3.2 The sources of strong nonlinearity

To understand the problem we are discussing in rigorous mathematical terms, it is important to distinguish between two phenomena occurring in the computation of the prediction confidence region and its approximations.

First, for asteroids observed only over a short arc, and lost since a long time, even the linear approximation results in confidence regions with a very elongated shape [Muinonen et al., 1997]. Both the confidence ellipsoid in the X space of the orbital elements, and the confidence ellipse in the observations plane, have semiaxes with a very large ratio, such as $10^5\simeq 10^6$and even more; that is, the corresponding normal and covariance matrices can be very poorly conditioned, with ratios between the largest and the smallest eigenvalue of the order of $10^{10}\simeq 10^{12}$ and more.

Second, the linear approximation fails whenever it is used to transform a region in the parameter space which is not small. The two phenomena are obviously acting together, that is nonlinearity is important because the longest axis of the ellipsoid is too long.

Both the wild increase in the size of the longest semiaxis, and the dominance of the nonlinear effects, can occur in each one of the three steps in the computation of an observation prediction; these three separate effects act together on the final result $Z(\sigma)$, but can be analysed separately.

The first step is the computation of the confidence region for the orbital elements X at epoch t₀: the full equations for the confidence region are approximated by the quadratic part, that is by the equations for the confidence ellipsoid:

$${\displaystyle m \over \displaystyle 2}\; \Delta Q \simeq (X-X^*)\cdot C_0 \, (X-X^*) \leq \sigma^2$$

with C₀ the normal matrix (for the elements at epoch t₀); if C₀ has a very large conditioning number, then the confidence ellipsoid is a poor approximation of the confidence region. What is needed is a way to describe a better approximation to the nonlinear confidence region, to be used in the cases where the covariance matrix is already badly conditioned at the epoch of the initial orbit determination.

The second step is the propagation of the uncertainty to the time t₁. If the integral flow $\Phi_{t_0}^{t_1}$ is expressed in terms of orbital elements, it is still nonlinear, but its nonlinearity is limited in the following sense. Excluding the cases in which close approaches of the asteroid with some major planet occur in the time span between t₀ and t₁, and the cases of resonant perturbations, the integral flow is well approximated by the 2-body integral flow. The integral flow of the 2-body problem is linear with respect to all elements but the semimajor axis. Thus the nonlinearity of the integral flow is large when the uncertainty of the semimajor axis is significant, when the time span (between t₀ and t₁) is long, and in few other cases, such as the close approach ones. What is needed is an efficient way to exploit the 2-body approximation, without losing the nonlinear effects.

The third step is the computation of the observation function, which is strongly nonlinear, and therefore the image of the confidence ellipsoid is not at all close to the confidence ellipse, unless the latter is quite small. The computation of the observation function, however, is very simple: no solution of differential equations is involved, the observation function R is known as a comparatively simple analytical algorithm, fully explicit apart from the solution of the Kepler equation. Thus it is possible to overcome the difficulty of the nonlinearity of R by brute force, by computing it in many points. The difficulty in doing this is that the confidence ellipsoid in the X space is a 6-dimensional region, and to sample it in an uniform way requires an enormous number of points, most of which would end up in predictions very close to each other when mapped into the 2-dimensional space of the observations Y. What is needed is an efficient way to find which points, in the confidence ellipsoid, will map onto points close to the boundary of the confidence prediction region.

This paper contains a solution -that is, an efficient approximation algorithm- for all three problems. These three solutions will be discussed in an order reversed with respect to the above list, namely the following subsection will define a new approximation to the boundary of the confidence prediction region, while the following sections will discuss the nonlinearity of the covariance propagation and the determination of multiple solutions.