Using 3D inversion schemes to solve 4D inverse problems

Seismic data observed at a given point in time are sensitive to the elastic parameters at this time. A Bayesian 3D inversion computes the posterior of the elastic parameters at that moment in time. In 4D inversion data at multiple time steps are collected, in order to investigate changes in elastic parameters. One option is to investigate each time step individually, and consider the differences, this approach ignore the temporal correlation of data. We use a Kalman filter approach to model the time correlations; in this approach we must separate the model for static and dynamic parameters. The current inversion methodology does not account for this separation. In the current note we show how we merge and split static and dynamic parameters, when conditioning to the current data set. This enables us to incorporate the standard inversion methodology for solving the 4D problem.