Download Supplementary Method 1

SUPPLEMENTARY INFORMATION, Moberg et al.
Supplementary Method 1
ESTIMATION OF UNCERTAINTIES IN THE RECONSTRUCTION
There are several reasons for uncertainties in the reconstruction. One is related to
the fact that the individual low-resloution proxy series used in the wavelet
transformation end in various years in the 20th century, so padding with surrogate
data was necessary in parts of the 20th century. Moreover, one low-resolution
record starts in the second century AD, so there is some lack of information also in
the first century. This naturally leads to uncertainties near the end points of the
reconstruction derived. For this reason, the low frequency component in our
reconstruction is shown in all figures only for the period AD 133-1925 (the period
when all series have data). The full reconstruction is shown for the period AD 11979, where the latter date is determined by a significant loss of tree-ring data after
this year. The uncertainties near the end points do not, however, affect the main
conclusion of this study; that there has been a large multicentennial temperature
change between around AD 1000-1100 and AD 1600-1700.
Other reasons for uncertainties relate to the variance among the individual proxy
series (henceforth referred to as uncertainty A), and to the determination of the
variance scaling factor (uncertainty B) and the constant adjustment term
(uncertainty C) that are derived from the 124-year overlapping period with
instrumental data. The methods for estimation of these three separate
uncertainties are described below.
UNCERTAINTY DUE TO VARIANCE AMONG LOW-RESOLUTION PROXIES
(UNCERTAINTY A)
There is a substantial variability among the eleven low-resolution proxies (see Fig.
S1). This is due to factors such as non-temperature influences on the proxies,
dating errors and differences in both regional and seasonal representativity.
Nevertheless, the main character of the low-frequency (>80-year) component of
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SUPPLEMENTARY INFORMATION, Moberg et al.
the reconstruction is quite robust as illustrated by the jack-knifed estimates (see
Figs. 2b and S3), where the reconstruction has been re-computed eleven times –
each time omitting one of the eleven low-resolution proxies. These eleven jackknifed series can be used to calculate a confidence interval for the mean value of
the low-frequency component in each year. Efron1 defines the jack-knife estimate
of the standard deviation, and we use their equation (1.4) to estimate 95%
confidence intervals for the mean value of the uncalibrated reconstruction (i.e. the
non-dimensional reconstruction derived from standardized proxy series) at each
year. An underlying assumption is that the data at each year consists of an
independent and identically distributed sample. In our case, this is not entirely true
because in each jack-knifed estimate of the low-frequency component there is
some identical influcence also from the tree-ring data used in the wavelet
transformation*. This influence, however, can only affect the highest frequencies in
the low-frequency component (i.e. the ‘wiggles’ with period lengths of
approximately 80-90 years), but not the longer timescales that are of the main
interest in this study. We consider the confidence intervals derived as reliable for
drawing inferences on variability at multicentennial timescales, although the
confidence level (95%) is only approximate. Fig. S3 shows the central estimate of
the low-frequency component (red), its jack-knifed estimates (blue) and the
confidence interval (yellow). The confidence interval (Fig. S3, yellow range) has an
average width of 0.23 (standardized units), which is small compared to the total
change of 1.1 (standardized units) between the peaks near AD 1100 and AD 1600
for the central estimate (Fig. S3, red curve). Hence, the difference between these
two time points is highly significant for the data used. This implies, because the
proxy data sites are geographically widely spread, that the proxy data reflect a
significant hemispheric-scale change between the two time points.
*
The separation of information from tree-ring data and low-resolution proxies is not abrupt at
exactly 80 years; there is rather a gradual transition between around 64 and 90 year scales.
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SUPPLEMENTARY INFORMATION, Moberg et al.
UNCERTAINTY IN THE DETERMINATION OF THE VARIANCE SCALING
FACTOR (UNCERTAINTY B)
We calibrated the Northern Hemisphere (NH) reconstruction by scaling its variance
and adjusting its mean value so that these become identical to those in the
instrumental record of NH annual mean temperatures in the overlapping period
1856-1979. Variance-adjusted combined marine and land temperatures2 were
used for the calibration. Because the reconstruction do not include variability at
timescales corresponding to Fourier periods less than four years, we removed
variability at these short timescales from the instrumental record to ensure
consistency with the reconstruction.
Formally, the calibration can be expressed as follows:
Tˆ t   f  Rt   c
,
(1)
where Tˆ t  is the estimated NH annual mean temperature, Rt  is the
dimensionless reconstruction, f is a factor used to scale the variance, c is a
constant that adjusts the mean and t is time. The factor f and constant c are
obtained as
f 
sT
sR
and
c T  f R
,
(2)
where sT and sR are the respective standard deviations of the instrumental NH
temperature data and the dimensionless reconstruction in the overlapping
(calibration) period, while T and R are the corresponding mean values.
Due to the relative shortness of the calibration period (124 years) compared to
the longest timescales of interest, there is an uncertainty in the determination of the
factor f. It is practically impossible to quantify this uncertainty directly from the data
used in the calibration period. Although it would in principle be possible to calculate
a confidence interval for the variance ratio sT2 sR2 by using the F-distribution, it turns
out that the resulting confidence interval becomes very large if one also accounts
for autocorrelation – which is nearly 0.9 for a lag of 1 year in both the
reconstruction and the instrumental data (after removal of <4-year variability). This
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SUPPLEMENTARY INFORMATION, Moberg et al.
strong autocorrelation reduces the efficient number of degrees of freedom to
approximately only 6 to 8, which is substantially smaller than the number of years
(124) in the calibration period. With such small numbers of degrees of freedom, the
F-distribution is very broad and not very useful here.
As a practically more useful way to estimate the uncertainty B, we use instead
the assumption that any long enough (a millennium or so) annually resolved time
series that can be considered as an approximate estimate of the NH mean
temperatures could be used as a surrogate for the instrumental data. If there exist
any such surrogate data, then these could be used to determine how much the
scaling factor f can vary between different 124-year long calibration periods. The
goal here is to determine the uncertainty due to the fact that only one 124-year
calibration period is available in reality, while another period (if it had existed) could
have led to a somewhat different result. Inherent in our philosophy, is the idea that
an appropriate surrogate series for NH temperatures and our reconstruction may
be regarded as two different realizations of NH mean temperatures in a climate
system (having a certain internal variability) responding to past changes in
radiative forcing.
We consider the following five time series as appropriate approximations of NH
mean temperature variability in the last millennium or so:

The multi-proxy NH annual mean temperature reconstruction by Mann et al.3
starting in AD 1000.

The multi-proxy NH summer temperature reconstruction by Jones et al.4
starting in AD 1000 .

The tree-ring based warm-season sensitive temperature reconstruction for
NH extratropical regions by Esper et al.5 starting in AD 831.

The warm season (mainly summer) temperature sensitive northern highlatitude tree-ring chronology average by Briffa6 starting in AD 1000

The NH annual mean temperatures from the transient run with the ECHO-G
model7-8, forced with reconstructed solar and volcanic radiative forcing and
greenhouse gas concentrations, starting in AD 1000. This is the same model
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SUPPLEMENTARY INFORMATION, Moberg et al.
run as shown in Fig. 2c in the main text. Here, we use the model data for the
period AD 1000-1910. The subsequent data were omitted because the
model experiment excluded all other anthropogenic effects than greenhouse
gases; hence the warming trend after around 1910 in the model run may be
overestimated.
It turns out that the resulting estimates of the uncertainty in f are very similar for
all five of the chosen surrogate series. The central estimate, obtained from the
instrumental data 1856-1979, is f = 0.590. An approximate 95% confidence interval
for f, determined using each of the five surrogate series are determined to be:
[ f 2.5
,
f97.5 ]
Mann et al.
[0.482
,
0.726]
Jones et al.
[0.489
,
0.712]
Esper et al.
[0.476
,
0.732]
Briffa et al.
[0.520
,
0.670];
ECHO-G
[0.481
,
0.726]
The strong agreement of the derived f 2.5 - and f97.5 -values illustrate that the
chosen way to determine uncertainty B is very robust as regards the choice of
surrogate data. For our final estimate of the uncertainty in f , we choose the
smallest f 2.5 - and largest f97.5 -value above, which are obtained when the Esper et
al. data are used as surrogates. In the text that follows below, we describe in detail
how the f 2.5 - and f97.5 -values were derived. We choose to use the ECHO-G model
data as an example, but the technique was identical for all five surrogate series.
By using ECHO-G data for the period 1000-1910, a sequence of 788 ‘surrogate
calibration periods’ (most of them are partly overlapping, but some are
independent) are obtained; the first being 1000-1123 A.D. and the last being 17871910. For each of these periods, we calculated the ratio of standard deviations
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SUPPLEMENTARY INFORMATION, Moberg et al.
~ sE
f 
sR
,
(3)
where sE is the standard deviation in the ECHO-G data and sR the standard
deviation of the dimensionless reconstruction as before. The 2.5 and 97.5
~
~
~
percentiles for these f -values are labelled f 2.5 and f97.5 .
The interval
~f
2.5

~
~
, f 97.5 contains 95% of all f -values and is therefore a 95%
~
confidence interval for the total uncertainty in the determination of f . With ‘total
~
uncertainty’, we mean the combined uncertainty in f originating from errors in
~
both our reconstruction and in the ECHO-G data. Because the full range of f -


~ ~
values in f 2.5 , f 97.5 is determined by errors in both series (i.e. not only in our
reconstruction), the ratio between the upper and lower limits of the interval
~ ~
f 2.5 , f 97.5 must be larger than the ratio between the upper and lower bounds of the


desired 95% confidence interval  f 2.5 , f 97.5  for the f -value. In other words, the
~
~
ratio f97.5 / f 2.5 must be larger than f97.5 / f 2.5 . It is not obvious how much smaller the
~
~
ratio f97.5 / f 2.5 should be compared to the ratio f97.5 / f 2.5 , but we find it reasonable
to assume that the surrogate series and our reconstruction contribute with equal
~
amounts to the total range of f -values and, as an approximation of the ratio
f97.5 / f 2.5 , we choose the value
~
~
f 97.5 / f 2.5 .
Because we are dealing with variance ratios, the original estimate of f ( f =
0.590) should be the geometric mean value of the upper and lower bounds of its
own confidence interval. Hence, following the argumentation above, a 95%
confidence interval for f can be defined as the interval  f 2.5 , f97.5  , where f 2.5 and
f97.5 are chosen such that f 97.5 / f 2.5 
~
~
f 97.5 / f 2.5 with the condition that f is the
geometric mean value of f 2.5 and f97.5 . It should be mentioned that this uncertainty
estimation is entirely unaffected by any systematic too large or too small variance
in the surrogate data used.
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To illustrate graphically in Fig. 2d the uncertainties derived so far, the following
technique was used. The central estimate, obtained with the equations (1) and (2)
(resulting in f = 0.509) is plotted with blue colour in Fig. 2d. Then, the range for
uncertainty A is plotted in medium-blue colour using the same f -value (and the
same constant term c) as for the central estimate.
Next, to illustrate also the uncertainty B graphically, the central estimate and the
upper and lower bounds for uncertainty A were recalculated for both the cases of
f 2.5 and f 97.5 . (This required recalculation also of the constant term c, to ensure
that the central estimate has the same mean level as the instrumental data in the
1856-1979 period). Finally, the range between the uppermost and the lowermost
value at each year, for either f 2.5 or f 97.5 , were plotted in Fig. 2d with light-blue
colour (but of course not overlying the medium-blue band for uncertainty A). Thus,
the area containing both the medium-blue and light-blue colour bands combines
the two uncertainties A and B. The fact that the two uncertainties A and B have
simply been added onto each other implies that the total range covered by the two
coloured bands together illustrate a confidence interval at a higher level than 95%.
UNCERTAINTY IN THE DETERMINATION OF THE CONSTANT ADJUSTMENT
TERM (UNCERTAINTY C)
Once the variance scaling factor f has been determined, the constant adjustment
term c is determined from the equation (2). The statistical uncertainty in this term
can be estimated from the residuals T t   Tˆ t  in the calibration period. The mean
of these residuals is zero by defintion, but its variance can be used to estimate the
uncertainty C by using the conventional technique for estimating confidence
intervals for the mean value of a sample drawn from an independent and identically
distributed random variable. However, it is necessary to adjust the number of
degrees of freedom because the residuals are strongly autocorrelated (0.86 for a
lag of 1 year). We use the approximation advised by Quenouille9;
Neff  N(1– r1)/(1+r1), where N is the number of observations, Neff is the effective
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SUPPLEMENTARY INFORMATION, Moberg et al.
number of observations and r1 is the lag-1 autocorrelation. For the data used here,
Neff  9; hence the effective number of degrees of freedom is 8. This leads, for the
central estimate of f , to a 95% confidence interval for c having a half-width of
0.042K. For the lower and upper bounds of f , the corresponding values are
0.039K (for f 2.5 ) and 0.047K (for f97.5 ). Taking the largest of these three values as
a conservative estimate valid for all f -values in the interval  f 2.5 , f97.5  , we plot in
Fig. 2d two dark-blue bands, each with a constant width of 0.047K, which surround
the light-blue bands that represent the uncertainty B. The resulting total uncertainty
interval, covering all three uncertainties A, B and C simply added onto each other,
is regarded as a very conservative confidence interval (at a much higher level than
95%) for the mean value of the low-frequency component in our reconstruction.
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