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Figure 7: Zero-offset redatuming to a horizontal interface separating two homogeneous media. The output source- receiver pair at n,z, and the desired traveltime t define the isochron z= %,(x;1,7), this being a circle with radius R,. A reflection at a point M* recorded at € would travel along the ray with length €+R,, which refracts at z, according to Snell’s law.

Figure 7 Zero-offset redatuming to a horizontal interface separating two homogeneous media. The output source- receiver pair at n,z, and the desired traveltime t define the isochron z= %,(x;1,7), this being a circle with radius R,. A reflection at a point M* recorded at € would travel along the ray with length €+R,, which refracts at z, according to Snell’s law.

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where R, = \/(z—zr)2 +(x—)2 (see Figure 1). Equation Figure 1: Zero-offset redatuming in a constant-velocity medium: The output source-receiver pair at 7,z, and the desired traveltime t define the isochron z = %,(x; 7,7), this being a circle with radius R,. A reflection at a point M* recorded at E would travel along the ray with length 2+ R,. For simple situations, analytic expressions for the stacking line and weight function can be found. As the most simple case, we consider the redatuming from a planar acquisition surface at depth z= 0 to a planar datum at depth z=z, ina homogeneous medium with velocity co. In this situation, the isochron z= %,(x;n,t) of point (n,t) as defined in equation (3) becomes
where R, = \/(z—zr)2 +(x—)2 (see Figure 1). Equation Figure 1: Zero-offset redatuming in a constant-velocity medium: The output source-receiver pair at 7,z, and the desired traveltime t define the isochron z = %,(x; 7,7), this being a circle with radius R,. A reflection at a point M* recorded at E would travel along the ray with length 2+ R,. For simple situations, analytic expressions for the stacking line and weight function can be found. As the most simple case, we consider the redatuming from a planar acquisition surface at depth z= 0 to a planar datum at depth z=z, ina homogeneous medium with velocity co. In this situation, the isochron z= %,(x;n,t) of point (n,t) as defined in equation (3) becomes
Figure 2: Model for the numerical experiment: A zero- offset experiment was simulated at z = 0 above a synclinal reflector. Also shown is the datum level (dotted line) at z= 200m.
Figure 2: Model for the numerical experiment: A zero- offset experiment was simulated at z = 0 above a synclinal reflector. Also shown is the datum level (dotted line) at z= 200m.
Figure 3: Modeled seismic zero-offset section. cy and cy (see Figure 7). Analogously to the previous case, the stationary ray defining the stacking line is the one that is normal to the output isochron z = &%,(x;n,T) and thus passes through the redatumed source-receiver point at n. Therefore, it connects S to S,, where it refracts according to Snell’s law (see Figure 7). Therefore, its length is again (R, +2£). From this consideration, it is clear that it’s traveltime, which defines the redatuming stacking line, is given by
Figure 3: Modeled seismic zero-offset section. cy and cy (see Figure 7). Analogously to the previous case, the stationary ray defining the stacking line is the one that is normal to the output isochron z = &%,(x;n,T) and thus passes through the redatumed source-receiver point at n. Therefore, it connects S to S,, where it refracts according to Snell’s law (see Figure 7). Therefore, its length is again (R, +2£). From this consideration, it is clear that it’s traveltime, which defines the redatuming stacking line, is given by
Figure 4: Seismic section resulting from redatuming operation 1 to z= 200 m.
Figure 4: Seismic section resulting from redatuming operation 1 to z= 200 m.
Figure 5: Seismic section resulting from Kirchhoff modeling at z= 200 m.
Figure 5: Seismic section resulting from Kirchhoff modeling at z= 200 m.
Figure 6: Comparison of the central traces of Figures 4 and 5. Notice the almost perfect coincidence between the redatumed (dashed line) and modeled (continuous line) traces.
Figure 6: Comparison of the central traces of Figures 4 and 5. Notice the almost perfect coincidence between the redatumed (dashed line) and modeled (continuous line) traces.
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