Determining the ionospheric electron density profile

We can measure the equivalent height of the ionosphere in a fairly straightforward manner, by timing how long it takes a radio pulse fired vertically upwards to return to ground level again. We can, therefore, determine the function $h(\omega)$ experimentally by performing this procedure many times over, using pulses of different central frequencies. But, is it possible to use this information to determine the number density of free electrons in the ionosphere as a function of height? In mathematical terms, the problem is as follows. Does a knowledge of the function

$$h(\omega) = \int_0^{z_0(\omega)} \frac{\omega}{[\omega^2-\omega_p^{~2}(z)]^{1/2}} \,dz,$$ (873)

where $\omega_p^{~2}(z_0) = \omega^2$, necessarily imply a knowledge of the function $\omega_p^{~2}(z)$? Note, of course, that $\omega_p^{~2}(z)\propto N(z)$.

Let $\omega^2 =v$ and $\omega_p^{~2}(z) = u(z)$. Equation (4.223) then becomes

$$v^{-1/2}\,h(v^{1/2}) = \int_0^{z_0(v^{1/2})} \frac{dz}{[v-u(z)]^{1/2}},$$ (874)

where $u(z_0) = v$, and $u(z)<v$ for $0<z<z_0$. Let us multiply both sides of the above equation by $(w-v)^{-1/2}/\pi$ and integrate from $v=0$ to $w$. We obtain

$$\frac{1}{\pi}\int_0^w v^{-1/2} \,(w-v)^{-1/2} \,h(v^{1/2})\,... ...0(v^{1/2})} \! \frac{dz}{(w-v)^{1/2}\, (v-u)^{1/2}}\right] dv.$$ (875)

Consider the double integral on the right-hand side. The region of $v$-$z$ space over which this integral is performed is sketched in Fig. 15. It can be seen that, as long as $z_0(v^{1/2})$ is a monotonically increasing function of $z$, we can swap the order of integration to give

$$\frac{1}{\pi} \int_0^{z_0(w^{1/2})} \left[ \int_{u(z)}^w \frac{dv} {(w-v)^{1/2}\,(v-u)^{1/2}}\right]\,dz.$$ (876)

Here, we have used the fact that the curve $z=z_0(v^{1/2})$ is identical with the curve $v=u(z)$. Note that if $z_0(v^{1/2})$ is not a monotonically increasing function of $v$ then we can still swap the order of integration, but the limits of integration are, in general, far more complicated than those indicated above. The integral over $v$ in the above expression can be evaluated using standard methods (by making the substitution $v=w\,\cos^2\theta+u\,\sin^2\theta$): the result is simply $\pi$. Thus, the expression (4.226) reduces to $z_0(w^{1/2})$. It follows from Eq. (4.225) that

$$z_0(w^{1/2}) = \frac{1}{\pi}\int_0^w v^{-1/2} \,(w-v)^{-1/2} \,h(v^{1/2})\,dv.$$ (877)

Making the substitutions $v=w\,\sin^2\alpha$ and $w^{1/2}=\omega$, we obtain

$$z_0(\omega) = \frac{2}{\pi}\int_0^{\pi/2} h(\omega\,\sin\alpha)\,d\alpha.$$ (878)

By definition, $\omega=\omega_p$ at the reflection level $z=z_0$. Hence, the above equation reduces to

$$z(\omega_p) = \frac{2}{\pi}\int_0^{\pi/2} h(\omega_p\,\sin\alpha)\,d\alpha.$$ (879)

Thus, we can obtain $z$ as a function of $\omega_p$ (and, hence, $\omega_p$ as a function of $z$) simply by taking the appropriate integral of the experimentally determined function $h(\omega)$. Since $\omega_p(z) \propto \sqrt{N(z)}$, this means that we can determine the electron number density profile in the ionosphere provided we know the variation of the equivalent height of the ionosphere with pulse frequency. The constraint that $z_0(\omega)$ must be a monotonically increasing function of $\omega$ translates to the constraint that $N(z)$ must be a monotonically increasing function of $z$. Note that we can still determine $N(z)$ from $h(\omega)$ for the case where the former function is non-monotonic, it is just a far more complicated procedure than that outlined above. Incidentally, the technique by which we have inverted Eq. (4.222), which specifies $h(\omega)$ as some integral over $\omega_p(z)$, to give $\omega_p(z)$ as some integral over $h(\omega)$ is known as Abel inversion.

Figure 15: A sketch of the region of $v$-$z$ space over which the integral on the right-hand side of Eq. (4.223) is evaluated