INTERNATIONAL JOURNAL OF CLIMATOLOGY Int. J. Climatol. 34: 3745–3751 (2014) Published online 12 February 2014 in Wiley Online Library (wileyonlinelibrary.com) DOI: 10.1002/joc.3941
Comparison of interpolation methods for estimating spatial distribution of precipitation in Ontario, Canada S. Wang,a G. H. Huang,a* Q. G. Lin,b Z. Li,a H. Zhangc and Y. R. Fana b
a Faculty of Engineering and Applied Science, University of Regina, Canada MOE Key Laboratory of Regional Energy and Environmental Systems Optimization, Resources and Environmental Research Academy, North China Electric Power University, Beijing, China c Department of Environmental Earth System Science, Stanford University, CA, USA
ABSTRACT: In this study, different interpolation techniques in a geographical information system (GIS) environment are analysed and compared for estimating the spatial distribution of precipitation in the province of Ontario, Canada. A high-resolution regional climate modelling system [Providing Regional Climates for Impacts Studies (PRECIS)] is used to simulate the present (1961–1990) and future (2071–2100) precipitation events for 12 meteorological stations over Ontario. The results verify that for the present case PRECIS simulates well the precipitation events when compared with observed data. The future precipitation events can be projected after the validation of PRECIS. Six interpolation methods are then used to generate spatial distribution of precipitation based on the projections of future precipitation of 12 meteorological stations; they include inverse distance weighting (IDW), global polynomial interpolation (GPI), local polynomial interpolation (LPI), radial basis functions (RBF), ordinary kriging (OK), and universal kriging (UK). Cross-validation is applied to evaluate the accuracy of interpolation methods in terms of the root mean square error (RMSE). The results indicate that LPI is the optimal method with the least RMSE for interpolating the PRECIS precipitation. LPI is then used to analyse spatial variations of the average annual precipitation for the period of 2071–2100 over Ontario. KEY WORDS
geographical information systems; geostatistical method; interpolation; precipitation; regional climate model; spatial variation
Received 9 September 2013; Revised 13 December 2013; Accepted 7 January 2014
1. Introduction Precipitation is one of the major climate variables. Accurate estimation of precipitation is essential for a number of applications such as natural resource management, agriculture management, ecosystem modelling, and hydrological modelling (Ashiq et al., 2009). Understanding the temporal and spatial distribution of precipitation is also a prerequisite to conduct climate change impact studies (Busuioc et al., 2001). Providing Regional Climates for Impacts Studies (PRECIS) is a regional climate modelling system that can help generate high-resolution precipitation data for any region of the world. Over the past decade, PRECIS has been extensively used for simulating the spatial distribution of precipitation in various regions (Zhang et al., 2006; Cao et al., 2007; Akhtar et al., 2008; Nazrul Islam et al., 2008; Marengo et al., 2009; Urrutia and Vuille, 2009; Campbell et al., 2011; Hadjikakou et al., 2011; Nikulin et al., 2012). The inherent variability in atmospheric processes and uncertainty in the modelling cannot be fully captured by the PRECIS model, resulting in * Correspondence to: G. H. Huang, Faculty of Engineering and Applied Science, University of Regina, Regina, Saskatchewan S4S 0A2, Canada. E-mail:
[email protected]
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difficulty using the PRECIS outputs directly for further applications. It is thus necessary to validate the PRECIS outputs with the observed precipitation data in terms of its ability to reproduce the general precipitation patterns over the study area. The precipitation data are usually available at certain locations; it is difficult and expensive to acquire spatial continuous data for the majority of locations, such as for mountainous and deep marine regions. Spatial interpolation techniques are thus essential for creating a continuous (or prediction) surface from sampled point values. In the past, a variety of interpolation techniques were studied for mapping climatic variables (Price et al., 2000; Vicente-Serrano et al., 2003; Hijmans et al., 2005; Hong et al., 2005; Attorre et al., 2007; Hiemstra et al., 2010; Eldrandaly and Abu-Zaid, 2011; Candiani et al., 2013). For example, Agnew and Palutikof (2000) developed a geographical information system (GIS)-based method for constructing the high-resolution maps of mean seasonal temperature and precipitation in the Mediterranean Basin. Lloyd (2005) compared the performance of five different interpolation methods for mapping monthly precipitation in Great Britain from sparse point data; the performance of each interpolation method was assessed through the examination of mapped estimates of precipitation and cross-validation. Zhang and Srinivasan (2009) developed
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a GIS program that incorporated different interpolation methods to facilitate automatic spatial precipitation estimation for the Luohe watershed. Among various interpolation methods, no method is always optimal; the best interpolation method for a specific situation can only be obtained by comparing their results (Sun et al., 2009). Some research has been conducted on comparing different interpolation methods in a variety of situations, and the GIS-based interpolation techniques have been recognized as a powerful tool for creating surfaces from measured points. Nevertheless, there is a lack of comparison of different interpolation methods for the spatial precipitation estimation in the province of Ontario, Canada. Therefore, the objective of this study is to conduct a thorough comparison of the GIS-based interpolation techniques for estimating the spatial distribution of precipitation in Ontario, Canada. This study has the following specific objectives: (1) validate the outputs of PRECIS over the period of 1961–1990 in terms of its ability to reproduce the observed precipitation events in Ontario; (2) project the future precipitation of 12 meteorological stations over the period of 2071–2100; (3) generate the average annual precipitation surfaces from measured point data for the period of 2071–2100 using different interpolation methods; (4) apply cross-validation to evaluate the accuracy of interpolation methods in terms of the root mean square error (RMSE) measure; and (5) use the optimal interpolation method to analyse spatial variations of the average annual precipitation for the period of 2071–2100 in Ontario.
2. 2.1.
Materials and methods Study area
Ontario, which is located in East Central Canada, is the second largest province with a total area of 1 076 395 km2 and is the most populous region with a population of over 13 million in Canada. Ontario consists of three main climatic regions, including Southwestern, Central and Eastern, and Northern Ontario. The southwestern parts of Ontario have a humid continental climate with hot, humid summers, and cold winters. Annual precipitation is well distributed throughout the year with a usual summer peak. Compared with Southwestern Ontario, Central and Eastern Ontario have a more severe humid continental climate with hot summers and longer colder winters. Annual precipitation is roughly equal to Southern Ontario. The northern parts of Ontario have a continental climate with long cold winters and short warm summers. Annual precipitation in this region is generally less than 700 mm. As recommended by the Ontario Ministry of Environment, Ontario can be divided into seven regions according to its administrative districts (Figure 1). 2.2.
Data collection
Two sets of precipitation data (simulated and observed) of 12 meteorological stations over Ontario were used in © 2014 Royal Meteorological Society
this study. Simulated precipitation data were obtained from the PRECIS runs with a resolution of 25 × 25 km over a 30-year period (1961–1990). Then the 30year precipitation data were averaged to generate the average annual precipitation data. HadRM3P was the model used in the PRECIS, which was the most recent regional climate model (RCM) from the Hadley Centre. Observed precipitation data, which could be downloaded from National Climate Data and Information Archive (http://climate.weatheroffice.ec.gc.ca), were used to validate the outputs of PRECIS in terms of its ability to reproduce the precipitation patterns in Ontario. Accordingly, the observed precipitation data of 12 meteorological stations were collected for the period 1961–1990. Figure 1 presents the locations of 12 meteorological stations over Ontario. The station names and their geographic coordinates are shown in Table 1. 2.3. Interpolation methods In this study, a variety of interpolation techniques, including inverse distance weighting (IDW), local polynomial interpolation (LPI), global polynomial interpolation (GPI), radial basis function (RBF), ordinary kriging (OK), and universal kriging (UK), were used to generate precipitation surfaces over Ontario. These techniques were divided into two groups: deterministic and geostatistical methods. The deterministic interpolation methods (i.e. IDW, LPI, GPI, and RBF) create continuous surfaces from measured points by taking advantage of mathematical formulae that determine either the extent of similarity or the degree of smoothing, whereas the geostatistical interpolation methods (i.e. OK and UK) utilize statistical models that quantify the spatial autocorrelation and the statistical relationships among measured points (Johnston et al., 2001). 2.3.1. IDW method IDW is a straightforward deterministic interpolation technique, which is based on an assumption that the interpolated values are influenced most by the nearby values and less by the distant observations. IDW weights the points closer to the interpolation location greater than those farther away. Thus, the weighting is a function of the distance between the point of interest and the sampling points (Sun et al., 2009). The usual expression of IDW is: n 1 Zi (di )p i =1 (1) Z = n 1 (di )p i =1
where Z is the predicted value of the interpolation point; Zi is the value of sampling point i (i = 1, 2, . . . , n); n is the number of sample points; di is the distance between the interpolated and sampled values; p represents the power parameter which is a positive real number. Int. J. Climatol. 34: 3745–3751 (2014)
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Figure 1. Study area and locations of meteorological stations.
Table 1. Station names and their geographic coordinates. Station name Windsor Airport London International Airport Toronto Pearson International Airport Toronto City Center Airport Ottawa International Airport Wiarton Airport North Bay Airport Sault Ste Marie Airport Sioux Outlook Timmins Victor Power Airport Big Trout Lake Moosonee Airport
Station number
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6139525 6144475 6158733 6158665 6106000 6119500 6085700 6057592 6037775 6078285 6010738 6075428
42.28 43.03 43.68 43.63 45.32 44.75 46.36 46.48 50.12 48.57 53.83 51.27
−82.96 −81.15 −79.63 −79.39 −75.67 −81.11 −79.42 −84.51 −91.90 −81.38 −89.87 −80.65
189.6 278 173.40 76.80 114 222.2 370.3 192 383.4 294.7 224.1 9.1
1961–1990 1961–1990 1961–1990 1961–1990 1961–1990 1961–1990 1961–1990 1961–1990 1961–1990 1961–1990 1961–1990 1961–1990
2.3.2. GPI method GPI uses a mathematical function to fit a smooth surface to the sample points. The GPI surface changes gradually from region to region over the area of interest and captures the global trend in the data. In contrast to IDW, GPI calculates predictions using the entire dataset instead of using the measured points within neighbourhoods. A first-order GPI fits a flat plane; a second-order GPI fits a surface, allowing for one bend; a third-order GPI allows for two bends; and so forth (ESRI, 2001). However, a single global polynomial can hardly fit a surface with a varying shape. Multiple polynomial planes are thus desired to better represent the surface. 2.3.3. LPI method Unlike GPI, LPI fits the local polynomial using points only within the specified neighbourhood instead of all © 2014 Royal Meteorological Society
the data. Then the neighbourhoods can overlap, and the surface value at the centre of the neighbourhood is estimated as the predicted value. GPI is useful for identifying long-range trends in the dataset, whereas LPI is capable of producing surfaces that capture the shortrange variation. 2.3.4. RBF method Different from GPI and LPI, RBF requires the surface to pass through each measured points. RBF fits a surface through the measured sample values while minimizing the total curvature of the surface. Compared with IDW, RBF can predict values above the maximum and below the minimum measured values, whereas IDW will never predict values above the maximum or below the minimum measured value (Johnston et al., 2001). RBF is capable of producing good results for smoothly varying Int. J. Climatol. 34: 3745–3751 (2014)
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surfaces such as elevation. However, RBF is ineffective when there is a dramatic change in the surface values within short distances (ESRI, 2001). 2.3.5. OK method In contrast with deterministic methods, kriging is a powerful statistical interpolation method which has the capability of giving unbiased predictions with minimum variance and taking account of spatial correlation and statistical relationships between measured points (Krige, 1966). As the most commonly used type of kriging family, OK which assumes that the mean is constant but unknown focuses on spatial components and uses only the sampling points within the local neighbourhood for the estimate of the predicted value (Luo et al., 2008). OK with measurement error models has advantages creating map smooth and having small standard errors.
the original sample is randomly partitioned into two datasets, with one used to train a model and the other used to validate the model. To reduce variability, the training and validation sets must cross-over in successive rounds such that each data point is able to be validated against. The RMSE was used in this study to evaluate the accuracy of interpolation methods, and it can be calculated as follows: n 1 RMSE = (zi − z )2 n i =1 (2) where Z is the predicted value; Zi is the observed value at sampling point i (i = 1, 2, . . . , n); and n is the number of sample points. 3. Results and discussion
2.3.6. UK method
3.1. Validation of PRECIS outputs
UK assumes the presence of a significant spatial trend in data values such as sloping surface or a localized flat region, which is a variant of the OK operation with no trend (ESRI, 2001). The trend or drift is a continuous spatial variation, which is too irregular to be modelled by a simple mathematical function. UK can thus be used to better describe the variation by a stochastic surface. UK is only effective when there is an obvious trend in the data with a great knowledge about spatial statistics.
In this study, three PRECIS runs corresponding to the baseline as well as the Intergovernmental Panel on Climate Change (IPCC) A2 (priority to economic issues) and B2 (priority to environmental issues) emission scenarios were carried out to simulate the precipitation events in Ontario. To validate the PRECIS outputs, the simulation data of nine RCM grids were first obtained for each meteorological station, and then the precipitation from the mean value of the nine RCM grids was calculated in order to diminish the possible random errors from the RCM simulating. Thus, the simulation results were the values of the averages of the nine grids for precipitation, and they were compared with the observed data of 12 meteorological stations. Figure 2 presents a comparison between simulated and observed average monthly precipitation patterns for four stations as an
2.4.
Cross-validation
Cross-validation is a statistical technique of evaluating and comparing the performance of different interpolation methods. It can be used to identify which method gives the best spatial interpolation. In a typical cross-validation, (a)
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Figure 2. Comparison of average monthly precipitation for stations: (a) Sioux Outlook Airport, (b) Timmins Victor Power Airport, (c) Big Trout Lake, and (d) Moosonee Airport. © 2014 Royal Meteorological Society
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example to validate the modelling results. The results show an overall good agreement between simulated and observed data. For the station of Sioux Outlook Airport, the PRECIS precipitation data under three scenarios are higher compared to the observed data except for January, August, and September. All the three simulation curves as well as the observation curve reach their peaks in June. For the station of Timmins Victor Power Airport, the results indicate that the PRECIS outputs match well with the observed data except for April, May, and June. For the station of Big Trout Lake, the results reveal that the PRECIS outputs are higher than the observed data in April, May, June, and July; apart from those months, the PRECIS results and the observed data have a good match. All curves reach their peaks in July, which is similar to those of Sioux Outlook Airport. As for the station of Moosonee Airport, the results indicate that the trend of the observation curve is similar to those of simulation curves under three scenarios. The highest precipitation occurs in July, which is the same as the situation of Big Trout Lake. In general, the simulated precipitation curves are higher than the observed ones during the summer time. This is because the short-term thunderstorm occurs more frequently in summer, and it has a significant impact on precipitation. However, climate model is incapable of capturing the short-term variations, resulting in the overestimation of precipitation during the summer time. Nevertheless, the overall trends of simulation and observation curves are a good match. Thus, the precipitation events are well simulated by PRECIS. 3.2. Comparison of interpolation methods After the PRECIS outputs were validated, the future precipitation of 12 meteorological stations could be projected for the period of 2071–2100 over Ontario. On the basis of the future precipitation data of 12 stations, six different interpolation techniques, including IDW, GPI, LPI,
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RBF, OK, and UK, were used to generate the average annual precipitation surfaces over Ontario. Figure 3 presents six interpolated annual precipitation surfaces for the period of 2071–2100 over Ontario. The results reveal that the precipitation surfaces generated by GPI and LPI methods are similar but different from those created by IDW and RBF. This is because interpolation techniques may be exact or inexact interpolators. IDW and RBF are exact interpolators, which generate a surface that passes through the control points; contrarily, GPI and LPI are inexact interpolators, which can predict a value at the point location that differs from its known value. The precipitation surfaces generated by inexact interpolators can be similar, but different from those created by exact interpolators. Figure 3(e) and (f) reveals that the precipitation surfaces generated by OK and UK are a perfect match. This is because UK is kriging with a local trend, which is a variant of the OK operation. UK with no trend is the same as OK. To reach the least RMSE, no local trend was considered in UK in this study. Thus, OK and UK may generate the same precipitation surfaces. Besides, the ranges of the average annual precipitation are 719–1071, 655–1024, 641–1055, 721–1069, 763–978, and 763–978 mm year−1 for six interpolation methods, respectively. The results indicate that OK and UK project the smallest range of future annual precipitation over Ontario; IDW and RBF project more precipitation falling in Southeastern Ontario compared with the other methods. On the other hand, interpolation results were compared on the basis of cross-validated RMSE. As shown in Table 2, the RMSE for different methods are in the order LPI < GPI < OK = UK < RBF < IDW, indicating that the minimum RMSE is obtained by LPI, which is the optimal method for interpolating future precipitation over Ontario. Overall, the interpolation performance of geostatistical methods is better than those of deterministic
(a) Inverse distance weighting (IDW)
(b) Global polynomial interpolation (GPI)
(c) Local polynomial interpolation (LPI)
(d) Radial basis functions (RBF)
(e) Ordinary kriging (OK)
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Figure 3. Average annual precipitation surfaces (for the period of 2071–2100) created by different interpolation methods. © 2014 Royal Meteorological Society
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methods, as both OK and UK in the geostatistical model perform well in terms of RMSE. Figure 4 presents a comparison of cross-validation between the optimal method (LPI) and the other methods. Moreover, as there is the largest number of meteorological stations in the southeastern region of Ontario, the spatial interpolation results may have a minor error when compared with the observed ones. 3.3.
Table 2. RMSE of different interpolation methods. RMSE (mm year−1 )
Interpolation method IDW GPI LPI RBF OK UK
111.18 94.87 94.74 104.30 95.35 95.35
Spatial variability
The average annual precipitation surface created by the best interpolation method (LPI) was used to analyse spatial variations of future precipitation over Ontario. Figure 3(c) shows that the average annual precipitation would be increasing gradually from Northwestern to Southeastern Ontario. The range of precipitation would be from 641 to 1055 mm year−1 . Thus, the Northwestern Ontario would receive the lowest average annual precipitation (approximately 641 mm year−1 ), whereas the Southeastern Ontario would receive the highest one (about 1055 mm year−1 ) over the period of 2071–2100. In terms of the spatial precipitation variations for 12 meteorological stations, the stations of Sioux Outlook and Big Trout Lake would receive precipitation of 700–800 mm year−1 . The average annual precipitation of the Sault Ste Marie Airport, Timmins Victor Power Airport, and Moosonee Airport stations would be 800–900 mm year−1 . As for the remaining six stations, namely, Windsor Airport, London International Airport, Toronto Pearson International Airport, Wiarton Airport, Ottawa International Airport, and North Bay Airport would receive precipitation of higher than 900 mm year−1 . Moreover, the highest precipitation would fall around the station of Ottawa International Airport. (a) LPI versus IDW
In terms of the spatial precipitation variations of the seven regions over Ontario (as shown in Figure 1), there would be lowest precipitation falling in regions 6 and 7 which are the driest areas in the province of Ontario. As for the regions 4 and 5 in central Ontario, the precipitation would be moderate (approximately 850 mm year−1 ). However, the precipitation would be high in regions 1 and 3, and increasing gradually from regions 1 to 3. In general, the future annual precipitation would be relatively low in the north of Ontario, and becoming higher from Northwestern to Southeastern Ontario.
4. Conclusions In this study, the PRECIS outputs were validated through comparing the simulated precipitation data with the observed data of 12 meteorological stations for the period of 1961–1990 over Ontario. The results indicated that the overall trends of simulation curves under three scenarios matched well with the observation curve. Thus, the precipitation events were well simulated by PRECIS. After the outputs of PRECIS were validated, the future precipitation events of 12 meteorological stations were
(b) LPI versus RBF
(d) LPI versus UK
(c) LPI versus OK
(e) LPI versus GPI
Figure 4. Comparison of cross-validation between the optimal method (LPI) and the other interpolation methods. © 2014 Royal Meteorological Society
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predicted by running PRECIS. On the basis of the projected precipitation data of 12 stations for the period of 2071–2100, six different interpolation techniques were in turn used to generate annual precipitation surfaces over Ontario. Then, the performance of interpolation techniques were compared and analysed in a systematic manner. The results indicated that LPI had the least RMSE, and thus LPI was the optimal method for interpolating the future precipitation over Ontario. The precipitation surface created by LPI was then used to analyse spatial variations of the future annual precipitation in Ontario. The results showed that the future annual precipitation would be increasing gradually from Northwestern to Southeastern Ontario. Both temporal and spatial variations of precipitation are important for regional climate change impact studies. However, only spatial variations of precipitation were analysed in this study. The analysis of temporal variations of precipitation is desired in future studies. Besides, the systematic errors of PRECIS can barely be improved by the interpolation techniques used. The accuracy of projected precipitation data needs to be improved in the PRECIS simulation for the study area. This methodology can also be applied for the fine-scale spatial distribution of other climatic variables such as temperature in the study area.
Acknowledgements This research was supported by the Program for Innovative Research Team (IRT1127), the Natural Science and Engineering Research Council of Canada, and the MOE Key Project Program (311013). The authors would like to express thanks to the editor and the anonymous reviewers for their constructive comments and suggestions. References Agnew MD, Palutikof JP. 2000. GIS-based construction of baseline climatologies for the Mediterranean using terrain variables. Clim. Res. 14: 115–127. Akhtar M, Ahmad N, Booij MJ. 2008. The impact of climate change on the water resources of Hindukush–Karakorum–Himalaya region under different glacier coverage scenarios. J. Hydrol. 355: 148–163. Ashiq MW, Zhao C, Ni J, Akhtar M. 2009. GIS-based high-resolution spatial interpolation of precipitation in mountain–plain areas of Upper Pakistan for regional climate change impact studies. Theor. Appl. Climatol. 99: 239–253. Attorre F, Alfo M, De Sanctis M, Francesconi F, Bruno F. 2007. Comparison of interpolation methods for mapping climatic and bioclimatic variables at regional scale. Int. J. Climatol. 27: 1825–1843. Busuioc A, Chen D, Hellstr¨om C. 2001. Performance of statistical downscaling models in GCM validation and regional climate change estimates: application for Swedish precipitation. Int. J. Climatol. 21: 557–578. Campbell JD, Taylor MA, Stephenson TS, Watson RA, Whyte FS. 2011. Future climate of the Caribbean from a regional climate model. Int. J. Climatol. 31: 1866–1878.
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