# Ontario digital levelling vertical control survey specifications

Learn the provincial specifications for digital levelling work performed in Ontario for projects that will be included in COSINE. These specifications replace the previous 2015 version.

April 2021

Prepared by the:

Geomatics Office

Ministry of Transportation, Ontario

St. Catharines

Geodetic Services Program

Office of the Surveyor General

Ministry of Natural Resources and Forestry

## Disclaimer

This technical documentation has been prepared by Her Majesty the Queen in right of Ontario as represented by the Ministry of Transportation Ontario, and the Ministry of Natural Resources and Forestry (the “ministries”). No warranties or representations, express or implied, statutory or otherwise shall apply or are being made by the ministries with respect to the documentation, its accuracy or its completeness. In no event will the ministries be liable or responsible for any lost profits, loss of revenue or earnings, claims by third parties or for any economic, indirect, special, incidental, consequential or exemplary damage resulting from any errors, inaccuracies or omissions in this documentation; and in no event will the ministries’ liability for any such errors, inaccuracies or omissions on any particular claim, proceeding or action, exceed the actual consideration paid by the claimant involved to the ministries for the materials to which this instructional documentation relates. Save and except for the liability expressly provided for above, the Ministries shall have no obligation, duty or liability whatsoever in contract, tort or otherwise, including any liability or negligence. The limitations, exclusions and disclaimers expressed above shall apply irrespective of the nature of any cause of action, demand or action, including but not limited to breach of contract, negligence, strict liability, tort or any other legal theory, and shall survive any fundamental breach or breaches.

## Introduction

This document defines the minimum standards that must be met when establishing 1^{st}, 2^{nd}, 2^{nd}-B and 3^{rd} Order Vertical Control (Geodetic Benchmarks) using Digital Levelling for the Ministry of Transportation, Ontario (MTO) and the Ministry of Natural Resources and Forestry, Ontario (MNRF), and for all projects destined for inclusion in the provincial geodetic database known as COSINE. It consolidates the relevant portions of Specifications and Recommendations for Control Surveys and Survey Markers (Natural Resources Canada, 1978), the Ministry of Natural Resources and Forestry Provisional Specifications for Digital Levelling (2017) and the Ministry of Transportation Ontario Vertical Control Survey Specifications (2018) into one comprehensive specification document for Ontario. However, any specific requirements in the contractual terms of reference for a project take precedence over the general requirements of this specification document.

## Definitions

For the purposes of this document the following definitions shall apply:

- Level loop
- A series of forward and backward level runs between Primary Benchmarks.
- Level run
- A one-way run starting on a published or new geodetic benchmark and ending on a different published or new geodetic benchmark.
- Primary benchmarks
- Existing geodetic benchmarks with published elevations that have been verified by performing benchmark stability checks (refer to Benchmark stability checks).
- Section
- A part of the level route between adjacent benchmarks.
- Set-up
- the level observations between two adjacent points (benchmarks or turning points). A set-up includes a backsight (BS) and a foresight (FS).

## General specifications — Applicable to all orders

- Digital levelling equipment must be used. The equipment accuracy specifications must meet or exceed the accuracy specifications for the order of levelling being performed.
- New benchmarks must be spaced no more than three kilometres apart.
- At least 2 levelling connections must emanate from each new or existing published benchmark along the level route between Primary Benchmarks.
- Maximum sight distance to the rod must not exceed 60 metres unless a shorter maximum sight distance is recommended by the equipment manufacturer.
- The minimum reading to the rod must not be less than 0.5 metres throughout the entire length of the loop. Also, the top stadia hair must be visible on the bar code when recording observations.
- For benchmarks mounted in horizontal surfaces, the level rod must touch the benchmark at the centre mark or top surface of the monument.
- For benchmarks mounted in vertical faces/walls, the level rod must be attached to a precise levelling “chisel” (or equivalent) that mounts on the base of the level rod and can be inserted into the slot at the centre of the monument.
- Avoid reading at distances known to manufacturers that can cause inaccurate rod readings.
- The difference between backsight and foresight distances at each set-up must not exceed 5 metres, and the sum of the differences for all set-ups in each section and level run must not exceed 10 metres.
- All level runs must have an even number of instrument set-ups. A single setup will be allowed when the distance between benchmarks is less than 100 metres and the same rod is used at both stations.
- Level loops must be observed between all Primary Benchmarks. Perform independent forward and backward level runs at different times of the day, and under different weather conditions where possible. See exception for 3
^{rd}Order levelling (refer to Third order specifications). - If the discrepancy between the forward and backward level runs exceeds the specified closure for that order, the level run must be relevelled until the discrepancy does not exceed that closure.
- Each reading of the rod (sighting) must be repeated twice. At each set-up, backsights and foresights must be read and recorded in the Backsight Foresight Foresight Backsight (BFFB) format.
- All existing geodetic benchmarks along the level route of order equal to or higher than the new benchmarks being established must be levelled through (refer to Appendix C, Appendix D — Part I and Appendix D — Part II — Network design).
- If no published benchmarks exist for 10 kilometres along the level route, at least one benchmark of equal or higher order within 3 kilometres of the level route, if available, must be tied in (refer to Appendix C, Appendix D — Part I and Appendix D — Part II — Network design).
- New benchmarks may be established between primary benchmarks and must be connected to them using forward and backward runs. See exception for 3
^{rd}Order levelling (refer to Third order specifications). - All junction points must be monumented (refer to Appendix A — Monumentation for types to be used).
- Stable concrete structures exclude sidewalks (unless authorized by the issuing agency for 3
^{rd}order vertical control), small concrete culverts, small pedestal or light-standard bases, and curbs. Suggested concrete structures include large box culverts, high-mast light bases, building foundations, and bridge abutments or pillars.

## Primary vertical control network integration

Primary benchmarks of order equal to or higher than the new benchmarks being established must encompass the project area. Prior to establishing new benchmarks, the published elevations of these primary benchmarks must be verified by Benchmark stability checks (refer to Benchmark stability checks below).

A minimum of three primary benchmarks for 1^{st} order levelling, or two primary benchmarks for 2^{nd}, 2^{nd}-B, and 3^{rd} order levelling are required at every limit of the project. Each pair of primary benchmarks must not be in the same structure and must be far enough apart (preferably a minimum of 0.5 kilometres apart) so that any disturbing influence in the area is not the same on these benchmarks. For 1^{st} order levelling it is recommended that the three primary benchmarks required at every limit of the project be spaced a minimum of 1.0 kilometres apart, if possible, to provide the most stable height references for the levelling network at this high level of accuracy. Refer to Appendix C for primary benchmark integration in a linear type of network. Refer to Appendix D — Parts I and II for primary benchmark integration in a block type or municipal network.

## Benchmark stability checks

A Benchmark stability check consists of levelling forward and backward between known primary benchmarks, at every limit of the network, and comparing the new observed elevation difference to the published elevation difference. The observed elevation difference is the average of the forward and backward level runs. The difference between the forward and backward level runs must meet the misclosure requirements for the order of the levelling being performed. If the misclosure requirements are met, the two observations are averaged and compared to the published elevation difference. If the misclosure between the observed (averaged) elevation difference and the published elevation difference is within the allowable closure specifications for the order of levelling being performed, the benchmarks are considered stable. If the misclosure exceeds the allowable closure specifications for the order of levelling, then one or more of the benchmarks are considered unstable and additional benchmarks must be checked using the same methodology until the closure specification between the published benchmarks is met. Refer to Appendix E for sample stability check calculations.

## Block type networks

When performing digital levelling for networks that do not form strictly linear connections, but rather have a block type design with connections emanating in various directions, certain additional checks must be performed to ensure accuracy. For block type networks (which are generally most often associated with municipal projects), additional stability checks must be performed within the area of the levelling network and a certain percentage of existing benchmarks within the network area must be occupied in the levelling exercises. The percentage of existing benchmarks to be occupied will be dependent on the size of the network, the accuracy/order of existing benchmarks within the area, as well as the number of existing benchmarks within the area of the levelling network. These requirements will be assessed based on the network design, access to existing stations and consultation with the issuing agency (the municipality). Typically, the network proposal should endeavor to occupy at least 20% to 25% of the existing benchmarks in the municipal/project area. Proposals must be submitted to the Geodetic Services Program, Office of the Surveyor General for all block type geodetic surveys, prior to field work being performed.

Block type (municipal) networks that cover an area of greater than 25 square kilometres (or approximately 5 kilometres x 5 kilometres) should consist of 4 (or more) interconnected loops that allow for independent loop closure checks and the isolation of possible observational errors within the network. Refer to Appendix D — Part I.

Block type (municipal) networks that cover an area of less than 25 square kilometres (or less than approximately 5 kilometres by 5 kilometres should consist of at least 2 (or more) interconnected loops that allow for independent loop closure checks and the isolation of possible observational errors within the network. Refer to Appendix D — Part II.

## First order specifications

- Monuments shall be rock posts set in stable rock or concrete structures, or caps affixed to 3.0 metres long X 25 millimetres diameter Round Iron Bars (RIB) driven a minimum of 30 cm below grade. The cap shall bear the full station name.
- A pair of calibrated invar rods must be used for all levelling. All invar rod pairs must have supporting documentation indicating that they have been calibrated within the last two years. Factory calibration of new invar rods is acceptable.
- A two-peg test must be performed at the beginning of the project, then daily, or additionally if the operator feels the instrument has been compromised. The collimation error shall not be greater than 0.05 millimetres per metre.
- Each reading of the rod (sighting) must be repeated four times. At each set-up, backsights and foresights must be read and recorded in the (Backsight Foresight Foresight Backsight) BFFB format, and then repeated again a second time.
- For Block type/municipal networks — Levelling networks shall consist of at least 4 loops of 25 km in length (or less).
- For Block type/municipal networks — A third valid stability check is required near the middle of the network or alternatively on the perimeter of the network.

### First order closure specifications

- Maximum section and level run misclosure between independent forward and backward level runs must be less than or equal to 4 millimetres multiplied by the square root of the distance levelled in kilometres. ≤4 mm * √km
- Maximum level loop misclosure between the sum of the forward and backward runs must be less than or equal to 4 millimetres multiplied by the square root of the distance levelled in kilometres. ≤4 mm * √km
- Where km equals the distance along the level run route in kilometres.

## Second order specifications

- Monuments shall be rock posts set in stable rock or concrete structures, or caps affixed to 1.8 metres long by 25 millimetres diameter Round Iron Bars (RIB) driven a minimum of 30 centimetres below grade. The cap shall bear the full station name.
- A pair of calibrated invar rods must be used for all levelling. All invar rod pairs must have supporting documentation indicating that they have been calibrated within the last two years. Factory calibration of new invar rods is acceptable.
- A two-peg test must be performed at the beginning of the project, then daily, or additionally if the operator feels the instrument has been compromised. The collimation error shall not be greater than 0.05 millimetres per metre.
- For Block type or municipal networks, levelling networks shall consist of at least 4 loops of 25 kilometres in length (or less).
- For Block type/municipal networks, a third valid stability check is required near the middle of the network or alternatively on the perimeter of the network.

### Second order closure specifications

- Maximum section and level run misclosure between independent forward and backward level runs must be less than or equal to 8 millimetres multiplied by the square root of the distance levelled in kilometres. ≤8 mm * √km
- Maximum level loop misclosure between the sum of the forward and backward runs must be less than or equal to 8 millimetres multiplied by the square root of the distance levelled in kilometres. ≤8 mm * √km
- Where km equals the distance along the level run route in kilometres.

## Second-B order specifications

- Monuments shall be rock posts set in stable rock or concrete structures, or caps affixed to 1.8 metres long X 25 millimetres diameter Round Iron Bars (RIB) driven a minimum of 30 centimetres below grade. The cap shall bear the full station name.
- A pair of invar, single piece or sectional fiberglass rods may be used. If the length of the levelling rods exceeds 3 metres, only the bottom 3 metres of the rods shall be used for observations.
- A two-peg test must be performed at the beginning of the project, then daily, or additionally if the operator feels the instrument has been compromised. The collimation error shall not be greater than 0.10 millimetres per metre.
- For Block type or municipal networks — Levelling networks shall consist of at least 2 loops of 25 kilometres in length (or less).
- For Block type or municipal networks — A third valid stability check is recommended near the middle of the network or alternatively on the perimeter of the network.

### Second-B order closure specifications

- Maximum section and level run misclosure between independent forward and backward level runs must be less than or equal to 16 millimetres multiplied by the square root of the distance levelled in kilometres. ≤16 mm * √km
- Maximum level loop misclosure between the sum of the forward and backward runs must be less than or equal to 16 millimetres multiplied by the square root of the distance levelled in kilometres. ≤16 mm * √km
- Where km equals the distance along the level run route in kilometres.

## Third order specifications

- Monuments shall be rock posts set in stable rock or concrete structures, or caps affixed to 1.8 metres long X 25 millimetres diameter Round Iron Bars (RIB) driven a minimum of 30 centimetres below grade. The cap shall bear the full station name.
- Invar rods, single-piece, sectional, or telescoping rods constructed of fiberglass or aluminum may be used. If the length of the levelling rods exceeds 3 metres, only the bottom 3 metres of the rods shall be used for observations.
- A two-peg test must be performed at the beginning of the project, then daily, or additionally if the operator feels the instrument has been compromised. The collimation error shall not be greater than 0.10 millimetres per metre.
- If a level run starts and ends on primary benchmarks, and the length of the run is less than 25 kilometres, one-way levelling is permitted.
- For Block type/municipal networks, levelling networks shall consist of at least 2 loops of 25 kilometres in length (or less).

### Third order closure specifications

- Maximum section and level run misclosure between independent forward and backward level runs must be less than or equal to 24 millimetres multiplied by the square root of the distance levelled in kilometres. ≤24 mm * √km
- Maximum level loop misclosure between the sum of the forward and backward runs must be less than or equal to 24 millimetres multiplied by the square root of the distance levelled in kilometres.≤24 mm * √km
- For one-way levelling: Maximum misclosure between the levelled elevation difference and the published elevation difference of the primary benchmarks must be less than or equal to 24 millimetres multiplied by the square root of the distance levelled in kilometres.≤24 mm * √km
- Where km equals the distance along the level run route in kilometres.

## Classification of new benchmarks

Every effort must be made to meet the specified order of accuracy throughout the vertical control network in the fully constrained adjustment. However, if it is not possible to achieve this level of accuracy for every station due to distortions in the existing network, a lower order of accuracy may be deemed acceptable by the issuing agency.

## Final deliverables

All levelling data must be adjusted through ministry-approved least squares adjustment software. Both minimally constrained and fully constrained least squares adjustments must be submitted. Closure statistics must be shown for all field level runs and level loops to verify misclosures are met for the required order of levelling. Closure statistics comparing field observations to published elevation differences between primary benchmarks, and between published benchmarks tied-in during the levelling, must also be shown.

All new and existing (published) benchmarks used in the survey must have their full (COSINE) name used in the least squares adjustments. All raw electronic files must be submitted.

### Other requirements

- Numerical evidence of all two-peg tests (raw digital files) and subsequent adjustments.
- A report detailing the equipment and methodology used, and explanations of any problems encountered, and their rectification.
- A clear high-resolution digital location reference sketch for each new benchmark. Refer to Appendix B — Monument position sketch — for sketch requirements.
- A digital copy of a text, Microsoft Word, or spreadsheet file containing a table of all new benchmarks and their proper location description.
- A digital copy of a text file containing approximate NAD83 coordinates (latitude, longitude, and ellipsoidal height) of all new benchmarks.
- A digital copy of the network diagram (to scale if possible) showing the actual observations and level routes followed between all new and existing benchmarks in the network.
- A digital copy of the level raw data with benchmarks identified with proper COSINE numbering.
- A digital copy of all field notes.

## References

- Natural Resources Canada, Specifications and Recommendations for Control Surveys and Survey Markers, 1978. Natural Resources Canada, Ottawa, 1978.
- Ministry of Transportation Ontario and Ministry of Natural Resources, 2004. Ontario Specification for GPS Control Surveys. The Queen’s Printer for Ontario, June 2004.
- Ontario Ministry of Natural Resources and Forestry Provisional Specifications for Digital Levelling, May 2017. The Queen’s Printer for Ontario, May 2017.
- Ministry of Transportation Ontario, 2018. Vertical Control Survey Specifications. The Queen’s Printer for Ontario, November 2018.

## Appendix A: Monumentation

### Other agency geodetic control screw-on bronze cap — dimensions and materials specifications — Type B

**Notes for engravings:**

- letter style is standard gothic
- letter sizes are:
- 5 millimetres stroke centre
- 6 millimetres edge to edge for outer inscription
- crosshair 3 millimetres stroke centre to stroke centre
- 3 millimeters edge to edge for inner inscription

- letter depth is 1 millimetre
- letters are tapered 20 degrees per side
- number size is 5 millimetres
- number depth is 1 millimetre

**Material notes:**

- use brass or bronze with not less than 83% copper content
- face of cap is buffed

### Ontario geodetic control screw-on bronze cap — Dimensions and materials specifications — Number B2

Image shows 3 views of bronze cap.

- First image is a top down view of cap which shows cross section A-A going from left to right and cross section B-B going from top to bottom. Inscription shown on top of cap shows the words “control survey” curving along the upper edge and the word “Ontario” curving along the lower edge. An 8 millimetre wide cross is shown at the centre of the cap.
- Image of cross section A-A shows the top of the cap being 55 millimetres in diameter with a 2 millimetres wide and 5 millimetres tall, angled shoulder. Cap is shown attached to a 25.4 millimetre diameter steel rod with a nut welded to bar for driving head seat below cap. Thread hole is 50 millimetres in height with 45 millimetres of thread, top 5 millimetres not threaded. Flat area shown on side being 15 millimetres wide and 35 millimetres tall and is mirrored on opposite side. Cap is threaded with standard thread to take 25.4 millimetre standard threaded rod with 8 threads per inch, minor diameter 0.8376 inch, and area at root 0.5510 square inch. Threaded hole for 5 millimetre Allan screw and 5 millimetre Allan screw shown 20 millimetre above base of cap. Base of cap is 41 millimetres in diameter and below shoulder shows radius of 30 millimetres.
- Image of cross section B-B shows the top is curved with a 90 millimetre radius. Threaded hole for 5 millimetre Allan screw shown 20 millimetres from base of cap.

**Notes for engravings:**

- letter stye is standard gothic and position as shown
- letter size is 5 millimetres and minimum letter depth is 1 millimetre

**Material notes:**

- brass or bronze with not less than 83% copper content

### Ontario vertical control slot-type concrete/rock bronze cap — Dimensions and materials specifications

Image shows 3 views of bronze cap.

- First image is a top down view of cap which shows cross section A-A going from left to right. Inscription shown on top of cap shows the word “Ontario” curving along the upper edge and “issuing agency” curving along the lower edge. The words “control survey” with a 2 millimetre wide horizontal slot below is shown at the centre of the cap. Cap is shown to have a diameter of 70 millimetres.
- Cross section A-A shows top of cap having a radius of 185 millimetres on it’s face and 6 millimeters thick at edge. 2 millimetre wide slot shown to be 35 millimetres in length. Base tapers from outer width of 20 millimetres, inner width of 16 millimetres to 16 millimetres outer width, 12 millimetres inner width. Bottom point has a radius of 10 millimetres. 4 countersunk holes per face 10 millimetres in diameter and 5 millimetres deep at 15 millimetre intervals approximately.
- Section B-B shown left to right looking down at centre. Section B-B shows an X shaped base with 10 millimetre radius on inside and 2 millimetre radius on outer points. Faces are 10 millimetres apart.

**Notes for engravings:**

- letter style is standard gothic
- letter sizes are:
- 5 millimetres stroke centre
- 6 millimetres edge to edge for outer inscription
- 3 millimetres stroke centre to stroke centre
- 3 millimeters edge to edge for inner inscription

- letter depth is 1 millimetre
- letters are tapered 20 degrees per side
- number size is 5 millimetres
- number depth is 1 millimetre

**Material notes:**

- brass or bronze with not less than 83% copper content
- face of cap buffed

## Appendix B: Sample geodetic control monument position sketch

Image shows a sample monument position sketch sheet.

Text from image:

- date: October 15, 2001
- job file: 2001-109
- type of monument: cap in rock
- relationship in ground: flush
- condition: new
- intervisible with: 00820010532
- location: point is located 15.1 kilometres south of a scenic lookout, 2.4 kilometres north of Mamainse Harbour Road, on the east side of Highway 17
- locate: yes
- position sketch: the example shows Highway 17 and the centreline with a measurement of 14.6 metres from centreline to the station. Shows an overhead hydro line and a rock cut. Shows a steel marker 0.13 metres south of station, nail in 80 centimetre diameter polar 18.45 metres southeast of station, and a nail in a bell pole 14.71 metres northeast of station. North arrow is shown in top right corner
- monument number with sample being 00820010533

## Appendix C: Vertical linear network design example

Image shows a linear network example in the shape of a T. There are 2 benchmarks at each end of the network, east, west, and south. An additional existing benchmark is shown part way along the east-west line and is labeled 10 kilometres or more along levelling route. Curved arrows are shown between stations to show direction of levelling both forward and back. A number of new stations are shown between the existing stations. At the intersection of the east-west line with the north-south line the station is labeled “Junction point must be monumented”. At the south end a new station is located between 2 existing benchmarks and is labeled “New benchmarks may be established between existing Primary Benchmarks”. Next is a Key showing the symbols used for existing published geodetic benchmark (PB denotes primary benchmark), new geodetic benchmark, and direction of level run. A note is shown stating where no published benchmark exist for at least 10 kilometres along the levelling route, at least one benchmark of equal or higher order within 3 kilometres of the levelling route, if available, must be tied in.

## Appendix D

### Part I: Block type/municipal vertical network design example

Second or 2-B orders (network area: 25 square kilometres and up).

Image shows a sample block type network consisting of 4 square blocks stacked 2 by 2. There are 2 existing benchmarks shown at each of the outside corners and labeled PB. The 2 on the top left are labeled stability check at beginning of network. The 2 at the top right are labeled optional stability check #1. The 2 at the bottom left are labeled optional stability check #2. The 2 at the bottom right are labeled stability check at end of network. 2 existing benchmarks are shown at the centre intersection and labeled stability check near middle of network. A combination of new and existing benchmarks are shown between the aforementioned benchmarks. Curved arrows are shown between the benchmarks to show direction of levelling in both directions. Below the sample network is a key showing the symbols used for existing published geodetic benchmark (PB denotes primary benchmark), new geodetic benchmark, and direction of level run.

**Notes:**

- stating optional stability check allows for better integration between new and existing networks
- for 2
^{nd}and 2^{nd}B Order levelling, each level run must be part of a level-loop (two-way levelling)

Stability Checks consist of two existing benchmarks for which closure requirements are met. Refer to Appendix E and Benchmark stability checks.

### Part II: Block type/municipal vertical network design example

Second or 2-B Orders (Network Area: less than 25 square kilometres).

Image shows a sample block type network consisting of 2 squares, one on top of the other. 2 existing benchmarks are shown on each of the corners at the top left, top right, and bottom right and marked PB. The 2 at the top left are labeled stability check at beginning of network. The 2 at the top right are labeled optional stability check. The 2 at the bottom right are labeled stability check at end of network. A combination of new and existing benchmarks are shown between the aforementioned benchmarks. Curved arrows are shown between the benchmarks to show direction of levelling in both directions. Below the sample network is a key showing the symbols used for existing published geodetic benchmark (PB denotes primary benchmark), new geodetic benchmark, and direction of level run.

**Notes:**

- stating optional stability check allows for better integration between new and existing networks
- stating for 2
^{nd}and 2^{nd}B order levelling, each level run must be part of a level-loop (two-way levelling)

Stability Checks consist of two existing benchmarks for which closure requirements are met. Refer to Appendix E and Benchmark stability checks.

## Appendix E: Diagram and table showing example calculations for the allowable misclosure and stability check of a sample vertical network

Image shows a sample diagram for calculating allowable misclosure error and stability check calculations. There is a sample levelling run showing benchmark “A” and benchmark “B” with curved arrows in both directions between representing the levelling measurements between. The forward direction shows a measurement of +4.8971 metres in height over a distance of 1.60 kilometres. The reverse direction shows a measurement of −4.8870 metres in height over a distance of 1.70 kilometres. Below this are notes about the measurements. Average forward and backward distance equals (1.60 kilometres plus 1.70 kilometres) divided by 2 equals 1.65 kilometres. Observed elevation difference from benchmark “A” to benchmark “B” equals +4.8971 metres. Observed elevation difference from benchmark “B” to benchmark “A” equals −4.8870 metres. Misclosure between forward and backward observations = 4.8971 metres subtract 4.8870 equals 0.0101 metres or 10.1 millimetres. Average observed elevation difference benchmark “A” to benchmark “B” equals (4.8971 metres plus absolute value of −4.8870 metres) divided by 2 equals 4.8920 metres. Benchmark “A” published elevation equals 100.101 metres. Benchmark “B” published elevation equals 104.989 metres. Published elevation difference benchmark “A” to benchmark “B” equals 4.888 metres.

Allowable misclosure error and stability check calculations

Calculation performed/comments | First order formula | Second order formula | Second order B formula | Third order formula |
---|---|---|---|---|

Allowable misclosure error | 1^{st} order (less than or equal to 4 millimetres multiplied by the square root of the distance in kilometres) | 2^{nd} order (less than or equal to 8 millimetres multiplied by the square root of the distance in kilometres) | 2^{nd} B order (less than or equal to 16 millimetres multiplied by the square root of the distance in kilometres) | 3^{rd} order (less than or equal to 24 millimetres multiplied by the square root of the distance in kilometres) |

Allowable misclosure error | 4 millimetres multiplied by the square root of 1.65 equals 5.1 millimetres | 8 millimetres multiplied by the square root of 1.65 equals 10.3 millimetres | 16 millimetres multiplied by the square root of 1.65 equals 20.6 millimetres | 24 millimetres multiplied by the square root of 1.65 equals 30.8 millimetres |

Misclosure between forward and backward observations | 10.1 millimetres | 10.1 millimetres | 10.1 millimetres | 10.1 millimetres |

Misclosure comparison check | Fails 1^{st} order | Meets 2^{nd} order | Meets 2^{nd}-B order | Meets 3^{rd} order |

Average observed elevation difference minus published elevation difference | 4.8920 metres minus 4.8880 metres equals 0.0040 metres or 4.0 millimetres | 4.8920 metres minus 4.8880 metres equals 0.0040 metres or 4.0 millimetres | 4.8920 metres minus 4.8880 metres equals 0.0040 metres or 4.0 millimetres | 4.8920 metres minus 4.8880 metres equals 0.0040 metres or 4.0 millimetres |

Stability check | Meets 1^{st} order stability check | Meets 2^{nd} order stability check | Meets 2^{nd}-B order stability check | Meets 3^{rd} order stability check |

Comments | Fails overall 1^{st} order stability check — failed misclosure comparison at 1^{st} order | Passes overall 2^{nd} order stability check — benchmarks are considered stable at 2^{nd} order | Passes overall 2^{nd}-B order stability check — benchmarks are considered stable at 2^{nd}-B order | Passes overall 3^{rd} order stability check — benchmarks are considered stable at 3^{rd} order |