《航海学》课程参考文献（地文资料）CHAPTER 23 NAVIGATIONAL ERRORS

CHAPTER 23NAVIGATIONALERRORSDEFININGNAVIGATIONALERRORS2300.Introductionthe plotting ofa reciprocal bearingA standard is avalueor quantity established by cus-Navigation is an increasingly exact science.Electronictom, agreement, or authority as a basis for comparisonpositioning systems give the navigator a greater certaintyFrequently,a standard is chosen as a model which approxi-than ever that his position is correct.However, the navigatormatesa meanoraveragecondition.However,thedistinctionmakescertainassumptionswhichwouldbeunacceptableinbetween the standard value and the actual value at any timepurelyscientificwork.should not beforgotten.Thus, a standard atmosphere hasFor example,when the navigator uses his latitudegrad-been established in whichthetemperature,pressure,anduations as a mile scale to computeagreat-circle course anddensity are precisely specifiedfor each altitude.Actual con-distance, heneglectstheflattening of the earthat thepoles.ditions, however,are generally differentfrom those definedWhenthenavigatorplots avisual bearing ona Mercatorby the standard atmosphere.Similarly,the values for dipchart, he uses a rhumb line to represent a great circle.Whengiven in the almanacs are considered standard by those whoheplots a celestial lineofposition,hesubstitutes a rhumbusethem, butactual dipmaybeappreciablydifferentfromlinefor a small circle.When he interpolates in sight reduc-thattabulatedtionorlatticetables,heassumesalinear(constant-rateAccuracy is the degree of conformance with the correctchangebetweentabulatedvalues.Alloftheseassumptionsvalue,while precision is a measure of refinement ofa valueintroduceerrorsThus, an altitudedetermined by marine sextant mightbe statedThere are so manyapproximations innavigation thatto the nearest 0.1', and yet be accurate only to the nearest 1.0'ifthere is a natural tendency for some of them to cancel oth-thehorizon is indistinct.ers.However, if the various small errors in a particular fixall have the same sign,the error might be significant.The2302.Systematic And RandomErrorsnavigator must recognize the limitations of his positioningsystems and understand the sources ofposition errorSystematic errors are those which follow some ruleby which they can be predicted.Random errors, on the2301.Definitionsotherhand,areunpredictable.Thelaws ofprobabilitygov-emrandomerrorsThe following definitions applyto thediscussions ofthis chapter:Ifanavigatortakes severalmeasurements thatare subjecttoError is the difference between a specific value and therandom error and graphs the results, the error values would becorrectorstandard value.Asused here,itdoesnot includemis-normallydistributedaroundamean,oraverage,value.Supposefor example,that a navigator takes 500 celestial observationstakes, but is related to lack of perfection. Thus, an altitudedeterminedbymarinesextant is correctedfora standardamountTable2302shows thefrequencyofeacherrorin themeasure-ofrefraction,but if the actual refraction at the timeofobserva-ment, and Figure 2302 shows a plot ofthese errors.The curve'stion varies from the standard, the valuetakenfrom thetable is inheight at anypointrepresentsthe percentageofobservations thaterrorbythedifferencebetweenstandard and actual refraction.can be expected to have the error indicated at that point. TheThis error will be compounded with others in the observed alti-probability of any similar observation having any given error istude.Similarly.depthdeterminedbyechosounderisinerrortheproportionofthenumberofobservationshavingthiserrortoamong other things, by the difference between the actual speedthetotal number of observations.Thus, the probability of an obofsound waves inthewaterandthespeedusedforcalibrationofservationhavinganerrorof-3'is:the instrument.This chapter is concerned primarily with the de-viation from standard values.Corrections can be applied for4010.08(8%)standard values oferror.It is the deviation from standard,as well500-12.5as mistakes, that produce inaccurate results in navigationA mistake is a blunder, such as an incorrect reading ofAn importantcharacteristic ofaprobabilitydistributionan instrument, the takingof a wrong value from a table,or351

351 CHAPTER 23 NAVIGATIONAL ERRORS DEFINING NAVIGATIONAL ERRORS 2300. Introduction Navigation is an increasingly exact science. Electronic positioning systems give the navigator a greater certainty than ever that his position is correct. However, the navigator makes certain assumptions which would be unacceptable in purely scientific work. For example, when the navigator uses his latitude grad￾uations as a mile scale to compute a great-circle course and distance, he neglects the flattening of the earth at the poles. When the navigator plots a visual bearing on a Mercator chart, he uses a rhumb line to represent a great circle. When he plots a celestial line of position, he substitutes a rhumb line for a small circle. When he interpolates in sight reduc￾tion or lattice tables, he assumes a linear (constant-rate) change between tabulated values. All of these assumptions introduce errors. There are so many approximations in navigation that there is a natural tendency for some of them to cancel oth￾ers. However, if the various small errors in a particular fix all have the same sign, the error might be significant. The navigator must recognize the limitations of his positioning systems and understand the sources of position error. 2301. Definitions The following definitions apply to the discussions of this chapter: Error is the difference between a specific value and the correct or standard value. As used here, it does not include mis￾takes, but is related to lack of perfection. Thus, an altitude determined by marine sextant is corrected for a standard amount of refraction, but if the actual refraction at the time of observa￾tion varies from the standard, the value taken from the table is in error by the difference between standard and actual refraction. This error will be compounded with others in the observed alti￾tude. Similarly, depth determined by echo sounder is in error, among other things, by the difference between the actual speed of sound waves in the water and the speed used for calibration of the instrument. This chapter is concerned primarily with the de￾viation from standard values. Corrections can be applied for standard values of error. It is the deviation from standard, as well as mistakes, that produce inaccurate results in navigation. A mistake is a blunder, such as an incorrect reading of an instrument, the taking of a wrong value from a table, or the plotting of a reciprocal bearing. A standard is a value or quantity established by cus￾tom, agreement, or authority as a basis for comparison. Frequently, a standard is chosen as a model which approxi￾mates a mean or average condition. However, the distinction between the standard value and the actual value at any time should not be forgotten. Thus, a standard atmosphere has been established in which the temperature, pressure, and density are precisely specified for each altitude. Actual con￾ditions, however, are generally different from those defined by the standard atmosphere. Similarly, the values for dip given in the almanacs are considered standard by those who use them, but actual dip may be appreciably different from that tabulated. Accuracy is the degree of conformance with the correct value, while precision is a measure of refinement of a value. Thus, an altitude determined by marine sextant might be stated to the nearest 0.1’, and yet be accurate only to the nearest 1.0’ if the horizon is indistinct. 2302. Systematic And Random Errors Systematic errors are those which follow some rule by which they can be predicted. Random errors, on the other hand, are unpredictable. The laws of probability gov￾ern random errors. If a navigator takes several measurements that are subject to random error and graphs the results, the error values would be normally distributed around a mean, or average, value. Suppose, for example, that a navigator takes 500 celestial observations. Table 2302 shows the frequency of each error in the measure￾ment, and Figure 2302 shows a plot of these errors. The curve’s height at any point represents the percentage of observations that can be expected to have the error indicated at that point. The probability of any similar observation having any given error is the proportion of the number of observations having this error to the total number of observations. Thus, the probability of an ob￾servation having an error of -3' is: An important characteristic of a probability distribution 40 500 - 1 12.5 = -= 0.08 8%( )

352NAVIGATIONALERRORSsymmetrical about the line representing zero error. ThisErrorNo. of obs.Percent of obs.means that in the normalized plot every positive error is-10°00.0matched by a negativeerror of the same magnitude.The av--9'1I0.2erageof all readings iszero.Increasingthe number of-8'240. 4 readings increases the probability that the erors will fit the-7'0.8normalizedcurve-6'91.8When both systematic and random errors are present in-5'173.4-4'285.6aprocess,increasingthenumberofreadingsdecreasesthere--3'408.0sidual random error but does not decreasethe systematic-2'5310.6error.Thus,if, for example, a number ofphase-difference-16312.6readings are made at a fixed point, the average ofall the read-06613.2ings should be a good approximation ofthe true value ifthere+1'6312.6is no systematicerror.But increasing the number of readings+ 2'5310.6will notcorrecta systematicerror.Ifaconstanterror is com-+ 3'408.0bined with a normal random error, the error curve will have+4'285.6the correct shape but will be offset from the zero value+ 5'173. 4+6'91.82303.Navigation SystemAccuracy+7'42100.8+8°0.4+ 9°0.2Ina navigationsystem,predictabilityisthemeasuref+10°0.0the accuracy with which the system can define the positionin terms of geographical coordinates; repeatability is the0500100.0measureofthe accuracy with which the system permits theTable2302.Normal distributionofrandom errorsuser to return to a position as defined only in terms of thecoordinates peculiar to that system.Predictable accuracy,is the standard deviation.For a normal error curve, squaretherefore,is the accuracy ofpositioning with respect togeo-eacherror, sum the squares,and dividethe sumby onelessgraphical coordinates; repeatable accuracy is the accuracythan thetotal number of measurements.Finally,takethewith which the user can return to a position whose coordi-square root ofthe quotient. In the illustration, the standardnateshavebeenmeasuredpreviouslywiththesamesystem.deviation is:Forexample,thedistancespecifiedfortherepeatableaccu-racy of a system, such as Loran C,is the distance betweentwo Loran C positions established using the same stations4474/8.966=2.99andtime-differencereadingsatdifferenttimes.Thecorre-499lation between thegeographical coordinatesandthe systemcoordinatesmayormaynotbeknownOne standard deviation on either side of themean de-Relative accuracy is the accuracy with which a user canfines the area under the probability curve in which lie 67determine his position relative to another user ofthe same nav-percent of all errors. Two standard deviations encompassigation system, at the same time. Hence, a system with high95 percent of all errors, and three standard deviations en-relative accuracy provides good rendezvous capability for thecompass 99 percent of all errors.users ofthesystem.Thecorrelationbetween thegeographicalcoordinates and the system coordinates is not relevantThe normalized curve of any type of random error is2304.MostProbablePositionSome navigators have been led by simplified defini-RORtions and explanationstoconcludethatthe line of positionis almost infallible and that a good fix has very little error.Amorerealisticconcept isthatofthemostprobablepo-sition (MPP).This concept whichrecognizes the probabilityoferror in all navigational information anddeterminesposi-tion by an evaluationofall available informationSuppose avessel wereto startfrom a completely accu-TERRORrateposition andproceed ondead reckoning.If courseandspeed over the bottom were ofequal accuracy,the uncertain-Figure2302.Normal curveofrandomerrorwith50percentty of dead reckoning positions would increase equally in allofarea shaded.Limitsofshadedarea indicateprobableerror.directions, witheitherdistance or elapsedtime (forany one

352 NAVIGATIONAL ERRORS is the standard deviation. For a normal error curve, square each error, sum the squares, and divide the sum by one less than the total number of measurements. Finally, take the square root of the quotient. In the illustration, the standard deviation is: One standard deviation on either side of the mean de￾fines the area under the probability curve in which lie 67 percent of all errors. Two standard deviations encompass 95 percent of all errors, and three standard deviations en￾compass 99 percent of all errors. The normalized curve of any type of random error is symmetrical about the line representing zero error. This means that in the normalized plot every positive error is matched by a negative error of the same magnitude. The av￾erage of all readings is zero. Increasing the number of readings increases the probability that the errors will fit the normalized curve. When both systematic and random errors are present in a process, increasing the number of readings decreases the re￾sidual random error but does not decrease the systematic error. Thus, if, for example, a number of phase-difference readings are made at a fixed point, the average of all the read￾ings should be a good approximation of the true value if there is no systematic error. But increasing the number of readings will not correct a systematic error. If a constant error is com￾bined with a normal random error, the error curve will have the correct shape but will be offset from the zero value. 2303. Navigation System Accuracy In a navigation system, predictability is the measure of the accuracy with which the system can define the position in terms of geographical coordinates; repeatability is the measure of the accuracy with which the system permits the user to return to a position as defined only in terms of the coordinates peculiar to that system. Predictable accuracy, therefore, is the accuracy of positioning with respect to geo￾graphical coordinates; repeatable accuracy is the accuracy with which the user can return to a position whose coordi￾nates have been measured previously with the same system. For example, the distance specified for the repeatable accu￾racy of a system, such as Loran C, is the distance between two Loran C positions established using the same stations and time-difference readings at different times. The corre￾lation between the geographical coordinates and the system coordinates may or may not be known. Relative accuracy is the accuracy with which a user can determine his position relative to another user of the same nav￾igation system, at the same time. Hence, a system with high relative accuracy provides good rendezvous capability for the users of the system. The correlation between the geographical coordinates and the system coordinates is not relevant. 2304. Most Probable Position Some navigators have been led by simplified defini￾tions and explanations to conclude that the line of position is almost infallible and that a good fix has very little error. A more realistic concept is that of the most probable po￾sition (MPP).This concept which recognizes the probability of error in all navigational information and determines posi￾tion by an evaluation of all available information. Suppose a vessel were to start from a completely accu￾rate position and proceed on dead reckoning. If course and speed over the bottom were of equal accuracy, the uncertain￾ty of dead reckoning positions would increase equally in all directions, with either distance or elapsed time (for any one Error No. of obs. Percent of obs. - 10′ 0 0. 0 - 9′ 1 0. 2 - 8′ 2 0. 4 - 7′ 4 0. 8 - 6′ 9 1. 8 - 5′ 17 3. 4 - 4′ 28 5. 6 - 3′ 40 8. 0 - 2′ 53 10. 6 - 1′ 63 12. 6 0 66 13. 2 + 1′ 63 12. 6 + 2′ 53 10. 6 + 3′ 40 8. 0 + 4′ 28 5. 6 + 5′ 17 3. 4 + 6′ 9 1. 8 + 7′ 4 0. 8 + 8′ 2 0. 4 + 9′ 1 0. 2 +10′ 0 0. 0 0 500 100. 0 Table 2302. Normal distribution of random errors. Figure 2302. Normal curve of random error with 50 percent of area shaded. Limits of shaded area indicate probable error. 4474 499 - 8.966 2.99 = =

353NAVIGATIONALERRORSspeed these would be directly proportional, and therefore ei-equidistant from the sides.If the lines are of unequal error,ther could be used).A circle of uncertainty would growthedistanceofthemostprobablepositionfromeachlineofaround thedead reckoning position as the vessel proceededpositionvaries asafunctionoftheaccuracyof eachLOpIfthenavigatorhad fullknowledgeofthedistributionand na-Systematic errors are treated differently.Generally,theture of the errors of course and speed, and the necessarynavigatortriestodiscovertheerrors and eliminatethem orknowledgeof statistical analysis,he could compute theradi-compensateforthem.Inthecase ofaposition determinedus ofa circle ofuncertainty,using the50percent,95percent,bythree ormore linesof position resultingfromreadingsorotherprobabilities.This technique isknown asfix expan-with constanterror,theerrormightbeeliminated byfindingsion when done graphically. See Chapter 7 for a moreand applying that correction which will bring all linesdetaileddiscussionoffixexpansion.through a common point.In ordinary navigation,statistical computationis notpracticable.However,thenavigator might estimate at anytime the likely error of his dead reckoningor estimated posi-tion.Withpractice,considerable skillinmaking this estimateis possible.Hewould take into account,too,thefactthat thearea ofuncertaintymight better berepresented by an ellipsethan a circle,with the major axis along the course line if theestimated error of the speed were greater than that of thecourse and the minoraxis along the course line ifthe estimat-ed error of the courseweregreater.He would recognize,too,thatthesizeoftheareaofuncertaintywouldnotgrowindi-rect proportion to the distance or elapsed time, becausedisturbingfactors,suchas windandcurrent,could notbeexpected to remain of constant magnitudeand direction,Also,Figure2304b.Ellipseof uncertaintywith linesof positionshe wouldknowthat the startingpoint ofthe dead reckoningofequalprobableerrorscrossingat anobliqueanglemight notbe completelyfreefrom error.The navigator can combine an LOP with either a deadLines of position whichareknown to beof uncertainreckoningorestimatedpositiontodetermineanMPP.De-termining the accuracy ofthe dead reckoning andestimatedaccuracy might better be considered as“bands of position"positions from which an MPP is determined is primarily awithaband withoftwicethepossibleamount of error.In-judgment call by the navigator.SeeFigure2304a.tersecting bands of position define areas of position,It ismost probablethat the vessel is near the center of the areaIfa fix is obtained from two lines of position, the areabut the navigator must realize that he could be anywhereof uncertainty is a circle if the lines are perpendicular andhave equal error.If one is considered moreaccuratethanwithin the area, and navigate accordinglythe other, the area is an ellipse.As shown in Figure 2304b,it is also an ellipse ifthelikelyerror ofeach is equal and the2305.Mistakeslines crossat anobliqueangle.If theerrors are unequal, themajor axis of the ellipse is more nearly in line with the lineTherecognition ofa mistake,as contrasted with anerror,of position having the smaller likely eroris not alwayseasy,sincea mistake may have anymagnitudeIfafix is obtained fromthree ormore lines of positionandmay be either positive ornegative.A large mistake shouldwith a total bearing spread greater 180°, and the error ofbe readily apparent if the navigator is alert and has an under-eachline is normallydistributed and equal to that ofthe oth-standing ofthesizeoferrortobe reasonably expected.Asmallers, themostprobablepositionisthepointwithin thefiguremistake is usually not detected unless the work is checked.If results bytwomethods are compared,such as a deadreckoningposition and a lineof position, exact agreementis unlikely.But,if thediscrepancyis unreasonablylarge,aDRmistake is a logical conclusion.If the99.9percent areas ofthe two results just touch,it is possible that no mistake hasOMPEbeen made. However, the probability of either one havingLine of Positianso great an error is remote if the errors are normal. Theprobabilityof bothhaving99.9percenterrorof oppositesign at the same instant is extremely small.Perhaps a rea-Figure2304a.Amostprobablepositionbased uponadeadsonable standard is that unless the most accurate result lieswithin the 95 percent area of the least accurate result, thereckoningposition and line of position having equalprobable errors.possibilityof a mistake should be investigated

NAVIGATIONAL ERRORS 353 speed these would be directly proportional, and therefore ei￾ther could be used). A circle of uncertainty would grow around the dead reckoning position as the vessel proceeded. If the navigator had full knowledge of the distribution and na￾ture of the errors of course and speed, and the necessary knowledge of statistical analysis, he could compute the radi￾us of a circle of uncertainty, using the 50 percent, 95 percent, or other probabilities. This technique is known as fix expan￾sion when done graphically. See Chapter 7 for a more detailed discussion of fix expansion. In ordinary navigation, statistical computation is not practicable. However, the navigator might estimate at any time the likely error of his dead reckoning or estimated posi￾tion. With practice, considerable skill in making this estimate is possible. He would take into account, too, the fact that the area of uncertainty might better be represented by an ellipse than a circle, with the major axis along the course line if the estimated error of the speed were greater than that of the course and the minor axis along the course line if the estimat￾ed error of the course were greater. He would recognize, too, that the size of the area of uncertainty would not grow in di￾rect proportion to the distance or elapsed time, because disturbing factors, such as wind and current, could not be ex￾pected to remain of constant magnitude and direction. Also, he would know that the starting point of the dead reckoning might not be completely free from error. The navigator can combine an LOP with either a dead reckoning or estimated position to determine an MPP. De￾termining the accuracy of the dead reckoning and estimated positions from which an MPP is determined is primarily a judgment call by the navigator. See Figure 2304a. If a fix is obtained from two lines of position, the area of uncertainty is a circle if the lines are perpendicular and have equal error. If one is considered more accurate than the other, the area is an ellipse. As shown in Figure 2304b, it is also an ellipse if the likely error of each is equal and the lines cross at an oblique angle. If the errors are unequal, the major axis of the ellipse is more nearly in line with the line of position having the smaller likely error. If a fix is obtained from three or more lines of position with a total bearing spread greater 180°, and the error of each line is normally distributed and equal to that of the oth￾ers, the most probable position is the point within the figure equidistant from the sides. If the lines are of unequal error, the distance of the most probable position from each line of position varies as a function of the accuracy of each LOP. Systematic errors are treated differently. Generally, the navigator tries to discover the errors and eliminate them or compensate for them. In the case of a position determined by three or more lines of position resulting from readings with constant error, the error might be eliminated by finding and applying that correction which will bring all lines through a common point. Lines of position which are known to be of uncertain accuracy might better be considered as “bands of position”, with a band with of twice the possible amount of error. In￾tersecting bands of position define areas of position. It is most probable that the vessel is near the center of the area, but the navigator must realize that he could be anywhere within the area, and navigate accordingly. 2305. Mistakes The recognition of a mistake, as contrasted with an error, is not always easy, since a mistake may have any magnitude and may be either positive or negative. A large mistake should be readily apparent if the navigator is alert and has an under￾standing of the size of error to be reasonably expected. A small mistake is usually not detected unless the work is checked. If results by two methods are compared, such as a dead reckoning position and a line of position, exact agreement is unlikely. But, if the discrepancy is unreasonably large, a mistake is a logical conclusion. If the 99.9 percent areas of the two results just touch, it is possible that no mistake has been made. However, the probability of either one having so great an error is remote if the errors are normal. The probability of both having 99.9 percent error of opposite sign at the same instant is extremely small. Perhaps a rea￾sonable standard is that unless the most accurate result lies within the 95 percent area of the least accurate result, the possibility of a mistake should be investigated. Figure 2304a. A most probable position based upon a dead reckoning position and line of position having equal probable errors. Figure 2304b. Ellipse of uncertainty with lines of positions of equal probable errors crossing at an oblique angle

354NAVIGATIONALERRORS2306.Conclusionorrecordsdata,hecanobtainonlyanapproximatepositionHe must understand his systems' limitations and use thisNopractical navigatorneed understand themathemat-understanding to determine the positioning accuracy re-ical theoryof error probability to navigate his ship safely.quired to bring his ship safely into harbor. In making thisdetermination, sound,professional, and conservative judgHowever, hemust understand that his systemsand process-es aresubjectto error.Nomatter howcarefully hemeasuresmentisofparamountimportance

354 NAVIGATIONAL ERRORS 2306. Conclusion No practical navigator need understand the mathemat￾ical theory of error probability to navigate his ship safely. However, he must understand that his systems and process￾es are subject to error. No matter how carefully he measures or records data, he can obtain only an approximate position. He must understand his systems’ limitations and use this understanding to determine the positioning accuracy re￾quired to bring his ship safely into harbor. In making this determination, sound, professional, and conservative judg￾ment is of paramount importance