Optical measurements of range and elevation angle are distorted by the earth's atmosphere. High precision refraction correction equations are presented which are ideally suited for surveying because their inputs are optically measured range and optically measured elevation angle. The outputs are true straight line range and true geometric elevation angle. The 'short distances' used in surveying allow the calculations of true range and true elevation angle to be quickly made using a programmable pocket calculator. Topics covered include the spherical form of Snell's Law; ray path equations; and integrating the equations. Short-, medium-, and long-range refraction corrections are presented in tables

Lear, W. M.

English

NASA Technical Reports Server

79-FM-U JSC-16064 
Refrrction Corrections 
for Su rveyino .b' 
(NASA-TB-80803) BEPRACTf:  . CORRECTIONS FOR N8O- IO907 
SURVEYIYG (NASA) 23 F !T ' ~ . 2 / f l P  A 0 7  
CSCL ZC-' 
Unclas  
63/74 39008 
Mission Planning and Analysis Division 
August 1979 
NASA 
National Aeronautics and 
Space Administration 
Lyndon 8. Johnson Space Center 
Houston. Texas 
https://ntrs.nasa.gov/search.jsp?R=19800002662 2020-03-21T20:19:34+00:00Z
7944-27 
SHU"LE PROGRAM 
REFRACTION CORRECTIONS FOR SURVEYING 
By U i l l i a m  H. Lear 
TRU 
JSC Task Monitor: Paul R .  Pixley Ff l  
Mathematical Physics Branch 
B 
Approved : 
E m i l  R .  Schiesser, Chief 
Hathematical Physics Branch 
Approved : R&la L.  Berry, Chief 
Mis/sion Planning and Analysis Division 
Mission Planning and Analysis Division 
National Aeronautics and Space Administration 
Lyndon B.  Johnson Space Center 
Houston, Texas 
August 1979 
79FW7 
JSG 1 6064 

79FM27 
CONTENTS 
Section 
1 .o INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . .  1 
2 .o 
3 -0 
THE SPHERICAL FORM OF SNEU'S LAW . . . . . . . . . . . . .  
WiE RAY PATH EQUATIONS . . . . . . . . . . . . . . . . . . .  
2 
5 
4 .O INTM;RATING THE EQUATIONS . . . . . . . . . . . . . . . . .  10 
5 .O E X A X F L E S . . . . . . . . . . . . . . . . . . . . . . . . . .  11 
6 .O REFPRENCES...... 15 . . . . . . . . . . . . . . . . . . .  
t'2r. . 
79FM27 
TABLES 
Table Page 
I SHORT- AND MEDIUM-RANGE REFRACTION CORRECTIONS . . . . 13 
11 LONG-RANGE REFRACTION CORRECTIONS . . . . . . . . . . . . . 14 
l v  
79FH27 
Figure 
1 
2 
3 
4 
5 
FIGURES 
Ref'raction effects . . . . . . . . . . . . . . . . . . . . .  
Sne1l.s law . . . . . . . . . . . . . . . . . . . . . . . .  
Ray path across spherical surfaces . . . . . . . . . . . . .  
R a y  path geometry . . . . . . . . . . . . . . . . . . . . .  
Differential geometry . . . . . . . . . . . . . . . . . . .  
Page 
1 
2 
3 
5 
7 
V 
79FM27 
1.0 INTRODUCTION 
Opt ica l  measurements o f  range and e l eva t ion  angle are d i s t o r t e d  by the  Ea r th ' s  
atmosphere as shown i n  f igu re  1. 
Figure 1 .- Refract ion effects. 
For long ranges,  the measured e l eva t ion  angle  EM may dev ia t e  from the t r u e  ele- 
vat ion angle  E by seve ra l  mi l l i r ad ians ,  and measured range p~ may d i f f e r  
f r o m  t rue  range by s e v e r a l  meters. Even short-range measurements are 
affected by r e f r a c t i o n ,  and t h e  effects are cumulative. 
I n t u i t i o n  may lead one t o  th ink  that r e f r a c t i o n  effects are minor when the eleva-  
t i on  angle is near zero.  Exactly the oppos i te  is t rue .  EM-E inc reases  wi th  a 
decreasing e leva t ion  angle ,  and the ray pa th  becomes more curved. Also t o  be 
noted is that a laser ranging device can not  be compensated t o  g ive  a t r u e  Pange 
measurement by using the speed of l i g h t  i n  the  atmosphere, at the instrumsnc. 
This w i l l  not inc lude  the  effects o f  t he  ray  pa th  bending and the passage 
through higher or lower l aye r s  o f  atmosphere, where the speed of  l i g h t  differs 
from t h a t  a t  the ins t rLmeP.  
range is obtained by using the speed o f  l i g h t  i n  a vacuum ( index of r e f r a c t i o n  
n = l ) .  
The work presented i n  t h i s  r epor t  is a sp inof f  of the  Space S h u t t l e  Program. 
The high prec is ion  r e f r a c t i o n  co r rec t ion  equat ions i n  t h i s  r epor t  are being used 
to evaluate  approximate r e f r a c t i o n  co r rec t ion  equat ions developed a t  the NASA 
Johnson Space Center. 
i d e a l l y  su i t ed  f o r  surveying s ince  t h e i r  i npu t s  are o p t i c a l l y  measured range and 
o p t i c a l l y  measured e l eva t ion  angle.  The outputs  are t r u e  s t r a i g h t - l i n e  range 
and true geometric e levdt ion  angle. The "shortt t  d i s t ances  used i n  surveying 
allow the  ca l cu la t ions  of t rue  range and t r u e  e l eva t ion  angle  t o  be quickly made 
u s i n g  a programable pocket ca l cu la to r .  
The equat ions i n  t h i s  r epor t  assume t h a t  measured 
It was determined that t h e  equat ions  i n  t h i s  r epor t  were 
2.0 THE SPHERICAL FORM OF WELL'S L A W  
Let n be the index of refraction. A l ight ray traveling from one media (ni)  
t o  =,other (n i+l )  is bent according to Sne l l ' s  law (fig. 2 ) .  . 
Figure 2.- Snell's law. 
Figure 3 shows how Snell's  law can be applicable to concentric spherical 
surfaces. 
2 
/ / Ri+l 
R; 
I 
I /  
\ 
\ 
Ri+l 
Ri 
C 
Figure 3. -  Ray path across spherical. surfaces. 
From the ABC triangle i n  figure 3 and from the law of sines, 
or 
But fi*om Snell's law of refraction, 
3 
79FM27 
Thus, 
or 
n i+ l  R i + l  s i n  4i+1 = n iRi  s i n  4 i  
In  the  next l a y e r ,  it is easily seen that 
ni+2 Ri+2 s i n  4 i+2  n i R i  s i n  4 i  
or,  i n  gene ra l ,  
nR s i n  4 = niRi  s i n  41 
Elevat ion angle EM is defined by 
& = g o - +  
As a r e s u l t ,  the spherical  form of S n e l l ' s  law is developed: 
1 1 
I nR cos EM = n i R i  cos Em I 
The e leva t ion  angle EM a t  r a d i u s  R I s  a funct ion only of n a t  R and the  
shape of the ray path "t any crazy curve) from R i  t o  R .  i n i t i a l  condi t ions:  , R i ,  and Em. T h a t  I s ,  EM is independent of the 
4 
79FM27 
3.0 THE RAY PATH EQUATIONS 
Figure 4 shows the ray path geometry from the initial altitude h i  
a1 titude hf . to the final 
Observer 
hi 
/ 
I I 
Target 
Figure 4 .- Ray path geometry. 
5 
The following parameters a r e  defined: 
Ro = re ference  radius o f  Earth = 6 378 165 meters; value not  c r i t i c a l  
R = % + h  
h = alt i tude above Ro 
hi = i n i t i a l  value of h 
hf = final value of h 
0 o c e n t r a l  angle 
01  = zero 
0f = value at target 
EM = e leva t ion  angle of ray path 
Qf = value of EM at target, f i n a l  value 
 EM^ = i n i t i a l  value of EM as measured by equipment 
ds = d i f f e r e n t i a l  p a t h  length  
p = t r u e ,  s t r a i g h t - l i n e ,  geometric range 
E = t r u e ,  geometric e l eva t ion  angle  
c = speed of  l i g h t  i n  a vacuum 
n = index of r e f r a c t i o n  a t  h, equal  t o  c/(speed of l i g h t  a t  h) 
no = value of n a t  Ro 
PM = measured range 
(2 )  
The value N = n-1 is the modulus of r e f r a c t i o n .  
assumed. That is, 
An exponent ia l  atmosphere is 
N = No exp(-h/HS) (3)  
The symbol HS is defined as atmospheric s c a l e  height. 
Values of No and HS are suppl ied  by meteorologis ts .  However, the  fol lowing 
empirical. r e l a t i o n s h i p  may be used f o r  HS (ref. 1) :  
1000 
Hs = meters 
No - 7.32=10'6 exp(5577No) 
NO 
a n  
(4) 
6 
This  value of HS 
data. However, it may not be the best for o p t r c a l  survey data. 
is used a t  Johnam Space Center for processing radar t r ack ing  
Figure 5(a) shows the d i f f e r e n t i a l  pa th  l end th  element 6s and its two 
components. Figure 5(b) shows the  photon ve loc i ty  along d s  and its two 
components. 
dh 
C 
- C O S  E,, n 
Figure 5 .- D i f f e r e n t i a l  geometry. 
Figure 5 shows that 
d s  c 
dt n 
- t -  
dh c - = - s i n  EM 
d t  n 
de 1 c 
d t  R n 
- = - - c o s %  
where, from equat ion ( 3 1 ,  n is giver. as a func t ion  of h by 
7 
n = 1 + N = 1 + exp(-h/Hig) 
Note from equat ion  (2 )  t h a t  . 
dpM = nds 
And, using equat ion (5) 
@M 
- = c  
d t  
79FM27 
(8 )  
(9) 
hl - -'*es;ion for EM is now requi red .  Equation (11 ,  the spherical form of  
S n e l l ' s  law of r e f r a c t i o n ,  could be used; however, i t  is f r augh t  with numerical  
p rob lem I For example, i f  Em = 0, then h (eq. ( 6 )  1 w i l l  remain at its 
i n i t i a l  ;;the, It has been determined tha t  it is more accura t e  to  develop 
a d i f fe re t t ia l  equat ion for EM and i n t e g r a t e  i t .  I n  equat ion  ( 1 1 ,  
h i .  
nR cos % = cons tan t  
D i f f e r e n t i a t i n g  w i t h  respect to time g i v e s  
iR cos F & .cos EM - nP k~ s i n  EM = 0 
But,  from e q u a t i m s  (8)  and (3 )  
Thus, 
N 
-R - cos F& + & cos EM - nR k~ s i n  EM = 0 
*S 
Using equat ion ( 6 )  for 
a 
79FM27 
N c  -R - - s in  sH COS E~ + c s i n  E~ COS E~ - n~ iH s in  E~ = o 
HS * 
And the equation for & is 
d% N c  - =(; - -)- cos qq 
d t  nfis 
Now, let 
a = ct  
B 
Then da = cdt and the resulting equations are slllamrrrized as follows. 
! 
! 
! 
! 
! 
! 
! 
I 
! 
! 
! 
I 
I 
! 
! 
! 
! 
I 
PM = a t 
d h 1  ! - = - sin EM 
da n ! 
de 1 1 I 
- = - -  COS En ! 
da R n ! 
I d"-(l-L)l - -  C M E H  1 I
da R nHs n ! 
(12) 
(13) 
(15) 
The i n i t i a l  conditions are 
a = O  
h = h i  
e r e p o  
EM = %i 
And where 
9 
R = % + h  
N = No exp(-h/HS) (17) 
n = N + 1  ( 18) 
The equat ions are i n t e g r a t e d  from a = o All tha t  re- 
maim is to ob ta in  the  expressions for p and E. It can be seen i n  figure 4 
that 
t o  a = measured range. 
I 
! 
! 
! 
! 
! 
! 
1 
I E t a r c t a n  ( T l / T z )  ! 
4.0 INTEGRATING TIE EQUATIONS 
A fou r th  order Runge-Kutta-Gill i n t e g r a t o r  has been found to  be very sui table  
for  most purposes. 
K u t t a - G i l l  i n t e g r a t o r  a l lows a maximum i n t e g r a t i o n  s t e p  size of about 10 000 
meters f o r  even the most p rec i se  surveying work. 
Other i n t e g r a t o r s  may be found i n  r e fe rence  2. me Runge- 
Let the  state vector  g be defined by 
10 
Let 
Then the fourth order Runge-Kutta integrator equations (ref. 2) ape: 
The Ruage-Kutta-Gill const, ?ts (ref. 2) are: 
a1 = 1/2 
c1 = 0 
d l  z 1/6 
5.0 MAHPLES 
Tables I and II shor examples of refra 
b2 = (2- *)/2 
c3 = ( 2 + f l ) / 2  
d3 = (2+ *)/6 d4 = 1/6 
tion corrections for 
No = 0.000395 
€IS t 5446 meters 
1 1  
79FM27 
h i  = 0 meters 
Qi = -0.239 degrees 
In tables I and 11, Ap = p~ - p meters and AE = & - E milliradians. 
12 
79FM27 
TABLE I.- SHORT- AND HEDIUM-RANGE REFRACTION CORRECTIONS 
20 000 
Pnv - 
100 
200 
300 
400 
500 
600 
700 
800 
900 
1000 
0.0395 
.0790 
.1185 
.1580 
.I975 
.2370 
.2765 
.3160 
.3555 
3950 
AX, mad 
0.00 357 
.OO 721 
-01 084 
.01 44a 
.01 810 
.02 173 
.02 536 
.02 899 
.03 261 
.03 624 
h p r  m 
0.3950 
'. 7903 
1.1859 
1.5817 
1.9779 
2.3743 
2.7711 
3.1680 
3.5653 
3.9628 
AE, mrad 
0.03 624 
.07 250 
.10 879 
.I4 509 
.18 141 
.21 774 
.25 409 
.29 046 
.32 684 
.36 324 
79FM27 
TABLE II.0 LONG-RANGE REFRACTION CORRECTIONS 
10 
20 
30 
40 
50 
60 
70 
80 
90 
100 
AE, mad 
-37.5 
-66.6 
-87.4 
-99 09 
-104.1 
-100.0 
-87.6 
-66.9 
-37.9 
-0.5 
6.1909 
-. I430 
- -0953 
-. 0478 
-.0003 
.0472 
.0947 
.1424 
.I903 
.2384 
3.9628 
7.9512 
11.9606 
15 9863 
20.0236 
24.0678 
28.1141 
32.1577 
36.1938 
40.2176 
0.36 324 
'.72 806 
1.09 418 
1.46 133 
1.82 923 
2.19 759 
2.56 615 
2.93 461 
3.30 271 
3.67 015 
14 
79FM27 
6.0 REFERENCES 
1 .  Bear., B .  R . ;  Tbayer, S. D . ;  and Cahoon, B .  A . :  Methods of Predicting the 
Atmospheric Bending of Radio Waves. 
6056, May 18, 1959. 
National Bureau of Standards Report 
2 .  Lear, W i l l i a m  M . :  Accuracy and Speed of 38 Self-starting Integrators. JSC 
I N  78-FM-39, June 1978. 
15 


Refraction corrections for surveying

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