Spatial Interpolation Accuracy of GIS for the Development of Climatic Maps:
A Review of Literature
Ghamar Fadavi1 and Javad Bazrafshan2 1MSc student in Agricultural Meteorology, University of Tehran, Karaj, Iran Irrigation Deptt., College of Agriculture, Karaj, Iran
Tel. 0098 261 224 1119 Fax: 0098 261 224 1119 e-mail: [email protected] 2 Assistant professor, University of Tehran, Karaj, Iran Irrigation Deptt., College of Agriculture, Karaj, Iran Tel. 0098 261 224 1119 Fax: 0098 261 224 1119 e-mail: [email protected] Key words: Spatial interpolation; Climatic data; Temperature and precipitation
ABSTRACT
Interpolation is a way of reconstructing continuous fields from variables measured at point locations. The spatial interpolation of climatic variables such as temperature and precipitation is increasingly important in the development of agricultural, hydrological, and ecological models. Many interpolation methods used by previous studies are purely mathematic ways without geographical significance being considered. Since 1970, researchers have adopted lots of spatial interpolation methods, such as Kriging, Inverse Distance Weighted (IDW) and Spline method. Recently the spatial interpolation methods have been improved by taking into account geographical factors like latitude, longitude and altitude as predictor variables using multiple linear regression. Recently, one general approach named as Cross-Validation (CV) is used to assess the interpolation results. The CV is still regarded as one useful tool to assess the relative accuracy of different interpolation algorithms with the same quality of the known points. Deeper research can be implemented in future work and sensibility to data changes of the mentioned interpolation methods can become the next focus. This study was aimed to review different work on the mapping climatic data using various interpolating methods. Introduction
Climate plays a significant role in flora and fauna distributions; it is usually a key to understand the interdependence between environmental and biological factors and is widely used in developing ecological zones and biodiversity. ( Woodward, 1987; Skirvin et al., 2003; Hong et al., 2005 ). A complete and accurate source of climate data is a prerequisite for the efficient modelling of a wide variety of environmental processes. In fact, observational records are typically incomplete, making it difficult to construct a continuous climate record. In particular, such data may: (1) be recorded for discrete periods, not spanning the entire time period of interest; (2) contain short intermittent periods where data have not been recorded; and (3) contain either systematic or random errors (Peck, 1997; Jeffrey, 2001). In mapping environmental variables, two main steps must be considered, 1) the sampling step, during which measured data are taken of the environmental variable at selected locations, and 2) the estimation step, during which the observations are interpolated to a fine grid. The quality of the resulting map is determined by both steps (Brus and Heuvelink, 2007). Interpolation is a way of reconstructing continuous fields from variables measured at point locations. The spatial interpolation methods are commonly used for estimating temperature or precipitation when climate stations are not dense. The spatial interpolation of climatic variables such as temperature and precipitation is increasingly important in the development of agricultural, hydrological, and ecological models. This study was aimed to review different work on the mapping climatic data using various interpolating methods.
Progress in climate interpolation methods
The progress of methods to interpolate climatic observations from meteorological stations networks has been the focus of many research studies (Thiessen, 1911; Peck and Brown, 1962; Kruizinga and Yperlaan, 1978; Hutchinson and Bischof, 1983; Phillips et al., 1992; Price et al., 2000; Dogan, 2007; Ordu and Demir, 2009; Rabah et al., 2011). Some researchers used simple methods, which include the latitude and longitude of meteorological stations. Because of the strong influence of topography on temperature vertical variations ( lapse rates), researchers showed that the inclusion of altitudes was crucial to generating sophisticated data for meteorological parameters through interpolation (Hong et al., 2005). Interpolation may have resulted from simple mathematical models (e.g. inverse distance weighting, trend surface analysis, and Thiessen polygons), or more complex models (e.g. geo- statistical methods, such as Kriging), (Negreiros et al., 2010). There are many studies showing the application of interpolation methods for estimating temperature or precipitation. The methods include distance weighting (Eischeid et al., 1995; Lennon and Turner, 1995; Ashraf et al., 1997; Dodson and Marks, 1997; Baltas, 2007 , interpolating polynomials (Stewart and Cadou, 1981; Tabios III and Salas, 1985; Eischeid et al., 1995), kriging (Phillips et al., 1992; Hammond and Yarie, 1996; Holdaway, 1996; Ashraf et al., 1997; Chen, et al., 2010) and splines (Hulme et al., 1995; Lennon and Turner, 1995). Among the methods, the geo-statistical interpolation method has found grounds in climatology because it is based on the spatial variability of variables of interest and can quantify the estimation uncertainty (Martinez-Cob 1996; Holawe and Dutter 1999; Serbin and Kucharik, 2008). In applications to climate data, Hancock and Hutchinson (2006), used the method of minimizing the generalised cross validation (GCV). The GCV is a measure of the predictive error of the fitted surface and is effectively calculated by removing each data point in turn and summing, with appropriate weighting, the square of the discrepancy of each omitted data point from a surface fitted to all other data points The value of the smoothing parameter that gives the smallest GCV is chosen.
Methods Features
Most major data sets in use today have been developed using one of six interpolation techniques: inverse distance weighting (IDW), various forms of kriging, ANUSPLIN tri-variate splines, local regression models Daymet and PRISM, and regional regression models (Daly, 2006). We will discuss some characteristics of these methods at the following titles: Thiessen polygons (Thiessen)
A simple way for interpolating is to assign to every unsampled point the value at its nearest control point using proximity polygons. This technique was introduced by Thiessen (1911), who was interested in how to use rain gauge records to estimate total rainfall across a region. The Thiessen polygon approach has the great virtue of simplicity, but it does not really produce a continuous field of estimates; consequently, processing nominal data in this way is not usually called interpolation.
Regression
A straightforward and simple way of spatial prediction with the help of co-variates is to make use of linear regression. For mapping with a regression model, the sampling problem boils down to selecting the locations that lead to the most accurate estimates of the regression coefficients. For spatial interpolation there is an extra reason for spreading of observations in geographical space. When regression residuals are spatially correlated, the map may become more accurate by interpolating these residuals and adding them to the predicted values from the regression model, as in regression kriging (Knotters et al., 1995; Hengl et al., 2004; (Stahl, et al., 2006).
Geostatistical Models
Geostatistical interpolation methods are based upon the structure of a variable’s spatial continuity. The equations and their derivations will not be repeated here, but the reader can find them in many excellent references (Isaaks and Srivastava, 1989). The spatial structure of each variable is summarized by its variogram, which is estimated by computing 1/2 the average squared difference between pairs of data points separated by some multiple of a given lag, or separation, distance. Kriging, the geostatistical interpolation process, uses the model parameters in computing a weighted sum of data values within a moving neighborhood for each point to be estimated. Kriging of climate data has been performed over a range of spatial scales, from that of a river basin or region, 1000–50,000 km2 (Bastin et al., 1984; Phillips et al., 1992), to very large regions, 75,000 to more than 1 million km2 ( Beek et al., 1992; Michaud et al., 1995). Multi- variate geostatistical methods provide a means of incorporating additional information into the interpolation process, from one or more secondary and usually more intensively sampled variables (Yates & Warrick, 1987). The addition of secondary information reduces the smoothing of estimated values produced by ordinary kriging (Goovaerts, 1997, p. 202). Cokriging, the multi-variate extension of kriging, poses additional requirements of variogram modeling for each secondary variable, as well as modeling of cross-variograms between the primary and each secondary variable (Myers, 1982). Among the Kriging methods that do not make use of an external variable are the following: Simple Kriging (SK), Ordinary Kriging (OK), Kriging with varying local means (Klm) and Block Kriging (BK), while among the methods that make use of a secondary variable are Factorial Kriging (FK), Kriging with External Drift (KED) and Cokriging (Cok).
Carrera-Herna´ndez and Gaskin, (2007) compared different interpolation methods in the Basin of Mexico for daily climatological variables including rainfall, minimum temperature and maximum temperature. The effect of considering elevation as a secondary variable to interpolate daily rainfall, minimum and maximum temperature was analyzed in this study using approximately 200 climatological stations for an area of 16,800 km2. In this study, the use of elevation as a secondary variable improved the spatial variation of all climatological fields even when they exhibited low correlation with elevation. According to the analyses presented, the use of Kriging with External Drift on a local neighborhood (KEDl) is recommended to undertake the spatial interpolation of rainfall, (a highly heterogeneous variable) while Kriging with External Drift (KED) is recommended to undertake the interpolation of a more continuous field such as minimum and maximum temperature. Skirvin et al., 2003 revealed that in the San Pedro River watershed area in south-eastern Arizona, high-resolution spatial patterns of long-term precipitation and temperature were better reproduced by kriging climate data with elevation as external drift (KED) than by multiple linear regression on station location and elevation as judged by the spatial distribution of interpolation error. Mean errors were similar overall, and interpolation accuracy for both methods increased with increasing correlation between climate variables and elevation. Stahl, et al., 2006, compared twelve methods for interpolating daily maximum and minimum temperatures over British Columbia, Canada, a region with complex topography and highly variable station density and elevational distribution. The simplest method, which uses the nearest station with an adjustment for elevation, generally had the greatest errors. All models performed better for years with greater station density, particularly in relation to higher-elevation stations.
Inverse distance weighted (IDW)
Inverse distance weighted (IDW) interpolation is based on the assumption that the value at an unsampled point can be approximated by a weighted average of observed values within a circular search neighbourhood, whose radius can be defined by the range of a fixed number of closest points. The weights used for averaging are a decreasing function of the distance between the sampled and unsampled points The common weighting function is the inverse of the distance squared (Philip and Watson, 1982).
Thin plate smoothing splines (spline)
The thin plate spline method is popular due to its efficiency, and ability to generate accurate predictions with a minimal number of guiding covariates. This is a mathematically elegant model for surface estimation that fits a minimum-curvature surface through the input points. Conceptually, it is like bending a sheet of rubber to pass through the points, while minimizing the total curvature of the surface. It fits a mathematical function to a specified number of nearest input points, while passing through the sample points. This method is best for gently varying surfaces such as elevation, water table heights, or pollution concentrations. It is not appropriate if there are large changes in the surface within a short horizontal distance, because it can overshoot estimated values. (Zheng and Basher,1995; Hancock and Hutchinson, 2006). Table 1 shows the summary of strengths and weaknesses of major interpolation techniques. Table 1. Summary of strengths and weaknesses of major interpolation techniques used to produce today’s popular spatial climate data sets, and Web location of a major associated data set. If an entry is a specific model, the general interpolation approach it employs is given in parenthesis after the name (Daly, 2006)
Interpolation Strengths Weaknesses Example data Technique Description set URL
IDW/2D General method that Readily available; Very simple; accounts http://climate.geog. Kriging uses horizontal IDW for distance effects distance to a station to very easy to apply; only; determine weight in both account for kriging requires udel.edu/∼climate/ averaging function distance relationships domain-wide semivariogram, which limits size and heterogeneity of domain
ANUSPLIN Specific model that fits Readily available; Simulates elevation http://biogeo.berkeley. (thin plate smoothing splines to relatively easy to relationship only; edu/worldclim/ splines) the station data in three apply; accounts for difficulty handling worldclim.htm dimensions spatially varying sharp elevation relationships spatial gradients in relationship Daymet (local Specific model that fits Local regression Not readily available; http://www.daymet. regression) local linear regressions accounts for spatially simulates elevation org of climate versus varying elevation relationship only; elevation relationships cannot handle nonlinear and nonmonotonic elevation relationships PRISM (local Specific model that fits Local regression Not readily available; http://www.ocs. regression) local linear regressions accounts for spatially requires significant oregonstate.edu/prism/ of climate versus varying elevation effort elevation, with slopes relationships; also to take advantage of that vary with accounts for full elevation effectiveness of terrain capability as barriers, terrain-induced climate transitions, cold air drainage and inversions, and coastal effects
Regional General method that Accounts for effects of A single, domain-wide http://geog.arizona. regression develops domain-wide, multiple variables relationship limits size multivariate functions (usually latitude, and heterogeneity of longitude, and modeling domain; may edu/∼comrie/ elevation) on climate not reproduce station patterns; stable values statistical relationship climas/anim.htm Conclusion
It can be concluded the following results from the different studies: 1. In comparisons with kriging methods that incorporate information from surrounding stations, linear regression tended to have larger prediction errors (Goovaerts, 2000). In order to add nearby station information, Daly et al. (1994) used regression within a moving window that implicitly incorporates local spatial continuity of climate variables; however, this algorithm is substantially more complex than the usual linear regression.
2. A basic difference between regression and geostatistical interpolation is the relationship between data points and the interpolated surface: in regression, the distance of data points from the interpolated surface is minimized by the method of least squares (Hogg & Tanis, 1977) but the surface may not pass through the points. Kriging, in contrast, is an exact interpolator and the kriged surface will pass through data points (Deutsch & Journel, 1992, p. 65).
3. Prediction can be further improved if correlated secondary information, such as a digital elevation model, is taken into account (Creutin and Obled, 1982; Goovaerts, 2000).
4. For a region without surface emissivity or satellite data, Kriging interpolation is recommended due to its considerations of prediction confidence in error map and spatial autocorrelation between sampling sites. Cokriging is suggested for areas with rough terrains and large variation in elevations (Yang et al., 2004).
References
1. Ashraf, M., J.C., Loftis, and K.G., Hubbard, 1997. Application of geostatistics to evaluate partial weather station networks. Agric. For. Meteorol., 84: 225-271.
2. Baltas, E., 2007. Spatial distribution of climatic indices in northern Greece. Meteorol. Appl. 14: 69–78. 3. Bastin, G., Lorent, B., Duque‘, C. & Gevers, M. (1984). Optimal estimation of the average areal rainfall and optimal selection of rain gauge locations. Water Resources Research, 20: 463-470.
4. Beek, E.G., Stein, A. & Janssen, L.L.F. (1992). Spatial variability and interpolation of daily precipitation amount. Stochastic Hydrology and Hydraulics, 6: 304-320.
5. Brus, J. B., and G. B. M., Heuvelink, 2007.Optimization of sample patterns for universal kriging of environmental variables. Geoderma. 138: 86–95.
6. Carrera-Herna´ndez, J.J., and S.J., Gaskin, 2007. Spatio temporal analysis of daily precipitation and temperature in the Basin of Mexico.Journal of Hydrology. 336: 231– 249. 7. Chen, D., T., OU, L., Gong, C.,Yu XU, L. Weijing, C., Hoi HO, and Q., Weihong, 2010. Spatial Interpolation of Daily Precipitation in China, 1951–2005. Advances in atmospheric sciences. 27( 6): 1221–1232.
8. Creutin, J.D., Obled, C., 1982. Objective analyses and mapping techniques for rainfall fields: an objective comparison. Water Resour. Res. 18 (2), 413–431.
9. Daly, C., 2006. Guidelines for assessing the suitability of spatial climate data sets. Int. J. Climatol. 26: 707–72.
10. Deutsch and A.G. Journel. Geostatistical Software Library and User's Guide. Oxford University Press, New York, 1992.
11. Dodson, R., and D., Marks, 1997. Daily air temperature interpolated at high spatial resolution over a large mountainous region. Clim. Res., 8(1): 1-20.
12. Dogan, H.M., 2007. Climatic portrayal of Tokat Province in Turkey; Developing climatic surfaces by using LOCCLIM and GIS. J. Biol. Sci., 7(7): 1060-1071.
13. Eischeid, J.K., F.B., Baker, T.R., Karl, and H.F., Diaz, 1995. The quality control of long- term climatological data using objective data analysis. J. Appl. Meteorol., 34: 2787-2795.
14. Goovaerts, P., 2000. Geostatistical approaches for incorporating elevation into the spatial interpolation of rainfall. Journal of Hydrology. 228:113–129.
15. Goovaerts, P., (1997). Geostatistics for Natural Resources Evaluation. Oxford: Oxford University Press. 483 pp.
16. Griffith, D., 1987. Spatial Autocorrelation: A Primer. Association of American Geographers, Washington, DC.
17. Hammond, T., and J., Yarie, 1996. Spatial prediction of climatic state factor regions in Alaska. Ecoscience, 3(4): 490-501
18. Hancock, P.A., M.F., Hutchinson, 2006. Spatial interpolation of large climate data sets using bivariate thin plate smoothing splines. Environmental Modelling & Software. 21: 1684 – 1694.
19. Hengl, T., Heuvelink, G.B.M., Stein, A., 2004. A generic framework for spatial prediction of soil variables based on regression-kriging. Geoderma. 120: 75–93.
20. Hogg, R. V., and E. A. Tanis. 1977. Probability and statistical inference. New York: Macmillan.
21. Holawe F, and R., Dutter, 1999. Geostatistical study of precipitation series in Austria: Time and space, Journal of Hydrology: 219(1–2): 70–82. 22. Holdaway, M.R., 1996. Spatial modeling and interpolation of monthly temperature using kriging, Clim. Res., 6; 215-225.
23. Hong, Y., H. A., Nix, M. F., Hutchinson, and T. H., Booth, 2005. Spatial interpolation of monthly mean climate data for China. Int. J. Climatol. 25: 1369–1379.
24. Hulme, M., D., Conway, P.D., Jones, T., Jiang, E.M., Barrow, and C., Turney, 1995. Construction of a 1961-1990 European climatology for climate change modelling and impact applications. Int. J. Climatol, 15: 1333-1363.
25. Hutchinson, M.F., and R.J. Bischof, 1983. A new method of estimating mean seasonal and annual rainfall for the Hunter Valley,New South Wales. Aust. Meteorol. Mag. 31, 179-184
26. Issaks, E. H., and Srivastava, R. M. 1989. Applied Geostatistics. Oxford University Press. 561p.
27. Jeffrey, S. J., J. O., Carter, K. B., Moodie, and A. R., Beswick, 2001. Using spatial interpolation to construct a comprehensive archive of Australian climate data. Environmental Modelling & Software. 16: 309–330.
28. Knotters, M., Brus, D.J., Oude Voshaar, J.H., 1995. A comparison of kriging, co-kriging and kriging with regression for spatial interpolation of horizon depth with censored observations. Geoderma. 67: 227–246.
29. Kruizinga, S., and G.J., Yperlaan, 1978. Spatial Interpolation of daily totals of rainfall. J. of Hydroligy. 36: 65-73.
30. Lennon, J.J., and J.R.G., Turner, 1995. Predicting the spatial distribution of climate: temperature in Great Britain. J. Anim. Ecol., 64: 370-392.
31. Martinez-Cob A., 1996. Multivariate geostatistical analysis of evapotranspiration and precipitation in mountainous terrain. Journal of Hydrology. 174:19–35
32. Michaud, J.D., Auvine, B.A. & Penalba, O.C. (1995). Spatial and elevation variations of summer rainfall in the southwestern United States. Journal of Applied Meteorology, 34: 2689–2703.
33. Myers, D.E. (1982). Matrix formulation of co-kriging. Mathematical Geology, 14: 249–257.
34. Negreiros, J., M., Painho, F., Aguilar, and M., Aguilar, 2010. Geographic information systems principles of ordinary kriging interpolator. J. Applied Sci., 10(11): 852-867.
35. Ordu, S., and A., Demir, 2009. Determination of land data of Ergene Basin (Turkey) by planning geographic information systems. J. Environ. Sci. Technol., 2(2): 80-87. 36. Peck, E.L., and M.J., Brown, 1962. An approach to the development of isohyetal maps for mountainous areas. J. Geophys. Res. 67: 681-694.
37. Peck, E.L., 1997. Quality of hydrometeorological data in cold regions. Journal of the American Water Resources Association. 33: 125–134. Piazza, A. D., F. Lo, Conti, L.V., Noto, F., Viola, and G., La, Loggia, 2011. Comparative analysis of different techniques for spatial interpolation of rainfall data to create a serially complete monthly time series of precipitation for Sicily, Italy. International Journal of Applied Earth Observation and Geoinformation. 13: 396–408.
38. Phillips, D. L., J., Dolph, and D. Marks, 1992. A comparison of geostatistical procedures for spatial analysis of precipitation in mountainous train. Agric. For Meteorol. 58: 119-141.
39. Philip G. M., and D. F. Watson, 1982. A precise method for determining contoured surfaces. Aust Pet Explor Assoc J. 22:205–212.
40. Price, D. T., D. W., Mckenney, I. A., Nalder, M. F., Hutchinson, and J. L., Kesteven, 2000. A comparison of two statistical methods for spatial interpolation of Canadian monthly climate data. Agr. For Meteorol. 101: 81-94.
41. Rabah, F.K.J., S.M. Ghabayen and A.A. Salha, 2011. Effect of GIS interpolation techniques on the accuracy of the spatial representation of ground water monitoring data in Gaza strip. J. Environ. Sci. Technol. 4(6): 579-589.
42. Serbin, S. P., C. J., Kucharik, 2008. Spatiotemporal mapping of temperature and precipitation for the development of a multidecadal climatic dataset for Wisconsin. J. of Applied Meteorol. And Climatol. 48: 742-757.
43. Skirvin, S. M., E. M., Stuart, M. P., McClaran, and D. M., Meko, 2003. Climate spatial variability and data resolution in a semi-arid watershed, south-eastern Arizona. Journal of Arid Environments. 54: 667–686.
44. Stahl, K., R. D., Moore, J. A., Floyer, M. G., Asplin, I.G., McKendry, 2006. Comparison of approaches for spatial interpolation of daily air temperature in a large region with complex topography and highly variable station density. Agricultural and Forest Meteorology. 139: 224– 236.
45. Stewart, R.B., and C.F., Cadou, 1981. Spatial estimates of temperature and precipitation normals for the Canadian Great Plains. Research Branch, Agriculture Canada, Ottawa, Canada. 46. Tabios III, G.Q., and J.D., Salas, 1985. A comparative analysis of techniques for spatial interpolation of precipitation. Water Resour. Bull., 21(3): 365-380.
47. Thiessen, A. H., 1911. Precipitation averages for large areas. Mon. Wea. Rev. 39: 1082-
1084.
48. Tobler, W, 1970. A computer movie simulating urban growth in the Detroit region. Economic Geography .46: 234–240.
49. Vasiliev, I., 1996. Visualization of Spatial Dependence: An Elementary View of Spatial Autocorrelation. CRC Press, Boca Raton, USA.
50. Woodward, F. I., 1987. Climate and plant distribution. Cambridge university press. pp.174.
51. Zheng, X. and Basher, R., 1995. Thin-plate smoothing spline modelling of spatial climate data and its application to mapping South Pacific rainfalls. Mon. Wea. Review, 123, 3086– 3102.
52. Yang, J. S., Y. Q., Wang, and P. V., August, 2004. Estimation of Land Surface Temperature Using Spatial Interpolation and Satellite-Derived Surface Emissivity . Journal of Environmental Informatics. 4 (1): 37-44.