3363 HYDROGRAPHIC MANUAL PaGE 230 



3363. Distance by Subtended Vertical Angle 



When the horizon is indistinct, or there appears to be abnormal refraction, or if 

 for any other reason the depression angle method cannot be used, distances to near- 

 by objects or buoys may be determined from vertical angles measured between the 

 waterline and the top of the object, the height of the object being known. In con- 

 structing survey buoys, the banners on each type of buoy should always be placed 

 at the same height above the barrel for possible use in the measurement of such angles. 

 The form.ula for determining the distance is 



j^_h cot a 

 3.28 



in which D is the distance in meters to the object, h is the height in feet of the object, 

 and a is the observed angle. This formula includes no correction for curvature or 

 refraction. Table 9 in Bowditch (page 131) is based on this formula. 



The variables in the formula are the observed angle and the height of the object, 

 neither the dip nor the height of eye entering into it. The accuracy of the observa- 

 tion is increased by having the eye as close to the water as practicable and, of course, 

 the object or buoy must be vertical at the time of the observation. Such observations 

 are often impracticable on buoys because they are usually canted more or less by 

 current or wind. If the object is nonfloating and its elevation above mean high water 

 is used as the height, it must be corrected for the difference between mean high water 

 and the height of the tide before applying the formula; the difference being additive 

 when the height of the tide is below the plane of mean high water. 



3364. Distance by Vertical Angle Above Horizon 



Where the object is comparatively distant and the formulas in 3362 and 3363 

 are inapplicable, the distance may sometimes be determined by a vertical angle measured 

 to the object above the visible horizon. Distances so determined are rarely of sufficient 

 accuracy to be used in conjunction with control by three-point fixes or R.A.R. They 

 may occasionally be of value to supplement sounding lines controlled by astronomic 

 observations or dead reckoning. 



The formula is 



D=1.15 



(v^-«+(yj-^8) 



in which a is the vertical angle in minutes of arc; D is the distance in nautical miles; 

 d (or dip) equals 0.98-ylH; h is the elevation of object in feet above sea level at the 

 time; and H is the height of eye in feet above sea level. 



The formula is adequate for most distances and convenient to use; refraction and 

 curvature are included. The formula is applicable whether the visible horizon is 

 between the observer and the object or the object is between the observer and the 

 visible horizon. Table 10 in Bowditch (page 133) covers this case but has not been 

 computed from this same formula and does not cover all possible conditions. 



3365. Ship Stations 



In launch or small-boat hydrography it may sometimes be desirable to supplement 

 the existing control, because of its distance, by using the foremast of the survey ship as 



