Amplitude Correction

Seismic amplitude correction has an unachievable, idealized

goal. The correction strives to provide a seismic section in

which the seismic amplitudes accurately portray the values

of the reflection coefficients. Under a more modest goal, but

still rarely achieved goal, the displayed amplitudes are

locally proportional to the reflection coefficients’ values.

The first section reveals why these are difficult goals and the

later sections show examples of amplitude correction

algorithms.

What causes the seismic amplitude problem?

The real world is a messy place. A multitude of phenomena

determines the recorded seismic amplitudes. Figure 1

summarizes many of the phenomena that determine the

recorded amplitudes. The following paragraphs review these

illustrated phenomena.

Figure 1: Factors that alter amplitudes. After (Sheriff,

1975)

Geometric Dilution Through the application of the conservation of total flux

(energy), the further the observation point is from the source,

the weaker the appearance of the source. The more distant

you are from a light bulb, the dimmer it appears. In a

constant-velocity world, the amplitude decreases linearly

with distance as the power decreases with the square of the

distance.

In the world of velocity gradients and directional sources,

the actual geometric fall-off depends upon the velocity

gradient.

Absorption There are different hypotheses for the origin of the

absorption of sound waves in the Earth. Prominent

hypotheses include the role of the movement of fluids in the

pores, the friction of microfractures and scattering. It is

difficult to confirm the relative influence of different

mechanisms because laboratory measurements operate at a

case, “close” means separated by a distance equal to the

characteristic wavelength of the wavefront. A later chapter

explaining the role of wavelets elaborates this point.

AVO The change of the reflection coefficient with offset depends

upon a ratio of six elastic parameters at each interface. (The

elastic parameters are P-wave velocity, S-wave velocity, and

density.) Each interface has a unique amplitude variation

with offset (AVO) as determined by these ratios. The

situation becomes even more complex if the seismic speed

becomes dependent on the angle of propagation (“seismic

anisotropy”) or if the medium is attenuative.

Changes in source strength and coupling

Surface sources, such a vibroseis, have different coupling

characteristics on hard versus compliant material. (With the

vibroseis source, an oscillatory mass is the energy source.)

In the marine case, there may be air-gun “drop out” because

of mechanical problems. Thus, not all air guns in the array

are necessarily in service at all times.

Changes in receiver coupling The geophone’s coupling to the ground depends on the

nature of that ground. With some near-surface conditions,

there may be difficulties in properly placing the geophones.

The receiver coupling can vary significantly within a seismic

recording line. The coupling of a single receiver can change

while the survey is acquired due to wind, rain or temperature

changes.

Source and receiver arrays Even in the ideal case, the source and receiver arrays create

amplitude directivity. This amplitude directivity is termed

the radiation pattern. (This is similar to a television

antenna’s directional sensitivity. A Rabbit-ear antenna, for

example, is often re-oriented in order to strengthen the

received signal.)

What is the appearance of raw shot profiles?

The net effect of the above-listed phenomena creates

recoding time-dependent amplitude decays. The following

figure shows a series of raw shot profiles.

Figure 2: Raw shot profiles. (Yilmaz, Seismic Data

Processing, 1987) p. 44, Figure 1-36. With author’s

permission.)

Yilmaz reference:

(1987) Page 44,

Figure 1-36

Similar to:

(2001) Page 96,

Figure 1.5-3

Raw shot gathers show

amplitude decay.

What are the realistic goals of amplitude correction?

Many migration algorithms assume the input trace

amplitudes are in direct proportion to the reflection

coefficients. Other, amplitude-correcting migration

algorithms assume the trace amplitudes originated from a

transparent Earth, one with only geometric amplitude

effects. In either case, the recorded trace amplitudes violate

either of these assumptions. With the multitude of

phenomena that alter amplitudes, it might be tempting to

give up and do nothing. Geophysicists try to at least partially

correct seismic wave energy loss with application of an

amplitude correction algorithm.

With the above-listed, amplitude-altering phenomena, we do

not know enough about the specific details of these

processes to apply a full inversion, directly solving for the

values of the reflection coefficients. In other words, the task

of inverting each of these phenomena is overwhelming. In

the face of this daunting task, we have goals that are more

modest:

1. Keep the data visible.

2. At the completion of all processing, trace

amplitudes are “representative” of the relative

strengths of the reflection coefficients in

comparison to other nearby amplitudes.

Evidence that we have not met these goals:

1. There are amplitude streaks.

2. There are dim or bright regions.

3. Amplitudes do not tie between different surveys.

Tim

e (

s)

xOffset

Figure 3 shows seismic data before the application of a

poststack amplitude correction. Figure 4 shows the same

seismic data after the application of a poststack amplitude

correction. This example illustrates the advantages of

keeping the data visible.

Figure 3: Seismic data before the application of a poststack

amplitude correction. (Processed by Hill from data

provided by Parallel Geoscience.)

Figure 4: Seismic data after the application of a poststack

amplitude correction. (Processed by Hill from data

provided by Parallel Geoscience.)

Yilmaz reference:

(1987) Page 51 & 55,

Figure 1-48 & 53

Similar to:

(2001) Page 115 & 116,

Figure 1.5-22 & -23

These figures show

the stacked section

before and after

application of gain

function shows

importance of

keeping data visible.

What are amplitude-correction solutions?

Amplitude-correction solutions can be divided into

deterministic and statistical categories. The deterministic

processes use the understanding of a dominant amplitude-

decay mechanism to invert the amplitudes. The statistical

approach depends upon the statistics in the traces

themselves.

In fact, the division between “deterministic” and “statistical”

is not quite so clean-cut because deterministic methods also

typically use the data to determine an essential theoretical

parameter.

The following provides both deterministic and statistical

amplitude-correction examples.

Deterministic The deterministic amplitude-decay compensation procedure

selects an algorithm and its parameters that best mimics the

decay of seismic amplitudes.

Spherical divergence Spherical divergence correction is a very common

deterministic amplitude process. For the constant-velocity

Earth, the spherical divergence decay is:

(1)

This relationship follows from the inverse-square law for

energy.

In the presence of a vertical velocity gradient, the curved-ray

effects further attenuate the amplitude according to the

following formula.

. (2)

Figure 5 cartoons the reason for the stronger decrease in

amplitude with distance. The total flux of energy between

the two ray-traced curves is constant. In the presence of a

velocity gradient, the separation between the two curved

rays increases at a faster rate than in the constant-velocity,

straight-ray case. Thus, for this 2D case, the decay of the

energy (and, associated amplitude) with distance is more

severe in the case with the vertical velocity gradient.

Figure 5: Geometric spreading decay.

While these formulas are theoretically satisfying, they may

do an inadequate job of matching the decay of real field data

because of the presence of additional decay phenomena. For

this situation, a statistical correction often follows the

application of these deterministic corrections.

Figure 6 and Figure 7 show a shot profile before and after

the application of spherical divergence correction. (Data

before the application of an amplitude correction is often

termed “raw” data.) Because this shot profile contains only

1.1 s of data, the differences in the two figures is modest.

Tim

e (

s)

X

Tim

e (

s)

X

ConstantVelocity

IncreasingVelocity

Figure 6: “Raw” shot profile. (Processed by Hill from data

provided by Parallel Geoscience.)

Figure 7: Shot profile after application of spherical

divergence correction. (Processed by Hill from data

provided by Parallel Geoscience.)

The next figure (Figure 8) is another example of shot

profiles before and after spherical divergence correction. In

this case, the comparison is significantly more dramatic.

Figure 8: Shot profile before (left) and after (right)

spherical divergence correction. (Yilmaz, Seismic Data

Analysis, 2001) Page 212, Figure 2.4-36. By author’s

permission.)

Yilmaz references:

(2001) Page 212,

Figure 2.4-36

Raw shot before and after

spherical-divergence

correction.

(1987) Page 44,

Figure 1-36 & 37

Similar to:

(2001) Page 82,

Figure 1.4-1, -2

Raw shot before and after

spherical-divergence

correction.

(1987) Page 58,

Figures 1-58 & 59

(2001) Page 84,

Figure 1.4-4, -.5

Shot profiles before and

after spherical-divergence

correction illustrates the gain

application enhanced the

visibility of the ambient

noise.

Attributes of deterministic gain functions The following are attributes of deterministic gain functions.

Data

independence.

Other than the specification of a

constant proportionality factor

through inspection of the

amplitudes in the data, the

deterministic gain processes are

ignorant of the actual amplitudes

in the data.

Tim

e (

s)

xOffset

Tim

e (

s)

xOffset

AfterBefore

xOffsetxOffset

t t

Amplitude

contrast

preserving.

The processes preserve the

amplitude contrasts between

nearby amplitudes.

Noise

susceptibility.

If the noise is stronger than the

signal, the deterministic gain

functions preserve the noise.

Reversibility. Because we define the

deterministic gain functions by a

very small number of parameters,

we can save those parameters and

apply an inverse to the gain at a

subsequent stage in processing.

Follow-up

gain.

Because a deterministic gain

algorithm may fail to meet its goal,

a statistical gain process may

follow.

Smooth. The time dependence of

deterministic gain functions is very

smooth; it does not have

discontinuities.

Streaks. A very straightforward application

of a deterministic gain algorithm,

with an identical gain function

applied to all traces, can result in a

vertically streaked seismic section.

The streaks arise through

systematic, location-dependent

variations in the input trace

amplitudes.

Statistical Observing that there is not a single, dominant amplitude-

determining phenomenon leads to the adoption of statistical

amplitude-correction algorithms. The statistical algorithms

use a simple statistical measure, such as an average, to

determine the gain function. Figure 9 idealizes the statistical

approach with the observed time dependence of the

amplitude decay specifying the gain function.

Figure 9: A statistically designed gain function attempts to

compensate for amplitude loss observed along a trace.

Trace-by-trace statistical gain algorithms As the name implies, “trace-by-trace” statistical gain

algorithms determine trace-dependent gain functions for

each trace from its observed amplitudes. The algorithms then

apply the respective gain functions to each individual trace.

Windowed AGC

An example of a trace-by-trace gain function is the

commonly used, windowed AGC (Automatic Gain Control)

process.

Using Figure 10 as a guide, the following are the steps for a

particular implementation of a windowed AGC operation.

1. Window the trace. User specifies a time

interval, the calculation

“window.”

2. Determine window

“power.”

Calculate the sum of the

absolute values of the

amplitudes in each time

window. Some

practitioners term the sum

of the absolute value of

the trace amplitudes as

the trace’s “power.”

3. Reciprocal of

power.

For each window interval,

calculate the reciprocal of

the calculated power.

4. Interpolate

power’s reciprocal

as expansion

function.

Linearly interpolate the

reciprocal of the

summation of the

absolute value of the trace

amplitudes. This

interpolated value is the

expansion function.

5. Apply gain

function.

For each time sample,

multiply the trace

amplitude by the

expansion function,

creating the gained trace.

Figure 10: Windowed AGC operation.

Figure 11 shows a shot profile after the application of a

trace-by-trace, windowed gain function. Figure 6 shows the

same data set before the application of a gain function and

Figure 7 shows the same input data, but after the application

of a deterministic gain function. Using a short window in the

statistical gain algorithm, the amplitudes are more

homogeneous than in these previous two figures.

Observed

Amplitude

Gain

Function

Data

determines

gain function

t t

Original

Trace

Expansion

Table

Expansion

Function

Gained

Trace

iAmpl

1

t

Figure 11: Shot profile after the application of a windowed

gain function. (Processed by Hill from data provided by

Parallel Geoscience.)

Yilmaz reference:

(2001) Page 87

Figure 1.4-8

Illustrates a time-dependent gain

function applied to one trace.

Notice that the time windows in

the middle display does not

exactly correspond to the upper

and lower illustration. This is

probably an illustration error.

Attributes of windowed AGC:

Not reversible. In practice, the algorithm

discards the applied gain

function immediately after its

application to the data.

Therefore, it is not possible to

remove that gain function at a

later stage in processing. This

is unlike the application of a

velocity-dependent, spherical

divergence correction. It is

possible to remove the

spherical divergence

correction in later processing

because we typically save the

velocity functions through the

entire processing. We could

later remove the applied

windowed AGC if the

algorithm saved its large,

applied-gain file.

Shadow zone. Because the algorithm uses

some window-based statistic,

such as the average of the

absolute value of the

amplitudes, very large

amplitude values may

dominate the determination of

the gain function over a

distance of the size of the

user-specified window. The

presence of these large

amplitudes creates small

values in the gain function

over the length of the window.

The gain function’s small

values diminishes the

amplitudes of adjacent small

values. (SeeFigure 12:

Shadow zone. Figure 12)

Regularizes

amplitudes.

The application of a statistical

gain correction causes the

value of its statistic, such as

the sum of the absolute value

of the amplitudes, to be

approximately the same in all

windows for all traces.

Because of this, all traces in

CMP have equal contributions

in the creation of the stacked

trace, providing maximum

signal-to-noise improvement.

Window length. The smaller the window

length, the greater will be the

degree of trace equalization

homogenization. The larger

the window, the lesser will be

the vote by any one amplitude.

Misleading

around salt.

Typically, there will be large

contrast in trace amplitudes at

the edges of salt, for example.

The salt interface itself will

have high amplitudes and the

region within the salt will

have very low amplitudes.

Through a modification of the

shadow zone effect, these low

amplitudes may dominate in

the determination of the

expansion function, causing

large values of the expansion

function if the majority of the

window is within the salt and

only a small portion outside of

the salt. Figure 13 cartoons

this situation for traces

adjacent to a salt dome. In this

model, the horizontal

reflection coefficients outside

of the salt do not vary

laterally, but after the

application of a windowed

gain, these amplitudes

adjacent to the salt are now

misleadingly larger.

Typically, there will be large

contrast in trace amplitudes at

the edges of salt, for example.

The salt interface itself will

have high amplitudes and the

region within the salt will

have very low amplitudes.

Through a modification of the

shadow zone effect, these low

amplitudes may dominate in

the determination of the

expansion function, causing

large values of the expansion

function if the majority of the

window is within the salt and

only a small portion outside of

the salt. Figure 13 cartoons

this situation for traces

adjacent to a salt dome. In this

model, the horizontal

reflection coefficients outside

of the salt do not vary

laterally, but after the

application of a windowed

gain, these amplitudes

adjacent to the salt are now

misleadingly larger. Similarly,

first arrival amplitudes are

amplifies and arrivals

Tim

e (

s)

xOffset

immediately behind first

arrivals are diminished.

Figure 12: Shadow zone. Amplitudes in the vicinity of salt

reflections get boosted.

Figure 13: Amplitude gain adjacent to salt.

The windowed AGC method has many different variations.

For example, the windowed amplitude statistic may be the

sum of the squares of the trace amplitudes or the windowing

function may be tapered.

Yilmaz reference:

(1987) Page 60,

Figure 1-63

Similar to:

(2001) Page 89,

Figure 1.4-11

As we decrease the size of

the window for the gain

function, the degree of

amplitude homogenization

is increased.

Figure 14: Deterministic gain correction. (Displayed by Jim

Reusser, Conoco. Shown by permission of ConocoPhillips.)

shows stacked land seismic data whose processing includes

the application of a deterministic gain correction. The

vertical oval surrounds one of many vertical amplitude

streaks. This is marine data whose onboard log noted

sporadic source problems. The vertical amplitude streaks are

most likely due to malfunctioning air guns in the source

array. As strictly applied, this deterministic amplitude

correction does not correct for the effects of source air gun

malfunctions.

Figure 15 shows the same data after the application of the

windowed AGC gain correction, among other processing

changes. In the quest to improve the data, the processing

geophysicist implemented many variations from the prior

processing represented by Figure 14. Notice the improved

top-to-bottom amplitude visibility and the reduced amount

of vertical amplitude streakiness at numerous lateral

locations. In spite of the increased vertical streaking, it may

very well be that Figure 14 provides the more accurate

representation of the differences in the average value of the

reflection coefficients between the upper and lower halves of

the displayed data. With the improved visibility in the lower

half of Figure 15, it is apparent that the character of the data

in the lower half is quite different from that of the upper

half.

Figure 14: Deterministic gain correction. (Displayed by

Jim Reusser, Conoco. Shown by permission of

ConocoPhillips.)

Figure 15: Windowed AGC gain correction. (Processed by

Jim Reusser, Conoco. Shown by permission of

ConocoPhillips.)

The following series of four figures compare the results of

the application of deterministic versus statistical gain

corrections for land and marine data. In these cases, the

data’s amplitudes clearly did not satisfy the assumptions of

the applied deterministic gain correction.

In the first land data example (Figure 16), there might have

been lateral variability of the source coupling or receiver

coupling. For a more optimal deterministic amplitude

correction process, it would be necessary to address those

variations in addition to the spherical divergence corrections.

Time

Am

plit

ude

Original amplitudes

Windowed gain function

Gained amplitudes

x

t

x

t

Figure 16: Land data after application of deterministic

gain correction. (Processed by Conoco. Shown by

permission of ConocoPhillips.)

Figure 17: Land data after application of statistical gain

correction. (Processed by Conoco. Shown by permission of

ConocoPhillips.)

In the following marine case, there might have been a gun

dropout problem. Thus, the source strength varied from shot

to shot. This variation produced the amplitude steaks visible

in Figure 18. A more successful deterministic approach

would have included a correction for the gun strength

variations in addition to the spherical divergence correction.

Figure 18: Marine data after application of deterministic

gain correction. (Processed by Conoco. Shown by

permission of ConocoPhillips.)

Figure 19: Marine data after application of statistical gain

correction. (Processed by Conoco. Shown by permission of

ConocoPhillips.)

Other processes A large number of seismic processing algorithms alter the

time-dependent gain of a trace. Because some of these

processes alter the amplitudes in a deleterious fashion, it

may be appropriate to follow that process with an

application of a statistical gain process. The following is an

itemization of various processes and the advisability of the

use of a following gain process.

X

Tim

e (

s)

X

Tim

e (

s)

x

t

x

t

Process Alters

gain?

Comments

Acquisition Yes As indicated in Figure 1, all

aspects of acquisition

determine the gain of the

recorded trace.

De-spiking Removal of noise spikes.

Deconvolution Yes Because decay is a function

of frequency, re-balancing

the amplitude spectrum

alters the overall time-

dependent amplitude decay.

A following chapter

explains deconvolution.

Frequency

Filtering

Yes Because amplitude decay is

a function of frequency,

altering the amplitude

spectrum through frequency

filtering alters the overall

time-dependent amplitude

decay. A later chapter

explains frequency filtering.

NMO

Correction

No Alters the times of events on

traces. It effect on the trace

amplitude times is most

dramatic for shallow, far

offset traces. Even as such,

we customarily neglect this

effect on amplitudes.

Statics

Solutions

No In general, trace shifts

applied by statics solutions

are not large. However,

stacked trace amplitudes can

increase with the application

of a proper statics solution.

De-multiple Yes Considering the average

multiple and primary

amplitude over a time

window, such an average

amplitude of multiples has a

slower decay rate than

primaries. This occurs

because the number of

multiples in a time window

increases with time in

comparison to the number

of primaries. With

increasing time, the total

number of independent,

multiples paths increases. In

the general case, the time-

dependent decay of a

window-based amplitude

measure (such as the

average of the absolute

value of the amplitude)

shows slower decay for

multiples than primaries.

After demultiple, we may

wish to gain the remaining

primaries to improve their

visibility or to take out the

effect of geometry (1/r)

amplitude decay.

Stack Yes Because stack improves the

signal-to-noise ratio and

because the signal-to-noise

ratio is time dependent,

stack alters the time

dependent gain. In addition,

necessary trace muting

alters the time-dependent

fold-of-stack and with that,

the time-dependent

amplitudes.

Coherency Yes Post-stack coherency

improvement alters the

signal-to-noise ratio and the

frequency content. Through

that, coherency algorithms

alter the apparent amplitude

decay.

Migration Yes By collapsing diffractions

and moving steep dips to a

shallower portion of the

seismic section, migration

alters the amplitude decay.

Yilmaz reference:

(1987) Page 40,

Figure 1-33,

Record 40

(2001) Page 80,

Figure 1.3-40

Example of noise spikes from

recording system.

Well calibration The availability of well information provides an opportunity

to estimate an appropriate gain function under the criterion

of matching the seismic amplitudes to a synthetic trace

created from the density and velocity logs. This method is

distinctly different from the previous, deterministic methods

because it does not require a theoretical hypothesis of the

origin of the amplitude decay. Instead it uses ground truth to

furnish the criterion for the creation of a gain function that

would tie the synthetic’s amplitude with that of a coincident

trace. This gain function may then be applied to all traces,

under the assumption that the amplitude attenuation is not a

function of lateral position.

Lateral homogeneity is not the only limiting assumption of

the well calibration approach. Density and velocity logs are

measured at frequencies that are much higher than that of

seismic waves recorded at the surface. The log data requires

careful upscaling and blocking to emulate the subsurface

properties sensed by propagating seismic waves.

Display of amplitudes A wiggle-trace display can reveal a dynamic range of up to

20 to 1. The values of the reflection coefficients in the

ground have a much greater variation. This explains the

popularity of color displays that are not limited by a

representation of a lateral trace excursion. However, the

storage requirements on interpretation workstations demand

that we reduce the dynamic range of the amplitudes before

display. The high amplitudes may be clipped and the low

amplitudes may become invisible.

To make informed decisions about trace amplitudes, it is

vital to have a standard of comparison. For this reason, some

prefer the use of the calibrated color bar along with the

expansion. A robust statistic, such as the median of the

amplitudes, may be the basis of the expansion and the

calibrated color bar. With the application of these tools, the

color display may portray the amplitude in terms of the

median of the absolute value of the amplitude for the entire

survey.

Interpreters’ role Raw seismic data are often un-interpretable and flat-out

ugly. Seismic records require processing to be displayed for

meaningful analysis. Here are some issues that the

interpreter should keep in mind:

Be aware of what the processed amplitudes

represent. Are these amplitudes processed with the

goal of being most visually pleasing? Are the

amplitudes balanced to bring out weak (but

important) reflection events or are the amplitudes a

representation of the subsurface reflectivity?

True amplitude processing aims at preserving the

reflection coefficient in the data. Is your data “true

amplitude”? If not, are the seismic processing steps

deterministic and reversible? Does your

interpretation depend upon the successful retrieval

of the true amplitudes through processing?

What were possible amplitude related problems of

the raw data? What processing-tools were

employed to fix these issues?

Be aware of the amplitude correction’s

assumptions. Does your data meet these amplitude

correction’s assumptions?

Does the shallower section introduce

uncompensated amplitude problems? The presence

of salt or gas serves as a couple of examples.

Be caution in interpreting amplitudes close to

strong reflection events such as water bottom or salt

bodies. High amplitude reflections often dominate

statistical amplitude-gaining algorithms and may

cause side effects.

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