1. Properties of Covariance and Variogram Functions CWR 6536 Stochastic Subsurface Hydrology

2. The Covariance Function The covariance function must be positive definite which requires that: positive definiteness guarantees that all linear combinations of the random variable will have non-negative variances. This implies:

3. The Variogram Function The negative semivariogram function must be conditionally positive definite which requires that: conditional positive definiteness guarantees that all linear combinations of the random variable will have non-negative variances. This implies:

4. Positive-definiteness is related to the number of dimensions in space over which the function is defined. Positive definiteness in higher order dimensional space guarantees positive definiteness in lower order dimensional space, but not vice-versa Must fit functions to sample covariances/ variograms which are positive definite in the appropriate dimensional space

5. Behavior of Covariance/Variogram functions near the origin Parabolic behavior Linear behavior

6. Behavior of Covariance/Variogram functions near the origin The nugget effect Pure nugget effect

7. Behavior of Covariance/Variogram functions near the infinity The presence of a sill on the variogram indicates second- order stationarity, i.e. the variance and covariance exist If the variogram increases more slowly than h 2 at infinity, this indicates the process may be intrinsically stationary If the variogram increases faster than h 2 this suggests the presence of higher order non-stationarity

8. The hole effect A variogram (covariance) exhibits the hole effect if its growth (decay) is non-monotonic The hole effect is often the result of some ordered periodicity in the data. If possible take care of this deterministically

9. Example of the hole effect

10. Nested Structures Nested structures are the result of observation of different scales of variability, i.e. - measurement error - pore-to-core scale variability - core-to-lens scale variability - lens-to-aquifer scale variability Variogram of total random field is represented by the sum of variograms at each scale

11. The Cross-Covariance & Cross- Variogram Functions In general the cross covariance can be an odd function, i.e. The cross variogram is always a symmetric even function because it incorporates only the even terms of the cross-covariance function

12. The Cross-Covariance & Cross- Variogram Functions In practice the asymmetry of the cross-covariance function is often neglected because: –Geostatistical applications generally use the direct and cross-variogram which are symmetric –Lack of data typically prevents asserting the physical reality of the asymmetry –Fitting valid models to asymmetric cross-covariances is difficult However in stochastic modeling asymmetric cross-covariances often arise.

13. Cross-covariance and Cross- variogram models Use of N multivariate random fields requires modeling N*(N+1)/2 direct and cross covariance (or variogram) models if asymmetry is ignored These models cannot be fit independently from one another because entire covariance matrix must be positive definite (positive semi-definite for variograms)

14. Cross-covariance and Cross- variogram models Ensuring that the cross-covariance (variogram) matrices for multivariate random fields are positive (semi) definite can be tedious when fitting models to data. Goovaerts (p. 108-123) outlines one technique (linear co-realization) for doing so Stochastic modeling techniques ensure that the resulting matrices are positive definite

15. Rules for Linear Model of Co- regionalization Every structure appearing in the cross semi-variogram must be present in all auto- semivariograms If a structure is absent on an auto-semivariogram it must be absent on all cross semivariograms involving this variable Each auto- or cross-semi variogram need not include all structures Structures appearing in all auto-semivariograms need not be present in all cross semivariograms There are constraints on the coefficients of the structures to ensure overall positive definiteness