[{"data":1,"prerenderedAt":4002},["ShallowReactive",2],{"blog-\u002Fblog\u002Fthe-weisbach-triangle-mastering-orientation-via-single-shaft-correlation":3,"blog-surround-\u002Fblog\u002Fthe-weisbach-triangle-mastering-orientation-via-single-shaft-correlation":3995},{"id":4,"title":5,"author":6,"body":7,"category":3964,"description":3965,"extension":3966,"image":3967,"meta":3968,"navigation":3987,"path":3988,"publishedAt":3989,"readTime":3990,"seo":3991,"sitemap":3992,"stem":3993,"__hash__":3994},"blog\u002Fblog\u002F1.the-weisbach-triangle-mastering-orientation-via-single-shaft-correlation.md","The Weisbach Triangle: Mastering Orientation via Single-Shaft Correlation","Surveyiiing Expert",{"type":8,"value":9,"toc":3941},"minimark",[10,15,33,37,40,44,226,230,376,380,387,591,926,1377,2140,2144,3105,3109,3435,3599,3603,3730,3734,3861,3865,3881,3885,3927,3931],[11,12,14],"h2",{"id":13},"overview","Overview",[16,17,18,19,23,24,28,29,32],"p",{},"In underground engineering, the most critical challenge is ",[20,21,22],"strong",{},"orientation",": ensuring the underground coordinate system perfectly aligns with the surface National Grid ",[25,26,27],"span",{},"1",". When a project is accessed via a single vertical shaft, the ",[20,30,31],{},"Weisbach Triangle"," is the primary method for transferring that bearing .",[11,34,36],{"id":35},"why-this-matters","Why This Matters",[16,38,39],{},"If your underground orientation is off by just a few seconds of arc, a tunnel driven for 1 km could miss its target by several decimeters. This leads to catastrophic structural failures and legal liabilities. The Weisbach method provides a mathematically rigorous way to minimize this risk .",[11,41,43],{"id":42},"background","Background",[16,45,46,47,153,154,225],{},"Correlation involves connecting a surface survey to an underground one. In a single shaft, two wires (",[25,48,51,80],{"className":49},[50],"katex",[25,52,55],{"className":53},[54],"katex-mathml",[56,57,59],"math",{"xmlns":58},"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML",[60,61,62,75],"semantics",{},[63,64,65],"mrow",{},[66,67,68,72],"msub",{},[69,70,71],"mi",{},"W",[73,74,27],"mn",{},[76,77,79],"annotation",{"encoding":78},"application\u002Fx-tex","W_1",[25,81,85],{"className":82,"ariaHidden":84},[83],"katex-html","true",[25,86,89,94],{"className":87},[88],"base",[25,90],{"className":91,"style":93},[92],"strut","height:0.8333em;vertical-align:-0.15em;",[25,95,98,103],{"className":96},[97],"mord",[25,99,71],{"className":100,"style":102},[97,101],"mathnormal","margin-right:0.1389em;",[25,104,107],{"className":105},[106],"msupsub",[25,108,112,144],{"className":109},[110,111],"vlist-t","vlist-t2",[25,113,116,139],{"className":114},[115],"vlist-r",[25,117,121],{"className":118,"style":120},[119],"vlist","height:0.3011em;",[25,122,124,129],{"style":123},"top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;",[25,125],{"className":126,"style":128},[127],"pstrut","height:2.7em;",[25,130,136],{"className":131},[132,133,134,135],"sizing","reset-size6","size3","mtight",[25,137,27],{"className":138},[97,135],[25,140,143],{"className":141},[142],"vlist-s","​",[25,145,147],{"className":146},[115],[25,148,151],{"className":149,"style":150},[119],"height:0.15em;",[25,152],{}," and ",[25,155,157,176],{"className":156},[50],[25,158,160],{"className":159},[54],[56,161,162],{"xmlns":58},[60,163,164,173],{},[63,165,166],{},[66,167,168,170],{},[69,169,71],{},[73,171,172],{},"2",[76,174,175],{"encoding":78},"W_2",[25,177,179],{"className":178,"ariaHidden":84},[83],[25,180,182,185],{"className":181},[88],[25,183],{"className":184,"style":93},[92],[25,186,188,191],{"className":187},[97],[25,189,71],{"className":190,"style":102},[97,101],[25,192,194],{"className":193},[106],[25,195,197,217],{"className":196},[110,111],[25,198,200,214],{"className":199},[115],[25,201,203],{"className":202,"style":120},[119],[25,204,205,208],{"style":123},[25,206],{"className":207,"style":128},[127],[25,209,211],{"className":210},[132,133,134,135],[25,212,172],{"className":213},[97,135],[25,215,143],{"className":216},[142],[25,218,220],{"className":219},[115],[25,221,223],{"className":222,"style":150},[119],[25,224],{},") are suspended to form a very short baseline. Because this baseline is so short (often \u003C 3m), any measurement error in the position of the wires is magnified. The Weisbach triangle arrangement is designed specifically to mitigate this magnification .",[11,227,229],{"id":228},"theory","Theory",[16,231,232,233,305,306,375],{},"The principle relies on setting up a theodolite at a station (",[25,234,236,255],{"className":235},[50],[25,237,239],{"className":238},[54],[56,240,241],{"xmlns":58},[60,242,243,252],{},[63,244,245],{},[66,246,247,249],{},[69,248,71],{},[69,250,251],{},"s",[76,253,254],{"encoding":78},"W_s",[25,256,258],{"className":257,"ariaHidden":84},[83],[25,259,261,264],{"className":260},[88],[25,262],{"className":263,"style":93},[92],[25,265,267,270],{"className":266},[97],[25,268,71],{"className":269,"style":102},[97,101],[25,271,273],{"className":272},[106],[25,274,276,297],{"className":275},[110,111],[25,277,279,294],{"className":278},[115],[25,280,283],{"className":281,"style":282},[119],"height:0.1514em;",[25,284,285,288],{"style":123},[25,286],{"className":287,"style":128},[127],[25,289,291],{"className":290},[132,133,134,135],[25,292,251],{"className":293},[97,101,135],[25,295,143],{"className":296},[142],[25,298,300],{"className":299},[115],[25,301,303],{"className":302,"style":150},[119],[25,304],{},") nearly in line with the two wires. This creates a very \"thin\" or \"distorted\" triangle. By measuring the tiny angle at ",[25,307,309,326],{"className":308},[50],[25,310,312],{"className":311},[54],[56,313,314],{"xmlns":58},[60,315,316,324],{},[63,317,318],{},[66,319,320,322],{},[69,321,71],{},[69,323,251],{},[76,325,254],{"encoding":78},[25,327,329],{"className":328,"ariaHidden":84},[83],[25,330,332,335],{"className":331},[88],[25,333],{"className":334,"style":93},[92],[25,336,338,341],{"className":337},[97],[25,339,71],{"className":340,"style":102},[97,101],[25,342,344],{"className":343},[106],[25,345,347,367],{"className":346},[110,111],[25,348,350,364],{"className":349},[115],[25,351,353],{"className":352,"style":282},[119],[25,354,355,358],{"style":123},[25,356],{"className":357,"style":128},[127],[25,359,361],{"className":360},[132,133,134,135],[25,362,251],{"className":363},[97,101,135],[25,365,143],{"className":366},[142],[25,368,370],{"className":369},[115],[25,371,373],{"className":372,"style":150},[119],[25,374],{}," and the distances between the instrument and the wires, we can solve the triangle to find the bearing of the wire baseline .",[11,377,379],{"id":378},"mathematical-principles","Mathematical Principles",[16,381,382,383,386],{},"The core of the Weisbach solution is the ",[20,384,385],{},"Sine Rule",".",[16,388,389,390,520,521,590],{},"Given a triangle with wires ",[25,391,393,421],{"className":392},[50],[25,394,396],{"className":395},[54],[56,397,398],{"xmlns":58},[60,399,400,418],{},[63,401,402,408,412],{},[66,403,404,406],{},[69,405,71],{},[73,407,27],{},[409,410,411],"mo",{"separator":84},",",[66,413,414,416],{},[69,415,71],{},[73,417,172],{},[76,419,420],{"encoding":78},"W_1, W_2",[25,422,424],{"className":423,"ariaHidden":84},[83],[25,425,427,431,471,475,480],{"className":426},[88],[25,428],{"className":429,"style":430},[92],"height:0.8778em;vertical-align:-0.1944em;",[25,432,434,437],{"className":433},[97],[25,435,71],{"className":436,"style":102},[97,101],[25,438,440],{"className":439},[106],[25,441,443,463],{"className":442},[110,111],[25,444,446,460],{"className":445},[115],[25,447,449],{"className":448,"style":120},[119],[25,450,451,454],{"style":123},[25,452],{"className":453,"style":128},[127],[25,455,457],{"className":456},[132,133,134,135],[25,458,27],{"className":459},[97,135],[25,461,143],{"className":462},[142],[25,464,466],{"className":465},[115],[25,467,469],{"className":468,"style":150},[119],[25,470],{},[25,472,411],{"className":473},[474],"mpunct",[25,476],{"className":477,"style":479},[478],"mspace","margin-right:0.1667em;",[25,481,483,486],{"className":482},[97],[25,484,71],{"className":485,"style":102},[97,101],[25,487,489],{"className":488},[106],[25,490,492,512],{"className":491},[110,111],[25,493,495,509],{"className":494},[115],[25,496,498],{"className":497,"style":120},[119],[25,499,500,503],{"style":123},[25,501],{"className":502,"style":128},[127],[25,504,506],{"className":505},[132,133,134,135],[25,507,172],{"className":508},[97,135],[25,510,143],{"className":511},[142],[25,513,515],{"className":514},[115],[25,516,518],{"className":517,"style":150},[119],[25,519],{}," and instrument station ",[25,522,524,541],{"className":523},[50],[25,525,527],{"className":526},[54],[56,528,529],{"xmlns":58},[60,530,531,539],{},[63,532,533],{},[66,534,535,537],{},[69,536,71],{},[69,538,251],{},[76,540,254],{"encoding":78},[25,542,544],{"className":543,"ariaHidden":84},[83],[25,545,547,550],{"className":546},[88],[25,548],{"className":549,"style":93},[92],[25,551,553,556],{"className":552},[97],[25,554,71],{"className":555,"style":102},[97,101],[25,557,559],{"className":558},[106],[25,560,562,582],{"className":561},[110,111],[25,563,565,579],{"className":564},[115],[25,566,568],{"className":567,"style":282},[119],[25,569,570,573],{"style":123},[25,571],{"className":572,"style":128},[127],[25,574,576],{"className":575},[132,133,134,135],[25,577,251],{"className":578},[97,101,135],[25,580,143],{"className":581},[142],[25,583,585],{"className":584},[115],[25,586,588],{"className":587,"style":150},[119],[25,589],{},":",[592,593,594,768,800],"ol",{},[595,596,597,598,629,630,153,699,386],"li",{},"Let ",[25,599,601,615],{"className":600},[50],[25,602,604],{"className":603},[54],[56,605,606],{"xmlns":58},[60,607,608,613],{},[63,609,610],{},[69,611,612],{},"w",[76,614,612],{"encoding":78},[25,616,618],{"className":617,"ariaHidden":84},[83],[25,619,621,625],{"className":620},[88],[25,622],{"className":623,"style":624},[92],"height:0.4306em;",[25,626,612],{"className":627,"style":628},[97,101],"margin-right:0.0269em;"," be the distance between wires ",[25,631,633,650],{"className":632},[50],[25,634,636],{"className":635},[54],[56,637,638],{"xmlns":58},[60,639,640,648],{},[63,641,642],{},[66,643,644,646],{},[69,645,71],{},[73,647,27],{},[76,649,79],{"encoding":78},[25,651,653],{"className":652,"ariaHidden":84},[83],[25,654,656,659],{"className":655},[88],[25,657],{"className":658,"style":93},[92],[25,660,662,665],{"className":661},[97],[25,663,71],{"className":664,"style":102},[97,101],[25,666,668],{"className":667},[106],[25,669,671,691],{"className":670},[110,111],[25,672,674,688],{"className":673},[115],[25,675,677],{"className":676,"style":120},[119],[25,678,679,682],{"style":123},[25,680],{"className":681,"style":128},[127],[25,683,685],{"className":684},[132,133,134,135],[25,686,27],{"className":687},[97,135],[25,689,143],{"className":690},[142],[25,692,694],{"className":693},[115],[25,695,697],{"className":696,"style":150},[119],[25,698],{},[25,700,702,719],{"className":701},[50],[25,703,705],{"className":704},[54],[56,706,707],{"xmlns":58},[60,708,709,717],{},[63,710,711],{},[66,712,713,715],{},[69,714,71],{},[73,716,172],{},[76,718,175],{"encoding":78},[25,720,722],{"className":721,"ariaHidden":84},[83],[25,723,725,728],{"className":724},[88],[25,726],{"className":727,"style":93},[92],[25,729,731,734],{"className":730},[97],[25,732,71],{"className":733,"style":102},[97,101],[25,735,737],{"className":736},[106],[25,738,740,760],{"className":739},[110,111],[25,741,743,757],{"className":742},[115],[25,744,746],{"className":745,"style":120},[119],[25,747,748,751],{"style":123},[25,749],{"className":750,"style":128},[127],[25,752,754],{"className":753},[132,133,134,135],[25,755,172],{"className":756},[97,135],[25,758,143],{"className":759},[142],[25,761,763],{"className":762},[115],[25,764,766],{"className":765,"style":150},[119],[25,767],{},[595,769,597,770,799],{},[25,771,773,787],{"className":772},[50],[25,774,776],{"className":775},[54],[56,777,778],{"xmlns":58},[60,779,780,785],{},[63,781,782],{},[69,783,784],{},"c",[76,786,784],{"encoding":78},[25,788,790],{"className":789,"ariaHidden":84},[83],[25,791,793,796],{"className":792},[88],[25,794],{"className":795,"style":624},[92],[25,797,784],{"className":798},[97,101]," be the distance from the instrument to the front wire.",[595,801,597,802,925],{},[25,803,805,829],{"className":804},[50],[25,806,808],{"className":807},[54],[56,809,810],{"xmlns":58},[60,811,812,826],{},[63,813,814],{},[815,816,817,823],"mover",{"accent":84},[66,818,819,821],{},[69,820,71],{},[69,822,251],{},[409,824,825],{},"^",[76,827,828],{"encoding":78},"\\hat{W_s}",[25,830,832],{"className":831,"ariaHidden":84},[83],[25,833,835,839],{"className":834},[88],[25,836],{"className":837,"style":838},[92],"height:1.0968em;vertical-align:-0.15em;",[25,840,843],{"className":841},[97,842],"accent",[25,844,846,917],{"className":845},[110,111],[25,847,849,914],{"className":848},[115],[25,850,853,900],{"className":851,"style":852},[119],"height:0.9468em;",[25,854,856,860],{"style":855},"top:-3em;",[25,857],{"className":858,"style":859},[127],"height:3em;",[25,861,863,866],{"className":862},[97],[25,864,71],{"className":865,"style":102},[97,101],[25,867,869],{"className":868},[106],[25,870,872,892],{"className":871},[110,111],[25,873,875,889],{"className":874},[115],[25,876,878],{"className":877,"style":282},[119],[25,879,880,883],{"style":123},[25,881],{"className":882,"style":128},[127],[25,884,886],{"className":885},[132,133,134,135],[25,887,251],{"className":888},[97,101,135],[25,890,143],{"className":891},[142],[25,893,895],{"className":894},[115],[25,896,898],{"className":897,"style":150},[119],[25,899],{},[25,901,903,906],{"style":902},"top:-3.2523em;",[25,904],{"className":905,"style":859},[127],[25,907,911],{"className":908,"style":910},[909],"accent-body","left:-0.25em;",[25,912,825],{"className":913},[97],[25,915,143],{"className":916},[142],[25,918,920],{"className":919},[115],[25,921,923],{"className":922,"style":150},[119],[25,924],{}," be the tiny measured angle at the instrument.",[16,927,928,929,1042,1043],{},"The internal angle at the wire 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is derived as:\n",[25,1044,1046,1098],{"className":1045},[50],[25,1047,1049],{"className":1048},[54],[56,1050,1051],{"xmlns":58},[60,1052,1053,1095],{},[63,1054,1055,1058,1061,1071,1074,1081,1083,1085],{},[69,1056,1057],{},"sin",[409,1059,1060],{},"⁡",[815,1062,1063,1069],{"accent":84},[66,1064,1065,1067],{},[69,1066,71],{},[73,1068,27],{},[409,1070,825],{},[409,1072,1073],{},"=",[1075,1076,1077,1079],"mfrac",{},[69,1078,784],{},[69,1080,612],{},[69,1082,1057],{},[409,1084,1060],{},[815,1086,1087,1093],{"accent":84},[66,1088,1089,1091],{},[69,1090,71],{},[69,1092,251],{},[409,1094,825],{},[76,1096,1097],{"encoding":78},"\\sin \\hat{W_1} = \\frac{c}{w} \\sin 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are very small, we can use the approximation ",[25,1606,1608,1657],{"className":1607},[50],[25,1609,1611],{"className":1610},[54],[56,1612,1613],{"xmlns":58},[60,1614,1615,1654],{},[63,1616,1617,1619,1621,1624,1627,1640,1642,1644],{},[69,1618,1057],{},[409,1620,1060],{},[69,1622,1623],{},"θ",[409,1625,1626],{},"≈",[1628,1629,1630,1632],"msup",{},[69,1631,1623],{},[63,1633,1634,1638],{},[409,1635,1637],{"mathvariant":1636},"normal","′",[409,1639,1637],{"mathvariant":1636},[69,1641,1057],{},[409,1643,1060],{},[1628,1645,1646,1648],{},[73,1647,27],{},[63,1649,1650,1652],{},[409,1651,1637],{"mathvariant":1636},[409,1653,1637],{"mathvariant":1636},[76,1655,1656],{"encoding":78},"\\sin \\theta \\approx \\theta'' \\sin 1''",[25,1658,1660,1686],{"className":1659,"ariaHidden":84},[83],[25,1661,1663,1667,1670,1673,1677,1680,1683],{"className":1662},[88],[25,1664],{"className":1665,"style":1666},[92],"height:0.6944em;",[25,1668,1057],{"className":1669},[1110],[25,1671],{"className":1672,"style":479},[478],[25,1674,1623],{"className":1675,"style":1676},[97,101],"margin-right:0.0278em;",[25,1678],{"className":1679,"style":1196},[478],[25,1681,1626],{"className":1682},[1200],[25,1684],{"className":1685,"style":1196},[478],[25,1687,1689,1693,1727,1730,1733,1736],{"className":1688},[88],[25,1690],{"className":1691,"style":1692},[92],"height:0.7519em;",[25,1694,1696,1699],{"className":1695},[97],[25,1697,1623],{"className":1698,"style":1676},[97,101],[25,1700,1702],{"className":1701},[106],[25,1703,1705],{"className":1704},[110],[25,1706,1708],{"className":1707},[115],[25,1709,1711],{"className":1710,"style":1692},[119],[25,1712,1714,1717],{"style":1713},"top:-3.063em;margin-right:0.05em;",[25,1715],{"className":1716,"style":128},[127],[25,1718,1720],{"className":1719},[132,133,134,135],[25,1721,1723],{"className":1722},[97,135],[25,1724,1726],{"className":1725},[97,135],"′′",[25,1728],{"className":1729,"style":479},[478],[25,1731,1057],{"className":1732},[1110],[25,1734],{"className":1735,"style":479},[478],[25,1737,1739,1742],{"className":1738},[97],[25,1740,27],{"className":1741},[97],[25,1743,1745],{"className":1744},[106],[25,1746,1748],{"className":1747},[110],[25,1749,1751],{"className":1750},[115],[25,1752,1754],{"className":1753,"style":1692},[119],[25,1755,1756,1759],{"style":1713},[25,1757],{"className":1758,"style":128},[127],[25,1760,1762],{"className":1761},[132,133,134,135],[25,1763,1765],{"className":1764},[97,135],[25,1766,1726],{"className":1767},[97,135],":\n",[25,1770,1772,1828],{"className":1771},[50],[25,1773,1775],{"className":1774},[54],[56,1776,1777],{"xmlns":58},[60,1778,1779,1825],{},[63,1780,1781,1799,1801,1807],{},[1628,1782,1783,1793],{},[815,1784,1785,1791],{"accent":84},[66,1786,1787,1789],{},[69,1788,71],{},[73,1790,27],{},[409,1792,825],{},[63,1794,1795,1797],{},[409,1796,1637],{"mathvariant":1636},[409,1798,1637],{"mathvariant":1636},[409,1800,1073],{},[1075,1802,1803,1805],{},[69,1804,784],{},[69,1806,612],{},[1628,1808,1809,1819],{},[815,1810,1811,1817],{"accent":84},[66,1812,1813,1815],{},[69,1814,71],{},[69,1816,251],{},[409,1818,825],{},[63,1820,1821,1823],{},[409,1822,1637],{"mathvariant":1636},[409,1824,1637],{"mathvariant":1636},[76,1826,1827],{"encoding":78},"\\hat{W_1}'' = \\frac{c}{w} \\hat{W_s}''",[25,1829,1831,1957],{"className":1830,"ariaHidden":84},[83],[25,1832,1834,1838,1948,1951,1954],{"className":1833},[88],[25,1835],{"className":1836,"style":1837},[92],"height:1.2386em;vertical-align:-0.15em;",[25,1839,1841,1920],{"className":1840},[97],[25,1842,1844],{"className":1843},[97,842],[25,1845,1847,1912],{"className":1846},[110,111],[25,1848,1850,1909],{"className":1849},[115],[25,1851,1853,1898],{"className":1852,"style":852},[119],[25,1854,1855,1858],{"style":855},[25,1856],{"className":1857,"style":859},[127],[25,1859,1861,1864],{"className":1860},[97],[25,1862,71],{"className":1863,"style":102},[97,101],[25,1865,1867],{"className":1866},[106],[25,1868,1870,1890],{"className":1869},[110,111],[25,1871,1873,1887],{"className":1872},[115],[25,1874,1876],{"className":1875,"style":120},[119],[25,1877,1878,1881],{"style":123},[25,1879],{"className":1880,"style":128},[127],[25,1882,1884],{"className":1883},[132,133,134,135],[25,1885,27],{"className":1886},[97,135],[25,1888,143],{"className":1889},[142],[25,1891,1893],{"className":1892},[115],[25,1894,1896],{"className":1895,"style":150},[119],[25,1897],{},[25,1899,1900,1903],{"style":902},[25,1901],{"className":1902,"style":859},[127],[25,1904,1906],{"className":1905,"style":910},[909],[25,1907,825],{"className":1908},[97],[25,1910,143],{"className":1911},[142],[25,1913,1915],{"className":1914},[115],[25,1916,1918],{"className":1917,"style":150},[119],[25,1919],{},[25,1921,1923],{"className":1922},[106],[25,1924,1926],{"className":1925},[110],[25,1927,1929],{"className":1928},[115],[25,1930,1933],{"className":1931,"style":1932},[119],"height:1.0886em;",[25,1934,1936,1939],{"style":1935},"top:-3.3997em;margin-right:0.05em;",[25,1937],{"className":1938,"style":128},[127],[25,1940,1942],{"className":1941},[132,133,134,135],[25,1943,1945],{"className":1944},[97,135],[25,1946,1726],{"className":1947},[97,135],[25,1949],{"className":1950,"style":1196},[478],[25,1952,1073],{"className":1953},[1200],[25,1955],{"className":1956,"style":1196},[478],[25,1958,1960,1964,2032],{"className":1959},[88],[25,1961],{"className":1962,"style":1963},[92],"height:1.4336em;vertical-align:-0.345em;",[25,1965,1967,1970,2029],{"className":1966},[97],[25,1968],{"className":1969},[1217,1218],[25,1971,1973],{"className":1972},[1075],[25,1974,1976,2021],{"className":1975},[110,111],[25,1977,1979,2018],{"className":1978},[115],[25,1980,1982,1996,2004],{"className":1981,"style":1231},[119],[25,1983,1984,1987],{"style":1234},[25,1985],{"className":1986,"style":859},[127],[25,1988,1990],{"className":1989},[132,133,134,135],[25,1991,1993],{"className":1992},[97,135],[25,1994,612],{"className":1995,"style":628},[97,101,135],[25,1997,1998,2001],{"style":1249},[25,1999],{"className":2000,"style":859},[127],[25,2002],{"className":2003,"style":1257},[1256],[25,2005,2006,2009],{"style":1260},[25,2007],{"className":2008,"style":859},[127],[25,2010,2012],{"className":2011},[132,133,134,135],[25,2013,2015],{"className":2014},[97,135],[25,2016,784],{"className":2017},[97,101,135],[25,2019,143],{"className":2020},[142],[25,2022,2024],{"className":2023},[115],[25,2025,2027],{"className":2026,"style":1282},[119],[25,2028],{},[25,2030],{"className":2031},[1288,1218],[25,2033,2035,2114],{"className":2034},[97],[25,2036,2038],{"className":2037},[97,842],[25,2039,2041,2106],{"className":2040},[110,111],[25,2042,2044,2103],{"className":2043},[115],[25,2045,2047,2092],{"className":2046,"style":852},[119],[25,2048,2049,2052],{"style":855},[25,2050],{"className":2051,"style":859},[127],[25,2053,2055,2058],{"className":2054},[97],[25,2056,71],{"className":2057,"style":102},[97,101],[25,2059,2061],{"className":2060},[106],[25,2062,2064,2084],{"className":2063},[110,111],[25,2065,2067,2081],{"className":2066},[115],[25,2068,2070],{"className":2069,"style":282},[119],[25,2071,2072,2075],{"style":123},[25,2073],{"className":2074,"style":128},[127],[25,2076,2078],{"className":2077},[132,133,134,135],[25,2079,251],{"className":2080},[97,101,135],[25,2082,143],{"className":2083},[142],[25,2085,2087],{"className":2086},[115],[25,2088,2090],{"className":2089,"style":150},[119],[25,2091],{},[25,2093,2094,2097],{"style":902},[25,2095],{"className":2096,"style":859},[127],[25,2098,2100],{"className":2099,"style":910},[909],[25,2101,825],{"className":2102},[97],[25,2104,143],{"className":2105},[142],[25,2107,2109],{"className":2108},[115],[25,2110,2112],{"className":2111,"style":150},[119],[25,2113],{},[25,2115,2117],{"className":2116},[106],[25,2118,2120],{"className":2119},[110],[25,2121,2123],{"className":2122},[115],[25,2124,2126],{"className":2125,"style":1932},[119],[25,2127,2128,2131],{"style":1935},[25,2129],{"className":2130,"style":128},[127],[25,2132,2134],{"className":2133},[132,133,134,135],[25,2135,2137],{"className":2136},[97,135],[25,2138,1726],{"className":2139},[97,135],[11,2141,2143],{"id":2142},"field-workflow","Field Workflow",[2145,2146,2147,2152,2279,2283,2548,2552,2668,2672,3026,3030],"steps",{},[2148,2149,2151],"h3",{"id":2150},"surface-setup","Surface Setup",[16,2153,2154,2155,2278],{},"Hang two plumb wires (",[25,2156,2158,2183],{"className":2157},[50],[25,2159,2161],{"className":2160},[54],[56,2162,2163],{"xmlns":58},[60,2164,2165,2181],{},[63,2166,2167,2173,2175],{},[66,2168,2169,2171],{},[69,2170,71],{},[73,2172,27],{},[409,2174,411],{"separator":84},[66,2176,2177,2179],{},[69,2178,71],{},[73,2180,172],{},[76,2182,420],{"encoding":78},[25,2184,2186],{"className":2185,"ariaHidden":84},[83],[25,2187,2189,2192,2232,2235,2238],{"className":2188},[88],[25,2190],{"className":2191,"style":430},[92],[25,2193,2195,2198],{"className":2194},[97],[25,2196,71],{"className":2197,"style":102},[97,101],[25,2199,2201],{"className":2200},[106],[25,2202,2204,2224],{"className":2203},[110,111],[25,2205,2207,2221],{"className":2206},[115],[25,2208,2210],{"className":2209,"style":120},[119],[25,2211,2212,2215],{"style":123},[25,2213],{"className":2214,"style":128},[127],[25,2216,2218],{"className":2217},[132,133,134,135],[25,2219,27],{"className":2220},[97,135],[25,2222,143],{"className":2223},[142],[25,2225,2227],{"className":2226},[115],[25,2228,2230],{"className":2229,"style":150},[119],[25,2231],{},[25,2233,411],{"className":2234},[474],[25,2236],{"className":2237,"style":479},[478],[25,2239,2241,2244],{"className":2240},[97],[25,2242,71],{"className":2243,"style":102},[97,101],[25,2245,2247],{"className":2246},[106],[25,2248,2250,2270],{"className":2249},[110,111],[25,2251,2253,2267],{"className":2252},[115],[25,2254,2256],{"className":2255,"style":120},[119],[25,2257,2258,2261],{"style":123},[25,2259],{"className":2260,"style":128},[127],[25,2262,2264],{"className":2263},[132,133,134,135],[25,2265,172],{"className":2266},[97,135],[25,2268,143],{"className":2269},[142],[25,2271,2273],{"className":2272},[115],[25,2274,2276],{"className":2275,"style":150},[119],[25,2277],{},") in the shaft, as far apart as the shaft diameter allows .",[2148,2280,2282],{"id":2281},"instrument-positioning","Instrument Positioning",[16,2284,2285,2286,2355,2356,2468,2469,2547],{},"Set up the theodolite at station ",[25,2287,2289,2306],{"className":2288},[50],[25,2290,2292],{"className":2291},[54],[56,2293,2294],{"xmlns":58},[60,2295,2296,2304],{},[63,2297,2298],{},[66,2299,2300,2302],{},[69,2301,71],{},[69,2303,251],{},[76,2305,254],{"encoding":78},[25,2307,2309],{"className":2308,"ariaHidden":84},[83],[25,2310,2312,2315],{"className":2311},[88],[25,2313],{"className":2314,"style":93},[92],[25,2316,2318,2321],{"className":2317},[97],[25,2319,71],{"className":2320,"style":102},[97,101],[25,2322,2324],{"className":2323},[106],[25,2325,2327,2347],{"className":2326},[110,111],[25,2328,2330,2344],{"className":2329},[115],[25,2331,2333],{"className":2332,"style":282},[119],[25,2334,2335,2338],{"style":123},[25,2336],{"className":2337,"style":128},[127],[25,2339,2341],{"className":2340},[132,133,134,135],[25,2342,251],{"className":2343},[97,101,135],[25,2345,143],{"className":2346},[142],[25,2348,2350],{"className":2349},[115],[25,2351,2353],{"className":2352,"style":150},[119],[25,2354],{},", as close to the wires as focusing permits (usually 2–4m) and nearly in line with them (angle ",[25,2357,2359,2380],{"className":2358},[50],[25,2360,2362],{"className":2361},[54],[56,2363,2364],{"xmlns":58},[60,2365,2366,2378],{},[63,2367,2368],{},[815,2369,2370,2376],{"accent":84},[66,2371,2372,2374],{},[69,2373,71],{},[69,2375,251],{},[409,2377,825],{},[76,2379,828],{"encoding":78},[25,2381,2383],{"className":2382,"ariaHidden":84},[83],[25,2384,2386,2389],{"className":2385},[88],[25,2387],{"className":2388,"style":838},[92],[25,2390,2392],{"className":2391},[97,842],[25,2393,2395,2460],{"className":2394},[110,111],[25,2396,2398,2457],{"className":2397},[115],[25,2399,2401,2446],{"className":2400,"style":852},[119],[25,2402,2403,2406],{"style":855},[25,2404],{"className":2405,"style":859},[127],[25,2407,2409,2412],{"className":2408},[97],[25,2410,71],{"className":2411,"style":102},[97,101],[25,2413,2415],{"className":2414},[106],[25,2416,2418,2438],{"className":2417},[110,111],[25,2419,2421,2435],{"className":2420},[115],[25,2422,2424],{"className":2423,"style":282},[119],[25,2425,2426,2429],{"style":123},[25,2427],{"className":2428,"style":128},[127],[25,2430,2432],{"className":2431},[132,133,134,135],[25,2433,251],{"className":2434},[97,101,135],[25,2436,143],{"className":2437},[142],[25,2439,2441],{"className":2440},[115],[25,2442,2444],{"className":2443,"style":150},[119],[25,2445],{},[25,2447,2448,2451],{"style":902},[25,2449],{"className":2450,"style":859},[127],[25,2452,2454],{"className":2453,"style":910},[909],[25,2455,825],{"className":2456},[97],[25,2458,143],{"className":2459},[142],[25,2461,2463],{"className":2462},[115],[25,2464,2466],{"className":2465,"style":150},[119],[25,2467],{}," should be ",[25,2470,2472,2494],{"className":2471},[50],[25,2473,2475],{"className":2474},[54],[56,2476,2477],{"xmlns":58},[60,2478,2479,2491],{},[63,2480,2481,2484],{},[409,2482,2483],{},"\u003C",[1628,2485,2486,2488],{},[73,2487,27],{},[409,2489,2490],{},"∘",[76,2492,2493],{"encoding":78},"\u003C 1^\\circ",[25,2495,2497,2510],{"className":2496,"ariaHidden":84},[83],[25,2498,2500,2504,2507],{"className":2499},[88],[25,2501],{"className":2502,"style":2503},[92],"height:0.5782em;vertical-align:-0.0391em;",[25,2505,2483],{"className":2506},[1200],[25,2508],{"className":2509,"style":1196},[478],[25,2511,2513,2517],{"className":2512},[88],[25,2514],{"className":2515,"style":2516},[92],"height:0.6741em;",[25,2518,2520,2523],{"className":2519},[97],[25,2521,27],{"className":2522},[97],[25,2524,2526],{"className":2525},[106],[25,2527,2529],{"className":2528},[110],[25,2530,2532],{"className":2531},[115],[25,2533,2535],{"className":2534,"style":2516},[119],[25,2536,2537,2540],{"style":1713},[25,2538],{"className":2539,"style":128},[127],[25,2541,2543],{"className":2542},[132,133,134,135],[25,2544,2490],{"className":2545},[2546,135],"mbin",") .",[2148,2549,2551],{"id":2550},"angular-measurement","Angular Measurement",[16,2553,2554,2555,2667],{},"Measure the angle ",[25,2556,2558,2579],{"className":2557},[50],[25,2559,2561],{"className":2560},[54],[56,2562,2563],{"xmlns":58},[60,2564,2565,2577],{},[63,2566,2567],{},[815,2568,2569,2575],{"accent":84},[66,2570,2571,2573],{},[69,2572,71],{},[69,2574,251],{},[409,2576,825],{},[76,2578,828],{"encoding":78},[25,2580,2582],{"className":2581,"ariaHidden":84},[83],[25,2583,2585,2588],{"className":2584},[88],[25,2586],{"className":2587,"style":838},[92],[25,2589,2591],{"className":2590},[97,842],[25,2592,2594,2659],{"className":2593},[110,111],[25,2595,2597,2656],{"className":2596},[115],[25,2598,2600,2645],{"className":2599,"style":852},[119],[25,2601,2602,2605],{"style":855},[25,2603],{"className":2604,"style":859},[127],[25,2606,2608,2611],{"className":2607},[97],[25,2609,71],{"className":2610,"style":102},[97,101],[25,2612,2614],{"className":2613},[106],[25,2615,2617,2637],{"className":2616},[110,111],[25,2618,2620,2634],{"className":2619},[115],[25,2621,2623],{"className":2622,"style":282},[119],[25,2624,2625,2628],{"style":123},[25,2626],{"className":2627,"style":128},[127],[25,2629,2631],{"className":2630},[132,133,134,135],[25,2632,251],{"className":2633},[97,101,135],[25,2635,143],{"className":2636},[142],[25,2638,2640],{"className":2639},[115],[25,2641,2643],{"className":2642,"style":150},[119],[25,2644],{},[25,2646,2647,2650],{"style":902},[25,2648],{"className":2649,"style":859},[127],[25,2651,2653],{"className":2652,"style":910},[909],[25,2654,825],{"className":2655},[97],[25,2657,143],{"className":2658},[142],[25,2660,2662],{"className":2661},[115],[25,2663,2665],{"className":2664,"style":150},[119],[25,2666],{}," between the wires with extreme precision (multiple sets on both faces) .",[2148,2669,2671],{"id":2670},"linear-measurement","Linear Measurement",[16,2673,2674,2675,2791,2792,2908,2909,3025],{},"Measure the distances ",[25,2676,2678,2702],{"className":2677},[50],[25,2679,2681],{"className":2680},[54],[56,2682,2683],{"xmlns":58},[60,2684,2685,2699],{},[63,2686,2687,2693],{},[66,2688,2689,2691],{},[69,2690,71],{},[73,2692,27],{},[66,2694,2695,2697],{},[69,2696,71],{},[73,2698,172],{},[76,2700,2701],{"encoding":78},"W_1W_2",[25,2703,2705],{"className":2704,"ariaHidden":84},[83],[25,2706,2708,2711,2751],{"className":2707},[88],[25,2709],{"className":2710,"style":93},[92],[25,2712,2714,2717],{"className":2713},[97],[25,2715,71],{"className":2716,"style":102},[97,101],[25,2718,2720],{"className":2719},[106],[25,2721,2723,2743],{"className":2722},[110,111],[25,2724,2726,2740],{"className":2725},[115],[25,2727,2729],{"className":2728,"style":120},[119],[25,2730,2731,2734],{"style":123},[25,2732],{"className":2733,"style":128},[127],[25,2735,2737],{"className":2736},[132,133,134,135],[25,2738,27],{"className":2739},[97,135],[25,2741,143],{"className":2742},[142],[25,2744,2746],{"className":2745},[115],[25,2747,2749],{"className":2748,"style":150},[119],[25,2750],{},[25,2752,2754,2757],{"className":2753},[97],[25,2755,71],{"className":2756,"style":102},[97,101],[25,2758,2760],{"className":2759},[106],[25,2761,2763,2783],{"className":2762},[110,111],[25,2764,2766,2780],{"className":2765},[115],[25,2767,2769],{"className":2768,"style":120},[119],[25,2770,2771,2774],{"style":123},[25,2772],{"className":2773,"style":128},[127],[25,2775,2777],{"className":2776},[132,133,134,135],[25,2778,172],{"className":2779},[97,135],[25,2781,143],{"className":2782},[142],[25,2784,2786],{"className":2785},[115],[25,2787,2789],{"className":2788,"style":150},[119],[25,2790],{},", ",[25,2793,2795,2819],{"className":2794},[50],[25,2796,2798],{"className":2797},[54],[56,2799,2800],{"xmlns":58},[60,2801,2802,2816],{},[63,2803,2804,2810],{},[66,2805,2806,2808],{},[69,2807,71],{},[69,2809,251],{},[66,2811,2812,2814],{},[69,2813,71],{},[73,2815,27],{},[76,2817,2818],{"encoding":78},"W_sW_1",[25,2820,2822],{"className":2821,"ariaHidden":84},[83],[25,2823,2825,2828,2868],{"className":2824},[88],[25,2826],{"className":2827,"style":93},[92],[25,2829,2831,2834],{"className":2830},[97],[25,2832,71],{"className":2833,"style":102},[97,101],[25,2835,2837],{"className":2836},[106],[25,2838,2840,2860],{"className":2839},[110,111],[25,2841,2843,2857],{"className":2842},[115],[25,2844,2846],{"className":2845,"style":282},[119],[25,2847,2848,2851],{"style":123},[25,2849],{"className":2850,"style":128},[127],[25,2852,2854],{"className":2853},[132,133,134,135],[25,2855,251],{"className":2856},[97,101,135],[25,2858,143],{"className":2859},[142],[25,2861,2863],{"className":2862},[115],[25,2864,2866],{"className":2865,"style":150},[119],[25,2867],{},[25,2869,2871,2874],{"className":2870},[97],[25,2872,71],{"className":2873,"style":102},[97,101],[25,2875,2877],{"className":2876},[106],[25,2878,2880,2900],{"className":2879},[110,111],[25,2881,2883,2897],{"className":2882},[115],[25,2884,2886],{"className":2885,"style":120},[119],[25,2887,2888,2891],{"style":123},[25,2889],{"className":2890,"style":128},[127],[25,2892,2894],{"className":2893},[132,133,134,135],[25,2895,27],{"className":2896},[97,135],[25,2898,143],{"className":2899},[142],[25,2901,2903],{"className":2902},[115],[25,2904,2906],{"className":2905,"style":150},[119],[25,2907],{},", and ",[25,2910,2912,2936],{"className":2911},[50],[25,2913,2915],{"className":2914},[54],[56,2916,2917],{"xmlns":58},[60,2918,2919,2933],{},[63,2920,2921,2927],{},[66,2922,2923,2925],{},[69,2924,71],{},[69,2926,251],{},[66,2928,2929,2931],{},[69,2930,71],{},[73,2932,172],{},[76,2934,2935],{"encoding":78},"W_sW_2",[25,2937,2939],{"className":2938,"ariaHidden":84},[83],[25,2940,2942,2945,2985],{"className":2941},[88],[25,2943],{"className":2944,"style":93},[92],[25,2946,2948,2951],{"className":2947},[97],[25,2949,71],{"className":2950,"style":102},[97,101],[25,2952,2954],{"className":2953},[106],[25,2955,2957,2977],{"className":2956},[110,111],[25,2958,2960,2974],{"className":2959},[115],[25,2961,2963],{"className":2962,"style":282},[119],[25,2964,2965,2968],{"style":123},[25,2966],{"className":2967,"style":128},[127],[25,2969,2971],{"className":2970},[132,133,134,135],[25,2972,251],{"className":2973},[97,101,135],[25,2975,143],{"className":2976},[142],[25,2978,2980],{"className":2979},[115],[25,2981,2983],{"className":2982,"style":150},[119],[25,2984],{},[25,2986,2988,2991],{"className":2987},[97],[25,2989,71],{"className":2990,"style":102},[97,101],[25,2992,2994],{"className":2993},[106],[25,2995,2997,3017],{"className":2996},[110,111],[25,2998,3000,3014],{"className":2999},[115],[25,3001,3003],{"className":3002,"style":120},[119],[25,3004,3005,3008],{"style":123},[25,3006],{"className":3007,"style":128},[127],[25,3009,3011],{"className":3010},[132,133,134,135],[25,3012,172],{"className":3013},[97,135],[25,3015,143],{"className":3016},[142],[25,3018,3020],{"className":3019},[115],[25,3021,3023],{"className":3022,"style":150},[119],[25,3024],{}," to the nearest millimeter .",[2148,3027,3029],{"id":3028},"underground-replication","Underground Replication",[16,3031,3032,3033,3104],{},"Repeat the process at the bottom of the shaft at station ",[25,3034,3036,3055],{"className":3035},[50],[25,3037,3039],{"className":3038},[54],[56,3040,3041],{"xmlns":58},[60,3042,3043,3052],{},[63,3044,3045],{},[66,3046,3047,3049],{},[69,3048,71],{},[69,3050,3051],{},"u",[76,3053,3054],{"encoding":78},"W_u",[25,3056,3058],{"className":3057,"ariaHidden":84},[83],[25,3059,3061,3064],{"className":3060},[88],[25,3062],{"className":3063,"style":93},[92],[25,3065,3067,3070],{"className":3066},[97],[25,3068,71],{"className":3069,"style":102},[97,101],[25,3071,3073],{"className":3072},[106],[25,3074,3076,3096],{"className":3075},[110,111],[25,3077,3079,3093],{"className":3078},[115],[25,3080,3082],{"className":3081,"style":282},[119],[25,3083,3084,3087],{"style":123},[25,3085],{"className":3086,"style":128},[127],[25,3088,3090],{"className":3089},[132,133,134,135],[25,3091,3051],{"className":3092},[97,101,135],[25,3094,143],{"className":3095},[142],[25,3097,3099],{"className":3098},[115],[25,3100,3102],{"className":3101,"style":150},[119],[25,3103],{}," to transfer the bearing to the tunnel traverse .",[11,3106,3108],{"id":3107},"formula-breakdown","Formula Breakdown",[16,3110,3111,3112,1768,3201],{},"The precision of the orientation is governed by the ",[20,3113,3114,3115,3200],{},"Standard Error of the derived angle (",[25,3116,3118,3141],{"className":3117},[50],[25,3119,3121],{"className":3120},[54],[56,3122,3123],{"xmlns":58},[60,3124,3125,3138],{},[63,3126,3127],{},[66,3128,3129,3132],{},[69,3130,3131],{},"σ",[63,3133,3134,3136],{},[69,3135,71],{},[73,3137,27],{},[76,3139,3140],{"encoding":78},"\\sigma_{W1}",[25,3142,3144],{"className":3143,"ariaHidden":84},[83],[25,3145,3147,3151],{"className":3146},[88],[25,3148],{"className":3149,"style":3150},[92],"height:0.5806em;vertical-align:-0.15em;",[25,3152,3154,3158],{"className":3153},[97],[25,3155,3131],{"className":3156,"style":3157},[97,101],"margin-right:0.0359em;",[25,3159,3161],{"className":3160},[106],[25,3162,3164,3192],{"className":3163},[110,111],[25,3165,3167,3189],{"className":3166},[115],[25,3168,3171],{"className":3169,"style":3170},[119],"height:0.3283em;",[25,3172,3174,3177],{"style":3173},"top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;",[25,3175],{"className":3176,"style":128},[127],[25,3178,3180],{"className":3179},[132,133,134,135],[25,3181,3183,3186],{"className":3182},[97,135],[25,3184,71],{"className":3185,"style":102},[97,101,135],[25,3187,27],{"className":3188},[97,135],[25,3190,143],{"className":3191},[142],[25,3193,3195],{"className":3194},[115],[25,3196,3198],{"className":3197,"style":150},[119],[25,3199],{},")",[25,3202,3204,3247],{"className":3203},[50],[25,3205,3207],{"className":3206},[54],[56,3208,3209],{"xmlns":58},[60,3210,3211,3244],{},[63,3212,3213,3223,3225,3228,3234],{},[66,3214,3215,3217],{},[69,3216,3131],{},[63,3218,3219,3221],{},[69,3220,71],{},[73,3222,27],{},[409,3224,1073],{},[409,3226,3227],{},"±",[1075,3229,3230,3232],{},[69,3231,784],{},[69,3233,612],{},[66,3235,3236,3238],{},[69,3237,3131],{},[63,3239,3240,3242],{},[69,3241,71],{},[69,3243,251],{},[76,3245,3246],{"encoding":78},"\\sigma_{W1} = \\pm \\frac{c}{w} \\sigma_{Ws}",[25,3248,3250,3311],{"className":3249,"ariaHidden":84},[83],[25,3251,3253,3256,3302,3305,3308],{"className":3252},[88],[25,3254],{"className":3255,"style":3150},[92],[25,3257,3259,3262],{"className":3258},[97],[25,3260,3131],{"className":3261,"style":3157},[97,101],[25,3263,3265],{"className":3264},[106],[25,3266,3268,3294],{"className":3267},[110,111],[25,3269,3271,3291],{"className":3270},[115],[25,3272,3274],{"className":3273,"style":3170},[119],[25,3275,3276,3279],{"style":3173},[25,3277],{"className":3278,"style":128},[127],[25,3280,3282],{"className":3281},[132,133,134,135],[25,3283,3285,3288],{"className":3284},[97,135],[25,3286,71],{"className":3287,"style":102},[97,101,135],[25,3289,27],{"className":3290},[97,135],[25,3292,143],{"className":3293},[142],[25,3295,3297],{"className":3296},[115],[25,3298,3300],{"className":3299,"style":150},[119],[25,3301],{},[25,3303],{"className":3304,"style":1196},[478],[25,3306,1073],{"className":3307},[1200],[25,3309],{"className":3310,"style":1196},[478],[25,3312,3314,3318,3321,3389],{"className":3313},[88],[25,3315],{"className":3316,"style":3317},[92],"height:1.0404em;vertical-align:-0.345em;",[25,3319,3227],{"className":3320},[97],[25,3322,3324,3327,3386],{"className":3323},[97],[25,3325],{"className":3326},[1217,1218],[25,3328,3330],{"className":3329},[1075],[25,3331,3333,3378],{"className":3332},[110,111],[25,3334,3336,3375],{"className":3335},[115],[25,3337,3339,3353,3361],{"className":3338,"style":1231},[119],[25,3340,3341,3344],{"style":1234},[25,3342],{"className":3343,"style":859},[127],[25,3345,3347],{"className":3346},[132,133,134,135],[25,3348,3350],{"className":3349},[97,135],[25,3351,612],{"className":3352,"style":628},[97,101,135],[25,3354,3355,3358],{"style":1249},[25,3356],{"className":3357,"style":859},[127],[25,3359],{"className":3360,"style":1257},[1256],[25,3362,3363,3366],{"style":1260},[25,3364],{"className":3365,"style":859},[127],[25,3367,3369],{"className":3368},[132,133,134,135],[25,3370,3372],{"className":3371},[97,135],[25,3373,784],{"className":3374},[97,101,135],[25,3376,143],{"className":3377},[142],[25,3379,3381],{"className":3380},[115],[25,3382,3384],{"className":3383,"style":1282},[119],[25,3385],{},[25,3387],{"className":3388},[1288,1218],[25,3390,3392,3395],{"className":3391},[97],[25,3393,3131],{"className":3394,"style":3157},[97,101],[25,3396,3398],{"className":3397},[106],[25,3399,3401,3427],{"className":3400},[110,111],[25,3402,3404,3424],{"className":3403},[115],[25,3405,3407],{"className":3406,"style":3170},[119],[25,3408,3409,3412],{"style":3173},[25,3410],{"className":3411,"style":128},[127],[25,3413,3415],{"className":3414},[132,133,134,135],[25,3416,3418,3421],{"className":3417},[97,135],[25,3419,71],{"className":3420,"style":102},[97,101,135],[25,3422,251],{"className":3423},[97,101,135],[25,3425,143],{"className":3426},[142],[25,3428,3430],{"className":3429},[115],[25,3431,3433],{"className":3432,"style":150},[119],[25,3434],{},[3436,3437,3439],"callout",{"type":3438},"warning",[16,3440,3441,3442,3540,3541,3569,3570,3598],{},"This formula proves that the error in your transferred bearing is your observational error multiplied by the ratio ",[25,3443,3445,3463],{"className":3444},[50],[25,3446,3448],{"className":3447},[54],[56,3449,3450],{"xmlns":58},[60,3451,3452,3460],{},[63,3453,3454],{},[1075,3455,3456,3458],{},[69,3457,784],{},[69,3459,612],{},[76,3461,3462],{"encoding":78},"\\frac{c}{w}",[25,3464,3466],{"className":3465,"ariaHidden":84},[83],[25,3467,3469,3472],{"className":3468},[88],[25,3470],{"className":3471,"style":3317},[92],[25,3473,3475,3478,3537],{"className":3474},[97],[25,3476],{"className":3477},[1217,1218],[25,3479,3481],{"className":3480},[1075],[25,3482,3484,3529],{"className":3483},[110,111],[25,3485,3487,3526],{"className":3486},[115],[25,3488,3490,3504,3512],{"className":3489,"style":1231},[119],[25,3491,3492,3495],{"style":1234},[25,3493],{"className":3494,"style":859},[127],[25,3496,3498],{"className":3497},[132,133,134,135],[25,3499,3501],{"className":3500},[97,135],[25,3502,612],{"className":3503,"style":628},[97,101,135],[25,3505,3506,3509],{"style":1249},[25,3507],{"className":3508,"style":859},[127],[25,3510],{"className":3511,"style":1257},[1256],[25,3513,3514,3517],{"style":1260},[25,3515],{"className":3516,"style":859},[127],[25,3518,3520],{"className":3519},[132,133,134,135],[25,3521,3523],{"className":3522},[97,135],[25,3524,784],{"className":3525},[97,101,135],[25,3527,143],{"className":3528},[142],[25,3530,3532],{"className":3531},[115],[25,3533,3535],{"className":3534,"style":1282},[119],[25,3536],{},[25,3538],{"className":3539},[1288,1218],". To minimize error, you must make ",[25,3542,3544,3557],{"className":3543},[50],[25,3545,3547],{"className":3546},[54],[56,3548,3549],{"xmlns":58},[60,3550,3551,3555],{},[63,3552,3553],{},[69,3554,784],{},[76,3556,784],{"encoding":78},[25,3558,3560],{"className":3559,"ariaHidden":84},[83],[25,3561,3563,3566],{"className":3562},[88],[25,3564],{"className":3565,"style":624},[92],[25,3567,784],{"className":3568},[97,101]," (instrument to wire) as small as possible and ",[25,3571,3573,3586],{"className":3572},[50],[25,3574,3576],{"className":3575},[54],[56,3577,3578],{"xmlns":58},[60,3579,3580,3584],{},[63,3581,3582],{},[69,3583,612],{},[76,3585,612],{"encoding":78},[25,3587,3589],{"className":3588,"ariaHidden":84},[83],[25,3590,3592,3595],{"className":3591},[88],[25,3593],{"className":3594,"style":624},[92],[25,3596,612],{"className":3597,"style":628},[97,101]," (wire to wire) as large as possible .",[11,3600,3602],{"id":3601},"practical-tips","Practical Tips",[3604,3605,3606,3612],"ul",{},[595,3607,3608,3611],{},[20,3609,3610],{},"Use Heavy Plumb-bobs:"," Immerse them in oil or water to dampen oscillations caused by shaft air currents .",[595,3613,3614,3617,3618,3687,3688,3729],{},[20,3615,3616],{},"Focusing Matters:"," The instrument at ",[25,3619,3621,3638],{"className":3620},[50],[25,3622,3624],{"className":3623},[54],[56,3625,3626],{"xmlns":58},[60,3627,3628,3636],{},[63,3629,3630],{},[66,3631,3632,3634],{},[69,3633,71],{},[69,3635,251],{},[76,3637,254],{"encoding":78},[25,3639,3641],{"className":3640,"ariaHidden":84},[83],[25,3642,3644,3647],{"className":3643},[88],[25,3645],{"className":3646,"style":93},[92],[25,3648,3650,3653],{"className":3649},[97],[25,3651,71],{"className":3652,"style":102},[97,101],[25,3654,3656],{"className":3655},[106],[25,3657,3659,3679],{"className":3658},[110,111],[25,3660,3662,3676],{"className":3661},[115],[25,3663,3665],{"className":3664,"style":282},[119],[25,3666,3667,3670],{"style":123},[25,3668],{"className":3669,"style":128},[127],[25,3671,3673],{"className":3672},[132,133,134,135],[25,3674,251],{"className":3675},[97,101,135],[25,3677,143],{"className":3678},[142],[25,3680,3682],{"className":3681},[115],[25,3683,3685],{"className":3684,"style":150},[119],[25,3686],{}," should be as close to the front wire as focusing allows to keep the ratio ",[25,3689,3691,3710],{"className":3690},[50],[25,3692,3694],{"className":3693},[54],[56,3695,3696],{"xmlns":58},[60,3697,3698,3707],{},[63,3699,3700,3702,3705],{},[69,3701,784],{},[69,3703,3704],{"mathvariant":1636},"\u002F",[69,3706,612],{},[76,3708,3709],{"encoding":78},"c\u002Fw",[25,3711,3713],{"className":3712,"ariaHidden":84},[83],[25,3714,3716,3720,3723,3726],{"className":3715},[88],[25,3717],{"className":3718,"style":3719},[92],"height:1em;vertical-align:-0.25em;",[25,3721,784],{"className":3722},[97,101],[25,3724,3704],{"className":3725},[97],[25,3727,612],{"className":3728,"style":628},[97,101]," low .",[11,3731,3733],{"id":3732},"common-mistakes","Common Mistakes",[3604,3735,3736,3742],{},[595,3737,3738,3741],{},[20,3739,3740],{},"Assuming Verticality:"," Ventilation in the shaft can deflect wires. Ensure all fans are off during the correlation .",[595,3743,3744,3747,3748,3860],{},[20,3745,3746],{},"Poor Triangle Shape:"," Making the triangle too \"fat\" (large ",[25,3749,3751,3772],{"className":3750},[50],[25,3752,3754],{"className":3753},[54],[56,3755,3756],{"xmlns":58},[60,3757,3758,3770],{},[63,3759,3760],{},[815,3761,3762,3768],{"accent":84},[66,3763,3764,3766],{},[69,3765,71],{},[69,3767,251],{},[409,3769,825],{},[76,3771,828],{"encoding":78},[25,3773,3775],{"className":3774,"ariaHidden":84},[83],[25,3776,3778,3781],{"className":3777},[88],[25,3779],{"className":3780,"style":838},[92],[25,3782,3784],{"className":3783},[97,842],[25,3785,3787,3852],{"className":3786},[110,111],[25,3788,3790,3849],{"className":3789},[115],[25,3791,3793,3838],{"className":3792,"style":852},[119],[25,3794,3795,3798],{"style":855},[25,3796],{"className":3797,"style":859},[127],[25,3799,3801,3804],{"className":3800},[97],[25,3802,71],{"className":3803,"style":102},[97,101],[25,3805,3807],{"className":3806},[106],[25,3808,3810,3830],{"className":3809},[110,111],[25,3811,3813,3827],{"className":3812},[115],[25,3814,3816],{"className":3815,"style":282},[119],[25,3817,3818,3821],{"style":123},[25,3819],{"className":3820,"style":128},[127],[25,3822,3824],{"className":3823},[132,133,134,135],[25,3825,251],{"className":3826},[97,101,135],[25,3828,143],{"className":3829},[142],[25,3831,3833],{"className":3832},[115],[25,3834,3836],{"className":3835,"style":150},[119],[25,3837],{},[25,3839,3840,3843],{"style":902},[25,3841],{"className":3842,"style":859},[127],[25,3844,3846],{"className":3845,"style":910},[909],[25,3847,825],{"className":3848},[97],[25,3850,143],{"className":3851},[142],[25,3853,3855],{"className":3854},[115],[25,3856,3858],{"className":3857,"style":150},[119],[25,3859],{},") reduces the accuracy of the sine approximation .",[11,3862,3864],{"id":3863},"faq","FAQ",[3866,3867,3868,3875],"accordion",{},[3869,3870,3872],"accordion-item",{"label":3871},"What is the ideal angle for a Weisbach Triangle?",[16,3873,3874],{},"The angle at the instrument should be as small as possible, ideally less than 1 minute of arc, but certainly less than 1 degree .",[3869,3876,3878],{"label":3877},"Why not use a Gyro-theodolite instead?",[16,3879,3880],{},"A gyro-theodolite is excellent but expensive and requires specialized calibration. The Weisbach method uses standard equipment .",[11,3882,3884],{"id":3883},"conclusion","Conclusion",[16,3886,3887,3888,3926],{},"The Weisbach Triangle remains a fundamental skill for the engineering surveyor. By mastering the geometry and minimizing the ",[25,3889,3891,3908],{"className":3890},[50],[25,3892,3894],{"className":3893},[54],[56,3895,3896],{"xmlns":58},[60,3897,3898,3906],{},[63,3899,3900,3902,3904],{},[69,3901,784],{},[69,3903,3704],{"mathvariant":1636},[69,3905,612],{},[76,3907,3709],{"encoding":78},[25,3909,3911],{"className":3910,"ariaHidden":84},[83],[25,3912,3914,3917,3920,3923],{"className":3913},[88],[25,3915],{"className":3916,"style":3719},[92],[25,3918,784],{"className":3919},[97,101],[25,3921,3704],{"className":3922},[97],[25,3924,612],{"className":3925,"style":628},[97,101]," ratio, you ensure your tunnel projects meet with sub-centimeter precision.",[11,3928,3930],{"id":3929},"references","References",[16,3932,3933,3934,3938,3939,386],{},"Schofield, W. (2001). ",[3935,3936,3937],"em",{},"Engineering Surveying",". 5th ed. Butterworth-Heinemann. ",[25,3940,27],{},{"title":3942,"searchDepth":3943,"depth":3943,"links":3944},"",2,[3945,3946,3947,3948,3949,3950,3958,3959,3960,3961,3962,3963],{"id":13,"depth":3943,"text":14},{"id":35,"depth":3943,"text":36},{"id":42,"depth":3943,"text":43},{"id":228,"depth":3943,"text":229},{"id":378,"depth":3943,"text":379},{"id":2142,"depth":3943,"text":2143,"children":3951},[3952,3954,3955,3956,3957],{"id":2150,"depth":3953,"text":2151},3,{"id":2281,"depth":3953,"text":2282},{"id":2550,"depth":3953,"text":2551},{"id":2670,"depth":3953,"text":2671},{"id":3028,"depth":3953,"text":3029},{"id":3107,"depth":3943,"text":3108},{"id":3601,"depth":3943,"text":3602},{"id":3732,"depth":3943,"text":3733},{"id":3863,"depth":3943,"text":3864},{"id":3883,"depth":3943,"text":3884},{"id":3929,"depth":3943,"text":3930},"Underground Surveying","A technical guide to orienting underground surveys from a single shaft using the Weisbach triangle method.","md","\u002Fimages\u002Fweisbach-triangle.jpg",{"authorCredentials":3969,"authorBio":3970,"commentCount":3971,"tags":3972,"relatedTools":3976,"relatedArticles":3979,"sources":3982},"LS, P.Eng","Expert in engineering surveying and underground dimensional control.",0,[3964,3973,3974,3975],"Orientation","Mining","Tunnels",[3977,3978],"Total Station","Plumb Wires",[3980,3981],"gyro-theodolite-orientation","traverse-adjustment-math",[3983],{"title":3984,"type":3985,"note":3986},"Engineering Surveying (Fifth Edition)","Book","Primary source for the mathematical derivation and workflow.",true,"\u002Fblog\u002Fthe-weisbach-triangle-mastering-orientation-via-single-shaft-correlation","2024-03-20",12,{"title":5,"description":3965},{"loc":3988},"blog\u002F1.the-weisbach-triangle-mastering-orientation-via-single-shaft-correlation","RQxQAAHL5RUlV7-lpCRlnU3uWyoXp72qA2unjYnduqM",[3996,3997],null,{"title":3998,"path":3999,"stem":4000,"description":4001,"children":-1},"Combination of Errors: Calculating the Propagation of Uncertainty","\u002Fblog\u002Fcombination-of-errors-calculating-the-propagation-of-uncertainty","blog\u002F2.combination-of-errors-calculating-the-propagation-of-uncertainty","How to mathematically determine the total error in derived surveying quantities like areas, volumes, and coordinates.",1783851952142]