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shows that additional changes in thickness were mainly controlled by a second-
order stochastic process with phase differences between silt and clay. Finally, there
are weak cyclicities for the 22-year and 11-year sunspot cycles involving both silt
and clay and a 15-year cyclicity mainly restricted to the silt. The exponential
thickness decrease in most varve time series suggests a rapid linear retreat of the
land-ice. Superimposed on this trend there occurred two relatively rapid increases
in thickness. One of these falls at the “datum” in Fig.
6.4
coinciding with the
so-called Cochrane readvance; the other one occurred near the end of series 4.
On average, the stochastic model indicated a 126-year cyclicity superimposed on
the linear retreat. Thus, the rate of land ice retreat was accelerated and decelerated
periodically. It is possible that at the end of a deceleration, the ice-sheet not only
came to a stand-still but readvanced relatively rapidly during a short period of time.
This phenomenon is known as “surging”.
Beginning with the theoretical models of Weertman (
1969
) and Lliboutry
(
1969
), there has been much progress in the deterministic modeling of land-ice
retreat. A comprehensive review of these theoretical developments can be found in
Fowler (
2011
). Surges of glaciers and land-ice are modeled by Fowler (
2011
). They
can indeed be periodic. There may even have existed mega-surges (Heinrich events
including the Hudson Strait mega-surge) with a periodicity of about 7,000 years.
6.2 Spatial Series Analysis
With respect to space series, two geostatistical topics of practical interest are
existence of “sill” and “nugget effect” (see e.g. Journel and Huijbregts
1978
; Isaaks
and Srivastava
1989
; Cressie
1991
; or Goovaerts
1997
). Suppose
(
h
) represents the
semivariogram, which is half the variance of the difference between values sepa-
rated by lag distance
h
. Semivariogram values normally increase when
h
is
increased until a sill value is reached for large distances. If element concentration
values are subject to second-order stationarity,
ʳ
(
h
) ¼
˃
ʳ
2
repre-
sents variance and
ˁ
h
is the autocorrelation function. The sill is reached when there
is no spatial autocorrelation or
2
(1
ˁ
h
) where
˃
2
. If regional trends can be separately fitted
to, for example, element concentration values, the residuals from the resulting
regional, systematic variation may become second-order stationary because the
overall mean in the study area then is artificially set equal to zero. Within most rock
types such as granite or sandstone, randomness of chemical concentration is largely
restricted to microscopic scale and sills for compositional data are reached over
very short distances. The nugget effect occurs when extrapolation of
ʳ
(
h
)
¼
˃
ʳ
(
h
) towards
the origin (
h
!
0) from observed element concentration values yields estimates
with
1). A pseudo-nugget effect arises when there is strong local
autocorrelation that cannot be detected because locations of samples subjected to
chemical analysis are too far apart to describe it adequately.
If a segment of the Earth's crust is sampled and element concentration values are
determined on the resulting rock samples, the spatial variability of the chemical
ʳ
(
h
)
>
0 (or
ˁ
h
<
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