The elevations of each sphere bottom above the toe, z^, and horizontal 

 distances from an arbitrary starting point, x„, were determined using the potentiometer 

 measurements. The profiler sphere horizontal coordinate was computed as x^ = jc^ + 

 L(l-cos (J)), relative 10x^ = with (J) = 0, and the profiler sphere vertical elevation above 

 bottom was computed d&Za = Zp- Lsin 4) - Dll, where L = length of the profiler arm 

 from pivot point to center of sphere, (j) = angle of the arm from horizontal, D = sphere 

 diameter, x^ = transverse distance the carriage moved, and Zp = elevation of the arm 

 pivot point. Figure 4.8 shows a definition sketch for this transformation, where the 

 initial carriage position x^ = corresponds to the initial sphere position x^ = with (j) = 

 0. The measured sphere elevations were averaged over a 0.5-cm interval in the cross- 

 shore direction to remove small variations in elevations resulting from minor stone 

 settlements that occurred during the tests. This is an essential step as this "noise" will 

 cause a bias in the eroded area calculation. 



For each arm, the eroded area was calculated using a Simpson's Rule 

 numerical integration of the damaged profile below the undamaged profile. The eroded 

 depth was determined as the maximum slope-normal distance between undamaged and 

 damaged profiles. This slope-normal distance was calculated using a numerical 

 technique beginning at each point on the eroded profile and stepping along a line normal 

 to the slope of cotQ = 2 until the undamaged profile was crossed. Then the point at 

 which the crossing occurred was interpolated to obtain the slope-normal distance. The 

 slope-normal minimum depth of cover at a given time / was calculated as the minimum 



79 



