Implicit extrapolation

The basis for wavefield extrapolation is an operator, W(k), that marches the wavefield P, at depth z, down to depth z+1.  

$$P_{z+1}=W(k) \; P_{z}$$ (1)

Ideally, W(k), will have the form of the phase-shift operator ().

$$W(k)=e^{i \sqrt{a^2-k^2}}$$ (2)

where $a=\omega/v$, and for simplicity $\Delta x=\Delta z=1$.

However, to apply this operator directly requires spatial Fourier transforms, and an assumption of constant lateral velocity. To overcome this limitation, short finite-difference approximations to W(k) are applied in the $(\omega-x)$ domain.

An implicit finite-difference formulation approximates W(k) with a convolution followed by an inverse convolution. For example, a simple implicit approximation to equation () that corresponds to the Crank-Nicolson scheme for the 45$^\circ$ one-way wave equation, is given by  

$$W_{im}(k)=e^{ia} \; \frac{1-4a^2+iak^2}{1-4a^2-iak^2}= e^{i \phi}$$ (3)

where $\phi = a-\arctan \frac{ak^2}{4a^2-1}$. Since this operator represents a phase-shift only, energy is conserved, and the formulation is unconditionally stable for all values of a.

An explicit approach approximates W(k) directly with a single convolutional filter. For example, a three-term expansion of equation () yields  

$$W_{ex}(k)=e^{ia} \; \left( 1+ \gamma_1 k^2 + \gamma_2 k^4 \right)$$ (4)

where complex coefficients $\gamma_1$ and $\gamma_2$ can be calculated using a Taylor series, for example.

Although in practice stability is not usually a problem for explicit operators, they can never represent a pure phase-shift. Hence, stability cannot be guaranteed for all velocity models ().

Also in order to preserve high angular accuracy for steep dips, explicit filters need to be longer than their implicit counterparts. The advantage of finite-difference methods over Fourier methods is that the effect of the finite-difference convolution filters is localized, leading to accurate results for rapidly varying velocity models. This is less of an advantage for long filters.