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The importance of clearance of lines and triangulation in geodetic surveys for linear measurements. It covers topics such as selecting lines of least resistance, checking clearance of lines from side points, observing base lines, and using maps and prominent features for station sites. The document also mentions the use of vertical angles and the formula for computing curvature corrections.
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NOAA Repri.nt
for Triangulation
BY WILLIAM MUSSETER
National Geodetic Survey Rockviiie, Md. September 1977
Reprinted December 1985
U.S. DEPARTMENT O f COMMERCE National Oceanic and Atmosplrer(c Admirrktrath National Ocean Survey
NOAA Reprint
Manual of Reconnaissance
for Triangulation
BY WILLIAM MUSSETER
National Geodetic S,wey
RockviUe, Md
September 1977
U.S. DEPARTMENT OF COMMERCE National Oceanic and Atmospheric Administration National Ocean Survey
CONTENTS
p8m 1 3 3 4 8 7 8 9
10 10 11 12 13 14 14 16 16 17 X 8 18 20 21 25. 23 28 30 32 33 33
. 36 38 38 39 40 43 44 56 41
50 50 50 52
4a
amp* 2-contfnud heating positions on work ah&. ............................... Spssdometer-compess travene- ............................. Tmmm notes.. .......................................... Curvature e h t -. - ............................................ Correction for earth’s curvature and Rfraction ................
Eumplw of correctioM...................... ........... Rebation charactmid car of lines. ............................... cle!amnce of lines.... ...................................... Avoidance of obatructioas ...................................... Meaauring hei’ghtu of ....................................... Setation of station t..... ..................................... Inwrviewa with property o.ers.... ............................ rstbrs to property oanen .................................. Descriptiom of statio^ ........................................ Temporwp nuukin ... .sg ................................... BeeonnWnce sketch. ........................................ Standard symbols ......................................... Party organization and outfit ................................... Instnrments.............................................. Gened property- ......................................... cb.ptsr 3. Specid prOb 1. ........................................ Erne lines. ................................................... Ehit.. .. .b. ...........................................
. atskcd ............................................... &aa nets ..................................................... Recovery of old atationL ..................................... Conaectiona to m r k a of other surveys.... ....................... Connectioaa to trmverse........................................ Supplamenul stations.. ........................................ Stations on buildings,,.,,, ..................................... Boundam surveps. ............................................ N.ngs..o ns..... .......................................... ULozTrRATIONS mmn 1. Different typea of Bgureo used in triangulation.....................
6. Relative eleuatioaa by reciprocal vertical antlm 7. Relative elevations by nonriciprocd vertical angla.
2. Elbow 6gure... ...............................................
4. Example of horizon sketch,,,,.,,,,,,,,,-.--,,,,;,,,,,,,^^^, 8. Tppid ., ,,, ,reconnaissance p,,,,,,,,,,,,,,,,,-,,,,,,,.me,obl,r .................... ............... 8. Datermination of obstruoting height.............................. 9. Example of compound profiling. ................................. 10. Obstructing height determined by vertica1 anqies and altimeter,,.,., 11. Indirect profiling. .............................................. 12. Checking clearance of line from side point. ca8e 1 .................. 13. Checking clearance of line from side point. m e 2 .................. 14. &le of indirect profiling.................................... 16. Detennination of obstructing height by method of “least favonrble” condidon ................................................... 10. Distance determination by “prrrallruing”..........................
ks sa 34 56 85 55 50 57 5a 60 61 01 68 84 66 67 69
72 74 74 75 70 70 70 18 79 81
84 86 88 87 88
13 14 15 21 26 39 41 41 48 4 i 4s 45 46 46
48 61
MAIWAL OF RECONNAISSANCE 'FOR TRLANGULATXON
During the past few years, such progress has been rnade in t h methods of executing tisnguistion that it is desirable to revise some-
cial Publication Xo. 98, "Reconnaissance and Signal Building') and Specid Publication No. EO, "5Ianual of First Order Triangulation." It ia with this in mind that the following out.line of present practices
This manual is primarily a guide to engineers of the United States Coast and Geodetic Survey, and, BS such. presupposes some familiar-
detic Surpey. The changes introdaced by the us of the porfabb steel tower signals and the multiple-unit observing partieq and by the closing in of the triangulation net, have made increasing demands upon the
has dected the methods emplopd. T h e use of the steel tower signal, first introduced in 1 st. has
enabled the reconnaisscmcs engineex to specify signals of greater heights without involving undue espense, and has d t e d in the use of triangulationz to the exclusion of traverse. for nearly all hori- zontal control. It has permitted triangulation to be carried into regions where this type^ of survey was considered impossible 8 few y e g ~ ago. Since high signa& me now readily available, t.he demand
and cost of sted@ppcd triaquktion parties sre large, this makes
towers require3 that greater attention be paid to accessibility in the selection of stations to guard against hauling difficulties. The econ- omy of the steel signal favors the establishment of many estn, or supplemental, points for local use and os t.ies to surveys of other or- ganizations. T h e arailabilitF of steel towers and the low cost of 1
formerly, and this results in better control of the area. Triangulation parties now consist of from two to nine observing units and the limit has bp no means been reached. Monthly espensea of these parties ~ull tu high 85 315.000. compared with $ 1. 500 to !B,OOo
expensive than formerly. I t is now imperative that reconnaissance parties reduce such failures to a minimum.. n e rapidity of pmg- mss of the triangulation parties requires'that they be furnished with aII information obtainable concerning roads and local conditions. With the completion of the major Imp of triangulution has come
60-mile spacing. and then by arcs with %-mile spacing. This spoic-
very meager limits. Fornierlp it opm common practice to follow lines of least resistance br following valleys and keeping in contact
s t d e s it may encounter. On this account.. it is often necessaq to cover areas that ore conves in profile, or otherwise difficult, for con-
T h e problems of reconnaissance differ for almost every arc. and
in the field. I n the folIowing paragmphs an attempt will he made to consider and explain tlie principles of recannuisance and give
To snme readers the methods outlined may appear unneces~arily' elaborate and expensive for tlie proper requirements of the recon- naissance. However. a careful inredgation of the records shows that when skill and energy are used in prosecuting the work. the costs are no higher than were those of earlier and less precise meth-
the custom of the reconnaissance engineer to rely more on his o m er- perienca and native ability than oil instnimental work. Cery good results were thus .secured and they frilly met the requirements of the time. However. on& ii m a l l pmportion nf the trimmilation em- cuted during that period was in difficult tini1)erecl regions. and t.he few men engaged on reconnaissance were t hornuglily experienced. Also. the reconnaissance engineer ciistomarily had charge of the signal building. which was clone well in advance of the observing,
found in the reconnissance.
....
A cornpnrison of the three orders of harizontdJcontral a r d i n d p
for these orders :
Bequimnmtr for Irodzontal control ZIlrsomnou
I '- I
It mill be noticed that the standards o€ nccuracp prescribed zbove apply mainly to the Iicld observations. Other stnnclards 3rr3 used for the adjusted work. The process of adjusting obcerratioiis by the method of least squares ninkes the results consistent tliroiidiout but does not remove dl errors. If the observational errors are small and indiscriminately plus and minus, then the adjustment will prob- ably distribute them so that there mill be but a diglit nccumulntion 0 4
and systemtic errors of varying signs am not distributed correctly by the adjustment process. Under certain condit.ions the specified allowable e m in the
tion meets the other apedictations for that particular p d e of con-
size of the triangulation figure of which .they are (L part, the dis- tance between th- points may be in m r in escess of that indi-
the computed length of any line can be estimated by computing
the mR, from the base to that line in accordance with the formula for the strength of &gums 89 given on page 11.
ards for triangulation which should be applied. It is possibie b~ a lucky balancing of errors to secure small triangle closures in a
standard. It is also possible by omitting from the computations obserrations which M e r greatly from the mean to reduce trhnglie
computing a chin of triangles will result in a very small dixrep- ancy in length on the nest ked line. The a c c w of triangulation ia perhaps best indicated by t.he probable error of a direction, but since this gage of the work iS not avaaable until after the adjust-
length, as given by the preliminary computa.tions, am the best avail- able field criteria. TO insure that the requisite accuracy is main-
observing in order that the sFstematic and accidental errors may be
SPECXFICATIONS FOR RECONNAISWiCE
The general instructions for reconnaissance as &en in S p e d Publication Xo. BO, 9hnurrl of First Order Trimplation," are s t i l l in force escept aa modified on account of the present short spans between h e d lines and the desirability of a mom intensive control
are repeated below. 1. Character of &gme&--The chain of' triangulation between base nets shall be ma& up of completed qnadrllatemls and of central-polat flgures. with all stations occupied. It must not be allowed to degenerate eren for o single flgure~ to singie triangles, escept by specific permission by the DLrector. There
to secure a base site at the desired Imatioi~.ZRI may be allowed to approach but not esceed 110. There will be a n g e r when this larger limit is used that an lnterrening bme may be uecesmtry. for it the discrepmrq between adjacent flrst-order bases Is found in nn,r ease to esceed 1 pitn In 3. 00 an interrenhg buse must be mmsured.
5. Base sites and base nets.-111 selecting base sites it should be kept In mind
slte where the -de on nur Xkncter tape length cloes tiot exceed 10 percent. and that nitmow volleys or mr111es less than .u) meters wide in the dlrectlon OF the base are not ilrsnperitble obstncles to meirsurement. The length of e:ich base ia to be not less than 1 kilometers. In ench base net great care should be taken to - re as good geometric conditions as possible. There should be no hesttanw in placing the bnse on rowh ground, prodded the rmghiiess Is not grenter thnn that lndfcnted above, If by dohg so the geometric condl- tiona in the base net are Improved. Each bnse net should not be longer than two ordinary figures of the mdn chain between bnses. The %ase net m a y also be strengthened by obsembg over as many lines between stations of the net as can be made tntervtsible wtthout escf+s&e cost For building or cntting. Caution is necessary in thus strengthening II base net by obserrlnq estm lines to amld making the flgare so COmRlferrted as to be escessivelp dlWcult and
6. Conneetiom to existing triangdation.-In starting from or connecting with either flrst- or seeond-order triangulation. it Is essential that the connection be ma& to a line of proper strength. nnd also that observations be made from the two ends of that llne a w n II third point of the eslstlng triangulatloa the new angles agree closely wlth the. old obaerv:ttfone * the exact recovery of the old stations Ie asmred. Eren when connecting the triangulatton of the third order It 1s better If meihle to connect with a line rather than a point, for the comparison of the lengths of the llne common.to the two sptems Oi triangulatlon m a y glve Wormation of q e n t value In adlnsting the weaker scheme. Connection in pMttion alone. ~LNXIelY. to (1 single point. or in position and a m u t h. namely. to a single mint but wlth o directton obserred from that poht to mother old station. m y sometimes be odtantageoualy made ot intervals bemeen the connections In length. If the line used for a connectton Opposite n weak 'nntjle in the old triangulation the comparison in length wiu have llttle value.
co!$ly to adjust.
CHILNGES OF SPECIFICATIONS
The following modifications of the above specifications are mainly to permit use of single t r i q l e s because of the relatirely short arcs of new trinngdation now required between junctioiis with adjusted ma and to allow for the fact that the desirable masinnurn length of
laterals about 10 miles. m i l e single triangIes are nom sometimes permissible; it must not be inferred that they may be used indiscriminately. A f i p giving double deteiminntion of length must be emploFed whenever it is possible to do so without uiidue expense or delay. Tlie limits of R, md Rt of P fipre have been reduced to 15 and 25: respectively7 and
their desirable limits to 10 and 25, respectively. The t e p g h of szvs
linea and base nets will be taken up rmder those headings (See pp.
@urea, that is, .figures requiring more tlisln one side gnation, is closely limited. However, such fi,- hare always been used when necessuy to cover Q difficult mea and to maintain the required figure
chosen at mill, but that the jud-pent of the reconnaissance engineer
the instructions. This limit has varied somewhat with the character of the country and the purpose of the triangulation. Genernlly, the
miles, with few diagonals exceeding 15 mila. This limitation is a imposed for two reMons-to avoid hindrance to progress caused by
lines they can readily use. azimuth marks, turnow established at a11 occupied stations, have larply eliminated the second condition. As for the first, it is recog- nized that for different types of country there am different econom- ical lengths of line. I n order to obtain the proper intensit) of con- trol, supplemental stations may be established within the areas of the main scheme figures. .Since the requirements for these supple- mental stations are only that each shall form a part of at least one well formed triangle, more accessible and useful locations can be found for them than when all the conditions of carrying forward the scheme are imposed. Generally, instructions are^ issued authorizing
ond-order tricmplotion only. When the adjustment of the trim- gulation net is made, a number of the arcs am designated “second order,” if that is the accu~cpactually obtained for ;hem. In practice, the angles for second-order triangulation are observed under the same
10
I 14' I
74 0 i 01 5 i s( I 4 1 37 34 32 30 ?a n 2s 24 23 22 22 n 3 19 I 9 18 IS 17 I f 1: I? 18 19 21 P
I 0' I
PI 19 16 I I 11 10 9 8 1 7 6 6 a 5 I
16 ' I
: i
gr
11 LO
r 6 5 s 4 4 4 4 3 3 3 4 4 4
I
I
a a
i
1 1 L L 1 c I I 1 I I I I L L s
I 3 2 (^21) I 1 1 1 I I I I
I I I I I I 1 I I I
I
USE OF TABLE
carried forward is concerned, proceed as follow: ( a ) Fnt ench flmr& take ont the distance angles. to the nearest degree U possible. for the best and second-best chains of triangles through the figure. These chafns :ire to be selected at d n t by estimation. and the estimate Is to be checked larer hy rhe results of compnrfsan. ( b ) For each triangle in ench chain enter the table with the distance angles ;ia the twu arguments and take our the tabular value.
ULYOAL OF BECOXXALSSLTCE 11
( 0 ) For each chin, the best and m u d best, through each figure, take the gum of the rrrbnlar values. ( a ) Uultiply en& sum by the factor D-C D '
for that Qpre, where D is the number of directions oherred and C 1s the number ab coadltions to be satMed in the figure. The qunntitlea so obtained namely,
will for convenience be called ZZt and I& for the'beat and seeond-best chains, resgectivelY ia dependent mainly upon the* strength of thc
Qure. The second-best chain contributes somewhat to the t o m strength and the other weaker and pm,ereseiveb less independent chain^ contribute still rrmaller amounta In deciding between figures they should be classed nccordlng to their best chains, unless said best china are very nearly of e~nalstrength and their second-best chains di& greatly.
(e) The strength of the
NOTES ON S U M OF RGUBES
from the foregoingformulas axid table is affected only by the line which is used to carry the lene@ forward to the next figure. It will be found that a figure very short in the direction of progress, compared to its
used in carqmg the computations forward. However, the side lines in such figurn lie opposite we& angles, and are not satisfactorp as
Generally speaking, a rapid increase or decrease in the width of the scheme w i l l introduce weakness, dthough not necessarily so. Such nrriation in width may be necessary in passing from one area into ,onother of greatly dsering characteristics. The transition can usu- ally be made without emat loss of strength by using acute and obtuse angles in combination 'as distance angles. T h e successive triangles am laid out so that the side opposite the larger distance angle in one tri- nn@a becomes opposite the smaller distance angle of the nest. T l h prmciple can be carried even further by taking adrantage of the fact
Since the triangles are adjusted so os to close theoretically, an crmr which would make one distance angle too large, should e a similar negative residual in the other. Hence in B triangle in whch one angle esceecls 90°r if the colog sine of the first is small the log sine of the second sliould be Iiirge ancl errorc will tend to compensate. Br refer- ence to the table, it will be found that a triangle with distance angles 365164°41-
la permissible to amy the tmo lengths through the same triangle provided M e r e n t combinations of distnnce angles are employed. However. the fke and six-sided central-point polygons are the only ones that should receive consider= atlon, and thep are Inferior to the simpler quadrilaterals. The factom of strengtb are 0.67 for five sidea and 0.88 for slr
C
E(N0.2 1
P (^0) n Fromm I .-DIhrent mea of used In trianmlatfoo. UNOCCUPXED STATIONS
from adjacent stations in the main scheme which are preferably in a
BmowRcIIBEzl
circamstances sometima atise which require^ the scheme to be offset
cent 6gure alongside, so that a side line of one is common to a side line of the other. (See fig. 2.) The strangth of the two rigures ia, of course, computed through the common side he.
Fraum 2-mbOw dgam AREA TRIAIVGtrLATION
Area triangulation consists of a scheme espanded by adding ti- angIes in all directions to form 3 spider web effect. This method
introcluction of new methods of adjustment and new calculating machine equipment, the simultaneous solution of area triangulation is not a formidable problem. The a i m method was initiated when it became necessary to provide covecige over areas completely bounded by arcs of trinnplation. Wien r l i w which should not be elongated in any direction, 31-0reduced to il size of about four figures across, the area method of reconnniszunce and triangulation is employed. Single triangles shaulcl Le uscd (see fig. 3) and a complete connection made to all
lisiied arc triangulation, the medium lines indicate the main scheme area reconimiss;mce, and the light lines are for supplemental stations.
