Properties of Covariance and
Variogram Functions
CWR 6536
Stochastic Subsurface Hydrology
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:
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:
• 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
Behavior of Covariance/Variogram
functions near the origin
• Parabolic behavior
• Linear behavior
Behavior of Covariance/Variogram
functions near the origin
• The nugget effect
• Pure nugget effect
Behavior of Covariance/Variogram
functions near the infinity
• The presence of a sill on the variogram indicates secondorder 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
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
Example of the hole effect
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
The Cross-Covariance & CrossVariogram Functions
• In general the cross covariance can be an odd
function, i.e.
Pk ( xi , x j )  Pk ( x j , xi ) but Pk ( xi , x j )  Pk ( x j , xi )
Pk (h)  Pk (h) but Pk (h)  Ph (h)
• The cross variogram is always a symmetric even
function because it incorporates only the even
terms of the cross-covariance function
1
 k (h)  Pk (0)  Pk (h)  Pk (h)
2
The Cross-Covariance & CrossVariogram 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.
Cross-covariance and Crossvariogram 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)
Cross-covariance and Crossvariogram 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
Rules for Linear Model of Coregionalization
• 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