Attenuation calculations

**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	2
Pages	7 - 46
DOI	http://dx.doi.org/10.1190/1.9781560801733
ISBN	ISBN 9781560801153
Store	SEG Online Store

Problem 2.18

A refraction seismic wavelet assumed to be essentially harmonic with a frequency of 40 Hz is found to have amplitudes of 5.00 and 4.57 mm on traces 2500 and 3000 m from the source. Assuming a velocity of 3200 m/s, constant subsurface conditions, and ideal recording conditions, what is the ratio of the amplitudes on a given trace of the first and fourth cycles? What percentage of the energy is lost over three cycles? What is the value of the damping factor $$ h $$ ?

Background

As a wave travels through a medium, the energy of the wave is gradually absorbed by the medium. This results in attenuation of the wave, the decrease in amplitude being approximately exponential with both distance and time. For a fixed time $$ t $$ , we have

{\begin{aligned}A_{1}=A_{0}e^{-\eta x},\end{aligned}}

(2.18a)

where the initial amplitude $A_{0}$ has decreased to $A_{1}$ after the wave travels a distance $x;\eta$ is the absorption coefficient. On the other hand, at a fixed location, the amplitude varies with time according to the equation

{\begin{aligned}A_{1}=A_{0}e^{-ht},\end{aligned}}

(2.18b)

$$ h $$ being the damping factor. During a period $$ T $$ , the wave travels a distance $\lambda$ , hence equations (2.18a) and (2.18b) show that

{\begin{aligned}hT=\eta \lambda .\end{aligned}}

(2.18c)

A damped harmonic wave can be written

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} A_{1} =A_{0} e^{-ht} \cos \omega t. \end{align}

The logarithmic decrement (“log dec”) 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/":): \delta is defined 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} \delta = \ln \left(\frac{\mathrm{amplitude}}{\hbox{amplitude one cycle later}}\right). \end{align}

(2.18d)

If 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 is the period, equations (2.18b) and (2.18d) show 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/":): \begin{align} \delta=hT=h/f=2\pi h/\omega =\eta \lambda. \end{align}

(2.18e)

The quality factor 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/":): Q is another attenuation constant; it is defined by the relation

${\begin{aligned}Q=2\pi /\left(\Delta E/E\right),\end{aligned}}$

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/":): \Delta E/E is the fractional wave energy loss/cycle. 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/":): E is proportional, 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/":): A^{2} , 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/":): E=E_{0} e^{-2ht} . Since 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/":): \Delta E={} energy loss in one period $$ T $$ ,

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} \Delta E/E=\Delta \left(e^{-2ht} \right)/e^{-2ht} =2h\left(\Delta t\right)=2hT=2\delta. \end{align}

(2.18f)

(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/":): \Delta E is the loss per cycle, so we have dropped the minus sign and set 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/":): \Delta t=T ). 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/":): Q becomes

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} Q=\pi /hT=\pi /\delta. \end{align}

(2.18g)

Solution

The wavelength 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/":): \lambda =3200/40=80 m. From equation (2.18a) we get

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}\eta =\left(1/x\right)\ln \left(A_{0} /A_{1} \right)=\left(1/0.50\right) \ln \left(5.00/4.57\right)=0.180\ \mathrm{km}^{-1}. \end{align}

From equation (2.18e),

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} \log \mathrm{dec} =\delta=\eta \lambda =0.180\times 0.080=0.0144. \end{align}

Equation (2.18b) shows that the amplitude ratio decreases by the same fraction over each interval 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 , hence the decrease in the ratio from the first to the fourth cycle 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} \ln \left(A_{1} /A_{4} \right)=3hT=38=0.0432, \end{align}

so

${\begin{aligned}A_{1}/A_{4}=e^{3\delta }=e^{0.0432}=1.044,\end{aligned}}$

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} A_4 = 0.958\ A_1. \end{align}

The fraction of the energy lost per cycle, 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/":): \Delta E/E , is equal 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/":): 2\delta from equation (2.18f). For 3 cycles, the fractional energy loss 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/":): 6\delta=0.0864=8.64\% .

From equation (2.18e), 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=f\delta=40\times 0.0144=0.576 sFailed 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/":): ^{-1} .

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