Diffraction imaging and time-migration velocity analysis using oriented velocity continuation

Oriented velocity continuation

Velocity continuation (Fomel, 2003c) is the imaginary process of a continuous transformation of seismic time-migrated images as they are propagated through different migration velocities. In the most general terms, the kinematics of velocity continuation can be described by an equation of the Hamilton-Jacobi type

$$\frac{\partial \tau}{\partial v} = F(v,\tau,\mathbf{x},\nabla \tau)\;,$$ (1)

where $\tau(\mathbf{x},v)$ is the location of a time-migrated reflector with time-domain coordinates $\mathbf{x}=(x_1,x_2,t)$ imaged with spatially constant time-migration velocity $v$. The particular form of function $F$ in equation (1) depends on the acquisition geometry of the input data. For the case of common-offset 2D velocity continuation for data with half-offset $h$,

$$F(v,t,x,p\old{,h})=v t p^2 + \frac{h^2}{v^3t},$$ (2)

and equation (1) corresponds to the characteristic equation of the image propagation process which describes a propagation of the time-migrated image $I(t,x,v)$ in velocity $v$ (Fomel, 2003c). Time-domain imaging can be performed effectively by extrapolating images in velocity and estimating velocity $v_m(t,x)$ of the best image (Fomel and Landa, 2014; Fomel, 2003b; Larner and Beasley, 1987).

As shown by Fomel (2003a), it is possible to extend the formulation of a wave propagation process from the usual time-and-space coordinates to the phase space consisting of time, space, and slope. Applying a similar approach to equation (1), we first employ the Hamilton-Jacobi theory (Evans, 2010; Courant and Hilbert, 1989) to write the corresponding system of ordinary differential equations for the characteristics (velocity rays), as follows:

$$\frac{d\mathbf{x}}{dv} = - \nabla_p F\;,$$ (3)

$$\frac{d\mathbf{p}}{dv} = \nabla_x F + \frac{\partial F}{\partial t} \mathbf{p}\;,$$ (4)

$$\frac{dt}{dv} = F - \nabla_p F \cdot \mathbf{p}\;,$$ (5)

where $\mathbf{p}$ stands for $\nabla \tau$, the gradient or slope of time-migrated wavefield energy.

If the image $I(t,\mathbf{x},v)$ is decomposed in slope components $\widehat{I}(t,\mathbf{x},\mathbf{p},v)$ so that

$$I(t,\mathbf{x},v) = \int \widehat{I}(t,\mathbf{x},\mathbf{p},v) d\mathbf{p}\;,$$ (6)

we can then look for an equation that would adequately describe a continuous transformation of $\widehat{I}$. To preserve the geometry of the transformation, it is sufficient to require that $\widehat{I}$ transports along the characteristics described by equations (3-5). Applying partial derivatives and the chain rule, we arrive at the equation analogous to the Liouville equation (Engquist and Runborg, 2003):

$$\frac{\partial \widehat{I}}{\partial v} = \left(F - \nabla_p... ... \mathbf{p}\right) \cdot \nabla_p \widehat{I}\;\old{.}\new{,}$$ (7)

Equation (7) describes, in the most general form, the process of oriented velocity continuation, image propagation in velocity in the coordinates of time-space-slope. It is a linear first-order partial differential equation of convection type which operates in the phase space.

Diffraction imaging and time-migration velocity analysis using oriented velocity continuation