The modified generalized moveout approximation: a new parameter selection

Abstract

Non‐hyperbolic generalised moveout approximation is a powerful tool to approximate the travel‐time function by using information obtained from two rays. The standard approach for parameter selection is using three parameters defined from zero‐offset ray and two parameters obtained from a reference ray. These parameters include the travel time and travel‐time derivatives of different order. The original parameter selection implies more fit at zero offset compared with offset from a reference ray. We propose an alternative approach for parameter selection within the frame of generalised moveout approximation by transferring more fit from the zero offset to a reference ray by changing in parameter selection. The modified approximation is tested against the original one in few analytical model examples, including the multi‐layered model.

A new parameterization for generalized moveout approximation, based on three rays

“…The parameters B and C are defined from the traveltime and its first derivative (ray‐parameter) at a large reference offset. Noticing that this method of parameter definition puts extra weight on the zero‐offset ray, Stovas and Fomel () proposed a modified version of GMA with a new parameter selection. In the modified GMA, they avoid the fourth‐order derivative of traveltime at zero offset, but use the second‐order derivative (curvature) at the far‐offset reference ray.…”

“…To approximate nonhyperbolic traveltime with a limited number of independent parameters has been the subject of a large amount of research. Shifted hyperbola approximation (Malovichko ; de Bazelaire ), rational approximation (Tsvankin and Thomsen ; Alkhalifah and Tsvankin ) and generalized moveout approximation (GMA; Fomel and Stovas ; Stovas ; Stovas and Fomel ) are more commonly employed, because of their simplicity, applicability or higher accuracy. Other known explicit moveout approximations include methods of Alkhalifah (, ), Zhang and Uren (), Taner, Treitel and Al‐Chalabi (), Ursin and Stovas (), Blias (, , ), Aleixo and Schleicher (), Ravve and Koren () and Abedi and Stovas ().…”

“…Traveltime approximations with higher degrees of freedom can be reduced to three‐parameter VTI moveout approximations, by explicit definition of their parameters based on the three VTI parameters. Using an asymptotic match, the five‐parameter equations of Ravve and Koren () are reduced to the rational approximation (Alkhalifah and Tsvankin ); the five‐parameter GMA (Fomel and Stovas ; Stovas and Fomel ) and the six‐parameter Extended GMA (Abedi and Stovas ) are reduced to a single equation, which is called VTI GMA.…”

“…There are two methods for general finite offset parameter definition of GMA, proposed by Fomel and Stovas () and Stovas and Fomel (). These methods use different order of traveltime derivatives obtained from two rays to define five parameters, which results in a model‐dependent higher accuracy.…”

Slowness‐domain kinematical characteristics for horizontally layered orthorhombic media. Part I: Critical slowness match

“…VTI), Fomel and Stovas (2010) and Stovas and Fomel (2012) requires five parameters: three parameters from the zero-offset ray (traveltime, NMO velocity and nonhyperbolic coefficient) and two parameters from an additional long-offset (wideangle) ray. In a later study by Stovas and Fomel (2017), the generalized moveout approximation has been modified by defining the fourth-order (nonhyperbolic) parameter from the reference ray, rather than from the zero-offset ray. In a recent study by Abedi and Stovas (2018), the accuracy of the moveout approximation has been essentially improved by the cost of an additional (sixth) parameter: the curvature of the ray at the non-zero reference offset.…”