Doppler Tomography for Vector Elds

The problem considered is to reconstruct a vector eld in the case when for each line the distribution of the vector eld components along the line is known. Such data are obtained from spectral analysis of signals, aaected by Doppler shifts, caused by reeections on moving particles (\velocity spectra"). In the general setting, the problem is unsolved. Using only integrated data along lines, one encounters the same problem as when reconstructing vector elds by means of time-of-ight measurements. Reconstruction formulas are given in this case, establishing the known fact that only the curl of the vector eld can be reconstructed without additional information. Inspired by the application to which the paper is most devoted, ultrasound Doppler measurements on blood ows, the case where the ow takes place in narrow channels is investigated in simulations and experiments.