Of weather prediction without models

After decades of model-driven weather forecasting which perpetuated the legacy of Abbe, Bjerknes [1] and Richardson [2], we hereby propose a novel scheme of weather prediction, freed of modelization. We set ourselves free of the burden of hydro-(and thermo-) dynamics which Abbe deemed necessary. We retain what Bjerknes called the diagnostic step of prediction, preferring not to engage into Bjerknes' prognostics. We reinforce the very state-of-the-art which Richardson found regretful: "the past history of the atmosphere is used, so to speak, as a full-scale working model of its present self".

Of course, this approach requires different tools than that which were available in Richardson's times. In fact, Farmer et al. [3] much later proposed the existence of such methods when describing the prediction of chaotic time series solely based on the extraction of information from the state space embedding of past values. It is further recognised that a universal persistence law [4], if not fractal dimension characteristics [e.g., 5], governs long-term atmospheric variability and thus climate change. It appears that sensitive dependence on initial conditions, far from being a handicap to prediction, constitutes the very basis on which our scheme functions. The statistical self-similarity which characterises such signals plays a definite role in reverting a volatility handicap into a major predictive strength. Works by Robert [6, 6bis] hint to the invalidity of this famous handicap, coined as the butterfly effect. Not only can the latter not apply to atmospheric sciences, but the very geometrical solutions of Navier-Stokes equations engender chaos which may not be equally present in the atmospheric system. Models may be chaotic but may not necessarily imbue the system with chaos.

The failure of Richardson's own attempt stemmed from requirements on the input data which should match the smoothness expected by the models. Richardson speculated that had the initial data been smoothed, his attempt would have succeeded. Though an important step in the process, achieving balance in the initial data cannot preclude important losses of prediction information. We therefore propose the existence of an approach where only observations on past raw data values are used to establish a forecast of present and future values. Pure mathematical reasoning [7] already supports the feasibility of such a structural approach. The reasons and heuristics of this reasoning are explained by way of a prisoners' game and analogy. Furthermore, some intervals into the future are found to exist, intervals which correspond to predictability bubbles.

Von Neumann recognized [8] the weather forecasting problem as a problem par excellence for an automatic computer. We believe that today Von Neumann would take into account the worthy challenge posed by our computer-theoretical interrogations. This approach may help tackle the challenge of rain prediction without any pretensions as to its underlying mechanism [9]. With such a scheme, we may also explore the limits [10] of seasonal prediction for temperate regions. A mathematical seer conceivably foretells the weather.

[1] Lynch P., The Emergence of Numerical Weather Prediction, Cambridge University Press, 2006

[2] Richardson LF., Weather Prediction By Numerical Process, Cambridge University Press, 1922

[3] Farmer JD. et al., Predicting Chaotic Time Series, Physical Review Letters, Vol. 59 (1987), pp. 845--848

[4] Bunde E. et al., Indication of a Universal Persistence Law Governing Atmospheric Variability, Physical Review Letters, Vol. 81 (1998), pp. 729--732

[5] Bodri L., Fractal Analysis of Climatic Data, Theoretical and Applied Climatology, Vol. 49 (1994), pp. 53--57

[6] Robert R., L'effet papillion n'existe plus, Gazette des Mathématiciens, Société Mathématique de France, No. 90 (Oct. 2001), pp. 11--25

[6bis] Robert R. et al., Long range predictability of atmospheric flows, Non Linear Processes in Geophysics, No. 8 (2001), pp. 55--67

[7] Hardin CS. et al., A Peculiar Connection Between the Axiom of Choice and Predicting the Future, American Mathematical Monthly, Feb. 2008, pp. 91--96

[8] Goldstine HH., The Computer from Pascal to Von Neumann. Reprinted with new Preface, Princeton University Press, 1993

[9] Falkovich G., Acceleration of Rain Initiation by Cloud Turbulence, Nature, Vol. 419 (2002), pp. 151--154

[10] Kirtman B. et al., WCRP Position Paper on Seasonal Prediction, Report from the First WCRP Seasonal Prediction Workshop, 4-7 June 2007, Barcelona, Spain