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Theory of Kriging Theory of Kriging W'ô Theory of random fields Random functions First-order stationarity Spatial covariance Second-order stationarity Theory of Kriging The intrinsic hypothesis Ordinary Kriging Optimization criterion D G Rossiter Computing the kriging variance Computing OK weights Nanjing Normal University, Geographic Sciences Department The OK system W¬'f0ffb Solution of the OK system 30-November-2016 Theory of Kriging W'ô Theory of random fields 1 Theory of random fields Random functions First-order Random functions stationarity Spatial covariance First-order stationarity Second-order stationarity Spatial covariance The intrinsic hypothesis Second-order stationarity Ordinary Kriging The intrinsic hypothesis Optimization criterion Computing the kriging variance Computing OK weights 2 Ordinary Kriging The OK system Optimization criterion Solution of the OK system Computing the kriging variance Computing OK weights The OK system Solution of the OK system Theory of Kriging Overview W'ô Theory of random fields Random functions First-order stationarity Spatial covariance Second-order • Kriging is a Best Linear Unbiased Predictor (BLUP) of the stationarity The intrinsic value of an attribute at an unsampled location. hypothesis Ordinary • “Best” is defined as the lowest prediction variance Kriging Optimization among all possible combination of weights for the criterion Computing the weighted sum prediction. kriging variance Computing OK • Derivation of the weights from this optimization weights 1 The OK system criterion depends on a theory of random fields. Solution of the OK system • This is a model of how the reality that we observe, and which we want to predict, is structured. • The model represents a random process with spatial autocorrelation. 12nd section of this lecture Theory of Kriging Theory of random fields W'ô Theory of random fields Random functions First-order stationarity Spatial covariance Second-order stationarity The intrinsic hypothesis Ordinary Presentation is based on R. Webster and M. Oliver, 2001 Kriging Optimization Geostatistics for environmental scientists, Chichester etc.: John criterion Computing the Wiley & Sons, Ltd.; ISBN 0-471-96553-7 kriging variance Computing OK weights Notation: A point in space of any dimension is symbolized by a The OK system Solution of the OK bold-face letter, e.g. x. In 2D this is (x ; x ). system 1 2 Theory of Kriging Theory of random fields – key idea W'ô Theory of random fields Random functions First-order 1 Key idea: The observed attribute values are only one of stationarity Spatial covariance many possible realisations of a random process (also Second-order stationarity called a “stochastic” process) The intrinsic hypothesis 2 This random processes is spatially auto-correlated , so Ordinary Kriging that attribute values are somewhat dependent. Optimization criterion 3 At each point x, an observed value z is one possiblility of Computing the kriging variance a random variable Z(x) Computing OK weights The OK system 4 There is only one reality (which is sampled), but it is one Solution of the OK system realisation of a process that could have produced many realities. µ and variance σ 2 etc. 5 Cumulative distribution function (CDF): FfZ(x; z)g = Pr[Z(x ≤ zc)] 6 the probability Pr governs the random process; this is where we can model spatial dependence Theory of Kriging Random functions W'ô Theory of random fields Random functions First-order stationarity • Each point has its own random process, but these all have Spatial covariance Second-order the same form (same kind of randomness) stationarity The intrinsic hypothesis • However, there may be spatial dependence among Ordinary points, which are therefore not independent Kriging Optimization • As a whole, they make up a stochastic process over the criterion Computing the whole field R kriging variance Computing OK • i.e., the observed values are assumed to result from some weights The OK system random process but one that respects certain Solution of the OK system restrictions, in particular spatial dependence • The set of values of the random variable in the spatial field: Z = fZ(x), 8x 2 Rg is called a regionalized variable • This variable is doubly infinite: (1) number of points; (2) possible values at each point Theory of Kriging Simulated fields W'ô Theory of random fields Random functions First-order stationarity Spatial covariance Second-order stationarity The intrinsic • Equally probable realizations of the same hypothesis spatially-correlated random process Ordinary Kriging Optimization • The fields match an assumed variogram model criterion Computing the • These are equally-probable alternate realities, assuming kriging variance Computing OK the given process weights The OK system Solution of the OK • We can determine which is most likely to be the one reality system we have by sampling. • Next pages: simulated fields on a 256x256 grid Theory of Four realizations of the same field – before sampling Kriging W'ô Unconditional simulations, Co variogram model sim3 sim4 25 Theory of random fields Random functions First-order stationarity Spatial covariance 20 Second-order stationarity The intrinsic hypothesis Ordinary 15 Kriging Optimization criterion Computing the kriging variance Computing OK 10 weights sim1 sim2 The OK system Solution of the OK system 5 0 −5 Theory of Four realizations of the same field – after sampling Kriging W'ô Conditional simulations, Jura Co (ppm), OK sim3 sim4 25 ● ● Theory of ● ● ● ● random fields ● ● ● ● ● ● ● ● Random functions ●● ● ●● ● ● ● ● ● ● ● ● ●● ● ●● ● ● ● ● First-order ● ● ● ● ● ● stationarity ● ●● ● ●● ● ● ● ● ●● ● ● ●● ● ● ● ● ● ● ●●●● ● ● ●●●● ● ● 20 Spatial covariance ● ● ● ● ● ●● ●● ● ●● ●● ●● ● ●● ● ●●● ●● ●●● ●● Second-order ● ● ● ● ● ● ● ● ● ● ● ● ● ● stationarity ● ● ● ● ● ● ● ● ● ●● ● ● ● ●● ● ● ● ● ● ● ● ● The intrinsic ●● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● hypothesis ●● ● ● ● ●● ● ● ● ●● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ●● ● ● Ordinary ● ● ● ● ● ● ● ● 15 ● ● ●● ● ● ● ●● ● ● ●● ●●● ● ● ●● ●●● ● ● ● ● ● Kriging ● ● ●● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● Optimization ●● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ●● criterion ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ●● ● ●●● ● ● ● ● ●●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ●● ● ● ● Computing the ● ● ● ● ● ● ● ● ● ● kriging variance ● ● ● ● ● ● ● ● ● ● Computing OK sim1 sim2 10 weights ● ● ● ● The OK system ● ● ● ● ● ● ● ● Solution of the OK ● ● system ●● ● ●● ● ● ● ● ● ● ● ● ●● ● ●● ● ● ● ● ● ● ● ● ● ● ● ●● ● ●● ● ● ● ● ●● ● ● ●● ● ● 5 ●●● ● ● ●●● ● ● ● ● ● ● ● ● ● ●● ●● ● ●● ●● ●● ● ●● ● ●●● ●● ●●● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ●● ● ● ● ● ● ● ● ● ●● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ●● ● ● ● ●● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ●● ● ● ● ● ● ● ● ● 0 ● ● ● ● ● ● ●● ● ● ● ●● ● ● ●● ●●● ● ● ●● ●●● ● ● ● ● ● ● ● ●● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ●● ● ●●● ● ● ● ● ●●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ●● ● ●● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● −5 Theory of Fields with same variogram parameters, different models Kriging W'ô sim.sph sim.pen Theory of random fields Random functions 20 First-order stationarity Spatial covariance Second-order stationarity The intrinsic 15 hypothesis Ordinary Kriging Optimization criterion Computing the 10 kriging variance Computing OK sim.exp sim.gau weights The OK system Solution of the OK system 5 0 −5 Theory of Fields with same model, different variogram parameters Kriging W'ô sim.sph.0 sim.sph.20 25 Theory of random fields Random functions First-order 20 stationarity Spatial covariance Second-order stationarity The intrinsic hypothesis 15 Ordinary Kriging Optimization criterion 10 Computing the kriging variance Computing OK sim.sph.40 sim.sph.60 weights The OK system Solution of the OK 5 system 0 −5 Theory of Kriging W'ô Uncorrelated field with first-order stationarity (same expected Theory of value everywhere) random fields Random functions Corresponds to pure “nugget” variogram model. First-order stationarity Spatial covariance white noise Second-order stationarity 8 The intrinsic hypothesis 6 Ordinary Kriging 4 Optimization criterion Computing the 2 kriging variance Computing OK weights 0 The OK system Solution of the OK system −2 −4 −6 −8 Theory of Kriging First-order stationarity W'ô Theory of random fields Random functions • First-order Problem: We have no way to estimate the expected stationarity Spatial covariance values of the random process at each location µ(x) ... Second-order stationarity • . since we only have one realisation (what we actually The intrinsic hypothesis measure), rather than the whole set of realisations that Ordinary could have been produced by the random process. Kriging Optimization criterion • Solution: assume that the expected values at all Computing the kriging variance locations in the field are the same: Computing OK weights The OK system x = 8x 2 Solution of the OK E[Z( i)] µ; i R system • This is called first-order stationarity of the random process • so µ is not a function of position x. • Then we can estimate the (common) expected value from the sample values and their spatial structure. Theory of Kriging Problems with first-order stationarity W'ô Theory of random fields Random functions First-order stationarity Spatial covariance • It is often not plausible: Second-order stationarity 1 We observe the mean value to be different in several The intrinsic hypothesis regions
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