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<br />329 <br /> <br />JOURNAL OF APPLIED METEOROLOGY <br /> <br />VOLUME 12 <br /> <br />tioned on Qc* is given by <br /> <br />220 <br /> <br /> <br />19: <br /> <br />Var[Qt* I Qc*] = X/.Ett Xt- (X/.Etc Xc)2/ (Xc'.Ecc Xc). (5) <br /> <br />200 <br /> <br />Given the convariance matrix .E (a regional character- <br />istic) Var[Qt*IQc*] can be minimized by the proper <br />choice of the vectors Xt and Xc. Given Xc the minimiza- <br />tion of Var[Q/IQc*] with respect to Xt gives the <br />obvious solution Xt=O and the variance is zero. Clearly <br />the minimization problem is meaningless unless restric- <br />tions are imposed on X. Normalization conditions, <br />however, can be imposed on Xt and on Xc, for example: <br /> <br />180 <br /> <br />i 160 <br />... <br />~ <br />~ 140 <br /> <br />1913 <br /> <br />'0 <br /> <br />~ <br />i 120 <br /> <br />... <br />s 100 <br />~ <br />;;: <br /> <br />(6) <br /> <br />I:Xi=n, <br />i=l <br /> <br />19:1 <br /> <br />I 80 <br />V> <br />~ 60 <br />~ <br /> <br />n+m <br />I: Xi=m. <br />i=n+l <br /> <br />. Seeded Seoson <br />. Unseeded Season <br /> <br />(7) <br /> <br />In the case of the total target and control runoffs these <br />normalization constraints are satisfied since all the Xi <br />are equal to 1. However, the normalization constraints <br />can be satisfied by other choices of values of the Xi <br />than the values 1. In the problem at hand of detecting <br />a runoff increase due to cloud seeding, one can find more <br />physically meaningful constraints than the normaliza- <br />tion ones. <br /> <br />Target: South Fork RIO Grande 01 South Fork, Colorado <br />Conlrol: Arllmas River 01 Howordsvllle, Colorado <br />RegreSSion Line Based on 28 Years of Record <br />Significance Level: 98% (one - toiled fest 1 <br />Apparent Relahve Increase: 24 % <br />Apporent Mean Seasonal Increase: 27,000 acre-fl. <br /> <br />40 <br /> <br />20 <br /> <br />o <br />o <br /> <br />40 60 80 100 120 140 16') <br />Seasonal (April- August) Flow In Thousands of Acre-Feet <br /> <br />FIG, 6. Regression line between target seasonal runoff <br />and control seasonal runoff. <br /> <br />4. The hydrologic constraints <br /> <br />Explicitly in terms of the X the simple-looking <br />formula given by Eq. (2) takes the form <br /> <br />where tiQi is the expected change of mean runoff iin <br />basin i due to cloud seeding and AQ is the vector with <br />components tiQi. The first question regarding the!;e <br />tiQi is: how are they known? If they were precise:iy <br />known what is the need for a statistical evaluation? <br /> <br />[X/.Ett Xt- (X/.Etc Xc)2/ (Xc'.Ecc Xc)] <br />~- _ , W <br />(X/ AQ)2 <br /> <br />280 <br /> <br />:: 240 <br />... <br />.. <br />u <br />.. <br />_ 200 <br />0 <br />~ <br />1 160 <br />2 <br />>- <br />= <br />~ 120 <br />0 <br />G: <br />g <br />~ 60 <br />.. <br />VI <br />;; <br />'" <br />:s <br />>- <br /> <br />.... <br />. <br /> <br />1952 <br />. <br /> <br />1948 <br />. <br /> <br />. Seeded Season <br />. Unseeded Season <br /> <br />1962 <br />. <br /> <br />1967.19~ ~47 <br />19~4 :~6 _1939 1~43 <br /> <br />t953 _1955 <br />'9i79 _1946 -1954 <br />;-1940 <br />19S1 <br />"" <br /> <br />Target: South Fork Rio Grande <br />Control: Piedra River neor Piedra. Colorado <br />Regression Line Based on 27 Years of Record <br />Stgniftcance Level: 96% (one- toiled test) <br />Apparent Relative Increase: 18'% <br />Apparent Mean Seasonal Increase: 58,OOOacre-ft. <br /> <br /> <br />120 160 200 240 260 320 360 <br />Control Seasonal (April- August) Flow In Thousands of Acre - Feet <br /> <br />, <br />440 <br /> <br />480 <br /> <br />400 <br /> <br />o <br /> <br />40 <br /> <br />60 <br /> <br />FIG. 7. Regression line between target seasonal runoff and control seasonal runo/T, <br />