Derivation and error analysis of echo phase parameters for the dynasonde

Ionospherically reflected echoes are received with a four-element multiplexed interferometer array and two phase-matched receivers in implementations of the National Oceanic and Atmospheric Administration dynasonde. From these data, six phase parameters (Φ₀, Φ_x, Φ_y, Φ_t, Φ_p, and Φ_f) are obtained and used to derive echo location, Doppler velocity, wave polarization, and virtual range. Since 2π aliasing is an inherent feature of interferometric spaced antenna phase measurements, the phase parameters cannot be derived directly from the measured phase values using the method of least squares. In this work, we introduce a general procedure for the derivation of these parameters that (1) employs a "zero-freedom" technique to derive initial estimates of the phase parameters, (2) derives shifted values of the measured phases from the six estimates, and (3) uses the method of least squares in conjunction with the shifted phases to improve the phase parameter estimates. This procedure minimizes the phase ambiguity inherent in interferometric phase measurements and derives phase parameters that approach the ideal least squares result. Furthermore, the ionospheric echo "quality" is quantified by two error parameters, defined as ŝ, the least squares of the measured phase errors, and EP, the RMS phase error. It is also shown that the value of ŝ relative to the square of the standard deviation of measured phase is equal to the number of degrees of freedom in the phase parameters. Applying these techniques to data acquired with the Utah State University dynasonde, we derive a standard deviation of <2° in the measured echo phase.