Assessment of areal interpolation methods for spatial analysis of SPI and EDI drought indices

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1 INTERNATIONAL JOURNAL OF CLIMATOLOGY Int. J. Climatol. 29: (29) Published online 12 March 28 in Wiley InterScience ( Assessment of areal interpolation methods for spatial analysis of SPI and EDI drought indices Rouhangiz Akhtari, a Saeed Morid, b * Mohammad Hossain Mahdian c and Vladimir Smakhtin d a Research Scholar, Tarbiat Modares University, College of Agriculture, Tehran, Iran b Tarbiat Modares University, College of Agriculture, Tehran, Iran c Soil and Watershed Conservation Research Institute, Tehran, Iran d International Water Management Institute, Colombo, Sri Lanka ABSTRACT: Drought monitoring is an essential component of drought risk management. Drought indices functions of precipitation showing the severity of dryness during a particular time period are often used for monitoring purposes. These indices may only be calculated originally at a limited number of sites where observations on climate variables are available. However, what is required for monitoring is to estimate the spatial distribution of drought severity over larger areas in the form of maps. In this article, several geostatistical methods including kriging, co-kriging and thin plate smoothing splines with and without secondary variables, as well as Thiessen polygons and weighted moving average were assessed for the derivation of maps of drought indices. The techniques are evaluated using 1-month Standardized Precipitation Index (SPI) and Effective Drought Index (EDI) at 43 climatic stations in the Tehran province of Iran. The results indicate that although kriging is the most accurate method, weighted moving average provides a reasonable accuracy combined with the simplicity and speed of the procedure, and thus, can be recommended for operational drought monitoring system in the region. Copyright 28 Royal Meteorological Society KEY WORDS spatial interpolation; kriging; WMA; TPSS; Thiessen; drought indices; Iran Received 23 March 27; Revised 31 December 27; Accepted 4 January Introduction Drought monitoring systems are essential tools in managing the risks associated with this natural disaster. Such systems make it possible to track drought onset, progress, severity and areal extent and can be used to trigger drought contingency plans, if those are available. Monitoring is normally performed using drought indices. Many drought indices have been developed to date. These include the Palmer Drought Severity Index (PDSI) (Palmer, 1965), which is widely used in the USA, the Deciles Index (Gibbs and Maher, 1967), which is operational in Australia, the China-Z Index (CZI), which is used by the National Metrological Center of China (Wu et al., 21), Standardized Precipitation Index (SPI) (McKee et al., 1993), which has gained world popularity, etc. The review of drought indices can be found in several sources (e.g. Morid et al., 26). Most of these indices are calculated using climate data from the meteorological stations, which are point measurements. For monitoring purposes, it is necessary to operationally produce the maps of drought severity from point measurements to trace drought development in the entire region or country. * Correspondence to: Saeed Morid, Tarbiat Modares University, College of Agriculture, Tehran, Iran. morid sa@modares.ac.ir A number of methods have been proposed for surface interpolation of climate variables like rainfall and temperature. The Thiessen Polygon (TP) method (Thiessen, 1911) is the oldest and simplest, and is being widely used (Driks et al., 1998; McCuen, 1998; Goovaerts, 2). Other methods include the Weighted Moving Average (WMA) (Bedient and Huber, 1992) and the Geostatistics (GS) approach. The latter is based on the theory of regionalized variables. Unlike the Thiessen and WMA methods, the GS method allows users to capitalize on spatial correlation between neighbouring observations to predict attribute values at un-gauged locations. This approach includes a family of methods like kriging, cokriging, kriging with an external drift, Thin Plate Smoothing Splines (TPSSs) and some others (Zheng and Basher, 1995). The comparison of these surface interpolation methods has been the subject of many meteorological studies. Among the climate variables, rainfall attracts the greatest attention. Several authors have shown the superiority of the GS methods over the conventional methods for estimation of rainfall at un-gauged locations. Goovaerts (2) compared the TP, WMA, ordinary kriging with varying local means, kriging with external drift and collocated co-kriging for spatial interpolation of monthly and annual rainfalls. The results showed large prediction errors of the TP and WMA, while ordinary kriging was more accurate. There are also studies that show that Copyright 28 Royal Meteorological Society

2 136 R. AKHTARI ET AL. the performance of the kriging method in the region with high density of observational networks (e.g. 13 rain gauges over 35 km 2 area) is not significantly better than that of simpler techniques (Borga and Vizzaccaro, 1997; Driks et al., 1998). In GS methods, a semivariogram plays the central role in the analysis. Skirvin et al. (23) showed the relation between the sill, threshold and the semivariogram model and topography and rainfall pattern. Holawe and Dutter (1999) evaluated spatial and temporal rainfall variation using GS methods. They illustrated the ability of the semivariogram to present the temporal variability of rainfall. More details about similar studies can be obtained in (Tabios and Salas, 1985; Sun et al., 2; Skirvin et al., 23; Touazi et al., 24). For spatial drought monitoring, some authors applied the WMA technique (Smakhtin and Hughes, 27; Svoboda, 24). Others suggest using a simple multiple linear regression-based model (Loukas and Vasiliades, 24; Livada and Assimakopoulos, 27). Less research, however, has been done on the comparison of different approaches for spatial interpolation of drought indices. This article intends to partially fill this gap and evaluate the performance of the TP, WMA, kriging, co-kriging and TPSS approaches for mapping of drought indices in the Tehran province of Iran. The province, which is frequently hit by droughts, is located in the northern part of Iran, has a total area of km 2 and a population of 14 million people. Annual precipitation varies from 7 mm in the north to 12 mm in the south. The comparison of spatial interpolation techniques is carried out using the time series of two drought indices SPI and Effective Drought Index (EDI) which were shown to perform well in the study area in the context of drought monitoring (Morid et al., 26). 2. Data and methods 2.1. Meteorological data There are more than 1 meteorological stations in the province, but due to the short period of records at some of them, the precipitation records from only 43 stations have been used (Figure 1). The record length at these stations is from January 197 to December 21. Figure 1 also shows the boundaries of 55 provincial cities administrative subdivisions, which are used later in the analysis Drought indices The SPI was designed by McKee et al. (1993) to quantify the precipitation deficit for multiple time scales. The index may be computed with different time steps (e.g. 1 month, 3 months, 24 months). This allows the effects of a precipitation deficit on different water resource components (groundwater, reservoir storage, soil moisture, streamflow) to be assessed. The SPI is defined for each of the above time scales as the difference between monthly precipitation (x i ) and the mean value (x), divided by the standard deviation (s), SPI = x i x (1) s Considering that the monthly precipitation data may consist of many zero values it is expected that precipitation values do not follow the normal distribution. Thus, Figure 1. A schematic map of the Tehran province showing the locations of the stations and the boundaries of the cities.

3 ASSESSMENT OF AREAL INTERPOLATION METHODS FOR SPATIAL ANALYSIS 137 Table I. Categorization of SPI and EDI values into classes. Drought class Class definition SPI EDI 3 Extremely wet Very wet Moderately wet Normal.99 to Moderately dry 1. to Severely dry 1.5 to Extremely dry one must perform initially a transformation of the data in order to follow a normal distribution. Therefore, a theoretical cumulative probability distribution function is adjusted on the precipitation data. Edwards and McKee (1997), applied the two-parameter gamma distribution for the calculation of SPI, while Guttman (1999), applied the Pearson type III distribution. The EDI is an attempt to more accurately determine the exact start and end of a drought period. Unlike most drought indices, the EDI in its original form is calculated with a daily time step (Byun and Wilhite, 1996). The EDI is a function of precipitation needed for a return to normal conditions (or to recover from the accumulated deficit since the beginning of a drought), which, in turn, is related to effective precipitation (EP). EP for any day is a function of precipitation of the current day, as well as previous days but with lower weights: [( i n ) / ] EP i = P m n (2) n=1 m=1 where i = duration of summation and P m = precipitation of m 1 days previously. The EP is the core of the EDI concept. The full EDI calculation procedure is rather complex; a simplified description of it is available in Morid et al. (26). The EDI varies in the range of 2.5 to 2.5. Similar to the SPI, it has thresholds indicating the range of wetness from extreme drought to extremely wet conditions. Categorization of SPI and EDI values is shownintablei Spatial interpolation procedures The spatial interpolation methods used in this study include TPs, WMA, Ordinary Kriging Method (OKM), co-kriging and TPSSs Thiessen polygons In this classical method, the study area is divided into regions (around the data points i.e. observation stations), which are defined by lines orthogonal to those connecting each nearest pair of stations. TPs result in a tiled surface rather than a continuous one (Figure 2) Weighted moving average The WMA is based on a weighting scheme so that closer observations tend to be more alike than those which are further apart. So, observations closer to the position x should receive larger weight. This can be calculated as: Z(x) = λ i = n λ i Z(x i ) (3) i=1 h u i n i=1 h u i h i R and λ i = h i >R (4) where n is the number of surrounding observations to be used for interpolation of each point within the selected radius (R), h is the distance between the observations (Z(x i )) and estimated values for unknown positions (Z (x)), u is the power of the equation (the higher values of u, the lower the effect of remote observations) Ordinary kriging method This method is a stochastic technique and local interpolator that uses the semivariogram as a measure of dissimilarity between observations. The general equation is similar to (2) above, but the weights are different. The weights associated with sample points are determined by Figure 2. The Thiessen polygons of Tehran province.

4 138 R. AKHTARI ET AL. direction and distance to other known points, so as to minimize the estimation variance between observations and estimations [i.e. Var(Z (x i ) Z(x i ))], while ensuring that the estimator [i.e. E(Z (x i ) Z(x i ) = )] is unbiased. These weights are obtained by solving a system of linear equations, which is known as ordinary kriging system. More details can be found in Isaaks and Srivastava (1989) or Goovaerts (1997). This method relies on the semivariogram to measure the dissimilarity between observations: γ(h) = 1 2n n {Z(x i ) Z(x i + h)} 2 (5) i=1 where n is the number of pairs of data point, which are separated by h distance; Z(x i ) and Z(x i + h) are the amounts of the variable Z at x i and x i+h locations. The structure of data may be described by four parameters: the sill, range, nugget and anisotropy. The variance value at which the curve reaches the plateau is called the sill. The total separation distance from the lowest variance to the sill is known as range. The nugget refers to variance at separation distance of zero. In theory, it should be zero. However, noise or uncertainty in the sample data may produce variability that is not spatially dependent. Anisotropy of the dataset describes spatial continuity with respect to the defined direction. It may be equal in all directions, which is known as an omnidirectional semivariogram. A more detailed description of kriging is available in Goovaerts (1997) Co-kriging method This method is an extension of the kriging interpolator that uses a second set of points of different attributes (e.g. elevation) to assist in the prediction process. The two attributes must be highly correlated with each other to derive any benefit from the process (Goovaerts, 1997) Thin plate smoothing splines This is a mathematically elegant model for surface estimation that has been progressively developed over the last decade. The model smoothes the data through minimizing the function, which combines the meansquare residuals and the roughness of the signal surface (Zheng and Basher, 1995). This method uses a covariancefunction to calculateλ(equation (3)) as follows, and it is possible to apply a different power (k): C(h) = h k log(h) k = m 1 (6) where C(h) is co-variance function, h is distance between the points, and m is order of relative derivation from the observed points. More details on TPSS can be found in Wahba (199) or Hutchinson (1991). 3. Results and discussion The performance of the five methods was evaluated and compared using the monthly data from the recent severe drought spell of and the cross-validation technique (Isaaks and Srivastava, 1989). The comparison criterions are mean absolute error (MAE) and mean biased error (MBE): MAE = 1 n MBE = 1 n n Z (x i ) Z(x i ) (7) i=1 n (Z (x i ) Z(x i )) (8) i=1 where n is the number of observations. The value of these criteria should be close to zero if the algorithm is accurate Semivariogram of indices During the calculation of drought indices, they get standardized so as to make them comparable between different parts of the region. To ensure that their spatial characteristics are not lost, monthly experimental semivariograms of the SPI, EDI and monthly rainfalls were drawn. The effect of changes in the directions was also examined that was not found to be significant. Therefore, omnidirectional semivariograms were applied for further analysis. The resulted experimental semivariograms showed a clear spatial structure of the indices. Figures 3 and 4 show example semivariograms of the indices for October 1998 and February 1999 along with the respective monthly rainfall and adjusted theoretical models. Different theoretical models (e.g. Gaussian and Spherical) were fitted to the experimental semivariograms and the best model was selected based on cross-validation procedure as explained in the previous section. Another criterion used was the ratio of the nugget to the sill (N : S), which is normally expressed as a percentage (Isaaks and Srivastava, 1989) (Table II). This ratio is referred to as the relative nugget effect. Values of the relative nugget effect near 1% indicate that a large degree of the variability is associated with the within sample measurements, and that relatedness between spatially separated measurements is limited. A relative nugget effect near zero indicates that the relatedness of spatially separated measurements within the range is strong. Table II shows the main variogram features from October 1998 to September 1999 (Iran water year). It can be seen that while average N : S ratios for rainfall in 1999 is 31.2%, they are 14.6 and 27% for the EDI and SPI, respectively. This indicates that relatedness of indices is higher than that of rainfalls, and that the EDI has higher relatedness than SPI. The EDI efficiency can be related to the concept of EP that this index uses. Similar results were observed in other years of the drought. Another interesting point is that even monthly ranges of the theoretical semivariogram of

5 ASSESSMENT OF AREAL INTERPOLATION METHODS FOR SPATIAL ANALYSIS 139 (a) (b) (c) Figure 3. The semivariograms of the drought indices [SPI (a), EDI (b)] and rainfall (c) for October (a) (b) (c) Figure 4. The semivariograms of the drought indices [SPI (a), EDI (b)] and rainfall (c) for February 1999.

6 14 R. AKHTARI ET AL. Table II. The main variogram features during October 1999 to September 2. Parameter Rainfall Effective drought index Standard precipitation index Month Model Nugget Sill Range (K M) N:S (%) Model Nugget Sill Range (K M) N:S (%) Model Nugget Sill Range (K M) N:S (%) Oct. EX GA SP Nov. SP GA GA Dec. GA GA SP Jan GA Feb GA SP Mar GA SP Apr GA May GA EX Jun. SP EX Jul GA GA Aug. GA EX Sep. GA GA Average SP.:Spherical model EX. Exponential model GA. Gaussian model. MAE-KG MAE-TPSS-2 MAE-WMA-2 MBE-KG MBE-TPSS-2 MBE-WMA-2.4 SPI.3.2 MAE.1 MBE Oct. Nov. Dec. Jan. Feb. March April May Jun July Aug. Sep..2 Figure 5. Evaluation of selected interpolation methods for the SPI monthly values. indices are higher than rainfall (column 5, 1 and 15 in Table II). This confirms that the indices have kept the spatial characteristics of rainfall Evaluation of interpolation methods For TPs, there is no specific parameter to be calibrated. But, WMA needs relevant amounts of u and n (Equations (3) and (4)). Using the cross-validation method and examining different values of these parameters, results in optimal values of 2 and 6, respectively. For kriging and co-kriging methods, the normalization of data is the initial requirement. To do this, monthly drought indices were normalized, using the Box-Cox transformation. However, normalization was not possible for a few months (e.g. April 1999) and application of the kriging methods was not feasible. The next step was evaluation of semivariograms and adjusting relevant theoretical models. For this, the number of neighbouring data points was set to be 16. Also, for the co-kriging method, elevation was used as a co-variable. But, due to low correlation between elevation and the drought indices (maximum R 2 =.2), this method was omitted from the subsequent analyses. In the TPSS method, the main parameter is k (Equation (6)). Again, performance of this method was evaluated while using a co-variable (elevation). On the basis of the resulting values for MAE and MBE, the power (k) was set to 2. Elevation did not improve the results significantly Comparison of the methods In addition to cross-validation with MAE and MBE, the maps resulted from the above methods have been evaluated visually. Figures 5 and 6 show the performance

7 ASSESSMENT OF AREAL INTERPOLATION METHODS FOR SPATIAL ANALYSIS 141 Table III. Performance of kriging, TPSS and WMA methods for spatial interpolation of drought indices during the drought spell. Indices SPI EDI Method Kriging TPSS WMA Kriging TPSS WMA Month MAE MBE MAE MBE MAE MBE MAE MBE MAE MBE MAE MBE Oct Nov Dec Jan Feb Mar Apr May Jun Jul Aug Sep MAE-KG MAE-TP-2 MAE-WMA-2.8 MBE-KG MBE-TP-2 MBE-WMA EDI MAE MBE Oct. Nov. Dec. Jan. Feb. March April May Jun July Aug. Sep..2 Figure 6. Evaluation of the selected interpolation methods for the EDI monthly values. of different methods for interpolation of the SPI and EDI, respectively. The TPSS has had better performance for the SPI based on the MAE and MBA. However, it has the highest fluctuation in the MBE (Figure 5). The performance of kriging can be ranked as second in terms of the MAE, but its MBE shows the lowest values among all the methods. The figure also illustrates close results of WMA and kriging. If the MAE is used as the main criterion, the TPSS will be selected as the best method. However, if both criteria are considered, kriging will outperform all. Similar results emerge for the EDI analyses (Figure 6). To visualize the results better, the SPI and EDI maps were drawn for the drought period of Figures 7 and 8 show the example drought maps for February Again, the similarities between kriging and WMA methods are clear. Despite the best results shown by TPSS in terms of MAE, it can be seen from the figures that drought detection for significant portions of the province is unrealistic. While the province suffers from the severe drought, the TPSS maps show very wet to extreme wet classes, especially in the eastern parts. This problem occurs in areas where the availability of rain gauges is scarce, like in the eastern part of the province, or some sections of the boundaries. The method is thus very weak in extrapolation. For instance, Figure 9 shows the status of the February 1998 drought the same as in Figure 7 but including areas beyond the boundary of the province. This behavior of TPSS can also be seen in other months. Thus, relying only on cross-validation is not always sufficient for evaluation of surface interpolation methods. Table III summarizes the performance of the GS methods for the 3 years of the drought spell Comparison of drought classes for the cities One of the main objectives of developing drought maps is tracking the status of drought in the cities administrative subdivisions of the province. The boundaries of the cities (Figure 1) were overlaid onto

8 142 R. AKHTARI ET AL. Figure 7. The SPI maps for February 1998 based on selected interpolation methods. Figure 8. The EDI maps for February 1998 based on selected interpolation methods. Table IV. The frequency of differences in detected drought classes by kriging and other surface interpolation methods for the cities of the Tehran province. Method Kriging TPSS Kriging WMA Kriging Thiessen DC a Standard precipitation index Entire drought period Effective drought index Entire drought period a Difference in classes. drought maps to examine the drought class of each city. Figure 1 shows detected SPI and EDI drought classes for all cities of the province in August 1999, using WMA and kriging methods. The same procedure was applied for other months and methods, and their respective maps were created to compare the detected drought classes by each interpolation method. Figure 11 shows one of these comparisons for February. The close detected drought classes by kriging and WMA and the differences by the TPSS are clear. This comparison was done for the entire drought spell, the differences between different methods having been compared with kriging, which yielded the best result (Table IV). In this table, DC (column 1 in Table I) stands for difference in classes. A DC of zero means that the detected drought class by kriging and the other method is the same, whereas 1 means that there is 1 class difference between kriging and others (e.g. normal is detected by kriging and moderate drought is detected by other methods). The table shows that over the entire period of the drought spell, the WMA and kriging detected EDI or SPI classes are the same for about 9% of the time for the cities of the province, and only 1 class difference has been observed. The next rank belongs to the Thiessen method. But in the case of TPSS, the differences are very high. Less than 65 and 55% of the time, the results are close to kriging, and up to 5 classes of differences have been observed with the results of kriging.

9 ASSESSMENT OF AREAL INTERPOLATION METHODS FOR SPATIAL ANALYSIS 143 WMA TPSS Kriging Figure 9. Detected drought status for February 1998 within and outside the province. Figure 1. Detected SPI and EDI drought classes for the cities of the Tehran province in August 1999, using WMA and kriging methods. 4. Conclusions This article compared a number of methods for spatial interpolation of drought indices to create drought maps of the Tehran province of Iran. The methods investigated include Thiessen, WMA, kriging and TPSS. The following conclusions can be drawn from this study: Based on the statistical criteria that were used to evaluate the performance of the methods, the TPSS and kriging appear to be the best techniques. However, the TPSS showed the highest sensitivity to inconsistent density of the stations in the province, which is a serious limitation of the method. Based on the statistical criteria and the accuracy of drought maps that resulted from kriging, this method could be the method of choice. However, it still has serious limitations including the high amount of required calculations, the need for expert judgment and the impossibility of normalizing the indices for a few months. Comparison of the WMA and kriging using the detection of drought classes for the cities of the province reveals very close results, such that for about 9% of all months the drought classes detected by the two methods are exactly the same. However, the WMA has an additional benefit of simplicity and quick calculations. On the basis of the analyses in this article, the WMA is recommended for spatial mapping of drought extent. The results of this study showed that relying solely on statistical criteria like MAE or MBA is not sufficient to decide which approach is superior. It is also necessary

10 144 R. AKHTARI ET AL. (a) The SPI drought class KRIGING TPSS WMA Thiessen The City No. of Tehran Province (b) The EDI drought class KRIGING TPSS WMA Thiessen The city No. of Tehran province Figure 11. The detected SPI and EDI drought classes for the cities of Tehran province in February 1999 based on the different interpolation methods (3 to 3 in the vertical axis refer to extremely wet, very wet, moderately wet, normal, moderately dry, severely dry, and extremely dry drought classes). that the created maps showing spatial distribution of drought indices are also examined. Despite the standardization approaches used in calculating drought indices, the variography analysis shows that the indices retain the spatial structure. Acknowledgements The authors are grateful to two anonymous reviewers for their helpful comments that substantially improved the manuscript. References Bedient PB, Huber WC Hydrology and Floodplain Analysis, 2nd edn. Addison-Wesley: Reading, MA. Borga M, Vizzaccaro A On the interpolation of hydrologic variables: formal equivalence of multiquadratic surface fitting and kriging. Journal of Hydrology 195(1 4): Byun HR, Wilhite DA Daily quantification of drought severity and duration. Journal of Climate 5: Driks KN, Hay JE, Stow CD, Harris D High resolution studies of rainfall on Norfolk Island. Part II: Interpolation of rainfall data. Journal of Hydrology 28(3 4): Edwards DC, McKee TB Characteristics of 2th century drought in the United States at multiple time scales. Climatology Report Number Colorado, State University: Fort Collins, CO. Gibbs WJ, Maher JV Rainfall Deciles as Drought Indicators, Bureau of Meteorology Bulletin No. 48. Commonwealth of Australia: Melbourne; 29. Goovaerts P Geostatistics for Natural Resources Evaluation. Oxford University Press: New York. Goovaerts P. 2. Geostatistical approaches for incorporating elevation into the spatial interpolation of rainfall. Journal of Hydrology 228: Guttman NB Accepting the Standardized Precipitation Index. A calculation logarithm. Journal of the American Water Resources Association 35: Holawe F, Dutter R Geostatistical study of precipitation series in Austria: Time and Space. Journal of Hydrology 219: Hutchinson MF The application of thin plate splines to continent-wide data assimilation. In Data Assimilation Systems, BMRC Research Report No. 27, Jasper JD (eds). Bureau of Meteorology: Melbourne; Isaaks EH, Srivastava RM Applied Geostatistics. Oxford University Press: New York. Livada I, Assimakopoulos VD. 27. Spatial and temporal analysis of drought in Greece using the Standardized Precipitation Index (SPI). Theoretical and Applied Climatology 89(3 4): Loukas A, Vasiliades L. 24. Probabilistic analysis of drought spatiotemporal characteristics in Thessaly region, Greece. Natural Hazards and Earth System Science 4: McCuen RH Hydrologic Analysis and Design, 2nd edn. Prentice Hall: Englewood Cliffs, NJ. McKee TB, Doesken NJ, Kleist J The relationship of drought frequency and duration to time scales. In Proceeding of the 8 th Conference on Applied Climatology, January 1993, Anaheim: California; Morid S, Smakhtin V, Moghaddasi M. 26. Comparison of seven meteorological indices for drought monitoring in Iran. International Journal of Climatology 26: Palmer WC Meteorological drought. Research Paper No. 45. U.S. Department of Commerce Weather Bureau: Washington, DC. Skirvin S, Stuart M, Marsh E, Mcklaran MP, Meko DM. 23. Climate spatial variability and data resolution in a semi-arid watershed, southeastern Arizona. Journal of Arid Environments 54: Smakhtin VU, Hughes DA. 27. Automated estimation and analyses of drought characteristics from monthly rainfall data. Environmental Modelling & Software 22(6): Sun X, Mein RG, Keenan TD, Elliott JF. 2. Flood estimation using radar and rain gauge data. Journal of Hydrology 239: 4 18.

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