Assumed Position

The assumed position (labeled AP) is an estimated position (latitude and longitude) of a vessel derived from dead reckoning (DR). Dead reckoning is the periodic updating of a vessel’s position based on it’s course and speed.

What is in the Rendering

The animated rendering aims to show how the fix is influenced by the assumed position. The animation revolves an imaginary assumed position from a celestial sight, AP-1 (red dot), 360 degrees around a stationary observer labelled OB (green dot). A second assumed position from a second celestial sight, AP-2 (red dot), is stationary and serves to define two LOPs (red lines) where the fix is designated by the white “target” at the LOP’s intersection. AP-1 and the observer share GP-1 (yellow dot) while GP-2 is from the AP-2 sight. The two circles of equal altitude around GP-1 and GP-2 (yellow circles) intersect at the observer.

The observer’s LOP (green line) is tangent to the observer’s circle of equal altitude around GP-1 at the observer’s position. The Zn bearings of the observer and both APs are cyan. As AP-1 revolves, its Zn bearing, LOP, a-value (magenta line) and intercept (black dot) all move accordingly. The current a-value can be read in the moving label. The remaining objects are stationary.

AP Accuracy

So, how accurate does the estimated dead reckoned position, and therefore the AP, need to be to derive an accurate celestial fix?

Using the sliders, change the AP-1 and GP-1 distances from the observer to see what effect the distances have on the fix shown as the white target. The nautical mile distance from the observer to the fix is shown in the text box and other data is updated in the print area at the bottom of the page. Observe how the fix distance changes with the AP-1 to observer bearing and distances. The animation speed can be changed with the Speed slider, and the animation can be paused and resumed with the Pause button. The Reset button redraws the initial view.

The view angle and zoom level can be changed with the arrow buttons. Alternately, use the preset buttons 1-4 to make the following predefined changes:

Preset 1: The AP-1 distance is 60 nm away from the observer with GP-1 far from the observer. The fix drifts nominally.

Preset 2: The AP-1 distance is 60 nm away from the observer with GP-1 very close to the observer. The Ho and AP-1 Hc altitudes are very high. The AP-1 LOP rocks and the fix drifts wildly away.

Preset 3: The AP-1 distance is only 6 nm away from the observer with GP-1 far from the observer. The fix does not drift.

Preset 4: The AP-1 distance is only 6 nm away from the observer with GP-1 very close to the observer. The Ho and AP-1 Hc altitudes are very high, but the fix drifts nominally.

Conclusion

It is generally observed that AP does not have to be very accurate as the a-value that determines where to plot the LOP will change to maintain the LOP tangent to the observer’s circle of equal altitudes. This behavior can be seen in the animation as the magenta-colored a-value line shrinks and grows. There are limits however to how well the a-value can maintain the LOP. The animation illustrates how the fix drifts away from the observer’s position under the following conditions, and is magnified if these conditions coexist:

In the animation, the AP’s distance to the observer is exaggerated for clarity. In practice, we can usually get the DR close enough so that the AP does not have to be very accurate. In order for the AP and the observer’s position to both fall on the plotting sheet, the AP should be within 30 nautical miles or so from the observer.

The Basics

The AP’s Function

The AP serves two main functions in celestial navigation. First, it serves as a base position from which to calculate the radius of the vessel’s circle of equal altitudes on a plotting sheet. The circle of equal altitudes is a circle centered on the geographical position (GP) of the sighted celestial body with a circumference that passes through the vessel’s position (here the vessel is labeled OB for ‘observer’). The circle’s radius is determined by the altitude or height of the celestial body in relation to the observer’s horizon as measured with the sextant. The same altitude could have been measured anywhere around the circle hence the name ‘circle of equal altitudes’. Accordingly, the observer could be anywhere around the circle’s circumference.

While we can determine the circle’s radius directly from the altitude, we cannot plot the entire circle on the plotting sheet. The circle may have a radius of thousands of nautical miles, but the plotting sheet often has a radius of about 120 nautical miles. Furthermore, the geographic position will likely be off the plot.

A way around the problem is to use the Intercept Method. The Intercept Method enables us to plot a short segment of the circle’s diameter that falls within the plotting area. The procedure begins by defining a position (latitude and longitude) that is near the observer known as the assumed position or AP. The AP is a derivation of the vessel’s dead reckoning (DR) position as the DR should be near the observer.

The AP, being a position on the surface of Earth, also has an altitude to the same celestial body that was sighted by the observer and thus also has a circle of equal altitudes. The AP’s altitude wasn’t sighted with a sextant on a sea horizon, but it can be computed for the AP in the sight reduction tables based on the body’s altitude in relation to the AP’s ‘apparent horizon’. The AP’s apparent horizon is a plane passing through the center of the Earth at 90 degrees to the AP’s ‘zenith’ which is a point directly above the AP.

Comparing the altitudes of the observer and the AP, the difference in radii of the two circles can be determined – this is called the ‘a-value’. Along with the a-value, we also need to know the direction to the GP from the AP. But since the geographic position of the celestial body can’t be plotted on the small plotting sheet, we obtain it’s direction from the sight reduction tables. Now we have two datapoints to locate the observer’s circle from the AP reference position.

Plotting

On the plot, the AP position is marked, and a line is plotted from the AP in the direction of the GP denoted ‘Zn’ which is the bearing of the GP relative to North (the angular distance from the AP to the GP is called the ‘zenith distance’ or ‘z’ – Zn is the north-relative bearing to this line). The a-value distance is marked-off along the Zn line from the AP either toward or away from the GP. If the AP’s altitude is greater than the observer’s altitude (the observer is farther from the GP than AP is), it is plotted ‘Away’ from the GP because the observer’s circle is larger; otherwise (the observer is closer to the GP than the AP is), it is plotted ‘Toward’ the GP because the observer’s circle is smaller.

The end point of the a-value distance is called the ‘intercept’. The intercept marks a point on the observer’s circle of equal altitudes. At the intercept, a straight line of position (LOP) is drawn perpendicular to the Zn line which causes it to be tangent to the observer’s circle of equal altitudes. Because the circle’s diameter is so great in relation to the ‘zoomed-in’ plotting area, the straight LOP in effect describes a segment of the observer’s circle that falls within the plotting area.

Where is the Fix?

To determine the vessel’s position or ‘fix’ (technically an ‘observed position’), two or more LOPs from two or more sights of different celestial bodies, or the same body sighted at a later time, must be plotted. If the vessel was underway between sights, the earlier LOPs must be ‘advanced’ along the vessel’s DR course. The vessel’s position will be close to where the LOPs intersect.

A Secondary Function

The AP is also used to provide whole-degree lookup values that are needed for using the sight reduction tables. To keep the sight reduction tables to a manageable size, they cannot provide infinite solutions to every possible position on Earth. To limit the resolution of the solutions, the input values required to enter the tables are in whole degrees. These values are the AP latitude, the local hour angle (LHA) and the declination of the celestial body. The DR latitude is rounded to the nearest whole degree to specify the AP latitude, and the AP longitude is specified in a manner that causes the local hour angle (LHA) computation to result in whole degrees.