Introduction
With mounting evidence indicating that Earth's climate is changing
and references therein, it is becoming
increasingly important to understand the potential impacts of climate change
on society. Impacts assessment requires projections of future climate under
increased concentrations of greenhouse gases (GHGs). For example,
understanding climate effects on food supply would require simulations of
future temperature and precipitation for use in agricultural yield models.
Crop yields, however, are highly nonlinear with temperature and precipitation
and therefore are sensitive not only to climatological means but also to
short-term extremes
e.g.,. In this
context, climate must be understood as an underlying, multivariate,
spatiotemporal probability distribution, for which weather is a random
realization. Human societies can be impacted by changes of not only the mean,
but of the entire probability distribution.
Comparison of modeled and observed global mean temperatures.
(a) CO2 concentrations used in “baseline” and “scenario”
runs with the CCSM3 model. (See Appendix for description
of experiments and observations.) Figures here truncate output after less
than 600 years but the scenario run extends for 6000 years.
(b) Corresponding annual model GMT (∘C) for the two runs
and observed GMT from the Global Historical Climatological Network. Model
output reproduces trends in global temperature well but with a systematic
offset from observations. To better show the similarity in trend we also plot
the observational record minus a 2 ∘C offset.
Changes in variability in both temperature and precipitation are physically
plausible. For precipitation, standard physics would suggest increases in
both spatial and temporal variability, with dry areas drier, wet areas
wetter, and rainfall occurring in more intense events .
Changes in temperature variability are less robustly predicted. Some
empirical studies suggest that temperature variability may already be
changing in particular contexts, though some studies argue for increases and
others for decreases . For instance, in large
portions of North America, a weakened polar jet stream (in part driven by
reduced latitudinal temperature gradients) may or
may not lead to prolonged
weather patterns that shift variation to lower frequencies. Studies of
variability changes in climate models remain limited, and it is not yet clear
which if any predictions are robust across models see,
e.g., or physical parameterizations see, e.g.,
. The area remains one of active research.
One complication to analyses of potential future changes in climate
variability is that while the deterministic climate models used for long-term
climate forecasts appear to capture trends, they do not accurately reproduce
observed current climate. These models, known as atmosphere–ocean general
circulation models (AOGCMs), are physically based numerical simulations of
transport of energy and moisture in the atmosphere and ocean, typically with
separate submodels for the atmosphere, ocean, sea ice, and vegetation. Many
AOGCMs successfully reproduce observed large-scale circulation, atmospheric
structure, latitudinal temperature gradients, storm tracks, and
quasi-periodic interdecadal phenomena such as the El Niño–Southern
Oscillation. When driven with historical records of CO2 and aerosol
emissions due to human and volcanic activity, they also reproduce well the
observed temperature trend of the last 2 centuries.
Figure demonstrates this ability to capture trends in
the widely used Community Climate System 3 (CCSM3) model
, which we use in examples throughout this
manuscript. (See Appendix for description of model and
experimental runs, as well as observational data used in comparisons.) CCSM3
and other AOGCMs do not, however, perfectly reproduce either the mean or
distribution of current climate. Model present-day global mean temperature
(GMT) can be offset by several degrees from observations (again, see
Fig. ) and probability distributions of temperature and
precipitation at individual locations do not match those of weather
observations (Fig. , which shows marginal
distributions in CCSM3 temperature output and observations for three
representative locations whose time series are given in
Fig. ; see also ,
for discussion).
The comparisons above suggest that climate models may be informative about
changes in climate, even while failing to capture certain current
characteristics. This is well-demonstrated for means (again, see
Fig. ), and the fact that AOGCMs capture trends in mean
climate well suggests that their physics may be sufficiently realistic to
provide a guide to trends in variability. We therefore seek a method of
producing simulations of future climate that combines model output with data
to incorporate both observational ground truth and model forecasts of trends.
An appropriate method should simply reproduce current climate when models
suggest no changes. When models do predict changes, the desired “data-driven
simulation” should reproduce model changes in second-order moments (e.g.,
covariance) of climate but retain most non-Gaussian characteristics of data,
rather than of model output, when changes in variability are relatively
small. Our motivation in this work is to develop an empirically driven
approach to simulating future climate that modifies existing observations in
terms of means and second-order moments (including covariances) based on
changes in model simulations.
(Left) Three locations (individual model pixels) used as examples
throughout the manuscript, chosen to represent different combinations of
seasonality, variability, and expected future changes: Illinois,
mid-continental with a strong seasonal component (green, 38.97∘ N,
90∘ W); Gulf of Guinea, near-equatorial with little seasonal
cycle (red, 1.86∘ S, 0∘ E); and Southern Ocean, which
has strong projected changes in both mean temperature and in variability
(blue, 61.2∘ S, 33.8∘ E). Annual standard deviation of
daily temperatures σ and projected temperature change Δ
(scenario–baseline) are Illinois: σ=10.81, Δ=3.87; Gulf
of Guinea: σ=1.97, Δ=2.43; Southern Ocean: σ=4.67,
Δ=8.10. (Right) Time series of the 3 years of daily temperature
(∘C) from the NCEP-DOE (National Centers for Environmental Predictions
– Department of Energy) Climate Forecast System Reanalysis at those
locations. (See Appendix for a description of the
observational data set.)
Marginal densities (by season) of daily mean temperature
(∘C) for the pixels in Illinois (top row), the Gulf of Guinea
(middle row), and the Southern Ocean (bottom row) for reanalysis data (solid blue
line), baseline model output (dashed blue line), and scenario model output
(dashed red line). The model output does not replicate the marginal
distributions of the reanalysis observations. Furthermore, the marginal
distributions in the model output change from the baseline to scenario
periods.
Many methods for combining observations with model output in climate
projections have been developed for use in impacts studies, especially those
involving hydrology and agriculture see, e.g.,. In these
cases, impacts models typically require inputs of temperature and
precipitation at finer spatial resolution than is provided by AOGCMs, whose
typical state-of-the-art resolution is on the order of 1∘ (111 km or 69 miles). For this reason, approaches for simulating future climate by
combining model output and data are often intertwined with methods for
downscaling to higher spatial resolutions, and are described in the
literature on statistical downscaling. We provide a brief summary of existing approaches, along with what
we consider to be the primary shortcomings of each approach.
All approaches that combine observations and model output in simulating
future climates correct in some way for model–observation discrepancies. One
approach is a simple “bias correction” in which any offsets between current
observed and modeled present-day climate are assumed to be systematic model
errors. Model simulations of future climate are then “corrected” by adding
the present-day bias (determined by comparing observations to a baseline
run). Bias corrections can be made on annual mean temperatures or, more
commonly, on monthly mean temperatures or annual harmonics, since models may
not perfectly capture observed seasonal variation. One drawback of this
approach is that all higher-order moments of the marginal and joint
probability distributions (variability, skewness, stationarity, etc.) are
provided by the future model output. As we have seen in
Fig. , climate models may not adequately capture
higher-order characteristics in the data.
A variant on this approach, typically termed “bias correction/spatial
disaggregation” (BCSD), attempts to provide a better approximation of
observed climate distributions by separately bias-correcting the different
quantiles of model output e.g.,.
This approach is also termed “quantile mapping” and involves computing a
transfer function between model simulations of present-day climate and actual
observations based on the ranked model output. The transfer function is then
applied to AOGCM projections of future climate. This approach accommodates
errors in higher-order moments of the model – in the most extreme case, the
procedure results in a full transformation of the empirical cumulative
distribution function (CDF) – but only corrects the marginal distributions
of the model, and takes no account of differences in the covariance structure
of the model output and the observed climate. Since human societies are
sensitive to climate variation at different timescales (e.g., to changes in
duration of droughts or rainfall that produces flooding), BCSD is not ideal
for estimating the societal impacts of climate change.
While the previous two approaches are model-based, i.e., they quantify
present-day model–observation discrepancy and apply it to future model
output, the “change-factor” or “delta” method see, e.g.,and
references therein is
observation-based: future climate projections are generated by modifying
observations based on present–future differences in AOGCM simulations.
Specifically, the delta method involves quantifying the difference (e.g., in
annual or monthly means) between model output from a baseline run driven with
present-day or preindustrial GHG concentrations and that from a “scenario”
run under future GHG concentrations, then adding this difference onto some observation set. As a
result, higher-order moments (in terms of the marginal and joint PDFs) will
be derived from observations. showed that
delta-method approaches may provide a better forecast of future climate than
bias-correction approaches. Delta-method approaches do not however generally
involve representing changes in variability in future climate regimes. Recent
advances have been developed to accommodate changing marginal variances
see, e.g.,; however, such
approaches ignore joint dependence characteristics (e.g., covariances).
In this work, we adopt the observation-based approach of the delta method
(modifying observations based on changes suggested by model output) but
extend the method to account for possible changes in variability and temporal
correlations. While recent work has extended the delta-method approach to
accommodate some aspects of changing variability
, these methods do not
account for changes in third-order or higher moments of the marginal
distribution, or in the covariance of the joint distribution. Changing
covariance structures in particular are a critical component of simulating
future climate for impacts assessment.
A delta- or change-factor approach that involves modifying covariance
structures poses substantial challenges. The approach requires modifying a
vector of random variables with a given joint dependence structure to produce
a new vector of random variables with a different dependence structure. To
achieve this goal, it helps to think about modifying quantities that are
independent (or close to independent) under both present and future climates.
In this regard, spectral-based approaches provide a natural framework. We
propose an approach that modifies the discrete Fourier transform (DFT) of
observations based on an estimated ratio of spectral densities of model
output. Under a large class of stationary processes, the DFT is a
transformation to approximate independence . This
approach shares an important quality with the delta method that when the
model suggests no changes (in either first- or second-order moment
characteristics), the simulations equal the observations.
One caveat is that the procedure is designed to transform model simulations
of an assumed equilibrium climate to another equilibrium climate while,
during foreseeable human timescales, climate will continue to remain in a
transient state. This approach does not directly address the important
problem of simulating transient climate behavior in the covariance structure.
However, it is likely that the method would remain an improvement over the
delta method even in predicting future transient climate states, with certain
extensions related to nonstationary time series. We do not explore the issue
in this paper, but point out a potential approach in
Sect. .
In the remainder of the paper, Sect. outlines the
methodology, explaining how to estimate the ratio of spectral densities and
use it to modify observations. We also explain an approach to account for a
limited type of temporal nonstationarity in the data as brought about by
differences in intraseasonal variability across seasons.
Section applies the method to generating simulations of
daily mean temperature for a higher-CO2 world, and
Sect. discusses results and future research needs. We
provide supplemental materials that give further details, a numerical study,
and information on how to access the code and data used to reproduce the
analysis.
Methods
Our method produces data-driven simulations of future climate that combine
observed climate with model predictions of changes to climate means,
variability and temporal correlation. To do this we need to take account of
changes in variability of model output over all temporal scales.
In the sections below, we first demonstrate the principle of our approach for
an idealized situation: we assume an infinite length observational time
series with known changes in the spectral process. We then develop the method
for the more practical setting in which
the time series of both observations and model output are finite
we do not know the explicit form of the spectral process
we do not know the explicit form of changes to the spectral process
climate exhibits a strong seasonal cycle in both first and second-order moments.
Motivation
We demonstrate here that given an infinite length Gaussian time series
representing present-day climate with a known spectral process and known future
changes in the spectral process, we can modify the continuous Fourier
transform separately at each frequency to produce output that has the correct
joint distribution for the future process. Let {Z0,t;t=0,±1,±2,…} represent a time series of an observable process of interest.
Furthermore, suppose {Z0,t} is a stationary Gaussian process with
E(Z0,t)=0 and covariance function γ0(h)=Cov(Z0,t,Z0,t-h)=E(Z0,tZ0,t-h). Let
{Z1,t;t=0,±1,±2,…} represent the future process that we
wish to simulate. Suppose that {Z1,t} is also a stationary Gaussian
process with E(Z1,t)=0, but with covariance function
γ1(h)=Cov(Z1,t,Z1,t-h)=E(Z1,tZ1,t-h), where γ1(h) is not necessarily
equivalent to γ0(h).
We are interested in modifying {Z0,t} in order to generate a random
process that is equal in (joint) distribution to {Z1,t}. The temporal
correlations in {Z0,t} makes this nontrivial. However, the orthogonal
nature of the spectral representation makes it the natural domain in which to
modify random quantities. For example, writing ı for -1,
Z0,t has the representation Z0,t=∫-0.50.5exp(2πıωt)dZ^0(ω) where Z^0(ω) is a
complex-valued Gaussian random measure with mean of 0 and for disjoint sets
[a,b]∩[c,d]=∅,
E(Z^0([a,b]),Z^0([c,d])‾)=0, and
x‾ represents the complex conjugate of x. That is,
Z^0(ω) (also referred to as the spectral process associated
with {Z0,t}) is an orthogonal Gaussian measure that can be modified
separately at each frequency to produce a process with a different covariance
structure. The spectral distribution associated with {Z0,t},
G0(ω), is the positive finite measure given by
E(|dZ^0(ω)|2)=dG0(ω). Assuming absolute summability of the covariance,
i.e., ∑h=-∞∞|γ0(h)|<∞, the
spectral distribution is absolutely continuous: dG0(ω)=g0(ω)dω and g0 is called the spectral density for
{Z0,t}. The spectral density can be obtained from the covariance
function γ0 by g0(ω)=∑h=-∞∞exp(-2πıωh)γ0(h). If γ1 is summable, we can
similarly consider the second-order characteristics of {Z1,t} based on
its spectral density g1(ω)=∑h=-∞∞exp(-2πıωh)γ1(h).
The spectral densities g0(ω) and g1(ω) provide
information regarding the covariance structure of {Z0,t} and
{Z1,t}, respectively, for frequencies, ω∈(-0.5,0.5]. Then,
given a spectral density ratio ρg(ω)=g1(ω)/g0(ω) we can modify the spectral process associated
with Z0,t to generate
Z1,t=∫-0.50.5exp(2πıωt)ρg(ω)dZ^0(ω),
which is a stationary, Gaussian process with E(Z1,t)=0 and
covariance γ1(h). In this way, we derive the future, unobservable
process in terms of the present process, modified by the ratio of their
spectral densities. If g1(ω)=g0(ω), for all ω∈(-0.5,0.5], then Z1,t=Z0,t, for all t (because in this special
case the procedure reduces to taking the DFT and then the inverse DFT of the
observations). In particular, the temporal covariance structure of the
simulations equals that of the observations.
Outline of approach
When working with real time series of climate observations and model output,
the spectral densities in the past and future, g0(ω) and
g1(ω), are not known. While g0(ω) can be estimated from
data, clearly we cannot provide an observation-based estimate of
g1(ω). A central question then becomes how to best represent the
spectral ratio ρg(ω). Let f1(ω) represent the future
spectral density associated with the computer model output. For AOGCMs,
f1(ω) may differ substantially from g1(ω). However, given
the model's suggested covariance structure under a baseline period,
represented by f0(ω), the estimated change in covariance
structure may be a reasonable approximation for the real changes in the
covariance structure, especially if those changes are relatively small. We
therefore do not assume that model output has the correct covariance
structure for a given GHG scenario, but assume that the computer model
provides a reasonable approximation to the changes in the spectral density
across all frequencies (i.e., ρg(ω)=ρf(ω) for all
ω∈(-0.5,0.5], where ρf(ω)=f1(ω)/f0(ω)).
Carrying out the simulation on real data then requires the following steps,
starting with {Z0,t} (observations), {Y0,t} (model base period time series), and {Y1,t} (model scenario period time series):
Preprocess the observations and model output to produce Z0,t*=(Z0,t-μ^z,t)/Dz,t, Y0,t*=(Y0,t-μ^0,t)/D0,t, and Y1,t*=(Y1,t-μ^1,t)/D1,t, which have mean of 0 and are stationary. See
Sect. for details on the estimation of the seasonal
cycle (i.e., μ^z,t, μ^0,t, and μ^1,t) and
see Sect. and Sect. S2 in the
Supplement for details on the estimation of
seasonal variation (i.e., Dz,t, D0,t, and D1,t).
Estimate the ratio of spectral densities of Y1,t*,
and Y0,t*, following the steps given in
Sects. and S1.
Then, use the estimated spectral densities to modify the discrete Fourier
transform of Z0,t*, producing
Z1,t*, following the instructions in
Sect. .
Reverse preprocessing to produce simulations Z1,t=μ^z,t+(μ^1,t-μ^0,t)+Dz,t(D1,t/D0,t)Z1,t*.
In the following subsections we describe in detail these primary steps:
estimating the spectral ratio and modifying the discrete Fourier transform;
removing the seasonal cycle; and modulating the deseasonalized time series.
Spectral-based conditional simulation
Let {Z0,t;t=0,…,T-1} represent the observations of the
process of interest, observed at regular time points. For now, assume that
the process is stationary with E(Z0,t)=0. We discuss how we
account for any trend in Sects.
and .
In the previous section T=∞ whereas, in practice, our observations are
observed discretely over a finite period and the spectral process associated
with Z0,t is unknown. First we approximate the true
spectral process by using the discrete Fourier transform of the observations
Z^0,k=T-1/2∑t=0T-1Z0,texp(-2πıωkt) for ωk=k/T and k=-T/2+1,…,T/2. Here,
Z^0,k are complex-valued quantities that provide an approximation
to Z^0(ω), with Z^0,k=Z^0,-k‾.
We can similarly define the cosine transform Z^0,kc=T-1/2∑t=0T-1Z0,tcos(2πωkt) and sine
transform Z^0,ks=T-1/2∑t=0T-1Z0,tsin(2πωkt), which relate
to the DFT through the equality Z^0,k=Z^0,kc-iZ^0,ks. Because Z^0,kc and Z^0,ks are
linear combinations of Gaussian processes, Z^0,kc and
Z^0,ks are also Gaussian and, under the condition that β=∑h=-∞∞|h||γ0(h)|<∞, the following
asymptotic results hold in terms of the covariance structure for the cosine
transform
Cov(Z^0,jc,Z^0,kc)=g0(ωj)/2+ϵT,j=kϵT,j≠k,
the sine transform
Cov(Z^0,js,Z^0,ks)=g0(ωj)/2+ϵT,j=kϵT,j≠k,
and also Cov(Z^0,jc,Z^s,ks)=ϵT,
for all j,k. Seefor details. Here,
ϵT is a generic remainder that varies with j and k and can be
shown to obey a bound |ϵT|≤β/T. As a result, our methodology is modifying
nearly independent quantities in order to produce simulations with a
different covariance structure than the observations. Let
ρ^f(ωk) be an estimate of the ratio of the spectral
densities at ωk. Then, the simulations (not accounting for changes
in mean) under a given scenario can be represented as
Z1,t=T-1/2∑k=-T/2+1T/2ρ^f(ωk)Z^0,kexp(2πıωkt);
so, when ρ^f(ωk)=1, k=-T/2+1,…,T/2, then Z1,t=Z0,t, t=0,…,T-1. This suggests the following covariance
structure for {Z1,t} for a given estimate
ρ^f(ωk):
E(Z1,t+hZ1,t)=T-1∑k=-T/2+1T/2ρ^f(ωk)g0(ωk)exp(2πıωkh).
A brief numerical study in Sect. S3 illustrates the
efficacy of this approach even for fairly small T when ρg=ρf is
known. In the following section, we provide the details of a penalized
likelihood approach to estimate ρf(ωk). Finally, although we
have motivated this methodology in terms of Gaussian processes, the resulting
simulation of Z1,t in Eq. () will tend to retain any
non-Gaussian characteristics of Z0,t, at least if
ρ^f(ωk) is nearly constant.
Estimation of the ratio of spectral densities (ρfωk)
We propose a penalized likelihood approach for estimation of
ρf(ωk), similar to the approach given in
for the estimation of one spectral density.
Let fj,k=fj(ωk), θj,k=log(fj,k), j=0,1,k=1,…,K, and θj=(θj1,…,θjK)′. Then, a penalized likelihood can be generally
written as
L0(θ0)+L1(θ1)+δJ(θ0,θ1),
where L0(θ0) and
L1(θ1) represent the Whittle likelihood
for j=0,1, respectively. So, L0(θ0) and
L1(θ1) provide an objective function that
determines the fit to the data, J(θ0,θ1) is a function
that penalizes lack of smoothness, and δ is a smoothness parameter.
Likelihood
Let {Yi,0,t;t=0,…,T-1} represent the ith realization of
AOGCM output (i=1,…,n0) of the baseline run and let
{Yi,1,t;t=0,…,T-1} represent AOGCM output for the ith
realization of the scenario run (i=1,…,n1). Here we introduce
the possibility of having multiple independent realizations, i.e., the AOGCM
output that was run under identical forcings but with different initial
conditions. Let Yi,0=Yi,0,0,…,Yi,0,T-1′ and Yi,1=Yi,1,0,…,Yi,1,T-1′. We assume that Y1,0,…,Yn0,0 are independent and identically distributed, that
Y1,1,…,Yn1,1 are also independent and
identically distributed, and finally that Yi,0 and
Yi′,1 are independent, for all i,i′.
Let Y^i,j,k=T-1/2∑t=0T-1Yi,j,texp(-2πıωkt) represent the DFT of the ith realization of the model
output at frequency ωk (for either the model baseline or scenario run).
Note that when Yi,0,t and Yi,1,t
follow stationary, Gaussian distributions, the periodograms Ii,j,k=Y^i,j,k2, for j=0,1, follow (asymptotically)
independent exponential distributions such that
EIi,j,k/fj,k→1 as
T→∞. As a result, the Whittle negative-log-likelihood
approximation L(θj;Ii,j)=∑k=-T/2+1T/2θj,k+Ii,j,kexp(-θj,k) is a reasonable approximation for the
likelihood in the objective function . In the
event that we have multiple realizations, we can take the average periodogram
Ij,k=∑i=1njIi,j,k which follows asymptotically a
gamma distribution with E(Ij,k)=fj,k as before but with
Var(Ij,k)=fj,k/nj (as opposed to Var(Ij,k)=fj,k).
We further linearize the log likelihood and carry out estimation using an
iterative, weighted least squares approach .
Let L(θj)≈∑k=-T/2+1T/2wj,k(mj,k-θj,k)2, with
mj,k=θj,k0+(Ij,k-exp(θj,k(0)))dθj,kdfj,kθj,k(0)=θj,k0+Ij,kexp(-θj,k(0))-1,andwj,k-1=dθj,kdfj,kθj,k(0)2Var(Ij,k)=nj-1.
Thus, this framework can handle the situation in which there is a different
number of independent realizations for the baseline and scenario runs. For
given initial conditions, these computations are iterated until convergence.
Penalty function
Although penalties could be placed on the individual spectral densities
themselves, for our analysis we only need an estimate of the ratio; therefore, we
place the penalty on the log ratio of the spectral densities
θ1-θ0 so that
J(θ0,θ1) can be
written as J(θ1-θ0).
Because we expect the ratio of spectral densities to be smoother than the
individual spectral densities themselves, it makes sense to place the penalty
on this ratio, enabling us to obtain a low-variance estimate of the ratio
while increasing the bias less than we would by smoothing each spectral density
individually. Our penalty function is then placed on the ℓth derivative
of λ(ω)=θ1(ω)-θ0(ω):
J(λ)=(2π)-2ℓ∫-0.50.5λ(ℓ)(ω)2dω. Using
Parseval's identity, this can be written as J(λ)=∑k=-∞∞k2ℓΛk*2,
where Λk* is the kth Fourier coefficient of
λ(ω), Λk*=∫-0.50.5λ(ω)exp(-2πıkω)dω. We then approximate the penalty
function by J(λ)≈∑k=-T/2+1T/2k2ℓΛk2, where Λk is the discrete
Fourier coefficient of λ, Λk=T-1/2∑j=-T/2+1T/2λ(ωj)exp(-2πıkωj).
The objective function that we minimize can then be written as
∑k=-T/2+1T/2n1(m1,k-θ0,k-λk)2+n0(m0,k-θ0,k)2+δk2ℓΛk2,
where, for a given smoothing parameter δ, we can iterate back and
forth between estimates of θ0,k and λk until
convergence. The ratio of spectral densities can be estimated using the
algorithm provided in Sect. S1.
We do not develop an automated method for choosing the smoothing parameter
δ in this paper. In a situation in which multiple realizations of a
climate scenario exist, it may be desirable to choose δ based on a
cross-validation study. Here, we chose δ=e-7≈9.12×10-4, which appears to give good visual results. Using the formula for
effective degrees of freedom given by yields
an approximate bandwidth for this smoother of 0.078 day-1, which is
quite broad considering that we are defining the spectral density on
(-0.5,0.5] day-1. We believe this degree of smoothing is acceptable
given that the estimated log-spectral ratios are quite flat. (As mentioned previously, one advantage
of smoothing on the ratio of spectral densities is that the ratio is flatter
than are the individual spectral densities.) However, we do see some evidence
that the ratio is less flat at the lowest (below-annual) frequencies. For
studies of interannual variability, there could be some advantage in using a
penalty function that allows for more flexibility in λ near the origin by
defining J(λ)=(2π)-2ℓ∫-0.50.5η(ω){λ(ℓ)(ω)}2dω for some
positive, even function η that takes on smaller values near 0. Such a
technique could resolve different changes at different interannual frequencies.
Seasonal cycle and long-term trend
The previous section assumed that the process of interest was a stationary
process with constant mean. Daily mean temperature however involves a strong
seasonal component. So, before estimating the spectral ratio and modifying
the DFT of the observations, we remove the seasonal cycle in the observations
and AOGCM output. The empirical mean of the observations and present–future
difference in the AOGCM output are then added back on at the end of the
algorithm. This part of our approach is analogous to the delta method and in
fact reduces to the delta method when the present and future spectral
densities are equal.
(Top) Log (base 10) of averaged periodograms for the Illinois location,
by season, for the reanalysis (left), model baseline period (middle), and
model scenario period (right). Note strongest variability in winter, weakest
in summer. (Middle) Identical to top but now for the demodulated time series.
Seasonal differences in variability are effectively removed, suggesting we
can treat these time series as stationary in time. (Bottom) Modulation
constants used for the reanalysis (left), model baseline period (middle) and
model scenario period (right), showing smallest values in summer, as
expected. See Figs. S1 and S2 for similar plots for other locations used as
examples; results are similar.
As mentioned previously, the delta method uses model output for changes in
first-order characteristics (e.g., overall mean and seasonal cycle) estimated
from the model output. This method typically involves adding the difference in the
overall mean (usually including the seasonal cycle) of the base and scenario
time slices for the AOGCM to the observations. Let μ^0,t and
μ^1,t represent monthly means or annual harmonics, i.e.,
μ^j,t=μ^j+∑k=1KR^j,kcos(2πωkt+ϕ^j,k) for j=z,0,1, and
ωk=k/365.25. The parameters μ^z, μ^0, and
μ^1 are the estimated long-term averages for the observations, base,
and scenario periods, respectively, and R^z,k, R^0,k, and
R^1,k are the estimated amplitudes at ωk for the
observations, base, and scenario periods, respectively. Lastly,
ϕ^z,k, ϕ^0,k, and ϕ^1,k are the estimated
phase shifts for ωk. All parameters are estimated using least
squares. Seasonal demodulation (Sect. )
is performed on Z̃0,t=Z0,t-μ^z,t,
Ỹ0,t=Y0,t-μ^0,t, and Ỹ1,t=Y1,t-μ^1,t, in order to account for seasonal difference in
second-order moments.
In our example, our AOGCMs have been run far past the point of CO2
stabilization and, therefore, can be considered to be nearly in an equilibrated
state. However, there is evidence of a long-term trend in temperature in the
observations {Z0,t}. We remove this long-term trend from the
observations using a simple linear regression of the observations against the
logarithm of CO2.
Accounting for seasonal nonstationarity
Thus far we have assumed that the deseasonalized observations and model
output, Z̃0,t=Z0,t-μ^z,t,
Ỹ0,t=Y0,t-μ^0,t, and Ỹ1,t=Y1,t-μ^1,t are (temporally) stationary. However, this need
not be the case and in applications involving daily mean temperature it
likely is not the case. Figure shows
the log-averaged periodograms by season for the Illinois pixel for the base
and scenario period, as well as for the observations (similar plots are
provided for the Southern Ocean and Gulf of Guinea in Figs. S1 and S2 in the
Supplement). Clearly, the seasonal spectral density functions for the base
period are different for the different seasons, with the winter months
showing the greatest variability and the summer months the lowest variability across
all frequencies. Note that in the case of the scenario period, variability
across frequencies in the winter has decreased, and is now roughly the same
as in spring and fall, but the summer variability is still roughly the same, and
is lower across most frequencies. Thus, the assumption of temporal
stationarity is not reasonable and, furthermore, the form of the
nonstationarity is somewhat different for the base and scenario periods.
However, the log periodograms for the different seasons are nearly parallel
for both periods, suggesting that it may be reasonable to treat the processes
as uniformly modulated .
Following Priestley, we consider Z̃0,t=Dz,tZ0,t*,
Ỹ1,t=D1,tY1,t*, and Ỹ0,t=D0,tY0,t*, after deseasonalizing, where Zt*,
Y1,t*, and Y0,t* are stationary
processes (corresponding to the observations, model output under scenario
period, and model output under base period, respectively). Then,
Dz,t, D1,t, and
D0,t are modulation constants to be estimated. Thus, if
we can find suitable values for the modulation constants, then we can perform
the spectral density estimation on Y1,t* and
Y0,t*, in order to modify the DFT of
Z0,t*, and then multiply by the constants D1,t as a last
step to account for the nonstationarity across seasons. Our approach to
estimating the modulation constants is provided in Sect. S2.
Figure and Figs. S1 and S2 show
the log-averaged periodograms for the modified process. The log-periodograms
are much closer together than they were originally, suggesting that this
approach accounts for most of the seasonal nonstationarity.
Application
In this section, we continue to illustrate our methodology using NCEP Climate
Forecast System Reanalysis observations and the CCSM3
output described in Appendix . Because the observations and
model output used are of different lengths (32 years and 100 years,
respectively) the Fourier frequencies will be different. As a result, after
estimating the ratio of spectral densities for the model output, we do a
simple linear interpolation on the log-spectral ratio of the model output to
the Fourier frequencies of the observations.
Although we are only modifying the temporal covariance structure, we can
produce maps that show how variability is changing at different locations and
different frequencies (e.g., see Fig. ). In general, at
most locations and at most frequencies, variability is decreasing in the
CCSM3 output for this particular scenario. Variability increases occur only
in a few regions. Increases in lower-frequency (periods close to 3.2 years)
variability appear primarily on land at low latitudes. Increases in
higher-frequency (periods of roughly 2 days) variability occur primarily at
low latitudes over water near coastlines.
Variability clearly changes differently at different locations
(Figs. , ) and, furthermore,
variability changes at a given location can differ with frequency
(Fig. , top panel). In Illinois and the Gulf of Guinea,
there is a modest decrease in low-frequency variability. At high frequencies,
there is a slightly greater suppression of variation in Illinois, whereas in
the Gulf of Guinea high-frequency variation is actually larger for the
future scenario than the present. The decreasing variability at high
frequencies in Illinois may be consistent with suggested changes in the polar
jet stream that impacts weather at the middle latitudes. For the Gulf of
Guinea, the slight suppression of low-frequency variation and the
amplification of high-frequency variation may suggest fundamentally changing
weather patterns at this location. These results show that the manner in
which variability changes is nontrivial and is dependent on the temporal scale.
As a result, an approach that considers changes across a variety of timescales is necessary (as opposed to a simple rescaling of the observations
based on changes in model output).
(Top) Estimated log (base 10) ratio of spectral densities for model
scenario vs. baseline at low and high frequencies. The low-frequency results
are the estimated log ratios at 1168 days and the high-frequency results at 2 days; however, due to the large degree of smoothing, it is best to think of them as
representing low- and high-frequency behavior. Both long- (left column) and
short-term (right column) variability decreases in nearly all locations.
Remainder of rows: estimated log-spectral densities at these frequencies for
reanalysis (second row), model baseline period (third row) and model scenario
period (bottom row), using the demodulated and deseasonalized time series.
The pattern of enhanced variability over land vs. ocean and high vs. low
latitudes is as expected.
(Top) Logarithm (base 10) of the estimated spectral density ratios in
the Southern Ocean (blue), Illinois (green), and Gulf of Guinea (red).
(Bottom) Logarithm of the estimated spectral densities of the reanalysis data
(solid line), base period (dashed line), and scenario period (dashed and
dotted line). The spectral density estimation was performed on the
deseasonalized and demodulated time series.
In contrast to those locations, however, are pixels such as the Southern
Ocean, where the change in variability remains relatively constant across all
frequencies (with approximately a 60 % decrease in overall variability).
For locations that exhibit this type of change in the spectral ratio, a
simple scaling of the observations may be acceptable. However,
Fig. indicates that in all three locations, the
across-frequency variation of the spectral density is greater than the
across-frequency variation of the spectral ratio, supporting our claim that
the spectral ratio is smoother than the spectral densities themselves.
All three locations used as examples show evidence of a mean shift (see
Fig. ). Mean shifts in Illinois and the Gulf of Guinea
are of a few degrees, similar to changes in global mean temperature, though in
the Illinois location the shift is small relative to temperature variability.
In the Southern Ocean location, the mean shift in local Winter (JJA,
June–August) is nearly 10∘, which is likely due to the loss of sea ice in the
future scenario. (Ice cover allows for lower temperatures than are possible over
open ocean). All locations show physically reasonable characteristics in
variability and in changes in variability: variability is stronger in the the
non-equatorial locations (Southern Ocean and Illinois) than near the Equator
and stronger in winter than in summer, and variability reductions are also
greater in winter.
Time series of daily mean temperature (∘C) for 3 years of reanalysis (blue) and simulations (red) at Illinois (top row),
Gulf of Guinea (middle row), and Southern Ocean (bottom row).
An important aspect of our approach is that it does not significantly alter
the non-Gaussian aspects (e.g., tail behavior) of observed climate. In fact,
in our method, when the model does not show changes in mean or covariance,
the simulations are simply the observations and, thus, non-Gaussian features
of the data are retained exactly. When the observations are not significantly
changed, the non-Gaussian features of the data are largely retained. For
instance, JJA in the Gulf of Guinea shows a marginal distribution that is
positively skewed. In this case, the simulation shows a slight decrease in
marginal variability, as well as an increase in mean temperatures, but we
maintain the positive skewness of the observations (see Fig. ).
We consider this to be a strength of our approach: in the event that there
are non-Gaussian features of the data, the simulations will retain these
features, at least when the change in the dependence structure is relatively
small.
Preserving the shapes of distribution of the observations (e.g., skewness,
kurtosis) would be a problem if the actual shapes of distributions changed
from present to future. For locations in or near bodies of water, changes in
temperature means can alter climate variability distributions because those
distributions are sensitive to the freezing point of water. As long as water
is liquid, temperature variability is constrained because air temperatures
cannot drop significantly below freezing. This property is evident in the
marginal distributions for both observations and model output in
Fig. , which show upward bumps in the marginal
densities at near-freezing temperature, i.e., a clustering of temperatures
near freezing. When open water freezes, however, air temperatures can become
very cold. The left-skewed tail in the Southern Ocean local winter in
reanalysis data (Fig. , bottom row JJA) indicates
that the location is sea-ice covered for part of the season. This freezing
point effect introduces further problems when model output is biased relative
to observations. The example of the Southern Ocean location in winter shows
this clearly. The wintertime base period CCSM3 temperatures show a
distribution characteristic of complete sea ice cover rather than the partial
cover implied by observations, with a strong cold bias (a mode over 10∘ below the freezing point) and a wide spread. Because sea ice is lost
in the warmer future scenario, the temperature change between model base and
future periods is very large. The modeled changes in both mean temperature and
temperature variability are therefore physically inappropriate when applied
to observations and produce unphysically high wintertime temperatures and
low wintertime variability in resulting data-driven simulations
(Fig. ). We note that this limitation applies not just to
our approach but to statistical downscaling methods in general. Systematic
offsets in the mean between climate model and data make it difficult to
adequately simulate future climate, so that simulations are inherently
limited by the skill of the AOGCM being used.
Marginal densities (by season) of daily temperature (∘C)
for the pixels at Illinois (top row), Gulf of Guinea (middle row), and
Southern Ocean (bottom row) for reanalysis (blue line) and the simulations
(red line). The simulations display marginal tail behavior similar to the
reanalysis observations.
Discussion
Detailed characterization of the nature in which climate is changing (mean
shifting, tail behavior, spatial and temporal covariance structures) is still
a relatively open area of inquiry. One of the best ways of studying how
future climate might change is by first investigating the nature in which the
statistical properties of the output of AOGCMs change from the present to
possible future scenarios. We have provided a method of quantifying how
temporal covariance is changing in these AOGCMs at different temporal scales.
Our results show that variability is changing differently at different
locations. At a given location, the changes in variability may be different
(in both magnitude, direction of the change, or both) across different
frequencies. We used this estimate of changing variability to produce
simulations that modify the temporal covariance structure of the
observations. In this way, we extend the delta method to be able to account
for changes in the mean and covariance structure.
Our method for producing simulations relies on modifying the discrete Fourier
transform of the observations and, as such, the length of the simulations in
this manner is currently limited by the available data. However, it is
possible that by recycling old observations, one could generate simulations
of longer length. Another possibility is to modify the observations by phase
scrambling and then appending these newer
pseudo-observations to the true observations.
We point out that we have not accounted for any changes in spatial and
spatiotemporal covariance structures. Accounting for changes in spatial
covariance is complicated by the nonstationarity present in the observations
(due to geography, land–ocean contrasts, etc.) and remains the subject of
future research. However, we do note that, due to the use of the
observations, we are mimicking any spatial structure in the present climate
regime.
Next, while we have provided a method for producing simulations of daily mean
temperature, most impacts assessments also rely on simulations of daily
precipitation. The methodology presented here is not fit to handle the highly
non-Gaussian, nonlinear nature of daily precipitation directly; however, a
popular approach in the statistics literature is to model precipitation using
a latent Gaussian process seefor an
example. The approach presented could be applied to such
a latent process. Latent processes might be further extended to consider
joint bias correction and downscaling of temperature and precipitation.
Perhaps most importantly, the methodology presented here is based on the
assumption of stationarity in the model output and the data. While we did
incorporate the concept of a uniformly modulated process to deal with
seasonal nonstationarity, this methodology is still limited to simulating
equilibrium climate. Because for the foreseeable future our climate will be
in a transient state, we must consider ways of extending this methodology to
be able to simulate transient climate. We point out that there is the
potential for this methodology to be extended by considering an evolutionary
spectral approach .
Lastly, our methodology is limited to generating simulations for those GHG
scenarios for which an AOGCM has been run. We cannot produce simulations for
arbitrary GHG emissions scenarios without first running the AOGCM to obtain
the necessary output. However, we note the potential to consider
“emulating” higher-order characteristics in the general circulation models
in order to generate simulations for arbitrary emissions scenarios. For
transient climates, it may be possible to relate changes in the covariance
structure to the past trajectory of CO2.
We believe that our approach provides a general framework for high-resolution
future climate simulation. Two critical features of our approach are that (1) it
is observation-driven, using the model output to suggest how to modify the
existing observations, and (2) it considers changes in both mean and
covariance structures; and this modification of covariance structure, by
taking place in the frequency domain, involves modifying quantities that are
at least approximately independent. We anticipate many opportunities to
extend our framework to generate more realistic simulations for use in
impacts assessment, and suggest that any extensions should seek to maintain
these features when feasible. Society's obvious need for better impacts
assessment begins with a better understanding of how climate will
change in the future. Our conceptual approach provides a valuable framework
for quantifying climate change and simulating future climate in order to meet
that need.