NPS-63-87-008

NAVAL POSTGRADUATE SCHOOL

CORRECTION OF THE WIND FIELD
MEASURED BY THE NCAR ELECTRA DURING FASINEX
FOR INERTIAL NAVIGATION SYSTEM DRIFTS
WILLIAM J. SHAW
GAIL T. VAUCHER
OCTOBER 1987

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FASINEX Contribution No. 18

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Washington, DC 20550

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NPS-63-87-008

NAVAL POSTGRADUATE SCHOOL
Monterey, California

Rear Admiral R. Austin K. Marshall

Superintendent Provost (Acting)

The work reported herein was supported in part by the National Science
Foundation (Ocean Sciences) with funds provided by Grant OCE-86-03050 .

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Correction of the Wind Field Measured by the NCAR Electra During FASINEX for Inertial
Navigation System Drifts

: PERSONA. aj"mcr(S)

william J. Shaw and Gail T. Vaucher

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October 1987

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FASINEX

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Aircraft Meteorological MeasTirements

Atmospheric Boundary Layer

9 aBSTRaC'^ {Continue on reveae it nectissry snd Kjtntity by bioctc number)

This report describes a technique for removal of
inertial navigation system drifts from winds measured by the
NCAR Electra during the Frontal Air-Sea Interaction
Experiment (FASINEX) in 1986. Without correction,
horizontal wind divergences calculated from these data may
contain errors as large as 10“^ s“^ resulting from the
Schuler oscillation. After correction, the divergence error
is reduced by at least an order of magnitude. Correction
coefficients and a subroutine to use them were provided for
the five of six FASINEX flights for which correction was
possible.

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William J. Shaw

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TABLE OF CONTENTS

Abstract i

page

1. INTRODUCTION 2

1.1 Role of the NCAR Electra 2

1.2 Need for Accuracy in Aircraft Winds 2

2. PROCESS OF AIRCRAFT WIND MEASUREMENT 3

2.1 General Method 3

2.2 Sources of Error in Aircraft Winds 3

2.2.1 Gust probe errors 3

2.2.2 Inertial navigation errors 3

3. CORRECTION OF INS WIND ERRORS 4

3 . 1 General scheme 4

3.2 Results from FASINEX 5

3.2.1 Position errors based on LORAN-C 5

3.2.2 Velocity errors from cubic splines 6

3.2.3 Uncorrected and corrected wind fields 7

Figure 3.1 8

4. REFERENCES 9

APPENDICES

A. Raw and Adjusted INS--LORAN Position Differences 10

B. Cubic Spline Fits to Position Errors 18

C. Velocity Errors Derived from Cubic Splines 24

D. Uncorrected and Corrected Velocity Fields 30

E. Sets of Spline Coefficients 36

E.l Spline Evaluation Subroutine 36

E.2 Spline Coefficients 37

F. Divergence Error From the Schuler Oscillation 42

Distribution List

■itn

ABSTRACT

This report describes a technique for removal of inertial
navigation system drifts from winds measured by the NCAR Electra during
the Frontal Air-Sea Interaction Experiment (FASINEX) in 1986. Without
correction, horizontal wind divergences calculated from these data may
contain errors as large as 10"^ s~ resulting from the Schuler
oscillation. After correction, the divergence error is reduced by at
least an order of magnitude. Correction coefficients and a subroutine
to use them were provided for the five of six FASINEX flights for which
correction was possible.

1

1. INTRODUCTION: THE FASINEX FIELD PROGRAM

I. I Role of the NCAR Electra

The Frontal Air-Sea Interaction Experiment (FASINEX) is a field
measurement program which occurred January through March of 1986 in the
Sargasso Sea. The objectives and experimental plan for this project
have been explained in detail by Stage and Weller (1985,1986). In
general, the purpose of the work was to investigate the structure of and
the interactions between the oceanic and marine atmospheric boundary
layers in the presence of the relatively simple inhomogeneity of a
subtropical ocean front. Investigators from the University of
California (Irvine), the National Center for Atmospheric Research
(NCAR), and the Naval Postgraduate School (NPS) participated in this
effort using the Electra aircraft operated by NCAR. The Electra 's
primary use was for the measurement of the horizontal variation of mean
atmospheric variables and turbulence fluxes in the atmospheric surface
layer. To accomplish this, the Electra flew a pattern of overlapping
boxes at an altitude of approximately 35 m as described by Stage and
Weller (1986).

1.2 Need for Accuracy in Aircraft Winds

The mean and slowly varying behavior of the horizontal wind field
is of particular importance to the FASINEX analysis. The cross-isobaric
angle of the momentum budget and the advection terms of that budget (and
other budgets as well) require that the mean wind be determined with the
greatest possible accuracy. It is expected that the horizontal momentum
flux and stability variations resulting from sea-surface temperature
changes across the ocean front will also induce secondary circulations
in the ABL. These secondary circulations should be evident in the
surface divergence and vorticity fields, which we intend to obtain from
the aircraft wind measurements.

Divergence and vorticity measurements may be obtained from
aircraft by line integral calculations or by differentiating surfaces
fit to the horizontal velocity fields. For either method, rather
accurate winds are required. Lenschow and Spyers -Duran (1987) have
considered the uncertainty which a velocity error growing linearly with
time would cause in line integral calculations of these two quantities.
As subsequent sections will show, it is more realistic to consider
sinusoidally varying wind component errors with periods of slightly
greater than 84 min.

2

2. PROCESS OF AIRCRAFT WIND MEASUREMENT

2.1 General Method

The measurement of the wind velocity from an aircraft is a vector
subtraction process in which the aircraft motion relative to the
atmosphere is subtracted from the aircraft motion relative to the
earth's surface. The difference is then the desired air motion vector.
Lenschow and Spyers -Duran (1987) describe in detail the methods by which
these measurements are made aboard the NCAR Electra. Basically, the
aircraft motion relative to the air is sensed by a gust probe system
which consists of fixed vanes and a pitot- static probe mounted on a boom
5 m in front of the aircraft. Aircraft motion relative to the earth's
surface is determined by integrations of the output from a Schuler - tuned
accelerometer. (The Electra uses the LTN-51 by Litton Systems, Inc.)
This will be discussed further in section 3.1.

2.2 Sources of Error in Aircraft Winds

2.2.1 Gust probe errors

Each of the motion measurement systems on the aircraft is subject
to error. The gust probe measurements are susceptible to biases in the
measurement of the airspeed vector due to upwash effects around the gust
probe. Since it is not feasible to calculate these biases based upon
ground-based measurements, the Electra is put through the "Lenschow
maneuvers" on each flight day (Lenschow and Spyers -Duran) . During these
maneuvers, which are commonly done on the ferry home, the aircraft
attitude is varied over large pitch and yaw angles. Additionally, the
airspeed is varied through speed runs from near -stall to maximum
cruising speeds. Other biases may be removed by short reverse -heading
runs under the assumption that the wind velocity field remains constant.
For cases in which reverse -heading runs are not performed, Grossman
(1977) has outlined a method which takes advantage of ordinary flight
geometry to accomplish the same purpose.

Another source of potential gust probe error in the calculation of
airspeed involves the determination of specific heat at constant
pressure c^ for air. Lenschow and Spyers -Duran (1987) have argued that
Cp variations arising because of changes in flight level humidity can
cause significant errors in the airspeed determination from the pitot-
static probe. The above problems are noted but will not be discussed
further in this report.

2.2.2 Inertial navigation errors

The inertial navigation system (INS) on board the Electra is
essentially a gimballed set of accelerometers which measures the

3

accelerations experienced by the aircraft in three dimensions. These
accelerations are then integrated once to obtain aircraft motion and
again to obtain aircraft position. In order to cope with the relatively
large acceleration due to gravity, the accelerometers are maintained in
a fixed orientation relative to the local vertical. This process is
accomplished by the application of appropriate torques based on position
changes computed from accelerations measured during the flight and is
called Schuler tuning (e.g., Broxmeyer, 1964).

It is impossible to perfectly align an INS so that its vertical
coordinate is precisely parallel to the gravity vector at the beginning
of a flight. As a result, a small portion of the acceleration of
gravity is perceived by the INS as a horizontal acceleration, and the
integrations of this error produce errors in velocity and position of
the aircraft relative to the earth. The error which results from the
initial misalignment of the platform is known as the Schuler oscillation
and is shown by Lenschow and Spyers- Duran (1987) to have a period of
84.4 min. Accelerometer saturation (as in sharp turns) and other noise
combine to give errors which unpredictably modify the phase and
amplitude of the Schuler oscillation and produce other drifts as well.

3. CORRECTION OF INS WIND ERRORS

3.1 General Scheme

It is not possible to correct INS errors in aircraft data without
recourse to independently determined position information. However, if
an aircraft records accurate position measurements at intervals which
are short compared to the Schuler period, it is possible to almost
completely remove the effects of the Schuler oscillation and other
inertial drifts from the horizontal wind vector.

The NCAR Electra now records position information from the Loran-C
navigational aids operated by the US Coast Guard. The data, which have
a specified accuracy of .25 n mi --or about 500 m--in the vicinity of
the FASINEX measurement area (U. S. Coast Guard, 1980), are sampled at a
rate of 1 s' and recorded as part of the standard meteorological data
stream on the aircraft.

The utility of using the Loran-C information to remove inertial
drifts in aircraft data has previously been demonstrated by Nicholls
(1983). Nicholls applied a correction, similar to that which will be
described here, to the data collected by the United Kingdom's C-130
meteorological research aircraft during the 1978 Joint Air-Sea
Interaction Experiment (JASIN) in the North Atlantic. He claimed
accuracies for his corrected winds of roughly 0.3 m s' .

The general approach to the correction of the aircraft winds for
FASINEX is described as follows. The motion of the aircraft relative to

4

the earth may be expressed as the vector sum of the aircraft motion
relative to the air and the air motion relative to the earth, or

G = A + V
or

V = G - A

where G is the aircraft ground velocity vector, A is
vector aircraft motion relative to the atmosphere, and
V is the net atmospheric motion.

In terms of position.

^INS “

X

at ^

and

*^TRUE

a

d

^ERROR

“ at ^^iNs

^RUE^

and it follows that

V = V
TRUE INS

G

ERROR

The aircraft winds may thus be corrected by subtracting the ground
velocity error vector from the NCAR- supplied wind vector.

3.2 Results from FASINEX

3.2.1 Position errors based on LORAN-C

The figures of Appendix A show raw position differences in km
between the Electra's INS and the Loran-C receiver as a function of time
for directions east and north of the 28^N, 70°W reference point
(approximately the center of the Electra's FASINEX box pattern). Also
shown for each flight is the aircraft track, which indicates the north-
south and east-west legs of the flight pattern. These data are included
to illustrate one difficulty in using Loran-C data at the edges of
coverage areas. There are sharp jumps as large as 5 km occasionally and
simultaneously in both components of the position differences. This
effect appears to be unrelated to aircraft turns and is more than likely
a result of atmospheric effects on Loran-C signal transmission.
Specifically, Loran-C positions are determined by precisely measured
differences in the reception time of signals from the geographically
widely separated stations of a particular Loran-C chain. On the edges
of a particular chain's reception area, a sky wave refracted to a
particular receiver may be as strong as or even stronger than the
directly transmitted ground wave. If a receiver alternately locks onto
a ground wave and a sky wave, jumps such as in the plots for the flight
of February 18 would result.

5

The removal of the jumps in position differences described above
is the only subjective portion of the Electra's wind correction and was
a significant requirement only for the flight of February 18. The
flights of February 16, 17, 21, and 24 required very little correction,
while the flight of February 20 contained so many large discontinuities
that wind correction was not attempted. It is notable that weather
conditions were the least disturbed of the experiment on the former days
and were highly disturbed on the latter. The flight of February 20
occurred while a significant (but not forecast!) storm was over the
FASINEX area. Since the atmospheric refractive index is a function of
temperature and humidity (Bean and Dutton, 1966; Battan, 1973), such
disturbed conditions would lead to significant variations in propagation
path. Appendix A also contains the plot of the adjusted east and north
position differences between the INS and the Loran-C data for February
18.

While position data were recorded at a sampling rate of 1 s" ,
the position data presented here are 10 s averages. The averaging was
done for convenience to reduce to number of data involved in the
computation of function fits described in the next sections. An obvious
effect of the averaging, however, was to greatly reduce the 500 m point-
to-point variability in the 1 s position difference time series (Figure
3.1). This gives reassurance in the determination of the slowly varying
Schuler oscillation.

3.2.2 Velocity errors from cubic splines

As shown in section 3.1, the velocity error in each component of
the INS -derived wind field is the derivative of the position error in
that component. Although the 84.4-minute Schuler oscillation is
obviously present in the position data, it is also apparent from the
position error plots that its phase and amplitude are not constant in
time. Therefore, it is not satisfactory to fit a simple sinusoid of the
Schuler period to the data in order to obtain a differentiable function.
We have chosen instead to use cubic splines [specifically the routine
ICSICU from the International Math and Statistics Library (software
product of IMSL, Houston, Texas)] . This procedure yields a function
with continuous first and second derivatives.

The smoothness of the spline function depends upon the number of
individual cubic segments which are joined to form the total function.
Endpoints of the individual segments are termed "knots”. Since
endpoints are shared except at the beginning and end of the overall
function, the number of knots is one more than the number of cubic
polynomial segments involved in the fit. Our primary interest was
resolving and removing the 84.4-min Schuler sinusoid. Since a sinusoid
has two inflection points per cycle, we chose a number of knots in each
fit which would yield slightly more than two inflection points in 84.4
min. This left the derivatives virtually unaffected by the higher
frequency position differences, which are more likely to be caused by
variations in Loran-C propagation path. Appendix B contains plots
(excluding the flight of February 20) of the cubic spline functions

6

17 FEB 86 - FLIGHT 2

Figure 3.1 (a) Raw position differences (east) for February 17 with
no averaging, (b) Raw position differences, but with 10 s
averaging.

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7

TiMIMO (SECONDS)

These curves are derivatives of the functions presented in Appendix B.
The combination of the Schuler oscillation and a longer-term drift yield
velocity errors which are typically as large as 3 m s‘ in each
component. If one assumes a sinusoidal velocity error of this amplitude
in a line integral divergence calculation using uncorrected data
collected from a 100 km square box, it can be shown (Appendix F) that
the error in the divergence may be as large as 1 x 10’ s~ ^ , This is
an order of magnitude larger than typical divergence values encountered
in the undisturbed atmosphere.

It is sobering to note that because of the oscillatory nature of
the position errors, there are times late in each day's flight when the
position error is near zero. In the years before the Electra recorded
INS -independent position information, it was common to use airport
position upon landing as the single (and generally only available)
external position reference for a flight. If the position error was
small, the conclusion was usually that the INS drift was small and that
the wind measurements, therefore, would be acceptably accurate. It is
obvious from the data presented in the appendices that this is not a
safe assumption.

3.2.3 Uncorrected and corrected wind fields

The figures of Appendix D show uncorrected and corrected wind
vectors for the FASINEX flights (excluding February 20). These vectors
are averages over 100 s intervals of the flight, which correspond to
very nearly 10 km in space. Inspection of the figures shows the
substantial difference the correction makes. For the flight of February
17, for example, the uncorrected winds are southeasterly with a
magnitude as large as 12 m s’ on the northern side of the box. On the

south side, winds are nearly easterly and have a magnitude closer to 5 m

s' . A vorticity calculation based on these data would result in
significant cyclonic vorticity. After correction, the winds still show
cyclonic vorticity, but of a smaller value. Maximum windspeeds on the
north side of the box are roughly 10 m s’ and minimum windspeed on the

south side are closer to 7 m s’"^. Further, the wind vectors on the

south side show much less change between the first and the second passes
of the aircraft through the center of the center of that area.
Uncorrected and corrected wind vector plots for the other FASINEX
flights are also contained in Appendix D.

Acknowledgments: This work was performed with the support of

NSF Grant OCE-86-03050 . Ms. Penny Jones expertly
transformed the manuscript into report form.

8

4. REFERENCES

Battan, L. J., 1973: Radar Observation of the Atmosphere. Chicago,
University of Chicago Press, 324 pp.

Bean, B. R. , and E. J. Dutton, 1966: Radio Meteorology. National Bureau
of Standards Monograph 92, U, S. Department of Commerce.

(Available from U. S. Government Printing Office, Washington, D.

C. ) . 435 pp.

Broxmeyer, C., 1964: Inertial Navigation Systems. New York, McGraw-

Hill, 254 pp.

Grossman, R. L. , 1977: A procedure for the correction of biases in

winds measured from aircraft. J. Appl . Meteor., 16, 654-658.

Lenschow, D. H., and P. Spyers -Duran , 1987: Measurement techniques: Air

motion sensing. NCAR Technical Bulletin No. 23, 49 pp.

Nicholls, S., 1983: An Observational Study of the Mid-latitude, Marine

Atmospheric Boundary Layer, Ph.D. dissertation. University of
Southampton, U.K. , 307 pp.

Stage, S. A. and R. A. Weller, 1985: The Frontal Air-Sea Interaction

Experiment (FASINEX); Part I: Background and scientific
objectives. Bull. Am. Meteor. Soc., 66, 1511-1520.

and , 1986: The Frontal Air-Sea Interaction Experiment

(FASINEX); Part II: Experimental plan. Bull. Am. Meteor. Soc.,

67, 16-20.

U. S. Coast Guard, 1980: Loran-C User Handbook. Department of
Transportation, COMDTINST M16562.3 (old CG-462), 63 pp.

9

A. RAW AND ADJUSTED INS--LORAN POSITION DIFFERENCES

This appendix contains plots of the position differences between
the INS and the LORAN-C signal which was received by the NCAR Electra.
Data are presented in an east-north geographic reference frame so that
position differences are independent of flight track. Also included in
each set of plots is the aircraft track versus time. Because of
discontinuities the data of February 18, a plot of the adjusted data for
that day immediately follows the raw data plot.

10

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12.0 13.0 14.0 1S.0 18.0 17.0 18.0 1S.0 20.0

B. CUBIC SPLINE FITS TO POSITION ERRORS

This appendix contains plots of the cubic spline curves fitted to
the data of Appendix A. They are plotted on the same scale so that
overlays can verify the fits.

18

CUBIC SPUME - tS FEB 88

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AV

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((UU5fl) XV

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12.0 13.0 14.0 16.0 16.0 17.0 16.0

ClliaiC SPLINE = 24 FEB 86

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TiyE past iDiO'OO^)

C. VELOCITY ERRORS DERIVED FROM CUBIC SPLINES

The curves which follow are the derivatives of the curves
presented in Appendix B.

24

25

12.0 13.0 14.0 1S.0 1S.0 17.0 18.0 1S.0 20 0 21.0 22.0

VIELOCITY ERROR - 17 FEB aS

26

12.0 13.0 14 0 Iti.O 13.0 ir.O 18.0 1& 0 111 0 21.0

27

12.0 13.0 14.0 15.0 16.0 17.0 16.0 1^.0 20.0 21 0 22.0

TiyE Oyooi'J

VEI.OCITY ERROR - 21 FEB 86

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28

12.0 13.0 14.0 ISt-O 10.0 17.0 18.0 10 0 0 21 0

29

D. UNCORRECTED AND CORRECTED VELOCITY FIELDS

This section contains plots of wind vectors obtained from 100 s
averages (approximately 10 km in distance) of the uncorrected and
corrected wind data. Scales are identical in both cases to allow for
comparison of the vector fields by overlay. Wind vectors are scaled so
that 0.1 degrees of latitude or longitude is 10 m s’ of windspeed.
Fields are corrected only with respect to ins drifts, other potential
errors, such as biases, have not been removed. also, no attempt has
been made here to account for non- stationarity of wind field itself.

30

LPTIT'JI'F (Lir'-RCESl LRTnU'JE IDCSRt£SI

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31

LnTHL'pr lOrGRPESt LPTITUDE (DEGREES!

rx

32

LRiiTJor (D'~gr!:e:s) lptitude idegreesi

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Uncorrected winds

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Corrected winds

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35

E. SETS OF SPLINE COEFFICIENTS

Following are the sets of spline coefficients for the correction
of INS-derived position and horizontal wind velocity components in an
east-north coordinate system. These coefficients have been determined
for the five (out of six) correctable Electra flights in FASINEX. With
the coefficients supplied, the output is in km for position error and in
m s" for the velocity errors.

E.l Spline Evaluation Subroutine

SUBROUTINE SPLINE(XK , C , CINT , N , T , VAL , DER)

C

C SUBROUTINE TO EVALUATE THE A CUBIC SPLINE AND ITS FIRST

C DERIVATIVE AT A PARTICULAR ABSCISSA VALUE T

C

c

XK --

INPUT

ARRAY OF N KNOTS FOR THE N SPLINES

c

SENT TO THIS ROUTINE

c

c --

INPUT

ARRAY OF (3,N) CUBIC SPLINE

c

COEFFICIENTS

c

CINT --

INPUT

ARRAY OF N SPLINE INTERCEPT VALUES

c

N --

INPUT

NUMBER SETS OF CUBIC SPLINE

c

COEFFICIENTS IN FUNC

c

T --

INPUT

ABSCISSA (E.G. TIME) VALUE AT WHICH

c

THE CUBIC SPLINE FUNCTION AND ITS

c

FIRST DERIVATIVE ARE TO BE

c

EVALUATED (IN SECS)

c

VAL --

OUTPUT

VALUE OF SPLINE AT TIME T (IN KM)

c

DER --

OUTPUT

VALUE OF FIRST DERIVATIVE AT TIME T

c

(IN M/SEC)

C

DIMENSION XK(1) ,C(3,N) ,CINT(1)

C

DO 10 1=1, N

IF(XK(I) .GT.T) GO TO 20
10 CONTINUE
C

20 IF(I.GT.l) GO TO 30
VAL = -9999.

DER = -9999.

RETURN

C

30 I = I-l

D =(T - XK(D) / 1000

VAL = CINT(N) + D*(C(1,N) + D^(C(2,N) + D*C(3,N)))
DER = D*(2*C(2,N) + D*3*C(3,N)) + C(1,N)

RETURN

END

36

E.2 Spline Coefficients

February 16 - DELTA X COEF:

XK

(SECS PAST OOOOZ)

CINT

C(1,N)

C(2,N)

C(3,N)

48535

0.005

-0.996

-0.256

0.087

50065

-1.804

-1.165

0.145

0.088

51360

-2.878

-0.345

0.488

-0.288

52889

-3.297

-0.877

-0.835

0.238

54966

-6.592

-1.273

0 . 644

0.040

56638

-6.731

1.219

0.846

-0.414

57714

-4.957

1.603

-0.489

-0.177

59242

-4.284

-1.138

-1.303

0.920

59750

-5.075

-1.749

0.096

0.135

62074

-6.919

0.895

1.041

-0.455

63858

-4.590

0.270

-1.391

0.342

65393

-6.218

-1.585

0.183

0.175

67162

-7.482

0.703

1.111

-0.424

69063

-5.044

0.328

-1.308

0.307

71634

-7.633

-0.317

1.057

-0.269

XK

(SECS PAST OOOOZ)

February

CINT

16 - DELTA Y COEF;
C(1,N) C(2,N)

C(3,N)

48535

1.381

0.909

-0.436

0.105

50270

2.193

0.342

0.108

0.083

52003

3.541

1.463

0.539

-0.192

53736

6.697

1.602

-0.459

-0.273

54852

7.534

-0.441

-1.372

0.484

55902

6.117

-1.721

0.155

0.115

58442

4.625

1.286

1.029

-1.095

59141

5.652

1.122

-1.264

0.192

62533

2.406

-0.825

0.690

-0.106

64180

2 . 446

0.586

0.167

-1.168

64602

2.635

0.103

-1.312

0.447

65673

1.789

-1.170

0.124

-0.013

69973

-2.000

-0.839

-0.047

-0.035

37

February 17 - DELTA X COEF:

XK CINT C(1,N) C(2.N) C(3,N)

(SECS PAST OOOOZ)

49800

-0.347

1.479

-2.212

0.474

51419

-1.737

-1.957

0.088

0.209

53037

-3.790

-0.034

1.101

-0.511

54656

-3.130

-0.488

-1.381

0.270

56145

-6.029

-2.807

-0.175

0.804

56981

-8.028

-1.415

1.841

-0.365

58673

-6.918

1.681

-0.013

-0.530

60014

-5.965

-1.213

-2.145

0.706

62294

-11.506

0.024

2.687

-0.797

64429

-6.964

0.600

-2.417

0.521

66725

-12.026

-2.265

1.169

-0.006

68046

-12.990

0.795

1.147

-0.472

69936

-10.576

0.070

-1.530

0.176

70839

-11.630

-2.262

-1.053

0.655

71944

-14.531

-2.191

1.117

-0.168

February 17 - DELTA Y COEF;

XK CINT C(1,N) C(2,N)

SECS PAST OOOOZ)

C(3.N)

49800

0

,322

-0.

.028

0,

.158

-0.

.003

51419

0,

.677

0.

.457

0,

.141

-0.

.091

53037

1,

.399

0.

.197

-0,

.302

0.

.145

55151

1,

.837

0.

.867

0

.619

-0.

.374

56065

2,

.861

1.

,061

-0.

.406

-0.

.048

57884

3,

,159

-0.

.892

-0

,668

0.

.368

59671

1,

.531

0.

.245

1

.304

-0.

.436

61641

3

.742

0.

,306

-1

.274

0.

.065

62514

3,

,082

-1.

.768

-1.

.103

0.

.588

64320

-0,

.247

0,

.001

2.

.082

-0.

.639

66166

2,

.832

1,

.160

-1.

.455

0.

,014

67705

1,

.224

-3,

.217

-1,

.390

1.

.015

69102

-3,

.219

-1.

.158

2,

,864

-0.

.653

71819

1,

.676

-0,

.060

-2.

.460

0.

.564

74787

-5,

,431

0.

.234

2.

,559

-0.

.828

38

February 18 - DELTA X COEF:

XK CINT

(SECS PAST OOOOZ)

C(l.N)

C(2,N)

C(3,N)

50582

-0,

.125

-0

.869

-0.

.735

0

.146

51884

-2.

.182

-2.

.041

-0.

.165

0.

.309

53188

-4,

.438

-0.

.900

1.

.041

-0.

.293

54490

-4.

.493

0.

.319

-0.

.105

-0.

.215

55154

-4,

.391

-0.

.105

-0.

.533

0.

.028

56988

-6,

.206

-1.

.781

-0.

.381

0.

.605

57876

-7,

.664

-1.

.028

1.

.229

-0.

.426

58591

-7,

.926

0.

.077

0.

.315

-0.

.162

60973

-8,

.143

-1.

. 178

-0.

.842

0.

.506

62600

-10.

.108

0.

.098

1

.627

-0

.399

63843

-8.

.240

2.

.290

0.

.136

-2.

.161

64121

-7.

.638

1.

.864

-1.

.670

0.

.196

65781

-8.

.245

-2.

.055

-0.

.692

0

.324

68059

-12.

.690

-0

.161

1.

.523

-0.

.469

70229

-10.

.666

-0.

.178

-1.

.531

0

.318

February 18 - DELTA Y COEF:

XK CINT C(1,N) C(2,N)

(SECS PAST OOOOZ)

C(3,N)

50582

0.

.274

-0.

,209

-0,

,342

0.

.262

51884

0.

.002

0.

,235

0,

,682

-0.

.004

53188

1.

.454

1.

,989

0.

,665

-0,

.973

53968

2.

.950

1.

,246

-1.

,615

0.

.456

54796

3.

.134

-0.

,490

-0.

.485

0.

.229

56723

2.

.028

0.

195

0.

.840

-0.

.410

57391

2.

.412

0.

.769

0.

.019

-0,

.079

60488

2.

.641

-1.

.374

-0.

.711

0.

.792

61072

1.

.754

-1.

.394

0.

.676

-0.

.199

62654

0

.453

-0.

.748

-0.

.268

1.

.406

63182

0.

.190

0.

.140

1.

.955

-1.

.201

63748

0.

.678

1.

.198

-0.

.088

-0

.199

65604

1

.329

-1.

.182

-1

,195

0,

.917

66492

-0

.022

-1.

.136

1

.248

-0

.252

69352

1

.033

-0,

.189

-0,

,917

0

,230

39

February 21 - DELTA X COEF:

XK CINT

(SECS PAST OOOOZ)

C(l.N)

C(2,N)

C(3,N)

50652

-0.

.584

0

.227

-1.

.965

0,

,577

52188

-2.

.775

-1

.724

0.

.693

-0.

.003

53722

-3,

.797

0,

. 387

0.

.682

-0.

360

55513

-2.

.983

-0

,631

-1,

.250

0.

.250

56791

-5.

.311

-2,

,601

-0.

.291

1,

,685

57244

-6.

.391

-1,

.830

1.

.996

-0.

501

59223

-6.

.076

0,

.188

-0.

.976

0,

,271

60205

-6.

,577

-0,

,946

-0.

,178

0.

105

63227

-8.

,175

0

.845

0.

.770

-1,

.014

64044

-7,

.524

0

,078

-1.

.710

0.

,895

64707

-7.

.964

-1,

.009

0.

.073

0.

.044

66845

-9.

,358

-0

.097

0,

.353

-0.

.180

68873

-9,

,604

-0,

,887

-0

.743

0.

.220

71639

-13.

,081

0,

.058

1,

.084

-0.

.368

73796

-11.

,610

-0

.404

-1.

,298

0.

.379

February 21 - DELTA Y COEF:

XK CINT C(1,N) C(2,N)

(SECS PAST OOOOZ)

C(3,N)

50652

0,

,914

-1.

,338

0,

.883

-0.

.116

52188

0.

,521

0,

,553

0,

,349

-0.

.116

53873

1.

.888

0.

,740

-0,

,237

-0.

.038

55257

2.

,357

-0.

136

-0,

.396

0.

.262

56005

2.

,143

-0.

,288

0,

.193

-0.

.008

58727

2.

,639

0,

596

0,

.132

-0.

.252

60339

2.

.887

-0.

,942

-1,

,086

0.

.596

61421

1.

,352

-1.

,197

0,

.850

-0.

.067

63037

1.

,355

1,

026

0,

.525

-0,

.576

64378

2.

.288

-0.

668

-1,

.789

0,

.710

65845

-0.

,302

-1.

,333

1,

.336

-0.

.210

67731

0.

,527

1.

,463

0

,146

-0.

.362

69089

1.

.875

-0.

,145

-1,

.331

0.

.397

71018

-0.

,505

-0.

,845

0,

.968

-0,

.196

74402

0,

,144

-1.

Oil

-1,

.017

0.

.409

40

February 24 - DELTA X COEF:

XK CINT

(SECS PAST OOOOZ)

C(1,N)

C(2.N)

C(3,N)

45404

-0,

.714

1.

,912

-1.

.624

0.

.248

46809

-0.

.549

-1.

.186

-0.

,579

0.

.247

48216

-2,

.673

-1.

,347

0,

.464

0.

.191

49621

-3.

.117

1.

.094

1.

.272

-0.

.806

50768

-1.

.406

0.

,832

-1.

.500

0,

,203

52434

-3.

.243

-2,

.473

-0.

.484

0.

770

52815

-4,

.213

-2.

.506

0.

,397

0,

.125

55223

-6.

.203

1.

.578

1.

.300

-0.

.687

56360

-3.

.734

1.

,869

-1.

.045

-0.

.171

57839

-3.

.806

-2.

.340

-1.

,804

0.

,990

59085

-7.

.610

-2.

.220

1.

.899

-0.

.089

59977

-8.

.143

0,

,954

1.

.660

-0.

.548

61703

-4.

.369

1.

.791

-1.

.175

-0.

,158

63074

-4.

531

-2.

,326

-1.

.826

1.

.326

63882

-6,

.905

-2,

,678

1.

.391

-0.

.076

66634

-5.

,317

3.

,259

0.

.767

-6.

.039

66817

-4.

,732

2.

,933

-2.

,549

0.

,373

February 24 - DELTA Y COEF:

XK CINT C(1,N) C(2,N)

(SECS PAST OOOOZ)

C(3.N)

45404

0.

.539

-1.

,279

2.

,188

-0.

,591

46975

1.

,637

1.

.216

-0.

600

0.

.069

48546

2.

.333

-0.

.159

-0.

.275

0.

,097

49718

1.

.925

-0.

.403

0,

.067

0,

.085

51751

2,

.100

0.

.926

0.

.587

-0,

.298

53382

3.

878

0.

.464

-0.

.870

0.

.122

54768

3.

.177

-1.

.243

-0.

,362

0.

.213

57005

0.

.968

0.

,336

1 .

.066

-0,

,082

57837

1.

.937

1.

.938

0.

860

-0.

.796

59184

4,

.162

-0.

.082

-2,

.359

0.

.649

61384

-0.

,530

-1.

.043

1.

,923

-0.

.373

63128

1,

.522

2.

.263

-0,

.026

-0,

,471

64927

2.

,7/0

-2.

,397

-2.

.565

1.

.218

66457

-2.

.541

-1,

.682

3.

.031

-0.

.701

69029

1.

.255

-0.

,003

-2.

.378

0.

.407

41

F. DIVERGENCE ERROR FROM THE SCHULER OSCILLATION

It is useful to consider what error an 84.4-min sinusoidal
oscillation may induce in a line integral calculation of the horizontal
divergence. The divergence is expressed as

where V is the horizontal wind vector, n is a unit vector
normal to the flight track, and dl is an increment of
distance along the track.

For simplicity, we restrict our flight geometry to a square box with a
perimeter of length C which is flown counterclockwise at a constant
groundspeed S beginning at the southeastern corner. We further assume
that the box sides are oriented east-west and north-south and that the
earth's curvature may be neglected. Considering only the error in the
east component of velocity, we may represent the oscillation as

(F.l)

TT • / 27T t . .

= a sin(— + 4 >)

(F.2)

where a is the amplitude, T is 84.4 min, and ^ is an
arbitrary phase. Noting that contributions to the integral
from the east component of the error come only from the
north- south legs and that dl = S dt, we may write

C/4S

C/S

€

u

-► -►
= V • u

=x USdt-^ USdt
€ A € A €

(F.3)

0

3C/4S

C/4S

C/S

[cos(^ + 4 >) - cos(^ + 4 >) ]

0

3C/4S

= ^ cos(B+i^) - cos(3B/4+^) - cos(B/4+<^)]

42

where B = 27 tC/ST.

Since the error depends on the random phase <f> at the beginning of the
box, we determine the maximum divergence error by maximizing the error
with respect to <f> as follows. (F.3) may be rewritten

a ST

™ 27tA ^ cosBcos(^ - sinBsin(^ - cos( 3B/4)cos(^

+ sin(3B/4)sin<^ - cos(B/4)cos<^

+ sin(B/4)sin(^ + cos(^] (F.4)

To maximize the error,

0

d4>

aST r

L -cosBsin<^ - sinBcos^ + cos( 3B/4) sin(^
+ sin(3B/4)cos<^ + cos(B/4) sin<^

+ sin(B/4)cos<^ - sin-^ ]

(F.5)

This is true if the argument in square brackets is zero. A little
rearranging yields

tan<^

[ sin(-B) -I- sin(3B/4) + sin(B/4) ]
t c^osB ccT^ cos(B/4) + 1 ]

For C * 400 km,

S = 100 m s

and T = 84.4 min.

4> = 0.66 rad (or 37.82")

Substituting these values into F.4 yields
€„ = 5.4 X 10'^ s'^

(F.6)

This error is additive with the contribution to the divergence from the
error in the north component of velocity. Therefore, the error in the
divergence could be as large as 10" s' . The position difference data
quality appears high enough to expect an effective reduction of at least
an order of magnitude in the Schuler amplitude a. As a result, the
error in divergence due to the INS drifts after correction should be no
more than 10’^ s' .

43

icsA -

Distribution List

No. of Copies

Dr. Robert F. Abbey 1

Office of Naval Research
800 N. Quincy Street
Arlington, VA 22217

Dr. Joost Businger 1

NCAR

P. 0. Box 3000
Boulder, CO 80307

Dr. Kenneth Davidson 1

Department of Meteorology Code 63Ds
Naval Postgraduate School
Monterey, CA 93943-5000

Dr. Carl Friehe 2

Mechanical Engineering
University of California, Irvine
Irvine, CA 92717

Dr. Gary K. Greenhut 1

NOAA/ERL R/E2
325 Broadway
Boulder, CO 80303

Dr . Howard Hanson 1

CIRES

P. 0. Box 449
Boulder, CO 80309

Dr. Warren Johnson 1

NCAR

P. 0. Box 3000
Boulder, CO 80307

Dr. Siri Jodha Singh Khalsa 1

CIRES

P. 0. Box 449
Boulder, CO 80309

Dr. Bill Large 1

NCAR

P. 0. Box 3000
Boulder, CO 80307

Dr. Don Lenschow 1

NCAR

P. 0. Box 3000
Boulder, CO 80307

1

Dr. F. K. Li

Jet Propulsion Laboratory
MS 183-701

4800 Oak Grove Drive
Pasadena, CA 91109

Dr. William Plant
Code 7913S

Naval Research Laboratory
Washington, DC 20375

Dr. Robert J. Renard, Chairman
Department of Meteorology, Code 63Rd
Naval Postgraduate School
Monterey, CA 93943-5000

Dr. William J. Shaw

Department of Meteorology, Code 63Sr
Naval Postgraduate School
Monterey, CA 93943-5000

Dr. Steve Stage
Department of Meteorology
Florida State University
Tallahassee, FL 32301

Dr. Robert A. Weller

Woods Hole Oceanographic Institution

Woods Hole, MA 02543

Library, Code 0142
Naval Postgraduate School
Monterey, CA 93943-5000

Research Administration, Code 012
Naval Postgraduate School
Monterey, CA 93943-5000

Defense Technical Information Center
Cameron Station
Alexandria, VA 22304-6145

1

1

12

2

1

2

1

2

National Science Foundation
Washington, DC 20550

1

DUDLEY KNOX LIBRARY

I iiiiiiMii hill iiiiiiii iiih; ill I mil

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