R.J. Sobey, S.A. Hughes r Coastal Engineering 36 (1999) 1736
28
with the least squares algorithm, was adopted as partial accommodation of the larger
error bands of the UV observational equations. The least squares objective function was
normalized as:
1
1
1
2
2
2
2
2
OZ v , ka , ka xa ; A j ;hn . s
Z f K q f D . q
fP q
Z f U q f V . ,
2N
NP
2 NUV
Z 33.
such that K, D, P and UV equation groups have equal weight. NP is 1 q 4 L and NUV is
1 q 8 L, where L is 1 Zsingle-width window. or 2 Zdouble-width window..
There are 3 q 20 L s 23 Zfor a single width window. or 43 Zfor a double width
window. observational equations. There are a total of 2 N q 3 q 20 L physical and
observational equations in 4 q J q N unknowns, where 2 N q 3 q 20 L must equal or
exceed 4 q J q N. A typical application has J s 3 and N s 3, so that there are 10
unknowns: v , ka , ka xa , A1, A2 , A3 ,h1,h2 ,h3. For a single-width window, there are 29
equations, and 49 for a double-width window. Experience suggested that this level of
overspecification was generally necessary to accommodate the observational error
bands. For theoretical wave traces with no observational error bands Zsee Section 6.,
only a closed set of KD and PUV equations was necessary for credible solutions;
overspecification did not compromise this success.
The problem formulation requires the least-squares solution of a system of simultane-
ous, nonlinear, implicit algebraic equations. Numerical solutions routinely used the
Dennis et al. Z1981b. NL2SOL code; an updated code is available from the internet
Netlib repository of scientific subroutines in fortran. The algorithm ZDennis et al.,
1981a. is a variation on Newton's method in which part of the Hessian matrix is
computed exactly and part is approximated by a secant updating method. To promote
convergence from poor starting guesses, a modelrtrust-region technique is used along
with an adaptive choice of the model Hessian. In operation, the algorithm sometimes
reduces to the classical GaussNewton or LevenbergMarquardt methods.
The number of unknowns, typically 10, is relatively large. In any such high order
nonlinear optimization problem, the crucial elements of successful solutions are invari-
ably accurate estimates of the Jacobian and good initial solution estimates. Accurate
estimates of the Jacobian require analytical estimates of the partial derivatives of all f
equations with respect to all the unknown parameters. These are the quantities
E f KrEv ,E f KrEk1,E f KrEk 2 , . . . through . . . , E fu2rEhN . The analytical derivatives were
evaluated and confirmed against finite difference approximations.
Establishing good initial solution estimates always requires extensive experience with
the particular physical problem and system of equations. The methodology finally
adopted was a two-stage algorithm. The initial step identified three overlapping record
sub-segments, the half-wave extract centered on the leading trough, the half-wave
centered on the central crest and the half-wave centered on the following trough. Each
half-wave is equivalent to an extra-wide window of width about 0.5Tz . In each
half-wave of, say, N discrete PUV observations, h points were located at the same
times such that there are 4 q J q N unknowns. K, D, P, U and V equations are assigned
at the same times, giving 5 N equations. For a PUV record sampled at s Hz, there
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