modeling ocean-bottom reflection loss measurements with the plane-wave reflection coefficient

Modeling ocean-bottom reflection loss measurements with the plane-wave reflection coefficient

N. Ross Chapman Defence Research Establishment Pacific, FMO Victoria, British Columbia, Canada VOS 1BO

{Received 17 May 1982; accepted for publication 22 December 1982) Ocean-bottom reflection loss has been measured at an abyssal plain site in the northeast Pacific Ocean. The layer of unconsolidated sediment at this site is thin {less than 200 m), and the near- surface sediments are composed of strongly reflecting sandy turbidites. In consequence, the surticial sediments and the relatively shallow substrate must be accounted for in describing the bottom loss measurements. The bottom loss has been modeled in this paper using the plane-wave reflection coefficient calculated for a geoacoustic model of the bottom developed from seismic profiling experiments carded out at the site. In the model the sediment column is described by fluid layers with a relatively small value of attenuation and a large value of sound-speed gradient in the surficial layer, and the substrate is assumed to be a solid. The bottom loss measurements are presented in 1/3 octave bands at 40, 80, and 160 Hz. The results are well-modeled at small grazing angles {0<20 ø) where the interaction is confined to the surficial sediments. At larger grazing angles the importance of including a solid substrate is demonstrated.

PACS numbers: 43.30.Dr, 43.30.Bp, 91.50.Ey, 92.10.Vz

INTRODUCTION

Measurement of bottom-reflection loss using small ex- plosive charges is a convenient and well-established method to study the interaction of low-frequency sound with the ocean bottom. Bottom loss is defined as the ratio of the re-

flected to the incident (plane-wave) sound intensities, and is usually expressed in decibels {dB) as a function of frequency and grazing angle. Since this definition is difficult (if not impossible) to apply experimentally, bottom loss estimates are derived in practice from measurements of the propagation loss of bottom bounce paths, assuming that the reflection field from the sea floor is specular. Although the con- cept is attractively simple, the bottom loss model does not provide an adequate description of the interaction in general, and several authors have pointed out that the contribution of sound propagating via subbottom paths should be included in the interpretation of the data. 1-6 When the caustic formed by these bottom-refracted paths is taken into account, Spof- ford has shown that bottom loss measurements can, in fact, be used to evaluate the average sound-speed gradient in thick layers of unconsolidated sediment found in deep water abyssal plains. •

In some abyssal plains, however, the presence of a strong reflector near the sea floor prevents the formation of a causticjn the subbottom sediments. This situation is found, for instance, in regions where the covering of unconsolidated sediments is thin and the underlying consolidated sediment or the basement rock presents a strongly reflecting layer relatively close to the ocean bottom. For this case the plane- wave reflection model provides a reasonable approach for interpreting bottom loss measurements.

This paper describes the analysis and interpretation of bottom loss measured in an experiment carried out over the Tufts Abyssal Plain in a region where a seismic refraction profile indicated that the layer of unconsolidated sediment was indeed thin. In addition, sediment cores obtained in this

part of the plain consist mainly of turbidite sequences of terrigenous sand or sandy silt. 7'8 The presence of these sandy deposits suggests that the sediments near the sea floor are strongly reflecting. Consequently, it is reasonable to expect that the contribution from the surficial layers, and also the relatively shallow substrate beneath the unconsolidated sediment must be taken into account in modeling the bottom loss measurements.

The bottom loss measurements have been interpreted using the plane-wave reflection coefficient calculated for a simple but realistic geoacoustic model of the ocean bottom. The structure of the model was developed from seismic reflection and refraction profiles taken at the site, and the parameters were evaluated from these data, or established values were used. 9 In the model the bottom was approximated by fluid sediment layers of constant density, attenuation, and sound-speed gradient overlying a solid substrate. The plane-wave reflection coefficient was calculated by numerical integration of the wave equation in each region of the model. 10,11

The experimental techniques for obtaining the measurements are described in the next section. Following this, the numerical method used to calculate the plane-wave reflection coefficient is outlined, and then the geoacoustic model of the bottom is described. Finally the bottom loss measurements are presented in 1/3 octave bands centered at 40, 80, and 160 Hz, and are interpreted in terms of the plane- wave reflection coefficient calculated for the geoacoustic bottom model. •

I. EXPERIMENTAL TECHNIQUES

The experimental measurements were carried out along a track originating from a site 46øN, 143.5øW over the Tufts Abyssal Plain in the northeast Pacific Ocean. The average depth at this location was 4500 m, and the bottom was flat, rising monotonically less than 100 m over the chosen track.

1601 J. Acoust. Soc. Am. 73 (5), May 1983 0001-4966/83/051601-07500.80 1601

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The shot run was carried out with two ships; one ship, CFAV ENDEAVOUR, monitored a hydrophone suspended from a surface float, and the second ship, MV PANDORA, opened range from the receiving system and deployed two relatively shallow 1.8-lb SUS charges at intervals of about 1.8 km. The receiver depth was 430 m {near the deep sound channel axis}, and the average shot depths were 23 and 190 m. The signals received at the hydrophone were digitized at a sampling rate of 1562.5 Hz and transmitted to the ship via an rf data link. The frequency response of the system was fiat over the range 5-650 Hz.

The bottom loss was determined from the propagation loss of the first bottom bounce path which interacted only with the ocean bottom, using the expression

BL = H m -- H c = (SL - RL} -- Hc. (1} Here BL is the bottom loss in dB, Hc is the ray theory esti- mate of the propagation loss, and H a is the measured propagation loss of the first bottom bounce path. This specific path was isolated in processing the data to eliminate interference effects of the surface-interacting first bottom paths typical of the shallow sources and receivers deployed in the experiment. The source levels SL, were measured in recent experiments at DREP, 12 and the received energy RL, was obtained via the FFT of the bottom bounce signal. Only the data from the 190-m shots were used to determine the estimates of bottom loss.

The 23-m shots, with their richer low-frequency out- put, were used to obtain a seismic refraction profile along the track. The shot records were filtered in a 1/3 octave band centered at 12 Hz, and the first head wave arrival was identified at each range. The travel time of the head wave was measured using a tone transmitted from the shooting ship to mark the explosion time.

II. NUMERICAL CALCULATION OF THE PLANE-WAVE REFLECTION COEFFICIENT

The plane-wave reflection coefficient was calculated numerically for a geoacoustic model developed from the experiments carried out at the abyssal plain site. The model consists of three horizontally stratified regions including the water column, the sediments, and the substrate as shown in Fig. 1. The water column is considered to be a constant sound-speed half-space, and the bottom is modeled by an arbitrary number of inhomogeneous fluid sediment layers overlying a homogeneous solid substrate half-space. In all layers absorption is taken into account by assuming the wavenumber k, to be complex.

The plane-wave reflection coefficient was calculated numerically by integration of the wave equation in each region of the geoacoustic model. 10.11 This method is capable of handling not only the multilayered structure but also any arbitrary variation of sound-speed, density, and attenuation in the layers. A summary of the mathematical formulation is outlined below.

The acoustic field at an angular frequency co in the water and in the sediment layers is described by the pressure P, given by the expressions

Po =/•o( ei%z q- Re - i'cøz) eilkøx cøs O-øt ), z•<O (2)

• /•/ REFLECTED •

• WA• Po' const. INCIDENT • •\ Co= const.

d 2 p2(z) ,c 2 (z),0:2 (z)

FLUID

•EDIMENT LAY E R

dN- I Pn-m (z), C N-I (Z),O• N.i(Z)

SOLID SUBSTRATE :::)N const. Cp = const.

O•p= const.

C S = const.

O• S = const.

FIG. 1. The multi-layered bottom model. Density, sound speed, and attenuation are denoted by p, c, and a, respectively, and the sediment layer thickness by h.

for the water, and for the sediments

p• = p• (z)e•{•o• •os o- cot ), j: 1, 2, ..., N, (3) Ao is the amplitude of the incident wave, % = ko sin 0 = {co/ Co)sin 0, Co is the water sound speed, and 0 is the grazing angle. R is the plane-wave reflection coefficient and is related to the bottom loss by

BL- - 20 ]ogl I. ' (4)

In the sediment layersœ• (z) satisfies the depth separated wave equation

c• 2

•-• pj(z) .hi_ JFj •pj(z) .ql_ [•(z) -- • 02 cos2O ] pj(z) where k 02 c0s20 is the separation constant, k•(z)= + ia•(z) is the complex wavenumber in the jth layer, and

a•(z) is the attenuation. This form of the wave equation in- eludes the effects of gradients in the density p via the term F• (• /&), where F• = -- (d /dz)ln p• (z).

The fields in the substrate are given by the compressional and shear wave velocity potentials

q• = •pei%Zei(køx cøs ø-•øt }, (6) and

tp = Ase'•'% '1•ø• •os o- tot ), (7) respectively. Here A v and As are amplitudes of the waves

1602 J. Acoust. Soc. Am., Vol. 73, No. 5, May 1983 N. Ross Chapman' Modeling loss with reflection coefficient 1602


2 -- k 02 c0s20, K•s = k 2 _ k 02 c0s20, •=k•, s k•, = co/c•, and ks = cO/Cs. At the fluid-fluid interfaces the boundary conditions of

continuity of normal stress {pressure} and normal compo- nent of velocity must be satisfied. Applying these conditions at the water-sediment interface yields the equation for eval- uating R

R = itro --(po/pl)[aPl(O)/aZ/Pl(O)] . (8) i% + ( po/p•)[•P•(O)/&/P•(O) ]

At the sediment-substrate interface the additional con-

straint of continuous tangential stress is required to satisfy the fluid-solid boundary conditions.

The numerical solution of R was achieved by first evalu- ating P2v- 1 and (c7/•z)P2v_ • at the bottom of the sediment layer adjacent to the substrate, using the fluid-solid boundary conditions. Starting at the bottom of this layer, the wave equation for P was integrated using a numerical routine described by Shampine and Gordon. 13 At each interface j/ j q- 1, the fluid-fluid boundary conditions were used to obtain the values of Pj and (•3/&)Pj from the computed values of P•. + • and (•/&)P•. + •. This procedure was repeated until the water-sediment interface was reached, and Eq. (8) was used to evaluate R.

III. DEVELOPMENT OF THE GEOACOUSTIC MODEL

Quantitative geophysical information about the abyssal plain site was obtained from the seismic refraction profile, and by deconvolution processing of the bottom reflected signals from the 190-m shots to determine the impulse response of the ocean bottom. The former method was used to probe the deeper sediment layers and the crustal rocks, while the latter method was used to determine the structure of the

surficial sediments. This information was used to develop the geoacoustic model of the bottom. The evaluation of the model parameters is discussed in the following paragraphs.

A. Seismic refraction profile

The seismic refraction profile was carried out using the 1.8-lb charges as described previously. The travel time of the first head wave arrival was plotted versus range, and the curves were analyzed using procedures described by Of- ficer TM to determine the thickness and the average sound speed of each subbottom layer. The bottom profile is shown in Fig. 2. The values of sound speed for the first two layers of crustal rock agree closely with the measurements reported by Wrolstad•5 for the eastern part of the Tufts Plain. For the sediment column, 'the average sound speed was determined to be 1.9 km/s and the thickness was estimated to be about

0.2 km. These values wer. e obtained from a secondary head wave which could be identified over several consecutive

ranges. It is likely that this sound speed tends to the maxi- mum value associated with deeper sediments rather than with the sediments at the sea floor. Head waves from the

slower near-surface sediments could not be resolved from

the strong ocean-bottom reflection. The value of sediment thickness agrees well with the measurements of Morton and

WATER

COLUMN

UNCONSOLIDATED

SEDIMENT

CRUSTAL

ROCK

Cw= 1.53 km/S

C1- 1.9 km/S

C2:4.0 krn /S

C 3- 6.1 krn /S

hw = 4.6 km

h 1 = 0.2 krn

h 2 = 2.0 krn

FIG. 2. Bottom profile for the Tufts Abyssal Plain site.

Lowtie who report a value of about 0.25 km from continuous reflection profiling experiments. 8

B. Deconvolution processing of bottom reflections

The detailed structure of the surficial sediment was de-

termined from an analysis of the first bottom bounce arrivals. An example of the broadband bottom-reflected signal for a grazing angle of 18.5* is shown in the upper part of Fig. 3. This signal can be represented as the convolution of the ocean bottom impulse response and the waveform of the shot. Since the shot waveform consists of a series of bubble

1.00.

0.50.

0.00'

-0.50

-1.oo

o

1.00-

0.50-

0.00-

-0.50-

-1.00-

, .

15 30 45 60 75 90 105

TIME (me)

120

O, 15 30 45 60 75 90 105 120

TIME (ma)

FIG. 3. Unprocessed bottom bounce signal (upper trace) and deconvolved ocean bottom impulse response (lower trace}.

1603 J. Acoust. Soc. Am., Vol. 73, No. 5, May 1983 N. Ross Chapman: Modeling loss with reflection coefficient 1603


pulses following the initial shock pulse, it is necessary to eliminate the effects of bubble pulse interference from the signal. This interference can be observed in the broadband signal beginning about 23 ms after the initial arrival. The ocean bottom impulse response was deconvolved by a curve- fitting technique using the/1-norm (least absolute value) cri- terion. a6 The la technique was used .because it provides a more robust procedure than conventional methods for the extraction of a sparse spike train. a7 The impulse response, plotted in Fig. 3 beneath the unprocessed signal, shows that deconvolution has indeed removed the bubble pulse interference.

The approach to identifying the mechanisms responsi- ble for the subbottom arrivals observed in the data is illus-

trated in Fig. 4. The broadband impulse responses deconvolved from consecutive bottom-reflected signals between grazing angles of 6 ø and 29 ø are plotted on the left, and the corresponding impulse responses, low-pass filtered at 200 Hz, are plotted on the fight. The arrivals from consecutive shots are stacked to align the initial reflection from the sea floor. At least three subbottom arrivals can be identified fol-

lowing within a few ms of the initial sea floor reflection,

although the first arrival apparent in the broadband traces is not resolved in the low-frequency results. These arrivals are either reflections from surficial layers, or refractions within the near-surface sediments. There is not evidence, however,

of deep refracted arrivals associated with the caustic formed in thick sediment columns. •-a The time differences observed

by Dicus 2 and Santaniello et al. 3 between those arrivals and the sea floor reflection (Ate80-100 ms at 30 ø) are much larger than the time separations observed in the data presented in Fig. 4.

The first and second subbottom arrivals in the broad-

band traces have been interpreted as reflections from near surface layers. (Only the second reflection is resolved in the filtered traces.) Assuming that the layer thickness and sound speed are independent of range, and that the layer thickness is small compared to that of the water column, the arrival time difference between the subbottom and the sea floor arri-

vals is related to the grazing angle by the expression

At • = (2h/c•) • -(2h/Co) • cos•O. (9)

Here h and c• are the sediment layer thickness and sound speed, and Co is the water sound speed at the ocean bottom.

6'5o 7'50 •- - - _

8'5ø • 9.5 ø

10.5 ø

12.0 ø _

13.5 ø

14.5 ø •_ 16.5 ø

18.5 ø •.•

23.0 ø •

2•).o ø •

0. 10 20 30 4 50 0 0 80 0 100

! I I I I I I I I I I

0, 10 20 30 40 50 60 70 80 90 100

TIME (ms) TIME (ms)

FIG. 4. Broadband and low-pass filtered impulse responses for grazing angles between 29 ø and 6.5 ø .



WATER SEA FLOOR

SUBBOTTOM

REFLECTOR

FIG. 5. Reflected and refracted sound paths in the near-surface sediments. The three

types of paths are observed for grazing angles 02 > 20ø; 20ø> 02 > 13.5ø; 03 < 13.5 ø.

The values of h and Cl were determined from a least-squares fit ofAt 2 plotted versus cos2•. For the first reflector, h - 6 m and c• -- 1.57 km/s, and for the second reflector (using the low-pass filtered traces), h - 35 m and Cl -- 1.59 km/s. The assumption of constant sound speed is not likely an adequate description of the actual sound-speed variation with depth; however, the values determined are reasonable estimates of the average sound speed in the layers.

At grazing angles less than about 20 ø, a third arrival is observed, which merges with the second subbottom reflection at about 13.5 ø and subsequently becomes the dominant arrival. This arrival was interpreted as a sediment refracted arrival, and was used to evaluate the sound-speed gradient in the near-surface sediments. Assuming that the depth depen- dence of the sound speed is given by

c(z) = c,(1 -- 2gz/½,) -1/• (10) Dicus has shown that the arrival time difference between the

refracted and the sea-floor-reflected arrivals is given by

At = (2/3g)[ 1 -- (c1/co) 2 c0s20 ]3/2, (11) where g is the sound-speed gradient in the sediment at the water-sediment interface. 2 The values of g and the ratio Cl/ Co were determined from a least-squares fit to a plot of (3/ 2 At )2/3 vs cos2•. Using the filtered impulse responses for grazing angles less than 13.5 ø, the values obtained were g = 2.9 s-• and c•/co -- 0.98.

The refracted arrival was observed only at these shallow grazing angles owing to the magnitude of the sediment sound-speed gradients and the presence of the shallow reflector about 35 m beneath the sea floor. At grazing angles larger than about 20 ø the near-surface gradient is not strong enough to return the incident acoustic energy by refraction, and consequently the sound is reflected from the subbottom layer. The refracted arrival begins to be observed as a third arrival following the reflection from the 35-m layer at angles less than about 20 ø . At these angles the gradients in the deeper sediments return the energy by refracted paths which turn just beneath the reflector. At grazing angles shallower than about 13.5 ø, the energy is refracted entirely within the near- surface sediments and the reflection is no longer observed. This model of the behavior is illustrated in Fig. 5 where the refracted and reflected ray paths are sketched.

C. The geoa½ousti½ model

The geoacoustic model developed from the data just presented is shown in Fig. 6. For the interpretation of the behavior at low frequencies, it is unlikely that the effect of

the shallow reflector at a depth of 6 m is significant. Conse- quently, the near-surface sediments have been modeled by a single layer 35-m thick, with sound-speed gradient 2.9 s -1 and an initial sound speed of 1.49 km/s. The large value for the gradient lies within the range of values reported by Ham- ilton for the sandy turbidite deposits found in the Tufts. Abyssal Plain. 9 The initial sound speed is slightly less than the water sound speed at the sea floor, and the value agrees well with the reported values of around 1.5 km/s measured from piston cores obtained in this area. 8 The remainder of the sediment column beneath this layer is also modeled by a constant sound-speed gradient, with a value of 1.2 s-1. This value is, however, smaller than that in the upper layer, and was taken from Hamilton's estimates for deeper sediments. 9 Hamilton's reported values were also used for the density in all •'egions, the shear wave speed in the substrate and the attenuation of the compressional and shear waves in the substrate. 9 The value used for sediment attenuation was taken from earlier work on bottom loss data. 18 The value is lower

than that reported by Hamilton, but is in good agreement with recent measurements of attenuation in deep water abyssal plains by Mitchell and Focke. 19

IV. INTERPRETATION OF BOTTOM LOSS

MEASUREMENTS

The bottom loss measurements determined in 1/3 oc-

tave bands centered at 40, 80, and 160 Hz are presented in Figs. 7-9 for comparison with the modeled losses calculated at the center frequency of each band. In these figures the measured values are indicated by closed circles, and the solid and broken curves show the calculated bottom loss for oa

solid and a fluid substrate, respectively. In all frequency bands the signal-to-noise ratio was greater than 10 dB. The large oscillations in the modeled results are due to interference between the refracted and reflected sound paths. These

Cw = 1.53 km/s P= 1.03 gm/cm 3 hl= 35mC1= 1'49km/•,• gl = 2.9s -1 Pl = 1.55gm/cm 3

C2 =1.62 km/s • -1 2 = 1.2S P2 = 1.7 gm/cm 3 h2=165 rn

O: = 0.015 f (kHz)

0.02 f 0.07 f

(kHz) Cp= 4.0 km/s

Cs= 1.9 km/s

WATER

UNCONSOLIDATED

SEDIMENT

SUBSTRATE

FIG. 6. Geoacoustic model of the bottom at the Tufts Abyssal plain site.

1605 J. Acoust. Soc. Am., ¾ol. 73, No. 5, May 1983 N. Ross Chapman: Modeling loss with reflection coefficient 1605


45- 45-

40-

35-

30-

,• 25-

O

•20- O

O

15-

10-

0

0

F=40HZ

i i i I i

•'; * % * •'0 ,'0 •0 •0 •0 80 90 GRAZING ANGLE

FIG. 7. Bottom loss versus grazing angle at a frequency of 40 Hz. Measure- ments are indicated by the closed circles, and the calculated losses for solid and fluid substrates by ( ) and (---), respectively.

effects are, of course, averaged out in the 1/3 octave band m•asurements.

The predicted bottom loss curves are in good agreement

45-

40-

35-

30-

en25_

O

20-

O

O

15-

10-

5-

F=80HZ

-- - i - ,11 -i i ! i

10 20 30 40 50 60

GRAZING ANGLE

FIG. 8. Bottom loss versus grazing angle at a frequency of 80 Hz. Measure- ments are indicated by the closed circles, and the calculated losses for solid and fluid substrates by ( ) and (---), respectively.

4O-

35-

3O-

•25-

O

•20- O

O

15-

10-

0

0

F=160HZ

' l ?

i ßß I I I I I I 10 20 30 40 50 60 70 8'0 GRAZING ANGLE

FIG. 9. Bottom loss versus grazing angle at a frequency of ! 60 Hz. Measure- ments are indicated by the closed circles, and the calculated losses for solid and fluid substrates by ( ) and (---), respectively.

with the measurements at angles less than 20* where the interaction is confined to the near surface sediments. The mea-

sured loss is small, about 1 dB or less, and is well modeled by the large sound-speed gradient in the first layer, and the small values of attenuation in the sediments reported in recent studies. 18,19 At grazing angles greater than about 30* the incident sound interacts with the relatively shallow substrate, and the measured loss increases abruptly to an average value of 8 dB. This observation compares favorably with the modeled results if the substrate is assumed to be a solid, and the effects of shear waves in the basement rock are taken

into account. The sudden increase in loss occurs very close to the predicted critical angle of 37 ø for shear waves propagating at a speed of 1.9 km/s. It is important to note that the fluid substrate model is unable to account for the relatively large losses measured between 35 ø and 65 ø . For grazing angles greater than 65 ø the observed loss is due to the excitation of compressional waves in the substrate. In this case the pre- dictions for both models show an increase in loss at the criti-

cal angle of 68 ø for compressional waves propagating at 4.0 km/s.

V. SUMMARY

The plane-wave reflection coet•icient has been used to model low-frequency bottom loss measurements obtained over the Tufts Abyssal Plain. In this region the sediment column is relatively thin, and the effects of the caustic com- monly formed in thicker sediments by deep refracting paths were observed only at very low grazing angles. Consequent- ly, the plane-wave reflection model provides an adequate



approach for interpreting the measurements. At small grazing angles the bottom loss is sensitive to the near-surface sound-speed gradient and the value of attenuation in the sediment. The importance of modeling the substrate as a solid is demonstrated by the qualitative agreement with the measurements at larger grazing angles when the effects of shear waves in the basement rock are taken into account.

IR. E. Christensen, J. A. Frank, and W. H. Geddes, "Low-Frequency Propagation via Shallow Refracted Paths Through Deep Ocean Unconso- lidated Sediments," J. Acoust. Soc. Am. 57, 1421-1426 (1975).

2R. L. Dicus, "Preliminary Investigation of the Ocean Bottom Impulse Re- sponse at Low Frequencies," US Naval Oceanographic Office TN 6130-4- 76 (1976).

38. R. Santaniello, F. R. DiNapoli, K. W. Dullea, and P. D. Herstein, "Stu- dies on the Interaction of Low Frequency Acoustic Signals with the Ocean Bottom," Geophys. 44, 1922-1940 (1979).

4D. C. Stickler, "Negative Bottom Loss, Critical Angle Shift, and the Inter- pretation of the Bottom Reflection Coefficient," J. Acoust. Soc. Am. 61, 707-710 (1977).

5G. R. Frisk, "Determination of Sediment Sound Speed Profiles Using Caustic Range Information," in Bottom Interacting Ocean/tcoustics, edited by W. Kuperman and F. Jensen (Plenum, New York, 1980), pp. 153- 157.

6C. W. Spofford, "Inference of Geoacoustic Parameters from Bottom Loss Data," in Bottom Interacting Ocean/tcoustics, edited by W. Kuperman and F. Jensen (Plenum, New York, 1980), pp. 159-171.

?D. R. Horn, B. M. Horn, and M. N. Delach, "Sedimentary Provinces of the North Pacific," in Geological Investigations of the North Pacific, edited by J. D. Hays (Geological Society of America, Boulder, CO, 1970), Mem- oir 126, pp. 1-21.

8W. T. Morton and A. Lowrie, "Regional Geological Maps of the No•th- east Pacific," Report NOO-RP 16, US Naval Oceanographic Office (1978).

9E. L. Hamilton, "Geoacoustic Modeling of the Sea Floor," J. Acoust. Soc. Am. 68, 1313-1340 (1980).

løK. E. Hawker and T. L. Foreman, "A Plane Wave Reflection Loss Model Based on Numerical Integration," J. Acoust. Soc. Am. 64, 1470-1477 (1978).

I IN. R. Chapman and B. Huber, "Numerical Calculation of the Plane Wave Reflection Coefficient for Realistic Geoacoustic Bottom Models," DREP Technical Memorandum 81-10 (June 1981).

12N. R. Chapman, unpublished data. 13L. F. Shampine and M. K. Gordon, Computer Solution of Ordinary Differ-

ential Equations (Freeman, San Francisco, 1975). 14C. B. Officer, The Theory of Sound Transmission (McGraw-Hill, New

York, 1957). 15K. Wrolstad, "Interval Velocity and Attenuation Measurements in Sedi-

ments from Marine Seismic Data," J. Acoust. Soc. Am. 68, 1415-1435 (1980).

•6N. R. Chapman and I. Barrodale, "Deconvolution of Marine Seismic Data using the/•-norm," Geophys. J. R. Astron. Soc. 72, 93-100 (1983).

•7j. F. Claerbout and F. Muir, "Robust Modeling with Erratic Data," Geophys. 38, 826-844 (1973).

•8N. R. Chapman, "Low Frequency Bottom Reflectivity Measurements in the Tufts Abyssal Plain," in Bottom Interacting Ocean/tcoustics, edited by W. Kuperman and F. Jensen (Plenum, New York, 1980), pp. 193-207.

198. K. Mitchell and K. C. Focke, "New Measurements of Compressional Wave Attenuation in Deep Ocean Sediments," J. Acoust. Soc. Am. 67, 1582-1589 (1980).



modeling ocean-bottom reflection loss measurements with the plane-wave reflection coefficient

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