# Horizontal resolution

Series Geophysical References Series Problems in Exploration Seismology and their Solutions Lloyd P. Geldart and Robert E. Sheriff 6 181 - 220 http://dx.doi.org/10.1190/1.9781560801733 ISBN 9781560801153 SEG Online Store

## Question

Assume that a salt dome can be approximated by a vertical circular cylinder with a flat top of radius 400 m at a depth of 3200 m. If the average velocity above the top is 3800 m/s, what is the minimum frequency that will give a recognizable reflection from the dome?

### Background

Huygens’s principle (see problem 3.1) states that waves are reflected from all illuminated parts of a reflector, the phase varying with the two-way traveltime from source to reflecting point to receiver. Thus the receiver records energy from all points of the reflecting area, the “reflection” being the sum of all of the increments, each with a different phase.

The first Fresnel zone (often referred to as “the Fresnel zone”) is the portion of the reflector from which the reflected energy arrives more-or-less in-phase so that it adds constructively. For constant velocity, it is a circle centered at the reflecting point ${\displaystyle P_{o}}$ and extending out to where the slant distance ${\displaystyle h_{1}}$ is such that ${\displaystyle h_{1}=h_{0}+\lambda /4}$ (see Figure 6.2a). Because ${\displaystyle h_{0}\gg \lambda /4}$, the Fresnel zone radius ${\displaystyle R}$ is

 {\displaystyle {\begin{aligned}R=(h_{1}^{2}-h_{0}^{2})^{1/2}\approx (\lambda h_{0}/2)^{1/2}.\end{aligned}}} (6.2a)

The annular ring defined by ${\displaystyle h_{1}}$ and ${\displaystyle h_{2}}$, where ${\displaystyle h_{2}=h_{1}+\lambda /4=h_{0}+2(\lambda /4)}$, is the second Fresnel zone, the outer radius being ${\displaystyle R_{2}=(\lambda h_{0})^{1/2}}$, and so on for successive zones. The amplitude of the total reflected energy as a function of ${\displaystyle R}$ is plotted in Figure 6.2b (see Sheriff and Geldart, 1995, section 6.2.3 for more details). The amplitude depends mainly on the first zone, the contributions of successive pairs of the other zones effectively cancelling each other. The first Fresnel zone is usually taken as the limit of the horizontal resolution for unmigrated seismic data, reflectors smaller than this appearing almost as point diffractors.

Aliasing is discussed in problem 9.4.

Figure 6.2a.  Geometry of first Fresnel zone.
Figure 6.2b.  Amplitude versus reflector radius.

### Solution

For a recognizable reflection (as opposed to a diffraction) on an unmigrated section, the radius of the dome should be at least as large as that of the first Fresnel zone, that is,

{\displaystyle {\begin{aligned}R>(\lambda h_{0}/2)^{1/2}=(Vh_{0}/2f)^{1/2}.\end{aligned}}}

Solving for the frequency ${\displaystyle f}$, we have

{\displaystyle {\begin{aligned}f=Vh_{0}/2R^{2}=3.8\times 3.2/(2\times 0.4^{2})=38\ {\rm {Hz}}.\end{aligned}}}

For frequencies lower than 38 Hz, the top of the dome is smaller than the Fresnel zone and the reflection energy falls off so that the reflection may not be recognized as such.