Uniqueness

For the sake of simplicity, we shall only consider the steady state equation (3.3.1). A discussion on the uniqueness of a solution for transmissivity T requires a clear separation between the continuous problem of solving the first order PDE (3.3.1) in T and the discrete formulations (3.3.7) and (3.3.10) associated with direct and indirect methods, respectively.

For the continuous inverse problem (3.3.1), uniqueness of a solution for T is often given in terms of regularity assumptions on V(p and A(p plus conditions on the availability of transmissivity data (often referred to as Cauchy data). Chicone and Gerlach (1987) have provided the most recent analysis of this question and provide criteria that ensure uniqueness that do not specifically require the existence of Cauchy data. For example, they prove uniqueness of the solution on all subdomains of the aquifer where all entering characteristics stay within the domain. In such cases, the Cauchy data is implicitly provided by the flow equation.

As regards the discrete version of the problem (3.3.1) with model and measurement errors included, the situation is usually too complex to provide general criteria which allow conclusions to be drawn about uniqueness from the properties of the available data alone. Usually such criteria most often also assume a fixed a priori discretization for the unknown transmissivity (see for example Carrera and Neuman, 1986). To illustrate this point, consider the discrete problem (3.3.4) or (3.3.7) associated with a direct method. The solution for 7/ is obviously unique if and only if the matrix D has full rank. However, such a criterion is of limited use since it is only valid for the particular form of the discretization

chosen for T.

In general, one can only hope to provide statements about the asymptotic properties of the discretization scheme at hand. For example, if the data is exact and the continuous problem has a unique solution, then

for any reasonable discretization scheme, such as the spectral expansions used in C h ap ter 6, the solution will be unique an d converge to th e tru e solution as th e discretization size goes to zero. If th e d a ta are corrupted w ith noise, it m ay be still possible to make useful statem en ts provided some fu rth e r a ssu m p tio n s are invoked. An exam ple of th is can be found in R ichter (1981b) an d F alk (1983) w here proofs of existence and uniquenes based on assu m p tio n s involving th e d a ta an d b o u n d ary conditions are given. F u rth e rm o re , R ich ter an d F a lk provide e rro r b o unds on th e estim ates. A lthough extrem ely useful for the in sig h t R ich ter a n d F alk's analyses provide, th e ir resu lts are u n fo rtu n ately not g en eral enough to cover situ a tio n s often en co u n tered in p ractice. F or exam ple R ich ter requires Cauchy d a ta on the inflow portion of th e aquifer b oundary while Falk assum es th a t prescribed flow boundary conditions are know n exactly.

In tu itiv ely , we can see th a t uniq u en ess of a solution for p ractical problem s is going to be linked to the difference betw een th e am o u n t of available d a ta and the num ber of transm issivity coefficients the m odeller is attem p tin g to reconstruct. Roughly speaking, uniqueness can be ensured if the num ber of independent pieces of inform ation in the head d a ta is a t least as large as the level of param eterization of th e unknow n transm issivity. In other words, a solution m ay be unique if the essen tial dim ension of the problem , also called th e ran k , is a t le a st as large as th e n u m b er of unknow n tran sm issiv ity param eters (Newsam, 1982, 1984, 1985). Q uoting Newsam (1984) we have th a t

" Once the rank is known, it should serve as a guide to the size of discretizations of the problem and of experimental data collections. One should expect that if either is much larger than the rank then the extra effort yields no improvement in results, whereas if either is much smaller, then significant information is not retrieved. In extreme cases it should help settle decisions such as whether the information recoverable is worth the cost of obtaining it. A prerequisite for this is that the rank be estimable without the need to carry out the experiment or to perform extensive numerical calculations."

U n f o r tu n a te ly , due to th e n o n -lin e a r d e p e n d e n c e b e tw e e n transm issivity T and piezometric head (p in (3.3.1), the com putation of the essen tial dim ension following the procedure proposed by N ew sam is not obvious. C onsequently, it appears th a t for th e tim e being only tria l and e rro r approaches can provide an ap p ro x im atio n for th e r a n k of th e problem. This is for exam ple the case in Yeh et al (1983), Yeh and S un (1984) and C arrera and N eum an (1986) where decisions on tran sm issiv ity z o n atio n a re m ad e from tr ia ls on th e g eo m etry of th e z o n atio n supplem ented w ith statistics to select the best p attern . Incidently, we would argue th a t instead of zonation, it m ight be easier to use generalized Fourier expansions for transm issivity and search serially for the lower order term s

as m entioned in C hapter 2.