Horizontal resolution

**Problems in Exploration Seismology and their Solutions**
Series	Geophysical References Series
Title	Problems in Exploration Seismology and their Solutions
Author	Lloyd P. Geldart and Robert E. Sheriff
Chapter	6
Pages	181 - 220
DOI	http://dx.doi.org/10.1190/1.9781560801733
ISBN	ISBN 9781560801153
Store	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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): P_{o} and extending out to where the slant distance Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): h_{1} is such that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): h_{1} = h_{0} +\lambda /4 (see Figure 6.2a). Because Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): h_{0} \gg \lambda /4 , the Fresnel zone radius Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): R is

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} R = (h_{1}^{2} -h_{0}^{2} )^{1/2} \approx (\lambda h_{0} /2)^{1/2}. \end{align}

(6.2a)

The annular ring defined by Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): h_{1} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): h_{2} , where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): h_{2} =h_{1} +\lambda /4=h_{0} +2(\lambda /4) , is the second Fresnel zone, the outer radius being Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): R_{2} =(\lambda h_{0})^{1/2} , and so on for successive zones. The amplitude of the total reflected energy as a function of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): 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.

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,

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} R>(\lambda h_{0} /2)^{1/2} =(Vh_{0} /2f)^{1/2}. \end{align}

Solving for the frequency Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): f , we have

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} f=Vh_{0} /2R^{2} =3.8\times 3.2/(2\times 0.4^{2} )=38\ {\rm Hz}. \end{align}

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.

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Characteristics of different types of events and noise	Reflection and refraction laws and Fermat’s principle
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Horizontal resolution