Traveltimes

For a given impulse at (h₁,t₁) the length of the raypath that connects the shot and receiver with any possible reflector point is constant, in a constant velocity medium. The locus of reflector points constitute an ellipse, the foci of which coincide with the source and receiver location:

 

$$0 = \frac{x^{2}}{a_{1}^{2}} + \frac{z^{2}}{b_{1}^{2}} -1 .$$ (1)

The lengths a₁, b₁ of the major and minor axis are defined as

 

$$a_{1}^{2} = \frac{V^{2} \: t_{1}^{2}}{4}$$ (2)

 

$$b_{1}^{2} = a_{1}^{2} - h_{1}^{2} \;,$$ (3)

where V refers to the constant subsurface velocity.

Similarly, the migration of an impulse at (t2,h2) in shot gather s₂ yields an ellipse of the form

 

$$0 = \frac{(x+d)^{2}}{a_{2}^{2}} + \frac{z^{2}}{b_{2}^{2}} -1 ,$$ (4)

where

 

$$a_{2}^{2} = \frac{V^{2} \: t_{2}^{2}}{4},$$ (5)

 

b₂² = a₂²-h₂².

The variable d denotes the difference between the midpoints of the two ellipses:

 

d = (s₂ + h₂) - (s₁ + h₁) .

Extrapolation of shot gather s₁ requires the construction of a shot gather s₂ which when migrated yields the same subsurface image as the migration of the s₁ impulse: the ellipse of (1). Such a shot gather s₂ consists of a family of events, migrated impulse responses of which are tangent to the s₁ ellipse (1).

When shot gather s₂ is migrated, this tangency condition provides constructive superposition along the original isochron of s₁ (1). Anywhere else the events will cancel each other.

The tangency of the ellipses, (1) and (4), requires that the ellipses coincide and have equal slope at some point (x,z). Differentiation of equations (1) and (4) with respect to x and elimination of dx / dz yields an expression for x:

 

$$d = (1 - \frac{a_{2}^2 \: b_{1}^2}{a_{1}^2 \: b_{2}^2}) \: x .$$ (8)

After multiplication of equations (1) by b₁², and (4) by b₂², z is eliminated by subtracting the two equations. Substitution of x from equation (8) yields

 

$$d^2 = \frac{b_{1}^2 - b_{2}^2}{b_{1}^2 \: b_{2}^2} \: (a_{1}^2 \: b_{2}^2 - a_{2}^2 \: b_{1}^2) .$$ (9)

Finally, replacement of a₂, b₂ by expressions (5) and (6) provides the desired quadratic relation for t₂²:

 

$$0 = a \: t_2^4 + b \: t_2^2 + c ,$$ (10)

where

 

a = a₁² - b₁²

 

$$b = - a_1^2 \: b_1^2 + b_1^4 + b_1^2 \: d^2 - 2 \: a_1^2 \: h_2^2 + b_1^2 \: h_2^2$$ (12)

 

$$c = a_1^2 \: b_1^2 \: h_2^2 - b_1^2 \: d^2 \: h_2^2 + a_1^2 \: h_2^4 .$$ (13)

In the derivation above, transforming equations (8) to (9) requires a division by $a_{1}^2 \: b_{2}^2 - a_{2}^2 \: b_{1}^2$. If

 

$$0 = a_{1}^2 \: b_{2}^2 - a_{2}^2 \: b_{1}^2$$ (14)

then it follows from equation (8) that the midpoints of the ellipses (1) and (4) coincide (d=0). Because of this symmetry, the point of tangency lies at the hyperbolae's minor vertex. Since the lengths b₁, b₂ of the ellipses' minor axes are equal, condition (14) requires that also the major axes of the two ellipses agree: the two ellipses have to be identical. Hence in the case of identical or reciprocal positioning of geophone and source, the original impulse at (t₁, h₁) does not generate a set of events in shot gather s₂, but results in an identical impulse at (t₂, h₂).

Note, that in the case b₂ = a₂ = R the second ellipse (4) collapses to a circle. The relationships described in equation (8) and (9) then describe the partial prestack migration to zero offset as described by Forel 1988.

The Appendix discusses a Mathematica script which derives this result `automatically'. This Mathematica script starts from the basic ellipse equations and performs the transformations as they are described above.