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CALIBRATION OF EDMI AND RECOMMENDATIONS
FOR A BASE LINE NETWORK IN VIRGINIA
by
Dennis Ray Varney
Thesis submitted to the Graduate Faculty of the
Virginia Polytechnic Institute and State University
in partial fulfillment of the requirements for the degree of
MASTER OF SCIENCE
APPROVED:
A. C. Kellie
in
Civil Engineering
S. D. Johnson, Chairman
January, 1982
Blacksburg, Virginia
E. A. Taylor
ACK~WWLE:CG}!ENTS
I would like to thank the following faculty and staff for
their assistance in the work and ~riting of this thesis:
Mr. W. H. Byrne, Sr., Mr. A. C. Kellie, Dr. D. P. Hajela,
Mr. J.B. Sutphin, and Mr. E. A. Taylor. A special thank you
goes to Prof. S. D. Johnson for giving his time and effort when
they were needed in so many places by so many people. I would
also like to thank Ann Crate and Janell Frymyer for their
excellent typing of this thesis and their kind consideration to
a meagerly paid graduate student.
I would also like to express my great appreciation to my
classmates, Rich McDearmon and Tony Moraco for the many hours
they spent helping me in the field. Without them, this thesis
would never have been finished.
Finally, I would like to thank my wife, Teresa, and hly son,
Jared, for the great sacrifices they have made and for the
tremendous encouragecent they have been throughout my graduate
career.
TABLE OF CONTENTS
ACKNOWLEDGMENTS ••
LIST OF FIGURES •
LIST OF TABLES
CHAPTER 1. INTRODUCTION
CHAPTER 2. PRINCIPLES OF ELECTRONIC DISTANCE MEASUREMENT.
Introduction Measurement Signals ••••
Distance Measurement Phase Shift Measurement •.
Geometric Reduction Atmospheric Effects •..
Instrumental Errors Constant Offset Errors Scale Errors •••••• Cyclic Error ••••• Pointing Errors ••
CHAPTER 3. Introduction
DETERMINATION OF RESOLUTION.
Test Procedures •• Analysis of Data Results
CHAPTER 4. DETERMINATION OF REFRACTION AFFECTS , . Introduc~ion Test Procedures • Analysis of Data Results
C:IAPTER 5. DETERMINATION OF CYCLIC ERROR Introdu~tion Test Procedures ••••• Analysis ~f Data •.•.••••• Least S~uares Application to Cyclic Error Data Linear~~a:ion .••••• Iru.cia.i. .\r.proximations, Convergence, and Weights Otier P~5iible Models • • ••.
CHA?TE'P. 6 • 'JE:''.:R}!INATION OF SCALE AND CONSTANT OFFSET ER..~ORS • • • •
I:itrodact l:n: Test ?rc:,::edure
iii
Page ii
V
vi
1
3 3 3 6 6 8
11 16 17 19 20 22
24 24 24 26 28
29 29 29 30 32
35 35 35 ~O .:.1 .:.1 ... 3 .;.4 --5
52 = '
TABLE OF CONTENTS (cont.)
Analysis of the Data. Results •••••••
CHAPTER 7. ESTABLISHMENT OF THE VIRGINIA TECH BASE LINE . Introduction • • • • • • Physical Requirements Construction of the Base Line Monuments Base Line Calibration The Virginia Tech Base Line •.•.••
CHAPTER 8. A PROPOSAL FOR A STATEWIDE NETWORK ON CALIBRATION BASE LINES
Introduction .•••• Questionnaire •••.•.•••• Number and Location of Base Lines Sponsor Cost ••••.
CHAPTER 9. CONCLUSIONS AND RECOMMENDATIONS
REFERENCES •
APPENDIX A. NUMERICAL LISTING OF CYCLIC ERROR DATA
APPENDIX B. CYCLIC ERROR PROGRAM LISTING
APPENDIX C. PROGRN1 OUTPUT MODEL 1
APPENDIX D. PROGRAM OUTPUT MODEL 2
APPENDIX E. PROGRAM OUTPUT MODEL 3
APPENDIX F. HORIZONTAL ADJUSTMENT OF THE VIRGINIA TECH
Page 53 54
59 59 59 63 66 67
69 69 69 70 72 74
i5
77
80
83
95
1')3
1..11
CALIBRATION BASE LINE. • . • • . • . :19
APPENDIX G. ADJUSTED ELEVATIONS OF THE VIRGINIA TECH CALIBRATION BASE LINE MONUMENTS . . • 1:5
APPD:DIX H. LEAST SQUARES APPLICATION TO CONSTANT OFFSET AND SCALE ERRORS .•.•.
APPE~;)IX I. EDMI BASE LINE QUESTIONNAIRE WITH TABULATED RESPONSES
.. !T"T'-' 'w ...... ~.i • •·
ABSTF.AIT
iv
, -..
]_.: 1
Figure
2.1
2.2
3.1
4.1
5.1
5.2
5.3
5.4
5.5
7.1
8.1
LIST OF FIGrRES
Amplitude Modulation of Laser Light
Digital Method of Phase Neasurement in the Ranger IV. • • • ••.
Resolution Bar
Ranger IV Refraction Data ••
Equipment Setup for Cyclic Error Test
Least Squares Fit of Cyclic Error Data
Residuals vs Observed Distance
Residuals vs Observed Distance
Residuals vs Cyclic Correction
Typical Base Line Configuration
Proposed Statewide Network of Calibrated Base Lines ..•••.••.••..••
V
5
9
25
33
36
39
47
48
49
62
73
Table
3.1
4.1
5.1
6.1
6.2
LIST OF TABLES
Resolution Data and Reduction.
Refraction Test Data
Least Squares Program Results for Models 1, 2, and 3. , . . .•••
Data Reductions for Constant Offset and Scale Errors
Corbin Base Line Data
·;i
27
31
46
55
56
CHAPTER l
INTRODUCTION
The impact of modern technology is very evident in the field
of surveying and mapping. One of the major technological
developments in use today is electronic distance measuring
instruments (EDMI). Surveyors have come to rely on the speed and
accuracy of these instruments. However, if reliable and accurate
measurements are to be obtained, surveyors must understand the
operational principles of EDMI, and they oust calibrate their
instruments periodically. One means of monitoring and
calibrating an EDMI is with a calibrated base line. In the state
of Virginia, there is only one base line open for public use.
Its location, Norfolk, makes calibrating an EDMI inconvenient for
many of the state's surveyors. The purpose of this thesis is to
examine the establishment of a statewide network of calibrated
base lines. The thesis is also intended to be a learning tool
that illustrates the concepts of electronic distance measuring.
In addition, the thesis outlines tests perforned to deterr:iine
EDMI measuring errors.
For the feasibility study, much attention will be given to
the base line establishment procedures of the National Geodetic
Survey (NGS). The NGS policy of establishing base lines on a
national level will provide consistency in calibrating EDMI. The
Virginia Polytechnic Institute and State University Geodetic
1
2
Division of the Department of Civil Engineering is establishing a
base line at the Virginia Tech Airport adhering to NGS specifi-
cations. Much of what has been learned in this experience will
be applied in the study.
The literature review discusses basic concepts of electronic
distance measurement. These concepts are: 1) the use of carrier
and modulated waves, 2) distance measuremer.t techniques, 3)
geometric reductions, and 4) atmospheric effects. The review
also includes a discussion of the systematic errors encountered
in electronic distance measurement.
Monitoring and testing EDMI is often neglected by many
users. All but one of the tests cited herein are tests that can
be done on a base line. As an illustration of the test
pro~edures involved, all the tests described herein were actually
conducted using a K&E Ranger IV laser EDMI. The tests included
were:
(1) refraction,
(2) resolution,
(3) scale error,
(4) constant offset error, and
(5) cyclic error.
The results of these tests are i~cluded in separate chapters.
C}Lt,.PTER 2
PRINCIPLES OF ELECT~ONIC DISTAI~CE MEASUREMENT
Introduction
Like most electronic equipment, EDM instruments have a
limited life and some component of error in their measurements.
The errors that must be periodically monitored are the systematic
errors which are fully discussed in later sections. A full
understanding of these errors requires a knowledge of the
principles of electronic distance measurement.
Measurement Signals
The measurement signal generated by the EDM device consists
of two components. The first is a high frequency, short wave-
length carrier signal. The second signal is a low frequency,
long wavelength measurement signal. Most EDMI's used by
surveyors and engineers are short to medium range instruments
using either infrared (IR) or red laser light as the carrier
signal. Both IR and red laser light have wavelengths of less
than 1 cm. Microwaves which have a somewhat longer wavelength
can also be used. The basic measurement principles involved are
the same regardless of the type of carrier signal involved
(Laurila, 1976).
Carrier signals are high frequency waves that require little
power to generate. The carrier signal is supplied by a solid
state oscillator circuit in the EDM device. However, the carrier
3
4
signal wavelength is coo short to be useful for distance
measurement. Instead, another sine wave of longer wavelength is
superimposed on the carrier wave. This second signal is called
the measurement signal. The process of modulation is used to
superimpose the measurement signal onto the carrier wave.
Modulation may be done in one of three ways:
1) amplitude modulation (.AM),
2) frequency modulation (FM), and
3) phase modulation (PM).
Electro-optical instruments use amplitude modulation and
microwave instruments use frequency modulation to produce their
measurement signal. Phase modulation is not used in EDM
instruments.
Since most EDMI's used in practice are electro-optical
instruments and since all the tests performed for this thesis
were conducted with an electro-optical instrument, .only amplitude
modulation will be discussed.
In amplitude modulation, the frequency and phase of the
carrier wave remain unaltered and the amplitude of the carrier
wave is altered sinusoidally. An amplitude modulated signal is
shown in Figure 2.1. The superimposed measurement signal has a
wavelength of some integral multiple of ten meters (Laurila,
1976).
It should be noted that only the phase shift of the measure-
ment signal can be determined. It i3 not possible to determine
5 ' ' ' /
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I 4
QJ s
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ell 11 ~,
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,-<
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,, ,16'
0:::: /
r:,:J
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~
..-I 0
. N
z 0 r:,:J
!-' i:i::
~ ;::; c..:,
,.J H
:::>
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:>
;::; t'l s :::3:
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6
the number of full wavelengths along the measurement path.
Hence, a series of different wavelength measurement signals must
be used. Usually the measurement wavelengths used are in
~ultiples of ten: for examp:e, 20, 200, and 2,000 meters. This
process is known as decade modulation. Theoretically, only the
phase shift of the longest measurement signal would need to be
measured. Practically, the precision of phase shift measurement
is fixed. Therefore, the longer measurement wavelengths are
considered "coarse measurements". The result of such coarse
measuremer.ts are combined with the more precisely measured phase
shift of the shortest wavelengths to obtain a precise measure-
ment.
Distance Measurement. The most common way to determine the
length of a line using a ~odulated wave is to measure the phase
shift between the transmitted and the received wave (Laurila,
1976). The basic distance equation is
where
D =
V =
T =
1 D = - (V•T) 2
single path distance,
velocity of the light' and
double path travel time.
The velocity of light is given as
C V = --n
(2 .1)
(2.2)
7
where
C = the speed of light in a vacuum, 299,792.458 km/sec, and
n = the refractive index of air.
The travel time of the light is found by measuring the phase
shift of the measurement signal. Mathematically,
T = _E_ • .!_ (2.3) 360° f
where
T = travel time,
f = frequency of the measurement signal, and
p = phase shift.
Phase Shift Measurement
There are three methods used to determine the phase shift: 1)
manual phase delay, 2) manual or automatic phase shift, and 3)
digital phase measurement. The digital method of phase measurement
is the most common method used in "light" instruments, and is the
only one discussed here. The other two may be found in (Laurila,
1976).
According to the K&E Ranger IV Maintenance Manual, the
digital, or pulse-counting, method of phase measurement utilized in
the Ranger IV uses a precise time base to determine the phase shift
of the modulated wave. As the oscillator begins the modulation of
the carrier wave, a reference intermediate frequency (IF) signal,
or wave, is produced. The reference IF signal has a much lower
8
frequency than the modulated wave and thus has a much longer
wavelength. As the modulated wave returns to the receiver
circuits, it is converted to the same intermediate frequency as the
reference IF signal. Whe~ this return IF signal, or wave, passes
from positive to negative, a zero cross circuit generates a start
pulse and a very accurate clock is started. The clock continues to
produce pulses until the reference IF wave crosses from negative to
positive. The pulses are then counted and used to determine the
phase shift, and ultimately the distance. Figure 2.2 illustrates
the digital method of phase measurement.
Geometric Reduction. Distances measured with EDMI, probably
without exception, are slope distances ano must be reduced to their
horizontal component. The most reliable method for reducing a
slope distance to horizontal is the difference in elevation method.
This method requires that the elevations of the station occupied
and the station sighted are known. If the heights of both the
distance meter and the reflector are then measured above their
respective stations, the vertical distance between the end points
of the measurement can be computed. The horizontal distance is
calculated using the Pythagorean Theorem. By the following
equation:
2 2 1/2 D = (s - ((I+H) - (T+h)) ) (2.4)
~- Reference IF Wave (f=l. 498 Hz) ~ · Stop Pulse
+ Zero -1-f
Transmitted Light
(f=l4. 98 MHz)
-----_.- .... '--...
[µReturn IF Wave ~'""-. / (f=l .498 Hz) ,,,
------
11111111111
Axis
+ ~ Zero Axis
~ I I
I
Start Pulse _/ ' \ \ .
\ '"'- . ght , ... _., "'----Reflecte9~ ~~z)
(f=l4.
LEGEND > Reflector
[::::> EDMI Transmitting Circuits
l:> EDMI Receiving Circuits
11111 Clock Pulses
DIGITp.L METHOD OF PHASE MEASUREMENT IN THE RANGER IV FIGURE 2.2
I.O
where
10
D = horizontal distance,
s = PPM corrected slope distance,
I= elevation of the EDMI monument,
H = height of the EDMI above the monument,
T = elevation of the target monument, and
h = height of the target above the monument.
In EDM calibration, if the horizontal distances between base
line monuments are given at the elevation of the base line,
reduction of the measured distance to sea level or grid datum is
not necessary. Likewise, curvature corrections to the measured
distances are not necessary if the calibrated distances are not
corrected for the earth's curvature.
Since EDMI are calibrated using horizontal distances, the
reliability of the horizontal distance is an important factor.
The reliability of horizontal distances determined using the
difference in elevation method is found by an error analysis of
Equation 2.4, wherein
or (2.5)
Partial derivatives of Equation 2.5 a~e tak2n with respect to the
parameters s, 6E, and 6t. The partials are:
ds, (2.6)
11
dDE = (dE+dH) d(dE), and
(s 2-(dE+dH) 2) 1/ 2 (2.7)
dDH = (dE+dH)
(s 2 (dE+dH)2) 1/ 2 d(dH). (2.8)
The accuracy of the hcrizontal distance is thus given by the
equation
(2.9)
Given specific data from a base line, the reliability of any
particular horizontal distance may be determined (Mikhail, 1980).
Atmospheric Effects
Atmospheric density changes continuously. As the density
changes, the velocity of electromagnetic energy changes. This
causes measurement errors unless appropriate corrections are
applied. Temperature and pressure are the most important
atmospheric conditions affecting air density. Temperature and
pressure effect measurements made with light and microwave
carrier signals. Microwaves are also affected by humidity.
These density parameters--temperature, pressure, and
humidity--can be measured and used to determine the atmospheric
correction.
Equation 2.2 shows that the speed of light depends on the
atmospheric conditions at the time of distance measurement. The
effects of the atmosphere are extremely important in instrument
12
calibration and all EDMI ~easurements should be corrected for
them. Corrections fo~ the atmosphere are expressed as parts per
million (ppm) corrections. This is because the magnitude of the
correction is in direct proportion to the length of the line.
The sign of the correction·is also a function of existing
atmospheric conditions.
Wavelength is related to velocity and frequency by the
formula
A =--c __ u (n)(f)2 '
(2.10)
where
c = 299,792,500 m/sec (the speed of light in a vacuum),
f = modulation frequency,
n = atmospheric refractive index, and
A = unit wavelength. (Meade, 1972) u
The unit wavelength is the length a light wave would have if it
traveled through a vacuum with a velocity equal to the speed of
light.
The refractive index, n, also called the preset or assumed
index, is based on standard atmospheric conditions of
1. 0°C or 273.2°K,
2. 760mm of mercury or 29.92 in. of mercury, and
3. 0% humidity.
13
Very rarely will these conditions exist. Therefore, the unit
wavelength corresponding to the tiOn-standard atmospheric
conditions will not be the unit wavelength. However, using
~easured atmospheric parameters, corrections can be applied to
the unit wavelength to determine the wavelength used to measure
the distance (Meade, 1972).
The correction to the unit wavelength can be determined from
measured atmospheric parameters. First, the group refractive
index, n, i~ described by the relationship g
n· = l·+ (287.604 + 4.8~64 + 0.018)10-6, g A A
n = 1 + (287.604 + 4.8864 + 0.068 )10-6, g (.910) 2 (.910) 4
(2.11)
n = 1.000293604. (Meade, 1972) g
The group refractive index is a constant for any group of
instruments using the same light wave. For example, the Cubic
HDH-70, the Zeiss SM4, and the Geodimeter 14 all have a carrier
wavelength of 910 nm infrared light, and therefore have the same
group refractive index (Tomlinson, 1971).
The re~ractive index varies with changes of temperature,
pressure, and humidity. The actual refractive index, na' is
determined from the relationship
n -1 -8 n = 1 + ~~g--~-•....E_ _ 5.5 elO
a t 760 t l + 273.2 l + 273.2
(2.12)
14
where
t = temperature, °C,
p = pressure, mm of Eg, and
e = vapor pressure, mm of Hg. (Meade, 1972)
The reduction of this formula is as follows:
n -1 n = 1 + __ g __ _ a 273.2 + t
273.2
_L_ 760
5.5 elO-S 273.2 + t
273.2
n -1 !la = 1 + 273. 2 (J!o 273~2
273.2(5.5)e10-S 273.2 + t , and (2 .13)
(n -l)p na = 1 + 0.359474 273~2 +
1. 502600el0- 5 273.2 + t
If n x 10- 6 = 0.359474 (n - 1), then N can be determined for any g
carrier frequency. Using N and the preset index, a correction
can be derived for any measurement wavelength thus:
/J) .. =n- (l + N X 10-\ _ 1. 502600e10- 5 273.2 + t 273.2 + t
In this form, n = (1 + ppm's). The formula may be further
reduced to yield a ppm correction, where
ppm N° p 15e = I - 273.2 + t + 273.2 + t
in which
I= (R.I. - 1) x 10- 6 ,
N = constant for a particular carrier wavelength\,
p = barometric pressure (in mm),
(2.14)
(2.15)
15
t = temperature (in °C), and
e = vapor pressure (in mm of Hg).
If the barometric pressure is measured in feet with an altimeter,
the altitude reading can be converted into mm of Hg by the
formula
where
P = 25.4 X a
e '
e = base of the nature logarithm,
3.3978 - R(3.6792 x 10- 5), and a =
R = altitude reading.
Likewise, the vapor pressure can be readily computed by the
formula,
e = e' + de,
where
e = vapor pressure (in mm of Hg), 6 e' = 4.58 X 10 ,
de= -0.000660(1 + 0.00115t')P(t-t'),
b = 7.5t'/(237.3 + t),
P = pressure (in mm),
c' = wet bulb temperature (°C), and
t = dry bulb temperature (°C). (Fronczek, 1977)
(2.16)
(2.17)
Finally~ when the ppm correction has been calculated, it is
applied to the slope distance as follows:
where
16
-6 s' = s(l + ppm x 10 ),
s' = corrected slope distance,
s = measured slope distance, and
ppm= part per million correction.
(2.18)
Instrumental Errors. Four possible sources of measurement error
are
(1) modulation of the carrier wave,
(2) demodulation of the carrier wave,
(3) conversion of the carrier wave into the return IF
wave, or
(4) comparison of the reference IF wave to the return
IF wave.
Both the modulated wave and the return IF wave are sinusoidal, so
that distortions in the modulation, demodulation, or conversion
of these waves should produce sinusoidal errors (Kelly, 1979).
The reference IF wave produced by the Ranger IV is a square wave
since it is used in the precise measurement of time (K&E Mainten-
ance Manual). Incorrect ceasurement of the time from negative
alternation of the return IF wave to the positive alternation of
the reference IF wave will also lead to errors in the
determination of the distance, though not necessarily sinusoidal.
These errors are collectively called instrumental errors.
17
Instrumental errors are separated into three categories, all
of which are systematic and thus correctable. The categories are
1. constant offset error,
2. scale error, and
3. cyclic error.
Constant offset error and scale error can be measured using
a calibrated base line. Without additional monumentation on the
typical base line, cyclic error will go undetected. Cyclic error
determination requires a series of ten to twelve monuments spaced
equally apart over the distance meter's measurement wavelength.
Constant Offset Errors
The constant offset error has two components: instrument
offset and reflector offset. In most distance meters, the
electrical center of the instrument is not the physical center of
the instrument. The distance between the electrical center and
the physical center of the distance meter is called the
instrument offset (Meade, 1972). The reflector offset results
from the transmitted light going from air to glass and back to
air. The glass has a larger index of refraction than air, thus
requiring more time for the light to pass through the glass than
the light would require to pass through the same distance cf air
(Wolfe, 1974), The EDMI senses this "extra" distance, thus
resulting in a measurement that is too long. This error can be
eli~inated by properly centering the prism. Both offset errors
18
are functions of the particular EDHI and reflector combination
and must be determined accordingly.
One technique for determining constant offset error is to
measure a line, AC, and its two constituent parts, AB and BC.
The offset error is constant for all three measurements and can
be represented mathematically as
AC - e = (AB - e) + (BC - e) (2.19)
where
AC = total line length,
AB = line length from point" A to point B,
BC = line length from point B to point C, and
e = constant offset error. (Green, 1977)
Solving for the constant offset error
e =(AB+ BC) - AC (2.20)
This method provides a unique solution for the constant offset
error.
A second technique for determining constant offset error
uses a calibrated base line. This method produces a least
squares estimate of error (Fronczek, 1977). The least squares
solution is the preferred solution when there are redundant
measurements (Mikhail, 1981). In this case, more than one base
line measurement is needed to provide redundancy. Redundancies
19
provide checks against m~asurement blunders - blunders that might
go undetected using Equation 2.19. Statistically, as the number
of redundancies made on a base line increases, the estimates of
the errors more accurately define the actual errors (Mikhail,
1981).
Scale Errors
The modulation frequency of electro-optical instruments is
fixed by an oscillator. The modulation frequency will slowly
shift as that oscillator ages. The change in modulation
frequency changes the length of the modulated wave and thus
produces an error in the measured distance. The error is
proportional to the distance measured and is termed scale error
(Greene, 1977).
One way of detecting scale error is with an electronic
frequency counter. However, there are also field procedures
available to determine scale error. The first field technique
requires comparison of an uncalibrated distance meter to one
which has a modulation frequency that is known to be accurate.
The procedure is to measure a line of approximately 100 meters
with both distance meters. Both distances are reduced for
atmospheric effects, for constant error, and for slope Jistance.
Any residual discrepancy is attributed to scale error (Greene,
1977).
20
A second technique for determining scale error is to use a
calibrated base line (Fronczek, 1977). If a calibrated base line
is used, no other distance meter is required. Only one
observation is needed. If more than one measurement is made,
redundancies exist and a least squares solution can be obtained.
Cyclic Error
Cyclic errors are those errors in measurement that occur due
to incorrect measurement of phase shift. The most significant
source of incorrect measurement of phase shift arises from
spurious coupling between the transmitting and receiving
channels. This results in an error that varies sinusoidally with
distance (Green, 1977).
The detection of cyclic error is accomplished by measuring a
series of distances that span the distance meter's nominal wave
length (Davis, 1981, and Moffitt, 1975). According to Kelly
(1979), the nominal wave length should be broken down into ten or
twelve equally spaced, precisely measured segments. After all
other previously mentioned reductions have been ~ade, the
remaining error is cyclic error. These errors are then plotted
against the series distance, or the distance along the wave, to
produce a cyclic error curve. This curve is then used to predict
cyclic error in all measured distanc2s (Moffitt, 1975).
There is another method given to determine cyclic error
(Robertson, 1976). In this method, the measuring procedure is
21
the same as the one given above. However, only one value is
given for the cyclic error as opposed to a continuous sinusoidal
error. Robertson further states that this single quantity, et,
also contains "resolution error", e , and since these errors are r
independent the actual cyclic error, ec, is
A single valued quantity for the cyclic error would probably
suffice if the amplitude of the error's sine wave is as small as
the instrument's resolution. When the cyclic error varies
between a large positive and a large negative quantity, one value
will not adequately describe the error.
An alternative is to consider the least squares estimate
derived from the observed data to be the best estimate of the
cyclic error. This alternative requires that the error be a
continuous function dependent only on the reflector's distance
from the EDMI. Furthermore, resolution is not considered an
instrumental error, it is considered an instrumental limitation.
Here the magnitude of the cyclic error is not a function of the
instrument's resolution. Other investigators, (Moffit and Davis,
1981), use these same two conc~pts, but a least squares reduction
cf the error is not used. Rather, one complete wavelength is
divided into ten or less equal parts, ~nd the errors are
computed, plotted, and connected with straight lines. These
investigations have shown cyclic error, of 6.5 t1m maximum. The
22
cyclic error is a system cyclic error. The error measured
includes both EDMI cyclic error and constant offset errors of the
EDMI and the reflector.
Since cyclic error requires a series of relatively short
distances, it cannot be measured on the typical four monument
base line. In fact, most base lines are designed so that cyclic
error is, or is close to, zero at the calibration distances
(Dracup, 1977). NGS requires that base line monuments be an
integral number of ten meters. At integral numbers of ten
meters, cyclic error is theoretically zero and will not cause
error in a measurement.
Pointing Errors
Pointing errors result from beam divergence. Infrared light
and microwaves can have a divergence angle of 1/3°. At this
angle, the beam diameter soon be£omes considerably larger than
·the reflector. Consequently the reflector samples only a portion
of the radiated energy and the particular portion sampled depends
upon the precise pointing of the instrument (Greene, 1977).
According to Kelly (1979), pointing error does not exist in
properly operating instruments. To insure proper operation, EDMI
should be tested for this error. The test is to vary the
paintings on a reflector for a single setup. If the readings
vary more than the instrument's resolution, pointing error may
be suspected and the instrument should be returned to the
23
manufacturer for repair (Kelly, 1979). The test is readily
adaptable to a calibrated base line, but a base line is not
essential.
Introduction
CHAPTER 3
DETERMINATION OF RESOLUTION
The resolution of an EDMI is the instrument's capability to
distinguish the individual divisions of a unit of length. Most
EDMI have a resolution of one centimeter or less and a least
count readout of one millimeter (Tomlinson, 1971). The
resolution of an instrument is used to analyze variance in test
data.
Each of the tests performed in this thesis used the K & E
Ranger IV to collect data. The resolution of the Ranger IV is
determined in this test, and will be adopted as the instrument's
measurement precision for the analysis of data in this thesis.
Test Procedures
The Ranger IV's resolution was tested with a resolution bar
at distances of 150 m and 1400 m from the instrument. The
resolution bar is shown in Figure 3.1. Each of the distance
intervals on the bar were measured and then compared to the bar
distances. The bar intervals were machined to within 0.02 mm
which is two orders of magnitude less than the least reading of
the Ranger IV. The close tolerance on the bar insures any
difference between the EDMI interval and the bar interval is due
to the resolution of the EDMI.
24
Dimensions between holes accurate to+ 0.00071 in.
-13 .oo 7 .oo · - 12.oof- 6.oo ---1- 6.oo -j -~ -=~ ,:,--,\ . --: I ' ,. • ___ __J
,, ,, -- 4 ', --.:' .00 _J_
- 0.7500 + 0.0005 for each hole.
30.00 1 ~ 1; li I rJ-----o.5 I 0.5
:--....__ o. 6 All dimensions are inches.
RESOLUTION BAR
FIGURE 3.1
t-> U1
26
The actual distance measurecent to each position on the bar
was made from the sane setup of the distance meter to the same
height of reflector under closely monitored atmospheric
conditions. This procedure was used at both bar stations to
insure all measurements were made with equal precision.
Thirteen readings were made for each position on the bar. The
last 10 readings were recorded and used in the determination of
the horizontal distance; the first 3 were used for instru~ent
warmup. Temperature and barometric pressure readings were taken
at the EDMI and reflector before and after each set of 13
readings. A humidity reading was made at both stations before
and after each series of 13 readings. The two temperature
readings and the two barometric reaaings for each measure~ent
were averaged and these two averages were used with the single
humidity average to calculate the ppm correction.
Analysis of Data
The observed slope distances were corrected for the ppm
error and then reduced to horizontal by the method of elevation
differences. Corresponding horizontal distances were then
subtracted to produce a measured bar interval. The differences
between the bar intervals and the measured intervals were summed
and the standard deviation computed. The standard deviation is
considered to be the instrument's resolution (Kelly, 1979).
Table 3.1 contains the data and the computed standard deviation.
27
Table 3.1
Resolution Data and Reduction
Measured Measured Bar Interval Distance Interval Interval Difference
(m) (m) (m) (mm)
149.6986 0.0706 0.0762 -5.6
149.7692 0.1799 0.1778 2.1
149.9491 0.0439 0.0508 -6.9
149.9930 0.1540 0.1524 L6
150.1470 0.1527 0.1524 0.3
150.2997
1399.6786 0.0796 0.0762 3.4
1399.7582 0.1712 0.1778 -6.6
1399.9294 0.0523 0.0508 1.5
1399.9817 0.1503 0.1524 -2.1
1400.1320 0.1507 0.1524 -1. 7
1400.2825 !: = -14.0 mm
µ = -1.4 mm
(j = +3. 8 mm
The resolution is equal to the standard deviation, cr, or
+ 3.8 mm.
28
Results
If no bias exists in the instrument, then the computed
value for the mean, X, must be statistically equal to zero,
Since the actual stand.a rd deviation of the instrument's
measurements is a sample standard deviation, at-test is used to
test for instrument bias. A 95% confidence level is used to
test the hypothesis,
H: X = µ = 0.0 0
Accept H if 0
x-a -t < -- < t df=(n-1),a/2 S- df=(n-1),1-a/2 X
-1.1-0 -2.262 < 1.2733 < 2.262
- 2.262 < -0.8639 < 2.262
Therefore, the hypothesis that X = 0 is accepted as is the
hypothesis that there is no instrument bias.
The manufacturer states the repeatibility of the Ranger IV
is 5.0 mm. The measured resolution is+ 3.82 mm. Since the
resolution is less than the repeatibility, the EDMI is operating
correctly.
Introduction
CHAPTER 4
DETERMINATION OF REFRACTION AFFECTS
The atmospheric parameters defined previously can
significantly alter the length of a measured distance. Under
field conditions these parameters can, at best, be measured only
at the ends of the measured line. These two measurements may
not be representative of the actual parameters.
The Barrel and Sears Formula, equation 2.11, is the most
often used predic~or of atmospheric refraction of light. The
formula was used in this experiment to evaluate the
effectiveness of the two line-end atmospheric measurements in
predicting the actual atmospheric parameters. The procedures
used and the results obtained are given in the remainder of this
chapter.
Test Procedures
The Ranger IV was used to measure the distance between the
0-meter and 1400-meter monuments of the Virginia Tech base line.
The Ranger IV was set up over the 0-meter monument from 9:00
a.m. to 12:30 p.m. The setup was not changed so that the
instrument height and the centering error would be constant
during the experiment. A reflector was set up ever the
1400-meter monument for the same time interval and ~c: noved
during the neasuring sequence. Beginning at 9:25 a.~., distance
29
30
measurements were made every 15 ~inutes until 12:10 p.m.
resulting in 12 measurements. Eefore and after each measurement
the temperature and barometric pressure were recorded at both
ends of the line. Each measurement consisted of 13 readings,
the last 10 of which were recorded. The 10 recorded readings
were averaged. The average was adopted as the measurement.
Similarly, the four temperature and four pressure readings were
averaged. In addition, the humidity was measured with a sling
psychrometer. Only one measurement was made for reduction
purposes since the humidity correction rarely exceeds 1 ppm.
The instruments used to measure the atmospheric parameters,
except the sling psychrometer, were either calibrated at Corbin,
Virginia, by NGS or calibrated here, at Virginia Tech, against
the Corbin calibrated instruments. The data gathered is
given in Table 4.1.
Analysis of Data
Figure 4.1 gives a graphic representation of the results
obtained. The graphs show the observed measured distance and
the reduced slope distance plotted against time. Theoretically,
as time increases the temperature and pressure change results in
changes in the wavelength. Thus, the observed distance changes
continuously and, in general, in the direction opposite the
temperature change (Fronczek, 1977). Conversely, as time
31
Table 4.1
Refraction Test Data
Time Temn/Altitude PPM
9:25 58/2840 53.9
9:40 58/2840 53.9
9:55 57/2838 53.4
10:10 56/2837 52.9
10:25 55/2826 52.3
10:40 56/2823 52.8
10:55 56/2819 52.7
11:10 57/2819 53.2
11:25 56/2815 52.7
11:40 56/2818 52.7
11:55 56/2816 52.7
12:10 57/2815 53.1
32
increases, the reduced slope distance remains constant within
certain limits.
As seen in the graph, the data is almost linear for the
morning observations. The largest ppm correction, i.e. the
smallest wavelength, was 53.99 ppm versus 52.38 ppm for the
longest wavelength. This small difference is due mainly to the
small temperature range.
Another interesting feature of the graph is the similarity
in the shapes of the measured slope and reduced slope distance
lines. In fact, the reduced slope distance line seems to be the
observed slope distance plus a translation. According to the
Barrel and Sears formula, this occurrance is expected for small
differences in the ppm correction, i.e., small temperature
changes. If the ppm are the same for each measurement and the
difference in elevation is constant, then the reduced slope
distance over time must be the measured sloped distance plus a
constant. Under this assumption the reduced slope distance
should also be constant. The differences in measured and
reduced slope distances exhibited in the data can be contributed
to the instrument's resolution.
Results
As seen in Figure 4.1, the observed slope distances show a
maximum difference of 7.4 mm which falls within the measured
resolution of the instrument, (13.82 mu). More importantly, the
s 0 0 0 Q) . °' °' C') r-1
(/l
4.0 cm
3,0 cm
::, 2. 0 cm i:: •n ;:;:: (1) CJ i:: cu µ (/l •n A (1) p. 0
r-1 (/)
'"Cl (1) ,._. ::, (/l cu 1.1 ;:;::
1.0 cm
0.0 cm
-(!J - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -0 0 0
0 0 0 0 0 . 0 0 ---------------- -- -G---------- ---- -- - ---- ---- -- ----
--------------------. - ------------ ----- -- ----- - ----A A A
A A A
A A 8 l:l -----------------------------
Q Measured slope distance.
6 Reduced slope distance. Resolution interval of the Ranger IV ( _±_3.8 mm).
30 60 90 120 Elapsed Time (minutes)
RANGER IV REFRACTION DATA ------FIGURE 4.1
150
6 ll
180
10.0 cm ~ (I) p. i:: n (I) p.
(/)
9.0 cm ..... 0
'd (l)
d I-'• en rt Al ::,
8.0 cm
n w 11) w ~ I-'• ;:1 i:: C/l
..... w •.o '° .
7.0 cm co 0 0 0
a
6.0 cm
34
maximum difference between reduced slope distances has decreased
to 5.1 mm. Theoretically, the difference after correction
should be zero. However, the corrections being applied are at,
or below, the instrument's resolutio~. From a practical point
of view, the results are satisfactory, but, emphasis must be
placed on the fact that the test was carried out under overcast
skies with very small temperature changes. Entirely different
results may be obtained on a sunny day when the temperature has
a wider range.
Introduction
CHAPTER 5
DETEIDiINATION OF CYCLIC ERROR
One of the three systematic errors occurring in EDMI is a
non-linear error. This non-linear error is cyclic in nature and
occurs when the phase comparison technique is in error (Greene,
1977). If the pulse counting mechanism were to deviate from
linearity at a certain phase value, the resulting error would
repeat each time a distance resulted in that phase, and would
vary sinusoidally with phase shift over one wavelength.
Test Procedures
In order to determine the cyclic error, a series of short
distances were measured over one conplete wavelength. The
measured distances were then corapared with the known distances to
determine the error. The procedure was discussed by Robertson
(1976). AK & E Ranger IV was tested for cyclic error.
A steel tape, calibrated by the National Bureau of
Standards, was stretched along a steel rail under a tension of
ten pounds, developed with a calibrated tension handle. A
calibrated tape thermometer was attached to the tape. The Ranger
IV was mounted over the tape. It's height above the rail and its
position on the tape were measured. Figure 5.1 shows the equip-
ment setup for this test.
35
l'lWFILE .VIEW
Tensi1,n Ilandle
I I I I I I I I I I I ~ I I I 11 CI I I I I I I I I I I LI I I I ~Jlil'l' Cinder Block Wall
~C-Clamp Calibrated Steel Tape
EQUIPMENT SETUP FOR CYCLIC ERROR TEST .
Frcurrn s. 1
Tape Thermometer
l,l
°'
37
A reflector, fabricated such that its constant offset error
was 0.00 rem, was centered over consecutive one-foot marks.
Measurement began on the 16-foot mark and ended on the 66-foot
mark. The reflector was centered with a mechanist's center
locator. The height of the reflector above the rail was measured
at each foot mark.
Since the distances were short, a signal filter had to be
used during the experiment. The signal filter blocks some of the
transmitted light, or signal, so the external signal and internal
signal may be balanced. The distance was read at each foot mark
13 times with the last ten readings being recorded. The atmos-
pheric temperature and pressure were read before and after each
foot measurement with a calibrated thermometer and altimeter.
Laser light transmitted by the Ranger IV is both temporally
and spatially coherent making it a very narrow, concentrated beam
over long distances (Laulira 1976). Thus, when the Ranger IV is
used with the 1 1/2 inch diameter reflector and a signal filter
at distances less than 70 ft., the laser beam has about 1/2 the
diameter of the reflector. In this test the difference in
elevation from the distance mecer to the center of the reflector
was 1.27 ft. If the beam was pointed at the lower half of the 1
1/2 inch diameter reflector while the prism was on the 16-foot
mark, the error in the slope distance would be 2000 ppm and with
the reflector at the 70-foot mark, the error would be 100 ppm.
38
Thus, extreme care had to be taken to insure that distance
observations were made with careful pointing.
To reduce the data, th~ temperacure and pressure were used
to calculate a ppra correction for each measurement. The slope
distance was corrected for the ppm calculated. The corrected
slope distance was then reduced to horizontal using the relative
heights of the distance meter and reflector. The constant offset
of the distance meter was then added to the horizontal distance.
The distances along the tape were calculated by subtracting
the EDMI position from the reflector position. A temperature
correction was applied to the tape distance, and the tape
distance was adopted as the true distance. No tension, sag, or
length corrections were necessary. The cyclic correction was
calculated by subtracting the EDMI distance from the tape
distance. The correction was then plotted against the tape
distance to yield Figure 5.2. A numerical list of the observed
data may be found in Appendix A.
The data in Figure 5.2 was gathered in three independent
cests. On three separate days, the apparatus was mounted on the
rail and new readings made. On day one, the consecutive foot
marks from 16 to 49 were measured. On day two, the consecutive
foot marks from 46 to 66 were measured. On day three, the even
foot marks from 16 to 50 were measured. The data on days one and
two were gathered by the same personnel while the data on day
three were gathered by different personnel.
10
8
6
4
2
-12.
-14 ·
-16
Meters Feet
LEGEND O Day 1 Data D. Day 2 Data 0 Day 3 Data
Model 1 Model 2
-·- Model 3
Q) [;)
0
l:J 6 8
15 2C 25
39
[]
0 / ...... '\ 0
10 12
30 35 40 Distance From EDMI
(meters)
14
45 50
LEAS7 SQUARES FIT OF CYCLIC ERROR DATA
FIGURE 5.2
16 18 20
55 60 65
40
Analysis of Data
As explained in an earlier section, several different waves
are used to actually measure a distance. Errors may occur in the
modulation or demodulation of the carrier wave, in the return IF
wave, or in the pulse counting technique. These errors will be
present in the measurement. The test is designed to detect
cyclic error in the Ranger IV, but not to reveal the source of
the error. The source is not important if the error is
systematic and a correction can be applied.
An examination of Figure 5.2 reveals that the data has a
sinusoidal shape along the distance axis. Thus, the data should
follow the general form of a sine wave. The most general form is
e =A~ sin (L + R) + T (5.1)
where, for the cyclic error data,
A= amplitude of the error wave,
L = observed distance along the error wave,
R z translation along the distance axis, and
T = translation along the amplitude axis.
The translation parameter along the distance axis, R, accounts
for the origin of the cyclic error wave while the other
translation parameter, T, accounts for asymmetry in the error
wave about the am?litude axis.
41
Least Squares Application to Cyclic Error Data
Equation 5.1 provides the mathematical model to be used in a
least squares program to fit a sine wave to the cyclic error
data. The parameters in the model are the amplitude, A, the
distance axis translation, R, and the amplitude axis translation,
T. This model will be used as the observation equation for a
least squares solution of these parameters by the observation
equation method.
Linearization
Since the observation equation for this data is not linear,
it must be linearized and iterated to converge to a solution for
the parameters. Linearization can be accomplished by applying
Taylor's series expansion, and neglecting second and higher order
terms. The linearized form of Equation 5.1 is
e = e0 +::(DA)+:; (DR)+:; (DT)
where
e = A sin (L+R) + T
e = e evaluated for each observation at the initial 0
(5.2)
approximations of the parameters, de dA = partial derivatives of e with respect to the paraneters,
evaluated at the initial approximations of the
parameters, and
DA= change in the parameters.
where
42
In matrix form, Equation 5.2 is
1 e = el m mo + Dl n
m = number of observations,
n = number of parameters,
(5.3)
B = matrix of partial derivatives of e with respect to the
· parameters, and
D = correction matrix.
Let e - e = F, then Equation 5.3 becomes 0
BD = F (5.4)
The D matrix represents changes in the parameters that are
to be applied to the initial approximations of those parameters.
The updated parameters are then used to re-evaluate Equation 5.2
and thus produce a new correction for the parameter to be used
with the updated parameter. This iteration process is continued
until the corrections are less than a predetermined value.
The weighted least squares solution for Dis
(5. 5)
where Wis the weight matrix for the observations.
A weighted least squares program has been written to
evaluate the data gathered in this test. A program listing may
43
be found in Appendix B. The following section provides some
information about this program.
Initial Approximations, Convergence, and Weights
The first parameter, amplitude, is a measure of the
magnitude of the cyclic error. The initial approximation for the
amplitude used in this program is
A = E - ((E - E . )/2) o max max min (5.6)
where
A = initial approximation for the amplitude, 0
E = largest measured max positive cyclic correction, and
E = largest measured negative cyclic correction. min The next parameter, amplitude translation, is an estimate of
the average value of the sine wave. Thus, in this case, it must
estimate the average cyclic correction and is initially
approximated in the program as
(5. 7)
where T, E , and E . are as previously defined. max min The last parameter, distance translation, accounts for a
translation from the cyclic error origin. It ~ay be approximated
as zero units. The actual units of the three para~eters depend
on the unit system used. Metric units are csed in this tzst.
The unit of amplitude is millimeters. The unit of distance
translation is meters. The units of amplitude translation is
44
millimeters. The units of the sine function argument must be
converted from meters to radians. This is accomplished by
multiplying the sum, L+R, by PI radians and dividing by one-half
the single path wave length, 5.00 meters. This conversion
defines
as a measure of the phase shift of the transmitted light.
For ·this program, iteration continues until all four
-10 parameters change by less than 1.0 x 10 units. This -10 convergence represents 1.0 x 10 mm for both the amplitude and
-10 amplitude translation and 1.0 x 10 for the distance -5 translation, or 1.3 x 10 seconds of arc for the phase shift.
The program will do 10 complete iterations before an iteration
stop is reached.
All the data observed in this experiment, 73 observations,
are assumed to have equal weight. Equal weights are justified
since all observations were obtained using the same equipment and
procedures.
Other Possible Models
From Equation 5.1, it is possible to obtain other models.
The distance t~anslation, R, can be removed to yield the
following model.
W = A sin (L) + T. (5.8)
45
The distance translation can be considered zero since the cyclic
error is theorized to be zero at ~he distance neter and at ten
meter intervals from the distance meter. Next, the distance
translation is replaced and the amplit~de translation, T, is
removed. This yields
W = A sin (L+R). (5. 9)
A fourth model is possible if neither translation is considered.
An examination of data shows that there is a translation of some
kind, so this fourth model will not be considered.
Hodel Analysis
Henceforth, the models will be numbered as follows:
Model 1: w = A sin (L") + T, 5
Model 2: w = A sin [ (L+R) 1T] + T, and 5
Model 3: w = A sin [ (L+R) ·;] + r. 5
Program results for each model may be found in Appendices C, D,
and E. Model results may be found in Table 5.1.
To begin the model analysis, techniques described by Draper
and Smith (1966) are applied. These are graphical techniques
that examine the residuals generated from a least squares
solution. The residuals for each nodel have been graphed and may
be fou~d in Figures 5.3, 5.4, and 5.j.
46
Table 5.1
Least Squares Program Results for Models 1, 2, and 3
Parameters ~fodel 1 Hodel 2 Model 3
Amplitude (mm) 7.366 -9.270 8.598
Amplitude Translation (mm) -2.988 * -2.667
Distance Translation (m) * 3.818 3.860
Reference Variance (mm)2 18.903 18.989 9. 969
Degrees of Freedom 71 71 70
* Denotes parameters not included in the model.
8
4
~
-4
-8 ,-,.
Cl) Cl) 1-l
·~ i 12 ~ Q)
'"Cl f3 •r-i •ri Cl)~ 8 Q) ~ ~ •r-i
f3 ~
4
0
-4
-8
.a-------
•
. .
------ Instrument Resolution ( '± 3,8 mm)
~
Model 3 a2 = 9.969 mm2
0
Model 1 a2 = 18.903 mm2
0
..!
~---------------------------------------------,-r~---------------------------· •.
5 6 7 8
. . .
-------------·----- -~----------------------~1-~-
9· 10 11 12 13 14 Distance from EDMI · (meters)
RESIDUALS .vs OBSERVED DISTANCE! FIGURE 5.3
15 16 17 18 19 20
p.. -....J
,,..... CJ)
CJ) µ M Q) (lj .µ ::l Q)
'U s ·rl •rl Ul M Q) M ~ •rl s
'-'
12 Model 2 a2 = 18.989 mm2
8 • 0 . . . . . . . 4 t_ - - _._ - - ' ' - - - - - f - - L •' - - - - - - - - - -•- ,- - - - - - - - - - --. 0
-4
n -()
5 6
I
7 8 9
. ..
10 11 12 13 14
Distance from EDMI
(meters)
RESIDUALS vs OBSERVED DISTANCE
FIGURE 5.4
15
Instrument Resolution ( + 3.8 mm)
- ---- - - -- -------. . . .
. .
16 17 18 19
,l::-CX)
UJ r-i C1l ;:l
'U ·rl UJ (I) p::
10
5
0
-5
-10
.• I • • . . .
-16 -12 -8
. . . . . . ..
·'
-4
. .
Model 1
. .
0 4
a2 0
. :• ..
8 Observed Cyclic Correction
NOTE: Units for residuals and cyclic corrections are millimeters.
8 ~-18.903 mm2
UJ 4 r-i C1l ;:l
'Cl •rl UJ
0 (I) p::
-4
-8
12 10
UJ r-i 5 C1l ;:l
'Cl •rl UJ (I) 0 p::
-5
. . .
-16
Model 3
a 2 = 9.969 mm2 J) . ,, . . . . . . . . .. . .. . . . . . . . . . . . . .. ... . . . . .. . . . .
-12 8 4 0 I~ 8 12 Observed Cyclic Correction
... ... . ,, .... ..
Model 2
. . ..
. .. . . . : . . . .
.. . . ...
a 2 = 18.989 mm2 0
.. . . . .
-16 -12 -8 -4 0 4 8 12 Observed Cyclic Correction
RESIDUALS vs CYCLIC CORRECTION
FIGURE 5.5
.po. I..O
50
The residuals for each model are assumed to be independent,
to have mean zero, to have a constant variance, cr2 , and to follow 0
a normal distribution. Under these assumptions, the residuals
plotted against the independent variable, distance, should be a
horizontal band. A non-horizontal band would indicate:
(1) the variance is not constant,
(2) errors were made in the calculations, and/or
(3) extra terms are needed.
In Figure 5.3, the residuals of Model 1 do not plot as a
horizontal band when plotted against the observed distance. The
graph actually looks somewhat sinusoidal. The sinusoidal
property leads to the conclusion that Model 1 needs an extra
parameter and that it is not an acceptable model. This
conclusion is further supported by Model l's graph of its
residual versus the observed cyclic error. This graph may be
found in Figure 5.5. This plot exhibits a sloped band
correlation between its residuals and the observed cyclic error.
Thus by graphical means, Model 1 is not an acceptable model.
Figure 5.3 gives graphic proof that Model 3 fits the data
better than Model 1. Likewise, Model 3's variance of unit weight
is only half as large as Model l's which indicates that the
residuals are smaller since all observations in all models have
unit weight. The residual graphs of Hodel 3 more closely exhibit
horizontal bands than those of Model 1. The graphs are defi-
nitely banded but not perfectly horizontal.
51
Model 2 is very much like Model 1. Its reference variance
is 18.989 mm2 where as Model l's reference variance is 18.905 2 mm. The similarity is also evident in their respective residual
graphs. Thus Model 3 should be selected as the best model.
Statistical analysis may also be done on the models. If the
residuals are still assumed to follow a normal distribution,
F-tests can be used to analyze the models' fit. The test will be
a two tailed test at a 95% confidence level with the hypothesis
that Model 3's reference variance is equal to those of
Models 1 and 2. Thus, for Models 1 and 3;
Accept if
H : 0
18.905 = Si
Hl: S2 j S2 1 3
F(71 70) 0.025 < ' '
= S2 = 3 9.969
S2 2 -<F s1 (71,70) o.975' 3
or .621 < 1.8964 < 1.605.
Therefore, reject the hypothesis that the two variances are
equal.
From these tests the conclusion can be drawn that Model 3 is
the best fitting model. Still, the relatively small difference
betveen the test statistics and the rejection limit indicates
that the models, though statistically different, may not be
operatively different. However, a choice of models is needed for
future use with the Ranger IV. Considering both the residual
graphs and the statistical tests, Model 3 is the best choice.
GP.APTER 6
DETER1'1INATION OF SCALE AND CONSTANT OFFSET ERRORS
Introduction
Of the three systematic errors, scale and constant offset
errors are normally the only errors determined with a calibrated
base line. Their determination can best be made by a least
squares reduction of data observed on the base line. The least
squares solution is readily adaptable to hand held calculators.
Test Procedure
The data used to determine the constant offset and scale
error of the Ranger IV was collected on the NGS base line at
Corbin, Va. The Corbin base line differs from a typical base
line in that the distance meter and reflector are mounted on
permanent metal stands as opposed to being mounted on a
transportable tripod. The elevation of the stands are known to+
0.1 mm so the instrument and reflector heights were taken above
the stand not actually above the monument. Instrument and
reflector heights were measured to+ 2 mm for each instrument and
each reflector setup. Ten readings were recorded for each
distance observat~on, not all base line intervals were measured.
The average of the te~ readings was adopted as the distance.
Temperature readings were made with a thermistor during the
series of ten measurenents. The thermistor gives an instan-
taneous temperature readou:. Unless changes of greater
52
53
than 1°C were observed dcring the series, only the initial and
final temperatures were recorded. The average temperature in all
cases was adopted as the temperature of the series. The pressure
was measured before and after each series with a calibrated
altimeter. Again, the average was adopted as the measured
parameter. Humidity was measured prior to the calibration
procedure and assumed constant throughout the day. For each
series of readings, a ppm correction was computed and applied to
the average slope distanc~. No value of instrument constant
offset was applied to the measured distance but reflector
constant offsets were applied. The slope distance was reduced to
horizonal using the method of elevation differences.
Analysis of the Data
The least squares solution equations for the constant offset
and scale errors are given by.
E(D 2)Eo - ED E(D o) C = __ a _____ a ___ a_
nE(D2) - (ED ) 2 a a
and (6 .1)
nE(D o) - rn Eo S = __ ....,a,_ ___ a __
nE(D 2) - (ED )2 (6. 2)
a a
where
C = constant offset error,
S = scale error,
n = number of distances measured,
54
D = calibrated distance, and a 8 = difference between the calibrated and measured distance.
These equations are derived in Appendix H.
In the Ranger IV calibration, 12 distances were observed.
The distances ranged from 50 m to 1000 m. The actual values of
the parameters listed in Equations 6.1 and 6.2 are given in
Table 6.1. The Corbin base line data is given in Table 6.2.
Results
The Ranger IV was calibrated at the Corbin, Virginia, base
line July 24, 1981. Using the Corbin Base Line data and
Equations 6.1 and 6.2, the findings are:
(1) the scale error for the Ranger IV is -7.92 ppm and
(2) the instrument constant offset is 0.1577 meters.
These two values must be tested against the hypothesis that they
are equal to zero (Fronczek, 1977). The following unknowns must
be determined: ~ 1) the estimated standard error of the scale, cr; s A 2) the estimated standard error of constant offset, a ; 3) the
C
reference variance, & ; 4) and the test statistics t and t . 0 S C
The equation for the reference variance is (Mikhail, 1981)
(6.3)
where
V = residual of an observation and
DF = cegrees of freedom.
55
Table 6.1
Data Reductions for Constant Offset and Scale Errors
Parameter Value
n 12
i: 0 1. 8430 i
i: (D 0 ) 952.4124 ai i
i:D 6,250.1143 ai
i: (D 2) 4,202,667.602 ai
(rn )2 39,063,928.76 ai
(4,202,667.602)(1.8430) - (6,250.1143)(952.4124) C =
12(4,202,667.602) - 39,063,928.76
C = 0 .1577 meters
12 (1. 8430) - (6,250.1143)(1.8430) s =
12(4,202,667.602) - 39,063,928.76
s = -7.92 ppm
56
Table 6.2
Corbin Base Line Data
Calibrated Measured Distance Distance Difference
1,000.0195 999.8705 0.1490
1,000.0]95 999.8700 0.1495
700.0189 699.8649 0.1540
700.0189 699.8671 0.1518
500.0265 499.8695 0.1570
500.0265 499.8756 0.1509
499.9928 499.8388 0.1540
499.9928 499.8396 0.1532
300.0005 299.8418 0.1587
300.0005 299.8461 0.1544
199.9923 199.8401 0.1522
50.0056 49.8473 0.1583
~ = 6,250.1143 m ~ = 1. 8430 m
The number of observation, n, equals 12.
57
For this particular observation equation, Equation 6.3 becomes
(Fronczek, 1977)
E (6-6) 2 2- [nE (D o) - ED Ui] "2 n a a cr =--------------~ o n-2 (6.4)
where
o = average of the differences between the calibrated and
measured distances and
o, S, n, and D are as previously defined. a
The estimated standard deviations of the scale and constant
offset are (Fronczek, 1977).
/\ [ cr2 n Jl/2 cr = 2 s 0 (ED ) 2 nED -a a and 2
" [ cr2 ma Jl/2 cr = C 0 nrn 2 - (ED ) 2
a a
(6.5)
(6.6)
Using these equations, the values for these three unknowns are
"2 1. 7063 X 10-5 2 cr = m 0
A 4.2440 X 10- 6 cr = m s
J\ -3 cr = 2.5115 X 10 m. C
The test is at-test at a 99% confidence level to test that
the scale and offset errors are zero (Fronczek, 1977).
Accept if
58
H : 0
-6 -7.92 X 10 = S = 0
t0.005, 10 < ts t 0.995, 10
-6 - 3.169 < - 7•92 X lO _6 < 3.169
4.2440 X 10
- 3.169 < 1.8657 < 3.169
For the constant offset error,
Accept if
H : 0.1577 = C = 0 0
t 0.005, 10 < tc = ec < t0.995, 10 C
- 3.169 < 0 •1577 < 3.169 2. 5115 X 10- 3
- 3.169 < 62.7912 < 3.169
Therefore the results show the scale error of the instrument
is not statistically different from zero and need not be applied.
The constant offset is statistically different from zero and
should be added to all measured distances.
CliAPTER 7
ESTABLISHMENT OF THE VIRGINIA TECH BASE LINE
Introduction
Recognizing the professional and academic need for a
calibrated base line, the Geodetic Division of the Department of
Civil Engineering began the establishment of the Virginia Tech
Base Line in early 1981. The base line is located at the
Virginia Tech Airport and is presently awaiting calibration by
NGS. The following chapter relates the theory and procedures
used to install the base line. These procedures are actually
those of NGS. NGS requires that these procedures be adhered to
if calibration is to be performed by their organization.
Virginia Tech recognizes NGS as being the organization that has
state-of-the-art knowledge and technology in the field of
precise surveying, and thus feels that no better procedures
could be found to govern the proper establishment of a
calibrated base line.
Physical Requirements
A calibrated base line is a series of stable colinear
monuments, between which the distances are precisely known.
Base lines must exhibi~ two physical features in order for them
to function properly. The two features are: 1) complete
intervisibility between all the stations on the line and
2) geologic stability 1f the site itself. Obviously any
;9
60
monument that is not visible from another monument could not be
measured using light waves, visible or infrared, resulting in an
unusable distance interval in the line. Likewise, monuments set
in ground which is unstable could not possibly be used to
mair.tain a fixed distance between monuments.
Other physical features are desired in addition to the two
essential conditions. The next important feature is the ground
cover. The ground cover should be as uniform as possible so
that the atmosphere above the ground will be uniform too. The
atmosphere is the medium through which the EDMI must measure and
is the most difficult variable in any measurement to determine
accurately. Any variation in ground cover will vary the
temperature over the measured line and thus introduce error into
the determination of the refractive index.
The effects of ground cover can be greacly reduced by
establishing a base line which has· a concave shape, i.e., higher
on the ends than it is in the middle. The dip in line allows
the line of sight between the instrument and the reflector to be
elevated above the normal height of the instrument. The
atmosphere is most turbulent in the first few feet above the
ground and the dip allows the extreme turbulence to be avoided
(Fronczek, 1977).
The line of sight must be totally clear of all obstructions
that could block the EDMI signal; fences, trees, high voltage
lines, etc. Highly reflective surfaces, such as po~ds or metal
61
buildings, must be avoided to protect microwave units from
ground swing error. Power lines and radio equipment in the
vicinity will also interfere with microwave equipment.
The ideal length of a base line is between 1000 and 1400
meters. The base line needs about 1000 meters to insure that
scale errors in the range of one part per million can be
detected in medium and long range EDMI (Greene, 1977). The line
is limited to 1400 m by calibration procedures. NGS uses a
}fA-100 Tellurometer to calibrate base lines. This instrument
has a range of 2 km, a resolution of+ 0.1 mm and an accuracy of
+ 1.5 mm+ 2 ppm (Romaniello, 1977). Therefore, at lengths
greater than fourteen hundred meters the accuracy of the
calibrated length is reduced to one-hundredth of a foot.
Typical land surveying practice is to measure distances to
one-hundredth of a foot. If surveyors wish to calibrate their
instruments to the nearest one-hundredth of a foot, they must do
so on base lines which are calibrated to better than
one-hundredth of a foot.
If the physical requirements listed above can be met, NGS
suggests the base line configuration follow the scheme shown in
Figure 7.1. The 430 m monument is set so that it does not
extend beyond the typical 500 m range of short range EDMI. The
short range instruments also require a third monument in order
to establish their constant offset. Obviously, this monument
must be between the Om and 430 m monument. Furtherr.ore, if thi3
8 G G G Om 150m 430m 1400m
TYPICAL BASE LINE CONFIGURATION
FIGURE 7.1
0-, N
63
third monument is set at or near the 150 meters, the longer
range instruments will have two li~es, i.e., 150 meter and a
1400 meter line, such that one is approximately 10 times longer
than the other. This arr2ngement is quite desirable for
determining the scale error of the instrument (Greene, 1977).
The absolute requirements on the distances between monuments are
that the distances be a multiple of ten meters and that
somewhere in the line there is a 150 meter distance accurate to
within 0.02 m. Most distance meters have a smallest measuring
wavelength of 20 meters (Meade, 1972). Therefore, to avoid any
cyclic error in calibrating the line or in calibrating the EDMI,
the distance between monuments should
be a whole multiple of ten meters.
Construction of the Base Line Monuments
The monuments of the base line are usually set in concrete
pillars. These pillars must be designed to be stable and
permanent. The stability of the pillars is a function of two
things: 1) the depth of frost penetration, 2nd 2) the stability
oi the ground it·self. Frost action will cause movement of the
pillar if the pillars are not extended below the frost line. To
further insure that frost heave has no effect on the pillar, the
bottom of the pillar is belled so that any uplifting force CTust
also ?USh against the ground.above the bell as well as the
concrete itself. The frost line varies from locality to
localicy, so that the pillar depths of a particular base li~~
64
will depend on its geographic location. As an exanple, the
maximum depth of frost penetration here in Blacksburg is 35
inches while the maxi~um depth of frost penetration in Virginia
Beach is only 17 inc~es (Geraghty, 1973). Thus, the Blacksburg
pillars should be 40 inches deep while the Virginia Beach
pillars need only be 24 inches deep. In addition, soil types
will effect pillar depth. To further the example, the sandy
soil in Virginia Beach may require the concrete pillar to be 36
inches deep while the cohesive clay in Blacksburg could support
a 34-inch pillar just as securely. Therefore, NGS has suggested
that all pillar depths be between three and one-half and five
feet deep and that all diameters be at least 14 inches in
diameter to insure property stability. The second variable
affecting monument stability is the stability of the ground in
which the concrete is cast. Areas of recent fill are not
suitable for a base line. As the ground settles and compacts,
the monuments will shift causing both distance and alignment
problems. Other totally unacceptable sites are sidewalks,
concrete or macadam surfaced roads, and landing strips, all of
which wear with age and eventually must be replaced or
resurfaced. Some sites that are acceptable are existing or
abandoned small airfields, strips along interstate highways, or
public land. All of these are acceptable as long as there are
no plans for future construction.
65
The construction of the base line also includes alignment.
As stated in the definition, the base line monuments should be
colinear thereby remov!ng any relia~ce on neasured angles to
compute the instrument constants. Again, this requirement is
flexible but NGS suggests no deviation of greater than 5° from
the line. Dracup (1977) gives a typical procedure for obtaining
the proper alignment and the correct distances between monu-
ments.
Another consideration in the construction of a base rine is
its cost. The easiest cost to delineate is that of the
materials. The materials needed are concrete, lumber, nails,
and small hand tools. The most difficult cost to estimate is
the cost of labor. Construction begins with field recon-
naissance and site selection. Field reconnaissance must yield
at least one, and preferable three, sites having the afore-
mentioned characteristics. At least one alternative site should
be available in case permission cannot be acquired to use a
particular site. Acquiring permission from the proper
authorities can be a long and tedious process. The granting
authority must realize the base line is to be open for public
use. After permission is granted, the monuments must be aligned
and positioned. The typical base line is nearly a mile long and
requires careful work to insure proper positioning. At least
one full day is required to set and to check the ~onurnents'
positions. At this time a map of the base line should be drawn
66
and kept f6r future use. Following proper positioning, the
holes may be dug. Much time and effort can be saved if the
holes are augered by a truck or tractor. Finally, the concrete
is poured and NGS brass disks are set.
Base Line Calibration
The actual calibration of the base line is done by the NGS
with assistance from the people establishing the base line. NGS
requires that experienced personnel be present to aid in
obtaining their observations. They also require that a map or
diagram of the base line be completed and sent to them prior to
the actual calibration of the base line. In addition, they
require that the site be readily accessible to the public. A
cooperative agreement is signed that defines each party's
responsibilities and costs.
Prior to the calibration field work, NGS requires that the
following checklist be completed by the applicant.
(1) Design
a. Straight line configuration.
b. Range between 1,000 m and 1,400 m.
c. All marks are intervisible.
d. 7errain - slope of 150 m section not exceeding 1
percent and that of other sections not exceeding
3 percent.
67
(2) Location
a. Not closer than 1/4 mile to radio masts, high
voltage power lines, radar domes, or microwave
towers.
b. Lines do not cross waterways, structures, or
fences; not closer than 100 feet to metal mesh
fences.
c. Site is accessible to the public; no plans for
future construction on the site.
(3) Monumentation
a. Two permanent reference marks for each base line
monument.
b. Base line monuments are poured concrete type.
The Virginia Tech Base Line
The Virginia Tech Base Line was completed in June, 1980 and
has an ideal location at the Virginia Tech Airport. It is 150
feet from, and runs parallel with, the north-south runway. The
monuments are arranged in the normal base line fashion and have
the typical nominal lengths. The monuments are brass NGS base
line disks set in circular concrete pillars 20 inches in
diameter and 48 inches in depth. The bottom two feet of each
pillar are belled out to a diameter of approximately thirty-two
inches. The base line has an excellent concave profile with
compiete intervisibility. The normal calibration fee was ~aived
by NGS since the base line was established for an educational
68
facility. At the time of this writing, the base line had not
been calibrated by NGS, but Appendices F and G contain the
calibration results of this thesis. Both horizontal and
vertical calibration results are given.
Introduction
CHAPTER 8
A PROPOSAL FOR A STATEWIDE
NETWORK OF CALIBRATION BASE LINES
Surveyors and engineers need to calibrate their EDMI
periodically so that they can have confidence their instrunents
are operating properly. To keep calibration time to a minimum,
each surveyor should have reasonable access to a base line.
Hereafter, follows a proposal for a statewide network of base
lines that should provide ample access for any EDMI user. The
proposal is based on:
(1) the base line installed on the Virginia Tech campus,
(2) the existing NGS calibration program, and
(3) a questionnaire completed by a sample of the state's
surveyors.
Questionnaire
The questionnaire was entitled "EDMI Base Line
Questionnaire" and was completed during a workshop, called
"Advanced Problems in Surveying Practice," held at the
Blacksburg, Marriott Inn on October 8-9, 1981. The workshop was
sponsored by the Geodetic Division of the Department of Civil
Engineering, VPI&SU. A complete questionnaire may be found in
Appendix I. The surveyors polled comprise a small sample and may
not be representative of all the state's surveycrs. :he
69
70
surveyors polled belong to a professional surveyors organization,
a third of the state's surveyors do not. Similarly, the
surveyors polled were attending a c~ntinuing education progran,
probably half of the state's surveyors do not. However, the ones
most likely to use a base line were the ones polled.
Two major conclusions may be drawn from the questionnaire:
1) the surveyors in the state see the need for calibration base
lines; they are willing to spend their time to both establish and
use a base line, and 2) they also see the need for tape cali-
bration and would like to see a taping monument included on all
base lines. A 100 foot taping monument was the length most
requested and could be used to calibrate 200 and 300 foot tapes.
The questionnaire has produced one major source for concern.
One half of those questioned indicated that they did not have a
thermometer or an altimeter with which to make meteorological
reductions. without these instruments, measurements of the
1000 m base line distance could differ fron the calibrated
length, by as much as 5.0 cm on a warm day, i.e. 70°F at 28.5 in
of Hg. The neglect of meteorological errors is unacceptable in
calibration procedures, and this point needs to be stressed to
those unfamiliar with the details of EDMI.
Number and Location of Base Lines
The controlling factor in the location of the base lines is
the allowable distance between individual bas2 lines. At present
there is a base line in Norfolk and one in 3lacksburg. New base
71
lines must be at least 75 miles apart according to NGS
specifications. If 2ncther base line is needed west of
Blacksburg, the most convenient location for southwest Virginia
surveyors and engineers would most likely be Abingdon.
The locations east of Rlacksburg are not nearly as easy to
choose. Most likely, as the population densifies, the number of
surveyors increases. Under this assumption, the Petersburg-
Richmond area needs a base line. This area does fall within 75
miles of Norfolk which, under a strict 75 mile rule, would
probably mean a base line at Ashland. However, if the NGS
realizes the state is going to install a statewide network, they
might be inclined to relax the rule and the base line could be
moved to Richmond.
From the population standpoint, the Arlington-Alexandria
area needs a base line. Thus, the central portion of the state
is without a base line. The North-South extent of the state will
probably require 2 additional base lines. On the basis of
population, Lynchburg is probably the best choice for the south
central area. However, if this site was moved south to Chase
City or South Boston, the last site could be put in Staunton.
Staunton is the most convenient location for the Staunton-
Harrisonburg-Charlottesville area.
To summarize, this location scheme calls for 7 base lines
located at:
72
(1) Abingdon,
(2) Blacksburg,
(3) South Boston
(4) Norfolk,
(5) Ashland,
(6) Staunton, and
(7) Alexandria.
The locations at Blacksburg and Norfolk are unchangeable. Figure
8.1 shows the recommended locations on a county map of the state.
Sponsor
The sponsor of the base lines should be the users. From the
public sector, the most frequent user will most probably be the
Virginia Department of Highways and Transportation (VDH&T). From
the private sector, the most freque~t user will most likely be
the land surveyor and engineer. Of these two sectors, the
private sector would probably use the base lines more frequently.
In other states, Montana, Alabama, North Dakota, and Oregon, the
state's professional surveying society has sponsored the base
line program (Pietras, 1981). Virginia has a professional
surveyors' society, the Virginia Association of Surveyors (VAS),
which has a statewide membership. VAS is technically qualified
to undertake such a task. The VAS chapter in which the base line
is to be established could take the responsibility of that parti-
cular base line. With a co!::I!littee approach, the chapter could
delegate the various jobs involved in establishing a base line to
I I
I
., .,
•
.,, /
Proposed base line site • 75 mile radius from base line.
/ _,,----..... ,/ I',
, /
/
, I
.,,
I I
I
,, /
,, ., .,, -- ---
I ' ' I
County Map of Virginia
PROPOSED STATEWIDE NETWORK OF
CALIBRATED BASE LINES
FIGURE 8.1
/ I
/ /
--- --......
' ' \
' ' \ \ \
I
' I ' I
I
\ I \ /
t
' \ \
'
...... w
74
different committees. By so doing, the time involved per person
could be lessened to the extent that a particular surveyor could
justify the time involved.
Cost
Based on experience with the Virginia Tech Base Line, the
real cost for a base line is time. The procedures outlined in
Chapter 7 are time consuming and may be somewhat of a burden for
one person. A committee approach might be the most equitable way
of distributing the time among chapter members. One
responsibility could be delegated to a two or three man committee
and the chapter president could serve as chairman.
CHAPTER 9
CONCLUSIONS AND RECOHMENDATIONS
Calibrated base lines exhibit two important features.
First, they provide the user with a means of testing his EDMI.
Th2 user can determine if a simple correction needs to be
applied to his measurements or if his instrument needs to be
repaired. Second, calibrated base lines provide the user with
checks that the test was performed properly. Other methods of
t~sting EDMI do not provide these checks. In addition, if all
base lines are calibrated by NGS, then the state, and even the
nation, will have a standard for calibrating EDMI.
Based on the work accomplished in this thesis, the
following conclusions are made:
1. The resolution of the Virginia Tech Ranger IV is
3.8 !I1I!l, its offset error is 0.1577 m, and its scale
error is 0.0 ppm.
2. Cyclic error in this Ranger IV is above the
instrument's significance level and should be
corrected for in each measurement using cyclic error
Model 3.
3. The Barrel and Sears formula is an adequate mcdel for
the prediction of atmospheric effects involved in
electronic distance measuring.
4. A system of seven base lines would cover the State of
Virginia in compliance with NGS criteria.
75
76
Based on the results obtained in this research, the
following recommendations are made:
1. Each EDMI used by Virginia Tech's Geodetic Division
should be tested on a calibrated base line for scale
and constant offset errors and in the laboratory for
cyclic error.
2. More work should be done in the field of cyclic error.
New models should be developed. Both new and existing
models should be tested using geometrically redundant
figures and least squares techniques.
3. The State of Virginia should establish a network of
calibrated base lines and adopt a minimum set of
standards requiring all the state's EDMI users to
calibrate the instruments at regular intervals.
4. VAS should actively seek and financially sponsor the
implementation of both this base line network and this
set of minimum standards.
REFERE:~CES
1. Raymond W. Tomlinson, "History of EDr!I," P.O.B., Point of Beginning, Feb-Mar. 1981. p. 34.
2. John R. Greene, "Accuracy in Eleccro-Optic Distance Measuring Instruments," Surveying and Mapping, (Sept., 1977), pp. 251-255.
3. Joseph F. Dracup, Charles, J. Fronczek, and Raymond W.
4.
5.
Tomlinson, Establishment of Calibration Base Lines (NOAA Technical Memorandum NOS NGS-8, 1977), pp. 3-13.
Charles J. Fronczek, Use of Calibration Base Lines (NOAA Technical Memorandum NOS NGS-10, 1977), p. 5.
Guy Borr.ford, Geodes:z, (Oxford, England: Clarendon Press, 1980), p. 47.
6. Charles G. Romaniello, "Advancing Technology in Electronic Surveying," Journal of the Surveying and Mapping Division, Proceedings of the American Society of Civil Engineers, Sept. 1970, p. 292.
7. Charles G. Romaniello, "EDM 1976," Surveying and Mapping, xxxvii, No. 1, (1977), p. 28.
8. Raymond W. Tomlinson and Thomas C. Burger, "Electronic Distance Measuring Instruments," Proceedings of the American Congress on Surveying and Mapping, (Washington, I). C., Spring 1971), p. lA.
9. Buford K. Meade, "Precision in Electronic Distance Measuring," Surveying and Mapping. xxxii. No. 1 (1972), p. 72.
10. James J. Geraghty, David W. Miller, Fritz van der Leeden, Fred L. Troise, "Depth of Frost Penetration," The Water Atlas of the United States, (Water Information Center, Port Washington, New York, 1973), Plate 9.
11. Personal Correspondence. John D. Bessler, Directer of NGS, to Steven D. Johnson, Assistant Professor, VPI&SU, April 8, 1982.
12. Darrell G. Bryan and Steven D. Johnson, Electronic Distance Measuring Equipment, Vi~ginia Polytechnic Institute and State University, 19, p. 16.
77
78
13. A. J. Robinson, 11Field Investigation into the New Hewlett-Packard Distance Meter," Proceedings of the ACSM, Washington, D.C., 1974, pp. 377-378, 380.
14. Kenneth D. Robertson, The Use and Calibration of Distance Measuring Equipment for Precise Mensuration of Dams, U.S. Army Engineer Topographic Laboratories, Fort Belvoir, VA, 1976, p. 8.
15. Clair E. Ewing and Michael M. Mitchell, Introduction to Geodesy, (Elsevier, New York, New York, 1976), p. 113.
16. Edward M. Mikhail and Gordon Gracie, Analysis and Adjustment of Survey Measurements, (Van Nostrand Reinhold Co., New York, 1981).
17. Simo H. Laurila, Electronic Surveying and Navigation, (John Wiley & Sons, Inc., New York, New York, 1976), pp. 29-37.
18. Maintenance Manual - Ranger IV Electronic Distance Measuring Instrument~ (Keuffel & Esser Co.), pp. 1.3-1.10.
19. Al Shenk, Calculus and Analytic Geometry, 2nd ed., (Goodyear Publishing Co., Santa Monica, CA, 1979), p. 130.
20. Mackenzie L. Kelly, "Field Calibration of Electronic Distance Measuring Devices," Proceedings of the American Congress on Surveying and Mapping, (Washington, D.C., March 1979), pp. 426-428.
21. Paul R. Wolf, Elements o-f Photogrammetry, (McGraw-Hill, Inc., New York, New York, 1974), p. 536.
22. Raymond E. Davis, Francis S. Foote, James M. Anderson, and Edward M. Mikhail, Surveying, Theory and Practice, 6th ed., (McGraw-Hill Book Company, New York, New York, 1981).
23. Francis H. Moffit, "Calibration of E:!:lMI's for Precise Measurement," Surveying and Mapping, xxxv, No. 2 (June 1975).
24. Norman Draper and Harry Smith, Applied Regression Analysis, (John Wiley and Sons, Inc., New York, New York, 1966).
25. Edward M. Makhail, Observations and Least Squares, (IEP, New York, New York, 1976).
26. R. A. Hirvonen, Adjustment by Least Sq~ares in Geodesv and Photogrammetrv, (Frederick Ungar Publi£hing Co. Inc., New York, New York, 1979).
79
27. Wilfrid J. Doxon, Frank J. Massey, Jr., Introduction to Statistical Analvsis, (McGraw-Hill Book Company, New York, New York, 1969).
28. Penelope Pietras, "A POB Survey of 45 State Surveying Organizations", Point of Beginning, Vol. VI, No. 6, August-September, 1981, pp. 44-53.
29. Doug Johnson, "Reflective Prisms for Surveying Use," Point of Beginning, Vol. VI, No. 3, February-March, 1981, pp. 26-33.
30. Hornbeck, R. W., Numerical Methods, New York: Quantum Publishers, 1975.
31. Votila, V. A., "Statistical Tests as Guidelines in Analysis of Adjustment of Control Nets," Surveying and Mapping, Vol. xxxv, No. 1, March, 1975, pp. 47-52.
APPENDIX A NUMERICAL LISTING OF
CYCLIC ERROR DATA
80
81
Cvclic Error Data
Tape Distance EDHL Distar.ce Cyclic Correction (m) (m) (mm)
~ay One :
4. 7205 4.7221 -6.6 5.0253 5.0345 -9.2 5.3300 5.3416 -11.6 5.6348 5.6460 -11. 2 5.9396 5.9533 -13. 7 6.2444 6.2583 -13.9 6. 5492 6.5598 -10.6 6.8540 6.8655 -11. 5 7.1587 7.1685 -9.8 7.4635 7.4755 -12.0 7.7683 7.7786 -10.3 8.0731 8.0822 -9.1 8.3779 8.3861 -8.2 8.6827 8.6943 -11. 6 8.9874 8. 9966 -9.2 9.2922 9.3007 -8.5 9.5970 9.6022 -5.2 9.9018 9.9042 -2.4
10.2066 10.2075 -0.9 10.5114 10.5105 0.9 10.8161 10.8145 l. 6 11.1209 11.1164 4. 5 11.4257 11.4210 4.7 11. 7305 11. 7249 5.6 12.0353 12.0298 5.5 12.3401 12.3340 6. 1 12.6448 12.6412 3.6 12.9496 12.9487 0.9 13. 2544 12.2543 0. 1 13.5592 13.5604 -i. 2 13.3640 13.8668 -2.8 14.1688 14.1725 -3.7 14.4735 14.4794 -5.9 14. 7783 14.7873 -9.0
~wo:
13.8222 13.8187 -3.5 14.1270 14.1347 -7.7 14.4318 14.4360 -4.2
82
Tape Distance Em!I DistancE: Cyclic Correction (m) (m) (mm)
Day Two (cont.)
14.7366 14.7449 -8.3 15.0414 15. 0492 -7.8 15.3461 15.3612 -15.1 15.6509 15.6578 -6.9 15.9557 15.9716 -15.9 16.2605 16.2741 -13.6 16.5653 16. 5776 -12.3 16.8701 16.8810 -10.9 17.1748 17.1849 -10.1 17 .4796 17.4903 -10.7 17.7844 17.7953 -10.9 18.0892 18.0989 -9.7 18.3940 18.4023 -8.3 18.6988 18.7057 -6.9 19.0035 19.0094 -5.9 19.3083 19.3118 -3.5 19.6131 19.6158 -2.7 19.9179 19.9162 . 1. 7
Day Three :
A.8 348 4.8492 -14.4 5' ~ 4444 5.4609 -16.5 6.0540 6.0623 -8.3 6.6636 6.6739 -10.3 7.2731 7.2862 -13.1 7 • 8828 7.8932 -10.4 8.4924 8.4991 -6.7 9.1020 9.1066 -4.6 9. 7116 9. 7102 1. 4
10.3212 10.3200 1.2 10.9308 10.9249 5.9 11. 5404 11.5301 10.3 12.1500 12.1404 9.6 12.7956 12.7508 8.8 13.3692 ~J.3647 4.5 13 .9 789 13. 9777 1. 2 14.5884 14.5917 -3.3 15.1981 15.2017 -3.6
APPENDIX B CYCLIC ERROR
PROGRAM LISTING
83
(.$JUB VAk~EV,NOLIST (*********************************************************************** C ~
C * (. * (. * C l E A S T S Q U A R ~ S A O J U S T H E N T * C C C L C ( (. (. (.
u ¥
C f- C Y C L 1 C E k R L R 0 A T A
0 B S E R V A 1 I C N E C U A T I O N i'1ETHUD
* * * lNTROUUCllD~ * * *
C THE fuLUhHN.; PROGkAM FITS A Slf-..E FUNCTION TO CYCLIC ERROR DATA C USIN~ AN lll:RATIVE CBSE~VATlON EQUAllON LEAST SQUARES ALOGRITHM. THE C ~ATHEMATlCAL HUOfl USED AS T~E OBSERVATICN EUU~TION IS (
C C (
C (.
C C (.
C C
1,HltJE
E = A*SIN(((OISTll)-R~FDlS)/~.O)*Pl)-TRANS
E = A = i;ISf(I) = Kl:fUIS = l' I = TRM~S =
CVCLJC E~ROR, AM~LITUOE, MtASURcD DIS1ANCE, TRANSLATION ALLING THE DISTANCE AXIS, NUMBEK OF HADIA~S JN !SC DEGkEES, AND TRANSLATIO~ ALONG lH[ CYCLIC ERROR AXIS.
C A, f~LF1l1S, Af,O TRANS ,I\IU lHE PARAMllf:J<S FOi< ~HICH A SOLUTION IS C S(jUGHT. ulST( I) IS THE CBSE~\IABL[ WhlCH IS USEu AS TH[ INPUT DATA.
* * * "' * * * .. ... * * * * * " * * * * * * * * * * *
(X) .p.
(, E IS Ttif. VARIABLE .. HICH IS TO MODELED. C ( (.
L C C C C C C (.
C
* * * l~PUT CATA***
Th[RE ARE THkE[ INPUT VARIAULLES:
LJ1ST( I) = Mt:ASUKEU CISTANCt, CYCEkk(l) = MEASURED CYCLIC EKRGR, ANO h(l) = WllCHT UF THE COSERVABLE.
C THE kcAO fLJRMAT FCR THE INPUT IS: C L lQl FCRMAT(f4,f7.4,Ti6,f5.1,T26,flu.4). L
" * * * * * * * * * * * * * * * *
L lHUS lHf. lHE INPUT DATA MUST ~E ARK.ANGEC IN THE FOLLmdN~ ORDER UN * l tAU~ tA,~U: * C * C VARJAHLE CLLLM~ N0. 1 S FOKMAT * C. Cl~f(I) 4 - 10 f7.4 * L C.YlLHR(1) lb - 20 FS.l * (. h(I) 26 - 35 Fl0.4 * C * C llil lJAfA uUES NLf HAVt TU 8[ SORTED BEfCRE lNPUTTH11G Il INTO THE Pk!J-* C Gf<AM. * (
C ( (.
C **~OUTPUT l~FOkMATJON * * *
* "' * * *
(X) v,
C * C lH[ fULLUWl~G IS A LIST, IN PROPER SEQUE~CE, OF THE OUTPUT INFOR-* L MATlU~: * C * C C C C C C C C C L C C C C C C C C
1. INPUT CATA IN lhE CkDEH IT kAS READ IN, 2. NUMrlER Cf ceSERVATICNS, 3. I~ITIAL APPKUXl~AllCNS
CGMPUTEk PROGRAM, 4. ADJUSTED PARAMtlERS,
OF THE PAkAMETERS GENERATED OY TH~
~. NU~DER GF ITERATIO~S NEEUED FCK CONVERGENCE, 6. PCST[RILRI CCFACTCR MATRIX, ETWB, 7. VAKIANCE Of UNIT hlEGHT, AND d. INPUT QATA AkKANGEC IN ASSE~OU~G OROEk fKOM THE SMALLEST
MtASm~EO LlISTMJCi.: lO lHE LARGEST MEASURED DISTANCE wlTtC THE CORRESPONDING CYCLIC EHROR, CYCLIC ERMUK RESIDUAL, A~J CBSEkVATIU~ ~IEGHT.
.:0: * v PROGR4M CAUlIC~S * * * (. fhlS PLlGHAM RUNS I~ ~ATFIV. THE PRUPER COMMAND TO EXECUTE THIS L ~~CG~At' J~: L l~ l. C
GLL .. IAl. MACtlB lt'S1..0P 'liATLIB r.ATfl\J URV4
C r11F L1ST1l~1; fCK Hli:: P"OGKAM ~Ill fJE FOUND IN A FILE CALLED< DRV4 L LlSI lNG A>. Tht PROGRAM LISTING Will NGT BE PRINTED. THE PROGRAM C UOLS lXECUTE ~IIH LGGUh SlCKAGE C
* * * * * * * * * * ~
* * "' * -~-.... * * * * * * * t
* * * *
co Q'\
L C (
* * *
C***********~****-****************************************************** C * C * C UIMl~SIO~ THE ARRAYS * C
C
li'1Pl1(,lf PF.Al* a (A-H,C-Z) HEAL•u OISTCM(UO),CYCERR(UO),EXCE~R(80),EXCDIS(80),LARERR(d0) hLAL*8 ll(3,3),ul(3,3),F(3),N(3,3),T(3),DELTt\(3),BTwF(3) R[AL * 8 AOJE~R(BO),V(HC),~(8Ul,BT~8(3,3J,8TW(3,3),EXCW(~O)
C h[AD I i\PUT CAl A C
C
,<= l t,UMCHS=C
1 READ(5,1Ul,END=7) DISTDM(K),C\'tEkR(K),~(K) lJl fUPMAT(T4,F7.4,Tl6,FS.l,T26,flC.4)
I\U,1.1c13s=f\Ut-'CBS+ l K=K+l GC IL l
C PkINl THE I~PUT CATA C
L
7 103
l 02 7 7
wRITU6,lOJ) F C 1{;'~ AT ( T l , ' l ' , T 5 , 1 0 l ST At-. C E ' , T 16 , 1 EK R lJ R • , T 3 0 , 1 W [ I G HT 1 / )
CL 77 I=l,t\LMGi:jS lhHITE(6, 102) DIS TOM( l) ,C\'CERK( I ),h( 1) f(KMAT(T~,f7.4,Tl6,F5.l,T25,rlu.4) CU-. TI i~lJ[
*
* , . ... *
• * ~
*
00 .....
C PKINT THE NUMBER CF GBSER~ATICNS C
~~ITE(6,l00) NUMOBS 100 fCkMAT(///////T5,'NUMBER OF OBSERVATIONS= ',13)
C C 5GRT THE INPUT DAlA C
CL 2 l=l,NUMCOS CG 3 J-=l,NUMGBS lf(OJSTDM(I).LE.DISTOM(J)I GO TL 3 IF(I.G[.J) GC TC 3 EXCOIS(I)=DISTDM(J) OISTDM(J)=DISTDM(I) CISIDM(l)=[)COIS(l) t:.XC[Rk( I )=C'rC£:RR( J) CYC[kR(J)=CYCtRR(l) C\'CERtU l )=EXCERR( I) l X L , , ( I ) = ~. ( 1 ) ~(J)=~dl) vd I )=EXOd I)
3 LLi\Jli,UE. 2 LCt-.TII\UE
" *
* * ... ....
C * C FIND THE LARGEST CYCLIC ERRO~, HlGERR, AI\U THE SMALLEST CYCLIC ERROR,* C SMEHR. • C *
B IGtkK=(YCE:kR( l) S~fRR=CVCEPR( l) CC 6 l=l,f'.UrJ.0!3S IF( olC.Et-R.GE.CYCEPR( I)) GC TO 8 HIGERR=C'tC[RR(I)
H lf(SME.kR.LE.C\'CERR(I)) GC TC 6
Cl) co
t L
S1"1El<k=C'VCERR( I) CCi'.1 lNlJE
L lNlllALIZl THi NUMBER OF ITERATIONS (
I J·: l C C CLNEHAl[ CNlTIAL APPROXIMATIONS FOR THE PAKAMETEKS C
116 C
TRANS=HIGERK-({blG(~M-S~ERRJ/2.C) AMP=eIGER~-lHANS REFCIS=4.0 PI=4.iluO•JAlA~(l.OOO) hfll[(6,ll~) hRITE(6,105) A~P,TRAf\S,kEFDIS FURMAf(Tl,'1',/////T5,'1~I1lAL APPRCXIMATIONS')
C INl[IALILE THEN C
ANO T MATRICES
2.1
l l
10 (.
t..:C 10 I=l,J cc. 11 J:;:;ld f\(I,J)=C .. CDC (CN11f\UE l(I)=O.OCO CU~T INUE
* * *
* * *
* * ~
* C FURM THE C~NfRlLlUTIUN TG THE NGRMAL ECUATIONS FOR EACH OtlSERVATION * L *
lJ C 9 I = 1 , ~ U fJ C e S B(l,l)=USIN((lDISTOM(l)-kEFDIS)/5.0DO)*Pl) e(l,2)=1.0CO d(l,J)=-ANP*Pl/~.ODO*DCOS((DISTLM(l)-REFUIS)*PI/5.UDO)
00
'°
f(l)=CYCERK(l)-(AMP*OSIN((DISTO~(l)-REFDIS)*Pl/5.000)+TRANSl ~(ll=tdl) CALL MfRANS(B,81,1 1 3) CALL MVMULT{BT,h,3,1,BTh) CALL MMM~Ll(Blh,B,3,1,3,BT~B) CALL MAOD(~,el~U,3,3,t\) CALL MVMULT{BTk,F,3,1,BlhF) CALL VADO(l,Blhf,3,T)
~ CCI\T1NuE C * C lNVlKT 8T~b ANU COMPUTE UELTA * C *
CALL .'!Alf\'V(t\,3,3) CALL MVMULl(N,1,3,3,0EllA}
L C ~POAT~ APPRuXIMAflCNS TU ThE PAl{AMETERS A~O NUHBER Of ITERATIONS C
A~P=AMP+CELTA(l) THAf\S=TRA~S+CELTA(2) REF01S=REfDIS+DELTA(J) ll=IT+l
* ¥
*
C * L lt::51 fLH. CUl\vtkGt:t\CE * C *
lt{IT.E;.11) GC TC 25 If(D[LTA(l).Gl.(1.00(-10)) GU TC 23 lf(CELTA(2).GT.(l.OOE-1C)) GO TC 23 IF(tJtLTA(:J).Gl.ll.GGE-10)) GC TC 23
C * C Ph UH T h i:: A LJ JU S IE D P J\ KAM E l E ~ S * C *
iiRllE(6,ll7)
\0 0
117 FCRMAT(///////T~,'ADJUSTED PARA~ETERS1 )
~k1TE(6 1 105) AMP,fRAI\S,REFDIS 105 FCRNAf(f5 1 1 AMPLITUUE = •,F14.5/,15,'TRAI\SLAT10N = ',F14.5,
L/15,'kEFEREI\CE DISTANCE= ',fl4.~) C C i'tnNT Th[ NUMIH:k Of ITERAlIONS FCK COI\V[HGENCE C
~.RITUo,lC7> IT 107 FCRMAT(//////15,'lltRATIOI\S = 1 1 11/)
C C PHlt'-IT TliE PCSTEkluRI CCFACTCR MAllH>i, tHhB C
C
ir. ~ I H: ( l:, 10 9) 109 fLRMAT(/15,'~T~B 11\VERSE MATRIX IS 1 )
CC 95 1=1,3 hHITL(6,133) (NCl,J),J=l,3J
<J5 CCl\Tll\lJE 133 fl~~AT(/1~,3(Ll5.4,5X))
C LALCULATE THE RESIDUALS C
SLMKlS-= C.OCC CC 13 f:1,1\LMOOS ADJ(kR( I )=AMP ~JSIN( (CISTOM( I )-IU:fDIS)*Pl/5.0DO)+TRANS \I (I) =ADJ(Rk ( 1 )-C)'CERR (I) Sl.Jr-iPt.)=SUMiU~+vll l**2*"' I)
13 LLtdll\Uc C C LALCUL~l~ A~C PRII\T THE VARIANCE Uf UI\IT hl[GHT C
VARU~T=SUMRES/(NU~CbS-3 ) ~RITEC~,108) VAMUhl
~
* ~
* * *
* * *
* * *
\0 r-'
108 FCRMAT(/////T5, 1 VARIAI\CE Cf UNIT WEIGHT= ',Fl2.5) C * C PRlhf THE UBSEkVED DISTANCE, THE GUSERVEU CYCLIC COKRECTION, THE ADJ-* C USftC tYCLIC CURRECTION, A~D THE RESIDUAL FOK THE CYCLIC CORRECTION. * C *
C
hRll((o,115) ll~ FL~MAT(Tl,'l',////T7,'CIST',Tl6 1 1 ERKOR',T24,'AOJD',T32,'RESD')
CC 33 l=l,~~~(US WKl rt(6,ll0) DISTOH(IJ,CYCERR(I),ADJERR(ll,V(l)
110 fGkMAT(T5,F7.4,3X,3(f5.l,3X)) JJ CCNllNUE
lf(IT.LT.10) GO TC 26
L PRINT Al\ EH~GH MESSAGt IF CONVERGENCE IS ~OT REACHED C
2~ 106 26
10
10
hRITl:(6,106) FCHMAT(T51 '1TERATION STCP REACHED') STGF E I\C SU8KGUTI1\E ~ACD(A,8,M,l\,Cl KEAL* E Al3,3l,B(3,3),C(3,3) CC 10 I=l,M CC 10 J·:l,N C(I,Jl=A(l,J)•B(l,J) HETLKI\ EI\ C SLURDUTI~E VAOD(V,~,M,X) REAL* 8 V(3,1),~(3,1) 1 X(3,1J CC IC l=l,M X (I, l J=\I (I, l) t~ ( 1, l) RETLKI\ [f\0
* * *
'° N
SUbKLUTII\E ~M~ULT(A,B,M,K,N,C) REAL* E A(J,3),0(3,3),C(J,3) CC 10 J:1,M CL 10 J=l,I\ C(J ,J)=G.GOC CG 10 L=l,K
10 C(I,J)=C(I,J)tA(I,L)*Otl,J) ~l:lLk.f\ [ I\ C SLriR~UJlNE MV~UlTlA,V,M,K,~) ~lAL * E A(3,J),Vl3,1),~(3,1) CL lO I=l,M hlI,l>=C.OCC CC lC l=l,I\
1 ,J h ( I , l } = W ( 1 , 1 l + A ( 1 , L ) * \i ( l , 1 ) kLlLRI\
lG
I:. I\ I) SLd~0UTINE MlkANS(A,e,P,N) KEAL* E A(3 1 3),B(J,3) U. 10 l=l,M Cl 10 J=l,N tdJ,I)=A(I,J) l~ElUd, 1:: r,c S L b KL lJ T 1 N E M A I N V ( A , N I{ C r, , N D R A ) UCuulE PKECISICI\ A(l),C I\A =i~t;HA * I\HC h CL 3 u O I : l , ~. µ C h f\,= I tl\DKA* ( 1-1) C=l./A(I\) A(i\)=l. CL lGO fl.:a:J,~A,~iORA
\D vJ
0 0 ;'!'I
u ..J:
I-..!:
u -u
0:. ....
...J ;;::;,
..::: I <..:,
:z: --
·-......
..::; .... --
.... I -
II * ::.::
II -, <t
w
::.:: <t •
-, *
:::, X
'J •
~
;..J z 0
::i l.!.J
00 <
II -o..:::
• 2.
II 0 ~
-t-
rq + -
--1"\J
Cl
..:: L. :.::. -<
L. .z:
-·...JO
II .. 11
c.:.J II < u
u 7-
-w
<t: w ~
...I ...I
+ +
:£. -!I II ..::: ~ -...I <t
'II' u I 4
LUW
-:::>:J
< z: L. L.
11 --~
-... I-;J
L. z: ..::: I-,J u
'...J <'t
'...) ;...) ~.::;
00 00 :""\I~,
94
a ..::: ,,..J >-x ....
~
:.:.J ..,... ;...)
APPENDIX C PP.OGRAM OUTPUT
MODEL 1
95
DI ST A NL I: EHKOK wl:IGHT
4.l1348 -14.4 1.oouo 5. 44't4 -16.5 1.0000 6. 0~'t0 -8.3 1.0000 6.6b36 -10.3 1.couo 7.2lJl -13.l 1.0000 -,.ao2.u -10.4 1.0000 8.492't -6.7 1.0000 9.1020 -4.b 1.0000 r;.Jllb 1.4 1.0000
10.3212 1.2 1.0000 lU.CJJui3 5.9 1.0000 ll. • .54 04 10.J 1.0000 12.1500 ', • 6 1.0000 12. ·,s,..u, a.u 1.cooo 13.3692 4.5 1.0000
\0 l.:i.C7!J9 1.2 1.ocuo 0\
14. 5 tHi4 -3.3 1.0000 l5.l9Hl -3.6 1.0000 l3.d2~~ 6.6 1.0000 14.4.1'.)2 -1.0 1.0000 l4.LHB -4.!> 1.0000 l 1t.l40U -5.l 1.0000 l::>.044'} -4.4 1.0000 18 • .;9tl2 -4.2 1.0000 l i3. 70.l l -2.1 1.0000 l<.J.0078 -1. 5 1.0000 19.JllJ':.i 1.0 1.0000 l~.6141 1.8 l.LOOO 19.9102 6. 2. 1.0000 l 5.3ielJ7 -11.6 1.0000
15.6546 -3.3 1.0000 15.<3595 -12.3 1.0000 llJ.2643 -10.0 1.0000 lt.5ti~Z -U.6 1.0000 16.t!/40 -1.2 1.0000 l1.17tl'1 -6.2 1.0000 11.i.a:rn -6.7 1.0000 u. 7dti6 -6.q 1.0000 LJ.OY35 -5.6 1.0000
,,.7208 -6.3 1.0000 ~.v256 -d.9 1.0000 5.3304 -11.2 1.0000 j.6352 -10.6 1.0000 5.939'1 -13 • .3 1.0000 t.,.244a -13.5 1.0000 o.5496 -10.2 1.0000 6. 85 1,4 -11.1 1.0000 7.l5G2 -9.J 1.0000 7.4ti't0 -11.5 1.0000
\.0 ......,
7.7088 -9.8 1.0000 8.07.:>5 -8.6 1.0000 ~ .3 7 84 -1.1 1.0000 a.csj2 -11.1 1.0000 H .CJUf30 -8.6 l .OJOO 9.2'128 -1.q 1.0000 9.5Ylb -'t .6 1.ut;uo 9.9024 -1.H l.CuOO
10.2012 -0.] 1.0000 llJ.5.llO 1.5 1.0000 lU.tHo8 2.3 1.0000 11.1217 5.3 1.0000 ll.4265 5.5 1.0000
ll.7H3 6.4 1.0000 12.0360 6.2 1.0000 lL.3't01:i 6.a 1.0000 12.64~b 4.4 1.0000 12.':i:>04 1.a 1.0000 l2.2S:i2 0.9 1.0000 lJ.~601 -0.3 1 .GGOO 13.8t.4B -2.0 1.0000 l4.16c:;6 -2.9 1.0000 lit.47'"t'.J -4.9 1.0000 l4.77Y2 -8.l 1.0000
NUMbEK Uf CBSERVATIU~S = lJ
INITIAL APPKDXI~ATIUNS A~PLITUilE = l3.40GOO TRANSLArIO~ = -3.10000
~ 00
AUJUS1ED PARAMETERS AMPLITUDE= 7.36643 TkA~SLATION = -2.98782
IlEkATIONS = 3
UThB INVEkSE ~ATRIX IS
O.l9Hl.D-Ol
LJ.332JU-02
V1\Rl~M~f dF UNIT hflGHT =
OISJ ~.7208 4.834U
EIH<OR -6.3
-14.4
ADJD -1.7 -2.2
0.33270-02
0.14070-Cl
l8.'i0265
I< E SD 4.6
12.2
'° '°
5.0i56 -8.'i -3.l 5.8 5.J3G4 -11.2 -4.? 6.7 5.4444 -16.5 -5.0 11.s :i. 6 J 52 -10.8 -5.9 4.9 5. <JJ':J9 -lJ. 3 -7.l 6.2 6. C5 '10 -tl. 3 -1. 'j 0.8 6.L448 -1.1.5 -d.L 5.3 6. '.j4 SL -10.2 -<.J .1 l. l 6.6636 -10.3 -<J. 4 0.9 o.c1~44 -11. l -9.8 1.3 7.l~'i2 -9.3 -10.2 -0.9 7.2/Jl -13.1 - l C • .3 2.8 7. 1t b 1t 0 -11. S -10.'t 1.1 7. 7681) -'). 8 -10.2 -0.4 /.Ud28 -10.'t -10.1 0.3 H.u7J:, -8.6 -<).') -1.J cJ.J7tJ4 -7.7 -(}. J -1.6 t-'
0 u. 4·) 24 -6.7 -9.0 -2.3 0
d.68J2 -11.1 -fj. 4 2 • ., ti. 'J d tW -8.l - /. 4 1.2 ~.l(j20 -4. {:; -t..Y -2.3 <-J.2928 -7.9 -6.2 1.7 g.SlJlb -4.6 -4.8 -0.2 <.J.711& 1.4 -4.3 -5.7 9.'J024 -1.d -3.4 -1.6
10.2072 -G.3 -2.0 -1.7 10.3212 1.2 -1. 5 -2.7 10.5120 1.5 -o., -2.2 lll.Blt8 2 • .3 O.l -1.7 l0.430ti ~.s 1. 1 -4.8 11.1217 :i.3 l.H -3.5 11.4265 :i • :i 2.8 -2.1
il.5404 10.3 3.1 -7.2 11.7.HJ 6.4 3.5 -2.9 12.0360 6.2 4.1 -2.1 12.1500 9.6 4.2 -5. '• 12.2552 0.9 4.3 J.4 12.3408 6.B 4.3 -2.5 12.6456 4.4 4.3 -0.1 12.759/.J 8.8 4.] -4.5 l.2."'504 1.8 4.1 2.3 1:.07ti<J 1.2 3.9 2.1 13.369?. 4.5 3.3 -1. 2 LJ.~bUl -0.3 2.8 3. l lJ.82~5 o.6 2.0 -4.6 l3.8L'ti.l -2.0 1. 8 3.U 14 .1 J OJ -4.5 0.8 5.3 14.ll.'16 ,;_ 2. c; 0.1 3.6 i't. 4 3 5 2 -1.0 -C.4 0.6 l4.47<t:, -4.9 -0.6 '•. J f-' 14.~88'1 -:::.J -1.1 2.2 0
f-' 14.7400 -5.l -1. iJ 3.3 l4.77'i2 -H.l -2.0 6.1 l ':l. C449 -4.4 -3.2 1.2 l~.l~lll -3.6 -3. SI -0.3 l 5. 3 4 <; 7 -11.6 -4.6 1.0 1 '.l • c 5 1tb -3.] -5.~ -2.6 15.C,5'15 -12.J -7.2 5.1 l6 • .co43 -10.0 -8.2 1.8 16.5692 -A.6 -<;.l -0.5 16.1.!740 -1.2 -9.8 -2.6 l7.l7'd') -6.2 -10.2 -4.0 17.4J]8 -o.7 -10.4 -3. 7 l7.7d86 -6.S -LC.2 -3.J
NO
-Ooo
.... ,a-.o • • • • • • •
...r!.t'l~ll'\,....-00" I
I I
I I
I I
.::ON
~rri ....
c::,~ I
e I
I e
I •
C" C
' C
XJ ,...
..:, -:r i,-, I
I I
I I
I I
::, N
,.._ '-" ~ ~
N
• • • • • • • ~ -:r /\I
............ ·-O
I
I I
I
..nl'\J ....
'::l;J' ....
(",J ~C
.0-"<1r-~..J-O
0"0'00 ............
0 ~ -
,:.) .., '° lJ' • • • •
• • •
'..:0 :0 .:0
~"" ~"" :?" ::,. ............................
102
APPENDIX D PROGRAM OUTPUT
MODEL 2
103
UISTANCE ER RGJ< i.\ EIGHT
4.83~U -14.4 1.0000 5.4'1.lt4 -16.5 1.0000 6.0540 -8.3 1.0000 6. 66 36 -lC.3 1.0000 7.2731 -13.l 1.0000 7.8828 -10.4 1.cooo 8.4924 -6.7 l.GOOO 9.1020 -4.6 1.0000 9.7116 L.4 1.0000
lC.3212 1.2 1.0000 LU.9308 5.9 1.0000 ll .5404 lC.3 1.cooo 12.ljQQ 9.b 1.0000 12.75Sb 8.8 1.0000 1J.3o92 4.5 1.0000 13.C7cl<J 1.2 1.0000 I-'
0
14.5884 -3.3 1.ocoo -1-'
15.l<JBl -3.6 1.0000 13.8255 6.6 1 .oaoo 14.4352 -1.c 1.0000 14.1303 -4.5 1.0000 14. 7400 -5 .1 1.0000 15.044'7 -4.4 1.0000 l B. 3 9 82 -4.2 1.0000 lt.i.7031 -2.7 1.0000 19.C078 -1.5 1.cooo l<J.Jlu9 1.0 1.0000 19.6141 1.u 1.0000 l<J.9102 6.2 l.OGOO 15.349/ -11.6 l.OCGO
l~.o:>46 -3.3 l.GOOO 15.S~'1~ -12.3 1.0000 16.2643 -10.0 1.0000 lc.56<,2 -8.6 1.0000 16.87 1,0 -7.2 1.caoo 17.1789 -6.2 1.0000 17.4838 -6.7 1.0000 17.7dd6 -6.9 1.0000 lU.0935 -5.6 1.0000
1t.72Ctl -b.3 1.0000 '.:i.U2~t. -8.9 1.0000 5.3304 -11.2 1.0000 5.t3~2 -lC.8 1.0000 S.'.'399 -13.3 1.oaoo 6.2448 -13.5 1.ccoo 6.54<;6 -10.2 1.0000 6.E544 -11.1 1.0000 7.1~~2 -9.3 1.ocoo I-' 7 .46 1t 0 -11.s l.OOGO 0
V1 7.76H8 -9.a 1.ocoo t3.C735 -8.6 1.ocoo 8.3784 -7.7 1.0000 tl .683.2 -11.l 1.0000 ~.9tH30 -t,.6 1.0000 ~.2lj2d -7. r; 1.0000 <;.S-;/6 -4.6 1.0000 9.902'• -1.a 1.0000
10.20/? -0.3 1.0000 lC.5120 l. :> 1.0000 10.a1co 2.3 1.0000 11.1217 '"- A 1.0000 _.,. -· ll.42u:i 5. '.) 1.ocoo
11.7313 6.4 1.ocoo 12.0360 6.2 1.0000 12.3400 6.8 1.ocoo 12.l.456 4.4 1.ocoo 12.45()4 1.a 1.0000 12. 25 52 C.9 1.0000 1::,. 560 l -0.3 1.0000 13.8648 -2.0 l.OGOO l4.16<J6 -2.9 1.0000 14.474~ -4.9 1.ccoo l'1.17Y2. -8.l 1.0000
~UHUEk UF OHSERVATIUNS = 73
INITIAL APPROXl~ATIUf\S Af'PlllUDE = 10.30000 HEfEREf\C..E' D1STAt-;C£ = 4.00000
I-' 0
°'
AOJLSllU PA~A~ETERS A~PLITUOt = -9.26966 HlfEKE~CE CISTA~CE = J.tll806
lTEl<ATILf\S = 2
PlSllRlORl COVARIANCE MATRIX, HT~e INVERSE, IS
0.2uo3U-Ol
O.l'J'.>4D-03
VARIAt\CE Cf UNIT wEIGHT -
LISJ 4.7208 4.1.JJ',H
Efd~CH -6.3
-14. ,t AUJC -5.0 -S.5
0.19540-C3
0.62470-03
18.C,89
RE.SO 1.3 8.9
I-' 0 -...J
5. 02 5t.l -8.9 -6.4 2.5 5. 3:.i 04 -11.2 -7.5 3.7 5. 4 4'• '• -16.5 -7.9 8.6 5 .6352 -10.8 -8.4 2.4 5.93'-.19 -13. 3 -s.o 4.3 6.Qj40 -8.3 -<;. l -0.8 6.2448 -1.:i. 5 -<;.3 4.2 6.5'tS6 -10.2 -<;,. 2 t.o o.66J6 -10.3 -',. l 1.2 t.8544 -11.1 -H. 7 2.4 1. l'.>92 -Y.3 -8.0 1.3 1.2731 -13.l -7 •. , 5.4 7.4ti4U -11.5 -7.0 4.5 I. -lbf-!8 -9.8 - 'j. 7 4.1 7.t!d28 -10.4 -5.l 5.3 u.0735 -8.6 -4.2 4.4 t.37t14 -1.7 -2.5 5.2 I--'
0 6.4924 -6./ -1. 9 4.8 Cl)
d.l8J2 -11. l -0.8 10.3 B.SH8U -8.6 1. u 9.o '). l U2 0 -4.6 1.6 t.. 2 s.2s2a -7. 9 2.1 10.6 lj • 5 c;, (.; _,,. l 4.4 9.0 ').7116 1.4 4.4 3.5 9.9024 -1.& 5.8 7.6
lC.2072 -0.3 1. l 7.4 10.)212 1.2 -, • 5 o.3 lC.512C 1., B. l 6.o lC.tllod 2.3 d.8 tJ • 5 lC.':l30fl 5.9 s.c 3.1 ll.1217 5.3 <,. 2 3.9 ll.4265 5.5 g.2 3.7
11.5404 10.3 9.2 -1.1 11.7313 6.4 9.0 2.b 12.0300 6.2 8.3 2.1 12.1500 9.6 8.0 -1.6 12.2~52 0. <; 1.1 6.8 12.3408 6.8 7.4 0.6 12.6456 4.4 (; • 2 1.a l2.7~S6 8.8 5.7 -3.1 12.9504 1.8 4.8 3.0 l3.C78'j 1.2 4.2 3.0 13.36~2 4.5 2.6 -1. 9 lJ.5601 -0.J 1.5 1.8 13.8255 6.6 -o.o -6.6 13.8648 -2.0 -C.3 1.7 14.1303 -4.5 -1. 8 2.1 14.1696 -2.s -2.0 0.9 14.4:,52 -1. G -~.5 -2.~ l'e.4745 -4 .9 -3.1 1.2 ..... 14.!;H84 -J.3 -4.3 -1.0 0
\.0
14.7400 -5.l -5.1 o.o 1i..-/7'J2 -8.l -5. 3 2.8 l :.i. Vt •1 ~ -4.4 -6.5 -2.1 1,.1c;a1 -3.6 -7. l -3.5 15.3'eS1 -11.6 -7. 6 4.0 l~.t54o -3.3 -u.s -5.2 l~.<;5<6 -12.3 -<.; .o - ~ _j • _,
l6.2l.i43 -10.c -9.3 0.7 lc.5t92 -8.6 _..,. 2 -0.6 l O. 8 7 '• LJ -1. 2 -d.1 -1.5 17.1784 -b.2 -7.9 -1.7 17.•1838 -6.7 -6.9 -0.2 ll.7Jb6 -6." -5.6 1.3
18.093~ -5.6 18.3')82 -4.2 18.7031 -2.1 l<i.C078 -1.5 l<;.JlU9 1.a l~.61'11 1.a l~.9102 6.2
-4.l ·-2. 4 -0.1
1.1 2.8 4.4 5. Ci
1.5 1.8 2.0 2.6 1. 8 2.6
-0.3
I-' I-' 0
APPENDIX E PROGRAM OUTPUT
MODEL 3
111
DlSIANCf. El<~ OR ~E:lGHl
4.i.l3'td -14.4 1.0000 5.4'144 -16.5 1.0000 (.i.0540 -8.3 1.0000 tJ.663b -10.3 1.ocoo 7.27H -13.l 1.0000 7.8828 -10.4 1.0000 8.4gz4 -6.7 1.0000 <J.1020 -4.6 1.0000 9.7116 1.4 1.0000
10.3212 1.2 1.c.000 10.'i3C8 5.9 1.0000 11.5404 lC.3 1.0000 12.1500 <;. 6 1.0000 12.7:J';6 8.8 1.0000 l3. 36Y2 '•., 1.0000 I-'
I-'
l3.C78g 1.2 1.0000 N
14.58b't -3.3 1.0000 1s.1sa1 -3.t 1.ccoo lJ.8255 6.6 1.0000 14.4.352 -1.c 1.0000 14.1303 -4.5 1.cooo l't.7400 -5.l 1.0000 15. 04'-tC, -4.4 1.ccoo 16.3<;02 -4.2 1.caoo 1H.7U3l -2.7 1.0000 19.0078 -1.5 1.0000 19.)109 1.c 1.0000 19.l:l<tl l.d 1.0000 l lJ. S l 02 6.2 1.0000 l':>.,491 -11.<.i 1.cooo
15.6546 -3.3 1.0000 l5.95S5 -12.3 1.0000 16.2643 -10.0 l.OGGO 1t.56'n -8.6 1.0000 16.8740 -7.2 l.lJOOO 17.l"/89 -6.2 l.OJOO l-/.4838 -6.7 1.0000 17.7086 -6.9 1.cooa l6.CY35 -5.6 1.ccoo
4.72C8 -c.3 l.Oi.JOO 5. 02 5ti -8.9 1.0000 5.33C<t -11.2 l.OGOO 5.6352 -lC.8 1.ccoo 'J.s3-,s -13.3 1.ococ &.2't48 -lJ.5 1.ocoo l.54S6 -10.2 l .GCOO l. c.::>44 -11.1 1.ocoo 7.1592 -',.3 1.0000 ...... 7.4u40 -11. 5 1.0000
...... L,.)
7.1088 -<;; .8 1.0000 8.0735 -8.6 1.0000 8. J 7 84 -1.1 1.ocoo 8. 6 d 32 -11.1 1.0000 b.9880 -8 .6 1.ocoo 4.2920 -1.9 1.0000 S.5'i7& -4.6 1.0000 'J. SC 2 4 -1.d l.OGOO
10.2012 -o.J 1.ccoo 10.5120 l • 5 1.0000 10.816d 2.3 1.0000 ll.1217 " -~-j 1.0000 ll.it2o5 S.5 1.0000
11.7313 6.4 1.0000 12.0.360 6.2 1.ocoo 12.3408 6.8 1.0000 12.645b 4.4 1.0000 l2.<i5C4 1.8 1.0000 12.2552 c.g 1.occo 13.5601 -C.3 1.ocoo 13.8648 -2.c 1.0000 l4.lt~6 -2.9 1.0000 14.474~ -4.<J 1.ocoo 14.7792 -8.l 1.0000
~UMDER Of CBSERVATIUNS = 73
INITIAL APPKOXl~AllU~S A~PLITUOf = 13.40000 IKAI\SLAflGI\ = -3.10000 1<EFER(I\CI.: DlSTANCL = 4.00000
...... I-' ~
AUJLS1LU PARA~ElERS A~PLITUDE = -0.597d8 TRANSLATION= -2.t6710 RfFEKfNCE CISTPNCE = 4.20b27
lTU<ArlONS = 4
POSTEKIGRI COVA~IANCE MATRIX, BTWB INVERSE, IS
0.30030-01
- C. 39 jj l D- C 2
-G.lt,790-03
-c. 3lJ 81 o-cz C.14230-Cl
-o.3oc;oo-c4
VARIANCE Of UNIT WllGHT = 9.<;6'H2
-0.1679D-03
-a. 3oqoo-04
0. 93550- 03
f-' f-' u1
UISI EHRGR ADJC RESD 4.7208 -6.3 -~.4 0.9 4.8348 -14.4 -6.0 8.4 5.0256 -8. 'i -6.~ 2.0 5.3304 -11.2 -8.2 3.0 5.44La4 -16.5 -8.7 7.8 5.l352 -10.8 -<J.4 1. '• 5.<1)()') -13.3 -10.3 ~--0 6.C54U -8.3 -10.6 -2.3 6.2't48 -13.5 -10.9 2.b 6.54S6 -10.2 -11.2 -1.0 6.6636 -10.3 -11.3 -1.0 6.e544 -11.1 -11.2 -0.1 1.l~lJ2 -S.3 -10.9 -1.6 7.273l -13.l -10.7 2.4 ~
7.4640 -11.5 -10.3 1.2 ~
°' t.76c8 -<J.E -9.4 C.4 7.8828 -10.4 -CJ.a 1.4 l:!.C735 -a. c -8.3 o.3 b.Jll:14 -1.1 -6.9 0.8 ti.4924 -6.7 -6.4 0.3 8.6tl.32 -11.1 -5.4 5.7 b. 9 880 -8.6 -3.8 4.8 9.1020 :..4 .c -3.2 1.4 9.2'i28 -1.c; -2.2 5.7 g.5<;16 -4.c -0.6 4.C 'i.7116 1.4 o.c -1.4 9.<;Gi4 -1.8 1.0 2.8
lC.2072 -U.3 2.4 2.1 10.3212 1.2 2.c, 1.7 10.512G 1.5 3.6 2.1
10.8168 2.3 4.6 2.3 l0.'1308 5.9 4.9 -1.0 11.1217 5.3 5.4 0.1 ll.426~ 5.5 5.8 0.3 11.5404 10.3 5. g -4.4 11.7313 6.4 5.9 -0.5 12.0360 6.2 5.7 -0.5 12.1500 ',. t 5.6 -4.0 12.2552 a. li 5.4 4.5 12.3408 6.8 5.3 -1.5 12.6456 4.4 4.5 0.1 12.7596 a.e 4.1 -4.7 12.9504 l.8 3 • '1 1.6 l3.C78<J 1.2 2.9 1.7 13.3692 4.5 1.6 -2.9 13.5601 -0.3 c.1 1.0 13.8255 6.6 -C.6 -1.2 13.864ti -2.0 -0.8 1.2 I-'
I-' 14.1303 -4.5 -2.3 2.2 -..J
14.1696 -2.s -2.5 0.4 14. 4 3 ::,2 -1.0 -3.9 -2.9 14.'d45 -4.~ -4.l 0.8 14.5!:Hl4 -3.3 -4. 7 -1.4 14.7400 -5.l -5.5 -0.4 14. 7 1 ·~2. -8.l -5.7 2.4 15.0449 -4.4 -7. 0 -2.6 15.1981 -3.6 - -, • 7 -4.1 15 •. HC,7 -11.6 -E.3 3.3 15.c54t; --:. . --. - -9.5 -6.2 l 5. c; ~CJ5 -12.3 -lC.3 2.0 ll • .2643 -10.c -lG.9 -0.9 l c. ~ 6 <j2 -8 .a -11.2 -2.6
lb.8740 -7.2 17.1789 -6.2 17.4838 -6.7 17.7886 -6.~ 18.C93S -5.a 18.39tl2 -4.2 18.7031 -2.1 1S.C078 -1.5 l'-).:HOS 1.c 19.t.l'tl L.8 l9.SlC2 6.2
-11. 2 -1 O.'i -10.3
-9.4 -8.2 -6.6 -5. 3 -3. 7 -2.1 -o.5
1.0
-4.C -4.7 -3.6 -2.5 -2.6 -2.6 -2.6 -2.2 -3.l -2.3 -5.2
..... ..... co
APPENDIX F HORIZONTAL ADJUSTMENT OF THE
VIRGINIA TECH CALIBRATION BASE LINE
119
u ~ II ~
V')
z 11
::> -
V,
~
..::: .,::
~·: >
:.,J
c:: z
:..u ~
V')
2 0
::, 0
'-'-\..I.
0 w
(~
~
i.U
i.:J '.l)
~
~
~
::> ::,
z z:
120
0000000000000000000000000 0000000000000000000000000 0~00~~0000000000~00000~00 • • • • • • • • • • • • • • • • • • • • • • • • •
oooo-0000-0000-0000-oooo-
000000000000000000~000000 000000~0000000000000000?0 0000000000000000000000000 . . . . . . . . . . . . ' . . . . . . . . . . . .
ooo--o~o--~~0--000------0
00000000000000000~0000000 ooo~oouoooooooooo~ooooo~u 0000000000000~000000~0000 • • • • • • • • • • • • • • • • • • • • • • • • •
oo---oa---00------000---0
-o~ouooooouooooooaoooooooo 0~00000000000000000000000
xooooooo~~OO
OO
UO
OO
OQ
OO
OC
OO
... . . . . . . . . . . . . . . . . . . . . . . . . . xo~---o------aoo~--0000--0 - <t ~ - zoocooooooo~ooo~~~oooooo~o ~uoooooooooooooooo~ooo~ooo -uocoooo~0ooo~~~oaoooooo~o t..J
• •
• •
• •
• •
• •
• •
• •
• •
• •
• •
• •
• •
• --~----?oooo-o~?o~-00000-0 .L 1.1. ,J,J
8 u
0.000 0.000 O.OGO 0.000 1.000 0.000 0.000 0.000 1.000 1.000 0.000 0.000 1.000 1.000 1.000 u.ooo 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000
ubSt:iWAT IUN wE!GIH CBSERVATION CONSTANT p ( 1) = 1.0 L( l) = 149.9999 p ( 2) = 1.0 L( 2) = 429.'.J56d P( 3) .::: 1.0 L ( 3) = 999.q415 p ( 4) = 1. a L ( 4) = 1399.9757 t' ( 5) = 1.c L( 5) = 1462. 84 74 f>( 6) = 1.0 L( 6) = 15U.Ou02 p ( 7) = 1.0 L ( ·1) = 279.9537 p ( 8) = 1.0 L ( l:i) = 849.93l:38 P( <;) = 1.0 L ( 9) = 1249.9783 P( 10) = 1.c L(lO) = l3l2.B52d p(ll) = 1.0 L(ll) = 279.9568 P( l2J = 1.c L(l2) -= 429.9532 I-'
N
P(l3) = 1.0 L(l3) = 56~.9780 I-'
P( 14) = l. o L ( 14) = 970.0118 Pl 15) = 1.0 L(l5) = 1032.8912 p ( lb) = 1.c l( 16) = 569.9052 P' n, = 1.0 L(l7) = 849.9450 Pl HI) = 1.c l(l8) = ',99.'1~15 P( 19) = 1.c L(l<Jl = 400.u32b PUU) = l -, ·"' l(20) = 462.9041 P(21) = 1.c L(2l) = 400.0287 P(22J = 1.c L(22) = <;JO. Ol '10 P(23) = 1.0 L(23) = 1249.9760 P(2't) = 1.c l( 24) = L399.9l 1t9 P(2~) := 1.c l(25) = 62.B/!j6 P(2&) = l.G L ( 26 J = 62.8807
P(27) = 1.c l(27) -:: P(28) = 1.0 l(28) = p ( 2':J) = 1.c L(2Y) = P(30) = 1.c L(30) =
SCLUTICt\ MATl{IX ( X) X( l) = l4S.<.J'1<H DIST FRC~ O+uO TC 1*50
X( 2) = 279.~Sdt DIST FRCM l+~O TC 4+30
X( 3) = ?6~.9E5l 0151 fRC~ 4+30 TC 10+00
X( 4) = 400.0310 UIST FRLi~ lutOO TC 14+00
X( 5) = 62.8771 DIST FRGM l4+UU TC THE AST~ONGMIC PILLAK
P0STl::k1Lt<l CGVAklANCE MATRIX - N INVERSE 0 .1 o 7 -O.GU3 -o.uoo -o.uoo
-0.083 O.lo7 -0.083 -o.oou -t).OOu -o.ceJ 0.167 -J.083 -0.DOO -o.occ -0.083 0.161 -u.ooo -o.uuo -U.JCO -0.083
462.9084 1032.8925 1312.8573 1462.8542
-0.000 -0.000 -0.000 -0.063
0.167
I-' N N
Rt::SIDUAL MATl{IX V( 1) = -o.coc V ( 2) = u.002 V ( 3) = O.ll02
"' 4) = -0.001 V ( ~) = 0.004 V ( 6) = -o.ouo V( 7) = 0.005 V( tl) :: o.oos V ( 9) = -U.GC4 V(lO) = -0.001 V(ll) = o .0(.)2 V(l2) = U.005 V(l3) = 0.007 V(l4) = 0.004 V(l~) = 0.002 V( lo) = -o.uuo I-' V(ll) = -0.001 N w V( ltl) = -o. o a a V( 19) = -0. Oll 2 V(lO) = O.OC4 V(21) = 0.002 V(22) = -0.001 V(23) = -0.0Cl V(24) = -o.ooc V(25) = 1).002 V(2o) = -0.CC4 V(27) = -ll.OOC V ( 2d) = 0.001 V(2(J) = -U.005 Vl3C) = -o.uo:;
STANUAKO DEVIATION CF UNIT WEIGHT= 0.004
1--' N -I>
APPENDIX G ADJUSTED ELEVATIONS OF THE VIRGINIA TECH CALIBRATION
BASE LINE MONUMENTS
125
126
Monument Adjusted Vertical Elevation
0 + 00 2130.523
1 + so 2124.181
4 + 30 2112.538
10 + 00 2097.695
14 + 00 2112.857
NOTE:
The unit for the monument stationing is meters. The unit for the
adjusted elevations is feet.
APPENDIX H LEAST SQUARES APPLICATION
TO CONSTANT OFFSET AND SCALE ERRORS
127
128
The least squares adjustment of calibration base line data to
yield constant offset and scale errors may be readily accomplished with
two equations that are easily adapted to hand held calculators. The
mathematical model for this adjustment is given by Fronczek (1977).
The model is
where
D = D + SD + C a h a
D = the calibrated distance, a
D = the measured slope distance reduced to horizontal, h
S = the scale error, and
C = the constant offset error.
(A.l)
Let the number of observations made be n. Then from equation A.1,
the observation equation for the ith observation is
where
V = D i ai
- D hi
-SD C ai
th v = the residual for the i
i observation and
D D ' S, and Care as previously defined. ai hi
For n observations,
v = D - D SD - C, 1 al hl al
v = D - D - SD - C, 2 d2 h2 a2
(A.2)
129
V = D - D SD - c. n an hn an
Rearranging the above equations and putting them in matrix form yields
l D 1 Is cJ 1Dal
- D cS V
1 al hl 1
V D 1 D - D cS 2 + a2 = a2 h2 = 2
. V D 1 D - D \I n an an hn
These matrix equations are now in the form
V + M = F
where
V = the matrix of residuals,
B = the matrix of coefficients for the unknowns,
C!. = the matrix of unknowns, and
F = the matrix of constants.
Equation A.3 is general form for a least squares adjustment
the Method of Indirect Observations (Hikhail,1981).
this method is
where, assuning all equations have equal wieght, t N =BB, and
Thus, from equation A.5,
The solution
(A. 3)
by
for
(A. 4)
(A.5)
(A.6)
130
n N = D D D D 11 = L (D 2)
al a2 an al i=l ai n
1 1 1 D 1 L D a2 i=l ai
D 1 an
n Letting L .. i:,
i=l
t = D D D 0 = L(D 0 ) al a2 an 1 ai i
1 1 1 0 Ee 2! i
0 n
-1 The solution requires N which way be found from
where
-1 N
1 = - Adj (N)
Adj(~~) = (N t
) and cofactor
N = n -LD 1 · cofactor ai
-rn I:(D _2)1 ai a1.
Since N is a symmetric matrix, t
N = (N ) = Adj(N). cofactor cofactor
n L D
i=l ai
n
(A. 7)
131
Also,
!NI = nI(D 2) - o::D )2, ai ai
so that
-1 1 n -ID N = ai
nI(D 2) - (ID )2 -ID_ I(D 2) ai ai aL ai
Recalling that
t = I(D 0 ) and t, = s ' ai i
0 C i
the solution for the unknowns, N-l t, is shown to be
1 nI(D o) - ID Io t, = ai i ai i
nI(D 2) - (ID )2 I(D 2Ho - rn I(D 0 ) ai ai ai i ai ai i
Thus, the least squares estimate 0~ the scale ani ""'~PS!: 1.TI ~ ')ffa~!:
errors is
nI(D 0 ) - rn Io ai i ai i
s = ' and (A. 8)
nI(D 2) - cm )2 ai ai
I(D 2Ho - rn I(D 0 ) ai i ai ai i
C = (A. 9)
nI(D 2) - cm )2 ai ai
APPENDIX I EDMI BASE LINE QUESTIONNAIRE
WITH TABULATED RESPONSES
132
1.
2.
133
EDMI Base Line Questionnaire Department of Civil Engineering
Geodetic Division Virginia Polytechnic Institute and State University
a. Do you own/rent or use an EDMI? own but do not use 1 own/rent and use 23 do not own/rent but use 7
b. What kind(s)?
1. 2. 3.
What length of tape do you use most? 100 I 15
.200 I 9 300' 12 other(s) 1. 0
2. 0
3. Would you like both your tape(s) and your distance meter(s) calibrated?
tape(s) only 0 distance meter(s) only 5 both 27
4. How much time can you afford to spend calibrating your distance meter?
0-2 hrs. 2 2-4 hrs. 7 4-6 hrs. 4 6-8 hrs. 15
5. What is the most convenient time for you to calibrate your distance ~eter(s) and/or tape(s) during the day?
early morning - before 9:00 a.m. 19 midday - 9:00 a.m. to 4:00 p.m. 8 late evening - after 4:00 p.m. 9
During the week?
Monday-Thursday 19 Friday-Saturday 12
134
6. How far do you think a surveyor should be asked to drive to a calibration baseline?
35-50 miles 50-60 miles other
17 8 7
7. How often do you feel an EDMI should be calibrated? 1/month 2 1/year 13 1/2 years 13 When it begins yielding erroneous results 5 Never after it is purchased other
8. How would you prefer to reduce your observations? NGS 12 Va. Tech 15 Yourself 15
9. Do you have a TI-58 2 TI-59 4 HP-65 3 HP-97 4 HP-41C 4 other 1. HP 9815 8
2. HP 45 9 3. HP 85 3 4. Other 16
10. Do you have two thermometers and either 2 barometers or 2 altimeters for making meteorological reductions?
no thermometer 15 1 thermometer 8 2 thermometers 7 no barometer/altimeter 16 1 barometer/altimeter 8 2 barometers/altimeters 2
135
11. Where would you like to see a calibration base line established?
12. Are you personally willing to devote your time and effort to establish a calibration base line in your area?
Yes No
Maybe, if
26 0
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The vita has been removed from the scanned document
CALIBRATION OF EDMI AND RECOMMENDATIONS
FOR A BASE LINE NETWORK IN VIRGINIA
by
Dennis Ray Varney
(ABSTRACT)
Three systematic instrumental errors exist in electronic
distance measuring instruments (EDMI): 1) scale error,
2) constant offset error, and 3) cyclic error. The potential
magnitude of these errors requires that each EDMI should be
calibrated for all three errors. The calibration constants of
EDMI may be monitored on an arbitrary base line; however, a
calibrated base line is required to perform an accurate EDMI
calibration for constant offset and scale errors. Calibration
of cyclic error requires monumentation not normally found on a
calibration base line. Cyclic error can be measured on a short
base lin~ in the laboratory.
The surveyors in the State of Virginia would benefit from a
statewide network of calibrated base lines. A network covering
the state would provide convenient access for the state's
surveyors. A unified network of base lines would give the
states surveyors a standard of comparison for their EDMI. This
standard of comparison would be nationwide if Virginia would
choose the National Geodetic Survey (NGS) to calibrate the base
lines in its network. Base line calibration by NGS would
require that the state abide by NGS specifications for
establishing its base line network.
