92 



Fishery Bulletin 105(1) 



PAf) = Py(f)-pM) 



because the power spectrum for white noise is constant 

 with frequency: p^{F)=2o~. Because power spectra are 

 nonnegative definite and p/f) does not depend on f, the 

 maximum possible observation noise variance K corre- 

 sponds to the minimum of p^,{f) over /"(see Fig. 1). Thus, 

 Equation 9 can be recast as 



(10) 2p which we denote as Elzjy), is a symmetric moving 



average filter of {y,| (Cleveland and Tiao, 1976; Box et 

 al., 1978): 



E{z,\y) = (0(B}y„ 



K'= min {pAf)]/2. 



0<f<l/2 l- ' 



such that 



CO(B}: 



(11) 



< a(B)a(F) 



of, r\(B)ri(F) 



al (p(B)(p(F) 

 a] ri(B)ri(F) 



(13) 



(14) 



Note also that Equation 10 can be recast again as 



p^f) 



^-^Ap^f)=(o(f)p^f), 



Py^f^ 



(12) 



where w (/) represents the term in parentheses. It will 

 be seen below that taif) is identical to the polynomial of 

 smoothing weights (Eq. 14) on the unit circle. 



Returning to the estimation problem now, knowledge 

 of af, then, together with the ARIMA model for the ob- 

 served process, is sufficient to estimate |2,| (Box et al., 

 1978). When t is not close to the endpoints of the time 

 series, the smoothed (maximum likelihood) estimate of 



5 



o 



Q. 



where the second equality follows from equation 8. Note 

 that o){B) is identical to o)(f) (Eq. 12) on the unit circle 

 (B=e ''•^'^'), so that Equations 13 and 14 are also inter- 

 pretable in terms of relations among power spectra. 

 Equation 13 is equivalent to Equation 2 in Pennington 

 (1985). It turns out ( Cleveland and Tiao, 1976) that equa- 

 tion 14 is also valid for t near the endpoints of the time 

 series; thus, one merely uses the ARIMA model for the 

 observed time series to hindcast or forecast additional 

 "observations" as needed. Box et al. (1978) describe a 

 method for calculating the coefficients of loiB); because 

 this reference may be difficult to obtain, we repeat their 

 description in the Appendix. 



In the case where the model for the underlying pro- 

 cess is (0,1,0) (i.e., a random walk model), the model 

 for the observed process is (0,1,1) (i.e., a 

 RWPUN, model). For this case, Penning- 

 ton (1985) noted that the value of the MA 

 parameter jjj of the observed process (6 in 

 his notation) is 



Frequency (Hz) 



Figure 1 



Example power spectra illustrating relationships between p^^f) (the 

 power spectrum of the observed time series), p^if) (the power spectrum 

 of the white observational noise, a constant), and p^(f) (the power 

 spectrum of the unobserved, underlying time series). In this example, 

 Pi,(f) = 0.25 so that pjf) has the same basic shape asp,,(/"), but is shifted 

 downwards at all frequencies by 0.25. p*^(f) represents the maximum 

 possible level, consistent with pjf) (and the ARIMA model for the 

 observed time series ly,)), for the power spectrum of the assumed 

 observational noise. Note that in this case, maximal smoothing (i.e., 

 taking p^(/")= p*(f)) eliminates all high frequency energy and would 

 result in over-smoothing. 



fJi 



(15) 



so that Equation 14 is completely deter- 

 mined by the ARIMA model for the 

 observed process and the observation error 

 variance, o~, can be estimated. 



Application of the noise reduction 

 algorithm to bottom trawl survey data 



We applied Box et al.'s (1978) noise reduc- 

 tion algorithm to 18 time series of abun- 

 dance indices for finfish (nine speciesxtwo 

 seasons; species are listed in Table 1) on 

 Georges Bank in the northwest Atlantic. 

 Time series for the fall survey spanned 40 

 years (1963-2002), and the spring time 

 series spanned 36 years (1968-2003). 



Stratified random bottom trawl sur- 

 veys have been conducted annually on 

 the northeastern continental shelf of the 

 United States from Cape Hatteras to the 

 Gulf of Maine in the fall since 1963 and 

 in the spring since 1968 by the National 

 Marine Fisheries Service (NMFS), North- 

 east Fisheries Science Center (NEFSC) 

 (Azarovitz, 1981; Anonymous, 1988; Reid 



