Let us formally express the real heights for the F region for the first equation from formula (1) as
Analogously, for the second expression from formula (1),
We required that in the F region, solutions xo and xx coincide, i.e.,
Hence we obtain the overdetermined system of equations relative only to xV
In order to solve system (15) by LSM it is necessary to minimize the function
Let us find the weight matrix W. For this purpose, let us determine the covariance matrix
Taking into account the definition of vector q (equation (15)),
and considering errors in ho′ and hx′ as uncorrelated, one can get
Consequently, the weight matrix W can be found from expression [Plackett, 1960]
which gives us W = N−1. This LSM solution for each fixed value of fV is
A value of fVL similar to that of method 1 is found by the numerical minimization of function (16). Taking into account the nonlinearity of our problem for parameter fV and carrying out the linearization procedure of the initial system of equation (15) relative to this parameter, for vector zLT = (ΔfV, xVLT) we finally obtain
where Re is the extension of matrix R due to linearization of the elements of matrix F relative to parameter fV.
 After determination of the valley parameters the reconstruction of N(h) profiles for the F region is provided according to formulas (13) and (14) for O and X ionogram traces, respectively. Let us determine the covariance matrix D(xo) for real heights of the N(h) profile calculated in the F region using the O trace. For this purpose, let us present δ xo in the form
with covariance matrix D(xo) determined as
It can then be shown that
The covariance matrix D(xx) is determined in an analogous way, and it can be obtained from formula (22) by replacement of the “O” index by the “X” index.
 The REG solution is found from minimization of the function
where S is a function from equation (16) and the symmetric stabilizer matrix U is 2 × 2 dimensional. The minimization procedure for equation (23) gives us a REG solution in the form
The REG parameter γ for each single value of fV is obtained from the condition of the function minimum:
Note that the method of minimum determination is the same as for F(γ).
 Covariance matrices for real heights in the F region for O and X ionogram traces are obtained from formulas (20) and (21), respectively. After all necessary calculations for covariance matrix D(xo) we come to an expression similar to formula (22), where matrices Bo and Qo are
The covariance matrix D(xx) can be obtained from formulas (22) and (27) by replacement of the index “O” by the index “X.”