Abstract

This paper reviews three methods of interpolation to be used on air temperature data from the Phoenix Metropolitan area. The air temperature measurements were taken at thirty-six discrete locations. Much of the geographical spatial analysis requires a continuous data set and this study is designed to create that surface. This study identifies the best spatial interpolation method to use for the creation of continuous data for air temperatures. The reviewed techniques include SPLINE, Inverse Distance Weighting (IDW) and KRIGING. A statistical assessment of the resultant continuous surfaces indicates that there is little difference between the estimating ability of the three interpolation methods with KRIGING performing better overall.
 
 

Introduction

It has been shown that there is no single preferred method for data interpolation . Aspects of the algorithm selection criteria need to be based on the actual data, the level of accuracy required, and the time and/or computer resources available . In the absence of criteria for selecting among the available techniques, this paper compares three spatial interpolation - SPLINE, Inverse Distance Weighting (IDW), and KRIGING – with the goal of determining which method creates the best representation of reality for measured air temperatures . The benefits and limitations of these commonly used interpolation methods are discussed in this paper. The data used in this assessment are air temperature measurements taken at thirty-six fixed climate stations. This assessment is important because much of geographic research includes the creation of data for spatial analysis. Selecting an appropriate spatial interpolation method is key to surface analysis since different methods of interpolation can result in different surfaces and ultimately different results . Statistical techniques are used to evaluate the three interpolation methods against independently collected data (Willmott 1984).

The research presented here is part of a larger study to develop a spatio-temporal integration model (STIM). The ultimate goal of STIM is to develop an interpolation method that incorporates environmental, anthropogenic, and geographical data with one application of the STIM to create air temperature surfaces at the detailed spatial resolution representing the heterogeneity of the urban area. A thorough understanding of a continuous surface of temperature data and the techniques used to develop it, are essential to understand the resulting output of the STIM. This paper assesses three different methods for spatial interpolation for air temperature and identifies which method will be used as input into the STIM.

Spatial Interpolation

Interpolation is a method or mathematical function that estimates the values at locations where no measured values are available . Interpolation can be as simple as a number line, however, most geographic information science research involves spatial data. Spatial interpolation assumes the attribute data are continuous over space. This allows for the estimation of the attribute at any location within the data boundary. Another assumption is the attribute is spatially dependent, indicating the values closer together are more likely to be similar than the values farther apart . These assumptions allow for the spatial interpolation methods to be formulated .

Spatial interpolation is widely used for creating continuous data when data are collected at discrete locations (i.e., at points). For example, precipitation maps provided by the National Weather Service (NWS) are generated from NWS stations. An interpolation method is used to create precipitation maps covering all of the United States. These point data are displayed as interpolated surfaces for qualitative interpretation. In addition to qualitative research, these interpolated surfaces can also be used in quantitative research from climate change to anthropological studies of human locational responses to landscape . However, when an interpolated surface is used as part of larger research project (e.g., STIM) both the method and accuracy of the interpolation technique are important. The goal of spatial interpolation is to create a surface that is intended to best represent empirical reality thus the method selected must be assessed for accuracy for these larger studies.

The degree of importance placed on spatial interpolation is reflected in the attention it receives in the literature. Reviews of spatial interpolation appear as early as the 1970s with updates every decade . In addition to review papers, entire books on the subject of spatial interpolation as well as textbooks with chapters on the subject are available . These publications and the inclusion of a Geostatistical extension in the new ESRI ArcInfo package, ArcGIS, represent the acceptance of interpolation as a legitimate and valuable method for data creation. In addition to the number of papers and books published on spatial interpolation, this topic is ubiquitous on the web. One search yielded 3,930 hits with the search term "spatial interpolation". Included in the results were numerous lectures, including the US National Center for Geographic Information and Analysis (NCGIA) core curriculum chapters, modifications of the chapters, and a variety of course materials.

Methods

Study Site

The Phoenix metropolitan area is located in the Sonoran desert in central Arizona. The population of metro Phoenix grew from 2.24 million in 1990 to 3.25 million people in 2001, representing a growth rate of 45.3% percent . All cities contained within this area have increased in population ranging from 32% to 4407% over the past 20 years. The metropolitan area is located in a valley with prominent mountain ranges rising up to 1000 meters above the valley floor. This area provides many desired traits that continue to draw new residents.

One of the reasons for this growth rate is the comfortable climate with mild winters and low humidity. The enticing climate can be modified by human impact . These impacts are continually being studied, especially in arid environments like Phoenix . Specific boundaries for my research are identical to the boundaries of the Central Arizona/Phoenix Long Term Ecological Research (CAP LTER) (Figure 1). The CAP LTER boundary includes all of the Maricopa Association of Governments (MAG) Planning Area and the US Census Bureau Urbanized Area, although it is about half the physical size of Maricopa County . The CAP LTER boundary includes an area almost 110 km east/west and 60 km north/south and encompasses approximately 6,387.46 km². The majority of the permanent climate stations used in this investigation are within or adjacent to this area.

In Figure 1, the temperature image derived from 1998 satellite imagery shows the heterogeneity of the urban area. The available air temperature data from NWS and local government stations used to create different interpolated surfaces should attempt to resemble the heterogeneity produced by the satellite image. Enormous amounts of research link urban change to climate conditions - particularly temperature . In the case of Phoenix, comfortable climate with mild winters and low humidity is countered by the summer heat and desert conditions. Therefore, any changes resulting from human activities or vegetation patterns that force temperatures to rise, adding exponentially to heat-related stresses, need to be better understood.

Figure 1: Study Area of Phoenix Metropolitan Area within CAP LTER boundaries

Data

The temperature data used for this application were collected at thirty-six permanent climate stations in the study area. These stations are sporadically distributed within the study area with a concentration in the central city (Figure 1). The requirements of the larger research project stipulate the data are from May 24, 1998 at 10:30am. Depending on the data collection method at the climate station, these data are either hourly averaged measurements of the air temperature at shelter height (1.5 meters) or a derived temperature from maximum and minimum daily temperature records. The actual range of temperature on this day was 23.3° C to 29.4° C.

The interpolated surfaces were created by estimating temperature from twenty-five sampled points. The created surfaces represent the temperature of the study area. The remaining eleven sites were withheld randomly from the interpolation to assess the interpolated surfaces.

Interpolation Method

The techniques assessed here include the deterministic interpolation methods of SPLINE and Inverse Distance Weighting (IDW) and the stochastic method of KRIGING in an effort to retain actual temperature measurement in a final surface. Each method selected requires that the exact data values for the sample points are included in the final output surface.

These spatial interpolation methods have various decision parameters. The descriptions below include the options and values used in the different modules of ArcGIS. For display purposes, all images are grouped to nine classes. The images show the colder values in lighter colors with the warmest values in the darkest color. The selected techniques, SPLINE, IDW and KRIGING, are not all of the interpolation methods, nor are they a comprehensive review of the ArcGIS Geostatistical extension. ArcGIS was employed as the software for this study, although the techniques are available in other software products as well.

SPLINE

The SPLINE method can be thought of as fitting a rubber-sheeted surface through the known points using a mathematical function. In ArcGIS, the spline interpolation is a Radial Basis Function (RBF). These functions allow analysts to decide between smooth curves or tight straight edges between measured points. Advantages of splining functions are that they can generate sufficiently accurate surfaces from only a few sampled points and they retain small features . A disadvantage is that they may have different minimum and maximum values than the data set and the functions are sensitive to outliers due to the inclusion of the original data values at the sample points. This is true for all exact interpolators, which are commonly used in GIS, but can present more serious problems for SPLINE since it operates best for gently varying surfaces, i.e. those having a low variance.

Inverse Distance Weighting (IDW)

Inverse Distance Weighting (IDW) is based on the assumption that the nearby values contribute more to the interpolated values than distant observations . In other words, for this method the influence of a known data point is inversely related to the distance from the unknown location that is being estimated. The advantage of IDW is that it is intuitive and efficient. This interpolation works best with evenly distributed points. Similar to the SPLINE functions, IDW is sensitive to outliers. Furthermore, unevenly distributed data clusters results in introduced errors.

KRIGING

Similar to IDW, KRIGING uses a weighting, which assigns more influence to the nearest data points in the interpolation of values for unknown locations. KRIGING, however, is not deterministic but extends the proximity weighting approach of IDW to include random components where exact point location is not known by the function. KRIGING depends on spatial and statistical relationships to calculate the surface. The two-step process of KRIGING begins with semivariance estimations and then performs the interpolation. Some advantages of this method are the incorporation of variable interdependence and the available error surface output. A disadvantage is that it requires substantially more computing and modeling time, and KRIGING requires more input from the user.

Assessment Methods

Evaluation of the interpolation methods follows the approach described by Willmott . This evaluation calculates error statistics on the eleven control stations with the recorded temperatures as the observed data and the interpolated temperatures as the predicted values. Summary univariate measures include the mean of the observed (Obar), mean of the predicted (Pbar), and their standard deviations (s_o, s_p). Another indicator of the models potential is how closely s_p approaches s_o such that the closer to the standard deviation the observed (s_o) is to the predicted standard deviation (s_p) the better the method is at reproducing the observed variance. Willmott cautions that these statistical measures should not be over analyzed .

Willmott (1984) also used five difference measures that are useful in evaluating the performance of the interpolation methods. These measures are: 1) mean absolute error (MAE), 2) root mean square errors (RMSE), 3) systematic root mean square errors (RMSE_s), 4) unsystematic root mean square errors (RMSE_u), and 5) the index of agreement (d). Equations from Willmott (1984) are in Appendix A. The ordinary least-squares (OLS) simple linear regression coefficients of a and b are used to compute the difference measures systematic and unsystematic root mean square errors (RMSE_s, RMSE_u). MAE is sometimes preferred over the RMSE as an evaluator because it is less sensitive to extreme values; however, RMSE is the error measure commonly computed in geographic applications. The systematic RMSE_s assesses whether the model errors are predictable, whereas the unsystematic RMSE_u identifies those errors that are not predictable mathematically. The final error measure, d, varies between 0.0 and 1.0. Therefore, the closer d is to 1.0 the better the agreement between O and P with 1.0 conveying perfect agreement and 0.0 complete disagreement.

Results

SPLINE

Figure 2: SPLINE surface

The results of a tension SPLINE with weight set at 1.0, number of points at six and cell size of 200 are in Figure 2. The smooth surface created by the spline function provides the correct general temperature trends for the area. It correctly shows a cooler area in the center of the city, surrounded by a warmer area with a cooler area to the southeast where substantial agriculture is present. The input parameters for the spline function are the input sampled points, the interpolation attribute (i.e., temperature), the type (regularized or tension), the weight, number of points to consider for each new value and the output cell size. The default settings provided by the software are the regularized for function and the default weight of 0.1. The software help suggests using these unless other criteria are applied. This project required the bounding of the SPLINE surface to the observed data range. The default regularized option resulted in a range of temperatures from –54.56 to +64.31 degrees C, which is impossible for Phoenix in May. The selection of TENSION was made to keep the range within the actual data range and the weight was increased to 1.0. When using the tension SPLINE, the higher the weight, the more the values conform to the range of sample data. In addition, for all of the interpolation methods the number of sample points to be used in the analysis of new locations was set to six points. This was due to the limited number of observed data points, the default value of twelve meant that about half the points were contributing to the area regardless of how far they were from the location being estimated. Reducing the number of points used, assures the use of the closest values in the calculation process. A cell size of 200m² was selected based on the ability to identify different microclimates within a 200m² area .

Inverse Distance Weighting (IDW)

Figure 3: Inverse Distance Weighting (IDW) surface

The distance-decay principle is shown by IDW surface in Figure 3. This surface shows more diversity in the central area than the tension SPLINE and is not as smooth, which is one of the general characteristics of a SPLINE surface. It may be inappropriate, however, to use the smoothing SPLINE functions for a highly heterogeneous area since it provides an unrealistic view of reality by reducing spatial variance. The IDW parameters specified in ArcGIS are the power option, search radius (variable or fixed), number of points, a possible maximum distance for influence on the data, and output cell size. The input points, sampled locations contained within a shapefile, are the same for all three methods. As are the use of six points and a cell size of 200 m². The power was set to the most commonly used value of two. When using a power of two, it is known as inverse distance squared weighted interpolation. The search radius was set to variable due to the sparse and irregularly spaced sampled locations.

KRIGING

Figure 4: KRIGED surface

In a KRIGED surface, there is more variation within the central city than with either the SPLINE or IDW surface (Figure 4). This surface also has cooler values on the west / southwestern side which are more consistent with reality than the other modeled surfaces. There are several parameters to be set when using the KRIGING option for creating an interpolated surface. First, is the selection of whether to use the ordinary or universal method. In this study, ordinary kriging was selected because there is no known overriding trend in the data. Ordinary analysis is most widely used in KRIGING. For this data set, the assumption that the constant means are unknown is true. The default spherical model with a variable radius was selected due to the sparse and irregular location of the weather stations. For this surface, the advanced options were not used. The output of the selection criteria described above is Figure 4.

Assessment

The summary univariate measures, the mean of the observed (Obar), mean of the predicted (Pbar), difference between Obar and Pbar, and their standard deviations (s_o, s_p) are displayed in Table 1.
 
 
 

		Summary univariate measures
Method			Obar	Pbar	Obar – Pbar	so	sp	n
Spline			28.09	26.89	1.20	0.94	1.27	11
IDW			28.09	26.86	1.23	0.94	0.85	11
KRIGING			28.09	26.88	1.21	0.94	0.48	11

Table 1: Summary Univariate Statistics of Three Interpolation Methods.

The terms are in degrees Celsius except n, b and d, which are dimensionless.

The summary statistics Obar and Pbar show that, on average, all surfaces under predict the air temperature. The IDW data has the largest average error (1.23) and the SPLINE data the smallest (1.20). The average predicted data (Pbar) for all of the methods fall within 0.03 degrees C of each other. The difference between standard deviations suggest the IDW method is most similar, followed by the SPLINE method, with the KRIGING method resulted in s_p furthest from the s_o. These suggest that the KRIGING method is slightly less able to reproduce the observed variance. Willmott (1984) suggests that these statistical measures should not be over analyzed. In general, all of the methods are the same.

Table 2 shows the calculation using Willcott equations for the five difference measures. The MAE and RMSE measures disagree with the univariate statistics on the potentially better interpolation methods with KRIGING having both a lower MAE and RMSE. The RMSE_s and RMSE_u measures have a similar response as the MAE and the RMSE in that the KRIGING method has the lowest RMSE_u. Since the majority of the RMSE error for KRIGING is in the systematic RMSE then there is potential to improve the method by tuning or refining the parameters. The final error difference measure, d, also indicates that KRIGING (d=0.44) is a bit better than both SPLINE (d=0.06) and IDW (d=0.41).
 

		Simple linear OLS coefficients		Difference Measures
Method	n	a	b	MAE	RMSE	RMSEs	RMSEu	d
SPLINE	11	43.95	-0.59	1.87	2.34	0.55	2.27	0.06
IDW	11	41.56	-0.52	1.63	1.96	0.95	1.39	0.41
KRIGING	11	28.82	-0.69	1.41	1.62	1.44	0.50	0.44

Table 2: Difference Measures between Three Interpolation Methods

The terms are in degrees Celsius except n, b and d, which are dimensionless.

Discussion

The five difference measures consistently identified KRIGING as the best method for interpolating surfaces, followed by IDW and then SPLINE. Nevertheless, as illustrated in Figure 5 none of the interpolation methods did a reasonable job of estimating the actual temperatures. This is because the variation in temperature with eleven sample points is too small to assess temperature over 110 km² study area. It is also true that only having twenty-five points to interpolate across that area is also a significant limitation because there are little differences in temperatures between the climate stations. It is impossible, however, to increase the sample size of either because we are constrained by the number of existing climate stations in a given area.

Figure 5: Scatterplot of the Observed Temperatures to Estimated (predicted) Temperatures

Conclusion

This study has shown that KRIGING is most likely to produce the best estimation of a continuous surface of air temperature. Nevertheless, the research presented here illustrates that regardless of the approach taken these interpolation methods do not adequately address the temperature variability inherent in an urban setting. As a result, it is critical that additional factors, unique to the urban environment be incorporated into spatial interpolation methods for a more realistic representation of this area. Consequently, an estimated surface of temperature based on KRIGING can be used as input into a more complex spatio-temporal integration model (STIM) to generate a better representation of temperature.
 
 

Appendix A: Equations for Difference Measures

References

2000. Hits and Misses: Fast growth in metropolitan Phoenix.: Morrison Institute for Public Policy.

Balling Jr., R. C. and Nasrallah, H. A., 1991. The impact of rapid urbanization on summer weather stress. Journal of Arid Environments 21:261-265.

Benedict, K. K., 2001. The application of meteorological interpolation methods to dendroclimatic data from Northeastern Arizona: Methods, assumptions and results, In Annual Meeting of the Society of American Anthropologist, New Orleans.

Brazel, A.; Selover, N.; Vose, R.; and Heisler, G., 2000. The tale of two climates: Baltimore and Phoenix urban long-term ecological research sites. Climate Research 15:123-135.

Bureau, U. S. Census, 2001. Census 2000 PHC-T-3. Ranking Tables for Metropolitan Areas: 1990 and 2000. URL: http://blue.census.gov/population/cen2000/phc-t3/tab03.pdf.

Burrough, P. A. and McDonnell, R. A., 1998. Principles of Geographical Information Systems., Spatial Information Systems and Geostatistics: New York, Oxford University Press Inc, 333 p.

Chou, Y.-H., 1997. Exploring Spatial Analysis in Geographic Information Systems.: Santa Fe, OnWord Press, 474 p.

Collins, J. P.; Kinzig, A.; Grimm, N. B.; Fagan, W. F.; Hope, D.; Wu, J.; and Borer, E. T., 2000. A new urban ecology. American Scientist 88:416-425.

Johnston, K.; Ver Hoef, J. M.; Krivoruchko, K.; and Lucus, N., 2001. Using ArcGIS Geostatistical Analyst.: Redlands, ESRI, 300 p.

Lam, N. S.-N., 1983. Spatial interpolation methods: a review. The American Cartographer 10:129-149.

Meyers, D. E., 1994. Spatial interpolation: an overview. Geoderma 62:17-28.

Oke, T. R., 1988. The Urban Energy Balance. Progress in Physical Geography 12:471-508.

Owen, T. W.; Carlson, T. N.; and Gillies, R. R., 1998. An assessment of satellite remotely-sensed land cover parameters in quantitatively describe in the climatic effects of urbanization. International Journal of Remote Sensing 19:1663-1681.

Robeson, S. M. and Willmott, C. J., 1996. Spherical spatial interpolation and terrestrial air temperature variability. In GIS and modeling: progress and research issues, ed. M. F. Goodchild, L. T. Steyaert, B. O. Parks, C. Johnston, D. Maidment, M. Crane, and S. Glendinnin, pp. 111-115. Fort Collins, CO: GIS World Books.

Schut, G., 1976. Review of interpolation methods for digital terrain models. The Canadian Surveyor 30:389-412.

Shumaker, L. L., 1976. Fitting surfaces to scattered data. In Approximation theory II, ed. G. Lorentz, and e. al, pp. 203-67. New York: Academic Press.

Snell, S. E.; Gopal, S.; and K., K. R., 2000. Spatial interpolation of surface air temperatures using artificial neural networks: Evaluating their use for downscaling GCMs. Journal of Climate 13:886-895.

Stein, M. L., 1999. Interpolation of Spatial Data: Some theory for kriging., Springer Series in Statistics.

Willmott, C. J., 1984. On the evaluation of model performance in physical geography. In Spatial Statistics and Models, ed. G. L. Gaile, and C. J. Willmott, pp. 443-460.