\log_{10} (p(\r))$. Two different scenarios are considered, namely, *nominal* and *perturbed*, to evaluate the proposed robust power distribution scheme for the ultrasonic array. In the first scenario, nominal, we assume that the array steering vectors are precisely modeled, i.e., $\atil(\r) = \mathbf{0}$. In Fig. \[fig:nominal\], the beampattern generated by the array is plotted for the nominal scenario. This figure represents how power is spatially distributed over the organ at risk in an idealistic situation. Here, the covariance matrix of the waveforms is optimized under the assumption that the steering vectors are accurately modeled by , and the performance is evaluated using exactly the same steering vectors without any perturbations. The power is noticeably concentrated in the tumor region and importantly the power in the healthy tissue is several decibels lower.

![Power distribution (transmit beampattern in dB) for the nominal scenario, i.e., using $\R_{nr}$ and $\atil(\r) \equiv 0$.[]{data-label="fig:nominal"}](nominal_05_07lines "fig:"){width="\columnwidth" height="5.5cm"}\

In the second scenario, perturbed, the idealistic assumptions are relaxed and model uncertainties and imperfections are taken into account. The second scenario represents the case where the true steering vectors are perturbed versions of the nominal steering vectors $\ahat(\r)$, i.e., the true steering vector equals $\ahat(\r) + \atil(\r)$ where $\atil(\r) \in 