Refractive index formula

The refraction index formula used in the MARINI-MURRAY formalism was derived by BARREL and SEARS [Barrel and Sears(1939)] for a limited range of wavelengths ($0.4\mu m$ to $0.6\mu m$). The nowadays recommended refraction index formulas of OWENS[Owens (1967)] or the most recent one derived by CIDDOR [Ciddor (1996)] encompass a wider range of wavelengths ($0.3\mu m$ to $2.0\mu m$) as well as an improoved accuracy of $10^{-8}$, due to the modeling of the non ideal gas behaviour of the atmospheric constituents. But a straightforward application of such a modern formula in the analytical approach of MARINI and MURRAY is not reasonable due to the following reasons:

• the MARINI-MURRAY approach relies on hydrostatic equilibrium and the barometric equation, i.e ideal gas behaviour.
• water vapour, which shows the worst non ideal gas behaviour, plays a secondary role for wavelengths in the visible spectrum
• the non hydrostatic water vapour distribution requires a water vapour profile, which in general shows no strong dependece on the surface conditions (see MENDES [Mendes (1999)])

These items lead to the conclusion, that the application of a state of the art refractive index formula for refractive delay modeling requires height profiles, which do not depend on the barometric equation. Even radiosonde profiles don't hold for that, since the height is derived by measuring the total pressure during the ascent of the radiosonde. Therefore it is proposed, as long as no detailed atmospheric data is available, to use a hybrid approach in which the dry and wet refraction delays are treated seperately. The dry refraction delay should be modeled with a new dispersion formula, leading to a modified MARINI-MURRAY model.

$$\Delta R = f_{Gr}(\lambda)\frac{g_1+g_2}{\sin(\theta_w)+\frac{g_2/(g_1+g_2}{\sin(\theta_w)+0.01}} +\frac{g_3}{\sin(\theta_w)}$$ (13)

The dispersion formula, taken from [Ciddor (1996)] normalized to the wavelength of $0.6943 \mu m$ reads:

$$f_{Gr}=\frac{k_1(k_0+(1/\lambda)^2)}{(k_0-(1/\lambda)^2)^{2}}+\frac{k_3(k_2+(1/\lambda)^2)}{(k_2-(1/\lambda)^2)^2}$$ (14)

The involved constants are:

$$\begin{array}{r@{\quad:=\quad}l r@{\quad:=\quad}l} k_0 & 238... ... k_1 & 205.0638\\ k_2 & 57.362 & k_3 & 5.944936\\ \end{array}$$