Python Implementation of Batch Least-Squares Filter for Satellite Orbit Determination

Accurate orbit determination techniques are fundamental to the maintenance and execution of any ongoing space-based mission. This project serves as a guide and demonstration of a batch sequential least-squares filter for Earth-orbiting satellites using exclusively open-source technologies. The target audience for this project was an academic institution aiming to keep track of an irregularly documented satellite. The observation function mimics a telescope, accepting right ascension and declination as measured values. State propagation was handled using the Poliastro library. This package boasts FORTRANlevel speed by utilizing the DOPRI8 integrator, explicitly calling FORTRAN code. Matrix inversion was solved using the SciPy banded solver function, a wrapper for the LAPACKdgbsv function, also written in FORTRAN. Frame conversions between ITRS (ECEF), GCRS (ECI), and ICRS (J2000) were handled using Astropy. A suite of tests with a range of noise were run to verify appropriate convergence of algorithm. In each case, the algorithm converged as expected with reasonable variances that changed in an anticipated fashion. These tests demonstrated that it is possible to achieve sub km accuracy for LEO satellites with 10 observations given 1 arcminute uncertainty and noise. Despite the interface requiring manual, the backend has been optimized to save memory supporting large batches of observations. As a result, the project detailed in this report requires little adaptation to support a much larger scale use such as tracking orbital debris. Any such changes are outlined in the designing a system subsection 3.3.1 or future expansion chapter. A GUI was assembled to support users with a limited coding background using Kivy

A Fast Bjo ̈rck-Pereyra-type Algorithm for Solving Complex-Vandermonde Systems

Modeling phenomenon of the interpolation problems can be seen in propagation of waves, weather conditions, real-time traffic patterns, signal processing, etc. There are different interpolation methods like polynomial interpolation, spline interpolation, rational interpolation, exponential interpolation, trigonometric interpolation, etc. In this situation, a fast Bj rck-Pereyra-type algorithm can be derived to solve this problem. In this poster we will present the most general trigonometric interpolation problems to solve complex-Vandermonde system. We present a fast algorithm for solving a system where the coefficient-matrix is a complex Vandermonde matrix. This method is much more favorable than the Gaussian elimination, which ignores the structure of the Vandermonde Matrix. The new algorithm applies to a fairly general new class to solve trigonometric interpolation problems. We present numerical experiments while elaborating better forward error bound than the Gaussian elimination. Moreover, we analyze, compare, and contrast the connection of trigonometric interpolation problems to phase polynomials interpolation problems through complexity, forward error, and stability of algorithms

Fast Split-Radix and Radix-4 Discrete Cosine Transform Algorithms

The Discrete Fourier Transform (DFT) has a plethora of applications in applied mathematics and electrical engineering. Discrete Cosine Transform (DCT) is a real-arithmetic analogue of DFT. DCTs with orthogonal trigonometric transforms have been especially popular in recent decades due to their applications in digital video technology and high efficiency video coding. One can say that DCT is the key transform in image processing, signal processing, finger print enhancement, quick response code (QR code), multi-mode interface, etc. In this talk, we first introduce sparse and scaled orthogonal factorization for the DCT and inverse DCT. Afterwards, we present fast split-radix and radix-4 DCT and inverse DCT algorithms. We show that the proposed algorithms attain the lowest theoretical multiplication complexity and arithmetic complexity for 8-point DCT II/III matrices. We perform execution time of the proposed algorithms while verifying the connection to the order of the arithmetic complexity. Finally, the language of signal flow graph representation of digital structures is used to describe potential for real-world circuit implementation

Complexity in water and carbon dioxide fluxes following rain pulses in an African savanna

The idea that many processes in arid and semi-arid ecosystems are dormant until activated by a pulse of rainfall, and then decay from a maximum rate as the soil dries, is widely used as a conceptual and mathematical model, but has rarely been evaluated with data. This paper examines soil water, evapotranspiration (ET), and net ecosystem CO2 exchange measured for 5 years at an eddy covariance tower sited in an Acacia–Combretum savanna near Skukuza in the Kruger National Park, South Africa. The analysis characterizes ecosystem flux responses to discrete rain events and evaluates the skill of increasingly complex “pulse models”. Rainfall pulses exert strong control over ecosystem-scale water and CO2 fluxes at this site, but the simplest pulse models do a poor job of characterizing the dynamics of the response. Successful models need to include the time lag between the wetting event and the process peak, which differ for evaporation, photosynthesis and respiration. Adding further complexity, the time lag depends on the prior duration and degree of water stress. ET response is well characterized by a linear function of potential ET and a logistic function of profile-total soil water content, with remaining seasonal variation correlating with vegetation phenological dynamics (leaf area). A 1- to 3-day lag to maximal ET following wetting is a source of hysteresis in the ET response to soil water. Respiration responds to wetting within days, while photosynthesis takes a week or longer to reach its peak if the rainfall was preceded by a long dry spell. Both processes exhibit nonlinear functional responses that vary seasonally. We conclude that a more mechanistic approach than simple pulse modeling is needed to represent daily ecosystem C processes in semiarid savannas

Efficient Split-Radix and Radix-4 DCT Algorithms and Applications

This project developed efficient, self-recursive, and fast split-radix and radix-4 algorithms for the Discrete Cosine Transforms (DCT) based on different boundary conditions. The project also addresses the self-recursive and stable aspects of split-radix and radix-4 DCT algorithms having simple, sparse, and scaled orthogonal factors. Moreover, the developed split-radix and radix-4 algorithms attain the lowest theoretical multiplication complexity and flop counts for 8-point DCT matrices in the literature. Numerical results are presented for the arithmetic complexity comparison of the proposed algorithms with the known fast and stable DCT algorithms. Software implementations have been written based on the proposed DCT algorithms. These results show that the proposed algorithms have attained low arithmetic complexity. The execution time of the proposed algorithms is presented while verifying the connection to the order of the flop counts. It is shown that the execution time of the proposed split-radix and radix-4 algorithms are more efficient than the existing radix-2 DCT algorithms. Finally, spectral analysis has been conducted based on the proposed DCT algorithms

The Intersection of Community and Place in an Outdoor Orientation Program

This study investigated outcomes of a college outdoor orientation program that utilized 3, 4, 5, and 12-day adventure based trips. Sense of community and sense of place were measured using a one-sample pre/post-test design (n = 118). Paired sample t-tests were implemented to explore differences and, in general, students reported significant changes following participation. Students demonstrated increases in the number of their friends, trusted individuals, and 6 of the 7 factorial components measuring sense of community and sense of place (p \u3c .05). The importance of community and place is discussed, highlighting suggestions and implications for future consideration in outdoor orientation programming