where a is a column vector
of regression parameters, W= [[X.sub.0] [W.sup.+]] and [W.sup.+] = [EDFATHER EDMOTHER FAMINCOM CIGTAX] with
Each row vector represents a fracture that contains the four variables, and each column vector
represents one of the variables that contains m (or n) rows.
An FFT of a column vector
v with length n is Wv, where W is given in (4).
where [theta] = [[PSI].sup.T]x is the n x 1 column vector
of projection coefficients, [[theta].sub.i] = <x, [[psi].sub.i]> = [[psi].sup.T.sub.i]x is the projection coefficient.
Here the column vector
[x.sub.i] (j) = [[x.sub.1](j), [[x.sub.2] (j), ..., [x.sub.m] (j)].sup.T], j = 1, 2, ..., n.
Product Aw is a column vector
whose components are sums of many addends if n is sufficiently large natural number.
This column vector
l(r), referred to as the lead field vector, represents the sensitivity of the sensor array in the direction of [eta] at the location r.
where a column vector
[d.sub..j] and a row vector [d.sub.i.] have appropriate sizes.
where [[??].sub.i_cal] is the ith column vector
of the standardized calibration flight matrix [[??].sub.cal]; [A.sub.i_cal] is the ith column vector
of the calibration flight matrix [A.sub.cal]; [[bar.A].sub.i_cal] and [s.sub.i_cal] are, respectively, the mean and standard deviation of the corresponding observed data.
Each column vector
of Y corresponds to either a reservoir model or an encoded reservoir model because [N.sub.para] is determined depending on the type of assimilation algorithms in this study.
For this purpose, [D.sub.i] (i = 1, ..., K) will be transformed to an n-dimensional column vector
in the finite field.