Summing di for all theodolites
n
.Ld= F(Xo, Yo, Zo) =
i=l
~ ~ [(1-4')
Xo - l,m,Yo - l,n,Zo - ... ,+ !;P<J'
+[ -
l,mXo+
(l-mll Yo - m,n,Zo - y,+
m,p,J+[ -
!;n,Xo - m,n,Yo+
(l-nil Zo - E,+ niP,J!
(9)
(10)
( 11)
Taking a partial derivative with respect to X 0 and sim-plifying
n n n
aa%
= .L(1-1D Xo+ .L(
-limi) Yo+ .L(
-lini) Zoo i=l i=l i=l
11 n
- .LXi
+
.LhPi = O. (12)i=l i=l
n n
- . LYi
+
.LmiPi = O. (13)i=1 i=l
121
(14) Rewriting (12), (13), and (14)
n n n
.L(1-1D XO +.L( -hmi) Yo +.L( -lini) Zo
i=l i=l i = l
(15)
n n n
.L(
-limi) Xo+
.L(1-mf) Yo +.L( -mini) Zoi = l i=l i=l
n n n
.L(
-lini) Xo+ .L(
-mini) Yo+
.L(l-n1) Zoi=l i=l i = l
Thus, an array of three symmetric simultaneous linear equations in the variables Xo, Yo, Zo is obtained. All other values are obtained from theodolite data.
Ah Outline of the Abbreviated Doolittle Method of Solution of a System of Symmetric Linear Equations
There are many methods for solving systems of linear equations such as those given in equation (15), and it was felt, after several methods were investigated-inasmuch as automatic calculating equipment would be available-that the abbreviated Doolittle method was the most economical in time for the desired accuracy. Dwyer's method, known as the Abbreviated Method of Multiplication and Subtrac-tion-Symmetric or, as he calls it, the Compact Method, is somewhat shorter than the abbreviated Doolittle method, but it involves more difficult 602-A control panel wiring.1 This method is applicable to any number of theodolites, greater than one, with no changes. Because, occasionally, tracking operators lose the object, this is a very necessary condition.
Approximately 45,000 arithmetical steps, depending upon the number of observations, are involved in the data reduc-tion for one test.
CHART I
OUTLINE OF ABBREVIATED DOOLITTLE METHOD OF SOLUTION I
=
cos H cos S'11'1,
=
cos H sin S n=
sinHp
=
Ix+
my+
nzA
(lA) =};(l-12)*
2 same as (IB) 3 same as (IC) 4 (4A) = (lA)
\
S (SA) = 1
B (lB) = -};[m
(2B) =};(l-m2)*
same as (2C) (4B) = (lB) (SB) = (lB) (lA)
C (lC) = -};In
(2C) = -};mn
(3C) = };(l-n2)*
(4C) = (lC) (SC) = (lC) (lA)
D
(4D) = (lD) (SD) =(lD)
(lA)
6 (6B) = (2B)-(SB) (lB)* (6C) = (2C)-(SC) (lB) (6D) = (2D)-(SD) (lB)
7 (7B) = 1
S 9
10 Z=(9D)
Y = (7D) - (7C)Z
X = (SD) - (5C)Z - (SB) Y
*
AL WAYS POSITIVE.The time required to compute by hand a reduction, of all the data obtained from a complete test, by the abbreviated Doolittle method is prohibitive. It is roughly estimated that two experienced persons might be able to compute the co-ordinates for one complete test in about a month. By use of the IBM equipment now at hand, which includes two type 602-A calculating punches, this process, starting with the film, can be completed in approximately three days. How-ever, by use of the Telecomputing Askania Reader and the IBM card-programmed electronic calculator, now on order, it is estimated that a reduction of one complete test from data of five theodolites may be completed in a matter of hours.
In regard to a two-theodolite solution versus a five-theodolite solution, a study has been made to answer the questions as to which method produces the better solution, and how much better is this solution. In using a comparison of third differences as a measure of the random errors
(7C) = (6C) (6B) (8C) = (3C)-(lC)
(SC)-(6C) (7C)*
(9C) = 1
(7D) =(6D) (6B)
(SD) = (3D)-(lC) (SD) -(6C)(7D) (9D) =(8D)
(SC)
present, it was found that a five-theodolite solution was considerably smoother: in fact, it had only about SO per cent as much random error as did the two-theodolite solution.
1. PAUL S. DWYER, "The Solution of Simultaneous Equations,"
Psychometrika, Vol. 6, No.2 (April, 1941).
DISCUSSION
Mr. Rich: l\1r. Schutzberger's problems are very similar to those of the Naval Ordnance Test Station, and I'm sure that similar methods are used by many groups in the coun-try. The concern of groups involved in data reduction is the speed with which results may be obtained after the tests and then placed in the hands of the interested parties.
Professor Tukey: A number of years ago we were trying to do things like this with two theodolites, using Mitchell
S E M I N A R 123
CHART II
OUTLINE OF COMPACT METHOD OF SOLUTION (Abbreviated Method of Multiplication and Subtraction-Symmetric)
I
=
cos H cos Sm
=
cosH sinSn
=
sin Hp =
Ix+
my+
nz2 3
4
A lA
~(l-l2)
4A=lA
B
2B
~(1-m2)
4B= IB
C D
~ (1-n3C 2 )
4C= IC 4D=ID
S 5B = (lA) (2B)-(lB)2 SC = (lA) (2C)-(lB) (lC) SD = (lA) (2D)-(lB) (lD) 6
7
[(1A) (3C)-(IC)2] SB-(SC)2 6C=
Z= 6D 6C
y = S~ [SD- (SC) (Z)]
X = I~ [lD- (lC) (Z) - (lB) (Y)]
cameras. With the Mitchell, if your film is read on a Recor-dak with about three special curves on it, the refraction corrections could not be entered, but most of the instru-mental corrections could be entered. I have never handled an Askania, but I would think there might be some possi-bility of this.
In some circumstances might not it be desirable to have a computing procedure that would reject the worst of the five theodolites on each point and take the least squares solution of the other four, or reject the worst of three and keep the other two?
Mr. Rich: We have the problem at Inyokern of having a three-station reduction system. Essentially, we obtain checks on the accuracy of the three stations used before we even place it into a data reduction scheme.
6D=
[(lA) (3D)-(lC) (lD)] SB-(SD) (SC)
Dr. Lotkin: At Aberdeen we are doing exactly the same type of problem, among others. As far as the method of re-duction is concerned, we have found, by comparison, that it is better to use the method of minimizing the squares of the sides of the triangle involved, rather than the squares of the distances from the lines of sight-one reason being that the normal equations become more simple, and we are able to put in more checks on the computing procedure as we go along. We have been able to mechanize the whole procedure on the IBM relay calculators where we start with the smoothing operation on the angles, then, by means of least squares, compute average position, smooth the position by means of least squares again, and obtain velocities and ac-celeration. It takes about two minutes per point for this whole process; and trajectories containing as many as 300 points may be reduced within an hour and a half to two hours.