Uncertainty, Emergence, and Statistics in Dendrochronology - Dendroclimatology - page 80

perhaps as determined by COFECHA (Holmes 1983 ) , this is an explicit indication
that some perhaps very complicated common environmental signal exists in those
tree rings. In turn, we can expect the averaging process to concentrate the common
signal in the mean-value function by averaging out the noise. The act of averag-
ing is a powerful way of getting rid of unwanted noise in tree rings, but it is never
perfect; i.e., the noise is never completely eliminated. Therefore, some measure of
empirical signal strength is useful because it tells us how well we have estimated
the underlying common signal and eliminated the unwanted noise.
The pioneering dendroclimatologist Edmund Schulman was aware of signal
strength issues and suggested that the ratio of the mean sensitivity of the average
chronology (MSc) to the average mean sensitivity of the individual series (MSs) in
the chronology be used as a signal strength diagnostic (Schulman 1956 , pp. 20-
24). This ratio he defined as coefficient R . With no noise present in the series,
MSc 1.0. With any noise present between series, MSc < MSs

because the averaging process reduces variance, so R < 1.0. Thus, the lower the
average chronology mean sensitivity is relative to the average of the individual
series, the lower the empirical signal strength. In Schulman's examples for semi-
arid site conifers, R > 0.80 was common. Schulman even described how to use R
for evaluating the signal strength in subsets of tree-ring series from a site to assess
the long-term stability of the chronology signal. This idea presaged the subsample
signal strength (SSS) statistic derived by Wigley et al. ( 1984 ) .
Mean sensitivity has continued to be used as a diagnostic signal strength statistic,
along with tree-ring chronology standard deviation and first-order autocorrelation
(e.g., Fritts and Shatz 1975 ; DeWitt and Ames 1978 ) . These statistics do not provide
explicit estimates of statistical uncertainty in the chronology mean-value function,
however. So, with the growing power and availability of computers in the 1960s,
Harold C. Fritts introduced the use of analysis of variance (ANOVA) to quantita-
tively describe the sources of tree-ring chronology uncertainty (Fritts 1963 ) , and
through his %Y term, its relative signal strength. An excellent example of the inter-
pretive use of ANOVA in tree-ring research can be found in Fritts ( 1969 ) . Later,
the average correlation between series (RBAR) was shown to be effectively equiv-
alent to %Y as a measure of percent variance in common between series (Fritts
1976 , p. 294). These results were in turn extended to include explicit estimates of
signal-to-noise ratio in tree-ring chronologies (Cropper 1982 a ). Shortly thereafter,
Wigley et al. ( 1984 ) explicitly derived the theory underlying the use of RBAR as an
estimate of percent common variance between series, demonstrated its mathemati-
cal equivalence to %Y, and extended those results to the derivation of the expressed
population signal (EPS), which provides an estimate of how closely a mean chronol-
ogy based on a finite number of trees expresses its hypothetically perfect chronology
based on an infinite number of trees. In addition, Wigley et al. ( 1984 ) derived the
subsample signal strength statistic, which quantifies the changing uncertainty in a
tree-ring chronology due to changing sample size. For more details and additional
extensions of these extremely valuable and widely used measures of empirical sig-
nal strength, see Briffa and Jones ( 1990 ) . A complementary method of expressing
tree-ring chronology uncertainty is the development of annual confidence intervals

= MSs, so R =