e

U.S. Department of the Interior

U.S. Geological Survey,

Amplitude Variation of Bottom Simulating Reflection with 

Respect to Frequency—Transitional Base or Attenuation?

AP2

ep

2

Scientific Investigations Report 2007–5091
Amplitude Variation of Bottom Simulating 

Reflection with Respect to Frequency—

Transitional Base or Attenuation?

By Myung W. Lee

Scientific Investigations Report 2007–5091

U.S. Department of the Interior

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

DIRK KEMPTHORNE, Secretary

U.S. Geological Survey

Mark D. Myers, Director

U.S. Geological Survey, Reston, Virginia: 2007

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Suggested citation:

Lee, M.W., 2007, Amplitude variation of bottom simulating reflection with respect to frequency—Transitional base or 

attenuation?: U.S. Geological Survey Scientific Investigations Report 2007–5091, 12 p.
iii

Contents

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

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

Theory and Models........................................................................................................................................2

Velocity of Partially Gas Saturated Sediments ................................................................................2

Transitional Base Model......................................................................................................................2

Attenuation Model................................................................................................................................4

BSR Amplitude from Real Seismic Data ....................................................................................................4

Analysis and Results .....................................................................................................................................4

Strength of BSR Amplitude .................................................................................................................4

Analysis and Interpretation of Observed BSR Amplitude..............................................................4

Attenuation and Transitional Base.....................................................................................................6

Preferred Model....................................................................................................................................7

Constraints on the Model ....................................................................................................................8

Conclusions.....................................................................................................................................................9

References Cited............................................................................................................................................9

Appendix. Reflections from Transition Zones......................................................................................11

Figures

1. Graph showing relationship among P-wave velocity, reflection coefficient, 

and frequency ...............................................................................................................................2

2. Graph showing seismic model for bottom simulating reflections ........................................3

3. Synthetic bottom simulating reflection (BSR) seismograms with different 

frequencies using a transitional base model...........................................................................3

4. Synthetic bottom simulating reflection (BSR) seismograms with different 

frequencies using attenuation models......................................................................................4

5. Seismic profiles showing detailed bottom simulating reflections (BSR) ............................5

6. Cross plot of the observed bottom simulating reflection (BSR) amplitude.........................6

7–10. Graphs showing:

 7. A theoretical bottom simulating reflection (BSR) amplitude reduction 

without velocity dispersion with respect to the seismic frequency range..................6

 8. A theoretical bottom simulating reflection (BSR) amplitude reduction 

with velocity dispersion with respect to the seismic frequency range .......................7

 9. A theoretical bottom simulating reflection (BSR) amplitude reduction for 

a preferred model with respect to the seismic frequency range..................................8

 10. Relationships between reflection amplitude and transitional thickness...................12
Abstract

The amplitude of a bottom simulating reflection (BSR), 

which occurs near the phase boundary between gas hydratebearing sediments and underlying gas-filled sediments, 

strongly depends on the frequency content of a seismic signal, 

as well as the impedance contrast across the phase boundary. 

A strong-amplitude BSR, detectable in a conventional seismic 

profile, is a good indicator of the presence of free gas beneath 

the phase boundary. However, the BSR as observed in lowfrequency multichannel seismic data is generally difficult to 

identify in high-frequency, single-channel seismic data.

To investigate the frequency dependence of BSR amplitudes, single-channel seismic data acquired with an air gun 

source at Blake Ridge, which is located off the shore of 

South Carolina, were analyzed in the frequency range of 

10–240 Hz. The frequency-dependent impedance contrast 

caused by the velocity dispersion in partially gas saturated 

sediments is important to accurately analyze BSR amplitude. 

Analysis indicates that seismic attenuation of gas hydratebearing sediments, velocity dispersion, and a transitional base 

all contribute to the frequency-dependent BSR amplitude 

variation in the frequency range of 10–500 Hz. When velocity 

dispersion is incorporated into the BSR amplitude analysis, the 

frequency-dependent BSR amplitude at Blake Ridge can be 

explained with gas hydrate-bearing sediments having a quality 

factor of about 250 and a transitional base with a thickness of 

about 1 meter.

Introduction

Gas hydrates, or clathrates, are ice-like crystalline solids 

composed of water molecules surrounding a gas molecule. 

The importance of gas hydrates as a potential energy source, 

as a greenhouse gas, and as a hazard for sea floor stability has 

been investigated (Kvenvolden, 1988; Dillon and others, 1991; 

Paull and others, 1991; Collett, 1996).

Gas hydrates form at appropriate temperature and pressure conditions when water is saturated with gas, and they are 

stable within the upper several hundred meters of sediments 

below sea floor (Claypool and Kaplan, 1974). The presence 

of methane gas hydrates in ocean sediments has generally 

been inferred from the bottom simulating reflection (BSR) in 

seismic profiles (Markl and others, 1970; Shipley and others, 

1979). The BSR simulates the surface of sea floor because 

the base of hydrate phase stability or phase boundary parallels 

the sea floor.

Hyndman and Spencer (1992) and Hyndman and Davis 

(1992) presented a model in which a BSR is generated by 

impedance contrast between sediments that have about 30 percent of the pore space filled with gas hydrate and the underlying 

sediments containing neither hydrate nor seismically detectable 

free gas. Their conclusion is based on an analysis of the reflection coefficient, vertical-incidence waveform, amplitude-versusoffset, and velocity structure. The best fit model for the seismic 

data is gas hydrate-bearing sediments (GHBS) having a sharp 

base and transitional top.

Except in the Hyndman and Spencer (1992) model, 

however, the high-amplitude BSR is generally interpreted as 

a reflection from a boundary between GHBS and underlying 

gas-filled sediments near the phase boundary (Shipley and 

others, 1979; Andreassen and Berteussen, 1990; Miller and 

others, 1991; Dillon and others, 1994). The Ocean Drilling 

Program (ODP) Leg 164, Sites 995 and 997 (Shipboard 

Scientific Party, 1996), proved the existence of free gas 

beneath the GHBS and confirmed that the BSR is caused by 

a marked impedance contrast across the methane hydrate 

phase boundary.

Lee and others (1994) demonstrated that the amplitude of 

BSR depends largely on the frequency content of the source 

signal. When a very high frequency seismic source is used, the 

continuous looking BSR in the low-frequency, multichannel 

seismic data appears as discontinuous and weak reflections 

(Lee and others, 1994; Gettrust and others, 1999). Chapman 

and others (2002) attributed the frequency-dependent amplitude of BSR, observed in the offshore of western Canada, to a 

transitional layer 4 to 8 m thick.

In order to examine the BSR amplitude characteristics 

with respect to the frequency content, single-channel seismic data from the Blake Ridge area acquired with an air gun 

source were analyzed. Two predictive models, one having a 

transitional layer at the base of GHBS and one with seismic 

attenuation within GHBS, were analyzed in terms of BSR 

amplitude and frequency.

Amplitude Variation of Bottom Simulating Reflection with 

Respect to Frequency—Transitional Base or Attenuation?

By Myung W. Lee
Theory and Models

Velocity of Partially Gas Saturated Sediments

Because the BSR occurs at the impedance contrast 

between GHBS and partially gas saturated sediments (PGSS), 

it is important to examine the behavior of P-wave velocities 

of PGSS with respect to frequency. The P-wave velocity of 

partially gas saturated sediments with respect to frequency 

can be predicted using a model proposed by White (1975). 

The essence of this model is that it includes coupling between 

fluid-flow waves and seismic body waves; also, it assumes that 

the medium consists of concentric spheres, where the outer 

part of the sphere is water saturated and the inner part is gas 

saturated. One important parameter of the model is the radius 

of the outer spherical shell, indicated as parameter b, which is 

a free parameter that can be adjusted to fit the measured dispersion relation. The radius of the inner sphere depends on the 

gas saturation and parameter b. For typical seismic frequencies, the White (1975) model predicts large P-wave dispersion 

when free gas saturation is low.

Figure 1 shows calculated P-wave velocities for the 

partially gas saturated sediments (PGSS), with the following parameters: porosity (), 0.55; clay volume content (Cv

), 

0.4; differential pressure (p), 6 MPa; gas saturation (Sg), 0.04; 

b = 3 cm; and permeability, 100 mD. The bulk and shear 

moduli of dry sediments were computed using the modified 

Biot-Gassmann theory proposed by Lee (2002).

The P-wave velocity at low frequencies, < 30 Hz, is 

1.1 km/s, whereas at high frequencies, > 200 Hz, it is about 

1.5 km/s. At logging frequencies in the range of 10 kHz, the 

P-wave velocity is about 1.6 km/s, which agrees with the 

well log velocities acquired at the Blake Ridge (Guerin and 

others, 1999). The transition frequency between high and 

low velocities depends on parameter b: as b decreases, the 

transition frequency increases. Because the P-wave velocity varies with frequency, the impedance contrast also varies 

with frequency.

Figure 1 also shows reflection coefficients with respect to 

frequency, assuming that the P-wave velocity of upper GHBS 

is 1.9 km/s. The reflection coefficient is about 0.3 at frequencies that are less than about 20 Hz, and then decreases as the 

frequency increases. The reflection coefficient is less than 0.1 

at frequencies greater than about 400 Hz. This implies that 

without any additional factors controlling the BSR amplitude 

with frequency, velocity dispersion alone decreases the BSR 

amplitude by about 3 times.

Transitional Base Model

Reflection characteristics from the transitional layer are 

given by Justice and Zuba (1986). To analyze BSR amplitudes, 

a symmetrical Ricker wavelet is used. A second-order transition 

zone, which is defined in such a way that the starting and ending velocities of the transition layer are equal to the upper- and 

lower-layer velocities respectively, is considered. The detailed 

reflection characteristics from a second-order transitional layer 

using the Ricker wavelet are given in the Appendix.

The reflected waveform (S(t)) from the transitional 

boundary is given by

S t V V

T ( ) r d ln( / ) = ( ) ∫ 1 02

τ τ (1)

where

V0 is the velocity at the top of the transition 

zone,

V1 is the velocity at the bottom of the transition 

zone, 

and

T is the width of the transition zone in twoway time.

Figure 1. Relationship among P-wave velocity, reflection 

coefficient, and frequency. The bulk and shear moduli of dry 

rock for this model are 0.7 GPa and 0.5 GPa, respectively, 

which were calculated using the modified Biot-Gassmann theory 

proposed by Lee (2002). , porosity; Cv , clay volume; p, differential 

pressure; GPa, gigapascal; Sg, gas saturation; b, adjustable 

parameter.

P-WAVE VELOCITY, IN KILOMETERS PER SECOND AND 

REFLECTION COEFFICIENT

FREQUENCY, IN HERTZ

Velocity

White (1975) model parameters;

φ = 0.55, Cv = 0.4

p = 6 MPa, Sg = 0.04

Solid line: b = 3 cm, Dotted line: b = 1 cm

Reflection coefficient

2

1

0.1

1 10 100 1,000 10,000

 Amplitude Variation of Bottom Simulating Reflection with Respect to Frequency
By integration of equation 1, the maximum amplitude 

expected from the transitional layer can be derived. Three different cases are considered:

The duration of the wavelet is similar to or longer than 

the transitional layer thickness. In this case, the extreme 

occurs at t = –T / 2 and the extreme is given by

A cG T ce T

max

( / ) = = ( / ) − 2 2 2 α (2)

where c = ln(V1 / V0).

The duration of the input wavelet is much shorter than the 

transition thickness. In this case, there is no interference 

between the top and bottom reflections. Therefore, two 

extreme amplitude points exist, as given by the following 

equation:

A c

T

e max

/ = 1 −

2

1 2

α (3)

If the duration of the input wavelet is between case 1 and 

case 2, there is no simple analytic solution.

In order to investigate the BSR amplitude with respect 

to the input frequency, a simple BSR model, qualitatively 

similar to the observed BSR at the Blake Ridge, was generated, as shown in figure 2. The model consists of (1) a section of GHBS about 150 m thick with a variable second-order 

transitional base; (2) velocity increased for the GHBS from 

mean values of 2.05 km/s at depth of 150 m to 2.25 km/s at 

1.

2.

3.

Figure 2. Seismic model for bottom simulating reflections (BSR). 

Model consists of gas hydrate-bearing sediments (GHBS) having 

a linearly increasing velocity with depth and a transitional base at 

the bottom of GHBS and underlying gas-filled sediments having 

a constant velocity. High-frequency random velocity fluctuations 

are superimposed on the basic BSR model and are ±0.1 km/s 

for gas hydrate-bearing sediment and ±0.2 km/s for gas-filled 

sediments. Q, quality factor.

Figure 3. Synthetic bottom simulating reflection (BSR) seismograms 

with different frequencies using a transitional base model. The 

left four plots are synthetic waveforms from a sharp boundary, 

and the right four plots are those from a 5-m thick transition layer. 

A, 10–60 Hz; B, 10–120 Hz; C, 10–240 Hz; D, 10–480 Hz.

VELOCITY, IN KILOMETERS PER SECOND

DEPTH, IN METERS

2.8

2.4

2.0

1.6

1.2

0 100 200 300 400

Q = 2000

Q = 50

Q = 100, 250

Hydrated interval

Transitional base

Free gas

Transitional Base Model

Sharp boundary Transitional thickness 5 m

 

A B C D A B C D

BSR

TIME, IN MILLISECONDS

Sea floor

Theory and Models   

depth of 300 m; and (3) a PGSS section with a velocity of 

1.6 km/s underlying the transitional layer. To mimic variable 

velocities of GHBS and PGSS, random velocity fluctuations 

of 0.1 km/s and 0.2 km/s are superimposed on GHBS and 

PGSS, respectively.

Figure 3 shows synthetic seismograms for a sharp 

boundary and for a second-order transitional layer having 

5 m thickness. The frequency contents of the synthetics are 

10–60 Hz, 10–120 Hz, 10–240 Hz, and 10–480 Hz for A, B, 

C, and D respectively. For reflections from a sharp boundary 

(fig. 3), the BSR maintains the same waveform and amplitude 

except narrowing of the waveform with increasing frequency 

content. The basic trough waveform can be readily identified irrespective of the frequency content. However, the 

characteristics of a BSR from the transitional layer are more 

complicated: as the frequency content increases, the amplitudes of the BSR decrease and it becomes more difficult to 

identify the BSR. Because the only difference between the two 

models is the characteristics of the reflector at the BSR level, 

other reflections are almost identical for each frequency band 

(fig. 3). Note that a small difference does exist because the 

interbed multiple contributions are slightly different.
Attenuation Model

Another mechanism of reducing amplitude of the BSR 

with respect to frequency is seismic attenuation. Two different 

attenuation models (quality factors of 100 and 250 for GHBS) 

are shown in figure 2, and the synthetic seismograms including attenuation are shown in figure 4. In this case, the amplitudes of BSR decrease as the frequency content increases, 

like the model for the transitional layer (fig. 3), but the basic 

waveforms are almost identical irrespective of the frequency 

content. The broadened waveform of the BSR observed for the 

transitional model is not apparent in the attenuation model.

BSR Amplitude from Real Seismic Data

Single-channel, vertical-incidence reflection data acquired 

at Blake Ridge using an air gun source are shown in figure 5 

for three different frequency bands: 10–60 Hz, 10–120 Hz, 

10–240 Hz. Qualitatively, the amplitude ratio of the water bottom 

reflection to BSR decreases as the frequency content increases, 

as demonstrated in the BSR model study shown in figure 4.

Both a root-mean-square (RMS) amplitude and a troughpeak amplitude within a 20-ms to 40-ms window were measured in order to quantify the amplitude variation of BSR with 

respect to the frequency content. For the frequency ranges 

of 10 to 60 Hz, 10 to 120 Hz, and 10 to 240 Hz, the ratios 

of RMS amplitude of water bottom reflection to the BSR 

reflection are 1.09 ± 0.24, 0.79 ± 0.19, and 0.6229 ± 0.15; and 

those when trough-peak amplitude is used are 1.07 ± 0.28, 

0.63 ± 0.18, and 0.51 ± 0.16 for 10–60 Hz. As the dominant 

frequency changes from 30 to 120 Hz, the strength of BSR 

reflection with respect to the water bottom reflection decreases 

about 40 percent when the RMS amplitude measurements are 

used and about 50 percent when the peak-trough amplitude 

measurements are used.

Analysis and Results

Strength of BSR Amplitude

The strength of the BSR amplitude chiefly depends on 

the impedance contrast, the characteristics at the base of the 

GHBS, and attenuation of the GHBS. Analyses indicate that 

the strong BSR amplitudes observed in many low-frequency, 

multichannel stacked seismic profiles are caused by a marked 

impedance contrast between GHBS and PGSS (Miller and 

others, 1991; Dillon and others, 1994; Lee and others, 1994).

The first-order effect on BSR amplitude is the impedance 

contrast. The BSR reflection coefficient in the Blake Ridge 

area is on the order of 0.1 to 0.2 (Lee and others, 1994), which 

was estimated from conventional seismic data with a dominant frequency of 30 Hz. An important question regarding the 

BSR amplitude has been: Why is it difficult to detect the BSR 

when a high-frequency source greater than about 200 Hz is 

used? (Or, similarly, why does the strength of BSR amplitude 

change significantly with respect to the frequency?) Because 

the reflection coefficient for the BSR depends on frequency, as 

demonstrated in figure 1, some of the BSR amplitude variation 

can be attributed to the velocity dispersion of PGSS, but this 

frequency-dependent reflection coefficient is not sufficient to 

fully explain the observed amplitude variation of the BSR.

Analysis and Interpretation of Observed 

BSR Amplitude

Figure 6 shows a cross plot of the BSR amplitude 

observed at the dominant frequency of 60 Hz versus that 

observed at 120 Hz, both of which were calculated using the 

RMS amplitudes. A dominant frequency of band-pass filtered 

seismic data was assigned by the method suggested by Lee 

(1987), whereby the dominant frequency of the band-pass 

filtered data is about half of the high end of the frequency 

spectrum. Therefore, the dominant frequencies for the seismic 

section filtered with 10–60 Hz, 10–120 Hz, and 10–240 Hz 

are assigned as dominant frequencies of 30 Hz, 60 Hz, and 

Q = 100 Q = 250

A B C D A B C D

Attenuation Model

BSR

Sea Floor

TIME, IN MILLISECONDS

Figure 4. Synthetic bottom simulating reflection (BSR) seismograms 

with different frequencies using attenuation models having a 

sharp boundary. The left four plots are synthetic waveforms with 

Q = 100, and the right four plots are those with Q = 250. A, 10–60 Hz; 

B, 10–120 Hz; C, 10–240 Hz; D, 10–480 Hz. Q, quality factor.

   Amplitude Variation of Bottom Simulating Reflection with Respect to Frequency
120 Hz, respectively. The cross plot (fig. 6) indicates that the 

BSR amplitude with respect to the dominant frequency is consistent among the traces and varies by a factor of 2.

The BSR amplitudes shown in figures 7, 8, and 9 are 

normalized by the BSR amplitude measured at the dominant 

frequency of 30 Hz. The average normalized BSR amplitudes 

calculated from the RMS amplitude are 0.73 ± 0.11 and 

0.58 ± 0.10 at the dominant frequency of 60 Hz and 120 Hz, 

respectively. When the peak-trough amplitude is used, they 

are 0.60 ± 0.15 and 0.48 ± 0.14 at the dominant frequency of 

60 Hz and 120 Hz, respectively.

One way to explain the difference of the BSR amplitude 

with respect to the dominant frequency is by assuming attenuation within GHBS sediments. The result of an attenuation 

model for the BSR is shown in figure 7A, using the following 

amplitude relationship:

A e

f t

Q

p ∝  −











π (4)

where

Q is the quality factor 

and

t is the two-way travel time.

To compare the amplitude variation with frequency for sediments having different Q and t, a parameter Λ (defined as Λ 

= t / Q) is used. Figure 7A shows the predicted normalized 

reflection amplitude using various values for Λ (5, 2.5, 1.25, 

0.5, and 0.25 ms) with measured normalized BSR amplitudes. The amplitudes of BSR analyzed by Chapman and 

others (2002), also shown in figure 7A for comparison, are 

normalized by the amplitude at 30 Hz, but, because theirs are 

estimated from the plot, they are not very accurate. Note that 

two-way travel time for the Blake Ridge data is ≈ 500 ms and 

≈ 250 ms for data analyzed by Chapman and others (2002). 

If the attenuation model is applicable to the observed BSR 

amplitude variation, figure 7A indicates that values for Q

between 100 and 300 for Blake Ridge amplitude-frequency 

data and between 50 and 150 for offshore western Canada data 

fit the amplitude-frequency data quite well.

Another way to explain the difference of the BSR amplitude with respect to the dominant frequency is by assuming 

a transitional base for the GHBS. The normalized reflection 

amplitudes predicted from the transitional base having an 

average velocity of 1.7 km/s are shown in figure 7B, which 

indicates that a transitional base of about 3 to 8 m would best 

fit the measured amplitude.

For the above modelings, the frequency-dependent 

reflection coefficients are not incorporated. Therefore, figure 7 

shows only the effects of attenuation and a transitional base on 

the BSR amplitude in the frequency range of 30 to 200 Hz.

Figure 5. Seismic profiles showing detailed bottom simulating reflections (BSR) acquired at different 

frequencies in the Blake Ridge area of offshore South Carolina using a single air gun source.

10–60 Hz 10–120 Hz 10–240 Hz

BSR

TIME, IN MILLISECONDS

FREQUENCY

SEA

FLOOR

1,500

2,000 2,000

1,500

Analysis and Results   
Attenuation and Transitional Base

Near the Black Ridge study area, Wood and others (2000) 

investigated the longitudinal wave (P-wave) attenuation from 

high-frequency single-channel and VSP data using a spectral modeling technique. The estimated Q is 90–500, which 

is only slightly lower than the Q values expected for nongas hydrate-bearing sediments. One of the results from the 

analysis of single-channel data indicates that the average Q for 

GHBS exceeds 300 (Wood and others, 2000). Brienzo (1992) 

estimated a Q value of 660 at the sea floor to 175 to 750 m 

below the sea floor in the Monterey, Calif., deep-sea fan for 

possible GHBS. On the basis of vertical seismic profile data at 

the Mallik 5L-38 well, Mackenzie Delta, Canada, Lee (2006) 

estimated attenuation of GHBS to be Q ≈150.

The results shown in figure 7A indicate that Q values 

between 100 and 300 for Blake Ridge data and Q values 

between 50 and 150 for offshore western Canada data appear 

to be close to, or possibly a little lower than, most GHBS 

analyzed by various investigators. Considering uncertainties 

associated with estimated attenuation and the amplitude of the 

BSR, the attenuation is still a feasible mechanism to explain 

the frequency-dependent BSR amplitudes.

NORMALIZED AMPLITUDE AT 120 HERTZ

NORMALIZED AMPLITUDE AT 60 HERTZ

0.8

0.7

0.6

0.5

0.4

0.3

0.4 0.5 0.6 0.7 0.8 0.9 1.0

Observed normalized BSR amplitude

Least-squares fit: A120 = 0.76A60– 0.002

Figure 6. Cross plot of the observed bottom simulating reflection 

(BSR) amplitude at the dominant frequency of 60 Hz relative to 

the observed BSR amplitude at 120 Hz. All BSR amplitudes are 

normalized to 30 Hz and root-mean-square amplitudes were 

used to compute BSR amplitudes. A linear regression line with a 

correlation coefficient r = 0.915 is shown as a solid line.

Figure 7. A theoretical bottom simulating reflection (BSR) amplitude reduction without velocity dispersion with respect to the frequency 

in the range of 20 to 200 Hz. All BSR amplitudes are normalized to the BSR amplitude at 30 Hz. Observed BSR amplitudes (root-meansquare (RMS) amplitude and peak-trough amplitude) and amplitudes analyzed by Chapman and others (2002) are also shown. A, 

attenuation model.  is defined as t/Q, where t is two-way travel time in milliseconds and Q is quality factor; B, transitional base model. NORMALIZED AMPLITUDE NORMALIZED AMPLITUDE

DOMINANT FREQUENCY, IN HERTZ DOMINANT FREQUENCY, IN HERTZ

Λ = 0.5 ms

Λ = 0.25 ms

Λ = 1.25 ms

Λ = 5 ms

Λ = 2.5 ms

Observed BSR amplitude (RMS)

Observed BSR amplitude (trough-peak)

Chapman and others (2002)

Observed BSR amplitude (RMS)

Observed BSR amplitude (trough-peak)

Chapman and others (2002)

16 m 10 m 6 m

4 m

Transitional layer

 thickness = 2 m

A B

1.0

0.8

0.6

0.4

0.2

0.0

1.0

0.8

0.6

0.4

0.2

0.0

0 50 100 150 200 0 50 100 150 200

   Amplitude Variation of Bottom Simulating Reflection with Respect to Frequency
Well logs, along with acoustic and electrical resistivity logs, indicate that Ocean Drilling program (ODP) site 995 

implies a transitional base with a thickness of about 20 m, 

but sites 997 and 994 imply a much shorter transitional base 

(Shipboard Scientific Party, 1996). Travel time inversion from 

the vertical seismic profiles (Holbrook and others, 1996) implies 

a large transitional layer, on the order of 50 m, but the seismic 

resolution is not high enough to determine the accuracy of this 

assessment. The chloride anomaly at site 994 implies a transitional base about 40 m thick and a much shorter or abrupt base 

at site 997 (Paull and Matsumoto, 2000), but data are insufficient 

to warrant a detailed analysis of the base of the GHBS.

Although the accuracy of the transitional base is debatable, it appears that all available information favors some kind 

of transitional base for GHBS. Therefore, the transitional layer 

with a thickness of 3–8 m is a reasonable assumption to interpret 

the behavior of the BSR amplitude with respect to frequency. 

Chapman and others (2002) also interpreted the amplitude variation of BSR by using a transition layer with 4–8 m thickness 

at the base of GHBS; however, they used a smooth transitional 

base rather than the sharp transitional base used in this report. 

Note that similar transitional layer thicknesses are interpreted 

for both the Blake Ridge and the offshore western Canada area.

Preferred Model

Although it is not clear which model—transitional base 

or attenuation—best fits the available data with respect to 

BSR, the following discussions, interpretations, and conclusions are derived from the present study. Figure 8 shows 

model results for attenuation and transitional base including 

the frequency-dependent reflection coefficients in the frequency range between 30 and 500 Hz. The amplitude decay 

with respect to frequency caused by the frequency-dependent 

reflection coefficient is much smaller than the measured BSR 

amplitude variations. The model using Λ = 1 ms (fig. 8A), 

which corresponds to Q = 500 for Blake Ridge data and 

Q = 250 for offshore western Canada data, predicts a much 

higher amplitude for the BSR, although the amplitude decay 

with respect to frequency is close to the observed decay. The 

predicted result using both Λ = 1 ms and velocity dispersion 

agrees well with measured BSR amplitude variation except for 

the amplitude at 450 Hz. Λ = 1 ms appears to be a reasonable 

value for frequency less than about 200 Hz, but appears to be 

high for frequency greater than 400 Hz (fig. 8A).

Figure 8B shows the model result using both the transitional base and velocity dispersion. Amplitude variation 

predicted using both a transitional base of 3 m and the velocity 

NORMALIZED AMPLITUDE

NORMALIZED AMPLITUDE

FREQUENCY, IN HERTZ FREQUENCY, IN HERTZ

1

0.1

0.01

1

0.1

0.01

10 100 1,000 10 100 1,000

Peak-trough amplitude (Blake Ridge)

RMS amplitude (Blake Ridge)

From Chapman and others (2002)

Based on velocity dispersion

Based on Λ = 1 ms

Based on Λ = 1 ms and velocity dispersion

Peak-trough amplitude (Blake Ridge)

RMS amplitude (Blake Ridge)

From Chapman and others (2002)

Based on velocity dispersion

Based on 3 m transitional layer

Based on 3 m transitional layer

 and velocity dispersion

A B

Figure 8. A theoretical bottom simulating reflection (BSR) amplitude reduction with velocity dispersion with respect to the 

frequency in the range of 20 to 500 Hz. Frequency-dependent reflection coefficient, which is caused by the P-wave velocity 

dispersion of partially gas saturated sediments, is used. All BSR amplitudes are normalized to the BSR amplitude at 30 Hz. Observed 

BSR amplitudes (root-mean-square (RMS) amplitude and peak-trough amplitude) and amplitudes analyzed by Chapman and others 

(2002) are also shown. A, attenuation model. Λ is defined as t/Q, where t is two-way travel time in milliseconds and Q is quality 

factor; B, transitional base model.

Analysis and Results   
Figure 9. A theoretical bottom simulating reflection (BSR) amplitude reduction for a preferred model with respect to the 

frequency in the range of 20 to 500 Hz. Frequency-dependent reflection coefficient, which is caused by the P-wave velocity 

dispersion of partially gas saturated sediments, is used. All BSR amplitudes are normalized to the BSR amplitude at 30 Hz. 

Observed BSR amplitudes (root-mean-square (RMS) amplitude and peak-trough amplitude) and amplitudes analyzed by Chapman 

and others (2002) are also shown. A, using the attenuation with b = 3 cm,  = 1 ms, and a transition zone (tr) with a thickness of 

1 m; B, using the attenuation with b = 1 cm, Λ = 0.6 ms, and a transition zone with a thickness of 1.5 m.  = t/Q, where t is two-way 

travel time in milliseconds and Q is quality factor. NORMALIZED AMPLITUDE

NORMALIZED AMPLITUDE

FREQUENCY, IN HERTZ FREQUENCY, IN HERTZ

1

0.1

0.01

1

0.1

0.01

10 100 1,000 10 100 1,000

Peak-trough amplitude (Blake Ridge)

RMS amplitude (Blake Ridge)

From Chapman and others (2002)

Based on velocity dispersion (b = 3 cm)

Based on Λ = 1 ms, tr = 1 m & velocity dispersion

Peak-trough amplitude (Blake Ridge)

RMS amplitude (Blake Ridge)

From Chapman and others (2002)

Based on velocity dispersion (b = 1 cm)

Based on Λ = 1.7 ms, tr = 1.5 m & velocity dispersion

A B

dispersion appears to fit the measured amplitude variation 

reasonably well except for the amplitude at 450 Hz, where the 

model overestimates the amplitude decay.

Without considering the amplitude at 450 Hz, the 

attenuation model appears to best fit the measured amplitude. 

Figure 8 indicates that Λ = 1 ms appears to be high and the 

thickness of 3 m for the transitional layer appears to be too 

thick for high frequency around 450 Hz.

Figure 9 shows that predicted amplitude variations using 

both attenuation (Λ = 1 ms) and a transitional base (thickness 

of 1 m) agree with measured amplitudes for frequencies less 

than 500 Hz. The velocity dispersion of PGSS is a wellknown phenomenon, and seismic waves are always attenuated. 

Therefore, the velocity dispersion and attenuation should always 

be included in the amplitude analysis of BSR. The interpreted 

transitional base is thin, if both attenuation and velocity dispersion are included in the analysis. Based on the modeling and 

analysis, attenuation is the most significant effect on the BSR 

amplitude variation for all frequencies, and a transitional base 

plays a minor role only at frequencies greater than about 200 Hz.

Constraints on the Model

Model parameters shown in figure 9A (b = 3 cm, 

Λ = 1 ms, thickness of transition zone of 1 m) are interdependent. The velocity dispersion derived from the White (1975) 

model depends strongly on the parameter b. If b = 1 cm is 

used in the model, all other parameters should be changed 

to fit the observation. The P-wave velocity and reflection 

coefficient using b = 1 cm are shown in figure 1 as dotted 

lines. The velocity dispersion in the seismic frequency band 

less than 100 Hz is negligible, whereas the velocity dispersion using b = 3 cm is substantial (fig. 1). According to White 

(1975), the velocity dispersion of PGSS is noticeable in the 

seismic frequency band, so b = 3 cm may be the more appropriate parameter to use. The model result using b = 1 cm with 

other parameters adjusted to match the observed amplitude of 

the BSR, which are Λ = 1.7 ms and a transition zone thickness of 1.5 m (fig. 9B), indicates that when attenuation is 

incorporated into the model, the thickness of the transition 

zone is small.

   Amplitude Variation of Bottom Simulating Reflection with Respect to Frequency
The attenuations of Q = 150 or Q = 500 are plausible 

values for GHBS in the Blake Ridge and western Canada. 

However, although velocity dispersion using b = 1 cm and b

= 3 cm can be considered as extreme values, the actual value 

may be between b = 1 cm and b = 3 cm. Therefore, the thickness of the transition zone using plausible values of Q and b is 

between 1 m and 1.5 m, which is much less than the thicknesses estimated by Chapman and others (2002).

The transitional characteristics of the base of the GHBS 

may be related to the amount of gas hydrate saturated in the 

pore space. If a “recycle model” of methane gas hydrate to 

enrich the base of GHBS (Paull and others, 1994) is correct, the thickness of the transitional layer should be small, 

because the highest concentration of hydrate may exist near 

the phase boundary. However, the phase boundary is dynamic, 

reflecting effects such as sediment loading, sea level changes, 

thermal condition, and flux of methane gas. Therefore, the 

GHBS near the phase boundary are unstable or metastable; 

dissociation of gas hydrate and (or) formation of gas hydrate 

occur near the phase boundary. Consequently, the highest gas 

hydrate concentration may be a little above the phase boundary and a transitional velocity at the base of the GHBS may 

be possible.

According to Xu and Ruppel (1999), the rate of methane flux at the base of the gas hydrate stability zone controls 

the thickness of the transition zone. As the flux of methane 

increases, the transition zone thickness decreases. A sharp 

boundary is possible if the flux of methane exceeds a critical value. The interpretation of a 1- to 1.5-m thick transition 

zone for offshore western Canada, where a high average flux 

rate of 1.3 mm/yr was estimated by Hyndman and others 

(1993), differs from the 4–8 m of transition zone interpreted 

by Chapman and others (2002). Although it is unknown if 

the transition zone thickness of 1 m to 1.5 m is theoretically 

possible using the average methane flux estimated for offshore 

western Canada, modeling that includes attenuation restricts 

the thickness of the transition zone to around 1 m. Because the 

model parameters are interdependent, the interpretation of the 

characteristics of BSR amplitude requires caution.

Conclusions

The amplitude reduction of the bottom simulating 

reflection (BSR) with increasing frequency content is due to 

the seismic attenuation, the transitional characteristics of the 

base of gas hydrate-bearing sediments (GHBS), and velocity 

dispersion of partially gas saturated sediments (PGSS). The 

amplitude analysis of BSR with respect to the frequency and 

the model analysis favors the interpretation that attenuation is 

the dominant effect on the BSR amplitude variation. At Blake 

Ridge, a model based on Q between 300 and 500 for GHBS 

and a transitional thickness of ≈ 1 to 1.5 m fits well with the 

observed amplitude variation, whereas Q between 150 and 

300 with a transitional thickness of ≈ 1 to 1.5 m fits well with 

the amplitude measured offshore western Canada. Without 

incorporating attenuation and velocity dispersion in the analysis, a thicker transitional layer would need to be estimated. 

Because velocity dispersion and attenuation are inherent to 

GHBS and PGSS, all BSR amplitude analyses should include 

these effects regardless of a transitional layer. However, 

because parameters used to fit the amplitude characteristics 

of the BSR amplitude are interdependent, a judicious choice 

of parameters or a restricted range of possible parameters 

is important.

The BSR amplitude analysis in the present study is limited because only single-channel seismic data acquired with 

an air gun source in a limited frequency range < 250 Hz are 

used, although deep-towed seismic data with high frequency 

by Chapman and others (2002) are incorporated. To better 

constrain the analysis, multichannel seismic data with a lowfrequency source combined with single-channel seismic data 

with a variety of high-frequency sources shot along the identical line are preferred.

References Cited

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in the Southern Barents Sea, indicated by a shallow seismic 

anomaly: First Break, v. 8, p. 235–245.

Brienzo, R.K., 1992, Velocity and attenuation profile in the 

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America, v. 92, p. 2109–2125.

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G.D., Wood, W.T., and Hyndman, R.D., 2002, Highresolution, deep-towed, multichannel seismic survey of 

deep-sea gas hydrates off western Canada: Geophysics, 

v. 67, p. 1038–1047.

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ed., Natural gases in marine sediments: New York, Plenum 

Press, p. 99–139.

Collett, T.S., 1996, Gas hydrate resources of the United States, 

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K.L., eds., 1995 National assessment of United States oil 

and gas resources on CD-ROM: U.S. Geological Survey 

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D.R., and Swift, B.A., 1991, Mapping sub-seafloor reservoirs of greenhouse gas—Methane hydrate, in Kumar, 

M., and Maul, G.A., eds., Proceedings of International 

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Marine Geodesy Committee, Marine Technology Society, 

p. 545–554.

References Cited   
Dillon, W.P., Lee, M.W., and Coleman, D.F., 1994, Identification of marine hydrates in situ and their distribution off 

the Atlantic coast of the United States, in Sloan, E.D., Jr., 

Happel, J., and Hnatow, M.A., eds., Natural gas hydrates: 

Annals of the New York Academy of Sciences, v. 715, 

p. 346–380.

Gettrust, J.F., Wood, W., Lindwall, D., Chapman, R., Walia, 

D., Spence, G., Macdonald, R., and Hyndman, R.D., 

1999, New seismic study of deep sea gas hydrates results 

in greatly improved resolution: Eos, September 12, 

p. 439–440.

Guerin, Gilles, Goldberg, D., and Melster, A., 1999, Characterization of in situ elastic properties of gas hydrate-bearing 

sediments on the Blake Ridge: Journal of Geophysical 

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vertical seismic profiling: Science, v. 273, p. 1840–1843.

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methane hydrate marine bottom simulating reflectors: Journal of Geophysical Research, v. 97, p. 6683–6698.

Hyndman, R.D., Wang, K., Yuan, T., and Spence, G.D., 1993, 

Tectonic thickening, fluid expulsion and the thermal regime 

of subduction zone accretionary prism—The Cascadian 

margin off Vancouver Island: Journal of Geophysical 

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and permafrost: Geophysics, v. 51, p. 1075–1086.

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of carbon in shallow geosphere: Chemical Geology, v. 71, 

p. 41–71.

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borehole from incident plane waves: Geophysics, v. 52, 

p. 1290–1296.

Lee, M.W., 2002, Modified Biot-Gassmann theory for calculating elastic velocities for unconsolidated and consolidated sediments: Marine Geophysical Researches, v. 23, 

p. 403–412.

Lee, M.W., 2006, Is amplitude loss of sonic waveforms 

due to intrinsic attenuation or source coupling of the 

medium?: U.S. Geological Survey Scientific Investigations 

Report 2006–5120, 13 p.

Lee, M.W., Hutchinson, D.R., Agena, W.F., Dillon, W.P., 

Miller, J.J., and Swift, B.A., 1994, Seismic character of 

gas hydrates on the Southeastern U.S. Continental margin: 

Marine Geophysical Researches, v. 16, p. 163–184.

Markl, R.G., Bryan, G.M., and Ewing, J.I., 1970, Structure 

of the Blake-Bahama Outer Ridge: Journal of Geophysical 

Research, v. 75, p. 4539–4555.

Miller, J.J., Lee, M.W., and von Huene, R., 1991, An analysis 

of a reflection from the base of a gas hydrate zone offshore 

Peru: American Association of Petroleum Geologists Bulletin, v. 75, p. 910–924.

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Paull, C.K., Matsumoto, R., Wallace, P.J., and Dillon, W.P., 

eds., Proceedings of the Ocean Drilling Program, Scientific 

Results: College Station, Tex., v. 164, p. 3–10.

Paull, C.K., Ussler, W. III, and Borowski, W.S., 1994, Sources 

of biogenic methane to form marine gas hydrates, in Sloan, 

E.D., Jr., Happel, J., and Hnatow, M.A., eds., Natural gas 

hydrates: Annals of the New York Academy of Sciences, 

v. 715, p. 346–380.

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of glaciation limited by marine gas-hydrates?: Geophysical 

Research Letters, v. 18, p. 432–434.

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(Leg 164), 1996, in Paull, C.K., Matsumoto, R., Wallace, 

P.J., and others, eds., Proceedings of the Ocean Drilling 

Program, Initial Reports: College Station, Tex., v. 164, 

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McMilen, K.J., Ladd, J.W., and Worzel, J.L., 1979, Seismic 

evidence for widespread possible gas hydrate horizons on 

continental slopes and rises: American Association of Petroleum Geologists Bulletin, v. 61, p. 698–707.

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p. 224–232.

Wood, W.T., Holbrook, W.S., and Hostkins, H., 2000, In 

situ measurements of p-wave attenuation in the methane 

hydrate- and gas-bearing sediments of the Blake Ridge, in

Paull, C.K., Matsumoto, R., Wallace, P.J., and Dillon, W.P., 

eds., Proceedings of the Ocean Drilling Program, Scientific 

Results: College Station, Tex., v. 164, p. 265–272.

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p. 5081–5095.

10 Amplitude Variation of Bottom Simulating Reflection with Respect to Frequency
The amplitude from a transitional base is calculated using 

the Ricker wavelet. The Ricker wavelet r(t) is given by the 

following formula

r t t e t ( ) = − ( ) − 1 2 2 2

α α (5)

with α π = 2 2 f p , where f

p is the dominant frequency of the 

Ricker wavelet in hertz.

Defining G t e t ( ) = −α 2, then

r t

d G

dt ( ) = − /( )

2

2 2α and G t '( ) ( = −2α tG t) (6)

The reflected wavefield (S(t))from the first-order transitional 

boundary is given by Justice and Zuba (1986):

S t V V

T ( ) r d ln( / ) = ( ) ∫ 1 02

τ τ (7)

where

V0 is the velocity at the top of the transition zone,

V1 is the velocity at the bottom of the transition 

zone, 

and

T is the width of the transition zone in two-way 

time.

The integration shown in equation 7 can be written as follows 

using equation 6:

r d t T G t T tG t

t

t T

( ) τ τ ( ) ( ) ( )

+

∫ = + + − (8)

Therefore, the reflection waveform from the transition zone is 

given by

S t V V

T ( ) t T G t T tG t ln( / ) = [( ) + ( ) + − ( )] 1 02 (9)

It is instructive to see the waveform in the following limit

T T

S t

c t T G t T tG t

T c d tGdt cr t → →

= [ + + − ] ≈ = 0 0 lim l ( ) im ( ) ( ) ( ) ( ) ( ) (10)

where c is a constant given by ln (V1/V0)/2 , which is the 

reflectivity of the transition boundary. Equation 10 indicates 

that, as the thickness of the transitional layer approaches zero, 

the waveform of the reflection approaches the input waveform 

and its magnitude is proportional to the reflection coefficient. 

When T is much greater than the period of the input, two 

reflections, one with (t+T)G(t+T) and the other with tG(t),

are separated in time, so the waveform is proportional to tG(t).

Because tG(t) is proportional to G'(t), it can be shown, using 

equation 6, that the reflected waveform from a thick transitional layer is the integrated waveform of the input.

The extremes of the reflection amplitude are at dS(t)/dt = 0, 

which is

dS

dt = = 0 ( G t + + T t ) ( + + T G) '( ) t T − − G t( ) tG'(t). (11)

Case 1. The duration of the wavelet is longer than or similar 

to the transitional layer thickness. In this case, the extreme 

occurs at t = –T/2 and the extreme is given by

A cG T ce T

max

( / ) = = ( / ) − 2 2 2 α (12)

Case 2. The duration of the input wavelet is much shorter 

than the transition thickness. In this case, there is no interference between the top and bottom reflections. Therefore, two 

extreme amplitude points exist and the extreme condition 

is G(t) + tG'(t) = 0 or G(t + T) + (t + T)G'(t + T) = 0 . The 

extreme point occurs at t = 1 2 /( α) or t = 1 2 /( α) – T. The 

extreme amplitude at these points is given by

A c

T

e max

/ = 1 −

2

1 2

α

(13)

Case 3. The duration of the input wavelet is between case 1 

and case 2. There is no simple analytic solution.

The Fourier amplitude spectrum of the Ricker wavelet,

R(), is given by

R Ge

p

p ( ) ( ) / ω

ω ω

α

πω

ω

ω ω = = − 2 2

2 2

4 2 2

(14)

where

P is the angular dominant frequency of the 

Ricker wavelet.

The amplitude spectrum of the reflection from a transition zone can be written, using equations 8 and 14 and from 

Justice and Zuba (1986),

S c T

T

e p ( ) sin( / )

( / )

/ ω

ω ω

ω

ω ω = − 2 2

2

2 2

(15)

Equation 15 indicates that when T approaches zero, the 

spectrum of the reflection is proportional to the input wavelet. On the other hand, when T becomes larger, the amplitude 

spectrum approaches the spectrum of the integrated Ricker 

wavelet. Therefore,

S e p ( ) / ω ω ω ω ∝ − 2 2 (16)

The maximum of equation 16 is at ω ω= p / 2 . This implies 

that the amplitude spectrum of a reflection from a thick transitional layer shifts towards a lower frequency and approaches 

about 70 percent of the dominant input frequency.

Appendix. Reflections from Transition Zones

Appendix. Reflections from Transition Zones 11
Figure 10A shows the peak amplitudes of transition zone 

reflections with respect to the transition zone thicknesses and 

the dominant frequency of the Ricker wavelet, and figure 10B

shows the actual waveforms. The reference peak amplitude 

is the reflection amplitude from the discontinuous (sharp) 

boundary, which is equivalent to the transition zone thickness 

of zero. When , which is defined as the ratio of the transition 

zone thickness in two-way time (T) to the dominant period of 

the input wavelet (1/ P), is less than about 0.8 (Region I), the 

peak amplitude is given by equation 12 and drops off rapidly 

with respect to : its value is about –12 db at  = 0.8. In this 

Figure 10. Relationships between reflection amplitude and transitional layer thickness. A, relationship between maximum reflection 

amplitude and transition zone thickness; B, reflection waveforms from transition zones. Parameter  is the ratio of transition zone 

thickness in two-way time to the period of the dominant symmetrical Ricker wavelet. RELATIVE AMPLITUDE, IN DECIBELS

0

–5

–10

–15

–20

–25

–30

0.0 0.5 1.0 1.5 2.0 2.5 3.0

–0.5

0.0

0.5

1.0

0 10 20 30 40

PARAMETER λ

RELATIVE REFLECTION AMPLITUDE

TIME, IN MILLISECONDS

Region I

Region II

Region III

λ = 0.25

λ = 0.50

λ = 0.75

λ = 1.0 λ = 2.0

region, the reflected waveform is close to the input waveform 

except that the dominant frequency of reflection shifts to the 

lower frequency as  increases.

In Region II, where  is between 0.8 and 1.1, there is 

no simple solution for the peak amplitudes or the waveforms. 

The peak amplitude in this zone varies between about –12 db 

and –17 db compared to the reference amplitude, and its waveform varies according to the amount of interference between 

the reflection from the top and bottom of the transitional layer.

In Region III, where  is greater than 1.1, the waveform is similar to the integrated input waveform, and the peak 

amplitude is given by equation 13 and decays slowly with .

12  Amplitude Variation of Bottom Simulating Reflection with Respect to Frequency