Climate reconstructions are means to extract the signal from uncertain
paleo-observations, so-called proxies. It is essential to evaluate these
reconstructions to understand and quantify their uncertainties.
Similarly, comparing climate simulations and proxies requires approaches
to bridge the temporal and spatial differences between both and to
address their specific uncertainties. One way to achieve these two goals
is so-called pseudoproxies. These are surrogate proxy records within
the virtual reality of a climate simulation. They in turn depend on an
understanding of the uncertainties of the real proxies including the
noise characteristics disturbing the original environmental signal.
Common pseudoproxy approaches so far concentrate on data with high
temporal resolution over the last approximately 2000 years. Here we
provide a simple but flexible noise model for potentially low-resolution
sedimentary climate proxies for temperature on millennial timescales,
the code for calculating a set of pseudoproxies from a simulation, and
one example of pseudoproxies. The noise model considers the influence of
other environmental variables, a dependence on the climate state, a bias
due to changing seasonality, modifications of the archive (for example
bioturbation), potential sampling variability, and a measurement error.
Model, code, and data allow us to develop new ways of comparing simulation
data with proxies on long timescales. Code and data are available at
10.17605/OSF.IO/ZBEHX.
Introduction
Proxy records and derived reconstructions are our only observationally
based information about past climates before the period covered by human
observations, that is before we have documentary or instrumental
evidence. Climate reconstruction methods statistically process the
information in the proxy records to extract the recorded climate signal.
However, multiple variables influence the signal recorded, and we are
often only interested or able to extract the contribution of one single
climatic parameter.
All other imprints of climate are noise with regard to this variable of
interest. Furthermore, part of the variability in the proxy records is
not caused by the climate but other factors influencing the original
generation of the proxy record. Thus, there are climatic and
non-climatic noise contributions to the proxy variability. This proxy
noise may cause biases and uncertainties in the resulting climate
reconstructions. Evaluating the quality and reliability of
reconstructions and of proxy records requires an understanding of the
noise in the proxy records. Only this knowledge allows us to obtain
reliable estimates of the errors in reconstructed properties.
Some aspects of statistical climate reconstruction methods can be
evaluated in so-called pseudoproxy experiments. In these experiments,
the reconstruction methods are mimicked for example in the controlled
conditions provided by climate simulations with Earth system models.
However, for these tests surrogate proxy records have to be produced,
which are compatible with the climate simulated by these models – the
pseudoproxies. In testing the reconstruction methods, pseudoproxies
eventually replace the real paleo-observations in the method and the
virtual climate of the Earth system simulation stands in for the real
climate.
Our use of the term pseudoproxy follows the literature since
. That is, a pseudoproxy represents a modification of
observational data, reanalysis data, or simulation output. It replaces
real-world proxies in an application. The term does not necessarily
refer to substitutes for specific proxy records or particular proxy
types. That is, the term pseudoproxy does not by itself imply that the
modifications of the input data validly represent the uncertainties or
characteristics of real-world data. This view of the term pseudoproxy is
in line with the past literature
compare, for example,.
Modifications of the input data may be as simple as adding white or
coloured noise or they may invoke more complex forward approaches
for example mechanistic proxy system models;see below.
Studies of the climate of the past 2000 years regularly use such
pseudoproxy approaches mimicking annually resolved proxies such as
dendroclimatogical ones. reviews the approach of
using pseudoproxy experiments to evaluate reconstruction methods with a
focus on the last millennium. Such methods basically originated in the
work of focussing on climate-field reconstructions. The
review by emphasizes the essential contribution of
pseudoproxy experiments to our understanding of past climates and to
evaluating our methods of studying past climates. To date, most studies
using pseudoproxies concentrated on the last few millennia. Few studies
considered periods further in the past
e.g..
For a useful test of reconstruction methods, the pseudoproxies should be
as realistic as possible, with statistical properties similar to the
real proxies. This is achieved by contaminating the climate variables
simulated by the Earth system model with statistical noise with a
certain amplitude and statistical characteristics. These properties
ideally are based on estimates of a realistic or at least plausible
noise to successfully mimic the behaviour of real-world proxies.
In our understanding there are various approaches to obtain such
pseudoproxies. These range from most comprehensive to most simplified.
We can try to obtain a comprehensive representation of a so-called proxy
system from the environmental influences
on a sensor to our measurement and formulate this into a mechanistic
forward model of the system of interest. Such models can be very complex
or they may concentrate solely on a core set of processes
compare the full and reduced implementations of the Vaganov–Shashkin approach to modelling tree rings presented by.
That is, the first approach to obtaining pseudoproxies is process-based.
Other, more reduced approaches potentially ignore this mechanistic
process understanding and focus on stochastic expressions of the noise
that influence our inferences about past climates. Such an approach can
try to formulate mathematically tractable expressions for statistical
noise terms, which represent the different processes or effects
influencing the stages from the original environmental conditions to our
final observation (Andrew M. Dolman, personal
communication, 2018, Thomas Laepple, personal communication, 2017).
Another way of producing pseudoproxies by focussing on stochastic noise
expressions uses simple estimates of plausible errors. These different
approaches can be specific for certain proxy types or very general. They
can focus on one stage of the proxy system from the environment to
measurement or consider multiple stages.
All these approaches fit into the concept of a proxy system model as
described by . The idea of forward models
to study the behaviour of proxies and proxy systems is not new
e.g. but
were the first to clearly delineate
the modelling of proxy systems. A proxy system represents the
biological, chemical, geological, and possibly also documentary system
that translates environmental influences into an archived state on which
researchers make observations. We usually refer to these observations
when speaking of climate proxies. A proxy system model is a
representation of how the proxy system translates the environmental
influences into our observations based on our understanding.
present a generalized concept of this
modelling approach, which consists of three components. First, a sensor
model reacts to the environmental influences. Second, an archive model
transforms these sensor recordings into archive units. A third model
translates the archive into representations of what we usually observe
on an archive. For example, the sensor “tree” records the environmental
influences in its archive “wood”, and we can make measurements on this
archive in the form of tree ring counts, widths, etc. The full system from
recording to observation is the proxy system.
Each stage in this system and its model representations adds
uncertainty, and each stage omitted in a generalization also increases
uncertainty. For example, the environment and the final reconstruction
process can be additional stages, but we can try to include the
associated uncertainties in any of the three stages proposed by
. That is, considering the
reconstruction stage, the calibration introduces additional uncertainty,
which is not a priori captured by the stages sensor, archive, and
measurement. We can argue to include this additional source of error in
the measurement stage. We can also argue that these uncertainties are de
facto uncertainties resulting from processes at the sensor stage or at
the archiving stage and include them there. Similarly, the sensor model
does not necessarily account for all uncertainties of the environmental
influences. An additional environmental stage could provide weighted
data of various environmental influences
compare, e.g.. These processes, however, can also be
included in the sensor model or uncertainties can be assumed to mostly
affect the measurement model. In short, inferences about past climates
from proxy data are based on observations of an archive that accumulated
a property of a system. This (property of the) system was sensitive to
and recorded an environmental process at some date. From the recording
stage to our inference there are multiple sources of error in our
inference.
The potential errors include different sources of noise related to
laboratory uncertainties like measurement errors and reproducibility,
local disturbances, dating uncertainty, time resolution, serial
autocorrelation, and all possibly dependent on the overall climate
state. Further uncertainty includes habitat preferences, seasonal
biases, the variability in the relation between sensor and environment,
long-term changes in this relation, long-term modifications of the
archive, sampling variability and sampling disturbances, and not least
generally erroneous assumptions on the researcher's side about the relation
between recording sensor and environment, i.e. the calibration
relation. A recipe for calculating pseudoproxies may include potential
error estimates for not only parts of the assumed proxy system but also the relation between the “observed” data and time, that is the
anchoring of the data in time.
Regarding dating/age uncertainty, there are various approaches to
dealing with it
e.g.
of which a number try to transfer the dating uncertainty towards the
proxy record uncertainty
e.g.. Our interest
explicitly is to include the uncertainty from the dating in a
statistical noise term for a pseudoproxy time series. Therefore, we do
not consider Bayesian or Monte Carlo methods but take a simple approach
to develop an error term for the uncertainty in the dating. We also do
not include explicit age modelling
compare, e.g..
In addition to evaluating reconstruction methods, a plausible estimate of noise
within the proxies can also assist in comparison studies between
model simulations and the proxy records or among different
model simulations. This increases our understanding about past climate
changes by consolidating information from all available sources, which
are proxy records and model simulations. The lack of high-quality
observations with small uncertainty is always going to hamper efforts to
assess the quality of model simulations of past climates. Such
comparisons have to rely on the paleo-observations from proxies, and
even the highest-quality proxy records have an irreducible amount of
uncertainty. Most often data–model comparisons take place in the virtual
reality of the model and use the modelled variables. In the case of
proxies, the comparison is between, for example, a temperature
reconstruction and a model. The alternative is to compare both in the
proxy space using a proxy representation of the model climate.
Pseudoproxies or a recipe how to compute them may be part of an
interface between the data on the one side and the model simulations on
the other side.
Recent years saw an intensification in the research on forward modelling
proxies for understanding proxies, testing reconstruction methods, and
evaluating simulation output
see, for example,.
Many of these approaches follow the concept of considering sensor,
archive, and observations as distinct steps in the process. Still, few
of these approaches consider transient timescales beyond the late
Holocene. Nevertheless, particularly the work by and
also allows for the calculation of different sedimentary
proxies over multi-millennial timescales based on knowledge of certain
processes in the respective proxy systems.
In this paper, we adopt the conceptual subdivisions of
to present a formal but still simple
noise-based approach to describe the disturbances masking the signal in
proxy records. This approach can also be applied to produce
pseudoproxies for timescales longer than the last few millennia, that is
proxies with coarser time resolutions than interannual and afflicted by
larger degrees of dating uncertainty. Thereby this work extends
previous pseudoproxy approaches, which often concentrated on well-dated
proxy systems affected by fewer sources of uncertainty.
The following presents a set of assumptions on proxy noise and estimates
for some of the mentioned error sources. We further provide
pseudoproxies based on these assumptions for the TraCE-21ka simulation
, which covers the last 21 000 years. We concentrate on
proxies which are subject to some kind of sedimentary process. Thus,
our work appears to be particularly similar to the proxy system model
for sedimentary proxies implemented by .
also consider the long timescales since the last
glacial maximum and rely on output from the TraCE-21ka simulation for
their forward modelling. Both the present paper and
follow the concept outlined by
. The main difference between
and the present study is that they provide a simple
process-focussed model of the proxy system, whereas we try to provide a
simple characterization of the noise in the proxy system that finally
influences the proxies. The process-based formulation of
concentrates on two types of marine proxies whereas
our noise-based approach tries to generalize over sedimentary proxy
types. We regard both approaches as complementary and want to emphasize
the value in having a multitude of methods to assess model simulations
and reconstruction methods.
Our approach contributes to the existing proxy system modelling and
pseudoproxy computation applications by being an intermediate step
between complex forward modelling approaches and the noise-based
approaches, of which the latter may ignore the proxy system workings.
Our code simplifies and generalizes more complex assumptions. The
noise focus and the generalizations allow us to provide global
pseudoproxy data and an ensemble of pseudoproxy data using the
TraCE-21ka simulation over the timescale of the last 21 000 years.
The paper assets at 10.17605/OSF.IO/ZBEHX
provide the generated pseudoproxy data and also include sample code.
Thereby the paper provides the data for one simulation to make an
informed comparison with real proxies and the data to evaluate
reconstruction techniques. Code and assumptions enable any interested
user to produce similar pseudoproxies for their simulation of interest.
We consider the measurement error, local changes to the original
proxy recording compare, e.g., the basic climate
state, a potential bias, and a simple estimate of the effect of dating
uncertainty. All noise expressions are coded in a way to flexibly allow
for different colours and types of noise.
Input data
We use the annual mean temperature at each grid point of the TraCE-21ka
simulation . To date, this is the only available
interannual transient Earth system model simulation covering the last
21 000 years. Specific technical considerations, for example, related to
freshwater pulses and sea-level adjustments lead to some artefacts in
the simulation output data fields. A brief description of the simulation
can be found at http://www.cgd.ucar.edu/ccr/TraCE/ (last access: 29 July 2019), and the
PhD dissertation of provides more details.
The presented results and figures are generally for one grid point at
150∘ E, 38.97∘ N. The simulation output at this grid point
has the benefit of representing a rather smooth evolution of temperature
over the last 21 000 years. Conversely, the less extreme climate
variations to be captured in a subsequent pseudoproxy can be seen as a
disadvantage. The document assets provide figures equivalent to those in
this document for a grid point at 105∘ W, 45.39∘ S in the
South Pacific.
On multi-millennial timescales we have to consider changes in the
insolation caused by changes in Earth's orbital elements. Global
insolation data are calculated using the R package palinsol
. We use simple Gaussian noise for most noise processes.
However, as the code is flexible, the user can easily change this.
Considerations and results
In defining what we consider as noise, we first have to state the
signal, which we assume the proxy system records. That is, do we assume
that the proxy records local or regionally accumulated signals? Here, we
take the signal of interest to be local; that is non-local influences
enter the noise term and are not part of the signal. In addition, there
are further local factors which affect the recording of the signal but
are not part of the signal of interest. The Appendix provides tables
(Tables to ) summarizing the considered
parameters and noise models in the various steps of our considerations.
In the following, we distinguish between different sources of errors
related to the concepts of sensor, archive, and measurements of
. Figure summarizes our
procedure. Each section contains a discussion of the implications of the
respective error term. Afterwards we discuss the results of applying the
respective step in the framework to the output of the TraCE-21ka
simulation.
Conceptual flow of the procedures.
Assumptions on essential error sources 1: sensorNoise and bias
The sensor, that is for example an organism or a physical or
biogeochemical process, reacts to multiple parts of its environment.
Researchers' interest is often only in one of the environmental
variables. The sensor, S, is likely a nonlinear function of the
environment, S(E), where E={ei}, with ei being
components of the environment. If our interest is only in the sensor's
reaction to one variable, T,
S(E)≈S^(T,ηi).
Under this assumption, further components of the
environment besides T contribute only noise components ηi to
the reaction of the sensor. Errors due to noise are not necessarily
additive but can also be multiplicative or could bias the estimate. In a
first step we, here, assume the sensor reaction to be
S(E)≈S^(T)+ηi.
Any of these errors or noise processes may show autocorrelation in
either space or time or both. Any such process may, in turn, add memory
to the sensor system. Indeed this memory effect and spatial or temporal
correlations may be large. For example, if a process takes place in an
environment with slowly and fast varying components, and our interest is
in one of the fast components, the low-frequency variations add a noise
or error with high autocorrelation in time.
The sensor reacts to all, potentially highly frequent, changes in its
environment. This local environment is unlikely isolated from the larger-scale system. Thus, additional noise may be due to the sensor reacting
to advected environmental properties instead of “local” ones or due to
the environment redistributing the sensor or the record. In the marine
realm but also in lake domains, currents may influence the sensor, while
in many domains the wind may affect the recording of the signal.
Furthermore, small- and large-scale spatial variations in the process may
affect the signal and contribute to the record. Our approach regards
these contributions as noise. All these influences may introduce spatial
and, here considered to be of more importance, temporal correlations in
those environmental properties, which we here consider as part of the
noise term. We assume that advection from other regions by currents and
wind is especially important in contributing autocorrelation to our
noise process. One can see these non-local factors as noise in the
archive rather than the sensor.
In addition to simple noise, redistributions of the environmental signal may
also introduce biases in our estimate of the environment. Any bias is
likely not fully time-constant but evolves with the environment on
interannual, multi-decadal, and multi-centennial to millennial
timescales. The different timescales result from the different
timescales of the environment. This is relevant for recent climate
changes and interannual to interdecadal climate variability, but it
becomes even more important for multi-millennial timescales where we
have to deal with the effects of changing seasons, glaciation,
deglaciation, changes in bathymetry, and lithospheric adjustments. All
of these processes may lead to biases, and such biases also lead to
autocorrelation in the error.
One example of such time-evolving biases is changes in the seasonality
of the environmental sensor. While one can see this again as a source of
uncertainty in a narrowly defined proxy system from sensor to
reconstruction, it is in the end a bias of our attribution of the
measurement to one season. We consider this bias on the sensor level.
There are other potentially erroneous attributions besides the
processes' seasonality. These are the location of the process in all
three dimensions, for example, the habitat of living organisms, and a
generally only partially correct calibration relationship. Again, these
are environmental factors influencing the sensor and we consider them to be
noise here. However, they reflect a non-stationarity of our
reconstruction–calibration relation. Nevertheless, the idea that the
modern relations between environment and proxy system worked over the
full period of interest is a fundamental assumption of paleo-climatology
e.g..
In the following, we assume three components to be important
disturbances of the signal at the sensor level: the environmental noise,
the redistribution, and the attribution errors. We reduce the latter to
the potential biases due to changes in the seasonality. Taking all three
components the sensor record becomes
S(E)≈S^(T)+ηenv+ηredistr+ηseason,
where we for the moment replace ηi by
ηenv. In the following, we reduce these three components to
two terms in our modifications of the input data.
Noise
First, we assume that ηi includes both the effects of
environmental dependencies and of redistribution. That is,
ηi=ηenv+ηredistr. This is the first error term.
This in fact implies that we should consider autocorrelated
noise processes. If we only modify the model output and concentrate on
one parameter T, for example, temperature data, our pseudoproxy at
this point becomes
P(x,y,t,T)=PT=T+ηi.
The current version of ηi is only a weakly
correlated autoregressive (AR) process of order one, which we
additionally scale by an ad hoc scaling factor. It thereby only includes
a small part of the potential correlations among errors due to
redistribution and other processes. The innovations are sampled
dependent on time and climate background from
N(0,S(t)2), where S(t) is a time-dependent standard
deviation. The time dependence mimics a dependence of the noise on the
background climate variability on long timescales. Here, we use a 1000-year moving standard deviation
S(ti)=σ(T(ti-499:ti+500)). Our general formulation
assumes that noise variability increases with increasing variability in
the parameter T. Obviously, it could also be that noise variability
reduces or reacts totally differently relative to the variability of
T. The code includes a switch to invert the moving standard
deviation about its mean or to randomize the orientation.
Bias
We can consider the changes of the seasonality, ηseason, as an
orbitally influenced bias term. We compute it for any latitude of
interest. We apply the orbital bias term as additive but one may see it
as a multiplicative or a nonlinear effect in many cases. Therefore the
code uses it after the noise term ηi. The bias is the second
error term in our formulation of modifications at the sensor level. The
bias term is a scaling of the changes in annual latitudinal insolation
but it is possible to choose different sub-annual time periods of
interest. The scaling is arbitrary and we refer to the provided code for
details. The bias is zero in the year 0 BP. We set it to be positive if
the insolation is larger; this can be randomized in the code. The
amplitude of the bias is scaled by an ad hoc constant. The bias becomes
notable at some latitudes but may be rather negligible elsewhere. We
take the bias as bias(t)=β⋅In, where β is the
scaling constant, and In is a normalized and shifted insolation.
In is calculated as
In=((I-I¯)/σ(I)⋅q0.25-I(t=0BP)+1)u-1 for a
chosen period. The chosen time period influences the statistics included
in the scaling. We consider the insolation since 150 000 BP.
q0.25 is the 25th percentile of the insolation data, u is
generally 1, but can be sampled from U={-1,1}.
The pseudoproxy becomes
PT(t)=T(t)+ηi(t)+bias(t).
Results
Figure a shows an example of the initial noise ηi.
The dependence on the background state is clearly visible for the
visualized grid point data. There is an increase during the deglaciation
and a multi-millennial reduction over the Holocene. Indeed, diagnose a reduction in temperature
variability from the Last Glacial Maximum to the Holocene by studying
centennial to millennial timescales.
Panel (b) of Fig. compares three potential amplitudes of
the orbitally induced bias. We use the version with the smallest
amplitude. Panel (c) of Fig. presents the grid point temperature of
the TraCE-21ka simulation and a simple 501-year running mean. The
comparison with Fig. d highlights that the effect of the
bias is rather small given our choice of its amplitude. Nevertheless,
comparing the panels also clarifies that our implementation of the bias
results in a colder annual record over most of the considered time
period while the record becomes slightly warmer in the very early
portion of the simulated data.
Visualizing considered error sources at the sensor stage: (a) the initial noise and the underlying moving window standard deviations of the input signal, (b) three versions of a potential bias as function of the local insolation, (c) the input data and their 501-point moving mean, (d) the input data and their 501-point moving mean plus noise and bias. The unsmoothed initial temperature is effectively hidden behind the unsmoothed temperature plus bias.
Assumptions on essential error sources 2: archive
So far our approach describes a record of an environmental influence
plus two error terms. This record becomes subsequently integrated in an
archive. Afterwards, various processes may modify the archive or
redistribute it. Modifications include selective destruction of parts of
the record by processes acting all the time or by sparse random events
or continually acting random processes. Examples are bioturbation or
resuspension. These processes may result in either a correlated noise
in time and space or simply white noise. Other de facto white noise
errors may result from our finite and random sampling of the archive.
However, this may be rather part of the observational noise.
Such modifications of the archive and sampling issues represent an
important step in using inverse reconstruction methods because it is a
priori not clear how the archive is generated and whether an individual
measurement represents mean environmental states or relates to single
events. In this context, forward models and pseudoproxy approaches of
sedimentary proxies are a crucial tool in disentangling the controlling
climatic environmental factors in the generation of sediment cores and
their interpretation.
Smoothing and noise
Because we focus on sedimentary proxies, we argue that the archiving
process foremost is a filter of variability above a certain frequency
level, for example, by diffusive processes or bioturbation
compareand their references. Dependent on the
system in question this may only affect the very high frequencies but
for other systems it may extend to multi-decadal or even centennial to
millennial frequencies. On top of this smoothing of the archive, there
may be additional noise as the smoothing function is unlikely
homogeneous. We assume such a filtering to be the fundamental
modification of the record in the archive, and, thus, only consider this
process in our archive modelling.
Inspired by the simple proxy forward formulations of
see also , we produce five
different versions of the archived pseudoproxy series. The first and
second series are simple running averages of the sensor record to which
we add a highly autocorrelated AR process of order one. The two versions
differ in the length of the averaging window, the AR coefficients, and
the standard deviations of the innovations. Versions three and four
similarly differ in the amount of average smoothing, but we use random
window lengths for each date. The rationale for the two different
smoothing lengths is to represent both strongly and only slightly
smoothed proxies.
The fifth version aims to mimic the behaviour of proxies when researchers
use only a small part of an available proxy, e.g. pick only a certain
number of samples. An example is the simple forward formulation for
Mg/Ca proxies by see also .
Smoothing lengths and random factors in this approach could depend on
the background climate. Indeed, the code includes options for the random
smoothing lengths to depend on the mean climate or the climate
variability. The provided data use an approach where the random
smoothing lengths follow an autoregressive process around a climate-dependent reference smoothing length, where, considering
, warmer climates result in shorter smoothing
intervals. The smoothed archive records are then either
PT(t)=g1(T(t)+ηi(t)+bias(t),t),
where g1(t) is the time-dependent filter, or
PT(t)=g2(T(t)+ηi(t)+bias(t))+AR,
where g2 is the constant smoothing and we add an
AR process to account for the inhomogeneities in the smoothing.
The fifth version of the pseudoproxy subsamples over the random filter
interval and adds a noise term to mimic a seasonal uncertainty. That is, we
sample n years within the filter interval, and take the mean over
the temperature and the noise for these years. We add another noise term
to represent the intra-annual seasonal uncertainty. PT in this case
becomes
PT=h(T(t),t)+h(ηi(t),t)+ηs,
where h(t) represents the subsampling and ηs
the intra-annual noise. We do not include the bias term for the
subsampled proxies. We apply the bias only for the mean
annual temperature; i.e. individual seasons show different biases. While
we could account for this by sampling the biases over the different
seasons or even months in producing h(t) or ηs, we prefer to
keep our model simpler. Excluding the bias term may be interpreted as
the seasonal subsampling cancelling out the bias. In reality any
cancellation would not result in a convergence on the simulated climate
state but more likely on a recorded value between the biased and the
“true” climate. The coded version of the subsampling still includes the
bias term as a comment.
Visualizing considered error sources at the archive stage: (a) 501-year moving mean of the input data, the pseudoarchive series with longer average smoothing lengths, and the subsampled record; (b) 501-year moving mean of the input data, the pseudoarchive series with shorter average smoothing lengths, (c) 501-year moving mean of the input data, the pseudoarchive series with longer average smoothing lengths, and the version with constant smoothing and added AR(1) process; (d) 501-year moving mean of the input data, the pseudoarchive series with shorter average smoothing lengths, and the version with shorter constant smoothing and added AR(1) process.
Results
The biased moving average already shows the differences between the
target temperature and the pseudoproxy record (compare Fig. ). The pseudoarchive series in Fig. a shows
this more clearly. Here we use a randomized smoothing interval.
Differences are less visible for shorter random smoothing intervals
(compare Fig. b). Further panels of Fig.
add the constant smoothing archive approximations which we modify by an
additional highly correlated AR process (Fig. c and d).
This procedure randomly amplifies, dampens, or inverts certain biases in
the presented case. That is, while the simple random smoothing may
emphasize the bias, the AR procedure overlies this bias with additional
variations.
The panels highlight an apparent offset between the randomly smoothed
archive series, the constantly smoothed archive series, and the smoothed
input data. The smoothed version of the input data as well as the
constant filtering use a centred approach, that is they are symmetric
about their date. The time-varying smoothing tries more realistically to
imitate a bioturbation approach
compareand their references and thus provides a
shift in the series.
Figure a also shows the seasonally subsampled
pseudoarchive proxy. The data ignore the bias term and the resulting
series is by construction symmetric around the original data, our
target. Nevertheless, there are pronounced deviations from the original
data. Considering only the deviations from the target temperature moving
mean highlights that this approach is notably more noisy than the
filtered data but preserves pronounced longer-term excursions of the
input data (not shown).
Assumptions on essential error sources 3: measurements
The archiving represents also a transformation from time units to
archive distance units, to depths, rings, distances. The proxy becomes a
tuple of date and data. Now the dates are uncertain as each data point
includes information from different original dates due to the smoothing
function. The sampling may lead to additional uncertainties due to
disturbances of the archive, and the dating of our samples is a
profoundly uncertain process.
Measurement error
Prior to dealing with errors due to dating uncertainty, we take an
additional noise term to represent measurement errors and apply this for
each date to account for the potentially imperfectly measured series.
The term includes not only the errors introduced by our assumed methods
of measuring the proxies but also the methods' potential to make mistakes.
This true measurement error may result in biases due to limits of
what our methods can detect or systematic offsets due to a
laboratory-specific, potentially erroneous approach to the measurement.
Potential offsets imply that we should generally expect a certain amount
of autocorrelation in this noise. The term has further to account for
the accidental handling of the records in the laboratory, for example,
influences from storage or from other processing of the samples and the
data, which may result in autocorrelated errors if these influences have
a systematic component. Thus, it is not necessarily the case that we can
consider inter-laboratory reproducibility to be white noise. However, the
intra-laboratory repeatability is likely indeed a white random process.
We also assume repeatability and reproducibility to be part of our
measurement error term. While we just mentioned various reasons to
assume autocorrelation in this error term, we only provide a white noise
term for the measurement noise. Again, the code allows modification of this.
We apply the measurement error term at the end. However, we introduce
this term before dealing with the dating uncertainty since we provide
proxies without dating uncertainty. The measured proxy series becomes
MT=PT+ηM.
In reality, we do not have a continuously sampled series,
but obtain only samples at certain intervals. Assuming N samples the
sampled pseudoproxy becomes
PPT=PT(t={t1,…,tN}).
The sampling of the archive likely produces errors in the
samples. We assume these are included in the measurement uncertainty. We
provide at each grid point sampled series of the pseudoproxies detailed
above. We do not distinguish between different sampling techniques. We
simply sample the records at certain dates and add the described noise
term.
Dating uncertainty
Dating uncertainty represents a big part of our overall uncertainty for
many proxies, especially for sedimentary proxy records. In our
framework, the smoothing function already redistributes information from
one date across the archive. Usually one considers this temporal
uncertainty separately from the proxy record error. For assessing
reconstruction methods and simulations, it would be beneficial to be
able to include dating uncertainty within the proxy error. That is, if
we consider proxies as tuples of data and date, we have to transform the
uncertainty of the date into an error term for the data. In the
following we distinguish between the dating uncertainty, that is the
uncertainty that a sample is from a certain date, and the dating error,
by which we mean the potential error in our (pseudo)proxy due to the
uncertain dating.
There are a number of approaches to transfer the dating uncertainty
towards the proxy record error
e.g.. Ensemble and
Bayesian age–depth modelling approaches also allow us to infer an
additional error term e.g.. However
in the present application, we want to capture the error in a
time series. Thus, we take a very simple approach, which assumes that
the error due to dating uncertainties is related to the climate state
over the period of the dating uncertainty. Nevertheless, since we
provide sample dates and random sampling uncertainties, the application
of age modelling to the pseudoproxies is in principle possible
e.g. following the approach of.
The code includes several variations in our estimation of an effective
dating error. These reflect different amounts of dependence between
subsequent samples. In all variants, we only consider dependence between
two subsequent samples while for real proxies the correlations may
extend across larger portions of the proxy record. The following general
approach is common to all variations in our procedure. First, we sample
uncertainties in time for each sample date. We take these as dating
uncertainty standard deviations. These uncertainties can be sampled
fully randomly or dependent on the available smoothing interval data
from the archive stage. Then we take the effective dating error at each
sample date and depth to be a random sample from a normal distribution.
The mean of this distribution is the difference between the sample data
and the mean over the data within plus and minus 2 dating uncertainty
standard deviations. The standard deviation of the distribution is the
standard deviation of the differences between the individual data points
within this interval and this mean. The effective dating error is then
ϵD=N(PTD‾,σD2),
where
PTD‾=PT(tS={ti-2σdating,…,ti,…,ti+2σdating})‾-PT(t=ti)
is the mean over the region of influence and
σD2=E[(PT(tS)-PTD‾)2]
is the variance of the distribution.
In the simplest formulation ignoring the dependence between subsequent
dates, the sampled pseudoproxies become
PPT(t1,…,tN)=g(T(t)+ηi(t)+bias(t),t)(t1,…,tN)+ϵD(t1,…,tN).
Alternative formulations of the pseudoproxy become
PPT(t1,…,tN)=g(T(t)+ηi(t)+bias(t))(t1,…,tN)+AR(t1,…,tN)+ϵD(t1,…,tN)
or
PPT(t1,…,tN)=h(T(t),t)(t1,…,tN)+h(ηi(t),t)(t1,…,tN)+ηs(t1,…,tN)+ϵD(t1,…,tN).
This initial formulation of the effective dating
uncertainty error ignores potential correlation between the dating
errors. The most simple way to account for this makes subsequent errors
dependent:
ϵDi=ρ⋅(ϵξDi-1+(PPTi-1-PPTi))+ϵξDi.
This formulation has only a minor influence on the
results. It is included in the code via a binary switch.
A slightly more complex formulation makes the error term at each date
dependent on the previous sample's age uncertainties and mean data.
Previous refers to archive units instead of time units. Then the dating
error becomes
ϵDi=ρ⋅(ϵDi-1+(PPTi-1-PPTi))+ϵξDi,
where ϵξDi represents the random innovations
for date i. Our initial choice of ρ=0.9 can give large
effective dating uncertainty errors. A switch in the code allows the use of
this interdependent error. Another switch allows the consideration of the
dependence between samples as a function of their dates and the dating
uncertainty,
ρ(t)=1-(ti-ti-1)/(2⋅σd(i-1)).
The time-dependent dating uncertainty for each date
σd(t) is generated randomly (compare above σD). We
provide data for the case with a time-dependent ρ(t).
Alternative simple formulations may include different noise processes
like noise generated from gamma distributions. The available smoothing
interval data can inform the sampled dating uncertainty. We could
further use this information to provide a deterministic, not random,
error for each sampled date; that is we could take a bias based on all
dates influencing the selected date within the dating uncertainty.
In our current setup the age uncertainty does not depend on the
measurement noise. The measurement error is added afterwards to the
series including the effective dating uncertainty error. This decision
is arbitrary. On the one hand a classical dating uncertainty affects the
measured value. Then, PPT above should also already include the
measurement error. On the other hand, the dating uncertainty affects the
archived values independent of the measurement noise. Therefore we keep
both independent.
The measured proxy series becomes
MT=PPT+ηM.
The final proxy is in temperature units as are the initial
input data. We ignore a separate term for potentially non-linear and
climate-state-dependent errors in our calibration relationship and
assume the measurement noise term accounts for this as well. A separate
term could again be a state-dependent Gaussian noise. It could also be a
noise from a skewed distribution whose mode depends on the background
climate. Conversely, a state-dependent bias term could simulate a
mis-specified calibration relation while a time-dependent bias term
could simulate a degenerative effect over time within the archived
series. None of these are included in the current version.
Visualizing considered error sources at the measurement stage for the full series: (a) 501-year moving mean of the input data, the pseudoarchive series with longer average smoothing lengths, and the constant smoothing plus AR series with added measurement noise; (b) 501-year moving mean of the input data, the pseudoarchive series with shorter average smoothing lengths, and the constant smoothing plus AR series with added measurement noise; (c) 501-year moving mean of the input data, the subsampled record, and the subsampled record with added measurement noise.
Results
Figure shows versions of an archived proxy plus
interannual measurement noise. The panels give an impression of how a
proxy would look from measurements on a perfectly annually sampled
archive. The final amplitude of the noisy proxy is generally slightly
smaller for all versions of our pseudoproxies than the amplitude of the
interannual variations for the chosen location. This may be different at
other locations. The different versions of the smoothing and of the
smoothing plus AR approaches are shown in Fig. a and b,
respectively. Figure c plots the seasonally subsampled
pseudoproxy. The final versions of the pseudoproxies generally preserve
previously included biases.
Visualizing the sampled records: (a) input data and their 501-year moving mean, the pseudoarchive series with longer average smoothing lengths plus the effective dating error and plus the effective dating error and measurement noise; (b) input data and their 501-year moving mean, the constantly smoothed record with longer smoothing length plus AR series with added effective dating error and with added effective dating error and measurement noise; (c) input data and their 501-year moving mean, the pseudoarchive series with shorter average smoothing lengths plus the effective dating error and plus the effective dating error and measurement noise; (d) input data and their 501-year moving mean, the constantly smoothed record with shorter smoothing length plus AR series with added effective dating error and with added effective dating error and measurement noise; (e) input data and their 501-year moving mean, the subsampled record with added effective dating error and with added effective dating error and measurement noise.
Figure presents a number of series sampled at N=200
dates. All panels include the original temperature data sampled at these
200 dates. The figure emphasizes how the initial temperature variability
at the chosen grid point is generally slightly larger than any of our
uncertainty estimates. Our effective dating uncertainty error seldom
results in large deviations from the archived record. The subsequently
applied measurement error also only seldom leads to large offsets
compared to either the original data or the effectively date-uncertain
record. Thus, for our chosen parameter settings and the shown
grid point, the pseudoproxies fall within the range of the initial
estimates. In turn, if we assume we have reliable calibration
relationships, our calibrated proxy series should also be reliable
estimates of the past states.
Nevertheless, the biased estimates occasionally are only bad matches for
the original data. This is also the case for the subsampled data where
we did not include the bias. Comparing the sampled pseudoproxy series to
the smoothed original temperature data (compare Fig. a)
highlights that estimates for past climates may well fall within the
range of the original interannual temperature variability but may
nevertheless strongly misrepresent the mean climate represented by the
sample.
Considering the effective dating uncertainty error, the discrepancies
between input data and pseudoproxy are rather small for uncorrelated or
weakly correlated age uncertainties. However, in the case of strong
dependencies between subsequent data, pronounced biases and mismatches
may occur (not shown). The assumed co-relation between two dates has a
strong influence on the size of these mismatches. We show the case for a
time-dependent co-relation between subsequent dates, which gives
intermediately sized mismatches.
General results
Figures to present the different versions of
the pseudoproxies for the chosen location. Under our assumptions, the
influence of the orbital bias term is notable. The approaches using
time-dependent smoothing or simple smoothing plus an AR process may
nearly or fully cancel the bias. This effect is less prominent for the
time-dependent filter. Generally, both approaches seem to have similar
effects.
Figure includes the effect when we hypothetically add
measurement noise at every date. Under our assumptions this noise is
still smaller than or only as large as the original interannual
variability but, including biases, mean estimates may be outside of the
interannual variability of the original data. In these examples, the
variability of the subsampled proxies is comparable to the smoothed ones
after a measurement error is added. It is interesting to note that for
the smaller smoothing the AR process seems to cancel the orbital bias
more strongly in Fig. . Figure shows the
data sets sampled at N=200 dates. It clarifies the error described
for the interannual data. The document assets provide equivalent
visualizations for another grid point. These generally confirm the above
descriptions.
Wavelet-based power spectral densities . Densities are weighted following to smooth the records for ease of comparison. Lines are for records split up by the first 10 k years of the records and the last 12 k years of the records. Input data refer to the input data at 10-year intervals. All panels include the late input data from the TraCE-21ka simulation as black lines; red lines are in all panels for a full period record; blue lines are in all panels for the last 12 k years of the version of a pseudoproxy. In addition to the input data from the TraCE-21ka simulation the panels show: (a) the sampled TraCE-21ka simulation input data, (b) the sampled pseudoarchive series with long average smoothing plus the effective dating error and the measurement noise (long random smoothing MT), (c) the constantly smoothed record with a longer smoothing plus an AR(1) process and including the effective dating error and the measurement noise (long constant smoothing MT), (d) the sampled pseudoarchive series with short average smoothing plus the effective dating error and the measurement noise (short random smoothing MT), (e) the constantly smoothed record with a shorter smoothing plus an AR(1) process and including the effective dating error and the measurement noise (short constant smoothing MT), (f) the subsampled data plus the effective dating error and the measurement noise (MT from subsampling).
Spectral power
Figure adds a comparison of power spectral densities
computed from a wavelet-based approach similar to the weighted wavelet
Z-transform of . The approach is described by
and McKay and colleagues provide a compiled version
at https://github.com/nickmckay/nuspectral (last access: 11 March 2019) . Due to the length of computation, we do
not show the density for the full 22 040-year input data but only for a
record sampled every 10 years. Results may be specific for the chosen
grid point.
Point by point correlation maps between input
data and the smoothed record plus AR(1) process plus effective dating
error and measurement noise for the sample dates within the first (a),
second (b), and third (c) subsequent 5000-year windows of the record and
the samples within the remaining years (d).
Left, standard deviation ratios of the
sampled 501-year moving mean input data relative to the smoothed record
plus AR(1) process and the effective dating error and the measurement
noise for the samples in the first 5000 years of the record (a), the last
7040 years of the record (b), and the full record (c). Right,
differences between the mean of the sampled input data and the mean of
the smoothed record plus AR(1) process and the effective dating error
and the measurement noise for the samples in the first 5000 years of the
record (d) and the last 7040 years of the record (e).
The figure shows estimates for the full records and for the data of the
last 12 000 years of the records. Spectral densities for the
regularly sampled original temperature data in Fig. a
highlight that the differentiation between full and late records results
in prominent differences for multi-centennial to millennial periods. Conversely, differences are smaller for the irregularly sampled
input temperature data but still notable for millennial periods.
However, there is an offset between the irregularly sampled data and the
regularly sampled input data.
Spectra for full and late records of the various pseudoproxies are
generally similar to the irregularly sampled input data spectra (Fig. b–f) but the offset to the input data can be smaller than
in Fig. a. Differences between sampled late and full
records are often largest at intermediate millennial periods. Deviations
are largest for the subsampled pseudoproxy approach at long periods
(Fig. f) but they also become notable for the constant
smoothing approaches at shorter periods in the centennial band (Fig. c, e). This is mainly due to the characteristics of the full
period spectra for the constant smoothing, which show an increase in
power spectral density for shorter and longer periods. That is, the
constant smoothing full period spectra are similar to grey noise spectra.
Despite these differences and the apparent offset to the input data
spectra, the irregularly sampled spectra for all cases are rather
similar.
Global data
The supplementary assets for this paper include plots of selected
series from our analyses at all grid points starting from the south
towards the north (supplementary document 1 Fig. 1 at
10.17605/OSF.IO/ZBEHX/). These series are the
input data at the grid point, the smoothed-plus-AR-process series at the
grid point, and its subsampled version including all uncertainties.
These plots highlight three main points. First, peaks and troughs at
some locations are clearly attributable to the specific implementation of
the forcing in the TraCE-21ka simulation (; see also). That is,
these signals are not realistic but due to technical decisions in the
production of the simulations. Furthermore there is potentially
unrealistic variability at some grid points for some periods. Second,
the bias term in its current version may have only a small influence at
certain latitudes. Third, our noise model shows often larger effects in
the midlatitudes and the tropics. There is also a longitudinal
dependence. Supplementary document 2 Fig. 1
(https://doi.org/10.17605/OSF.IO/ZBEHX/) emphasizes the regional
differences in the long-term climate evolutions by selecting only
grid points in equal intervals to provide a more intuitive view of the
globe. Similarly, supplementary document 2 Fig. 2 adds scatter plots
of the pseudoproxy on the y axis against the original data on the x axis
for a small selection of grid points, highlighting the common lack of a
clear relation besides the deglaciation.
Figure provides correlation coefficients between the
sampled interannual grid point data and the pseudoproxies including all
uncertainties for the strong smoothing plus AR. The four panels show
correlations for those samples within the first, second, and third 5000-year chunks of the original data, and those samples in the remaining
years. We choose to present the data this way to avoid detrending the
data over the deglaciation interval. Relations between original data and
pseudoproxies are generally weakest in the tropical belt. In the period
until present, correlations are overall weak. High-latitude correlations
are most notable during the deglaciation and slightly less notable
during the first millennia of the Holocene. In these periods,
correlations appear to be largest in areas with glacial remnants.
Figure adds for the first, the last, and the full period
the relative standard deviation σT21k/σP in the left
column and the bias T¯T21k-T¯P in the right column.
T21k refers to the simulation, P to the pseudoproxies. For the standard
deviation ratios, we use 501-year moving averages of the TraCE-21ka
data. Variability is generally larger in the pseudoproxies except for
the North Atlantic and the northern high latitudes in the early period,
and it is larger in the pseudoproxies more or less everywhere in the
late period. Over the full period, variability is notably larger mainly
in the tropics and the Southern Hemisphere; it is about equal over
Antarctica and wide regions of the Northern Hemisphere. The variability
is clearly larger in the input data only over a small region in the
northern Pacific.
The overall largest bias occurs off the coast of southeastern Greenland
in the early period in Fig. . Otherwise there is a
spatial separation between the mid-latitudes to high latitudes and the tropics
and subtropics for both periods. The bias is more prominent in the
higher latitudes where it is predominantly positive in the early period
but predominantly negative in the late period. Obviously, the general
latitudinal bias pattern is by construction because we construct the
bias as a function of latitudinal insolation.
On generalizations of the errors
While we already chose comparatively simple procedures for our approach
to obtain pseudoproxies from a model simulation, it is likely possible
to simplify these to a higher degree. Such a general expression for the
error in proxies over multi-millennial timescales may be more usable in
a number of ad hoc model evaluations and model–data comparisons. Most
importantly, such a generalized approach also allows us to quickly produce
ensembles of pseudoproxies.
Following our previous assumptions, the easiest way to obtain such a
generalized error model would be to assume a simple, potentially
correlated noise model for the sensitivity of the sensor to the
environment. Here, we use an AR process of order one with AR coefficient
ϕ=0.7. Either here or later one scales the series or adds a bias
term to account for changing seasonality over multi-millennial
timescales. The sum of the input data and this error are then subject
to a simple moving averaging function. On top of this another simple
correlated noise process mimics that the redistribution in the archive
is not constant in time. Another random component accounts for the
measurement error. Thus, simple correlated noise may be enough to catch
the essence of the error. In short, the generalized pseudoproxy becomes
MT(t1,…,tN)=g(T(t)+ηit+bias(t))+ϵD(t1,…,tN)+ηM,
where g is the smoothing, ηi is the initial
noise, bias is the bias term, ϵD is the effective dating
error, and ηM is the measurement error. This is conceptually
identical to the smoothing plus AR approach presented above. Its
derivation is less grounded in real proxies. The provided data differ
only in the amount of autocorrelation in the noise terms.
Visualizing the simplified essence of the surrogate proxy calculations: (a) input data and 501-point moving mean; (b) input data plus initial noise and bias term; (c) moving mean of input data plus noise plus bias and the same record plus an AR(1) process; (d) smoothed temperature plus noise plus bias plus AR process sampled at 200 dates, this record plus the effective dating error, and this record plus the effective dating error and measurement noise; (e) smoothed temperature plus noise plus bias plus AR process sampled at 100 dates, this record plus the effective dating error, and this record plus the effective dating error and measurement noise.
Figure summarizes results for the generalized approach. It
clarifies that while an error may mask certain features of the past
climate evolution, this simple generalized pseudoproxy generation is
unlikely to distort the proxy completely if we take the assumptions made
above to be approximately appropriate. Interestingly, the generalization
appears to modify the input signal slightly less than the more complex
approach. However, as we display slightly different data comparisons
here, it is more appropriate to note that the dating uncertainty has
only a minor effect compared to the initial bias and AR process
modifications and compared to the subsequent addition of the measurement
noise.
While researchers may validly wish for such simplified recipes for
producing pseudoproxies, using a full or at least more complex
process-based approach is advisable, if it is necessary to account for
effects of biology, environmental long-term changes, and other weakly
constrained uncertainties. More complex approaches further allow to
better mimic non-linearities between the climate and sensor and thus a
truly non-linear pseudoproxy.
Ensemble of pseudoproxies
In the following we present an ensemble of pseudoproxies. At 144
locations we compute 500 pseudoproxy records each. For this, we make
slight modifications to the generalized approach. These adjustments
relax our assumptions and result in larger differences between members
of the ensemble than would be possible without the modifications. The
locations are the grid points, which are close to proxies
included in , , or
. Figure shows the locations. Using the
generalized approach provides an ensemble based on the most reduced
formulation. The provided code allows users to produce ensembles for
their input data of interest.
Modifications to the code are as follows. First, we use a number of
parameter values sampled from either uniform distributions around the
otherwise fixed value or a list of values. Second, we consider
random orientations for bias and moving standard deviations; that is we
take S as Su where we sample u from U={-1,1}. We
provide the script for the ensemble production as supplementary example
code at https://doi.org/10.17605/OSF.IO/ZBEHX. As mentioned above,
these changes relax our assumptions on the effect of changes in the
background climate.
Map of the locations for the ensemble of
surrogate proxies.
Visualizing the surrogate proxy ensemble at
selected locations (longitude and latitude in top left corners of the
left column panels). The left column shows the input data plotted as
grey lines, and two random members of the ensemble as blue and purple
lines. The right column plots the range of the ensemble transparently shaded
brown, and blue and purple lines are the same two random members.
The x axes are years BP. The panel on the bottom right shows the figure
legend.
For Fig. we select eight locations to represent the locally
diverse representations of the climate in the TraCE-21ka simulation and
how the ensemble of pseudoproxies modifies this. The figure provides an
impression of the range of the local ensembles and of two random
ensemble members around the original temperature series. The diversity
of the local climates in TraCE-21ka carries over to individual
pseudoproxies and their ensembles. In addition to this, Fig.
mainly reflects the results of previous sections regarding how
constrained our pseudoproxies are. However, we commonly see
pseudoproxies and ensembles exceeding the variability of the original
temperature data, not least because of our modifications to the
selection of parameters and the orientation of the bias about its mean.
Provided
data
Tables to detail the provided data files.
All files are in netcdf format. These are generally gridded files on the
original TraCE-21ka grid. Only the ensembles of pseudoproxies are
provided at their respective individual grid points. The data repository
at https://doi.org/10.17605/OSF.IO/ZBEHX provides instructions on how
to access the file structures.
List of files provided, the variables included, their description, the category (full surrogate proxy field, essence field, or ensemble), and size of the ensemble. All files have the same stem Bothe_Trace21k_Pseudo_Proxies_ and the ending _annual.nc.
FilenameVariable nameVariable descriptionCategoryGridEnsemblesizesizeBothe_Trace21k_Pseudo_Proxies_noise.savenoise.saveinitial environmental noisefield96×481bias.noise.data.savebias.noise.data.savedata + noise + biasfield96×481smooth.savesmooth.savesmoothed data + noise + biasfield96×481meas.noise.smooth.savemeas.noise.smooth.savesmoothed data + noise + bias plus measurement noisefield96×481ar.smooth.savear.smooth.saveconstantly smoothed plus AR processfield96×481meas.noise.ar.smooth.savemeas.noise.ar.smooth.saveconstantly smoothed plus AR plus measurement noisefield96×481short.smooth.saveshort.smooth.savesmoothed date + noise + bias for shorter smoothingfield96×481meas.noise.short.smooth.savemeas.noise.short.smooth.savesmoothed data + noise + bias plus measurement noise for shorter smoothingfield96×481short.ar.smooth.saveshort.ar.smooth.saveconstantly smoothed plus AR process for shorter smoothingfield96×481meas.noise.short.ar.smooth.savemeas.noise.short.ar.smooth.saveconstantly smoothed plus AR plus measurement noise for shorter smoothingfield96×481subsampled.savesubsampled.saveseasonally subsampled data + initial noisefield96×481meas.noise.subsampled.savemeas.noise.subsampled.saveseasonally subsampled + noise plus measurement noisefield96×481
Continued list of files provided, the variables included, their description, the category (full surrogate proxy field, essence field, or ensemble), and size of the ensemble. All files have the same stem Bothe_Trace21k_Pseudo_Proxies_ and the ending _annual.nc.
FilenameVariable nameVariable descriptionCategoryGrid sizeEnsemble sizesampledsamp.subsampled.save, samp.meas.noise.smooth.save, samp.input.save, samp.input.save.short, samp.input.save.ar, samp.input.save.ar.short, samp.noise.save, samp.noise.save.short, samp.noise.save.ar, samp.noise.save.ar.short, samp.bias.noise.data.save, samp.bias.noise.data.save.short, samp.bias.noise.data.save.ar, samp.bias.noise.data.save.ar.short, samp.ar.smooth.save, samp.smooth.save, samp.short.smooth.save, samp.short.ar.smooth.save, samp.meas.noise.short.smooth.save, samp.dates.save, samp.dates.save.short, samp.dates.save.ar,samp.dates.save.ar.short, samp.meas.noise.ar.smooth.save, samp.meas.noise.short.ar.smooth.save, samp.meas.noise.subsampled.savesampled versions of the various variables and the dates of the samplesfield96×481
Continued list of files provided, the variables included, their description, the category (full surrogate proxy field, essence field, or ensemble), and size of the ensemble. All files have the same stem Bothe_Trace21k_Pseudo_Proxies_ and the ending _annual.nc.
FilenameVariable nameVariable descriptionCategoryGrid sizeEnsemble sizedating-errorsamp.dates.save, samp.dates.save.short, samp.dates.save.ar,samp.dates.save.ar.short, unc.date.samp, unc.date.samp.short, unc.date.samp.ar, unc.date.samp.ar.short, unc.date.subsampled.save, unc.date.meas.noise.smooth.save, unc.date.noise.save, unc.date.bias.noise.data.save, unc.date.ar.smooth.save, unc.date.smooth.save, unc.date.short.smooth.save, unc.date.short.ar.smooth.save, unc.date.meas.noise.short.smooth.save, unc.date.meas.noise.ar.smooth.save, unc.date.meas.noise.short.ar.smooth.save, unc.samp.meas.noise.subsampled.savedate uncertain versions of the various variables and the dating uncertaintiesfield96×481
Continued list of files provided, the variables included, their description, the category (full surrogate proxy field, essence field, or ensemble), and size of the ensemble. All files have the same stem Bothe_Trace21k_Pseudo_Proxies_ and the ending _annual.nc.
FilenameVariable nameVariable descriptionCategoryGridEnsemblesizesizeEssence_gen.noise.envgen.noise.envgeneralized environmentalnoise termessence96×481Essence_noise.gen.datnoise.gen.datinput data + generalized environmental noiseessence96×481Essence_bias.noise.gen.datbias.noise.gen.datinput data + generalized noise + bias termessence96×481Essence_smooth.bias.noise.gen.datsmooth.bias.noise.gen.datsmoothed input + noise + biasessence96×481Essence_ar.smooth.bias.noise.gen.datar.smooth.bias.noise.gen.datsmoothed input + noise + bias plus AR processessence96×481Essence_uncertain-sampledsamp.ar.smooth.bias.noise.gen.dat, unc.samp.ar.smooth.bias.noise.gen.dat, meas.unc.samp.ar.smooth.bias.noise.gen.dat, unc.date.samp.gen, samp.dates.save.gendate uncertain versions of generalized data, generalized dating uncertainty, sample datesessence96×481essence_ensemblePseudoproxy, Dates, DateUncertaintysurrogate proxy data, dating, uncertainty of datingensemble144500Lat, Lonlatitude, longitudeensemble1441Code and data availability
The TraCE-21ka simulation data are available from
http://www.cgd.ucar.edu/ccr/TraCE and were obtained via the Earth System Grid
(http://www.earthsystemgrid.org/project/trace.html, ). Our results as described
in Sect. are available from the Open Science
Framework (OSF) at 10.17605/OSF.IO/ZBEHX/. There,
one also finds sample code for computing proxies and the script for
computing the ensemble at 144 locations.
Conclusions and outlook
This publication presents a flexible yet simple approach for describing
the error originating from climatic and non-climatic sources in
proxy records over multi-millennial timescales including the last
deglaciation. The assumptions are relatively simple but they are based
on similar assumptions for process-based proxy system forward models.
The approach can be easily extended to compute ensembles of proxies for
single locations. We chose to give one set of pseudoproxies for each
grid point of the TraCE-21ka simulation and an ensemble of pseudoproxies
at locations close to real proxy locations. This simulation has a
specific climatology but a comparison to real proxy data
may easily be achieved by only considering anomalies
as performed by. The provided pseudoproxy
data and the code to compute further pseudoproxies allow the
application of our pseudoproxy approach for the evaluation of models,
the comparison of models to paleo-data, and the testing of
reconstruction and data-assimilation methods.
We choose only one possible set of parameters in our pseudoproxy model,
but we sample around this set for the ensemble of pseudoproxies. We
choose these specific parameters to provide some disturbance to the data
but not to get anywhere too far away from the original state. For
example, it is quite likely that we have to face larger biases in
reality than represented by our choice. Users should make their own
choice of parameters according to their assumptions on the various
noise contributions.
One can easily extend the chosen approach to even longer timescales.
Some modifications may be advisable considering the dating uncertainty
to account for the likely sparser data further back in time, to better
accommodate the increasing uncertainty, and especially to be more
realistic in considering an effective dating uncertainty error for the
pseudoproxy data. Similarly, we do not consider spatial correlations in
the noise. Such correlations between locations are probably relevant for
some noise terms while they are probably less important for others.
We focused on the time series approach and did not choose a
probabilistic approach like, for example, or
. Neither does our approach as of now explicitly
link to probabilistic age modelling approaches as described by
, , or .
There are a variety of other potential approaches to obtaining simple
pseudoproxies from the model data. One such example would be to consider
an envelope around the model state, to randomly select a set of dates
from the original data, fit a smooth through this set, and then sample
again around this uncertain smoothing. Similarly, Gaussian process
models or generalized additive models may be valuable means in producing
pseudoproxies for paleo-climate studies over timescales longer than the
common era of the last 2000 years. For example,
shows the benefits of generalized additive models for studies on
paleo-environmental time series.
The present approach ignores a variety of possible complications. For
example, we currently do not consider hiatuses in the sensor.
Furthermore, the dependency on the background climate is small.
Nevertheless, we are confident that this approach is of value for the
comparison of simulation data and proxy data over long periods, for
testing reconstruction methods, and for evaluating different model
simulations against each other.
Tables of parameters
Tables to summarize the considered
parameters and noise models. They also clarify whether the parameter
settings are used for a global field of surrogate proxies, a more
generalized approach, an ensemble calculation, or all.
List of parameters used.
DescriptionParameterValueCategorySeason limits for insolation biasmon1.for.insol, mon2.for.insol1, 12allNumber of samples along the full recordn.samples200allScaling of initial noise amplitudeamp.noise.env0.5field, essenceSwitch for proportionality of initial noiseswitch.orient.runsd.noise.env0allModel for the initial noisemodel.noise.1c(0.3)field, essenceStandard deviation of innovations for initial noisesd.noise.1not usedfield, essenceLength of window influencing initial noiselength.window.runsd1000field, essenceSwitch for orientation of biasswitch.orient.bias.seas0allScaling of bias termamp.bias.seas4field, essence
Continuation of list of parameters used.
DescriptionParameterValueCategorySwitch for smoothing variantswitch.smoothing3fieldSecondary switch for smoothing, see codeswitch.sm.21fieldScaling for climate dependence of smoothingscale.sm1/10fieldMean smoothing length for longer random smoothingrand.mean.length.smooth350fieldStandard deviation for longer random smoothingrand.sd.length.smooth75fieldModel for longer alternative smoothingmodel.smooth.1c(0.99)fieldModel for longer alternative climate dependent smoothingmodel.clim.smooth.1c(0.9)fieldBasis long smoothing length for alternative approachrand.length.smooth.mean.1500fieldStandard deviation for longer alternative smoothing approachessd.model.smooth.110fieldFixed longer smoothing lengthfix.length.smooth501fieldMinimum allowed longer random smoothing lengthmin.rand.length.smooth40fieldAR coefficient for added AR(1) processcoeff.ar.smooth0.999fieldStandard deviation for the innovationssd.ar.smooth0.01fieldMean smoothing length for shorter smoothingrand.mean.length.smooth.231fieldStandard deviation for shorter random smoothingrand.sd.length.smooth.25fieldModel for shorter alternative smoothingmodel.smooth.2c(0.7)fieldModel for shorter alternative climate dependent smoothingmodel.clim.smooth.2c(0.9)fieldBasis short smoothing length for alternative approachrand.length.smooth.mean.231fieldStandard deviation for shorter alternative smoothing approachessd.model.smooth.24fieldFixed shorter smoothing lengthfix.length.smooth.231fieldMinimum allowed shorter random smoothing lengthmin.rand.length.smooth.25fieldAR coefficient for added AR(1) processcoeff.ar.smooth.20.9fieldStandard deviation for the innovationssd.ar.smooth.20.15field
Continuation of list of parameters used.
DescriptionParameterValueCategoryNumber of picked samples for subsamplingn.samp.pick30fieldStandard deviation of innovations for subsampling noisesd.noise.pick0.5fieldModel of subsampling noisemodel.noise.pickc()field1.96 sigma of measurement-noiselim.noise.meas1.5field, essenceNoise model for measurement noisemodel.noise.measc()field, essenceNoise model for measurement noise for subsampled recordmodel.seas.pick.noise.measc()field1.96 sigma for measurement noise for subsampled recordlim.seas.pick.noise.meas1.5fieldSwitch for correlated effective dating errorswitch.cor.date.unc1allSwitch for weakly correlated onlyswitch.weak.cor.date.unc1allSwitch for time dependent correlatedswitch.delta.cor.date.unc1allFixed correlated dating error coefficientcor.date.unc0.9allMean of distribution of dating uncertaintymean.date.unc350allStandard deviation of distribution of dating uncertaintysd.date.unc100allSwitch for length of influence on dating uncertaintyswitch.cor.length1allSwitch for date samplingswitch.sampling1allSwitch for dating uncertainty samplingswitch.sampling.unc1allModel for initial noise for generalized casemodel.gen.noisec(0.7)essenceModel for initial noise for generalized casecoeff.gen.ar.smooth0.999essence, ensembleStandard deviation for AR process innovations, generalized casesd.gen.ar.smooth0.01essence, ensembleSmoothing length generalized caselength.filter.uniform501essence
Continuation of list of parameters used.
DescriptionParameterValueCategoryEnsemble sizesize.ensemble500ensembleAmplitude of scaling of initial noiseamp.noise.envU(0.4,1.5)ensembleScaling of biasamp.bias.seasU(3,10)ensembleStandard deviation of measurement noiselim.noise.measU(0.75,3)/1.959964ensembleAR coefficient of measurement noise modelrand.model.coeffU(0.3,0.8)ensembleAR coefficient of initial noise modelrand.model.coeff.genU(0.6,0.8)ensembleWindow of influence of background climate – not usedrand.width.background.sdU(500,2000)ensembleWindow of influence of background climaterand.width.background.sd1000ensembleWidth of window of filter influencelength.filter.uniformis random sample fromensembleL={301,303,305,…,1001}Competing interests
The authors declare that they have no conflict of interest.
Author contributions
OB designed and conducted the study and was the main author. All three authors discussed the methods, the results, and their implications.
Special issue statement
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.
Acknowledgements
This work contributes to PalMod, German Climate Modeling Initiative:
From the Last Interglacial to the Anthropocene – Modeling a Complete
Glacial Cycle. It is funded through the German Federal Ministry of
Education and Research's (BMBF) Research for Sustainability initiative
(FONA). We acknowledge discussions with Andrew Dolman, Thom Laepple, and
Nils Weitzel as influences in our approach. We want to thank the three anonymous referees and the editor for their comments, which helped to improve this paper.
Financial support
This research has been supported by the Bundesministerium für Bildung, Wissenschaft, Forschung und Technologie (grant no. 01LP1509A).
Review statement
This paper was edited by Michal Kucera and reviewed by three anonymous referees.
ReferencesAnchukaitis, K. J. and Tierney, J. E.: Identifying coherent spatiotemporal
modes in time-uncertain proxy paleoclimate records, Clim. Dynam., 41,
1291–1306, 10.1007/s00382-012-1483-0,
2013.Annan, J. D. and Hargreaves, J. C.: A new global reconstruction of temperature changes at the Last Glacial Maximum, Clim. Past, 9, 367–376, 10.5194/cp-9-367-2013, 2013.Blaauw, M. and Christen, J. A.: Flexible paleoclimate age-depth models using
an autoregressive gamma process, Bayesian Anal., 6, 457–474,
10.1214/ba/1339616472, 2011.Boers, N., Goswami, B., and Ghil, M.: A complete representation of uncertainties in layer-counted paleoclimatic archives, Clim. Past, 13, 1169–1180, 10.5194/cp-13-1169-2017, 2017.Bothe, O., Wagner, S., and Zorita, E.: Simple noise estimates and pseudoproxies for the last 21k years, 10.17605/OSF.IO/ZBEHX, 2018.Bradley, R. S.: Chapter 1 – Paleoclimatic Reconstruction, 1–11,
Academic Press, 10.1016/B978-0-12-386913-5.00001-6, 2015.Breitenbach, S. F. M., Rehfeld, K., Goswami, B., Baldini, J. U. L., Ridley, H. E., Kennett, D. J., Prufer, K. M., Aquino, V. V., Asmerom, Y., Polyak, V. J., Cheng, H., Kurths, J., and Marwan, N.: COnstructing Proxy Records from Age models (COPRA), Clim. Past, 8, 1765–1779, 10.5194/cp-8-1765-2012, 2012.
Brierley, C. and Rehfeld, K.: Paleovariability: Data Model Comparisons, Past
Global Changes Magazine, 22, 57–116, 2014.Carré, M., Sachs, J. P., Wallace, J. M., and Favier, C.: Exploring errors in paleoclimate proxy reconstructions using Monte Carlo simulations: paleotemperature from mollusk and coral geochemistry, Clim. Past, 8, 433–450, 10.5194/cp-8-433-2012, 2012.Clark, P. U., Shakun, J. D., Baker, P. A., Bartlein, P. J., Brewer, S., Brook,
E., Carlson, A. E., Cheng, H., Kaufman, D. S., Liu, Z., Marchitto, T. M.,
Mix, A. C., Morrill, C., Otto-Bliesner, B. L., Pahnke, K., Russell, J. M.,
Whitlock, C., Adkins, J. F., Blois, J. L., Clark, J., Colman, S. M., Curry,
W. B., Flower, B. P., He, F., Johnson, T. C., Lynch-Stieglitz, J., Markgraf,
V., McManus, J., Mitrovica, J. X., Moreno, P. I., and Williams, J. W.:
Global climate evolution during the last deglaciation, P.
Natl. Acad. Sci. USA, 109, E1134–E1142,
10.1073/pnas.1116619109, 2012.Comboul, M., Emile-Geay, J., Evans, M. N., Mirnateghi, N., Cobb, K. M., and Thompson, D. M.: A probabilistic model of chronological errors in layer-counted climate proxies: applications to annually banded coral archives, Clim. Past, 10, 825–841, 10.5194/cp-10-825-2014, 2014.Crucifix, M.: palinsol: Insolation for Palaeoclimate Studies,
available at: https://CRAN.R-project.org/package=palinsol (last access: 29 July 2019), r package
version 0.93, 2016.Dee, S., Emile-Geay, J., Evans, M. N., Allam, A., Steig, E. J., and Thompson,
D.: PRYSM: An open-source framework for PRoxY System Modeling, with
applications to oxygen-isotope systems, J. Adv. Model.
Earth Sy., 7, 1220–1247, 10.1002/2015MS000447, 2015.Dee, S. G., Russell, J. M., Morrill, C., Chen, Z., and Neary, A.: PRYSM v2.0:
A Proxy System Model for Lacustrine Archives, Paleoceanography and
Paleoclimatology, 33,
1250–1269, 10.1029/2018PA003413, 2018.Dolman, A. M. and Laepple, T.: Sedproxy: a forward model for sediment-archived climate proxies, Clim. Past, 14, 1851–1868, 10.5194/cp-14-1851-2018, 2018.Evans, M. N., Reichert, B. K., Kaplan, A., Anchukaitis, K. J., Vaganov, E. A.,
Hughes, M. K., and Cane, M. A.: A forward modeling approach to paleoclimatic
interpretation of tree-ring data, J. Geophys. Res.-Biogeo., 111, G03008, 10.1029/2006JG000166, 2006.Evans, M. N., Tolwinski-Ward, S. E., Thompson, D. M., and Anchukaitis, K. J.:
Applications of proxy system modeling in high resolution paleoclimatology,
Quaternary Sci. Rev., 76, 16–28,
10.1016/j.quascirev.2013.05.024, 2013.Foster, G.: Wavelets for period analysis of unevenly sampled time series,
Astron. J., 112, 1709, 10.1086/118137, 1996.Goswami, B., Heitzig, J., Rehfeld, K., Marwan, N., Anoop, A., Prasad, S., and Kurths, J.: Estimation of sedimentary proxy records together with associated uncertainty, Nonlin. Processes Geophys., 21, 1093–1111, 10.5194/npg-21-1093-2014, 2014.Graham, N. and Wahl, E.: Paleoclimate Reconstruction Challenge: Available for
participation, PAGES news, 19, 71–72, 10.22498/pages.19.2.71, 2011.Haslett, J. and Parnell, A.: A simple monotone process with application to
radiocarbon-dated depth chronologies, J. Roy. Stat.
Soc. C.-App., 57, 399–418,
10.1111/j.1467-9876.2008.00623.x, 2008.He, F.: Simulating Transient Climate Evolution of the Last Deglaciation with
CCSM3, PhD thesis, University of Wisconsin-Madison, available at:
http://www.cgd.ucar.edu/ccr/TraCE/doc/He_PhD_dissertation_UW_2011.pdf (last access: 29 July 2019),
2011.Hind, A., Moberg, A., and Sundberg, R.: Statistical framework for evaluation of climate model simulations by use of climate proxy data from the last millennium – Part 2: A pseudo-proxy study addressing the amplitude of solar forcing, Clim. Past, 8, 1355–1365, 10.5194/cp-8-1355-2012, 2012.Jones, P. D., Briffa, K. R., Osborn, T. J., Lough, J. M., Van Ommen, T. D.,
Vinther, B. M., Luterbacher, J., Wahl, E. R., Zwiers, F. W., Mann, M. E.,
Schmidt, G. A., Ammann, C. M., Buckley, B. M., Cobb, K. M., Esper, J.,
Goosse, H., Graham, N., Jansen, E., Kiefer, T., Kull, C., Küttel, M.,
Mosley-Thompson, E., Overpeck, J. T., Riedwyl, N., Schulz, M., Tudhope,
A. W., Villalba, R., Wanner, H., Wolff, E., and Xoplaki, E.: High-resolution
palaeoclimatology of the last millennium: A review of current status and
future prospects, Holocene, 19, 3–49, 10.1177/0959683608098952, 2009.Konecky, B., Dee, S. G., and Noone, D.: WaxPSM: A forward model of leaf wax
hydrogen isotope ratios to bridge proxy and model estimates of past climate,
J. Geophys. Res.-Biogeo., 124, 2018JG004708,
10.1029/2018JG004708, 2019.Kopp, R. E., Kemp, A. C., Bittermann, K., Horton, B. P., Donnelly, J. P.,
Gehrels, W. R., Hay, C. C., Mitrovica, J. X., Morrow, E. D., and Rahmstorf,
S.: Temperature-driven global sea-level variability in the Common Era,
P. Natl. Acad. Sci. USA, 113, E1434–E1441,
10.1073/pnas.1517056113, 2016.Kurahashi-Nakamura, T., Losch, M., and Paul, A.: Can sparse proxy data constrain the strength of the Atlantic meridional overturning circulation?, Geosci. Model Dev., 7, 419–432, 10.5194/gmd-7-419-2014, 2014.Laepple, T. and Huybers, P.: Reconciling discrepancies between Uk37 and Mg/Ca
reconstructions of Holocene marine temperature variability, Earth
Planet. Sc. Lett., 375, 418–429, 10.1016/j.epsl.2013.06.006,
2013.Lehner, F., Raible, C. C., and Stocker, T. F.: Testing the robustness of a
precipitation proxy-based North Atlantic Oscillation reconstruction,
Quaternary Sci. Rev., 45, 85–94,
10.1016/j.quascirev.2012.04.025, 2012.Liu, Z., Otto-Bliesner, B. L., He, F., Brady, E. C., Tomas, R., Clark, P. U.,
Carlson, A. E., Lynch-Stieglitz, J., Curry, W., Brook, E., Erickson, D.,
Jacob, R., Kutzbach, J., and Cheng, J.: Transient simulation of last
deglaciation with a new mechanism for Bolling-Allerod warming, Science, 325, 310–314, 10.1126/science.1171041,
2009.Mann, M. E. and Rutherford, S.: Climate reconstruction using “Pseudoproxies”,
Geophys. Res. Lett., 29, 1501, 10.1029/2001gl014554,
2002.
Marcott, S. A., Shakun, J. D., Clark, P. U., and Mix, A. C.: A reconstruction
of regional and global temperature for the past 11,300 years, Science, 339, 1198–201, 10.1126/science.1228026, 2013.Marsicek, J., Shuman, B. N., Bartlein, P. J., Shafer, S. L., and Brewer, S.:
Reconciling divergent trends and millennial variations in Holocene
temperatures, Nature, 554, 92–96, 10.1038/nature25464,
2018.Mathias, A., Grond, F., Guardans, R., Seese, D., Canela, M., and Diebner,
H. H.: Algorithms for Spectral Analysis of Irregularly Sampled Time Series,
J. Stat. Softw., 11, 1–27, 10.18637/jss.v011.i02,
2004.McKay, N., Graham, M., Zhu, F., and Emile-Geay, J.:
available at:
https://github.com/nickmckay/nuspectral, last access:
11 March 2019.Osborn, T. J. and Briffa, K. K.: The real color of climate change?, Science,
306, 621–622, 10.1126/science.1104416, 2004.R Core Team: R: A Language and Environment for Statistical Computing, R
Foundation for Statistical Computing, Vienna, Austria,
available at: https://www.R-project.org/ (last access: 29 July 2019), 2017.Rehfeld, K. and Kurths, J.: Similarity estimators for irregular and age-uncertain time series, Clim. Past, 10, 107–122, 10.5194/cp-10-107-2014, 2014.Rehfeld, K., Münch, T., Ho, S. L., and Laepple, T.: Global patterns of declining temperature variability from the Last Glacial Maximum to the Holocene, Nature, 554, 356–359, 10.1038/nature25454, 2018.Schmidt, G. A.: Forward modeling of carbonate proxy data from planktonic
foraminifera using oxygen isotope tracers in a global ocean model,
Paleoceanography, 14, 482–497, 10.1029/1999PA900025, 1999.Shakun, J. D., Clark, P. U., He, F., Marcott, S. A., Mix, A. C., Liu, Z.,
Otto-Bliesner, B., Schmittner, A., and Bard, E.: Global warming preceded by
increasing carbon dioxide concentrations during the last deglaciation.,
Nature, 484, 49–54, 10.1038/nature10915,
2012.Simpson, G. L.: Modelling Palaeoecological Time Series Using Generalised
Additive Models, Front. Ecol. Evol., 6, 149,
10.3389/fevo.2018.00149, 2018.Smerdon, J. E.: Climate models as a test bed for climate reconstruction
methods: pseudoproxy experiments, Wires Clim. Change, 3, 63–77,
10.1002/wcc.149,
2012.Steiger, N. and Hakim, G.: Multi-timescale data assimilation for atmosphere–ocean state estimates, Clim. Past, 12, 1375–1388, 10.5194/cp-12-1375-2016, 2016.Thompson, D. M., Ault, T. R., Evans, M. N., Cole, J. E., and Emile-Geay, J.:
Comparison of observed and simulated tropical climate trends using a forward
model of coral δ18O, Geophys. Res. Lett., 38, L14706,
10.1029/2011gl048224, 2011.Tolwinski-Ward, S. E., Evans, M. N., Hughes, M. K., and Anchukaitis, K. J.: An
efficient forward model of the climate controls on interannual variation in
tree-ring width, Clim. Dynam., 36, 2419–2439,
10.1007/s00382-010-0945-5, 2011.Trachsel, M. and Telford, R. J.: All age–depth models are wrong, but are
getting better, Holocene, 27, 860–869, 10.1177/0959683616675939,
2017.Vardaro, M. F., Ruhl, H. A., and Smith, K. L.: Climate variation, carbon flux,
and bioturbation in the abyssal north pacific, Limnol. Oceanogr.,
54, 2081–2088, 10.4319/lo.2009.54.6.2081, 2009.
Von Storch, H., Zorita, E., Jones, J. M., Dimitriev, Y.,
González-Rouco, F., and Tett, S. F.: Recontructing past climate from
noisy data, Science, 306, 679–682, 10.1126/science.1096109, 2004.