FOREWORD

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The previous issue of this document published in August, 1973, introduced the confidence region concept for the classification of horizontal control. This new issue retains that concept, but extends it tocover short lines (less than three km), treatment of which was not considered in the 1973 specifications. The previous formula r = Cd defining the limit of the semi-major axis of the 95 percent confidence region, and applicable only to long lines, is replaced by r = C(d+0.2), which can be applied to lines of any length.

This is the only major change made. In addition, some refinements have been made to the earlier specifications. Short sections on vertical control by altimetric traversing, ground elevation meter and satellite Doppler have been added in Part 1. In Part 2, the portion dealing with measurement guidelines is included in a separate appendix (Appendix C). Appendices D and E have been added to facilitate understanding and use of the specifications for horizontal control. Appendix D indicates how simple computations can serve as an aid in network design and analysis, should computer facilities not be available. Appendix E lists numerous examples of standard deviations for various instruments and methods. The final addition is Appendix F, which is a resumé of the use of photogrammetry in the interpolation of control. Minor rewriting has been done of many parts of the text to improve explanation.

All control surveys conducted by the Branch and those that now exist in Branch records will, henceforth, be classified according to these specifications.

R. E. Moore,
Director-General,
Surveys and Mapping Branch.

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CONTENTS

Foreword

PART 1--VERTICAL CONTROL

Levelling

• Specifications
• Loops and Closures
• Individual Section Runnings
• Bench Mark Stability
• Instruments
• Procedures

Other Methods

• Vertical Angles
• Altimetric Traversing
• Ground Elevation Meter (GEM)
• Satellite Doppler

PART 2--HORIZONTAL CONTROL

• Introduction
• Specifications
• Network Design
• Photogrammetic Methods

PART 3--SURVEY MARKERS

• Terminology
• Recommendations

APPENDICES

A - The Concept Of Confidence Region

• Introduction
• Standard Deviation
• Weights
• Confidence Interval
• Confidence Region
• Summary Of The Confidence Region Concept
• Example Of The Application Of Confidence Regions

B- Comparison Of 1961 And 1978 Horizontal Control Specifications

C- Measurement And Check Guidelines For Conventional Methods

• Introduction
• Triangulation
• Trilateration
• Traversing
• EXPLANATORY NOTES ON MEASUREMENTS
• Directions
• Lengths
• Azimuths
• EXPLANATORY NOTES FOR CHECKS
• Directions Vs Lengths
• Azimuths

D- Simple Computations As An Aid In Network

• Design And Analysis
• Confidence Level Of Misclosure

E- Examples Of Standard Deviations For Various Instruments And Methods

F- Densification Of Horizontal Control By Photogrammetric Methods

• General
• GUIDELINES
• Planning Considerations
• Procedures
• Expected Accuracy

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TABLES

I	VALUES OF C FOR HORIZONTAL CONTROL SURVEYS ACCORDING TO ORDER, USING r = C(d+0.2)
II	ACCURACY STANDARDS FOR HORIZONTAL CONTROL SURVEYS
III	SURVEY MARKER RECOMMENDATIONS
A-I	FACTORS FOR CONFIDENCE REGIONS
B-I	MAXIMUM ANTICIPATED ERRORS IN ADJUSTED HORIZONTAL CONTROL
B-II	COMPARISON OF STANDARD DEVIATIONS
B-III	COMPARISON OF 1961 AND 1978 GUIDELINES FOR MEASURED LENGTHS
B-IV	TRAVERSE WORK ACCURACY GUIDELINES
C-I	MEASUREMENT AND INTERNAL CHECK GUIDELINES
C-II	ARC-SINE CORRECTIONS (METRES)
C-III	ARC SINE CORRECTIONS (FEET)
C-IV	SINE EQUATION TEST (FIRST-ORDER)
C-V	SINE EQUATION TEST (SECOND-ORDER)
D-I	VARIANCES FOR TRAVERSE E-D-C-L-K-J
E-I	LENGTH BY TAPE AND SUBTENSE BAR
E-II	LENGTH BY VARIOUS TYPES OF EDM INSTRUMENTS
E-III	STANDARDS FOR METEOROLOGICAL MEASUREMENTS,FOR CALIBRATION TESTS AND FOR ELEVATION DIFFERENCES AND MEAN ELEVATION DETERMINATIONS
E-IV	DIRECTIONS
E-V	AZIMUTHS
E-VI	POSITION DIFFERENCES

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FIGURES

1	Ellipse showing the 95% Confidence Region of one Station Relative to another
2	Accuracy Standards for Horizontal Control Surveys
3	Survey Tablet Marker (Type 1)
4	Survey Bolt Marker (Type 2)
5	Survey Iron Pipe Marker with Base (Type 3)
6	Survey Post Marker (Type 4)
7	Pile Driven Survey Pipe Marker (Type 5)
8	Cylindrical Concrete Survey Marker with Subsurface Post (Type 6)
9	Survey Marker in Shallow Overburden (Type 7)
10	Cylindrical Concrete Survey Marker (Type 8)
11	Cylindrical Concrete Survey Marker (Type 9)
12	Sleeve Type Survey Marker (Type 10)
13	Sleeve Type Survey Marker (Type 11)
14	NRC Type Deep Bench Mark (Type 12)
A-1	Normal Distribution with Mean µ and Standard Deviation σ
A-2	Traverse Loop of 10 Sides
C-1	Triangulation, Single-Chain Network - First-Order
C-2	Triangulation, Single-Chain Network - Second to Fourth-Order
C-3	Triangulation, Cross-braced Quadrilateral Network - Second to Fourth-Order
C-4	Trilateration Network - First to Fourth-Order
C-5	Trilateration Network - First to Fourth-Order
C-6	Traverse - First to Fourth-Order
D-1	Example of Simple Survey Tie
D-2	Example of Simple Survey Tie
D-3	Example of a More Complicated Survey

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PART 1--VERTICAL CONTROL

LEVELLING

Specifications

The different orders of vertical control are defined in terms of the allowable discrepancy between independent forward and backward levellings between bench marks as follows:

Special Order	± 3 mm or 0.012 ft. .
First-Order	± 4 mm or 0.017 ft. .
Second-Order	± 8 mm or 0.035 ft.
Third-Order	± 24 mm or 0.10 ft. .
Fourth-Order	±120 mm or 0.5 ft. .

Note:

K (kilometres) or M (miles) = the distance between bench marks measured along the levelling route. By using equipment and procedures as recommended for a particular order of levelling, it is expected that the discrepancy between the first forward and first backward runnings will not exceed the allowable discrepancy listed above in approximately 84 percent of the sections over the course of a level line. Those sections that exceed the allowable discrepancy must be relevelled and treated as specified under Individual Section Runnings.

Loops and Closures

Loop misclosures should seldom exceed the limits given above for various orders of levelling, taking K as the length in kilometres or M as the length in miles along the level route around the loop. To maintain the specified accuracy, long narrow loops should be avoided. The distance between any two bench marks, measured along the actual level route, should not exceed four times the straight-line distance between them. Branch, spur, or open-ended lines should be avoided because of the possibility of undetected gross errors.

Individual Section Runnings

* Special Order

a) Level each section once forward and once backward independently, using different instrument men, and if possible, different instruments under different weather conditions and at different times of the day.
b) Reject and relevel any running with a suspected gross error. c) The section is complete if the discrepancy between the original forward and backward runnings is not greater than 3 mm or 0.012 ft. . The difference of elevation is the mean of the two runnings, where:
Mean =
d) Relevel when the discrepancy between the original forward and backward runnings is greater than 3 mm or 0.012 ft. .
e) A relevelled section is complete if the discrepancy is not greater than 3 mm or 0.012 ft. between at least one forward and one backward running, afterthe rejection test (item h) is satisfied.
f) A relevelled section is complete if, after several relevellings, there are at least two forward runnings and two backward runnings surviving the rejection test (item h), even though there is not a check between a forward running and a backward running.
g) The difference of elevation for a relevelled section is the mean of the separate means of the acceptable forward runnings and the acceptable backward runnings.
h) Rejection Test: where is the mean of the separate means of the forward and backward runnings for a section, a running x is rejected when:
Ix - l > 3.7 mm or 0.015 ft. .

* First-Order

a) Level each section once forward and once backward, independently, using different instrument men, and, if possible, under different weather conditions and at different times of the day.
b) Reject and relevel any running with suspected gross errors.
c) The section is complete if the discrepancy between the original forward and backward runnings is not greater than 4 mm or 0.017 ft. . The difference of elevation is the mean of the two runnings,where:
Mean =
d) Relevel when the discrepancy between the original forward and backward runnings is greater than 4 mm or 0.017 ft. .
e) A relevelled section is complete if the discrepancy is not greater than 4 mm or 0.017 ft. between at least one forward and one backward running, after the rejection test (item h) is satisfied.
f) A relevelled section is complete if, after several relevellings, there are at least two forward runnings and two backward runnings surviving the rejection test (item h), even though there is not a check between a forward running and a backward running.
g) The difference in elevation for a relevelled section is the mean of the separate means of the acceptable forward runnings and the acceptable backward runnings.
h) Rejection Test: where is the mean of the separate means of the forward and backward runnings for a section, a running x is rejected when:
Ix - l > 4.7 mm or 0.020 ft. .

* Second-Order

A) Level each section once forward and once backward independently, using different instrument men, and, if possible, under different weather conditions and at different times of the day.
b) Reject and relevel any running with suspected gross errors.
c) The section is complete if the discrepancy between the original forward and backward runnings is not greater than 8 mm or 0.035 ft. . The difference of elevation is the mean of the two runnings, where:
Mean =
d) Relevelwhen the discrepancy between the original forward and backward runnings is greater than 8 mm or 0.035 ft. .
e) A relevelled section is complete if the discrepancy is not greater than 8 mm or 0.035 ft. between at least one forward and backward running, after the rejection test (item h) is satisfied.
f) A relevelled section is complete if, after several relevellings, there are at least two forward runnings and two backward runnings surviving the rejection test (item h), even though there is not a check between a forward running and a backward running.
g) The difference in elevation for a relevelled section is the mean of the separate means of the acceptable forward runnings and the acceptable backward runnings.
h) Rejection Test: where is the mean of the separate means of the forward and backward runnings for a section, a running x is rejected when:
|x - | > 9.4 mm or 0.040 ft. .

Bench Mark Stability

Before starting a line of levels, the stability of the starting bench mark must be proven by carrying out two-way levelling between the starting and an adjacent bench mark, and comparing the new difference of elevation with the original difference. If this check is within the allowable discrepancy for the order of levelling being carried out, both bench marks are assumed to be stable. If the check is greater than the allowable discrepancy, then one or both bench marks are considered unstable and additional bench marks must be checked until an agreement is obtained. The bench marks being checked must be located far enough apart so that any disturbing influence is not the same on both marks.

Instruments

To obtain the standards of accuracy set out under "Bench Mark Stability", the following equipment is recommended:

* Special Order

A) Self-levelling instrument equipped with parallel-plate micrometer, telescope magnification of at least 40x, and a high speed compensator with sensitivity equal to or better than a 10"/2 mm level vial; or spirit-level instrument equipped with parallel-plate micrometer, telescope magnification of at least 40x, and a 10"/2 mm or better level vial.
b) Invar, double-scale rods with line graduations of width 1 mm to 1.6 mm.
c) Rod supports.
d) Circular levels permanently attached to the rods.
e) Foot plates or steel pins for turning points.
f) Sun shade and instrument cover.

* First-Order

A) Self-levelling instrument with parallel-plate micrometer, telescope magnification of at least 32x, and a compensator with sensitivity equal to or better than a 10"/2 mm level vial; or spirit-level instrument with or without parallel-plate micrometer, telescope magnification of at least 32x, and a 10"/2 mm or better level vial.
b) Invar, double-scale rods with line graduations of width 1 mm to 1.6 mm; or invar rods of checker-board design with smallest graduations not less than 1 cm or 0.01 yard and with check graduations on the reverse side.
c) Circular levels permanently attached to the rods.
d) Foot plates for turning points.
e) Sun shade and instrument cover.

Note:

When the parallel-plate method of levelling is employed, for special-order or first-order levelling, double scale line-graduated rods must be used. The spacing of the smallest graduations must be equivalent to the displacement of the parallel-plate micrometer. When the 3-wire method is employed for first or second-order levelling, rods with checker-board design must be used.

*Second-Order

It is recommended that the equipment for the first-order levelling be used also for second-order levelling. However, instruments with a level vial sensitivity of 20"/2 mm or equivalent compensator, may be used.

* Lower-Order

The sensitivity of the level vial or equivalent compensator, should be in the 40" to 50"/2 mm range. Rods with the graduations on wood or metal alloy other than invar are satisfactory, however, it is important that the length graduations be accurate.

Procedures

To obtain the standards of accuracy set out above, the following procedures are recommended:

* Special Order

A) All sections must have an even number of set-ups.
b) Difference between backsight and foresight distances at each set-up and their total for each section not to exceed 5 metres or 15 feet.
c) Alternate readings of backsight and foresight at successive set-ups.
d) Maximum length of sight 50 metres or 165 feet with weather conditions and terrain permitting.
e) Line of sight not less than 0.5 metre or 1.5 feet above the ground.
f) Rod reading consists of mean of centre-wire reading on each scale after applying scale constant.

* First-Order

A) All sections must have an even number of set-ups.
b) Difference between backsight and foresight distances at each set-up and the total for each section not to exceed 10 metres or 33 feet.
c) Alternate readings of backsight and foresight at successive set-ups.
d) Maximum length of sight 60 metres or 195 feet with weather and terrain conditions permitting.
e) Line of sight not less than 0.5 metre or 1.5 feet above the ground.
f) Rod readings consist of:
1. Parallel-plate method-- mean of the centre wire of each scale after applying scale constant.
2. 3-wire method--mean of the readings for the three wires.

*Second-Order

First-order procedures are recommended. If second-order levelling forms part of a closed loop and the new levelling is less than 40 kilometres or 25 miles, single one-way levelling is acceptable. In this case, first-order procedures are essential. If single, one-way levelling is used, the misclosure must not exceed 8 mm or 0.035 ft. .

* Lower-Order

On third and fourth-order work, only one wire need be read. It is preferable that the difference of elevation between successive bench marks be determined twice by two independent levellings.

If the levelling does not form part of a closed loop, or if the length of new levelling exceeds 40 kilometres or 25 miles for third-order work, or 80 kilometres or 50 miles for fourth-order work, some form of double levelling must be employed. Such double levelling may consist of: two independent levellings in opposite directions or in the same direction, levelling with one instrument and two sets of turning points, or simultaneous levellings with two instruments and one set of turning points. In every case, the rod readings must be recorded as two separate levellings, and the two differences of elevation between successive bench marks compared. If two sets of turning points are used with one instrument, two rods graduated in different units should be employed.

In all lower-order levelling, it is extremely desirable to use the two-rod system and to keep balanced foresights and backsights.

OTHER METHODS

Vertical Angles

Elevations of an accuracy equivalent to lower-order spirit levelling may sometimes be obtained by vertical angles in conjunction with traverse, trilateration or triangulation. Highest accuracy can be obtained only on short lines (a few hundred metres). For best accuracy on long lines, the angles at each end must be read simultaneously using one-second or better instruments. To be considered simultaneous, the observations at each end must start within a minute. A minimum of three simultaneous sets is recommended, where a set consists of readings taken in the direct and reverse positions.

The sets should be taken preferably at different times of day, but should not be taken during periods when atmospheric conditions are unstable, such as at dawn or dusk. The order of accuracy which can be achieved depends largely on the care in taking simultaneous readings and on the stability of the atmosphere. However, results equivalent to fourth-order levelling accuracy can seldom be secured if lines are greater than about 15 kilometres (10 miles).

In regions such as mountainous or coastal areas, the effect of the geoid and deflections of the vertical may be significant and should be considered.

Altimetric Traversing

In areas where other methods of survey are difficult to use due to terrain or forest cover, elevations by altimetric traverse may be obtained rapidly with a standard deviation of (0.7 + 0.01 K) metres, where K is the distance in kilometres between elevation control points. If the traverse is repeated, the errors are reduced by a factor of 1/ . If accuracy better than 1.5 metres is required, all traverses should be repeated. A battery of at least two precise altimeters must be carried on the traverse.

Best results are obtained when the weather is stable, the winds light and steady and not over 30 kilometres per hour. Traverse points should be in a relatively straight line between the elevation control points; a corridor not wider than 0.1 K is suggested. Traverses should be run as rapidly as possible at a steady rate of progression (70 kilometres per hour forward progress is recommended and is obtainable in practice).

Care must be exercised to obtain air temperature readings representative of the vertical air column being measured. It is recommended that the traverse profile not depart in elevation by more than 150 metres from the starting control elevation.

Ground Elevation Meter (GEM)

The GEM is a dampened pendulum device which is mounted in a specially prepared motor vehicle. It is equipped with sensors which continually monitor changes in slope, the velocity of the vehicle, and the distance travelled. The incremental differences of elevation obtained are integrated by an on-board computer to produce total elevation differences between stations.

Fast, economical elevation determination is possible with GEM. On double-run lines, standard deviations (for the mean) in the order of 0.12 < metres (where K = distance in kilometres between elevation control points) or 0.30 metre, whichever is the greater, can be obtained. Single-run lines are not acceptable since small deviations in drift rate due to variations in speed, slope and road characteristics cannot be compensated for.

Maximum accuracy can only be achieved if proper procedures are employed. These include:

A) good control at both ends of a line;
b) line lengths not excessive, preferably less than 50 kilometres;
c) double runs made on the same day with the same weight and weight distributions on each run. Generally, every attempt should be made to maintain the same conditions on the forward and back runs;
d) speeds must be adjusted to suit road conditions with a maximum of 30 kilometres per hour;
e) drift rates should be examined and compared to past rates since calibration;
f) the drift for a portion of the route which exhibits the same surface (e.g., paved) should be linear; and
g) calibrations should be made at regular intervals (at least semi-monthly) or whenever the results indicate a need (e.g., large changes in the drift rate, non-linear drift, or unacceptable agreement between the mechanical and electronic counters).

Satellite Doppler

Elevation differences can be determined to an accuracy of 0.5 metre (one standard deviation) from observations of the U.S. Navy Navigation Satellite System satellites. Fifty satellite passes should be observed simultaneously at two (or more) stations up to about 200 kilometres apart. Receiver timing yields best results; either broadcast or precise ephemeris will yield the same results. Data reduction must be in simultaneous mode. (Position differences can be determined at the same time from the same data). The elevation differences produced are with respect to the reference ellipsoid and, hence, at least one station in a series must have known orthometric elevation, and the differences of geoid undulation must be known between all stations. Antennae should be calibrated for phase centre and should preferably have a ground-plane component. The antennae should be placed at a constant height above ground, and the ground at the various stations should have similar reflective properties. The final accuracy of orthometric elevations will also depend upon the accuracy of the initial orthometric elevation and the accuracy of geoid undulation differences.

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PART 2--HORIZONTAL CONTROL

INTRODUCTION

In August, 1973, the Surveys and Mapping Branch published "Specifications and Recommendations for Control Surveys and Survey Markers". Those specifications were specifically designed for the most common types of control surveys carried out by the Branch and did not contain specific provision for short lines. As a result of greater interest in urban control surveys, the Branch prepared "Specifications and Recommendations for Horizontal Control Surveys with Short Lines" in June 1975, as a provisional supplement to the 1973 publication. These specifications combine the 1973 and 1975 specifications.

SPECIFICATIONS

Horizontal control surveys are classified as first, second, third or fourth-order according to standards of accuracy.

The statistical concepts of standard deviation and confidence region are used to define standards of accuracy. These statistical concepts replace the concept of maximum anticipated error used in the Branch specifications issued in 1961 (See AppendicesA and B).

A survey station of a network is classified according to whether the semi-major axis of the 95 percent confidence region, with respect to other stations of the network, is less than or equal to: r = C (d+0.2), where r is in centimetres, d is distance in kilometres to any station and C is a factor assigned according to the order of survey. An ellipse bounding the 95 percent confidence region is shown in Figure 1. For first-order, the value assigned to C is 2. This means that for a station to be classified as first-order, the semi-major axis of the 95 percent confidence region must be less than or equal to r = 2d + 0.4.

For two stations 10 km apart, r = 20.4 cm. For these stations to be classified as first-order, the semi-major axis of the 95 percent confidence region of one station relative to the other must be less than or equal to 20.4 cm. The values of C assigned to various orders of survey are shown in Table I (Figure 2 is a graph of r against distance. See also Table II).

As noted in Table II, the use of r = C (d+0.2) causes the parts per million (ppm) and ratio values to change significantly with distance, for short lines; this reflects practical considerations. Experience shows that with most modern methods of establishing closely-spaced control, the overall pattern of error propagation -- the combination of instrumental and centering errors, theeffects of network configuration, and a host of other contributing errors, most of which defy individual identification--is not proportional to distance.

Figure 1

Ellipse Showing the 95% Confidence Region of One Station Relative to Another (the area within which there is a 95 percent probability of the true relative position being situated).

TABLE I

VALUES OF C FOR HORIZONTAL CONTROL
SURVEYS ACCORDING TO ORDER,
USING r = C(d+0.2)
(r is in cm, d is distance in km)

ORDER	C
1st	2
2nd	5
3rd	12
4th	30

The errors of measurement contributing to this pattern can be divided into two groups; those proportional to distance and those that are independent of distance. As lines become shorter, the second group becomes dominant. For the commonly used short-distance measuring instruments, the first group is dominant above three kilometres, and the second group is significant within the range zero to three kilometres. Therefore, these specifications are useful for surveys with points either closely or widely spaced or with a mixture of both.

Figure 2
Accuracy Standards for Horizontal Control Surveys (based on r = C(d + 0.2), where r is in cm and d in km)

Where a survey network is distorted by constraint (inaccuracies in positions held fixed), examination of the adjustment results should be made beyond merely observing whether the error ellipses are within these accuracy standards. This examination should include a study of the residuals and the relative shift in positions between free and constrained adjustments. In computing standard error ellipses for networks under constraint, the computed standard deviation for unit weight from the adjustment should be used. Sometimes this means that stations, which would be classified as a first-order survey by an unconstrained adjustment, must be classified as lower-order until a general readjustment removes the distortion.

Guidelines on network design and measurements are given in "Network Design" to assist in achieving the various orders of accuracy. However, it is stressed that by merely following the guidelines one does not ensure the achievement of the order of accuracy desired. The order can only be confirmed by an analysis of the survey results.

NETWORK DESIGN

The size and shape of the confidence region is dependent not only on the accuracy of the field measurements but also on the configuration of the control network.

For a network to fulfill its basic role as a strong and reliable reference framework, it must be homogeneous, feature a reasonable number of redundancies, and the individual figures should be well-shaped. Stations should be as evenly spaced as possible, and all adjacent pairs of stations in the network should preferably be connected by direct measurement. The ratio of the longest length to the shortest should never be greater than five and usually should be much less.

A basic principle of control surveys is to work from the large to the small; therefore, the spacing of higher-order control stations should generally be greater than that of lower-order stations. In addition, there should always be a sufficient density of higher-order control to govern the establishment of lower orders.

Frequently, these ideals cannot be realized. Reality is often a network that has adjacent points which cannot be conveniently connected, that has large variation in lengths, and that has been measured with various instruments with significantly different accuracies. The surveyor must design the network with these factors in mind .

To design a network to achieve required accuracies, good a priori estimates of the accuracies of various instruments used with various techniques must be available. These estimates must reflect not only the consistency of several measurements of the same quantity by the same instrument, over a short interval of time under ideal conditions, but must also reflect normal random errors likely to occur in normal field use, under normal operating conditions by personnel who take only normal precautions. In addition, the estimates must take into account systematic errors that may not be evident in a normal survey; for example, an uncorrected zero error in Electronic Distance Measuring (EDM) instruments, systematic meteorological errors due to imperfect measuring techniques, etc. Appendix E lists typical standard deviations that may be expected under normal circumstances and which may be used to compute weights in network design programs. Higher accuracies should be estimated if extraordinary precautions are taken in calibration and measurement.

The accuracy of a horizontal control survey can be assessed properly from the results of a rigorous least-squares adjustment of the measurements. Since this assessment can only be made after the field work has been completed, something more helpful is needed for those who wish to design networks and prepare measurement guidelines, and who require some reasonable assurance that a particular order of accuracy will be obtained when the field work is done.

TABLE II
ACCURACY STANDARDS FOR HORIZONTAL CONTROL SURVEYS

(showing the variation in proportional accuracy over short distances)

SEMI-MAJOR AXIS OF 95% CONFIDENCE REGION, r = C(d+0.2); WHERE d IS THE DISTANCE BETWEEN ANY TWO STATIONS
Order	C	for d = 0.03 km			for d = 0.1 km			for d = 0.3 km
cm	pm	ratio	cm	pm	ratio	cm	pm	ratio	cm	pm	ratio
1	2	0.5	153	1/6500	0.6	60	1/16700	1.0	33	1/30000
2	5	1.2	383	1/2600	1.5	150	1/6700	2.5	83	1/12000
3	12	2.8	920	1/1100	3.6	360	1/2800	6.0	200	1/5000
4	30	6.9	2300	1/430	9.0	900	1/1100	15.0	500	1/2000

SEMI-MAJOR AXIS OF 95% CONFIDENCE REGION, r = C(d+0.2); WHERE d IS THE DISTANCE BETWEEN ANY TWO STATIONS
Order	C	for d = 1.0 km			for d = 3.0 km			for d = 10 km
		cm	pm	ratio	cm	pm	ratio	cm	pm	ratio
1	2	2.4	24	1/41700	6.4	21	1/46900	20	20	1/50000
2	5	6.0	60	1/16700	16.0	53	1/18800	50	50	1/20000
3	12	14.4	144	1/6900	38.4	128	1/7800	120	120	1/8300
4	30	36.0	360	1/2800	96.0	320	1/3100	300	300	1/3300

The best course of action is to simulate the proposed network in a suitable computer program such as GALS* using a priori estimates for the standard deviations of the proposed measurements (see Appendix E). The results of such a simulation study, tempered with the wisdom of practical experience, usually provide a reliable indication of the accuracy likely to be obtained in the field.

For those not able to conduct computer simulation studies, some aids are provided in this publication:

• Appendix C provides measurement guidelines for the conventional methods--triangulation, traversing and trilateration--based on practical experience, and the results of computer simulation studies of simple idealized networks. At best, these guidelines are a general guide only and must be treated with caution. The reader should pay particular attention to the characteristics of the idealized networks depicted therein, to determine whether extrapolation can reasonably be made from the guidelines to the task at hand. The reader should also pay attention to the explanatory notes which follow the network sketches in Appendix C.
• Appendix D demonstrates some simple calculations that can be of benefit in estimating the accuracy of points in a network.
• Appendix E lists typical standard deviations, stemming from practical experience, for distances, directions, azimuths and position differences measured using various instruments and methods of observation.

*A Geodetic Survey computer program.

PHOTOGRAMMETRIC METHODS

On occasion, horizontal control can be densified effectively using photogrammetric methods (see Appendix F).

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PART 3--SURVEY MARKERS

TERMINOLOGY

Usually, objects marking control survey stations are called monuments or markers. A study of the terminology indicates that survey marker is the most common term used in North America to identify stations that may have any or all of the following properties; prominence, permanence, stability and definite location. Since the practice of having two names for the same thing can cause ambiguities, it is recommended that the term marker supplant the term monument in all future references to control survey stations. However, bench mark, a distinctive, well-understood term reserved for vertical control markers, may continue to be used.

RECOMMENDATIONS

Table III lists types of survey markers for various types and conditions of terrain. The type of marker best suited for a given type or condition of terrain depends on factors such as local conditions, transportation, materials available, equipment available and cost. For these reasons, the following statements can only be considered as recommendations.

• Several types of markers may be suitable for a particular situation;in such cases, the recommended markers are listed in order of preference.
• Several of the markers listed are illustrated with variable dimensions such as the length of the Survey Pipe with Base (Type 3) and the length and diameter of the Cylindrical Concrete Survey Markers (Types 6, 8 and 9). In cases where a range has been given for the variation, surveyors should choose dimensions to suit the particular situation and local conditions.
• Because of frost action on rough concrete, it is recommended that forms be used for all concrete markers. Manufactured cylindrical forms in various diameters and lengths are readily available and serve the purpose well.
• It is also recommended that concrete should not be used for vertical control markers because of frost action on the rough surfaces, even when forms are used.

The choice of the Tablet Marker (Type 1) or the Bolt Marker (Type 2) is optional. Because the tablet is larger than the bolt, it can be found more easily, but it is more liable to destruction in settled areas.

TABLE III
SURVEY MARKER RECOMMENDATIONS

(Markers listed left to right in order of preference)

TYPE OF CONDITION OR TERRAIN	ORDER OF SURVEY	RECOMMENDED TYPE OF MARKER (see Figures 3 to 14)
		VERTICAL	HORIZONTAL	HORIZONTAL REFERENCE MARKS
Bedrock, rock outcrops, large boulders, concrete structures	All	1 or 2	1 or 2	1 or 2
Granular soils (sand and gravel)	1st 2nd Lower	3,101 3,8,9,101 4,3,8,9	6,101 3,5,8,9,101 4,3,8,9	3,8,9
Till (glaciated cohesive soils)	1st 2nd Lower	3,101 3,8,9,101 4,3	6,101 3,8,9,101 4,3	3,8,9,4
Fine-grained soils (silts and clays of low plasticity)	1st 2nd Lower	12,3,101 3,101 3,4,5	6,101 3,8,9,101 5,4	3,5,4,8,9
Low bearing strength soils (very fine silts and clays of medium to high plasticity)	1st 2nd Lower	12,42,5 3,42,5 3,42,5	5,6 3,5 4,5	3,5,4,8,9
Muskeg, bog, swamp (highly organic wet soils)	1st 2nd Lower	12,42,5 42,5 5,42	5,42 5,42 5,42	5,4
Shallow overburden	1st 2nd Lower	43 43 43	7,11 7,43 43	43
Weathered rock, soft rock (shale)	1st 2nd Lower	43 43 43	43 43 43	43
Construction fill	1st 2nd Lower	12 4,5 4,5	6 5,8,9,4 4	4,5
Permafrost	1st 2nd Lower	12 12,4 4	4 4 4	4

Notes:
¹ The Sleeve Type Survey Marker (Type 10) is designed for use on municipal control surveys where one marker would be suitable for both vertical and horizontal control. Because material and installation costs are high for this type of marker, consideration should be given to two separate markers. By doing this, better locations can be selected for the vertical control markers, which don't have to be intervisible.
² If a longer rod is necessary for stability, 16 mm. copper covered steel rod in 3 m. sections (such as that normally used forelectrical grounding) can be driven to any depth.
³ Set in drilled hole

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APPENDIX A

THE CONCEPT OF CONFIDENCE REGION

INTRODUCTION

The concept of confidence region was not used before 1973 in Canada for control survey specifications, and may be unfamiliar to many users of these specifications. For this reason, the following pages are devoted to an explanation of the concept.

No attempt is made to provide rigorous definitions, nor to show the mathematical derivation of confidence regions, nor to explain the statistical tests referred to. Rather, these notes are intended to convey general principles on which the specifications are based.

STANDARD DEVIATION

Standard deviation is also referred to as standard error or root mean square error. Standard deviation squared, σ², is referred to as variance. In this publication, the computed mean and standard deviation of a group of observations of a physical quantity are denoted by m and s respectively, and the theoretical or true values are denoted by µ and σ.

Standard deviation is a statistical measure of accuracy. It is a measure of the dispersion of the observations of a quantity such as an angle or distance from the mean of the group of observations. The standard deviation, s, of a group of observations can be computed using the formula:

s =

where: n is the number of observations, x_i is the i^th observation, and m is the mean. This formula gives an estimate of the theoretical standard deviation.

If observational errors of a physical quantity are assumed to be random and distributed according to the normal distribution with mean µ and standard deviation σ, then we can expect 68 percent of the observations to lie within one standard deviation of the mean, and 95 percent within two standard deviations. In other words, the probability that an observation falls within one standard deviation of the theoretical mean is 0.68, and the probability that it fails within two standard deviations is 0.95. Other probability values are given in tables of the standard normal distribution found in books on statistics. Another commonly used value is 2.6σ where the range -2.6σ to +2.6σ represents a probability of 0.99. Figure A-1 represents the normal distribution.

Figure A-1
Normal Distribution with Mean µ and Standard Deviation σ

It is useful to note that the standard deviation of the mean of n measurements is the standard deviation of a single measurement divided by , or σ_n = σ .

WEIGHTS

The weight p of an observation is, by definition, inversely proportional to the variance. That is, p = k/ σ², where σ is the standard deviation of the observation and k is called the variance factor, which can be any convenient number. The square root of the variance factor is equivalent to the standard deviation for an observation having a weight of 1. In a least squares adjustment, it is important that individual variances (and hence weights) be in proper proportion. This means that reasonably good estimates of the standard deviations for observations must be made. When the a priori standard deviations (i.e., the standard deviations estimated before adjustment) adequately represent the physical accuracy of measurement, the standard deviation of unit weight computed from an adjustment is close to .

CONFIDENCE INTERVAL

This is defined as an interval within which we have a specified degree of confidence (expressed as a percentage) that an actual value lies. For example, if we know the theoretical mean µ and standard deviation σ, the 95 percent confidence interval for a random measurement is (µ - 2σ, µ + 2σ), i.e., from µ - 2σ to µ + 2σ. For the mean of n measurements a is replaced by σ/ . The idea is usually reversed, because we want to know the 95 percent (for example) confidence interval for the mean, m, of a set of n observations. In this case, the probability (or degree of confidence) is 0.95 that the interval (m - 2σ/ , m + 2σ/ ) contains µ, the theoretical mean or true value. When σ, the theoretical value, is not known, we compute an estimate s, and then the confidence interval is based on what is known as the t distribution. The confidence interval is not developed or used in these specifications, but serves as a general introduction to the concept of confidence region.

CONFIDENCE REGION

This is defined as a region within which we have a specified degree of confidence (expressed as a percentage) that an actual value lies. For normally distributed observations in two dimensions, a confidence region is bounded by an ellipse.

Hence, assuming normally distributed observations, a confidence region indicating the accuracy of horizontal control survey coordinates is bounded by an ellipse.

The ellipse that is sometimes referred to as the standard ellipse (or standard error ellipse) is based on the standard deviation of unit weight and the network configuration. It is computed in a least squares adjustment from the inverse matrix of the normal equations.

The standard ellipse bounds a confidence region of from 30 to 39 percent, depending on the number of redundant measurements (or degrees of freedom) in the adjustment. Thus for a survey network with 8 degrees of freedom, the probability is 0.38 that the true coordinate values for a point lie within the standard ellipse centred about the position determined by a least squares adjustment. The confidence region for any point can be computed relative to any other point in the adjustment.

The size and orientation of a standard ellipse depend on the design of the survey network and on the standard deviation of unit weight used to establish the scale of the ellipse. The design consists not only of the configuration of the survey network but also the various measurements with appropriate weights. Some suggestions for survey network design are made under a separate heading in Appendix C.

It is very important that weights in a least squares adjustment be in proper proportion to each other. Otherwise results including standard ellipses will be inaccurate. This means that good a priori estimates of the standard deviations for the various observations should be available. Usually, experience together with inspection and checking of the observations will produce good estimates (see Appendix E for examples of standard deviations).

Nevertheless, the standard deviation of unit weight:

,

(where p is weight, v is residual, f is degrees of freedom), as computed from the unconstrained adjustment, should be tested against the a priori standard deviation of unit weight, , using the chi-square test. If the test indicates a significant difference, an effort should be made to arrive at better estimates of the weights. When the estimates are judged satisfactory, the a priori standard deviation of unit weight should be used to determine the standard ellipse. In effect this equates experience and careful estimates to infinite degrees of freedom. Where experience is lacking and good estimates of weights cannot be arrived at, it is advisable to use s, the computed standard deviation of unit weight; and it should be pointed out that the fewer the degrees of freedom, the less reliable is the value of s.

After a survey has been analyzed from the results of an unconstrained adjustment, it is often necessary to constrain it to two or more fixed positions. In this case, it is advisable to inspect changes in residuals resulting from the constraint and to determine the relative changes in positions. In computing standard ellipses for surveys under constraint, it is advisable to use the computed standard deviation for unit weight since it reflects the degree of distortion.

For classifying surveys, a standardized confidence region (that is, a fixed percentage) is used. For these specifications, the 95 percent confidence region has been selected. The 95 percent confidence region about an adjusted point represents the region within which the probability is 0.95 that the true coordinate position of the point lies relative to a selected point in the survey network. The 95 percent confidence region is an enlargement of the standard ellipse obtained by multiplying the axes of the latter by the appropriate factor from Table A-l. Assuming adequate experience and good a priori estimates of weights, the factor 2.45 corresponding to infinite degrees of freedom should be used. Where these conditions are not present, the factor from Table A-l corresponding to the actual degrees of freedom should be used. Table A-l is based on the statistical F distribution and lists factors for the 90 and 99 percent as well as for the 95 percent confidence regions.

TABLE A-l
FACTORS FOR CONFIDENCE REGIONS

F	C90	C95	C99
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 25 30 40 50 60 80 120 x	9.95 4.24 3.31 2.94 2.75 2.63 2.55 2.50 2.45 2.42 2.39 2.37 2.35 2.34 2.32 2.31 2.30 2.29 2.28 2.28 2.25 2.23 2.21 2.20 2.19 2.18 2.17 2.15	19.97 6.16 4.37 3.73 3.40 3.21 3.08 2.99 2.92 2.86 2.82 2.79 2.76 2.73 2.71 2.70 2.68 2.67 2.65 2.64 2.60 2.58 2.54 2.53 2.51 2.49 2.48 2.45	99 99 14.07 7.85 6.00 5.15 4.67 4.37 4.16 4.00 3.89 3.80 3.72 3.66 3.61 3.57 3.53 3.50 3.47 3.44 3.42 3.34 3.28 3.22 3.19 3.16 3.12 3.09 3.03

NOTES:

f = degrees of freedom in the adjustment

C₉₅ = factor by which axes of standard ellipse are to be multiplied to obtain 95 percent confidence region (derived from the F distribution; where f = degrees of freedom in the adjustment)

Summary of the Confidence Region Concept

• Where good estimates of standard deviations of observations are available, the a priori standard deviation of unit weight should be used to determine the standard ellipses for unconstrained adjustments, and the factor 2.45 for infinite degrees of freedom should be used from Table A-l for the 95 percent confidence regions.
• Where a network is distorted by constraint, the computed standard deviation for unit weight should be used to determine the standard ellipses, and the factor 2.45 may be used from Table A-l.
• Where good estimates of standard deviations of observations are not available, the computed standard deviation for unit weight should be used to determine the standard ellipses and the factor corresponding to the actual degrees of freedom should be used from Table A-l for the 95 percent confidence region.

EXAMPLE OF THE APPLICATION OF CONFIDENCE REGIONS>

To illustrate the application of confidence regions in the classification of a survey, a traverse loop of 10 sides as in Figure A-2 is considered.

Figure A-2
Traverse Loop of 10 Sides

At previously established station 1, an azimuth is obtained from another previously established point, and following procedures recommended for second-order traversing, angles are measured six times, and the lengths twice. If the closing angle is measured, there are three degrees of freedom, and if it is not measured, there are two degrees of freedom. This assumes that measurements of each side and angle are meaned.

Measurements that are independent may be counted as individual observations. In this example, the six observations of each angle are made at one instrument set-up and in a short time span, and are not considered to be independent. The same applies to the length measurement of each line; the two observations are consecutive, using the same instruments, with no time interval between, and no re-centering.

To begin with, the 95 percent confidence region may be computed for the unclosed traverse to check the actual misclosure against what is expected for that level of confidence. The 95 percent confidence region should enclose the starting position (with a probability of 0.95). If this is not the case, a blunder or bias should be suspected. It does not mean that a blunder or bias exists; it means that one or the other probably exists. Stated in other words: if the observations are normally distributed without bias or blunder, then the probability is low (0.05) that the true value (misclosure) falls outside the 95 percent confidence region. Other confidence levels such as the standard ellipse or the 90 or 99 percent may also be used in checking misclosures (see also Appendix D). Where a blunder or bias is indicated and a study of results does not disclose and correct the error, then of course re-observing should be considered.

Good estimates of the standard deviations of the measurements are usually available, and in this case the a priori standard deviation of unit weight should be used in determining the standard ellipse, and the factor 2.45 from Table A-l should be used to arrive at the 95 percent confidence region. However, if good estimates of the standard deviations of the measurements are not available, because of inexperience, malfunctioning equipment, etc., the standard deviation of unit weight computed from a least squares adjustment should be used, provided there is sufficient redundancy to give a reasonably reliable value.

In the case of the traverse of Figure A-2, with only two or three degrees of freedom, the computed standard deviation for unit weight would be quite unreliable. It would be better to use the a priori value for the standard deviation of unit weight to determine the standard ellipses, and the factor from Table A-l corresponding to the actual number of degrees of freedom to arrive at the 95 percent confidence regions.

The order of the survey is confirmed by checking the semi-major axis of each 95 percent confidence region against the second-order standard r = 5(d+0.2). The semi-major axes of the 95 percent confidence regions for station 6, for example, relative to all other stations must be less than the standard r = 5(d+0.2), where d is the distance in kilometres between station 6 and each of the other points.

After one has acquired some experience in judging where weakness lies in survey configurations, only confidence regions between selected stations need be checked. Care should be taken to ensure that neighbouring stations, particularly those not directly connected by measurements, meet the criterion. In addition, residuals of the adjusted network should be inspected to detect gross errors and because confidence regions do not reveal small pockets of distortion.

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APPENDIX B

COMPARISON OF 1961 AND 1978 HORIZONTAL CONTROL SPECIFICATIONS

This Appendix compares the principal specification requirements used in the texts of the 1961 and 1978 editions of the Horizontal Control Specifications. Its purpose is to emphasize that although the accuracy concept is different in the two documents, the classification of horizontal control is basically the same, (except for lines shorter than about three kilometres). Classification in both documents is based on the level of accuracy between any two stations after adjustment. The 1978 specifications require that the semi-major axis of the 95 percent confidence region of one station relative to any other station must be less than or equal to r= C(d+0.2).
The 1961 specifications state:

"Horizontal control shall be classified as first, second, third, or fourth-order without regard to the method of survey. Special instructions may be issued for work that does not fit into the standard classification. Any order of control may apply to triangulation, trilateration, or traverse, or to any combination of these, implying in all cases the same order of accuracy."

"Each order of control implies a certain standard of accuracy, as tabulated in Table B-l. The accuracy is expressed as the maximum anticipated error in the computed length or azimuth of the line joining any two points of the survey (after adjustment)."

"The term maximum anticipated error is used to indicate the greatest error that may reasonably be expected, taking into account the systematic as well as the accidental errors. While this error cannot be accurately computed, it is believed that it may be more meaningful than a previously calculated probable or standard error, in the computation of which some sources of error may be neglected" (Table B-l).

TABLE B-l
MAXIMUM ANTICIPATED ERRORS IN ADJUSTED HORIZONTAL CONTROL

HORIZONTAL CONTROL	LENGTH	AZIMUTH
First-order
Distance greater than 400 miles		4"
Distance less than 400 miles	1 in 50 000	4"
Second-Order	1 in 20 000	10"
Third-Order	1 in 10 000	20"
Fourth-Order	1 in 5 000	40"
*M is the distance in miles.

The italics used here, are to draw attention to some aspects of the 1961 specifications that are often overlooked. It is clearly stated that classification is without regard to the method of survey and is based on accuracy between any two points of the survey after adjustment. In other words, classification is to be based on the analysis of adjusted results and applies to any two points, which means that design must be considered. These principles are in complete accord with the 1978 specifications.

The 1961 specifications also include a table of "detailed requirements" for attaining the necessary accuracy to qualify the work for the "various orders of control" with the qualification: "these detailed requirements should not be taken too rigidly but should be regarded as examples of the class of work necessary to attain the specified accuracy". This same approach is followed in the 1978 specifications by the provision of measurement guidelines.

The 1961 specifications use maximum anticipated error as a criterion of accuracy whereas the 1978 specifications use a 95 percent confidence region. To compare classification standards of the 1961 and 1978 specifications, these criteria must be brought to some common reference. Since maximum anticipated error is defined as the greatest error that may reasonably be expected, it can be interpreted to be 2.5 times the standard deviation which, in the linear sense, represents 99 percent probability. The 95 percent confidence region may also be related to standard deviation since its axes are approximately 2.5 times longer than those of the standard error ellipse.

The 1978 specifications require the semi-major axis of the 95 percent confidence region to be less than or equal to a value representing 20 ppm for first-order, 50 ppm for second-order, 120 ppm for third-order, and 300 ppm for fourth-order. (It is assumed here that reasonably long lines are being dealt with and that the formula r = C (d+ 0.2) can be approximated by r = Cd). Both specifications refer to accuracy between any two points and these can be compared by reducing the accuracy to standard deviation in length and azimuth expressed as a proportion (four seconds of arc expressed as a proportional part is 1 part in 50 000 for example). The following Table B-ll compares the standard deviations between the 1961 and 1978 specifications (for first-order, M is taken as less than 400 miles).

First and second-orders are identical, and the 1978 specifications relax the standards for third and fourth-order.

TABLE B-ll
COMPARISON OF STANDARD DEVIATIONS

A comparison of the 1961 and 1978 guidelines for measurement specifications is of some interest. It is relatively simple to compare guidelines for the accuracy of length measurement. For lengths in triangulation (taking maximum anticipated error as equivalent to 2.5 times the standard deviation) we have the guidelines shown in Table B-lIl.

The 1961 specifications prescribe lengths more precise than the guidelines for the 1978 specifications. However, the 1961 specifications refer to the accuracy of short, taped baselines whereas the 1978 specifications refer to the accuracy of the main sides of the triangulation. In the older type of first-order triangulation and some second-order, the baselines were usually quite short compared to the average side of the main triangulation network. The high accuracy of the baseline was largely dissipated in the baseline network expansion, however, with the result that the final side would be only about one third to one fifth as accurate as the original baseline measurement. When allowances are made for this, the length accuracies prescribed in the 1961 and 1978 specifications appear comparable.

TABLE B-IV
TRAVERSE WORK ACCURACY GUIDELINES

The traverse length accuracy suggested in the 1961 specifications for third-order would require a fairly strong traverse grid to achieve the 1961 third-order adjusted accuracy of 1 in 25 000. It would meet the 1978 third-order standard of 1 in 21 000 with less difficulty. The suggested 1961 accuracy for fourth-order lengths is the same accuracy as the fourth-order position accuracy standard, namely a standard deviation of 1 in 12 500. This would be impossible to achieve in traverse work, but with a very strong grid, it could meet the 1978 fourth-order length accuracy standard of 1 in 8333.

It is difficult to compare traverse misclosure guidelines since they are stated in quite different terms. The 1961 allowable misclosures are given for azimuth and for positional misclosure in proportional parts after azimuth adjustment. The 1978 specifications advise that the 95 percent confidence region of the closing position should enclose the starting position, otherwise, a blunder should be suspected. Tests on simulated and actual traverses show that the 1978 specifications are realistic.

As to trilateration, the authors of the 1961 specifications admitted a lack of knowledge and experience. The 1978 specifications are based on more experience and on simulation and analysis. Closer azimuth spacing and less length redundancy per station are suggested.

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APPENDIX C

MEASUREMENT AND CHECK GUIDELINES FOR CONVENTIONAL METHODS

INTRODUCTION

The guidelines given in Table C-1 are based on practical experience and on the results of computer simulation studies of simple, idealized networks depicted in Figures C-1 to C-6 following. At best they are a genera/ guide only and must be used with caution.

TRIANGULATION

First-Order--single chain; average length 20 km; all angles and sides measured; 10 courses between azimuth control (see Figure C-1).

Figure C-1

Triangulation, Single-Chain Network--First-Order

Second, Third and Fourth-Order -- single chain; average lengths 15 km, 10 km and 5 km respectively; all angles measured; 1 side measured every fourth triangle; 12, 15 and 20 coursesrespectively between azimuth controls (see Figure C-2).

TABLE C-I
MEASUREMENT AND INTERNAL CHECK GUIDELINES

(for long lines only)

CHARACTERISTICS			ORDERS
			1st	2nd	3rd	4th
M E A S U R E M E N T S	DIRECTIONS	Least count of theodolite	0.2"	1.0"	1.0"	10.0"
		Minimum number of sets	24 (4x6)	6	2	2
		Standard deviation (mean of all sets)	0.7"	2.0"	4.0"	8.0"
	LENGTHS	Standard deviation (complete determination)
		* Triangulation and Traverse	4 ppm	10 ppm	20 ppm	40 ppm
		* Trilateration	3 ppm	5 ppm	10 ppm	20 ppm
	AZIMUTHS	Minimum number of sets	32	12	4	4
		Standard deviation (mean of all sets)	0.4"	1.5"	3.0"	6.0"
		Maximum number of courses between control azimuths
		* Triangulation	10	12	15	20
		* Trilateration	6	8	10	12
		* Traverse	5	6	9	12
C H E C K S	DIRECTIONS	Standard deviation of a triangle misclosure	1.7"	5.0"	10.0"	20.0"
	DIRECTIONS VS. LENGTHS	Consistency checks	See explanatory notes on page C-4.
	AZIMUTHS	Maximum difference between control azimuths	2"	5"	10"	20"

Figure C-2
Triangulation, Single-Chain Network--Second to Fourth-Order

Second,Third and Fourth-Order -- cross-braced quadrilaterals; average lengths 15 km, 10 km and 5 km respectively; all angles measured; one length measured every second quadrilateral; 12, 15 and 20 courses respectively between azimuth controls (see Figure C-3).

Figure C-3
Triangulation, Cross-braced Quadrilateral Network-- Second to Fourth-Order

TRILATERATION

First, Second, Third and Fourth-Orders-- patterns shown in Figures C4 and C-5; one redundant line per point; average lengths 20 km, 15 km, 10 km and 5 km respectively; 6, 8, 10 and 12 courses respectively between azimuth controls.

Figure C-4
Trilateration Network--First to Fourth-Order

Figure C-5
Trilateration Network--First to Fourth-Order

TRAVERSING

First, Second, Third, and Fourth-Order -- simple, reasonably straight traverses as shown in Figure C-6 average lengths 20 km, 15 km, 10 km and 5 km respectively; 5, 6, 9 and 12 courses respectively between azimuth controls.

Figure C-6
Traverse--First to Fourth-Order

EXPLANATORY NOTES ON MEASUREMENTS

Directions

A set (formerly called a position) consists of a series of pointings of the theodolite such that:

• the setting of the horizontal circle remains constant;
• the pointings are made successively in clockwise or anti-clockwise order on one face followed by corresponding successive pointings on the opposite face in reverse order.

For first-order work, sets are observed in four groups of six sets each (4x6), spread over at least two days. Starting times for successive groups must be separated by at least two hours and observation times on different days should be staggered.

For all orders of work, circle and micrometer settings must be varied to minimize the effects of graduation errors.

Lengths

The value given is the standard deviation in ppm of a complete length determination using EDM equipment (examples for various instruments and methods are given in Appendix E). For each order a complete determination will consist of one or more "double" measurements where a double measurement is:

For microwave equipment -- the mean of two measurements, each comprising one set of coarse readings and a sufficient number of sets of fine readings for ground swing to be properly determined. For the two measurements, each instrument is to be employed alternately as master and remote.

For electro-optical equipment -- the mean of two measurements, each of which must consist of sets of readings in all phase positions (including any internal reference-path readings, when applicable) and readings on all available modulation frequencies. The observing procedures must be such that all length ambiguities are resolved. For some instruments an offset reflector bar may prove useful for this purpose.

*First-Order

A complete length determination will be the mean of at least three double measurements. For microwave equipment these double measurements must be made at least four hours apart. At least one set of meteorological readings is to be taken at each end of the line before and after each measurement. In making microwave measurements, particular attention should be paid to wet and dry bulb thermometer readings which should be taken in the wind, in the shade and preferably well away (two metres or more) from the ground. Under certain conditions an error of 1°C in determining the wet bulb depression can produce a corresponding error in the distance of up to 10 ppm. For electro-optical measurements, dry bulb temperature readings are important: an error of 1°C can produce an equivalent distance error of about two ppm. Crystal frequency calibrating should be done at least once very two weeks during continuous field operations.

* Second-Order

A complete length determination will consist of at least one double measurement accompanied by at least one set of meteorological readings at each end of the line before and after each double measurement.

* Third and Fourth-Orders

At least one double measurement similar to that mentioned in the second-order above should still be made. This is required not for accuracy but as a safeguard against blunders.

Azimuths

It is assumed that the Polaris-at-any-hour-angle method will be used for astronomic azimuth determinations (except in the far North--above about latitude 65°). Based on this assumption, a set will comprise successive pointings on the reference object and Polaris on one face followed by successive pointings in reverse order on the other face. For surveys involving first-order and second-order work, the use of a stride level is essential. In third and fourth-order work, a stride level is not required but careful centering of the plate level is very important, particularly in high latitudes.

In all cases, circle and micrometer settings should be varied for the various sets, to minimize the effects of graduation errors.

All first-order azimuth and longitude observations must be spread over at least two nights. Reciprocal azimuth determinations are very desirable in first-order work.

The standard deviation (Table C-l) is a measure of the internal precision of the work only. The standard deviations of azimuth determinations (external) in first and second-order surveys are about twice the internal values.

In all first-order work, accompanying astronomic longitude determinations must also be made. These should also be made in second-order work when the prime vertical component of the plumbline deflection is likely to be large. In all these cases (Laplace stations), the standard deviation value quoted in Table C-l is also the criterion for a complete longitude determination.

The value given is the maximum number of courses permitted along the most direct route between azimuth control points. Azimuth control may consist of a line in a previously established and adjusted survey network of equal or higher order.

For traverses, the value applies only to those that are simple and reasonably straight (see Figure C-6). Interlocking traverse grids are a different matter and a proper assessment of the need for azimuth control in these can only be made by a computer simulation study of the whole network.

EXPLANATORY NOTES FOR CHECKS

Directions vs Lengths

For first and second-order triangulation consisting of single-chain networks in which all angles and sides are measured, the consistency of measured lengths may be checked against measured angles using the sine-equation test. Before this test can be applied, all sea-level lengths must be corrected by means of the arc-sine correction so that the sine-law relationship is still valid when the ellipsoidal (observed) angles are used. Tables C-ll and C-III list the corrections for various line lengths in metres and feet respectively.

The sine-equation test compares the value of (a sinB - b sinA), for a particular triangle, with the value cr. where:

σ = α . sinB. 1O^-5

and σ_a, σ_b are the standard deviations of the lengths a and b in ppm, and σ_A, σ_B are the standard deviations of the angles A and B in seconds. Usually a priori values based on past experience are used (see Appendix E).

The result should be σ in 68 percent of the cases and 2σ in 95 percent of the cases.

Table C-IV gives values of:

For various values of A and B. assuming σ_A = σ_B = 1.0" and σ_a = σ_b = 4 ppm (first-order criteria for long lines) while Table C-V gives values of this function for various values of A and B. assuming σ_A = σ_B = 2.85" and σ_a = σ_b = 10 ppm (second-order criteria for long lines).

For first and second-order trilateration, the consistency of measured lengths is checked using least squares electronic computer programs because the computations are too elaborate for hand computing in the field.

Note:

The usual side-equation test has been omitted. Most modern first and second-order triangulation networks feature numerous measured lengths making the side-equation test unnecessary.

Azimuths

The expression given is the maximum permissible difference between a control azimuth and the azimuth carried through the network from previous control using the observed angles and following the most direct route. It is based on 95 percent confidence limits derived from external accuracy estimates for direction and azimuth observations. N is the number of angles.

TABLE C-ll
ARC-SINE CORRECTIONS (METRES)

LENGTH (km)	CORRECTION (metres)	LENGTH (km)	CORRECTION (metres)	LENGTH (km)	CORRECTION (metres)
7	0.001	26	0.072	45	0.372
8	.002	27	.080	46	.398
9	.003	2	.090	47	.424
10	.004	29	.100	48	.452
11	.005	30	.110	49	.481
12	.007	31	.122	50	.511
13	.009	32	.134	51	.542
14	.011	33	.147	52	.575
15	.014	34	.161	53	.609
16	.017	35	.175	54	.644
17	.020	36	.191	55	.680
18	.024	37	.207	56	.718
19	.028	38	.224	57	.757
20	.033	39	.242	58	.798
21	.038	40	.262	59	.840
22	.044	41	.282	60	.883
23	.050	42	.303
24	.057	43	.325
25	.064	44	.348
The correction is always subtracted from the sea-level length.

TABLE C-lIl
ARC-SINE CORRECTIONS (FEET)

LENGTH (1000's feet)	CORRECTION (FEET)	LENGTH (1000's feet)	CORRECTION (FEET)	LENGTH (1000's feet)	CORRECTION (FEET)
20	0.003	100	0.380	180	2.217
25	0.006	105	0.440	185	2.407
30	0.010	110	0.506	190	2.608
35	0.016	115	0.578	195	2.819
40	0.024	120	0.657	200	3.042
45	0.035	125	0.743
50	0.048	130	0.835
55	0.063	135	0.935
60	0.082	140	1.043
65	0.104	145	1.159
70	0.130	150	1.283
75	0.160	155	1.416
80	0.195	160	1.557
85	0.233	165	1.708
90	0.277	170	1.868
95	0.326	175	2.038
NOTE: The correction is always subtracted from thesea-level length.

TABLE C-lV
SINE EQUATION TEST (FIRST-ORDER)

Angle A or B	Angle A or B
	05	10	15	20	25	30	35	40	45
10	6.21	3.93
15	5.86	3.34	2.62
20	5.73	3.11	2.32	1.97
25	5.67	2.99	2.16	1.78	1.58
30	5.63	2.93	2.07	1.67	1.45	1.32
35	5.61	2.89	2.02	1.60	1.37	1.23	1.13
40	5.60	2.87	1.98	1.56	1.32	1.17	1.06	.99
45	5.59	2.85	1.96	1.53	1.28	1.12	1.02	.94	.89
50	5.59	2.84	1.94	1.50	1.25	1.09	.98	.91	.85
55	5.58	2.83	1.93	1.49	1.23	1.07	.96	.88	.82
60	5.58	2.82	1.92	1.47	1.22	1.05	.94	.86	.80
65	5.57	2.82	1.91	1.46	1.21	1.04	.92	.84	.78
70	5.57	2.81	1.90	1.46	1.20	1.03	.91	.83	.77
75	5.57	2.81	1.90	1.45	1.19	1.02	.90	.82	.76
80	5.57	2.81	1.90	1.45	1.19	1.02	.90	.81	.75
85	5.57	2.81	1.90	1.45	1.18	1.01	.90	.81	.75
90	5.57	2.81	1.90	1.45	1.18	1.01	.89	.81	.75
95	5.57	2.81	1.90	1.45	1.18	1.01	.90	.81	.75
100	5.57	2.81	1.90	1.45	1.19	1.02	.90	.81	.75
105	5.57	2.81	1.90	1.45	1.19	1.02	.90	.82	.76
110	5.57	2.81	1.90	1.46	1.20	1.03	.91	.83	.77
115	5.57	2.82	1.91	1.46	1.21	1.04	.92	.84	.78
120	5.58	2.82	1.92	1.47	1.22	1.05	.94	.86	.80
125	5.58	2.83	1.93	1.49	1.23	1.07	.96	.88	.82
130	5.59	2.84	1.94	1.50	1.25	1.09	.98	.91	.85
135	5.59	2.85	1.96	1.53	1.28	1.12	1.02	.94	.89
140	5.60	2.87	1.98	1.56	1.32	1.17	1.06	.99
145	5.61	2.89	2.02	1.60	1.37	1.23	1.13
150	5.63	2.93	2.07	1.67	1.45	1.32
155	5.67	2.99	2.16	1.78	1.58
160	5.73	3.11	2.32	1.97
165	5.86	3.34	2.62
170	6.21	3.93
e.g..For A = 30 degrees and B = 40 degrees, table gives

Angle A or B	Angle A or B
	50	55	60	65	70	75	80	85	90
10
15
20
25
30
35
40
45
50	.81
55	.78	.74
60	.75	.72	.69
65	.73	.70	.67	.65
70	.72	.68	.66	.63	.62
75	.71	.67	.64	.62	.61	.59
80	.70	.67	.64	.62	.69	.59	.58
85	.70	.66	.63	.61	.59	.58	.57	.57
90	.70	.66	.63	.61	.59	.58	.57	.57	.57
95	.70	.66	.63	.61	.59	.58	.57	.57
100	.70	.67	.64	.62	.60	.59	.58
105	.71	.67	.64	.62	.61	.59
110	.72	.68	.66	.63	.62
115	.73	.70	.67	.65
120	.75	.72	.69
125	.78	.74
130	.81
135
140
145
150
155
160
165
170
e.g.. For A = 30 degrees and B = 40 degrees, table gives 1.17

TABLE C-V
SINE EQUATION TEST (SECOND-ORDER)

Angle A or B	Angle A or B
05	10	15	20	25	30	35	40	45
10	17.69	11.17
15	16.67	9.49	7.43
20	16.30	8.82	6.56	5.55
25	16.13	8.50	6.11	5.02	4.42
30	16.04	8.31	5.86	4.71	4.06	3.67
35	15.98	8.20	5.70	4.51	3.83	3.41	3.13
40	15.94	8.13	5.59	4.37	3.67	3.23	2.93	2.72
45	15.92	8.08	5.52	4.28	3.56	3.10	2.79	2.57	2.41
50	15.90	8.05	5.47	4.21	3.48	3.01	2.69	2.46	2.29
55	15.89	8.02	5.43	4.17	3.42	2.94	2.61	2.38	2.20
60	15.88	8.00	5.41	4.13	3.38	2.89	2.56	2.31	2.13
65	15.87	7.99	5.39	4.10	3.35	2.85	2.51	2.26	2.08
70	15.86	7.98	5.37	4.08	3.32	2.82	2.48	2.23	2.04
75	15.86	7.97	5.36	4.07	3.30	2.80	2.46	2.20	2.01
80	15.86	7.97	5.35	4.06	3.29	2.79	2.44	2.18	1.99
85	15.86	7.96	5.35	4.05	3.29	2.78	2.43	2.17	1.98
90	15.86	7.96	5.35	4.05	3.28	2.78	2.43	2.17	1.98
95	15.86	7.96	5.35	4.05	3.29	2.78	2.43	2.17	1.98
100	15.86	7.97	5.35	4.06	3.29	2.79	2.44	2.18	1.99
105	15.86	7.97	5.36	4.07	3.30	2.80	2.46	2.20	2.01
110	15.86	7.98	5.37	4.08	3.32	2.82	2.48	2.23	2.04
115	15.87	7.99	5.39	4.10	3.35	2.85	2.51	2.26	2.08
120	15.88	8.00	5.41	4.13	3.38	2.89	2.56	2.31	2.13
125	15.89	8.02	5.43	4.17	3.42	2.94	2.61	2.38	2.20
130	15.90	8.05	5.47	4.21	3.48	3.01	2.69	2.46	2.29
135	15.92	8.08	5.52	4.28	3.56	3.10	2.79	2.57	2.41
140	15.94	8.13	5.59	4.37	3.67	3.23	2.93	2.72
145	15.98	8.20	5.70	4.51	3.83	3.41	3.13
150	16.04	8.31	5.86	4.71	4.06	3.67
155	16.13	8.50	6.11	5.02	4.42
160	16.30	8.82	6.56	5.55
165	16.67	9.49	7.43
170	17.69	11.17
e.g.. For A = 30 degrees and B = 40 degrees, table gives 3.23

Angle A or B	Angle A or B
	50	55	60	65	70	75	80	85	90
10
15
20
25
30
35
40
45
50	2.17
55	2.07	1.97
60	2.00	1.89	1.81
65	1.94	1.83	1.75	1.68
70	1.90	1.79	1.70	1.63	1.58
75	1.87	1.75	1.67	1.60	1.55	1.51
80	1.84	1.73	1.64	1.57	1.52	1.48	1.46
85	1.83	1.72	1.63	1.56	1.51	1.47	1.44	1.42
90	1.83	1.71	1.62	1.55	1.50	1.46	1.44	1.42	1.41
95	1.83	1.72	1.63	1.56	1.51	1.47	1.44	1.42
100	1.84	1.73	1.64	1.57	1.52	1.48	1.46
105	1.87	1.75	1.67	1.60	1.55	1.51
110	1.90	1.79	1.70	1.63	1.58
115	1.94	1.83	1.75	1.68
120	2.00	1.89	1.81
125	2.07	1.97
130	2.17
135
140
145
150
155
160
165
170
e.g.. For A = 30 degrees and B = 40 degrees, table gives 3.23

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APPENDIX D SIMPLE COMPUTATIONS AS AN AID IN NETWORK DESIGN AND ANALYSIS

Usually, it is only on large projects that a rigorous design analysis by computer program is feasible; for many surveys a rigorous mathematical design is neither feasible nor necessary. Figures D-1 and D-2 illustrate examples of simple survey ties which can be analyzed easily and accurately.

Figure D-1
Example of Simple Survey Tie

In Figure D-1, if the azimuth a were to be measured with a technique having a standard deviation of 5", the 95 percent confidence region of the position of B relative to A would have a semi-axis perpendicular to AB of:

2.5 sin 5" x 2000 m = 0.121 metre.

If the distance AB was to be measured with a technique having a standard deviation of (+ 1 cm + 3 ppm), the 95 percent confidence region of the position of B relative to the position of A would have a semi-axis in the direction of AB of:

2.5 x √(0.01² +(3 x 2000 x 10^-6)²) = 0.029 metre.

Therefore, the semi-major axis of the 95 percent confidence region would be 0.121 metre for the difference in position between A and B. From Figure 2, point B would be classed as third-order relative to point A. The reliability of this tie is very low because there is no redundancy that would isolate or at least indicate the presence of a blunder. Without such a check, the tie could not be considered adequate for control purposes.

Figure D-2
Example of Simple Survey Tie

In Figure D-2 if point C was being positioned from A and B by measured lengths each having a standard deviation

σ = ,

this would give:

σ_AC = 0 005 m, and σ_BC= 0.005 m.

An approximate formula for the semi-major axis of the 95 percent confidence region of a point positioned by two measured distances separated by an angle C is:

r = 2.5 cosec C:

In this example, r = 0.036 metre. From Figure 2 (for a distance of 250 metres), the positional accuracy of point C would be classed as third-order.

As in the previous example, the reliability of such a tie would be quite unsatisfactory because of the lack of any redundancy. A third measurement of either direction or distance would be a minimum requirement.

Figure D-3 illustrates a more complicated survey in which some of the practical considerations in designing survey networks to meet a specific order classification can be examined. A and B are higher order points and a fourth-order traverse C. . .L is proposed to be Sun with ties to A from C and G. Clearly there are some weaknesses in the net. The position differences HB and JE are weakly determined, and the orientation of the net, being wholly dependent on the sighting AB, has no check against a blunder in that sighting. The line FG is so short that special precautions, such as accurate forced centering at F and G. are needed to preserve directional accuracy.

Figure D-3
Example of a More Complicated Survey

Because the net has so many elements, simplifying approximations must be made to permit any approximate analysis. For such a net, experience is necessary both to estimate measurement accuracies reasonably attainable under field conditions and to make approximations of the survey structure. Such estimates must stem from rigorous computer analysis using actual measured data on real survey projects.

To estimate roughly the accuracies of H with respect to B. and J with respect to E, suppose that all angles were measured with a standard deviation of 8" and that all distances were measured with an instrument having an accuracy of:

σ = metres.

The standard deviation of the distance AGH, σ_AH can be approximated by simple combination of the standard deviations of the two parts:

i.e., σ_AH = ,

where σ_AG = = 0.092 metre,

and σ_GH = 0.032 metre.

Thus σ_AH = = 0.097 metre.

An estimate of the standard deviation of the position of H relative to A in the direction perpendicular to AH could be computed from the standard deviation of the angle at A and distance AH, i.e.,

11 500 sin 8" = 0.446 metre.

The actual value would be slightly larger because the error in the angle at G was neglected. On the other hand, the traverses through CDE, etc., and CLK will add strength. A reasonable estimate would be 0.5 metre. Since the distance AB forms part of a higher order survey the error it contributes to the closing course HB is negligible. Thus, the semi-major axis of the 95 percent confidence region for the position difference HB is 2.5 x 0.5 = 1.25 metres in a distance of 3.2 km. A rigorous analysis gave 1.02 metres; from Figure 2. This is exactly the fourth-order accuracy limit.

To estimate the accuracy limit of the difference of positions E and J. a more sophisticated analysis is necessary, but one that can still be done quickly with a slide rule or hand held calculator. The basic analysis for any leg of a traverse from station i to station j with bearing β_ij and length L_ij km can be derived using three simple formulae. The standard deviation of the northing of the leg, σ_DNij can be derived from:

where σ_bij is the standard deviation of the angle between the bearing β and the previous leg, and σ_Lij is the standard deviation of the measured length L_ij.

The standard deviation of the easting of the leg, σ_DEij can be derived from:

σ²_DEij = sin² β_ijσ_Lij + cos² β_ij σ_bij (cm²).

and a term σ_DNDEij that takes into account errors in other directions can be found from:

σ_DNDEij = sin β_ij cos β_ij (cm²)

The value of k depends on the units used to express the various quantities. For L in km, σ_L in cm, σ_β in seconds of arc, k = 2.062648.

These three terms can be added for successive legs of a traverse, to give the accuracy of the terminal relative to the initial point. Thus for a three-legged traverse through stations 1, 2, 3, 4:

σ²_ΔN14 = σ²_ΔN12 + σ²_ΔN23 + σ²_ΔN34

σ²_ΔE14 = σ²_ΔE12 + σ²_ΔE23 + σ²_ΔE34

σ²_DNTΔE14= σ²_DNΔE12+ σ²_DNΔE23 + σ²_DNΔE34

The semi-major axis of the standard error ellipse, σ_max, of point n relative to point i can be derived by computing the square root of the larger of the two roots of the quadratic equation:

aλ²+bλ+ c = 0.
The roots are given by λ =

where:

a = 1,

b = -- (σ² _DNin + σ² _DEin),

c = σ²_DNin σ²_DEin - σ²_DNDEin

σ_{max =}

Considering now the network in Figure D-3, the most direct survey route from E to J is through the traverse E D C L K J. We have postulated angles measured with a standard deviation of 8", i.e.: σ_β = 8". We can derive the standard deviation of each leg and analyze the traverse from E to J using the preceding formula (see Table D-l for the various values).

TABLE D-l
VARIANCES FOR TRAVERSE E-D-C-L-K-J

Log	Scaled L (km)	Derivedσ2L (cm2)	ScaledBearingβ	σ2DN(cm2)	σ2DE (cm2)	σ2DNDE (cm2)
E D	1.8	7.24	63°	40.2	15.8	-16.8
D C	1.8	7.24	125°	35.1	20.9	+19.5
C L	2.2	8.84	148°	26.8	54.8	+28.8
L K	2.8	11.84	256°	111.7	18.0	-24.9
K J	2.3	9.29	316°	43.2	45.7	+35.1
E J = σ	10.9			257.0	155.2	+41.7

From this:

λ²-(257.0 + 155.2) λ + (257.0 x 155.2-41.7 x 41.7) = 0.

The larger root λ = 271.9, and σ_max = 16.5 cm. This only takes into consideration the survey data by the most direct route from E to J. There is also the route through F G H I. If we assume a σ_max of 16 cm for that route, a good estimate of the semi-major axis of the 95 percent confidence region resulting from the consideration of both routes will be 2.5 x 16/ = 28 cm. From Figure 2 for the one-kilometre distance between E and J. this is within the fourth-order accuracy limit. A rigorous analysis of this net gave 25 cm for this line and this is well within the fourth-order accuracy limit. Thus, this approximate analysis of the network gives a value which is reasonably good and indicates that the line meets fourth-order standards.

Clearly, the best method to strengthen the network would be to measure the lines HB and EJ. These measurements would probably bring the network within third-order specifications. This would also be a check for any blunder in the sighting AB. However, if for some reason these distances could not be measured, a substantial improvement in the accuracy of the measured angles would strengthen the network, but to a lesser degree. The unreliability of the orientation of the net could be improved by measuring angles BAG and BAC independently at different times with different instruments.

CONFIDENCE LEVEL OF MISCLOSURE

The computation procedure outlined for analyzing traverses facilitates the analysis of traverse misclosures with the chi-square test (see any standard text on statistics). If the misclosure of a traverse in northing and easting is C_{Δ N} and C_{Δ E} respectively, we may compute:

x²=

where the i(i) subscript signifies the summation of the terms around the whole traverse to the point of commencement. Compare the computed with the theoretical _2,0.05 = 5.99. If < _2,0.05 the misclosure is acceptable at the 95 percent confidence level.

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APPENDIX E

EXAMPLES OF STANDARD DEVIATIONS FOR VARIOUS INSTRUMENTS AND METHODS

TABLE E-I LENGTH BY TAPE AND SUBTENSE BAR

METHOD	STANDARD DEVIATION (metres)	REMARKS
Invar Tape		Techniques described in Geodetic Survey Pub. 73. L = line length in metres.
Steel Tape		N = number of tape lengths. P = length of each tape in metres. Very careful slope, sag and temp. corrs. applied.
Steel Tape		N and P as above. Clinometer used for vertical angles up to 5°; alignment by picket; air temp. used for corrections; tension handle used for spans over 30 m.
Steel Tape		N and P as above; no tension handles; nominal temp. correction.
Subtense Bar		Standard deviations of approx. 1" for angle measurements and of 1 mm for plumbing at each end of line. L = line length in metres.

TABLE E-II LENGTH BY VARIOUS TYPES OF EDM INSTRUMENTS

Note: The proportional part of the accuracy of all EDM measurements is limited by the accuracy of the modulation frequency, by the accuracy of the meteorological measurements and by the reliability of the samples measured as representative of the conditions along the measured line. The nonproportional part is dependent on the accuracies of plumbing and of the zero corrections of the instrument and reflector and the resolution of the instrument. The modulation frequency and zero correction of the instrument must be checked whenever the instrument is repaired, or after rough treatment as well as on a regular schedule. Accuracy standards for recording meteorological measurements, for frequency of calibration tests and for elevation differences are given in Table E-III and are referred to in this table.

Type	Standard Deviation (metres)	Standard*	Remarks
Mekometer		M1	Forced centering.
Tellurometer MA100 (infrared)		M2 M5	Mean of 8 measurements** anytime. Mean of 2 measurements anytime.
Ranger 3 and 4 (Laser)		M2 M5	Mean of 8 measurements anytime Mean of 2 measurements anytime
Geodimeter (visible light)		M2 M5	Mean of 6 measurements. Mean of 2 measurements.
Tellurometer MRA2		M3 M4	37 cavities, three double measurements.*** 10 cavities, single measurement.
Tellurometer MRA3		M3 M4	13 cavities, three double measurements.*** 10 cavities, single measurement.
Tellurometer MRA4		M3 M4	10 cavities, three double measurements.*** 10 cavities, single measurement.
*Standards as listed in Table E-III. **For definition of measurement, see APPENDIX C - Lengths. ***Successive measurements alternating master end. Successive doubles separated by at least four hours.

TABLE E-III STANDARDS FOR METEOROLOGICAL MEASUREMENTS, FOR CALIBRATION TESTS AND FOR ELEVATION DIFFERENCES AND MEAN ELEVATION DETERMINATIONS

S T D *	Meteorological Measurements				Calib. Schedule (additional to calib. after repair or rough usage)		Elevation Determinations
	Temp. at one of both ends	Standard Deviations					Standard Deviations for a line L metres long
		Pressure (mm)	Dry Bulb Temp (°C)	Wet Bulb Temp (°C)	Zero	Freq.	Elev. Diff h** (m)	Mean Elev. H (m)
M1	both	3	0.1	0.1	2 mo	2 mo		2
M2	both	3	0.2	0.2	4 mo	1 mo		2
M3	both	3	0.2	0.2	4 mo	2 wks		2
M4	both	3	0.5	0.5	2 yr	6 mo		10
M5	one only	3	1	-	6 mo	1 yr		10
*These standards are those referred to in Table E-ll. **For example: to achieve the accuracies listed in Table E-ll for measurements requiring standard M2, for a line 1000 m long between terminals 25 m different in elevation, the elevation difference should be measured by a method giving a standard deviation not greater than:

TABLE E-IV DIRECTIONS

Instrument Least Count	STANDARD DEVIATIONS		Remarks
Seconds	Metres perp. to L
0.2"			L = Line length in metres. P is std. dev. of plumbing of instrument and target. Normal values: 0.0014m. Four groups of 6 sets during 2 days Starting time of each group differing by at least 2 hours.
1.0"			Mean of 6 sets.
1.0"			Mean of 4 sets.
1.0"			Mean of 2 sets.
10.0"			Mean of 2 sets.
20.0"			Mean of 2 sets.
30.0"			Mean of 2 sets.
Note: The angle between two directions with standard deviations σ1 and σ2 as derived above will have a standard deviation . If the lengths of the lines are about the same in the most accurate case above, (σ1 = σ2), the standard deviation of the angle between them will be about:

TABLE E-V AZIMUTHS

INSTRUMENTS	STD. DEV.*		REMARKS
First-Order Astronomic	1"		Azimuth and longitude observed.
Second-Order Survey Theodolite with stride level (Polaris observations) Second-Order Survey Theodolite without stride level (Polaris observations	3" 6"	}	No observed longitude. Non-mountainous terrain where small deflections in prime vertical can be expected and estimated or assumed zero.
Second-Order Survey Theodolite with stride level (Polaris observations) Second-Order Survey Theodolite without stride level (Polaris observations)	about 10" about 15"	}	No observed longitude. Mountainous terrain where large deflections can be expected and which can only be roughly estimated.
Gyro Theodolite	3" - 10"		No observed longitude. Using special techniques. Ultimate accuracy depends on local deflection.
Gyro Theodolite	20"		No observed longitude. Using normal techniques.
Second-Order Survey Theodolite (solar observations)	{	10" - 20" 30" - 45"	With Roelofs Solar Prism, depending on method. Without Roelofs Solar Prism, depending on method.
*Standard deviation of azimuth transferred to a line between two stations.

TABLE E-VI POSITION DIFFERENCES

Method	Standard Deviation	Remarks
Satellite Doppler	0.5 m	50 passes over at least 48 hours; broadcast or precise ephemeris; simultaneous observations at two or more stations simultaneous (multi-station) solution for positions and orbital biases.
Satellite Doppler	1.5 m	50 passes, precise ephemeris, non-simultaneous observations at the two stations.
Satellite Doppler	7.5 m	50 passes; broadcast ephemeris; non-simultaneous observations at the two stations.
Inertial Survey System (Litton Autosurveyor)	0.5 m	Double run in straight line by helicopter between control spaced at 80 km.
Notes: Position Differences by Satellite Doppler Differences of geographic position between two or more stations may be obtained by Doppler observations of satellites. The values in this table refer to observations of the United States Navy Navigation Satellite System, also known as the TRANSIT system. Application of the precise ephemeris to the solution will give a position whose standard deviation relative to datum is 1 m, whereas for the broadcast ephemeris this standard deviation is 5 m. When only the broadcast ephemeris is used, two known stations must be observed simultaneously with each group of unknown stations; this is to provide adequate accuracy and orientation relative to both datum and existing stations. In either case (broadcast or precise ephemeris) a receiver with internal timing for Doppler counts will give best results. The standard deviation of a 30-second Doppler count should not exceed 20 cm. Meteorological observations (temperature, pressure, and relative humidity) should be taken every six hours (more frequent during weather changes such as front) for use in the tropospheric refraction correction model. Two days (48 hours) of observing are needed to provide a desirable range of meteorological conditions. Also, 48 hours of observing is needed to provide the complete range of satellite to ground geometry. Ionospheric refraction effects are largely removed through the use of two frequencies. Receivers should be warmed up at least three days before measurement, and antennae should be mounted at survey-tripod height over surfaces that are not radio-reflective. Satellite passes whose approach elevation angle is less than 15° at closest approach should not be used. Also Doppler counts observed below 7.5° should not be used because of the inadequacy of the tropospheric refraction correction model near the horizon.

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APPENDIX F DENSIFICATION OF HORIZONTAL CONTROL BY PHOTOGRAMMETRIC METHODS

GENERAL

Photogrammetric instrumentation and methods have progressed in recent years to the point where accuracy, reliability and cost saving make the photogrammetric control extension an attractive adjunct to field control surveys. This is especially so in projects which require the establishment of moderate to high control density. In general terms the establishment of control points by photogrammetric methods is accomplished in two ways:

• By establishing x, y, z coordinates of well identifiable points in a single stereo model which is properly controlled by means of geodetic points surveyed on the ground.
• By "bridging" between known geodetic control points employing aerial-triangulation methods.

GUIDELINES

Planning Considerations

Where projects are planned to include control supplementation by photogrammetric means, the objectives of the project and its overall size (whether multi-model blocks or single model jobs) will dictate consideration of the following important factors:

• Required accuracy of the final product.
• Scales of photography.
• The type of camera and its characteristics.
• Flying height of aircraft.
• Type of terrain.
• Percentage of forward and side overlaps.
• Accuracy, density and distribution of field survey control points within project area.
• The density of supplementary control points required.
• Precision of the mensuration instruments.
• The method of photogrammetric adjustment to be employed.

Procedures

It is recommended that:

• All field control and the required supplementary control points be suitably targeted for best photo-identification purposes.
• When possible, targeting precede the flight program.
• When identification photography of the target points is flown, the transfer of these points from the identification to the mapping photography must be done using a precision-point transfer instrument such as Wild PUG4.
• All pass points and tie points be "pugged" on diapositives using a point-transfer instrument such as Wild PUG4.
• Mensuration for fully analytical aerial triangulation be performed on well-calibrated mono or stereocom-parators having a least-count precision of 1 to 2 µm.
• Mensuration for semi-analytical aerial triangulation be carried out on well-calibrated stereorestitution instru-ments of fixed or determinable perspective centres and of precision equal to or better than such instruments as the Wild A7 and A10.
• Each photogrammetric point in the stereo model be measured at least two times in order to maintain a check on reliability of observations,
• Methods of bundle adjustment, or the simultaneous adjustment of independent models by similarity transformations, be used for aerial triangulation.

Expected Accuracy

Expected accuracy depends on the factors previously mentioned under guidelines, and also such additional factors as:

• The accuracy of performing photogrammetric measurements.
• The competence and experience of personnel involved in photogrammetric data acquisition, analysis and adjustment.
• Technology available to the executing agency.
• Economic and environmental factors.

From the experience of the Topographical Survey, the expected standard deviation is 8-10 µm (photo scale) if the photogrammetric models are measured on a stereocomparator and 12-14 µm if a stereorestitution instrument (e.g., Wild A7) is used for mensuration.