An alternate power-law form for estimating action spectrum weighted irradiance change is commonly used for large changes in ozone amount, which empirically matches data quite well (Booth and Madronich, 1994; Blumthaler et al., 1995). This form is given in Eq. (5.3).
where A and A are two values of ozone that correspond to two values of irradiance, F1 and F2, respectively. The RAF(#) is an empirically selected set of constants to match Eq. (5.3) to observed data. For small changes in A equation (5.3) is approximately the same as Eq. (5.1) by expanding F1/F2 = [1 + (A - A) /A)]-RAF = 1 - RAF [(A - A) /A)] + - , so that (F1 - F2)/F2 = - RAF (A - A)/A + — for (A - A) «A.
The primary use for the power-law form is for processes that span a wide wavelength range that includes UV-B and UV-A. The best-known example is the erythemal action spectrum weighted irradiance AerY(A)F(2), where the weighting is given by the McKinlay and Diffey (1987) action spectrum AErY(A) (see
Section 5.7). Since the power law form is empirical, measurements or radiative transfer calculations, as a function of ozone amount, must be made to estimate RAFery(£). For large SZAs the UV-B portion of the irradiance diminishes rapidly and the product AERY(X)F(X) is dominated by UV-A, which has almost no sensitivity to ozone change. As a function of latitude, the mid-day RAFERY(#) is approximately constant in low and middle latitudes, and then decreases at high latitudes (high SZA) (Bodhaine et al., 1997). This behavior appears to be quite different from the monochromatic RAF in Eq. (5.1) (see Section 5.7).
The existence of two definitions of RAF (Beer's Law, equation (5.1) and Power Law, Eq. (5.3)) can cause some confusion. The physically based Beer's Law formulation can also be used to derive monochromatic AF/F for large changes in ozone AH/n. Let F1 = F0 exp(-a/2i sec(#)) and F2 = F0 exp(-a/22 sec(#)), then define AF12/F2 = (F1 - F2)/F2 and A^12/^2 = -^2)/.^2. The result is given in Eq. (5.4).
In Section 5.7, we show that the use of Eq. (5.4) Beer's Law leads directly to the Power Law after integrating over the wavelength range corresponding to an action spectrum. In the following sections, A^2/^2 is small, so that the change for near-monochromatic dF/F can be estimated from Eq. (5.1) or (5.4).
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