# NMO in a horizontally stratified earth

Series Investigations in Geophysics Öz Yilmaz http://dx.doi.org/10.1190/1.9781560801580 ISBN 978-1-56080-094-1 SEG Online Store

We now consider a medium composed of horizontal isovelocity layers (Figure 3.1-6). Each layer has a certain thickness that can be defined in terms of twoway zero-offset time. The layers have interval velocities (v1, v2, …, vN), where N is the number of layers. Consider the raypath from source S to depth point D, back to receiver R, associated with offset x at midpoint location M. [1] derived the traveltime equation for this path as

 ${\displaystyle t^{2}=C_{0}+C_{1}x^{2}+C_{2}x^{4}+C_{3}x^{6}+\cdots ,}$ (3)

where ${\displaystyle {{C}_{0}}={{t}_{0}}^{2},\ \ {{C}_{1}}={1}/{v_{rms}^{2}},}$ and C2, C3, … are complicated functions that depend on layer thicknesses and interval velocities (Section C.1). The rms velocity vrms down to the reflector on which depth point D is situated is defined as

 ${\displaystyle v_{rms}^{2}={\frac {1}{t_{0}}}\sum _{i=1}^{N}v_{i}^{2}\Delta \tau _{i},}$ (4a)

where Δτi is the vertical two-way time through the ith layer and ${\displaystyle {{t}_{0}}=\sum \nolimits _{i=1}^{N}{\Delta {{\tau }_{i}}.}}$ By making the small-spread approximation (offset small compared to depth), the series in equation (3) can be truncated to obtain the familiar hyperbolic form

 ${\displaystyle t^{2}=t_{0}^{2}+{\frac {x^{2}}{v_{rms}^{2}}}.}$ (4b)
Figure 3.1-1  The NMO geometry for a single horizontal reflector. The traveltime is described by a hyperbola represented by equation (1).

When equations (1) and (4b) are compared, we see that the velocity required for NMO correction for a horizontally stratified medium is equal to the rms velocity, provided the small-spread approximation is made.

How much error is caused by dropping the higher order terms in equation (3)? Figure 3.1-7a shows a CMP gather based on the velocity model in Figure 3.1-8. Traveltimes to all four reflectors were computed by the raypath integral equations [2] that exactly describe wave propagation in a horizontally layered earth model with a given interval velocity function. We now replace the layers above the second shallow event at t0 = 0.8 s with a single layer with a velocity equal to the rms velocity down to this reflector — 2264 m/s. The resulting traveltime curve, computed using equation (4b), is shown in Figure 3.1-7b. This procedure is repeated for the deeper events at t0 = 1.2 and 1.6 s as shown in Figures 3.1-7c and d. Note that the traveltime curves in Figures 3.1-7b, c, and d are perfect hyperbolas. How different are the traveltime curves in Figure 3.1-7a from these hyperbolas? After careful examination, note that the traveltimes are slightly different for the shallow events at t0 = 0.8 and 1.2 s only at large offsets, particularly beyond 3 km. By dropping the higher order terms, we approximate the reflection times in a horizontally layered earth with a small-spread hyperbola.