Proxy records represent an invaluable source of information for reconstructing past climatic variations, but they are associated with considerable uncertainties. For a systematic quantification of these reconstruction errors, however, knowledge is required not only of their individual sources but also of their auto-correlation structure as this determines the timescale dependence of their magnitude, an issue that has been often ignored until now. Here a spectral approach to uncertainty analysis is provided for paleoclimate reconstructions obtained from single sediment proxy records. The formulation in the spectral domain rather than the time domain allows for an explicit demonstration and quantification of the timescale dependence that is inherent in any proxy-based reconstruction uncertainty. This study is published in two parts.

In this first part, the theoretical concept is presented, and analytic expressions are derived for the power spectral density of the reconstruction error of sediment proxy records. The underlying model takes into account the spectral structure of the climate signal, seasonal and orbital variations, bioturbation, sampling of a finite number of signal carriers, and uncorrelated measurement noise, and it includes the effects of spectral aliasing and leakage. The uncertainty estimation method, based upon this model, is illustrated by simple examples. In the second part of this study, published separately, the method is implemented in an application-oriented context, and more detailed examples are presented.

The central issues of climate sciences include the estimation, understanding, and prediction of climatic variations across ranges of space and timescales that are relevant to the specific field of study. From an inductive perspective, such studies are necessarily based on observational data which the variability may be estimated from, whereas from a deductive perspective observational data are needed in the course of the validation of theories and models. For certain fields of study, instrumental or satellite data may provide a useful data source. Nonetheless, once processes are studied that involve climate states or variations at times before the instrumental era or that involve timescales longer than this, reconstructions obtained from paleoclimate proxies become indispensable. Such proxy records reveal imprints of past climatic conditions created by, for example, impacts on the calcification of the shells of marine organisms

Reconstruction uncertainties may arise, for example, during the sampling and measurement procedure associated with measurement errors occurring in the laboratory

It turns out that a careful and systematic investigation of these reconstruction uncertainties is indispensable if we are to properly exploit the source of information contained in proxy archives, for such important issues like the estimation of the future evolution of natural and forced climate variability. Until now, however, reconstruction uncertainty estimates have often lacked the required accuracy

One possibility to estimate the auto-correlation structure of reconstruction uncertainties is in the application of proxy forward models that generate proxy time series from climate (model) time series

Accordingly, the aim of this paper is to provide a conceptual approach and, based thereon, an analytically derived method to estimate timescale-dependent reconstruction uncertainties for the example of sediment archives. Specifically, the method yields uncertainty estimates having been given a set of parameters that specify (i) the spectral structure of a supposed true climate signal, (ii) seasonal and orbital variations, (iii) proxy seasonality, (iv) bioturbation, (v) archive sampling parameters, (vi) sampling of a finite number of signal carriers, and (vii) uncorrelated measurement noise, and it takes into account the effects of spectral aliasing and leakage. The fact that archive smoothing is represented by bioturbation limits the validity of the method in its current form to proxy archives from sea and lake sediments. However, it has the potential to be generalized to other sedimentary archives such as ice cores by modifying the smoothing operator to represent isotopic diffusion.

The pivotal idea of our approach to address the timescale dependence of the uncertainty is in the derivation of its power spectrum since the spectrum is directly related (by the Wiener–Khintchine theorem; see

Part 1 of this study provides the theoretical basis of the uncertainty estimation method. In Sect.

Before we can formulate our timescale-dependent uncertainty estimation method, we have to provide a precise definition of the underlying reconstruction uncertainty model, including our assumptions and simplifications that allow for an analytic treatment of the problem. Specifically, in order to define the uncertainty model, we need to

suppose a structure of the true climate signal, which the final uncertainty estimates will be based upon, because some uncertainty components and their timescale dependence are subject to that structure;

make simplifying assumptions regarding the archive formation, concerning proxy seasonality, the climate–proxy relationship, the sediment accumulation rate, and the effects of bioturbation mixing;

specify the archive sampling and measurement procedure;

define the reconstruction error as the difference between the obtained climate reconstruction and a suitable reference climate;

define the reconstruction uncertainty in terms of the expected value of the squared reconstruction error.

We assume that the supposed true climate signal consists of two components: a stochastic signal

The stochastic signal

The deterministic signal

Schematic illustration of the reconstruction uncertainty model.

To reflect proxy seasonality, we assume a seasonally confined time window during which the proxy is abundant. Thus, the climate signal and, in particular, the seasonal cycle are recorded only during those seasons. The length of this proxy abundance window is specified by the parameter

In the following, we also neglect any uncertainties regarding the climate–proxy relationship, including calibration errors. Furthermore, we assume a known and constant sediment accumulation rate. Thus, we will treat all signals simply as a function of time and assume that the constant time–depth relationship is given as independent information.

Signal smoothing by sediment mixing caused by bioturbation is assumed to occur instantly and uniformly within the uppermost layer of the sediment. The thickness of this layer, the bioturbation depth, can be divided by the sediment accumulation rate to obtain the corresponding bioturbation timescale

We assume that the archive is sampled by taking slices of sediment, the thickness of which corresponds to time intervals of length

Because signal carriers are retrieved from arbitrary positions within each slice, the effect of the sediment sample timescale

Finally, we need to include the effect of proxy seasonality as defined in the previous subsection. This is accomplished through multiplying

In practice, a finite number,

In general, each laboratory measurement is associated with a measurement error

The reconstruction error can now be defined as the difference between the obtained climate reconstruction (Eq.

The reconstruction error

As will be shown in Sect.

In addition to the uncertainty caused by the stochasticity of

The individual components are to be interpreted as follows: the component

Now, in order to formulate our timescale-dependent uncertainty estimation method, we need a spectral representation of the expected power

The reconstruction uncertainty model, defined in the previous section, is now translated into the spectral domain. Since the two components of the supposed true climate signal have different properties – in the sense that

A spectral representation of the stochastic signal component

Following the approach of

The proof that the jitter PDF

To obtain the power spectral density of the reconstruction error components of

Note that the square bracket term in Eq. (

Schematic illustration of the auto-covariance function of the discrete process

With these properties of the abovementioned components

By analogy with Eq. (

From this we can now obtain a spectral representation of the squared reconstruction uncertainty components

The deterministic signal

Again following the approach of

The structure of

To understand the structure of

First, if there is no amplitude modulation of the seasonal cycle (

The white noise variance

Second, if the seasonal cycle is modulated by orbital variations (

Third, if we consider the same case but with

With these properties of the abovementioned components

To obtain spectral representations of

The discrete time Fourier transform of the rectangle function is given by the Dirichlet kernel

With Eq. (

The reconstruction uncertainty components

First is the uncertainty component

Second is the white noise uncertainty component

Third are the reconstruction bias

In practice, during the process of data analysis, climate reconstructions are often smoothed by some linear filter either because one is explicitly interested in time averages of the reconstructed climate variable or because one may hope to reduce the reconstruction uncertainty by averaging out short-timescale noise. However, the extent to which the uncertainty actually shrinks depends on the auto-correlation structure of the reconstruction error, which, by the Wiener–Khintchine theorem, is directly related to the power spectral density of the error. Thus, from the expressions of the error power spectral densities, derived in Sect.

If the reconstruction error time series is smoothed by, for simplicity, a discrete moving average filter of width

Illustration of the method for estimating timescale-dependent reconstruction uncertainties in the spectral domain, shown for the uncertainty components

This power spectral density is shown in Fig.

This component (blue dots), as well as the white noise component (green dots), is shown again in Fig.

Finally, the sum of the abovementioned components, given by

Likewise, one can define other timescale-dependent uncertainty metrics. For example, one might be interested in the uncertainty of the difference between the time averages over two periods of length

To conclude this section, we briefly present an example of the time series and power spectra of the reconstruction bias

Example of the reconstruction uncertainty based on the deterministic component

To allow for an analytic treatment of the problem, the method for estimating timescale-dependent reconstruction uncertainties, presented in Sects.

We assume a fixed proxy seasonality in the sense of applying every year the same seasonal timing of a prescribed proxy abundance period, characterized by the parameters

We neglect calibration errors representing uncertainties regarding the climate–proxy relationship. Assuming this relationship is linear and is obtained by linear regression, errors of this type may have two effects. Uncertainties in the intercept parameter will introduce a reconstruction uncertainty that is constant in time like the bias uncertainty

We assume a constant sediment accumulation rate and a constant bioturbation depth, and we also assume regular sampling from the sediment core and neglect dating uncertainties, although relaxing these assumptions may generate additional uncertainties of noticeable magnitude. For example, the relevance of dating uncertainties is demonstrated by

Furthermore, the timescale-dependent uncertainties obtained from our method depend explicitly on assumptions regarding the structure of the supposed true climate signal

Finally, although our method provides an advancement in the quantification of reconstruction uncertainties, it also introduces a number of model parameters which are associated with their own uncertainty. However, if we are to improve quantitative uncertainty estimates, our reconstruction uncertainty model helps to identify those parameters which are most important and, therefore, need to be determined at higher precision. For example, how much seasonality is imposed on a certain proxy at a given geographical location within a specific local ecological system? On the other hand, it is possible to investigate how parameter uncertainties translate into reconstruction uncertainties, as was shown for the seasonal phase parameter

The present study introduces a method, the so-called Proxy Spectral Error Model

The method proves useful in different ways. First, it can serve to obtain quantitative uncertainty estimates for practical applications in paleoclimate science. This is demonstrated in Part 2 of this study

The reconstruction uncertainties can be decomposed into two components. The first is a component whose variance is obtained by multiplying the power spectrum of the supposed true climate signal by a transfer function and then integrating. This so-called error transfer function has a structure corresponding to a bandpass filter with its cut-off timescales given by the longest applied archive smoothing timescale and by a suitably chosen reference smoothing timescale

In the presence of proxy seasonality such that the climate signal is recorded by the proxy only during a limited seasonal window each year, the abovementioned error transfer function has additional high-frequency peaks at the seasonal cycle frequency and at its higher harmonics, and, thus, it corresponds to a multi-bandpass filter in this case. In consequence of this, a certain amount of variance is reallocated from the abovementioned white noise uncertainty component to the first component, although it appears there at the lowest frequencies because of spectral aliasing. Thus, proxy seasonality may generate uncertainties that are highly correlated in the time domain. In most cases, this low-frequency uncertainty will be dominated by the seasonal cycle and its amplitude modulation caused by orbital variations

If, in addition, the proxy abundance window is known to have a preferred seasonal timing throughout the year, then the contribution that the seasonal cycle signal (with its deterministic phase) makes to both of the abovementioned two uncertainty components is further modified. The white noise component can be larger or smaller than for random seasonal timing, and, in particular, the first uncertainty component may include a (potentially time-varying) deterministic bias in this case. Moreover, the sum of their variances may change because of the in-phase subsampling from a deterministic signal.

Uncertainties caused by laboratory measurement errors are independent of the abovementioned components, and, thus, the associated power spectral density can simply be added to the error power spectrum obtained from our method. In practice, this uncertainty component is assumed to be white noise such that it scales inversely with any averaging timescale.

Another interesting and future application of the derived analytic expressions would be the inference of the power spectrum of the true climate signal. Specifically, by setting the reference climate in our method to zero and then repeating the entire derivation, one obtains the analytic expressions for the power spectrum of the climate reconstruction itself rather than of its error. Thus, one obtains an operator that transforms the power spectrum of the supposed true climate signal into a spectrum subject to the distortions caused by the processes included in our reconstruction uncertainty model. Then, given all of the parameters of the uncertainty model, and assuming a parametric form for the true climate signal, it might be possible to estimate its parameters by means of a maximum likelihood approach (that investigates the likelihood, under a given set of parameters, of the power spectrum estimated from a specific proxy record). This essentially amounts to inverting the aforementioned operator, which is similar to the correction technique used by

The variance of

From the abovementioned expressions, we can write the variance of

No data sets were used in this article.

TL developed the underlying idea of the Proxy Spectral Error Model. TK, AMD, and TL designed the conceptual outline of the research. TK laid out and performed the mathematical derivation of the analytic expressions and wrote the paper based on numerous discussions with AMD and TL.

The authors declare that they have no conflict of interest.

This article is part of the special issue “Paleoclimate data synthesis and analysis of associated uncertainty (BG/CP/ESSD inter-journal SI)”. It is not associated with a conference.

This is a contribution to the European Research Council (ERC) project SPACE. The work profited from discussions at the Climate Variability Across Scales (CVAS) working group of the Past Global Changes (PAGES) program. We thank Cristian Proistosescu (University of Illinois Urbana-Champaign) for bringing to our awareness the formalism to describe the effects of jittered sampling in signal processing and Dan Amrhein (University of Washington) for discussions on the spectral uncertainty approach. We also thank Shinya Nakano and one anonymous referee for their useful comments which helped to improve this work.

This research has been supported by the European Research Council (ERC) project SPACE, which has received funding from the ERC under the European Union’s Horizon 2020 research and innovation program (grant agreement no. 716092). Andrew M. Dolman was supported by the German Federal Ministry of Education and Research (BMBF) through the PALMOD project (FKZ: 01LP1509C).

This paper was edited by Denis-Didier Rousseau and reviewed by Shinya Nakano and one anonymous referee.