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Additional closure equations can be provided by observational and free surface bound-

ary equations at neighboring times within a short local window of duration t 0 that is

centered on *t*i. At least 4 q *J *q *N *independent equations must be provided to solve for

4 q *J *q *N *unknowns.

Apart from this strict mathematical closure requirement, there is an additional

constraint. The K and D free surface boundary equations are exact, but the P, U and V

observational equations have measurement error bands. Also, the P and U, V sensors are

unlikely to have the same accuracy. Present technology has much larger error bands on

the U, V traces than on the P traces; this is seen quite clearly in Fig. 2b, where the

sampling rate is 4 Hz. These realities of the problem formulation are accommodated by

a significant overspecification of the problem, especially with the UV observational

equations, and the adoption of a least squares rather than an exact solution. Flexibility in

the time location of the PUV observational equations was also introduced through cubic

The local LFI-PUV theory has three free parameters, the truncation order *J *and the

number of h points *N *in each local window together with the width t 0r*T*z of the local

windows. The truncation order has much the same authority as order in an analytical

wave theory ZStokes, conoidal. or truncation order in Fourier wave theory. A window

width of t 0r*T*z s 1 would be a global solution. A particular solution will be designated

LFI-PUVw *J*, *N*,t 0r*T*z x. For example, LFI-PUVw3,3,0.1x has *J *s 3, *N *s 3 and t 0r*T*z s

0.1.

As in Sobey Z1992., the analysis segment routinely adopted was a double wave

sequence centered about a crest suggested by the pressure trace. Solutions are sought in

narrow local windows, with a target width of order t 0 s 0.1*T*z , where *T*z is the zero-up

crossing period of the record segment.

In all applications of the present theory, it has proven convenient to assign an odd

number of local water surface elevation hZ *t*n .; with *N *s 1 or 3 Zfor a single width

window. or 5 Zfor a double width window., such that there is always a computed water

surface elevation centrally located in the window. With respect to the time at the center

of a window of width t 0 , hZ *t*n . points are located at *t*rt 0 s 0; "0.5; "1.0. K and D

equations are applied at each of these hZ *t*n . points, depending on the value of *N*. At

window. or 10 Zfor a double-width window. of these water surface equations.

P, U and V observational equations are located at the center of the window.

Additional PUV equations within the window are located such that UV observational

equations are not assigned a weighting that was inconsistent with their routinely larger

error bands. With respect to the time at the center of a window of width t 0 , P equations

are located at *t*rt 0 s "0.25, "0.5 in a single width window and also at "0.75, "1 in

a double width window. The associated U and V equations are located at *t*rt 0 s "0.125,

"0.25, "0.375, "0.5 in a single-width window and also "0.625, "0.75, "0.875,

"1 in a double-width window. The explicit higher density of UV equations, together