Sivaramakrishnan Lakshmivarahan John M. Lewis - Forecast Error Correction using Dynamic Data Assimilation

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As neophytes in science, we wondered about predictability. On the one hand, we knew that an eclipse of the sun could be predicted with great accuracy years in advance; yet, some of us wondered why the 23 day forecasts of rain or snow were so suspect. Even before we took university courses on the subject, we could speculate, and often incorrectly, as to the reasons that underlie marked difference in predictability of these events.

In this regard, it is instructive to examine the thoughts and ideas of one of the twentieth centurys most celebrated contributors to the theory of dynamical-system predictabilityEdward Lorenz. He was drawn to mathematics as a young man and was headed for a Ph.D. in mathematics at Harvard after completing his M.A. degree under the direction of George Birkoff in 1940 (Fig.).

WW II interrupted his career plan. With the likelihood of being drafted into military service, he opted for training as a weather forecaster in the U.S. Army Air Corps. Upon completion of the 9-month training in the so-called Cadet program at MIT, he served as a weather officer in the Pacific Theatre. At wars end and with honorable discharge from the Army, he contemplated his futurecontinuation for the doctorate in mathematics at Harvard or a change of direction toward meteorology. After a long conversation with Professor Henry Houghton, chair of the meteorology department at MIT, he decided to enroll as a graduate student in meteorology at MIT. His view of weather prediction in 1947 follows:

And Lorenz was referring to the subjective method of forecasting where experience and synoptic typingcategorization of flow structureswere the basis of forecasting. But shortly after he entered graduate school at MIT, the first numerical weather prediction (NWP) experiments took place at Princetons Institute of Advanced study under the direction of Jule Charney. And indeed, by 1950, Charney and his team succeeded in making two successful 24-h forecasts of the transient features of the large-scale flow using a simplified dynamical model. The success set the meteorological world abuzz where speculation ran the gamut from unbridled optimism to skepticism. By the mid-1950s, operational prediction of these large-scale waves took place in both Sweden and the USA. But identification and correction of systematic errors became the major theme of operational forecasting during the late 1950s into the early 1960s. Among the most obvious problems was the unrepresentativeness of observations used to initialize the modelsfor example, the presence of only single upper-air observations over the oceanic regions bounding the USAfrom ship Papa (approximately 400km south of Adak, Alaska) and ship Charlie (between Greenland and Iceland). Nevertheless, there was hopefulness that a more representative set of observations would go far to improve the forecasts and extend the useful range of prediction. But by the early 1960s, this hopefulness was dealt a severe blow.

As reviewed in Lorenzs scientific biography The Essence of Chaos (Lorenz ).

The chaotic systems have come to be called unstable systemssystems unforgiving of inaccuracy in initial conditions. Those systems that are forgiving of modest uncertainty in initial conditions are referred to as stable systems. It is the basis for understanding the very accurate prediction of solar eclipses, for example, and the imprecise prediction of rainfall. As we look at the contributions to the predictability question in meteorology, there is a preponderance of papers dealing with forecast uncertainty in response to uncertainty in the control vector (initial conditions, boundary conditions, and parameterizations). Little attention has been paid to errors in prediction that result from physical deficiencies in the modelsinexact parameterizations for example. We consider this question in Chap.

Determination of predictability limits from realistic atmospheric model simulations came in the mid- to late-1960s. The plans for extended-range prediction (the order of weeks) came with the goals of the Global Atmospheric Research Program (GARP), and this dictated predictability tests with the existing general circulation (GC) models (models at Geophysical Fluid Dynamics Laboratory (GFDL), Lawrence Livermore Laboratory (LLL), and University of California Los Angeles (UCLA)). The growth rate of error was found by creating an analogue pairthe control or base state and the forecast that stemmed from a perturbation to control (to the base state initial condition). Divergence of this analogue pair produced the growth rate. The doubling time for error was 5 days when averaged over the three models (Charney ). These models produced results that were better than climatology out to about 2 weeks. Akio Arakawa, one of the modelers involved in the study, remembers the situation as follows:

One of the most impressive and influential studies of predictability limit for weather prediction models came with Lorenzs study in the early 1980s that made use of operational NWP products at the European Centre for Medium-range Weather Forecasts (ECMWF) (Lorenz ).

We have already mentioned the unrepresentativeness of observations as a source of error in specifying the initial conditions of a dynamical model that in turn contribute to the models forecast error. Using the 2.5-day doubling time for error as found by Lorenz () that compares the predictability limits that came with Lorenzs monumental study at ECMWF in the early 1980s and the latest operational predictions. Most obvious is the extension of the predictability limit to 2 weeks in the southern hemisphere in response to the availability of more and better satellite observations over the ocean dominated hemisphere.