AD-Ri48 393 MULTIPLE SCOTTERING EFFECTS IN RADAR OBSERVATIONS OF i/i.JSR-84-203B Fi9628-84-C-001

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Multiple Scattering Effects inRadar Observations-of Wakes

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Multiple Scattering Effects in Radar Observationsof Wakes

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C. G. CallanK. M. Case F19628-84-C-0001

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20. ABSTRACT (Continue on reverse side if necessary and identify by block number) SThe large amplitude waves observed in ship wakes and the large radar returns, *.including sharp angular dependencies, suggest that first order Bragg scatteringtheory is inadequate to describe the experimental data.

This report considers a simple theory which concludes that there can be lookangles such that second order scattering can be larger than first order. Theseconclusions are very tentative because of the lack of knowledge of the basicwave fields. k

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.. .. .. . .

Multiple Scattering Effects inRadar Observations of Wakes

C. G. CallanK. M. Case

August 1984 EL C

JSR-84-203BSD CO 8Approved for public release; distribution unlimited.B

JASONThe MITRE Corporation

1820 Dolley Madison BoulevardMcLean, Virginia 22102

-, .. ,..- '. ..

MULTIPLE SCATTERING EFFECTS IN RADAR OBSERVATIONS OF WAKES

I. Introduction

The large amplitude waves observed in ship wakes and the

large radar returns Including sharp angular dependences suggest that

first order Bragg scattering theory is inadequate to describe the -

experiments.

Here we consider a simple theory to qualitatively investigate

- higher order effects. There are two important limitations.

(a) We restrict our consideration to a simple scalar problem

with simple boundary conditions. A full scale theoretical treatment

of the true electromagnetic problem can readily be carried out.

However, it is much more complicated, probably obscures the central . -

points, and in view of (b) probably unwarranted at this point.

(b) The detailed nature of the scattering wave field is very -

poorly known. To get some insight we have applied the formula to an

'- . '2

. . . . . . . . . . . . . . . . . . . . . . ... . . . ,,. -.

," ]. .o.. . . . . . . ... .. . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . ... ,. .. :

* . .- ..--- o ,,

idealized mathematical description of a Kelvin wake and to a model

"derived" from the Dabob Bay experiments.

The essential conclusions are that there can be look angles

such that second order scattering can be larger than first order and

comparable in magnitude with the first order scattering when that is

significant.

We emphasize the weakness of the conclusions because of the

lack of detailed knowledge of the basic wave fields. Recommendations .0

to obtain the needed information are made.

PAccession For

NTIS GRA&IDTIC TAB EU13arMOlmCedutrit-.ition.

By. -_____'_

Dint

-/1 2

...................*.-.......

. . . . . . . . •. - -.

. . . . .. . . . . . . . .. . . . . . . . . . . . . . . . . . . . .

II. A Simple Theory

We consider the electromagnetic field to be described by a

simple scalar field satisfying the wave equation.0

(V' +k') 0- 1

An incident field

i r r kv Z)ie h(2)

2 2 2(kh+ k ) k

is to be scattered by a patch of a wavy surface centered on the plan-

Z -0, i.e., the surface S is

*Z -h(x,y) (3)

where h is to be small.

As a simple boundary condition we choose *-0 on S

We decompose the total field *into

3

All

+6

0

Here is the field which would result if h -0.

i.e. * 0 eik~ e vi, e i (5)

Further we introduce a Green's function G (r', r) satisfying

VI G (r, r) 6Q8r r') (6)

and G (x.y,o; r 0 (7)

Applying Green's theorem we obtain

(r) Ar' (r',r) -G(r,r 2a (r') dS . (8)

For a perturbation theory we assume h e and consider a

formal expansion inc.

Then &+(r) - *(r) + C2() + ***(9)

4

- -0

.... .. .. .. .............................................

Restricting ourselves up to second order terms this can be

wri tten

dx'dy A#(r) G(') rr) (10)0

We readily construct

-i d2 Kei (K x- + K y(y-y')

1 e + e -e -e

Then to first order we have

(r If dx'dy' *(x,y,O) 2- G(x,y0,;r) (11)

But SO (x', y', h) + *(x', y-, h) -0S

ik1 *01(' y', 0) - 2ik h e

and Gxyo, o; r)

i [k (x-x') + k (Y -)- 1 2ff d 2ke x (e1JZ e }U (12) 0.(2w)2

5

. . . . . . . . . .

Thus

(r 2ik ff dx'dy f d 2e iexky

i i

i[(k k x'+ (k - k Y]x h(x', y') e x >' y (13)

Now if we write 0.

h(x', Y') - f.. d q h(~ e (14)

2(2w)

FrmEuain(8hetenranofta

2 If xdy *1 x% , ) r]- G (, %) y, 0, r) %

#,Q 2 d ae-v(w) 2,0 ~x ~0

+ h (x y', 0) Gr)' y', 0;(,,0

Note: With our boundary condition

6

2

az-2

Also CO i(1, y-, h) + e 2 *(x' y1 0) + * 0 0

22But to order c2 whave3'2Z

*(x', y', 0) -- h ~ ~ ,y', 0)

and so

*2-2 ff dx dy*4(xo, yo, 0) G~x', y', 0; r)(16)

which yields ~-

-81k- l~ 2 2'

d qd p h(2') R(kp) k71

i[k 1L - -]*r iWzk -ak)Z -ip(k 1 - k)Zx e {e -e

In particular if we ask for the ampitude of the wave in the

back scattered direction, we obtain

(a) From equation (15)

7

(2w)

(b) From Equation (17)

-8ikv 2 ,- 2 2

02 ff dq h (kL h lK )k2 -q

Thus

4 f 2 q (k 1 -)hk 1L 2

1 (21)2

8

. .

0

III. Application to "Mathematic.l" Kelvin Wakes

Let us now apply these results to the Kelvin wake as computed

by simple stationary phase methods. There are many things wrong with 0

this simplest picture of the wake, but we believe that it gives a good

enough picture of the wave heights for an analysis of the importance

of multiple scattering. 0

The stationary phase picture asserts that at a point (x,y) in

the wake, making an angle e with resper-t to the wake axis, the •

dominant wave vectors in the surface height. h, are

k g g x 0

x 2U 2 2U 2 y

2

y 4U2 2 4U2 Y

where U is the ship velocity which is in the x direction. This means

that the ocean wave height function is

h(x,y) A(x,y) cos 0 (x,y)

O(x,y) = kx(xy) x + k (x,y) y 0

Y

S- ..- . o •

- . . . ...

where A(x,y) is a slowly varying function, poorly calculable from

stationary phase arguments, and 0 is rapidly varying and, of course,

determined by the stationary phase arguments. There are actually two

stationary phase solutions for k the one given, corresponding to

the diverging wave train, and another one corresponding to the

transverse wave train. The two trains are displayed in Figure 1. The

short wavelengths of direct interest in radar backscatter are to be

found in the diverging wave system, so we will not concern ourselves

here with the transverse waves. We will later resort to a combination

of theoretical argument and direct observation to determine reasonable

values for A in various parts of the wake.

To model a SAR observation of the ocean surface, we assume

that the radar processing in effect forms a beam which illuminates a -

patch on the surface of linear dimension b centered on the point

(Xoyo). The radar return is therefore computed from the formula of

the previous section by extending the spatial integrals only over theb +b b b ;

window x - -< x

S

S

0

- DivergingS.

Transverse

- S

Figure 1. Kelvin Waves

S

0

S

S

0

11

0

window function. The result for a patch centered on a point at wake

angle 0 is

H~~~ c>-Ao b/2h(-) A(6) dedn cos (- ( - - ) + (-4U2- 2 - qy ) n)

b22U 20 4U 20

In writing this expression we have simplified the variation of (k x,k )

across the radar patch. The omitted effects would contribute terms

quadratic in c and n to the phase of the cosine. We have verified

that for parameter values appropriate to the SEASAT or Dabob Bay

problems, this approximation is reasonable. The result is

h(q,O) - A(O)b 2 sinc [(q + sinc [(q g24U 2 22

For our purposes, it suffices to know that h(q,O) has a maximum

2amplitude of b A(O) and that it is peaked at the appropriate -

stationary phase wave number with a width in wave number space of

2w/b

We are now able to compute the radar backscatter amplitude in

first and second Born approximation. According to the previous .

section,

SC h (2kh,0)

h0

.12

. .- . . . . .- . .. " . . . . . . . . . . . . . .

2

d - .

J" d q_ 2 4!k q2 h, 0h q , h k 0h "q ':-

(21r)

where k is the magnitude of the radar wave vector, k is the

horizontal projection of that wave vector and C is a common factor of

dimension I/L which we would need to know in order to get the

absolute scattered intensity.

Since h is a sharply peeked function of its argument, 1,2

will be large only for a narrowly-defined band of e If we define

-9

kKELVIN(O) 2 U 2 e 22 -2U0 '4U0 2

then the condition for first and second order scattering to be large

is

first order: 2kh kKELVIN("

second order: k h k KELVIN( 0)

The angular width of the first-order Born pattern is determined by the

width in momentum space of h and the rate of variation with 8 of 0

kKELVIN. A bit of algebra shows that

13

• . o , o ,. • • . o~o~oq .o ...... o. ........ ... . . . -. •. .... ,. . ......-

68 b2wkKELVIN

If b 6 m and 'ULVIN .3 m (typical values for SEASAT), this 0

gives 66/8 - 1/20 . This is very narrow indeed, but not out of line

with observations.

Let us finally consider the relative sizes of first and

second order scattered amplitudes. Due account being taken of the

peek amplitude of h and its momentum space width, we find that the

maximum backscatter amplitudes in first and second Born approximations

are

NAX 20 -C bA(61

-1 * C - b •.)

0 2MAX -(2kA(B 2)) C b 2A( 2 ) . _.

81 and 82 are the (different!) wake angles at which the two types of

scattering reach their maximum, k is the radar wave vector and b is 0

the radar patch size. The integral over q in the expression for

is, because of the narrowness in wave vector space of the function

h concentrated at q -0, leading to a considerable simplification

in the formulas.- ." .o ,. -.

14

M'OM..ro.' o "° ooO~OOo'~o ,-o ."° Oo "o °- s°-.. . .... ..- .- .--.-.. . . . . . .'.. . . . . . . .... . . . . . -. . . . . . ..•. . ..~l ,~o % % oO'of °'-.Z. % ., ° "=

We expect that A(8) is a reasonably slowly varying function

of e , except in the near neighborhood of 8 = 0 Therefore, to

compare first and second order scattering we will assume that

A(81) 1 A(02) obtaining

2 4kA - 8 A/Xrada r

From slender ship theory we extract the result that the amplitudes of

the diverging and transverse wave trains are related by

3/2Adiverging Atransverse

for 9 not too small. One readily finds transverse wave amplitudes of ..

.5 m a kilometer behind a large ship. This would correspond to a

diverging wave amplitude of 1.5 cm at e = 60. We will, therefore,

take 1 cm as a representative value for A( 8 ). This is confirmed by A -

the in situ wave height measurements of the Dabob Bay experiment. For

a radar wavelength of 23.5 cm (appropriate to all the measurements

under discussion) we therefore have 02/1/0 1.

Our conclusion is that surface ship Kelvin wakes are strong

enough that multiple scattering from the wake itself cannot be

" neglected.

............................................. .. '.. . . .. . . . . . . . . . . . . . . . . . . . . . . ................. ................

S

At a minimum, this means that multiple returns, due to S

various orders of scattering ought to be seen. This might be the .

explanation of the multiple V structures seen in some SEASAT pictures.

0

S

S

S_

S_

S

S

16

1.*.~.*. *. .......

... ** .*................................................ .................................... - ...........................S. . ...-............. . ..........S.~%

' ..,-. .- .

0

IV. Dabob Bay

In the Dabob Bay experiment, the radar return from a wake was

measured and the actual wave height was measured in a one-dimensional

transverse cut across the wake. The maximum radar return came from

points at wake angle 8 - 3.60 . The direct measurement of the wave

height in this region indicates that there is little energy at wave.0 .

numbers appropriate to first order scattering and a substantial peak

at wave numbers appropriate to second order scattering. This peak

does not occur at the wave number predicted by simple Kelvin theory.

.0This is a hydrodynamic puzzle about which we can say nothing. The

question of interest to us is whether, given the measured surface

height, it is reasonable that second order scattering should overwhelm

first order scattering.

The r.m.s. displacement recorded at the Interesting wake

angle appears to be about 1 cm, and corresponds to surface wavelengths

of order 40 cm. As was noted in the previous section, the quantity

2kA, which determines the relative amplitude of second and first Born

scattering for equal rms surface displacement, is of order I for -

A 1 cm and a = 24 cm. An examination of the power spectralradar

density of the surface slope shows that the p.s.d. is down by a factor

of 10-2 from the peak value at the wave number appropriate to first

17

"?. - .

____.______,_,'.______.,___."_..__._" ..... ,...,.,..".. . . . . . . .

order Born scattering. Consequently, the intensity of second order .

Born scattering should be about 102 times that of first order Born

scattering.

So, if we modify our scattering theory to include higher

order effects, there is no inconsistency between the location of the

intense radar returns within the wake and the directly observed wake 0

structure. Obviously, one should combine the observed wake structure

with a radar scattering model to obtain a predicted radar wake

structure and attempt a detailed comparison with the observed radar S

wake images. An important missing element is a two-dimensional map of

the wake structure since that is what really enters the radar cross-

section. We have used a one-dimensional wake crossing measurement for

rough comparison purposes, but must be aware that surprises may occur

when we confront the two-dimensional data derived from photographic

images of the wake.

. ..

18" " • "

. . ~ . . .-.. . . . . . . . . .. . •...

.. . . . . . . . . . . . . . . . . . . . . . .. . .

V. Conclusions S

These are based on a simplified scattering model and some . -

questionable assumptions as to the nature of the scattering field.

The scattering model can be readily improved. Thus with an increase

in the complexity of the formula intrinsically electromagnetic effectsp

(such as polarization, Fresnel diffractions, and dielectric properties

can be included. This hardly seems warranted till more is known about

the hydrodynamic structure of the wake.

Two hydrodynamic descriptions have been used.

(a) The idealized mathematical Kelvin wake. 0

(b) Some estimates have been made using the Dabob Bay data.

Our conclusion is that first order Bragg theory is not

generally adequate. At some viewing angles there will be strong first

order Bragg scattering with a sharp angular dependence. At other S

angles first order Bragg scattering will be small. However, there can

be second order Bragg scattering of an intensity comparable to that

where first order is large. The angular width of the second order S

observations should be greater than that of first order, but not

necessarily dramatically so.

19

.. .. .... . .... '...-.....

. " .°. "-'." I

. . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . .

-.. r ro

The main limitations on our present work is that only special 0

directions of look are adequately described. Dabob Bay experiments

are closest to what are considered.

To compute what should be observed at other aspect angles

empirical two dimensional wave height spectra are needed. Given these

the conclusions drawn about angular sharpness might be modified.

These also might help us to understand the experiment observations at

different aspect angles.

Some of the needed information could perhaps be obtained by

reworking the Dabob Bay data. Also some of this might be obtained

from the NOSC data.-'

20.

S

20 .,:.:.::.,

- . . . . . . .. . . . . . . . . . . ... .

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•.-.".-.

.~.............,.

S* .. '.

FILMED

1-85

DTIC