Pattern dynamics, pattern hierarchies, and forecasting in complex multi-scale earth systems
Catastrophic disasters afflicting human society are often triggered by tsunamis, earthquakes, widespread flooding, and weather and climate events. As human populations increasingly move into geographic areas affected by these earth system hazards, forecasting the onset of these large and damaging events has become increasingly urgent. In this paper we consider the fundamental problem of forecasting in complex multi-scale earth systems when the basic dynamical variables are either unobservable or incompletely observed. In such cases, the forecaster must rely on incompletely determined, but "tunable" models to interpret observable space-time patterns of events. The sequence of observable patterns constitute an apparent pattern dynamics, which is related to the underlying but hidden dynamics by a complex dimensional reduction process. As an example, we examine the problem of earthquakes, which must utilize current and past observations of observables such as seismicity and surface strain to produce forecasts of future activity. We show that numerical simulations of earthquake fault systems are needed in order to relate the fundamentally unobservable nonlinear dynamics to the readily observable pattern dynamics. We also show that the space-time patterns produced by the simulations lead to a scale-invariant hierarchy of patterns, similar to other nonlinear systems. We point out that a similar program of simulations has been very successful in weather forecasting, in which current and past observations of weather patterns are routinely extrapolated forward in time via numerical simulations in order to forecast future weather patterns.