a a a a

#  GDCA: GAN-based single image super resolution with Dual discriminators and
Channel Attention

Thanh Nguyen  Hieu Hoang  Chang D. Yoo  Korea Advances Institute of Science
and Technology (KAIST)

###### Abstract

Single Image Super Resolution (SISR) is a very active research field. This
paper addresses SISR by using GAN-based approach with dual discriminators and
incorporate with attention mechanism. The experimental results show that GDCA
can generate sharper and high pleasing images compare to other conventional
methods.

###### keywords:

Deep Learning, Machine Learning, GAN, Attention

##  1 Introduction

Figure 1: The generator architecture consists of channel attention blocks and
local extraction blocks with one long skip

Single Image Super Resolution (SISR) is the problem of reconstructing an
accurate high-resolution (HR) image from its low-resolution (LR) counterpart.
The reconstructed image is referred as the super-resolution (SR) one. Recent
approaches using deep learning show the impressive result with much higher
quality compare to conventional approaches. This result engages more
researchers to work on and make deep learning SISR to be an active research
field along with other canonical topics [5, 4, 6] .

CNN based approach has achieved significant improvements over conventional
methods. Dong et al. [1] is a pioneer with three-layer CNN (SRCNN) for the
SISR. After that, VDSR [2] was introduced with an increase of network depth to
20 layers, which outperforms SRCNN… These approaches aim to maximize Peak
Signal Noise Ratio (PSNR) with the more efficient deeper network. The problem
with these approaches is that the SR images are blurry result in visually
unpleasing.

Recently, GAN-based methods have emerged as an effective solution to overcome
the blurry limitation. Taking advantage of GANs enables to reconstruct SR
images with high-frequency details and high perceptual quality. GAN based
approach usually consists of a generator and a discriminator. Discriminator
try to identify HR or SR image whereas generator try to fool discriminator to
classify its generated SR image as HR image. SRGAN [3] employs an adversarial
loss term to increase visually pleasing quality. SRFeat [7] used two
discriminators and adopts the adversarial loss terms in both image and feature
domains, resulting better perceptual quality.

Although previous approaches achieve high quality results, they did not take
attention into account.. With SISR, attention can help the network focus on
importance regions which have high texture in other to get higher performance.
In this work, we thoroughly incorporate attention with previous approaches and
prove the effectiveness of attention mechanism in SISR. To be more specific,
we propose “GAN-based single image super resolution with Dual discriminator
and Channel Attention.” (GDCA). GDCA takes advantage of GAN. GDCA introduce
new generator with integrated attention module to boost performance. we also
employ two discriminators which are image discriminator and feature
discriminator. With these two discriminators, network discriminates in both
image space and feature space resulting in a better perceptual quality. Our
contributions are listed below:

  * •

Propose a new generator architecture with a combination of skip connection,
channel attention, batch norm removal, and use mean absolute error loss rather
than conventional mean square loss.

  * •

Dual discriminators are utilized: image discriminator and feature
discriminator. The network is trained under GAN framework.

  * •

We thoroughly evaluate our network on Root Mean Square Error (RMSE),
“Perceptual Index” based on Ma + NIQE (PIMR2018). Experimental results show
that our GDCA method is superior to has high performance on several benchmarks
proving the effectiveness of attention in SISR. GDCA also generated much more
pleasure image. The finding will engage people apply attention to their
network in other to get high result on SISR.

##  2 METHODOLOGY

In this section, we present our proposed generator architecture as well as the
explanation for our design. Then we explain the training loss functions used
to train this network efficiently.

Figure 2: The discriminator architecture. The number above a convolution
represents the number of filters

###  2.1 Network architecture

As the GAN-based architecture, our architecture includes two parts: generator
and discriminators. Therein, the discriminators includes image discriminator
and feature discrimination. The generator network is shown in Figure 1. The
network receives the low resolution (LR) image and produces corresponding
super resolution (SR) image. The overall network consists of multiple channel
attention (CA) blocks and local extraction (LE) blocks with a weighted long
skip connection. Firstly, a 5x5 convolution layer to extract low-level
features are applied to input to extract course feature representation.
Secondly, multiple CA blocks and LE blocks are employed to learn higher-level
features with more non-linearity and larger receptive fields. Lastly, two sub-
pixel convolution layers (SPL) proposed in [8] are used to up-sample the
feature map to produce the target output. The discriminators include feature
discriminator and image discriminator. Both discriminators have the same
network architecture but receive different inputs. The network architecture is
combination of feed forward convolution neural network and fully connected
layer as shown in Figure 1. Image discriminator inputs are SR image and HR
image whereas feature discriminator inputs are the feature map of SR image and
HR image. These feature maps are extracted from corresponding Conv5 layer of
VGG19 pre-train network of SR image and HR image. Both of the discriminators
try to classify the image into real and fake class. Utilizing the dual
discriminator is the crucial factor of the whole framework. Image
discriminator distinguishes the image space domain whereas feature
discriminator distinguishes image in the feature domain. Both of
discriminators make network stronger conventional discriminator result in a
better output.

###  2.2 Loss function

There are three loss terms contributing to the total loss: perceptual
similarity loss, image GAN loss, and feature GAN loss. Our losses follow the
general GAN loss framework and aim to improve the perceptual quality.

$\min_{g}\max_{d}\left(E_{y\sim p_{\text{data }}(y)}[\log(d(y))]+E_{x\sim
p_{\text{data }}(x)}[\log(1-d(x))]\right)$

Where g(x) is the generator network, d is a discriminator network, x is the
input, and y is the sample from real data distribution.

##  3 EXPERIMENTAL AND RESULT

###  3.1 Quantitative result

We evaluated our GAN-based final generator on test set: Set 5. The final
generator here was obtained by training the pre-train generator with GAN-based
losses. Root Mean Square Error (RMSE) and Perceptual Index (PI) are chosen.
These methods measure the perceptual quality of the output image. The higher
PI, the better perceptual quality. Contrarily, the lower RMSE, the better
reconstruction quality. The detail result is show in Table 1.

|  Super-SR[1] |  SRFeat [7] | Ours
---|---|---|---
Perceptual Index | 1.98 | 2.25 | 2.47
RMSE | 15.30 | 14.95 | 13.95
Table 1: Result of our method compared to Super-SR and SRFeat on Set 5
evaluation set. Perceuptual Index and RMSE are measurement metrics.

From the table, our generator produces better RMSE than other state of the art
models, while obtains a com-parable perceptual index (2.47).

###  3.2 Qualitative result

Our method show impressive qualitative result as show in Figure 3. Our GAN-
based method achieves sharper output while mean square-based methods produce
blurry result.

(a) Mean Square

(b) GDCA

(c) Ground truth

Figure 3: Qualitative result performing on Set 5 test set. GDCA shows sharper
result comparing to Mean Square-based method

##  4 CONCLUSION

GDCE addresses SISR by using dual discriminators and incorporate with
attention mechanism. The experimental results show that GDCA can generate
sharper and high pleasing images compare to other conventional methods.

## References

  * [1] Chao Dong, Chen Change Loy, Kaiming He, and Xiaoou Tang.  Learning a deep convolutional network for image super-resolution.  In European conference on computer vision, pages 184–199. Springer, 2014.
  * [2] Jiwon Kim, Jung Kwon Lee, and Kyoung Mu Lee.  Accurate image super-resolution using very deep convolutional networks.  In Proceedings of the IEEE conference on computer vision and pattern recognition, pages 1646–1654, 2016.
  * [3] Christian Ledig, Lucas Theis, Ferenc Huszár, Jose Caballero, Andrew Cunningham, Alejandro Acosta, Andrew Aitken, Alykhan Tejani, Johannes Totz, Zehan Wang, et al.  Photo-realistic single image super-resolution using a generative adversarial network.  In Proceedings of the IEEE conference on computer vision and pattern recognition, pages 4681–4690, 2017.
  * [4] Thanh Nguyen, Tung Luu, Trung Pham, Sanzhar Rakhimkul, and Chang D. Yoo.  Robust maml: Prioritization task buffer with adaptive learning process for model-agnostic meta-learning.  In ICASSP 2021 - 2021 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), pages 3460–3464, 2021.
  * [5] Thanh Nguyen, Tung M Luu, Thang Vu, and Chang D Yoo.  A pre-training framework for learning feature representation in visual observation reinforcement learning.  Conference of IEIE, pages 1890–1893, 2021.
  * [6] Thanh Nguyen, Tung M Luu, Thang Vu, and Chang D Yoo.  Sample-efficient reinforcement learning representation learning with curiosity contrastive forward dynamics model.  arXiv preprint arXiv:2103.08255, 2021.
  * [7] Seong-Jin Park, Hyeongseok Son, Sunghyun Cho, Ki-Sang Hong, and Seungyong Lee.  Srfeat: Single image super-resolution with feature discrimination.  In Proceedings of the European Conference on Computer Vision (ECCV), pages 439–455, 2018.
  * [8] Wenzhe Shi, Jose Caballero, Ferenc Huszár, Johannes Totz, Andrew P Aitken, Rob Bishop, Daniel Rueckert, and Zehan Wang.  Real-time single image and video super-resolution using an efficient sub-pixel convolutional neural network.  In Proceedings of the IEEE conference on computer vision and pattern recognition, pages 1874–1883, 2016.


A simple conceptual model of abrupt glacial climate events.
A conceptual model of Dansgaard-Oeschger events
H. Braun et al.
H. Braun

H. Braun
Heidelberg Academy of Sciences and Humanities, c/o Institute of
Environmental Physics, University of Heidelberg, Im Neuenheimer Feld 229,
69120 Heidelberg, Germany.
A. Ganopolski
Potsdam Institute for Climate Impact Research, P.O. Box 601203, 14412
Potsdam, Germany.
M. Christl
PSI/ETH Laboratory for Ion Beam Physics, c/o Institute of Particle
Physics, ETH Zurich, 8093 Zurich, Switzerland.
D. R. Chialvo
Department of Physiology, Feinberg Medical School, Northwestern
University, 303 East Chicago Ave. Chicago, IL 60611, USA.

Here we use a very simple conceptual model in an attempt to reduce essential
parts of the complex nonlinearity of abrupt glacial climate changes (the
so-called Dansgaard-Oeschger events) to a few simple principles, namely (i)
the existence of two different climate states, (ii) a threshold process and
(iii) an overshooting in the stability of the system at the start and the end
of the events, which is followed by a millennial-scale relaxation.
By comparison with a so-called Earth
system model of intermediate complexity (CLIMBER-2), in which the events represent
oscillations between two climate states corresponding to two fundamentally
different modes of deep-water formation in the North Atlantic, we demonstrate
that the conceptual model captures fundamental aspects of the nonlinearity of
the events in that model. We use the conceptual model in order to reproduce
and reanalyse nonlinear resonance mechanisms that were already suggested in
order to explain the characteristic time scale of Dansgaard-Oeschger
events. In doing so we identify a new form of stochastic resonance (i.e. an
overshooting stochastic resonance) and provide the first explicitly
reported manifestation of ghost resonance in a geosystem, i.e. of a
mechanism which could be relevant for other systems with thresholds and with
multiple states of operation. Our work enables us to explicitly simulate
realistic probability measures of Dansgaard-Oeschger events (e.g. waiting
time distributions, which are a prerequisite for statistical analyses
on the regularity of the events by means of Monte-Carlo
simulations). We thus think that our study is an important advance in
order to develop more adequate methods to test the statistical
significance and the origin of the proposed glacial 1470-year climate cycle.

DO events as seen in the GISP2 ice-core from Greenland [Grootes et al., 1993, Grootes and Stuiver, 1997]. The figure shows Greenland
temperature changes over the interval between 10000 and about 40000 years
before present. DO events (0-10) manifest themselves as saw-tooth shaped warm
intervals.  Dashed lines are spaced by 1470 years.

Time series of North Atlantic atmospheric / sea surface temperatures during the last ice age
reveal the existence of repeated large-scale warming events, the so-called Dansgaard-Oeschger
(DO) events [Dansgaard et al., 1982, Grootes et al., 1993].
In climate records from the North Atlantic region the events have a
characteristic saw-tooth shape (Fig. <ref>): They typically
start with a warming by up to 10-15 K [Severinghaus and Brook, 1999, Leuenberger et al., 1999] over only a few years/decades. Temperatures remain high for
centuries/millennia until they drop back to pre-events values over a century
or so. A prominent feature of DO events is their millennial time scale: During
Marine Isotope Stages (MIS) 2 and 3, successive events in the GISP2 ice core
were reported to be often spaced by about 1470 years or multiples thereof
[Alley et al., 2001, Schulz, 2002, Rahmstorf, 2003], compare
Fig. <ref>. A leading spectral peak corresponding to the 1470-year
period was found [Grootes and Stuiver, 1997, Yiou et al., 1997], and this spectral component was
reported to be significant at least over a certain time interval [Schulz, 2002].
We note, however, that the statistical significance of this pattern is still under
debate [Ditlevsen et al., 2007], in particular because of the lack of adequate
nonlinear analysis methods.

It was proposed that DO events represent rapid transitions between two fundamentally different
modes of the thermohaline ocean circulation (THC) [Oeschger et al., 1984, Broecker et al., 1985], most
likely corresponding to different modes of deep-water formation [Alley and Clark, 1999, Ganopolski and Rahmstorf, 2001]. The origin of these transitions is also under debate:
In principle they could have been caused by factors from outside of the Earth
system [Keeling and Whorf, 2000, Rial, 2004, Clemens, 2005, Braun et al., 2005], but they
could also represent internal oscillations of the climate system
[Broecker et al., 1990, Sakai and Peltier, 1997, van Kreveld et al., 2000].
Several nonlinear resonance mechanisms have been suggested in order
to explain the characteristic timing of DO events, including coherence
resonance [Ganopolski and Rahmstorf, 2002, Timmermann et al., 2003, Ditlevsen et al., 2005] and stochastic resonance [Alley et al., 2001, Alley et al., 2001, Ganopolski and Rahmstorf, 2002, Rahmstorf and Alley, 2002].

§ SPECTRUM OF MODELS

DO events have already been simulated by a variety of models, ranging from simple conceptual
ones to Earth system models of intermediate complexity (EMICs). Conceptual models are most
suitable to perform large numbers of long-term investigations because they require very
little computational cost. However, they are often based on ad-hoc assumptions and only
consider processes in a highly simplified way. In addition to that, the number of adjustable
parameters is typically large compared to the degrees of freedom in those models. This implies
that seemingly good results can often be obtained merely by excessive tuning. Nevertheless,
conceptual models can provide important help for the interpretation of complex climatic

The gap between conceptual models and the most comprehensive general circulation
models (GCMs), which are not yet applicable for millennial-scale simulations because of their
large computational cost, is bridged by EMICs [Claussen et al., 2002]. EMICs include
most of the processes described in comprehensive models (in a more reduced form), and
interactions between different components of the Earth system (atmosphere, hydrosphere,
cryosphere, biosphere, etc.) are simulated. The number of degrees of freedom typically
exceeds the number of adjustable parameters by orders of magnitude. Since many EMICs are fast
enough for studies on the multi-millennial time scale, they are the most adequate tools for the
simulation of DO events.

§ THE CONCEPTUAL MODEL

The simple conceptual model which we use here is an extended version of the model
described by Braun et al., 2005 (in the Supplementary Material of that publication).
Here we use the model to demonstrate and analyse two apparently counterintuitive
resonance phenomena (stochastic resonance and ghost resonance)
that can exist in a large class of highly nonlinear systems. Due to the
complexity of many of those systems it is often impossible to precisely
identify the reasons for the occurrence of these resonance phenomena. Our
conceptual model, in contrast, has a very clear forcing-response relation as
well as a very low computational cost and thus provides a powerful tool to
explore these phenomena and to test their robustness. Furthermore, we describe
and discuss the applicability of the model for improved statistical analyses
(i.e. Monte-Carlo simulations) on the regularity of DO events. In the
following the key assumptions of the conceptual model are first discussed. In
the Supplementary Material we then compare the model performance under a
number of systematic forcing scenarios with the performance of a more
comprehensive model (the EMIC CLIMBER-2), compare Supplementary Information
File and Supplementary Figs. 1-6. In the framework of the conceptual model we
finally demonstrate and interpret two hypotheses that were previously
suggested in order to explain the recurrence time of DO events, and we discuss
how these hypotheses could be tested.

§.§ Model description

Our conceptual model is based on three key assumptions:

* DO events represent repeated transitions between two different climate states,
corresponding to warm and cold conditions in the North Atlantic region.

* These transitions are rapid compared to the characteristic life-time of
the two climate states (i.e. in first order approximation they occur
instantaneously) and take place each time a certain threshold is crossed.

* With every transition between the two states the threshold overshoots
and afterwards approaches equilibrium following a millennial-scale
relaxation process. This implies that the conditions for a switch between
both states ameliorate with increasing duration of the cold and warm

Our three assumptions are supported by paleoclimatic evidence and/or by
simulations with a climate model:

* Since long, DO events have been regarded as repeated oscillations between two
different climate states [Dansgaard et al., 1982]. It has been suggested that
these states are linked with different modes of operation of the THC
[Oeschger et al., 1984, Broecker et al., 1985]. This seminal interpretation has since then
influenced numerous studies and is now generally accepted [Rahmstorf, 2002].
Indirect data indicate that
the glacial THC indeed switched between different modes of operation [Sarnthein et al., 1994, Alley and Clark, 1999] which, according to their occurrence during cold and warm
intervals in the North Atlantic, were labelled stadial and interstadial modes. A third
mode named Heinrich mode (because of its presence during the so-called Heinrich events)
is not relevant here.

* High-resolution paleoclimatic data show that transitions from cold conditions in
the North Atlantic region to warm ones often happened very quickly, i.e. on the
decadal-scale or even faster [Taylor et al., 1997, Severinghaus and Brook, 1999]. The opposite
transitions were slower, i.e. on the century-scale [Schulz, 2002], but
nevertheless still faster that the characteristic life-time of the cold and
warm intervals (which is on the centennial to multi-millennial time scale, compare Fig. 1). The
abruptness of the shifts from cold conditions to warm ones has commonly been
interpreted as evidence for the existence of a critical threshold in the
climate system that needs to be crossed in order to trigger a shift between
stadial and interstadial conditions [Alley et al., 2003]. Such a threshold could be
provided by the THC (more precisely, by the process of deep-water formation): When warm
and salty surface water from lower latitudes cools on its way to the North Atlantic,
its density increases. If the density increase is large enough (i.e. if the surface gets
denser than the deeper ocean water), surface water starts to sink. Otherwise,
surface water can freeze instead of sinking. The onset of deep-water formation can
thus hinder sea-ice formation and facilitate sea-ice melting (due to the
vertical heat transfer between the surface and the deeper ocean). A switch between two
fundamentally different modes of deep-water formation can thus dramatically change sea
ice cover and can cause large-scale climate shifts. Such nonlinear, threshold-like
transitions between different modes of deep-water formation are at present considered as
the most likely explanation for DO events [Alley et al., 1999, Ganopolski and Rahmstorf, 2001].

* The time-evolution of Greenland temperature during the warm phase of DO
events has a characteristic saw-tooth shape
(Fig. <ref>). Highest temperatures typically occur during the
first decades of the events. These temperature maxima are followed by a
gradual cooling trend over centuries/millennia, before the system returns to
cold conditions at the end of the events. This asymmetry supports the idea
that the system overshoots in some way during the abrupt warming at the
beginning of the events and that the subsequent cooling trend
represents a millennial-scale relaxation towards a new equilibrium
[Schulz et al., 2002, Centurelli et al., 2006]. We note that the time-evolution of
Greenland temperature provides no clear evidence for an overshooting during
the opposite transitions (i.e. from the warm state back to the cold
one). This, however, is not necessarily in contradiction to our assumption:
This lack of an overshooting in the temperature fields does not necessarily
mean that the ocean-atmosphere system did not overshoot, since Greenland
temperature evolution in the stadial state might have been dominated by
factors other than the THC (respectively its stability), e.g. by Greenland
ice accumulation, which would mask the signature of the THC in the ice core

We will show later (in Sect. 3.3.) that the assumption of an overshooting in the
stability of the system is in fact strengthened by the analysis of model
results obtained with the coupled model CLIMBER-2. In that model the
overshooting results from the dynamics of the transitions between the two
climate states: In the stadial state deep convection occurs south
of the Greenland-Scotland ridge (i.e. at about 50 $^\circ$N). In the
interstadial state, however, deep convection takes place north of the ridge
(i.e. at about 65 $^\circ$N). The onset of deep convection north of the
Greenland-Scotland ridge, which releases accumulated energy to the atmosphere
(i.e. heat that is stored mainly in the deep ocean), in first place starts DO
events in the model. This heat release leads to a reduction of sea ice, which
in turn further enhances sea surface densities between 50 $^\circ$N and 65
$^\circ$N (e.g. by increased local evaporation and reduced sea ice transport
into that area). As a result deep convection also starts between 50 $^\circ$N
and 65 $^\circ$N, and much more heat can be released to the
atmosphere. Without a further response of the THC the system would return
quickly (within years or decades, i.e. with the convective time scale) to its
original state. In CLIMBER-2, however, the changes in deep convection trigger
a northward extension and also an intensification of the ocean circulation
(i.e. an overshooting of the Atlantic meridional overturning circulation;
compare Ganopolski and Rahmstorf, 2001), which maintains the interstadial climate
state since it is accompanied by an increase in the salinity and heat flux to
the new deep convection area (at about 65 $^\circ$N). In response to the
overshooting of the overturning circulation, the system relaxes slowly (within
about 1000 years, i.e. with the advective time scale) towards the stadial
state. We note that the advective time scale corresponds to the millennial
relaxation time in our conceptual model. The model CLIMBER-2 also supports the
validity of our overshooting assumption during the opposite transition (from
the warm state back to the cold one), as we will show in Sect. 3.3. We would
like to stress that our interpretation of the processes during DO events is,
of course, not necessarily true since we can only speculate that the
underlying mechanism of the events is correctly captured by CLIMBER-2.

§.§ Model formulation

We implement the above assumptions in the following way (compare
Fig. <ref>): First we define a discrete index s(t) that indicates
the state of the system at time t (in years). Since we postulate the existence
of two states, s can only take two values (s=1: warm state, s=0: cold
state). We further define a threshold function T(t) that describes the
stability of the system at time t (i.e. the stability of the current model

Dynamics of our conceptual model. Shown is the time evolution of the model, in response to
a forcing that is large enough to trigger switches between both model states. Top: Forcing
f (black) and threshold function T (red). Bottom: Model state s (grey; s=0 corresponds to
the cold state, s=1 to the warm one) and state variable S (green). At time $t''$ the forcing
falls below the threshold function and a shift from the cold state into the warm one is
triggered. With this transition, the threshold function switches to a non-equilibrium value
(representing an overshooting of the system) and afterwards approaches equilibrium following
a millennial-scale relaxation. At time $t'$ the forcing exceeds the threshold function, and a
transition from the warm state back into the cold one is triggered. With this transition,
the threshold function switches to another non-equilibrium value and approaches equilibrium
following another millennial-scale relaxation, until the forcing again falls below the
threshold function and the next switch into the warm state is triggered. Note that the
state variable S is chosen to be identical to the threshold function T. For
convenience, discontinuities in T and S are eliminated by linear interpolation. T, S
and f are normalised in the figure.

Second we define rules for the time evolution of the threshold function T. When the
system shifts its state, we assume a discontinuity in the threshold function: With the switch
from the warm state to the cold one (at time t' in Fig. <ref>) T takes the value
A$_{0}$. Likewise, with the switch from the cold state into the warm one
(at time t” in Fig. <ref>) T takes the value A$_{1}$. As long
as the system does
not change its state the evolution of T is assumed to be given by a relaxation process:

\begin{equation}
\frac{dT}{dt} = - \frac{(T-B_s)}{\tau_s}
\end{equation}

(s labels the current model state, $\tau_s$ denotes the relaxation time in
that state, B$_{s}$ is
a state-dependent constant that labels the equilibrium value of T in each model
state). These assumptions result in the following
expression for the threshold function T:

\begin{equation}
T(t) = (A_s-B_s) \cdot \exp(- \frac{t-\delta_s}{\tau_s})+B_s.
\end{equation}

Note that in the above expression the index s again denotes the current state of the model
(i.e. s=0 stands for the cold state and s=1 for the warm one),
$\delta_0$ labels the time of the
last switch from the warm state into the cold one, and $\delta_1$ indicates the time of the last switch
from the cold state into the warm one.

Third we assume that transitions from one state to the other are triggered each time a given
forcing function f(t) crosses the threshold function T. More
precisely, we assume
that when the system is in the cold state ($s[t''] = 0$) and the forcing is
smaller than the
threshold value ($f[t''+1] < T[t''+1]$) the system switches into the warm
state ($s[t''+1] = 1$). This
shift marks the start of a DO event. Likewise, when the system is in the warm
($s[t'] = 1$) and the forcing is larger than the threshold value ($f[t'+1] >
T[t'+1]$) the system
switches into the cold state ($s[t'+1] = 0$). That shift represents the
termination of a DO event. If none of these conditions is fulfilled, the
system remains in its present state (i.e. $s[t+1] = s[t]$).

To simplify the comparison of the model output with paleoclimatic
records we further define a state variable S, which represents anomalies in
Greenland temperature during DO events. For simplicity we assume that the
state variable is equal to the threshold function: $S(t) = T(t)$ (i.e. we
assume that Greenland temperature evolution during DO events is closely
related to the current state of the THC,
respectively to its stability). We stress that
this assumption is of course highly simplified, because Greenland
temperature is certainly not only influenced by the THC but also by other
processes such as changes in ice accumulation during DO oscillation. However,
this assumption is not crucial for the dynamics of our model, since the
timing of the switches between both model states is solely determined by the
relation between the forcing function f and the threshold function T. This
means that even if we included a more realistic relation between T and S, the
timing of the simulated climate shifts would be unchanged and the model
dynamics would thus essentially be invariant.

Note that $\delta_0$ and $\delta_1$ are not adjustable; they rather represent
internal time markers. Thus, six adjustable
parameters exist in our model as described here, namely $A_{0}$, $A_{1}$,
$B_{0}$, $B_{1}$, $\tau_{0}$ and $\tau_{1}$. Our choice for these parameters
is shown in Table <ref>. With these parameter values the system is
bistable (i.e. no transition is ever triggered in the absence of any forcing,
since $B_{0} \le 0$ and $B_{1} \ge 0$) and almost symmetric. That
means that the average duration of the simulated warm and cold intervals is
almost equal. When compared with Greenland paleotemperature records this
situation most likely corresponds to the time interval between about 27000 and
45000 years before present, during which the duration of the cold and warm
intervals in DO oscillations was also comparable (Fig. 1). The
model can, however, also represent an
unstable (for $B_{0} > 0$ and $B_{1} < 0$) or a mono-stable system, in which the stable
state is either the warm one (for $B_{0} > 0$ and $B_{1} \ge 0$)
or the cold one (for $B_{0} \le 0$ and $B_{1} < 0$); when compared to the ice
core data this situation is closer to the time interval between 15000 and
27000 years before present, since during that time the system was preferably
in its cold state and the forcing apparently crossed the threshold only
infrequently and during short periods of time.

Parameters of the conceptual model. All parameters have the same
values as in the publication of Braun et al., 2005. $A_{0}$, $A_{1}$, $B_{0}$ and $B_{1}$ are
given in freshwater units (i.e. in mSv = milli-Sverdrup; $1~\unit{mSv} = 10^{3}~\unit{m^{3}/s}$), since the
conceptual model was originally designed to mimic the response of the THC to
an anomaly in
the surface freshwater flux.

Parameter     Chosen value

A$_{0}$     -27 mSv

A$_{1}$     27 mSv

B$_{0}$     -9.7 mSv

B$_{1}$     11.2 mSv

$\tau_{0}$     1200 years

$\tau_{1}$     800 years

§.§ Comparison with a coupled climate model

In order to test our conceptual model we compare its performance under a
number of systematic forcing scenarios with the performance of the far more
comprehensive model CLIMBER-2 (a short description of that model is given in
the Appendix; a detailed description exists in the publication of
Petoukhov et al., 2000).

Analogous to Braun et al., 2005, we investigate the response of both models to
a forcing that consists of two century-scale sinusoidal cycles. In the
conceptual model, the forcing is implemented as the forcing function f. In the
EMIC, the forcing is added to the surface freshwater flux in the latitudinal
belt 50-70 $^{\circ}$N, following Ganopolski and Rahmstorf, 2001 and
Braun et al., 2005. This anomaly changes the vertical density gradient in the
ocean and can thus trigger DO events. Switches from the cold state into the
warm one are excited by sufficiently large (order of magnitude: a few
centimetre per year in the surface freshwater flux into the relevant area of
the North Atlantic) negative freshwater anomalies (i.e. by positive surface
density anomalies that are strong enough to trigger buoyancy [deep]
convection), and the opposite switches are triggered by sufficiently large
positive freshwater anomalies (i.e. by negative surface density anomalies that
are strong enough to stop buoyancy [deep] convection). This justifies our
choice for the logical relations that govern the dynamics of the transitions
in the conceptual model (i.e. $f(t) < T(t)$ as the condition for the switch
from the cold state to the warm one, $f(t) > T(t)$ for the opposite switch).

A detailed comparison between both model outputs is presented in the
Supplementary Material.
We here only summarise the main results: We find a general agreement between both
models, which is robust when the forcing parameters are varied over
some range
(Supplementary Figs. 1-6). The conceptual model reproduces the existence of three
different regimes (cold, warm, oscillatory) in the output of the EMIC
and also their approximate position in the
forcing parameter-space. By construction only the
nonlinear component in the response of the EMIC to the forcing is reproduced by the
conceptual model (this component represents the saw-tooth shape of DO
events). A second,
more linear component is not included in the conceptual model (this component represents
small-amplitude temperature anomalies which are superimposed on the saw-tooth
shaped events
in the EMIC). In particular, the conceptual model very well reproduces the
timing of the onset of DO events in the EMIC. The fact that our conceptual model, despite its
simplicity, agrees in so many aspects with the much more detailed model
CLIMBER-2 suggests
that it indeed captures the key features in the dynamics of DO events in that

We would like to stress that the output of the EMIC indeed supports our
assumption of an overshooting in the stability of the system during the
transitions between both climate states: When driven by a periodic forcing
(with a period of 1470 years), the EMIC can show periodic oscillations during
which it remains in either of its states for more than one forcing period
(i.e. for considerably more than 1470 years, compare Supplementary Fig.
4a). This implies that (at least in the EMIC) the conditions for a return to
the opposite state indeed ameliorate with increased duration of the cold or
warm intervals. If the thresholds in the model were constant (or gradually
increasing with increasing duration of the simulated cold / warm intervals),
in contrast, the duration of the cold and warm intervals during the simulated
oscillation could never be longer than 1470 years: If a periodic forcing does
not cross a constant (or gradually increasing) threshold within its first
period, it never crosses the threshold, due to the periodicity of the forcing.

§ NONLINEAR RESONANCE MECHANISMS IN THE MODEL

Strongly nonlinear systems can show complex and apparently counterintuitive
resonance phenomena that cannot occur in simple linear systems. In this
section we use our conceptual model to demonstrate and to discuss two of these
phenomena, i.e. stochastic resonance (SR) and ghost resonance (GR). Since the
explanation of the 1470-year cycle (and in fact even its significance) is
still an open question, we further discuss how future tests could distinguish
between the proposed mechanisms.

§.§ Stochastic resonance (SR)

Stochastic resonance. The input consists of: 1. a
sub-threshold sinusoidal signal with a period of 1470 years and an amplitude
of 4.5 mSv (about 40 percent of the threshold value B$_{1}$ above which
DO events occur in the model), 2. a random Gaussian-distributed signal with
white noise power signature (standard deviation $\sigma$ = 8 mSv) and a cutoff
frequency of 1/(50 years). The cutoff is used since no damping exists in the
model and it thus shows an unrealistically large sensitivity to high-frequency
(i.e. decadal-scale or faster) forcing. A: Total input (black), periodic
input component (grey), model output (green). Dashed lines are spaced by 1470
years. B: Relative frequency to obtain a spacing of 1470 years $\pm$10% (triangles)
respectively $\pm$20% (squares) between successive events, as a function of the noise level $\sigma$.

In linear systems that are driven by a periodic input, the existence of noise
generally reduces the regularity of the output (e.g. the coherence between the
input and the output). This is not necessarily the case in nonlinear systems:
Excitable or bistable nonlinear systems with a threshold and with noise,
which are driven by a sinusoidal sub-threshold input, can show maximum
coherence between the input and the output for an intermediate noise level,
for which the leading output frequency of the system is close to the input
frequency. This phenomenon is called stochastic resonance (SR)
[Benzi et al., 1982, Gammaitoni et al., 1998]. SR has been suggested to explain the
characteristic timing of DO events [Alley et al., 2001], i.e. the apparent
tendency of the events to occur with a spacing of about 1470 years or integer
multiples thereof. It has further been demonstrated that DO events in the
model CLIMBER-2 can be subject to SR [Ganopolski and Rahmstorf, 2002].

Here we apply our conceptual model to reproduce these results and to reanalyse
the underlying mechanism. We use an input that is composed of:
(i) a sinusoidal signal with a
period of 1470 years, (ii) additional white noise. Figures <ref>
and <ref> show that for a suitable noise level the model can indeed show DO
events with a preferred spacing of about 1470 years or integer multiples
thereof. The reason for this pattern in the output is easily understandable in
the context of the model dynamics: DO events in the model are triggered by
pronounced minima of the total input (total input = periodic signal plus
noise). These minima generally cluster around the minima of the sinusoidal
signal, and the start of the simulated events thus has a tendency to coincide
with minima of the sinusoidal signal (Fig. <ref>a). Some minima of
the sinusoidal signal, however, are not able to trigger an event, because the
magnitude of the noise around these minima is too small so that the threshold
function is not reached by the total input. Consequently, a cycle is sometimes
missed, and the spacing of successive events can change from about 1470
years to multiples thereof.

Unlike the model CLIMBER-2 (which has a complex relationship between the input
and the output and also a large computational cost) our conceptual model can
be used for a detailed investigation of the SR, e.g. because the
dynamics of the model is simple and precisely known and because probability
measures (such as waiting time distributions) can be explicitly computed. In fact, the
resonant pattern in the conceptual model (Fig. <ref>) is due to
two time-scale matching conditions: The noise level is such that the average
waiting time between successive noise-induced transitions is comparable to
half of the period of the periodic forcing, and also comparable to the
relaxation times $\tau_{0}$ and $\tau_{1}$ of the threshold function (compare
Fig. <ref>b). This situation is different from
the usual SR, in which thresholds (or potentials) are constant in time (apart
from the influence of the periodic input). In the
usual SR, only one time-scale matching condition exists
[Gammaitoni et al., 1998], namely the one that the average waiting time between
successive noise-induced transitions (i.e. the inverse of the so-called
Kramers rate) is comparable to half of the period of the periodic forcing.

Distribution of the spacing $\Delta$T between successive events. The input in
a and b consists of noise only, with a standard deviation $\sigma$ of 8 mSv
(as in Fig. <ref>a). In c and d, a sinusoidal input component
(amplitude = 4.5 mSv, period = 1470 years; compare Fig. <ref>) is
added to the noise. In a and c the threshold function is constant in each
state (20 mSv in the warm state and -20 mSv in the cold one), while the
overshooting relaxation assumption is used in b and d (with threshold
parameters as shown in Table 1). Thus, c corresponds to the “usual” SR while
d shows our “overshooting” SR.

In order to investigate the implications of the second condition we simulate
histograms for four different scenarios in the conceptual model
(Fig. <ref>): 1. noise-only input, constant threshold
(Fig. <ref>a); 2. noise-only input, overshooting threshold
(Fig. <ref>b); 3. noise plus periodic input, constant threshold
(Fig. <ref>c); 4. noise plus periodic input, overshooting
threshold (Fig. <ref>d). We note that 3. corresponds to the usual
SR, while 4. describes our overshooting stochastic resonance. As can be
seen from the histograms, the existence of the millennial-scale relaxation
process leads to a synchronisation in the sense that the waiting times between
successive events are confined within a much smaller time interval (about
1000-4500 years with the overshooting, compared to about 0-10000 years without

This confinement is plausible since the transition probability
between both model states strongly depends on the magnitude of the threshold,
which declines with increasing duration of the cold or warm
intervals: When the
standard deviation of the noise level is chosen such that
the average waiting time between successive noise-induced transitions is
comparable to the relaxation times $\tau_{0}$ and $\tau_{1}$, as in
Fig. <ref>, the overshooting relaxation strongly reduces the
transition probability for waiting times much smaller than the relaxation time
(since the corresponding values of the threshold function are large) and
increases the transition probability for waiting times of the order of the
relaxation time or larger (since the corresponding values of the threshold function are
considerably smaller). The probability to find an only century-scale spacing between
successive events is thus small, because the corresponding transition
probabilities are small. On the other hand, the probability to find a
multi-millennial spacing is also small, because the states are already
depopulated before (i.e. the probability to obtain lifetimes considerably
larger than the relaxation time is almost zero). This explains why the
possible values for the spacing between successive DO events are restricted to
a much smaller range than in the usual SR (i.e. in the case with constant

This synchronisation effect is indeed not unique to the conceptual model: The
output of the coupled model CLIMBER-2 shows a similar pattern (with
possible waiting times between successive DO events of e.g. about 1500-5000 years
or about 1000-3000 years, depending on the noise level; compare Fig. 4a-d in
the publication of Ganopolski and Rahmstorf, 2002). This similarity is of course not
surprising, since the conceptual model is apparently able to mimic the
events in the EMIC and since an overshooting in the stability of both states clearly also
exists in CLIMBER-2 (compare Sect. 3.3). We note that in the GISP2 ice core data, DO
events in the time interval 27000-45000 years before present (which, as
discussed in Sect. 3.2, is the best analogue to the “background climate
state” in our conceptional model, since the duration of the cold and warm
intervals in the data is comparable in that interval) have spacings of about
1000-3000 years (compare Fig. 1) and were reported to cluster around values of
either about 1470 years or about 2940 years [Schulz, 2002]. Because the SR
mechanism could explain such a pattern (compare Fig. <ref>d) it
has originally been proposed. However, this mechanism requires a sinusoidal
input with a period of about 1470 years, which has so far not been detected.

§.§ Ghost resonance (GR)

In linear systems which are driven by a periodic input, the frequencies of the
output are always identical to the input frequencies. This is not necessarily
the case in nonlinear systems. For example, nonlinear excitable (or bistable)
systems that are driven by an input with  frequencies corresponding to
harmonics of a fundamental frequency (which itself is not present in the
input) can show a resonance at the fundamental frequency, i.e. at a frequency
with zero input power. This phenomenon, which was first described in order to
explain the pitch of complex sounds [Chialvo et al., 2002, Chialvo, 2003] and later
observed experimentally e.g. in laser systems [Buldu et al., 2003], is called
ghost resonance (GR). GR and SR can indeed occur in the same class of systems,
e.g. in bistable or excitable systems with thresholds. However, unlike SR, GR
requires a periodic driver with more than one frequency. Although many
geophysical systems might be subject to GR (since the relevant processes often
have thresholds), the occurrence of this mechanism has so far not expressly
been demonstrated in geoscience.

Ghost resonance. Top: Forcing (black) and model response (green). Middle:
Amplitude spectrum of the forcing. Bottom: Amplitude spectrum of the model response. We use two sinusoidal forcing
cycles, with frequencies of 7/(1470 years) and 17/(1470 years), respectively, and with equal
amplitudes. These two cycles coincide every 1470 years and create peaks
of particularly pronounced magnitude, spaced by exactly that period. Thus,
despite the fact that there is no spectral power at the corresponding
frequency (see middle panel), the forcing repeatedly crosses the threshold at
those intervals. Consequently, the response of the conceptual model (i.e.
the time evolution of the state variable S) shows strictly repetitive DO events with
a period of 1470 years (as indicated by the dashed lines, which are spaced by
1470 years). Despite the lack of a 1470-year spectral component in the forcing, the
output shows a very prominent peak at the corresponding frequency.

Here we discuss a hypothesis that was recently proposed to explain the
1/(1470 years) leading frequency of DO events [Braun et al., 2005]. The underlying mechanism of
the hypothesis is in fact the first reported manifestation of GR in a
geophysical model system. According to that hypothesis the 1/(1470 years)
frequency of DO events could represent the nonlinear climate response
to forcing cycles with frequencies close to harmonics of 1/(1470 years). Our
conceptual model illustrates the plausibility of this mechanism: We use a
bi-sinusoidal input with frequencies of 7/(1470 years) and 17/(1470 years), i.e. with
frequencies corresponding to the 7th and the 17th harmonic of a 1/(1470 years)
fundamental frequency, and with equal amplitudes. A spectral component corresponding
to the fundamental frequency is not explicitly present in the input. Since the two
sinusoidal cycles correspond to harmonics of the missing fundamental, the input signal
repeats with a period of 1470 years. For an appropriate range of input amplitudes, the
output of the conceptual models shows periodic DO events with a period of 1470 years
(Fig. <ref>). Unlike the input, the model output exhibits a
pronounced frequency of 1/(1470 years), corresponding to the leading frequency
of DO events and to the fundamental frequency that is absent in the
input. This apparent paradox is explained by the fact that the two driving
cycles enter in phase every 1470 years, thus creating pronounced peaks spaced
by that period. Because the magnitude of these peaks results from constructive
interference of the two driving cycles, it is indeed robust that a threshold
process can be much more sensitive to these peaks than to the two original driving
cycles [Chialvo, 2003].

The main strength of the GR mechanism is that – unlike the SR mechanism – it
can relate the leading frequency of DO events to a main driver of natural
(non-anthropogenic) climate variability, since proxies of solar activity
suggest the existence of solar cycles with periods close to 1470/7 (=210)
years (De Vries or Suess cycle) and 1470/17 ($\approx$86.5) years (Gleissberg
cycle) [Stuiver and Braziunas, 1993, Wagner er al., 2001, Peristykh and Damon, 2003]. So far, however, no
empirical evidence for this mechanism has been found [Muscheler and Beer, 2006],
nor has it been shown yet that changes in solar activity over the solar cycles
are sufficiently strong to actually trigger DO events.

In order to investigate the stability of this mechanism we further add a
stochastic component (i.e. white noise) to the forcing. In this case the
events are – of course – not strictly periodic anymore. Similar to the SR
case, an optimal (i.e. intermediate) noise level exists for which the waiting
time distribution of the simulated events exhibits a maximum at a value of
1470 years, corresponding to the period of the fundamental frequency of the
two input cycles (Fig. <ref>). In contrast to the SR case, in which a fairly
simple waiting time distribution with a few broad maxima of century scale
width is obtained (compare Fig. <ref>d), we now find a much more
complex pattern with a large number of very sharp lines of only decadal scale
width. Since the waiting time distributions of both mechanisms are
considerably different, it could – at least in principle – be possible to
distinguish between both mechanisms by analysing the distribution of the
observed DO events. In practise, however, this approach is complicated by the
fact that only about ten events appear to be sufficiently well dated
for this kind of analysis, and even their spacing has an uncertainty of about
50 years [Ditlevsen et al., 2007], which is already of the same order as the
width of the peaks in Fig. <ref>b. We note that the mechanism that
is described in Fig. <ref> is known as ghost stochastic resonance (GSR), and its
occurrence and robustness has already been reported before in other systems
with thresholds and multiple states of operation [Chialvo, 2003]. At least
in our system, however, this mechanism is even more complex than the other two types of
resonance (SR, GR). It is beyond the scope of this paper to describe the GSR
mechanism in more detail.

Ghost stochastic resonance. The input consists of: 1. two sinusoidal forcing
cycles with frequencies of 7/(1470 years) and 17/(1470 years), respectively,
and with an amplitude of 8 mSv, 2. a random Gaussian-distributed signal with
white noise power signature and a cutoff frequency of 1/(50 years), as in
Fig. <ref>. In a, the relative frequency to obtain a spacing
of 1470 years $\pm$1% (triangles) respectively $\pm$2% (squares) between
successive events is shown as a function of the noise level $\sigma$. B shows
the distribution of the spacing $\Delta$T between successive events (standard
deviation of the noise: 5.5 mSv).

§.§ Testing the proposed mechanisms

The most direct way to test which of the proposed mechanisms – if any
– provides the correct explanation for the timing of DO events would be to
reconstruct decadal-scale density anomalies in the North Atlantic in
connection with the events. This is not possible since even the most highly
resolved oceanic records do not allow to reconstruct the variability of the
glacial ocean on that time scale. Thus, only indirect tests can be
performed. The identification of the postulated 1/(1470 years) forcing
frequency, which has so far not been detected, would certainly give further
support for the SR mechanism. And in order to support the GR mechanism, it
remains crucial to demonstrate that century-scale solar irradiance variations
are indeed of sufficiently large amplitude to trigger repeated transitions
(with a preferred time scale of about 1470 years) between the two glacial
climate states. This could be tested with climate models.

An elegant and simple test is to make use of the observation that DO
events in the Earth system model CLIMBER-2 represent the nonlinear response to
the forcing, and that an additional (and much smaller) linear response is
superimposed on the events. In the absence of any threshold crossing (e.g. in
the Holocene, during which DO events did not occur) the response to the
forcing, in contrast, does not show a strong nonlinear component. This
suggests that Holocene climate archives from the North Atlantic region might
be able to reveal what triggered glacial DO events. This approach has two
major advantages: First, more reliable (e.g. better resolved and dated)
records are available to solve this issue. Second, linear analysis methods can
be used for that purpose, e.g. linear correlations. In the context of the GR
mechanism, for example, the existence of a pronounced correlation between
Holocene climate indices from the North Atlantic and solar activity proxies
(reconstructed e.g. from $^{14}$C variations in precisely dated tree rings)
would be expected. Up to now at least one study exists that supports this
prediction of a linear relationship between century-scale solar forcing and
North Atlantic climate variability throughout the Holocene: Proxies of drift
ice anomalies in the North Atlantic show a persistent correlation and a
statistically significant coherency with “rapid (100- to 200-year),
conspicuously large-amplitude variations” in solar activity proxies
[Bond et al., 2001], in accordance with the proposed GR mechanism.

The most challenging test, however, is the direct analysis of the glacial climate records. We are
convinced that one of the main difficulties in this approach is the high
degree of nonlinearity of the events, which – according to our interpretation
– has so far not been adequately addressed in many previous studies. For example,
several attempts have already been made in order to investigate the 1470-year
cycle by means of linear spectral analysis methods, and significance levels
have commonly been calculated by assuming a red noise background. To us this
assumption seems to be oversimplified, since the system responds at a
preferred time scale even when driven by white noise (compare
Fig. <ref>b). We thus suspect that the significance levels obtained
by this method are unrealistically high. We further think that the
reported lack of a clear phase relation between solar activity proxies and DO
events [Muscheler and Beer, 2006] cannot rule out the idea that solar forcing
synchronised DO events, since in the case of an additional stochastic
forcing component (i.e. in the GSR case) the events are triggered by the
combined effect of solar forcing and noise. Thus, the observed lack could also
imply that only some of the events were in first place triggered by the Sun,
whereas others were caused mainly by random variability (e.g. by noise).

A new and promising approach, which is based on a Monte-Carlo method, has
recently been proposed in order to test the significance of the glacial 1470-year
climate cycle: Ditlevsen et al., 2007 define a certain measure in order to distinguish
between different hypotheses for the timing of DO events, and they explicitly
calculate the value of this measure for the series of events observed in the
ice core. They then compare the calculated value with the values obtained by
several hypothetic processes, e.g. by a random process (for which
assumptions concerning the probability distribution of the recurrence times of
the events have to be made). Significance levels are obtained from the
(numerically estimated) probability distributions of the measure as generated
by the considered process. Although we do not share their conclusions (because
we think that more adequate measures can be chosen, which give considerably
different results) we think that this approach is elegant because significance
levels are not calculated based on linear theories. The method is thus also
applicable to highly nonlinear time series. A major hurdle in this method is that for each considered
process the probability distribution of the waiting times – which is unknown
for almost all processes – somehow has to be specified. For example,
Ditlevsen et al. use a simple mathematical (i.e. an exponential) distribution
in order to mimic random DO events. In order to improve their novel
approach, some method is thus needed to calculate waiting time distributions
in response to any possible input. Comprehensive models are not applicable,
due to their large computational cost. Our conceptual model, in contrast, is
well designed for that purpose because it is combines the ability to mimic the
complex nonlinearity of DO events as described by an accepted Earth system
model with the extremely low computational cost of a very simple (zeroth
order) model. We thus think that our work is an important step in
order to develop improved statistical analysis methods which are able to cope
with the extreme nonlinearity of DO events.

[Discussion and conclusions]

We here discussed DO events in the framework of a very simple conceptual model
that is based on three key assumptions, namely (i) the existence of two
different climate states, (ii) a threshold process and (iii) an overshooting
in the stability of the system at the start and the end of the events, which
is followed by a millennial-scale relaxation. These assumptions are in
accordance with paleoclimatic records and/or with simulations performed with
CLIMBER-2, a more complex Earth system model. In a couple of systematic tests
we showed (in the Supplementary Material) that despite its simplicity, our
model very well reproduces DO events as simulated with CLIMBER-2, whose
dynamics is based on a (albeit reduced) description of the underlying
hydro-/thermodynamic processes. The correspondence between both models thus
strengthens our interpretation that the conceptual model can successfully
mimic key features of DO events, and that these can be regarded as a new type
of non-equilibrium oscillation (i.e. as an overshooting relaxation
oscillation) between two states of a nonlinear system with a threshold.

Although we discussed our model dynamics in the context of the (thermohaline)
ocean circulation, our model does not explicitly assume that DO events are
linked with changes in the ocean circulation: Threshold behaviour and multiple
states exist in many compartments of the climate system (not only in the
ocean, but e.g. also in the atmosphere and in the cryosphere). Our model thus
cannot rule out a leading role of non-oceanic processes in DO
oscillations. The millennial time scale of the events (which is represented in
our model by the assumption of a millennial-scale relaxation), however,
corresponds to the characteristic time scale of the thermohaline circulation
and thus points to a key role of the ocean in DO oscillations.

The main strength of our model is its simplicity: Due to the obvious relationship
between forcing and response, the model can demonstrate why even a simple
bistable (or excitable) system with a threshold can respond in a complex way
to a simple forcing, which consists of only one or two sinusoidal inputs and
noise. We applied our model to discuss two highly nonlinear and apparently
counterintuitive resonance mechanisms, namely stochastic resonance and ghost
resonance. In doing so we reported a
new form of stochastic resonance (i.e. an overshooting stochastic
resonance), in which the overshooting of the system leads to a further
synchronisation effect compared to the usual stochastic resonance. Our
study provides the first explicitly reported manifestation of ghost resonance
in a geophysical (model) system. Since threshold behaviour and multiple
equilibria are not unique to DO events but exist in many geophysical
systems, we would indeed expect that ghost resonances could be inherent in
many geosystems and not just in our model.

In addition to its applicability to demonstrate and interpret nonlinear
resonance mechanisms, and to test their stability, we further illustrated the
ability of our conceptual model to simulate probability measures (e.g. waiting time distributions, which
are required in order to test the significance and the cause of the proposed
glacial 1470-year climate cycle by means of Monte-Carlo simulations). Because
it combines the ability to reproduce essential aspects of DO
events with the extremely low computational cost of a conceptual model (which
is up to 10$^{7}$ times lower than in the Earth system model CLIMBER-2), we
think that our model represents an important advance in order to develop
adequate nonlinear methods for improved statistical analyses on DO events.

§ APPENDIX: DESCRIPTION OF CLIMBER-2

The Earth system model CLIMBER-2, which we used for our analysis, has
dynamic components of the atmosphere, of the oceans (including sea
ice) and the vegetation. Dynamic ice sheets were not included
in our study. CLIMBER-2 is a global model with coarse
resolution: For the atmosphere and the continents the spatial
resolution is 10$^{\circ}$ in latitude, and 7 sectors are considered in
longitude. The ocean is zonally averaged with a latitudinal
resolution of 2.5$^{\circ}$ for the three large ocean basins. A detailed
description of the model is given in the publication of Petoukhov et al., 2000.

DO events in the model represent abrupt switches between two different climate
states (stadial [i.e. cold] and interstadial [i.e. warm]),
corresponding to two different modes of the THC: In the interstadial
mode, North Atlantic deep water (NADW) forms at about 65 $^\circ$N and much of
the North Atlantic is ice-free. In the stadial mode, NADW forms
at about 50 $^\circ$N and a considerably larger area of the North Atlantic is
ice-covered. We note that for the climatic background conditions
of the Last Glacial Maximum (LGM) only the stadial mode is stable in the
model whereas the interstadial mode is excitable but unstable
[Ganopolski and Rahmstorf, 2001]. Moreover, the stability of both modes depends on the
actual climate state (e.g. on the configuration of the Laurentide ice sheet
and on the freshwater input into the North Atlantic), and the stability
properties of the system change when the background conditions are modified
(more precisely, the system can be bistable or mono-stable).

Transitions between both modes can be triggered by anomalies in the
density field of the North Atlantic, for example by variations in the
surface freshwater flux (since the density of ocean water increases
with increasing salinity). In our study we thus implement the forcing
as a perturbation in the freshwater flux (in the latitudinal belt
50-70 $^{\circ}$N): We start the model with the climatic background conditions
of the Last Glacial Maximum (LGM). Following earlier simulations
[Braun et al., 2005] we then add a small constant offset of 17 mSv
($1~\unit{mSv} = 10^{3}~\unit{m^{3}/s}$) to the  freshwater flux. For this
climate state (which we label perturbed LGM) the THC is in fact bistable
and DO events can be triggered more easily than for LGM conditions. This
perturbed LGM state gives us the background conditions for the model
simulations as presented in this paper.

The authors thank R. Calov, A. Mangini, S. Rahmstorf, K. Roth and A. Witt for
discussion, and P. Ditlevsen (in particular for observing the difference
between the usual stochastic resonance and our overshooting stochastic resonance) and
two anonymous reviewers for helpful comments. H. Braun was funded by Deutsche
Forschungsgemeinschaft, DFG project number MA 821/33.

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