Transformation to one dimension by slant stack

A rich literature (c.f. FGDP) exists on the one-dimensional model of multiple reflections. Some authors develop many facets of wave-propagation theory. Others begin from a simplified propagation model and develop many facets of information theory. These one-dimensional theories are often regarded as applicable only at zero offset. However, we will see that all other offsets can be brought into the domain of one-dimensional theory by means of slant stacking.

The way to get the timing and amplitudes of multiples to work out like vertical incidence is to stop thinking of seismograms as time functions at constant offset, and start thinking of constant Snell parameter. In a layered earth the complete raypath is constructed by summing the path in each layer. At vertical incidence $p \ = \ 0$, it is obvious that when a ray is in layer j its travel time t_j for that layer is independent of any other layers which may also be traversed on other legs of the total journey. This independence of travel time is also true for any other fixed p. But, as shown in Figure 12, it is not true for a ray whose total offset $\sum \ f_j$, instead of its p, is fixed.

 

multangle Figure 12 Rays at constant-offset (left) arrive with various angles and hence various Snell parameters. Rays with constant Snell parameter (right) arrive with various offsets. At constant p all paths have identical travel times.

Likewise, for fixed p, the horizontal distance $\ f_j \$ which a ray travels while in layer j is independent of other legs of the journey. Thus, in addition, $t_j\ +\$const$\ f_j \$ for any layer j is independent of other legs of the journey. So $t_j' \ =\ t_j \,-\,pf_j$ is a property of the $j^{\hbox{th}}$layer and has nothing to do with any other layers which may be in the total path. Given the layers that a ray crosses, you add up the t_j and the f_j for each layer, just as you would in the vertical-incidence case. Some paths are shown in Figure 13.

 

intervalve1
Figure 13 A two-layer model showing the events ( t₁ , 2 t₁ , t₂ , t₂ + t₁ ). On the top is a ray trace. On the left is the usual data gather. On the right the gather is replotted with linear moveout t' = t - pf. Plots were calculated with ( v₁ , v₂ , 1 / p) in the proportion (1,2,3). Fixing our attention on the patches where data is tangent to lines of slope p, we see that the arrival times have the vertical-incidence relationships--that is, the reverberation period is fixed, and it is the same for simple multiples as it is for peglegs. This must be so because the ray trace at the top of the figure applies precisely to those patches of the data where dt/dx = p. Furthermore, since $\delta_1=\delta_2$, the times ( t₁' , 2 t₁' , t₂' + t₂' ) also follow the familiar vertical-incidence pattern. (Gonzalez)

To see how to relate field data to slant stacks, begin by searching on a common-midpoint gather for all those patches of energy (tangency zones) where the hyperboloidal arrivals attain some particular numerical value of slope $p \ = \ dt/df$.These patches of energy seen on the surface observations each tell us where and when some ray of Snell's parameter p has hit the surface. Typical geometries and synthetic data are shown in figures 13 and 14.

Both the t_j and the t_j' behave like the times of normal-incident multiple reflections. While the lateral location of any patch unfortunately depends on the velocity model v(z), slant stacking makes the lateral location irrelevant. In principle, slant stacking could be done for many separate values of p so that the (f,t)-space would get mapped into a (p,t)-space. The nice thing about (p,t)-space is that the multiple-suppression problem decouples into many separate one-dimensional problems, one for each p-value. Not only that, but the material velocity is not needed to solve these problems. It is up to you to select from the many published methods. After suppressing the multiples you inverse slant stack. Once back in (f,t)-space you could estimate velocity and further suppress multiples using your favorite stacking method.

Figure 14 is a ``workbook'' exercise.

 

intvel2
Figure 14 The same geometry as Figure 13 but with more multiple reflections.

By picking the tops of all events on the right-hand frame and then connecting the picks with dashed lines, you should be able to verify that sea-bottom peglegs have the same interval velocity as the simple bottom multiples. The interval velocity of the sediment can be measured from the primaries. The sediment velocity can also be measured by connecting the $n^{{\rm th}}$ simple multiple with the $n^{{\rm th}}$ pegleg multiple.

Transformation to one dimension by slant stack for deconvolution is a process that lies on the border between experimental work and industrial practice. See for example Treitel et al [1982]. Its strength is that it correctly handles the angle-dependences that arise from the source-receiver geometry as well as the intrinsic angle-dependence of reflection coefficient. One of its weaknesses is that it assumes lateral homogeneity in the reverberating layer. Water is extremely homogeneous, but sediments at the water bottom can be quite inhomogeneous.