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10 Apr 2019

10 Apr 2019

Projecting the climate using quantile maps

^{1}Department of Statistics, University of Chicago, 5747 S. Ellis Ave, 60637 Chicago, IL, USA^{2}Department of Atmospheric Sciences, University of Illinois at Urbana-Champaign, Urbana, IL, USA

^{1}Department of Statistics, University of Chicago, 5747 S. Ellis Ave, 60637 Chicago, IL, USA^{2}Department of Atmospheric Sciences, University of Illinois at Urbana-Champaign, Urbana, IL, USA

Abstract

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The study of climate change and its impacts depends on generating projections of future temperature and other climate variables. For detailed studies, these projections usually require some combination of numerical simulation and observations, given that simulations of even the current climate do not perfectly reproduce local conditions. We present a methodology for generating future climate projections that takes advantage of the emergence of climate model ensembles, whose large amounts of data allow for detailed modeling of the probability distribution of temperature or other climate variables. The procedure gives us estimated changes in model distributions that are then applied to observations to yield projections that preserve the spatiotemporal dependence in the observations. We use quantile regression to estimate a discrete set of quantiles of daily temperature as a function of seasonality and long-term change, with smooth spline functions of season, long-term trends, and their interactions used as basis functions for the quantile regression. A particular innovation is that more extreme quantiles are modeled as exceedances above less extreme quantiles in a nested fashion, so that the complexity of the model for exceedances decreases the further out into the tail of the distribution one goes. We apply this method to two large ensembles of model runs using the same forcing scenario, both based on versions of the Community Earth System Model (CESM), run at different resolutions. The approach generates observation-based future simulations with no processing or modeling of the observed climate needed other than a simple linear rescaling. The resulting quantile maps illuminate substantial differences between the climate model ensembles, including differences in warming in the Pacific Northwest that are particularly large in the lower quantiles during winter. We show how the availability of two ensembles allows the efficacy of the method to be tested with a “perfect model” approach, in which we estimate transformations using the lower-resolution ensemble and then apply the estimated transformations to single runs from the high-resolution ensemble. Finally, we describe and implement a simple method for adjusting a transformation estimated from a large ensemble of one climate model using only a single run of a second, but hopefully more realistic, climate model.

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Haugen, M. A., Stein, M. L., Sriver, R. L., and Moyer, E. J.: Future climate emulations using quantile regressions on large ensembles, Adv. Stat. Clim. Meteorol. Oceanogr., 5, 37-55, https://doi.org/10.5194/ascmo-5-37-2019, 2019.

1 Introduction

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While numerical simulations of the Earth's climate capture many observed
features of daily temperature, they cannot capture all details of observed
temperature distributions with perfect fidelity. Studies of climate impacts
that require detailed simulation of future climate on fine temporal and
spatial scales therefore often make use of a combination of model output and
observations. There are two fundamental approaches to combining model output
and observations to simulate future climate: *bias correction*, which
adjusts future model output based on the difference between present model
output and observations, and *delta change*, which adjusts present
observations based on the difference between present and future model output
(e.g., Wood et al., 2004; Teutschbein and Seibert, 2012; Gudmundsson et al., 2012).

Any method for simulating future climate under forcing levels for which no
observational data are available (e.g., elevated CO_{2} concentrations)
must make some assumptions about the relationship between the climate model
and the observations. With the bias correction method we would effectively
assume that the error between model and observations is invariant over time
(see for example Li et al., 2010; Wang and Chen, 2014; Cannon et al., 2015). In
contrast, with the delta change method we assume that observations and
climate model would follow the same relationship between the current and
future. Thus, in this latter case, we assume that a map between present and
future that applies to the model output is also appropriate to apply to the
observational data. Specifically, the delta change method only uses the model
output to learn about how the climate will change, not the absolute state of
the climate in either the present or the future (Déqué, 2007; Graham et al., 2007; Olsson et al., 2009; Willems and Vrac, 2011; Themeßl et al., 2012; Leeds et al., 2015). As a consequence, future
simulations corrected with the delta change method will largely retain
properties such as spatiotemporal dependence in the observations that models
may have trouble capturing, insofar as observations and gridded data products
capture the true climate (Dee et al., 2014). To the best of our knowledge, there
is no strong empirical evidence showing when one method or the other will
give more realistic future climate simulations, but we believe that the
current dependencies in the observational record will often provide a more
accurate representation of future dependencies than will future climate model
output. Accordingly, in this work, we focus on methods for simulating future
climate, such as delta change, that adhere to the principle of using model
output only to learn about changes in climate, avoiding any reliance on the
model accurately capturing detailed characteristics of joint distributions of
many climate variables (e.g., Leeds et al., 2015;
Poppick et al., 2017). Nevertheless, we recognize that bias
correction methods are also worth pursuing and consider their strengths and
weaknesses relative to delta change when one has a large ensemble. For either
approach, it is critical to have tools for assessing how well models do in
estimating these changes.

The emergence of large initial-condition ensembles opens up many possibilities for the use of more complex statistical analyses to describe the climate in greater detail. The availability of large ensembles has important, but different, impacts on bias correction and delta change approaches. We can think of both methods as having two steps: estimation of a transformation and then application of that estimated transformation to some set of data to obtain a simulated future. The estimation step for bias correction does not benefit much from the large ensemble because it requires estimation of the distribution of the observations as a function of season and year from the limited observational record, which will remain the weak link in the estimated transformation no matter the size of the ensemble. The simulation step does benefit from the large ensemble, because now we can trivially simulate many futures by applying the estimated bias correction transformation to each member of the ensemble. The situation is reversed for delta change. Now, estimation of the transformation is greatly enhanced by the large ensemble because the transformation is from the model present to the model future and we have both for every ensemble member. On the other hand, for the simulation step, we have to apply the transformation to the observational record, so we have no direct way of producing multiple simulated futures. Which method will be more useful may depend on the available information, the quantity being simulated, and the use to which the simulation will be put. When considering far tails of climate variable distributions, there is some value in being able to make many simulations. Nevertheless, we maintain that it will often be more important to have fewer projections with a well-estimated transformation than more projections with a poorly estimated transformation.

The availability of more than one large ensemble of the same forcing scenario under different models or versions of a model is particularly valuable for multiple reasons. Obviously, but importantly, we can look at how estimated transformations differ across ensembles as a measure of potential model bias. We can also provide a more direct evaluation of the delta change method by using one of the ensembles to estimate the transformations and the other as repeated realizations of a set of pseudo-observations. Specifically, we can apply the transformation estimated from one model to the present data of a second model (i.e., pseudo-observations) and then directly compare their statistical characteristics to the observed future pseudo-observations from this second model. Here, we have two ensembles for the same forcing scenario, both using the Community Earth System Model (CESM, Hurrell et al., 2013), and treat the higher-resolution ensemble as pseudo-observations on which we validate a statistical model built using the other ensemble. This approach is similar to one described in Vrac et al. (2007), which is called the “perfect model” approach in Dixon et al. (2016) and has mainly been used to evaluate statistical downscaling methods. Using two models with one model as pseudo-observations also allows for the estimation of the long-term changes in the bias between models, as discussed in Maraun (2012). The large volume of pseudo-observations allows for detailed evaluations even for fairly extreme quantiles.

One common approach for producing projections of future climate is to use
transformations of quantiles
(Hayhoe et al., 2004; Li et al., 2010; Swain et al., 2014). For $\mathrm{0}\le p\le \mathrm{1}$, a
*p*th quantile for a random quantity *X* is a value *x* for which the
probability that *X* does not exceed *x* equals *p*, so we will call *p* a
non-exceedance probability. The quantile function of *X* is then just a
function from these non-exceedance probabilities *p* to values for *X* and
equals the inverse of the cumulative distribution function (CDF) of *X*.
Quantiles of climate variables are often modeled as fixed within a certain
seasonal range and time span and are assumed to vary smoothly with the
non-exceedance probabilities, either through a linear fit of the
quantile–quantile plot (qq-plot)
(Stoner et al., 2013; Räisänen and Räty, 2013) or through a
parametric or non-parametric fit of the CDF (Watterson, 2008; McGinnis et al., 2015; Miao et al., 2016; Diffenbaugh et al., 2017). To study
seasonality, it is common to apply this approach separately to each season
(Boberg and Christensen, 2012; Maraun, 2013), which can lead to a
discontinuous trend estimate, whereas the true trend should change smoothly
in time. Although efforts have been made to eliminate unrealistic jumps at
certain time points (Piani et al., 2010; Themeßl et al., 2012), we
present an approach that does not require breaking up the data into seasons
or time periods. Specifically, we develop a quantile estimation technique
that accounts for seasonality by exploiting the expectation that quantiles
vary smoothly (both across days within a year and across years for a given
day). In other words, the smoothness constraint is explicitly applied across
time rather than across probabilities. As in prior work by
Haugen et al. (2018), for each quantile, we fit a single model to all
ensemble members, but we extend on that work to better fit extreme quantiles
by modeling them as exceedances over less extreme quantiles. This nesting
allows the use of simpler functions of time for these exceedances, with
smoothness across non-exceedance probabilities still exploited through a
linear interpolation between the estimated quantiles. Combined with the
wealth of information available in a large ensemble, using simpler functions
with fewer degrees of freedom allows for stable estimates even in the
extremes of the temperature distribution, where the number of exceedances (or
effective observations) is lower than in the bulk.

Using this new approach to quantile estimation, we then develop an algorithm for transforming available observations to produce a future simulation. Importantly, given the limited observational record, the algorithm requires only minimal processing of the observational data, just a simple linear rescaling to make the ranges of the observational data comparable to that of the model output. Although the primary algorithm we show is a delta method approach, we describe how the quantile estimation method could be modified to estimate the transformations needed in a bias correction approach. We describe this modification in Sect. 3.1, but, as already noted, such estimates will have much larger variability due to the limited observational record.

In the present study, the two ensembles are of comparable sizes, but it could commonly happen that lower-resolution models would have larger ensembles. This situation is reminiscent of the problem of multi-fidelity surrogate modeling in the computer modeling literature (Kennedy and O'Hagan, 2000). In this case, if one were to assume that the higher-resolution model did a better job of capturing changes in climate, it would be natural to use the results from the limited number of higher-resolution runs to adjust quantile transformations estimated from the larger, lower-resolution ensemble. We develop a simple method for carrying out such an adjustment and apply it using our entire lower-resolution ensemble and a single run from the higher-resolution ensemble.

In the following sections, we describe our data, methods, and results. Section 2 describes the model output and observational data used in this work. Section 3 gives details on our approach to quantile mapping using regression quantiles and how we estimate uncertainties by treating the multiple runs in an ensemble as independent and using jackknifing. Section 4.1 describes the results of our methods for eight cities in the United States, focusing on comparing the estimated transformations for the two ensembles. Section 5 summarizes what we see as the advantages of the proposed methodology over alternative approaches and discusses further thoughts and challenges for producing accurate simulations of future climate.

2 Data

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We consider surface temperature from two climate model ensembles, both
generated with the CESM
(Hurrell et al., 2013). The first is the CESM Large Ensemble
(LENS) (Kay et al., 2015) which is composed of 40 simulations at
∼1^{∘} horizontal resolution from 1920 to 2100 with historical
atmospheric forcings until 2005 and Representative Concentration Pathway 8.5
(RCP 8.5) beyond (Van Vuuren et al., 2011). The second ensemble,
described in Sriver et al. (2015) and referred to here as SFK15, uses
the same forcing scenario and is composed of 50 simulations each at ∼3.75^{∘} horizontal resolution. For simplicity, we demonstrate our
approach using only a subset of the data, from the locations of eight major
US cities (Seattle, Denver, Chicago, New York City, Los Angeles, Phoenix,
Houston, and Atlanta). Parts of both of these ensembles run from 1850 to
2100,
but we only use the period after 1920. For each ensemble, the only difference
between the multiple runs within an ensemble are the initial conditions. As
is common practice, we will assume that the initial-condition sensitivity of
the climate models yields statistically independent runs. There are known
issues with the independence between the LENS simulations, as discussed in
Sriver et al. (2015), along with other minor differences between the
ensembles, but for the purposes of our study, we will assume that all
simulations can be treated as statistically independent. For more information
on the variability in model ensembles, see
Deser et al. (2012), Deser et al. (2014), Vega-Westhoff and Sriver (2017), and Hogan and Sriver (2017).

For the observational record, we use the ERA reanalysis data product (Dee et al., 2011), which is derived from a combination of observations and data assimilation. This reanalysis uses a forecast model run at 12-hourly intervals and is validated against observations. We use the ERA-Interim dataset for the period 1979–2016, since 1979 is when satellite data became available, greatly improving the quality of reanalysis products. For each day we average the 06:00, noon, 18:00, and midnight temperature values (UTC). The daily averages for the model output are averages over 48 30 min time steps across each day.

In this work, we show results for eight cities in the United States with widely varying climate characteristics. To give some idea of how results for the two models compare to each other and to the reanalysis data, Fig. 1 shows average daily temperature distributions over the entire year for these eight cities. The aggregation across all seasons can produce some unusual patterns but, generally speaking, the LENS ensemble provides a better match to the reanalysis than does the SFK15 ensemble. The agreement with reanalysis is perhaps the worst in Los Angeles and Phoenix, both of which have dry climates.

3 Methods

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We present a method that projects observed climate variables into the future by building a quantile map from the ensemble data and avoids complex modeling of the observations due to their relative sparsity compared with the ensemble data. The only processing of the observational data is a linear normalization to put it on approximately the same scale as the model output. The climate model output is also first normalized, but is then nonlinearly transformed by using estimated quantiles from a statistical model of the quantiles as functions of two temporal dimensions, seasonal and long-term, including interactions between them to allow for slowly changing seasonal patterns. Finally, the initial normalizations are inverted to obtain the simulated future temperatures. In the end, every observed temperature is subject to a monotonic transformation into a future temperature so that the transformed temperature time series will largely retain the temporal and spatial dependencies of the observational data.

We first describe the quantile mapping procedure through which we project a set of observations into the future. For now, assume that all relevant quantile functions and normalizing functions are known; see the next subsection for details on how we estimate these functions.

The basic goal of the transformation is to shift each quantile of the present
time series to its corresponding future quantile value. Let *X*(*d*,*t*) be the
observed temperature on the *d*th day of the year in year *t* and write
*F*_{d,t} for the corresponding CDF. Similarly, let *Y*_{j}(*d*,*t*) be the
equivalent model value obtained from simulation *j* in an ensemble of
simulations and write *G*_{d,t} for the corresponding CDF, which is the same
for all *j*. Assume that *t*_{p} is a year for which the observational
record is available, in which case, the model output would also be generally
available, and *t*_{f} is some future year for which only the model
output is available. To transform *X*(*d*,*t*_{p}) to the year
*t*_{f}, we would ideally use
${F}_{d,{t}_{\mathrm{f}}}^{-\mathrm{1}}\left({F}_{d,{t}_{\mathrm{p}}}\left({X}_{d,{t}_{\mathrm{p}}}\right)\right)$.
Unfortunately, we do not have any direct information about
${F}_{d,{t}_{\mathrm{f}}}$. Following the principle of using the model output only
to learn about how the climate changes rather than the absolute state of the
climate, we could instead project *X*(*d*,*t*_{p}) into the future by
using
${G}_{d,{t}_{\mathrm{f}}}^{-\mathrm{1}}\left({G}_{d,{t}_{\mathrm{p}}}\left({X}_{d,{t}_{\mathrm{p}}}\right)\right)$.
This approach is especially attractive in the current setting in that we do
not need to make any inferences about *F*_{d,t} for either the present or the
future, but instead only have to estimate *G*_{d,t}, which we should be able
to estimate well for *t*=*t*_{p} and *t*_{f} using the large
ensemble. However, because the ranges of the observations and model output
may differ substantially at some locations, we choose to first normalize the
observations and model output to put them on comparable scales. Specifically,
we first normalize both observed and model distributions by their respective
median and scale as a function of *d* and *t*, using as a scale measure the
difference between the 0.9 and 0.1 quantiles. We choose a scale measure
based on quantiles somewhat far out into the tails because we wish to make
the ranges of the observations and the model output to be similar, but other
measures of scale could be used. After having transformed the normalized
temperature from *t*_{p} to *t*_{f}, we undo the original
rescaling, taking into account information from the model output on changes
in median and scale.

We now describe this procedure mathematically. Write *m*_{x}(*d*,*t*) and *r*_{x}(*d*,*t*) for, respectively, the estimated median and scale of the observations as
functions of (*d*,*t*). Define the normalized observations

$$\begin{array}{}\text{(1)}& \stackrel{\mathrm{\u0303}}{X}(d,t)={\displaystyle \frac{X(d,t)-{m}_{x}(d,t)}{{r}_{x}(d,t)}}\end{array}$$

and write ${\stackrel{\mathrm{\u0303}}{F}}_{d,t}$ for the corresponding CDF. Denote the equivalent
median and scale functions for the model output by *m*_{y}(*d*,*t*) and *r*_{y}(*d*,*t*)
and write ${\stackrel{\mathrm{\u0303}}{G}}_{d,t}\left(y\right)$ for the CDF of the normalized model output.
Using the rescaled observation $\stackrel{\mathrm{\u0303}}{X}(d,{t}_{\mathrm{p}})$, our projection to year
*t*_{f} is given by

$$\begin{array}{ll}{\displaystyle}\widehat{X}(d,{t}_{\mathrm{f}})& {\displaystyle}={r}_{x}(d,{t}_{\mathrm{p}}){\displaystyle \frac{{r}_{y}(d,{t}_{\mathrm{f}})}{{r}_{y}(d,{t}_{\mathrm{p}})}}{\stackrel{\mathrm{\u0303}}{G}}_{d,{t}_{\mathrm{f}}}^{-\mathrm{1}}\left({\stackrel{\mathrm{\u0303}}{G}}_{d,{t}_{\mathrm{p}}}\left(\stackrel{\mathrm{\u0303}}{X}(d,{t}_{\mathrm{p}})\right)\right)\\ \text{(2)}& {\displaystyle}& {\displaystyle}+{m}_{x}(d,{t}_{\mathrm{p}})+{m}_{y}(d,{t}_{\mathrm{f}})-{m}_{y}(d,{t}_{\mathrm{p}}).\end{array}$$

As we will describe in the next subsection, the quantile function is estimated at a discrete set of non-exceedance probabilities and the quantiles at other probabilities are estimated by linear interpolation. The estimated CDF is then obtained by inverting this estimated quantile function. If an observation is located outside the range of discrete non-exceedance probabilities, we transform it according to the nearest estimated non-exceedance probability. For example, if 0.001 is the lowest non-exceedance probability directly estimated and $\stackrel{\mathrm{\u0303}}{X}(d,{t}_{\mathrm{p}})<{\stackrel{\mathrm{\u0303}}{G}}_{d,{t}_{\mathrm{p}}}^{-\mathrm{1}}\left(\mathrm{0.001}\right)$, then we would define $\widehat{X}(d,{t}_{\mathrm{f}})$ by using Eq. (2) with ${\stackrel{\mathrm{\u0303}}{G}}_{d,{t}_{\mathrm{f}}}^{-\mathrm{1}}\left({\stackrel{\mathrm{\u0303}}{G}}_{d,{t}_{\mathrm{p}}}\left(\stackrel{\mathrm{\u0303}}{X}\right(d,{t}_{\mathrm{p}}\left)\right)\right)$ replaced by $\stackrel{\mathrm{\u0303}}{X}(d,{t}_{\mathrm{p}})+{\stackrel{\mathrm{\u0303}}{G}}_{d,{t}_{\mathrm{f}}}^{-\mathrm{1}}\left(\mathrm{0.001}\right)-{\stackrel{\mathrm{\u0303}}{G}}_{d,{t}_{\mathrm{p}}}^{-\mathrm{1}}\left(\mathrm{0.001}\right)$. Other schemes could also be employed outside the quantile domain. For example, an extreme value distribution could be incorporated outside some threshold. The large ensemble allows us to obtain stable estimates of fairly extreme quantiles without making specific assumptions about the tail behavior of the distributions as required by methods based on extreme value theory, so we choose to avoid these methods here. For temperature distributions, which generally follow distributions with thin tails (Brown et al., 2008) so that ${\stackrel{\mathrm{\u0303}}{G}}_{d,{t}_{\mathrm{p}}}^{-\mathrm{1}}\left(\mathrm{0.001}\right)-\stackrel{\mathrm{\u0303}}{X}(d,{t}_{\mathrm{p}})$ will be very small with high probability, we believe our approach should work well.

This procedure implicitly makes the assumption that the observational quantile map from the present to the future is the same as the climate model's map from the present to the future, which can be broken down into the following three components:

$$\begin{array}{}\text{(3)}& {\stackrel{\mathrm{\u0303}}{F}}_{d,{t}_{\mathrm{f}}}^{-\mathrm{1}}\left({\stackrel{\mathrm{\u0303}}{F}}_{d,{t}_{\mathrm{p}}}(\cdot )\right)={\stackrel{\mathrm{\u0303}}{G}}_{d,{t}_{\mathrm{f}}}^{-\mathrm{1}}\left({\stackrel{\mathrm{\u0303}}{G}}_{d,{t}_{\mathrm{p}}}(\cdot )\right),\end{array}$$

$${m}_{x}(d,{t}_{\mathrm{f}})-{m}_{x}(d,{t}_{\mathrm{p}})={m}_{y}(d,{t}_{\mathrm{f}})-{m}_{y}(d,{t}_{\mathrm{p}}),$$

and

$$\frac{{r}_{x}(d,{t}_{\mathrm{f}})}{{r}_{x}(d,{t}_{\mathrm{p}})}}={\displaystyle \frac{{r}_{y}(d,{t}_{\mathrm{f}})}{{r}_{y}(d,{t}_{\mathrm{p}})}}.$$

We see that all three of these assumptions only require that the model correctly describes changes in distributions between the present and future, with the first corresponding to changes in shape, the second to changes in location and the third to changes in scale. We have no direct empirical method for checking these assumptions. However, by treating the higher-resolution ensemble as pseudo-observations, we will be able to gain indirect evidence on their accuracy.

A pictorial representation of the steps of the transformation for the LENS
pixel including Chicago is given in Fig. 2. The first
panel of this figure shows a set of wintertime observations from the
reanalysis in black, so essentially a set of *X*(*d*,*t*_{p}) values for a
range of days and years. LENS values for this same range of days and years
are shown in red, or *Y*_{j}(*d*,*t*_{p}) values, and LENS values for the
same days but a set of future years, or *Y*_{j}(*d*,*t*_{f}) values, are
shown in green. The middle panel pictures normalized temperature values. The
black histogram in the middle panel shows the normalized reanalysis values,
or the $\stackrel{\mathrm{\u0303}}{X}(d,{t}_{\mathrm{p}})$ values. The histogram of transformed
normalized temperature, or ${\stackrel{\mathrm{\u0303}}{G}}_{d,{t}_{\mathrm{f}}}^{-\mathrm{1}}\left({\stackrel{\mathrm{\u0303}}{G}}_{d,{t}_{\mathrm{p}}}\left(\stackrel{\mathrm{\u0303}}{X}\right(d,{t}_{\mathrm{p}}\left)\right)\right)$ values,
is given in dark blue. The dark blue and black histograms are difficult to
distinguish, suggesting that the estimated shape of the transformed
observations does not greatly change, at least for the bulk of the
distribution. The light blue curve adjusts for changes in location and scale,
viz. ${\stackrel{\mathrm{\u0303}}{G}}_{d,{t}_{\mathrm{f}}}^{-\mathrm{1}}\left({\stackrel{\mathrm{\u0303}}{G}}_{d,{t}_{\mathrm{p}}}\left(\stackrel{\mathrm{\u0303}}{X}\right(d,{t}_{\mathrm{p}}\left)\right)\right){r}_{y}(d,{t}_{\mathrm{f}})/{r}_{y}(d,{t}_{\mathrm{p}})+{m}_{y}(d,{t}_{\mathrm{f}})-{m}_{x}(d,{t}_{\mathrm{p}})$,
and shows a substantial warming as well as some increase in spread in
normalized temperature. The right panel shows, in dark green, the transformed
temperature distribution, or $\widehat{X}(d,{t}_{\mathrm{f}})$ values. The qq-plot
for $\widehat{X}(d,{t}_{\mathrm{f}})$ vs. *X*(*d*,*t*_{p}) values in the right
panel shows the coldest temperatures increasing substantially more than the
rest of the distribution, so that in fact there is a noticeable nonlinear
aspect to the transformation.

Our procedure for carrying out this transformation is novel in several ways. First, for any given day of the year, it naturally allows a separate transformation from any one year during which both model output and observations are available to any year in which model output is available. Second, it separately accounts for linear transformations in both the observations and model output, with the consequence that if the results from the model output only show linear transformations in temperature between the present and the future, then the transformation applied to the observations will also be linear. Third, it transforms even highly extreme observations in a natural way without making strong assumptions about the tails of temperature distributions.

Although outside the scope of this paper, our technique can be modified to swap the assumptions in Eq. (2) with the assumption that the discrepancy between model and observations is time-independent, as would be the case in a bias correction approach. That is, following Li et al. (2010) for example, we would replace the assumption of Eq. (2) with ${\stackrel{\mathrm{\u0303}}{F}}_{d,{t}_{\mathrm{p}}}^{-\mathrm{1}}\left({\stackrel{\mathrm{\u0303}}{G}}_{d,{t}_{\mathrm{p}}}(\cdot )\right)={\stackrel{\mathrm{\u0303}}{F}}_{d,{t}_{\mathrm{f}}}^{-\mathrm{1}}\left({\stackrel{\mathrm{\u0303}}{G}}_{d,{t}_{\mathrm{f}}}(\cdot )\right).$ The transformation recommended by Li et al. (2010) is, leaving out normalization,

$$\begin{array}{ll}{\displaystyle}\widehat{X}(d,{t}_{\mathrm{f}})& {\displaystyle}=Y(d,{t}_{\mathrm{f}})+{F}_{d,{t}_{\mathrm{p}}}^{-\mathrm{1}}\left({G}_{d,{t}_{\mathrm{f}}}\right(Y(d,{t}_{\mathrm{f}}))\\ \text{(4)}& {\displaystyle}& {\displaystyle}-{G}_{d,{t}_{\mathrm{p}}}^{-\mathrm{1}}\left({G}_{d,{t}_{\mathrm{f}}}\left(Y\right(d,{t}_{\mathrm{f}}\left)\right)\right)).\end{array}$$

The fundamental difference in these two approaches is that in
Eq. (3), we modify model output instead of projecting observations
as is done in Eq. (2). When a large ensemble of model output is
available, a drawback of Eq. (3) is the need to estimate the
observational CDF *F*, which has fewer data to support its estimation.
Nevertheless, both techniques Eqs. (2) and (3) allow for
a smooth seasonal trend fitted on all data simultaneously. As discussed in
Leeds et al. (2015), another important difference between these two
approaches is that Eq. (2) retains characteristics of spatial and
temporal dependence in the observations, whereas Eq. (3) assumes
model output captures these dependencies appropriately.
Leeds et al. (2015) and Poppick et al. (2017) describe methods
for transforming observed daily temperature time series that allows for
changes in means, variances, and temporal correlations in the future, but does
not consider changes in the shapes of distributions. We discuss the
possibility of combining this approach with the methods described here to
allow for changes in both marginal distributions and temporal dependence of
daily temperatures in Sect. 5.

A large body of work has been dedicated to fitting quantiles (Koenker, 2005), mostly from a frequentist perspective (Koenker, 2004; McKinnon et al., 2016; Rhines et al., 2017), although there is a Bayesian literature for quantile regression (Yu and Moyeed, 2001; Reich et al., 2011; Kozumi and Kobayashi, 2011; Geraci and Bottai, 2014). We are particularly interested in fitting quantiles while imposing a smoothness constraint on the estimates (Koenker and Schorfheide, 1994; Yu and Jones, 1998; Oh et al., 2011; Reiss and Huang, 2012).

The methods for estimating quantiles used here expand on the approach used in Haugen et al. (2018), where a fixed set of spline functions were used as a basis for fitting each quantile, similar to Wei et al. (2006). The main novelty over Haugen et al. (2018) is to estimate more extreme quantiles as exceedances beyond less extreme quantiles. The idea of using estimates at less extreme quantiles to constrain estimates at more extreme quantiles also appears in Liu and Wu (2009), with the added constraint that estimated quantiles should not cross. Subject to this non-crossing constraint, Liu and Wu (2009) use the same set of regressors for all quantiles, whereas we reduce the number of regressors for estimating exceedances beyond certain quantiles. Due to the large number of observations available (180 years and 50 simulations), the quantiles are fairly well estimated for the number of spline functions used and we did not have any problems with estimated quantiles crossing. Nevertheless, especially when estimating a large number of quantiles, methods that guarantee no crossings may be advisable. For a more in-depth discussion of quantiles crossing, see for example Bondell et al. (2010) and references therein.

We model the CDF of temperature by assuming each quantile is smoothly varying
in both temporal dimensions, day of the year $d\in [\mathrm{1},\mathrm{365}]$ (there is no leap
year in the CESM), and year $t\in [\mathrm{1920},\mathrm{2099}]$, using splines of different
complexity as predictors of the quantiles. The seasonal basis functions are
constrained to periodic splines, while the long-term basis functions are
constrained to natural splines (Friedman et al., 2001). We also include
historical zonal volcanic forcings as a regressor, *Z*_{volc}
(Sriver et al., 2015). For quantiles between 0.1 and 0.9, we largely
follow the approach taken in Haugen et al. (2018). Specifically, we
find that seasonal effects for quantiles between 0.1 and 0.9 are well
described by a periodic spline with 14 degrees of freedom, and long-term
yearly effects are modeled by a natural spline with 6 degrees of freedom. We
include interaction terms between seasonal and long-term effects to allow
seasonal patterns to change slowly over time, but only use 3 seasonal basis
functions with the same 6 long-term basis functions resulting in $\mathrm{6}\times \mathrm{3}=\mathrm{18}$ degrees of freedom to capture this effect.

Write $Z=[{Z}_{\mathrm{1}},{Z}_{\mathrm{2}},\mathrm{\dots},{Z}_{k},{Z}_{\mathrm{volc}}]$ for the matrix whose columns are
the values for these spline basis functions for each day and year. Each row
of *Z* corresponds to a day of the year, *d*, and a year, *t*, and is denoted
$Z(d,t)\in {\mathbb{R}}^{k+\mathrm{1}}$. If we denote each temperature observation
as *X*(*d*,*t*), then quantiles are estimated by $Z{\widehat{\mathit{\beta}}}_{p}$ at the *p*th
percentile where *β*_{p} is obtained by solving the following optimization
problem (Koenker and Bassett, 1978),

$$\begin{array}{ll}{\displaystyle}{\widehat{\mathit{\beta}}}_{p}& {\displaystyle}=\underset{\mathit{\beta}}{\text{min}}\left\{\sum _{d,t:X(d,t)\ge Z(d,t)\mathit{\beta}}p\left|X\right(d,t)-Z(d,t\left)\mathit{\beta}\right|\right.\\ \text{(5)}& {\displaystyle}& {\displaystyle}+\left.\sum _{d,t:X(d,t)<Z(d,t)\mathit{\beta}}(\mathrm{1}-p)\left|X\right(d,t)-Z(d,t\left)\mathit{\beta}\right|\right\}.\end{array}$$

When we are fitting quantiles to model output, the sum in Eq. (3)
should be understood as summing over all simulations as well as all years and
days. It is possible to use more formal methods to select the complexity of
the model, as in Koenker and Schorfheide (1994), Oh et al. (2011),
Daouia et al. (2013), and Fasiolo et al. (2017) for example. However, the basis
functions we have chosen allow considerable flexibility in the evolution of
seasonally varying quantiles, with the resulting estimates having quite small
variances (see Sect. 4.1), so there may not be much to be
gained by trying to optimize the selection of the number of basis functions.
All computations are performed in R (R Core Team, 2013) and the quantile regressions
are done with the package `quantreg`

(Koenker, 2017).

We are particularly interested in estimating fairly extreme quantiles. The further out in the tail of the distribution one goes, the less effective information there is in the data about the corresponding quantile. Thus, we choose to use simpler models for exceedances beyond two well-estimated quantiles, here the 0.1 and 0.9 quantiles. Specifically, after first using a set of predictors to normalize the data, we fit another set of coefficients on the normalized data with the same predictors for quantiles between the 0.1 and 0.9 quantiles inclusive. Next, we fit more extreme quantiles of the distribution as exceedances outside the 0.1 and 0.9 quantiles, using a similar model but with fewer basis functions: a 3 degree of freedom seasonal spline and a linear time trend without interaction between these two dimensions. The effect of modeling the tails as exceedances is that the bulk trend is added on top of the tail trend and thus we retain a high level of detail in the tails in spite of the relatively lower degrees of freedom in the predictors. The tail quantiles estimated in this way are 0.001, 0.010, 0.025, 0.050, and 0.075 and 1 minus each of these values.

When projecting the observations, we consider the time-window 1979–2016, which we project 80 years into the future (years 2059–2096). As previously mentioned, we do not require a detailed modeling of the CDF of the observations, merely an estimate of the location and scale as a function of the two temporal dimensions. Because of the reduced time span of these observations we use fewer degrees of freedom in the splines, namely 10 seasonal and 1 temporal (i.e., linear) without any interaction terms. This normalization step is only included to put the observations and model output on the same scale, so it is not essential that the model used to normalize the observations captures finer details of the median and scale functions.

As already noted, we include zonal volcanic forcings in the statistical model
as they are included in the CESM climate model for the historical period and
cause substantial short-term changes in temperature. We also tried including
the CO_{2} forcings in our model, but they did not improve the
predictions when used in addition to the spline in the yearly parameter, *t*,
and were thus not included in our final model.

The availability of a large ensemble makes it possible to quantify the
uncertainty of our estimates based on treating the *n* different ensemble
members as independent and identically distributed realizations of the same
stochastic process and thus avoid any modeling of dependencies within a
single realization. Castruccio and Stein (2013) and Haugen et al. (2018),
among others, have similarly exploited this assumption to produce uncertainty
estimates based on climate model output using bootstrap procedures. Here, we
choose to use jackknife estimates of bias and variance
(Davison and Hinkley, 1997), another standard resampling approach that only
requires re-estimating the model as many times as the number of ensemble
members. Specifically, holding out each simulation in the ensemble one at a
time and calculating a quantile map with the remaining simulations yield
*n*=40 different quantile maps for the LENS ensemble from which the jackknife
bias and variance are estimated. For each held-out simulation *j* we calculate
an “influence function,”
${l}_{\mathrm{jack},j}=(n-\mathrm{1})(T-{T}_{-j})$,
where *T* and *T*_{−j} are in our case the projected temperature quantiles
with all simulations and with one simulation held out, respectively. Then,
the jackknife estimate of variance is given by

$$\begin{array}{}\text{(6)}& {v}_{\mathrm{jack}}={\displaystyle \frac{\mathrm{1}}{n(n-\mathrm{1})}}\left(\sum {l}_{\mathrm{jack},j}^{\mathrm{2}}-n{b}_{\mathrm{jack}}^{\mathrm{2}}\right),\end{array}$$

where ${b}_{\mathrm{jack}}=-{\sum}_{j=\mathrm{1}}^{n}{l}_{\mathrm{jack},j}$ is the jackknife estimate of bias (Davison and Hinkley, 1997). As we will see in Sect. 4.1, the jackknife standard errors are quite small and, in particular, are generally much smaller than the noteworthy differences between the estimated transformations based on the two different ensembles. Thus, we will not generally report these standard errors, since they are likely a smaller source of uncertainty than unknown model biases. Nevertheless, the jackknife is a promising technique to assess the uncertainty of the projection due to the simulation variability. Although it is not obvious from Fig. 3, the jackknife estimate of variance scales inversely with the ensemble size, ${v}_{\mathrm{jack}}=O(\mathrm{1}/n)$ (Efron, 1981).

There is, of course, no direct way to evaluate the accuracy of a future climate projection using currently available data. However, if we are willing to assume that the discrepancies in the statistical characteristics of some quantity of interest between two models are comparable to the discrepancies between the output from some model and reality, then we can at least get a qualitative idea as to how well the procedure is working. Perhaps more realistically, we may be able to obtain something like a lower bound on the actual errors in the statistical characteristics of our climate projections. Specifically, by treating the output from a second model as pseudo-observations, we can apply our transformation method and then directly compare the statistical characteristics of these transformed pseudo-observations to the available future model output from this second model. This basic idea has been used frequently for evaluating statistical downscaling procedures, where it is called the perfect model approach (Dixon et al., 2016), but it is equally appropriate in the present setting of projecting observations into the future.

There are a number of ways one could summarize the results of such an
analysis. One method is to fix a particular quantile, *q*, and apply the
transformation from year *t*_{p}=1920 to some future year *t*_{f} for each day of
the year. Both quantile maps are then plotted as a function of *t*_{f} and day
of the year and differences between the two quantile maps can be taken for a
succinct comparison, viz.

$$\begin{array}{}\text{(7)}& \left({\widehat{G}}_{d,{t}_{\mathrm{p}}}^{\mathrm{L}}{)}^{-\mathrm{1}}{\widehat{G}}_{d,{t}_{\mathrm{f}}}^{\mathrm{L}}\right(q)-({\widehat{G}}_{d,{t}_{\mathrm{f}}}^{\mathrm{S}}{)}^{-\mathrm{1}}{\widehat{G}}_{d,{t}_{\mathrm{p}}}^{\mathrm{S}}\left(q\right),\end{array}$$

where the superscript L refers to estimates based on the LENS ensemble and S to the SFK15 ensemble. Another approach is to first apply a projection based on one of the models forward in time and then apply the second model's projection backwards in time to the original time point. To be explicit, we perform the following operation,

$$\begin{array}{}\text{(8)}& \left({\widehat{G}}_{d,{t}_{\mathrm{p}}}^{\mathrm{S}}{)}^{-\mathrm{1}}{\widehat{G}}_{d,{t}_{\mathrm{f}}}^{\mathrm{S}}\right({\widehat{G}}_{d,{t}_{\mathrm{f}}}^{\mathrm{L}}{)}^{-\mathrm{1}}{\widehat{G}}_{d,{t}_{\mathrm{p}}}^{\mathrm{L}}\left(q\right).\end{array}$$

If there is a close correspondence between the models, this double quantile map will be close to the identity map for all days and years.

4 Results

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We report here results for eight cities in the United States representing a
broad range of geographical settings, where we build a detailed statistical
model using both the LENS and the SFK15 ensembles. These statistical models
define quantile maps that we use to project present observational data into
the future. We see in Fig. 1
that LENS and SFK15 ensembles differ both from each other and from
observations, but those present-day biases do not mean that the projected
future changes for the two ensembles will necessarily be very different. The
statistical model helps to illuminate evolving differences between the two
ensembles. As a preliminary illustration, we show in Fig. 3
a simple median quantile fit of ensemble output at an example grid cell
containing Seattle in historical and future years. The magnitude of the
differences between the median fits is substantial, with LENS being warmer in
all seasons other than winter. Temperature differences grow in the future
projection, suggesting a greater warming in the LENS ensemble. Since both
ensembles use the same forcing scenario, the discrepancy between the future
temperatures could be attributed to the different model resolutions used in
the two ensembles (∼1^{∘} vs. ∼3.75^{∘}). For example,
different resolutions capture the land–sea contrast and land topography
differently.

For both ensembles, we estimate a quantile map as a function of day of the year and long-term time for each of the eight locations using the methods in Sect. 3. For visualization purposes we choose to show the temperature quantile shift for quantiles between 0.001 and 0.999 from 1989 to 2099 as a function of day of the year, as seen in Fig. 4. This figure captures the nuances and complexities of the local temperature distributions. Notice the gradual yet distinct change from season to season. In the tails there is sometimes a large divergence of the temperature shift, particularly for Chicago in the winter, which shows the complexities in how quantiles change over time and the necessity of a detailed statistical model. Each location has unique features, but there are nontrivial differences between the two ensembles' quantile maps for all eight locations. Maybe the most important difference is the increased overall warming of the LENS compared to the SFK15 ensemble. Both ensembles show larger warming in the winter in the northern cities with the exception of Seattle. Thus, for at least Denver, Chicago and New York City, either ensemble will yield transformations that will especially increase the coldest wintertime temperature values.

We next examine the evolution in time of quantiles implied by these quantile
surfaces. Consider, for each day of the year, the difference between a given
quantile in 1920 and the same quantile for every year 1920+*t* for
$t=\mathrm{0},\mathrm{\dots},\mathrm{180}$. Note this difference is necessarily 0 for *t*=0. As
expected, all quantiles monotonically warm in *t* for all seasons and for
both ensembles (Fig. A1). Of perhaps greater
interest is the difference between these two transformations, using
Eq. (4) and plotted in Fig. 5.
We see that the LENS ensemble
mostly shows increasingly greater warming than SFK15 as *t* increases, with
differences of up to 5 ^{∘}C for non-exceedance probability 0.01 during
the winter; see for example Maraun (2013) for related discussion.

Another way to assess how well the transformation approach might work in practice is the double transformation given by Fig. (5). We see significant seasonally dependent discrepancies between the two projection models derived from the LENS and the SFK15 ensemble (Fig. 6). The more prominent discrepancies are many times larger than the standard errors for the LENS transformation (Fig. 7) given by the jackknife procedure described in Sect. 3.3 and, therefore, would be statistically significant by any reasonable measure. Thus, to avoid obscuring the figures, we do not display statistical uncertainties in our figures for quantile maps.

To show explicitly how our procedure benefits from the large ensemble, we
split up the 40-run LENS ensemble into 8 ensembles of size 5 and estimate the
transformation for each of them. Variations between these mini-ensemble
temperature projections are on the order of 1 ^{∘}C as shown in
Fig. 8 where we show, relative to the results
for the full 40-simulation ensemble, 8 individual quantile maps using 5 LENS
simulations each for Chicago. There is some notable variability in the
mini-ensemble results. For example, quantile maps 1 and 6 show substantial
differences from each other and from the full ensemble results, especially at
more extreme quantiles. Nevertheless, comparing to the quantile map for the
full ensemble in Fig. 4, these variations are small enough
that the results from a 5-member ensemble are still informative about some
aspects of the seasonally varying quantile transformation.

We next employ the perfect model approach described in Sect. 3.4. Specifically, using the model built from SFK15 ensemble for the whole simulation period 1920–2100, we treat the portion of the LENS ensemble from 1979–2016 as our pseudo-observational data and project this forward in time by 80 years and compare with the LENS data during this future time, namely 2059–2096. Each of the 40 LENS simulations are treated as independent pseudo-observational data and are normalized individually using 10 seasonal and 1 temporal degree(s) of freedom, without interaction terms. Using this normalization and the quantile map from the SFK15 ensemble, the LENS data from 1979–2016 are transformed using Eq. (2). For comparison, we also give results applying the quantile map estimated from the full LENS ensemble to a single LENS run treated as pseudo-data, which should work very well since we are testing the LENS model on itself.

An example comparison between the two models is shown in Fig. 9, where we show winter temperature projections near Seattle with histograms and qq-plots. Figure 10 compares projections for all eight locations (see Fig. A2 for summertime results); it is clear that simulating data from the same ensemble as used in building the quantile map yields much better agreement than using a quantile map from the other ensemble. This difference was already foreshadowed when we compared the quantile maps in Fig. 4. The difference is not a function of the particular ensemble member chosen. Since the LENS ensemble includes 40 simulations, we can apply this perfect model test to all 40 of them, yielding a much larger sample of projections. Resulting histograms and qq-plots are smoother, but look very similar to Fig. 8 (Fig. A3). One might suspect that the difference between ensembles stems from their different resolutions. While resolution may play some role, a simple upscaling of the LENS data to match the SFK15 data does not ameliorate this discrepancy (results not shown).

In the present setting we have roughly equal numbers of runs for both
resolutions of CESM. In many instances, one might have many more runs at
lower resolution because of the lower cost of those runs. This situation
bears some resemblance to what is known as multi-fidelity surrogate modeling
in the computer modeling literature; see for example Kennedy and O'Hagan (2000), Forrester et al. (2007), Le Gratiet and Garnier (2014), and He et al. (2017). The goal in these
works is generally to predict the value of some fairly low-dimensional result
of some process, perhaps even univariate, as a function of some input
parameters, by using a well-chosen combination of observations, a small
number of high-fidelity computer model simulations and a larger number of
lower-fidelity simulations. In the present work, we are focusing on a single
scenario of forcings, so that the problem of interpolating in parameter space
is not addressed here. The present situation's difficulty is caused by the
highly multivariate and stochastic nature of the process we seek to simulate
(the space–time climate system) and that we are most interested in the
simulation for a setting for which we have no observational data (a future
with much higher CO_{2} concentrations than exists in the historical record).
There could be scope for developing analogs for the types of methods first
introduced in Kennedy and O'Hagan (2000) that assume that the computer models of
various fidelity and the observed process can be modeled as jointly Gaussian
processes that depend on the input parameters of interest, but here we just
consider a very simple method for combining the full ensemble of SFK15
simulations with a single run of the LENS ensemble as a tool for estimating
the quantile transformations we need to simulate future climate.

To fit the SFK15 quantile map followed by a LENS re-calibration, we first use
the detailed statistical model (or quantile map) applied in previous sections
to the 50 SFK15 simulations, so including an intercept, 14 seasonal terms,
6 long-term temporal terms, 18 interaction terms, and a volcanic forcing term. We
then reduce these 40 predictors to 4 components: component 1, an intercept;
component 2, temporal (including the volcanic term); component 3, seasonal;
and component 4, interactions. Specifically, using the notation from
Sect. 3.2 and *S*(*j*) as the set of indices for each of
the four components,

$$\begin{array}{}\text{(9)}& {W}_{j}(d,t)=\sum _{i\in S\left(j\right)}{Z}_{i}(d,t){\widehat{\mathit{\beta}}}_{i},j=\mathrm{1},\mathrm{\dots},\mathrm{4},\end{array}$$

where ${\widehat{\mathit{\beta}}}_{i}$ represents the fitted coefficient of basis function
*Z*_{i} from the original quantile map using the full SFK15 ensemble. Thus,
${\sum}_{j=\mathrm{1}}^{\mathrm{4}}{W}_{j}(d,t)$ is the quantile map for day *d* and time *t* based on
the SFK15 ensemble. We recalibrate this quantile estimate using an estimate
of the form ${\mathit{\gamma}}_{\mathrm{1}}{W}_{\mathrm{1}}(d,t)+{\mathit{\gamma}}_{\mathrm{2}}{W}_{\mathrm{2}}(d,t)+{W}_{\mathrm{3}}(d,t)+{W}_{\mathrm{4}}(d,t)$,
estimating *γ*_{1} and *γ*_{2} using quantile regression applied to a
single LENS run. That is, only the temporal and the intercept components are
re-estimated for each quantile simulation using Eq. (3); the seasonal
and the interaction terms are kept at their estimated values from the SFK15
fit. Other choices are certainly possible here, but even this simple
procedure is sufficient to show the benefit of this type of re-calibration.

Using this quantile map, we can then project observations into the future using the procedure described in Sect. 3. For Seattle, we projected each year of reanalysis data into the single future year 2099 based on the full SFK15 ensemble, the full LENS ensemble and the full SKF15 ensemble re-calibrated using one LENS simulation. We chose winter in Seattle because estimated shape transformations for SFK15 and LENS showed large differences for this location and season and so should provide a difficult challenge for our approach. Nevertheless, the results, shown in Fig. 11, demonstrate that we can largely correct for the differences between the projections based on the full SFK15 and LENS ensembles by using just one LENS simulation in addition to the full SFK15 ensemble, although we can see that the coldest temperature values are not warmed sufficiently. The method should work even better for Seattle summers or any time of year for Phoenix.

5 Discussion and conclusions

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Taking advantage of two large ensembles of climate model output, we project observations into the future based on their seasonality and long-term time evolution from 1920 to 2099. By projecting the whole observational record simultaneously, temporal and spatial dependencies in the observational record that a climate model may not accurately capture are maintained in the future projections. Our method, building on the quantile regression approach in Haugen et al. (2018), differs from other methods by computing quantile maps simultaneously for all data available in the ensemble, instead of using a window of nearby data or separating data by season or month. Using two temporal dimensions in the statistical model for the quantile map yields insight into the way temperature transitions across seasons. While this technique is primarily focused toward projecting current observations into the future, it could be adapted to bias-correct future model output (see for example Li et al., 2010).

The ensemble of simulations makes it possible to estimate the uncertainty of the quantile projections based on the jackknife variance, which shows that the simulation standard deviation is approximately one order of magnitude lower than the larger model discrepancies as seen when comparing projections between the LENS and the SFK15 ensemble. Thus, efforts to realistically assess the full uncertainty in estimated transformations based on large ensembles should focus on the impact of model biases on these estimates.

To assess how well our method works, we use two ensembles, one treated as pseudo-observations and the other to build our quantile map. Since the LENS data have marginal distributions that resemble the ERA-Interim data more closely (Fig. 1), this ensemble is treated as pseudo-observations. We see substantial and statistically significant differences between the two ensembles, where the higher-resolution ensemble shows greater warming, especially at lower quantiles in some locations.

The nested statistical model and the large ensemble used in building the quantile map enable stable estimates of quantiles in the extreme tails, e.g., the 0.001 and the 0.999 quantiles. When nesting models, we first estimate the 0.1 and the 0.9 quantiles and then consider only the exceedances beyond these quantiles to model any more extreme quantiles. These further quantiles are modeled with fewer degrees of freedom to match the fewer observations available beyond these thresholds, providing a more stable estimate. Extreme quantiles beyond the 0.999 quantile are all projected by the same amount as a function of seasonality, avoiding further parametrization using extreme value distributions.

Following Leeds et al. (2015) and Poppick et al. (2017), we advocate for the use of delta change approaches to future climate simulation when feasible. The goal in these works was to use model output to learn about changes in temporal means and covariances in daily temperature and then make transformations in the spectral domain to modify observational data to capture these changes in means and covariances. The transformations used were linear in the observations and, hence, did not address possible changes in the shapes of temperature distributions. As the present work shows, both CESM ensembles show clear changes in the shapes of daily temperature distributions. Thus, it would be of interest to develop a delta change method that could account for changes in both shapes and in covariances. One possibility would be to first employ the methods in Poppick et al. (2017) for transforming means and covariances in transient climates and then apply the methods developed here to change shapes of distributions.

Having estimated how every year in the observational record would be projected into any future year spanned by the climate model, we can produce projections spanning different numbers of years than the original observational record. For example, if we take the observational record to be of 40 years length, which is the approximate length of the post-satellite reanalysis era, we are not just restricted to simulating a single 40-year period but can for example instead produce four projections of the same future decade by separately projecting each of four 10-year periods in the observational record into that decade. Similarly we can project every observed year into one future year, e.g., 2099 (see Fig. 11). Same-year projections open up the possibility of quantifying the uncertainties in making projections for a given future year due to having a limited observational record by treating each of these projected years as an independent sample from the relevant distribution.

If it is expensive to run a high-quality or high-resolution model, an alternative we propose is to first build a quantile map using a less expensive model, then reduce the dimensionality of the prediction space and recalibrate the quantile map with the limited data from the expensive model by re-estimating the regression parameters in this lower-dimensional space. While the results show a clear improvement in the projections after the recalibration, there is clearly scope for further development using this approach.

In this work, we have only modeled marginal distributions of a single quantity, average daily temperature. The general methodology can be applied to other quantities, although some care would be needed in extending the approach to daily precipitation, which will generally have a large fraction of zeroes (Friederichs and Hense, 2007; Wang and Chen, 2014). A simple way to handle transformations of multiple variables is to apply individual univariate transformations to each variable separately. Since the transformations will be applied to observational data, the transformed data will largely retain the dependencies between the quantities that exist in the observations, in keeping with the principle of relying on the data rather than the model output to capture complex dependencies in the climate system. Note that this approach assumes these spatiotemporal dependencies do not change over time. Thus, if the climate model projects a credible change in this dependency structure, it would not be present in the projection (Zscheischler and Seneviratne, 2017). As an extension, one could consider an approach inspired by Bürger et al. (2011) and Cannon (2016) to match the multivariate correlation structure with the future (changed) correlation structure.

Code and data availability

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Code and data availability.

Instructions to reproduce and extend the results from this paper are available at https://github.com/matzhaugen/future-climate-emulations-analysis (last access: 18 March 2019), from where the raw data can also be accessed.

Appendix A

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Competing interests

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Competing interests.

The authors declare that they have no conflict of interest.

Acknowledgements

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Acknowledgements.

This work was supported in part by STATMOS, the Research Network for Statistical Methods for Atmospheric and Oceanic Sciences (NSF-DMS awards 1106862, 1106974 and 1107046), and by RDCEP, the University of Chicago Center for Robust Decision-making in Climate and Energy Policy (NSF-SES awards 0951576 and 146364). We acknowledge the University of Chicago Research Computing Center, whose resources were used in the completion of this work. Ryan L. Sriver acknowledges support from the Department of Energy Program on Coupled Human and Earth Systems (PCHES) under DOE Cooperative Agreement no. DE-SC0016162.

Review statement

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Review statement.

This paper was edited by Seung-Ki Min and reviewed by two anonymous referees.

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Short summary

This work uses current temperature observations combined with climate models to project future temperature estimates, e.g., 100 years into the future. We accomplish this by modeling temperature as a smooth function of time both in the seasonal variation as well as in the annual trend. These smooth functions are estimated at multiple quantiles that are all projected into the future. We hope that this work can be used as a template for how other climate variables can be projected into the future.

This work uses current temperature observations combined with climate models to project future...

Advances in Statistical Climatology, Meteorology and Oceanography

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