Many geophysical applications of the array sonic measurements require the knowledge of the true source-time function. Recovery of the source-time function from the borehole sonic P head waves is different from the source inversion problem in exploration seismology. The difficulty in the inversion of borehole sonic measurements arises due to the inexact knowledge of the impulse response, e.g., the inability to model the whole problem including the tool body and transducers. The random noise often encountered in seismic signals is not the key obstacle here. The inverse source problem is ill posed due to the interference of P head-wave multiples. Using waveforms from a laboratory scale model we have applied two deconvolution methods, one using a Wiener filter and the other the time-domain least-squares method. As expected, without constraints on the solutions, one cannot recover a satisfactory source-time function. An unconventional smoothness constraint is applied in the source spectrum (instead of the usual smoothness in the time-domain signals), which corresponds to a finite-duration pulse in the time domain (instead of the usual band-limited spectrum). This technique is thus called the “duration-limited” inversion. The inverted results, obtained by Wiener filtering combined with this “duration limiting” process and multichannel stacking, agree well with an independent free-field measurement. Furthermore, reconstructed receiver waveforms using the inverted source function match the measured ones. The inversion procedure is robust and potentially useful for field measurements.