A. Using Multiple Harmonics as a Proxy for the Seasonal Cycle

In our example we use four harmonics of the annual cycle (a linear combination of four sine and four cosine waves with periods that are integral factors of 12 months). These multiple harmonics capture more of the structure of the seasonal cycle, allowing for the fact that the seasonal cycle is not a perfect sine wave. The seasonal proxy is given by

seasonal proxy(t) = a1sin(2pit) + a2cos(2pit) + a3sin(4pit) +

a4cos(4pit) + a5sin(6pit) + a6cos(6pit) + a7sin(8pit) + a8cos(8pit)

where t is time measured in years.

Here we will examine how the four annual harmonics individually and collectively capture the seasonal cycle of ozone. Figure 9.13a shows the total ozone seasonal cycle at 40° to 50°N estimated from our model using four annual harmonics to represent the seasonal cycle. Figure 9.13b shows the estimated seasonal cycle in the Equator to 10°S latitude band.

The first harmonic has a period of 12 months (n = 1; period = 12/n; purple line), the second a period of 6 months (n = 2; period = 12/n; blue line), the third a period of 4 months (n = 3; period = 12/n; green line), and the fourth a period of three months (n = 4; period = 12/n; red line). The estimated seasonal cycle, which is the sum of each of the harmonics, is denoted by the black line. At 40-50°N the annual harmonic (period of 1 year) is by far the largest component of the seasonal cycle. In this band the other harmonics capture the finer structure of the cycle. In the Eq-10°S latitude band, shown in Figure 9.13a-b, the seasonal cycle has a more complicated pattern. Here the annual and semiannual (period of 6 months) harmonics are both large components of the seasonal cycle. Recall from Chapters 3 and 8 that the seasonal cycle in the tropics is small, and there is a semiannual oscillation of the circulation at these latitudes as well. The higher harmonics capture these variations. For more information on harmonics and harmonic analysis, see Bloomfield (1976).

B. Functional Form of the Seasonally Varying Trend Proxy

The functional form of D is the same as that of the seasonal proxy, except we only use 2 harmonics (the first 5 terms). That is,

D(t) = d1sin(2pit) + d2cos(2pit) + d3sin(4pit) + d4cos(4pit)