Moveout approximation for vertical seismic profile geometry in a 2D model with anisotropic layers

Abstract

A B S T R A C TI introduce a new explicit form of vertical seismic profile (VSP) traveltime approximation for a 2D model with non-horizontal boundaries and anisotropic layers. The goal of the new approximation is to dramatically decrease the cost of time calculations by reducing the number of calculated rays in a complex multi-layered anisotropic model for VSP walkaway data with many sources. This traveltime approximation extends the generalized moveout approximation proposed by Fomel and Stovas. The new equation is designed for borehole seismic geometry where the receivers are placed in a well while the sources are on the surface. For this, the time-offset function is presented as a sum of odd and even functions. Coefficients in this approximation are determined by calculating the traveltime and its first-and second-order derivatives at five specific rays. Once these coefficients are determined, the traveltimes at other rays are calculated by this approximation. Testing this new approximation on a 2D anisotropic model with dipping boundaries shows its very high accuracy for offsets three times the reflector depths. The new approximation can be used for 2D anisotropic models with tilted symmetry axes for practical VSP geometry calculations. The new explicit approximation eliminates the need of massive ray tracing in a complicated velocity model for multi-source VSP surveys. This method is designed not for NMO correction but for replacing conventional ray tracing for time calculations.

High‐accuracy two‐interval approximation for normal‐moveout function in a multi‐layered anisotropic model

“…Finally, we can write the traveltime equation as (Fig. c): where t 1 (x) is defined by equation with parameters A, B and C defined from equations derived by Blias () (Appendix A). Function t 2 ( x ) is defined by equation with coefficients A 1 , B 1 and C 1 defined by equations .…”

“…One of the main applications of the two-interval approximation would be in traveltime inversion. The number of rays to calculate is important for 3D Monte-Carlo type traveltime inversion, where the number of calculated traveltimes may be hundreds of millions (Pei et al 2009;Blias 2013). It implies that we are interested in an approximation with the minimum number of rays.…”

The modified generalized moveout approximation: a new parameter selection

“…The idea of using the second-order derivative of travel time with respect to offset obtained from the reference ray was proposed by Blias (2013) * E-mail: alexey@ntnu.no and used in another moveout approximation with focus on vertical seismic profile acquisition geometry. One can see the non-symmetry with respect to the order of travel-time derivatives used from the zero offset and a reference ray offset.…”

“…The generalised moveout approximation (GMA) proposed by Fomel and Stovas (2010) becomes a very convenient tool in various kinematic problems (Stovas 2010;Fomel and Kazinnik 2013;Blias 2013;Asgharzadeh, Nadri, and Bona 2014;Xu and Stovas 2015). The GMA is a five-parameter approximation, where three parameters are computed from the zerooffset ray and two parameters are computed from a reference ray.…”

“…One can see the non-symmetry with respect to the order of travel-time derivatives used from the zero offset and a reference ray offset. The idea of using the second-order derivative of travel time with respect to offset obtained from the reference ray was proposed by Blias (2013) * E-mail: alexey@ntnu.no and used in another moveout approximation with focus on vertical seismic profile acquisition geometry.…”