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Introduction to kriging The Best Linear Unbiased Estimator BLUE for space time mapping Definition of Space Time Random Fields Spatiotemporal Continuum Spatiotemporal Field p s t denotes a location in the space time domain E SxT A field is the distribution c across space time of some parameter X Space Time Random Field S TRF A S TRF is a collection of possible realizations c of the field X p p c X p Space s X p Space s Realization c 1 Realization c 2 Time t Time t The collection of realizations represents the randomness uncertainty and variability in X p Multivariate PDF for the mapping points Defining a S TRF at a set of mapping points We restrict Space Time to a set of n mapping points pmap p1 pn Each field realization reduces to a set of n values xmap x1 xn The S TRF reduces to set of n random variables Xmap X1 Xn The multivariate PDF The multivariate PDF fX characterizes the joint event Xmap xmap as Prob xmap Xmap xmap dxmap fX xmap dxmap hence the multivariate PDF provides a complete stochastic description of trends and dependencies of the S TRF X p at its mapping points Marginal PDFs The marginal PDF for a subset Xa of Xmap Xa Xb is fX xa dxb fX xa xb hence we can define any marginal PDF from fX xmap Statistical moments Stochastic Expectation The stochastic expectation of some function g X p X p of the S TRF is E g X p X p dx1 dx2 g x1 x2 fX x1 x2 p p Mean trend and covariance The mean trend mX p E X p and covariance cX p p E X p m p X p m p are statistical moments of order 1 and 2 respectively that characterizes the consistent tendencies and dependencies respectively of X p Homogeneous Stationary S TRF A homogeneous stationary S TRF is defined by A mean trend that is constant over space homogeneity and time stationarity mX p mX A covariance between point p s t and p s t that is only a function of spatial lag r s s and the temporal lag t t t cX p p cX s t s t cX r s s t t t A homogeneous stationary S TRFs has the following properties It s variance is constant i e sX Proof sX It s covariance can be written as 2 p sX 2 2 p E X p mX p 2 cX p p cX r 0 t 0 is not a function of p cX r t E X s t X s t s s r t t t mX 2 This is a useful equation to estimate the covariance Experimental estimation of covariance When having site specific data and assuming that the S TRF is homogeneous stationary then we obtain experimental values for it s covariance using the following estimator c X r t 1 rN t rN t 1 i x head i x tail i m X 2 where N r t is the number of pairs of points with values xhead xtail separated by a distance of r and a time of t In practice we use a tolerance dr and dt i e such that r dr shead stail r dr and t dt thead ttail t dt Spatial covariance models cX r co exp 3r2 ar 2 Gaussian model co sill variance ar spatial range Very smooth processes Exponential model cX r co exp 3r ar more variability Nugget effect modelcX r co d r purely random Nested models cX r c1 r c2 r where c1 r c2 r etc are permissible covariance models Example Arsenic cX r 0 7sX 2 exp 3r 7Km 0 3sX 2 exp 3r 40Km where the first structure represents variability over short distances 7Km e g geology the second structure represents variability over longer distances 40Km e g aquifers Space time covariance models cX r t is a 2D function with spatial component cX r t 0 and temporal component cX r 0 t Space time separable covariance model cX r t cXr r cXt t where cXr r and cXt t are permissible models Nested space time separable models cX r t cr1 r ct1 t cr2 r ct2 t Example Yearly Particulate Matter concentration ppm across the US cX r t c1 exp 3r ar1 3t at1 c2 exp 3r ar2 3t at2 1st structure c1 0 0141 log mg m3 2 ar1 448 Km at1 1years is weather driven 2nd structure c1 0 0141 log mg m3 2 ar1 17 Km at1 45years due to human activities The simple kriging SK estimator Gather the data xhard x1 x2 x3 T and obtain the experimental covariance Fit a covariance model cX r to the experimental covariance Simple kriging SK is a linear estimator Xk SK l0 l T Xhard SK is unbiased E Xk SK E Xk Xk SK mk l T Xhard mhard SK minimizes the estimation variance sSK 2 E xk xk SK 2 sSK 2 lT 0 lT Ck hard Chard hard 1 Hence the SK estimator is given by xk SK mk Ck hard Chard hard 1 xhard mhard T And its variance is sSK 2 sk 2 Ck hard Chard hard 1 Chard k Example of kriging maps Run Kriging Example introToKrigingExample m Example of kriging maps Observations Only hard data are considered Exactitude property at the data points Kriging estimates tend to the prior expected value away from the data points Hence kriging maps are characterized by islands around data points Kriging variance is only a function to the distance from the data points Limitations of kriging Kriging does not provide a rigorous framework to integrate hard and soft data Kriging is a linear combination of data i e it is the best only among linear estimators but it might be a poor estimator compared to non linear estimators The estimation variance does not account for the uncertainty in the data itself Kriging assumes that the data is Gaussian whereas in reality uncertainty may be non Gaussian Traditionally kriging has been implemented for spatial estimation and space time is merely viewed as adding another spatial dimension this is wrong because it is lacking any explicit space time metric


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UNC-Chapel Hill ENVR 468 - Introduction to Kriging

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