International Journal of Geophysics

Volume 2016, Article ID 2848750, 6 pages

http://dx.doi.org/10.1155/2016/2848750

## Influence of Error in Estimating Anisotropy Parameters on VTI Depth Imaging

Center of Seismic Imaging, Universiti Teknologi PETRONAS, 32610 Seri Iskandar, Malaysia

Received 11 March 2016; Revised 5 April 2016; Accepted 20 April 2016

Academic Editor: Alexey Stovas

Copyright © 2016 S. Y. Moussavi Alashloo et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Thin layers in sedimentary rocks lead to seismic anisotropy which makes the wave velocity dependent on the propagation angle. This aspect causes errors in seismic imaging such as mispositioning of migrated events if anisotropy is not accounted for. One of the challenging issues in seismic imaging is the estimation of anisotropy parameters which usually has error due to dependency on several elements such as sparse data acquisition and erroneous data with low signal-to-noise ratio. In this study, an isotropic and anelliptic VTI fast marching eikonal solvers are employed to obtain seismic travel times required for Kirchhoff depth migration algorithm. The algorithm solely uses compressional wave. Another objective is to study the influence of anisotropic errors on the imaging. Comparing the isotropic and VTI travel times demonstrates a considerable lateral difference of wavefronts. After Kirchhoff imaging with true anisotropy, as a reference, and with a model including error, results show that the VTI algorithm with error in anisotropic models produces images with minor mispositioning which is considerable for isotropic one specifically in deeper parts. Furthermore, over- or underestimating anisotropy parameters up to 30 percent are acceptable for imaging and beyond that cause considerable mispositioning.

#### 1. Introduction

It is well-known that hydrocarbon reservoirs and overlying strata are commonly anisotropic [1, 2]. In reality, it is rare to have media with elliptical or weak anisotropy properties. However, anellipticity (deviation of wavefield from ellipse) has been commonly observed in the Earth’s subsurface, and it is a significant characteristic of elastic wave propagation [3, 4].

Another challenging issue in depth imaging is the computation of the travel time taken by a seismic wave from source to receiver. An efficient method to compute travel times is solving the eikonal equation by employing finite differences [5, 6]. Different techniques have been introduced to solve the eikonal equation, such as embedding methods, single-pass methods, sweeping methods, and iterative methods [7]. The main difference of these techniques is in how they cope with the complication of multivalued solutions and in finding solutions in the vicinity of cusps and discontinuities [8]. Anisotropy was initially added to an eikonal solver algorithm by Dellinger [9]. The embedding and iterative methods are both time consuming, particularly in heterogeneous and anisotropic conditions [7]. Fast sweeping methods are originally proposed for isotropic media [10]; however, a modification is executed to handle the anisotropic condition [11]. Single-pass or fast marching method (FMM) is another tool for computing travel times but is not generally applicable for anisotropic medium [5]. This algorithm has since been modified to work for anisotropy [12, 13].

In this study, a prestack depth migration algorithm is developed based on an anelliptic VTI compressional wave equation. Fomel’s anelliptic approximation [14] for both phase and group velocity of P-wave are employed to derive the eikonal equation. The fast marching finite difference approach is used as our eikonal solver since it is fast and stable for travel time computation. In anisotropic study, four anisotropic models are used: a true model which is exactly similar to the model employed for forward modelling, a model with values 30 percent less than the true model, a model with values 40 percent less than the true model, and a model with values 30 percent more than the true model. The calculated travel times are compared and employed in a standard Kirchhoff migration to obtain the image of the subsurface. The Marmousi model, as a complex model, is used to test the algorithm. Finally, we analyze both isotropic and VTI images qualitatively.

#### 2. Methodology

We develop a new algorithm to incorporate anelliptic VTI travel times into prestack depth imaging. Our workflow for PSDM consists of the following: (1) travel time computation and (2) Kirchhoff depth migration. Step (1) provides travel times for the Kirchhoff migration. The algorithm is discussed in detail below.

##### 2.1. VTI Fast Marching Eikonal Solver

The anisotropic wavefield propagation under a high frequency assumption is defined by the eikonal equation:where is the travel time, , , are the Cartesian coordinates, and is the VTI phase velocity. The FMM solves (1) by considering the fact that the direction of energy propagation follows the group velocity equation. This method is similar to a ray that is perpendicular to wavefronts defined by phase velocity. This ray is called the travel time gradient. A wave equation is needed as a kernel of fast marching algorithm. Fomel [14] enhanced the anelliptic P wave approximation proposed by Muir and Dellinger [15] through replacing the linear approximation with a nonlinear one. By using the shifted hyperbola approximation, he obtains the following equation for P wave phase velocity:where , , where are the density-normalized components of the elastic tensor, is the phase angle, and and are the anellipticity coefficient and the elliptical component of the velocity, respectively, defined by where and are Thomsen’s parameters, and . Similarly, for approximating the group velocity, the shifted hyperbola approach is applied on the Muir’s approximation to unlinearize the equation. The new group velocity approximation is where , , , is the group angle, and is the elliptical part:Approximations in (2) and (5) are used for ray tracing in locally homogeneous cells needed in this algorithm. An attentively selected order of travel time evaluation is the main advantage of the FMM. This method is an upwind method which means if a wave propagates from left to right, a difference scheme should be applied for reaching upwind to the left to collect information to construct the solution downwind to the right.

According to Fomel [16], although the algorithm follows a certain procedure, the grid points are divided into three classes, namely,* Alive*, points which are behind the wavefront and have been already computed;* NarrowBand*, points on the wavefront awaiting assessment; and* FarAway*, which remains untouched ahead of the wavefront (Figure 1). In other words, the source positions at the beginning of the evaluation are considered* Alive.* Given that they are initial points, their travel time is zero. All points that are one grid point away are taken as* NarrowBand*, and their travel times are computed analytically. All other grid points are marked as* FarAway* and have an “infinitely large” travel time value [5, 16]. The algorithm includes the following main steps:(1)Find the point with the minimum travel time among the* NarrowBand* points.(2)Tag the point as* Alive* and remove it from* NarrowBand*.(3)Check the neighbours of the minimum point that are not* Alive.* If any of them are categorized as* FarAway,* update them as* NarrowBand*. It means the wavefront is advanced, and the minimum point is behind it.(4)Update travel times for points on the NarrowBand (wavefront) by solving (1) numerically.(5)Repeat the loop until all points are behind the wavefront.For updating, one to three neighbour points are selected and their travel time values need to be less than the current value. After choosing the points, the quadratic equation, should be solved for which is the updated travel time value. is the travel time at the neighbouring points, is the slowness () at the point , and is the grid size in direction.