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7 AD-A079 302 WISCONSIN UNIV-MADISON DEPT OF STATISTICS F/6 12/1IS SOE NEW MATH4EMATICAL METHODS FOR VARIATIONAL OBJECTIVE ANALYSI--ETC(U)

OMSEP 79 A WAH8A ,J WENOELBERGER N0OOON-77-C-0675UNCLASSIFIED UDS-578N L

Department of Statistics

University of WisconsinMadison, Wisconsin 53706

i7

O c "i JECHNICAL REPORT VO. 578

- 6 SOME NEW MATHEMATICAL METHODS

FOR VARIATIONAL OBJECTIVE ANALYSIS

USING SPLINES AND CROSS-VALIDATION.

by

/ 1 ~Grace ahba'

gJames Wendelberger

--WI Department of StatisticsUniversity of Wisconsin

- OPCbiCr2

Typist: Debbie Dickson

1Research supported by the Office of Naval Research under Contract No.N00014-77-C-0675.

2Research supported by the National Science Foundation under Grant No.ATM75-23223

SOME NEW MATHEMATICAL METHODS FOR

VARIATIONAL OBJECTIVE ANALYSIS

USING SPLINES AND CROSS-VALIDATION

Grace WahbaJames Wendelberger

Department of StatisticsUniversity of WisconsinMadison, Wisconsin 53706

ABSTRACT

Let t(x,y,p,t) be a meteorological field of interest, say height,

temperature, a component of the wind field, etc. We suppose that data

- N{oil = concerning the field of the form ti = Lio + ei are given,i1I

where each L is an arbitrary continuous linear functional and ci is

a measurement error.

The data i may be the result of theory, direct measurements,

remote soundings, or a combination of these. We develop a new mathe-

matical formalism exploiting the method of Generalized Cross Validation,

and some recently developed optimization results, for analyzing this

data. The analyzed field, tN,m,x' is the solution to the minimization

problem: Find * in a suitable space of functions to minimize

I N (L1 -s) 2 (*)t z + Jm(,D)N

where

m~!~ ! L t! ff f m 2

rf2f ( dxdydpdta--\ax ay ap a t4

C11+a2+n34"a4*M

The a* 2 are assumed mean square errors. Functions of d =1,2, or 3 of

11ii

the four variables x,y,p,t are also considered. Under rather general

conditions, we give an explicit representation for the minimizer of

(*). The parameter X controls the tradeoff between the infidelity of

the analyzed field to the data, and the roughness of the analyzed

field as measured by J m(.). Alternatively X may be thought of as

controlling the half-power point of the implied data filter. m controls

the number of continuous derivatives that 4N ,m will possess, alternatively,

m may be thought of as controlling the steepness or "roll-off" of the

data filter. High m correspQnds to a steep roll-off. The parameters

X and m are chosen by the method of Generalized Cross Validation (GCV).

This method estimates that X and m for which the implied data filter has

maximum predictive capability. This predictive capability is assessed

by the GCV method by (implicitly) leaving out one data point at a time

and determining how well the missing point can be predicted from the

remaining data. The results extend those of Sasaki and others in

several directions. In particular, no preliminary interpolation or

smoothing of the data is required and it is not necessary to solve a

boundary value problem or even assume boundary conditions to obtain a

solution. Prior covariances are not assumed. The parameters X and

m play the role of signal to noise ratio and "order" of the covariance,

these being the two most important parameters in the prior, and are

estimated from the immediate data rather than historical data or guess-

work. The numerical algorithm is surprisingly simple for any N with

N2 somewhat less that the high speed storage capacity of the computer.

The approach can be used to analyze temperature fields from

radiosonde measured temperatures and satellite radiance measurements

simultaneously, to incorporate the geostrophic wind approximation and

other information. In a test of the method (for d = 2) simulated 500mb

height data was obtained at discrete points corresponding to the U.S.

radiosonde network, by using an analytic representation of a 500mb wave

and superimposing realistic random errors. The analytic representation

was recovered on a fine grid with what appear to be impressive results.

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