[14] 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 **x**_{o} and **x**_{x} coincide, i.e.,

Hence we obtain the overdetermined system of equations relative only to *x*_{V}

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 **h**_{o}′ and **h**_{x}′ 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 *f*_{V} is

A value of *f*_{VL} 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 *f*_{V} and carrying out the linearization procedure of the initial system of equation (15) relative to this parameter, for vector **z**_{L}^{T} = (Δ*f*_{V}, **x**_{VL}^{T}) we finally obtain

where **R**_{e} is the extension of matrix **R** due to linearization of the elements of matrix **F** relative to parameter *f*_{V}.

[15] 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**(**x**_{o}) for real heights of the *N*(*h*) profile calculated in the *F* region using the *O* trace. For this purpose, let us present δ **x**_{o} in the form

with covariance matrix **D**(**x**_{o}) determined as

It can then be shown that

The covariance matrix **D**(**x**_{x}) is determined in an analogous way, and it can be obtained from formula (22) by replacement of the “*O*” index by the “*X*” index.

[16] 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 *f*_{V} is obtained from the condition of the function minimum:

Note that the method of minimum determination is the same as for *F*(γ).

[18] 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**(**x**_{o}) we come to an expression similar to formula (22), where matrices **B**_{o} and **Q**_{o} are

The covariance matrix **D**(**x**_{x}) can be obtained from formulas (22) and (27) by replacement of the index “*O*” by the index “*X*.”