The mean extent of Arctic sea ice for September 2011, calculated from the University of Bremen time series (website; technical paper), was 4.6 million km2. A Gompertz curve that I estimated last April based on Uni Bremen data for 1972 through 2010 gave a predicted September 2011 mean of ... 4.6 million km2. Encouraged by this lucky guess (and that’s what it was!), I here offer an even earlier and more naive prediction for the UB September 2012 mean ... 4.3 million km2, with a confidence interval of ±.8 (Figure 1).
Like linear and quadratic models, Gompertz curves are an un-physical, simple way to describe past changes in Arctic ice measures, and to suggest near-term predictions about the future. Extending them more than a few years ahead, as done in Figure 1, is purely speculative. If the curve in Figure 1 accurately describes the path of Arctic sea ice, we would see virtually ice-free September conditions, defined as extent below 1 million km2 (the dark red line) before 2030.
Linear and quadratic trend models are more widely used, but the Gompertz more realistically fits past data. It is the only one of these three that can match both the slow decline seen in early years of the satellite era (especially visible in the longer UB dataset, which goes back to 1972), and the more recent accelerating decline. Linear models decline at a constant rate, which the ice clearly has not done. Quadratic (inverted-U) curves impose a slight but counterfactual increase in the early years.
Both linear and quadratic models also predict extent below zero, in time. An appealing feature of Gompertz models is that they never drop below zero, instead approaching this physical limit asymptotically (although rapidly). While the curve approaches zero smoothly, its implication for real ice might be different. To illustrate an alternative, Figure 2 shows four simulated trajectories based on the model of Figure 1.
For each simulation in Figure 2, I added white noise to represent year-to-year variations in weather. Deviations around the Gompertz curve in Figure 1 pass statistical tests for white noise (Ljung–Box Q), so this seems a reasonable starting point. The added noise has a standard deviation set equal to that of the 2007–2011 period; in other words, it mimics the unpredicted behavior of year-to-year variations in the recent past. Noise-driven predictions below 0 were truncated. Figure 2 suggests the possibility that even if the Gompertz model proves prescient, and virtually ice-free conditions arrive before 2030, we still could often see “recovery” years with more ice.
These Gompertz versions of the “death spiral” are not grounded in physical argument, but provide a visualization of how recent trends might continue. As with the September 2011 results, their predictions are eminently testable. They could provide a challenging benchmark for evaluating the skill of physical (or more substantive statistical) models, as those are improved.