Nonlinear methods of statistic simulation of virtual parameter values for investigating uncertainties in orbits determined from observations

Abstract

Determination of orbital parameters from observations is formally a nonlinear inverse problem for solving which evidently nonlinear methods are required. Meanwhile, an accompanying stage in solving the inverse problem is the evaluation of parametric accuracy to which, however, linear methods are conventionally applied. This is quite justified if parametric errors caused by observation errors are rather small, otherwise this is not at all since the nonlinearity of the inverse problem can be considerable to influence on the evaluations of parametric accuracy especially when the observations are very few. With the advent of quick-operating and multiprocessor computers, recently one tends to employ statistic simulation of virtual parameter values for investigating uncertainties in orbits determined from observations. In the paper are just discussed the methods designed specially for nonlinear statistic simulation of virtual parameter values. Their efficiency is investigated in application to estimating uncertainties in the orbit of Jovian satellite S/2003 J04 whose orbital parameters are ill-determined owing to scanty available observations. Indices of nonlinearity are introduced for making decision in the choice between linear and nonlinear methods.

Construction of confidence regions in problem on probabilistic study into motion of minor bodies of the solar system

“…(21) are determined through the appropriate objective function by means of the relation σ = The theoretical substantiation of using the factors calculated from objective functions can be found in (Avdyushev, 2010;2011). The corresponding factors determined on the basis of rms deviations represent more accurately the difference in sizes of confidence regions constructed using the linear and nonlinear methods, especially if the topography of surfaces of the objective function is complex and its changes are large with small variations in parameters.…”

“…where s is the number of points deter mined by the length of the sample of the vectors , that is simulated by the random number generators; is the number defined for each point by the rela tions (Avdyushev, 2009(Avdyushev, -20102011) If the measurement sample is not small (n > 20), the multiplier and we may leave it undeter mined, reducing thus the number of arithmetic opera tions to be executed in Eq. (12), which leads to an increase in speed of the algorithm used.…”