How does index of refraction changes with horizontal range I am reading this ITU recomendation, and it says that the refractive index varies mainly with altitude as shown below:

What does it mean "mainly"? I understand how vertical change is achieved, as it is governed by the Snell's Law, but does refractive index also changes with range as a radio wave is travelling? How big is the effect of the range on the refractivity index? Does anyone have some articles/books where I could find more information about how does the index of refraction changes horizontally?
 A: As interactions between the molecules in a gas are weak, optical effects of gases are primarily driven by the interaction with the individual molecules in the gas, so within typical ranges the index of refraction will be proportional to the density as a good first approximation and since the interactions are weak and the coefficients are small, effects of different components of the gas will simply add up.
The composition of air, up to the content of water vapor is pretty consistent (at least within the troposphere). So we expect the formula can be written as a sum of the refractive index of the dry air plus the refractive index of water vapor in terms of their densities:
$$ n - 1 = N_d(\rho) + N_w(\rho) $$
Of course, for real materials we have to add temperature dependencies, as the interactions of the waves with the molecules itself may depend on the temperature.
The density can be determined from the partial pressures and the temperature, via the equation of state of the ideal gas ($R$ is the universal gas constant, $M$ the molar mass of the gas in question):
$$ pV = nRT \Rightarrow  \frac{p}{T} = \frac{nR}{V} = \frac{R}{M} \rho $$
The partial pressures of the dry air and water vapor can be computed from the relative humidity, the total air pressure and the vapor pressure of water at the temperature in question.
With a short google search, I found the following paper giving formulas for the index of refraction of air for radio waves https://www.fig.net/resources/proceedings/fig_proceedings/fig_2002/Js28/JS28_rueger.pdf.
It gives the following formula (I've truncated the precision of the coefficients – there are more precise equations in the paper for more restricted situations) for the index of refraction in terms of the partial pressure of the dry air $p_d$ and the partial pressure of the water vapor $p_w$:
$$ (n - 1) \cdot 10^6  \approx 77.7\,\mathrm{\frac{K}{hPa}} \cdot \frac{p_d}{T} + 71.3 \, \mathrm{\frac{K}{hPa}} \cdot  \frac{p_w}{T} + 3.7 \cdot 10^5 \, \mathrm{\frac{K^2}{hPa}} \cdot  \frac{p_w}{T^2} $$
Except for the last term, this fits our expectation. I am not sure what's the mechanism for the last term but it seems to be a molecular mechanism, and at typical temperatures $T = 300K$ it's of comparable size as the $\propto\,\frac 1 T$ term.
In summary, the refraction in horizontal direction is complex, as it depends on the changes of pressure, temperature and humidity of the air through which the radio wave moves. While the horizontal change of density is (more or less) the similar everywhere, because it (more or less) follows the equation $p \propto p(h_0) e^{-(h-h_0)Mg/RT}$. (Of course there are variations, and the equation is not precise due to temperature changes in dependency of the height, etc.).
