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At the Azimuth Code Project, we are aiming to produce educational software that is relevant to the Earth sciences and the study of climate. Our present software takes the form of interactive web pages, which allow you to experiment with the parameters of models and view their outputs. But to fully understand the meaning of a program, we need to know about the concepts and theories that inform it. So we will be writing articles to explain both the programs themselves and the math and science behind them.

In this two-part series, I’ll cover the Azimuth stochastic resonance example program, by Allan Erskine and Glyn Adgie. In the Azimuth blog article Increasing the Signal-to-Noise Ratio with More Noise, Glyn Adgie and Tim van Beek give a nice explanation of the idea of stochastic resonance, which includes some clear and exciting graphs. My goal today is give a compact, developer-oriented introduction to stochastic resonance, which will set the context for the next blog article, where I’ll dissect the program itself. By way of introduction, I am a software developer with research training in computer science. It’s a new area for me, and any clarifications will be welcome!

The concept of stochastic resonance

Stochastic resonance is a phenomenon, occurring under certain circumstances, in which a noise source may amplify the effect of a weak signal. This concept was used in an early hypothesis about the timing of ice-age cycles, and has since been applied to a wide range of phenomena, including neuronal detection mechanisms and patterns of traffic congestion.

Suppose we have a signal detector whose internal, analog state is driven by an input signal, and suppose the analog states are partitioned into two regions, called “on” and “off” – this is a digital state, abstracted from the analog state. With a light switch, we could take the force as the input signal, the angle as the analog state, and the up/down classification of the angle as the digital state.

Consider the effect of a periodic input signal on the digital state. Suppose the wave amplitude is not large enough to change the digital state, yet large enough to drive the analog state close to the digital state boundary. Then, a bit of random noise, occurring near the peak of an input cycle, may “tap” the system over to the other digital state. So we will see a probability of state-transitions that is synchronized with the input signal. In a complex way, the noise has amplified the input signal.

But it’s a pretty funky amplifier! Here is a picture from the Azimuth library article on stochastic resonance:

Stochastic resonance has been found in the signal detection mechanisms of neurons. There are, for example, cells in the tails of crayfish that are tuned to low-frequency signals in the water caused by predator motions. These signals are too weak to cross the firing threshold for the neurons, but with the right amount of noise, they do trigger the neurons.

See:

Stochastic resonance, Azimuth Library.

Stochastic resonance in neurobiology, David Lyttle.

Bistable stochastic resonance and Milankovitch theories of ice-age cycles

Stochastic resonance was originally formulated in terms of systems that are bistable – where each digital state is the basin of attraction of a stable equilibrium.

An early application of stochastic resonance was to a hypothesis, within the framework of bistable climate dynamics, about the timing of the ice-age cycles. Although it has not been confirmed, it remains of interest (1) historically, (2) because the timing of ice-age cycles remains an open problem, and (3) because the Milankovitch hypothesis upon which it rests is an active part of the current research.

In the bistable model, the climate states are a cold, “snowball” Earth and a hot, iceless Earth. The snowball Earth is stable because it is white, and hence reflects solar energy, which keeps it frozen. The iceless Earth is stable because it is dark, and hence absorbs solar energy, which keeps it melted.

The Milankovitch hypothesis states that the drivers of climate state change are long-duration cycles in the insolation – the solar energy received in the northern latitudes – caused by periodic changes in the Earth’s orbital parameters. The north is significant because that is where the glaciers are concentrated, and so a sufficient “pulse” in northern temperatures could initiate a state change.

Three relevant astronomical cycles have been identified:

• Changing of the eccentricity of the Earth’s elliptical orbit, with a period of 100 kiloyears

• Changing of the obliquity (tilt) of the Earth’s axis, with a period of 41 kiloyears

• Precession (swiveling) of the Earth’s axis, with a period of 23 kiloyears

In the stochastic resonance hypothesis, the Milankovitch signal is amplified by random events to produce climate state changes. In more recent Milankovitch theories, a deterministic forcing mechanism is used. In a theory by Didier Paillard, the climate is modeled with three states, called interglacial, mild glacial and full glacial, and the state changes depend on the volume of ice as well as the insolation.

See:

Milankovitch cycle, Azimuth Library.

Mathematics of the environment (part 10), John Baez. This gives an exposition of Paillard’s theory.

Bistable systems defined by a potential function

Any smooth function with two local minima can be used to define a bistable system. For instance, consider the function V(x)=x 4/4x 2/2V(x) = x^4/4 - x^2/2:

To define the bistable system, construct a differential equation where the time derivative of x is set to the negative of the derivative of the potential at x:

dx/dt=V(x)=x 3+x=x(1x 2)dx/dt = -V'(x) = -x^3 + x = x(1 - x^2)

So, for instance, where the potential graph is sloping upward as x increases, -V’(x) is negative, and this sends X(t) “downhill” towards the minimum.

The roots of V’(x) yield stable equilibria at 1 and -1, and an unstable equilibrium at 0. The latter separates the basins of attraction for the stable equilibria.

Discrete stochastic resonance

Now let’s look at a discrete-time model which exhibits stochastic resonance. This is the model used in the Azimuth demo program.

We construct the discrete-time derivative, using the potential function, a sampled sine wave, and a normally distributed random number:

ΔX t=V(X t)*Δt+Wave(t)+Noise(t)=X t(1X t 2)Δt+α*sin(ωt)+β*GaussianSample(t)\Delta X_t = -V'(X_t) * \Delta t + \mathrm{Wave}(t) + \mathrm{Noise}(t) = X_t (1 - X_t^2) \Delta t + \alpha * \sin(\omega t) + \beta * \mathrm{GaussianSample}(t)

where Δt\Delta t is a constant and tt is restricted to multiples of Δt\Delta t.

This equation is the discrete-time counterpart to a continuous-time stochastic differential equation.

Next time, we will look into the Azimuth demo program itself.

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