2.6 Two-dimensional continuous interpolation 3: Kriging - introduction to geostatistics. References - geostatistics. References geostatistics (cntd.

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1 .6 Two-dimensional continuous interpolation 3: Kriging - introduction to geostatistics Spline interpolation was originally developed or image processing. In GIS, it is mainly used in visualization o spatial data, where the appearance o interpolated surace is important. In geology and geomorphology, on the other hand, a dierent interpolation method called kriging is widely used which was developed by a South Arican geologist D. G. Krige. Though originally developed or geological analysis, kriging is now widely used not only in geology and geomorphology but also in other ields related to GIS, say, human geography, epidemiology, biostatistics, and archaeology. Reerences - geostatistics Kriging is a part o geostatistics, a statistics that treats spatially distributed (and usually countinuous) stochastic phenomena. Geostatistics includes kriging, spatial modeling o continuous surace, spatial continuous processes, spatiotemporal statistics, and so orth.. Isaak, E. H. and Srivastava, R. M. (989): An Introduction to Applied Geostatistics, Oxord University Press.. Cressie, N. (993): Statistics or Spatial Data, nd Edition, John Wiley. 3. Wackernagel, H. (995): Multivariate Geostatistics, Springer. 4. Christakos, G. and Hristopulos, D. T. (998): Spatiotemporal Environmental Health Modelling, Kluwer. Reerences geostatistics (cntd.).6. Variogram, covariogram, correlogram 5. Chiles, J.-P. and Deliner, P. (999): Geostatistics: Modeling Spatial Uncertainty, John Wiley. 6. Christakos, G. (000): Modern Spatiotemporal Geostatistics, Oxord University Press. 7. Webster, R. and Oliver, M. A. (00): Geostatistics or Environmental Scientists, John WIley. 8. Mallet, J.-C. (00): Geomodeling, Oxord University Press. S: Study region A: Area o S (x): Surace unction deined in S Our objective is to estimate (x) in S rom observed data at sample points. S

2 Deinition o variogram In kriging we use one o three unctions deined rom the surace unction to be estimated.. variogram. covariogram 3. correlogram To explain those unctions, or the present, we assume that the surace unction (x) is known. γ ( h) x S t S, x t h { } x S t S, x t h x t dd t x dtx d Variogram unction indicates the average (square) dierence o the surace value between two points o distance h. Thereore, rom the variogram unction, we can see how (x) luctuates in S γ (h) h Figure: Example o variogram In usual, variogram unction increases monotonically with h, because the surace values at two near locations are more similar than those at two distant locations. This gives one theoretical basis or various spatial interpolation methods However, we oten ind variogram unctions that do not increase monotonically with h. Anisotropic variogram Linear Spherical Exponential Variogram is a unction o h, the distance between two points in S. This implies that variogram does not consider anisotropy in the luctuation o surace unction. This type o variogram is thus oten called isotropic variogram. On the other hand, anisotropic variogram explicitly considers anisotropy o the surace unction. It is a unction o both the distance h and the direction θ. Quadratic Wave Power Figure: Typical variograms

3 Deinition o anisotropic variogram Covariogram and correlogram γ ( h, θ) ( x t, u) x S t S, x t h, cosθ x t u { } ( x t, u) x S t S, x t h, cosθ x t u x t dd t x dtx d Covariogram and correlogram are also unctions o h, the distance between two points in S, and they also represent the luctuation o the surace unction. u: Unit vector parallel with the X-axis. The unction γ(h, θ) indicates the luctuation o the surace unction in the direction o angle θ measured counterclockwise rom the X-axis. Covariogram { }{ }, μ μ x S t S x t h dd S tx x t S, x t h C h μ: Mean o the unction (x) in S x t dd t x Covariogram corresponds to covariance used in general statistics. I we assume that the surace unction (x) ollows a stochastic process, its covariance o two points o distance h is given by covariogram. d S μ x x x A Relationship between variogram and covariogram σ γ ( h) C h σ : Variance o the unction (x) x σ S { μ} A dx This equation indicates that variogram and covarioram are equivalent in the sense that one completely determines the other. It also shows that, i a variogram is an increasing unction o h, the covariogram is a decreasing unction o h. 3

4 Correlogram Relationship between variogram and correlogram ρ ( h) x S t S, x t h C h σ σ { μ} { μ} x t dd t x x S t S, x t h dd tx ρ ( h) ( h) γ σ As well as covariogram unction, correlogram unction is usually a monotonic decreasing unction o h. The three unctions, variogram, covariogram, and correlogram are equivalent and interchangeable; we can calculate any o the three unction rom the other unctions..6. Outline o kriging P i : The ith sample point in S (i,,..., n) z i : The locational vector o P i (x): Surace unction to be interpolated in S Kriging interpolates the value at a certain location by a weighted summation o the values at surrounding sample points. Estimator unction o (x) is thus given by n ( z ) ˆ wi i w i (x): Weight unction or P i at x i For simple explanation, vector and matrix notation is introduced. ( z) ( z ) ( z ) n w x w w wn Then the estimator unction is written as T ˆ x w x The problem is how we determine the weight unction, which is the main issue in kriging. There are various kriging methods that use dierent methods o determining the weight unction. 4

5 In kriging, to speciy the weight unction w(x), we assume that the surace unction (x) ollows a stochastic process, and we treat the surace values at sample points as observed data obtained rom the stochastic process. The surace unction (x) is not a deterministic unction. This claims that we cannot speciy the surace unction (x) by observation because o measurement error. Kriging uses the ramework o statistics, an this is why it is oten called geostatistics. Besides this assumption, every kriging method imposes its own conditions (assumptions) that the weight unction w(x) has to satisy. We then calculate w(x) based on the conditions, and inally estimate the surace unction or the whole region by using n ˆ wi ( z ) i i.6.3 Simple kriging Simple kriging puts two assumptions on the behavior o (x). Assumption A: The expectation o the surace unction, E[(x)], is constant in S. In practice, instead o Assumption A, simple kriging uses the assumption below. Assumption A : The expectation o the surace unction is zero in S. Though this assumption is not realistic, and because o this disadvantage simple kriging is not used in GIS, it makes the methodology o kriging easier to understand. Assumptions A and A are equivalent because o the ollowing reason. Let μ be the expectation o (x), that is, E[(x)]. We then deine g(x) by g μ Instead o estimating (x) directly, we obtain the same result by estimating g(x) and add μ to the estimated g(x). This greatly reduces the amount o calculation. Assumption A: The covariance o the surace unction o two locations is given by a unction o only the distance between the locations. The covariance o (x) o two locations x i and x j is { } ( xi) ( x j) { ( xi) ( xi) } ( x j) ( x j ) E ( xi) ( x j) C, E E E 5

6 By Assumption A the covariance unction becomes ( xi) ( x j) ( xi) ( x j) C ( xi x j ) C, E Covariances are usually represented as a matrix called covariance matrix: ( z), ( z) ( z), ( z) ( z), ( zn ) ( z ), ( z ) ( z ), ( z ) ( z ), ( z ) C C C C C C n C C ( n), C ( n), C ( n), ( n) z z z z z z In simple kriging, covariance matrix is then written as σ C( z z ) C( z zn ) ( z z ) σ C( z zn ) C C C( n ) C( n ) σ z z z z σ C ( 0) Similarly, or plain explanation, covariance vector is introduced:,, ( ) ( ) C x z C x z C x z C x z c C, ( n) C( n ) x z x z Desirable properties o estimator unctions Expectation o estimator unction In kriging, not limited to simple kriging, it is desirable that estimator unctions have the ollowing properties. Property P: unbiasedness The expectation o estimator unction is equal to the expectation o the original surace unction. Expectation o the estimator unction: n E ( z ) E ˆ wi i i E Property P: eiciency The variance o estimator unction is smaller than any other estimator unction. 6

7 Variance o estimator unction As ar as we use a linear combination o observed data, the estimator unction is unbiased independently o the weight unction w(x). We do not have to take Property P into account in estimation o the weight unction w(x). We can ocus only on property P to speciy w(x). Instead o the variance o estimator unction, we usually discuss the mean square error (MSE) o estimator unction, because it is equivalent to the variance but easier to derive analytically. MSE ˆ { ˆ } E { ˆ } { } ˆ E + E E n n n ( z z ) σ ( x z ) w w C + w C i j i j i i i j i σ + T T w x Cw x w x c x We choose the weight unction w(x) that minimizes the variance o estimator unction represented by the mean square error. Mathematically, we solve wx ˆ σ min MSE wx min + T T w x Cw x w x c x We can solve this minimization problem by solving w x The result is C c w x MSE ˆ 0 Consequently, estimator unction o the surace is given by n ( x ) ˆ w i i ( C c ) T c T x C i 7

8 An example But how do we calculate covariance unctions? In simple kriging we arbitrarily choose a covariogram unction rom typical ones, such as C h 00e C h h and calculate the covariances ? μ To use Assumption A (E[(x)]0 in S), we substitute μ rom the observed data o (x) at sample points ? μ0 We then assume a covariogram unction given by C h 3h 0exp ˆ 50.5 Figures indicate the weight or sample points 8

9 Limitations o simple kriging. The choice o covariogram unction is arbitrary.. The sum o weight w(x) is not equal to one. 3. It is not realistic to assume that the expectation o (x) is constant in S. There is usually at least a slight variation in E[(x)] among locations. The irst shortcoming can be partly corrected as ollows. We irst choose a theoretical (typical) covariogram unction that contains several ree parameters. We then it the unction to the observed data at sample points and estimate the parameter values by a statistical procedure, or instance, the least square method..6.4 Ordinary kriging The second and third problems, on the other hand, cannot be resolved i we continue to use simple kriging. Ordinary kriging overcomes the second limitation o simple kriging, that is, the sum o weight w(x) is not equal to zero. In ordinary kriging the sum o weight w(x) is equal to zero at any location in S. The weight w(x) is estimated in the same way as the simple kriging: min MSE ˆ wx Ordinary kriging add one constraint to the minimization problem: T w Since the new minimization problem is an optimization problem with a constraint, it cannot be solved by only dierentiating the mean square error o the estimator unction o (x) by w(x). To solve optimization problems with constraints, we use the method o Lagrange multipliers. 9

10 We introduce a new unction λ(x), which is called a Lagrange multiplier. The original problem with a constraint min MSE ˆ wx T s.t. w x then becomes a problem without constraints: T min MSE ˆ + λ, λ x wx x w x x The result is w x C c x where w C ( z), ( z) C ( z), ( zn ) w+ C wn + C ( n), C ( n), ( n) z z z z λ x 0 C, ( z) C( x z ) c+ C, ( n) C x z ( x z n ) An example In ordinary kriging covariogram unction is arbitrarily chosen rom typical theoretical unctions, or estimated rom the observed data at sample points by the least square method ? μ57.8 To estimate the surace unction, we again speciy a covariogram unction arbitrarily which is given by C h 3h 0exp ˆ 50.5 Figures indicate the weight o sample points 0

11 Limitations o ordinary kriging The surace value estimated by ordinary kriging is identical to that by simple kriging. This, however, happened only by chance. These two methods usually yield dierent results. Ordinary kriging inherits one disadvantage rom simple kriging: It is not realistic to assume uniormity in the expectation o (x). There is usually at least a slight variation in E[(x)] among locations..6.5 Universal kriging Universal kriging overcomes the limitation o ordinary kriging. Universal kriging assumes heterogeneity in the expectation o (x) in S. Universal kriging is based on the ollowing our assumptions. Assumption A: The surace unction (x) is not deterministic but ollows a stochastic process. Assumption A: The expectation o the surace unction, E[(x)], is given by a unction o location x indicated by μ(x). Assumption A3: The covariance o (x) between two locations is given by a unction o only the distance between the locations. Assumption A4: The sum o the weight unction w(x) is equal to one at any location. The mean square error is then given by MSE ˆ x E { ˆ ( x )} T T T T w Cw + σ w c + w μ w μ w μ where μ ( x ) μ ( x ) 0 μ 0 0 μ ( n ) x

12 .6.6 Advanced kriging We then solve min MSE ˆ wx, μ T s.t. w x by the method o Lanrange multipliers to estimate both w(x) and μ(x) simultaneously. ) Block kriging It oten happens in spatial analysis that we need only the mean value o (x) in a subregion in S, instead o (x) or the whole region S. Block kriging estimates the average value o (x) in a region in S without estimating the whole surace. Block kriging is thus more eicient than universal kriging. ) Cokriging Cokriging interpolates two suraces simultaneously, say, (x) and g(x), by using two sets o measured values at sample points. In short, to interpolate (x) or g(x), cokriging uses twice as much inormation as universal kriging does. Though we expect that the result is twice as accurate as that o universal kriging, whether or not cokriging works successully depends on the correlation between the two unctions. 3) Disjunctive kriging Instead o linear combination o observed data, n ˆ wi ( zi) i disjunctive kriging uses a nonlinear unction o the data to estimate the surace value at a sample point. This increases lexibility o the surace unction estimated, but decreases the eiciency o computation..6.7 Comparison o spline and kriging Spline Kriging Applications in GIS Visualization Preprocessing Spatial analysis Observed data Correct Plausible Evaluation o result Appearance Goodness o it as a model Theoretical background Mathematics (?) Statistics Sotware packages Available Available Kriging may seem rather complicated and its mathematical background diicult to understand. However, today s GIS and its add-in extensions can estimate the weight unction and interpolate the surace unction automatically. You can do kriging without complete understanding o its theoretical background.

13 .6.8 Cross validation.7 Areal interpolation One o the diiculties in spatial interpolation is that we cannot tell to what degree an interpolated unction is accurate. To evaluate its accuracy, we estimate the unction value at a sample point rom observed data at other sample points, and calculate the estimation error at the sample point. We repeat this process or all the sample points, and call this process cross validation. Areal interpolation is a process o transerring attribute data aggregated by a certain zonal system to another system. It is necessary when, or example, we use two spatial datasets simultaneously, one reported by census districts and the other by a square lattice. We have to transer the attribute data o the ormer dataset into the latter one ? Figure: Data reported by census tracts Figure: How many people are there in the circle? Terminology.7. Areal weighting method Source zones: Source zones are spatial units used or aggregating spatial data and reporting the aggregated attribute data. Target zones: Target zones are spatial units in which we want to know the attribute data. Source zones 7? 6 9 Areal weighting method assumes that spatial objects are uniormly distributed in each source zone. This method thus allocates attribute data in proportion to the size o subregions. Target zone 3

14 Estimation o attribute data in a target zone Because o uniormity assumption, areal weighting method does not work when the distribution o spatial objects is not uniorm, say, when they are intensely clustered. In urban area population distribution is globally smooth and oten uniorm but it has a great variation at a local scale. Thereore, in estimation o population distribution, areal weighting method is appropriate or global interpolation but is not suitable or local interpolation..7. Point-in-polygon method Point-in-polygon method can be used when every source zone has its own representative point, which is usually located at the centroid (gravity center) o the zone. Point-in-polygon method assumes that in each source zone all the spatial objects are located exactly at the representative point, and assigns the attribute value to the representative point The attribute value o the target zone is then calculated by summing up all the attribute values o the representative points contained in the target zone. Estimation o attribute data in a target zone In contrast to areal weighting method, point-in-polygon method works successully when spatial objects are clustered around representative points In Japan, point-in-polygon method is used to create the census data in raster ormat (mesh data) rom the original census data aggregated by census tracts. 4

15 .7.3 Advanced methods o areal interpolation Kernel method Kernel method assumes that in each source zone spatial objects are distributed around the representative point ollowing a kernel unction, a decreasing unction o the distance rom the representative point. Kernel method puts small bumps called kernels centered at representative points, whose summation represents density distribution o spatial objects. We calculate attribute data in target zones by integrating the density unction. Intelligent methods Intelligent methods use additional inormation about the distribution o spatial objects. In estimation o population distribution, land use and land cover data are oten used. This drastically improve the accuracy o attribute data estimated in target zones. Homework Q..3 Homework Q..4 Delaunay triangulation is used in two-dimensional interpolation where sample points are irregurally distributed. Describe the properties o Delaunay triangulation and the procedures o constructing Delaunay triangulation rom a set o given points. Universal kriging is deined as a mathematical optimization problem with constraints: min MSE ˆ wx, μ T s.t. w x Describe the detailed process o solving this problem and give explicit orms o w(x) and μ(x). 5