Is Amplitude Loss of Sonic Waveforms Due 

to Intrinsic Attenuation or Source Coupling 

to the Medium?

Scientific Investigations Report 2006-5120

U.S. Department of the Interior

U.S. Geological Survey
Is Amplitude Loss of Sonic Waveforms Due 

to Intrinsic Attenuation or Source Coupling 

to the Medium?

By Myung W. Lee

Scientific Investigations Report 2006–5120

U.S. Department of the Interior

U.S. Geological Survey
U.S. Department of the Interior

Dirk Kempthorne, Secretary

U.S. Geological Survey

P. Patrick Leahy, Acting Director

U.S. Geological Survey, Reston, Virginia: 2006

Posted online June 2006

Version 1.0

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Suggested citation:

Lee, M.W., 2006, Is amplitude loss of sonic waveforms due to intrinsic attenuation or source coupling to the medium?: 

U.S. Geological Survey Scientific Investigations Report 2006-5120, 13 p.
iii

Contents

Abstract...........................................................................................................................................................1

Introduction.....................................................................................................................................................1

Measured Sonic Amplitudes at the Mallik 5L-38 Well.............................................................................2

Source Radiation (Coupling) ........................................................................................................................3

Real Data Analysis.........................................................................................................................................4

Prediction of Source Radiation at the Mallik 5L-38 Well................................................................4

Calculation of Apparent Attenuation.................................................................................................5

Discussion.......................................................................................................................................................5

Various Attenuation Measurements..................................................................................................5

Gas Hydrate Saturation and Attenuation..........................................................................................7

Intrinsic Attenuation and Source Coupling ......................................................................................8

Conclusions.....................................................................................................................................................8

Acknowledgments.........................................................................................................................................8

References Cited............................................................................................................................................8

Appendix 1. Amplitude Loss of Compressional Waves in Brine Due to Source Coupling ...........11

Appendix 2. Attenuation at Mallik 3L-38...............................................................................................12

Traveltime Inversion ...........................................................................................................................12

Figures

1. Measured sonic and Stoneley-wave amplitudes at Mallik 5L-38 well,

Mackenzie Delta, Northwest Territories, Canada...................................................................2

2. Normalized amplitudes of P-waves, S-waves, and Stoneley waves...................................5

3. Intrinsic attenuation and apparent attenuation.......................................................................6

Appendix Figures

1–1. P-wave amplitude loss due to source coupling compared to amplitude loss

inside brine ..................................................................................................................................11

2–1. Spectral ratios at Mallik 3L-58 well .........................................................................................12

2–2. Velocities derived from VSP data at Mallik 3L-58 well and well-log velocities

measured at Mallik 5L-38 ..........................................................................................................13
iv

Conversion Factors

Multiply By To obtain

Length

meter (m) 3.281 foot (ft) 

kilometer (km) 0.6214 mile (mi)

Multiply By To obtain

Length

foot (ft) 0.3048 meter (m)

mile (mi) 1.609 kilometer (km)
Abstract

Sonic waveforms acquired in gas-hydrate-bearing 

sediments indicate strong amplitude loss associated with an 

increase in sonic velocity. Because the gas hydrate increases 

sonic velocities, the amplitude loss has been interpreted 

as due to intrinsic attenuation caused by the gas hydrate in 

the pore space, which apparently contradicts conventional 

wave propagation theory. For a sonic source in a fluid-filled 

borehole, the signal amplitude transmitted into the formation depends on the physical properties of the formation, 

including any pore contents, in the immediate vicinity of the 

source. A signal in acoustically fast material, such as gashydrate-bearing sediments, has a smaller amplitude than a 

signal in acoustically slower material. Therefore, it is reasonable to interpret the amplitude loss in the gas-hydrate-bearing 

sediments in terms of source coupling to the surrounding 

medium as well as intrinsic attenuation. An analysis of sonic 

waveforms measured at the Mallik 5L-38 well, Northwest Territories, Canada, indicates that a significant part of the sonic 

waveform’s amplitude loss is due to a source-coupling effect. 

All amplitude analyses of sonic waveforms should include the 

effect of source coupling in order to accurately characterize 

the formation’s intrinsic attenuation.

Introduction

In sediments, the presence of gas hydrate—a crystalline substance composed of a “cage” of water molecules that 

surround gas molecules—significantly changes the sediments’ 

physical properties; for example, their elastic velocities are 

higher than the elastic velocities of sediments without gas 

hydrates. Various seismic survey methods utilize this velocity 

increase as a means of identifying and quantifying gas hydrate 

concentrations.

The amplitude loss of sonic waveforms, measured by a 

Dipole Shear Sonic Imager (DSI) (12 kHz for monopole source 

and 2 kHz for dipole source), has been interpreted to be due 

to intrinsic attenuation. For example, (1) Guerin and Goldberg 

(2002) analyzed the DSI sonic waveforms measured at the 

Mallik 2L-38 well, Mackenzie Delta, Canada, and suggested 

that the P- and S-wave attenuation increases as the gas hydrate 

saturation increases. (2) Later, Guerin and others (2005) 

applied the same analysis techniques to the sonic waveforms 

acquired at the Mallik 5L-38 well and thereby confirmed the 

validity of the previous analysis at the neighboring Mallik 

2L-38 well. (3) In addition, Matsushima (2005a) arrived at the 

same conclusion as Guerin and others (2005) on the basis of 

an analysis of sonic waveforms acquired at the Nankai Trough 

exploratory well offshore of Tokai, central Japan.

Inversion of hole-to-hole seismic data acquired near 

the Mallik 5L-38 well site allowed Bauer and others (2005) 

to conclude that the amplitude loss in the frequency range 

of 200–2,000 Hz increases as the gas hydrate saturation 

increases, but the magnitude of P-wave attenuation is about 

one-half of that determined by the sonic waveform analysis 

(Guerin and others, 2005). Bauer and others attributed these 

P-wave data to frequency-dependent attenuation, whereby the 

amplitude loss increases with increasing frequency.

Several investigators proposed various mechanisms to 

explain the peculiarly large amplitude loss inferred from the 

sonic waveform analysis of gas-hydrate-bearing sediments 

(GHBS) and from the hole-to-hole seismic data. These 

investigators then applied the amplitude loss as a parameter 

to quantify the amount of gas hydrate (Guerin and Goldberg, 

2002; Chand and Minshull, 2004; Dvorkin and Uden, 2004; 

Guerin and others, 2005).

However, the present study uses surface vertical seismic 

profile (VSP) data to demonstrate that the amplitude loss of 

sonic waveforms measured for GHBS is less than that measured in water-saturated sediments. Rossi and others (2005) 

used tomographic analysis of three-component seismic data 

from an ocean-bottom seismograph (OBS) array offshore 

northwest Svalbard to conclude that GHBS manifest themselves with high P-wave velocities and low P-wave attenuations, in agreement with the analysis discussed in this report 

(see appendix 2). Gei and Carcione (2003) suggested that 

attenuation should decrease with increasing gas hydrate 

saturation, and they argued that gas hydrate makes sediments 

more cohesive.

Thus, two opposing views and contrasting analyses 

exist for the amplitude loss of sonic waveforms measured for 

GHBS. Attenuations estimated at low frequency in sediments 

with various amounts of gas hydrate (in surface VSP data and 

Is Amplitude Loss of Sonic Waveforms Due to Intrinsic 

Attenuation or Source Coupling to the Medium?

By Myung W. Lee
OBS data) suggest that gas hydrate presence in sediments 

decreases their intrinsic attenuation. However, attenuations 

estimated at higher frequencies (in sonic and hole-to-hole 

seismic data) suggest that gas hydrate presence in sediments 

increases their intrinsic attenuation. These different views have 

been based on data acquired with different source environments as well as different input frequencies.

Theories predict that the strength of radiated energy 

strongly depends on the elastic parameters of the surrounding 

medium. For example, the displacement from a given point 

source acting on a half space decreases as the density and elastic velocities of the half space increase. The source environment did not change during the acquisition of VSP and OBS 

data, but continuously changed during the acquisition of sonic 

and hole-to-hole seismic data as the sources were located at 

different depths. Therefore, the measured amplitude loss of 

sonic waveforms could have been the effect of the source 

coupling to the surrounding medium as well as changing the 

intrinsic attenuation.

The seismic responses of borehole sources (monopole 

and dipole) inside a fluid-filled hole can be calculated by 

using numerical methods (for example, Tsang and Rader, 

1979; Cheng and Toksöz, 1981; Kurkjian and Chang, 1986). 

However, in the present report, closed-form solutions under 

the assumption of far-field and low-frequency approximation 

of borehole sources by Lee and Balch (1982) and Lee (1986) 

are used to explain the amplitude loss observed in sonic waveforms at the Mallik 5L-38 well.

Measured Sonic Amplitudes at the 

Mallik 5L-38 Well

Sonic waveforms were measured by the DSI tool, and 

examples of monopole and dipole waveforms are shown in figure 1, modified from Guerin and others (2005). For comparison, the Stoneley-wave amplitude is shown (fig. 1) with a fullbore formation microimager (FMI) log, modified from Collett 

and others (2005). Zones with dark color in the FMI image 

represent low-resistivity zones, which correspond to little, if 

any, gas hydrate, whereas zones with light color indicate zones 

highly saturated in gas hydrate.

Monopole

waveform

Dipole

waveform

High

gas hydrate 

saturation

Low

gas hydrate

saturation

Stoneley

waveform

FMI

static image

1,000

1,050

DEPTH, IN METERS

DEPTH, IN METERS

850

900

950

Figure 1. Measured sonic and Stoneley-wave amplitudes at the Mallik 5L-38 well, Mackenzie 

Delta, Northwest Territories, Canada. The monopole and dipole waveforms are modified from 

Guerin and others (2005). The Stoneley-wave amplitude and borehole formation microimager (FMI) 

static image are modified from Collett and others (2005).

 Is Amplitude Loss of Sonic Waveforms Due to Intrinsic Attenuation or Source Coupling to the Medium?
In the depth intervals 890–895 m, 910–930 m, and 

950–960 m, which are highly saturated in gas hydrate, the 

monopole and dipole amplitudes are significantly reduced 

compared to the amplitudes in other intervals (Collett and 

others, 2005). Based on the sonic waveforms, shown in figure 

1, and gas hydrate saturations estimated from the nuclear magnetic resonance (NMR) tool (Kleinberg and others, 2005), the 

following observations can be made:

1. Both monopole and dipole waveform amplitudes are much lower in the intervals that are 

highly saturated in gas hydrate.

2. The monopole amplitude loss appears to be 

slightly greater than the dipole amplitude loss.

3. The Stoneley-wave amplitude is higher in intervals highly saturated with gas hydrate than in 

intervals with little or no gas hydrate.

Guerin and Goldberg (2002) attributed some energy loss 

to the greater velocity contrast between the borehole fluid and 

the formation, yet their estimated intrinsic attenuation has 

to be appreciably higher in GHBS than in non-gas-hydratebearing sediments (NGHBS) to model the amplitude loss 

observed at the Mallik 2L-38 well. They also noted that the 

intrinsic attenuation increase suppresses later arrivals such as 

the Stoneley mode in dipole waveforms. This amplitude prediction for the Stoneley wave by Guerin and Goldberg (2002) 

is opposite to the results shown by Collett and others (2005) in 

the Mallik 5L-38 well (fig. 1). Because the propagation mode 

of the Stoneley wave is different from that of the monopole Pwave, it is also questionable to attribute amplitude loss of the 

Stoneley wave to the intrinsic attenuation of the formation.

This report attempts to explain the amplitude variations 

of P-waves, S-waves, and Stoneley waves for GHBS (as 

shown in fig. 1) by using source coupling to the surrounding 

medium rather than by using intrinsic attenuation.

Source Radiation (Coupling)

In order to semiquantitatively analyze sonic log amplitudes, the radiation pattern valid under the assumption of the 

low-frequency and far-field approximation derived by Lee and 

Balch (1982) and Lee (1986) is used. The responses of various 

waves in real data obtained from monopole and dipole sources 

are more complicated than those indicated in this report. However, this simple approach can be used to predict a significant 

part of the measured amplitude loss, as shown later.

A brief theoretical analysis appropriate for this investigation follows. Let subscript 1 denote the borehole fluid and 

subscript 2 denote the surrounding solid; α, β, and ρ represent 

the P-wave velocity, S-wave velocity, and density, respectively; R is the distance from the source to the receiver, and ω

is the frequency. In a spherical coordinate system (R, θ, φ), a 

radial displacement field (Ur) due to an initial displacement 

source (V0) that acts on the axis of a fluid-filled borehole is 

given by the following formula in the frequency domain (Lee 

and Balch, 1982):

U i V

T

e

R

ik R

r =  ́ -

+

 ́

- w r

pr a

b f a

a r r

a

0 1

2 2

2

2 2

2

2

4 1 2

1 2 2 cos /

/ , (1)

where

Ta

b

a

a f

a = - æ

è

ç

ç

çç

ö

ø

÷

÷

÷

÷

2

2

1

2

1

2 2

2

2 1 cos

,

ka

w

a 2

2

=

,

and

i = -1 .

When a symmetrical source (like a monopole source) is 

applied inside a fluid-filled borehole, tube waves (that is, the 

Stoneley-wave type) are generated and travel with a velocity 

of CT. Note that a tube wave is a boundary wave propagating 

along a fluid-filled borehole. The pressure response of a tube 

wave (PT) is given by the following formula from Lee and 

Balch (1982):

P i V C

a T C T = μ T

w r

p

0 1

2 2 , (2)

where a is the borehole radius and CT is the tube-wave velocity, which is given by

CT = +

æ

è

ç

ç

çç

ö

ø

÷

÷

÷

÷

-

1

1

2

1

2 2

2

1 2

a

r

r b

/

. (3)

When a radial stress source with a constant magnitude Trs

is applied to the wall of a fluid-filled borehole, the azimuthal 

component of displacement (Uθ) is written as follows (Lee, 

1986):

U T i a e

R

ik R

q q w q f

b b a

b

= + -

é

ë

ê

ê

ù

û

ú

ú

-

rs sin sin sin

( / )

2

2 2 1 22

2

2

, (4)

where

T aT rs = 0

2 2

2 4pr b ,

Source Radiation (Coupling)   
and kb w b 2 = / 2 .

The monopole response of the sonic tool is simulated by 

using equation 1, and the amplitude response of the monopole 

(Ur) is given by equation 5 when φ = π/2 (perpendicular to the 

borehole):

Ur μ

+

1

2 2 2

2

1

2 r a b a r r 1 2 ( / / ) . (5)

A dipole source can be approximated by using two 

opposing radial sources 180° apart in the azimuthal direction. 

From equation 4, the dipole response is given by

Uq r bμ

1

2 2

2

. (6)

The azimuthal variation for the acoustic field (the 

radiation pattern on a plane perpendicular to the borehole or 

radiation pattern at φ = π/2), as calculated from equation 1 for 

the monopole and equation 4 for the dipole, agrees with the 

radiation pattern for the multiple acoustic sources of the logging system (Tang and others, 2002).

The Stoneley waves are dispersive, so the velocity 

depends on the frequency. The tube wave is the low-frequency 

component of the Stoneley wave and is nondispersive as shown 

in equation 3. In this analysis, it is assumed that the pressure response of the tube wave given in equation 2 is a good 

approximation of the pressure response of the Stoneley wave.

The amplitude loss (that is, the attenuation) (Q–1) of each 

wave is derived from the following equation (Schön, 1998):

Q V

f x x

A x

A x

- = -

æ

è

ç

ç

çç

ö

ø

÷

÷

÷

÷

1

2 1

1

2

1

p log ( )( ) , (7)

where V is the velocity, f is the linear frequency, A(x1) is 

amplitude at location x1, and A(x2) is the amplitude at location 

x2. The amplitude at a distance R is given by

A e R μ -a , (8) 

where α = πfQ–1/V.

A relative amplitude loss due to source coupling is independent of the distance the wave traveled, whereas the amplitude loss due to intrinsic attenuation is dependent on the travel 

distance. In order to meaningfully compare amplitude loss due 

solely to source coupling with the measured attenuation, an 

apparent attenuation coefficient (αa) is defined as the attenuation that reduces the amplitude as much as the amplitude loss 

due to source coupling. In other words, the wave amplitude 

due to the coupling effect (Ac) is defined as

A e R

c μ -aa

. (9)

If a constant excitation frequency and a constant strength 

of source are assumed, the apparent attenuation (Qa,j) due to 

source coupling can be written as

Q V a,j j Ac,j

- μ1 log( ) .

Therefore,

Q

Q

V A

V A

a,j j j

a

c,

c,

-

- =

1

1

1

, 1 1

log( )

log( )

. (10)

If amplitudes normalized by Ac,1 are used, equation 10 can be 

written as

Q Q

V G C

V G C Q

V G C

V C j a

j j j j

a, a,

- - - = = 1

1

1

1 1

1

1

1

,

log( )

log( )

log( )

log , (11)

where Gj is the normalized amplitude and C is a scalar to 

convert Gj back to the true amplitude in equation 10. Equation 

11 indicates that if the apparent attenuation at j = 1 is known, 

apparent attenuations corresponding to the source coupling 

can be evaluated with known C and normalized amplitudes.

In order to estimate apparent attenuation, the scalar C

in equation 11 is computed through the use of the following 

equation:

C A e Q f R V = = - -

c,

a,

1

1

1 p 1 /

. (12)

Real Data Analysis

Prediction of Source Radiation at the

Mallik 5L-38 Well

Figure 2 shows the normalized amplitude responses of 

P-waves, S-waves, and tube waves calculated by using the 

geophysical logs at the Mallik 5L-38 well and the source 

radiations (or source coupling to the formation) given in equations 2, 5, and 6 for the Stoneley-wave, P-wave, and S-wave 

responses, respectively. Like sources, receivers would have 

receiver coupling to the surrounding medium. However, in all 

amplitude computations, the receiver coupling is ignored.

As shown in the observed amplitude variation, the source 

radiation predicts amplitude loss to be largest for the monopole (P-wave response) and smallest for the tube wave. However, the amplitude loss of the S-wave due to source coupling 

is as great as that for the P-wave. As the P-wave velocity (or 

S-wave velocity) increases, the amplitude response of the Pwave (or S-wave) decreases. The tube-wave velocity increases 

as the formation S-wave velocity increases. Figure 2 indicates 

that the tube-wave amplitude is higher where the S-wave 

 Is Amplitude Loss of Sonic Waveforms Due to Intrinsic Attenuation or Source Coupling to the Medium?
velocity (the S-wave velocity is proportional to the P-wave 

velocity) is higher, in agreement with the measured amplitude 

variation of the Stoneley wave shown in figure 1.

The details of the monopole and dipole responses inside 

a fluid-filled borehole could be different from the radiation 

pattern shown in this report. However, the amplitude variation 

with respect to formation velocities using the low-frequency 

and far-field approximation of the source radiation pattern 

agrees qualitatively with the observed amplitude behavior of 

sonic waveforms measured by DSI at the Mallik 5L-38 well.

P-wave amplitude loss due to source coupling is not 

limited to borehole sources. Appendix 1 illustrates the effect 

(determined in laboratory studies by Prasad and Dvorkin, 

2004) of source coupling on the measured amplitude inside a 

brine; as indicated, about half the measured amplitude loss is 

due to source coupling.

Calculation of Apparent Attenuation

The amplitude variation shown in figure 1 can be interpreted from the intrinsic attenuation point of view, and this 

view has been adopted, for example, by Guerin and others 

(2005) and Matsushima (2005a). The amplitude can also be 

analyzed and interpreted as a consequence of the source radiation, defined here as the apparent attenuation to differentiate it 

from the intrinsic attenuation. The amplitude losses due to the 

intrinsic attenuation can be compared with those due to source 

coupling by using the apparent attenuation calibrated with the 

intrinsic attenuation at j = 1.

Guerin and others (2005) used 12 kHz and 2 kHz for 

P-wave and S-wave attenuation model studies, respectively, 

and these frequencies were used for the apparent attenuation 

in equations 10 and 11. The distances from the monopole 

and dipole sources to receiver number 5 are approximately 

3.5 m and 4.5 m, respectively; these distances are used for 

equation 12.

Guerin and others (2005) derived the following relationships between the intrinsic attenuation and the gas hydrate 

saturation:

QP Sh

- = + 1 0. . 0342 0 076 : for monopole (13a)

QS Sh

- = + 1 0. . 0785 0 135 : for upper dipole (13b)

where Sh is the gas hydrate saturation. In order to calibrate the 

apparent attenuation from the intrinsic attenuation, the intrinsic attenuation at j = 1, or at depth of 843.84 m, with Sh

 = 0 is 

used as the apparent attenuation at j = 1. In other words, the 

apparent attenuations of P-waves (QaP,1

–1) and S-waves (QaS,1–1) 

are 0.0342 and 0.0785, respectively. Because the gas hydrate is 

present at depths greater than 891 m at the Mallik 5L-38 well, 

the intrinsic attenuations calculated by using equations 13a 

and 13b are due to attenuation in water-saturated sediments, 

and these values are used to calibrate the amplitude loss due to 

the source coupling.

Figure 3A shows the calculated apparent attenuation with 

respect to gas hydrate saturations estimated from the nuclear 

magnetic resonance tool (NMR) (Kleinberg and others, 2005). 

The calculated P-wave apparent attenuation is approximately 

one-half that of the intrinsic attenuation estimated by Guerin 

and others (2005), and the S-wave apparent attenuation 

appears to be comparable to the intrinsic attenuation.

If the source-coupling effect is used to account for the 

amplitude loss shown in figure 3A, the difference between the 

apparent attenuation and the attenuation measured by Guerin 

and others (2005), shown in figure 3B, can be interpreted as 

the intrinsic attenuation of GHBS. 

Discussion

Various Attenuation Measurements

As mentioned previously, two opposing views exist for 

explaining the amplitude loss of the sonic waveform due to the 

presence of gas hydrate. One view is based on the sonic waveform analysis and suggests that the amplitude loss increases 

as gas hydrate saturation increases and is much greater than 

for non-gas-hydrate-bearing sediments (NGHBS). Arguments 

and evidence supporting intrinsic attenuation as the primary 

cause of the amplitude loss of the sonic waveform include the 

following:

DEPTH, IN METERS

NORMALIZED AMPLITUDE

P-WAVE VELOCITY,

IN KILOMETERS PER SECOND

0.1

1 1

0.1

850 900 950 1,000 1,050 1,100 1,150

Normalized amplitude

P-wave, S-wave, Tube wave

P-wave velocity

Mallik 5L-38

Figure 2. Normalized amplitudes of P-waves, S-waves, and 

Stoneley waves calculated by using the source coupling with 

measured P-wave velocities at the Mallik 5L-38 well. A monopole 

is simulated by using a symmetrical volume source, a dipole 

source is simulated by using two opposite radial sources 180° 

apart, and the Stoneley wave is approximated by using the tube 

wave due to a symmetrical volume source.

Discussion   
1. From the analysis of hole-to-hole seismic surveys with frequencies ranging from 200 to 2,000 

Hz at the Mallik well sites, Bauer and others 

(2005) demonstrated that there is frequencydependent P-wave attenuation with a magnitude 

about one-half of that estimated from the DSI 

sonic logs (12 kHz monopole and 2 kHz dipole 

sources) by Guerin and others (2005). Compared to the waveform recorded at the source 

and receiver located in NGHBS, the high-frequency part of the waveform from formations 

highly saturated in gas hydrate decays more 

than the low-frequency part. The smaller attenuation in hole-to-hole seismic data compared 

to that estimated from the sonic waveforms is 

attributed to the lower frequency of the seismic 

source data compared to the sonic waveforms. 

Because the amplitude decay based on the lowfrequency approximation of the source radiation 

is independent of input frequency, it is not clear 

whether the source radiation can explain the 

observed frequency-dependent amplitude loss at 

the Mallik sites. In order to analyze the frequency-dependent amplitude loss of sonic waveforms by using the source radiation, numerical 

modeling techniques proposed by Tsang and 

Rader (1979), Cheng and Toksöz (1981), or 

Kurkjian and Chang (1986) could be used.

2. From the analysis of waveforms measured by 

using resonant-column experiments, Priest and 

others (2005) reported that P-wave attenuation increases with the increase of gas hydrate 

saturation. They interpreted the observed attenuation on the basis of an intercrack squirt flow. 

However, it should be noted that the P-wave 

attenuation is a maximum when the pores have 

approximately 3–5 percent gas hydrate saturation and the P-wave attenuation is higher than 

the S-wave attenuation. Neither result agrees 

with observations at the Mallik sites.

3. Several authors have proposed various attenuation mechanisms to explain the amplitude loss 

in sonic waveforms measured for gas-hydrate–

B bearing sediments (GHBS):

–0.02

0.00

0.02

0.04

0.06

0.08

0.10

GAS HYDRATE SATURATION, IN FRACTIONS

A GAS HYDRATE SATURATIONS, IN FRACTIONS DIFFERENCE OF ATTENUATION

0.00

0.0 0.2 0.4 0.6 0.8 0.10

0.0 0.2 0.4 0.6 0.8 0.10

0.05

0.10

0.15

0.20

0.25

0.30

RELATIVE APPARENT ATTENUATION

Qap

–1: P-wave coupling

Qas

–1: S-wave coupling

Qp

–1: P-wave attenuation

Qa

–1: S-wave attenuation

Qp

–1 – Qap–1, LSF

Qs

–1 – Qas–1, LSF

Figure 3. A, Intrinsic attenuation (QP

–1, QS–1) and apparent 

attenuation (QaP

–1, QaS–1) with respect to gas hydrate saturations 

estimated from the nuclear magnetic resonance (NMR) tool at the 

Mallik 5L-38 well. The dashed lines are the least-square fits to the 

intrinsic attenuation estimated by Guerin and others (2005). B, The 

difference between the intrinsic attenuations estimated by Guerin 

and others (2005) and the apparent attenuations calculated by 

using the source coupling at the Mallik 5L-38 well.

 Is Amplitude Loss of Sonic Waveforms Due to Intrinsic Attenuation or Source Coupling to the Medium?
(a) Guerin and Goldberg (2005) applied the 

three-phase Biot equation proposed by 

Carcione and Tinivella (2000) with a squirt 

flow and friction between matrix and gas 

hydrate.

(b) Chand and Minshull (2004) applied the 

squirt flow mechanism proposed by Parra 

(2000) to interpret the sonic amplitude loss 

at the Mallik well sites.

(c) Dvorkin and Uden (2004) proposed that 

macroscopic squirt flow due to the heterogeneity in the formation is responsible for the 

large amplitude loss in GHBS. 

Because causality requires a very specific 

relationship between velocity and attenuation, 

a large attenuation is generally associated with 

high velocity dispersion (Dvorkin and Uden, 

2004). However, none of these three theories 

mentioned velocity or velocity dispersion except 

the theory of Guerin and Goldberg (2005), who 

discussed velocities only at particular frequencies. A problem with the Guerin and Goldberg 

(2005) approach is that at 100 percent gas 

hydrate saturation, the theory predicts the smallest P-wave attenuation (zero), but the highest 

S-wave attenuation.

As shown in this report, some attenuation mechanisms 

have been proposed to explain the amplitude loss of the sonic 

waveform due to gas hydrate in the sediments, but none of the 

theories accurately predict both elastic wave velocities and 

amplitude losses for GHBS.

Arguments and evidence advanced for the opposing 

view that the amplitude loss for sonic waveforms measured 

in GHBS is less than for those in NGHBS include the 

following:

1. It is generally accepted that as sediments 

become more compacted or velocities become 

larger, the intrinsic attenuation becomes 

smaller. On the basis of this general perception, Gei and Carcione (2003) proposed that the 

attenuation in GHBS data is less than that in 

NGHBS data because gas hydrate, being stiffer 

than pore water, makes the porous medium 

more cohesive.

2. Intrinsic attenuation generally accompanies 

velocity dispersion. VSP data at the 

Mallik 2L-38 well indicate that the velocity 

dispersion at the Mallik site in the frequency 

range 20–12,000 Hz (surface seismic to sonic 

frequency) is negligible or very small (appendix 

2). This result implies that the intrinsic attenuation of GHBS could be small. 

3. The P-wave attenuation in GHBS, as estimated 

from the near-offset and far-offset VSP data, is 

less than that in NGHBS at the Mallik 3L-38 

well (appendix 2). Also, Rossi and others (2005) 

demonstrated that the P-wave attenuation of 

GHBS is less than that in NGHBS at offshore 

northwest Svalbard on the basis of the analysis 

of ocean-bottom seismograph data. Attenuation analysis of two VSPs at the Nankai Trough 

exploratory wells, offshore Japan, indicated 

no significant P-wave attenuation in GHBS, in 

contrast to high P-wave attenuations estimated 

from the sonic waveforms at the same wells 

(Matsushima, 2005b).

On the basis of data measured by varying the source 

environment (for example, well log, hole-to-hole seismic), 

it has been shown that amplitude loss increases as the gas 

hydrate saturation increases. However, data measured by 

holding a constant source environment (for example, surface 

VSP and marine ocean-bottom seismograph) indicate that 

amplitude loss decreases as gas hydrate saturation increases. 

These observations imply that the measured amplitude loss 

from a sonic waveform appears to be associated with source 

environments.

Gas Hydrate Saturation and Attenuation 

The relationship between attenuation and gas hydrate 

saturation is given in equation 13, which indicates that the 

difference of attenuation at Sh = 0 percent and Sh = 100 percent 

is ∆QP

–1 = 0.076 for P-waves and ∆QS–1 = 0.135 for S-waves. 

The least-squares fit to the difference of attenuation shown in 

figure 3B is

QP Sh

- = + 1 0. . 003 0 02 : for monopole (14a)

QS Sh

- = + 1 0. . 0009 0 04 : for upper dipole. (14b)

In equations 14a and 14b, the intercepts of the linear function have no meaning because these are relative quantities, 

but the slopes do represent the attenuation dependence on 

hydrate saturation. Equations 14a and 14b yield the difference 

of attenuation at Sh = 0 percent and Sh = 100 percent, which is 

∆QP

–1 = 0.02 for P-waves and ∆QS–1 = 0.04 for S-waves. These 

values are three or four times smaller than those calculated by 

using equations 13a and 13b.

The difference of attenuation plotted in figure 3B shows 

much scattering, so the estimates are not accurate. However, 

figures 3A and 3B suggest that the measured attenuation with 

hydrate saturation given by Guerin and others (2005) is possibly overestimated by at least a factor of two.

Discussion 
Intrinsic Attenuation and Source Coupling

The objective of source-coupling analysis is determining 

whether the observed amplitude loss of sonic waveforms at the 

Mallik 5L-38 well site is primarily due to intrinsic attenuation 

or to source coupling. If source coupling is the dominant cause 

of the amplitude loss, the attenuation of the sonic waveforms 

measured for GHBS may not increase with increasing gas 

hydrate saturation. As discussed previously in this report, attenuation measurements using low-frequency sources with constant 

source environments support the view that the attenuation of 

the sonic waveforms measured for GHBS is less than that in 

NGHBS, whereas measurements using high-frequency sources 

with varying source environments support the opposite view. 

The results of source-coupling analysis still support the analysis of Guerin and others (2005), but the magnitude of intrinsic 

attenuation corrected for the source coupling is, at most, less 

than one-half that estimated by Guerin and others (2005).

The present study demonstrates that the amplitude variation observed at the Mallik 5L-38 well qualitatively agrees 

with that predicted by using source coupling. Furthermore, 

this study suggests that all analyses of amplitudes of sonic 

waveforms should include the effect of source coupling in 

order to accurately estimate the intrinsic attenuation of GHBS. 

The source-coupling formulation as presented here can be 

improved by (1) using high-frequency and near-field equations to model the source coupling, (2) including receiver 

coupling, and (3) analyzing frequency dependence of source 

and receiver coupling. These improvements could yield more 

accurate assessment of the intrinsic attenuation of GHBS.

If the measurements of the two opposing views relative 

to attenuations are both correct, the mechanism of attenuation 

in the high-frequency range may be different from that in the 

low-frequency range. In that case, the source coupling may 

only reduce the amount of attenuation, but may not change the 

slope of the plot in figure 3B. However, if it is assumed that the 

analysis of high-frequency measurement is erroneous owing to 

effects other than the intrinsic attenuation, other effects such as 

a source coupling should be investigated. The source coupling 

presented in this report could not have accounted for all the 

amplitude losses in the sonic waveforms because of approximated solutions. Although it is uncertain whether an improved 

formulation of the source coupling can resolve the controversy 

over the attenuation of sonic waveforms measured for GHBS, 

it is worth pursuing further, because there are evidences—

although only in the low-frequency ranges—that high-frequency 

attenuation measurements in the borehole may be erroneous.

In order to reconcile the different interpretations of 

amplitude loss of sonic waveforms measured for GHBS and to 

investigate detailed mechanisms of attenuation of GHBS, more 

controlled experiments are required. As indicated in Hearst 

and others (2000), the dispersive nature of propagation modes 

due to the presence of a fluid-filled borehole makes data 

analysis of sonic waveforms complicated; one complication is 

the introduction of amplitude loss that is difficult to relate to 

the intrinsic attenuation of the formation.

Conclusions

This study demonstrates that the observed amplitude 

loss of sonic waveforms measured at the Mallik 5L-38 well 

is due to source coupling as well as to intrinsic attenuation of 

GHBS. The source coupling accounts for at least half of the 

observed amplitude loss due to gas hydrate at this well site. A 

consideration of the source coupling when analyzing a sonic 

waveform’s amplitude loss indicates that both P-wave and 

S-wave attenuations are not as high as previously estimated 

by Guerin and others (2005). The analysis presented herein 

emphasizes the importance of source coupling in analyzing 

waveforms that were acquired either by using borehole acoustic/seismic sources, such as monopole and dipole sources for 

the logging, or by using borehole sources for hole-to-hole 

seismic experiments.

The attenuation of GHBS is an important attribute to 

consider when attempting to identify and quantify in place 

gas hydrate through the use of seismic methods. Regardless of 

whether the attenuation increases or decreases with increasing gas hydrate saturation, the fact that such changes do occur 

significantly affects how the seismic data acquired for GHBS 

are interpreted. Data analysis of sonic waveforms is complicated by the dispersive nature of the propagation modes, 

which introduces amplitude loss that is difficult to relate to 

the intrinsic attenuation of the formation. Therefore, to fully 

understand the attenuation mechanism and to accurately measure the intrinsic attenuation of GHBS, controlled laboratory 

experiments are required.

Acknowledgments

Well logs at the Mallik 5L-38 gas hydrate research well 

and VSP data at the Mallik 3L-38 well were acquired by Schlumberger Ltd. The well was drilled to investigate gas hydrates 

in a collaborative research project among Japan National Oil 

Company, Japan Petroleum Exploration Company, Geological 

Survey of Canada, U.S. Geological Survey, GeoForschungsZentrum Potsdam, India Ministry of Petroleum and Gas, 

BP-ChevronTexaco-Burlington, U.S. Department of Energy, 

and the International Continental Scientific Drilling Program. 

I thank William Waite and John Miller for many discussions 

and helpful suggestions.

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10 Is Amplitude Loss of Sonic Waveforms Due to Intrinsic Attenuation or Source Coupling to the Medium?
To understand the elastic properties of frozen sediments 

and to get some insight into the attenuation mechanism of 

gas-hydrate-bearing sediments, Prasad and Dvorkin (2004) 

performed pulse transmission experiments in brine for temperatures ranging from 18° to –21°C. Appendix 2 analyzes 

the amplitude loss of unfrozen brine (temperatures from 18° 

to –3°C) in order to assess the source-coupling effect on the 

amplitude of P-waves.

The radiation of a P-wave from a point source in 

unbounded water (UP) can be written as (Tolstoy and Clay, 

1966)

UP

w

w

μ

a

r (1–1)

where ρw and αw are the density and P-wave velocity of water, 

respectively. As temperature increases, the density of brine 

decreases and velocity increases.

Figure 1–1 shows the normalized measured amplitude and 

normalized amplitude variation due to source coupling to the 

brine. In this calculation, measured velocities by Prasad and 

Dvorkin (2004) and calculated densities accounting for temperature variation were used. Figure 1–1 indicates that about 

half of the measured amplitude loss is due to source coupling.

Appendix 1. Amplitude Loss of Compressional Waves in Brine Due to

Source Coupling

P-WAVE VELOCITY,

IN KILOMETERS PER SECOND

0.96

0.98

1.42 1.44 1.46 1.48 1.50

1.00

NORMALIZED AMPLITUDE

Measured amplitude

Amplitude using coupling

Figure 1–1. P-wave amplitude loss due to source coupling is 

compared to the amplitude loss inside brine measured by Prasad 

and Dvorkin (2004).

Appendix 1. Amplitude Loss of Compressional Waves in Brine Due to Source Coupling 11
Attenuation is assumed to be proportional to the first 

power of frequency (Schön, 1998) and is given by

A f j HA f fj R H j A f jRj ( )= 0 ( )exp(- o l ) (0 )exp( ) -a , (2–1)

where Aj(f) is the amplitude spectrum at distance Ri, f is the 

linear frequency, αj is the attenuation coefficient (αj = λj f), A0

is the input or reference amplitude spectrum, and H is a constant representing frequency-independent amplitude loss such 

as loss due to the geometrical spreading.

If a constant-input wavelet is assumed, the attenuation 

coefficient can be computed with the following equation:

ln ( ) ln A f

A f H fR a bf j

j j

0 ( )

æ

è

ç

ç

çç

ö

ø

÷

÷

÷

÷

= -l o -

, (2–2)

where a and b are the intercept and the slope, respectively, of 

the least-squares fit of the amplitude ratio.

The quality factor (Q), which is the inverse of attenuation, is calculated as (Schön, 1998) follows:

Q f

V

R

bV = = pa

p

, (2–3)

where V is the interval velocity.

Figure 2–1 shows the spectra ratios of A875 to A560 (a 

line connected with open circles) and A1115 to A560 (a line 

connected with dots), where A560, A875, and A1115 are the 

amplitude spectra at depths of 560 m, 875 m, and 1,115 m, 

respectively. Sediments at depths between 560 m and 875 

m are water-saturated sediments or non-gas-hydrate-bearing 

sediments (NGHBS), and sediments at depths between 875 

m and 1,115 m are mostly gas-hydrate-bearing sediments 

(GHBS) (actually 890 m and 1,109 m are the exact top and 

bottom of GHBS at Mallik 3L-38).

By using an average velocity of 2.07 km/s and equation 2–2, with b1

 = 6 × 10–3 and R = 315 m, the quality factor 

determined for NGHBS is 80. The attenuation between 875 m 

and 1,115 m can be calculated with the spectra ratios shown 

in figure 2–1. By using the average velocity of 2.54 km/s, R = 

240 m, and b = 1.9 × 10–3, equation 2–1 yields the coefficient 

of b for GHBS as b = b2 – b1, and the quality factor of GHBS 

is 156. Thus, the attenuation (Q–1) of NGHBS (0.0125) is 

higher than that of GHBS (0.0064).

Traveltime Inversion

If a layered medium is assumed, the first arrival times 

of two offset VSP data can be used to derive interval velocities and dip angles (Lee, 1990). The arrival times and offsets 

used in the present study are from Milkereit and others (2005). 

Vertical components of the zero-offset and offset P-wave VSP 

data were used to derive P-wave velocity, assuming that the 

dip angles are negligible.

As shown in figure 2–2, estimated P-wave interval 

velocities from the near-offset VSP data (offset = –22 m) are 

3.26, 2.09, and 2.56 km/s for the permafrost, NGHBS, and 

GHBS, respectively. Figure 2–2 also shows the interval velocities derived from the traveltime inversion by using the three 

offset VSP data (–22-, 83-, and 257-m offsets) and by using 

sonic log data at the Mallik 5L-38 well. The general trend 

of the interval velocities from the offset VSP data agrees 

well with that from the sonic log. The average velocity for 

GHBS derived from the offset VSP data is 2.56 km/s, which 

is the same as the velocity derived from the near-offset data, 

whereas the average velocity is 2.66 km/s for the sonic log at 

the Mallik 5L-38 well. The P-wave velocity from the sonic 

log is about 4 percent faster than that derived from the VSP 

data. Although velocity dispersion is assumed to cause the 

discrepancy between the sonic log and VSP results, the velocity dispersion in this case is small. Heterogeneity within the 

medium may cause the disagreement, but this is not certain.

Appendix 2. Attenuation at Mallik 3L-38

FREQUENCY, IN CYCLES PER SECOND

0.1

0 20 40 60 80

1

RATIO OF AMPLITUDE SPECTRA

Ratio A875 / A560

Ratio A1115 / A560

b2 = 7.9 × 10–3 or α2 = (1.43 × 10–5)f

b1 = 6 × 10–3 or α2 = (1.9 × 10–5)f

Figure 2–1. Spectral ratios at the Mallik 3L-58 well, 45 m 

southwest of Mallik 5L-38 well. A560, A875, and A1115 are amplitude 

spectra at the depths of 560 m, 875 m, and 1,115 m, respectively.

12  Is Amplitude Loss of Sonic Waveforms Due to Intrinsic Attenuation or Source Coupling to the Medium?
DEPTH, IN METERS

400 600 800 1,000 1,200

4

3

2

1

P-WAVE VELOCITY, IN KILOMETERS PER SECOND

Well-log velocity at Mallik 5L-38

VSP inversion using –22-, 83-, and 257-m offset

Average velocity from –22-m offset

Water saturated

Permafrost

Gas-hydrate bearing

Figure 2–2. A graph showing velocities derived from the VSP 

data at the Mallik 3L-58 well and well-log velocities measured at 

the Mallik 5L-38.

Appendix 2. Attenuation at Mallik 3L-38 13