Umbilic points of slowness surface in elastic orthorhombic media

For 3D seismic data processing, azimuth plays an important role, especially in complex subsurface regions. In such regions, elliptical or anelliptical orthorhombic models are commonly used to describe wave propagation. In these models, the behaviour of the slowness surface needs more detailed analysis. Umbilic points defined by the equal principal curvatures exist in a complex 3D model. In the vicinity of the umbilic points, the traveltime surface has the shifted hyperbola form that needs to be considered in processing operations like velocity analysis. Fractured media characterized by the orthorhombic model are more likely to have umbilic points, and it is important to address their positions. If exists, umbilic points can provide additional constraints in inverting for model parameters. Through a defined condition, we examine the position of the umbilic point and derive their explicit formulas. We analyse umbilic points for elliptical and anelliptical orthorhombic models in the numerical example. For the elliptical orthorhombic model, the formulas for the umbilic point position on different symmetry planes are derived and corresponding conditions are also identified. Further, we numerically examine umbilic point positions for anelliptical orthorhombic models and observe that the umbilic points are located out of two vertical symmetry planes. Moreover, caused by interference from two neighbouring umbilic points, a more significant deviation in traveltime is found in the anelliptical orthorhombic model.