Difference between revisions of "Causes of high-frequency losses"

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  | isbn    = ISBN 9781560801153
 
  | isbn    = ISBN 9781560801153
 
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== Problem ==
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== Problem 6.20 ==
 
Denham’s (1980) empirical high-frequency limit is related both to the loss of high frequencies and to the dynamic range of the recording system. Reconcile this limit with losses by absorption of <math>0.15\ {\rm dB}/\lambda</math>, spreading, and high-frequency loss because of peg-leg multiples (as illustrated in Figures 6.20a,b). Take 84 dB as the dynamic range of the recording-system.
 
Denham’s (1980) empirical high-frequency limit is related both to the loss of high frequencies and to the dynamic range of the recording system. Reconcile this limit with losses by absorption of <math>0.15\ {\rm dB}/\lambda</math>, spreading, and high-frequency loss because of peg-leg multiples (as illustrated in Figures 6.20a,b). Take 84 dB as the dynamic range of the recording-system.
  
[[file:Ch06_fig6-20a.png|thumb|{{figure number|6.20a.}} Peg-leg mechanism.]]
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[[file:Ch06_fig6-20a.png|thumb|center|{{figure number|6.20a.}} Peg-leg mechanism.]]
  
 
=== Background ===
 
=== Background ===
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Denham’s relation is <math>f_{{\rm \; max\; }} =150/t</math>, where <math>f_{{\rm \; max\; }}</math>is the maximum usable frequency and <math>t</math> is the traveltime. We assume two depths 100 and 4100 m and find the average velocity <math>\bar{V}</math> for this depth interval using the South Louisiana curve in Figure 5.5b; the result is <math>\bar{V}=2600\ {\rm m/s}</math>. Thus, <math>t\approx 3.1\ {\rm s}</math> and Denham’s relation gives <math>f_{{\rm \; max\; }} =48</math> Hz.
 
Denham’s relation is <math>f_{{\rm \; max\; }} =150/t</math>, where <math>f_{{\rm \; max\; }}</math>is the maximum usable frequency and <math>t</math> is the traveltime. We assume two depths 100 and 4100 m and find the average velocity <math>\bar{V}</math> for this depth interval using the South Louisiana curve in Figure 5.5b; the result is <math>\bar{V}=2600\ {\rm m/s}</math>. Thus, <math>t\approx 3.1\ {\rm s}</math> and Denham’s relation gives <math>f_{{\rm \; max\; }} =48</math> Hz.
  
[[file:Ch06_fig6-20b.png|thumb|{{figure number|6.20b.}} Attenuation because of peg-legs (from Schoenberger and Levin, 1978).]]
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[[file:Ch06_fig6-20b.png|thumb|center|{{figure number|6.20b.}} Attenuation because of peg-legs (from Schoenberger and Levin, 1978).]]
  
 
For a frequency of 48 Hz,<math>\lambda =2600/48= 54\ {\rm m}</math>. The total distance traveled by the wave <math>=8000\ {\rm m} =148\lambda</math>. From Figure 6.20b, we take <math>0.08\ {\rm dB}/\lambda</math> as a median value of <math>\eta \lambda</math> for the loss due to peg-leg multiples, the total loss being <math>0.08 \times 148 = 12\ {\rm dB}</math>. Taking absorption losses as <math>0.15\ {\rm dB}/\lambda</math> gives 22 dB for absorption. Spreading causes the amplitude to decrease inversely as the distance and the distance ratio is 4100/100 = 41, giving a loss of <math>20 {\rm \; log\; } 41 = 32\ {\rm dB}</math>. Thus, of the total losses of 66 dB, about 20% is due to peg-leg multiples, 30% to absorption, and 50% to spreading. Before generalizing these conclusions, we must realize that spreading losses decrease rapidly with distance compared to peg-leg multiples and absorption losses.
 
For a frequency of 48 Hz,<math>\lambda =2600/48= 54\ {\rm m}</math>. The total distance traveled by the wave <math>=8000\ {\rm m} =148\lambda</math>. From Figure 6.20b, we take <math>0.08\ {\rm dB}/\lambda</math> as a median value of <math>\eta \lambda</math> for the loss due to peg-leg multiples, the total loss being <math>0.08 \times 148 = 12\ {\rm dB}</math>. Taking absorption losses as <math>0.15\ {\rm dB}/\lambda</math> gives 22 dB for absorption. Spreading causes the amplitude to decrease inversely as the distance and the distance ratio is 4100/100 = 41, giving a loss of <math>20 {\rm \; log\; } 41 = 32\ {\rm dB}</math>. Thus, of the total losses of 66 dB, about 20% is due to peg-leg multiples, 30% to absorption, and 50% to spreading. Before generalizing these conclusions, we must realize that spreading losses decrease rapidly with distance compared to peg-leg multiples and absorption losses.
 
[[file:Ch06_fig6-21a.png|thumb|{{figure number|6.21a.}} Ricker wavelet (i) in time domain and (ii) in frequency domain.]]
 
  
 
The total loss of 66 dB over the 4000 m interval is well within the 84 dB dynamic range of the system.
 
The total loss of 66 dB over the 4000 m interval is well within the 84 dB dynamic range of the system.

Latest revision as of 15:33, 8 November 2019

Problem 6.20

Denham’s (1980) empirical high-frequency limit is related both to the loss of high frequencies and to the dynamic range of the recording system. Reconcile this limit with losses by absorption of , spreading, and high-frequency loss because of peg-leg multiples (as illustrated in Figures 6.20a,b). Take 84 dB as the dynamic range of the recording-system.

Figure 6.20a.  Peg-leg mechanism.

Background

Denham’s high-frequency limit is discussed in problem 7.4 and absorption in problem 2.18. Absorption versus spreading (divergence) is the subject of problem 3.8. Spreading causes the amplitude to fall off inversely with distance. Peg-leg multiples (illustrated in Figure 6.20a,b) are described in problem 3.8.

The dynamic range of a recording system is the ratio (usually expressed in decibels) of the maximum signal that can be amplified without distortion to the background noise of the system.

Solution

Denham’s relation is , where is the maximum usable frequency and is the traveltime. We assume two depths 100 and 4100 m and find the average velocity for this depth interval using the South Louisiana curve in Figure 5.5b; the result is . Thus, and Denham’s relation gives Hz.

Figure 6.20b.  Attenuation because of peg-legs (from Schoenberger and Levin, 1978).

For a frequency of 48 Hz,. The total distance traveled by the wave . From Figure 6.20b, we take as a median value of for the loss due to peg-leg multiples, the total loss being . Taking absorption losses as gives 22 dB for absorption. Spreading causes the amplitude to decrease inversely as the distance and the distance ratio is 4100/100 = 41, giving a loss of . Thus, of the total losses of 66 dB, about 20% is due to peg-leg multiples, 30% to absorption, and 50% to spreading. Before generalizing these conclusions, we must realize that spreading losses decrease rapidly with distance compared to peg-leg multiples and absorption losses.

The total loss of 66 dB over the 4000 m interval is well within the 84 dB dynamic range of the system.

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