Basics in Geostatistics 2 Geostatistical interpolation/estimation: Kriging methods. Hans Wackernagel. MINES ParisTech.

Transcription

1 Basics in Geostatistics 2 Geostatistical interpolation/estimation: Kriging methods Hans Wackernagel MINES ParisTech NERSC April

2 Basic concepts Geostatistics Hans Wackernagel (MINES ParisTech) Basics in Geostatistics 2 NERSC April / 35

3 Concepts Geostatistical model The experimental variogram is used to analyze the spatial structure of the data from a regionalized variable z(x). It is fitted with a nested variogram model, thus providing the structure function of a random function. The regionalized variable (reality) is viewed as one realization of the random function Z(x), which is a collection of random variables. Kriging: a linear regression method for estimating point values (or spatial averages) at any location of a region. Conditional simulation: simulation of an ensemble of realizations of a random function, conditional upon data for non-linear estimation.

4 Concepts Geostatistical model The experimental variogram is used to analyze the spatial structure of the data from a regionalized variable z(x). It is fitted with a nested variogram model, thus providing the structure function of a random function. The regionalized variable (reality) is viewed as one realization of the random function Z(x), which is a collection of random variables. Kriging: a linear regression method for estimating point values (or spatial averages) at any location of a region. Conditional simulation: simulation of an ensemble of realizations of a random function, conditional upon data for non-linear estimation.

5 Concepts Geostatistical model The experimental variogram is used to analyze the spatial structure of the data from a regionalized variable z(x). It is fitted with a nested variogram model, thus providing the structure function of a random function. The regionalized variable (reality) is viewed as one realization of the random function Z(x), which is a collection of random variables. Kriging: a linear regression method for estimating point values (or spatial averages) at any location of a region. Conditional simulation: simulation of an ensemble of realizations of a random function, conditional upon data for non-linear estimation.

6 For the top series: Stationarity stationary mean m and covariance function C(h) For the bottom series: mean and variance are not stationary, actually the realization of a non-stationary process without drift. Both types of series can be characterized with a variogram.

7 Kriging of the mean Kriging of the mean of a random function Hans Wackernagel (MINES ParisTech) Basics in Geostatistics 2 NERSC April / 35

8 Spatially Correlated Data Sample locations x α in a spatial domain: With spatial correlation we need to consider that: each sample point plays a different role in estimating the mean of the spatial domain, distances to neighboring points play a role. How should samples thus be weighted in an optimal way?

9 Estimation of the Mean Value Using the arithmetic mean: all samples get the same weight: M = 1 n We rather need an estimator: M = 1 n Z(x α) α=1 w α Z(x α ) α=1 with weights w α reflecting the spatial correlation.

10 Stationary random function We assume translation-invariance of the mean: [ ] x D : E Z(x) = m and of the covariance: x α, x β D with x α x β = h : cov(z(x α), Z(x β )) = C(x α x β ) = C(h)

11 Unbiased estimator The estimation error in our statistical model: M }{{} estimated value m }{{} true value should be zero on average: [ ] E M m = 0 No bias: the estimator M does not on average yield a value that is different from m.

12 No bias Bias is avoided using weights of unit sum: w α = 1 α=1 Consider: [ ] E M m [ ] = w α E Z(x α ) m α=1 }{{ m } = m m = 0 α=1 w α }{{} 1 [ = E α=1 ] w α Z(x α ) m

13 Variance of the estimation error The variance of the estimation error is: [ σe 2 = var(m m) = E (M m) 2] [ ] 2 E M m }{{} σ 2 E = α=1 β=1 [ ] = E M 2 2 mm + m 2 = α=1 β=1 2 m w α w β C(x α x β ) [ ] w α w β E Z(x α ) Z(x β ) [ ] w α E Z(x α ) +m 2 } {{ } m α=1 0

14 Minimal estimation variance Aim: weights w α that produce a minimal estimation variance: minimize σe 2 subject to w α = 1 α=1 The objective function ϕ has n+1 parameters: ( ϕ(w 1,..., w n, µ) = var(m m) 2 µ }{{} α=1 criterion where µ is a Lagrange multiplier. ) w α 1 } {{ } condition Setting partial derivatives to zero, we obtain n+1 equations: α : ϕ(w 1,..., w n, µ) w α = 0, ϕ(w 1,..., w n, µ) µ = 0

15 Kriging equations The method of Lagrange yields the equation system for the optimal weights wα KM of the estimation of the mean: β=1 w KM β C(x α x β ) µ KM = 0 for α = 1,..., n β=1 w KM β = 1 The variance at the minimum: σ 2 KM = µ KM is equal to the Lagrange multiplier.

16 Special case: no spatial correlation When the covariance model is only nugget-effect: { σ 2 if x C(x α x β ) = α = x β 0 if x α x β the kriging of the mean system simplifies to: wα KM σ 2 = µ KM for α = 1,..., n wβ KM = 1 β=1 The solution weights are all equal: w KM α = 1 n M = 1 Z(x α ) with variance µ n KM = σkm 2 = 1 n σ2 α=1

17 Special case: no spatial correlation When the covariance model is only nugget-effect: { σ 2 if x C(x α x β ) = α = x β 0 if x α x β the kriging of the mean system simplifies to: wα KM σ 2 = µ KM for α = 1,..., n wβ KM = 1 β=1 The solution weights are all equal: w KM α = 1 n M = 1 Z(x α ) with variance µ n KM = σkm 2 = 1 n σ2 α=1

18 Ordinary Kriging Ordinary Kriging Hans Wackernagel (MINES ParisTech) Basics in Geostatistics 2 NERSC April / 35

19 Estimation at an unsampled location Sample locations x α (blue dots) in a spatial domain D: x 0 Aim: estimate Z at an unsampled location x 0.

20 Ordinary kriging The estimate Z is a weighted average of data values Z(x α ): Z (x 0 ) = w α Z(x α ) α=1 with w α = 1 α=1 The weights wα OK of the Best Linear Unbiased Estimator (BLUE) are solution of the system: β=1 w OK β γ(x α x β ) + µ OK = γ(x α x 0 ) for α = 1,..., n β=1 w OK β = 1 Minimal variance: σ 2 OK = µ OK + α=1 w OK α γ(x α x 0 )

21 Ordinary kriging The estimate Z is a weighted average of data values Z(x α ): Z (x 0 ) = w α Z(x α ) α=1 with w α = 1 α=1 The weights wα OK of the Best Linear Unbiased Estimator (BLUE) are solution of the system: β=1 w OK β γ(x α x β ) + µ OK = γ(x α x 0 ) for α = 1,..., n β=1 w OK β = 1 Minimal variance: σ 2 OK = µ OK + α=1 w OK α γ(x α x 0 )

22 Kriging weights The behavior of kriging weights Hans Wackernagel (MINES ParisTech) Basics in Geostatistics 2 NERSC April / 35

23 Isotropic model: Geometric anisotropy Spherical variogram, weights sum up to 100% 25% 25% L 25% 25%

24 Isotropic model: Geometric anisotropy Spherical variogram, weights sum up to 100% 25% 25% L 25% 25% Anisotropic (ranges horizontal 1.5L, vertical.75l): 17.6% 32.4% L 32.4% 17.6%

25 The screen effect Spherical variogram with range 2L Left sample is at 1L and right at 2L from target σ 2 OK = % 34.4% A B

26 The screen effect Spherical variogram with range 2L Left sample is at 1L and right at 2L from target σ 2 OK = % 34.4% A B Introducing an extra sample σ 2 OK = % A 48.2% C 2.7% B Adding the sample C screens off the sample B.

27 Case-study Filtering noisy images by cokriging Hans Wackernagel (MINES ParisTech) Basics in Geostatistics 2 NERSC April / 35

28 Application: metallography Trace elements are usually masked by instrumental noise. Data: image of phosphorus (P) trace elements. Images for chrome (Cr) and (Ni) are less noisy. Geostatistical filtering is used to remove the noise.

29 Structural analysis Image of phosphorus Nested variogram γ(h) = 384 nug(h) + 75 exp(h) + 13 h

30 Filtering the nugget-effect Raw image of phosphorus Filtered image

31 Multivariate data P Cr Ni

32 Multivariate structural analysis Direct and cross variograms Matrix variogram model: G(h) = B 0 nug(h) + B 1 exp(h) + B 1 h

33 Filtering the nugget-effect Phosphorus Filtered by kriging Filtered by cokriging

34 Case study Space-time filtering Séguret & Huchon, JGR 1990 Hans Wackernagel (MINES ParisTech) Basics in Geostatistics 2 NERSC April / 35

35 Earth magnetism Magnetic anomalies are essential to study earth history. Magnetism is influenced by several external factors like: solar wind explaining daily fluctuations (period: 24 hours) rotation of the moon around the earth (period: 28 days) solar perturbartions (half-year cycle) Available data: SEAPERC campaign (Ifremer, 1986) Data from a research vessel about magnetism over a fractured area of 111 km² off Peru. Fluctuations of earth magnetism Measurements at a Peruvian observatory for the time period of the campaign.

36 Daily fluctuation of earth magnetism Huancayo observatory (Peru): 22 to 28/08/1986 Time series (6 days) Variogram

37 SEAPERC campaign Ship moves along a profile in 12 hours Map Study area

38 Measurements at observatory and along ship route

39 Filtering the daily fluctuations of magnetism Space-time model: Z(x, t) = Y(x) + m(t) With time perturbations After geostatistical filtering

40 Conclusion Summary We have seen: how to set up a variogram model and a corresponding random function model with several components, that these components can be extracted by kriging, that this applies to multivariate or space-time filtering problems. These methods are based on estimators that are linear combinations (weighted averages) of data. However we often are asked to estimate statistics that are not linearly related to data. We will see next how to provide answers by geostatistical simulation. Hans Wackernagel (MINES ParisTech) Basics in Geostatistics 2 NERSC April / 35

41 References JP Chilès and P Delfiner. Geostatistics: Modeling Spatial Uncertainty. Wiley, New York, 2nd edition, G Matheron. The Theory of Regionalized Variables and its Applications. Number 5 in Les Cahiers du Centre de Morphologie Mathématique. Ecole des Mines de Paris, Fontainebleau, S Séguret and P Huchon. Trigonometric kriging: a new method for removing the diurnal variation from geomagnetic data. J. Geophysical Research, 32(B13): , H Wackernagel. Multivariate Geostatistics: an Introduction with Applications. Springer-Verlag, Berlin, 3rd edition, 2003.