TM No. 377 



(i.e., a finite length sample) in the ensemble. (See Lee, I960..) For example, 

 let an experiment last T seconds and he repeated N times. Let X,°j he a sample 

 from the (Tl* experiment at time J . If the statistical properties of X/i , for 

 fixed j and f from O to N, are the same as those for fixed »' and J from to T 

 (as both N and T tend to infinity) the process is ergodic. Thus, in a sense, 

 the ergodic hypothesis is an extension of the concept of stationarity applied 

 to both time and space for the complete data ensemble. 



Note that for the definition the spatial distribution concept has been 

 applied to ensemble. What has been described is sometimes referred to as 

 a homogeneous statistical environment! i.e., a system which contains both 

 time and spatial stationarity Tne physical implications of this concept 

 should be emphasized. One can learn about a fundamental ergodic process 

 occurring over a large physical region by sampling only a relatively small 

 volume (or, in fact, a single point) within tne region as a function of 

 time. Conversely, one can theoretically learn as much about this process 

 by an instantaneous sampling over the whole region. From an experimental 

 point of view, the first method of making observations is obviously more 

 applicable to ocean wave study. However, both types of measurements have 

 been applied to free surface wave studies. The first method is exemplified 

 by wave staff observations at a point in the ocean as a function of time. 

 The second method is demonstrated by observations (stereo photographs) of 

 waves on a large area of ocean at a given instant of time (see Chase, et al, 

 1957)o 



The existing mathematical models for time series analysis incorporate 

 both stationarity and ergodicity concepts. It is obvious, however, that 

 no geophysical process can be stationary in the mathematical sense. The 

 geophysicist must be satisfied with the statistics of the process under 

 study if he can state that they change very little in the duration of the 

 sampling. If this condition holds, then the mathematical abstraction will 

 assist his analysis! albeit, somewhat imperfectly. 



Planning Data Sampling for Spectral Meas urements — Tne plans for the 

 spectrum analysis were guided by discussions with Professors V, P. Starr 

 and E. N. Lorenz of the Department of Meteorology at M.I.T., and with per= 

 sonnel in the NUWS computer laboratory. Also consulted were the references 

 of Blackman and Tukey (1958), Miller and Kahn (1962), Kinsman (i960), and 

 Munk and Mac Donald (i960). 



The value of a spectral measurement program is heavily dependent upon the 

 following; (l) the quality and quantity of data.; and (2) the desired pre- 

 cision, resolution, and cutoff frequency. Each estimate of the energy (or 

 contribution to the total variance) obtained from a finite data record for 

 a particular frequency is actually an approximation of the average value 

 contained within a bank centered at the nominal frequency. The spectral 

 values are evenly spaced on a frequency interval from zero to the so-called 

 "Nyquist" frequency. This is defined by °. 



*" = IKr » (111-2) 



50 



