22

A prediction of the local kinematics is sought throughout the water column in the

immediate neighborhood of the PUV gauge at horizontal position *x*a . The pressure

sensor is at known elevation *z * P and the directional current meter at known elevation

o

times *t*i , as are measured velocity component traces *u*abs s *u*a Z *t*i; *x*a , *z * UV . at the same

o

discrete times. The local water depth *h *and depth-uniform current *U*a are also known.

The current can be estimated from the UV traces as a time-average over sufficient time

ZEq. Z3.. plus the three observational equations ZEqs. Z11. and Z12... These are the

simultaneous implicit algebraic equations:

2

cosh *k*Z h q *z * P .

h y *p*d bs s 0

o

cosh *kh*

Z 14.

cosh *k*Z h q *z * UV .

h y *u*1bs s 0

o

Z v y *k*a Ua .

sinh *kh*

cosh *k*Z h q *z * UV .

h y *u*obs s 0,

Z v y *k*a Ua .

2

sinh *kh*

In principle, a unique local solution at each time *t*i can be computed. An analytical

solution does not seem feasible. A direct numerical algorithm is the NewtonRaphson

method. Writing Eq. Z14. as *f*iZ *x * j . s 0 where *i*, *j *s 1,2,3,4 and vector *x * Z*j n*. as the

solution estimate at iteration *n*, the correction D *x * Z*j n*. is suggested by the local Taylor

series expansion:

Z *n*.

E *f*i

D *x * Z*j n*. q . . . s 0.

Z 15.

E *x*j

The Jacobian E *f*irE *x * j is evaluated analytically and Eq. Z15. is solved by matrix

inversion.

The predictive potential of this algorithm was evaluated for a theoretical steady wave

train of period 10 s and height 10 m, directed at qpr10 to the *x*-axis in 20 m of water

and an opposing current of y1 mrs. The PUV gauge is located at *z * P s *z * UV s y10 m.

The initial theoretical PUV trace was computed from Airy theory at a sampling time

interval of 0.5 s. With this trace as the measured PUV record, the local linear algorithm

predicted the local wave frequency, the local wave number components and the local

water surface elevation. Except in the immediate neighborhood of zero-crossings, where

profile curvature is minimal and Eq. Z14. is ill-conditioned, there was consistent

agreement with Airy theory to four significant figures.

A second theoretical PUV trace Zrecord `Twenty'; see Table 1. was computed from

near-exact Zglobal. Fourier wave theory ZSobey, 1989., also at a sampling time interval

of 0.5 s. With this trace as the measured PUV record, the local linear algorithm was

significantly less successful.