148 A NEW METHOD OF ESTIMATING STREAM-FLOW 
EXAMPLE OF OBSERVATION EQUATION FOR NORMAL STREAM-FLOW 
The first form of observation equation for a least-square solution to determine 
the normal flow of a stream is that shown in equation (56). Since it is the intention 
here to present an example from the final solution which served to fix the unknowns 
for Stream A, equation (56) must be extended to include all the rR' terms indicated 
in equation (33) and otherwise modified in the manner to be indicated. The 
inclusion of all the rR"s is possible in the final solution because the laws of freezing 
and melting are assumed to be known, as derived in such steps as (1) to (6) 
above. Also — ' is assumed to be known. With these quantities known the com- 
plete observation equation for Stream A can be written in the following form: 
S.+r 1 R\+rJl\+rJl\+rtf\+rJt\+rtf\+r 1 R' 1 +rJl\+r, a R\ a +r n R' tl ,+r l o 
R\ -D' = v (58) 
The only differences between this equation and equation (33) are that D' has 
been placed on the left of the equality sign, the residual, v, indicated on the right of 
it, and the term r,R't has been divided into two parts, r ia R\ a and r, b R\ b . The 
last-named alteration was made in this particular case merely to determine whether 
the effect of the change in the storage as defined in the first half of r, (r, a ) was 
different from the effect as defined in the last half (r 96 ), on the stream-flow on the 
current day. The final numerical values to be presented later show that the dif- 
ference is small for Stream A, and the splitting up of the -f^R', term into two 
parts is not justifiable. 
Besides the alterations mentioned with reference to equation (58), one other 
alteration of equation (56) is advantageous. Instead of computing directly the 
unknowns S c , R' h R\, . . . R'i from the least-square solution in which the 
observed D' is used as absolute term, it is advantageous to make a preliminary 
substitution in observation equations, using approximate values of S c , R' h R',, 
. . . R'io as determined from previous solutions — as indicated in steps (1) to (6) 
above, or as estimated, or as obtained in any manner whatsoever — and to compute 
corrections to the assumed values of S e , R\, R'), . . . R\ from the least-square 
solutions, using the residuals in the preliminary substitution in observation equa- 
tions as the absolute term in the observation equations for determining the correc- 
tions. 
Thus, let the assumed values of S c , R'i, R' h R',, . . . R\ be designated by 
R", R"i, R"i, R",, . . . R"io, respectively, and let the residuals resulting from a 
preliminary substitution in observation equations be designated by — D. Then, 
using the r's, an example of the computation of which will be presently presented, 
the substitution in observation equation takes the form 
R''+r l R\+r i R\+rtf\+rtf\+rtf\+rtf\+r 7 R%+r i R\+r, a R\a+r, b R\ b +r 10 
R" 10 -D'=-D (59) 
— D is the discrepancy between the stream-flow computed from the assumed 
values of S c , R\, R\, . . . R' 10 ; that is from R", R\, R\, . . . R\ (with the 
use of the r's) and the observed stream-flow, D'. (Note that this is not the same as 
the — D in equations (42), (45) and (46).) Using — D as the absolute term, the 
observation equation, (58), now takes this form 
R+nRi+rtRi+rtRt+r^+riRi+rtRt+riRi+rsRs+r, a R„ +r tb R»b+rio 
R l0 -D = v (60) 
