D irect vs Indirect methods

I t is in terestin g to note th a t in the reviews by Yeh and Kool et al , there is no com parison betw een direct and in d ire ct m ethods. Indeed, th e re appears to be no work reported in the literatu re th a t attem p ts to analyse the m erits and disadvantages of these two approaches.

The lite ra tu re dedicated to aquifer identification clearly indicates th a t m uch more research h a s been u n d e rta k e n on in d irect m ethods th a n on direct ones. For example, in the tables provided by Yeh, only two papers on direct m ethods a re referenced for th e period 1977-1986 w hile tw elve publications on indirect m ethods appear in the sam e period. This is also reflected in work rep o rted by p ractisin g hydrogeologists w here in d irect m ethods are clearly favoured. T his m ay be p a rtly due to th e fact th a t indirect m ethods are conceptually simple. F u rth erm o re, w ith decreasing costs in com puter tim e and availability of off-the-shelf packages to solve the

forw ard problem (3.3.1) and the non-linear optim ization (3.3.10), indirect m ethods m ay appear, a t first sight, easier to im plem ent.

Clearly, a m ajor difference betw een direct and indirect approaches is th a t th e form er m inim izes an equation erro r criterion th a t involves errors in the w ater balance a t each node of the discretized aquifer while th e la tte r m inim izes the erro r betw een the observed an d the computed h ead data. In doing so, the m ain advantage of direct m ethods is th a t they are linear in rf and lin e arity will be k ep t if th e m ethod is applied to th e general equation (3.2.2). A disadvantage is th a t th ey do not g u a ra n tee optim um m atch in g betw een m easu red an d com puted d a ta . C onversely, in d irect m ethods provide a best m atch. However, they are not lin e ar and therefore require an iterativ e search for the solution.

In order to fu rth e r examine the differences betw een both approaches, it is useful to single out th e steps req u ired in the im plem entation of each m ethod:

Direct method:

A

1) Recover a surface (p from the head m easurem ent vector <p* ;

2) D iscretize th e d istrib u ted p a ra m eters an d the flow equation and construct th e m atrix equation (3.3.4);

3) Use prior inform ation to select a functional L to be used in the stabilizing constraint ;

4) Compute an optim um value for a ; and

5) Solve the lin ear least squares problem (3.3.7).

In d irect m ethods:

1) Discretize the distributed param eters and the flow equation in order to solve the forward problem (3.3.1);

2) Use prior inform ation to select the stabilizing functional L ; 3) Compute an optim um value for a ; and

4) Solve iteratively the non-linear least squares problem (3.3.10).

The following rem arks m ay now be made:

Direct m ethods

A

Indeed th e success or failure of th e m ethod will depend strongly on how close the interpolant (p is to the tru e head (p.

2) I t is not n ecessary to know th e p iezo m etric h e a d b o u n d ary conditions in order to construct equation (3.3.7). Since in practice, b o u n d ary conditions are often only a n o th e r form of h e a d and tran sm issiv ity data, they can be included in th e d a ta base used for

A

th e reconstruction of (p. This im p o rtan t point is fu rth e r analysed in C hapter 6.

3) B eing lin e a r, d irect m ethods lead to sim ple a lg o rith m s. For exam ple, an estim ate for th e reg u larizin g p a ra m e te r a can be obtained by general cross-validation (Golub et a l, 1979).

4) If in an y of th e steps, n o n -lin e a rity is in tro d u c ed , ite ra tiv e algorithm s will be required. In such a case, th e direct m ethod loses a m ajor advantage. This would occur for in stan ce, if in ste a d of attem p tin g to estim ate tran sm issiv ity T, one was to estim ate the logarithm of transm issivity.

5) For the direct method, errors in the head m easu rem en ts ap p ear as model errors in th a t they affect th e m a trix D in (3.3.7). As a consequence, it may be difficult to invoke sta n d ard statistical theory to analyse properties of the solution of equation (3.3.7).

In d irect m ethods

1) A m ajor step is to choose and im plem ent an optim ization technique to solve the nonlinear least squares problem (3.3.10).

2) The piezometric head boundary conditions need to be chosen from th e o u tse t in order to solve th e fo rw ard problem . Since, as m en tio n ed above, b o u n d ary con d itio n s are o ften in p ractice constructed from scattered and noisy h ead and tran sm issiv ity data, th eir full specification requires some form of extrapolation th a t m ay introduce significant model errors.

3) The estim ation of a requires ad hoc sta tistic a l procedures which have only lim ited validity.

4) M odelling e rro rs such as in a p p ro p ria te b o u n d a ry conditions introduce errors in the computed head (p. This is likely to affect the noise term (p- qf* in such a way th a t its d istrib u tio n m ay d ep art seriously from norm ality.

From th is discussion it could be argued th a t th e m ain disadvantage of direct m ethods is th a t they do not provide p a ra m eter estim ates th a t ensure

optim um m atching of m easured and computed data. F u rth erm o re, despite th e lin e a rity of the form ulation in term s of the unknow n tran sm issiv ity , e rro rs in th e h e ad in te rp o la n t ap p ear in a n o n lin ear fash io n as model errors affecting D in (3.3.7). Therefore these errors m ay not be am enable to sta n d a rd sta tistic a l treatm en ts. An advantage en su in g from lin e a rity is th a t classical m a trix decom position techniques can be used to analyse p ro p e rtie s of th e p a ra m e te r e stim a te s an d s e p a ra te th e d iffe re n t components in the d a ta th a t affect th eir accuracy.

On th e o th er h an d , indirect m ethods yield p a ra m e te r e stim ates th a t provide a b est m atch betw een m easured and computed data. However, due to th e ir nonlinearity, it is difficult to m ake general statem en ts w ith regard to th e ir properties and, as is the case for direct m ethods, ad hoc statistica l procedures m ay be required. F u rth erm o re, the solution of th e no n lin ear regression (3.3.10) m ay depend strongly on the in itial p a ra m e te r estim ate

0 0 from which the iterative search is initialized. This is p articu larly tru e for aquifer identification where d a ta are scarce and levels of u n c ertain ty are high. Finally, it is not easy to separate the role th a t th e different d a ta com ponents play in the estim ates. For example, while th ere m ay be very little inform ation in the head d ata, a solution to the inverse problem of aq u ifer id en tificatio n m ay still be possible because enough ad d itio n al in fo rm atio n h a s been specified in th e b o u n d ary conditions. T his is p a rtic u la rly tru e for N eum ann type boundary conditions since in such cases, the sensitivity of transm issivity to changes in head values is greatly enhanced in the neighbourhood of the boundary (C a rre ra and N eum an, 1986). Indeed, in practice, m odellers alw ays a tte m p t to specify as m any N eu m an n b o u n d ary conditions as possible to tak e a d v an tag e of th is enhanced sensitivity.

As already m entioned, no q u an titativ e comparisons betw een direct and indirect approaches have been reported in the hydrogeological lite ra tu re . However, some in terestin g results relevent to both m ethods can be found in the m athem atical literatu re. They are presented in the next section.