Totals for Compartments 



CT^ = compartment total of {character} in all strata 



F, = count of photo points in stratum "/z" in compartment "p" (exclusive of 



hp 



ground- sampled subcompartments) 



X = acres of forest land in compartment "p" (exclusive of ground-sampled 

 P acreage) 



CT = Y ^ V • + y TVA.. yA.. 6,., (5) 



p ^ p 1 F^^ h ^ hx %k. n%K ^ ' 



Z i k 



In expression (5) , i ranges over only the sampled subcompartments within 

 compartment "p." 



Forest Totals 



Combining all the preceding estimation formulae into an estimate of the forest 

 grand total {FGT) , 



TV A 



FGT 



^hp ^ ^hi E n, 6 



h 



^ hzk ^ kj 'kj 



k hik 



I 



n^ 



5 '^^^hi ^ ^ik ^hik 

 ^ k 



(6) 



where 



J indexes individual trees 

 k indexes the stands 

 i indexes the subcompartments 

 p indexes the compartments 

 /Zj Z index the strata 



For purposes of error calculation , th ere are two random variables in the samplin 

 procedure. These are designated W and TVA^ in expression (6). 



W is the estimated stratum weight. The expression for the variance of FGT must 

 include a component arising from W because it is an estimate that is subject to 

 sampling variation. The usual derivations of variance-estimating formulae for two- 

 stage sampling plans assume that the stratum weights follow a multinomial distribution 

 In the design we have described, the systematic nature of the photo-point grid will 

 result in estimates having somewhat lower variance. Williams (1956) has demonstrated 

 how the variance of the mean of systematic samples such as these can be evaluated. 

 However, the planned spacing between photo points in the compartments is large enough 

 that the correlation between strata identified at points separated by 4/10 mile should 

 be quite low. Accordingly, the multinomial distribution should be a reasonable basis 

 for evaluating the variance of the first-stage sample of stratum weights. 



14 



