Error propagation bounds in dual and triple beam vector Doppler ultrasound

Measurement of the velocity components along two (three) directions enables the two (three)dimensional velocity vector to be estimated exactly. However, in practical systems employing such multiple beam techniques, there will usually be errors in the measured velocity components along each beam, which will lead to errors in the estimated velocity magnitude and direction. This error propagation problem is analyzed in both two and three dimensions by decomposition of the velocity estimation matrix, and exact upper and lower bounds are derived for both the magnitude and angle bias as a function of the angle between the beams. The bias in triple beam systems is shown to have identical bounds to that in dual beam systems with an equivalent interbeam angle. It is found that small errors in the individual beam velocity components can be magnified in the final determination of velocity magnitude and angle. Plots are presented to assist system designers to specify the interbeam angle(s) to avoid gross velocity estimation errors.