Differential operators 1: The first derivative
John C. Bancroft
Differential operators are used in many seismic data processes such as triangle filters to reduce aliasing, finite difference solutions to the wave equation, or wavelet correction when modelling with diffractions or migrating with Kirchhoff algorithms. Short operators (two to nine samples) may be quite accurate when the data are restricted to locally low order polynomials, but may be inaccurate in other applications.
Because of the size of the contents, this topic is divided into three papers; the first reviews operators for the first derivative, the second paper examines the second derivative, and the third paper examines the square-root derivative that is also referred to as the rho filter. Theses papers examine various methods for computing the operators and compare their size with their accuracy.
Four approximations to the derivative are evaluated; an ad hoc windowed spectral definition, a polynomial approximation, the Remez approximation, and a recursive approximation. Significant details of the derivations have been included for clarification.