[{"data":1,"prerenderedAt":3492},["ShallowReactive",2],{"blog-\u002Fblog\u002Freciprocal-levelling-and-curvature":3,"blog-surround-\u002Fblog\u002Freciprocal-levelling-and-curvature":3481},{"id":4,"title":5,"author":6,"body":7,"category":3450,"description":3451,"extension":3452,"image":3453,"meta":3454,"navigation":3473,"path":3474,"publishedAt":3475,"readTime":3476,"seo":3477,"sitemap":3478,"stem":3479,"__hash__":3480},"blog\u002Fblog\u002F8.reciprocal-levelling-and-curvature.md","Reciprocal Levelling: Defeating Earth Curvature and Refraction","Surveyiiing Expert",{"type":8,"value":9,"toc":3425},"minimark",[10,15,86,90,93,181,185,294,298,303,573,577,1378,1382,1767,1771,1838,2165,2741,2745,3170,3174,3246,3250,3269,3273,3404,3408,3411,3415],[11,12,14],"h2",{"id":13},"overview","Overview",[16,17,18,19,76,77,81,82,85],"p",{},"In standard differential levelling, we keep backsights and foresights equal to cancel out errors. But what happens when you need to transfer a level across a ",[20,21,24,51],"span",{"className":22},[23],"katex",[20,25,28],{"className":26},[27],"katex-mathml",[29,30,32],"math",{"xmlns":31},"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML",[33,34,35,46],"semantics",{},[36,37,38,42],"mrow",{},[39,40,41],"mn",{},"500",[43,44,45],"mtext",{}," m",[47,48,50],"annotation",{"encoding":49},"application\u002Fx-tex","500\\text{ m}",[20,52,56],{"className":53,"ariaHidden":55},[54],"katex-html","true",[20,57,60,65,69],{"className":58},[59],"base",[20,61],{"className":62,"style":64},[63],"strut","height:0.6444em;",[20,66,41],{"className":67},[68],"mord",[20,70,73],{"className":71},[68,72],"text",[20,74,45],{"className":75},[68]," wide river? You cannot set up in the middle. ",[78,79,80],"strong",{},"Reciprocal Levelling"," is the solution: a specialized field procedure that mathematically eliminates Earth curvature, atmospheric refraction, and collimation error without needing equal sight lengths ",[20,83,84],{},"34",".",[11,87,89],{"id":88},"why-this-matters","Why This Matters",[16,91,92],{},"Over long distances, the Earth's surface curves away from the horizontal line of sight, and the atmosphere \"bends\" the light ray (refraction).",[94,95,96,139],"ul",{},[97,98,99,135,136,85],"li",{},[78,100,101,102,134],{},"Curvature (",[20,103,105,120],{"className":104},[23],[20,106,108],{"className":107},[27],[29,109,110],{"xmlns":31},[33,111,112,118],{},[36,113,114],{},[115,116,117],"mi",{},"c",[47,119,117],{"encoding":49},[20,121,123],{"className":122,"ariaHidden":55},[54],[20,124,126,130],{"className":125},[59],[20,127],{"className":128,"style":129},[63],"height:0.4306em;",[20,131,117],{"className":132},[68,133],"mathnormal","):"," Always makes the staff reading too high ",[20,137,138],{},"35",[97,140,141,174,175,177,178,85],{},[78,142,143,144,134],{},"Refraction (",[20,145,147,161],{"className":146},[23],[20,148,150],{"className":149},[27],[29,151,152],{"xmlns":31},[33,153,154,159],{},[36,155,156],{},[115,157,158],{},"r",[47,160,158],{"encoding":49},[20,162,164],{"className":163,"ariaHidden":55},[54],[20,165,167,170],{"className":166},[59],[20,168],{"className":169,"style":129},[63],[20,171,158],{"className":172,"style":173},[68,133],"margin-right:0.0278em;"," Usually bends the ray toward the Earth, making the reading too low ",[20,176,138],{},".\nTogether, these can cause errors of several centimeters over just a few hundred meters, which is unacceptable for bridge or dam construction ",[20,179,180],{},"35, 36",[11,182,184],{"id":183},"theory","Theory",[16,186,187,188,260,261,290,291,85],{},"By taking observations from both sides of the obstacle, the error ",[20,189,191,218],{"className":190},[23],[20,192,194],{"className":193},[27],[29,195,196],{"xmlns":31},[33,197,198,215],{},[36,199,200,205,207,210,212],{},[201,202,204],"mo",{"stretchy":203},"false","(",[115,206,117],{},[201,208,209],{},"−",[115,211,158],{},[201,213,214],{"stretchy":203},")",[47,216,217],{"encoding":49},"(c - r)",[20,219,221,247],{"className":220,"ariaHidden":55},[54],[20,222,224,228,232,235,240,244],{"className":223},[59],[20,225],{"className":226,"style":227},[63],"height:1em;vertical-align:-0.25em;",[20,229,204],{"className":230},[231],"mopen",[20,233,117],{"className":234},[68,133],[20,236],{"className":237,"style":239},[238],"mspace","margin-right:0.2222em;",[20,241,209],{"className":242},[243],"mbin",[20,245],{"className":246,"style":239},[238],[20,248,250,253,256],{"className":249},[59],[20,251],{"className":252,"style":227},[63],[20,254,158],{"className":255,"style":173},[68,133],[20,257,214],{"className":258},[259],"mclose"," plus any residual collimation error (",[20,262,264,278],{"className":263},[23],[20,265,267],{"className":266},[27],[29,268,269],{"xmlns":31},[33,270,271,276],{},[36,272,273],{},[115,274,275],{},"e",[47,277,275],{"encoding":49},[20,279,281],{"className":280,"ariaHidden":55},[54],[20,282,284,287],{"className":283},[59],[20,285],{"className":286,"style":129},[63],[20,288,275],{"className":289},[68,133],") affects the \"forward\" sight and \"backward\" sight in opposite directions relative to the true difference in height ",[20,292,293],{},"34, 37",[11,295,297],{"id":296},"mathematical-principles","Mathematical Principles",[299,300,302],"h3",{"id":301},"_1-combined-curvature-and-refraction","1. Combined Curvature and Refraction",[16,304,305,306,358,359,389,390,526,527,530,531],{},"The combined error (",[20,307,309,327],{"className":308},[23],[20,310,312],{"className":311},[27],[29,313,314],{"xmlns":31},[33,315,316,324],{},[36,317,318,320,322],{},[115,319,117],{},[201,321,209],{},[115,323,158],{},[47,325,326],{"encoding":49},"c-r",[20,328,330,349],{"className":329,"ariaHidden":55},[54],[20,331,333,337,340,343,346],{"className":332},[59],[20,334],{"className":335,"style":336},[63],"height:0.6667em;vertical-align:-0.0833em;",[20,338,117],{"className":339},[68,133],[20,341],{"className":342,"style":239},[238],[20,344,209],{"className":345},[243],[20,347],{"className":348,"style":239},[238],[20,350,352,355],{"className":351},[59],[20,353],{"className":354,"style":129},[63],[20,356,158],{"className":357,"style":173},[68,133],") in meters for a distance ",[20,360,362,376],{"className":361},[23],[20,363,365],{"className":364},[27],[29,366,367],{"xmlns":31},[33,368,369,374],{},[36,370,371],{},[115,372,373],{},"D",[47,375,373],{"encoding":49},[20,377,379],{"className":378,"ariaHidden":55},[54],[20,380,382,386],{"className":381},[59],[20,383],{"className":384,"style":385},[63],"height:0.6833em;",[20,387,373],{"className":388,"style":173},[68,133]," (in km) is:\n",[20,391,393,429],{"className":392},[23],[20,394,396],{"className":395},[27],[29,397,398],{"xmlns":31},[33,399,400,426],{},[36,401,402,404,406,408,410,412,415,418],{},[201,403,204],{"stretchy":203},[115,405,117],{},[201,407,209],{},[115,409,158],{},[201,411,214],{"stretchy":203},[201,413,414],{},"=",[39,416,417],{},"0.0673",[419,420,421,423],"msup",{},[115,422,373],{},[39,424,425],{},"2",[47,427,428],{"encoding":49},"(c - r) = 0.0673 D^2",[20,430,432,453,476],{"className":431,"ariaHidden":55},[54],[20,433,435,438,441,444,447,450],{"className":434},[59],[20,436],{"className":437,"style":227},[63],[20,439,204],{"className":440},[231],[20,442,117],{"className":443},[68,133],[20,445],{"className":446,"style":239},[238],[20,448,209],{"className":449},[243],[20,451],{"className":452,"style":239},[238],[20,454,456,459,462,465,469,473],{"className":455},[59],[20,457],{"className":458,"style":227},[63],[20,460,158],{"className":461,"style":173},[68,133],[20,463,214],{"className":464},[259],[20,466],{"className":467,"style":468},[238],"margin-right:0.2778em;",[20,470,414],{"className":471},[472],"mrel",[20,474],{"className":475,"style":468},[238],[20,477,479,483,486],{"className":478},[59],[20,480],{"className":481,"style":482},[63],"height:0.8141em;",[20,484,417],{"className":485},[68],[20,487,489,492],{"className":488},[68],[20,490,373],{"className":491,"style":173},[68,133],[20,493,496],{"className":494},[495],"msupsub",[20,497,500],{"className":498},[499],"vlist-t",[20,501,504],{"className":502},[503],"vlist-r",[20,505,508],{"className":506,"style":482},[507],"vlist",[20,509,511,516],{"style":510},"top:-3.063em;margin-right:0.05em;",[20,512],{"className":513,"style":515},[514],"pstrut","height:2.7em;",[20,517,523],{"className":518},[519,520,521,522],"sizing","reset-size6","size3","mtight",[20,524,425],{"className":525},[68,522]," ",[20,528,529],{},"36, 37",".\n",[532,533,534,535,572],"em",{},"(Assuming refraction is ",[20,536,538,560],{"className":537},[23],[20,539,541],{"className":540},[27],[29,542,543],{"xmlns":31},[33,544,545,557],{},[36,546,547,550,554],{},[39,548,549],{},"1",[115,551,553],{"mathvariant":552},"normal","\u002F",[39,555,556],{},"7",[47,558,559],{"encoding":49},"1\u002F7",[20,561,563],{"className":562,"ariaHidden":55},[54],[20,564,566,569],{"className":565},[59],[20,567],{"className":568,"style":227},[63],[20,570,559],{"className":571},[68],"th of curvature).",[299,574,576],{"id":575},"_2-the-reciprocal-formula","2. The Reciprocal Formula",[16,578,579,580,659,660,730,731,862,863,526,1283,1286,1287,659,1347,1375,1376,85],{},"If ",[20,581,583,603],{"className":582},[23],[20,584,586],{"className":585},[27],[29,587,588],{"xmlns":31},[33,589,590,600],{},[36,591,592],{},[593,594,595,598],"msub",{},[115,596,597],{},"x",[39,599,549],{},[47,601,602],{"encoding":49},"x_1",[20,604,606],{"className":605,"ariaHidden":55},[54],[20,607,609,613],{"className":608},[59],[20,610],{"className":611,"style":612},[63],"height:0.5806em;vertical-align:-0.15em;",[20,614,616,619],{"className":615},[68],[20,617,597],{"className":618},[68,133],[20,620,622],{"className":621},[495],[20,623,626,650],{"className":624},[499,625],"vlist-t2",[20,627,629,645],{"className":628},[503],[20,630,633],{"className":631,"style":632},[507],"height:0.3011em;",[20,634,636,639],{"style":635},"top:-2.55em;margin-left:0em;margin-right:0.05em;",[20,637],{"className":638,"style":515},[514],[20,640,642],{"className":641},[519,520,521,522],[20,643,549],{"className":644},[68,522],[20,646,649],{"className":647},[648],"vlist-s","​",[20,651,653],{"className":652},[503],[20,654,657],{"className":655,"style":656},[507],"height:0.15em;",[20,658],{}," and ",[20,661,663,681],{"className":662},[23],[20,664,666],{"className":665},[27],[29,667,668],{"xmlns":31},[33,669,670,678],{},[36,671,672],{},[593,673,674,676],{},[115,675,597],{},[39,677,425],{},[47,679,680],{"encoding":49},"x_2",[20,682,684],{"className":683,"ariaHidden":55},[54],[20,685,687,690],{"className":686},[59],[20,688],{"className":689,"style":612},[63],[20,691,693,696],{"className":692},[68],[20,694,597],{"className":695},[68,133],[20,697,699],{"className":698},[495],[20,700,702,722],{"className":701},[499,625],[20,703,705,719],{"className":704},[503],[20,706,708],{"className":707,"style":632},[507],[20,709,710,713],{"style":635},[20,711],{"className":712,"style":515},[514],[20,714,716],{"className":715},[519,520,521,522],[20,717,425],{"className":718},[68,522],[20,720,649],{"className":721},[648],[20,723,725],{"className":724},[503],[20,726,728],{"className":727,"style":656},[507],[20,729],{}," are readings from side A, and ",[20,732,734,762],{"className":733},[23],[20,735,737],{"className":736},[27],[29,738,739],{"xmlns":31},[33,740,741,759],{},[36,742,743,750,753],{},[593,744,745,748],{},[115,746,747],{},"y",[39,749,549],{},[201,751,752],{"separator":55},",",[593,754,755,757],{},[115,756,747],{},[39,758,425],{},[47,760,761],{"encoding":49},"y_1, y_2",[20,763,765],{"className":764,"ariaHidden":55},[54],[20,766,768,772,814,818,822],{"className":767},[59],[20,769],{"className":770,"style":771},[63],"height:0.625em;vertical-align:-0.1944em;",[20,773,775,779],{"className":774},[68],[20,776,747],{"className":777,"style":778},[68,133],"margin-right:0.0359em;",[20,780,782],{"className":781},[495],[20,783,785,806],{"className":784},[499,625],[20,786,788,803],{"className":787},[503],[20,789,791],{"className":790,"style":632},[507],[20,792,794,797],{"style":793},"top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;",[20,795],{"className":796,"style":515},[514],[20,798,800],{"className":799},[519,520,521,522],[20,801,549],{"className":802},[68,522],[20,804,649],{"className":805},[648],[20,807,809],{"className":808},[503],[20,810,812],{"className":811,"style":656},[507],[20,813],{},[20,815,752],{"className":816},[817],"mpunct",[20,819],{"className":820,"style":821},[238],"margin-right:0.1667em;",[20,823,825,828],{"className":824},[68],[20,826,747],{"className":827,"style":778},[68,133],[20,829,831],{"className":830},[495],[20,832,834,854],{"className":833},[499,625],[20,835,837,851],{"className":836},[503],[20,838,840],{"className":839,"style":632},[507],[20,841,842,845],{"style":793},[20,843],{"className":844,"style":515},[514],[20,846,848],{"className":847},[519,520,521,522],[20,849,425],{"className":850},[68,522],[20,852,649],{"className":853},[648],[20,855,857],{"className":856},[503],[20,858,860],{"className":859,"style":656},[507],[20,861],{}," are readings from side B:\n",[20,864,866,942],{"className":865},[23],[20,867,869],{"className":868},[27],[29,870,871],{"xmlns":31},[33,872,873,939],{},[36,874,875,878,891,893],{},[115,876,877],{"mathvariant":552},"Δ",[593,879,880,883],{},[115,881,882],{},"H",[36,884,885,888],{},[115,886,887],{},"A",[115,889,890],{},"B",[201,892,414],{},[894,895,896,937],"mfrac",{},[36,897,898,900,906,908,914,916,919,921,927,929,935],{},[201,899,204],{"stretchy":203},[593,901,902,904],{},[115,903,597],{},[39,905,425],{},[201,907,209],{},[593,909,910,912],{},[115,911,597],{},[39,913,549],{},[201,915,214],{"stretchy":203},[201,917,918],{},"+",[201,920,204],{"stretchy":203},[593,922,923,925],{},[115,924,747],{},[39,926,549],{},[201,928,209],{},[593,930,931,933],{},[115,932,747],{},[39,934,425],{},[201,936,214],{"stretchy":203},[39,938,425],{},[47,940,941],{"encoding":49},"\\Delta H_{AB} = \\frac{(x_2 - x_1) + (y_1 - y_2)}{2}",[20,943,945,1014],{"className":944,"ariaHidden":55},[54],[20,946,948,952,955,1005,1008,1011],{"className":947},[59],[20,949],{"className":950,"style":951},[63],"height:0.8333em;vertical-align:-0.15em;",[20,953,877],{"className":954},[68],[20,956,958,962],{"className":957},[68],[20,959,882],{"className":960,"style":961},[68,133],"margin-right:0.0813em;",[20,963,965],{"className":964},[495],[20,966,968,997],{"className":967},[499,625],[20,969,971,994],{"className":970},[503],[20,972,975],{"className":973,"style":974},[507],"height:0.3283em;",[20,976,978,981],{"style":977},"top:-2.55em;margin-left:-0.0813em;margin-right:0.05em;",[20,979],{"className":980,"style":515},[514],[20,982,984],{"className":983},[519,520,521,522],[20,985,987,990],{"className":986},[68,522],[20,988,887],{"className":989},[68,133,522],[20,991,890],{"className":992,"style":993},[68,133,522],"margin-right:0.0502em;",[20,995,649],{"className":996},[648],[20,998,1000],{"className":999},[503],[20,1001,1003],{"className":1002,"style":656},[507],[20,1004],{},[20,1006],{"className":1007,"style":468},[238],[20,1009,414],{"className":1010},[472],[20,1012],{"className":1013,"style":468},[238],[20,1015,1017,1021],{"className":1016},[59],[20,1018],{"className":1019,"style":1020},[63],"height:1.355em;vertical-align:-0.345em;",[20,1022,1024,1028,1280],{"className":1023},[68],[20,1025],{"className":1026},[231,1027],"nulldelimiter",[20,1029,1031],{"className":1030},[894],[20,1032,1034,1271],{"className":1033},[499,625],[20,1035,1037,1268],{"className":1036},[503],[20,1038,1041,1057,1068],{"className":1039,"style":1040},[507],"height:1.01em;",[20,1042,1044,1048],{"style":1043},"top:-2.655em;",[20,1045],{"className":1046,"style":1047},[514],"height:3em;",[20,1049,1051],{"className":1050},[519,520,521,522],[20,1052,1054],{"className":1053},[68,522],[20,1055,425],{"className":1056},[68,522],[20,1058,1060,1063],{"style":1059},"top:-3.23em;",[20,1061],{"className":1062,"style":1047},[514],[20,1064],{"className":1065,"style":1067},[1066],"frac-line","border-bottom-width:0.04em;",[20,1069,1071,1074],{"style":1070},"top:-3.485em;",[20,1072],{"className":1073,"style":1047},[514],[20,1075,1077],{"className":1076},[519,520,521,522],[20,1078,1080,1083,1129,1132,1172,1175,1178,1181,1222,1225,1265],{"className":1079},[68,522],[20,1081,204],{"className":1082},[231,522],[20,1084,1086,1089],{"className":1085},[68,522],[20,1087,597],{"className":1088},[68,133,522],[20,1090,1092],{"className":1091},[495],[20,1093,1095,1120],{"className":1094},[499,625],[20,1096,1098,1117],{"className":1097},[503],[20,1099,1102],{"className":1100,"style":1101},[507],"height:0.3173em;",[20,1103,1105,1109],{"style":1104},"top:-2.357em;margin-left:0em;margin-right:0.0714em;",[20,1106],{"className":1107,"style":1108},[514],"height:2.5em;",[20,1110,1114],{"className":1111},[519,1112,1113,522],"reset-size3","size1",[20,1115,425],{"className":1116},[68,522],[20,1118,649],{"className":1119},[648],[20,1121,1123],{"className":1122},[503],[20,1124,1127],{"className":1125,"style":1126},[507],"height:0.143em;",[20,1128],{},[20,1130,209],{"className":1131},[243,522],[20,1133,1135,1138],{"className":1134},[68,522],[20,1136,597],{"className":1137},[68,133,522],[20,1139,1141],{"className":1140},[495],[20,1142,1144,1164],{"className":1143},[499,625],[20,1145,1147,1161],{"className":1146},[503],[20,1148,1150],{"className":1149,"style":1101},[507],[20,1151,1152,1155],{"style":1104},[20,1153],{"className":1154,"style":1108},[514],[20,1156,1158],{"className":1157},[519,1112,1113,522],[20,1159,549],{"className":1160},[68,522],[20,1162,649],{"className":1163},[648],[20,1165,1167],{"className":1166},[503],[20,1168,1170],{"className":1169,"style":1126},[507],[20,1171],{},[20,1173,214],{"className":1174},[259,522],[20,1176,918],{"className":1177},[243,522],[20,1179,204],{"className":1180},[231,522],[20,1182,1184,1187],{"className":1183},[68,522],[20,1185,747],{"className":1186,"style":778},[68,133,522],[20,1188,1190],{"className":1189},[495],[20,1191,1193,1214],{"className":1192},[499,625],[20,1194,1196,1211],{"className":1195},[503],[20,1197,1199],{"className":1198,"style":1101},[507],[20,1200,1202,1205],{"style":1201},"top:-2.357em;margin-left:-0.0359em;margin-right:0.0714em;",[20,1203],{"className":1204,"style":1108},[514],[20,1206,1208],{"className":1207},[519,1112,1113,522],[20,1209,549],{"className":1210},[68,522],[20,1212,649],{"className":1213},[648],[20,1215,1217],{"className":1216},[503],[20,1218,1220],{"className":1219,"style":1126},[507],[20,1221],{},[20,1223,209],{"className":1224},[243,522],[20,1226,1228,1231],{"className":1227},[68,522],[20,1229,747],{"className":1230,"style":778},[68,133,522],[20,1232,1234],{"className":1233},[495],[20,1235,1237,1257],{"className":1236},[499,625],[20,1238,1240,1254],{"className":1239},[503],[20,1241,1243],{"className":1242,"style":1101},[507],[20,1244,1245,1248],{"style":1201},[20,1246],{"className":1247,"style":1108},[514],[20,1249,1251],{"className":1250},[519,1112,1113,522],[20,1252,425],{"className":1253},[68,522],[20,1255,649],{"className":1256},[648],[20,1258,1260],{"className":1259},[503],[20,1261,1263],{"className":1262,"style":1126},[507],[20,1264],{},[20,1266,214],{"className":1267},[259,522],[20,1269,649],{"className":1270},[648],[20,1272,1274],{"className":1273},[503],[20,1275,1278],{"className":1276,"style":1277},[507],"height:0.345em;",[20,1279],{},[20,1281],{"className":1282},[259,1027],[20,1284,1285],{},"34, 38",".\nThis formula cancels out ",[20,1288,1290,1311],{"className":1289},[23],[20,1291,1293],{"className":1292},[27],[29,1294,1295],{"xmlns":31},[33,1296,1297,1309],{},[36,1298,1299,1301,1303,1305,1307],{},[201,1300,204],{"stretchy":203},[115,1302,117],{},[201,1304,209],{},[115,1306,158],{},[201,1308,214],{"stretchy":203},[47,1310,217],{"encoding":49},[20,1312,1314,1335],{"className":1313,"ariaHidden":55},[54],[20,1315,1317,1320,1323,1326,1329,1332],{"className":1316},[59],[20,1318],{"className":1319,"style":227},[63],[20,1321,204],{"className":1322},[231],[20,1324,117],{"className":1325},[68,133],[20,1327],{"className":1328,"style":239},[238],[20,1330,209],{"className":1331},[243],[20,1333],{"className":1334,"style":239},[238],[20,1336,1338,1341,1344],{"className":1337},[59],[20,1339],{"className":1340,"style":227},[63],[20,1342,158],{"className":1343,"style":173},[68,133],[20,1345,214],{"className":1346},[259],[20,1348,1350,1363],{"className":1349},[23],[20,1351,1353],{"className":1352},[27],[29,1354,1355],{"xmlns":31},[33,1356,1357,1361],{},[36,1358,1359],{},[115,1360,275],{},[47,1362,275],{"encoding":49},[20,1364,1366],{"className":1365,"ariaHidden":55},[54],[20,1367,1369,1372],{"className":1368},[59],[20,1370],{"className":1371,"style":129},[63],[20,1373,275],{"className":1374},[68,133]," entirely, provided the atmospheric conditions remain constant during the two sets of observations ",[20,1377,84],{},[11,1379,1381],{"id":1380},"field-workflow","Field Workflow",[1383,1384,1385,1389,1534,1538,1685,1689,1724,1728],"steps",{},[299,1386,1388],{"id":1387},"instrument-at-a","Instrument at A",[16,1390,1391,1392,1461,1462,1531,1532,85],{},"Set up the level near Point A. Take a precise reading ",[20,1393,1395,1412],{"className":1394},[23],[20,1396,1398],{"className":1397},[27],[29,1399,1400],{"xmlns":31},[33,1401,1402,1410],{},[36,1403,1404],{},[593,1405,1406,1408],{},[115,1407,597],{},[39,1409,549],{},[47,1411,602],{"encoding":49},[20,1413,1415],{"className":1414,"ariaHidden":55},[54],[20,1416,1418,1421],{"className":1417},[59],[20,1419],{"className":1420,"style":612},[63],[20,1422,1424,1427],{"className":1423},[68],[20,1425,597],{"className":1426},[68,133],[20,1428,1430],{"className":1429},[495],[20,1431,1433,1453],{"className":1432},[499,625],[20,1434,1436,1450],{"className":1435},[503],[20,1437,1439],{"className":1438,"style":632},[507],[20,1440,1441,1444],{"style":635},[20,1442],{"className":1443,"style":515},[514],[20,1445,1447],{"className":1446},[519,520,521,522],[20,1448,549],{"className":1449},[68,522],[20,1451,649],{"className":1452},[648],[20,1454,1456],{"className":1455},[503],[20,1457,1459],{"className":1458,"style":656},[507],[20,1460],{}," on a staff at A (short sight) and ",[20,1463,1465,1482],{"className":1464},[23],[20,1466,1468],{"className":1467},[27],[29,1469,1470],{"xmlns":31},[33,1471,1472,1480],{},[36,1473,1474],{},[593,1475,1476,1478],{},[115,1477,597],{},[39,1479,425],{},[47,1481,680],{"encoding":49},[20,1483,1485],{"className":1484,"ariaHidden":55},[54],[20,1486,1488,1491],{"className":1487},[59],[20,1489],{"className":1490,"style":612},[63],[20,1492,1494,1497],{"className":1493},[68],[20,1495,597],{"className":1496},[68,133],[20,1498,1500],{"className":1499},[495],[20,1501,1503,1523],{"className":1502},[499,625],[20,1504,1506,1520],{"className":1505},[503],[20,1507,1509],{"className":1508,"style":632},[507],[20,1510,1511,1514],{"style":635},[20,1512],{"className":1513,"style":515},[514],[20,1515,1517],{"className":1516},[519,520,521,522],[20,1518,425],{"className":1519},[68,522],[20,1521,649],{"className":1522},[648],[20,1524,1526],{"className":1525},[503],[20,1527,1529],{"className":1528,"style":656},[507],[20,1530],{}," on a staff at B (long sight across the obstacle) ",[20,1533,84],{},[299,1535,1537],{"id":1536},"instrument-at-b","Instrument at B",[16,1539,1540,1541,1611,1612,1682,1683,85],{},"Move the instrument to the other side, near Point B. Take a reading ",[20,1542,1544,1562],{"className":1543},[23],[20,1545,1547],{"className":1546},[27],[29,1548,1549],{"xmlns":31},[33,1550,1551,1559],{},[36,1552,1553],{},[593,1554,1555,1557],{},[115,1556,747],{},[39,1558,549],{},[47,1560,1561],{"encoding":49},"y_1",[20,1563,1565],{"className":1564,"ariaHidden":55},[54],[20,1566,1568,1571],{"className":1567},[59],[20,1569],{"className":1570,"style":771},[63],[20,1572,1574,1577],{"className":1573},[68],[20,1575,747],{"className":1576,"style":778},[68,133],[20,1578,1580],{"className":1579},[495],[20,1581,1583,1603],{"className":1582},[499,625],[20,1584,1586,1600],{"className":1585},[503],[20,1587,1589],{"className":1588,"style":632},[507],[20,1590,1591,1594],{"style":793},[20,1592],{"className":1593,"style":515},[514],[20,1595,1597],{"className":1596},[519,520,521,522],[20,1598,549],{"className":1599},[68,522],[20,1601,649],{"className":1602},[648],[20,1604,1606],{"className":1605},[503],[20,1607,1609],{"className":1608,"style":656},[507],[20,1610],{}," on the staff at B (short sight) and ",[20,1613,1615,1633],{"className":1614},[23],[20,1616,1618],{"className":1617},[27],[29,1619,1620],{"xmlns":31},[33,1621,1622,1630],{},[36,1623,1624],{},[593,1625,1626,1628],{},[115,1627,747],{},[39,1629,425],{},[47,1631,1632],{"encoding":49},"y_2",[20,1634,1636],{"className":1635,"ariaHidden":55},[54],[20,1637,1639,1642],{"className":1638},[59],[20,1640],{"className":1641,"style":771},[63],[20,1643,1645,1648],{"className":1644},[68],[20,1646,747],{"className":1647,"style":778},[68,133],[20,1649,1651],{"className":1650},[495],[20,1652,1654,1674],{"className":1653},[499,625],[20,1655,1657,1671],{"className":1656},[503],[20,1658,1660],{"className":1659,"style":632},[507],[20,1661,1662,1665],{"style":793},[20,1663],{"className":1664,"style":515},[514],[20,1666,1668],{"className":1667},[519,520,521,522],[20,1669,425],{"className":1670},[68,522],[20,1672,649],{"className":1673},[648],[20,1675,1677],{"className":1676},[503],[20,1678,1680],{"className":1679,"style":656},[507],[20,1681],{}," on the staff at A (long sight) ",[20,1684,1285],{},[299,1686,1688],{"id":1687},"simultaneous-observations-best-practice","Simultaneous Observations (Best Practice)",[16,1690,1691,1692,1720,1721,85],{},"For maximum precision, use two instruments and two observers to take readings at A and B at exactly the same time. This ensures the atmospheric refraction (",[20,1693,1695,1708],{"className":1694},[23],[20,1696,1698],{"className":1697},[27],[29,1699,1700],{"xmlns":31},[33,1701,1702,1706],{},[36,1703,1704],{},[115,1705,158],{},[47,1707,158],{"encoding":49},[20,1709,1711],{"className":1710,"ariaHidden":55},[54],[20,1712,1714,1717],{"className":1713},[59],[20,1715],{"className":1716,"style":129},[63],[20,1718,158],{"className":1719,"style":173},[68,133],") is identical for both sights ",[20,1722,1723],{},"39",[299,1725,1727],{"id":1726},"reduction","Reduction",[16,1729,1730,1731,526,1765,85],{},"Average the height differences from both setups to find the true ",[20,1732,1734,1750],{"className":1733},[23],[20,1735,1737],{"className":1736},[27],[29,1738,1739],{"xmlns":31},[33,1740,1741,1747],{},[36,1742,1743,1745],{},[115,1744,877],{"mathvariant":552},[115,1746,882],{},[47,1748,1749],{"encoding":49},"\\Delta H",[20,1751,1753],{"className":1752,"ariaHidden":55},[54],[20,1754,1756,1759,1762],{"className":1755},[59],[20,1757],{"className":1758,"style":385},[63],[20,1760,877],{"className":1761},[68],[20,1763,882],{"className":1764,"style":961},[68,133],[20,1766,84],{},[11,1768,1770],{"id":1769},"step-by-step-example","Step-by-Step Example",[16,1772,1773,1776,1777,1837],{},[78,1774,1775],{},"Problem:"," Observations across a river (",[20,1778,1780,1801],{"className":1779},[23],[20,1781,1783],{"className":1782},[27],[29,1784,1785],{"xmlns":31},[33,1786,1787,1798],{},[36,1788,1789,1791,1793,1796],{},[115,1790,373],{},[201,1792,414],{},[39,1794,1795],{},"780",[43,1797,45],{},[47,1799,1800],{"encoding":49},"D = 780\\text{ m}",[20,1802,1804,1822],{"className":1803,"ariaHidden":55},[54],[20,1805,1807,1810,1813,1816,1819],{"className":1806},[59],[20,1808],{"className":1809,"style":385},[63],[20,1811,373],{"className":1812,"style":173},[68,133],[20,1814],{"className":1815,"style":468},[238],[20,1817,414],{"className":1818},[472],[20,1820],{"className":1821,"style":468},[238],[20,1823,1825,1828,1831],{"className":1824},[59],[20,1826],{"className":1827,"style":64},[63],[20,1829,1795],{"className":1830},[68],[20,1832,1834],{"className":1833},[68,72],[20,1835,45],{"className":1836},[68],").",[94,1839,1840,2002],{},[97,1841,1842,1843,1948,1949,85],{},"Inst. at A: ",[20,1844,1846,1873],{"className":1845},[23],[20,1847,1849],{"className":1848},[27],[29,1850,1851],{"xmlns":31},[33,1852,1853,1870],{},[36,1854,1855,1863,1865,1868],{},[593,1856,1857,1860],{},[115,1858,1859],{},"h",[115,1861,1862],{},"i",[201,1864,414],{},[39,1866,1867],{},"1.472",[43,1869,45],{},[47,1871,1872],{"encoding":49},"h_i = 1.472\\text{ m}",[20,1874,1876,1933],{"className":1875,"ariaHidden":55},[54],[20,1877,1879,1883,1924,1927,1930],{"className":1878},[59],[20,1880],{"className":1881,"style":1882},[63],"height:0.8444em;vertical-align:-0.15em;",[20,1884,1886,1889],{"className":1885},[68],[20,1887,1859],{"className":1888},[68,133],[20,1890,1892],{"className":1891},[495],[20,1893,1895,1916],{"className":1894},[499,625],[20,1896,1898,1913],{"className":1897},[503],[20,1899,1902],{"className":1900,"style":1901},[507],"height:0.3117em;",[20,1903,1904,1907],{"style":635},[20,1905],{"className":1906,"style":515},[514],[20,1908,1910],{"className":1909},[519,520,521,522],[20,1911,1862],{"className":1912},[68,133,522],[20,1914,649],{"className":1915},[648],[20,1917,1919],{"className":1918},[503],[20,1920,1922],{"className":1921,"style":656},[507],[20,1923],{},[20,1925],{"className":1926,"style":468},[238],[20,1928,414],{"className":1929},[472],[20,1931],{"className":1932,"style":468},[238],[20,1934,1936,1939,1942],{"className":1935},[59],[20,1937],{"className":1938,"style":64},[63],[20,1940,1867],{"className":1941},[68],[20,1943,1945],{"className":1944},[68,72],[20,1946,45],{"className":1947},[68],", Staff at B ",[20,1950,1952,1971],{"className":1951},[23],[20,1953,1955],{"className":1954},[27],[29,1956,1957],{"xmlns":31},[33,1958,1959,1968],{},[36,1960,1961,1963,1966],{},[201,1962,414],{},[39,1964,1965],{},"1.835",[43,1967,45],{},[47,1969,1970],{"encoding":49},"= 1.835\\text{ m}",[20,1972,1974,1987],{"className":1973,"ariaHidden":55},[54],[20,1975,1977,1981,1984],{"className":1976},[59],[20,1978],{"className":1979,"style":1980},[63],"height:0.3669em;",[20,1982,414],{"className":1983},[472],[20,1985],{"className":1986,"style":468},[238],[20,1988,1990,1993,1996],{"className":1989},[59],[20,1991],{"className":1992,"style":64},[63],[20,1994,1965],{"className":1995},[68],[20,1997,1999],{"className":1998},[68,72],[20,2000,45],{"className":2001},[68],[97,2003,2004,2005,2106,2107,530,2159,526,2162,85],{},"Inst. at B: ",[20,2006,2008,2033],{"className":2007},[23],[20,2009,2011],{"className":2010},[27],[29,2012,2013],{"xmlns":31},[33,2014,2015,2030],{},[36,2016,2017,2023,2025,2028],{},[593,2018,2019,2021],{},[115,2020,1859],{},[115,2022,1862],{},[201,2024,414],{},[39,2026,2027],{},"1.496",[43,2029,45],{},[47,2031,2032],{"encoding":49},"h_i = 1.496\\text{ m}",[20,2034,2036,2091],{"className":2035,"ariaHidden":55},[54],[20,2037,2039,2042,2082,2085,2088],{"className":2038},[59],[20,2040],{"className":2041,"style":1882},[63],[20,2043,2045,2048],{"className":2044},[68],[20,2046,1859],{"className":2047},[68,133],[20,2049,2051],{"className":2050},[495],[20,2052,2054,2074],{"className":2053},[499,625],[20,2055,2057,2071],{"className":2056},[503],[20,2058,2060],{"className":2059,"style":1901},[507],[20,2061,2062,2065],{"style":635},[20,2063],{"className":2064,"style":515},[514],[20,2066,2068],{"className":2067},[519,520,521,522],[20,2069,1862],{"className":2070},[68,133,522],[20,2072,649],{"className":2073},[648],[20,2075,2077],{"className":2076},[503],[20,2078,2080],{"className":2079,"style":656},[507],[20,2081],{},[20,2083],{"className":2084,"style":468},[238],[20,2086,414],{"className":2087},[472],[20,2089],{"className":2090,"style":468},[238],[20,2092,2094,2097,2100],{"className":2093},[59],[20,2095],{"className":2096,"style":64},[63],[20,2098,2027],{"className":2099},[68],[20,2101,2103],{"className":2102},[68,72],[20,2104,45],{"className":2105},[68],", Staff at A ",[20,2108,2110,2129],{"className":2109},[23],[20,2111,2113],{"className":2112},[27],[29,2114,2115],{"xmlns":31},[33,2116,2117,2126],{},[36,2118,2119,2121,2124],{},[201,2120,414],{},[39,2122,2123],{},"1.213",[43,2125,45],{},[47,2127,2128],{"encoding":49},"= 1.213\\text{ m}",[20,2130,2132,2144],{"className":2131,"ariaHidden":55},[54],[20,2133,2135,2138,2141],{"className":2134},[59],[20,2136],{"className":2137,"style":1980},[63],[20,2139,414],{"className":2140},[472],[20,2142],{"className":2143,"style":468},[238],[20,2145,2147,2150,2153],{"className":2146},[59],[20,2148],{"className":2149,"style":64},[63],[20,2151,2123],{"className":2152},[68],[20,2154,2156],{"className":2155},[68,72],[20,2157,45],{"className":2158},[68],[532,2160,2161],{},"(Note: Instrument height acts as the 'short sight' reading if set directly over the point).",[20,2163,2164],{},"40, 41",[2166,2167,2168,2330,2490],"ol",{},[97,2169,2170,526,2173,2329],{},[78,2171,2172],{},"Height diff from A:",[20,2174,2176,2213],{"className":2175},[23],[20,2177,2179],{"className":2178},[27],[29,2180,2181],{"xmlns":31},[33,2182,2183,2210],{},[36,2184,2185,2187,2193,2195,2197,2199,2201,2203,2205,2208],{},[115,2186,877],{"mathvariant":552},[593,2188,2189,2191],{},[115,2190,1859],{},[39,2192,549],{},[201,2194,414],{},[39,2196,1867],{},[201,2198,209],{},[39,2200,1965],{},[201,2202,414],{},[201,2204,209],{},[39,2206,2207],{},"0.363",[43,2209,45],{},[47,2211,2212],{"encoding":49},"\\Delta h_1 = 1.472 - 1.835 = -0.363\\text{ m}",[20,2214,2216,2274,2293,2311],{"className":2215,"ariaHidden":55},[54],[20,2217,2219,2222,2225,2265,2268,2271],{"className":2218},[59],[20,2220],{"className":2221,"style":1882},[63],[20,2223,877],{"className":2224},[68],[20,2226,2228,2231],{"className":2227},[68],[20,2229,1859],{"className":2230},[68,133],[20,2232,2234],{"className":2233},[495],[20,2235,2237,2257],{"className":2236},[499,625],[20,2238,2240,2254],{"className":2239},[503],[20,2241,2243],{"className":2242,"style":632},[507],[20,2244,2245,2248],{"style":635},[20,2246],{"className":2247,"style":515},[514],[20,2249,2251],{"className":2250},[519,520,521,522],[20,2252,549],{"className":2253},[68,522],[20,2255,649],{"className":2256},[648],[20,2258,2260],{"className":2259},[503],[20,2261,2263],{"className":2262,"style":656},[507],[20,2264],{},[20,2266],{"className":2267,"style":468},[238],[20,2269,414],{"className":2270},[472],[20,2272],{"className":2273,"style":468},[238],[20,2275,2277,2281,2284,2287,2290],{"className":2276},[59],[20,2278],{"className":2279,"style":2280},[63],"height:0.7278em;vertical-align:-0.0833em;",[20,2282,1867],{"className":2283},[68],[20,2285],{"className":2286,"style":239},[238],[20,2288,209],{"className":2289},[243],[20,2291],{"className":2292,"style":239},[238],[20,2294,2296,2299,2302,2305,2308],{"className":2295},[59],[20,2297],{"className":2298,"style":64},[63],[20,2300,1965],{"className":2301},[68],[20,2303],{"className":2304,"style":468},[238],[20,2306,414],{"className":2307},[472],[20,2309],{"className":2310,"style":468},[238],[20,2312,2314,2317,2320,2323],{"className":2313},[59],[20,2315],{"className":2316,"style":2280},[63],[20,2318,209],{"className":2319},[68],[20,2321,2207],{"className":2322},[68],[20,2324,2326],{"className":2325},[68,72],[20,2327,45],{"className":2328},[68]," (B is lower).",[97,2331,2332,526,2335,2329],{},[78,2333,2334],{},"Height diff from B:",[20,2336,2338,2375],{"className":2337},[23],[20,2339,2341],{"className":2340},[27],[29,2342,2343],{"xmlns":31},[33,2344,2345,2372],{},[36,2346,2347,2349,2355,2357,2359,2361,2363,2365,2367,2370],{},[115,2348,877],{"mathvariant":552},[593,2350,2351,2353],{},[115,2352,1859],{},[39,2354,425],{},[201,2356,414],{},[39,2358,2123],{},[201,2360,209],{},[39,2362,2027],{},[201,2364,414],{},[201,2366,209],{},[39,2368,2369],{},"0.283",[43,2371,45],{},[47,2373,2374],{"encoding":49},"\\Delta h_2 = 1.213 - 1.496 = -0.283\\text{ m}",[20,2376,2378,2436,2454,2472],{"className":2377,"ariaHidden":55},[54],[20,2379,2381,2384,2387,2427,2430,2433],{"className":2380},[59],[20,2382],{"className":2383,"style":1882},[63],[20,2385,877],{"className":2386},[68],[20,2388,2390,2393],{"className":2389},[68],[20,2391,1859],{"className":2392},[68,133],[20,2394,2396],{"className":2395},[495],[20,2397,2399,2419],{"className":2398},[499,625],[20,2400,2402,2416],{"className":2401},[503],[20,2403,2405],{"className":2404,"style":632},[507],[20,2406,2407,2410],{"style":635},[20,2408],{"className":2409,"style":515},[514],[20,2411,2413],{"className":2412},[519,520,521,522],[20,2414,425],{"className":2415},[68,522],[20,2417,649],{"className":2418},[648],[20,2420,2422],{"className":2421},[503],[20,2423,2425],{"className":2424,"style":656},[507],[20,2426],{},[20,2428],{"className":2429,"style":468},[238],[20,2431,414],{"className":2432},[472],[20,2434],{"className":2435,"style":468},[238],[20,2437,2439,2442,2445,2448,2451],{"className":2438},[59],[20,2440],{"className":2441,"style":2280},[63],[20,2443,2123],{"className":2444},[68],[20,2446],{"className":2447,"style":239},[238],[20,2449,209],{"className":2450},[243],[20,2452],{"className":2453,"style":239},[238],[20,2455,2457,2460,2463,2466,2469],{"className":2456},[59],[20,2458],{"className":2459,"style":64},[63],[20,2461,2027],{"className":2462},[68],[20,2464],{"className":2465,"style":468},[238],[20,2467,414],{"className":2468},[472],[20,2470],{"className":2471,"style":468},[238],[20,2473,2475,2478,2481,2484],{"className":2474},[59],[20,2476],{"className":2477,"style":2280},[63],[20,2479,209],{"className":2480},[68],[20,2482,2369],{"className":2483},[68],[20,2485,2487],{"className":2486},[68,72],[20,2488,45],{"className":2489},[68],[97,2491,2492,2495,2698,526,2738,85],{},[78,2493,2494],{},"True Difference:",[20,2496,2498,2549],{"className":2497},[23],[20,2499,2501],{"className":2500},[27],[29,2502,2503],{"xmlns":31},[33,2504,2505,2546],{},[36,2506,2507,2509,2511,2513,2537,2539,2541,2544],{},[115,2508,877],{"mathvariant":552},[115,2510,882],{},[201,2512,414],{},[894,2514,2515,2535],{},[36,2516,2517,2519,2521,2523,2525,2527,2529,2531,2533],{},[201,2518,204],{"stretchy":203},[201,2520,209],{},[39,2522,2207],{},[201,2524,214],{"stretchy":203},[201,2526,918],{},[201,2528,204],{"stretchy":203},[201,2530,209],{},[39,2532,2369],{},[201,2534,214],{"stretchy":203},[39,2536,425],{},[201,2538,414],{},[201,2540,209],{},[39,2542,2543],{},"0.323",[43,2545,45],{},[47,2547,2548],{"encoding":49},"\\Delta H = \\frac{(-0.363) + (-0.283)}{2} = -0.323\\text{ m}",[20,2550,2552,2573,2680],{"className":2551,"ariaHidden":55},[54],[20,2553,2555,2558,2561,2564,2567,2570],{"className":2554},[59],[20,2556],{"className":2557,"style":385},[63],[20,2559,877],{"className":2560},[68],[20,2562,882],{"className":2563,"style":961},[68,133],[20,2565],{"className":2566,"style":468},[238],[20,2568,414],{"className":2569},[472],[20,2571],{"className":2572,"style":468},[238],[20,2574,2576,2579,2671,2674,2677],{"className":2575},[59],[20,2577],{"className":2578,"style":1020},[63],[20,2580,2582,2585,2668],{"className":2581},[68],[20,2583],{"className":2584},[231,1027],[20,2586,2588],{"className":2587},[894],[20,2589,2591,2660],{"className":2590},[499,625],[20,2592,2594,2657],{"className":2593},[503],[20,2595,2597,2611,2619],{"className":2596,"style":1040},[507],[20,2598,2599,2602],{"style":1043},[20,2600],{"className":2601,"style":1047},[514],[20,2603,2605],{"className":2604},[519,520,521,522],[20,2606,2608],{"className":2607},[68,522],[20,2609,425],{"className":2610},[68,522],[20,2612,2613,2616],{"style":1059},[20,2614],{"className":2615,"style":1047},[514],[20,2617],{"className":2618,"style":1067},[1066],[20,2620,2621,2624],{"style":1070},[20,2622],{"className":2623,"style":1047},[514],[20,2625,2627],{"className":2626},[519,520,521,522],[20,2628,2630,2633,2636,2639,2642,2645,2648,2651,2654],{"className":2629},[68,522],[20,2631,204],{"className":2632},[231,522],[20,2634,209],{"className":2635},[68,522],[20,2637,2207],{"className":2638},[68,522],[20,2640,214],{"className":2641},[259,522],[20,2643,918],{"className":2644},[243,522],[20,2646,204],{"className":2647},[231,522],[20,2649,209],{"className":2650},[68,522],[20,2652,2369],{"className":2653},[68,522],[20,2655,214],{"className":2656},[259,522],[20,2658,649],{"className":2659},[648],[20,2661,2663],{"className":2662},[503],[20,2664,2666],{"className":2665,"style":1277},[507],[20,2667],{},[20,2669],{"className":2670},[259,1027],[20,2672],{"className":2673,"style":468},[238],[20,2675,414],{"className":2676},[472],[20,2678],{"className":2679,"style":468},[238],[20,2681,2683,2686,2689,2692],{"className":2682},[59],[20,2684],{"className":2685,"style":2280},[63],[20,2687,209],{"className":2688},[68],[20,2690,2543],{"className":2691},[68],[20,2693,2695],{"className":2694},[68,72],[20,2696,45],{"className":2697},[68],[532,2699,2700,2701],{},"B is lower than A by ",[20,2702,2704,2720],{"className":2703},[23],[20,2705,2707],{"className":2706},[27],[29,2708,2709],{"xmlns":31},[33,2710,2711,2717],{},[36,2712,2713,2715],{},[39,2714,2543],{},[43,2716,45],{},[47,2718,2719],{"encoding":49},"0.323\\text{ m}",[20,2721,2723],{"className":2722,"ariaHidden":55},[54],[20,2724,2726,2729,2732],{"className":2725},[59],[20,2727],{"className":2728,"style":64},[63],[20,2730,2543],{"className":2731},[68],[20,2733,2735],{"className":2734},[68,72],[20,2736,45],{"className":2737},[68],[20,2739,2740],{},"41",[11,2742,2744],{"id":2743},"formula-breakdown","Formula Breakdown",[16,2746,2747,2778,2779,2848,2849,2932,2933,530,3031,526,3168,85],{},[78,2748,2749,2750,134],{},"Collimation Error (",[20,2751,2753,2766],{"className":2752},[23],[20,2754,2756],{"className":2755},[27],[29,2757,2758],{"xmlns":31},[33,2759,2760,2764],{},[36,2761,2762],{},[115,2763,275],{},[47,2765,275],{"encoding":49},[20,2767,2769],{"className":2768,"ariaHidden":55},[54],[20,2770,2772,2775],{"className":2771},[59],[20,2773],{"className":2774,"style":129},[63],[20,2776,275],{"className":2777},[68,133]," You can solve for the instrument's error after the level is established.\nActual reading ",[20,2780,2782,2799],{"className":2781},[23],[20,2783,2785],{"className":2784},[27],[29,2786,2787],{"xmlns":31},[33,2788,2789,2797],{},[36,2790,2791],{},[593,2792,2793,2795],{},[115,2794,597],{},[39,2796,425],{},[47,2798,680],{"encoding":49},[20,2800,2802],{"className":2801,"ariaHidden":55},[54],[20,2803,2805,2808],{"className":2804},[59],[20,2806],{"className":2807,"style":612},[63],[20,2809,2811,2814],{"className":2810},[68],[20,2812,597],{"className":2813},[68,133],[20,2815,2817],{"className":2816},[495],[20,2818,2820,2840],{"className":2819},[499,625],[20,2821,2823,2837],{"className":2822},[503],[20,2824,2826],{"className":2825,"style":632},[507],[20,2827,2828,2831],{"style":635},[20,2829],{"className":2830,"style":515},[514],[20,2832,2834],{"className":2833},[519,520,521,522],[20,2835,425],{"className":2836},[68,522],[20,2838,649],{"className":2839},[648],[20,2841,2843],{"className":2842},[503],[20,2844,2846],{"className":2845,"style":656},[507],[20,2847],{}," contains ",[20,2850,2852,2878],{"className":2851},[23],[20,2853,2855],{"className":2854},[27],[29,2856,2857],{"xmlns":31},[33,2858,2859,2875],{},[36,2860,2861,2863,2865,2867,2869,2871,2873],{},[201,2862,204],{"stretchy":203},[115,2864,117],{},[201,2866,209],{},[115,2868,158],{},[201,2870,214],{"stretchy":203},[201,2872,918],{},[115,2874,275],{},[47,2876,2877],{"encoding":49},"(c - r) + e",[20,2879,2881,2902,2923],{"className":2880,"ariaHidden":55},[54],[20,2882,2884,2887,2890,2893,2896,2899],{"className":2883},[59],[20,2885],{"className":2886,"style":227},[63],[20,2888,204],{"className":2889},[231],[20,2891,117],{"className":2892},[68,133],[20,2894],{"className":2895,"style":239},[238],[20,2897,209],{"className":2898},[243],[20,2900],{"className":2901,"style":239},[238],[20,2903,2905,2908,2911,2914,2917,2920],{"className":2904},[59],[20,2906],{"className":2907,"style":227},[63],[20,2909,158],{"className":2910,"style":173},[68,133],[20,2912,214],{"className":2913},[259],[20,2915],{"className":2916,"style":239},[238],[20,2918,918],{"className":2919},[243],[20,2921],{"className":2922,"style":239},[238],[20,2924,2926,2929],{"className":2925},[59],[20,2927],{"className":2928,"style":129},[63],[20,2930,275],{"className":2931},[68,133],".\nTrue reading should be ",[20,2934,2936,2960],{"className":2935},[23],[20,2937,2939],{"className":2938},[27],[29,2940,2941],{"xmlns":31},[33,2942,2943,2957],{},[36,2944,2945,2951,2953,2955],{},[593,2946,2947,2949],{},[115,2948,597],{},[39,2950,549],{},[201,2952,918],{},[115,2954,877],{"mathvariant":552},[115,2956,882],{},[47,2958,2959],{"encoding":49},"x_1 + \\Delta H",[20,2961,2963,3019],{"className":2962,"ariaHidden":55},[54],[20,2964,2966,2970,3010,3013,3016],{"className":2965},[59],[20,2967],{"className":2968,"style":2969},[63],"height:0.7333em;vertical-align:-0.15em;",[20,2971,2973,2976],{"className":2972},[68],[20,2974,597],{"className":2975},[68,133],[20,2977,2979],{"className":2978},[495],[20,2980,2982,3002],{"className":2981},[499,625],[20,2983,2985,2999],{"className":2984},[503],[20,2986,2988],{"className":2987,"style":632},[507],[20,2989,2990,2993],{"style":635},[20,2991],{"className":2992,"style":515},[514],[20,2994,2996],{"className":2995},[519,520,521,522],[20,2997,549],{"className":2998},[68,522],[20,3000,649],{"className":3001},[648],[20,3003,3005],{"className":3004},[503],[20,3006,3008],{"className":3007,"style":656},[507],[20,3009],{},[20,3011],{"className":3012,"style":239},[238],[20,3014,918],{"className":3015},[243],[20,3017],{"className":3018,"style":239},[238],[20,3020,3022,3025,3028],{"className":3021},[59],[20,3023],{"className":3024,"style":385},[63],[20,3026,877],{"className":3027},[68],[20,3029,882],{"className":3030,"style":961},[68,133],[20,3032,3034,3070],{"className":3033},[23],[20,3035,3037],{"className":3036},[27],[29,3038,3039],{"xmlns":31},[33,3040,3041,3067],{},[36,3042,3043,3045,3047,3050,3052,3055,3057,3059,3061,3063,3065],{},[115,3044,275],{},[201,3046,414],{},[43,3048,3049],{},"Actual",[201,3051,209],{},[43,3053,3054],{},"True",[201,3056,209],{},[201,3058,204],{"stretchy":203},[115,3060,117],{},[201,3062,209],{},[115,3064,158],{},[201,3066,214],{"stretchy":203},[47,3068,3069],{"encoding":49},"e = \\text{Actual} - \\text{True} - (c - r)",[20,3071,3073,3091,3113,3135,3156],{"className":3072,"ariaHidden":55},[54],[20,3074,3076,3079,3082,3085,3088],{"className":3075},[59],[20,3077],{"className":3078,"style":129},[63],[20,3080,275],{"className":3081},[68,133],[20,3083],{"className":3084,"style":468},[238],[20,3086,414],{"className":3087},[472],[20,3089],{"className":3090,"style":468},[238],[20,3092,3094,3098,3104,3107,3110],{"className":3093},[59],[20,3095],{"className":3096,"style":3097},[63],"height:0.7778em;vertical-align:-0.0833em;",[20,3099,3101],{"className":3100},[68,72],[20,3102,3049],{"className":3103},[68],[20,3105],{"className":3106,"style":239},[238],[20,3108,209],{"className":3109},[243],[20,3111],{"className":3112,"style":239},[238],[20,3114,3116,3120,3126,3129,3132],{"className":3115},[59],[20,3117],{"className":3118,"style":3119},[63],"height:0.7667em;vertical-align:-0.0833em;",[20,3121,3123],{"className":3122},[68,72],[20,3124,3054],{"className":3125},[68],[20,3127],{"className":3128,"style":239},[238],[20,3130,209],{"className":3131},[243],[20,3133],{"className":3134,"style":239},[238],[20,3136,3138,3141,3144,3147,3150,3153],{"className":3137},[59],[20,3139],{"className":3140,"style":227},[63],[20,3142,204],{"className":3143},[231],[20,3145,117],{"className":3146},[68,133],[20,3148],{"className":3149,"style":239},[238],[20,3151,209],{"className":3152},[243],[20,3154],{"className":3155,"style":239},[238],[20,3157,3159,3162,3165],{"className":3158},[59],[20,3160],{"className":3161,"style":227},[63],[20,3163,158],{"className":3164,"style":173},[68,133],[20,3166,214],{"className":3167},[259],[20,3169,529],{},[11,3171,3173],{"id":3172},"practical-tips","Practical Tips",[94,3175,3176,3224,3233],{},[97,3177,3178,3181,3182,3220,3221,85],{},[78,3179,3180],{},"Targeting:"," Over ",[20,3183,3185,3202],{"className":3184},[23],[20,3186,3188],{"className":3187},[27],[29,3189,3190],{"xmlns":31},[33,3191,3192,3199],{},[36,3193,3194,3197],{},[39,3195,3196],{},"200",[43,3198,45],{},[47,3200,3201],{"encoding":49},"200\\text{ m}",[20,3203,3205],{"className":3204,"ariaHidden":55},[54],[20,3206,3208,3211,3214],{"className":3207},[59],[20,3209],{"className":3210,"style":64},[63],[20,3212,3196],{"className":3213},[68],[20,3215,3217],{"className":3216},[68,72],[20,3218,45],{"className":3219},[68],", a standard staff becomes hard to read. Use a specialized \"target staff\" with a movable vane that the observer can signal to be raised or lowered ",[20,3222,3223],{},"42",[97,3225,3226,3229,3230,85],{},[78,3227,3228],{},"Time of Day:"," Avoid reciprocal levelling during the \"heat shimmer\" of midday. Observations are most stable on cloudy days or just after sunrise ",[20,3231,3232],{},"12",[97,3234,3235,3238,3239,3242,3243,85],{},[78,3236,3237],{},"Instrument Choice:"," Use a ",[78,3240,3241],{},"Tilting Level"," with a parallel-plate micrometer for sub-millimeter precision on the staff ",[20,3244,3245],{},"43, 44",[11,3247,3249],{"id":3248},"common-mistakes","Common Mistakes",[94,3251,3252,3260],{},[97,3253,3254,3257,3258,85],{},[78,3255,3256],{},"Non-Simultaneous Readings:"," If the sun comes out between the first and second setup, the air density changes, refraction changes, and the error will not cancel ",[20,3259,3232],{},[97,3261,3262,3265,3266,85],{},[78,3263,3264],{},"Touching the Tripod:"," On soft riverbanks, any pressure on the tripod legs will cause \"settlement error,\" ruining the level ",[20,3267,3268],{},"14, 45",[11,3270,3272],{"id":3271},"faq","FAQ",[3274,3275,3276,3290,3339],"accordion",{},[3277,3278,3280],"accordion-item",{"label":3279},"Does reciprocal levelling work for Total Stations?",[16,3281,3282,3283,3286,3287,85],{},"Yes. This is known as ",[78,3284,3285],{},"Reciprocal Trigonometrical Levelling",". It uses vertical angles and slope distances and follows the same logic to cancel refraction ",[20,3288,3289],{},"39, 46",[3277,3291,3293],{"label":3292},"How far can I sight with reciprocal levelling?",[16,3294,3295,3296,3335,3336,85],{},"With a high-quality tilting level and target staff, sights of up to ",[20,3297,3299,3316],{"className":3298},[23],[20,3300,3302],{"className":3301},[27],[29,3303,3304],{"xmlns":31},[33,3305,3306,3313],{},[36,3307,3308,3310],{},[39,3309,549],{},[43,3311,3312],{}," km",[47,3314,3315],{"encoding":49},"1\\text{ km}",[20,3317,3319],{"className":3318,"ariaHidden":55},[54],[20,3320,3322,3326,3329],{"className":3321},[59],[20,3323],{"className":3324,"style":3325},[63],"height:0.6944em;",[20,3327,549],{"className":3328},[68],[20,3330,3332],{"className":3331},[68,72],[20,3333,3312],{"className":3334},[68]," are possible, though precision decreases with distance ",[20,3337,3338],{},"47",[3277,3340,3342],{"label":3341},"Why is refraction 1\u002F7th of curvature?",[16,3343,3344,3345,3400,3401,85],{},"This is an empirical \"average\" coefficient (",[20,3346,3348,3369],{"className":3347},[23],[20,3349,3351],{"className":3350},[27],[29,3352,3353],{"xmlns":31},[33,3354,3355,3366],{},[36,3356,3357,3360,3363],{},[115,3358,3359],{},"K",[201,3361,3362],{},"≈",[39,3364,3365],{},"0.14",[47,3367,3368],{"encoding":49},"K \\approx 0.14",[20,3370,3372,3391],{"className":3371,"ariaHidden":55},[54],[20,3373,3375,3378,3382,3385,3388],{"className":3374},[59],[20,3376],{"className":3377,"style":385},[63],[20,3379,3359],{"className":3380,"style":3381},[68,133],"margin-right:0.0715em;",[20,3383],{"className":3384,"style":468},[238],[20,3386,3362],{"className":3387},[472],[20,3389],{"className":3390,"style":468},[238],[20,3392,3394,3397],{"className":3393},[59],[20,3395],{"className":3396,"style":64},[63],[20,3398,3365],{"className":3399},[68],"). In reality, it varies with temperature and pressure, which is why simultaneous observations are critical ",[20,3402,3403],{},"39, 48",[11,3405,3407],{"id":3406},"conclusion","Conclusion",[16,3409,3410],{},"Reciprocal levelling is a fundamental technique for crossing major geographic gaps. By doubling the observations, the surveyor turns the Earth's \"distortions\" against each other, leaving only the true geometric difference in height.",[11,3412,3414],{"id":3413},"references","References",[16,3416,3417,3418,3421,3422,85],{},"Schofield, W. (2001). ",[532,3419,3420],{},"Engineering Surveying",". 5th ed. Butterworth-Heinemann. ",[20,3423,3424],{},"16, 17",{"title":3426,"searchDepth":3427,"depth":3427,"links":3428},"",2,[3429,3430,3431,3432,3437,3443,3444,3445,3446,3447,3448,3449],{"id":13,"depth":3427,"text":14},{"id":88,"depth":3427,"text":89},{"id":183,"depth":3427,"text":184},{"id":296,"depth":3427,"text":297,"children":3433},[3434,3436],{"id":301,"depth":3435,"text":302},3,{"id":575,"depth":3435,"text":576},{"id":1380,"depth":3427,"text":1381,"children":3438},[3439,3440,3441,3442],{"id":1387,"depth":3435,"text":1388},{"id":1536,"depth":3435,"text":1537},{"id":1687,"depth":3435,"text":1688},{"id":1726,"depth":3435,"text":1727},{"id":1769,"depth":3427,"text":1770},{"id":2743,"depth":3427,"text":2744},{"id":3172,"depth":3427,"text":3173},{"id":3248,"depth":3427,"text":3249},{"id":3271,"depth":3427,"text":3272},{"id":3406,"depth":3427,"text":3407},{"id":3413,"depth":3427,"text":3414},"Vertical Control","A technical deep dive into reciprocal levelling techniques to eliminate instrumental and environmental errors across wide obstacles like rivers and valleys.","md","\u002Fimages\u002Freciprocal-levelling.jpg",{"authorCredentials":3455,"commentCount":3456,"tags":3457,"relatedTools":3462,"relatedArticles":3465,"sources":3468},"LS",0,[3458,80,3459,3460,3461],"Levelling","Curvature","Refraction","Precision",[3463,3464],"Digital Level","Automatic Level",[3466,3467],"precise-levelling-techniques","digital-levelling-advantages",[3469],{"title":3470,"type":3471,"note":3472},"Engineering Surveying (Fifth Edition)","Book","Primary source for the mathematical proofs of error elimination in levelling.",true,"\u002Fblog\u002Freciprocal-levelling-and-curvature","2024-03-27",18,{"title":5,"description":3451},{"loc":3474},"blog\u002F8.reciprocal-levelling-and-curvature","_G_KCM-TlKwSg0EyyHNbU3dwvu4a5m4WuriFBzd9gNs",[3482,3487],{"title":3483,"path":3484,"stem":3485,"description":3486,"children":-1},"The Clothoid Spiral: Engineering Seamless Transitions in Road Design","\u002Fblog\u002Fclothoid-transition-curves","blog\u002F7.clothoid-transition-curves","A technical guide to the geometry, design, and setting out of clothoid transition curves for high-speed road and rail projects.",{"title":3488,"path":3489,"stem":3490,"description":3491,"children":-1},"EDM Calibration: Eliminating Zero, Scale, and Cyclic Errors","\u002Fblog\u002Fedm-errors-and-calibration","blog\u002F9.edm-errors-and-calibration","A technical guide to understanding and correcting the three primary error sources in Electromagnetic Distance Measurement (EDM) equipment.",1783851951263]