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True-Amplitude Processing Techniques for 

Marine, Crustal-Reflection Seismic Data 

By MYUNG W. LEE and DEBORAH R. HUTCHINSON 

U.S. GEOLOGICAL SURVEY BULLETIN 1897 
DEPARTMENT OF THE INTERIOR 

MANUEL LUJAN, JR., Secretary 

U.S. GEOLOGICAL SURVEY 

Dallas L. Peck, Director 

Any use of trade, product, or firm names in this publication is for descriptive 

purposes only and does not imply endorsement by the U.S. Government. 

UNITED STATES GOVERNMENT PRINTING OFFICE: 1990 

For sale by the 

Books and Open-File Reports Section 

U.S. Geological Survey 

Federal Center 

Box 25425 

Denver, CO 80225 

library of Congress Cataloging-in-Publication Data 

True-amplitude processing techniques for marine, crustal-reflection seismic data, 

by Myung W. Lee and Deborah R. Hutchinson. 

p. cm.-(U.S. Geological Survey bulletin ; 1897) 

Includes bibliographical references. 

Supt. of Docs. no.: 119.3:1897 

1. Seismic reflection method-Data processing. 2. Earth-Crust. 3. Submarine 

geology-Superior, Lake. 4. Signal processing. I. Hutchinson, Deborah R. II. Title. 

Ill. Series. 

QE75.89 no. 1897 

[QE33) 

557.3 s-dc20 

(551.1'3) 89-600269 

CIP 
CONTENTS 

Abstract 1 

Introduction 1 

Acknowledgments 2 

Signal-to-noise ratio consideration 2 

Automatic trace editing 2 

Weighted stack 3 

Seismic data 4 

True-amplitude processing 5 

Comparison of the methods 10 

Post-stack adjustment 14 

Discussion 15 

Conclusions 20 

References cited 21 

Appendix 1 22 

FIGURES 

1. Graph showing signal-to-noise ratio improvement by an edited and weighted 

stacking for two-channel noise model 5 

2. Map of the Great Lakes area showing location of GLIMPCE seismic reflection lines and simplified geology 6 

3. Stacked profiles with and without true-amplitude processing 8 

4-8. Traces showing: 

4. Comparison of vertical amplitude variation with and without trueamplitude processing 9 

5. Examples of true-amplitude processing for part of line A on figure 2 

with and without post-stack amplitude adjustment 11 

6. Example of true-amplitude processing for part of line A on figure 2 by 

single-trace, root-mean-square equalization with and without post-stack 

amplitude adjustment 12 

7. Example of true-amplitude processing for part of line A on figure 2 by 

a common-depth-point gather, root-mean-square equalization with and 

without post-stack amplitude adjustment 13 

8. Comparison of vertical amplitude variation for true-amplitude-processed 

traces 14 

9. True-amplitude-processed stacked section of line Con figure 2 without 

post-stack amplitude adjustment 16 

10. Traces showing comparison of migration result for true-amplitude-processed 

section for line A on figure 2 17 

11. True-amplitude-processed section with post-stack amplitude adjustment for 

line C on figure 2 18 

12. True-amplitude-processed section with post-stack amplitude adjustment for 

line A on figure 2 19 

Contents Ill 

True-Amplitude Processing Techniques for 

Marine, Crustal-Reflection Seismic Data 

By Myung W. Lee and Deborah R. Hutchinson 

Abstract 

Preserving the relative true-amplitude of seismic traces 

provides significant advantages for deep crustal investigations, 

particularly for the identification and analysis of deep reflections 

originating near the crust-mantle boundary. Preserving latera I 

variations along deep reflections is also important for resolving 

amplitude changes due to geologic causes rather than to 

artifacts from recording or source conditions. 

Three relative true-amplitude processing methods (automatic edit, single-trace equalization, and a common-depthpoint gather equalization) were applied to the 20-second seismic data of the Great lakes Multidisciplinary Program on 

Crustal Evolution in order to evaluate the performance of the 

different techniques. Particular attention was paid to the improvement of signal-to-noise ratio and to the preservation of 

latera I continuity of the deep reflections. 

This study indicates that the automatic editing procedure based on the median amplitude of a common-depth-point 

gather provided substantial signal-to-noise ratio improvement 

of the seismic profiles. Post-stack amplitude balancing was 

commonly necessary to maintain the lateral continuity of the 

deep reflections and to reduce migration noise. The results 

from these processing strategies confirm the existence of large 

differences in crustal reflectivities and in the shape and geometry of crust-mantle transition beneath central and western lake 

Superior. 

INTRODUCTION 

Conventional seismic data processing techniques 

using automatic gain control (AGC) work well for imaging the gross structural component of upper to middle 

crustal regions from deep crustal seismic data. However, it is sometimes difficult to make visual interpretations of deeper reflections; that is, reflections from 

the lower crust and upper mantle, owing to the lack of 

amplitude contrast between signal and back-scattering 

noise. In order to differentiate signal from noise for deep 

reflections, it is advantageous to have information about 

both the relative true amplitude and lateral coherency 

of primary reflections. In contrast to AGC processing, 

relative true-amplitude (RTA) processing (sometimes 

referred to as true-amplitude (TA) processing) yields 

much more information about the relative strength of 

the reflections, thereby providing pertinent information 

about acoustic impedance and, potentially, about lithologic properties causing deep reflections. 

Manuscript approved for publication, June 7, 1989. 

The RTA processing technique has been used for 

direct hydrocarbon detection from the early 1970's, 

and Anstey (1977) provided many examples of trueamplitude sections. Recently, amplitude variation with 

offset distance has been used to characterize hydrocarbons, stratigraphic trap exploration, and reservoir 

analysis (Backus, 1987). Yu (1985) provided a detailed 

analysis of amplitude variation with offset to evaluate 

reservoir quality. 

Because of additional processing efforts (and 

therefore additional cost), TA processing has not been 

routinely applied to deep reflection data, although its 

use is becoming more common. Grow (1980) used TA 

processing to interpret two sets of anomalously high amplitude events between 6 and 7.5 s (seconds) as continental basement and possible synrift volcanic units associated with the East Coast Magnetic Anomaly. Highamplitude events on the COCORP (Consortium for 

Continental Reflection Profiling) Death Valley profile 

were interpreted as evidence of a mid-crust magma 

chamber (de Voogd and others, 1986). RTA processing has been used to estimate reflection coefficients and 

to interpret strong reflections within the upper mantle (Warner and McGeary, 1987). Recently, Mayrand 

and Milkereit (1988) analyzed the Vancouver Island 

deep crustal profiles and concluded that the amplitude anomaly near the top of the subducting Juan de 

Fuca plate cannot be explained by changing recording conditions alone, implying some significant geological cause. These examples emphasize the significance 

of true-amplitude processing both for exploration and 

deep crustal study. 

In order to preserve relative true amplitude, we 

must consider two amplitude correction procedures for 

TA processing. The first is to eliminate the propagation effect on the vertical amplitude variation, such 

as spherical divergence, transmission, and attenuation. 

The second is to minimize the lateral amplitude variation due to the source strength changes, receiver coupling variation, or near-surface inhomogeneities. In 

the present paper, we focus on preserving, rather than 

correcting, vertical and lateral amplitude variations. 

Two of the major problems in true-amplitude processing are as follows: (1) degradation of signal-to-noise 

ratio (SNR) due to anomalously high noise bursts, and 

(2) abrupt lateral amplitude changes due to the nearsurface effects. Therefore, we placed particular emphaIntroduction 1 
sis on pre-stack processing to increase SNR for stacking 

and on post-stack amplitude adjustment to suppress the 

near-surface effects. 

Three different techniques-automatic editing, 

single-trace equalization, and a common-depth-point 

(CDP) gather equalization-are discussed and applied 

to the GLIMPCE (Great Lakes Multidisciplinary Program on Crustal Evolution) seismic data. Trueamplitude processing using an automatic editing procedure was discussed by Mayrand and Milkereit (1988); 

our approach is similar to theirs. Yu (1985) discussed 

different equalization methods for true-amplitude processing; the single-trace equalization method is similar to 

the windowed-trace equalization method discussed by 

Yu. A CDP gather equalization method is also investigated here in an attempt to improve lateral continuity 

of the deep reflections. 

All three methods preserve the vertical amplitude 

variation quite satisfactorily, but lateral amplitude 

preservation is difficult to obtain. Post-stack amplitude 

adjustment is generally required in order to avoid 

abrupt lateral amplitude variation. 

In this paper, we initially take a theoretical 

approach and develop a trace model that can be used 

to evaluate SNR improvement by either an edited 

or weighted stack. A hypothetical two-trace stack 

of varying SNR provides a simple case to illustrate 

the range of possible SNR improvement. We then 

describe the G LIMPCE seismic data and the three TA 

processing techniques. Finally, we compare the results 

and discuss post-stack amplitude adjustment. 

Acknowledgments 

The data described in this paper were collected 

by Geophysical Services, Inc., and the GLIMPCE 

project was funded jointly by the Geological Survey of 

Canada and the U.S. Geological Survey. Our sincere 

appreciation is extended to all the G LIMPCE members, 

and, in particular, to William Cannon, John Behrendt, 

and Alan Green. We thank Jill McCarthy and John 

Miller for reviewing the manuscript. All data were 

processed on a VAX 11/780 computer using CogniSeis' 

DISCO software with additional programs developed by 

U.S. Geological Survey personnel. 

SIGNAL-TO-NOISE RATIO CONSIDERATION 

Improvement of SNR is always desirable for seismic data processing, and it is particularly important for 

deep crustal studies where signal strength is generally 

weak, mainly due to low-reflectivity of lower crustal material and attenuation of the seismic signal with propagation time. In order to analyze the SNR improvement 

via true-amplitude stacking, the following trace model 

is defined. 

where nj: amplitude at ith sample at jth trace in 

a CD P gather. 

Sij: signal in the jth trace assumed to be 

stationary. 

nii: noise in the jth trace assumed to be 

stationary. 

ai, /3j: scales controlling signal or noise amplitude, respectively, in the jth trace. 

(1) 

Noise is assumed to be statistically independent 

between traces that have zero mean and variance o- 2 , 

and it is also assumed to be independent of the signal. 

Furthermore, it is assumed that only the signal and 

noise strength vary within a CDP gather; but signal 

and noise statistical properties are identical. 

The SN R after N -trace stacking is given by: 

N 

s I: O:j 

SNR = i=l 

uJj~Pj (2) 

where S is the ensemble average of a signal or the 

amplitude of a signal. As mentioned in Robinson 

(1970), statistically optimum stacking can be achieved 

by equalizing the signal on each trace and weighting the 

resulting trace with the respective signal-to-noise ratio. 

Here we studied SN R improvement by an automatic trace editing quite similar to that of Mayrand 

and Milkereit (1988) and a weighted stacking similar to 

that of Robinson (1970). 

AUTOMATIC TRACE EDITING 

The main objective is to find a subset N1 that 

improves the SN R by N1-trace stacking as compared 

with the straight N-trace stacking. 

Using equation (2), we can define a modified 

SN R, (SN R)*, by: 

(3) (SNR)* = lf!i. Nt 

0" I: /3~ j=l J 

Then, we wish to find N1 such that 

(SN R)* > (SN R). (4) 

2 True-Amplitude Processing Techniques for Marine, Crustal-Reflection Seismic Data 
Mayrand and Milkereit (1988) proposed a method 

of searching N1 by comparing the stacked traces generated by different subsets. Here, we investigated the 

range of the subsets that satisfy equation 4 under the 

following simple signal or noise model. Appendix 1 

treats a more general trace model for SN R improvement. 

A. Signal model.-A signal model in a CDP 

gather is defined so that N1 trace has identical signal 

strength (S) and the remaining N2 traces, N2 = N -Nt, 

have identical signals whose strength is scaled by a 

under the same ambient noise strength. In other words, 

Iii = Sij + nii (for) j = 1, 2 ... N1 (reference traces) 

= asij + nii (for) j = N1 + 1, ... N . 

(5) 

In this signal model, equation 4 can be written as: 

The solution of equation 6, ac, is given by: 

where 

0 < ac < ~ ( ji+q- 1) 

N2 

q=-. N1 

(6) 

(7) 

In other words, under constant ambient noise, 

rejecting the trace that has signal strength less than 

! ( .;r+q- 1) of the reference signal will improve 

q 

S N R during the stacking process. 

B. Noise model.-Likewise, the noise model is 

defined by: 

Tii = Sij + nij for j = 1, ... N1 

= Sij + f3nij for j = N1 + 1, ... N (8) 

In this case, equation 4 can be written as 

(9) 

and its solution, f3c, is given by 

Equation 9 indicates that dropping the traces that 

contain noise strength greater than f3c of the reference 

trace will increase SN R more than the straight N -trace 

stacking. 

As shown in equations 7 and 9, the SN R improvement by dropping traces is a function of q, which is 

the ratio of number of rejected traces to number of retained traces during stacking. This analysis indicates 

that by dropping either the high noise trace (such as 

a noise burst) or the low-level trace (such as a weak 

source), or both, always increases the SN R. The range 

of the signal-to-noise ratio improvement will be discussed later. 

WEIGHTED STACK 

Now consider the SN R improvement for the 

weighted stack under the same model conditions shown 

in equation 1. Using equations 1 and 2, the SN R for a 

weighted stack can be written as: 

N 

S L: wiai 

(SN R)* = ---=-i;::=:=

1

== 

N 

u L: w~f3~ j=l 1 1 

where Wj is a weighting function for jth trace such that 

Wj = 1 for j = 1, 2 ... N1 

Wj = w for j = N1 + 1 ... N2 . 

A. Signal model.-ln the case of varying signal 

strength with constant ambient noise (equation 5), the 

condition for SN R of weighted stack is greater than the 

straight stack and is given by: 

1 + qa 1 + aqw --- < ---;:==== y'i"+q y'1 + w2q 

(10) 

One of the choices for the weight function can 

be derived by requiring the ensemble average of trace 

energy to be equal among a CDP gather. This choice 

of weight is given by: 

Weighted Stack 3 
w= (11) 

where 1 = u / S, the inverse of amplitude SN R for the 

reference trace. With a choice of w given in equation 

11 with f3 = 1, it can be shown that there is no solution 

for equation 10. This observation implies that in the 

signal model, weighted stack does not improve SN R 

but rather degrades SN R. 

B. Noise model.-When the noise strength 

changes while the signal strength remains constant in a 

gather, the condition of the S N R of the weighted stack 

is greater than the straight stack given by: 

1 + q 1 + wq 

../1 + qf32 < ..j1 + w2f32q . (12) 

The solution of equation 12, f3c, can be written as: 

When Wj > 1 

f3c = qw + q + 2 < 1 

2qw+w+ 1 ' (13) 

and when Wj < 1, then: f3c > 1. 

With this choice of Wj, given in equation 11, the 

solution in equation 13 is an implicit relation between 

f3c and q. It can be shown that the solution exists for all 

values of f3c· This means that the weighted stack always 

increases SN R more than the straight stack when the 

signal strength is constant throughout the CDP gather 

under variable, ambient noise. 

For the particular case in which a = f3, the 

statistically optimum stacking by Robinson (1970) is 

quite similar to this approach. 

The general behavior of S N R improvement by 

editing and weighting can be inferred from the behavior of a two-trace stacking. In the signal model, as 

shown in equation 7, dropping the trace whose signal 

strength is approximately less than 0.5 of the reference signal strength will increase the S N R, whereas 

weighted stacking will degrade the S N R. The S N R improvement by editing is about 30 percent when a = 0.1 

and the SN R improvement is independent of SN R of 

the reference trace. In the noise model, the SN R improvement can be analyzed by the SN R improvement 

factor defined by the ratio of S N R of the edited or 

weighted stack to the straight stack. 

This relation can be written as: 

v'i+J32 

qe = 2 for edited stack, 

and (14) 

(1 + w)v'i+J32 for weighted stack. 

qw = 2..j1 + w2f32 

As shown in equation 14, the SN R improvement 

for editing is independent of the S N R of the reference 

trace, whereas the SN R improvement by weighting 

depends strongly on the SN R of the reference trace. 

The relationship given in equation 14 for SN R 

improvement a of two-trace stack with various SN R of 

the reference trace ranging from 10 to 0.1 is shown in 

figure 1. In a two-channel stacking, SN R improvement 

by editing can be achieved when f3 > .J3 and an 

increasing function of f3 (curve a of fig. 1). When 

f3 ~ 2.5, the edited stack is always better than the 

weighted stack in the SN R range of 10-0.1. This 

observation is important in processing very noisy deep 

crustal data where SN R is generally small. Also, SN R 

improvement by weighted stacking is better for low 

S N R than high S N R data. 

SEISMIC DATA 

About 1,350 km of G LIMPCE seismic data were 

acquired during September 1986 with the objectives of 

better understanding the deep structure and tectonics 

of the Midcontinent rift system and the Grenville 

tectonic province (fig. 2). The energy was a 127.5-L 

(7,780-in3) tuned airgun array, and 20 seconds of data 

were recorded by a 120-channel streamer. Detailed 

descriptions of data acquisition and data processing 

with an AGC application are found in Lee and others 

(1988). Some of the geological interpretations of this 

data set are in Behrendt and others (1988) for deep 

crustal structure of the Midcontinent rift system, Green 

and others (1988) for Grenville tectonics, and Cannon 

and others (1989) for detailed rift structure. 

Most of the G LIMPCE data were shot in shallow 

water underlain by variable thicknesses of glacial deposits and Precambrian clastic, volcanic, or crystalline 

bedrock, resulting in contamination by abundant multiples and side-scattering noise. This paper does not address methods of suppressing coherent noise and these 

noises have been eliminated as much as possible prior 

to TA processing. 

4 True-Amplitude Processing Techniques for Marine, Crustal-Reflection Seismic Data 
w 

U) 

<( 

w 

a: 

5 

4 

() 3 

~ 

a: 

z 

U) 

2 

0 

0 2 3 4 5 

f3 

6 7 8 9 10 

Figure 1. Signal-to-noise ratio (SN R) improvement by an edited and weighted stacking for two-channel noise model. /3 

controls the amount of noise content, and 'Y is the inverse of amplitude SN R. /3 = 1 is a reference trace model, and increasing 

f3 indicates increasing noise. a, Edited stacking; b, weighted stack with 'Y = 10 (low SN R); c, weighted stack with 'Y = 1; d, 

weighted stack with 'Y = 0.1 {high SN R). 

All the data were processed at 24- and 30-fold (for 

shots spaced at 62.5 and 50 m, respectively), and four 

adjacent traces were vertically summed resulting in a 

50-m CDP interval. 

TRUE-AMPLITUDE PROCESSING 

Seismic amplitude decays with increasing recording time due to geometrical spreading, attenuation 

(a combination of intrinsic and apparent attenuation), 

and other propagation effects. Also, seismic amplitude 

varies laterally owing to the recording conditions in addition to the geological causes. The purpose of TA processing is to preserve the amplitude variation vertically 

as well as horizontally. In this paper, we focus on how to 

reduce the amplitude artifacts due to source or recording conditions and increased SN R. In order to preserve 

the vertical variation of relative amplitude, the amplitude decay should be corrected in a systematic manner. 

The generally accepted power law was applied to the 

GLIMPCE seismic data in order to correct this amplitude decay. The gain function we chose is a second 

power of two-way traveltime and was applied up to 18 

seconds of two-way traveltime. 

Generally, the lateral variation of deep reflections 

can be caused by the following reasons: 

1. Changes in source strength and source coupling, 

2. Changes in receiver coupling or recording condition, 

3. Near-surface inhomogeneity such as shoals or 

channels, 

4. Changes in noise levels, 

5. Different propagation and attenuation properties 

of the inhomogeneous upper crustal material, and 

6. Lateral lithologic changes of deep reflectors. 

Even though the relative contribution of items 1-

6 depends highly on local geology and can vary significantly from data set to data set, the purpose of 

true-amplitude processing is to preserve lateral lithologic changes (6) as much as possible. 

In order to do a comparative study for TA 

True-Amplitude Processing 5 
Figure 2. location of GUMPCE (Great lakes Multidisciplinary Program on Crustal Evolution) seismic reflection lines (A-C, 

F-J, 1986) on a simplified geological map of the Great lakes area, U.S.A. 

processing techniques, we attempted the following three 

methods: 

1. Automatic editing based on a median root-meansquare (RMS) amplitude in a CDP gather, 

2. Single-trace RMS amplitude equalization, and 

3. A CDP-gather RMS amplitude equalization. 

Let's define RMS amplitude Ai for the j-channel 

for a given time window as: 

for i = 11,h + 1, . .. 12 

j = 1,2, ... N 

(15) 

with Tii as the ith sample of jth trace in a CDP gather 

and h and 12 as the starting and ending time window. 

Automatic editing procedure for TA processing is 

defined here as such that traces whose RMS amplitude 

does not satisfy the following relationship are omitted 

during the stacking. The condition is: 

(16) 

where Am is the median RMS amplitude in a CDP 

gather and cl and c2 are lower and upper threshold 

constants, respectively. Consider the time window as 

an important parameter. For example, if the RMS 

amplitude analysis is done for the ambient noise part 

such as the time window before the first arrival or later 

part of the records for the deep crustal study, then we 

can choose cl = 0 and c2 as a non-zero constant, 

depending on the data set. This case is similar to the 

noise model discussed in the earlier section, in which 

we edited out high ambient noise for TA processing. 

However, if the window is designed for the signal part, 

6 True-Amplitude Processing Techniques for Marine, Crustal-Reflection Seismic Data 
for example the first arrivals, we can select C1 as some 

non-zero constant and C2 as a large number. This 

method is similar to that of the signal model discussed 

earlier in which low-signal traces are edited out for TA 

processing. If the window contains both the signal part 

and noise part, we can use non-zero constants of C1 and 

c2. 

If the trace model discussed previously (equations 

5 and 8) can be applied to the median amplitudebased automatic editing procedure, we can write the 

automatic editing procedures using equations 7 and 9 

as: 

and when q is quite small, it can be approximated by 

The justification of the editing based on the RMS 

amplitude is as follows. One basic assumption for this 

approach is that the time average of seismic traces is 

close to the ensemble averages of the seismic traces. 

In other words, Ai is close to the ensemble average 

of the trace model shown in equation 1. Lee (1960) 

provided an excellent discussion on the justification for 

the interchange of time and ensemble averages, and 

Robinson (1970) used this property for his investigation 

of optimum stacking. 

Thus, for the purpose of automatic editing, we 

treat the median RMS amplitude Am as the ensemble 

average of a reference trace model shown in equation 

5 or equation 8. Therefore, under the assumption 

of small q, dropping the traces whose amplitude is 

approximately less than 0.5 Am corresponds to editing 

out low-signal components in a CDP gather, possibly 

caused by a weak source strength. Also, dropping 

the traces whose amplitude is approximately greater 

than J2Am corresponds to editing out the high-noise 

component in a CDP gather such as a noise burst. 

Suppressing the high noise component of the data is 

the most important procedure in removing the outliers 

(high episodic random noise burst) in TA processing. 

In summary, in the automatic editing method, 

lateral amplitude preservation will be accomplished 

through the assumption that the median RMS amplitude in a CDP gather is not affected significantly by 

the source (or recording) condition, noise contamination, or near-surface inhomogeneity; and SN R will improve through dropping out high-noise and low-signal 

traces. 

An example ofTA processing of GLIMPCE line C 

in Lake Superior by the automatic editing procedure is 

shown in figure 3. One part (fig. 3A) shows the stacked 

profile without any editing and the other (fig. 38) shows 

the same section after the automatic editing procedure 

(C1 = 0.5, C2 = 1.45), and a time window of 8-14 s. 

During this editing, a total of about 15 percent of the 

traces were edited out. A noise burst (N B) in figure 

3A has been mostly removed after editing. This kind of 

NB problem is a common nuisance to all TA processing 

techniques to a varying degree. Figure 38 indicates 

the SN R improvement and the better lateral continuity 

of the reflections, although subtle, by the automatic 

editing procedure. 

The RMS amplitude variation with two-way traveltime for differently stacked traces is shown in figure 

4. In this example, the RMS amplitude was computed 

using a window of 40 ms and applying a five-sample 

median filter (200 ms) before plotting. The heavy line 

in figure 4A represents the RMS amplitude of trace 60 

shown in figure 3A (no edit), and the light line represents the RMS amplitude of trace 60 in figure 38 

(edit). Also, the light line near 0 dB (decibel) represents the amplitude difference between the heavy and 

light RMS amplitude curve. Figure 48 shows the same 

result except that the editing was done using cl = 0.5 

and c2 = 2.0. 

Some interesting observations can be made from 

data shown in figures 3 and 4: 

1. The positive large difference (on the order of 

10 dB) before the onset time on figure 3A indicates that 

automatic edit using c2 = 1.45 reduced the ambient 

noise level quite a lot. 

2. The negative difference on the order of 3 dB 

between 11"" 13 seconds in figure 4A implies the signal 

enhancement of deep reflections, which occurs in the 

vicinity of the crust-mantle transition described by 

Behrendt and others (1988). 

3. These empirical results confirm the theoretical 

prediction that SN R improves by editing. They also 

support the prediction of c2 ~ V2 for small q. 

4. In general, S N R improvement increases with 

increasing time. This observation also confirms the 

previous analysis that the SN R improvement by editing 

is greater for large noise content, because SN R is 

decreasing with increasing traveltime. 

In addition to the overall SN R improvement, 

the amplitude increase on the order of 3 dB for the 

crust-mantle transition zone (shown as E1 and E2 in 

figure 4A could be significant for quantitative analysis 

of the origin of deep reflections. We emphasize that 

the preceding observations and remarks will vary from 

data set to data set. However, the general behavior of 

automatic editing procedure based on simple noise or 

signal model is established. 

True-Amplitude Processing 7 
Trace 

8 

10 

(/) 

Q 

z 

0 

u 

w 

(/) 

z 

u..i 

~ 

~ 

12 

14 

2KM--~ 

20 40 60 80 20 40 60 80 

Figure 3. A comparison of true-amplitude processing. A, Processing without automatic editing; 8, processing with 

automatic editing. Editing parameters are C1 = 0.5, C2 = 1.45, and 8-14 seconds time window. KM, kilometer; NB, noise 

burst. 

Also, TA processing by editing the traces as described is not an optimum processing method as far as 

SN R improvement is concerned. The SN R improvement by editing depends highly on the threshold constants ae or f3e· Because the noise or signal statistics 

of the real data in a CDP gather is much more complex than the simple model shown in equation 1 and 

the trace statistics vary among CDP gathers, the optimum selection of ae or f3e cannot be achieved unless 

some trial-and-error method is performed at every CDP 

gather. Our methods simply illustrate that the SN R 

improves by either editing out l9w-signal or high-noise 

traces. 

Single-trace RMS amplitude balancing can be 

done by replacing nj by iij' which is given by: 

(18) 

where R is some reference amplitude. 

If the ensemble average of a trace is very close 

to time average, f:. is similar to the weighting factor 1 

defined in equation 11. This process is similar to 

windowed-trace equalization by Yu (1985). The basic 

assumption of our single-trace balancing method is that 

8 True-Amplitude Processing Techniques for Marine, Crustal-Reflection Seismic Data 
0 

-10 

-20 

-30 

CJ) 

....J 

UJ 

co 

-40 

u 

UJ 

0 

z -50 

w 0 0 

::::> 

!= 

....J 

0.. 

~ -10 

<X: 

-20 

-30 

-40 

B 

-50 

0 2 4 6 8 10 12 

TIME, IN SECONDS 

14 16 18 20 

Figure 4. Comparison of vertical amplitude variation of true-amplitude-processed traces. Heavy line indicates 

root-mean-square amplitude without editing, and light line indicates root-mean-square amplitude with editing. 

Light line near 0 decibel denotes amplitude differences between heavy and light lines. A, Editing with C1 = 0.5, 

c2 = 1.45, and 8-14 seconds time window; 8, editing with cl = 0.5, c2 = 2.0, and 8-14 seconds time 

window. E1 and E2 represent large amplitude reflections in the lower crust. 

the effect of amplitude variation in a CDP gather caused 

by non-geological effect can be suppressed by balancing 

the RMS amplitude of each trace. 

SN R improvement of single-trace RMS equalization can be analyzed based on S N R improvement of 

the weighted stack mentioned previously. If the signal 

strength is somewhat constant in a gather and noise 

strength varies among the traces, single-trace RMS 

equalization always improves SN R compared to the 

straight stack. On the other hand, when signal strength 

changes in uniform ambient noise level, this process always degrades SN R. If the RMS amplitude level within 

a given time window is constant in all traces, irrespective of source or receiver location, this process could 

True-Amplitude Processing 9 
also preserve the lateral amplitude variation. 

The third method, which uses the RMS amplitude 

equalization of a CDP gather, is considered in an 

attempt to preserve the lateral amplitude variation. 

The method replaces 7i; by rfi;, that is: 

(19) 

where 

and R is constant. 

As can be seen from equation 19, there is no SN R 

improvement during stacking (same as the straight 

sum), but the RMS amplitude level between stack traces 

is somewhat balanced. The idea behind this approach is 

the assumption that the average total power of a CDP 

gather in a given large time window is constant if there 

is no change in recording or source conditions. 

COMPARISON OF THE METHODS 

The TA stacked section of a part of line A 

(reflection line A on figure 2) with an automatic edit 

is shown in figure 5A. Edit parameters are C1 = 0.5, 

C2 = 2.0, and 8-14 s time window. The choice of 

C2 = 2.0 is conservative and is chosen mainly to remove 

the effect of high-amplitude noise burst. Amplitude loss 

in the middle of the section (trace 50-80) below 4-5 sis 

accompanied by high-amplitude coherent noise in the 

top 0-4 s that is possibly caused from the irregular 

water bottom. For this dim zone, lateral changes in 

the strength and continuity of reflections below 4-5 s 

represents artifacts of near-surface noise effects rather 

than actual subsurface changes in the reflectivity. 

The same result with a single-trace RMS amplitude equalization using a time window of 5-20 s is 

shown in figure 6A. Similar dim spots are observed, and 

the near-surface effect on the lateral continuity of reflections is more pronounced in figure 6A than in figure 

5A. 

The stacked profile using a CDP-gather equalization and a time window of 8-20 s is shown in figure 7 A. 

The result also shows a dim spot similar to that observed in figures 5A and 6A. In addition to dim spots, 

figure 7 A shows the contamination of an outlier (shown 

as an arrow near trace 120). The whole trace shown by 

the arrow looks like a dead trace, but a large-scale plot 

indicates that this outlier is an isolated high-amplitude 

noise burst (amplitude highly reduced due to equalization) around 14 s. This kind of noise presents a 

problem during the migration process, creating a migration smile. Also notice the reduced amplitude of the 

right-dipping events near 15 s compared with the events 

shown in figure 5A. 

The dim spot below 4-5 s at traces 50-80 is a 

typical example of deterioration of the lateral continuity 

of deep reflections owing to a dramatic change in SN R 

caused by near-surface inhomogeneities. On the basis of 

the results of the three methods, we interpret that this 

abrupt SN R change is due to both a loss of transmitted 

signal and an increase of coherent noise rather than to 

an increase of ambient noise level. 

The vertical amplitude variation with time for 

four consecutive stacked traces of part of line C is shown 

in figure 8. Each dot represents each RMS amplitude 

computed within a 40-ms time window, and the solid 

line denotes five sample (200 ms) median RMS amplitudes of the mean of the four traces. The amplitude 

variation of a stacked trace using an automatic edit process and cl = 0.5, c2 = 2.0, and 8-16 s time window is 

shown in figure 8A. The amplitude variation for a CDPgather equalization is shown in figure 8 B. Both figures 

show a great deal of scattering, but their mean values 

of four consecutive traces are similar. Similar amplitude variations are observed for the single-trace RMS 

equalization method. 

We examined about 700 km of TA-processed 

G LIMPCE data (lines A, C, and G, and part of lines F, 

B, H, and J; see figure 2). The performance of the TA 

methods, on the basis of the 700-km GLIMPCE data, 

is as follows. 

1. All three methods preserve the vertical amplitude 

variation except for a few traces contaminated by 

high-amplitude episodic events. For both amplitude 

equalization methods, the time window should be large 

in order to have better statistics, and the window should 

avoid upper crustal regions (approximately less than 

5 s) because of the significant lateral inhomogeneity of 

the upper crust and coherent noise contamination. 

2. All three methods failed to preserve true lateralamplitude variation of deep reflections to a different degree and manner. The performance varies from line to 

line and from location to location. Generally, the failure to preserve lateral continuity comes from the data 

itself and not from the technique used. This conclusion 

necessitates a need for a post-stack amplitude adjustment. 

3. The automatic editing technique provides an overall SN R improvement for the deep reflections (greater 

10 True-Amplitude Processing Techniques for Marine, Crustal-Reflection Seismic Data 
I 

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0 

5 

(/) 

0 

z 

0 

u 

w 

(/) 10 z 

u.i 

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i= 

15 

20 

---12KM---

120 80 40 120 80 40 

Figure 6. An example of true-amplitude processing for a part of line A (reflection line A on fig. 2; see same part of line 

on fig. 5) by a single-trace, root-mean-square equalization using 5-20 seconds time window. Every third trace was plotted 

for clarity. A, Without post-stack amplitude adjustment; 8, with post-stack amplitude adjustment using 15-20 seconds 

time window. KM, kilometer. 

12 True-Amplitude Processing Techniques for Marine, Crustal-Reflection Seismic Data 
T 

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co 

u 

w A 0 

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<t: -10 .... 

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... ..... . · ... , . .. ._.; 

-30 "···· .. 

. ... .. .._ 

-40 .. . .,., '· .. 

B 

-50 

0 2 4 6 8 10 12 14 16 18 20 

TIME, IN SECONDS 

Figure 8. Comparison of vertical amplitude variation for true-amplitude (TA) processed traces. Each dot represents a rootmean-square (RMS) amplitude of 40-ms time window. Amplitudes of adjacent four traces are plotted. A solid line represents 

a five-sample median amplitude of a mean of the consecutive four traces. A, TA processing using an automatic edit; 8, 

TA processing using a common-depth-point-gather RMS equalization. E1 and E2 represent large-amplitude reflections in the 

lower crust. 

than approximately 8 s), and a CDP-gather equalization presents more difficulties in handling outliers. 

POST-STACK ADJUSTMENT 

As observed previously, the lateral continuity of 

the deep reflections is difficult to preserve and 

the degree of lateral amplitude preservation is datadependent. During the GLIMPCE data acquisition, 

the 60-airgun array performed quite satisfactorily, and 

array volume was nearly constant throughout the survey. Also, there were no obvious problems related 

to the streamer except for the reduced buoyancy of 

14 True-Amplitude Processing Techniques for Marine, Crustal-Reflection Seismic Data 
the fresh water in the lakes. Thus, the lateral amplitude 

variation of the deep reflections, such as those shown 

in figures 5-7, is probably not caused by changes in 

source strength or receiver coupling. We found three 

key elements for the deterioration of lateral continuity. 

1. High impedance near surface inhomogeneities 

(such as channels and shoals) commonly prevents the 

transmission of seismic signal and generates abundant 

coherent noise. This kind of example is shown in fig-· 

ures 5A, 6A, or 7 A near trace 60. We call this a "type 

1" effect. 

2. Substantial changes within the near-surface geology affect the transmission of the seismic signal dramatically ("type 2" effect). An example of this kind of 

problem is shown in figure 9. Cannon and others (1989) 

interpreted the Douglas fault, near trace 1200 in figure 

9, as a major fault bounding the rift basin within Lake 

Superior. The amplitude characteristics north of this 

fault are remarkably different from those south of it; 

notice how traces 400-1200 are dim relative to traces 

1200-2180. The RMS amplitude within the upper 1 s 

does not change much across the fault. However, the 

RMS amplitude below 2-3 s has an average difference 

of about 10 dB. This amplitude change across the fault 

is observed in the stacked traces by automatic editing, 

the median amplitude of the CDP gathers, and the near 

traces. For this problem (fig. 9), RMS amplitude equalization techniques worked better than the editing technique. We interpreted this amplitude variation to be 

caused by the changes of transmission characteristics of 

the shallow upper layers and not from changes in the 

data acquisition. This kind of lateral amplitude change 

should be corrected also. 

3. Strong lateral heterogeneity of upper crustal material affects the amplitude of the deep reflections. The 

discontinuous appearance of deep reflections shown under the brace in figure 9 reflects the lateral heterogeneity 

of the upper crustal material. We call this propagation 

effect a "type 3" effect. 

The migration method partially corrects for type 3 

effect, but type 1 and 2 effects should be corrected by 

some other method in order to investigate the deep reflections. 

We propose an RMS amplitude adjustment after 

stack as a quick solution. The amplitude adjustment 

factor ( 6) is defined as 

R 

6 =A~ J 

(20) 

where R is a constant, A; is the RMS amplitude of 

the stacked trace within a specified time gate, and n 

is an exponential constant that controls the degree of 

adjustment. Notice that this amplitude adjustment 

does not alter the vertical amplitude variation. 

Figures 5 B, 6 B, and 7 B show the after-stack amplitude adjustment of figures 5A, 6A, and 7 A, respectively, in which n = 1 and time window is 15-20 s. 

Without any doubt, post-stack amplitude adjustment 

produces a better and more geologically sound seismic 

section. Another advantage of post-stack amplitude adjustment can be found in the migration process. Warner 

(1987) mentioned some of the problems associated with 

migrating deep crustal seismic data, that is, synformal migration artifacts (smiles). He attributed most 

of the artifacts to an abrupt amplitude change caused 

by structures in the top few kilometers of the surface. 

Thus, balancing the amplitude variation in a reasonable 

way is important in order to reduce migration artifacts. 

The result of migration of data stacked by an automatic edit without post-stack amplitude adjustment 

is shown in figure lOA, and the migration result with 

a post-stack amplitude adjustment is shown in figure 

108. Part of the unmigrated section is shown in figure 

5. The result indicates that post-stack amplitude adjustment not only reduces migration noise but also increases overall S N R. Because migration artifacts tend 

to increase with increasing two-way traveltime, reducing abrupt lateral amplitude changes by post-stack amplitude adjustment is extremely important f~r imaging 

deep crustal reflections. 

The selection for the post-stack RMS window is 

more critical than the pre-stack RMS window for TA 

processing. Our experience with the GLIMPCE seismic 

data indicates that the time window should not include 

the most heterogeneous upper crustal region (less than 

about 5 s). The window that worked best was 15-20 s, 

in which coherent back-scattering energy dominated the 

section. 

DISCUSSION 

Examples ofTA-processed GLIMPCE deep crustal 

profiles using the automatic editing procedure are 

shown in figures 11 and 12 (line C and line A, fig. 2, 

respectively). We will not discuss the detailed geological implications of these profiles, but rather the TA 

techniques and the usefulness of TA processing for deep 

crustal study. 

Post-stack amplitude adjustment was applied to 

both lines C and A (fig. 2) using a 15- to 20-s window. 

Neither line is migrated. Figure 9 is identical to figure 

11 except that no post-stack amplitude adjustment has 

Discussion 15 
1-

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<( 

u. 

Cf) 

<( 

_J 

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::::> 

0 

0 

l 

z 

0 

0 

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C\1 

0 

0 

<0 

T"'"' 

0 

0 

co 

0 

0 

~ 

a.. 

0 

(.) 

0 LO 0 

..-

SONOJ3S Nl '3JAII.l 

16 True-Amplitude Processing Techniques for Marine, Crustal-Reflection Seismic Data 

LO 

..- 0 

C\1 

.s::; 

... 

0 

c:: 

c:: 

0 

E c::n E 

Q1 8 

= 

b.O~ . ·- Cl u..u 
(fJ 

0 

z 

0 

u 

w 

(fJ 

z 

u.i 

~ 

i= 

COP 

0 

5 

10 

15 

!-1oKM-j 

2000 1800 1600 2000 1800 1600 

Figure 10. Comparison of migration result for true-amplitude-processed section for line A (reflection line A, fig. 2). A, Migration 

without post-stack amplitude adjustment; every fourth trace was plotted for clarity. 8, Migration with post-stack amplitude 

adjustment. KM, kilometer; CDP, common depth point. 

Discussion 17 
-, 

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zo () 

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,... 

SONOJ3S Nl '311\11.1 

18 True-Amplitude Processing Techniques for Marine, Crustal-Reflection Seismic Data 

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Discussion 19 
been applied. Comparing these two sections, we can 

conclude that post-stack amplitude adjustment compensated for the type 2 effect reasonably well. On the 

lower part of line C, the dim zone from trace 400-

1200 in the section without post-stack amplitude adjustment (fig. 9) is nearly eliminated in the section with 

it (fig. 11). Some dim spots still remain throughout line 

C and are probably related to the type 3 effects-the 

propagation effect-rather than to the type 1 or type 2 

effects. 

The type 1 effect is easily recognized owing to its 

association with a localized and short duration (in the 

range of 5 s) amplitude burst. Recognizing the type 

2 effect, however, needs a detailed analysis. Only line 

C clearly shows the type 2 effect. All other lines are 

contaminated mostly by the type 1 effect. 

Both figures 11 and 12 indicate that the proposed 

post-stack amplitude adjustment is not an ideal solution 

for preserving lateral continuity of deep reflections, but 

that it works reasonably well in reducing type 1 and 2 

effects on the deep reflections. 

One of the important aspects of the TA section 

is its ability to identify deep reflections based on 

the strength of the reflection in addition to lateral 

coherency. The banded reflections shown in figure 11 

within the brackets lie above the inferred upper mantle 

(M in figure 11) and have been interpreted as reflections 

from the crust-mantle transition zone (Behrendt and 

others, 1988). The amplitude contrast shown in figure 

11 and arrival times of these reflections support this 

interpretation quite well. Detailed vertical amplitude 

analysis shown in figure 8 indicates that an average 

amplitude of reflections within this zone is more than 

6 dB higher than that of back-scattering noise in the 

upper mantle. The amplitude analysis indicates two 

distinct reflection arrivals (E1 and E2 in figure 8), 

and these reflections are similar to the double band of 

reflections at the base of the crust noted elsewhere in 

the continental U.S. (Klemperer and others, 1986). The 

reflection strength of E1 or E2 can be estimated from 

the TA-processed section. On the basis of amplitude 

variation shown in figure 8 and water-bottom reflection 

coefficient (about 0.6), the reflection coefficient of the 

E1 and E2 reflections are estimated to be in the range 

of 0.05. This reflection coefficient was estimated near 

CDP 890 shown in figure 11. This reflection strength 

is somewhat less than that calculated for bright spots 

within the crust (de Voogd and others, 1986) and 

upper mantle (Warner and McGeary, 1987). Reflection 

coefficients vary along the crustal-mantle transition and 

are estimated to be about 0.1 near CDP 700 of line 

C. These reflection coefficients, taken in conjunction 

with velocities determined from coincident refraction 

data recorded during the GLIMPCE experiment (fig. 2), 

will provide a critical constraint in interpreting possible 

lithologic boundaries at the crust-mantle transition. 

In figure 12, M indicates reflections near the 

Moho discontinuity interpreted by Behrendt and others 

(1988). The average amplitude of the interpreted 

reflection Moho near CDP 3800 is about 10 dB higher 

than the background noise level, and the amplitude 

contrast between reflection M and the background noise 

supports the interpretation that M is a real reflection 

and could be at or near the Moho. Also, the high 

amplitude anomaly in the lower crust represented by 

B in figure 12 could be due to actual lithologic changes 

rather than to processing or recording artifacts. 

Another use of TA processing is to investigate 

variations in crustal reflectivity. Figures 11 and 12 are 

excellent examples of differences in crustal reflectivity. 

Observe the high reflectivity of the middle crust between 6-11 seconds in figure 12 and the general lack 

of reflectivity in that region in much of figure 11. The 

Moho is considerably more disruptive, discontinuous, 

and complicated on figure 12 compared with its banded 

linear geometry in much of figure 11. This kind of reflectivity of the crust provides significant information for 

understanding and comparing different types of crust. 

These are only a few examples of the advantages of TA 

processing for deep crustal studies. We presented three 

different methods for TA processing. In TA processing, high-noise bursts outside the time window for either editing or balancing a trace commonly degrade the 

overall quality of the seismic profile. In order to circumvent this kind of problem, we suggest some kind of 

surgical mute, such as automatic surgical blanking of 

burst noise by Bernie (1987), be applied to the trace 

before TA processing. 

CONCLUSIONS 

In summary, we can make the following conclusions based on the TA processing results from 

GLIMPCE seismic data: 

1. All three methods can preserve the vertical amplitude variations for the deep reflections if the window 

for amplitude balancing is large and avoids the most 

heterogeneous upper crustal region. 

2. Automatic editing procedure based on using the 

median RMS amplitude provided the best overall improvement of SN R. 

3. All three methods generally failed to preserve true 

lateral amplitude variations of the deeper reflections. 

These variations are probably caused by changes in the 

transmission and attenuation properties of the nearsurface and upper crustal material rather than by 

shortcomings of the editing or equalization techniques. 

20 True-Amplitude Processing Techniques for Marine, Crustal-Reflection Seismic Data 
4. A post-stack amplitude adjustment is necessary 

in order to preserve the lateral variation of reflection 

amplitude and to reduce migration artifacts. Careful 

choice of the time window for post-stack adjustment is 

extremely important. 

5. TA processing provides significant information 

about deep crustal seismic reflections. In addition 

to the conventional AGC processing, we recommend 

processing all deep crustal seismic data in such a 

way that the reflection amplitude is preserved both 

vertically and laterally. 

REFERENCES CITED 

Anstey, N.J., 1977, Seismic interpretation-The physical 

aspects: Boston, Mass., International Human Resources 

Development Corp., 637 p. 

Backus, Milo M., 1987, Amplitude versus offset-A review: 

57th Annual International Meeting and Exposition, 

Society of Exploration Geophysicists, New Orleans, 

expanded abstract, p. 359-364. 

Behrendt, J.C., Green, A.G., Cannon, W.F., Hutchinson, 

D.R., Lee, M.W., Milkereit, B., Agena, W.F., and 

Spencer, C., 1988, Results from GLIMPCE deep seismic 

reflection profiles over the mid-continent rift system: 

Geology, v. 16, p. 81-85. 

Bernie, A.J., 1987, Automatic surgical blanking of burst 

noise in marine seismic data: 57th Annual International 

Meeting and Exposition, Society of Exploration Geophysicists, New Orleans, expanded abstract, p. 477-478. 

Cannon, W.F., Green, A.G., Hutchinson, D.R., Lee, M.W., 

Milkereit, B., Behrendt, J.C., Halls, H.C., Green 

J.C., Dickas, A.B., Morey, G.B., Sutcliffe, R., and 

Hinze, W.J ., 1989, The Midcontinent rift beneath Lake 

Superior-New views from GLIMPCE seismic reflection profiling: Tectonics, v. 8, p. 305-332. 

Green, A.G., Milkereit, B., Davidson, A., Spencer, C., 

Hutchinson, D.R., Cannon, W.F., Lee, M.W., Agena, 

W.F., Behrendt, J.C., and Hinze, W.J., 1988, Crustal 

structure of the Grenville front and adjacent terranes: 

Geology, v. 16, p. 788-792. 

Grow, J.A., 1980, Deep structure and evolution of the 

Baltimore Canyon trough in the vicinity of the COST 

no. B-3 well, United States mid-Atlantic continental 

slope area: U.S. Geological Survey Circular 833, p. 117-

125. 

Klemperer, S.L., House, T.A., Hansen, E.C., Oliver, J.E., 

and Potter, C.G., 1986, The Moho in northern Basin 

and Range Province, Nevada, along the COCORP 

40°N seismic-reflection transect: Geological Society of 

America Bulletin, v. 97, p. 603-618. 

Lee, M.W., Agena, W.F., and Hutchinson, D.R., 1988, 

Processing of the GLIMPCE multichannel seismic data: 

U.S. Geological Survey Open-File Report 88-225, 46 p. 

Lee, Y.N ., 1960, Statistical theory of communication: London, John Wiley and Sons, 509 p. 

Mayrand, L.J., and Milkereit, B., 1988, Automated editing 

and true amplitude stacking of seismic data: Canadian 

Journal of Earth Science, v. 25, p. 1811-1823. 

Robinson, J.C., 1970, Statistically optimal stacking of seismic data: Geophysics, v. 35, p. 436-446. 

de Voogd, B., Serpa, L., Brown, L., Hansen, E., Kaufman, S., 

Oliver, J., Troxel, B.W., Willemin, J., and Wright L.A., 

1986, Death Valley bright spot-A midcrust magma 

body in the southern Great Basin, California: Geology, 

v. 14, p. 64-67. 

Warner, Mike, 1987, Migration-Why doesn't it work for 

deep continental data?: Geophysical Journal of Royal 

Astronomical Society, v. 89, p. 21-26. 

Warner, M., and McGeary, S., 1987, Seismic reflection coefficients from mantle fault zones: Geophysical Journal 

of Royal Astronomical Society, v. 89, p. 223-230. 

Yu, G., 1985, Offset-amplitude variation and controlledamplitude processing: Geophysics, v. 50, p. 2699-2708. 

References Cited 21 
APPENDIX 1. 

The condition of SN R improvement by an automatic editing process for a general trace model shown 

in equation 1 can be derived as follows. 

Let 

Nt 

L O:j = iimNl ' 

i=l 

N L O:j = iicN2 , 

i=N1+l 

N1 

""""" 2 -2 L-J {3i = /3mN1 ' 

i=l 

N 

""""" 2 -2 L-J {3i = f3c N2 . 

Then, equation 4 can be written as 

(A -1) 

(A- 2) 

A. Signal model.-Let /3i = 1 in equation A-1, 

then equation A-2 can be written as 

(A- 3) 

If the distribution of o:i is the same as in equation 5, then equation A-3 is identical to equation 7. 

B. Noise modeL-Letting O:j = 1 in equation 

A-1, we can show that: 

(A- 4) 

where Pc or /3m is a square root of an average of !3]. If 

the distribution /3i is the same as in equation 8, equation 

A-4 is identical to equation 9. 

Thus, using the average of a: or f3 defined in A-1, 

we can have an interpretation similar to that discussed 

in the main text. 

22 True-Amplitude Processing Techniques for Marine, Crustal-Reflection Seismic Data 

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