Knowledge about crystal anisotropy is mainly provided by crystal orientation fabric (COF) data from ice cores. To gain a broader understanding about the distribution of crystal anisotropy in ice sheets and glaciers we use seismic measurements. Two effects are important: (i) sudden changes in crystal orientation fabric (COF) lead to englacial reflections and (ii) the anisotropic fabric induces an angle dependency on the seismic velocities and thus also recorded traveltimes. We present a framework to connect COF data with the elasticity tensor and thus determine seismic velocities and reflection coefficients for cone and girdle fabrics from ice-core data. These results are compared to VSP measurements form Antarctica to validate the overall approach. Normal spread reflection seismic data are used to show the large influence of crystal anisotropy on the normal moveout (NMO) velocity of P-waves. These anisotropic NMO-velocities can be determined from the elasticity tensor with help of the Thomsen parameters for P-, as well as SH-waves. The discrepancy introduced by an isotropic assumption using stacking velocities as depth conversion velocities is in the range of 7-8 % for P-waves, but only about 1% for SH-waves. Hence, using velocities for the depth conversion, which were derived during the stacking process is no longer valid for compressional waves. However, with knowledge about the depth of reflections from other independent data sets, such as radar data or borehole depth, it is possible to determine the Thomsen parameters. We demonstrate that the analysis of normal spread reflection seismic data in combination with radar data gives a tool of determining the anisotropic ice fabric of glaciers and ice sheets. This is an important contribution to constrain results from the upcoming generation of anisotropic ice-flow models by remotely sensed data.