Data Break 8: Kriging the Meuse RiverBIOS 737 Spring 2004 p.1/27

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1 Data Break 8: Kriging the Meuse River BIOS 737 Spring 2004 Data Break 8: Kriging the Meuse RiverBIOS 737 Spring 2004 p.1/27

2 Meuse River: Reminder library(gstat) Data included in gstat library. data(meuse) makes the data available. Soil samples from flood plain of Meuse River, near village of Stein (Belgium). Geostatistical data: locations s and attributes Z(s). Data Break 8: Kriging the Meuse RiverBIOS 737 Spring 2004 p.2/27

3 Kriging steps Empirical semivariogram. Fit theoretical semivariogram to empirical semivariogram values. Krige with best fitting semivariogram values. Data Break 8: Kriging the Meuse RiverBIOS 737 Spring 2004 p.3/27

4 What do we have? (Zinc) zinc concentrations (ppm) y x Data Break 8: Kriging the Meuse RiverBIOS 737 Spring 2004 p.4/27

5 What do we want? (Predictions here). Prediction Grid y x Data Break 8: Kriging the Meuse RiverBIOS 737 Spring 2004 p.5/27

6 Step 1: Data Gaussian? Recall from Data Break 6, log(zinc) closer to Gaussian than zinc. We will krige the log(zinc) values. (Note: biased prediction of exp(log(zinc)) = zinc, but we ignore this in the example). Data Break 8: Kriging the Meuse RiverBIOS 737 Spring 2004 p.6/27

7 Step 2: Empirical Semivariogram(s) logzinc.var <- variogram(log(zinc) 1, loc= x+y, data=meuse) plot(logzinc.var$dist, logzinc.var$gamma, xlab="distance", ylab="semivariance") Data Break 8: Kriging the Meuse RiverBIOS 737 Spring 2004 p.7/27

8 Resulting plot 0.6 semivariance distance Data Break 8: Kriging the Meuse RiverBIOS 737 Spring 2004 p.8/27

9 Some other cool plots The variogram cloud consists of the plot of all contrasts Z(s i ) Z(s j ) plotted versus s i s j plot(variogram(log(zinc) 1, loc= x+y, data=meuse, cloud=true)) Also some need interactive commands here... Data Break 8: Kriging the Meuse RiverBIOS 737 Spring 2004 p.9/27

10 Variogram cloud 3 semivariance distance Data Break 8: Kriging the Meuse RiverBIOS 737 Spring 2004 p.10/27

11 Step 2: Fit Theoretical Semivariograms fit.variogram function in gstat Not entirely clear what optimization rule, based on practical experience rather than theoretical motivation. To get best fitting spherical semivariogram... model.1 <- fit.variogram(logzinc.var, vgm(psill=1,model="sph", range=300,nugget=1)) plot(logzinc.var, model=model.1) title("best Fit Spherical") Data Break 8: Kriging the Meuse RiverBIOS 737 Spring 2004 p.11/27

12 Best spherical Best Fit Spherical 0.6 semivariance distance Data Break 8: Kriging the Meuse RiverBIOS 737 Spring 2004 p.12/27

13 Step 2: Fit Theoretical Semivariograms To get best fitting exponential semivariogram... model.1 <- fit.variogram(logzinc.var, vgm(psill=1,model="exp", range=300,nugget=1)) plot(logzinc.var, model=model.1) title("best Fit Exponential") Data Break 8: Kriging the Meuse RiverBIOS 737 Spring 2004 p.13/27

14 Best Exponential Best Fit Exponential 0.6 semivariance distance Data Break 8: Kriging the Meuse RiverBIOS 737 Spring 2004 p.14/27

15 Step 2: Fit Theoretical Semivariograms To get best fitting Gaussian semivariogram... model.1 <- fit.variogram(logzinc.var, vgm(psill=1,model="gau", range=300,nugget=1)) plot(logzinc.var, model=model.1) title("best Fit Gaussian") Data Break 8: Kriging the Meuse RiverBIOS 737 Spring 2004 p.15/27

16 Best Gaussian Best Fit Gaussian 0.6 semivariance distance Data Break 8: Kriging the Meuse RiverBIOS 737 Spring 2004 p.16/27

17 Step 2: Fit Theoretical Semivariograms To get best fitting Matern semivariogram... model.1 <- fit.variogram(logzinc.var, vgm(psill=1,model="mat", range=300,nugget=1)) plot(logzinc.var, model=model.1) title("best Fit Matern (kappa=0.25)") Data Break 8: Kriging the Meuse RiverBIOS 737 Spring 2004 p.17/27

18 Best Matern (κ = 0.25) Best Fit Matern (kappa=0.5) 0.6 semivariance distance Data Break 8: Kriging the Meuse RiverBIOS 737 Spring 2004 p.18/27

19 What s best? We ll use the spherical theoretical semivariogram. (Skipping formal AIC or other comparison). What are the best fit values? > model.1 model psill range 1 Nug Sph Nugget: 0.051, Partial sill: 0.591, Range: Data Break 8: Kriging the Meuse RiverBIOS 737 Spring 2004 p.19/27

20 Kriging # define the theoretical semivariogram m <- vgm(psill=0.591,"sph", range=896.97,nugget=0.051) # ordinary kriging: logzinc.krige <- krige(log(zinc) 1, x+y, model = m, data = meuse, newd = meuse.grid) # make levelplot levelplot(var1.pred x+y, data=logzinc.krige, aspect = mapasp(logzinc.krige), main = "ordinary kriging predictions") Data Break 8: Kriging the Meuse RiverBIOS 737 Spring 2004 p.20/27

21 Prediction levels ordinary kriging predictions y x Data Break 8: Kriging the Meuse RiverBIOS 737 Spring 2004 p.21/27

22 Kriging Variance ordinary kriging variance y x Data Break 8: Kriging the Meuse RiverBIOS 737 Spring 2004 p.22/27

23 Prediction levels Prediction (log zinc) Prediction x y Data Break 8: Kriging the Meuse RiverBIOS 737 Spring 2004 p.23/27

24 Prediction levels (zinc) Prediction (zinc) Prediction x y Data Break 8: Kriging the Meuse RiverBIOS 737 Spring 2004 p.24/27

25 Kriging variance Kriging variance surface Kr. var x y Data Break 8: Kriging the Meuse RiverBIOS 737 Spring 2004 p.25/27

26 Notes MSPE highest where fewest observations. Back-transforming to original zinc levels gives idea of prediction, but is no longer BLUP (we lose L and U ). gstat does much more complicated analyses too. Data Break 8: Kriging the Meuse RiverBIOS 737 Spring 2004 p.26/27

27 Summary of geostatistical methods We ve really only seen the whirlwind tour of the basic elements. Semivariogram. Ordinary kriging. Many other options too. Questions? Data Break 8: Kriging the Meuse RiverBIOS 737 Spring 2004 p.27/27

28 HW 4 Repeat this analysis for the Smoky Mountain ph data. Should be able to use the data break code to do this. Due April 28, graphs, output. Data Break 8: Kriging the Meuse RiverBIOS 737 Spring 2004 p.28/27

29 Project 2 Write up results of HW 4 as a report with Introduction, Methods, Results, and Discussion sections. Due May 3, 5pm. Data Break 8: Kriging the Meuse RiverBIOS 737 Spring 2004 p.29/27