Modeling of Dynamic Migration of z -Curves for GNSS Signals: A Geometric and Numeric Approach

A z-curve is the set of points on Earth’s surface that have identical pseudoranges to Global Navigation Satellite Systems (GNSS) satellites (A,B). The zero Time Difference of Arrival (TDOA) model eliminates common mode errors in signal transmission and the clock bias of GNSS receivers. Building upon a static model of z-surfaces and z-curves, this paper presents a dynamic model for migration of z-surfaces and z-curves with respect to satellite motions and atmospheric changes. Specifically, we use geometric transformation matrices and numerical solvers to describe the migration of the satellite Terrestrial Service Volume (TSV) model, z-surface, and z-curve. Affine transformations are applied to TSV and z-surface migrations, while for z-curve migration, we adopt a modified Newton’s method and a 4th-order Runge-Kutta method to align z-curve points on ground and within satellite coverage. These migration functions are validated using Global Positioning System (GPS) data.