The lower thermosphere–ionosphere system (LTI) at altitudes between about 100 and 200 km is characterized by transitions of several atmospheric attributes. It is the lower part of the heterosphere where atmospheric constituents are no longer mixed by turbulence and start to follow separate barometric laws e.g.,. As part of the thermosphere, the temperature profile shows a significant increase with altitude throughout the whole LTI e.g.,. As part of the ionosphere, it includes the E-layer peak in electron density and the bottom side of the F layer e.g.,. With strongly altitude-dependent neutral–ion and neutral–electron collision frequencies, the LTI supports an anisotropic conductivity tensor that gives rise to a complex interplay of electric fields and currents. The conductivity tensor components affecting the directions perpendicular to the ambient magnetic field, namely Pedersen and Hall conductivities, show pronounced maxima in the LTI e.g.,. A key variable quantifying its energetics is the Joule heating rate. Particular rich dynamics can be observed in the auroral region at high latitudes where energy and momentum from the magnetosphere are fed into the ionosphere through currents flowing parallel to the ambient magnetic field lines e.g.,. A comprehensive review of LTI features, measurement techniques, and models is provided by .

Since the early 20th century, the LTI has been studied extensively using ground-based remote sensing facilities such as ionosondes and radars, but in all aspects requiring in situ observations, it remains underexplored territory. Rocket flights e.g., can offer only local and temporally confined information. Major technical challenges have so far prevented a satellite mission to the deep, dense part of the LTI, despite scientific interest, community proposals, and feasibility studies by major space agencies e.g.,. An early conception of the TIMED mission e.g., considered dipper options for in situ investigations of the LTI. A recent initiative along this line is the Daedalus mission proposal , submitted to ESA in response to the Explorer 10 call under the Earth Observation Program and selected together with two other proposals for a Phase-0 science and technical study. Daedalus aims to perform in situ measurements in the LTI from an elliptical orbit, with a nominal perigee of 150 km and an apogee on the order of 2000 km. Very low altitudes down to 120 km will be sampled by use of propulsion, through a series of short excursions in the form of perigee descent maneuvers. These are planned to be performed at high latitudes (> 65∘ magnetic latitude), where Pedersen conductivity and Joule heating maximize. The highly elliptical orbit of Daedalus leads to a natural precession of the orbit's semi-major axis, in both magnetic latitude and magnetic local time; this means that Daedalus will perform measurements along its elliptical orbit down to the nominal perigee of 150 km throughout all magnetic latitudes. The geophysical observables sampled by Daedalus will enable a series of derived products to be obtained, as described in Table 1 of the Daedalus Report for Assessment , which, among many other geophysical variables, include the calculation of Pedersen conductivity and Hall conductivity.

The range of accessible perigees will be particular critical for any future LTI satellite mission, with severe impact on the propellant budget and other mission performance parameters . With nominal perigees not much less than 150 km, peak conductivities and currents controlling E-region electrodynamics typically lie below the orbits. Physically meaningful downward continuation of in situ satellite measurements is desired, ideally using state-of-the-art models of the LTI e.g.,. Another critical element of LTI mission conception is the number of spacecraft . The Daedalus Ionospheric Profile Continuation (DIPCont) project is concerned with vertical profiles of LTI variables and their reconstruction from dual-spacecraft and single-spacecraft observations. More specifically, the focus of the project is on the quality of profile continuation towards the lower LTI with its maxima in conductivities and current intensities, as given by the accuracy, the resolution, and the coverage of the reconstructions obtained from in situ measurements. Inspired by early work to extrapolate vertical profiles carried out under the Daedalus Phase-0 Science Study, DIPCont introduces a systematic probabilistic approach to the problem.

The DIPCont procedure to assess the quality of in situ measurement downward continuation is detailed in Sect. . In brief, after choosing an LTI model, representative ensembles of altitude profiles are generated by means of Monte Carlo simulations. The altitude profile ensembles give rise to statistical measures of relative deviation, which in turn allow for estimation of extrapolation horizons, effectively capturing the altitude range where deviations remain within given error thresholds. The basic ideas are illustrated in Fig. , displaying electron density and Pedersen conductivity extrapolation horizons for a range of relative error thresholds along the orbits of a dual-satellite mission. Horizontal distance corresponds to the latitudinal (north–south) direction, with the origin of the horizontal axis centered at the highest latitude along the satellite orbits. In the LTI model runs leading to Fig. , latitudinal inhomogeneity parameters are set to reproduce the two electron density maxima observed by a polar-orbiting satellite when crossing the auroral oval. See Sect.  and Appendix  for details. It is important to note that the filled contour representations of electron density and Pedersen conductivity model distributions mainly serve to provide contextual information, while the essential results of the DIPCont modeling procedure are the extrapolation horizons represented as plain contour lines, in response to the satellite orbit configuration (white lines). The extrapolation horizons of the model run shown in Fig.  suggest that for a dual-satellite mission as anticipated in the Daedalus Report for Assessment , downward continuation yields relative errors in the range of about 10–50 % at altitudes where electron density and Pedersen conductivity maximize. Implications are discussed in more detail further below in Sect.  and contrasted with the single-satellite case.

The LTI model used to introduce and demonstrate the DIPCont methodology in this paper is presented in Sect. . The parametric model captures the whole LTI temperature range and thus addresses a main source of variability. To limit the number of model parameters and thus also instabilities during model inversion in this initial DIPCont study, LTI variables showing less pronounced changes and ionization source mechanisms are treated in a simplified manner. Furthermore, since the quality of downward continuation is the focus of our study, the LTI model is restricted to E-region physics, with the influence of the F region left for future work.

Further first results are presented in Sect. , including a brief comparison between the single-spacecraft and the dual-spacecraft scenario. In Sect. , our findings are discussed in the context of important technical parameters and constraints relevant for a low-altitude mission. The body of the paper is concluded in Sect.  with prospects for upcoming work. Model derivations and technical details are presented in the Appendices, with particular emphasis on the incorporation of a non-isothermal temperature profile varying linearly with altitude.

Probabilistic measures of extrapolation quality produced by the DIPCont procedure detailed in Sect.  are based on synthetic in situ observations predicted by a model of the LTI. As emphasized in space physics textbooks and reviews of the LTI e.g.,, the full complexity of LTI variability and dynamics calls for a full multi-species description, taking into account source and loss processes varying in importance and efficiency as functions of magnetic latitude and local time and further factors. In the future, DIPCont functionality is planned to be included in the Daedalus MASE (Mission Assessment through Simulation Exercise) toolset , designed with the purpose to assess and demonstrate the closure of the mission objectives of the proposed Daedalus mission.

The more complex the LTI model of choice, however, the larger the number of parameters that are to be estimated with a downward continuation of in situ satellite measurements, which in turn tend to negatively affect the stability of model inversion. With these implications in mind, the initial version of the DIPCont package contains a simplified LTI description based on a limited set of parameters. Extrapolation quality of a single but important process, namely the formation of Pedersen conductivity σP, is supposed to be studied in a self-consistent manner. To this end, only a single particle species is considered, and classical photoionization physics is applied to parametrize ionospheric layer formation. Furthermore, as explained in reviews of ionospheric physics e.g.,, contributions from electron–neutral collisions to the Pedersen conductivity σP peak in the D region but are unimportant at higher altitudes; see also Fig. 4 in . We thus arrive at the expression 1σP=Nee2miνinνin2+Ωi2 (e: elementary charge; mi: ion particle mass; Ωi: ion gyrofrequency), suggesting that the altitude variabilities in electron number density Ne and ion–neutral collision frequency νin need to be modeled carefully. Less critical is the dependence of ion gyrofrequency Ωi on magnetic field strength as it does not vary much over the LTI altitude range and is captured with sufficient accuracy by well-established empirical models. Different parametrizations exist for the ion–neutral collision frequency νin e.g.,. In general the expressions are directly proportional to the number density Nn of neutral particles. As presented in Appendices  and , also the self-consistent construction of electron density Ne rests prominently on the Nn profile, which in turn is conveniently modeled in terms of the density scale height HnN. This aspect is chosen as a starting point below in Sect.  to further explain and motivate the LTI modeling approach.

The LTI model can be summarized in the form m=m(z|p), with a vector m of LTI observables and derived functions and a vector p=p(x) of model parameters, separating the primary (strong) dependence on altitude z from the secondary (weak) dependence on horizontal location x in a numerically efficient manner. Note the model functions are local representations of altitude profiles in the sense that they refer to a flexible reference altitude, z0, that can be adapted to the locations where measurements are taken. In the DIPCont development phase it was observed that parameters of model functions in local representations typically showed weaker correlations and could be estimated more reliably, in particular as compared to the regional representations, relying on parameters at some fixed altitude, like the peak electron density height.

As in the Daedalus Report for Assessment , the vertical boundaries of the LTI region are assumed to be at zB=100 km (base or bottom-side altitude) and zT=200 km (topside altitude).

The DIPCont modeling procedure is as follows.

Synthetic noise-free measurements mj=m(zj|p) are created along anticipated Daedalus satellite orbit sections around perigee at altitudes zj=z(tj) and horizontal distances xj=x(tj). The chosen model parameters are defined by vectors p=p(x#) on a grid of horizontal distances x#. The integration and approximation methods employed for constructing the satellite orbits are described in Sect.  and in Appendix .

The major ingredients of the DIPCont processing chain, namely generation of synthetic in situ measurements along satellite orbits, Monte Carlo simulations of vertical profiles, and construction of extrapolation horizons, are summarized in Figs. –, displaying electron density Ne and Pedersen conductivity σP as two variables of key importance for the structure and the dynamics of the LTI. As indicated by Eq. () and the respective profiles in Fig. , electron density makes the main contribution to the peaked height variation in Pedersen conductivity, with secondary contributions of neutral density and possibly ion temperature through the parametric form chosen for the ion–neutral collision frequency; see Sect.  and also Fig. S3. Furthermore, Pedersen conductivity controls the height variation in Joule heating, whose characterization is one of the main scientific targets of the proposed Daedalus mission . In the neutral wind reference frame, Joule heating is j⟂⋅E⟂=σPE⟂2, where the subscript ⟂ indicates a vectorial component perpendicular to the ambient magnetic field direction B^. Height variations in E⟂ are negligible according to the following rationale; see, e.g., . Due to high parallel conductivity, the electric field component E‖=Es parallel to B^ vanishes, i.e., 0=Es=-∂Φ∂s, where s is the magnetic field line coordinate, and Φ denotes the electric potential. The electric field component Eq in a direction perpendicular to B^ captured by a coordinate q then satisfies ∂Eq∂s=-∂∂s∂Φ∂q=-∂∂q∂Φ∂s=∂Es∂q=0.

When instead of two selected horizontal locations as in Fig.  an equidistant grid of horizontal coordinates is defined for DIPCont simulations and the construction of extrapolation horizons, the results can be displayed together with the underlying model distributions and satellite orbits as in Fig. . In the following examples, such displays are used to visualize DIPCont results for different spacecraft configurations. Section  offers a first qualitative assessment of extrapolation quality in terms of varying inter-spacecraft distance. Section  contrasts the performance of the dual-spacecraft configuration considered so far with the results of the single-spacecraft case.

Note that the horizontal axis corresponds to the latitudinal (north–south) direction. In the simulations that led to Figs. –, horizontal variations were disregarded for better comparability. In Fig.  and in the following, latitudinal inhomogeneity of electron density is meant to reproduce the two maxima observed by a polar-orbiting satellite when crossing the auroral oval. The highest latitude corresponds to the origin of the horizontal axis. Since the physics of energetic particle precipitation is not incorporated in this initial version of the DIPCont package, the horizontal variation in electron density expected for an auroral oval crossing is prescribed through ad hoc choices of horizontal electron density peak parameter profiles; see the option LTIModelType='NeAuroralZoneCrossing' in the DIPCont code as part of the Supplement to this report. The functional forms of horizontal electron density peak parameters are given in Appendix .

Our first results suggest that altitude profiles of key LTI variables can be reconstructed with sufficient accuracy from in situ measurements if the effective altitude range covers relevant physical scales such as the local density scale height Hn0N. This is the case for a dual-spacecraft configuration with an inter-spacecraft separation of 20 km at perigee; see Figs.  and –. By two-point sampling, one can retrieve the vertical profiles of electron density and Pedersen conductivity essentially down to the bottom of the LTI region, a few scale heights under the lower satellite and including the peak altitudes. For Pedersen conductivity, errors are expected in the range of about 10–50 %, with the peak altitude and most of the conductivity within the 32 % extrapolation horizon in the chosen example, consistent with rocket observations .

Given the current knowledge of key LTI variables, error levels of about 10–50 % may well improve the situation. An important motivation behind the Daedalus proposal was the large error margin in Joule heating estimates, with a major contribution by errors in conductance (height-integrated conductivity). Thus, pointed out that for a substorm event investigated by , there were differences of up to 500 % between three proxies for the Joule heating rates integrated over the Northern Hemisphere. Even if this setup cannot be directly compared to our virtual environment, the order-of-magnitude difference between the two error margins looks encouraging for follow-up work on ionospheric profile continuation.

The DIPCont framework allows economical and technical questions regarding the impact of different LTI mission cost factors to be addressed. On the one hand, a dual-spacecraft mission seems to automatically imply higher costs because a second satellite needs to be built. On the other hand, a major cost driver of any deep LTI mission is the necessary amount of propellant that is required in order to maintain a spacecraft in orbit due to enhanced atmospheric drag at very low perigee altitudes. Since in a dual-spacecraft setup the role of the lower perigee satellite can be shared, each of the two probes would have to carry half of the total amount of propellant required to maintain the same total observation time required by a single-satellite mission. Moreover, the necessary amount of thermal shielding depends as well on perigee altitude, and each of the two probes would have to withstand the maximum thermal stress at perigee less often. Our findings show that the two-point setup allows for a more effective extrapolation to lower altitudes, which in turn means that a higher perigee may well be a meaningful option.

Data processing would also benefit from raising the perigee. As shown by simulations carried out for the technical assessment of Daedalus , a hydrodynamic shock develops in front of the spacecraft at altitudes under ∼120–130 km, complicating the retrieval of unperturbed data from the observed ones. Another LTI mission parameter considered in this paper is the apogee altitude controlling the proximity to the Van Allen belts and thus the necessary amount of radiation shielding but also affecting the available LTI observation time near perigee. The analysis presented in Sect.  shows that the amount of data gathered for statistical studies depends only moderately on apogee altitude.

The current version of the DIPCont framework concentrates on the E layer, assuming that contributions from the F layer can be disregarded or subtracted before processing, e.g., using the NeQuick approach to model topside ionospheric sounding data . The DIPCont package contains a parameter NeF to study the effect of F-layer residuals on extrapolation quality in future work.

The first results presented here are planned to be validated and extended in more extensive studies. Besides varying orbital parameters such as perigee altitude and inter-spacecraft distance, the impact of numerical parameters such as the horizontal selection window, 2Δx, needs further investigation. As already commented in Sect. , alternative functional forms for modeling ion–neutral collision frequencies or other variables may also be considered.

The DIPCont methodology introduced in this paper is designed to assess the quality of downward continuation of LTI variables using in situ satellite measurements and parametric models. While first results have been obtained with a simplified LTI description based on a single particle species, the Monte Carlo simulation machinery in DIPCont is not constrained to a particular model setup. By quantifying the quality of extrapolated in situ measurements, DIPCont can help to assess the science return of specific configurations and thus to optimize the parameters of upcoming LTI missions.

First DIPCont tests, performed on electron density and Pedersen conductivity, show promising results, to be consolidated by further parametric studies. Application of DIPCont to a modeled event, like the geomagnetic storm event of March 2015 addressed in the Daedalus Report for Assessment , is an upcoming target. This could be performed using the capabilities of the Daedalus MASE toolset . Future studies are planned to include Joule heating, which was a major driver of the Daedalus mission proposal. To investigate auroral processes and the electrodynamics of magnetosphere–ionosphere coupling, ionization through energetic particle precipitation needs to be incorporated. The Hall current nature of auroral electrojets calls for including electron–neutral collisional interaction as a major contributor to Hall conductivity formation.

Coordination between an LTI mission, like Daedalus, and a topside mission, like Swarm or DMSP (Defense Meteorological Satellite Program), would enhance the return of both missions. As an example, reconstruction of vertical profiles of ionospheric conductivity based on LTI observations could help to calibrate topside estimates of the conductance, while topside electron density could provide upper continuation and constrain the height-integrated total electron content (TEC) inferred from LTI data. Combination with ground-based observatory data such as ionosondes would offer further valuable constraints to DIPCont and thus enable more comprehensive modeling of the LTI.