Texas A&M University. Zachary Department of Civil Engineering. Instructor: Dr. Francisco Olivera. CVEN 658 Civil Engineering Applications of GIS

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1 1 Texas A&M University Zachary Department of Civil Engineering Instructor: Dr. Francisco Olivera CVEN 658 Civil Engineering Applications of GIS The Use of ArcGIS Geostatistical Analyst Exploratory Spatial Data Analysis and an Integrated Regionalization of Colorado Precipitation and Elevation Data Lacey Bodnar December 6, 2010

2 2 Contents Abstract Introduction Literature Review Methodology... 5 The reference material for all information about, and procedures regarding, the Geostatistical Analyst extension is the ESRI manual ArcGIS 9: Using ArcGIS Geostatistical Analyst by Johnson et al Geostatistics The Model for Ordinary Kriging and Cokriging ArcGIS Geostatistical Analyst Exploratory spatial data analysis Validation Application, Results, and Discussion Description of the Study Area and Datasets Colorado Topographic Features Colorado Climate Datasets Results Exploratory Spatial Data Analysis Regionalization Maps Discussion Conclusion Works Cited Appendix A: Exploratory Spatial Data Analysis Graphs Appendix B: Regionalization Maps ArcGIS Default Map: Inverse Distance Weighted ESDA Map: Ordinary Kriging with Outlier Removal and 2nd Degree Local Trend Removal Elevation Map: Ordinary Cokriging with Outlier Removal and 2 nd Degree Local Trend Removal Elevation and World Image Map... 27

3 3 Abstract The primary goal of this regionalization analysis was to determine which interpolation methodology resulted in the least prediction error for precipitation in the mountainous state of Colorado. This error was systematically quantified for three models (inverse distance weighted, kriging, and cokriging) and nine variations, including combinations of logarithmic transformations, trend removal, anisotropy, and outlier removal. A secondary objective was to account for the effect of elevation on spatial variation in precipitation. The results of method comparisons demonstrated that most method variations are improvements in certain error measurements, but not in all. Variations from the default were better at improving mean error than maximum or minimum error. By combining several variations, it was possible to arrive at a prediction map with all error values lower than the default setting. Using the Ordinary kriging and cokriging methods in conjunction with outlier and 2 nd degree local trend removal resulted in the lowest error. The fact that the cokriging map, of which elevation is a secondary variable, produced the least error confirms the expectation that precipitation and elevation are positively correlated. 1 Introduction Regionalization procedures allow scientists and water managers to take point data, such as precipitation and stream flow, and extrapolate values for locations where no monitoring is in effect. This is essential because it is not possible to put a gauging station at every possible location. Additionally, large scale analysis, such as watershed modeling or soil quality models, often requires knowing aggregate values, such as total runoff (Carrera-Hernandez et al. 2007, Shen et al. 2001, Langella et at. 2010). In turn, calculating runoff requires first knowing the depth of precipitation across a surface. This report demonstrates how the application of regionalization procedures to point data allows for the creation of a continuous predicted rainfall surface map.

4 4 This project explores the uses of ArcGIS Geostatistical Analyst for Exploratory Spatial Data Analysis (ESDA) and large scale regionalization of average monthly precipitation for the year 2000 across the state of Colorado. A secondary objective of the project is to account for the effect of elevation on precipitation across the Rocky Mountains. An advantageous feature of geostatistical methods is that they allow for the statistical determination of the accuracy of the predicted surface. This capability, called validation, was applied to compare multiple regionalization methods and arrive at a final recommendation for the method with least error. 2 Literature Review A great variety of regionalization methods exist, and their complexity tends to increase with time. In 1998, Frei and Schar used a moderately simple distance weighted deterministic method to assess precipitation climatology across the Eurpoean Alps. In 2001, Shen et al. used a method of nearest-station assignment to average a dense network of gridpoints into polygons for assessment of soil quality models in Alberta, Canada. More recently, Langella et al. (2010) investigated the application of neural computing for matrix analysis in regionalization of climate data in the Campaina regions of Italy. The link between elevation and precipitation has also been studied by several authors. In 2007, J.J. Carrera-Hernandez and S.J. Gaskin used kriging algorithms to analyze the spatial and temporal characteristics of minimum and maximum temperature, precipitation, and its correlation with elevation. Additionally, in 2010, Diodato et al. used cokriging in GIS to estimate rainfall in the Eastern Nepalese Mountains. This report goes beyond that research by incorporating exploratory spatial data analysis in the selection of numerous variations in kriging models. A validation of each variation allows for the quantification of error in a large scope assessment of model prediction accuracy, and enables the author to make recommendations for the highest accuracy modeling.

5 5 3 Methodology The reference material for all information about, and procedures regarding, the Geostatistical Analyst extension is the ESRI manual ArcGIS 9: Using ArcGIS Geostatistical Analyst by Johnson et al. 3.1 Geostatistics Geostatistical interpolation is based on the assumptions that locations that are closer together will be more similar than locations that are farther apart. As a result, the values of closer points are weighted more heavily than those far away. The empirical semivariogram is the key link between point data and regionalized surfaces. It relates the distance between points to the difference squared between their values; it is the graphical representation of the similarity of points with distance. The model that is best fit to the semivariogram is then used in the interpolation method that creates a surface prediction. The geostatistical interpolation methods chosen for this report were kriging and cokriging The Model for Ordinary Kriging and Cokriging In kriging, the statistical weighting is based not only on the distance between measured points, but also the overall spatial relationships between locations. Cokriging uses the same procedure, but allows for the assessment of multiple variables which have an effect on the variable of interest. The model for kriging is based on the following equations: Eq 1) Eq 2) The value at location s [Z(s)] is equal to a constant mean (µ) plus random errors with spatial dependence *ε(s)+.

6 6 The predicted value [Ź(s 0 )] is equal to the sum from one to N of the product of an unknown weight (λ i ) of the observed value at the ith location and the value at the ith location [Z(s i )]. Eq 3) The lowest error occurs when the difference between the true value [Z(s 0 )], and the predictor *Σλ i Z(s i )] is as small as possible. This occurs when the result of Eq 3 is minimized. Eq 4) The purpose of kriging is to solve for all the weights (λ). The gamma (Г) matrix is populated with semivariogram values based on a given distance between two locations i and j. The vector g is a list of semivariogram values between the predicted location and each known point. The procedure for cokriging is conceptually similar, but it complicated by the fact that the covariance of two or more variables must be accounted for in predictions. By solving these kriging equations for the given sample points, the ArcGIS Geostatistical Analyst was able to produce regionalized surface maps.

7 7 3.2 ArcGIS Geostatistical Analyst For the purposes of the geostatistical analysis used in this project, the raw data for each point (precipitation station) must have at least the following five attributes: STATION ID, LATITUDE, LONGITUDE, AVERAGE ANNUAL PRECIPITATION, and the station ELEVATION. The station ID wasthen used to identify each point and relate it to its record in the precipitation database. Latitude and Longitude was needed for the Display XY Data function in ArcGIS to plot each point on the map. The precipitation record was the primary spatial parameter analyzed by the Geostatistical Wizard. Elevation was used as the second spatial parameter in the cokriging operation. The Geostatistical Analyst extension allows for the exploration of spatial data. An improved understating of the characteristics of a data set can be used to make improvements to the regionalization model. The extension also provides a Geostatistical Wizard for prediction mapping. The user has the option of accepting the pre-defined (default) settings, or making adjustments to better represent the spatial properties of the data. The default settings were used in this report to create the first regionalization map. Then, multiple adjustments were assessed based on information obtained from the spatial data exploration. A comparison of the error associated with these models, to the error of the default model, allowed for a quantifiable assessment of the advantages and disadvantages of each attempted model Exploratory spatial data analysis The tools available for ESDA in the Geostatistical Analyst include: Histogram, Normal QQ Plot, Trend Analysis, Voronoi Map, Semivariogram / Covariance Cloud, General QQ Plot, and Crosscovariance Cloud. Each tool is defined in greater detail on the following pages.

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10 Validation Geostatistical validation required that the points be split into two groups, one to be used for the prediction surface (training), and the other to contain known values used to test the accuracy (test). Using the Geostatstical Analyst function Create Subsets, the data was split 70:30 and put in a Training_Testing geodatabase. The validation function was then invoked after the generation of a regionalized map by simply clicking on the map layer and selecting Validation. Validation calculated an error associated with each point in the test group. The statistics of the error column of the test station attribute table was used as a quantifiable measure of the various interpolation methods. 4 Application, Results, and Discussion 4.1 Description of the Study Area and Datasets The scope of the project included the entire state of Colorado. Colorado extends from 37 to 41 N latitude and from 102 to 109 W longitude (Doesken et al, 2003). It is the eighth largest American state, and has an area of over 104,000 mi 2. As a mountainous state, Colorado was chosen in order to develop a method for integrating elevation spatial data with predictions of precipitation. Records for average monthly precipitation in 2000 were used from 185 precipitation stations distributed across the state. The relative locations of the sites are shown on the map below. Figure 1: Colorado Study Sites

11 Colorado Topographic Features The average elevation of Colorado is 6,800 feet above sea level, making it the highest contiguous state (Doesken et al, 2003). It has 59 mountains over 14,000 feet, and 830 mountains between 11,000 and 14,000 feet. Along Colorado s eastern border, elevations range from approximately 3,350 feet to 4,000 feet. Elevations increase westward, and reach between 5,000 and 6,000 feet where the plains transition to the front range of the Rocky Mountains. In the Rocky Mountain foothills, elevation rise sharply to 7,000 to 9,000 feet. Across the mountains, elevations reach 9,000 feet, with the highest points over 14,000 feet. These topographic features affect temperatures, wind patterns, and storm tracks during all seasons. Though Colorado is well known for its mountains, the eastern high plains account for almost 40 percent of the state area. High ground is also present along the eastern border to the south along New Mexico, and to the north along Nebraska and Wyoming (Doesken et al, 2003) Colorado Climate The overall climate of Colorado is semi-arid (Doesken et al, 2003). It is highly affected by variations in elevation, and to a lesser extent, by the orientation of the mountains and valleys in regards to typical air movements. The average annual precipitation of Colorado is 17 inches, but that number varies significantly with location. Precipitation tends to decline gradually from the eastern border, and reaches the lowest point in the state near the Rocky Mountain foothills. Precipitation then increases rapidly across the mountain range as elevation increases. To the west of the Rockies, elevation declines, and precipitation gets progressively lower (Doesken et al, 2003).

12 Datasets The foundational dataset for the project was the monthly and annual precipitation data from climate gauging stations across Colorado. This data was retrieved in a raw format, meaning in an excel table only, with no link to ArcGIS. For the year 2000, there were 185 records available. The information for accumulated precipitation per month was averaged to find the average monthly precipitation. The information was provided by the Colorado Decision Support System, and may be found online at Three additional datasets were also used to facilitate the regionalization analysis. One was a Shapefile of the locations of Colorado Precipitation Stations, used to plot the stations in ArcMap. Important attributes of this dataset were: NAME, STATION_ID, LATDECDEG, and LONGDECDEG. The raw average annual precipitation data was joined to the precipitation stations Shapefile by the field STATION_ID. This allowed the stations with precipitation data to be projected in ArcGIS, using the Display XY function, based on the latitude and longitude record of the Shapefile. The joined table of precipitation station records was then split into a two datasets for the validation process. Seventy percent of the records (129 stations) were included in the training set. The remaining thirty percent (56 stations) were put in the test set, and stored in a personal geodatabase. The remaining two datasets, State Boundaries and World Imagery, were used for presentation purposes only, to help display the location and characteristics of the project site. Table 2: Description of Data Sets Data Source Format URL Monthly and Annual Precipitation Data Colorado Precipitation Stations Colorado Decision Support System Colorado Decision Support Excel table Shapefile System State Boundaries USGS Shapefile es/state_bounds/state_bounds.zip World Imagery ESRI JPEG Imagery/MapServer

13 Results Exploratory Spatial Data Analysis The Geostatistical Analyst was used to search for spatial trends in the state-wide distribution of precipitation. Graphs were generated for each of the ESDA tools described in section Please refer to Appendix A for the complete sequence of graphs. For a normal distribution, the mean will equal the median, the Skewness will equal one, and the Kurtosis will be equal to three. The histogram generated from the complete data set had a mean of and a median of , with a difference of The Skewness factor was , and the Kurtosis was Station 9181 appears to be a positive outlier. Positive outliers will contribute to a high positive skew factor. After removing the highest and lowest values (possible outliers), the mean became 1.21, the median , the Skewness , and the Kurtosis Also, a logarithmic transformation resulted in a mean of , and a median of The Skewness changed to , and the Kurtosis to The Normal QQ Plot was also used to assess data normalcy. The closer the data s quantile is to the straight line, the more normal the distribution. The data quanitles were high for both low (below -1.11) and high (above 1.67) standard normal values. For mid-range values, the quantiles fit the line well. After a log transformation, the low (below -1.67) standard normal values were below, rather than above, the line of normalcy. Trend analysis was used to study spatial patterns in the data. Two trends were discovered in the data. The green line projected against the XZ (north-south) plane represented a 2 nd order trend. With a rotation angle of 117:, the trend stretched from the N-NW to the S-SE. The blue line projected against the YZ plane at 45: is also a second order trend from the NE to the SW.

14 14 The standard deviation Voronoi map displayed areas of high standard deviation relative to neighboring values. The map showed that the greatest differences in precipitation are concentrated in western half of the map. The south western quadrant of the western half had especially high deviations Regionalization Maps The primary goal of this regionalization analysis was to determine which variation in methodology resulted in the least prediction error. A secondary objective was to account for the effect of elevation on spatial variation in precipitation. In order to accomplish these goals, three maps were created (See Appendix B: Regionalization Maps). The first map was generated using all the default ArcGIS Geostatistical Analyst settings. This map was called ArcGIS Default Map: Inverse Distance Weighted. The second map was produced using kriging and adjustments chosen from the exploratory spatial data analysis. Many variations of models were tested, and their error was validated. A comparison of the model errors is provided in Table 3. The second map, called ESDA Map: Ordinary Kriging with Outlier Removal and 2nd Degree Local Trend Removal was made using the model variations which produced the least error. The third map was generated using cokriging, with elevation of the stations as the second dataset. It was called Elevation Map: Ordinary Cokriging with Outlier Removal and 2nd Degree Local Trend Removal.

15 15 Table 3: Comparison of Regionalization Models Default Inverse Distance Weighted Minimum: Maximum: Mean: SD: Model Type Ordinary Kriging Ordinary Cokriging Model Variation Error Advantage Disadvantages Default MIN: MAX: Mean: SD: Log Transformation Global 2 nd degree trend removal Local 2 nd degree trend removal MIN: MAX: Mean: SD: MIN: MAX: Mean: SD: MIN: MAX: Mean: SD: Anisotropy MIN: MAX: Mean: SD: Outlier removal MIN: MAX: Mean: SD: Outlier Removal 2 nd degree local trend removal MIN: MAX: Mean: SD: Default MIN: MAX: Mean: SD: Outlier Removal 2 nd Degree Local Trend Removal MIN: MAX: Mean: SD: The mean error is reduced compared to the default IDW. Mean error and error SD are less than default. Mean error is reduced compared to default. Mean error is reduced compared to default. Mean error is lower than default. SD is slightly reduced. MAX, Mean, and SD are improvements over the default setting. All values are improvements over the default setting. Minimum and mean errors are better than the default IDW map. All values are improvements over the default setting of all maps. MIN, MAX, and SD measurements are higher than the default. Minimum and maximum error is slightly higher. Minimum and maximum error and standard deviation are slightly higher than default. Minimum and maximum error and standard deviation are higher. Minimum and maximum errors are higher. Minimum error is slightly larger. NA The MAX error and standard deviation of the error is slightly higher. NA

16 Discussion The results of method comparison demonstrated that most methods are improvements in certain respects, but not in all. A log transformation changed the error in ways consistent with what was observed on the log transformation of the Normal QQ Plot. As seen below, the transformation was an overall improvement. The maximum values and middle values are all better fit. However, the minimum values on the log graph deviated farther away from the line. This is consistent with slightly higher minimum error, and reduced maximum, mean, and standard deviation error of the log transformation regionalization map. Figure 2: Normal QQ Plot, No Transformation Figure 3: Normal QQ Plot, Log Transformation Removing trends proved to be effective at reducing average error. However, that came at a trade-off with higher minimum and maximum errors, and as a result, higher standard deviation. Trends are the non-random aspect of a spatial model, meaning they are deterministic. Recall the first kriging equation: Eq 1)

17 17 The value at location s [Z(s)] is equal to a constant mean (µ) plus random errors with spatial dependence *ε(s)+. Kriging methods are intended to account for random errors. Trend removal allows for the separation of deterministic variation from random variation in the ε(s) factor. As described earlier in section 4.2.1, two second-order trends were identified in the data. One stretched from the N-NW to the S-SE. The other extended from from the NE to the SW. Figure 4: Global Trend Removal Figure 5: Local Trend Removal Anisotropy was also effective in reducing mean error, and to a slight degree, standard deviation of error. However, the minimum and maximum errors increased. Anisotropy occurs when spatial autocorrelation changes with both distance and direction. Adding anisotropy to the regionalization method allowed for the incorporation of directional influences. In a situation such as this, where geographical variation produces large scale physical patterns that are related to the variable of interest, it seems that quantifying the effect of direction on the predicted surface will result in greater accuracy. Figure 6: Accounting for Anisotropy in Geostatistical Wizard

18 18 Method comparison showed that outlier removal is a strong technique for minimizing error. In the ordinary kriging default, log transformation, and trend removal error distributions, the highest positive error was always station number Investigating the map showed that the station with the highest positive error was located closest to the station with the highest recorded value (9181). When examining the histogram, station 9181 appears to be a definite positive outlier. Removing the outlier reduced the skew, as well as the maximum, average, and standard deviation of error. Figure 7: Spatial Relationship of Positive Outlier and Highest Positive Error The station with the highest negative error, station 1959, is also located in close proximity to the station with the lowest recorded average annual precipitation, station 797. This close spatial relationship, between the lowest value station and the most underestimated location from the kriging analysis, suggests that station 797 is skewing the results too low. However, removing the minimum value increased the minimum error from to Thus, it does not seem justified to say that station 797 is an outlier. Rather, it likely contributes to an accurate prediction of that area. The cokriging map was an improvement over the first two, because it produced the lowest

19 19 prediction error. The unique aspect of cokriging was factoring in the autocorrelation between elevation and precipitation. This reinforces the fact that precipitation increases with increasing elevation. By inspection, it is evident that the cokriging elevation map fits the geographic features of the state (See Elevation and World Imagery Map, Appendix B). Precipitation is highest (darkest red) over the mountain peaks. It tapers off moving east and west. Precipitation increases again noticeably along the eastern border. This is consistent with the topography of the state, where higher elevations occur to the south along New Mexico, and to the north along Nebraska and Wyoming (refer to Section 4.1.1). Conclusion The primary goal of this regionalization analysis was to determine which interpolation methodology resulted in the least prediction error for the mountainous state of Colorado. A secondary objective was to account for the effect of elevation on spatial variation in precipitation. The results of method comparisons demonstrated that most methods are improvements in certain error measurements, but not in all. Model variations from the default were better at improving mean error than maximum or minimum error. Every variation, including default kriging, log, trend, anisotropy, and outlier removal, resulted in a lower mean error than the default map, and either higher minimum or maximum error, or both. By combining several variations, it was possible to arrive at a prediction map with all error values lower than the default setting. Using the ordinary kriging and cokriging methods in conjunction with outlier and 2 nd degree local trend removal resulted in lower error. The fact that the cokriging map produced the least error confirms the expectation that precipitation increases with increasing elevation.

20 20 Works Cited Carrera-Hernandez, J.J., and Gaskin, S.J. (2007). Spatio temporal analysis of daily precipitation and temperature in the Basin of Mexico. Journal of Hydrology, 336, Diodato, N., Tartari, G., Bellocchi, G. (2010). Geospatial rainfall modeling at Eastern Nepalese Highland from ground environmental data. Water Resources Management, 24, Doesken, N.J., Pielke, R.A., Bliss, O.A.P. (2003). Climate of Colorado. Colorado State University, < (Nov. 13, 2010). Frei, C., and Schar, C. (1998). A precipitation climatology of the Alps from high-resolution rain-gauge observations. International Journal of Climatology, 18, Johnson, K., Ver Hoef, J.M., Krivoruchko, K., Lucas, N. (2003). ArcGIS9: Using ArcGIS Geostatistical Analysis. < (Nov. 13, 2010). Langella, G., Basile, A., Bonfante, A., Terribile, F. (2010). High-resolution space-time rainfall analysis using integrated ANN inference systems. Journal of Hydrology, 387, Shen, S.S., Dzikowski, P., Li, G., Griffith, D. (2001). Interpolation of daily temperature and precipitation data onto Alberta Polygons of Ecodistrict and Soil Landscapes of Canada. Journal of Applied Meteorology, 40,

21 21 Appendix A: Exploratory Spatial Data Analysis Graphs Histogram: No Transformation Histogram: Log Transformation Histogram: Outlier Removal (Highest and Lowest Value) Normal QQ Plot: No Transformation Normal QQ Plot: Log Transformation

22 22 Trend Analysis Voronoi Map: Mean Voronoi Map: Standard Deviation* *used to look for local variation Voronoi Map: Cluster used to look for local outliers

23 23 Semivariogram Covariance Cloud General QQ Plot* (High and Low Outlier Removed) *X-axis is precipitation, Y-axis is elevation Crosscovariance Cloud (High and Low Outlier Removed)

24 24 Appendix B: Regionalization Maps ArcGIS Default Map: Inverse Distance Weighted

25 ESDA Map: Ordinary Kriging with Outlier Removal and 2nd Degree Local Trend Removal 25

26 Elevation Map: Ordinary Cokriging with Outlier Removal and 2 nd Degree Local Trend Removal 26

27 Elevation and World Image Map 27