-
I
OF TEC
NOV 3 1964
-19RARp1-
ON THE LINEARIZED ATMOSPHEIC C0NT.IIBUTIONS
TO
REENTRY VEHICLE CEP
F2ED MARVIN SHINNICK III
S.B., Massachusetts Institute of Technology
1959
SUBMITTED IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE
DEGREE OF MASTER OF SCIENCE
at the
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
June, 1964
Signature of Author
Signature redacted
Department of Aeronautics
and Astronautics, June 1 9 64
Certified by
Signature redacted
Thesis Supervisor
Signature redacted
Accepted by
Chairman, Departmental
Graduate Committee
~
U
2
ON THE LINEARIZED ATMOSPHERIC CONTRIBUTIONS
TO REENTRY VEHICLE CEP
by
Fred M. Shinnick III
Submitted to the Department of Aeronautics and
Astronautics on May 22, 1964 in partial fulfillment
of the requirements for the degree of Master of Science.
ABSTRACT
The adjoint method of Bliss is explored as a means of computing the influence coefficients required for the computation
of the atmospheric components of reentry vehicle CEP. It is
found to be practical and straightforward when implemented on a
modern digital computer, such as the IBM 7094.
During the course of the investigation it was found that the
statistical description of the atmospheric uncertainties affecting CEP is incomplete, and some of the possible effects of this
incompleteness are detailed, while others are suggested.
The author concludes that, by use of the adjoint method of
Bliss, the development of detailed information concerning the
reentry vehicle's response to statistical uncertainties in its
environment is made sufficiently practical as to justify technically (but perhaps not financially) a more rigorous examination
of the statistics of that environment from the relatively new
(circa 1957) standpoint of interlevel correlations.
Some inadequacies of the mathematical model used to describe
the reentry vehicle are pointed out, together with some possible
inadequacies of CEP itself as a design number, as possible areas
of further investigation.
iii
ERRATA
Page 10 line 2
replace 3 with 5
Page 17 line 3
omit ellipsis of five dots between "density"
and "lat"
Page 19 line 6
(from the bottom) Sentence beginning "A
brief note . . ." should be replaced
with:
"A brief note on the effect of
varying P (K) is in order."
Page 32 line 13
replace "washed" with "averaged"
Page 55 line 19
add to sentence ending in "show" the
following phrase: ". . . since 50 ft/sec.
is not greatly larger than the standard
devistion pf winds at the altitude in
question and since the winds extend over
a comparitively small altitude region."
Page 87 line 6
replace "addition" with "additional"
Page 129 Reference 27 replace "statical" with "statistical".
iv
ACKNOWLEDGEMENTS
The help, the courtesy, the generosity of many people made
this thesis possible.
are too many.
I cannot thank them all by name here; they
My first and deepest thanks go to those, not
mentioned by name, who took time from their own lives to help me
along.
Many of these are employees of the AVCO Corporation's
Research and Advanced Development Division, where I was privileged
to work during part of the time this thesis was in preparation.
Much of this work was supported by contractual work for the
Ballistic Systems Division, Air Force Systems Comand.
Among individuals, greatest thanks must go to Prof. W. E.
VanderVelde, my thesis advisor, whose patience and help was
beyond calculation, to Dr. F. W. Diederich, Vice-President,
Engineering, AVCO/RAD, who suggested the topic, and Mr. Norman
Sissenwine, of the Air Force Cambridge Research Laboratories,
who gave freely of his knowledge of statistical
climatology.
Last, but very far from least, special thanks are due four
people:
Mr. J. DeNatale, Chief, Digital Prcgranming Section,
AVCO/RAD, who sweated out the debugging of the programs with me,
Miss J. Chapman, who put up with rough drafts beyond number,
Mr. R. Rice, who was kind enough to look them over and suggest
better English in places and D. A. Rogers who provided the
inspiration for my attempting graduate work at all.
Final responsibility for this thesis rests with myself, and my
conclusions and recommendations are independent of any official views
which the Defense Establishment may have on matters which are mentioned
here.
I
vi
TABLE OF CONTENTS
Chapter No.
Page No.
1
Introduction
1
2
Theoretical Discussion
5
3
The Computer Model
28
4
The Statistics of the Environment and Vehicle
46
5
Results, Conclusions and Recommendations
69
A
Sample Computer Output from Adjoint Program
90
B
Meteorological Data Used in Analysis
Appendices
116
Figures
1
Probability of No-Kill vs. Kill Radius/CEP
for Several Numbers of Missiles Arriving in
7
a Circular Normal Probability Distribution
2
Radius of Severe Damage for Underground
Structures and Groundburst vs. Yield
3
Radius for Given Overpressure vs. Weapon
Yield for 100,200 and 300 psi Overpressures
11
Sketch of the Normal Bivariate Probability
Density Function
20
5
CEP for Unbiased Bivariate Normal Distribution
22
6
CD vs. M Used for the Study
30
7
Coordinate Uystemas Used W
of Motion
35
Derive Equations
9
vii
8
The Matrix B
36
9
The Matrix C
37
10
Computational Scheme
38
11
Sketch of Wind-Trajectory Coordinate System
47
12a,b,c
Estimate of Wind Bias Induced by Selective
Loss of Radiosondes
13
Lines of Constant Range Over a Spherical, NonRotating Earth, from 300,000 ft Altitude to
Altitude
14a,b
Percentage of Density Influence Coefficient
Above 32 km
72,73
15a,b
Impact Mach Number vs. V &
75,76
16a,b
Ratio of Density Influence Coefficient (all
77,78
bands) to Downrange Wind Influence Coefficient.
(all bands),
17a,b
Atmospheric CEP vs. V &
60,61,62
'
70
80,81
Tables
1
Effects of Simplified Earth Model on Dispersion
29
2
Comparison of Dispersion Influence Coefficients
Obtained by Adjoint Method With Those From
Standard Trajectories
41
3
Altitudes fov.Commonly Used Pressure Levcls
57
4
Accuracy of U.S. Density M4easurements
58
5
Atmospheric CEP at r= -200 for V./CDA of 1000
82
6
Maximum Effect on Atmospheric CEP of Altering
Certain Elements of the Covariance Matrix V
83
References
128
viii
LIST OF SYMBOLS AND ABBREVIATIONS
DEFINITION
SYMBOL
SEE:
Eq. 2.3
A
vector of dispersion influence coefficients
a
speed of sound, feet per second
B
matrix of partial derivatives shown in Fig. 8
Eq. 2.19
C
matrix of partial derivatives shown in Fig. 9
Eq. 2.19
CD
vehicle drag coefficient
C q+CMA vehicle dynamic damping deriVative
Chap. 1
circular error probable
Chap. 1
D
dispersion vector, feet
vehicle aerodynamic drag, pounds
Eq. 2.1
Eq. 3.2.
E
expectation of event C
F
vector of functions in Eq. 2.16
Eq. 3.2
G
vector of functions in Eq. 2.17
Eq. 3.3
C EP
GMT
2
Eq. 2.6
Greenwich Mean Time
g
2
surface acceleration of gravity, feet per second
h
altitude, feet
K
constant used to calculate CEP
m
vehicle mass, slugs
total number of perturbations, in Sec. 2.4
P
J will occur
probability that the event
vector of quantities to be perturbed
Eq. 2.9
p
vector of perturbations in P
Eq. 2.1
Fig. 5
U
ix
DEFINITION
SYMBOL
Q
QI
SEE:
relative wind, i.e. velocity of R/V with respect
to air, feet per second
Eq. 3.2.3
"Quantity of Interest;" in this thesis, the
downrange and crossrange components of miss
distance in feet
Eq. 2,24
radius of the earth, feet
R
R/V
reentry vehicle
S
covariance matrix of impact points
Eq. 2.9
S*
orthogonalized matrix of impact points
Eq. 2.15
machine time required to solve various
parts of the problem
Sec. 2.4
T, Tj, T
t
t
time, seconds
interval of integration, seconds
u
downrange component of wind, feet per second
v
crossrange component of wind, feet per second
w
vertical component of wind, feet per second
V
vehicle velocity relative to earth, feet per second
W/CDA
Sec. 3.6
Fig. 7
R/V ballistic performance parameter,
pounds per square foot
Eq. 3.1.1
dispersion influence coefficient due to wind,
feet per foot per second
Eq. 4.6
downrange distance, feet
general random variable used to define expectation
Eq. 3.1.4
Eq. 2.6
y
crossitange distance, feet
vector of perturbations in Y
Eq. 3.1.5
Eq. 2.19
Y
vector of state quantities
weapon yield, equivalent tons of TNT
Eq. 2.16
Sec. 2.1
flight path angle
Fig. 7
angle between vehicle velocity and relative wind
Fig. 7
reentry bearing angle
Fig. 10
8W
x
77
H
x
iYMBOL
A
DEFINITION
SEE:
angle relating vehicle velocity and relative wind
vector of adjoint variables
Fig. 7
Eq. 2.20
covariance
Eq. 2.5
atmospheric density, slugs per cubic foot
0-'
standard deviation
Eq. 2.5
transformation matrix from S to S*
Eq. 2.14
rotation angle defining g
Eq. 2.13
Fig. 4
angle between the nominal trajectory plane
and the perturbed trajectory plane
Fig. 7
SUBSCRIPTS ETC.
speed of sound
Eq. 4.1
drag coefficient
Eq. 4.1
E
east wind
Eq. 4.1
I
number of density bands
Eq. 4.6
influence coefficient bands
Eq. 4.6
J
number of wind bands
Eq. 4,6
k
target station
Eq. 2.9
K
number of target stations
Eq. 2.9
N
north wind
Eq. 4.1
WX
downrange wind
Eq. 4.6
WY
crossrange wind
Eq. 4.7
x
downrange
Eq. 4.6
y
crossrange
Eq. 4.6
a
CD
ij
xi
DEFINITION
SYMBOL
pe.rtur 'ation
in
(
&( )
SEE:
density
( )
derivative with respect to time
( )'
derivative with respect to altitude
()
expectation of ( )
( )
transpose of matrix or vector
Eq. 2.6
CHAPTER 1
INTRODUCTION
The defense of the United States in the nuclear age is largely
based on the retaliatory power of the long range ballistic missile.
It follows that an accurate analysis of such missiles is essential to
national security, for the stakes, while simple to state, are beyond
imagination and comprehension in any but a strictly numerical sense.
If the deterrent succeeds, i.e. is not used, and an unnecessarily
large part of the national defense budget is spent on it, the mistake
will only be very expensive. If not enough is spent on it, and it is
therefore too small, then the deterrent has an increased probability of
failure, i.e. of being used. The question Herman Kahn
uses to judge
the resultant situation is: "Will the survivors envy the dead?"
If the deterrent is larger than necessary and still fails,
(for the probability of failure is never eliminated, only reduced)
then there is a markedly increased chance that there will be no survivors
to envy the dead.
The stakes involved in the accurate estimation of ballistic
missile system performance are awful.
Technical performance receives a further dimension of importance
if the possibility of arms control agreements is raised. As is pointed
out by Kent
2
,
improvements in system accuracy will be among the last
2
to come under arms control, and therefore will receive great emphasis
after increasing weapon yields and numbers have been legislated out of
existence as means of increasing system efftctivenes3s.
How good is the state of the art of technical performance
estimation? Jerome Weisner, former Presidential Science Advisor, writes
the following rather illuminating passage :
is typical of many in which the methods are
simple and obvious, but is one which should be held in considerable suspicion because of the unreliability of the assumptions. In particular, estimates of exchange ratios are very
sensitive to estimates of missile accuracy, a bit of information that is very hard to get and dangerous to trust completely, and one that is subject to change as missiles develop. This is not to imply that such calculations are not
valuable, but rather that judgement and care should be applied when making and using them.
This calculation*xV
This thesis is an attempt to improve the confidence in which these
estimates may be held safely.
The measure of technical performance of a ballistic missile
System is the probability thati it zill .kiAnits t&rgct complex,, The
problem of computing this probability considered in its entirety is so
complex as to defy description. In order to reduce the problem to
analyzable size, it usually is cut along the physical joints of the
system, e.g. the launch vehicle, the reentry vehicle, the defense
system, etc., and the separated parts studied in a relatively independent fashion0
This thesis is concerned with one of the numbers used to obtain
this probability, the CEP or circular error probable. CEP is defined
as the radius of a circle, centered on a desired target point, within
which the reentry vehicle's warhead has a probability of
A simple CEP weapons allocation problem
see 2.3.
-
of
n
3
The problem of determining CEP is capable,
in part, of linearization
in that many of the causes of reentry vehicle dispersion may be analyzed
on the basis of linearized equations with their attendant virtue of
superposition. Those to be considered here are the effects of the
reentry process, specifically the variations in the atmosphere over the
target, i.e. the atmospheric density, the speed of sound, the winds;
each a function of altitude, and a vehicle parameter, the drag coefficient, a function of Mach number.
It is the decrease in computation time resulting from linearization which is the center of interest of this paper. In principle, a
machine time speedup of at least a factor of 4 is likely; 10 is
possible*, This is not to say that nonlinear components are unimportant.
To mention briefly just one example, Murphy
has described a species of
nonlinearity in the aerodynamic damping coefficient Cmq+Cmd which can,
in a reentry situation, generate a divergent angle of attack oscillation.
This, in turn, will produce an increase in CD large enough to drive the
effect of the resultant additional aerodynamic acceleration on impact
point completely out of the linear range.
However, the problems associated with computing such nonlinear
effects are such that no appreciable speedup seems possible over the
parametric or the Monte Carlo approaches of simply running many trajectories to discover what the influence on impact point of the nonlnear
components is.
A second area of concern emerged in the course of investigation,
* As events turned out, a speedup of almost a factor of 20 was
demonstrated as possible; the actual benefit is limited by the adequacy of statistical data to about 10.
'4
The derivation of the model produces a requirement for certain statistical parameters defining what is known about the vehicle and its
environment. Frequently, estimates of these parameters did not exist,
and the question arises: How important are these parameters and what is
the cost of neglecting them? This question is answered in part by this
thesis.
The other contributions of the reentry process to the technical
performance of the missile system are the reentry system reliability
and the airburst fuzing altitude error. These will not be discussed
beyond this introduction because, in the case of the reliability, the
errors required to induce catastrophic failure are such as to surpass
the limitations of linearity, or, in the case of the airburst fuzing
altitude error, the the current thinking with regard to fuzing
systems is classified, and therefore the pertinent influence functions
cannot be investigated here. Once these functions are determined, they
may be manipulated by techniques analogous to those developed here. It
should be noted that, in the case of airburst fuzing, there is a
definite relationship between detonation altitude and detonation
location. Obviously, an early detonation will be both high and short.
Further, the importance of detonating at the altitude for maximum
damage is such that, for airburst fuzing, the accuracy with which CEP
need be determined may become less, since the airburst fuzing altitude
error may become an important contributor to the determination
of the
kill probability,
For an unclassified discussion of the importance of accurate
determination of airburst altitude, Glasstone5 is recommended. For the
remainder of this thesis, groundburst will be assumed in order to avoid
5
the problem. This also implies that a hard target is under attack,
which in turn increases the stringency of the CEP requirements,
6
CHAPTER 2
THEORETICAL DISCUSSION
2.1 Order of Magnitude of CEP From Weapons Effects
The sizes of CEP which are important may be obtained by considering first the probability of escaping kill given the size of CEP
relative to the kill radius and then estimating the kill radius from
studies of weapons effects. The circular normal probability distribution
provides the worst requirement on CEP of any bivariate normal distribution, i.e. for any given kill probability, the ratio of CEP to kill
radius is highest for this distribution. Therefore, this distribution
will be assumed for the remainder of this section because it will
result in a conservative (i.e. small) size for the minimum CEP of
interest.
Using the tables of Rosenthal and Rodder
ility of no kill as a function of kill radius
6
we may show the probab-
/
CEP for several numbers
of missiles fired at the target (Figure 1), From this figure, it may be
seen that the advantage of reducing CEP relative to kill radius is not
pronounced for weapons systems for which kill radius
/
CEP is much over
two. This number will be considered the minimum CEP of interest. On the
other hand, if kill radius
/
CEP is much less than 0.7, an acceptable
kill probability produces unacceptable requirements on the number of
reentry vehicles fired. This defines the maximum CEP of interest*
43
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Two additional assumptions used in the preparation of Figure 1
should be noted: First, the probability of kill follows the "cookie
cutter" hypothesis, i.e. the probability of survival inside the kill
radius is zero; outside it is one. Second, if more than one reentry
vehicle is fired at a given target, the attacking R/V's are statisti-
cally independent. This point will be discussed further in the next
section.
To give even a rough estimate of kill radius requires three specifications: weapon yield, burst altitude and the kill criterion. A
cursory survey of current unclassified
estimates of US weapon yields
suggests a range from 600 kilotons for Minuteman to ten megatons for
Titan II, with a possible escalation of the latter to 35 megatons
Secretary of Defense MacNamara7 indicates that there is good reason for
going to many weapons of comparitively low yield which would tend to
indicate that 35 megatons is probably an upper limit. The data of
Reference 7 are used to provide bench marks for the following with the
warning that they should not be trusted very far. Minuteman estimates
in these sources range from 600 kilotons to one megaton, for example.
The burst altitude is fixed by the assumption of groundburst;
further it is quite arbitrarily restricted to sea-level.
Kill criteria based on blast effects can be defined in two ways,
by assuming the damaging effect to be related first to crater dimensions,
and second to the maximum distance from ground zero at which a given
overpressure occurs. Glasstone 5 , on page 300, suggests 1.25 crater
radii as appropriate for hard underground structures. Within that radius,
structural collapse is likely. The resultant kill radii are plotted
against yield for rock and earth in Figure 2. Figure 3 gives, for the
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same yields, maximum overpressure radii for three overpressures of
common interest.* The assumptions behind the data of Reference 3 are
largely unstated. Owing to the extremely rough cut in which we are
interested in this section, no effort has been made to track these
assumptions down.
Putting these considerations together, we may reach the following
conclusions:
1. The order of magnitude of CEP which is of interest to the
ballistic missile systems analyst is from a few hundreds to a few
thousands of feet.
2, Missile accuracy is a good place to attack the problem of
increasing kill probability since kill radius
/
CEP is, to first order,
the parameter of interest, and kill radius is weakly dependent on yield.
(Roughly proportional to yl/3).
3.The above conclusion is relaxed slightly owing to the possibility
that yield is not proportional to R/V weight. In fact, Kent2 suggests
that yield scales as the )4!/3 power of weapon weight; therefore, if one
assumes that R/V weight is proportional to weapon weight, yield will
scale as the 4/3 power of R/V weight. This assumption will be used
here.
2.2 Targeting Considerations
CEP must not be considered simply as a design number, constant for
a given reentry configuration. The purpose for which CEP is to be used,
the target complex, the targeting philosophy and the deployment philos-
ophy all can have an effect on how CEP is computed and on what the
Although qverpressures of greater than 300 psi can be designed
against, Weisner indicates that it is not practical.
-
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12
final value is.
What will CEP be needed for? The following brief list is intended
to be suggestive, not comprehensive. (1) Weapons allocation: The fundamental question here is: "How many reentry vehicles are needed to
destroy this target with an adequate probability of success?" (2) Systems procurement: "How many reentry vehicles should be bought in all?"
(3) Maintainance cycle determination: Owing to seasonal changes in the
atmosphere, the CEP required for weapons allocation will fluctuate. As
a result, the total allocation of reentry vehicles also may fluctuate,
By taking advantage of this fact, one can time the down periods of
missiles so that more are down for maintenance during periods when the
atmospheric variability is small, and fewer missiles are needed4 This
reduces the number of reserve missiles required to cover for the ones
which are down. So we ask: "How does overall CEP vary with season?"
(4) Systems design: "Given a preliminary shape, how big should it be
for minimum system cost?" Figure 1 indicates that the larger the ratio
of kill radius to CEP, the fewer reentry vehicles need be fired at a
given target. For a given W/CDA and CD vs. Mach number, there are two
tradeoffs: For a larger vehicle (up to the maximum size allowed by the
booster) a larger and therefore higher yield and kill radius warhead
may be accomodated. This, however, limits one to the immediate vicinity
of the minimum energy combination of reentry velocity and flight path
angle for a given
ne. By moving
,
to a higher velocity and steeper
reentry angle for the same range, (a "lofted" trajectory) the
atmos-
pheric contributions to CEP can be reduced. (The guidance error, however, may increase as Wheelon1 2 suggests.)
This lofting implies a
reduction in available weight and therefore in available yield6 It also
13
implies that less yield is required. Thus, the optimal size of the reentry vehicle (assuming booster thrust allows a choice) is not immediately obvious. Further, considerations of guidance system design and
of attitude control system designll will be affected by CEP. How much
weight is allocated to making the vehicle more accurate and how much is
allocated to making the warhead bigger is the question to be asked. The
nature of the sample over which the statistics are to be taken is different for each of the above uses; therefore CEP may be expected to be
different.
It should be noted that factors outside the scope of this study
may minimize the effect of any of the above, e.g. the costs associated
with providing a variable maintenance
cycle could outweigh the savings
from not needing as many reserve vehicles. The question of target com-
plex enters owing to the fact that various types of targets may have a
tendency to cluster either in given geographical locations or in given
sorts of geographical locations. (Industrial sites near raw materials,
command sites near government centers or buried in mountains, etc.)
This in turn affects the input statistics in a manner which may be different from a purely random selection of targets. Therefore, it may be
possible to predict a priori what the effect of the inclusion or
exclusion of a given class of targets may be for the overall system
procurement CEP.
The targeting philosophy enters for many reasons. A few might be:
(1) How are targets assigned to launch vehicles? If this assignment is
random over the particular ensemble to be evaluated, then the bearing
angle becomes random, with effects upon the statistics required which
are discussed in Section 4.1, The advantages of retargeting on a random
14
basis
are of a security nature: the enemy, not knowing or being able
to predict which sites apply to his most critical targets, must attack
them all; further, he will have to build a more elaborate defensive
capability. because he will not know from which direction his threat
will come.
On the other hand, this procedure may select a set of bearing
angles too close together or too far apart for optimum saturation
of the enemy defense installation, and may limit performance in some
cases. (2) How often is targeting changed? The fewer times retargeting
is performed, the less likely it is that a mean target atmosphere will
match the atmosphere on which targeting was predicated, thereby introducing a known bias which, in turn, affects CEP unfavorably. On the
other hand, the more frequently the guidance system is readjusted, the
less reliable the system becomes, owing to human error. A tradeoff
analysis will be required in the design of the targeting procedure to
maximize system performance. (3) How many R/V's arrive on target?*
Curiously enough, the way in which CEP is calculated is affected by this
number.
There are two kinds of uncertainties which must be dealt with:
those which .affect each R/V equally, and those which are random from
R/V to R/V. An example of the first kind of error might be the target
location. We think the target is at point A. In fact,.owing to imperfect
geodesy, mapmaking, etc., there is an uncertainty as to where the target
It is desireable at this time to note the'fact that the target
considered is undefended. This simplifies the problem of defining the
phrase "arrive on target," in that this event is defined in terms of
two independent probabilities, the missile accuracy and the missile
reliability relative to catastrophic failure for the undefended target.
The defended target makes this a less simple problem0
*
15
really is. If we aim at point A with a perfect missile, and hit it,
there is only a probability that the target is there.
Further, if many
shots are fired at point A with perfect missiles, and hit it, that
probability is not changed, which, assuming the "'cookie cutter" kill
probability description, means that the kill probability is not improved. This error has the nature of a random bias: it does not change
from one sample to the next, but it is known only in a statistical
sense. When dealing with this sort of uncertainty, one must spread one's
shots to increase the probability of kill.
The second kind of error is exemplified by the guidance system
errors. Each R/V's reentry conditions represent a sample from the set
of possible guidance responses to the launch environment. Here, if one
fires at point A, one probably will miss in a random fashion, and the
probability of kill will be improved when one fires more than once, even
if the cookie cutter kill model is used. Since the aim point may be
changed relative to the target in a random fashion, the relative sizes
of the two sorts of error is subject to a degree of control, and the
optimum ratio is not immediately obvious.
To conclude this section, it is desireable to repeat what was written at the beginning: CEP must not be considered simply as a design
number, constant for a given reentry configuration. A few of the considerations required before an evaluation can be made realistically have
been mentioned; no attempt to be exhaustive or authoritative has been
made; neither was an attempt made to check against current targeting
philosophy.
This section is included simply to emphasize the true nature of CEP:
a design number, yes; but a design number which relates the system's
hardware both to its software (i.e. its employment doctrine) and to
specific environment; it cannot be rationally evaluated without examining both of these things in detail.
2.3 Basic Probability
The sources of impact location error (or dispersion) which may be
linearized are primarily of a continuous nature, such as perturbations
the atmospheric density about its mean value. These perturbations are a
function of altitude, as is the density itself. For a given trajectory,
these linear components may be written as a vector equation
impact
c
Dk
(2.1)
reentry
Dk is a column vector of two components, the downrange and crossrange components of the vector from the aim point to the impact point.
P is a column vector of six components; the variables which are to
be perturbed. Five of these are explicit functions of altitude: the
atmospheric density, speed of sound, and the three components of wind;
downrange, crossrange and vertical. The sixth, the vehicle's drag
coefficient, is an implicit function of altitude athrough its dependence
on Mach number. The perturbations themselves are denoted by p.
The partial derivative
4
is a 2x6 matrix of altitude dependent
functions, and the index k (k a 1,
..
,
K) refers to a particular
nominal trajectory. The convention that the partial derivative of a
scalar with respect to a column vector is a row vector should be
recalled.
Statistical data on the Pk generally are not available as continuous functions of their independent variable, but as discrete
17
functions over finite bands of the independent variable.
Typically, the
sort of statement one may find is (using density as an example), "The
standard deviation of density
.
.
at 4 kilometers is 0"
This
value will be assumed in this thesis to be valid for all altitudes above
the given altitude & below the next higher altitude at which a value is
given.
One may recognize this fact by redefining P
K
as the perturbations
of the six variables, band by band, i.e., Plk is the density perturbation
from zero km. to 2 km., P2 k from 2kmo to 4 km., etc.; pik is the CD
perturbation from zero to .5 Mach, pi+ Ik from -5 to 1.1 Mach, etc.
Now PK is a column vector of M components where M is the sum of the number
of bands used to approximate each of the six components of the old perturbation vector.
Equation 2.1 becomes
=
(2.2)
As
where
A
range
p(2-3)
The range is the size of one band of one independent variable.
The assumption is now made that each of the variables making up
the vector PK is random, with mean
-
)
(2.4)
I4~
T6
L
MO
-ae
-(25)
.
and with covariance matrix
18
where superscript T indicates the transpose; the bar superscript
indicates the expectation:
L
where
.;
PCXdis the probability that X
(2.6)
Xe PL;<j
is the value )( takes.
If x
is continuous
/a)
.'(0(2.7)
where
(2.8)
Equation 2.4 simply implies a perfect targeting process, i.e.,
one in which the mean impact point is the aim point.
This does not
exclude the unknown biases which were discussed in section 1.2; only
the known biases are eliminated by this assumption.
The further assumption will be made that Dk is a random variable
with a normal bivariate probability distribution function.
be exactly true if each of the components of p
k
This will
is normally distribu-
ted; it will be approximately true (by virtue of the Central Limit
Theorem) if none of the non-Gaussian components of Pk is of over-
whelming importance.
This is an exceedingly imprecise statement of
the circumstances in which the central limit theorem is valid, however, Parzenl5, indicates that cases where the random variables are
dependent are not yet completely analyzed.
Therefore, the precise
limits on the validity of this assumption should be investigated
further.
19
The computation of CEP is, with these assumptions, a straightforward process.
K
L
where
PL4 is the probability that the kth target station will be se-
lected.
The bar indicates the expectation of the quantity beneath.
Inserting the definition of Dk, we obtain
Sz
(2.10)
by making use of the fundamental matrix identity of transposes
Recalling that the expectation of a sum is the sum of the expectations
of the terms, and that the expectation of a constant times a random
variable is that constant times the expectation of the :random variable,
2.10 is simplified to
Is
P
A
1Q F
A
(2.12)
All the quantities in Equation 1.12 may be evaluated in principle,
although it is not appropriate to discuss P[Ehere.
The P[A]fall out-
side the area of this paper as they are based on highly classified
intelligence information; for our purposes, a single station will be
investigated, thereby making the point moot.
reason for error is in order.
A brief note on the
The word "station" implies not only
a location, e.g., a hard command post, but a time, e.g., winter,
Neither of these factors is distributed on an equi-likely basis
among its possibilities, the first because one target simply is
more important than another and the second because, for a variety of
20
reasons, war is not equally likely at all times.
The matrix S is the covariance matrix for two normally distributed, correlated random variables.
This may be visualized as the
hill" of the bivariate normal probability distribution being skewed
through an angle
:
Figure 4
By rotating the coordinate frame through p
become uncorrelated.
,
the variables
For the bivariate case, this process may be
written
(2.13)
Z/U2
(2,14)
s4 C)S C - I
0
Q
(2-15)
where S* is the desired uncorrelated matrix.
Having the principal axes of the probability ellipse and their
n rrn
orientation, one then locates the aim point in the 0 max and 7 'n
coordinate system and applies to Reference 6 for the CEP.
If there
21
is no determinate bias (as is assumed here) the CEP may be read
immediately from Figure 5.
for 0
/ O~
If a quick approximation is desired,
<-.- 5 the approximation CEP
is valid to within 2.3%.
: 0,589 x (Qemax+ Tmin)
It must be recalled, however, that, owing
to the scaling laws described before, this represents a 7.06% error
in yield required, and a warhead weight error of 5.25%.
This strong
dependence on CEP accuracy of design conditions will be emphasized
in the comparisons to follow.
The next section describes the calcu-
lation of A;.
2.4
The Calculation of the Ak
We may write the basic non linear equations of motion in brief
as a vector equation
c
F (Y, P)
(2.1:3)
in the state vector, Y, the vector P of quantities to be perturbed,
Since we are concerned with
and the independent variable, time.
fairly steep reentries of ballistic vehicles altitude will be monotonic with time and may be used as the independent variable by
dividing 2.1 by the equation defining
(2.17)
Equations 2.16 - 2.19 are defined in detail in Chapter 3.
To do so
here would becloud the issue, which is the development of the adjoint
equations to follow.
This division results in a new set of equations in which Y is
reduced by one.
(2.18)
A
'4
I
I
I
L
.4 1144-
7V7'
I'
i1'1
'''~
t$
t{14fi'4t{{jiI}I
~
777t
-
+
--
r~rrr-rr~r~rr~r~r
4+f~rTTr~~r7 8
1
t
4
4~~
+
I
r
t
4'
t
*v
tt
4--+
4
14
t
4tI
T1
4T_ 1
.4
t H T I tlit
It
11+
tT
T~dnrtft~it~tttt'itttII
1
I+I
1
44
{
T
rf 'r{ I'''
1114 'I~t'1-r'n~tIFV*t
4
II
i
rt
4-4
-I :I
-4-
r1iT1fI
4
-i
I
4 +II
l
+ 4'
1
+ 4
1
23
In principle these five non-linear equations (one for each of
,
X
the five state variables V,
x, y, would be solved for one
perturbation after another, a nominal impact point subtracted from
each of the perturbed ones, and the results collected into the
m x 2 influence coefficient matrix defined before as Ak'
(The sub-
script k, which refers to trajectories, will be omitted in this
section to avoid confusion with the components of the vector p.)
The requirement on the computer to do this is approximately
KT (Z +#1)minutes for m perturbations, assuming m is large so
that program loading time is small compared to running time,
The
m/2 arises because one may be clever and not run the perturbed
trajectory above the point where the perturbation exists.
K is the
number of nominal trajectories to be investigated and T is the time
required to run one of these trajectories from reentry,
On a 7094,
if all the trajectories are stacked on the machine as one job to
minimize the program loading time, this line of attack requires
about ten minutes for 100 trajectories, and results in insufficiently
accurate answers, since taking the difference between two nearly
equal numbers removes many significant figures from the answer.
Some improvement may be obtained by making explicit use of the
assumed property of linearity.
If we take the collection of partial
derivatives represented by:
13 =
y
,c
P
1
24
we may write the linearized perturbation equations based on Eq. 2.18 in
matrix form as:
(2.19)
Here B and C are 5x5 and 5x6 matrices, respectively, of altitude
varying quantities. The lower case quantities y', y and p are the
perturbations of the corresponding upper case quantities, and
is
not computed because h is not a random variable in this problem,
+
T ) where TJ
2
is the new time required to compute the nominal trajectory ( which will
The computing time required has now become K(T
is the
be increased owing to increased output requirements,) and T*
2
time required to solve Eqs. 2.19 from reentry to the ground. This approach eliminates the loss of significant figures by dealing directly
with the perturbations. Therefore, this method represents an improvement.
By taking a further step, one may make great savings in time,
16
which is the
however, and it is this step, first described by Bliss
key to this section.
Closely related to Equation 2.19 is a set of equations
'
A
=
6
_
called the adjoint equations. The adjoint functions,
to be influence functions upon the
-A(2.20)
, will be shown
'quantity of interest' of the state
variables Y, i.e. the value of 'A at a given altitude can be made to
represent the change in the quantity of interest resulting from a unit
perturbation in the state variables Y at the altitude in question.
To show this, first premultiply Eq. 2.19 by 9T and 2.20 by yT
and add:
-rT
17-
13T(2,,21)
wx-
25
By making repeated use of Eq. 2.11, this becomes
which may be integrated from h
(2.22)
A- Cf_
( ;\T y),
to h 2 > hi to obtain
f
2t
GL&
(2.23)
Z
If we neglect the integral term by assuming no perturbations p,
we may obtain the effect of a change in the state variables at h 2 on
the quantity of interest at h
by writing
~~))
~()
L
7c(QL)
Comparison of Eq. 2.24 with Eq. 2.23 shows that
that the proper initial conditions for '
and therefore,
will be simply the partial
derivatives of the quantity of interest with respect to the state
variables Y at the altitude of interest. For the dispersion problem
the altitude of interest is h =0, and the quantities of interest are
two of the state variables, yl, and y5 , the down and cross range perturbations of the trajectory. Because of this fact, the initial conditions
on
\ may be written by inspection; they are:
0
0
[=
0
0
0 0
1 0
10 1-
(2.25)
Thus, if one wished, one could obtain the perturbations owing to
small changes in reentry conditions by running a nominal trajectory
from reentry to the ground, to obtain the altitude dependent matrix B,
and then solving Equations 2.20 from the ground back up, using the
initial conditions 2.25. The product of
at reentry and the
I
26
'perturbations in state conditions at reentry would be the desired dispersions.
The problem of perturbations in the environment, ratper than of
state conditions directly is only slightly more complex, and is solved
by using the integral term in Eq. 2.23. Rewriting Eq. 2.24, one obtains
7
_(2.26)C)-c
L
if one assumes no perturbations in reentry conditions (i.e. state
variables), Recalling that the p are constant perturbations in environmental variables, such as CD and (0, over relatively small altitude
bands, hl to h2 , we may rewrite Eq. 2.23 as:
j
~J~O~L
(2,27)
where all terms on the right hand side except the third are zero.
C a)
(2.28)
Another way of seeing what has been done here is to recall that
and therefore, that Eq. 2.28 could be written
Now, to obtain the -p
~pJ~Cd
f(
2
,2 9
)
~y
desired, simply obtain B and C histories
from a solution of the nominal trajectory, as was done before when B
was all that was needed; solve the adjoint equations exactly as before,
using the same initial conditions 2.25 (QI has not changed, only the
perturbations causing it) and then, using the stored histories of
'A
and C, integrate Eq. 2.28 back down to the ground, storing each integral from the top of a band to the bottom of a band as the appropriate
27
components of
, and resetting the integral to zero before pro-
ceeding to the next band. The computational task now reduces to K
nominal trajectories, 2K solutions of the adjoint equations and 6K
integrals to evaluate over several non-overlapping limits. The requirement for only 6K integrals rather than 12K is the result of the
fortunate circumstance tha .t cch rtoc urbation affects only the downrange component of miss distance or the crossrange component, never
both. This is a computational task much smaller that the 7K trajectories required to obtain this information for the crudest possible
definition of the perturbation quantities p (which assumes that a
single number defines the perturbations over the entire range of the
independent variable) and very much smaller that the time required if
one makes an analysis of the situation to the level that the available statistics would warrent. (This would be on the order of 75K to
100K trajectories.) Some detailed comments on running time will be
presented in Section 3.6.
I
28
CHAPTER 111
TBE COMPUTER MODEL
3.1
Introduction
The purpose of this chapter is to describe the atmospheric and
planetary data used in the simulation model, and the model itself.
First, the planet-target assumptions will be considered, then the
equations of motion, their derivatives and their programming.
3.2
The Planet Model
The planet will be considered to be a non-rotating sphere of radius
2.0902286 x lO
ft. with an inverse square gravitational force whose
surface value is 32.21852 ft./sec 2 .
These values derive from the work
of Guess and Pelinel.
The effect of neglecting oblateness and rotation in the problem is
of measurable extent.
Utilizing a trajectory program which allows va-'
ation in the earth model in both of these effects 18,
runs were made with
the following initial conditions:
Reentry latitude,
52oN*
Reentry altitude,
3.0 x 10 5ft.
*This latitude gives impacts near 500 North for a trajectory reentering
due South at the conditions quoted. This latitude is typical of the
USSR, but not of China.
29
Reentry velocity
2.5 x 10
Reentry azimuth
900, 1800, 2700*
Reentry flight path angle
-200
W/CDA (hypersonic value)
lo3 lb/ft2
(CD/CD hypersonic) max
8.0
with and without a 5% density perturbation.
ft./sec.
Comparison of the rotating
and non-rotating oblate earth models and of the oblate and spherical
earth models shows maximum errors listed in Table 1.
Table 1 also
shows the percentage increase in yield and the percent increase in
weapon weight implied by this amount of change in a kill radius.
The
scaling laws employed are those of Sec. 2.1
TABLE 1
Effects of Simplified Earth Model on Dispersion
dispertion
error
yield
change
weight
change
rotation neglected
1.127%
3.42%
2.56%
oblateness neglected
1.345%
4.o8%
3.05%
3.3 The Target Model
The target, as has been indicated before, is a hard, point type
target; an example might be a critical command post.
its location is exactly known, and it is at sea level,
It is undefended,
The justifica-
tion for the last two statements is that the analysis is not greatly
'i.e. reentry due east, south and west, respectively
Fig. 6 gives the detailed shape of this drag curve
TI
W
I
t-
t-
-r-
Ii
44
ITE
-i ---
if
I T
I
-44t
44-
I~
II
-iTi
-- --
tI
-T
-
f
31
affected by the inclusion or exclusion of either item; target location
uncertainties (i.e. geodetic and geographic uncertainties) are independent of other error sources, and always will fall into the "random bias"
category rather than the "random from vehicle to vehicle" category;
impact altitude error is much the same, although the theoretical form
of the initial conditions on the adjoint equations will change slightly.
Further, these errors are steadily being reduced with the increasing
sophistication of geodetics techniques.
Finally, the magnitude of
such errors for realistic cases is classified.
The justification of the assumption of no defense lies first in
the complication that a defense would introduce (just as one example,
missile system reliability in the sense of catastrophic failure could
become a strong function of miss distance, rather than being an independent parameter as before) and second, in the fact that no such system
is now known to exist, Soviet protestations to the contrary, as far as
the open literature is concerned.
3.4
The Nominal Atmospheric Model
The nominal atmosphere used in the United States. Standard Atmosphere,
1962 19*
This atmosphere assumes the barostatic equation, and therefore,
is defined by a temperature-altitude profile, the nominal atmospheric
conditions at sea level, the atmospheric composition as a function of
altitude, and the gravitational potential.
The assumption of any
standard atmosphere is, in part, one of convenience, and ideally one
Some checkout runs used the earlier ARDC 1959 atmosphere.
difference is not significant.
The
should use a set of parameters defined as the mean for each target staThe differences between such profiles and the nominal atmosphere
tion.
are not completely negligible, at least by the criteria of linearity
indicated in Section 3.7, and the effect of such an assumption on CEP
should be investigated.
Examples of comparisons between target station
mean profiles and standard profiles are given by Sissenwine, Ripley and
Cole
3.5
21
,
and more recently, by Cole and Court
22
The Vehicle Simulation
The vehicle is assumed to be adequately simulated by a non-lifting
particle.
This implies that the vehicle's attitude is controlled at
reentry so that the reentry angle of attack and lateral rates are
small and the vehicle is spin stabilized, so that any residual lift
vector is rotated about the flight path, "washing out" any preferential
direction for lateral acceleration.
Kresa
describes a simple scheme
for doing this and a conceptually similar scheme is assumed to be used
on the vehicle of this study.
The vehicle's characteristics can be summed up in three parts.
First is the nominal W/CDA (lb/ft2 ), which is calculatedu using the
vehicle weight and the hypersonic drag coefficient of the vehicle,
together with its reference area.
efficient variation.
several variables,
Second is the vehicle's drag co-
While this could be considered as a function of
(Mach number, Reynoldb- number and ablation rate) for
the purposes of this study it will be a function of Mach number only.
There is partial justification for this assumption in two facts:
First,
the portion of the trajectory where Reynolds number effects are significant is above the region where influence coefficients are large and also
33
re apt to be masked by atmospheric
in a region where Cuncertainties
Second, the complexity of the interaction between abla-
uncertainties.
tion and C
precludes further analysis at this time.
The vehicle equations of motion using time as the independent
variable are as follows:
t
(3.1.3)
an(3.1.4)
/are
2(
ra
(3.1.5)
(3.1.6)
where
D
=
/c0(4)
M =
Q
=
aerodynamic drag, lb.
(3.2.1)
Mach No.
(3.2.2)
velocity of vehicle relative to the air
V = velocity of vehicle relative to the earth
g = surface acceleration of gravity
R = radius of earth
h = vehicle altitude
m * vehicle mass
7-o
1~>
( VC(V*
Q ((V
oA~
G
( V
-U C&AC&dL
ca* A 4,A)/-(A
zY/z
(3.2.4)
(3.2.3)
34
A=
-
s(/Ur
u, v, w
=
4
(3.2.5)
(3.-2.6)
components of wind relative to earth in
x, y, z directions, ft/sec.
are defined in Figure 7.
The nominal trajectory is simulated on the computer with the
simplif ications
Q
V
0
to Equations 3:and the results written on tape.
This part of the
procedure requires .14 minutes per case or about 8.4 seconds.
Write-
out was every 2 sec.ond and the results would be interpreted as altitude
dependent functions in the following.
3.6
The Adjoint Problem
Dividing Equations 3.JX by3,.l.6
one bbtains
(3.-3.2)
jkA
-WV2,44
(3.3.1)
(3.3.3)
-
q~~9"R(3.3.4)
(3.3.5)
35
Figure 7
h -altitude
V
-
inertial velocity
7A
yn
crossrange distance
-
S-flig" at path ang le
-
7B
x
-
downrange
distance
altitude
Q-vel city
\
x-y plane
plane normal t
V
relative to air
The matrix B
Figure 8
+ g
~/3 Cvc64
)
4G_
-- S o If 0
2?~
(R~
0
0
V2. R-4-1 T-
+1
I?
0
0
0
w.
)
V
Y~7VWI
3
0
~4~-C44
0
0
0
0
0
0
-, RC4,C r
0
0
0
R Rcat
0
0
0
0
the mat-ix C
Figure 9
pS
2xoo Mnf
2
a,
2ooVVn AC4
0
0
J co
Am
-fSVCn
2X o -M 44A4
0
Ss cd
Y~v
M 4 C,
(
*-C D)
Z d/A
-w
0
0
0
0
0
- M+ q)
s 5C, ce
- psC,
0
\ .
0
0
Ke~
M
38
Forming the appropriate partial derivatives one may write Equations
2.19 in detail as:
By + Cp
y
F S7V S1
p
=
[
91P
1oo
7
CD
1
C)
Lk
-
where
y=
and B and C are shown as Figures 8 and 9.
These equations were programmed on an IBM 7094 using FORTRAN II
as Avco digital computer program 1562.
The basic block diagram is as
shown in Figure 10.
Figure 10:
COMPUTE
I
NOMINAL
TRAJECTORY
Computational Scheme
Store on
Tape
compute
B&C
matrices
integrate
Ad joint
Eqs. up.
integrate
fundamental
formula
down
program 1562
Program
1213X
Print
Out
The tape storage was used for two reasons:
Influence
Coefficients
First, it allowed use of
an existing trajectory program with minimal modification. CThis program
used time rather than altitude as the independent variable, but this
has no effect on the problem as a whole since the nominal trajectory
history is of interest, not the way in which it was obtained.
The
program is AVCO's digital computer 1213X; no reference has been written
for this program as yet.
The tape input serves another purpose; the program can be halted
while the trajectory output is checked to verify that the program has
worked adequately to that time.
The following steps are straightforward; a modified AdamsBashforth 4 point predictor-corrector was used for both integrations;
39
the reason for this usage on the integration of the fundamental formula
was that it had capability of selecting its own integration interval,
which is desirable in cases where the function to be integrated is of
a rapidly changing character.
This scheme is not self-starting; a standard Runge-Kutta scheme
is used to load the predictor corrector with the first four values it
requires.
The interval of integration on the Runge-Kutta scheme is
kept small to minimize the unknown truncation error.
Once the procedure has been started the scheme is as follows:
1.
a set of values of derivatives are computed at t (using the
equations of 1213X as an example)
2.
using the last value of At, and the values of the integrals
at the last four times, a value for the integral is extrapolated at t
3.
t
Using the extrapolated values of the integral, new derivatives,
at t +
4.
+ _
A
t,
are computed.
Using the four most recent values of the derivatives, the
values of the integrals are recalculated at t +
A
t and
compared with the values obtained in step 2.
5.
If this error is above a preset tolerance, the value at t + A t
is discarded,
the value of f.
t is reduced by a preset
percentage and one tries again from it.
6.
If the error is below a second preset tolerance, the value of
,,
t is increased by the same preset percentage and one pro-
ceeds to the next time step.
7.
This test is for economy.
If neither bound is passed, one proceeds to the next integration with unchanged
A
t.
40
8.
This
At preselected time intervals, results are printed out.
is accomplished by computing the next print time and comparing
t +
t with it.
.
If print time is less than t +
integration interval,
and print times coincide.
A
t, the
t, is reduced so that the integration
The case in which one is forced to
n, t by this process (t is slightly less than a
a very small
print time) is dangerous from an economic standpoint, since
many integrations must be performed to bring
A
t up to an
economically acceptable level by the process described above.
This is guarded against in the following way:
between t I+
t and the next print time is checked.
A
value is less than
Thus,
A
the difference
A t,
If this
A t is set to half this value.
t is never reduced to less than half its previous
value by the accident of print time coming inconveniently
clost tot+
3.7
A t.
Checkout of the Influence Coefficients
Using an early version of the program, two check cases were run
and compared with the influence coefficients obtained by running nonlinear equations of motion.
The vehicle properties used were as
follows:
W/CDA
=
1000 lb/ft2
CD vs M
=
Figure 6
The reentry conditions were
- reentry
=
300,000 ft
reentry
=
-200
V reentry
=
25,000 ft/sec (case 1)
15,000 ft/sec (case 2)
'
=
-1
41
The influence coefficients were, in each case, integrated over the entire range of the variable so that there was, in effect, just one band.
The results are shown in Table 2.
Table 2
Test Cases for Influence Coefficient Generator
25,000 feet per second
Trajectory XFalues
590
ft.
0
205
ft.
0
39
ft.
0
Perturbation
Adjoi at Values
603
ft.
0
207
ft.
0
u
39.1
ft.
0
38.3 ft.
v
0
20.25
0
w
20.17
D,, -wn
Cross
'
0
S
s a/a
Down
500
ft.
0
225
ft.
0
44
ft.
0
0
17
Down
ft.
Perturbation
E/(
1
Cross
:
Adjoint Values
gCD/CD
500
ft,
0
Sa/a
226
ft.
0
43.8 ft.
0
42 ft.
V
0
w
Cross
0
er second
15,000 feet
Trajectory Values
38.oft.
0
16.9 ft.
Down
42.lft.
0
Cross
42.
The discrepancies are felt to exist mostly in the "Trajectory Values"
column, owing to the inherently superior accuracy of the adjoint
approach.
At present this matter cannot be resolved completely, how-
ever, because the adjoint technique's sensitivity to its several numerical integration parameters has not been sufficiently examined.
In the earliest operational form of the problem, running time was
approximately 26 seconds per case.
The excess over the 15 seconds
predicted in Chapter 2 is partly due to tape handling and partly due to
the inefficient programming which is the curse of early production versions of any program.
In addition the requirement of providing extra
output from the nominal trajectory program (0.5 sec steps rather than
entry and impact only) increased the running time of the nominal program
from six seconds to approximately eight seconds.
Thus, the maximum ex-
penditure of time for one combination of vehicle and reentry condition
will be
34 seconds; enough for approximately 11 non-linear trajectories
if one is "clever" and runs only from the start of the perturbation.
It also is possible to be "clever" with the adjoint technique by
running only up to the highest altitude for which statistics exist
(generally about 32 km..).
By so doing, a further speedup of between
two and three can be achieved.
Therefore, the 34 seconds can be reduced
to less than 25 seconds, enough for about eight non-linear trajectories.
The resultant speedup is approximately a factor of seventeen over
the previous method, with a concurrant increase in data accuracy.
primary purpose of this investigation is thus fulfilled.
The
(The factor of
17 assumed the data was dividedointoL26 bands fort thO altitude dependent
quantities and 6 bands for CD. The sitata was later brought intto line
with the level of data available, which represented 71 bands in all or
43
a speedup of only about 9.
In Chapter 5 suggestions will be made for
the further increase of speed; a further increase of a factor of 1.5 or
2 seems possible.)
Limitations Imposed by the Linearity Hypothesis
3.8
The question of the maximum perturbation which would result in an
acceptably linear relationship between perturbation and dispersion was
examined by assuming a constant value for each perturbation in question
and increasing that value until a) it was larger than any reasonable
value for a standard deviation of that perturbation of b) the nonlinear relationship between the perturbation and dispersion showed
clearly on a plot.
These studies were performed at the shallowest
0
reentry flight path angle used in the study, i.e. -20 ,
since dispersion
is at a maximum for shallow angles, and since those quantities which
showed least linearity, e.g. density, become less important at steep
angles relative to the more linear components, e.g. wind.
The results of this sub-investigation seem satisfactory in broad;
no such non-linearities were found to be significant.
In detail, three
substudies were performed.
a)
Winds were examined at V = 25,000 ft/sec, 15,000 ft/sec and
at W/CDA = 1000 lb/ft 2 and at 2500 lb/ft 2 , for a total of four sets of
trajectory runs.
described earlier.
Two of these were used for the adjoint check cases
Checks were made for both, down and cross range
winds out to 100 ft/sec.
the program were detected.
No non-linearities above the noise level of
In light of this, and considering the simi-
larity of the manner in which downrange and vertical winds enter the
problem, vertical winds were run only to obtain the adjoint check cases.
b)
Density and 0D were similarly examined by means of a single
set of perturbations since they enter the C matrix in the same way, the
results differing only from the definition of the influence coefficient
bands.
The maximum deviation from linearity obtained were for the
W/CDA = 2500 cases, where plus and minus, 10% perturbation impact points
were used to compute a nominal impact point by means of a linear interpolation.
This was compared with the true nominal impact point, and
the difference divided by the separation of the two perturbed impact
points.
The result was on the order of 3.2%; for 5% perturbations, the
error was in the noise level.
Hence it was concluded that, since the
standard deviation of density is smaller than 5% at almost all times
(see Figures 3 and 4, Reference 20, for examples), this problem may be
neglected with complete safety.
In the case of drag coefficient, for which statistical uncertainties
are much less well known, the effect of non-linearities can be neglected
by realizing that the division into bands also will tend to prevent
non-linearities from becoming significant.
d)
Speed of sound was investigated only at 1000 = W/CDA.
Since
this dispersion effect is dependent on the existance of a non-zero
d C /dM, it will become very small for large W/CDA because such vehicles
impact supersonically.
At W/C DA = 2500 lb/ft , and -2 0 O
Mach numbers range from above 3 to above 6.
i
,
impact
Inspection of Figure 6
will show that the higher values of d CD/dM are not achieved by such a
trajectory.
Here + 1% differences were compared with +}% differences.
Again, the error is in the noise level.
The range -1% to +1% was not extended for the large values of C a
shown in Appendix B since these occur at higher altitude s and 1) are
4i5
not considered entirely accurate owing to reasons discussed in the next
chapter,
2)
the influence coefficients at such altitudes are very
small, as was discussed above.
46
CHAPTER 4
THE STATISTICS OF THE ENVIRONMENT AND VEHICLE
4.1
A Priori Considerations
The quantities which are assumed variable and which can effect the
mathematical model described in Chapter 3 were listed in Chapter 2.
They form a partitioned covariance matrix as follows:
CCODC7
IAtC~rC
(4.1)
P4?b'
-
where each element of the matrix represents a sub-matrix giving the
correlations between the various bands into which the data was broken.
For example, if there are two such bands, the density-speed of sound
terms become
0
=k
C ,a,~
(4-2)
Even though this matrix is 'symmetric, it takes the definition of only a
few bands for its size to get out of hand.
Are there simplifications?
In 4.1, certain terms may be seen a priori to be zero, as a result
of the statistical independence between variations in the vehicle parameters and the atmpsphere.
Therefore, the matrix 4.1 may be simplified
to read:
Ot00r
ore
'E !YN
-,
.......-
[-A-&
scc
E
-00
0
0
0
6
(4-3)
K
f~h~r
0Mur~
f~&r~.
Arm
It can be shown that, for certain targeting philosophies a good
many more of the off diagonal terms may be made to vanish.
This comes
about because wind statistics exist in an earth-fixed coordinate system rather than a down-range-cross range coordinate system.
To trans-
form from one to the other requires the reentry a imuth angle, and depending on the targeting philosophy, this may be random from trajectory
to trajectory.
Defining the two coordinate systems, one obtains
N
y
x
E
Figure 11
W
= WN cos
W
= WN sin 7
+ WE sin7
WE cos
(4.4)
where winds are defined positive blowing toward the north and toward the
east.
(The specific definition of wind directions used is not import-
ant to the following, and meterological usage is by no means consistant
48
during the period over which the references used were written.
Compare
23, 24k
CAVEAT EMPTORI)
.
What are the statistical implications of the addition of
list of possibly random variables?
c. +
÷
A wox~
(WNC
7
%w~
for one of the possible trajectories.
L-
45)
+-
-
4l.6)
WE
c
ez
(4-7)
There are I density bands and
Forming the expectations over all trajectories of DDT
J wind bands.
7
and assuming
to the
Again omitting the subscript k
Dz~~
x=
7
,
for example, the two reports of France '
to be statistically independent of the perturbations
making up V, there results
z :
Ti~~
~~ = J pA
S
ZZAVWr
40
0+
W(4e
4 . 8I
e
*
+
Here, p
entirely
a,
u
swz8w2-
could be any or all of the perturbations which result in
.downrang dispensiqn,.i.e
titshowswwhat
ill happen to,
and
w.
49
ZZ7
(4.9)
-
# S JW cnzr
S WW
~/&~
A wyj ((S
+2Z
Wi
If' one can say that
~WAI
6 WN 2
7/is
415yW~
9,
4
We -
J
WeW
S
r-
(4.i10)
z
I0
uniformly distributed from 00 to 360
7
-.
2
0
(4.11)
and
~7Z
-~~
/-1-
(4.12)
Substituting these simplifications into Equations 4.8 - 4.10 yields
r
"tT7~~
/
Al, A
A(5C-v-J74(l.13
A y
4/
J
-
(
.
3
50
(4.15)
This reduces the requirements for statistics to the following, since the
values in V which will not be used may be replaced with zeroes without
affecting the outcome:
O':c o
O0
(4.16)
~tO
O ~.tQ-o~0,
O
0
0
0Tf4 N
C)
The type of weapons pystem which would show this
-
distribution
most clearly would be a Polaris type 03rtemirith-iCBMirang9, capaableof
reaching its target from any direction simply by sailing to the proper
geodesic and shooting.
For stationary systems, the width of the
distribution which can
be obtained by reassigning targets and silos at random is limited, and
therefore, the simplifications derived above cannot be applied a priori.
51
wa8 determinate and
For this study, the assumption was made that
0
equal to 90
,
corresponding to an eastward reentry.
This choice was made
to permit retention of all terms for which statistics had been computed,
to allow evaluation of their effect upon CEP.,
4.2
The Covariance Matrix V Used in The Study
The basic problem concerning V is that it does not exist in full.
Many studies have investigated atmospheric statistics; comparatively few
of these deal with covariances of atmospheric statistical variables between altitudes.
Among those which do, one may mention as being of par-
ticular interest the work of Sissenwine et al
21
concerning atmospheric
22
to higher alti-
densities, which was later extended by Cole and Court
tudes.
The former has the particular interest of including pressure
covariance data at 20 and 22 km in an attempt to use pressure statistics
near the top of the region of available statistics as a substitute for
The latter goes
the integrated effects of density above that altitude.
higher (30 km vs 24-25 km) and gives a very limited amount of data of
the wind-density correlation coefficient variety.
Charles
has provided an excellent set of horizontal wind statis-
tical data for the US, including North-East wind correlations at different altitudes, and correlations between different stations at different
times.
This is a great extension on Court's
26
pioneering work.
Charles
gives statistics at constant pressure levels rather than constant alti-tude levels as did Court.
These data have one great shortcoming:
target areas.
they do not cover potential
For such areas, the work of France
23, 24
* is of interest,
At this writing, this is all that is published in this series; more
reports are in preparation.
52
although it
fails from the standpoint of altitude,
stopping at 100 mb.
Here horizontal wind statistics are given only for the same component
at different pressure levels.
The density statistics are given for var-
ious altitude levels.
The above description is about all that is available in the published literature.
Upon consultation with the Applied Climatology Branch of the Air
Force Cambridge Research Laboratories, Hanscom Field Bedford, Mass., a
convenient set of raw data for 56 stations in the Old World was made
available, including several in European Russia.
These data, compiled
and described in detail by Beelitz et. al. of the Free University of
27
Berlin, are available as Air Force Tape AF 322; extracted from it in
brief were the following data at each station:
surface pressure, al-
titude at selected constant pressure surfaces (700, 500, 300, 200 and
100 mb).
Temperature, u and v components of wind and density were also
available at each of the isobaric levels and the surface.
360 samples
were available, one daily for the months January, February and March of
1955-8.
Of these stations, Moscow was chosen.
Every third record was taken, to attempt to avoid the problem of
non-independent samples, so that the data used consists of 120 samples.
Most of the references cited are in agreement that a three day interval
is sufficient to eliminate this problem, although the proper period
varies as a function of season and of the specific quantity.
Linear interpolation from one sample height to the next was used
for temperature and the two wind components.
In the event of a missing
53
data point,* the data were simply interpolated on a linear basis, as was
all data in these categories above 100 mb.
The speed of sound was computed from a given temperature by means
20
The density was
.
of the formulae defining the ARDC 1959 atmosphere
computed similarly, using the reference density at the bottom of a given
band and the assumed linear temperature profile above.
Since the atmospheric density can be computed using the barostatic
equation, a temperature profile and a single density value, the data
used here are redundant.
The effects of such redundancy were investi-
gated by comparing the computed value of density at the top of a band,
which is at the same altitude.
The percent differences appeared to be
consistant with the instrument accuracies quoted in the next section,
so no further adjustment was made.
Using the 120 profiles per station values were computed for the
means and variances at two km intervals of all four quantities.
resultant
The
64 x 64 matrix provided a set of all statistical data invol-
ving the four quantities, density, speed of sound, north wind and east
wind, for the given stations.
23
These data were compared with those of France
tion, to investigate consistency.
couraging.
for the same sta-
The result is not particularly en-
While the density and wind stand deviations seemed to be of
the same order of magnitude as existing data
many detailed differ-
The density-density covariances show a change from
21
but their numerical
positive to negative at or near the proper point
ences are evident.
Missing data were indicated on AF322 by a reading-of 0 C. Since these
data were taken at night or early morning, in the winter this proved
to be fairly unmbiguouiefbrnMos-ow.
54
values, particularly at near-equal altitudes (e.g . 0 and 2 km) seem too
high in places (e.g. .8T as compared with .68).
The wind-density num-
bers agree qualitatively with those of Cole and Court
couraging.
either
which is en-
The wind data are not directly comparable with those of
Charles or France, since their data is reported on a pressure
level, rather than an altitude level basis, however, the standard deviations appear to be somewhat high, as are the means.
The conclusions
resulting from this data are to be considered as tentative until such
time as the data are analyzed in more detail.
Two causes of error were
suggested by Sissenwine et al;first, that the data used in many other
reports relies on more altitudes.
In addition to the "standard" pres-
sure levels, data are also evaluated at so-called "critical" levels,
where the temperature lapse rate is observed to change.
The lack of
this data would tend to explain the systematically high numerical
values of correlation coefficient since there are fewer causes for
statistical independence in the reported data.
error lies in the treatment of missing data.
The second source of
It would be wiser, per-
haps, simply to concede that the data is gone, since the linear extrapolations will spread apart as one increases altitude.
This results
in abnormally large values of standard deviation, which are observed
particularly for density, at high altitudes in this data.
Both of
these phenomena indicate that the numerical values reported in Chapter
5 will be too large.
The computed results are displayed in Appendix B.*
*It will be noted that, rather than the matrix V itself, the normalized version of V, i.e.
the correlation coefficient matrix R was
used, where Rij= Fij/ C-i
a-j
0=1
55
The problem of what to use for vertical winds is very difficult.
Data on vertical winds taken simultaneously with horizontal winds is
just beginning to be available.*
In conversation with N. Sissenwine,
A. Cole and I. Lund of AFCRL, two causes of vertical winds were suggested:
a)
Clear air turbulence occurs from 20,000 ft to 40,000 ft in
a layer 2000 to 5000 ft thick.
Within this layer, the air rotates in
vortices whose axes are horizontal.
20 to 50 ft/sec.
The maximum wind speed is from
This is to be considered the maximum.
This descrip-
tion must be considered as very tentative, and open to controversy.
It is not more than a place to start, to get order of magnitude effects.
b)
Fortunately, order of magnitude effects suffice.
Convective conditions, such as thunderclouds, cause gusts
which may be described as occurring anywhere from 2000 ft to the
tropopause.
They will be 5000 to 10,000 ft thick, and will have a
magnitude on the order of 50 ft/sec.
Vertical winds in these altitude regions produce negligible
effect on dispersion as the "row 7" influence coefficients in Appendix A. show.
This conclusion is strengthened when one realizes
that the probability is far from one that these vertical wind profiles
will be encountered.
It is only the possibility of moderate to strong correlation
with other components that motivates the decision to include the
vertical wind influence coefficients in the adjoint program.
work done on this problem is contained :W an article
*A summary
by Crutcher , which the author was made aware of too late for
specific inclusion here.
56
For this study, the contribution of vertical winds will be neglected in the computation of CEP.
The case of drag coefficient uncertainties is also very difficult
to describe; unfortunately, there is no evidence that will be small.
The author has chosen arbitrarily a 2% value as one
(7',
with the Mach
number bands broken into a subsonic band, extending to M=O.5, a transonic band extending from M=O.5 to M=l.4, a low supersonic band extending from M=1 . 4 to M=4, a high supersonic band extending from M=4
to M=12 and a hypersonic band from M=12 on.
These were chosen
by
eye from Figure 6 as being bands within which the character of the
CD vs M curve does not seem to change much.
These sets of conditions
are assumed statistically independent, to ease the computational
problem.
In any realistic case the statistical nature of the uncertainties
in the drag coefficient curve must be analyzed in detail, considering
theoretical prediction techniques and their uncertainties, wind
tunnel, shock tube and similar tests, their uncertainties, flight
tests, both successful and not and the imponderable, "experience".
This is a task beyond the scope of this work.
57
4.3
Adequacy of the Data
Since no reasonable drag coefficient date could be obtained, the
adequacy of the assumption made in the last section cannot be commented
on.
With the knowledge of vertical winds in its present state, one
must restrict one's attention to four of the components listed originally:
density, speed-of-sound, north wind and east wind.
Two questions must be asked concerning the adequacy of these data:
first, how accurate are the data available, and second, what is the
best way of covering the blank spaces in the data.
As a matter of con-
venience for the discussion which follows, since atmospheric data will
be reported both in terms of pressure levels and in terms of altitude,
the following table based on Reference 19 gives a comparison between
the methods of reporting.
TABLE 3
"
Altitudes for Commonly Used Pressure Levels
p (Millibars)
h(ft) (to nearest 100 ft)
850
700
500
300
200
100
50
30
4,800
9,900
18,300
30,100
38,700
53,200
67,600
78,500
20
10
87,200
104,000
No attempt has been made to overcome or evaluate the effect of the
possible inadequacies listed here; neither has an attempt been made to
be comprehensive.
It is highly probable that some of these problems
58
are already solved in the meterological literature, but it is equally
certain that some are very much open questions.
sented in the same spirit as was Section 2.2:
This section is prean unsolved problem is
here, and work may be worthwhile.
There are two places where such data may break down:
the accur-
acy of the device which takes the measurement and the adequacy of the
way in which the atmosphere was sampled.
4.3.1
Instrumentation Problems
The overwhelming majority of measurements are taken by radio-
sonde measurements.
Current U.S. instrument accuracies are listed in
Reference 22, from which Table 4 was prepared.
TABLE 4
Accuracy of U.S. Density Measurements
altitude (kn)
6 km
1
Density Error
0.3%
12 km
18 km
0.5%
0.7%
24 km
0.9%
30 km
1.2%
23
The case for winds is claimed to be worse, since France
indi-
cates that + 10 knots may be an upper limit on uncertainty but that
this is not a well known area.
23
although Beelitz
The accuracy of Soviet equipment is not known
27
indicate that it is biased relative to Western equipment, a
et al
fact which should be kept in mind.
The second problem relative to equipment was first investigated
59
26
by Court
and is that wind data at high altitudes are biased in fav-
or of low winds because high winds blow the balloon out of range of the
ground tracking equipment.
Court considers the magnitude of this bias
by observing mean winds and their standard deviations at zero and one
kilometer altitudes at Patrick AFB, Florida by decreasing his sample
size in a selective manner.
This selection was made by including only those values resulting
from soundings that had dats for progressively higher altitudes.
ing at autumn, one finds data as shown in Figure 12.
Look-
While these data
indicate a serious problem, it should be noted that the last few years
has seen a great improvement in tracking procedures, and this problem
should be reviewed in light of this fact.
The next problem is that in general, there is no data over 100 mb
for areas of interest.*
This was handled in this paper by assuming a
constant temperature lapse rate from wherever that data stopped to
30 km.
Two other suggestions have been made for filling in data which
may have merit, and should be tried.
First, for densities, it has been suggested that the standard
deviation of pressure at an altitude might be used to simulate the
standard deviations of densities above that altitude, since pressure
at an altitude results from the integration of density
of the
A notable exception is the data supplied during the IGY.
.
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63
influence function curve may be worked out.
Sissenwine et al have
generated data which may be used to check this method by comparing
data from Reference 21 with those of Reference 22 for the same statThe comparison will not be perfect since the periods of record
ion.
differ, but a start can be made.
Second, it has been suggested that the missing data be filled
in above 100 mb by substituting data from friendly stations where
data at higher levels is available at approximately the same latitude.
The results obtained by Cole and Nee
indicate that this idea is
worthy of consideration.
Sampling Problems
4.3.2
The measurement of statistical parameters of the atmosphere may
be looked on as an experiment in which measurements are to be taken.
As such, the question of proper selection of the measurements to be
taken and of a proper interpretation of their statistical significance is important.
There are at least seven ways to go wrong in sel-
ection of observations.
First, the overall period of record may not be long enough.
The
data of Hering and Salmela 30 suggests the possibility of long term
variations in atmospheric parameters.
"Although the flow pattern in
this altitude range (100 mb-25 mb) is characteristically persistent
for rather long time periods, marked differences may be noted in the
prevailing circulation for winter months of various years."
The
example chosen is the 50 mb map for January 1953 and for January
1957.
ERR
64
Second, individual measurements may be spaced too far apart, .usualy, radiosonde measurements are taken twice a day, at the same time
each day.
For example, Charles
25
has taken measurements at 1500 GMT.
Considering the diurnal variation in the atmosphere, this may not be
often enough.
As an example of this, the low level jet stream, long
known to light plane pilots in the American Southwest, was not well
known to meteorologists because it existed just before dawn and radiosonde measurements did not exist at that time.
It was only the con-
struction of an extremely tall television antenna in Texas, with continuously operating wind and temperature recorders which made Iszumi's
description 31 of the phenomenon possible.
The diurnal variation problem is worst at the ground and such
variations will become weaker as one rises.
This problem will achieve
its maximum effect in the case of moderate performance R/V-trajectory
combinations, where impact is in the transonic to low supersonic region
and influence coefficients for this altitude region are strongest.
Here again something must be done; whether showing that, for a
few typical stations, the problem is negligible and extrapolating this
conclusion to all stations (which has the advantage that there will be
no intelligence problem) or that the variations at a few stations are
still easily extended to all stations even though not negligible, or
having to mount a massive attack on the problem, is not known.
tainly our understanding of this problem area is in its infancy.
CerOne
possible way of determining the significance of this diurnal variation
would be to take a series of measurements once every third day (to
65
eliminate persistence) but taken at a random time of the day and comparing the statistical parameters resulting with those resulting from conventional sampling techniques.
This would produce statistics which are
adequate for CEP purposes assuming that vehicles are not retargeted on
an hourly basis, at cost much less than the means indicated by Court and
Salmela 32 (which incidentally Will give an idea of what is involved in
neglecting this parameter, although the period of record precludes meaningful statistical data.
It also gives an idea of what is available in
the area of short (less than 6 hours) interval rawinsonde measurements.)
Third, simultaneously, in a sense, the interval between observations
may be too short owing to problems of persistence.
Persistence of an
observation has been well studied and seems to continue for 24 to 36
hours (Ref. 21).
To be on the safe side, in counting the number of in-
dependent observations in a sample, one should not assume observation to
be independent until at least 48 hours have passed.
this study, 72 hours were allowed to pass.
For the purpose of
This is most important in
ascribing confidence levels to statistical parameters generated.
As
previously indicated, the values 24 to 36 hours are very rough, and depend strongly on season and on the specific quantity investigated.
Fourth, there is the question of how fine to cut the statistics.
Should they be annual, seasonal, monthly or what?
ing is:
The tradeoff govern-
As one gets finer and finer increments, the size of the stan-
dard deviation decreases because more and more of the uncertainty in the
atmosphere becomes bias:
The January mean is better for estimating the
January 17 density profile than the Winter mean, which in turn is better
66
than the annual mean.
On the other hand, the number of measurements
available for January is only 1/3 of those available for winter, so
that while the standard deviation decreases, one's confidence in it
also decreases.
Presently the tradeoff point seems to be at the seasonal level,
judging from the available published data (seefor example, France 23,
24 )and that is what will be used here for numerical purposes.
As more
data and more understanding of required data is gained, this may change.
Fifth, the selection of break points can be significant.
The break-
ing of statistics into seasons of exactly three months (December, January, February, = winter, etc.) is arbitrary and is not always the best
way to minimize the statistical variation within a time period.
This is
most evident when one is faced with swift changes in climate such as
the onset of the monsoon in India and other parts of Asia or the less
well known, much more significant for our purposes and descriptively
titled phenomenon of "explosive warming" found in the Canadian stratosphere and described by Craig and Hering.
mentions this problem.
In Reference 24, France also
How far one wished to go in this matter is some-
thing for research to decide.
Probably people should select with too
much care at least once.
Sixth, collections of raw data always have holes in them resulting from equipment malfunction, etc.
The "raw" date reported on are
elaborately smoothed to the extent that such missing holes are not infrequently filled in, using the weather maps of the day to make estimates of such date. Charles 25 and Beelitz 2
and in Beelitz the prodess -used is given.
indicate this is done,
67
The
Seventh, as was noted, radiosondes drift with the wind.
reentry vehicle reenters over a track on the order of 200 miles long
for a reentry flight path angle of 200.
es:
Therefore, the question aris-
what is the effect of the fact that the reentry vehicle exper-
iences a different atmospheric sample from the sample experienced by
the radiosonde, because of the different bearings the two objects take
as they move relative to the target?
34i
This problem probably is negligible, as Buell,
has shown that,
to a first approximation, correlations are quite high (in excess of
.8 at the minimum) between winds at the same isobaric surface for distances of this order.
One may hope that the correlations for densit-
ies have the same order of magnitude of distance in their scaling law.
4.4
An Alternate Approach to the Problem
A second method of approaching the wind and density problem exists:
to generate a synthetic wind profiles'uused in missile guidance system
analysis and structural response analysis (e.g. the Trembath wind profile 3 5 ).
These, being designed to provide a bad wind shear environ-
ment, are not applicable to the CEP problem.
The author does not feel
this is a useful method, since it is difficult if not impossible to
design a synthetic profile for CEP analysis which is independent of
the R/V, and therefore it is necessary to reevaluate the profile every
time the R/V is significantly altered.
For the purpose of obtaining
a design number, one may find this procedure attractive since it requires less computer input and less release of sensitive information
to contractors, also since it makes proposal-type numbers more
68
directly comparable in design evaluation.
The present method allows one to uncouple the reentry vehicle influence from the atmosphere's statistical quality.
phere need be described only once.
Thus, the atmos-
Papers such as that of Beiber36
indicate the increasing favor which the present method has for guidance and structural analysis, especially for research vehicles, where
there is effectively only one launch site and therefore relatively few
statistics required to describe the atmosphere.
MR
69
CHAPTER 5
RESULTS, CONCLUSIONS AND RECOMMENDATIONS
5.1
Scope of the Study
rectangular grid of 15 reentry trajectories was chosen, with
A
velocities from 15,000 ft/sec to 25,000 ft/sec and with flight path
angles
(
'
e) from -200 to -400, as shown by the circles on Figure 13.
As will be seen, these conditions correspond to ranges above the
atmosphere (spherical, non-rotating earth) of from less than 1500
nautical miles to more than 7,500 nautical miles.
From a launch site
in the North Central U. S., no part of the Soviet Union is out of
range of these trajectories.
Further, most of these trajectories
are on the lofted side of the minimum energy line.
Two well separated W/CDA's were chosen, to shown the effect
They are 1000 lb/ft 2 and 2500 lb/ft2
.
of changing this parameter.
Only one drag curve was used., shown in Figure 6.
to a slender conical shape with a sharp nose.
It corresponds
The vehicle was
assumed to be non-ablating as the ablative portions of the influence
coefficient program have not yet worked adequately.
This is a serious
fault of the implementation of the method detailed.
Ablative effects
are three in number:
First, as material ablates the vehicle looses
mass, thereby reducing W/CDA
the CD changes owing to
and increasing dispersion.
Second,
the increased bluntness of the tip, again
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71
reducing W/CDA.
Third, the mass injection will have some effect on
CD directly.
These effects are most important for a pointed shape, such as
was chosen here.
tially less.
For a blunt nosed shape, their effect. is substan-
Also, the magnitude of the effect will be dependent in
large part on the nature of the heat shield.
Influence coefficients
were obtained for the five quantities of interest to the study for
each of 30 trajectories;* the drag coefficient in the bands indicated
in Section 4.2 and the altitudes in 2 kilometer bands beginning at
the ground and continuing to 32 kilometers.
Above this point atmos-
pheric statistical data are not available generally,
5.2
Effects of Neglecting Influence Coefficient
The effect of completely neglecting the statistical variation
of the atmosphere above 32 km. was investigated at both values of
W/CDA and was found to be strongest for density and next strongest
for winds.
This was done by dividing the influence coefficient
for the region reentry to 32 km altitude by the influence coefficient
from reentry to ground.
The maximum values for density were found
in the upper right hand corner of the V- 1 map and were approximately
4.5% for the W/CDA
W/CDA
1000 lb/ft
= 2500 lb/ft2 0
and approximately 101 for the
These results are shown in Figure 14a and b.
The reason for this trend, and for the dramatically different charac-
ter of Figures 14a and
*Actually 29 cases.
1 4b,
is that there is a large increase in
The W/CDA = 1000 lb/ft2 Ve = 15,000 ft/sec and
te % -35O case was lost due to an input error.
This error was not
retrieved since it was discovered that the plots did not require the
inclusion of the point.
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r
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.
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1
74*
dispersion due to density at low altitudes which result from the
CD
factor and the d CD/dM term in B(ll) (See Figure 9).
The correlation
between impact Mach number and the percentage of density dispersion
shown below 100,000 feet is striking.
Cases 4, 5 and 10 (Figure 13)
are the only ones which showed supersonic impact in tne W/CDA = 1000
series, and there is a slight increase in impact Mach number toward
the lower left which corresponds closely with the shape of Figure 14a.
In Figure
shown.
1 4b,
all impacts are supersonic, and no irregularities are
The Mach numbers at impact are shown in Figures 15a and 15b.
The lower left hand portion of Figure 15a was virtually flat, and
plotting was not possible, however, a very slight increase was observed as one moved toward Case 11.
The correlation between high impact velocities and high percentages of density influence at altitudes where the statistics are
unknown is fortunate, for it is at these high impact velocities
that density has its smallest relative contribution to the whole
of the downrange dispersion.
To show this, the effect of perturbing
density by a uniform 1% was compared with the effect of a one foot
per second downrange wind and the resultant ratio shown in Figures 1 6 a
and 16b.
The data of Figures Pa
statistics of the
and Pb are optimistic since the actual
atmosphere are not considered.
These will increase
the indicated percentages for two reasons:
First,
generally increases with altitude at high altitudes
Second, the use of several bands in lower altitude ranges will tend
to decrease the contribution of the lower altitude ranges.
Therefore,
r~-
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79
the true percentage of influence above 32 km may be expected to be
higher than reported.
The percentage differences for speed of sound
were found to be negligible, as would be expected.
5.3
CEP for Various Combinations of Statistics
The basic CEP's shown in Figures 17a and l7b are for the full set
of Moscow statistics and 2% variation in CD as one standard deviation.
It will be noted that, as would be expected, CEP's increase with decreasing velocity, with the exception of a small region in the lower
left hand corner.
The reason for this effect is bound up with the
the increase with decreasing velocity will become less important at
,
large values of influence coefficient at low Mach numbers; therefore,
higher W/CDA and also for the same W/CDA for a lower drag rise shape0
The values of CEP for cases 1, 6 and 11, at the far left of the
figure are tabulated in Table 5.
Figures 17a and 17b and Tables 5 and 6 include only the effects
studied here, i.e. horizontal winds, density, drag coefficient and
speed of sound.
Therefore, the CEP's reported in Figure 17 and in
Table 6 are to be taken with a grain of salt.
tending effects at work:
There are two con-
first, the atmospheric statistics probably
are biased to give too large estimates of CEP and second, there are
many excluded contributors to CEP which tends to make any estimate
of CEp computed here too small.
Some idea of the effect on CEP of changing the accuracy with
which CD is known may be gathered by changing the percentaogo used u5
a 1 Crvalues of CD uncertainty0
The improvement resulting from re-
ducing 2% to 1o was investigated, as was the degradation resulting
4_
+1
-1
r
>'4'"
._
4444
7W77YVF
1
t'
-
__
-.4
AC
T_
H
44-4444 bi4jrnjj
T
-
4
p
+ I 'T
,14:T
r-TJ+
4-4
44 211
''1LL2i~it~A~
T11 I L I
7VFI17z
4t
_TI7FT
4-
-i-
9
f-t
It $1
*1 -
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l+tU
1
i fj
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2-Li'
rwf7N+~- 1
[a-'P4
br4 LW tl~
itt jz4j
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7
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I~
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~
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Wi#~Ht4zUI_4zi4
t ft
Wflt-4#7 Iii
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S
71442-t 1171171
-1 A
4
-
2
5'
44
4H
T
f
.--.
-
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a
I
8
tiii
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1
k4141
14
4;a
JK
4W4 4 1
j,
4
t
4
~4
T
1
t
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4
4_1
h1f
TI
I t
211
4'
{ 4
L44
t
1
4t4
L4
f4
_
Ii
82
from an increase to 4>.
Considering the drastic change in CEP re-
2
2
sulting from changing W/CA from 1000 lb/ft to 2500 lb/ft , the
results were comparatively constant.
Their maxima are reported in
Table 6.
TABLE 5
Atmospheric CEP at
Ve (ft/sec) W/CA
1000 lb/ft 2
=
-200
W/CA
2500 lb/ft2
25000
2481
237
20000
2500
278
15000
2433
304
The effect of not using certain of the less commonly available
(and therefore, less well studied) atmospheric parameters was investigated by first blocking out the terms in the atmospheric matrix
pertaining to speed of sound and second by additionally blocking
out wind-density correlations.
These data also are reported in
Table 6, as the percentage increase in CE
resulting from including
these terms if they had not been included previously.
As before,
the percentage change in weapon yield and in vehicle weight (assuming
R/V weight is proportional to weapon weight) which is required to
keep the value of kill radius/CEP constant in the face of these
changes are included.
5.4
Conclusions
5.4.1
General:
While the adjoint method of Bliss is not yet fully worked
out for all
cases of interest to which it
is well adapted (notably
dispersion due to ablation) it has shown itself to be a quick,
TABLE 6
CAUSE
W/CDA
Decrease in CD
1000
4
-2.86
-8.45
-6.26
uncertainty to 1:
2500
11
-2.21
-6-78
-5.05
Increase in CO
uncertainty to 4%
1000
2500
3
11
10.97
8.37
36.7
27.3
26.4
19.7
Inclusion of speed
of sound terms for
the first ti.me
1000
2500
11
11
2.05
1.09
6.29
3.31
Inclusion of speed
of sound and of
wind- density correlation terms for
the first time
1000
2500
6
8
CASE
CEP
CHANGE,
10.4
6.9
%
YIELD CHANGE, %
33.1
22.15
R/V WEIGHT CHANGE,
4.84
2.48
23.9
16.2
%
MAXIMUM EFFECT ON ATMOSPHERIC CEP OF ALTERING CERTAIN
ELEMEN'S OF THE COVARIANCE MATRIX V
00
84
economical and accurate way of computing several components of reentry
For the thirty cases of this study (including, for this
vehicle CEP.
purpose, the improperly run one) the total machine time (using an
IBM 7O94-I) used for getting the required influence coefficients
was of the order of 9.5 minutes as opposed to about 95 minutes using
the approach of subtracting impact points gained from non-linear
trajectories.*
Further, experience gained in the checkout phases of
this job, together with an understanding of how Equation 2.27 is
integrated indicates that the only expense involved in increasing the
number of bands pointed out is in the printing out itself and the
additional choices of 4 t required to make print steps coincide with
integration steps.
From this, one may conclude that, while much
still can be done and should be done toward improving the efficiency
and accuracy of the adjoint program, the greatest gains will be made
by investigating the other causes of CEP and the missin,. atmospheric
statistics.
It is concluded that the statistics available in the open
literature are not adequate to do an acceptable job of computing
CEP without making many assumptions, the effects of which are only
beginning to be analyzed from a CEP viewpoint.
Comparison of the
data in Appendix B with data obtained by others shows that the use
of constant lapse rates to extrapolate temperature and winds is
not adequate, also that not including the data at "critical" levels
has a startling effect on correlation coefficient.
*Fortunately, the latter number is only an estimate.
85
5.4.2
Design:
The amount of drag rise in the transonic region is of overriding importance to CEP except for very high performance vehicles;
it may well be that there are shapes which, while they do not have
as good performance at the hypersonic end of the CD curve, where
W/CDA is defined, have sufficiently better performance where drag
rise is concerned to make up for the reduction in W/CDA.
5.4.3
Procedure:
Which of the atmospheric parameters may be neglected depends
on the uncertainty with which kill radius is }nown.
Obviously, if
kill radius is only known to, say, :+ 10, a reduction in th
uncer-
tainty with which CEP is known from 35o to 24 isn't ";oin; v
:e
much in terms of how well kill probability is known.
However, Iere
are some conclusions which may be reached as to the order of attack:
Probably the CD uncertainty (1 C-) at which the point of diminishing CEP returns is reached is on the order of 3%, although more can
be tolerated if analysis shows that the Mach number regime from 4 on
up can be divided into more statistically independent bands.
The
next statement that can be made is that, winds are the most important
contributors to CEP, and greatest effort should be concentrated on
getting good wind information.
5.5
Recommendations for Further Study
5.5.1
The trajectory Model:
The first cut at the effects of ablation should be included.
The level to which this may be done probably will be limited by
economics as a full ablation analysis is long, complex and costly.
86
There is a possibility that winds induce a residual lift that
is not "averaged out" by the vehicle's spin.
This should be investi-
gated, particularly for high L/D shapes such as this study used.
5.5.2
The Computer Model
For various reasons (convenience in plotting, etc.) it
has
been customary to print out trajectory histories in equal times or
altitude increments.
This practice was carried over into the data
storage of the programs reported here.
and in fact it may be undesireable.
There is no need for this,
By printing (or storing) at
each integration interval, one does not perform unnecessary computation to match print and integration steps, one does not create data
that are not needed
for accuracy, and conversely one does not
ignore data that are needed where the data to be integrated vary
rapidly.
Some idea of the cost involved in matching print steps
and integration steps can be seen when it
is realized that, in comnu-
ting the nominal trajectories with printout every 1 second, the
machine time increased by about 50% over what was required when only
the initial conditions and impact conditions were printed out,
If a special purpose operational CEP calculation program is
to be written, the segmentation into two parts linked by tape
probably is not desireable.
It is not needed on a 32,768 word
machine, and tape is a slow storage medium.
The precomputation of the matrices B and C should be reviewed,
as should the specific integration scheme used.
Lastly, the usual
cleaning up of the programs should be made, such as removing
unnecessary instructions and the adjustment of integration controls
87
to optimize the program's running.
should be relatively sinple from an analytical standpoint
to expand the analysis to include the airburst case
(
It
conditions) essentially by changing the initial conditions on
.
it to compute the perturbations in any interestin. -erminal
(Perhaps one may have to introduce addition state variables, as
well, such as t.).
Finally, as a more remote possibility, the adjoint process
described here is one half of the interactive optimization scheme
known as
'the meghod of steepest ascent."
Should it prove
feasible to determine the exchange coefficients between zmall
changes in vehicle shape and CD vs M curve, the possibility of
closing the loop by shaping the drag curve for minimum CEP may
prove fruitful.
This possibility rests on the potential tradeoff
between W/CDA and the amount of drag rise noted in Section 5,4.2.
5.5.3
Computation of Meteorological Matrices
Suggestions for improving the raw data will be found in Section
4.2 and 4.3.
There are three things which should be done in the
area of reducing this data.
First, there is nothing sacred about constant width altitude
bands, just as there is nothing sacred about three month seasons.
Other band widths should be tried, both from the standpoint of reducing the number of bands required and from the standpoint of
improving the accuracy of the data.
Second, the means of extrapolating data above the altitudes
where it is available should be reviewed.
As was indicated the
88
constant lapse rate assumption used here probably will not suffice.
Third*, while these data are correlated, there is a linear
combination of them which are not correlated, obtained as the
components of the eigenvectors of the covariance matrix V.
The
possibility exists that these data will present a comparatively
few independent random variables, which may be more meaningful
particularly if
noise.
the less important ones can be interpreted as
The implications of this possible condition on the validity
of applying the Central Limit Theorem to the problem should not be
overlooked.
The Target Model:
5.5.4
Three things were assumed in order to make this analysis
feasible.
First:
The restriction to ground burst fuzing does not present
any great mathematical difficulty to remove as far as computing CEP
itself goes.
However, the inclusion of airburst error into CEP
does present a problem in that it
cally.
links CEP and kill radius statisti-
Since CEP itself is of interest only as an intermediary to
obtaining kill probability, consideration should be given to determining kill probability directly from the detonation point probability
ellipsoid, something which should be entirely feasible with the
adjoint method of analysis.
Second:
In connection with the above, the possibility of
The author is indebted to N. Sissenwine and A. Cole for raising
this point in conversations with the author.
*
11
89
including area targets with variable shapes, sizes and priorities
of regions within (a power plant is more important than a suburb)
raises considerable room for investigation.
Finally, the most challenging relaxation analytically results
when the target is allowed to shoot back.
The tradeoffs involved
for the offense are essentially how much payload goes to increasing
kill radius, how much to reducing CEP (high W/CDA shape, attitude
control system etc.) and how much to keeping the defense from
knocking you down (penetration aids:
decoys etc.).
The various
combinations of offensive and defensive doctrines available should
keep high-speed computer stockholders happy for a long time.
90
APPENDIX A
SAMPLE COMPUTER OUTPUT FROM ADJOINT PROGRAM
Four cases were included to show the output and its format.
They are from the W/CDA ~ 1000 set, and do not include effects
above 32 km.
TABLE A-1
REENTRY CONDITIONS FOR CASES SHOWN
CASE NO.
Ve
e
1
25,000 ft/sec
-200
5
25,000 ft/sec
-40
11
15,000 ft/sec
-200
15
15,000 ft/sec
-4O
For each case, 6 pages of output are presented.
The fourth
row is reserved for ablation when it becomes available.
For each case the column KK is simply an output count;
the column ALTITUDE (FT)
is the altitude of the bottom if the
appropriate band, and DOWNRANGE and CROSSRANGE
coefficients for the respective bands.
are the influence
The upper altitude of the
91
uppermost band is 104992.0 ft. (32 km).
In the case of CD vs M, the altitude printed out corresponds
to the altitude at which one passes out of the bottom of the
corresponding Mach number band.
The Mach number bands are slightly
discrepant from their description in Section 3.2 in that the transonic band is here subdivided in two at M ~ 1.1.
Therefore there are
two transonic components to the CD influence coefficients rather
than one, and the fourth and fifth contributions should be added
together.
Finally, it should be noted that these data do not extend to
300,000 ft., as did the check runs in Chapter 3, so cases 1 and 11
here are not directly comparable to the data presented in the check
cases.
ROW NUMBER
ALTITUDE(FT.)
1
DENSITY
DOWNRANGE
CROSSRANGE
98430.
-0. 3637E+01
-0.
91868.
-0.4805E+01
-0.
85306.
-0.6384E+D1
78744.
-0.8527E+01
72182.
-0.1 14 E+02
65620.
-0.1-534E+62
.7
59058.
-0.2060E+02
8
52496.
-0.2756E+02
9
4 59344
-0.36862+02
10
39372.
-0.4933E+02
32810.
-0.6522E+02
26248.
-0.3176E+02
19686.
-09952E+02
14
13124.
-0.1111E+03
15
6%2.
-0.2302E+02
16
C.
-0.6180E+02
KK
-0.
-0.
3-
-0.
-0.
-0.
-0.
-0.
-0.
-0.
12
-0.
CASE 1
V
25,000 ft/sec
= -2 0 0
92
-0.
0.
-0.
SPEMi
ROW NLMOER 2
~
SCUND
DUWNRANGE
CROS SRANW
98430.
-0.2685E-C1
-0.
18
91868.
-0.3524F-01
0.
19
853)6.
433E -01
--
20
78744.
-0 .6088E-01
21
72182.
-0 .756E-01
22
65626.
-0 .1033E-0
23
59058.
.1317E-00
-0.
24
52496.
^ .?42RE-00
-3.
KK
ALTITUDE(FT.)
17
_
-0.
-
-
OF
45934.
26
39372.
l-0.
.17OF2
27
3281C,.
28
2629.
29
19686.
30
13124.
31
6562.
32
0.
0.
.P914
F
-12~52Er+01
CASE I
V = 25,0 00 ft/ sec
=
-200
93
-0
ROW NUMBER 3
DRAG COEFFICIENT
DOWNRANGE
CROSSRANGE
33843.
-0.2365E+03
-0.
34
17790.
-0.2378E+03
-0.
35
13196.
-0.7766E+02
-0.
36
5298.
-0.5093E+02
-0.
37
0.
-0.2153E+01
-0.
KK
ALTITUDE(FT.)
33
CASE 1
V
25.000 ft/sec
=-20
94
ROW NLMBER 5
DOWNRANGE WIND
KK
ALTITUDE(FT.)
42
98430.
0.105&E-O0
-0.
43
91868.
0.134!E-00
-0.
44
85306.
0.171!E-00
-0.
45
78744.
0.2189E-00
-0.
46
72182.
0.2802E-00
-0.
47
656?0.
0.3609E-00
-0.
48
59058.
0.4680t-00
-0.
49
52496.
0.6094E+00
-0.
50
45934.
(.809E+0O
-0.
51
39372.
0.1089E+01
-0.
52
32810.
0.1496E+01
-0.
53
26248.
0.1874E+01
-0.
54
19686.
0.2430E+01
-0.
55
13124.
0.6455E+01
-0.
56
6562
0.2747E+02
-0.
-0.3202E+02
-0.
57
DOWNRANGE
0.
CASE I
V
25,000 ft/sec
=-200
95
CROSSRANGE
CROSSRANGE WINO
ROW NUMBER 6
DOWNRANGE
ALT ITUDE C T. T)
CROSSR ANGE
98430.
-p.
6 .1030F-0W
91868.
-0.
0.1316E-0
85306.
-C,.
0.916,80E-00
78744.
-0.
0.2156E-60
7Z182.
-p.
65620.
-0.
0.3546E-00
0.4567E200
59028
-p.
67
-0.
52496.
45934.
0.5872E+O0
O w76C4:E+00
3932.
-0.
0.9944E+00
68
32810'.
-0.
0.1325E 01
69
26248.
-0.
0.1780E+01
19686.
-0.
O.2857E+'Ol
13124.
-0.
0.7137ME01
ILa
6562.
, ,.0
0 *15 71E
-0.
0.
CASE 1
V = 25,000 ft/see
2= -20
96
0.4567E+01
ROW NMBER
KK
ALTITUDE(FT.
I
VERTICAL
COWNRANGE
CROSSRANGE
98430.
0.202 8E-00
-0.
75
91868.
0.2518E-00
-0.
76
85306.
O.3121PE-00
-0
77
78744.
0.3859E-00
-0.
78
72 82.
0.4731E-00
-0.
79
65620.
0.5770E+00
-0.
80
59C058.
0.6978E +00
-0.
81
52496.
0.8284E+00
-0.
82
45934.
0.9598E+00
-0.
83
39372.
0.1101E+01
-0.
84
32810.
0.1237E+01
-0.
85
26248.
0. 1423E+01
-0.
86
19686.
0.1989E+01
-0.
87
13124.
0.3384E+01
-0.
88
6562.
-0.5870E+01
89
C.
0.2341E+02
CASE I
25 000 ft/sec
=-20
97
.
74
-0.
-0.
ROW NUIER 1
KK
2
3
ALTITUDE
DENSITY
T)
DOWNRANGE
430*
036:9--
91868.
CROSSRANGE
0-0
-0.
-0.1729E-00
-0.
6.
-0. 2200F-00.
4
78744.
-0.2754E-00
-0.
6
656200.
-044B4E00
-0.
...
~056
7
8
52496.
9
45934
0#58 62 E *0
-0.7144E+00
0.
-0.
M63E+00
10
39372.
-0.1099E+01
-0.
11
32810.
-0.1326E+01
-0.
12
26248.
-0.1502E+01
-0.
3
1984
-0
14
13124.
-0.1588E+01
-0.
15
65620.
-0.1336E-+01
-0.
16
0.
-0*4833E-00
-0.
CASE 5
V = 25,000 ft/seC
0
= -4 0
98
00
ROW NUMBER
SPEED OF SOUND
2
0 OSRANGE
DOWNRANGE
198430.
-0
9 8
-0.
.1026E-
-0.
*-6,1294E-02
~-0,1640E-020
*
-0.
*-0.2041E-02
~-0.2570'E-02-0
*
-0. 3972E-O2-.
9058'
Z.
24
0.
-6.3232-02
6562*
52496.
-0.4806C-02
-0.
25494*
-0.566
-0.
-0
.0
1 -* ....
Z63
0.
321D
188 7E-0
-0.2087-0-0
29---- 9
660-169nE+0
31
0.1454E+
3*0
CASE 5
V = 25,000 ft/s-ec
=
4
99
-0.
-0.
[
RO NU
KK
ER 3
ALTITUDE(FT.)
DOWNRANE
3082.
-Q.4015F+O1
6 .966
CASE 5
V = 25,000ft/sec
0
= -4 0
0o0
CROSSRANGE
-. 5
090:
*
34
ORAG COEFIC ENT
-4
-0.
-0
ROW NUMBER 5
KK
DOWNRANGE WIND
DOWNRANGE
ALTITUDE(FT.)
CROSSRANGE
40
98430.
0.3280E-01
-0.
41
91868.
O.4186E-01
-0.
4
85306.
0.5384E-01
-0.
43
78744.
0.6781E-01
-0.
44
72182.
0.8687F-01
-0.
45
65620.
0.1124E-00
-0.
46
59058.
Q.1433E-00
-0.
47
52496.
0.1842E-00
-0.
48
459134.
02313E-00
-0.
49
39372.
0.2964E-00
-0.
50
32810
0.3732E-00
-0.
51
26248.
0.4487E-00
-0.
52
19686.
0.5417F+00
-0.
53
13124.
0.6331E+00
-0.
54
6562.
O.7597E+00
-0.
55
0.
0.8281E+00
-0.
CASE 5
V = 25,000 ft/sec
=-400
101
CROSSRANGE WIND
ROW NUMBER 6
DOWNRANGE
ALTITUDE(FT.)
I
LRUSSRANGE~
98430.
-0.
0.3232E-01
91868.
-0.
0.41469-01
85306.
-0.
0.530E-01
78744.
-0.
0.6783E-01
T2i82.
-0.
0.866SE-01
61
65620.
-0.
S0 1109E-0
62
59058.
63
52496.
64
45934
560
0*e142 E -00
-0.
0.1810E-00
0.2303E-00
-0.
-p.
39372.
0,2932E-00
32810.
0 - 36'97E-00
67
26
0.4469E-00
68
19686.
69
13124.
-0.
-0
0.-6259 E +M
70
6562.
-0.
O.7622E+00
't8
0e5283E+00
0.7028E+00
CASE 5
V
25,000 ft/sec
-4 = 40
102
PEP
ROW NUMBER 7
VERTICAL
CROSSRANGE
ALTITUDE(FT.)
72
98430.
0.3732E-01
-0.
73
91868.
0.4764E-01
-0.
74
85306.
0.6128E-01
-0.
75
78744.
0.7715E-01
-0.
76
72182.
0.9880E-01
-0.
77
65620.
0.1278E-00
-0.
78
59058.
0.1627E-00
-0.
79
52496.
0.2088E-00
-0.
80
.
DOWNRANGE
KK
4593 4
0.2616E-00
-0.
81
39372.
0.3344E-00
-0.
82
32810.
0.4195E-00
-0.
83
26248.
0.5C17E+00
-0.
84
19686.
0.6047E+00
-0.
85
13124.
0.7133E+00
-0.
86
6562.
0. 8702E+00
-0.
87
0.
0.9C58E+00
-0.
CASE 5
V = 25s000 ft/sec
= -400
103
r
ROW NUt4BER I
KK
ALTITUDEF
194
30
2
91868.
3
853060
4
78744.
OENS1TY
DOWNRANGE
CROSSRANGE
-0.3794E01'
-0.
-0.4933E+O1
-0.
0.6447E+0
-0.
-0.8480e+01
-0.
0.11)8E+0Z
-00
6
65620.
-0.1483E+02
-0.
7
51050.
-0.1966 E-02
-0.
8
52496.
-0.2601E+02
-0.
9
45934.
-O.3444E+02
0.
10
39372.
-0.4549E+02
-0.
-0.5936E+02
-0.
12
26248.
-0.7404E+02
-0.
13
14686*
-0. 8700E+02
-0.
14
13124.
-0.8530E+02
-0.
16
6562.
-0.28975+02
0.
0.7945E+01
CASE 11
V = 15,000 ft/sec
S= -2 0 0
104
-0.
ROW NUMBER 2
KK
ALTITUDE(FT.)
SPEED OF SOUND
DOWNRANGE
CRO$SRANGE
17
98430.
-0. 4549E-01
-0.
is
91868.
-0.5887E-01
-0.
19
85306.
-0.6963E+0O0
20
78744.
-0.1275E+01
-0.
72182.
-0o.1644E+01
-0.
-0. 2109E+01
-0.
-0. 2658E+O1
-0.
23
52496.
-0. 3269E+01
-0.
Z5
45934.
-0. 440 8E+01
-0.
26
39372.
-0.2161E+02
-0.
-0
27
32810.
-0. 3626E+02
28
26248.
-0.5361E+02
29
19686.
-0.7229E+02
30
13124.
U.1646E+U3
31
6562.
-0.1068E+03
32
0.
0.5732E+00
.
24
-0.
-0.
-0.
CASE 11
V = 15,000 ft/sec
= -200
105
-0.
DRAG COEFFICIENT
ROW NUMBER 3
KK
ALTTUDE(FT.)
S51956.
34
23057.
DOWNRANGE
-0.9768E+02
-O.2548E+03
37
5271.
0.
-0.
0.
692 180.8622E+02
36
CROSSRANGE
-0.6727E+02
-O.1596+2
CASE 11
V = 15 000 ft/sec
= -200
106
-0.
0.
ROW NLMBER 5
KK
ALTITUDE(FT.)
42
843.
43
91868.
4
DOWNRANGE WIND
DOWNRANGE
CROSSRANGE
0.1402E-00
-0.
O.177CE-OC
-0.
530.
0.2203E-00
-0.
45
78744.
0.2778F-00
-0.
46
72182.
0.3534E-00
-0.
47
65620.
0.4529E-00
-0.
48
59058.
0.5829E+00
-,
49
52496.
O.7561E+00
-0.
50
45934.
0.9951E+00
-0.
51
39372.
0.1205E+01
-0.
52
32810.
0.1604E+01
-0.
53
26248.
0.2407E+01
-0.
19686.
0.3537E+01
-0.
55
13124.
0.1538E+02
-0.
56
6562.
0.6972E+01
-0.
57
0.
0.1410E+02
-0.
4
-
CASE 11
V = 15,000
.=
-20
107
ft/sec
CROSSRANGE WIND
ROW NUMBER 6
KK
9186 8
60
85.6
61
78744.
2rfl ]o 2
62
63
652.-0.O.14E
65
52,496.-.063EO
-0.
0. 1615E-0
0.
092044E-00
-0.
092593M-00
-0.
0*3290E-00
O0 6983E+O0
66
4 15934:,i
-0.
67
39312.
0
68
3280
-0.
69t
1 ~
.14EO
.15E
0, 2389E+01
26248.-0
04483E+01
-00
1968bu .
71
0., 1451E+OZ
132.-0.
6562*
'3
0. 1276E-00
-0.
898430
59
CROSSRANgE
DOWNRANGE
ALTITUDE(T.
V.
-00
L~?U I
.'tV~
IJ
vel+V;JILCT
L
-U.
U.
CASE JII
IS 000
V
-y
200
. .. 1
%* 1U
9
ROW
7
NUMBER 7
RD NUMBER
KK
ALTITUDG(FT.)
VEf~~ICAL
VERTICAL
DOWNRANGE
CROSSRANGE
98430.
O.2053E-O0
-0.
75
91868.
O.2517E-00
-0.
76
85306.
0.3091E-00
-0.
77
78744.
0.3776E-00
-0.
72182.
0.4553E-00
-0.
79
65620.
0.5448E+00
-0.
80
5905$
0.6413E+00
-0.
81
52496.
0.7384E+00
-0.
82
45934o
0.8370E+00
-0.
83
39372.
0.9880E+00
-0.
84
32810.
0.1142E+01
-0.
85
26248.
0.1432E+01
-0.
86
19686.
0.2394E+01
-0.
87
13124.
0.9332E+01
-0.
88
6562.
0.4124E+01
-0.
89
0.
-0.1259E+02
-0.
CASE 11
V
15,000 ft/sec
X = -20'
109
ROW NUMBER I
KK
ALTITUDE(FT.)
DENSITY
DOWNRANGE
CROSSRANGE
-
98430.
-0. 54985+00
-0.
2
91868.
-0.7211E+00
-0.
3
85306.
-0.9481E+00
-0.
4
78744.
-O.1243E+01
-0.
5
72182.
-O.1660E+01
-0.
6
6562Q.
-0.2198E+01
-0.
7
05,.
-0.2922E+01
-0.
8
52496.
-O.3879E+01
-0.
9
43934.
-0.5129E+01
-0.
10
39372.
-0.6779E+01
-0.
3280.
-0.8879E+01
-0.
12
26248.
-0.1111E+02
-0.
13
19686.
_0.1357F+02
-0.
14
13124.
-0.1626E+02
-0.
15
6562.
-0.1761E+02
-0.
16
0.
-0.9100E+01
-0.
1
CASE 15
V = 15,000
=-400
-110
ft/sec
ROW NUMBER
KK
2
AL! ITUUE (F T.)
SPEED OF SOUND
DOWNRANGE
CRSSR ANGE
17
98430.
-0.6674E-02
-0.
18
91868.
-0.8749E-02
-0.
19
85306.
-0.1147E-01
-0.
26'
78744.
-0.1497F-01
-0.
21
72182.
-0. 3293E-00
-0.
22
65620.
-0.3326E6-00
-0.
23
59058.
-0. 4290E-00
-0.
-0. 5447 E+OO
-0.
24
25
45934.
-0. 6786E+00
-0.
26
3 937 2.
-0. 1029E+01
-0.
27
32810m
-0. 2419E+01
-0.
28
26248.0
-0.3704E+U1
-p.
19686.
-0.1350E+02
-0.
13124.
-0.9491E+01
-0.
-0.
30
31
6562.
-0.1979E+02
32
p.
0.7972E+OOU
CASE 15
V= 15,000
T=
~J40
i40
ft/sec
-0.
ROU'
KK
A LTITUDOET)
DOWNRANGE
NUtER 3
DRAG COEFFICIENT
CROSSRANGE
40874s"
-0.2408E+02
-0.
34
11781.
-0.5468E+02
-0.
35
5298
-0. 1760E+O2
-0.
-0.5408E+01
-0.
36
0.
CASE 15
V = 15,000 ft/sec
= -40'
112
DOWNRANGE
ROW NUMBER 5
KK
DOWNRANGE
ALTITUDE(FT.)
WINC
CROSSRANGE
~41
98430.
0.5101E-01
-0.
42
91868.
0.6524E-01
-0.
43
85306.
0.8334E-01
-0.
44
78744.
0.1057E-OC
-0.
45
72182.
0.1348E-00
-0.
46
65620.
0.1725E-00
-0.
47
59058.
0.2214E-00
-0.
48
92496.
0.2845E-00
-0.
49
45934.
0.3642E-00
-0.
50
39372.
0.4658E-00
-0.
51
32810.
0.5998E+00
-0.
52
26248.
0.7692E+00
-00
53
19686.
0.9268E+00
-0.
54
13124.
0.1512E+01
-0.
55
65620
0.2448E+01
-0.
0.5366F+01
-0.
56
0.
CASE 15
V = 15,000
r=
-40'
113
ft/sec
ROW NUMBER 6
KK
AL TUD (FT.
DOWNRANGE
*430.
5
9
58
9168
CROSSRANGE WIND
CROSSR&NGE
-0.
0.492lf-Ol
-0.
0.62949-01
-0.
0.8015E-01
6
874.
-0.
0.1027E-00
61
S
2182
-0.
0.1307E-00
62
65620.
-0.
0. 1675E-00
63
59058.
-0.
0.2138E-00
64
52496.
-0.
0.2716E-00
65
45934.
-0.
0.3470E-00
66
39372.
-0.
0.4443E-00
67
32810.
-0.
0.5714E+00
68
26248.
-0.
0.7244E+00
0.
0*9558E+00
-0.
0.1458E+01
-0
0.2641E01
-0.
0.4777E+01
6
7013124
71
6562
72
0
CASE 15
V
15,000 ft/sec
= -40
114
ROW NUMBER 7
KK
ALTITUDE(FT.)
VERTICAL
COWNRANGE
CROSSRANGE
73
98430.
0.4734F-01
-0.
74
91868.
0.6001E-01
-0.
75
85306.
0.7587E-01
-0.
76
78744.
0.9498E-01
-0.
77
72182.
0.1221E-O0
-0.
78
65620.
u.1528E-00
-0.
79
59058
0.1920E-00
-0.
80
52496.
0.2402E-C0
-0.
81
45934.
O.2971E-00
-0.
82
39372.
0.3649F-00
-3.
83
32810.
0.4576F-00
-0.
84
26248.
O.5626E+00
-0.
85
19686.
0.7572E+00
-0.
86
13124.
0.1104E+01
-0.
87
6562.
0.1973E+I1
-0.
88
C.
0.3496E+01
-0.
CASE 15
V = 15,000 ft/sec
= -400
115
APPENDIX B
METEOROLOGICAL DATA USED IN ANALYSIS
The data here are reasonably self-explanatory.
The RHOBAR
are the mean values of density, in slugs per cubic foot, for 0
(2) 30 kilometers, and are read in ascending order.
The ABAR are
the mean values of the speed of sound in feet per second, for the
same altitudes.
The UBAR are the mean values of the west wind in
feet per -second and the VBAR are the means of the south wind.
The standard deviations of density and speed-of-sound are
given as fractions of the mean, while the winds are given in feet
per second.
The off diagonal submatrices (RHO A MATRIX, etc.) are
the upper right hand members; the lower left hand members are
omitted.
I.
.- e0o1E-Ow
0.1040*E-09
0.16116E-01
0.1890RE-02
RHOBAR
0.10161E-01
0.75660E-03
0.54tt4E-03
0.39vtE-03
UE4IiE-03
0. O'tSE-03
0.lS*44E-03
0.11264E-03
O.W3W3OE-04
O.6Z46SE-04
0.46959E-04
U.54',tLE-4
0.98136E+03
0.994feE+03
0.10141f+64
'.
ABAR
I
0.104i3E+04
O.10654E+04
0.1060?E+04
0.10953E+94
0.11093E+04
0.1122SE+04
0.115'vf+f
1t,',tE+0t
O.Z67?E+2
0.-2Ot3E+02
0.2666?E+02
UBAR
0.32821E+02
0.3837E+02
0.4eESE+02
0.444k4E+G0
0.4bZ60E+02
0.48055E+0Z
0.49651E+02
0.51646E+02
0.53442E+02
0.5523E+02
0.57033E+02
0.5682vE+0
3.346iE+01
0.11841E+O1
0.144&ZE+01
0.12344E+01
0.85312E+00
-0.11566E+01
-0.392o7E+o1
-0.5779(E+01
-U.'3O04E+01
-0.44813E+01
-0.11332E+02
-0.13183E+02
-0.15034E+02
-0.16885E+02
-0.16735E+02
-0.20586E+02
-1O3U
SE+04
VBAR
117
RHO RHO
STANDARD DEVIATIONS
0.30001E-01
0.1149tE-0i
0.15883E-01
0.14401E-01
0.19437E-01
0.55142E-01
0.1111&E-00
0.12693E-00
1.12t06E-00
0.11766E-00
0.*.177E-01
0.66337E-01
0.34335E-01
0.45050E-01
0.10044E-00
0.17101E-eG
RHO RHO MATRIx
1.0000 0.4034 0.b62 0.S1S4 0.4456-0.2206-0.ZS21-O.2972-0.3113-0.3149-0.3205-O.3245-0.2605 0.1368 0.2460 0.2720
0.$034 1.0000 0.6937 0.5937 0.1?15-0.3121-0.3363-0.3392-0.3464-0.3447-0.3450-0.3448-0.2758 0.15C0 0.2667 0.2913
0.f901 0.-437 1.0000 0.7637 0.1795-0.3243-0.3426-0.3339-0.3339-0.3252-0.311-0.3039-0.2044 0.1974 0.2756 0.2092
0.5154 0.5 9 37 0.7637 1.0000 0.6915 0.0126-0.1690-0.2082-0.2311-0.2317-0.2257-0.1948-0.0263 0.3142 0.3237 0.31(4
0.2456 0.1715 0.1795 0.6925 1.0000 0.4172 0.1141 0.0263-0.0159-0.0329-0.0370-0.0097 0.1403 0.2835 0.2300 0.2103
-O.Z200-0.3121-0.3243 0.0126 0.4172 1.0000 0.9355 0.8924 0.0642 0.8454 0.8291 0.8041 0.5933-0.4297- 0.6500-0.7035
-0.2021-0.3363-0.3426-0.1698 0.1141
0.9355 1.0000 0.9910 0.9776 0.9644 0.9469 0.9040 0.5916-0.6232- 0.6515-0.0900
-0.2972-0.3392-0.3339-0.202 0.0283 0.8924 0.9910 1.0000 0.9960 0.9894 0.9771 0.9301 0.6215-0.6409- 0.8833-0.9268
-0.3113-0.3464-0.3339-0.2311-0.0159 0.8642 0.9776 0.9960 1.0000 0.9974 0.9094 0.9561 0.6502-0.6204-0.0841-0.9321
-0.3149-0.3447-0.3252-0.2317-0.0329 0.8454 0.9644 0.9894 0.9974 1.0000 0.9968 0.9717 0.6071-0.5971-0.8694-0.9235
-0.3205-0.3450-0.3101-0.2257-0.0370 0.8291 0.9469 0.9771 0.9094 0.9968 1.0000 0.9870 0.7381-0.5412-0.8364-0.8992
-0.3245-0.3440-0.3039-0.1940-0.0097 0.0041 0.9040 0.9381 0.9561 0.9717 0.9870 1.0000 0.8362-0.4022-0.7413-0.6218
-0.2665-0.2752-0.2044-0.0283 0.1403 0.5933 0.5916 0.6215 0.6502 0.6871 0.7381 0.8362 1.0000 0.1651-0.2534-0.3773
0.1360
0.1560 0.1974 0.3142 0.2835-0.4297-0.6232-0.6409-0.6204-0.5971-0.5412-0.4022
0.1651 1.0000 0.9121 U.S30t
0.2460 0.2667 0.2756 0.3237 0.2300-0.6506-0.8515-0.8833-0.0841-0.8694-0.8364-0.7413-0.2534
0.9121 1.0000 0.9914
0.2720 0.2913 0.2892 0.3104 0.2103-0.7035-0.8908-0.9268-0.9321-0.9235-0.8992-0.8218-0.3773
0.8506 0.9914 1.0000
118
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V
V STANDARD DEVIATIONS
0.41981E+02
0.50663E+02
0.49194E+02
0.4359$E+02
0.59014E+02
0.30450E*04
0.4206?E+02
0.46947E+02
0.5?939E+02
0.68213E+02
0.90829E+02
0.10272,E+03
-
O.90493E+01
1.%)00
0.29
0.3905 0.5sst
0.3?t0 o.?d7
0.79272E+02
V
V
MATRIx
0.2257 0.1686 0.0662 0.0013 -0.0656-0.1117-0.1427-0.1640-0.1790-0.1901
0. 29%2 1.0000 0.9007 0.9849 0.9255 0.9103 0.6678 0.0125 0.6646 0.4618 0.2715 0.1234 0.0149-0.064f-0.1240-0.lV4
0. 3905 0-900' 1.0000 0.9434 0.8767 0.8523 0.8006 0.744? 0.6035 0.4125 0.2350 0.0976-0.0029-0.0763-0.1310-0.1728
0. 3365 0.9649 0.9434 1.0000 0.9657 0.9463 0.8*55 0.623? 0.6675 0.4563 0.2599 0.1079-0.0032-0.0844-0.1449-0.1911
0. 3260 0.9255 0.8767 0.9657 1.0000 0.9800 0.8856 0.8169 0.6574 0.4423 0. 24 39 0.0909-0.0206-0.1019-0.1(24-0.2085
0. 2872 0.9103 0.8523 0.9463 0.9800 1.0000 0.9544 0.8871 0.7181 0.4899 0.2781 0.1141-0.0057-0.0932-0.1584-0.2083
0. 2257 0.8678 0.8006 0.8855 0.8858 0.9544 1.0000 0.9569 0.8081 0.5897 0.3789 0.2120 0.0082-0.0031-0.0717-0.1243
0. 1686 0.8125 0.7447 0.8237 0.8189 0.8871 0.9569 1.0000 0.9443 0.7988
0.6312
0.4865 0.373E 0.2873 0.2209 0.1t90
0.0062 0.6648 0.6035 0.6675 0.6574 0.7181 0.8081 0.9443 1.0000 0.9523 0.8513 0.7470 0.6580 0.5865 0.5296 0.4840
0. 0013 0.4618 0.4125 0.4563 0.4423 0.4899 0.5897 0.7980 0.9523 1.0000 0.9708 0.9143 0.8565 0.8058 0.7633 0.7280
-0.0656 0.2715 0.2350 0.2599 0.2439 0.2781 0.3789 0.6312 0.8513 0.9708 1.0000 0.9847 0.9553 0.9243 0.8959 0.8712
-0. 1117 0.1234 0.0976 0.1079 0.0909 0.1141 0.2120 0.4865 0.7470 0.9143 0.9847 1.0000 0.9922 0.9766 0.9596 0.9433
-0. 1427 0.0149-0.0029-0.0032-0.0206-0.0057 0.0882 0.3736 0.6580 0.8565 0.9553 0.9922 1.0000 0.9958 0.9672 0.9774
-0. 1640-0.0646-0.0/63-0.0844-0.1019-0.0932-0.0031
0.2873 0.5865 0.8058
0.9243
0.9766 0.9958 1.oou L.994
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-0. 1901-0.1694-0.1720-0.1911-0.2085-0.2003-0.1243 0.1690 0.4840 0.7280 0.8712 0.9433 0.9774 0.9926 0.998f 1.0000
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0,0640 0.312i
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MATRIX
0.3504 0.326? 0.2*72 0.2914 0.2920 0.2597 0.1705 0.0001 0.0022-0.0493-0.0630-0.1059-0.1221-0.1340
01840 0.1003 0.1450 0.1176 0.0*93 0.0722 0.02?S-0.01 6 5-0.0466-0.0649-0.0760-O.0S31-0.0660-0.owi4
-0.et*s-0.0043
0.0244-0.0099-0.031-0.070.1006-0.105-0.1010-0.0792-0.056-0.039-0.027-0.0191-0.0127-U.U07$
-0-1642 0.02U1 0.0536 0.0075-0.0376-0.0608-0.0965-0.1029-0.0946-0.0729-0.0511-0.047-0.0232-0.0149-0.0086-0U.04?
0.0479 0.1143 0.1427 0.0960 0.0272 0.0063 0.0126-0.0226-0.0619-0.0862-0.0994-0.1029-0.1036-0.1032-0.1024-0.101
0.0347 0.0695 0.0917 0.1064
-0.0439-0.22t9-0.2195-0.2174-0.1849-0.148-0.1944-0.1606-0.0942-0.0200
0.11E7 0.1241
-0-1087-0.2754-0.2736-0.2677-0.2231-0.2269-0.2430-0.2020-0.1200-0.0280 0.0402 0.0036 0.1113 0.1298 0.1427 0.1521
-0.1282-0.2606-0.2769-0.2740-0.2322-02389-0.2554-0.2111-0.1238-0.026
0.0456 0.0913 0.1204 0.1397 0.1532 0. 131
-0.1370-0.2825-0.2780-0.2764-0.2359-0.2440-0.2607-0.2149-0.1253-0.0254
0.0481 0.0947 0.1244 0.1441 0. 1 579 0.1679
-0.1423-0.2835-0.2785-0.2778-0.2381-0.2469-0.2637-0.2171-0.1261-0.0249
0.0495 0.0968 0.1267 0.1467 0. 1606 0.1707
-0.1459--0.2842-0.2789-0.2787-0.2395-0.2489-0.2657-0.2186-0.1266-0.0245
0.0505 0.0981 0.1283 0.1484 0.1624 0.1726
-0. 1485-0.2647-0.2792-0.2794-0.2406-0.2504-0.2673-0.2197-0.1271-0.0243
0.0513 0.0991 0.1295 0.1497 0.1638 0.1741
-0.1507-0.2651-0.2795-0.2799-0.2414-0.2515-0.2685-0.2206-0.1274-0.0241
0.0518 0.0999 0.1305 0.1507 0.1649 0.1752
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0.0523 0.1006 0.1313 0.1516 0.1658 0.1761
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0.0527 0.1012 0.1319 0.1523 0.1666 0.1769
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0.0531 0.1017 0.1325 0.1529 0.1672 0.177f
125
V MATRIX
A
0.4330 0.1*20 0.1004 0.217S 0.1497 0.2324 0.1664 0.0955 0.0052-0.0791-0.1401-O.1792-0.2040-0.2200-0.230g-0.23$5
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