CPClimate of the PastCPClim. Past1814-9332Copernicus PublicationsGöttingen, Germany10.5194/cp-13-545-2017Assimilation of pseudo-tree-ring-width observations into an atmospheric general circulation modelAcevedoWalterFallahBijanbijan.fallah@met.fu-berlin.dehttps://orcid.org/0000-0003-3302-2030ReichSebastianCubaschUlrichInstitut für Meteorologie, Freie Universität Berlin, Carl-Heinrich-Becker-Weg 6–10, 12165 Berlin, GermanyInstitut für Mathematik, Universität Potsdam, Karl-Liebknecht-Strasse 24–25, 14476 Potsdam, GermanyBijan Fallah (bijan.fallah@met.fu-berlin.de)31May201713554555714September201626September201627March201730March2017This 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/13/545/2017/cp-13-545-2017.htmlThe full text article is available as a PDF file from https://cp.copernicus.org/articles/13/545/2017/cp-13-545-2017.pdf
Paleoclimate data assimilation (DA) is a promising technique to
systematically combine the information from climate model simulations and
proxy records. Here, we investigate the assimilation of tree-ring-width (TRW)
chronologies into an atmospheric global climate model using ensemble Kalman
filter (EnKF) techniques and a process-based tree-growth forward model as an
observation operator. Our results, within a perfect-model experiment setting,
indicate that the “online DA” approach did not outperform the
“off-line” one, despite its
considerable additional implementation complexity. On the other hand, it was
observed that the nonlinear response of tree growth to surface temperature
and soil moisture does deteriorate the operation of the time-averaged EnKF
methodology. Moreover, for the first time we show that this skill loss
appears significantly sensitive to the structure of the growth rate function,
used to represent the principle of limiting factors (PLF) within the forward
model. In general, our experiments showed that the error reduction achieved
by assimilating pseudo-TRW chronologies is modulated by the magnitude of the
yearly internal variability in the model. This result might help the
dendrochronology community to optimize their sampling efforts.
Introduction
The low-frequency temporal variability in the
climate system cannot be estimated from the available time span of
instrumental climate records. Accordingly, paleoclimate reconstruction must
necessarily rely on the use of the paleoclimate proxy records. These natural
archives exhibit several problematic features, e.g., low time resolution,
sparse and irregular spatial distribution, complex nonlinear response to
climate, and high noise levels. Therefore the proper extraction of the
climate signal contained therein can often remain opaque .
To date, many different ideas have been proposed in order to link proxy
records to the paleoclimate conditions where they were created, e.g.,
data-driven statistical techniques, climate model hindcasts and Bayesian
probabilistic methods (see , for a recent review).
Among this plethora of approaches, data assimilation (DA) methodologies are
today particularly appealing as they deliver estimates of paleoclimate
quantities by systematically combining the information of paleoclimate
records with the dynamical consistence of climate simulations
.
So far, several very diverse paleo-DA schemes have been investigated,
including pattern nudging , forcing singular vectors
, 4D-Var ,
particle filters
and
ensemble Kalman filter techniques (EnKF;
; see
for further references).
An important difference between paleo-DA and traditional meteorological DA is
that the assimilation period might be very long compared to the timescales of
the dynamical model. Under these conditions, the randomizing action of the
chaotic model dynamics becomes dominant and consequently the forecast appears
completely de-correlated from the previous analysis state. This phenomenon,
currently referred to as an “off-line regime”, has been observed in several
paleo-DA studies .
Furthermore, some recent studies have assumed the presence of the off-line
condition and accordingly have removed the re-initialization step after
assimilation altogether . These types of
DA methodologies will be referred to in this paper as “off-line DA” techniques, in order to contrast them with traditional “online DA” techniques, where the state of the model is updated after the assimilation
of observations. Note that despite their lack of accumulation of
observational information over time, off-line DA methodologies have already
been shown to be more robust than traditional climate field reconstruction
(CFR) techniques based on orthogonal empirical functions and stationarity
assumptions .
A typical assumption in most of the paleo-DA studies so far conducted is that
the climate–proxy relation is linear. Nonetheless, currently it is widely
recognized that climate proxies are the result of complex recording
processes, which can be of a physical, chemical and biological nature. More
realistic methodologies have been recently sculpted by the paleoclimate
community in order to investigate the climate–proxy relationship, considering
the distinct processes whereby the climate signal is recorded in proxy
archives. Proxy forward modeling appears to be
one of the most promising methodologies in this area. In a proxy forward
model the climate forcing is used as input data for producing the artificial
proxy records which can be directly compared with the actual ones. One
application of proxy forward models is the prediction of the evolution of
proxy archives . They can also be applied within climate
reconstruction strategies by using probabilistic inversion methods like
Bayesian hierarchical modeling , Markov Chain Monte
Carlo (MCMC) and DA .
Several recent studies have investigated the applicability of process-based
forward models in a paleo-DA setting
. (AC15,
hereafter) utilized the process-based tree-ring-width (TRW) forward model Vaganov–Shashkin Lite
(VSL) and an online EnKF scheme to assimilate
pseudo-TRW records into a chaotic two-scale dynamical system. They found that
the nonlinearities of the forward model may deteriorate the performance of
the EnKF. Furthermore, they observed that this loss of skill may be
ameliorated by means of a fuzzy logic (FL)-based extension of the VSL model.
compared the off-line and online implementations of a
“degenerate particle filter” applied to a low-resolution Earth system
model. They found similar skill for both methods on the continental and
hemispheric scales. Nonetheless, they concluded that in the off-line method
the temporal consistency of the model is lost. On the other hand,
used three different nonlinear proxy forward models
(including VSL) and an off-line EnKF scheme to assimilate TRW, coral and ice
core records into two different isotope-enabled atmospheric general
circulation model (AGCMs). They demonstrated that the linear–univariate models
for tree-ring width may not capture the AGCM's climate, especially for
regions where the tree's growth is dominated by moisture.
This paper follows the rationale of AC15 but within a more realistic
scenario, where an AGCM is used as a dynamical system and the observational
network resembles the currently available TRW chronologies. The purpose of
this study is then to contribute to the present knowledge of paleo-DA
techniques by addressing the following two questions:
Does the off-line regime naturally appear for the assimilation of
TRW records into an AGCM?
Is the FL-based extension of the VSL model still useful to improve the
performance of a time-averaged EnKF technique when a climate model is used?
This study is structured as follows. In Sect. we describe
the DA technique, the TRW forward model and the climate model as well as the
experimental setting used. Our numerical results are shown in
Sect. , followed by a discussion in Sect. .
Materials and methodsData assimilation basics
In this paper, the term DA designates the process of estimating the state of a
system using observations and the physical laws governing the evolution of
the system as represented in a numerical model . In a
typical DA scheme, a dynamical model is integrated in time until observations
become available. Afterwards, the predicted state, also known as forecast, is
“updated” using the observational information in order to obtain a
corrected state, also known as the analysis. Finally, the model is
re-initialized from the analysis state and propagated in time until the next
assimilation time, completing the “analysis” cycle. DA methods
have evolved from very empirical approaches, such as Newtonian relaxation, to
probabilistic ones that attempt to estimate the probability density function
(PDF) of the model state conditional on the observations (see
, and , for reviews).
Among the currently available DA techniques, EnKF
occupies an outstanding position for several reasons. It offers an
appealing trade-off between accuracy and computational expenses. It works
robustly for sparse observational networks and a moderate number of ensemble
members . Furthermore, EnKF's implementation does not
require any modification of the model's code and uncertainty estimates can be
directly obtained from the ensemble spread . The main
disadvantage of EnKF, within a paleoclimate setting, is its inability to
handle non-Gaussian PDFs, which can easily arise from the nonlinearities of
climate models and observation operators. Recently, there have been several
developments in the field of nonlinear DA for high-dimension systems
; however, at the present fully non-Gaussian DA
techniques are still prohibitively expensive to run for general circulation
climate models.
Kalman filter
Within the Kalman filter (KF) , the PDF of forecast state
p(x) is assumed to be given by a Gaussian function with mean
xf and covariance Pf:
p(x)∝exp-12x-xfTPf-1x-xf.
The observations y(tj) are also assumed to have Gaussian errors
and therefore the conditional probability of the observation vector y
given the state x is
py∣x∝exp-12y-H^xfTR-1y-H^xf,
where H^ is the observation operator and R is
the observation error covariance matrix. Following the Bayes
theorem, the conditional probability of the state given the observations,
i.e., the analysis PDF, is
px∣y∝exp-12x-xfTPf-1x-xf-12y-H^xfTR-1y-H^xf.
Finally, assuming that H^ is a linear function,
px∣y is also a Gaussian function whose
mean and covariance can be calculated by the Kalman update
equations :
xa=xf+Ky-H^xf,Pa=I-KH^Pf,
where the Kalman gain matrix K is given by:
K=PfH^†H^PfH^†+R-1.
Ensemble Kalman filter
For realistic geophysical models, the dimensionality of the model state can
be very high and then the calculation and storage of the covariance matrices
can be prohibitively expensive. A solution to this problem is provided by the
EnKF , which uses an ensemble of model states
(X(t)=x1,…,xm) to
approximate KF equations.
Following this approach, the mean and covariance of the forecast take the following form:
〈Xf〉=1m∑i=1mxif,Pf=1m-1Xf′Xf′T.
Here Xf′∈Rn×m denotes the forecast ensemble
deviation matrix:
Xf′=Xf-〈Xf〉eT,
where e=(1,…,1)∈Rm. An analysis ensemble whose
covariance satisfies Eq. () can be generated in
different ways, which can be classified into two main families: stochastic
and deterministic filters .
In the stochastic approach an observational ensemble Y is
generated by adding realizations of the observational noise to the
observation vector y. The analysis ensemble is then created by the
following update equation:
Xa=Xf+KY-H^Xf.
In the deterministic approach, instead of creating an ensemble of
observations, the analysis mean and deviations are calculated using update
formulae which do not involve random numbers (see for further references).
A practical problem of EnKF schemes is that due to the limited ensemble
size, the forecast uncertainty is usually underestimated. This leads to an
excessive confidence in the forecast, and after several assimilation cycles
the observations may be completely ignored. This situation is normally
avoided by means of an ad hoc procedure known as “covariance inflation”,
where the forecast covariance matrix is multiplied by a constant greater than 1. Another undesired consequence of the limited ensemble size is that the
ensemble state at any grid point will present non-negligible spurious
correlations with observations located far apart in space. This difficulty is
solved using another ad hoc procedure known as “covariance localization”.
Here, we utilize the R localization , where the
entries of the observation error covariance matrix are multiplied by a
function that increases exponentially with distance.
Time-averaged ensemble Kalman filter
The EnKF algorithm was initially designed to estimate the instantaneous state
of a model given instantaneous observations. As a consequence, EnKF cannot be
directly applied to paleoclimate data given that the observational
information present in proxy records is typically the average of a function
of the state over long time periods. A solution to this conflict is provided
by the time-averaged EnKF , where the instantaneous forecast
is decomposed into its time-averaged part and the anomalies around it.
Afterwards, the original EnKF update formula is used to assimilate the
time-averaged observations into the time-averaged forecast, obtaining the
time-averaged analysis. Finally, the instantaneous analysis is form by adding
the unaltered time-averaged forecast anomalies to the time-averaged analysis.
This approach is based on the assumption that the observations can only
contain time-averaged information . If the time-averaged period is considerably longer than the timescales of the dynamical model,
which may easily be the case for paleoclimate records, after assimilation the
ensemble spread can reach climatological levels, leading to a complete lack
of estimation skill for the forecast quantities. This behavior, currently
referred to as the off-line regime, was first observed for the time-averaged
EnKF applied to a quasi-geostrophic atmospheric jet model
. Afterwards, several studies used the
simplified off-line time-averaged EnKF approach with global climate models
assuming the presence of the off-line
regime. However, to our knowledge, there had not been numerical evidence of
the onset of off-line conditions for a full time-averaged EnKF algorithm
applied to an AGCM. As mentioned in the introduction, filling this knowledge
gap is one the objectives of this paper.
TRW forward model
The VSL model for TRW chronologies offers an intermediate-complexity approach
between ecophysiological and completely data-driven models
, where the climate-driven component of
tree-ring growth is parameterized by way of a simple representation of the
principle of limiting factors (PLF) . The biological
concept of PLF states that the pace at which a plant develops is controlled
by the single basic growth resource, typically either energy or water, that
is in shortest supply. Within VSL the limiting factors considered are
near-surface air temperature (T) and soil moisture (M). These variables
influence tree-ring growth via the “growth response” functions
gT=ΨT-TLTU-TL
and
gM=ΨM-MLMU-ML,
where Ψ is the piece-wise linear “standard ramp” function
Ψ(u)=0if0⩾uuif0<u⩽11ifu>1,TL and ML denote minimum thresholds for temperature
and moisture below which there is no growth, and TU and
MU are upper thresholds above which tree growth is optimal.
Afterwards, the growth rate GMIN is determined by the smallest
growth response, i.e.,
GMIN=mingT,gM.
Finally, the yearly TRW values W are obtained by integrating in time the
growth rate modulated by the relative insolation I:
Wn=∫tn-τtnGMIN(t)I(t)dt.
Here n stands for the number of the year, tn is the time at the end of the year n and τ is the
length of the year (12 months).
Regarding DA, VSL presents two challenging nonlinear aspects:
(i) a “thresholded response” , arising from the
insensitivity of trees to climate variability during dormancy and optimal
growth, and (ii) “abrupt shifting of recorded variable”
, coming from the structure of
Eq. (). There, the use of the minimum function
implies that tree growth at a particular time can be limited by either
temperature or moisture. Accordingly, transitions between growth limitation
regimes necessarily happen in a sudden manner.
VSL from the fuzzy logic viewpoint
The term fuzzy logic was coined by and refers to a
mathematical theory which has been very successful in modeling complex
systems involving imprecise data and vague knowledge of the underlying
mechanisms. Since its introduction, FL has greatly influenced
many disciplines, most notably control theory . Within the
environmental sciences, FL has been applied in ecological and hydrological
modeling . Regarding climate proxy
forward modeling, AC15 recently showed that the VSL model can be completely
embedded into the framework of FL. Within this reinterpretation, the growth
response function gT (gM) corresponds to the
membership function to the set ST (SM) of optimal
temperature (moisture) conditions for tree growth. Temperature (moisture)
values lying below TL (ML) present null values for
gT (gM) and accordingly do not belong to
ST (SM). On the other hand, temperature (moisture)
values lying above TU (MU) lead to gT
(gM) values equal to 1, meaning they belong completely to
ST (SM). All the other temperature (moisture)
conditions present growth responses between 0 and 1, and consequently they
are considered to belong partially to ST (SM). This
idea of partial membership is the basis of fuzzy logic and the sets defined
this way are called fuzzy sets. Furthermore, the intersection of the fuzzy
sets ST and SM is again a fuzzy set
ST∧M, whose membership function can be
calculated by evaluating the minimum between GT and
GM:
gT∧M=mingT,gM.
Equation () is completely equivalent to the
Eq. (). Then VSL's growth rate function can be interpreted as the
membership function for the fuzzy intersection set ST∧M. In FL theory, the minimum function (Eq. ) is
one of the most popular representations of the intersection operation;
however, it is not the only one as a whole family of appropriate
functions actually exist referred to as t-norms (see ). In AC15 a number
of t-norms was tested as replacement for VSL's growth rate function within a
highly simplified paleo-DA setting. In particular, it was found that the
product t-norm gT∧M=gT⋅gM might significantly improve the performance of the
time-averaged EnKF technique. Accordingly, beside the minimum t-norm, we
also consider the product growth response VSL-Prod in this paper:
GPROD=gT⋅gM.
An important aspect of the product growth response VSL-Prod is the presence
of an additional growth limitation regime where T and M concurrently limit
tree-ring growth. This “co-limitation” regime allows a gradual transition
between temperature- and moisture-limited growth limitation regimes and
accordingly a progressive alternation of the recorded variable. Growth
co-limitation was initially recognized by , who called
attention to the limitations of the original formulation of the PLF due to
. Within the ecological research community, co-limitation
has been widely acknowledged as an crucial resource limitation phenomenon
, with abundant observational support both in terrestrial
and aquatic environments.
Interestingly, within the vegetation modeling community the original PLF
formulation is still the predominant approach to model photosynthesis
.
Experimental design
Following the rationale used in the experiments of AC15, we conducted a set
of DA experiments using the Simplified Parameterizations, primitivE-Equation DYnamics (SPEEDY) model as a dynamical system
and the VSL forward model as an observation operator. The time-averaged state of the
atmosphere is estimated via the EnKF approach of . In the
following, we describe in detail each of the components of our experimental
setting.
Atmospheric general circulation model
The SPEEDY model
is an intermediate-complexity atmospheric general
circulation model (AGCM) comprising a spectral dynamical core and a set of
simplified physical parameterizations, based on the same principles as
state-of-the-art AGCM but tailored to work with just a few vertical levels.
Regarding the interaction with the ocean, SPEEDY offers two possible
configurations: (i) prescribed ocean – sea surface temperature is
directly imposed as forcing; (ii) slab ocean – the atmospheric
model is coupled to a slab ocean model (q flux adjusted mixed layer model)
forced by climatological ocean dynamics. Despite its low resolution and the
relatively low complexity of its parameterizations, SPEEDY still captures
many observed global climate features in a realistic way, while its
computational cost is at least 1 order of magnitude lower than the one of
sophisticated state-of-the-art AGCMs at the same horizontal resolution
. The latter makes SPEEDY especially suitable for studies
involving long ensemble runs, like the ones necessary for this study.
DA technique
The SPEEDY model was embedded by into the
SPEEDY-LETKF framework, which offers a parallel implementation of a local
ensemble transform Kalman filter (LETKF) . Among the
different types of EnKF, LETKF is particularly promising for high-resolution models given that the calculation of the analysis for a particular
grid point requires only the information of the neighboring observations.
Therefore, LETKF offers outstanding scalability properties. SPEEDY-LETKF is
an open-source software which has already been used for several DA
studies . Here,
SPEEDY-LETKF was extended to allow the assimilation of pseudo-TRW
observations by means of either online or off-line time-averaged EnKF
methods.
For the experiments presented in this paper, we employed ensembles of 24
members due to computational constraints. We used a constant multiplicative
inflation of 1 % after the ensemble update and R localization via the
following formula:
Rloc=R⋅exp(rh/2λh2+rv/2λv2),
where rh and rv stand for the horizontal and vertical
distances, respectively. Their corresponding scaling parameters were set to
the values of λh=500km and λv=0.4lnp.
Schematic of a typical observation system simulation experiment
(OSSE) with ensemble online (with cycling) and
off-line (no-cycling) DA
methods. t designates the time axis and X(Y) denotes the model state
(observation) space. Boxes with sharp (rounded) corners represent data
(processes). Red (green) vertical bars indicate the forecast
(analysis) spread. Vertical dotted lines represent the assimilation
steps.
Numerical experiments
We performed a set of Observation System Simulation Experiments (OSSEs) (see
Fig. ), consisting of (i) a single model trajectory
xNature, referred to as “true” run or “nature” run,
that is used as a prediction target, (ii) pseudo-observations created by
applying the observation operator to xNature and adding
simulated observational noise, (iii) an observationally constrained ensemble
run XDA, where the pseudo-observations are assimilated,
and (iv) a free ensemble run XFree, where no
observations are assimilated and then the ensemble just freely evolves under
the action of the model dynamics. XFree is intended to
provide a benchmark of performance, against which it is possible to assess
the added value of the DA scheme.
Initially, a 1-year-long spin-up run is performed starting from 1 January
1860. Afterwards, the final state of this model trajectory is used as the
initial condition for a 150-year-long nature (“true”) run. The ensemble
runs with and without DA are identically initialized from a set of states
gathered from the last 2 months of the spin-up run (lagged 2-day
initialization). Note that the nature run and the different ensemble runs are
generated with the same time-varying forcing fields. Regarding the
atmosphere–ocean coupling, we used SPEEDY's slab ocean configuration,
motivated by the fact that the slow variability in the slab ocean may lend
predictability to the atmosphere. In these conditions, the online DA
technique should have higher chances to outperform the off-line technique.
Observation generation
Pseudo-TRW observations are produced following VSL's formulation plus a
final white noise addition step, in which random draws from a Gaussian
distribution are imposed on the time-averaged observations. Noise levels are
assessed by means of the signal-to-noise ratio (SNR), given by the ratio of
the standard deviation of the unpolluted pseudo-TRW observations to the
standard deviation of the additive white noise. Most of the results reported
in Sect. correspond to the optimistic value SNR =10,
with the exception of the last sensitivity study, which analyzes the
dependency on observational noise levels. Regarding the geographical
distribution of the observations, we place a pseudo-TRW chronology at every
grid box where at least one actual TRW chronology from the database of
is present. This strategy yields a realistic
observational network comprising 257 stations (see Fig. ).
Station set resembling real TRW network from
.
Concerning the configuration of the observation operator, we focus our study
on the role of the growth rate function by configuring VSL in such a way that
no thresholded response takes place. This is done by setting the upper and
lower response thresholds to the maximum (minimum) values during the nature
(true) run so that the response functions reduce to linear rescaling
operators. We consider three different growth rate functions leading to three
VSL configurations: (i) VSL-T, where the growth rate is directly given by the
growth response to temperature (GT=gT); (ii) VSL-Min,
where the original “minimum” t-norm GMIN is used; and
(iii) VSL-Prod, where the FL-based “product” t-norm GPROD is
utilized. Note that within the setup described above, VSL-T generates purely
temperature-limited pseudo-chronologies, while VSL-Min and VSL-Prod generate
mixed ones, where temperature and soil moisture determine tree-ring growth
pace.
Finally, respecting the soil moisture fields used to drive the VSL model, we
consider two options: (i) extracting soil moisture time series from the
climatological surface boundary conditions of the SPEEDY model and (ii) using
the precipitation and temperature output of SPEEDY as input for the leaky
bucket model (LBM)
The LBM code was extracted from VSL v2_3
(ftp://ftp.ncdc.noaa.gov/pub/data/paleo/softlib/vs-lite/).
, as it is normally done with the VSL model. Note that with
the first option, the time series obtained present only intra-annual
variability (yearly periodicity), while with the second option moisture time
series do exhibit interannual variations.
Diagnostics
Thanks to the availability of the truth model evolution for our OSSEs, the
forecast and analysis skill of the ensemble runs can be directly assessed.
Given the annual resolution of TRW chronologies, we study the filter
performance for yearly averaged values of near-surface temperatures. We focus
our analysis on near-surface temperature due to the larger error reduction in
this field as compared to other variables (e.g., humidity, u wind,
v wind) when DA is applied. The behavior of ensemble runs is monitored by
means of the root mean square error (RMSE) for the ensemble means. The
results are shown as (1) time series of globally averaged temperature RMSE,
(2) histograms of these time series and (3) maps of time-averaged (150 years)
temperature RMSEs.
ResultsUnconstrained ensemble run
An AGCM is an example of a nonautonomous system, and accordingly the
evolution of its state is determined by both the atmospheric dynamics and the
external forcing. The influences of these two distinct factors can be
disentangled to some extent by considering atmospheric variability to be a
superposition of an internal component, caused by the intrinsic dynamics, and
an external one, resulting from the variations in the boundary conditions
. Under this assumption, internal and external variability
can be separated by way of a free ensemble run, using the ensemble mean as an
estimate of the forced variability and the ensemble spread as an estimate of
the internal variability. Following this train of thought, Fig. a
shows the magnitude of the yearly internal variability in near-surface
temperature. Geographical regions around the Equator present negligible
yearly internal variability, which can be explained by the fact that these
areas are dominated by the daily cycle, and accordingly the 1-year-long
averaging strongly attenuates their internal variability. On the other hand,
maximum spread values occur at high latitudes around ±70∘. These
yearly internal-variability maxima can be related to planetary-scale patterns
fluctuating over timescales longer than 1 year, such as the “annular modes”
and displacements of the jet stream
. Regarding the error of the free ensemble run,
Fig. b shows that the RMSE maxima clearly coincide with the
internal-variability maxima, given that quantities without internal
variability can be well described by an ensemble run without DA. From that,
it becomes clear that in our setting, DA can only be beneficial to estimate
quantities displaying significant yearly internal variability.
Assimilating linear univariate time-averaged observations
In order to focus on the comparison between online and off-line DA
techniques, we first consider the assimilation of temperature-limited
pseudo-chronologies produced with VSL-T. Note that this observation operator
generates linear univariate time-averaged observations, and accordingly the
time-averaged EnKF must be in good conditions to operate, given that no
nonlinearities are present in the observation operator.
Figure b and c show how the assimilation effectively reduces the
error of the analysis for both online and off-line methods. All areas
adjacent to the observational network show low RMSE values; however, it is
important to highlight that stations located in areas of strong yearly
internal variability are more efficient than the others at reducing the error
of the analysis quantities. An example of this is the chronologies placed in
Alaska, where RMSE reductions of up to 1.5∘ take place. Conversely,
stations situated in regions with weak yearly internal variability do no lead
to significant error reductions, as is the case of the chronologies located
in the Himalaya area, where SPEEDY presents no significant yearly temperature
internal variability to be constrained via DA. Therefore, we claim that the
error reduction for analysis quantities is modulated by the magnitude of the
yearly internal variability at a specific site.
Yearly near-surface temperature spread (a) and
RMSE (b) for the free ensemble run.
Yearly near-surface temperature RMSE for the ensemble run
constrained by VSL-T pseudo-TRW observations. (a) Online forecast
quantities, (b) online analysis quantities and
(c) off-line analysis quantities
Global near-surface temperature RMSE for the forecast ensemble run
constrained by VSL-T pseudo-TRW observation (red) and the free ensemble run
(black) for the analysis of online (green) and off-line (blue) DA. Panel
(a) shows time series and panel (b) their corresponding histograms. Horizontal
lines represent the mean values.
Yearly near-surface temperature RMSE for the analysis of the
ensemble runs using off-line DA
and climatological soil moisture. (a) Assimilating VSL-T pseudo-TRW
observations, (b) assimilating VSL-Min pseudo-TRW observations and
(c) assimilating VSL-Prod pseudo-TRW observations
Global near-surface temperature RMSE for the off-line analysis of
the ensemble runs constrained by VSL-Min (red), VSL-Prod (green) and VSL-T
(blue) pseudo-TRW observations. Climatological soil moisture is used to drive
the VSL model. Horizontal lines represent the mean values. Panel (b)
shows the histograms of the time series.
Global yearly near-surface temperature RMSE box plots for the free
ensemble run forecast (free) and the analysis of the off-line DA
runs using VSL-Min observation operator and climatological soil moisture
(DA_Min), VSL-Min observation operator and leaky bucket model
(DA_Min_LBM), VSL-Prod observation operator and climatological
soil moisture (DA_Prod), and VSL-Prod observation operator and
leaky bucket model (DA_Prod_LBM).
Averaged global yearly near-surface temperature RMSE for the
analysis of the ensemble run with off-line DA, VSL-Min observation operator and different signal-to-noise
ratios. The green star shows the corresponding value for the free
run.
On the other hand, online forecast quantities do not present significant
error reductions, and consequently the online time-averaged EnKF appears to
work under the off-line regime. This situation can be seen in
Fig. a, which presents a strong resemblance with Fig. b,
corresponding to the RMSE of the free ensemble run. Furthermore, looking at
global RMSE time series (Fig. ), it is evident that the errors for
the online technique increase with time, as opposed to the corresponding
off-line quantities, which presented stable behavior all along the running
time interval. Overall the off-line DA methodology leads to lower error
levels on the global scale. The existence of this upward trend for online errors
might be attributed to an undesired effect arising from the re-initialization
of the ensemble after each assimilation step. For all our online DA
experiments we observed the aforementioned lack of forecast skill.
Consequently, for the rest of this section we will focus on the off-line DA
technique.
Assimilating mixed TRW chronologies
Here we analyze the role of the structure of the growth rate function in the
performance of the off-line DA scheme. Given that both GMin and
GProd are nonlinear functions, their use within the observation
operator degrades the operation of the time-averaged EnKF, as it is
illustrated by Fig. . Some areas displaying very low RMSE values
for VSL-T, such as Alaska, Siberia and central Asia, present considerably
higher error levels when VSL-Min or VSL-Prod are utilized. This loss of skill
to nonlinearities in the observation operator is also apparent in the global
RMSE time series, as can be seen in Fig. , where VSL-T clearly
outperforms the other two observation operators.
Regarding the dependency on the representation of the PLF, Fig.
shows that the use of VSL-Prod leads to considerably lower RMSE values in
central Europe and the east coast of the United States. This edge of VSL-Prod
over VSL-Min is also observed on the global scale, as can be seen in
Fig. . These results concur with the ones of AC15, where in a
highly simplified paleo-DA setting the use of VSL-Prod instead of VSL-Min
appeared also beneficial to the performance of the time-averaged EnKF
technique. Concerning the soil moisture fields, Fig. shows how
using the soil moisture calculated by the LBM reduces the error levels of
both VSL-Min and VSL-Prod DA runs. However, these improvements in filter
performance are more significant for VSL-Min than for VSL-Prod.
Finally, respecting the dependency of the filter performance on the
observational noise levels, Fig. shows how the off-line DA scheme
is considerably effective for SNR values higher than 1. Increasing further
the observational noise (reducing SNR) leads to a fast increase for the
averaged global RMSE curve, which reaches the error levels of a simulation
without DA (free ensemble run) at SNR =0.03.
Conclusions
Using the time-averaged EnKF methodology and a
process-based proxy forward model (VSL), we assimilated pseudo-TRW
chronologies in an AGCM (SPEEDY). Using a set of perfect model experiments we
studied two different aspects of the paleo-DA problem: (i) the onset of the
off-line regime in the assimilation of observations averaged during long time
periods and (ii) the impact of a nonlinear observation operator on the
performance of EnKF-based time-averaged DA approaches.
Our online DA experiments in general showed no forecasting skill, and accordingly they appear to operate under the off-line regime. Moreover, they
exhibited a detrimental increasing error trend not present for our
experiments with off-line DA schemes, where no re-initialization of the model
was performed. In these conditions, the off-line time-averaged EnKF appears
to outperform its online counterpart. This result complements the studies of
and , where the off-line regime was
observed to emerge within a DA setting comprising the time-averaged EnKF and
a quasi-geostrophic atmospheric jet model. From our point of view, the
off-line regime can arise either from the dynamical model or from the DA
scheme. Regarding the dynamical model, the main reason for losing forecast
skill is that the period between consecutive observations considerably
exceeds the predictability horizon of the model, and accordingly the ensemble
forecast completely forgets the observational information assimilated at the
analysis steps. For SPEEDY, due to its purely atmospheric nature, the 1-year
averaging period of TRW chronologies appears to be considerably longer than
its dynamical timescales. Therefore, the appearance of the off-line regime
is to be expected. On the other hand, for more realistic climate models,
there is already evidence of processes with timescales longer than 1 year
. Examples of these sources of slow variability are (i) the
“annular modes” , which strongly affect climatic
conditions in high latitudes, (ii) the latitudinal oscillation of the cell
structure, which influences the position of the jet streams, and (iii) the
oscillations of the Intertropical Convergence Zone (ITCZ), which greatly
impacts humidity in several parts of the globe , as well as the El Niño–Southern Oscillation (ENSO), which affects the climate of a large
portion of the tropical and subtropical areas. Consequently, we consider that
for more realistic paleo-DA settings using more comprehensive Earth system
models, the off-line regime is less likely to arise. Regarding the DA scheme,
a possible reason for the appearance of the off-line regime is the time-averaged
update strategy , which assumes complete decorrelation
between time-averaged and instantaneous variables. This condition might not
be satisfied by SPEEDY, and accordingly the estimation of instantaneous
quantities might be badly affected. Another limitation of the time-averaged
EnKF is its reliance on Gaussianity. This assumption might be easily violated
in a climate model, for example by definite positive quantities such as
humidity. Consequently non-Gaussianity might be also one of the culprits of
the onset of the off-line regime in our experiments.
Concerning the influence of nonlinearities in the observation operator, the
performance of the off-line time-averaged EnKF appeared to be significantly
sensitive to the selection of the t-norm used to represent the PLF. In our
experiments, the product t-norm outperformed the original minimum t-norm, as
previously observed for a two-scale model in AC15.
described trees as fundamentally lossy
This
adjective is currently used in information technology to designate
data encoding methods that lead to information loss from the original version
for the sake of reducing the amount of data needed to store the content.
recorders of climate, due to the integrated nature of the information
contained in them and the standardization process used to minimize the
non-climatic effects on growth. In the same vein, we argue that the “abrupt
shifting” of the recorded variable (temperature or moisture) – implied by the
minimum function used in VSL's original formulation – might constitute an
additional source of lossiness, which can be reduced by resorting to an FL-based
representation of the PLF. In particular, for the product t-norm, the
existence of an additional co-limitation regime makes a smoother shifting
of the recorded variable possible. As a cautionary remark, we want to highlight
that the pseudo-observations assimilated in our experiments present several
important limitations: (i) the thresholded response of trees to temperature
and moisture was not considered in order to focus on the role of the growth
rate function; (ii) VSL's parameters were set in a completely homogeneous
fashion for all the observational stations, whereas actual TRW networks are
strongly heterogeneous, comprising chronologies generated under highly
dissimilar growth limitation regimes. More realistic TRW assimilation
experiments will probably have to address these issues as well as the
necessity of considering model errors by conducting imperfect model OSSEs.
Finally, we want to mention that the translation of VSL into the FL language
suggests other possible extensions for VSL. (i) Growth response functions can
be generalized using the extensive knowledge on membership functions gathered
in the FL research community . In particular, it might be
possible to tailor the shape of the growth response functions so as to
optimize the performance of VSL regarding the particular application at hand,
e.g., Gaussian DA, as was done in this paper. (ii) Additional limiting
factors, e.g., CO2 concentration, can be easily incorporated into the
fuzzy inference system by adding new rules designed to express expert
knowledge about the influence of these factors on tree growth. (iii) The
intrinsic uncertainty regarding VSLs parameters might be taken into account
via the emerging theory of stochastic FL . Furthermore,
this ability of the FL approach to efficiently simulate complex processes
involving vaguely understood mechanisms can be used in the development of new
proxy forward models as well as in the extension of the existing ones.
No data sets were used in this article.
The authors declare that they have no conflict of
interest.
Acknowledgements
This work was supported by the German Federal Ministry of Education and Research
(BMBF) as part of the Research for Sustainable Development initiative (FONA;
www.fona.de) through the PalMod project (FKZ: 01LP1511A). The computational resources were
made available by the High-Performance Computing Center (ZEDAT) at Freie
Universität Berlin and the German Climate Computing Center (DKRZ). Walter
Acevedo wishes to acknowledge partial financial support by the Helmholtz
graduate research school GeoSim.
Edited by: H. Goosse
Reviewed by: two anonymous referees
ReferencesAcevedo, W., Reich, S., and Cubasch, U.: Towards the assimilation of
tree-ring-width records using ensemble Kalman filtering techniques, Clim.
Dynam., 46, 1909–1920, 10.1007/s00382-015-2683-1, 2015.Amezcua, J., Ide, K., Kalnay, E., and Reich, S.: Ensemble transform
Kalman-Bucy filters, Q. J.Roy. Meteor. Soc., 140, 995–1004,
10.1002/qj.2186, 2014.Annan, J. D. and Hargreaves, J. C.: Identification of climatic state with
limited proxy data, Clim. Past, 8, 1141–1151, 10.5194/cp-8-1141-2012,
2012.Barkmeijer, J., Iversen, T., and Palmer, T. N.: Forcing singular vectors and
other sensitive model structures, Q. J. Roy. Meteor. Soc., 129, 2401–2423,
10.1256/qj.02.126, 2003.Bhend, J., Franke, J., Folini, D., Wild, M., and Brönnimann, S.: An
ensemble-based approach to climate reconstructions, Clim. Past, 8, 963–976,
10.5194/cp-8-963-2012, 2012.
Blackman, F. F.: Optima and limiting factors, Ann. Bot., 19, 281–295, 1905.Boucher, É., Guiot, J., Hatté, C., Daux, V., Danis, P.-A., and
Dussouillez, P.: An inverse modeling approach for tree-ring-based climate
reconstructions under changing atmospheric CO2 concentrations,
Biogeosciences, 11, 3245–3258, 10.5194/bg-11-3245-2014, 2014.Breitenmoser, P., Brönnimann, S., and Frank, D.: Forward modelling of
tree-ring width and comparison with a global network of tree-ring
chronologies, Clim. Past, 10, 437–449, 10.5194/cp-10-437-2014, 2014.
Brönnimann, S.: Towards a paleoreanalysis?, ProClim-Flash, 51, p. 16, 2011.Burgers, G., van Leeuwen, P. J., and Evensen, G.: Analysis scheme in the
ensemble Kalman filter, Mon. Weather Rev., 126, 1719–1724,
10.1029/94JC00572, 1998.Crucifix, M.: Traditional and novel approaches to palaeoclimate modelling,
Quaternary Sci. Rev., 57, 1–16, 10.1016/j.quascirev.2012.09.010, 2012.Dee, S. G., Steiger, N. J., Emile-Geay, J., and Hakim, G. J.: On the utility
of proxy system models for estimating climate states over the common era,
J. Adv. Model. Earth Syst., 8, 1164–1179, 10.1002/2016MS000677, 2016.Deza, J. I., Masoller, C., and Barreiro, M.: Distinguishing the effects of
internal and forced atmospheric variability in climate networks, Nonlin.
Processes Geophys., 21, 617–631, 10.5194/npg-21-617-2014, 2014.Dirren, S. and Hakim, G. J.: Toward the assimilation of time-averaged
observations, Geophys. Res. Lett., 32, L04804, 10.1029/2004GL021444,
2005.Dubinkina, S. and Goosse, H.: An assessment of particle filtering methods and
nudging for climate state reconstructions, Clim. Past, 9, 1141–1152,
10.5194/cp-9-1141-2013, 2013.Dubinkina, S., Goosse, H., Sallaz-Damaz, Y., Crespin, E., and Crucifix, M.:
Testing a particle filter to reconstruction climate over the past centuries,
Int. J. Bifurcat. Chaos, 21, 3611–3618, 10.1142/S0218127411030763,
2011.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.Evensen, G.: Sequential data assimilation with a nonlinear quasi-geostrophic
model using Monte Carlo methods to forecast error statistics, J. Geophys.
Res., 99, 10143–10162, 10.1029/94JC00572, 1994.
Fritts, H. C.: Tree rings and climate, Academic Press, New York, 1976.Hakim, G., Annan, J., Broennimann, S., Crucifix, M., Edwards, T., Goosse, H.,
Paul, A., van der Schrier, G., and Widmann, M.: Overview of data assimilation
methods, PAGES news, 21, 2 pp., 2013.
Hakim, G. J., Emile-Geay, J., Steig, E. J., Noone, D., Anderson, D. M.,
Tardif, R., Steiger, N., and Perkins, W. A.: The last millennium climate
reanalysis project: Framework and first results, J. Geophys. Res.-Atmos.,
121, 6745–6764, 10.1002/2016JD024751, 2016.Hamill, T. M.: Ensemble-based atmospheric data assimilation, in:
Predictability of Weather and Climate, edited by: Palmer, T. and Hagedorn,
R., Cambridge University Press, 10.1017/CBO9780511617652.007, 2006.Harpole, W. S., Ngai, J. T., Cleland, E. E., Seabloom, E. W., Borer, E. T.,
Bracken, M. E., Elser, J. J., Gruner, D. S., Hillebrand, H., Shurin, J. B.,
and Smith, J. E.: Nutrient co-limitation of primary producer communities,
Ecol. Lett., 14, 852–862, 10.1111/j.1461-0248.2011.01651.x, 2011.Holton, J. and Hakim, G. J.: An Introduction to Dynamic Meteorology, Academic
Press, http://books.google.de/books?id=hLQRAQAAIAAJ, last access:
15 May 2017, 2013.Huang, J., van den Dool, H. M., and Georgarakos, K. P.: Analysis of
Model-Calculated Soil Moisture over the United States (1931–1993) and
Applications to Long-Range Temperature Forecasts, J. Climate, 9, 1350–1362,
10.1175/1520-0442(1996)009<1350:AOMCSM>2.0.CO;2,
1996.Hughes, M. and Ammann, C.: The future of the past – an earth system
framework for high resolution paleoclimatology: editorial essay, Climatic
Change, 94, 247–259, 10.1007/s10584-009-9588-0, 2009.
Hughes, M., Guiot, J., and Ammann, C.: An emerging paradigm: Process-based
climate reconstructions, PAGES news, 18, 87–89, 2010.Hunt, B. R., Kostelich, E. J., and Szunyogh, I.: Efficient data assimilation
for spatiotemporal chaos: A local ensemble transform Kalman filter,
Physica D, 230, 112–126, 10.1016/j.physd.2006.11.008, 2007.Huntley, H. and Hakim, G.: Assimilation of time-averaged observations in a
quasi-geostrophic atmospheric jet model, Clim. Dynam., 35, 995–1009,
10.1007/s00382-009-0714-5, 2010.
Kalman, R. E.: A New Approach to Linear Filtering and Prediction Problems,
J. Basic Eng.-T. ASME, 82, 35–45, 1960.
Kalnay, E.: Atmospheric modeling, data assimilation, and predictability,
Cambridge University Press, 2003.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.Lahoz, W., Khattatov, B., and Menard, R.: Data Assimilation: Making Sense of
Observations, Springer, http://books.google.de/books?id=KivkFpthm1EC,
last access: 15 May 2017, 2010.Li, H., Kalnay, E., Miyoshi, T., and Danforth, C. M.: Accounting for Model
Errors in Ensemble Data Assimilation, Mon. Weather Rev., 137, 3407–3419,
10.1175/2009MWR2766.1, 2009.Lien, G.-Y., Kalnay, E., and Miyoshi, T.: Effective assimilation of global
precipitation: simulation experiments, Tellus A, 65, 19915,
10.3402/tellusa.v65i0.19915, 2013.Lorenc, A. C.: Analysis methods for numerical weather prediction, Q. J.Roy.
Meteor. Soc., 112, 1177–1194, 10.1002/qj.49711247414, 1986.
Lorenz, E. N.: Predictability, a problem partly solved, in: Proceedings of
ECMWF Seminar on predictability, ECMWF, Reading, UK, 4–8 September 1995,
1–19, 1996.Luhandjula, M K.. and Gupta, M. M.: On fuzzy stochastic optimization, Fuzzy
Set. Syst., 81, 47–55, 10.1016/0165-0114(95)00240-5, 1996.
Marchini, A.: Modelling Ecological Processes with Fuzzy Logic Approaches, in:
Modelling Complex Ecological Dynamics, edited by: Jopp, F., Reuter, H., and
Breckling, B., Springer Berlin Heidelberg, 133–145, 2011.Mathiot, P., Goosse, H., Crosta, X., Stenni, B., Braida, M., Renssen, H., Van
Meerbeeck, C. J., Masson-Delmotte, V., Mairesse, A., and Dubinkina, S.: Using
data assimilation to investigate the causes of Southern Hemisphere high
latitude cooling from 10 to 8 ka BP, Clim. Past, 9, 887–901,
10.5194/cp-9-887-2013, 2013.Matsikaris, A., Widmann, M., and Jungclaus, J.: On-line and off-line data
assimilation in palaeoclimatology: a case study, Clim. Past, 11, 81–93,
10.5194/cp-11-81-2015, 2015.
Miyoshi, T.: Ensemble Kalman filter experiments with a primitive-equation
global model, PhD thesis, University of Maryland, College Park, 197 pp.,
2005.Miyoshi, T.: The Gaussian Approach to Adaptive Covariance Inflation and Its
Implementation with the Local Ensemble Transform Kalman Filter, Mon. Weather
Rev., 139, 1519–1535, 10.1175/2010MWR3570.1, 2010.
Molteni, F.: Atmospheric simulations using a GCM with simplified physical
parametrizations. I: model climatology and variability in multi-decadal
experiments, Clim. Dynam., 20, 175–191, 2003.
Nguyen, H. T., Prasad, N. R., Walker, C. L., and Walker, E. A.: A First
Course in Fuzzy and Neural Control, 1 Edn., Chapman and Hall/CRC, 2002.Niinemets, Ü. and Kull, K.: Co-limitation of plant primary productivity
by nitrogen and phosphorus in a species-rich wooded meadow on calcareous
soils, Acta Oecol., 28, 345–356, 10.1016/j.actao.2005.06.003, 2005.Paul, A. and Schäfer-Neth, C.: How to combine sparse proxy data and coupled
climate models, Quaternary Sci. Rev., 24, 1095–1107,
10.1016/j.quascirev.2004.05.010, 2005.Pendergrass, A., Hakim, G., Battisti, D., and Roe, G.: Coupled Air-Mixed
Layer Temperature Predictability for Climate Reconstruction, J. Climate, 25,
459–472, 10.1175/2011JCLI4094.1, 2012.Reich, S. and Cotter, C.: Probabilistic Forecasting and Bayesian Data
Assimilation, in: Cambridge Texts in Applied Mathematics, Cambridge
University Press, https://books.google.de/books?id=xVpiCAAAQBAJ, last
access: 1 May 2017, 2015.Ruiz, J. J., Pulido, M., and Miyoshi, T.: Estimating Model Parameters with
Ensemble-Based Data Assimilation: A Review, J. Meteorol. Soc. Jpn., Ser. II,
91, 79–99, 10.2151/jmsj.2013-201, 2013.
Saito, M. A., Goepfert, T. J., and Ritt, J. T.: Some thoughts on the concept
of colimitation: three definitions and the importance of bioavailability,
Limnol. Oceanogr., 53, 276–290, 2008.
Salski, A.: Ecological Applications of Fuzzy Logic, in: Ecological
Informatics, edited by: Recknagel, F., Springer Berlin Heidelberg, 3–14,
2006.
Se, Z.: Fuzzy Logic and Hydrological Modeling, CRC Press, 2009.
Singh, B. N. and Lal, K. N.: Limitations of Blackman's law of limiting
factors and Harder's concept of relative minimum as applied to
photosynthesis, Plant Physiol., 10, 245–268, 1935.Smith, D. M., Scaife, A. A., and Kirtman, B. P.: What is the current state of
scientific knowledge with regard to seasonal and decadal forecasting?,
Environ. Res. Lett., 7, 015602, 10.1088/1748-9326/7/1/015602, 2012.Steiger, N., Hakim, G., Steig, E., Battisti, D., and Roe, G.: Assimilation of
Time-Averaged Pseudoproxies for Climate Reconstruction, J. Climate, 27,
426–441, 10.1175/JCLI-D-12-00693.1, 2014.
Talagrand, O.: Assimilation of observations, an introduction,
Journal-Meteorological Society of Japan Series 2, 75, 81–99, 1997.Thompson, D. W. J. and Wallace, J. M.: Annular Modes in the Extratropical
Circulation. Part I: Month-to-Month Variability, J. Climate, 13, 1000–1016,
10.1175/1520-0442(2000)013<1000:AMITEC>2.0.CO;2,
2000.Tippett, M. K., Anderson, J. L., Bishop, C. H., Hamill, T. M., and Whitaker,
J. S.: Ensemble Square Root Filters*, Mon. Weather Rev., 131, 1485–1490,
10.1175/1520-0493(2003)131<1485:ESRF>2.0.CO;2, 2003.Tolwinski-Ward, S. E.: Inference on Tree-Ring Width and Paleoclimate Using a
Proxy Model of Intermediate Complexity, PhD thesis, The University of
Arizona, http://hdl.handle.net/10150/241975, last access: 22 May 2017,
2012.Tolwinski-Ward, S. E, Tingley, M., Evans, M., Hughes, M., and Nychka, D.:
Probabilistic reconstructions of local temperature and soil moisture from
tree-ring data with potentially time-varying climatic response, Clim. Dynam.,
44, 791–806, 10.1007/s00382-014-2139-z, 2014.Tolwinski-Ward, S. E., Evans, M. N., Hughes, M., 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.
Vaganov, E., Hughes, M., and Shashkin, A.: Growth Dynamics of Conifer Tree
Rings: Images of Past and Future Environments, in: Ecological studies,
Springer, New York, 183, 358 pp., 10.1007/3-540-31298-6, 2006.van der Schrier, G. and Barkmeijer, J.: Bjerknes' hypothesis on the coldness
during AD 1790–1820 revisited, Clim. Dynam., 25, 537–553,
10.1007/s00382-005-0053-0, 2005.
Van Leeuwen, P. J., Cheng, Y., and Reich, S.: Nonlinear data assimilation,
Springer, 2015.
von Storch, H., Cubasch, U., González-Ruoco, J., Jones, J., Widmann, M.,
and Zorita, E.: Combining paleoclimatic evidence and GCMs by means of data
assimilation through upscaling and nudging (datun), in: Proceedings of the
11th Symposium on Global Change Studies, AMS, Long Beach, CA, USA,
9–14 January 2000, 28–31, 2000.Whitaker, J. S., Compo, G. P., and Thépaut, J.-N.: A Comparison of
Variational and Ensemble-Based Data Assimilation Systems for Reanalysis of
Sparse Observations, Mon. Weather Rev., 137, 1991–1999,
10.1175/2008MWR2781.1, 2009.Widmann, M., Goosse, H., van der Schrier, G., Schnur, R., and Barkmeijer, J.:
Using data assimilation to study extratropical Northern Hemisphere climate
over the last millennium, Clim. Past, 6, 627–644,
10.5194/cp-6-627-2010, 2010.Woollings, T., Pinto, J. G., and Santos, J. A.: Dynamical Evolution of North
Atlantic Ridges and Poleward Jet Stream Displacements, J. Atmos. Sci., 68,
954–963, 10.1175/2011JAS3661.1, 2011.Yin, X. and Struik, P.: C3 and C4 photosynthesis models: An overview from the
perspective of crop modelling, NJAS-Wagen. J. Life Sc., 57, 27–38,
10.1016/j.njas.2009.07.001, 2009.
Zadeh, L. A.: Fuzzy logic and approximate reasoning, Synthese, 30, 407–428,
1975.