PIOMAS is the Pan-Arctic Ice Ocean Modeling and Assimilation System (version 2.1) by the Polar Science Centre at the University of Washington. That’s a data assimilation model tracking the total volume of ice floating on the Arctic Ocean and surrounding seas and tidal waterways. That is ice *volume* as opposed to ice *area* (actual frozen surface) or ice *extent* (sea surface with at least some ice). A model is needed because, while area and extent can be readily measured by satellite remote sensing, until recently ice thickness (and hence volume) could not be. PIOMAS uses a numerical model representing the physics of ice generation, movement and melting, constrained by observations such as ice extent, sea surface temperature in ice free areas, and polar weather effects such as wind (affecting ice structure) and cloud (affecting incident solar radiation). It’s also calibrated against historical observations like those from nuclear submarines.

The graph shows two traces:

- The light blue-filled total ice volume curve, plotted on the left axis. That varies through a huge range with the annual freeze-thaw cycle, currently between about 5 000 and 25 000 cubic kilometres (yes, cubic kilometres – or, if you like, billion tonnes).
- The dark blue line, plotted on the right axis, which shows the same results expressed as anomalies (differences) from the annual average curve. For the average (mean) curve, I use the 30 year interval 1981 — 2010, a slightly different interval to that used for the Polar Science Centre graph.

Note that the scale is the same on both vertical axes, and the right (anomaly) axis is positioned so that the horizontal axis crosses it at about -12, which is the anomaly at which there will be zero ice at the date of the annual average minimum (12 September). So if the dark blue trace reaches the bottom axis around mid September some year soon, that’s it, there is no ice left (according to PIOMAS).

The red dots show the latest volume estimates from the CryoSat-2 satellite direct measurement of ice thickness and volume (as obtained by ‘Wipneus’ from the Cryosat-2 data team¹). The agreement is surprisingly good, except right in the middle of winter when Cryosat-2 volume estimates are significantly higher than PIOMAS. (CryoSat-2 estimates thickness by using a high resolution radar altimeter to accurately measure the height of the top of the floating ice above sea level, a somewhat challenging task from 700 km up. Volume is then estimated as ice thickness x extent x concentration.)

The graph also shows two fitted and extrapolated curves, which is a matter of some contention…

### Choosing the fitting function

There’s been criticism of fitting parabolas (*second order polynomials* or *quadratics*) to the PIOMAS sea ice volumes: “over-fitting”, “no basis”; that sort of thing. To me a second order fit is the logical and defensible choice.

What is the most commonly fitted function? That would be the straight line, also called a *first order polynomial*. Why? Often it’s because there’s an underlying understanding — a justifiable expectation of a linear process, or of a linear trend for a time series. In other cases the choice seems more prosaic. It’s *what you do* (rather, what everyone else does), or, more generously, it’s seen to be the parsimonious choice. There can be no justification for excess complexity (William of Ockham), and a straight line is the simplest choice.

That is plainly correct. We have:

*y = mx + c*

or, as a function of time:

*ƒ(t) = mt + ƒ(0)*

The extreme simplicity is obvious in the differential form, where the equation is just:

*ƒ′(t) = m*

The rate of change with time is a constant. There is no simpler time-varying function.

The problem with that for the PIOMAS output is twofold:

- Except over some narrow time window, there is no reasonable expectation of a linear trend. The basic expectation is of an accelerating process via the likes of albedo feedback and decreasing ice structural strength.
- It doesn’t fit.

So parsimony is good, but if the function is inadequately complex to fit the expectation and the data, then it is unsuitable. We need adequate, preferably justifiable, complexity.

In the polynomials, the next most complex option is the second order:

*y = ax² + bx + c*

Or, as a time function:

*ƒ(t) = at² + ƒ′(0).t + ƒ(0)*

That is also a very simple function, again most obviously in its differential form. It’s just the function for which the acceleration in the rate of change is a constant:

*ƒ′′(t) = 2a*

That is the simplest of all functions with a time-varying rate of change, which is the same as saying that it is the parsimonious choice in a case where there is a sound basis to expect acceleration. My view is that PIOMAS is such a case.

Others have been fitting more complex things to PIOMAS, including exponential functions and various S-curves. While there is a suggestion of S-shaped behavior in some global climate model simulations, in my view the evidence is insufficient at present for any particular S-function choice to be considered parsimonious.

### Extrapolation

The next common objection to plots like this lies with the extrapolation to zero ice volume. It is argued that is done without justification or validity. Extrapolation of a fitted function amounts to statistical prediction, a process that is often misapplied. A key point is that correlation by itself is never reliably predictive. The risk of death by car crash has been steadily decreasing at the same time as mobile phone use has been exploding, so does that mean that if I text while driving, I’ll be safer? There are countless absurd examples.

What’s needed for statistical prediction to be reasonable is *causation*: some quite fundamental understanding leads us to expect that *a* is caused by *b*; we observe that *a* and *b* are correlated as expected; then given *b* we can predict *a*, and estimate the statistical uncertainty. (The nature of the causal relationship will often be incompletely known, so there’ll be additional *systematic* uncertainly.)

So what of PIOMAS? Ice loss is not caused by the elapse of time, but we know that global warming processes have been inexorably increasing over recent decades, that those processes are continuing, and that they should cause sea ice loss — accelerating sea ice loss. I think the case for “secondary” causation is unusually clear here, and therefore statistical prediction is strongly justified. It follows that *so is extrapolation* (although obviously not without limit).

### The future

How soon will the ice be gone? On the model results the summer ice will all be gone sometime around ~~2015 to 2022~~ 2025 (PIOMAS v2.1 update + slight post-2012 recovery). At the resolution of the PIOMAS model, “gone” might not mean every last bit, but it surely means gone for practical purposes. More interestingly perhaps — and on a much longer extrapolation — the results suggest that we should expect zero northern sea ice for about half the year sometime in the 2030s.

### Reference:

1. Laxon S. W., K. A. Giles, A. L. Ridout, D. J. Wingham, R. Willatt, R. Cullen, R. Kwok, A. Schweiger, J. Zhang, C. Haas, S. Hendricks, R. Krishfield, N. Kurtz, S. Farrell and M. Davidson (2013), CryoSat-2 estimates of Arctic sea ice thickness and volume, Geophys. Res. Lett., 40, 732–737, doi:10.1002/grl.50193

[…] information on this chart is available at Gerg’s Net. Ice loss so far in 2014 is now tracking very close to 2013 levels. It is quite possible that the […]