4.3. Variogram Models and Their Fitting
32
Let Z(x) be an intrinsic random function with semi-variogram γ(h). The two
main characteristics of a stationary variogram are:
(i) its behavior at the origin, the three types of which are shown in Figure 4.2
(parabolic, linear and nugget effect);
(ii) the presence or absence of a sill in the increase of γ(h), i.e., γ(h) = constant
when |h| > a.
Thus, the currently used theoretical models can be classified as:
Models with a sill (or transition models) and a linear behavior at the origin
(a) spherical model
(b) exponential model
and a parabolic behavior at the origin
(c) Gaussian model
Models without a sill (the corresponding random function is then only intrinsic
and has neither covariance nor finite a priori variance)
(a) models in |h|θ , with θ ∈ (0, 2)
(b) logarithmic model
Nugget effect: It has been seen that an apparent discontinuity at the origin of
the semi-variogram, i.e., a nugget constant Co , can be interpreted as a transition
structure reaching its sill value Co at a very small range compared with the available
distances of observation.
Remark: For the moment, only isotropic models will be considered.
4.3.1
Models with a Sill, or Transition Models
The sill value of a transition structure is the a priori variance of the random function
Z(x) which is second-order stationary and has a covariance C(h) = γ(∞) − γ(h).
The models presented below have been normed to 1, i.e., they correspond to
random functions with unit a priori variance:
V ar{Z(x)} = C(0) = γ(∞) = 1 .
To obtain a model with a sill C(0) = C 6= 1, it is enough to multiply the given
expressions for γ(h) or C(h) by the constant value C.
4.3. Variogram Models and Their Fitting
33
Linear Behavior at the Origin
This is the most frequent type of behavior encountered in mining practice (variogram of grades and accumulations) and it is most often accompanied by a nugget
effect.
(a) Spherical model:
γ(h) =
(
C[ 32 ha − 12 ( ha )3 ] for h ≤ a
C
for h > a,
(b) Exponential model:
γ(h) = C[1 − e−h/a ]
Note that the spherical model effectively reaches its sill for a finite distance
r = a = range, while the exponential model reaches its sill only asymptotically, cf.
Figure 4.10. However, because of experimental fluctuations of the variogram, no
distinction will be made in practice between an effective and an asymptotic sill.
For the exponential model, the practical range a′ can be used with a′ = 3a, for
which γ(a′ ) = C(1 − e−3 ) = .95C
Figure 4.10: Variogram Models.
The difference between the spherical and exponential models is the distance
(abscissa) at which their tangents at the origin intersect the sill, cf. Figure 4.10:
r = 2a/3, two-third of the range for the spherical model;
r = a = a′ /3, one third of the practical range for the exponential model.
Thus, the spherical model reaches its sill faster than the exponential model.
34
4.3. Variogram Models and Their Fitting
Parabolic Behavior at the Origin
This very regular behavior at the origin is seldom found in mining practice. In
the absence of nugget effect, it corresponds to a random function Z(x), which
is very continuous in its spatial variability. Thickness variograms of continuous
sedimentary beds sometimes display such a parabolic behavior at the origin, but
are usually accompanied by a slight nugget effect due to errors of measurement.
(c) Gaussian model:
γ(h) = C[1 − e−h
2 /a2
] .
The √
sill is reached asymptotically and a practical range can be considered with
a′ = 3a, for which γ(a′ ) = C(1 − e−3 ) = .95C, cf. Figure 4.10.
4.3.2
Models without a Sill
These models correspond to random functions Z(x) with an unlimited capacity
for spatial dispersion; neither their a priori variances nor their covariance can be
defined. The random functions Z(x) are only intrinsic.
(a) Models in rθ :
γ(r) = rθ , with θ ∈ (0, 2) ,
the limits 0 and 2 being excluded.
These models have a particular theoretical and pedagogical importance (they show
a whole range of behaviors at the origin when the parameter θ varies and they are
easy to integrate).
In practice, only the linear model is currently used:
γ(r) = ωr ,
with ω the slope at the origin.
Remark 1 For small distances (r → 0), the linear model can be fitted to
any model that has a linear behavior at the origin (e.g., spherical and exponential
models).
Remark 2 As θ increases, the behavior of γ(r) = rθ at the origin becomes
more regular and corresponds to a random function Z(x) of more and more regular
spatial variability.
Experimentally, models in rθ for θ ∈ (1, 2) are often indistinguishable from a
parabolic drift effect. The choice of interpretation, as a drift (non-stationarity) or
a stationary model in rθ with θ close to 2, depends on whether or not it is desired
to make a drift function m(x) = E{Z(x)} obvious.
35
4.4. Fitting of a Spherical Variogram Model
4.4
Fitting of a Spherical Variogram Model
In this section we present suggestions for exercises, no ready solutions for fitting
problems.
(i) Samples (v) of an ore body (V ) of size 750 m in direction EW and 330 m
in N S on a grid of 6m are available. Because the thickness of the deposit
varies, the accumulation of the analyses in [%.m] (grade times thickness) is
considered. For this data the variogram γ̂(h) has been calculated (see Table
4.3).
Table 4.3: Variogram Data: Accumulation.
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
h γ̂(h)EW γ̂(h)N S
6m
0.14
0.17
12
0.20
0.26
18
0.23
0.32
24
0.29
0.42
30
0.33
0.42
36
0.40
0.58
42
0.47
0.68
48
0.48
0.54
54
0.53
0.56
60
0.60
0.45
66
0.65
0.60
72
0.44
0.72
78
0.64
0.48
84
0.54
0.40
90
0.67
0.52
96
0.60
0.57
102
0.72
0.44
108
0.44
0.48
114
0.66
0.58
120
0.45
0.64
The statistical variance of the samples (v) of the ore body (V ) is σ 2 (v/V ) =
0.55[m.%]2 .
– The points γ(h) depending on h are presented in the next diagram Figure
4.11. Obviously, we have a geometrical anisotropy.
– Parallel to the h-axis we draw a straight line at 0.55 in respect to the
y-coordinate. This represents the empirical variance measured from the volumes v in the ore body V , namely
σ 2 (v/V ) ∼
= γ(∞) = 0.55 = C
.
4.4. Fitting of a Spherical Variogram Model
36
Figure 4.11: Variogram Fitting on Accumulation Data.
– For each of the two variograms a tangent through the first points towards
the origin (h = 0) is drawn. It is obvious that γ(h) for h → 0 is not equal
0. There is a nugget variance Co which has to be determined. Because of the
hypothesis of geometrical anisotropy there can be only one nugget variance.
We then have C1 = C − Co = .55 − .09 = .46.
– From the crossing points of the tangents with the horizontal line γ(∞) =
0.55 we find in the Matheron-model 2/3 of the ranges. We can read:
2/3a1 = 55 for the EW -direction, which means: a1 = 82.5 = a.
2/3a2 = 37 for the N S-direction, which means: a2 = 55.5.
– With the relation q = a1 /a2 = 1.49 the N S-variogram may be geometrically transformed to the EW -variogram, such that only an isotropical EW variogram must be computed.
4.4. Fitting of a Spherical Variogram Model
37
– The Matheron-model then is:
γ(h) =
(
− 12 ( |h|
)3 ] for 0 ≤ |h| ≤ a
Co + C1 [ 23 |h|
a
a
Co + C1
for |h| > a .
(ii) Values of an empirical variogram of a deposit of porphyry-molybden are
given, the variance of the samples is .81, see Table 4.4.
Table 4.4: Variogram Data of a
h
200’
282’
400’
488’
564’
600’
800’
1000’
1200’
1400’
1600’
Deposit of Porphyry-molybden.
γ̂(h)
0.43
0.57
0.63
0.75
0.85
0.85
0.87
0.88
0.87
0.85
0.80
It is to determine: Co , C, a and the theoretical variogram model. The fitting
is left to the reader.
(iii) Let be given some data of an ore deposit of nickel, see Table 4.5.
It should be determined if we have the case of a simple or a nested spherical
model.
The data should be graphically presented, the sills and the ranges estimated
and the variogram model computed and compared with the empirical values.
– The given variogram data are presented in the next Figure 4.12. Because
of the obvious two bends we decide for a twice nested variogram model
3 h
1 h
3 h
1 h
γ(h) = Co +γ1 (h)+γ2 (h) = Co +C1 [ ( )− ( )3 ]+C2 [ ( )− ( )3 ], h < a1 < a2 .
2 a1 2 a1
2 a 2 2 a2
- am- A straight line through the first two points of the empirical variogram
which crosses the ordinate at 0.4 lets us decide for Co = 0.4.
- bm- A horizonal line through the highest, more or less stable part of the
variogram yields to the total sill
C = Co + C1 + C2 = 2.55 .
38
4.4. Fitting of a Spherical Variogram Model
Table 4.5: Variogram Data of a Nickel Deposit.
Distance
between the
Samples
2
4
6
8
10
12
14
16
18
20
22
24
26
28
30
32
34
36
38
40
42
44
46
48
50
52
54
56
58
60
62
64
66
68
70
Empirical
Variogram
0.74
1.10
1.34
1.58
1.72
1.81
1.87
1.90
1.93
1.92
1.95
2.01
2.09
2.16
2.25
2.29
2.38
2.35
2.36
2.39
2.48
2.52
2.56
2.55
2.49
2.59
2.61
2.64
2.68
2.62
2.52
2.59
2.53
2.47
2.56
Number
of
Pairs
1222
1194
1186
1152
1137
1120
1095
1077
1055
1026
1011
990
969
950
919
899
886
860
848
825
814
787
779
767
750
736
722
705
689
675
657
639
628
612
597
4.4. Fitting of a Spherical Variogram Model
39
Figure 4.12: Fitting of a Twice Nested Variogram Model.
The range of γ2 is between 46 und 54 m and we choose a2 = 50m, i.e.
2/3a2 = 33.3m, which fixes the point bm
.
- cm- The approximate tangent of γ2 , moved parallel to bm
, shows the crossing
point with the ordinate at
Co + C1 = 1.4 .
It follows that C2 = C −Co −C1 = 2.55−1.4 = 1.15 and C1 = 1.4−0.4 = 1.0.
- dm- The part of γ2 on the total variogram subtracted from the tangent am
,
gives the crossing point of the line Co + C1 the value 23 a1 = 7m and therefore
a1 = 10.5m. With the so found parameters the first trial of the fitting is
finished:
Co = 0.4, C1 = 1.0, C2 = 1.15, a1 = 10.5m, a2 = 50 m.
Now one should compute the fitted variogram, draw it in the diagram and
modify the parameters if necessary.