Introduction to applied geostatistics. Short version. Overheads

Transcription

1 Introduction to applied geostatistics Short version Overheads Department of Earth Systems Analysis International Institute for Geo-information Science & Earth Observation (ITC) < March 21, 2007

2 Introduction to applied geostatistics 1 Topic: Resources There are many resources, at various mathematical levels, some aimed at particular applications. These lists are not comprehensive but should be good starting points: Texts Web pages Computer programmes

3 Introduction to applied geostatistics 2 Texts: Mathematical Chilès, J.-P. and Delfiner, P., Geostatistics: modeling spatial uncertainty. Wiley series in probability and statistics. John Wiley & Sons, New York. Christakos, G., Modern spatiotemporal geostatistics. Oxford University Press, New York. Cressie, N., Statistics for spatial data. John Wiley & Sons, New York. Ripley, B.D., Spatial statistics. John Wiley & Sons, New York.

4 Introduction to applied geostatistics 3 Texts: In the context of a particular application field Davis, J.C., Statistics and data analysis in geology. John Wiley & Sons, New York. Fotheringham, A.S., Brunsdon, C. and Charlton, M., Quantitative geography : perspectives on spatial data analysis. Sage Publications, London ; Thousand Oaks, Calif. Stein, A., Meer, F.v.d. and Gorte, B.G.F. (Editors), Spatial statistics for remote sensing. Kluwer Academic, Dordrecht. Kitanidis, P.K., Introduction to geostatistics : applications to hydrogeology. Cambridge University Press, Cambridge, England.

5 Introduction to applied geostatistics 4 Texts: Application-oriented but mathematical Webster, R., and Oliver, M. A., Geostatistics for environmental scientists. Wiley & Sons, Chichester. Goovaerts, P., Geostatistics for natural resources evaluation. Oxford University Press, Oxford and New York. Isaaks, E.H. and Srivastava, R.M., An introduction to applied geostatistics. Oxford University Press, New York.

6 Introduction to applied geostatistics 5 Texts: Emphasis on computational methods Venables, W.N. & Ripley, B.D., Modern applied statistics with S, 4 th edition. Springer-Verlag, New York. Deutsch, C. V., & Journel, A. G., GSLIB: Geostatistical software library and user s guide. Oxford University Press, Oxford.

7 Introduction to applied geostatistics 6 Web pages R: R spatial projects: gstat: gslib: GEOEAS: ILWIS: ArcGIS Geostatistical Analyst: http: // Geostatistical analysis tutor [Colorado (USA) School of Mines]:

8 Introduction to applied geostatistics 7 Computer programmes ILWIS 3.3 (ITC) R open-source environment for statistical computing and visualisation; includes several relevant libraries, including * gstat, by Pebesma * spatial, by Ripley * geor, by Ribeiro & Diggle * spdep, by Rowlingson & Diggle * spatstat, by Baddeley & Turner (point pattern analysis) * sp, underlying spatial data structures (used by others) ArcGIS Geostatistical Analyst (ESRI) [requires ArcGIS base] PCRaster + gstat (Utrecht) [free] GeoEAS, GSLIB, Variowin, VESPER...

9 Introduction to applied geostatistics 8 Topic: Introduction to Spatial Analysis 1. Concepts of space: geographic and feature spaces 2. What is special about spatial data? 3. Key concepts in spatial analysis 4. Measuring spatial correlation

10 Introduction to applied geostatistics 9 What is space? A set of n continuous dimensions; dimension i has range [x imin x imax ] Points are mathematical n-dimensional vectors: x = (x 1, x 2,, x n ) Depending on how we choose the axes, we can speak of both geographic and feature spaces...

11 Introduction to applied geostatistics 10 Feature space This space is not geographic space, but rather a mathematical space formed by any set of variables: Axes are the range of each variable Coordinates are values of variables, possibly transformed or combined Not included in the common use of the term spatial data or analysis But the observation may be related in this space and we often plot variables in this space, e.g. 2-D scatterplots This is the space in which univariate, bivariate, or multivariate analysis are carried out.

12 Introduction to applied geostatistics 11 Geographic space Axes are 1-d lines One-dimensional: coordinates are on a line with respect to some origin (0): (x 1 ) = x Two-dimensional: coordinates are on a grid with respect to some origin (0, 0): (x 1, x 2 ) = (x, y) = (E, N) Three-dimensional: coordinates are grid and elevation from a reference elevation: (x 1, x 2, x 3 ) = (x, y, z) = (E, N, H) Must transform latitude-longitude to grid coordinates in some 2-d projection; distortions occur over large areas Can work directly with geographic coordinates, but not as a grid

13 Introduction to applied geostatistics 12 What is special about spatial data? (1) 1. The location of a sample is an intrinsic part of its definition. 2. All data sets from a given area are implicitly related by their coordinates models of spatial structure 3. Values at sample points can not be assumed to be independent 4. That is, there may be a spatial structure to the data Classical statistics assumes independence, at least within sampling strata Major implications for sampling design and statistical inference 5. Data values may be related to their coordinates spatial trend

14 Introduction to applied geostatistics 13 Key Concepts Spatial dependence: the value of a variable at a point in space is related to its value at nearby points; knowing the value of these points allows us to predict (with some degree of certainty) the value at the chosen point Spatial structure: the nature of the spatial relation: how far, and in what directions, is the spatial dependence? How does the dependence vary with distance and direction between points? Support of a sample: the physical dimensions it represents (n.b. may try to predict to coarser or finer resolutions)

15 Introduction to applied geostatistics 14 Topic: Exploratory spatial data analysis Since spatial data were collected at known points in geographic space, we should visualise them in that space. Distribution of sample points Postplots (values vs. locations): where are which values? Geographic postplots: with images, landuse maps etc. as background: do there appear to be any explanation for the distribution of values? Spatial structure: range, direction, strength... Is there anisotropy? In what direction(s)? Do there seem to be several populations with distinct geographic distribution?

16 Introduction to applied geostatistics 15 Point distribution This shows how sample points are distributed in space. What was the sampling plan? Random or clustered? Are some areas over or under sampled?

17 Introduction to applied geostatistics 16 Example: Walker Lake: Distribution of points All points

18 Introduction to applied geostatistics 17 The Postplot: distribution of values in space The so-called postplot shows how the data values are distributed in space. Are values of closeby points similar to each other, or do the values appear to be random? Does there appear to be a trend? Are there distinct clusters of high or low values? Is there any directional difference in clustering? (anisotropy)

19 Introduction to applied geostatistics 18 Meuse Distribution of Log(Cadmium) in soils

20 Introduction to applied geostatistics 19 Geographic postplot This shows the postplot against a background that may explain the distribution of samples or values. Examples: land cover or land use geologic or soil units structural geology

21 Introduction to applied geostatistics 20 Meuse Log(Cadmium) on a false-colour composite

22 Introduction to applied geostatistics 21 Topic: Spatial correlation 1. What is spatial auto-correlation? 2. Evidence of spatial correlation 3. Computing spatial correlation and covariance 4. Summarizing and visualising spatial covariance; the empirical variogram Topics for later units: 1. modelling spatial correlation 2. predicting using the modelled structure

23 Introduction to applied geostatistics 22 Spatial Correlation Question: are nearby points in geographic space also nearby in feature space? That is, does knowing the value of some variable at some location give us information on the value at nearby locations? The concept of correlation between variables can be applied to correlation within a variable, using distance to model the relation

24 Introduction to applied geostatistics 23 Covariance and Correlation Recall: for two non-spatial variables X and Y : Sample covariance: s XY = 1 n 1 n (x i x) (y i y) i=1 Sample correlation coefficient: the covariance normalized by sample standard deviations; range [ ]: r XY = s XY s X s Y = (xi x) (y i y) (xi x) 2 (y i y) 2 Can we extend this idea to a single variable, which is then correlated with itself?

25 Introduction to applied geostatistics 24 Auto-correlation We want to apply the idea of correlation to one variable (auto-correlation); the prefix auto- means self, here referring to the single variable. Here, the correlation is controlled by some other dimension: time if the variable is collected as a time series space if the variable is collected at points in space So we will get a measure of how much the variable is correlated to itself, considering the other factor (time or space).

26 Introduction to applied geostatistics 25 Auto-covariance The spatial auto-covariance is computed within the same variable, using pairs of observations. Each pair of observations (x i, x j ) has a covariance, showing how they jointly differ from the variable s mean x: (x i x)(x j x) There are (n (n 1))/2 point pairs for which this can be calculated This is a large number! For example, with 200 points this is 19,900 point pairs.

27 Introduction to applied geostatistics 26 Modelling the auto-covariance By themselves the individual auto-covariances are not usefull; they just quantify the covariance of each point pair. We need to summarize the individual covariances as a covariance function of spatial separation Theory: the covariance depends only on the separation between point. If we can model this function we can then predict the covariance between any two locations in space.

28 Introduction to applied geostatistics 27 Semivariances It is easier to model semivariances than covariances: Each pair of observation points has a semivariance, usually symbolized by the Greek letter gamma, i.e. γ, defined as: γ(x i, x j ) = 1 2 [z(x i) z(x j )] 2 Each point pair is separated by a known distance, so... We can plot the semivariances against distance as a variogram cloud, with (n (n 1))/2 points in the graph Can also summarize in a variogram (The semi refers to the factor 1/2, because there are two ways to compute for the same point pair)

29 Introduction to applied geostatistics 28 The gstat package of R We illustrate the concepts of spatial correlation with the gstat package of the R environment and the meuse example data set. The meuse data frame has coördinates in fields x and y; these are used to promote the object to class SpatialPointsDataFrame. > # view package information > library(help=gstat) > # load the package > library(gstat) >?meuse > # load sample data > data(meuse) > # as loaded is a data frame > summary(meuse) > # promote to class SpatialPointsDataFrame > coordinates(meuse) <- ~ x+y > # now has explicit coordinates > summary(meuse)

30 Introduction to applied geostatistics 29 > summary(meuse) Object of class SpatialPointsDataFrame Coordinates: min max x y Is projected: NA proj4string : [NA] Number of points: 155 Data attributes: cadmium copper lead zinc elev Min. : 0.20 Min. : 14.0 Min. : 37.0 Min. : 113 Min. : st Qu.: st Qu.: st Qu.: st Qu.: 198 1st Qu.: 7.55 Median : 2.10 Median : 31.0 Median :123.0 Median : 326 Median : 8.18 Mean : 3.25 Mean : 40.3 Mean :153.4 Mean : 470 Mean : rd Qu.: rd Qu.: rd Qu.: rd Qu.: 674 3rd Qu.: 8.96 Max. :18.10 Max. :128.0 Max. :654.0 Max. :1839 Max. :10.52 dist om ffreq soil lime landuse dist.m Min. : Min. : :84 1:97 0:111 W :50 Min. : 10 1st Qu.: st Qu.: :48 2:46 1: 44 Ah :39 1st Qu.: 80 Median : Median : :23 3:12 Am :22 Median : 270 Mean : Mean : 7.48 Fw :10 Mean : 290 3rd Qu.: rd Qu.: 9.00 Ab : 8 3rd Qu.: 450 Max. : Max. :17.00 (Other):25 Max. :1000 NA s : 2.00 NA s : 1

31 Introduction to applied geostatistics 30 The empirical variogram To summarize the variogram cloud, compute average semivariance at various separations ( lags ); this is the empirical variogram γ(h) = m(h) 1 [z(x i ) z(x j )] 2 2m(h) i=1 m(h) is the number of point pairs separated by vector h In practice, we have to define the set of vectors in each bin (to have enough points); that is, we collect a distance range into one bin. (Note: there are other ways to estimate the variogram from the variogram cloud; in particular so-called robust estimators.)

32 Introduction to applied geostatistics 31 Example of an experimental variogram > (v <- variogram(log(cadmium)~1, data=meuse)) np dist gamma np are the number of point pairs in the bin; dist is the average separation of these pairs; gamma is the average semivariance in the bin.

33 Introduction to applied geostatistics 32 Plotting the experimental variogram This can be plotted as semivariance gamma against average separation dist, along with the number of points that contributed to each estimate np: > plot(v, plot.numbers=t) (Note: gstat defaults to 15 equally-spaced bins and a maximum distance of 1/3 of the maximum separation. These can be over-ridden with the width= and cutoff= arguments, respectively; or explicit bin limits can be set with the boundaries= argument.)

34 Introduction to applied geostatistics 33 Default variogram of Log(Cd) semivariance distance

35 Introduction to applied geostatistics 34 Features of the experimental variogram Later we will look at fitting a theoretical model to the experimental variogram; but even without a model we can notice some features, which we define here only qualitatively: Sill: maximum semi-variance * represents variability in the absence of spatial dependence Range: separation between point-pairs at which the sill is reached * distance at which there is no evidence of spatial dependence Nugget: semi-variance as the separation approaches zero * represents variability at a point that can t be explained by spatial structure In the previous slide, we can estimate the sill 1.9, the range 1200 m, and the nugget 0.5 i.e. 25% of the sill.

36 Introduction to applied geostatistics 35 Defining the bins (1) Distance interval, specifying the centres. E.g. (0, 100, 200,...) means intervals of [ ], [ ],... All point pairs whose separation is in the interval are used to estimate γ(h) for h as the interval centre Narrow intervals: more resolution but fewer point pairs for each sample > v <- variogram(log(cadmium)~1, meuse, boundaries = seq(50, 2050, by = 100)) > plot(v, pl=t) > par(mfrow = c(2, 3)) # show all six plots together > for (bw in seq(20, 220, by = 40)) { v<-variogram(log(cadmium)~1, meuse, width=bw) plot(v$dist, v$gamma, xlab=paste("bin width", bw))

37 Introduction to applied geostatistics 36 Variograms of Log(Cd) with different bin widths v$gamma v$gamma v$gamma bin width 20 bin width 60 bin width 100 v$gamma v$gamma v$gamma bin width 140 bin width 180 bin width 220

38 Introduction to applied geostatistics 37 Defining the bins (2) Each bin should have > 100 point pairs; > 300 is much more reliable > v <- variogram(log(cadmium)~1, meuse, width=20) > plot(v, plot.numbers=t) > v$np [1] [16] [31] [46] [61] [76] > v <- variogram(log(cadmium)~1, meuse, width=120) > v$np [1] > plot(v, plot.numbers=t)

39 Introduction to applied geostatistics 38 Topic: Modelling the variogram From the empirical variogram we now derive a variogram model which expresses semivariance as a function of separation vector. The model allows us to: Infer the characteristics of the underlying process from the functional form and its parameters; Compute the semi-variance between any point-pair, separated by any vector which is used in an optimal interpolator ( kriging ) to predict at unsampled locations.

40 Introduction to applied geostatistics 39 A variogram model, with parameters

41 Introduction to applied geostatistics 40 Authorized variogram models Only some functional forms can be used to model the variogram (theoretical and mathematical constraints) The permitted forms are called authorized models Simplest: The exponential model; sill c, effective range 3a γ(h) = c{1 e ( h a ) } E.g. if the effective range is estimated as 120, the parameter a is 40. Another common model: The Spherical model; sill c, range a γ(h) = c { 3h 2a 1 2 ( ) } h 3 a : h < a c : h a

42 Introduction to applied geostatistics 41 Graphs of authorized variogram models Linear with sill variogram model Circular variogram model Spherical variogram model semivariance sill nugget range semivariance sill nugget range semivariance sill nugget range separation distance separation distance separation distance Pentaspherical variogram model Exponential variogram model Gaussian variogram model semivariance sill nugget range semivariance sill nugget range semivariance sill nugget range separation distance separation distance separation distance

43 Introduction to applied geostatistics 42 MComparaison Mseparation Msemivariance MExponential MSpherical MGaussian MPentaspherical MCircular MLinear-with-sill distance of variogram models Comparaison of variogram models m0m0m 2M2M 4M4M 6M6M 8M8M m0.0 M0.0 M0.2 M0.4 M0.6 M0.8 M1.0 Msill Mnugget Mrange semivariance sill Exponential Gaussian Circular Spherical Pentaspherical Linear-with-sill 0.2 nugget 0.0 range separation distance 6 8 Models vary considerably, from origin to range

44 Introduction to applied geostatistics 43 Comparaison of models available in gstat > show.vgms()

45 Introduction to applied geostatistics 44 Choosing a model (1) The empirical variogram should be one realization of a random process. So, what do we expect from the process that is supposed to be responsible for the spatial structure represented in the variogram? Exponential: First-order autoregressive process: values are random but with dependency on the nearest neighbour; boundaries according to a Poisson process Gaussian: as exponential, but with strong close-range dependency, very smooth at each point.

46 Introduction to applied geostatistics 45 Choosing a model (1) continued Spherical, circular, pentaspherical: Patches of similar values; patches have similar size range) with transition zones (overlap of processes); These differ mainly in the shoulder transition to the sill

47 Introduction to applied geostatistics 46 Choosing a model (2) Which has been successfully applied with this kind of data? (This is evidence for the nature of this kind of process) What do we expect from the supposed process? if we have some other evidence of its spatial behaviour. For example, a Gaussian model might be expected for a phenomenon which physically must be very continuous, e.g. the surface of a ground-water table. Visual estimate of functional form from the variogram (Fit various models, pick the statistically-best fit)

48 Introduction to applied geostatistics 47 Fitting the model Once a model form is selected, then the model parameters must be adjusted for a best fit of the experimental variogram. By eye, adjusting parameters for good-looking fit * Hard to judge the relative value of each point * This is all that s possible in ILWIS Automatically, looking for the best fit according to some objective criterion * Various criteria possible in gstat In both cases, favour sections of the variogram with more pairs and at shorter ranges (because it is a local interpolator). Mixed: adust by eye, evaluate statistically; or vice versa

49 Introduction to applied geostatistics 48 Fitting a variogram model in gstat We ve decided on a spherical + nugget model: > # Calculate the experimental variogram and display it > v1 <- variogram(log(cadmium)~1, meuse); plot(v1, plot.numbers=t) > # Fit by eye, display fit > m1 <- vgm(1.4, "Sph", 1200, 0.5); plot(v1, plot.numbers=t, model=m1) > # Let gstat adjust the parameters, display fit > m2 <- fit.variogram(v1, m1); m2 model psill range 1 Nug Sph > plot(v1, plot.numbers=t, model=m2) > # Fix the nugget, fit only the sill of spherical model > m2a <- fit.variogram(v1,m1,fit.sills=c(f,t),fit.range=f); m2a model psill range 1 Nug Sph In this case, the eyeball did a pretty good job...

50 Introduction to applied geostatistics semivariance semivariance distance distance By eye: c 0 = 0.5, c 1 = 1.4, a = 1200; total sill c 0 + c 1 = 1.9 Automatic: c 0 = 0.548, c 1 = 1.340, a = 1149; total sill c 0 + c 1 = The total sill was almost unchanged; gstat raised the nugget and lowered the partial sill of the spherical model a bit; the range was shortened by 51 m.

51 Introduction to applied geostatistics 50 What sample size to fit a variogram model? Can t use non-spatial formulas for sample size, because spatial samples are correlated, and each sample is used multiple times in the variogram estimate Stochastic simulation from an assumed random field with a known variogram suggests: 1. < 50 points: not at all reliable to 150 points: more or less acceptable 3. > 250 points: almost certaintly reliable More points are needed to estimate an anisotropic variogram. This is very worrying for many environmental datasets (soil cores, vegetation plots,... ) especially from short-term fieldwork, where sample sizes of are typical. Should variograms even be attempted on such small samples?

52 Introduction to applied geostatistics 51 Topic: Approaches to spatial prediction This is the prediction of the value of some variable at an unsampled point, based on the values at the sampled points. This is often called interpolation, but strictly speaking: Interpolation: prediction is only for points that are geographically inside the (convex hull of the) sample set; Extrapolation: prediction outside this geographic area (Note: same usage as in feature-space predictions)

53 Introduction to applied geostatistics 52 A taxomomy of spatial prediction methods Strata divide area to be mapped into homogeneous strata; predict within each stratum from all samples in that stratum Global predictors: use all samples to predict at all points; also called regional predictors; Local predictors: use only nearby samples to predict at each point Mixed predictors: some of structure is explained by strata or globally, some locally

54 Introduction to applied geostatistics 53 Which approach is best? No theoretical answer Depends on how well the approach models the true spatial structure, and this is unknown (but we may have prior evidence) Should correspond with what we know about the process that created the spatial structure

55 Introduction to applied geostatistics 54 Polynomial trend surfaces A global predictor which models a regional trend The value of a variable at each point depends only on its coödinates and parameters of a fitted surface This is modelled with a smooth function of position, z = f (x, y) = f (E, N) for grid coördinates; this is called the trend surface Simple form (plane, 1 st order): z = β 0 + β x E + β y N Higher-order surfaces may also be fitted (beware of fitting the noise!)

56 Introduction to applied geostatistics 55 Fitting trend surfaces The trend surface is predicted by linear regression with coödinates as the predictor variables and the response variable to be predicted, using data from all sample points. All samples participate equally in the prediction We can measure the goodness of fit of the trend surface to the sample by the residual sum of squares The same cautions as in feature-space regression analysis! Ordinary Least Squares (OLS) is often used but is not really correct, since it ignores possible correlation among closely-spaced samples; better is Generalised Least Squares (GLS)

57 Introduction to applied geostatistics 56 Predictions of 1st and 2nd order Trend Surfaces in the study area M1st M M M M M M M M M M M M-2.0 M-1.5 M-1.0 M-0.5 M0.0 M0.5 M1.0 M1.5 and 2nd order trend surfaces, study area M xmxm ymym MTS1 MTS2 1st and 2nd order trend surfaces, study area y x

58 Introduction to applied geostatistics 57 Predictions of 1st and 2nd order Trend Surfaces in the bounding box -4M1st M M M M M M M M M M M M-4M-4M -3M-3M-3M -2M-2M-2M -1M-1M-1M 0M0M0M 1M1M1M 2M2M2M 3M3M3M 4M4M4M and 2nd order trend surfaces, bounding box -4M xmxm ymym MTS1 MTS2 1st and 2nd order trend surfaces, bounding box y x

59 Introduction to applied geostatistics 58 Approaches to prediction: Local predictors No strata No regional trend Value of the variable is predicted from nearby samples * Example: concentrations of soil constituents (e.g. salts, pollutants) * Example: vegetation density

60 Introduction to applied geostatistics 59 Local Predictors Each interpolator has its own assumptions, i.e. theory of spatial variability Nearest neighbour (Thiessen polygons) Average within a radius Average of the n nearest neighbours Distance-weighted average within a radius Distance-weighted average of n nearest neighbours... Optimal weighting Kriging

61 Introduction to applied geostatistics 60 Local predictor: Nearest neighbour (Thiessen polygons) Predict each point from its single nearest sample point Conceptually-simple, makes the minimal assumptions about spatial structure No way to estimate prediction variances, ignores other nearby information Maps show abrupt discontinuities at boundaries, so don t look very realistic But may be a more accurate predictor than poorly-modelled predictors

62 Introduction to applied geostatistics 61 Local predictor: Average within a radius Use the set of all neighbouring sample points within some radius r Predict by averaging : ˆx 0 = 1 n n i=1 x i, d(x 0, x i ) r Although we can calculate prediction variances from the neighbours, these assume no spatial structure closer than the radius Problem: How do we select a radius?

63 Introduction to applied geostatistics 62 Local predictors: Distance-weighted average Inverse of distance to some set of n nearest-neighbours: ˆx 0 = n i=1 x n i d(x 0, x i ) / i=1 1 d(x 0, x i ) Inverse of distance to some set of n nearest-neighbours, to some power k ˆx 0 = n i=1 x i k d(x 0, x i ) k/ i=1 1 d(x 0, x i ) k Implicit theory of spatial structure (a power model), but this is not testable Can select all points within some limiting distance (radius), or some fixed number of nearest points, or... How to select radius or number and power?

64 Introduction to applied geostatistics 63 Inverse distance in gstat The idw method is used. There is no model of spatial variability, so there is no way to estimate a prediction variance. > kid <- idw(log(cadmium) ~ 1, meuse, meuse.grid) [inverse distance weighted interpolation] > levelplot(var1.pred ~ x+y, as.data.frame(kid), aspect="iso") The weights are computed only from the inverse distance; they do not account for spatial structure nor for the relative positions of the sample points. Compare inverse distance (linear) to Ordinary Kriging with a spherical model (range = 1150 m): OK gives a much smoother map.

65 Introduction to applied geostatistics 64 Inverse distance y x y x Ordinary kriging y x y x

66 Introduction to applied geostatistics 65 Approaches to prediction: Mixed predictors For situations where there is both long-range structure (trend) or strata and local structure * Example: Particle size in the soil: strata (rock type), trend (distance from a river), and local variation in depositional or weathering processes One approach: model strata or global trend, subtract from each value, then model residuals Regression Kriging. Another approach: model everything together Universal Kriging or Kriging with External Drift

67 Introduction to applied geostatistics 66 Topic: Ordinary Kriging The theory of regionalised variables leads to an optimal interpolation method, in the sense that the prediction variance is minimized. This is based on the theory of random functions, and requires certain assumptions.

68 Introduction to applied geostatistics 67 Kriging A Best Linear Unbiased Predictor (BLUP) that satisfies a certain optimality criterion (so it s best with respect to the criterion) It is only optimal with respect to the chosen model and the chosen optimality criterion Based on the theory of random processes, with covariances depending only on separation (i.e. a variogram model) Theory developed several times (Kolmogorov 1930 s, Wiener 1949) but current practise dates back to Matheron (1963), formalizing the practical work of the mining engineer D G Krige (RSA). * Should really be written as krigeing (Fr. krigeage) but it s too late for that.

69 Introduction to applied geostatistics 68 What is so special about kriging? Predicts at any point as the weighted average of the values at sampled points * as for inverse distance (to a power) Weights given to each sample point are optimal, given the spatial covariance structure as revealed by the variogram model (in this sense it is best ) So, the prediction is only as good as the model of spatial structure! The prediction error at each point is automatically generated as part of the process of computing the weights.

70 Introduction to applied geostatistics 69 How do we use Kriging? 1. Sample, preferably at different resolutions 2. Calculate the experimental variogram 3. Model the variogram with one or more authorized functions N.b. the variogram model may already be known from other studies or theoretical considereations 4. Apply the kriging system of equations, with the variogram model of spatial dependence, at each point to be predicted Predictions are often at each point on a regular grid (e.g. a raster map) 5. Calculate the variance of each prediction; this is based only on the sample point locations, not their data values.

71 Introduction to applied geostatistics 70 OK in gstat The krige method is used with a variogram model: # compute experimental variogram v <- variogram(log(cadmium) ~ 1, meuse) # estimated model m <- vgm(1.4, "Sph", 1200, 0.5) # fitted model m.f <- fit.variogram(v, m) data(meuse.grid); coordinates(meuse.grid) <- ~ x +y # interpolation grid kr <- krige(log(cadmium)~ 1, loc=meuse, newdata=meuse.grid, model=m.f) [using ordinary kriging] # visualize interpolation; note aspect option to get correct geometry levelplot(var1.pred ~ x+y, as.data.frame(kr), aspect="iso") # visualize prediction error levelplot(var1.var ~ x+y, as.data.frame(kr), aspect="iso") Note the model specification (model=m.f); this gives the assumed covariance structure with which to compute the optimal weights.

72 Introduction to applied geostatistics 71 Ordinary kriging (OK) results for Meuse log(cd) y y x x

73 Introduction to applied geostatistics 72 Kriging prediction errors for Meuse log(cd) y y x x

74 Introduction to applied geostatistics 73 How realistic are maps made by Ordinary Kriging? The resulting surface is smooth and shows no noise, no matter if there is a nugget effect in the variogram model So the field is the best at each point taken separately, but taken as a whole is not a realistic map The sample points are predicted exactly; they are assumed to be without error, again even if there is a nugget effect in the variogram model

75 Introduction to applied geostatistics 74 Non-parametric geostatistics A non-parametric statistic is one that does not assume any underlying data distribution. For example: a mean is an estimate of a parameter of location of some assumed distribution (e.g.mid-point of normal, expected proportion of success in a binomial,... ) a median is simply the value at which half the samples are smaller and half larger, without knowing anything about the distribution underlying the process which produced the sample. In geostatistics, non-parametric refers to methods that make no assumptions about the distribution of the data values, only about spatial structure.

76 Introduction to applied geostatistics 75 Non-parametric geostatistics: Motivation (1) There is some positive motivation... In some applications, we may be most interested in finding areas with values above a certain threshold (e.g. polluted areas), and not really care if we get accurate predictions in other areas (as long as we are sure they are below the threshold) So the form of the distribution is not important, just whether a value is above or below some threshold. In these applications, we often want a probability that an interpolated point exceeds the threshold; this is directly useful for probabilistic decision-making * e.g. whether or not to clean up a polluted site

77 Introduction to applied geostatistics 76 Non-parametric geostatistics: Motivation (2)... and there is also some negative motivation: The outlier problem: a dataset may contain a few very high values These can make the area mean arbitrarily high (n.b. not the median) These contribute a disproportionate amount to the total variance as well These can make the experimental semivariogram unreliable for typical values and useless for unusual values: * the point-pairs where the outliers are included will have very high semivariances * these contribute disproportionately to the average semivariance in a bin... *... so that the variogram is very difficult to model

78 Introduction to applied geostatistics 77 E.g. a random sample of 15 N(10, 1) variates with one outlier at 100 (i.e. 10x the expected value) replacing the last value: 1. Without outlier: x = 9.95, s 2 x = With outlier: x = 15.98, s 2 x = The one point with value 100 accounts for (100 x) 2 /15 = 470 of the variance, i.e. 470/540 = 87% of it So, it will make semi-variances of point pairs involving this point much higher than others; these can be seen in the variogram cloud Note: the median is only slightly affected: if the outlier replaces a value above the median, the next-highest value is now the median

79 Introduction to applied geostatistics 78 Solutions to the outlier problem 1. Ignore (assume that they represent a different population and remove from the dataset before further analysis) under-estimation, can t find hot spots 2. Set to some arbitrary maximum, nearer the bulk of the population; same problem 3. Transform the variable to logarithms for modelling; transform back for the final maps and estimates Good solution if the whole distribution is lognormal Not optimal if the aim is just to bring some outliers closer (i.e. the rest of the distribution is not lognormal) 4. Transform to indicator variables, interpolate by Indicator Kriging (IK)

80 Introduction to applied geostatistics 79 Lognormal Kriging 1. Transform the data to their (natural) logarithms: y( x) i = log z( x i ); this should be approximately normally distributed 2. Model and interpolate with the transformed variable (OK, block kriging, UK, KED, trend surfaces... ) 3. Optional: Back-transform to original units of measure Back-transformation is not required if we don t care about the original variable, e.g. if the logarithm itself is a useful index. (Back-transformation of prediction variances is only possible for SK.)

81 Introduction to applied geostatistics 80 Indicator kriging This is a simple non-parametric (also called distribution-free) method ofinterpolation. It is used primarily to estimate the probability of exceeding some pre-defined threshold value. It can also be used to estimate an entire cumulative probability distribution (CDF). Note that there are other non-parametric methods, e.g. disjunctive kriging; these have a much more difficult theory.

82 Introduction to applied geostatistics 81 Distribution-free estimates So far we have assumed an approximately normal or lognormal distribution of the target spatially-correlated random variable. But this may be demonstrably not true. A non-parametric approach does not attempt to fit a distribution to the data, but rather works directly with the experimental CDF, by dividing it into sample quantiles. To work with these, we introduce the idea of indicator variables.

83 Introduction to applied geostatistics 82 Indicator variables Binary variables: Take one of the values {1, 0} depending on whether the point is in or out of the set; i.e. if it does or does not meet some criterion * These are suitable for binary nominal variables, e.g. { urban, not urban }; { land use changed, land use did not change } A continuous variable can be converted to an indicator z t by a threshold or cut-off value x t : z t = 1 x x t * e.g. x t = 350 to cut-off at 350 mg kg -1 * Formally: I( x i, z t ) = 1 iff Z( x i ) z t ; 0 otherwise * By convention 1 indicates values below the threshold (to model the CDF); inverting reverses the sense

84 Introduction to applied geostatistics 83 Setting up indicators in gstat > mind <- as.data.frame(meuse)[c("x","y","cadmium")]; str(mind) data.frame : 155 obs. of 3 variables: $ x : num $ y : num $ cadmium: num > attach(mind) > quantile(cadmium, seq(0,1,.1)) 0% 10% 20% 30% 40% 50% 60% 70% 80% 90% 100% > for (q in seq(.1,.9,.1)) mind <- + cbind(mind, as.numeric(cadmium<=quantile(cadmium,q))) > names(mind)[4:12] <- paste("q",seq(1:9),sep="") > mind$q5[1:30] [1] So field q5 of data frame mind contains a 1 if the corresponding value of field cadmium is 2.10, the fifth decile (i.e. the median).

85 Introduction to applied geostatistics 84 Indicator map Every sample point is either 1 ( in ) or 0 ( out ); a binary map No measure of how far in or out Prepare a series of indicator maps, with increasing thresholds, to visualise the cumulative sample distribution A common strategy is to divide the range of the sample values into quartiles or deciles and prepare an indicator for each The proportion of 1 s will increase with increasing quantile.

86 Introduction to applied geostatistics 85 The Indicator variogram Compute as for a parametric variogram; every sample point has either value 1 (below the cutoff, in the set) or 0. The semivariance of each point pair is either 0 (both above or below; both out or in) or 0.5 (one above, one below; one out, one in). For a quantized continuous variable, each indicator variable (quantile) might well have different spatial structure Variograms near the two ends of the CDF have few 1 s or 0 s (depending on the end), so few point-pairs will have semivariance 0.5 hard to model (fluctuates) Model as for parametric variogram; however the total sill must be < 0.5 (generally it s a lot lower)

87 Introduction to applied geostatistics 86 Probability kriging using indicator variables 1. Calculate the indicator at the required threshold 2. Calculate the empirical variogram for that indicator (not the median) (May have to use a threshold closer to the median if there are too few 1 s so that the variogram is erratic) 3. Model the variogram 4. Solve the kriging system at each point to be predicted, using Simple Kriging (SK) with the quantile proportion as the expected value (e.g., in the 6th decile, 0.6 of the values are expected to be 1 s) Note! this is only true if the original sampling scheme was unbiased! If not, also estimate the mean (use OK). 5. If necessary, limit the results to the range [0... 1] 6. This may be interepreted as the probability that the point does not exceed the threshold

88 Introduction to applied geostatistics 87 Indicator kriging in gstat > # convert to spatial object > coordinates(mind) <- ~ x +y > #compute the variogram for the 90th percentile > vq9 <- variogram(q9~1, mind) > plot(vq9, plot.numbers=t) > mq9 <- vgm(0.05,"sph",500,0.04) > plot(vq9, plot.numbers=t, model=mq9) > mq9f <- fit.variogram(vq9, mq9) > plot(vq9, plot.numbers=t, model=mq9f) > # erratic around sill, leads to very short range variogram > mq9f model psill range 1 Nug Sph > # krige this quantile; note expected proportion of 1 s is known > k9 <- krige(q9~1, mind, meuse.grid, beta=0.9, model=mq9f) [using simple kriging] > levelplot(var1.pred~x+y, as.data.frame(k9), aspect="iso") > # this is the probability of being *below* the cutoff of 8.26ppm

89 Introduction to applied geostatistics Indicator variogram (9th decile); semivariance semivariance Estimated model distance distance Fitted model; note unrealistic nugget semivariance y Probability < 8.26mg kg distance x 0.2 0

90 Introduction to applied geostatistics 89

91 Introduction to applied geostatistics 90 Summary: Advantages of IK Makes no assumption about the theoretical distribution of the data values, yet still give realistic probability estimates Outlier-resistent: these can not increase the estimate or prediction variances of an indicator arbitrarily; for data values they only affect one quantile Simple Kriging is used at each quantile, which improves the estimate.

92 Introduction to applied geostatistics 91 Summary: Disadvantages of IK Variograms may be difficult to model, especially at the highest and lowest quantiles (few pairs with different 0/1 values)