14J042 SUPPLEMENTARY MATERIAL
Peak-Identification Uncertainty
To begin, we assigned a probability to each 2 cm segment of each firn core, which
reflects the likelihood that this segment is associated with a mid-summer peak. We refer to the
segments within a given ice core as 𝑆𝑖 where 𝑖 = 1, 2, … , 𝑁𝑠 , and 𝑁𝑠 is the number of 2 cm
segments. For the vast majority of segments, this probability value 𝑝𝑖 is either a 0 (definitely not
associated with an annual mid-summer peak) or a 1 (definitely an annual mid-summer peak).
Other segments are assigned a probability pi which is between 0 and 1. For each segment, we
then draw a realization 𝑋𝑖 from a Bernoulli distribution with probability 𝑝𝑖 , and treat the ice core
segments with 𝑋𝑖 = 1 as one possible collection of mid-summer peak segments. If we believe
that each peak segment was truly associated with the same date (e.g., January 1), then the annual
accumulations for years 2010, 2009,…, 1966 could then be obtained by summing the segmentwise accumulations between peaks.
In cases where there are multiple potential peaks within a small range, a more complex
probability structure must be specified before generating a sequence of possible datings. For
example, in one range of segments in SEAT-10-4 (the noisiest record), we cannot simply assign
Bernoulli probabilities 𝑝𝑖 ∈ [0, 1] to the various potential peaks near depths 1922 cm and 1964
cm because it was determined that while it is possible that peaks reside near one or both of these
two depths, it is not reasonable that beginning-of-year peaks are absent at both depths.
Consequently, the following probabilities were specified:
Peak Locations
Probability
1922 cm and 1964 cm
50%
1922 cm only
25%
1964 cm only
25%
Neither 1922 cm nor 1964 cm
0%
The simulated accumulation records can then be generated based on these probabilities.
Peak-Date Uncertainty
As discussed in the main text, we assigned a 90% probability range of ±δ days to each
potential peak, representing the probability of deviation of the summer maximum from Jan. 1,
assuming a zero-mean normal distribution for the peak-date errors. This method accounts for the
peak-date uncertainty.
Probability-Generated Accumulation Time Series
To create an accumulation time series for a given ice core, these steps are followed:
1. Obtain a collection of mid-summer peak segments as outlined in Supplemental Materials:
peak-identification Uncertainty.
2. For each peak segment date (denoted 𝐷𝑖 ), specify the standard deviation for the peak-date
errors (𝜎𝑖 ) described in Supplemental Materials: Peak-Date Uncertainty.
3. For each peak segment 𝑆𝑖 , draw a realization 𝜖𝑖 from a Normal (0, 𝜎𝑖 ) distribution and let
the new date equal 𝐷𝑖 + 𝜖𝑖 .
4. For each peak segment, find the segment date closest to 𝐷𝑖 + 𝜖𝑖 and denote this new
segment 𝑆𝑖′ . Replace the original peak segment 𝑆𝑖 from the collection of mid-summer
peak segments in Step 1 with the new segment 𝑆𝑖′ .
5. Sum accumulation between mid-summer peaks.
The steps above are used to general 1000 accumulate time series accounting for peak-date and
peak-identification uncertainties.
Sensitivity Analysis
The choice of peak-date and peak-identification uncertainty estimates will impact the resulting
interannual variability and trend analysis. One of the strengths of this model is that it is open to
sensitivity analysis via changing uncertainty input values. This approach allows the user to
hypothesize different optimistic or pessimistic views of the two types of dating errors and then
explore the impacts on trend inferences and interannual variability. Sensitivity analyses were run
for these data analyses including the exploration of the distribution for the uncertainty associated
with the identification of Jan 1 (normal vs. heavy-tailed) and the relative assumed peak-date for
this distribution (conservative vs. realistic vs. optimistic estimates).
Relative to the first assumption, we began by concluding that the error distribution is symmetric
since (for these data) mislabeling “Jan 1” as d days before the true Jan 1 is just as likely as
mislabeling “Jan 1” as d days late. Further, a bell-shaped distribution is logical because we
expect that dating errors will mostly be clustered near zero with increasingly extreme errors
becoming increasingly less likely. Thus, our sensitivity analyses focused on comparing the
normal distribution with heavy-tailed distributions such as the t distribution with 3 degrees of
freedom (or “t3”). However, except for quantiles in the top and bottom 1% of the distribution of
accumulations (which do not affect the estimated central 95% intervals calculated for the plots),
the choice of normal vs. t3 makes little difference. Note that despite the t3 distribution being
more spread, our approach adjusts for this fact. When using the Normal model, we take the
estimated measurement uncertainty for identifying Jan 1 (e.g., δ=30 days with 90% confidence)
and use δ = 1.645 sigma in order to estimate sigma. Possible dating errors are then drawn from a
Normal(0,1) distribution multiplied by sigma (i.e., a draw from a Normal(0,sigma) distribution).
If we assumed a t3 distribution for the dating errors, we would then use δ = 2.35 sigma to
estimate a smaller sigma, but then draw possible dating errors from a t3 (more dispersed than the
normal) multiplied by the smaller sigma. Thus, the simulated dating errors are only subtly
affected by the choice of the symmetric distributional model chosen. Figure S1 shows the
distribution of dating uncertainties (in years) for a normal model and for a t3 model, assuming
that δ =30 days. Thus, distributional form can subtly impact conclusions about trend, but not as
dramatically as when a more/less conservative approach is used in quantifying possible dating
uncertainty, for example δ =30 days versus δ =15 days.
Figure S1. Distribution of dating uncertainties (in years) for a normal model (black line) and for
a t3 model (red line), assuming that δ =30 days
With respect to the choice for the size of δ, this choice can have substantial impact upon
estimated accumulation trends and interannual variability, particularly when uncertainty for an
ice core is dominated by peak-date uncertainty as opposed to year-count uncertainty. However,
shrinking δ will have a relatively predictable impact on the estimated accumulation trend and
interannual variability. Specifically, as δ decreases, the variability of the estimated trend and
interannual variability will also decrease.
Detrending
The yearwise detrending approach begins with the original series of 2 cm ice core
segments which are denoted 𝑆𝑡1 , … , 𝑆𝑡𝑛𝑡 , where the year 𝑡 is in the range 1, . . . , 𝑇, and 𝑛𝑡 is the
number of segments within year 𝑡. Also, let 𝑎𝑡𝑗 , 𝑡 = 1, . . . , 𝑇, 𝑗 = 1, . . . , 𝑛𝑡 , be the accumulation
associated with the segment 𝑆𝑡𝑗 . The segments are assigned to a year using the best assessment
of mid-summer peaks. We denote the mean annual accumulation for a core with 𝜇 and each
∗
year’s accumulation total for year 𝑡 with 𝑦𝑡 . We obtain new segment accumulations 𝑎𝑡𝑗
=
𝑎𝑡𝑗 𝜇/𝑦𝑡 so that when we sum the new segment accumulations within each year we obtain a new
𝑛
∗
𝑡
annual accumulation 𝑦𝑡∗ = ∑𝑗=1
𝑎𝑡𝑗
that is equal to exactly 𝜇. That is, before introducing any
perturbation due to dating uncertainty, the series 𝑦1∗ , … , 𝑦𝑇∗ is constant at 𝜇. This allows us to
investigate the potential for spurious changes from flat accumulation (no variability) due to the
kinds of dating uncertainties.
The linear detrending approach is similar to the yearwise detrending, but instead of
completely flattening the series to a constant, we detrend the original series of annual
accumulations using linear regression. Specifically, we regress 𝑦1 , … , 𝑦𝑇 on time (year) 𝑥1 , … , 𝑥𝑇
to obtain predicted values 𝑦̂1 , … , 𝑦̂𝑇 . We then create the new series of linearly detrended annual
accumulations 𝑦̃𝑡 = 𝑦𝑡 − 𝑦̂𝑡 + 𝜇. Next, we create new segment accumulations using 𝑎̃𝑡𝑗 =
𝑎𝑡𝑗 𝑦̃𝑡 /𝑦𝑡 so that when we sum the segments within each year we obtain new annual
𝑛
𝑡
accumulations 𝑦̃𝑡 = ∑𝑗=1
𝑎̃𝑡𝑗 that vary around the core’s mean 𝜇 in a realistic fashion, but have a
linear slope of exactly 0.