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

2. 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)

3. 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)

4. 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)

5. 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

6. 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

7. 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.

8. 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

9. 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

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

11. 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)