at the origin when separation distance is zero is called
natier 1996). According to ASTM standards, variogram
the nugget effect. Measurement error and/or natural
analysis should have at least 20 pairs of data (ASTM D
occurrences can cause this discontinuity at the origin.
1996).
These values are illustrated in Figure 13.
Spherical Variogram Model
N(h)
[
]
∑ zxi - z( xi + h)
1
2
γ (h) =
(2)
h
3
h if h ≤ a, otherwise
2 N(h) i =1
1.5 - 0.5
(3)
a
γ (h) = a
where γ(h)
1.
=
variance
h
=
separation distance
xi
=
position of one sample in the pair
Exponential Variogram Model
xi + h
=
position of other sample in the pair
3h
z
=
thaw depth
γ (h) = 1 - exp - .
(4)
a
N
=
number of pairs of data.
The basic variogram models are divided into two
Gaussian Variogram Model
types; those that reach a plateau, which are called tran-
sition models, and those that do not. Theoretical models
3h2
such as spherical, exponential, Gaussian, and power (or
γ (h) = 1 - exp - 2 .
(5)
a
linear) models are used to fit the thaw depth on the vario-
gram plot (Fig. 14 and 15). The spherical model is the
Power Model
most commonly used transition model (eq 3) and has a
a
γ (h) = h .
(6)
linear behavior at small separation distance near the
origin but flattens out at larger distance and reaches a
Alaska North Slope
sill at a. The tangent at the origin reaches the sill at
The variogram model that best fit the North Slope
about two-thirds of the correlation range. The exponen-
thaw depth data was the spherical model. Spatial cor-
tial model is another commonly used transition model
relations were found on Atquasuk, Betty Pingo, Happy
(eq 4). The exponential model rises more steeply near
Valley, and West Dock data sets. These data grids are
the origin and flattens out more gradually. Like the expo-
situated on the coastal plain except for Happy Valley
nential model, the Gaussian model reaches its sill
(near the foothills). Atquasuk shows the correlation
asymptotically with a parabolic behavior near the ori-
range of approximately 660 m and with sill ranging
gin, and the sill is defined as the "practical range" or
from 230 to 440 depending on the sampling date (Fig.
the distance at which the variogram value is 95% of the
16a). The correlation range for the three sampling dates
sill (eq 5). The power or linear model is not a transition
is constant. Betty Pingo exhibits a correlation range of
model since it does not reach a sill but continuously
up to 450 m and sill varying from 140 to 370 (Fig. 16c).
increases with separation distance h (eq 6). The soft-
The thaw depth correlation range for Betty Pingo data
ware we used for variogram analysis is Variowin (Pan-
140
540
Barrow
480
8/27
120
Atquasuk
8/02
420
7/17
8/30
7/01
100
8/04
360
7/18
80
300
γ (h)
240
γ(h)
60
180
40
120
20
60
0
0
200
400
600
800
1000
0
200
400
600
800
1000
0
h
h
a. Atquasuk.
b. Barrow.
Figure 16. Variogram plots for Alaska North Slope.
13
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