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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

spline interpolation among the measured points.

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

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