On the representation of multiplicative noise in modeling Dansgaard–Oeschger events

The interpretation of multiplicative noise in a stochastic differential equation in the context of data-driven inverse modeling is discussed. Application to the well-known paleoclimate phenomenon of Dansgaard–Oeschger events leads to qualitatively different ‘climate potentials’ in the case of the Itô or the Stratonovich interpretation of the stochastic integral. While a physical model is endowed with an interpretation from construction, whether implicitly or explicitly, inverse models derived from data do not afford such a luxury. In this case, a physical model must accompany the mathematical model equation in order to be able to choose a stochastic interpretation. This case study illustrates the differences between the two representations of stochastic noise and demonstrates the need for input from physical constraints when constructing conceptual stochastic models of the observed climate records.


Introduction
One of the most famous examples of abrupt climate changes observed in the paleoclimatic record are the Dansgaard-Oeschger (D-O) events.The climate record of the Last Glacial Period (LGP), which spanned approximately 120 to 11 kiloyears before year 2000 (kyr b2k), is measured in the ice-cores of the Greenland ice sheet and marked by distinct and abrupt transitions between colder stadial and warmer interstadial periods [1].These climatic changes are known as D-O events, and occurred about 24 times in the LGP.The D-O events correspond to approximately 10-15 Kelvin of warming in Greenland over the course of a few decades, with subsequently incremental cooling to the fully glacial conditions of the stadial [2].While there is only direct evidence of D-O events in the LGP because the ice-core record of Greenland only extends to the end of the last interglacial period, they may not be unique to this time period.Coupling with Antarctic ice-cores [3], evidence in marine sediment cores [4] and speleothems [5] (see also references therein) suggest they may have occurred in previous glacial periods as well.
D-O events are interesting in the context of the present climate primarily because of their temporal scale.They are an example that the climate can change on time scales that could be of consequence in the near future, namely of decades to centuries.They are additionally intriguing because there is no universal agreement on their cause and transition mechanism.The transitions between stadial and interstadial

Additive noise
The majority of previous studies assume the intensity of the noise is constant or additive [12,16,17,29].The Fokker-Planck equation associated with the additive noise SDE  =  () +   (2) is When the stationary probability distribution is obtained from the observed time series record, the potential that drives the dynamics of the additive noise SDE is obtained from the Fokker-Planck equation by having    = 0: where   = − ().The potential can be uniquely determined from the stationary probability density  () up to a constant factor of the noise strength.Due to the monotonicity of the logarithm the maxima of the probability density function  () coincide with the minima of the potential  () and thus with the stable equilibria of the deterministic dynamics.Therefore the number of equilibria will always be identical to the number of maxima in probability density.

Multiplicative noise
From the paleoclimatic record it is observed that the intensity of the fast fluctuations constituting the noise as indicated in Eq. ( 1) depends on the climate state [40].State dependent noise is termed multiplicative noise.In this case deriving the resulting potential is not completely straightforward.This is due to the fact that when integrating the noise term by way of generalized functions, the resulting Riemann-Stieltjes integral where   ∈ [ −1 ,   ], has a different expected value depending on where in the interval   is chosen.The two most common choices are the left endpoint, named the Itô interpretation [41], and the midpoint, named the Stratonovich interpretation [42].Often the -convention is used to designate the different interpretations, where the value of  in the interval   = (1 − ) −1 +   is 0 for Itô and 1∕2 for Stratonovich.
In theory, any value of  ∈ [0, 1] is a valid choice for the stochastic integral, but these are by far the two most common.For a more general function  ( () ) , the definition of the integral is and the -convention is  * (  ) = (1 − )( −1 ) + (  ).
Due to the difference of the stochastic integrals, Eq. ( 1) is incomplete and an interpretation of the noise term must be specified [43].As a consequence the same SDE can result in different stochastic processes depending on whether the Itô or Stratonovich interpretation is applied.A corollary is that two different SDEs, one interpreted as Itô and the other as Stratonovich, can result in the same stochastic process.Thus solving the inverse problem of deriving the SDE, and especially the potential, from a stochastic realization requires a specification of the noise.Here we perform the derivation of the SDE from the data for both the Itô and Stratonovich integrals.
As SDEs they are distinguished by the notation  =   () + () ⋅   for Itô and  =   () + ()•  for Stratonovich, where   and   are different potential functions.The associated Fokker-Planck equation using the  convention is As can be seen from this equation, a simple relation exists between the drift terms of the two interpretations, so any Stratonovich integral may be converted to an Itô integral and vice versa.This relation is especially useful when numerically integrating an SDE since the commonly used Euler-Maruyama method is only applicable to Itô SDEs.A Stratonovich SDE can be converted to one of the Itô type and integrated using the Euler-Maruyama method, or alternatively integrated with a predictor-corrector scheme such as a Heun method.Itô and Stratonovich integrals have some other differences as well, the most notable being that differentiation under the Itô interpretation requires the Itô lemma [41], On the other hand the Stratonovich interpretation uses the chain rule of regular calculus.
One previous study includes state-dependent noise in the form of a piecewise constant noise term, where the amplitude is a lower constant value in the interstadials than in the stadials [44].However their state dependent noise function () still has a derivative that is zero except for a single point, so there is no difference between the stochastic interpretations.

Data
The paleoclimate data studied is a time series of the  18 O in permille as measured in the Greenlandic ice-core extracted as part of the North GReenland Ice-core Project (NGRIP) [45].The 20-year average values on the GICC05modeltext time scale are used [46].The time series is also truncated at 85 kyr b2k as the resolution decreases further back in time due to the thinning of layers of ice in the ice core.For a further distinction of the two states, the data is detrended.Insolation trends due to orbital variations are removed through subtracting a 25 kyr running mean [17] and the resulting  18 O anomaly is analyzed.This 25 kyr running mean corresponds to the highest frequency of orbital variations, namely precession, which has a period of approximately 20 kyr.This method is effectively a rectangular kernel, which has the important property of not filtering out impulses, i.e. the D-O events themselves.Fig. 1 shows the time series that is the starting point of this study.

Derivation of multiplicative noise 𝜎(𝑥)
A heuristic method is used to derive the multiplicative noise term () from the data.Since the fluctuations are larger in the stadials than in the interstadials [40] we prescribe a linearly decreasing function of () with respect to the  18  anomaly.Physically, if the noise term is to represent the influence of the atmosphere, an increase in Greenlandic temperatures corresponds to a decrease in the meridional temperature gradient which in turn decreases atmospheric forcing.Following the definition of the stadial and interstadial periods [47], the data is separated into the two states.The values of ⟨⟩ and  in each of these two states is derived, and the linear function is constructed from these values.To measure  it is assumed that the signal in either of the two states follows an Ornstein-Uhlenbeck (O-U) process, following [40] For a stationary O-U process the variance is given by the fluctuationdissipation relation and the term  may be recovered from the autocorrelation

Derivation of the non-linear potentials F(x)
Once the function () is determined the multiplicative Fokker-Planck Eq. ( 7) is solved for  () for the two stochastic calculi.Fig. 2 shows the two potentials  () = ∫  () in the Itô and Stratonovich interpretations, along with the potential for the additive noise case (2).When comparing the drift in the multiplicative noise cases to that of the additive noise case, the stability of the interstadial is much reduced.Further, for the Itô case, the interstadial has in fact lost stability such that the resulting climate potential is mono stable.

Discussion
Due to the qualitative difference of the potential obtained by the two different stochastic interpretations, careful consideration is required when deriving physical properties of the system from observations.The question to be answered is whether the climate potential that underlies the D-O events is monostable, as in the Itô calculus, or bistable, as in the Stratonovich calculus.How is a choice of a stochastic integral made?The data itself is agnostic to interpretation, and mathematically the problem is inconsequential: either interpretation works, as long as they are applied consistently.The choice is ultimately a modeling problem.In this case the SDE inversely modeled by the data requires a conceptual physical model of the phenomenon by which to interpret the results.Conventional understanding is that a physical system is better suited to Stratonovich interpretation.The Wong-Zakai theorem [48] gives that the limit of a sequence of stochastic processes with finite autocorrelation that goes to zero is interpreted as Stratonovich.In this sense it is derived from a continuous process.For a fully discrete system, for example in financial analysis, the Itô interpretation is more appropriate.
However, the one dimensional SDE is generally the result of simplification of dynamics that occur on multiple time scales.Through this reduction, multiplicative noise of the Itô type can be seen in physical systems as well.The most prominent example are inertial systems with colored noise, where for convenience, we use the notation, ẋ ≡ ∕ and   =   .
To reduce the complexity of Eq. ( 13), two limits are taken: one is an adiabatic elimination of fast inertia (  = () −1 → 0) into the overdamped regime (also known as the strong dissipation or Smoluchowski regime) and the second is the white-noise approximation of the unresolved dynamics (  → 0).If the time scale of the inertial relaxation is greater than that of the noise autocorrelation (  ≫   ), then the multiplicative noise is Itô in the limit as both go to zero.If the noise autocorrelation in greater than the relaxation time scale of the inertia (  ≪   ), the Stratonovich interpretation for the multiplicative noise is used [49][50][51][52][53][54].
The system of Eq. ( 13) in the white noise limit is For additive noise () = , adiabatic elimination is equivalent to setting the left hand side of Eq. ( 14) equal to zero [55,56].This is not the case for multiplicative noise, where the limits must be taken carefully.Eq. ( 13) in the white noise and adiabatic limit is where  * = ( + ) + (1 − )() [56].In the case where the fluctuation-dissipation theorem applies, the system obtained by setting the left hand side of Eq. ( 13) to zero is equivalent to the system interpreted in the anti-Itô ( = 1) sense.This is the result of Volpe et al. [57] and Lançon et al. [58], see also [56,59].
Another example where we see Itô multiplicative noise is in an SDE with noise feedback delay [60].This is a circuit system which has been designed to effectively act in an Itô manner by implementing an explicit dependence of the multiplicative noise on the previous time step through introducing a delay of the feedback of the state on the noise term, i.e. the  in the () term.One dimensional systems can also be either Itô or Stratonovich in the case where multiple times scales are involved [61,62].Now understanding the situation, we return to the case of the  18 O anomaly time series.To accompany the SDE derived directly from the data, a conceptual physical model is required, which will not only provide an idea of the underlying mechanisms but also provide a stochastic interpretation.The stochastic bistable Stommel-type model [36] has been extended to include multiplicative noise by Timmermann, Lohmann and Monahan [63,64].Their multiplicative noise term arises due to a stochastically parameterized eddy transport in salinity and temperature.As in Cessi [36], the temperature relaxes quickly to some mean value, and a white noise approximation may be made.In this case the white-noise limit is taken via the Wong-Zakai theorem, so that the resulting SDE uses Stratonovich calculus.Then the thermohaline circulation remains bistable based on the model derived from the data.
In another study, Kwasniok and Lohmann [65] fit the D-O event time series data to a stochastic oscillator ẍ = − ẋ +  () +   (18) which is Eq. ( 14) with constant damping and additive noise.The variable  represents a temperature proxy and its derivative is the change in temperature, but otherwise the system is not physically defined.They find it is in the strongly dissipative regime and can be reduced to a first-order equation by adiabatic elimination of the second derivative.By augmenting the system with multiplicative noise, a reduction to first order would result in an SDE with the Itô interpretation.However, this interpretation still relies on the fact that the autocorrelation of the noise is assumed to be decaying faster than the relaxation of the fast variable, i.e   ≫   .This could be understood as a temporal scale of the thermal relaxation time of the surface temperature of Greenland and the autocorrelation time of the atmospheric variability.The stochastic interpretation is still a result of modeling choices.
Various studies model the D-O events using a two-dimensional model generally assume the form of a van der Pol or FitzHugh-Nagumo type model [15,17,22].These have a form similar to Eq. ( 14), but the variable of interest is on the fast timescale.This means the system is in the underdamped regime and cannot be adiabatically reduced to one dimension.While the models in these studies include only additive noise, we again consider their stochastic interpretation in the possible case of multiplicative noise.In the underdamped regime of Eq. ( 14) the difference between the Itô and Stratonovich integrals is smaller than , so they are equivalent.First, the second order SDE is split into a system of first-order equations Using the -convention, and the difference between the interpretations given in the  term is of the order () 3∕2 <  and thus vanishes faster than the time scale of the dynamics.If instead there is a stochastic component to both variables, Then the stochastic term is in the fast variable is instead and the difference between the Itô and Stratonovich interpretations is now of the order  and must be taken into account.
The previously mentioned study of Krumscheid et al. [44] has statedependent noise in a data-driven model derivation and assumes Itô calculus.Their multiplicative noise term takes the form of a piecewise constant function, and as such has zero derivative except at the jump.Due to this, the difference between the Itô and Stratonovich interpretations vanishes except at a single point, which can be safely disregarded.Their study does include continuous functions as candidate noise terms but find the additive noise model outperforms them with regard to their parameter fitting routine.

Conclusions
In this study, we have derived a data-driven conceptual model of the D-O events including multiplicative noise and seen the reduced stability of the interstadial state when compared to the stadial.We also describe the need to specify a stochastic calculus to be able to interpret the climate potential of the model.We have outlined models which may be interpreted as either Itô or Stratonovich in limiting cases and suggest that the interpretation depends on a physical understanding of the system.For example, if the system derived from the data were meant to represent the stability of the thermohaline circulation but the stochastic integral interpreted as Itô, one would arrive to the erroneous conclusion that the data shows the overturning circulation was monostable, when there is evidence beyond the ice-core record that it is bistable.
The result that the Itô interpretation leads to monostable dynamics is one that is echoed by other conceptual models [17,33,65].These monostable excitable models with fast-slow dynamics, which require at least 2 dimensions, mirrors the scenario in which the Itô interpretation is applicable.That is, the Itô interpretation comes about due to reduction of a strongly dissipative inertial system, which can be represented as a 2D system with different timescales, to a system in 1D.
Ultimately, this work shows that when deriving stochastic models, the stochastic interpretation can fundamentally affect the results.Interpreting whether the underlying nature of the climate that gave rise to D-O events is a mono-or bi-stable system is vital step to understanding the phenomenon.This is affected by the noise interpretation, which seems to be a completely non-physical, mathematical formalism but is in fact determined by the timescales of the dynamics of the system.

Declaration of competing interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Fig. 1 .
Fig. 1.Detrended  18 O signal from NGRIP from years 20 to 85 kyr b2k (note that time runs from right to left).