The quantity {Vi'-\-l'^) in equation 5 is tlie 

 mean sum of the squares of the distances between 

 the primary interpolation axes. Several choices 

 are possible for secondary interpolation axes 

 (Stearns, 1968). The correct choices are those axes 

 actually used by the cartographer in drawing the 

 map. A human being, however, in his subjective 

 approach to contouring, is seldom fully aware of 

 just what axes he has used. Therefore, in com- 

 puting the reliability of the present maps, I 

 assumed that the shortest axes were used, which, 

 in the rectangular trackline siu-veys of the Middle 

 Atlantic Continental Shelf, equaled the distance 

 between soundings. These distances are usually 

 equal within any unit area, so the quantity 

 (V,. + l") equals 61 



The topographic slopes. — Positional and inter- 

 polation errors (which are expressed in distance 

 units) are converted into depth errors, by multi- 

 plying them by g Cos y, where g is the positive 

 topographic slope in the vicinity of the soundings 

 (or the interpolated point) and y is the angle 

 between the errors and the local slope. 



The slopes g,,, g,, and g^ were assumed to be 

 equal and were estimated from five systematic 

 samples taken in each 5-minute imit area. Each 

 sample consisted of the maximum slope measured 

 in a circle 1 nautical mile in diameter. The mean 

 slope, g, was taken as the mean of the five samples, 

 and the variance of the slopes, Vg, was approxi- 

 mated by (0.43 ci))-, where w was the range of the 

 five samples. This estimate of the variance assumed 

 that the slopes have a normal distribution within 

 each 5-minute unit area and was used as a com- 

 putational expedient (see Dixon and Massey, 

 1957, pp. 273, 404). 



The cosines. — In estimating the cosines in 

 equations 1 and 2, I assumed that all angles had 

 an equal probability of occurrence; thus yp, 7;, 

 and 7;/ range from zero to ir radians (from 0° to 

 180°), and the probability functions of the angles 

 equal l/ir. Hence, 



Cos- 



-r 



Cos7d7=0 



and 



Vc. 



i7 — - <- 



tJo 



0S=7d7=}^ 



This assumption may be true for positional errors, 

 because I assumed that these errors are unbiased. 

 It is not strictly true, however, for interpolation 



errors, and a more accurate, although more time- 

 consuming method could have been used ; i.e., the 

 final map could have been matched with the inter- 

 polation networks actually used and the angles 

 measured. 



The Source Diagram 



The source diagram on each map shows the 

 number, scale, and date of the USCGS hydro- 

 graphic surveys and nautical charts used in the 

 construction of tlie maps. 



The maps depict the sea floor at the dates of the 

 \'arious surveys, and the user must draw his own 

 conclusions as to changes that may have taken 

 place since then. Significant changes are likely 

 only along some portions of the coast above about 

 10 fm., in offshore shoal areas, and along the 

 upper Continental Slope where slumping may 

 have occurred (see Lucke, 1934a, 1934b; Howard, 

 1939; Heezen, 1963; Miller and Zeigler, 1964; 

 Stewart and Jordan, 1964; Uchupi, 1967). In 

 such areas the maps and their reliability diagrams 

 sliould be used with caution. 



Those who wish to study the actual soundings 

 may examine or purchase copies of the original 

 hydrographic survey sheets from the USCGS, 

 Washington, D.C. 



Diagram of the Mean Distance Between Track Lines 



The mean distance between track lines is a 

 common device for indicating the reliability of 

 bathymetric maps. Reliability is usually assumed 

 to be better where the lines are closely spaced. 

 Trackline spacing also indicates the resolution of 

 a survey; i.e., the minimum size of features con- 

 sistently discoverable from the survey. Surveys 

 with many different trackline spacings were used 

 in drawing the maps. Consequently, the isobaths 

 are more detailed in some areas than in others. 



Diagram of the Standard Deviation 

 of the Isobath Depth Error 



The standard deviations in the isobath depth 

 error diagram are estimates of liow much and how 

 frequently the depths indicated on the maps may 

 depart from the true depths. Tlie diagram is based 

 on the square root of the variance given by equa- 

 tion 2 and shows the average standard deviation 

 in unit areas of 5 geographical minutes to a side. 

 It applies to depths as indicated on the maps, not 

 to the original soundings. A more detailed dia- 

 gram of the entire mapped area is reproduced in 

 figure 2. 



BATHYMETRIC MAPS AND GEOMORPHOLOGY OF JIIDDLE ATLANTIC CONTINENTAL SHELF 



43 



