CWR 6536 Stochastic Subsurface Hydrology Optimal Estimation of Hydrologic Parameters using Kriging
Purpose of Kriging • To estimate regional distribution of a spatially variable parameter • To estimate accuracy of regional distribution • Need scattered point measurements of the variable of interest • Need knowledge of the spatial correlation structure
Purpose of Kriging • Estimate value at an unmeasured point • Use estimated values to: • produce map of variable • use as input parameter for deterministic or stochastic groundwater flow/transport model
Desirable Characteristics of Estimated Values • Linear, i.e. weighted linear combination of observed values: • Unbiased, i.e. • Efficient, i.e. minimum estimation variance for given number of observed points is minimized • Kriging is sometimes referred to as a BLUE estimate
Simple Kriging • Simple kriging is optimal estimation of a random field, e.g. T(x), with a known mean, m(x), and a known covariance PTT(x,x’). • Assume random field of interest is T(x). Define a zero mean random field Y(x) as Y(x)=T(x)-m(x). • Since the expected value of T(x) is m, Y(x) is a zero mean random variable with covariance, PTT(x,x’), and variance, sY2(x)=sT2(x).
Simple Kriging • Define kriging estimate as: • Check for bias: • Choose li so that estimation variance is minimized:
Simple Kriging • Minimize estimation variance • Recall definition of covariance function:
Simple Kriging • Therefore • To minimize this function of litake partial derivative of equation with respect to all li (for i=1,N) and set equal to zero:
Simple Kriging • This results in a linear system of N equations with N unknowns: • Or in matrix notation: • Onceli are determined construct estimate from
Simple Kriging • Can also now calculate the kriging variance • Plugging in solution of kriging equations gives: • Or in terms of original variable gives:
Properties of Kriging Process • System of kriging equations has a unique solution for li if PTT(x,x’) is a positive definite function and if the measurement locations xi are unique. • Solution of equations is required for each x0 at which an estimate is desired. • Estimation or kriging variance is always smaller than prior unconditional variance. Degree to which it is smaller depends on the spatial correlation between estimation point x0 and the observation points xi. Higher spatial correlation leads to lower kriging variances.
Properties of the Kriging Process • At observations points kriging equations assigns l=1 to observation and l=0 to all other measurements. Thus kriging recovers exact measurement values and gives an estimation variance of zero at observation points. • The kriging equations and estimation variance equation depend only on the covariance function P and the spatial configuration of the data points xi. Thus before any wells are drilled or soil cores taken the kriging system can be solved and estimation variances can be evaluated at each point in the domain. Thus the amount of information gained from alternative monitoring networks can be evaluated.
Properties of Kriging Process • Kriging system is applicable to isotropic or anisotropic covariances, and also to stationary or non-stationary covariances. • In general if estimate covariances from field measurements there is rarely enough data to deduce anisotropy or non-stationarity • If derive covariances from physical flow and transport equations will usually have anisotropic, non-stationary covariance functions
Summary • Given a particular realization of a random field T(x) from an ensemble with known mean, m(x), and covariance function PTT(x)…. • Prior to any site specific measurements the best estimate of the state of the random field is the unconditional mean, m(x). The uncertainty of using this estimate is the unconditional variance, sT2(x). • Once measurements are taken can improve on this estimate. BLUE is and the uncertainty of using this estimate is • If the underlying random field is normally distributed the BLUE estimator is identical to the conditional mean of the random field and the kriging variance is identical to the conditional variance
Summary • If underlying field normally distributed can use kriging variance to contsruct confidence intervals around the optimal kriging estimate: • 67% confident data value falls within +/- 1 krigings. • 95% confident data value falls within +/- 2 krigings. • 99% confident data value falls within +/- 3 krigings. • If underlying field normally distributed can generate an ensemble of random realizations that are conditioned on (i.e. kriged with) the observations using for use in a conditional Monte Carlo simulation (Delhomme, 1979)