Ongoing work in paleoclimate reconstruction prioritizes understanding the origins and magnitudes of errors that arise when comparing models and data. One class of such errors arises from assumptions of proxy temporal representativeness (TR), i.e., how accurately proxy measurements represent climate variables at particular times and time intervals. Here we consider effects arising when (1) the time interval over which the data average and the climate interval of interest have different durations, (2) those intervals are offset from one another in time (including when those offsets are unknown due to chronological uncertainty), and (3) the paleoclimate archive has been smoothed in time prior to sampling. Because all proxy measurements are time averages of one sort or another and it is challenging to tailor proxy measurements to precise time intervals, such errors are expected to be common in model–data and data–data comparisons, but how large and prevalent they are is unclear. This work provides a 1st-order quantification of temporal representativity errors and studies the interacting effects of sampling procedures, archive smoothing, chronological offsets and errors (e.g., arising from radiocarbon dating), and the spectral character of the climate process being sampled.

Experiments with paleoclimate observations and synthetic time series reveal that TR errors can be large relative to paleoclimate signals of interest, particularly when the time duration sampled by observations is very large or small relative to the target time duration. Archive smoothing can reduce sampling errors by acting as an anti-aliasing filter but destroys high-frequency climate information. The contribution from stochastic chronological errors is qualitatively similar to that when an observation has a fixed time offset from the target. An extension of the approach to paleoclimate time series, which are sequences of time-average values, shows that measurement intervals shorter than the spacing between samples lead to errors, absent compensating effects from archive smoothing. Nonstationarity in time series, sampling procedures, and archive smoothing can lead to changes in TR errors in time. Including these sources of uncertainty will improve accuracy in model–data comparisons and data comparisons and syntheses. Moreover, because sampling procedures emerge as important parameters in uncertainty quantification, reporting salient information about how records are processed and assessments of archive smoothing and chronological uncertainties alongside published data is important to be able to use records to their maximum potential in paleoclimate reconstruction and data assimilation.

Paleoclimate records provide important information about the variability,
extremes, and sensitivity of Earth's climate to greenhouse gases on
timescales longer than the instrumental period. As the number of
published paleoclimate records has grown and the sophistication of
numerical model representations of past climates has improved, it
has become increasingly important to understand the uncertainty with
which paleoclimate observations represent climate variables so that
they can be compared to one another and to model output. Additionally,
quantifying uncertainty is important for ongoing efforts to assimilate
paleoclimate data with numerical climate models

Paleoclimate records can have errors arising from many different sources:
biological effects

Much of the previous study of errors arising from sampling in time
has focused on aliasing, whereby variability at one frequency in a
climate process appears at a different frequency in discrete samples
of that process.

A second area of previous focus stems from chronological uncertainties,
whereby times assigned to measurements may be biased or uncertain.
In some cases, such as for radiocarbon dating, estimates of these
uncertainties are available from Bayesian approaches that incorporate sampling procedures

This paper synthesizes effects contributing to TR errors in an analytical model and explores their amplitudes and dependence on signal spectra and sampling timescales. Extending results from time-mean measurements to time series demonstrates how sampling practices can lead to aliasing errors when records are not sampled densely, e.g., when an ocean sediment core is not sampled continuously along its accumulation axis. While we do not claim that TR error is the most important source of uncertainty in paleoclimate records, it does appear to be large enough to affect results in some cases. Moreover, this work is a step towards reducing the number of “unknown unknowns” in paleoclimate reconstruction.

Our focus is first on errors arising when a mean value computed over
one time period is used to represent another time period –
for instance, when a time average over 20–19 ka (thousand years
ago) is used to represent an average over 23–19 ka, the nominal timing
of the Last Glacial Maximum

the duration over which an observation averages (

the observation is offset from the target by a time

the paleoclimate
archive was smoothed prior to sampling, whether by bioturbation, diagenesis, residence times in karst systems upstream of speleothems

Several factors can contribute to temporal representativeness errors, defined here as the difference

This list is not exhaustive and neglects, for instance, effects from small numbers of foraminifera in sediment core records and other errors that are
inherited from the construction of

Because in paleoclimatology we do not have complete knowledge of the underlying climate signal

In practice, though we do not know

The power transfer function

While the details are left to the Appendix, it is noteworthy that in many practical cases, TR errors can be straightforwardly attributed to signal variability within a particular frequency band. This frequency band behavior emerges because

Here we explore the procedure for estimating TR errors described in the previous section in the context of estimating mean properties at the Last Glacial Maximum (LGM), the period roughly
20 000 years ago that is associated with the greatest land ice extent
during the last glacial period. Following

Temporal representativeness error in the time and frequency
domains.
Errors in representing a 4000-year mean by a 1000-year mean
are estimated by computing the difference

How large is the TR error in representing

A prominent feature in Fig.

Next, we will extend our analysis of NGRIP to cover a range of different values of

Figure

Error variances
are equal if

The succinct expression of TR errors in terms of power spectra in Eq. (

Error-to-signal variance fractions

Similar to Fig. 4, Fig.

Error-to-signal fractions

Dependencies on

To the extent that these simple experiments reflect actual paleoclimate sampling procedures, one could attempt to sample time-mean intervals to avoid TR errors. In the absence of archive sampling, the (trivial) result is that

Having explored how choices of

For all values of

A small amount of oversmoothing is present at

Same as Fig.

Same as Fig.

When the dating of a measurement is uncertain, a range of

Integrating Eq. (

Paleoclimate time series are sequences of time-mean values; here, we discuss how the TR errors discussed for time-mean estimation affect transient records of climate variability. We show that in the absence of archive smoothing, dense sampling (i.e., setting the averaging interval equal to the spacing between measurements) is a nearly optimal approach to minimize TR errors.

The sampling theorem of

In the process
of constructing a paleoclimate time series, sampling time-mean values
serves a moving average and thereby an anti-aliasing filter. Thus,
we expect sample averaging procedures to affect aliasing errors in
time series, as also discussed by

Sampling a paleoclimate archive nonuniformly in time could better approximate the ideal filter and reduce errors, but this may not be practical given the challenges of recovering and sampling paleoclimate data.

To demonstrate sensitivities to sampling parameters we again compute noise-to-signal
ratios. In keeping with our local measure of TR error, we take the signal strength to be the standard deviation of the time series that would result if

As in the time-mean case, the effects of archive smoothing are large in a regime of sampling parameter space (

Same as Fig.

This paper presents a framework for quantifying temporal representativeness (TR) errors in paleoclimatology, broadly defined as resulting when one time average is represented by another. A simple model illustrates interacting effects from record sampling procedures, chronological errors, and the spectral properties of the climate process being sampled.

We find that TR errors for time-mean estimates can be large relative to climate signals, with noise-to-signal standard deviation
ratios greater than 1 in some cases, particularly those in which the observational interval

Though not the principal goal, these analyses provide a basis for
sampling practices that minimize errors, for instance for avoiding oversmoothing that can arise
through the combined effects of sampling and archive smoothing. When constructing paleoclimate time series,
it is important to bear in mind not just the Nyquist frequency but
also the role of sampling and smoothing timescales as anti-aliasing filters; these considerations
point to dense sampling (i.e., without space between contiguous samples)
in order to minimize error in the absence of effects from archive smoothing (Sect.

To some extent, the simple model for TR error can be generalized to
more complex scenarios. If samples are nonuniform in time –
for instance, due to large changes in chronology or because material
was sampled using a syringe or drill bit with a circular projection
onto an archive – then the sinc function in (Eq.

Several caveats apply to the uncertainty estimates given. First, the model neglects some effects that may be important, such as inhomogeneities in preserved climate signals owing to, e.g., diagenesis or scarcity of biological fossils. Second, nonstationarity in record spectra
leads to time variations in errors, as illustrated in Fig.

Results for time series (Sect.

Representativity errors due to aliasing are not limited to the time domain, and similar procedures
may be useful for quantifying errors due to spatial representativeness
by considering how well proxy records can constrain the regional and
larger scales typically of interest in paleoclimatology. An analogous
problem is addressed in the modern ocean by

The hope is that these procedures may prove useful for 1st-order
practical uncertainty quantification.
A challenge is estimating the signal spectrum

This Appendix describes an analytical approach for estimating temporal representativity errors in the context of estimating time means.
These errors have a compact representation in the frequency domain
that rationalizes interactions
between sampling procedures, time uncertainty, and signal spectra in contributing to errors. Fore more on the theorems and properties of Fourier analysis that are referenced, see, e.g.,

Define a mean value

Illustration of the frequency dependence of errors in representing
an instantaneous measurement of a process

The Fourier transform will be written using both the operator

The integrand of Eq. (

When

We can expect that the presence of archive smoothing might reduce errors originating from high frequencies in

Using Eq. (

To study the error contribution from a time offset

This Appendix extends the analytical approach for estimating temporal representativity errors from estimating time means to time series. Define the associated moving average time series that would result if all of

The NGRIP oxygen isotope record used is available at

The author declares that there is no conflict of interest.

Thanks to LuAnne Thompson, Greg Hakim, Lloyd Keigwin, Cristi Proistecescu,
Carl Wunsch, and Thomas Laepple for useful conversations. Comments from two anonymous reviewers helped improve the paper.
Wavelet software was provided by Christopher Torrence and Gilbert Compo, and it is
available at the following URL:

This research has been supported by the National Science Foundation, Division of Atmospheric and Geospace Sciences (postdoctoral fellowship grant) and the National Oceanic and Atmospheric Administration (grant no. NA14OAR4310176).

This paper was edited by Denis-Didier Rousseau and reviewed by three anonymous referees.