2.3 Improvements and modifications
2.3.1 Integration method
SATIRE-S originally employed the int tabulated IDL routine which uses a five-point Newton-Cotes integration formula. When spectral comparisons between model data and real observations (see Chapter 5) were initially performed, the sum of spectral sub-regions did not add up to a direct integration over the whole spectral interval. To function effec-tively, the Newton-Cotes method requires the interval being integrated over to be divided into equal parts, but the wavelength scale used for the spectra in both model and observa-tions is non-linear.
In contrast, the trapezium rule integration method is simple in its approach, can be well understood and the summation over sub-regions does equal the integration of the entire spectral region. The use of polynominals to approximate the tabulated data in the Newton-Cotes method introduces errors, the source of which is not explicity clear, and thus, when comparing the outputs of both methods, a difference emerges.
2.3 Improvements and modifications Chapter 2: SATIRE Model
Figure 2.8: A demonstration of the different results gained by using the trapezium rule compared to int tabulated by integrating over different spectral widths. The summation of integrals over different sized sub-interval of the full 200-1600 nm region are plotted.
The red, crossed line is for int tabulated and the black, crossed line for the trapezium method. Integrating over 200-1600 nm directly produces the red dotted line when using int tabulated, whereas using the trapezium method gives the same result as the black, crossed line.
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Chapter 2: SATIRE Model 2.3 Improvements and modifications
By integrating over sub-intervals within an arbitrarily large spectral region the difference in output from the two integration methods can be shown. For example, the integral of the spectral region 200-1600 nm in SATIRE-S (using data on 1996 May 20) equals ∼1201.1 Wm−2 when using the trapezium rule method. The summation of 140 integrals over 10 nm sub-intervals, for example, should (and does) give the same result. In Fig. 2.8, the summation of equal-width sub-intervals of 11 different sizes is performed over 200 to 1600 nm using the trapezium method (black) of integration and using int tabulated (red).
The dotted red line, at 1201.6 Wm−2, is the value of a direct integration over 200-1600 nm using the int tabulated method. This is 0.5 Wm−2higher than when using the trapezium rule method. From this figure the summation of sub-intervals when using the trapezium rule is the same whatever interval size is used to integrate over the full 200-1600 nm spectral region. This is not the case using int tabulated. There does not appear to be any correlation between interval size and the difference relative to integration using the trapezium rule, though generally larger bins give less reliable results. The 200-1600 nm region considered here, and the chosen sub-intervals, are just examples. In this case the deviation of int tabulated from the trapezium rule can exceed 0.7 Wm−2 and the range of possible integral values varies up to twice that (i.e. the difference between 200 nm intervals and 700 nm intervals). Although this is only about 0.1%, a difference of 1.7%, 23 Wm−2, has been found when integrating over the full range from 9-160 000 nm.
While these differences are small in most cases, it means that additional flux is being introduced in an unclear way that can change the relative variation of the different spec-tral regions being considered. It is also unsatisfactory to have the sum of the parts not equal the whole. These problems disappear when using the trapezium rule. Therefore, the trapezium rule replaces int tabulated in SATIRE-S and is used for all integrations performed in this thesis.
The change of integration method has an effect on the reconstruction, most notable in the long-term gradient. Because Bsatis fixed by varying it to gain a fit to TSI data, the change in integration method requires that the free parameter is reconsidered. In the top panel of Fig. 2.9, model reconstructions between February 2003 and October 2009 are plotted, normalised to the SORCE/TIM TSI data (not shown) during the 2008 minimum. Both integration methods have Bsat = 443 G, chosen so that the trapezium method (blue) will agree with SORCE/TIM with a unity regression (see Chapter 3). From this plot, it can be seen that a reconstruction using int tabulated (red) produces a lower cycle gradient for a given Bsatthan using the trapezium rule.
2.3 Improvements and modifications Chapter 2: SATIRE Model
Figure 2.9: The effect of changing integration method on TSI reconstructions. (top) The long term trend of each method using the same model settings and normalised to SORCE/TIM at December 2008. (bottom) To show the effect on the short-term variabil-ity, the free parameter is set to gain a unity regression with SORCE/TIM and normalised to the shorter period shown. The shorter period is between the dotted lines of the top plot.
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Chapter 2: SATIRE Model 2.3 Improvements and modifications
By decreasing Bsat, a unity regression with a reconstruction using int tabulated is found at 390 G. The correlation coefficient between the two is, as expected, very good (Rc = 0.997), but there are minor changes to the short-term variation. The period between the vertical dotted lines in the upper plot is replotted in the lower plot of Fig. 2.9, along with SORCE/TIM (green). In this plot, both reconstructions are normalised to SORCE/TIM over the shorter period plotted. There are clearly differences, but these are all very small4.
Therefore, ultimately, once the free parameter is set to give the best agreement with TSI observations the difference between the two approaches is minor. The benefit of chang-ing integration method is mainly found when dochang-ing spectral comparisons. TSI integra-tion within SATIRE-S should be consistent with the integral of SSI and for this reason the change is implemented. The result is that the chosen value of the free parameter is different to, and cannot be compared with, previously published reconstructions using SATIRE-S. Due to the nature of fitting to TSI, the actual irradiance trends are very similar.