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4 hydrology geostatistics-part_2

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A more technical presentation about Kriging by Alberto Bellin. It covers simple and ordinary kriging, and other topics

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4 hydrology geostatistics-part_2

  1. 1. GEOSTATISTICS: Part II geostatistics as spatial interpolation technique Alberto Bellin Department of Civil, Environmental and Mechanical Engineering University of Trento Alberto Bellin Department of Civil, Environmental and Mechanical Engineering University of TrentoGEOSTATISTICS: Part II geostatistics as spatial interpolation technique 1 / 25
  2. 2. Outline 1 The kriging paradigm 2 Simple and Ordinary Kriging 3 Secondary Information Alberto Bellin Department of Civil, Environmental and Mechanical Engineering University of TrentoGEOSTATISTICS: Part II geostatistics as spatial interpolation technique 2 / 25
  3. 3. General structure of the model Z⇤ (x) m(x) = n(x) X i=1 i (x) [z(xi ) m(xi )] Z⇤(x): best (in some sense) estimate of the SRF Z at the position x; z(xi ): measurement of the SRF Z at the location xi ; m(x): local mean (at the position x); n(x): number of measurements used for the estimation (not necessarily all measures available) Alberto Bellin Department of Civil, Environmental and Mechanical Engineering University of TrentoGEOSTATISTICS: Part II geostatistics as spatial interpolation technique 3 / 25
  4. 4. The Kriging paradigm (continued...) SEVERAL FORMS OF THE GEOSTATISTICAL INTERPOLATION ARE AVAILABLE DEPENDING ON 1 model of spatial variability for the mean; 2 number of points used in the interpolation To compute the weights i we impose two conditions: 1 unbiased estimation: E [Z⇤(x) Z(x)] = 0 2 minimize the variance of the error, i.e.: 2 E = E h (Z⇤(x) Z(x))2 i = min Z(x): is the true (unknown) value of the SRF Z at the location x. Alberto Bellin Department of Civil, Environmental and Mechanical Engineering University of TrentoGEOSTATISTICS: Part II geostatistics as spatial interpolation technique 4 / 25
  5. 5. Simple Kriging We assume: m(x) = m = const and known, therefore: Z⇤ SK (x) = n(x) X i=1 SK i (x) [z(xi ) m] + m = n(x) X i=1 SK i (x) z(xi ) + 2 41 n(x) X i=1 SK i (x) 3 5 m We can see immediately that: E [Z⇤ SK (x) Z(x)] = m m = 0 Alberto Bellin Department of Civil, Environmental and Mechanical Engineering University of TrentoGEOSTATISTICS: Part II geostatistics as spatial interpolation technique 5 / 25
  6. 6. Simple Kriging (continued...) By replacing the above expression of Z⇤ SK into the general expression of the error variance and setting equal to zero the derivatives with respect to i , we obtain after a few manipulations: n(x) X i=1 SK i (x) CR(xi , xj ) = CR(xj , x); j = 1, ..., n(x) and 2 E = CR(0) n(x) X i=1 SK i (x) CR(xi , x) where CR is the covariance function of the residual: R((x)) = Z((x)) m Alberto Bellin Department of Civil, Environmental and Mechanical Engineering University of TrentoGEOSTATISTICS: Part II geostatistics as spatial interpolation technique 6 / 25
  7. 7. Ordinary Kriging We assume: m(x) = const, within a given searching area centered on x, but unknown, therefore: Z⇤ OK (x) = n(x) X i=1 OK i (x) z(xi ) + 2 41 n(x) X i=1 OK i (x) 3 5 m(x) By imposing the following non-bias condition: n(x) X i=1 OK i (x) = 1 We obtain: Z⇤ OK (x) = n(x) X i=1 OK i (x) z(xi ) with n(x) X i=1 OK i (x) = 1 Alberto Bellin Department of Civil, Environmental and Mechanical Engineering University of TrentoGEOSTATISTICS: Part II geostatistics as spatial interpolation technique 7 / 25
  8. 8. Ordinary Kriging (continued...) The minimum of the error variance should be computed under the condition that Pn(x) i=1 OK i (x) = 1: L = 2 E (x) + 2 µOK (x) 2 4 n(x) X i=1 OK i (x) 1 3 5 The minimum of the above expression is obtained for: ( Pn(x) i=1 OK i (x) CR(xi , xj ) + µOK (x) = CR(xj , x); j = 1, ..., n(x) Pn(x) i=1 OK i (x) = 1 while the error variance assumes the following expression: 2 E = CR(0) n(x) X i=1 OK i (x) CR(xi , x) µOK (x) Alberto Bellin Department of Civil, Environmental and Mechanical Engineering University of TrentoGEOSTATISTICS: Part II geostatistics as spatial interpolation technique 8 / 25
  9. 9. Ordinary kriging written with the semivariogram ( Pn(x) i=1 OK i (x) (xi , xj ) µOK (x) = (xj , x); j = 1, ..., n(x) Pn(x) i=1 OK i (x) = 1 2 E = n(x) X i=1 OK i (x) (xi , x) µOK (x) The semivariogram allows to filter out the local mean m(x) that is constant but unknown, over the local neighborhood W (x). Alberto Bellin Department of Civil, Environmental and Mechanical Engineering University of TrentoGEOSTATISTICS: Part II geostatistics as spatial interpolation technique 9 / 25
  10. 10. Kriging the local mean Estimation of the local mean m⇤ OK (x) = n(x) X i=1 OK i,m z(xi ) (1) with ( Pn(x) i=1 OK i,m (x) CR(xi , xj ) + µOK m (x) = 0; j = 1, ..., n(x) Pn(x) i=1 OK i,m (x) = 1 This system is similar to that of the OK except for the right-hand side of the first n(x) equations, which is set to zero. 2 E = Var {m⇤ OK (x) m(x)} = n(x) X i=1 n(x) X j=1 OK i (x) CR(xi , xj ) Alberto Bellin Department of Civil, Environmental and Mechanical Engineering University of TrentoGEOSTATISTICS: Part II geostatistics as spatial interpolation technique 10 / 25
  11. 11. Kriging the local mean: An example ● ● ● ● ● ● ● ● ● ● 1 2 3 4 5 6 01234 Cd Estimates − SK vs OK Distance [km] CdConcentration[ppm] SK OK Mean ● ● ● ● ● ● ● ● ● ● 1 2 3 4 5 6 01234 Trend Estimates − SK vs OK Distance [km] CdConcentration[ppm] u1 u2 u3 u4 u5 u6 u7 u8 u9 u10 OK SK Alberto Bellin Department of Civil, Environmental and Mechanical Engineering University of TrentoGEOSTATISTICS: Part II geostatistics as spatial interpolation technique 11 / 25
  12. 12. Relationship between Ordinary and Simple Kriging Ordinary Kriging estimation is equivalent to: 1 Compute the local mean m⇤ OK (x) by using OK with the data within the neighborhood of x; 2 apply the SK estimator to the residuals R(x) = Z(x) (m⇤ OK (x) m) Therefore: Z⇤ OK (x) = n(x) X i=1 SK i (x) [z(xi ) m⇤ OK (x)] + m⇤ OK (x) n(x) X i=1 SK i (x) z(x) + SK m (x)m⇤ OK (x) Z⇤ OK (x) = Z⇤ SK (x) + SK m (x) [m⇤ OK (x) m] Alberto Bellin Department of Civil, Environmental and Mechanical Engineering University of TrentoGEOSTATISTICS: Part II geostatistics as spatial interpolation technique 12 / 25
  13. 13. Simple vs Ordinary Kriging: An example Alberto Bellin Department of Civil, Environmental and Mechanical Engineering University of TrentoGEOSTATISTICS: Part II geostatistics as spatial interpolation technique 13 / 25
  14. 14. How to infer the residual covariance First the residual semivariogram R is inferred, then the pseudo residual covariance is computed as follows: CR(r) = A R(r), where A is an arbitrary constant 2 (r) = E n [Z(x) Z(x + r)]2 o 2 R(r) + [m(x) m(x + r)]2 This calls for selecting data pairs that are una↵ected (or slightly a↵ected) by the trend: in this case m(xi ) ⇡ m(xi + r) for i = 1, ..., n(x), and therefore R = . The residual semivariogram can be inferred directly from the data Alberto Bellin Department of Civil, Environmental and Mechanical Engineering University of TrentoGEOSTATISTICS: Part II geostatistics as spatial interpolation technique 14 / 25
  15. 15. Cross validation 1 Remove a measurement and perform kriging with the remaining data on the point removed. An estimation of the error is given by the di↵erence between the estimate and the measurement; 2 re-insert the measurement into the dataset and repeat the procedure with another measurement; 3 repeat the procedure with all the other measurements to obtain an estimation of the error at each measurement’s location. An example of error statistics estimated from cross-validation is shown in the following slide Alberto Bellin Department of Civil, Environmental and Mechanical Engineering University of TrentoGEOSTATISTICS: Part II geostatistics as spatial interpolation technique 15 / 25
  16. 16. Cross validation (continued) Statistics of true and estimated concentrations at 100 test locations Cd Cu Pb Co Mean True value 1.23 23.2 56.5 9.8 OK estimates 1.36 24.0 55.4 9.4 Std deviation True values 0.69 25.8 40.3 3.5 OK estimates 0.41 7.5 11.7 2.4 % contamination True values 63.0 8.0 42.0 0.0 OK estimates 92.0 0.0 66.0 0.0 % misclassification OK estimates 35.0 8.0 36.0 0.0 Alberto Bellin Department of Civil, Environmental and Mechanical Engineering University of TrentoGEOSTATISTICS: Part II geostatistics as spatial interpolation technique 16 / 25
  17. 17. Cross validation: graphical representation of the errors Alberto Bellin Department of Civil, Environmental and Mechanical Engineering University of TrentoGEOSTATISTICS: Part II geostatistics as spatial interpolation technique 17 / 25
  18. 18. Use secondary information: Classification of the approaches Exhaustive secondary information ! 8 < : Kriging within strata Simple Kriging with varying local mean Kriging with external drift Primary data z(x), i = 1, ..., n are supplemented by secondary information at all primary data locations xi plus the locations x where estimation is performed Type of secondary information: - a categorical attribute s with K mutually exclusive states - a smoothly varying continuous attribute y, for example the concentration of another metal Non-exhaustive secondary information ! Cokriging Alberto Bellin Department of Civil, Environmental and Mechanical Engineering University of TrentoGEOSTATISTICS: Part II geostatistics as spatial interpolation technique 18 / 25
  19. 19. Simple Kriging with varying local mean The mean of the field, which is variable in space, is obtained from the secondary information 1 if the SI is related to a categorical attribute s with K non-overlapping states: m⇤ SK (x) = m|sk with s(x) = sk. The conditional mean is given by: m|sk = 1 nk nX i=1 i(xi ; sk) · z(xi ) nk = Pn i=1 i(xi ; sk) is the number of primary data locations belonging to the category sk 2 if the SI is a continuous attribute y: m⇤ SK (x) = f (y(x)), where f is a regression function. Alberto Bellin Department of Civil, Environmental and Mechanical Engineering University of TrentoGEOSTATISTICS: Part II geostatistics as spatial interpolation technique 19 / 25
  20. 20. Simple Kriging with varying local mean (continued...) Simple Kriging is then applied to the residuals: Z⇤ SKlm(x) = m⇤ SKlm(x) + n(x) X i=1 SK i (x) [z(xi ) m⇤ SKlm(xi )] An alternative to regression: The secondary attribute is discretized into K classes: (yk, yk+1] and the local mean is computed as follows: m⇤ SK (x) = m|sk = 1 nk nX i=1 i(xi ; sk) · z(xi ) i(xi ; sk) = ⇢ 1 if y(xi ) 2 (yk, yk+1] 0 otherwise Alberto Bellin Department of Civil, Environmental and Mechanical Engineering University of TrentoGEOSTATISTICS: Part II geostatistics as spatial interpolation technique 20 / 25
  21. 21. Simple Kriging with varying local mean: an example 1 2 3 4 5 6 -1.0-0.50.00.51.01.52.02.5 Simple Kriging of residuals and Trend component Distance [Km] Cdconcentration[ppm] Trend SK 1 2 3 4 5 6 01234 SK varying mean Distance [Km] Cdconcentration[ppm] distance semivariance 0.2 0.4 0.6 0.8 0.5 1.0 1.5 Alberto Bellin Department of Civil, Environmental and Mechanical Engineering University of TrentoGEOSTATISTICS: Part II geostatistics as spatial interpolation technique 21 / 25
  22. 22. Simple Kriging with varying local mean: an example (continued) Alberto Bellin Department of Civil, Environmental and Mechanical Engineering University of TrentoGEOSTATISTICS: Part II geostatistics as spatial interpolation technique 22 / 25
  23. 23. Kriging with external drift KED is a variant of the kriging with a trend model (KT). The trend m(x) is modeled as a linear function of a smoothly varying secondary variable y(x): m(x) = a0(x) + a1(x) y(x) Note that KT uses: m(x) = a0(x) + KX k=1 ak(x) fk(x). The KED estimators is written as follows: Z⇤ KED(x) = n(x) X i=1 KED i (x) z(xi ) Alberto Bellin Department of Civil, Environmental and Mechanical Engineering University of TrentoGEOSTATISTICS: Part II geostatistics as spatial interpolation technique 23 / 25
  24. 24. Kriging with external drift (continued...) The weight are computed by solving the following system of equations: 8 >>>>>>>>>>>>>>>< >>>>>>>>>>>>>>>: n(x) X j=1 KED j (x) CR(xi , xj ) + µ0(x) + µ1(x) y(xi ) = CR(xi , x), i = 1, ..., n(x) n(x) X j=1 KED j (x) = 1 n(x) X j=1 KED j (x) y(xj ) = y(x) Alberto Bellin Department of Civil, Environmental and Mechanical Engineering University of TrentoGEOSTATISTICS: Part II geostatistics as spatial interpolation technique 24 / 25
  25. 25. Kriging with external drift (continued...) The conditional variance (the minimized error variance) becomes 2 E = CR(0) n(x) X i=1 KED i (x) CR(xi , x) µKED 0 (x) µKED 1 (x) n(x) X i=1 KED i (x) y(xi ) = CR(0) n(x) X i=1 KED i (x) CR(xi , x) µKED 0 (x) µKED 1 (x) y(x) Alberto Bellin Department of Civil, Environmental and Mechanical Engineering University of TrentoGEOSTATISTICS: Part II geostatistics as spatial interpolation technique 25 / 25

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