SECT. 5] LONG-TERM VARIATIONS IN SEA-LEVEL 601 



4. The Analysis of Observations 



Variations in sea-level, both of a periodic and a secular nature, are in many 

 cases of great interest to scientific disciplines other than oceanography. It 

 behoves the oceanographer, therefore, to be prepared to speak with some 

 authority upon the information that can be extracted from his observations of 

 sea-level. This requires that he use the best possible methods of treating his 

 data numerically and, of the utmost practical importance to all concerned, 

 provide an estimate of the accuracy of his findings. 



An excellent example of this arises in the analytical treatment of annual 

 mean heights of sea-level to determine the relative magnitudes of the eifect 

 of barometric pressure and wind stress, secular variations and the nodal tide. 

 Previous attempts to determine the last two mentioned quantities have all 

 suffered by ignoring the contributions of wind and pressure, as will be shown. 



Let 



Zt ^ /{Bt, x) + cT + a cos OT + b sin OT + cf){T), (4) 



where Z is the annual height of observed sea-level, T is time in years, /{Bt. x) 

 is a term representing the contributions from variations in air pressure with 

 time and distance (this includes wind effect), c is the secular variation, and the 

 trigonometrical terms represent the nodal tide, 6 being the annual increment of 

 — 19°. 33 per year. <^(T) represents contributions to Zt from all other causes. 



Due to the strong correlation between the terms cT and b sin dT, the secular 

 variation and the nodal tide cannot be computed by independent processes. 

 Thus it may be shown that, if c is computed directly from Z using the method 

 of least squares, the answer will contain a contribution amounting to approxi- 

 mately 0.16 from the nodal tide; conversely, if b is computed directly from Z 

 by harmonic analysis, the answer will contain a contribution of 5.7c from the 

 secular variation. Using least squares with the three variables T, sin 6T and 

 cos 6T, complete separation can be assured, and this has been done for stations 

 in European waters for the years 1940 to 1958 inclusive. Table III illustrates 

 the results. Although there is good agreement between stations for the quantity 

 c, and for the amplitude R{ = \/[*^ + ^^J) ^^^ t^^® phase lag g of the nodal tide, 

 the latter is some 90° in advance of the equilibrium value of zero, and the 

 amplitude is many times larger than equation (2) would indicate. It will be 

 seen below, however, that regional agreement is no proof of the reality of the 

 results. 



Consider the probable accuracy of these calculations. If each value of Z is 

 subject to a random error such that the standard deviation of Z is a, it is shown 

 in standard text-books on statistics that the standard error of the amplitude 

 is a\/{2ln) and that of the phase lag is {alR)\/{2ln), n being the number of 

 observations. The standard deviations of Z are given in Table III, and it will 

 be seen that they vary from 24 mm at Tregde (Norway) to 79 mm at Furuogrund 

 (Sweden). 1 For an average value of a equal to 40 mm, it follows that with 



1 These values also contain contributions from the secular variations. 



