WMOD00545

• Project Name: Colorado River Basin Pilot Project
• Title: Test of Runoff Increase Due to Precipitation Management for the Colorado River Basin Pilot Project
• Date: 3/3/1973
• Weather Modification - Doc Type: Report

MARCH 1973

H, J. MOREL-SEYTOUX AND F. SAHELI

332

b. Hydrologic significance of Q*

From its definition [Eq. (3)J Q* is a combination of
runoff LV. However, Q* per se has no hydrologic mean-
ing. It is an "artificial" runoff, since the Qi are both
runoff quantities and random variables but Q* is only
a random variable. Hydrologic meaning may be given
to one parameter in its distribution, namely its mean,
by requiring of the Xi that they satisfy the constraints:

n n
Q/= L XiQi= L Qi, (16)
i=l i=l

n+m n+rn
Qc*= L XiQi= L Qi' (17)
i=n+l i=n+l

The hydrologic interpretation of (16) and (17) is that
the expectation of the random variable Q,* is the mean
of the total runoff for the group of n basins in the target
area and similarly in the control area. The above
equations are more meaningful constraints than the
normalization ones defined by (6) and (7). Similarly,
in the target area, one can impose the constraint that

n n
~Qt*=L Xi~Qi=L ~Qi.
i=l i=l

The hydrologic interpretation of (18) is that the mean
increase of Q/ is that of the total runoff for the group

of n basins. This constraint is crucial because there
would be no point in reducing the variance of Q/
given Qc* if at the same time the quantity ~Q/ was
also reduced considerably below that of the mean
increase of the total runoff, In addition, because the
~Qi used in (18) are only rough estimates, some may
be underestimated and some overestimated. The
calculated ~Q/ from (18) may be somewhat different
than the observed total increase in runoff during the
years of cloud seeding. That chance becomes partic-
ularly serious if some of the Xi, solutions of the minimi-
zation of .\1* subject to the constraints defined by
(16)-(18), turn out to be negative. In the extreme case
the ~Q7' calculated on the basis of the observed ~Qi
during the years of seeding rather than the estimated
ones could even be negative. For this reason the Xi
in the target area are subjected to a non-negativity
condition, namely:

Xi~O, for i=1,2,"'n.

(19)

(18)

5. Minimal time detection test design

In summary, the most powerful test of the effect of
precipitation management on runoff in the Colorado
River Basin Pilot Project can be deduced from the
solution to a minimization problem. The formulation
of the minimization problem is the following.
We minimize the objective function lV*, defined as

n n+m n+m n+m
3.84{L L XiXjaij-[(L L X;Xkb'k)2/( L L XkXICkl)J}
i=1 j=l i=l k~n+1 k=n+1 l=n+l

.V*=

(20)

(where ai) is an element of the covariance matrix in
the target area, bij an element of the covariance matrix
between target and control areas, and Ckl an element of
the covariance matrix in the control area) with respect
to the Xi, where i=l, 2, .. 'n, n+1, ., 'n+m, subject
to the constraints:

n n n
L XiDiQi+L (l-Di)Qi=L Qi,

i=l

i=L

i=l

n n n
:C XiDi~Qi+ L (l-Di)~Qi= L ~Qi,

i=l

i=l

i=l

n+m n+m n+m
~= XiOiQi+ L (1-Di)Qi= L Qi,
1=n+l i=n+t i=n+L

Xi~O, for i=1,2,.. .n,

(L Xi~Qi)2
i=l

n
L Di=lIt
i=l

(25)

n+m
L Di=lIc,
i=n+l

(26)

(21)

where Oi is a variable that takes only two values, zero
or one, depending upon whether the corresponding
variable Xi is zero or non-zero, respectively, and where
IIt( ~ n) and IIc( ~ m) are the maximum number of
target and control runoff variables to be used in the
combinations Q/ and Qc*, respectively. The solution
of this minimization problem will systematically select
which basins should be used in the significance test and
with which weights.
The mathematical programming problem mentioned
here is not a standard one. It was solved successfully
by a modified version of the Jacobi differential algorithm
(Wilde and Beightler, 1967). The details of the method

(22)

(23)

(24)