A New Gradient Operator with Applications in Geophysics
We introduce a numerical first-order derivative operator (gradient) that produces more accurate results than known traditional operators, such as Ando, Sobel and the so-called Isotropic operators. The derivation of the operator here draws from the concept of plane waves, a mathematical representation of wave propagation that is widely used in wave physics, and in particular, exploration seismology. The motivations for using plane waves are that: (a) they naturally possess a propagation direction (directivity), which intuitively relates to the notion of anisotropy, and (b) they usually are periodic in space, therefore the link to spatial frequency (wavenumber). Though in the generic sense anisotropy relates to the dependence of some physical property on direction, in the context of numerical analysis it refers to the inaccuracy associated with a numerical operation, such as by a gradient operator, as a function of direction. Unlike customary gradient operators, which are derived primarily on the basis of amplitude with little or no regard to frequency or direction, the one we introduce here honors explicitly both wavenumber and directivity. It follows, naturally by design, that this operator assimilates spatial bandwidth and adapts for numerical isotropy intrinsically. Aside from suffering from numerical artifacts, all existing operators (except for the Finite Difference operator) are limited in handling only 2D scenarios, although in the literature some authors believe that extendibility to 3D is possible. Here, we provide 2D and 3D gradient formulations capable of producing accurate (artifact-free) results when tuned to the specific bandwidth of the application being used. For generality and computational efficiency, however, we give approximate versions of these operators that are entirely independent of bandwidth and directivity and still outperform the more traditional types of operators, in terms of accuracy.
We limit our analysis to second-degree operators (3 x 3 for 2D, and 3 x 3 x 3 for 3D) which, in addition to their computational efficiency compared to higher degrees, conveniently yield simple, exact analytical 2D and 3D solutions. We illustrate our points by applying the operators to synthetic and real seismic data. Though not covered in this work, and seemingly more laborious to achieve, extending the derivation to higher degrees and/or higher orders may be possible.
AAPG Search and Discovery Article #90141©2012, GEO-2012, 10th Middle East Geosciences Conference and Exhibition, 4-7 March 2012, Manama, Bahrain