Geomagnetic Data Assimilation: A Method for Determining Error Covariances

We present initial testing of a technique for assimilating geomagnetic data into numerical geodynamo modeling. The technique is analyzed using a one-dimensional system consisting of coupled non-linear equations for momentum, induction and energy variations in a periodic domain. We construct a synthetic assimilation test by making magnetic field observations on a "nature" (or true solution) which differs from the model by some random errors with known statistics. We wish to understand how the magnetic observations can be used to constrain all of the system variables. The coupling in the model between the different variables can be used to obtain correlations between their errors. The ensemble method involves carrying out a number of model runs, each with a different perturbed initial state. The resulting ensemble is then used to determine error correlations between each of the state variables. In this way, the difference between the numerical and observed values of surface magnetic flux indicates corrections to velocity and temperature within the core. This technique leads to a "balanced" forcing on the numerical solution which should lead to a more consistent solution. Results of assimilation in this system can then be used to understand how to carry out geomagnetic assimilation in our MoSST core dynamics model.