Here is the statistical analysis from the paper:
The method of Roe et al. is summarized as follows. Let 1L be the change in
glacier length over the past 130 years (∼1.9 km), and let σL be the standard
deviation of glacier length due to stochastic fluctuations in mass balance, b, from
natural, interannual climate variability. The signal-to-noise ratio is defined by
sL =1L/σL
. Likewise, sb =1b/σb
. Ref. 12 demonstrates that the two are related via
sL =γ sb
, where γ is an amplification factor that depends only on the duration of
the trend and the glacier response time, τ . The probability density function (PDF)
for sb
is generated by combining the signal-to-noise ratios of the observed
melt-season temperature and annual-mean precipitation trends, normalized by the
summer (bs) and winter (bw ) mass-balance variability (Supplementary Fig. 1a,b,c),
respectively. We take σbw = 0.3 myr−1
and σbs =0.5 myr−1
, based on the observed
mass-balance variability at Gulkana Glacier and the analysis of the global datasets
of glacier mass balance33. The glacier response time is given by τ =−H/bt
, where
H is a characteristic glacier thickness, and bt
is the (negative) net mass balance at
the terminus. We set H=590 m, based on the scaling relationship for glacier
geometry suggested by Haeberli and Hoelzle34 and measured cross-sections35; and
we set bt =−7 myr−1
, estimated by extrapolating the vertical mass-balance profiles
calculated by Flowers et al.8
, thus giving a central estimate for τ of ∼80 years. A
PDF is estimated assuming τ follows a gamma distribution incorporating a broad
uncertainty of στ =τ /4 (Supplementary Fig. 1d). The PDFs for γ and sb are
combined to give a PDF for σL
from the relation σL =γ 1L|
obssb
. This, in turn, is
used to evaluate the null hypothesis that 1L|
obs occurred due to natural variability.
Supplementary Fig. 1e shows our estimate that there is only a 0.5% chance
that the observed retreat of Kaskawulsh Glacier happened in the absence of a
climate trend.