Comparison of mapping functions

In order to assess the adequacy of the MARINI-MURRAY model in terms of the envisaged conditions in charter item 1, the model is compared with the results of a raytracing algorithm, which uses the quite complex refractive index formula derived by [Owens (1967)]. The atmosphere for the raytracing algorithm is modeled in the same way as it is done in the derivation of the MARINI-MURRAY model, i.e. a lapse rate of $6.5 K/km$, atmospheric form factors according to equations 9 and 10 and a tropospheric height of 10 km. Figure 1 shows the results obtained for the original MARINI-MURRAY model (equation 12), for the modified MARINI-MURRAY model (equation 13) and a third mapping function derived from the GARDNER mapping function [Gardner and Rowlett (1976] namely

$$\Delta R = \frac{1}{g(\phi,H)} \left [f_{Gr}(\lambda) \frac{... ...{\sin(\theta_w)+0.017}}} + \frac{g_3}{\sin(\theta_w)} \right ]$$ (16)

with the coefficients

$$\begin{array}{r@{\quad:=\quad}l} A & g1+10^{-6}\frac{80.343 ... ... 10^{-13} P T^2\frac{K(\phi,T,P)}{2-K(\phi,T,P)}\\ \end{array}$$

The term 0.017 in the last denominator is again an empirical adjustment to optimise convergence at low elevation angles. The number appearing in coefficient C is not yet deduced to physical constants. Here again the wet component of the correction was taken out of the continued fraction expansion to ensure a proper treatment of dispersion.

Figure 1: Comparison of mapping functions with raytracing for various wavelengths and water vapour partial pressures.

As expected the MARINI-MURRAY mapping function serves best at it's design wavelength of $0.6943 \mu m$ with an error of up to 1.5 cm at 10 degrees elevation and a humidity of 80 %. Generally speaking the highest deviations (up to 4.5 cm) from the raytracing results show up at wavelengths far apart from $0.6943 \mu m$ and high humidities.
The modified MARINI-MURRAY mapping function compensates partly the errors incurred by the incorrect dispersion model. Here the maximum deviations are of about 1.5 cm.
It is to note, that the modified GARDNER mapping function performs very well with deviations smaller than 2mm if we restrict ourselves to elevations larger than 20 degrees.