3-D extension

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3-D extension

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

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

(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

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

(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.

2013-07-26