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Mathematical techniques used in minimum variance orbital navigation and guidance, midcourse phase, lunar parking, descent orbits, landing, ascent, and rendezvous
A first-order linear and numerically non-diffusive Eulerian transport algorithm, minVAR, was recently developed for preservation of correlations between interrelated tracers during advective transport. The present study extends this work by: (1) providing further investigation of several interesting geometric constructions found in contours of constant minVAR, short for minimum variance, through extension to three coordinate dimensions. These contours capture point-by-point representations of thousands of individual atmospheric aerosol and/or cloud particles as they evolve and are rendered on Eulerian grids at a level of sub-grid resolution limited only by numerical precision; and (2) exploration of geometric similarities between the Arakawa C-grid, used to obtain interpolated values of the wind field at grid scale and minVAR. In particular, we consider interpolation of the u and v horizontal components of wind velocity from grid to sub-grid scales. The last results are motivated by recent applications of the Weather Research and Forecasting (WRF) model applied in the coastal Houston region, where the recent TRacking Aerosol Convection Interactions ExpeRiment (TRACER) field campaign was organized. A unique and fully consistent mapping is obtained between particles moving along meteorological wind trajectories and the non-diffusive, non-dispersive representation of such trajectories on an Eulerian grid.
The inclinations of the neutral line at the ecliptic plane derived from source surface model maps of coronal fields are measured for the interval from June 1976 to March 1992. The mean and median values of 53 deg and 57 deg are close to the average inclinations determined earlier from minimum variance analyses of solar wind measurements at sector boundaries, but the mode falls in the 80 deg - 90 deg bin. This result, which is based on the model assumptions implicit in deriving the source surface maps, predicts that the heliospheric current sheet typically intersects the ecliptic plane nearly at right angles, even without steepening by stream interaction regions. High inclinations dominate the solar cycle for about 7 years around solar maximum. Dips to lower inclination occur near solar minimum, but high variance admits a wide range of inclinations throughout the cycle. Compared to the smooth solar cycle variation of the maximum latitudinal excursion of the neutral line, often treated as the tilt angle of a flat heliospheric current sheet, the noisy variation of the inclinations reflects the degree to which the neutral line deviates from a sine wave, implying warps and corrugations in the current sheet. About a third of the time the neutral line so deviates that it doubles back in longitude.
The design of an optimal dynamic compensator for a multivariable discrete time system is studied. Also the design of compensators to achieve minimum variance control strategies for single input single output systems is analyzed. In the first problem the initial conditions of the plant are random variables with known first and second order moments, and the cost is the expected value of the standard cost, quadratic in the states and controls. The compensator is based on the minimum order Luenberger observer and it is found optimally by minimizing a performance index. Necessary and sufficient conditions for optimality of the compensator are derived. The second problem is solved in three different ways; two of them working directly in the frequency domain and one working in the time domain. The first and second order moments of the initial conditions are irrelevant to the solution. Necessary and sufficient conditions are derived for the compensator to minimize the variance of the output.
The dynamic structure of large-amplitude Alfven disturbances of the interplanetary magnetic field is examined by transforming one-hour intervals of Explorers 33 and 35 magnetometer data from the solar ecliptic coordinate system to a coordinate system defined by the principal axes of the variance matrix. It is demonstrated how some interplanetary magnetic field fluctuations observed by both Explorers are consistent with local properties theoretically predicted for plane large-amplitude Alfven waves by Barnes and Hollweg (1974). The different types of angular motion of the magnetic field component normal to the direction of minimum variance may be indicative of the detailed conditions of the solar coronal plasma in the regions generating the Alfven waves, or some aspect of local generation.
The detection of primordial B modes of the cosmic microwave background (CMB) could provide information about the early stages of the Universe's evolution. The faintness of this signal requires exquisite calibration accuracy and control of instrumental systematic effects which otherwise could bias the measurements. In this work, we study the impact of an imperfect relative polarisation gain calibration on the recovered value of the tensor-to-scalar ratio r for the LiteBIRD experiment, through the application of the blind Needlet Internal Linear Combination (NILC) foreground-cleaning method. We derive requirements on the relative calibration accuracy of the overall polarisation gain (Δg ν ) for each LiteBIRD frequency channel. Our results show that minimum variance techniques, as NILC, are less sensitive to systematic gain calibration uncertainties compared to a parametric approach, if the latter is not equipped with a proper modelling of these instrumental effects. In this study, the most stringent requirements are found in the channels where the CMB signal is relatively brighter, with the tightest constraints at 166 GHz (Δg ν ≈ 0.16%). This differs from the outcome of an analogous analysis performed with a parametric method, where the tightest requirements are obtained for the foreground-dominated channels. Gain calibration uncertainties, corresponding to the derived requirements, are then simultaneously propagated into all frequency channels. By doing so, we find that the overall impact on estimated r is lower than the total gain systematic budget for LiteBIRD approximately by a factor 5, due to the correlations of the impacts of gain calibration uncertainties in different frequency channels. In order to decouple the systematic effect from the specific choice of the model, we derive the requirements assuming constant spectral parameters for the foreground emission. To assess the robustness of the obtained results against more realistic scenarios, we repeat the analysis assuming sky models of intermediate and high complexity. In these further cases, we adopt an optimised NILC pipeline, called the Multi-Clustering NILC (MC-NILC). We find that the impact of gain calibration uncertainties on r is lower than the LiteBIRD gain systematics budget for the intermediate-complexity sky model. For the high-complexity case, instead, it would be necessary to tighten the requirements by a factor 1.8.
We estimate the magnitude of the bias due to non-Gaussian extragalactic foregrounds on the optimal reconstruction of the cosmic microwave background (CMB) lensing potential and temperature power spectra. The reconstruction is performed using a Bayesian inference method known as the marginal unbiased score expansion (MUSE). We apply MUSE to a minimum variance combination of multifrequency maps drawn from the Agora publicly available simulations of the lensed CMB and correlated extragalactic foreground emission. Taking noise levels appropriate to the SPT-3G D1 release, we find non-Gaussian foregrounds may bias the MUSE reconstruction of the lensing potential amplitude at the level of (0.7 ± 0.3)σ when using modes up to ℓ max = 3500. We do not detect a statistically significant bias, finding a value of (-0.4 ± 0.3)σ, when restricted to lower angular multipoles, ℓ max = 3000. This work is a first step toward understanding the impact of extragalactic foregrounds on optimal reconstructions of CMB temperature and lensing potential power spectra.
Analysis of microwave sky signals, such as the cosmic microwave background, often requires component separation using multifrequency methods, whereby different signals are isolated according to their different frequency behaviors. Many so-called blind methods, such as the internal linear combination (ILC), make minimal assumptions about the spatial distribution of the signal or contaminants, and only assume knowledge of the frequency dependence of the signal. The ILC produces a minimum-variance linear combination of the measured frequency maps. In the case of Gaussian, statistically isotropic fields, this is the optimal linear combination, as the variance is the only statistic of interest. However, in many cases the signal we wish to isolate, or the foregrounds we wish to remove, are non-Gaussian and/or statistically anisotropic (in particular for the case of Galactic foregrounds). In such cases, it is possible that machine learning (ML) techniques can be used to exploit the non-Gaussian features of the foregrounds and thereby improve component separation. However, many ML techniques require the use of complex, difficult-to-interpret operations on the data. We propose a hybrid method whereby we train an ML model using only combinations of the data that , and combine the resulting ML-predicted foreground estimate with the ILC solution to reduce the error from the ILC. We demonstrate our methods on simulations of extragalactic temperature and Galactic polarization foregrounds and show that our ML model can exploit non-Gaussian features, such as point sources and spatially varying spectral indices, to produce lower-variance maps than ILC—e.g., reducing the variance of the B-mode residual by factors of up to 5—while preserving the signal of interest in an unbiased manner. Moreover, we often find improved performance even when applying our ML technique to foreground models on which it was not trained. Published by the American Physical Society 2025
We propose a novel statistical method to measure cosmic birefringence and demonstrate its power in probing parity violation due to axions. Exploiting an empirical correlation between the integrated radio polarization direction of a spiral galaxy and its apparent shape, we devise an unbiased minimum-variance estimator for the rotation angle, which should achieve an uncertainty of 5°–15° per galaxy. In conclusion, large galaxy samples from the forthcoming SKA continuum surveys, together with optical shape catalogs, promise a comparable or even lower noise power spectrum for the rotation angle than in the CMB Stage-IV (CMB-S4) experiment, with different systematics.
We present a measurement of the full shape of the power spectrum of the thermal Sunyaev–Zel’dovich (tSZ) effect down to arcminute scales using cosmic microwave background (CMB) data from the South Pole Telescope (SPT) over a roughly 100 deg 2 field. The analysis incorporates data from the 2019–2020 seasons of the SPT-3G survey in bands centered at 95, 150, and 220 GHz; from the full SPTpol dataset at 150 GHz; and from the Herschel–SPIRE survey in bands centered at 600 and 857 GHz. We combine data from all the above bands using linear combination (LC) techniques to produce a tSZ or Compton-y map. We modify the LC weights to produce multiple versions of the Compton-y map, including minimum-variance (MV) and foreground-minimized (-min) maps. We measure the auto- and cross-power spectra of a subset of these maps in the range ℓ ∈ [500, 5000]. While this power spectrum includes contributions from signals other than tSZ, we present numerous checks to show that the most challenging foreground signal, the cosmic infrared background (CIB), is much lower than the desired tSZ signal in the scales of interest in this work. The final tSZ power spectrum is measured at 9.3σ with both the MV and CIB-min maps. Our results are consistent with those reported in other CMB surveys across the literature. Using the difference in the tSZ power spectrum from the MV and CIB-min maps, we reconstruct the scale-dependent tSZ–CIB cross correlation $ρ^{\textrm{tSZ}}_{ℓ}$ x CIB, finding 3.1σ evidence for a nonzero correlation coefficient that is positive on large scales and approaches zero for ℓ > 2500. This result represents the deepest tSZ maps ever produced and provides new constraints that can help refine astrophysical feedback mechanisms and models of the intracluster medium.
Unmanned spacecraft guidance policy and orbit determination, using linear perturbation theory and minimum variance estimation
Minimum variance method for estimating state vector with acceptable accuracy constraints in orbit calculation
Minimum variance simulation for satellite attitude determination reliability using magnetic and solar measurements
Differential equations for minimum variance linear filter separating signals from additive correlated noise, using discrete time optimum formulas
Discrete parameter covariance stationary stochastic process in rotation sampling, deriving minimum variance unbiased linear population mean estimator by constrained optimization procedure
An iterative method, which is characterized as a maximum-likelihood minimum-variance technique, was used to extract the aerodynamic parameters of a Navion airplane from flight data. The purposes were to compare the results with parameters obtained from wind-tunnel tests and with results obtained by analog matching the same data, and to develop techniques for application of the parameter extraction program. Results from the study showed that the parameter-extraction program can produce aerodynamic parameters which will permit close estimation of the aircraft time histories used in the extraction process. The program determined an estimate of the standard deviations of the states and parameters. These estimates were used to indicate how well the calculated states fit the flight data and the confidence in the values of the estimated parameters. The study also showed that the values of the parameters were affected by the data and mathematical model used during the extraction process. Because of the lack of confidence in the parameters extracted by use of some of the sets of data, several parameters were estimated by other methods. By using a combination of methods, a set of parameters which gave a fit to the data was obtained.
A method is proposed for dealing with the divergence phenomenon problem encountered in applications of minimum-variance recursive estimators when the error covariance calculated by the estimator becomes inconsistent with the actual error covariance. The proposed method differs from previous ones in that the state model and statistics are accepted as given. The form of the optimal estimator is used, but a constant check is made on the consistency of the calculated and actual error covariances. The method is independent of the source of error, whether it be inaccuracies in the system model, incorrect values of the a priori and random process statistics, approximations required in the case of nonlinear systems, or computational roundoff. Several simulated examples, in which inconsistencies in the calculated and actual error covariances exist, show a significant improvement in the performance of the estimator when the given procedure is applied.
For life testing procedures, a Bayesian analysis is developed with respect to a random intensity parameter in the Poisson distribution. Bayes estimators are derived for the Poisson parameter and the reliability function based on uniform and gamma prior distributions of that parameter. A Monte Carlo procedure is implemented to make possible an empirical mean-squared error comparison between Bayes and existing minimum variance unbiased, as well as maximum likelihood, estimators. As expected, the Bayes estimators have mean-squared errors that are appreciably smaller than those of the other two.