A number of data sets were studied. We list the data sets analyzed. Most of these are available through the American National Ocean and Atmosphere Administration’s web site.
http://www.ncdc.noaa.gov/paleo
A number of these data sets were discarded from the study sometimes after a significant amount of analysis had been undertaken. These are listed first with the reasons they were discarded.
10.2.1 Discarded Data Sets
Baffin Island: A 1241 year summer temperature reconstruction between 752 and 1992 AD based on laminae thickness of sediments from Donard Lake, Baffin Island, Canada by Moore et al. (2001). The data set was problematic in that it had high skewness and was clearly non-normal data. This could have affected the reliability of the Whittle estimator.
Burgundy: A 633 year temperature reconstruction based on grape harvest records in the Burgundy region of France by Chuine et al. (2004). It covered a period from 1370 to 2003 AD. This data set showed no evidence of long memory. The ACF decayed quickly, the AIC selected an AR(4) model as adequate and the periodogram was almost flat with only a small rise in power at low frequencies.
Moberg: A 1980 year Northern Hemisphere temperature reconstruction by Moberg
et al. (2005) based on composite low and high resolution proxies. It covered the period 1 to 1980 AD. The p-value reported by the Beran (1992) test was zero. It was previously analyzed by Mills (2007) who considered an ARFIMA(2,0.45,2) model. The Beran test also reported a p-value of zero for this ARFIMA model. This data was discarded as we could not find a model where the Beran test reported a p-value above zero. Also, the periodogram was atypical of long memory time series. Mills (2007) only considered the long memory properties of this series in the time domain. His paper is a clear example of why long memory time series must be considered in both the time and frequency domains.
West Greenland: A 1230 year reconstruction of temperatures in West Greenland
from ice core data by Fisher et al. (1996) which covered the period 753 to 1982 AD. This was not examined in detail as the periodogram was atypical of long memory series.
Yamal: A 4,000 year summer temperature reconstruction from larch tree data by Hantemirov and Shiyatov (2002) which covered the period 2067 BC to 1996 AD
from the Yamal Peninsula, Siberia. This was discarded as the authors stated that temperature variability at time scales longer than a few centuries was removed by their reconstruction techniques. This was confirmed by spectral analysis which showed exceptionally low power in the low frequencies.
10.2.2 Data Sets Analyzed
Campito Mountain: An unbroken set of 5405 annual tree ring widths from bristle- cone pines (speciespinus longeava) on Campito Mountain, California. The trees were at an altitude of over 11,000 feet. The data covered the period 3435 BC to 1969 AD. The rings were measured to the nearest 0.01 mm.
The Campito Mountain data set is regarded as one of the standard examples of the concept of a long memory process. The data set was produced by the Tree Ring Laboratory at the University of Arizona. It is available as the data setcamp in thetseriespackage withinR.
Colorado: A 2247 year temperature reconstruction based on tree ring data by Salzer and Kipfmueller (2005) which covered the period 250 BC to 1996 AD for the Colorado Plateau region. A reconstruction of precipitation was available in the same data file but was not considered.
Elk Lake: The Elk Lake varve data was prepared by Dean (1994) of the United States Geological Survey. It was a series of 10,224 annual varves from Elk Lake in north-eastern Minnesota within the upper Mississippi River Basin. Varves are, essentially, layers of debris which collect in the bottoms of lakes and are part of a larger class of data known as cyclostratigraphic records, see Weedon (2003). The Elk Lake data set was unusually long as the lake was formed after the retreat of glaciers at the end of the last ice age.
In the data there were a number of places where the varve thicknesses were constant. Dean (2006) stated that at the base of the varved section there were several short disturbed zones where the varve couplets could not be measured
but the number could be estimated. Each of the estimated varves was then assigned a constant thickness based on the thickness of the disturbed zone. The accumulation rate was not strictly constant but was close to being constant.
Northern Hemisphere: A 1283 year Northern Hemisphere temperature reconstruc-
tion which covered the period from 713 to 1995 AD by D’Arrigo et al. (2006). The authors offered two reconstructions, one based on standard methodologies and the other based on “Regional Curve Standardization” (RCS). The authors stated that they believed the RCS better captured the low frequency tempera- ture fluctuations. This belief was supported by a higherH estimate in the RCS reconstruction than the reconstruction which used standard methodologies.
Scotland: A data set containing three stalagmite thickness series from a cave in North- west Scotland by Proctor et al. (2002). The thickness data extends to 3600 years before the present.
Shihua: Several data sets were available from the Shihua Cave near Beijing, China. We examined the 2650 year record of stalagmite thicknesses and the associated warm season temperature which was reconstructed from the same stalagmite by Tan et al. (2001). Both data sets cover the period 665 BC to 1985 AD. This data was studied by Rea et al. (2007a,b).
TheH estimates and Beran (1992) p-values are presented in Table (B.3).
Tasmania: A 3592 year reconstruction of warm season temperatures since 1600 BC by Cook et al. (2000) based on Tasmanian Huon pine tree rings.
Tornetr¨ask: A 1993 year reconstruction of temperatures at Tornetr¨ask, Sweden from tree ring data by Briffa et al. (1992) which covered the period from 1 to 1993 AD. The data file appears to have been updated since the publication of Briffa et al. (1992).
Urals: A 1077 year summer season temperature reconstruction based on Siberian Larch tree ring data by Briffa et al. (1995) which covered the period from 914 to
1990 AD.
Western USA: A 1780 year reconstruction of temperatures based on tree ring data in the Western USA by Mann et al. (1998) which covered the period 200 to 1980 AD. Note: The reference to Mann et al. (1998) for this data set was given in Jones and Mann (2004) but this data set seems too long for the reconstruction discussed by Mann et al. (1998).
We present a detailed analysis of the Campito Mountain data in this chapter as the data are regarded as standard example of long memory time series and are of exceptionally high quality. This data set was often attributed in the literature to the late Valmore LaMarche. The earliest paper of his we could find which discusses the Campito data is LaMarche (1974).
Unless otherwise indicated allH estimates were obtained with the Whittle estima- tor.
A number of other results can be found in Appendix (B).