Introduction to kriging: The Best Linear Unbiased Estimator (BLUE) for space/time mapping

Definition of Space Time Random Fields Spatiotemporal Continuum p=(s,t) denotes a location in the space/time domain E=SxT Spatiotemporal Field 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} The collection of realizations represents the randomness (uncertainty and variability) in X(p) X(p) Space s Time t Realization c(1) X(p) Space s Time t Realization c(2)

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 Mean trend and covariance 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. sX2(p)= sX2 Proof: sX2(p)= E[(X(p)- mX(p))2] = cX(p, p) = cX( r=0, t=0 ) is not a function of p It’s covariance can be written as cX(r , t)= E[X(s,t)X(s’,t’)] ||s-s’|| =r, |t-t’| =t- mX2 ,  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 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 Gaussian model: cX(r) = co exp-(3r2/ar2) co = sill = variance ar = spatial range Very smooth processes Exponential model: cX(r) = co exp-(3r/ar) more variability Nugget effect model cX(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) = 0.7sX2 exp-(3r/7Km) + 0.3sX2 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 sSK2 = E[(xk - xk(SK) )2] ∂sSK2 / ∂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 sSK2 = sk2 - Ck,hard Chard,hard-1 Chard,k

Example of kriging maps introToKrigingExample.m 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)