|
<br />a. ,
<br />1J
<br />
<br />ta2
<br />Qsi
<br />
<br />Cov(Q ., Q .)
<br />S1 SJ
<br />
<br />otherwise.
<br />
<br />for i=j
<br />
<br />(19)
<br />
<br />The increase of spring runoff from grouped basins,
<br />6Q*, is given by
<br />s ,
<br />
<br />6Q; = a16Qsl + aZ6QsZ+"'+ an Qsn
<br />
<br />n
<br />1: ai6Qsi'
<br />i=l
<br />
<br />(ZO)
<br />
<br />where 6Qsi (i=l,Z,. ..,n) represents the increase in
<br />
<br />spring runoff from an individual basin. Now impose
<br />the restriction that
<br />
<br />Q*
<br />s
<br />
<br />n
<br />1:
<br />i=l
<br />
<br />n
<br />1: Qsi
<br />i=l
<br />
<br />(Zl)
<br />
<br />a,Q.
<br />1 S1
<br />
<br />where Q; is the mean of the
<br />the mean of the
<br />
<br />values and Qsi is
<br />impose the restric-
<br />of the 6Qsi values,
<br />
<br />Q*
<br />s
<br />Also
<br />
<br />tion that 6Q;
<br />i.e. ,
<br />
<br />Q . values.
<br />S1
<br />is equal to the sum
<br />
<br />6Q*
<br />s
<br />
<br />n
<br />1: ai6Qsi
<br />i=l
<br />
<br />(ZZ)
<br />
<br />n
<br />1: 6Qsi
<br />i=l
<br />
<br />Finally the number of years, N*, for evaluation of
<br />grouped basins is given by the following expression:
<br />
<br /> n n
<br />3. 84aQ* 3.84 1: 1: a, ,a.a.
<br /> i=l j=l 1J 1 J
<br />N* s
<br />(6Q*)2 (6Q*)2
<br />s s
<br />
<br />n n
<br />1: 1:
<br />i= 1 j =1
<br />
<br />a. ,a.a. (Z3)
<br />1J 1 J
<br />
<br />where the a, and a. are as yet arbitrary but sub-
<br />1 J
<br />ject to the constraints expressed by equations (Zl) and
<br />(ZZ). Choose the ai's such that the number of years,
<br />
<br />N*, is kept to a minimum value. Setting
<br />
<br /> n n
<br />f(a1,Ctz' ...J an) L L a. ,Ct.Ct.
<br /> i=l j=1 1J 1 J
<br /> n n
<br />gl (Ct1,CtZ' Ctn) I (Q .Ct) ( \ Qsi)
<br />.. ..J L
<br /> i=1 S1 1 i=1
<br /> n n
<br />gZ(Ct1,aZ' an) \" C3QsiCti) - ( \' 6Q .)
<br />.. .. . , i L
<br /> i=1 i=1 S1
<br />
<br />a new function is defined
<br />
<br />F(al,CtZ" ,Ctn,Al,AZ)=f(al'CtZ'" ,Ctn)-Algl (al,aZ' .. ,an)-
<br />
<br />AzgZ(al,Ctz,..an)
<br />
<br />The ai's that make the objective function F(al,aZ,..,ad
<br />
<br />in equation (Z4) minimum give the minimum value for N*
<br />in equation (Z3).
<br />
<br />By taking the partial derivative of F(al,aZ,..an,
<br />Al,AZ) with respect to the Cti'S, AI' and AZ and setting
<br />each derivative equal to zero, one obtains the system
<br />of equations:
<br />
<br />n n
<br />of _ \' \' Q
<br />o L akJ,CtJ. + l. aikc\ ~ Al sk - AZ6Qsk
<br />a; - j=1 i=1
<br />
<br />Z
<br />
<br />n
<br />L akiCti - QskAl
<br />i=1
<br />
<br />o
<br />
<br />6QskAZ
<br />
<br /> for k 1,Z,..,n
<br />of n n
<br />~ L Q .a. + ( L Qsi) 0
<br />i=1 S1 1 i=1
<br />of n n
<br />as ~ 6Qs i Cl i + ( L 6Qsi) 0
<br />i=1 i=1
<br />
<br />~I
<br />
<br />I
<br />
<br />or in matrix notation
<br />
<br />Zall
<br />Za2l
<br />
<br />2a12
<br />2a22
<br />
<br />2aln - Qsl - 6Qsl a1
<br />2a2n - QsZ - 6Qs2 Ct2
<br />
<br />o
<br />
<br />o
<br />
<br />2an1 2an2 2a - Qsn - ':] a 0
<br />nn n
<br />Q~1 QsZ Qsn 0 Al d <l;,J ~
<br /> 1=1
<br /> n _
<br />~QSl 6QsZ 6Qsn 0 AZ ( L IIQ .)
<br /> i=1 S1
<br /> (25)
<br />
<br />The system of equation (25) is linear and its resolution
<br />for the unknown ai's is obtained by the Gaussian
<br />
<br />elimination procedure. Thus a procedure is described
<br />that objectively selects the optimal group of basins of
<br />a given size among a larger set. The procedure also
<br />determined the optimal parameters of the combination of
<br />runoff variables for minimum time evaluation.
<br />
<br />(Z4)
<br />
<br />It remains to apply this technique in practice to
<br />the Upper Colorado River Basin. Before doing so,
<br />Chapter IV describes the data used in the analysis.
<br />
<br />16
<br />
|