Attenuation calculations

ADVERTISEMENT
From SEG Wiki
Jump to: navigation, search

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 ?

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 , we have


(2.18a)

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


(2.18b)

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


(2.18c)

A damped harmonic wave can be written

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


(2.18d)

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


(2.18e)

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 ,


(2.18f)

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


(2.18g)

Solution

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

so

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.

Continue reading

Previous section Next section
Relation between nepers and decibels Diffraction from a half-plane
Previous chapter Next chapter
Introduction Partitioning at an interface

Table of Contents (book)

Also in this chapter

External links

find literature about
Attenuation calculations
SEG button search.png Datapages button.png GeoScienceWorld button.png OnePetro button.png Schlumberger button.png Google button.png AGI button.png