As a condition for further generalization of the migration to zero-offset in variable velocity media, I develop the theory for 2-D migration to zero offset (MZO) in constant velocity media, starting from prestack migration in midpoint-offset coordinates. At the end of this chapter I arrive at an integral formulation for the MZO operator, analytically derived from the double square root (DSR) prestack migration equation. The integral formulation for the MZO is very similar in form to the DSR equation, suggesting a generalization to variable velocity media via a phase-shift algorithm. Further chapters deal with offset separation and the depth variable v(z) and laterally variable v(x,z) velocity media.
Introducing the Double Square Root Equation The theory for the double square root (DSR) equation is discussed in detail in the first chapter of Yilmaz's (1979) thesis. Without going into mathematical detail I will sketch the path of the basic theory for obtaining the DSR migration equation in offset and midpoint coordinates starting from the wave equation. Readers familiar with the DSR equation can skip directly to the next section.
The scalar wave equation in a 2-D medium of constant density can be written as
For a constant velocity we write
To find the solution to equation (7) we would need to have two initial or boundary conditions. We only have the pressure field at z=0 as a boundary condition, but we can still solve the problem if we decide to resolve only the upgoing waves, in other words to use the exploding reflectors principle.
If we know the pressure field (or wavefield) at a certain depth we can propagate it forward in time or backward in time. We can also propagate it up in depth (along the z-axis) or down. To understand how we determine the propagation direction we have to analyze the values and sign of kz. The function
represents a plane wave. If we ignore kx x, which determines the lateral variation, we can introduce a function which we call phase(z,t) defined as
The phase is constant along a plane wave, and we write
for the phase of a particular plane wave. The plane wave is moving downward when kz has the same sign with because z increases with t in order to keep the phase constant. So for the upward moving waves we need to have opposite signs of kz and (z is decreasing when t is increasing). We have now figured out that in order to have only upgoing waves we have to look at the sign of and assign to kz the opposite sign. Therefore equation (7) becomes:
Setting z=0 in equation (9) we identify P0 as the data recorded at the surface:
In this form we can use the data recorded at the surface to propagate the wavefield to any depth level
Knowing the wavefield at any depth z0 we can find the wavefield at any other depth z0+z. For positive values of z we have to propagate the wavefield back in time (toward t=0) because we know that the wavefield travels upward. If the known wavefield is at depth z0 and we want to find the wavefield at depth z0-z, then we are propagating the wavefield forward in time. This is the direction we use for modeling.
However for depth varying velocity v(z) we have kz approximately constant only for small depth intervals () where we can consider the velocity constant. Therefore equation (10) becomes
There are several restrictions on the values of kz. Equation (6) has the solution (7) only for real values of kz which imposes the condition
The solution represented in equation (10) is for a single Fourier transform component of the wavefield. The general solution in time-space coordinates is obtained by summing all the Fourier coefficients obtained from equation (10)
In the case of a seismic experiment with many shots and receivers we can downward continue separately the shots and the receivers to any depth level. The total phase shift to the same depth level z becomes the phase shift of the shots plus the phase shift of the receivers.
where y and h are respectively the midpoint and offset coordinates, while xs and xg are the shot and geophone surface coordinates. Note that the variable h represents half the total distance between the source and geophone. The total phase shift in the new wavenumber coordinates becomes
Isolating the Zero-offset migration The basic concept for analytically deriving the MZO from prestack migration is to separate the latter into two processes:
We start with the constant velocity prestack migration in offset-midpoint coordinates (Yilmaz, 1979) formulated as:
The phase is defined in the dispersion relation as
The constant velocity zero-offset migration (Gazdag, 1978) can be formulated as:
In order to convert equation (15) into a form similar to equation (17), I use a change of variables from to such that after integrating over the variable kh, equation (15) will be transformed into the form:
where represents the zero-offset data field. The rationale for casting the prestack migration equation in this form is to identify the operations needed to obtain the zero-offset field from the common-offset field. The assumption is that the output of zero-offset migration and prestack-migration is the same image. By dissecting the prestack-migration and separating the zero-offset migration operator, we isolate the migration to zero-offset (MZO) operator.
Using Hale's (1983) derivation, a new variable is introduced in order to isolate the zero-offset migration operator. The expression for the new variable is found by equating the dispersion relation for prestack migration to the dispersion relation of zero offset migration,
and squaring the two equations twice. This algebra is demonstrated in detail in Hale's thesis (1983), Appendix 3.A, and therefore I did not repeat it here. The final expression for the variable is found to be
In order to isolate the zero-offset migration operator, after substituting the expression of given by equation (19) in the prestack migration equation (15, the order of integration is changed between and kh. The integration boundaries have to be observed carefully as they are modified after each change of variables and integration order. However, for the sake of simplicity, I will ignore in the following demonstration the integration limits, which are discussed in Appendix D.
By substituting the variable in equation (15) with its new expression (19) as a function of ,and changing the integration order between and kh the prestack migration equation becomes
The field defined as
The last equation in (21) is of course the zero-offset migration equation (17), the classic zero-offset downward continuation and imaging described by Gazdag (1978) or Stolt (1978). Equation (22) represents a way of obtaining the zero-offset section from prestack data in midpoint-offset coordinates.
So far the operations needed to obtain the zero-offset stacked section from the prestack field are:
MZO as phase shift The interpolated field in equation (22) represents the values of the field after remapping from to . It is obtained by first Fourier transforming the initial prestack field along all three (time, midpoint and offset) axes: ,and second interpolating from to .As in Stolt migration (Popovici et al. 1993), we can replace the two steps of
Formally we inverse Fourier transform in time equation (22) to have
The only drawback so far to equation (26) is that it performs a Fourier transform and later a summation over the offset variable. I will show in the next chapter how the offset variable can be separated and as a result MZO can be applied to distinct common-offset sections. Once MZO is applied to separate common-offset sections I isolate the conventional NMO and DMO processes. I will further show how equation (26) can be applied to variable velocity media, via a phase shift algorithm similar to Gazdag migration, and PSPI or split-step.
A In this appendix I show that by writing the variable function of , the double square-root (DSR) phase used in prestack migration is transformed to a new form corresponding to the phase used for zero-offset migration. The transformation from to as defined in equation (19) is:
where vh and vy are defined as
The DSR phase is transformed from
Hale (1983) in the Appendix A of his thesis proves an equivalent assertion, with a different logic. Comparing the DSR phase with the phase of the zero-offset migration (defined as a single square root), Hale finds the expression of which transforms the former into the latter. I was tempted to refer the reader to his appendix as an indirect proof, but decided to include a thorough derivation, for completeness.
Using the identity
I rewrite the DSR phase as
Examine the expression under the second square root (SSR) in equation (27)
and substitute for the expression in .The expression under the second square root becomes:
B The purpose of this appendix is to evaluate the Jacobian of the transformation from to :
Starting with the transformation of variable
and differentiating we have
and therefore the Jacobian is:
C The purpose of this appendix is to find the inverse of the transformation , or to express function of .Starting with the original transformation of variable we have
and note that has always the same sign as .Square the equation to obtain
and after isolating the terms in we have the equation
The existence conditions for kz
ensure that is always positive and therefore is always real within the existence limits. The choice of a positive sign for the discriminant in equation (31) is assisted by the observation that for vh=0, the case of a zero-offset data field, the equation becomes an identity as it is expected. Chosing the positive sign for the discriminant, equation (31) becomes
and using the observation that has the same sign as we have:
D In this appendix I follow the integration boundaries for all the integral transformations from equation (15) to equation (26). In equation (15) the values of the constant kz, given by equation (16), have to be real. This requires the conditions
to be satisfied simultaneously. Considering all four possible sign cases for vy and vh represented in Figure , and the interval of existence for displayed in the shaded area, the two requirements can be reduced to the condition
In Figure the shaded area represents the region of integration established by equation ((34) for a constant ky. khkyomegaheight=2.in,width=6.in. Regions of integration. The existence condition for vh in equation (34) requires the integration boundaries in equation (15) to be as follows:
After the change of variable from to in equation (19)
we need to determine the new integration boundaries. In equation (21) the new variable takes values from to , but the boundary values for kh have to be expressed now function of the new variable .Starting with the initial boundary equation (34) and squaring it we have
and replacing with its expression in we obtain
After multiplying by and grouping the terms we have
which is transformed in the condition for kh:
Therefore the second line in equation (21) should have the integration boundaries:
and subsequently equation (22) has the same integration boundaries in kh
Finally, the change of integration variable from back to from equation (24) to equation (26) will restore the initial condition for kh:
However, since in equation (26) there is an interchange in the order of integration variables, the integration boundaries become