2.4 Covariance propagation

The vector of solve for parameters X represents the state vector at some time t, and is solution of a system of ordinary differential equations, the equation of motion. In the asteroid case the equation of motion is the N-body problem, the asteroid orbit being perturbed by the gravitational attraction of the planets; for some comets, non-gravitational effects are also relevant.

Let us assume these differential equations can be solved, as it is possible theoretically (and practically by a numeric approximation), as a function of the initial conditions X₀=X(t₀), for some epoch t₁: this function is the integral flow

$$X_1=X(t_1)=\Phi_{t_0}^{t_1}[X(t_0)]\ .$$

The differential of the integral flow is expressed by a matrix of partial derivatives, the state transition matrix

$${\displaystyle \partial X_1 \over \displaystyle \partial X_0}=D\Phi_{t_0}^{t_1}=D\Phi_{t_0}^{t_1}[X_0]$$

which is in turn solution of the variational equation, a system of linear ordinary differential equations. Also the variational equation has a solution (which can be computed numerically, simultaneously with the solution of the equation of motion).

By the use of the state transition matrix, the normal and covariance matrix can be propagated from time t₀ to an arbitrary time t. We shall indicate with subscript 0 the quantities referring to the epoch t₀, with subscript t the quantities for epoch t; for the normal matrix

$$C_0=\left({\displaystyle \partial \xi \over \displaystyle \pa... ...laystyle \partial \xi \over \displaystyle \partial X_0}\right)$$

the propagation to time t can be computed by

$$C_t=\left({\displaystyle \partial \xi \over \displaystyle \pa... ...splaystyle \partial \xi \over \displaystyle \partial X}\right)$$

with X=X(t); computing the Jacobian matrix by the rule of the composite differential:

$$C_t=\left({\displaystyle \partial \xi \over \displaystyle \pa... ...aystyle \partial X_0 \over \displaystyle \partial X}\right)\ .$$

The covariance matrices are just the inverse of the normal matrices for the same epoch, thus

$$\Gamma_0=C_0^{-1}\ \ \ ;\ \ \\ Gamma_t=C_t^{-1}=\left({\displ... ...style \partial X \over \displaystyle \partial X_0}\right)^T\ .$$

In conclusion to propagate the normal and covariance matrix, and to compute the confidence ellipsoid for another epoch, it is not necessary to solve again the least square problem, it is not necessary to go back to the observations and to recompute the residuals, but only to solve the variational equation. However, as we shall see in Section 4, the assumption of linearity is often questionable for this step of the computation.