AD-A174 235 ACOUSTIC SCATTERING KERNELS FROM

ARCTIC SEA ICE(U) I/i

SCIENCE APPLICATIONS INTERNATIONAL CORP MCLEAN VAR "0 RUBENSTEIN OCT 86 SAIC-8b.'i@77 N8@@14-B4-C-8i88

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N ACOUSTIC SCATTERING KERNELSU FROM

ARCTIC SEA ICE

SAIC-86/1077

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I ACOUSTIC SCATTERING KERNELSFROM

ARCTIC SEA ICE

SAIC-86/1077

Post Office Box 1303 1710 Goodridge Driv'e, McLean, Virginia 22102, (703) 814300

m

ACOUSTIC SCATTERING KERNELS

FROM ARCTIC SEA ICE

SAIC-86/1 077

October 1986

Prepared b y:

David Rubenstein

Prepared for:

Mr. E.D. Chaika OLI Fol

Mr. B.N. Wheatley

AEAS Program . T;

ONR Detachment, Code 1 32

[ i t i- , ,':

Contract No. N00014-84-C-0180 Av'i .iyCodej

Dis.t

SCIENCE APPLICATIONS INTERNATIONAL CORPORATION

1710 Goodridge Drive

P.3. Box 1303 A5McLean, Virginia 22102 d :fl(703) 821-4300d li

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REPORT DOCUMENTATION PAGE AEOR F AD INS TRC'RlON SB E FORE C OMP LET-II NG F 0RSI1. 8 ;1 ; 1E:P .jAkR2 GOVT ACCESSION "C0. .1 REC(P!EeT% CA-AL:Z -. ;3EqSAIC-86/1077 1AD -A 1- V.7-3.1

' -- '-Eland S~fillo) TYPE CF Ri7F;-)T & PE=10 :ZVEREO

Auoustic Scattering Kernels from Arctic Technical ReportSea Ice 6. PERFORMING CtG. REP)RT NI MBER

SAIC-86/10777. AUTwOR(s, 8. CONTRACT OR GRANT NUMBERr)

David Rubenstein N00014-84-C-0180

S. PErfORMING ORGANIZATION NAME AND AOMRSSS 10. PROGRAM ELEMENT. PROJECT. TASK

Science Applications International Corp. AREA & WORK UNIT NUMERS

1710 Goodridge Dr., P.O. Box 1303 Task 27.2McLean, VA 22102

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October 1986AEAS Program, ONR Detachment, Code 132 13. NUMBEROF PAGESNSTL, MS 39529

4 MONITORING AGENCY NAME A

AOORESS(.I dittornf Irom Contolling Otfite) IS. SECuA;TY CLASS. (of thes report)

Unclassified

5s,. DECLASSIFlCATION'OOW GRADING

SCHEDULIE

16. DISTRI3UTION STATEMENT (of this Report)

Unlimited

17. DISTRIBUTION STATEMENT (of th. abstrac entered in Stock 20. it differenE trom Reporf)

r 1g. SOPPLEMENTARY NOTES

19, KEY WORDS eContinue on reveae side it neces sary and identWy by block number)

Arctic, Scatter, Acoustic, Ice, Keel, Ridge

20 ABSTRACT (C'rntsmue on rvevere side it ntceissry and jaentiy bV block "umbt,)

/ A new algorithm for computing scattering kernels from Arctic icekeels is described. The. basic equations are adapted from Burke andTwersky (1966) but the algorithm does not involve approximations inthe limits of low or high frequencies, and is therefore more accurate.Calculations over a range of frequencies and grazing angles arepresented. An important result is obtained. At high frequencies(greater than 200 Hz) a significant component of the incoherent

'scattered energy is directed into shallow Dropagation anales (less

DDJ, , r 1473 ECI'IN OF I soe UNCLASSIFIEDSECURITY CL..ASSIFICATION OF T'eS P AGE 0)'11 Uci ts 'd

,CC;. RITY CLASSIVICATIC06 OF T141% P&Gt(hen 0410. 1Mteted

Z -*1 :tract (Continued)

a15degrees). This component must be accounted for in estimates ~of scattering loss.r

UNCLAS SI F IED-

I

YADLI OF U T

Section Page

1 INTRODUCTION .... *..........*..********** **** 1-1

2 MODEL EQUATIONS ............ *................................ 2-1

2.1 Mathieu's Equations and Mathieu Functions .............. 2-12.2 Scattering Solution .................................... 2-52.3 Grid of Randomly Distributed Scatterers ................ 2-8

3 METHOD OF CALCULATION ...... ............................. 3-1

4 RESULTS ........................... 4-1

i

b -

A ., J.J

LIST OF FIGiJEIS

Figure pg

2.1 Elliptic Coordinate System, uiand 0 .................... 2-2

2.2 Diagram of elliptical cross section of cylinder in the loweerhalf-plane, incoming grazing angle u and outgoing angle *.The angle -u corresponds to virtual image plane wave ..... o.. 2-6

4.1 Reflection Coefficient R for Frequencies 50, 100, and 200 Hz 4-5

4.2 Scattering cross sections at 20 Hz. Incoming grazing anglesare 5*, 100, 15*, and 200. Keel parameters: Width -21.61m,Depth - 4.3 m, Average Separation - 92 m .......... 4-6

4.3 As in Figure 4.2, but at 50 Hz .................. 4-8

4.4 As in Figure 4.2, but at100 Hz ................. o. 4-10

4.5 As in igure 4.2, but at150 Hz ................... 4-12

4.6 As in Figure 4.2, but at 200 Hz ...................... 4-14

4.7 As in Figure 4.2, but at 250 Hz ................. .0 ..... 4-16

4.8 As in Figure 4.2, but at 300 Hz ............................ 4-18

4.9 As in Figure 4.2, but at 400 Hz .................. ....... 4-20

4.10As n Fiure4.2,butat 80 H ....................... 4-.

4.10 As in Figure 4.2, but at8100 Hz ...... o...................... 4-22

4.11 As in Figure 4.2, but at 1600 Hz .............................. 4-24

4.13 Reflection coefficient R (solid curve) and 1-SL (dashedcurve) f or 5* grazing angle. The quantity 1-SL (SL isscattering loss) represents the sum of coherent energy plusthe propagating component of incoherent scattering energy ... 4-28

4.14 Reflection coefficient R (solid curve) and 1-SL (dashedcurve) for 100 grazing angle. The quantity 1-SL (SL isscattering loss) represents the sum of coherent energy plusthe propagating component of incoherent scattering energy ... 4-29

4.15 Scattering cross sections at 20 Hz. Incoming grazing anglesare 50, 100, 150, and 200. Keel parameters: width =38.7 m,Depth =7.7 m, Average Separation =62 m ........................ 4-30

iv

LIST OF FIGURES (Continued)

F e

4.16 As in Figure 4.15, but at 50 Hz ............................. 4-32

I 4.17 As in Figure 4.15, but at 100 Hz ........................... 4-344.18 As in Figure 4.15, but at 150 Hz ............................ 4-36

4.19 As in Figure 4.15, but at 200 Hz ............................ 4-38

4.20 As in Figure 4.15, but at 250 Hz ....................... 0.... 4-40

4.21 As in Figure 4.15, but at 300 Hz ............................ 4-42

4.22 As in Figure 4.15, but at 400 Hz .......................... 4-44

4.23 As in Figure 4.15, but at 800 Hz ........... ,.. 4-46

4.24 As in Figure 4.15, but at 1600 Hz ........................... 4-48

'

vU

i

V

section 1INRlqODUCTIONI

The Arctic ice-covered environment presents a sort of surface

scattering that is quite different from that in the open ocean. Under-ice

acoustic propagation undergoes incoherent scattering in the presence of

ice keels. Greene and Bowen (1983) reviewed the special environmental

conditions that differentiate acoustic propagation in the Arctic from open

ocean regions. The under-ice roughness associated with ice keels has a

very important effect on sound propagation. Greene (1984) developed a

statistical model of under-ice roughness, based on a distribution of

randomly oriented cylindrical keels. Using this model, Greene developed

an interim scattering model (SISM/ICE), which is a hybrid of theories

terization of scattering is important for the Navy's propagation models.

Rubenstein et al. (1986) described an implementation of SISM/ICE for the

ASTRAL and PE models. The concepts are also relevant to other models,

including FACT, FFP, MPP, and RAYMODE.

The SISM/ICE model is probably adequate for low frequencies,

below about 200 Hz. However, at higher frequencies, SISM/ICE ignores the

scattered, incoherent component of energy that is directed within the

sound channel (Greene and Rubenstein, 1986). Therefore, a more realistic

scattering kernel is needed for higher frequency applications. In this .4-

report, an algorithm is described for computing realistic scattering

kernels for a field of ice keels.

The under-ice surface is considered to be flat, with cylindrical

bosses of elliptical cross section. The basic assumptions are that the

ice keels

(1) have a constant size, and a half-width-to-depth ratio of I1.6, as suggested by Diachok (1976),I-

~0 - - 4 ~* ~ :~ '

(2) have random orientation,

(3) have random spacing with uniform distribution along a

track, and that

(4) pressure release boundary conditions are appropriate.

Section 2 of this report describes the model equations in some

detail. The basic equations are from Burke and Twersky (1966). A new

method of calculation is described in Section 3. The method involves a

direct calculation of powr series of Mathieu functions, and is more

accurate than the approximate methods of Burke and Twersky. Section 4

presents results of applying the model to determine reflection coeffi-

cients, scattering cross sections, and scattering losses for particular

fields of ice keels.

SV

1-2

U

Section 2

SNODE, EQUATIONS

2*1 NATHIEU'S EQUATIONS AND MATRIEU FUTIONS

The elliptic coordinate system is a natural one to use for the

problem of scattering off an elliptical cylinder. Following the notation

from Morse and Feshbach (1953), the coordinates u and e are related to

Cartesian coordinates by

x - 1/2 a cosh u cose,

y 1 1/2 a sinh , sine, (2.1)

where a curve U = constant is an ellipse and a curve 8 = constant is a

hyperbola, each with foci at x 1 _ 1/2 a. See Figure 2.1 for a schematic

drawing. For reference, the elliptic coordinates are related to polar

coordinates by

r 1/2 a [cosh 2 , sin2e]l /2

. tan- [tanh U tane . (2.2)

In Cartesian coordinates, the Helmholtz wave equation is

written

j+ a + k2 ) 0 (2.3)

3x2 3y2

and transforming to elliptic coordinates, the equation becomes

___ a2 + h2 (cosh 2V _ cos 2e)o : 0 , (2.4);u 82

2-I.

where

h - ak , (2.5)2

2-1

Ve

,. __.... lw=2

I!I0_

yUZ.

44

Figure 2.1. Elliptic Coordinate Systes, U and e.

4 2-2

k - 2w/X is the wavenumber, and X is wavelength. Eq. (2.4) separates into

two ordinary differential equations

d2H + (b - h2 cos2e)H - 0 , (2.6)

de 2

- d2-M + (b - h2 cosh2 u)M = 0, (2.7)dp

2

with

H(8) M(l) ( (2.8)

Eq. (2.6) is Mathieu's equation, and (2.7) is known as Mathieu's modified

equation.

The solutions to (2.6) are periodic in 6. The solutions

Sem (h,e) MnO An (h,m) cos (nO)

n=0

So m (h,e) = 7 Bn (h,m) sin (nO), (2.9)n=1

are even and odd functions, respectively, about O=O. Those functions with

even order m have period w, and those with odd order m have period 2w.

Also, the even order functions are even about e = w/2 and 3w/2, and the

odd order functions are odd there.

The Fourier coefficients in (2.9) are normalized as

n

l-n B= . (2.10)

2-3.m

In addition, normalization constants are defined by

2, me (h) - f Sem (h,8)2 do

0

-2= 1. A (h,m)2

n=0 en n

2w

Mo (h) - f So (h,e) 2 dom0

ir B (h,m)2 (2.11)

n=1I

where I for n -0, and -2 for n > O.whr n n

Solutions to the modified equation (2.7) may be written

- Go

Je 2 (h,) - / 2 (-1)nm A2 J n(h o shp) ,

J° (h,) = 2 (-1) A J (h cosh)2m+1 2 0- 2n+1 2n+1

-

Jo (h,p.) / - tanhiu (-1)n- 2nB J (h coshU),2m 2 72n 2nn=1

- n --(

Jo2 +1 (h,p) -/ tanhp (-1) n(2n+1)B J (h coshu), (2.12)

n=O 2n+1 2n+1

where J are Bessel functions, and Je and Jo are radial Mathieu functionsn

of the first kind. Radial functions of the second kind Ne and No can be

obtained by replacing the J in (2.12) with Neumann functions N n , and byn

changing the lower summation limit for No from 1 to 0. Radial functions

of the third kind He and Ho are analogous to Hankel functions,

2-4

N=-p

Hen = Jen + i Nen o

U Ho = Jo + i No • (2.13)n n n

2.2 SCARTXNG SOLWOU

We develop the solution for plane waves scattered by a semi-

elliptical cylinder. In terms of Mathieu functions, a plane wave is

expressed by

ikr cos(u-0) ik (xcosu + ysinu)$~i=e =

" n (n h u IS J n (h 'e)in .j"-hu2 Mj n(h) Jj n (hU), (2.14)

n=O j=e,o n

where the summation variable j indicates a summation over even and odd

Mathieu functions, and the subscript i denotes a wave that is incident on

a cylindrical surface.*

The scattered wave * s (see Figure 2.2) is represented in terms

of outgoing radial Mathieu functions,

Sjn(h,u)Sj (h,e)Os in X8-- n in (h)n Hj n(h, 1 ) , (2.15)

n=O j=e,o n Mjn

where the coefficients Xjn are determined by the boundary conditions. We

use a free surface boundary condition,

*i + *S = 0. (2.16)

This expression for *i is the same as that quoted by Morse and Feshbach

(1953), but is different from that quoted by Burke and Twersky (1964)by a factor /w72. However, a second discrepancy in Burke and Twersky(1964) in their asymptotic expression for radial Mathieu functions Hemand Hom compensates exactly, so that the later results fall into agree-ment.

ZJ. 2-5

7NO

-6

Therefore, the coefficients become

3 Ji n (h,uo)n HJ n(h,o (21

where 0 - Uo is the surface of the elliptical cylinder.

To obtain the solution for a semi-elliptical cylinder, we use

Rayleigh's image technique. we superpose two incident waves from direc-

tions u and -u, and compute the total scattered field. Because of the

symmetry properties of periodic Mathieu functions,

Se (u) Sen (- u)

So (u) = -So (-u) , (2.18)n n

S- *s(u) *8(-u)

-i = So (h,u) So (h,9)S n n n(h) Ho n(hu) . (2.19)

n=O n

3 We can derive an approximation to (2.19) which is accurate inthe far field. Using the asymptotic formulas

So (h,e) + So (h,4)n n

I ir .-nHo (h,U) 1r ei , (2.20)

we write

"'"2 ikr/ i kr f( ,u), (2.21)= C ikr

2-7

where

G So (h, u) So (h,*)f(*,u) = 4w 0 0on o o (h)" (2.22)

n=O no(h

We have written the function f(#,u) in a form comparable to the f_ func-

tion used by Burke and Twersky (1964, 1966), with u and * being the incom-ing and outgoing grazing angles, respectively. The normalization con-

stants Mo (h) are defined as in Morse and Feshbach (1953) and Burke andn

Twersky (1966), but are different from those used by Burke and Twersky

(1964) by a factor 2w. We also note that our angles here are grazing

angles, whereas in Burke and Twersky (1964, 1966) they are incident.

We call the case treated here oblate, because the ground plane

y-O is aligned parallel to the elliptical cylinder's major axis. Because

of the symmetry of this case, the even Mathieu functions have dropped outof (2.19) and (2.22). The prolate case, in which the major axis is

aligned perpendicular to the ground plane, is also treated by Burke and

Twersky (1964, 1966). Because it is not specifically of interest to the

ice keel scattering problem, it is not included here. The expressions

analogous to (2.19) and (2.22) for the prolate case involve summations

over odd-order even periodic Mathieu functions and even-order odd periodic

Mathieu functions.

2.3 GRID OF RANDOMLY DIS1RIBUTED SCATr!ZUS

Here the equations for a grid of randomly distributed ice keels

(semi-elliptical cylinders) are summarized, from Burke and Twersky (1966).

We define a quantity

n (223Z k sinu f(u,u) , (2.23)

where n is the number density of scatterers per unit length. Then the

coherent power reflection coefficient is given by

2-8

5%%

R 1+Z 2 (2.24)1 1 -ZThe incoherent scattering cross section per unit length is written

wn f(-gU)1 (2.25)

I Conservation of energy may be expressed by+csc(u) faud#- 1(2.26)

0

that is, coherent reflected energy, plus the integral of incoherent

scattered energy, equals the total incident energy.

2-9

Section 3

NXinMo oF CILCUL&'flON

s'P Burke and Twersky (1966) used approximate methods for computing

the scattering solutions. Their approximations are valid in the limits of

low and high frequencies.

Instead of using these approximate methods, wecalculated the

reflection coefficient R and scattering cross section a directly, by

explicitly evaluating the power series (2.22).

The algorithm developed by Clemm (1969) was used to compute the

periodic and radial Mathieu functions. This algorithm uses trigonometric

function power series for computing periodic Mathieu functions, and Bessel

function power series for computing radial Mathieu functions. In comput-

ing these power series, a tolerance level of 10-13 is set. The power

series is truncated when the magnitudes of two successive terms, relative

to the magnitude of the largest term, is less than or equal to the toler-

C ance level. Test cases were run, comparing computed values of Mathieu

K. functions with values tabulated in Abramowitz and Stegun (1964). Agree-

ment was good to seven decimal places. In order to achieve agreement with

Morse and Feshbach (1953), it is necessary to choose the Ince normaliza-

tion option in Clemm's (1969) algorithm for the radial Mathieu functions,

and to multiply the computed values by a factor Vw12.

The power series in (2.22) was explicitly evaluated, up to a

truncation order. This truncation order was reached when its addition

changed the summation value, in a relative sense, less than a given toler-

ance level. This tolerance level was 10-4.•U

i.% As a check on the correctness of the solutions, qualitative

comparisons of R and a results with figures in Burke and ?wersky (1966)

3-1

%q

"'" , % % ° -% % *% "-" ". -"- ."A, " % , % , ' % % .% % % * - • % ,%

.- . .,• o

were performed. As a quantitative check, the expression (2.26) for energy

conservation was evaluated. The integral over outgoing angles was

evaluated using a simple trapezoidal numerical integration. The average 3resolution from 0 to w was about 50, with 20 resolution in the vicinity of

the main scattering lobe. The error was found to be on the order of 1%.Better accuracy might be achieved by improving the angular resolution in

the numerical integration, or by decreasing the tolerance level in the

summation of (2.22). Of course, both approaches would lead to increased

computational expense.

i ~ ~3- 2 :

MN Sectiton 4

3 REULTS

This section presents results for calculations of coherent power

reflection coefficient R, and scattering cross section a. For these cal-

culations we make certain assumptions concerning ice keel shapes and

distributions. We assume that keels have a half-width-to-depth ratio of

1.6, that they have a random orientation, and that they are spaced random-

ly along a track. Consider the assumption that keels have random orienta-

tion. Even if the ice field contains keels of only a single size, the

projected keel width intercepted by any particular track will take on a

range of values. The average projected keel width will be equal to the

actual keel width times a factor w/2 . If the keels are spaced randomly,

then we can attribute an average separation distance to an ice keel field.

Figure 4.1 shows reflection loss as a function of grazing angle,

for three frequencies; 50, 100, and 200 Hz. The keel depth is 4.3 m.

Therefore the keel width is 13.76 m, and the average projected width is

21.61 m. We assume an average keel separation distance of 92 m. The 200

Hz curve increases most rapidly with grazing angle, because its small

wavelength (7.5 m) is best able to resolve the keel shape. The curve

reaches a peak at 30*, and then falls as shadowing decreases and more flat

surface becomes illuminated. Qualitatively similarly shaped curves were

presented by Burke and Twersky (1966).

Figures 4.2 - 4.12 show scattering cross section a as a function

of scattered angle, for different frequencies, ranging from 20 Hz to 3500

Hz. Each figure shows four incoming grazing angles; 50, 100, 150, and

20', signified by small arrows on the left side of each plot. The reader

should note that the radial coordinate axes have different scales on each

S,of the plots.

4-1

111

At 20 Hz and 50 Hz, a single main scattering lobe appears in

each plot. Secondary lobes begin to appear at 100 Hz and above. At high

frequencies, the secondary lobe structures become quite complicated. The

main scattering lobes increase in magnitude with increasing frequency and

with increasing grazing angle. They also become more narrow with increas-

ing frequency.

The orientations of the main lobes decrease in angle with

increasing frequency. Above about 250 Hz, the orientation angle also

increases with incoming grazing angle. At the highest frequencies, the

lobes are oriented at angles nearly equal to their respective incoming

grazing angles. In other words, at high frequencies, most of the

scattered energy is directed into the same angle as that of coherent

specular reflection.

We computed the surface scattering loss coefficient SL, given

by

SL- 1 - R - csc(u) f a(*,u)d* - csc(u) •(*,u)d . (4.1)0

Note the difference between the ranges of integration in (4.1) and (2.26).

The range from 0 to L is the angular range of propagation in the Arctic

Ocean. Energy directed below fL = 158, represented by the last term in

(4.1), is eventually absorbed by bottom loss. Figures 4.13 and 4.14 show

R and 1-SL as a function of frequency, for grazing angles u = 50 and u =

100, respectively. These figures show that at frequencies below 200 Hz,

the coherent power reflection loss is a good estimator of scattering loss.

However, above 200 Hz R and 1-SL begin to diverge. The reason is that a

significant fraction of the scattered energy is directed within the

angular range of propagation, from 00 to 150. Therefore, it is necessary

to evaluate the full expression (4.1) for an accurate estimate of scat-

tering loss at high frequencies.

4-2

.,

Figures 4.15 - 4.24 show the scattering cross section a, in a

manner analogous to Figs. 4.2 - 4.12, for a different set of keel param-

eters. The frequencies range from 20 Hz to 1600 Hz. The parameters for

this case are: keel depth = 7.7 m, average projected width - 38.7 m,

average separation = 62 m. This case gives results that are qualitatively

similar to those of the first case. Quantitatively, the magnitude of a is

greater for this second case, because of the larger size and denser

concentration of keel cylinders.

The results may be summarized as follows:

1) At low frequencies, below about 200 Hz, most of the incoher-

ent, scattered energy is directed into large angles with

respect to the horizontal, and therefore exits from the

sound propagation channel.

2) With increasing frequency, the orientation angle of the main

lobe of the scattering cross section asymptotically

approaches the incident grazing angle. Above 200 Hz, a

significant fraction of the incoherent, scattered energy is

directed into shallow propagation angles, less than 150.

For realistic predictions, acoustic propagation models must

incorporate these effects. This was done, for example, in an energy

transport model described by Greene and Rubenstein (1986). This transport P

model, in fact, includes azimuthal as well as vertical angle scattering

cross sections. This approach allows a more general three-dimensional

prediction of propagation.

4-3

,% ,

- - ~-vUv t~-w L-~ ~

4

UI

I-4

-% (This is ~pageV

N.

4.

4.

4.'

4.,

S.

4-4

- ~. ~r~- : ~

PON rV

, JA,.: r r .=,,. - - - --

5.

coC,,

0 100 Hz0!

-Jz03-

wLL

z

0

*m 0

0*0 30 60 90

GRAZING A NG L E

F-qure 4.1. Reflection Coefficient R for Frequencies 50, 100, and 200 Hz.

4-5

Sigma24.0 18.0 12.0 6.0 6.0 12.0 180 240

1- 20. -- C)Angie- 5.00 Width- 21.61 mnAigsepw 92. in Deptht- 4.30 mn

10° Sigma 1010.0 7.$ 5.0 2.5 *.5 5.0 75 1O.0

120 HaAngie 1000 Width- 2161 mIAigsep- 92 M Depth- 4 30 mr

Figure 4.2. Scattering cross sections at 20 Ia. incoming grazing anglesare 5, 100, 15, and 200. Keel paraueters: Width - 21.61a, Depth - 4.3 a, Average Separation 92 a.

4-6

.9w

"Ur

"10""S'ig mazc *10"

20.0 150 10.0 5.0 5.0 10.0 1S, '0.0

f -20. Hz NYAngie 15.0 tdthz- 21 61 mz

Avgsep- 92. in Depth- 4 30 10t 6

*0 " Sigmz *to-$

4.0 3.0 20 1.O 1.0 2.0 3.0 40

II

F

-20. zW4h261iAngie- 20.00 ~t- f6Awgsep- 92. In Depth- 430 m

riqure 4.2. (Continued)

'I-, 4-7

M .

C !gF~j

10"" iSigma "20.0 15.0 tO1O 5.0 5.0 10.0 15.0 20,0

I- 50. HzAngle- 500 Wtdin 21.61 inAIvgsp- 92 mI Depth- 4.30 m,

8.0 6.0 4.0 2.0 2.0 4.0 6.0 8.0

50. Hs:An- 0.00 Width- 21.61 mAvgsep- 92 m Depth- 4 30 m "

Fiqure 4.3. As in FPi ure

4.2, but at 50 Es.

4-8

i

_ . ,r , ,% 1 , .. , , t ".r ,- - . -

, ,- , - , .r ,2,% ". " .%

' ," .

Sim0.0*0 0.015 0.010 0.005 0.005 0 0.01 OY5 0.020

1 - 50. IAngle- 15.00 Wtdth- 2f61 mAtvgsep- 92. mn Depth- 4 30 rn

Sigma0.03* 0.024 0.076 0.008 0.008 0.016 0.024 0.03*

f-50. If:Angle- 20.00 Width- 21 61 mn

Avgsp- 9. mDepth- 4.30 mn

Figure 4.3. (Continued)

4-9

* 4

IA

Sigma0.012 0.009 0 006 0.003 0.003 0 006 0 009 0 012

f - 100. HNAnzge- 5.0 With 2161 mAalgsop- 92. 7m Depthi- 4 30 m

*0.04 0.03 0.0* 0.01 0.01 Sigma0. 0 0. 03 0.04

f.100. HzAn~gle- 10 00 WdA 16Avgsep- 92 m Wdth- 4.30 M

Figure 4.4. An in Figure 4.2, but at 100 Hz.

4-10

.4.%

Sigma0.08 0.06 004 00 o.o0 004 006 008

In

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Sigma0.16 0.12 008 004 004 0.08 0.12 0.16

-

3

f-100. ffzAngie- 200 Widtht- 21.61 mn.Avgsep- 92. m Depth- 430 ,

FiLgure 4.4. (Continued)4-11

- - - -

" " " " " "." " " " .' .''" "". ',' ,. t,""..'. " ", ," " .' -''," " - " " " " " r" . °

' '. " ',. , ' ' 4.

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, 4-.

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Sigma008 0.06 0.04 00 0.0* 0.04 0.06 0.08

II

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Fiqure 4.5. As in Fiqur. 4.2, but at 150 Bz.

4-12

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F- 50. Heu"Anglea 20 00 II/Wt'::Avgsep- 92. 7n jeptt= )e 00 t

rigure 4.5. (Continued)

4-13

IL"0-- ~- ~ I

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Figure 4.6. As in Figure 4.2, but at 200 O.

tA*F . -,4-1 4

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j

Figure 4.6. (Continued) £1

4-15

F~v.4

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f - 250. HzAngle- 70 00 Width- 2767 MAugsop- 9Z m Depth- 4.30 mn

Figure 4.7. As in Figure 4.2, but at 250 Hz.

4-16

Sigmac0.32 0.24 0.16 008 0.08 0.I 0.24 0 32II

f-250 HzAngle- 150_0 idth- 2161 nAvgsep= 92 m Depth- 430 m

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4-.

Fiqure 4.7. (Continued)

4-17

.-. %

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in

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300so Hz_ T, jAng It- 1000 Widtht- 2161 InAvgsep- 92 m Depth- 4 30 rn

rigure 4.8. he in Piqure 4.2, but at 300 a.

4-18

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Figure 4.8. (Continued)

4-19

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Figure 4.9. As in Fiqure 4.2, but at 400 Hz.

4-20

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riqure 4.9. (Continued)

4-21

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fJ- 800. ffaAngle- 5.00 With- 2161 mnAvgsop- 92. m Depth- 4 30 M

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f- 800. HsAngle- O.00 Wdth- 2.161 mAvgsep- 92. in Depth= 4 30 mt

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Figure 4.10. AU in Figure 4.2, but at 800 fz.

4-22

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Figure 4.10. (Continued)

4-23

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Fi re 4.11. As in FIqure 4.2, but at 1600 Sz.

4-24

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Figure 4.11. (Continued)

4-25

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Figure 4.12. (Continued)

4-27

N.. N

Ln

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6 GRAZING ANGLE

12" eee SCATTERING LOSS-- COHIERENT REFLECTION LOSS

0

z* '2%

LU 4-

LU

Lu 0

0 20 500 70 1000

FREQUENCY CHz)

Figure 4.13. Reflection coefficient t (solid curve) and 1-SL (symbols)

for 50 grazing angle. The quantity 1-SL (SL is scatteringloss) represents the am of coherent energy plus the props-gating component of incoherent scattering energy.

4-28

Jil .,.- - .- . , il.l.. . . , fl* .

1414

10 O AZWG ANGLE

12" • SCATTERING LOSS- COHERENT REFLECTION LOSS

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FREQUENCY (Hz)

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Figiure 4.14. Reflection coeff icient R (solid curve) and 1-SL (symbols)for 100 grazing angle. The quantity 1-SL (S is scatteringlose) represents the am of coherent energy plus the propa-gating component of incoherent scattering energy.

4-29

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Figure 4.15. Scattering cross sections at 20 Hz. incoming grazing anglesare SO, 100, IS*, and 200. Meel parameters: Width -38.7a, Depth - 7.7 a, Average Separation -62 .

4-30

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Figure 4.15. (Continued)

4-31',,%

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Figure 4.16. As in Figure 4.15, but at 50 Hz.

4-32

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Figure 4.16. (Continued)

4-33

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4-34

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Figure 4.17. (Continued)

4-35

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Figure 4.18. As in Figure 4.15, but at 150 as.

4-36

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Figure 4.18. (Continued)

4. 4-37• -y

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Figure 4.19. An in iqgure 4.15, but at 200 Es.

4-38

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Fiqure 4.19. (Continued)

4-391

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Figure 4.20. As in Figure 4.15, but at 250 Hz.

4-40

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Frigure 4.20. (Continued)

4-41

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4-42

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riqure 4.21. (Continued)

$ 4-43

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Figure 4. 22. An in Figure 4.15, but at 400 Oz.

4-44

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.

Fiqure 4.22. (Continued)

4-45

i• , , . .T ; - - , . . . . ...

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4-46

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Figure 4.23. (Continued)I

4-47

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Figure 4.24. (Continued)

4-49

Ii

Abramowitz, M., and I.A. Stegun (Eds.), 1964: Handbook of MathematicalFunctions. NBS App. Math. Ser. 55, U.S. Govt. Printing Off., Washington,D.C.

Burke, J.E., and V. Twrsky, 1964: On scattering by an elliptic cylinderand by a semi-elliptic protuberance on a ground plane. J. Opt. Soc. Am.,54, 732-744.

Burke, J.E., and V. Twersky, 1966: Scattering and reflections byellip-tically striated surfaces. J. Acoust. Soc. Am., 40, 883-895.

Clemm, D.S., 1969: Characteristic values and associated solutions of" Mathieu's differential equations. Comm. ACM, 12, 399-407.

a Diachok, 0.1., 1976: Effects of sea-ice ridges on sound propagation in

the Arctic Ocean. J. Acoust. Soc. Am., 59, 1110-1120.

Greene, R.R., and K.E. Bowen, 1983: "Introduction to the Arctic environ-ment. Environmental conditions relevant to acoustic propagation andmodeling." SAI-84-232-WA, 67 pp.

Greene, R.R., 1984: "Ice statistics and acoustic scattering in the ArcticBasin." SAI-84/1132.

Greene, R.R., and D.M. Rubenstein, 1986: Models of under-ice scattering.To be submitted to J. Acoust. Soc. Am.

Morse, P.M., and H. Feshback, 1953: Methods of Theoretical Physics,McGraw-Hill, New York.

Rubenstein, D., M. Blodgett, and R. Keenan, 1986: "HP 9020 ArcticModeling Capability." SAIC-86/1906.

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