NCEO - National Centre for Earth Observation

4D-var double pendulum

Author: Ross Bannister

Software requirements: Fortran 77

Numerical weather prediction models are complicated, cumbersome, and require extensive resources to work. To demonstrate the technique of 4d-Var, we resort to a very simple physical system that has itself little to do with atmospheric science - the double pendulum. Unlike a weather prediction model, which could contain over a million degrees of freedom, the double pendulum is described by just four variables. These the are two angles (see Fig.) and their time rates of change of these angles.

The equations of motion are comprised of coupled ordinary differential equations, rather than of partial differential equations as an atmospheric model is. We cannot therefore speak of a number of spatial dimensions in the same way that we can with an atmospheric model. The term "4d-Var" is somewhat a misnomer, but the distinguishing thing about 4d-Var is that there is a time dimension - which we have. We should more aptly call the weather forecasting "4d-Var" technique "3+1d-Var", and perhaps for the double pendulum system, "1+1d-Var".

Intermittent data assimilation (as is done by this model) is performed in cycles. Each cycle spans a time interval, within which all observations are analysed, and merged with the model. The objective of this procedure is to estimate of the optimal initial conditions (relevant to the start of the cycle) which will reproduce, via the model, the observations. The idea is that the integration can then be advanced further to forecast future states of the system.

The code and documentation can be found at