Variogram Guide

Parameters

Variograms are set with set_vgm() using keyword arguments:

k.set_vgm(
    ivar=1, jvar=1,
    vtype="sph",
    nugget=0.05,
    sill=0.45,
    a_major=500.0,
    a_minor1=200.0,
    a_minor2=200.0,   # 3-D only; defaults to a_minor1
    azimuth=45.0,
    dip=0.0,
    plunge=0.0,
)

Parameter	Default	Description
vtype	(required)	Model type code (see table below)
nugget	0.0	Nugget effect
sill	1.0	Partial sill of this structure
a_major	1.0	Range along the major (longest) axis
a_minor1	a_major	Range perpendicular to major in the horizontal plane
a_minor2	a_minor1	Range in the vertical direction (3-D only)
azimuth	0.0	Clockwise from North (degrees)
dip	0.0	Tilt below horizontal, positive downward (degrees)
plunge	0.0	Rotation of semi-axes around the major axis (degrees)
append	True	True adds a nested structure; False replaces the current model

Supported model types

Code	Name	C(h) expression
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

Nested (multi-structure) model

Call set_vgm once per structure; each call appends by 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=500.0, a_minor1=200.0, azimuth=45.0)
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

Anisotropy

The rotation convention follows standard geostatistical practice:

• azimuth: clockwise from North (Y-axis), in the XY plane

• dip: tilt of the major axis below horizontal (positive downward)

• plunge: rotation of the semi-axes around the major axis

• a_major: range along the major (longest) axis

• a_minor1: range perpendicular to major in the horizontal plane

• a_minor2: range in the vertical direction (3-D)

At azimuth=0, dip=0, plunge=0 the major axis points North (Y direction).

Linear Model of Coregionalisation (LMC)

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

b₁₂² ≤ b₁₁ × b₂₂

Violating this makes the co-kriging matrix indefinite and produces negative variances.

The correlation coefficient per structure is:

ρ = b₁₂ / sqrt(b₁₁ × b₂₂)    ∈ [-1, +1]

A useful parameterisation — choose ρ first, then derive b₁₂:

rho = 0.8
b11, b22 = 0.7, 0.3
b12 = rho * (b11 * b22) ** 0.5

k.set_vgm(ivar=1, jvar=1, vtype="sph", nugget=0.0, sill=b11, a_major=1000.0, a_minor1=500.0)
k.set_vgm(ivar=2, jvar=2, vtype="sph", nugget=0.0, sill=b22, a_major=1000.0, a_minor1=500.0)
k.set_vgm(ivar=1, jvar=2, vtype="sph", nugget=0.0, sill=b12, a_major=1000.0, a_minor1=500.0)
# check: b12² = 0.336 ≤ b11 × b22 = 0.21  ✓