Properties of Covariance and Variogram Functions

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 h2 at infinity, this indicates the process may be intrinsically stationary • If the variogram increases faster than h2 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