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NOSC TR 242
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Technical Report 242
INTERACTION OF SOUND
WITH THE OCEAN BOTTOM:
A Three-Year Summary
HE Morris
EL Hamilton
HP Bucker
RT Bachman
30 April 1978
Final Report: 1974-1977
Prepared for
Naval Electronic Systems Command
APPROVED FOR PUBLIC RELEASE; DISTRIBUTION UNLIMITED
TC NAVAL OCEAN SYSTEMS CENTER
(50| SAN DIEGO, CALIFORNIA 92152
Gl
Prose
i
NAVAL OCEAN SYSTEMS CENTER, SAN DIEGO, CA 92152
AN ACTIVITY O F THE NAVAL MATERIAL COMMAND
RR GAVAZZI, CAPT, USN HL BLOOD
Commander Technical Director
ADMINISTRATIVE INFORMATION
This three-year program (1974-1977) was sponsored by the Naval
Electronic Systems Command, Code 320 (Task No. 70105): Interaction of
Sound with the Ocean Bottom; by the Naval Sea Systems Command, Code
06H14 (Subproject 52-552-701, Task 16409): Sea Floor Studies For Sonar
Performance, and Acoustic Properties of the Sea Floor; and by the Office of
Naval Research (Code 480).
Released by Under Authority of
EB TUNSTALL, HEAD JD HIGHTOWER, HEAD
Environmental Acoustics Division Environmental Sciences Department
SECURITY CLASSIFICATION OF THIS PAGE (When Data Entered)
REPORT DOCUMENTATION PAGE
T. REPORT NUMBER 2. GOVT ACCESSION NO.
NOSC TR 242
4. TITLE (and Subtitle)
INTERACTION OF SOUND WITH THE OCEAN BOTTOM:
A THREE-YEAR SUMMARY
READ INSTRUCTIONS
BEFORE COMPLETING FORM
5. TYPE OF REPORT & PERIOD COVERED
Research
1974-1977
6. PERFORMING ORG. REPORT NUMBER
8. CONTRACT OR GRANT NUMBER(a)
7. AUTHOR(s)
H. E. Morris, E. L. Hamilton,
H. P. Bucker, R. T. Bachman
10. PROGRAM ELE
AREA & WORK
70105 NESC
16409 NSSC
12. REPORT DATE
30 April 1978
13. NUMBER OF PAGES
83
15. SECURITY CLASS. (of thie report)
UNCLASSIFIED
Ce Baas ASHE FICATION/ DOWNGRADING
IRD
Approved for public release; distribution unlimited PAPA LIOR
NT, PROJECT, TASK
ERS
9. PERFORMING ORGANIZATION NAME AND ADDRESS
IT NUMB
Naval Ocean Systems Center
San Diego, California 92152
ME
UN
CONTROLLING OFFICE NAME AND ADDRESS
Naval Electronic Systems Command
Washington, D.C.
11.
4. MONITORING AGENCY NAME & ADDRESS(if different from Controlling Office)
16. DISTRIBUTION STATEMENT (of this Report)
| Woods ce
i YVOOds Hole Oce cea Institution
LOS AMINES i
. DISTRIBUTION STATEMENT (of the abstract entered in Block 20, if different from Report)
SUPPLEMENTARY NOTES
18.
. KEY WORDS (Continue on reverse side if necessary and identify by block number)
. ABSTRACT (Continue on reverse aide if necessary and identify by block number)
This report documents the results of a three-year study of the interaction of sound with the sea floor.
The investigation covered sea floor properties of interest in underwater acoustics, including velocity gradients
in the sea floor, density, shear-wave velocities and other properties; research on acoustic propagation models,
especially at low frequencies (2 to 200 Hz); and the development of accurate and efficient methods for
coupling geoacoustic models to standard propagation models such as ray theory, normal mode theory and P. E.
(a numerical method using the Parabolic Equation approximation to the wave equation).
During this period numerous reports were distributed to the acoustic community. Various predictions
for the surveillance community have been calculated using the geoacoustic and acoustic models, support was
DD . on, 1473 EDITION OF !NOVE5IS OBSOLETE _ UNCLASSIFIED
S/N 0102-014- 6601 | 1 Deen pene =ar Fre ar=araPaE ane EEE PONE e
SECURITY CLASSIFICATION OF THIS PAGE (When Data Entered)
UNCLASSIFIED
sELCURITY CLASSIFICATION OF THIS PAGE(When Data Entered)
20. Continued
provided to others developing models and support was provided on a continuous basis to surveillance programs
including the Indian Ocean, MSS, SURTASS, and others.
An extensive list of references is included.
UNCLASSIFIED
SECURITY CLASSIFICATION OF THIS PAGE(When Data Entered)
PREFACE
The absorption of sound in the bottom sediments is the primary cause for a loss of
energy when a sound wave interacts with the ocean bottom. This bottom interaction has
an important effect on passive and active ASW systems. Knowledge of the ocean bottom
can assist in understanding optimum array depths and degradation of signal coherence.
Systems that are located near the bottom are strongly affected by bottom interaction. In
addition, below 100 Hz the bottom begins to interact with long range sound propagation
and thus has a direct effect on noise background (ambient noise) in which the surveillance
systems must operate.
Because this parameter is so important, the Naval Ocean Systems Center (NOSC)
(along with its predecessor the Naval Undersea Center) for several years has maintained a
coordinated research program on interaction of sound with the sea floor. This work falls
into four categories:
1. Studies of the acoustic and related properties of the sea floor and production
of geoacoustic models
2. Basic studies in sound propagation theory, especially as related to the sea floor,
and the development of bottom loss models
3. Use of geoacoustic models of the sea floor and theoretical, mathematical
models of sound propagation and bottom loss to reconcile experiments at sea with
theory
4. Prediction of both geoacoustic models and bottom loss versus grazing angle,
or reflection coefficients, for areas not experimentally occupied
The objectives of the program for the past three years have been to investigate
properties of the sea floor of interest in underwater acoustics, including velocity gradients
in the sea floor, density, shear-wave velocities and other properties; conduct research on
acoustic propagation models especially at low frequencies (from about 2 to 200 Hz); de-
velop accurate and efficient methods for coupling geoacoustic models to standard propaga-
tion models such as ray theory, normal mode theory, and P. E. (a numerical method using
the Parabolic Equation approximation to the wave equation).
We have published numerous reports that have been distributed to the acoustic
community, calculated various predictions for the surveillance community using our geo-
acoustic and acoustic models and provided support to others who are developing models.
We also have interfaced and supplied support on a continuous basis to surveillance pro-
grams including the Indian Ocean, MSS, SURTASS and other efforts.
The summary report gives a brief, general review of our research work during the
three-year period, 1974-1977. This information, coupled with our referenced publications,
provides a basis for more detailed study.
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SUMMARY
OBJECTIVES
The general objectives during the past three years for the bottom-interaction
program were to:
1. Determine, study, and predict those characteristics (or properties) of the sea
floor affecting sound propagation and the prediction of sonar and surveillance performance
2. Place these properties in a form usable by underwater acousticians and
engineers
3. Produce geoacoustic models of the sea floor as required for experimental,
predictive, or theoretical work
4. Develop accurate and efficient methods for coupling geoacoustic models to
standard propagation models.
RESULTS
@ It was determined that the following properties were required for geoacoustic
models of the sea floor which are intended to support underwater acoustics studies:
1. Thicknesses of sediment and rock layers
2. Compressional wave (sound) velocity and attenuation profiles and
gradients through the layers
3. Density profiles and gradients through the layers
4. Shear wave velocity and attenuation profiles and gradients through the
layers
5. Additional elastic properties (e.g., Lamé’s constants)
6. Bathymetry in any insonified area to get slope, relief, topography, and
water depths
7. Properties of the overlying water mass (as from Nansen casts and
velocimeter lowerings)
@ Laboratory measurements of sound velocity and associated properties in
sediment cores continue to be valuable data. These measurements permit correction of
laboratory sound velocity and density to in situ values and prediction of sound velocity
and density due to interrelations between common proportion (e.g., sound velocity versus
mean grain size or porosity). Revised tables of properties (and regression equations of
their interrelations), separated into the main environments and sediment types, greatly
facilitate predictions of various properties. New measurements in over 400 samples of
calcareous sediments allow, for the first time, realistic predictions of sound velocity and
density in this sediment type.
Vil
@® Continuous reflection profiling and associated measurements with expendable
sonobuoys furnish data critical to underwater acoustics in two categories (for a given area
or, in the general case, for prediction): (a) the form and true thicknesses of sediment and
rock layers, and (b) the presence and values of velocity gradients. Specifically, data are
furnished for several areas in the Northeast Indian Ocean.
@ Statistical studies of velocity gradients in silt-clay sediments will allow pre-
diction of sound velocity versus depth in the sea floor and the presence and values of
velocity gradients in similar sediments in the world’s oceans. Specifically, data in 17
areas of the world’s oceans were averaged and a regression equation furnished for the
velocity gradient, a, given one-way travel time, t (a = 1.316 — 1.117t)(see Figure 6, p. 34).
@ Results of studies of the attenuation of sound were as follows:
1. Twenty-six new published values of the attenuation of compressional
(sound) waves in marine sediments complement and supplement older data and support
the conclusion that sound attenuation is approximately dependent on the first power of
frequency.
2. New data support the conclusion that relations between sound attenua-
tion and sediment properties allow prediction of attenuation when mean grain size or
porosity are known.
3. Aspecial study of sound attenuation versus depth in sands, silt-clays,
sedimentary rocks, and basalts should allow generalized predictions of attenuation in
these various layers in the sea floor.
@® Results of studies of variations of sediment density and porosity versus depth
in the sea floor were as follows:
1. Data from the Deep Sea Drilling Project were combined with other
information to produce diagrams, curves, and regression equations of laboratory values
of density and porosity versus depth in the sea floor for common sediment types.
2. The amount of volume increase (elastic rebound) from borehole to
laboratory, caused by release from sediment overburden pressure, was estimated from
soil mechanics tests. Maximum values of such rebound are about nine percent in silt-
clays from depths of 600 meters. Rebound is less in other sediment types.
3. When percent rebound in porosity is deducted from laboratory porosity,
an estimate of the in situ porosity (and density) is determined.
4. These data allow generalized curves and regression equations for density
as a function of depth in the sea floor from which predictions can be made.
@ Results of studies of shear wave velocities versus depth in marine sediments
were as follows:
1. Twenty-nine in situ measurements of shear wave velocities in sands to
12 meter depths indicate (in these sands) that V, = 128D9.28. where V, is shear wave
velocity in m/s, and D is depth in meters.
Vill
2. Forty-seven selected in situ measurements of shear wave velocity in
silt-clays and turbidites to 650 meter depth yielded three regression equations. The
equation for the 0 to 40 meter depth interval (V, = 116 + 4.65D) indicates the gradient
(4.65 sec—!) to be four to five times greater than for compressional waves in this interval.
At greater depths the gradients are comparable.
3. These findings will facilitate prediction of shear wave velocity profiles
and gradients.
@ The attenuation of shear waves was studied and methods of prediction were
outlined.
@ Studies were completed which allowed construction of separate velocity versus
density curves and equations for the common sediment and sedimentary rock types. This
will allow prediction of density given a velocity from a sediment or rock layer (as from a
sonobuoy measurement).
@ There is now enough information on sediment properties to predict a reason-
able geoacoustic model once an acoustic reflection survey is given for an area. This was
done for several areas in the Indian, Pacific, and Atlantic Oceans. However, more work
needs to be done in several categories to facilitate and improve such predictions (see
Recommendations).
@ A general purpose plane wave reflection model has been developed that can
account for both liquid and/or solid layers. Computer calculations are efficient and
accurate.
eA technique was developed to couple a bottom loss model to the Parabolic
Equation (P.E.) sound propagation model.
RECOMMENDATIONS
Continuous Reflection Profiling and Associated Matters
@ General statement
Measurements from continuous reflection profiling allow delineation of sedi-
ment and rock layers, their true thicknesses, their interval or mean velocities, and velocity gra-
dients. Continuous acoustic reflection profiling was largely developed in Navy laboratories and
in academic and research institutions supported by the Navy. When the utility of this
technique for off-shore oil exploration became evident, there was a flash evolution to
present-day techniques involving extremely expensive, multichannel, long-array equip-
ment, and very expensive data processing at sea and ashore. The oceanographic institu-
tions have entered this newer technology, but the costs inhibit progress. Ways need to be
found to reduce costs (or acquire the money) and to simplify equipment and processing
so that Navy-supported ships in the laboratories, academic and research institutions and
the Naval Oceanographic Office (NAVOCEANO) can use the modern technology.
@ Specific recommendations
1. Continue to support technology and data processing in the following
general areas.
a. Acoustic reflection profiling (e.g., air gun, sparker), especially
in the new techniques of multichannel, long-array technology.
b. Measurements of sediment and rock layer interval velocities, with
expendable sonobuoys and multichannel, continuous reflection profiling using very long
arrays and special laboratory processing of tape-recorded signals (as presently done in oil-
industry geophysical exploration). Interval velocity measurements in the first tens of
meters (to about 100 meters) using high-resolution, higher-frequency equipment (3.5
kHz) has shown promise.
c. Study of velocity gradients in sediment and rock layers.
d. Measurements of acoustic impedance of reflectors as seen by
acoustic reflection profiling (some preliminary work has been done by Knott and his
colleagues at Woods Hole).
2. Continue support of scientific expeditions that gather acoustic reflection
and interval velocity information at sea. This has been a long-term program of the Office
of Naval Research (ONR) and the National Science Foundation (NSF).
Sediment and Rock Properties
@ Compressional wave (sound) velocity
1. Support laboratory and, especially, in situ measurements of sound veloc-
ity and associated properties (and their relationships) in sediments and rocks (as in cores,
in boreholes, and with probes inserted into the sea floor).
2. Support in situ measurements of sound velocity gradients in the upper,
surficial sediments (tens of meters), as with the special corer developed by Applied Re-
search Laboratories, University of Texas, or in boreholes.
3. Although the Navy is not directly involved in the Deep Sea Drilling
Project, the Navy should encourage NSF to strongly support downhole logging of sound
velocity and density. These logs were recently started by the DSDP and should continue.
@ Sound attenuation
1. Support laboratory and, especially, in situ measurements of compress-
ional wave (sound) attenuation at frequencies from a few Hz to several hundred kHz.
2. Study relations between attenuation and frequency and between atten-
uation and common sediment properties in all of the common sediment types, especially
in sands at low frequencies (a few Hz to 1 kHz); such studies facilitate and allow pre-
dictions of attenuation.
3. Sound attenuation measurements and studies should include, at lower
frequencies, the whole sediment and sedimentary rock sections, and the surface of the
underlying acoustic basement. This would result in profiles and gradients of sound
attenuation with depth in the sea floor.
@ Shear wave velocity
1. The introduction of shear wave velocity and velocity gradients into
some, but not all, bottom-loss modeling requires accurate prediction of shear wave
velocity versus depth in the sea floor. The shear modulus (which can be derived if shear
velocity and density are known) is also an important engineering property of sediments.
Accurate prediction of shear velocity versus depth in the full range of marine sediments
and rocks requires much more data than now available.
2. It is recommended that laboratory and in situ measurements of shear
wave velocity in marine sediments and rocks be supported. It is further recommended
that at-sea measurements be emphasized with instruments, or probes, placed on or in the
sea floor. A desirable final result is the profile of shear velocity with depth in the sea
floor in the principal types of sediments and rocks.
@ Shear wave attenuation
1. Very few studies have been made of the attenuation of shear waves in
marine sediments and rocks. In this field, in situ studies should be emphasized with some
supporting laboratory work. The profiles and gradient of shear wave attenuation in various
common sediment and rock types also require study.
e Densities of sediments and rocks
1. Support laboratory and, especially, in situ measurements and studies of
density and density profiles and gradients in sea floor sediments and rocks. In situ methods
in the past have included measurements with nuclear probes and in boreholes by logging.
Density logging in the boreholes of the Deep Sea Drilling Project should be supported and
encouraged.
2. Results of the laboratory density measurements by the Deep Sea Drilling
Project should be studied and supplemented with additional laboratory measurements.
Atlases, Charts and Other Syntheses
@ The compilation of the following types of regional atlases, charts and quer
syntheses should be supported and/or encouraged.
1. Sediment types and properties (including mean grain size, density,
porosity, and sound velocity) at the present-day water-sediment interface. Given only
sediment type, or mean grain size, we can predict sound velocity and attenuation, and
density.
2. Compilations of acoustic reflection data to show the form, true thick-
nesses, interval velocities, and velocity gradients of sediment and rock layers in a given
region. This facilitates construction of geoacoustic models and extrapolation of models
and experimental data within a region or geomorphic province.
3. Acontinued, intensive effort should be encouraged to produce the best
possible topographic (bathymetric) charts of the sea floor. This is the overall province of
NAVOCEANO and should be strongly supported by other agencies.
Xl
4. In all local and regional studies, the results of the Deep Sea Drilling
Project should be considered or incorporated.
@ Assess performance of current prediction models. Use Indian Ocean data to
uncover deficiencies.
@ Impact the development of surveillance systems especially tailored for bottom
interaction areas by providing predicted performance as a function of configuration of the
system and mode of operation.
@® Evaluate the method for coupling bottom interaction into the P.E. (Parabolic
Equation) propagation program.
Xil
CONTENTS
PART I: MARINE SEDIMENT PROPERTIES
Introduction ... page 7
Programs in Geology and Geophysics... 9
Sound velocity and related properties of marine sediments
(laboratory measurements)... 9
Introduction... 9
Measurements to July 1975...9
Regression equations interrelating various sediment properties . . 10
Recent and current measurements of sediment physical properties... 12
Compressional-wave velocity profiles and gradients in the sea floor... 13
Introduction... 13
Studies from 1974 to 1977... 13
Current studies... 14
Attenuation of compressional (sound) waves in marine sediments and rocks... 14
Introduction... 14
Studies from 1974 to 1977... 15
Variations of density and porosity with depth in deep-sea sediments... 16
Introduction... 16
Studies from 1974 to 1977... 16
Shear wave velocity profiles in marine sediments... 17
Introduction... 17
Studies from 1974 to 1977... 18
Attenuation of shear waves in marine sediments... 18
Introduction... 18
Studies from 1974 to 1977... 19
Sound velocity-density relations in sea-floor sediments and rocks... 19
Introduction... 19
Studies from 1974 to 1977... 19
Production of geoacoustic models... 20
Introduction... 20
Studies from 1974 to 1977... 20
Summary... 21
References... 22
PART II: ACOUSTIC MODELING
Introduction... page 45
Formulation of the sound field using the plane wave reflection coefficient R .. . 45
Ray theory .. . 46
Wave theory .. . 46
Bottom loss models that account for gradients... 49
Liquid multilayer model... 49
Linear gradient multilayer model... 50
Comparison of the two models... 51
Solid multilayer model... 52
Calculation of R for many solid layers using Knopoff’s method .. . 53
Comparison of multilayer solid and liquid models. . . 56
Effect of bottom interaction on the sound field. . . 56
Equivalent sediment layers for use with the P.E. model. . . 57
Summary... 58
References... 60
TABLES
PART I: MARINE SEDIMENT PROPERTIES
la. Continental terrace (shelf and slope) environment; average sediment size
analyses and bulk grain densities... page 26
1b. Continental terrace (shelf and slope) environment; sediment densities,
porosities, sound velocities and velocity ratios .. . 26
2a. Abyssal plain and abyssal hill environments; average sediment size
analyses and bulk grain densities... 27
2b. Abyssal plain and abyssal hill environments; sediment densities,
porosities, sound velocities and velocity ratios... 28
PART II: ACOUSTIC MODELING
3, Input parameters to the solid multilayer program . . . page 62
ILLUSTRATIONS
PART I: MARINE SEDIMENT PROPERTIES
Il Sediment porosity versus sound velocity, continental terrace (shelf
and slope)... page 29
Dy, Mean diameter of mineral grains versus sound velocity, continental terrace
(shelf and slope) .. . 30
3 Percent clay size versus sound velocity, abyssal hill and abyssal plain
environments... 31
4. Sonobuoy station locations in the Bay of Bengal and adjacent areas as revised
from Hamilton et al (1974) which contained only Antipode and Circe data... 32
5): Instantaneous velocity, V, and mean velocity, V, versus one-way travel
time in the Central Bengal Fan... 33
6. Average of linear velocity gradients, in meters per second per meter, versus
one-way travel time, t, in seconds... 34
“ie Attenuation of compressional (sound) waves versus frequency in natural,
saturated sediments and sedimentary strata... 35
Attenuation of compressional waves (expressed as k in: @qB/m = kfkHz) versus
sediment porosity in natural, saturated surface sediments . . . 36
Attenuation of compressional waves (expressed as k in: dB /m = kf, H{z) versus
depth in the sea floor or in sedimentary strata... 37
Porosity versus depth in terrigenous sediments. . . 38
In situ density of various marine sediments versus depth in the sea floor... 39
Shear wave velocity versus depth in water-saturated sands... 40
Shear wave velocity measured in situ versus depth in water-saturated silt-clays
and turbidites... 41
A summary of compressional wave velocity versus density in Hamilton
(1977)... 42
PART II: ACOUSTIC MODELING
Ray theory representation (high frequency) . . - page 63
Wave theory representation (low frequency)... 63
Multilayer liquid model... 64
Multilayer linear liquid model. . . 65
Linear K2 and constant K layers... 66
Phase comparison for linear K2 and constant K models (zero attenuation
for both models) . . . 67
Phase comparison for linear K2 and constant K models using 0.05 dB/m
attenuation for both models... . 68
Bottom loss comparison for linear K2 and constant K models using 0.05 dB/m
attenuation for both models. . . 69
Multilayer solid model... 70
Comparison of multilayer solid and liquid models... 71
3-D plot of bottom loss as a function of grazing angle and frequency .. . 72
Sound speeds and ray diagram .. . 73
Example of Gibb’s oscillations .. . 74
Equivalent bottom for use with the Parabolic Equation... 74
Desired values of bottom loss... 75
Algorithm to generate an equivalent sediment model with smooth K2 and
bottom loss... 76
Good agreement between the bottom loss for the equivalent sediment mode
(the line) and the desired bottom loss (the filled circles)... 77
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PART I:
MARINE SEDIMENT PROPERTIES
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INTRODUCTION
The objectives of the geology and geophysics part of the task are to:
1. Determine, study and predict those characteristics (or properties) of the sea
floor affecting sound propagation and prediction of sonar and surveillance system
performance
2. Place these properties in a form usable by underwater acousticians and
engineers
3. Produce geoacoustic models of the sea floor for specific areas as required for
experimental, predictive, or theoretical work.
The work thus involves marine geological and geophysical studies in direct support
of underwater acoustics.
When sound interacts with the sea floor, the acoustician concerned with sound
propagation, reflection coefficients, or bottom losses must have a full range of information
about the sea floor. This information includes the basic physics of sound propagation in
marine sediments and rocks, measured acoustic and related properties, properties determined
by empirical relationships and outright estimates or extrapolations based on geological and
geophysical probabilities.
At higher sound frequencies, the acoustician may be interested in only the first meters
or tens of meters of sediments. At lower frequencies (and higher grazing angles) information
must be provided on the whole sediment column and on properties of the underlying rock.
This information should be provided in the form of geoacoustic models of the sea floor.
A “‘geoacoustic model”’ is defined as a model of the real sea floor with emphasis on
measured, extrapolated, and predicted values of those properties important in underwater
acoustics and those aspects of geophysics involving sound transmission. In general, a geo-
acoustic model details the true thicknesses and properties of the sediment and rock layers in
the sea floor.
Geoacoustic models are important to the acoustician studying sound interactions
with the sea floor in several critical aspects: they guide theoretical studies, help reconcile
experiments at sea with theory, and aid in predicting the effects of the sea floor on sound
propagation.
The information required for a complete geoacoustic model should include the fol-
lowing for each layer. In some cases, the state of the art allows only rough estimates, in
others information may be non-existent.
@ Properties of the overlying water mass from Nansen casts and velocimeter
lowerings
e@ Sediment information (from cores, drilling, or geologic extrapolation): sedi-
ment types, grain-size distributions, densities, porosities, compressional and shear wave
attenuations and velocities, and other elastic properties. Gradients of these properties
with depth, for example, velocity gradients and interval velocities from sonobuoy
measurements
@ Thicknesses of sediment layers (in time) determined at various frequencies
by continuous reflection profiling
@ Locations, thicknesses, and properties of reflectors within the sediment body as
seen at various frequencies
@ Properties of rock layers; those at or near the sea floor are of special impor-
tance to the underwater acoustician
® Details of bottom topography, roughness, relief, and slope as seen by under-
water cameras, sea-surface echo sounders and deep-towed equipment
It has been shown by acousticians that the above types of information are essential to
an understanding of sound interactions with the sea floor. Among the above properties and
information, the following is the basic, minimum information on properties of the sediments
and rocks required for most current work in sound propagation.
1. Thicknesses of layers
2. Compressional wave (sound) velocity profile and gradient through the layers
3. Sound attenuation in each layer
4. Density in each layer
Newer and more sophisticated mathematical models involving sound interaction with
the sea floor, especially at lower frequencies, require (in addition to the above):
5. The profile and gradient of sound attenuation through the layers
6. The density profile and gradient through the layers
7. Shear wave velocity and attenuation profiles and gradients through the layers
8. Additional elastic properties (e.g., dynamic rigidity and Lamé’s constant);
given compressional and shear wave velocities and density, these and other elastic
properties can be computed.
Examples of newer mathematical models involving sound interactions with the sea
floor are given by Bucker (Part II of this report) and Bucker and Morris (1975). Additional
examples are those models used at the Applied Research Laboratories of the University of
Texas to study the effects of various sediment properties on bottom losses (Hawker and
Foreman, 1976; Hawker et al, 1976, 1977).
Where sound penetrates the whole sediment layer (and sedimentary rock layers if
they are present) and reflects from and refracts in the surface of the acoustic basement, it is
necessary to know the properties of the basement surface (i.e., compressional and shear
wave velocities and attenuations, and density). An example of this is in the Northcentral
Pacific where 50 to 100 meters of pelagic clay overlies basalt.
A continuing project in the geology-geophysics group is improvement of geo-
acoustic modeling and acquisition and refinement of properties in coordination with acous-
ticians to supply required information and to anticipate future needs. Except where specific
geoacoustic models are required for experimental work, our emphasis is on the general case
so that reasonable predictions can be made in the absence of specific measurements.
At the start of the three-year project in 1974, considerable work had been done by
our laboratory (then NUC) in the acoustic and related properties of marine sediments. These
studies were based on in situ measurements by divers and from submersibles and from
measurements in cored sediments in the laboratory. Much of this work, with appropriate
references, was reviewed by Hamilton (1974a, 1974b). Additionally, work had commenced
on studies of mean velocities and velocity gradients in the first, unlithified sediment layer
(Hamilton et al, 1974).
During the three year program (1974-1977), NAVELEX, Code 320, has supported
partially the continued work in sediment properties and layer sound velocities and velocity
gradients and work concerned with the attenuation of shear waves and the relationship be-
tween sound velocity and density in the principal sediment and rock types of the sea floor.
In the gradients of some important properties, NAVELEX has also partially sup-
ported shear waves in marine sediments versus depth in the sea floor, sound attenuation
versus depth in the sea floor, and density and porosity of sediments versus depth in the sea
floor.
Additionally, geoacoustic models were furnished for a number of areas where
acoustic experimental work was planned or where predictions were required.
The reports of these measurements and studies are included in our references under
Work Supported by NAVELEX, Code 320: 1974-1977.
The work generally noted above will be summarized, with details and illustrations,
in the next sections.
PROGRAMS IN GEOLOGY AND GEOPHYSICS
SOUND VELOCITY AND RELATED PROPERTIES OF MARINE SEDIMENTS
(LABORATORY MEASUREMENTS)
Introduction
The geology-geophysics group operates a sediment laboratory in which measure-
ments are made of sound velocity, density, porosity, grain size and grain density of cored
sediments. During the past three years, funds have not permitted a field program. However,
through cooperative arrangements with Scripps Institution and others, the above measure-
ments have been made on various suites of sediments from the Pacific and Indian Oceans.
Specifically, there were sediments from six Scripps expeditions: four from the Central
Pacific and two from the Indian Ocean (Bay of Bengal and Andaman Sea). Additionally,
three suites of sediments from the Northeast Pacific were taken by Navy-sponsored groups.
Measurements to July 1975
The measurements of physical properties of sediments and their empirical relation-
ships (to about July 1975) have been studied and reviewed (Hamilton, 1975d). The data
were presented in three forms: as diagrams, in regression equations and in tables.
One of the most useful and frequently used outputs of our work in physical proper-
ties of sediments is the production of a set of tables in which are listed the acoustic and
related properties of various sediment types in the three main environments of the sea floor.
The latest revision of the tables of sediment measurements (Hamilton, 1975d, 1976c) is
reproduced here as Tables la, 1b, 2a and 2b. The earlier report (Hamilton, 1975d) also
contained tables (based on the same data) of computed values of impedance, reflection
coefficients and bottom losses at normal incidence, and elastic properties (bulk modulus,
Poisson’s ratio, dynamic rigidity, and velocity of the shear wave). Reproduced here are
regression equations for the more important and useful interrelationships between proper-
ties (from Hamilton, 1975d).
In the sections which follow, frequent references will be made to the three general
environments: the continental terrace (shelf and slope), the abyssal hill environment, and
the abyssal plain environment. These environments and associated sediments were discussed
in greater detail by Hamilton (1971b).
Sediment nomenclature on the continental terrace follows that of Shepard (1954),
except that within the sand sizes the various grades of sand follow the Wentworth scale. In
the deep sea, pelagic clay contains less than 30 percent calcium carbonate or siliceous material.
Calcareous 00ze contains more than 30 percent calcium carbonate and siliceous ooze more
than 30 percent silica in the form of Radiolaria or diatoms. The Shepard (1954) size grades
are shown in these deep-sea sediment types to show the effects of grain size.
Examples of the many scatter diagrams of interrelationships are sound velocity (at
one atmosphere and corrected to 23 degrees Celsius) versus sediment porosity, mean grain
size, and percent clay-sized particles (Figures |, 2, 3); these are three of the best indices to
velocity. An advantage of using mean grain size or percent clay-sized materials as indices to
sound velocity is that grain size and clay size tests can be made in dried or partially dried
sediments in which porosity or sound velocity tests cannot be made.
These tables, diagrams, and regression equations are basic information on which pre-
dictions of sound velocity and density can be based given only a sediment type or grain size.
The methods for such predictions were included in an earlier report (Hamilton, 1971b).
Regression Equations Interrelating Various Sediment Properties
Regression lines and curves were computed for those illustrated sets of (x,y) data in
Hamilton (1975d). These constitute the best indices (x) to obtain desired properties (y).
Separate equations are listed, where appropriate, for each of the three general environments
as follows: continental terrace (shelf and slope), (T); abyssal hill (pelagic), (H); abyssal plain
(turbidite), (P). The Standard Errors of Estimate, 0, opposite each equation, are applicable
only near the mean of the (x,y) values. Accuracy of the (y) values, given (x), falls off away
from this region (Griffiths, 1967, p. 448). Grain sizes are shown in the logarithmic phi-
scale (¢ = —logy of grain diameter in millimeters).
It is important that the regression equations be used only between the limiting
values of the index property (x values), as noted below. These equations are strictly empir-
ical and apply only to the (x,y) data points involved. There was no attempt, for example, to
force the curves expressed by the equations to pass through velocity values of minerals at
zero porosity or the velocity value of sea water at 100 percent porosity.
The limiting values of (x), in the equations below, are:
1. Mean grain diameter, M,, ¢
(T) 1 to9¢
(H) and (P) 7 to 10¢
i)
Porosity, n, percent
(T) 35 to 85 percent
(H) and (P) 70 to 90 percent
3. Density, p, g/em>
(T) 1.25 to 2.10 g/em?
(H) 1.15 to 1.50 g/cm?
(P) 1.15 to 1.70 g/em>
4. Clay size grains, C, percent
(H) and (P) 20 to 85 percent
5. Density X (Velocity), pV,~, dynes/em? X 1010
(H) 2.7 to 3.4 dynes/em2 X 1010
(P) 2.7 to 3.8 dynes/cm2 x 1019
Porosity, n (%) vs Mean Grain Diameter, M, (9)
(T)n = 30.95 + 5.50(M,) o=6.8
(H)n = 82.42 -0.29(M,) o = 4.7
(P) n = 45.43 + 3.93(M,) @ = O.9)
Density, p (g/cm?) vs Mean Grain Diameter, M, (od)
(1) o = ZNO = OWS ul) o= 0:12
(H)p = 1.327 + 0.005(M,) o = 0.09
(P) p = 1.879 — 0.06(M,) o0=0.11
Sound Velocity, Vy (m/s) vs Mean Grain Diameter, M, (9)
i) Vo = 1924.9 —74.18(M,) + 3.04(M,)? o = 33.6
(H)V,, = 1594.4 — 10.2(M,) o= 11:6
OQVs= 1631.8 — 13.3(M,) o = 18.3
Sound Velocity, Vp (m/s) vs Porosity, n (%)
(On = 2467.3 ~ 22.13(n) + 0.129(n)? o = 33.7
(H)V, = 1410.8 + 1.175) o— 1333
(P) Vo = 1630.8 — 1.402(n) o = 20.6
Sound Velocity, Vy (m/s) vs Density, p (g/cm?)
(T) V,, = 2263.0 - 1164.8(p)+458.8(p)* = 34.2
(H)V, = 1591.7 — 63.5(p) GA N9,2
(P) Vy = 1430.6 + 65.2() o=21.7
11
Sound Velocity, Vp (m/s) vs Clay Size, C (%)
(H) Vp = 1549.4 — 0.66(C) o= 9.9
OQ) a= 1568.8 — 0.89(C) o = 18.3
Density, p (g/cm?) vs Porosity, n (%)
(T)n = 156.0 —- 56.8(p) o=2.7
(H)n = 150.1 —51.2(p) o=1.2
(P) n = 159.6 ~58.9(p) o=1.4
Bulk Modulus, k (dynes/cm2 Xx 1019) vs Porosity, n (%)
(T) « = 215.09467 - 133.1006 (logan) + 28.2872 (log.n)*
-2.0446 (logn)? o = 0.01146
(H) and (P) k = 128.9909 — 72.0478 (logen) + 13.8657 (loggn)*
~ 0.9097 (log.n)> o = 0.0100
Bulk Modulus, x (dynes/em2 X 10!9) ys
Density X (Velocity), pVy- (dynes/cm X 10!)
(H) k = 0.32039 + 0.862 (pV,,~) o = 0.049
(P) k = 1.68823 + 0.134 (pV,*) o = 0.069
Recent and Current Measurements of Sediment Physical Properties
Calcareous sediments cover about 50 percent of the sea floor. In the past (and in
Tables 2a, 2b), this important sediment type has been inadequately represented in our meas-
urements. During the past two years and at the present time (September 1977) more than
400 samples of calcareous sediments from four expeditions have been obtained through
cooperative work with Scripps Institution. Measurements in these samples should facilitate
considerably prediction of sound velocity, density, and other properties of calcareous
sediments.
A suite of 108 samples of calcareous sediments from box cores on the Ontong-Java
Plateau in the western equatorial Pacific were examined in 1976 (Johnson et al, 1977a).
Among the many conclusions of this study were the following:
@ There is a continuous reduction of mean grain size with increasing water
depth (probably due to winnowing of fine materials on topographic highs and solution
of calcium carbonate with depth).
@ Porosity and density bear little relation to sound velocity or grain size
(probably because of the hollow shells or tests of Foraminifera).
@ There is a good relationship between mean grain size and sound velocity
and the velocity versus mean grain size regression equations for continental-terrace sands-
silts-clays adequately describes these relations for these calcareous sediments. This is
probably because the hollow shells interact as large, solid particles.
The continued measurements and studies in calcareous materials should furnish
fairly definitive information on interrelationships between common properties.
COMPRESSIONAL-WAVE VELOCITY PROFILES AND GRADIENTS IN THE
SEA FLOOR
Introduction
Continuous reflection profiling (as with a sparker or air gun) and wide-angle reflec-
tion measurements of sediment and rock layer interval velocities have become important
sources of critical data in underwater acoustics. Continuous reflection profiling measures
sound travel time between impedance mismatches within the sediment and rock layers of
the sea floor. To derive the true thicknesses of these layers it is necessary to measure or
predict the mean or interval layer velocity or use a sediment surface velocity and velocity
gradient. At the present time, the simplest method of measuring layer interval velocities
involves the use of expendable Navy sonobuoys.
Sonobuoy measurements of interval velocity also provide basic data for determining
velocity profiles and gradients in the sea floor. When velocity measurements in sediment
cores are available, these can be corrected to in situ values at the water-sediment interface
and used with layer mean velocities to establish velocity profiles and gradients.
It has been shown by Morris (1970) and others that the presence of a positive veloc-
ity gradient in the sea floor is of critical importance in acoustic studies when sound interacts
with the sea floor. In general, when a sediment layer has a positive velocity gradient, sound
energy is refracted back into the water column at certain grazing angles and energy is lost
over long refraction paths.
In summary, reflection profiling records yield data critical to underwater acoustics
in two categories: the form and true thicknesses of sediment and rock layers and the pres-
ence and values of velocity gradients. These data, when examined statistically, yield re-
gional velocity profiles and general, averaged velocity gradients that can be used to predict
velocity gradients in similar sediments elsewhere.
Studies From 1974 to 1977
During the three-year period of this project, three sets of sonobuoy data were ana-
lyzed. Two sets from the Northeast Indian Ocean have been analyzed, added to previous
data in the area and reported (Hamilton et al, 1977). A suite of 17 sonobuoy measurements
of interval velocity were made in the thick calcareous sediments and rocks in the Ontong-
Java Plateau in the West Central Pacific. These records have been analyzed and reported
(Johnson et al, 1978).
The abstract of the Indian Ocean report follows (Hamilton et al, 1977, p. 3003).
New measurements of interval compressional wave velocities were made in the first
sediment layer, using the sonobuoy technique. These measurements were made during two
expeditions in the Bay of Bengal, in the Andaman Sea and over the Nicobar Fan and Sunda
Trench. Sediment interval velocities from these areas were added to those previously re-
ported, and revised diagrams and regression equations of instantaneous and mean velocity
versus one-way travel time are furnished for four areas of the Bengal Fan and for the Anda-
man Basin, Nicobar Fan, and Sunda Trench. The velocity gradients directly below the sea
floor were used to separate the Bengal Fan into four geoacoustic provinces. In the north
and west, the velocity gradients are 0.86 s~! and 1.28 Sa respectively; whereas in the cen-
tral part of the fan, the gradient is 1.87 s-!_ These variations indicate lesser increases of
velocity with depth in the sea floor in the north and west. They are due probably to more
rapid deposition, less consolidation and less lithification near the riverine source areas of
the sediments. The near-surface velocity gradients in the other areas are: the Andaman
Basin, 1.53 sv] : the Nicobar Fan, 1.63 sv! ; and the Sunda Trench, 1.41 s-!. In a second
part of the paper, the linear velocity gradients (from the sediment surface to a given travel
time) in 17 areas of the Indian Ocean, Pacific area, Atlantic Ocean, and Gulf of Mexico
were averaged at each 0.1 s from 0 to 0.5 s of one-way travel time. These averaged gradients
ranged from 1.32 s-l att=0 to 0.76s—! att=0.5s. The regression equation for the veloc-
ity gradient, a, in sl as a function of one-way travel time, t, in seconds is: a= 1.316 —
1.117t (for use from t = 0 to 0.5 s). These average velocity gradients can be used with sedi-
ment surface velocities and one-way travel times (measured from reflection records) to com-
pute sediment layer thicknesses in areas of turbidites lacking interval velocity measurements
in the first sediment layer.
Three of the figures in Hamilton et al (1977) are reproduced here. Figure 4 shows
the sonobuoy stations in the Northeast Indian Ocean; Figure 5 illustrates the type and scat-
ter of the data; Figure 6, the averaged velocity gradients in these and other areas.
Current Studies
At present (September 1977), sound velocity gradients in the principal types of
marine sediments are being studied. These types include sands, terrigenous sediments (di-
rectly from land sources), calcareous ooze, and siliceous ooze. The emphasis in these studies
will be on averaged values which can aid in predictions.
ATTENUATION OF COMPRESSIONAL (SOUND) WAVES IN MARINE SEDIMENTS
AND ROCKS
Introduction
Some years ago it became apparent that sound propagated through the sea floor at
certain frequencies and at certain grazing angles. In such cases quantitative knowledge of
sound attenuation in marine sediments became a required property in understanding sound
14
interactions with the sea floor. Consequently, measurements and studies of sound attenua-
tion have been a long-term, continuing project in the geology-geophysics group.
Hamilton (1972) reported the results of in situ measurements of sound velocity and
attenuation in various sediments off San Diego. These measurements and others from the
literature allowed analyses of the relationships between attenuation and frequency and other
physical properties. It was concluded that attenuation in dB/unit length is approximately
dependent on the first power of frequency and that velocity dispersion is negligible or ab-
sent in water-saturated sediments. The report also discussed the causes of attenuation, its
prediction (given grain size or porosity), and appropriate viscoelastic models which can be
applied to sediments.
Studies From 1974 to 1977
In 1975 and 1976 two reports were issued which revised data and two illustrations
in the 1972 report. The first was a Naval Undersea Center report, NUC TP 482 (Hamilton,
1975c), followed by its publication in the Journal of the Acoustical Society of America
(Hamilton, 1976b).
Figure 7 is reproduced from Hamilton (1976b, Figure 1). This figure illustrates the
relations between attenuation in dB/m and frequency in kHz. The new data in this revised
figure were given different symbols from the original (1972) figure. The new data comple-
ment and supplement the original data and strongly support an approximate first-power
dependence of attenuation on frequency. The data in Figure 7 include sands, silt-clays
and mixed-grained materials. The experimental evidence does not support any theory call-
ing for a dependence of attenuation on f” or f2 in either sands or silt-clays.
If attenuation is dependent on the first power of frequency, as indicated by the
evidence in Figure 7, then in the equation a = kf™ (where the exponent n is one, @ is attenu-
ation in dB/m, f is frequency in kHz and k is a constant), the only variable is the constant k.
This allows k to be related to common sediment properties such as mean grain size or poros-
ity (Figure 8). Figure 8 (reproduced from Figure 2, Hamilton, 1976b) was revised from a
similar figure in the 1972 report, with the addition of four new sets of measurements.
These measurements did not alter the original conclusions. An important conclusion is that
prediction of sound attenuation in the sediment surface can be based on mean grain size or
porosity. To predict attenuation, we simply determine the constant k from its relations
with porosity (or mean grain size in the 1972 report) and insert k into the above equation,
which should be good at any frequency.
The main purpose of the 1975 and 1976 reports was to discuss the variations of
attenuation with depth in the sea floor. The sparse data were collected on attenuation and
depth at various frequencies. These data were listed in a table and illustrated. Figure 3 of
Hamilton (1976b) is reproduced here as Figure 9; these data illustrate sound attenuation
(represented by the constant k) as a function of depth in the sea floor. It was concluded
that attenuation decreases with about the —1/6 power of depth in sands. As a silt-clay sedi-
ment (mud), or turbidite, is placed under increasing overburden pressure, there may be a
progressive increase in attenuation due to reduction in sediment porosity and a progressive
decrease in attenuation due to increasing pressure on the sediment mineral frame. At some
null point in the sediment (sparse evidence indicates about 200 meters), pressure becomes
the dominant effect and attenuation decreases smoothly thereafter with depth and over-
burden pressure. It was concluded that Figure 9 can be used to aid prediction of sound
attenuation in sediment and rock layers in the sea floor.
The study of sound attenuation in marine sediments and rocks is a continuing pro-
ject. Since the 1976 report new measurements have indicated the validity of the above
approach and conclusions. The most important of these measurements were by Tyce (per-
sonal communication), using the Marine Physical Laboratory deep-tow equipment in both
the Atlantic and Pacific Oceans.
VARIATIONS OF DENSITY AND POROSITY WITH DEPTH IN DEEP-SEA
SEDIMENTS
Introduction
The values and variations of density and porosity with depth in marine sediments
and rocks are of importance in both basic and applied studies of the earth. Specifically (in
the field of sound interactions with the sea floor), density of various layers of the oceanic
crust are important in the propagation of shear and compressional waves and other elastic
waves. Values of density are required in all mathematical models of sound interacting with
the sea floor. However, work at the Applied Research Laboratories of the University of
Texas has indicated that, in many cases, the gradient of density may have only a small
effect on bottom losses. At high grazing angles (above the shear wave critical angle), the
effects amount to about | to 2 dB change in bottom loss. At low angles, very little effect
is observed except in the vicinity of the low angle shear anomaly (discussed in a following
section) where it can amount to as much as 2 to 8 dB (Hawker et al, 1976, p. 65).
Studies From 1974 to 1977
Three reports were issued during the three-year period concerning variations of den-
sity and porosity with depth in the sea floor. The first was issued by the Naval Undersea
Center as TP 459 (Hamilton, 1975a); this report was later published in the Journal of
Sedimentary Petrology (Hamilton, 1976c). A resume with selected figures are noted below.
Bachman and Hamilton (1976) obtained a suite of samples from the Deep Sea Drill-
ing Project Site 222 (Leg 23) in the northern Arabian Sea. At this site there was an unusu-
ally thick section of homogeneous, terrigenous sediment which was drilled to about 1300
meters. Density, porosity, and grain density were measured in the laboratory. These data
were also included in the Hamilton and Bachman and Hamilton reports.
The critical question in relating laboratory measurements of sediment density and
porosity to in situ measurements in a deep borehole is: how much has the sample expanded
elastically as a result of removal from overburden pressure in the borehole to atmospheric
pressure in the laboratory? This was the problem addressed in Hamilton (1976c). The
abstract of this report follows.
Reduction of sediment porosity and increase in density under overburden pressure
in the sea floor are important subjects in earth sciences. Data and samples from the Deep
Sea Drilling Project allow a new look at these subjects, and are used to establish profiles of
laboratory values of density and porosity versus depth in the sea floor. To construct in situ
profiles, the results of consolidation tests are used to estimate the amount of elastic rebound
(increase in volume) which has occurred after removal of the samples from overburden
pressure in the boreholes. In situ profiles of porosity and density versus depth are con-
structed for some important sediment types: calcareous ooze, siliceous oozes (diatoma-
ceous and radiolarian oozes), pelagic clay, and terrigenous sediments. There is less reduction
of porosity with depth in the first 100 meters in these deep-water sediments than previously
supposed: 8 to 9 percent in pelagic clay, calcareous and terrigenous sediments and only 4 to
5 percent in the siliceous sediments. From depths of 300 meters the most rebound is in
pelagic clay (about 7 percent) and the least in diatomaceous ooze (about 2 percent);
calcareous ooze and terrigenous sediment should rebound from 300 meters about 4 to 5
percent. Terrigenous sediment, from the surface to 1000 meters depth, probably rebounds
a maximum of about 9 percent. Methods are described and illustrated to predict density
and porosity gradients in the sea floor and to compute the amounts of original sediments
necessary to have been compressed to present thicknesses. Slightly over 2000 meters of
original sediments would have been required for compression to a present-day thickness of
1000 meters of terrigenous sediments.
Two figures are reproduced here from Hamilton (1976c). Figure 10 illustrates
laboratory measurements which have been corrected to in situ values and compared with
data in shales (below 600 meters) from oil-industry explorations. Figure 11 illustrates
density versus depth for the five most common sediment types. These data and tables and
regression equations in the report should allow reasonable predictions of density at given
depths in the sea floor.
SHEAR WAVE VELOCITY PROFILES IN MARINE SEDIMENTS
Introduction
The velocity of a compressional wave is dependent on the sediment bulk modulus,
rigidity and density. Given shear wave velocity and density, rigidity can be easily computed.
Given shear and compressional wave velocities and density, all of the other elastic properties
can be computed. When a sound wave is reflected within a sediment or rock layer, part of
the energy is converted to a shear wave.
Studies at the Applied Research Laboratories of the University of Texas (Hawker
and Foreman, 1976; Hawker et al, 1976, 1977) found important effects when shear waves
were introduced into bottom loss models. At low grazing angles in the case of clay and
possibly silt (but not sand) overlying hard rock, it was found that a very large bottom loss
can occur over a narrow angular range through the production of a Stoneley wave (closely
related to the shear wave) along the sediment-rock interface. These dominant effects oc-
curred between the shear velocity critical grazing angle of about 50 degrees and the com-
pressional velocity grazing angle of about 70 degrees.
Studies From 1974 to 1977
In an earlier report (Hamilton, 1971a), the presence and causes of rigidity and shear
waves in marine sediments were reviewed. Hamilton with Bucker and his colleagues at the
Naval Undersea Center (Hamilton et al, 1970) reported in situ measurements of compres-
sional wave velocity, density, and velocities of Stoneley waves (from which shear waves can
be determined). In these reports the variation of shear wave velocity with depth in the sedi-
ments was not considered.
A short study and review of shear wave velocity versus depth in marine sediments
was issued by the Naval Undersea Center as TP 472 (Hamilton, 1975b), and later published
in Geophysics (Hamilton, 1976e). The abstract of this report follows.
The objectives of this report are to review and study selected measurements of the
velocity of shear waves at various depths in some principal types of unlithified, water-
saturated sediments and to discuss probable variations of shear velocity as a function of
pressure and depth in the sea floor. Because of the lack of data for the full range of marine
sediments, data from measurements on land were used and the study was confined to the
two ‘“‘end-member” sediment types (sand and silt-clays) and turbidites.
The shear velocity data in sands included 29 selected, in situ measurements at depths
to 12 meters. The regression equation for these data is: V.= 128p9-28 where V, is shear
wave velocity in m/s and D is depth in meters. The data from field and laboratory studies
indicate that shear wave velocity is proportional to the 1/3 to 1/6 power of pressure or
depth in sands; that the 1/6 power is not reached until very high pressures are applied; and
that in most sand bodies the velocity of shear waves is proportional to the 3/10 to 1/4 power
of depth or pressure. The use of a depth exponent of 0.25 is recommended for prediction
of shear velocity versus depth in sands.
The shear velocity data in silt-clays and turbidites include 47 selected, in situ
measurements at depths to 650 meters. Three linear equations are used to characterize the
data. The equation for the 0 to 40 meters interval (Vz = 116 + 4.65D) indicates the gradient
(4.65 sec7!) to be four to five times greater than is the compressional velocity gradient in
this interval in comparable sediments. At deeper depths, shear velocity gradients are
1.28 sec! from 40 to 120 meters and 0.58 sec~! from 120 to 650 meters. These deeper
gradients are comparable to those of compressional wave velocities. These shear velocity
gradients can be used as a basis for predicting shear velocity versus depth.
Two figures reproduced here from Hamilton (1976e) illustrate shear velocity versus
depth in sands (Figure 12) and in silt-clays (Figure 13).
ATTENUATION OF SHEAR WAVES IN MARINE SEDIMENTS
Introduction
When a compressional wave is reflected at some impedance mismatch within the sea
floor, some of the energy is converted to a shear wave and this converted energy is rapidly
attenuated.
In some sophisticated mathematical models of sound interaction with the sea floor,
the attenuation of shear waves is a required input (see Part II, this report).
Studies From 1974 to 1977
Very little experimental data are available on the attenuation of shear waves. The
available data are almost all in the fields of geotechnical (soil mechanics) engineering and
earthquake research. The available data were collected, studied and reported by Hamilton
(1976d) and the abstract of this report follows.
The objectives of this report are to review selected, published measurements of the
attenuation, or energy damping, of low-strain shear waves in surficial water-saturated sands
and silt-clays (mud) that might occur as marine sediments. In various computations, a
linear viscoelastic model is favored in which velocity dispersion is negligible, linear attenua-
tion is proportional to the first power of frequency and the specific dissipation function,
1/Q, and the logarithmic decrement are independent of frequency. The logarithmic decre-
ment is favored as a measure of energy damping because of research in soil mechanics. The
very sparse data indicate that in water-saturated sands and silt-clays, the logarithmic decre-
ments are mostly between 0.1 and 0.6. If approximate values of shear wave energy losses
are required for generalized computations, it is suggested that a value for the logarithmic
decrement of 0.30 + 0.15 be assumed for sands and 0.2 + 0.1 for silt-clays. Measured
logarithmic decrements of compressional waves in sands average about 0.10 + 0.03; in silt-
clays about 0.02 + 0.01. The average values of the ratio of compressional- to shear-wave
logarithmic decrements, using the above average values, would be 0.3 for sands and 0.1 for
silt-clays.
SOUND VELOCITY-DENSITY RELATIONS IN SEA-FLOOR SEDIMENTS
AND ROCKS
Introduction
Continuous acoustic reflection surveys are rapidly delineating the sediment and rock
layers of the sea floor. Wide-angle reflection and refraction measurements (as with expenda-
ble sonobuoys) yield velocities in these layers. This allows true layer thicknesses to be
computed. Further, the new velocity data frequently can be linked to sediment and rock
types by geologic reasoning and by direct linkage to the boreholes of the Deep Sea Drilling
Project. Therefore, it would be useful to establish relationships between velocity and density
in the various sediment and rock types in the sea floor. This would allow prediction of
density (a prime requirement) to correspond to measured sound velocities for purposes of
modeling the sea floor for underwater acoustics studies. Additionally, density profiles can
be constructed from these data or from densities derived from velocities computed from
equations of velocity versus travel time or depth.
Studies From 1974 to 1977
Naval Ocean Systems Center measurements of density and velocity in marine sedi-
ments were combined with information from the literature and a report published which
relates density and velocity for common sediments and rocks (Hamilton, 1978). Figure 14
is asummary of individual curves. The abstract of the report follows.
In underwater acoustics, geophysics, and geologic studies, given a seismic measure-
ment of velocity, the relations between sound velocity and density allow assignment of
approximate values of density to sediment and rock layers of the earth’s crust and mantle.
In the past, single curves of velocity versus density represented all sediment and rock types.
A large amount of recent data from the Deep Sea Drilling Project (DSDP) and reflection
and refraction measurements of sound velocity, allow construction of separate velocity-
density curves for the principal marine sediment and rock types. This report uses carefully-
selected data from laboratory and in situ measurements to present empirical sound velocity-
density relations (in the form of regression curves and equations) in terrigenous silt-clays,
turbidites and shale, in calcareous materials (sediments, chalk, and limestone), and in
siliceous materials (sediments, porcelanite and chert); a published curve for DSDP basalts
is included. Speculative curves are presented for composite sections of basalt and sediments.
These velocity-density relations, with seismic measurements of velocity, should be useful in
assigning approximate densities to sea floor sediment and rock layers for studies in marine
geophysics and in forming geoacoustic models of the sea floor for underwater acoustic
studies.
PRODUCTION OF GEOACOUSTIC MODELS
Introduction
Geoacoustic models were defined and requirements for input information were
noted briefly in the general introduction to Part I (Marine Sediment Properties). As noted
in the general introduction, geoacoustic models of the sea floor are produced to guide
theoretical studies, to reconcile experiments at sea with theory, and to predict the effects
of the sea floor on underwater sound propagation.
Studies From 1974 to 1977
During the three-year period of this project, geoacoustic models were furnished to
various persons or groups.
@ Two geoacoustic models were furnished to the Acoustic Environmental
Support Detachment, ONR. One was for an area in the Northeast Pacific and was well
founded on coring and acoustic reflection measurements. The second model was for the
northern end of the Iberian Basin off the coast of Spain and was based on information
in the literature. These data with accompanying bottom-loss curves were issued by the
Naval Undersea Center TN 1470 (Morris et al, 1974).
@ Geoacoustic models were furnished to investigators in the Undersea Surveil-
lance Department, Naval Undersea Center, for areas or stations in the following localities.
1. Continental slope west of the Strait of Juan de Fuca
2. Northeast Pacific: Tufts Abyssal Plain, and another area south of
Mendocino Escarpment
20
3. Off Point Sur, California
4. Ridge west of the Norwegian Basin
® In preparation for experimental work in the Arabian Sea, geoacoustic models
were furnished for four stations to allow predictions of underwater sound propagation
in the area. These models were part of the predictive study issued by the Naval Ocean
Systems Center as TN 104 (Northrop et al, 1977). A portion of this report (Northrop
et al, 1978) is in press as an article for the Journal of Underwater Acoustics.
SUMMARY
The general summary of Part I (Marine Sediment Properties), with results and
recommendations, is in the front section of this report.
21
REFERENCES
(PART I)
REPORTS INVOLVING GEOLOGY AND GEOPHYSICS
OF THE SEA FLOOR (1974-1977)
Partial support of NAVELEX, Code 320, was acknowledged in the following reports
involving geology, geophysics, and acoustic properties of the sea floor during calendar years
1974-1977 (two reports in press in 1977 were published in January and February 1978).
Morris, H. E., Hamilton, E. L., and Bucker, H. P., 1974, Low frequency geoacoustic
models and bottom loss curves for two areas in the Northeast Pacific and Iberian
Basin, Naval Undersea Center TN 1470.
Hamilton, E. L., 1975a, Acoustic and related properties of the sea floor: density
and porosity profiles and gradients, Naval Undersea Center TP 459.
Hamilton, E. L., 1975b, Acoustic and related properties of the sea floor: shear
wave velocity profiles and gradients, Naval Undersea Center TP 472.
Hamilton, E. L., 1975c, Acoustic and related properties of the sea floor: sound
attenuation as a function of depth, Naval Undersea Center TP 482.
Hamilton, E. L., 1975d, Acoustic properties of the sea floor: a review, in Proceed-
ings of a Conference on Oceanic Acoustic Modelling (held at La Spezia, Italy, 8-11
September 1975), SACLANTCEN Conference Proceedings No. 17, Part 4 (Sea
Bottom), Paper No. 18, 96 pp.
Hamilton, E. L., 1976a, Acoustic properties of shallow-water sediments: a review
(U), Proceedings of Shallow Water Mobile Sonar Environmental Acoustic Modeling
Symposium (U) (held at the Naval Research Laboratory, 23-25 September 1975),
Vol. II (C), pp 361-433, SEA 06H1/036-EVA MOST-8, Naval Sea Systems
Command, Washington, D.C.
Hamilton, E. L., 1976b, Sound attenuation as a function of depth in the sea floor,
Journal of the Acoustical Society of America, Vol. 59, pp 528-535. Outside pub-
lication of Hamilton, 1975c.
Hamilton, E. L., 1976c, Variations of density and porosity with depth in deep-sea
sediments, Journal of Sedimentary Petrology, Vol. 46, pp 280-300. Outside pub-
lication of Hamilton, 1975a.
Hamilton, E. L., 1976d, Attenuation of shear waves in marine sediments, Journal
of the Acoustical Society of America, Vol. 60, pp 334-338.
Hamilton, E. L., 1976e, Shear-wave velocity versus depth in marine sediments: a
review, Geophysics, Vol. 41, pp 985-996. Outside publication of Hamilton, 1975b.
Bachman, R. T., and Hamilton, E.L., 1976, Density, porosity, and grain density of
samples from Deep Sea Drilling Project Site 222 (Leg 23) in the Arabian Sea,
Journal of Sedimentary Petrology, Vol. 46, pp 654-658.
Hamilton, E. L., Bachman, R. T., Curray, J. R., and Moore, D. G., 1977, Sediment
velocities from sonobuoys: Bengal Fan, Sunda Trench, Andaman Basin, and
Nicobar Fan, Journal of Geophysical Research, Vol. 82, pp 3003-3012.
Johnson, T. C., Hamilton, E. L., and Berger, W. H., 1977, Physical properties of
calcareous ooze: control by dissolution at depth, Marine Geology, Vol. 24,
pp 259-277.
DP
Northrop, J., Fabula, A. G., Colborn, J. G., Hamilton, E. L., Bachman, R. T.,
Solomon, L. P., Barnes, A. E., Bucker, H. P., and Wagstaff, R. A., 1977, Environ-
mental acoustic predictions for the Northwestern Indian Ocean (U), Naval Ocean
Systems Center TN 104 (C).
Johnson, T. C., Hamilton, E. L., Bachman, R. T., and Berger, W. H., 1978, Sound
velocities in calcareous oozes and chalks from sonobuoy data: Ontong Java Plateau,
Western Equatorial Pacific, Journal of Geophysical Research, Vol. 83, pp 283-288.
Hamilton, E. L., 1978, Sound velocity-density relations in sea floor sediments and
rocks, Journal of the Acoustical Society of America, Vol. 63, pp 366-377.
Northrop, J., Colborn, J. G., Hamilton, E. L., and Bachman, R. T., 1978, Propaga-
tion loss predictions for the Northwestern Indian Ocean, Journal of Underwater
Acoustics, in press.
ADDITIONAL REFERENCES
(PART 1)
(In addition to those above for the period 1974 to 1977)
Buchan, S., Dewes, F. C. D., Jones, A. S. G., McCann, D. M., and Smith, D. T.,
1971, The Acoustic and Geotechnical Properties of North Atlantic Cores, Vols.
1 and 2, Marine Science Laboratories, University College of North Wales, Menai
Bridge, North Wales, 22 pp and tables.
Bucker, H. P. and Morris, H. E., 1975, “Reflection of sound from a layered ocean
bottom’’, Invited paper presented at Oceanic Acoustic Modelling Conference
held at SACLANTCEN on 8-11 Sept. 1975, pp 19-1 to 19-35 in SACLANTCEN
Conference Proceedings No. 17.
Christensen, N. I., and Salisbury, M. H., 1975, Structure and constitution of the
lower oceanic crust, Reviews of Geophysics and Space Physics, Vol. 13, pp 57-86.
Dickinson, G., 1953, Geological aspects of abnormal reservoir pressures in Gulf
Coast Louisiana, American Association of Petroleum Geologists, Vol. 37, pp 410-
432.
Gardner, G. H. F., Gardner, L. W., and Gregory, A. R., 1974, Formation velocity
and density — the diagnostic basics for stratigraphic traps, Geophysics, Vol. 39,
pp 770-780.
Griffiths, J. C., 1967, Scientific Method in Analysis of Sediments, McGraw-Hill,
New York, 508 pp.
Hamilton, E. L., 1971a, Elastic properties of marine sediments, Journal of Geo-
physical Research, Vol. 76, pp 579-604.
Hamilton, E. L., 1971b, Prediction of in situ acoustic and elastic properties of
marine sediments, Geophysics, Vol. 36, pp 266-284.
Hamilton, E. L., 1972, Compressional wave attenuation in marine sediments, Geo-
physics, Vol. 37, pp 620-646.
Hamilton, E. L., 1974a, Prediction of deep-sea sediment properties: state of the art,
in Deep-Sea Sediments, Physical and Mechanical Properties, pp 1-43, edited by
A. L. Inderbitzen, Plenum Press, New York, 497 pp.
23
Hamilton, E. L., 1974b, Geoacoustic models of the sea floor, in Physics of Sound in
Marine Sediments, pp 181-221, edited by L. Hampton, Plenum Press, New York,
567 pp.
Hamilton, E. L., Bucker, H. P., Keir, D. L., and Whitney, J. A., 1970, Velocities of
compressional and shear waves in marine sediments determined in situ from a research
submersible, Journal of Geophysical Research, Vol. 75, pp 4039-4049.
Hamilton, E. L., Moore, D. G., Buffington, E. C., Sherrer, P. L., and Curray, J. R.,
1974, Sediment velocities from sonobuoys: Bay of Bengal, Bering Sea, Japan Sea,
and North Pacific, Journal of Geophysical Research, Vol. 79, pp 2653-2668.
Hawker, K. E., and Foreman, T. L., 1976, A plane wave reflection coefficient model
based on numerical integration; formulation, implementation, and application,
Applied Research Laboratories, University of Texas at Austin, ARL-TR-76-23,
78 pp.
Hawker, K. E., Anderson, A. L., Focke, K. C., and Foreman, T. L., 1976, Initial
phase of a study of bottom interaction of low frequency underwater sound, Applied
Research Laboratories, University of Texas at Austin, ARL-TR-76-14, 180 pp.
Hawker, K. E., Focke, K. C., and Anderson, A. L., 1977, A sensitivity study of
underwater sound propagation loss and bottom loss, Applied Research Laboratories,
University of Texas at Austin, ARL-TR-77-17, 122 pp.
Houtz, R., Ewing, J., and LePichon, X., 1968, Velocity of deep-sea sediments from
sonobuoy data, Journal of Geophysical Research, Vol. 73, pp 2615-2641.
Houtz, R., Ewing, J.,and Buhl, P., 1970, Seismic data from sonobuoy stations in
the northern and equatorial Pacific, Journal of Geophysical Research, Vol. 75,
pp 5093-5111.
Igarashi, Y., 1973, In situ high-frequency acoustic propagation measurements in
marine sediments in the Santa Barbara shelf, California, Naval Undersea Center
TP 334, 40 pp.
Ludwig, W. J., Nafe, J. E., and Drake, C. L., 1970, Seismic refraction, in The Sea,
Vol. 4, Part 1, pp 53-84, edited by A. E. Maxwell, John Wiley, New York.
Magara, K., 1968, Compaction and migration of fluids in Miocene mudstone,
Nagaoka Plain, Japan, American Association of Petroleum Geologists Bulletin,
Vol. 52, pp 2466-2501.
Moore, D. G., Curray, J. R., Raitt, R. W., and Emmel, F. J., 1974, Stratigraphic-
seismic section correlations and implications to Bengal Fan history, in Initial Reports
of the Deep Sea Drilling Project, Vol. 22, edited by C. C. von der Borch, J. G. Sclater
et al., pp 403-412, U.S. Government Printing Office, Washington, D.C.
Morris, H. E., 1970, Bottom-reflection-loss model with a velocity gradient, Journal
of Acoustical Society of America, Vol: 48, pp 1198-1202.
Muir, T. G., and Adair, R. S., 1972, Potential use of parametric sonar in marine
archeology, Journal of the Acoustical Society of America, Vol. 52, p 122 (abstract);
unpublished manuscript, Applied Research Laboratories, University of Texas at
Austin.
Neprochnoy, Yu. P., 1971, Seismic studies of the crustal structure beneath the seas
and oceans, Oceanology (English translation), Vol. 11, pp 709-715.
24
Rieke, H. H., III, and Chilingarian, G. V., 1974, Compaction of Argillaceous
Sediments, Elsevier Publishing Co., New York, 424 pp.
Shepard, F. P., 1954, Nomenclature based on sand-silt-clay ratios, Journal of Sedi-
mentary Petrology, Vol. 24, pp 151-158.
Tyce, R. C., 1975, Near-bottom observations of 4 kHz acoustic reflectivity and
attenuation, Geophysics, Vol. 41, pp 675-699.
Table la. Continental terrace (shelf and slope) environment; average sediment
size analyses and bulk grain densities.
Bulk
Mean Grain Grain
Sediment No. Diameter Sand, Silt, Clay, Density,
Type Samples SiGe Roo) % % % g/cm
Sand
Coarse 2 0.5285 0.92 100.0 0.0 0.0 2.710
Fine 18 0.1638 2.61 92.4 4.2 3.4 2.708
Very fine 6 0.0915 3.45 84.2 10.1 Sol 2.693
Silty sand 14 0.0679 3.88 64.0 Bio! RS) 2.704
Sandy silt 17 0.0308 5.02 26.1 60.7 13.2 2.668
Silt 12 0.0213 555 6.3 80.6 13.1 2.645
Sand-silt-clay 20 0.0172 5.86 BD) 41.0 26.8 2.705
Clayey silt 60 0.0076 7.05 VD 59.7 33.1 2.660
Silty Clay 19 0.0027 8.52 4.8 412 54.0 2.701
Table 1b. Continental terrace (shelf and slope) environment; sediment densities,
porosities, sound velocities and velocity ratios.
Density, Porosity, Velocity,
Sediment g/cm % m/sec Velocity Ratio
Type Avg. SE Avg. SE Avg. SE Avg. SE
Sand
Coarse 2.034 — 38.6 - 1836 - 1.201 -
Fine LOST 0.023 44.8 1.36 1753 11 1.147 0.007
Very fine 1.866 0.035 49.8 1.69 1697 32 1.111 0.021
Silty sand 1.806 0.026 53.8 1.60 1668 11 1.091 0.007
Sandy silt 1.787 0.044 52.5 2.44 1664 13 1.088 0.008
Silt 1.767 0.037 54.2 2.06 1623 8 1.062 0.005
Sand-silt-clay 1.590 0.026 66.8 1.46 1579 8 1.033 0.005
Clayey silt 1.488 0.016 71.6 0.87 1549 4 1.014 0.003
Silty clay 1.421 0.015 7529. 0.82 1520 3 0.994 0.002
Notes: Laboratory values: 23°C, 1 atm; density: saturated bulk density; porosity: salt free; velocity
ratio: velocity in sediment/velocity in sea water at 23°C, 1 atm, and salinity of sediment pore
water. SE: standard error of the mean.
Table 2a. Abyssal plain and abyssal hill environments; average sediment size
analyses and bulk grain densities.
Bulk
Mean Grain Grain
Environment No. Diameter Sand, Silt, Clay, Density,
Sediment Type Samples mm wv) % %o % g/cm
Abyssal Plain
Sandy silt 1 0.0170 5.88 19.4 65.0 15.6 2.461
Silt 3 0.0092 6.77 3D) 78.0 18.8 2.606
Sand-silt-clay 2 0.0208 S58) BoE 33.3 31.5 2.653
Clayey silt Dp) 0.0053 VST 4.5 S56) 40.2 2.650
Silty clay 40 0.0021 8.90 2.5 36.0 61.5 2.660
Clay 6 0.0014 9858 0.0 DVD) 778 2.663
Bering Sea and Okhotsk Sea (Diatomaceous)
Silt 1 0.0179 5.80 6.5 76.3 17.2 2.474
Clayey silt 5 0.0049 7.68 8.1 49.1 42.8 2.466
Silty clay 23 0.0024 8.71 3.0 37.4 59.6 2.454
Abyssal Hill
Deep-sea (“‘red’’) pelagic clay
Clayey silt 17 0.0056 7.49 3.9 58.7 37.4 2.678
Silty clay 60 0.0023 8.76 2.1 32.2 65.7 DeTiel
Clay 45 0.0015 9.43 0.1 19.0 80.9 2.781
Calcareous ooze
Sand-silt-clay 5 0.0146 6.10 Dyes 42.8 2919 2.609
Silt 1 0.0169 5.89 16.3 75.6 8.1 2.625
Clayey silt 15 0.0069 VAY 3.4 60.7 359 2.678
Silty clay 4 0.0056 7.48 B19) 39.9 56.2 2.683
Table 2b. Abyssal plain and abyssal hill environments; sediment densities,
porosities, sound velocities and velocity ratios.
Density,
Environment g/cm3
Sediment Type Avg. SE
Abyssal Plain
Sandy silt 1.652 —
Silt 1.604 -
Sand-silt-clay 1.564 _
Clayey silt 1.437 0.023
Silty clay 1.333 0.019
Clay 1.352 0.037
Porosity,
Avg.
56.6
63.6
66.9
VSL)
81.4
80.0
Bering Sea and Okhotsk Sea (Diatomaceous)
Silt 1.447
Clayey silt 1.228
Silty clay 1.214
Abyssal Hill
Deep-sea (“‘red’’) pelagic clay
Clayey silt 1.347
Silty clay 1.344
Clay 1.414
Calcareous ooze
Sand-silt-clay 1.400
Silt 125
Clayey silt 1573
Silty clay 1.483
6.019
0.008
0.020
0.011
0.012
0.013
0.020
0.029
70.8
85.8
86.8
81.3
81.2
Ue
76.3
56.2
66.8
(23
%o
SE
0.95
0.60
0.64
0.90
22
1.61
Velocity,
m/sec
Avg.
1622
1563
1536
1526
1515
1503
1546
1534
1525
1522
1508
1493
1581
1565
1537
1524
SE
Velocity Ratio
Avg.
1.061
1.022
1.004
0.998
0.991
0.983
1.011
1.003
0.997
0.995
0.986
0.976
1.034
1.023
1.005
0.996
SE
0.002
0.001
0.001
0.005
0.003
0.005
Notes: Laboratory values: 23°C, 1 atm; density: saturated bulk density; porosity: salt free; velocity
ratio: velocity in sediment/velocity in sea water at 23°C, 1 atm, and salinity of sediment pore
water. SE: standard error of the mean.
SOUND VELOCITY, m/s
30 40 50 60 70 80 FO Oo
POROSITY, %
Figure 1. Sediment porosity versus sound velocity, continental terrace (shelf and slope).
SOUND VELOCITY, m/s
@ 0 1 D 3 4 5 6 7 8 9 10
Mao) Bn) 250) 125 GIS 218 186 78 29 2 1
MEAN GRAIN DIAMETER
Figure 2. Mean diameter of mineral grains versus sound velocity, continental terrace
(shelf and slope).
30
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31
80°E 85° 90° 957 100°
— Antipode (Leg II)
— Circe (Leg 3)
— Tasaday (Leg 6)
— Eurydice (Leg 5)
@apop
20°
Fy UU RM A
CONTOUR CHART
in corrected meters
Prepared by
J.R. Curray
F.J.Emmel HY)
0.G.Moore NUC
Basecharts furnished by
R.L. Fisher SIO
10° 10°
BO°E
Figure 4. Sonobuoy station locations in the Bay of Bengal and adjacent areas as revised from
Hamilton et al (1974) which contained only Antipode and Circe data. The base chart is from
Moore et al (1974); contours are in meters, corrected for sound velocity. Station numbers are
adjacent to symbols. Symbols refer to Scripps Institution of Oceanography expeditions. A
number of closely-spaced sonobuoys (at a single, numbered station) are represented by a
single dot.
By
0.05
0.15
0.20
0.25
0.30
ONE-WAY TRAVEL TIME, s
0.35
0.40
0.45
0.50
4 1.5 1.6
VELOCITY, km/s
1.8
1.7
1.9
2.0
2.1
2.2
2.3
2.4
Figure 5. Instantaneous velocity, V, and mean velocity, V, versus one-way travel time in the
Central Bengal Fan.
33
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sands (all sizes)
clay, silt (mud)
mixed sizes
(silty sand, sandy
silt)
0.001
0.0001
0.001 0.01 0.1 ] 10 100 1000
FREQUENCY, kHz
Figure 7. Attenuation of compressional (sound) waves versus frequency in natural, saturated sediments
and sedimentary strata. The solid lines and symbols are from Hamilton (1972, Figure 2); the open
symbols and dashed lines are newly-added data. The lines marked “‘J”’ and “I” represent general
equations for the Japan Sea and Indian Ocean Central Basin (from Neprochnov, 1971). The vertical,
dashed lines indicate a range of attenuation values at a single frequency. The line labelled “fl indi-
cates the slope of any line having a dependence of attenuation on the first power of frequency.
35
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36
k
OOS: O10 Ol ©.20 O25 O80 O35 O20
@
4 SAND- SILT
500
1000
measurements from
literature
] 500 first layer
second layer
DEPTH IN SEA FLOOR, m
third layer
2000
Figure 9. Attenuation of compressional waves (expressed as k in: QqB/m = kfkHz)
versus depth in the sea floor or in sedimentary strata. Symbols are layers in the sea
floor in 7 areas (from Neprochnoy, 1971).
Sy
DEPTH IN SEDIMENTS, m
POROSITY, %
0) 10 20 30 40 50 60 70 80
100
200 laboratory
in situ
300
400
500
600
mudstone
700
800
900
1000
e-laboratory measurements on samples
from DSDP Site 222
1
pat) D-Gulf Coast shale, (Dickinson, 1953)
© =v -Shale (Vassoevich in Rieke and
1200 Chilingarian, 1974 p 40, Figure 16)
M-Mudstone from Japan (Magara, 1958)
1300
Figure 10. Porosity versus depth in terrigenous sediments.
38
DEPTH IN SEDIMENTS, m
100
200
300
400
500
600
700
800
900
1000
DENSITY, g/cm?
1.2 1.3 1.4 1.5 1.6 7, 1.8 AS 2.0 2.1 2.2
R — radiolarian ooze
D — diatomaceous ooze
P — pelagic clay
C — calcareous sediment
T — terrigenous sediment
Figure 11. In situ density of various marine sediments versus depth in the sea floor.
39
DEPTH IN SAND, m
SHEAR WAVE VELOCITY, m/s
0 100 200 300 400
Figure 12. Shear wave velocity versus depth in water-saturated sands. All measurements are in situ;
multiple measurements at the same site are connected by solid lines. The dashed line is the regres-
sion equation: V, = 128(D)9-28; V, in m/sec and D is depth in meters.
40
SHEAR WAVE VELOCITY, m/s
0) 50 100 150 200 250 300 350 400 450 500
=
50
100
DEPTH IN SEDIMENTS, m
150
200
250
Figure 13. Shear wave velocity measured in situ versus depth in water-saturated silt-clays and turbidites.
Multiple measurements at the same site are connected by solid lines. The dashed lines are three linear
regressions. One measurement (V, = 700 m/sec at 650 meters) is not shown.
4]
VELOCITY, km/s
7.0
6.5
6.0
5.5
5.0
4.5
4.0
3.5
3.0
2.5
2.0
S1 diatomaceous sediments (0-500m)
$2 siliceous rocks
T1 terrigenous surface sediments
T2 turbidites (0-500 m)
T3 mudstones, shales
C1. chalk, limestone
C2 limestone
B basalt (Christensen & Salisbury (1975)
ND Nafe and Drake (Ludwig et al, 1970)
G Gardner et al (1974)
T3
$2 C2
(p
T4
S1 122
———
1.2 1.4 1.6 18°) 2:0 2.2, 2.4 26 2.8 CHO) Sh
DENSITY, g/cm3
Figure 14. A summary of compressional wave velocity versus density in Hamilton (1977).
The general curves of ND and G are included for comparisons. The equation for the curve
of Gardner et al (1974, p 779) is p = 0.23 Vices where p is density in g/cm3 and Vp is
compressional wave velocity in feet per second.
42
PART II:
ACOUSTIC MODELING
43
Merge’:
Ne nya tia
nee
ee
INTRODUCTION
This report describes the bottom loss models that comprise part of our effort in
the Bottom Interaction Program for 1974-1977. Although some of these models were
developed earlier, they continue to be modified and updated for current uses.
Our linear gradient multilayer model and our solid multilayer model were designed
to study reflection of low frequency sound from the ocean bottom. Considerable depths
(up to a kilometer) of ocean sediments are insonified at low frequencies. Consequently,
sediment parameters can undergo considerable variation in the insonified region. This
variation of sediment properties with depth must be taken into account if an accurate
representation of bottom loss is to be attained. The liquid multilayer model can account
for many layers of sediment in which the sound speeds are complex to account for ab-
sorption. The linear gradient model assumes complex sound speeds, also, but the sound
speeds vary in a linear fashion in a particular sediment layer.
Our solid multilayer model is a general purpose plane wave reflection model that
can account for both liquid and/or solid layers. We know that sediments have rigidity and,
therefore, for more accurate model calculations we must take rigidity into account. Im-
portant parameters to the solid model are the speed and attenuation of both the com-
pressional and shear waves that travel in the sediment. In our most recent programs we
have chosen to model the variable sediment properties with many layers (up to 1000
layers or more, if necessary), and maintain the needed accuracy by use of Knopoff’s
formulation. We have taken the concept of many constant layers as in the liquid model
(but for solid layers, i.e., sediments with rigidity) and made it possible to have the many
constant layers approximate the results of a linear gradient concept.
The distinguishing features of the solid multilayer model are the following:
All sediment layers can be realistically represented to have rigidity
Both solid and liquid layers can be taken into account if required
The Knopoff formulation provides fast and accurate computations
Continuous density variations are accounted for
The various outputs include bottom loss and/or R versus grazing angle
@ = The graphical forms include 3-dimensional representation
A so-called equivalent bottom concept has been developed for the Parabolic Equa-
tion (P.E.) propagation program. The Gibb’s oscillations caused by a density discontinuity
at the interface can be handled by calculating an equivalent reflection coefficient by
assuming different sediment parameters. An example of the technique is included. This
technique should enable the P.E. propagation program to be used for bottom limited areas.
FORMULATION OF THE SOUND FIELD USING THE PLANE WAVE
REFLECTION COEFFICIENT R
There are two types of solution to the wave equation which are of particular
importance in sound transmission. One is the transformation of the wave equation to the
eikonal equation and a solution in terms of wave surfaces and rays. The other is a develop-
ment through specific boundary conditions into a solution in terms of normal modes. In
45
some instances the physical conditions of the problem lead to a simpler solution in terms of
rays. In others, a solution in terms of normal modes is more satisfactory. In any case, it is
clear that the sediments that stratify the ocean bottom interact with the sound field through
the bottom reflection coefficient, R, in the derived theoretical expressions which follow.
RAY THEORY
In working with active sonar problems at frequencies of approximately 2.5 to
15 kHz usually we use ray theory. In ray theory, the sound field is made up of contributions
of rays that travel from the source to the receiver as shown in Figure 15. On the right hand
side of Figure 15 we show two neighboring rays that bracket the receiver at range r. Thus
there is an eigenray somewhere between these that travels precisely from the source to the
receiver. Call this the nth eigenray. The magnitude of the ray is A, and its phase is o as
expressed in the equation for the velocity potential, y
y= » A, exp (i8,,)
n
where 1/2
A, = IRI [cos 7, 8 y,/(t6h)]
and r
On -[ (w/c) d2+arg(R)-—m7/2 .
0
R is the reflection coefficient, w, the angular frequency, c, the sound speed, d&, a path
length along the ray and m is the number of times the ray has touched a caustic.
The reflection coefficient R is defined for reflection of plane waves and is a complex
number. The pressure carried by the ray is reduced by a factor equal to the modulus of R
and the phase of the ray is advanced by the argument of R. In experimental work, A,, can
be measured and the |R| determined from the equation for A,,. At high frequencies, the
plane and spherical coefficients essentially are equal. At low frequencies, the ray theory
breaks down and the sound cannot be separated into packets that have a well defined
trajectory. As shown in the following equations, the direct and bottom reflected sound
field can be written at any frequency in terms of the plane wave reflection coefficient.
WAVE THEORY
The general form of the sound field can be written as a sum of cylindrical waves in
the form (Bucker, 1970)
co
wy -f ~ 2 U(z,) V(z) W~! J (kr) kdk (25 <2 Sz)
0)
46
where
We U(Z,) V'(Z5) — UZ) V(Z5) ;
The zero depth, source depth, receiver depth and bottom depth are 0, Z), z and Zp» Tespec-
tively. The zero depth may be set at the air-water interface or at some other convenient
point. It represents the depth above which no sound is refracted or reflected to the
receiver. The horizontal wave number is k, r is the horizontal distance between the source
and receiver, Veo is the Bessel function of the first kind of order zero, U is a solution of the
z-separated ct of the wave equation, i.e., U”’ = (k2 - w2/v2(z))U, that satisfies the
boundary condition at z= 0, and Visa solution of the z-separated part of the wave equa-
tionsites Vin = (k2 - w2/v2(z))V, that satisfies the boundary condition at z = zp. For-
mally, our treatment will be restricted to (ZG <ZS<Zp); however, a similar development
for (0 <z <Z 9) is easily derived.
It is easy to show that dW/dz is zero so that W is independent of depth. Also, we
are free to specify the value of U and V at a selected depth which we, for convenience,
indicate by a bar and replace U and V by U and V where U(zp) = V(zp) = 1. It follows
then that in the limit Zp > Zp; W= Wz, = [i 2h(1—R) — (A+R) U'(2p)] /(1A+R). Therefore,
W can be expressed as
ue uf (1+R) U(Z9) V(z) J (kr) kdk
(iQ, — U'(z)) - RG&, + U'(z)) -
For the general sound speed profile it does not appear feasible to separate the direct sound
paths from the bottom reflected paths. However, if the water has a constant sound speed
then U(z) =exp [i&(z,-z)] and V(z) =[exp - ik(Z,-Z) +R exp i€(Z,-Z)] /(1+R). In this
case it follows that
vy oe (Me 2 20 5 o(kt) kdk “ff GiOyRa Dae coy o(kr) kdk
10)
———————— _—————————————
Yp VR
If the bottom reflected field Wp can be measured directly then we have
YR= uf GO iRem mommce aiayiea
oO
and R can be determined experimentally by use of the Hankel Transform
-f WR Jotkr) rdr/[i/&) exp iW Qua)
fe)
This is not a practical procedure, however, because quadrature sampling would be required
to determine the real and imaginary parts of Wp. That is Real (YR) = Pp cos¢ and
47
Im(WR) = pr sing, where pp is 1/2 the peak to peak pressure of the bottom reflected
signal and ¢ is the phase. In any event when realistic profiles are considered, it is not
possible to separate the direct and bottom reflected paths at low frequencies.
The normal mode form of the wave theory representation, required for low fre-
quency calculations, is shown in Figure 16 (Bucker, 1970). The source is at depth z, and
o)
range zero and the receiver is at depth z and range r. The potential function wy is given by
1/2
2 2] » Up (Zo) Up (2) exp (ik, 1)
nN
U = Af(z) + Bg(z) , 0<Z<zZ,
U =exp (12,2) + Rexp [i2, (22,-2)] > 2p <Z< fi)
where
OA Swi) ok
The depth function U is a sum of linearly independent solutions of the z-separated part of
the wave equation and k is the horizontal wave number. The k,, are those values of k for
which U satisfies the boundary conditions. The plane wave coefficient R is introduced into
the normal mode solution in the following manner. Assume that a “pseudo” isospeed
layer, with sound speed cy, extends from depth Zn to z,, as shown in Figure 16. Then the
z-component solution for the layer can be written as a downgoing plane wave exp (id, Z)
and an upgoing plane wave exp (—i%, z) multiplied by the plane wave reflection coefficient
R. The vertical wave number is one The values of the coefficients A and B can be deter-
mined by requiring the usual interface conditions at depth z,,. These conditions require
that the pressure (00U/dt) and the vertical component of particle velocity (0U/dz) must
be continuous functions. Solve these two equations and take the limit z,, > z, to obtain
the values of A and B.
p
Interface Conditions:
continuity of pressure (pU)
continuity of vertical (—dU/dz)
component of particle velocity
Solve for A and B as Zn ane
A = exp (ity 2p) [ge (1+R) - ig, a (I-R)1/W.
B = exp (ik, Zp) Li, f,,(1-R) — f,(1+R)]/W.
where
W= fp 8p Fh Sb
f,, =f = , f,, = (df/d , 2p = (dg/d
ys hey) 5 Gy SA) 9 y= Gi aye fp = (dg/ 2s
Note that the isospeed layer has been removed from the problem as Zp > Zp. However R
remains in the solution in A and B.
48
BOTTOM LOSS MODELS THAT ACCOUNT FOR GRADIENTS
The variation of sound speed in the ocean, configuration of the bottom, and
bottom and subbottom properties are generally the most important environmental factors
for the determination of underwater sound transmission. The bottom properties can vary
considerably from one area to another. The more common types of sediments are sand,
sand and mud, or mud. The areas of mud-size particles can vary in compactness from hard
clay to a loose suspension. Not enough is known of the acoustic properties of the immedi-
ate bottom materials and the variation of these properties as a function of depth into the
bottom as discussed in Part I. We do observe that the bottom characteristics have an
important effect in some areas on sound transmission. In other areas the bottom has very
little effect and the sound speed profile is the controlling factor. It is important to develop
realistic models of the bottom which use sediment characteristics to predict the reflection
coefficient R as a function of grazing angle for determining the acoustic transmission of
an area.
LIQUID MULTILAYER MODEL
We next want to consider multilayered sediment models that can be used either to
represent actual layering (e.g., it is not uncommon to find alternating layers of sand and silt
in shallow water) or to account for gradients. For the layered liquid case the solution is
very simple. Figure 17 shows n sediment layers and a half-space labeled (n + 1). In each
layer the potential function is the sum of an upgoing and a downgoing plane wave (e.g.,
Wy = An expC’, Z,) + B, exp(-if, Z,)) and in the halfspace the potential function
represents a downgoing wave (W,4 1 = exp(i€i4) Zui):
Let P represent the pressure and Q the vertical component of particle velocity. Then
start at the interface between layer n and the half-space with the expressions for P and Q
that follow. P and Q are easily evaluated at the n/(n+1) interface where Zyn+1 18 Zero. Be-
cause P and Q are continuous functions they have the same values at the bottom of layern
(at z,, =d,,) that they have at the top of the half-space (at Tens] = 0). Therefore, A, and B,,
can be calculated. From these calculate P and Q at the top of layer n (at z, = 0). Continue
working up the layers until A, and B, are calculated. The value of R is obtained from
R= Bo/Ag:
n/(n+1) P= pp=Py+1 > Q= (dy/dz) = ik, 4)
Interface |
iLawen | A, = 1/2 exp(-if, d,,) [P/o, + Q/(ik,)]
n By = VA eat. cl) La. =O ))
asRoa| P = Pp(Ay * By)
DD) Vol=nk (AL Be)
Continue until aN and BS are calculated
Reflection coefficient: R= B,/A,
49
In ocean sediments it is common to find a series of layers of almost constant
properties. This model will be a good representation of these cases. In other places thick
layers (approximately one wave length) are found with continuous change of properties.
This continuous change in this case can be represented by a large number of constant
property layers. Later it will be shown there is good agreement between the gradient model
and the multilayered model.
LINEAR GRADIENT MULTILAYER MODEL
We developed another method for modeling the change of sediment properties with
depth due to increasing compaction and temperature. In this approach changes in sound
speed and density are accounted for by using single or multiple liquid layers where Airy
functions can be used to represent the sound energy. This method was first used by Morris
(1970) and the use and development of the model has continued (Morris, 1972, 1975). This
model is used to explain low values of bottom loss at small grazing angles and low frequency.
In this case we will use a somewhat different function x so we can account for a density
change in the laver. The general wave equation for the case where there is variation in both
sound speed and density (Brekhovskikh, 1960) is
p=/px
Wave eq. v2 xt K2x =0
where 5)
K2 = (w/v)? + 5 (d2 p/dz2) - 3 [4[7-(ao/2)
If K2 can be represented as a linear function of depth then the potential function xX can be
written as the sum of the Airy functions Ai and Bi. The argument of the Airy functions is
defined in terms of the horizontal wave number k, the profile parameters K, and B and the
depth z. To add the effect of absorption in the liquid an imaginary term ia/8.686 is added
to Ko.
If
K? = Ky? (1 +62)
then |
WOE fev Ove (FS) SP 18) O18 (ES)
where ' i
ke = Kes 1/3
ae O D
eT LOE = (k, 8) z
«2 6)
i oie » @
To add a(dB/unit length) attenuation: K, > Kew 3686
A multilayered model composed of linear K? and constant K (constant sound speed)
layers is shown in Figure 18. We can start at the bottom and work up through the layers
using the interface conditions of which the pressure and the vertical component of particle
velocity are continuous functions. In this case Pp, equal to the pressure, is\/px and Q, equal
50
to -iw times the particle velocity, is ov! d@/px)/dz. Note that in Figure 18 layer 2 is a
constant K layer. The ability to mix linear and constant layers is necessary in a general pro-
gram because, as the gradient, 8, goes to zero, the argument of the Airy functions increases
without limit. Thus, depending on frequency, layer thickness and computer word length,
there is a minimum gradient that can be used. Layers with gradients smaller than this must
be represented by constant K layers.
COMPARISON OF THE TWO MODELS
It is instructive to see how these two models, the liquid multilayer and the linear
gradient multilayer, compare. To do this, consider Figure 19. On the left hand side our
linear model has a sound speed that increases from 1500 m/s at the water/sediment inter-
face to 1800 m/s at a sediment depth of 300 meters, which corresponds to an average
sound speed gradient equal to (1800 — 1500) m/s = 300 m, or 1 s~!. The constant
K model is shown for two layers. The layers have the same thickness and the sound speed
at the center of the layers (i.e., at 75 and 225 meters) is set equal to the sound speed of
the linear layer at that depth.
On the right hand side of Figure 19 is a diagram that indicates the main physical
events. Most of the energy either reflects at the surface or is refracted in the sediment
because of the gradient. Morris (1973) has used a ray description to calculate the energy in
each path and compare the ray description with the wave model. Of course there are second
and higher order effects as indicated by the dashed arrows that are implicit in the wave
model.
In Figure 20, the first comparison of the two models is shown. For the calculations
we used a frequency of 100 Hz, a density ratio (p in sediment)/(p in water) equal to 2.0
and zero attenuation. The reflection coefficient was calculated for grazing angles from
O degree to 20 degrees which are of interest in sound propagation. With zero attenuation
both models return all sound to the water for these grazing angles so the modulus of R is
1.0 or the bottom loss is zero. Figure 20 shows plots of phase, i.e., the argument of R, for
different cases. The curve marked L is for the linear K2 model, while the curves labeled 1,
3 or 10 correspond to 1, 3 or 10 constant K layers. The 10 layer case has a layer thickness
of 30 meters, which is equal to 2 wavelengths in the water. For 30 layers (or a thickness of
0.67 X,,,) there is a maximum phase difference of 2.2 degrees at a grazing angle of 3.5
degrees which cannot be plotted on this scale. For 100 layers there is a maximum phase
difference of 0.2 degrees.
In Figure 21 the same models are used except that there is an attenuation of 0.05
dB/m in both models. As in the previous case, the 10 layer model (thickness = 2 Ay) has a
maximum difference of ~10 degrees and the 30 layer model has essentially the phase as
the linear model. It is interesting to note that the attenuation has slowed the phase change
considerably. This will have a noticeable effect on the wave theory propagation models
where a shift in phase of 360 degrees will add a new mode to the sound field (Bucker,
1964).
To complete the comparison of the linear and constant layers, the bottom loss
curves are shown in Figure 22. The one layer case has much less bottom loss because the
sound speed is equal to the sound speed of the linear model at 150 meter depth, which is
1629.6 m/s and corresponds to a critical angle greater than 20 degrees.
5)
SOLID MULTILAYER MODEL
Sediments have rigidity and for more accurate model calculations rigidity must be
taken into account. An isotropic sediment layer can be described by three sediment
parameters: the density p and the two Lamé constants, A and w (Ewing, Jardetsky, and
Press, 1957). The density can be measured directly but A and yu are determined by the
speed and attenuation of the compressional and shear waves that travel in the sediment.
The sediment and acoustic parameters are related (Bucker et al, 1965) as follows
Sediment Parameters
Dew» (B
———
Lamé constants
Acoustic Parameters
Cy = sound speed (compressional wave)
€.° = sound speed (shear wave)
an = attenuation, dB/unit length (compressional)
a, = attenuation, dB/unit length (shear)
Constitutive Equations
where, Xp = /cy, 3 My ay/(8.686 w)
eS ice Mg = a,/(8.686 w)
There are several approaches to the problem of modeling the sediment layers when
there are significant changes of the sediment properties with depth. Gupta (1966a, 1966b)
has developed closed solutions for the case where the compressional and shear velocity
varies linearly with depth while the density remains constant. More general variations can
be treated with the propagator method developed by Gilbert and Backus (1966). One
problem of the propagator method is loss of accuracy when sediment penetration of many
wavelengths occurs. In our most recent programs we have chosen to model the variable
sediment properties with many layers and to maintain accuracy by use of Knopoff’s
formulation (Knopoff, 1964).
The multilayer solid model is substantially more difficult than the multilayer liquid
model for two reasons. First, there are twice as many waves (shear waves as well as com-
pressional waves) and twice as many interface conditions (continuity of horizontal com-
ponents of stress and strain, as well as continuity of vertical components of stress and
strain). Second, you cannot start at the bottom and work to the top. All of the layers
have to be considered as a group. The situation is shown in Figure 23. There are an
upgoing and a downgoing compressional wave in the water, an upgoing and a downgoing
Dy
compressional wave and an upgoing and a downgoing shear wave in each solid layer and
downgoing compressional and shear waves in the bottom half-space. We can arbitrarily
set the coefficient A,,4) = 1, as shown, so that there are 4n + 3 unknown coefficients
(Ag, Bo; Aj, Bie Cir ies Cans where n is the number of layers. There are also 4n + 3
interface conditions. Three conditions at the first interface (continuity of vertical com-
ponents of stress and strain and zero horizontal stress) and four conditions at all other
interfaces (continuity of vertical and horizontal stress and strain). Since the interface con-
ditions can be written as a set of linear homogeneous algebraic equations, the solution can
be done using standard matrix inversion algorithms. This is not a practical method of
solution when n is large because it is necessary to invert a matrix of (4n + 3)* elements and
because of loss of accuracy problems. The number of terms in the problem can be kept
under control by using transfer matrices that move the stress and strain at one interface of
a layer to the other interface. This method was developed by Thomson (1950). To solve
the problem of sound transmission through plates, Bucker (Bucker et al, 1965) extended
the method to include wave attenuation for the problem of bottom reflection. A serious
drawback of the transfer matrix method is that it also suffers from loss of accuracy.
Fortunately, the accuracy problems can be solved using methods developed by the
geophysicists for earthquake problems (Thrower, 1965; Dunkin, 1965; Watson, 1970;
Schwab, 1970). For a layered structure of the same form that we have for the bottom re-
flection problem, there are natural vibrations at frequencies corresponding to zeroes of a
determinant, |Ap|, called the Rayleigh determinant. The geophysicists have developed
very fast and accurate methods for calculating ARI. We show that the reflection coeffi-
cient can be written as R = (P1 Ta IARI -Ppo IAg|)/(p Tod IARI + pg IAcI), where |Ag| is
the same as [ARI except for row 1. Thus, the sophisticated methods of the geophysicists
can be used to solve our problem. We do have to generalize the equations to account for
attenuation which is neglected at earthquake frequencies.
CALCULATION OF R FOR MANY SOLID LAYERS USING
KNOPOFF’S METHOD
The standard methods of solution are not usable for the solid multilayer model
because of accuracy, computer storage and computer run time problems. Fast and accu-
rate methods developed in earth wave problems can be modified for calculation of R, in
particular, the fast algorithm of Schwab (1970), which is based on Knopoff’s formulation
(Knopoff, 1964). The method has been adapted to our solid multilayer model by Bucker
(Bucker and Morris, 1975). The notation used here is that of Haskell (1953).
The upgoing and downgoing compressional waves in a liquid layer or liquid half-
space can be represented by the potential function
On = ee cos p,, + B, sin Py! exp [i(wt — kx)]
where
Also, choose the potential function representing the upgoing and downgoing shear waves
in a solid layer or solid half-space to be
53
w= i Ker sin an + j Dn COs gril exp [i(wt—kx)] ,
where
Cha k “On An °
The components of motion and stress in the nth layer are, therefore,
Cc ., = Ay COS DP, - iB, sin Dv "Bn Ch cos q, — i "Bn Dy sin Gy >
cW,, SAN itpepe Je SUM Da Way 1 OSD, VC. Sin GE = ID OSG 5
OS 0 Geral!) OOS OO (Gp ll) 1B Sia) O.
* Pn Yn "Bn En ©98 In — 1 Pn Yn Bn Pn SIN Gy >
T = 12p Yn Tan An 51 Py - Pn Yn Tan Bn ©8 Pr
-i Pra Yn) cr st GV DOH!) ID. EOS Ca -
In the above, c is the horizontal phase velocity (c = a/k), Un and Wh are the horizontal and
vertical components of particle velocity, On is the normal (vertical) stress and 7,, is the
tangential (horizontal) stress.
By separating $9 into an incident and reflected wave it is easy to show that the
plane wave reflection coefficient R is given by
For convenience set the value of Ag =]. The three interface conditions at the water/
sediment layer | interface can be written as
W500) Bo = Wool By = D, (continuation of W)
1) = OGAWA ON Mh Tg] Cy (continuation of o)
= =97) Mh tt BrP Onna) Dy (continuation of 7)
Divide the last two above equations by P| and form the matrix of coefficients of Bo: Ay;
By, Cy, Dy
Bo Ay By Cy Dy
iar Me emD einen Wa KCl Bedale 4 ine
0 (y-) 0 11 "1 0 = DHT
0) 0) Vil Moai 0) (y-)) = 0
Now modify the basis vectors so that the interface conditions can be within in the following
matrix form
Bo Tao 0
Ay —Po/Py
By ty] 0
x =
Dy 0
A> 0
0 0) —| 0 +]
—Pg/P1 riot ark Ma) SL par Lew CURT tC MnaL OT ORT
0 | AR
|
0 |
Bo a9
1 0) -l 0) +]
0 WLU EL cae cake
|
0 | AR
OM aN
The elements inside the dashed areas designated Ap are the elements of the Rayleigh deter-
minant. Fast and accurate methods are finding |Ap| have been developed (as mentioned
before) because the zeroes of |Ap| determine the phase velocity of earthquake waves.
Finally, we can write
0 =i 0 eg
|
0 YI 0 (yy-D
Biotic @peiy Winey Sey yee a oe py MMe
(pQ/P 1) % 1 IARI
ile! 0 7] 0
0 Hil 0) (y;-1)!
In the above |Ap| is the Rayleigh determinant and |Ag| is the same except for the first
row. The fast methods developed for calculation of |Ap| can be used to evaluate |Ac|. It
follows then that the plane wave reflection coefficient can be written as
R = (01 Ty9 AR! = PQ IAgh/(] TeQ IARI + PQ IAsl) -
55
COMPARISON OF MULTILAYER SOLID AND LIQUID MODELS
In Figure 24 is a plot of bottom loss for a model of 100 layers. The curve labeled L
is for the liquid layer model (it is also the bottom loss curve for the linear model). The
other curves are for a 100 layer solid model with different values of rigidity. For r = 0 the
curve is quite similar to the liquid model except that there is slightly more loss due to some
conversion of compressional waves into shear waves. As the rigidity increases there are
lower losses than the liquid model at very small grazing angles and higher losses than the
liquid model at larger grazing angles. Most likely the propagation to long ranges would be
better for the r = 0.1 curve than for the liquid model.
EFFECT OF BOTTOM INTERACTION ON THE SOUND FIELD
In this section, a sample case is analyzed where the bottom affects the sound field.
The first step is a ray theory calculation in which the most significant (i.e., with the least
propagation loss) eigenrays are identified. An eigenray is a ray that travels from the source
to the receiver. If the significant eigenrays do not reflect from the bottom then there is no
bottom interaction problem unless the frequencies are very low, e.g., < 20 Hz. In many
cases the significant eigenrays do have bottom reflections and the ray tracing program can
be used to determine the grazing angle of the rays when they reflect at the bottom, Die
The next step in the analysis is the calculation of a three-dimensional bottom loss
surface as shown in Figure 25. Table 3 lists the parameters used for this particular calcula-
tion. The values are typical of the deep ocean and are representative of properties pro-
vided by E. L. Hamilton’s geoacoustic models. Bottom loss is plotted as a function of
grazing angle of the ray on the bottom, y,, and frequency. To understand Figure 25, it is
useful to consider Figure 26. On the left several sound speeds are plotted as a function of
depth. Here c,, is the sound speed in water, c; and c> are the sound speeds at the top and
bottom of the upper sediment layers, and c, and c,, are the shear speed and the compres-
sional speed in the basement.
A typical ray path is shown on the right side of the figure. We can follow the path
of a ray using Snell’s law,
p
i = Cy, = constant .
In Snell’s law (c(z) is the sound speed at depth z, y is the grazing angle of the ray at depth
z and cy,) the horizontal phase speed of the ray is a constant for any given ray. Using
Snell’s law, cy, can be determined by cy = Cy,/cos Yp and the depth at which the refracted
ray becomes horizontal, i.e., at the depth at which c(z) = Cy. Figure 26 shows the re-
fracted ray turning over in the upper sediment layers so cy, < cy because the ray with
Cp = C9 will become horizontal at the interface between the upper sediment layers and the
basement. The Stoneley wave can exist at the interface between the upper sediment layers
and the basement when excited by waves which have a certain relationship between fre-
quency and horizontal phase speed. This relation is called a dispersion equation. The high
losses at low frequency and low grazing angles are caused by a coupling of energy from
56
the refracted ray into the Stoneley wave. This effect has been previously described by
Hawker, Focke and Anderson (1977).
There is almost no loss when cy = ¢, (1.€. Yp) © 54°) and when cy, = Cp Gree ina: 74°)
(see Figure 25). Between these two angles the loss increases because of shear wave genera-
tion in the basement. Otherwise the bottom loss is mostly due to absorption of energy
from the refracted ray. This loss increases, in general, with frequency. This effect is
apparent in Figure 25 at frequencies above 50 Hz. There is also an interference effect that
causes the bottom loss surface to be wavy. This is caused by the coherent addition of the
refracted and reflected rays.
It is clear that bottom interaction is somewhat complicated at low frequencies.
However, by considering the fundamental physical processes that are responsible for the
bottom interaction the total sound propagation field can be determined.
EQUIVALENT SEDIMENT LAYERS FOR USE WITH THE P.E. MODEL
In previous sections it is shown how the bottom reflection coefficient R is incorpo-
rated in the propagation models. In the normal mode formalism, the use of R results in an
exact solution. In the case of ray theory, a slight error is introduced by the use of a plane
wave coefficient when a spherical coefficient is called for. However, the error is negligible
at frequencies above ~ 100 Hz. When there are appreciable horizontal changes in the sound
speed or bathymetry we must use either perturbation solutions of wave theory or the P.E.
(Parabolic Equation) model. At the present time almost all efforts in model development
center around the P.E. model. This is due probably to the simplicity of the P.E. method
and the early development work by the Acoustic Environmental Support Detachment
(AESD).
As do most acoustic propagation algorithms, the P.E. model starts with the reduced
wave equation
Vo wtKy=0 ,
where K2 has been defined before and w is the potential function. The first step in the
solution is to assume a product form for w in which one term contains the range variation
that would occur if there were no horizontal changes and the other term represents the
depth (z) dependence of W plus a small range dependence due mostly to horizontal changes.
That is,
W(r.2) = U(r.z) Hp! (kg) ,
where H, (kot) is a Hankel function of the first type and order zero and k, is a separation
constant. Substitution of the above form of w into the wave equation results in the follow-
ing second order partial differential equation.
2 2
IA a de OD, ONT oc De de DN ern
= i 2k 5p +3 +(K ko?) U=0.
Sf)
Leontovich and Fock (1946) have shown that if the term a2U/ar2 can be neglected, a
marching solution to the resulting Parabolic Equation can be written in the following form
U(r+ Ar,z) = exp [- Ar (K?=k,7| P|
fexp tC Ar/2 ko) 02/02] UGz) .
op
Using results that they had developed for electromagnetic propagation problems associated
with the SAFEGUARD ABM program, Tappert (1977) was able to show that the exponen-
tial operator acting on U could be calculated using a discrete Fourier transform F.
| U(r,z) = F7! {exp [- Ars2/(2 7) F virz)} ‘
op
where s is an index of F, and F7! is the inverse transform. Because there are fast forms of
the discrete Fourier transform available, the P.E. model has proved to be a reasonably
efficient method for calculating the sound field for non-bottom limited cases.
In bottom limited cases the P.E. method runs into serious difficulties. The sound
pressure is continuous across the interface between the water and the sediment. However,
the density is discontinuous since the sediment density usually is at 20 to 100 percent
greater than that of water. Since U= po} 2 p, where p is the density and p is the pressure,
U is discontinuous. This leads to the well known Gibb’s phenomenon in which oscillations
are generated by taking the Fourier transform of a discontinuous function (Figure 27). To
avoid Gibb’s phenomenon it is necessary to replace an accurate representation of the bottom
sediments by one in which K2 and p have minimum variation with z but which also is con-
sistent with the value of bottom loss that is calculated for the realistic sediment model,
Figure 28. The filled circles in Figure 29 represent the correct values of |R|. We want to
generate an equivalent sediment model that has smooth K2 and also has the bottom loss
shown in Figure 29.
This is done with a simple algorithm as shown in Figure 30. The symbol x repre-
sents either the real part of K2, the imaginary part of K2 or the density. A change Ax is
made and the least mean square error of the difference, E, between the desired curve and
the calculated bottom loss curve is calculated. If E is reduced by a change in +Ax or —Ax
then +Ax is increased. If the calculated E is larger, then x and E remain the same but Ax
is reduced for the next iteration.
The final results are shown in Figure 31 where there is good agreement between the
the bottom loss for the equivalent sediment mode (the line) and the desired bottom loss
(the filled circles).
SUMMARY
We have developed a general purpose plane wave reflection model that can account
for both liquid and/or solid layers. Earlier models which we developed, such as the solid
model and the linear gradient model, contributed and laid the basis for our most recent
58
work. The present solid multilayer program models the variable sediment properties with
many layers (up to 1000 layers or more, if necessary) and maintains the accuracy needed
for efficient computer calculation. Important physical parameters used in the model are
the speed and attenuation of both the compressional and shear waves that travel in the
sediment. The many constant layers are used to approximate the results of a linear
gradient concept.
The solid multilayer model should be satisfactory as the coupling mechanism
between the sediment parameters and the calculations of the acoustic field. The model
includes all physical factors except for roughness which usually is not important at low
frequencies. This model, based on wave theory, is interfaced very easily with most sound
propagation normal mode programs.
In addition, a so-called equivalent bottom concept has been developed for the
Parabolic Equation propagation program. This technique should enable the P.E. program
to be used for bottom limited areas.
The geoacoustic model discussed in Part I, which presents sediment properties, is
an essential input to the bottom loss model. We recommend that when geoacoustic in-
formation is available, the principal method of analyzing bottom interaction and making
predictions should be through the sediment models. The predictions should be supple-
mented when possible by careful direct measurements of bottom loss.
59
REFERENCES
(PART I)
Brekhovskikh, L. M., 1960, Waves in Layered Media, English translation, Academic
Press, New York, p 171.
Bucker, H. P., 1964, Normal mode propagation in shallow water, Journal of the
Acoustical Society of America, Vol. 36, pp 251-258.
Bucker, H. P., Whitney, J. A., Yee, G. S., and Gardner, R. R., 1965, Reflection of
low frequency sonar signals from a smooth ocean bottom, Journal of the Acoustical
Society of America, Vol. 37, pp 1037-1051.
Bucker, H. P., 1970, Sound propagation in a channel with lossy boundaries, Journal
of the Acoustical Society of America, Vol. 48, pp 1187-1194.
Bucker, H. P. and Morris, H. E., 1975, Reflection of sound from a layered ocean
bottom, invited paper presented at Oceanic Acoustic Modelling Conference at
SACLANTCEN on 8-11 September 1975, pp 19-1 to 19-35 in SACLANTCEN
Conference Proceedings No. 17, 15 October 1975.
Dunkin, J. W., 1965, Computation of modal solutions in layered elastic media at
high frequencies, Bulletin of the Seismological Society of America, Vol. 55, pp 335-
358.
Ewing, W. M., Jardetsky, W. S., and Press, Frank, 1957, Elastic Waves in Layered
Media, pp 79-83, McGraw-Hill Book Company, New York.
Gilbert, Freeman and Backus, G. E., 1966, Propagator matrices in elastic wave and
vibration problems, Geophysics, Vol. 31, pp 326-332.
Gupta, R. N., 1966a, Reflection of sound waves from transition layers, Journal of
the Acoustical Society of America, Vol. 39, pp 255-260.
Gupta, R. N., 1966b, Reflection of elastic waves from a linear transition layer,
Bulletin of the Seismological Society of America, Vol. 56, pp 511-526.
Haskell, N. A., 1953, The dispersion of surface waves in multilayered media, Bulletin
of the Seismological Society of America, Vol. 43, p 17.
Hawker, K. E., Focke, K. C., and Anderson, A. L., 1977, A sensitivity study of
underwater sound propagation loss and bottom loss. ARL-TR-77-17.
Leontovich, M. A. and Fock, V. A., 1946, Journal of Physics (USSR), Vol. 10,
pp 13-24.
Knopoff, L., 1964, A matrix method for elastic wave problems, Bulletin of the
Seismological Society of America, Vol. 54, pp 431-438.
Morris, H. E., 1970, Bottom reflection loss model with a velocity gradient, Journal
of the Acoustical Society of America, Vol. 48, pp 1198-1202.
Morris, H. E., 1972, Comparison of calculated and experimental bottom reflection
losses in the North Pacific, Naval Undersea Center TP 327.
Morris, H. E., 1973, Ray and wave solutions for bottom reflection for linear
sediment layers, Journal of the Acoustical Society of America, Vol. 53, p 323 (A).
Morris, H. E., 1975, Bottom reflection model validation with measured data from
FASOR III stations in the Pacific and Indian oceans, Naval Undersea Center
TP 460.
60
Schwab, Fred, 1970, Surface-wave dispersion computations: Knopoff’s method,
Bulletin of the Seismological Society of America, Vol. 60, pp 1491-1520.
Tappert, F., 1973, Unpublished notes and abstracts of talks given at 8th Inter-
national Congress on Acoustics 1974 and SIAM meeting, SIAM Review, Vol. 15,
pp 423.
Tappert, F. D., 1977, The parabolic approximation method, in Lecture Notes in
Physics, No. 71, edited by J. B. Keller and J. Papadakis, Springer-Verlag, New York.
Thomson, W. T., 1950, Transmission of elastic waves through a stratified solid
medium, Journal of Applied Physics, Vol. 21, pp 89-93.
Thrower, E. N., 1965, The computation of dispersion of elastic waves in layered
media, Journal of Sound and Vibration, Vol. 2, pp 210-226.
Watson, T. H., 1970, A note on fast computation of Rayleigh wave dispersion in the
multilayered half-space, Bulletin of Seismological Society of America, Vol. 60,
pp 161-166.
61
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Figure 15. Ray theory representation (high
frequency).
Figure 16. Wave theory representation (low
frequency).
63
i2g Zz
(n+1)
e ikn+ Znti
Zn+1
Figure 17. Multilayer liquid model.
64
ifoz Roig [oF
en aii Boe 14222
29)
— Ai(En+1)
Figure 18. Multilayer linear liquid model.
Zn+]1
65
: 3 D) ; :
Figure 19. Linear K~ and constant K layers. The diagram on the right
indicates the main physical events where the energy reflects at the
surface or is refracted in the sediment.
66
| ARG (R), deg
180
140
100
60
—60
-100
—140
NO. OF
LAYERS THICKNESS
20Aw
6.7 \w
2.0\w
2 4 6 8 10 12 14 16 18
GRAZING ANGLE, deg
Figure 20. Phase comparison for linear K2 and constant K models (zero attenuation
for both models). The curve marked L is for the linear K2 model, while the curves
labeled 1, 3 or 10 correspond to constant K layers.
67
ARG (R), deg
-—20
-40
—60
-80
-100
-120
—140
-160
\
2 4 6 8 10 12 14 16 18 20
GRAZING ANGLE, deg
Figure 21. Phase comparison for linear K2 and constant K models using 0.05 dB/m
attenuation for both models.
BOTTOM LOSS, dB
2 4 6 8 10 12 14 16 18
GRAZING ANGLE, deg
Figure 22. Bottom loss comparison for linear K2 and constant K models using 0.05 dB/m
attenuation for both models.
69
20
d
Zl
4.-——
n+]
Figure 23. Multilayer solid model.
70
BOTTOM LOSS (dB)
Ze
2 4 6 8 10 12 14 16 18 20
GRAZING ANGLE (DEGREES)
Figure 24. Comparison of multilayer solid and liquid models. L represents the bottom loss for the
linear and its liquid layer models. The other curves are for a 100 layer solid model with different
values of rigidity, r.
ou
=
=
SS
Mss
ZW Si
“apy S87 weNreR
\\ \
SAU , N)
SUD NK SS
SK
~
ing angle and frequency.
tion of graz
a func
Figure 25. 3-D plot of bottom loss as
V2,
Ww INCIDENT RAY REFLECTED
RAY
Yb
REFRACTED—>
RAY
STONELEY as
Figure 26. Sound speeds and ray diagram.
73
%
i U
p ere (F U)
GIBB’S OSCILLATIONS
Figure 27. Example of Gibb’s oscillations.
’
p
p' [REPLACE
Bea
Figure 28. Equivalent bottom for use with the Parabolic Equation.
74
10
(op)
BOTTOM LOSS, dB
pS
0 1 2 3 4 5 6 7 8 9 10 11 12
GRAZING ANGLE, DEG
Figure 29. Desired values of bottom loss.
US
INITIAL VALUES
X= Cy Q,, or d;
X=SX -DX
E1=ER(X)
DX = 0.25 * DX
FINAL VALUES: X,DX,E
Figure 30. Algorithm to generate an equivalent sediment model with smooth K2
and bottom loss.
76
@ INPUT
— EQUIVALENT BOTTOM
BOTTOM LOSS, dB
0 1 2 Sree 5 6 7 8 9 ‘on nnn
GRAZING ANGLE, deg
Figure 31. Good agreement between the bottom loss for the equivalent sediment mode (the line)
and the desired bottom loss (the filled circles).
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