Raytrace migration of normal-incidence seismograms

4.2.2 Raytrace migration of normal-incidence seismograms

Most modern migration techniques are based on wavefield concepts that treat the recorded data as a boundary value. This was not always the case; however, because raytrace migration was very popular before 1980. As is often true, raytrace methods provide a great deal of insight into the problem and can be adapted to almost any setting. Understanding raytrace migration can help considerably in understanding the corresponding wavefield technique. For example, the very word “migration”

4.2. FUNDAMENTAL MIGRATION CONCEPTS

Datum (recording surface)

∆x

∆l

Figure 4.8: Two normal-incidence rays from a dipping reflector beneath a v(z) medium. The delay between the rays allows the ray parameter to be estimated.

comes from a popular, though somewhat incorrect, view that the process just “moves events around” on a seismic section. This encourages a fundamental, and very pervasive, confusion that time and depth sections are somehow equivalent. A better view is that an unmigrated time section and a migrated depth section are independent (in fact orthogonal) representations of a wavefield. Raytrace migration emphasizes this latter view and makes the role of the velocity model very clear.

Figure 4.8 illustrates the link between reflector dip and the measurable traveltime gradient (for normal-incidence rays) on a ZOS. The figure shows two rays that have normal incidence on a dipping reflector beneath a v(z) medium. More general media are also easily handled, this just simplifies the discussion. The two rays are identical except for an extra segment for the lower ray in the bottom layer. From the geometry, this extra segment has a length ∆x sin δ where ∆x is the horizontal distance between the rays. If there were receivers on the interface between layers three and four, then for the upcoming rays, they would measure a traveltime delay, from left to right, of

where the factor of two accounts for two-way travel. This implies a horizontal traveltime gradient (horizontal slowness) of

Inspection of the geometry of Figure 4.8 shows that equation (4.26) can be rewritten as

where p n is the ray parameter of the normal-incidence ray. In the v(z) medium above the reflector, Snell’s law ensures that p n will remain equal to twice the horizontal slowness at all interfaces including the surface at z = 0. Thus, a measurement of ∆t/∆x at the surface of a normal-incidence seismogram completely specifies the raypaths provided that the velocity model is also known. In fact, using equation (4.26) allows an immediate calculation of the reflector’s dip.

Of course, in addition to the reflector dip, the spatial position’s of the reflection points of the normal rays are also required. Also, it must be expected that more general velocity distributions

88 CHAPTER 4. ELEMENTARY MIGRATION METHODS

Datum (recording surface)

Figure 4.9: The migration of a normal-incidence ray is shown. Knowledge of the ray parameter allows the ray to be traced down through the velocity model.

than v(z) will be encountered. In v(x, y, z) media, the link between reflector dip and horizontal slowness is not as direct but knowledge of one still determines the other. Normal-incidence raytrace migration provides a general algorithm that handles these complications. This migration algorithm will be described here for 2D but is easily generalized to 3D. Before presenting the algorithm, a formal definition of a pick is helpful:

A pick is defined to be a triplet of values (x 0 ,t 0 , ∆t/∆x)measured from a ZOS having the following meaning

x . . . inline coordinate at which the measurement is made. t 0 . . . normal-incidence traveltime (two way) measured at x. ∆t/∆x . . . horizontal gradient of normal-incidence traveltime measured at (x, t 0 ).

To perform a raytrace migration, the following materials are required:

1. (x i ,t 0ij , ∆t/∆x ij ) . . . a set of picks to be migrated. The picks are made at a the locations x i and at each location a number of t 0ij and ∆t/∆x ij values are measured.

2. v(x, z) . . . a velocity model is required. It must be defined for the entire range of coordinates that will be encountered by the rays.

3. Computation aids. . . one or more of: a calculator, compass, protractor, computer. Finally, the algorithm is presented as a series of steps with reference to Figure 4.9. For all

locations x i , i = 1, 2, . . . n then each ray must be taken through the following steps: Step 1 . . . Determine the emergence angle at the surface of the ij th ray according to the formula

Step 2 . . . Denote the current layer by the number h. Project the ray down to the next interface and determine the incident angle φ hij .

89 Step 3 . . . Determine the length of the ij th raypath in the current (h th ) layer, l hij and compute the

4.2. FUNDAMENTAL MIGRATION CONCEPTS

layer traveltime

Step 4 . . . Determine the refraction angle into the (h + 1) th layer from Snell’s law:

v h+1

sin(θ (h+1)ij )=

sin(φ hij )

Step 5 . . . Repeat steps 2 → 4 for each velocity layer until the stopping condition is fulfilled

which simply says that the raypath traveltime must be half the measured traveltime. Step 6 . . . Draw in a reflecting segment perpendicular to the raypath at the point that the stopping

condition is satisfied.