Attenuation calculations

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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 ?


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 , we have


where the initial amplitude has decreased to after the wave travels a distance is the absorption coefficient. On the other hand, at a fixed location, the amplitude varies with time according to the equation


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


A damped harmonic wave can be written

The logarithmic decrement (“log dec”) is defined as


If is the period, equations (2.18b) and (2.18d) show that


The quality factor is another attenuation constant; it is defined by the relation

where is the fractional wave energy loss/cycle. Because is proportional, to , . Since energy loss in one period ,


( is the loss per cycle, so we have dropped the minus sign and set ). Thus, becomes



The wavelength is m. From equation (2.18a) we get

From equation (2.18e),

Equation (2.18b) shows that the amplitude ratio decreases by the same fraction over each interval , hence the decrease in the ratio from the first to the fourth cycle is


The fraction of the energy lost per cycle, , is equal to from equation (2.18f). For 3 cycles, the fractional energy loss is .

From equation (2.18e), s.

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