An Improved Method for the Inversion of Backscatter Ionograms by Using Neighborhood-Aided and Multistep Fitting

Abstract

To solve the problem that a parameter search easily falls into a local optimum and the two-dimensional electron density profile construction error is large in the process of backscatter ionogram inversion, an improved method using neighborhood-aided and multistep fitting is proposed. The ionospheric parameter inversion results in the adjacent space are combined and reconstructed by using the neighborhood-aided correction method. The introduction of auxiliary information sources addresses the defects of the conventional genetic algorithm. The local region multistep fitting method is used to describe the local uniformity and global inhomogeneity of the two-dimensional electron density profile by dividing the fitting region. The experimental results show that the proposed method can improve the accuracy of backscatter ionogram inversion and provide reliable support for tracking radio ray trajectories.

1. Introduction

Studying the characteristics of ionospheric media is one of the most important areas of ionospheric scientific research and geospatial environment monitoring [1]. This subject is closely related to frequency selection and equipment deployment in modern radar, communication and navigation systems [2]. At present, ionospheric detection methods based on ground equipment mainly include vertical detection, oblique detection and backscatter detection. Compared with the other two methods, backscatter detection has the advantages of a long detection distance and wide coverage. It plays a crucial role in areas where vertical sounders and oblique sounders cannot be placed, such as oceans and seas [3]. During operation, backscatter sounding systems carry out frequency sweep detection at a fixed azimuth and obtain high-frequency (HF) backscatter ionograms (BSIs), which reflect how the group path changes with the detection frequency. Then, the ionospheric media characteristics in the detection direction are deduced from the BSIs, which is called ionization inversion [4]. Due to effects, such as time focusing and spherical focusing, the BSI has a clear, steep leading edge (also known as the minimum group path). In addition to the influence of the electron density distribution, the leading edge is almost unaffected by other factors, such as the antenna beam and ground characteristics, so it can be accurately interpreted. Therefore, it is often used for ionospheric inversion research and plays an important role in analyzing the characteristics and propagation conditions of HF radio transmission media.

In recent years, many scholars have carried out relevant research on the inversion method of ionograms. Benito used a genetic algorithm (GA) and a simulated annealing algorithm (SAA) to invert the oblique ionogram of an elevation scan [5]. These two algorithms can easily use prior information to constrain the solution and reduce the nonuniqueness of the solution of the inversion problem. Li, Zhao and Jiang et al. used the SAA to invert the measured sweep frequency oblique ionogram [6,7,8], which was inspired by Benito. Compared with the self-scaling technology of ionograms based on the template matching method proposed in reference [9], the SAA improves the stability of the system and outputs reliable ionospheric inversion information with satisfactory efficiency, especially when the ionosphere is in a quiet state and changes slowly. However, the global search ability of the SAA is poor, and many iterative operations are needed to obtain good inversion results. As Benito noted, the time required by GA inversion to obtain the best solution is approximately nine times less than that of the SAA [5]. In order to obtain the unique solution of the inversion problem, Ponomarchuk proposed a method based on the comparison of experimental and calculated minimum delays of scattered signals with corresponding distances to the skip zone border [10]. This method has better mathematical interpretability, but compared with GA and SAA methods, its performance is more affected by the recognition accuracy of backscatter sounding signal leading edge. This method is only applicable to the inversion of the F2 layer. Guo proposed a method to estimate ionospheric parameters by using reference source information and over-the-horizon radar measurements [11]. However, without prior knowledge, it is difficult to find natural reference sources or arrange artificial cooperative reference sources near the target of interest. Song proposed applying the hybrid GA based on the SAA to the parameter inversion of the ionization map to obtain the complementary advantages of the GA and SAA [12,13]. Experiments have shown that the hybrid algorithm is superior to the GA and SAA in terms of the optimization and stability of inversion. However, the GA is still limited by its own defects; when the initial population selection is not appropriate, it will lead to the rapid increase in good individuals in the initial stage of the search, resulting in a loss of diversity in the population and a risk of falling into local optima.

It should be noted that to improve the effectiveness of the inversion algorithm, other scholars have used multisite radar to provide additional valuable information. Other methods of improving the inversion method are also possible. For example, Feng proposed an inversion algorithm based on solution space constraints [14]. In this method, the electron density profile information over the transmitting station is used as the known condition by coordinating the vertical and backscatter stations. Lou proposed a comprehensive ionospheric detection technique based on the minimum mean square error criterion and global search [15]. The improvement in inversion accuracy also depends on multisource information provided by multibase radar. Feng and Lou’s method places higher demands on experimental equipment but inspires us to assist the GA in inverting self-correcting errors by introducing additional valuable information. In addition, the electron density profile obtained by the above ionogram inversion method is discretely distributed in geographical space. Simple approximations introduce large errors when using information regarding missing parts. On the basis of considering the variation in the horizontal gradient of the electron density distribution, a reasonable fitting method should be constructed to obtain a more realistic two-dimensional electron density profile.

To solve the above problems, an improved method for the inversion of BSIs by using neighborhood-aided and multistep fitting is proposed. The goals of this study include the following:

• Considering the horizontal gradient of the electron density distribution, the ionospheric parameter inversion results in the adjacent space are combined and reconstructed. Unlike the traditional genetic inversion method, which regards each inversion process as an independent event, this method introduces auxiliary information sources, which can effectively compensate for the defects of the traditional GA, such as easily falling into local optima and having a poor local search ability. At the same time, the improved algorithm reduces the evolution time of the GA, thus improving the speed of calculating the inversion results.
• The discrete one-dimensional electron density profile is fitted by the local region multistep fitting method to recover the local uniformity and global inhomogeneity of the two-dimensional electron density profile. The range of the fitting region is determined by the fitness value calculated in the GA. With the idea of the moving least-square method, in the region of fit, the gradient of the electron density does not obviously change, so a smooth two-dimensional profile can be fitted. In the whole large region, several independent fitting regions can reflect the horizontal gradient of the electron density distribution. Compared with simple approximate processing, the error is effectively reduced. Compared with the conventional subsection fitting method, the actual ionospheric variation law is better matched, and the problem of discontinuous and uneven fitting curves on adjacent subsections is avoided.

The rest of this article is organized as follows: Section 2 introduces the principle of BSI inversion. Then, an improved inversion algorithm is introduced. The experimental results and analyses are presented in Section 3, and conclusions are drawn in Section 4.

2. Materials and Methods

2.1. Principle of BSI Inversion

2.1.1. Quasi-Parabolic Distribution

In the process of ionospheric parameter inversion, the quasi-parabolic (QP) model is often used to describe the distribution of ionospheric electron density [16]. For example, the electron density profile of the single-layer ionosphere is expressed as

$$f_N^2(r)=\{\begin{array}{cc}{f}_c^2\left[1-{\left(\frac{r-r_m}{y_m}\right)}^2{\left(\frac{r_b}{r}\right)}^2\right],& r_b<r<r_m\left(\frac{r_b}{r_b-y_m}\right)\\ 0,& \mathrm{otherwise}\end{array}$$

(1)

where $r$ is the distance to the center of the Earth, and $f_N(r)$ is the plasma frequency at $r$. The relationship between $f_N(r)$ and the electron density here, $N_e(r)$, is $f_N^2(r)=80.6N_e(r)$, and $f_c$ is the critical frequency of this layer. The relationship between $f_c$ and the electron density here, $N_m$, is $f_c^2=80.6N_m$. In addition, $r_b$ is the distance from the bottom of the ionosphere to the center of the Earth, $y_m$ is the half-thickness of the ionosphere and $r_m$ is the distance from the maximum electron density of the ionosphere to the center of the Earth. The above three parameters satisfy $r_m=r_b+y_m$.

In the QP model, the ground range between the receiving site and the transmitting point and the group range of the wave can be calculated accurately. The ground range $D$ and group range $P^{\prime }$ are determined by

$$D=2r_0\left(\gamma -{\beta }_0\right)-\frac{2r_0^2\cos {\beta }_0}{\sqrt{C}}\ln \left(\frac{\sqrt{B^2-4AC}}{\frac{2\sqrt{C}}{r_b}\sqrt{A{r}_b^2+B{r}_b+C}+\frac{2C}{r_b}+B}\right)$$

(2)

$$P^{\prime }=2\left(r_b\sin \gamma -r_0\sin {\beta }_0\right)-\frac{2}{A}\left\{r_b\sin \gamma +\frac{B}{4\sqrt{A}}\ln \left[\frac{B^2-4AC}{{\left(2A{r}_b+B+2r_b\sqrt{A}\sin \gamma \right)}^2}\right]\right\}$$

(3)

where $r$ is the incidence angle of the radio ray at the bottom of the ionosphere, $r_0$ is the radius of the Earth and ${\beta }_0$ is the elevation of the radio ray emission.

$$A=1-\frac{1}{F^2}+{\left(\frac{r_b}{F{y}_m}\right)}^2$$

$$B=-2r_m{\left(\frac{r_b}{F{y}_m}\right)}^2$$

$$C={\left(\frac{r_b{r}_m}{F{y}_m}\right)}^2-r_0{\cos }^2{\beta }_0$$

$$F=f/f_c$$

$$\cos \gamma =\left(r_0/r_b\right)\cos {\beta }_0.$$

The real ionosphere consists of several layers, such as D, E, F1 and F2. Considering that the real ionosphere cannot be accurately described by a simple QP model, the QP model is improved to a multi-quasi-parabolic (MQP) model [17]. In this paper, an MQP model containing E and F layers (composed of F1 and F2 layers) is taken as an example to describe and test the inversion algorithm. It should be noted that the current ionogram interpretation algorithm is developing rapidly and largely meets the needs of the pattern discrimination of the ionogram and the extraction of the leading edge. Although the application of the method in this paper is described only in E–F mode, it can meet the requirements of multimode inversion when combined with the intelligent interpretation technique. The MQP model is given as follows:

$$\{\begin{array}{cc}{N}_E=a_E-b_E{\left(1-\frac{r_{mE}}{r}\right)}^2,& \frac{d{N}_E}{dr}=-\frac{2r_{mE}{b}_E}{r^2}\left(1-\frac{r_{mE}}{r}\right)\\ N_{EF}=a_{EF}+b_{EF}{\left(1-\frac{r_{mEF}}{r}\right)}^2,& \frac{d{N}_{EF}}{dr}=\frac{2r_{mEF}{b}_{EF}}{r^2}\left(1-\frac{r_{mEF}}{r}\right)\\ N_F=a_F-b_F{\left(1-\frac{r_{mF}}{r}\right)}^2,& \frac{d{N}_F}{dr}=-\frac{2r_{mF}{b}_F}{r^2}\left(1-\frac{r_{mF}}{r}\right)\end{array}$$

(4)

where the subscripts E, EF and F indicate that the parameters are evaluated at layer E, the joining layer and layer F, respectively. In addition, $a_E=N_{mE}$, $b_E=a_E{(r_{bE}/y_{mE})}^2$, $a_F=N_{mF}$ and $b_F=a_F{(r_{bF}/y_{mF})}^2$. According to the principle that the electron density and gradient at the connection point of each layer are equal and assuming that the joining layer is connected with the maximum electron density of the E layer (i.e., $a_{EF}=a_E$ and $r_{EF}=r_{mE}$), $b_{EF}$ can be deduced as follows:

$$\{\begin{array}{c}{b}_{EF}=-\frac{b_F{r}_{mF}\left(1-r_{mF}/r_c\right)}{r_{mE}\left(1-r_{mE}/r_c\right)}\\ r_c=\frac{r_{mF}{b}_F\left(r_{mF}/r_{mE}-1\right)}{a_F-a_E+b_F\left(r_{mF}/r_{mE}-1\right)}\end{array}$$

(5)

where $r_c$ is the height of the intersection between the joining layer and the F layer.

2.1.2. Inversion Principle Based on the GA

The curve corresponding to the detection frequency $f$ and the minimum group range ${P^{\prime }}_{\min }$ is called the leading edge. The essence of the inversion process is to analyze the leading edge of the HF BSI to obtain three parameters of the ionosphere: the critical frequency $f_c$, the ionosphere height $r_b$ and the half-thickness $y_m$. Before the inversion, it is assumed that the leading edge has been preprocessed, that it is complete and that the boundary between the E layer and F layer is obvious. There is a prerequisite condition that the leading edge of the F layer can be inverted using the inversion results of the E layer as the known conditions.

Because there are many combinations of parameters in the parameter space of the model, the analytical method is time-consuming to solve. It is a reasonable choice to use the GA to search for the optimal result. The GA is a common method of nonlinear geophysical inversion. It simulates the process of population evolution in nature, does not depend on the selection of the initial model and retains high accuracy while improving the inversion speed. The specific steps are as follows:

• The ionospheric parameter space $\left\{f_c,y_m,r_b\right\}$ is constructed based on the accumulated experience of long-term observation. Different parameter search spaces are given by different geographical locations, seasons and times of year. Generally, the variation range of the given parameters will affect the efficiency of the GA, and other knowledge should be used to narrow the search space as much as possible.
• $n_{GA}$ groups of parameters $\left(f_{c0},y_{m0},r_{b0}\right)$, $\left(f_{c1},y_{m1},r_{b1}\right)$, $\cdots$ and $\left(f_{c{n}_{GA}},y_{m{n}_{GA}},r_{b{n}_{GA}}\right)$ are randomly selected from the ionospheric parameter space as the initial ionospheric model. Each parameter is encoded with $n_{BIN}$ bits of binary code, and the three parameters are connected in a cascading way. Each group of parameters is called an individual, and the combination of $n_{GA}$ groups of parameters is called the initial population. $n_{GA}$ is called popsize.
• Three points are selected from the leading edge of the original BSI, and their corresponding frequencies are denoted as $f_1$, $f_2$ and $f_3$. $f_1$, $f_2$ and $f_3$ are substituted into the ionospheric model constructed by the $n_{GA}$ groups of parameters $\left(f_{c0},y_{m0},r_{b0}\right)$, $\left(f_{c1},y_{m1},r_{b1}\right)$, $\cdots$ and $\left(f_{c{n}_{GA}},y_{m{n}_{GA}},r_{b{n}_{GA}}\right)$. The corresponding minimum group range $P_i^{\prime }\left(f_{c0},y_{m0},r_{b0}\right)$, $P_i^{\prime }\left(f_{c1},y_{m1},r_{b1}\right)$, $\cdots$, $P_i^{\prime }\left(f_{c{n}_{GA}},y_{m{n}_{GA}},r_{b{n}_{GA}}\right)$ $(i=1,2,3)$, namely, the theoretical leading edge, is calculated.
• The sum of the standard variances of the theoretical leading edge and the measured BSI leading edge is calculated as the objective function $\mathsf{\Phi }(f_c,y_m,r_b)$. Let the fitness function be $F(f_c,y_m,r_b)=C_{\max }-\mathsf{\Phi }(f_c,y_m,r_b)$, where $C_{\max }$ is any large number.
• Selection, crossover and mutation are performed. Selection refers to taking the ratio of the fitness of an individual to the total fitness of the population as the probability of the individual being selected and performing $n_{GA}$ selection operations to obtain the next-generation population. Crossover means setting the probability of crossing to $P_c\in \left(0,1\right)$. A crossover position is selected at a random position in the binary code of two sets of parameters, and the tail codes after the crossover position are swapped. Mutation means that the mutation probability is set to $P_m\in \left(0,1\right)$ and the symbol of the mutation position is changed from 0 to 1 or from 1 to 0. At this point, the next-generation population is obtained, and steps 1 to 4 are repeated until the set evolution algebra $m_{GA}$ is reached.
• The individuals with the highest fitness in the last-generation population are the results of the inversion of ionospheric parameters.

2.1.3. Limitations of Inversion Based on the GA

• The GA itself has two defects. The first is “premature convergence”. In other words, in the initial stage of search, if the initial population is not reasonably constructed, the diversity of the population will be lost due to the rapid increase in good individuals, which will cause the program to fall into a local optimum and fail to find the global optimal solution. The second defect is a poor local search ability. It is found that the GA can reach 90% of the optimal solution quickly, but it takes a long time to find the real optimal solution. An effective method of addressing these two defects is to perform many repeated experiments, but this will greatly reduce the efficiency of the algorithm. A more reasonable approach is to incorporate the spatial gradient variation of the electron density to correct it.
• The existing method of discrete to continuous processes is unreasonable. To recover the nonuniformity of the two-dimensional electron density profile, the ionospheric information needs to be inverted by intensive sampling at the leading edge. This is a discrete process, and the obtained ionospheric parameters are also distributed discretely at different distances. When discrete incomplete information is used, approximate substitution is often used for processing. This method will introduce approximation error or conflicting information and reduce the accuracy of subsequent processing. Considering the strong local correlation of ionospheric parameters, we aim to construct a more realistic two-dimensional electron density profile by combining the moving least-square method for data fitting.

2.2. Improved Inversion Algorithm

2.2.1. Neighborhood-Aided Correction

When the ionospheric detection direction has a north–south component, the geographical latitude of the radio wave reflection region will change with the propagation distance, and this latitude change has a great influence on the ionospheric critical frequency $f_c$. Therefore, unlike the ideal condition that only three sampling points can be taken on the leading edge to invert the complete ionosphere, for detection across latitudes, the whole leading edge should be sampled at small intervals many times. Specifically, to obtain the ionospheric reflection region information corresponding to different ground ranges, it is necessary to group the leading edges according to the change in the detection frequency (from small to large). When the frequency interval is small enough, the reflection region corresponding to a set of data (a small leading edge) is also small enough to be approximately considered to be in the same state. The reflection regions corresponding to different sets of data are in different states. This not only conforms to the MQP model selected but also reflects the ionospheric changes in the whole detection direction through the inversion results of different groups, that is, local uniformity and overall nonuniformity.

The potential numerical continuity of each set of inversion results can be used to assist the GA in correcting the inversion results. Conventional genetic inversion methods regard the inversion process of each frequency band as an independent event and do not consider the continuity of the overall electron density change. In fact, when the frequency steps are small, the ground range changes little, as does the corresponding ionosphere reflection region. According to experience, for radar or communication equipment, the variation trend of the three parameters is relatively stable, and there will not be many drastic changes in a small range. Therefore, the information used by the conventional inversion method based on the GA is incomplete, and its anti-error ability needs to be improved. In this regard, we propose a neighborhood-aided correction method, and the specific steps are as follows:

• A total of 5 consecutive GA inversions are carried out. Taking the $i$th iteration as an example, the grouped ionospheric parameters obtained are denoted as $\{\left(f_c^{i-2},y_m^{i-2},r_b^{i-2}\right),$$\left(f_c^{i-1},y_m^{i-1},r_b^{i-1}\right),\cdots ,\left(f_c^{i+2},y_m^{i+2},r_b^{i+2}\right)\}$.
• Combination and reconstruction are performed for ionospheric parameters in the adjacent space. Considering that multiple evolution takes a long time for the GA and the algorithm performance is greatly affected by the random selection of the initial population, the proposed method reduces the number of GA evolutions. Based on the continuity of the ionospheric parameter space, we use a combination of ionospheric parameters in the adjacent space instead of multiple evolution to increase the ability of optimal parameter search. Through reconstruction, we can obtain a more satisfying solution to the objective function and fully explore the value of potential auxiliary information. Here, the adjacent space is defined as the four other leading edges closest to the leading edge of the segment after segmented processing of the leading edge. The objective function $\mathsf{\Phi }(f_c,y_m,r_b)$ is set as the sum of the standard variances of the theoretical and real leading edges, and the parameter combination that minimizes $\mathsf{\Phi }$ is the optimal parameter $\{f_{c\_best},y_{m\_best},r_{b\_best}\}$. As an example of a combination reconstruction, the optimal parameter can be expressed as:

  $$\mathsf{\Phi }(f_{c\_best},y_{m\_best},r_{b\_best})=min\mathsf{\Phi }(f_{c\_fn},y_{m\_yn},r_{b\_rn})$$

  (6)

  where $f_{c\_fn}$ is the critical frequency of the $fn$ group of the 5 groups of parameters involved in the combination reconstruction process, $fn=1,2,\dots ,5$. Similarly, $y_{m\_yn}$ and $r_{b\_rn}$ are the corresponding half-thickness of the ionosphere and distance from the bottom of the ionosphere to the center of the earth, respectively.
• Outlier detection is performed. Through residual analysis, the anomalous theoretical leading-edge points with large residuals are removed so that the corresponding real leading-edge points are no longer involved in inversion. For abnormal theoretical leading-edge points with relatively large residuals, steps 1 and 2 are repeated for an additional round of enhanced GA inversion, and the relatively optimal parameters are retained according to the calculation results of the objective function $\mathsf{\Phi }(X)$. The residual $\delta$ refers to the difference between the leading-edge position of the real ionization diagram and the leading-edge position of the theoretical derivation, which follows the normal distribution $N\left(0,{\delta }^2\right)$ and is expressed as

  $$(\delta -\overline{\delta })/S={\delta }^{\ast }$$

  (7)

  where $\overline{\delta }$ is the average residual error, $S$ is the standard deviation of the residual error and ${\delta }^{\ast }$ is the standardized residual error, which obeys the normal distribution $N\left(0,1\right)$. When the normalized residual of the leading-edge points falls outside the confidence interval of 0.95, they can be judged as abnormal leading-edge points and are not used for ionospheric parameter inversion.
• $n$ groups of ionization parameters are obtained, that is, the result of the inversion of the ionospheric parameters.

The flow chart of the neighborhood-aided correction algorithm is shown in Figure 1.

2.2.2. Local Region Multistep Fitting

The common method for addressing an incomplete ionogram is approximate substitution; that is, the missing value is replaced by the adjacent value of the missing parameter, and only the existing information is used for subsequent calculation. This method does not consider the gradient change of the real electron density distribution, which will introduce conflicting information or artificial error and reduce the inversion accuracy. To solve this problem, we design a multistep fitting method based on the moving least-square method. The two-dimensional electron density region to be fitted is divided into several local subregions to be fitted, and the weighted least-square method is applied to fit all nodes in each local subregion. Weighted and segmented processing is carried out in each subregion until global fitting is completed. According to this idea, in the region of fit, the gradient of electron density does not obviously change, so a smooth two-dimensional profile can be fitted. In the whole large region, several independent fitting regions can reflect the horizontal gradient of the electron density distribution. The multistep fitting method can improve the reconstruction accuracy of the two-dimensional profile and better match the real situation. The specific steps are described below.

The two-dimensional electron density region to be fitted is divided into several local subregions to be fitted, and the fitting function is constructed in the subregions, which can be expressed as

$$f_{\mathrm{fit}}(x)={\displaystyle \sum p_j}(x)a_j(x)=p^T(x)a(x)$$

(8)

where $a(x)$ is the coefficient to be solved and $p(x)$ is the basis function. For two-dimensional problems, a linear basis $p(x)={\left[1,x,y\right]}^T$ with $m=3$ and quadratic basis $p(x)={\left[1,x,y,x^2,xy,y^2\right]}^T$ with $m=6$ are generally selected, where $m$ is the number of basis function items.

$n$ discrete points are assumed in the subregion, with $x=\left[x_1,x_2,\cdots ,x_n\right]$ and $y=$ $\left[y_1,y_2,\cdots ,y_n\right]$. The weighted least-squares method is used to solve the coefficient vector $a\left(x\right)$ with the sum of the squares of the weighted residuals of the nodes as the objective function, which can be expressed as

$$J={\displaystyle \sum _{i=1}^{n}w}\left(\left|x_0-x_i\right|/r_w\right){\left[{\displaystyle \sum _{j=1}^m{p}_m^T}\left(x_i\right)a_m\left(x_i\right)-y_i\right]}^2,$$

(9)

where $x_0$ is the midpoint of the subregion, $r_w$ is the radius of the subregion and $w\left(\left|x_0-x_i\right|/r_w\right)$ is the weight of the nodes in the subregion. In this way, the problem of finding the coefficient matrix is transformed into the problem of determining the minimum value of the above equation. We take the derivative of $J$ with respect to $a$, and we have that

$$\frac{\partial J}{\partial a}=A_{\mathrm{fit}}(x)a(x)-B_{\mathrm{fit}}(x)y^T=0,$$

(10)

where

$$A_{\mathrm{fit}}(x)={\displaystyle \sum _{i=1}^{n}w}\left(\left|x_0-x_i\right|/r_w\right)p\left(x_i\right)p^T\left(x_i\right),$$

(11)

$$B_{\mathrm{fit}}(x)=\left[w\left(\left|x_0-x_1\right|/r_w\right)p\left(x_1\right),w\left(\left|x_0-x_2\right|/r_w\right)p\left(x_2\right),\cdots ,w\left(\left|x_0-x_n\right|/r_w\right)p\left(x_n\right)\right].$$

(12)

We substitute Equations (10) and (11) into Equation (9) to obtain

$$f_{\mathrm{fit}}(x)=p^T(x)A_{\mathrm{fit}}^{-1}(x)B_{\mathrm{fit}}(x)y^T.$$

(13)

The weight function $w$ plays a very important role in fitting, and the value of $w\left(\left|x_0-x_i\right|/r_w\right)$ decreases with increasing $\left|x_0-x_i\right|/r_w$. The weights of discrete points are 0 when they are outside the subregion and 0 to 1 when they are inside the subregion, which is the key to the local approximation property of the fitting profile.

The weight function is an exponential function, expressed as

$$w\left(\left|x_0-x_i\right|/r_w\right)=\{\begin{array}{cc}\frac{\left(e^{-g{\left(\left|x_0-x_i\right|/r_w\right)}^2}-e^{-g}\right)}{1-e^{-g}},& w\left(\left|x_0-x_i\right|/r_w\right)\le 1\\ 0,& \mathrm{otherwise}\end{array}$$

(14)

where the parameter $g$ can be selected according to the degree of approximation.

In the above treatment, the midpoints of each subregion are treated as equally important. However, according to the fitness value in the GA, the leading-edge reconstruction effects are different according to the different inversion results, and the importance of the corresponding midpoint needs to be adjusted. Here we reduce or increase the importance of the midpoint by scaling the radius of the fitting subregion. The scaling degree depends on the fitness value of the parameters corresponding to the midpoint. The higher the fitness is, the higher the reliability of the parameters obtained by inversion, and then the midpoint can radiate more areas. After scaling, the radius of the subregion can be expressed as

$$r_{w\_new}=\frac{F}{\overline{F}}{r}_w$$

where $F$ and $\overline{F}$ are, respectively, the fitness and average fitness of the parameters corresponding to the midpoint of all the fitting regions.

The flow chart of the local region multistep fitting algorithm is shown in Figure 2.

3. Results

To verify the effectiveness of the proposed method, the international reference ionosphere (IRI) model data were used as the reference basis for real ionospheric parameters and leading-edge calculations. The IRI project, a joint initiative of the international committee for space research (COSPAR) and the international union of radio science (URSI), aims to establish an empirical standard model of the ionosphere. At present, the model can be used to calculate the electron density, electron temperature, ion temperature, ion composition and other parameters of the ionosphere at a given location and time. It has been widely used in many research fields, such as ionospheric data analysis, HF over-the-horizon radar applications and radio communication applications.

The experimental parameters are set as follows: The detection point is at (35.0° N, 114.0° E). The direction of propagation is southward, and the local time at the transmitter is 13:00 on 20 August 2015. The detection method is sweep frequency detection, the sweep frequency range is 3.2 to 28 MHz and the detection elevation is 15° to 65°. The transmitter location is shown in Figure 3.

The population number of GA $n_{GA}$ is set to 100, the conventional evolution times are set to 300, the optimized evolution times $m_{GA}$ in this paper are set to 100 and the binary coding length $n_{BIN}$ is set to 4.

3.1. Leading-Edge Inversion Results and Analysis

After the interpretation of the leading edge, the leading-edge frequency range is 3.2 to 28 MHz. With a 0.4-megahurtz interval, there are 63 data points in total, which are expressed as $(f_1,P_{r1}^{\prime })$, $(f_2,P_{r2}^{\prime })$, $\cdots$, $(f_i,P_{ri}^{\prime })$ $(i\le 63)$ in ascending order of detection frequency, where $f_i=3.2+0.4(i-1)$. Every three adjacent points are regarded as a group of data, and the frequencies of the groups do not overlap; there are 21 groups of data. After the front is divided into 21 segments, each segment is inverted separately. Using the ionospheric parameters obtained from the inversion, the variation in electron density with height in the radio wave reflection region corresponding to the detection data is obtained. The position of the reflection area is 1/2 of the ground range $D$ on the ground.

Figure 4 shows the inversion results of the electron density profile at (31.0° N, 114.0° E) by different methods. The blue dotted line represents the electron density profile of the IRI model, which is regarded as the real profile. The solid yellow line represents the profile inversion results of the conventional GA, where the population number is 100 and the number of evolutions is 300, and the optimal value is obtained through five experiments. The solid green line represents the profile inversion result of many repeated GA inversions, where the population number is 100, the number of evolutions is 300 and the optimal value is obtained through 20 experiments. The solid red line represents the profile inversion results after neighborhood-aided correction by the proposed method.

As shown in Figure 4, the gap between the conventional GA inversion results and the real profile is the largest. The peak frequency error of 0.3 MHz in the F layer leads to a larger overall electron density. The reason is that the initial population of the GA is limited. Once the ionospheric parameters corresponding to the initial individuals differ greatly from the real values, it is difficult to generate qualified individuals through selection, crossover and mutation. This situation can be improved by multiple GA inversion or population enlargement. As shown in the green curve in the figure, the repeated GA inversion simulates the defects to a certain extent, but it also takes a great deal of time. In contrast, the results obtained by using the proposed method are in good agreement with the values of the whole real ionospheric profile. This shows that the neighborhood-aided correction method can obtain more accurate ionospheric parameters by using the regularity of gradient variation in adjacent space.

In terms of time consumption, repeated GA inversion requires an additional $(25-5)n_{GA}{m}_{GA}$ frontier calculation iterations (600,000 iterations), while the method proposed in this paper only needs ${\left(n_{BIN}\right)}^9$ frontier calculation iterations (260,000 iterations) even without any repetition of parameter values in adjacent space. Due to the spatial continuity of the electron density distribution in the real inversion, the number of calculations will be far less than this value due to the repeated parameters in the adjacent space. Figure 5 shows the comparison between the theoretical and measured leading edges synthesized from the inversion results of the electron density profile using the proposed method. It can be seen that the two are very close, which indicates that the inversion results of the method in this paper are accurate and reliable.

Without loss of generality, other transmitting sites are randomly selected to conduct 10 experiments within 150 km of the original radio transmitting location (the other conditions remain unchanged). The root-mean-square error (RMSE) and mean relative error (MRE) are used to calculate the error of the ionospheric profile inversion results, which can be expressed as

$${\delta }_{\mathrm{RMES}}={\left\{\frac{1}{M}{\displaystyle \sum _{i=1}^M{\left[f_p^{\prime }\left(h_i\right)-f_p\left(h_i\right)\right]}^2}\right\}}^{1/2}$$

(15)

$${\delta }_{\mathrm{MRE}}=\frac{1}{M}{\displaystyle \sum _{i=1}^M\frac{\left|f_p^{\prime }\left(h_i\right)-f_p\left(h_i\right)\right|}{f_p\left(h_i\right)}}$$

(16)

where $f_p$ is the reconstructed profile, $f_p$ is the real profile and $M$ is the set of height points calculated by 1 km steps from 130 to 400 km.

Table 1. RMSE and MRE of the inversion results.

	GA	GA	Proposed
	(5 Simulations)	(20 Simulations)
RMSE	0.523	0.207	0.159
MRE	5.22%	3.93%	2.27%

shows the RMSE and MRE values of the inversion results using each method. Through this comparison, it can be found that the calculation results of the proposed method are more accurate than those of other methods in terms of the RMSE value and MRE value.

An example using measured data is shown in Figure 6, which occurs at 10:14 LT on 19 February 2016. Similar to the previous simulation results, the proposed method reliably reconstructs the leading-edge data, thus proving its potential to process real data.

3.2. Two-Dimensional Profile Fitting Results and Analysis

The one-dimensional electron density profile is calculated based on the ionospheric parameters obtained from all leading-edge data inversions, and then the two-dimensional electron density profile over the detection path is obtained by combining the one-dimensional profiles, as shown in Figure 7. It can be seen that with the increase in ground range, the ionospheric electron density presents a trend of horizontal gradient change. In Figure 7b, the conventional ionospheric inversion results are discrete.

When uncertain information between discrete points needs to be used in subsequent processing, simple approximate processing will introduce systematic errors. In Figure 7c, the local region multistep fitting method is used in this paper to better recover the true gradient change in electron density. The proposed method has a strong effect on the numerical points in the fitting subregion but has no effect on the numerical points outside the subregion, which is consistent with the real ionospheric state.

Without loss of generality, 100 experiments are conducted at other transmitting points within 150 km of the original radio transmitting position (the other conditions remain unchanged). The results before and after fitting are compared with the real two-dimensional section, and the error statistical results are obtained, as shown in Figure 8. When the critical frequency error exceeds 0.5 MHz, the inversion results are judged to be incorrect. When the critical frequency error is between 0.5 and 0.2 MHz, the inversion results are basically accurate. When the critical frequency error is less than 0.2 MHz, the inversion results are very accurate. As shown in Figure 8, 99.3% of the inversion results are controlled within the error range after the fitting method in this paper is applied, which is 8.7% higher than before fitting. In addition, 56.5% of the inversion results are very accurate, 18.1% higher than before fitting. The method presented in this paper improves the inversion ability of the two-dimensional electron density profile and has application value in processing complex and changeable backscatter data.

In order to highlight the superiority of this method, three groups of comparative experiments are carried out, and each group of experiments are repeated 100 times by changing the transmitting point. Random errors with mean square error of 0, 10 and 20 km are added to the leading-edge points, respectively, to simulate the influence of severe weather on the electromagnetic environment. The methods of this paper, literature reference [10] and literature reference [12] are compared. In the hybrid GA in literature reference [12], the parameters of the GA algorithm are consistent with the proposed method. The initial temperature of SAA is 2000 °C, the termination temperature is 0.01 °C and the temperature decay factor is 0.8. The average critical frequency errors of the inversion results are shown in

Table 2. Comparison of average critical frequency errors (MHz).

Leading-Edge Points Error	Method [10]	Method [12]	Proposed
MSE 0 km	0.194	0.243	0.183
MSE 10 km	0.567	0.373	0.258
MSE 20 km	0.979	0.558	0.417

.

It can be seen from Table 2 that the proposed method and method [10] have better performance when the leading-edge data are accurate. However, with the increase in leading-edge data error, the performance of method [10] decreases faster. The reason is that in method [10], the inversion accuracy of ionospheric parameters directly depends on the accuracy of leading-edge identification, used to recover the frequency dependence of the distance to the skip zone border. In contrast, the proposed method reduces the interference caused by leading-edge errors to a certain extent through mutual correction of neighborhood information. The performance of method [12] is relatively poor because the rationality of the initial population selection of GA is still one of the important factors affecting the inversion results. After repeated experiments for many times, the strong randomness leads to the special case of great result error, which leads to the increase in the average error level.

3.3. Ray Tracing in the Ionosphere

After the distribution of the ionospheric electron density is obtained, the propagation process of radio waves can be accurately recovered by ray tracing. Ray tracing plays an important role in determining the blind area of radio coverage, studying irregular phenomena, calculating the height of ray reflection and correcting the azimuth of the detected target. When the geomagnetic effect and collision effect are ignored, the refractive index $\mu$ of the radio wave ray is expressed as

$${\mu }^2=1-f_p^2/f^2,$$

(17)

where $f_p$ is the plasma frequency and $f$ is the radio frequency.

After the refractive index change is known, only the reflection height needs to be calculated, and the idea of solving the ground range in layers by Snell’s law can be used to quantitatively describe the ray trajectory. Snell’s law determines the ground range corresponding to the height by dividing the ionosphere into equal step sizes.

Figure 9 shows the ray tracing results when the wave frequency is 13 MHz and the elevation angle ranges across 15° to 65° (5° apart). Figure 9a is the ray tracing result using the real ionospheric profile, Figure 9b is the ray tracing result inversion using the conventional GA (without neighborhood-assisted correction and local step fitting) and Figure 9c is the ray tracing result after neighborhood-aided correction only. Figure 9d is the ray tracing result improved by using both neighborhood-aided correction and local region multistep fitting, which is the final inversion result of this paper.

Table 3. Ground distance of the ray tracing results (km).

	Elevation Angle (°)
	15	20	25	30	35	40
Real data	1457	1176	1051	979.3	912.4	1206
Conventional GA	1741	1405	1046	965	947	/
Neighborhood-aided correction	1649	1258	1050	895.4	928.5	1075
Neighborhood-aided correction and local region multistep fitting	1658	1213	999.7	920	959.3	1217

shows the ground distance of the ray tracing results.

The comparison between Figure 9b,c shows that neighborhood-aided correction increases the inversion accuracy of ionospheric parameters, making the electron density distribution more consistent with the real value, and its ray tracing results are also preliminarily corrected. For example, the rays originally emitted at an elevation of 40° in Figure 9b are not reflected correctly and are corrected in Figure 9c. Within the range of elevation angles (15–35° in the figure) from which the radio wave can be reflected to the ground, the average distance error from the ground range in Figure 9a decreases from 113.4 to 84.3 km and the average error percentage decreases from 8.9 to 6.9%.

The comparison between Figure 9c,d shows that local region multistep fitting further refines the electron density distribution. Within each step interval height of equal length, it is ensured that slight changes in the refractive index can be reflected. Due to the long distance of the HF rays, this slight error in the refractive index will be magnified at the end of the long-distance propagation, bringing a significant effect. Within the range of elevation angles (15–40° in the figure) from which the radio wave can be reflected to the ground, the average distance error from the ground range in Figure 9a decreases from 84.3 to 67.8 km and the average error percentage decreases from 6.9 to 5.7%.

The comparison between Figure 9a,d shows that the ray tracing results after inversion by the proposed method are basically consistent with those obtained by using real data and the accuracy of ray tracing is improved. Due to the restriction of the MQP model, the one-dimensional electron density profile is not completely consistent with the real profile.

Notably, we consider the ionospheric medium to be approximately isotropic, regardless of the effect on the geomagnetic field. At the same time, the collision effect can be ignored without considering the influence of energy absorption. In the study of ionospheric inversion, the main parameters of interest are the group path and direction of radio wave propagation. The electron collision only causes the loss of radio wave energy and has little effect on the inversion results. However, under the influence of the geomagnetic field, once radio waves are split into O waves and X waves, they will have two possible propagation paths, characterized by different phases and group velocities. This will have a great impact on the inversion results, which should be considered in the future research.

4. Conclusions

An improved method for the inversion of BSIs by using neighborhood-aided and multistep fitting is proposed in this paper. The ionospheric parameter inversion results in adjacent space are combined and reconstructed by using the neighborhood-aided correction method. The introduction of auxiliary information sources corrects the defects of conventional GA, such as falling into local optima and having a poor local search ability. The local region multistep fitting method is used to describe the local uniformity and global inhomogeneity of the two-dimensional electron density profile by dividing the fitting region. Combined with the idea of the moving least-square method, the problem of discontinuous and unsmooth fitting curves on adjacent sections is avoided while the horizontal gradient variation of the overall electron density distribution is reflected. The experimental results show that the proposed method can improve the accuracy of BSI inversion and provide reliable support for tracking radio ray trajectories.