Surface temperature determination from Borehole measurements: Regularization and error estimates

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V()I.. 18 N()o (1993)

60l

SURFACE TEMPERATURE DETERMINATION

FROM

BOREHOLE

MEASUREMENTS: REGULARIZATION AND

ERROR ESTIMATES

TRANTHI LE and MILAGROSNAVARRO

HoChiMinhCity University DepartmentofMathematics

HoChiMinhCity, VIETNAM

University ofthePhilippines DepartmentofMathematics

Diliman,QuezonCity,PHILIPPINES

(ReceivedJuly15, 1993andinrevised tbrmMarch 30. 1994)

ABSTRACT. Theauthors consider theproblem of determiningthetemperaturedistribution

u(x,

t) on

thehalf-linex 0,

>

0, frommeasurementsat aninteriorpoint,forall

>

0.Asis well-known, thisis

anill-posedproblem Using theTikhonovmethod, the authors givearegularized solution, and assuming the(unknown)exact solution is inH(R),a

>

0 They giveanerror estimateoftheorder

fore 0,wheree

>

0is abound on themeasurementerror

KEY

WORDS AND PHRASES. Surface temperature distribution, Tikhonov regularization, error

estimate.

1991AMSSUBJECT CLASSIFICATION CODES. 35R25, 35K05

1. INTRODUCTION.

We consider the problem of determining the temperature distribution

u(z,t)

on the half line z 0,

>

0 frommeasurements at z 1for all

>

0 Theproblemisof importanceinGeophysics. It has been consideredbyvariousauthors

(see

references) Of particular relevancetothepresent workare

thepapersof Anderssen and Saull

[1]

and of Anderssen

[2],

andthe paper ofCarasso

[3].

Thepaperof Anderssen and the one of Anderssen and Saull. give the physical background and concentrate on numerical and stabilization problems. The work ofCarasso makesuseof the Tikhonovregularization method. The difference betweenCarasso’sresult andours canbest be seeninRemark2atthe end of this paper. Thepaperof Talenti and Vessella

[4]

isalso of interest,itgives stability estimates between the

two solutions We should note that the problemunder consideration is ill-posed inthe sensethat a solutiondoesnotalways exist, and that whenever they exist,solutionsdonotdependcontinuouslyonthe given

(measured)

data. The ill-posednessis discussedbyvariousauthors

(see

[1 ],

[3]

and

[5]).

Weshall regularize the problemusing the Tikhonov method. Weshallgiveerrorestimates, assuming the exact solution canbe extendedtoafunction inH

(/),

a>0.

2. MAIN"RESULTS.

We proposetosolvethe heat equation

ut-

uxx

0 0

<

x

<

c,

>

0,

(2.1)

subjecttothe following conditions

(2)

andadecaycondition,which,following[5].wetake as

li x-’2l

f-’

(.r, dt -0 forany F> 9 (232

Ifwe write

v(t)

for

u(0,

t), thenwe have the following representation formula, .alid lbv:_l.r.0/=0.

which,withoutloss of generality,we assumehere

(z,

tl

v(slK(x.t-

s)ds

where(seeWidder

[6]

andCannon

[7])

K(x,t)

(47r)

--1xt-3/2

exp(

z"/4t)

if t>0

=0

Tofind

v(t),

it istherefore sufficienttosolve the followingintegralequation

(4r)

-1 v(s)(t-

s)-a/ezp(-

1/4(t-

s)) 9(t),

(24)

whichisconvenientlywritten as

V/

v(s)K(t-

s)

K(t)

t-3/2exp(

1/4t),

iftO

(25)

where

>0 (,20)

>

O, t,2 7)

t>0

=0, t<0

Weassumeg(t)tobe anL2-function, and, hence,

h(t)

is anL,2-function. Taking theFourier transform

of bothsidesof

(2.7)

gives

where

R(p)(p)

(p)

(2.8)

(2.9) 1

K(t)e

-’pt_dt

K(p)

and similarly forh and v, whichareunderstoodtovanish for

<

0. We shall considerthe following regularized equation

/v+

IRlZa

=R

,

;>o,

(2.1o)

Notethatasolution

O

of

(2.10)

alwaysexistsandisLipschitzianin h The inverse Fouriertransform

v

of’0a,whichweshalltakeasourregularizedsolutionforanappropriate/3, isthereforeLioschitzianinh

sincethe

L2-Fourier

transform isaunitary operatoron

L2

(1R).

Now,

letvbetheexactsolutionof

(2.7)

for h h0, and supposethe error between the measured fight handside of

(2.7)

andthe

(unknown)

exactvalue is less thane

>

0, i.e.,

]h- h0

12

<

e where

12

L2-norm

(2.11)

Then,wehave the followingerrorestimate,taking/3 e

THEOREM2.1. Let(2.11 hold,andsupposevis inH (IR),a>0 Then

(3)

’LE,’,IPERATURE I)ISTRIBUI1() ()N i\ HA1,F-I,1NE b03

where

/,

re(t)

V/

[p) exp(ipt)dp

andc 2/2BwithBdefined in(220)

PROOF. By (2 8)and(29),wehavefor

L

>0

/(-’)

+

R I"(-)

-+R()-o!

Taking the

L2-inner

productwith

"

and estimating,we havethe inequality

(2 3)

(2.14)

Letting/3 e

<

1, andnotingthat

wethen have. atlersomerearrangements

In particular,

and

(2 15)

(2.16)

IR(-)1

1)

(2 17)

Now,taking the

L-inner

product of bothsidesof(2.13)with pl

(v -’)

and estimating,wehave

(for/5

e)

the inequality

e

II

p

l’(

-’)12

+11

P

R(

-)12

<_

e

II

p

Vl

II

p

(v

-)1 +

II

p

Rloola K01=

II

p

O(V

-V)l

Inviewof thefollowingresult

(see [8])

IR(p)

2emp

(-

(I

p

1

’/2)

we have

pl

1

<

.

From (2.18),wehave,inpaaicular

llPl

(%-v)IIlpl

o(v,-v)I2

([Ipl

gl+llpl ole)

where

Let

Then,wehave

llvll

lpll.

(the H%norm).

(2.18)

(2.19)

B

2(lip

]’/]2oo2

v

2)

+1 (2.20)

The foregoingstep establishes anH%bound on

v

v. Asa final step in ourproof, weestimate the smallness of

lye

vl=

fore+0. Tothisend,wehave foranya

>

0

le

-(P)

12

dp

<-

fv

IR(p)Ie/

R(a)I2I(’" -’)(P)I2

dp

<a I<_a

(4)

60/4 ’1. ’1. I,E ANI) N. N,\VARRO

Onthe other hand

I(’Oe

-)(p)l’dp

<_

a

-’

ip!(._

-)l’dp

<_

B"

/a

"’

Now, leta _{Pe be}thepositivesolutionof the equation

pi

’’-

exp

Pl/2)

’’

lie

Then

(2 23)

Taking the logarithm of eachsideof(2 22)gives

(2.24)

Since_Pe oo for e 0, it followsthat,for alle

>

0 sufficientlysmall, theleft-hand sideof(2.24)is

lessthan

p/

orequivalently

(1/p)

<

1/(lnl/e)

(2

25)

Hence,

for all smalle

>

0,wehave from

(2.23)

(226)

Thiscompletestheproof

REMARK

1. Amongall choices

of/

e

,

6

>

0,asTikhonov parameters,our choice e, in a

sense, is, from the point ofview ofour analysis, the optimal one, except for the coefficient c inthe inequality

(2.12).

First,if6

>

1, thenouranalysisbreaksdown,sincethe steps leadingto

(2.19)

areno longervalid. If0

<

6

<

1, then, retracingthe steps from

(2.13)

to

(2.24),

wewouldhave,

for/

e

,

an estimateof theform

(2.27)

where

c’

isboundedas e 0. Weomitthedetailsthere.

REMARK 2. In [3], Carasso (under weaker conditions on the exact solution) produced a regularizedsolutionthatapproximatestheexactsolutiononlyatinteriorpoints:r

>

0.

In closing,wewould liketopoint out that multidimensional analoguesofourproblemaremuch more complicated. They are considered,e.g., in[9], where theproblem istreated numerically, andin

10],where theproblemisformulatedas a momentproblem.

ACKNOWLEDGEMENT. The authors would like to thank the referee for the most valuable

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Boreholemeasurements,MathematicalGeoh)g)’5(1973)

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