JACOBIAN FOR NMO

Let us denote $\bold N$ the NMO operator and $\bold N^{\bold T}$ its transpose operator. The matrix ${\bf N^T N}$ is diagonal for nearest neighbor interpolation (Claerbout, 1989). To see how the coefficients of the diagonal of ${\bf N^T N}$ look like, let us compute the operator ${\bf N^T N}$ in the continuous domain. For any traces $\bold d_1$ and $\bold d_2$ it holds

$$\begin{eqnarray} (\bold N \bold d_1,\bold N \bold d_2)&=&\int \bold N \bold d_1(... ..._2(t) {t \over \sqrt{t^2-x^2m}} dt \\ &=&(\bold d_1,{\bf N^TNd}_2)\end{eqnarray}$$ (41) (42) (43) (44)

so that we have

$${\bf N^TNd}_2(t) = {t \over \sqrt{t^2-x^2m}} \bold d_2(t).$$ (45)