The Application of Analytical Photogrammetryto Missile Trajectory Measurement*

GEORGE H. ROSENFIELD,

Photogrammetric Analyst,RCA Missile Test Projectt

:\BSTRACT: This paper concems the philosoph·y of data reduction of opticalsystems data. A discussion is given of the mathematical techniques used toproduce the final optical information .. and the peculiarities of the data from thel'ariOlls instrumental systems are presented.

PREFACE

T HIS paper presents a discussion of themathematical techniques used in re­

ducing missile test photogrammetric data.It does not enter into a presentation of themathematical formularization, since that isadequately coyered in pertinent technicalreports. The fundamental concepts of Analyt­ical Photogrammetry, both I ntersection andStereo Photogt'ammetry, as they apply tomissile testing are explained.

The data deri ved from the various types ofphotogrammetric instrumentation retain thepeculiarities of the random and systematicerrors associated with the particular recordinginstrument and the particular data reductioninstruments used in the preparation of themissile flight data. Because of this, thedescription of the peculiarities of the differ­ent types of optical instruments is presentedas a description of the random and systematicerrors affecting a photogrammetric reduction.A chart outlining many of the sources of errorassociated with a photogrammetric instru­mentation system is included with this paper.The chart was prepared by D. C. Brown andpublished in reference (3). The paper alsopresents a discussion of the mathematicalpropagation of both the random and system­atic errors through the data reduction proc­ess, and indicates a method of determiningthe error influence on the reduced data.Since a familiarization with the photogram­metric instrumentation used in missile testingis needed as background for this presentation,the followi ng brief descri ption of the generaltypes of cameras used is gi Yen.

Photogrammetric missile trajectury dataare obtained from several types of instrumen-

GEORGE I-I. ROSE FIELD

tation cameras, which are generally catego­rized as being either fixed cameras or trackingcameras. Fixed cameras are those operated ina given fixed orientation, the missile tra­jectory passing through the field of view ofthe camera. Tracking cameras are those whichtrack the missile in flight, the camera orien­tation changing for each photographic frame.The high-precision fixed camera is the Ballis­tic Camera, which utilizes a single glass-platenegative upon which is recorded the entireevent of interest. Normally, this is photo­graphed against a background of stars, fromwhich an accurate orientation can be cali­brated. All other cameras of somewhat lowerprecision lIsed in a fixed orientation arecategorized as fixed cameras. These camerasnormally use til m as the photographic base.Either ground surveyed target boards or the

* Presented at the Society's 27th Annual Meeting, The Shoreham Hotel, Washington, D. c., March19-22, 1961.

t Atlantic Missile Range, Patrick Air Force Base, Fla.

547

548 I'HOTOGRAMMETRIC ENGINEERING

mount dials are used to determine thecamera orientation.

Tracking cameras can be categorized asmetric or non-metric. The metric trackingcameras are of the Cine-theodolite type whichrecord Azimuth and Elevation angles to­gether with the event photographed for eachframe. The non-metric tracking telescopes donot record camera orientation, but can beused for certain metric applications such asmissile-attitude determination, where thetelescope orientation is computed from theknown missile-position data.

I :-.ITRODUCTIOK

Analytical Photogrammetry is the art orscience of applying mathematical methods toobtaining reliable measurements by meansof photographs. For many years, the applica­tion of analytical photogrammetry remaineda laboratory curiosity and a classroom exer­cise. The increased utilization of the elec­tronic computer in recent years has led tosophisticated developments of numericaltechniques in analytical photogrammetrythat were not possible in the past.

The basic information of all photogram­metry is the image-space measurement of theobject-space phenomena. The measurementsare normally in the form of linear or polarcoordinates on the image-space medium, andmay be made and utilized either directly orindirectly in the resulting photogrammetricreduction. A major advantage of analyticalphotogrammetry is that the basic measure­ments can be considered independently, andcan be adjusted for all perturbations of thephotogrammetric model, to the fullest desiredextent of the state of human knowledge.Since it is impossible to make a perfectmeasurement, a mature instrumental systemincludes knowledge of the nature of themeasuring error. Proper computational tech­niques allow propagation of the measuringerror through the adjustment to obtain thevariances of the reduced data. Proper tech­niques also allow an indication of the residual,uncorrected, systematic error remaining inthe system. The fundamental mathematicalconcept applicable to analytical photogram­metry is the Method of Least Squares, in whichthe numerical adjustment aims to minimizethe weighted sum of the squares of themeasuring residuals.

ANALYTICAL PHOTOGRAMMETRY

The direct mathematical solution of aphotogrammetric adjustment results in aquadratic form of the residuals, the solution

of which requires complicated proceduresthat are somewhat impractical in nature. Thepopular methods for the solution of photo­grammetric adjustments are based uponNewton's method of successive approxima­tion. A set of variational equations is estah­lished from the initial photogram metriccondition equations. These equations de­scribe the photogrammetric model in amathematical format. The usual method ofestablishing the variational equations is tolinearize the condition equations by a Taylor'sexpansion about the observations and theparameters. The solution requires approxima­tions for the parameters which are sufficientlyclose to the true value to allow convergence.

The Method of Least Squares results in thecomputation of the most probable values forthe unknown parameters, and thus reducesthe effects of random errors. The least squaresadjustment cannot properly allow for theinfluence of systematic errors, and these mustbe compensated in the reduction before enter­ing into the adjustment.

The major photogrammetric reductionsperformed by analytical methods are thosefor interior and exterior orientation of thephotogrammetric camera, triangulation tocompute object space position information,and computations to determine the attitudeof a body in object space. Interior orientationof a photogrammetric camera refers to thecalibration of the optical elemen ts of thecamera body. These elements normallyinclude the establishment of the principal­distance c and the position of the principal­point with respect in the fiducial marks of thecamera x p , YP' The exterior orientation is thatset of quantities which fixes the position ofthe camera station and the angular orienta­tion of the photograph. The position is usu­ally expressed in terms of three rectangularcoordinate distances Xc, ye, ZC. The elementsof angular orientation are usually expressedas three angular rotations. In aerial photo­grammetry the rotations may be designatedtj>, W, K, while in terrestrial photogrammetrythey are (x, w, K. In the usual photogrammetriccase, given any three of the nine elements oforientation, it is possible to determine theremaining six, if sufficient observations andcontrol information are available.

The solution of a photogrammetric prob­lem may be performed using the tenets ofeither Intersection Photogrammetry or ofStereo Photogrammetry. Analytical StereoPhotogrammetry is based upon a mathemati­cal formulation for the precepts of stereos­copy which attends the si multaneous con-

).IISSILE TRAJECTORY MEASUREMENT 549

sideration, in three dimensions, of an objectfrom different perspectives. Discussion of thestereo procedures will be given later.

The concept of Intersection Photogram­metry requires first, the independent andprecise orientation of each camera to be usedin the photogrammetric net; and second, thedetermination of the most probable solutionto the photogrammetric problem by means ofa rigorous least squares adjustment usingredundant information, considering the basicorientations to be error free. Propagation ofthe random error effects by co-varianceanalysis techniques, and determination of thesystematic error effects by overdeterminingthe solution, indicates the accuracy of thereduced data as a by-product of the adjust­ment. This philosophy of computation can beapplied to solution of the problems of missile­position determination (triangulation) andmissile-atti tude determination.

FUNDAMENTAL MEASUREMENTS

The basic data for missile-trajectory meas­urements consist of observations on thephotographic records from the several cam­eras to be used in the reduction. Theseobservations are in the form of measurementsmade to an image on the photographic film orplate; the form of the measuremen t dependsupon the type of instrumentation and thedata required. For example: Ballistic cameradata are obtained from x. y, coordinateobservations made to the symmetric imagesof point sources of light: star images fororientation, strobe light or Ran images fortriangulation. Fixed camera orientation dataare obtained from x, y, coordinate observationmade to the symmetric center of the imagesof surveyed target boards. Fixed Cameraposition data are obtained from x, y, coordi­nate observations made to the images of theimages of the nose of the missile, the leadingedge of the Rame, or to the symmetric imagesof a missile-borne light source. Cine-theodo­lite position data are obtained from measure­ments made on the recorded images of theAzimuth and Elevation dials together withlinear measurements in machine counts of thetracking error: l1A t, l1E t . The Tracking errormeasurements are made to the images of thenose of the missile, the Rame, or the blob onthe film, depending upon the aspect-angle anddistance of the missile from the camera site.

Attitude data, pitch and yaw, are pre­dominantly obtained using measurements ofthe so-called V-angle; the angle between theimage of the missile-body vector and the axespf the film coordinate system. For a tracking

camera attitude reduction, only the V-anglemeasurements for two or more cameras arerequired. For a fixed camera attitude reduc­tion, the V-angle measure men ts togetherwith the x, y, plate coordinates of some pointon the image of the missile are required fortwo or more cameras. For convenience, thereis generally considered the plate coordinatesfor the tracking point used in the Positionreduction. Ballistic Attitude reduction refersto the simultaneous determination of thethree elements of attitude: the pitch, yaw,and roll, of the missile. These data are ob­tained from the x, y, plate coordinate meas­urements made to the symmetric images ofeither paint pattern intersections or specialmissile-borne light sources.

A distinction has been made between thex, y, coordinate observations and the x, y.plate coordinates. The x, y, observations arerecorded pointings made in the coordinatesystem of the measuring comparator to thespecified image on the emulsion of the photo­graphic medium. These observations must betransformed to the coordinate system of theoperational negative (the glass-plate or photo­graphic-film) and then corrected for theperturbations introduced by the peculiaritiesof the particular optical instrument. Thefinal corrected values are then commonlyreferred to as plate coordinates, regardless ofthe composition of the photographic base.

RA:-IDOM ERRORS

The primary sources of random error in thephotogrammetric reduction occur in a two­fold manner. \i\Then optimum images arebeing considered, such as with BallisticCamera data or missile-borne light sources,the most important random error is due tothe differences in position between the latentand the developed images. These image shiftsdue to random emulsion creep are caused bystrain occurring in the emuision duringprocessing of the negative;;. The other pri­mary source of random errOl is the measuring­error due to:

1) setting-error, the inability of themeasuri ng operator to repeat his meas­urement; and

2) identification-error, the inability of theoperator to set on corresponding pointsof di fferent images.

Other sources of contributing random errorare caused by residual uncorrected environ­mental anomalies and also by random errorsof the measuring comparator. The composite

550 PHOTOGRAMMETRIC ENGINEERING

rsources of error in 8 ballistic cameral- system emDloyin~ star cslibrations

Bentificstionerrors

FIG. 1

Random temperaturefluctuations during

measuring

Disturbance of cameraorientation betweenstar calibration sndtriangulation event

of all these random errors are commonlycalled the reading-error.

The magnitudes of the random errors forthe different types of optical instrumentationdata are given below. Except for the randomerror of the V-angle attitude measurements,the values are obtained from reference (1).

For measurements made on Ballistic Cam­era plates to optimum-size symmetric images(approximately 50 microns diameter) usingprecision comparators, the average values forrandom errors as indicated by time seriesanalysis of series of light-source images aretypieally(Tx' (T~, ""'3 microns.

The random error for Cine-theodolite data

is considered to lie in the ability of the opera­tor to interpolate the azimuth and elevationdial recordings together with the ability tomeasure the tracking-error. For measure­ments made with improved Theodolitemeasuring equipment, the combined valuesvary typically in the range: OA, (TE, "",5 sec­onds to 10 seconds of arc.

The random error values for Fixed Cameradata vary depending upon image quality andthe measuring instrument. Typical equipmentfor measuring Fixed Camera film are thehigh-speed film reader and the precisioncomparator.

Table 1 summarizes the results of extensive

MISSILE TRAJECTORY MEASUREMENT 551

TABLE I

Measured on Measured onQuality of image film-reader com/Jarator

<Ix (microns) <Iy (microns) <Ix (microns) <Iy (microns)

poor (nose of missile in poor 40-60 12-25 30-45 14-20lighting)

average (nose of missile under 20-30 10-15 17-22 6-9average condition)

excellent (missile-borne light- 10-15 10-15 -1-7 -1-7source)

arc,4. for V-angle attitude data: 30 minutes

of arc.

These standard deviations of the reducedparameters are commonly referred to as theGDOP (the Geometric Dilution of Precision)

The unit standard deviation (T IS normallyconsidered to be:

1. for Ballistic Camera data: 3 microns,2. for Fixed Camera data: 10 microns,3. for Cine-theodolite data: 10 seconds of

(1)

(1.2)

<I2

Wj =_"Uj2

propagated through the adjustment bycovariance analysis techniques to obtain thestandard deviation of the reduced trajectorydata. Appropriate weighting factors for thevarious measurements are obtained from theinverse square ratio of the respective measur­ing errors as related by an arbitrary constantof proportionality referred to as the unitvariance (T2. (As a rule (T2 is chosen to repre­sent the measuring variance.)

Thus the weight of any observation j iscomputed from:

I t has been shown by Brown (3), that therelative co-variance matrix of the parametersdetermined by a least squares adj ustment isgiven by the inverse of the coefficient matrixof the normal equations. It follows that thematrix N-l multiplied by the unit variance(T2 would represent the co-variance matrix ofthe red uced parameters. In particular, fortriangulation, the N-l matrix is of the order3 X3, since the X, Y, Z, coordinates of thetriangulated position point are the onlyunknowns. If nkk-1 denotes the diagonalelement of N-l corresponding to the kthunknown element of position, denoted hereby Uk the standard deviation of Uk would begiven by:

investigation by time series analysis.In Table 1, the x coordinate is assumed to

coincide with the general direction of themissile travel. For an extremely poor imagesuch as the leading edge of the flame, therandom measuring-error will be correspond­ingly high.

The random error associated with theV-angle measurements for determining atti­tude data has been investigated by Reed (2).His experience indicated that the measuring­error ranged from .1 to 5 degrees dependingupon the missile-image; and that averageimages had a measuring-error of .5 degrees.The random error associated with the meas­urements on symmetric images for BallisticAttitude data approximate 5 to 10 microns inthe x and y coordinates.

The photogrammetric determination toobtain the most probable trajectory data ofthe object under investigation is performed ina rectangular Cartesian coordinate system.The reduction usually considers the coordi­nate positions of the camera stations, theorientations of the camera optical axes, andthe measured plate coordinates of the imagesfor Ballistic Camera and Fixed Cameraposition data; and the direction of the rays inspace to the object being tracked for Cine­theodolite position data. The measuredV-angles or x, y, plate coordinates are usedfor attitude data. The mathematics for thesereductions have been widely distributed inreferences (3 through 8), and will not bepresented herein. The reduction proceduresutilize rigorous least squares adjustmenttechniques and a redundancy of data. Forexample: a 2-station reduction for positiondata already contains an overdeterminationwith one degree of freedom, and usually 3 sta­tions or more are utilized in obtaining trajec­tory position data. The adjustment proce­dures for attitude reduction are analogous tothat described above.

The random error of the measurements are

552 PHOTOGRAMMETRIC ENGINEERING

(1.3)

and are a function of the geometry of thephotogrammetric model, and of the randomerror of the fundamental measurements. Forposition data, the co-variance matrix definesthe confidence or error ellipsoid associatedwith the computed most probable position inspace. A direct relationshi p exists betweenthe GDOP and the random error of theobservation. Thus, if the measuring error ischanged by a given factor, the GDOP is alsochanged by the same factor, provided thatother considerations of the photogrammetricmodel remain constant.

The nondiagonal elements of N-l deter­mine the correlations existing between theerrors in the computed elements. The coeffi­cient of correlation for the gth and kth ele­ments is:

ngk-1

pgk = y'ngg 1 nkk 1

where nuk-1 denotes the element in the gthrow and kth column of N-l. The coefficient ofcorrelation provides an indication of thedegree of mutual variation in a pair of statis­tical variables.

SYSTEMATIC ERRORS

The source and influence of systematicerrors affecting a photogrammetric model arewell known to the qualified optical engineerand photogrammetrist, and steps are usuallytaken to eliminate the influence of these errorsin the final product of most photogram­metric operations. A comprehensive treat­ment of the systematic errors has beenpresented by Brown (3) in a manner ame­nable to analytical photogrammetric tech­niques.

Two groups of systematic errors are con­sidered, the first group being those ordinarilycorrected in the mathematical reduction.This first group may be further subdividedinto those errors corrected directly and thosecorrected indirectly. Of the systematic errorscorrected directly, radial-lens distortion andatmospheric refraction have the greatesteffect upon the photogrammetric model, andelaborate reductions must be performed toeffect their compensation. Another majorsource of systematic error is the comparatoror film reader used for making the observa­tional measurements. Not only must thecomparator be a precision instrument inproper adjustment, but it must also beaccurately calibrated to remove the effects ofresidual systematic errors. The major com­parator errors to be considered are: non­perpendicularity of the comparator axes or

ways, weave of the ways, periodic errors, andsecular errors. Other systematic errors di­rectly correctable in the data reductionprocess are those introduced by the peculiar­ities of the particular type of instrumentation.For example, certain cameras utilize a filmmoving across either a plane surface or adrum. The shutter may also be of the movingfocal-plane type which must of necessity belocated a finite distance from the actualfocal-plane of the camera. Time series photog­raphy also introduces its own peculiaritiesinto the data reduction process.

The other class of systematic errors arethose indirectly corrected and whose com­pensation comes about as a by-product of thereduction. One such error is the constantmeasuring error of the operator. Selection ofall measuring points to be of the same size andquality will eliminate this effect; but isobtainable only under rare and specialcircumstances, as with Ballistic Cameraplates of light flashes against a star back­ground. For normal conditions, the average ofa set of direct and reversed measurements(reverse measurements are made with theplate rotated 1800 from the original measur­ing position) will eliminate the operator bias.Scale errors are another class of indirectlycorrectable systematic errors. These includeany errors which are linear in function andwhich can be corrected by a constant multi­plication factor. Errors of this type include:dimensional changes in the emulsion and/orphotographic base due to temperature andhumidity fluctuations; the linear componentof radial lens distortion; and linear compara­tor errors. The combined effect of all sucherrors can be compensated by use of acalibrated principal-distance. For differentialchanges in dimension, it might be necessaryto utilize two calibrated principal-distances,one for the x coordinate and one for the ycoordinate.

The second group of systematic errors arethose not normally corrected in the datareduction process. Control of such systematicerrors is mostly a prevention problem, anddepends upon the establishment of suitablefacilities for the camera, upon sufficientlyrefined instrumentation, upon proper photo­graphic exposures, and upon reasonable care inthe field work. A listing of systematic errors ofthis type would incl ude: survey errors for thecamera stations and control points, unflatnessof the photographic plate, timing errors,identification bias, and camera instabilitybetween orientation and trajectory data re­cording. The causes of camera instability are

MISSILE TRAJECTORY MEASUREMENT 553

The least squares adjustment must con­sider the total number of independent condi­tion equations only. Consideration of non­independent condition equations will lead tosingularity of the determinant of the normai

The mllllmum number of observations 110

required to solve the existing condition canhe determined from a consideration of thecondition equations of the adjustment. Theminimum number of observations are thoserequired to solve the existing conditionprovided that no unknown parameters existin the system.

The total number of independent conditionequations rno arising from the system is equalto the degrees of freedom plus the number ofparameters p:

eliminate this major disadvantage of Inter­section Photogrammetry. In Stereo-Photo­grammetry, the camera orientations anddesired trajectory computations are carriedsimultaneously in the adjustment, and thecomputed orientations are forced to fit withinthe photogrammetric model. The computa­tions for such adjustments and the associatederror propagations are more complicated thanfor Intersection Photogrammetry, and havebeen derived by Brown (9). A compromisetechnique can be utilized in which only aminimum of selected trajectory determina­tions are carried in the Stereo solution. Themass of the trajectory data can then bedetermined by the intersection method usingthe orientations resulting from the initialStereo solution. In this manner, the orienta­tions must fit the photogrammetric model.

The least squares adjustment consists ofdetermining the vector of measuring resid­uals, and the vector of parameter corrections,which satisfy the given condition equationswhile minimizing the weighted quadraticform of the residuals. The vector of residualsis computed from the elements of the leastsquares adjustment after the final correctionshave been applied to the unknown param­eters. The mean square error (the estimatedunit variance) arising from the adjustment isobtained by dividing the weighted quadraticform of the residuals s by the statisticaldegrees of freedom f involved in the adjust­ment.

The degrees of freedom f for the system isequal to the total number of observations nminus the minimum no required for solution:

many, but include: physical disturbance(accidental and deliberate) by the operatingpersonnel; and environmental disturbancesby the effects of wind, and temperaturechanges, and the effects of solar radiation.

The magnitudes of the systematic errorsfor the different types of optical instrumenta­tion data are given below. The comments andvalues are obtained from reference (1).

"With good data and a star calibration,Ballistic Camera systematic error is signifi­cantly less than the standard deviation of therandom error and probably ranges from 1 to 2microns. This assertion is based upon athorough system analysis and is supported bythe residuals resulting from orientation cali­brations and from triangulations. With dataof poor quality (over-sized, unsymmetricimages) biases of the order of 5 to 10 microns(and occasionally even more) have beenindicated by triangulation residuals. Signifi­cant accuracy dilution has also been found toarise from camera instability caused by windand temperature change."

"The systematic error in reduced Cine­theodolite Postion data has been determinedin practice to be of the order 30 to 40 secondsof arc. Proper calibration procedures whichare in process of being instituted at theAtlantic Missile Range will reduce this biasto the values 5 to 10 seconds of arc."

"The systematic error in the Fixed Camerainstrumentation system depends largely uponthe method for determining the orientation ofthe individual cameras used in the triangula­tion reduction. With typical target boardorientation (three to six targets), systematicerror is equivalent to 20 to 40 microns. Thesefigures may be doubled if target board dis­tribution is especially poor. Bias in a dialorientation generally ranges from 180 to 240seconds. Star orientation of the Fixed Camerawill result in a lower bias than from the targetboard orientation, and proper calibration andoperating procedures would reduce the bias ofthe dial orientations."

The concept of Intersection Photogram­metry, as stated earlier, considers that theorientations of the cameras are error-free.Obviously, this is not the case in actualpractice, the error of the orientation varyingfrom less than the 3 micron random error ofBallistic Camera, to the excessive amount forthe uncalibrated Fixed Camera dial orienta­tion. Thus the major source of bias in thereduced trajectory-measurements are causedby the influence of the incorrect orientationsof the respective cameras. The utilization ofStereo-Photogrammetric procedures will

= n - no.

tno =f+ p.

(1.4)

(1.5)

554 PHOTOGRAMMETRIC ENGINEERING

equation coefficient matrix, and thus preventa solution to the problem.

The unit variance 1I02 estimated from theadjustment therefore becomes:

The Bias Factor is a convenient tool forcomparing systematic errors between in­strumentation systems, and between differenttrajectory computations from the same typeof instru mentation. The Bias Factor is also

useful in propagating the systematic errorthrough derivative data computations.

In order to obtain full benefit of the leastsquares adjustment as regards the associatederror propagation, it is necessary that aredundancy of data be utilized for an over­determined solution. A unique solution carry­ing only the minimum number of observa­tions required for solution would lead to zerodegrees of freedom, and make it impossible toestimate the degree of systematic error in thereduced data.

REFERENCES

1. "The Accuracy of AMR Instrumentation,"RCA Quality Analysis TN No. 31, 15 December1958.

2. Reed, F. c., "Three Least Squares AttitudeReductions," Naval Ordnance Test StationTech. Memo 293, November 1953.

3. Brown, D. c., "A Treatment of AnalyticalPhotogrammetry with Emphasis on BallisticCamera Application," RCA Data ReductionTechnical Report No. 39, 20 August 1957.

4. Rosenfield, G. H., "Fixed Camera Data Re­duction Analysis Procedures," RCA Data Re­duction Technical Report No. 41, 1 February1958.

5. Davis, R. G., "Techniques for the StatisticalAnalysis of Cine-theodolite Data," avalOrdnance Test Station Report No. 269(NAVORD 1299),22 March 1951.

6. Rosenfield, G. H., "A General Least SquaresAdjustment for Computing Ballistic MissileAttitude," RCA Data Processing TechnicalReport No. 53, September 1959.

7. Rosenfield, G. H., "A Least Squares Pitch andYaw Reduction for Narrow Angle TrackingCameras," RCA Data Processing TechnicalReport No. 57, December 1959. (To be pub­lished)

8. Rosenfield, G. H., "A General Least SquaresPitch and Yaw Reduction for Wide Angle FixedCamera," RCA Data Processing TechnicalReport No. 59, February 1960. (To be pub­lished)

9. Brown, D. c., "A Solution to the General Prob­lem of Multiple Station Analytical Stereotri­angulation," RCA Data Reduction TechnicalReport No. 43, 10 February 1958.

(1.6)

(1.8)

(1.7)

0"0{3, = _.

0"

Comparison of this with an accurate prede­termined value of 1I2 based on the randomerror in the observations, provides an indi­cation of the goodness of the adjustment, andin particular, of the presence of a significantdegree of systematic error. This comparisonmay be made quantitative by utilization ofthe fact that the weighted quadratic form ofthe resid uals di vided by the uni t variance hasa chi square x2 distribution with f degrees offreedom.

Hence, if one computes:

the probability of obtaining a value of X2 asgreat as x02 with f degrees of freedom can beobtained from a table of the cumulative chisquare distribution. If this probability isexcessively small, the presence of a significantdegree of systematic error would be indicated.

If one is interested in the mean error (theestimated standard deviation) arising fromthe adjustment of the kth unknown param­eter, one should substitute IIO obtained from(1.6) for II in equation (1.2).

The ratio of the mean error of the adjust­ment to the unit error results in the quantityknown as the Bias Factor: