## Improving Weather Forecasting: Shadowing in the Atmosphere

M.S. Thesis, 2009, Mathematics, University of Vermont**Abstract:**Modeling Earth's atmospheric conditions is difficult due to the size of the system, and predictions of its future state suffer from the consequences of chaos. As a result, current weather forecast models quickly diverge from observations as uncertainty in the initial state is amplified by nonlinearity. One measure of the strength of a forecast is its shadowing time, the period for which the forecast is a reasonable description of reality. The present work uses the Lorenz '96 coupled system, a simplified nonlinear model of atmospheric conditions, to extend a recently developed technique for lengthening the shadowing time of a dynamical system. An ensemble of initial states, systematically perturbed using knowledge of the local dynamics, is used to make a forecast. The experiment is then repeated using inflation, whereby the ensemble is regularly expanded along dimensions whose uncertainty is contracting. The first goal of this work is to compare the two forecasts to reality, chosen to be an imperfect version of the same model, and determine whether variance inflation succeeds. The second goal is to establish whether inflation can increase the maximum shadowing time for a single member of the ensemble. In the second experiment the trajectory of reality is known a priori, and only the closest ensemble members are considered at each time step. When inflation is introduced to this technique, it is called stalking. Variance inflation was shown to have the potential to be successful, with the extent dependent upon algorithm parameters (e.g. size of state space, inflation amount). Under idealized conditions, the technique was shown to improve forecasts over 50\% of the time. Under these same conditions, stalking also exhibited the potential to be useful. When only the best ensemble members were considered at each time step, the known trajectory could be shadowed for an entire 50-day forecast 50-75\% of the time. However, if inflation occurs in directions incommensurate with the true trajectory, inflation can actually reduce stalking times. Thus, utilized appropriately, inflation has the potential to improve predictions of the future state of atmospheric conditions, and possibly other physical systems.

**Talks:**

- Shadowing in the Atmosphere: Student Research Conferences, University of Vermont. April 16, 2009

- Aggressive Shadowing of a Low-Dimensional Model of Atmospheric Dynamics: Thesis Presentation, University of Vermont. May 18, 2010

**Publications:**

- Lieb-Lappen, R. M.; Danforth, C. M.

**2012**. Aggressive Shadowing of a Low-Dimensional Model of Atmospheric Dynamics.

*Physica D*, 241: 637-648.

- M.S. Thesis: Aggressive Shadowing in the Atmosphere.

**Advisor:**Chris Danforth