Proof of the ABC refraction equation

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	11
Pages	415 - 468
DOI	http://dx.doi.org/10.1190/1.9781560801733
ISBN	ISBN 9781560801153
Store	SEG Online Store

Problem 11.4

Prove the ABC refraction equation [equation (11.4a)].

Background

The ABC equation is often used to calculate the weathering thickness. Assuming reversed profiles as shown in Figure 11.4a and writing 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/":): t_{AC} , 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/":): t_{BC} for the traveltimes from the sources to a geophone at 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/":): C 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/":): t_{AB} for the traveltime from 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/":): A to 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/":): B , the ABC equation gives the depth 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_{C} as

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} h_{C} =\frac{1}{2} \left(t_{AC} +t_{BC} -t_{AB} \right)\left[V_{1} V_{2} /\left(V_{2}^{2} -V_{1}^{2}\right)^{1/2} \right]. \end{align}

(11.4a)

Solution

Assuming 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/":): A , 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/":): B , 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/":): C are coplanar and that elevation corrections have been made, we can write

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} V_{1} \left(t_{AC} +t_{BC} -t_{AB} \right) &=V_{1} \left(t_{MC} +t_{NC} -t_{MN} \right)=2V_{1} t_{MC} -V_{1} t_{MN} \\ &=2h_{C} /\cos \theta _{c} -MN\left(V_{1} /V_{2} \right)\\ &=2h_{C} /\cos \theta _{c} -\left(2h_{C} \tan \theta _{c} \right)\sin \theta _{c} \\ &=2h_{C} /\cos \theta _{c} -\left(2h_{C} \sin^{2} \theta _{c} /\cos \theta _{c} \right)\\ &=\left(2h_{C} /\cos \theta _{c} \right) (1 - \sin^{2} \theta _{c}) = 2h_{C} \cos \theta _{c}. \end{align}

Figure 11.4a.  The ABC method.

Thus,

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} h_{C} &=\frac{1}{2} \left(t_{AC} +t_{BC} -t_{AB} \right)V_{1} /\cos \theta _{c}\\ &=\frac{1}{2} \left(t_{AC} +t_{BC} -t_{AB} \right)V_{1} /[1-(V_{1} /V_{2} )^{2} ]^{1/2} \\ &=\frac{1}{2} \left(t_{AC} +t_{BC} -t_{AB} \right)V_{1} V_{2} /[V_{2}^{2} -V_{1}^{2} ]^{1/2}. \end{align}

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Table of Contents (book)

Also in this chapter

• Salt lead time as a function of depth
• Effect of assumptions on refraction interpretation
• Effect of a hidden layer
• Proof of the ABC refraction equation
• Adachi’s method
• Refraction interpretation by stripping
• Proof of a generalized reciprocal method relation
• Delay time
• Barry’s delay-time refraction interpretation method
• Parallelism of half-intercept and delay-time curves
• Wyrobek’s refraction interpretation method
• Properties of a coincident-time curve
• Interpretation by the plus-minus method
• Comparison of refraction interpretation methods
• Feasibility of mapping a horizon using head waves
• Refraction blind spot
• Interpreting marine refraction data

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