Autoregressive estimation of the splitting matrix of free-oscillation multiplets

Recent, large earthquakes, recorded by the rapidly growing global seismic n etworks, have produced a vast amount of high-quality data. These new data a llow us to develop new techniques to determine the coupling and splitting c haracteristics of multiplets, and hence determine the 3-D structure of the Earth. Current techniques tend to be computationally intensive and non-line ar and require detailed models of earthquake sources (which might be unavai lable for the large and often complicated events used in free-oscillation r esearch). Here, we introduce a new technique that allows us to solve for th e most general form of the splitting matrix in a few steps without knowledg e of the earthquake sources. The method takes advantage of the fact that certain linear combinations of seismograms for a single earthquake have an autoregressive property for whi ch the propagator matrix is related to the exponentiated splitting matrix. To retrieve the propagator matrix, it is necessary to use only displacement scalars from a reference earth model and instrument calibrations to perfor m a two-step linear inversion. Multiple events can (and, generally, must) b e used to allow retrieval of the propagator matrix. It is straightforward t o recover the splitting matrix from the propagator matrix and it is the ele ments of the splitting matrix that provide linear constraints on the 3-D st ructure of the earth. We illustrate the technique by using nearly 900 vertical-component recordin gs from 10 large earthquakes to recover the splitting matrices of a variety of multiplets. The results are presented as splitting functions, including some of the first robust anelastic splitting functions to be determined.