N“ AD A061 971 PCTCOROLOSY INTERNATIONAL INC MONTEREY CALIF F~S *0/1scam PROPAGATION FOR*LATION FOR 3—0 RAY—TRACt CO UTATIINS SP—tYc (U)

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METEO - RQ LOGY I NTERNAT IONAL ft ~C O R P O R A T E D

P . 0 . s o x 3 4 ~ • M O N T E R E Y , C A L I F O R N I A 9 3 ~~4 O

LEVEL~)~~~ 4~ OUND PRO PAGATIONFORMULATION FOR 3-D RAY-TRACE COMPUTATIONS

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SPHERI CAL EARTH4

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Project M-135 , Technical Note One

December 1966

/ \/ 1 :~~ J~/’ /~~,—

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Prepare y: M. MI Holl 17* IMts $ótu..

Under Contract ~~T~~~227 1-67-M- 1Ol2 kfl UI$Isp r-~i. 1 UAIUSscD o

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I~1~~ 9Th~1Slit AVAIL suie WtsTar

For the Officer-in-ChargeFleet Numerical Weather FacilityU. S. Naval Postgraduate SchoolMonterey , California

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~robIem :

Given the speed-of-sound distribution in the sea the problem is to

calculate the trace of a sound t a y which leaves 1 a prescribed locati on in a

prescribed d i rec t i on . The formulation inc ’udes the curvature—of — th e-ear th

: factor and other curvature terms arlsi nq ’i n the Select ion of hori zontal

coordinates.

Pa rt A: rormu lauon of the Equations

I . The Governing Vector Equation

The ray is directed normal to , and in the direction of advancement

of , the wave trout. We as sign the un i t vector t t o this direction . At a

point in the pro qres sion ot the sound ray the unit vector t and the vecto~V c , being the local ascendent of the sound speed , toqetht ’r ieneral ly

def ine a plane . This plane is the plane of rig . 1 .

t ô t

\ \

)

• ,

~~~~•~~~__J

_ _

~~‘,

— ~~ —

Fiq . 1 Refract ion of the Ray

1or arrives at (The integration may also ho performed backwauls In t im e . )k ieomet t i c vectors are denoted by curl underscore .

— 1 —

~- - - - .~~~~ .-— .~

[ ~~~~~~~~~~~~

. . _______ _ _ _ _ _ _ _ _ _ _ _ _

_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _

t ’t it ’ component ot V c which ts normal to t and hence l ies in the

wave trout • may be ox pies sod by

— t x ( t x \‘c ) — t t . Vt. ’ ~~ . . ( 1)

It is th i s comp on ent ot V c which causes the d i i t ei ou t ta l p i o ] r e S s iou of the

wave t i ou t ~t i id the ~t ssociat ed bending 01 the ray .

l o t t h e pai a i n ot et s he a I m eat i l l ea sul e in the pio qi ession ot the r a y

in t ime ~uct cu to u t Ot the t o u q t h I ucrenient traversed is

c ot . (.~~)

In t o t i t i s ot the q~’o mc ’tt y ot l’ tq . 1 the bending ot the ray may be expressed

1 y If

~~~~~ (3)01

The’ cot i es poudi u~j qo ve t u i n ~j equation , expu ’ssed in vector form , is

I ~t~ t x ( t x V c ) . (4)

The d i t t e t e i t U a l ope at ot , 1 ‘l’~t , implies t . l t t f eren t i aUon In time moving with

the ray . l q u a t i o t t (4) t e a d i ly fol lows from Eq. (3) in that Ot = — ii 6

The han q t ’ in a unit ve t .’t oi is always directed normal to the vector .

It the ’ speed ot soun d is relatively constant in time over the period

of prop agat ion ot the ray we may choose to integrate along path length

rather than in t e ims of t i m e . in this case Eq. (4) may be expr essed by

_ _ _ _ A~~~LJ j

- ...--~~~ .-. ,, - ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~~~~~~~~

-~~

- -~~~

--~~~

— -

~~

.

~~

.

D t= t x ( t x V K ) , (5)

where

K i n c . (6)

2. Vertical arid Horizontal Components

We pioceed by re la t ing the vector equation , Eq . (5) , to the natural

rt.’ t o rence or ientat ions of hor izontal and ver t ica l . The un i t vector t

def ined as before as the local di i oction of the r ay , and the u n i t vector k

def ined as the d i re c t ion of local g rav i ty , togethet genera l ly def ine a

vertical plane . Thi s plane is the plane of Fig . 2 .

Fig . 2 Vertical and Horizontal References

The horizontal component of t defines the orientation 01 the unitvector . The unit vector j completes a r ight—handed t r ip le :

x~~~ (7)

— 3—

-

~ .

,v~ - ~~~~~~~ ~~~~~ —~~~~

- --- -~~~~~~- .-- -—

- ~~

.— -

~~~

-

And ! points out of the plane of Fig . 2 , toward the reader. Wh ereas the

vector k is locally defined (i. e. a function of position only) the vectors

and i depend on the particular ray in that i is parallel to the horizontal

component of t . The angle V is defined as shown in Fig . 2. The

parameter Z denotes the depth below sea level of the location.

I ~ Directional change in the unit vector t along the ray , s ,

Involves only two components , and these lie in the p lane of the wave front.

The plane of the wave front may be defined by the vector pair j ari d j x t

The vectors t and j x t may be expressed by

t = i c o s Y + k sin V ,

x t = — k cos V + i sin )‘ . (9)

If’

We proceed by developing the left-hand side of Eq. (5) :

Dt D cos V + k sin Y )

DY Dl Dk(- i sin V + k cos Y ) — + cos )‘ —~~~

- + sin V- -

~~ Ds Ds Ds

DV Di D i Dk= - (~~ x t) + cos V (~~~~ j . .~~.i- + k k . fl.— ) + s inY

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ (10)

We approximate the earth-curvature factor by

D k cos Y— = - 1 t i lDs - R - z

-,~~~~~ - —, _______________ -,—-

— J

where R is the local radius of curvature ot the earth . Subs t i tu t io n of

Eq. ( 11) in Eq. ( 10) yields:

D t .~ D i -- D~ —

= — ( x t ) ~ cos V j . ~~~~— — x t R ~

( ]2 )

Making use of Eq. (12) we now take the ( i x t ) component of

Eq. (5) :

DY _ _ _ _

= —

R — z — (~~ x t ) . t x ( t x \ k )

= - cos~~ + ( j x t ) • V K

= - cos Y

- cos V + sin V i . V K . (13)

And the j component:

D icos V . —

~~~ = . t x ( t x “ K) = — j . \ K . ( 14)

- Ds

Equations ( 1 3) and (14) are the desired pair of components of the governing

equation , Eq. (5) .

Provided the ray does not become vertical , we may also use a l inear

measure , x , along i , as the independent argument for the integration:

Dx = Ds cos V . (15)

We may also introduce the linear measure y along j . With these defini t ions ,

Eqs. (13) arid (14) may be expressed by

—5—

__ __ -~ - -______ - .—,

~~ —‘ ~‘

~1

D) 1— — tan ) ( lu )

D i 2= — sec ) . (17)

p

3. lion :ontal Co otd tnat e s

Let ( and ~ be a suitable pair of orthogonal coordinates for the

hon ~onta1 expanse . The a scendents define the corresponding uni t vectol s

I and I

E l

\V J . ( 19)

IfWe choose that I and j foi m a l e f t—handed system with k :

I x J - k . (20)

At an arbi trar y loca t ion the un it vectors i and i , defi ned eat l ic t

form angle ~ wi th I as shown in F ig . .

J

Il- I

rig . 3 Def init ion of the Anglo ~

- . ~~ _ _ _ _ _ _ _ —~~- -- ~~_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _

~~~~~~~~~ - - -~~~~~~~~ -.—-~~~~~~

We proceed as follows:

D i D= j . - (cos ~~I + sin ~ J )

= j . (-~~i n $ ~ + Dx Dx

D I= — ~~ ~ (cos ~ j — sin ~ x k )

PL- Dx

-

Dx

= - - j . (~ . v I ) . (2 1)

Subst i tu t ion in Eq. (17) y i e l d s If

+ sec 2 ) ~ K ~ \- (22 )

where

Q : . _ j . ( i . \’ I ) (23 )

is d ue to the curvature of the horizontal coor dinat es. Equat ion ( 1 u) is the

other of our pair of governing equations.

-I . Ton g it ude and Lati tu de Coordina tes

Longitude , 9 , and latitude , , for m a p air of orthogonal coot d in —

atcs 3 . The asceridents define the uni t vectors:

3Longitucl c is measured cast of Greenwich only.

-7-

LA

________

I - -

“ 9 = I / (R— :.) cos Q (24)

= ~~~~~~ . (2~i)

These e xp r e s s t ons are s nq u l at at the poi es . The cut va tu t e t erm , Eq . (2 3)

develops as lollows :

= — ] ~‘05 ~~ I ~- sin ~~ J . V I

- j .J~ cos $ I . V ~ sin $ J . V I

_ j .~~~~@s $ [ ~~~~~~~

-

~~~

_ }

- — cos $ tan~~’ R — ~~ . ( 2 t ~

rht.~ t.~ - nunq equations tot the r a y t r a c e may be w r i t t e n :

If

2 — — tan ) (cOs ~~i si u $ j ) \ K (2 7)

- cos~~~ tan ~ +

2 (s in $ 1 - cos $~~ ) . \~~~ . (2~ )

The dire ct ion al d e r i v a t i v e s m~ v be expr es sod by

1“K — ( ‘9)

- (1~— ~) cos ~

• 1 (30)

— R —: ~

Also t equ t ro d , as increments ot iut e~i t a t ion , are

59 c~ s8 Ox , (3 1)

(R— z) cos ~~

— 8 —

- - - s_p

r

-

~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~~~~~~~~~~~~~ ~~~~~~~~~~~~ ~ i

= O x , (32)

where Ox is tru e hori zontal trac e—length increment .

5. Cartesian Coordinates in a Polar Stereographic Projection

The coordinates

X L cos 9 (33)

Y — L sin B , (34)

where

L (R—z) M cos c~ (35) It

1 + sinM ° ( 36)

1 + sin ~

map into Cartesian coordinates in a polar stereographic projection , true at

latitude . The coordinate origin is at the pole.

The ascendents define the unit vectors:

M I , (37)

E M ! • (38)

The curvature term , Eq. (23) , transforms as follows:

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

~~

Q = - ! . [c o s$r • V i + sin~~~J . V t 3.

= {- cos $ sin 9 + sin $ cos (R-z) cos~~ •

This can be shown by expressing the unit vectors in terms of the polar

vector s , Eqs. (24) and (25) . Elimination of 8 and o from Eq. (39) , using

Eqs. (33) through (36) , leads to

— sin BX — cos (40)(R~z)~ (1 + Si- n c D )

The governing equations for the ray trace may be written: -.

= - - -i--- - + tan )’ fcos s M ’~~ + sin e M~~~ } (4 1) If

= s i n $ X-

+ sec2 V ~~sin~~ M~~~ - cos ~~~~~~~~~ (4 2)

6. Integration Formulae

We choose to carry on with the coordinates introduced in Section 5:

Cartesian coordinates in a polar stereographic projection.

The governing equations , Eqs. (4 1) and (42) , may be developed for

numerical integration by introducing finite increments:

8)’ -{~~

+~~~~~~Ox + t a f l) ’ {~~~ OX + ~~ SY ) (4 3)

= M ~~ - Y O X + sec2

~~{

~~~~ö Y _ ~~~ o x ) . (44)(R—z ) (1 + sin ( p )

-10-

--~~~~- ~~~~~~~~ - - - .-_

~ ppr ---~~~~~

- -~~~~~~~~~~~~~~~~

--

~

-

~~~~~~~~~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

To these we may add:

OX = M cos $Ox (4~ )

OY = M sin $Ox (46)

O Z = t an YOx (4 7)

For integrations in which the trace is permitted to become vertical

we must reintroduce

Ox = cos Y Os , (4 8)

and u se O s , the linear progression of the ray, as the argument of inte—

grati on.

For integration with time as the argument we reintroduce

Os = c O t . (-4 9)

In order to succintl y handle surface and bottom reflections the

argument may be geared to depth increment Oz , unless an upper bound

on Ox Is exceeded in which case Oz must be reduced.

At the surface , and at the bottom , reflections are performed by

reversing the sign of V . At the bottom this involves the approximation

that the bottom is f lat .

—11 —

- - .-

~

- - ---- --- -~~~~~-.- _ _ _ -—

r ---—---—-- —------ -

Part B: A Scheme for the Numerica l Integration

We herewith develop a proposed integration scheme based on the

geometric visualization of refraction and the adjustments for earth and

coordinate curvatures. Equations (4 1) and (42) express the ray propagation

as differential equations in terms of Cartesian coordinates in a polar

stereographic projection . In order to visualize the geometry we rewrite

these equations as follows:

curvature term refraction term

R -Z + j x t . V~~( Os (50)

= -

Ox + sec V j . ~K O s . (5 1)(R— z) (1 + sin ( p )

The vectors j x t and may be recollected from Fig . 2: The vector j x t

lies in the vertical pl ane and is normal to the ray. The vector j lies in the

horizontal plane and is normal to the ray .

A segment of the ray, in which segment the refraction terms are

taken as constant , de scribes a circular arc in space. We propose to

integrate the ray along chords for such arc segments. Representative values

for the refraction terms are each numerically evaluated for the segment.

Along a chord the curvature terms may also be considered steady.

This is apparent for the term — Ox / R—z if we neglect the minor local

variation of R—z . For the other curvature term we introduce the argument

along the chord increment Os. That is ,

X X + c o s $ V (52)

Y Y0 ÷ si n ~~ V . (5 3)

—12 -.

__________________________

r -

~~~~

--—-— -‘-

~~~~~~~~~~~~

Substitution i-ri the coordinate curvature term yield

sin ~ X - cos ~ Y - sin $X 0 - COS

2 — 2 (5 4)

(R— z) (1 + s l n Q ) (R—z) (1 + s inç , )

p

and shows that this term may also be considered steady along a chord .

We assume that K , the natural logarithm of the speed of sound , is

given at an array of grid points in the polar stereographic proj ect ion , to t

each of a number of depth levels.

We shall describe the integration In terms of the incremental step

from position X~ ‘

~~~~

, z , at which point the ray is directed at angles

$ , to the next position where these values are indicated by sub- . -

scripts r n .

Let be the R-th estimate of the angles with which It

the ray is directed at position T+ 1. The first guess of these a ngl e S may be

approximated by

(55)

~T+ 1 . (5t~)

Linear extrapolation from r - 1 and r values may yield a better f i r s t guess.

F The chord Is directed from position r with the angles

(R) =

1 (V 1. +

~~ ‘

(57)

= 2 -r - r + 1 ) . (58)

—13—

- - -

~

__ _ _ _ _

— -—-----~—.,,— — —__ — -_---— -_-- —_~- _-_ _

~~

—

‘

l

C hoost ‘ig a ~-hord t e~rigth , ~ s , the R — tl i e s t im a t e ot the i ciy ’ S advance is

g iv en by

— NI cc’s N

~ N Os (~ ‘~~~

p

-‘ 1~

(R) ~

(R) o , (~~O)

6 a sin ) N ös . (~~

I )

~vt i e i e N I is an ci p ptc ’p t t a t c ’ly ~- - t i t e t ed v a l t i i - .‘~l t h e m a p p t i t ~j ta~’tc ’t , I: q. ( 3 t )~~

We citso t e ’t I u i i e

I ) I.Ii’)cc’s ,Ss .

We nius t now ca l c u lat e t lie L~ ’ii di nq exp e t let ic ed b y the t a I ~V( ‘i hi i s If

increment , as e x pr e s s e d by l qs . ( SO) and (5 1)

The cut v atu t e te rms at e eva uctted by

(l)~ox

sin — cc’s ~t’ ö- (~ -l)

(R - - ~~ ( 1 s in ç’ )0

wh e t e an ap~ o p i t a t e l y ct ’t i t e t t ’d v a l t i e ’ oh R — . i s used.

F ot evaluat ing t he ’ amount oh t e ’ t t act b i t we i equi t e i eL~ese n t c t I t v e

va tue of the rt ’tt act iou te r ins • We ’ ‘rop e ‘se th at th i s he ’ done ’ h’~ h i i t t i t ’ —

d i t t e r e - u c e ev a lua t ion ot the r e q u i t e d d i r e c t i o n a l de ’t tv at l v e s ot K. The ’

s pace i ncr emeu t to t evaluctti r i g I hre ’si ’ sho uld hi ’ comnn’usur ate ’ w i t h th e ’

— 1 1—

- - . .

-- _ _ ~~~~~~~~~~

—

~~~~~~~~

_ _ _

.0

length ot the chord; In t ac t We ’ ~~~1 opost ’ th at i t be made t ’qu a I to Os I t ) ‘ I I V t -

the iii ’ s i t e d i t ~ pr t ’ se nt at t vent ’

/ 1~A /.—

I

, ~~~~~~~

vt - i t tc ~~ l p l a t t e ’ hot i ~or i t ~t L p lan t ’ If

r i g . -I Pos i t ions A , 11, R and I. b r t~v a l i t a t i n t i th e ’ I) i rect ~c ’rial l~ e ’t i v t t i \ - c sc ’ t K

l’he I oca l ion s at w h r cl i K shou Ed b ’ in t e r polated at t ’ t nd it -a ted i i i

- I . Iii the ’ Vet heal p l an t ’ the ’ p o s i t i o n s ar t ’ A , above , and t~, below;

the i r sepal at i o n is Os , to t un r ig a cross wi th the chor e !, in the hot L’c ’nta l

~l ant ’ the pc’s i t ions a r t ’ R , r i ght , and 1. , l e f t ; t h e i r s~ p ara t ion i s

t e r m i n g a c ioss with the ’ chord . The R — t h e ’s t t m a t e s oh these p o s i t r o n s a t e :

,\ + I

~ ~ (R) •

~ N ) cc’s NI Os

~~(R) a ~j (c c’s ) (R) s i n ) (R) ) sin $~~~~

NI Os

(R~ ( ~

(R) — c c ’ S ) Os

_ _ _ _ _ _

- -~- -—-n ~—y--~ —

— -- —.-- -- -- __________________________

x~~ ~ X ~ 1 (t ’0S

~

(R) — ~~ )

~ (R~) cc’s ~ (R)

M Os

y 0~ y e -~ (cc’s ‘i

N — sin ~

N) ~~~~

N M Os

+ —~ (sin )’ N c~05 ~

N) Os

X - e ~in $N

) M Oso

~~R)

~~ 1 (cc’s ,,N

~~ ~~ N

— c ’os ~~‘~) M Os

(R) 1 (R)~ s i n Os

0

~ (R) ~~~

(cc’s ~~0~) cc’s — sin ~ (R) ) M Os

~~(k) V I (cc’s )N sin ÷ cc’s ~~~~~~ M Os If

- -

I

(R) = ~ sin~~~

’~ Os . (t;s)

Let the values ot K int~ rpc’iated at these loca t ion s be denoted by

Kf~ , K~~ , K~1~ and K~~~ .

r et r ac t ion ter ms are acc -c’r c l iny ly c~ lyo n by

( j x t . v K Os) (R) KJ~R)

— K~R) ( ‘ 7)

sec ~,(R) . VK Os) (R)

sec ~ N (K~~ — ~~(R~) (~~~)

— 1 t ; —

- ~~~~~~

T~~~ ~~~~~~ :i~~~~~~~ -~~~~~

—I

The R— th estimate of the bending of the ray In the coordinate system is

given by

~ ,,,

(R) ~~~~

+ K,~R)

- K E~R) (( 9)

6~ (R) =

s1n~~~ x - cos ~ (R) 0

6~~N sec ~ N ~~(R) - K 1

N ) . (70)

These give the R+l estimate of the angles with which the ray is d i rec t e d

at position r+l:

‘IT

+ O V N (71)

If

~~~ (R+ 1) fl + 6~~(R) (7 2)

With these new estimates the R-l cycle Is performed beginning again with

Eq. (57) . The cycle is repeated until convergence is attained ( i .e. wi th in

some finite tolerance) . Upon such convergence the desired chord has been

found , the Incremental step is completed , and the ray moves to position

T+ 1.

It should be noted that in evaluating the coordinate curvature term ,

Eq. (64) , the origin for X and Y is at the pole. Also , the ang le ~ is the

angle made by the ray with the X axis; that is ,

cos~~ i • . (73)

.4’

-17-

- V

_ _ — - _ _

— .. V

-

~

-

Iii the su qeje ’ si ee l scheme ’ , the chc ’r d I Oti c i l i t , Os , may he chic ’s ott

tot each s tep. We ’ tool t h a t the opt n mum choice ’ tot Os would be such

~ that cc’uve’l qe nc ’t ’ r s a lt at irec! at R 3 . I ‘wov or , I t may be’ w or t h exper t —

mer i t ing with Os to see how the ~‘otivei elt ’ilct ’ is at I oc t eel

Ret lect ieri c’t the r a y , at the s e ’a sun t a ct ’ and at I t i e ’ bottom a t so

t n t tue ’nct’ s the’ chi c ’t ce’ 01 Os . I e ’t I t ie ’ dl st ar let ’ to the boundct r y bet r ig

approached by the t a y be’

-

-

— —

~~~- I (74)

where ~~ is :.oro , or the depth , h i . I t

then ~

‘

(h~ I) V

Os ,

sin )

sheru Id hi’ ii Se t! as ~qt~i1 d I ont j t h l i t th~’ l i e ’ xl -y ~- le ’ so th at t he’ st e ’p t ~r Le ’ s th e ’

I ~iy t ’xac tI ~ to the bounitiar y . At t h i t : ; hune ’tul 0 , ) is r eplaced by — )

as su n i t ncr th e ’ bottom Is I ron t : onrt a I

Aiio t hie ’n ~ c ’ n i t i i i e2e ’ue ’, must be pr c ’vtde’cl t ot . In appr oaching the ’

uppe ’t c’n t ower boundary , ou t ’ er the ’ o t h e ’u oh the A , ah~ ve , or 11, below ,

Lc’c~r I t o r t s i r n a y I a l l beyond the boundary . Pr c’v is t or i should ho nitatit ’ so t hat

the tr i t er polatt oni c’t K wi l t y ield a suitab le ex I t apc ’Iated va tue in such

cases.

-18-

_ _

Flow Diagram i n Terms of Operations

Position: X Y Z

Angles of Ray: T~~

~T

Set estimates: Eqs. (55) and (56)

Compute chord angles: Eqs . (5 7) and (58)

Choose Os

Is Oz too large ? No; proceed with Os.

Yes; replace Os according to Eq. (75) .

Compute curvature terms: Eqs. (63) and (64)

Compute locations A , B , R , L: Eq. (65)

V Interpolate K at A , B , R and L

Compute refraction: Eqs. (67) and (68)

Compute total bending: Eqs. (69) and (70)

Compute ‘IT 1) and ~~~ 1) : Eqs. (7 1) and (72)

Test angles for convergence:

No; recycle

-. . Yes; advance step (Including any reflection) to r+ 1

position and angles

—19—

---—— - --

~

-- - - ~~~~~~~~~ - - ~~~ - ‘ V — -‘ V

F- -

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~

-

~~

--- -V.” “

~~

‘ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

The program requires an interpolation subroutine for interpol at ing to

specific locations in the three—dim ensional distribution of K In c , where

c is the speed of sound . Computationally , the program requires zero-order

continuity of the interpolation; hence even linear interpolation between grid-

point values could be used . However in the interests of better accuracy

highe r-order interpolation is preferabl e.

The three-dimensional distribution of K may be expressed by the

- V two-dimensional distributions for a number of discrete depth levels . These

depth levels should be concentrated where the physical ly and operation ally

significant variabili ty of K is greatest . The set of fields is represented in

terms of a polar— stereographic gr id—point array. It is quite feasible that

the three-dimensional distribution of K , for an ocean region of interest ,

could be stored in the core memory of the CDC 6400 computer.

The interpolation , for location , z~ for example , may be

pe rformed by interpolating to ~A at the four level s which include the

i - V - - depth ZA in the middle layer. Cubic interpolation in z would then corn—

plete the interpolation . It should be recalled that ZA

may lie above the

sea surface ; in this case linear extrapolation of K from the surface and

first sub—surface depth may be adequate .

VV ____ T•~~ 7T~~~~~~~~~~ V~~~~ . _ ______________— ii