolation accuracy among the three methods, with its MAE and RMSE less than half of cokriging and IDW in IHQ or AOQ. TPSS underestimated the annual average temperature in IHQ with a mean bias of −0.4°C. On the contrary, cokriging and IDW overestimated the annual average temperature in IHQ (MB = 1.0°C for cokriging and 1.1°C for IDW) and underestimated in AOQ (MB = −0.6°C for cokriging and −0.2°C for IDW). As for the interpolation for the annual average precipitation, TPSS similarly performed better than cokriging and IDW with lower MAE and RMSE in IHQ and AOQ. In addition, TPSS slightly overestimated the annual average precipitation in IHQ with the mean bias of 5.2 mm and underestimated the precipitation in AOQ with the mean bias of −4.2 mm, the absolute value of which was only sixth of cokriging and IDW.

Fig. 3.
Fig. 3.

Absolute errors of TPSS (green bars), cokriging (yellow bars), and IDW (red bars) interpolation for annual average (a) temperature and (b) precipitation at 60 interest points in HMR from 1961 to 2018.

Citation: Journal of Hydrometeorology 24, 1; 10.1175/JHM-D-22-0039.1

Table 3

Error metrics (MB, MAE, and RMSE) for the annual average temperature and precipitation interpolated by TPSS, cokriging, and IDW in IHQ and AOQ, where the IHQ is the intersection of the QTP and the HMR and the AOQ is the areas in the HMR but out of the QTP (see Fig. 1 for details).

Table 3

2) Interpolation effects

We assessed the interpolation effects of TPSS, cokriging, and IDW by comparing their capability to depict the temperature (precipitation) gradient with elevation in the hillslope scale and to illustrate the characteristics of the typical natural landscape in HMR (i.e., the dry valleys).

(i) Temperature and precipitation gradient along elevation

The temperature surfaces interpolated by TPSS, cokriging, and IDW showed that the annual average temperature decreased from the southeast to the northwest in HMR (Figs. 4a–c), which is consistent with the increasing elevation in the same direction. Therefore, all three interpolation methods reflected the temperature distribution pattern in HMR, that is, temperature decreases with the increasing elevation from the southeast to the northwest on the regional scale.

Fig. 4.
Fig. 4.

Annual average temperature interpolated by (a) TPSS, (b) cokriging, and (c) IDW in HMR from 1961 to 2018. The (d) elevation (extracted from Fig. 1) and annual average temperature interpolated by (e) TPSS, (f) cokriging, and (g) IDW in the northern Shaluli Mountains (the area circled by the purple dotted line) exemplify the differences in the capability of three methods to depict the temperature gradient along elevation in complex terrain. In (e)–(g) the enlarged temperature distributions extracted from (a)–(c) are shown.

Citation: Journal of Hydrometeorology 24, 1; 10.1175/JHM-D-22-0039.1

Besides, the TPSS interpolation could well exhibit the vertical temperature gradient with elevation at the hillslope scale, in sharp contrast with cokriging and IDW failing to reflect (Figs. 4d–g). We enlarged the elevation (Fig. 4d) and the interpolated temperature (Figs. 4e–g) in the northern Shaluli Mountains (the area marked by the dashed purple circle in Figs. 4a–c), roughly located at 29°–32°N and 97°–99°E, between the upper reaches of the Jinsha River and the Yalong River. The temperature interpolated by TPSS decreased from the floor to the top of the northern Shaluli Mountains with an approximate range from −6° to 9°C. In contrast, the cokriging and IDW could not display such a vertical temperature gradient with elevation. Furthermore, TPSS illustrated a steep temperature variation from IHQ to AOQ, with the regional average temperature of 2.5° and 11.7°C, respectively. The temperature difference between IHQ and AOQ of TPSS was 9.2°C, larger than cokriging (6.2°C) and IDW (6.3°C).

To explore the reasons for the better temperature interpolation performance of TPSS than cokriging and IDW, we analyzed the spatial distribution of their differences by taking the temperature surface interpolated by TPSS as the baseline and subtracting it from that generated by cokriging (TCokringing−TPSS) and IDW (TIDW−TPSS) (Fig. 5). The pixels with the positive values were painted with warm colors (red), meaning the temperature estimated by cokriging (IDW) was higher than TPSS. The pixels with negative values were painted with cold colors (blue), representing the estimation of cokriging (IDW) less than TPSS.

Fig. 5.
Fig. 5.

The differences between the temperature interpolated by TPSS with (a) cokriging (TCokriging−TPSS) and (b) IDW (TIDW−TPSS). The pixels with positive values are painted with warm colors (red), while the other pixels with negative values are painted with cold colors (blue). Region A, circled by the black dashed line, is around the Gongga Mountains, with the maximum temperature estimation differences of cokriging and IDW. Regions B and C, marked by the green dashed lines are valleys with negative estimation differences in temperature. Region B is located in the Anning River Basin, and region C is situated near the Lijiang section of the Jinsha River.

Citation: Journal of Hydrometeorology 24, 1; 10.1175/JHM-D-22-0039.1

The results indicated that the cokriging and IDW overestimated the annual average temperature compared with TPSS in most HMR, especially in IHQ (Figs. 4 and 5). The regional average temperature in HMR estimated by cokriging (10.8°C) and IDW (11.0°C) was twice as high as TPSS (4.8°C). The estimation differences for temperature between TPSS with cokriging (6.8°C) and IDW (7.0°C) were even more remarkable over IHQ, nearly 1.7 times that over AOQ (3.8° and 4.1°C, respectively) (Table 4). In addition, the positive estimation differences of temperature increased with the elevation, with a maximum over 15.0°C around the Gongga Mountains, the summit of the HMR (region A in Fig. 5). Although the temperature estimated by cokriging and IDW was larger than TPSS in most HMR, the estimation differences (i.e., TCokriging−TPSS and TIDW−TPSS) were negative in valleys and their surroundings, such as regions B and C in Fig. 5. Therefore, taking the temperature surface interpolated by TPSS as the baseline, the cokriging and IDW overestimated the temperature in high mountains and underestimated it in valleys with low elevation. Such estimation differences between cokriging and IDW with TPSS overshadowed the hillslope scale’s vertical temperature gradient along elevation.

Table 4

Regional average temperature (precipitation) and estimated deviations of TPSS, IDW, and cokriging over IHQ, AOQ, and HMR.

Table 4

As for the precipitation interpolation, the TPSS, cokriging, and IDW displayed the consistent spatial distribution of precipitation in HMR at the regional scale (Figs. 6a–c). The HMR had more precipitation in the south than the north, roughly bounded by 29°N. There was more rainfall on the two flanks than in the middle area south of 29°N, opposing the spatial distribution in the area north of 29°N. In addition, the precipitation estimated by TPSS increased with the elevation at the hillslope scale (Figs. 6d,e). This precipitation gradient with elevation was not reflected by cokriging and IDW interpolation (Figs. 6f,g).

Fig. 6.
Fig. 6.

Annual average precipitation interpolated by (a) TPSS, (b) cokriging, and (c) IDW in HMR from 1961 to 2018. The (d) elevation and annual average precipitation interpolated by (e) TPSS, (f) cokriging, and (g) IDW in the southern Shaluli Mountains (the area circled by the purple dotted line) exemplify the differences in the capability of three methods to depict the vertical precipitation gradient in complex terrain. The enlarged elevation distribution in the southern Shaluli Mountains extracted from Fig. 1 is shown in (d). In (e)–(g) the enlarged precipitation distributions extracted from Figs. 4a–c are shown.

Citation: Journal of Hydrometeorology 24, 1; 10.1175/JHM-D-22-0039.1

Taking the precipitation interpolated by TPSS as the baseline, the cokriging and IDW underestimated the precipitation (i.e., PCokriging/IDW−TPSS < 0) in most HMR (Fig. 7). The regional average precipitation in HMR interpolated by cokriging (763.2 mm) and IDW (767.2 mm) was 80.3 and 76.3 mm less than TPSS, respectively (Table 4). However, the estimation differences of precipitation were negative in region D–F and valleys with lower elevations (Fig. 7). It is worth noting that the maximum estimation differences of precipitation interpolation were detected in IHQ, whether for cokriging or IDW. Moreover, regions with the most positive estimation differences roughly corresponded to areas with low precipitation, while the regions with negative estimation differences were spatially consistent with high precipitation over IHQ. Therefore, compared with TPSS, cokriging and IDW weakened the spatial discrepancy of precipitation by overestimating the minimum and underestimating the maximum over IHQ.

Fig. 7.
Fig. 7.

The differences between the precipitation interpolated by TPSS with (a) cokriging (PCokriging−TPSS) and (b) IDW (PIDW−TPSS). The pixels with negative values (PCokriging/IDW−TPSS < 0) are painted with warm colors (orange) and pixels with positive values (PCokriging/IDW−TPSS > 0) are painted with cold colors (blue). The regions D–F enclosed by red dashed lines are areas with positive estimation differences for precipitation of cokriging and IDW. Region D lies to the south of the Shaluli Mountains. Region E and F are located in the upper reaches of the Nu River and Min River, respectively.

Citation: Journal of Hydrometeorology 24, 1; 10.1175/JHM-D-22-0039.1

(ii) Depiction of the typical natural landscape: Dry valleys

Contrasting the spatial distribution of dry valleys with the temperature and precipitation surfaces interpolated by TPSS, cokriging, and IDW, we found that TPSS interpolation better reflects the characteristics of dry valleys than the other two methods (Fig. 8). Take the III 3–1 of the Jinsha River (Batang and Detuo section), III 3–2 of the Lancang River (Yanjing section), and III 3–3 of the Nu River (Nujiang Bridge section) as examples. The TPSS interpolation demonstrated higher temperature and lower precipitation in valleys than their surroundings (Fig. 8c), which was not shown by cokriging and IDW interpolation (Figs. 8d,–e).

Fig. 8.
Fig. 8.

Comparison of the capability for (a)–(c) TPSS, (d) cokriging, and (e) IDW to depict the typical features of dry valleys, i.e., higher temperature and less precipitation than their surroundings. The boundary and codes of dry valleys are extracted from Yang (2000). The comparison is magnified and displayed in dry valleys III 3–1, III 3–2, and III 3–1, the upper reaches of the Jinsha River, the Lancang River, and the Nu River, respectively in (c)–(e).

Citation: Journal of Hydrometeorology 24, 1; 10.1175/JHM-D-22-0039.1

Comparing the mean temperature and precipitation interpolated by three methods in dry valleys III 3–1, III 3–2, and III 3–3 (Table 5), we discovered that the TPSS had a higher temperature and lower precipitation (10.4°C and 511.6 mm, respectively) than cokriging (9.5°C and 575.4 mm, respectively) and IDW (9.5°C and 547.9 mm, respectively). In addition, the TPSS interpolation reveals greater climatic regional variability with larger interval length (i.e., maximum–minimum) and SDEV of temperature and precipitation than cokriging and IDW in complex terrain (Table 5).

Table 5

Statistical summary of annual average temperature and precipitation interpolated by TPSS, cokriging, and IDW in dry valleys III 3–1, III 3–2, and III 3–3 from 1961 to 2018.

Table 5

Except for the statistical analysis for the regional average, we also plotted the vertical profiles of temperature and precipitation interpolated by TPSS, cokriging, and IDW across one valley located in the upper reaches of the Nu River, comparing their interpolation capability on a small scale (Fig. 9). We found a significant negative correlation (p < 0.01) between elevation and temperature interpolated by TPSS and IDW [Fig. 9c(1)], and a significant positive correlation (p < 0.01) between elevation and precipitation interpolated by TPSS [Fig. 9c(2)].

Fig. 9.
Fig. 9.

Comparison of interpolation performance of three methods in a section of dry valleys located in the upper reaches of the Nu River, Lie River, and Yu River, which is marked with a red dashed line between two red pins in Figs. 8a and 8b. (a) Backgrounds: (a1) elevation, (a2) annual average temperature, and (a3) precipitation; (b) satellite images derived from Google Earth Pro (https://earth.google.com/); (c) vertical profiles of annual average (c1) temperature and (c2) precipitation interpolated by TPSS (red lines), cokriging (green lines), and IDW (purple lines) from the starting point to the ending point.

Citation: Journal of Hydrometeorology 24, 1; 10.1175/JHM-D-22-0039.1

4. Discussion

Regional climate conditions play an important role in theoretical studies and the operational applications of weather and disasters. However, the gridded climate datasets generally contain much uncertainty (Newman et al. 2019), which comes from the source of observations, the density of gauges, data homogenization, choice of interpolation methods, and methods of gap filling. However, reliably interpolating climate data will significantly improve weather forecasts and disaster warnings in complex terrain areas. Hence, our study illustrates the satisfactory applicability of TPSS for climatic spatial interpolation in complex terrain. We screened the optimal TPSS model in MRB and HMR to explore factors influencing the optimal TPSS model in different regions. The MRB and HMR are close geographically, with 51.8% of the former (3.1 × 104 km2) located within the latter (Fig. 10). Furthermore, the spatial interpolation in HMR and MRB share the temperature and precipitation data at 22 meteorological stations. Nevertheless, the optimal TPSS models were remarkably different in independent spline variables, covariates, the transformation of the dependent variable, and the smoothing directives at MRB and HMR (Table 6).

Fig. 10.
Fig. 10.

Spatial distribution of 125 (blue and green dots) and 47 (orange and red pentagrams) meteorological observation stations in and around HMR and MRB, respectively.

Citation: Journal of Hydrometeorology 24, 1; 10.1175/JHM-D-22-0039.1

Table 6

Comparison of optimal TPSS spatial interpolation model in HMR and MRB.

Table 6

We try to explain the differences between the two optimal TPSS models by comparing the features of topography and meteorological stations in HMR and MRB (Fig. 10). Despite the neighborhood of HMR and MRB, they have an apparent discrepancy in topographic features. The average elevation of HMR (3435.8 m MSL) is nearly 1.5 times that of MRB (2005.2 m MSL). For the area in MRB but out of HMR, the mean elevation is only 707.4 m MSL, much lower than the part located within HMR (3212.1 m MSL). The area with an elevation over 3000 m MSL in HMR accounts for 67.5%, contrasting with only 35.0% in MRB. In addition, MRB has a higher density of meteorological stations (5.37 per 1000 km2) than the HMR (1.67 per 1000 km2). Therefore, we deduce that the optimal TPSS interpolation model is susceptible to terrain and the density of meteorological stations.

Apart from elevation, other topographic factors such as slope, aspect, and exposure also have a substantial impact on the spatial distribution of temperature (Bindajam et al. 2020; Saavedra et al. 2020; Stewart and Nitschke 2017) and precipitation (Kumari et al. 2017; Tang et al. 2018). Therefore, we added slope and aspect as independent spline variables or covariates to grid temperature and precipitation in HMR. Although adding two independent variables did not improve the interpolation accuracy in our experiment, we cannot infer that the influence of slope and aspect is neglectable in the climatic spatial interpolation in HMR. The reason should be that the meteorological stations are often located in areas with open terrain and a slight slope. More than half the meteorological stations have a slope lower than 5° in HMR. Thus, intensive observations are essential to explore the influence of topographical factors on the spatial climate distribution in HMR.

5. Conclusions

We evaluate the climatic spatial interpolation capability of TPSS, cokriging, and IDW from a new perspective of estimation accuracy and interpolation effects in HMR with extremely complex terrain and scarce gauges. On the one hand, TPSS has superior interpolation accuracy than cokriging and IDW, with higher correlation (CC and R2) and more minor errors (MB, MRE, and RMSE), whether for temperature or precipitation. Despite the poorer interpolation behavior of the three methods in mountainous regions than lowlands, TPSS has higher interpolation accuracy in mountainous regions than cokriging and IDW in flat areas. On the other hand, TPSS vividly mirrors the vertical temperature (precipitation) gradient along elevation and the unique natural landscape of dry valleys characterized by high temperature and low precipitation in HMR. TPSS illustrates larger climatic regional differentiation between IHQ and AOQ than cokriging and IDW. Compared with TPSS interpolation, the cokriging and IDW overestimate the temperature with the regional mean deviation of 6.0° and 6.2°C, respectively, and underestimate the precipitation with −80.3 and −76.3 mm, respectively. However, both methods estimate lower temperature and higher precipitation in valleys, considerably overshadowing the features of dry valleys in HMR.

Our work indicates that the satisfactory climatic spatial interpolation performance of TPSS is closely related to the screening of the optimal TPSS models varying with the topographic feature and density of ground gauges. Therefore, partitioning large-scale areas could help enhance climatic spatial interpolation accuracy by considering meteorological observation gauges’ terrain and spatial distribution. Meanwhile, our exploration in the evaluation perspective of interpolation performance suggests that it is indispensable in future research to take the typical natural landscape into account in mountainous regions with complex terrain and scarce gauges. Our empirical study in HMR will provide practical and constructive spatial interpolation methods for improving climatic model accuracy, precisely nowcasting extreme weather events, and scientifically preventing and mitigating mountain disasters at a regional and global scale.

Acknowledgments.

This work was supported by the Strategic Priority Research Program of the Chinese Academy of Sciences (XDA23090302) and the Second Tibetan Plateau Scientific Expedition and Research Program (STEP) (2019QZKK0903).

Data availability statement.

The datasets used in this study are freely available as follows: The daily climate data from the China Meteorological Administration are available at http://data.cma.cn/site/index.html; the DEM data (GMTED2010) from the U.S. Geological Survey (USGS) and the National Geospatial-Intelligence Agency (NGA) are available at https://topotools.cr.usgs.gov/gmted_viewer/viewer.htm; the coastline data from the National Catalogue Service for Geographic Information are available at https://www.webmap.cn/commres.do?method=result100W.

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