where , necessarily imply a knowledge of the function ? Recall that .

Let and . Equation (1133) then becomes

(1132) |

where , and for . Let us multiply both sides of the previous equation by and integrate from to . We obtain

Consider the double integral on the right-hand side of the previous equation. The region of - space over which this integral is performed is sketched in Figure 782. It can be seen that, as long as is a monotonically increasing function of , we can swap the order of integration to give

Here, we have used the fact that the curve is identical with the curve . Note that if is not a monotonically increasing function of then we can still swap the order of integration, but the limits of integration are, in general, far more complicated than those indicated previously. The integral over in the previous expression can be evaluated using standard methods (by making the substitution ): the result is simply . Thus, expression (1136) reduces to . It follows from Equation (1135) that

(1135) |

Making the substitutions and , we obtain

(1136) |

By definition, at the reflection level . Hence, the previous equation reduces to

(1137) |

Thus, we can obtain as a function of (and, hence, as a function of ) by simply taking the appropriate integral of the experimentally determined function . Because , this means that we can determine the electron number density profile in the ionosphere provided that we know the variation of the equivalent height with pulse frequency. The constraint that must be a monotonically increasing function of translates to the constraint that must be a monotonically increasing function of . Note that we can still determine from for the case where the former function is non-monotonic, it is just a far more complicated procedure than that outlined previously. Incidentally, the mathematical technique by which we have inverted Equation (1133), which specifies as some integral over , to give as some integral over , is known as