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A transversely isotropic (TI) model in which the tilt is constrained to be normal to the dip (DTI model) allows for simplifications in the imaging and velocity model building efforts as compared to a general TI (TTI) model. Although this model cannot be represented physically in all situations, for example, in the case of conflicting dips, it handles arbitrary reflector orientations under the assumption of symmetry axis normal to the dip. Using this assumption, we obtain efficient downward continuation algorithms compared to the general TTI ones, by utilizing the reflection features of such a model. Phase-shift migration can be easily extended to approximately handle lateral inhomogeneity using, for example, the split-step approach. This is possible because, unlike the general TTI case, the DTI model reduces to VTI for zero dip. These features enable a process in which we can extract velocity information by including tools that expose inaccuracies in the velocity model in the downward continuation process. We test this model on synthetic data corresponding to a general TTI medium and show its resilience.

Migration velocity analysis (MVA), despite the many developments in recent years, is still a challenging process especially in complex media. MVA is even more of a challenge in anisotropic media in which the medium is described by several parameters, all of which can change as a function of position. Anisotropy introduces flexibility to the model to better simulate the Earth subsurface, but it also introduces a null space to the parameter estimation process or MVA. As a result, we need to use anisotropy to allow for more freedom up to the point where data seize to influence the model, or even part of it. This anisotropy null-space tradeoff has recently guided us to using a transversely isotropic (TI) medium with a tilted axis of symmetry (TTI). To avoid the null space, such tilt is assumed to be in the direction of the dip [

In transversely isotropic with vertical symmetry axis (VTI) media, the acoustic problem can be described by three parameters [

Alkhalifah and Sava [

Downward continuation, with the double-square equation in the DTI framework, utilizes the equal incidence and reflection angles imposed by the constraint. As shown by Alkhalifah and Sava [

Considering that angle gather extraction is a localized process relying on the plane wave behavior around the scattering point, it is applicable, within the limits of high frequencies compared to medium variations, to complex media. To allow the DSR-based downward continuation to honor lateral inhomogeneity at least approximately, we can utilize the phase-shift-plus-interpolation concept [

Wavefield reconstruction for multioffset migration based on the one-way wave equation under the survey-sinking framework [

Having an analytical representation for the migration operator allows us to develop migration analysis tools. The dependency of the migration operator on medium parameters is at the heart of such developments. Using the Sava and Vlad [

However, our desire is to relate the wavefield perturbation directly to medium perturbation, not just the phase velocity. For the DTI model,

One question that arises is how does the DTI constraint be imposed on a model? Specifically, what happens when we have conflicting dips? For the equations developed here and especially those of Alkhalifah and Sava [

Part of BP anisotropic velocity model that contains a salt body. The abrupt change in velocity magnitude can be interpreted as reflections, and the arrows point to examples of the possible directions of TI symmetry tilt to accommodate a DTI model.

Thus, in the BP model shown in Figure

The response of imaging to a dataset that includes pulses reveals some of the features of the operator involved in the imaging process. Here, our input data to the migration includes five pulses at times 0.6, 1.2, 1.8, 2.4, and 3 seconds at zero offset under the common midpoint (CMP) location of 4 km. The medium is vertically inhomogeneous with velocity increasing linearly with depth and

The impulse response for the prestack DTI phase-shift migration of 5 pulses at zero offset in a vertically inhomogeneous DTI model with

As expected, the response is symmetric despite the DTI nature of the medium. Unlike the TTI case, where the symmetry axis is set to a direction, the symmetry axis here is set to be normal to the reflector, and since the response includes all dips, it will also includes all possible symmetry directions. The angle gather behavior of the impulse response completes the saddle shape of the 3D operator.

In the following example, we use for simplicity a vertically inhomogeneous model, although nothing in the development of processes for DTI requires that. We consider the reflector model in Figure

A reflector model depicting a salt flank with three parallel reflectors laying alongside the flank. The angle of symmetry is normal to the three reflectors at 30 degrees from the vertical.

Prestack synthetic data generated using Kirchhoff modeling for the TTI model in Figure

Conventional phase shift downward continuation requires that no lateral velocity variation be present. Since the synthetic model has no lateral velocity variation, we use the phase-shift approach to migrate the data. However, prior to applying the zero-lag imaging condition, we map the offset wavenumbers to angle and, thus, obtain angle gathers. Figure

Migrated section after an isotropic migration with velocity of 2 km/s of the TTI syntheticdata in Figure

If we downward continue using a VTI phase-shift migration followed by an anisotropic angle gather mapping [

Migrated section after a VTI migration with velocity of 2 km/s and

Migrated section after DTI-based migration with velocity of 2 km/s and

This synthetic test shows an example of the usefulness of the DTI model for analysis of key reflections. Usually, for migration velocity analysis purposes, the symmetry axis is set to be normal to the reflector dip for the reflections used in the analysis, and this is the case even for isotropic layers, which is a special case of DTI where

Constraining the symmetry axis of a transversely isotropic medium to be normal to the reflector dip (DTI) allows for explicit formulation of plane waves around the scattering point. These formulations form the basis for angle decomposition and simplified downward continuation. As a result, DTI is a convenient model for anisotropy parameter estimation in media in which such models are applicable. This model also allows us to use the general TTI assumption in a simplified form that better fits the information embedded in the recorded data. A simple synthetic example demonstrated the potential features of this model.

The authors thank KAUST and the Center of Wave Phenomena at Colorado School of Mines for their support.