Paper-ID: VGI 198001
A New Determination of the Height of the World’s Highest Peak
Jing-Yung Chen1 , Dan-Sun Gun2
1
Research Institute of Geodesy and Photogrammetry, Peking, China, derzeit Institut
für Landesvermessung und Photogrammetrie, Technische Universität Graz,
Rechbauerstraße 12, A-8010 Graz
2
Research Institute of Geodesy and Photogrammetry, Peking, China
Österreichische Zeitschrift für Vermessungswesen und Photogrammetrie 68 (1), S.
1–19
1980
BibTEX:
@ARTICLE{Chen_VGI_198001,
Title = {A New Determination of the Height of the World’s Highest Peak},
Author = {Chen, Jing-Yung and Gun, Dan-Sun},
Journal = {{\"O}sterreichische Zeitschrift f{\"u}r Vermessungswesen und
Photogrammetrie},
Pages = {1--19},
Number = {1},
Year = {1980},
Volume = {68}
}
ÖZNuPh
1
68. Jahrgang / 1 980 / H eft 1
A New Determination of the Height of the World's Highest Peak
Von Jing-Yung Chen1), Dan-Sun Gun, Peki n g , China
Vorbemerkung
Die VR China hat 1 975 eine wissenschaftliche Expedition auf den höchsten Berg der Welt
durchgeführt, bei welcher erstmals auch geodätische Ziele verfolgt wurden. Um die genaue Höhe
des Quomolangma Feng genannten Mount Everest zu bestimmen, wurden auf diesem ein 3 , 5 m
hohes, trigonometrisches Signal errichtet u nd von umliegenden, etwa 6.000 m hohen Stationen
trigonometrische Messungen ausgeführt. Die Höhen u n d Positionen dieser Stationen wurden
durch Nivellements, Schweremessungen und trigonometrische Messungen ermittelt und a n das
geodätische G rundsystem der VR China angeschlossen.
An der Ausarbeitung dieser außerordentlichen geodätischen und bergsteigerischen Lei­
stung hat Herr Jing-Yung Chen, wissenschaftlicher M itarbeiter des Forschu ngsinstitutes für
Geodäsie und Photogrammetrie des chinesischen N ationalbüros für Geodäsie u n d Kartographie
i n Peking und der Verfasser des folgenden Beitrags, maßgebend mitgearbeitet.
Herr Jing-Yung Chen hat sein Studium 1 960 in Wuhan abgeschlossen, war bis 1 974
Forschungsassistent und Lehrer in der Abteilung für Astronomische Geodäsie und ist seit d ieser
Z eit wissenschaftlicher Mitarbeiter im Forschungsinstitut für Geodäsie und Photogrammetrie in
Peking. Herr J.-Y. Chen beschäftigt sich mit wissenschaftlichen Problemen der Vermessungs­
grundlagen in der VR China u n d hat dazu eine Reihe von Beiträgen publiziert. Seit Beginn 1 979
absolviert er an der TU in G raz einen zweijährigen Studienaufenthalt mit dem Z iel, westliche
geodätische Verfahren näher kennenzulernen und Vorschläge für eine Verbesserung der
chinesischen Vermessungsgrundlagen zu erarbeiten. Der Aufenthalt von J .-Y. Chen i n G raz ist ein
erster Schritt zu einer vom Unterzeichneten durch Besuche in der VR China angestrebten
wissenschaftlichen Zusammenarbeit auf dem Gebiet der Geodäsie zwischen der VR C h i n a und
Österreich.
Karl Rinner,
G raz
Abstract
The Quomolangma Feng') (Mt. Jolmo Lungma) is the world's highest peak. lts accurate
height above mean sea-level has long been the concern of surveyors and geographers the world
over. I n 1 975, members of a Chinese mountaineering expedition for the first time erected on the
summit of the QF3) a metallic target and also measured there the thickness of the covering snow.
At the same time, and in coordination with the expedition, a surveying group carried out leve l l i n g ,
triangulation, astronomical and gravimetric measurements within a close range of t h e QF. T h e
nearest g ravity station was at an elevation of 7 7 90 m and a t a distance of o n l y 1 .9 km f r o m the
summit. I n this paper the computation for determ i n i n g the height of the peak according to rigorous
geodetic theory is described in detail, and som e q uestions regarding atmospheric refraction,
deviation of the vertical, gravity and the geoid are discussed.
Most of the short-comings have now been overcome which existed in the previous height
d eterminations. The height of the QF as obtained i n 1 975 is 8848.1 3 m ± 0.35 m above mean sea-
')Presently at Institut für Landesvermessung und Photogrammetrie, Technische Universität Graz, Österreich.
')lt was called Mount Everest before.
')QF is used as an abbreviation of "Quomolangma Feng" in this paper.
2
ÖZIVuPh
68. Jahrgang / 1 980/ Heft 1
level. This may be the most accurate value that has so far been obtained for the h i ghest peak of
the world.
lntroduction
The Quomolangma Fen g is the world's h i g hest peak. lt is situ ated on the
border between China and Nepal. lts prec ise hei g h t h as long been a m atter of
interest among the su rveyors and geog raphers the world over [4], [7], [8]. I n
1 975, a Chi nese mou ntaineering exped ition erected for t h e fi rst time a meta llic
target on top of the peak (Fi g . 1 ), and measured the th ickness of the coveri n g
snow a t the spot. At the same time a Chinese su rvey i n g g ro u p carried out
coord inated geodetic measu rements with i n a c lose range of the QF (about
30 km x 30 km).
F i g u re 1
As a resu lt of these efforts, it became possi b le to com pute the height of
the QF by rigorous geodetic meth ods. The heig h t value (8848 . 1 3 m ± 0.35 m )
m ust be considered t h e m ost accu rate one thus f a r o btained [5] [1 7].
ÖZfVuPh
3
68. Jahrgang / 1 980/ Heft 1
West Rongbuk Glacler
East Rongbuk Glacler
e
,/"
�
Es
r:.
I
I
( North Col
'
'
'r
7600
I
I
@Oomolangma Feng
0
second-order station
third-order side
0
third-order station
9
EDM side
gravimetric station
e
astronomical station
--
"'-.../
second-order side
e
astronomical azimuth
astronomical levelling line
levelling line
stations occupied for interseeting the QF
The determination of the QF in
Fig ure 2
1975.
{,
4
ÖZIVuPh
68. Jahrgang / 1 980 / Heft 1
1. Field Work
Beg i n n i n g from the national 1 st order traverse at D i n g ri, in the south of
Tibet, a 2nd order tria n g u l ation chain abo ut 60 km l an g was laid out toward
the d i rection of the QF. A 3rd order chain along the East Rongbuk G lacier,
and an EDM traverse along the West Rongbuk G lacier, were tied to the 2nd
order stations, and control led by Laplace azi m uths at their end points (Fig . 2).
The stan dard erro r (s. e.) of angle measurement of these 3rd order stations
after adjustment is ± 1 '.'73. Among the 3rd order stations were nine from
where the target was visible; th ese were used for i ntersecting the QF. They
were 8.5 km to 21 . 2 km away from the peak ( 1 2 . 8 km on the average); eleva­
tions ran ged from 5600 m to 6240 m (5784 m on the ave rage). The m axi m u m
angle o f i ntersection was 6 9 ° .
Astronom ical latitude a n d l o n g itude were d eterm i ned on 1 5 stations i n
t h i s su rveyed reg i o n , their s. e . are ± 1 " and ±0�06 respectively. A m o n g these
astronom ical statio n s the nearest one to the QF, I I I 29, is at an elevation of
6336 m and o n ly 5 km away from the peak. In ad d ition, five Laplace azi m u ths
were determ ined with a s . e . of ±2".
G ravimetrie meas u re m ents were made on 20 poi nts i n th is reg i o n , their
s. e. is ±3 mgal. The two h ig h est points are at elevatio n s of 7050 m and
7790 m , respective ly, the latter be i n g o n ly 1 . 9 km apart trom the s u m m it. They
are the h i g hest g ravity points on land that h ave ever bee n meas u red i n the
h istory of geodetic s u rvey i n g .
Starting from a n ational ben c h m ark ( B M ) a t D i n g ri, a 2 n d order leve l l i n g
line was measu red southward u p to a B M s o u t h of the Rongbuk Lamasery.
The triang u lation station I I I 7 , abo ut 5683 m above mean sea-level and
1 3 . 6 km north of the s u m mit, was c o n n ected to this B M by means of spirit
levelling; the mean squ are value of random e rrors of mean height d ifference
per km is ±0.71 mm. The height of I I I 7 was used as the basis to r t h e succee­
ding height determ i natio n .
Reciprocal trigo n ometric leve l l i n g was carried out between t h e 3rd order
stations to determ i n e their own height. As soon as the m etal l i c target was
erected on the s u m m it, zen ith d istances to the target were observed from all
the n i n e stations d u ri n g t h ree consecutive d ays.
I n the meanti m e, a n u m ber of s o u n d i n g bal loons were l ifted to m easure
the temperat u re g rad ient of the upper atm osphere; the val u es were u sed later
to d eterm ine a reliable coefficient of refractio n .
T h e height o f t h e target a n d t h e thickn ess of t h e covering s n ow a t the
spot were measured with 3 . 5 1 m and 0.92 m , respectively.
ÖZfVuPh
5
68. Jahrgan g / 1 980 /Hefl 1
2. Height Computation
Restrictions d u e to the specific geographical cond itions in the area led to
the following general proced u re for determ i n i n g the heig ht of the QF:
- Extension of the national geodetic a n d leve l l i n g nets to a range as c lose to
the QF as possible;
- Observation of the horizontal ang les and zenith d istances to the peak from
a number of selected triangu lation and traverse stations, so as to dete rmine
the geographical coord inates of the Q F and the height differences between
the observation stations and the s u m m it;
- Finally red u ction of the measu red height to mean sea level (orthometric
heig ht) utilizing the d ata of the astro n o m ical and g ravi m etric measureme nts
in the su rveyed reg i o n .
T h e topography aro u n d the QF is very rou g h . The g e o i d i n this reg ion is
deemed to be rath er complicated and can not be consid ered as bei n g close to
a reference ellipsoid. Consequently, the gen eral princi ple i n a rigorous
com putation has to be that, the su rface of an ellipsoid is taken as the basic
reference su rface to which all the observed data m ust be red u ced [2], [1 4];
then the horizontal and height com putat ions are carried out.
The pri ncipal steps of data process i n g in this com putation are as fol lows:
2. 1 The Normal Height of Station III 7, HJ17
Station I I I 7 is con nected with the national datu m 0 at Tsi n gtao t h ro u g h
t h e n ational 1 st and 2 n d o rder leve l l i n g net. I n order t o obtain1117
HJ17, two co rrection terms should be added to the level led height d iffere n ce, 2: l.ih, fro m 0 to I I I
0
7 , by the following form ula [1 1 ], [1 6]
HJ11
=
III7
III 7 g
7
� l.ih + � e + �
III
0
0
0
_
y
--
y
Lih
(1 ),
where e is the correction caused by the non-parallelism of the level su rfaces i n
t h e normal g ravity field; g a n d y are values o f measu red g ravity a n d n o rmal
g ravity along the line of leve l l i n g , res pective ly. The s. e . of H h M 1, is esti­
mated to be
M1
± 0.14 m
(2),
=
accord i n g to the usual accu racy esti m ation form u la for spirit level l i n g in a
normal height system [1 6].
2. 2 Geodetic Height of III 7, Hii11
The geodetic height Hf is the s u m m ation of normal height H; a n d h e i g ht
anomaly �1 for any station i . As reg ards I I I 7, we have
(3).
6
ÖZfVuPh
68. J a h rgang / 1 980/ Heft 1
I n the process of com puting the height of the QF, �111 1 w i l l be cancelled as
shown in equation (1 4) below, hence the error in �1111 h as n o effect o n the
results, and an approxi m ate value of it will be sufficient.
2.3 Geodetic Heights of the Triangulation and Traverse Stations, H�
Corrections for the d eviation of the vertical are applied to the observed
value of the zen ith d istance Z, red u c i n g it to the geodetic zenith d istance z•
referred to the ell ipsoid [1 1 ]:
z·
= Z + u, u = � cosA +
ri
sinA
(4),
where u is the component of the astrog eodetic deviation of the vertical in the
observed d i rection, the g eodetic azi m uth of which is A, and �. ri are the
components of the deflections of the verti cal.
Then the common form u l a of rec iprocal trigono metric leve l l i n g is used to
compute the differences of geodetic heig hts, Lih1111 1 between I I I 7 a n d any 3rd
order trian g u l ation and traverse station i .
T h e geodetic height for station i can n o w be written as
(5),
2.4 Geodetic Height of the QF, H'Q
The zenith d istances to the target o n the peak observed from station i a re
corrected with Eq u . (4). The d ifference of g eodetic height between the Q F and
station i , Lih 10, is com puted accord i n g to the fo l lowi n g precise fo r m u l a [12]
(6),
where
H•
_o_ +
H0
LiH
=
Li K
= 21R <1 - sink
3 R2
'
-'
·
6 R2
z·
);
"I" is the height of the i nstru ment at an observed stat i o n ; "a" is the height
of the target at the QF; "s" is the distance o n the ellipsoid from station i to the
OK ; "R" is the rad i u s of c u rvat u re of t h e n o rm al section of the e l l i psoid along
the l i n e of observation; "k" is the coeffi cient of refraction .
The geodetic heig ht of the QF is then
(7).
ÖZfVuPh
7
68. Jahrgang / 1 980/ Heft 1
The resu lts obtai ned from each station are presented i n Table 1.
Assi g n i n g to the res u lt of each station a weig h t i nversely proportional to
the square of its d istance to the peak, the weig hted m ean of the geodetic
height of the QF, H 0, is fo u n d .
T h e maxi m u m difference between H 0; derived from various stations is
1.4 m . M 2 is the s. e. estimated from the residuals:
M2
=
(8),
± 0.18 m
M 2 contains the errors i n ßh 11171 and ßh 10, but does not include the error o f the
H �11. therefore it really presents a measu re for the error of tri g o n o m etric
leve l l i n g .
Table 1
The Heig ht Values of the Q F from D iffe rent Stations
Observation
Distance
Station
s (km) 1 )
Height of
Station
H ; (m)1)
Coefficient of Normal H e ig ht
Refraction
of the QF
H 6, (m)
k
8
II
7
III
East 2
E ast 3
West 1
West 2
West 3
West 4
West 5
21.2
13.6
10.0
8.5
11.3
11.7
12.0
12.5
14.7
6120
5683
6242
6168
5602
5748
5750
5772
5798
0.0744
0.0810
0.0758
0.0761
0.0793
0.0772
0.0828
0.0804
0.0820
8846.21
8846.59
8846.88
8846.31
8846.39
8847.36
8847.61
8847.63
8846.85
M ean
12.8
5874
0.0788
8846.852)
') only approximate values are given here
Res i d u a l
(m)
-0.64
- 0.26
+ 0.03
- 0.54
- 0.46
+ 0.51
+ 0.76
+ 0.78
0.00
') weighted mean
2.5 Normal Height of the QF, H 6
Accord i n g to (3), we h ave
H6
=
H ö - �o
H6
=
H ö - �1 111 - (�o - �1 d
or
l
(9),
where the difference of the height anom aly from III 7 to the peak ( �0 - �1 d , can
be computed by the following prec ise fo r m u la of astro n o m ical leve l l i n g [16],
[12]
Q
Q
�o - �1111 = � U S -� ßE
1117
1117
(10).
g - 'f
ßE =
ßH
'(
l
8
ÖZfVuPh
68. J ah rgang / 1 980/ Hefl 1
The seco nd term of e quation (10) relates to g ravity anomalies. When the
d ifference of height is g reat, then the value of this term always su rpasses that
of the first term, re lating to the d eviation (com pare Table 2). Hen ce, i n ge neral
speaki ng, the second term sh o u ld not be neg lected i n mou ntain reg ions.
Fro m the last column of Table 2 o n e g ets the value of (�0 - �1111), as -1.825 m.
Table 2
Com putation of D ifferences of Height Anomalies (unit: m eter)
Point Along
the Une
s
H
III 7
East 2
East 3
East 5
East 6
North Go i
No. 7600
�E
u s - �E
- 0.012
+ 0.180
- 0.192
-0.008
- 0.026
+ 0.018
-0.117
+ 0.047
-0.164
-0.091
+ 0.101
-0.192
-0.058
+ 0.200
-0.258
- 0.065
+ 0.321
-0.386
- 0.176
+ 0.475
- 0.651
- 0.527
+ 1.298
- 1.825
5683
4279
6242
2656
6168
3042
6301
2819
6565
1329
7050
1394
7790
1947
The Q F
Sum
u s1)
8849
17466
') u is the deviation of the vertical of station along the astronomical levelling line
s is the distance between two consecutive stations
The g ravity val ues at all points but the QF along the l i n e of astro n o m ical
leve l l i n g g iven in Table 2 were actu ally m eas u red. The g ravity valu e of point
No. 7600, mentioned above, is partic u l arly im portant for a reliable p red iction
of g ravity val ue of the QF. The values of the d eviation of the vertical at points
III 7. East 2 and East 5 were o btained d i rectly by astro n om ical observations,
but those of the othe r stations are ded u ced by p red ictio n . The s. e., M3, of the
d ifference of height anomalies (�0 - �1d is estimated accord i n g to the follo­
wing formula
M3
m2
= ± /g2
"
QF
QF
�H
1117
1117
y
Sm :2: s + :2: ( - m 9) 2
(11) ,
where m " and m 9 are the s . e . for the val u e of d eviation of th e vertical and of
the g ravity anomaly at the co rrespo n d i n g station (it will be disc ussed later i n
detail).
ÖZfVuPh
9
68. Jahrgan g / 1 980 / H eft 1
M3
= ±
0. 1 0 m
( 1 2).
The resu lt obtai ned for the n o rmal height of the QF from e quation (9) is
Hö
=
8846 .85 m
( 1 3).
By su bstitutin g e quations (7), (5), (3) into (9) successively, it is easy to
obtain
(1 4).
lt can be seen from this form u l a that an error i n �1111 will not affect the
resu lt of H (i.
Accord i n g to e quatio ns (1 4), (1 2), (8), (2) the s . e . , MH(i, of the normal
hei g ht of OF can be written as
MHö = ./M�
+
M�
+
M�
=
±
0.25 m
(1 5).
2.6 The Orthometric Height of the QF, Hß
The relation between n o rm a l height and orthom etric heig ht of the QF is
[1 1 ] , [1 6]
Hß = 'Ym H YQ
( 1 6),
9m
ellipsoid
The relation of the basic surfaces
at the QF ,
Fig u re 3
10
ÖZtVuPh
68. Jahrgan g / 1 980/ Hett 1
where yrr,, the mean normal g ravity, can be obtained by s i m ple comp utatio n .
The meth od for com puti n g the mean actual g ravity g m will b e outlined below.
By form ula (1 6), the difference between the orth ometric h e i g ht a n d the n o rmal
heig ht of Q F i s comp uted to be 1 . 28 m . That means the q u asigeoid u n der the
QF l ies 1 .28 m above the geoid (Fi g . 3).
Conseq u ently the o rtho m etric height of the Q F is 8846.85 + 1 .2 8
8848 . 1 3 m above m ean sea-level of the Yel low Sea, o r
Hö
=
8848 . 1 3 m
=
(1 7).
The error for computi n g g m at QF is about ±28 mgal (refe rred to 3 . 6). lts
effect on the conversion from Hö to Hö may be estim ated, accordi n g to (1 6), as
M4
=
±
0.25 m
(1 8).
So, the s. e„ M Hö• of the o rthometric height of the QF determ i n ed i n 1 975
can be estimated as
M Hö
=
± ...; M M + M �
± 0.35 m
=
±
...; M t
+ M� + M� + M�
(1 9).
3. Discussion
3.1 The Target
The heig ht of the target was designed i n acco rdance with the topography
on the summit as described by C hi nese mountaineers who scaled the park in
1 960, and verified by aerial photos of the QF area. Estim ates based o n the
elevations and positions of the obse rvation stations re lative to the peak
showed that a 3.5 meter-h i g h target on the s u m m i t would be visible trom all
the observation stations. The peak is always covered by wh ite snow a n d the
backg rou n d for o bservations is a bright blue sky. M an y p ractical tests were
needed to f i n d that the targ et m ust be painted in red to provide good co ntrast
for accu rate point i n g . Besides, the target had to be made of l i g h t alloy with
high strength, weig h i n g o n ly 3.74 kg , with flexible legs and foldi n g cylinder for
convient carry i n g du ri n g c l i m b i n g . Yet its struct u re was designed to withsta n d
stron g gales always preva i l i n g on the s u m m it . The central r o d of the target i s
g raduated i n centimeters, so that an accu rate readi n g for the h e i g h t of t h e
target c a n easi ly be take n . W h e n erected, the targ et was sec u re d w i t h t h ree
nylon ropes fastened to ice picks firmly ancho red i nto the icy g ro u n d. As a
matter of fact, it took less than half an h o u r to erect it on t h e peak secu rely,
and in spite of the storms and g ales preva i l i n g on the peak, the target h as
remained standing for at least more than 3 years, for in May 1 978 some
Austrian mou ntaineers took a s u m m it photo showing th emselves together
ÖZfVuPh
11
68. Jahrgang / 1 980/ Heft 1
with this target (Fig. 6). Practice has thus proved that such a targ et is fit for
extreme conditions and wo u l d be worth i ntroducing elsewhere.
3. 2 Field-data and their accuracy
I n the close vin icity of the QF area besides the performance of geodetic
meas u re ments a lot of astro n o m ical and g ravi m etric data have been g a i n ed.
Same featu res of the data a n d their accu racy are listed in Table 3 a n d 4 .
Table 3
Features of the Field-data in the QF s u rveyed Area
target
height of the targ et itself on the s u m m it
depth of the covering snow u n der the target
height of the h i g h est BM (2n d orde r), I I 8
distance of the nearest BM (4th o rder) to the Q F ,
III 7
height of the h i g h est astronomic statio n , I I I 29
astronom ical distance of the nearest astro n o m ic station to the
measurement
QF, 11129
sum of the astro n o m i c stations
average distance between two co nsecutive
astro n o m i c stations along the astronom ical
leve l l i n g l i n e
h e i g h t o f the h i g h est g ravi m etric point, No. 7600
g ravi m etric
distance of the nearest g ravim etric point
to the QF, No. 7600
meas u rement
sum of the g ravimetric points
stations for
i ntersecti n g
the QF
height of the h i g hest station
average height
the shortest distance to the QF
average dista nce to t h e Q F
sum o f stations
maxi m u m angle of i ntersection
days lasted for i ntersection
sou n di n g
balloon
days lasted for so u n di n g balloon measurement
meas u rement
3.51 m
0.92 m
61 20 m
1 3 . 6 km
6336 m
5 km
15
2.5 km
7790 m
1 . 9 km
20
6240
5784
8.5
1 2.8
9
69
3
m
m
km
km
days
9 days
12
Table 4
ÖZIVuPh
68. J ah rgang / 1 980/ Heft 1
Standard Errors of the Field-data
mean height difference for one km spirit leve l l i n g from BM to I I I 7 ±0. 7 1 m m
mean height difference i n rec i procal trigonometric leve l l i n g for
one side of a triangle
astro nom ical latit u de (Talcott)
±0.04 m
± 1 '.'0
astronomical longitude (Tsinger)
±0'.06
astronomical azimuth (Ho u r Angle of Polaris)
±2'.'0
angle meas u rement after adjustment
± 1 '.'8
distance meas u rement
± 1 0 ppm
g ravi metric measurement
±3 mgal
F i g . 6 T h e target on t h e summit o f the QF
(photo by Austrian mountaineer Robert Schauer, May 9, 1 978)
ÖZ!VuPh
68. Jahrgang / 1 980 / H eft 1
13
3.3 The Vertica/ Refraction of the Atmosphere
I n o rder to u n derstand correctly the ru les concern i n g the vertical refrac­
tion of the atmosphere in the QF area, many experim ents were carried o u t on
"th e roof of the world" before t h e h e ig ht determination . One kin d of experi­
ment was the reciprocal trigo n o m etric leve l l i n g on some stations (con nected
by the spi rit leve l l i n g li ne) u n der different m eteorolog ical conditions. The
followi n g resu lts were o bta ined for the refraction coefficient i n different
situations:
1 . The value k of the coefficient of refraction is smal lest an d m ost stable
between 1 2h-1 5h (local ti m e), consequently this is the most favo u rable time
for o bserving zen ith distances.
2. When the trigon ometric leve l l i n g was carried out in the triang u lation a n d
traverse, the mean val ue of the refraction coefficient f o r the time from
1 2h-1 5h was about 0.1 0.
3. The amplitude of di u rnal variation of the coefficient of refraction a l o n g the
s i des of triangulation l i nes has a maxi m u m value of 0.20 a n d a mean value
of 0.07.
Another kind of experiment was carriad out by i ntersecting the s u m m it of
the QF from some trian g u lation statio n s, knowing their heig hts. The aim of
this experi ment was to ve rify that the fo llowing form u l a [3] is su itable to
calcu late the refraction coefficient for sig hts to the Q F :
ßh
p
1
(20),
k = 6 . 706
(3. 4 2 + 7) [1 - 3 (3. 4 2 + 7) T] sin Z
T2
where T is the tem peratu re of the air at a station in °K; P is the atmospheric
pressur'e at the observation station i n m m of a merc u ry co l u m n ; 7 is the mean
vertical g radient of the air tem peratu re i n °C / 1 00 m ; ßh is the difference of
the height i n 1 00 m; Z is the zen ith distance of the line of sight.
Form ula (20) is derived accordi n g to the physical pro perties of a free
atm osphere and by ass u m i n g the vertical g radient 7 of the air temperatu re as
constant; in theory it is therefo re especially su itable for sights to t h e Q F for
the path of ray of such sig hts always lies h i g h above the g ro u n d.
.
By using the value of k calcu lated accordi n g to fo rmula (20), the h e i g ht of
the QF derived fro m experi mental statio n s located 8 to 77 km away fro m the
peak showed no apparent systematic difference relating to the length of sig ht.
Bes i des, for these sig hts, the amplitude of di u rnal variation of the coefficient
of refraction o btai ned by this form u l a is always less than 0 . 03, with an average
of o n ly 0.01 7 . Th is is about one q u arter of the amplitude for an o rdin ary
terrestrial sight mentioned above. The stabil ity of the refraction coefficient for
sig hts to the QF really co rrespo n ds to the practical situation i n the area of Q F .
These resu lts demonstrate the a daptabil ity o f t h e form ula f o r t h e sig hts t o t h e
QF. Conseq uently form u l a (20) was em ployed t o com pute the values o f k for
the sights to the QF i n 1 975; the values of k were arou n d 0.08 (Tab . 1 ).
14
ÖZfVuPh
68. J a h rgang / 1 980/ Heft 1
I n order to get an accu rate val u e of k it is essential that the vertical
g radient of air temperatu re, T , m u st be determined correctly. Same of the
following experiences m ay be useful i n this context :
1.
m ust be measu red in situ practically by radio sou n di n g of the m eteorologi­
cal conditions at different altit u de of the atmosphere.
2. lt is i m portant to identify the lowest a ltitu de from which t h e vertical g radient
of air temperatu re beg ins to enter i n the com putation of T . O u r experiments
showed that the lowest altit u de is 500 m above the g ro u n d i n the Q F area.
3. In form ula (20) T is a mean value for the m eteorological period i n which the
temperatu re, pressure, etc . , of the atmosph ere are s i m i l ar. lt the o bserva­
tion take an extended time, then o n e or more different values of T s h o u l d be
used to co mpute k for this o bservation, dependi n g on th e n u m be r of the
local meteorological periods .
T
3.4 The Prediction o f the Deviation o f the Vertical
I n poi nts without astronom ical observations the values of the astrogeode­
tic deviatio n of the vertical were predicted on the basis of the known astro­
geodetic deviations of the vertical by way of topog rap h i c isostatic deviations
of the vertical [1 1 ], [1 5] . In computing the topog rap hic isostatic deviations the
outm ost radius of i nteg ratio n was 670 km and the depth of isostatic compen­
sation was 1 1 3. 7 km [6], [9], [1 O] .
Same trial computations were made to check the real accu racy of predic­
tion by this method. In one such comp utation using a total of 1 5 astronom ical
stations in the su rveyed reg ion, 8 statio n s were arbitrarily chosen as co ntrol
poi nts, and the values for the remai n i n g , 7 stations, were predicted. The
differences between the predicted an d the o bserved values are listed in Table
5 . The s. e. of the prediction as computed from these differences is ±2'.'8 .
At fo u r of the n i ne 3 rd o rder stations t h at were used for the observatio n of
the QF, the val ues of � a n d 11 were derived di rectly from astro n o m ical observa­
tions, while in the other five stations they were predicted as described a bove.
With an average distance of 1 1 .5 km from the station to the QF, the effect of
the prediction erro r i n o n e station o n the height of the Q F is esti m ated to be
±0. 1 6 m. After averag i n g the res u lts from all stations, the effect o n the mean
height of the QF will be not more than ±0.06 m. A n esti m ate based on equa­
tion (1 1 ) shows that the error of the pre dicted deviation of th e vertical effects
the resu lts of astro nom ical leve l l i n g (�0 - �,d by about + 0 .09 m (here, m"
±2'.'8, s � s
4 .36x1 0-1). Hence it can be seen that even i n such a h ig h ly
mou ntainous area, the p rediction m ethod for the astrogeodetic deviation of
the vertical by way of to pographic isostatic deviation of the vertical is j ustified.
The accu rracy o btained m ay not be com parable to that in a flat area, h owever,
its effect on the fin al resu lts is l i m ited.
=
=
ÖZfVuPh
Table 5
15
68. Jahrgan g / 1 980/ Heft 1
The E rror of th e P red i c tion of the Dev i a tion of th e Ver tical
S ta tion
V�
v 11
2
3
II
II
4
III
8
III
25
29
III
Eas t 2
+ 0 '.'37
+ 2. 51
+ 1 . 84
- 3 .94
- 3 . 38
- 0 . 56
- 0 . 53
+ 3 '.'74
+ 2. 1 8
+ 2. 29
+ 1 . 34
- 1 . 85
- 3 . 84
+ 2. 0 2
Sum
- 5 . 25
+ 5 . 88
[vv]
37 . 37
4 8 . 03
± 2'.'31
± 2'.'6 2
± V[vv] / n
v: the d ifferences between th e
predi c tion and the observed
values of th e devi a tion of th e
vertica l a t a s ta ti o n .
3.5 Gravity Prediction
We h ave com p u ted the free-air anom aly, Bouguer anom aly, B o u g u e r
anomaly wi th terrain correc ti o n , and iso s ta ti c anomaly 1 ) a t 20 measured
g ravi ty poi n ts in th e vici n i ty of th e peak. As expec ted, th e vari a ti o n .of th e f ree­
air anomal ies is larges t, while tha t of th e isos ta tic anomal ies is smallest, w i th
an ampl i tude of variation of o n ly abo u t 40 m g a l . Hence the g rav i ty val u es of
the Q F was pred i c ted o n th e basis of the meas ured g ravi ty value by way of
isos ta ti c anomal ies [1 2] , [1 4]. I n c 6m p u ti n g the isos ta tic anomalies we too k
1 66 . 7 km a s the ou tmost rad ius of i n tegration and 1 1 3 . 7 km a s th e dep th of
isos ta ti c com pensa tion .
I n o rder to es ti m a te the accu racy of th e pred i c ted g ravi ty , we also made a
trial compu ta tio n . The g ravi ty values a t two poi n ts neare s t to th e s u m m i t, i . e .
the North Go i (7 .050 m) and N o . 7600 (7. 790 m ) , w h i c h had been measur �d b y
g rav i m e te rs, were considered as u n known and pred i c ted on th e basis o f the
o ther m easured g ravi ty values. The d ifferences b e tween th e pred i c ted and
observed va lues were fou nd to be 1 5 mgal for N o r th Go i and 1 9 mgal for po i n t
No. 7600. Gonseq u e n tly the s . e . for th e pred i c ted g rav i ty value a t the to p of
the Q F is es ti m a ted a t ± 20 mgal. We also made some tes ts o n selec ti ng a
proper depth of isos ta tic compensa tion for g rav i ty and deviation pred i c tion i n
th is area. T h e tes ts showed , th a t th e accu racy of pred i c ti o n wo uld d e teriorate,
if the depth was less than 50 km o r more th an 200 km .
') Obtained alter Bouguer. Terrain and isostatic corrections.
ÖZfVuP h
16
68. Jah rgang / 1 980/ Heft 1
3.6 The Mean Gravity Va/ue 9m
In order to compute 9m at the QF, the gravity values at the QF from the
ground surface to the geoid had to be determined. As it was impossible to
measure the gravity underground, it was calculated on the hypothesis of
isostasy: the plumbline from the summit to the geoid was evenly divided into n
sections, the dividing points being denoted by 0, 1, 2, . . . (n-1), n (Fig. 4).
geoid
The plumb line at the QF is
divided into n sections.
Figure 4
Let A, and B; be the vertical components of gravitational attraction to the
ith point on the plumbline due to topography and isostatic compensation,
respectively, counting positively downwards; and ßg, is the free-air reduction
from the summit to point i. We use the following expression with second order
terms [13]:
ß
ßg; = 0.3086 [(n - i) �] - 0. 72 x 10-1 [(n - i) H ]2
(21),
n
n
The gravity value gi at the ith point is:
(22),
where g" is the gravity value at the top of the summit. The mean gravity value
9m is then derived by the following formula:
9m
=
9n - An + Am + Bn - Bm + 0.1543 Hß - 0.12
in which
n
1
Am = -- L.A,
n+1
Bm
=
-1
n+1
°
n
L.B
o
'
X
)
Hg
1o-7 Hß ( 2 Hß + �)
n
(23),
(24).
ÖZIVuPh
68. Jahrgang / 1 980 / H eft 1
17
H(km)
Figure 5
8.85
7 . 08
1.77
---'-"'----'----'---'---'----'----'----'--600 -400 -200
c
+200 +400 +600 +800
The veriation of A;
-the
A mgal )
�
vertica l attraction
due to topography,
Fig. 5 shows the variation of the attraction due to topography, Au with
respect to the height of point i. lt is seen clearly that there is no linearity in the
gravity values along the plumbline under such a high peak. An error of about
100 mgal would be introduced if the calculations were based on linear predic­
tion. According to the trial computation, when n >30, the amplitude of the
variation of the mean gravity 9m or the mean normal gravity Ym was only within
3 to 5 mgal, so we took n = 50 for computing 9m and Ym ·
There are two sources of error in calculating the gravity value 9m by the
above method: the error of the summit gravity value 9n and that of deducing 9m
on the basis of the hypothesis of isostasy. For the value of g"' a s. e. of about
±20 mgal due to prediction has been estimated in 3.5. The latter may also be
regarded as prediction, however, along the vertical direction on the hypothe­
sis of isostasy. So the same s. e. of ± 20 mgal may well be adapted. By
combining the two sources of errors, the s. e. of 9m obtained is estimated to be
±28 mgal.
3. 7 Use of Normal Height as a Medium
The fact that normal height is used as a medium to compute the orthome­
tric height from the geodetic height needs explanation. The geodetic height is
the sum of orthometric height and undulation, that means the separation of
the geoid from the ellipsoid. In order to determine the undulation, the values
of deviation of the vertical on the geoid (not on the earth surface) must be
known so that the geoidal undulation could be computed by the simple
formula of astronomical levelling. But the deviation on the geoid cannot be
obtained directly from astronomical observation unless a correction for the
curvature of the plumbline is applied. The curvature of the plumbline is
dependent on the mass distribution ·in the crust of the earth which is not
exactly known. Therefore the usual practice is to correct only for the normal
curvature of the force line of the normal spheroid, when the elevation is not
ÖZfVuPh
18
68. Jahrgan g / 1 980 / Heft 1
high. But this simplification would introduce a serious error when the eleva­
tion is high, such as in the Q F area. Therefore in the computation we have to
use normal height as an intermediate result. The unique feature of the normal
height system is that it is entirely independent of the mass distribution in the
crust of the earth. The normal height can be obtained by applying to the
levelled height a correction for gravity anomalies on earth surface. The height
anomalies can be obtained from astronomical levelling, using values of � and
'f\ on the earth surface and supplementing them by gravimetric data. Hence in
the process of our computation no assumption is needed on the mass distri­
bution in the earth's crust to obtain the normal height of the peak. The formula
used is rigorous, and the accuracy of the result is affected only by the errors
of observation and of prediction, but no error exists due to hypothesis. Only in
the reduction from the normal height to orthometric height of the Q F an
assumption on the mass distribution in the earth's crust has to be made. The
assumption serves to compute the mean value of gravity gm along the vertical
between the earth surface and the geoid. The error in the last step of the
computation is estimated to cause a s. e. of ± 0 . 25 m of the value of orthome­
tric height of the QF (see 2.6 and 3.6 ).
4. Conc/usion
The new determination of the height of the Q F in 1975 has the following
specific characteristics:
1. A target was set up on the top of the summit and the thickness of the
covering snow on the spot during the observation was measured. This was
necessary to reduce an otherwise serious error of sighting the summit and
of computing the height of the peak.
2. The normal height system was used as a medium in this computation. As no
assumption on the mass distribution in the earth's crust was needed in the
steps for computing the normal height of the QF. Therefore the result of
this new determination avoided a part of the error caused by the approxi­
mation in these assumptions.
3. Sufficient astronomical and gravimetric measurements were made in the
close vicinity of the QF area, such that the height of the Q F obtained from
spirit levelling and trigonometric levelling could be reduced rigorously to
the geoid. The resulting height is thus the real orthometric height. The sum
of corrections relating to gravity anomalies in our computation amounts to
about 2 . 5 m. That is to say, if computation is made disregarding the gravity
data, as it was in the practice of all previous determinations, the resulting
error in the computed value of the height of the Q F would be more than
2 m. This value itself showes clearly that the field work and the procedure of
computation used are both proper and necessary.
ÖZfVuPh
19
68. Jahrgang / 1 980/ Heft 1
The result of this new determination, obtained in 1 975, is the orthometric
height of the QF with a value of 8848 . 1 3 m ± 0.35 m above mean sea-level of
the Yellow Sea. This result is claimed to be the most accurate and scientific
one that has so far been obtained. Another result of the effort consists of the
data processing procedure which proved that it could also be necessfully
applied to compute accurately the height of other high mountains.
Acknow/edgement
The precise height measurement of the QF could not have been obtained
without the exceptional achievements by Chinese mountaineers and the
dedication of the accompanying surveying group.
We wish to thank Prof. Dr. Yun-Lin Chen, Chief engineer of the General
Bureau of Surveying and Mapping of China, for his valuable suggestions and
beneficial instructions during all the course of the work. For the useful and
exact calculations we would also like to express our appreciation to our
colleagues who took part in this computation.
Finally, it is with gratitude that we acknowledge the support and encoura­
gement given to us by Prof. Dr. Karl Rinner for publishing this paper and for
many useful discussions during its preparation. We also wish to thank Dr.
Franz Leberl and Mr. Herbert Lichtenegger for their help.
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