The approach to aq u ifer identification p resen ted la te r in Section 6.5 requires th a t the head surface be expanded in term s of ten so r products of Chebyshev polynomials over the subdom ain (W. W ith h ead m easurem ents available a t each node of a reg u la r grid contained in the expansion of the su itab ly scaled h ead surface in term s of C hebyshev polynom ials is straig h tfo rw ard .
Before p re s e n tin g a n ex p an sio n of th e piezom etric h e a d over a tw o-dim ensional subdom ain ‘W in term s of tensor products of Chebyshev polynomials, let u s first consider a one-dim ensional expansion over the interval [-1,1]. To do so, let L 2([ -l,l] ,w ) denote the H ilb ert space of real functions defined on the interval [-1,1], endowed w ith the inner product
l
(f>g) = jw (x) f(x) g(x) dx (6.4.7)
- l
where w(x) is a weighting function given by
w(x) = (1-x2) 1/2
The set of Chebyshev polynomials {Cn(x) I n = 0,1,...} defined by
c < y 1
(6.4.8)
CJx) = -j=r cos[n arc cos(x) ] for n = 1 ,2 ,...
-Ik/2
constitute a complete set of orthonormal polynomials in L 2([-l,l],w ) (Fox and Parker, 1968). Orthonormality of the set means that
l
^w(x) C Jx) Cn(x) dx = Smn (6.4.9)
- l
where 5mn is the Kroenecker delta. Furthermore, with completeness of the set, for any function f in L 2([-l,l],w ) we have
N
lim
X (f-CJ
C Jx) = f(x) (6.4.10)N ^ ° ° n =0
where the convergence is in the norm induced by the inner product (6.4.7). Chebyshev polynomials have many properties that are very useful in approximation theory. A detailed presentation of the use of Chebyshev polynomials in numerical analysis can be found in Fox and Parker (1968) while a considerable number of useful formula are given in Abramovitz and Stegun (1972). For the sake of presentation, further properties of Chebyshev polynomials required in this Chapter have been relegated to Appendix A7.
The extension of the one-dimensional case to real functions defined on
scaled so th at the square subdomain W is [~ l,l]x [-l,l] as illustrated in Figure 6.3.
Mmi
• head measurements
Figure 6.3 Two dimensional aquifer Q with a square subregion <W included in it and containing head measurements on a regular grid.
In this case, let L 2( [ - l, 1 ] x [ -l, 1 ], w) denote the Hilbert space of real functions defined on [-l,l]x [-l,l], endowed with the inner product
l l
(f,g) - ^ w (x )w (y ) f(x,y) g(x,y) dxdy (6.4.11)
- l - i
The set of tensor products of Chebyshev polynomials
°mn(X’y) = CrrfX) Cn ^ m >n = °> 1> 2’ ”* (6.4.12) forms a complete set of orthonormal functions in L 2( [ - l , l ] x [ - l , l ] , w)
(Courant and Hilbert, 1953). In other words, for any function f(x,y) in
L 2([-l, 1 ]x[-l, 1 ], w) we have
M N
lim X = K*y) (6-4.13)
M ,N->oo m=( ) n =Q
where the convergence is in the norm induced by the inner product (6.4.11). With head measurements available at each node (x^yß of the aquifer, equation (6.4.13) is used to expand the piezometric head surface over the subdomain as follows. Upon division of the domain M into the squares
% = [Xi,xi+1] x ^ y j+1 ], the head (p(x,y) over W- is assumed to be a bilinear function of x and y with (p(xityj) = (p*tj where (p*tj is a head measurement at node (xity-). With this, we obtain the finite and hence approximate representation
(KW) =
X X
^pqCp(x)Cq(y) (6.4.14) p = 0 q =0where the generalized Fourier coefficients t; are given by
l l
Zpq = jjw(x)w(y) y(x,y) Cp(x)Cq(y) dxdy (6.4.15)
- l - i
The details of the computation of (6.4.15) are given in Appendix A8.
At this stage we would like to emphasize two points. First, the above procedure to compute %pq is certainly not the most efficient. For example, techniques based on the Fast Fourier Transform (Press et al, 1986) would be better from a numerical point of view. However, the location of the measurement points in this latter case has to follow a certain pattern which complicates the computation. Secondly, one may wonder if the use of (6.4.2), i.e. a 'pointwise' approach, to compute t;pq in (6.4.14) would not be somehow more appropriate than the 'global' approach given by (6.4.15). We have performed several numerical experiments to test (6.4.2) with the smoothing functional L[(p(z)] involving the linearized curvature of the head surface (p. Insofar as using such a head interpolant for the estimation of transmissivity, the results were much less accurate than those obtained when the head surface was reconstructed from (6.4.15).
Note finally th at a Chebyshev expansion for the interpolation of the recharge q can be constructed following exactly the above methodology. Recall, however, th at for the sake of presentation the aquifer recharge q is assumed known.
6.5 ESTIMATION OF TRANSMISSIVITY BY GALERKIN
METHODS
With the aquifer recharge q assumed known and availability of a
A
Chebyshev expansion (p of the head over the subdomain (W, we are now in a position to reconstruct the transmissivity T over W by use of the aquifer
flow equation (6.1.1).
A
As m entioned previously, w ith th e presence of errors in (p we can only recover low frequency com ponents of tran sm issiv ity . C onsequently, th e second step in our identification procedure is to expand tran sm issiv ity in term s of a complete set of orthonorm al functions over W , and a ttem p t to recover th e first coefficients. For the tran sm issiv ity expansion, we select again ten so r products of C hebyshev polynom ials since, as will be shown later, this simplifies the com putational b u rden associated w ith the m ethod. W ith this T(x,y) I J