Error analysis on radar tracking of falling sphere

Brockman, W. E.

Hain, J. L.

English

NASA Technical Reports Server

REQUIREMENTS FOR TRACKING RADAR FOR FALLING SPHERES
By John L. Hain and William E. Brockman
Booz, Allen Applied Research, Inc.
SUMMARY
As a part of the system design of an upper air synoptic sounding system for NASA
Langley Research Center, a technique was developed for expressing explicitly the effect
of pertinent radar accuracy limits on the uncertainties in the meteorological data pro-
duced by the system. The results of this aspect of the study are briefly reviewed herein.
INTRODUCTION
The passive falling sphere is the lightest and least expensive payload which has
been shown to provide satisfactory data in the upper atmosphere, i.e., 30 to 100 km alti-
tude. The payload and the means of lofting it have been well developed and their costs
are well understood and manageable. The usable data, however, come from a high-cost
ground tracker of limited availability. With few exceptions the data have been obtained
using rather expensive trackers procured for and dedicated to significantly different
purposes. The purpose of this paper is to explore the relation between the requirements
for data on the motion of the sphere and selected sources of error in a radar tracker.
The discussion will consist of three parts, covering: first, the sphere trajectory;
second, some pertinent radar accuracy limits; and third, the consequent limits on density
and wind accuracies. Since only certain limiting conditions are considered, the results
are essentially boundaries rather than explicit statements of accuracies applicable to
specific configurations.
A detailed derivation of the equations used in developing these boundaries is
included in reference 1.
SYMBOLS
B Receiver Bandwidth
C D Drag Coefficient
E Elevation Angle of Sphere from Tracker
F o Receiver Noise Figure
175
https://ntrs.nasa.gov/search.jsp?R=19700009275 2020-03-17T01:22:11+00:00Z
g Gravitational Constant
176
Gt,Gr
L
N
Pt
R
S/N
At
X_Z
W
W z
OL_
_q
a
%
Gain of Transmitting and Receiving Antennas
Tracker System Losses
Number of Independent Data Points
Radiated Power of Tracker
Slant Range Between Sphere and Tracker
Radial Velocity Between Sphere and Tracker
Effective Signal to Noise Ratio
Time Interval in Seconds
x and z Coordinates of Sphere Position
Wind Vector
Vertical Component of Wind
Angles Defining Line of Sight Between Sphere and Tracker
Bias Error in Any Parameter q
Antenna Beamwidth
Operating Wavelength of Tracker
Density
Radar Cross Section of Sphere
Standard Deviation of Any Parameter q
Variance of Random Error in q
_" Tracker Pulse Length
Dots over a symbol denotethe degreeof the derivative with respect to time.
THE SPHERETRAJECTORY
Assuming a rocket-lofted sphere, a skewedtrajectory envelopesuch as that
depicted in Figure 1was used. Wind profiles of +50% and +99% were included to assure
adequacy of spatial coverage. A collocated launcher and tracker were assumed to mini-
mize personnel, logistic and real estate costs for a synoptic system. The initial condi-
tions (at the top of the sphere trajectory) are as follows:
Altitude
Horizontal displacement
Horizontal velocity
Gravitational acceleration
Area/Mass ratio
Radius of the earth
140 km
40 km
200 meters/sec
- 9.8 meters/sec
6.54
6,378,388 meters
The resultant theoretical descending trajectory, using the +50% wind profile, is shown in
Table 1. If such an actual trajectory can be observed as a suitable set of coordinates vs.
time, both density and wind data may be determined.
The density of the atmosphere in the immediate vicinity of the falling sphere may be
derived as a function of:
Vertical velocity of the sphere
Vertical acceleration of the sphere
Drag coefficient of the sphere.
The local wind vector may be derived as a function of:
Vertical and horizontal velocity of the sphere
Vertical and horizontal acceleration of the sphere
Gravitational constant.
Thus, if our trajectory is measured as a set of spatial coordinates vs. time, it is
apparent that the falling sphere technique is as sensitive to errors in the first and second
time derivatives of the coordinates as it is to errors in the coordinates themselves. For
ease in exploring these relationships, a two-dimensional flight profile for the sphere was
assumed. This is equivalent to aligning the launcher inclination and the effective plane of
one of the tracker's angular sensors with the prevailing wind.
If the first and/or second derivatives are obtained by fitting a function to the data
points and taking the derivative, the error can be separated into two parts. One part is
177
the error due to noise in the data, the other dueto lack of fit by the function to the
physical laws that producedthe data points.
The error dueto noise is a function of:
Fitting length (numberof points)
Data frequency
Noise in measuredparameter
Methodused (polynomial anddegree).
The error dueto lack of fit is a function of:
Fitting length (numberof points)
Data frequency
Numerical characteristics of the function that producedthe data points
Methodused (polynomial and degree).
Sincethe purpose of this discussion is to explore the impact of radar errors, only
the error dueto noise in the measurementswill bepursued.
The expression for the error in density which has beenderived (ref. 2) is
\_w,/ \_Wz/ _z-g/ + g _Wz %/
The bias error is the error due to lack of fit and will, as previously stated, not be con-
sidered. Uncertainty in the drag coefficient, probably one of the most significant problems
relative to the falling sphere technique, is, fortunately, not germane to the tracking accu-
racy exploration.
rewritten as
Eliminating these terms, the expression for error in density may be
2
(__P.P) = 4 a2+ 1 or.2+ 4 2(__Wz)2z (__g)2 z (__w_)2%z
Similarly, the variance in the horizontal wind is
where the expression for the horizontal wind is
W = _ _
E-g
178
PERTINENT RADAR ACCURACY LIMITS
The geometry of a generalized tracker operating on a passive falling sphere is
shown in Figure 2. As shown, the definitions are those commonly used for a phased-
array tracker. The two-dimensional analysis merely assumes that the wind lies along
either the a or ]9 axes. The elevation angle is thus the complement of a (or /_).
In the case of the usual electromechanical tracker, the analysis assumes no change in
azimuth, so that all angular data is again in the elevation angle. The other two measure-
ments of which a radar tracker is capable, range and range rate (or radial velocity), have
been shown (ref. 3) to be interrelated through pulse width so that the corresponding accu-
racy limits are not independent. The standard deviation of range varies directly with the
pulse width while that of radial velocity varies inversely. Although not strictly true in
the general case, for many applications either may be computed from the other with an
accuracy comparable to that which could be obtained by direct measurement. Therefore,
the radar measurements which were explored in detail were those of slant range and
elevation angle.
The classical radar range equation may be written (ref. 4) in the form
S PtGtGr _2_
N R4BFoL
where Pt is in watts, )_ is in centimeters, R is in nautical miles, c; is in meters 2,
and B is in hertz. The achievable accuracy is a function of the effectivesignal to noise
ratio,and that varies inversely as the fourth power of the slant range. Thus, the quality
of the meteorological data will degrade very rapidly with increasing distance to the falling
sphere. Thermal noise, an inseparable part of every real signal, establishes a limit
beyond which no hardware can extract usable data. The standard deviations for range
and angle measurements on a single pulse basis (ref.4) are
a
_E 2 S_
where eR is in meters, _E and 8 are in milliradians, and T is in microseconds.
Using these basic tools of sphere dynamics and of radar accuracy limits it was then
possible to establish certain limits on the quality of meteorological observations.
179
The obvioustrade-offs were thosebetweenmaintaining vertical resolution and
enhancingapparentaccuracy by smoothingover a large number of data points. Thus,
the length of the smoothing interval becamea goodindicator of relative merit.
ESTIMATIONOF ACCURACYLIMITS
Onefinal simplification: Fully acceptabletechniquesfor separating vertical winds
from density variance haveyet to be developed,andthe radar does not offer a solution.
Therefore, the vertical componentof the wind vector, like the uncertainty in the drag
coefficient, does not appear in the final error model. It simply is not a part of the radar
error contribution.
Sincethe errors in velocities and accelerations are smaller when a quadratic poly-
nomial is fitted to the data points thanwhensuccessive linear polynomials are used, the
quadratic fit wasused throughout the study. Velocities were then evaluatedfrom the first
derivative and acceleration from the secondderivative of this smoothcurve.
The error in the first derivative as a function of the error in the parameter is given
by
2 12 _2
_Cl N(N + I)(N + 2)At2 q
and the error in the second derivative is given by
¢.2 = 720 a 2
q (N - 1)N(N + 1)(N + 2)(N + 3)At 4 q
The error model consisted of expressions for the error in density and in horizontal
winds as functions of the variance of range, range rate, range acceleration, elevation
angle, elevation rate, and elevation acceleration, as well as values dependent on the tra-
jectory and smoothing interval.
But since only the range and elevation angle were measured, with the rates and
acceleration being derived mathematically, the model was modified by substituting the
derived variances of 1_, R, l_, and E. The model then took the form:
2 Fla2 + F2CrE
aW2= GlCr2 + G2(_ 2
180
where the F's and G's are lengthy functions of the coordinates and of the fitting func-
tions, but not of errors in determining the coordinates.
Since angular accuracy is the most cost-sensitive parameter of a tracker (aside
from attaining a workable signal to noise ratio) the relationship of meteorological func-
tion error to angular error was explored.
The first method of analysis was to assume a slant range error of 5 meters and to
compute the smoothing interval when the elevation angle error was 0.05 mil and the den-
sity error was 2%. The density error limit was then increased to 3% and various values
of elevation angle error were tried until the resulting smoothing interval was approxi-
mately equal to the 0.05 mil, 2% result. The process was repeated for density errors of
4% and 5%. The corresponding wind error was computed at each altitude level, for each
combination of elevation angle error and density error. The complete profiles are given
in Figure 3.
The following combinations of density error and elevation error yield approximately
equal smoothing intervals. The maximum horizontal wind error for each combination is
as shown.
Elevation, Density,
(7E (mils) ap (%)
P
Wind,
crw (meters/sec)
0.05 2 19
0.i0 3 37
0.15 4 54
0.20 5 70
The second method was to allow the elevation angle error to assume successively
larger values, the only other parameter which was allowed to change as a consequence
was the smoothing interval. This has the net effect of increasing the uncertainty as to the
altitude at which the computed density was valid and, thus, results in a net uncertainty as
to the density profile. The data are shown in Figure 4.
C ONC LUSIONS
A method has been developed, programed, and tested for quickly determining the
error contours applicable to a passive falling sphere, upper-air sounding system. It was
then apparent that a synoptic system using the given trajectory placed stringent require-
ments on the tracker.
181
A stated design goal of a vertical resolution of 500 meters, with a standard devia-
tion of 2% in density data and of 5 meters/second in wind velocity (below 70 km altitude)
can be met, but it requires a sufficiently high signal to noise ratio in the tracker that the
uncertainty due to thermal noise is no greater than 5 meters in range and 0.05 mil in
angle.
Relaxation of vertical resolution permits longer smoothing intervals with conse-
quent dramatic reduction in requisite angular accuracy.
REFERENCES
1. Booz, Allen Applied Research, Inc.: System for Upper Atmospheric Sounding (SUAS).
Vol. II - Technical Report. NASA CR-66753-2, 1969.
2. Luers, J.K.: Estimation of Errors in Density and Temperature Measured by the
High Altitude ROBIN Sphere. Proceedings of the Third National Conference on
Aerospace Meteorology, Amer. Meteorol. Soc., 1968, pp. 472-477.
3. Skolnik, Merrill I.: Radar Systems. McGraw-Hill Book Co., Inc., 1962.
4. Barton, David K.: Radar System Analysis. Prentice-Hall, Inc., c.1964.
182
z
o
i.i j _ _ _ _ _ p,,,,, _ _ _ _ _
i,ml
i=_
=E
C'_
Z_--.
o z
r.J I--
i,,,JI
z
._1 __ ¢_
I I I I
_, _, ,.,., I I I I I I
i.,l=l
X
Z 0
_" I._ _,- I I I" I" _'_ '_ I I
,_ c._ ,.,., i I
,_c i,.J
X
w
o =_. I.,J
z I--
L..J
X
_L.J •
w
z
::_ _'-
,_[ I--
..-a ix
•._ L_J
¢_ ¢_
L_
-- I,,,,,,
=_, Q-
X ..,-
t(.
183
70
dO_
L,,J
_.- 60
L,.JI
._, 50 --_--
40 \
_30 "J\
80° LAUNCH
loo I--L_
,0 UP IJ, NGI WIIlD/ ,DOVrH RLNGE\ /_ WINI
,o r )
/ _'_,.,/
2o_-_-\ / / i
10 _ "i
0 _-- ,_ k I IAtA_I.K AIiU
LJ_UHC'1 sr [E
.60 -40 .20 o 20 40 60
J
80 100 120 140 160 180 200 220 240
RAHGE (KILOMETERS)
Figure 1.- Flight profile for inclined launch.
iCEHTERLIHEOF
FIELDOF VIEW
MUTUALLY /
,0 INFLATED
SPHERE
f
/
z/" "R & R
PERPENDICULARAxES _ //'
_" I_ x-a.Jll '/ l /"*-I: , lt i J//_/IV !"_'I /I" __i_ __I lllilli __'.._
a AXIS
Figure 2.- Geometry of measured quantities.
184
100
8o
_E
_7o
_. 6O
so p,
40
30
0 10 20 30 40 50 60 70
WIND ERROR (METERS/SEC), 500 METER VERTICAL RESOLUTION
Figure 3.- Wind errors corresponding to various density errors.
100
90
80
_70
i-
_, 6O
5O
40 --_--
30
0 1000
...........
2000 3000 4000 5000 6000 7000
SMOOTHING INTERVAL (METERS)_ ap =2%
P
8000 9000 10,000
Figure 4.- Requisite smoothing intervals for apparent 2% density error.
185



Requirements for tracking radar for falling spheres

Abstract

Similar works

Full text

Available Versions

NASA Technical Reports Server