Accuracy of speed trials on the measured mile

(Hydrographic News Letter)

ACCURACY OF SPEED TRIALS ON THE

IASURED MILE

by

J.Th.

Verstelle; Hydrographic Office Royal Netherlands Navy

April

_1958.

S

U M M A

R Y

Speed over the ground S is determined from observations of time interval between perpendicular crossing of two parallel lines of beacons, usually exactly one nautical mile apart.

The cmbined effect of observational errors is investigated. It is shown that - contrary to a widely held opinion - these effects are not negligea'ble.

The required speed through the water V is determined by the usual method ôf taking the'means of means' of at least ₄ determinations cf S on opposite courses. The reliability of this method of elimination of drift is not a subject 'f discussion in this News Letter it is however shown that the standard error in V is considerably smaller than in each of the 4 determina-tions of S.

Trials in daytime are to be preferred, because night trials are likely to be considerably less accuratee

Other methods of speed trials are disdussed. in News Letter no ₂₇

1958.

NOTATIONS

S = hourly speed rver the ground. )

same unit as s V = hourly speed through the water

= time interval in seconds between perpendicular crossing of two parallel lines of beacons.

s = length f measured mile = perpendicular distance between the ex-tensions of two lines cf beacons; when S and V are expressed in (nautical) miles, s = 1.

Note: In this News Letter the unit used in the computations is the meter. The final value of V is always converted into nautical miles, the British mile being

1853

m and the continental mile eoualling

1852 m.

COMPUTATION OF S AND V.

6OO

S-

xs

V is computed by the method of determining the so-called 'means of means'

of S (at least ₄ runs, alternatively in opposite directions). Example: S

38003

o

i

43578

O393

40592

40618

rn/h _{21.93 naut.mi./hour.} S

40893

40643

S4 44578

The formula for the means of means (4 runs) reads: S

_i

+ 3E + 3S + S

2 3 4

De!U

NIEUWSBRIEF HYDROGRAFIE No.

28

(1)

SYSTEMATIC ERRORS.

Any systematic erro in the perpendicular distance s between the two extensions of the lines of beacons (the 'mile' being too short or too

long) has a times as large efiect n S and on V.

As the user has no posible means of control, in actual practice a measured mile is always assumed to be correct; systematic errors therefore will not be discussed in this News Letter.

The stopwatch used for measuring At should be well calibrated and a rate exceeding 0.2 seconds in, say 10 minutes cannot be tolerated.

EFFECT OF RANDOM ERRORS.

From differentation of formula (1):

dS 3600 ds

3600

d At

t

_At

s and At are non-orrelated cuantities, hence:

2

(3600 )2

+ 6OO S )2 2 ₍ m5

---

At

mit L Differentiating formula (2):

dV =d.S +cL +

dB +dS

8 8 5

84

2 1 2

92

92

12

m 1fl5 + m, + ₊ m5 2 3 4

21

2

12

12

12

tm + +

+rn

i

2 3 4

ifl

these formulae, m = standard error.

ESTIMATION OF MAGNITUDE OF RANDOM OBSERVATIONAL ERRORS.

5.1. The r'bserver makes a certain error in his estimate of the exact

moment of being on the extension of the line of beacons, or in other words in the realisation of this line at sea.

The errer is dependent on the 'sensitivity' of the line and a measure for this sensitivity is the smallest observable angle 6 (fig. 1). From a paper presented by the author to the 1955

'International Conference on Lighthouses and other Aids to Navigation' it follows that - usin binoculars of about 6 times magnification - in day time the smallest observ'able angle S is of the order of 10 seconds of arc X), provided:

a) beacons of suitable design;

) good visiility;

e) an experienced observer.

In the above mentioned paper the following formula is derived f,r the distance p (fig. 1):

p d tang. 6( - I) . . .

2

X) At night this angle is considerably larger, dependant on the vertical angle between the two beacon lights; 30" seems to be

a fair estimate (using binoculars), which would increase p by a factor

_3.

(4)

3

fig. i

see figure 1

As sirning:

b = 400 mT

₍₄₃₇

yards)

d - b = 2

naut.mi.(Er.)

= 3706

m, hence d

4106 ni.

O = 10"

it follows from

₍₅₎

that:

= 1.9 in..

Assuming:

b =

400 ni.

d - b

_{= 3}

naut.mi.

_{= 5559 ni,}

hence d

_{= 5959 na}

8 _10"

it follows that: = 4.1 ni.

Conclusion: p increases rapidly with increasing distance to the shore.

Note: 1. For shorter distance b between beacons, p increases rapidly.

2.

Only a very careful observer can detect an opening angle of

10".

The realisation at sea of each of the two extensions therefore must 1e expected to be uncertain to an amount p and consecuently p \J' is a

fair estimate of the standard error in the distance s. Conseuently, in the two assumptions made above, we find:

= 1.9 x 1.4 = 2.66

m ) 2 miles ni

4.1 x 1.4 = 5.74

m.) _miles 2 from n from ni

=7m.

) ni

=33m.

) s shore s shore

5.2

The interval of time ¿t is determihed by means of a stopwatch. Any observer has a personal error, being in fact a reaction time in pressing the knob f the stopwatch at the moment when the beacons are bserved to be in line, which - strange as it may seem - may be either positive or negative. For any particular observer, part

f his reaction time is constant and onsecjuently does not affect the difference in time t. The reaction time however partly also is of a randomly variable character and elaborate trials have shown that this random error for an exerience observer is of the order of + or -

_0.4

seconds in Lt.

= 0.4

sec.

2

= 16

x l0 sec.

6.

EXAMPLES.

6.1

Distance to shore 2 naut.miles. S = 18 miles(br.)

_{= 35354}

m./h.

b = 400 ni.;

6= 10"; it =

200

seconds.

Substituting the standard errors of section

_5.1

and

5.2

in formula (3):

= 2268 + 4450 = 6718

niS = 82

m./h. =

0.044

naut.mi./h.

= 123

m./h. = 0.066 naut.mia/h.

33554

0037% of S

4732

= 69 m./h. =

_0.037

naut.mi./h.

69

0.21 of V.

333.54

Note: It should be realised that actual conditions rften are less favourable than the assumptions made here, i.e. h may be shorter and 6 larger, especially at night.

7.

CONCLUSIONS.

i. Observations must be made with great care and every possible

pre-caution has to be taken to keeo errors as small as rossible. Accuracy decreases rapidly with increasing distance of trial runs

to the shore.

Although no general figure can be given as to accuracy to be expec-ted, even for an experienced observer and for fairly 'sensitive' lines of beacons, the effect of unavoidable observational errors in steed trials on the measured mile is not negligeable.

Trials during daytime must he preferred, as night trials are likely to be considerabay less accurate; a decrease by

a factr

₃ might be expected.

8. CONSULTD LITER?TURE.

Paper no.

_443,

_{International Conference on Lighthuses and other} aids to Navigation, Scheveningen 1955:

J.Th. Verstelle; Lateral sensitivity of leading lights(ranges) Netherlands Hydroraphic Office News Letter no. _{27 1958:} J.Th. Verstelle; Ships' acceptance trials.

's-Gravenhage, 25 april

_1958.

Ge z ien:

De Chef der Hydrgrafie, = 2099

46

mi/h. = 0.025 naut.mi./h. _0.14% of V.

-

333.54

6.2 Distance to shore ₃ naut. miles.

De ca\ ograaf-hoofdar,

\\\\\.

S 18 miles(Br.) =

_{33354 m./h.}

b

= 400 md;

0= 10" = 200 sec.

m = 10692 +

₄₄₅₀

= 15142

Th.IK.brn y n Asbeck

_{J.Th. Vero11e.}

Dis tributie: Bibliotheek CHYD. Sous-Chef Hydrografie. Cartografen Hydr3grafie. Cdtn. HDG en ZFK. Hr.Ms. tSnefliusU Hr.Ms. "Luymes'. Kltz.Ir. W. Langeraar. Ltz. SD i J.E. de Wit. Ltz SD i R.van den Oever.)

Ltz. SD i H.van Aalderen. _{) persoonhijk}

Ltz i H.K. den Daas. Ltz i J.M. Schalkwijk. Ltz 2 c.c. J.C. Kreffero Hfd.Bur. Wetensch.Onderzoek (2) Staf Smaldeel (2). Navgis. G-ezaghebber "Dewakem1ûar"

Bibliotheek Kementerian Pelajaran.

c-ezaghebber "Birnasakti".

Hr. J.P.J. de C-root (K.P.H. persoonlijx). Hr. J. van Roon.

Marid

₍₃

ex.)

P.H. I.

Instituut voor Scheepvaart & Luchtvaart (i).