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A0 A119 4&96 NAVAL SURFACE WEAPONS CENTER DAHLGAEN VA F/6 4/1 METHODTO CORRELATE THE UPPER AIR DENSITY WITH SURFACE DENSIT--ETC(U) I JUL 82 L J MCANELLT. UNCLASSIFIED NSWC/TR-82-81 NL E1"EhhEEE ELE~hEEE i IEhh~~E
Transcript of apps.dtic.milUNCLASSIFIED SECUMITY CLASSIFICATION REPORT OF DOCUMENTATION THIS PAGE (When Data...
A0 A119 4&96 NAVAL SURFACE WEAPONS CENTER DAHLGAEN VA F/6 4/1METHOD TO CORRELATE THE UPPER AIR DENSITY WITH SURFACE DENSIT--ETC(U)I JUL 82 L J MCANELLT.
UNCLASSIFIED NSWC/TR-82-81 NL
E1"EhhEEEELE~hEEE
i IEhh~~E
.,.u *~-*1
UNCLASSIFIED
SECUMITY CLASSIFICATION OF THIS PAGE (When Data Entered)
REPORT DOCUMENTATION PACE READ INSTRUCTIONS
I. REPORT NUMBER 2. GOVT ACCESSION NO. 3. RECIPIENT'S CATALOG NUMBER
NSWC TR 82-81
4. TITLE (and Subtitle) S. TYPE OF REPORT & PERIOD COVERED
A METHOD TO CORRELATE THE UPPER AIR DENSITY FinalWITH SURFACE DENSITY AND ESTIMATE THEBALLISTIC DENSITY FOR AIR OR SURFACE LAUNCHED S. PERFORMING ORG. REPORT NUMBER
ISSILES7. AUTHOR(,) 6. CONTRACT OR GRANT NUMBER(e)
Loren J. McAnelly
9. PERFORMING ORGANIZATION NAME AND ADDRESS 10. PROGRAM ELEMENT. PROJECT. TASKAREA 6 WORK UNIT NUMBERS
Naval Surface Weapons Center (KI) SF 32-391-694Dahlgren, VA 22448 SF 32-399-694
11. CONTROLLING OFFICE NAME AND ADDRESS 12 R PORTDATE
Naval Surface Weapons Center
Dahlgren, VA 22448 13.ir MBER OF PAGES
14. MONITORING AGENCY NAME & ADDRESS(Il different from Controlfing Office) IS. SECURITY CLASS. (of thie report)
UNCLASSIFIED
150. DECLASSIFICATION/DOWNGRADINGSCHEDULE
16. DISTRIBUTION STATEMENT (of this Report)
Approved for public release; distribution unlimited.
17. DISTRIBUTION STATEMENT (of the abetract entered in Block 20, it different from Report)
18. SUPPLEMENTARY NOTES
19. KEY WORDS (Continue on reverse side If neceeery end Identify by block number)
Atmospheric densityBallistic densityExponential functions of altitudeLeast-squares techniqueNomograms
20. ABSTRACT (Continue on reverse aide if neceeeary and Identify by block number)
Atmospheric data collected twice daily from five ocean stations over aperiod of I years were used in a study to correlate ballistic densities at vari-ous levels. The st-idy indicates that a good correlation does exist and thattables, nomograms, or data in a suitable format should be produced for use withnew and existing fire control computers and range tables prepiared for theFleet.
DD , o, 1473 EDITION OF NOV is OSOETE UNCLASSIFIEDS 'N 0102-_ F-014.6601_ UNLASSIFED
SECURITY CLASSIFICATION OF THIS PA~t (men Date fntered)
UNCLASSIFIEDSECURITY CLASSIFICATION OF THIS PAGE (When Deem Entered)
20 ABSTRACT (Cont.)
Two exponential functions of altitude were used to express the density-onefrom surface to 36,000 ft, the other from 36,000 to 65,000 ft. The coefficientiwere selected in a manner that produces a continuous function from sea level to65,000 ft. The coefficients of the two functions were determined using a non-linear least-squares technique.--
\ Fluctuations in the density profiles below 5,000 ft altitude were noted andwere investigated by fitting a sample of temperature and pressure data with aleast-squares technique.
Ballistic densities for firing at surface targets were computed foraltitudes to 50,000 ft. While the current projectiles in the Fleet may notreach an altitude of 50,000 ft, this may become a reality in the near future.The procedure presented in this report could very easily be modified to computeballistic density when firing at air targets at altitudes up to 50,000 ft.
UNCLASSIFIEDSECURITY CLASSIFICATION OF THIS PAOE( men Dae anteroed.
r
TR 82-81
FOREWORD
The range tables for Navy guns include a nomogram for computing ballistic
density, which may be used when only the surface density is known. While these
nomograms are intended for use when better estimates of ballistic density are not
available, they are based on data prepared in 1922, despite the advent of radio-
sondes and the establishment of ocean weather stations.
Since there frequently are times when a radiosonde observation cannot be made
in the immediate operating area just before firing, or a timely and applicable
meteorological message from the Fleet Numerical Weather Center (FNWC) is not avail-
able, another method for obtaining ballistic densities would serve a useful
purpose.
This report documents a study that shows that ballistic densities can be
estimated from the surface density with a reasonable degree of accuracy. Data are
also presented about the upper atmosphere which may be useful when further studies
pertaining to the atmosphere are conducted.
The study was funded by the Naval Sea Systems Command Surface Weapons
Aerodynamics and Structures Research and Development Block Program.
Released by:
0. F. BRAXTON, Head
Strategic Systems Department
DTIc
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ACKNOWLEDGEMENT
The author would like to thank Dr. W. A. Kemper and Mr. Henry E. Castro for
their help in preparing this report. Dr. Kemper provided valuable assistance in
preparing the text of this report. Mr. Castro derived a nonlinear least-squares
technique which was used to fit the observed density data. The technique is
described in Appendix A.
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CONTENTS
Page
INTRODUCTION ................................................................. 1
DETAILED PROCEDURE ........................................................... 4
FITTING A SAMPLE SET OF DATA WITH AN ALTERNATE PROCEDURE ........................ 7
FITTING THE U.S. STANDARD ATMOSPHERE ......................................... 9
RESULTS ...................................................................... 13
DISCUSSION OF RESULTS ........................................................ 14
CONCLUSIONS .................................................................. 17
RECOMMENDATIONS .............................................................. 18
REFERENCES ................................................................... 19
APPENDIX A - A NONLINEAR LEAST-SQUARES TECHNIQUE .............................. A-i
APPENDIX B - AN ALTERNATE METHOD FOR COMPUTING DENSITY ........................ B-i
APPENDIX C - COEFFICIENTS OBTAINED BY FITTING OBSERVED DENSITY .............. C-i
APPENDIX D - DENSITY RATIO PROFILES FOR STATION E ............................. D-1
APPENDIX E - BALLISTIC DENSITY PROFILES FOR STATION .......................... E-I
APPENDIX F - SELECTED BALLISTIC DENSITIES FOR STATIONS E, V, D, N, AND C ..... F-I
DISTRIBUTION ................................................................... (1)
vii
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INTRODUCTION
One of the basic premises of fire control systems in use in the Fleet today is
a "Standard Atmosphere." Obviously, the same premise is also used in the prepara-
tion of ballistic tables for the Fleet. Since the conditions used as a basis for a
standard atmosphere are rarely encountered on earth, corrections for nonstandard
conditions are always in order. A procedure frequently used to correct for a non-
standard density involves a ballistic air density in the zolution of fire control
problems. The ballistic air density may be defined as a scale factor that, when
multiplied by the standard air density at each altitude, produces a density struc-
ture that would cause the same displacement of a projectile impact point as the
actual density structure.
The most reliable method of determining ballistic density is to use accurate,
representative, and timelj meteorological observations in the computation. There
are, however, occasions when these measurements are not available, and alternate
methods are necessary to provide the requisite information. It is a standard pro-
cedure to include a nomogram in the ballistic tables for the Fleet. The use of the
nomogram is an alternative that is available to the Fleet for estimating the upper
air density for naval gunfire when a ballistic meteorological forecast is not
available. The nomograms may be used to estimate the ballistic density when the
air temperature, pressure, and relative humidity at the surface are known.
The basic data used to prepare the nomograms were obtained from Table IV, The
Ballistic Density Factor,1 which was published in 1935. A footnote to this table
states, "This table has been extracted from Table C of a pamphlet entitled Method
for Determining Air Temperature and Ballistic Air Density, published by the Signal
Corps (Meteorological Service), U.S. Army in 1922."
Fire control systems have been improved to the point where an accurate deter-
mination of the upper air density is required if the potential of the systems is to
be fully utilized. The need for improving the nomograms or providing an alternate
prncedure to determine the ballistic density has long been recognized.
The work described in this report, stated briefly, consisted of (1) obtaining
Rawinsonde data from ;, rious weather stations, (2) sorting the data according to
LI
TR 82-81
surface densities and seasons, (3) fitting the upper air densities with a least-
squares technique, and (4) computing ballistic densities.
Rawinsonde data used in the study were obtained from five maritime stations
which took soundings daily at 0000 and 1200 hr G.m.t. over a period from 1968 to
1973. The data, consisting of both significant and mandatory levels, were provided
by the National Climatic Center, Asheville, North Carolina. Mandatory levels are
specified pressure altitudes, which are obtained by interpolation. Significant
levels are the altitudes that are considered essential in defining a weather pro-
file. Soundings that did not include surface density were not used in the study.
Some of the obvious errors were eliminated in processing the data prior to perform-
ing the least-squares fit, but no attempt was made to eliminate errors in the basic
data beyond ttai point.
The geodetic coordinates of the five stations used in the study are listed
below:
Station Name Latitude Longitude
E 350 0' N 48" 0' WV 340 0' N 1640 0' EC 520 45' N 350 30' WD 440 0' N 41" 0' WN 300 0' N 140" 0'W
A map of the stations is shown in Figure 1.
No land-based stations were used in the analysis, even though naval gunfire
support is conducted on beach areas. Since the density correlations are markedly
affected by the nature of the terrain,2 there is more than a probability that
somewhat different results would be obtained for different littoral areas of the
earth.
All computer programs used in the study were coded in FORTRAN EXTENDED and
were executed with a Control Data Corporation (CDC) 6700 computer.
The purpose of performing the work described in this report was to establish a
procedure to determine the correlation between upper air and surface densities and
2
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TR 82-81
to devise a method for estimating the ballistic air density based on surface
conditions.
DETAILED PROCEDURE
The first step in the procedure was to compute the air density using the tem-
perature, pressure, and relative humidity. The geopotential altitudes were con-
verted to geometric altitudes. Sondes that did not include surface conditions were
omitted. The data for each station were sorted into two groups according to the
time of day and stored in permanent files with the Indexed-Sequential File
Organization. This procedure permits accessing the data in either a sequential or
random order. The sondes were assigned indices which were stored in separate
sequential files. One file of indices was created with the corresponding surface
densities in ascending numerical order, and a second file was created with the sur-
face densities also sorted into the four seasons. A follow-up program was designed
to skip specified segments when the files were read. This made it convenient to
group the data in various ways without rearranging the Indexed-Sequential files.
The data were then selected in groups where the median surface density of each
group varied by a percentage point from the U.S. Standard surface density as
defined in Reference 3 (i.e., into classes by surface density ratios where the
class interval was one percent). Since large variations were encountered in the
frequency of the surface densities, it was necessary to eliminate some of the
sondes in the mid-ranges. Several fits were made using various numbers of samples
in each fit. No significant differences were obtained by using more than 25 sam-
ples; consequently, a further restriction was placed on each group that limited the
number of sondes to 25. While some of the groups outside the mid-ranges had varia-
tions in surface densities of one percent, the actual number of samples was, in
some cases, less than 25. In contrast, some of the groups of 25 samples in the
mid-ranges had variations in surface densities of only a fraction of one percent.
Various functions were investigated to represent a mean density profile for
each group of data, Comparisons were made of three different functions for rep-
resenting the density in the altitude range from 0 to 36,089 ft:
4
TR 82-81
1. a third-degree polynomial
2. a third-degree polynomial for the ratio of the observed density to the
U.S. Standard density
3. an exponential function plus a linear term
While these three functions produced similar results, the exponential function was
selected. The fitting technique is described in Appendix A. The following expres-
sion was used in the altitude range from 0 to 36,089 ft:
P ce(c 2h) +CP ficle 2 + ch (I)
wmere,
p is the density (lb/ft 3 )
h is the altitude (ft)
ci, c 2 , and c3 are constants determined by the fit
A similar expression was used to fit the data in the region from 36,089- to
65,000-ft altitude:
Cc4*(h - 36089))
P = P36089 e + c 5*(h - 36089) (2)
where,
P36089 is the density at 36,089 ft, computed using equation (1) and the
data up to 36,089 ft
c 4 and c5 are coefficients determined in the fit above 36,089 ft
A bias was noted in the lower altitudes with each of the three methods. After
some experimenting, data below 3,500 ft were excluded from the fit to determine the
coefficients. The fit was then extrapolated to mean sea level, and a linear cor-
rection was added from 0 to 3,500 ft, which made P equal to PO at the surface:
5
TR 82-81
(c h)
P= c1 e 2 + c3h + (P0 - c1 )(3500 - h)/3500 (3)
where,
P is the density (lb/ft3 )
ci, c2, and c3 are the coefficients obtained with equation (1)
h is the altitude (ft)
p0 is the median of the observed surface densities (lb/ft3 )
Although this procedure does not use any of the available data between surface
and 3,500 ft in the least-squares fit to determine the coefficients cl, c2, and
c residuals from the fit in this region are included in the analysis. Adding the
linear correction made a significant improvement in the residuals from surface to
3,500 ft.
The upper boundary of 36,089 ft was selected from the zone where equation (1)
was used in order to coincide with the breakpoint in the U.S. Standard atmosphere.
While there are some seasonal, latitudinal, and diurnal variations in the isopycnic
level where the density remains nearly constant, no attempt was made to explore
this facet.
After fitting the observed density data, using equations (1) Men (2), density
profiles were computed using equations (2) and (3) with the coefficients obtained
in the least-squares fit. Ballistic densities for each profile were computed with
the equation,
nBallistic Density = (WF. * R.) (4)
i= I 1
where,
n is the number of zones
WF. is the density weighting factor for the ith zone1
| " II • - . .. i _ " : rr .. . - • . ..
TI 82-81
R. is the ratio of the mean density of the ith zone of the profile to the U.S.IStandard density at that altitude, which is the mid-point of the ith zone.
The density weighting factors used to compute the ballistic density were
obtained by computing trajectories with ballistic parameters for the 5-Inch High
Fragmentation Projectile fired with a Mk 73 charge, which produces an initial
velocity of 2,950 ft/sec. The altitudes covered by each zone are presented in
Table 1, and a tabulation of the weighting factors are presented in Table 2.
TABLE 1. ALTITUDE OF ZONES
Zone Altitude (ft) Zone Altitude (ft) Zone Altitude (ft)
1 0 - 1,000 10 9,000 - 10,000 19 26,000 - 28,0002 1,000 - 2,000 11 10,000 - 12,000 20 28,000 - 30,0003 2,000 - 3,000 12 12,000 - 14,000 21 30,000 - 32,0004 3,000 - 4,000 13 14,000 - 16,000 22 32,000 - 34,000
5 4,000 - 5,000 14 16,000 - 18,000 23 34,000 - 36,0006 5,000 - 6,000 ]5 18,000 - 20,000 24 36,000 - 38,0007 6,000 - 7,000 16 20,000 - 22,000 25 38,000 - 40,0008 7,000 - 8,000 17 22,000 - 24,000 26 40,000 - 45,0009 8,000 - 9,000 18 24,000 - 26,000, 27 45,000 - 50,000
FITTING A SAMPLE SET OF DATA WITH AN ALTERNATE PROCEDURE
An alternate method was used to fit a sample set of data in the region from
surface to 36,000 ft. In this procedure, the virtual temperature was computed,
using the observed temperature and relative humidity. Separate least-squares fits
were then made to the virtual temperatures and pressures. The temperature and
pressure profiles obtained in this manner were then used to compute the density.
This procedure and the results are described in detail in Appendix B. The sample
set of data was obtained from 25 sondes that had been selected previously for fit-
ting with equation (1). The sondes were observed at Station E during the spring
season, 1200 hr G.m.t., on days that had a surface density close to the U.S.
Standard density at mean sea level.
7
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The purpose of using the alternate method was to evaluate the method described
in the section Procedure and to provide additional information about the atmo-
sphere, particularly in the lower altitudes.
While this method is a good procedure for providing temperature, pressure, and
density profiles, it is a more time-consuming method for obtaining density profiles
than the method of making a least-squares fit to the densities.
The density profile obtained with this method did not result in a noticeably
different profile, but it did provide additional information about the temperature
and pressure. The pressure data from surface to 36,000-ft altitude were fit with a
single function that has a continuous derivative. The temperature data were
divided into three altitude zones. Separate least-squares fits were made to each
zone in a manner that made the curve continuous but with a different lapse rate in
each zone. The improvement in using the correction term in equation (3) to fit the
density is largely due to the variations in the temperature lapse rate at the lower
altitudes.
FITTING THE U.S. STANDARD ATMOSPHERE
In order to further test the least-squares method and the exponential repre-
sentation of the density, a fit was made to the U.S. Standard density, using equa-
tions (1) and (2). Since the U.S. Standard temperature has a constant lapse rate
from surface to 36,089-ft altitude, the correction term in equation (3) was not
used. It should be noted that equations (1) and (2) were not intended to define
the U.S. Standard density but were selected to fit the observed data. However, the
standard deviations in the fit were only 0.14 and 0.28 percent in the zones from 0-
to 36,089-ft, and 36,089- to 65,000-ft altitude, respectively. Since the density
varies with altitude, the residuals were expressed as ratios of the U.S. Standard
density at the corresponding altitude. A tabulation of the fit is presented in
Tables 3 and 4.
9
TABLE 3. LEAST-SQUARES FIT OF U.S. STANDARD DENSITY0 - 36,000 ft
1 2 3 4U.S. Standard L. S. Fit Col. 2 - Cal. 3
Altitude Density Density Col. 3
(ft) (lb/ft3) (ib/3) _ _
0. 0.076474 0.07665845 -0.00240610500. 0.075362 0.07551388 -0.00201122
1,000. 0.074262 0.07438439 -0.001645331,500. 0.073174 0.07326977 -0.001307122,000. 0.072098 0.07216982 -0.00099517
2,500. 0.071035 0.07108433 -0.000693953,000. 0.069984 0.07001309 -0.000415533,500. 0.068945 0.06895591 -0.000158244,000. 0.067917 0.06791259 0.000064944,500. 0.066902 0.06688293 0.00028510
5,000. 0.065898 0.06586675 0.000474505,500. 0.064906 0.06486384 0.000649916,000. 0.063925 0.06387404 0.000797836,500. 0.062956 0.06289715 0.000935687,000. 0.061998 0.06193299 0.00104969
7,500. 0.061051 0.06098138 0.001141598,000. 0.060116 0.06004216 0.001229868,500. 0.059191 0.05911513 0.001283389,000. 0.058278 0.05820014 0.001337769,500. 0.057375 0.05729702 0.00136106
10,000. 0.056483 0.05640559 0.0013724510,500. 0.055602 0.05552569 0.0013743111,000. 0.054731 0.05465717 0.0013508511,500. 0.053871 0.05379985 0.0013224112,000. 0.053022 0.05295360 0.00129174
12,500. 0.052182 0.05211824 0.0012233613,000. 0.051353 0.05129363 0.0011574513,500. 0.050534 0.05047962 0.0010773414,000. 0.049725 0.04967605 0.0009853914,500. 0.048926 0.04888278 0.00088407
15,000. 0.048137 0.04809968 0.0007759815,500. 0.047358 0.04732658 0.0006638916,000. 0.046589 0.04656336 0.0005506716,500. 0.045829 0.04580987 0.0004175417,000. 0.045079 0.04506598 0.00028881
17,500. 0.044338 0.04433156 0.0001452618,000. 0.043606 0.04360647 -0.0000107218,500. 0.042884 0.04289057 -0.00015329
10
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TABLE 3. LEAST-SQUARES FIT OF U.S. STANDARD DENSITY
0 - 36,000 ft (Continued)
1 2 3 4U.S. Standard L. S. Fit Col. 2 - Col. 3
Altitude Density Density Col. 3
(ft) b/ft3) (lb/ft 3 )
19,000. 0.042171 0.04218375 -0.0003023219,500. 0.041468 0.04148588 -0.00043087
20,000. 0.040773 0.04079682 -0.0005837620,500. 0.040087 0.04011645 -0.0007341321,000. 0.039410 0.03944466 -0.0008786721,500. 0.038742 0.03878132 -0.0010138722,000. 0.038083 0.03812631 -0.00113606
22,500. 0.037432 0.03747953 -0.0012680323,000. 0.036790 0.03684084 -0.0013799523,500. 0.036156 0.03621014 -0.0014951624,000. 0.035531 0.03558732 -0.0015825124,500. 0.034914 0.03497226 -0.00166590
25,000. 0.034306 0.03436486 -0.0017127925,500. 0.033705 0.03376501 -0.0017772426,000. 0.033113 0.03317260 -0.0017966826,500. 0.032529 0.03258753 -0.0017961427,000. 0.031952 0.03200970 -0.00180254
27,500. 0.031384 0.03143900 -0.0017494228,000. 0.030823 0.03087534 -0.0016950828,500. 0.030270 0.03031861 -0.0016032329,000. 0.029725 0.02976872 -0.0014685829,500. 0.029187 0.02922557 -0.00131974
30,000. 0.028657 0.02868907 -0.0011178630,500. 0.028134 0.02815913 -0.0008922731,000. 0.027619 0.02763564 -0.0006022431,500. 0.027110 0.02711853 -0.0003146632,000. 0.026610 0.02660771 0.00008623
32,500. 0.026116 0.02610307 0.0004952433,000. 0.025629 0.02560455 0.0009550133,500. 0.025149 0.02511204 0.0014716534,000. 0.024676 0.02462548 0.0020516134,500. 0.024210 0.02414477 0.00270177
35,000. 0.023751 0.02366983 0.0034294036,000. 0.022853 0.02273694 0.00510442
Mean of Column 4 -0.00000178
Standard Deviation of Column 4 0.00140643
11
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TABLE 4. LEAST-SQUARES FIT OF U.S. STANDARD DENSITY36,200 - 65,000 ft
1 2 3 4U.S. Standard L. S. Fit Col. 2 - Col. 3
Altitude Density Density Col. 3
(ft) (lb/ft3) (lb/ft3)
36200. 0.022666 0.02253574 0.0057803237000. 0.021814 0.02169457 0.0055049038000. 0.020794 0.02068713 0.0051658239000. 0.019822 0.01972648 0.0048423340000. 0.018895 0.01881043 0.00449577
41000. 0.018012 0.01793693 0.0041854142000. 0.017170 0.01710398 0.0038596843000. 0.016367 0.01630972 0.0035119044000. 0.015602 0.01555234 0.0031928745000. 0.014873 0.01483014 0.00289033
46000. 0.014178 0.01414147 0.0025834447000. 0.013516 0.01348478 0.0023154148000. 0.012884 0.01285858 0.0019766449000. 0.012282 0.01226147 0.0016744750000. 0.011709 0.01169208 0.00144691
51000. 0.011162 0.01114914 0.0011536652000. 0.010641 0.01063141 0.0009024053000. 0.010144 0.01013772 0.0006197654000. 0.0096701 0.00966695 0.0003254555000. 0.0092186 0.00921805 0.00005944
56000. 0.0087882 0.00879000 -0.0002043757000. 0.0083780 0.00838182 -0.0004556158000. 0.0079870 0.00799260 -0.0007001859000. 0.0076142 0.00762145 -0.0009510560000. 0.0072589 0.00726754 -0.00118830
61000. 0.0069202 0.00693006 -0.0014226062000. 0.0065973 0.00660825 -0.0016575163000. 0.0062895 0.00630139 -0.0018871964000. 0.0059961 0.00600878 -0.0021103565000. 0.0057164 0.00572976 -0.00233130
Mean of Column 4 0.00003016
Standard Deviation of Coltmn 4 0.00284086
The following equations were used in the least-squares fit of the U.S.
Standard density:
12
Ti 82-81
0 h 36089
(c2 h)p =cle + c3 h
36,089 _ h 65,000
(c * (h - 36089)S.36089 e + c * (h - 36089)
where
h is the altitude (ft)
The coefficients obtained in the least-squares fit are:
c1 0.0766584476 (lb/ft3)
c 2 -0.282539693789 E-4 (ft - 1 )
c 3 -0.138466611596 E-6 (lb/ft4 )
P360 89 " 0.0226549960 (lb/ft3)
c 4 -0.475504250122 E-4 (ft- 1 )
c5 0.459663274698 E-11 (lb/ft 3)
RESULTS
The coefficients that were obtained by fitting the density data, and the root
mean squares (rms) or standard deviation of the residuals and their algebraic means
are given in Appendix C for the five stations analyzed. Data are presented for two
times of day (0000 and 1200 hr G.m.t.) for the four seasons and for the seasons
combined for each station.
Profiles of the density ratios for station E are given in Appendix D. The
ratios were obtained by evaluating the least-squares fit of the observed densities
at specified altitudes and dividing the density in each case by the U.S. Standard
13
TR 82-81
density for the sane altitude. Profiles are shown for the two times of day for
each of the four seasons and for the seasons combined.
Profiles of the ballistic densities obtained by multiplying the ratios with
the density weighting factors using equation (4) are shown in Appendix E for sta-
tion E for the same conditions. Density ratios and ballistic densities were com-
puted for each of the five stations, but the profiles are shown only for station E
in Appendixes D and E.
While the ballistic densities were computed for the 27 zones listed in
Table 1, only an abbreviated tabulation is given in Appendix F. Data are presented
for only two zones, 30,000 and 50,000 ft. The ballistic densities were computed
with equation (4), using the density weighting factors given in Table 2. Data for
the two zones are given for each of the five stations and for each surface condi-
tion where sufficient data were available to make a least-squares fit to the
observed data.
DISCUSSION OF RESULTS
One question is of primary interest-is there sufficient correlation between
surface density and that of the upper atmosphere to accurately predict the bal-
listic density for various altitudes from surface measurements? To answer this
question, we first note that the density ratio curves given in Appendix D for sta-
tion E are families of very similiar curves. If some of the curves were missing,
we could do a fair job of reproducing the curves by interpolation. This would seem
to indicate, qualitatively at least, that one could predict the density ratios
aloft from the starting point of the curve on the x-axis, which is the surface
density ratio. The same is true of the curves given for ballistic density in
Appendix E. It deserves mentioning that the points plotted in Appendix D are not
the mean values of individual sondes but are values corresponding to the functional
fit.
The sondes were groupe.1 in order that a more reliable mean and standard devia-
tion could be established. It is recognized that the curves in Appendix D do not
reflect the errors of the means or the variations in the upper atmosphere within a
14
TR 82-81
group of sondes. A total of 14,800 sondes were processed in this study in an
attempt to establish reliable means and standard deviations. In every fit, the
mean and standard deviation were computed. These are tabulated in Appendix C. A
large majority of the means are only a small fraction of one percent.
The standard deviations of the residuals are given in Table 5 according to percen-
tiles. A total of 300 fits was made in each of the two altitude zones, 0 to 36,000
and 36,000 to 65,000 ft. The standard deviations of the residuals were arranged in
numerical order without regard to station, season, or time of day. The standard
deviation of the 50th percentile is 1.114 percent in the lower altitude zone.
TABLE 5. STANDARD DEVIATIONS OF THE RESIDUALS
Altitude Altitude0 - 36,000 (ft) 36,000 - 65,000 (ft)
Percenttle (%) (%)
10 0.675 2.01920 0.799 2.29330 0.908 2.58640 1.000 2.83250 1.114 3.067
60 1.227 3.29570 1.424 3.58480 1.577 3.95990 1.703 4.298100 2.257 6.377
Note: The standard deviations in this table were obtained by multiplying thestandard deviations (SIQMA) in Appendix C by 100 in order to express them as apercentage instead of as a ratio.
It should also be remembered that there are variations in the surface densi-
ties because of grouping the data, which probably has a neglible effect on the means
but makes a small contribution to the standard deviations. The noise level in the
data appears to increase with altitude, which is to be expected. The residuals are
much larger in the fits above 36,000 ft than they are at the lower altitudes, while
the density should be more stable at the higher altitudes.
It may be noted that the ballistic density represents a weighted average of
the density ratios. Accordingly, any fluctuations that are random, and that may be
15 I
TR 82-81
present in the ratios of the true density to the mean density, would tend to aver-
age out in computing the ballistic density.
Another question that deserves consideration is-how important is geographic
location, season, and time of day? Only five stations used in the study are
located in wide ocean areas, far from land, and between 30@ and 50* N latitude.
Data were available for only two times of day, 0000 and 1200 hr G.m.t., which makes
it difficult to determine the diurnal variations. Based on the data available,
there does not seem to be much payoff for preparing separate tables for different
times of day and different seasons for the geographic areas that might be repre-
sented by the five stations.
There does appear to be a significant difference between the correlates of
stations E, N, and C, North latitudes 35, 50, and 52* 45', respectively. The
ballistic density at 30,000 ft, corresponding to the same surface density, is about
one-half percent higher at the lower latitude stations than at station C. The sur-
face densities run higher at station C. The upper air density may decrease more
rapidly than it does at the other stations, resulting in lower ballistic densities
for the same surface density.
Data by season do not appear to correlate differently, except at station C.
The largest difference noted here was for surface density 1.03, where the ballistic
density at 30,000 ft was about one percent lower in summer than in other seasons.
While the procedure presented in this report has been used on only five sta-
tions that have an oceanic climate, it should be possible to use essentially the
same procedure for processing data from stations where the climate is more affected
by land masses. Some modifications might be required, since the temperature lapse
rate might be quite different. The diurnal and seasonal variations in pressure and
temperature are normally much larger over land masses than they are over large
bodies of water.
A similar but earlier study 2 was made by the Army Electronics Command, Fort
Monmouth, New Jersey. Data from land-based stations were used in this study and a
number of tables were prepared for each geographic area, season, and time of day.
16
4
TR 82-81
CONCLUSIONS
1. There does appear to be a good correlation between the surface density and
the density at various altitudes over the mid-ocean areas studied.
2. The resulting correlations are different for different latitudes. They do
not change appreciably with diurnal time or season over the areas studied.
3. The differences between surface and upper atmospheric conditions are much
larger over littoral areas.
4. The procedure described in this report for determining the correlation
between upper air density and the surface density may be used for a variety of
stations with only minor modifications to the procedure.
5. While the procedure for estimating ballistic density was developed prima-
rily for firing Navy guns, it may also be useful to predict the ballistic den-
sity for air-launched missiles.
6. It is estimated that the density ratios at altitudes not to exceed
36,000 ft may be predicted with an error not to exceed 1.0 percent, using the
correlation between the surface density and the density of the upper atmo-
sphere.
7. It is estimated that the ballistic density may be predicted with an error
not to exceed 0.5 percent in 90 percent of the cases where the altitude does
not exceed 36,000 ft.
8. It is estimated that the ballistic density may be predicted with an error
between one and two percent where the altitude is between 36,000 and
50,000 ft. This error could be reduced somewhat if the data used in the study
had been collected with present day equipment.
17
• I
TR 82-81
RECOMMENDATIONS
1. Correlations should be obtained for other latitudes, including both mari-
time and beach areas of strategic importance.
2. More data should be obtained for the extremely low and high surface densi-
ties.
3. A further study should be conducted to determine how the procedure of
estimating the ballistic density, as described in this report, could be imple-
mented in fire control systems and in preparation of range tables.
4. Consideration should be given to using the analytic expression for density
in computer programs that simulate the trajectories of unguided missiles.
18
TR 82-81
REFERENCES
1. U.S. Naval Academy, Range and Ballistic Tables, 1935, p. 84.
2. Marvin J. Lowenthal, The Accuracy of Ballistic Density Departure Tables 1934-
1972, ECOM-5436, OSD 1366 (April 1972).
3. Committee on Extension to the Standard Atmosphere, U.S. Standard Atmosphere,
1976, (Washington, D.C.: Government Printing Office, 1976).
19
Ti 82-81
APPENDIX A
A NONLINEAR LEAST-SQUARES TECHNIQUE
A-I
TR 82-81
A NONLINEAR LEAST-SQUARES TECHNIQUE
Statement of the Problem
Let (xi, yi), i = 1,2, ..., N be a set of N data points and
f(x, al, a2 , ... , al4) (A-1)
be a function with x as the independent variable. (a.), j = 1,2, ... , M are
parameters of the function and the function may not be linear in these parameters.
We would like to determine values of the M parameters such that
i=N 2F(aI , a2 , .... aM) i Wi[Y i - f(xi, a a 2, ..., a (A-2)
is a minimum. This is the least-squares solution, which is the same as minimizing
the sum of the squares of the residuals. A weight W. is assigned to each data1
point.
For simplicity of notation, we let
fi(a) = f(xi, a,, a2 1 .... , aM), i = 1,2, ..., N (A-3)
where a = (al. a2, ... , aM). The only restriction on the function fi(a) is that
the partial derivatives of the function with respect to the M parameters be
continuous in all of the arguments.
This implies that f.(a) is a function that can be differentiated.1
We assume that these partial derivatives exist at a point 1, and in some
neighborhood about i, which includes the point a. Let
3= + Aa (A-4)
A-3*1.li E
Ti 82-81
where i (31, a 23 ' am) and Aa- (Aal, Aa2, ..., AaM).
If f.(a) can be differentiated at 1, then the differential at 3 is an1
approximation to f.(a) - f.(i) and f.(a) - f.-) O(Aa)(a-i).
Thus, j=M af.(i)
fiC(a) f f(a) I Aa. + E(Aa) (A-5)j. i
where E(Aa)-*O as Aa+O for i = 1,2, ... , N.
If the approximation i is close enough to the solution a, then the problem is
considered solved when
tj - ! I j T, j - 1,2, ... ', (A-6)
where T - desired precision for the solution.
Formulation of Solution
We now derive the formulation for finding Aa. The solution a is then equal to
1 + Aa.
Using equation (A-5), we form the overdetermined system of equations (N>M)
Y jinM afi(M)f+ I Da. aa. (A-7)
for data points i = 1,2, ... , N.
A--4
I
TR 82-81
The system of equations used to find a becomes
a f ( ) f( ) f ( )a-1 Aa + - -a + + - Aa = Y - fC) CA-8)
a1 1 a 2 ... 14 i
(i f 1,2, ... , N).
The system of observational equations (A-8) can now be written in matrix form
as given by
AX = Y (A-9)
where,
aa I 3a2 ... 3aM
f 2(a) f 2(a) af2(i)a aa2 .. a
f N (a) af N () af N()a 3a 2 ...
Aa1
A2
XA-
A-5
TR 82-81
Yl - fl (I
Y2 - f 2
YN- fN(a)
The column matrix Y is called the residual matrix. Solving matrix equation
(A-9) yields the normal equations in matrix form
AT Ax ATY (A-10)
If the residuals are to be weighted, we can include the weight matrix W
where,
'W1
W2
o20w .(A-Il)
WN
Completing the solution to the matrix equation (A-10) and upon including the
weight matrix, we have
X- (AT WA) - 1 ATWY (A-12)
X is the least-squares solution to the system of equations (A-8) with the
weights includ. Elements of the X matrix are now tested in absolute value
against the tolerance T. If the tolerance test is satisfied, we are done and the
least-square solution to equation (A-2) is given by
a. a. + ta, j - 1,2, ... , M (A-13)
A-6
Ti 82-81
Whenever the tolerance test fails, equation (A-13) gives the next approxima-
tion and the iteration process is repeated with matrix equation (A-12) until the
tolerance test is satisfied.
Remarks
Once an acceptable solution has been found, it may be desirable to know how
well the function fits the given data. Analysis for how well the solution fits can
usually be made by examining the residuals for each observation.
The variances for the solution (a) can be found from the variance, covariance
matrix.
If the variances and residuals are large, a different approximating function
f.(a) may be tested.1~
When polynomials are used for testing, the residuals will drop off rapidly as
the degree of the polynomial increases from one. Further increase in the degree of
the polynomial will show little improvement of the residuals.
A-7
TR 82-81
APPENDIX B
AN~ ALTERNATE METHOD FOR COMPUTING DENSITY
B-1
TR 82-81
AN ALTERNATE METHOD FOR COMPUTING DENSITY
Since the procedure used to fit the observed densities provided little infor-
mation as to why the correction term for altitudes below 3,500 ft improved the fit,
an investigation was made using a sample set of data obtained from 25 sondes. The
sondes were taken consecutively from the group that had been sorted according to
season after the surface densities were arranged in numerical order. The median of
the surface densities in the sample was equal to the U.S. Standard density at mean
sea level. The sondes were released at station E during the spring season,
1200 hr G.m.t.
Separate least-squares fits were made to the pressure and temperature in the
altitude range from 0 to 36,089 ft. No attempt was made to fit the temperature or
pressure above 36,089-ft altitude.
The equation used to fit the pressure is similar to equation (1)
( P2h)(-)P = p 1 e + p 3h (B-1)
where
P is the pressure (mb)
h is the altitude (ft)
Pit P21 and p3 are constants determined in the least-squares fit
The observed temperatures were first converted to virtual temperatures with
the equation
TV T (B-2)
1.0 -0.3783*RH*V P
where
TV is the virtual temperature (°K)
T is the observed temperature (K)
RH is the observed relative humidity (dimensionless)
B-3
TR 82-81
V is the saturation vapor pressure (mb)
P is the observed pressure (mb)
Several least-squares fits were made with the virtual temperature in the alti-
tude range from 0 to 36,089 ft using polynomials that, in every case, produced
large residuals below 5,000 ft. A big improvement was made by segmenting the data
into three zones, 0 to 3,500, 3,501 to 6,000, and 6,001 to 36,089 ft. A one-degree
polynomial least-squares fit was used in the two lower zones and a third-degree
polynomial was used in the upper zone. The intersections of the equations are
located at 3,731 and 5,270 ft which were used to define the boundaries of the
zones.
The density was computed with the equation
p - P/(R*T) (B-3)
where
p is the density (lb/ft3 )
P is the pressure determined by a least-squares fit (mb)
T is the temperature determined by a least-squares fit (OK)
R is a constant, 45.981610868
Computing the density in this manner made a small reduction in the mean of the
residuals but no improvement in the standard deviation. While a single function
fits the pressure from sea level to 36,089 ft, the nonlinearity in the temperature
profile is quite apparent. The lapse rate in the temperature based on the least-
squares fit from sea level to 3,731 ft is 2.60"C per 1,000 ft and 0.358C per
1,000 ft from 3,731 to 5,270 ft using the intersections of the equations to define
the zone boundaries. The average lapse rate based on the least-squares fit in the
zone from 5,270 to 36,089 ft is 1.99"C per 1,000 ft. The latter value is in good
agreement with the lapse rate of 1.98C per 1,000 ft from sea level to 36,089 ft
that was used to define the U.S. Standard atmosphere.
Profiles of the temperature and pressure obtained with the least-squares fit
are shown in Figures B-1 and B-2, respectively. A profile of the density computed
B-4
TR 82-81
0CD0 __________
(0
CD~
CDC0
C3.
LJ
0: C
I-C3 ___ __ _ _ _
C)0
C)
(0D:
!2
03C)_ __ _ __ _ __ _
C'J
C
C) _ ___ _ _ _
C0
03
C300 290 280 270 260 250 240 2310 220 21'0 200'
TEMPERATURE DEC K
FIGURE B-1. STATION F SPRING 1200 hr G.m.t.Temperature profile based on three least-squares fits, linear 0 -3,500, linear3,500 - 6,000, and third-degree polynomial 6,000 -3,600 ft. The equations werespliced at 3,730 and 5,270 ft.
B-5
TR 82-81
00
0
00
C7
CD0 1
C
o ',
c)
o
200 300 400 500 600 700 800 900 1000 110OO 1200
PRESSURE MB
FIGURE B-2. STATION E 1200 hr G.m.t.Pressure profile based on a least-squares fit with an exponential function.
B-6
TR 82-81
with equation (7) is shown in Figure B-3. For comparison, Figure B-4 is a profile
of the density computed with equations (1) and (3) where the coefficients were
obtained by making a least-squares fit to the observed densities. Very little dif-
ference can be seen when curves of Figures B-3 and B-4 are superimposed.
From these results-(l) the small difference in the residuals between the two
methods of calculating densities, and (2) indiscernible difference in the two
curves obtained-it appears that there is no advantage in computing density from
smoothed temperature and pressure data versus smoothed density data. It may be
noted that the first method requires 11 undetermined constants, the latter
only 4.
The method of making separate least-squares fits to the temperature and
pressure does provide a much clearer explanation as to why the correction term in
equation (3) improved the fit to the density. The reason is the large variations
in the lapse rate of the temperature at the lower altitudes.
The equations used to fit the temperature are
Altitude (ft)
0 - 3,500 Temperature = a0 + a h
3,500 - 6,000 Temperature = a; + a'h
6,000 - 36,000 Temperature = a" + a"h +a"h +0 1 2 3
where
temperature is the virtual temperature °K
h is the altitude in feet
The altitudes of intersections are 3,731 and 5,270 ft.
B-7
W
T1 82-81
10
C3w
(n
0
0
0 C__ __
w(i,
0L
0
CD
t-0
(0
0D0
Co
0.80 0.74 0.68 0.62 0.56 0.50 0.44 0.38 0.32 0.26 0.20
DENSITY LB/CU FT M10-1
FIGURE B-3.Density profile computed from the data of Figures B-1 and B-2.
B-8
TR 82-81
C30)
C)
C)4
CD4
C-)
4
LUC0
C!
- oi l
CD C-J
CD
C!
C3
CD_ __ _ __ _ _ _ _C30r
0DOD__
0.8 0.4 0680.62 0.56 0.50 0.44 0.38 0.32 0.26 0.20
OENSIT% LB/CU FT *10-
FIGURE B-4. STATION E SPRING 1200 h~r G.m.t.Density profile computed directly from a least-squares fit to the density.
B-9
TR 82-81
The coefficients obtained in the least-squares fit are
a0 = 0.29098019920169 E+3
a, = -0.26010921095663 E-2
a; = 0.28261034232183 E+3
a' = -0.35814511053088 E-3
a" = 0.28474414363984 E+30a" = -0.42144325442356 E-31
a" = -0.69303432938534 E-72a" = 0.85246672752231 E-123
The mean and standard deviations of the residuals divided by the computed
values are
Altitude (ft) Mean Sigma N
0 - 3,500 -0.00000002 0.00603867 983,500 - 6,000 0.00000000 0.01019354 766,000 - 36,000 -0.00000008 0.01654041 357
0 - 36,000 -0.00000362 0.01433263 531
N is the number of data points in the fit.
The temperature profile is shown in Figure B-i.
The equation used to fit the pressure is
P = c 1* e(c 2h)*e1 +c 3 h
where
P is pressure in mb
h is the altitude in feet
The coefficients obtained in the least-squares fit are
B-10
TR 82-81
cI = 0.1025894426895 E+4
c2 = -0.3512226501974 E-4
c3 = -0.1469124958633 E-2
The mean and standard deviation of the residuals divided by the computed
values are
Altitude (ft) Mean Sigma N
0 - 36,000 -0.00000270 0.01171363 531
N is the number of data points in the fit.
The pressure profile is shown in Figure B-2.
The density profile shown in Figure B-3 was computed with the equation
p = P/(R*t)
where
p is the density (lb/ft3)
P is the pressure (mb)
T is the virtual temperature (°K)
R is a constant, 45.981610868
The mean and standard deviation of the residuals divided by the computed value
are
Altitude (ft) Mean SigmaN
0 - 36,000 0.00009937 0.01129087 531
The density profile shown in Figure B-4 for altitudes below 3,500 ft was com-
puted with the equation,
B-l
mi - l I n r " :" ,' ... -- I II .. . I II iii - • . . . . .. . '" " . . ..
TR 82-81
(c2 h)
2 3500-P =cle +ch +(0 C) 3500-h
and for altitudes equal to or greater than 3,500 ft was computed with the equation
(c h)p = cle + c3h
which is the same procedure described preyiously in the report. The coefficients
used to create the profile are
cI = 0.076640263730 (lb/ft3 )
c2 = -0.304780287061 E-4 (ft-1 )
P0 = 0.076474 (lb/ft3 )
c3 = -0.517405589378 E-7 (lb/ft4 )
The mean and standard deviation of the residuals divided by the computed value
are
Altitude (ft) Mean Sigma
0 - 36,000 0.00048651 0.01160715
B-12
TR 82-81
APPENDIX C
COEFFICIENTS OBTAINED BY FITTING OBSERVED DENSITY
C-1 im . .111 1 Illl~llllimt . .. . " --
TR 82-81
In this appendix, four lines of data appear for each group of sondes that were
used in the least-squares fit.
Lines 1 and 2:
Item Identification Description
1 N Number of data points that met toler-ances
2 JE Number of points used in fit3 AMEAN Mean of the residuals4 SIGMA Standard deviation of the residuals
Line I is for the altitude range from 0 to 36,089 ft, and Line 2 is for the alti-
tude range 36,089 to 65,000 ft. The first five items in Lines 3 and 4 are coef-
ficients CI, C2, and C3 for equation (1), which were obtained by making a least-
squares fit in the altitude range 3,500 to 36,089 ft, and coefficients C4 and C5
for equation (2), which were obtained by making a least-squares fit in the altitude
range 36,089 to 65,000. The coefficients are listed in the order of their sub-
scripts. The next item on Line 4 is the median of the surface densities of the
sondes used in the fit. The last item is a counter and has no significance.
Data points refer to the individual altitudes where temperature, pressure, and
humidity were recorded. The residuals were expressed as a ratio of the residual to
the density obtained with the least-squares fit as stated previously. No data
points were eliminated in the least-squares fit; but tolerances on the residuals
were specified, and the number of points that met the tolerances were recorded.
The tolerance used in the lower altitude region was 0.025, and in the higher region
it was 0.150. A few tests were made to eliminate bad data points before the least-
squares fit was performed.
C-3
4i
TR 82-81
w w w wto 0 - .V 4 1 N k0
'0U% N m' No 9%0 4D In Iw. 0Y t0 0
5N T4 9%Nr4 P2 t. 1- W 4P
N 09 49 913 49 44U V 4 KC d* ta a V%
E C X:5 Z .4 0. x ~r0, X Z P. wE X aLD .4 U% o LD to U% Lo LDW% QL 9%Di Lo Lo WLD w bL J0%6-4 " * ".4* " .4 4 " " -4 614 $-4 ""U on-%f H.9P -4In(A (A. (A (n ( ( ( Aif (AIf (Aif CAif i/f 19(AtA CA(
t-* N' 4nS 4' is P.. 041 N% c N V- ntP.% NU 4k .4p C2L %a ca Na= 4 a sc C pC2 0 ,
0.* 1 W100% g.to U t Iu'mn * 1 40 SD4 we408 1 Nee0 inW N f" W 40 t W 0' O W 9iJ L'n w t. %V 1a wcm N C .4P2 0 P a . PP aNo c; F
f% a0 * 4D 0e 6.9% 0%10 InO * .* . 4P.) N0 v4 V4 (lJ _Y
U1% l P *0N-
44 14 so P74m1 V1 % a,% 0N4
9,, w 4 off of w0 of 96ao)9m Md - 0O r # - f3 0 it 0' P1 96):T 0 X r- 0 Z0 0
ot 4 1 49 49 d d 14 1409 1 5 8 taCd I q i
.4 t 0 tN .4% N 01 0at c.4 m2 C3 "N 11% .4N% .4 0D-t i'0 WciO m .40 N M I W%
III *N ONe .4N P0"I .4* NNS I 2Ito0D
e C, co a2 -* . Fuj 40 W 960C2 W C = *4 # C W4 1 4 0a %Da W D 4 * M O ) 3 m 00a OO2 on4 @CN% 0 0%
=(P ca 4014 Wca 4 0a C 0 aOa a00N c aV 0@ 42 004. 0 0 .0 4 * N a% * 04's * *o ' UN .) * o * .9LA
(AU o 1*0*v V40 e~ d 146 941um . 440 1040 %Dz '4 in' 0 oC2r.. If% N% 100 N..Y4 %a
OD Nt 0* U9% C20 %aN -# 0 UN. doCV' ofN .. -tN SN# o l wo 'ON W). V4
4 f 0 10 W 0 to to 0
a MP 4 N iiPO (P m WP24 93 409 .w1 4 If% V4 91.4 L9%C %C' tv PN
00 cm 0' c 0 = 1%0 10a 0 0 91%%a8 I1 do I NII* I 2 NI 91%iV-4LIW W4 w mW NW =W 0% w I1% W N
a 00 PON. do (P 0440 wN 40 0 tNO .4D 0%?N. 50% 86)0' *cj00 C2
W do 0 0: 0' ~ N P21 -tmUN f
InAi )L 2N o4 nmC . 4N Q- cc% U% %0 % 24 J4 W21) %D.4
1.- .-4 042 % .4 4 I .cmNA N - QN C3C24 t- S- 2 0 P.N 2Z Wk1 e N .0 0 wt*4 ce fn
8.. Cm4 C2 5% N 0I-if co 91 *010% CfN
94 V4 0'% V4N
c-4
TR 82-81
I P I I I I S Iw w Ww W W W
5. 4 p1.. 0%e N :1 0%
V-t 4 0% V.. 0%NN 0 UN f n51V4 4 5..15%l
in m o w% 0 to
C Y I I c m I
NN 00Q 1 r, 0Mt' * 0l *01) o1cP€: .4N Ot) olwP ! 4 I Am o .4.40 o o, I at 4 a w N NU 9. t= v 1. v r 40 m i In% r %0 v x 0r v. =a X% xiLo @ 0Jt Wca Lo IS w J0O Lo1 J I .44i. #aL0 W U% L N
N-* .0 "1 , 4" 0 n "S .. " W6 " , t " In.e e ,.
O0%W o 0% • • 1 NL- 5 • ,UJ *t..W'P •*0W,,tl U% •, o • .4.J 0% .O
( A N (A40 (A t( A(A 1 A A( C0 V)N*(
* *4 N 4 N ..o 0 .N' . do C C2 w N qD .1
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TR 82-81
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Ti 82-81
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C-34
TR 82-81
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C-36
TR 82-81
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c-37
TR 82-81
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C-38
TR 82-81
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C-40
TR 82-81
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r'. 4 ~ 4m P. 05.r 9940 U's. 05'.Le '.* -+ VF '. tN 0S2 .4.4 - W P P0 a.mt f
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CD'. o .a, '.0QI0 OKdo0 0 mm 0-0 0 %0 US u' 0 S&W 0OD&0 0 ONcPON P' n I c I- aN co. cc t I ILUN 0%N- 1 '1
In %D C %.0 o N o CID 40 '. 0 0 . 4 0 0 .UCP 4'.0.~C- 942 P. in0 N LAS N# 4 o'.
C2. ca %D cm .4 WC3 J cm W Z3WW3 0 C3 D VW0 W 0.
%a 0 9 0 0 0, CA a4 * NC
%aO CY Q %a v WN UN.94 0 V4. U% 94 w CN &A Vo00 LA1%. NO 0 94 so .N W 4 U%0 0 0 to 4
CX 00 40 C3 o C3 ~ W) =0 cm 0m GoCP 0c
C3oW t OW %0W '0 %Dw tW N NW u'.w -4 W OW94* NP- rl' M0 -tm Sb- 04 N o 94* f M to .6%bf.90- OD. P- NP. *0 9 00 LA%' I 4%P. N0 P4 -V '.0 too NO. CD W m 04%0
wlNZZLA0ZZ'oNZzLA,4zZwm z NZ4 2 2ZZ% N ZZ P,-0z z r- 094m NO CD 9Y pn~. '.0%aP. Z w440 In~ 4 NY* m'.
094 omr0 .94r 0* MQ M 1h. w1 at so 40MNMC .
NY CY N40 1.- W% Vol4UA c0 In mA V) U%' t 94 NY
C-4 1
Ti 82-8i
w w wU w
0' w N0lP.. 4 NY 0 NY
cN we 0
3cI c"z Z.* ZE z E za z z t
",c, b- -* " - * I"lU " V%~ f " %4 f "U%
aD'% BU UN %0 %D1 04pv w 94,0 t- %o D 00P. in PW. 0'01P. 0% UN N t.a w 0Pt-" "I0 a 44 VA.p Pft = a-r= 0%
U% UN g W'a irU' m u 1W 04 4Wg-t".4 wt0 to? .4 g in u
0 P.. q O 0 0
%o N0 '0 %D
zZiD Zzo Z %D zz"~ Z z v
w Wl %U% DwWjNp..w~w440ww m'Ir * zZ *r ZE 0 VE I* x a
0%N w N I~n w 2N .4 m%aW* n& nN **M A -1 V4.4 -U% 94 C 0%iO w = tU'o %~v APcm N0YcW) c U D 4p'0 0'* %D F,a I in a,V4 0%W '4 W 4 V-w w w -W~ *0l
,4~ " .4 4 oNUm =Pf4 m 00=d
C24005 in ~ 0 M0 ID000n N 3 V NI Io f * *a1 4o I top " *N' m * t%
0' 0 F'P. Fn UN UN " %0 40 cmE~ mO V40
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to V4 %a' U N c BAN fnN -'4
00m P.0 a P- Ot3 P ~.. 16
0u w~a lA% W% 6% mw
%D0 fN fuO V% 0%0% P.. 0v wZ a. M' Zo Z' M0 ZA Z 0 -4 .4 .0 N- 0%a - 0% Z 0 M PN 4D U% 0 LA' P4 0 MO 0' Y 0a A. a a CY w- Ny CY 0 N D 400 N% ca40 40
* ' 4 0 0%
C-42
TR 82-81
APPENDIX D
DENSITY RATIO PROFILES FROM STATION E
D-1
TR 82-81
DENSITY RATIO PROFILES FOR STATION E
Station E is in the mid-Atlantic east of Bermuda. The uniformity and the
shape of the density profiles indicate that the U.S. Standard atmosphere does not
represent the atmosphere over station E too well for the period sampled. Almost
all the curves above 3,500 ft are fishhook shaped. Below 3,500 ft, the density
ratios increase with altitude. The lapse rate is larger than standard, which is
referred to in meteorology as a "superadiabat." From 3,500 to 35,000 ft, the den-
sity ratio curves are all concave, as seen from the right. That is, density to
about 15,000 ft decreases with altitude more than in the standard atmosphere.
Above 15,000 ft, the density ratios increase with altitude. Density ratios are all
greater than one above 30,000 ft, and they grow larger, regardless of surface den-
sity ratios. Overall, when density is two percent below standard at the surface,
it averages three percent above standard at 35,000 ft. Above 35,000 ft, the
curves are more erratic, which is probably due to the higher noise level in the
observations.
Some seasonal differences are apparent in this region. For example, in sum-
mer, both at 2100 hr and 0900 hr local time, the density ratios increase linearly
from 35,000 to 45,000 ft to a value of 11 percent above standard. This seasonal
behavior may be associated with the seasonal change in the height and temperature
of the tropopause.
The diurnal differences are very small. When the 9 a.m. and 9 p.m. curves are
superimposed, the coincidence of the curves is remarkable. This could mean that
there are no diurnal changes in density; or that, as the surface temperature and
density change diurnally, the density aloft changes in a corresponding manner, and
the correlation is preserved. It could also be due to the fact that 9 a.m. and
9 p.m. temperatures are nearly the same. At stations N and V, the sonde times
would likely correspond to greater differences in surface temperature and density.
The values for surface density reflect the seasonal changes in temperature.
Surface density class medians ranged from 0.95 to 0.98 in summer; in winter, they
-anged from 0.97 to 1.02.
D-3
TR S2-81
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TI 82-41
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TR 82-81
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TR 82-81
APPENDIX E
BALLISTIC DENSITY PROFILES FOR STATION E
A AII9 496 NAVAL SURFACE WEAPONS CENTER DAHLOREN VA FIR 4/1I A METHOD TO CORRELATE THE UPPER AIR DENSITY WITH SURFACE DENSIT -ETCtU)I~ UL 82 L J MCANELLY.
UNCLASSIFIED NSWCITRA82-8 NL
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Ti 82-81
Station ESpring 0000 hr G.m.t.
(20 hr 48 min local time)0 _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _
00 U-3
CD
0D0w
I*
CDCDa0 .
0DQ
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CC
0 L
0.94 .S 09 .7 09 .9Jo .1 10 .3 10 .5 10
BALSI0EST
0-
Ti 82-81
Station ESpring 1200 hr G.m.t.
(8 hr 48 min local time)
C3C3
CD
U)
V
0C3~~0M
00*
QU3
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Li
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0. CDCDCD
C3
0D0CD ___ _
C3
w
Q0 _3__
U3
0-940.95 0.96 0.97 0.96 0.99 1.00 1.01 1.02 1.03 1.04 1.05 1.06.
BALLISTIC DENSITY
E-4
Ti 82-81
Station EStmer 0000 hr G.m.t.
(20 hr 48 =in local time)
C
Lo
0OV,
U)
0.0j
Cc _D-
CD)
0I
Lii
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TR 82-81
Station ESumer 1200 hr G.s.t.
(8 hr 48 win local time)a _ _ _ _ _ __ __3_ _ __ _ _ _ _ _ _C3C5-------------------------------
CM0 'I I
to-e
C2C3m. __ )__- 103
42
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C3LO0W
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0303
UO
-----------------------------0.940.96 0.96 0.97 0.96 0.99 1.00 1.01 1.02 1.03 1.04 1.05 1.06
BALLISTIC DENSITY
1-6
TR 82-81
Station EFall 0000 hr G.m.t.
(20 hr 48 mini local time)0 _ _ _ __ _ _ _ _ __ _ _ _ __ _ _ _ _
0
U3
C3
0-01
4.
0C3W,
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BALSTCDEST
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Ti 82-81
Station 3Fall 1200 hr G.m.t.
(8 hr 48 min local time)
a _ _ _ _ __ __2_ _ __ _ _ _ _ _
C3C3 _ - - -ao
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C _ _ - -
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M
0
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0.409 0._09_.8 09 .0 1-0 . .3 10 -S 10
BRLSTCDEST
Ti 82-81
Station EWinter 0000 hr G.m.t.
(20 hr 48 =in local tine)o10
U , 3C
I
0 - -, _- - -"0I
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- -CDc0
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R-9
...'... ... ' . .. . '" ' . .. " " ..
.. .. . ...
0
. . ..
0
Ti 82-81
Station BWinter 1200 hr G.u.t.(8 hr 48 mini local time)
C
C
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0
0
0
CDI-U.3 .
-J C
03Q.--
0
40
0.9A0."6 0.96 0.97 0.96 0.99 1.00 1.01 1.02 1.03 1.04 1.01 1.06
BALLISTIC DENSITY
9-10
TR 82-81
Station E
All Seasons 0000 hr G.m.t.(20 hr 48 min local time)
C3
U3
CID'
CD i i/iQ
0o
0a0
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0D
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.D .
0.940.96 0.96 0.9'7 0.0 0.99 1.00 1.01 1.02 1.03 1.;4 1-05 1-;G
BALLISTIC DENSITY
TR 82-81
Station EAll Seasons 1200 hr G.m.t.(8 hr 48 min local time)
0Q -
C3
Q0O
Q ---
030D
(w~
U)
-LJ
0
CcCC3
0Sn
0
0.94 0.95 0.96 0.97 0.90 0.99 1.00 1.01 1.02 1.03 1.04 1 .05 1 .06
BALLISTIC DENSITY
1-12
TI 82-81
APPENDIX F
SELECTED BALLISTIC DENSITIES FOR STATIONS E. - N, AND C
F-1
Ti 82-81
Local time corresponding to:
Station 0000 hr G.m.t. 1200 hr G.m.t.
E 20 hr 48min 06 hr 48minV 10 hr 54 min 22 hr 54 minC 21 hr 38min 09 hr 38minD 21 hr16 min 09 hr16 minN 14 hr 40 min 02 hr 20 min
(To be used with the following tables.)
F-3
- - - - -
Ti 82-81
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Ti 82-81
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TR 82-81
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TI 82-81
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TR 82-81
Un 00co 0 n OC4 r- - cn* in ~%0r = %0 Q% 0000t 0% 80 .- i000- -400%0-4 .- i80091~~oo o00 0 0 00 000%00 0000
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Ti 82-81
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University of Dayton Research K44 (Masters) IInstitute
Dayton, OH 45409 1 K50 5
National Weather Service G 18060 13th StreetSilver Spring, MD 20910 1 GlO 1
Metropolitan State College G12 1014th and AraphoeAttn: Dr. W. A. Kemper (Box 69) 1 N20 IDenver, CO 80203
N30 IMr. Loren J. McAnelly209 Walnut Drive N50 1Fredericksburg, VA 22405
N51 1
N52 1
N53 1
U 1
E431 10
(4)
