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Distribution limited to DoD and DoDcontractors only; Specific Authority; 22Apr 87. Other requests must be referred toDirector, Defense Nuclear Agency,Washington, DC 20305-1000.
AUTHORITYDSWA ltr., 13 Apr 1998; Same
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~'~~TED ATA CONFIDENTIALATOMIC ENERGY ACT 1954 OTE F1 LE C06, DNA-5826F-SUP
p, DYNAMIC PRESSURE IMPULSE FOR NEAR-IDEAL ANDNON-IDEAL BLAST WAVES (U)Height of Burst Charts (U)Supplement to DNA 5826 (U)
C) E. I. Bryant DTICF. I. Allen ELECTEKamanTempo etP.O . Drawer QQ AUG 2 6~8Santa Barbara, CA 93102
31 December 1983
Technical Repoit
CONTRACT No. DNA 001-82-C-0287Distribution autorzed to the Department of Defenseand IIIIIDoD contractors only; Specific Authority (PublicLaw 79-565), 22 April 1967. Other requests shall be
referred to Director, Defense Nuclear Agency,Washington, DC 20305- 10101.
THIS WORK WAS SPONSORED BY THE DEFENSE NUCLEAR AGENCYUNDER RDT&E RMSS CODE B344082466 Y99QAXSGO0039 H2590D.
Prepared forDirector
DEFENSE NUCLEAR AGENCYWashington, DC 20305-1000 I
CLASSIFIED BY DD Form 254, 15 June 1982, ContractNo. DNA 001-82-C-0287, and CG-W-4, Rev 1, 'ioint DOE/ -
DoD) Nuclear Weapon Classification Guide" August 1982(SRD(N)).
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CONFIDENTIAL. FORMERLY RESTRICTED DATAZ& 56CUffY ""SWILAflOM AUT1011TY 3. 04MTM)~I AVA11AUUTY OF 11EPORT
D ForCA WTM 54 D15 u1Uact 4 Distributiongathorized to the Department ofNJ ince Formerly Restricted Data Defense and ~.. DoD contractors only;
ORGANZATIN 11111OR1T WNUMS) S. mofoluTOW onRGA11MZA11 REPOltrT MUMMA"()
KT-83-032 DNA-5826F-SUPFEL.do. MAWd OF PERFORMIN ORANZTION ft 0010C SYMOOG.0 1111 "AW Of MONT01111111 ORG"ANO
Kama Tempo~ DirectorKama Temo IDefense Nuclear Agency
ft *006155 KRYli SOO. OW~ WC0111111 Mb A00615 (ftO~j.s. &W wVc60P.O. Drawer QQSanta Barbara, CA 93102 Washington, DC 20305-1000
8L MAWE OP FUNOV@ISF164ONSORG103 SW. OFFIE SYNSOO. t PRO11CUMENT IMNUMNT 10EN11IPICAT1N NUMEER
ISPSS/Ullrich DNA 001-82-C-0287 I
OIL AD0615 XW SOW. 401111 WCW IQ SORC OF PUNwN NU copMUEM111
POGINAM o110m TAM SKK N
_______________________________ NO O NO. G lIO
DYNAMIC PRESSURE IMPULSE FOR NEAR-IDEAL AND NON-IDEAL BLAST WAVES (U)Height of Burst Charts (U) Supplement to.DNA-5826 (U)Bryent, E. J. and Allen, F. J.
Tehnca I E 'P "33 To 8 31130""" 14rAEO EOT(~~~D~ .P;AG CO9IL JPUMNTNYNOTATIO
This work was sponsored by the Defense Nuclear Agency under RDT&E RMSS Code8344082466 Y99QAXSGO0039 H25900.
I?. cosASI comR I& SUISCI TM~~~gs asw
PRw aou Ssjil~mup Nular Weapoins E fects yai esueoAi 'blast (nuclear) Heiqht of Burst
1 9 11Precursor (nuclear) Sachs Scaling O- LM*.55" C (ox go flswmiiiii # MIGNMP a" O*6 6MW
>-Blast wave data for all past field tests have been reviewed. Some previously unused gagedata of sufficiently good quality for determining dynamic pressure impulse have beenfound and the da-ta-rtduced. Dyn~amic presiture impulse has been inferreid from tank dis-placement data-(a-s was done previously using truck displacement data, Reference 1 Pusinqthe curve of dynamic pressure impulse versus displacement as a calibration curve (withthe tank playing the role of a gage). This allowed extension of dynamic pressure impulseresults to higher values than previously available and also provided data points atadditional scaled burst heights. Following this, Height of Burst Charts, i.e., iso-scaled dynamic pressure impulse contours in the scaled height of burst -- scaled groundrange plane, were constructed using thj totality of the useable dynamic pressure impulse 1data. The charts4' Figures 11 and 12, ire for two cases: Near-Ideal -- Lightly/ModerateDust and Near-Ideal -- Moderate/Heavy Dust. The former is the better determined of thetwo. It is believed that these charts are the most accurate obtainable from the totality
20. OISTR111IT1ON11I AWAILASUTY OF AETRACr I11. ATRACr MEUNITV CLASS11FICATION -.4110Wu~lASSWEJuMff40 13 SAM AS W'v C3 or Unclassified
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00 FORM 1473. &4MA 5 A R 0~Mylb n ryo ftswl. SICU111r CLASIWIATION Of iWIS PAGE
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27 . SECdRITY CLASSIFICATION AUTHORITY (Continued)
DNA 001.82-C-0287 and CG-W-4, Rev 1, "Joint DOE/DoD Nuclear Weapon Classification
Guide,4 August 1982 (SRD(N)).
3. DISTRIBUTION/AVAILABILITY OF REPORT (Continued)
Specific Authority (Public Law 79-585), 22 April 1987. Other requests shall bereferred to Director, Defense Nuclear Agency, Washington, DC 20305-1000.
18. SUBJECT TERMS (Continued)
Light Dust Blast WaveHeavy Dust Blast Wave
.19. ABSTRACT (Continued)
of all existent blast wave measurements. These charts supersede the Height of
Burst Charts in Reference 1.
(
41
SECUiTY CLASSIF&CTION OF THIS PAQE
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n u horiz e di sclo sure subject to admin-
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nistrative end criminal senctions. Handle asR e t i t d D t e in fo r e ig n dlis e m in t io n .
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Seton 1 .b, Atomic Enrgy Act 1954. C0 N FIDENTIA II.Reticel '° L'Mm ' n rg c l L CO N FIDENTIA L
,
Ts page 19 u.. ..fl.d;,Pami ore r r
CONFIDENTIAL
SUPPLEMENT TODYNAMIC PRESSURE IMPULSE INVESTIGATIONS
PREFACE(This Preface is Unclassified.)
We wish to thank ls. Jeanne Rosser for her careful attention to themany details of tabulation, curve plotting and computational checking involvedin the preparation of this report. We wish to thank lr. John Keefer, BRL, forproviding IRL unpublished pressure-time records for several nuclear events;these data proved quite useful.
Accesio°i For
NTIS CRA&IDTIC 7fABU;,anno, , ,ed El
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This page Is UNCLASSIFIED. Resicied Dala in foreign diaseminstio.Section 144.b, Atomic Energy Act,.1914,
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r4
UNCLASSIFIED
TABLE OF CONTENTS
(This Table of Contents is Unclassified)
Section Page
PREFACE . . . . . . . . . . . . . . . . . . . . . . . . . . ..
LIST OF ILLUSTRATIONS ................... 3
LIST OF TABLXS . . ....... . . . . .......... 4
GENERAL NOMENCLATURE . ..... .............. S
1 INTRODUCTION . . . . . . . . . . ............... 7
2 ANALYSIS OF BLAST MASUREMENTS ..... ................. 8
2.1 Measurements For Events Not Previously Considered . . . 82.2 Analysis of Tank Data ......... ........ 9
2.2.1 Dynamic Pressure Impulse Versus Displacement ForTanks ........ . . . . . . . . . ...... 10
2.3 Discussion Of and Minor Improvements Upon PreviouslyUsed Blast Data . . . . . . . . . . . . . . . ...... 12
2.4 Dynamic Pressure Imlse Versus Displacement For Self-Propelled Howitzers .................. 13
3 RESULTS AND CONCLUSIONS ... .................... 1S
4 RECOMMENDATION ... .................... . . . . 17
REFERENCES . . . ...................... 18
APPENDIX - The Least Squares Procedures ............... 19Appendix
A The Least Squares Procedures ... ..................... 19B Figures and Tables ...... .. ....................... ... 29
2
UNCLASSIFIED
:.y.i *
UNCLASSIFIED
LIST OF ILLUSTRATIONS(This Llst of Illustrations is Unclassified)
FIGURE NO. PAGE
I Dynamic Pressure Impulse vs. Displacement and Displace- 40ment vs. Dynamic Pressure Impulse for Tanks
2 Dynamic Pressure Impulse vs. Displacement and Displace- 41ment vs. Dynamic Pressure Impulse for Howitzers
3 Scaled Dynamic Pressure Impulse vs. Scaled Ground Range 42
4 Scaled Dynamic Pressure Impulse vs. Scaled Ground Range 43
S Scaled Dynamic Pressure Impulse vs. Scaled Ground Range 44
6 Scaled Dynamic Pressure Impulse vs. Scaled Ground Range 4S
7 Scaled Dynamic Pressure Impulse vs. Scaled Ground Range 46
7 Scaled Dynamic Pressure Impulse vs. Scaled Ground Range 47
9 Scaled Dynamic Pressure Impulse vs. Scaled Ground Range 48
10 Scaled Dynamic Pressure Impulse vs. Scaled Ground Range 49
11 Dynamic Pressure Impulse - Height of Burst Chart for soNear-Ideal -- Light/Moderate Dust (Scaled to 1 KT)
12 Dynamic Pressure Impulse - Height of Burst Chart for SlNear-Ideal -- Moderate/Heavy Dust (Scaled t: 1 KT)
3
UNCLASSIFIED
• ,.
UNCLASSIFIED
LIST OF TABLES(This List of Tables is Unclassified)
TABLE NO. PAGE
I General Information - Nuclear Tests 29
2 Dynamic Pressure IMPUlse Versus Range 30
3 Sumary of Displacement Data for Tanks and Howitzer.. 34
4 Least Squares Fit Results - Dynamic Pressure Impulse 37vs. Displacement and Displacement vs. Dynamic PressureINNIlS*
5 Least Squares Fit Results - Scaled Dynamic Pressure 38IMpulse vs. Scaled Ground Range
4
UNCLASSIFIED
UNCLASSIFIED
GENERAL NOMENCLATURE
I - dynamic pressure impulse
D * displacement
R ground range
x a scaled ground range
HOB m height of burst
Sd , Sit Sp, St a scaling factors
Po 1/3 1sd -(T4
T .(*Q + 273 1/2 14,7 2/314.7 1
00S 14.7"
To(C) * 273 1/2 Po 1/3 11/3st % 2-,33 '
P0 a ambient pressure (lbs/sq in)
T ambient temperature (degrees C)
V a weapon yield (KT)
aO, a 1 , a2, b, t, a, A, B a constants in least squares fits
x,y - independent and dependent variables in least squares fits
a standard deviation
a varianceai
N * number of data points
r a correlation coefficient
R • multiple linear correlation coefficient
UNCLASSIFIEDmt
' ' .Vm~a *.m~a~a*V~#r
UNCLASSIFIED
GENERAL NOMsENCLATURE (Continued)
1.,R coefficient of determination
S *sun of squares of deviations of data points from fitted curve
S' *sum of squares of fractional (relative) deviations o~f data points fromfitted curve
nubo 1/2 w standard deviation of(N4 - nubrof regression coefficieonts fractional error
St1/2I * -i-) x 100 '~root mean square percent error
SO - side on
FO a face an
NO a rear on
See Appendix, "The Least Squares Procedures", for definitions/defining equa-
tions of the following specific examples of the above general quantities:
02;r, R, r2, R 2; S~ I S S;~D St 1 So St~ ' C 51 Ct qp St x q n q S iq D;'nIq tSIq
Oct El.'a~r~ %c 2 Es~ ED at
6
UNCLASSIFIED
UNCLASSIFIED
SECTION I(This Section is Unclassified)
INTRODUCTION
The objective was to construct improved height of burst contours fordynamic pressure impulse. In particular, we wished to extend the set of con-tours previously generated (Reference 1) to higher values of dynamic pressureimpulse and to obtain data points along the contours at additional scaledburst heights, i.e., to better determine the contour shapes. To do this weanalyzed all of the data available. This includes dynamic pressure-time wave-forms for shots not used in Reference I (because no trucks were exposed on theseshots): TEAPOT Hornet-S, Post-ll, and Zucchini-14; PLIMBBOB Franklin-2, Wilson-4,Hood-6, Kepler-9, and Owens-tO. It also includes data for shots in which tanksand howitzers were exposed but no dynamic pressure measurements were made:GREENHOUSE Easy-2; TUMBLER-SNAPPER Fox-6 and How-?; UPSHOT-INOTHOLE Annie-l*, lpNancy-2, Badger-S, and Simon-?.
-• .-
I
There was one dynamic pressure measurement on this shot. ,
7.A
UNCLASSIFIED
b h k, r ..
UNCLASSIFIED
SECTION 2(This Section is Unclassified)
ANALYSIS OF BLAST MEASUREMENTS
2.1 Measurements For Evtv.tp Not Previously Considered.
We consider first the shots not previously considered for which somedynamic pressure measurements were available. Some of the waveforms were notvery good owing to noise (mixed inextricably with real physical irregularities),baseline movement, or gage record cutoff. However, many of the waveforms werefound useable, being a quality similar to, or not significantly inferior to,those used in Rr.ference 1. General information on all of the nuclear tests ofconcern in this report is given in Table 1. Dynamic pressure impulse, scaledand unscaled, is given in Table 2 for each Operation/Event, scaled height ofburst, and ground range for which we have obtained results. Also given inTable 2 is the manner in which we obtained the values of dynamic pressureimpulse.
We discuss the data for the shots in the order listed in Section 1.
TEAPOT, Hornet-5. As indicated in Table 1 we used BRL gage data.*There were two gage results at each of the three ground ranges. In two instan-ces the two results are in good agreement. At the closest-in range, 256 meters,the two gages differ by about a factor of two but one of the two appears to becorrect, the other badly in error. We used the average of the two results forthe 329 and 460 metre ranges, and the apparently correct results for the 256metre ground range.
TEAPOT, Post-ll. For this shot dynamic pressure waveforms are avail-able in Reference 2. We used a planimeter to obtain the area under each curveand then used the data reduction procedure described in detail in Reference 1.A detailed discussion of accuracy of this procedure is also given in Reference1. For this shot waveforms are given at four ground ranges in Reference 2.The waveforms are not good but are of sufficient quality to be useful. As
indicated in Table 2 we used the average of BRL's result and our result forthe two ground ranges at which BRL provided results. Our resulto and BRL'sresults agreed to within about 3% which is quite good; (the two results arebased on the same gage data but the data reduction procedures are indepeident;we have analyzed errors inherent in data reduction in Reference 1, and, inmany instances, errors can be quite large).
For the two ground ranges at which BRL did notreduce the data, we used the result of our own data reduction process.
TEAPOT, Zucchini-14. For this shot we averaged our result withthe result of BRL's data reduction process for the three listed ground ranges.
* BRL gage data used in this report are published In DNA-TR-85-161.Furnished by Mr. J. Keefer, BRL.
!K
8
UNCLASSIFIED
) %7=
UNCLASSIFIED
In each case the difference was between 3 and S%. Results are based on twogages at the 610 and 700 metro ground ranges and upon one gage at the 794metre range.
PLUMBBOB, Franklin-2. For this shot we used the BRL result for thesingle ground range for which there appeared to be a good measurement.
PLU BBOB, Wilson-4, For this shot BRL also provided values ofdynamic pressure impulse as listed in Table 2. (We integrated the waveformcurves as a check, however.)
PLUMBBOB, Hood-6. Of the 6 ranges at which there were gage resultsthe data appeared to be valid at 4 locations. There were two gages at each ofthese locations. We used the BRL results as listed in Table 2 (again after anintegration check).
PL14BBOB, Kepler-9. In this case waveforas obtained at three groundranges were quite poor. In one case there was a large disparity between twoBRL gage results (almost a factor of 4); in a second case there was almost afactor of 2 difference between our gage-reduced result and BRL's result. Atthe third ground range, 762 metres, the waveform was somewhat better. Thepercent difference between our result and BRL's result was rather large, about20%; in Table 2 we list the average of the two results. We discarded theresults at the other two ground ranges.
PLUMOBOB, Owens-lO. For this shot we again used the average of ourgage-reduced result and the BRL result at two of the three ground ranges, 305and 518 metres. At the 305 metre range the results are in reasonably goodagreement (within 9%); at the 518 metre range the two results differ by afactor of 1.4 but the waveforms are quite poor. At the 457 metre range ourvalue of dynamic pressure impulse was exactly equal to BRL's value of scaleddynamic pressure impulse; the BRL value appeared anomalous on a data plot,probably as a result of failure to scale the value obtained. In this case weused our gage-reduced result.
2.2 Analysis of Tank Data.
We next consider the set of shots listed in Section 1 on which tankswere exposed but there were no dynamic pressure measurements. Our rationale ellhere is based on a finding we have discussed in detail in Reference 1: the 1P
displacement which a vehicle exposed to a blast wave sufers can be used asa measure of the dynamic pressure impulse it receives. ,hat is, the vehiclecan serve as a gage for dynamic pressure impulse if we are able to "calibrate"this "gage". The calibration is the curve of dynamic pressure impulse versusdisplacement. (We are ignoring low yield devices where diffraction effectsplay a role.) We have analyzed the procedure and results for 1/4 ton and 2ton trucks in considerable detail in Reference 1.
We would not be able to use the procedure for tanks if we had no dynamicpressure impulse data for the ground ranges at which tanks were exposed. Thatis, we would have no way of obtaining the necessary calibration curve. For-tunately, we have the necessary dynamic pressure impulse data. It is supplied
%9
UNCLASSIFIED
%v %% %/ I?%W, 1
- -- tqlJWI1~W1II~u-u~I~lPW IIWUWVW IYWMIMY WI( V'" 1W V WM%. 11W VIM VW 11W MW UrW 1W
UNCLASSIFIEDby information from TEAPOT, Met-12 and Apple 11-13 and from UPSHOT-KNOTHOLE,Annie-i and Grable-lO. The data for the pair of UPSHOT-MOTHOLE events aregage data supplied by BRL; for Grable-lO we also have results based on bothgage and (truck) displacement from Reference 1, but at larger values of groundrange than those at which tanks were exposed. The BRL data and the Reference 1data are very compatible, i.e., very well represented by a single curve ofscaled dynamic pressure impulse versus scaled ground range (as shown in Figure7, to be discussed later). The results are listed in Table 2.
For Met-12 we have the results from Reference 1, listed in Table 2, basedon both gage and (truck) displacement data. (There is a considerably largerbody of truck displacement data than of tank displacement data so that thedynamic pressure impulse-displacement curve, i.e., the calibration curve, isbetter determined for trucks.)
For Apple 11-13 the results listed in Table 2 are taken only partly fromReference 1. In this instance, values of dynamic pressure impulse at the 518and 625 metre ranges listed in Reference 1 (based only upon gage data) havebeen averaged with BRL gage results, i.e., the two results given equal weight.At the 625 metre range this.makes a negligible difference while at the S18metre range the value of dynamic pressure impulse listed in Reference 1 differsfram that of Table 2 by about 16%. Finally at the 808 metre range the resultlisted in Table 2 differs from that listed in Reference 1; the latter is inerror as a check of our (previous) data reduction reveals.
2.2.1 Dynamic Pressure Impulse Versus Displacement For Tanks.
We plotted dynamic pressure impulse versus displacement for thetanks exposed on events Met-12, Apple 11-13, Annie-1 and Grable-10. Thisprovides a total of 14 data points* listed in Table 3 along with the tankdisplacement data for other shots.
There are several variables which cause deviations of the datarelative to a smooth curve through the points: (1) vehicle orientation withrespect to the bomb -- 4 data points correspond to side-on orientation, 7 toface-on orientation, 1 to rear-on orientation, 1 to face-on 45 orientation,and 1 to face-on 3/4 left orientation; (2) there are three different tanks --M4A3, M24, and M48; (3) there are two surface conditions -- rough sand andfine sand. There obviously are not sufficient data to sort out the effectsof the variables. However, when we plotted the data we found that: (1)(initial) orientation appears to make little difference, any effect beingsubmerged in effects of the other variables; (we intuitively expect initial T.,;1orientation to make less difference for tanks than for trucks); (2) anysystematic deviation which could be attributed to type of vehicle could aswell be attributed to inaccuracy in dynamic pressure impulse -- for example, . L
the Met-12 points (points B, 6, and 17 in Table 3) are a little high and are
- I-We havenot used Point 8 of Table 3, a Smoky-IS point. Placing this pointon Figure 1 shows that there is clearly something wrong with it -- and whatis wrong with it involves the displacement, not the dynamic pressure impulse, ',
even though the latter involves extrapolation on Figure 6.
10
UNCLASSIFIED
L , €- I
,., ,,'..'...... , .. .... ...,:...
UNCLASSIFIED
M48 points while the Apple 11-13 points are a little low and pertain to bothM48's and 24's. (In fact, if we use an average dynamic pressure impulseversus scaled ground range curve for the two shots, Met-12 and Apple 11-13,these deviations disappear.)
Plots of the data and the fits obtained are shown in Figure i.*
We used these data to infer dynamic pressure impulse from themeasured displacements on events GREENHOUSE Easy-2; UPSHOT-KNOTHOLE Nancy-2,Badger-5, and Simon-7; TUMBLER-SNAPPER Fox-6 and How-7.
We have noted in Reference I that, despite rather wide fluctuationsin displacement owing to the several variable factors mentioned, the centralcurve following the trend of the data points provides a rather good calibrationwhich enables us to use a measured displacement to determine the dynamic pres-sure impulse to which the vehicle was subjected. Since displacements vary owingto uncontrolled factors, the inferred dynamic pressure impulses will exhibit fconsiderable dispersion. (However, in many cases described in Reference 1, thedisplacement-inferred values are about as reliable as the gage-inferred values,especially when there are two or more values which can be averaged.)
The calibration curve for tanks is less well determined than thatfor trucks (Reference 1), there being fewer data points. Also the displacementsare smaller resulting in greater errors especially from small yield. Nonetheless,when the displacement-inferred dynamic pressure impulse values are plotted versusscaled ground range (along with a few deta peints for which there also are gagedata), we see that the curve obtained is reasonably well determined. See Figure3.
We used an eye-drawn curve rather than a proportional fit in Figure 1in inferring dynamic pressure impulse from measured displacements for the follow-ing reasons: J
(1) point 20, very small displacement, is much less importantthan the other data points in application of this fit, i.e., inference ofdynamic pressure impulse from measured displacement;
(2) point 21 has virtually the same displacement as the cen-troid so that it can be ignored in drawing a straight line through the datapoints; [the line must pass through the centroid:
r-IT= 2.448, r-T- = 3.768; "
q
on the abscissa and ordinate scales from Figure 1 we see that both fits dopass through the point D(nD) - 11.56, Iq(Zn lq) = 43.29; the lines do not
q
pass through the point = 21.85, I-= 47.95];
q
* Because of the mentioned variables whose effects we cannot disentangle owingto the small amount of data, we have numbered the points on Figure 1. Cor-responding numbers are listed in Table 3. By comparison one can verifythat the data do not exhibit any marked effects owing to differences amongthe above variables.
UNCLASSIFIED
-. ".. " .
%.- - ~ .
UNCLASSIFIED(3) the Grable-lO points 21 and 22 are badly inconsistent with
one another and points 9 and 10 are somewhat inconsistent. Thus our eye-diawncurve was a little higher than Fit 1 of Figure 1 at the small displacement endand, like Fit 1, passed through the centroid of the data points. The differ-ence, however, is nowhere greater than a few percent. We also note fromFigure 1 that a curve through the data points would do no better than astraight line and is therefore not warranted. It is quite possible, however,that if the data covered a wider range a straight line would no longer beadequate.
2.3 Discussion Of and Minor Improvements Upon Previously Used Blast Data.
Most of the dynamic pressure impulse data listed in Table 2 which wehave not yet discussed is taken from Reference 1. In a few instances a valuetaken from Reference 1 (gage only - no vehicle displacement data available)was averaged with a BRL gage-reduced result as shown in the table. In oneinstance, Turk-4, ground range 59S metres, we reduced the gage data eventhough the gage had cut off near the end. We estimated the shape and time ofpulse completion from other Turk-4 data and believe that the inaccuracy sointroduced is small - not more than a few percent. Reference to Figure 3shows why this is desirable: Turk-4 has the lowest scaled burst height forwhich we have gage data (except for a single Annie-1 point and except forsurface burst data); its scaled burst height is only moderately greater thanthose of other events shown on Figure 3; the Turk-4 data are consistent withthe data for the other events shown on Figure 3 and extend to a much greatervalue of scaled ground range. Thus the curve fit of Figure 3 is not diminishedin reliability by inclusion of Turk-4 data which, however, enable its use overa much greater range in constructing height of burst charts, the object of thisreport.
For events Yuma-4 and Wasp Prime-9 we used our values from Reference 1,therein described as "first iteration" results; i.e., when available, truckdisplacement-inferred dynamic pressure impulse values were averaged with gagevalues in obtaining our best estimates of dynamic pressure impulse. (Themanner of averaging, justification, and discussion of accuracy and reliabilityare considered in detail in Reference 1.) We originally considered Yuma-4 tobe an ideal event, but we now believe the values achieved by averaging displace-ment-inferred dynamic pressure impulse values with the gage values represent animprovement over the gage values alone; the results achieved are certainly morecompatible with the other data in the same scaled height of burst region. SeeFigure 9 (and compare with gage only data listed in Reference 1, page 11S).
For event Encore-9, a near-ideal event, we use the gage results -- eventhough in Reference 1 we sought to improve upon these results by use ofdisplacement-inferred dynamic pressure impulse. The reasoning used in Reference1 was that since the data scatter on a displacement versus dynamic pressureimpulse plot was just as great for ideal/near-ideal shots as for non-ideal andsince the displacement-inferred dynamic pressure impulse for non-ideal shotsis, on the average, as accurate and reliable as the gage values -- thenaveraging of displacement-inferred with gage-inferred dynamic pressure impulseshould improve the values for near-ideal events just as it does for non-ideal.The points at issue are discussed in detail in Reference 1. One point, however,
UNCLASSIFIED
UNCLASSIFIEDis that there is a much greater variety of ideal/near-ideal shots than ofnon-ideal, especially regarding the range of weapon yields in the data base.'When good gage data are available, these data should produce better resultsthan can be obtained by averaging in displacement-inferred values -- owing tothe effects of the several uncontrolled variablez previously described. Thatis, while, on the average, displacement-inferred values are reliable, consider-able dispersion is to be expected. Compare, for example, Figure 3; here withthe exception of the Turk-4 data points and a single Annie-1 point, all of theplotted points are based entirely upon displacement-inferred values of dynamicpressure impulse. While the central curve following the general trend of thedata is reasonably well determined, the data point deviations are rather large N.-- considerably larger than the deviations in Figures 4-10.
Near-ideal shots also have (small) real differences relative to oneanother. This is shown, for example, by"Figure 3.13 of Reference 1 which isa plot of scaled dynamic pressure impulse versus scaled ground range. Datafor the near-ideal (surface burst) shots do not completely coalesce under thescaling -- there are clearly small but real differences owing to variablefactors not controlled in the experiments. In Figures 3-10 the data for theevents plotted on each figure seem to coalesce quite well with respect to asingle curve. In Figure 7 some deviation can be seen. In general, deviationstend to appear when data for several events are shown on a single plot, whenthere are several data points for each event, and where the data for severalevents overlap, i.e., cover the same domain of the abscissa, scaled groundrange. The data in Figures 3-10 per:ain to a much smaller range of weaponyields than do the surface burst dati of Reference 1 (Figures 3.13 and 3.15). %In general, when the data for sever; & shots coalesce we can attach a highdegree of reliability to the data.
.J4i
In constructing height of burst charts in this report we used the resultsdescribed and listed in Table 3. For the surface burst data we used Figure3.13 of Reference 1.
Finally we analyzed field data for many more shots, but found the dataunusable. In some instances the waveforms were very bad; in other instancesthe gages were placed at elevations other than 3 feet above the ground. Theeffect of gage elevation is non-negligible.- (Different types of gages alsoexhibit somewhat different responses which must be accounted for in order toachieve consistent results.)
NOTE: The BRL data used in Table 3 have not been published.
2.4 Dynamic Pressure Impulse Versus Displacement for Self-Propelled Howitzers.
Figure 2 is a plot of displacement - dynamic pressure impulse data for
For a single (extrapolated) point on the charts for which the scaled dynamicpressure impulse is IS kPa-sec and the sealed height of burst "is zero, wealso used Figure 3.15 of Reference 1 and took account of the differencebetween Figures 3.13 and 3.15 in this region. Figure 3.15 extends to slight-ly higher values of dynamic pressure impulse than does Figure 3.13 so thatless extrapolation is required. -:
'3
UNCLASSIFIED
,... y . .- ,; .. ... . .. . . ,.:.
UNCLASSIFIED
self-propelled howitzers. Here we used values of dynamic pressure impulsealready described. There are only eight data points. These involve 2 vehicles,3 orientations, 2 soil types and 7 nuclear events. Three of the data points(2 Badger-S and 1 Simon-7) result from shots for which there were no dynamicpressure measurements. Therefore, the values we used, taken from Figure 3 andfrom the scaled height of burst chart, Figure 11, are actually based upon the Ndisplacement-inferred dynamic pressure impulse values for tanks.
We did not attempt to use the howitzer displacement data to infer values
of dynamic pressure impulse. To do so, we should proceed in the same manner
as for the tank data. Referring to Figure 2, we see that of the eight datapoints only five (points 31-35) could be used. No curve of any reasonabledegree of reliability can be drawn based on these points.
W.4
W .
14~
4...-,
UNCLASSIFIED .,,:.,
UNCLASSIFIED
SECTION 3(This Section is Unclassified)
RESULTS AND CONCLUSIONS
The results described are given in Tables 2 and 3. Least squares fitsare given in Tables 4 and 5. A description of the methods used in obtainingthese fits and the definitions and explanations of the several measures ofgoodness of fit listed are given in the Appendix. The simplest measure ofgoodness of fit to understand is the root mLan square percent error (ED andElq ). From Table 4 we see that for the displacement - dynamic pressure impulse
fits, the errors are quite large.* The error in estimating dynamic pres-sure impulse from given displacement is fairly large (35-40%) but tolerableand as we have seen the values obtained, on the average, are reasonable and of
considerable value especially when gage data are poor or lacking. If we were,
however, to attempt to infer displacement from a given dynamic pressure impulse
the errors would be much larger (150%) -- though again the results would repre-
sent reasonable approximations to the average displacement to be expected. (The
reason for the wide dispar A ty between the errors for the two types of inference
is seen to be simply due to the slope of the fitted lines on the log-log plots,
Figures I and 2, or equivalently from the exponents in the fits listed on the
Figures and in Table 4.)
The percent errors, E1 , shown, in Table 5 for the dynamic pressure impulse
versus scaled ground range q fits are quite modest, varying from about 4 to 25%.
These errors are quite comparable to those obtained in Reference 1 for the var-
ious fits therein.
The results described are shown along with the fits, in Figures I - 10.
These results were then used, along with the surface burst data in Reference 1
(Figure 3.13 and in one instance Figure 3.15) to construct the desired scaled
dynamic pressure impulse contours as shown in Figures 11 and 12. Also shown
is the locus of points separating the regular and Mach reflection regions.
We have drawn the contours to conform as accurately as possible to the
plotted points while maintaining smoothness and a continuous variation in
contour shape as we proceed from low to high values of scaled dynamic pressure
impulse. a
Figure 11 is the better and more reliable of the two charts. The curves
fit the data points much better than for Figure 12. In the latter case we have
not attempted to force the curves through the data points as this would lead to %
structure in the contour shapes which is not justified by the data.
There are various degrees of dust**, the Met-12 data corresponding to the
heaviest dust case while the Apple 11-13 and Bee-6 data represent more moderate
* The errors here are much larger than in Reference I because there are much Nmore data for trucks '(of concern in Reference 1) than for the tanks and
howitzers of concern here. ,*%.** We now believe that tle degree of "dustiness" is an indicator of the severity
of the precursor which is related in turn to the temperature of the pre-
shock thermal layer.
15
UNCLASSIFIED
%a.. ,
UNCLASSIFIED
dust. This is the principal cause of the difficulty in Figure 12. However, wedo not believe that separate charts for moderate dust and for heavy dust arewarranted. This distinction is simply too fine considering the quality of thedata.
In conforming the contours to the data points in as reasonable a manneras possible, we note that the two charts differ for the higher scaled burstheights, i e., the comparable contours do not coincide, even though the samedata are used for the two highest scaled burst heights on Figures 11 and 12.We cannot regard the contours in this region as well determined; our preferencein this region is for Figure 11, because of the greater smoothness of the con-tours and because we believe the data at the third highest scaled burst heightare better for Figure 11 than for Figure 12.
Note: In applications of Figures 11 and 12 the scaling factor S (seeGeneral Nomenclature) should be used rather than W-1/3 for scaling grgund rangeand height of burst; Si should be used in conjunction with contours rather thanW-1 /3 for instances in which ambient pressure and temperature are specified anddiffer significantly from the standard values of 14.7 psi (101.4 kPa) and 15"C,respectively.
e
16
UNCLASSIFIED-I.
I.'
. - .J. I
UNCLASSIFIED
SECTION 4(This Section is Unclassified)
RECOWIENDATION
It is recomended that the Height of Burst Chart, Figure 11, be acceptedas standard, i.e., as the most accurate obtainable from the totality ofexistent blast wave measurments. Figure 12 which includes Iq for moderate/heavy dust environments is for unique conditions. The description of theseconditions was discussed in Reference 1. Thus, Figure 12 should be used onlyfor si ilar conditions or be ignored.
17
UNCLASSIFIED-t%
UNCLASSIFIED
REFERENCES(This List of References is Unclassified)
1. E. Bryant and F. Allen, "Dynamic Pressure Impulse for Near-Ideal and Non-
Ideal Blast Waves - Height of Burst Charts (U)". XT-81-04(R), IS May
1981. Final Report, Part I, IS Feb 1980 to IS Apr 1981 on Contract No.
DNAOO1-80-C-OS6. (SECRET)
2. E. J. Bryant, N. H. Ethridge and J. H. Keefer, "Measurements of Air-Blast
Phenomena with Self-Reccruing Gages, Operation Teapot - Project 1.14b",
WTF-IlSS, 16 July 1959.
3. P. G. Guest, Numerical Methods of Curve Fitting, Cambridge University
Press, 1961.
4. Paul G. Hoel, Introduction to Mathematical Statistics, John Wiley and
Sons, Inc., New York, 1954.
S. Irvin Guttzan and S. S. Wilks, Introduction to Engineering Statistics,
John Wilty and Sons, Inc., New York, 1965.
6. Ya-lun Chou, Statistical Analysis, Holt, Rinehart and Winston, New York,
Second Editio--7
.. '.
18
UNCLASSIFIED
wil 4r, I
UNCLASSIFIE0APPENDIX A
(This Appendix is Unclassified)
THE LEAST SQUARES PROCEDURES
1. INTRODUCTION.
All of the least squares fits in this report were obtained with the TI-59calculator using library programs supplied by Texas Instruments.
All of the fits were either of the form
y - mx + b(1y-ax~b axor Y a a0 + a x1 + a x 2
or ya. 1X 1 2 2where y was the natural logarithm of the desired quantity; x was either the
independent (controlled) variable or its logarithm and the least squares
variables x1, x2 involved only the special case x I x and x2 M x2 .
The least squares equations are obtained by minimizing the sum of the
squares of the deviations of the data from the curve. If the experimental
points have a variable scatter for a given small range of x, weights should
be applied in the procedures so that the least squares equations correspond-
ing to Equation (1), for example, are obtained by setting the derivatives of
S a Mtiy I - (Mxi + b)] 2 (3)
with respect to a and b equal to zero, thus obtaining two linear algebraic
equations which can be solved for a and b. Here S is the weighted sum of the
squares of the residuals about the fitted curve andl • a a2 /a i 2
where a is a constant (to be determined from the deviations of the data from
the fitted curve) and ai 2 is a measure of the expected deviation from the ktrue value for an observation y1 (of unit weight).
In the Texas Instruments programs Wi 1 I. In individual cases to be
discussed we will point out that the progr=s used are nonetheless quite
adequate for our needs. This is mainly due to tho fact that, in all cases,
we minimized the sum of the squares of the logarithm of the desired function.
This is equivalent to minimizing the sum or the squares of the percentage
deviations from the curve since (tt ) - A . (4)
19
UNCLASSIFIED
v', '~ _
UNCLASSIFIED
That is, for small deviations from the eux., the fractional error isyequal to A(bt y). For large deviations this is n.ot exact and in our tabula-
tion of results we show both
E(tn Yi - bt y. r (- )
where y. 4 the value of y(xi) corresponding to the curve fit.*
The statistical analog of Equation (4) (Reference 3) is
Var (.n y) * Var y (Var a variance) . (4a)
Now in all of the cases the values of y covered a large range. If we con-
sider, for example, a displacement measurement of 100 metres to be in error
by 2%, the square of its deviation from the true value is 4; if we consider a
displacement Measurement of I metre to be in error by 20%, the square of its
deviation from the true value is 0.04. So even though the percent errors in
our data are not necessarily uniform (as best these errors are known), assum-
ing them to be uniform is much closer to reality than any other assumption we
can make. From the example just given we readily see that the coefficieots
in the least squares fits would be determined almost entirely by the data
with large values of y it we were to minimize absolute rather than percent
deviations from the fitted curves.
2. DYNAMIC PRESSURE IMPULSE VERSUS DISPLACEMENT (a)DISPLACEMENT VERSUS DYNAMIC PRESSURE IMPULSE (b)
In application of our results we need fits of the data for cases (a) and
(b), i.e., with each quantity used in the role of independent (controlled)
and of dependent variable.
In each subcase of this case the data are fitted very well by straight
lines on log-log plots. Thus for fitting on the TI-S9 calculator the fit is
of the form y amx b (1)
with y -n q, x - tD D for Case (a)
and y -Zn D, x - Zn lq for Case (b)
and b=Z.nB . (s) -
• In most cases we omit the subscript i denoting the ith data point; the
summations are taken over the data points in all cases so that omission
of the subscript i will not lead to any confusion.
20.
U NCL A IF! ED
* iq~%-,
UNCLASSIFIED
Equation (I) can then also be written [for Case (a)], Iq - BDm (ia)
with a similar relation for Case (ib). However, after we havo fitted Case (a)
we can calculate (b) directly (without fitting it) provided we recall certain
data summations from the machine memory. For Equation (1) we obtain from the
TI-59 fit: rx Z -x Evi ,slope a m - 2 (6)
Ex. (rx) /N
Yintercept " b" Y N' (7)
(2)1/2
and the correlation coefficient, r m (8)
Y rr 2 is called the coefficient of determination. The various summations can be
recalled from the machine memory. ax 2 and a y2 are the variances of the
x-irray and y-array data and are given by *x2 1 ~I
a2 12 ) (9)x
a y . r(y, 1 (10)
with oar t . (11)
and r' y (12)
N is the number of data points.
Equations (6) to (8) are not symmetrical in x and y and the algebraic
inverse of the least squares fit Iq- BDm (la)
I 1/m n/is D T but this is not a least
squares fit to the data with the roles of Iq and D reversed.
However, manipulation of Equations (6) to (10) leads to the equation
r1 2x ) 1 3 (1x'. 2 ]1,2 12
which is symmetrical in x and y, i.e., unchane when x and y are interchanged.
Bar over a quantity indicates an average value.
21,
UNCLASSIFIED
x. '% , %- 4
UNCLASSIFIED
Thus, if we have obtained the least squares fit Iq - BD M we know that
the correlation ratio r has the same value with the roles of Iq and D
reversed. It can be. shown (by taking the origin at the data centroid),
(References 3 and 4), that the least squares fits
Iq = BD m Ca)
D = Alq " (ib)
have values of Z and m which satisfy C m. (13)
Thus, having determined m (and r) in the least squares fit to case la
we obtain e in the least squares fit to Case (ib) from Equation (13). The
constant A in Equation lb is then determined by the fact that both least
squares fits (la) and (Ib) pass through the centroid of the data distribu-
tion. (Compare Reference S.) Thus, from Equation (Ib) we havr
t& D -ZJ A + f-AI q (lb)
and in particular "-aD - Zt A C + 7 - (14)
where from Equations (11) and (12)
Z (.eA D) (15)
and these values are available from the machine memory after running case
(1a). Thus from Equation (14) we determine In A, hence A and the least
squarei fit to case (lb) is fully determined. -"--
Having obtained the least squares fits, the coefficient of determination
provides a measure of the goodness of fit. By manipulation of Equation (3),
with W. a 1, and Equations (6) tlrough (12) it can be shown that
l1r 2 S (16)
Noy
In this report we are concerned only with positive correlation, so that the
nearer r is to +1, the smaller is S, the sum of the squares of the deviations
from the fitted curve, and hence the better the fit to the data. The expres-
sion for 1 - r2 can also be put in the form
22
UNCLASSIFIED
V €A. ,
.-
UNCLASSIFIED2
•, EC " J ( -
l- r r - (7
thus expressing I - 2 as a ratio of the sum of the squares of the deviations
from the curve to the sum of the squares of the deviations from the mean (of
the data). The denominator may be regarded as a normalizing factor. It is
independent of the functional form used for the fitting function. Without
such a denominator 1 - r2 would tend to increase with the number of data points
- even if the data were excellent.
Now letting S ., Iq and S n D be the sum of the squares of the deviations
from the fitted curve when Zt Iq and Zn D, in turn, play the role of dependent
variable, we have from Equation (16)
(1- r2 ) Sbt Iq SZn D (18)
N Na
Here Z Iq =q)' ,(...1Iq))(19)
2 [ (Z D) 2 _ "D) 2
7ZnD N
in accord with Equations (9) through (12). Since we have seen that r is
unchanged when the roles of D and Iq ore reversed, Equation (18) shows that
the sums of the squares of the deviations from the curves are not the same
for the two corresponding least squares fits (to the same data) but that
Zn I Zn I
Zn D a2 D InIDIn all cases treated in this report we found Sbt Iq to be considerably
less than S. a D"
In an application to be made of the results of this case we wish to
know the standard deviation of the value of the slope of the curve as well as
the standard deviation of our observation. We use the following notation
with the subscript c referring to the curve fit in each instance:
23
UNCLASSIFIED
.' .w " "' " ,° ,#'.V.r , , ', ''=.,r~ tw'='=" " ." / .v •"••
' •"%'• ,"" " "". . ""°"". . " . .' " " .".""' v ' '
UNCLASSIFIED
2A Iqs C 2 variance of an observation ofZn lq relative to the
curve; (note that we have assumed the variance of
the percent error to be independent of the abscissa
value)
2 m a variance of slopes of curve fits.
Then it can be shown that (Reference 6)
a2 S /N-2*& I qC = & Iq/
(21)2 s /N-2
't DC b D,
S-N n 1 N-2 1 -2 a 2
S - -.N N-
variablez Iv the Iihterrlsrvrew loseta
q q~2 N-2 a(22)
SA I N-2 I-r24nI
Na2 N-2 a2 4'in D 'inD
Using Equation(S) twice, first with D, I qbeing independent and dependentvariable an~d then with their roles reversed, we also see that
am m(3)m~
In the tabular results for the Displacement - Dynamic Pressure Impulse
data curve fits, we list for each subcase the quantities N, S a Iz q tAIq
S n D' aen D,C' t, m, at. am in addition to r and the fitting functions.
To gain further insight into the reliability of the data fits we have
also tabulated the following additional quantities defined as follows:
-A'
Here there are N-2 degrees of freedom; the two degrees of freedom lost
correspond to the number of regression coefficients (two).
24
UNCLASSIFIED
NF%
- %~ .
UNCLASSIFIED
s6 E(-DC
-( (24)
q 'qC
1/2
D
So 1/2 (25)
I ,C a-
and finally the root mean square percent error which is
ED (S.C/N)I/2 x 100%
(26)
E - (S I/N) 1 / 2 x 100%Iq Iq.
The a' quantities, Equation (25), are standard deviations of the fractional
error (percent error apart from a factor.of 100). Thus when the number of
data points N is fairly l-,ge (large enough so that the fit is fairly reli-
able) the values of qC are almost the same as the values of ED and
Eiq, respectively, (aside from the factor of 100 expressing the latter quan-
tities in percent).
The reason we have chosen to list these additional quantities, in this
and in other data fits to be discussed shortly, is that in many cases the
data scatter is rather large (irrespective of the functional form selected
for fitting the data). This means that a deviation 2n y in Zt y for a given
data point may differ considerably from Ay/y. To illustrate, suppose the INW
curve fit and data point values of a displacement are 10 metres and 6 metres.
The contribution of this point to the sum of the squares of the deviations
of Ln D is then (tn 10- t 6) 2 . 0.261 (irrespective of which value is thecurve fit value and which is the data point value).
The contribution to S6 is 2
(10-6).,160P
if the curve fit value is 10I and the data point value is 6; if, however, the
25UNCLASSIFIED
.~~N
UNCLASSIFIED
curve fit valut is 6 and the data point value is 10, the contribution to S
0 6 0.444.
Since the least squares curve tends to pass through the data region with
a fairly uniform distribution of points on either side, on the average the
curve fit will be below the data point about as often as above it. Averagingth e bov re ult yi ld s 0.160 * 0.444
the above results yields 0 a 0.302 which is to be compared with
the value 0.261 for the logarithmic deviation. Had we chosen the values 10
and 9.8 instead of 10 and 6, the two compared numbers would be virtually
identical.
It is inherent in the nature of the data we are dealing with in this
report that there is large data scatter while the number of data points is
quite small. We found (Reference 1), using the same least square procedures
described herein, that, when the scatter of the data is small, S' 1 fairly
close in value to S tn D and similarly Si is fair;r close to S Usually
S< S6 nd SD < S . The situatlon is sometimes reversed,%owever,bt0 q q
usually because one or two data points with large deviations from the curve
fit are below the curve. Conversely, for a case in which one or two data
points are far above the curvt it can happen that S >" S t since there is --
no limit to the contribution such a point can make to S', y here indicating
any quantity whose Lt has been fitted. (For a data point below the curve fit
the maximum contribution to S; is 2 2Y'Yc"C- c) - C-.) = 1 .)
The results shown in Table 4 are consistent with these statements taken from
Refererce 1. [The large effect which a single data point can have may be
seen by calculating D-D 2 "- for point 31 of Figure 2. The value, 18.52, is
Uc
almost as large as S' (a 19.06) for the 8 data points of Figure 2.)
Finally, quantities such as a' and Eq are directly associated with1q,C
the plotted data and are easily visualized. Since they pertain to fractional
or percent errors they apply equally well at all points along the curve fit
(although the curve fit has greater predictive accuracy in the vicinity of
the data centroid than toward the ends of the data range). On the other A
26
UNCLASSIFIED
,...- , ', w'', ..' ' V ' , ' ",,',;, ,.',; ;., ,'-., .' " .' . . " .'dr. '\ ,". % -. .. .' "i .''. " '"'. "" -''-.-''- '.."" U-
UNCLASSIFIED
hand aft q refers to the natural logarithm of the plotted data (ordinate)
and the latter may be quite small or large (or even negative), so that while
ae., t q is a standard measure of the data scatter with respect to the fitted
curve, it is not as simple to interpret as are the additional quantities
which we have tabulated.
In sum, the tabulated measures of goodness of fit provide some insight
into the assessment of the reliability and utility of the data and the
curve fits and help with the interpretation of results.
The quantities described here are also tabulated for various curve fitsof data other than displacement-dynamic pressure impulse data. We discuss
the least squares treatment of the remaining data in the following section.
3. ADDITIONAL LEAST SQUARES FITS.
Scaled Dynamic Pressure Impulse versus Scaled Ground Range:
The fitted curve is given byy a a0 a x + a x 2 (27)
where x - scaled ground range, y - b S I. %N.Sq*
In running this case on the TI-S9 calculator we feed in the natural
logarithm of each data point. Again, the machine generates a fit and all of '-'
the relevant summtions over the data are available from the machine memory.
For this type of fit (trivariate), however, the machine provides a quantity
R2 rather than r as a measure of the goodness of fit. R is the multiple
linear correlation coefficient between y and the other least squares variables
x1 a x and x2 = x ; R2 is called the coefficient of determination (see Ref-
erence 6) and is given by
r 2r2 2r r IR2 ayl r 2 2 r12 l r 2 (28)
- r .
The subscripts 1 and 2 refer to x and x2; ry1 r, and r12 are given by
equations analogous to Equation (8a).
(,)],2-1/2 ...
y- Y. " ( J( - - ( (29) %
27 *~..~
UNCLASSIFIED
..
UNCLASSIFIED
r , x 2 y - 3xN ] ( C r x 4 - 2 ]XN ( y 2 . c N 2--] 1 / 2
2-22 .1/2r~__ a Ex3 r.x)1)][X" :_1 . -N
(The superficial lack of symetry between r and ry, results from the fact
that the least squares variables x I and x2 as used here are x and x2 , respec--.,.
tively.) The quantities ryI, ry2, and r 12 are the coefficients of correlation(also called simple correlation or zero-order coefficients) between y and x1 ,
y and 2. and x1 and x2. respectively. The important point here is that
2( - R ) y (17a)
R2so that R provides a measure of the goodness of fit precisely similar to that
provided by r2 in the previous discussion: R2 is a measure of the closeness
of fit of the regression plane (in It y, x 1 , x2 space)* to the data points.
There are two further changes: replacement of N-2 by N-3 in Equations
(21) and (2S) and replacement of I by S q
it /N-3
SI S I qC ' n SjIq/
(21a)
tA o,c n Dnand
/N-3)1/2 (2Sa)
I'~ q c a (S~ Ii~q
Referring to the results listed in Table 5 we see that usually
S. iI itq < SiIq (and in fact this relation holds for all of the cases where
the number of data points > 11). The two values are usually quite close,
however, as might be txpected, since the data dispersion is fairly modest.
x , x • scaled ground range x2 a x.
28
UNCLASSIFIED ::.-,.'.
J1€e -Wrr.rV. •
, ,._,,.,,,_ ,... ,... , . ,, -......... ,.,.......,,..., .,.. ._,.-., . . ,. , , ,,. ,.. %,_.
CONFIDENTIAL!M4IX B
FIG~role.. AND TABLES
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29
CONFIDENTIAL
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