23 



Mathematically, this is an ill posed problem, as the flow is governed by an elliptic 

 equation (Laplace, Eq. 2.2), in which the solution is determined by the boundaries, 

 but there are only data at a single location. Some of these difficulties can be sur- 

 mounted through the application of the free surface boundary conditions. While there 

 are no data measured near the surface, the free surface boundary conditions remain 

 appropriate, and necessary to define the solution at that boundary. 



When the LFI method was applied to a water surface trace (Sobey 1992), the 

 location of the the water surface was known, and the boundary conditions could 

 be directly applied at that location. Unfortunately, when working from a subsurface 

 pressure record, the location of the water surface is not known. While the free surface 

 boundary conditions are well defined, the fact that the location at which they must 

 be applied is not known makes the problem more difficult. This is the complication 

 that leads to the difficulties in finding full nonlinear solutions to all free surface flow 

 problems. In order to apply the free surface boundary conditions when working with 

 a subsurface record, the location of the water surface must be found, together with 

 the potential function, in each window. 



In order to locate the free surface, the water surface is defined at A'^ surface 

 nodes equally spaced in time throughout the window. The elevation of these nodes 

 is unknown, and will be sought as part of the solution. Including the water surface 

 nodes as part of the sought solution introduces A'^ additional unknown parameters for 

 a total of 3 + J + A'^ unknowns in each window (k, kx, u, Ai . . . Aj, 7?i . . . t/at). The 

 free surface boundary conditions, Eq. 3.3 and 3.2, are applied at each surface node. 



3.2 Formulation of the Optimization 



Finding the unknown parameters in a nonlinear system of algebraic equations is 

 known as nonlinear optimization. The system in this case consists of the nonlinear 

 Bernoulli equation and the nonlinear free surface boundary conditions. The free 

 surface boundary conditions are nonlinear in two respects, involving second order 

 terms in dynamic free surface boundary condition (the v? terms and B in Eq. 3.3), 

 and second and third order terms in the modified kinematic boundary condition (the 



