Theory of interval traveltime parameter estimation in layered anisotropic media

Appendix: Traveltime and offset as functions of ray parameters

In this appendix, we show the derivation of the offset (equation 9) and traveltime (equation 10) functions in terms of two ray parameters ($p_1$ and $p_2$ ) in 3D. The total offset is constituted of offset increments from each individual layer and can be expressed as

$$h_i = \sum\limits^N_{n=1} D_{(n)} \frac{dh_i}{dz}~,$$ (76)

where the derivative $\frac{dh_i}{dz}$ represent the change in the offset $h_i$ direction with respect to the vertical direction $z$ , and $D_{(n)}$ denote the thickness of the $n$ -th layer. According to the ray theory (Cervený, 2001), this derivative can be related to the derivative of the Christoffel equation with respect to the ray slownesses as follows

$$\frac{dh_i}{dz} = \frac{\partial F_{(n)} /\partial p_i}{\partial F_{(n)} /\partial Q_{(n)}}~,$$ (77)

where $F_{(n)}(p_1, p_2, Q_{(n)}) = 0$ and $Q_{(n)}$ denote the Christoffel equation and the vertical ray slowness of the interested wave mode in the $n$ -th layer. Equation A-2 can be simplified further due to implicit differentiation in the Christofel equation as

$$\frac{dh_i}{dz} = \frac{\partial F_{(n)} /\partial p_i}{\partial F_{(n)} /\partial Q_{(n)} } = -\frac{\partial Q_{(n)}}{\partial p_i}~,$$ (78)

which after substitution in equation A-1 results in the function of $h(p_1,p_2)$ given in equation 9. Analogously, we can follow the same line of reasoning and derive an expression of traveltime. We start from the total traveltime expression given by

$$t = \sum\limits^N_{n=1} D_{(n)} \left( \frac{\partial t}{\partial... ...rtial t}{\partial h_2}\frac{dh_2}{dz} + \frac{\partial t}{\partial z} \right)~,$$ (79)

where the derivatives $\frac{\partial t}{\partial h_1} = p_1$ and $\frac{\partial t}{\partial h_2} = p_2$ represent the ray parameters in the two directions of the local coordinates. Since the ray parameters are conserved in the sequence of horizontal layers due to Snell's law, we can further transform equation A-4 into equation 10 as follows:

$$t(p_1,p_2) = \sum\limits^N_{n=1} D_{(n)} \left( p_1 \frac{dh_1}{dz} + p_2\frac{dh_2}{dz} + \frac{dt}{dz} \right)$$

$$= p_1 \sum\limits^N_{n=1} D_{(n)}\frac{dh_1}{dz} + p_2 \sum\limits^N_{n=1} D_{(n)}\frac{dh_2}{dz} + \sum\limits^N_{n=1} D_{(n)}\frac{dt}{dz}$$

$$= p_1 h_1 + p_2 h_2 + \sum\limits^N_{n=1} D_{(n)}Q_{(n)}~,$$ (80)

where $\frac{dt}{dz} = Q_{(n)}$ denotes the vertical slowness in the $n$ -th layer.

Theory of interval traveltime parameter estimation in layered anisotropic media