CPClimate of the PastCPClim. Past1814-9332Copernicus PublicationsGöttingen, Germany10.5194/cp-12-1375-2016Multi-timescale data assimilation for atmosphere–ocean state estimatesSteigerNathannsteiger@ldeo.columbia.eduhttps://orcid.org/0000-0003-3817-9168HakimGregoryhttps://orcid.org/0000-0001-8486-9739Lamont-Doherty Earth Observatory, Columbia University, Palisades, NY, USADepartment of Atmospheric Sciences, University of Washington, Seattle, WA, USANathan Steiger (nsteiger@ldeo.columbia.edu)24June20161261375138822July201519August201521May201627May2016This work is licensed under a Creative Commons Attribution 3.0 Unported License. To view a copy of this license, visit http://creativecommons.org/licenses/by/3.0/This article is available from https://cp.copernicus.org/articles/12/1375/2016/cp-12-1375-2016.htmlThe full text article is available as a PDF file from https://cp.copernicus.org/articles/12/1375/2016/cp-12-1375-2016.pdf
Paleoclimate proxy data span seasonal to millennial timescales, and Earth's
climate system has both high- and low-frequency components. Yet it is
currently unclear how best to incorporate multiple timescales of proxy data
into a single reconstruction framework and to also capture both high- and
low-frequency components of reconstructed variables. Here we present a data
assimilation approach that can explicitly incorporate proxy data at arbitrary
timescales. The principal advantage of using such an approach is that it
allows much more proxy data to inform a climate reconstruction, though there
can be additional benefits. Through a series of offline
data-assimilation-based pseudoproxy experiments, we find that
atmosphere–ocean states are most skillfully reconstructed by incorporating
proxies across multiple timescales compared to using proxies at short
(annual) or long (∼ decadal) timescales alone. Additionally,
reconstructions that incorporate long-timescale pseudoproxies improve the
low-frequency components of the reconstructions relative to using only
high-resolution pseudoproxies. We argue that this is because time averaging
high-resolution observations improves their covariance relationship with the
slowly varying components of the coupled-climate system, which the data
assimilation algorithm can exploit. These results are consistent across the
climate models considered, despite the model variables having very different
spectral characteristics. Our results also suggest that it may be possible to
reconstruct features of the oceanic meridional overturning circulation based
on atmospheric surface temperature proxies, though here we find such
reconstructions lack spectral power over a broad range of frequencies.
Introduction
Paleoclimate proxies sample widely different timescales. High-resolution
paleoclimate proxies such as tree rings or corals have annual or seasonal
resolution, whereas lower resolution proxies such as sediment cores can
provide anywhere from annual- to millennial-scale information depending on
the core and its location . Additionally, high-resolution
proxies tend to be short, and are mostly limited to the past two millennia,
whereas some low-resolution proxies can reach the Cenozoic
e.g.,. In addition to the many timescales of proxies,
the climate system itself varies across a large range of timescales: from
atmospheric blocking to ocean overturning circulation to ice-age cycles.
Thus,
any faithful reconstruction of past climate must account for as many of these
timescales, captured by both proxies and climate models, as possible.
Few paleoclimate reconstruction methods have been created that specifically
incorporate multiple proxy timescales. Most reconstructions use either low-
or high-resolution proxies alone. If multiple scales of proxy data are used
together, researchers often resort to coarsening high-resolution proxies
e.g., or linearly interpolating low-resolution proxies
to a “higher resolution” e.g.,. One major reason for
this is that many traditional multivariate regression methods are not
constructed to easily calibrate in both low- and high-frequency domains;
, for example, present a such a modified method and discuss
these and related challenges. Additionally, including multiple timescales is
not entirely a methodological problem but partly a temporal sampling issue:
given that instrumental temperature data only span the past 150 years or so,
low-frequency reconstruction techniques have few degrees of freedom on which
to be calibrated and validated if the timescale is longer than about a
decade. Specific methods that have been developed with multiple timescales
in mind include the time-series reconstruction methods of and
. use a Bayesian hierarchical model
approach, while use an approach based on pairwise
comparisons that is particularly flexible and was used extensively, for
example, in a recent high-profile paper that reconstructed continental-scale
temperatures over the common era . Space–time
reconstruction methods include and
. developed a spectral analog
approach, where analogs are drawn from an instrumental temperature data
product, that they used to reconstruct European April–September temperatures
over the past 1400 years. developed a generalized
neural network approach and used it to reconstruct winter precipitation in
the Mediterranean over the past 300 years.
Data assimilation (DA) provides a flexible framework for combining
information from paleoclimate proxies with the dynamical constraints of
climate models. In principle, DA can provide reconstructions of any model
variable, from surface temperature to sea water salinity to atmospheric
geopotential height. Among DA techniques, we are unaware of any method
specifically designed for the challenges of paleoclimate reconstructions that
incorporates proxies across an arbitrary range of timescales. DA-based
reconstructions have so far used only a single uniform timescale
e.g., or have performed separate reconstructions at
different uniform timescales . Traditional DA adjoint
methods as applied in weather forecasting can and do include multiple timescales p. 181–184, but these timescales are very short
by comparison to the timescales involved in paleoclimate. Here we develop a
DA-based algorithm for space–time climate reconstructions that can assimilate
proxies at any time resolution. Because of the limited time span of
observational data sets, we explore the features and skill of this technique
within a synthetic, pseudoproxy framework. The ensuing pseudoproxy
experiments use an “offline” DA implementation, wherein prior ensembles
are drawn from a previously run climate model simulation and are not
integrated forward in time after each DA update step. This allows us to test
the algorithm over long time spans, perform carefully controlled experiments,
and unambiguously define errors.
Multiproxy reconstructions can potentially overcome some limitations of
single proxy reconstructions, such as filling in for the missing frequency
components of a particular proxy . However, the primary,
pragmatic benefit to incorporating proxies at multiple temporal resolutions
is that more information can inform the reconstruction. By comparison with
current weather observations, paleoclimate proxies are more expensive and
time-consuming to gather and their spatial distribution is far less
extensive. Therefore, including any additional unbiased information should
meaningfully improve the reconstructions. In addition, it is possible that
particular reconstruction methods could benefit from multi-scale proxy data.
Within a coupled atmosphere–ocean DA framework, suggest
that assimilating time-averaged observations of atmospheric variables may
improve present-day estimates of ocean circulation. They argue that these
improvements arise from the fact that time averaging high-frequency
observations improves the signal over noise in the covariance relationship
between the atmosphere and the slowly varying ocean overturning circulation.
We test this hypothesis within a paleoclimate context and assesses whether or
not atmosphere–ocean state estimates can be improved by including proxies
and climate states at multiple timescales. Therefore this test goes beyond
the benefit of simply being able to include more proxies in climate
reconstructions.
Assimilation technique
Data assimilation refers to a mathematical technique of optimally combining
observations (or within this context, proxy data) with prior information,
typically from a model. The model, in this case a climate model, provides an
initial, or prior, state estimate that one can update in a Bayesian sense
based on the observations and an estimate of the errors in both the
observations and the prior. The prior contains any climate model variables of
interest and the updated prior, called the posterior, is the best estimate of
the climate state given the observations and the error estimates. The basic
state update equations of DA e.g., are given by
xa=xb+K[y-H(xb)],
where K can be written as
K=cov(xb,H(xb))[cov(H(xb),H(xb))+R]-1,
and cov represents a covariance expectation. xb is the
prior (or “background”) estimate of the state vector and
xa is the posterior (or “analysis”) state vector.
Observations (or proxies) are contained in vector y. The observations
are estimated by the prior through H(xb), which
is, in general, a nonlinear vector-valued observation operator that maps
xb from the state space to the observation space. For
example, tree-ring width may be estimated from grid-point values of
temperature and moisture in the prior. Matrix K, the Kalman gain,
weights y-H(xb) (which is called the
innovation) and transforms it into state space. Matrix R is the
error covariance matrix for the observations. We note that
Eq. () assumes that xb and
H(xb) are Gaussian-distributed and that their
errors are unbiased. The DA update process involves computing
Eq. () to arrive at the posterior state; within the context
of the climate reconstruction problem, the posterior state is the
reconstructed state for a given time. Space–time reconstructions are
obtained by iteratively estimating the posterior state for each year or time
segment of the reconstruction.
From the first covariance term of Eq. (), we can interpret
K as “spreading” the information contained in the observations
through the covariance between the prior and the prior-estimated
observations. This implies that, other things being equal, larger values of
cov(xb,H(xb)) will weight the
innovation more heavily; thus, this new information not contained in the
prior has a bigger influence. One way to improve this covariance relationship
may be to use time-averaged observations, particularly if the model or
climate system has better covariance relationships at longer timescales.
For the update calculations we employ an ensemble square-root Kalman filter
with serial observation processing, applied to time averages (see
, for a detailed DA algorithm and fuller discussion of
DA terminology). We extend the technique of ,
, and by iteratively applying the
state-update equations across multiple timescales by leveraging the serial
observation processing approach to the Kalman filter .
The previous related approaches have only considered annual timescales or
less and cannot be trivially applied to proxies of arbitrary timescales: the
prior must be constructed so as to provide meaningful time averages and the
algorithm must be able to handle irregular proxy timescales.
The following general algorithm allows one to assimilate any collection of
observation or proxy data, including time averages with irregular duration:
Construct a prior (“background”) ensemble xb at
the highest temporal resolution of interest (e.g., monthly or annual), or a
collection of them with one for each time step (e.g., monthly or annual ensembles assigned to particular months or years).
Loop over observations and assimilate each at their own timescale:
Decompose the prior ensemble(s) that overlap in time with the observation y
into time averages (overbar) and deviations from this average (prime) via
xb=x‾b+xb′,
such that the time average of x‾b matches the timescale of y.
Estimate the observation via the “proxy system model” H(x‾b), and update the time-averaged
ensemble(s) x‾b⟶DAx‾a, with x‾a as
the posterior (or “analysis”) time-mean ensemble(s).
Add back the time deviations, xb=x‾a+xb′,
which can serve as the prior(s) for another observation.
Note that if y shares the same timescale as xb, then
the method is the same, but with xb′=0.
After all observations have been assimilated, the ensemble mean of xb
provides the best estimate of the state for each analysis time.
We now discuss an illustrative implementation of this general algorithm that
we employ for the experiments in this paper. Consider a paleoclimatic
situation where the observations are a collection of annual proxy data and
also proxy data representing irregularly averaged climatic information. Let
the prior ensembles be constructed, as we do in our experiments, by a random
sample of annually averaged climate states from a long climate simulation and
initially assigned to each year of a reconstruction (Fig. ).
Following the steps outlined above, proxies representing differently averaged
time intervals can be assimilated by averaging over the prior ensembles for
the time intervals defined by the proxy. For example, a proxy value
representing information over the years 1700–1720 would update the prior
ensembles averaged in time over that same interval. Annual proxies can simply
be assimilated by updating the ensembles for each year of available proxy
data (Fig. ). This approach proceeds by assimilating each
proxy over its full time extent, and after every proxy is assimilated, one is
left with an updated version of what one started with: a time sequence of
ensemble state estimates at annual resolution.
Schematic of the general reconstruction method using an offline
approach. Prior ensembles of m state vectors, χ, are assigned to each
of the n years. To retain some temporal coherency, the rows are composed of
time-coherent blocks drawn from a climate model simulation (arbitrarily
illustrated here as a 3-year block, or three consecutive annual states). The
method updates prior ensembles for specific years corresponding to annual
proxy data points, while for long-timescale proxies prior ensembles are
computed by time averaging across the rows corresponding to the years of a
proxy data point.
In the example above and all of the experiments shown here we use a
“no-cycling” or “offline” DA approach, where the prior ensembles are
drawn from existing climate model simulations. This approach has vast
computational benefits over a “cycling” or “online” approach, where one
must integrate an ensemble of climate model simulations forward in time after
each DA update step. Indeed, for the paleoclimate reconstruction problem, it
is infeasible to cycle an ensemble of tens to hundreds of CMIP5-class
coupled-climate models (as used here) for hundreds or thousands of years. Moreover,
in the offline case one may use hundreds to thousands of ensemble members
from multiple models and simulations, reducing the potential for model bias
and sampling error. It is also advantageous to use an offline approach when
the predictability time limit of the model is shorter than the timescale of
the observations: for example, if observations are only available at annual
resolution yet the model cannot skilfully forecast the climate state a year
into the future, then no useful information is gained by cycling the model.
recently compared online and offline approaches to
paleoclimate DA with a fully coupled Earth system model and found no
improvement with the online method, suggesting that the model was unable to
provide useful information at analysis times. Nevertheless, one way the
approach outlined here can generalize to the online approach is by cycling
on the shortest timescale (e.g., annual or seasonal) and updating longer
timescales at the end of the appropriate interval without cycling.
Also note that for the sake of simplicity in the illustrative example and
throughout the paper, we are assuming that an irregular, long-timescale
proxy is just an average of some climate variable over a given time interval.
Real proxies are nearly always more complex than this and would necessitate a
more sophisticated proxy system model (H(xb) in
Eqs. and ); however, the algorithm described
above is general and covers the case when such models are available.
An important detail of the algorithm outlined above is that when updating the
long-timescale states, the deviations from the time mean are not also
updated. For most applications of this algorithm, we presume that there would
be essentially two types of proxy timescales involved, annual and decadal
timescales. Thus, we assume here that the decadal proxies would not inform
annual climate variability. However, there is no formal or practical obstacle
that stops one from updating these deviations from the time mean
. If, for example, one were working with proxies from two
very similar timescales where there may be important shared covariance
information, the deviations could also be updated by appending them to the
mean states, x‾b, and then proceeding with the
algorithm.
One key point about the method outlined above is that if a reconstruction
uses the offline approach together with multiple timescales, then a random
sample of annually averaged climate states will not have meaningful
multi-year averages. Temporal consistency of the priors will need to be
ensured in order to have coherent long-timescale covariance relationships.
One way to account for this is to draw priors in random blocks of consecutive
years from the employed climate model simulation (see Fig. ).
The length of these blocks can be determined based on the needs of the
specific reconstruction problem (e.g., the length of the longest proxy timescale) and the length of available model simulations. If multiple long
simulations are available (they need not be from the same model), different
rows in Fig. could be different model simulations and the
block length could be the length of the reconstruction; this option avoids
any discontinuities in time that result from small block lengths.
The DA technique of state space augmentation (as discussed in, for example,
) has long been used for handling arbitrary
observation operators and can also be used for updating time-averaged
quantities or parameter estimation e.g.,. Such
an approach does have some important similarities to what we present here,
particularly the ability to update time-averaged information. However, we are
not aware of any paper that has worked out a multi-scale state augmentation
approach for the paleoclimate reconstruction problem. We think this could
represent a viable approach, but it remains untested for paleoclimate
reconstructions.
Experimental frameworkModels and variable characterizations
For the experiments presented here, we are interested in (1) how the
reconstruction methodology proposed in Sect. performs in
both the atmosphere and ocean, (2) how the differing timescales of the
atmosphere and ocean may be leveraged in the reconstruction process, and
(3) how these results vary with two different models having quite different
spectral characteristics in their coupled-climate systems. To this end we
choose two long preindustrial control simulations (part of the Coupled Model
Intercomparison Project Phase 5, available for download at
http://www.earthsystemgrid.org/), one from the climate model GFDL-CM3
(800 years in length) and the other from CCSM4 (1051 years in length). We
also choose two illustrative reconstruction variables, global-mean 2 m air
temperature and the Atlantic meridional overturning circulation (AMOC).
Figures and characterize the global-mean
temperature and an AMOC index for each simulation (defined here as the
maximum value of the overturning stream function in the North Atlantic
between 25 and 70∘ N and between depths of 500 and 2000 m),
respectively. In these reconstructions the state vector,
xb, only contains global latitude–longitude gridded values
of 2 m air temperature together with global-mean temperature and the AMOC
index as single-dimension appended state variables (rather than deriving them
from the state vector itself). H(xb) simply uses
the surface temperature values of the state vector at the proxy locations.
Note that even though these are only single-dimension variables, the DA
framework proposed here can trivially reconstruct spatial variables as well
. From Figs. and
we see that these two models display different spectral characteristics for
both global-mean temperature and the AMOC index.
Characterization of the global-mean 2 m air temperature variables
used in this paper. Panels (a) and (b) show the global-mean
temperature time series for the preindustrial control simulations of
GFDL-CM3 and CCSM4, respectively. Panels (c) and (d) show
their respective power spectra (GMT) with a best-fit red noise (RN) spectrum
(computed as in ) and an estimated 95 %
confidence interval.
Characterization of the Atlantic meridional overturning circulation
(AMOC) index variables used in this paper. Panels (a) and
(b) show the AMOC index time series (defined in the text) for the
preindustrial control simulations of GFDL-CM3 and CCSM4, respectively.
Panels (c) and (d) show their respective power spectra with
a best-fit red noise (RN) spectrum (computed as in
) and an estimated 95 % confidence
interval.
We next assess whether there are strong covariance relationships between the
observation variables and the reconstruction variables at different time
averages. Recall that the key covariance relationship in the DA update
equations is between the prior variables and the prior estimate of the
observations (Eq. ). A simple assessment of this is shown in
Fig. , which shows the correlation between the prior
variables and the surface temperature time series at the pseudoproxy grid
points for both climate simulations at a range of time averages. (Note that
the correlation of two time series is simply the covariance normalized by the
product of the standard deviations of the two time series.)
Figure indicates that there is increased covariance
information (or more locations with higher correlations) between surface
temperature and the prior variables at longer timescales. This information
is leveraged by the equations of DA to potentially improve the low-frequency
components of the reconstructed variables. An important point about computing
correlations at increasing time averages is that the number of degrees of
freedom in the time series is also reduced, potentially spuriously inflating
the correlations in Fig. . Accounting for these reduced
degrees of freedom by performing a test of statistical significance would
not, however, be particularly germane: the DA equations do not “know” about
95 % confidence intervals, just the covariance information. If, after
performing the reconstructions and computing several different skill metrics,
we see an increase in reconstruction skill, then we can infer that the
information was in fact useful for the reconstructions.
Panels (a) and (b) show the distribution of
correlation values between the global-mean 2 m air temperature (GMT) and the
2 m air surface temperatures (T2m) at the proxy locations for GFDL-CM3 and
CCSM4 at a range of time averages. Panels (c) and (d) show
similar correlation distributions but for the correlation between the
Atlantic meridional overturning circulation (AMOC) index and the 2 m air
surface temperatures at the proxy locations. The correlations are computed
for proxy grid points at a given time average, with the distribution of these
correlations shown in gray dots and also summarized by box plots, where the
blue boxes enclose the 25th and 75th percentiles and the whiskers extend to
±1.5 times this interquartile range.
Pseudoproxy construction
The pseudoproxy experiments employed here follow the general framework of
many previous studies (see , for a summary and review) but
with some important modifications. Generally, after one or more climate model
simulations are chosen to represent nature, a pseudoproxy network is chosen
that mimics real-world proxy availability, similar to the network chosen here
and shown in Fig. a; this particular network is composed of
a spatially thinned version of the proxy collection of
(thinned over Asia and North America, where the proxy density is high) and all
of the proxy locations in and .
Pseudoproxies are typically generated by adding random white noise to the
chosen network of climate model temperature series. We note that the choice
of white noise (as opposed to “red noise”, for instance) is a simplification
of the “real” noise in proxies. However, we consider this to be a reasonable
choice because the purpose of the present work is primarily to illustrate a
new reconstruction method. The added noise is usually assumed to be the same
value for all proxy locations, with a common signal-to-noise ratio (SNR)
being 0.5 (where SNR≡var(X)/var(N),
and where X is a grid-point temperature series drawn from the true state, N is an additive noise series, and var is the variance.). Following
recent work by , we instead randomly draw SNR values from a
distribution characteristic of real proxy networks (Fig. b).
This distribution is a shifted gamma distribution (shape
parameter = 1.667, scale parameter = 0.18, shifted by 0.15) with a
mean SNR of 0.45 and is modeled after Fig. 3 from .
(a) Pseudoproxy locations used in this study (n=274),
drawn from the predominantly high-resolution (annual) proxy collection of
and all the comparatively low-resolution (decadal to
centennial) proxy locations in and .
(b) The signal-to-noise ratio (SNR) distribution for the
pseudoproxies, based on a real-world estimate of . For a given
Monte Carlo experiment, the SNR for each pseudoproxy was randomly drawn from
this distribution.
Also, in contrast to nearly all pseudoproxy experiments, we use pseudoproxies
at two different timescales for each model. Importantly, we use the same SNR
distribution for both timescales and add the noise to the time series
after averaging. Within the DA framework, the additive error for
each proxy is accounted for in the entries of the diagonal matrix
R. The SNR equation above is related to R in that each
of these entries is equal to var(N) for a given proxy. The process
of adding the noise after averaging ensures that R is
statistically identical for each reconstruction. This process isolates the
role of the covariance relationships in Eq. (). By drawing from
the same SNR distribution for all pseudoproxy timescales we are also
assuming that the distribution is an appropriate characterization of the
error in long-timescale proxies; we assume this for simplicity and also
because we are not aware of a systematic assessment of SNR values for
low-resolution proxies as have done for annual-resolution
proxies.
We also note that an important idealization of the present pseudoproxy
experiments, which we share with all pseudoproxy experiments heretofore
published, is that we use a perfect model approximation where the
pseudoproxies from one model simulation are used to reconstruct that same
simulation; for example, pseudoproxies from the CCSM4 simulation are used to
reconstruct the CCSM4 simulation. In a real DA-based reconstruction the
climate model will never be a perfect description of the real climate system
from which the assimilated observations are derived. Since the purpose of the
present work is to illustrate a new algorithm, we have not considered this
additional layer of complexity. This additional aspect could only be fully
assessed within a study of real proxy climate reconstructions; using one
simulation to reconstruct another can assess inter-model differences, but it
is unclear how these results would relate to model–nature differences.
Pseudoproxy experiments
The primary results of this paper are presented in a series of 12 experiments
using only atmospheric surface temperature pseudoproxies to reconstruct the
global-mean temperature and AMOC index of the two climate model simulations
discussed previously. For each variable, and each model, three experiments
are performed: (1) short (annual) pseudoproxies only, (2) long (5- or 20-year
time averages) pseudoproxies only, and (3) both short and long time-averaged
pseudoproxies. We have chosen the long timescale for the CCSM4 simulation to
be 20 years, and we note that an alternative choice of one to several decades
gives similar results (not shown). The situation is more complex with the
GFDL-CM3 simulation because of the presence of an approximate 22-year
periodic signal in the AMOC (Fig. a and c). A choice of 20
years for GFDL-CM3 would effectively undersample the AMOC variability, and so
we have chosen a long timescale of 5 years for GFDL-CM3. Unfortunately, a
long timescale of 5 years for CCSM4 shows little difference in the results
over the annual timescale reconstructions (not shown), as would be suggested
by the small difference in correlation (covariance) between 1 and 5 years
(Fig. b, d).
Both the short-only and long-only reconstructions use 100 pseudoproxies
randomly drawn from the network of 274 proxy locations shown in
Fig. a. For the mixed-resolution reconstructions, 100
pseudoproxies are randomly drawn from the network for each timescale, giving
a total of 200. This is an approximation of the real-world setting, where one
usually has proxies at multiple timescales and would like to use all of
them. Following the algorithm outlined in Sect. , for the
multi-scale reconstructions, we assimilate the long-timescale pseudoproxies
first, followed by the annual timescale pseudoproxies; we also performed
these reconstructions by swapping which timescale was assimilated first and
found statistically identical results (not shown), as would be expected from
the linearity of this approach. For these mixed-resolution reconstructions,
we have also ensured that there is no overlap between locations associated
with the two timescales.
We have reconstructed the first 400 years of each simulation while drawing
the priors from the following 400 years of the simulations. Each year had a
prior size of 1000 (e.g., from Fig. , m=1000), while the
blocks were randomly drawn in 20-year continuous segments. This uniform block
length was chosen because it was the longest timescale of the pseudoproxies
and because the pseudoproxies were constructed over regular long intervals
and thus discontinuities at block edges were not a concern (see
Fig. and the discussion in Sect. ). Because
the prior ensemble size was 1000, we did not employ covariance localization,
a common DA practice for controlling sampling error. Each of the 12
reconstructions is repeated 100 times in a Monte Carlo fashion where new
proxy networks and SNR values are randomly chosen each iteration; the new
pseudoproxy networks are randomly drawn from the network shown in
Fig. a and the SNR values are randomly drawn from the
distribution shown in Fig. b. All the reconstruction
figures show the mean of 100 of these Monte Carlo reconstruction iterations
along with error bars indicating ±2σ of the “grand ensemble” of
analysis ensembles for all the Monte Carlo iterations (with an ensemble size
of 1000 and 100 iterations, the grand ensemble has 1×105 members).
Global-mean temperature reconstructions (mean of 100 Monte Carlo
iterations, with error bars indicating ±2σ of the iterations and
analysis ensembles) for the three types of experiments discussed in the text
and for each climate model simulation. Black lines indicate the true time
series, while red lines indicate the reconstructed time series for only short-timescale (annual) pseudoproxies, only long timescale (5 or 20 years)
pseudoproxies, and both long- and short-timescale pseudoproxies. Skill
metrics of the reconstructions, correlation (r), coefficient of efficiency
(CE), and mean continuous ranked probability score (CRPS) are shown at the
top of each subpanel.
Cross spectra of the reconstructed global-mean temperature time
series with the true global-mean temperature time series, for the
reconstructions shown in Fig. . For reference, the dashed
gray line indicates the cross spectra of the true time series with itself, or
equivalently its own power spectrum.
Reconstruction results
The reconstructions of global-mean temperature are shown in
Fig. along with their associated ±2σ error
estimates. In Fig. , panels (a) and (b) show the
reconstructions with the annual pseudoproxies, (c) and (d) show the
reconstructions with the long-timescale proxies, and (e) and (f) show the
reconstructions for both timescales. Skill metrics, computed at annual
resolution, are shown for each reconstruction: correlation (r), coefficient
of efficiency (CE), and mean continuous ranked probability score (CRPS). The
coefficient of efficiency for a data series comparison of length N is
defined as
CE=1-∑i=1N(xi-x^i)2∑i=1N(xi-x‾)2,
where x is the “true” time series, x‾ is
the true time-series mean, and x^ is the reconstructed time series.
The metric CE has the range -∞<CE≤1, where
CE=1 corresponds to a perfect match and CE<0 generally
indicates no reconstruction skill or a bias in the reconstruction. The CRPS
is a “strictly proper” scoring metric that accounts for the skill of both
the mean and the spread of an ensemble forecast or state estimate
. The CRPS measures the area of the squared
difference between the cumulative distribution functions of the posterior
ensemble state estimate and the true state (a Heaviside function centered on
the true value). A smaller area would indicate a more skillful
reconstruction, so smaller values of CRPS are better. All CRPS values shown
in the figures are temporal means of CRPS over the reconstruction interval.
Comparing the reconstructions in Fig. we see that the
bottom panels with both timescales have the best skill and the smallest
error bars, indicating greater confidence. For GFDL-CM3, there is a 25 %
reduction in the mean standard deviation over the annual-only experiment and
a 17 % reduction over the long timescale only; for CCSM4, there is a 2 %
reduction in the mean standard deviation over the annual timescale only and
a 28 % reduction over the long timescale only.
AMOC index reconstructions (mean of 100 Monte Carlo iterations, with
error bars indicating ±2σ of the iterations and analysis ensembles)
for the three types of experiments discussed in the text and for each climate
model simulation. Black lines indicate the true time series, while red lines
indicate the reconstructed time series for only short-timescale (annual)
pseudoproxies, only long-timescale (5 or 20 years) pseudoproxies, and both
long- and short-timescale pseudoproxies. Skill metrics of the
reconstructions, correlation (r), coefficient of efficiency (CE), and mean
continuous ranked probability score (CRPS) are shown at the top of each
subpanel.
Note that the long-timescale reconstructions shown in
Fig. c and d have sharp edges at 5- (for GFDL-CM3) or 20-year (for CCSM4) intervals. This is due to the simplified experimental design
we have employed where all the long-timescale pseudoproxies are averages
over a given 5- or 20-year period. As discussed in Sect. ,
this experimental design is only a single illustrative example of the general
algorithm. The data from real proxies are not always apportioned into
specific time frames but can be scattered irregularly in time (e.g., fossil
coral records). Using many irregular proxies will act to smooth the long-timescale reconstructions. As long as the timescales can be estimated and
an appropriate proxy system model is used, the algorithm of
Sect. can handle any real proxy data. While not dealt with
explicitly here, real long-timescale (low-resolution) proxies also have
dating uncertainty, which will also tend to smooth the reconstructions. The
algorithm can account for dating uncertainty through the Monte Carlo
framework by repeating the reconstructions many times and sampling from an
age model for a given proxy.
One assessment of skill as a function of timescale is to compute the cross
spectrum of the reconstructed time series with the true time series
(Fig. ). The cross spectra in this case reveal the
relationship between the two time series as a function of frequency or
period. As a point of reference, the dashed gray lines in this figure
indicate the cross spectra of the true time series with itself, which is the
same as its own power spectrum.
Following a common technique to
reduce noise in the cross spectra, they are computed using Welch's averaged
periodogram method, which samples segments of the time series and averages
the power spectra of these samples to arrive at the cross power spectral
densities. As a result, the gray line spectra in Figs. and
should not be expected to precisely match up with
Figs. and
Considering
Fig. b we see that the annual-only reconstruction does a
better job of matching the power at short periods than the 20-year-only
reconstruction; however, the 20-year-only reconstruction performs better at
longer periods. The mixed timescale reconstruction, 20+1, does better or
just as well as the single-timescale reconstructions at both short and long
periods. This same general result holds for Fig. a, though
it is more difficult to see because of the much larger power at longer
periods in the GFDL-CM3 simulation.
Figure shows three timescale reconstructions of the
AMOC index for the two model simulations, similar to
Fig. . In these AMOC index reconstructions, we see the
same general patterns as with the global-mean temperature reconstructions,
where the multi-scale reconstructions provide the most skill (r, CE, CRPS)
as well as the smallest error bars. However, while the multi-scale
improvements in skill for the AMOC reconstructions are significant, these are
generally smaller than for the global mean temperature reconstructions. For
GFDL-CM3 there is a 3 % reduction in the mean standard deviation of the
error bars over the annual-only experiment and a 1 % reduction over the
long timescale only; for CCSM4, there is a 2 % reduction in the mean
standard deviation over the annual timescale only and a 4 % reduction over
the long timescale only. Figure shows the corresponding
cross spectra for the reconstructions shown in Fig. .
Given that the pseudoproxies are of surface air temperature, it is not
surprising that the absolute skill values of the AMOC reconstructions are
reduced relative to the reconstructions of global-mean temperature. Though it
is striking how much power is lost in the reconstructions
(Fig. ) considering that these are “perfect model”
experiments. This result is most likely due to the fact that the covariances
between surface temperatures and the AMOC index are quite small when
compared with the covariances for temperature (Fig. ); note
that even though the CCSM4 mean correlation value for temperature,
(Fig. b), is comparable to that for the AMOC,
(Fig. d), the absolute correlation values are much larger,
including a cluster of positive correlation values between 0.6 and 0.4. Thus
surface temperatures at the global proxy locations are relatively
uninformative about the AMOC.
Cross spectra of the reconstructed AMOC index time series with the
true AMOC index time series, for the reconstructions shown in
Fig. . For reference, the dashed gray line indicates the
cross spectra of the true time series with itself, or equivalently its own
power spectrum.
An additional result from Fig. is the improved
low-frequency components of the AMOC reconstructions when time-averaged
surface temperature pseudoproxies are used. We argue that this result follows
from the fact that the annual observations of atmospheric surface temperature
are essentially noise to the slowly varying ocean. One may improve the
information content relevant to the ocean by averaging over the atmospheric
noise. This interpretation may also be seen in Fig. , where
the correlation (covariance) information between the atmosphere and the ocean
is particularly low at annual averages but improves at longer time averages
(as also seen in ). To further test this idea we
performed mixed-resolution experiments in which the 100 pseudoproxies for
each timescale were taken from the same locations. Therefore, here the long-timescale proxies are just the time-averaged versions of the annual proxies. We
found essentially identical results with those shown in
Figs. –: all CRPS values were
identical to three decimal places, r and CE were either identical to two
decimal places or had changes of only ±0.01, and there were no
discernible differences in the cross spectra.
We note that all the cross spectra of the reconstructions shown in
Figs. and show a decrease in power
relative to the true state, though this need not always be the case. In
additional experiments we performed using global ocean heat content, we found
that this reconstructed variable tended to have more power than the true
state and was thus higher than the respective dashed gray lines (not shown).
Therefore the reduced power relative to the true state in
Figs. and should not be interpreted as
saying something general about the nature of DA-based reconstructions or the
particular approach employed here.
As an approximation of a real reconstruction scenario, the experiments shown
in Figs. and with two timescales
use twice as many pseudoproxies as the single-timescale experiments (200 vs.
100). Therefore the improved skill might simply be a consequence of having
more observation information. We tested this idea by repeating all the
experiments shown here but instead increasing the number of observations to
200 for each experiment: the single-timescale reconstructions used 200
randomly drawn pseudoproxies and the multi-scale reconstructions used 100
randomly drawn pseudoproxies each for the two timescales (the same as in the
previous multi-scale reconstructions). Figure is a
characteristic example of the results of these additional tests.
Figure a shows the reconstructions of the AMOC
index with the CCSM4 model output and Fig. b shows
the respective cross spectra. In (a), the skill is best for the multi-scale
reconstructions and in (b) the cross spectra show the same general result of
improved low-frequency power for the time-averaged pseudoproxies. However,
the cross spectra for the 20+1 reconstruction are not always closest to the
true spectrum, suggesting that the number of pseudoproxies does play a role
in improving the spectrum of the reconstructions. Indeed, it should be the
case that, as long as the proxies are unbiased, adding more of them will
improve a DA-based reconstruction.
AMOC index reconstructions (mean of 100 Monte Carlo iterations, with
error bars indicating ±2σ of the iterations and analysis ensembles)
and corresponding cross spectra similar to those shown in
Figs. b, d, f and b but for the case
where each experiment uses 200 pseudoproxies: the single-timescale
reconstructions use 200 pseudoproxies each, while the multi-timescale
reconstructions use 100 pseudoproxies for the short timescale and 100
pseudoproxies for the long timescale.
Conclusions
This paper presents a data assimilation approach for paleoclimate
reconstructions that can explicitly incorporate proxy data on arbitrary
timescales. This approach generalizes previous data assimilation techniques
in the sense that many scales of both proxies and climate states can be
included explicitly in a single reconstruction framework. The primary
interest in such a reconstruction technique is that it allows for the
inclusion of much more proxy data in climate reconstructions. Given the
spatially sparse and noisy nature of proxies, more information will tend to
improve the quality of the reconstructions. Besides this benefit, using
multi-scale proxy data may be particularly useful given the many inherent
timescales of the climate system, such as the fast timescales of the
atmosphere and the slow timescales of the ocean. We performed three types of
realistic atmosphere–ocean pseudoproxy reconstructions to assess the impact
of using observations at multiple timescales: (1) short (annual)
pseudoproxies only, (2) long (∼ decadal) pseudoproxies only, and
(3) both short and long time-averaged pseudoproxies. We found for both
global-mean temperature and an index of the AMOC that the reconstructions
that incorporated proxies across both short and long timescales were more
skillful than reconstructions that used short or long timescales alone
(Figs. and ). This result holds even
when the number of pseudoproxies for the single-timescale reconstructions are
doubled (Fig. a). Multi-scale reconstructions would
be expected to perform better than single-scale reconstructions because they
can include information at multiple timescales and because the prior can be
better conditioned as it is used
from one timescale to the next.
We found that reconstructions incorporating long-timescale pseudoproxies
improve the low-frequency components of the reconstructions over
reconstructions that only use annual timescale pseudoproxies (Figs. , , and
b). This result may at first seem surprising because
the annual pseudoproxies should contain the low-frequency information. It is
helpful to recall that the data assimilation algorithm outlined here proceeds
by sequentially finding the optimal state at each time segment of interest
given the prior, the observations, and their respective errors. This state
update critically relies on the covariance between the prior and the model
estimate of the observations (Eq. ). If, for example, surface
temperature observations do not covary well with the AMOC at annual
resolution, then the posterior AMOC estimate will be little changed compared
to the prior . But if the time average of surface
temperatures has a large covariance with the AMOC, the posterior will be more
influenced by the observations. This result is not controlled by the noise
added to the pseudoproxies because, as noted in Sect. , we
ensured that R from Eq. () remains fixed for both
timescales.
These results indicate that DA-based atmosphere–ocean state
estimates may be improved by including proxies and climate states from
multiple timescales. The general results outlined above are consistent
across the employed climate models. These results also show, as suggested by
, that given a representative prior ensemble,
features of the AMOC may be
reconstructed using observations of surface variables. However, the
reconstructions lack spectral power across all frequencies, which we
attribute to the relatively small covariances between surface temperatures
and the AMOC in the models we
employed (Fig. c and d). Therefore observations of ocean
quantities such as salinity or indirect measures of ocean circulation may be
better suited to reconstructing the AMOC.
Data availability
The GFDL-CM3 simulation is publicly available at
http://nomads.gfdl.noaa.gov/.
The CCSM4 simulation is publicly available at
https://www.earthsystemgrid.org/home.html.
Acknowledgements
We acknowledge the Program for Climate Model Diagnosis and Intercomparison
and the WCRP's Working Group on Coupled Modeling for their roles in making
available the CMIP5 data set. Support of the CMIP5 data set is provided by
the US Department of Energy (DOE) Office of Science. This work was supported
by the National Science Foundation (grant AGS-1304263) and the National
Oceanic and Atmospheric Administration (grant NA14OAR4310176). We thank James Annan and three anonymous reviewers for their very helpful comments on previous versions of the paper.
Edited by: H. Goosse
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