4. Reconstruction of N(h) Profiles From the Vertical Sounding Data

[13]  Determination of n_e(h) profiles from the data on the frequency dependency of the absorption requires solution of the accompanying problem: reconstruction of N(h) profiles. This problem was solved by us in the following way.

4.1. D Region

[14]  The ionization vertical profile in the D region was found by the modified method of Beynon and Rangaswamy [1968]. The essence of the modification was the following [Danilkin et al., 1977]. Instead of model (6) the approximation

(8)

was used which agree well with rocket measurements. The parameter N0 was fixed and taken equal to 100 cm^-3. The values h₀ and H were considered as unknown. They were found from the requirement of coincidence of the left-hand and right-hand sides of (7a) at the condition of minimum of the functional

(9)

where h'_theor are the calculated values of the virtual heights of the "o" trace of the ionogram at the given h₀ and H parameters. In other words, the problem of a search for conventional minimum was solved. The working frequencies f_i were taken from the range from f _min to f₁+0.2 MHz with a step of about 0.1 MHz. Such procedure made it possible to reduce by several times the impact of random errors on the measurements of the virtual heights [Danilkin et al., 1981].

4.2. E Region

[15]  For the calculation of the vertical profile of the electron concentration in this layer both traces in the ionogram were used. The plasma frequency range from f₁ to f_oE was split by the net frequencies f_i to elementary intervals D f_i=f_i+1-f_i about 0.2 MHz wide. It was assumed that in all these intervals except the last one the electron concentration depends on height in a linear way. A parabolic distribution was accepted in the last interval. First, the part of the virtual heights D h'(f) corresponding to the E layer was found by a subtraction of the input of the ionospheric region located below the level of the reflection of the wave f₁, used for the measurement of the absorption. Then by the least squares method the true heights h_i corresponding to the net frequencies f_i were determined. To do that, the functional was minimized:

(10)

where the T index means transposing and vectors z and D h' are

The matrix M has a structure

where the elements of each matrix M_o and M_x correspond to the accepted approximation of N(h) in the elementary intervals. The solution has the form

(11)

4.3. Valley and F Region

[16]  The calculation of the nonmonotonous N(h) profile above the maximum of the E region was performed by the same method as for the E region. First the input of the region located below into the virtual heights of the "o" and "x" signals reflected from the F region was excluded. Then by the least squares method the N(h) profile was calculated. The only difference was that the first columns of the M_o and M_x matrix depended on minimum electron concentration N_v in the valley. In other words the minimized functional has the form

(12)

For the region unseen by ionosondes the models of monotonous distribution, plateau with N_v=N_mE, and triangle dependencies of N on h within the valley at N_vmE were accepted in a sequence. For every model and each value of N_v decreasing with a small step the current solution was found using formula (11). The solution providing in (12) the minimum possible value was accepted as a final solution.