Resume of Aerial Triangulation Adjustment at the Army Map ...RESUME OF AERIAL TRIANGULATION...

RESUME OF AERIAL TRIANGULATION ADJUSTMENTAT THE ARMY MAP SERVICE

Robert S. Brandt, Photogrammetric Engr., Chief, Triangulation Branch,Photogrammetric Division, Army Map Service

INTRODUCTION

. AS THE demand for more large-scale maps increases so the need for practical.f1.. sy~tems to produce these maps increases. The application of photogram­metric surveys to the problems of producing economical map coverage is ac­cepted as effic:ient; but even though present photogrammetric survey methodscan satisfy the majority of map requirements, it is generally recognized thatthere is a need for improvement. This improvement in survey methods is usuallymeasured in terms of the cost of a map because it is the cost of the map whichnormally determines how the mapping will be done. It is recognized that thecost is governed somewhat by necessity or the use to which the map may be put,but basically it is the cost of doing a job which is the basis for comparison ofinstruments, methods, and accuracy. Consequently, the Army Map Service con­tinues a limited amount of research with particular emphasis on stereoscopictriangulation, assuming that it is the control network which to a large extentlimits the cost and the accuracy of a map. Similarly it might be stated that it isthe control network which may determine whether a map is made at all.

Throughout the past twenty years, great sums of money and possibly greateramounts of effort have been spent in attempting to increase the precision of thevarious photogrammetric systems. As the precision increased, it was logicallyexpected that the possibilities of using aerial triangulation would increase inproportion. It is believed that this potential is still available, but that in orderto realize it more fully, additional emphasis must be placed upon the analysisand adjustment of aerial triangulation if there is to be any significant gain inefficiency. .

There are basically two approaches to the problem of photogrammetric tri­angulation. These can be stated simply as the stereoscopic and the non-stereo­scopic. Essentially, both courses employ various degrees of mathematics andmechanics to relate the photographic geometry to the surface geometry of theearth. It is evident that there are infinite possibilities using either approach, andlikely that they may converge at some point and in combination give the mostefficient performance. The user of photogrammetric triangulation is the only onein position to judge when and where to use either or both.

At the Army Map Service the greater concentration has been given to stereo­scopic triangulation. During the past five years a great deal of practical stereo­t~iangulation experience has been accumulated in connection with normal pro­duction operations. The accumulation of experience through these years hasmade quite a dent in our triangulation methods. But unfortunately, those en­gaged in the actual job of production do not always have sufficient time for thenecessary academic approach to the problem. However, there have been briefperiods when it has been possible to test and study some of these problems.These studies have provided a more solid basis for the methods used. Moreover,these studies have indicated that the classical stereo-triangulation theory ofProfessor O. V. Gruber! generally fits the empirical facts a'S we have found them.

1 Gruber, O. V., Beitrag zu Theorie und Praxis von Aeropolygonierung lind Aeronivellment,Bildmessung und Luftbildwesen, Number 3, 1935.

806

AERIAL TRIANGULATION ADJUSTMENT AT ARMY MAP SERVICE 807

PURPOSE

Inasmuch as the Army Map Service is concerned prima'rily with producingmaps using basically a "production line" system, various adjustment techniqueswere devised simple enough for a large group to handle efficiently. I t is the

,purpose herein to present some of the procedures that have been tested with theresults obtained, and to indicate the thinking behind their use.

INSTRUMENTS USED

In the photogrammetric surveys at the Army Map Service, the Multiplexand the Stereoplanigraph are used in conjunction with six inch metrogon wide­angle photography. Forward lap is usually 53%-60% giving a base altituderatio of 2/3. Adjacent strip side lap generally approximates 20%. There is a fargreater degree of difference between the conditions one might find in any givenproject area than in the analytical methods one might use with the Multiplexas compared to those employed with the Stereop-Ianigraph. Hence, in practicaloperations, the methods are the same. These methods will be described as theyapply to both instruments and the results given for each instrument separately.

The first step in the triangulation is the instrument phase. At this stage thehorizontal and the vertical are carried together. It is the prime function of theinstrument and operator to establish three-dimensional geometry of a strip as anintegrated whole. This requires precision and a thorough understanding of theproblem. Once the geometry of a photographic strip is established the next stepis to relate that geometry to the geometry of the earth's surface. This is a func­tion of mathematical analysis and adjustment.

Conventionally, it is easier to think about aerial triangulation adjustmentif one separates the horizontal problems from the vertical problems. This ap­proach is taken in practical operations but the correlation between X, Y and Zis retained. Similarly, the horizontal analysis will be described first and then thevertical, realizing that both are integral parts of the system and dissected foranalysis only.

HORIZONTAL ANALYSIS AND ADJUSTMENT

For the horizontal analysis a coordinate system is used to record deviationsin terms of X and Y. The X axis of a strip is generally positioned parallel to theline of flight. Then the procedure is to calculate the various adjustments basedentirely on the errors found. For .illustration of the process, the method de­scribed will involve the use of control at the ends of a triangulated strip. Various

Ax;tS

LEGEND

~ rlorilontol and Vertical Geodelic Control

Unadjusted Position of Control os Ploftea· From

The Stereo BridgePhoto Centers

FIG, 1

808 PHOTOGRAMMETRIC ENGINEERING

FIG. la

++

ramifications of the system are possible as various distributions of control areen'countered. But these will not be treated here.

For a hypothetical case, field established control can be used as illustratedin Figure 1; two control points in the starting model and two control points inthe ending model. The first model is usually oriented to the field- control, al-,though this is not essential to the method if the X, Yand Z coordinates of thecontrol points are recorded as measured in the stereo-model. Consider a twelve

model bridge as shown in Figure 1.The starting model (1 and 2) is ori­ented to plane coordinates for thetwo geodetic stations. Successivemodels are added until model twelve(12-13) is reached. X and Y devia­tions in the ending model are meas­ured and recorded as Figure 1. I t isevident from these measurementsthat model twelve is out of positionin X and in Yand also is rotated. Inaddition its scale is not the same asthe scale of model one.

In order to adjust these devia­tions, the horizontal analysis and correction is broken down to four mutuallydependent subdivisions. The X deviation is divided into two components called~x and "Sw" where X is the algebraic sum of ~x and "Sw", when ~x is the mag­nitude of translation error parallel to the X axis. "Sw" is the variation in thatshift to rotate the model into proper position. The Y deviation is divided simi­larly into ~y and Az (azimuth), where Y is the algebraic sum of ~y and Az, when~y is the scale change normal to the X axis and Az is translation normal to theX axis at any given point. See Figure 2.

b.x=total translation error parallel to the X axis for a point on the X axis"Sw" = total X translation difference parallel to X axis between b.x and points

lateral to the X axis, and this difference due to rotation is directly pro­portional to the perpendicular distance from the X axis to a point con­sidered. Similarly, "Sw" is proportional to the Y differences given by theazimuth curve. Thus, the azimuth curve may be defined by determining"Sw", or the rotation (Sw) rna be determined from the slope of theazimuth curve. Points on either side of the X axis carry the appropriatesign ±, depending upon the direction of rotation.

Az (azimuth) = total translation error perpendicular to the X axis for a point on theX axis .

b.y = total translation difference perpendicular to the X axis between Az andpoints lateral to the X axis, and this difference due to scale is directlyproportional to the perpendicular distance from the X axis. b.y is like­wise proportional to the b.x differences given by the b.x curve. Points oneither side of the X axis carry appropriate signs ± depending upon thescale deviation present.

From consideration of the fundamental type of systematic variables in thetriangulation, (viz. residual radial distortion, tangential distortion, differentialfilm change etc.) the plane geometry of a triangulated strip is consideredanalogous to the condition, Figure lA, where a graduated series of a similartrapezoids are joined. If the area of each trapezoid is consistently greater thanits neighbor, the relationship of this geometry to a grid coordinate system would

AERIAL TRIANGULATION ADJUSTMENT AT ARMY MAP SERVICE 809

COMPONENTS OF HORIZONTAL ADJUSTMENT

~"-"0

x = .6.x ~ V&""V + Sw

f},.. ·• ..El

Y - Az JJ~l

Clif 6

6

+ D.y

LEGEND

f:, . Horizontal and Vertical Geodetic Controlo '" Unadjusted Position of Control as Plotted

from the Stereo Bridge

FIG. 2

I

£::. Horizontal and Vertical Geodetic Control

o Unadjusted Position of Control asPlotted From The Stereo Bridoe

+ ..... Photo Centers

LEGEND

Absolute Position'of EndinQ Model --""

n:-~__ m __ ;

, ,~ :

}/ "odels 2 'hru 12 IftCIUln~:___ t F ~ _' +_--1-__ - X Axis

'LJ ~

. FIG. 3

810 PHOTOGRAMMETRIC ENGINEERING

be as follows: The X deviation Llx, i.e. lengthening of the strip, is directly pro­portional to the square of the number of trapezoids joined. The Y deviation Lly,i.e. the expansion of the strip, is directly proportional to the number of trape­zoids joined. Deviation Lly can also be computed as the first derivative of theequation for Llx or the slope of the Llx curve. The Y deviation Az (azimuth) wouldbe circular and is computed from the equation of a circle. Moreover, the Xdeviation "Sw" (rotation or swing) would be computed as the first derivativeof the equation of that circle or from the slope of the azimuth curve.

I

100

Segment Nsl I Segment Ns2 I Segment N:3 I Segment Ns4 I25 50 75 100

% of Total X Distance Between Control

FIG. 4

The graphic representation of Llx becomes quite simple in practice if one usesN2+N

the geometric progression Llx = K· 2 ,where N is the number of seg-

ments of the total length of the strip and K is the constant of proportionali(y.2Plotting the Llx curve is illustrated, Figure 4, where the triangulated strip isdivided into four segments. Percentage of Llx error is plotted against per cent oftotal X distance from the following:

N2+ N~x= K--­

2

Segment

N=l

N=2

N=3

N = 4

Per cent of Maximum ~x

12+1~x = K -- = 1 = 10%

2

22 + 2~x = K -- = 3 = 30%

2

32 + 3~x = K -- = 6 = 60%

2

42 + 4~x = K -- = 10 = 100%

2

It can be seen that if the Llx deviation at the end of the fourth segment isconsidered to be 100%, then the percentage errors fOt: the 1st, 2nd and 3rdsegments will be 10, 30 and 60 per cent respectively. This is quite a convenientmethod of plotting the Llx curve on graph paper and then interpolating all cor-

2 Ibid., pp. 136-141.

AERIAL TRIANGULATION ADJUSTMENT AT ARMY MAP SERVICE 811

rections for pass points graphically. Similarly, the Y direction scale deviation.1y can be computed as the first derivative of the equation of the .1x curve orthe slope of the .1x curve. As a simple graphical expedient in finding this curve(called .1y curve) one needs merely to measure the slope of the .1x curve overshort segments and plot resulting values. For example, if the difference alongthe slope of the X curve for a 200 mm. segment of the X curve is one millimeterthen the Y deviation .1y, for 200 mm. of Y is equal to one millimeter along aY line which bisects the 200·mm. segment ofX. Thus reading a few values offof the .1x curve, the .1y curve can be plotted on graph paper and the correctionsfor pass points interpolated.

In most triangulated strips it is evident that Yazimuth deviation (Az) ispresent. Often times, this azimuth deviation is a curve. When it is evident thatthe ending model is rotated relative to the starting model as illustrated, Figure 1,experience has shown that the azimuth deviation is circular. Rotation of theending model is illustrated by the measurement of X direction deviations, whichare not equal for the two control points used, Figure 1. Assuming that the ac-

.cumulated azimuth errors are systematic, these deviations are considered to beapproximately equal in magnitude for each model. They have the general shapeof several of the conic sections but the conic sections which fit the conditions·best are the circle and the parabola.

From analytical geometry an equation of a circle is determined when thereare given three geometric conditions which lead to three simultaneous equations.The simplest set of conditions consists of three non-collinear points. It is knownfrom plane geometry that a unique circle may be passed through any three non­collinear points. Thus, using the equation of a circle:

x 2 + y2 + D x + Ey + F = 0

it is possible to establish the Azimuth ourve merely by analysis of the swing orrotation deviation in the ending model relative to the beginning model. Thisanalysis is shown in Figure 3. Because the rotation of the ending model can bemeasured by the.difference between .1x at the top of a strip and .1x at the bot­tom of a strip, it follows that the x and y values for three non-collinear pointscan be determined and used as the constants for the circle which passes throughthem. After the constants D, E, and F are determined, it is relatively a simplematter to compute the values of three more points on the circle. Then the valuesof azimuth deviation can be plotted and graphically connected by using a spline.In this manner the azimuth curve is defined and used to interpolate the Az cor­rection for all pass points.

To find the swing deviation (Sw) the approach is similar to the manner inwhich.1y is found. As previously noted, the X direction deviations are not equal.This rotation is caused by the non-linear form which azimuth deviation takes,and can be computed as the first derivative of the azimuth curve and/or theslope of this curve. Hence, the swing deviation may also be found graphically,simply by measuring short segments of the slope of the azimuth curve, andplotting the results for some convenient unit of measure to give "Sw."

These four curves, .1x, .1y, Az and"Sw" are normally superimposed on avellum or graph paper which also contains a copy of the instrument-establishedpass-point positions and the geodetic stations to be used. From the four curvesas plotted on the graph paper it is then possible to interpolate the X and Ycorrection for all photogrammetric established control.

The procedures as outlined herein provide the adjustment for single stripanalysis. In practical operations the conditions are rarely such that a single strip

812 PHOTOGRAMMETRIC ENGINEERING

wiIl' cover the area to be mapped. Consequently, it is necessary to adjust aseries of triangulated strips to each other in order to achieve a homogeneoussolution for the area. This inter-strip adjustment is analyzed and adjusted byareas in a manner similar to the "Block_Adjustment Method" used by ProfessorM. ZeIler. 3 After the interstrip adjustment is completed, pass points common totwo strips have only one position. Use of this type of adjustment, where threeor more strips are examined, generaIly reduces the maximum deviations andusually has little effect on the root mean square er'ror for the area.

VERTICAL ANALYSIS AND ADJUSTMENT

It is generaIly known from aposteriori statistics that deviations in verticaldatum for stereo triangulated strips are quadratic. Most compensations forthese Z deviations have in on~ way or another involved the parabola or thecircle, but from practical operations it is widely conceded that additional air­borne control is needed to complement the data derived from stereo-instrumenttriangulation alone. This need has resulted in the use of the barometric altim­eter, and/or the statoscope. At the Army Map Service, studies have been madeusing barometric altimeter recordings as a complement to the geodetic control,to provide additional data for analysis and adjustment of vertical triangulation.

As mentioned previously, the horizontal and vertical instrument triangula­tion are carried together. Use of the altimeter data, stereo-instrument data andgeodelic data to establish the absolute vertical geometry of the strip is againthe function of mathematical analysis and adjustment.

For the vertical analysis, a rectangular coordinate system i's used (graphpaper) to record the deviations in Z, and to provide the medium for a flexibletype of procedure readily adaptable to production use. To illustrate the verticalanalysis and adjustment consider a strip, Figure 1, where the available verticalfield control is in lines perpendicular to the line of flight at the terminal models ofthe strip. The starting model is usually oriented to the available ground eleva­tions given for the two control points. Successive models are added by relativeorientation only until after the twelfth model is reached. For epch camera posi­tion in the bridge a bi recording is made. In addition the Z deviations relativeto ground elevation in the ending model are recorded. From experience it isknown that the middle ordinate of the vertical datum curve is approximatelydefined by the relative (bz) plotting of camera positions in the bridge.4 To definemore exactly the middle ordinate of the "bz" curve, the altimeter recordings areused to adjust the "bz" recordings so that most of the variation resulting fromchange of height of the aircraft is removed. This adjustment is illustratedgraphically, Figures S and Sa, for a forty-six model stereo bridge run with theStereoplanigraph and the same bridge. using the Multiplex. It is evident thatthere is much less difficulty defining the "bz" curve after altimeter variationsare compensated. Then to a graphic plot of the adjusted "bz" recordings, a meancurve is drawn which is accepted as the approximate vertical datum curve forthe bridge. This curve must be correlated to the geodetic control available forthe strip. When the adjusted "bz" curve is set at zero for the starting model thedeviation in this "bz" curve at the ending model will not normally be equal tothe Z deviation from true geodetic datum for that model, Figure 6. Thus, twocomponents for the adjustment of Z deviations must be considered. The devia-

3 Zeller, M. Die Bestimmung von Punktnetzen mittels Luftriangulation und deren Ausgleichung, Schweizerischen Zeitschriftfur Vermessung und Kulturtechnik, Jahrgang 1950, Heft Nr. '10.

• War Department Technical Manual 5-244 Multiplex Mapping Equipment, June 1943, pp.60-62.

AERIAL TRIANGULATION ADJUSTMENT AT ARMY MAP SERVICE 813

GGRAYLING TEST

Sfereoplonigroph "b z· Oeviafion

o ·· .. Unadjusted "bz" RecordinQs

o ·...Altimeter Adjusted ·bz" Recordings

··Mean -bi" Curve

FIG. S

GRAYLING TESTMultiplex -."bz" Deviation

5

..:.o2

00..

... 0.. 0IL

'"<>0

'".~

o

Q.g'

r'..:; 0.. 0

.cot'-'

"N.....oo...og

20 40 60 80. 100 120 140X Distance Per 20,000 Feet

.. Unadjusted "bz· RecordinQIAltimeter Adjusted 'bz" Recordinqa

... Mean "bz" Curve

FIG. Sa

FIG. 6

\ .160 180

'\\> •

\

814 PHOTOGRAMMETRIC ENGINEERING

tion Z (error in projected vertical datum) becomes the algebraic sum of "bz" and~z when "bz" is the recorded mean plotting camera station position and ~z isthe divergence between deviations Z and "bz". Solving for ~z becomes simpleif one again considers the horizontal deviations. If the horizontal scale is chang­ing, it follows that projection distance is changing proportionally. Accordingly,if the scale of the bridge becomes progressively larger, the projected photogram­metric datum will "drop-off" more as the projection distance increases. In thismanner, it is assumed that the vertical datum curve diverges from the mean"bz" curve. Proceeding from this assumption, ~z is derived as the proportionalvertical quadratic scale error. Therefore, like the formula used to find ~x onecan use:

PD (N2 + N)toz = - K - ----

X 2

where K is a constant of proportionality" and should be equal to K as determinedfrom the X correction for the same strip, PD is the projection distance of thestrip, X is the total length of the strip and N is a segment of X. A practicalexpedient for defining the ~z curve on a graphical plot, is the same as that usedto plot ~x. Percentage of ~z error is plotted against percentage of X distance asshown above for the ~x curve, Figure 4. To correct pass points for Z datumdeviations it is necessary to take only the algebraic sum of the interpolatedvalues from the bz and ~z curves. In addition to the error in vertical datum that.one finds in a stereo-triangulated strip, it is also usual to find an accumulationof tilt perpendicular to the flight line. Evidence of this is apparent from thedifference in Z datum deviations for the two control points which lie lateral tothe line of picture centers in the ending model. This tilt is assumed to be directlyproportional. to X distance between lines of field control and is computed from:

tilt correction = w(;) ( ~S)

where the constant w is the tilt in millimeters perpendicular to the X axis atterminal control, X is the total length of the strip, N is a segment of X, Y is thehorizontal distance perpendicular to the flight line between the field controlpoints, and s is the perpendicular distance from the X axis of the strip to anygiven point, and carries the appropriate sign (plus or minus) relativ to themean error in vertical datum for a point on the axis of the strip with the samevalue of N. Considered as an integral part of the w tilt analysis, is an adjustmentof the differences for carry-over points along the strip. Usually these differencesare removed for the entire strip algebraically, but often-times they are onlymeaned, considering individual points depending upon what is considered to bethe basic cause for such differences with any given photography.

RESULTS OF TESTING THE METHODS

In order to evaluate the methods in quantitative terms, and to comprehendtheir effect on over-all project planning, a specific test is described below. Thistest was divided into two phases involving the Stereoplanigraph in the firstphase and the Multiplex in the second. The instrumental methods were slightlydifferent for the two instruments, but the adjustments for both phases wereidentical with those described above.

• Gruber, O. V., Beitrag zu Theorie und Praxis von Aeroploygonierung und Aeronivellnent,Bildmessung und Luftbildwesen, Nr. 3, 1935, pp. 136-141.

AERIAL TRIANGULATION ADJUSTMENT AT ARMY MAP SERVICE 815

The area selected for test purposes was part of a regular production projectcalled Camp Grayling, Michigan. Three adjacent strips of continuous 6" Metro­gon photography were selected. Two of the strips were unbroken for forty-sixstereo pairs, the third extended only thirty-eight models. The flight height was6,200 feet. This test area was roughly rectangular in shape, 35 miles by 4t milesor approximately 145 sq. miles considering the broken rectangle formed by theshorter strip. Flight lines were in the north-south direction so that the twolonger strips covered about 30', and the shorter strip about 22t' of latitude alongthe strip. Across the three strips the coverage was about 5.6' of longitude.

TABLE I. STEREOPLANIGRAPH AND MULTIPLEX HORIZONTAL TRIANGULATION RESULTS

Grayling, Michigan-Strips 18, 19, and 20Aerial Camera T-5 41-4164Metrogon Lens MF 2,398Flight Height 6,200 FeetTriangulation Scale 1 :5.000

Instrument Type Multiplex Stereoplanigraph

Total number of stereo models covering area 130 130

Number of square miles in area 145 145

Average length of horizontal stereo-bridge 43 Models 43 Models35 Miles 35 Miles

Number of horizontal geodetic stations used for area 9 9

umber of horizontal geodetic stations used per stereo-bridge 4 4

Number of horizontal geodetic stations used to check areaaccuracy 25 25

Mean maximum error, three worst check-point observations 1.7 mm. 0.96 mm.27.9 feet 15.7 feet

Root mean square error (Standard deviation), all check- 1.2 mm. 0.65 mm.points 19.7 feet 10.6 feet

PROCEDURE FOR EACH INSTRUMENT

For the Stereoplanigraph test, all strips ~ere triangulated on one instrumentusing one experienced operator. The three strips were set in relative orientationto the limits. Instrument coordinates were recorded for all photo-identifiedpoints. Each strip was tied relatively by systematic removal of all parallax inthe same positions for all models. In addition to recording instrument coordi­nates in millimeters, the instrument position of all identified points was recordedon Dyrite material which contained a U.T.M. grid at 1 :5,000 scale. No attemptwas made to use barometric altimeter data in the instrument work.

In order to triangulate forty-six models using a fifteen projector Multiplexdouble frame, it was necessary to alter normal instrument procedures somewhat.Each strip was set in four segments, all succeeding segments were oriented totheir predecessors, by resetting the last two models of the preceding segment.In this manner, the last two models of one segment became the first two modelsof the next segment. Using this method of "lapping" two models, it is assumedthat the three dimensional geometry of the strip is precisely related. Multiplexcoordinates were recorded on Dyrite material carrying a U.T.M. grid at 1 :5,000

816 PHOTOGRAMMETRIC ENGINEERING

TABLE II. STEREOPLANIGRAPH AND MULTIPLEX VERTICAL TRIANGULATION RESULTS

Grayling, Michigan-Strips 18, 19, and 20Aerial Camera 1'-5 41-4164Metrogon Lens MF 2,398 .Flight Height 6,200 FeetTriangulation Scale 1: 5,000

Instrument Type Multiplex Stereoplanigraph

Total number of stereo-models coveringarea 130 130 130 130

------N umber of square miles in area 145 145 145 145

------.'\verage length of vertical stereo-bridge 43 Models 10 Models 43 Models 10 Models

35 Miles 8 Miles 35' Miles 8 Miles

Number of vertical geodetic stationsused for area 9 23 9 23

Number of vertical geodetic stationslIsed per stereo-bridge 4 4 4 I 4

------------N umber of vertical geodetic stations

used to check area accuracy 248 218 248 218-- ----Mean maximum error, three worst 1.77 nim. 1.17 mm, 0.68 mm. 0.23 mm.

check-point observations 29.0 feet 19.2 feet 11.2 feet 3.8 feet

Root mean square error (Standard ,786 mm, 0.47 mnl. 0.38 mm. 0.11 mm.deviation), all check-points' 12.9 feet 7.7 feet 6.2 feet 1.8 feet

Ratio of maximum error to flight height 1/214 1/323 1/554 1/1,632-------------------------

Ratio of mean square error to flight

Iheight 1/481 1/805 1/1,000 1/3,444

scale. The Multiplex system of relative orientation was identical to that used inthe Stereoplanigraph. Like the Stereoplanigraph phase,. the analysis of theMultiplex results was conducted for the complete forty-six model bridge. Allerrors introduced at the break between segments were considered homologousto all the other variations in the strip as a whole.

A summary of the results of a'ccuracy testing is shown in' Tables 1 and 2.The lines of check control were running east-west almost perpendicular to ther'orth-south flying. The ground distance between lines of horizontal check con­trol was approximately nine miles so that there were three check-lines betweenthe control bands used to adjust the work, Figure 7. These "in between" horizon­tal lines were used only to test horizontal bridging accuracy and did not providedata for the triangulation. Vertical check control als9 crossed the flights in aneast-west direction at a spacing of three to four miles, Figure 7a, giving ninevertical check lines for the forty-three model analysis and approximately five

. vertICal check lines in the ten-model analysis for the area. These "in between"vertical lines were used in both the ten model evaluation and the forty-threemodel evaluation as shown, Figure 7a, only to test vertical bridging accuracyand not as control for the triangulation.

There were nine horizontal stations used to control the area horizontally.For testing the horizontal accuracy there were distributed in the area, as indi­cated above, twenty-five field established check points. Vertically, nine eleva­tions were used to control the triangulation in the first analysis for the full

Strip 20

Strip 19

Strip 18

AERIAL TRIANGULATION ADJUSTMENT AT ARMY MAP SERVICE 817

HORIZONTAL CONTROL DISTRIBUTION(Table Na.l)

,0 1 \ :1! \- \~ \ /

(.0

\ (~

i( )

0 I, I 0.J \

LEGEN.D

o .... Terminal Model Horizontol Control__ . ·Bondl of Horizontal Cheek Control

FIG. 7

VERTICAL CONTROL DISTRIBUTION(Tabl. NO.2)

43 Model Bridge

oStrip 20

Strip 19

Strip 18o

10 Model Bridge

Strip 20

Strip 19

Strip 18

,0 J 0 \ 0

! J ~I/ I I

~ }I ; u u

I I~

0 \ 0 \ 0 I 0)

0 I 0

LEGEND

0"" .. T.rminal Nod.1 V.rticol COlltrol

--.. Bonds of V.rtical Chick Control

FIG.7a

length of the strips, and two hundred forty eight field established elevations asaccuracy check points. In the second analysis for ten model bridges, twenty­three elevations were u~ed as control and two hundred eighteen elevationsserved as vertical accuracy check points. It should be understood that the testfor different length bridging involved merely adding control for analysis pur­poses and did not involve additional instrument work.

818 PHOTOGRAMMETRIC ENGINEERING

FIG. 8

DISCUSSION

It is possible that the procedures presented may appear to be quite laborious.However, numerous shortcuts are available in the mathematics and in thegraphical curve plotting. For the past two years these methods have been usedin various degrees by the Army Map Service on regular project triangulation.More specifically, they have been applied to approximately twelve thousandstereo-triangulated models, or approximately six hundred aerial photographicstrips. More than twenty wide-angle cameras equipped with six-inch Metrogonlenses have been utilized. he instrument triangulation has employed eightStereoplanigraphs and twenty units of approximately fifteen projector Multiplexdouble frames, containing three hundred Multiplex projectors. In addition, tothe sixteen Metrogon lenses used as distortion compensators with the Stereo­planigraph, six reduction printers were used in conjunction with the Multiplexwork. All of the work was performed by a group of thirty people with an eight­year average span of photogrammetric experience. This quantity of work couldnot have been produced if these procedures were not adaptable to a wide varietyofconditions, and were not simple enough to be applicable to "continuous-flow"type of operations.

/

AERIAL TRIANGULATION ADJUSTMENT AT ARMY MAP SERVICE 819

Consider the test area again. It would take approximately four hundredoffice man-hours to establish the horizontal and vertical position of the neededphotogrammetric control for the photography used. This means that the posi­tions and elevations for more than two hundred picture-identified points cover­ing an area of 145 sq. miles could be established in that time. Three men couldcomplete the job easily in less than four weeks, and all of this control data wouldbe available on the compilation media, ready for use.

From the horizontal accuracy checks tabulated in Table I, it is clear thateithel the Stereoplanigraph or the Multiplex would meet the National Standardsfor horizontal map accuracy at a scale of 1-: 25,000 where the horizontal toleranceis forty-two feet. It is likewise evident that the Stereoplanigraph horizontalresults would satisfy the National Standards requirement for a 1: 10,000 .scalemap where the horizontal tolerance is sixteen feet. Moreover, it is also apparentthat the Stereoplanigraph horizontal accuracy is such that it would only benecessary to triangulate alternate strips, to achieve the needed horizontal ac­curacy for a scale of 1: 25,000.

From Table I it can be seen that the standard deviation for the Stereoplani­graph established horizontal control is 10.6 feet. Although there are no quantita­tive data available for the standard deviation of photo-identification process, itmay well be that a man's ability to photo identify a group of field-establishedpoints at a photo scale of 1: 12,000 approximates something around ten feet.Furthermore, the picture-point control used was principally third-order traversestations. These stations are established under specifications which require thatthe extension of third-order traverse should not exceed eight miles of trav­ersed distance from a triangulation station of third-order accuracy or higher

.or from traverse stations of second-order accuracy. In the stereo-triangulationthis control is related across ground distances of thirty-five miles and there isno absolute assurance that these lines of photo-ties are as precisely related atthat distance as one may safely assume for their accuracy in shorter grounddistances. Within the scope of the test, it was not considered necessary to evalu­ate the horizontal results for distances shorter than forty-six or thirty-eightmodels, because it was intended to determine whether a thirty minute quad­rangle could be accurately bridged, using geodetic control at the northern andsouthern limits only.

In Table II, results are given for an average length of forty-three models andfor ten models, first because there was a much greater density of vertical geo­detic check control, and second because of the greater variation in map require­ments for vertical accuracy. For example, if it is a reasonable assumption to saythat the contour accuracy in Stereoplanigraph compilation is four times theroot mean square error for photo-tie points, then from the Stereoplanigraphresults a ten-foot contour could be drawn from the results achieved using geo­detic vertical control spaced at eight-mile intervals. Similarly it is apparent thata twenty-five foot contour is feasible with the Stereoplanigraph using geodeticvertical control at thirty-five mile spacing. If one examiFles the distribution ofcontrol used to check results, Figure 7a, it is evident that those areas whichlogically could be expected to be weak are checked. More specifically, thosepositions farthest from geodetic control in either the forty model evaluation orthe ten model evaluation are checked. Moreover, in the ten model evaluation thecheck lines are only at the center of each bridge, giving a root mean square erroronly in those areas which could be expected to be the weakest.

It might be stated that there are some general concepts which emerge as aresult of the accumulation of Army Map Service experience. First, it is clearthat a quasi-statistical type of approach to the analysis and adjustment of

- 820 PHOTOGRAMMETRIC ENGINEERING

stereo-triangulation is basically sound. Therefore, the more observ:}tions madethe more reliable the statistics. If this is true then the larger the stereo-bridgearea, the better the analysis can be. For example, it isnow common practice at theArmy Map Service, to complete all the instrument work for a flight in a con­tinuous integrated setting, so that the geometry is related for the full limits ofany given strip. This approach is taken with work on the Multiplex and theStereoplanigraph. Secondly, aneroid barometric altimeter recordings, if usedstatistically, define the "bz" curve more consistently than any other knownmethod for the two stero-systems rp.entioned herein. If one examines the bzcurves illustrated, Figures Sand Sa,- it is evident that more precise relativealtimetric data would have little or no effect in the resultant mean "bz" curve.More specifically, reducing the spread between relative differences in bz, aftercorrected by barometric data, would not affect the position of a mean curve inthe curve fitting process. Therefore, it would appear that beyond what is givenby barometr-ic altimeter recording, more precise relative altimetric data need.not be considered too important. Much greater source of error is the absolutealtimetric data that is necessary to relate the stereo-datum to sea level. More­over, using the methods described with the equipment mentioned herein, it isapparent that one has a "photogrammetric profile recorder" which will giverelative accuracies of a very high order using nothing more than a barometricaltimeter in the aircraft.

Furthermore, the methods of analysis and adjustment provide a very usefultool for evaluating instrument performance, and in this respect provide the mapmaker a very useful gauge as to the pncticallimits for his equipment. To illus­trate this last point, consider again the geometrical analysis of the stereo-bridgeas a whole. It is obvious from the nature of the needed adjustments that eachmodel has a systematic warpage. The magnitude of these various warpages canbe rather precisely determined by detailed examination of the adjustmentcurves. For example, using proportionality formulae and comparing the resultsof the Stereoplanigraph with those of the Multiplex in the test area, one findsthe order of magnitude of the proportionality constant for the Stereoplanigraph10-8 as compared to 10-6 for the Multiplex. This could very plausibly mean thatthe Stereoplanigraph model is less warped than the Multiplex for the areatested. Similarly this could indicate that the possible Stereoplanigraph contourline is four times the mean square error of spot-height readings as compared tofive times the mean square error for the possible Multiplex contour line.

Finally, in the continual attempts to deGrease map costs by increasing thedistance between-geodetic control stations, the greater emphasis in the UnitedStates has been given to increased flying heights. This concentration has con­sidered the geometry of only one model as the practical photogrammetric limit.Experience at the Army Map Service has shown that the photographic geom­etry of an area which includes a great many models, can be precisely estab­lished, and that along with any emphasis on higher flying, the photogrammetricengineer would do well to consider as an equally important alternate, the infi­nite possibilites of aerial triangulation.

AUTHOR'S NOTE: Photogrammetrists reading this article should consider that themethods presented in this paper are by no means perfected in terms of cause and effect.The practical photogrammetrist will recognize, however, that there are all degrees ofcompromise in map production, and that any practical method is an approximation con­cerned more with satisfying map requirements than exact theory. Furthermore, itshould be noted that these methods have been in use for the past two years and arecontinually being revised in terms of theoretical cause and effect, though the basic ap­proach generally remains the same.