Intercept Method - Methodology

Methodology

Suitable bodies for celestial sights are selected, often using a Rude Star Finder. Using a sextant, an altitude is obtained of the sun, the moon, a star or a planet. The name of the body and the precise time of the sight in UTC is recorded. Then the sextant is read and the altitude (Hs) of the body is recorded. Once all sights are taken and recorded, the navigator is ready to start the process of "sight reduction" and plotting.

The first step in sight reduction is to correct the sextant altitude for various errors and corrections. The instrument may have an error, IC or index correction (See article on adjusting a sextant). Refraction by the atmosphere is corrected for with the aid of a table or calculation and the observer's height of eye above sea level results in a "dip" correction, (as the observer's eye is raised the horizon dips below the horizontal). If the Sun or Moon was observed, a semidiameter correction is also applied to find the centre of the object. The resulting value is "observed altitude" (Ho).

Next, using an accurate clock, the observed celestial object's geographic position (GP) is looked up in an almanac. That's the point on the Earth's surface directly below it (where the object is in the zenith). The latitude of the geographic position is called declination, and the longitude is usually called the Hour angle.

Next, the altitude and azimuth of the celestial body are computed for a selected position (assumed position or AP). This involves resolving a spherical triangle. Given the three magnitudes: local hour angle (LHA), observed body's declination (dec.), and assumed latitude (lat), the altitude Hc and azimuth Zn must be computed. The local hour angle, LHA, is the difference between the AP longitude and the hour angle of the observed object. It is always measured in a westerly direction from the assumed position.

The relevant formulas (derived using the Spherical trigonometric identities) are:

sin(Hc) = sin(lat) · sin(dec) + cos(lat) · cos(dec) · cos(LHA)

or, alternatively,

Where

Hc = Computed altitude

lat = Latitude

dec = Declination

LHA = Local Hour Angle

Since recently these computations can be easily done using electronic calculators or computers but traditionally there were methods which used log or haversine tables. Some of these methods were H.O. 211 (Ageton), Davies, haversine, etc. The relevant haversine formula for Hc is

Where Hc is the zenith distance, or complement of Hc: Hc = 90° - Hc.

The relevant formula for Z is

When using such tables or a computer or scientific calculator, the navigation triangle is solved directly, so any assumed position can be used. Often the dead reckoning DR position is used. This simplifies plotting and also reduces any slight error caused by plotting a segment of a circle as a straight line.

With the use of astro navigation for air navigation, faster methods needed to be developed and tables of precomputed triangles were developed. When using precomputed sight reduction tables, selection of the assumed position is one of the trickier steps for the fledgling navigator to master. Sight reduction tables provide solutions for navigation triangles of integral degree values. When using precomputed sight reduction tables, such as H.O. 229, the assumed position must be selected to yield integer degree values for LHA (local hour angle) and latitude. West longitudes are subtracted and east longitudes are added to GHA to derive LHA, so AP's must be selected accordingly. When using precomputed sight reduction tables each observation and each body will require a different assumed position.

Professional navigators are somewhat split in usage between sight reduction tables on the one hand, and handheld computers or scientific calculators on the other. Either method is equally accurate. It is simply a matter of personal preference which method is used. An experienced navigator can reduce a sight from start to finish in about 5 minutes using nautical tables or a scientific calculator.

The precise location of the assumed position has no great impact on the result, as long as it is reasonably close to the observer's actual position. An assumed position within 1 degree of arc of the observer's actual position is usually considered acceptable.

The calculated altitude (Hc) is compared to the observed altitude (Ho, sextant altitude corrected for various errors). The difference between Hc and Ho is called "intercept" and is the observer's distance from the assumed position. The resulting line of position (LOP) is a small segment of the circle of equal altitude, and is represented by a straight line perpendicular to the azimuth of the celestial body. When plotting the small segment of this circle on a chart it is drawn as a straight line, the resulting tiny errors are too small to be significant.

Navigators use the memory aid "computed greater away" to determine whether the observer is farther from the body's geographic position (measure intercept from Hc away from the azimuth). If the Hc is less than Ho, then the observer is closer to the body's geographic position, and intercept is measured from the AP toward the azimuth direction.

The last step in the process is to plot the lines of position LOP and determine the vessel's location. Each assumed position is plotted first. Best practise is to then advance or retire the assumed positions to correct for vessel motion during the interval between sights. Each LOP is then constructed from its associated AP by striking off the azimuth to the body, measuring intercept toward or away from the azimuth, and constructing the perpendicular line of position.

To obtain a fix (a position) this LOP must be crossed with another LOP either from another sight or from elsewhere e.g. a bearing of a point of land or crossing a depth contour such as the 200 metre depth line on a chart.