The Least-Squares Process of MEDIA for Computing DRVID ... · least-square s polynomial approximation of data points in which the optimum degree of the polynomial is automatically

N A T I O N A L A E R O N A U T I C S A N D S P A C E A D M I N I S T R A T I O N

Technical Memorandum 33-542

The Least-Squares Process of MEDIA for Computing

DRVID Calibration Polynomials

R. K. Leavitt

E

J E T P R O P U L S I O N L A B O R A T O R Y

C A L I F O R N I A I N S T I T U T E O F T E C H N O L O G Y

P A S A D E N A , C A L I F O R N I A

May 15, 1972

https://ntrs.nasa.gov/search.jsp?R=19720017913 2020-06-02T01:41:42+00:00Z

N A T I O N A L A E R O N A U T I C S A N D S P A C E A D M I N I S T R A T I O N

Technical Memorandum 33-542

The Least-Squares Process of MEDIA for Computing

DRVID Calibration Polynomials

R. K. Leavitt

J E T P R O P U L S I O N L A B O R A T O R Y

C A L I F O R N I A I N S T I T U T E O F T E C H N O L O G Y

P A S A D E N A , C A L I F O R N I A

May 15, 1972

PREFACE

The work described in this report was performed by the Data Systems

Division of the Jet Propulsion Laboratory in support of the Tracking System

Analytic Calibration activities.

JPL Technical Memorandum 33-542 iii

CONTENTS

I. Introduction 1

II. Method 3

A. Least-Squares Smoothing 3

B. Finite Differences of the Smoothed Data 6

C. Statistical Runs Test 7

III. Application 10

A. Simulated Data 10

B. Real Data 12

C. Derivatives of the Least-Squares Polynomials 15

.0- St-at-i-stic-a-l-R-un•s-T-es-t.-.-.-.-.-. -.-.-. 1_7-

References 18

Appendix A. Plots Illustrating MM'71 Charged-ParticleCalibration Polynomials and Their Derivatives 30

Appendix B. Comparing Derivatives of Faraday Rotation andDRVID Calibration Polynomials 45

TABLES

1. Dispersion about the least-squares fits 19

2. Statistics of the differences between the derivatives ofDRVID and Faraday rotation calibration polynomials 20

FIGURES

la. Cubic with step function 21

Ib. Cubic with 5-ns noise 22

Ic. Cubic with 10-ns noise 23

2a. Tenth-degree polynomial 24

2b. 125 points sampled from tenth-degree polynomialwith 10-ns noise 25

2c. 250 points sampled from tenth-degree polynomialwith 10-ns noise 26

JPL Technical Memorandum 33^542-. v

CONTENTS (contd)

FIGURES (contd)

2d. 500 points sampled from tenth-degree polynomial•with 10-ns noise 27

3a. Least-squares fit without runs test 28

3b. Least-squares fit with runs test applied 29

vi . JPL Technical Memorandum 33-542

ABSTRACT

This document describes and evaluates a process for computing a

least-square s polynomial approximation of data points in which the optimum

degree of the polynomial is automatically determined. An iterative smooth-

ing technique is used to replace every point with the value taken on by a

moving least-squares polynomial computed from a subset of points centered

at the point to be replaced. The optimum degree of the resulting polynomial

~a~pp^r^xirnatio'n~rs~deterrnined-by--a-nal-yzing-the-finite-differê.nces_of_ea_ch_

successive set of smoothed points. To evaluate the process, this document

uses both artifically constructed data and actual Mariner Mars 1971 (MM'71)

tracking data.

This process has been incorporated into the Transmission Media Cali-

bration computer program (MEDIA), which calibrates radiometric data to

overcome the effects on the tracking signal of charged-particle media.

MEDIA was used in support of MM'71.

JPL. Technical Memorandum 33-542 vii

I. INTRODUCTION

The radio signal going to and from a spacecraft at planetary distances

travels through several media containing charged particles. These particles

affect the propagation of the radio signal and, in turn, the radiometric data

recorded at the DSN tracking station. The charged-particle content of the

media through which the radio signal passes is determined at the tracking

stations by differencing range and integrated doppler (DRVID) observables

(-Ref-.—1-)-.— -T-o-eo-r-r-e-c-t-the-r-ecorded-radiome.tric_data_for ;the.s_e_effects_,_the_digi_r .

tal computer program MEDIA (Refs. 2 and 3) computes calibration polynomials

from the DRVID data for each tracking pass of the stations. To compute these

polynomials, MEDIA employs a least-squares curve fitting process that auto-

matically determines the appropriate degree required to optimize the fit.

This curve fitting process uses an iterative technique in which the

data points are smoothed with a moving least-squares polynomial. In each

successive iteration, every datum is replaced by the value taken on by this

polynomial, which is computed from a subset of k points centered at the

point to be replaced. Its degree and the number of points k of the subsets

are determined from the total number of data points in the pass.

After each smoothing iteration, the f i rs t , second, etc. , finite differ-

ences of the resulting smoothed points and each of their variances are

computed until the minimum variance is determined. When the minimumJ.-L

variances from two successive iterations occur for the same (say, the m )

finite differences, then a statistical F-test is used to compare them.

The smoothing process is ended when the F-test is satisfied. A least-

squares polynomial of degree m is then fitted to the smoothed points. A

statistical runs test is performed on the original data relative to this fit to

detect those cases in which the DRVID data do not resemble a polynomial.

When the test fails, the degree is increased, a new fit is computed, and the

JPL Technical Memorandum 33-542

runs test is again performed with the new fit. This process is repeated

until either the test has been satisfied or the degree of fit exceeds its

maximum allowable value.

The following two conditions must be satisfied by the data in order to

use this process:

1. The data must occur at equal time intervals.

2. The data should usually resemble a polynomial.

The f i r s t condition is imposed to simplify and to speed up the process-

ing logic. Without it, the computer run time required to process a pass of

data would be prohibitive.

The second condition is obvious, since the objective of the process is

to fit a polynomial to the points. Violations of this condition should be

detected and overcome by the runs test that is applied. Such occurrences,

however, should be infrequent; otherwise, the processing time would become

excessive.

This curve fitting technique is not applied unless there are at least

16 data points. If there are fewer then 16 points but at least three, a least-

squares polynomial is fit directly to the points. Its degree is set at the

largest integer contained in the square root of the number of points.


II. METHOD

A. Least-Squares Smoothing

A moving least-squares-fitting iterative technique is used to smooth

the raw data. Each successive iteration smooths the data until the smoothed

points resemble closely the resulting polynomial that is to be fit to the data.

Each smoothing iteration replaces every data point with the value taken on

by the moving polynomial, which is computed from a subset of k points

centered at the point to be replaced.

The number of points k in this subset is computed from the total

number of points in the pass. In general, the more points there are in a

pass, the greater the sample rate, and the more likely that low-frequency

noise will appear in the data. Increasing k results in more smoothing of

- the data-j- whie-h -is necessary_to_filter J:he_low^fj:_equencyjipis_e._ Hence,_k is

increased as the number of points in the pass increases. It is set equal to

the largest integer contained in the square root of the total number of points

in the pass and then forced to be odd by adding one if necessary. For

computing ease, k must always be odd.

To preserve local trend in the data, the degree of the moving poly-

nomial is increased also as the total number of points increases. It is set

equal to the largest integer contained in the square root of k. This compen-

sates somewhat for the extra smoothing that results from increasing k.

The independent variables t., i = 1, 2, • • • , k, in each subset are

expressed in seconds past January, 0, 1950. They are scaled so that they

range in value from -1 to +1 within each subset to assure a solution to the

least-squares equation and to simplify the computations. These scaled

quantities x. are defined as follows:

t. -i

X =1 t k - * !


The x take on the same values in all subsets once k is determined.

This follows from the condition that the data points occur at equal time inter-

vals. Thus, f o r k = 3, x = {-1, 0, l}; for k = 5, x = {-1, -1 /2 , 0, 1/2, l};

for k = 7, x = {-1, -2 /3 , -1/3, 0, 1/3, 2/3, l}; etc.

The least-squares equation can be expressed as follows (Ref. 4):

where

1.

2.

3.

V nZ-,x \

Cf»» C^ • • • , C are the unknown coefficients of the least-

squares polynomial. J

n is the degree of the least-squares polynomial.

All sums are from 1 to k, e .g . ,

k

Z n+1x x.

4. The set of values y represent the dependent variables in each

of the subsets, i. e. , i-

{y} = y l> YZ , '"

.JPL Technical Memorandum 33-54.2

By substituting into the above equation

/ " ' " l \Xl X2 Xk

V n n n /X 1 X 2 "• Xk/

and transposing terms, we can express the coefficients as follows:

-1

Ex • • • E x n1 1

E n+1x

T nY, Lî-x. L~I~L n+1

X2 xk

Jn n . .. n

C V "V1 2 xk

The matrix inside the brackets can be computed before entering the main

smoothing loop. All of the values included are known and depend only on the

total number of points in the pass.

Since each of the subsets is centered at the point corresponding to

x = 0, only the constant term coefficient CQ need be calculated. At x =0,

all of the other terms of the polynomial vanish; i.e. , y(0) = C^,. Thus, in

the smoothing process, the coefficient C,, replaces the y-value at the center

of each subset. It is computed by simply taking the dot product of the first

row of the matrix in the brackets times the vector of y-values contained in

each subset.

The end points of the pass are replaced by the values taken on by

least-squares lines that are fit to the f i rs t k-points and the last k-points.

The additional smoothing that results is necessary to prevent the smoothed

function from curling at its ends.

JPL Technical Memorandum 33-.S42

B. Finite Differences of the Smoothed Data

The f i rs t , second, etc., finite differences of the smoothed points are

analyzed after each smoothing iteration to determine (1) when the smoothing

process is completed and (2) the degree of the resulting least-squares

calibration polynomial. The key parameters are the variances of these

finite differences and, in particular, the minimum variance.

The i finite differences 6.y. are defined as follows:

.y. = 6. i y. , i - 6 . , y.,j'l j- l/i + l ]-1 'ifor i = I, 2, • • • , d.

.th ,. .,where by. = y. and d. is the number of j finite differences and is always j

less than the number p of points in the tracking pass; i.e. ,

d. = p-j

The nature of the finite differences in the case of polynomials is that each

successive set (j = 1, 2, etc. ) of finite differences only has meaning

up to the degree of the polynomial. For example, in the case of a cubic,

only the f irst , second, and third finite differences have any meaning. In an

absolute (or ideal) sense, the fourth and higher differences would all vanish.

Practically, however, almost all numbers are truncated. Therefore, the

fourth differences of a cubic are random truncation errors , and each

succeeding set of differences tends to double in value and to alternate in

sign.

The variances .s. of the j finite differences can be estimated with '

the usual equation: ' . ,

d. .

2 i = l

d.

!k=l

JPL Technical Memorandum-33-542

This parameter is a good measure of the finite differences. It decreases in

value as j increases as long as the finite differences are meaningful, and

then it increases. Therefore, the minimum variance of this set is the

dividing point between meaningful and not meaningful finite differences. It

is also a convenient measure of the amount of smoothing that has been

accomplished.

The analysis of the finite difference is summarized as follows. The

finite differences and their corresponding variances are computed for

j = 1, 2, 3, etc. , until the minimum variance is determined. This is done

for each smoothing iteration until the minimum variances for two successiveJ.-L

iterations occur for the same, say the m , finite differences. The F-test

is then used to compare these two minimum variances. If there is a signifi-

cant difference between them, the smoothing process is continued; if there is

-no-significant-differencie,_At_is_terminajted.^ The value of m is used as the

degree of the least-squares polynomial that is fit to the set of smoothed

points. This polynomial becomes the charged-particle calibration equation

for the pass if it satisfies the statistical runs test described below.

C. Statistical Runs Test

A statistical runs test is performed on the data relative to the fit to

ensure that the resulting polynomial does indeed represent the raw data.

This test is necessary in the event that the raw data do not resemble a

polynomial. Although such occurrences could be tolerated occasionally,

too many could increase the computer processing time to an unacceptable

level.

A run is defined as a set of consecutive points, all lying on the same

side of the polynomial (either above or below). It must contain at least one

point. The total number of runs is one plus the number of times two con-

secutive points lie on opposite sides of the polynomial and can range from

one, if all points lie on one side, to the total number of data points in the

pass. In the latter case the points would have to alternate from one side to

the other.


The cumulative sampling distribution z of the number u of runs can be

approximated from the normal tables using the following equation (Ref. 5):

u - "Z + 1/2z = ———s

z

where

u = number of runs

z" = q + 1

s2 = q (q-1)z p - 1

and

In these equations,

p = total number of points

p = number of points on one side of the polynomial

and

p_ = number of points on the other side of the polynomial

Note that p1 + p_ = p.

The runs test is satisfied if z is greater than the lower 5% level

of the cumulative normal distribution, i. e. , if z > -1. 654. This is a one-

sided test, since we are only interested in detecting runs that are too long.

Long runs (i. e. , too few runs) indicate that the fit does not adequately

follow the flow of the data points.


If the runs test is not satisfied, then the degree of fit is increased by

one, a new polynomial is fit to the smoothed points, and the runs test is

performed again. This process is repeated until the test has been satisfied.

The resulting polynomial then becomes the calibration equation.

JPL..Technical Memorandum 33-542

III. APPLICATION

The least-squares process described above was applied to various sets

of both simulated and real data to evaluate and to demonstrate its curve-

fitting ability. The simulated sets provided a controlled situation in which

the parameters were known. . The real sets were used to demonstrate how

the process would function in a real-life situation. A comparison with

Faraday rotation calibration polynomials was also performed.

The following paragraphs describe various statistical tests that are per-

formed on the least-squares fits that result from this process. These tests

include the chi-square over degree of freedom, linear regression, and anal-

ysis of covariance. The primary parameter used in these tests is the variance

of the points about the polynomial fit to them. For simulated data, this vari-

ance is compared to the known variance that was artificially constructed in

the data. For real data, it is compared with the variance of the actual

noise in the radiometric data. In comparing the resulting polynomials with

Faraday rotation calibration polynomials, the mean and variance of the

differences of the derivatives of the corresponding polynomials are analyzed.

A. Simulated Data

Two equations were used to synthesize DRVID data: a cubic and a

tenth-degree polynomial. These equations are defined as follows:

Ap(t) = 1 + t + t2 + t3

Ap(t) = 0. 15 + 0. 15t + 0. 15t2 + • • • +0. 15t10

The cubic equation is used to produce three cases of 100 points each

by superimposing on it 0-, 5-, and 10-ns Gaussian noise. In the first case,

where no noise is present, a step function results. Because of format

constraints imposed on the DRVID values, such functions are anticipated

when the spacecraft is very close to earth. The noise level will be so

small that it will be below the imposed precision level of the data and,

hence, undetectable. The 5- and 10-ns noise levels are considered repre-

sentative of the noise that will be present as the spacecraft gradually moves

away from earth.

10 JPL Technical Memorandum 33-542

Figures la, Ib, and Ic, respectively, are graphs showing the data

points for the three cases and the least-squares polynomials that were fit to

them. The derivatives of these polynomials are also shown in each of the

corresponding figures. The fit to the step function is presented to demon-

strate that such functions are adequately processed. However, step func-

tions could create problems for the statistical runs test because it assumes

random noise. For this reason the runs test has been made a program option

for the user.

For the 5- and 10-ns noise cases, the variance s of the data points

about the least-squares polynomial is an efficient parameter for testing the2 2

fit (Ref. 4). The statistic used is s /cr , where cr is either 5 or 10 ns. The

distribution of s /cr is the chi-squared over degree of freedom (X /df)

density function (Ref. 5). The 10% level of confidence is used for this test

so^jthat the requirements for rejecting the fit are not too stringent.

The acceptance criteria for 100 points are

0. 777 < s2/o-2 < 1. 24

The following tabulation summarizes the results of this test:

£. s5 4.53

10 10.41 1.083

Both are within the acceptance criteria; therefore, both fits are acceptable

according to the test.

The tenth-degree polynomial was chosen to produce a curve with a

relatively steep slope. Figure 2a illustrates this equation and its derivative.

A 10-ns noise was superimposed on this curve to obscure its slope. It is

believed that DRVID data may behave in a similar manner in extreme cases.

Samples of 125, 250, and 500 evenly spaced points over the interval

-1 < t < 1 were taken, converted to DRVID data, and used as input. Fig-

ures 2b, 2c, and 2d illustrate each set of points, their resulting least-

squares polynomials, and their derivatives. The variances about these

JPL Technical Memorandum 33-542 11

polynomials were subjected to the X /df test. The results are summarized

below.

Points

125

250

500

s

.!!•10.

10.

23

24

38

2s ,

1.

1.

1.

/ 2'or

26

05

08

Criteria

o.0.

0.

801 -

858 -

898 -

1.

1.

1.

22

15

11

2 2Although s /cr is within the acceptance criteria for the samples of 250 and

500, it is not for the 125-point sample. These results seem to indicate that

the sample rate should be increased as the noise increases to detect local or

irregular trends in the data.

In conclusion, the results from the simulated data demonstrate that

the process is performing in an acceptable manner, except possibly for

extreme and unlikely cases. In such cases, the user will have to override

the process by specifying the degree of fit. Input options built into the

process make such action possible.

B. Real Data

MM'71 is the primary source of real data used to evaluate the curve-

fitting ability of the process in an actual data-processing environment. The

early part of the flight, when MM171 was relatively close to earth, was selected

for the evaluation to minimize the number and effect of the variables involved

and to make the analysis easier. Appendix A contains the graphs of the

least-squares fits for the passes used in this evaluation. They are included

to illustrate the operation of the process on typical data.

The least-squares polynomials in these graphs appear to represent

reasonably the data points in each pass, and their derivatives seem

believable, except possibly at some of the ends of the passes. The data

points in these cases exhibit slight trends at the ends of the passes that prob-

ably would disappear in the smoothing process if the passes were extended.

This is particularly apparent in the relatively short pass of day 195 (Fig. A-4),

where the effects of the end points are magnified and result in what would

appear to be an unrealistic derivative. This "end points phenomenon" should

be kept in mind during the numerical evaluation of the fits that follows.


The variances of the deviations of the points from the fitted polynomials

are used to evaluate the fits as in the case of simulated data. They are

a measure of the dispersion of the data caused by noise in the received signal.

The fit is considered to be good if the resulting variance is an accurate measure

of this dispersion. Table 1 lists the statistics for the tracking passes early

in the flight that are used in this evaluation. In the last column on the right,

the variances have been converted to dBm units (Ref. 6), -which are compa-

rable to units of power, i . e . ,

-2

where [i? is the inverse of the variance in dBm units and s is the variance.

The dispersion due to noise is related to the power received at the

~tf a~clsing~stations";~the~less"power-receivedi— the -greate-r -the-dispersion --- -—

(assuming everything else equal). The following tabulation lists the expected

power received at Station 1 2 at 5 -day intervals:

Day

186

191

196

201

206

211

Power, dBm

-163. 7

-165.3

-166.8

-168. 9

-170. 9

-173.0

The relationship between these values and the variances of the deviations

from the least-squares polynomials is what will be evaluated.

Linear regression lines and correlation coefficients (Ref. 5) -were

computed for both the inverse of the variances (dBm units) and the power

received vs. days. Their slopes b, correlation r, and 95 percentiles pq-

of the r distribution are listed below:

Power

Variance

Points

6

14

b

-0. 3737

-0.4163

r

-0.9977

-0. 9931

P95

0. 729

0.457


Since |r| > Pqr in both cases, it is concluded that both sets are correlated

•with days.

The slopes of the two regression lines were compared using the analy-

sis of covariance technique (Ref. 7). If there is no significant difference

between these slopes, then the variances are related to the dispersion of the

tracking data received at the stations. The F-distribution is used to test

the slopes. The required equations are summarized below.

Sum of squares Degrees of freedom

Denominator

Numerator

ni

n.

- x ) 2( b . - b ) 2

(n, -2)

k - 1

where

and

k = 2 is the number of data sets

.thn. = 6, 14 is the number of points in the i set

(x . . , y..) are the pointsij ij

.thx. is the mean of the x-values in the i seti

a. is the y-intercept of the i set

b. is the slope of the i set

' k n.

b = •• 1J= 1

(x.. - x . ) 2 .b .

k n.

(x.. - x . ) 2


- The F-ratio that results from these equations is

1 993TP - • ' ' ' ° — 7 Q O,

0. 655 ~ J - ? - 5 •

From the F-distribution tables at the 95% confidence level,

F(16, 1) .= 4. 50

Since 3. 93 < 4. 50, it can be concluded that the two slopes are the same.

Thus, the dispersion of the deviations about the fit is an accurate measure

of the noise in the data, and the fits adequately represent DRVID data.

C. Derivatives of the Least-Squares Polynomials

The derivatives of the least-squares polynomials must accurately

represent the rate of change~~6"f"tHe ~cha~r~ged-particle media.-—I-t—is-the—rate-

of change that is important in the calibration of. the doppler tracking data.

The accuracy of these derivatives is evaluated here using Faraday rotation

data (Ref. 8), which measure the char ged-particle content of the ionosphere.

They are obtained by passing a signal from a stationary satellite through

the ionosphere to a tracking station and mapping the results to the line-of-

sight of the spacecraft. They should be comparable to DRVID data when the.

spacecraft is close to earth and the ionosphere is practically the only source

of charged particles. The Faraday calibration polynomials used for this

comparison were computed by the digital computer program ION (Ref. 8).

Appendix B contains the graphs that illustrate the derivatives that

result from the MM'71 DRVID and Faraday rotation data and the differences

between them for the days included in this evaluation. The letter F identi- - -

fies the polynomials that represent the Faraday derivatives. The ends of

the Faraday derivatives have been deleted from the graphs and from the

evaluation because of their erratic and unrealistic .movements. The oscilla-

tions that are so pronounced in the graphs of. the differences are caused

almost entirely by the Faraday derivatives. The relatively large differences

at the ends of the curves probably result from the end points phenomenon of

the DRVID derivatives, as noted earlier. This effect is magnified again in

the relatively short pass of day 195 (Fig. B-4). Table 2 lists the statistical

JPL Technical Memorandum 33-542 1.5

parameters from these graphs that are used in the numerical analysis that

follows. The last column on the right presents the inverses of the variances

in dBm units. A linear regression line and correlation coefficient -were

computed for these variances (dBm units) vs. days to determine whether

there is any relation between them and the dispersion of DRVID data. The

results are summarized below:

N =14

b = -0.141

r = -0.261

P95 = 0.457

Since |rj < Pqc» the variances are not correlated with days; i .e., the

slope b of the regression line is not significantly different from zero.

Therefore, the differences of the derivatives are not related to the disper-

sion of the DRViD data or to the signal noise, both of which (as noted above)

are correlated with days and have the same slope. This result indicates

that the least-squares process has successfully filtered the noise from the

data.

From the values in Table 2, the total mean and standard deviation of

the differences of derivatives are -0.0031 and 0.0881 mm/s, respectively.

Although the mean is relatively small, the standard deviation does indicate

a rather large spread considering the size of the derivatives. However,

computing error percentages at this time does not seem worthwhile since

it is not known whether DRVID or Faraday contributes the greater error. A

more profitable approach might be to apply these calibrations to a simplified

orbit determination process and evaluate its convergence.

In conclusion, the derivatives of the resulting DRVID calibration

polynomials are acceptable provided the tracking passes are not overly

short in duration. The successful filtering of random noise from the data

has minimized the dangers inherent in computing derivatives of raw data

such as these. This accomplishment is one of the most important results

of the process. Another significant attribute of the process is the fact that it

has not introduced into the fit such distortions as "ringing" or other similar

deformations, which too often appear in least-squares polynomial approxima-

tions. Such distortions are often amplified in the derivatives of these


polynomials to the extent that the subsequent application of the derivatives is

impractical. Although the differences between the derivatives of DRVID and

Faraday polynomials have what seems to be a relatively large dispersion,

they do average out as indicated by the small total mean. This anomaly is

not yet fully understood; however, there are several likely causes that might

contribute to it:

1. The end points phenomenon.

2. The mapping of the Faraday rotation calibrations to the space-

craft line-of-sight.

3. The lack of pronounced trends in the DRVID data. (The data

used in this evaluation were all sampled during the "flat"

portion of the ionosphere's diurnal charged-particle content

curve.)

D. Statistical Runs Test ~ —

The need for a statistical runs test is demonstrated in Fig. 3a. As

shown in this figure, the least-squares process determined that a fourth-

degree polynomial would fit the points. Their configuration, however, is

such that this degree is not adequate. Probably, an exponential function

would fit these points better than a polynomial.

The runs test, upon detecting such situations, causes the degree to be

increased until the test is satisfied. In this example, a tenth-degree poly-

nomial was required to fit the points adequately enough to satisfy the test.

The resulting fit is shown in Fig. 3b.


REFERENCES

1. MacDoran, P. F. , "A First Principles Deviation of the DifferencedRange vs. Integrated Doppler (DRVID) Charged Particle CalibrationMethod," Space Programs Summary 37-62, Vol II, Jet PropulsionLaboratory, Pasadena, Calif. , March 1970.

2. Madrid, G. A. , TSAC Software Requirements Document for theMM'71 Era, Internal Report 900-391, Rev. A, Jet PropulsionLaboratory, Pasadena, Calif. , Aug. 1970.

3. Leavitt, R. K. , DEPP, Transmission MEDIA, A Subassembly of theTracking System Analytical Calibration Assembly (TSAC), Phase A,Internal Document DEPD No. 315-59, Jet Propulsion Laboratory,Pasadena, Calif. , Jan. 1971.

4. Hildebrand, F. B. , Introduction to Numerical Analysis, McGraw-HillBook Co., Inc., New York, N. Y. , 1956.

5. Dixon, W. G. , and Massey, F. J. , Introduction to Statistical Analysis,McGraw-Hill Book Co., Inc., New York, N. Y. , 1957.

'6. Ryder, J. D. , Electron Engineering Principles, Prentice--Hall, Inc. ,Englewood Cliffs, N. J. , 1952.

7. Mood, A. M. , Introduction to the Theory of Statistics, McGraw-HillBook Co., Inc., New York, N. Y. , 1950.

8. Mulhall, B. D. , et al. , Tracking System Analytic CalibrationActivities for the Mariner Mars 1969 Mission, Technical Report32-1499, Jet Propulsion Laboratory, Pasadena, Calif. , Nov. 15, 1970.


Table 1. Dispersion about the least-squares fits

Day

188

189

191

195

198

199

200

201

204

205

207

208

209

210

N

417

245

370

146

415

423

209

363

423

424

401

384

423

227

s

0.5183

0.5639

0.6387

0. 8261

0.8681

0.9024

0.9500

0.9992

1. 1967

1. 1850

-1...3.83.9

1.4145

1.5839

1.4439

i / 21/s

3.7225

3. 1448

2.4514

1.4653

1.3270

1. 2280

1. 1080

1. 0016

0.6983

0.7121

0,_522JL

0.4998

0. 3986

0.4796

dBm

35.71

34.98

33. 89

31.66

31.23

30. 89

30.45

30. 01

28.44

28. 53

_ _27^18_

26.99

26. 01

26. 81


Table 2. Statistics of the differences between the derivativesof DRVID and Faraday rotation

calibration polynomials

Day

188

189

191

195

198

199200

201

204

205

207

208

209

210

N

438

250

377

150

432

432

215

368

233

123

436

436

436

298

X

0. 026

0. 023

0.022

-0. 176

0. 022

0.005

-0.053

0.042

-0. 105

0. 132

-0.061

0.071

-0..036

-0. 032

s

0.046

0. 097

0. 105

0. 150

0.066

0. 043

0. 132

0.077

0. 159

0.077

0.090

0.077

0.061

0.084

iA2

472.59

106. 28

90.70

44.44

229. 57

540. 83

57.39

168.66

39.56

168.66

123.46

168.66

268.74

141.72

dBm

56. 74

50. 27

59. 58

46.48

53.61

57. 33

47. 59

52. 27

45. 97

52.27

50.91

52. 27

54. 29

51.51


RAP .25

AA

1 ! -20

HA

f -15

1

-10H

8S .05

-oo12

4.Q

3.5

RA

ej 3-D

; ...

'I "•"1.5

K

H

l.D

.5

-0 <12

__

HI

r—

"s

—

___

sV

_..-

__

""*s,

"~

•

^_

• -

—

— — .

^ ~~ • --

14. 16

,

,

yQ^1

—

f£ —

_

r**

.

—

jpKlj

—

—

—

—

i j—

—cat.

— -t—r»

—

•ex

—

f&.

_.. . . .. _

^~~

— -

^

• —

——

^-•

- -

—

s

—

.X'

— -

—

---

—

~~KS

-

—

___ , /

Z_

s

/

4 .4 ....

•-"- f ''"" 1 ~-

//

__ /i/J/

/

2$

j

_1_

!

i

—

"

::j/.[ -.j.. :/4_ _j I 4

/ T I ji I 1... . _r ._,

j f1 j

„. I ™1-i f

j I '_ ! _ _ ' _tll_— }— t T~1

i t ij ! ii f i

"! - ^ •-1 I '1 ! - 1I |i i •

i -i i * is i f t

1 <i

: f- * ii I i

18. 20- ZZ- 24.

HOURS

rtEL

.ffff

WM

— -

__

i

—

(S®

—

_Zj —

5&

—

J'

r^J

Ofy*^

,v(j,r,i

-JO-.'

I

ii ' *

- —— -} — { — ]

id—:

i

—

j— ^ _ ^

I

f#

E

— *b~ef

I—

i

/•yf~pr-i.

1__

L_

_Lj

i

- ,.(

'1 "i

iJO

Lfi-P~cSKor._

.

~

1

^™

::

.. 4..... 4.. .

•-$..£

-^—-i" • j -f- -

,-ir i *.„ yi_. ,1 j |^Pl-^ i T2^- I-- - - i - - ,! 1

f s |

— J -L-L. ' " J.I l li ' I

j '

: i "t

i i -i i.. ... 4 . . * - - . » *.

. . ,_i_.. ,*.,„, * ^-^_ . .* . * ^-

— .* i • 4- •*

i - + -* *-

f- » » *•

* + - » * -

' i t t r. •If. 16- 18- 20. 22. 24.

HOURS NPTS = lOOi NOEO = 9» RNS = ,0t& « . 303 NS

CHARGED PARTICLE CALIOBATION - DAY = 3i<i STA = 11. s/c «

Fig. la. Cubic with step function


.30

.25

• SO

.15

~T"

EiE

.05

! .OD

13 ^Lj.--r--t-::pT

T—T

tzjiz20. 22-

HOURS NCTS ' 100« NOEO = it RK8 = . JT1P =

CHARGED PARTICLE C A L I B R A T I O N - DAY » 3^» STA = Mi S/C = ?2

SZ.

NS

Fig. Ib. Cubic with 5-ns noise


• 35

R

N -30

fR

I , '25

SN -20

1 -

S.10

S1

• OS

• OD~12

5.

4.atAM

f 3.

SA

F -

I "«

" o.

' -i .

-z.12

s—

—

__

—

s

-

--

- - -

—

N

— -

- — •

^x^

• —

___

'"s^Nw^

—

—

~^ -=99

.....

—

!«_•

——

mu ff

14. 16

'

=»-'

—

—

-o-

—

—

c

1 —

o

—

—

-o-Q^*

v c

_o-

^ —

—

=w

Jn

n

JJ-•"<

o

—

—

,_Q

*J

o

—

—

—

—

=«—•""^^

—

^^

—

~s

—

^^

18.

HOURS

(

=»

.

~)

—

~~T3

!•! "•

o

—

•e—

'~d•r—

o

—

——

J

- — *-1

—

—

_h_

,">

—

—

d

^—

o~ o

—

j/

—

—

//

— --

.-_——

v^r

20

h _

O

^

*>-G

~>U

J

—

^

-O .

CP1

^

_ . 1 .

-._

.. .1 —1

— -

—

//

Iii 1

—

/Y

--—I—

//

xl -/ \'

'—_4__ii

---— -

__/

r

—

22

C

,.S

_o.

iii

—±-^_ |f

, s'yoi

-.d

o-O 1_4Q Of—&-

i=?±!z±i:ol ^

O !

I

J

_ - !

1i

._ 1 J

i-j"1 i" " "

1

1" -i i

J

- d

'::1 t 1 1 •I . ....4. --, . . ,. ..j * j *~Xl'"i* ~r ""f

E

- . - ...i 1 !— (- • j- 1' -

" J" J""T""-j. H- -,.. . .._ .| ..-. — _ .j. ,.. -

i j i iT"" t "T -

!_4i *j |

_ ... , T-1 j1 f1 i

| ?] |

ii 1 <i i i

r

1

i i i i.-sn—

j i :t j . -

Of t i i^Si.«. i - i .*VY

o

_~L.j__4_i__J "_j_J " "t""1 i •

- 4 ! I\"~~

_-l__Jl^_ 4._.

_ _

- . .

' f !

. ...--;. . ^ .- .,_ _

~±dr± -1 :

— t •• T" -r -•

- } • ( • • 4 .•- 4 t 4

. .jt1

•* -i.

t 4 i

; t - -14. 16. 18- 20. 22. z«.

HOURS NPtS ° lOOi NOEO « * RH5 « ,7(00 = law US

C H A R G E D P A R T I C L E C A L I B R A T I O N - DAY = 34«« STA •= Hi S/C •= 72

Fig. Ic. Cubic with 10-ns noise


.0

12- 16. 18.

HOURS OF DAY

20. 22-

I !

_i

drd

14. 16.

OF OAVCHARGED PARTICLE CALIBRATION - STA - M» S/c •

20-

Fig. 2a. Tenth-degree polynomial


.12

.06

— * '™™,4 —I

._.j.™™»,j_4..,~|_

SEE

~i—»-

EJEE~r—•r~~tr~i~_._1—_4_—..

14. 16. 18.

HOURS

2I2T

1.5

HOUB8 _ _ . . NPTS = IZOi HOIO • 3»

CHARGED PARTICLE CALIBRATION - DAY « 347» STA • M» S/C «

Fig. 2b. 125 points sampled from tenth-degree polynomial with 10-ns noise


m.10

.08

16. 18.

HOURS

20. 22.

— ej —

•®H

d*

—

—&•

, — -1

.0

-*Vr

5 — zJ

!»-32

t-0-

__j

rs^

igrH

H

_

1 O

^

O ""

K-0|

€

-— -r-rr^

o

—

—

>— c

0~©-!>—

7T"

i

rr

i

-O™

-{T-

3

Jn

W^-r>-

O—

-O—y

>

3=d

1=

=5

b

__

— O'

1! £

331t,

3333

—

r~— €

•Ji~ t

__

— sô-

o_<

—

i

t?—

sden

o

—O

=

u

<J

~~— .

>" — "L

:i

""""

~H-

—A— n

3 —

O —

K>—i

-%

•>—

flOCJ

j <;— &

0

•V

n

3T, rt

e —

ro~s»-t

~O-

..,., <;

1

—

-69-

u_

55ôn— c

cc;

y ,-„„„—

—

-V

PFfc-e-Z35

—

5

Si?

FU

J33™™-J

...... JLi.~

=

__

»—

^^dZ__

—ff-

Trr

iO

i2S

K"

d•0 ;

aro~jQ.|~'

1 '"j

- t:

<[

.,—.,—...}...-....„„„

~i

EE_-

( '• ' "• • • r— |L. -rr-:! —

-0 . —^-r-

^l-. I. 1

E3={— T— r-igz±rz|_z4— r|_ .

° tZIlj — br

"j t tlj "

__j__i__ :

4rz}""~"4 v-

-a-Zâ":14. 16. 18. 20. 22. 84.

HOURS NP18 « EBOt MKO « » UN* • .?«?! « 1 Ok ZM H8

2-5

1-5

N

f

-.5

-1-0

-1.5

-2-D

CHARGED P A R T I C L E C A L I O R A T I O N - DAY • 34?* STA - HI s/c • 7?

Fig. 2c. 250 points sampled from tenth-degree polynomial with 10-ns noise

Z6 JPL Technical Memorandum 33-54Z

•2O

R

? .-.15

aA

I

fi

05

Ift

nn

s

— 12

R 2"ASP

5 '•A

>/

r

j

X1

-

*^

A

of>r.1 O

,X^

O

ft

1°

^"

{

oo

oo

^

-~

"14

>

> coM

o~~<n<>-

> r

•••••

> -"O

°rrtt

•nn "«

n S

o

>o0

3

-O-

•K

^16

O

Jnoc^

-i

y*-,'

11 •-" -i

r?u

ou<^>-<-

u

oon>

o

?'

^v*1

^—-

-~18

HOU

^

(1)sb/ciA

— -J

9S

o ot

^J"O-rl

r» 1

)

ooc^

^ 1

0<-i

6>n cJo

X

o *•

*#>

s

20

oGO

01)

^ (

^X

n.

/ir

O^J

^L

/X"

O

J

O-

n .sk

//

(

">

osSd

uX3"

/

^,/

22

Onp.,

-O-, (.fflV

0^

f^0

/'

-O—

r,u0

&V>o

24.

CHARGED P A R T I C L E C A L I B R A T I O N - DAV - 34?i STA = MI s/c « ?a

Fig. 2d. 500 points sampled from tenth-degree polynomial with 10-ns noise


-10

HOURS NPIS = 140- NDEG = 1. RMS = '. H398 = lr-3. -\

CHARGfO P A R T I C L E C A L I B R A T I O N - DAY = I, STA = I 2- S/C = 75

Fig. 3a. Least-squares fit without runs test


-25- !r^

HUIIRS NP1S - UO. NUEG = IU. RMS = 1. 11 I S = ^3. 3?1 NS

CHARGED P A R T I C L E r.AL IBRAT I ON - DAY = I, STA = I a. S/_ = ?5

Fig. 3b. Least-squares fit with runs test applied


APPENDIX A

PLOTS ILLUSTRATING MM'71 CHARGED-PARTICLE CALIBRATION POLY-

NOMIALS AND THEIRDERIVATIVES


=!=

MM/ -.20

•t-/"i—'—r-

.05

.00

-.05

4—r—i-EjEjEjE

nji

10. 11.

HOUSS

12. 13.

2.0

-2.5

HOURS NPIS = II 7t NOEG = 1i RMS = .3183 = «.°t« N3

CHARGED PARTICLE CALIBRATION - DAY = IS8< STA = \z, s/C = 75

Fig. A-l.

JPL Technical Memorandum 33-54Z 31

cVis

— t"tlzizj"_i i }_ j !~r-1—i—r

~~l~i~J~i i i

±

-.3zone:

FT

6.0 7.Q 7.5 8.0 8.5

HOURS

9.5 ta.o 10.5

<i

v

~— "

r', — |_j

A^

n•n

» ^

-*

n\\

~f\

n

^(SI

^_

— c

•Hr

""C>»rr

•J

• ;

s•AT

~

n

'N/•

—

n

•y •

S^

c

—

)

3.

._jp

±.

^

o

ff

<>

—

j

*'n

n

"S.

"i n

;

" ,

tc\

1

Nf_

n

i

a.

i-L,

n

h.

r1

c

,i-j-,

V

u

(„

• — .

f l

-a

•v_

^^fn

£

O

O

nj

i,_

~c

|O.J L

C

u

— ,

t^

-(_ .

_

i —

^

M*O

Uc

V

c

"I i

fTH

(•

J-

oj

o-c

V.

T

fj

)

X^_

~o_L

—

o

r

tr ~c

_

,>

-XI

n^

—

om—

—

—

n

D

U""7

ro

— a.

j

> —

i

-4

~

—

o

r

frf-

-H-t>-ie-

—

—

; {—

1 i

j

i— i — i

iII11

3*~

,

I

— L-i

i "i —i I

| I

!.

â-.

O-

<x

)ji

^3i— U_— j_. }

—f-c

O-O-Cfccfc»:>H"5™H"

zjs i EE!

-oh-

r-bjaJ I

i" 1 i"I I i

iiT r ; , u, ! ....

.-{-j \ j.

- 1 -!--i '—i i i

1 ! ., t... |... j.,

t~i —ii

j i i i

i f

— j— i1 °! ii ! JfOI j

_j__p_{_~ I 1 !

__

_!_

4— •p-ld~Hr% <» i '^H^H-l-

t i l l

i, 4-.->i-_„! i,, |._t Irt 1

_'__

;:±dd_-±JO i ! '1 ' 'f T'i

11

0 6.5 7.0 1.5 8.0 8.5 9.0 9.5 10.0 IO.5

HOURS N»1S = 3«3i NOEO = 3, RMS = . 3<3« = f, !tf« NS

-1.5

CHARGED PART I CUE .CAL I3RAT ION - DAY = 18Si STA = \ Z, S/C = 75

Fig. A-2.


.15

.10

.05

.00

-.05

-.15

-.20

-.as

ntrrz!

-1,5

-2.0

-2.5

-3.0

-3.5

-4.5

k*

i—H--H-}=!~1—j—i-

fqq:

d:

3pq=t:

—i.

1——l—~. j i

H

12.

EJEgE

10. 12.

HOURS NCTS = 3fOi MOEG = 3, RKS = . <38/ = 8.3C3 N3

CHARGED P A R f - C U C A L I O R A I I O N - DAY = 191 . STA = i^, S/C - ?$

Fig. A-3.


4

3

S

1

0

.1

.z5.

=L..-y-1

\ 1<i\

• \\\\

\\\

ss\\s

Sk

^ s.\

•

vS

V,

—

"

^^•H—,

^

^

Ss/'

s/s

I 1 "".

1 —

XS

////

//

—

—

— •/

—" 11. .:i::.

1i

———

i tt 1

i-

i

5 6.0 6.5 7.0 7.5 8.O 8.5

HOURS

1.5

c i.o

B .0

-1.5

6.0 6.5 7.0 8.0

HOURS ' NP1S = M«i NOEG = «> HMS = . 8e«t = IS. 0?3 NS

CHARGED P A R T I C L E C A L I B R A T I O N - DAY = MSi SIA = I e. S/C - 75

Fig. A-4.


.06

.03

.no

-.as

-.06

-EL- =X2t±I-«-

—V-

-.12

~t~

T=t-}-r=|=p=|=i:

5. 8. 9.

HOURS

10.

10. H. 12.

HOURS NPTS = «13. NOEG = •>! RMS = .8481 = i 1.38? HZ

CHARGED P A R T I C L E C A L I B R A T I O N - DAY = 1 «. 8. STA = 1 ?., 5/C - 75

Fig. A-5.


-.3

-40-

Eflfc•^^v

2K

&-

&\

9 &*LI

ee

3^•[O

i<$

=r

teiztkbs

C^.—£

M i

x£

-3.

qSFH^

_ 1—a5 L_ I. 16. 7. S. 9. 10.

HOURS

CHARGED P A R T I C L E C A L I O R A I I O N - DAY =3i NOEG = ^> »MS = ,SO

STA = iei s/c = 75

11. 12.

.^ = \Z, 0<C NS

Fig. A-6.

36 JPL Technical Memorandum 33-54Z

? .1y£ .0

*• A

E -.1

H

£ --2f& ' - ; 3

0

M« . -.5

•/

-.6

-.75.

3.

2.

*AN

r ,.c

f 0.

R

8 -i-?

« -2.

-3.

-4.5.

—

1

4-t-

—

t1f

—

/f

. —

/

u'1

—

——

//

/

...

f

-

f—

——

f'

•

—

—

"•—fc

—

—

—

—

—

--J^ - i

^ 1

— .

—

—

—

— •

—

—

—

—

—

-

—

—

— 1i""I

—

— 1

—

—

— jj1

— i i__

^2

— b"M=

1II

i-j-

±—

/

-HP1-

—

~™" ,

—

•

, —V~.. — .

- ,

fe±ppr————

, —

—

—

bpq

O 5.5 6.0 6.5 7.0 7.5 8.O 8.5

HOURS /

—

1 (

—

^

—

~>r>

\ roV

o

—

o

o.

<^

o

I

—

—

1

<A

O

——

<

-

•s

y *•

U

—

—

h

|7|)

__ *— •«.

•V-k

—

—

<,

ff=••.

v>

„„_„

—

O<:

j

c

>

u

n<-

>p-

— <

—

0

0J

(J

—

U

>

(•)

~C

o

^

U

r

0

—

—

- r

1

_u

> u

.

—

—

—

—

-}c

o

f)Jo

—

J<

-iJ

^

—

—

—

o

• — (°

I)

0

—

—

—

L|

=er

u

—

——

)

nii(

c.

—

o

r<u

—

9>Or,

i

Q_

V

si.

—

|

=

}l

^"

.

_

1

rTi"~~~

—

•-h

V

—

—

—

V

7=»?

—

1

q

_

ua.0-

—

*-

_1)_

r>

—

ii

1

f

—

j>0 noat

—

<E>

ii

")

— -«!°ii

O j,,±jz:=T=

o....i.y.,!— giA'C

—_

=!

j

o-

~

a

• i?r±f -..* —

~ Ji '..._. ..

_.

E~""

——

—

tz——=d

...—,_:——

L..Ji —

— i—_j —

j

P• j i—

—

~i —

—

_J—

i

9.0

—

=

^

n—

—

—> I

u r

^

,

_ .

"

0 5.5 6.0 6.5 7.0 7.5 8.0 8.5

i!

•>-<

^

2TO'

-

--

^<j

cP:

:«:

._ , i» -

i •

s—

N-i1_

"•••

' -'

9.0

CHARGED PARTICLE C A L I B R A T I O N - DAY = 200, 5TA = i 2i S/C = 75

Fig. A-7.


-2.

-3.

9. 10.

MOUHS NP1S = 3t3< NOEG = «i RMS =

CHARGED P A R T I C L E C A L I B R A T I O N - DAY * 801 1 STA = 13. S/C =

12.

NS

Fig. A-8.


NP1S • NOEO NS

- O'Y

Fig. A-9.

S/C =


.05

itf

if

-.T-.!.

ir-r-r

'-.05

_j_

rr-I

T^

E^=4—4-_,—j_

~r~>—|-4-H-

^B3_j_|-

5. 6. 7. 8.• • ' • ' . ' '

; HOURS

9. 10.

5.

MOU9S

PAR '.sett CA

6. 7.

•?*< KOEO = <i «HS

- O*Y * ^05l su * \e< s/c

9. 111.

*HS = 1,1830 = 18,810 NS

Fig. A-10.

.40 JPL Technical Memorandum 33-542

Fttt*.20

TT qq-4T-H-

-H

.00

-.05 -H-

5. 8.

HOURS

HDU"»5 NP1S ' <01i NOEO

- O.'.Y = aO?i SI* *

10.

«« "IS = 1,3831 = 18. ««« NS

?i 5/C = 75

Fig. A-11.


CH.«.a»GE3 C A L I B R A T I O N - DAY = ?08t ST» = l & S/C 75

Fig. A- 12.


-.3

-|—:F -J_

c EF:zi:

pqq-I-

6. 8. 9.

HOURS

10.

*Z3< NOEB = •>! RKS =

CHARGED P A R T I C L E C A L I B R A T I O N - DAY = 301* STA = 1?i S/C = 75

Fig. A-13.


NOEO

- DAY = 2 I O i SIA = 1 ?, s/C « 75

Fig. A- 14.


APPENDIX B

COMPARING DERIVATIVES OF FARADAYROTATION AND DRVID CALIBRA-

TION POLYNOMIALS

JpL Technical Memorandum 33 -542 45

SLOPE • . 00?i RMS • . 0*1, COBR - .?«!««

HOURS Of OAT IBS

O R V I O VS F A R A D A Y R O T A T I O N C A L I 3 R A T I O N S IMM/SECI - STA = 1Z, S/C = 75

Fig. B-l .


6. 8.

NP1S • ?50i SOEV •

10. 12. 14.

SLOPE • -. OOfi RMS • . OS3i CDSR • -. 307*

-1.010. 12.

HOURS Of OAt 18<*

D3VID VS F A R A D A Y R O T A T I O N C A L I B R A T I O N S I M M / S E C ) - STA « 13. S/C = 75

Fig. B-2.


g . - H

HOURS OF OAT i siDRVIO VS F A R A D A Y R O T A T I O N C A L I B R A T I O N S ( M M / S E C I - STA = 1Z, S/C = 75

Fig. B-3.


NPTS • 160. SDEV • .160. SLOPE • .oati RMS -

HOURS OF DAY 119

O R V t O VS F A R A D A Y R O T A T I O N C A L I 3 R A T I O N S I MM/SEC) - 5TA = 1 S/C - 75

Fig. B-4.


NP1S • 13Z, SDEV . .066, SLOPE •

ID. 12.

HOURS OF DAY 198

O R V I O V5 F A R A D A Y R O T A T I O N C A L I B R A T I O N S I M M / S E C I - STA = I ?t S/C = 75

Fig. B-5.


* .OP '•

10. 12. 14.

SLOPE • .003i RMS • .037i CORR • .5H«

10. iZ.

HOURS OF OKI lit

O R V I D vs F A R A D A Y R O T A T I O N C A L I B R A T I O N S IMM/SECI - STA = \z, s/c = 75

Fig. B-6.


NPTS • 216» SOEV • . I3a, SLOPE • -. 003« RMS • . I32i CDRR • -. OfS

HOURS OF 0*1 ZOO

O R V I O VS F A R A D A Y R O T A T I O N C A L I B R A T I O N S ! M M / S E C ) - STA » 12, S/C =

Fig. B-7.


.20

.15

.05

.00

6. 8. • 10. 12.

NPTS • 368. SOEV • .077, SLOPE • . 00<f, RMS • . 053i CORR •

-1.0

DRV

HOURS OF OAV ôl

ID VS f A R A O A Y R O T A T I O N C A L I B R A T I O N S I M M / S E C I - STA = 1Z \ S /C = 75

Fig. B-8.

JP:L Technical Memorandum 33-54Z 53

6. 8. 10. 12.

NPTS - Z33, SOEV . .IBS, SLOPE • .036, RMS - .066, CORR • .<

_i--i- _i- -i-

_i 1 : i_

3=3—S—1

i i j..--i:~hr -h

i. i ij q —j—.

I-H—f--l6. 8. to. IS.

HOURS OF OAt ZQ1

DRVID VS F A R A D A Y R O T A T I O N C A L I B R A T I O N S IMM/SEC! - STA = 13, S/C = 75

Fig. B-9.


SLOPE • .035i RMS • .OM> CORR • .183"

HOURS OF DAY 309

O R V I O VS F A R A D A Y R O T A T I O N C A U 3 R A T I O N S I M M / S E C ) - STA = 13. S/C - 75

Fig. B-10.


Nf>TS • *3«i SOEV • .DID. SLOPE • .008* RMS •

HOURS of OAV eo?ORVID V S F A R A D A Y R O T A T I O N C A L I 3 R A T I O N S I M M / S E C I - S T A s/c

Fig. B- l l .


5 . . ~I i I

SOEV • .Q7?i SLOPE • .ooti *MS • .art, cons • .orr

HOURS OF OAT eos

O R V I D VS MRAOAY R O T A T I O N C A L I B R A T I O N S IMM/SECI - STA = 1 ?i S/C = 75

Fig. B-12.

JPL Technical Memorandum. 33-54Z 57.

HOURS OF DAY Z01

D R V I D VS F A R A D A Y R O T A T I O N C A L I B R A T I O N S I M M / S E C I - STA = 12, S/C = 75

Fig. B-13.


f

*f»<iDA»

•1IN

SLOPE • -. Old RMS • . 0««> CORR • ~.t*a*

-1.0 is.

HOURS OF DAY ?to

O R V I O VS F A R A O A Y R O T A T I O N C A L I 3 R A T I O N S IMM/SEC! - STA S/C = 75

Fig. B-14.

JPL Technical Memorandum 33-542NASA - JPL - Coml., I.A., Colif.

59

Documents

The Least-Squares Process of MEDIA for Computing DRVID ... · least-square s polynomial approximation of data points in which the optimum degree of the polynomial is automatically