Theory of interval traveltime parameter estimation in layered anisotropic media

Traveltime expansion

Assuming the Einstein repeated-indices summation convention, we can expand the one-way traveltime $t$ into a Taylor series of half offset $h_i$ ($i=1$ or 2 in 3D) around zero offset as follows:

$$t(h_i) = t_0 + t_i h_i + \frac{1}{2}t_{ij} h_i h_j + \frac{1}{6} t_{ijk} h_ih_jh_k+ \frac{1}{24}t_{ijkl}h_ih_jh_kh_l + ...~,$$ (1)

where $t_0$ is one-way vertical traveltime, $t_{ij}=\frac{\partial^2 t}{\partial h_i \partial h_j}$ and $t_{ijkl}=\frac{\partial^4 t}{\partial h_i \partial h_j \partial h_k \partial h_l}$ are second- and fourth-order derivative tensors, respectively. Both tensors are symmetric thanks to the symmetry of mixed derivatives. Analogously, we can also derive, for the negative half offset $-h_i$ ,

$$t(-h_i) = t_0 - t_i h_i + \frac{1}{2}t_{ij} h_i h_j - \frac{1}{6} t_{ijk} h_ih_jh_k+ \frac{1}{24}t_{ijkl}h_ih_jh_kh_l + ...~.$$ (2)

Assuming pure-mode reflections with source-receiver reciprocity, we can sum the two expansions (equations 1 and 2) for the two legs of rays to derive the expansion of the two-way traveltime as follows (Al-Dajani and Tsvankin, 1998):

$$2t(h_i) = 2t_0 + t_{ij} h_i h_j + \frac{1}{12}t_{ijkl}h_ih_jh_kh_l + ...~.$$ (3)

Equation 3 can be additionally transformed into the series of the squared two-way traveltime in terms of the full offset $x_i = 2 h_i$ as follows:

$$4t^2(x_i) \approx 4t_0^2 + a_{ij} x_i x_j + a_{ijkl} x_i x_j x_k x_l + ...~,$$ (4)

where

$$a_{ij} = t_0 t_{ij}~,$$ (5)

$$a_{ijkl} = \frac{1}{16} \left( t_{ij} t_{kl} + \frac{t_0}{3}t_{ijkl} \right)~.$$ (6)

In consideration of the symmetry of the time derivative tensors, the quadratic and quartic terms in equation 4 reduce to the following known expressions (Al-Dajani et al., 1998):

$$a_{ij}x_ix_j = t_0\left( t_{11}x^2_1 + 2t_{12}x_1 x_2 + t_{22} x^2_2 \right)~,$$ (7)

$$a_{ijkl}x_ix_jx_kx_l = \left( \frac{t^2_{11}}{16} + \frac{t_0t_{1111}}{48}\right)x^4_1 + \left( \frac{t_{11}t_{12}}{4} + \frac{t_0t_{1112}}{12}\right)x^3_1 x_2 ~$$ (8)

$$+ \left( \frac{t_{11}t_{22}}{8} + \frac{t^2_{12}}{4} + \frac{t_0t_{1122}}{8}\right)x^2_1 x^2_2$$

$$+ \left( \frac{t_{22}t_{12}}{4} + \frac{t_0t_{1222}}{12}\right) x_1 x^3_2 + \left( \frac{t^2_{22}}{16} + \frac{t_0t_{2222}}{48}\right)x^4_2~.$$

In the derivation of the general formulas for moveout coefficients in the next section, we keep the tensor notation, which simplifies the use of tensor operations. We also use the fact that, in the case of horizontally stacked layers, the half-offset $h_i$ and reflection traveltime $2t$ can be expressed in terms of horizontal slownesses (ray parameters) $p_1$ and $p_2$ in $h_1$ and $h_2$ directions as follows:

$$h_i(p_1,p_2) = -\sum\limits^N_{n=1} D_{(n)} \frac{\partial Q_{(n)}(p_1,p_2)}{\partial p_i}~,$$ (9)

$$2t(p_1,p_2) = 2\left(p_1 h_1 +p_2 h_2 + \sum\limits^N_{n=1} D_{(n)} Q_{(n)}(p_1,p_2)\right)~,$$ (10)

where $D_{(n)}$ and $Q_{(n)}(p_1,p_2)$ denote the thickness and the vertical slowness of the $n$ -th layer. The derivation of equations 9 and 10 is included in the appendix. The general dependence $Q_{(n)}(p_1,p_2)$ follows directly from the Christoffel equation. Throughout the text, we use the subscript index in parentheses to indicate the corresponding layer. The upper-case and lower-case letters denote interval and effective parameters respectively.

Theory of interval traveltime parameter estimation in layered anisotropic media