Evaluation of ${\partial k_{x_h}}/{\partial \phi}$and ${\partial k_{y_h}}/{\partial \phi}$

Equations (15) and (19) express the partial derivatives of the offset wavenumbers in the rotated coordinates. We need to evaluate the partial derivatives of the offset wavenumbers in the data coordinates. Therefore, we need to differentiate the expressions defining the inverse rotation; that is, differentiate with respect to $\phi$ the following expressions:

$$\begin{eqnarray} k_{x_h} &=& \cos \phi k_{x_h}'+ \sin \phi k_{y_h}', \\ k_{y_h} &=& -\sin \phi k_{x_h}'+ \cos \phi k_{y_h}'.\end{eqnarray}$$ (19) (20)

The partial derivatives are then given by the following expressions:

$$\begin{eqnarray} \frac{\partial k_{x_h}}{\partial \phi} &=& \cos \phi \frac{\par... ..._h}'}{\partial \phi} - \cos \phi k_{x_h}'. - \sin \phi k_{y_h}'.\end{eqnarray}$$ (21) (22)