The reconstruction of a h ead surface from sc attere d and noisy head m easu rem en ts for the purpose of tran sm issiv ity estim atio n is a difficult challenge for several reasons. The first obvious one is th a t w ith aquifer transm issivity, recharge and/or boundary conditions only p artially known, th e p ro cess g e n e ra tin g th e d a ta is p a r tia lly u n k n o w n . Second, m easurem ents are usually only available on irreg u lar grids which are not dense enough to contain all the featu res of the continuous head surface. This m ay induce significant interpolation errors. For example, if the d a ta is sp arse and irreg u larly d istrib u ted , sin g u laritie s in the h ead surface generated by point sources or sinks can be difficult to reproduce. Finally, the use of the in terp o lated h ead surface for su b seq u en t tran sm issiv ity estim ation requires th a t the interpolation be as accurate as possible.
Two different approaches to h ead in terp o latio n are possible. The first one is practical. After discretization of the aquifer, a flow n et is draw n from th e s c a tte re d h e a d m e a s u re m e n ts a n d from a n y a v a ila b le p rio r in fo rm a tio n w hich m ay in clu d e know ledge of g e n e ra l a q u ife r flow direction, presence of m ounds, im pervious vertical boundaries, etc. From the contour lines, head values are inferred a t each node of the aquifer grid. An exam ple of th is type of in terp o latio n can be found in G hassem i et al (1987).
In c o n tra st, au to m atic in te rp o la tio n ro u tin e s often s ta r t w ith an expansion of th e piezom etric h ead surface in term s of a set of b asis functions 0 p (z) , p = l,...yP i.e.
<p(z)
2 W
p-1(6.4.1)
and th en solve the m inim ization problem
m in X (<P*m-< P (zjy ^P aL[(p(z)] m = 1 (6.4.2) w here
(p*m are h ead m easurem ents;
z m a re th e lo c a tio n s of th e h e a d m e a s u re m e n ts w ith
Zjn (3cmym) in Cl ,
a is a positive scalar; and
L[(p(z)] is a positive functional th a t am algam ates prior inform ation about the system.
The disadvantage of autom atic interpolation m ethods over m an u al ones is th a t the experience and knowledge th a t the hydrogeologist h as about the aquifer system m ay not be easily quantified so th a t its autom atic insertion into th e interpolation algorithm m ay be very difficult if not impossible. In addition, th e selection of a n optim um set of basis functions [&p) and p a ra m e te r a for th e p a rtic u la r problem a t h a n d is u su a lly fa r from trivial. On the other hand, m an u al m ethods are likely to be contam inated by a subjective bias caused by th e hydrogeologist's own preferences and beliefs. They are also tim e consuming.
In order to assess th e perform ance of in terp o latio n m ethods in th e context of aquifer identification, it is not apposite to tu rn to th e lite ra tu re since, in m ost cases, th e q uestion of h e ad in terp o latio n receives little m ention. For example, in th e review by Yeh (1986), out of a to tal of eleven papers th a t deal w ith direct approaches, only th ree indicate th e type of interpolation technique th a t was used. In view of th eir rarity , they deserve some m ention. They are
1) G eneralized orthogonal regression
by N elson (1968) w here th e h e ad surface is recovered by orthogonal regression. However, Nelson does not provide any detail about th e featu res and properties of th is interpolant, referring instead the in terested read er to a report w ritten by O ster (1963). Having not yet been able to obtain a copy of
th is report, we cannot comment on th is interpolation technique. 2) M ultid im en sio n al splines
reported in S agar et al (1975). However, a t the tim e of publication, Sagar et al stated th a t
"...the use of splines in multidimensional interpolation has not been analyzed and appears to be a difficult subject".
In ad d itio n to m u ltidim ensional splines, S ag ar et al refer very briefly to L agrange polynom ials. In both cases, only very lim ited in fo rm atio n is given w ith reg ard to the perform ance of the interpolation schemes.
3) K rig in g
by Yeh et al (1983) w here interpolation is perform ed following techniques borrow ed from geostatistics. In a few words, kriging is a n in terp o latio n technique based on lin ear estim ation theory (Journel and H uijbregts, 1978) w hich e stim a te s values a t points by th e lin e a r com bination of a ctu al m ea su re m e n ts h a v in g m inim um variance (the m ethod is described in detail in C h ap ter 8). In m ost cases, kriging algorithm s are constrained to be exact in te rp o la n ts. C onsequently, they provide no sm oothing in th e neighbourhood of m e a su re m e n t points. Typically, for one-dim ensional cases (K itanidis and Vomvoris, 1983), the interpolated function will have the profile given in Figure 6.2.
• point measurements ________ interpolation via kriging
---- smooth interpolation
Figure 6.2 Kriging interpolation vs smooth interpolation.
Therefore, kriging m ay introduce spurious effects in th e neighbourhood of the m easu rem en t locations th a t m ay resu lt in high frequency noise. This is
clearly highly undesirable for ill-posed problems. However, Yeh et al. give no assessem en t of th e effect of such erro rs in th e aquifer identification problem .
W ith regard to the rem aining 8 papers surveyed by Yeh (1986) in which no m ention of specific in terp o latin g algorithm s were m ade, our conclusion is th a t the head d a ta h ad already been provided a t each node of the aquifer g rid .