Variogram ellipsoid
The variogram ellipsoid is used to define any parameter which is not isotropic, the most The variogram ellipsoid is used to define any parameter which is not isotropic, the most common example being the range of the spherical model.
common example being the range of the spherical model.
The variogram ellipsoid is defined using fields
The variogram ellipsoid is defined using fields VANGLE1VANGLE1,, VAXIS1VAXIS1, etc in an identical manner, etc in an identical manner to the sear
to the search ellipsoid, described ch ellipsoid, described in the Searcin the Search Volume section. h Volume section. An example usAn example using theseing these rotation fields is given further on in this section.
rotation fields is given further on in this section.
As you will se
As you will see, the default values e, the default values specify no rotation. specify no rotation. Therefore if you Therefore if you want to define thewant to define the variogram ellipsoid to have the same orientation as the search ellipsoid you must make sure variogram ellipsoid to have the same orientation as the search ellipsoid you must make sure the fields
the fields SANGLE1SANGLE1,, SAXIS1SAXIS1, etc in the, etc in the Search Volume Parameter Search Volume Parameter file are the same as fieldsfile are the same as fields VANGLE1
VANGLE1,,VAXIS1VAXIS1, etc in the, etc in the Variogram Model Parameter Variogram Model Parameter file.file.
Variogram
Variogram
model
model typestypes Field STs defines the mField STs defines the model type for structure odel type for structure s. s. The options for The options for STs are:STs are:
1
Examples of the 5 model types are shown in the diagrams that follow, which also include a Examples of the 5 model types are shown in the diagrams that follow, which also include a two-structure spherical model.
two-structure spherical model.
1:
1: Spherical Spherical model model 2: 2: Power Power modelmodel
3:
3: Exponential Exponential model model 4: 4: Gaussian Gaussian modelmodel
5:
5: De De Wijsian Wijsian model model 6: 6: Two-structure Two-structure sphericalspherical
Spherical model - type 1 Spherical model - type 1
The spherical model is defined by the range a, and the spatial variance C:
The spherical model is defined by the range a, and the spatial variance C:
ã
ãii(h) (h) = = C C [1.5 [1.5 x x h h / / a a - - 0.5 0.5 x x (h (h / / a)a)33 ] ] if if hhaa
=
= C C if if h>ah>a The five fields you need to specify are:
The five fields you need to specify are:
STsSTs =1 for a spherical model=1 for a spherical model
STsPAR1STsPAR1 range in direction 1 (X axis after rotation)range in direction 1 (X axis after rotation)
STsPAR2STsPAR2 range in direction 2 (Y axis after rotation)range in direction 2 (Y axis after rotation)
STsPAR3STsPAR3 range in direction 3 (Z axis after rotation)range in direction 3 (Z axis after rotation)
Power model - type 2
The five fields you need to specify are:
The five fields you need to specify are:
STsSTs =2 for a power model=2 for a power model
STsPAR1STsPAR1 power in direction 1 (X axis after rotation)power in direction 1 (X axis after rotation)
STsPAR2STsPAR2 power in direction 2 (Y axis after rotation)power in direction 2 (Y axis after rotation)
STsPAR3STsPAR3 power in direction 3 (Z axis after rotation)power in direction 3 (Z axis after rotation)
STsPAR4STsPAR4 slope Cslope C
Exponential model - type 3 Exponential model - type 3
The exponential model is defined by parameter a and spatial variance C:
The exponential model is defined by parameter a and spatial variance C:
ã
ãii(h) (h) = = C C [1 [1 - - exp(-h exp(-h / / a)]a)]
The five fields you need to specify are:
The five fields you need to specify are:
STsSTs =3 for an exponential model=3 for an exponential model
STsPAR1STsPAR1 parameter a in direction 1 (X axis after rotation)parameter a in direction 1 (X axis after rotation)
STsPAR2STsPAR2 parameter a in direction 2 (Y axis after rotation)parameter a in direction 2 (Y axis after rotation)
STsPAR3STsPAR3 parameter a in direction 3 (Z axis after rotation)parameter a in direction 3 (Z axis after rotation)
STsPAR4STsPAR4 spatial variance Cspatial variance C Gaussian model - type 4
Gaussian model - type 4
The gaussian model is defined by parameter a and spatial variance C:
The gaussian model is defined by parameter a and spatial variance C:
ã
ãii(h) (h) = = C C [1 [1 - - exp(-hexp(-h22 / a/ a22)])]
The five fields you need to specify are:
The five fields you need to specify are:
STsSTs =4 for a gaussian model=4 for a gaussian model
STsPAR1STsPAR1 parameter a in direction 1 (X axis after rotation)parameter a in direction 1 (X axis after rotation)
STsPAR2STsPAR2 parameter a in direction 2 (Y axis after rotation)parameter a in direction 2 (Y axis after rotation)
STsPAR3STsPAR3 parameter a in direction 3 (Z axis after rotation)parameter a in direction 3 (Z axis after rotation)
STsPAR4STsPAR4 spatial variance Cspatial variance C
De Wijsian model - type 5
The De Wijsian model is defined by parameter C:
ãi(h) = C x loge(h) h>1
= 0 h1
The four fields you need to specify are:
STs =5 for a De Wijsian model
STsPAR1 parameter C in direction 1 (X axis after rotation)
STsPAR2 parameter C in direction 2 (Y axis after rotation)
STsPAR3 parameter C in direction 3 (Z axis after rotation)
Rotation example
This example illustrates the case where you need 3 rotations to describe the anisotropy. The first rotation is a conventional azimuth rotation of 20o around the Z axis, the second rotation is a dip of 40o around the new X axis, and the final rotation is 60o around the new Y axis.
Ranges for structure 1 are 100m, 200m and 300m in the new X, Y and Z directions. Fields are:
E3 VAXIS1 VAXIS2 VAXIS3 ST1PAR 1
The definition of the variogram model in early Datamine grade estimation processes permitted both the nugget variance Co and the spatial variance C to be anisotropic i.e.
different values in different directions. In theory, this allowed zonal anisotropy to be modelled, where the variogram sill is different in different directions. However, this often lead to problems with the stability of the kriging matrix, which meant that many cells were left unestimated.
In order to prevent problems with the kriging matrix the definition of the variogram model in ESTIMA does not allow anisotropic Co or C values for model types 1 to 4. An anisotropic C value for the De Wijsian model (type 5) is acceptable because the model equation does not have any other anisotropic variables.
In ESTIMA zonal anisotropy can be represented by a multi-structure model. The following example shows a two structure spherical model with a sill of 30 for an azimuth of 0o and a sill of 45 for the 90o direction.