Antialiasing of Kirchhoff operators by reciprocal parameterization

3-D antialiasing

The proposed method of antialiasing is easily generalized to the case of a three-dimensional integral operator. In this case, we need to consider three different parameterizations: $t(x,y)$, $x(t,y)$, and $y(t,x)$ and switch from one of them to another according to the rule:

• if $\Delta t \geq {{\Delta x} {\vert\partial t / \partial x\vert}}$ and $\Delta t \geq {{\Delta y} {\vert\partial t / \partial y\vert}}$, use $t(x,y)$,
• if $\Delta x \geq {{\Delta t} {\vert\partial x / \partial t\vert}}$ and $\Delta x \geq {{\Delta y} {\vert\partial x / \partial y\vert}}$, use $x(t,y)$,
• if $\Delta y \geq {{\Delta t} {\vert\partial y / \partial t\vert}}$ and $\Delta y \geq {{\Delta x} {\vert\partial y / \partial x\vert}}$, use $y(t,x)$.

Following Biondi (2001), I illustrate 3-D antialiasing by applying prestack time migration on a single input trace. The results are shown in Figures 13, 14 and 15. The result without any antialiasing protection (Figure 13) contains clearly visible aliasing artifacts caused by the steeply dipping parts of the operator. Antialiasing by temporal filtering (Figure 14) removes the artifacts but also attenuates the steeply dipping events. Antialiasing by the proposed reciprocal parameterization (Figure 15) removes the aliasing artifacts while preserving the steeply dipping events and the image resolution.

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Figure 13. Prestack 3-D time migration of a single input trace. Top: time slice at 1 s. Bottom: vertical slice. No antialiasing protection has been applied. As a result, aliasing artifacts are clearly visible in the image.

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Figure 14. Prestack 3-D time migration of a single input trace. Top: time slice at 1 s. Bottom: vertical slice. Antialiasing by temporal filtering has been applied. Aliasing artifacts are removed, steeply dipping events are attenuated.

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Figure 15. Prestack 3-D time migration of a single input trace. Top: time slice at 1 s. Bottom: vertical slice. The proposed reciprocal antialiasing has been applied. Aliasing artifacts are removed, steeply dipping events are preserved.

Antialiasing of Kirchhoff operators by reciprocal parameterization