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Formulas for interval NMO velocities

Equations 3043 and 44 lead to the following Dix-type inversion formula

$\displaystyle A_{ii(N)} = \bigg(t_{0(N)}-t_{0(N-1)}\bigg)\left(\frac{t_{0(N)}}{a_{ii(N)}}-\frac{t_{0(N-1)}}{a_{ii(N-1)}}\right)^{-1}~,$ (48)

where $ a_{ii(N)}$ and $ a_{ii(N-1)}$ with $ i=1,2$ denote the coefficients for the reflections from the top and bottom of the $ N$ -th layer respectively, and $ A_{ij(N)}$ denotes the interval coefficient in the $ N$ -th layer. $ t_0$ is vertical one-way traveltime with the similar notation rules. We can convert equation 48 to a familiar form (Dix, 1955) by expressing it in terms of interval moveout velocity $ V^2_1 = 1/A_{22}$ and $ V^2_2 = 1/A_{11}$ as follows:

$\displaystyle V^2_{i(N)} =\frac{v^2_{i(N)}t_{0(N)}-v^2_{i(N-1)}t_{0(N-1)}}{t_{0(N)}-t_{0(N-1)}}~.$ (49)

Equation 49 is a scalar form of the tensor equation 30 for the special case of aligned orthorhombic layers.


next up previous [pdf]

Next: Formulas for interval quartic Up: Coefficients of traveltime expansion Previous: General formulas for coefficients

2017-04-14