Short time prediction of sea surface height: prediction of a degenerate stochastic process

CHAPTER 6.

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SHORT TIME PREDICTION OF SEA SURFACE' HEIGHT:*

PREDICTION OF A DEGENERATE STOCHASTIC PROCESS

by

Louis J. Cote.

ABSTRACT

The problem is to give a prediction of sea surface height at

a given point a short time in the future making use of knowIecge

of past heights at that point

and.

surrounding points.

In parti.

cular it is desired to find what improvement comes from using

the surrounding points.

' ,

-The sea surface height is a.

ndomfunction of time and

position.. It can be worked with mathematically using .a "random

ariplitude" representation.

Using the latter, the mean square error of prediction i

easily wr.tten down as an integral.

The solution of the

pre-diction problem, were it knbwn, would be equivalent to inninmizing

this integral.

The fact that the sea surface height function can be found

as a solution of an initial value problem makes the random

function of three variables degenerate.

The problem

correspond-ing to Wiener's statememit of the one dimensional prediction

problem is to assume. knowledge Of the entire plane for aU the

past. This corresponds to the inital value problem and the

prediction is without error.

The statistically meaningful

problems

re concerned with the best use of infonriation in a

finite range.

The solution,,of this type of problem is nOt yet

known. ' , .

Long rangé"redictiqn of sea si.u!face heIght (ocean waves) has been discussed.by Pierson [1952, 1954] and by Pierson, Neumann and James [1954]. 'TIis sort of prediction

isessen-tially the estimation of long time averages associated with the process, such as the average "period" and the average height of the one-third highest waves. These forecasts are freuent1y made many hours ina4vance using meteorological

* Paper prepared for...the Research Division of the College of Engineering, New York UniversitY, under Contract

Nonr

285(17)

as"sponsored by the Office of Naa1 Research.

**pdue University.

CORRECTION: This research was

the sea surface over a certain point at a future time was left open. Indeed such a prediction could not be made using only weather data; it would require bowledge of therecent _sea

surface heights in the' area where the forecast is to be made. To ththk of forecasting the exact form of the sea surface at. all requires a mathematical picture of the sea surface

height as a function of location and time. This picture should, to be, realistic,. .take account of' the complexity and irregularity

observed on the actual sea surface, and yet it should take ad-vantage of 'any stable features of the sea surface so as to

avoid too much complexity in the theory. Such a representation has been presented by PIerson [1952,1954]. In this work, the .surface height. Is described as a time series or random function.

It is regarded as the resultant of the superpositiOn of many iñfinite'imal contributions from different -regular plane waves. A plane wave of frequency , traveling in the direction,

e,

has a well Iiown mathematical form. To déscrlbe it completely, it'

is necessary to.know, in addition to the frequency and direction, the amplitude and phase. The amplitude of each elementary wave

is infinitesimal, arid an amplitude density is assumed which de-pends on the frequency and direction. The phases of the elementary waves are assumed to be random. The result is written as an

integral but must be'regarded as a limit of a sum of random ele-ments. It is written as equation (1).

9i (s,t) =1

Tc4,4ixc

s-u

E(M.e)}

VCAiM, e)Jt

e

-'..

..

(1)

The elementary wave traveling in the direction 9, at fre-quency,.M , is given by the cosine term. [ACM,e)i2 is the square

of the amplitude density and i.frequently. referred to as the energy spectrum of the waves.

- & mathematicall 'more 'convenient way of doing this (for

áertain applications) is to write the intera1 as the limit of a sum of 'elementary waves with random amplitudes, The ran-doth amplitu4es will not be

uniformly

distributed for all

fre-quencies and directions as were the hasés in the preceding paragraph, but will have, mean values 'depending on the frequency and direction. The integral for the random amplitude representa-tion of the seasurface is given by equarepresenta-tion (2).

j.Jc44f(X

e

&)

_,at3[ACs6J c1

, (M,O)

-o

IAt][I(MM]

U2(M,e)

SHORT TIME PREDICTION OF SEA. SURFACE HEIGHT:

PREDICTION OF A DEGENERATE STOCHASTIC PROCESS

In thisexpression the phase of the .siné and cosine term is zero.

ACM,e)

here is everywhere positive suchthat'JA(J4,e)12 i the square of the amplitude density. The dC.M,e)..are random;vari-ables of _a standard type (the so-called Wiener process) so that

ACM

,e)

d

3(

,e).

are . independent Gaussian random variables with variances

IA(4,e)Id.ade.

For the approximatiOn of equation (2) over a finite net, the volume bounded by the vertical sides of the. eiementa] areas d.itdG imder the energy density spectrum is computed to obtain an elemental variance, and the ampitudeS of the cosine and

sine

terms are then chosen at random from ,a nor-ma]. distribution with this variance.

This integral expression is used to frame the short time prediction problem. To be specific, this problem is to estimate 11(O,O,a) for aO from a knowledge of (x,y,t) for t<O and some range of x and y, say,

x1<x'X

and y]..cyc.y. The choice of prediction methods is restricted o linear combinations of known data: for the case of discrete knowledge, weighted aver-ages such as equaton .(3)'

YL(0,0,&) =

£ £

K(X,'i,tJi)

(x,I3,,tA)

,

and for the continuous case integrals such as equation (4) are.. used.

'(°,°)

=1 1

f

7(x,cs,*):c4'vtt

-. ,,

_,,

In both cases the problem is to make a juicious choice of K(x,y,t) so as to take fU].1 advantage of the information. The mean square error of the prediction method can be written

(uslng the continuous case to illustrate) as-equation (5).

yjh

. ..

Square = ensemble

'frU00,0Jf J

K(X,%t)

7j(x,, f)xd141

error average of ' .

This is the expected value of a random quantity since both functiOns 'k(O,O,a). and (x,y,t) are random quantities. The

size of an érr'or is generally measured by its mean square. The prediction problem is to minimize this mean square by a proper choice of IC(x,y,t). The mean square error can.be written

as equation (6):

mean square error

Ge3.('

(tc...

.)..4t)4xSi

at]

(6)

4.

ct_f r'i'

mathematical problem reduces to minimizing this: integral. For the case where our knowledge of )z (x,y,t) is limited to all of the past at one point of the sea urface, the solu-tion of the minimizasolu-tion problem is given by Wiener (1949)

It is obvious that we can only improve on this by using know-ledge of the past of other poInts.. The exact amount of Im-provement and also such matters as which points will give the most improvement are the questions which this Investigation set out to answer. The result.of a straight attempt to generalize Wiener's won' by assuming complete knowledge of the past 7of the whole sea surface (i e., x1 =

-00

, X

= +O0

Yi =

-00

Y2 = +co) is that a prediction can be made without

error. Indeed much less knowledge is sufficient for this be-cause this prediction is simply an initial value problem. When. the waves all travel from the same half plane, it is

sufficient to know the entire surface for one past time point. Thus the random function is degenerate.because although its nominally a function of three variables, it is mathematically :equivalent to a function of two variables.

This emphasizes the fact that we do not even approximately know the entire sea surface, and we must make a prediction from partial owledge.. For the case where knowledge is in a

con-tinuous region, the problem reduces to minimizing the mean square error integral by a choice of K(x,y,t) in equation (6). But the interior integral has finite limits, and this makes it dif-fer essentially froth the one dimensiopal integral of the prob-lem solvedby Wiener The solution to this problem is not yet known.

When the past is known for a few--four or five--points on the surface, the, problem takes a different mathematical form. This problem has been solved partially and is discussed in.

several places1. However, tlie partial solution does not apply. to the wave problem because of the type of spectrum which is assumed. ..

To summar1zethe State of solution of the short time pre-diction problem. The problem is set into mathematical form in two ways. It is possible to evaluate the mean square pre-diction error for any predictor, though with considerable effort. It is hoped that mathematical work will lead to

optimal predictors jrj one or. both forms, and with this it will be possible to 3udge how much advantage is gained from observ-ing at additional points, and where best to place these points.

SHORT TIME PREDICTION OF SEA SURFACE HEIGHT:

PREDICTION OF A DEGENERATE STOCHASTIC PROCESS

REFERENCES

Pierson,. W.J., Jr., [1952]: A Unified Mathematical Theory for the Analysis, Propagation and Refraction of Storm Generated Ocean Surface Waves. Research Department, College' of Engineering, New York University.

Pierson, W.J., Jr., [1954].: Wind Generated Gravity Waves in Advances in GeoDhysics, vol. 2, Academic Press, 'Inc., (in press).

Pierson, W.J., Jr., G. Neumann, and EL W. James,

[1954]:

Practical Methods for Observing and Forecasting Ocean Waves by Means of Wave Spectra and Statistics. U. S.

Hydrographic Office (In press).

Wiener, N., [1949]: The and

Smoothing of Stationary Time Series. John Wiley and sons, New York.