Revision of the MARINI-MURRAY model

Based on an expansion with hypergeometric functions, MARINI and MURRAY [Marini and Murray (1973)] obtained the following integrals for the atmospheric refraction correction $\Delta R$:

$$\begin{array}{rcl} \Delta R & = & \frac{1}{\sin(\theta_0)} ... ...r_1} (N(r)N_{Gr}(r) - \frac{1}{2}N(r)^2) dr \right] \end{array}$$ (1)

The phase and group refractive indexes are defined by

$$N := 10^{6} (n - 1)$$ (2)

$$N_{Gr} := 10^6 (n_{Gr} - 1).$$ (3)

By expanding the apparent angle of elevation $\theta_0$ in terms of the true angle of elevation $\theta_w$ and retaining only the first term, the third integral in equation 1 is cancelled. The remaining three integrals were recomputed in order to treat the correction terms for dry air and water vapour separately and to visualize the physical constants involved.

$$10^{-6} \int\limits_{r_0}^{\infty} N_{Gr}(h)dh = \frac{1}{g(\phi,H)}\left [ f_{Gr}(\lambda)g_1 + g_3 \right ]$$ (4)

$$\frac{10^{-6}}{r_0} \int\limits_{0}^{\infty} h N_{Gr}(h)dh +... ...- \frac{1}{2} N(h)^2)dh = frac{f_{Gr}(\lambda)}{g(\phi,H)} g_2$$ (5)

where we have defined the following terms:

$$g_1 = 80.343 \times 10^{-6} \left[ \frac{\cal{R}}{M_d \bar{g... ...{M_w}{M_d}\right) \frac{\cal{R}}{4 M_d \bar{g}} P_w(0) \right]$$ (6)

$$g_2 = 10^{-6} \frac{80.343 {\cal{R}}^2}{R_E M_d^2 \bar{g}^2}... ... P(0)^2}{4M_d\bar{g}T(0)\left(3-\frac{1}{K(\phi,T,P)}\right)}$$ (7)

$$g_3 = - 10^{-6} \frac{11.3 \cal{R}}{g(\phi,H) 4M_d \bar{g}} P_w(0)$$ (8)

$$g(\phi,H) = 1 - 0.0026 \cos(2\phi) - 0.00031 H\\$$ (9)

$$K(\phi,T,P) = 1.163 - 0.00968 \cos(2\phi) - 0.00104 T(0) + 0.00001435 P(0)\\$$ (10)

The following table explains the input values:

$$\begin{array}{r@{\quad:=\quad}l r@{\quad:=\quad}l} {\cal R} ... ...P_w & \mbox{water vapour partial pressure in mb}\\ \end{array}$$

The refraction correction is now reduced to the simple formula

$$\Delta R = \frac{f_{Gr}(\lambda)}{g(\phi,H)} \left [\frac{g_1 + g_3}{\sin(\theta_w)}+\frac{g_2}{\sin(\theta_w)^3} \right ]$$ (11)

where the dispersion of water vapour was set equal to the dispersion of dry air. This reduces the validity of equation 11 to wavelengths in the vincinity of $0.6943 \mu m$, since the dispersion function $f_{Gr}$ is normalized to this wavelength.
To ensure a better convergence of equation 11 it expanded into a continued fraction:

$$\Delta R = \frac{f_{Gr}(\lambda)}{g(\phi,H)}\frac{g_1+g_2+g_3}{\sin(\theta_w)+\frac{g_2/(g_1+g_2+g_3)}{\sin(\theta_w)+0.01}}$$ (12)

The empirical term $0.01$ was obatined by comparing the reduction formula to values obtained by raytracing of radiosonde profiles, which served also as validation of the refduction formula.