R.J. Sobey, S.A. Hughes r Coastal Engineering 36 (1999) 1736
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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 ti. 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 0rTz 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 0rTz s 1 would be a global solution. A particular solution will be designated
LFI-PUVw J, N,t 0rTz x. For example, LFI-PUVw3,3,0.1x has J s 3, N s 3 and t 0rTz s
0.1.
5. Numerical implementation
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 0.1Tz , where Tz 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 tn .; 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 tn . points are located at trt 0 s 0; "0.5; "1.0. K and D
equations are applied at each of these hZ tn . points, depending on the value of N. At
N s 5, the local window width is doubled. There are 2 N s 2 or 6 Zfor a single-width
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 trt 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 trt 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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