Direct methods

According to th e term inology established by N eum an (1973), th ere are essen tially two approaches to p a ra m e te r identification: th e direct a n d

indirect m ethods. For a b rief account of these two m ethods, let (p* denote th e vector c o n ta in in g th e a v aila b le h e a d m e a s u re m e n ts <P*(z m)> m = where are the m easurem ent locations in the dom ain Q.

D irec t m eth o d s f ir s t reco v er a c o n tin u o u s su rfa c e <jo fro m th e m easu rem en ts (p* via the use of interpolators and th en solve (3.3.1) as a first order p a rtial differential equation in T. W ith q known, a solution for

A

T along any characteristic p a th defined by d z/d 6 = V(p m ay be form ally d e riv e d a n d sh o w n to be g iv en by th e fo llo w in g e x p re s s io n (Dietrich et al, 1986a)

0 T(6) = (3.3.2) with A defined by V 0 A(v) (3.3.3)

Use of equation (3.3.2) for inference purposes is very lim ited for several reasons. F irst, T(6) in (3.3.2) is of infinite dim ension and we already know from th e discussion in C hapter 2 th a t only a finite p aram eterizatio n of T

can be recovered in the presence of noise. Second, to be of p ractical use, aquifer identification algorithm s ought to reconstruct T globally over the aquifer so th a t tran sm issiv ity estim ates can be used to solve th e forw ard problem (3.3.1). In (3.3.2), T(0) is only defined along lines an d therefore does not yield global estim ates. Finally, even if th e h ead surface was known exactly, use of (3.3.2) w ith (3.3.3) would be com putationally expensive as it re q u ire s n u m erica l co n stru ctio n of th e c h a ra c te ris tic lin es from th e

A

d ifferen tial eq u atio n d z / d 6 = V(p, and th e n n u m erical in te g ratio n s of (3.3.2) and (3.3.3).

However, (3.3.2) illu stra te s some of th e fe a tu re s en co u n tered w hen e stim atin g T from indirect head m easurem ents. For exam ple, it shows th a t if r is a p a rtic u la r characteristic curve solution of th e eq u atio n

A

d z / dO = V(p, th e solution for T exists and is unique along r if there is exactly one piece of inform ation about T on T which allows th e constant

T ( 0) to be d eterm in ed . This inform ation is u su a lly refe rred to as the Cauchy d a ta . In addition, (3.3.3) shows th a t th e estim ation of T involves th e in te g ra l of a L aplacian along lines. W ith th is, one m ay be led to th e co n clu sio n t h a t th e m e a su re of ill-c o n d itio n in g a s s o c ia te d w ith tran sm issiv ity estim ation is equivalent to one differentiation. As noted by R ichter (1981a), th is is only true if the dim ension of the aquifer is one. For h ig h e r dim ensions, th e ill-conditioning asso ciated w ith tra n s m iss iv ity estim atio n is equivalent to two differentiations of th e d ata. This point is pursued fu rth e r in the next section.

In ste ad of u sin g (3.3.2), direct approaches are in practice obtained by discretizing (3.3.1) as

q = D Tj + e (3.3.4)

where q an d tj are vectors associated w ith the discretization of q and T,

resp ec tiv e ly ; D is th e m a trix re su ltin g from a d isc re tiz a tio n of th e

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differential equation V»TV(p ; and e represents errors.

Note in passing th a t if q is only partially known, it can be decomposed as

%

= +

. 0

.

where q t is known and q 2 is unknown, and 0 is a null vector. With this, equation (3.3.4) can be written in partitioned form as

+ e (3.3.6)

where J is composed of null and identity matrices. Nevertheless, for the sake of presentation we shall use (3.3.4) only and assume th at q is known exactly.

A

Direct methods usually derive an estimate 77 for the unknown ij from the minimization of llg - Drill2 where INI2 is a weighted sum of squares. Since the estimation of T is an ill-posed problem, the computation of ij may require some form of stabilization. This is often done by minimizing

\lq - Drill2 subject to additional constraints obtained from prior knowledge, i.e.

min llq-Drilf + aL(r]) (3.3.7)

n

where

a is a positive scalar representing the trade-off between minimizing llq -D7]l\2 and stabilizing the solution; and

L is a positive functional representing prior knowledge about 77. If prior knowledge of 77 is available in the form of a vector of point measurements 77* obtained by pump or slug tests (Bardsley et al, 1985; Sageev, 1986), L is often chosen to be of the form

L(n) = l|T7 - T/*l|2 (3.3.8) On the other hand, the prior information may be th a t certain components in 77 should not be too large. In this case, one may choose for L the quadratic form

L( 77; = 77V 77 (3.3.9)

where W is a weighting matrix that is positive definite; and riT denotes the transpose of 77. In this case (3.3.7) is often referred to as a ridge

regression (Golub et a l, 1979). This form is used in C hapter 6 w ith tjt W T]

m easuring the linearized curvature of transm issivity.