We propose to tackle the critical issue of designing boundary conditions for TTI elastic wave equations by formulating a stable low-order Absorbing Boundary Condition (ABC). The construction principle is based on ellipticity which simplifies the slowness surfaces into ellipsoids of revolution. Then, a change of variables is used to deduce the TTI ABC from the isotropic one. Moreover, we derived a source formula which generates P-waves only on an elliptic TTI medium. Numerical experiments are performed to illustrate the performance of the ABC in both time-based and time-harmonic 3D cases and to show the long time stability of the condition.


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