Introduction. Semivariogram Cloud

Transcription

1 Introduction Data: set of n attribute measurements {z(s i ), i = 1,, n}, available at n sample locations {s i, i = 1,, n} Objectives: Slide 1 quantify spatial auto-correlation, or attribute dissimilarity typically expressed as: 1 2 [z(s i) z(s j )] 2 as a function of separation distance between sample pairs s i and s j introduce the sample semivariogram, its characteristics, and provide some examples NOTE: Spatial auto-correlation is a second-order characteristic of spatial variation, and hence the sample semivariogram should be computed from data whose spatial variation is not explained by first-order effects justify the need of going beyond the sample semivariogram to a semivariogram model introduce parametric functions of distance that can be used as formal theoretical semivariogram models discuss issues of fitting semivariogram models to sample semivariogram values Semivariogram Cloud Definition: A scatter-plot of attribute squared semidifferences between all possible pairs of samples measured at different locations, versus their separation distance Computational procedure: Slide 2 1 construct Euclidean distance matrix D = [d ij, i = 1,, n, j = 1,, n] between all n 2 pairs of data locations, where d ij is defined as: d ij = h ij = s i s j 2 construct squared semidifference matrix E = [e ij, i = 1,, n, j = 1,, n] between all n 2 pairs of attribute values, where e ij is defined as: e ij = 1 2 [z(s i) z(s j )] 2 3 plot each distance value d ij against the corresponding squared semidifference e ij ; in other words, plot e = vec(e) versus d = vec(d) The plot of all pairs {d ij, e ij } is termed a semivariogram cloud

2 Semivariogram Cloud Example Bay Area rain gauge precipitation Semivariogram cloud semidifferences squared Slide distance A measure of dissimilarity between attribute values measured at different locations, ie, a spatial measure of attribute dissimilarity Expected graph pattern: As the distance d ij between sample pairs increases, the corresponding squared semidifference e ij should also increase Difficult to interpret, so we consider groups of sample pairs separated by similar distances ie, average squared semidifferences within distance classes (x-axis bins in the right graph above) Semivariogram Cloud Versus Plot 9 Semivariogram cloud 12 Sample semivariogram 8 7 squared semidifference semivariance values Slide distance Going from the first to the second: distance class or midpoint define a set of L distance classes; the l-th class has limits: (d l t l, d l + t l ], where d l is the class midpoint and t l is half the class width (or distance tolerance) for a given distance class (d l t l, d l + t l ], the semivariogram value ˆγ(d l ) is the average of n(d l ) << n 2 squared attribute semidifferences computed from sample pairs whose inter-distances d ij satisfy: d l t l < d ij d l + t l in other words, the semivariogram plot can be regarded as a summary of the semivariogram cloud, according to some distance-based grouping of samples

3 Computing Sample Semivariograms 1 compute distance matrix D = [d ij, i = 1,, n, j = 1,, n] and squared semidifference matrix E = [e ij, i = 1,, n, j = 1,, n] between n 2 data pairs D = d 12 d 13 d 14 d 15 d 12 d 23 d 24 d 25 d 13 d 23 d 34 d 35 d 14 d 24 d 34 d 45 d 15 d 25 d 35 d 45 E = e 12 e 13 e 14 e 15 e 12 e 23 e 24 e 25 e 13 e 23 e 34 e 35 e 14 e 24 e 34 e 45 e 15 e 25 e 35 e 45 Slide 5 2 for a given distance class (d l t l, d l + t l ], find entries of E that correspond to entries of D falling in that distance class, eg: d 12 d 13 d 14 d 15 d 12 d 23 d 24 d 25 d 13 d 23 d 34 d 35 d 14 d 24 d 34 d 45 d 15 d 25 d 35 d 45 e 12 e 13 e 14 e 15 e 12 e 23 e 24 e 25 e 13 e 23 e 34 d 35 e 14 e 24 e 34 e 45 e 15 e 25 e 35 e 45 3 sample semivariogram ˆγ(d l ) for that class is the average of the n(d l ) squared semidifferences, e-values, whose corresponding distances, d-values, fall in class (d l t l, d l + t l ]; ie, the mean of all e-values in boxes in the matrix on the right above Examples of Semivariogram Computation Locations separated by distance class (5 15] Histogram of squared semi differences (5 15] 7 y coordinates frequency # of data 68 mean 262 std dev 475 minimum 1 q(25) 1 median 39 q(75) 22 maximum Slide x coordinates Locations separated by distance class (15 25] squared semi-difference eij Histogram of squared semi differences (15 25] 7 y coordinates frequency # of data 16 mean 386 std dev 733 minimum 1 q(25) 3 median 129 q(75) 49 maximum x coordinates squared semi-difference eij ˆγ((5 15]) = 262, ˆγ((15 25]) = 386 = averages of values displayed in histograms Map views linking sample pairs that contribute to such histograms are extremely informative

4 Sample Semivariogram Plots Consider a set of L distance classes with midpoints {d l, l = 1,, L} and tolerances {t l, l = 1,, L} The plot of semivariance values {ˆγ(d l ), l = 1,, L} versus the average sample inter-distance for each class is called a sample semivariogram ˆγ(d l ) = 1 n(d l ) e c = 1 n(d l ) 2n(d l ) c=1 n(d l ) d ij (d l t l,d l +t l ] [z(s i ) z(s j )] 2 Slide 7 Bay Area rain gauge precipitation Sample semivariogram semivariance γ ( h) lag distance h numbers above bullets denote # of sample pairs contributing to ˆγ(d l ) at each lag distance could also graph variances of e-values within the distance classes; ˆγ() =, always Semivariogram Characteristics 2 Variogram, γ(h) Increasing Variability Slide Lag Distance (h) sill: limit semivariogram value (plateau) is approximately equal to sample variance (for representative sample) range: distance at which semivariogram reaches (or starts oscillating around) sill = distance of influence of any datum on another nugget effect: discontinuity at origin (ˆγ() > ); sum of measurement error and micro-structures (variability at scales smaller than sampling interval) watch out for sparse data, outliers and positional or attribute errors transformation of Euclidean distance into statistical distance bearing imprint of specific phenomenon

5 Sample Semivariogram Shape & Interpretation (1) Quadratic shape near origin: Image Semivariogram: quadratic shape at origin North γ 8 Slide East Distance Interpretation: highly continuous (extremely smooth) spatial attribute variability spatial attribute is differentiable typical variables: elevation, temperature, Sample Semivariogram Shape & Interpretation (2) Linear shape near origin: Image Semivariogram: linear shape at origin North γ 8 Slide East Distance Interpretation: continuous variability (not extremely smooth) of spatial attribute attribute is not differentiable typical variables: ore grades,

6 Sample Semivariogram Shape & Interpretation (3) Discontinuous near origin: Image Semivariogram: discontinuous at origin Slide 11 North -1 γ East Distance Interpretation: highly irregular (quasi-random) spatial variability at small scales typical variables: precipitation, Sample Semivariogram Shape & Interpretation (4) Oscillating (around sill): 2 DN (image intensity) values Semivariogram of DN values North 2 15 γ 8 4 Slide East 2 Distance Interpretation: periodic variability of spatial attribute yields sinusoidal semivariogram semivariogram shape possibly due to limited sampling need to provide physical evidence for periodicity frequently encountered in time series

7 The Need for Semivariogram Models Problems: (i) sill, range, and relative nugget, cannot be determined directly from the sample semivariogram plot, (ii) a continuum of semivariogram values γ(d) for any distance vector d is required in interpolation, but sample semivariogram values {ˆγ(d l ), l = 1,, L} are typically calculated only for few (L) distances {d l, l = 1,, L} Slide 13 Semivariogram model definition: parametric function γ(d; θ) fitted to sample semivariogram values {ˆγ(d l ), l = 1,, L}; θ denotes parameter vector with, eg, range, and sill (for a given semivariogram function) Sample semivariogram of precipitation Sample and model semivariogram of precipitation γ 8 γ sample variogram model variogram Distance (degrees) Distance (degrees) semivariogram modeling is more than a curve fitting exercise; Warning: cannot use any curve as semivariogram model!!! Valid Semivariogram Models: Pure Nugget Effect 12 Nugget effect variogram model semivariance γ(d) Slide lag distance d, if d = γ(d; θ) = σ, if d > θ = [σ], where σ denotes attribute variance indicates complete absence of spatial correlation could occur due to measurement error and microstructure, ie, features occurring at scales smaller than sampling interval

8 Valid Semivariogram Models: Spherical 15 Spherical variogram model (sill=, range=3) semivariance γ(d) 5 Slide lag distance d 3 σ d 2 r 1 d 3, if d < r 2 r γ(d; θ) = σ, if d r θ = [σ r], where r is the model range linear behavior at origin clearly defined range parameter r Valid Semivariogram Models: Exponential 15 Exponential variogram model (sill=, range=3) semivariance γ(d) 5 Slide lag distance d γ(d; θ) = σ 1 exp 3d r θ = [σ r] linear behavior at origin; rises faster than spherical; reaches sill asymptotically effective range parameter r; distance at which 95% of sill reached

9 Valid Semivariogram Models: Gaussian 15 Gaussian variogram model (sill=, range=3) semivariance γ(d) 5 Slide lag distance d γ(d; θ) = σ 1 exp 3d2 r 2 θ = [σ r] quadratic behavior at origin; implies smooth spatial variability of attribute values; reaches sill asymptotically effective range parameter r; distance at which 95% of sill reached Valid Semivariogram Models: Nugget + Exponential 1 Standardized nugget+exponential variogram model 8 γ Slide 18 Distance, if d = γ(d; θ) = a + ([σ a][1 exp( 3d )]), if d r θ = [σ a r] discontinuous at origin; reaches sill asymptotically practical range parameter r; distance at which 95% of sill reached a/σ = relative nugget contribution = proportion (to total sill) of purely random spatial variability more complex models can be built by adding or multiplying valid models

10 Fitting Semivariogram Models to Sample Data Or fitting valid semivariogram functions (curves) to sample semivariogram values Manual fitting: select number of semivariograms, their type (functional form), sill, and range model behavior at origin (nugget effect, shape of semivariogram at distances smaller than first lag) using prior knowledge about phenomenon Slide 19 Automatic fitting: least squares fit (ordinary, generalized, weighted): choose semivariogram model parameters (typically iteratively) so as to minimize discrepancy between model and sample semivariogram values over all lags; other methods also available treat with caution, especially with sparse data and outliers Cross-validation: given a proposed parameter set, ie, a semivariogram model, perform cross-validation using geostatistical interpolation, and record resulting error statistics repeat with different model parameters, and select as optimal model the one whose parameters yield best cross-validation error statistics Summary Spatial auto-correlation can be quantified by looking at attribute dissimilarity as a function of separation distance The semivariogram cloud is too cloudy for detecting meaningful patterns Slide 2 The semivariogram plot is constructed by averaging squared semidifferences within distance bins to smooth out the variability in the semivariogram cloud NOTE: Watch out for trends (first-order effects) in the data; a sample semivariogram quantifies second-order effects and might be contaminated by variations due to trends/drifts A quantitative way to encapsulate a sample semivariogram is through a parametric semivariogram model Fitting procedures exist for estimating the parameters of semivariogram models, ie, for fitting model semivariograms to sample semivariograms The final semivariogram model can be used for simulation (pattern generation) and geostatistical interpolation NOTE: A semivariogram model is a spatial process model, whose parameters are inferred from the sample data through the sample semivariogram

11 Introduction Data: set of n attribute measurements {z(s i ), i = 1,, n}, available at n sample locations {s i, i = 1,, n} Objectives: (i) predict or interpolate unknown attribute value z(s p ) at location s p from the n sample data, and (ii) assess reliability of predicted value Slide 1 Geostatistical spatial interpolation: predicted attribute value = weighted linear combination of sample data values + attribute mean, if known (non-linear methods also exist) a semivariogram model is used to determine the weights, which account for: spatial auto-correlation between sample data and unknown value spatial auto-correlation between sample data themselves (data redundancy) in addition, and contrary to most interpolation algorithms, geostatistics offers a measure of reliability (prediction error variance) regarding the attribute prediction SK prediction: ẑ(s p ) = m + Simple Kriging (SK) n w p (s i )[z(s i ) m] = wp T r i=1 w p = [w p (s i ), i = 1,, n] T : (n 1) vector of SK-weights assigned to n sample data for prediction at location s p ; superscript T denotes transposition r = [z(s i ) m, i = 1,, n] T : (n 1) vector of residual data from known mean m Slide 2 z(s 1 ) m ẑ(s p ) = m + w p (s 1 ) w p (s i ) w p (s n ) z(s i ) m wp T z(s n ) m r use semivariogram model to determine weights at each prediction location; typically, it is the covariogram model that is used due to computational reasons Lecture Notes Introduction to Geostatistical Spatial Interpolation total # of slides = 22

12 Semivariogram / Covariogram / Correlogram Model 15 Semivariogram model 15 Covariogram model 15 Correlogram model variance γ( ) = σ() σ() variance ρ() 1 unit correlation semivariance γ(d) 5 covariance σ(d) 5 correlation ρ(d) 5 Slide 3 range lag distance d range lag distance d range lag distance d Conversion between models, with σ() = γ( ) being the sill of the semivariogram model: Semivariogram covariogram: σ(d) = σ() γ(d) Covariogram correlogram: Semivariogram correlogram: Covariogram semivariogram: ρ(d) = σ(d) σ() ρ(d) = 1 γ(d) σ() γ(d) = σ() σ(d) Requisites for Geostatistical Interpolation I Slide 4 Data-to-data and data-to-unknown distances: d 1j d 1n D = d i1 d in d n1 d nj and d p = d 1p d ip d np Comments: as any other interpolation method, one accounts for the proximity of the n sample locations to the prediction location s p Note: Vector d p changes from one prediction location s p to another, hence the subscript p unlike other interpolation methods, one also accounts for the proximity between sample locations themselves (sample configuration or data layout) Note: Matrix D of sample-to-sample distances is the same for all prediction locations Lecture Notes Introduction to Geostatistical Spatial Interpolation total # of slides = 22

13 Requisites for Geostatistical Interpolation II From distance matrices to model covariance matrices: Take any distance value d ij and d ip, ie, any entry in D and d p, and transform it, via the covariogram model, to a covariance value σ(d ij ) and σ(d ip ) Slide 5 Data-to-data and data-to-unknown model covariances: σ() σ(d 1j ) σ(d 1n ) Σ = σ(d i1 ) σ() σ(d in ) and σ p = σ(d n1 ) σ(d nj ) σ() σ(d 1p ) σ(d ip ) σ(d np ) data-to-data covariance matrix Σ: (n n) matrix with model covariance values σ(d ij ) between any two sample locations separated by distance d ij data-to-unknown covariance vector σ p : (n 1) vector with model covariance values σ(d ip ) between the n sample locations and the prediction location s p Note: Vector σ p changes from one prediction location s p to another, hence the subscript p Requisites for Geostatistical Interpolation III Slide 6 Data-to-data and data-to-unknown model covariances: Comments: Σ = σ() σ(d 1j ) σ(d 1n ) σ(d i1 ) σ() σ(d in ) σ(d n1 ) σ(d nj ) σ() and σ p = σ(d 1p ) σ(d ip ) σ(d np ) data-to-data covariance matrix Σ: encapsulates the redundancy between the sample data; for positive spatial auto-correlation, the more clustered is the sample layout, the more redundant are the sample data (less information content); a clustered sample layout typically translates into larger entries in Σ data-to-unknown covariance vector σ p : encapsulates the statistical proximity (correlation) between the sample data and the unknown attribute value z(s p ) at the prediction location s p ; that correlation is a function of distance between sample and prediction locations, not of the actual (unknown) value z(s p ); The larger the entries of vector σ p, the stronger the predictive power of sample data Lecture Notes Introduction to Geostatistical Spatial Interpolation total # of slides = 22

14 Simple Kriging (SK) System & Weights σ() σ(d 1n ) σ(d n1 ) σ() w p (s 1 ) w p (s n ) = σ(d 1p ) σ(d np ) Σw p = σ p Comments: Slide 7 the SK system is a (disguised) version of the normal equations for the case of regression with no intercept term: X T Xb = X T y, where X is the design matrix and y is the vector of data on the dependent variable; in regression, the data-to-data covariance is estimated as X T X/n, and the data-to-unknown covariance as X T y/n the weights vector w p is obtained by solving the SK system, as w p = Σ 1 σ p, anew at each prediction location s p since the entries of σ p change entries of w p do not depend on data values or on sill, σ(), of covariogram model: σ() ρ() ρ(d 1n ) ρ(d n1 ) 1 w p (s 1 ) w p (s n ) = σ() ρ(d 1p ) ρ(d np ) w p (s 1 ) w p (s n ) Interpreting the Simple Kriging Weights 1 ρ(d 1n ) = 1 σ() ρ(d n1 ) 1 1 ρ(d 1p ) σ() w p = Σ 1 σ p ρ(d np ) Slide 8 if sample interdistances d ij are larger than correlogram range, then ρ(d ij ) =, and Σ = σ()i, the (n n) identity matrix; this entails that w p (s i ) = ρ(d ip ), ie, weights are equal to correlogram values but in general, Σ = σ()i, ie, sample interdistances are within correlation range, in which case Σ 1 modulates σ p : influence of samples in clusters is downplayed the closer the sample data to the prediction location, and the more spread out the data over the study region, the better the SK prediction is expected to be for sample data far away (beyond correlation range) from the prediction location s p, ρ(d ip ) = and w p (s i ) = : all weighs are equal to for prediction at a sample location s p s i, data-to-unknown covariance vector σ p = σ i is same as i-th column of Σ; this yields w p (s i ) = 1 if s i = s p, otherwise: only sample co-located with prediction location receives non-zero (= 1) weight Lecture Notes Introduction to Geostatistical Spatial Interpolation total # of slides = 22

15 Simple Kriging Prediction and Error Variance Once the SK weights are computed as w p = Σ 1 σ p, they are substituted in the following equations to compute the SK prediction ẑ(s p ) and associated error variance ˆσ(s p ) Slide 9 SK prediction does not depend on sill σ() of covariogram model: z(s 1 ) m ẑ(s p ) = m+wp T r = m+[w p (s 1 ) w p (s n )] = m+ n w p (s i )[z(s i ) m] i=1 z(s n ) m SK prediction error variance does depend on covariogram model sill σ(): σ(d 1p ) ˆσ(s p ) = σ() wp T σ p = σ() [w p (s 1 ) w p (s n )] = σ() n w p (s i )σ(d ip ) n=1 σ(d np ) which can also be written as: ˆσ(s p ) = σ() 1 n i=1 w p(s i )ρ(d ip ) Interpreting the SK Prediction and Error Variance ẑ(s p ) = m + Comments: n w p (s i )[z(s i ) m] i=1 n ˆσ(s p ) = σ() w p (s i )σ(d ip ) i=1 Slide for sample data far away (beyond correlation range) from the prediction location s p, w p (s i ) =, i: all weighs are equal to In this case, the SK prediction equals the known mean m and the SK error variance equals the known covariogram sill: ẑ(s p ) = m and ˆσ(s p ) = σ(); away from the sample data, SK yields back the (assumed known) attribute overall mean and variance for prediction at a sample location s p s i, w p (s i ) = 1 if s i = s p, otherwise: the SK prediction identifies the known sample datum and the SK error variance is zero: ẑ(s i ) = z(s i ) and ˆσ(s i ) = ; SK is an exact interpolation algorithm for all other prediction locations, the SK predictions depend on the sample data configuration and their values, while the SK error variances depend only on the sample data configuration; both SK predictions and error variances depend on the covariogram model σ(d) adopted Lecture Notes Introduction to Geostatistical Spatial Interpolation total # of slides = 22

16 Determining the SK Weights: Step Local data configuration (1) (2) (5) (3) (4) 646 (6) 783 (7) Slide (n x n) matrix of data-to-data inter-distances: D = i, j-th element of D: d ij = s i s j (n x 1) vector of prediction-to-data-location distances: T d p = i-th element of d p : d ip = s i s p Determining the SK Weights: Step Local data configuration 1 Correlogram model (1) (2) (5) (3) 8 6 ρ (d) ρ(d) = exp( 3d ) (4) 646 (6) 783 (7) 2 Slide d 361 exp( 3 361/) exp( 3 447/) exp( 3 671/) exp( 3 86/) = exp( 3 894/) exp( 3 949/) exp( /) 2 d p σ p =sill exp( 3d p /range) These would be the weights if one ignored auto-correlation between sample data Lecture Notes Introduction to Geostatistical Spatial Interpolation total # of slides = 22

17 Determining the SK Weights: Step Local data configuration 1 Correlogram model (1) (2) (5) (3) 8 6 ρ (d) ρ(d) = exp( 3d ) (4) 646 (6) 783 (7) 2 Slide SK system: Σ d w p (s 1 ) 34 w p (s 2 ) 26 w p (s 3 ) 13 w p (s 4 ) = 9 w p (s 5 ) 7 w p (s 6 ) 6 w p (s 7 ) 2 w p σ p i, j-th element of matrix Σ: σ ij = 1 exp( 3 d ij /) Determining the SK Weights: Step 4 Slide 14 w p (s 1 ) w p (s 2 ) w p (s 3 ) w p (s 4 ) w p (s 5 ) w p (s 6 ) w p (s 7 ) wp = Σ 1 σp 145 SK weights (1) (2) -1 2 (5) (3) prediction = variance = (4) 28 7 (6) (7) original weights vector (w p = σ p ) modified by Σ 1 to account for sample redundancy; eg, w p (s 1 ) = 27 instead of ρ(d 1p ) = 34 Lecture Notes Introduction to Geostatistical Spatial Interpolation total # of slides = 22