On the characteristics of Coefficients in SST Estimation Functions by Split-Window Method

In t h e sea s u r f a c e temperature (SST) estimation by t h e window method f o r AVHRR da ta , two kinds of s t r u c t u r e are popularly assumed in t h e regression analysis. DVF: Y = (c-b)X4 + b(X4-X5) + a SWF: Y = X 4 + g ( x 4 X ~ ) + d where Y , X 4 and X 5 are t h e est imated SST. t h e br ightness tanmatures of ch. 4 and ch. 5 , respectively. The degree of freedom in the coeff ic ient of DVF is two, bu t t h a t of SWF is one. In many empirical resu l t s , however, DVF has similar coeff ic ients t o those in SWF. In th i s paper, t h e s imilar i ty of coeff ic ients i s investigated and showed t h a t i t comes from t h e specif ic charac te r i s t ics of t h e d a t a set provided by t h e split-windows. The assertion was confirmed by t h e simulation by using Mutsu Bay match-up d a t a set. 1 I " X 1 0 N The split-window method is known as an excellent algorithm f o r the correct ion of t h e atmospheric e f f e c t s in t h e sea s u r f a c e tempera ture (SST) estimation. Ch. 4 and ch. 5 of AVHRR occupy t h e ad jacent thermal in f ra red bands in t h e atmospheric window. The SST i s es t imated by a l inear function of those br ightness temperatures . For t h e purpose, two typical s t r u c t u r e s have been used, i. e., DVF (dcuble variable function): Y = cX4 bX5 + a SWF (split-window functicn) : = ( c b ) X ~ + b(X4-X.) + a Y = X 4 + g ( x 4 X ~ ) + d where Y i s t h e estimated SST, and X I and X 5 are t h e brightness tempera tures of ch. 4 and ch. 5, respectively. The s t r u c t u r e of SWF comes f rom t h e anlysis to t h e rad ia t ive t r a n s f e r equation. That is, t h e re lat ion between ( Y x , ) and ( X 4 X 5 ) can be approximated to be linear (Maclain 1981. McMillin and Crosby 1984) . Y i s meant t h e s e a t r u t h SST. There are o t h e r dis turbances in t h e SST estimation. Then DVF is introduced as a generalized s t r u c t u r e of SWF. Both DVF and SWF are linear functions, bu t they have d i f fe ren t degrees of freedom in the coefficients. DVF has of two f o r b and c. but SWF has of one f o r g only. The higher degree of freedom can be more flexible t o compensate collaborating e r r o r s totally. The coeff ic ients of SST estimation funct ions a r e usually calculated by applying the regression analysis t o match-up sets of Y. X 4 and X 5 observed direct ly o r indirectly. Table I is a l is t of SST estimation functions proposed by various authors . Both DVF and SWF exis t in the list , but i t is very interest ing that the var iat ions of the i r coeff ic ients remain v e r y small. That is. (1 ) ( c b ) remains near ly eqlual t o one, (2) b and g are r e s t r i c t e d in a narrow range, and (3) a and d are r e s t r i c t e d in a narrow range. Those s imilar i t ies have been understood as a supplementary proof of t h e e f fec t iveness of t h e split-window method (McMillin and Crosby 1 9 8 4 ) . 2 (XIEFFcms WE To m m m ANALYSIS Asaune that a set of mat&-ups IS = { ( Y , , x4 ,. xs d I i l l -NI i s given, where y , , x 4 , and x5 , are observed values of Y , X I and X 5 at a c e r t a i n point and time. i i s the identification number and N is the total number of the match-ups. For general var iables of V and W. descr ibe t h e covariance and t h e cor re la t ion coeff ic ient of them as SV w and RV W . respectively. But f o r t h e convenience, t h e descriptions will be abbreviated as follows. Sx4.x4 4 S . 4 . S X 5 . X S 4 s 5 5 , su Y 4 s w , s x 4 .x5 4 s4 5 . su. x 4 4 s u 4 , s u x 5 4 sus, R x 4 .x5 4 R 4 s . R y . x 4 4 Rv4 , R u . x 5 4 R u 5 The next formulas are obviious f rom t h e s ta t i s t ica l theory. S45 = R 4 5 E X E , Sv4 = R u 4 / X X T , ( 1 )