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[1] The problem of reconstruction of daytime nonmonotonic height profiles of electron density N(h) from ionograms of ground-based vertical sounding is considered. The regularization (REG) scheme has been proposed for determination of valley parameters. A comparison of the least squares method (LSM) with the regularization (REG) scheme was carried out using computer simulations. It is shown that LSM results are very sensitive to random errors in virtual heights. This fact leads to an unpredictably large deviation of the reconstructed N(h) profile from the real one. The application of REG methods makes the results more stable to random errors in measurements and allows us to decrease the deviation of the restored real height and valley depth from their model values by several times in comparison with LSM results.

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[2] The ionospheric plasma is a sensitive indicator of processes in the dynamic system “Sun–solar wind–magnetosphere–ionosphere–thermosphere” [Rodger, 1999]. Global changes in the ionosphere can be effectively monitored via measurements of space-time variations of electron density N [Pulinets and Benson, 1999]. For this purpose, it is necessary to calculate a series of height N(h) profiles from virtual height measurements along satellite orbits or from a network of ground-based ionospheric stations. Errors in measurements inevitably lead to errors in N(h) profiles, which, among other factors, depend on the geomagnetic coordinates of the observation point [Denisenko et al., 1998a]. Therefore in order to reconstruct the global distribution of electron density N it is necessary to take into account statistical weights of individual N(h) profiles. It is straightforward to do this using vertical sounding data because reconstruction of monotonic N(h) profiles is practical [Danilkin et al., 1988] and any errors can be estimated reliably enough [Denisenko et al., 1998a].

[3] When using data from a network of ground-based ionospheric stations, the reconstruction of ionospheric spatial structure below the F region maximum is a more complicated problem. Any ionospheric regions invisible to the ionosonde can lead to large errors in reconstructed N(h) profiles [Danilkin et al., 1988]. Modern acceptable methods of N(h) profile calculation exist which use the least squares method (LSM) [Reinisch and Huang, 1983; Titheridge, 1985; Wright and Pitteway, 1998]. However, they do not solve this problem completely because systematic and random errors are not under control in such a method.

[4] This paper is targeted toward the demonstration that the replacement of the LSM with the regularization (REG) method for N(h) profile calculation allows us to obtain results stable to random errors in virtual height measurements.

2. Model

[5] The problem is solved only for the valley area and the bottom part of the F region by computer simulation methods. It is assumed that the N(h) profile is known up to the E region's maximum height h_{m} (e.g., calculated by the method of Beynon and Rangaswamy [1968]). Therefore, as initial data, frequency dependencies of virtual heights h_{o}′(f) and h_{x}′(f) for ordinary (O) and extraordinary (X) traces are assumed with the ionospheric contribution below the E region maximum subtracted. It is assumed that reflections of O and X waves from the F region are observed beginning from some level where plasma frequency f_{N} = f_{min} is greater than the critical frequency f_{c} for the E region. It is assumed that virtual heights for the O trace are known for frequencies f_{i}, i = 1, 2, …, n, f_{1} = f_{min} and virtual heights for the X trace are known for equivalent frequencies

where f_{H} is the linear gyrofrequency of electrons.

[6] For the f_{N}^{2}(h)profile in the valley and in the adjacent invisible region the following model is assumed (Figure 1):

where f_{V} is the minimal plasma frequency in the valley at height h_{V}, H_{1} is the half-thickness of the lower parabola describing the decreasing electron density, H_{2} is the half-thickness of the upper parabola describing the increasing electron density,

For height intervals between reflection levels of signals h_{i−1} ≤ h ≤ h_{i} an exponential distribution is assumed:

After substitution of the expressions listed above into the formula for calculation of virtual heights

and corresponding transformation, finally one gets the expression (1) shown below. In the previous formula, μ′ is group refractive index, and h_{r} (f) is refraction height that depends on frequency.

[7] For inverse problem solving, the searching parameters are real heights for nodal plasma frequencies f_{V}, f_{F}, f_{1}, f_{2},…, f_{n}. There are two methodologies for searching the solution presented. For the first case, the N(h) profile is obtained in the valley and in the F region simultaneously. The solution is obtained by the regularization method from the optimum criterion of the residual square sum jointly for h_{o}′(f) and h_{x}′(f) curves. In the second case, the problem is solved in two stages. In the first stage the system of equations for O and X traces is reduced to overdetermine the system of equations that contains only parameters of the invisible area as unknown values. This part of the solution for the valley is obtained by the REG method from the optimum criteria of the residual square sum of real heights reconstructed in the F region separately for O and X traces of the ionogram. In the second stage the monotonic part of the profile is obtained for the bottom of the F region.

3. Inverse Problem: Method 1

[8] Let us present n-dimensional vectors of virtual heights for O and X traces as follows:

where x_{V} is a two-dimensional vector of real heights for the invisible part of the ionosphere relevant to plasma frequencies f_{V} and f_{F}, x is an n-dimensional vector of real heights for the F region relevant to plasma frequencies f_{1},…,f_{n}, F_{o} and F_{x} are matrices with dimensions n × 2 corresponding to the Fredholm operator, and V_{o} and V_{x} are matrices with dimensions n × n corresponding to the Volterra operator. By combining both expressions from formula (1) into one, we can get the overdetermined system of equations

where h′^{T} = (h_{o}′^{T}, h_{x}′^{T}), h^{T} = (x_{V}^{T}, x^{T}), index “T” means transposition, and M is a matrix with dimensions (2n) × (n + 2):

[9] We obtained the solution of the overdetermined system of equations (2) using two methods: the least squares method (LSM) and the regularization (REG) method [Denisenko et al., 1998b]. The LSM solution is used to obtain the REG solution as well as for comparison with REG results. It is assumed that the covariance matrix (matrix of errors) for virtual heights is

where σ^{2} is the variance of error in virtual heights independent of wave frequency and polarization and I is the unitary matrix with dimensions n × n.

[10] For the LSM solution determination the following function must be minimized:

where, because of equation (4), the weight matrix W = I with dimensions 2n × 2n. The LSM solution depending on parameter f_{V} is [Plackett, 1960]

The value of f_{VL} is obtained numerically from the minimum condition for function (5) by an exhaustive search of value h = h_{L} (f_{v}) from formula (6).

[11] The important stage of problem solving is the estimation of error in real heights h and the minimal plasma frequency f_{V} in the valley. The theory of error transfer from measured characteristics toward deduced parameters was developed for linear models only [Plackett, 1960]. For an application of this theory it is necessary to make a linearization of the problem in the neighborhood of the obtained solutions. Therefore the initial system of equations (2) in the neighborhood of the LSM solution must be present in the form

where Δf_{V} = f_{V} − f_{VL} and a is a vector column whose components are obtained for f_{V} = f_{VL} and they are

Finally, the error matrix for vector z_{L} is obtained as [Plackett, 1960]

where s^{2} is the estimator of variance σ^{2}. Diagonal elements of matrix D give us estimators of variances of deduced parameters:

[12] To obtain the REG solution instead of equation (5), the following function must be minimized:

where stabilizer matrix C is symmetric and only elements C_{11} and C_{22} are different form zero. The minimization of equation (9) for linear parameters leads to the REG solution

The parameter of regularization γ for any value of f_{V} is searched from the minimum condition [Denisenko et al., 1998b]

where the angle brackets mean statistical averaging procedure, and δ h_{R} and δ h_{L} are vectors of random error in REG and LSM solutions, respectively. Components of these vectors are proportional to . The REG solution is considered as found when a minimum for equation (9) is reached during an exhaustive search for parameter f_{V}.

[13] As for LSM, the estimation of errors in f_{V} and h in the REG method requires the linearization of the initial system of equation (2) in the neighborhood of the REG solution f_{VR} and h_{R}. Matrix elements a_{i} are calculated for extended matrix M_{e} (equation (7)) for f_{V} = f_{VR} and h = h_{R}. These calculations finally give us the following covariance matrix for error in vector z_{R}:

4. Inverse Problem: Method 2

[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} = (Δf_{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).

[17] The covariance matrix for the REG solution for the valley point including linearization for parameter f_{V} has the form:

[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.”

5. Computer Simulation

[19] For solving the direct problem, i.e., for calculation of accurate values of virtual heights above the E region maximum, the following values of N(h) profile parameters were assumed: f_{c} = 3.0 MHz, f_{V} = 2.8 MHz, h_{m} = 110 km, H_{1} = 10 km, H_{2} = 10 km, h_{i} − h_{i−1} = 3 km, and i = 2, 3,…, n. Ten working frequencies were used for waves at each polarization. The frequency range for O waves was assumed to be from 3.2 to 5.0 MHz with steps of 0.2 MHz. The working frequencies of X waves f_{i}^{x} were chosen such that their reflection levels in the ionosphere coincide with the reflection heights of corresponding O waves. The dipole geomagnetic field approximation was assumed. Calculations were provided for various angles Θ between the vertical and the vector of geomagnetic field intensity.

[20] To assist in the construction of the stabilizer matrices C from equation (9) and U from equation (21), the random errors ε_{i} were added to the accurate values of virtual heights. These added errors have an even distribution, such that the average values 〈ε_{i}〉 = 0, and various values of RMS deviation σ(ε_{i}). The calculation was carried out for various values of Θ with various sets of random errors ε_{i}. Finally, to search the regularizing solution by method 1 (REG-1), the stabilizer matrix C from equation (9) was constructed from eigenvalues λ_{i} of matrix B from equation (6). The result that was most stable for random errors was obtained with C_{11} = λ_{2}^{2} and C_{22} = λ_{1}^{2}. To search the regularizing solution by method 2 (REG-2), the stabilizer matrix U from equation (22) was constructed from eigenvalues λ_{i} of matrix B_{V} from equation (18). The optimal result was obtained with U_{12} = U_{21} = 0, , and .

[21] A typical example of the N(h) profile reconstruction for various angles Θ is presented in Tables 1 and 2. For all cases, model values of virtual heights are disturbed by fixed vectors of errors ε_{o}^{T} = (ε_{1}^{o}, ε_{2}^{o},…, ε_{n}^{o}) and ε_{x}^{T} = (ε_{1}^{x}, ε_{2}^{x},…, ε_{n}^{x}) for O and X traces, respectively. For all cases presented in the tables, σ(ε) = 5 km. The systematic error is defined as the difference between the reconstructed value of reflection height for frequency f_{1} and the accurate (model) value of h_{min}. The random error as the RMS deviation of real height for plasma frequency f_{N} = f_{1} is indicated.

Table 1. Results of Calculation for Method 1 (Jointly Using O and X Ionogram Traces)^{a}

Θ, deg

f_{H}, MHz

LSM-1

REG-1

F_{v}, MHz

σ(F_{v}), MHz

Systematic Error, km

σ(H), km

F_{v}, MHz

σ(F_{v}), MHz

Systematic Error, km

σ(H), km

a

Model value of valley depth f_{V}= 2.8 MHz.

90

0.84

2.92

0.72

−3.97

3.56

2.78

0.18

2.44

2.01

75

0.86

2.91

0.69

−3.83

3.69

2.78

0.18

2.42

1.96

60

0.93

1.99

1.28

15.96

3.72

2.79

0.17

2.12

1.93

45

1.06

2.97

0.78

−13.58

4.32

2.83

0.21

−2.45

1.22

30

1.27

2.03

1.35

16.11

1.62

2.86

0.24

−2.33

0.93

15

1.54

1.99

1.24

16.23

1.61

2.88

0.29

−2.54

0.45

Table 2. Results of Calculation for Method 2 (O and X Ionogram Traces Used Separately)^{a}

Θ, deg

f_{H}

LSM-2

REG-2

F_{v}, MHz

σ(F_{v}), MHz

Systematic Error, km

σ(H), km

F_{v}, MHz

σ(F_{v}), MHz

Systematic Error, km

σ(H), km

a

Model value of valley depth f_{V} = 2.8 MHz. First values in systematic and RMS error columns correspond to O wave, and second values correspond to X wave.

90

0.84

2.92

0.72

−5.90, −5.87

2.36, 2.32

2.83

0.18

−7.69, −8.20

1.97, 1.95

75

0.86

2.91

0.69

−5.99, −5.95

2.43, 2.39

2.83

0.19

−7.77, −8.28

1.97, 1.95

60

0.93

1.99

1.28

23.98, 24.02

2.43, 2.44

2.84

0.23

−7.05, −7.33

2.28, 2.25

45

1.06

2.97

0.78

−5.59, −5.60

2.35, 2.37

2.85

0.25

−7.32, −7.47

1.75, 1.79

30

1.27

2.03

1.35

24.26, 24.29

1.90, 1.82

2.87

0.28

−7.05, −6.79

1.68, 1.74

15

1.54

1.99

1.24

24.48, 24.51

1.78, 1.69

2.91

0.31

−0.90, −0.87

1.14, 1.20

[22] One can see from Tables 1 and 2 that in all cases the regularization methods REG-1 and REG-2 gave us results that are more stable to random errors in virtual heights than those provided by LSM. LSM can compete with the REG method in accuracy of real height determination only in the angle Θ interval from 90° to 75°. However, the LSM error in minimal valley plasma frequency for this angle interval is approximately more than 4 times the REG error. In the auroral ionosphere (Θ = 15°), systematic LSM error can be up to tens of times greater than corresponding REG errors. The high sensitivity of the LSM solution to random errors then leads to the situation for Θ = 60° and Θ = 45° where a tendency for sign changing of systematic errors in real heights, from negative values on the magnetic equator (Θ = 90°) to positive values near the pole (Θ = 15°), breaks. For REG methods the systematic error smoothly changes from 2.44 km (Θ = 90°) to −2.54 km (Θ = 15°) for REG-1 and from approximately −8 km (Θ = 90°) to approximately −1 km (Θ = 15°) for REG-2.

[23]Figure 2 provides a comparison between N(h) profiles obtained by LSM-1 and REG-1 methods with an initial distribution of electron density for geomagnetic conditions that correspond to the angle Θ = 30°. An analogous comparison between results obtained for Θ = 60° by LSM-2 and REG-2 is presented in Figure 3. Figures 2 and 3 demonstrate the advantage of the regularization technique over that of the LSM technique. In general, the method REG-1 is the most acceptable by means of providing uniform data processing over a wide interval of angles Θ. However, the method REG-2 can be useful when large horizontal gradients exist in the ionosphere, where O and X waves propagate through the plasma with different height profiles of electron density N.

[24] The inverse problem solution for other sets of random errors gives us N(h) profiles somehow different from the profiles in Figures 2 and 3. However, in all cases the application of REG algorithms gives results that have smaller systematic and random errors compared to LSM results.

6. Conclusions

[25] In this paper a comparison was provided between the use of LSM and the use of regularization when applied to nonmonotonic N(h) profile reconstruction. It was demonstrated that the LSM results are very sensitive to random errors in virtual height. When the geomagnetic latitudes of observation points change, this fact leads to unpredictably large deviations of reconstructed N(h) profiles from the accurate one. Using the regularization technique allows us to largely remove these deviations. Moreover, REG methods decrease the deviation of real heights and valley depth from their accurate values by several times relevant to LSM results. Therefore it is advisable to use regularizing algorithms for N(h) profile reconstruction when determining the spatial-time variations of electron density above big areas.