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Ionospheric electron density modelling using B-splines and constraint optimization

Abstract

 Many modern applications, such as precise point positioning, autonomous driving or precision agriculture would benefit significantly if a high-precision and high-resolution model of electron density in the ionosphere and the plasmasphere would be globally available. Since the development of such a model still relies on data with insufficient and uneven global coverage, the consideration of background information and the introduction of equality and inequality constraints on Chapman key parameters are essential. In this work, we develop a multi-layer Chapman model based on B-spline expansions of selected key parameters of the electron density. The unknown series coefficients of the key parameters are subject to equality and inequality constraints. Finally, the developed model is applied to a combination of real and semi-simulated input data; the results are validated through ionosonde measurements.

Graphical Abstract

Introduction

The Sun’s radiation has important impact on different components of the Earth system, for instance on the sub-components plasmasphere and ionosphere. The ionization of neutral gas molecules results in ionized plasma and free electrons. The four-dimensional (4D) space and time-dependent electron density need to be accurately known for satellite navigation and telecommunication, since electromagnetic waves are strongly affected on their path through the charged part of the atmosphereFootnote 1. The 4D electron density is a highly variable function: amongst others it reflects plasma fluctuations with periods less than a few minutes, diurnal and seasonal variations, long-period changes following the solar cycle of 11 years as well as the impact of space weather events, such as solar flares and coronal mass ejections (CME).

If the 4D electron density would be known everywhere within the plasmasphere and the ionosphere at any moment in time, we would be able to correct every measurement of space-geodetic observation techniques, such as the Global Navigation Satellite Systems (GNSS), Satellite Altimetry, the French satellite tracking system Doppler Orbitography and Radiopositioning Integrated by Satellite (DORIS) or Very Long Baseline Interferometry (VLBI) for the plasmaspheric and ionospheric impact independently if we use single or multi-frequency measurements. Vice versa, all these observation techniques including, e.g. the space-based GNSS technique of measuring Ionospheric Radio Occultations (IRO) as well as the Langmuir probe measurements on-board CHAMP and Swarm provide valuable information on the state of the plasmasphere and the ionosphere, and thus for modelling the electron density. To be more specific, in this paper we aim on modelling the electron density function

$$\begin{aligned} N_e = N_e(\varphi , \lambda , h, t) \end{aligned}$$
(1)

depending on latitude \(\varphi\) and longitude \(\lambda\)—both defined with respect to a geocentric Earth-bounded coordinate system \(\Sigma _E\)—on the height \(h = r - R_E\) above the surface of a spherical Earth with a constant radius \(R_E\) and r being the radial distance and on the time t with high accuracy and high resolution. Since modelling a 4D function is rather challenging, the electron density is usually transferred to the 3D Vertical Total Electron Content (VTEC) depending on latitude and longitude as well as the time. Integrating the electron density (1) along the ray-path, e.g. between a transmitting GNSS satellite \(P^S\) and a ground-based receiver \(P_R\) the Slant Total Electron Content (STEC)

$$\begin{aligned} STEC(P^S, P_R, t) = \int _{P^S}^{P_R} \, N_e(\varphi , \lambda , h, t) \, ds \end{aligned}$$
(2)

is obtained. Due to the dispersive property of the plasmasphere and the ionosphere the geometry of the ray-path with the differential line element ds depends on the carrier frequency f of the electromagnetic wave, and thus deviates from the straight line between \(P^S\) and \(P_R\). To simplify the notation and bring vector notation into play, we introduce the 3D position vector

$$\begin{aligned} {{\varvec{x}}} = r \cdot \left[ \cos \varphi \cos \lambda , \cos \varphi \sin \lambda , \sin \varphi \right] ^T \end{aligned}$$
(3)

defined in \(\Sigma _E\) and rewrite Eq. (2) as

$$\begin{aligned} STEC({{\varvec{x}}}^S, {{\textbf {x}}}_R, t) = \int _{P^S}^{P_R} \, N_e({{\varvec{x}}}, t) \, ds \;. \end{aligned}$$
(4)

Multiplying the right-hand side of Eq. (4) with the factor \(\pm 40.3/f^2\), we obtain the ionospheric delay \(d_{\mathrm {ion}} = ({\pm 40.3}/{f^2}) \cdot STEC\). This relation is an approximation, valid for standard conditions of the geomagnetic field and signal frequencies f larger than 1 GHz. It neglects the higher-order effects which may reach up to 0.2 cm in zenith direction for GPS signals. Note, the sign of \(d_{\mathrm {ion}}\) changes with reference to carrier phase observations (”−”) or for pseudorange measurements (”\(+\)”). The conversion of STEC into VTEC is achieved by introducing an isotropic mapping function M(z) depending on the zenith angle z. It performs the projection

$$\begin{aligned} VTEC({{\varvec{x}}}_{IPP}, t) = M(z) \cdot STEC({{\varvec{x}}}^S, {{\varvec{x}}}_R, t) \end{aligned}$$
(5)

under the assumption that all electrons are concentrated in a spherical shell \(\Omega _{H}\) of an infinitesimal thickness with fixed height \(H = r - R_E\) above the spherical Earth. This procedure is denoted as the single layer model (SLM) approach; the spatial reference for the VTEC observation is given by the ionospheric piercing point \(P_{IPP}\) with position vector \({{\varvec{x}}}_{IPP}\). It is defined as the intersection point of the straight line between \(P^S\) and \(P_R\) and the sphere \(\Omega _{H}\); see e.g., Schaer (1999). From Eq. (4), we obtain from the integration of the electron density along the height h the result

$$\begin{aligned} VTEC({{\varvec{x}}_{IPP}, t}) = \int _{\mathrm {height}} \, N_e(\varphi _{IPP}, \lambda _{IPP}, h, t) \, dh \;. \end{aligned}$$
(6)

Note, that if the electron density is precisely known the result (6) usually outperforms the corresponding VTEC value from Eq. (5) because applying the mapping function M(z) can introduce a significant error.

Several approaches exist for representing VTEC globally by using input data from GNSS. The Ionospheric Associated Analysis Centers (IAAC) of the International GNSS Service (IGS) routinely provide Global Ionosphere Maps (GIM) representing VTEC based on different mathematical approaches, e.g. the series expansion or the discretization technique. Whereas in the latter case, usually the voxel-based tomography approach is chosen (see e.g. Hernández-Pajares et al. (1999)), in the case of a 2D series expansion spherical harmonics (SH) are often used (see e.g. Schaer (1999)). However, since SHs are oscillating over the sphere and, thus, are characterized by a global behaviour, they cannot represent data of heterogeneous density and quality in a proper way; see e.g. Schmidt et al. (2011). Consequently, at DGFI-TUM, we rely for more than two decades on 2D B-spline basis functions for modelling VTEC, see e.g. Zeilhofer (2008), Schmidt (2007), Schmidt et al. (2015), Limberger (2015) and Erdogan et al. (2017a) for two main reasons, namely (1) they are characterized by their localizing feature and (2) they can be used to generate a multi-scale representation (MSR) based on wavelet theory; see e.g. Schmidt et al. (2015) and Goss et al. (2019). In the case of global modelling, the 2D B-spline basis functions are defined as tensor products of 1D endpoint-interpolating B-spline functions depending on the latitude \(\varphi\) and 1D trigonometric B-spline functions depending on the longitude \(\lambda\); see Schmidt et al. (2011), Schmidt et al. (2015), and Goss et al. (2019); general introductions to both kinds of B-spline basis functions can be found in Schumaker and Traas (1991) as well as Lyche and Schumaker (2000).

Multiplying a 2D series expansion, e.g. in terms of SHs or 2D tensor product B-spline functions dependent on latitude \(\varphi\) and longitude \(\lambda\) with a vertical profile function dependent on the height h, a mathematical model is obtained for describing a 3D function, such as the spatial part of the electron density \(N_e\), i.e. \(N_e(\varphi , \lambda , h)\). Howe et al. (1998) and Gao and Liu (2002) realized this concept—also known as the separability approach Hernández-Pajares et al. (2000)—by combining a SH expansion with empirical orthogonal functions (EOF); Schmidt et al. (2008) transferred this global approach to a regional application by multiplying the 2D tensor product endpoint-interpolating B-splines with the first five EOFs obtained from a principal component analysis (PCA) of a reference model, namely the International Reference Ionosphere (IRI). As an alternative to the aforementioned empirical approach, in ionosphere research often so-called “physics-motivated” functions, e.g. the Chapman function and the Epstein function are used; see Feltens (2007) and Rawer (1988). As will be shown by Eq. (8), we rely in this work on the Chapman function. A further approach is defining the vertical profile function as another 1D series expansion in terms of endpoint-interpolating B-spline functions along the height. This way the electron density \(N_e(\varphi , \lambda , h)\) is spatially modelled by a series expansion in terms of 3D tensor product B-spline functions; This approach increases the number of B-spline coefficients to be estimated significantly, but remains free from any assumptions about the physical distribution of the electrons along the vertical. A synergy between this 3D B-spline representation and the voxel-based approach mentioned before was shown by Olivares-Pulido et al. (2019). An extensive discussion of the aforementioned and further approaches to model the electron density (1) only spatially or even spatio-temporally, i.e. in three or four dimensions, can be found in the PhD thesis of Limberger (2015).

Each Chapman function (8) is defined by three key parameters, two of them are included in a strongly non-linear manner and are also highly correlated with each other. Since such strong correlations also occur between key parameters of different ionospheric layers, the application of the classical least squares estimation often leads—if at all—to physically unrealistic estimation results for the unknown key parameters, e.g. a negative value for the maximum of the electron density within a specified layer; see Liang (2017). Such problems are often handled by introducing “inequality constraints” for the unknown parameters and solved by using optimization algorithms, such as the ones described in Koch (1985), Roese-Koerner (2015), Nocedal and Wright (2006), Boyd and Vandenberghe (2004). Inequality-constrained optimization problems are relatively difficult to solve compared to either unconstrained or equality constrained optimization problems; see Chapter 16 of Nocedal and Wright (2006).

Although there are optimization procedures available (see e.g. Lötstedt (1984), Gill et al. (1984), Coleman and Li (1996) and Mead and Renaut (2010)), the detail of their applied algorithms, the applied assumptions and the implementation along with the possibility to obtain accuracy information, i.e. to estimate standard deviations are either not described or not always explicitly transparent to the user or not available at all. In many cases these algorithms are made for smaller dimensional problems—within 100 to 500 unknown parameters—and are not easily “scalable” for relatively higher-dimensional problems, e.g. a global 4D electron density modelling.

In this paper, due to the consideration of inequality constraints, the estimators of the selected Chapman key parameters are lying within physically realistic limits, for example, non-negative values for the maximum value of the electron density within a specified ionosphere layer. We will develop in the following a general concept where both equality and inequality constraints can be included.

This paper is organized as follows: in the second Section titled “Electron density modelling”, we derive the multi-layer Chapman model of the global electron density. In the following third Section, we set up the series expansions in 2D B-spline basis functions for the selected key parameters and define the inequality and equality constraints related to given bound functions in matrix and vector form. In the next section titled “optimization approach”, we describe the optimization algorithm in detail and present a recipe for applying our developed ICOA. In the next section, we then apply the ICOA to a combination of real and semi-simulated input data and describe the validation of the estimated key parameters using the IRI model as well as ionosonde observations. Finally, we provide a summary and outlook for follow-up work.

Electron density modelling

As can be seen in Fig. 1 the ionosphere defined between the height values \(h_{min}\) and \(h_{max}\) is stratified into the D-, the E-, the \(F_1\)- and the \(F_2\)-layer with a smooth transition into the plasmasphere. Following Limberger (2015), we may set \(h_{min} = 50\) km and \(h_{max} = 1000\) km. Each layer has its physical significance, due to the dominating atmospheric processes, such as ionization, recombination and the overall energy balance of the ionosphere; see e.g. Chapman (1931).

Fig. 1
figure 1

An approximate representation of vertical electron density profile between \(h_{min}\) and \(h_{max}\); the quantities \(N_m^Q( =\) maximum value of the electron density, peak density), \(h_m^Q( =\) peak height) and \(H^Q( =\) scale height) are the key parameters of the ionosphere layer Q with \(Q \in \{D, E, F_1, F_2 \}\). Figure is adapted from Limberger (2015) and is not to accurate scale, but is merely meant for illustration purpose. For a more detailed description of ionospheric profile shape, see e.g. Davies (1990)

In this work, we approximate according to Feltens (2007) the vertical electron density profile by the so-called multi-layer approach

$$\begin{aligned} N_e(h)\,=\,& {} N_e^{D}(h) + N_e^{E}(h) + N_e^{F1}(h) + N_e^{F2}(h) + N_e^{P}(h) \nonumber \\= & {} \sum _{Q=1}^4 N_m^Q \ p^Q(h) + N_0^P p^P(h) \;, \end{aligned}$$
(7)

where the summation index \(Q \in \{1 = D, 2 = E, 3= F_1, 4 = F_2\}\) refers to the four layers of the ionosphere and P means the plasmasphere. We use the Chapman function

$$\begin{aligned} N_e^Q(h) = N_m^Q \ {\exp } \left( \frac{1}{2} \left( 1 - \frac{h - h_m^{Q}}{H^{Q}} - {\exp } \left( - \frac{h - h_m^{Q}}{H^{Q}} \right) \right) \right) \end{aligned}$$
(8)

to describe the vertical electron density distribution of the Q-layer. Herein, the three quantities \(N_m^Q( =\) peak density), \(h_m^Q( =\) peak height) and \(H^Q( =\) scale height) are the key parameters of the Q-layer. The peak density \(N_m^Q\) corresponds to the maximum value of the electron density within the Q-layer and the peak height \(h_m^Q\) means the altitude corresponding to the peak density. The scale height \(H^Q\) refers to the thickness of the profile around the peak height. Whereas \(N_m^Q\) means a scale factor in Eqs. (7) and (8), the key parameters \(h_m^Q\) and \(H^Q\) define the shape of the profile function \(p^Q(h)\) introduced in Eq. (7). Note, each profile function \(p^Q(h)\) is defined within the full height range of the ionosphere, i.e. between \(h_{min}\) and \(h_{max}\), and not restricted to the sub-range of the Q-layer. This way no jumps or discontinuities occur at the transition points of two adjacent layers along the vertical.

The exponential decay of the electron density within the plasmasphere P above the height value \(h_m^{F_2}\) will be described by the function

$$\begin{aligned} N_e^P(h) = N_0^P \ {\exp } \left( - \frac{|h - h_m^{F_2}|}{H^P} \right) , \end{aligned}$$
(9)

where the quantities \(N_0^P( =\) basis density) and \(H^P(=\) scale height) are the two key parameters of the plasmasphere. Whereas \(N_0^P\) means a scale factor in Eqs. (7) and (9) the parameters \(h_m^{F_2}\) and \(H^P\) define the shape of the profile function \(p^P(h)\) introduced in Eq. (7). From Eqs. (7), (8) and (9), we conclude that our multi-layer Chapman model (MLCM) is based on the set

$$\begin{aligned} {\mathcal {K}}\,=\, \{ N_m^{D}, h_m^{D}, H^{D}, N_m^{E}, h_m^{E}, H^{E}, N_m^{F_1}, h_m^{F_1}, H^{F_1}, N_m^{F_2}, h_m^{F_2}, H^{F_2}, N_0^P, H^P \} \end{aligned}$$
(10)

of altogether 14 key parameters \(\kappa _r\) with \(r = 1,2, \ldots , 14\). This set can be further extended, e.g. by choosing different scale heights for the top and the bottom side of each layer. In the following we will stay with the set as defined in Eq. (10).

Both the D-layer and the E-layer are thin compared to the \(F_1\)-layer and to the \(F_2\)-layer; see Berkner and Wells (1934) and Bremer (1998). Whereas the peak densities \(N_m^D\) and \(N_m^E > N_m^D\) of the D- and the E-layer, respectively, are at least two orders of magnitude lower compared to the peak density \(N_m^{F_2}\) of the \(F_2\)-layer under nominal conditions, i.e. \(N_m^D < N_m^E \ll N_m^{F_2}\), for the \(F_1\)-layer the definition \(N_m^{F_1} = 0\) holds during nighttime; see Liang (2017). The 14 key parameters \(\kappa _r \in {\mathcal {K}}\) defined in Eq. (10) vary on spatio-temporal scales and our aim is their global modelling using an appropriate mathematical approach. The unknown model parameters of this approach should then be estimated by a sufficient number of observations which are assumed to be functionals of the 4D electron density \(N_e(\varphi , \lambda , h, t)\) as defined in Eq. (1), e.g. STEC measurements as introduced in Eq. (4).

Earlier studies, e.g., by Limberger (2015) and Liang (2017) highlighted the two main challenges for the selection of an appropriate estimation procedure, namely (1) an inhomogeneous and insufficient global coverage of observations and (2) significant correlations between most of the key parameters. The first item refers to large spatial gaps of observations, from ground-based GNSS receivers, in particular over oceans and deserts as well as dense clusters of observations, for example, over Europe, the United States and Japan; see Erdogan et al. (2017b) and Goss et al. (2020). The different measurement techniques we use in this study and the spatial distribution of their observations will be discussed briefly in Section  “Application to real-data”.

The second item states that most of the key parameters \(\kappa _r \in {\mathcal {K}}\) are physically correlated to each other, e.g. due to the electron–ion recombination and different coupling processes in the thermosphere–ionosphere system; see e.g., Stankov et al. (2003), Stankov and Jakowski (2006) as well as Belehaki et al. (2006). Statistical correlations between the key parameters arise due to their observability and sensitivity to electron density. This was shown in Limberger et al. (2014a) with regard to regional ionosphere modelling considering only the \(F_2\)-layer in the South American region by using a combination of real and simulated observations. The incorporation of inequality constraints partially decorrelates the unknown parameters by localizing them to be within specific magnitude bounds. Accordingly, Limberger (2015), Liang (2017) highlighted the need for stochastic modelling and the consideration of inequality constraints on the key parameters. We address this issue in detail in Section “Optimization approach”.

Keeping in view the aforementioned challenges, in this paper we use (1) electron density modelling considering the D-, the E-, the \(F_1\)-, and the \(F_2\)-layer as well as the plasmasphere; (2) the multi-layer Chapman approach for modelling the altitude dependency of the electron density; (3) 2D B-spline tensor product basis functions for the representation of the selected key parameters along latitude and longitude, and (4) the combination of Formosat-3/COSMIC and GRACE radio occultation measurements as well as so-called semi-simulated observations based on the separability approach developed by Hernández-Pajares et al. (2000). Furthermore, the two most important novelties of this paper are (1) the consideration of inequality constraints on selected key parameters of the multi-layer Chapman model (MLCM) and (2) the application of a constraint optimization algorithm for parameter estimation. To the knowledge of the authors, this approach has not yet been applied to ionosphere modelling.

Parameterization of the multi-layer Chapman model (MLCM)

Each key parameter \(\kappa _r \in {\mathcal {K}}\) of the set \({\mathcal {K}}\) introduced in Eq. (10) of the MLCM as defined by Eqs. (7), (8) and (9) can be represented globally by the 2D B-spline expansion:

$$\begin{aligned} \kappa _r(\varphi , \lambda , t) = \sum _{k_1 = 0}^{K_{J_1} - 1} \sum _{k_{2} = 0}^{K_{J_2} - 1} {d^{J_1, J_2}_{k_1,k_2; \kappa _r}}(t) \; N_{J_1,k_1}^{d_N}(\varphi ) \; T_{J_2,k_2}^{d_T}(\lambda ) + v_{\kappa _r}(\varphi , \lambda , t) \end{aligned}$$
(11)

in terms of the \(K_{J_1} \cdot K_{J_2} = u\) 2D tensor products \(N_{J_1,k_1}^{d_N}(\varphi ) \; T_{J_2,k_2}^{d_T}(\lambda )\) of the endpoint-interpolating polynomial B-spline functions \(N_{J_1,k_1}^{d_N}(\varphi )\) of degree \({d_N}\) and the trigonometric B-spline functions \(T_{J_2,k_2}^{d_T}(\lambda )\) of degree \({d_T}\). The shift parameters \(k_1 \in \{0,1, \ldots , K_{J_1} - 1 \}\) and \(k_2 \in \{0,1, \ldots , K_{J_2} - 1 \}\) for the endpoint-interpolating and the trigonometric B-spline functions define the position of the 2D tensor product B-spline on the sphere \(\Omega _H\). The total numbers \(K_{J_1} = 2^{J_1} +2\) and \(K_{J_2} = 3 \cdot 2^{J_2}\) of the B-spline functions depend on the B-spine levels \(J_1\) and \(J_2\) with respect to latitude \(\varphi\) and longitude \(\lambda\). Whereas the function \(v_{\kappa _r}(\varphi , \lambda , t)\) denotes the truncation error, the u time-dependent series coefficients \(d^{J_1,J_2}_{k_1,k_2;\kappa _r}(t)\) are the unknown parameters of the expansion (11) and have to be determined. In the sequel we select \(d_N = d_T = 2\) for the degree values \({d_N}\) and \({d_T}\). Consequently, Fig. 2 shows in the right panel exemplarily the 2D quadratic B-spline basis function \(N_{4,11}^2(\varphi ) \cdot T_{3,2}^2(\lambda )\). Due to its strong localization feature only observations available within the so-called influence zone, i.e. the non-zero area of this 2D B-spline function, contribute to the estimation of the related series coefficient \(d^{4,3}_{11,2;\kappa _r}(t_i)\) at the time moment \(t = t_i\) for the selected key parameter \(\kappa _r \in {\mathcal {K}}\).

Fig. 2
figure 2

2D tensor product B-spline representation of a key parameter \(\kappa _r\), as introduced in Eq. (11). The endpoint-interpolating B-splines (top left panel) and the trigonometric B-splines (bottom left panel) are shown for the selected level values \(J_1 = 4\) and \(J_2 = 3\), respectively. Accordingly, there are \(K_{J_1} = 2^{J_1} + 2 = 18\) B-splines \(N_{4,k_1}^2(\varphi )\) with \(k_1 = 0,1,\ldots , 17\) defined along the latitude and \(K_{J_2} = 3 \cdot 2^{J_2} = 24\) trigonometric B-splines \(T_{3,k_2}^2(\lambda )\) with \(k_2 = 0,1,\ldots , 23\) defined along longitude between \(-180^{\circ }\) and \(180^{\circ }\). As one of altogether \(432 = 18 \cdot 24\) 2D tensor product B-splines functions the right panel shows exemplarily the function \(N_{4,11}^2(\varphi ) \cdot T_{3,2}^2(\lambda )\); the non-zero region is referred to as its influence zone. Both the polynomial and trigonometric B-splines basis functions are unitless and hence their product also remains unitless in this figure

Although different B-spline level values can be considered for each key parameter \(\kappa _r \in {\mathcal {K}}\), we choose for all of them the same level values, resulting in the same number of coefficients for each key parameter. This is due to the fact that according to Goss et al. (2019) the numerical values for the levels \(J_1\) and \(J_2\) are depending on the sampling intervals \(\Delta \varphi\) and \(\Delta \lambda\) of the input data and on the spectral content of the individual key parameters.

Since the key parameters of the set \({\mathcal {K}}\) describe the electron density of the ionosphere and the plasmasphere according to the MLCM, significant correlations \(\rho _{r,s} = \rho _{s,r}\) exist between the estimated sets of B-spline coefficients \(d^{J_1,J_2}_{k_1,k_2;\kappa _r}(t_i)\) and \(d^{J_1, J_2}_{k_1,k_2; \kappa _s}(t_i)\) of various pairs \((\kappa _r, \kappa _s)\) of two key parameters \(\kappa _r \in {\mathcal {K}}\) and \(\kappa _s \in {\mathcal {K}}\) with \(r,s = 1,2, \ldots , 14\) at the time moment \(t = t_i\), e.g. for \(\kappa _r = N_m^{F_2}\) and \(\kappa _s = H^{F_2}\); see Limberger et al. (2014a).

For demonstrating the proposed optimization approach, a subset \({\mathcal {K}}_1 \subset {\mathcal {K}}\) of altogether \(r_1\) key parameters \(\kappa _r \in {\mathcal {K}}_1\) will be chosen in our electron density modelling as also done earlier by Limberger et al. (2014b), Liang et al. (2015) and Tsai et al. (2016). For that purpose we will neglect the dependence on time t in the sequel. As already mentioned in the context of the definition of the set \({\mathcal {K}}\) in Eq. (10) we will assume now that all \(r_1\) key parameters \(\kappa _r \in {\mathcal {K}}_1\) are subject to inequality constraintsFootnote 2, such as

$$\begin{aligned} - \kappa _{r;u}(\varphi ,\lambda ) + \kappa _r(\varphi ,\lambda )&\le 0, \end{aligned}$$
(12a)
$$\begin{aligned} \kappa _{r;l}(\varphi ,\lambda ) - \kappa _r (\varphi ,\lambda )&\le 0, \end{aligned}$$
(12b)

where the lower bound function \(\kappa _{r;l}(\varphi ,\lambda )\) and the upper bound function \(\kappa _{r;u}(\varphi ,\lambda )\) are given as physically realistic limits. Examples for the key parameters \(\kappa _r \in {\mathcal {K}}_1 = \{ \ldots , N_m^{F_2}, h_m^{F_2}, \ldots \}\) according to Eqs. (12a) and (12b) with \(220 \text { km} \le h_m^{F_2}(\varphi ,\lambda ) \le 500 \text { km}\) and \(0.02 \text { EDU} \le N_m^{F_2}(\varphi ,\lambda ) \le 2.50 \text { EDU}\) are exemplar inequality constraints

$$\begin{aligned} -500 \text { km} + h_m^{F_2}(\varphi ,\lambda )&\le 0&\text { km} \\ 220 \text { km} - h_m^{F_2}(\varphi ,\lambda )&\le 0&\text { km} \\ -2.50 \text { EDU} + N_m^{F_2}(\varphi ,\lambda )&\le 0&\text { EDU} \\ 0.02 \text { EDU} - N_m^{F_2}(\varphi ,\lambda )&\le 0&\text { EDU} \;. \end{aligned}$$

For convenience, we define electron density units (EDU) as \(1 \ \text {EDU} = 10^{12} \ \text {el}/\text {m}^3\). The example realized by the four equations above for the upper and lower bounds on the \(F_2\)-layer peak density \(N_m^{F_2}(\varphi ,\lambda )\) and the corresponding peak height \(h_m^{F_2}(\varphi ,\lambda )\) at a specific spatial location \((\varphi ,\lambda )\) is illustrated in Fig. 3. The region enclosed by the constraint bounds \(N_{m;l}^{F_2}\), \(N_{m;u}^{F_2}\), \(h_{m;l}^{F_2}\) and \(h_{m;u}^{F_2}\) is called the feasible region \(\mathcal{F_R}\) for the key parameters \(N_m^{F_2}\) and \(h_m^{F_2}\). The final solution of the estimation process must be a pair \((N_m^{F_2}, h_m^{F_2}) \in \mathcal{F_R}\). Figure 3 also shows that the lower and upper bounds for the two selected key parameters can be chosen differently for day and night, and thus are time-dependent.

Fig. 3
figure 3

\(F_2\)-layer electron density profile with the lower bound functions \(\kappa _{r;l}(\varphi ,\lambda ) = N_{m;l}^{F_2}(\varphi ,\lambda ) = 0.2\) EDU and \(\kappa _{r;l}(\varphi ,\lambda ) = h_{m;l}^{F_2}(\varphi ,\lambda ) = 220\) km as well as the upper bound functions \(\kappa _{r;u}(\varphi ,\lambda ) = N_{m;u}^{F_2}(\varphi ,\lambda ) = 2.5\) EDU and \(\kappa _{r;u}(\varphi ,\lambda ) = h_{m;u}^{F_2}(\varphi ,\lambda ) = 500\) km of the 2 key parameters \(N_m^{F_2}(\varphi ,\lambda )\) and \(h_m^{F_2}(\varphi ,\lambda )\) at the spatial position \((\varphi ,\lambda )\); the region within the bounds—highlighted in yellow colour—is denoted as the feasible region \(\mathcal{F_R}\). The figure shows alongside also the day and night side electron density profiles, indicating that the bound functions are actually time-dependent

Note, that we did not consider any inequality constraint among the same type of key parameter, but related to different layers, e.g. the condition \(h_m^{F_1} < h_m^{F_2}\) for the two peak heights of the \(F_1\)- and the \(F_2\)-layer. This condition must be fulfilled and could be realized by two non-overlapping constraints, such as \(h_{m}^{F_1} < h_{m;u}^{F_1} = h_{m;l}^{F_2}\) and \(h_{m;l}^{F_2} \le h_m^{F_2}\).

Besides the inequality constraints, such as introduced in Eqs. (12a) and (12b), the equality constraints

$$\begin{aligned} \kappa _r(\varphi ,\lambda ) = \kappa _{r;e}(\varphi ,\lambda ) \end{aligned}$$
(13)

must hold for the remaining \(r_2 = 14 - r_1\) key parameters \(\kappa _r \in {\mathcal {K}}_2\) with given physically realistic values \(\kappa _{r,e}(\varphi ,\lambda )\). Note, that the subset \({\mathcal {K}}_2 = {\mathcal {K}} \backslash {\mathcal {K}}_1\) means the complementary set of \({\mathcal {K}}_1\).

Inserting the series expansion (11) into the inequality constraints (12a), and (12b) and neglecting the truncation error yields:

$$\begin{aligned} - \kappa _{r;u}(\varphi ,\lambda ) + \sum _{k_1 = 0}^{K_{J_1} - 1} \sum _{k_{2} = 0}^{K_{J_2} - 1} d^{J_1,J_2}_{k_1,k_2;\kappa _r} \; N_{J_1,k_1}^2(\varphi ) \; T_{J_2,k_2}^{2}(\lambda )&\le 0, \end{aligned}$$
(14a)
$$\begin{aligned} \kappa _{r;l}(\varphi ,\lambda ) - \sum _{k_1 = 0}^{K_{J_1} - 1} \sum _{k_2 = 0}^{K_{J_2} - 1} d^{J_1,J_2}_{k_1,k_2;\kappa _r} \; N_{J_1,k_1}^{2}(\varphi ) \; T_{J_2,k_2}^{2}(\lambda )&\le 0 \;. \end{aligned}$$
(14b)

Defining the \(u \times 1\) vector

$$\begin{aligned} \pmb {d}_{\kappa _r} = \big [d^{J_1,J_2}_{0,0;\kappa _r},\, d^{J_1,J_2}_{1,0;\kappa _r},\, \ldots , \, d^{J_1,J_2}_{K_{J_1} - 1,K_{J_2} - 1;\kappa _r} \big ]^T \end{aligned}$$
(15)

of the B-spline series coefficients \(d^{J_1,J_2}_{k_1,k_2;\kappa _r}\) as well as the \(u \times 1\) vector

$$\begin{aligned} \pmb {\phi }_{J_1,J_2}(\varphi , \lambda ) = \big [N_{J_1,0}^2(\varphi ) \; T_{J_2,0}^{2}(\lambda ),\, N_{J_1,1}^2(\varphi ) \; T_{J_2,0}^{2}(\lambda ),\, \ldots , \, N_{J_1,K_{J_1} - 1}^2(\varphi ) \; T_{J_2,K_{J_2} - 1}^{2}(\lambda ) \big ]^T \end{aligned}$$
(16)

of the 2D tensor product B-spline basis functions \(N_{J_1,k_1}^2(\varphi ) \; T_{J_2,k_2}^{2}(\lambda )\) Eqs. (14a) and (14b) can be rewritten as:

$$\begin{aligned} - \kappa _{r;u}(\varphi ,\lambda ) + \left( \pmb {\phi }_{J_1,J_2}(\varphi , \lambda ) \right) ^T \cdot \pmb {d}_{\kappa _r}&\le 0, \end{aligned}$$
(17a)
$$\begin{aligned} \kappa _{r;l}(\varphi ,\lambda ) - \left( \pmb {\phi }_{J_1,J_2}(\varphi , \lambda ) \right) ^T \cdot \pmb {d}_{\kappa _r}&\le 0. \;. \end{aligned}$$
(17b)

In the next step, we evaluate the lower bound function \(\kappa _{r;l}(\varphi ,\lambda )\) and the upper bound function \(\kappa _{r;u}(\varphi ,\lambda )\) at altogether \(m_1\) points \(P(\varphi _i, \lambda _i) = P_i\) with \(i = 1, \ldots , m_1\). Applying Eq. (11) to \(\kappa _{r;l}(\varphi _i,\lambda _i)\) and assuming that with \(m_1 > u\) the points \(P_i\) are defined, for instance, on a regular grid or a Reuter grid on the sphere \(\Omega _H\), the linear equation system

$$\begin{aligned} \pmb {A}_{\kappa _{r}} \; \pmb {d}_{\kappa _{r;l}} = {\varvec{\kappa }}_{r;l} + \pmb {e}_{r;l} \end{aligned}$$
(18)

results, where the \(m_1 \times u\) matrix \(\pmb {A}_{\kappa _{r}} = [(\pmb {\phi }_{J_1,J_2}(\varphi _i, \lambda _i))^T]_{i=1, \ldots , m_1}\) is of full column rank, the \(m_1 \times 1\) vector \({\varvec{\kappa }}_{r;l} = [{\kappa _{r;l}(\varphi _i,\lambda _i)}]_{i=1, \ldots , m_1}\) collects the given lower bound function values and the \(m_1 \times 1\) vector \(\pmb {e}_{r;l} = [-v_{\kappa _{r;l}}(\varphi _i, \lambda _i)]_{i=1, \ldots , m_1}\) the corresponding negative truncation error values. The \(u \times 1\) vector

$$\begin{aligned} \pmb {d}_{\kappa _{r;l}} = \big [d^{J_1, J_2}_{k_1,k_2;\kappa _{r;l}},\, d^{J_1, J_2}_{1,0;\kappa _{r;l}},\, \ldots , \, d^{J_1, J_2}_{K_{J_1} - 1,K_{J_2}-1;\kappa _{r;l}} \big ]^T \end{aligned}$$
(19)

of the series coefficients \(d^{J_1, J_2}_{k_1,k_2;\kappa _{r;l}}\) is defined analogously to the vector \(\pmb {d}_{\kappa _{r}}\) in Eq. (15). Under the assumption that all lower bound function values \({\kappa _{r;l}(\varphi _i,\lambda _i)}\) are weighted equally the application of the least-squares estimation (LSE) to Eq. (18) yields the solution

$$\begin{aligned} {\overline{\pmb {d}}}_{\kappa _{r;l}} = (\pmb {A}_{\kappa _{r}}^T \pmb {A}_{\kappa _{r}})^{-1} \; \pmb {A}_{\kappa _{r}}^T {\varvec{\kappa }}_{r;l}, \end{aligned}$$
(20)

and thus the vector

$$\begin{aligned} {\overline{{\varvec{\kappa }}}}_{r;l} = \pmb {A}_{\kappa _{r}} \; {\overline{\pmb {d}}}_{\kappa _{r;l}} \end{aligned}$$
(21)

of the approximate values \({\overline{\kappa }}_{r;l}(\varphi _i,\lambda _i)\) for the lower bound function values \(\kappa _{r;l}(\varphi _i,\lambda _i)\) at the \(m_1\) grid points \(P_i\). If we perform now in the same manner as in the Eqs. (18) to (21) for the upper bound function \(\kappa _{r;u}(\varphi ,\lambda )\), we obtain the vector

$$\begin{aligned} {\overline{{\varvec{\kappa }}}}_{r;u} = \pmb {A}_{\kappa _{r}} \; {\overline{\pmb {d}}}_{\kappa _{r;u}} \end{aligned}$$
(22)

of the approximate values \({\overline{\kappa }}_{r;u}(\varphi _i,\lambda _i)\) for the lower bound function values \(\kappa _{r;u}(\varphi _i,\lambda _i)\) at the \(m_1\) grid points \(P_i\) from the estimation \({\overline{\pmb {d}}}_{\kappa _{r;u}}\) of the \(u \times 1\) vector

$$\begin{aligned} \pmb {d}_{\kappa _{r;u}} = \big [d^{J_1, J_2}_{k_1,k_2;\kappa _{r;u}},\, d^{J_1, J_2}_{1,0;\kappa _{r;u}},\, \ldots , \, d^{J_1, J_2}_{K_{J_1} - 1,K_{J_2}-1;\kappa _{r;u}} \big ]^T \end{aligned}$$
(23)

of the series coefficients \(d^{J_1, J_2}_{k_1,k_2;\kappa _{r;u}}\). In addition to the optimization algorithm (presented in the next section), the proposed approach also requires least-squares adjustment for determining the constraint bounds on B-splines coefficients; see Eq. (20). The equality and inequality constraints are originally applied to Chapman key parameters and are then transformed to the B-spline coefficients using least squares estimate of the constraint bound. Combining least square and optimization together is the most direct way to represent the unknown B-spline coefficients of Chapman key parameters as well as their constraint bounds in the Gauss–Markov Model (GMM)Footnote 3 described in the following section. Of course other options (e.g. spherical harmonics) to represent the transformed constraint bounds (20) are also possible. But for the sake of utilizing the advantages of B-spline basis functions, we wish to use it here. Defining the \((2 \cdot m_1) \times u\) constraint coefficient matrix \({\pmb {B}}_{\kappa _r} = \big [-\pmb {A}_{\kappa _{r}}^T, \, \pmb {A}_{\kappa _{r}} \big ]^T\) as well as the \((2 \cdot m_1) \times 1\) constraint bound vector \(\pmb {b}_{\kappa _r} = \big [- {\overline{{\varvec{\kappa }}}}_{r;l}^T, \, {\overline{{\varvec{\kappa }}}}_{r;u}^T \big ]^T\) the inequality constraints (12a) and (12b) can be expressed as the linear constraint matrix inequality (LMI)

$$\begin{aligned} {\pmb {B}}_{\kappa _r} \; \pmb {d}_{\kappa _r} \le \pmb {b}_{\kappa _r} \end{aligned}$$
(24)

for the \(u \times 1\) vector \(\pmb {d}_{\kappa _r}\) from Eq. (15) of the series coefficients \(d^{J_1,J_2}_{k_1,k_2;\kappa _r}\) for an unknown key parameter \(\kappa _r \in {\mathcal {K}}_1\). Since the Chapman function (8) and the exponential decay function (9) for the plasmasphere depend non-linearly on a large number of key parameters \(\kappa _r \in {\mathcal {K}}\), the coefficient vector

$$\begin{aligned} \pmb {d}_{\kappa _r} = \pmb {d}_{\kappa _{r}|0} + \Delta \pmb {d}_{\kappa _r} \end{aligned}$$
(25)

needs to be decomposed into the vector \(\pmb {d}_{\kappa _{r}|0}\) of the approximate, i.e., initial values \(d^{J_1,J_2}_{k_1,k_2;\kappa _r|0}\) of the series coefficients \(d^{J_1,J_2}_{k_1,k_2;\kappa _r}\) and the vector \(\Delta \pmb {d}_{\kappa _r}\) of the corresponding correction values \(\Delta d^{J_1,J_2}_{k_1,k_2;\kappa _r}\). Inserting Eq. (25) into Eq. (24) yields the LMI

$$\begin{aligned} {\pmb {B}}_{\kappa _r} \; \Delta \pmb {d}_{\kappa _r} \le \Delta \pmb {b}_{\kappa _r} \end{aligned}$$
(26)

for the \(u \times 1\) correction vector \(\Delta \pmb {d}_{\kappa _r}\) with \(\Delta \pmb {b}_{\kappa _r} = \pmb {b}_{\kappa _r} - {\pmb {B}}_{\kappa _r} \, \pmb {d}_{\kappa _{r}|0}\). In the same manner as we derived the LMIs (24) and (26), we obtain from Eq. (13) the linear constraint matrix equality (LME)

$$\begin{aligned} \pmb {E}_{\kappa _r} \; \pmb {d}_{\kappa _r} = \pmb {e}_{\kappa _r} \;, \end{aligned}$$
(27)

where \(\pmb {E}_{\kappa _r}\) is a \(m_2 \times u\) constraint coefficient matrix and \(\pmb {e}_{\kappa _r} = \pmb {E}_{\kappa _r} \; {\overline{\pmb {d}}}_{\kappa _{r;e}}\) is the \(m_2 \times 1\) corresponding constraint vector for the key parameter \(\kappa _r \in {\mathcal {K}}_2\) with \({\overline{\pmb {d}}}_{\kappa _{r;e}} = (\pmb {E}_{\kappa _{r}}^T \pmb {E}_{\kappa _{r}})^{-1} \, \pmb {E}_{\kappa _{r}}^T {\varvec{\kappa }}_{r;e}\) calculatedFootnote 4 analogously to Eq. (20) and \(m_2 \ge u\). Inserting Eq. (25) into Eq. (27) yields the LME

$$\begin{aligned} {\pmb {E}}_{\kappa _r} \; \Delta \pmb {d}_{\kappa _r} = \Delta \pmb {e}_{\kappa _r} \end{aligned}$$
(28)

for the \(u \times 1\) correction vector \(\Delta \pmb {d}_{\kappa _r}\) with \(\Delta \pmb {e}_{\kappa _r} = \pmb {e}_{\kappa _r} - {\pmb {E}}_{\kappa _r} \, \pmb {d}_{\kappa _{r}|0}\). Note, that in case of \(\pmb {d}_{\kappa _{r}|0} = {\overline{\pmb {d}}}_{\kappa _{r;e}}\) the right-hand side of Eq. (28) reduces to \(\Delta \pmb {e}_{\kappa _r} = \pmb {0}_{(m_2 \times 1)}\), where \(\pmb {0}_{(m_2 \times 1)}\) means the \(m_2 \times 1\) vector of zero values. Furthermore, in case \(m_2 = u\), we may set \({\pmb {E}}_{\kappa _r} = \pmb {I}_u\), where \(\pmb {I}_u\) means the \(u \times u\) identity matrix. However, in the following we will stay at the more general LME formulation (28).

To demonstrate the developed procedure, we introduce the subset

$$\begin{aligned} {\mathcal {K}}_1 = \{\kappa _1 = N_m^{F_2}, \kappa _2 = h_m^{F_2}, \kappa _3 = H^{F_2}, \kappa _4 = N_0^P, \kappa _5 = H^P \} \end{aligned}$$
(29)

of the \(r_1 = 5\) key parameters of the \(F_2\)-layer and the plasmasphere. Consequently, the set \({\mathcal {K}}_2 = {\mathcal {K}} \backslash {\mathcal {K}}_1\) contains the remaining \(r_2 = 9\) key parameters of the \(D-, E\)- and \(F_1\)-layer. For setting up the parameter estimation model, we define the \((14 \cdot u) \times 1\) vector

$$\begin{aligned} \pmb {d}_{{\mathcal {K}}} = \big [ \pmb {d}_{{\mathcal {K}}_1}^T, \pmb {d}_{{\mathcal {K}}_2}^T \big ]^T, \end{aligned}$$
(30)

with the \((5 \cdot u) \times 1\) vector \(\pmb {d}_{{\mathcal {K}}_1} = \big [\pmb {d}_{\kappa _1}^T,\, \pmb {d}_{\kappa _2}^T,\, \ldots , \, \pmb {d}_{\kappa _5}^T \big ]^T\) and the \((9 \cdot u) \times 1\) vector \(\pmb {d}_{{\mathcal {K}}_2} = \big [\pmb {d}_{\kappa _6}^T,\, \pmb {d}_{\kappa _7}^T,\, \ldots , \, \pmb {d}_{\kappa _{14}}^T \big ]^T\) for the coefficient vectors \(\pmb {d}_{\kappa _r}\) as defined in Eq. (15), with \(r = 1, 2, \ldots , 14\). In the same manner we define the two \((14 \cdot u) \times 1\) vectors

$$\begin{aligned} \pmb {d}_{{\mathcal {K}}|0}&= \big [ \pmb {d}_{{\mathcal {K}}_1|0}^T, \pmb {d}_{{\mathcal {K}}_2|0}^T \big ]^T, \end{aligned}$$
(31a)
$$\begin{aligned} \Delta \pmb {d}_{{\mathcal {K}}}&= \big [ \Delta \pmb {d}_{{\mathcal {K}}_1}^T, \Delta \pmb {d}_{{\mathcal {K}}_2}^T \big ]^T \end{aligned}$$
(31b)

of the vectors \(\pmb {d}_{\kappa _{r}|0}\) and \(\Delta \pmb {d}_{\kappa _r}\) of the initial series coefficient values and the corresponding correction values, respectively, as defined in Eq. (25). Consequently, we obtain from Eq. (26) the LMI \(\pmb {B}_{{\mathcal {K}}_1} \Delta \pmb {d}_{{\mathcal {K}}_1} \le \Delta \pmb {b}_{{\mathcal {K}}_1}\) with \(\pmb {B}_{{\mathcal {K}}_1} = \big [ \pmb {B}_{\kappa _1}, \,\pmb {B}_{\kappa _2}, \, \ldots , \, \pmb {B}_{\kappa _5} \big ]\) and \(\Delta \pmb {b}_{{\mathcal {K}}_1} = \big [\Delta \pmb {b}_{\kappa _1}^T,\, \Delta \pmb {b}_{\kappa _2}^T,\, \ldots , \, \Delta \pmb {b}_{\kappa _5}^T \big ]^T\) or

$$\begin{aligned} \pmb {B}_{{\mathcal {K}}} \; \Delta \pmb {d}_{{\mathcal {K}}} \le \Delta \pmb {b}_{{\mathcal {K}}_1}, \end{aligned}$$
(32)

with \(\pmb {B}_{{\mathcal {K}}} = \big [\pmb {B}_{{\mathcal {K}}_1} , \; \pmb {0}_{(10 \cdot m_1) \times (9 \cdot u)} \big ]\). In the same manner it follows from Eq. (28) the LME \(\pmb {E}_{{\mathcal {K}}_2} \Delta \pmb {d}_{{\mathcal {K}}_2} = \Delta \pmb {e}_{{\mathcal {K}}_2}\) with \(\pmb {E}_{{\mathcal {K}}_2} = \big [ \pmb {E}_{\kappa _6}, \,\pmb {E}_{\kappa _7}, \, \ldots , \, \pmb {E}_{\kappa _{14}} \big ]\) and \(\Delta \pmb {e}_{{\mathcal {K}}_2} = \big [\Delta \pmb {e}_{\kappa _6}^T,\, \Delta \pmb {e}_{\kappa _7}^T,\, \ldots , \, \Delta \pmb {e}_{\kappa _{14}}^T \big ]^T\) or

$$\begin{aligned} \pmb {E}_{{\mathcal {K}}} \; \Delta \pmb {d}_{{\mathcal {K}}} = \Delta \pmb {e}_{{\mathcal {K}}_2}, \end{aligned}$$
(33)

with \(\pmb {E}_{{\mathcal {K}}} = \big [\pmb {0}_{(9 \cdot m_2) \times (5 \cdot u)} , \; \pmb {E}_{{\mathcal {K}}_2} \big ]\).

To the knowledge of the authors, there are only a few relevant works, where an inequality constrained optimization algorithm (ICOA) is applied in geodesy, e.g. the geodetic levelling and datum definition by Koch (1985) and Koch (1988), the non-negative variance component estimation (VCE) as well as the zenith troposphere delay (ZTD) estimation by Roese-Koerner (2015). In the former two papers, the Lagrange multiplier and slack variable concepts are already used (described in detail in the following section) and the latter uses the so-called “Active-set methods”; see Boyd and Vandenberghe (2004), Nocedal and Wright (2006). Inequality-constrained least squares was used by Zhang et al. (2013) to avoid negative VTEC in global ionosphere maps.

Optimization approach

To estimate the vector \(\pmb {d}_{{\mathcal {K}}}\) as defined in Eq. (30) we need a sufficient number of observations, well distributed over the region of investigation, i.e. in our study given globally within the ionosphere and plasmasphere. To simplify the procedure we assume that electron density measurements \(N_e({\varvec{x}}_j)\) at observations points \(P({\varvec{x}}_j) = P_j\) are given. Thus, we introduce the \(n \times 1\) vector \({\varvec{y}}= \varvec{n}_e - \varvec{n}_{e|0}\) as the difference of the \(n \times 1\) vector \(\varvec{n}_e = [N_e({\varvec{x}}_j)]_{j=1, \ldots ,n}\) of the electron density observations \(N_e({\varvec{x}}_j)\) and the \(n \times 1\) vector \(\varvec{n}_{e|0} = [N_{e|0}({\varvec{x}}_j)]_{j=1, \ldots ,n}\) of the corresponding approximate values \(N_{e|0}({\varvec{x}}_j)\) computed from the vector \(\pmb {d}_{{\mathcal {K}}|0}\) of the initial B-spline coefficients as defined in Eq. (31a). Note, that in this study we deal exclusively with electron density observations \(y_j = N_e({\varvec{x}}_j)\) and not with measurements \(y_j = f_j(N_e({\varvec{x}}))\) which are functionals of the electron density, such as GNSS STEC observations as introduced in Eq. (4). In this latter case further linearization steps have to be performed as well as a numerical integration along the ray-path between the transmitting satellite \(P^S\) and the receiver \(P_R\); for more details concerning these steps we refer to Limberger (2015), Liang (2017) or Zeilhofer (2008). Since in case of electron density measurements \(y_j = N_e({\varvec{x}}_j)\) the observation equation is non-linearly depending on most of the key parameters, we have to set up the \(n \times (14 \cdot u)\) matrix \(\pmb {X}= \big [\pmb {X}_{\kappa _1}, \; \pmb {X}_{\kappa _2}, \;\ldots , \; \pmb {X}_{\kappa _{14}} \big ]\) of the partial derivatives of the electron density with respect to the B-spline coefficients of all 14 key parameters \(\kappa _1\) to \(\kappa _{14}\).

Following Koch (1999) and using the aforementioned matrices and vectors we introduce the Gauss–Markov model (GMM)

$$\begin{aligned} {\varvec{y}}+ \pmb {e}= \pmb {X}\pmb {\beta }\; \; \text { with } \; \; D({\varvec{y}}) = \sigma ^2 \pmb {P}^{-1}, \end{aligned}$$
(34)

where \(\pmb {\beta }= \Delta \pmb {d}_{{\mathcal {K}}}\) is the \((14 \cdot u) \times 1\) vector of the unknown corrections for the B-spline coefficients as defined in Eq. (31b); \(\pmb {e}\) is the \(n \times 1\) vector of the observation errors, D means the dispersion operator, \(\pmb {P}\) is the given \(n \times n\) positive definite weight matrix of the observations and \(\sigma ^2\) is the unknown variance factor. Note, that besides the B-spline coefficients of all 14 key parameters we do not consider any further unknown parameter in the vector \(\pmb {\beta }\), e.g. no bias parameter. We apply the LSE such that the objective function

$$\begin{aligned} J(\pmb {\beta }) = \frac{1}{\sigma ^2} \left( {\varvec{y}}- \pmb {X}\pmb {\beta }\right) ^T \pmb {P}\left( {\varvec{y}}- \pmb {X}\pmb {\beta }\right) \end{aligned}$$
(35)

needs to be minimized subject to the LMI (32) and the LME (33). With the \((10 \cdot m_1) \times 1\) vector

$$\begin{aligned} \pmb {s}_{{\mathcal {K}}_1} = \big [s_1^2,\, s_2^2,\, \ldots , \, s_{10 \cdot m_1}^2 \big ]^T \end{aligned}$$
(36)

of the non-negative slack variables \(s_i^2\) we rewrite the LMI (32) as the LME

$$\begin{aligned} \pmb {B}_{{\mathcal {K}}} \; \Delta \pmb {d}_{{\mathcal {K}}} + \pmb {s}_{{\mathcal {K}}_1} = \Delta \pmb {b}_{{\mathcal {K}}_1} \quad \text {with} \quad \pmb {s}_{{\mathcal {K}}_1} \ge \pmb {0}_{(10 \cdot m_1) \times 1} \;. \end{aligned}$$
(37)

To minimize the objective function (35) subject to the LMEs (33) and (37), we define the Lagrange function

$$\begin{aligned} L(\pmb {\beta }, \pmb {\lambda }, \pmb {s})\,=\,\frac{1}{\sigma ^2} \left[ \left( {\varvec{y}}- \pmb {X}\pmb {\beta }\right) ^T \pmb {P}\, \left( {\varvec{y}}- \pmb {X}\pmb {\beta }\right) + 2 \cdot \pmb {\lambda }^T \left( \pmb {B}\pmb {\beta }- \pmb {b}+ \pmb {s}\right) \right] \;; \end{aligned}$$
(38)

see e.g. Koch (1985) and Koch (1999). In Eq. (38) we introduced the \((10 \cdot m_1 + 9 \cdot m_2) \times (14 \cdot u)\) constraint coefficient matrix \(\pmb {B}= \big [\pmb {B}_{{\mathcal {K}}}^T, \, \pmb {E}_{{\mathcal {K}}}^T \big ]^T\), the \((10 \cdot m_1+ 9 \cdot m_2) \times 1\) constraint bound vector \(\pmb {b}= \big [\Delta \pmb {b}_{{\mathcal {K}}_1}^T, \, \Delta \pmb {e}_{{\mathcal {K}}_2}^T \big ]^T\), the \((10 \cdot m_1 + 9 \cdot m_2) \times 1\) vector

$$\begin{aligned} \pmb {s}= \big [\pmb {s}_{{\mathcal {K}}_1}^T, \, \pmb {s}_{{\mathcal {K}}_2}^T \big ]^T \end{aligned}$$
(39)

of the slack variables with \(\pmb {s}_{{\mathcal {K}}_2} = \pmb {0}_{(9 \cdot m_2) \times 1}\), and thus \(\pmb {s}\ge \pmb {0}_{(10 \cdot m_1 + 9 \cdot m_2) \times 1}\), as well as the \((10 \cdot m_1 + 9 \cdot m_2) \times 1\) vector

$$\begin{aligned} \pmb {\lambda }= \big [\pmb {\lambda }_{{\mathcal {K}}_1}^T, \, \pmb {\lambda }_{{\mathcal {K}}_2}^T \big ]^T \end{aligned}$$
(40)

of the Lagrange multipliers \(\lambda _i\). Setting \(\pmb {Q}= 2 \cdot \pmb {X}^T \pmb {P}\, \pmb {X}\) and \(\pmb {q}= 2 \cdot \pmb {X}^T \pmb {P}\, {\varvec{y}}\), Eq. (38) can be rewritten as

$$\begin{aligned} L(\pmb {\beta }, \pmb {\lambda }, \pmb {s}) =\, \frac{1}{\sigma ^2} \left[ \frac{1}{2} \cdot \pmb {\beta }^T \pmb {Q}\, \pmb {\beta }- \left( \pmb {q}^T - \pmb {\lambda }^T \pmb {B}\right) \pmb {\beta }- \pmb {\lambda }^T (\pmb {b}- \pmb {s}) + {\varvec{y}}^T \pmb {P}\,{\varvec{y}}\right] \; . \end{aligned}$$
(41)

It shall be noted that in the special case that for the vector \(\pmb {s}\) of the slack variables \(\pmb {s}= \pmb {0}\) holds, the problem (41) becomes an optimization problem with equality constraints only. In case that for the vector \(\pmb {\lambda }\) of the Lagrange multipliers \(\pmb {\lambda }= \pmb {0}\) holds, the problem means an unconstrained optimization problem.

To estimate the optimal solution \(({\widetilde{\pmb {\beta }}}, {\widetilde{\pmb {\lambda }}}, {\widetilde{\pmb {s}}})\), the derivatives of the Lagrange function (41) with respect to the three unknown parameter vectors \({\pmb {\beta }},{\pmb {\lambda }}, \text {and} \, {\pmb {s}}\) have to be set to the zero vector, i.e.

$$\begin{aligned} \frac{\partial L(\pmb {\beta }, \pmb {\lambda }, \pmb {s})}{\partial \pmb {\beta }}&= \pmb {0}_{(14 \cdot u) \times 1}, \end{aligned}$$
(42a)
$$\begin{aligned} \frac{\partial L(\pmb {\beta }, \pmb {\lambda }, \pmb {s})}{\partial \pmb {\lambda }}&= \pmb {0}_{(10 \cdot m_1 + 9 \cdot m_2) \times 1}, \end{aligned}$$
(42b)
$$\begin{aligned} \frac{\partial L(\pmb {\beta }, \pmb {\lambda }, \pmb {s})}{\partial \pmb {s}}&= \pmb {0}_{(10 \cdot m_1 + 9 \cdot m_2) \times 1} \;. \end{aligned}$$
(42c)

Evaluating these three derivatives, we obtain the so-called Karush–Kuhn–Tucker (KKT) conditions:

$$\begin{aligned} \pmb {Q}\, {\widetilde{\pmb {\beta }}} - \pmb {q}+ \pmb {B}^T {\widetilde{\pmb {\lambda }}}&= \pmb {0}_{(14 \cdot u) \times 1}, \end{aligned}$$
(43a)
$$\begin{aligned} \pmb {B}\, {\widetilde{\pmb {\beta }}} - \pmb {b}+ {\widetilde{\pmb {s}}}&= \pmb {0}_{(10 \cdot m_1 + 9 \cdot m_2) \times 1} , \end{aligned}$$
(43b)
$$\begin{aligned} {\widetilde{\pmb {\lambda }}}^T {\widetilde{\pmb {s}}}&= \,0 \quad \text {with} \quad {\widetilde{\pmb {s}}} \ge \pmb {0}_{(10 \cdot m_1 + 9 \cdot m_2) \times 1} \quad \text {and} \quad {\widetilde{\pmb {\lambda }}} \ge \pmb {0}_{(10 \cdot m_1 + 9 \cdot m_2) \times 1} \;; \end{aligned}$$
(43c)

see e.g. Karush (1939) and Kuhn and Tucker (1951). The consequence of Eq. (43c) is \(\widetilde{\lambda _i} \, {\widetilde{s}}_i^2 = 0\) subject to \({\widetilde{s}}_i^2 \ge 0\) and \({\widetilde{\lambda }}_i \ge 0\) for \(i = 1, 2, \ldots , (10 \cdot m_1 + 9 \cdot m_2)\), i.e. the product of each pair of a Lagrange multiplier \({\widetilde{\lambda }}_i\) and the associated slack variable \({\widetilde{s}}_i^2\) has to be zero. In other words, if the slack variable is different from zero, the corresponding Lagrange multiplier must be zero and vice versa. Furthermore, for the equality constraints (33) the slack variables are according to Eq. (39) zero by definition. Since the matrix \(\pmb {Q}\) is at least positive semi-definite, the KKT conditions (43a) to (43c) are not only necessary, but also sufficient. Consequently, the optimization problem (41) can be solved by evaluating the system (43a) to (43c); see e.g. Koch (1985) or Nocedal and Wright (2006).

There are different techniques to solve the KKT conditions (43a) to (43c); see e.g. Borwein and Lewis (2000), Gass and Harris (2001), Roese-Koerner (2015). We use the IPM ( Nocedal and Wright (2006), pg.481), where a solution pair \(\{ {\widetilde{\pmb {\lambda }}}, {\widetilde{\pmb {s}}} \}\) is estimated iteratively from the assumption \({\widetilde{\lambda }}_i \, {\widetilde{s}}_i^2 = \sigma _{\mathrm {cen}} \cdot \mu _{\mathrm {com}}\) for all \(i = 1, 2, \ldots , (10 \cdot m_1)\), using the so-called “complimentarity measure” \(\mu _{\mathrm {com}}\) and the ”centering parameter” \(\sigma _{\mathrm {cen}}\); see e.g. Mehrotra (1992), Tütüncü et al. (2003) and Nocedal and Wright (2006). \(\mu _{\mathrm {com}}\) and \(\sigma _{\mathrm {cen}}\) are tuning parameters and responsible for the intermediate solution path taken by the optimization parameters \({\widetilde{\pmb {\lambda }}}\) and \({\widetilde{\pmb {s}}}\). This way, according to Nocedal and Wright (2006), we obtain the perturbed KKT condition

$$\begin{aligned} \pmb {S}\, \pmb {\Lambda }\, \pmb {1}- \sigma _{\mathrm {cen}} \cdot \mu _{\mathrm {com}}\, \pmb {1}= \pmb {0}_{(10 \cdot m_1 + 9 \cdot m_2) \times 1} \;, \end{aligned}$$
(44)

also known as complimentary slack condition, where \(\pmb {1}= \big [1,\, 1,\, \ldots , \, 1 \big ]^T\) means an \((10 \cdot m_1 + 9 \cdot m_2) \times 1\) vector of ones. The other matrices and vectors in Eq. (44) are defined as

$$\begin{aligned} \begin{aligned} \pmb {S}= \begin{bmatrix} \pmb {S}_{{\mathcal {K}}_1} &{} \pmb {0}_{(10 \cdot m_1) \times (9 \cdot m_2)} \\ \pmb {0}_{(9 \cdot m_2) \times (10 \cdot m_1)} &{} \pmb {S}_{{\mathcal {K}}_2} \end{bmatrix} \ \ ;&\ \ \pmb {\Lambda }= \begin{bmatrix} \pmb {\Lambda }_{{\mathcal {K}}_1} &{} \pmb {0}_{(10 \cdot m_1) \times (9 \cdot m_2)} \\ \pmb {0}_{(9 \cdot m_2) \times (10 \cdot m_1)} &{} \pmb {\Lambda }_{{\mathcal {K}}_2} \end{bmatrix}, \\ \pmb {S}_{{\mathcal {K}}_1} = \text {diag}\left( \tilde{s}_{1}^2, \, \ldots , \, \tilde{s}_{10 \cdot m_1}^2 \right) \ \ ;&\ \ \pmb {S}_{{\mathcal {K}}_2} = \text {diag}\left( \tilde{{s}}_{10 \cdot m_1 + 1}^2 = 0, \, \ldots , \, \tilde{{s}}_{10 \cdot m_1 + 9 \cdot m_2}^2 = 0 \right) , \\ \pmb {\Lambda }_{{\mathcal {K}}_1} = \text {diag}\left( {\tilde{\lambda }}_{1}, \, \ldots , \, {\tilde{\lambda }}_{10 \cdot m_1} \right) \ \ ;&\ \ \pmb {\Lambda }_{{\mathcal {K}}_2} = \text {diag}\left( {\tilde{\lambda }}_{10 \cdot m_1 + 1}, \, \ldots , \, {\tilde{\lambda }}_{10 \cdot m_1 + 9 \cdot m_2} \right) \;. \end{aligned} \end{aligned}$$

For simplification we introduce the three vectors

$$\begin{aligned} {\overline{\pmb {\beta }}}&= \big [{\widetilde{\pmb {\beta }}}^T, \, {\widetilde{\pmb {\lambda }}}^T, \, {\widetilde{\pmb {s}}}^T \big ]^T, \end{aligned}$$
(45a)
$$\begin{aligned} {\overline{\pmb {\beta }}}_0&= \big [{\widetilde{\pmb {\beta }}}_0^T, \, {\widetilde{\pmb {\lambda }}}_0^T, \, {\widetilde{\pmb {s}}}_0^T \big ]^T, \end{aligned}$$
(45b)
$$\begin{aligned} \Delta {\overline{\pmb {\beta }}}&= \big [\Delta {\widetilde{\pmb {\beta }}}^T, \, \Delta {\widetilde{\pmb {\lambda }}}^T, \, \Delta {\widetilde{\pmb {s}}}^T \big ]^T = {\overline{\pmb {\beta }}} - {\overline{\pmb {\beta }}}_0 \;. \end{aligned}$$
(45c)

Whereas the vector \({\overline{\pmb {\beta }}}_0\) contains the given initial values, the vector \(\Delta {\overline{\pmb {\beta }}}\) of the corresponding corrections can be calculated from the linear equation system

$$\begin{aligned} \left[ \begin{array}{*{3}{c}} \pmb {Q}&{} \pmb {B}^T &{} \pmb {0}_{(14 \cdot u) \times (10 \cdot m_1 + 9 \cdot m_2)} \\ \pmb {B}&{} \pmb {0}_{(10 \cdot m_1 + 9 \cdot m_2) \times (10 \cdot m_1 + 9 \cdot m_2) } &{} \pmb {I}_{(10 \cdot m_1 + 9 \cdot m_2)} \\ \pmb {0}_{(10 \cdot m_1 + 9 \cdot m_2) \times (14 \cdot u)} &{} \pmb {S}_0 &{} \pmb {\Lambda }_0 \end{array}\right] \cdot \left[ \begin{array}{*{1}{c}} \Delta {\widetilde{\pmb {\beta }}} \ \\ \Delta {\widetilde{\pmb {\lambda }}} \\ \Delta {\widetilde{\pmb {s}}} \end{array}\right] = \left[ \begin{array}{*{1}{c}} - \pmb {Q}\, {\pmb {\beta }}_0 + \pmb {q}- \pmb {B}^T {\pmb {\lambda }}_0 \\ - \pmb {B}\, {\pmb {\beta }}_0 + \pmb {b}- \pmb {s}_0 \\ - \pmb {S}_0 \, \pmb {\Lambda }_0 \, \pmb {1}+ \sigma _{\mathrm {cen}} \cdot \mu _{\mathrm {com}}\, \pmb {1}\end{array}\right] , \end{aligned}$$
(46)

which follows from the KKT equations (43a), (43b) and (44) applying Newton’s method. In Eq. (46) the matrices \(\pmb {Q}\) and \(\pmb {B}\) as well as the vectors \(\pmb {q}\) and \(\pmb {b}\) are known, the \((10 \cdot m_1 + 9 \cdot m_2) \times (10 \cdot m_1 + 9 \cdot m_2)\) matrices \(\pmb {S}_0\) and \(\pmb {\Lambda }_0\) are evaluated at the initial positions \({\widetilde{\pmb {s}}}_0\) and \({\widetilde{\pmb {\lambda }}}_0\); see e.g. Nocedal and Wright (2006). Note, in the context of the definition of the GMM (34) we set \(\pmb {\beta }= \Delta \pmb {d}_{{\mathcal {K}}}\), and thus we may identify the vector \({\widetilde{\pmb {\beta }}}_0\), Eq. (45b), of the initial values for the B-spline coefficients as the zero vector.

As already mentioned the above presented solution of the KKT equations is based on the IPM. The reader is referred to Mehrotra (1992), Anderson et al. (1996), Billups and Ferris (1996), Potra and Wright (2000), Boyd and Vandenberghe (2004), Nocedal and Wright (2006), Mitchell et al. (2006), Forst and Hoffmann (2010) for alternative KKT solution techniques.

The choice of the step size values between two adjacent iteration steps it and \(it+1\) is one of the most important specifications, because they are needed to calculate the updated values of the elements of the parameter vector \({\overline{\pmb {\beta }}}\) defined in Eq. (45a). In an unconstrained optimization problem, a ”full step” can be taken, i.e. \({\overline{\pmb {\beta }}}_{\mathrm {it+1}} = {\overline{\pmb {\beta }}}_{\mathrm {it}} + \Delta {\overline{\pmb {\beta }}}_{\mathrm {it}}\) or in other words the step size matrix \(\pmb {T}_{\mathrm {it}} = \pmb {I}\) equals the identity matrix along the descent direction. However, the choice of the step size values in presence of inequality constraints is not trivial, e.g. due to the risk of overshooting the feasible region. In general, the solution of the linear equation system (46) has to be updated from the iteration step it to the next step \({it+1}\) by

$$\begin{aligned} {\overline{\pmb {\beta }}}_{\mathrm {it+1}} = {\overline{\pmb {\beta }}}_{\mathrm {it}} + \pmb {T}_{\mathrm {it}} \Delta {\overline{\pmb {\beta }}}_{\mathrm {it}} \end{aligned}$$
(47)

until convergence is reached. In Eq. (47) the update matrix \(\pmb {T}_{\mathrm {it}}\) is defined as

$$\begin{aligned} \pmb {T}_{\mathrm {it}} = \text {diag}\left( \pmb {T}_{\beta }, \, \pmb {T}_{\lambda }, \, \pmb {T}_{s} \right) , \end{aligned}$$
(48)

with the diagonal block matrices

$$\begin{aligned} \pmb {T}_\beta&= \text {diag}\left( t_{\beta _1}, \, \ldots , \, t_{\beta _{14 \cdot u}}\right) \;, \end{aligned}$$
(49a)
$$\begin{aligned} \pmb {T}_\lambda&= \text {diag}\left( t_{\lambda _1}, \, \ldots , \, t_{\lambda _{10 \cdot m_1}}, \, t_{\lambda _{10 \cdot m_1 +1}} , \, \ldots , \, t_{\lambda _{10 \cdot m_1 + 9 \cdot m_2}} \right) \;, \end{aligned}$$
(49b)
$$\begin{aligned} \pmb {T}_s\,&= \text {diag}\left( t_{s_1^2}, \, \ldots , \, t_{s_{10 \cdot m_1}^2}, \, t_{s_{10 \cdot m_1 + 1}^2} , \, \ldots , \, t_{s_{10 \cdot m_1 + 9 \cdot m_2}^2} \right) , \end{aligned}$$
(49c)

such that the diagonal elements are the respective step sizes used along \(\pmb {\beta }\), \(\pmb {\lambda }\) and \(\pmb {s}\) satisfying the conditions

$$\begin{aligned} \left[ t_{\beta _1}, \ldots , t_{\beta _{14 \cdot u}}\right]&\in (0, 1]; \, \\ \left[ t_{\lambda _1}, \, \ldots , \, t_{\lambda _{10 \cdot m_1}}, \, t_{\lambda _{10 \cdot m_1 +1}} , \, \ldots , \, t_{\lambda _{10 \cdot m_1 + 9 \cdots m_2}} \right]&\in (0, 1]; \, \\ \left[ t_{s_1^2}, \, \ldots , \, t_{s_{10 \cdot m_1}^2}, \, t_{s_{10 \cdot m_1 +1}^2} , \, \ldots , \, t_{s_{10 \cdot m_1 + 9 \cdot m_2}^2} \right]&\in (0, 1] \;. \end{aligned}$$

Although according to the definition (36) of the vector \(\pmb {s}\) the slack variables for the equality constraints are included in \(\pmb {T}_{\mathrm {it}}\), they can be omitted from Eq. (48). Since the non-negativity of the Lagrange multipliers \(\lambda _i\) and the slack variables \(s_i^2\) has to be checked in each iteration, the chosen step size values along the descent direction are an important consideration. This procedure is one of the main modifications in our algorithm compared to the classical Newton method; see e.g. Nocedal and Wright (2006).

The descent direction (45c) is estimated by solving the linear equation system (46) with the initial update matrix \(\pmb {T}_{\mathrm {it}} = \pmb {I}\). Further on, the step size estimation becomes a sub-problem where the goal is the estimation of the largest diagonal elements \(t_{\beta _j} \, \text {for all} \, j \in \{1, \ldots , 14 \cdot u\}, \; t_{\lambda _k}, t_{s_k^2} \, \text {for all} \, k \in \{1, \ldots , 10 \cdot m_1 + 9 \cdot m_2 \}\), of \(\pmb {T}_{\mathrm {it}}\) which when used in Eq. (47), satisfy the KKT conditions (43c), and thus

$$\begin{aligned} \begin{aligned} \pmb {T}_{\mathrm {it}}&= \text {maximize} \left\{ t_{\beta _j}, t_{\lambda _k}, t_{{\widetilde{s}}_k^2} \vert \, ({\widetilde{\pmb {s}}} + \pmb {T}_s \, \Delta {\widetilde{\pmb {s}}} \ge \pmb {0}) \text { and } ({\widetilde{\pmb {\lambda }}} + \pmb {T}_\lambda \, \Delta {\widetilde{\pmb {\lambda }}} \ge \pmb {0}) \right\} , \\&\text {such that } 0 < \{t_{\beta _j},\, t_{\lambda _k}, \, t_{{\widetilde{s}}_k^2} \} \le 1 \, \text {for all} \; j = \{1, \ldots , 14 \cdot u\}, \; k = \{1, \ldots , 10 \cdot m_1 + 9 \cdot m_2 \} \;, \end{aligned} \end{aligned}$$
(50)

where \(\Delta {\widetilde{\pmb {s}}}\) and \(\Delta {{\widetilde{\pmb {\lambda }}}}\) are the descent directions along the slack variables and the Lagrange multiplier vectors. Note, in the most general case, different step size matrices corresponding to each of the three vectors \(\pmb {\beta }\), \(\pmb {\lambda }\) and \(\pmb {s}\) could be chosen. However, in this work, it was found that using only two step size matrices, i.e. one for the unknown model parameter vector \(\pmb {\beta }\) and the other for the vector pair \(\pmb {\lambda }\) and \(\pmb {s}\), is sufficient, and thus \(\pmb {T}_\lambda\) and \(\pmb {T}_s\) are assumed to be equal. These choices are made as part of the “design” of the developed ICOA. Furthermore, the step sizes are estimated in each iteration. This aspect is one of the most computationally demanding properties of the developed procedure, i.e. it becomes one of the disadvantages of adding more inequality constraints to our modelling problem; see Nesterov and Nemirovskii (1994), Todd et al. (1998), Potra and Wright (2000), Halldórsson and Tütüncü (2003), Mitchell et al. (2006)). During this work, it was found that an approximate step size is also generally sufficient in the developed ICOA. Having estimated the step size matrix (48), the complementarity measure

$$\begin{aligned} \mu _{\mathrm {com}} = \frac{({\widetilde{\pmb {\lambda }}} + \pmb {T}_\lambda \, \Delta {\widetilde{\pmb {\lambda }}} )^T ({\widetilde{\pmb {s}}} + \pmb {T}_s \, \Delta {\widetilde{\pmb {s}}} )}{10 \cdot m_1 + 9 \cdot m_2} \end{aligned}$$
(51)

is computed; for details see Tütüncü et al. (2003).

In the following, we present the altogether 8 steps of the developed iterative algorithm for the combined step size and descent direction estimation within the ICOA:

  1. (1)

    Set the initial values \({\widetilde{\pmb {\beta }}}_0\), \({\widetilde{\pmb {s}}}_0\), \({\widetilde{\pmb {\lambda }}}_0\) from Eq. (45b) of the unknown optimization parameters and solve for the descent directions \(\Delta {\widetilde{\pmb {s}}}\) and \(\Delta {\widetilde{\pmb {\lambda }}}\) considering \({\widetilde{\pmb {s}}}_0 \ge \pmb {0}\) and \({\widetilde{\pmb {\lambda }}}_0 \ge \pmb {0}\).

  2. (2)

    Initialize the centering parameter \(\sigma _{\mathrm {cen}}\) and use the initial step size matrix \(\pmb {T}_{\mathrm {it}} = \pmb {I}\). Compute \(\mu _{\mathrm {com}}\) using Eq. (51); this value of complimentarity measure is called \(\mu _{\mathrm {com,full}}\) because it means the ”full step” assumption \(\pmb {T}_{\mathrm {it}} = \pmb {I}\).

  3. (3)

    Estimate \(\Delta {\widetilde{\pmb {\beta }}}\), \(\Delta {\widetilde{\pmb {\lambda }}}\) and \(\Delta {\widetilde{\pmb {s}}}\) from the linear equation system (46) with the initial value vectors and matrices \({\widetilde{\pmb {\beta }}}_0\), \({\widetilde{\pmb {\lambda }}}_0\), \({\widetilde{\pmb {s}}}_0\), \(\pmb {T}_{\mathrm {it}}\) and the tuning parameters \(\mu _{\mathrm {com, full}}\) and \(\sigma _{\mathrm {cen}}\).

  4. (4)

    Estimate the step size \(\pmb {T}_{\mathrm {it}}\) from Eq. (50).

  5. (5)

    Compute the complimentarity measure \(\mu _{\mathrm {com}}\) using Eq. (51) with the update matrix \(\pmb {T}_{\mathrm {it}}\) from Step 4.

  6. (6)

    Set the centering parameter \(\sigma _{\mathrm {cen}} = (\mu _{\mathrm {com,full}}/\mu _{\mathrm {com}})^3\) as proposed by Tütüncü et al. (2003)).

  7. (7)

    Compute the updated optimization parameters with Eq. (47) using \(\pmb {T}_{\mathrm {it}}\) computed in Step 4.

  8. (8)

    Repeat the Steps 4 to 7 until convergence is reached.

We have derived in this work the linear equation system (46) to solve the inequality constrained optimization problem, in which, \(\pmb {Q}\), \(\pmb {B}\) and \(\pmb {P}\) are known matrices; \({\varvec{y}}\) and \(\pmb {b}\) are given vectors or calculable from given values. We used Newton’s method with the initial value \({\overline{\pmb {\beta }}}_0\) and the step size matrix \(\pmb {T}_{\mathrm {it}}\). For our investigations, i.e. for the global electron density modelling problem, the choice of initial values defined within the feasible region is necessary. In the next section, we will apply our proposed constrained optimization procedure to real data.

Fig. 4
figure 4

F3C electron density profiles \(N_e = N_e({\varvec{x}}_{\mathrm {F3C}})\) (green) and GRACE electron density profiles \(N_e = N_e({\varvec{x}}_{\mathrm {GRACE}})\) (red) during the time interval 00:00 UT to 04:00 UT on March 12, 2015

Application to real-data, validation and discussion

In this section we apply the developed ICOA, as described in previous two sections, to model the electron density \(N_e\) from direct measurements \(y_j = N_e({\varvec{x}}_j)\) given at observation sites \(P({\varvec{x}}_j) = P_j\). To be more specific, we choose electron density profiles from GRACE and Formosat-3/COSMIC (F3C) ionospheric radio occultation (IRO) measurements profiles. As reported in Yue et al. (2014), F3C produced on average 1000 to 1500 occultations per day during the years 2014 and 2015. We refer to Tsai et al. (2016) for the role of the F3C mission and the follow-up mission Formosat-7/COSMIC-2 (F7C) in global electron density modelling and recommend the papers of Tsai et al. (2009), Hernández-Pajares et al. (1998), Limberger (2015) for detailed description of IRO retrieval methods. For the scope of this work, the focus was on the development of ICOA and therefore the IRO observations have been directly taken from the COSMIC Data Analysis and Archive Center (CDAAC); see Schreiner et al. (2003). Figure 4 shows the projections of the electron density profiles on the surface of Earth for a 4-h time interval at March 12, 2015. Note, in the context of IRO processing an electron density profile cannot be interpreted as a set of electron density values given along the vertical, i.e. \(N_e = N_e(h_j)\), but given along the tangent points \(P({\varvec{x}}_{T_j})\) which are defined as a series of the closest points to Earth along the line-of-sight between the occulting GPS and the LEO satellite at specific time moments; see e.g. Limberger et al. (2013). Since both the F3C mission and the GRACE mission provide only a relatively small number of electron density observations, we introduce a third type the electron density observations derived from the so-called separability approach Hernández-Pajares et al. (2000). In this approach electron density values at globally distributed 3D grid points \(P(\varphi _k, \lambda _k, h_j)\) at time moment \(t_i\) are defined as

$$\begin{aligned} N_e(\varphi _k, \lambda _k, h_j, t_i) = VTEC(\varphi _k, \lambda _k, t_i) \cdot {\overline{p}}_k(h_j, t_i) \;, \end{aligned}$$
(52)

where the VTEC values \(VTEC(\varphi _k, \lambda _k, t_i)\) are evaluated from VTEC models at the 2D points \(P(\varphi _k, \lambda _k)\) at time \(t_i\) and \({\overline{p}}_k(h_j, t_i)\) means a value of a vertical profile function at the horizontal grid point \(P(\varphi _k, \lambda _k)\) evaluated at height \(h_j\) at time moment \(t_i\) fulfilling the normalization condition

$$\begin{aligned} {\overline{p}}_k(h_j, t_i) = \frac{{p}_k(h_j, t_i)}{\int _{h_{\mathrm {min}}}^{h_{\mathrm {max}}} {p}_k(h, t_i) \, dh} \;. \end{aligned}$$
(53)

As VTEC model we choose DGFI-TUM’s high-resolution global ionosphere map (GIM) ’othg’ which comprises a much higher spectral content (up to spherical harmonic degree \(n = 33\)) as the GIMs from the Ionospheric Associated Analysis Centers (IAAC) of the International GNSS Service (IGS). As stated in the IGS Technical Report 2018 Villiger A (2019) ’othg’ is one of the worldwide best GIMs; it is based on a 2D B-spline representation according to Eq. (11) and characterized by a spatial resolution of \(2.5^\circ \times 5^\circ\) with respect to latitude \(\varphi\) and longitude \(\lambda\) as well as a temporal resolution of 10 minutes; for more details concerning ’othg’ we refer to Goss et al. (2019). The profile functions \({p}_k(h, t_i)\) in Eq. (53) are computed from the International Reference Ionosphere (IRI) 2012 Bilitza, Dieter et al. (2014) at around 90 discrete heights \(h_j\) between 100 km and 1000 km at the regular grid points \(P(\varphi _k, \lambda _k\) at time moments \(t_i\) between March 12, 2015, 0:00 UT and March 14, 2015, 24:00 UT. This is the reason for the large number of electron density observations from the separability approach (SA) compared to the GRACE and F3C input data, as shown in Fig. 5. Note, that the electron density observations (52) can be seen as semi-simulated data, since ’othg’ is computed from real GNSS data and could be supplemented by DORIS and satellite altimetry observations; the profile functions are in opposite generated by the empirical model IRI2012.

Fig. 5
figure 5

Number of electron density observations \(N_e = N_e({\varvec{x}}_{\mathrm {GRACE}})\) from GRACE, \(N_e = N_e({\varvec{x}}_{\mathrm {F3C}})\) from the F3C mission and \(N_e = N_e({\varvec{x}}_{\mathrm {SA}})\) from the separability approach during the time interval March 12, 00:00 UT to 24:00 UT, 2015

Fig. 6
figure 6

Maps of the lower bound of the three key parameters \(\kappa _{1; l} = N_{m; l}^{F_2}(\varphi , \lambda )\) for the peak density (top row), \(\kappa _{2; l} = h_{m; l}^{F_2}(\varphi , \lambda )\) for the corresponding peak height (middle) and the scale height \(\kappa _{3; l} = H_{l}^{F_2}(\varphi , \lambda )\) (bottom) at three different time epochs on March 12, 2015

Fig. 7
figure 7

Maps of the upper bound of the three key parameters \(\kappa _{1; u} = N_{m; u}^{F_2}(\varphi , \lambda )\) for the peak density (top row), \(\kappa _{2; u} = h_{m; u}^{F_2}(\varphi , \lambda )\) for the corresponding peak height (middle) and the scale height \(\kappa _{3; u} = H_{u}^{F_2}(\varphi , \lambda )\) (bottom) at three different time epochs on March 12, 2015

It needs to be mentioned that we chose this way of creating a sufficient large observation vector \({\varvec{y}}\) for the GMM (34), because the number of real electron density measurements from IRO, the Langmuir probe measurement system on CHAMP and Swarm, ionosonde data, etc., is much too less for estimating the 4D electron density function (1) by applying our developed ICOA. It is possible that in the near future, when using large amounts of data from GNSS (constellations), IRO missions and CubeSats, it will no longer be necessary to consider simulated or semi-simulated input data. As shown in Fig. 8, the input block top left contains electron density observations \(N_e = N_e({\varvec{x}}_{\mathrm {SA}})\) generated accordingly to Eq. (52) from GNSS observations. The next block top mid comprises the electron density observations \(N_e = N_e({\varvec{x}}_{\mathrm {F3C}})\) from the IRO measurements of the F3C mission. Finally, the input data block top right contain electron density observations \(N_e = N_e({\varvec{x}}_{\mathrm {GRACE}})\) derived from GRACE and/or GRACE-FO IRO measurements. Additionally, electron density information from GRACE K-band measurements, from CHAMP and Swarm Langmuir probe data can also be supported here. After running some pre-processing steps, such as outlier detection the data are stored in the \(n \times 1\) vector \({\varvec{y}}\) of the GMM (34). In a first run of the developed ICOA, we identify the weight matrix \(\pmb {P}\) with the identity matrix \(\pmb {I}\). In our modelling approach, we set the B-spline levels \(J_1\) and \(J_2\) in the series expansion (11) for the key parameter \(\kappa _r\) as well as for the corresponding upper and lower bound functions \(\kappa _{r;u}\) and \(\kappa _{r;l}\) to the values \(J_1 = 4\) and \(J_2 = 3\); cf. Fig 2. Consequently, with \(u = 432\) our model consists of \(14 \cdot 432 = 6048\) series coefficients comprised in the vector \(\pmb {d}_{{\mathcal {K}}}\) introduced in Eq. (30). Due to the decomposition of the set \({\mathcal {K}}\) into the subsets \({\mathcal {K}}_1\) and \({\mathcal {K}}_2\) according to Eq. (29), we have to estimate altogether \(5 \cdot 432 = 2,160\) unknown series coefficients collected in the coefficient vector \(\pmb {d}_{{\mathcal {K}}_1}\). The remaining key parameters collected in the subset \({\mathcal {K}}_2\), i.e. the peak densities, the corresponding peak heights and the scale heights of the D-, the E- and the \(F_1\)-layer are assumed to be given. More specifically, the functions \(N_m^{D}(\varphi , \lambda )\), \(h_m^{D}(\varphi , \lambda )\), \(N_m^{E}(\varphi , \lambda )\), \(h_m^{E}(\varphi , \lambda )\) \(N_m^{F_1}(\varphi , \lambda )\) and \(h_m^{F_1}(\varphi , \lambda )\) have been taken from IRI2012 and the scale heights \(H^{D}(\varphi , \lambda )\), \(H^{E}(\varphi , \lambda )\) and \(H^{F_1}(\varphi , \lambda )\) are derived from the empirical ionospheric slab-thickness relation from Limberger (2015), Liang (2017). The lower bound functions \(N_{m;l}^{F_2}(\varphi , \lambda )\), \(h_{m;l}^{F_2}(\varphi , \lambda )\), \(H_{l}^{F_2}(\varphi , \lambda )\), \(N_{0;l}^{P}(\varphi , \lambda )\) and \(H_{l}^{P}(\varphi , \lambda )\) as well as the upper bound functions \(N_{m;u}^{F_2}(\varphi , \lambda )\), \(h_{m;u}^{F_2}(\varphi , \lambda )\), \(H_{u}^{F_2}(\varphi , \lambda )\), \(N_{0;u}^{P}(\varphi , \lambda )\) and \(H_{u}^{P}(\varphi , \lambda )\) are also obtained from IRI 2012. Examples of the lower and upper constraint maps are shown in Figs. 6 and Fig. 7.

Fig. 8
figure 8

Flowchart of the developed approach to estimate the electron density of the ionosphere and plasmasphere following the ICOA to estimate the selected Chapman key parameters \(\kappa _r\) of the set \({\mathcal {K}}_1\) by applying inequality and equality constraints

Fig. 9
figure 9

Maps of the estimated key parameters \({\widetilde{\kappa }}_1 = {\widetilde{N}}_m^{F_2}(\varphi , \lambda )\) for the peak density (top left), \({\widetilde{\kappa }}_2 = {\widetilde{h}}_m^{F_2}(\varphi , \lambda )\) for the corresponding peak height (top mid) and the scale height \({\widetilde{\kappa }}_3 = {\widetilde{H}}^{F_2}(\varphi , \lambda )\) (top right) as well as their standard deviations \({\widetilde{\sigma }}(N_m^{F_2}(\varphi , \lambda ))\), \({\widetilde{\sigma }}(h_m^{F_2}(\varphi , \lambda ))\) and \({\widetilde{\sigma }}(H^{F_2}(\varphi , \lambda ))\) (bottom panels) at 17:00 UT on March 12, 2015

We have estimated the B-spline coefficients corresponding to the 5 key parameters \(\kappa _r\) of the set \({\mathcal {K}}_1\) between March 12 to 14, 2015 with a temporal resolution of 1 hour. As examples, maps of the estimated \(F_2\)-layer key parameters \({\widetilde{\kappa }}_1 = {\widetilde{N}}_m^{F_2}(\varphi , \lambda )\) for the peak density, \({\widetilde{\kappa }}_2 = {\widetilde{h}}_m^{F_2}(\varphi , \lambda )\) for the corresponding peak height and the scale height \({\widetilde{\kappa }}_3 = {\widetilde{H}}^{F_2}(\varphi , \lambda )\) at 17:00 UT on March 12, 2015, computed from the series expansion (11) at regular grid points, are shown in the top row of Fig. 9 along with maps of their estimated standard deviations \({\widetilde{\sigma }}(N_m^{F_2}(\varphi , \lambda ))\), \({\widetilde{\sigma }}(h_m^{F_2}(\varphi , \lambda ))\) and \({\widetilde{\sigma }}(H^{F_2}(\varphi , \lambda ))\) computed via a Monte Carlo approach; for details see Roese-Koerner et al. (2014)).

For the validation of these results, we evaluate the series expansion (11) for the key parameters \({\widehat{\kappa }}_1 = {\widehat{N}}_m^{F_2}\) and \({\widehat{\kappa }}_2 = {\widehat{h}}_m^{F_2}\) at the exact locations of the selected ionosonde stations using the estimated B-spline coefficients; see Fig. 10. For the same time period, we also obtained the \(N_m^{F_2}\) and \(h_m^{F_2}\) directly measured in those stationsFootnote 5 for validation. The prevailing space weather situation for the month of March 2015 is also shown using the daily averaged time series of Kp and F10.7 indices in Fig. 11. We define the deviations \(\Delta N_m^{F_2}\) and \(\Delta h_m^{F_2}\) as the differences between the estimated and ionosonde-derived key parameters at the station locations. As selected results, we present (1) the time series of the estimated key parameters \(N_m^{F_2}\) and \(h_m^{F_2}\) in comparison with them from the ionosonde station EB040 between March 12 and 14, 2015, in Fig. 12, (2) the time series of the key parameters deviations \(\Delta N_m^{F_2}\) and \(\Delta h_m^{F_2}\) at the ionosonde station EB040 during the same time interval, in Fig. 12 as well as (3) the average values of the deviations \(\Delta N_m^{F_2}\) and \(\Delta h_m^{F_2}\) between March 12 and 14, 2015 at the station GR13L for temporal interval of 6 hours in Table 1.

Fig. 10
figure 10

Locations of the ionosonde stations used for the validation. The two stations EB040 and GR13L for which the validation is performed are marked in “green”

Table 1 Mean deviations with respect to \(N_m^{F_2}\) (Columns 2–4) and to \(h_m^{F_2}\) (Columns 5–7) at the station GR13L between March 12, 00:00 UT to March 14, 24:00 UT, 2015

As shown in Fig. 4, there is an insufficient global observation coverage of F3C and GRACE IRO observations for modelling the key parameters in \({\mathcal {K}}_1\). This leads to a non-uniformity between different regions, thus leading to a unusually larger key parameter deviation. But the largest \(N_m^{F_2}\) deviation is observed during 0–6 UT on two of the three (13, 14 March 2015) days due to the possible inaccuracies of the approximate given values in \({\mathcal {K}}_2\), especially for the layers that are presumed inactive during the night. The daily averaged \(h_m^{F_2}\) deviation across all stations was determinedFootnote 6 to be ± 30 km without any systematic offset. Similar deviation values are also reported by Lei and Chuo (2014), Limberger et al. (2014b), Liang et al. (2015), McNamara et al. (2008). Furthermore, Hunsucker (1991), McNamara et al. (2008) mention that the effect of the ionosonde instrument and the technology types, e.g. digisonde, dynasonde, incoherent scatter radar is also significant for the analysis of deviation, which depends on several factors, such as the time of the day, season, location of ionosonde, the phase of the solar cycle, sunspot number and other modelling error(s). To analyse if there is a systematic offset in the deviation due to these reasons, ideally a longer validation campaign is required. Although we have assumed given values of the E-layer in our modelling approach but did not consider the sporadic E-layer, see Arras et al. (2009) and Tsai et al. (2018), which also needs to be considered in further work.

Fig. 11
figure 11

Space weather situation in March 2015 is shown using the daily averaged time series of Kp and F10.7 indices. While Kp index is unitless, F10.7 has the unit of solar flux units (SFU) with \(1 \ SFU = 10^{-22}W m^{-2} Hz^{-1}\). The values corresponding to the time window in Table 1 correspond to day of year (DOY) 71, 72 and 73 of year 2015. Data for Kp and F10.7 are downloaded from NASA omniweb explorer portal https://omniweb.gsfc.nasa.gov/form/dx1.html accessed on 26 June 2022

The difference between the estimated and the ionosonde-derived peak density as well as the peak height for the station EB040 is shown in Fig. 12, where the ionosonde captures the local variations of the electron density and the estimated key parameters follow a smooth transition with time. The ionosonde data has the higher temporal resolution of 10 minutes compared with our estimates at 1 hour. While it is necessary to use VTEC data for electron density modelling in our approach, its relative number of observations compared to that of F3C and GRACE is higher due to the larger number of discrete altitudes used for defining the profile function. This situation is expected to change with more observations from the F7C mission and the availability of other IRO observationsFootnote 7.

Fig. 12
figure 12

Comparison of estimated key parameters \({\widehat{N}}_m^{F_2}\) (top left panel) and \({\widehat{h}}_m^{F_2}\) (top right panel) with the ionosonde data for the station EB040 at March 14, 2015. The lower panels show the corresponding deviations \(\Delta {N}_m^{F_2}\) (left) and \(\Delta {h}_m^{F_2}\) (right)

For the GR13L station, the maximum deviation with respect to the peak density \(N_m^{F_2}\) is 0.49 ± 0.23 EDU which is approximately 22% of the peak value and for \(h_m^{F_2}\) we obtain -42.91 ± 7.97 km, which is approximately 9% of the peak value at this location. The corresponding mean values of the two key parameter deviations are 0.17 EDU and 14.3 km, respectively. Both the maxima and the mean deviations are computed during the validation period 12 to 14 March 2015. At the EB040 station, the corresponding maximum deviations are 0.77± 0.09 EDU, 48.4 ± 12.5 km, respectively, and the trend is in the same order of magnitude at the other stations as well. Furthermore, we also noticed a difference between the deviations during the day and during the night. As already discussed, they are likely caused due to a non-uniform observation coverage for ED modelling. Similar trends between the IRI model and GPS derived TEC values were also reported in Endeshaw (2020) and Bjoland et al. (2016). Due to the relatively large number of observations from the separability approach as well as a higher IRI model accuracy near the \(F_2\) peak height, IRI has a significant impact on all the modelled key parameters. Similar conclusions have been made by Limberger et al. (2014b) and Liang et al. (2016). It shall be noted that the larger deviations of the estimated peak density with respect to the ionosonde measurements are mainly due to the sensitivity, i.e. the linear relation between the electron density \(N_e\) and the peak density \(N_m^{F_2}\) according to the Chapman function (8), compared to the double exponential relation of \(h_m^{F_2}\) within the Chapman function. Also considering the truncation error involved in using B-spline expansions, the differences in the temporal resolution of the ionosonde measurements and the estimated key parameters, the choice of the profile function p(h) obtained from the climatology in IRI, the approximate given values for the elements of the set \({\mathcal {K}}_2\) as well as the ionosonde observation quality, we conclude the deviations are within acceptable limits.

Summary and outlook

Electron density is the most important geodetic parameters in modelling the upper atmosphere, since it has a significant impact on many applications, such as precise point positioning and satellite navigation. Therefore, its precise modelling is both essential for applications as well as challenging for atmosphere sciences including space weather studies. However, a limited number of observation techniques with global coverage, especially along the altitude and the mutual correlation among different Chapman key parameters are the two main limitations. That means robust estimation estimates are required for 4D global electron density modelling. In this paper, we have developed an inequality constrained optimization algorithm, the ICOA. We have used the multi-layer Chapman approach, where the electron density profiles of the different layers are defined by a set of altogether 14 key parameters. Depending on the chosen layers, their key parameters are then modelled using series expansions in terms of 2D tensor product B-spline functions; the corresponding coefficients are then estimated using ICOA. This approach allows for modelling and estimating multiple Chapman key parameters simultaneously, thereby considering their mutual correlations. Furthermore, we have applied inequality constraints to physically localize the selected key parameters in physically realistic bounds. This way we defined the so-called feasibility region in which we have to search for the final solution. A validation with both simulated and ionosonde data was performed to assess the overall performance. Note, that the results from the simulations are not shown in this paper. Some important findings from our investigations can be summarized as follows:

  • The estimated optimization parameters are sensitive to the initial values \(\pmb {\beta }_0\), the total number as well as the spatial distribution and quality of the observations, the number of inequality constraints, their chosen lower and upper bound functions as well as on the selected B-spline levels for the Chapman key parameters.

  • Particularly, within the optimization algorithm, the initial values \({\overline{\pmb {\beta }}}_0\) as defined in Eq. (45b) and the step size matrix \(\pmb {T}\) from Eq. (47) have significant impact on the estimated optimization parameters. Therefore, a systematic procedure is required and has to be applied for both aspects; cf. the presented recipe below Eq. (51). In this paper, initial values were chosen from within the feasible region \({\mathcal {F}}_R\).

  • The choice of the lower bound functions \(\kappa _{r;l}\) and the upper bound functions \(\kappa _{r;u}\) are of particular importance when key parameters from two or more ionospheric layers have to be considered in the set \({\mathcal {K}}_1\). As shown in Fig. 3, the bound functions are generally time varying and numerically in different orders of magnitude. As an example, while the peak density of the \(F_1\)-layer drops to zero at night, that of the \(F_2\)-layer and the plasmasphere still have a significant value; see e.g. Bremer (1998) and Berkner and Wells (1934). Therefore, if the \(F_1\)-layer key parameters are considered in the set \({\mathcal {K}}_1\), their corresponding constraint bounds need to be adapted with time. Accordingly, a reasonable balance between extremely restrictive or more relaxed constraints for both the upper and the lower bound functions is needed. Qualitatively, restrictive constraints are those characterized by relatively small differences \(\kappa _{r;u}(\varphi , \lambda ) - \kappa _{r;l}(\varphi , \lambda )\); relaxed constraints, however, allow for larger differences.

  • In our study, the key parameters of the D-, the E- and the \(F_1\)-layer are obtained from the IRI 2012 model. It shall be noted that IRI also makes assumptions, such as the existence of \(F_1\) layer only within mid- and low-latitudes; see Bilitza (2000), Coisson et al. (2005). Although these assumptions may be reasonable from a long-term climatology perspective, for precise global 4D electron density modelling we need improved models for these layers.

In future, the authors will further investigate the performance of the constraint optimization technique for modelling different sets of key parameters in \({\mathcal {K}}_1\), especially including the bottom layers. Furthermore, different B-spline levels for the individual key parameters as well as Formosat-7/COSMIC-2 (F7C2) electron density observations are also planned in near future. Since the scope of this paper is the ICOA, we have not investigated on the exact nature of mutual correlations between the key parameters of set \({\mathcal {K}}_1\), which depends on the source and duration of data, space weather conditions and also the choice of profile function p(h) in Eq. (52). These aspects shall be investigated in future work as well.

Availability of data and materials

The datasets used and/or analysed during the current study are available from the corresponding author on reasonable request.

Notes

  1. Availability of ionospheric corrections also improve convergence time in precise point positioning (PPP) solutions.

  2. As the key parameters in Eq. (10) are modelled globally on a sphere, the constraints (12a), (12b), (14a) and (14b) apply and must be satisfied at all latitude, longitude pairs on the global grid.

  3. Only then the constraint and unconstrained terms of the Lagrangian function (38) can be represented using the same unknown parameter vector \(\varvec{\beta }\) to be later used in (34).

  4. The equality constraint bounds transformation will lead to an approximation due to the truncation error in B-spline series expansion. This is perhaps one of the few disadvantages of transforming the constraint bounds from the space of key parameters to B-spline coefficients.

  5. The ionosonde observations are obtained from the NGDC open source repository ftp.ngdc.noaa.gov/ionosonde/

  6. Note, we show only results for the ionosonde station EB040 and not for the others.

  7. F7C data are already available since October 2019, but the time period we selected for our study is March 2015.

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Acknowledgements

This work was done in the scope of the project titled “Development of an operational prototype for the determination of the thermospheric density on the basis of a thermosphere–ionosphere coupling model” with the acronym TIK (original German title “Entwicklung eines operationellen Prototyps zur Bestimmung der thermosphärischen Dichte auf Basis eines Thermosphären-Ionosphären Kopplungsmodells” ). The authors would like to acknowledge the German Federal Ministry for Economic Affairs and Energy (BMWi) for sponsoring this project via the German Aerospace Center (DLR Space administration) under the research number: FKZ50LZ1605.

Funding

Open Access funding enabled and organized by Projekt DEAL. This work was done in the scope of the project titled “Development of an operational prototype for the determination of the thermospheric density on the basis of a thermosphere–ionosphere coupling model” with the acronym TIK (original German title “Entwicklung eines operationellen Prototyps zur Bestimmung der thermosphärischen Dichte auf Basis eines Thermosphären-Ionosphären Kopplungsmodells” ). The authors would like to acknowledge the German Federal Ministry for Economic Affairs and Energy (BMWi) for sponsoring this project via the German Aerospace Center (DLR Space administration) under the research number: FKZ50LZ1605.

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GLG developed the inequality constrained optimization algorithm described in this paper and applied it to model the 4D electron density using the estimated B-spline coefficients and wrote the major part of the manuscript. MS reviewed the manuscript and supported the study with possible validation campaigns. Both authors read and approved the final manuscript.

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Correspondence to Ganesh Lalgudi Gopalakrishnan.

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Lalgudi Gopalakrishnan, G., Schmidt, M. Ionospheric electron density modelling using B-splines and constraint optimization. Earth Planets Space 74, 143 (2022). https://doi.org/10.1186/s40623-022-01693-4

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Keywords

  • Ionosphere layers
  • Multi-layer Chapman model
  • Inequality constraints
  • B-spline expansions
  • Optimization approach