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LOS ALAMOS SCIENTIFIC LABORATORY

of the

UNIVERSITY OF CALIFORNLA

Report written:

January 20, 1955

LAMS-1902 This document consists of 36 pages

&3 -~

PHOTOGRAMMETRIC PRO,BLE MS ASSOCIATED

WITH

NUCLEAR DETONATION TEST EXPERIMENTS

by

D. F. Seacord. Jr.

All Los AlarncIs reports present the opinions of the author or authors and do not necessarily

reflect the views of the Los Alamos Scientific Laboratory. Furthermore, this LAMS report

has not been prepared for general distribution. It is accordingly requested that no distribution be made without the permission of the Director’s Office of the Laboratory.

$p’ -

Qy,

f$ &//“ Ek

/’

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qgglliimk

\

Report distributed~UN ~ 1955

-m

AE C Classified Technical Library

University of California Radiation Laboratory, Livermore

E dgerton, Germeshausen & Grier, Inc., Boston

Los Alamos Report Library

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LAMS-1902 “

1-7

8

9

10-40

4

.

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UIVCIASSIFIED

AESTRACT

Photogrammetricanalysisof objectpositionand displacement

in two and threedimensionsis outlined.

arisingin the photographicdocumentation

of an explosionfireball.are indicated.

Extensionto problems

of the rate of growth

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—

--

Cormm!rs

SECTIONI INTRODUCTION. . ..Oo. ,0000000 6

SECTION1:[ TWO-DIMENSIONALDISPL4CR4ENTOF A POINT ● 7

SECTION1:[1 THREE-DIMENSIONALDISPLACEMENTOF A

POINT.*..*..... ● 00000 13

SECTIONIV FIREBALLPHOI’(WUWMETRY..~. ..c. c 17

A. SurfaceBurst

B. Air Burst

a) Triangulation

b) MagnificationFactor

c) Apwent Radiusvs. True Radius

d) Distortionof ImageDue to Geometry

-m

. -4-

—

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FIGURE1

FIGURE2

FIGURE3

FIGURE~

FIGURE5

FIGURE6

FIGURE7

FIGURE8

FIGURE9

FIGURElCl

FIGURE11.

uNcfAssl/c

B

IIMJSTRATIC~

age

OBJECTDISPLACEMENTMM CRWXNG OPTIC

AXIS. . . . . . . . . . . . . . . . ..s. 26

OBJECTDISPLACEMENTACRCXW OPTICAXIS . . . 27

MEASUREDFIIM HEIGHTVS ● OBJECTHEIGHT.. ● 28

THREE-DIMENSIONAL

THREE-DIMENSIONAL

TRUE V6 . --

SURFACEBURST ● .

TRIANGUIJLTIONFOR

DISPLACEMENT,ELEVATION. @

DISPLACEMENT,PLAN. . . ● ~

FIREBALLRADIUS,

● *o***o* ● * ● ** 9

BuRsrpom. e. . ...32

TRUE HEIGHTC@’BURST. ● o

MAGNIFICATIONFACTOR. e .

APPARENTVs● TRUE RADIUS,

VERTICALPROJECTION. . .

APPARENTVs. TRUE RADIUS,

HORIZONTALPROJECTION, c

-5-

.. ****e .* 33

eeeemeee● 9$

AIR BURST,

● ☛ ““”””””3>

AIR BURST,

qgmmiqlm ~__““ d

,*. .e*b ..0 36 I

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I

,

-----m bIwrwJflEll

PHC7i’OGIWWETRIC PROBLEW ASSOCIATED

WITH NUCLEARDETONATIONTEST EXPEMMERTS

1. INTRODUCTSCIIJ

With the adventof the LASL mass-motion

Buster/Janglethe problemof photogreaumetric

importanceto GroupJ-10 in the reductionof

experimentson Operation

analysisbecaneof direct

photo~aphicdata.

The conversionof positionand displacement,as measuredon the

photographicrecord,to the true spatialpositionand displacement,involvedthe photogrammetrictechniquein twu dimensions(sincethe maasuredobjectswere approximatelyin the same horizontalplane as the

nucleardetonationand the camera)~ Sincethe objectdisplacements

were smallin comparisonto the camerafieldof view,a simplecosine

correctionsufficedfor translatingfilm measurementsto true displacement

This methodwas imccessfuU.yappliedto the resultsof both Operation

Buster/Jangleand OperationTumbler/Snapper.

However,large-scaledetonationssuch as thoseIn OperationIvy,

whereindisplacementswere comparableto the field of view,requireda

more refinedcorrectionfactor. Furthermozw,with the extenstonof the

mass-motionexperimentto high altitudes,theproblemcame to include

three-dimensional. photogrammetry.

The extensicmof this type of analysisto photographicdata in

f~eldsothertham mass motion becameapparentwith the applicationof

the analyticsolutionto fireballdata in the determinationof the total

hydrodynamicyield. Since

an accuracyof betterthan

the abovemethod LB theoretically capableof

~ 1$ in yield,the error in measure

~\Jw$’’5mw

diia%iaul UNCLASSIFIED

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-“

~h!C~(Q,qFIF~

diaM?termust be reducedb the orderof ~ 0.2$. As groundworkfor the ‘-

futureimprovementin basicfireballdata for the analyticsolution,it

was felt necessmryto derivethe photogramnetricequationsfor correcting

the observedfireballdiameterto its true diameter.

The derivationswhich follovare based on severalsimplifyingassumptions8 (1) lens distortionis negligible,(2) the ob~ectrange is such

that atmosphericrefractioneffectsmay be neglected,and (3) curvature

@ the earthis not Men into account(exceptwhere specificallynoted).

II. TUO-MNENSIIONALDISPLACE31ZNTOF A POINT

!l!hephetogramaetricanalysisfor four cases involvingthe displacement

of a point in two dimensions(i.e.,opticaxis horizontalaad displacement

in the horizontal plane)were involvedin the data analysisfrom Operation

Ivy, and are presentedin the followhg sections. In general,the optic

axis is net perpendicularto the line of objlectdisplac~nt in the region

where the displs,c~nt occurs;this is usuallydictatedby the restrictions

imposeden the locationof ob~ectand camera.

Figure1 is applicableto CasesA and B, The nucleardetonationpoint

is designatedZ, the camera,C, and the initLalpositionof the object,O.

(5heexperinenteJdescriptionmaybe statedbriefly: the shockwave, expaadingfrom Z, causesO to be displacedalongline D.L.; the recorded

film displacementxl is to be relatedto the true spatialdisplacementyl).

The problemis, of course,s~tricel about the opticaxis but “left”and

“right”will be used in a sensecompatiblewith the figure.

the

Case A: The objectis displacedfrom iibs initialpositiontowards

opticaxis but does net reach the opticmxis. Referriagto Figure1,

followingparametersare kaowa~

-7-

U

UNCLASSIFl~b

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f.

& =

xl =

$=

(-f)+y) =

a=

e=

It ie

came!ralens focti -~ ength

rangefrom camerato detonationpoint

displacementas measuredon film

anglebetweendisplacementline nnd line connecting

detonationpointand camera

angl,ebetweenopticaxis and lino from camerato

initialpositionof object

anglebetweenllne from camerato Initialposition

objectand linefrom camerato detonationpoint

of

anglebetweendisplacementline and line from inltiaJ

Wsition of ob~ectto c~ra

desiredto find yl, the true spatialdisplacement,

the aboveParameters.

Y1 RP2 W2

sinP ‘==V’ &? “

=1 [$x)- (~+y)J

or:

XL/xl =

RP2 and W2 are variables

them to knownparameters.

RP2 * Rc +22 *

y f

in terms of

(A-1)

and ~ybe replacedby equationsrelating

% +(7- 3q.)sin* (A-2)

f

sincez2=y2 sin* =( Y- Yl)sti* ,whmey=m+Y2

But :

lpgo-(p+y+e)

~= Rcsin(/3+y)

Wln e

(A-3) -

(A-4)

SubstitutimgA-3and A-h in A-2:

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and substitutingA-5 In A-1:

~ is relatedto Rz by:

~=qsingf

sin (a+ p+y+ @-J

s~ (A-6:) andreplaclng kby (A-7):

(A-7)

xl ~ sin {tcos @+~ [1 + sin @+~ ca6 @+y+ e ) csc @] (A-8)

‘1 “ ~(a+fl+y+ #) [f + xl .0s (@

It my be men that,if the opticaxia is perpendicularto the llne

of dlsplacenwnt$ .0s (~+ 7+6) 00 and A-8 reducesto:

(A-8a)

CASE B: TM obJectis displacedfrom its Initialpositienup to the

opticaxts. Figure1 appliesto Case B with:

Y = Y3+Y2

x = X1+X2

z*-&- (B-1)

(B-2)

(B-3)

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In contrastto A-88, the aboveequation:Lsvalidwhetheror not the

optic-s Is perpendicularto the llne of dioplacemeat.

CASE C: l!heob~ectis displacedfrom itw iaitlalpositionup to and

acrossthe opticaxisbut

Figure2 illustrates

the opticaxis designated

not to the perpendicular line of stght.

this casevlth the displace-t to the right of

by “V”. From the figure:

RP2

P

sin (a+ +y+fil f

(c-la)

ac - Z1

f

~ sinfif

sin

v cos (a+p+y+ #)

(C-lc)

(C-id)

(C.le)

SubstitutingC-1b throughC-le in C-la aad solvingfor V:

x’ Rz Sin$

(a+ +y+ $J

P [’U(s a+ +y+ p) P

+ x’ Cos B

(a+ +y+

(C-2)

From Case B:

AddingC-2 and

x q sin # Cos (gby)

Y=*8~g sin ( a +p +7~fi (C-3)

-1o-

-P

I

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CA6E D: Thg objectis displacedfrom its Initialposition, across

opticaxis,and up to the perpendicularMae of sight. ConsiderV?

X* of Figure2:

cQmbiningD-1 and D-2:

v~ .x” Rzsti@

f

w, from Case B:

Adding :

y+ v’= R:z Sill $

[

x Com p+

-~ “-% + ‘“

1

(D-1)

(D-2)

(D-3)

(D-4)

(D-5)

After crossingthe perpendicularline of sight,the conditionis identicaltQ tmt

!Ihefour

D=(y+ V)=

+

de16cribedby Case C.

caIBe6 may be generalized:

d

X IRz Sill @ COS (8+7) 1 + s~, Coa (a +~+y+ d) Csc e

ZG (a+p+y+fl f sin@ +x Cos ~+~ COB ( a+p +y+ $)

1

xl

Fsin (a+p+y+ f?) + x ‘ C084[a+~+y+

9’ J

where x and xt we the components

of the opticsad?, respectively.

The generalequationreduces

ing manner:

af displacenwnton the left and right

to the four specialcases in the followAPPROVED FOR PUBLIC RELEASE

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CaseB x! =0 and x = f “tan(p+y)

Case C x = f tan (fl+y)

Caee D x = f tan (~+y) and Xt = f cut (a +~+y+ ~)

the rigoroussolutionsmay, in practice<,be greatlys~lified. If

the opticaxis is very nearlyperpendicularw the line of displacement:

sin (a+/3+y+ r) :1

cos (a+~+y+ ~) & O

and the generalequationreducesto:

x Rz D~— sin # Cos (/3+y)

[

-— 1

f

+ X: sfLno 1

Dependingupon photographicfactorssuclLa6 fOcal lengthand object

distance,the simplifiedrelationattainsa degree of accuracywithinthe

limitsof camra resolution.

OtherspeciBl cases have

of obdectsin two dimensions.

arisenin connection with the displacmt

In

photographedvith the same csmera

on the same line of displacement.

displacementin termeof the same

several ixwtsacestwo obJectsham been

and, in general,the obJectsdo not lie

In relatingfilm displacementto spatial

knownparamters az

cases,the trigoneaetricequatlonebecomeexceedingly

the analyticprocedureoutlinedaboveaay be followed

the problem.

It is obviousthat s~try aboutthe ol~ticaxis

uzed in the foregoing

involved. Eowever,

for Variation of

and symmetryabout

a line of displacementperpendicularto the alpticaxis existand tlw derived equationsIBpplyin aU four “quadrants,”

Althougha discussionof obJectheightis in oppositionto the “two-

~nsional” hypothesis,the filtuenalysiaprovidesa checkon the

-12-

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~RB

supposedlyknown objectheight. Figure3 illustratesthe spatialrelation~

R

+

Rz l= f/g = Cos (fl+y) sin sine ‘

H = Rp h/g

Substituting:

H=h Rz sin g Cos (P+y~

f sine

(la,lb)

where:

h = measuredfilm hei@t of object

H . true l~pati~heightof object

III. !l!hree-l)imenoional Displacementof A Poin~

An extensionof the photogranmetricanalysisto threedimensionswata

necessitatedby c)xperimentson OperationIvy. I’hedisplacementof an ob-

~ect at an altitudehigh in comparisonto the locationof the nucleordetonation occursIn iWee dimensions,whereasthe recordeddisplacementis,

of course,in two dimensionson the film.

As in the two-dimensionalcase,the measuredand true displacements

uw to be rehtedlto readilydeterminedparamsterseFigures4 and 5 are

the elevationand plan views,respectively,applicableto the following

derivation.

ConsideringFigure4:

g/’f =

i

X+y (1)

(X+y) = ncos (~+e) (2)

or

i,- gn cos (e+@)

f

-13-

-m

(3)

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g, aandf are bwwnand (c+-m) ~y redeterminedM fol.xYIFs :

(c+8) = (v-8)

of tie verticalcameraaiming

The relstim betweenten

are separatedby (~+ y) MBY

tany

Relatingj to Z (Figure

j = “**f =

$ where ~ is known,but ~ is

Zngle, p , on the camera-ob~ect

p~-~~)wheretithetwo

be shownto be:

= ‘*F

4):

‘i?:(;+;’’::; +’l

the projection

verticalplanet

sub~ectplanes

(4)

(5)

(6)

zero-objectverticalangle,~ , on the c~ra-ob~ect verticalplanoe As

in the derivstiomof (4):

tanf

*7 “ —Cos ~

where

X= a- (p+y) +$ (~ F-e 5)

The angle 6 is knownfrom:

tam c - tan (6+0) - &/f (7)

The componentof motion,~, in the verticalcamera-burstplane is now

known. In a slmiJsrmanner,the componentof motion,m, in the horizontal

plane is determirnled (Figure5).

lc/f =

(r+s) =

L=

or:

L

G

(8)

o Cos (p+y:l (9)

KO COS (B+ 7’)_

.@ (lo)

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~m

RelatingL “tom:

(SL)

Anglesa and # are knownand Y may be determinedfrom:

twn y = tan (B+y) - K/f (12)

From the known components

gent, D, is determinedfrem:

D=

[

m2 +

Substitutingfor m and j (from

of displacement,j and m, the true displ.aceJ2

(5)

D ={[i%wl

1

1/2

ein2~ (13)

and (11)):

‘ +[::fl;+;::j’+o)] 2.,.27}1’2

,

(14)

SinceK and g are the horizontaland verticalcomponentsof displacement on the filmltheyare relatedto the measuredfilm displacement,d,

by:

K = d stip)

)

g=dcos~)

where~ is the ang,lebetweenthe displacementd and the verticalcomponentg.

SinceO = n cos 8 , (14)reduces,with substitutionof (15),to:

(15)

[[(~) 2

1/2

D.$

IIP

~~:+cosy Cos (~+y) + Cos cos~ 8in’l)cos

;* 1}

.8) 2

-Y

(16) sin (q+ ~ -e

where:

Y

= tan-l

[

tan (~+yf) - K/f’

1

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tat+ [tan(*-8) - ,/f]

“-’ [*]

‘n-’ [*I

a+fi - (gl+y)

d 1206 P

d sin ~

true spatialdisplace~nt

slant range, camera to initial position of obJect

measuredfilm displacement

lens focallength

anglebetweenfi~ displacementand verticalcomponentthereof

verticalangle,camerato initimlpositionof ob$ect

horizontalcameraaimingangle in relationto the initial

Poetitionof object

verticalcameraaimingangle

zero-objectverticalangle

horizontalcameraaimingangle

horizontalangle,zero-cameraand zero-ob~ect

In practiceequation(16)maybe greatlysimplifiedif long focal length

lensesare used and the camera16 at a largedistancefrom the object. Under

theseconditions,(16)may be shownto reducoto:

2

+

[~, )

Cos

1

sin ‘J_

B +7)

-16-

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IV. FireballPhc)togrammetry

-B

Photogramzetric analysis, as appliedto :eireballmeasurements,falls

into tvo main categories:surfaceburst,wheiielnthe true zeropoint is

knovn in advanceand camerasare aimedaccord:hgly,and

probabilityof the trueburstpointbeing off the optic

A. SurfaceBurst ——

Figure6 illustratesthe relation

and the apparentradius,

vatureis not taketiinto

ten

Cos

and

%

where:

%

‘a

R

f

dnta

such

a

a

as determinedfrom

account. Rom the

. @R - r@

= @a

= (mf/f) Cos (tan-l

botsreenthe

air bursts,with the

axis ●

true fireball

film measurements.Ikrth

f:lgure:

radius

cur-

(1)

(2)

= true fireballradius

= apparentfireball

= rangefrom camera

= lens focallength

radiuson film

to zero

For all practicalpurposes,this correctionneed be appliedonly to

from station6close to the fireball;in general,the camerais at

a larger-e that the correctionis of the orderof 0.05~,a negligiblepercentageat the presentstateof the techniquefor measuringthe

film data. Of course, for cameras used at long range,the surfaceat the

detonationpoint ,1snot seen and a correctionfor the curvatureof the

earthmust be appkledto the observedradius~

-17-

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A further sourceof discrepancy

radii is

ate such

be about

88 thoBe

zero

film

B.

a)

and

atmosphericrefraction~Ho

effects;hovever,the order

betweentrue

attemptwill

of zagnitude

and apparentfIreball

be made here to evaluhaB been observedto

5 feet displacementin Xl milesfor atmosphericconditions such

encounteredon OperationIvy.

Air Burst —.

The problemsassociatedvith an alz’burst are:

(a) l!hetrue locationof the burst,determinedby photographic

triangulation.

(b) !lhemagnificationfactorbetweenimageand ob~ectwhen the

olq)ect is not on the opticaxis.

(c) ~le true radius (or di-ter) versue the apparentradius

(or diameter).

(d) The distortionof the imagedue to the off-axisburst.

Triangulation

Figure 7 shows

the true: ground

measurements by:

two photostationsin relationto

zero● The angles~,~ and ~2 are

@Sllled ~ound

determinedfrom

Augles al and a 2 are knovn from the camera aiming; 71 and the

distancebetweenthe tuo stations,D, are knownfrom surveyedcoordinates.

$ . go-(a@/32+Y’1) (1)

N= (Yl + al +Bl) (2)

and

D sin X.sh $

(3)

(4)

-18-

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From which follow: =lmEa!nm

z = x cos (al+/31) (5)

v=xsti(al +/31) (6)

As a check, the secondstationprovides:

W=ysin(

U=yms(

Substituting(1) and (2)

(5) m (6)2 the full form of

(Y2 + &) (7)

a2+B2) (8)

in (3) and (4)and applying(3)and (4) to

the equationsexpressingthe coordinatedistancesof true groundzero from the two stationsare:

{

B

{ a.; ~~ ‘~~~~

Z = D cosc~sal+I

V=Dsti(al+@})cos(~z+62+ ~)

Cos (a2+&-alStation1

North

East

Station2

~ =Dsin (aZ+&) sin ( a!+ii+y,)

.0s( C22+p2 - a, -& r---

South

Havinglocated

may be determined.

x ifi

6 is

E ia

3

3

3

I

true groundzero,the heightof the burstabove this point

Referringto Figure8:

knownfrom the foregoinganalysis‘

the verticalcameraaimingangle (abovethe horizontal

referenceplane)

the changein elevationabove sea levelbetweentrue

groundzero and the camera

the earth curvaturein distanceX

the heightof the burst above the horizontalplane

the true heightof burst abovethe ground

MB

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face

[

-’ -A

‘=xl+ h/f tane 1

-p) = H/x (1)

(2)

H“=H+E+c (3)

Strictly speaking, the burstheightis the perpendicularto the earth~ssurbut} sincecurvatureis smallfor c~r~ rangesof the orderof several

miles, the clifferencebetweenthe true heightand that givenby Equation(3) is

negligible.

b) WgnificationFhctor

Figure9 showsplan and elevationviewsfor derivingthe magnification

factor. In the figure:

cameralocation

plnnnedburst point

actualburst point

groundrangeto planne~burst point

groundrangeto actualburstpoint

anglebetweenopticaxis and actualburst point,

in plan

vertical cemera aiming angle

The magnification factoris:

Roe

%3

R3

Rw/f where

~ +R3+R4

RI

T

Cos

A

Cos

-—

P

-zo-

(1)

(2)

-BP

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APPROVED FOR PUBLIC RELEASE
But :

x .&d!-

Cos

mp = !d&+%

Cos /3

R:L + A =R2cosa

Substituting(5) ,, (6), and (7) in (4) and simplifying:

%3 = R2 COS u Cos B + (HB - %) sin /3

(4)

(5)

(6)

(7)

(8)

c) Appment Radius vs. True Radius

Figurw 10 and 11 shov the verticaland horizontalprojections

for an off-axisair burst. The burst occursat a height A hv above the

plannedbursthefL@t$ a distance A E away from the plannedvertical

ob~ectplaneof fhe cameraand a distance A S to the right of the verticalplaneof the opticaxis. Theseerrorsin burst positionwill result

in a measured

nitudeof the

burst error.

film radius which is larger

imageerroris woportional

than the true radius. me magto themagnitudeof the off-axis

Furthemzore,an imageradiusmeasuredhorizontallydoes not equal

that insured vex’tically,againdue to the off-axischaracter of the burst

position.

Consideringthe verticalradii:

(1)

(a

-21-

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~Er

The fireballradius,as measuredon the film, is then:

thdv = dv2 - dvl

1

=f tan(ev -p). tan(Yv -m] (3)

But :

Substitu

A~=f

Cv =yv+& (4)

@ (4) m (3) and siIuFlifyIngt

(5)

tan pv[1 +tan%tanzyv+tanaylr + tanap]

[1 +tmlyvtsq [1 - tanY@an#b+tan&aaYv+t’a@an#v]

Solvingfor tan (#v:

(6) .-

A~ [l+ianfhnYv]2

tan~v=~

[1 + Wu@ tanayv + tanzyv + taml~’] -13dv[tan~ - * Yv] [1- yv -u?]

%

sin #v = G

where:

h + Ahv

Rsv” —

sin yv

Therefore:

& = sin-l

Since

a relation

tan f&and

the sine and tangentof

foundbetween A% and

sin fl,r:

I& Sln Yy

h+ hv Al

(7)

(8)

(9)

J

% - known,@v-y be ●laminatedand

Rfv. In the right triangledefinedby

(A A% DR@,)2 + (Bf - c A&)2 D2 R& = (A Q)2 (lo)

~Ib

-22-

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APPROVED FOR PUBLIC RELEASE
where,for algebraicsimplification:

A = (l+~~~Yv)2

B . (l+~2@~2Yv+~~n2Yv+ ~2~)

c = (tan@ -W Y.)(1 +tm yv tan/3 )

D= sin Yv

h + Ahv

Solving(10) for Rfv :

A Ah

% =

[ 1

D (A2 + c2)A~ - 2 ~f@, + B2 f2 1’2

(Xl)

Equation (11) relates the measured film fireball radius to the true

fireballradius. The fireballradiusin the c~bJectplane,as normlly determinedfrom the fIlm measurementis, however:

&v = M/f Ah (12)

where ~f is the magnificationfactorof sectionb. In tb notationused

in Figures10 and U:

M = (l~hcos ah + AE) cos ~ + (h + Ahv) sin# (13)

(U)

Relatingappnrentfireballradiusto true fireballradiusthrough

Ed (12):

(14)

Equation(14)may be simplifiedin practice: If, as is usuallythe

case, the

(i.e., M

camera is at a range

>> %,v),eqwtion

k

very largecomparedto the firebaU radius

(14)reducesto:

-23-

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~~

Moreover,if the burst occurson the o~bicaxis (i.e., Yv =/3),

the apparentradiusis equalto the true radius.

When applyingtie general equation(14) to a specific case it is

necessary to evd.uate M a8 a functionof $ :Lnorderto detemine whether

the simplifiedrelationmy be valid. In sozw instancestbs burstwill

be much closerto a givencamerastation,~“1 the data from that station

shouldbe correctedby the generalequationto givethe true radius$

whereas other camerastationsare far emwgb from the burst that tie

simplifiedeq=tion may be usede

Althoughthe equationshave been derivedfor a specific burst location,

synnnetryabout the opticaxis implies,~ mny readilybe shown,-t tie

equationsare validregardlessof the locationof the burstwi- respect

to the -d burst position.

The derivationfor the horizontalradiuoproceedsin the samefashion

as that for the verticalradius,resultingin:

b

where:

M = (Rsh cos ah + AE) cos~ + (h + Ahv) sin~

M’ M>> I&, equation(I-6) reducesto~

(16)

(17)

and, if the burst iS on-axis(ah - o)

Rfh=%h=Rfv”%f (M)

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d) Distortion Or Imge Dueto Geometry

uthough the general equations (14) and (16) iticate very

serious distorticms of the fireball image when the camera is close to

the burst$ the most frequently encountered comditions$ represented by

equations (15) and (17), also give evidence ar distortud fireball images.

A detailed studyof this distortionis felt tw be unnecessaryat this

tiw2 for sevaral reasons.

Lens distortion

-es; towardthe edge

ks been assumed negligible in the foregoing

of the film frame this distortionis probably

comparable to that produced by off-axis objects ● Moreover$ the current

mthod of measuring fireball radius consists of fitting a circular grid

to the fireball hage; since, at least in the early stages, the f Ireball

is not spherical, the smodhing afforded by the grid method probably oversbadowe the distortion. kical exmqples hme been workedwith the derived equations and, in general, shew a deviation of true radiusfram apparent radius of the orderof several tenths of

is greater than IMs error. Until such tim as

be reduced,the radiuscorrectionis felt to lbe

a per cent; scatter of data

the variation in data can

unnecessary fur most of

the film data; s~wcialoccasionsmay arise in which the full corrections

shouldbe applied.

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qmmmml!m

-.

csi -*

u.

ni

-26-

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-1

n

N

K& A-‘~ x

/ ./ \

-2i’-

91HKHBE

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n.

-1

5

w

I

— — — —

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+85

ii

6

(0

I

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-!!

1

I

1

1

1

1

-33-

IJNCMsSIFIED

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..q~o.

I

I

I

I

I

1

UNCIJVSFIEIJ

I

I

FIG. 9

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9*

I

I

I

-35-

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-t

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v I I I

1

/

/1 I

ww’$s\F\ED

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