Non-hyperbolic common reflection surface

3-D extension

In the case of 3-D multi-azimuth acquisition, both $d$ and $h$ become two-dimensional vectors. A natural way to extend approximation (8) is to replace it with

$$\widehat{\theta}_{CRS}(\mathbf{d},\mathbf{h};t_0) = \sqrt{F(\mathbf{d}) + \mathbf{h}^T \mathbf{B}_2 \mathbf{h}}\;,$$ (14)

where $F(\mathbf{d}) = (t_0 + \mathbf{d}^T \mathbf{a}_1)^2 + \mathbf{d}^T \mathbf{A}_2 \mathbf{d}$ , $\mathbf{a}_1$ is a two-dimensional vector, and $\mathbf{A}_2$ and $\mathbf{B}_2$ are two-by-two symmetric matrices (Tygel and Santos, 2007). A similar approach works for extending approximation (12) to

$$\widehat{\theta}(\mathbf{d},\mathbf{h};t_0) = \sqrt{\frac{F(\math... ...,\mathbf{h} + \sqrt{F(\mathbf{d}-\mathbf{h}) F(\mathbf{d}+\mathbf{h})}}{2}}\;,$$ (15)

where $\mathbf{C}=2 \mathbf{B}_2+\mathbf{a}_1 \mathbf{a}_1^T-\mathbf{A}_2$ . In the 3-D case, we have not found a simple connection between approximation (15) and the analytical reflection traveltime for a 3-D hyperbolic reflector.