The non-commutivity of shear wave splitting operators at low frequencies and implications for anisotropy tomography

S U M M A R Y Measurements of the splitting or birefringence of seismic shear waves constitute a powerful and popular technique for characterizing azimuthal anisotropy in the upper mantle. The increasing availability of data sets from dense broad-band seismic arrays has driven interest in the development of techniques for the tomographic inversion of shear wave splitting data and in comparing splitting measurements with anisotropic upper-mantle models obtained from other constraints, such as surface wave analysis. Two different theoretical approaches have been developed for predicting apparent shear wave splitting parameters (fast direction and delay time) for models that include multiple layers of anisotropy at depth, which is useful for comparing azimuthally anisotropic surface wave models with shear wave splitting measurements. These approaches differ in one key aspect, which is whether or not the shear wave splitting operator can be treated as commutative. In this paper, we investigate the theoretical source of this discrepancy, and show that at frequencies relevant to most studies of upper-mantle anisotropy, the term that results in the non-commutivity of the shear wave splitting operator in the expressions for multiple-layer splitting must be retained. In contrast, the quantity known as the splitting intensity, which is closely related to the apparent fast direction and delay time, does commute at these frequencies. We illustrate these inferences with forward modelling examples and discuss their implications for the tomographic inversion of shear wave splitting measurements, the comparison of surface wave models with shear wave splitting observations and the joint inversion of surface wave and shear wave splitting observations for upper-mantle anisotropic models.