  Due to the importance of satellites for society and the exponential increase
in the number of objects in orbit, it is important to accurately determine the
state (e.g., position and velocity) of these Resident Space Objects (RSOs) at
any time and in a timely manner. State-of-the-art methodologies for initial
orbit determination consist of Kalman-type filters that process sequential data
over time and return the state and associated uncertainty of the object, as is
the case of the Extended Kalman Filter (EKF). However, these methodologies are
dependent on a good initial guess for the state vector and usually simplify the
physical dynamical model, due to the difficulty of precisely modeling
perturbative forces, such as atmospheric drag and solar radiation pressure.
Other approaches do not require assumptions about the dynamical system, such as
the trilateration method, and require simultaneous measurements, such as three
measurements of range and range-rate for the particular case of trilateration.
We consider the same setting of simultaneous measurements (one-shot), resorting
to time delay and Doppler shift measurements. Based on recent advancements in
the problem of moving target localization for sonar multistatic systems, we are
able to formulate the problem of initial orbit determination as a Weighted
Least Squares. With this approach, we are able to directly obtain the state of
the object (position and velocity) and the associated covariance matrix from
the Fisher's Information Matrix (FIM). We demonstrate that, for small noise,
our estimator is able to attain the Cram\'er-Rao Lower Bound accuracy, i.e.,
the accuracy attained by the unbiased estimator with minimum variance. We also
numerically demonstrate that our estimator is able to attain better accuracy on
the state estimation than the trilateration method and returns a smaller
uncertainty associated with the estimation.
