In this paper, we investigate strategies for ambiguity resolution in Low Earth Orbit Positioning, Navigation, and Timing (LEO-PNT) systems, particularly across multiple frequency bands beyond the traditional L-band used in GNSS. By leveraging the rapid geometric change of LEO satellites, our study examines the feasibility of fast and reliable integer ambiguity resolution (IAR) in Precise Point Positioning with Ambiguity Resolution (PPP-AR). Using end-to-end simulations, we assess the impact of different signal bands on convergence time and positioning accuracy. The methodology incorporates the LAMBDA 4.0 toolbox for mixed-integer estimation models, providing insights into how LEO-PNT enhances PPP-AR performance. Results demonstrate significant improvements in ambiguity resolution success rates and positioning convergence when incorporating LEO signals, highlighting their potential in future PNT application.

In this contribution, we introduce some new theory for the classical GNSS ambiguity function (AF) method. We provide the probability model by means of which the AF-estimator becomes a maximum likelihood estimator, and we provide a globally convergent algorithm for computing the AF-estimate. The algorithm is constructed from combining the branch-and-bound principle, with a special convex relaxation of the multimodal ambiguity function, to which the projected-gradient-descent method is applied to obtain the required bounds. We also provide a systematic comparison between the AF-principle and that of integer least-squares (ILS). From this comparison, the conclusion is reached that the two principles are fundamentally different, although there are identified circumstances under which one can expect AF- and ILS-solutions to behave similarly.

In this contribution we introduce the dual mixed-integer least-squares problem and study it in relation to its primal counterpart. The dual differs from the primal formulation in the order in which the integer ambiguity vector a∈Zⁿ and baseline vector b∈R^p are estimated. As not the ambiguities, but rather the entries of b are usually the parameters of interest, the attractiveness of the dual formulation stems from its direct computation of b. It is shown that this potential advantage relies on the ease with which an implicit integer least-squares problem of the dual can be solved. For the convoluted cases, we introduce two methods of simplifying approximations. To be able to describe their quality, we provide a complete distributional analysis of their estimators, thus allowing users to judge whether or not the approximations are acceptable for their application. It is shown that this approach implicitly introduces a new class of admissible integer estimators of which we also determine the pull-in regions. As the dual function is shown to lack convexity, special care is required to be able to compute its global minimizer bˇ. Our proposed method, which has finite termination with a guaranteed ϵ-tolerance, is constructed from combining the branch-and-bound principle, with a special convex-relaxation of the dual, to which the projected-gradient-descent method is applied to obtain the required bounds. Each of the method’s three constituents are described, whereby special emphasis is given to the construction of the required continuously differentiable, convex lower bounding function of the dual.

In this contribution we consider mixed-integer least-squares problems, where the integer ambiguities a∈Zⁿ and real-valued parameters b∈R^p are estimated. Both a primal and a dual formulation can be considered, with the latter concerning the ambiguity resolution process taking place into the parameters' domain. We study the p = 1 case, where an ad hoc 'P1' algorithm is introduced, and some geometrical insights are provided. It is demonstrated how the algorithm's complexity (i.e. number of candidate integer solutions to be evaluated) grows linearly with the ambiguity dimensionality n, differently from the primal formulation where an exponential growth is observed. By means of numerical simulations, here based on Global Navigation Satellite System (GNSS) models, we show the efficiency of this proposed 'P1' algorithm, meanwhile also demonstrating its quasi-optimal statistical performance.

In this work we introduce the LAMBDA 4.0 toolbox, which provides an enhanced implementation for integer estimation, validation, and success rate evaluation. This free and open-source toolbox is a major update to LAMBDA 3.0 (2012), while it also integrates the functionalities from Ps-LAMBDA 1.0 (2013), thus respectively merging both estimation and evaluation capabilities. The new implementation provides redefined algorithms, such as an improved integer search strategy with a one-order reduction in the computational time, along with additional estimators: Vectorial Integer Bootstrapping (VIB), Integer Aperture Bootstrapping (IAB) and Best Integer Equivariant (BIE). This toolbox aims to become a valuable resource for researchers and/or practitioners dealing with mixed-integer models in high dimensions, e.g., terrestrial-based carrier-phase systems, multi-constellation Global Navigation Satellite Systems (GNSS), or other interferometric applications.

Enceladus, with its subsurface ocean, is amongst the top priority targets in the search for life beyond Earth. Following on discoveries from the Cassini mission that Enceladus possesses a global subsurface ocean containing salt and organic compounds, there are many unconstrained properties of the ocean and ice shell that must be investigated to further assess the habitability of Enceladus and begin the search for biosignatures on Enceladus. In this paper, we present a concept study for a New Frontiers class multi-lander and orbiter mission to Enceladus that investigates if there is or ever was a habitable environment on Enceladus. The mission architecture includes an orbiter for detailed chemical analysis of material erupted from Enceladus’ plumes and four impact landers for geophysical measurements. As part of our mission concept study, we explore key trades for orbital and surface science, as well as assess the scientific potential and hazards of candidate landing sites on Enceladus. The novelty of our mission architecture and consideration of both orbital and surface science elements makes this work directly relevant to a broad range of potential future mission architectures under consideration, such as those identified in the 2023–2032 Planetary Science and Astrobiology Decadal Survey.

In this contribution, we extend the principle of integer bootstrapping (IB) to a vectorial form (VIB). The mathematical definition of the class of VIB-estimators is introduced together with their pull-in regions and other properties such as probability bounds and success rate approximations. The vectorial formulation allows sequential block-by-block processing of the ambiguities based on a user-chosen partitioning. In this way, flexibility is created, where for specific choices of partitioning, tailored VIB-estimators can be designed. This wide range of possibilities is discussed, supported by numerical simulations and analytical examples. Further guidelines are provided, as well as the possible extension to other classes of estimators.

This study is focused on the effect of solar flux conditions on the dynamics of gravity waves (GWs) in the thermosphere. Air density and crosswind in situ estimates from the Gravity Field and Steady-State Ocean Circulation Explorer (GOCE) accelerometers are analyzed for the whole mission duration. The analysis is performed in the Fourier spectral domain averaging spectral results over periods of 2 months close to solstices. A new GW marker (called C³ _f) is introduced here to characterize GWs activity under low, medium, and high solar flux conditions, showing a clear solar damping effect on GW activity. Most GW signal is found in a spectral range above 8 mHz in GOCE data, meaning a maximum horizontal wavelength of around 1000 km. The level of GW activity at GOCE altitude is strongly decreasing with increasing solar flux. Furthermore, a shift in the dominant frequency with solar flux conditions has been noted, leading to larger horizontal wavelengths (from 200 to 500 km) during high solar flux conditions. The correlation between air density variability and GW marker allows to identify most of the large-amplitude perturbations below 67° latitudes as due to GWs. The influence of correlated error sources, between air density and crosswinds, is discussed. Consistency of the spectral domain results is verified in the time domain with a global mapping of high-frequency air density perturbations along the GOCE orbit. This analysis shows a clear dependence with geomagnetic latitude with strong perturbations at magnetic poles and an extension to lower latitudes favored by low solar activity conditions. These results are consistent with previous Challenging Minisatellite Payload (CHAMP) data analysis and with general circulation models.