Introduction
Since the pioneering studies of , modelling efforts with
general circulation models have routinely been used to understand, quantify
and identify the causes of past changes in monsoon dynamics.
One general approach to this end has been to perform snapshot experiments for
specific time slices in the past. The general circulation model is run with a
particular set of initial conditions for a perpetual year for a long
computational time until equilibrium is reached. The epoch used for defining
the astronomical forcing and boundary conditions is one for which specific
efforts are being undertaken to collect observations. This is the general
spirit of projects such as COHMAP and PMIP
. Specifically, the COHMAP project focused on a series
of time slices spaced every 3000 years throughout the deglaciation
, while PMIP historically focused on the
mid-Holocene and the Last Glacial Maximum, though on this basis an increasing
number of periods are being considered, including the Eemian
and the last interglacials .
Based on these experiments, it is now well understood that glacial boundary
conditions, typical, for example, of the Last Glacial Maximum, induce a
weakening of moisture transport over the Indian subcontinent and a reduction of
precipitation in East Asia see also. On the other hand, an increase in northern
summer insolation compared to a reference state strengthens monsoon dynamics, in
agreement with general considerations on the dynamics of heat transport and on
the location of the Intertropical Convergence Zone. These effects may be
combined. For example, showed the possibility of intense
Indian monsoon under glacial conditions, more specifically stage 6.5, when the
astronomical configuration is favourable.
These past climate simulations are often complemented with additional
sensitivity experiments. One classical experimental setup consists in
considering two end-member states, often the pre-industrial and one
well-defined past period, and intermediate configurations for which one
or several forcing components are “activated” while the others are left as the
pre-industrial configuration e.g..
Such sensitivity studies will be referred here as “local” approaches, in the
sense that only a small set of forcing conditions are explicitly considered out
of the space of possible forcings.
Palaeoclimate modelers are also concerned with the phase relationship between
forcing and climate. In particular, climatic precession may be seen as a
quasi-periodic rotation of the point of smallest Earth–Sun distance (it will be
referred to here as the perigee because we work in geocentric coordinates) and the
vernal equinox. By considering specific periods in the past for GCM experiments
one can only already develop a partial understanding of the phase
relationships. Specifically, showed that an “early-phase”
configuration (perigee reached in April) produces a stronger monsoon, which
occurs earlier in the year than a “late-phase” configuration
(perigee reached in September). Alternatively,
see also proposed the use of long transient simulations to
study the evolutionary response to orbital forcing of global summer monsoon
over the past 280 000 years. They showed that north tropical
sea-surface temperature leads June insolation by about 40∘. This
particular work did not consider CO2 and ice boundary condition effects.
At the time of writing, such experimental setups can only be afforded with fairly
low-resolution models (these authors used FOAM) with an acceleration technique:
one model year actually represents 100 years of simulation time.
Here, we will experiment with an alternative approach that will enable us
to simultaneously document the sensitivity of a general circulation
model (HadCM3); the independent and combined effects of different forcing
components on monsoon dynamics, namely astronomical forcing, CO2 and ice
boundary conditions; and, finally, estimate the phase relationship
between monsoon response and insolation forcing.
The starting point of this approach consists in performing an ensemble of
snapshot simulations. The ensemble is designed such that experiments span the
space of possible forcing configurations that the Earth encountered during
the late Pleistocene (ca. the last 800 000 years). For this reason the approach
will be qualified as “global”; more specifically, this is a global sensitivity analysis because we do not explicitly consider a reference state.
Thus, a statistical model is used to estimate the state of the system at any
input point within the space spanned by the experiment ensemble. To this end,
we consider a statistical model that is commonly referred to as an “emulator” in the
statistical literature . In particular, the term emulator
refers to the following properties :
it is derived from a small number of model runs filling the
entire multidimensional input space;
once the emulator is built, it is not necessary to perform any
additional runs with the model.
The emulator is then used to generate visual diagnostics and numerical indices
summarising the sensitivity of the model to the different elements of the
forcing.
This technique of emulation is beginning to be commonly used to estimate uncertainties
on climate model outputs, given probability distributions on uncertain
quantities such as model parameters or elements of the forcing
. Such approaches may also integrate information from
observations following a Bayesian formalism in order to construct posterior
distributions of model parameters and update current knowledge on predictive
quantities such as climate sensitivity . The
inference model may in particular include a statistical quantity called model
discrepancy, used to express the distance between the model and the real world
.
Compared to this series of works the present objective is a bit
different. As stated, we are interested in input quantities which we know
varied in the past, though we will assume that they varied sufficiently slowly
to justify a hypothesis of quasi-stationarity of the ocean–atmosphere system
with the forcing.
Our purpose is to estimate the contribution of input factors to the temporal
climate variance that can be observed in palaeoclimate records. To this end we
refer to the statistical theory of global sensitivity analysis with emulation
formalised by based on general principles of global
sensitivity analysis and experiment design ,
but adapted to our particular objective.
The paper is structured as follows. Section provides a
description of the emulator and the simulations used. The section is
admittedly technical and contains material that has been published before in
the statistical literature. However, following the practice of recent
articles of climate literature e.g., we choose to walk the
reader through the details of emulation design (see also video in the
supplementary material). This also gives us the opportunity to document in
detail technical statistical modelling choices. The hasty reader may,
however, jump to the Sect. , where the results of applying
the emulator on the Indian monsoon region are discussed. We focus, on the one
hand, on the performance of the emulator as such and, on the other hand, on
the climatic lessons emerging from this experiment. In particular, the
specific influence of ice sheet topographic forcing is quantified.
Conclusions follow in Sect. .
Methodology
Experiment design
The first task is to define the space of input configurations to be explored
with an ensemble of experiments. We consider five input factors: the three
elements of astronomical forcing (eccentricity e, longitude of perigee
ϖ, where ϖ=0 when perigee is in March, and obliquity
ε), the concentration in carbon dioxide (CO2), and a
variable called the ice or glaciation level, which combines ice and orography
forcings associated with the presence of continental ice in the Northern
Hemisphere.
The three elements of astronomical forcing are combined under the form of
esinϖ, ecosϖ and obliquity ε. This choice is
justified by the fact that these combinations produce orthogonal patterns in
the season–latitude space, and generally insolation at any point and time in
year is well approximated as a linear combination of those terms
. The factors esinϖ and ecosϖ are sampled
in the range [-0.05,0.05], while ε is varied in the range
22–25∘. Atmospheric CO2 concentration is sampled in the range
180–280 ppm.
Left panel: ice area, in normalized units, and maximum height (in
meters) in the region 45–75∘ N and 240–275∘ W
(Laurentide Ice Sheet), as a function of time in the boundary conditions
used in the experiment. Right panel: Height (in
meters) of the ice sheets. This shows that, although the volume of the ice
masses is quite different, their area is not. Red circles indicate the
boundary conditions used for this specific study.
Experiment plan design, optimised to maximise the minimum distance
between points and to achieve orthogonality (maximise the determinant of the
covariance of input factors). Right: ecosϖ–esinϖ space
distribution; middle: esinϖ–obliquity space distribution; right:
glaciation level–CO2 space distribution.
The glaciation level is determined as follows. Our purpose is to
select 11 realistic boundary conditions representative of
glacial–interglacial dynamics. Pragmatically, we sampled these
boundary conditions among the series prepared by
, and kindly supplied to us by Prof. Paul Valdes,
University of Bristol. Level 1 corresponds to present-day conditions, and
levels 2 to 11 are chosen as such to represent approximately 10 equally spaced
top altitudes of the North American Ice Sheet, within the glaciation phase. One
limitation of this design for the present purpose is that levels 3 to 11
effectively represent similar ice sheet areas – thus similar albedo
forcing – even though they sample very different ice sheet volume
(see Fig. ).
The next step is to define an ensemble of experiments to run with the climate
model in order to efficiently span the input space. The choice of the number
of experiments and, for each experiment, the choice of input parameters is
called the design. A design point refers in this context to a specific
experiment. The construction of the design should conform to rules of good
practice explained, for example, in . In particular, we want the
design to be space filling, and theoretical considerations and
experience point to the Latin hypercube design as a good starting point. The principle for a Latin
hypercube design of n elements is to divide the ranges covered by each input
factor into n distinct categories, each experiment sampling one of the n
categories without replacement. However, many Latin hypercubes could be
constructed in this way, and the design most appropriate for emulation should
satisfy additional constraints. Following p. 167 and
we combine two criteria. First, we select, among the possible
Latin hypercube designs, those maximising the minimum Euclidean distance found
between any two design points. This is called the maxi–min criteria. Among
those designs, we chose those maximising the determinant of X′X, so
that the resulting design is also near-orthogonal.
For this application, two additional constraints need to be accounted for in
order to avoid sampling unrealistic inputs that would be uninformative for the
sensitivity analysis of climate over the Pleistocene: exclude forcings with
e>0.05 and exclude combinations of high CO2 and high glaciation
levels (and conversely), delineated by an ellipse with large and small axes as
shown in Fig. . To satisfy these constraints, the design
points generated by the Latin hypercube sampling procedure lying in the
exclusion zone are geometrically projected on the allowed region. This
procedure may break some of the original properties of the design (maxi–min
and orthogonality), but it offers the practical advantage of enhancing the
coverage of the input space near its boundary.
Experiment setup: simulation name and number,
astronomical parameters (eccentricity, longitude of the perigee and
obliquity), CO2 concentration and glaciation level.
No.
Name
e
ϖ
ε
CO2
Ice level
No.
Name
e
ϖ
ε
CO2
Ice level
–
(∘)
(∘)
(ppm)
–
–
(∘)
(∘)
(ppm)
–
1
xadba
0.0527
53.52
23.6
277.3
1
32
xadfa
0.0383
334.53
23.8
257.8
6
2
xadbb
0.0520
211.44
22.9
267.5
1
33
xadfb
0.0417
139.99
24.5
214.1
6
3
xadbc
0.0309
218.44
23.1
262.6
1
34
xadfc
0.0480
215.67
23.2
225.0
6
4
xadbd
0.0201
350.24
23.2
271.2
1
35
xadfd
0.0404
140.60
22.1
225.0
6
5
xadka
0.0282
256.84
24.2
264.1
2
36
xadga
0.0301
194.43
22.4
254.1
7
6
xadkb
0.0466
228.06
24.2
263.4
2
37
xadgb
0.0261
208.55
22.9
189.8
7
7
xadkc
0.0411
88.21
23.3
273.5
2
38
xadgc
0.0503
202.65
24.3
260.8
7
8
xadkd
0.0077
358.66
22.3
255.1
2
39
xadgd
0.0389
122.16
22.3
257.8
7
9
xadaa
0.0403
316.14
22.1
270.6
3
40
xadge
0.0345
97.90
23.4
246.8
7
10
xadab
0.0263
271.85
22.2
270.7
3
41
xadgf
0.0362
299.18
22.2
246.8
7
11
xadac
0.0416
140.71
22.7
269.6
3
42
xadgg
0.0440
355.96
24.0
260.9
7
12
xadad
0.0257
167.54
22.6
256.1
3
43
xadgh
0.0422
287.83
24.7
203.2
7
13
xadae
0.0406
167.95
23.1
240.7
3
44
xadha
0.0436
51.20
22.5
192.6
8
14
xadaf
0.0460
305.89
23.9
224.9
3
45
xadhb
0.0333
26.49
22.7
254.3
8
15
xadag
0.0293
93.07
22.3
264.7
3
46
xadhc
0.0461
205.77
24.3
186.2
8
16
xadda
0.0244
323.78
22.8
214.1
4
47
xadhd
0.0386
246.02
23.1
214.1
8
17
xaddb
0.0421
114.71
23.7
214.2
4
48
xadhe
0.0405
38.22
24.8
225.0
8
18
xaddc
0.0253
23.96
23.6
235.9
4
49
xadhf
0.0491
221.00
23.6
235.9
8
19
xaddd
0.0469
1.20
24.9
235.1
4
50
xadia
0.0150
341.91
22.8
244.4
9
20
xadei
0.0000
0.00
23.0
230.4
5
51
xadib
0.0457
78.40
23.0
235.9
9
21
xadej
0.0500
90.00
23.0
230.4
5
52
xadic
0.0226
113.92
23.0
225.0
9
22
xadek
0.0500
0.00
23.0
230.4
5
53
xadid
0.0400
53.05
22.4
232.9
9
23
xadel
0.0000
0.00
24.0
230.4
5
54
xadie
0.0336
143.57
24.9
231.3
9
24
xadea
0.0155
217.23
23.4
205.9
5
55
xadja
0.0452
260.43
24.0
182.0
10
25
xadeb
0.0527
52.54
24.2
235.9
5
56
xadjb
0.0444
319.59
24.4
209.2
10
26
xadec
0.0456
4.52
24.1
206.6
5
57
xadjc
0.0463
192.48
24.7
191.0
10
27
xaded
0.0135
68.81
24.6
246.8
5
58
xadca
0.0350
305.63
24.1
190.5
11
28
xadee
0.0236
260.39
24.5
217.6
5
59
xadcb
0.0137
145.99
23.9
216.4
11
29
xadef
0.0396
285.78
25.0
246.8
5
60
xadcc
0.0250
136.64
23.3
186.4
11
30
xadeg
0.0251
276.28
24.3
271.0
5
61
xadcd
0.0243
75.55
22.9
197.7
11
31
xadeh
0.0404
359.97
23.5
206.9
5
Note that this design is in principle suitable for continuous factor
ranges only. The glaciation level used for experiments is an integer
obtained by rounding the value obtained by this process to the closest
integer. Designs specifically adapted for input spaces mixing
categorical and continuous variables could best be implemented in the
future see, for example,for an up-to-date review.
Table lists the simulations with their input
parameters. The choice of 61 members is a conservative implementation
of the recommendation of 10 experiments per input factors
. In fact, a first 57-member design was produced
using the method above, to which 4 members were added
(experiments 20–23). These experiments are idealised orbital changes that
were performed during the first phase of this project in order to locally
explore the model sensitivity to astronomical forcing.
Climate simulator
The climate model – referred to in this context as the simulator –
is the general circulation model HadCM3 , using the MOSES2
dynamic land surface scheme . The atmospheric component
dynamics and physics are resolved on a 3.75∘×2.5∘
longitude–latitude grid. The oceanic component has a horizontal resolution of
1.25∘×1.25∘.
Initial conditions are the final state of the PMIP2 0K experiment
featured in . Each simulation is run for
400 years, except for the xadk# set. Accidentally, the
first 200 years did not account for ice sheet topography. This
was corrected for the following 200 years. In the case of the
xadk# simulations, they were run for 300 years, accounting for
ice sheet topography from the beginning. Typical residual deep-ocean
temperature trends are of the order of 10-4 ∘Cyear-1.
The last 100 years of all simulations with orographic forcing
were retained for analysis. Over this interval, the
top-of-the-atmosphere imbalance ranges between -0.2 and
-0.1 Wm-2. The last 100 years of the
experiment section without orographic forcing are also used for an
investigation of the specific effect of the orographic forcing
(cf. Sect. ).
Emulator
At this stage we suppose that the simulator HadCM3 has been run for all design
points. We now show that it is possible to estimate, with quantified
uncertainty, the output that one would have obtained by running HadCM3 at any
input lying within the parameter space spanned by the design.
To this end, we need to develop a statistical model that can interpolate the
outputs obtained with the simulator at the design points. The procedure is akin to geospatial interpolation, except that the input field is here
five-dimensional, instead of two- or three-dimensional as in most geospatial applications
(cf. video in the supplementary material).
In particular, we follow and use a Gaussian process model,
with a Bayesian formalism. Although there is no strict practice, the term
emulator is often reserved to such Bayesian meta-models.
The calibration of the emulator is mathematically described as follows. Let
xj be the set of input values of the jth member of the design
(here: a vector of which the components are the astronomical forcing, ice level
and CO2). The output of the climate model is modelled as a stochastic process
combining a global response function (the regressors) with a local component.
It is fully specified by the mean m̃ and a covariance Ṽ
function, which have the following priors:
m̃(x)=h(x)′βṼ(x,x⋆)=σ2c(x,x⋆)
where c(x,x*) is the Gaussian process correlation
function, and σ2 its variance; h(x) is a (q×1) vector of a priori known regression functions; and β is the
vector of corresponding regression coefficients. Note that the
()′ is used to denote a horizontal vector. The definition of the
correlation function is given below.
Let f(x) denote the climate model output when run at input vector
x. In Bayesian language, we say that the fact of actually running
the model at the design n points allows us to
update our knowledge of f(x) at any input point.
We also need to make a choice regarding the values of β and σ2.
Given that we do not know their true value, we proceed, in the Bayesian way,
by defining prior probabilities for these quantities. We would like not to
introduce specific information on β and σ2. Given that
σ2 is a scale factor, theoretical considerations show that the
prior (β,σ2)∝σ-2 is appropriate as a vague
prior, i.e., all values of β are a priori equally plausible and the
probability density of σ2 decays in a way that preserves independence
on unit choices .
In these conditions, the posterior estimate of f(x) is a
Student t distribution with n-q degrees of freedom, with the following mean
and variance :
m(x)=h(x)′β^+T(x⋆)′A-1(y-Hβ^),V(x,x⋆)=σ^2[c(x,x⋆)-T(x)A-1T(x)′+P(x)(H′A-1H)-1P(x⋆)′],
respectively, with
σ^2=1n-q-2(y-Hβ^)′A-1(y-Hβ^)and β^=(H′A-1H)-1H′A-1y,
where y is a matrix of n lines, of which each line gathers the
input of the respective experiments; T(x)j=c(x,xj); and P(x)=h(x)′-T(x)A-1H. In the following, we conveniently approximate the
Student t distribution by a normal distribution. Although in principle is
true only as n→∞, is accurate enough in practice for
values of n-q larger than 20.
Remember that xj are the input parameters (astronomical
configuration, etc.) of experiment j of the design. Hence, for example,
T(x)j is a scalar, obtained by applying the so-called
correlation function defined below between the input vector x – at
which one wants to predict the simulator output – and the input xj
of the design. Consequently, the quantity T(x) is treated as an
n-component vector, of which the respective components are associated with
the different elements of the design. With this framework, the choices of the
regression functions h(x) and the Gaussian process correlation
function c(x,x⋆) are application-dependent. This is
where the user has the opportunity to inject knowledge on the expected
response of the simulator.
For this application, linear regression is an adequate choice because
the seasonal and annual forcings are almost linear with the input
factors, except possibly for glaciation level. Hence,
h(x)′=(1,x′).
The correlation function c(x,x*) is a linear measure of how
informative the simulator output at x is about the simulator output at
x*. It is thus a key component of the emulator. We use here the
classical exponential decay :
c(x,x*)=exp[-(x′Λ-2x*)].
The scaling matrix Λ is diagonal, with components λi called
the length scales. The interpretation is thus that the correlation between the
outputs of two experiments decreases exponentially as the normalised distance
between two input factors decreases. The normalisation factors are the length
scales. Intuitively, the length scale may thus be interpreted as a measure of
the roughness of the surface response: the larger the length scale, the smoother
the response surface (see video animation in the Supplement).
There is a further correction to be accounted for before using this function.
The quantity we are interested in emulating is the hypothetic mean of an
infinitely long experiment that has perfectly reached the stationary state.
In practice, we have to be content with the mean of a finite-length
experiment, obtained for a specific set of initial conditions and which may
not have perfectly reached the stationary state. The difference between the
output of an experiment and the ideal experiment average is expected to be
small yet impossible to predict exactly because it may chaotically depend on
initial conditions. It may effectively be accounted for in the emulator as
follows. Observe that the function c(x,x*) always appears as
filling the elements of a matrix (Eqs. and
). This matrix is further modified by adding a small
element along the diagonal called the nugget ν, which will absorb the
effects mentioned about the experiment sample being only an estimate of the
stationary state. The error tolerance will be of the order of σ^2ν.
The nugget has another benefit: it regularises the problem for large length
scales, and it may in particular be shown that posterior means converge to the
solution of a linear regression problem for
λi→∞ .
The remaining problem is to estimate the hyperparameters λi and ν
completely. Following , we maximise the emulator
likelihood the expression used here is from:
logL(ν,Λ)=-12log|A||HTA-1H|+(n-q)log(σ^2).
In order to guarantee that the emulator is at least no less informative than
would be linear regression, recommend the use of a
penalised likelihood as follows:
logLp(ν,Λ)=logL(ν,Λ)-2M‾(ν,Λ)ϵM‾(∞),
where M‾(ν,Λ) is the mean squared error between the
training points and the emulator's posterior mean at the design points, and
M‾(∞) is its asymptotic value at
λi→∞. We use ϵ=1.
It is worth noting that, in our case, using the normal likelihood or the
penalised one has practically no effect on the results.
Sensitivity measures
We are now in a position to estimate the simulator output at potentially any
input point spanned by the design. It is now possible to develop indices, of
which the purpose is to summarise the sensitivity of the simulator to
individual or combined factor throughout the whole input space. This is the
general idea of global sensitivity analysis.
In particular, one of the early applications of Bayesian emulators (as we use
here) was to estimate sensitivity measures to quantify the uncertainty on a
simulator output arising from the fact that the inputs are themselves uncertain
. In this context, the uncertain inputs may be quantified
by means of a multivariate probability density function ρ(x). The
problem of interest here is slightly different because we know how the inputs
varied in the past. The theory of global sensitivity analysis may, however, be
recycled by giving ρ(x) a frequentist interpretation. In other words, we use
ρ(x) to describe the time-wise occupation density of the input
space estimated by considering the history of the late Pleistocene.
Lines: 66, 90 and 95 % percentiles of the empirical
distribution used to describe the probability distribution in the CO2–ice space (Eq. ). Dots: observations of CO2
and estimates of ice level
assuming a linear relationship with the LR04 stack of benthic foraminifera
δ18O over the last 800 000 years. Based on
these observations, the empirical distribution appears to be slightly biased
towards high ice level at low CO2.
In particular, the occupation density along the components of the astronomical
forcing can be estimated with histograms of long time series generated with
known astronomical solutions, such as those presented by . We
then consider the following empirical distribution to broadly capture the
observed covariance between CO2 and glaciation level (see
Fig. ):
ρ(c*,i*)∝N0.5,38113-1132where 0<c*<1,0<i*<10elsewhere,
where c* and i* are inputs standardised as follows:
c*=(CO2-180ppm)/(100ppm),i*=(glaciation level-1)/10.
In order to relate output variances with input variances, we first define
what is known in the global sensitivity literature as the main effect
associated with an input p e.g.Chapter 1:
η(xp)=∫Xp‾f(x)ρ(xp‾|xp)dxp‾,
where we have denoted Xp‾ as the space spanned by all the
components of x but p, and ρ(xp‾|xp) is
the density of occupation of the space Xp‾ given the
vector p. The main effect is thus the expected mean of the simulator
output, given a known value of xp but no more information than the
prior on the other components of x.
Given that we cannot run the model at every point of the space
Xp,
this quantity is uncertain, but its mean and variance may be estimated with the
emulator:
mp(xp)=Ef(η(xp))=∫Xp‾ρ(xp‾|xp)dxp‾,Vpp(xp,xp⋆)=Varf(η(xp))=∬Xp‾×Xp‾V(x,x⋆)ρ(xp¯|xp)ρ(xp¯⋆|xp)dxp¯dxp¯⋆,
where Ef and Varf denote mean and variance due to using
the emulator instead of actually running the simulator at all points. On this basis, it is possible to define two measures of
sensitivity of the outputs to input xp:
Sp=EfVar(η(xp))andS¯p=EfVar(η(x))-Var(η(xp¯)).
The quantity Sp, called the main effect index is the loss in output variance that would occur assuming
that xp is known and constant, compared to a situation where all
factors vary. More precisely, this is the expected loss, averaged
over all possible values of xp e.g.Chapter 1. On
the other hand, S‾p is the output variance that occurs when
factor p is variable; all other factors assumed to be known and constant.
This is the total effect index. The distinction between
main and total effect is particularly important when there is a covariance
between input factors. This is the case here: CO2 and ice volume co-vary.
More precisely, the main effect index associated with, for example, ice
volume, includes an implicit contribution associated with the fact that
CO2 co-varies with ice level. The total effect index does not include this
contribution. Therefore, we use the total effect index.
In order to compute Sp and S‾p, we define the auxiliary
quantities:
Σp=∫Xp[mp(xp)2+Vpp(xp,xp)]dρ(xp),Σ0=[m0(x)2+V00(x,x)],Σ=∫χ[m(x)2+V(x,x)]dρ(x),
where the subscripts 0 and 00 imply that the space
Xp‾ referred to in the intergrals (12) and (13) is
the full input space. It may then be shown that
Sp=Σp-Σ0,S‾p=Σ-Σp‾.
JJAS sea-level pressure and surface temperature of the two regions
depicted: NI and IO. Units are in ∘C.
Results
In order to study the Indian monsoon, we define two regions: northern
India (NI), with coordinates 70–100∘ E, 20–40∘ N,
and the northwestern Indian Ocean (IO), with coordinates
55–75∘ E, 5–15∘ N see. The
chosen regions are depicted in Fig. , in which the
sea-level pressure and surface temperature of one of the simulations
are shown. The NI region covers the Indian subcontinent and part of the
Tibetan Plateau (which is dry today), while IO covers the northwestern part
of the Indian Ocean. In the supplementary material we explore another continental
region which does not include the Tibetan Plateau .
Diagnostic of emulator performance considering experiments 11 and
40. Shown are the mean and standard deviations of sea-surface temperature
(left panel) and mixed-layer depth (right panel). Clearly seen are the two
bad predictions, especially in the case of sea-surface temperature.
Sensitivity to glaciation level and esinϖ for sea-surface
temperature and mixed-layer depth. Top panels: the contour plots include the
experiments 11 and 40. The effect of these experiments are clearly visible in
both cases, ice level 3 in the case of sea-surface temperature and glaciation
level 7 for mixed-layer depth. Bottom panels: the removal of these
experiments a smooth response of the emulator, as clearly seen in the contour
plots.
We focus specifically on four physical variables representative of the
summer Indian monsoon process: June-July-August-September (JJAS)
temperature and precipitation on the continental box, and JJAS sea-surface temperature (SST) and mixed-layer depth on the Indian Ocean
box. Over the experiment design, continental temperature varies
between 15 and 21 ∘C. Precipitation varies between 72 and
230 mmmonth-1, SST between 25 and 31 ∘C, and
mixed-layer depth between 29 and 59 m. For emulation, the
logarithms of precipitation and mixed-layer depth are used, because
these distributions are more Gaussian than those of the absolute values.
Emulation validation
An emulator using all 61 experiments is calibrated using the procedure given
in Sect. , with scales λi (with i=1,…,5) and
nugget determined by maximisation of the penalised likelihood. The
performance of the emulator is then assessed following a leave-one-out cross-validation approach, that is, we construct 60 emulators to predict the
experiment being left out. Figure shows the result
of this leave-one-out cross-validation procedure for SST and mixed-layer
depth only, the other variables being discussed later.
This leads us to the following observations:
For esinϖ, ecosϖ and
ice volume, the length scales λ are of the same order
of magnitude as the range covered by the input factors.
This is the ideal scenario: the space between two experiments
is consistent with the decorrelation length of the simulator.
There are some instances where length scales are
much greater than the scale of the variables: this is observed on all
output variables for the response to CO2 and, to a lesser
extent, for obliquity. A large covariance scale implies that response
is linear with respect to the factor, which is indeed a realistic
outcome for CO2, in the range considered. This is not a problem
on its own. It simply informs the user that a sparser sampling of
this factor would have worked as well.
The leave-one-out cross-validation plot shows that two experiments are
not well captured by the Gaussian process model for
SST (experiments 11 and 40), and one for mixed-layer depth
(experiment 40). The emulator fails to predict the outputs within an error of
less than 3 standard deviations when they are left out of the calibration
procedure. The effects of these experiments on the emulator output are well
visible in Fig. (top panels). These plots, which will be
commented on in more detail in Sect. , represent the mean
model response (Eq. ) as a function of glaciation level and
esinϖ, and assuming CO2 fixed. The figure reveals departure
from smooth gradients contours, most notably the 26.25 and 26.5 ∘C
isotherms on the SST plot and the 38.5 m iso-depth that conflict with
our expectation of a smooth response structure.
At this stage one could consider an alternative emulator, calibrated on a
59-member experiment design in which the two problematic simulations are
omitted.
Emulator scales for the different fields under
study. In general, scales are commensurate with the range covered by the
input factors. However, for CO2 and sometimes obliquity, the scales
are much larger than the fields' scale. This simply indicates that the response
is linear with respect to the factor.
Length scales
Nugget
λecosϖ
λesinϖ
λε
λCO2
λice
–
–
(∘)
(ppm)
–
Land temperature
0.0704
0.0914
3.191
940
3.348
0.0047
Land precipitation
0.1153
0.3037
20.221
12 588
2.2807
0.0188
Sea surface temperature
0.1118
0.1142
600.
9786
7.307
0.0035
Mixed-layer depth
0.0767
0.0308
3.7724
411
10.6960
0.0439
This new emulator with new scales λi and nugget (see
Table ) presents a much more satisfactory performance
(Fig. ):
All ancillary emulators constructed for the leave-one-out diagnostic
capture between 38 (mixed-layer depth) and 43 (continental temperature) of
the leave-one-out experiments within 1 standard deviation, and between 56 and
58 within 2 standard deviations, which roughly correspond to the 66 and
95 % ratios expected for a normal distribution.
The normalised errors are compatible with
a normal distribution based on the Shapiro–Wilk normality test,
except for continental temperature (normality rejected with 97 %
confidence).
There is no error exceeding 3 standard deviations.
Finally, the suspicious anomalies generated on the
glaciation/precession plots are cleared (Fig. , bottom
panels).
Based on our experience with HadCM3 we are inclined to give more credit to this
new emulator as a predictor of HadCM3 outputs, rather than the one obtained
with simulations 11 and 40. Of course, this choice leaves us with the task of
explaining what went wrong with these two simulations. It seems that we have to
leave it as an open case. Further inspection of these particular experiments
reveals a clear warm–cold–warm pattern in the North Atlantic, and cooling over
the rest of the ocean, exemplified here by comparing experiments 11 and 15
(Fig. ). This pattern has been seen before in HadCM3, most
notably in early experiments of the Last Glacial Maximum . It
was associated with an enhancement of the North Atlantic Overturning
Circulation cell, and can be annealed by addition of freshwater in the North
Atlantic . Experiments 11 and 40 have, however, low to
moderate glaciation levels, and reasons why their behaviour should differ from
the other experiments are far from clear. Based on further inspection of time
series as well as that of longer experiments, we are left with the speculation
that the particular 100 years used to construct climatic averages correspond to
some meta-stable state of the ocean circulation, possibly excited by the
spin-up procedure.
Although we appreciate the difficulty, from a statistical inference
prospective, of rejecting problematic experiments for the calibration of the
emulator, we find it in fact positive that the emulator is effective in
identifying experiments that behave unexpectedly compared to the bulk of the
design.
Let us now consider the nugget.
As explained, this quantity quantifies the uncertainty of the simulation, i.e.
how representative of the mean model state are the 100-year simulations.
The residual error in the emulator is of the order of σ^2ν, but
it can be estimated precisely by looking at the posterior variance at design
points. Here, the obtained nuggets induce residual errors with
standard deviations of 0.04 ∘C on continental temperature,
2.3 % on precipitation, 0.05 ∘C on SST, and 0.7 % on
mixed-layer depth. All these values are consistent with the 100-year
variances of the corresponding quantities in HadCM3.
Thus, remarkably, the emulator calibration has successfully estimated model
internal variability using only 100-year means, which we take as one more
argument to use the recalibrated emulator.
Sensitivity measures
Figure summarises the sensitivities of the four
different variables to the external factors. ecosϖ and
esinϖ are grouped together under the term “precess”, for
climatic precession.
The figure shows that continental summer temperature is primarily determined
by precession, CO2 and, to a lesser extent, ice volume. It shows no
significant sensitivity to obliquity. Continental precipitation is also mainly
driven by precession and less to ice volume. In contrast to temperature, it
exhibits no sensitivity to CO2.
Similar to continental temperature, SST is primarily driven by precession and
CO2 and, to a lesser extent, ice volume. It also shows a larger
response to obliquity. Finally, mixed-layer depth shows a pattern similar to
precipitation, except that the response to obliquity is not significant
compared to the sources of uncertainty induced by the emulation and sampling
variance.
Sensitivity to precession
Figure displays the effects of precession on the four
variables retained for analysis. The choice here is to show the
effects by fixing ice and CO2 concentration at three distinct
levels representative of the course of glaciation (from top to
bottom): glaciation level 1/CO2=280 ppm, glaciation
level 5/CO2=230 ppm and glaciation level
11/CO2=180 ppm. Quantities are further averaged over
obliquity. In order to ease the interpretation, the months
representing the time at which perigee is reached are written on
the plots: June for ϖ=90∘, September for
ϖ=180∘, etc. That is, neglecting slow transient effects
that could be associated with the deep ocean response, this graphical
representation provides an indication of the phase lag between the
climate response and the precession forcing of insolation.
Sea surface temperature difference between simulations 11 and 15 (see
Table ). There is a clear warming pattern in the North
Atlantic, which affects the mean sea-surface temperature.
Diagnostic of emulator performance. Shown are the mean and standard
deviation of the simulated and the emulated data points for the all the
simulations with the exception of simulation number 11 and 40. Top left
panel: continental temperature; top right panel: continental precipitation;
bottom left panel: sea-surface temperature; bottom right panel: mixed-layer
depth.
Sensitivity analysis: shown is the standard deviation of model
outputs (S‾) of each variable, induced by variations in
input factors during the Pleistocene. From left to right, top to bottom:
continental precipitation, continental temperature, sea-surface temperature
and mixed-layer depth.
Sensitivity to ecos(ϖ) and esin(ϖ) for all fields.
Each panel, from top to bottom, shows the four fields with a different
configuration of glaciation level – CO2 concentration. Top panels:
glaciation level=1 and CO2=280 ppmv. Middle
panels: glaciation level=5 and CO2=230. Bottom panels:
glaciation level=11 and CO2=180. All fields were
integrated over obliquity.
Sea surface temperature difference for two idealised
simulations. CO2 concentration, glaciation level and precession
remained fixed, the only difference being obliquity (23 and 24∘).
Sensitivity to CO2 and glaciation level. From left to
right: continental temperature, sea-surface temperature and mixed-layer
depth. Fields were integrated over esin(ϖ), ecos(ϖ) and
obliquity.
Orography–no-orography difference. From top to bottom, left to
right: effect on continental temperature, precipitation, sea-surface
temperature, and mixed-layer depth, with orography forcing (black) and
without (red). The dotted lines show one standard deviation of the emulator
prediction. One may see a departure point from glaciation level 3 in all
four fields, as this is the point at which orography forcing becomes the most
significant.
We see that the temperature response is in phase with June
insolation at low glaciation levels, and in phase with July insolation
at mid- and high-glaciation stages.
This feature may physically be understood by considering the summer
precipitation response. Precipitation enhances latent heat cooling
when perigee is around July. This effect gradually weakens as
glaciation takes place and the total amount of precipitation declines,
hence the drift towards a more linear response. At higher glaciation
levels the JJAS temperature response phase also aligns with July
insolation.
The maximum precipitation is obtained when perigee is reached in
early July. Among the series of experiments shown by
, it is indeed the 126 000 yearBP
experiment (i.e. July perigee) experiment that shows the strongest
precipitation response over India.
Furthermore, continental precipitation and mixed-layer depth show opposite
response phases to precession. This result is consistent with the
earlier findings of , who identified a shoaling of the
mixed-layer depth in this region by about 6 m, consistent across
different models, in 6000-year experiments (September
perigee). examined also two nearly opposite
precession configurations with the IPSL model, corresponding to
perigee in April and October, respectively, and they found
a shoaling of the mixed-layer depth compared to the present-day (perigee
in January) in both cases.
attributed the mixed-layer depth shoaling to
a stratification effect involving the response of SST. On this point,
our analysis reveals that the maximum SST response occurs when
perigee is reached in May. This is not so surprising given that the
ocean thermal inertia generally imposes a lag of a few months between
the forcing and the response. This response, however, induces an
asymmetry between perigee in April and perigee in October, the
first one only showing anomalously high SSTs. This is consistent with
the analysis of seasonal cycle response provided by
.
Sensitivity to obliquity
The response of obliquity is mostly linear, as we can infer from the high values
of the length scales (see Table ).
The range of obliquity covered during the Pleistocene induces negligible
continental temperature response over the west Indian box. It also induces a
slight increase in precipitation. Regarding the Indian Ocean box, there is a
somewhat larger effect on SST compared to continental temperature, but not
significant. As for the mixed-layer depth, the response to obliquity is
negligible.
In order to better understand the effect of obliquity, we considered the four
idealised experiments (simulations 20–23; see
Table ). In particular, we discuss here experiments 22 and
23, termed OBL23 and OBL24. They use zero eccentricity, the same CO2
concentration and glaciation level, and differ by the configuration of
obliquity (24 and 23∘, respectively). The temperature difference map
for JJAS reveals the signature of obliquity-induced insolation changes, with
a warming of Northern Hemisphere continents, and slight cooling of
significant areas of the tropical oceans (see Fig. ).
Sensitivity to CO2 and glaciation level
The response of all variables to CO2 is best captured by
linear processes (optimal λi largely exceeds the range
covered by the experiment design). Hence, the contribution of
CO2 to the climate response may be estimated straightforwardly
from the coefficients β^, given by Eq. (). Specifically,
the continental temperature and SST responses to the 100 ppm range
covered by the experiment design are 2.03 and 1.40 ∘C, respectively.
This corresponds to CO2 doubling sensitivities of 3.20 and
2.21 ∘C, in line with the reported HadCM3 sensitivity in CO2
doubling experiments see, for example, Fig. 5 of The
responses of precipitation and mixed-layer depth are, again, opposite and
very moderate: +6 % of precipitation over 100 ppm and
- 0.5 % of mixed-layer depth.
Figure shows the response of continental temperature
(left panel), sea-surface temperature (middle panel) and mixed-layer
depth (right panel) to the variations of CO2 concentration and
glaciation level. The temperature ranges covered by CO2 and
glaciation levels are of the order of 1 and 2 ∘C
for the continent and ocean surface, respectively. The continental
ice effect is mainly present between glaciation levels 1 and 3. With
the ice sheet reconstructions used here, the ice area extent which is
responsible for the shortwave forcing almost reaches its maximum value
at glaciation level 3. Further increasing the glaciation levels
affects climate predominantly through the orography forcing
(cf. Sect. ).
Orographic effect
Finally, we consider the differences between the simulations with and
without orography forcing of the ice sheets. The latter is potentially
important given that mountains and elevated land masses affect the
atmospheric circulation and precipitation patterns, and then the whole
climate system. To this end, an emulator was calibrated on the
available present-day orography experiments.
The net effect orography can then be seen in
Fig. , where all four variables are plotted as
a function of the glaciation level. Black solid lines show the
respective variables obtained with the standard experiment design,
while red solid lines show the response obtained with the experiment
design assuming pre-industrial orography, regardless of the presence
of ice sheets. The value plotted is obtained from
Eq. (). Note that by construction this value is also
implicitly a function of CO2 concentration, which enters
Eq. () via the factor
ρ(x|xice). Dotted lines indicate
a 1σ deviation, in both cases, based on
Eq. (), using xp=xp*.
A clear deviation is seen around glaciation level 3. This effect is
due to the fact that, as explained in Sect. , levels
3–11 represent effectively similar ice sheet area, but significantly higher
orography (see Fig. ). Hence, the albedo forcing dominates
over the lower range of glaciation levels (1–3), with decreasing temperatures,
precipitation and mixed-layer depth shoaling. The orography–no-orography
differences appear more markedly above index 3: orography reduces the cooling
trends by as much as 1 ∘C on the continent at glaciation level 11,
and even reverses the precipitation trend. As stated in the Introduction,
it is known that ice orography
forcing may impact monsoon precipitation regimes,
but to our knowledge the specific effect of Northern Hemisphere ice sheet
orography on the Indian monsoon is yet to be documented. The warming signal caused
by orography may be understood by considering the increase in surface potential
temperature over elevated regions, similar to what is seen today over the
Tibetan Plateau. Because of these high potential temperatures, down-sloping air
is effectively warmer than it would be in the absence of orography forcing, and
contributes here to increasing the Northern Hemisphere continental surface
temperatures. Orographic forcing generally induces atmospheric circulation
anomalies and effects on ocean circulation and stratification. For example,
Fig. suggests a weak positive effect on mixed-layer
depth, quite small compared to the astronomical forcing effects. An in-depth
analysis of these effects falls beyond the scope of the present contribution.
Conclusions
We present a first application of a global sensitivity analysis theory to study
the climate response of the Indian monsoon to the climate factors which
evolved during the Pleistocene, namely the astronomical forcing
(esin(ϖ), ecos(ϖ), ε), CO2 concentration
and glaciation level.
We focus, in particular, on four variables: continental temperature,
continental precipitation, sea-surface temperature and mixed-layer
depth. These variables were averaged for the JJAS season over northern
India and northwestern Indian Ocean.
Similar to a number of recent studies based on statistical modelling for global
sensitivity analysis of computationally expensive simulators, the technical
implementation follows a three-step methodology:
Designing an experiment plan. We adopted a Latin hypercube
design, optimised following two constraints: maximisation of the minimum
distance between two points in the input space – this is called the maxi–min
property – and maximisation of the determinant of the matrix of covariance between
the input factors – this is a constraint of orthogonality. In addition, the
design excludes configurations with excessive eccentricity and unrealistic
combinations of CO2 and glaciation level.
Calibration and validation of the emulator.
The validation was performed following a leave-one-out cross-validation
approach. Two experiments were excluded of the design as presenting an
anomalous North-Atlantic SST patterns. The emulator calibrated on the
remaining 59 experiments overall validates the present statistical modelling
choices.
Quantifying and visualising the individual and combined effects of the different factors on the summer Indian monsoon, based
on sensitivity measures and cross-section plots.
This analysis yielded the following conclusions:
precession controls the response of four variables:
continental temperature in phase with June–July insolation; high
glaciation favouring a late-phase response; sea-surface temperature in
phase with May insolation; and continental precipitation in phase with July
insolation, and mixed-layer depth in antiphase with the latter.
The effect of CO2 on continental temperature and SST is of
similar size to that of precession on summer continental temperature and
SST.
Obliquity is a secondary effect, negligible on most variables
except sea-surface temperature.
The effect of glaciation is dominated by the albedo forcing,
and its effect on precipitation competes with that of precession.
The orographic forcing reduces the glacial cooling induced by
the albedo forcing, and even has a positive effect on
precipitation.
The present study confirms the high potential of emulation for exploring and
understanding the response of climate models. One originality of the present
work was to consider, as inputs, several elements of the climate forcing that
(have) varied in the past, and the emulator was used as a method to help us
quantify the link between forcing variability and climate variability. The
methodology may naturally be applied to other regions of focus and other
climate models.