universal program for adjustment of any geodetic network

測地学会誌,第26巻,第3号

(1980),147-171頁Journal of the Geodetic Society of Japan

Vol. 26, No, 3, (1980), pp. 147-171

Universal Program for Adjustment of Any Geodetic Network

(PAG-U)

Takehisa HARADA

Pacific Aero Survey Co., Ltd.**

(Received July 2, 1980)

測地観測量網平均汎用プログラムの現状

パシフィック航業株式会社原田健久*

(昭和55年7月2日受理)

要旨

　測地観測量網平均汎用プログラムに関する最初の報告を国土地理院英文報告(第12巻,1966)に

書いてから既に14年の才月が流れた.この間計算機の発達と学の進歩に応じてプログラムには何回

も改良が加えられた.大きな改良についてはその都度報告させていただいた.しかし報告洩れにな

っている事柄で,しかもこのプログラムに対する理解を深める上では極めて重要な事項が幾つかあ

る.例えば辺長と方向や角が一緒に観測されている場合には,なぜ辺長の観測方程式の左辺を単な

るv(s)でなくv(s)ρ"/sにする方が良いのか,とか,未知点の登録順序についてのサツマ芋輪切り

論等である.いろいろな意味でこのプログラムの改良も一段落の状態にある今,1979年春の学会に

おける「1点固定のFree Network解」の報告と併せて,これまでに書き洩らしてきた諸々の事に

ついて述べる.

ABSTRACT

Fourteen years have elapsed since the first paper on the Universal Program for ad

justing strictly any geodetic network, which consists of four kinds of observations such as side length, azimuth, angle and direction, was presented in the Bulletin of the Geo

graphical Survey Institute, Japan (vol. 12, 1966). The program had been frequently improved according to the development of computer and the advancement of science. Important improvements had been reported in the Journals of the Geodetic Society of Japan every time they were added to the program. There are, however, some important remarks which have not been mentioned in these papers. Nowadays, the improvement of the program seems to be sluggish. Under these circumstances reviewing total flow of the program including these remarks will be useful for persons using the program or considering introduction of the program.

Main features of the program are as follows

1. Available for any reference ellipsoid.

2. Available for the geodetic networks located anywhere on the earth.

3. There is no restriction in number of points.

It depends only on the ability and capacity of computers.

*昭和54年5月日本測地学会第51回講演会にて発表.**2-13-5,Higashiyama,Meguro,Tokyo,Japan

148 Takehisa HARADA

4. There will be no restriction in size of network, if maximum length of side is less

than 1,000 km. Of course, no matter how small the size is, there will be no

trouble.

5. There will be no restriction in any kind of survey and figural shape of network.

6. Blunders in input data can be checked briefly on the output printed after the

short-time test computation.

7. An advance estimate on the strength of a geodetic network can be obtained by

output-printing diagonal elements in the weight coefficient matrix of all unknown

values (east-west and north-south components in small increment to be added to

approximate position of every unknown point).

8. It is a matter of course that the network can be adjusted strictly by means of

the method of least squares. Network can be naturally adjusted according to

individual preferences, such as multi-fixed points or one-fixed point and one-fixed

azimuth etc..

9. In addition to the conventional methods mentioned above, a new technique of

Free Network solution developed recently is also available. It is based on that

the positions of all points are unknown in the Free Network solution. In this

program both methods are available by which sum of square of each displace-

ment vector for all points becomes minimum and sum of displacement vectors of

all points becomes zero.

10. If flexibility still remains in rotation and size of figure despite the fact that the

position of any single point in the network is known, the method by which sum

of square of each displacement vector becomes minimum can be applied in this

program.

11. Rigorous weight of every observation can be decided automatically according to

its accuracy.

12. Standard deviation of the position of each point found after net-adjustment can

be shown with an error ellipse in addition to both components in east-west and

north-south.

13. Displacement of every point is shown with a vector and both components in east-

west and north-south.

14. Any system of plane rectangular coordinates can be usable, if it is based on the

Gauss-Kruger's conformal projection principle.

15. Final result table for each point will be automatically made including many kinds

of geodetic quantities i, e. coordinates of the points, contraction ratio and the

direction of north at the points, azimuth, direction and distance of every geodetic

line connecting between each of the points and other surrounding points. Of

course some of them are expressed with both values on the ellipsoid and on the

plane. The final result table also contains a few supplemental elements by which

distances and directions can be easily transformed from the ellipsoid to the plane

Universal Program for Adjustment of Any Geodetic Network 149

and vice versa.

16. In case that high precision printing is desired in the output, numerical values of

longitude and latitude can be printed to 6 digits of decimals in arc second and

distances can be printed to 0.1 mm.

17. Radiating sides from a central point can be formed automatically refering code

numbers of the central and other surrounding points in input data. Both old and

new values and variances are shown for these radiating side-lengths and for

horizontal angles formed by the neighboring sides at the center. Especially their

standard deviations can be computed rigorously for side-lengths and horizontal

angles, even if they are not observed directly.

18. The local crustal strains such as rotation, dilatation, major and minor principal

strain axes, and maximum shear strain can be obtained by comparing new tri

angle with old one formed based on code numbers of three vertices in input data.

19. Computation and operation of matrices in this program can be conducted very

fast and efficiently due to the fact that zero-parts in every matrix are completely

neglected in any step such as in storing, adding, multiplying and introducing of

inverse matrix.

I. Outline

Geodetic surveying is concerned with two types of surveying, one for heights and

the other for horizontal elements. Concerning the latter, this program performs precise

adjustments of any geodetic network regardless of its shape or of its location on the earth

by using longitudes and latitudes as expressed by continuous coordinates over the earth's

surface and, in addition, for areas that had been surveyed earlier before the latest sur

veying, it computes land movements and ground strains that took place during the inter-

vening period.

1. Two Cases of Application

While a more detailed flow diagram is given in Figure 13, Figure 1 as shown below

indicates the two cases of application of this program. In order to determine land move

ments, both old and new longitudes and latitudes must be known for each point. There-

fore, when old data are not available, adjustment of a geodetic network is required as a

case shown on the left side of the flow diagram in Figure 1. When both old and new

data are available, land movements can be determined without performing such adjustment,

which is the case shown on the right side of the diagram. When old data are not avail-

able, the geodetic network adjustment alone is performed without computing land move

ments and crustal strains.

150 Takehisa HARADA

Fig. 1. Rough flow of the Program.

2, Outline of Program Structure

The program is structured as an overlay system composed of one main program and

11 sub-routines.

Program Language: FORTRAN IV

3. Computing Speed

Additions and multiplications of matrices take up the bulk of computing time . Figure 2 is a typical array of a matrix with sizeable zero areas in the upper right and the lower

left as represented by A and B respectively in the figure. As the number of points in-

creases, the zero areas rapidly grow in size. And unless something is done, such meaning-

less calculations as (X) +0, (X) X 0, 0+0, 0 X 0, should sharply increase in number adding

wasteful time simply to extend the total computing time.

Therefore, this program totally eliminates both A and B areas , which fall outside of

Fig. 2. An example of matrices in computation.

x : significant values

blank space: zero


the area enclosed by the broken lines, from calculations as well as from storage.

This reduces the total computing time significantly and thus makes it possible to

perform calculations for precise adjustments of a large geodetic network involving enormous

numbers of points, including computations of inverse matrices, which formerly could not

be accomplished due to the tremendous time required. And it also permits evaluation of

accuracies in various terms for the whole geodetic network, which would be otherwise

impossible.

4. Observations

The geodetic network adjustment involves the following four types of observations.

a. Side length

b. Azimuth

c. Direction observation

d. Horizontal angle

These must be expressed in values as projected on the surface of the reference ellip-

soid.

a. Side Length

This is defined as the length of a geodetic line connecting two points on the reference

ellipsoid. It is not a raw measurement value of slope distance but a value corrected for

curvatures of light and electromagnetic waves and heights of observation points.

b. Azimuth

This refers not to astronomical azimuth AQ but to geodetic azimuth Ag. The geodetic

azimuth, however, cannot be directly measured in observation and in application, there-

fore, Ag is obtained from the observed astronomical azimuth as modified by means of the

following Laplace's equation.

Ag=Aa-(ƒÉa-ƒÉg) sinƒÓ-tan h(sin A-cos A) (1)

where and are vertical deviations of the point which are given as follows.

•¬

where (AQ, pa), (Ag, cog) are astronomical and geodetic longitudes and latitudes of the point and

h represents the height angle of the azimuth point as seen from the observation point.

Therefore, the application of azimuths in geodetic network adjustment requires that

the astronomical azimuth and astronomical longitude and latitude of a certain point should

be known by observation and so be the geodetic longitude and latitude of the point.

For such an area where geodetic surveying has never been conducted before, since

geodetic longitudes and latitudes of each point are not known, an adjustment should be

152 Takehisa HARADA

performed initially without azimuths to obtain approximate geodetic longitudes and lati-

tudes of each point and then to find geodetic azimuths by Equation (1). Only after that,

the adjustment is to be performed using the azimuths.

c. Direction observation

One pair observation :

After fixing the horizontal graduated circle of the theodolite at a certain point, several

points in the vicinity are observed consecutively for their respective directions and their

measurements are read out from the horizontal graduated circle. In the next step, as the

telescope is moved back, the points are observed for directions this time in the reverse

order to return to the original position. The two read-out values thus obtained in both

ways, forward and backward, for each direction are averaged to complete one pair obser-

vation.

By changing the fixed position of the horizontal graduated circle, this one pair obser-

vation is repeated for a certain number of times. Observed values are averaged for each

direction and the resulting values are applied as observed values of directions. In PAG-U,

the value for one arbitrarily chosen direction is reduced to 0°0'0" and the rest of the

values are converted accordingly to form a new series of values which are used as input

data.

d. Horizontal angle

This is an angle formed by two observed directions. The value is obtained as the

difference between two directions. When the two directions are observed each with a

weight of p, the weight of the angle is p/2.

5. Geodetic Network Adjustment

Geodetic Network Adjustment:

The values of three interior angles of a plane triangle as measured in observation,

for example, do not usually add up precisely to 180°0'0". This is because of errors in

observation, and raw observation values do not provide any geodetic configuration of

primary significance that they are supposed to represent. In order to obtain a geodetic

pattern of primary significance, it is necessary to modify each observation value in steps

in a manner not to contradict each other. This process is called the geodetic network

adjustment.

Geodetic Network Adjustment by the Least Squares Method:

This is the most precise method for geodetic network adjustment. When v~ is assumed

as a small correction value to be added to each observed value and pb as a weight of

observed values, each value of vti is determined in a manner to satisfy the following formula.

•¬(3)


6. Geodetic Network Adjustment by Observation Equations

In this program, one observation equation is developed for one set of observation and

after theoretically evaluating the weight of observation , each equation is solved by the least squares method so as to satisfy Equation (3).

In so doing, all known points are represented by longitudes and latitudes as already

known but not only that, approximate longitudes and latitudes are assumed for all un-

known points. And observation equations are formed with small correction values as

unknowns to be applied to improve the approximate longitudes and latitudes assumed in

the above.

Observation Equation [1] :

The Observation Equation is defined as follows taking a side length for example.

Assuming the observed value of a side length between Points A and B as Sobs, its small

correction value as vs and the side length computed from the approximate longitudes and

latitudes adopted for Points A and B respectively as Sado (sub ado stands for `adopted')

and small increments to be applied to the adopted longitudes and latitudes of the points

as (ƒÂƒÉa, ƒÂƒÓa) and (ƒÂƒÉb, ƒÂƒÓb), then the observation equation for side length is formed as

follows.

•¬(4)

where coefficients Ca, Cb, da and db are functions of the average latitude of the two points . In practice, Equation (4) above is applied with Sobs transpositioned to make the equa-

tion as follows.

vs=CaƒÂƒÉa+daƒÂƒÓa+cbƒÂƒÉb+dbƒÂƒÓb+ (Sado-Sobs) (5)

I n the actual computation procedure, first, small increments of longitudes and lati

tudes (ƒÂƒÉ, ƒÂƒÓ) and then adjusted longitudes and latitudes (ƒÉ+ƒÂƒÉ,ƒÓ+ƒÂƒÓ) are found after

adjustment of the geodetic network. Subsequently, adjusted side length Sad; is calculated

from them. Finally, vs is obtained from vs=Sad;-Sobs .

7. Free Network Solution

Displacement Vector:

Fig. 3. Displacement of an unknown point.

154 Takehisa HARADA

In Figure 3, Pado is assumed as representing the adopted approximate location and

Pad; as the location after the adjustment of a geodetic network. The arrow from Pado to

Pad) is Displacement Vector as expressed by V. Thus, V represents the actual land

movement from the position as measured in the old surveying to the position as identi-

fied in the new surveying.

The perpendicular components of east-west and north-south of V can be expressed

as NaA cos cv and Mow where

N : Curvature radius in prime vertical

M : Curvature radius in meridian

Free Network Solution:

The conventional method of geodetic network adjustment has been to apply the least

squares method assuming one or more known points in the geodetic network. Recently,

a precise method of geodetic network adjustment has been developed which applies the

least squares method assuming all points as unknown. This method is called the Free

Network Solution.

Geodetic observation values with respect to side lengths, azimuths, directions, hori

zontal angles, etc., do not actually provide any information to fix the location of a geo

detic network. Observations of directions and angles determine its pattern but allows an

infinite number of similar figures. If one or more observed values of side lengths are

additionally available, the dimensions of a figure can be determined but still leaving freedom

of rotation. If azimuths are further given, rotation elements can be accommodated but

Fig. 4. Concept of free network solution is shown in the figure.


parallel movements of the network are left free. Refer to Figure 4. Quadrangle (a) is formed by Pi, P2, Pg and P4 whose locations are already determined by old surveying . Suppose the dimensions of the quadrangle are determined as (b) by the adjustment of the

geodetic network in paucity of azimuth observation in new surveying. In this instance,

if one point and one azimuth are known, the respective points of Quadrangle (b) can be

automatically determined in relation to the reference ellipsoid on which they are based . In the Free Network Solution, however, since all points are unknown , the location of

Quadrangle (b), for example, cannot be fixed on the earth. Now if figure (b) is arbitrarily

overlaid on figure (a) happening to result in figure (c), then a pattern of displacement

vectors for all four points is obtained as shown in figure (c). Depending on the manner

in which figure (b) is overlaid on figure (a), patterns of displacement vectors change in

an infinite variety. In cases like this where all points are unknown, theoretically, point

locations cannot be determined. Despite that, the concept of the Free Network Solution

is to attempt to determine the location of a figure on the basis of one selected pattern

of displacement vectors out of a maltitude of various patterns on certain qualifications.

If the condition assumed for displacement vectors between points is reasonable enough

in physical terms, the displacement vectors chart should be reasonably significant.

This program permits two methods of Free Network Solution assuming,

Square sum of displacement vectors is minimum, V2=min. [2]

and

Total sum of displacement vectors is zero, V=0 [3]

The Free Network Solution in terms of normal equations :

To solve a number of observation equations similar to Equation (5) by the least squares

method, the normal equation as follows derived from Equation (5) must be solved.

BX=L (6)

Fig. 5. Relation between kinds of observations and both methods of free network solutions.

156 Takehisa HARADA

where X represents column vector consisting of small longitude and latitude increments,

B is their coefficient matrix (square and symmetric), and L is real number column vector.

When there are known points in the geodetic network and, therefore, its location is

determined, Equation (6) can be solved as

X=B-1L (7)

In the case of Free Network, however, the determinant of B is

CBI=O (8)

where inverse matrix B-1 has no place and, therefore, Equation (6) cannot be solved.

The statement that a myriad of vector charts can be developed is thus explained in

mathematical terms as above.

Figure 5 shows the types of observations conducted in usually practised surveying.

In the case of a surveying that follows the left side of the diagram, at least either direc-

tions or horizontal angles are measured in observation and so are side lengths and azimuths,

with only freedom left being for parallel movement. Therefore, if the location of one

point in a geodetic network is known, the location of the geodetic network can be deter-

mined. This means that by eliminating arbitrarily selected symmetrical 2 rows and 2

columns in B, the determinant is no longer zero. The defect in rank of matrix B is de-

fined as 2 in this particular instance. As rotation becomes free in paucity of observed

azimuth and sizes of a figure become free due to the lack of observed values of side

lengths. the number of defects in rank of B increases by 1.

The important thing to note is that while the method based on Z Vz=min. applies to

all types of surveying, the method based on E V= 0, on the contrary, is applicable only

when the number of defects in rank is 2. This means, for example, in a figure where

rotation is free, an azimuth is fixed by arbitrarily setting a rotation and after that, the

figure can be moved only in parallel. Then a solution to satisfy E V=0 can always be

found and, therefore, for each varying locking azimuth, there is a solution to satisfy

V=0 and there can be no single solution to satisfy V=0 in a figure where rotation

is free.

This program as shown in Figure 5, makes the Free Network Solution applicable to

all types of conventional surveying. For example, in such a case of triangulation where

neither side lengths nor azimuths are available, all that can be determined from obser-

vations is a pattern of the network which allows as solutions an infinite number of similar

figures subject to free rotations. 4n the other hand, the E V Z = min, method finds only

one solution by appropriately expanding and rotating the figure on the basis of the positions

assumed for each point.

7' Free Network Solution with One Fixed Point

(Supplemented, May 1979)

In the case where one point in the network is known, the Free Network Solution


based on E V2-min. (i. e, the square sum of displacement vectors is minimum) can be

applicable.

Namely, when one point in the network is fixed,

(1) If observed values are available only of side lengths, angles and directions but

not of azimuths:

The geodetic pattern is fixed in terms of size and figure but it permits free

rotation around the fixed point.

(2) If observed values are available of azimuths, directions and angles, but not of

side lengths:

The geodetic pattern allows as solutions an infinite number of similar figures

of all sizes with the same azimuth:

(3) If observed values are available only of angles and directions but not of side lengths nor azimuths:

The geodetic pattern allows free rotation around the fixed point while varying

in size.

Out of these infinite numbers of solutions, the Free Network Solution provides only

one solution so as to minimize the square sum of displacement vectors.

The defects of rank in the normal equations are 1 in (1) and (2) and 2 in (3).

Method of Input Data Preparation:

No special consideration is required but simply to tollow the procedure given in

the manual.

PAG-U recognizes the number of known points and if the number is one and the

side lengths, azimuths, angles, or directions are observed, it performs regular geodetic

network adjustment and, for other cases, the program perceives which of above (1), (2)

or (3) is the case and then automatically performs the Free Network Solution.

8. Final Result Table

Final result tables can be developed for two types of coordinate systems, i. e., Japan's

plane rectangular coordinate system and any arbitrarily chosen coordinate system based

on the same principle (Gauss-Kruger's Conformal Projection Theory) where 2o, con, X0, 10

of the origin and the contraction ratio have to be given.

At each of all points, the vicinity is searched out automatically for such points that

are mutually connected by observations, and the relations between the point and those

connected points in the vicinity are fully represented in a table in terms both of longitude

and latitude coordinates and of the plane rectangular coordinate system as well as con-

versions of such elements as side lengths and directions from ellipsoid to plane and vice versa.

In Figure 6, assuming 0 as the origin of the plane rectangular coordinate system,

A as the point for which a final result table is to be developed, and B as one of the

surrounding points, the meridian can be shown as AN and the geodetic line AB as i.

s represents the distance of AB on the rectangular coordinate system while S is the geo-

158 Takehisa HARADA

detic line length of AB.

In the final result table are represented the following values.

(X, Y )

Contraction ratio: Extent of contraction of a small figure around Point A.

Direction of true north (direction of north reckoned clockwise from the positive direction

of X axis) =Y

(negative in the fig. 6)

Direction

Azimuth=A

Direction= T=A+Y

T-t

t (direction of line segment between A and B on the plane) = T - (T- t)

Distance

Length of geodetic line=S

' S/s

l The length of line segment AB on the plane coordinate system=s=S=-(S/s)

9. Variations of Radial Side Lengths and Sequential Angles around a Point

Refer to Figure 7. Specifying one certain point (P0) and several surrounding points

as P1, P2... P5 clockwise in input data, the following values with respect to radial side

lengths (Sl, S2... S5) and sequential angles (81, 02. .. 85) are represented in output.

With respect to side length :

Old value : Si,old

New value and its standard deviation: Si,new(SD)

Variance : ‡™Si=Si,new- Si,old

Relative variance : ‡™Si/Si

Average value of relative variance : •¬


Fig. 7. Radial side lengths and sequential

angles around a point.

With respect to angle:

Same as for the side length in the above except that S is substituted by d.

The important thing to note here is that these new values of S and 0 are calculated

in relation to a most precise possible location of each point as adjusted even if the side

lengths and angles are not directly measured by observation and that standard deviations

of new values are presented as results of strict computations to indicate levels of reliability

of these values.

10. Location of Point

Error Ellipse:

The accuracy of the location of an unknown point obtained from the network ad-

justment is usually expressed by standard deviations in terms of centimeters in the north-

south and east-west directions. Strictly speaking, however, it should better be defined in

terms of an error ellipse for the reason that the accuracy of the position of a point is

not equi-directional by nature, which means a point location can be determined more

accurately in one direction than in another depending on the figure of a network, method

of observation and its accuracy, and standard deviations due to directions are distributed

Fig. 8. Error ellipse of most reliable position found after net-adjustment for an unknown point.

160 Takehisa HARAVA

in a waisted figure like a cocoon surrounding the point . The maximum value occurs on

one direction of the perpendicular with the minimum on the other direction as represented

by a and b respectively in Figure 8. Since it is rather cumbersome to draw a waisted

figure, it is usually approximated to an ellipse with a and b as axes . Called 'the error ellipse', it is commonly used as a means to show directional variations of reliability of a

point location.

With respect to unknown points, the following values are expressed . (Refer to Figure 8.)Longitude and latitude of approximate position of P ado : ƒÉ, ƒÓ

Small increment for longitude and latitude : ƒÂƒÉ, ƒÂƒÓ

Longitude and latitude of most precise possible position of Pad;

(adjusted) : ƒÉ+ ƒÂƒÉ , ƒÓ+ ƒÂƒÓ

East-west component of standard deviation of Pad; position:

North-south component of standard deviation of P ad; position:Azimuth of major axis of error ellipse: A

Length of major axis of error ellipse :

Length of minor axis of error ellipse : b

East-west component of displacement vector: NƒÂƒÉ cos ƒÓ

North-south component of displacement vector : MƒÂƒÓ

Azimuth of displacement vector: A

Value of displacement vector : V

11. Strain

Strain Ellipse :

The method to express horizontal ground movements in terms of displacement vector

of each point is good so far as it is visual and easy to understand . But as pointed out earlier in Section I-7, charts of displacement vectors come in a variety of forms and fall

short of primary significance. This is a major shortcoming of the method to represent

horizontal tectonic movements in terms of displacement vectors . The strain method is another way to express horizontal changes of the ground by taking note of the changes

in the pattern of a geodetic network . Refer to the left figure in Figure 9. Let's assume a circle with point Po at the center

drawn on the ground at the time of the old surveying . Supposing the ground is subject to compression force in the direction of PoB, the circle distorts in shape as time lapses and land forms change. As a result, the figure turns out to be an ellipse in the new

surveying as shown on the right of Figure 9 . Let's take two axes, P0C and P0D , for example, that crossed each other perpendicularly at Po in the old surveying . In the new surveying, the two axes are no longer perpendicular to each other as seen in the ellipse

shown on the right in Figure 9. Among an infinite number of pairs of such perpendi-cular axes, there is one pair of axes that remain perpendicular . Such pairs of axes are (POA, PoB) and (P '0A', P'0B') as shown in the respective figures, They correspond to the


Fig. 9. When horizontal compression force works uniformly over the ground, a circle (on the left) drawn on the ground changes in form to an ellipse (on the right) after lapse of time.

major and minor principal axes of the ellipse. The rates of elongation or contraction of

the radiuses of both axes of the ellipse from those of the original circle are defined as

strains of major and minor principal axes of a strained ellipse, and they are usually ex-

pressed in the unit of 10-6 (micro-strain). The perpendicular pair of axes (POC, POD)

which are just in the middle between the two principal axes in the original circle loses

perpendicularity to the greatest extent in the strained ellipse. Let •¬C'PO'D' be right

angle -ƒÓ, then 0 as expressed in radian is defined as maximum shear strain. The word

shear implies movement of scissor blades. If a fault movement is anticipated for the future,

faults are likely to occur in parallel with POC or POD. The increase of area in the ellipse

from the original circle is called dilatation. The values of these various strains are not

independent from each other. The difference between the two principal axes is equivalent

to the maximum shear strain and the sum of the principal strains to dilatation. In prac

tice, instead of drawing a circle on the ground, the same changes in figure can be known

from the change in the figure of triangle formed by three points, P1, P2 and P3 which

were located on the periphery of a circle in the old surveying and, therefore, values of

above strains can be determined by computation for each triangle.

This program computes rotation of the region concerned with each nominated triangle,

dilatation, directions and values of strains in the major and minor principal axes and

maximum shear strain. [4]

By a combined use of the strains of primary significance and the displacement vectors

which are easy to see, the ground movements can be estimated more accurately.

12. Q (for evaluation of the strength of a geodetic network)

There should be various methods to evaluate the strength of a geodetic network but

the important thing is whether it provides values most relevant to evaluation of the ac-

162 Takehisa HARADA

curacy of the determined position of each point.

There are three factors that affect the accuracy in determing the position of a point . The first is the pattern of a geodetic network and the second is the type of observation

to be conducted, the third being the accuracy of observation . The last factor, namely, the accuracy of observation depends , on the instruments to be used and, therefore, it is left out of the consideration of the strength of a geodetic network.

In this program, the following values which indicate reliability of the position of each

point in a most straight-forward manner, or, in other words, the values which provide a

standard deviation of an unknown quantity when multiplied by standard deviation of an

observation with unit weight mo are obtained as •¬ (east-west component) , •¬ (north-

south component), and further compounded with east-west and north-south component

in the formulae below, the resulting values are shown in one table for all unknown

points.

•¬

The smaller the values in the above, the better the positions of points can be deter-

mined.

As for such points where Q values are too large, since it means either the array or

pattern of points is not set properly enough or observation is too weak, it should be considered to reselect points or to strengthen observations.

Note : As obvious from (9) above, the true value of an unknown that this program is

concerned with is not OA but OA cos cp.

II. Important Notes

There are two important things to bear in mind in preparation of input data. Four

additional points, though of lesser importance, are also made in the following.

1. Registration Sequence of Unknown Points

when the number of points that form a geodetic network is small, the computing

time is short and it does not matter in what sequence unknown points are registered.

But when the number of points amounts to tens and hundreds, the registration sequence

of unknown points makes a significant difference in the computing time. The good under-

standing of this section should help to plan for the shortest computing time.

As mentioned in Section I-3, the bulk of computing time is concerned with matrices.

And most of the matrices involved are band matrices with numeral values occurring in

a diagonal belt as shown in Figure 2. Computing time depends solely on the width of


Fig. 10. Approximate longitudes and latitudes of unkown

points should be inputed for the purpose of saving

time in computation in order of lemon slices as shown

in the figure. The figure encircled by the dotted line

is an enlargement of the portion of the network having

point No. 50 at center.

this zone. The smaller the width is, the shorter is the time required for computation.

Figure 10 shows a geodetic network. In this particular geodetic network, a number

is given to each point. Please note here that these numbers are serial numbers given as

unknown points are read for their longitudes and latitudes, and they are not related in

any manner to the identification numbers of points read at the beginning of input data.

In all matrices, composing elements are arranged in the sequence of these numbers which

are independent from the identification number of the points.

Unknown points should be registered in a sequence as shown in Figure 10. It looks

like a section of a lemon slice and it is important to cut it into as many such slices as

possible and to arrange them in an orderly manner from left to right and so on (or from

top to down and so on), what is good about this is that in Figure 10, if paying atten-

tion to Point No. 50, for example, 3 lengths, (50-42, 50-43 and 50-51 as shown in solid

line), one angle •¬43, 50, 51 and three directions (shown by arrows pointing to 59, 58 and

42) are observed. Therefore, the unknown points that relate to Point No. 50 occur among

the points from Point No. 42 through Point No. 59. This leads to

59-42=17 (10)

The number acquired in the above formula controls the width of the band in the form

as shown in Figure 2 in relation to Point No. 50 in the band matrix. Such numbers as

obtained in (10) above that control the width of a band matrix can be assumed for every

one of all points that comprise a geodetic network. The smaller the maximum of such

numbers is, the shorter the computing time becomes. If the above procedure is followed

in arranging unknown points, it should accomplish the purpose of reducing the comput-

164 Takehisa HARADA

Fig. 11. In such case as having branches in the network

lemon slices still should be cut without considering the

branches as shown in the figure.

in time.

The aforementioned analogy of a lemon slice to illustrate the arrangement of un-

known points is applicable to any geodetic network regardless of its pattern . Even for

a network with a pattern that branches off, as shown in Figure 11 , it still is comparable to a lemon which is sliced into segments a, b, c, .. , n, 0.. , and, importantly, for where

it branches off, d, d'; e, e'; ..., they are given sequential numbers as if they were con

nected.

2. Weight of Observed Values

Principle:

Observed values should be weighted in inverse proportion to the square of standard

deviations. This is a very basic principle of the least squares method. From this prin

ciple are derived the following.

When the instruments and the method used in observation are identical , the weight

is proportional to the number of repetitions of observation.

The weight of an angle is 1/2 of the weight of the directions that form the angle . (Refer to I-4-d.)

Unit Weight Observation:

It is widely practised that by specifying the weight of one certain observation as 1, the weight of another observation as p is determined according to the above principle . The first thing to be found after least squares computations is the standard deviation of

this unit weight observation.


Case of simultaneous solutions of different kinds of observation values by least squares

method:

Generally speaking, when the least squares method is applied simultaneously to

observed values of different natures (time, mass, length, etc.), each type of value is given

a weight according to the above principle by assuming an appropriate unit weight obser

vation for the respective types. Once observation equations are formed, all it takes is

dimensionless computation and it does not matter in what units the observation equations

are expressed.

When it is considered, however, that the standard deviation of unit weight observation

resulting from computations is something like an average of standard deviations of these

different type observations, it is desirable that the standard deviations of these unit weight

observations are similar in value. It means that vtias in the principle formula of the

least squares method (3), i, e.‡”pivi2=min. is made similar in value as well as in the

extent of their distribution. For this purpose, such a manipulation is required that by

multiplying both sides of an observation equation like (5) by appropriate constant k, vl

as in‡”pivi2 =min. is considered in terms of kvi instead of simple vi.

Case in which a geodetic network is comprised by side lengths and other observed

values:

It is most reasonable to express small correction values to be applied to observed values

of azimuths, directions and angles in the unit of arc seconds. Correction values for azi

muths, directions and angles are represented summarily as v(B). When the small correc

tion value v(s) to be applied to the observed values of side lengths is expressed in the unit

of centimeters, v(O) and v(s) become similar in value. However, when considering a geodetic

network in which there exist triangles of various sizes, v(O) does not change much due to

the sizes of the triangles while v(s) is subject to the following well known formula.

•¬

where a and b are constants to be determined by the type of a distance measurement in

strument used. Therefore, values of v (s) range fairly widely. This makes the standard

deviation of unit weight observation resulting directly from the geodetic network adjust

ment rather difficult for understanding. This problem can be averted almost completely

by using the value of {v(s)/S} which is a dimensionless number for v applicable to pv2

=min, instead of directly v(s) in the observation equation of side lengths . At the same

time, since in this instance the values of {v(s)/S} become very small compared with the

values of v(8), it is necessary to multiply this by large numbers like 106 or 106 to make

it similar to the value of v(e).

Also as can be seen from Figure 12, the uncertainty of side length AB is relative to

the direction or the angle as seen from Point C. In this sense, the correction value of

the side length relates closely to the correction of the directions. Hence an idea is obtained

that by selecting p" (1 radian as expressed in arc seconds which is approximately 2.106)

166 Takehisa HARADA

Fig. 12. The uncertainty of side length

AB is relative to the direction or the

angle as seen from point C.

as a large number multiplier as mentioned in the above, v (s) applicable to ‡”pv2 = min, is

applied as follows.

v(s)/ S•EƒÏ" (12)

When ƒÏ" is considered a dimensionless figure, (12) is a dimensionless value, but ƒÏ"

can also be considered a value in the unit of arc seconds and then (12) is expressed in

arc seconds. When viewed in relation to Figure 12, the latter might be better for under-

standing.

As in this case where side lengths and other values are observed together in a survey-

ing, by applying v (s) p"/S instead of v (s) as the correction value for side lengths applicable

to ‡”pv2=min., both tasks of narrowing the width of v(s) distribution and making v values

similar for a geodetic network can be accomplished simultaneously. Furthermore, in cases

where a direction or an angle is chosen for a unit weight observation , if (12) is applied,

the weight of (12) can be evaluated properly with relation to unit weight observation and,

therefore, the standard deviation of unit weight observation can be obtained from the

geodetic network adjustment in a manner perfectly to suit the purpose. Thus, the advan-

tage is comparable to a case of `getting three birds with one stone'.

In this instance, the observation equation for side lengths is expressed as follows by

multiplying the both sides of (5) by p"/S.

•¬(13)

where

Ca' =CaƒÏ"/S, .. .

When side lengths and other values are known by observation, this program applies

Equation (13). Its weight p is obtained from anticipated standard deviation of unit weight

observation of direction or angle, m", and a and b of (11), basing on the principle men

tioned in II-2, in the following formula.

•¬


When side lengths alone have been observed, on the one hand the procedure mentioned

above can be used, and on the other hand such an adjusting method is also applicable

where Equation (5) as expressed in the unit of length is directly employed as an obser-

vation equation, assuming S ,, as an average side length for unit weight observation. Weight p of observed length S is determined as follows.

•¬

The program allows two alternative`methods, i. e., automatic weighting by computer

performing the above procedure or assigning weights to each observed value in input data according to individual data producers' judgement.

3. Reference Ellipsoid

Two types of ellipsoid, Bessel and International (old), are built in the program. Their

specifications are as follows.

Radius of Equator Inverse number of flattening

m Bessel 6377397. 155 299. 15281285

International (old) 6378388.0 297.0

When an ellipsoid other than the above is used, the name of the ellipsoid, radius of

the equator and inverse number of flattening are to be specified in input data.

4. Approximate Longitudes and Latitudes of Unknown Points

When old data of longitudes and latitudes are available for unknown points, they can

be applied as their approximate longitudes and latitudes. Otherwise, approximate values

are assigined to the unknown points. It does not matter how rough the values are but the

same value should never be given to more than one point. However rough the values of

approximate longitudes and latitudes are, by replacing them with the better locations ob

tained from the geodetic network adjustment and repeating the adjustment two or three

times, the values converge in most cases.

5. Test Computation

Large errors in the input data can be checked by test computations. If the 'test' is

specified in the input data, the program refers the values computed from approximate

longitudes and latitudes of each point to all observed values and without performing a

geodetic network adjustment, it prints them out to terminate. If large discrepancies between observed values and computed values are found in the output, it will be known

168 Takehisa HARADA

whether the errors are in observed values, point numbers or longitudes and latitudes.

6. Number of Digits to be Printed out

In regular print-out:

Longitude, latitude...........The unit of second and 5 digits of decimals

Distance, plane coordinates. .. mm

Angle, direction.............. The unit of second and 3 digits of decimals

When observations are very accurate (where, for example, distances are measured to

0.1 mm), or the average observed distance is 1 km or less:

Longitude, latitude........... The unit of second and 6 digits of decimals

Distance, plane coordinates.. . 0.1 mm

Angle, direction.............. The unit of second and 3 digits of decimals

# Conditions of input data for print-out of high accuracy.

Print-outs of high accuracy can automatically result in the following cases.

(a) When assumed (or old) longitudes and latitudes of unknown points, or longitudes and latitudes of known points include one or more values that have a figure other

than 0 on the sixth digit of decimals.

(b) When the average value of observed side lengths is 1 km or less.

III. Flow Diagram of the Program

Figure 13 is a detailed flow diagram of this program. It shows how the points made

in Chapters I and II are incorporated in the program.

IV. Before Preparation of Input Data

This program has many ramifications, and which branch to take at each decision

point is determined by the instructions specified in the input data. Therefore, all necessary

instructions to choose between alternatives must be given in the input data.

Following are the alternatives to consider.

a. Whether it is test or real computation. (See II-5.)

b. Selection of a reference ellipsoid (See II-3.)

c. Whether to perform a geodetic network adjustment or to compute ground movements

alone without a geodetic network adjustment. (See I-1.)

d. Whether to produce a final result table after geodetic network adjustment or not .

(See I-8.)e. To assign weights to observed values according to individual input data producers'

judgement, or to have a computer do that automatically. (See II-2.)

f. When weights are assigned automatically, which to apply for unit weight observations,


average side length, directions or angles. (See II-2 .)

g. When only side lengths are observed, which to apply for unit weight observations , average side length, directions or angles . (See II-2.)

h. Are there any known points? When there is no known point, the Free Network

Solution must be applied.

i. In the Free Network Solution, is it to be solved based on?:

‡”V2=min., or

‡”V =0, or

Both (‡”V=0 and ‡”V2=min.)

(See I-7.)

j. Whether or not the prior evaluation of the strength of the geodetic network is the main objective. (See I-12.)

k. Whether to obtain variances of radial side lengths and sequential angles around each

point. (See I-9.)

1. Whether to compute strains, (See I-11.)

References

[1] Japanese text book of Geodesy.

[2] E. Mittermayer: A Generalisation of the Least-squares Method for the Adjustment of Free Networks . Bull. Geodesique, 104, 1912, 139-157.

[3] T. Harada: The net-adjustment ‡”V=0 is obtained automatically in the improved Universal Program . Journ.

Geod. Soc. Japan 14, 156-158, 1969.

[4] T. Harada: Calculation of strain was added to the Universal Program. Journ. Geod. Soc . Japan 17, 1-3, 1971.

170 Takehisa HARADA

Fig. 13 (A) Flow of the program.


Fig. 13. (B) Flow of the program.

universal program for adjustment of any geodetic network

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