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

2. Spatiotemporal Continuum  p=(s,t) denotes a location in the space/time domain E=SxT Spatiotemporal Field  A field is the distribution  across space/time of some parameter X Space/Time Random Field (S/TRF)  A S/TRF is a collection of possible realizations  of the field, X(p)={p,  }  The collection of realizations represents the randomness (uncertainty and variability) in X(p) Definition of Space Time Random Fields X(p)X(p) Space s Time t Realization   X(p)X(p) Space s Time t Realization  (2)

3. Defining a S/TRF at a set of mapping points  We restrict Space/Time to a set of n  mapping points, p map =(p 1,…, p n )  Each field realization reduces to a set of n values,  map =(  1,…,  n )  The S/TRF reduces to set of n random variables, x map = (x 1,…, x n ) The multivariate PDF  The multivariate PDF f X characterizes the joint event x map ≈  map as Prob.[  map < x map <  map + d  map ] = f X (  map ) d  map 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 x a of x map = (x a, x b ) is f X (  a ) = ∫ d  b f X (  a,  b ) hence we can define any marginal PDF from f X (  map ) Multivariate PDF for the mapping points

4. Stochastic Expectation The stochastic expectation of some function g(X(p), X(p’), …) of the S/TRF is E [g(X(p), X(p’), …)] = ∫ d  1 d  2... g(  1,  2,...) f X (  1,  2,...; p ; p’,...) Mean trend and covariance The mean trend m X (p) =E [X(p)] and covariance c X (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) Statistical moments

5. A homogeneous/stationary S/TRF is defined by  A mean trend that is constant over space (homogeneity) and time (stationarity) m X (p) = m X  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’| c X (p, p’) = c X ( (s,t), (s’,t’) ) = c X ( r=||s-s’||,  =|t-t’|  ) A homogeneous/stationary S/TRFs has the following properties  It’s variance is constant, i.e.  X 2 (p)=  X 2 Proof:  X 2 (p)= E[(X(p)- m X (p)) 2 ] = c X (p, p) = c X ( r=0,  =0 ) is not a function of p  It’s covariance can be written as c X (r,  )= E[X(s,t)X(s’,t’)] ||s-s’|| =r, |t-t’| =  - m X 2,  This is a useful equation to estimate the covariance Homogeneous/Stationary S/TRF

6. 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,  ) is the number of pairs of points with values (X head, X tail ) separated by a distance of r and a time of . In practice we use a tolerance dr and d , i.e. such that r-dr ≤ ||s head -s tail || ≤ r+dr and  -d  ≤ ||t head -t tail || ≤  +d  Experimental estimation of covariance

7. Gaussian model: c X (r) = c o exp-(3r 2 /a r 2 )  c o = sill = variance  a r = spatial range  Very smooth processes Exponential model: c X (r) = c o exp-(3r/a r )  more variability Nugget effect model c X (r) = c o  (r)  purely random Nested models c X (r) = c 1 (r) + c 2 (r) + …  where c 1 (r), c 2 (r), etc. are permissible covariance models Example: Arsenic c X (r) = 0.7  X 2 exp-(3r/7Km) + 0.3  X 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. Spatial covariance models

8. c X (r,  ) is a 2D function with spatial component c X (r,  ) and temporal component c X (r=0,  ) Space/time separable covariance model  c X (r,  ) = c Xr (r) c Xt (  ), where c Xr (r) and c Xt (  ) are permissible models Nested space/time separable models  c X (r,  ) = c r1 (r) c t1 (  ) + c r 2 (r)c t 2 (  ) + … Example: Yearly Particulate Matter concentration (ppm) across the US  c X (r,  ) = c 1 exp(-3r/a r1 -3  /a t1 ) + c 2 exp(-3r/a r2 -3  /a t2 )  1 st structure c 1 = 0.0141(log mg/m 3 ) 2, a r1 =448 Km, a t1 =1years is weather driven  2 nd structure c 1 = 0.0141(log mg/m 3 ) 2, a r1 =17 Km, a t1 =45years due to human activities Space/time covariance models

9. Gather the data  hard =[  1,  2,  3, …] T and obtain the experimental covariance Fit a covariance model c X (r) to the experimental covariance Simple kriging (SK) is a linear estimator  x k (SK)  0  T x hard SK is unbiased  E[x k (SK) ] = E[x k ] ═► x k (SK)  m k + T (x hard  m hard ) SK minimizes the estimation variance  SK 2 = E [( x k  x k (SK)  ) 2 ]  ∂  SK 2 / ∂ T  ═► T = C k,hard C hard,hard -1 Hence the SK estimator is given by  x k (SK)  m k + C k,hard C hard,hard -1 (x hard.  m hard ) T And its variance is   SK 2   k 2 - C k,hard C hard,hard -1 C hard,k The simple kriging (SK) estimator

10. Run Kriging Example introToKrigingExample.m Example of kriging maps

11. 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)