NNAAVVIIGGAATTIIOONNAALL AALLGGOORRIITTHHMMSS CCeelleessttiiaall FFiixx NNaauuttiiccaall AAllmmaannaacc SSiigghhtt RReedduuccttiioonn aallggoorriitthhmm ffoorr nn LLooPPss © Andrés Ruiz San Sebastián – Donostia 43º 19’N 002ºW http://www.geocities.com/andresruizgonzalez 2 Index Variables...................................................................................................................................... 3 Sight Reduction ........................................................................................................................... 3 Running Fix.................................................................................................................................. 4 Calculated Position ...................................................................................................................... 4 Estimated Position Error .............................................................................................................. 4 Plot .............................................................................................................................................. 5 Lines of Position ........................................................................................................................................................................ 5 Ellipse........................................................................................................................................................................................ 5 Position...................................................................................................................................................................................... 5 Mathematical basis ...................................................................................................................... 5 A1. Algorithms ............................................................................................................................. 7 A2. Examples............................................................................................................................. 10 A3. Software .............................................................................................................................. 12 A4. Source code ........................................................................................................................ 13 A5. References .......................................................................................................................... 15 Abstract This paper describes an analytic method for calculating the position by the observation of celestial bodies as effective alternative to traditional graphic methods used in celestial navigation. The algorithm is totally general, allowing the use of simultaneous sights, or observations taken at different times, including navigating a course at a certain speed. It uses the method of least squares to obtain the most probable position, and through successive iterations makes it possible to reduce the error to approximate the circle of equal altitude by the straight line of position. It also provides the error in the calculation of the position and the Confidence ellipse. © Andrés Ruiz, June 1999 San Sebastián – Donostia 43º 19’N 002ºW http://www.geocities.com/andresruizgonzalez Current version: 200804 http://www.geocities.com/andresruizgonzalez/ n Lop Fix Navigational Algorithms 3 It describes the algorithm officially Hc Calculated or Computed Altitude. adopted by the HM Royal Navy, UK, to Z Azimuth (true) calculate the situation by observations of Measured clockwise around the stars using the sextant. It was also included horizon from 0° to 360°, is the arc of in The Nautical Almanac published annually the horizon between the meridian of by the USA Naval Observatory. a place and the vertical circle passing through a celestial body. It is a robust and effective alternative to the traditional graphical methods to get the p Intercept of a sight position by intersection of lines of position or p = Ho – Hc bisectors. Towards = + / Away = – This algorithm replaces plotting lines of R Course or track position, LoP, for each observation, moving Measured as for azimuth from 0° to Lops if they are not simultaneous and 360°. obtaining of the position, by a calculation that systematizes this process and get the V Speed most probable position based on the method in knots. of least squares. n Number of observations For the simplicity of its calculations, especially in matrix form, it can be used with a calculator or spreadsheet. Sight Reduction The procedure uses the classic Marcq Variables Saint Hilaire method to reduce the sights as the mathematical link between the observer UT1 Universal Time and the celestial body. If you know your Approximately Greenwich Mean Time: GMT estimated latitude and longitude, you can predict the true bearing and the height of the In degrees: object above the horizon. This angle can B Latitude then be compared to your corrected sextant N (+) / S (–) angle to produce a position line. With L Longitude several sights, the method plots a fix through the statistical intersection of these E (+) / W (–) position lines. GHA Greenwich Hour Angle GHA = GHA(Aries) + SHA The following sight reduction formulae are used: W to E from 0° to 360°. SHA Sidereal Hour Angle SHA = 360° - Right Ascension. LHA = GHA + L DEC Declination Hc = asin( sin B sin DEC + cos B cos Dec cos LHA ) N (+) / S (–). LHA Local Hour Angle Bsin Hcsin - DECsin DECsin - Hcsin Bsin W to E from 0° to 360°. Z = acos( ) B cos Hc cos Hc cos B Ho Observed Altitude Is apparent altitude corrected for if( 0 < LHA < 180º ) Z = 360 – Z refraction and if appropriate corrected for parallax and semi- If the local hour angle is less than 180° diameter, [2]. then the azimuth is 360° less the product of the above expression: http://www.geocities.com/andresruizgonzalez/ n Lop Fix Navigational Algorithms 4 n n Running Fix 2 2 C = ∑sin Zi F= ∑ pi An estimate can be made of the position i=1 i=1 at the adopted time of fix. The position at the time of the observations can then be easily G = A C - B2 calculated provided that the course and speed has been constant. Using speed (V) As a checking: A+C = n in knots and the track (R) the equations are: An improved estimate of the fix is given by: t = UT1 - UT1 observation fix C D - B E B = B + dB = B + v e e G V t B B = B e + cos R A E - B D 60 Lv = L e + dL = Le + V t sin R G cos Be L L = L e + 60 cos Be The distance between the assumed position and the improved estimated L and B are the estimated longitude and e e position in nautical miles is: latitude at the time of fix and t is the time interval in hours. Do = 60 dL2 cos 2 Be + dB 2 Calculated Position The method substitutes the DR position The position lines for one or more with the calculated fix in order to converge observations can be plotted using the on a solution. Do < 20 nm. azimuth Z and the intercept p: BeB = Bv p = Ho - Hc Le = Lv • If p is positive the position line is drawn along the azimuth. Estimated Position Error • If p is negative, the position line is away If three or more position lines are from the assumed position by adding obtained an estimate of the error in position 180° to the azimuth, (Z+180º). may be calculated. In general as the number of observation increases the error in the Provided that there are no observation estimated position decreased. errors, the observer should be close to, or The standard deviation of the estimated along the position line. Two or more position position, in nautical miles is: lines are required to determine a fix. S The procedure uses the -Method of Least σ = 60 n − 2 Squares- to determine a fix from up to three observations. S = F - D dB - E dL COS Be If pi and Zi, (i=1,n), are the intercept and And the standard deviation in latitude and azimuth for the i observation: longitude is: C n n σB = σ 2 G A = ∑cos Zi D= ∑ pi cos Zi i=1 i=1 n n A σL = σ B = ∑cosZZi ⋅ sin i E= ∑ pi sin Zi G i=1 i=1 http://www.geocities.com/andresruizgonzalez/ n Lop Fix Navigational Algorithms 5 The Confidence Ellipse of axes (a,b) is: Ellipse σ k a = Giving to α values between 0 ° and 360 °, n B points on the confidence ellipse focused on + 2 sin2θ (Be, Le) are obtained: x = a cos α sin θ – b sin α cos θ + 60 dL cos B σ k e b = n B y = a cos α cos θ + b sin α sin θ + 60 dB − 2 sin2θ Position 2B tan 2θ = In each iteration, the origin is chosen in A − C the position obtained in the previous step. The scale factor is: Mathematical basis k= - 2Ln(1- Prob) The equation on the Cartesian plane of For a level of 95%, Prob = 0.95 the LoP, around the estimated position, is (see Two celestial LOPs Fix [1]): Plot p= xsin z+ y cos z Taking a system of Cartesian axes, it is For n observations, the most probable possible to draw the various elements that position, MPP, is the centre of gravity of the define the celestial fix. polygon formed with the intersection of the n • Origin: estimated or assumed LOPs. Mathematically this is obtained position. optimising the following equation, the distance between the fix and the LoP: • X axis: according to a parallel, n positive eastwards. 2 S=∑[] pi − ycos zi − x sin zi • Y axis: according to a meridian, i=1 positive towards the North. Minimizing the function S: ∂S Lines of Position = 0 ∂x Placed the plot within a square of 20 ∂S nautical miles, centred on the assumed = 0 position, the LoP defined by p and Z is ∂y determined by the intersection with the sides of this square: And solving the