If 



xZ = zX 

 then (2) 



zX 



In reality, both translations occur simultaneously with the result that 

 the closure point actually migrates upslope as the water level rises. Shift- 

 ing the closure point upward and shoreward will affect the outcome of volumet- 

 ric calculations, and there are at least two ways to account for this small 

 defect in the geometric justification just given for equation (1). 



First, a closure depth midway between the original and final depths could 

 be used with equation (1) to improve the accuracy of the calculation. The 

 horizontal translation of the profile would then imply a slight irregularity 

 or "step" where the new and old profile shapes meet. The step would consist 

 of a wedge of surplus sediment above the closure depth and an equal volume 

 deficiency below; therefore, a local exchange of sediment is easily imagined 

 which would eliminate the step and completely reestablish the identical smooth 

 profile shape without affecting the overall sediment balance expressed in 

 equation (1). This method of accounting for the migration of the closure 

 depth is easy to visualize and consistent with the geometric derivation given 

 for the predictive equation. / 



Second, a more formal development of the sediment balance would have 

 integrated between profiles, allowing the closure point to move in infinites- 

 imal steps with the water surface. This approach also eliminates the step 

 problem and results in the more precise relationship: 



x = X ;in ^^— (3) 



Z - z 



Neither method of adjusting equation (1) (by measuring the critical depth 

 from an intermediate water level elevation or using eq. 3) is generally neces- 

 sary because the change in water level, z, is usually so small relative to 

 the total height, Z, that all three methods provide essentially the same 

 results. For example, if z < O.IZ all results agree within less than 1 

 percent. 



Thus, the simple expression, x ~ zX/Z, is not only valuable as a close 

 approximation, but also most useful because it is easily (a) recalled by 

 visualizing the adjustment of two rigid translations, (b) explained in the 

 same manner, and (c) used as a quick mental check on the ultimate retreat 

 expected for various values of the independent variables. 



c. A Realistic Closure Depth (Item c). Determining a realistic closure 

 depth is usually extremely difficult. The most direct approach is to compare 

 historic bathymetric surveys of the site in question. Unfortunately, adequate 

 survey data of this type are rare. Neither pier nor stadia surveys extenc] 

 deep enough, and if a hydrographic survey does extend to deep water, allow- 

 ances must be made for the fact that both sounding errors and boat-positioning 

 errors usually increase significantly with depth and with distance from shore. 

 It is thus often impossible to substantiate apparent offshore changes. On the 



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