Variogram models#

Parameters#

Variograms are set with set_vgm(). All parameters are keyword arguments:

Parameter	Default	Description
`vtype`	(required)	Model type code (see table below)
`nugget`	`0.0`	Nugget effect (discontinuity at origin)
`sill`	`1.0`	Partial sill — variance contributed by this structure
`a_major`	`1.0`	Range along the major (longest) axis
`a_minor1`	`a_major`	Range along the first minor axis (defaults to isotropic)
`a_minor2`	`a_minor1`	Range along the vertical axis (3-D only)
`azimuth`	`0.0`	Azimuth of major axis, degrees clockwise from North
`dip`	`0.0`	Dip angle, degrees positive downward
`plunge`	`0.0`	Plunge angle, degrees
`append`	`True`	`True` appends a nested structure; `False` replaces the current model

Supported model types#

Code	Name	Covariance C(h)
`sph`	Spherical	`1 - 1.5(h/a) + 0.5(h/a)³` for h < a, else 0
`exp`	Exponential	`exp(-3h/a)`
`gau`	Gaussian	`exp(-3(h/a)²)`
`pow`	Power	`h^α` (α < 2)
`lin`	Linear	`1 - h/a`
`hol`	Hole effect	`sin(πh/a) / (πh/a)`
`bsq`	Bessel-squared spherical	—
`cir`	Circular	—
`nug`	Pure nugget	1 at h = 0, else 0

Single-structure model#

k.set_vgm(ivar=1, jvar=1,
          vtype="sph", nugget=0.05, sill=0.95, a_major=500.0)

Nested (multi-structure) model#

Call set_vgm multiple times. Each call appends one structure (append=True is the default):

k.set_vgm(ivar=1, jvar=1, vtype="nug", nugget=0.05, sill=0.0,  a_major=1.0)
k.set_vgm(ivar=1, jvar=1, vtype="sph", nugget=0.0,  sill=0.45, a_major=300.0)
k.set_vgm(ivar=1, jvar=1, vtype="exp", nugget=0.0,  sill=0.50, a_major=800.0)
# total sill = 0.05 + 0.45 + 0.50 = 1.0

Anisotropic model#

Specify different ranges along each axis and rotate with azimuth / dip:

k.set_vgm(ivar=1, jvar=1,
          vtype="sph",
          nugget=0.0, sill=1.0,
          a_major=1000.0, a_minor1=400.0,  # 2-D anisotropy
          azimuth=45.0)                     # major axis points NE

For 3-D add a_minor2 (vertical range) and dip / plunge.

Replacing a variogram on a reused object#

When reusing a Kriging object with a different variogram, pass append=False on the first set_vgm call to clear the previous model:

# first run
k.set_obs(...)
k.set_vgm(ivar=1, jvar=1, vtype="sph", sill=1.0, a_major=500.0)
k.set_grid(...)
k.set_search(ivar=1)
k.solve()

# second run — different variogram
k.set_vgm(ivar=1, jvar=1, vtype="exp", sill=1.0, a_major=800.0, append=False)
k.solve()

Without append=False the second run would accumulate structures from the first run, silently doubling (or tripling) the total sill.

Linear Model of Coregionalisation (LMC)#

For co-kriging with variables 1 and 2, every nested structure k must satisfy the LMC constraint:

$$b_{12,k}^2 \leq b_{11,k} \times b_{22,k}$$

where b denotes the partial sill for each variable pair. Violating this makes the co-kriging matrix indefinite and will produce negative variances.

The correlation coefficient per structure is:

$$\rho_k = \frac{b_{12,k}}{\sqrt{b_{11,k} \times b_{22,k}}}$$

A safe starting point is b12 = 0.8 * sqrt(b11 * b22) (ρ = 0.8).

Example LMC:

k.set_vgm(ivar=1, jvar=1, vtype="sph", sill=1.00, a_major=500)  # b11
k.set_vgm(ivar=1, jvar=2, vtype="sph", sill=0.80, a_major=500)  # b12 — ρ = 0.8
k.set_vgm(ivar=2, jvar=2, vtype="sph", sill=1.00, a_major=500)  # b22
# check: 0.80² = 0.64 ≤ 1.00 × 1.00 = 1.00  ✓